Nonholonomic Motion Planning Strategy for Underactuated Manipulator

This paper develops nonholonomic motion planning strategy for three-joint underactuated manipulator, which uses only two actuators and can be converted into chained form. Since the manipulator was designed focusing on the control simplicity, there are several issues for motion planning, mainly including transformation singularity, path estimation, and trajectory robustness in the presence of initial errors, which need to be considered. Although many existing motion planning control laws for chained form system can be directly applied to the manipulator and steer it to desired configuration, coordinate transformation singularities often happen. We propose two mathematical techniques to avoid the transformation singularities. Then, two evaluation indicators are defined and used to estimate control precision and linear approximation capability. In the end, the initial error sensitivity matrix is introduced to describe the interference sensitivity, which is called robustness. The simulation and experimental results show that an efficient and robust resultant path of three-joint underactuated manipulator can be successfully obtained by use of the motion planning strategy we presented. 1. Introduction In the past few years, nonholonomic motion planning has become an attractive research field. Much theoretical development and application have been exploited. This paper can be viewed as the further study on the basis of [1], which proposed -joint underactuated manipulator [2]. And inspired by [3–5], its physical design is carried out mainly focusing on the kinematic model with triangular structure, which can be converted into chained form and achieve the control simplicity. Although motion planning control law can steer the chained form system to desired state, there exists transformation singularity while chained form path is transformed back into actual coordinates. Even in the absence of singularity, planning nonholonomic motion is not an easy task. Transformation singularities add a second level of difficulty: we must take into account both transformation singularities and nonholonomic constraint. In order to solve transformation singularity problem, the path in chained form space should remain in the singularity-free regions. In other words, when a path does not belong to those regions in chained form coordinate, transformation singularity will appear in actual coordinate. Therefore, transformation singularity avoidance is equivalent to obstacle avoidance in chained form space. Singularity regions can be checked by diffeomorphism of chained form