In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem (IKP) of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions. Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method. The experimental results show that this method has a large advantage over the classical methods on geometric intuition, computation and real-time, and can be directly extended to all serial robot manipulators and completely automatized, which provides a new tool on the analysis and application of general robot manipulators.

This paper is concerned with the development of a near minimum-time suboptimal controller for a robot manipulator to follow a prespecified path which is designed to avoid obstacles in the workspace.A dynamic model for the manipulator which is simple to manipulate is established using the average dynamics method of linearization. The model has the advantage that it will be continuously updated over the control time periods. This makes it suitable for high speed or variable payload applications. Bang-bang control theory in conjunction with synchronization of executing time for each joint is then used to derive the suboptimal controller. Simulation results under different conditions of obstacles and loads indicate that the desired controller moves the manipulator around the obstacles in near minimum-time with an acceptable level of accuracy. JNTRODUC TIONIn general, two approaches are used for manipulator control. The first method employs off-line optimal trajectory planning followed by on-line path tracking. The second method involves the design of suboptimal controllers using realistic approximation models of manipulator dynamics. For the first approach, once the planned trajectory as a function of time is available, one can use one of the well known on-line trajectory tracking methods. However, there are no known trajectory planning methods which include the manipulator dynamics. Maximum speed and acceleration of the arm vary with its position, payload, and configuration. Since it may otherwise not be able to follow the preplanned trajectory with the prescheduled velocity and acceleration, the trajectory planning has to be made on the basis of the global least upper bound of all possible arm's speeds and accelerations. In most cases this global least upper bound of speed and acceleration is much less than that of the manipulator's capability. Therefore the manipulator's full capacity may not be efficiently utilized. Using the second approach, only a few attempts have been made due to the highly nonlinear nature of the manipulator's dynamics.For the solution of the near minimum-time control problem with obstacles in the workspace the second approach is desirable, in order to utilize the maximum torque for each joint. The solution for the minimum-time problem over one trajectory segment is bang-bang with a certain switching curve which is difficult to compute.