Brief paperRobust adaptive motion/force control for wheeled inverted pendulums☆
Introduction
Wheeled inverted pendulums have attracted a lot of attention recently (Brooks et al., 2004, Gans and Hutchinson, 2006, Grasser et al., 2002, Jung and Kim, 2008, Li and Luo, 2009, Nasrallah et al., 2007, Pathak et al., 2005) as shown in Fig. 1. Similar systems like the cart and pendulums have been studied in the literature Ibanez, Frias, and Castanon (2005) and Zhang and Tarn (2002). The differences from these systems are that the inverted pendulum’s motion in the present system is not planar and the motors driving the wheels are directly mounted on the pendulum body (Pathak et al., 2005).
Motion of wheeled inverted pendulums is governed by under-actuated configuration, i.e., the number of control inputs is less than the number of degrees of freedom to be stabilized (Isidori, Marconi, & Serrani, 2003), which makes it difficult to apply the conventional robotics approach to control Euler–Lagrange systems. Due to these reasons, increasing effort has been made towards control design that guarantees stability and robustness for mobile wheeled inverted pendulums.
Although wheeled inverted pendulums systems are intrinsically nonlinear and their dynamics are described by nonlinear differential equations, if the system operates around an operating point, and the signals involved are small, we can obtain a linear model approximating the nonlinear system in the region of operation. In Ha and Yuta (1996), motion control was proposed using a linear state-space model. In Grasser et al. (2002), dynamics was derived using a Newtonian approach and the control was designed based on the dynamic equations linearized around an operating point. In Salerno and Angeles (2003), dynamic equations of the inverted pendulum were studied involving pitch and rotation angles of the two wheels as the variables of interest, and in Salerno and Angeles (2004) a linear controller was designed for stabilization considering robustness as a condition. In Blankespoor and Roemer (2004), a linear stabilizing controller was derived by a planar model without considering yaw. In Kim, Kim, and Kwak (2005), the exact dynamics of a two-wheeled inverted pendulum was investigated, and linear feedback control was developed on the dynamic model. In Pathak et al. (2005), a two-level velocity controller via partial feedback linearization and a stabilizing position controller were derived.
Based on the idea of linearization, a model-based approach is generally utilized in dynamic control. If accurate knowledge of the dynamic model is available, the model-based control can provide an effective solution to the problem. However, wheeled inverted pendulum control is characterized by unstable balance and unmodelled dynamics, and subject to time varying external disturbances, in the form of parametric and functional uncertainties, which are generally difficult to model accurately. Therefore, traditional model-based control may not be the ideal approach since it generally works best only when the dynamic model is known exactly. The presence of uncertainties and disturbances would disrupt the function of the traditional model-based feedback control and lead to unstable balance.
Moreover, the wheeled inverted pendulum is definitely different from other nonholonomic systems subject to (i) only kinematic constraints which geometrically restrict the direction of mobility, i.e., wheeled mobile robot (Ge et al., 2001, Ge et al., 2003); (ii) only dynamic constraints due to dynamic balance at passive degrees of freedom where no force or torque is applied, i.e., the manipulator with passive link (Arai and Tanie, 1998, Luca and Oriolo, 2002); (iii) both kinematic constraints and dynamic constraints. It is obvious that the wheeled inverted pendulum is more complex than the former two cases, therefore, the controls suitable for (i) and (ii) cannot be directly applied for (iii).
A challenging problem is to control a mobile inverted pendulum system whose cart is not constrained by guide rail like cart–pendulum systems, but moves in its terrain while balancing the pendulum. Therefore, the nonholonomic constraint force between the wheels and the ground should be considered in order to avoid slipping or slippage. A similar application of interacting with environments can be found in Dong (2002), where motion/force control is considered for mobile manipulators under holonomic constraints. Recent works, including Grasser et al. (2002), Jung and Kim (2008), Li and Luo (2009), Nasrallah et al. (2007) and Pathak et al. (2005), where the wheeled inverted pendulum moves on the planar plane or on the incline plane, do not consider nonholonomic constraint force, i. e. friction force, while assuming beforehand that the ground can provide enough friction as needed, but in practical applications, this assumption is difficult to satisfy. When the ground friction cannot support motion, the control performance by these controllers will be degraded.
In this paper, by discovering and utilizing the unique physical properties of wheeled inverted pendulums, we separate the zero-dynamics subsystem to simplify the model. Then, we propose a robust adaptive motion/force control for wheeled inverted pendulums. Since the system except the zero-dynamics subsystem is still a MIMO nonlinear system, we propose adaptive robust controls to accommodate the presence of parametric and functional uncertainties in the dynamics of wheeled inverted pendulums.
The main contributions of this paper are that: (i) adaptive robust motion/force control is developed for wheeled inverted pendulums by using their physical properties with parametric and functional uncertainties; (ii) the nonholonomic constraint force between the wheels and the ground is considered in order to avoid slipping or slippage; (iii) based on Lyapunov synthesis, motion/force stabilities are achieved and the input-to-state stability properties of the zero dynamics are used to derive bounds on the the tracking errors.
Section snippets
Preliminaries
In the following study, let denote the 2-norm, i.e. given , .
Lemma 2.1 Letwithrepresenting a-dimensional strictly proper exponentially stable transfer function,anddenoting its input and output, respectively. Thenimplies that,is continuous, andas. If, in addition,as, then (Ge, Lee, & Harris, 1998).
Lemma 2.2 Forandwith, we have.
Proof If , we have .
Dynamics of mobile wheeled inverted pendulums
Consider the following wheeled inverted pendulum dynamics described by Lagrangian formulation: where with is the vector of generalized coordinates with as the position coordinates, as the heading angle, and as the tilt angle as shown in Fig. 1. is the inertia matrix, is the vector of Coriolis and centrifugal forces, is the vector of gravitational forces, is the vector of the bounded external
Control objectives
By appropriate selection of a set of vector , since the control input is only dimension, and , only variables of can be controlled, the control objective can be specified as: design a controller that ensures the tracking errors of from their respective desired trajectories to be within a small neighborhood of zero, i.e., where . Ideally, should be the threshold of measurable noise, while the constraint force error is
and -subsystems motion control
Let us define the following notations as , , , , where , is the reference signal described in internal state space, and is positive diagonal. Apparently, we have From the dynamic equation (17) together with (19), we have Let , , and be nominal parameter vectors which give the corresponding nominal functions and , respectively. There exist some finite
Simulation
Let us consider a mobile wheeled inverted pendulum as shown in Fig. 1. The following variables have been chosen to describe the vehicle (see also Fig. 1): : the torques of the left and right wheels; : the tilt angle of the pendulum; : the direction angle of the mobile platform; : the radius of the wheels; : the distance between the two wheels; : the length of the pendulum; : the mass of the mobile pendulum; : the mass of each wheel; : the moment of inertia of the mobile
Conclusions
In this paper, robust adaptive motion/force control design is carried out for dynamic balance and stable tracking of desired trajectories of a mobile wheeled inverted pendulum, in the presence of unmodelled dynamics, or parametric/functional uncertainties and nonholonomic constraint force. The control is mathematically shown to guarantee semi-global uniformly bounded stability, and the steady state compact sets to which the closed loop error signals converge are derived. The size of compact
Zhijun Li received Dr. Eng. Degree in Mechatronics, from Shanghai Jiao Tong University, PR China, in 2002. From 2003 to 2005, he was a postdoctoral fellow in Department of Mechanical Engineering and Intelligent systems at the University of Electro-Communications, Tokyo, Japan. From 2005 to 2006, he was a research fellow in the Department of Electrical and Computer Engineering at the National University of Singapore, and Nanyang Technological University, Singapore. Currently, he is an associate
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Zhijun Li received Dr. Eng. Degree in Mechatronics, from Shanghai Jiao Tong University, PR China, in 2002. From 2003 to 2005, he was a postdoctoral fellow in Department of Mechanical Engineering and Intelligent systems at the University of Electro-Communications, Tokyo, Japan. From 2005 to 2006, he was a research fellow in the Department of Electrical and Computer Engineering at the National University of Singapore, and Nanyang Technological University, Singapore. Currently, he is an associate professor in the Department of Automation, Shanghai Jiao Tong University, PR China. Dr. Li is IEEE Senior Member and his current research interests are adaptive/robust control, mobile manipulators, nonholonomic systems, etc.
Yunong Zhang received B.S., M.S. and Ph.D. degrees respectively from Huazhong University of Science and Technology (HUST), South China University of Technology (SCUT) and the Chinese University of Hong Kong (CUHK), respectively, in 1996, 1999 and 2003. He is currently a professor at the School of Information Science and Technology, Sun Yat-Sen University (SYSU), Guangzhou, China. Before joining SYSU in 2006, he had been with the National University of Ireland (NUI), University of Strathclyde, and National University of Singapore (NUS) since 2003. His main research interests include neural networks, robotics and Gaussian processes. His web-page is now available at http://www.ee.sysu.edu.cn/teacher/detail.asp?sn=129.
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This work is supported by Shanghai Pujiang Program under Grant No. 08PJ1407000 and the Natural Science Foundation of China under Grant Nos. 60804003, 60935001 and the New Faculty Foundation under Grant No. 200802481003. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic.