Robotic manipulation has shown to be a key technology in factory automation. In order for a robot to perform some specific tasks, the robot is required to move according to the commands from a motion controller. Most motion control applications of robot manipulators can be categorized into two main classes. The first is the point-to-point motion control or set-point regulation where the robot is required to move from an initial position to a final desired position in the workspace. Pick-and-place operations are typical examples of the point-to-point motion control applications. Other examples that require set-point control are spot welding and hole drilling. The second type is trajectory tracking applications, such as arc welding, machining, and painting, where the robot has to follow a desired trajectory.
Robot manipulator consists of rigid links connected by joints. One end of the manipulator is fixed to a base and an end effector or tool is connected to the other end. The vector space in which the joint displacements are defined is often referred to as the joint space, and the coordinates in which the manipulator task of the end effector is specified is referred to as the task space, which can be a Cartesian space or an image space depending on the task requirements. The motion control problem of a robot can be formulated either in the joint space (Kelly et al. 2005) or in the task space (Spong et al. 2006). In the joint-space control methodology, the desired position of the end effector is converted to a corresponding desired joint configuration by solving an inverse kinematic problem, and a feedback control law is designed so that the robot joints follow the desired joint position. To eliminate the problem of solving the inverse kinematics, the robot motion control problem can be directly formulated and designed in task space. A transformation matrix Transformation matrix or Jacobian matrix Jacobian matrix is used to transform the task-space feedback error to joint control inputs.
The most commonly used controllers in industrial applications are PD and PID controllers (Ziegler and Nichols 1942). The main advantages of such controllers are the simplicity and ease of implementation. However, the kinematics and dynamics of a robot manipulator are highly nonlinear with coupling between joints, and hence, the linear control theory cannot be applied directly to design PD or PID controllers for a robot manipulator. By exploring physical properties of the robot dynamics, Takegaki and Arimoto (Takegaki and Arimoto 1981) first showed using Lyapunov method (Slotine and Li 1991) that a simple PD controller with gravity compensation is effective for set-point control of a robot manipulator Robot manipulator. The result was an important landmark in robot control theory. Inspired by the original work (Takegaki and Arimoto 1981), much progress has been made in understanding the robot motion control problem, and various control methods have been developed for a robot manipulator (Arimoto and Miyazaki 1984, 1985; Arimoto et al. 1994; Ortega et al. 1995; Arimoto 1994; Wen and Bayard 1988; Niemeyer and Slotine 1991; Berghuis et al. 1993; Cheah et al. 1998, 2004, 2007, 2010; Cheah 2003; Dixon 2007; Liang et al. 2010; Braganza et al. 2005; Wang and Xie 2009; Garcia-Rodriguez and Parra-Vega 2012; Cheah and Liaw 2005; Kelly 1997; Wang et al. 2007).
This chapter focuses on motion control methods of robot manipulators that were developed based on the Lyapunov method. In particular, several set-point and adaptive tracking controllers are presented in both the joint space and task space, for a robot manipulator with uncertainty. The robot kinematics and dynamics are nonlinear with coupling between joints, and a good understanding of the structure and properties of the models is essential for the design of simple and effective controllers. The dynamics and kinematics of robot manipulators and their basic properties are first introduced in section “Dynamics and Kinematics of Robot Manipulators” and several examples are given to illustrate the properties. The next two sections of this chapter present the standard motion controllers in robotics, which serve as foundation works for the design of most robot controllers based on the Lyapunov method. Section “Set-Point Control by PD Plus Gravity Controller” introduces the set-point controllers based on the PD plus gravity control strategy. Section “Adaptive Control of Robot Manipulators” presents adaptive control methodology for tracking control of robot manipulators. Recent advances in sensing technology has led to the research and development of sensory task-space feedback control laws for robot manipulators. The use of task-space sensory feedback information such as visual information improves the endpoint accuracy in the presence of uncertainty. Section “Approximate Jacobian Set-Point Control” presents the basic task-space sensory feedback control problem of a robot manipulator with kinematic and dynamic uncertainty for set-point control applications. Moreover, the results are extended to deal with uncertain gravitational force. Section “Adaptive Jacobian Tracking Control” presents the adaptive Jacobian controller for task-space sensory feedback tracking control applications. Simulation results are presented in section “Simulation Results.” A brief review of the basic concepts and theories for stability analysis of nonlinear systems is also provided in the Appendix.