Minimizing Cable Swing in a Gantry Crane Using the IDA-PBC Methodology

Transcription

1 3 th National Conference on Mechanisms and Machines (NaCoMM7), IISc, Bangalore, India. December -3, 7 NaCoMM-7-65 Minimizing Cable Swing in a Gantry Crane Using the IDA-PBC Methodology Faruk Kazi, Ravi Banavar, Romeo Ortega, Narayan Manjarekar Systems and Control Engineering, II Bombay, Mumbai, India LSS, Supelec, Gif-sur-Yvette, France Corresponding author ( banavar@sc.iitb.ac.in) Abstract Precise payload positioning by an overhead crane is difficult due to the fact that the payload can exhibit a pendulum-like swinging motion. he stabilization of loads that are carried by cranes is tedious, and the lack of truly efficient control strategies implies a large economical loss due to the additional time involved in the process. From a control theoretical point of view, cranes are underactuated mechanical systems which give rise to challenging control issues. Motivated by the desire to achieve fast and precise payload positioning while minimizing swinging motion, several researchers have developed various controllers for overhead crane systems. In this paper, we apply a controller design techniue called interconnection and damping assignmentpassivity based control (IDA-PBC), that achieves stabilization for underactuated mechanical systems invoking the physically motivated principles of energy shaping and damping injection. IDA-PBC endows the closed-loop system with a Hamiltonian structure with a desired energy function that ualifies as a Lyapunov function for the desired euilibrium. he success of this method relies on the possibility of solving a set of partial differential euations (PDEs) that identify the energy functions that can be assigned to the closed-loop. In this paper, we use a partial feedback-linearization innerloop for explicit solution of these PDEs. Keywords: IDA-PBC, underactuated systems, cableoperated robot Introduction Gantry cranes are all pervasive in heavy engineering industry. A schematic representative of one such mechanism operating in two dimensions is shown in Figure. he objective is point to point positioning of the payload with minimum cable swing. here are two actuators- a linear actuator which actuates the cart and a rotary one which actuates the winch. For the purpose of the study here, we make the following assumptions:. he cable is massless and inelastic. PSfrag replacements θ Fx x θ M l m Figure : Overhead gantry crane. Dissipative forces on the cart and at the winch are negligible. Assumption simplifies the dynamic model and is a reasonable assumption given the comparatively large inertias of the payload and the cart. Assumption is used to simplify the modeling. A more general mechanism would involve two translational motions of the cart and a spherical pendulumlike motion of the payload. Several researchers have examined the control problem for the overhead crane system to achieve precise payload positioning with minimum swing. Fang et al [] utilize a simple proportional-derivative (PD) controller to asymptotically regulate the overhead crane system, the coupling between the planar gantry position and the payload angle is increased by the nonlinear controllers. In [], an overhead crane that exhibits double-pendulum dynamics is investigated by Weiping et al. Control of mechanical systems in a nonlinear setting has received much attention in the past decade. Amongst the techniues developed, a general and promising one has been the IDA-PBC methodology. he idea here is to synthesize a controller that stabilizes the closed loop system about a desired euilibrium and imparts certain characteristics to the closed loop response. In [3] and [4], passivity based interconnection and damping assignment control techniues are used to stabilize underactuated mechanical systems. he asymptotic stabilization of classical ball and beam system F l

2 3 th National Conference on Mechanisms and Machines (NaCoMM7), IISc, Bangalore, India. December -3, 7 NaCoMM-7-65 and a novel inertia wheel pendulum is achieved through a new parametrization of the closed loop inertia matrix. he matching conditions of controlled Lagrangian and IDA-PBC are discussed in [5]. he IDA-PBC methodology is extended to the class of underactuated mechanical systems with kinematic constraints in [6]. It also introduces the simplified matching euations on constrained manifold. his work is closely related to the controlled Lagrangian strategy for generalized matching euations for underactuated mechanical systems proposed in [7] and [8]. In [9], Kenji Fujimoto et al presented an asymptotic stabilization procedure of nonholonomic systems which are described in Hamiltonian framework. hese systems are then transformed into canonical forms with specified structure matrices using generalized canonical transformations. In [] Sorensen et al propose augmentation of kinematic inputs with standard Hamiltonian formulation. hese inputs change the internal structure of the mechanical system but do not change the stored total energy of the system. Potential energy shaping based controller for the point to point control of a gantry crane is discussed in [] wherein pulley dynamics is modeled as a holonomic constraint. his concept is further extended in [] for a combined flatness and energy based controller design which can robustly track an off-line computed trajectory. he paper is organized as follows: Section presents the dynamic model of the crane in port-hamiltonian framework begining with the Euler-Lagrange euations of motion. We then perform partial feedback linearization since it facilitates the controller design using IDA-PBC methodology. Section 3 starts with a brief introduction to the IDA-PBC theory applied to such systems. We then discuss the controller design strategy based on IDA-PBC methodology to gantry crane problem. Simulation results are discussed in Section 4. Finally, we wrap up this paper with some concluding remarks in Section 5. Dynamic Model In this section we develop the Euler-Lagrange euations of motion for the overhead gantry crane system. he configuration variables are θ x l where θ S denotes the payload angle about the vertical axis, x IR denotes the gantry position along the X coordinate axis and l IR denotes the cable length. he control u IR is defined as u where and represent the control-force inputs acting on the cart and the pulley, respectively. Note that the rotary actuation of the winch is translated as a tensile force on the cable for the purpose of control design. he control objective is to move the payload from any position to the desired position specified as: D x D l D () Notice that the desired configuration is a stable euilibrium and the control challenge is more of improving the transient performance in a large domain of attraction. he Lagrangian for the overhead gantry crane can be expressed as, where, and M L M V () ml ml cosθ ml cosθ M m msinθ msinθ m V mgl cosθ where M is the mass of the cart, m is the mass of the payload. Here M is the mass matrix and V represents the potential energy of the system. he resulting Euler-Lagrange euations are, ml θ mlcosθẍ ml θ l mgl sinθ ml θcosθ M m ẍ ml sinθ m l θcosθ ml θ sinθ msinθẍ ml ml θ mgcosθ hese Euler-Lagrange euations can be cast in the following form M C V Gu (3) where C, represent the centripetal-coriolis terms. Before proceeding with the controller design, we partially feedback linearize the system since this facilitates the design of the IDA-PBC controller [5, 3] and [4].. Partial Feedback Linearization We proceed as follows. With the vector IR n of generalized coordinates partitioned as R n m and a IR m, we may write the dynamic euations of the n degrees of freedom system as, where, m m a f (4) m m a f u M m m (6) m m is a partition of the symmetric, positive definite inertia matrix, the vector functions f and f contain Corioliscentrifugal and gravitational terms. Note that a represents actuated co-ordinates. For notational simplicity we will henceforth not write the explicit dependence on of these coefficients. Let v m u f m denote the new input and we have a partially linearized system given by a m f m m I (5) v (7)

3 3 th National Conference on Mechanisms and Machines (NaCoMM7), IISc, Bangalore, India. December -3, 7 NaCoMM-7-65 Recasting in terms of momentum coordinates to facilitate a Hamiltonian description we have p f (8) ṗ f m f m m I where p f represents momenta after feedback linearization. In explicit form (9) yields p f ṗ f gsin For future notational convenience we define cos B and F p f 3 3 gsin v cos In the next section we develop a control law based on the IDA-PBC methodology. 3 Stabilization of the Gantry Crane using IDA-PBC Methodology he basic philosophy of the IDA-PBC methodology is to assign the closed loop dynamics of the system to a desired Hamiltonian system characterized by the triple M d (9) v J V d standing for the desired inertia matrix, a skew-symmetric matrix and a desired potential energy respectively. We present a brief introduction from [4]. We start with a system whose Hamiltonian is H p p M p V () where IR n, p IR n are the generalized position and momenta, respectively, M M is the inertia matrix, and V is the potential energy. If we assume that the system has no natural damping, then the euations of motion can be written as ṗ I n I n H p H G u () he matrix G IR n m is determined by the manner in which the control u IR m enters into the system and is invertible in the case the system is fully actuated, that is, m n. But for the more difficult case of underactuated system the rank G m n. he desired (closed-loop) energy function is assumed to be having the following form: H d p p M d p V d () where M d M d and V d represent the (to be defined) closed-loop inertia matrix and potential energy function, respectively. We will reuire that V d have an isolated minimum at that is argminv d (3) In PBC, the control input is naturally decomposed into two terms u u es p u di p (4) where the first term is designed to achieve the energy shaping and the second one injects the damping. he desired port-controlled Hamiltonian dynamics are taken of the form ṗ where the terms J d p R d p H d p H d J d J d M M d M d M J p R d R d GK v G! (5) represent the desired interconnection and damping structures, respectively. he matrix R d is included to add damping into the system. his is achieved via negative feedback of the (new) passive output G p H d. Hence second term of (4) can be selected as u di " K v G p H d (6) where K v K v. he skew-symmetric matrix J (and some of the elements of M d ) can be used as free parameters in order to achieve the kinetic energy shaping. For the desired closed-loop dynamics, we state the following proposition from [4], which reveals the stabilization properties of IDA-PBC approach. Proposition 3.. he system (5) with () and (3) has a stable euilibrium point at. his euilibrium is asymptotically stable if it is locally detectable from the output G p H d p. An estimate of the domain of attraction is given by Ω c where Ω c # p IR n$ H d p c% and c sup# c H d 3. Energy Shaping $ Ω c is bounded % (7) o obtain the energy shaping term, u es, of the controller we replace (4) and (6) in () and euate it with (5) I n I n H p H M M d M d M J p G H d p H d u es (8) 3

4 - 3 th National Conference on Mechanisms and Machines (NaCoMM7), IISc, Bangalore, India. December -3, 7 NaCoMM-7-65 While the first row of the aforementioned euations is clearly satisfied, the second set of euations can be expressed as Gu es H M d M H d J M d p (9) Now, it is clear that if G is invertible, that is, if the system is fully actuated, then we may uniuely solve for the control input u es given any H d and J. In the underactuated case, G is not invertible but only full column rank, and u es can only influence the terms in the range space of G. his leads to the following set of constraint euations, which must be satisfied for any choice of u es : G&'# H M d M H d J M d p%( () where G& is a full rank left annihilator of G, that is, G& G. Euation (), with H d given by (), is a set of nonlinear PDEs with unknowns M d and V d, with J a free parameter, and p an independent coordinate. If a solution for this PDE is obtained, the resulting control law u es is given as u es G G G H M d M H d J M d p () he PDEs () can be naturally separated into the terms that depend on p and terms which are independent of p, that is, those corresponding to the kinetic and the potential energies, respectively. hus, () can be euivalently written as G&*) p M p M d M p M d p J M d p+ () G&'# V M d M V d %, (3) he first euation is a nonlinear PDE that has to be solved for the unknown elements of the closed-loop inertia matrix M d. Given M d, (3) is a simple linear PDE, hence the main difficulty is in the solution of (). Clearly, solution to () can be simplified if there exists a full rank left annihilator G& of G such that G& ) p M p + (4) his condition essentially imposes that the mass matrix M does not depend on the unactuated coordinate. It is satisfied by many well-known physical examples, for instance, the ball and beam, the VOL Aircraft and the Acrobot. But in our case, the mass matrix M was found to be dependent on the unactuated coordinate θ, the cable swing angle. o overcome this we employed partial feedback-linearization. Careful observation of (8) reveals that the new inertia matrix is identity and hence satisfying above condition. 3. he Matching Euations For Crane Dynamics Comparing the desired dynamics with actual dynamics after feedback linearization we get, p f F p f B M M d M d M J p v es H d p H d (5) his gives the following matching euation p f M d p f H d (6) F B v M d H d J p f H d (7) Pre-multiplying the above euation by B & (a full rank left annihilator of B such that B & B ) we have B & F B v B & M d H d J p f H d (8) Writing in coordinate form results in following explicit representation where B & F Q " g sin p f Q p f which gives rise to two PDEs - one for the kinetic energy and the other for the potential energy. Now, solving these two PDEs, (as detailed in [3, 4]), we can design a controller based on the IDA-PBC methodology. Here B & cos Solving the Potential Energy PDE he potential energy PDE can be written using the matching euation as g sin B & M d V d (9) Since mass matrix is identity for the feedback-linearized system, we propose desired mass matrix, M d, to be a constant matrix of the following form, M d k k k 3 k k 4 k 5 k 3 k 5 k 6 then the solution of (9) takes the form V d " gcos Φ z z / (3) where z and z sin. Note that the selection of Φ is governed by the condition(3). For this, the necessary condition V d is satisfied if and only if Φ z z, while the sufficient condition V d will hold if the Hessian of Φ at the is positive [4]. In our case we choose Φ to be a uadratic function which yields V d gcos α α sin 3 where α i i are used as tuning parameters. g denotes the euilibrium configuration and 4

5 3 th National Conference on Mechanisms and Machines (NaCoMM7), IISc, Bangalore, India. December -3, 7 NaCoMM Solving the Kinetic Energy PDE he KE-PDE is p f Q p f B & M d p f M d p f J M d p f Since M d is chosen to be a constant matrix, it is independent of. Hence, the KE-PDE gets converted to an algebraic euation as Solving this we get J as, p f Q p f B & J M d p f (3) J 4 k 6 p f k 6 p f 3.5 Energy Shaping Control he energy-shaping term v es of the control input is synthesized as follows. We have F B v es M d H d J p f H d (3) B v es F M d H d J M d p f (33) Now, it is clear that if B is invertible, that is, if the system is fully actuated, then we may uniuely solve for the control input v es given any H d and J. Since the system is underactuated, B is not invertible but only full column rank, and we have v es B 7 B B7 Here, his gives, v es M d F M d V d J M d P f k 6 65 (34) 98 3 : Z cos k 6 p p 3; cos : 3 cos 3 Z : 3 cos gcos k 6 3 α Z sin g k p. Note that the k 6 is the where Z α < sin free parameter which is available for tuning. 4 Simulations and Results Simulations were carried out with a twofold objective, first to show that the energy shaping controller proposed with IDA-PBC methodology ensures minimization of cable swing, and second to illustrate the robustness properties of the controller. he damping injection matrix was taken to be k v of the diagonal form K v. k v Effect of damping injection is illustrated in Fig. for k v and in Fig. 3 for k v. All other system parameters were kept the same while changing the damping k v. Mass of the cart, M was taken as kg and that of the payload, m was 5 kg. For the same initial conditions the settling time was observed to be increasing with increase in damping injection. Fig. 4 illustrates robustness property of the controller which is inherent in passivity based control. Here the parameters of the controller are the same as in the case of Fig. while taking M and m to 5 kg and 5 kg, respectively, for the simulations. he system response did not change significantly but the control effort was found to be more. he initial condition of the swing angle is deg for the results shown in Fig. to Fig. 4. We demonstrate the controller performance with the initial condition of 3 deg in Fig. 5 to emphasize the fact that our controller is effective for even large deviations of θ (outside linear regime). Here all other parameters are kept the same as in the case of Fig.. θ (t) (deg) x(t)(m) l(m) Initial: deg Desired: deg Initial:.5 m Desired:.5 m Initial:. m Desired:.8 m ime(sec.) θ (rad per sec) θ (rad) Damping Injection Damping is achieved via negative feedback of the passive output which is given as: where K v K v7 (35) we get v di " K v v di K v B 7 p f H d (35) is a damping injection gain. Solving p f cos p f p f 3 (36) Finally, the control input is calculated by combining v di and v es as v v es p v di p (37) Figure : Simulation results for K v. 5 Conclusions In this paper, we presented an IDA-PBC controller for an overhead gantry crane system. We employed partial feedback-linearization to simplify the solution of PDEs arising from the matching euation. So far, as found in literature the gantry crane problem has been considered with fixed cable length and hence has been modeled as a simple pendulum on a cart for designing control laws. In this paper we considered the cable length as a variable, which closely 5

6 3 th National Conference on Mechanisms and Machines (NaCoMM7), IISc, Bangalore, India. December -3, 7 NaCoMM-7-65 θ (t) (deg) x(t)(m) l(m) Initial: deg Desired: deg Initial:.5 m Desired:.5 m Initial:. m. Desired:.8 m ime(sec.) θ (rad per sec) θ (rad) θ(t)(deg) x(t)(m) l(m) Initial:3 deg Desired: deg 3 4 Initial:.5 m Desired:.5 m Initial:. m Desired:.8 m 3 4 ime(sec.) θ (rad per sec) θ (rad) Figure 3: Simulation results for K v. Figure 5: Simulation results for swing angle θ 3 deg. θ (t) (deg) x(t)(m) l(m).5.5 Initial: deg Desired: deg Initial:.5 m Desired:.5 m Initial:. m Desired:.8 m ime(sec.) θ (rad per sec) θ (rad) Figure 4: Simulation for Robustness M 5 kg, m 5 kg. replicates real-life crane systems. he control law so developed was found to perform well for the control objective of point to point control with swing minimization as shown in the simulations. he proposed controller was found to be robust for parameter uncertainties like change in mass of the cart as well as the payload. 6 Acknowledgement his work was partially supported by the Indo-French Centre for the Promotion of Advanced Research under a joint project between II-Bombay and LSS, Supelec. References [] Y. Fang, W. E. Dixon, D. M. Dawson, and E. Zergeroglu, Nonlinear Coupling Control Laws for an Underactuated Overhead Crane System, IEEE rans. on Mechatronics, vol. 8, no. 3, pp , 3. [] D. Liu, W. Guo, J. Yi, and D. Zhao, Passivity-Based Control for a Class of Underactuated Mechanical Sys- tems, in Proceedings of the 4 International Conference on Intelligent Mechatronics and Automation, (Chengdu, China), pp. 5 54, 4. [3] F. Gomez-Estern, R. Ortega, F. R. Rubio, and J. Aracil, Stabilization of a Class of Underactuated Mechanical Systems via otal Energy Shaping, in Proceedings of the 4 th IEEE Conference on Decision and Control, (Orlando, Florida USA), pp ,. [4] R. Ortega, M. W. Spong, F. Gomez-Estern, and G. Blankenstein, Stabilization of a Class of Underactuated Mechanical Systems via Interconnection and Damping Assignment, IEEE rans. on Automatic Control, vol. 47, no. 8, pp. 8 33,. [5] G. Blankenstein, R. Ortega, and A. J. van der Schaft, he matching conditions of controlled Lagrangians and IDA passivity-based control, International Journal of Control, vol. 75, no. 9, pp ,. [6] G. Blankenstein, Matching and stabilization of constrained systems, in Proceedings of the Mathematical heory of Networks and Systems, (Notre Dame, Indiana),. [7] J. Hamberg, Simplified Conditions for Matching and for Generalized Matching in the heory of Controlled Lagrangians, in Proceedings of the American Control Conference, (Chicago, Illinois), pp ,. [8] J. Hamberg, General Matching Conditions in the heory of Controlled Lagrangians, in Proceedings of the Conference on Decision & Control, (Phoenix, Arizona USA), pp , 999. [9] K. Fujimoto, Stabilization of a class of Hamiltonian systems with non holonomic constraints and its experimental evaluation, in Proceedings of the Conference on Decision & Control, (Phoenix, Arizona USA), pp ,

7 3 th National Conference on Mechanisms and Machines (NaCoMM7), IISc, Bangalore, India. December -3, 7 NaCoMM-7-65 [] M. J. Sorensen, J. D. Bentsen, P. Andersen, and. S. Pedersen, Asymptotic Stabilization of Non- Holonomic Port-Controlled Hamiltonian Systems, in Proc. of the 5 th IFAC/EURON Symposium on Intelligent Autonomous Vehicles (IAV ), (Lisbon, Portugal), 4. [] R. Banavar, F. Kazi, R. Ortega, and N. Manjarekar, he IDA-PBC Methodology Applied to a Gantry Crane, in Proceedings of the Mathematical heory of Networks and Systems, (Kyoto, Japan), pp , 6. [] F. Kazi, R. Banavar, R. Ortega, and N. Manjarekar, Point-to-Point Control of a Gantry Crane: A combined Flatness and IDA-PBC Strategy, in Proceedings of the European Control Conference, (Kos, Greece), 7. [3] J. A. Acosta, R. Ortega, A. Astolfi, and A. D. Mahindrakar, Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one, IEEE rans. on Automatic Control, vol. 5, no., pp , 5. [4] M. W. Spong, Energy Based Control of a Class of Underactuated Mechanical Systems, in Proceedings of 996 IFAC World Congress, (San Francisco, CA),

8 88