Tracking-error model-based predictive control for mobile robots in real time
Introduction
In recent years there has been an increasing amount of research on the subject of mobile robotics. Mobile robots are increasingly used in industry, in service robotics, for domestic needs (vacuum cleaners, lawn mowers, pets), in difficult-to-access or dangerous areas (space, army, nuclear-waste cleaning) and also for entertainment (robotic wars, robot soccer).
Several controllers were proposed for mobile robots with nonholonomic constraints, where the two main approaches to controlling mobile robots are posture stabilization and trajectory tracking. The aim of posture stabilization is to stabilize the robot to a reference point, while the aim of trajectory tracking is to have the robot follow a reference trajectory. For mobile robots trajectory tracking is easier to achieve than posture stabilization. This comes from the assumption that the wheel makes perfect contact with the ground, resulting in nonholonomic constraints, which means that not all the velocities are possible at a certain moment. An extensive review of nonholonomic control problems can be found in [7]. According to Brockett’s condition [2] nonholonomic systems cannot be asymptotically stabilized around an equilibrium using smooth time-invariant feedback. Completely nonholonomic, driftless systems are controllable in a nonlinear sense; therefore, asymptotic stabilization can be obtained using time-varying, discontinuous or hybrid control laws. An exponentially stable, discontinuous feedback controller was proposed by [3] and the point stabilization of mobile robots via state-space exact-feedback linearization using proposed coordinates was studied in [17].
Trajectory tracking is more natural for mobile robots. Usually, the reference trajectory is obtained by using a reference robot; therefore, all the kinematic constraints are implicitly considered by the reference trajectory. The control inputs are mostly obtained by a combination of feedforward inputs, calculated from reference trajectory, and feedback control law, as in [22], [10], [16], [1]. Lyapunov stable time-varying state-tracking control laws were pioneered by [5], [20], [21], where the system’s equations are linearized with respect to the reference trajectory, and by defining the desired parameters of the characteristic polynomial the controller parameters are calculated. The stabilization to the reference trajectory requires a nonzero motion condition. Many variations and improvements of this simple and effective state-tracking controller followed in latter research. An adaptive extension of this work was introduced in [18], where adaptive capabilities are included to increase the robustness to robot-model uncertainties. A fuzzy inference mechanism extension that compensates for environmental perturbations such as variable friction is proposed in [19]. A tracking control using modified input–output linearization providing a least-squares solution for a nonsquare system is dealt with in [6]. In [13] a trajectory-tracking state-feedback controller is combined with an observer that is used to estimate an unknown orientation error. Input–output linearization is used in [22] to follow reference trajectory dynamics considering a dynamic model. Some of the solutions to the controller design solve both the trajectory-tracking and posture-stabilization control problems, where the stabilization problem is usually converted to an equivalent tracking problem, as in [18], [16], [9]. In [9] a saturation feedback controller where saturation constraints of the velocity inputs are incorporated into the controller design is introduced. In [16] a dynamic feedback linearization technique is used to control a mobile robot platform.
Predictive control techniques are a very important area of research. In the field of mobile robotics predictive approaches to path tracking also seem to be very promising because the reference trajectory is known beforehand. Most model-based predictive controllers use a linear model of mobile-robot kinematics to predict future system outputs. In [15] a generalized predictive control is chosen to control a mobile robot, where a quadratic cost function penalizing the tracking errors and control effort is minimized. A generalized predictive controller using a Smith predictor to cope with an estimated system time delay is presented in [14]. In [8] a model-predictive control based on a linear, time-varying description of the system is used. The control law is again solved by an optimization of a cost function. The nonlinear predictive controller scheme for a path-tracking problem is proposed in [4]. Here, a multi-layer neural network is employed to model the nonlinear kinematic behavior of a mobile robot. However, the optimum solution of the control vector is still obtained by minimizing a cost function, like in previous studies.
This paper deals with a differentially driven mobile robot and trajectory-tracking control on a reference trajectory that is a smooth twice-differentiable function of time. The model-predictive control law is based on a linearized error dynamics model obtained around the reference trajectory. The main idea of the control law is to minimize the difference between the future trajectory-following errors of the robot and the reference robot with defined, desired dynamics. The proposed control law is analytically derived; therefore, it is computationally effective and can be easily used in fast real-time implementations. The main advantages over predictive control are an error model-based prediction and an explicitly obtained analytical control law. The model-predictive control obtained is compared to a well-known time-varying state-tracking control law [5], [20], [12], which is based on the literature review done in this work and works presented by [6], [18], [19], one of the most common and successful approaches in mobile robot tracking control. The design of the state-tracking control law in a discrete time domain is given. The experimental results for both control laws obtained for a real robot are evaluated and compared.
The remainder of the paper is organized as follows. In Section 2 is a description of the mobile robot, its control architecture and its kinematics. The concept of trajectory-tracking controller design, where the control strategy consists of feedforward and feedback actions, is introduced in Section 3. In Section 4 the proposed model predictive controller is derived. The experimental results for the predictive control obtained are presented in Section 5, and the conclusion is given in Section 6.
Section snippets
Mobile-robot control-system design
The control-system design proposed in this study, the experiments and the comparisons were performed on the small, two-wheeled, differentially driven mobile robot shown in Fig. 1.
Definition of the trajectory-tracking problem
There are two basic control approaches to solving the mobile robot’s motion task: stabilization to a fixed posture and tracking of the reference trajectory.
For nonholonomic systems, the trajectory-tracking problem is easier to solve and more natural than posture stabilization. According to Brockett’s condition [2] asymptotic stability of a nonholonomic system to a fixed posture is only possible with a time-varying or discontinuous feedback. Stabilization, therefore, cannot be achieved by a
Design of the trajectory-tracking controller
To design the controller for trajectory tracking the system (10) will be written in discrete-time form as where , is the number of the state variables and , and is the number of input variables. The discrete matrices and can obtained as follows: which is a good approximation for a short sampling time .
Experimental results
The experimental results were obtained on the real mobile-robot platform explained in Fig. 2. Two different design procedures which result in similar controller structures, given in Eqs. (25), (26), are tested and compared. To ensure a fair comparison, a similar dynamics, which gives comparable actuator actions for the two controllers, was designed. The first control law is obtained by the proposed model-predictive control, and the second by the state-tracking controller, common in the
Conclusion
The model-predictive trajectory-tracking control of a mobile robot is presented in this paper. The proposed control law minimizes the quadratic cost function consisting of tracking errors and control effort. The solution for the control is analytically derived, which enables fast real-time implementations. The proposed model-predictive control was tested on real mobile robots and the experimental results obtained were compared to a time-varying state-tracking controller. Both controllers
Acknowledgment
The authors would like to acknowledge the Slovenian Research Agency under CRP MIR M2-0116 project “Mobile Robot System for Reconnaissance, Research and Rescue” for funding this work.
Gregor Klančar received his B.Sc. degree in 1999 from the Faculty of Electrical Engineering of the University of Ljubljana, Slovenia, where he is currently employed as a member of the national young researcher scheme. His research interests are in the area of fault diagnosis methods, multiple vehicle coordinated control and mobile robotics.
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Gregor Klančar received his B.Sc. degree in 1999 from the Faculty of Electrical Engineering of the University of Ljubljana, Slovenia, where he is currently employed as a member of the national young researcher scheme. His research interests are in the area of fault diagnosis methods, multiple vehicle coordinated control and mobile robotics.
Igor Škrjanc received the B.Sc., M.Sc., and Ph.D. degrees, all in electrical engineering, from the Faculty of Electrical and Computer Engineering, University of Ljubljana, Slovenia, in 1988, 1991, and 1996, respectively. He is currently an Associate Professor with the same faculty. His main research interests are in adaptive, predictive, fuzzy, and fuzzy adaptive control systems.