A SIMPLE ITERATIVE SCHEME FOR LEARNING GRAVITY COMPENSATION IN ROBOT ARMS

Transcription

1 A SIMPLE ITERATIVE SCHEME FOR LEARNING GRAVITY COMPENSATION IN ROBOT ARMS A. DE LUCA, S. PANZIERI Dipartimento di Informatica e Sistemistica Università degli Studi di Roma La Sapienza ABSTRACT The set-point regulation problem for robot manipulators under gravity is usually solved by either model-based compensation or PID control. The former cannot be applied if an unknown payload is present or when model parameters are poorly estimated, while the latter requires fine and lengthy tuning of gains in order to achieve good performance on the whole workspace. A simple iterative scheme is proposed for generating exact gravity compensation at the desired set-point, without the knowledge of the robot dynamic model. The control law starts with a PD action on the joint error, updating at discrete instants an additional feedforward term. Global convergence of the scheme is proved under a mild condition on the proportional gains. Simulations are presented for a 3R rigid robot arm moving in a vertical plane. 1. Introduction It is well known that a rigid robot arm can be asymptotically stabilized around a given joint configuration via a PD controller on the joint errors, provided that gravity is exactly cancelled by feedback [1]. This result holds globally, i.e. starting from any joint configuration, and requires only positive definite gain matrices. However, online cancellation of gravity effects leads to a nonlinear control law, which may be difficult to implement. Under a mild condition on the proportional gain, this scheme can be simplified to constant gravity compensation, evaluated only at the desired configuration [2]; a purely linear feedback law with a feedforward action is then obtained. In this case, the proportional gain should be chosen so to dominate the gradient of the gravity forces in the whole robot workspace. Similar conditions have been found also for robots with elasticjoints [3] or with flexible links [4], under a further assumption on the arm stiffness. In all cases, an exact knowledge of the gravity vector is required. This condition is difficult to be realized, e.g. for a robot picking up multiple unknown payloads, and

2 would need anyway an identification procedure of the robot link parameters. As a result, a steady-state error is present in general with this type of control even for a simple point-to-point task. High-gain feedback reduces but does not eliminate this error, exciting on the other hand unmodeled dynamic effects. A standard remedy is the addition of an integral term to the linear control law but several problems arise with the design of a PID, partly due to the nonlinear nature of the robot. Typically, saturation will occurr during large transient phases and reset or anti-windup procedures have to be devised when starting far from the final position. Moreover, from a theoretical point of view, asymptotic stability of robot PID control has been proved only locally around the desired configuration and/or under complex inequalities among the proportional, derivative, and integral gains [5, 6]: tuning of the PID gains is in fact a difficult task for nonlinear systems. In practice, these drawbacks may be overcome by adding the integral action only near the final point, so that gross motion is performed with PD control, while fine positioning is achieved with PID. This is a common strategy in industrial robot controllers, as in the Unimation Puma 56 [7], but no formal proof of convergence has been given for this method. As an alternative, an interpolating trajectory can be generated between the starting and the final point, but this involves further computations which could be ackward to handle in case of on-line set-point corrections. In this paper, we propose a fast iterative scheme that builds up the required gravity compensation at the final configuration, with a very limited knowledge about the robot gravity terms. A PD-based control law is applied iteratively and the constant gravity feedforward is learned without the explicit introduction of an integral error term nor the use of high-gain feedback. An easy to check sufficient condition is given for the convergence of the scheme to zero steady-state error, taking into account in the analysis the full nonlinear robot equations. 2. Control Scheme Consider a rigid robot arm with N rotary joints, whose dynamicmodel is B(q) q + S(q, q) q + g(q) =u (1) with inertia matrix B, Coriolis and centrifugal forces S q, and gravity forces g. Since the gravity term contains only trigonometric functions of the configuration q, itis easy to derive the following bound [2] g α. (2) q Note that this inequality holds also for prismaticjoints, even if in that case the gravity term g(q) may be unbounded (at least, in principle). When the input torque u is chosen as a PD control u = K P (q d q) K D q, K P >, K D >, (3)

3 with a desired constant configuration q d, then the robot arm will be driven to a steady-state condition q = q, q =, which from (1) satisfies the following equation g( q) =K P (q d q), (4) implicitly defining the residual joint error. Usually, a decentralized approach is followed by choosing diagonal gain matrices K P and K D. An iterative compensation scheme is introduced as follows. Let q be the initial arm configuration. The control law during iteration i is defined as u = 1 β K P (q d q) K D q + u i 1, β > (5) for i =1, 2,..., and where the term u i 1 is a constant feedforward. If u =, which is a common initialization, the first iteration is performed with a simple PD. Indeed, one may collect the best information on the required gravity term by setting u =ĝ(q d ), where the hat denotes an estimate. System (1) under control (5) reaches the equilibrium state q = q i, q =, satisfying g(q i )= 1 β K P (q d q i )+u i 1. (6) Note that the unknown gravity term in q i is determined through the reading of the control effort at steady-state. For the next iteration, the feedforward is istantaneously updated as u i = 1 β K P (q d q i )+u i 1, (7) and control (5) is applied again starting from the current position q i. In order to prove that the sequence {q,q 1,...} converges to q d, starting from any initial value q, we assume that: (a) K P >α (b) <β 1 2 The first assumption is the same given in [2] for showing that the PD control law with constant gravity compensation u = K P (q d q) K D q + g(q d ) (8) is globally asymptotically stable, when the gravity vector is known. It follows from (a) that the robot arm under control (8) has a unique steady-state solution for any value of q d. When K P = diag {k Pi }, assumption (a) is satisfied iff min i {k Pi } >α. We remark that this condition is rather conservative for an openchain robot arm, requiring for each joint a proportional gain which is stiffer than

4 the one computed for the most loaded joint. Indeed, distal joints carry less weight than proximal ones; accordingly, they may be regulated using smaller gains. The second assumption guarantees the convergence of the iterative scheme (7), and in particular that lim i u i = g(q d ), with a priori knowledge limited to the bound (2) on the gravity term. Let e i = q d q i. At the end of the ith iteration, eqns. (6,7) imply that u i = g(q i ) at the steady-state q i, and so u i u i 1 = g(q i ) g(q i 1 ) α q i q i 1 α( e i + e i 1 ), (9) where inequality (2) has been used. On the other hand, from eq. (7) u i u i 1 = 1 β K P e i. (1) From eqns.(9,1), since K P is diagonal, it follows immediately 1 β α e i < 1 β K P e i α( e i + e i 1 ), (11) and then e i <β( e i + e i 1 ). (12) Thus, assumption (b) guarantees that e i < β 1 β e i 1 (13) defines a contraction mapping. Therefore, lim i e i =, and asymptoticconvergence is proved for any initial condition q. Merging conditions (a) and (b) into (5), it follows that the overall proportional gain matrix ˆK P = K P /β has to be chosen so to satisfy ˆK P > 2α. (14) Again, this general condition on the minimum eigenvalue of the symmetric matrix ˆK P is turned into a lower bound on the single elements for a diagonal proportional gain matrix. A series of remarks are now in order: The iterative scheme (5,7) is reminiscent of learning control algorithms that successfully achieve reproduction of repetitive trajectories for robot arms [8, 9, 1]. However, no repositioning of the arm into the initial configuration is performed (nor required) here, at any iteration.

5 The iterative scheme can be interpreted as a discrete-time PID, in which the integral term is updated only at fixed instants. Moreover, this approach combines in an automaticway the benefits of a PD control far from the destination and of an integral action close to the goal, avoiding wind-up effects. As a further merit of the scheme, one should consider that gains with guaranteed convergence properties are easily selected. The bound (2) on the gravity term may be evaluated directly taking into account the maximum admissible payload, so to ensure exact set-point regulation in all operating conditions. As a drawback, since each update of the feedforward term should be performed at steady-state, the control scheme converges to the desired position in a doubly infinite time. However, ultimate boundedness of the error is obtained in finite time, by updating the feedforward term as soon as joint variations definitely drop below a given threshold, even before a complete stop. An interesting aspect is to estimate the distance from necessity of the sufficient condition (14). This point has been investigated only through simulations, some of which are reported in the next section: from these, it follows that the above criterion (derived with unknown gravity term) is not more conservative than condition (a) of [2] (which assumes that an exact gravity compensation is performed). 3. Simulation Results The proposed control scheme has been simulated on a 3R rigid robot arm moving in a vertical plane. The three robot links have equal length l i =.5 m, with uniformly distributed masses of 3, 2, and 1 kg, respectively. Motor torque saturations are set at 8, 4, and 2 Nm (beyond the reduction elements). In this case, the gravity vector is given by g 1 (q) = g[(m 1 d 1 + m 2 l 1 + m 3 l 1 ) sin q 1 +(m 2 d 2 + m 3 l 2 ) sin(q 1 + q 2 ) +m 3 d 3 sin(q 1 + q 2 + q 3 )] g 2 (q) = g[(m 2 d 2 + m 3 l 2 ) sin(q 1 + q 2 )+m 3 d 3 sin(q 1 + q 2 + q 3 )] (15) g 3 (q) = gm 3 d 3 sin(q 1 + q 2 + q 3 ), where d i = l i /2 and q =(,, ) corresponds to the straight downward position (of minimum potential energy). For evaluating α, the following factorization is convenient: g q = 1 1 A 1 cos q 1 A 1 2 cos(q 1 + q 2 ) 1 1, 1 A 3 cos(q 1 + q 2 + q 3 ) (16)

6 where A 1 = g(m 1 d 1 + m 2 l 1 + m 3 l 1 ) A 2 = g(m 2 d 2 + m 3 l 2 ) (17) A 3 = gm 2 d 3. With the given data, a value α = 4.35 results, attained for q =. In the first example, a motion from q = to the straight arm position of maximum gravity force (π/2 of first joint rotation) is commanded, using as proportional and derivative gains ˆK P = diag {1, 6, 28}, K D = diag {2, 1, 2}. (18) The update (7) for u i is made at fixed intervals of 3 seconds. Figures 1 3 and 4 6 show the joint errors and the applied torques over 1 seconds. In this case, two updates are sufficient for regulating the error to zero within 7.5 seconds. Note that only the first position gain in (18) does satisfy the sufficient condition (14), while the gain of the third joint is even smaller than α. In the second example q d =(3π/4,, ) is the desired set point, while the same initial positions and the same gains (18) are used. Errors and torques over 16 seconds are displayed in Figs for the three joints. Five updates are now necessary for obtaining convergence, while a saturation occurs initially for joint 1 and 2. No special care was taken for minimizing the duration of the motion: a faster global transient could have been obtained by updating sooner the feedforward u 3, then u 4, and finally u 5. This example shows the capability of learning the exact gravity compensation also when the wall of maximum gravity force has to be overcome. Notice that intermediate torque steady-state values lie now on both sides of the final required values, indicating that feedforward terms can also be reduced when needed. In fact, the scheme was found to converge without problems also when resetting the desired set point back to the initial position q =, where the required compensation is zero (an equilibrium point). As a third example, the second motion was performed with smaller positional gains. In particular, ˆKP = 5 I was used, which satisfies the condition (a) of [2] but not our condition (14). Update intervals for u i are slowed down to 6 seconds, in order to enable the reaching of steady-state conditions. Figures show 4 seconds of position errors and torques. A persistent oscillatory behavior results as a consequence of the poor learning capabilities: the robot arm switches alternatively from an almost horizontal configuration, where the maximum torque effort is stored, to the upward straight configuration, where the error feedback torque counterbalances the learned feedforward term so to give a rather small net torque. Note also that, being ˆK P >α, there is still a unique equilibrium configuration for each applied feedforward. As a result, this choice of reduced gains gives a quantitative information on how much the sufficient condition (14) could be relaxed in general.

7 Figure 1. Position error for joint 1 q d =(π/2,, ) Figure 2. Position error for joint Figure 3. Position error for joint 3

8 4. Conclusions A simple iterative scheme has been presented for robot set-point regulation tasks under gravity, that does not require the knowledge of the robot dynamicmodel. The control scheme generates the exact gravity compensation at the desired set point, starting initially with a PD law, and updating at discrete instants an additional feedforward term. A lower bound condition on the magnitude of the proportional gain in the PD control part is sufficient to prove global convergence of the scheme. Simulation results have shown the effectiveness of the approach, pointing out that the convergence condition is also close to be necessary. The approach is designed in the joint space but can be extended to regulation of a desired cartesian pose, provided that a bound on the kinematic Jacobian is also known. More interestingly, it can be shown that the learning scheme works also in the presence of flexibility, both concentrated at the joint or distributed along the link, by exploiting the associated model properties found in [4]. We mention here that experimental results have already been obtained for the two-link planar lightweight robot prototype with a very flexible forearm available at the Robotics Laboratory of DIS [11], tilted from the horizontal plane to include gravity effects. 5. Acknowledgements This paper is based on work partly supported by Consiglio Nazionale delle Ricerche under contract PF67 (Progetto Finalizzato Robotica). 6. References 1. M. Takegaki and S. Arimoto, A new feedback method for dynamic control of manipulators, Trans. ASME J. on Dynamic Systems, Meas., and Control vol. 12, no. 2, pp , P. Tomei, Adaptive PD controller for robot manipulators, IEEE Trans. on Robotics and Automation, vol. 7, pp , P. Tomei, A simple PD controller for robots with elastic joints, IEEE Trans. on Automatic Control, vol. 36, pp , A. De Luca and B. Siciliano, An asymptotically stable joint PD controller for robot arms with flexible links under gravity, 31st IEEE Conf. on Decision and Control, Tucson, S. Arimoto and F. Miyazaki, Stability and robustness of PID feedback control for robot manipulators of sensory capability, in Robotics Research: 1st Int. Symp., M. Brady and R.P. Paul (Eds.), MIT Press, Boston, MA, pp , F. Khorrami and Ü. Özgüner, Decentralized control of robot manipulators via state and proportional-integral feedback, 1988 IEEE Int. Conf. on Robotics and Automation, Philadelphia, PA, pp , J. Wen and S. Murphy, PID control for robot manipulators, Tech. Rep. CIRSSE, no. 54, Rensselaer Polytechnic Institute, Troy, NY, S. Arimoto, S. Kawamura, and F. Miyazaki, Bettering operation of robots by learning, J. of Robotic Systems, vol. 1, pp , P. Bondi, G. Casalino, and L. Gambardella, On the iterative learning control theory of robotic manipulators, IEEE J. of Robotics and Automation, vol. 4, pp , A. De Luca, G. Paesano, and G. Ulivi, A frequency-domain approach to learning control: Implementation for a robot manipulator, IEEE Trans. on Industrial Electronics, vol. 39, pp. 1 1, A. De Luca, L. Lanari, P. Lucibello, S. Panzieri, and G. Ulivi, Control experiments on a two-link robot with a flexible forearm, 29th IEEE Conf. on Decision and Control, Honolulu, HI, pp , 199.

9 Figure 4. Applied torque for joint 1 q d =(π/2,, ) Figure 5. Applied torque for joint Figure 6. Applied torque for joint 3

10 Figure 7. Position error for joint 1 q d =(3π/4,, ) Figure 8. Position error for joint Figure 9. Position error for joint 3

11 Figure 1. Applied torque for joint 1 q d =(3π/4,, ) Figure 11. Applied torque for joint Figure 12. Applied torque for joint 3

12 Figure 13. Position error for joint 1 with reduced gains q d =(3π/4,, ) Figure 14. Position error for joint Figure 15. Position error for joint 3

13 Figure 16. Applied torque for joint 1 with reduced gains q d =(3π/4,, ) Figure 17. Applied torque for joint Figure 18. Applied torque for joint 3