Precise path tracking in Cartesian overhead cranes through an improved non-colocated Wave Based Control
Wave Based Control (WBC) has been proven to be very effective in combining position and oscillation control of underactuated flexible multibody systems [1]. This paper proposes an extension of the method aimed at ensuring accurate path tracking of a load suspended by a planar Cartesian crane, even in the presence of non ideal actuators and sensors. The basic configuration of WBC [1] uses the measured mechanical wave coming back from the flexible system to the actuator, the so called returning wave, to accomplish both position control and active swing damping. In Cartesian cranes, two independent controllers for the X and Y axis set the trolley position references, q T , to be the sum of two components (q denotes both the X and Y motion). The first component is set to half the load displacement reference, q L , and is the launch wave. The second component is the measured returning wave, denoted bq . The addition of this second component performs active vibration damping while moving the system the remaining second half of the target displacement. This achieves exact steady state final positioning in rest-to-rest motion. When WBC is instead employed in path tracking in the presence of delay or noise in sensor measurements, or of delay between the trolley reference and actual position, correct tracking of the spatial trajectory during transients is not always ensured. Indeed, the measured return wave will have inaccuracies in amplitude and in time delay (phase) with respect to the launch wave. To deal with this, two improvements of WBC are suggested in this paper. First, a new term, γ ( ) q s , is added to q T , leading to the following relation in the Laplace domain: 1 q ( ) q ( ) b ( ) γ ( ) 2 ref ref T L q q s s s s = + + (1) Second, a model based estimation of the return wave is computed using an Extended Kalman Filter. Such a state observer is based on an augmented dynamic model of the system, including a simplified model of the delay. The feedback and non-colocated correction γ ( ) C( ) ( ) q q s s s = ε is computed through the convolution between the contour error ( ) q s ε (i.e. the distance between the desired spatial contour to the actual position, regardless of the time required to reach such a position) and a linear regulator, C( ) s , which can be a PID regulator. In effect, an outer non-colocated loop is closed. Indeed, whenever the aim of the control is to ensure accurate load path tracking even in the presence of sensor disturbances or external excitations on the load, non-colocated actuator/sensor pairs can be effectively adopted to increase the external disturbance rejection. Besides being more suitable than tracking error in trajectory following tasks, the use of the contour error makes the controller get rid of the unavoidable delay introduced to perform swing control, which is due to the nature of the physical coupling between the trolley and the load. A numerical method suitable for the estimating ( ) q s ε in load swing control is also developed, by taking advantage of the general technique proposed in [3]. The reduced computational effort of the formulation proposed, ensures easy and effective real-time control. Finally, in order to decouple the return wave from the corrective term, γ ( ) q s , the computation of b ( ) q s is modified by removing the effect of γ ( ) q s as follows: ( ) ( ) ˆ b ( ) G( ) q G( ) γ ( ) a ( ) q L q q s s s s s = − + , with a ( ) q ( ) b ( ) q T q s s s = − (2) In Eq. (2) G( ) s is the wave transfer function, q̂ L is the output of the Kalman Filter and q T is the actual trolley position. Experimental results assessing the effectiveness of proposed method have been obtained through an Adept Quattro robot mimicking a Cartesian crane and driving a load suspended by a 1 meter cable. The non-colocated WBC has been implemented by closing an outer loop computing q T , and sending it to the robot inner position controller through using an Ethernet connection. The robot is also instrumented with two CCD cameras sensing the swing angles at a low sample rate, i.e. introducing delay in the loop. Some results are shown in Figure 1 with reference to three different paths: circular, 8-shaped and square. The plots in the first column show the path tracking in the case of no swing control, i.e. when only the native robot position controller is adopted in the inner loop. Considerable and persistent load oscillations prevent acceptable tracking of the suspended load. The second column shows the same tests performed using WBC [2]: effective swing damping is ensured, but the issues discussed above prevent exact tracking. Finally a great improvement is obtained with the noncolocated WBC here proposed: desired and actual load trajectories are much closer. No swing control WBC Non-colocated WBC Figure 1: Path tracking: experimental results. References [1] W. J. O’Connor. Wave-echo control of lumped flexible systems. Journal of sound and vibration, Vol. 298, No.4, pp. 1001-1018, 2006. [2] G. Boschetti, R. Caracciolo, D. Richiedei, A. Trevisani. Moving the suspended load of an overhead crane along a pre-specified path: A non-time based approach. Robotics and ComputerIntegrated Manufacturing, Vol. 30, No. 3, pp. 256-264, 2014. [3] M.A. El Khalick, N.Uchiyama. Contouring controller design based on iterative contour error estimation for three-dimensional machining. Robotics and Computer-Integrated Manufacturing, Vol. 27, No. 4, pp. 802-807, 2011.
Moving the suspended load of an overhead crane along a pre-specified path: A non-time based approach
This paper introduces a noncolocated non-time based regulator for the path tracking control of the suspended load of a four degrees of freedom gantry crane. The regulator is named Delayed Reference Noncolocated Control (DRNC). The DRNC control action is exerted by modifying the moving platform reference trajectory. Such a goal is achieved by both shifting trajectory execution in time, and modifying the platform path with respect to the reference path defined for the load. Simulation results prove the capability of the method to ensure extremely precise path tracking, while ensuring very simple controller implementation, which makes it well suited for industrial application.
Improving contour accuracy by Fuzzy-logic enhanced cross-coupled precompensation method
Abstract Contour accuracy is obtained by efforts in algorithms. Most motion control algorithms reduce either position error or contour error to gain accuracy. In this paper, we investigate the idea of constructing a Fuzzy-logic controller to a proven algorithm, the cross-coupled precompensation method (CCPM), and using both position and contour error to generate compensation term. Simulation comparison with known methods and experimental comparisons among the uncoupled system, the cross-coupled system, the original CCPM and the proposed method were given. It was shown that the performance of the original CCPM was further enhanced. The proposed method also showed a better performance when subjected to external load.