1
Command Shaped Closed Loop Control of Flexible Robotic Manipulators Ashish Singlaa⇤ , Ashish Tewari a
b
and Bhaskar Dasgupta
c
Mechanical Engineering Department, Thapar University, Patiala, INDIA b Department of Aerospace Engineering, IIT Kanpur, INDIA c Department of Mechanical Engineering, IIT Kanpur, INDIA
Abstract It is demonstrated that a combination of inverse dynamics feedforward control, command shaping and linear state feedback is sufficient to track large movements of flexible robotic manipulators. Due to large structural vibrations, inherent with flexible manipulators, precise control of such manipulators is a challenging task. Small elastic deformations are assumed along with large rigid-body movements and are measured relative to the large rigidbody motions. A novel closed-loop tracking system has been proposed with the aim of achieving large movement tracking of flexible manipulators along with vibration suppression. The nonlinear feedforward control is obtained by dynamic inversion for generating the nominal trajectory to obtain linearized models. The linear feedback is designed with a linear quadratic regulator (LQR) which ensures stabilization as well as desired performance of the flexible systems. Further, a reduced order observer (ROO) is used to estimate non-measurable states of a flexible manipulator and command shaping (CS) is utilized to reduce the vibration levels. The efficacy of the proposed control scheme is demonstrated with a case study of two-link flexible manipulator tracking a large square trajectory in Cartesian space. Large reductions in vibration levels as well as in input torque magnitudes are observed when compared to controllers implemented without command shaping. The paper also presents frequency analysis, which has been attempted to study the amount of nonlinearity present in the flexible systems. Index Terms Command shaping, vibration suppression, flexible links, finite element method, state estimation.
I. I NTRODUCTION Equirements of high-speed performance and low energy consumption have motivated the design of light-weight, flexible manipulators. Most of the conventional robots are designed for maximum stiffness so as to achieve good positional accuracy and non-oscillatory response. However, highly stiff manipulators are more bulky, consume more power and possess low payload-to-robot-weight ratio. The viable solution to these problems is to relax the stiffness constraint and seek flexible manipulators, blessed with unique features like light in weight, portable, faster movements, low power consumption and high payload-to-robot-weight ratio [1]. A typical flexible manipulator contains both rigid-body as well as flexible movements, resulting in a challenging control problem. The control scheme must take into account both the movements. The rigid-body movement control comes under the domain of conventional manipulator control whereas flexural movement control comes under the domain of vibration suppression of flexible systems. Due to large structural vibrations, inherent with flexible manipulators, precise control of such manipulators is a challenging task. In this work, small elastic deformations are assumed along with large rigid-body movements, in which small elastic displacements are measured relative to the large rigid-body motions. To control flexible manipulators efficiently, an accurate dynamic model of the system is required a priori. Various techniques have been developed in the past to obtain the dynamic model of flexible
R
⇤
Corresponding author. Email:
[email protected]
16
Fig. 8.
Two-link flexible manipulator following square trajectory.
on a two-link flexible manipulator. In the case study, effect of input shaping in vibration reduction, while tracking a square trajectory in Cartesian space by a two-link flexible manipulator has been demonstrated. Large reductions in vibration levels as well as in input torque magnitudes have been observed when compared to controllers implemented without command shaping. The paper also contributes to evaluate the degree of nonlinearity present in flexible systems by performing the frequency analysis, i.e. configuration dependence of the elastic frequencies. In has been demonstrated that, for different configurations of the manipulator, significant variations are found in the elastic frequencies of the flexible systems. These variations have been handled effectively by using robust shapers. R EFERENCES [1] Book WJ, Majette M. Controller design for flexible, distributed parameter mechanical arms via combined state space and frequency domain techniques. Journal of Dynamic Systems, Measurement, and Control. 1983;105(4):245–254. [2] Tzou HS. Nonlinear structural dynamics of space manipulators with elastic joints. International Journal of Analytical and Experimental Modal Analysis. 1989;4:117–123. [3] Usoro PB, Nadira R, Mahil SS. A finite element/Lagrange approach to modeling lightweight flexible manipulators. Journal of Dynamic Systems, Measurement, and Control. 1986;108:198–205. [4] De Luca A, Siciliano B. Closed-form dynamic model of planar multilink lightweight robots. IEEE Transactions on Systems, Man and Cybernetics. 1991;21(4):826–839. [5] Patil OM, Gandhi PS. On the dynamics and multiple equilibria of an inverted flexible pendulum with a tip mass on a cart. Journal of Dynamic Systems, Measurement, and Control. 2014;136(4):041017. [6] Mohan A, Saha S. A recursive, numerically stable, and efficient simulation algorithm for serial robots with flexible links. Multibody System Dynamics. 2009;21(1):1–35. [7] Tokhi MO, Mohamed Z, Shaheed MH. Dynamic characterisation of a flexible manipulator system. Robotica. 2001;19(5):571–580.
17
[8] Singer NC, Seering WP. Preshaping command inputs to reduce system vibration. Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME. 1990;112(1):76–82. [9] Hyde JM, Seering WP. Using input command pre-shaping to suppress multiple mode vibration. IEEE International Conference on Robotics and Automation. 1991;:2604–2609. [10] Singh T, Vadali SR. Robust time-optimal control: frequency domain approach. Journal of Guidance, Control, and Dynamics. 1994; 17(2):346–353. [11] Singhose W, Derezinski S, Singer N. Extra-insensitive input shapers for controlling flexible spacecraft. Journal of Guidance, Control, and Dynamics. 1996;19(2):385–391. [12] Pao LY, Singhose WE. Robust minimum time control of flexible structures* 1. Automatica. 1998;34(2):229–236. [13] Vaughan J, Yano A, Singhose W. Comparison of robust input shapers. Journal of Sound and Vibration. 2008;315:797–815. [14] Drapeau V, Wang D. Verification of a closed-loop shaped-input controller for a five-bar-linkage manipulator. IEEE International Conference on Robotics and Automation. 1993;:216–221. [15] Hillsley KL, Yurkovich S. Vibration control of a two-link flexible robot arm. Dynamics and Control. 1993;3(3):261–280. [16] Tewari A. Active Vibration Suppression of Spacecraft With Rate-Weighted Optimal Control and Multi-Mode Command Shaping. In: Aiaa/aas astrodynamics specialist conference and exhibit, providence, rhode island. 2004. [17] Yigit AS. On the Stability of PD Control for a Two-Link Rigid-Flexible Manipulator. Journal of Dynamic Systems, Measurement, and Control. 1994;116:179–185. [18] Feliu V, Rattan KS, Brown Jr HB. Adaptive control of a single-link flexible manipulator. IEEE Control Systems Magazine,. 1990; 10(2):29–33. [19] Shaheed MH, Tokhi O. Adaptive closed-loop control of a single-link flexible manipulator. Journal of Vibration and Control. 2013; 19(13):2068–2080. [20] Rattan KS, Feliu V. A robust control scheme for flexible manipulators. IEEE National Aerospace and Electronics Conference,. 1991; :1090–1095. [21] Zhang N, Feng Y, Yu X. Optimization of terminal sliding control for two-link flexible manipulators. IEEE 30th Annual Conference of Industrial Electronics Society. 2004;2:1318–1322. [22] Cambera JC, Chocoteco JA, Feliu V. Feedback linearizing controller for a flexible single-link arm under gravity and joint friction. In: Robot2013: First iberian robotics conference. Springer. 2014. p. 169–184. [23] Le Ballois S, Duc G. H-(infinity) control of an earth observation satellite. Journal of Guidance, Control, and Dynamics. 1996;19(3):628– 635. [24] Tzes A, Yurkovich S. An adaptive input shaping control scheme for vibration suppression in slewing flexible structures. IEEE Transactions on Control Systems Technology. 1993;1(2):114–121. [25] Gorius T, Seifried R, Eberhard P. Approximate end-effector tracking control of flexible multibody systems using singular perturbations. Journal of Computational and Nonlinear Dynamics. 2014;9(1):011–017. [26] Zain M, Tokhi MO, Mohamed Z. Hybrid learning control schemes with input shaping of a flexible manipulator system. Mechatronics. 2006;16(3-4):209–219. [27] Singhose W, Singer N. Initial investigations into the effects of input shaping on trajectory following. American Control Conference, 1994. 1994;3:2526–2532. [28] Banerjee A, Singhose W. Command shaping in tracking control of a two-link flexible robot. Journal of guidance, control, and dynamics. 1998;21(6):1012–1015. [29] Pel´aez G, Pelaez G, Perez J, Viz´an A, Bautista E. Input shaping reference commands for trajectory following Cartesian machines. Control Engineering Practice. 2005;13(8):941–958. [30] Singhose WE, Singer NC. Effects of input shaping on two-dimensional trajectory following. Robotics and Automation, IEEE Transactions on. 1996;12(6):881–887. [31] Lee JH, Lee BH, Lee SM, Chung CY. Preshaped trajectory command for fast repetitive PTP motion of PD-controlled flexible joint manipulators. In: Proceedings of the ieee/rsj international conference on intelligent robots and systems. 1997. p. 1546–1552. [32] Gorinevsky D, Vukovich G. Nonlinear input shaping control of flexible spacecraft reorientation maneuver. Journal of Guidance, Control, and Dynamics. 1998;21:264–270. [33] Kapila V, Tzes A, Yan Q. Closed-loop input shaping for flexible structures using time-delay control. Journal of Dynamic Systems, Measurement, and Control. 2000;122:454–460. [34] Liu KP, You W, Li YC. Combining a feedback linearization approach with input shaping for flexible manipulator control. In: International conference on machine learning and cybernetics. 2003. p. 561–565. [35] Shan J, Liu HT, Sun D. Modified input shaping for a rotating single-link flexible manipulator. Journal of Sound and Vibration. 2005; 285(1-2):187–207. [36] Romano M, Agrawal BN, Bernelli-Zazzera F. Experiments on command shaping control of a manipulator with flexible links. Journal of Guidance, Control, and Dynamics. 2002;25(2):232–239. [37] Biediger E, Lawrence J, Singhose W. Improving trajectory tracking for systems with unobservable modes using command generation. In: Proceedings of the american control conference. 2005. p. 513–518. [38] Kenison M, Singhose W. Concurrent Design of Input Shaping and Proportional Plus Derivative Feedback Control. Journal of Dynamic Systems, Measurement, and Control. 2002;124:398–405. [39] Usoro PB, Nadira R. Analysis of lightweight flexible manipulator dynamics. In: Proceedings of the asme international computers in engineering conference and exposition. Vol. 1. 1984. p. 167–174. [40] Swarup A, Gopal M. Comparative study on linearized robot models. Journal of Intelligent and Robotic Systems. 1993;7(3):287–300. [41] Tewari A. Modern control design with MATLAB and SIMULINK. Chichester; Wiley. 2002. [42] Abaqus/Standard Analysis Users Manual, Abaqus Inc., Version 6.9, Section 6.5.3. . 2009.
