Control of Robot. Ioannis Manganas MCE Master Thesis. Aalborg University Department of Energy Technology

Transcription

1 Control of Robot Master Thesis Ioannis Manganas MCE4-3 Aalborg University Department of Energy Technology

2 Copyright c Aalborg University 8 LATEXhas been used for typesetting this document, using the TeXstudio IDE. All plots have been created using either Matlab or Inkscape. Simulink has been used for all simulations. Measurement data have been acquired using LabVIEW.

3 Title: Control of Robot Semester: Semester theme: Master s thesis Project period:..8 to.6.8 ECTS: 3 Supervisor: Torben Ole Andersen & Lasse Schmidt Project group: MCE4-3 SYNOPSIS: Ioannis Manganas The focus of this thesis is the development of a backstepping controller that avoids the "explosion" of terms and for which boundedness results are based on Lyapunov stability theory. The controller is designed for application to symmetrical hydraulic cylinders, operating as joint actuators for a DOF manipulator. Each electro-hydraulic system is subject to parameter variations and external disturbances. Initially, the state of art is investigated for a simplified backstepping based design. The model of the system is developed and validated from experimental data. The controller selected is redesigned for the investigated electro-hydraulic system where it is found difficult to prove stability for the closed loop system. A different controller is designed, based on sliding mode disturbance observer, for which boundedness results are acquired for the closed loop system. Again, the high complexity of the backstepping technique is avoided. Efficacy of the designed algorithm is investigated by conducting simulation results and comparing selected performance indices with reference controllers. Sensitivity analysis suggests increased robustness of the proposed control algorithm.

4 Pages, total: 55 Appendix: 55 Supplements: - By accepting the request from the fellow student who uploads the study group s project report in Digital Exam System, you confirm that all group members have participated in the project work, and thereby all members are collectively liable for the contents of the report. Furthermore, all group members confirm that the report does not include plagiarism.

5 Contents Preface Summary vii ix Introduction. Scope of the report The backstepping control design procedure State Of Art Combination with adaptive techniques Combination with SMC Preliminary conclusions System Modeling. Nonlinear model of the system Model of the manipulator in joint space Model of the manipulator in the actuator space Modeling of the hydraulic actuators Model of the system Validation of the nonlinear model Verification of the gravitational terms Verification of the dynamics Linearization Chapter summary Simplified backstepping control design 9 3. Backstepping controller design Load pressure controller design Analysis of the closed loop system stability in the Lyapunov framework Chapter 3 summary iii

6 iv Contents 4 Problem formulation and control design Problem formulation - Hypothesis Challenges of the design Approach Controller design and analysis Summary of Approach Approach Summary of the first two steps of the adaptive backstepping design Third step based on disturbance observer Simulation of proposed algorithm in continuous time Chapter 4 summary Reference Controllers Linearization and plant transfer functions Analysis in frequency domain Pressure feedback Disturbance rejection Proportional-Integral controller design Anti-windup Proportional-Lead and Proportional compensator design Velocity feedforward Chapter 5 summary Performance and robustness comparisons 7 6. Performance indices Noise Velocity estimation Periodic trajectories Performance under nominal conditions Robustness comparisons Parameters that vary Variation of the initial volumes Variations in leakage coefficient and Coulomb friction Variation of the viscous friction coefficient Variations in effective oil bulk modulus Variations in trajectory and load Summary of simulation results and comments Conclusions from simulations

7 Contents v 7 Conclusions and Future Work 7. Conclusions Future Work Bibliography 3 A Manipulator kinematics 9 A. Forward kinematics A.. Homogeneous transformations A.. Tool tip frame {hx h y h } A..3 Center of mass frames {Rx cm y cm } and {Cx cm y cm }.... A. Positions and Velocities of centers of mass A.. Position vectors A.. Velocity vectors A.3 Dynamic modeling of the mechanical system A.3. The Euler-Lagrange equation A.3. Generalized coordinates and energy B Trajectory generation and inverse kinematics 5 B. Workspace of the manipulator B. Trajectory in Cartesian space B.3 Slow trajectory B.3. Trajectory in joint space B.4 Fast trajectory B.4. Fast trajectory in joint space B.4. Fast trajectory in actuator space B.5 Spectrum analysis of piston reference signals B.5. Periodic trajectories B.6 Realizability of trajectories and P L Q L diagrams B.6. Maximum power transfer B.6. P L Q L diagrams for the trajectories B.6.3 P L Q L diagrams with load at the tool tip C Linearization 9 C. Linearization of the valve flow equation C. Linearization of the mechanical system C.3 Block diagram representation of the linearized system C.4 Selection of the linearization point D Velocity estimation 33 D. Differentiator design D. Application

8 vi Contents E Discretization 37 E. Discretization of the linear reference controllers E.. Sampling E.. The bilinear transform E..3 PI reference controller E..4 P and P-Lead compensator E..5 Overview of reference control algorithms E. Nonlinear controllers E.. Integration E.. RABLIN - Chapter E..3 Proposed Approach - Section F Theorems 45 F. Assumptions F.. Lipschitz continuity F. Lyapunov stability theorems and extensions G Parameters 49 G. Parameters used in the simulation model G.. Mechanical and hydraulic model G.. Parameters used in the hydraulic model G. Linearization point G.3 Controller parameters G.3. Reference controllers G.3. RABLIN G.3.3 Approach

9 Preface This thesis documents the th semester project of MCE4-3. It investigates the possibility of designing a controller based on integrator backstepping, while avoiding the high complexity of the last design step and on the same time deriving stability conclusions based on Lyapunov theory. The thesis is divided in 4 main parts. In the first part, the integrator backstepping technique is studied and the state of art of application in electro-hydraulic servo systems is investigated. The works that present decreased complexity are selected as candidates for application to the selected physical setup. In the second part, the model of the system is briefly developed. The model is validated with laboratory data. The selected method is redesigned, based on the given system. In the third part, a controller that preserves the low complexity of the selected controller is designed, based however on the possibility to prove stability or boundedness for the closed loop system. Finally, in the fourth part, the selected, designed and two reference controllers are compared under nominal conditions and sensitivity analysis is performed. The conclusions from the simulation results and the design process follow. Aalborg University, June, 8 Ioannis Manganas <imanga6@student.aau.dk> vii

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11 Summary Electro-hydraulic servo systems combine fast response with high power density and are utilized in a plethora of industrial applications. However, their nonlinear dynamics, as well as time varying parameters render the control design process rather complex, if high tracking performance is required. In this thesis, control design for electro-hydraulic servo systems in the form of symmetrical cylinders is considered. Specifically, the integrator backstepping control design procedure is investigated for application in these servo systems. This technique can be combined with adaptive and robust control laws, promising increased performance under parameter variations. In the last step, Lyapunov stability for the closed loop system can be proven by proper selection of the control variable. The disadvantage of backstepping is the increased complexity, stemming from the "explosion" of terms. In this thesis, it is investigated whether this level of complexity can be decreased. A method found in the literature is adopter for the control design for the system into consideration. This system is a DOF manipulator whose joints are actuated by the symmetrical cylinders. Apart from the decreased complexity, proving stability using Lyapunov theory is required. Initially, in order to design the selected control algorithm and test proving stability, a model of the system is developed. The model is validated using experimental data. Afterwards, the selected reference controller is redesigned for application to the system. This controller reduces the complexity by replacing the final step of the backstepping procedure by a linear controller for the load pressure loop. It is however proven difficult to prove stability for the whole closed loop system using a Lyapunov function candidate that is widely used for the backstepping design. In order to keep the low complexity of the reference controller, but at the same time derive stability results, the third backstepping step is designed using sliding mode control techniques. In the first approach, the term that induces complexity is directly canceled using a first order differentiator. However, selecting the parameters for the adaptive parameters proves to be difficult and a second approach is made. In this second approach, a sliding mode disturbance observer is used. Boundedness region for the system parameters, apart from the adaptation error terms, are estimated. ix

12 x Summary As a next step, two linear controllers are designed as reference and all four controllers are compared in simulations based on selected performance indices. It is found that the proposed controller, while robust to parameter variations and showing increased tracking performance, is sensitive to measurement noise. Another issue is the successful implementation using velocity estimation.

13 Chapter Introduction Electro-hydraulic systems are widely used in industrial processes, owing to their high power density. However, the hydraulic parameters can vary with operating conditions and can be difficult to estimate. The range of variation can also be significant. Furthermore, their dynamics are highly nonlinear, which poses a challenge for control design. Linear controllers have been found to achieve good performance, however when dealing with systems with uncertainties and varying parameters, other approaches can provide better tracking results. These uncertainties can stem from unknown or time varying parameters, such as the effective oil bulk modulus changing with temperature, a time varying load and the initial control volumes. Another type of uncertainty is due to neglected dynamics and nonlinearities that have not been accounted for, eg. friction forces. In [5], linear and nonlinear controllers were applied on a hydraulically actuated manipulator and compared in regard to tracking error. For the nonlinear controllers and especially the adaptive inverse dynamics controllers, the position tracking performance was higher than that of the linear controllers. In [7] it was found that the nonlinear controllers yielded better accuracy results, with Variable structure controllers presenting the highest accuracy, albeit complex to design. Model Reference Adaptive Control (MRAC) was found to be accurate when the initial transients were absent, or when initial values of the estimated parameters could be established beforehand.. Scope of the report In this report, the design of a control algorithm for two electro-hydraulic systems is considered. Each of these systems represents an actuated link of a two degree of freedom (-DOF) manipulator, located at the Fluid Power and Mechatronic Systems laboratory at Aalborg University.

14 Chapter. Introduction Even though designing control algorithms for serial link manipulators is a well studied topic in literature, the specific application poses additional challenges. The majority of the control design techniques, mostly for the nonlinear case, are based on the selection of the links angles as generalized coordinates and consider as input the torque applied at each joint. However, in the specific application the actuators are linear and as a result their linear positions provide a more intuitive choice as state variables. Furthermore, the applied torque is substituted by the applied force from each actuator. The aforementioned peculiarities change the form of the kinematic and dynamic description of the -DOF manipulator to a more complex form. The second difference of the specific system is that the actuator dynamics are highly nonlinear, contrary to the dynamics of direct current (DC) motors, or the highly researched dynamics of alternating current (AC) motors, especially of the permanent magnet type. In order to take into consideration the actuator dynamics, alongside the dynamics of the manipulator, the system can be considered as two subsystems. The backstepping approach is preferred in this report, since it provides a way of constructing a Lyapunov function for the whole system, that can be rendered ideally negative definite by a suitable selection of the control input. Another advantage of this technique is that it can be augmented with different control design approaches, overcoming their specific restrictions. An example of this would be the matching condition. The difficulty of applying the backstepping approach is the high complexity of the resulting control law. As a result, in this project the feasibility of designing a nonlinear control law based on backstepping and applying it to the specific setup is studied. The aim is to decrease the complexity in the highest degree possible that still guarantees crisp stability conclusions in the sense of Lyapunov. A secondary objective is to make the resulting control law more robust to parameter variations. In this chapter, the main idea of the backstepping design procedure is introduced. The state of art of backstepping based designs in electro-hydraulic systems follows, that acts as a guide on the selection of the simplest algorithm to consider for the specific application of the report.. The backstepping control design procedure A method to design nonlinear controllers for electro-hydraulic systems is the backstepping approach [6]. The main advantages of the backstepping design are that the matching criterion is relaxed [4] and a Lyapunov function for the system is constructed. Furthermore, using the backstepping approach it becomes somewhat easier to discern the nonlinear terms to be canceled in each step, leading to reduced control effort. The disadvantages are the increased complexity, stemming from the

15 .. The backstepping control design procedure 3 differentiation of the intermediate control laws in each step. Example design The backstepping procedure is illustrated for a general system in strict feedback form [6]. This example system has a similar form to a hydraulic servo system and is described by: ẋ = x (.) ẋ = f (x, x ) + g(x, x )x 3 (.) ẋ 3 = h(x, x, x 3 ) + k(x, x, x 3 )u (.3) It is assumed that no external disturbances exist and the parameters of the system are exactly known. Furthermore, the functions g(x, x ) and k(x, x, x 3 ) are assumed to be nonsingular. The purpose of the design is that x tracks a reference signal r(t). Defining the tracking error variable z = x r, the Equation. can be rewritten as: ż = x ṙ (.4) The tracking error could be asymptotically stabilized by a state feedback law of the form α (z ) plus a feedforward term, if the state variable x is seen as the input variable. However, since x is a state variable, another controller is designed so that the error between the actual value of x and its desired value, α (z ), becomes zero. To this end, the error variable z is defined: z = x α (z ) (.5) In order to regulate z to zero, the state variable x 3 is seen as the input. A control law of the form α (z, z ) is designed by viewing the state variable x 3 as an input variable. The same reasoning leads to the definition of the error variable z 3 as: z 3 = x 3 α (z, z ) (.6) As a result, the error dynamics of the system described by Equations. -.3 are written as: ż = z + α (z ) ṙ (.7) ż = f (x, x ) + g(x, x )z 3 + g(x, x )α (z, z ) α (z ) (.8) ż 3 = h(x, x, x 3 ) + k(x, x, x 3 )u α (z, z ) (.9) It is also assumed that the functions g(x, x ) and k(x, x, x 3 ) never become equal to zero and the full state vector is available for measurement. From Equation.9, the actual input variable is available and a controller can be designed so that for the

16 4 Chapter. Introduction system described by the Equations the origin is a Globally Asymptotically Stable (GAS) equilibrium point. The procedure starts by initially making the origin GAS for the subsystem in Equation.7 by selecting the desired stabilizing feedback law α (z ), which is the desired value for the state variable x. The selection can be facilitated by using the following Control Lyapunov Function (CLF) 6 [6]: By selecting α (z ) as: V (z ) = z (.) V (z ) = z ż = z [z + α (z ) ṙ] (.) The derivative of the Lyapunov function V (z ) becomes: α (z ) = k z + ṙ (.) V (z ) = k z + z z (.3) If z becomes equal to zero, V (z ) < and tracking error approaches zero asymptotically, with a rate dependent on the value of k. This motivates the selection of a control law that drives z to zero, or in other words, makes the state variable x equal to its desired value α (z ). To achieve this the subsystem described by Equations is considered. A CLF for the second subsystem can be written as: V (z, z ) = V (z ) + z (.4) V (z, z ) = k z + z z + z [ f (x, x ) + g(x, x )z 3 + g(x, x )α (z, z ) α (z )] (.5) By selecting: Then, for V (z, z ): α (z, z ) = g(x, x ) ( z f (x, x ) + α k z ) (.6) V (z, z ) = k z k z + g(x, x )z z 3 (.7) To stabilize z 3 to the origin, the whole system described by Equations is considered. A CLF is selected as: V 3 (z, z, z 3 ) = V (z, z ) + z 3 (.8) V 3 (z, z, z 3 ) = k z k z + g(x, x )z z 3 + z 3 [h(x, x, x 3 ) + k(x, x, x 3 )u α (z z )] (.9)

17 .3. State Of Art 5 Now the actual input variable can be used to design a control law so that for the whole error dynamics system, the origin is rendered GAS. This can be achieved via the control law: u = k(x, x, x 3 ) [ α (z, z ) h(x, x, x 3 ) g(x, x )z k 3 z 3 ] (.) As a result, for the Lyapunov function V 3 (z, z, z 3 ): V 3 (z, z, z 3 ) = k z k z k 3 z 3 (.) and the origin is GAS. This means that the error variables z, z 3 approach zero and as a result, the tracking error also approximately becomes equal to zero. The procedure is illustrated in Figure.. The virtual control variables are used so that a Lyapunov function is designed for the whole system, which is rendered negative definite using the actual control input. r Int. Control Int. Control d d dt dt z z k α k g α z 3 d dt k3 u k k x 3 System g x x f g h h f Figure.: The backstepping procedure. In Figure., the functions f (x, x ), g(x, x ), h(x, x, x 3 ) and k(x, x, x 3 ) are written without arguments and without all their signal paths. They are assumed to be completely known and are canceled. The derivatives of the intermediate, or virtual, control laws α (z ) and α (z, z ) can be analytically computed. They are expressions of the system states and derivatives of the reference signal, which are assumed known. However, these expressions become more complex with every recursion of the control design..3 State Of Art Owing to the aforementioned advantages, nonlinear controllers based on backstepping have been designed and applied to electro-hydraulic systems. To account for the parameter variations, different techniques have been applied alongside the backstepping design.

18 6 Chapter. Introduction.3. Combination with adaptive techniques In [45], a backstepping controller is designed for a symmetrical cylinder for force and position control. Simulation results are performed to compare the tracking performance to different controllers. In [8] a backstepping controller is designed for pressure tracking of a symmetrical electro-hydraulic actuator. Full state feedback is assumed. An adaptation algorithm has been designed based on the Lyapunov function to account for lack of accurate knowledge of the valve s discharge coefficient. In the first part of [4], exact backstepping controller design is applied to an asymmetric cylinder for position reference tracking, while in the second part an adaptation algorithm for the estimation of the hydraulic parameters is established. This algorithm is based on the Lyapunov function. In the experimental results, the proposed controller, albeit complex, outperforms a PD controller. However, the mechanical system s parameters are assumed to be exactly known and that full state feedback is available. The problem of unknown mechanical and hydraulic parameters is solved in [] by an identification procedure based on the recursive least squares algorithm. This procedure is realized before the control design with a sinusoidal input plus low power white noise. Afterwards, the exact, full state backstepping algorithm is designed and compared with a PID controller. The nonlinear controller is found to require less control input power for better tracking performance, especially when the load is increased. The Lyapunov function is designed in a way that the state matrix of the resulting closed loop error dynamics is almost diagonal when the tuning parameters are selected appropriately. Then, the negative effects of the magnitude of the hydraulic elements is alleviated. In [49], the Adaptive Robust Control (ARC) method is applied to an asymmetrical cylinder. This method is based on backstepping and for each step, the virtual control law is composed of adaptive and robust terms. In [5], ARC is applied on a symmetrical cylinder. The discontinuous projection method is used alongside tuning functions for parameter estimation. Both uncertain parameters and uncertain nonlinearities are dealt with. The resulting controller is rather complex and many tuning parameters are required. In all the aforementioned works, the use of adaptive laws assumes that the system is linear in the parameters. However, this assumption does not hold when the initial control volumes are unknown. This is tackled in [5], where adaptive backstepping control is applied to a valve actuated asymmetrical cylinder. A modified Lyapunov function is introduced so that the uncertain initial volumes can be estimated and used for control design, even though the system is not linear in parameters in regard to these terms. Furthermore, the overall controller is composed of many tuning parameters which can be difficult to select. Experimental results show that the proposed controller provides superior tracking performance to a non-adaptive backstepping controller and to an adaptive controller assum-

19 .3. State Of Art 7 ing known initial control volumes. The same approach was applied to a pump controlled asymmetrical cylinder system in []. In [9] and [8] an adaptive backstepping controller is designed for a symmetrical cylinder servo system. The tuning functions approach is used. Simulations show that including the valve s second order dynamics results in increased tracking performance. The full state vector is considered available and the resulting control law is complex and computationally heavy. This is deduced by the number of assignments and mathematical manipulations of the algorithm. In order to overcome the increased complexity of the backstepping based controllers, mainly due to the differentiation of the virtual control laws, different approaches have been made. In [], the backstepping design stops at the first two steps and the load pressure error is regulated by a PID controller. The simulation results, comparing the developed scheme to the one proposed in [9] augmented with the LuGre friction model, present good tracking performance. The complexity of the control algorithm is reduced by a wide margin. However, stability is not proven for the whole system, with or without the inclusion of the valve dynamics. Another method, described in [4], is proposed to decrease the complexity. The system is a DOF manipulator, actuated by an asymmetrical, servo valve controlled cylinder. An adaptive backstepping controller is designed similar to [5]. In order for the complexity to be reduced and ease implementation, the final step in the backstepping procedure is omitted. The force controller is designed as a pressure feedback loop and a proportional controller. Furthermore, the backstepping controller for the mechanical part is reduced to a PD controller, by noticing the resulting virtual control law of the second step. The overall tuning of the PD controller is based on linear techniques. The resulting controller presents increased end-point trajectory tracking performance, compared to a reference linear controller with flow feedforward. Instead of adapting to an unknown disturbance, such as the external force in [4], a disturbance observer is utilized. The backstepping design procedure then follows. The controller is compared experimentally to a Sliding Mode Controller (SMC) and a proportional linear controller. In [], a backstepping observer is designed for the load pressure estimation. Uniform ultimate boundedness is shown for the closed loop system..3. Combination with SMC SMC is used in [6] for trajectory tracking of a symmetrical cylinder. An integral type sliding surface is selected for the tracking error. However, to avoid chattering, a robust controller without a switching is designed. As a result, only boundedness of the tracking error can be proven. The adaptation laws for the unknown parameters, which include the initial volumes of the system, are found via a modified Lyapunov function, similar to the one used in [5]. In this work, the disturbance

20 8 Chapter. Introduction forces include the not modeled nonlinearities, external forces and unmodeled friction. This function has been assumed to be smooth and bounded, as well as its first derivative. Furthermore, since the state vector does not contain the load pressure, boundedness of the load pressure is assumed. In [5], SMC is designed based on the backstepping technique for position tracking control of a symmetrical cylinder. The parameters are considered known. The controller is designed on the linearized system, and is compared to a PID and an SMC controller. SMC is also used in [37]. The bulk modulus is considered constant as well as the rest of the parameters of the system, which is a servo valve controlled symmetrical cylinder. The backstepping approach is followed for the mechanical subsystem. In the second step, an SMC is introduced for robustness. The selected sliding surface is linear. Moreover, to avoid chattering, the sign function is replaced with a hyperbolic tangent one. The inner, load pressure loop is designed based on making the derivative of the Lyapunov function of the error negative definite. The flow equation is avoided via an inverse model of the valve s characteristics, acquired experimentally. A feedforward friction compensation term, as well as a reduced order disturbance observer are designed. The controllers are tested on an experimental setup. In [44], control of a hydrostatic transmission system is investigated. A backstepping based controller is designed for velocity tracking of the hydraulic motor. A second order SMC is designed to tackle the unknown disturbances due to the load torque and a state observer, based on extended linearization is used. The observer also features a robust, switching term. To attenuate the chattering effect, the sign functions are substituted by the hyperbolic tangent function, for the experiment. The proposed scheme presents marginally increased tracking performance, compared to a backstepping controller without SMC. In [48], the position tracking of a hydraulic motor is investigated. The unmeasurable states as well as the derivatives of the virtual control laws are acquired via a differentiator. This differentiator is designed with singular perturbation techniques and is finite time convergent. Neither phase lag nor chattering is present. The square root function of the orifice flow equation, which contains the sign function, is approximated by the upper and lower bounds of the chamber pressures. No adaptation algorithms are present. The added complexity stemming from the design parameters is used to adjust the tracking performance. However, this increases the control effort. The results of the backstepping controller with the two differentiators are validated in simulations with added measurement white noise. A different approach was made in [47] for the tracking control of a symmetrical cylinder. Initially, for the model formulation, the state vector contains the position, velocity and acceleration, instead of the position, velocity and load pressure. This assumes however that the disturbance terms can be differentiated. The dependence of the control input gain on the load pressure is dealt with via a pre-compensator,

21 .4. Preliminary conclusions 9 since the load pressure can be measured. Through the appropriate definition of the position tracking error variable and three switching like error dynamics surfaces, the control input is designed as three different terms. The first term contains a stabilizing and an error integral term, the second term a feedforward term for the cancellation of the linear-in-parameters terms and finally, a third control term. This final term is an asymptotic-like SMC with variable gain. The interesting point of the adaptive term is that for the parameter update law, only the reference values of trajectory are needed, thus the effect of noise is attenuated. The discontinuous projection method is used for the parameter update law. The whole state vector is assumed known and the velocity and acceleration measurements are acquired via the backward differentiation of the position measurement, passed through a Butterworth filter. The designed controller is experimentally compared to a PI, and adaptive backstepping controller and a version of the proposed controller without adaptation. The average value and the standard deviation of the tracking error are shown to be reduced..4 Preliminary conclusions Given the time varying and nonlinear nature of the electro-hydraulic servo systems, the backstepping controller design technique can be a viable approach that ensures a stable design. However, control algorithms of high complexity are derived. This puts a computational burden on the real time computer, as well as makes debugging and tuning a difficult procedure. Moreover, combination with SMC can induce chattering for low complexity designs and adaptation laws give rise to more tuning parameters. It is therefore highly desirable that a stable controller can be designed without many computations, so that it is easily deployable. This aspect is also considered in [9]. Furthermore, the varying and unknown parameters have to be considered, either by adaptation or by the controller being robust against them. A design procedure similar to the ones proposed in [4] or [] seem to be promising in that the step presenting the highest complexity of the backstepping design has been replaced with a simpler algorithm. As a result, it is intended at this point that a similar design be applied for the studied -DOF manipulator system. After the design is complete, stability in the sense of Lyapunov is considered. In order to design a control algorithm, the model of the system is developed. This is the topic of the following chapter. Note about block diagrams In the entirety of the report, block diagrams are used that feature feedback loops. The feedback signals are provided by sensors, whose output is voltage. It is considered that the bandwidth of all sensors is very wide, comparable to the sampling frequency. As a result, the sensors can be represented

22 Chapter. Introduction by simple gains. Furthermore, since the gains have been adjusted in the data acquisition algorithm, in the block diagram each sensor s gain can be considered to be equal to one. The following block diagram represents an ideal general case, where as an example a position feedback loop is considered. In Figure., this example system with measurement units is illustrated. X (s) ref [V] [V] G (s) p X(s) [m] [V] [V] K= [m] [m] Figure.: Block diagram of position feedback loop. Using block diagram manipulation, the same system can be represented by the block diagram of Figure.3: X ref (s) [V] [m] [m] [V] K= [V] K = X ref (s) [m] [V] [m] [m] G (s) p X(s) [m] Figure.3: Manipulated block diagram of position feedback loop. The plant is represented by the transfer function G p (s), the control input is a voltage signal for an electric valve. In the case a compensator was present in cascade with the plant, its input can be in the unit of the physical variable, here in [m], and its output in [V]. This will be the case for all the block diagrams of the report. The shaded part will be taken into consideration, meaning that the reference input and the measured output of a system will be in the same units, those of the output. Note regarding existence and uniqueness of solutions In this report, existence and uniqueness of the solution that satisfy the initial conditions for the differential equations of the form: ẋ = f(x), x(t ) = x Existence and uniqueness of the solutions requires Lipschitz continuity [9], [4] of the function f(x). It is assumed to hold for all the systems of nonlinear differential equations in the report.

23 Chapter System Modeling The development of the selected control algorithms requires the development of a model. The behavior of this model should represent as close as possible the actual system. As a result, in this chapter the model of the system is developed. Validation of the model using experimental data follows, a procedure that shows whether the model, and in extension the simulations, can be trusted.. Nonlinear model of the system.. Model of the manipulator in joint space P s x p P T u v Figure.: Illustration of the rigid link manipulator. Figure.: Diagram of the symmetrical actuator. The nonlinear model of the rigid manipulator, illustrated in Figure., can be derived using the Euler-Lagrange equations, selecting the joint angles as coordinates. The derivation of the model for the planar DOF manipulator has been derived in the literature in works such as [43] and [3]. The equations of motion have the

24 Chapter. System Modeling form: D(φ) φ + H(φ, φ) φ + Ḡ(φ) = τ (.) where τ is the vector of applied joint torques and φ is the vector of the joint angles... Model of the manipulator in the actuator space Since the equations of motion are functions of the selected generalized coordinates, which are the joint angles, it is desirable that they are expressed using variables directly related to the actuators, such as the actuator length or the piston position. The reason for this transformation is that these variables are measured and directly controlled. The joint angles can be related to the actuator lengths via the relationships: ( L OQ + LOH φ = w v + arccos ) x (.) L OQ L OH ( L φ = π w + v arccos BW + L BG ) x (.3) L BW L BG Furthermore, the angular velocities and acceleration of the joints are related to the piston velocities and acceleration by: φ = J s (x)ẋ (.4) φ = J s (x)ẍ + J s (x)ẋ (.5) J s = x 4LOQ L OH (L OQ +L OH x ) x 4L BG L BW (L BG +L BW x ) (.6) The joint torques, τ, can be related to the forces applied by the actuators, F, by the relationship: τ =J D (x)f = ( J s (x) ) T F (.7) [ ] LOQ sin(α J D (x) = ) (.8) L BG sin(α ) ( L OQ + x α = arccos ) ( L OH L, α = arccos BG + x ) L BW (.9) L OQ x L BG x..3 Modeling of the hydraulic actuators The hydraulic actuators are two symmetrical hydraulic cylinders. The position of each cylinder s piston is controlled via a 4/3 flow control proportional servo valve.

25 .. Nonlinear model of the system 3 Furthermore, the supply and tank pressures present small variations throughout operation, so they are assumed to be almost constant. A diagram of one actuator is presented in Figure.. The two control volumes are highlighted in different colours. Servo valve The servo valve s response can be described as that of a second order system, using an approximation of the frequency response from the datasheet [33]: X v (s) = ω v s + ζ v ω v s + ωv U(s) (.) Pressure dynamics Flow continuity equations are used to model the pressure dynamics for the chambers of each cylinder. An assumption here is made regarding the effective oil bulk modulus of each chamber. It is assumed that for each cylinder, both are equal. Furthermore, since the cylinder is symmetric and the valve is matched and symmetric, the load pressure can be used instead of the each chamber s pressure. This decreases the order of the system. The load pressure dynamics for each cylinder is described by []: ( Ṗ L = β e f f V A (x p ) + ) (Q L Aẋ C V B (x p ) l P L ) (.) where the effective bulk modulus, considering constant operation temperature, is given by []: ( ) P β e f f = a β e f f,max log a + a 3 (.) P max Other models for effective oil bulk modulus, where the dissolved air in the fluid is taken directly into consideration are available []. However, knowledge of the percentage of dissolved air is difficult to acquire. In Equation., cross-port or internal leakage flow is assumed to be laminar and is characterized by the leakage coefficient C l. The volumes of each chamber, V A (x p ) and V B (x p ) respectively, contain the volume of the hoses and the volume of each chamber. Flow through the valve For the symmetric and matched servo valve orifices, the load flow equation is given by []: Q L = K v x v P s P t sign(x v )P L (.3)

26 4 Chapter. System Modeling assuming that the flow is turbulent. Motion of cylinder piston Applying Newton s second law for the cylinder piston, the equation of motion is acquired []: m p ẍ = AP L B v ẋ F C F (.4) The mass of the fluid in the cylinder s chambers has been neglected as well as any gravitational effects. Furthermore, the mass of the piston, m p, is much smaller than the elements of the inertia matrix and can also be neglected...4 Model of the system Expressing Equation. in actuator space, using Equations. -.3, and.7, the model of the manipulator in actuator space is derived: M(x)ẍ + C(x, ẋ)ẋ + G(x) = F (.5) where x = [ x x ] T, F = [ F F ] T and from Equation.7, for each actuator: F i = AP L,i B v,i ẋ F C,i i =, (.6) Finally, the relationship between the total actuator length, x i and the piston position x p,i can be derived using Figure.3: x i = x min,i + H B + x p,i (.7) x i,min Figure.3: Relationship between the total actuator length and the piston position. Summary of the nonlinear model model In actuator space, the model of the system can be written as: M(x)ẍ + [C(x, ẋ) + B v ] ẋ + [G(x) + F C ] = AP L (.8)

27 Load Pressure [bar] Load Pressure [bar].. Validation of the nonlinear model 5 and for each actuator: ( Ṗ L = β e f f V A (x p ) + ) (Q L Aẋ C V B (x p ) l P L ) (.9) Q L = K v x v P s P t sign(x v )P L (.) X v (s) = ω v s + ζ v ω v s + ωv U v (s) (.). Validation of the nonlinear model In order to proceed with the controller design, the derived model should be verified with data from the laboratory setup. This is an essential step, because simulation using a verified model can help with the selection of tuning parameters that are not easy to systematically derive. Moreover, in the case of linear controller design, the designed controller can be applied on the actual system with minimal retuning... Verification of the gravitational terms To verify the gravitational terms, a small input voltage is applied on the servo valve of one servo system, while the other remains on a fixed position. Then the cylinder moves with a low velocity from end to end. By doing this test, the hydraulic model is verified, as well as the gravitational term of the mechanical system, since owing to the low value of the velocity, only the gravitational term from the mechanical model has value other than zero. The gravitational terms are verified through the load pressure for each servo system. In Figures.4 and.5, the load pressures of each servo system when it is being retracted is shown. For both cases, the other servo system is in the extended position. 6 4 Cylinder 4 3 Cylinder Measurement Simulation Measurement Simulation Time [s] Time [s] Figure.4: Load pressure of servo system. Figure.5: Load pressure when of servo system. The discrepancies between the experimental values and the simulations are probably due to the parameters selected to model the effective bulk modulus. Using

28 Input voltage [V] Input voltage [V] 6 Chapter. System Modeling Equation., it is not easy to tune all parameters for an exact result. Other models for the effective oil bulk modulus, describe the physical process in a more natural way, however rely on parameters that are difficult to determine, such as the percentage of the air dissolved in the fluid. Another factor that can lead to differences from the actual system, is the angle of the center of mass of the second link, whose calculation is not considered in this report. All in all, the behavior of the force applied from the cylinders to the links is of the same form for both the model and the actual system... Verification of the dynamics Verifying the dynamics of the system is of crucial importance for the assessment of the plant to be controlled, as well as for the evaluation of the control design. To proceed with the verification, a series of step voltage inputs was applied to each servo valve. Then, the same inputs were applied to the simulation model at the same time instances. The point of interest is the magnitude and frequency of oscillations during the transients cause by the step inputs. The input voltages are shown in Figures.6 and.7. 3 Cylinder 3 Cylinder Time [s] Figure.6: Step voltage sequence for servo valve Time [s] Figure.7: Step voltage sequence for servo valve. Regarding the first servo system, that drives the first link of the manipulator, the load pressure response can be seen in Figure.8 The highest discrepancy between the response of the model and the laboratory setup happens when the voltage is stepped and the cylinder is close to its end positions, at 9.65 [s] and.75 [s]. At these time instances also, the input voltage was stepped with a different sign, leading to change in the direction of motion. This leads to the initial conclusion that the damping ratio varies with the piston position. Similar results were acquired when Cylinder was retracted. Regarding the second servo system, the step responses are shown in Figure.: The same conclusions can be made regarding the damping of the system. The constant difference on the load pressure of the second servo system may be due

29 Load Pressure [bar] Load Pressure [bar] Load Pressure [bar] Load Pressure [bar].3. Linearization 7 Cylinder Measurement Cylinder Simulation 9 Measurement Simulation Time [s] Time [s] Figure.8: Load pressure of servo system during step voltage inputs. Cylinder is extended. Figure.9: Enhanced part of Figure.8. Cylinder Cylinder 3 Measurement Simulation 5 Measurement Simulation Time [s] Figure.: Load pressure of servo system during step voltage inputs. Cylinder is extended Time [s] Figure.: Enhanced part of Figure.. to the values of the moment of inertia around the center of mass, as well as the position of the center of mass. Both topics were not considered for the report and the values of the parameters used were derived in previous reports, such as []. All things considered, the model can be trusted for simulations, however care should be taken before applying a control law to the actual system, if in the simulated model the magnitude of oscillation is high, since the model seems more damped in specific positions..3 Linearization Linearizing a nonlinear model can provide information about the system properties that can be important for the control design process, even if nonlinear controllers are to be applied. Information regarding damping ratio, amplitude peaks and eigenfrequencies can help explain phenomena observed during simulations or during the testing of the controller on the system. Furthermore, this piece of information can be used to explain in a more quantitative way intuitive observations made by an initial overview of the nonlinear model. The linearized system is

30 8 Chapter. System Modeling derived in Appendix C..4 Chapter summary In this chapter, the model of the system was derived using the Euler-Lagrange equations. The hydraulic system model was also derived, based on the assumptions of turbulent flow through the valves orifices, laminar leakage flow between each chamber of the cylinders and equal effective oil bulk modulus in both chambers of each cylinder. The model was validated against experimental data. In the following chapter, the selected control design are adapted to the DOF manipulator system.

31 Chapter 3 Simplified backstepping control design In this chapter, the results of the designs of reduced complexity, are adapted to the system in question. The design follows that of [4]. In the final section of the chapter, stability of the system in the sense of Lyapunov is investigated. It is deemed difficult to estimate the stability margins for the system in a systematic way. Nonetheless, this is an approach that greatly reduces the complexity of the backstepping controller, at the cost of global stability results. This controller will be denoted as RABLIN, from [4], standing for Robust Adaptive Backstepping LINear controller. 3. Backstepping controller design The goal of the control design is that each hydraulic servo system follows its piston position measurement. To this end each system is written as a SISO system and the couplings due to the mechanical model are considered as disturbances. As a result, the two mechanical systems can be written as: M (x p )ẍ p + M (x p )ẍ p + (C (x p, ẋ p ) + B )ẋ p + C (x p, ẋ p )ẋ p + G (x p ) + F C, (ẋ p ) = AP L, M (x p )ẍ p + M (x p )ẍ p + (C (x p, ẋ p ) + B )ẋ p + C (x p, ẋ p )ẋ p + G (x p ) + F C, (ẋ p ) = AP L, (3.) ( ) Ṗ L,i = β e f f V A,i (x p,i ) + ( C V B,i (x p,i ) l,i P L,i Aẋ i + K v,i ū v,i ) (3.3) (3.) i =, for each servo system. The dynamics of the valve have been neglected and furthermore, a pre-compensator has been used for the system since the load 9

32 Chapter 3. Simplified backstepping control design pressure is available for measurement and no parameters are needed: u v,i = ū P s P t v,i (3.4) sgn(u v,i )P L,i It is thus assumed that the model of the system, using the pre-compensator, is Lipschitz continuous. This means that the solutions to the differential equations exist and are unique. The controller design has the same objective for both servo systems, so the following applies equally to both. However, due to the differences in the parameters, any design parameters will not be equal. The backstepping controller design is shown for the first servo system in the following. The subscript i is therefore dropped as well as the arguments of the system parameters. Defining the state variables x = x p, x = ẋ p, the hydraulic servo system is written in the state space form: ẋ = x (3.5) ẋ = ϑ x + ϑ P L d(t) (3.6) Ṗ L = ϑ 3 P L ϑ 4 x + ϑ 5 ū (3.7) where the uncertain system parameters and disturbances due to coupling and the gravitational force: ϑ = C + B M, ϑ = A ϑ 3 = β e f f ( V A (x p ) + ϑ 5 = β e f f ( V A (x p ) + V B (x p ), d(t) = M ẍ p + C ẋ p + G (3.8) M M ) ( C V B (x p ) l, ϑ 4 = β e f f V A (x p ) + ) A (3.9) V B (x p ) ) K v (3.) The system described by Equations 3.5 to 3.7 is in strict feedback form. From [6], the error dynamics of the system are derived via the coordinate transformation: z = x r z = x a (z ) z 3 = P L a (z, z, ˆϑ) (3.) where ˆϑ = ϑ + ϑ is the estimate of the unknown parameters, r the piston position reference signal and a j, j =, are the virtual control laws. The parameter estimation error is represented by the symbol ϑ. The derived error dynamics in the new