of a Suspended Load with a Robotic Crane

Transcription

1 Proceedings of the American Control Conference Chicago, llinois June 2000 Anti-Swing Control of a Suspended Load with a Robotic Crane Jae Y. Lew Ahmed Khalil Dept. of Mechanical Engineering Ohio University Athens, OH USA Abstract This paper presents a feedback control algorithm for a robotic crane carrying a suspended load. The proposed controller suppresses the swing motion of the load while the crane carries the load to a desired position. The analysis includes the nonlinear dynamics of the crane and two dimensional swing motion of the load, but the controller does not require the exact model of the crane nor payload. Using joint acceleration and one sampled delayed torque, the crane system is decoupled and then a linear controller is designed for both the crane motion and the load swing independently. Stability of the controller is proved and a simulation study is performed to show its effectiveness. controlled velocity source which may not hold under a heavy load. n addition, it often requires an exact model of the overall system. The anti-swing feedback control introduced in the paper is proposed to overcome these issues. First, the proposed controller considers the nonlinear dynamics of the crane and multi-dimensional swing. Second, it does not require an accurate model of the crane nor the mass of load. Third, it has separate a position feedback loop and swing motion loop. Thus, it can be easily implemented to an existing industrial system. 1 ntroduction Cranes are widely used in industry to transport heavy loads. For example, on construction site, harbor or warehouse, it is often found that a crane carries a suspended load to the point of interest. However, its operational efficiency and safety are often limited by the swing motion of the load. To reduce the swing motion of the load while carying it, the crane requires a special control scheme. As a general case study, this paper presents an anti-swing feedback algorithm for the two link planar robotic crane carrying a suspended load as shown in Figure 1. Many researchers have worked on the issue of antiswing control of the crane system. Their work can be divided into largely two groups: open loop and closed control. The open loop method such as low pass filter, symmetric acceleration profile[ ], and input shaping (= notch filter) [2] [3] are intended not to excite the swing motion. They are simple to implement but can not suppress swing motion caused by external disturbance. On the other hand, the closed loop method of swing motion provides the ability to eliminate existing swing motion and rejects disturbance at the cost of extra sensors. Several closed loop control schemes have been developed based on root locus and loop shaping[4], LQR[5], fuzzy logic[l], and singular perturbation 161. Generally, the analysis is limited to only one or two dimensional linear overhead crane. Also, the crane dynamics is often ignored and considered only as a Figure1 Robotic Crane with a Suspended Load The outline of the paper is as follows: First, the dynamics of a load suspended to a two link planar robot is described. Second, the control algorithm is derived and its stability is proved. Third, a simulation study is performed to show the effectiveness of the proposed controller. Experiment with a physical system is planned as a future study. 2 Dynamics When a planar robot with two revolute joints carries a suspended load as shown in Figure 1, the dynamics of the system can be obtained using the Lagrange equation. Assuming the swing motion is very small and the load is a point mass, the overall dynamics can be simplified as $ AACC 1042

2 +[: KO]{;}={;] where q is the joint coorinates [qlt,q2t]t; Bis the swing angle coordinates [6',',B,']'; tis the joint torque; M, is the inertia matrix of the robot itself; C, is the nonlinear matrix due to centrifugal and Coriolis force without the load;m,and M/, are the coupling inertia matrices; C, and Cllr are the coupling nonlinear terms due to carrying a load; and Kl is the swing stiffness matrix due to the gravity. Detail description of each term can be expressed as n this paper, the feedback control algorithm is developed for the dynamic system described by Eq (2). 3 Control Scheme The control objective is to determine the input control 'ir such that the swing motion, 8, damps out as quickly as possible while the robot crane motion, q, follows the desired path. This is a difficult control problem because (1) one control input, z has to control two variables, q and 0, (2) an exact model of an inertia matrix and nonlinear terms is not available due to the unknown load and they vary as the configuration changes. A special robust damping controller is necessary to meet such a goal. The derivation of the controller is as follows: first, the crane motion is decoupled from the swing motion under the assumption that we know only an estimation of M,(~), Mr and that no information on nonlinear term'cr is provided. The first equation of Eq (2) is rewritten as. (3) M,ij+(M,(q)-M~)q+M,,,(q)ii+ M,,(q)8+ C,(4,q) = 5 where ci = cos(qj, si = sin(qi) c2 = cos(ql+qz), s12 = sin(ql+qz) > t al = Gl+mllcl-+m211, a2 = G ~ + m21,: a3 = m&ll, P = mj9 P3 = mj1l2, PS = mp1d p2 = mplj2 $4 = mjf Pb = mpl. Gi is the moment of inertia w.r.t the mass center of the i- th link; mi is the mass of the i-th link; lci is the length to the mass center; li is the link length; mp is the mass of the load; 1 is the length of the rope assumed to be fixed; and g is the gravitational acceleration. Term ai is defmed to represent the dynamic parameters of the robot in a simpler form and pi is to represent the coupling dynamic parameters from the load. The inertia matrix error term, nonliner term and coupling term with the swing can be combined as N(q,g,q,e). N is obtainable from Eq (3) as N(4i94,q98)= (M,(q)-Ml,)ii+M,,,(q)ij+M,/(q)8+C,(q,q)(4) = rp - M,q rp is the one-sample previous actuator torque assuming zp = tat high sampling rate, and the estimation of manipulator's inertia term Mr is known, and joint acceleration q is assumed to be measurable from the optical encoders by differentiating twice with respect to time. However, numerical noise in the differentiation could be a problem. t requires a careful design of a digital filter. This has been demonstrated successfully by [7] and [8]. n summary, using Eq (4) the uncertainty term can be computed from the acceleration measurement and one-sample delayed torque, and it is used as a feedforward term to linearize the system dynamics. Thus, the acceleration feedback will be implemented to the control system in the form of z =?,,-Mrq+u. (5) 1043

3 By applying the control input, the system equation from Eq (4) becomes where U is the new input for the feedback control. By taking the inverse of the inertia matrix of Eq (8), it may be rewritten as [ i}+[o" M;Kl][;}+ { M ;'c,(q,~~=[m~'~~~~~;' 0 where ij is the joint angle error between the desired and measured, qd - q ; ij is the joint velocity error between the desired and measured, qd - q ; qd is the desired joint acceleration; and K, and K, are the proportional and derivative matrix, respectively. When the composite controller Eq (10) is applied to Eq (8), the closed loop error dynamics for the crane becomes 1.. M rq + Kd $ + Kpq = -uslow. (12) As shown in the first row of Eq (7), the crane motion is mostly decoupled from the swing motion of the load. They are coupled only through the control input, U. Rewrite Eq (7) into two equations as Mrq=u (8) e + Mi1K16' + My1C,(q,q) = H(q)u (9) where H(q) = M;lM;(q)M;l. The physical meaning of H is the coupling dynamic relationship effect between the joint torque and the force at the tip that causes the swing motion. nterestingly, H is independent of the load mass. n fact, it is related to the Jacobian matrix, J(q) between the robot tip and the joints. This can be easily shown by multiplying them together symbolically H(q) = MilM:(q)Myl - p4'1 - pss12 - p5'"12][ p4'1 -k p5'12 b5'12 /08,] Since uslow changes slowly compared to the fast error dynamics, it can be treated more llke a constant. Then the error dynamics of Eq (12) converge to be very small with the proper choice of positive definite matrices Kp and &. f Mr is estimated as a diagonal matrix, gain <p and & also can be chosen as diagonal matrices related to the natural frequency and damping of the decoupled crane motion. Now, the slow control input, uslow, is defined for the swing motion in Eq (9) as uslow =-H-l(q )Ksde (13) where H-l(q) = rj-l(q)m, assuming that the kinematics of the robot crane is not singular. Only the derivative feedback of the swing with gain Ksd is added to increase the damping in the swing motion. As Eq (1 2) converges quickly with high gain K,, we may assume lj z 0 and 4" m 0, i.e., q U qd and q w qd for the slow dynamics, where q, is the desired crane joint angle. Then, after applying the composite controller, Eq (9) becomes Second, a composite (fast and slow) controller is designed based on the partially decoupled model in Eqs (8) and (9). Recall that the control objective is to 8+KSdb+M;'K,B = -M;'C,(q,,qd)+ H M,qd * (14) determine the control input, U, that stabilizes both Eqs (8) and (9). Define the control input, U, which has two parts, Using two-time scale theory, ufuf and uslow are chosen in two different time scales. t is intended that ufasf affects only decoupled rigid dynamics in Eq (8) and uslow is for swing motion in Eq (9). For ufasf, a linear tracking controller will work well with high gains such that the control bandwidth is much higher than the swing frequency. Therefore, ufasf can be chosen as. 1 t is easily shown that M;'Kl is a diagonal matrix of g/l and is always positive definite. The right hand side of Eq (14) is bounded since qd, qdand qd are bounded. Therefore, as long as Ksd is chosen as a positive definite matrix, Eq (14) remains asymptotically stable. We may adjust the damping of the swing by increasing gain Ksd. Finally, combining acceleration feedback, ujas, and uslow, the overall proposed damping controller is r = r, - Mrq + Kpf + K,$ + M,qd - N-'(q)M,KSdb (15) Ufasr = K,q+ Kdq +Mrqd (11) As shown in Eq (15), the proposed controller does not require information of the load mass. The only modeling information required is Jacobian, estimation of inertia matrix of the robot crane and the rope length. 1044

4 ~~~. 4 Simulation & Discussion A computer simulation study was performed to demonstrate the effectiveness of the proposed anti-swing control scheme. The simulation model was based on a physical system which is under construction at Ohio University. The test bed consists of a two-link rigid manipulator carrying a load as shown in Figure 1. As the robot moved, the suspended load oscillated in the XY plane. The parameters of the system are given in Table1. These parameters closely matched with the test bed. The system was simulated in SMULNKTM for a 20-second response. Mass of 1" link Mass of 2na link kg kg center Length of 1' llnk m Length of 2na link m Length to mass center of 1" link m Length to mass center of lin m Mass of load Length of Rope kg 1.2 m The controller gains for the crane are chosen as These values were selected based on Eq (12) assuming that there was no load. Then, the gains of the anti-swing gain controller are determined using Eq (14) as A desired trajectory was given as a straight line with a third order time polynomial equation and with 5 seconds travel time in Cartesian coordinates. ts initial position was given at [ 0.4, 0.21~1 in the XY plane and the final position was at [0.2, -0.2]m, which corresponds in the joint coordinates: joint 1 moved from -0.2 to rad and joint 2 moved from 1.34 to 2.1 rad. Figure 2 (a) and (b) show the desired and actual joint angles response during the straight line motion with 0.2 Kg load. Comparing two results, both the crane joints followed the desired joint path accurately with or without the antiswing controller. However, Figure 3 (a) and (b) show the improvement of the swing motion during that time. With anti-swing controller, the swing motion dissipated in less than 3 cycles after the robotic crane stopped. Next, to demonstrate the robustness with respect to the load change, the mass of the load was changed to 2 Kg while keeping the same feedback gains and following the same straight line as above. Figure 4 (a) and (b) show the swing motion. The anti-swing controller is still effective with 1000 % payload change. 5. Conclusion An anti-swing damping controller for a robotic crane carrying a suspended load was proposed in this paper. ts stability and design procedure were presented for a nonlinear crane with a multiple dimensional swing with an unknown load. The simulation study demonstrated that the significant damping effect was obtained throughout a large configuration change. The algorithm itself is relatively simple to be implemented under the assumption that the joint acceleration is measurable. Currently, a test bed is being built. Future study will discuss the experimental results and implementation issue of the proposed controller. Reference 1. Yoon, J.S., et al. Various Control Schemes for mplementation of the Anti-Swing Crane. in ANS 6th Topical Meeting on Robotic and Remote System Monterey, CA. 2. Noakes, M.W., R.L. Kress, and G.T. Appleton. implementation of Damped-Oscillation Crane Control for Existing AC induction Motor-Driven Cranes. in ANS Fifih Topical Meeting on Robotics and Remote Systems Knoxville, TN. 3. Parker, G.G., et al. Experimental Verification of a Command Shaping Boom Crane Control System. in American Control Conference San Diego, CA. 4. Lee, H.H., Modeling and Control of a Three-Dimensional Overhead Crane. Journal of Dynamic Systems, Measurement, and Control, : p Wen, B., et al. Modeling and Optimal Control Design of Shipboard Crane. in American Control Conference San Diego, CA. 6. Yu, J., F.L. Lewis, and T. Huang. Nonlinear Feedback Control of a Gantty Crane. in Proceedings of American Control Conference Settle, WA. 7. Bonitz, G. and T.C. Hsia. Robust nternal Force Based impedance Control for Coordinating Manipulators - Theory and Experiments. in EEE nternational Conference on Robotics and Automation Minneapolis, MN. 8. Lew, J.Y. and S.-M. Moon. Experimental Study of Compliant Base Manipulator Control. in EEE Conference on Decision and Control Pheonix, Az. 1045

5 G e -1 - r L.g \. \ \, '.: i -1.4r ',, L-., Figure 2 (a) Joint 1 Response with 0.2 Kg Load,,,,,,,, 0.02 Figure 3 (b) Swing Motion of 0.2 Kg Load in Y 22 2'3 i 1 W Dampino (soia \\' WO Damping (dolled) ' Desired Angle (dashed) 0.01-,,,,,, 1 r 7 1.3i-._, Figure 2 (b) Joint 2 Response with 0.2 Kg Load Figure 4 (a) Swing Motion of 2 Kg Load in X 0.01, 001, 1, ' i-.i - Time (Sec) Figure 3 (a) Swing Motion of 0.2 Kg Load in X Figure 4 (b) Swing Motion of 2 Kg Load in Y 1046