Eindhoven University of Technology MASTER. Output regulation for a nonlinear mechanical system from design to experiments. Janssen, B.R.A.

Transcription

1 Eindhoven University of Technology MASTER Output regulation for a nonlinear mechanical system from design to experiments Janssen, B.R.A. Award date: 2005 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain

2 Output Regulation For a Nonlinear Mechanical System: From Design to Experiments B.R.A. Janssen DCT Master s thesis Supervisor: Coaches: Prof. Dr. H. Nijmeijer Dr. A. Pavlov Dr. Ir. N. van de Wouw Eindhoven University of Technology Department of Mechanical Engineering Dynamics and Control Group Eindhoven, 19 April 2005

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4 Contents Preface Summary vii ix 1 Introduction The Output Regulation Problem Problem Statement Example Overview The TORA System A Benchmark Mechanical System TORA Setup Research Goal Outline Output Regulation Control Theory Linear Output Regulation: Basic Theory Problem Statement Solvability Analysis Linear Output Regulation: Controller Design Static State Feedback Controller Dynamic Output Feedback Controller Robust Output Feedback Controller Local Output Regulation for Nonlinear Systems Problem Statement Solvability Analysis Nonlinear Static State Feedback Control Summary Output Regulation on a TORA System TORA Dynamics Controller Design Simulations TORA Construction Design Specification TORA Dimensioning Pendulum Drive Drive Specifications Drive Selection TORA Configuration Design Considerations

5 iv CONTENTS Pendulum Bearings Support Structure Motor Attachment Material Construction Strength and Stiffness Torsional Stiffness Deflections Strength Pendulum Shaft Conclusions Summary Experimental Case Study Experimental Setup System Description Parasitic Phenomena System Modelling & Controllers Experiments Parameter Settings Experiments Experiments vs. Simulations Results Experimental Data Analysis Conclusions Conclusions and Recommendations Conclusions Recommendations A OR: Additional Theory 69 A.1 Observer-based Output Feedback Controller A.2 Robust Output Regulation A.2.1 Nominal System Parameters A.2.2 Perturbed System Parameters: Robustness Analysis B Optimal design 73 B.1 System, Forces and Torques B.2 Design Goals and Constraints B.2.1 Design Goal B.2.2 Design Constraints B.3 Qualitative Analysis B.4 Quantitative Analysis B.4.1 Constraints B.4.2 Objective B.5 Conclusions C Motor Selection 83 C.1 Direct Drive Motors C.2 Indirect Drive Motors C.3 Maxon Details

6 CONTENTS v D TORA Loads 87 D.1 Steady-state Pendulum Reaction Forces D.2 Collision Forces D.3 Design Load D.4 External Moments E Linear Elastic Beam Theory 93 E.1 Deflections E.2 Stiffness E.3 Strength F Component Specifications 97 G Detailed Drawings 105 H TORA and H-bridge parameters 125 Samenvatting 133 Acknowledgments 135

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8 Preface This report, Output Regulation For a Nonlinear Mechanical System: From Design to Experiments, covers the master s thesis study of the author, which has been performed within the division of Dynamics and Control of the faculty of Mechanical Engineering at the Eindhoven Technical University. The research was conducted within the wider scope of a PhD study on output regulation of the author s direct supervisor, Dr. A. Pavlov, titled: The Output Regulation Problem: A Convergent Dynamics Approach. The latter is mainly a theoretical work, whereas the study presented in this report covers an experimental approach of the output regulation problem. The report is accompanied by a DVD, which contains movies of the experiments and the experimental data. Eindhoven, March 2005

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10 Summary A mechanical benchmark system that is commonly used in nonlinear control theory is the socalled TORA system (Translational Oscillator with a Rotational Actuator). A TORA system is a nonlinear mechanical system consisting of an un-actuated cart, having one degree of freedom, a translation in the horizontal plane. A pendulum consisting of an eccentric mass rotating in the horizontal plane is mounted on the cart. Actuation of this pendulum results in a pendulum motion, which induces reaction forces on the cart. Hence the cart can be indirectly actuated via the pendulum. Also within the scope of the output regulation problem (ORP) the TORA system is often reverted to as a benchmark system. As many control problems, the output regulation problem deals with the tracking of reference signals or rejection of disturbances in the output of a dynamical system. In contrast to conventional tracking and disturbance rejection problems, in the output regulation problem the class of reference signals and disturbances consists of solutions generated by some autonomous system of differential equations, the so-called exosystem. The linear output regulation problem has been extensively studied in the past. More recently the nonlinear output regulation problem has been (and still is) subject of research. Sofar, only simulations have been used to validate nonlinear output regulation controllers. The aim of this study is to construct a physical TORA system in order to experimentally validate non-linear output regulation controllers. Moreover, one is interested in the applicability of output regulation controllers, i.e. in the practical problems related to the use of output regulation controllers in practice. It should be emphasized that the TORA system itself is not of any practical use, but only serves as a nonlinear mechanical benchmark system. To start with, the output regulation theory is studied. It is proven that solving the output regulation problem comes down to solving a special set of equations, the so-called regulator equations. In the case of linear output regulation this requires the solving of a matrix equation, whereas in the nonlinear case this involves the much more difficult task of solving a set of partial differential equations. Subsequently different controllers that solve the output regulation problem have been studied. First for the linear case and then for the nonlinear case. Since a TORA setup is neither available commercially nor in our lab, a suitable TORA setup has been designed and constructed by adapting an existing X-Y positioning setup, the H-bridge setup. An optimal TORA design analysis, with the objective to minimize the required motor torque, yields a set of design parameters that serve as a guideline for the structural TORA design. Moreover, a suitable motor to drive the rotational eccentric mass is selected and the best mounting configuration for the pendulum on the cart is determined. Based on the design parameters, the motor choice and the mounting configuration, a detailed structural TORA design has been made. The setup is designed such that it is suitable to conduct a variety of (tracking, disturbance rejection) experiments. The construction of the setup did not give rise to any problems. Despite the choice of a (technically inferior) indirect drive solution due to its low costs and ease of control, the setup performs satisfactory from a mechanical point of view. In order to experimentally validate output regulation controllers, disturbance rejection experiments have been performed. In these experiments there is a disturbance force acting on the cart

11 x (parallel to its degree of freedom), that is the output of an external autonomous system. The torque applied to the rotational eccentric mass on top of the cart is now to be controlled such, that the reaction force of the eccentric mass on the cart compensates the disturbance force and hence the cart comes to a standstill (for some set of initial conditions on the TORA en exosystem states). For the sake of simplicity the most basic output regulation controller has been implemented. This is a nonlinear static state feedback output regulation controller, which is not robust with respect to system and exosystem parameter uncertainties and requires knowledge on the full system and exosystem states. The experimental results show that disturbance rejection is achieved only approximately, i.e. the cart sticks in a position close, but not equal to, its equilibrium position. This is due to improper friction and cogging compensation in the cart motion, improper friction compensation in the arm motion, system parameter uncertainties and backlash in the gearbox. The experimental results show a good resemblance with results obtained from simulations on a qualitative level. On a quantitative level there are significant, but not very large, differences. Since approximate disturbance rejection is achieved despite the presence of parasitic phenomena, the experiments have shown that the application of output regulation controllers in mechanical systems is feasible, i.e. there are no practical limitations preventing the output regulation controller to work on a physical setup. In order to improve the controller performance the magnitudes of the parasitic phenomena need to be reduced, which needs a more detailed study of these phenomena. Moreover, the availability of the TORA setup paves the way for experiments on more advanced output regulation controllers such as output feedback controllers and controllers that are robust with respect to system (and exosystem) parameter uncertainties, as well as for tracking experiments.

12 Chapter 1 Introduction The output regulation problem (ORP) is one of the central problems in control theory. This problem deals with asymptotic tracking of prescribed reference signals and/or asymptotic rejection of undesired disturbances in the output of a dynamical system. The main feature that distinguishes the output regulation problem from conventional tracking and disturbance rejection problems is that in the output regulation problem the class of reference signals and disturbances consists of solutions generated by some autonomous system of differential equations. This system is called an exosystem. Reference signals and/or disturbances generated by the exosystem are called exosignals [32]. Many researchers have tackled the problem of output regulation, see for example [7, 18, 23, 35]. This has lead to procedures to construct controllers that solve the (nonlinear) ORP (either locally or globally), including robust and adaptive controller designs. So far the performance and characteristics of these controllers have only been studied using simulations. To the authors best knowledge, (nonlinear) output regulation controllers have never been practically or experimentally implemented on physical (mechanical) systems. A common benchmark system in nonlinear control, which is also very suitable in the setting of the output regulation problem is the so-called TORA system (Translational Oscillator with a Rotational Actuator). This is a nonlinear mechanical system, extensively described in [5] and often used for output regulation simulations (see for example [18, 35]). The purpose of this this master thesis study is to attain experimental experience in output regulation control by implementing output regulation on a physical, experimental TORA setup. 1.1 The Output Regulation Problem The output regulation problem deals with disturbance rejection and/or tracking of some signal that is a solution of a system of a known class. In general approaches to the disturbance rejection/tracking problem, one assumes complete knowledge of the past, present and future time history of the trajectory to be tracked or the disturbance to be rejected. In output regulation on the other hand, one only assumes knowledge of the class of the exosystem that generates the disturbance and/or reference signal. In particular, output regulation may concentrate on harmonic exosignals, generated by a (linear) harmonic oscillator. In case the exosystem is a linear harmonic oscillator, the ORP reduces to the rejection and/or tracking of signals consisting of sinusoids with known frequency, but unknown initial conditions, or, what is the same, with unknown amplitude and phase. In particular, output regulation may concentrate on such harmonic exosignals generated by (linear) harmonic oscillators.

13 2 1.1 The Output Regulation Problem Problem Statement In a more formal way, the output regulation problem mentioned above can be described as follows. Given a system ẋ = f (x,u,w), y = h y (x,w), e = h e (x,w), where x is the state, y the measured output, e the regulated output to be controlled, u a control input and w an external signal generated by the exogenous system: design a controller u such that: ẇ = s(w), a The closed-loop system is asymptotically stable for w (t) = 0. b e(t) = h (x(t),w (t)) 0 as t on solutions of the closed-loop system. Depending on the problem formulation one can demand that these two conditions hold either locally, globally or in some predefined set of initial conditions. Several different forms of the output regulation problem can be distinguished. The objectives for each form are different, but the resulting control goals, a and b are the same in each case. The first form of the output regulation problem is the disturbance rejection problem: an output of the exosystem acts as a disturbance on the system. The objective is to achieve a specific steady-state output (generally a zero output) of the system, regardless of the amplitude and phase of the disturbance. In this problem the exosystem ẇ = s(w) with output d (w) acts on the system: where the regulated output e = h e (x). ẋ = f (x,u,d(w)), A second form of the output regulation problem is the tracking problem. In the steady-state situation, the output of a system should follow the output of the exosystem. The exosystem, ẇ = s(w), generates an output r (w), which is a reference signal for the system: ẋ = f (x,u), with output g (x). In this case the regulated output is defined as e = g (x) r (w). The third form of the output regulation problem is the synchronization problem, which can be regarded as a special case of the tracking problem for which the dimension of the system equals the dimension of the exosystem and the system and exosystem outputs are defined as being their full states.

14 1. Introduction Example To illustrate the output regulation problem, consider the example of the (tracking) problem of a helicopter landing on the deck of a pounding ship. The helicopter is described by a (non-linear) system: ẋ = f (x,u), y = g (x), where x are the helicopter states, u are the helicopter input(s) (for example rotor pitch and/or throttle) and the output y is the helicopters absolute height. The motion of the ship pounding on the waves can be described by a system: ẇ = s(w) h = r(w), in which w are the ship states and the output h is the absolute ship deck height. The latter system is the exosystem. As one can imagine for a ship pounding on the waves, this exosystem is a harmonic oscillator producing harmonic outputs. The regulated output e for this problem can be defined as: e = y h = g (x) r (w), which is the height of the helicopter above the deck. The controller u should be such that the control goals a and b defined in the previous section are satisfied, i.e.: We want the helicopter to be an asymptotically stable system. The height e of the helicopter above the deck should tend to 0 as time t or, in other words, the helicopter lands on the deck, despite the motion of the deck. The example presented above is a simplification of reality (for example, the very important constraint y(t) h(t) (t), which ensures the helicopter does not crash into the deck, has not been taken into account). However, the example illustrates the type of problem that we are dealing with Overview The output regulation problem has been a subject of study for quite some time. The linear problem has been extensively studied in the 1970s in [9, 13, 14, 44], yielding necessary and sufficient conditions for the existence of a controller solving the problem and a procedure to construct such a controller. The incorporation of an internal model of the exosystem in the controller proved to be a very effective way of ensuring the asymptotic decay of the regulated output for every input signal (reference trajectory or disturbance force) from the class of signals generated by the exosystem. A solution to the local nonlinear output regulation problem, encompassing necessary and sufficient conditions for the solvability of the problem and a procedure for designing a controller that solves the problem (locally), was presented in the early nineties in [20] and [23] (although for specific classes of exogenous inputs the problem had been addressed before). Solving the nonlinear ORP involves the solution of a mixed set of partial differential/algebraic equations, which is not an easy task. Satisfactory results can also be obtained with an approximation of the solution of this set of equations. This so-called approximate output regulation problem was studied in [19, 21, 34, 42]. Other aspects of the non-linear output regulation such as the presence of uncertainties, (semi) global and adaptive output regulation were studied in the late nineties in [7, 22] and [38]. Recently, our research group within the Eindhoven University of Technology, has studied, among other output regulation issues, the problem of how to estimate the range of admissible initial conditions for the nonlinear local output regulation problem, [18, 32, 34, 35]. For a more elaborate overview on the developments in output regulation one is referred to the works of [7, 8, 33] and especially the recent work of [32], which addresses the global uniform ORP.

15 4 1.2 The TORA System 1.2 The TORA System A Benchmark Mechanical System In the early nineties Bupp et. al. in [5] proposed a benchmark problem for nonlinear control design. This benchmark problem consists of the control of the motion of a cart possessing one translational degree of freedom, which is actuated by an eccentric rotational mass actuator mounted on the cart. The system of cart and rotational actuator is referred to as the so-called TORA system (Translational Oscilator with a Rotational Actuator) or RTAC (Rotational/Translational ACtuator) system. Figure 1.1 depicts a schematic representation of the TORA system. cart x T u F d gravity θ eccentric mass Figure 1.1: Schematic representation of a TORA system. This nonlinear benchmark problem originates from simplified models used in the research into the resonance capture phenomenon occurring in a dual-spin spacecraft [41] and is in the same spirit as a linear benchmark problem presented earlier by Wie et. al [43]. The TORA system also serves as a benchmark for output regulation control aspects, see [18, 32, 35]. The ORP for the TORA system can be formulated as a disturbance rejection or a tracking problem. In the case of disturbance rejection there is a harmonic (disturbance) force F d acting on the cart and the regulated output e is defined as the cart displacement x. In the case of tracking there is no disturbance force acting on the cart, but the cart is to track a certain reference trajectory, which is the output of some exosystem, and the regulated output e equals the difference between actual and desired cart displacement x x des. In either case, the ORP consists of the problem of finding a controller u that drives the eccentric mass (from hereon referred to as a rotational link, arm or pendulum ) through the torque T u in such a way that: The closed-loop TORA system, in absence of a disturbance and/or reference signal, is an asymptotically stable system (either locally or globally). The regulated output e tends to 0 as time t (i.e. successful tracking and/or disturbance rejection occurs). In the scope of the ORP, the TORA system can either serve as: a benchmark problem for nonlinear control, or, from a more practical point of view, as a system to study the use of a rotational actuator for suppressing translational vibrations. The reason for the TORA system to serve as a benchmark system is that it is one of the simplest nonlinear mechanical system that encompasses two properties that ask for dedicated control strategies. Due to the non-linear coupling between the translational and rotational motion, the system is

16 1. Introduction 5 not globally feedback linearizable using a straightforward method such as feedback linearization. The system is underactuated, i.e. it has two degrees of freedom x and θ but only one actuated degree of freedom (θ). In a more general approach to (semi) global stabilization and tracking of the TORA system, many nonlinear controllers have been proposed and simulated: passivity-based feedback controllers that only require pendulum angle measurements for feedback [5, 6, 11, 12, 25], integrator backstepping based controllers, requiring the full state for feedback [6, 25], or only pendulum angle measurements [28] and other controllers that only require pendulum angle measurement for feedback [27]. Some of these controllers have even been implemented on an experimental TORA setup by Bupp et. al. (see [6]). Output regulation based controllers, however, have not been implemented on this setup TORA Setup In this study we pursue the physical implementation of an output regulation controller of some type on an experimental TORA system. To this end we will need a real TORA setup that is suitable to implement experimental controllers. Since such setup is neither available commercially, nor in our our lab, there is a need to construct such a setup. A significant part of this study has been devoted to the construction of such an experimental TORA setup. In order to save cost and to optimally use the available facilities in the lab, the TORA setup has been constructed using an existing X-Y positioning (H-bridge) setup. This H-bridge, a former pick-and-place robot, consists of two (sets of) perpendicular axes (X and Y 1,Y 2 ) in the horizontal plane, each driven by (a set of) Linear Motor Motion System(s) (LiMMS), as can be seen in Figures 1.2 and 1.3. Since the translational oscillator already exists in the form of the H-bridge X-carriage, to be referred to as cart from hereon, only an eccentric rotational actuator to be placed onto this cart needs to be constructed. The X-sled linear motor is used to emulate a spring and, if required, to apply a (predefined) disturbance force to the cart. The Y 1 and Y 2 carriages will be kept still at a certain position using some low level feedback control, thus leaving only the translation of the cart and the rotation of the arm attached to the cart as degrees of freedom. An electric motor will drive the arm attached to the cart, thus forming a TORA system. X DC motor LiMMS Y1 LiMMS Y2 LiMMS X Y1 Arm Y2 T Figure 1.2: H-bridge schematically. Figure 1.3: H-bridge picture. Except for the experimental TORA setup presented by Bupp et. al. (see [4 6]), there exists no operational TORA setup. Moreover, to the author s knowledge, a TORA setup on which output regulation is implemented will be unique.

17 6 1.3 Research Goal 1.3 Research Goal The (nonlinear) output regulation problem is a comprehensive problem that has been studied extensively. However, output regulation controllers have never been physically implemented. This lack of experimental experience gives rise to several questions that are yet unanswered. Are there for example (physical) limitations to the use of output regulation control and do experimental results compare with simulation results? To obtain experimental experience on output regulation, the aim of this study is to physically implement some form of output regulation control on the TORA system, a benchmark system in nonlinear control and output regulation theory. Since such a system is not available, a suitable TORA setup needs to be constructed. Summarizing, the research goals for this study can be stated as follows. Design and construct a TORA setup that is suitable to implement a wide variety of output regulation control strategies. Implement a (simple) form of output regulation on the TORA system to demonstrate the effectiveness of output regulation in practice. Test the applicability if the ORP in practice. i.e. identify the practical problems related to the use of output regulation control strategies. Compare experimental and simulation results. 1.4 Outline The outline of this thesis is as follows. To start with, Chapter 2 addresses the output regulation principles, theory and controller design. In Chapter 3 we construct a controller specifically for the TORA system and verify its performance by simulations. Then, Chapter 4 addresses the design and construction of the experimental TORA system. Chapter 5 contains the experimental results and a comparison of these experimental results with the results obtained from simulations. Finally, the thesis is concluded with conclusions and recommendations for further research in Chapter 6.

18 Chapter 2 Output Regulation Control Theory In this section, we address the basic ideas behind output regulation control and, associated with that, the construction of certain controllers solving the output regulation problem. First, for the sake of simplicity, the linear case is discussed, where solutions are valid globally. Then, since in the end we have a nonlinear TORA system to cope with, the linear case is extended to the case of nonlinear output regulation, where solutions are valid only locally. The works of [7], [33] and [8] are the main sources for the theory presented here. Throughout the next sections two types of controllers are considered: a simple controller based on static state feedback and a more complex (robust) general output regulation controller based on dynamic output feedback. From the point of view of output regulation, the latter controller is much more interesting than the first one, because it has two major advantages over the state feedback controller: robustness with respect to system parameters and requiring only the measurement of the output. However, for practical reasons, only the second type of controller has finally been physically implemented in the experimental TORA system. The standard output regulation theory assumes knowledge of the class of the exosignals. Moreover, it assumes the exosystem generating the tracking and/or reference signals to be completely known. This is not always realistic from a practical point of view, but it does significantly facilitate the controller design. It should be noted however, that there exist procedures to construct output regulation controllers that are robust with respect to variations in the exosystem parameters and therefore do not require the exosystem to be known exactly. However, these controllers are outside the scope of this study. Some robustness features of the controllers discussed here are addressed in Section Linear Output Regulation: Basic Theory In this section we address the basic theory behind the linear output regulation problem and controllers solving the problem. The next subsection, subsection 2.1.1, presents a more precise statement of the output regulation problem, which is followed by a solvability analysis in subsection In the latter subsection it shown that the basis of all controllers solving the ORP is in a specific set of equations. Section 2.2 addresses the design of various (linear) output regulation controllers, based on this set of equations. Finally in Section 2.3 the linear output regulation theory is expanded to the nonlinear output regulation theory and nonlinear output regulation controllers are designed.

19 8 2.1 Linear Output Regulation: Basic Theory Problem Statement The system description for the linear case is given by: ẋ = Ax + Bu + Pw y = C y x + Q y w e = C e x + Q e w, (2.1) and the exosignals w are generated by the linear exosystem: ẇ = Sw, (2.2) where x R n is the system state, u R k is the control input, w R m is the exosystem state, e R le is the regulated output, y R ly is the measured output and the matrices A, B, C y, C e, P, Q y, Q e and S are of corresponding dimensions. Apart from the distinction between the regulated output e and the measured output y, (2.1) and (2.2) are just the linear equivalents of the general system and exosystem given in Section The following assumptions are made on the exosystem: 1. The exosystem is assumed to generate harmonic exosignals, i.e. the matrix S has only simple eigenvalues on the imaginary axis. 2. The parameters of the exosystem are known, i.e. the frequency contents of the harmonic exosignal, which is defined by the matrix S, is known. These are two standard assumptions in the (linear) output regulation problem. They represent the practically important case of harmonic disturbances and/or reference signals with known frequencies, but unknown phases and amplitudes. Another important assumption in output regulation control is that the number of outputs l e is the same as the number of inputs k. The output regulation problem is defined as to find a controller of the form: ξ = Λξ + Υy, u = Θξ + Ψy (2.3) resulting in the (forced) closed-loop system: ż = A cl z + P cl w ẇ = S w, (2.4) with the regulated output e: where: z = [ x ξ such that: ] [ A + BΨCy BΘ, A cl = ΥC y Λ e = C e, cl z + Q e w, (2.5) ] [ P + BΨQy, P cl = ΥQ y ], C e, cl = [ C e 0 ], A1 The closed-loop system (2.4) is asymptotically stable for w = 0. A2 For all initial conditions z (0), w (0), it holds that e(t) = C e, cl z (t) + Q e w (t) 0 as t. Note that these control goals are the same as the goals a and b described in Section In the sequel two particular forms of the general output regulation controller 2.3 are addressed in more detail.

20 2. Output Regulation Control Theory 9 1. A static state feedback controller of the form: u = Kx + Nw. (2.6) In this case, Ψ = [K,N] and, since the feedback law is static, Λ = Υ = Θ = 0. All states x and w are measured and available for feedback, i.e. y = [ x T,w T] T. 2. A dynamic (regulated) output feedback controller of the form: ξ = Λξ + Υe u = Θξ. (2.7) In this case the regulated output equals the measured output: i.e. C y = C e = C, Q y = Q e = Q and Ψ = 0. y = e The more general case where the measured output y does not equal the regulated output e, nor the full state, i.e. the case of arbitrary C y and Q y, is not addressed in this study Solvability Analysis The solvability analysis presented in this section is performed for the general output regulation controller (2.3). Hence the results are also valid for the controllers (2.6) and (2.7), which are particular case of the general controller. We start the solvability analysis of the linear ORP with a reformulation of the control goals defined in the previous section. Such reformulation makes it easier to analyse the problem. We will show that the conditions A1 and A2 are equivalent to: AA1 The matrix A cl is Hurwitz. AA2 For every solution of the exosystem w (t) there exists a solution z (t) such that e (t) = C e, cl z (t) + Q e w (t) = 0 for all t R. It follows directly that AA1 implies A1 and vice versa. Also, AA1 and AA2 together imply A2. This can be seen from the following reasoning. Condition AA1 implies the closed-loop system is asymptotically stable. Since the system is linear, this means that any two solutions z 1 (t), z 2 (t) converge to each other as t. Furthermore, by AA2, for all w (t) there exists a solution z (t) on which ē (t) = 0. Because of the previous statement, any solution z (t) will now converge to z (t) as t. Consequently, as t, any regulated output e(t), corresponding to z (t), will converge to the regulated output ē(t) = 0, corresponding to z (t). The latter, in fact, is equal to condition A2. In the sequel of this section it is shown that A1 and A2 together imply AA2, which completes the proof that A1 and A2 are equivalent to AA1 and AA2. This involves a specific set of equations, the so-called regulator equations. The solvability of these equations proves to be a basic necessary condition for the existence of a solution of the ORP. Since the above sets of conditions (A and AA) are equivalent, the output regulation problem simplifies to finding a controller satisfying the set of conditions AA. Let us now derive the basic necessary conditions for the solvability of the output regulation problem, which also includes the last part of the proof that the sets of conditions A and AA are equivalent (i.e. that A1 and A2 together imply AA2). These necessary conditions are based on the following statement.

21 Linear Output Regulation: Basic Theory Lemma 2.1 Consider the forced closed-loop system (2.4) satisfying A1 and A2. Then there exist a matrix Π such that: ΠS = A cl Π + Pcl, (2.8) 0 = C e, cl Π + Qe. (2.9) Proof: A basis for [ the solution ] of the forced closed-loop system (2.4) is formed by the generalized eigenvectors of cl Acl P. Any solution of the closed-loop system consist of two parts. One 0 S part of the solution is in the subspace that is spanned by the generalized eigenvectors corresponding to the stable eigenvalues of A cl. Since these eigenvalues are in C, this part of the solution tends to zero. The other part of the solution is in the so-called center subspace that is spanned by the generalized eigenvectors corresponding to the eigenvalues of S. These eigenvalues, by assumption, are on the imaginary axis, thus the corresponding parts of the solution are marginally stable. Since the part of the solution in the stable subspace tends to zero, the whole steady-state solution is fully in the center subspace. The dynamics in the center subspace are determined by a relationship between z and w of the form z = Πw. If U = [ ] [ ] u 1 u 2... u m contains all the Acl P generalized eigenvectors of cl corresponding to the eigenvalues of S, then, by superposition, the steady state solution = U f (t), where f (t) reflects the time dependency 0 S ( ) zss (t) w (t) of z ss (t) and w (t). The matrix U can be decomposed into an upper part U z and a lower part U w such that: z ss (t) = U z f (t) and w (t) = U w f (t). Since the number of states of w equals the number of different eigenvalues of S, which in turn equals the number of independent eigenvectors in U w, the matrix U w is invertible. Hence, z ss (t) = U z Uw 1 w (t) = Πw (t). Substituting this expression for z in the forced closed-loop system (2.4) yields Π Sw (t) = A cl Πw (t)+pcl w (t), for all w (t). Since the choice of w (t) is arbitrary, this equation implies (2.8). Figure 2.1 gives a graphical representation of the theory mentioned above: any solution z (t) converges to the solution z (t) = Πw (t) on the center subset z = Πw. z = Πw Figure 2.1: Attractiveness of the invariant center subspace. It can also be proven that, when A1 and A2 are satisfied, for the steady-state solution (z ss (t),w (t)) it holds that e ss (t) = C e, cl z ss (t)+q e w (t) = 0. The essence of this proof is as follows: Since both the solution z ss (t) = Πw (t) and exosignal w (t) are harmonic, also e ss (t) = C e, cl z ss (t)+q e w (t) is harmonic. Moreover, under A2, e(t) 0 for t. Together with the property of being harmonic this yields that e ss (t) must be 0 for all t R (for details: see [7], page 30). Substituting solution z ss (t) = Πw (t) and e = 0 in (2.5) yields 0 = C e, cl Πw (t) + Qe w (t), which proves that,

22 2. Output Regulation Control Theory 11 for a system satisfying A1, and A2 there indeed exists a matrix Π satisfying (2.9). From the above it follows that the solvability of (2.8) and (2.9) means that for all w (t) there exists a solution z ss (t) = Πw (t) for the closed-loop system (2.4) along which e(t) = 0. In particular we now have shown that A1 and A2 together imply AA2. By substitution of A cl, C e, cl and P cl, and the creation of a separation between state- and input terms, equations (2.8) and (2.9) transform to: [ Πx Π ξ ] [ S = A 0 ΥC y Λ ] [ Πx Π ξ ] [ B + 0 [ Ce 0 ] [ Π x Π ξ ] ( [ ΨCy Θ ][ Π x Π ξ ] ) [ + ΨQ y + P ΥQ y ] (2.10) ] + Q e = 0, (2.11) Lemma 2.2 If we now only consider the parts of these equations (2.10) and (2.11) concerning the system states, i.e. the upper parts, the equations reduce to: ΠS = AΠ + B Γ + P (2.12) 0 = C e Π + Q e, (2.13) where Π = Π x and Γ = Ψ(C y Π x + Q y ) + ΘΠ ξ. The set of equations (2.12) and (2.13) are called the regulator equations In the case of static state feedback Ψ = [K,N] and Λ = Υ = Θ = { }, which yields that Γ equals KΠ + N. In the case of dynamic output feedback Ψ = 0, which yields that Γ equals ΘΠ ξ. Now if the ORP is solved, the control goals A1 and A2 are satisfied. Then, by Lemma 1.1 and 1.2, the regulator equations are solvable. This implies there is a solution x(t) = Πw (t), corresponding to the control input u(t) = Γw (t), along which e(t) = 0. Solvability of the regulator equations thus is a basic necessary condition for the solvability of the output regulation problem. Moreover, satisfaction of A1 i.e. asymptotic stability of the unforced closed-loop system, requires A cl to be Hurwitz (AA1). From standard linear control theory this requires stabilizability of the pair (A,B) and detectibility of the pair (A,C y ). Hence the necessary conditions for the output regulation problem to be solvable are: B1 B2 The pair (A,B) is stabilizable and the pair (A,C y ) is detectible. The regulator equations (2.12) and (2.13) are solvable. A set of sufficient conditions for the existence of a solution of the regulator equations (2.12) and (2.13) is ([7], page 20): the pair (A,B) is stabilizable, the pair (A,C y ) is detectable, [ A λi B for all λ that are the eigenvalues of S, rank C y 0 ] is maximal. 2.2 Linear Output Regulation: Controller Design In the previous section we addressed the basic theory behind the output regulation problem. In particular we showed that the regulator equations play an important role in the output regulation problem and their solvability is a necessary condition for the existence of a solution. In this section we will focus more specifically on the two types of controllers solving the ORP presented

23 Linear Output Regulation: Controller Design in Section We will address a static state feedback controller and two dynamic output feedback controllers, a non-robust observer based variant and a variant that is robust with respect to variations in the system parameters. The basic idea behind controllers solving the output regulation problem, is to construct a controller that consists of two parts. A first part that is capable of generating a control signal that keeps x(t) on the set x = Πw, once x is initiated on this set. From the solution of the regulator equations it then follows that on this set the regulated output e is zero. A second part is capable of generating a control signal that renders the set x = Πw globally asymptotically attractive. Together these parts yield a controller that, for all initial conditions x(0) and w (0), makes x(t) Πw and thus e(t) Static State Feedback Controller We start with the most simple form of a controller solving the ORP: a static state feedback controller. A nonlinear version of this controller, discussed in Section 2.3, will finally be implemented on the (nonlinear) experimental TORA setup. First the static state feedback controller is stated and its structure is discussed, next it will be shown that such a controller indeed solves the ORP. Lemma 2.3 Assume the necessary conditions for the existence of a solution of the ORP are satisfied, i.e. B1 and B2 hold. Then a static state feedback controller of the form u = Γw + K (x Πw), (2.14) where Π and Γ are solutions to the regulator equations and K is such that matrix A cl = A + BK is Hurwitz, solves the static state feedback ORP. Recall, that in the case of static state feedback, all states x and w are measured and available for feedback: y = [ x T,w T] T and Λ = Υ = Θ = { }. Proof: To show, that under conditions B1 and B2, there indeed exists a static state feedback controller of the above form that solves the regulator problem, first we address the control goal AA1, that is A cl = (A + BK) should be Hurwitz. Under condition B1 this is achieved by a proper choice of K. Second, we address the control goal AA2, that is, for every solution of the exosystem w (t) there exists a solution x such that ē (t) = C e x(t) + Q e w (t) = 0 for all t R. As follows from the solvability of the regulator equations (more specifically (2.13)), such a solution indeed exists and is equal to x(t) = Πw (t), corresponding to the control equals to u(t) = Γw (t) (see previous section). The reason to use this form of the controller, comprising the K (x Πw) part, is to realize a separation between a steady-state input part Γw and a stabilizing part K (x Πw), but in fact this controller is equivalent to the general linear state feedback controller u = Kx + Nw (with N = Γ KΠ). The first part of the controller, u 1 = Γw, provides a control action that makes x(t) stay in the set x = Πw (on which e = 0), once x has been initiated on this set. The second part of the controller, u 2 = K (x Πw), assures globally asymptotically attractiveness of the set x = Πw. This makes solutions from any initial conditions x(0) and w (0) converge to x(t) = Πw (t). In general, the controller (2.14) is not robust with respect to variations in the system parameters. The static state feedback controller (2.14) requires the full state of the system x and exosystem w for feedback. Under some circumstances it is possible however to construct an observer

24 2. Output Regulation Control Theory 13 with which the system states x and exosystem states w can be obtained from measurements of the regulated output only. Such a (dynamic) output feedback controller is discussed in the next section Dynamic Output Feedback Controller In general, neither the whole state of the system x, nor the entire state of the exosystem w is available for feedback. Usually only the system output is measurable. Under certain conditions however, it is possible to construct an observer that retrieves the system and exosystem states from the measured output y. In this section we address such an observer-based dynamic output feedback controller in more detail. This controller is nothing more than the static state feedback controller (2.14) discussed in the previous section, supplemented by an additional observer. Recall that we only considered systems for which the measured output equals the regulated output, i.e. y = e, C y = C e = C, Q y = Q e = Q and Ψ = 0. A commonly used observer construction technique is to take the original system matrices as the system matrices for the observer. The states to be observed are x and w. The open-loop system describing these states, (2.1), can be rewritten as: q = [ A P 0 S y = [ C Q ] q, ] [ B q + 0 ] u (2.15) where q T = [ q T 1, q T 2 by: ] = [ x T, w T ]. A full state observer for the system (2.15) is now given ξ = [ A P 0 S ŷ = [ C Q ] ξ, ] [ B ξ + 0 ] u + J (ŷ y) (2.16) where ξ T = [ ξ T 1, ξ 2 T ] = [ ˆx T, ŵ T ] is the composed observed system state in which ˆx and ŵ are the observed system and exosystem states, respectively, and ê is the observed (regulated) output. If now J is chosen such that the dynamics for the observer error δ = ξ q, given by δ = ([ A P 0 S ] + J [ C Q ]) δ, (2.17) are asymptotically stable, then the observed system states will asymptotically ([ ] converge to the A P real system states, as desired. However, this does require the pair, [ C Q ]) to be 0 S detectible. Now assume that the necessary conditions for ([ the existence ] of a solution of the ORP are satisfied, i.e. B1 and B2 hold and also the pair, [ C Q ]) is detectable. Then there A P 0 S exists an output feedback controller that solves the output regulation problem i.e. satisfies AA1 and AA2 which is of the form: ξ = Λξ + Υy (2.18a) u = Θξ. (2.18b) The first part of this controller (2.18a) in fact is the observer, whereas the control input (2.18b) in fact is of the same form as the static state feedback controller from the previous section (see

25 Linear Output Regulation: Controller Design (2.14)), where instead of the measured states, the observed states are used for feedback: u = Γŵ + K (ˆx Πŵ) = [ K, Γ KΠ ] ξ = Θξ. (2.19) Again Π and Γ are the solutions to the regulator equations and K is such that the matrix [A+BK] is Hurwitz. Moreover, J is such that the observer error system ([ matrix ] from (2.17) is Hurwitz. A P Such a choice is feasible due to the detectability of the pair, [ C Q ]). 0 S Substitution of (2.19) in the observer dynamics (2.16) yields the dynamic part of the observer based controller (2.18a): ([ ] [ ] A P B ξ = + Θ + J [ C Q ]) ξ J y 0 S 0 (2.20) = Λξ + Υy. For the proof that controller (2.18) based on the observer (2.16) indeed solves the output regulation problem the reader is referred to Appendix A.1. Just as the static state feedback controller discussed in the previous section, the observer based output feedback controller addressed in this section is in general not robust with respect to variations in the system parameters Robust Output Feedback Controller In the previous sections we addressed a static state feedback controller, which requires the whole state [ x T, w T] T to be available for feedback and a dynamic output feedback controller, which only requires the regulated output y to be measured for feedback. Both controllers however are not robust with respect to variations in the system parameters. Because in practice the physical system parameters never exactly equal the nominal values on which the controller is based, a controller which will also perform in the face of small deviations in the system parameters is desirable. In this section we discuss a dynamic output feedback controller that possesses such a robustness property. First the definition of robustness is stated in a more precise manner. Next, the controller and a procedure to construct it are addressed. The proof that the controller presented in this section indeed solves the output regulation problem is included in Appendix A.2. An additional remark should be made with respect to robust dynamic output feedback. As already mentioned in Section 2.1.1, in the following theory it is assumed that the measured output equals the regulated output (i.e. e = y, C e = C y = C, Q e = Q y = Q and Ψ = 0 ). This assumption is actually not necessary for a controller to exist, but makes the construction of such a controller much easier. One is referred to [44] for an extension of the robust output regulation theory in case the regulated output does not equal the measured output. It is a necessary requirement however that the number of outputs does not exceed the number of inputs and that the measured output includes the regulated variable. Here we assume the number of inputs to equals the number of outputs i.e. l e = l y = k. One needs to clearly distinct between robustness with respect to variations in system parameters and robustness with respect to variations in the exosystem parameters. The latter, i.e. robustness with respect to small deviations in the exosystem matrix S from the nominal matrix S 0, cannot be achieved using linear controllers. Using linear controllers the regulated output tends to zero on excitation frequencies corresponding to the zeroes of the closed-loop system. These zeros are determined by the controller parameters, which can be seen from the closed-loop transfer function H (jω) closed loop = H (jω) con H (jω) sys 1 + H (jω) con H (jω) sys.

26 2. Output Regulation Control Theory 15 Hence a linear controller can only yield a zero regulated output for a specific finite set of excitation frequencies. Deviations from this (set of) frequencies (or nominal values of S 0 ) will lead to a nonzero regulated output. Therefore, a linear controller cannot be robust with respect to deviations in the exosystem matrix S. Only if (special) non-linear controllers are allowed, this specific robustness property can be achieved. These controllers however, are outside the scope of this study (see [24] and [31] for further details). Robust ORP statement By only considering robustness with respect to the system parameters, the robust output regulation problem reduces to the problem of finding a controller that solves the output regulation problem, i.e. a controller that satisfies AA1 and AA2 for the nominal system parameters, i.e. for the system: as well as for the physical system parameters, i.e. for the system: ẋ = A 0 x + B 0 u + P 0 w (2.21) e = C 0 x + Q 0 w, (2.22) ẋ = Ax + Bu + Pw (2.23) e = Cx + Qw, (2.24) where matrices A, B, C, P and Q have small deviations from the nominal matrices A 0, B 0, C 0, P 0 and Q 0. Controller Design In the following lemma a controller design is proposed that, under certain conditions, solves the (linear) robust output regulation problem. Lemma 2.4 Assume that the following conditions hold: C1 the pair (A 0,B 0 ) is stabilizable, C2 the pair (A 0,C 0 ) is detectable, [ ] A0 λi B C3 for all λ that are the eigenvalues of S rank 0 is maximal. C 0 0 Then there exists a dynamic output feedback controller of the form: ξ 1 = Φξ 1 + Ne u 1 = Rξ 1, ξ 2 = Kξ 2 + Le, u 2 = Mξ 2, (2.25a) (2.25b) u = u 1 + u 2 = Rξ 1 + Mξ 2, (2.25c) that solves the robust output regulation problem, i.e. satisfies AA1 and AA2 for the nominal as well as for the deviated system parameters. The matrices for the controller (2.25) can be chosen as follows. In general, the matrix Φ is defined as a composition of matrices S min : S min Φ = 0 S min , S min

27 Local Output Regulation for Nonlinear Systems where the matrix S min is the matrix that appears when S is transformed into a block-diagonal form: [ ] 0 S =, 0 S min whose minimal polynomial coincides with the characteristic polynomial of S min. The size of matrix Φ is qk qk, where k is the number of input (and output) components and q is the dimension of S min (for details see [7], page 21]). The matrices R and N are defined (chosen) such that the pair (Φ,N) is controllable and the pair (Φ,R) is observable. There always exist such R and N because of the special structure of the matrix Φ. Furthermore, under conditions C1, C2 and C3, it is always possible to chose the matrices K, L and M such that the unforced closed-loop system is asymptotically stable, i.e. that A cl is Hurwitz. In this case the controller (2.25) and the system (2.21) together yield the nominal unforced closedloop system: ẋ A 0 B 0 R B 0 M x ξ 1 = NC 0 Φ 0 ξ 1 (2.26) ξ 2 LC 0 0 K ξ 2 or ż = A cl z The proof that, under conditions C1, C2 and C3, the controller presented above indeed solves the robust output regulation problem is included in Appendix A.2. Let us discuss the physical meaning of the parts of the robust output regulation controller (2.25) presented here. The first part of the controller (2.25a) is capable of generating the control input u 1 (t) such that a solution x(t) = Πw (t) stays on the set x = Πw on which e = 0, when it is initiated on this set. This requires the control input u 1 (t) = Γw (t). In the state feedback controller this is easily achieved because w (t) is available for feedback. However, since the exosystem output w (t) is not available for feedback here, an internal model of the exosystem is implemented in the controller (2.25a). The basic idea behind an internal model is that it is capable of generating an output equal to the output of the exosystem i.e. the system (2.25a) is capable of generating an output u 1 equal to the output v of the exosystem ẇ = Sw v = Γw. To this end, knowledge about the exosystem (which was one of the base assumptions) is required, which is contained in matrix Φ. The second part of the controller (2.25b) in fact is a dynamic output feedback part that assures attractiveness of the set x = Πw. For further details one is referred to the additional theory on the robust output regulation controller in Appendix A Local Output Regulation for Nonlinear Systems The previous sections addressed the linear output regulation problem, on basis of which the basic output regulation theory has been discussed. The goal however is to implement nonlinear output regulation on the nonlinear TORA system. This section briefly addresses the (local) nonlinear output regulation control theory, which in fact is just an extension of the linear theory discussed in the previous section. A major difference is that the solution and stability properties only hold locally and not globally. Here we only address a nonlinear state feedback controller, which is an extension of the linear state feedback controller discussed in Section 2.2.1, since we will implement such a controller on the TORA setup.

28 2. Output Regulation Control Theory Problem Statement The system description for the nonlinear ORP is given by: ẋ = f (x,u,w) y = h y (x,w) e = h e (x,w), (2.27) which is a generalization of the linear system (2.1). The exosystem description for the nonlinear ORP is given by: ẇ = s(w), (2.28) which is just a generalization of the linear exosystem (2.2). For the nonlinear ORP we assume that: 1. The system and exosystem right-hand sides are continuously differentiable, i.e. f, h y, h e and s are continuously differentiable woth respect to x, w and u. 2. The exosystem has a critically stable linearization. 3. The parameters of the exosystem are known. 4. f (0,0,0) = 0, h y (0,0) = 0, h e (0,0) = 0 and s(0) = 0. Analogously to the linear case (2.3), a general form of a nonlinear output regulation controller is given by: ξ = η (ξ,y) u = θ (ξ,y), (2.29) and η (0,0) = 0, θ (0,0) = 0. The control goals for the local nonlinear output regulation problem are analogue to A1 and A2: D1 The unforced closed-loop system: ẋ = f (x,θ (ξ,y),0) ξ = η (ξ,h y (x,0)), (2.30) has a local asymptotically stable (in the first approximation) equilibrium at (x,ξ) = (0,0). D2 The forced closed-loop system ẋ = f (x,θ (ξ,y),w) ξ = η (ξ,h y (x,w)) ẇ = s(w), (2.31) is such that: e(t) 0 as t for all initial conditions x(0),ξ (0) and w (0) in the neighborhood of the equilibrium.

29 Local Output Regulation for Nonlinear Systems Solvability Analysis A first necessary condition for the nonlinear output regulation problem to be solvable, just as in the linear case, follows from the required asymptotic stability (in first approximation) of the equilibrium (x, ξ) = (0, 0) of the unforced closed-loop system (2.30). The first approximation of the unforced closed-loop system (w = 0) is given by: ż = A cl z, (2.32) where: z = [ x ξ ] [ A + BΨCy BH, A cl = GC y F ], and A = [ ] [ ] f f, B = x (0,0,0) u,, C y = (0,0,0) [ ] hy, F = x (0,0) [ ] η, G = ξ (0,0) [ ] η, H = y (0,0) [ ] θ, Ψ = ξ (0) [ ] θ. y (0,0) From standard linear control theory it follows that system (2.32) is can be made locally asypmptotically stable, i.e. A cl is Hurwitz, by the proper matrices F, G, H and Ψ, if the pairs (A,B) and (A,C y ) respectively are stabilizable and detectible. Analogously to the linear case discussed in Section the local nonlinear output regulation problem is only solvable if there exist C 1 mappings π (w) and γ (w) with π (0) = 0 and γ (0) = 0, satisfying the nonlinear regulator equations: π s(w) = f (π (w),γ (w),w) w 0 = h e (π (w),w), (2.33) for all w in the neighborhood of w = 0. These nonlinear regulator equations (2.33) are the nonlinear counterparts of the linear regulator equations (2.12) and (2.13). In the linear case, solvability of the regulator equations imlpies the existence of a solution x(t) = Πw (t) that is on the invariant center subspace x = Πw, along which e(t) = 0, corresponding to a control input u(t) = Γw (t). In the nonlinear case, solvability of the regulator equations implies the existence of a solution x(t) = π (w (t)) that corresponds to the control input u = c(w (t)) and lies on the invariant manifold x = π (w) on which the regulated output e(t) = 0. Analogously to the linear case, based on the same reasoning, one can conlude that the control goals D1 and D2 are equivalent to the control goals: DD1 The matrix A cl is Hurwitz. DD2 For sufficient small solutions of the exosystem w (t) there exists a solution z (t) such that e (t) = h e (z (t),w (t)) = 0. Basic necessary conditions for the existence of a solution of the nonlinear ORP can thus be stated as: BB1 The pair (A,B) is stabilizable and the pair (A,C y ) is detectible. BB2 The nonlinear regulator equations (2.33) are solvable. The above set of conditions BB1 and BB2 is a set of necessary and sufficient conditions for the existence of a static state feedback controller solving the local nonlinear ORP. For conditions under which the nonlinear regulator equations are solvable one is referred to [7], page 67.

30 2. Output Regulation Control Theory Nonlinear Static State Feedback Control As mentioned before we address a nonlinear static state feedback controller, since this is the controller that is physically implemented in the TORA setup. In the case of static state feedback there is no dynamic feedback part i.e. u = θ (y) and all states x and w are measured and available for feedback: y = h y (x,w) = [ x T,w T] T. Now assume conditions BB1 and BB2 hold. Then there exists a static state feedback controller of the form: u = c(w) + K (x π (w)), (2.34) where π (w) and c(w) are local solutions to the nonlinear regulator equations (2.33) and K is such that matrix A cl = A + BK is Hurwitz. This controller solves the nonlinear ORP locally. From reasoning similar to those in the case of linear static state feedback one can conclude that under conditions B1 and B2, there indeed exists a nonlinear static state feedback controller (2.34) that solves the ORP locally. First we address the control goal DD1, that is A cl = (A + BK) should be Hurwitz. Under condition BB1 this is achieved by a proper choice of K. Second, we address the control goal DD2. As follows from the regulator equations there indeed exists a solution x(t) = π (w (t)), corresponding to the control input u = γ (w (t)), that is on the invariant center manifold described by the graph of the mapping x = π (w), on which the regulated output e(t) = h (x(t),w (t)) is zero,. The nonlinear static state feedback controller (2.34) discussed in this section is section is of the same form and based on the same reasoning as the linear static state feedback controller form Section There are two main differences however: 1. Instead of solutions to a set of linear regulator equations, a solution of a set of nonlinear (partial differential) regulator equations is required, which, in general, is valid only locally. 2. The nonlinear controller is valid only locally i.e. for w (t) and x(t) sufficiently close to zero. 2.4 Summary In this chapter we addressed the basic principles of the output regulation problem and the design of different controllers solving it. For the sake of simplicity we start with the case of linear output regulation and the design of linear output regulation controllers. In the end, the linear output regulation theory and controller design is expanded to the nonlinear case, since the aim is to implement output regulation on the nonlinear TORA. At the beginning of this chapter the (linear) output regulation problem has been specified in more detail. The problem is to find a controller that makes the closed-loop system asymptotically stable in absence of the exosystem and, for all initial conditions on the system and exo-system states, makes the regulated output tend to zero. The main condition for solvability of the ORP is the solvability of a specific set of equations, the regulator equations (2.12) and (2.13). If solvable, the regulator equations, in the linear case, ensure the existence of an invariant subspace, described by x = Πw, corresponding to the control input u = Γw, on which e = 0. Any output regulation controller consists of two parts. One part is capable of generating the steady state control input u = Γw, which ensures that x is on the invariant subspace x = Πw, on which e = 0, once it has been initiated on this set. The other part is capable of rendering the invariant subspace attractive for any initial conditions x(0), w (0). Two types of controllers have been addressed. First, a simple static state feedback controller that requires the full states x and w for feedback is discussed. This controller is neither robust

31 Summary with respect to system parameters uncertainties, nor with respect to exo-system parameter uncertainties. Secondly, two dynamic output feedback controllers have been proposed. One output feedback controller that is based on the static state feedback controller equipped with an observer and one internal model based output feedback controller. The latter is interesting within the scope of output regulation, since it is robust with respect to system parameter uncertainties whereas the observer-based output feedback controller is not. Next, the linear output regulation theory and controller design is expanded to the nonlinear case, since the aim is to implement an output regulation controller on the nonlinear TORA system. The theory and controller design based on the regulator equations for the nonlinear case are similar to the linear case. The main difference that needs to be taken into account is that solutions of the nonlinear regulator equations, in general, are only valid locally, which limits the validity of the controllers to sufficient small values of x(t) and w (t) only. Moreover, whereas in the linear case the solution of the regulator equations follows from a matrix equation, in the nonlinear case a solution of the regulation equations often involves solving a set of partial differential equations. One way to deal with this is to approximate the solutions of the regulator equations (see [20, 21] for details). For some nonlinear systems, and also for the TORA system, it is possible to solve the regulator equations taking advantage of the specific system structure (see [18] for details). The only nonlinear controller that has been addressed in detail is the nonlinear version of static state-feedback controller discussed in the linear case. This is the type of controller that is implemented on the TORA setup. As mentioned above, this controller is only valid locally. Previous research within the group of Dynamics and Control has resulted in estimates of the region of attraction of nonlinear output regulation controllers for the TORA system (see [35]). It would be interesting to experimentally validate these results. The assumptions at the basis of the theory discussed in this chapter are that the exosystem is generating harmonic signals and that the parameters of this exosystem are exactly known. From a practical point of view, however, it is not quite realistic to assume the exosystem parameters to be exactly known. It should be mentioned that there exist nonlinear output regulation controllers that are robust with respect to exosystem parameter uncertainties and hence not need exact knowledge of the exosystem states (see [24, 31] for more detail). These controllers, however, are outside the scope of this study.

32 Chapter 3 Output Regulation on a TORA System The previous section addresses the basic theory of the output regulation problem and some controllers solving it. In this section, an output regulation controller is applied to the benchmark TORA system, as it was introduced in Section 1.2. To start with, the TORA system dynamics are stated, after which the construction of a controller solving the output regulation problem is addressed. Then the characteristics of this controller are studied by simulations, which also give an idea of the control effort that is required to attain output regulation. This is important for the design and construction of the pendulum drive at a later stage. The form of output regulation that we address is the disturbance rejection problem, i.e. a disturbance force acting on the cart in the x-direction is compensated by actuation of the pendulum such that, in steady state, the cart displacement and velocity are zero. The controller that is implemented is a nonlinear static state feedback controller of the form (2.34). The disturbance force to be considered is a simple harmonic signal, generated by an exosystem being a linear harmonic oscillator. 3.1 TORA Dynamics Again, consider the TORA system as described in Section 1.2. Figure 3.1 depicts the TORA system with its characteristic parameters. k T u M x F d z y x l θ m, I Figure 3.1: Schematic TORA system with its characteristic parameters; gravity is in z-direction. The parameters in this figure have the following meaning:

33 Controller Design M = cart mass [kg] m = pendulum mass [kg] l = distance from the rotational joint to the center of gravity of the pendulum [m] I = moment of inertia of the pendulum around the z-axis with respect to [kgm 2 ] its own center of gravity k = spring stiffness [N/m] F d = (disturbance) force acting on the cart in x-direction [N] T u = (actuation) torque applied to the pendulum [Nm] Based on the generalized coordinates [ x θ ] T, the equations of motion of the ideal TORA system (no friction, no other parasitic effects) are derived using a Lagrangian approach: (M + m) ẍ + ml ( θcos θ sinθ) θ2 + kx = F d (3.1) ( ml 2 + I ) θ + (ml) ẍ cos θ = Tu. (3.2) For the sake of notational simplicity, specific mass and inertia terms in these equations of motion are lumped into 3 variables, yielding: ) M M ẍ + M L ( θ cos θ θ2 sinθ + kx = F d (3.3) where M M = (M + m), M L = ml and M J = ( I + ml 2). M J θ + ML ẍcos θ = T u, (3.4) In the particular case of a physical TORA system, in which the cart and the pendulum consist of multiple structural parts, M M, M L and M J can be regarded as: M M = the sum of all translational and rotational masses in the TORA system. M L = the sum of the ml terms of all rotating parts. M J = the sum of the inertias of all rotating parts with respect to the rotational axis. In some cases it is convenient to use a set of normalized equations of motions. A set of suitable normalizations is given, among others, in [4]: x = MM k 1 MM x, τ = t, F = F d, M J M M k M J T = M M k M J T u and ε = M L MJ M M. (3.5) These normalizations render the equations of motions (3.1) and (3.2) dimensionless: ) x + x + ε( θ cos θ θ2 sin θ = F (3.6) θ + ε xcos θ = T. (3.7) From (3.6) and (3.7) it can be seen that the coupling between the rotational and the translational motion in the TORA system is nonlinear and completely determined by the parameter ε. 3.2 Controller Design This section addresses the design of the output regulation controller that is physically implemented in the actual TORA setup. Since it is the first time we physically implement an output regulation controller, we take a relatively simple (nonlinear) static state feedback controller (2.34), which is

34 3. Output Regulation on a TORA System 23 an extension of the linear static state feedback controller discussed in Section As mentioned before, we solve the disturbance rejection problem. We start with the system and exosystem description. Since the output regulation theory discussed in the previous chapter is based on the assumption that the parameters of the exosystem are known, we need to exactly specify the exosystem generating the disturbance. Next, the construction of the static state feedback controller can be addressed, which involves the solution of the regulator equations. System The TORA system description is given by its equations of motion (3.3) and (3.4). The regulated output e = x is the translational displacement with respect to the equilibrium position of the cart. The control input u is the applied pendulum torque T u, and F d is the disturbance force generated by the exosystem (3.10) addressed in the next paragraph. Defining the state x T = [ x ẋ θ θ ], the system of equations regarding the definition of the regulated output and the equations of motion (3.3) and (3.4) can be transformed into a general state-space notation of (2.27): ẋ = x 2 ( ) M J M MM J ML 2cos2 (x 3) M L x 2 4 sin(x 3 ) ML M J cos(x 3 )T u kx 1 + F d M Lcos(x 3) M MM J M 2 L cos2 (x 3) e = [ ] x. x 4 ( ML x 4 2 sin(x 3 ) kx 1 + F d ) + M M M MM J M 2 L cos2 (x 3) T u, (3.8) Exosystem The harmonic disturbance force we initially want to reject is a simple harmonic signal: F d (t) = F 0 sin(ωt + φ), (3.9) which is described by the output of the linear harmonic oscillator: ẇ = Sw F d = [ 1 0 ] w, (3.10) where w = [ w1 w 2 ] [ 0 Ω, and S = Ω 0 ]. The amplitude F 0 and phase φ are determined by the initial conditions w 1 (0) and w 2 (0). Linearization In order to construct a suitable controller, we need the first approximation of the unforced closedloop system, i.e. the linearization of (3.8) for F d = 0. As one can observe from its physical structure, the unforced open-loop system does not have a unique equilibrium, but a set of equilibria that is given by x T eq = [ 0 0 θ eq 0 ]. In other words, standstill of the cart at zero position and standstill of the pendulum at any angle reflect equilibrium positions. The linearization is performed for the equilibrium point x T eq = [ ]. The linearization of the unforced closed-loop system is given by ẋ = A l x + B l T u, e = C l x, (3.11)

35 Controller Design where A l = kmj α km L α 0 0 0, B l = 0 ML α 0 M M α, and C l = [ ], in which α = M M M J M L 2, and is only valid for values of x around zero. Moreover, x and e in the linearized system are the same as in the nonlinear system. Controller Form As mentioned in the introduction of this section, a simple nonlinear static state feedback controller of the form (2.34) is considered. For the TORA system, this controller is given by T u = γ (w) + K (x π (w)), (3.12) where π (w) and γ (w) are solutions of the nonlinear regulator equations and K is such that the pair(a l,b l ) is Hurwitz. Solving the Regulator Equations To obtain π (w) and γ (w), the nonlinear regulator equations (2.33) need to be solved (locally). In other words, we need to find the mappings x = π(w) and T u = γ(w) with π(0) = 0 and γ(0) = 0 satisfying: π Sw = f (π (w),γ (w),w) w 0 = h e (π (w),w), (3.13) for all w in some neighborhood of w = 0. In equation (3.13), f (x,u,w) and h e (x,w) denote the system right-hand side and the output function given in (3.8). In general, solving a set of nonlinear regulator equations is not an easy task. A general procedure to solve nonlinear regulator equations for the class of systems to which the TORA system belongs is based on the systems zero dynamics and is presented in [17]. The TORA system, however, does not satisfy some conditions that are required for this procedure to be valid. In [18], the procedure from [17] is adapted to yield a solution for the TORA system using the specific TORA system structure. For the TORA system in this case the use of formal methods to solve the regulator equations as presented in [17] and [18] is actually not necessary. The solution to the regulator equations here straightforwardly follows from the system description thanks to the fact that the exosystem is a linear harmonic oscillator. The problem of solving the regulator equations (3.13) is nothing more than to find the system states x and input torque T u dependencies on the exosystem states w in the steady-state situation in which e is zero. Moreover, the regulator equations (3.13) are equivalent to the system (3.8) where x = π (w), T u = γ (w), e = 0 and F d = w 1. System (3.8), in turn, is just the state-space formulation of the equations of motion (3.3) and (3.4) supplemented with the expression for the regulated output e. Let x j in steady-state be π j (w), where j = 1,2,3,4 is the index of the states. Then substitution of x = π (w), T u = γ (w) and F d = w 1 in the equations of motion yields: ( ) M M π 2 (w) + M L π 4 (w)cos (π 3 (w)) (π 4 (w)) 2 sin (π 3 (w)) + kπ 1 (w) = w 1 (3.14) By definition: M J π 4 (w) + M L π 2 (w) cos (π 3 (w)) = γ (w). (3.15) π 1 (w) = π 2 (w) (3.16) π 3 (w) = π 4 (w), (3.17)

36 3. Output Regulation on a TORA System 25 and the regulated output: e = π 1 (w) = 0. (3.18) Now, from (3.18) and (3.16) it can be directly concluded that π 1 (w) = 0 and π 2 (w) = 0, which is obvious from a physical point of view: in the steady-state situation the cart is required to be at a standstill at zero position. Substituting the obtained π 1 and π 2 in (3.14) and (3.15) and using (3.17) to rewrite π 4 gives us ) M L ( π 3 (w)cos (π 3 (w)) ( π 3 (w)) 2 sin (π 3 (w)) = w 1 (3.19) M J π 3 (w) = γ (w). (3.20) From the latter equations we can see that π 3 (w), and thus π 4 (w) and γ (w), can easily be solved because of the special structure of the equations. Using the identity β cos β β 2 sin β = d2 dt (sin β) 2 and rewriting w 1 as w 1 = 1 d 2 Ω 2 dt (w 2 1 ), which follows from the structure of the exosystem (3.10), (3.19) reduces to: sin (π 3 (w)) = w 1 M L Ω 2, (3.21) (without loss of generality, integration constants are set to zero) and thus: ( ) w1 π 3 (w) = arcsin M L Ω 2. (3.22) Differentiating this expression with respect to time gives π 4 (w): Ωw 2 π 4 (w) =. (3.23) M 2 L Ω 4 w1 2 Differentiating the expression (3.22) with respect to time twice and multiplying by M J, according to (3.20) yields ( ) γ (w) = Ω2 w 1 M J M 2 L Ω 4 w1 2 w2 2. (3.24) (ML 2Ω4 w1 2) 3 2 Equation (3.24) clearly indicates that the disturbance force term w 2 1 should not exceed the magnitude of the M 2 L Ω4 term in order to prevent the occurrence of singularities. As one can imagine from a physical point of view, this means there exists a maximum to the disturbance force amplitude for which disturbance rejection can be achieved. The individual parts π j and γ presented above together form the complete mappings π (w) and γ (w) that satisfy the regulator equations. Setting Controller Gains To satisfy the requirement of asymptotic stability of the unforced closed-loop system, the gain matrix K has to be such that the system matrix of the first approximation of the unforced closedloop system, A l + B l K is Hurwitz. Since the controllability matrix P C = [ B l A l B l A 2 l B l A 3 l B l ] = M 0 L km α 0 JM L α 2 M L km α 0 JM L α 0 2 M 0 M km α 0 2 L α 2 M M km α 0 2 L α 0 2 is of full rank, the closed-loop poles can be placed anywhere. In practice, a standard pole placement procedure (Matlab: acker) is used to compute a matrix K = [ k 1 k 2 k 3 k 4 ] that places the closed loop-poles at a predefined location in the left-half plane.

37 Simulations Remarks on Output Feedback Since we consider a static state feedback controller, we assume that all system and exosystem states are available for feedback. Therefore, detectability of the pair (C y,a l ), is satisfied by definition. Recall that this detectability property, together with the stabilizability of the pair (A, B) are necessary conditions for the existence of a solution to the (nonlinear) output regulation problem (BB1). One might consider the implementation of an output feedback controller for the TORA system. In that case the detectability of pair (C y,a l ) depends on the specific measured output. If only the cart displacement x is measured and available for feedback, i.e. C y = [ ], the pair (C y,a l ) is not detectable and the output regulation problem cannot be solved using nonlinear counterparts of the output feedback controllers discussed in Sections and If one measures an additional output, for example the pendulum angle, the resulting output matrix C y is given by C y = [ which makes the pair (C y,a l ) detectable. Therefore it is possible to construct an observer-based and/or robust output feedback controller for the TORA system. However, the regulated output in this case no longer equals the measured output. The theory on output feedback controllers in Section and is based on the assumption that the measured output equals the regulated output. Therefore one should be careful when using this theory to construct a nonlinear output feedback controller for the TORA system. Summary We have addressed the disturbance rejection problem for the TORA system, where the disturbance force acting on the cart is a single sinusoidal signal with known frequency and the control input is the torque applied to the pendulum. The proposed controller solving this problem is given by ], T u = γ (w) + K (x π (w)), where: π (w) = 0 0( arcsin ) w 1 M LΩ 2 Ωw 2 M 2 L Ω 4 w1 2 (, γ (w) = Ω2 w 1 M J M 2 L Ω 4 w1 2 w2) 2, (3.25) (ML 2Ω4 w1 2) 3 2 and K is such that A l + B l K is Hurwitz. This controller solves the above mentioned disturbance rejection problem for all system and exosystem initial conditions sufficiently close to 0. If one wants to implement an output feedback controller on the TORA system one has to pay attention that this requires a measured output that is different from the regulated output, which somewhat complicates the construction of such a controller. 3.3 Simulations The TORA system with its output regulation controller discussed in the previous section is simulated using Matlab Simulink. The simulations have been performed for three reasons: 1. to obtain insight in the pendulum motion (amplitudes and velocities) that is to be expected in the experimental setup,

38 3. Output Regulation on a TORA System 27 Table 3.1: Key parameter settings and resulting steady-state ( ss ) and maximum ( max ) control torques. Sim Initial conditions poles F 0 [N] T ss [Nm] T max [Nm] [ #1 0.2, 0, 1 6 π, 0] [ 2 + 5i, 2 5i, i, 1.5 4i] [ #2 0.2, 0, 1 6 π, 0] [ 4 + 5i, 4 5i, i, 3.5 4i] [ #3 0.2, 0, 1 6 π, 0] [ 2 + 5i, 2 5i, i, 1.5 4i] to assess the required (transient) control torques in order to select a suitable pendulum actuator for the experimental setup, 3. to obtain a reference for the behavior of the experimental TORA setup. Because the simulation results are to be compared with the experimental data, simulations are performed using the actual (design) system parameters of m = 6 kg, l = 0.2 m and k = 500 N/m (see section 4.2). In the experimental setup there are additional mass and inertia terms due to the presence of mechanical components linking the mass m to the pendulum shaft and the presence of the pendulum support structure. These are also taken into account. The resulting fundamental TORA system parameters on which the simulations are based are: M M = kg, M L = 1.40 kg m, M J = 0.29 kg m 2. These parameters correspond to the experimental TORA system as it is constructed, loaded with a mass of 6 kg at a 0.2 m radius 1. The resulting dimensionless coupling between rotational and translational motion ε in this case is Note that at the time the simulations are performed, the motor type (and its corresponding dynamics) that will drive the pendulum is still unknown. Moreover, we will use the simulation results at the design stage to select a suitable pendulum drive. Hence the motor dynamics are not included in the above parameter set. Furthermore, we apply a sinusoidal disturbance force with a frequency of 1 Hz, and an amplitude of 10 N (this corresponds to the level of disturbance force prescribed in the design specifications on the experimental TORA setup, which are discussed later). A set of mild closedloop poles [ 2 + 5i, 2 5i, i, 1.5 4i] is chosen such that the resulting controller gains K = [ ] are moderate. We simulate a response for the initial condition of a 0.2 m cart displacement and a 30 deg pendulum angle with both zero cart and pendulum velocity i.e. x 0 T = [ π 0 ]. The results of this first simulation (simulation #1) are indicated with a solid line in Figure 3.2. Satisfactory disturbance rejection is achieved since the cart displacement tends to zero; after 6 seconds the amplitude of the translational oscillation has decreased to non-significant levels. The corresponding steady-state pendulum amplitude is about 10 deg and the amplitude of the steadystate applied control torque about 2.2 Nm, while the maximum control effort during transients is about 8 Nm. The set of parameters for which simulation #1 is performed is a very realistic set of parameters for the experimental setup. However, in order to be able to chose a suitable actuator for the pendulum at the stage of the setup design, we are interested in some more extreme, yet feasible setup configurations that may occur in the experimental TORA setup. 1 The TORA system is constructed such that the pendulum can accommodate different masses at different distances from the axis of rotation. A mass of 6 kg at a 0.2 m radius is the design mass and radius for the TORA system to be constructed. See also Chapter 4.

39 Simulations Simulation #2 (dashed line in Figure 3.2) is similar to Simulation #1, except for the controller gains. In simulation #2 some set of stronger closed-loop poles with smaller real parts, [ 4 + 5i, 4 5i, i, 3.5 4i], is chosen. The corresponding controller gains K = [ ], as expected, yield a faster decrease of regulated output x with respect to Simulation #1. This comes at the cost of an increase in maximum required control torque during transients, which reaches a peak value in excess of 30 Nm. Simulation #3 (dash-dotted line in Figure 3.2) again is similar to simulation #1, except for the disturbance force amplitude. Instead of the standard 10 N, a 20 N amplitude disturbance force is applied. This also yields a slight increase in maximum required control torque during transient, but the major effect is an increase in the amplitude of the required steady state control torque to about 4.5 Nm. Summarizing, one can state that in all three simulations satisfactory disturbance rejection is achieved. The settling time and required control torques during transients and in steady-state depend, as expected, on the controller gains and the magnitude of the disturbance force. Table 3.1 gives an overview of the main parameter settings and the resulting control torque levels for the three simulations. The steady-state pendulum amplitude is about 10 to 20 degrees (depending on the magnitude of the disturbance force), whereas during transients this amplitude can reach about 60 degrees. The pendulum velocities are within a /s range. As a closing remark to this chapter one should note that the construction of the controller and the simulations presented in this chapter are only valid for the ideal TORA system, i.e. a TORA system in which no friction and no other parasitic effects are present. It is obvious that these effects are present in the experimental TORA setup. The way in which these effects are dealt with is discussed in Chapter 5.

40 3. Output Regulation on a TORA System Simulation # 1 Simulation # 2 Simulation # x [mm] 0 x [mm] time [s] time [s] θ [deg] time [s] 100 θ [deg] time [s] 30 θ [rpm] 50 0 θ [rpm] time [s] time [s] Tu [Nm] 0 10 Tu [Nm] time [s] time [s] Figure 3.2: Output regulation for the TORA system based on a static state feedback controller. Cart subjected to harmonic disturbance force. Response for different disturbance force amplitudes and controller gains. Right plots are details of left plots.

41 Simulations

42 Chapter 4 TORA Construction The main aim of this study is to physically implement output regulation on a TORA setup in order to demonstrate the principle of output regulation and to test its applicability in practice. Since a TORA setup is neither commercially available, nor present in our lab, there is a need to construct such a setup. Figure 3.1 schematically depicts the TORA system which exactly represents the system that is to be physically constructed, i.e. the setup must satisfy the equations of motion (3.3) and (3.4). As mentioned in Section 1.2.2, the translational carriage from the existing H-bridge X-Y positioning unit will serve as the TORA cart (see Figures 1.2 and 1.3). This prevents the need to construct such a cart. Moreover, the cart linear motor is ideally suitable to apply a (user defined) disturbance force and to emulate the spring. In order to obtain a TORA system from the H-bridge setup, the cart only needs to be equipped with a suitable eccentric rotational actuator. In practice this demands the design and construction of a pendulum of suitable dimensions (i.e. suitable values of M, m, l and I), a rotational joint with its support structure and the choice and implementation of a suitable actuator for this pendulum. This chapter on the construction of the TORA setup starts with the setting of the design specifications. Subsequently, the, in some sense, optimal values of the design parameters (e.g. M, m, l and I) are determined. Based on these design parameters the torque requirements can be computed and a suitable pendulum drive can be selected. Then the possible TORA configurations (placement of the pendulum on the cart) are addressed, followed by a detailed mechanical design of the desired configuration. A strength and stiffness analysis, in the end, provide an indication of strength and stiffness of the structure. 4.1 Design Specification The main requirement to the TORA system can be stated as follows. The experimental TORA system must be a setup suitable to conduct disturbance rejection and tracking experiments with a variety of output regulation controllers for the largest possible set of initial conditions and disturbance force (or reference trajectory) amplitudes and frequencies. Moreover, the motion of the TORA system must be visually perceptible, i.e. the amplitudes and frequencies of the TORA motions should be such that they are clearly observable with the naked eye for demonstration purposes. Furthermore the system should be cost effective. To narrow this very wide qualitative set of requirements, we initially focus on a design for the case of disturbance rejection. This does not exclude the capability of performing tracking experiments, but it does yield a more feasible design problem. Moreover, a requirement is set on the magnitude and frequency of the harmonic disturbance force that needs to be compensated for. This requirement is the following: the minimum disturbance force acting on the cart, that the pendulum should be capable of compensating for, is such that, in case there is no controller active and the pendulum rotation is locked, the whole of cart and pendulum is in a translational oscillation with an amplitude of at least 25 mm and frequency of 1 Hz. The reason for this roundabout method to

43 TORA Dimensioning specify a disturbance force level, is that the motion of the cart and the pendulum in the TORA setup needs to be visually clearly perceptible. Summarizing, the experimental TORA setup design should meet the following requirements: The setup should be suitable to implement a variety of tracking and/or disturbance rejection controllers. The setup should be such that the pendulum motion has maximum effect on the cart motion, i.e. the setup is such that the a certain pendulum actuation torque yields a maximum pendulum reaction force in X-direction in steady state, as well as during transients. This implies that the range of initial conditions and disturbance force amplitudes that can be handled (from a mechanical point of view) are as large as possible. The setup should be capable of compensating at least a disturbance force resulting in an uncontrolled translational oscillation with an amplitude of 25 mm and a frequency of 1 Hz. The setup should be cost effective. 4.2 TORA Dimensioning From the equations of motion (3.1) and (3.2), it follows that the parameters that determine the TORA system dynamics are the cart mass M, the pendulum mass m, the pendulum inertia (with respect to its own center of gravity) I, the distance l of the center of gravity of the pendulum from the axis of rotation and the spring stiffness k. Moreover, in practice, the gear ratio i of the gearbox that is necessary to drive the pendulum (discussed later in Section 4.3) is a parameter of interest. The question that forms the basis for the TORA design is how the above (design) variables should be set in order to yield an optimal TORA system. To answer this question one needs to specify the desired TORA dynamics and calculate the corresponding system parameters. The optimal TORA design is mainly defined as a design in which the motor torque required for achieving disturbance rejection is minimal. This optimization criterium is explained by the fact that larger torques require larger, more expensive, motors. At the same time, one also needs to maximize the magnitude of disturbance force and the range of initial conditions that can be compensated for by the pendulum. We can thus state the following design problem: Find the values for the parameters M, m, l, I, k and i, such that: the required motor torque to drive the pendulum is minimal; the set of non-zero initial conditions for which output regulation can be attained is as large as possible; the magnitude of disturbance force that can be compensated for is as large as possible. A detailed analysis of this design problem is addressed in Appendix B. Here only the results and their consequences for the TORA design are stated. 1. Small cart mass M and pendulum inertia I are desirable in order to keep the required motor torque to a minimum. In the final TORA setup design the value of M is kg. The pendulum inertia I is minimized by constructing a pendulum consisting of a point-mass (compact mass of high-density material) connected at some distance to the pendulum shaft using a light, low-inertia structure. 2. The optimum of m and l is considered to be that of a 6 kg mass at a distance of 0.2 m from the pendulum shaft. However, the pendulum should be capable of accommodating a range of masses m and pendulum lengths l, as specified by table B.2.

44 4. TORA Construction The gear ratio i to be used in the transmission (see Section 4.3) that connects the motor to the pendulum shaft is 1: The spring stiffness k can be chosen virtually arbitrary. In the analysis of the design problem k is set to the realistic value of 500 N/m. 4.3 Pendulum Drive A very important part of the TORA setup that will largely determine its costs and performance, is the pendulum drive. This drive encompasses all hardware components between the existing dspace signal processing system which hosts the controllers and the shaft of the pendulum. The actuator for the pendulum is an electric motor. No other motor types (hydraulic, compressed air) have been considered because the use of an electric motor has two major advantages over its alternatives: good controllability and ease of practical implementation (clean, no need for oil/air hoses and compressors/pumps). Moreover, the pendulum is actuated at its shaft. Alternatives such as, actuation of the pendulum via a cam-shaft or via some linkage mechanism, have not been considered because of their constructional complexity and lack of distinct advantages over actuation at the pendulum shaft Drive Specifications In order to select a suitable pendulum drive, the pendulum load profile, i.e. the pendulum torquespeed characteristics are determined. Since the simulations discussed in Section 3.3 have been performed based on the parameters of the actual experimental TORA system (M M, M L and M J ) and a realistic value of the spring stiffness of k = 500 N/m, they form a good basis to obtain a realistic load profile. Therefore, the simulation results in figure 3.2 are extended with a graph of the required torque at the pendulum shaft versus the pendulum speed. Figure 4.1 displays the resulting load characteristics for the most torque-demanding simulations, simulation #2, with the somewhat faster poles and simulation #3 with the larger disturbance force amplitude. It simulation #2 simulation # torque [Nm] 0 10 torque [Nm] speed [rpm] speed [rpm] Figure 4.1: Pendulum load characteristics for simulation #2 and #3. should be noticed however, that the torque levels in figure 4.1 are realistic required torques at the pendulum shaft, without taking into account any motor. The actual required motor torques will be higher, because of the additional motor mass m mot, which contributes to the cart mass

45 Pendulum Drive M and the presence of rotor (and gearbox) inertia. This can be seen from the expression for the maximum required steady-state motor torque, which is given by: T m ss(max) = ( ) ml 2 + I + i 2 I mot F0 Ω 2, (4.1) i (ml) 2 Ω 4 F0 2 where F 0 = xspec, 0 ( k (M + m) Ω 2 spec ) and Imot is the motor and gearbox inertia with respect to the motor shaft. The angular frequency Ω spec and amplitude x spec, 0 are 2π rad/s and 25 mm, respectively, and correspond to the motion induced by the disturbance force for which disturbance rejection needs to be attained, as prescribed by the design specification (see Section 4.1). Equation (4.1) follows from the equation for the maximum required steady-state motor torque (3.24), which is derived in Appendix B (see formula (B.6)). Also the required peak torque during transient depends on the motor inertia and mass, but this dependency cannot be expressed analytically. Based on the load characteristics from simualtion #2 and #3 in Figure 4.1, one can conclude that the maximum required pendulum speed is 100 rpm, the maximum required peak torque is 35 Nm and the maximum required steady-state torque amplitude is about 5 Nm. In this case the latter corresponds to an RMS value of about 3.5 Nm 1. The raw peak torque and velocity requirements are quite extreme compared to capabilities of the range of motors we are looking at (with- or without the use of a transmission). We permit ourselves, however, to weaken the demands on peak torques and speed to 20 Nm and 60 rpm, respectively. From figure 4.1 one can conclude that this does not conflict with the steady-state torque requirements, but only yields an input saturation during transient in the more extreme situations of high controller gains. The input saturation during transients might limit the range of initial conditions for which output regulation can be attained; this, however, is considered to be acceptable. Additional requirements on the pendulum drive follow from the analysis of the TORA dynamics (see Section 4.2), which indicates that the mass and inertia of the pendulum drive should be kept to a minimum. From a control point of view, friction and backlash should be minimized, or if possible, avoided at all. Besides requirements on the pendulum actuator, there are also requirements on the electronic part of the pendulum drive: the amplifier. Since the pendulum torque is considered to be the TORA system input, we require to prescribe a torque to the pendulum shaft. Using an electric motor this implies the prescription of a certain current. The amplifier should therefore have a current control capability. Moreover, the amplifier should have a reference input (digital or analogue) in order receive a reference signal from the dspace system which hosts the controllers. For feedback purposes we need measurement of the pendulum angle θ and the pendulum velocity θ. The last, but important, requirement on the pendulum drive is that is should be easily controllable. Summarizing, the requirements to the pendulum drive are: 1. maximum speed of at least 60 rpm at the pendulum shaft, 2. minimum peak torque (around zero velocity) of 20 Nm at the pendulum shaft, 3. minimum RMS continuous torque of 3.5 Nm at the pendulum shaft, 4. capable of operating in 4 quadrants (positive and negative torque in both positive and negative direction of rotation), 5. minimum mass and inertia, 6. minimum friction and backlash, 1 Since electric motors are specified by the RMS value of their continuous torque capability, we need to specify the continuous pendulum torque requirement as a RMS value as well.

46 4. TORA Construction 35 Table 4.1: Comparison between a brushed and a brushless DC motor. Feature brushless DC motor brushed DC motor Commutation electronically, generally based on Hall position sensors mechanically, via brushes Life span long shorter (brushes subjected to wear) Power / weight ratio high moderate Efficiency high moderate Torque characteristics linear with current almost linear (torque loss at high speed due to brush friction) Control electronics 3 phase invertor standard current amplifier Control complexity complex (electronic commutation) simple Cost high low 7. torque (current) control capability, 8. easily controllable Drive Selection There are multiple questions that need to be answered in order to select a suitable pendulum drive. First, what type and size of electric motor is required? Second, is there a need of a transmission between the motor and the pendulum shaft, and if so, what type of transmission and transmission ratio is preferable? Third, the selection of a suitable amplifier and other considerations that play a role need to be taken into account. In the remainder of this section these questions will be addressed. The selection of a suitable motor is not a straightforward process of just looking for a motor that matches the (torque) requirements, because the requirements itself depend on the motor parameters. Recall that the required steady-state motor torque depends on the motor mass and motor inertia (see (4.1)). The way in which this difficulty is dealt with, is discussed in more detail in Appendix C. In this section, only the resulting options and a final selection is presented. Motor Type Commonly used in servo systems are brushed DC motors, brushless DC motors and stepper motors. For a more detailed description of the principles of these motors one is referred to [10] and [36]. A stepper motor, because of its physical structure, can be position- or velocity- controlled, but it is very uncommon to be torque-controlled. Since the pendulum needs to be torque driven, stepper motors are considered not suitable for application in the pendulum drive, which leaves us with the choice between a brushed or brushless DC motor. Table 4.1 gives an overview of the main advantages and disadvantages of brushless DC motor versus those of a brushed DC motor. From the advantages of a brushless DC motor over its brushed variant mentioned in table 4.1, only the higher torque to weight ratio and the exact linear behavior are of interest for the pendulum drive. The main advantages of the brushed DC motor are its low cost and simple control. The fact the in the DCT lab a wide variety of standard analogue and digital current amplifiers for brushed DC motors is available, as well as the experience on the use and control of these motors, adds to the advantages of the use of a brushed DC motor. A more complex and expensive 3 phase current amplifier to drive a brushless DC motor is not available and would have to be purchased.

47 Pendulum Drive Therefore, despite its lower torque to weight ratio and the fact that it is not exactly linear, a brushed DC motor is preferred over a brushless version (if the torque requirements allow the use of such a motor). Drive Configuration Requiring a peak and continuous torque of respectively 20 Nm and 3.5 Nm with a maximum speed of a 60 rpm makes the actuation of the pendulum a typical case of a high-torque low-speed application. If possible, one wants the pendulum shaft to be directly driven, i.e. the motor should be directly connected to the pendulum shaft without the use of any transmission (see Figure 4.2(a)). The advantages of a direct drive are huge: it avoids the introduction of friction, loss of stiffness High res Encoder Large Brushless DC Motor Low res Encoder Small Brushed DC Motor Low res Encoder Small Brushed DC Motor Self built Pre-loaded transmission Standard Gearbox Flexible Coupling Flexible Coupling Flexible Coupling High res Encoder Bearing Bearing Bearing Bearing Pendulum Shaft Pendulum Shaft Pendulum Shaft Pendulum Shaft Ringtoque motor Bearing Bearing Bearing Bearing (a) (b) (c) (d) Figure 4.2: Possible drive configurations. and, most important, the introduction of backlash - these effects are almost inevitably connected to the use of a transmission. However, because of the high-torque, low-speed application we are dealing with, a motor that directly drives the shaft needs to be capable of generating the required large peak torques. In general, this makes such a motor a large and heavy motor. In Appendix C, the selection of suitable motors for the use in a direct drive configuration is addressed. The conclusion from this analysis is that in direct drive configurations only brushless DC motors can be applied, since brushed DC motors have a too low torque to weight ratio. This conflicts with our preference for a brushed DC motor. Another disadvantage connected to the direct drive solution is that one needs a high resolution encoder because the motor shaft has the same (low) speed as the pendulum shaft. A special motor that is ideally suitable for the use in a high-torque low-speed direct drive configuration is a so called ring-torque motor as depicted in Figure 4.3. Ring-torque motors are so-called frameless motors; rotor and stator assembly come separately and need external bearings. In general, they are directly fitted onto the shaft that is driven; there is no need for a coupling.

48 4. TORA Construction 37 Because of their large diameter (typically 160 mm for this application) and extreme high torque to weight ratio they can smoothly generate high torques and provide excellent stiffness. They are available as brushed and brushless variants. Figure 4.2(d) schematically shows a pendulum drive configuration based on a ring-torque motor. Instead of the direct drive solution one can also choose to drive the pendulum via a transmission, introducing a transmission ratio. This reduces the motor torque requirements and thus the size, weight and cost of the required motor and, due to the lower torque demands on the motor, allows the use of a brushed DC motor. Disadvantages connected to the use of a (standard) transmission are the introduction of backlash, additional friction, additional inertia, and also the loss of stiffness. The problem of backlash in a standard transmission can be overcome by using a backlash-free preloaded transmission. Since such a transmission is not commercially available for the (large) torques we need to accommodate, it would have to be custom-built. Figure 4.2(c) and 4.2(b) schematically show a pendulum drive configuration based on small brushed DC motor with a standard gearbox and a custom-built transmission, respectively. Selection For both the direct drive configuration and the indirect drive configuration Appendix C addresses the selection of a motor that suits the torque and speed requirements. The resulting motors for the direct drive application are a standard brushless DC motor, the Kollmorgen AKM52G or a brushless DC ring-torque motor, the Kollmorgen F4309A. For the indirect drive a small brushed DC motor, the Maxon RE40 with a transmission ratio of 1:113 suits the torque requirements. The transmission can be a standard low-backlash planetary gearhead, the Maxon GP42C, or a custom-built backlash free transmission. Table 4.2 gives a comparison of the 4 resulting pendulum drive configurations. From Table 4.2 one can conclude that the direct drive configurations are preferable because of their technical superiority: the absence of backlash and minimum friction. The main disadvantages of a direct drive configuration are its cost and the compulsory use of a brushless motor. The only advantage that a custom-built preloaded transmission has over a standard transmission is its lack of backlash. However, the construction of such a transmission is very time consuming and expensive. Moreover, from simulations similar to those described in Section 3.3, it follows that the backlash of the standard low-backlash planetary gearhead does not significantly influence the performance of output regulation controllers for the TORA system. Concluding, we can state that the desired drive configuration is the configuration consisting of a small brushed DC motor with a standard gearbox, as depicted in Figure 4.2(c). In particular this configuration consists of the Maxon RE40 brushed DC motor in combination with a Maxon GP42C low-backlash planetary gearhead (detailed technical specifications are included in Appendix F), with a gear ratio of 1:113 (see also optimization procedure, Appendix B). Moreover, Appendix C discusses this specific motor and gearbox choice in more detail. Position measurement is provided laminated stator permanent magnet rotor Figure 4.3: The structure of a ring-torque motor.

49 TORA Configuration Table 4.2: Drive configurations comparison. Feature DD standard DD ring-torque IDD standard IDD custom motor Kollmorgen AKM52G Kollmorgen F4309A Maxon RE40 build Maxon RE40 + type DC brushless DC brushless DC brushed DC brushed transmission none none planetary gear custom build Maxon GP42C preloaded gear ratio - - 1:113 1:100 need of coupling yes no yes yes backlash none none at pendulum none friction limited very limited large (up to 30% large of torque input) weight [kg] a inertia (% of total inertia) ? structural complexity low high (very tight low high tolerances) amplifier 3-phase inverter 3-phase inverter 4-quadrant current 4-quadrant current amplifier availability purchase purchase in DCT lab in DCT lab encoder high resolution high resolution standard standard total cost indication a estimate by a Maxon optical incremental encoder, the Maxon HEDL55 (detailed technical specifications can be found in Appendix F). The motor, gearbox and encoder come as one unit, which does not need any other assembling than mounting the gearbox to the cart. A 4-quadrant digital or analogue amplifier available in the DCT lab is used to power the motor. 4.4 TORA Configuration As mentioned in the introduction of this chapter, the H-bridge cart (see figure 1.3) needs to be fitted with a pendulum. From the TORA dimensioning discussed in Section 4.2, it follows that this pendulum primarily consists of a 6 kg mass attached at 0.2 m from the axis of rotation. However, the pendulum should also be capable of accommodating masses varying from 2 to 8 kg at distances from the axis of rotation varying from 0.1 to 0.4 m (see table B.2). One can think of 4 possible ways to attach the pendulum to the cart. Figure 4.4 shows the cross section of the cart on its X-guidance (for θ = 0), including some basic dimensions in mm. In this figure the 4 different pendulum configurations are indicated. Three configurations consist of a vertical rotational shaft, attached to the front (or back) of the cart. The pendulum can be attached to this shaft such that it is completely below, aside, or above the whole structure of cart and X-guidance (configuration 1, 2 and 3 respectively). Another possible configuration is not to have a simple shaft, but a large hollow shaft, enclosing the whole cart (configuration 4) such that the center of rotation of the pendulum coincides with the cart center. It is desirable to minimize the asymmetric cart load; firstly, to prevent the cart from toppling

50 4. TORA Construction 39 p F p x F p y 3 m F m z Cart electronics b3 2 m F p y p F supp z Cart Linear Motor y p z Fx p x F p y 4 m F m z b4 F m z F p x X guidance 0 b b1 p F p x F p y 1 m F m z a 1,3 a 2 a 4 Figure 4.4: Cross section (for θ = 0) of different TORA configurations. Points p denote the centers of rotation, points m denote the centers of gravity of the different pendulum configurations. over, but also to prevent unequal friction in the left and right guidance. The main cause of an asymmetric cart load, is the moment Mx 0 at the same time the moments My 0 and Mz 0 (all around the origin O) also cause the cart load to be asymmetric. These moments are generated by the reaction forces Fx, p Fy p that the pendulum exerts on the cart as well as the gravitational forces Fz m and Fz supp of the pendulum and its support structure, respectively. The magnitude of these moments depends on the specific TORA configuration via the distances a i and b i. In this perspective configuration 2 is undesirable. In this configuration the moment Mx 0 is rather large with respect to the other configurations due to a large value of a 3 (see Appendix D and especially Section D.4 for a quantitative analysis). Moreover, in configuration 2, the pendulum has no full range. Configuration 3 has the advantage of full cart stroke in combination with a full pendulum range, because the pendulum is not bound by the Y-axis supports. However, this configuration is considered to be too dangerous. The pendulum is swinging at eye height above the laboratory floor (1.8m), without any physical end-stop. Another danger is that the cart, because of its high center of gravity, will topple from its guidance in case it violently hits its end-stops. Moreover, the safety cap that can be placed on top of the whole H-bridge does not fit anymore if the pendulum is placed on top. Configuration 4 is considered to be to structurally too complex. The large diameter shaft cannot

51 Design Considerations be supported by a large single bearing, because this bearing would be too heavy. Instead, a complex structure of a preloaded array of multiple smaller bearings along the shaft perimeter would be required to support the shaft. Configuration 1 does not have the disadvantages mentioned above. In configuration 1 the moments M 0 x, M 0 y and M 0 z are kept to a minimum because of small values of a 1 and a 2 (again, see Section D.4 for details) and the pendulum has full range. Moreover, in configuration 1 he pendulum is encaged within the H-bridge base frame (see Figure 1.3). This forms a solid physical end-stop, which makes the pendulum safer to operate. The only disadvantage of configuration 1 is the limited cart stroke. Because the width of the space between the two Y-axis supports is 1 m and the maximum pendulum length taht needs to be accommodated is 0.4 m, there is only a 0.2 m cart stroke available at which the pendulum has full range. Yet however, this is considered to be sufficient. Concluding, we can state that configuration 1 is the best option most desirable. The TORA system is designed according to this configuration. 4.5 Design Considerations Figure 4.5(a) depicts a 3D design of the TORA system based on configuration 1. The actual TORA system that has been built according to this design is depicted in Figure 4.5(b). Figure 4.6(b) depicts a cross section (Y-Z plane) of the design of the additional hardware mounted in the cart. An overview of the part numbers and their names is given in Table 4.3. Below follows a description of the general outline of the design. The next subsections address particular design features in more detail. Only the pendulum shaft is discussed in a separate section (Section 4.6) because the shaft mainly determines the pendulum strength and stiffness characteristics. For detailed component drawings, including exact dimensions, one is referred to Appendix G. For values of masses and inertias of the different components one is referred to Table H.2 in Appendix H. As a design tool the 3D CAD package UNIGRAPHICS (UG) has been used z x y (a) (b) Figure 4.5: The TORA system: design (a) and the actual setup (b).

52 4. TORA Construction 41 Table 4.3: Part names and numbers. Part No. Part Part No. Part Part No. Part 1 pendulum table 9 motor support 17 weight large 2 pendulum side 10 clamp-hub block 18 weight half 3 pendulum shaft 11 clamp-hub 19 bearing-lock-nut 4 bearing house 12 connection block 20 angular contact bearing 5 support plate 13 lower pendulum lock 21 deep groove bearing 6 base plate 14 torsion tube 22 lock ring 7 cart translator 15 connection tube 23 X-guidance 8 mounting block 16 motor-gearbox assembly 24 flexible coupling The pendulum itself (2, yellow) is clearly visible in Figure 4.5(a). Weight, consisting of a stack of steel plates (17, 18, red), can be mounted on the pendulum at different positions. The pendulum is attached to a vertical shaft (3, red) at its upper and bottom side. The shaft is supported by a set of ball bearings, which are housed in the bearing house (4, gray vertical square tube ). The shaft runs through the bearing house from the bottom to the top of the bearing house. The bearing house is mounted to the cart by means of a baseplate (6, brown) that is bolted on top of the linear motor compartment of the cart (7, black). A support plate (5, sea-green) connects the lower end of the bearing house to the base plate in order to obtain a stiff transfer of pendulum reaction forces and moments F p x and M p y to the cart. Two additional mounting blocks (8, blue) connect the bearing house directly to the side of the cart in order to accommodate reaction forces and moments F p y and M p x. The motor and gearbox assembly (16, dark green) is mounted up side down onto the motor support (9, green). Inside this motor support there is a flexible coupling that connects the gearbox output shaft to the pendulum shaft allowing small misalignments between the two Pendulum In the previous paragraph the general outline of the pendulum and its support structure design is discussed. This section lists some design details concerning the pendulum itself; the yellow structure in Figure 4.5(a). More detailed drawings of the pendulum structure are presented in Figure 4.7. The pendulum structure consists of a table (1) onto which steel plates can be stacked to a height corresponding to a desired total weight. The stack can be easily attached to the table at distinct distances from the shaft (3) (determined by the hole pattern in the table) with the use of long, 6 mm bolts, fitted with wing-nuts. The pendulum table consists of a 5 mm aluminium plate. Two versions of the steel weight plates are available to provide maximum flexibility in pendulum load configuration. The large plates (17) measure mm and weigh g. The half plates (18) measure mm and weigh 93.2 g. Underneath the pendulum table there are two vertical aluminium pendulum sides (2) to accommodate gravitational forces and related moments (see Figure 4.5(a)). The lower rims of the pendulum sides are folded inwards over a 90 0 angle. This provides higher strength and stiffness of the sides with respect to gravitational forces and their resulting moments and prevent the side plates from buckling. The connection between the pendulum table and the pendulum sides is provided by a hollow aluminium square tube (15). This square tube is riveted both to the pendulum table and to the pendulum sides (see the pendulum front view, Figure 4.7(c)). The open structure of table and pendulum sides is closed by an anti-torsion tube (14) that connects the two pendulum sides (see Figure 4.6(b)). This large diameter tube, which is kept in place by six 6 mm bolts, prevents rotation of one pendulum side with respect to the other. The upper interconnection of the pendulum and the shaft, which is depicted in detail in Figure 4.7(a), provides the transmission of all pendulum torques T p, reaction forces F p x and F p y and

53 Design Considerations B A p 1 F p x F p y 11 F p z (a) (b) Figure 4.6: Pendulum design cross section (Y-Z plane). Total of pendulum, support structure and motor (b), bearing and flexible coupling detail (a). gravitational forces F p z from the pendulum table and pendulum sides to the shaft. The cross section view in figure 4.7(a) clearly indicates the working principle of the clamp hub (11) that is applied to realize the upper interconnection. Conical surfaces pressed together by 6 bolts on the clamp perimeter provide a radial clamping pressure between the clamp hub and the shaft and between the clamp hub and the aluminium block (10) that houses the clamp hub. Appendix F contains the full technical specifications of the clamp hub. With a torque rating of 350 Nm and a permissable axial load of 28 kn, the clamp hub applied in the upper pendulum - shaft connection suffices to accommodate the maximum design loads (see table D.1). The lower interconnection of the pendulum and the shaft, which is depicted in Figure 4.7(b), only accommodates reaction forces in y (and x) direction resulting from the gravitational moments M p x (and M p y). It consists of two simple aluminium blocks (12) and (13) that enclose the shaft when bolted together.

54 4. TORA Construction A 11 A Bottom view Crossection AA 13 3 Bottom view 3 12 Front view (a) (b) (c) Figure 4.7: Upper pendulum-shaft connection bottom view and cross section (a), lower pendulumshaft connection bottom view (b), pendulum front view (c) Bearings To enable the pendulum shaft to rotate smoothly one needs (ball) bearings of suitable types applied in a suitable configuration. The bearing configuration that is applied in the TORA system is depicted in Figure 4.6(a). The size of the bearings is primarily determined by the shaft diameter, which is 25 mm (see Section 4.6.4). Since all bearing types of this size have load ratings far beyond the applied range of loads, bearing load capacity is not an issue of concern. The shaft is loaded radially (Fx, p Fy p ), axially (Fz p ), with moments Mx p and My p resulting from the gravitational pendulum force Fz m and with moments induced by the reaction forces F p i via the distance A p. The shaft needs to be radially supported by bearings at two positions as far apart from each other as possible, the lower of which is as close to the pendulum table as possible. This provides a bearing configuration with minimal bearing loads, that is as stiff as possible.the axial displacement of the shaft should be locked in both directions at either the upper (set of ) bearing (s) or the lower, but not at both. As can be seen in Figure 4.6(a), a set of two angular contact bearings (20) in X-configuration just above the pendulum table fixes the point A, thus supporting the shaft radially, as well as vertically (in two directions), while still permitting slant of the shaft with respect to the vertical. The latter is suppressed by the upper bearing, a deep-groove bearing (21). This bearing supports the upper shaft part radially but does not lock the shaft axially because of its loose fit onto the shaft. The bearing clearance of the lower set of angular contact bearings, which mainly determines the radial and axial pendulum clearance, can be adjusted by turning the bearing-lock-nut (19). This nut is screwed into the lower bearing house end. The upper bearing is axially locked with respect to the bearing house by means of a standard lock-ring (22). The bearing configuration consisting of a pair of angular contact bearings together with a deepgroove bearing is often used for one-end radial and axial loaded shafts (for example in a milling machine spindle). The advantage of the bearing configuration as applied in the TORA setup is that it neatly supports the shaft and that radial and axial clearance can be eliminated by pre-loading the lower set of angular contact bearings. A disadvantage of this configuration, however, is that it

55 Design Considerations requires tight (radial) alignment tolerances on the position of the upper and lower bearings, which slightly complicates manufacturing. The use of self aligning bearings, would have overcome this drawback because they do not require precise alignment. Self-aligning bearings, however, provide other disadvantages: larger bearing diameters (which would lead to larger and heavier bearing houses), less suitable for axial loads and larger (axial) clearances. Hence the bearing configuration depicted 4.6(a) is applied in the design Support Structure Figure 4.8 depicts the main components of the pendulum support structure that keeps the pendulum in place and provides the transfer of the pendulum reaction forces to the cart. The interface between the cart and the pendulum is formed by the aluminium baseplate (6) that is bolted on top of the lower cart part (7). The latter is a quite solid aluminium block that forms the linear motor translator and accommodates the linear motor coils. The cart upper structure (see Figure 4.5(b)) accommodates the on-board X-sled electronics. It is constructed from thin aluminium sheets and is bolted on top of the base plate. The presence of the baseplate allows to avoid the utilization of the poor mounting facilities 2 provided by the cart itself. Moreover it enables the user to easily (de)mount the pendulum in case one needs to operate the H-bride without a pendulum attached. Another advantage is that the use of a baseplate prevents the need for any machining operations on (parts of) the H-bridge itself. Because deep-groove and angular contact bearings only have a limited capability of accommodating misalignments, the upper and lower bearing(s) are housed in the same aluminium bearing house (4). In order to provide easy mounting interfaces with the other TORA components, the bearing house outside is square. Although difficult to machine, a one-piece large bearing house does facilitate the design, since there is no need for an additional structure connecting the upper and lower bearing house. 6 F B y B 8 F B x 4 x z 5 y 20 A F A y 20 F A x 19 F A z Figure 4.8: Pendulum support structure, shaft is not visible, bearing house is cut. 2 Mounting facilities on the translator are poor since the aluminium motor housing only provides some small M4 threaded holes, many of which are corrupted. The aluminium plates of the upper cart structure also provide poor mounting facilities because of the absence of suitable mounting holes, its non-stiff structure with respect to some load directions and its non-reproducible mounting on the translator.

56 4. TORA Construction 45 The starting point of the pendulum support structure design is the position of the lower and upper bearings (Figure 4.8, positions A and B, respectively). From these positions, the bearing loads Fi A and Fi B (resulting from the pendulum reaction forces) need to be transferred to the cart. The gravitational force Fz A is transmitted from point A to the base plate via the bearing house. Since the bearing house is bolted underneath the baseplate almost against the cart (7), the gravitational force virtually only induces shear stresses in the baseplate. The upper and lower bearing loads Fy A and Fy B and their resulting moments with respect to the cart are accommodated by the tensile and compressive stresses (in Y-direction) in the baseplate and the aluminium mounting blocks (8). These mounting blocks are bolted to the sides of the bearing house and onto the back of the cart using threaded holes that are present in the translator. Because of the vertical distance between the position A, where Fy A is applied, and the mounting blocks, the bearing house will be loaded with moments M x. This is rather undesirable, since the bearing house should guarantee the alignment of the upper and lower bearings. The bearing house, however, is stiff enough with respect to these moments and the resulting bearing house deflections are too small to endanger the bearing alignment. The lower bearing load Fx A and its resulting moments M y with respect to the cart is directly transferred to the cart via the aluminium support plate (5) Motor Attachment To attach the gearbox to the pendulum shaft, the use of a flexible coupling (24) is required to accommodate the inevitable radial and angular misalignments between the pendulum and gearbox shaft (see Figure 4.6(a)). A suitable coupling is considered to be a steel bellow coupling, type ROBA-DX , size 1, manufactured by Mayer power transmission. Its maximum torque capacity of 60 Nm suffices to accommodate the the maximum pendulum design torques of about 50 Nm (see Table D.1). Moreover, the coupling torsional stiffness of Nm/rad is approximately of the same magnitude as the stiffness of the other components in the pendulum drive-train (pendulum shaft, motor shaft, etc.). Therefore the coupling does not significantly deteriorate the overall pendulum drive-train stiffness (see Section for details on stiffness). The full technical specifications of the coupling are included in Appendix F. Figure 4.9: A ROBA-DX steel bellow coupling. Figure 4.6(a) depicts the attachment of the motor to the TORA structure. Since the distance A-B is required to be as large as possible (see Section 4.5.2), the deep-groove bearing (22) is placed just below the base plate. Because of the height of the flexible coupling, the motor cannot be directly mounted onto the baseplate. A motor support (9), consisting of a thin-walled aluminium cup, bridges the distance between the motor and the baseplate. The motor support provides a (torsionally) stiff transmission of the pendulum torques T p to the base plate Material The material that is used for all components, except for the shaft, is aluminium alloy (ST51). Since the strength requirements are not very demanding the use of the relatively weak aluminium (σ yield = 280 MPa) instead of steal (σ yield = 400 MPa) is allowed for most parts. The main reasons for the use of aluminium, however, are its low mass density (ρ Al = 2720 kg/m 3, ρ St = 7850 kg/m 3 ) and its good machining properties. Another advantage connected to the use of aluminium over the use of steel is the higher stiffness with respect to bending of plates. However the modulus of elasticity of aluminium is about three times less then that of steel (E Al =69 GPa, E St =207 GPa): with respect to a steel plate of the same mass and area, an aluminium plate is about three times as thick. From linear elastic (beam) theory it follows that that the plate stiffness with respect to bending (out of the plane) proportionally depends on the term Et 3, where E is the modulus of elasticity and t is the plate

57 Strength and Stiffness J 1 J 2 J 3 J 4 c 1 c 2 c 3 c 4 c 5 c 6 (a) c 2 c 3 c 4 c 5 J 4 (b) Figure 4.10: Dynamic model of the pendulum drive-train (all stiffness and inertias reduced on pendulum shaft). Full model (a), and the reduced model (b). See table 4.4 for parameter legend. thickness. Hence the bending stiffness of an aluminium plate is 9 times larger than that of a steel plate of the same mass and area. The latter fact is a consideration of special interest when selecting a suitable material (aluminium) for the pendulum sides (2) Construction All components of the pendulum and its support structure discussed in the previous section have been designed using UG. Based on the resulting detailed design drawings (included in Appendix G), the design has been realized by the central University workshop, the GTD, without any problem. 4.6 Strength and Stiffness This section briefly addresses the strength and stiffness of the TORA design as depicted in Figures It should be emphasized that the aim is not to exactly calculate the strength and stiffness of the TORA structure, but only to obtain estimates thereof. The estimates form an indication whether or not strength or stiffness problems are to be foreseen. The tools that are used are linear elastic beam theory and some simple straightforward rigid body dynamics. Special attention is paid to the pendulum shaft. The shaft material and shaft dimensions are mainly determined by its required (torsional) stiffness. The pendulum shaft is a solid steel shaft with a diameter of 25 mm, which forms the basis for all calculations in this section. A motivation for the selection of this shaft is included at the end of this section. Only the essence of the strength and stiffness calculations are included in this section. Some basic aspects of the linear elastic beam theory that are applied in this section are addressed in Appendix E. Moreover, in this appendix special attention is paid to calculations on the pendulum shaft Torsional Stiffness Since the pendulum rotation is the actuated degree of freedom in the TORA structure, the rotational stiffness of the pendulum and all components in the drive-train mainly determine the pendulum s dynamic behavior. Figure 4.10(a) depicts a dynamic model of the pendulum and

58 4. TORA Construction 47 Table 4.4: Values of torsional stiffness and inertia of the components in pendulum drive-train. Part Full name Scale Value Origin c 1 servo stiffness i 2 c servo Nm/rad unknown, considered rigid c 2 gearbox stiffness Nm/rad unknown, estimate c 3 gearbox shaft stiffness Nm/rad calculation (see Appendix E) c 4 coupling stiffness Nm/rad according to manufacturer spec. c 5 shaft stiffness Nm/rad calculation (see Appendix E) c 6 pendulum stiffness Nm/rad calculation (see Appendix E) J 1 motor and gearbox inertia i 2 I mot 0.20 kg m 2 according to manufacturer spec. J 2 coupling inertia kg m 2 according to manufacturer spec. J 3 pendulum shaft inertia kg m 2 according to CAD drawings J 4 pendulum inertia a kg m 2 according to CAD drawings a Including inertias of all structural pendulum parts: 1,2,10,11,12,13,14. its drive-train components, assuming these can be dynamically described by a discrete number of rigid bodies connected by linear elastic elements. All stiffness and inertias are reduced with respect to the pendulum shaft. Table 4.4 gives estimates of the stiffness and inertia parameters for the various components. The servo stiffness c 1, which scales with squared gear ratio i 2, depends on the controller gains. Because of the unknown controller gains, the value of c 1 is unknown beforehand. However, because of the large gear ratio (i 2 = ) it is assumed to be large enough to be considered rigid. Also the gearbox torsional stiffness c 2 is unknown (no data available). Its nominal value is set to be Nm/rad, which is an estimate 3. Since the torsional pendulum stiffness c 6 c 2,3,4,5, the pendulum can be considered rigid. Moreover the pendulum inertia J 4 J 2,3, which legitimates the neglect of the pendulum shaft and coupling inertias. The above simplifications yield the reduced dynamic model, depicted in Figure 4.10(b). The eigenfrequency of this reduced model is given by: f eig = 1 ceq [Hz], (4.2) 2π J 4 where the (overall) equivalent torsional stiffness c eq is given by: c eq = ( ) 1 = Nm/rad. c 2 c 3 c 4 c 5 Hence an estimate of the eigenfrequency of the pendulum drive is f eig = 18.4 Hz. For frequencies well below the eigenfrequency of 18.4 Hz, the pendulum and its drive-train thus behave like one inertia. Since the steady-state excitation frequency is only 1 Hz, which is far below the eigenfrequency, the pendulum drive train can be considered rigid and no complementary dynamic effects will occur. Even higher excitation frequencies that occur during transients will stay sufficiently far below the eigenfrequency to guarantee a proper dynamic response Deflections In order te get an idea of the stiffness of the pendulum support structure in all (lateral) directions we address the deflections δx m, δy m and δz m of the pendulum center of gravity, caused by the pendulum reaction forces and moments F p i and M p i. Figure 4.11 schematically depicts these deflections and the deformation of the components causing them. In addition to deflections caused 3 From a structural point of view one could argue that the gearbox stiffness must be at least smaller than the coupling stiffness c 4 of Nm/rad, because of its somewhat smaller diameter. Moreover, the gearbox is considered to be the weakest link in the pendulum drive train.

59 Strength and Stiffness by the torsion of the pendulum drive-train, deflections δx m are caused by bending of the shaft under moments and reaction forces My p and Fx, p and bending and shear of the bearing house and the support plate under the bearing reaction force Fx A (see Figure 4.11(a)). Deflections δy m are caused by bending of the shaft under moments and reaction forces Mx p and Fy p, and bending of the bearing house under the bearing reaction force Fy A (see Figure 4.11(b)). Deflections δz m are a direct result of the bending of shaft and bearing house as described above, but are also caused by bending of the pendulum structure itself under the weight Fz m of the steel mass. z y x y z x δ m x m m δ m z m δ m y m δ m z δ m y m m y z x δ m x (a) (b) (c) Figure 4.11: Deflections of the pendulum. Front view (X-Z plane) for θ = 90 0 (a), side view (Y-Z plane) for θ = 0 0 (b) and top view (X-Y plane) for θ = 0 0 (c). Since the TORA structure consists of components of simple basic shapes the deflections δi m can easily be estimated using linear elastic (beam) theory. For the pendulum shaft these calculations are included in Appendix E.2. Similar calculations have been performed for the other components (bearing house, support plate, pendulum). The total resulting deflections δx m, δy m and δz m under a nominal design load (see table D.1) are 0.26 mm, 0.02 mm and 0.12 mm, respectively. The main contribution to these deflections is torsion of the pendulum drive-train 4. The second largest effect is the bending of the pendulum shaft. Bending of the bearing house has the smallest effect on the deflection δx,y,x. m The magnitude of these deflections are admissible and the corresponding pendulum support structure stiffness is considered to be sufficient Strength In the previous paragraphs simple linear elastic (beam) theory is used to obtain estimates for the (torsional) pendulum stiffness and deflections of the pendulum and its support structure. The same tools can be used to obtain estimates of the material stress under maximal design load. A calculation of the stress levels under maximum design load in the pendulum shaft is included in Appendix E.3. The resulting maximum equivalent Von Mises stress σ eq in the shaft is 47.5 MPa, which is far below the yield stress σ yield of steel (400 Mpa). The stress levels in the other main TORA components (bearing house, support plate, pendulum) are assumed to be far below the material yield stress beforehand and need no further calculation.

60 4. TORA Construction 49 Table 4.5: Strength and stiffness characteristics for different pendulum shafts. Mat. denotes shaft material (St for steel, Al for aluminium). D [mm] Mat. Mass c eq f eig σ eq σ yield δ θ δ M i [mm] D o D i [kg] [Nm/rad] [Hz] [MPa] [Mpa] [mdeg] δ M x δ M y δ M z 10 0 St St St St St St St St Al Al Al Pendulum Shaft In the experimental TORA setup a solid 25 mm diameter steel shaft is applied. All calculations in the previous paragraphs are based on this shaft. In this section the selection of this particular shaft is briefly discussed. Table 4.5 gives the values of stiffness, stresses and deflections, as mentioned in the previous paragraphs, for shafts of different dimensions (hollow or solid) and material (steel or aluminium, denoted St and Al, respectively). From the data in Table 4.5 one can conclude that for all but the smallest shaft diameters, σ eq stays well below σ yield. Hence shaft strength is not a constraint for the selection of a suitable shaft. Aluminium shafts, however light, are not preferable because of their softness. This softness would complicate the coupling and pendulum interconnections, which actually require hard (steel) shaft surfaces. Another consideration concerning the pendulum shaft selection is the outer shaft diameter. Larger outer shaft diameters provide larger drive-train stiffness 5. Larger shaft outer diameters, however, come at the cost of larger (and heavier) bearings and a larger shaft weight. The latter can be compensated using hollow shafts, which do not suffer from a significant decrease in (torsional) stiffness, but do have a significant lower weight with respect to their solid counterparts. A good compromise is thought to be a 25/15 mm diameter hollow steel shaft, which has a sufficiently high torsional stiffness (corresponding to an eigenfrequency of 18 Hz), small displacements under nominal load δi m, and a significantly lower weight then its solid counter part. Two practical aspects prevent us, however, from applying this shaft. Firstly, the connection of the shaft to the coupling requires the outer shaft diameter to be locally reduced to 20 mm. With an inner diameter of 15 mm, this is considered to yield a too low wall thickness. Secondly, hollow shaft-grade steel shafts are expensive. This makes us choose the 25 mm diameter solid steel shaft, which is drilled at both ends to save some weight (see detailed shaft drawing in Appendix G) Conclusions Simple lineair elastic (beam) theory provides indications for the strength and stiffness of the different components of the pendulum and its support structure, especially the pendulum shaft. The resulting estimated values for the pendulum (torsional) eigenfrequency and expected deflections 4 Note that the magnitude of δx m is larger then that of δm y. The difference is caused by the torsion of the pendulum drive-train which for small angles θ only contributes to deflections δx m and not to deflections δy m. 5 A proportional increase in pendulum shaft stiffness yields a less then proportional increase in drive-train stiffness because of the limited stiffness of the other components (gearbox, gearbox shaft, coupling) in the pendulum drivetrain.

61 Summary of the pendulum may not be very accurate, but are far from critical. Hence we may conclude that the analysis in this section does not give rise to any concerns about strength or stiffness. 4.7 Summary Because the cart of the experimental TORA setup to be constructed consists of an existing H- bridge X-carriage, only a pendulum, its support structure, and its drive need to be designed. The problem of finding the pendulum dimensions that minimize the pendulum torque requirements combined with good TORA performance is addressed in Appendix B and yields a set of design parameters for the pendulum. Together with the design specifications on the TORA setup these design parameters form the starting point for the design. Despite the technical superiority of a direct drive concept to actuate the pendulum (no backlash, little friction, high stiffness), we choose to drive the pendulum through a gearbox. The reason for this choice is that a gearbox enables the use of a small brushed DC motor, which is a lot cheaper than its large (brushless) direct-drive counterpart, and is also easier to control. The backlash, loss of stiffness and additional friction that are introduced by the use of a gearbox will deteriorate the TORA performance, but are not considered to be critical. In order to minimize moments on the cart (to prevent undesired asymmetric cart loading and toppling of the cart when it hits its physical end-stop) the pendulum is places underneath the H-bridge X-carriage. This configuration is also considered to be the safest because in this case the solid Y-axis supports encage the pendulum. Based on these key-design decisions, the detailed structural pendulum design has been made. The design of the pendulum and its support structure, but also some more specific design considerations, are discussed in Section 4.5. A strength and stiffness analysis using straightforward linear elastic (beam) theory and basic rigid body dynamics indicates that no strengths or stiffness problems are to be expected. Finally, the design presented in this chapter has been built by the central university workshop, the GTD, without any problem.

62 Chapter 5 Experimental Case Study The aim of this study is to physically implement a form of output regulation control for the TORA system, which is a benchmark system in nonlinear control, and to study the controller performance in experiments. The design of the experimental TORA setup has been addressed in Chapter 4. The design of the controller that is implemented in this setup is discussed in Chapter 3. Simulations (see Section 3.3) indicate that the controller works for the ideal TORA system. The experimental TORA setup, however, is not an ideal setup because friction, cogging, and backlash phenomena are present. Moreover, the physical system parameters will differ from the nominal system parameters. Although some of these parasitic phenomena can be compensated for, the question is whether the output regulation controller still works on the experimental setup. More specifically, the question is how its performance on the experimental setup deteriorates with respect to simulations and whether or not the use of an output regulation controller is feasible from a practical point of view. Since the controller (3.12) that is implemented solves the output regulation problem only locally, another point of interest is the size of the region of attraction for this controller. In this chapter we first address the implementation of the designed controller on the experimental TORA system and, secondly, we study what parasitic phenomena are present in the experimental TORA system, and how we should cope with them. Next, the experiments are discussed and the experimental results are presented. Moreover, the experimental results are compared with simulation results. Furthermore, the experimental results are analyzed in more detail in order to answer the specific questions on the output regulation experiments. At the end of the chapter the main conclusions concerning experimental output regulation are stated. The experimental case study addressed in this section has been performed within the wider scope of a PhD study on the ORP. For a compact description of the experiments and the experimental results one is referred to the PhD thesis [32]. This chapter addresses the experimental case study in somewhat more detail. 5.1 Experimental Setup The constructed experimental TORA system is extensively discussed from a mechanical point of view in Chapter 4. This section addresses the experimental setup from a control point of view. First, the system hardware is discussed. Then, the parasitic phenomena that are present in the setup are listed. Finally, the actual practical controller implementation is addressed.

63 Experimental Setup System Description As discussed in the introduction (Section 1.2.2) the TORA system is constructed by adapting an existing X-Y positioning system, the H-bridge setup (see Figure 1.2 for a schematic representation and Figure 1.3 for an overview of the H-bridge setup). The Y1 and Y2 carriages of the H-bridge setup are controlled by a low level PID controller to maintain a fixed position. The X-carriage (cart), fitted with the pendulum and moving along the X-axis, forms the actual TORA system. The cart is actuated by a linear motor (LiMMS). A small DC motor actuates the pendulum via a gearbox. Figures 4.5 and 5.2 depict the parts of interest; the cart and pendulum structure. The force F applied to the cart by the linear motor is proportional to the current I F fed to the linear motor coils, i.e. F = κ M F I F, where κ M F = 74.4 N/A is the linear motor constant. The current I F is proportional to the (voltage) control signal u F, which is fed to the linear motor through a proportional current amplifier, i.e. I F = κ A F u F, where κ A F = 1 A/V is the amplifier constant. Hence F = κ F u F, where κ F = κ M F κa F = 74.4 N/V. The DC motor torque T m proportionally depends on the the motor current I T, i.e. T m = κ M T I T, where κ M T = N/A is the motor constant. The current I T is proportional to the (voltage) control signal u T which is fed to the motor through a proportional analogue current amplifier 1, i.e. I T = κ A T u T, where κ A T = 1.6 A/V is the amplifier constant. Hence T m = κ T u T, where κ T = κ M T κa T = N/V. The displacement x of the cart is measured using a linear incremental encoder (Heidenhain LIDA 201) with a 1 µm resolution. The angular position θ of the pendulum is measured at the motor shaft by a rotational incremental encoder (Maxon, HEDL55) with a (quadrature decoded) resolution of (which consequently implies a degree resolution at the pendulum shaft). The cart speed ẋ and pendulum angular velocity θ are obtained by numerical differentiation and filtering of the measured signals x and θ, respectively. A dspace real-time signal processor hosts the controller(s), performs the signal processing and provides the I/O connections to the setup. Moreover, a real-time interface with a standard PC provides the user with real-time data readings and real-time parameter tuning capabilities. The actual controllers are built using Matlab Simulink and converted to C-code to be hosted by the dspace system. For a schematic overview of the controller hardware one is referred to Figure H.1 in Appendix H. Table H.1, in this same appendix, lists the specifications of all hardware components Parasitic Phenomena As mentioned before, the experimental TORA system differs from the ideal TORA system because of the presence of parametric uncertainties and parasitic phenomena in the cart X-LiMMS and in the pendulum drive train. The X-LiMMS consists of a translator of multiple iron-core coils that translates along a stator base plate fitted with permanent magnets (see Figure 5.1). A properly (position dependent) commutated three-phase current through the coils induces a thrust force F on the translator. For more details on the working principle of the LiMMS in general and the commutation in particular 1 Originally the use of a Maxon digital current amplifier was foreseen. This amplifier better suits the motor characteristics and would yield better motor performance (higher possible motor torques). However, the electromagnetic noise resulting from the digital PWM signal appeared to interfere with the readings of the H-bride encoder signals. This problem, at the time the experiments were conducted, could not be solved quickly. Hence an analogue amplifier has been used, whose performance suffices for the experiments conducted in the scope of this case study. See [26] for more details.

64 5. Experimental Case Study 53 Translator Phase I Phase II Phase III Iron cores Encoder DC motor Gearbox Cable guidance Linear Encoder Cart Guidance X-axis stator base Roller bearings N S S N N S Stator base with permanent magnet array S Pendulum Figure 5.1: Schematic representation of the working principle of a three-phase linear motor. Figure 5.2: The adapted H-bridge setup with its main components. one is referred to the works [16] and [37], respectively. The physical structure of the LiMMS implies that besides the thrust force F two 2 (parasitic) forces are present: 1. The velocity dependent friction force F f (ẋ), which is introduced by a number of roller bearings supporting the translator on its guidance (bearing positions are indicated in Figure 5.2). 2. The position dependent cogging force F c (x), which is a result of the natural attraction between the permanent magnets and the iron-cores of the coils in the translator, even if no current flows through the coils. In the pendulum drive-train, the motor brushes, motor bearings, pendulum shaft bearings and the friction in the gearbox all contribute to a (angular velocity dependent) friction torque T f ( θ). The gearbox, however, causes the majority of this friction torque. Moreover, the gearbox introduces backlash in the pendulum drive-train. This backlash has a magnitude of approximately at the pendulum shaft (manufacturer specifications), which corresponds to a free displacements at the pendulum mass δ m x (see Figure 4.11) of 1.75 mm at zero pendulum angle. In order to be able to compensate for the undesired cogging and friction forces F f (ẋ) and F c (x) in the cart motion and the pendulum friction torque T f ( θ), these parasitic forces and torques are identified. The next subsections address the identification procedures that have been performed. Cart Cogging Force The cart cogging force F c (x) is identified using the methods presented in [3] and [39]. The essence of the identification procedure is as follows. The cart is moved along the X-axis at a constant (low) velocity of 50 mm/s using a PID-feedback controller; once in forward direction and once in reverse direction. The resulting control effort, scaled with the appropriate amplifier gain and motor constant to yield the actuation force F contr, is plotted in Figure 5.3(a). Because of the constant velocity the friction force can be considered constant and of the same magnitude, but opposite sign, for the forward and reverse run. If one now averages the control efforts required to obtain the constant speed in forward and reverse direction, the friction force cancels out. The resulting average control effort is a measure for the position dependent cogging force. The identified cogging 2 Actually a third parasitic force is present, a reluctance force. This force is mainly a result of imperfections in the commutation of the coils. The present coil commutation process assumes an ideal array of equidistantly spaced permanent magnets with the same magnetic field strength, whereas in the real setup the magnets are not perfectly aligned and may not all have the same magnetic field strength. The position dependant part of the reluctance force is taken into account as a part of the cogging force, whereas the remaining part is neglected.

65 Experimental Setup Fcontr [N] Fc [N] x[m] x[m] (a) (b) Figure 5.3: Controller actuation force F contr in forward (blue) and reverse (red) identification run (a). The identified cogging force F c (x) (b). force F c (x), plotted in Figure 5.3(b), results from this average of the forward and reverse run, depicted in Figure 5.3(a), after some filtering. The remarkable difference in cogging force peak amplitude in the position intervals 0.25 < x < 0 and 0 < x < 0.25 is caused by the fact that the stator base consists of two different magnet arrays. Cart Friction Force The cart friction force F f (ẋ) is also identified using the methods presented in [3] and [39]. The essence of the identification procedure is as follows. Again, the cart is moved along the X-axis at a constant velocity using a PID-feedback controller. At the same time, the cart cogging force is compensated using the previously identified values of the cogging force. The identified friction level for a specific velocity is obtained from the average control effort (excluding the cogging compensation part) over the trajectory that is covered with this constant speed. This procedure is repeated for several different velocities. The resulting identified friction force F f (ẋ) is plotted in Figure 5.4. The identified cart friction force shows two remarkable effects. First the Stribeck effect (decrease of friction for velocities just above or below zero) seems to be absent (even if one observes the friction force at very low speeds of about 1 mm/s). Secondly, for ẋ 0 the (viscous) friction force, as could be expected from a theoretical Stribeck curve, depends more or less proportionally on the cart speed, except for a local peak at around ± 0.15 m/s. These two peculiar phenomena are noticed, but cannot be explained from a physical point of view. Pendulum Friction Torque The pendulum friction torque T f ( θ) is identified using the same method as the one used to identify the cart friction force. The resulting identified pendulum friction torque T f ( θ) is plotted in Figure 5.5. The identified pendulum friction force also does not show any Stribeck effect System Modelling & Controllers Recall that the controller (3.12) that is implemented on the TORA setup is designed for an ideal TORA system as described by the equations of motion (3.3) and (3.4). The experimental setup,

66 5. Experimental Case Study Identified Compensated Identified Compensated Ff [N] 10 0 Tf [Nm] ẋ[m/s] Figure 5.4: Identified cart friction force F f (ẋ) and the friction compensation force ˆF f (ẋ) θ [deg/s] Figure 5.5: Identified pendulum friction torque T f ( θ) and the friction compensation torque ˆT f ( θ). however, taking into account all active (parasitic) forces and torques, can be modelled as follows: M M ẍ + M ) L ( θ cos θ θ2 sin θ = F F f (ẋ) + F c (x) (5.1) M J θ + ML ẍcos θ = it m T f ( θ), (5.2) where M M, ML and M J are the nominal system parameters, which are defined as M M, M L and M J in Section 3.1 with magnitudes corresponding to the actual TORA setup, F = κ F u F is the force acting on the cart, T m = κ T u T is the applied motor torque and i is the gear ratio. The values of MM, ML and M J are initially based on the TORA design, but have been tuned during tests preliminary to the actual experiments. They will be discussed in more detail in Section In order to make the controller (3.12) work on the experimental setup described by (5.1) and (5.2), we first need to compensate for friction in the cart and pendulum motion and for the cogging force in the X-axis. Moreover, a virtual spring action needs to be implemented and a proper disturbance force needs to be applied. For the cart we define the controller u F = 1 κ F ( ˆFf (ẋ) ˆF c (x) kx + F d ), (5.3) where ˆF f (ẋ) and ˆF c (x) are the cart friction and cogging compensations based on (but not necessarily equal to) the identified cart friction and cogging forces F f (ẋ) and F c (x), k is the (virtual) spring stiffness that can be set as desired, κ F is an amplification constant as defined in the previous paragraph and F d is the applied disturbance force. For the pendulum we define the controller u T = 1 ( T u + iκ ˆT f ( θ) ), (5.4) T where ˆT f (ẋ) is the pendulum friction compensation based on (but not necessarily equal to) to the identified pendulum friction ˆT f (ẋ), i is the gear ratio, κ T is an amplification constant as defined in the previous paragraph and T u is the applied pendulum control torque prescribed by the output regulation controller. If one implements the low level-controllers (5.3) and (5.4), the system takes the following form M M ẍ + M ) L ( θ cos θ θ2 sinθ + kx = F d + ε F (5.5) M J θ + ML ẍcos θ = T u + ε T,

67 Experiments where ε F and ε T are residual terms as a result of inaccurate friction and cogging compensation and due to deviations in the nominal system parameters M M, ML and M J from the real system parameters M M, M L and M J. If one does not take into account the residual terms ε F and ε T, the system (5.5) is now of the same form as the original system (3.3), (3.4), for which the controller (3.12) solves the local output regulation problem. Recall that the disturbance force F d applied to the cart is generated by the linear harmonic oscillator (3.9) ẇ 1 = Ωw 2, ẇ 2 = Ωw 1, F d = w 1, (5.6) which yields F d = F 0 sin(ωt + φ), where F 0 and φ are determined by the initial conditions w 1 (0) and w 2 (0). The output regulation controller that is implemented is a simple static state feedback (local) output regulation controller (3.12) of the form T u = γ (w) + K (x π (w)), (5.7) where π (w) and γ (w) are defined as in (3.25) and K is such that A l + B l K is Hurwitz (A l and B l as defined in (3.11)). An overall controller, that is hosted by the dspace system, comprises the controllers (5.3), (5.4) and (5.7) and computes the states of the exo-system (5.6). The full system state x = [ x ẋ θ θ ] is derived from measurements of x and θ. The friction and cogging compensations are implemented as look-up tables. The controller is running at a sample frequency of 4 khz. A schematic representation of the overall control layout can be found in Figure H.2 in Appendix H. 5.2 Experiments Before any experiment can be conducted, first the proper controller parameters need to be set. After addressing the controller parameter settings in the next subsection, the experiments that have been performed are discussed and the experimental results are presented. The actual discussion of these results is given in the next section, Section 5.3. In general, the nonlinear controller (5.7) that is implemented on the setup, as its linear counterpart (2.14), is not robust with respect to deviations in the system parameters. For proper controller performance, the nominal system parameters therefore need to match the parameters of the physical setup. To obtain values of the physical setup parameters, one, in fact, requires an identification process to be carried out. At the moment the identification experiments were conducted, however, fast results were more important than performance in order to meet the deadline on the PhD thesis [32] of my supervisor, in which the experimental results needed to be included. Within this perspective, most system parameters have not been identified. Instead, nominal values have been implemented and then tuned during tests in the run up to the actual experiments to obtain optimal performance. Only the vital factors such as friction and cogging forces have been identified (see Section 5.1.2). This approach with respect to the controller parameter settings is not an optimal one, but, as shown in the subsequent sections, does yield meaningful experimental results which allow for first conclusions on output regulation control on the physical TORA system Parameter Settings To start with the pendulum is loaded with 21 large steel plates, with a total mass of kg, at a distance from the rotational axis of m 3. The resulting nominal system mass M M = Note that the design load configuration consists of a mass of 6 kg at a distance of 0.2 m from the rotational axis. The load configuration used here was originally only intended to test the setup, but has never been changed

68 5. Experimental Case Study 57 kg. This value is obtained by adding the mass of all (pendulum) components to the mass of the original X-carriage. The first is obtained by weighing all components that have been added to the original X-carriage, whereas the latter has been identified in earlier studies on the H-bridge setup (see [16]). The nominal eccentric mass term M L = 1.25 kgm is neither measured, nor identified experimentally, but calculated using the CAD model of the pendulum design. A part of nominal inertia term M J = 0.54 kgm 2 also follows from the CAD model of the pendulum design, whereas the other part is based on the inertia of the components (motor, gearbox, coupling etc.) as specified by their manufacturers. Tables H.2 and H.3 list the (nominal) mass and inertia values for the individual TORA components. The parameters ML and M J have been tuned during tests in the run up to the actual experiments in order to obtain better performance. The resulting tuned parameters ˆM L = 1.34 kgm (7% larger than the nominal value) and ˆM J = 0.43 kgm 2 (21% smaller than the nominal value) give high performance in terms of desirable steady-state behavior, i.e. stand still of the cart instead of a small residual cart motion. These parameters have been used in the final experiments. The value of the spring stiffness k is set to its nominal design value of 500 N/m. The cart cogging force compensation ˆF c (x) is set equal to the identified cart cogging force ˆF c (x) presented in Figure 5.3(b). The cart friction force compensation ˆF f (ẋ), represented by the dotted line in Figure 5.4, is set equal to 90% of the identified friction force F f (ẋ) represented by the solid line in Figure 5.4. Moreover, for cart velocities ẋ < m/s the friction compensation is set to ˆF f (ẋ) = ẋ F f (ẋ). The under-compensation of the cart friction over the whole velocity range and especially around zero velocity reduces the friction induced limit cycling which is observed in experiments if the friction compensation is set to 100%. Moreover, due to imperfections in the cart friction force identification, a 100% compensation, as observed in experiments, leads (at some velocities) to friction overcompensation resulting in unstable system behavior. A drawback of the friction undercompensation, as we will conclude from the experimental data, is the existence of an equilibrium set for the cart. In other words, the cart sticks at some position x close, but not equal, to zero. The pendulum friction torque compensation ˆT f ( θ), represented by the dotted line in Figure 5.5, is set equal to 150% of the identified friction force T f ( θ) represented by the solid line in Figure 5.5. Again this is the result from a tuning procedure in the run up to the actual experiments. For lower values of pendulum friction compensation, a steady-state cart stand-still cannot be obtained. Since it is very unlikely from a physical point of view that we are actually over-compensating the pendulum friction, it may be that friction force T f ( θ) has been improperly identified. Another possible cause could be the fact that the gearbox friction torque, which forms the main part of the total pendulum drive-train friction, is not only speed dependent but also torque dependent. Since the friction identification has been performed for very low torques (constant speed experiments), this may cause the actual pendulum friction in experiments (which are conducted at higher torque levels) to be higher than the identified values. This would justify the proposed friction over-compensation. The controller gain matrix K is set to K = [ 29, 1.5, 11, 1.9 ]. The setting of the gains K, and correspondingly the placement of the closed-loop poles, is a trade-off between high gains K 1 and K 3 (corresponding to the cart position and pendulum angel, respectively) and real parts of the poles which are as small as possible. Large values of K 1 and K 3 are required to before starting the experiments.

69 Experiments adequately compensate for residual friction present in the system. Moreover, from experiments it follows that large values of K 1 and K 3 are essential to make the controller work at all, i.e. to observe any convergence from the transient to a steady-state situation. Small values of the real parts of the poles, on the other hand, are required to obtain fast convergence and sufficient robustness of the closed-loop system. A more detailed analysis of the controller gains indicates that the above demands are contradictory. The values of the final controller gains are obtained from some optimization. While the real parts of the poles are required to stay below the level of about -1, in this optimization the objective is to find values for K 1 and K 3 as large as possible. The resulting matrix K is mentioned above. In addition, the control signal resulting from this matrix K does not exceed, in most operating conditions, the bounds that are determined by the motor and current amplifier Experiments In this section the experimental settings are addressed, the type of experiments that have been conducted are discussed and the results are presented. A more detailed analysis of these results is presented in Section 5.3. The initial conditions on the exo-system (5.6) are set to w 1 (0) = 0 and w 2 (0) = F 0 for all experiments. This corresponds to the disturbance force F d = F 0 sin (Ωt). The disturbance force excitation frequency, as specified in the TORA design specifications, is set to 1 Hz. This corresponds to the value of Ω in the exosystem equal to Ω = 2π rad/s. The disturbance force amplitude F 0 is set to 15 N and 25 N for different experiments. Two types of experiments are performed. In the first type of experiments the system starts in a given initial condition x(0) = x 0 [mm], ẋ(0) = 0 [mm/s], θ (0) = θ 0 [deg] and θ(0) = 0 [deg/s]. Experiments are performed for three different sets of initial conditions for each of the two disturbance force amplitudes of F 0 = 15 N and F 0 = 25 N. The specific initial conditions are indicated in Table 5.1. This table also contains the corresponding experiment names. Under these names one can find the original experimental data, as well as movies of the experiments, on the DVD accompanying this thesis. Figure 5.6 presents the experimental results in terms of the displacement x(t) and the control effort represented by the current I or [A] that is fed to the DC motor. Note that I or only comprises the actual output regulation control effort (5.7), which does not include the control effort for friction compensation, i.e. I or = 1 iκ M T T u. From the results of this first set of experiments we can immediately conclude that the output regulation controller, in combination with the cogging and friction compensation, compensates a significant part of the disturbance force and stabilizes the cart in a position close to zero. Table 5.1: Names, initial conditions and disturbance force amplitudes for the different types and numbers of experiments. Type Number Name F 0 [N] x 0 [mm] θ 0 Type Type 2 # 1 experiment_set3_03_01_ # 2 experiment_set3_03_04_ # 3 experiment_set3_03_07_ # 1 experiment_set3_03_03_ # 2 experiment_set3_03_06_ # 3 experiment_set3_03_09_ experiment_set3_04_01_01 15 random random - experiment_set3_04_03_01 25 random random

70 5. Experimental Case Study 59 The second type of experiments is also performed for each of the two disturbance force amplitudes F 0 = 15 N and F 0 = 25 N. In this type of experiments, however, initially only the feedback part of the output regulation controller (5.7) is active, i.e. T u = Kx, whereas the disturbance force F d is activated. Since there is no disturbance force compensation, the cart and pendulum start oscillating. Then, at some arbitrary moment, the feed-forward part of the controller is activated, i.e. T u = γ (w) + K (x π (w)). In fact, this second type of experiment is similar to the first type for random initial conditions x 0, because the moment at which the feedforward part of the controller is activated is arbitrary. Figure 5.7 presents the experimental results for the type 2 experiments. Again, Table 5.1 gives the experiment names under which the original experimental data and movies can be found on the DVD. Moreover, from the results of the second type of experiments we can conclude that the output regulation controller, together with the cogging and friction compensation, compensates a significant part of the disturbance force and stabilizes the cart in a position close to zero Experiments vs. Simulations To obtain a better understanding of the experimental results we compare the experimental results with simulation results. A type 1 experiment with initial conditions x 0 = 200 mm and θ 0 = 20 deg and a disturbance force amplitude F 0 = 15 N is selected for such a comparison (experiment_set3_03_04_02). A simulation as discussed in Section 3.3 has been performed for the nominal system parameters M M, ML and M J. Moreover, the same simulation is performed for the tuned system parameters ˆM M, ˆML and ˆM J. Figure 5.8 presents the experimental results (solid line), as well as the simulation results for the nominal system parameters and the tuned system parameters (dashed and dotted lines, respectively). One can directly conclude from this comparison that there is a strong qualitative resemblance between the experimental and simulation results. A more detailed discussion of the similarities and differences and their possible causes is presented in the next section. 5.3 Results The results in this section represent the first steps in experimental output regulation. The implementation of the output regulation controller in the physical TORA setup involves many hardware and controller components (motors, amplifiers, friction/cogging compensations, etc.) to be combined. Since many of these components are used for the first time, there is little experience on their use. Therefore there is little knowledge on exact component parameters and characteristics. This implies there are many uncertainties in the output regulation experiments on the physical TORA setup. Therefore it is difficult to draw strong quantitative conclusions from the experimental results. However, the experimental results do allow for qualitative conclusions. In the next subsection the experimental data are analyzed in more detail. The conclusions on these experimental results are stated in the last subsection Experimental Data Analysis The results of the type 1 and type 2 experiments show that after transients the cart comes at a standstill at a position close, but not equal to zero. This means approximate output regulation is achieved. Exact output regulation is not achieved because of the presence of modelling uncertainties ε F and ε T (as in (5.5)) that are not taken into account at the stage of controller design. There are three causes for these uncertainties: improper friction and cogging compensation, (un-modelled) backlash in the gearbox, parametric uncertainties (in parameters M M, M L and M J ).

71 Results F 0 = 15 N F 0 = 25 N x [mm] Experiment # 1 Experiment # 2 Experiment # 3 x [mm] time [s] F 0 = 15 N time [s] F 0 = 25 N 2 I or [A] 1 0 I or [A] time [s] time [s] Figure 5.6: Experimental results for a 15 N (left) and 25 N (right) disturbance force amplitude for three different initial conditions (Experiment Type 1). F 0 = 15 N F 0 = 25 N x [mm] 50 0 x [mm] time [s] time [s] F 0 = 15 N F 0 = 25 N I or [A] I or [A] time [s] time [s] Figure 5.7: Experimental results for a 15 N (left) and a 25 N (right) disturbance force amplitude. Disturbance compensation is activated at some arbitrary moment during the experiment indicated by vertical line (Experiment Type 2).

72 5. Experimental Case Study 61 x [mm] x [mm] x [mm] 5 0 Experimental result Sim result nominal Sim result tuned ẋ [mm/s] time [s] time [s] ẋ [mm/s] time [s] time [s] ẋ [mm/s] time [s] time [s] θ [deg] θ [deg] θ [deg] time [s] time [s] time [s] θ [deg/s] θ [deg/s] θ [deg/s] time [s] time [s] time [s] I or [A] 1 0 I or [A] 1 0 I or [A] time [s] time [s] time [s] Figure 5.8: A comparison of the results of a type 1 experiment, with initial conditions x 0 = 200 mm, θ 0 = 20 deg and F 0 = 15 N, and simulations for nominal system parameters ( M M, ML and M J ) and tuned system parameters ( ˆM M, ˆML and ˆM J ).

73 Results These three causes are analyzed in more detail below. System Parameter Uncertainty To start with, recall that we are dealing with three sets of system parameters. The actual TORA system parameters M i, which are not exactly known. The nominal system parameters M i, obtained from weight measurements and from the CAD model, are assumed to be close to the actual TORA system parameters. The tuned set of system parameters ˆM i is the set of system parameters that is implemented in the physical TORA controller. From Figure 5.8 one can conclude that the results of the simulation based on the tuned system parameters match the experimental results more close than the results of the simulation based on the nominal system parameters. This indicates that the tuned system parameters ( ˆM i ) are closer to the real system parameters than the nominal system parameters ( M i ). Another fact that points in the same direction, is the fact that, in experiments in which the controller is based on the nominal system parameters, no satisfactory disturbance rejection is achieved on the TORA setup (no output regulation is achieved, cart keeps oscillating with a large amplitude). Moreover, simulation results for the nominal system parameters in Figure 5.8 show an almost similar behavior of the cart position x and cart velocity ẋ to simulation results for tuned system parameters. This indicates that the cart motion does not strongly depend on the values of the system parameters M i for the range or parameters we are looking at (the pendulum motion and the control effort I or, however, do show a significant difference between simulations for nominal and tuned system parameters). The cause for the difference between experimental and simulation results with respect to the cart motion therefore is unlikely to be caused by deviations of the tuned system parameters from the actual system parameters. For a more accurate set of nominal system parameters a proper identification procedure needs to be performed. Friction & Cogging Compensation The effect of undercompensation of the cart friction, which implies that the controlled TORA setup is not frictionless, is clearly visible in the experimental results. The fact that at the end of the transients, when the cart velocity becomes low, the cart sticks in position close, but not equal to zero, is a direct result of cart friction under-compensation (see Figures 5.6, 5.7 and 5.8). Another phenomenon that can be attributed to the cart friction undercompensation is the lower amplitudes of x and ẋ in experiments with respect to the simulation results. The residual friction between cart and X-guidance dissipates energy and damps the cart motion. As concluded from the system parameter uncertainty analysis above, this difference in amplitude is unlikely to be caused by the system parameter uncertainties. In some experiments the phenomena of limit-cycling has been observed: the cart does not come to a standstill but keeps oscillating with a small amplitude. In approximately 10% of the experiments this so-called limit cycling occurs. Figure 5.9 shows the limit cycling in the cart motion for such an experiment. The cause is thought to be in a complicated interaction of friction / friction compensation and cogging / cogging compensation. Moreover, the backlash phenomena discussed in the next paragraph may also contribute to the occurrence of limit cycling. Why some experiments result in limit cycling in the cart motion, whereas most do not, is unclear. The experimental results provide no specific clues to draw conclusions on the accuracy of the cart cogging compensation and the pendulum friction compensation, which was tuned at 150% of its identified value in order to yield satisfactory results.

74 5. Experimental Case Study x [mm] time [s] Figure 5.9: Limit cycling in the cart motion, observed in a type 1 experiment with initial conditions x 0 = 200 mm, θ 0 = 20 deg and F 0 = 15 N. What precedes shows that the inaccurate cogging and friction compensations in cart and pendulum motion require additional investigation, which is outside the scope of our current research. Backlash With the selection of an indirect drive configuration for the actuation of pendulum at the stage of the TORA design we accepted the presence of a backlash in the gearbox. The effect of this backlash on the approximate output regulation may be small in terms of its contribution to the steady-state cart position error, however, it clearly appears in the experimental results. At the moment the motor torque crosses zero, which in steady-state coincides with zero pendulum angle and maximum pendulum speed, the motor-gearbox combination decouples from the pendulum. While the controller reduces the motor speed, the pendulum keeps rotating freely at its maximum speed. When the pendulum has travelled the backlash zone it impacts the slower rotating motor shaft. This causes peaks in the pendulum velocity θ, which are clearly visible in the steadystate detail in Figure 5.8. The velocity peaks are amplified by the controller and fed back into the DC motor actuating the pendulum, causing a major part of the ripple on the control effort. As the experimental results in Figure 5.8 show, the impacts due to backlash also have effect on the cart velocity and as a result, on the cart position. The latter effect, however, in this case is relatively small Conclusions The first conclusion that can be drawn from the type 1 and type 2 experiments is that the output regulation controller (5.7) achieves approximate output regulation. This means that the regulated output x(t) does not tend to zero exactly, but either sticks in an equilibrium position close to zero or keeps oscillating with a small amplitude. These phenomena are caused by the presence of modelling uncertainties ε F and ε T, which result from improper friction and cogging compensation, the presence of backlash in the gearbox and system parameter uncertainties. Improvement of the performance of the controller (5.7) requires the reduction of the modelling and parameter uncertainties. For a more accurate set of system parameters one needs to perform a proper identification procedure. Also, an improvement of the friction and cogging compensation needs additional investigation. The backlash, however, cannot be eliminated. Only its influence may be reduced by a well-chosen controller gain matrix K. The choice of such an optimal set of controller parameters is a new problem in the field of nonlinear output regulation. The second conclusion that can be drawn from the experimental results is that output regulation control strategies, in general, seems to be feasible from a practical point of view. The

75 Results practical implementation of the output regulation controller (5.7) did not give rise to any significant problems. Moreover, the simple output regulation controller (5.7) is not necessarily robust for parameter and model uncertainties, whereas it does achieve approximate output regulation on the physical TORA setup in the presence of these uncertainties. Whether or not one considers this approximate output regulation to be successful depends on the demands on the controller performance. The third conclusion that can be drawn from the experimental results is that the region of attraction of the local output regulation controller (5.7) is quite large. As the experimental results show, approximate output regulation can be achieved from initial conditions as large as x 0 = 200 mm and θ 0 = 90 0, which are quite large initial conditions if one considers the dimensions of the setup. As a concluding remark we recall that the output regulation controller that is implemented on the TORA setup is a simple (nonlinear) static state-feedback controller that requires the full system and exo-system states. In general, an output regulation controller will only be of practical use if it does not require the system and exo-system states, but only some specific system output. One could think of dynamic output feedback controllers incorporating an internal model of the exosystem, as discussed in Chapter 2 for the linear case. Moreover, the implementation of nonlinear robust output regulation controllers would reduce the negative effects of the model and parameter uncertainties on the system performance.

76 Chapter 6 Conclusions and Recommendations The goal of this study is the implementation of a form of output regulation control on the TORA system, a benchmark system in nonlinear control and output regulation. Moreover, specific goals are the design and construction of a suitable TORA setup to demonstrate the effectiveness of output regulation techniques in experiments, to test the applicability of output regulation in practice and to compare experimental and simulation results. To this end the output regulation theory and controller design are studied and an experimental TORA setup has been designed and constructed. A simple (nonlinear) static state feedback controller is successfully implemented on this setup and experiments are performed. The conclusions of this study are presented below. Recommendations for further research are included at the end of this chapter. 6.1 Conclusions The main conclusion concerns the TORA setup. The design and construction of the TORA setup is considered to be successful. The TORA setup performs as desired: no strength or stiffness problems, the specifications with respect to torque and speed are met and the effect of the presence of backlash is limited. Moreover, no major design or constructional problems have been encountered. The TORA setup that is constructed not only serves as a setup for experimental output regulation, but has also to be regarded as a uniform platform that can be used for various experiments. At the moment, for example, experiments on under-actuated control are in preparation, i.e. to make the pendulum, which is made frictionless by applying the proper friction compensation torque, track a reference trajectory by actuation of the X and Y axis of the H-bridge only. Since a TORA setup is neither available commercially nor in our lab, a suitable TORA setup has been designed and constructed by adapting an existing X-Y positioning setup, the H-bridge setup. The TORA setup is designed such that it is suitable to conduct a variety of (tracking, disturbance rejection) experiments. Despite the choice of a (technically inferior) indirect drive solution due to its low costs and ease of control, the setup performs satisfactorily from a mechanical point of view. Of course there are some small practical hardware issues that can be improved. These are not vital, however, for the experiments that have been performed within the scope of this research. If one desires one could easily replace the current gearbox (with limited torque capacity) for a larger model (Maxon GP52C), without the need of any adaptations to the setup. Also, if studied in more detail, the digital amplifier noise problem can be solved. These and other issues are addressed in more detail in [26]. The second conclusion concerns the experiments on output regulation. Disturbance rejection experiments have been performed on the TORA setup using a simple static state feedback output

77 Conclusions regulation controller. From the experiments it follows that output regulation is obtained approximately. This means that the regulated output x(t) does not tend to zero exactly, but either sticks in an equilibrium position close to zero or keeps oscillating with a small amplitude. There are three causes for these phenomena: under compensation / incorrect compensation of friction and cogging, system parameter uncertainties, backlash in the gearbox. Improvement of the friction and cogging compensation requires a more detailed study of these phenomena. Moreover, the system parameter uncertainty can be reduced by performing proper system parameter identification procedures. The backlash cannot be eliminated from the setup, but its effect may be decreased by proper tuning of the gain matrix K. In addition to the overall experimental results, one can conclude that the region of attraction of the local output regulation controller that is implemented on the setup is quite large. Moreover, the experimental results correspond to the simulation results, at least on a qualitative level. On a quantitative level there is a good resemblance only in the steady-state situation. The third conclusion, as shown by the output regulation experiments that have been performed, is that output regulation does work on a physical setup, i.e. there are no fundamental practical limitations to its applicability. Despite the presence of parameter uncertainties and other unmodelled effects, which in this particular case may be of significant magnitude, approximate output regulation is achieved. This proves that output regulation is feasible from a practical point of view. Whether or not one considers the achieved approximate output regulation to be successful depends on the demands on the controller performance. Now let us look at this study within the larger framework of the output regulation problem. The experiments performed in this study contribute to the research on output regulation as these are the first experiments in which a nonlinear output regulation controller is physically implemented on a nonlinear mechanical system. (in contrast to the work of [29] on experimental output regulation on a linear system driven by non-linear actuators.) The controller that is implemented on the setup in this study is a simple static state feedback controller requiring full knowledge on the states x and w. Moreover, it is not proven to be robust with respect to system parameter uncertainties. In practical applications, however, a controller requiring the full system and exo-system states to be known is of not much use, since the full system states and especially the full exo-system states are often not available. Moreover, the specific advantage of the output regulation controller over a standard feed-forward controller design is that it does not necessarily require the reference / disturbance signal to be exactly known, but only the system generating the signal. To this end one needs to apply output regulation controllers based on (regulated) output feedback and comprising an internal model of the exo-system, as discussed in Section for the linear case. Despite the advantages output regulation controllers may offer, their (large scale) implementation in industry, machinery or equipment requires more research. Robustness, performance and straightforward criteria for its design are important characteristics for a controller to be successfully applied in practice. Although the experiments performed in this study show output regulation controllers do work in practice, the current output regulation control strategies in general do not possess these characteristics to an extent that makes them directly applicable on a large scale. Yet, the experiments performed in this study may bring the application of output regulation control in practice a small step closer to reality.

78 6. Conclusions and Recommendations Recommendations The study of experimental output regulation yields three main recommendations for further research. First, the effects of the parasitic phenomena that are present in the TORA setup need to be reduced as much as possible. For the current (non-robust) output feedback controller this means that the limit cycling and magnitude of the equilibrium set in which the cart sticks will decrease. In the ideal situation the influence of the parasitic phenomena would be negligible. In the resulting setup any (output regulation) controller could be implemented, without the need to worry about model and parameter uncertainties. There are four aspects that require a more detailed analysis. The friction and friction compensation in the cart, which seem to be the main cause of stick and limit cycling, need to be studied in more detail. Because of the small peculiarities in the identified cart friction that cannot be explained from a physical point of view, it is advisable to re-identify the friction. Moreover, the compensation itself needs a more detailed analysis. The effect of the friction and friction compensation in the pendulum is not very clear. The 150% overcompensation that is applied in the experiments, however, suggests that there is some error in the identified pendulum friction. A re-identification of this friction would therefore be advisable. Moreover, the possible torque dependency of the gearbox friction torque should be studied. Estimates of the system parameters M L and M J are obtained from the CAD model of the pendulum design. To obtain more exact values of these parameters a proper parameter identification procedure should be performed. The controller gain matrix K could be subjected to a more extensive tuning in order to obtain better performance for the current controller. In a more general approach one could investigate the problem of, in some sense, optimal choice of controller parameters, which is a new problem in the field of output regulation Secondly, a whole range of more advanced forms of output regulation controllers can be implemented on the TORA setup. As mentioned earlier in this chapter, the current controller requires full knowledge on the system and exo-system states. Moreover, the current controller is not robust with respect to (exo) system parameter uncertainties. A next step would be to implement a nonlinear output feedback controller, as discussed for the linear case in Section The resulting non-linear controller, however, is not proven to be robust with respect to system parameters. A second step would be to implement nonlinear output feedback controllers that do possess this robustness property. Finally it would be interesting to implement nonlinear output feedback controllers that are also robust with respect to exo-system parameter uncertainties. As one can imagine, the latter controllers would be of great use in practice. A good starting point for the construction of these controllers is the work of [7]. Thirdly, in contrast to the disturbance rejection experiments performed in this study, one could focus on tracking experiments. The pendulum, in this case, has to be controlled such that the cart follows a certain reference trajectory. The essence of tracking experiments, from an output regulation point of view, is the same as that of disturbance rejection experiments. The practical difficulties involved, however, may be different.

79 Recommendations

80 Appendix A OR: Additional Theory A.1 Observer-based Output Feedback Controller In Section 2.2.2, we discussed an observer-based dynamic output feedback ([ controller. ] We stated A P that under the assumptions B1, B2 and detectability of the pair, [ C Q ]) it is 0 S possible to construct a dynamic output feedback controller of the form (2.18) that solves the output regulation problem. Also the construction of this controller was addressed. This appendix contains the proof that this controller indeed solves the linear output regulation problem, i.e. that the corresponding closed-loop system satisfies AA1 and AA2 for the nominal system parameters. Before addressing this proof we consider the forced closed-loop system (2.4), which is composed of the dynamics of the system to be regulated and the observer dynamics. With the controller (2.18) where δ is defined as in (2.19) and Λ and Υ as in (2.20) the closed-loop system (2.4) is described by: ẋ ξ 1 = ξ 2 A BK B (Γ KΠ) J 1 C A + BK + J 1 C P + B (Γ KΠ) + J 1 Q J 2 C J 2 C S + J 2 Q x ξ 1 ξ 2 + P J 1 Q J 2 Q w. (A.1) Without loss of generality we can perform a transformation on the system (A.1), that replaces the observer dynamics ξ with the observer error dynamics δ. The transformation is introduced in two steps and is nothing more than applying proper row and column operations. Recall that: δ x = ˆx x = ξ 1 x and δx = ξ 1 ẋ, δ w = ŵ w = ξ 2 w and δw = ξ 2 ẇ. Now we replace the ξ terms by the error terms δ in the left-hand side of (A.1) and compensate with the corresponding terms (ẇ = Sw and x from the first row of (A.1)) on the right-hand side, which yields: ẋ δ x δ w = A BK B (Γ KΠ) J 1 C A A + J 1 C P + J 1 Q J 2 C J 2 C S + J 2 Q x ξ 1 ξ 2 + P J 1 Q P J 2 Q S Then we replace the ξ terms in (A.2) by the Θ terms by appropriate column operations: ẋ A + BK BK B (Γ KΠ) x P + B (Γ KΠ) δ x = 0 A + J 1 C P + J 1 Q δ x + 0 w, 0 J 2 C S + J 2 Q δ w 0 δ w w. (A.2) (A.3) which completes the transformation that is required to show the observer-based output feedback controller (2.18) indeed solves the linear output regulation problem.

81 70 A.2 Robust Output Regulation First, we address the control goal of asymptotic stability of the unforced closed-loop system (A1, AA1). The unforced closed-loop system is given by (A.1), in which w is set to 0. Since the above transformation does not affect stability properties, asymptotic stability of the system (A.3) implies asymptotic stability of the system (A.1). Hence the unforced closed-loop system is asymptotically stable if the system matrix [ in (A.3) is Hurwitz, ] which is equivalent to the condition that the A + J1 C P + J matrices [ A + BK] and 1 Q both are Hurwitz. Since the pairs (A, B) and J 2 C S + J 2 Q ([ ] A P, [ C Q ]) are respectively stabilizable and detectable, this is achieved by a proper 0 S choice of the matrices K and J T = [ ] J1 T, J2 T. Second, let us address the control goal AA2. We will show that the solution x = Πw, δ x = 0 and δ w = 0 is a solution of the closed-loop system. If we substitute this solution in the forced closed-loop system (A.3) we obtain the following equation: Πẇ = (A + BK)Πw + (BΓ BKΠ + P)w. (A.4) Together with ẇ = Sw, equation (A.4) reduces to: ΠS = AΠ + BΓ + P, (A.5) which is equivalent to the first regulator equation (2.12). By construction of the controller, matrices Π and Γ satisfy the regulator equations (A.5), and thus (A.4) holds. This shows that the solution x = Πw, δ x = 0 and δ w = 0, or equivalent: x = Πw, ˆx = Πw and ŵ = w, indeed is a solution of the closed-loop system. Moreover, by the regulator equations (2.12) and (2.13), along this solution the regulated output e(t) = 0 for all t R. From([ the above, ] it can be concluded that under conditions B1, B2 and detectability of the A P pair, [ C Q ]) there indeed exists a a controller of the form (2.18) that solves the 0 S linear output regulation problem. A.2 Robust Output Regulation In Section 2.2.3, we discussed the robust output regulation problem. We stated that under the assumptions C1, C2 and C3 it is possible to construct a dynamic output feedback controller of the form (2.25). Also the construction of this controller was addressed. This appendix contains the proof that the controller (2.25) indeed solves the robust output regulation problem, i.e. satisfies AA1 and AA2 for the nominal as well as the perturbed system parameters. A.2.1 Nominal System Parameters In this section, we show that, under conditions C1, C2 and C3 the controller (2.25) solves the output regulation problem for nominal system parameters. First, let us address the control goal AA1. In this case A cl equals A s0, so in order to satisfy AA1, A s0 must be Hurwitz. Under conditions C1, C2 and C3, this can be achieved by a proper choice of the matrices K, L and M, as follows from the reasoning given below. From [7], page 22, it can be shown that if the pairs (A 0,B 0 ) and (Φ,N) are stabilizable (the first is assured by C1, the second by the choice of N) and the pairs (A 0,C 0 ) and (R,Φ) are detectable (the first is assured by C2, the second by the choice of R), assumption C3 implies

82 A. OR: Additional Theory 71 stabilizability of the pair: ([ A0 0 NC Φ and detectability of the pair: ([ A0 B 0 R 0 Φ ] [ B0, 0 ]), (A.6) ], [ C 0 0 ]). (A.7) Since stabilizability is not affected by state feedback, stabilizability of the pair (A.6) is equivalent to E1 and for similar reasons the detectability of the pair (A.7) is equivalent to E2: ([ ] [ ]) A0 B E1 The pair 0 R B0, is stabilizable. NC Φ 0 ([ ] A0 B E2 The pair 0 R, [ C NC Φ 0 0 ]) is detectable. Under conditions E1 and E2 it is possible to choose K, L and M such that A s0 is Hurwitz (as follows from the closed-loop description, equation (2.26)). It now follows that conditions C1, C2 and C3 are sufficient to ensure the existence of a controller of the form (2.25) that satisfies control goal AA1 for nominal values of the system parameters. To prove the controller discussed here also satisfies the control goal AA2, we recall that conditions C1 - C3 imply solvability of the regulator equations for nominal system parameters, i.e. there exist Π 0 and Γ 0 such that: Π 0 S = A 0 Π 0 + B 0 Γ 0 + P 0 0 = C 0 Π 0 + Q 0. (A.8) Moreover, it follows from [7] (pages 23-26) that conditions C1 - C3 imply existence of Ξ 0 such that: [ ] Φ 0 Ξ 0 S = Ξ 0 K 0, (A.9) and [ R M ] Ξ0 = Γ 0. (A.10) x(t) Π 0 [ ] Now consider z (t) = ξ 1 (t) = Π0 Ξ 10 w(t) = w (t), which is shown to be a solution Ξ ξ 2 (t) Ξ 0 20 of the closed-loop system on which e = 0. Substituting this z in the forced closed-loop system (which is just an extension of (2.26)): for all w (t) yields: ẋ ξ 1 ξ 2 = A 0 B 0 R B 0 M NC 0 Φ 0 LC 0 0 K x ξ 1 ξ 2 + P 0 NQ 0 LQ 0 w, (A.11) Π 0 Sw (t) = A 0 Π 0 w (t) + B 0 (RΞ 10 + MΞ 20 )w(t) + P 0 w (t) Ξ 10 Sw (t) = ΦΞ 10 w (t) + N (C 0 Π 0 + Q 0 )w(t) Ξ 20 Sw (t) = KΞ 20 w (t) + L(C 0 Π 0 + Q 0 )w(t). (A.12) Since (A.9), (A.10) and (A.12), imply (A.8), z (t) indeed is a solution of the closed-loop system on which e = 0. Hence condition AA2 is satisfied. From above it can be concluded that conditions C1, C2 and C3, are sufficient to guarantee the existence of a controller of the form discussed above that does achieve the control goals AA1 and AA2 for nominal values of the system parameters.

83 72 A.2 Robust Output Regulation A.2.2 Perturbed System Parameters: Robustness Analysis To analyse the robustness of the controller, we study the controller properties in case the parameters in the system matrices A, B, C, P and Q are slightly different from those in the nominal system matrices A 0, B 0, C 0, P 0 and Q 0. For the stability of the unforced closed-loop system (A1, AA1), we again consider the matrix A s0 from the system (2.26). Due to the choice of Φ, K, L, M, N and R this matrix is Hurwitz. In general, if a matrix A (p) is continuous in p, also its eigenvalues λ (p) are continuous in its parameter p. If λ (p) is in C, then λ (p + δp) is still in C if variations δp are small enough. Since A s0 is continuous in its parameters, its property of being Hurwitz will not be affected by small deviations in the system parameters from their nominal values. Therefore the matrix A s, which is defined as A s0, but based on the deviated system matrices A, B and C, will be Hurwitz as well, ensuring robustness of the stability property (A1, AA1) for small deviations in the system parameters. To study whether the existence of a solution z, on which e(t) = 0 for all w (t) (AA2), is robust for small deviations in the system parameters we again use the result from [7], page 23-26, as discussed in the previous section. Notice that, due to continuity, if conditions C1 - C3 hold for the nominal system parameters, they also hold when small deviations are present in the system parameters. Hence properties (A.8), (A.9) and (A.10) hold for perturbed system parameters too; i.e. there exist matrices Π, Γ and Ξ satisfying the following equations: ΠS = AΠ + BΓ + P, (A.13) 0 = CΠ + Q, [ ] Φ 0 ΞS = Ξ, (A.14) 0 K [ ] R M Ξ = Γ, (A.15) where matrices A, B and C have small variations with respect to their nominal counterpart A 0, B 0 and C 0 and the matrices P and Q are arbitrary. x (t) From a similar analysis as performed in the previous section, it follows that z = ξ 1 (t) = ξ 2 (t) Π Ξ 1 Ξ 2 w (t) = [ Π Ξ ] w (t) is a solution of the closed-loop system for perturbed system parameters. On this solution the regulated output equals to zero. (e = 0). From above it follows that, under conditions C1, C2 and C3, a controller of the form (2.25) solves the robust output regulation problem. (i.e. it satisfies conditions A1 and A2 or, equivalent, conditions AA1 and AA2, for systems with nominal system parameters as well as for systems with slightly perturbed system parameters). In other words, this controller is robust with respect to perturbations in the system parameters. However, note that, since conditions C1, C2 and C3 are independent of the matrices P and Q and the proof of robustness is valid for any P and Q, robustness with respect to the parameters in P and Q is not restricted to small perturbations. A controller of the form (2.25) thus solves the output regulation for nominal and slightly perturbed values of the matrices A, B and C and for any values of the matrices P and Q. However, it should be noted that this linear control strategy is not robust with respect to perturbations in the exosystem parameters S. To achieve robustness for perturbations in the exosystem parameters S a special non-linear control strategy is required.

84 Appendix B Optimal design This appendix addresses the design problem introduced in Section 4.2 in more detail. The question is how to find an optimal design of the TORA system in terms of the cart mass M, the pendulum mass m, the pendulum inertia (with respect to its own center of gravity) I, the distance l of the center of gravity of the pendulum from the axis of rotation, the spring stiffness k and the gear ratio i of the gearbox that is needed to drive the pendulum (see Section 4.3). Preceding the optimization of the above mentioned parameters we update the TORA system model in order to include the motor dynamics. Moreover, the applied disturbance force is specified more exact, an expression for the required steady state motor torque is derived and the design goals and constraints are stated. Finally, a qualitative and quantitative analysis are performed, which allow to draw conclusions on the optimal TORA system parameters. Although we refer to the procedure to determine the optimal values of the TORA system parameters as an optimization procedure, one should note that the following analysis is not a complete mathematically sound optimization procedure. Since we are dealing with a design problem there are many practical issues that cannot be taken into account in a formal optimization procedure, but do play a role in the system parameter setting. The following analysis therefore has been used (and should be interpreted) as a guideline in the design process, rather then an objective and exact optimization process. B.1 System, Forces and Torques In order to formulate the design problem more specifically, the magnitudes of disturbance forces and required control torques need to be specified in more detail. Moreover, the TORA system dynamics need to be extended with the dynamics of the motor that will be used in the experimental setup to drive the pendulum. System Recall the TORA system equations of motion (3.1) and (3.2): (M + m) ẍ + ml ( θcos θ sinθ) θ2 + kx = F d ( ml 2 + I ) θ + (ml) ẍcos θ = Tu. So far we considered the control torque T u acting at the pendulum shaft. In the experimental TORA system however, this required control torque is generated by an electric motor fitted with a gearbox. This means that additional motor dynamics determined by the gear ratio i, the inertia I mot of the motor and gearbox (with respect to the motor shaft), and the motor mass m mot

85 74 B.1 System, Forces and Torques appear in the TORA system dynamics. The relationship between the required control torque at the pendulum shaft and the motor torque is given by: ( ) T u = T p = i T m I mot θm, (B.1) where θ = θ p = 1 i θm and the superscripts p and m denote the distinction between values at the pendulum shaft and at the motor shaft, respectively. The presence of the additional motor (and gearbox) mass and inertia effectively means that the term M J is increased with i 2 I mot and, since the motor is mounted to the cart, the motor mass contributes to M. At the same time, the required control torque is scaled down with a factor i. The equations of motion including the motor dynamics result from a substitution of (B.1) in the original equations of motion (M + m) ẍ + ml ( θcos θ θ2 sin θ) + kx = F d ( ml 2 + I + i 2 ) (B.2) I mot θ + (ml) ẍcos θ = it m. Applied Disturbance Force F The disturbance force F (t) is a single harmonic signal given by equation (3.9), with an amplitude F 0 [N], angular frequency Ω [rad/s] and phase φ [rad]. The design specifications in Section 4.1 prescribe this disturbance force to be such that it results in a 1 Hz (Ω = 2π rad/s), 25 mm amplitude translational oscillation of the whole cart-pendulum system, in case the pendulum rotation is locked (i.e. θ = θ = θ = 0). The desired motion of the cart and pendulum in the situation when the pendulum is locked is therefore thus given by x spec = x spec, 0 sin(ω spec t + φ spec ), where x spec, 0 = m, Ω spec = 2π rad/s and some φ spec rad. (B.3) The corresponding applied (disturbance) force F d rendering this motion is computed from the first equation in (B.2) with θ = θ = θ = 0, which results in F d (t) = x spec, 0 ( k (M + m + mmot ) Ω 2 spec) sin(ωspec t + φ spec ). (B.4) Consequently the harmonic disturbance force F ((3.9)) is determined by: the amplitude F 0 = ( xspec, 0 k (M + m + mmot )Ωspec) 2, frequency Ω = Ω spec, and phase φ = φ spec. One should note that, as a result of the requirement on the disturbance force to generate the motion (B.3) while θ is kept equal to zero, the (minimal) magnitude of the disturbance force we apply to the cart explicitly depends on the total mass of the cart, motor and pendulum and on the spring stiffness. Motor Torque T m To address the required motor torque one needs to distinguish between the required motor torque in steady-state and the required motor torque during transients. The required steady-state motor torque directly follows from the required steady-state control input γ (w) (3.24). If the presence of the motor and gearbox is taken into account, equation (3.24) results in: ( ) ( ) ml 2 + I + i 2 I mot Ω 2 w 1 T m ss (w) = γ (w) = ( i (ml) 2 Ω 4 w1 2 (ml) 2 Ω 4 w 2 1 w 2 2 ) 3 2, (B.5)

86 B. Optimal design 75 where the subscript ss denotes the steady-state situation and γ (w) results from the solution of the regulator equations for the system including the motor dynamics. Moreover, recalling that w 1 (t) = F (t) = F 0 sin(ωt + φ) (3.9), w 2 = 1 Ωẇ1 (3.10) and w2 1 + w2 2 = F0 2, it can be seen from (B.5) that the maximum required steady control torque appears when sin(ωt + φ) = 1 and has a magnitude of ( ) ml Tss(max) m 2 + I + i 2 I mot F0 Ω 2 =, (B.6) i (ml) 2 Ω 4 F0 2 where F 0 = xspec, 0 ( k (M + m + mmot ) Ω 2 spec). The required control torque during transients, as mentioned and shown in simulations in Section 3.3, depends on: the initial conditions x 0, the controller gains K. Since both controller gains and initial conditions are not known beforehand, one cannot calculate the required control torque during transients. However, simulations can provide an indication for such transient control torques. B.2 Design Goals and Constraints In Section 4.2 the design goals are stated qualitatively. This section addresses these goals in more detail. Moreover, the constraints on the TORA design are introduced and their dependency on the system parameters is formulated. B.2.1 Design Goal The three design goals mentioned in Section 4.2 (minimum required torque to drive the pendulum, maximum range of initial conditions and disturbance forces for which output regulation can be attained) are reduce one design goal: Minimize the maximum required steady-state motor torque T m ss(max), as given by Equation (B.6). In fact this one design goal is not equal to the three goals specified in Section 4.2. Minimum required steady-state motor torque does not necessarily imply minimum required motor torque during transients. A set of system parameters that minimizes the maximum required steady-state motor torque (for a certain disturbance force amplitude and frequemcy) does not necessarily result in the largest range of disturbance force amplitudes and frequencies for which disturbance rejection can be achieved with a certain available motor torque. A more correct design goal would be that of maximal torque efficiency, i.e. maximal pendulum reaction force at its bearings in X-direction ) Fx p = ml ( θcos θ θ2 sinθ, (B.7) for minimum required motor torque T m in steady-state and during transients. (Equation (B.7) directly follows from the equations of motion (B.2).) The influence of the system parameters on this design objective, however, is very difficult to evaluate. That is, there exist no explicit analytical expressions relating F p x and T m during transients of because their non-linear coupling via θ, θ and θ.

87 76 B.3 Qualitative Analysis Since the design goal of torque efficiency is too complex to evaluate, in this search for the optimal TORA design the design goal is set to minimize the required steady-state motor torque. Although this design goal not fully covers our demands on optimal system performance specified by the torque efficiency design goal, an approximate evaluation of the latter, based on the work of [4], indicates that the resulting sets of optimal system parameters for both design goals are similar. B.2.2 Design Constraints The TORA design is limited to some constraints that need to be taken into account: 1. For practical reasons the pendulum length l should not be too large. The pendulum will be constructed underneath the cart of the existing H-bridge X-LiMMS (see Section 4.4), where it is constraint to the space between the two Y-axis supports, which are 1 m apart (see Figure 1.3). The cart stroke for non-zero pendulum angles is thus determined by the pendulum length. To allow for a 0.2 m cart stroke, where the pendulum has a full range, the maximum total pendulum length is set to 0.4 m: l 0.4m. (B.8) 2. From a constructional point of view, it is desirable to keep the moments due to gravity on the rotational joint to a minimum. For a structure of the scale we are dealing with, the maximum moment due to gravity, T grav = g ml that is considered to be acceptable, is 20 Nm: g ml 20Nm. (B.9) 3. The steady-state pendulum rotation θ ss (t) is given by equation (3.22): ( ) F0 sin(ωt + φ) θ ss (t) = arcsin mlω 2. F 0 If mlω exceeds 1, singularities will occur. To ensure that 2 mlω does not exceed 1 we set 2 F 0 mlω Equivalent, but more feasible from a practical point of view, is the constraint 2 on the amplitude of the steady-state pendulum motion: 60 0 < θ ss, max < The constraint on the steady-state pendulum rotation amplitude can thus be defined as: ( ) arcsin F0 mlω (B.10) F 0 B.3 Qualitative Analysis In this section, we analyse the influence of the TORA system parameters on the required steadystate motor torque. Recall the expression for the maximum required steady-state motor torque (B.6): ( ) ml Tss(max) m 2 + I + i 2 I mot F0 Ω 2 =, (B.11) i (ml) 2 Ω 4 F0 2 where F 0 = xspec, 0 ( k (M + m) Ω 2 spec ). (B.12) From expression (B.11), one can directly conclude that the smaller I and I mot are, the smaller the steady state-state torque is. Moreover, T m ss(max) is monotonically decreasing for decreasing F 0. One thus also desires minimum values of F 0.

88 B. Optimal design 77 Inertias I and I mot As concluded above, minimal values of I (with respect to the pendulum center of gravity) and I mot are desirable. For the physical pendulum to be constructed this means two things. First, a motor/gearbox with minimal inertia should be applied. Secondly, for any given pendulum mass m and length l, the pendulum should consist of a compact-shaped mass which is of high density material having minimal dimensions in x and y direction in order to yield minimal inertia I. Spring Stiffness k The spring stiffness k and the total TORA mass (M + m) together determine the amplitude F 0 of the disturbance force via the above expression (B.12). Moreover, these parameters determine the eigenfrequency of the TORA system in case the pendulum rotation is locked. Since the spring stiffness is implemented virtually by means of a position feedback in the actuation of the linear motor, it is possible to set it to almost any desired value. A suitable value of the spring stiffness k is assumed to be 500 N/m 1. Cart Mass M The cart mass M does not directly effect the required steady-state pendulum torque B.11, but only indirectly via the disturbance force amplitude (B.12). The latter is required to be small in order to minimize the the required steady-state pendulum torque. Anticipating a total TORA mass of about 23 kg and taking into account the angular excitation frequency of Ω spec = 2π rad/s it follows that (M + m) Ω 2 spec will be larger than 500 N/m. Having set the spring stiffness to 500 N/m, (B.12) suggest that a minimization of F 0 leads to the desire of a small cart mass M. Gear Ratio i The influence of the gear ratio on the TORA dynamics is not univocally. On one hand, the higher the gear ratio, the more torque is available at the pendulum shaft. On the other hand, a higher gear ratio will give rise to a higher inertia torque loss because of higher accelerations of motor and gearbox inertias (see also equation (B.1)). Moreover, the expressions for the maximum required steady-state motor torque (B.11) does not indicate a mere monotone increase or decrease with respect to the gear ratio i. The influence of the gear ratio needs to be studied using a quantitative analysis, as described in Section B.4. Pendulum Mass m and Length l The influence of the pendulum mass m and length l on the TORA dynamics is also not univocally, as one can see from the expressions for the maximum required steady-state motor (B.11). Therefore, the influence of these parameters needs to be studied using a quantitative analysis, as described in Section B.4. B.4 Quantitative Analysis The qualitative analysis from the previous section yields the conclusion that the cart mass M, the pendulum inertia I and the motor inertia I mot all should be minimal in order to yield an optimal TORA design. Moreover, the choice of the remaining design parameters, m, l and i requires a further quantitative analysis in order to obtain insight in their influence on the design. The mass of the pendulum support structure (part of M) however, is determined by the mass m and length 1 One should note that a value of k equal to or around (M + m) Ω 2 would lead to an eigenfrequency of the pure mass-spring system that is close to the excitation frequency, which is not desirable. Taking into account the angular excitation frequency of Ω spec = 2π rad/s and anticipating a final total TORA mass of about 23 kg, which leads to (M + m) Ω 2 spec = 914 N/m, a spring stiffness of 500 N/m places the eigenfrequency far enough from the excitation frequency.

89 78 B.4 Quantitative Analysis Initialization: estimates on M and I no motor, I mot = 0 Optimization step m, l, i Simulations (as in Section 3.3) motor requirements Structural design M, I Motor selection I mot, motor mass Figure B.1: Iterative procedure to obtain optimal TORA dimensions. l of the pendulum that needs to be supported. Also the mass and inertia I mot of the motor and gearbox depend on the motor (and gearbox) that is selected, which in turn depends on torque requirements resulting from the pendulum characteristics m and l. The resulting procedure to obtain the optimal TORA design parameters is an iterative process, as illustrated in Figure B.1. In this section, we address the last step in this iterative procedure, i.e. we find a set of values for m, l and i, that: a satisfies the four constraints (B.8), (B.9) and (B.10) from Section B.2.2, b yields a minimum steady-state motor torque (B.11), while the remaining parameters are fixed at the values stated in Table B.1. Before we can start the quantitative evaluation of the constraints and the steady-state motor torque, three remarks need to be taken into account. 1. The pendulum inertia I is desired to be minimal. In practice, this is achieved by fitting the pendulum with a concentrated mass constructed of a stack of rectangular steel plates all measuring 100 x 120 mm. The plates can be stacked to a height corresponding to the desired mass m. This yields a direct relation between m and I: I = 1 12 m( 0, ,12 2), (B.13) Relation (B.13) is substituted in the expression for the steady-state motor torque (B.11), thus eliminating I. The pendulum inertia therefore does not appear in table B The support structure (see Figures 4.5 and 4.6 for an indiction) that connects the stack of steel plates to the pendulum shaft also contributes to the TORA mass and inertia terms. Parameter Value Remark k 500 N/m As set in Section B.3 M kg Final TORA mass, see Table H.3 I mot kg m 2 Final motor choice, specifications in Figures F.1-F.3 m p l p kg m From CAD model, see Table H.4 I p + m p l p kg m 2 From CAD model, see Table H.4 Ω 2π rad/s According to specification x spec, m According to specification Table B.1: Fixed parameters.

90 B. Optimal design 79 However this contribution (as desired) is small, it is incorporated in this quantitative analysis. The support structure mass m p is incorporated in the mass M and its inertia terms m p l p and I p + m p l p2 make the the optimization objective (B.11) slightly change to T m ss(max) = ( ) ml 2 + m p l p2 + I + I p + i 2 I mot F0 Ω 2. (B.14) i (ml + m p l p ) 2 Ω 4 F0 2 The values of m p l p and I p + m p l p2 are listed in Table B.1 and follow from the mass and dimension properties (based on the CAD model of the design) listed in Table H Considering the motor that is selected, the choice of a gear ratio is limited to either 1:113 or 1:156 (see Appendix C.3 for gearbox details). B.4.1 Constraints The three constraints from Section B.2.2 determine the feasible domain for the design variables m and l. The first constraint (B.8) just limits the pendulum length l to 0.4 m, whereas the second and third constraint are somewhat less transparent. A graphical representation of the moment due to gravity (B.9) and the maximum steady-state rotational pendulum amplitude (B.10) for different values of m and l is provided in Figure B.2. In Figure B.2, a black dashed line, indicating the constraints, bounds the feasible domains of possible combinations of pendulum mass m and pendulum length l. Solid lines indicate feasible and dotted lines indicate unfeasible combinations of m and l. The cross section of the feasible domains from the two graphs in Figure B.2 forms the net feasible domain of m and l for this optimization problem. Table B.2 lists this net feasible domain. From Table B.2 we can conclude that most combinations of m and l are feasible. Only large masses at large distances from the axis of rotation cause too high gravitational moments on the shaft. Moreover, small masses at small distances from the axis of rotation cause too large steady state pendulum amplitudes, coming close Tgrav [Nm] θss max [deg] l = 0.05 m l = 0.10 m l = 0.12 m l = 0.15 m l = 0.20 m l = 0.25 m l = 0.30 m l = 0.35 m l = 0.40 m m [kg] m [kg] Figure B.2: Gravitational moment T grav on pendulum shaft (left) and maximal steady-state rotational pendulum amplitude θss (right) for different values of m and l. Black dashed lines indicate the constraint values.

91 80 B.5 Conclusions to singularity. m [kg] l [m] v v v v v v v v v v v v v v v v v v v v v v 6 - v v v v v v v v v v v v v v v v Table B.2: Feasible domain, indicated by v. B.4.2 Objective Figure B.3 depicts a graphical representation of the maximal required steady-state motor torque (equation (B.14), in which I has been substituted by 1 12 m( ) ), for different values of m, l and i, whereas the other parameters have been fixed at the values stated in Table B.1. In Figure B.3, the overall feasible combinations of m and l from Table B.2 are indicated by solid solid, whereas the unfeasible combinations are depicted by dotted lines. A first, very convenient, phenomenon that is observed in Figure B.3, is that the minimal values of the required steady-state torque result from feasible combinations of m and l. One can also observe that, for feasible combinations of m and l, the larger the mass m, the lower the required stead-state motor torque T m ss. Moreover, if looked at from a qualitative point of view, the influence of the gear ratio on the required steady-state motor torque is small. B.5 Conclusions Let us summarize the results from the qualitative and quantitative analysis of the influence of the system parameters on the TORA design and their consequences for the experimental TORA system. As stated in the introduction of this appendix the results from the optimization procedure should be treated as a guideline for the TORA design, rather than a genuine and solid optimal solution. This should be taken into account regarding the following conclusions: 1. The nominal value of the spring stiffness k is set to 500 N/m. 2. From the qualitative analysis it follows that the cart mass M and pendulum inertia I need to be minimized in order minimize the maximum of the required steady-state motor torque. Because of the same reasoning, also minimum motor mass and motor inertia I mot are desirable. The value of M in the final TORA design is kg, whereas the mass and inertia of the motor-gearbox assembly is 1.04 kg and kg m 2, respectively. The pendulum inertia (given a certain pendulum mass m and length l) is minimized by constructing a pendulum consisting of a compact mass of high-density material that is attached to the pendulum shaft using a small low-mass structure. 3. The quantitative analysis of the constraints yields a net feasible domain for the pendulum mass m and pendulum length l, given by table B From results of the quantitative analysis (Figure B.3) it can be observed that there are no large differences in the magnitude of the required steady-state motor torque between the

92 B. Optimal design i = 1 : 113 i = 1 : T m ss [mnm] T m ss [mnm] l = 0.05 m l = 0.10 m 70 l = 0.12 m l = 0.15 m l = 0.20 m l = 0.25 m 60 l = 0.30 m l = 0.35 m l = 0.40 m m [kg] m [kg] Figure B.3: The maximum required steady-state motor torque for different values of m, l and i. Solid and dotted lines indicate feasible and unfeasible combinations of m and l, respectively. two gear ratios. Based on the results of the optimization procedure, a choice for one of the two gear ratios therefore is not obvious. One should remark, however, that a smaller gear ratio allows higher pendulum speeds for a given maximal motor speed. Compared to a gear ratio of i = 1 : 156, a gear ratio i of 1:113 therefore results in free additional speed, i.e. higher possible pendulum speeds, without loss of steady-state pendulum torque. Taking into account the desire to construct a TORA setup as versatile as possible, a gear ratio of i of 1:113 is selected. 5. From the results of the quantitative analysis (Figure B.3) for i = 113 it follows that the combination of m and l requiring the smallest steady-state motor torque is that of a mass of 6-8 kg at a distance of m from the pendulum shaft. Moreover, the absolute minimum steady-state motor torque is is reached for a mass of 8 kg at a distance from the pendulum shaft of 0.2 m. Within this perspective the following remarks should be noted: If one extrapolates the results in figure B.3 for larger masses, one could conclude that a mass of 12 kg at distance of 0.15 m would yield a global minimum of the steady-state motor torque. Masses of more than 8 kg have not been considered however, because they may possible give rise to other constructional problems. Although the optimum values of m and l are 8 kg and 0.2 m, the design values of m and l are set to 6 kg and 0.2 m, respectively. This does give rise to a somewhat larger steady-state motor torque, but it requires less mass, less material and less volume. The design values of m and l are therefor set to 6 kg and 0.2 m, respectively.

93 82 B.5 Conclusions

94 Appendix C Motor Selection In Section 4.3.2, it is mentioned that the selection of a motor that meets the torque and speed requirements is not a straightforward process of merely looking for a motor that matches these (torque) requirements because the requirements themselves depend on the motor mass and inertia. For the steady-state situation this can be concluded from the expression for the maximum required steady state motor torque given by (4.1). However, although no analytical expressions are available, also the required transient torques depend on the motor mass and inertia. In case of a direct drive configuration, where one needs a large and heavy motor, the motor mass is the dominant parameter in effecting the required torque, whereas the motor inertia is negligible with respect to the total pendulum inertia (see also Table 4.2). In the indirect drive configuration on the other hand, it is the other way around. The motor mass is relatively small because of the smaller motor, but the equivalent motor and gearbox inertia i 2 I mot (see equation (4.1)) in this case significantly contribute to the required motor torque. For both the direct drive and indirect drive configuration, this dependency of required torque on motor mass and inertia is discussed in the next sections in more detail, which enables us to select a motor that suits the (torque) requirements. C.1 Direct Drive Motors We study the possible use of three different kind of motors for the direct drive configuration: brushed DC motors, standard brushless DC motors and special ring-torque motors, as mentioned in Section Each category is characterized by its continuous and peak torque to weight ratios, which are more or less the same for all motors within a category. Figure C.1 shows the continuous torque and peak torque ratings for different motors from each category, with respect to the their mass (* for DC brushed motors, + for standard DC brushless motors, for ring torque motors). Assuming a constant torque to weight ratio we use the least square method to fit a linear torque-weight characteristic for each motor category (black, blue and red solid lines for peak torque characteristics and black, blue and red dashed lines for continuous torque characteristics). Also the dark and light shaded areas in Figure C.1 give the continuous and peak torque requirements, respectively, taking into account the motor mass. For zero motor mass these requirements comply with the specified 20 Nm peak torque and the continuous torque with an RMS value of 3.5 Nm, as prescribed in the drive specifications (Section 4.3.1). Based on the relationship for the maximum required steady-state motor torque, equation (4.1), one can extend the continuous torque requirements for increasing motor masses (thick dashed line in Figure C.1; virtually linear). Simulations give the required peak torques for different motor masses. These simulation results serve as a basis to set the peak torques demand to be approximately ( motor mass).

95 84 C.2 Indirect Drive Motors available peak torque available continuous torque brushed DC motors brushless DC motors ring-torque motors peak torque requirements torque [Nm] continuous torque requirements (RMS) motor mass [kg] Figure C.1: Torque match for direct drive configuration, each marker represents a motor. From figure C.1, conclusions can be drawn concerning the motor (types) that are suitable for application in a direct drive configuration. Brushed DC motors turn out to have a too low torque to weight ratio to be used at all. They meet neither the specified continuous torque nor the specified peak torque requirements. A standard brushless DC motor of 6.5 kg or more has the right torque rating to satisfy the peak and continuous torque requirements. If one uses a ring torque motor to drive the pendulum, one only needs a 2.3 kg motor to meet these requirements. Actual motors that satisfy the torque requirements are a 3.2 kg ring torque motor, the Kollmorgen F4309A, and a 7.4 kg brushless DC motor, the Kollmorgen AKM53G. However, a 5.8 kg version, the Kollmorgen AKM52G, which does match the continuous torque requirements but is a little short on peak torque, is also considered to be acceptable. C.2 Indirect Drive Motors In contrast to the direct drive configuration, in the indirect drive configuration the motor mass is relatively small because of the smaller motor. At the same time, however, the equivalent motor and gearbox inertia, i 2 I mot (see equation (4.1)), will significantly contribute to the required motor torque. If one considers the use of a small DC motor in combination with a standard gearbox, the Maxon company is a very convenient and good motor supplier. It has a wide variety of motors and gearboxes available. Moreover, it can supply complete units consisting of a motor, gearbox and an encoder. Considering the Maxon motor program we select the Maxon RE 40, a small 150 W standard brushed DC motor, to drive the pendulum. Together with a Maxon GP 42C planetary gearhead with a 1:113 gear ratio this motor forms a combination that satisfies the (torque) requirements (see the next paragraph for details). The gear ratio forms an integral part of the pendulum design optimization procedure, as discussed in Appendix B, which yields the optimum value of 1:113.

96 C. Motor Selection 85 In Figure C.2, the same speed and torque characteristics as in Figure 4.1 are depicted, but only projected in one quadrant and taking into account the additional mass and inertia introduced by the Maxon RE 40 motor and GP42C gearbox. The red lines indicates simulation # 2, based on the somewhat faster poles. The blue lines indicates simulation # 3, based on a higher disturbance force amplitude. The solid lines close to the origin represent the steady-state situation, whereas the dashed lines represent a typical torque-speed characteristic during transients. Moreover, the RMS values of the steady-state required torques are depicted by dotted lines. The thick solid lines indicate the motor limits, based on the motor specifications, taking into account a 30 % frictional torque loss in the gearbox. The light and dark shaded areas indicate the peak and continuous operation range of the motor. torque [Nm] stall torque max. power motor limitations steady-state sim # 2 steady-state RMS sim # 2 transient sim # 2 steady-state sim # 3 steady-state RMS sim # 3 transient sim # 3 20 max. cont. torque 10 max. speed speed [rpm] Figure C.2: Torque match for indirect drive configuration From figure C.2 one can conclude that in the steady-state situation, even in case of a higher disturbance force amplitude, the torque and speed capacity is sufficient. During transient, in some cases torque and speed saturation will occur, as can be seen from the torque-speed curve during transient for simulation # 2, which is outside of the motor operating range. C.3 Maxon Details The Maxon RE 40 motor, whose the technical specifications are given in Figure F.1, comes in 3 versions: a 12V, a 24V and a 48V version, differing in rotor winding resistance and inductance. The high voltage versions have a high winding resistance, resulting in a lower current level compared to the lower voltage versions. Moreover, the higher voltage versions have slightly higher continuous torque capacity. Because of the current limitations of the available amplifiers we are forced to use the 48V version. This gives us the advantage of a higher continuous torque capacity, but this comes at the cost of a higher inductance, which negatively effects the dynamic response of the motor. Simulations indicate, however, that the latter effect is not significant. The Maxon GP42C gearbox is a special low-backlash planetary gearbox with ceramic gearwheels and shafts (see Figure F.2 for technical specifications). Compared to standard steel gearheads, ceramic gearheads have a higher torque capacity and lower backlash (0.5 degree, compared to

97 86 C.3 Maxon Details up to 2 degrees). With respect to the requirements on the torque that needs to be transmitted through the gearhead, one needs to take into account that a considerable part of the motor torque is used to accelerate the motor and gearbox inertia itself which is not transmitted to the pendulum via the gearbox. Therefore only the torque levels resulting from the pure pendulum simulations, in which no motor inertia is taken into account (see Figure 4.1), need to be accommodated by the gearbox, rather than the torque levels from Figure C.2. The pure pendulum simulations show a continuous torque level of 3.5 Nm and a peak torque level of somewhere between 20 and 35 Nm, depending on the torque level at which the motor is saturated. These torque levels must be accommodated by the gearbox. At stall however, the full motor torque will be transmitted via the gearbox to the pendulum. Figure C.2 indicates that this stall torque can be as large as 50 Nm. The maximum continuous torque capacity of the GP 42C gearhead is 15 Nm at its output shaft, while intermitted peak torques of 22.5 Nm are permissable. The manufacturer claims though, that higher torque levels would not cause the gearbox to fail, but only shorten its life time. Since life time is not an issue in the TORA setup (average operation times are in the order of half an hour per week) it is assumed that, despite the rated peak torque exceeding that will occur, the GP42C suffices to drive the pendulum. The reason not to choose a larger gearbox is the larger weight of such a gearbox, which in turn would increase the torque requirements 1 For angular position measurement a standard optical incremental encoder is selected, the Maxon HEDL55, which is fitted on the motor shaft. This encoder has 500 physical lines, which results in a quadrature decoded resolution of 2000 counts per motor shaft revolution. With a gear ratio of 1:113 this results in counts per revolution of the pendulum shaft, or equivalent, a resolution of 1.6 mdeg. Figure F.3 gives the technical data of the Maxon HEDL55 encoder. 1 In the final experiments on the TORA system the torque levels, due to mild controller gains and not too large initial conditions, stay well within the gearbox torque specifications. However, due to some more torque demanding pendulum motions in the pendulum testing phase, or due to the oscillating characteristic of the transmitted torque during the experiments, the gearbox seems to have been damaged a little. The gearbox still functions as desired, only it sounds a bit worrying. As long as the gearbox keeps working there is no problem. In case it would fail, Maxon has provided us with a complete new spare motor-geabox assembly that can easily be mounted. Moreover, there is the possibility to mount a heavier gearbox, the Maxon GP52C, without the need to make any changes to the setup. The latter is not available as a spare unit, but would have te be purchased. For more detail on the operation of the gearbox one referred to the work of [26].

98 Appendix D TORA Loads In order to design a proper pendulum and its support structure, one requires the magnitudes of the loads to which it is exposed. This appendix contains an analysis of these loads, which are the reaction forces that the pendulum exerts on the cart. These reaction forces are assessed during normal operation (Section D.1) as well as in worst case conditions, i.e. a collision of the cart or pendulum with its physical end stop (Section D.2). The resulting nominal and maximal design loads, which are addressed in Section D.3, serve as a basis for several design consideration. Moreover, in addition to the analysis of the different TORA configurations in Section 4.4, the resulting moments on the cart are calculated and compared for different possible TORA configurations in Section D.4. D.1 Steady-state Pendulum Reaction Forces Figure D.1 depicts the reaction forces of the pendulum on the cart at the lower shaft end (point p in Figure 4.4). This is the point where the pendulum, via the shaft, connects to the cart, which is a point of interest from a constructional point of view. The forces F p i are the same in Figure D.1 and 4.4. If one considers the reaction forces at point p, however, the gravitational force F m z induces some additional moments M p x and M p y that need to be taken into account. In steady-state situation the resulting pendulum force in x direction, F p x exactly opposes the disturbance force F, given by equation (3.9). We can thus write: F p x = F 0 sin (Ωt). (D.1) -M p z F p x F p M p x -M p y y z x θ F p y F p z Figure D.1: Reaction forces and moments on lower shaft end.

99 88 D.2 Collision Forces Moreover, it holds that Fy p = Fx p cos (θ) sin (θ). ( ) Note that θ equals arcsin F0 sin(ωt) M LΩ in steady-state (see equation (3.22)). Substituting this 2 expression for θ in the expression for Fy p gives: F p y = M 2 L Ω4 F 2 0 sin2 (Ωt). The net steady-state pendulum reaction force in the horizontal plane F p = yields: F p = M L Ω 2. Then, there is also the gravitational force F z z the pendulum exerts on cart: (D.2) F p x 2 + F p y 2, which (D.3) F p z = mg. (D.4) Since the center of gravity of the pendulum is at a distance l from the pendulum shaft, the gravitational force induces the moments M p x = mgl cos (θ) = M L g cos (θ) and M p y = mgl sin (θ) = M L g sin(θ) at the pendulum shaft. Again, using the expression for θ in steady-state (equation (3.22)), these moments can be formulated as: M p x = M L g 1 F 0 2 sin 2 (Ωt) M 2 L Ω4, (D.5) and M p y = g Ω 2 F 0 sin(ωt). (D.6) The pendulum drive torque T p (which can be regarded as the moment M p z ) in steady-state situation follows from equation (4.1), in which i is set to 1 and I mot is set to 0. (Equation (4.1) actually describes the required steady-state motor torque, for the TORA system including the motor dynamics. In this case, however, we are interested in the required torque at the pendulum shaft, which does not depend on the motor dynamics. By setting i to 1 and I mot to 0 in Equation the required torques at the pendulum shaft are obtained from Equation (4.1).). The (design) pendulum mass m is 6 kg and the (design) pendulum length l is 0.2 m, which results in M L = 1.2 kgm. Furthermore, the gravitational acceleration g, is rounded at 10 m/s 2, Ω is 2π rad/s, and the disturbance force amplitude, as defined in Section B.1, for the actual TORA setup is about 10 N 1. Table D.1, among others, gives the maximum values of F p x, F p y, F p z, M p x, M p y and T p in the steady-state situation, which form a basis to set the nominal design loads, as discussed in Section D.3. D.2 Collision Forces The previous section addressed the pendulum reaction forces and moments in the steady-state situation. Besides higher force levels during transients, much higher forces and moments can be expected when the pendulum or cart violently hits its physical end stop. Such collisions do not occur during normal operation, but it is very unlikely that the H-bridge and pendulum controllers will never become unstable or that other control problems occur, causing the X and/or Y carriage to collide with the end stops. The TORA system is constructed such that it survives these collisions without any permanent damage. 1 Based on the actual values of M = kg and k = 500 N/m

100 D. TORA Loads 89 ẋ ẋ k1 x z y x z y F coll B k1 k2 FA coll F p x, A F p x, B k2 Figure D.2: Possible collision configurations. Pendulum hits end stop first (left), cart hits end stop first (right). Pendulum is parallel to x-axis. Figure D.2 shows the two possible collision configurations as the cart reaches the end of its stroke. When the pendulum is parallel to the x-axis ( θ = ±90 0), either the pendulum can hit its end stop first (left situation, or situation A in Figure D.2) or the cart can hits its end stop first (right situation, or situation B in Figure D.2). In order to obtain an estimate of the magnitude of the collision forces the maximum cart velocity 2 is assumed to be 1 m/s. Also the helical spring k 1 that forms the physical end stop of the cart is assumed to be linear and to deflect 10 mm when it is hit by the cart at 1 m/s 3. Further, the rubber/polycarbonate pendulum end stop k 2 is assumed to behave like a linear spring that deflects 15 mm when it is hit at 1 m/s. During collision, the cart and pendulum kinetic energy U k = 1 2 M Mẋ 2 is converted to potential spring energy U s = 1 2 F coll d, where d is the maximum spring deflection and F coll is the maximum force during collision. The total TORA mass M M consists of the (design) pendulum mass m of 6 kg and the mass of the cart and the pendulum support structure M of kg. The cart velocity ẋ is 1 m/s. Consider the case the pendulum hits its end stop first (case A). In this situation d A = m, which yields a maximum collision force FA coll = 1544N. The magnitude of the reaction force that the pendulum exerts on the cart as a result of this collision is given by F p x, A = F A coll M M+m, which in this case equals F p x, A = 1152N. In the case the cart hits its end stop first (case B), d B = m, which yields a maximum collision force FB coll = 2316 N. The magnitude of the resulting force that the pendulum exerts on the cart in this case is given by F p x, B = F B coll m M+m, which equals F p x, B = 600N. A similar analysis for a collision of the H-bridge Y-carriages with their end stops yields a worst case reaction force magnitude Fy coll = 1130N. The maximum torque in the pendulum shaft T p during collision, is the maximum torque the motor can deliver to the shaft. From Figure C.2, it follows that this maximum torque is about 50 Nm. A collision does not yield any additional forces F z and moments M x and M y other than the always present moments and forces due to gravity. Table D.1 gives the maximum magnitudes of the reaction forces and moments during collision. These form from a basis to set the maximum design loads, as discussed in the next section. 2 The cart velocity is limited to a speed of 1 m/s by a safety layer in the H-bridge controller. If the controller and the safety layer fail, the H-bridge is assumed to switch off. 3 Estimate based on the physical spring dimensions and H-bridge structure.

101 90 D.3 Design Load Load Steady-state max Worst-case max Design nominal Design max (collision) Fx p [N] Fy p [N] Fz p [N] Mx p [Nm] My p [Nm] T p [Nm] Table D.1: Pendulum reaction forces and moments at pendulum shaft. D.3 Design Load Table D.1 gives the magnitude of the TORA loads in the steady-state situation and during collision. The steady-state load situation forms a basis for the nominal design load of the TORA system. This nominal load level is chosen with some margin with respect to the steady-state load level. Table D.1 gives the specific nominal design loads. The maximum design load is determined by the maximum load the pendulum will experience, which is during collision. Under this load the pendulum is not required to have 100% performance, but it should not be permanently damaged. Table D.1 also gives the specific maximum design loads. The maximum design value of F p z is determined by an estimate of the peak forces which arise when mass is dropped onto the pendulum during the placement of the masses. The nominal and maximal design loads as listed in Table D.1 indicate what loads the TORA system should be able to accommodate and form starting point for a strengths and stiffness analysis which is addressed in Section 4.6. D.4 External Moments In addition to Section 4.4, in which the possible TORA configurations are addressed, in this appendix the external moments acting on the cart for the four different TORA configurations are analyzed. However not completely correct from a physical point of view, to obtain an indication of the magnitude of these moments, we only consider the situation in which θ = 0 0 combined with maximum steady-state values of the reaction forces 4 F p i. In this situation the external moments acting on the cart with respect to the origin 0 (see Figure 4.4) are given by: Mx 0 = Fz p a i + Fz supp a supp Fy p b i M 0 y = ±F p x b i M 0 z = ±T p ± F p xa supp, (D.7) where Fz supp is the weight of the pendulum support structure, which is about -70 N, and a supp is the distance of this support structure from the cart center 5, which equals to 140 mm for configurations 1, 2 and 3 and 0 mm for configuration 4. The plus / minus with Fx p and Tz p indicates that these forces can either be positive or negative, depending on the momentary direction of rotation. Table D.2 gives the lengths a and b for the different TORA configurations. Moreover table D.2 4 The maximum values of the reaction forces do not necessarily occur at θ = If one considers only Mx 0 supp for configuration 1,2 and 3, with Fz = 70 N and Fz p = 60 N a lower value of a supp would yield a lower value of Mx 0. Constructional limitations however, prevent asupp to be smaller than 140 mm.

102 D. TORA Loads 91 TORA configuration a [mm] b [mm] Mx 0 [Nm] My 0 [Nm] Mz 0 [Nm] # # # # Table D.2: Cart moments. contains an indication of the maximum values of external moments during steady-state according to equations (D.7), based on the maximum steady-state TORA load (first column in table D.1). These external moments don t show much difference for configurations 1, 2 and 3. The only conclusion that can be drawn here is that configuration 2 is undesirable because of it induces large moments M 0 x.

103 92 D.4 External Moments

104 Appendix E Linear Elastic Beam Theory In addition to Section 4.6, this appendix contains some standard linear elastic (beam) theory. In particular this theory is used to calculate stiffness, deflections and stresses for the pendulum shaft, which is the main component of the pendulum drive-train. Although not addressed in this appendix, the same calculations can be performed for the other components in the pendulum drive-train and support structure (bearing house, support plate, etc.). Figure E.1 depicts the pendulum shaft and its load configuration in the X-Y plane. The shaft is loaded axially (due to the weight of the pendulum F p z ), torsionally (due to the drive torques T p ), radially (due to the pendulum reaction forces F p y and with moments as a result of the radial loads and the gravitational moments M p x. Note that pendulum shaft load configuration in the X-Z plane is similar, except for the reaction forces and moments, which for the load configuration in the X-Z plane are given by F p x and M p y. From an equilibrium of forces and moments if follows that: F A y F B y = h 3F p y M p x h 2 = h 3F p y M p x h 2. + F p y (E.1) Moreover, from an analysis of the moments at point A it follows that M A x = h 3 F p y M p x. (E.2) The sequel of this appendix addresses the (torsional) stiffness, deflections of the shaft end, and corresponding material stresses of the pendulum shaft and under the load configuration depicted in E.1. E.1 Deflections The deflection of the pendulum shaft in y-direction at point p, δ p, shaft δy p, shaft y, is given by = αx A p, Ap h 2 + δy, (E.3) where αx A is the angle of the shaft part B A at position A resulting from only the moment Mx A p, Ap and δy is the deflection of the shaft part A p (assuming the part B A is rigid and fixed) resulting from only the moment Mx p and the reaction force Fy p. Both αx A and δy p Ap follow

105 94 E.1 Deflections T p Fy B B F p z A α A x α P x P F P y p, shaft δy α B x h 1 h 2 h 3 M p x F p z T p axial load [N] F A y F p z 0 torsional load [Nm] T p 0 shear load [N] 0 F B y bending load [Nm] M A x F P y 0 M p x Figure E.1: Pendulum shaft and its loads in Y-Z plane. The shaft lengths h i are defined as h 1 = 31.5 mm, h 2 = mm and h 3 = 64 mm. from linear elastic beam theory for beams with uniform cross-section (see for example [2]). The p, shaft resulting expression for δ is: y p, shaft δ y = MA x h 1 3EIx s h 2 + F p y h 3 3 3EI s x Mp xh 2 3 2EIx s, (E.4) where E [N/m 2 ] is the modulus of elasticity of the shaft material and Ix s [m 4 ] is the second moment of area of the shaft cross section with respect to the x-axis. For circular cross sections Ix s is given by Ix s = π ( D 4 64 o Di 4 ), (E.5) where D o is the shaft outer diameter and D i is the shaft inner diameter. The solid pendulum shaft is made of steel (E = Pa) and has a m diameter. Under a nominal design load (see Table D.1), the pendulum shaft deflection at point p in p, shaft y-direction, δy, according to (E.4), equals mm. A similar analysis can be performed to obtain the deflections in x-direction. Moreover, a similar analysis can be performed to obtain deflections of the other parts in the pendulum (support) structure (bearing house, support plate

106 E. Linear Elastic Beam Theory 95 etc.). In addition to Section 4.6.2, the above calculation of the shaft deflection only serves as an example of how the deflections δi m have been calculated. E.2 Stiffness As mentioned in Section the torsional stiffness of the pendulum shaft is important since the rotation is the actuated degree of freedom of the pendulum. Here the specific values for the torsional stiffness of the pendulum shaft as well as for the gearbox shaft are calculated. The torsional stiffness of a shaft of uniform cross-section is given by c = GI t L [Nm/rad], (E.6) where G [N/m 2 ] is the shear modulus of elasticity of the shaft material, L [m] is the shaft length, and I t is the polar moment of area of the shaft cross-section. For cylindric shafts the latter is given by: π ( I t = D 4 32 o Di 4 ), (E.7) where D o is the shaft outer diameter and D i is the shaft inner diameter. The solid steel (G = N/m 2 ) Maxon gearbox shaft, with an estimated length of 10 mm and a diameter of 12 mm, according to equation (E.6), has a torsional stiffness c 3 of Nm/rad. The 25 mm solid steel pendulum shaft is torsionally loaded over length of L = h 1 + h 2 + h 3 = 244 mm (from the top of the shaft to the point p where the pendulum is attached, see also Figure E.1) which, according to equation (E.6), yields a torsional stiffness of Nm/rad. The torsional stiffness c 6 of the pendulum itself is determined by the resistance of the pendulum table to bending induced by the pendulum reaction forces Fx,y. m To obtain an estimate of this stiffness, the pendulum table is approximated by a cantilever of length l, which is fixed at its base (the pendulum shaft) and laterally loaded at its end by the pendulum force Fx m. The torsional stiffness c 6 is then given by c 6 = F x m l 2 [Nm/rad], (E.8) m, pend δ x m, pend where δx is the deflection of the cantilever at point m induced by the pendulum force Fx m. m, pend The deflection δx is given by m, pend δx = F x m l 3 3EIz p, (E.9) where E is the modulus of elasticity of the pendulum table material and Iz p is the second moment of area of the cantilever with respect to the z axis. Since the pendulum table is of aluminium, E = Pa. Moreover Iz p = 1 12 tb3, where t is the pendulum table thickness, which is 5 mm, and b is the pendulum table width, which is set to 90 mm (minimum pendulum table width). The resulting estimate for the torsional pendulum stiffness c 6, according to (E.8), is Nm/rad. In addition to Section 4.6.1, we can therefore state that the torsional stiffness of the gearbox shaft is Nm/rad, that of the pendulum shaft is Nm/rad and that of the pendulum itself is Nm/rad. E.3 Strength From the shaft load configuration depicted in Figure E.1, one can conclude that the maximum shaft stress occurs at point A. At point A the shaft is subjected to a bending moment M A x, a

107 96 E.3 Strength shear force F A y, a torsional moment T p and an axial load F P z. The maximum material stresses (at the shaft outer surface) resulting from these loads are given by σ tor = T p 1 I t 2 D o, σ ax = F z p C, σ shear = F A y C and σ bend = MA x I x 1 2 D o, (E.10) ( ) where C = π 4 D 2 o Di 2 [m 2 ] is the area of the cross section of the shaft, Fy B and Mx A are given by equations (E.1) and (E.2) and I x and I t are given by equations (E.5) and (E.7), respectively. For maximum design load (see table D.1) the resulting stresses are σ tor = 16.1 N/m 2, σ ax = 1.0 N/m 2, σ shear = 3.1 N/m 2, σ bend = 38.2 N/m 2. Neglecting the axial and shear stresses, the remaining combined torsional and bending induced stress configuration yields an equivalent Von Mises stress σ e of 47.5 MPa.

108 Appendix F Component Specifications This appendix contains the manufacturer specifications of the components that are applied in the TORA setup. These are the motor, the gearbox, the encoder, the clamp-hub and the flexible coupling.

109 98 Figure F.1: Maxon RE40 technical specifications. Type is selected. Note that, although this type is not in the stock program anymore, it was in the stock program at the time of purchase.

110 F. Component Specifications 99 Figure F.2: Maxon planetary gearhead GP42C technical specifications. Type is selected

111 100 Figure F.3: Maxon incremental encoder HEDL55GP42C technical specifications.

112 F. Component Specifications 101 Figure F.4: Clamp hub specifications (Type DA-25-F).

113 102 Figure F.5: ROBA DX coupling drawing (Type , size 1).

114 F. Component Specifications 103 Figure F.6: ROBA DX coupling specifications (Type , size 1).