# Cross-Coupled Contouring Control of Multi-DOF Robotic Manipulator
## 1 Introduction  
Robotic manipulators are important for automation and industrial applications in contour tracking and trajectory tracking. Contour tracking is very popular in many industrial applications such as welding, laser cutting, and machining [1; 2; 3; 4], and contouring control is one of the most important tasks for robotics and computer numerical control machine tools [5; 6; 7; 8; 9]. One of the most important issues in machining is the reduction of contour errors to ensure the quality of machined parts. In robotic applications such as precision machining, a desired task is often defined in terms of the end-effector (task space) motion: contour tracking motion [10; 11]. Robotic technology can provide an excellent base to reduce the contour error because of its flexibility and affordability. A contour error is defined as the shortest distance between the current position and the desired contour curve, and contour error is a key factor used to evaluate the quality of a machined part. Conventional approach to deal with such a contouring problem is to treat it as a tracking control problem for each individual axis or joint. Unfortunately, reducing tracking error does not ensure reducing the contour error, because the contour error is a result of combined tracking errors of all axes or joints, and the dynamics of each axis is coupled with other axes for a nonlinear robotic system. To deal with contour tracking problems, the cross coupling control (CCC) method was first developed by [PERSON] [1] for translational motion systems, followed by a variable-gain CCC in [2]. A PItype cross coupling controller was developed for contour tracking of a linear bi-axis motion system based on linear control theory in [3]. A position loop-based cross-coupled control was proposed for a two-axis motion stage in [4]. [PERSON] [5] applied the fuzzy logic technology to deal with a free-form contour following problem by varying the feed rate of the system. Some surveys about the tracking and contour error control for CNC systems were presented in [6; 7]. For simple contours, the contour error can be calculated analytically as discussed in [1; 2]. To calculate contour errors of complex freeform contours, the concept of equivalent error was introduced for contouring control in [12]. A polar coordinate representation method was proposed in [13] to calculate contour errors for a biaxial translational motion system. Contour error estimation procedures for three-dimensional translational machining were developed in [8; 9], based on the linear dynamic model of a CNC machine system. An orthogonal global task coordinate frame [14] for the calculation of contour errors was proposed and applied to a biaxial system. It should be noted that the majority of contouring control is focusing on biaxial or multi-axial translational motion systems that are linear dynamic systems. There are two types of applications for robot manipulators in the task space level: the task space trajectory tracking control and task space contour tracking control. For task space trajectory tracking control, [PERSON] [10] developed a set of task space PD control laws based on the Jacobian regulation. An adaptive tracking control was proposed for robots with unknown kinematic and dynamic properties in [11]. In addition, the adaptive technology was applied to the contour coupling control for a CNC system in [15]. As for the contouring control, a cross-coupled control algorithm based on the Lyapunov stability criteria and the recursive updating technique was proposed and applied to a two-link direct drive robot [16]. In addition, a contour control strategy based on the coordinate transformation was developed in [17]. The contour tracking problem for 2-DOF robots was discussed in [16; 17; 18], but there is no research on cross-coupled contour tracking control for multi-DOF robot manipulators.