Singularity avoidance for acrobots based on fuzzy-control strategy☆
Introduction
An acrobot is a two-link manipulator moving in a vertical plane with an actuator at the elbow but no actuator at the shoulder. It is subject to a second-order nonholonomic constraint, and cannot be completely linearized in the whole motion space. So, an acrobot is a typical nonlinear mechanical system and is a focus of interest in the field of underactuated mechanical control systems.
Fuzzy control has proven to be a powerful tool for the control of complex nonlinear systems (e.g., [1], [2], [3]). Even though it does not employ a precise mathematical model, a fuzzy controller can make effective decisions on the basis of imprecise linguistic information [4]. For this reason, fuzzy control has found wide application in the field of control engineering. For example, it has been used to control industrial heat treatment furnaces [5], which had previously been controlled by conventional PID controllers. The use of fuzzy controllers not only produces much better control results than PID controllers do, but it also enables the establishment of very stable, fully automatic systems.
The complexity and nonlinearity of the characteristics make it almost impossible to control the motion of an acrobot with a single control law. Several pioneering studies use fuzzy control to try to solve this problem. For example, Smith et al. analyzed the stability issue from the standpoint of global asymptotic stability based on fuzzy characteristic matrices [6], and also showed that a well-tuned fuzzy controller could even reject unknown random exogenous disturbances. In addition, Brown and Passion devised fuzzy and adaptive fuzzy control laws for the control of an acrobot, and used genetic algorithms to tune the control parameters [7]. However, those methods are complicated and the settling time is relatively long.
In order to simplify the design of the control system, the motion space is usually divided into two subspaces: the attractive area, which is around the unstable inverted equilibrium point; and the swing-up area, which is the rest. Various control methods for the swing-up area are based on the idea of increasing the energy of the acrobot until it escapes from the swing-up area and moves into the attractive area [8], [9], [10], [11], [12], [13], [14], [15], [16]. However, since the characteristics of an acrobot are very complicated and nonlinear, taking only energy into consideration does not yield efficient control of the movement from the swing-up area into the attractive area. As can be seen in [7], with this kind of control law, it takes a fairly long time for the acrobot to get out of the swing-up area. To shorten the settling time, Lai et al. devised a control law that combines model-free and model-based fuzzy control [12]. The drawback in this case is that the control parameters are sensitive to the posture of the acrobot, which is closely related to the settling time; and unfortunately, they offered no systematic method of tuning them. Lai et al. then analyzed the problem of how to shorten the settling time and pointed out that only by properly managing both energy and posture could an acrobot be made to quickly enter the attractive area, i.e., the acrobot should stretch out naturally when the energy reaches the value for the unstable inverted equilibrium point. This dual problem was also solved through the use of fuzzy control in [16]. On the other hand, a singularity occurs when energy and posture are considered together [15], [18]. Xin and Kaneda proposed a method of solving this problem [15]: one parameter in the control law was made large enough to prevent the denominator of the term for control torque from becoming zero; but simulations showed that the settling time was fairly long. In contrast, the singularity was simply avoided in [18] by further dividing the swing-up area into two subareas. This strategy is useful in handling the singularity, but it also entails some drawbacks, namely, the division of the swing-up area increases the complexity of the control law and may even lengthen the settling time.
This paper proposes an integrated fuzzy-control strategy to solve the singularity problem and to quicken the system response. A fuzzy motion-control law is first designed based on a Lyapunov function, which is employed to achieve an integrated control objective for energy and posture. This fuzzy controller regulates one of the design parameters in the motion control law to solve the singularity problem. The validity of this method is demonstrated through simulations, which show that the settling time is much shorter than that in [15]. In order to further shorten the settling time and reduce oscillations in the control torque, another fuzzy controller is designed that improves control performance by regulating another design parameter in the motion control law. A combination of these two fuzzy controllers greatly reduces both oscillations in the control torque and settling time. In addition, it also suppresses the change in energy around the value of the potential energy that the acrobot has at the unstable inverted equilibrium point.
The paper is organized as follows: Section 2 explains the singularity problem in the control of an acrobot. Section 3 presents a solution employing one fuzzy controller. Section 4 describes an integrated fuzzy control strategy employing two fuzzy controllers that improve control performance. Section 5 discusses some simulation results, and Section 6 presents some concluding remarks.
Section snippets
Singularity problem in motion control
In this section, an acrobot is first modeled in the state space. Then, the singularity that arises in motion control is explained, and the cause is analyzed.
A fuzzy-control strategy that solves the singularity problem
A consideration of the denominator of reveals that one way of avoiding the singularity would be to choose the parameter such that
However, in general, the choice of a constant that satisfies (35) makes very large during the whole period of motion control. In addition, note that the acrobot swings up from the stable downward equilibrium point, , at the beginning of motion control; and the initial values of and are and
Integrated fuzzy-control strategy
Although the above fuzzy-control strategy solves the singularity problem and produces better results than those obtained previously, there is room for further improvement. In particular, the following points require investigation: the settling time is still too long, and the control torque changes abruptly. Note that the speeds at which the Lyapunov functions converge are related to the parameter , which is constant in the above fuzzy-control law. A different produces quite
Simulations
This section presents some simulation results that demonstrate the validity of the method. Matlab and Simulink were used in the design and the simulations. The reference position of the acrobot is ; and the control objective is to swing it up from that position to the upward equilibrium position, .
The parameters of the acrobot used in the simulations are [8]
Conclusions
The aim of this study was to solve the singularity problem that arises in the motion control of an acrobot when an integrated control objective for energy and posture is used. A fuzzy controller was first designed to regulate one of the parameters of the motion control law to solve the singularity problem. Another fuzzy controller was then designed that improves control performance by regulating another parameter in the motion control law to keep the values of the two parts of the control
Xu-Zhi Lai received B.S., M.Sc., and Ph.D. degrees in engineering from Central South University, China, Changsha, in 1988, 1991 and 2001, respectively. In 1991, she joined the School of Information Science and Engineering, Central South University, Changsha, Hunan, China, where she is currently a professor. She was a visiting scholar in the Department of Mechatronics, School of Engineering, Tokyo University of Technology from April 1998 to April 1999. She was also a visiting scholar in the
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2015, Applied Mathematics and ComputationCitation Excerpt :In [7], Spong firstly used a partial feedback linearization (PFL) approach to design a swing-up controller that makes the acrobot enter an attractive area around the upright position, and then the controller switches to a LQR-based balancing controller that stabilizes the acrobot at the upright position finally. To employ this kind of switching strategy on the stabilization of the acrobot better, Henmi et al. [8] designed a new reference trajectory for the linear variables of the acrobot appeared in the PFL approach, Xin [9] and Lai et al. [10] used an energy-based to design the swing-up controller, and Lai et al. presented a fuzzy swing-up controller in [11]. All these make the acrobot enter the attractive area more easily.
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2013, AutomaticaCitation Excerpt :Then, a controller is designed for each subspace; and switching from one to the other enables the objective to be reached. Design methods for the swing-up controller include a partial feedback linearization (PFL) method (Spong, 1995), an energy-based (EB) method (Lai, She, Yang, & Wu, 2009), and an intelligent method (Lai, Yang, She, & Wu, 2009). Design methods for the balancing controller include a linear quadratic regulator (LQR) method (Spong, 1995), a pseudo-linearization method (Davison & Bortoff, 1997), and a robust control method (Xin & Kaneda, 2001).
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Xu-Zhi Lai received B.S., M.Sc., and Ph.D. degrees in engineering from Central South University, China, Changsha, in 1988, 1991 and 2001, respectively. In 1991, she joined the School of Information Science and Engineering, Central South University, Changsha, Hunan, China, where she is currently a professor. She was a visiting scholar in the Department of Mechatronics, School of Engineering, Tokyo University of Technology from April 1998 to April 1999. She was also a visiting scholar in the Department of Mechanical and Engineering, University of Toronto and in the School of Engineering, University of Guelph from September 2004 to March 2006. Her research interests are wireless sensor networks, intelligent systems, nonlinear systems, and robotics.
Simon X. Yang received a B.Sc. degree in engineering physics from Peking University, Beijing, China, in 1987, an M.Sc. degree in biophysics from the Chinese Academy of Sciences, Beijing, China, in 1990, an M.Sc. degree in electrical engineering from the University of Houston, Houston, Texas, in 1996, and a Ph.D. degree in electrical and computer engineering from the University of Alberta, Alberta, Canada, in 1999. In 1999, he joined the School of Engineering, University of Guelph, Guelph, Ontario, Canada as an assistant professor. Currently, he is professor and the director of the Advanced Robotics and Intelligent Systems (ARIS) Laboratory at that university. His research interests are in the areas of robotics, intelligent systems, sensors, control systems, image and signal processing, neurocomputation, and bioinformatics. He has published over 160 refereed journal papers, book chapters, and conference papers. He is a technical editor of the journal Dynamics of Continuous, Discrete and Impulse Systems and the International Journal of Information Acquisition. He has been involved in the organization of many international conferences.
Jin-Hua She received his B.Sc. degree in control engineering from the Central South University, China, in 1983; and M.Sc. and Ph.D. degrees in control engineering from the Tokyo Institute of Technology, Japan, in 1990 and 1993, respectively. In 1993, he joined the Department of Mechatronics, School of Engineering, Tokyo University of Technology; and in April, 2004, he transferred to the University’s School of Bionics, where he is currently an associate professor. He received the paper prize in control engineering practice at IFAC in 1999 (jointly with M. Wu and M. Nakano). His current research interests include the application of control theory, repetitive control, expert control, Internet-based engineering education, and robotics. He is a member of the Society of Instrument and Control Engineers (SICE) and the Institute of Electrical Engineers of Japan (IEEJ).
Min Wu received his B.S. and M.S. degrees in engineering from the Central South University, Changsha, China in 1983 and 1986, respectively. In July,1986, he joined the staff of the university, where he is currently a professor of automatic control engineering. He was a visiting scholar in the Department of Electrical Engineering, Tohoku University, Sendai, Japan, from 1989 to 1990, and a visiting research scholar in the Department of Control and Systems Engineering, Tokyo Institute of Technology, Tokyo, Japan, from 1996 to 1999. He received his Ph.D. degree in engineering from the Tokyo Institute of Technology, Tokyo, Japan in 1999. Dr. Wu received the best paper award at IFAC in 1999 (jointly with M. Nakano and J.-H. She). His current research interests are robust control and its applications, process control, and intelligent control. He is a member of the Nonferrous Metals Society of China and the China Automation Association.
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This work was supported in part by the National Science Foundation of China under grants 60674044 and 60425310.