Elsevier

Automatica

Volume 43, Issue 7, July 2007, Pages 1226-1233
Automatica

Brief paper
Modified transpose Jacobian control of robotic systems

https://doi.org/10.1016/j.automatica.2006.12.029Get rights and content

Abstract

The simplicity of Transpose Jacobian (TJ) control is a significant characteristic of this algorithm for controlling robotic manipulators. Nevertheless, a poor performance may result in tracking of fast trajectories, since it is not dynamics-based. Use of high gains can deteriorate performance seriously in the presence of feedback measurement noise. Another drawback is that there is no prescribed method of selecting its control gains. In this paper, based on feedback linearization approach a Modified TJ (MTJ) algorithm is presented which employs stored data of the control command in the previous time step, as a learning tool to yield improved performance. The gains of this new algorithm can be selected systematically, and do not need to be large, hence the noise rejection characteristics of the algorithm are improved. Based on Lyapunov's theorems, it is shown that both the standard and the MTJ algorithms are asymptotically stable. Analysis of the required computational effort reveals the efficiency of the proposed MTJ law compared to the Model-based algorithms. Simulation results are presented which compare tracking performance of the MTJ algorithm to that of the TJ and Model-Based algorithms in various tasks. Results of these simulations show that performance of the new MTJ algorithm is comparable to that of Computed Torque algorithms, without requiring a priori knowledge of plant dynamics, and with reduced computational burden. Therefore, the proposed algorithm is well suited to most industrial applications where simple efficient algorithms are more appropriate than complicated theoretical ones with massive computational burden.

Introduction

Many approaches have been employed for the complex problem of controlling mechanical manipulators and robotic systems. A prime difficulty for all approaches is due to the strong non-linearities and time dependencies in the dynamics of such systems. Hence, a wide range of algorithms has been suggested to challenge this task, including Adaptive Control algorithms as proposed by Slotine and Li (1987), and Taira, Sagara, and Katoh (2000), Time-Delay Control as suggested by Youcef-Toumi and Ito (1987), Motion-rate Control as presented by Umetani and Yoshida (1989), Kelly and Moreno (2005), Artificial Neural Networks and Fuzzy Control as proposed by Meghdari, Naderi, and Alam (2005), Dominguez-Lopez, Damper, Crowder, and Harris (2004), Steil, Röthling, Haschke, and Ritter (2004), Mbede et al. (2005), Hybrid Motion/Force Control as suggested by Raibert and Craig (1981), Chiu, Lian, and Wu (2004), and Impedance Control as proposed by Hogan (1985), and Moosavian, Rastegari, and Papadopoulos (2005).

Transposed Jacobian (TJ) control is one of the simplest algorithms used to control motion of robotic manipulators. According to Craig (1989), the TJ algorithm has been arrived at intuitively, and is similar to classic PD-action controllers. In the case of using an approximate Jacobian, Miyazaki, Masutani, and Arimoto (1988) have shown that the damping matrix and the position gain matrix of this controller play an important role in system stability. Apparently, the algorithm can be applied to redundant manipulators as shown by Asari, Sato, Yoshimi, and Tatsuno (1993), and as discussed by Chiaverini, Sciavicco, and Siciliano (1991) it does not fail when a singularity occurs. Hootsmans and Dubowsky (1991) have developed an extended Jacobian transpose control algorithm to improve the performance of mobile manipulator systems. Subsequently, to fulfill simplicity requirements, Bevly, Dubowsky, and Mavroidis (2000) have developed Simplified Cartesian Computed Torque (SCCT) Control algorithm for highly geared climbing robots.

The performance of TJ-based algorithms has been experimentally compared to those of different algorithms using unit quaternions on a direct-drive spherical wrist by Garcia and Kelly (2002). Papadopoulos and Moosavian (1995) have compared the performance of this simple algorithm to those of various model-based algorithms. Both experimental and simulation results show the merits of the TJ algorithm in controlling of highly non-linear and complex systems with multiple degrees-of freedom (DOF), motivating further work on this algorithm. However, since the TJ is not dynamics-based, poor performance may result in fast trajectory tracking. Use of high gains can deteriorate performance seriously in the presence of feedback measurement noise. Another drawback is that there is no formal method of selecting its control gains, and a heuristic selection of gains makes it difficult to apply.

In this paper, a new Modified Transpose Jacobian (MTJ) algorithm is developed which employs stored data of the control command in the previous time step, as a learning tool to yield an improved performance. The gains of this new algorithm can be selected more systematically, and do not need to be large, hence the noise rejection characteristics of the algorithm are improved. Stability analysis, based on Lyapunov's theorems, shows that both the standard TJ and the MTJ algorithms are asymptotically stable. Simulation results show that tracking performance of this new algorithm is comparable to that of Model-Based (MB) algorithms, without requiring a priori knowledge of plant dynamics, and with reduced computational burden.

Section snippets

General motion control laws

Availability of a system dynamics is always helpful in the design of a control system. Using the expressions for the kinetic and potential energy, and applying Lagrange's equations for a robotic system, the dynamics model can be obtained and has the formH(q)q¨+C(q,q˙)=Q(q),where all gravity and non-linear velocity terms are contained in vector C, and H is a positive definite matrix, function of the generalized coordinates q. Gravity terms that cause static positioning errors in control, can be

MTJ control law

The TJ control law defined by Eq. (7) is now modified to achieve both precision and simplicity:Q=JCT{Kde˙+Kpe+h(t)},where Kp and Kd are positive definite gain matrices, e is the tracking error defined in Eq. (6), and h(t) is to be determined for feedback linearization. Substituting Eq. (8) into Eq. (3), yieldsKde˙+Kpe=H^q^¨+C^-h(t).Considering Eq. (3), this is equivalent toKde˙+Kpe=Q^-h(t).It is motivating to note that if the right-hand side (RHS) of Eq. (10) becomes equal to zero, then the

Obtained results

To focus on algorithmic aspects, a simple two link planar manipulator is considered under various conditions. Performing low-speed vs. high-speed tracking task, selection of higher gain for the TJ, and noise rejection characteristics of the proposed MTJ algorithm is investigated in these simulations, and compared to those of alternative algorithms.

The system is a 2-link planar manipulator on a horizontal plane, see Fig. 1(a). The task is tracking a trajectory defined byxdes=l12+l22cos(ωt+π/4)+

Conclusions

This paper presented the new MTJ control which yields a better performance (in terms of tracking errors, with the same requirements of actuator forces/torques) compared to the standard Transposed Jacobian (TJ) algorithm. The MTJ controller approximates a feedback linearization solution, using stored data of the control command in the previous time step as a learning tool, with no need to a priori knowledge of the plant dynamics. Therefore, unlike an MB algorithm, it is not affected by

S. Ali A. Moosavian received his B.S. degree in 1986 from Sharif University of Technology and the M.S. degree in 1990 from Tarbiat Modaress University (both in Tehran), and his Ph.D. degree in 1996 from McGill University, all in Mechanical Engineering. He is currently an Associate Professor at the Mechanical Engineering Department at K. N. Toosi University of Technology (Tehran). He teaches courses in the areas of robotics, dynamics, automatic control, analysis and synthesis of mechanisms. His

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    S. Ali A. Moosavian received his B.S. degree in 1986 from Sharif University of Technology and the M.S. degree in 1990 from Tarbiat Modaress University (both in Tehran), and his Ph.D. degree in 1996 from McGill University, all in Mechanical Engineering. He is currently an Associate Professor at the Mechanical Engineering Department at K. N. Toosi University of Technology (Tehran). He teaches courses in the areas of robotics, dynamics, automatic control, analysis and synthesis of mechanisms. His research interests are in the areas of dynamics modeling, and motion/impedance control of terrestrial and space robotic systems. He has published more than 75 articles in journals and conference proceedings. He is a Member of the IEEE, and one of the founders of the ARAS Research Center for Design, Manufacturing and Control of Robotic Systems, and Automatic Machineries.

    Evangelos Papadopoulos received his Diploma from the National Technical University of Athens (NTUA) in 1981, and his M.S. and Ph.D. degrees from the Massachusetts Institute of Technology (MIT) in 1983 and 1991, respectively, all in Mechanical Engineering. From 1985 to 1987, he was an analyst with the Hellenic Navy, Athens, Greece. In 1991 he was appointed a Lecturer at the Department of Mechanical Engineering at MIT. He then joined McGill University and the Centre for Intelligent Machines (CIM) as an Assistant Professor, and was tenured in 1997. Currently, he is an Associate Professor with the Mechanical Engineering Department at the NTUA. He teaches courses in the areas of systems, controls, mechatronics and robotics. His research interests are in the area of robotics, including space, mobile and underwater robotics, modeling and control of dynamic systems, haptic devices, mechatronics, and design. In 2003, he served as a Guest Editor for the Advances in Robot Dynamics and Control issue of the ASME/IEEE Transactions on Mechatronics. He has published more than 100 articles in journals and conference proceedings. He is a Senior Member of the AIAA and of the IEEE, and a member of the ASME, the NYAS, the Technical Chamber of Greece (TEE) and the Sigma Xi.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Masaki Yamakita under the direction of Editor Mituhiko Araki.

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