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In this paper, an efficient method for calculating the joint rates of manipulators is proposed. The method realizes desired velocity of the end effector expressed in Cartesian coordinates. By the method, the analytical solution of the joint rates for a 6 d.o.f. manipulator can be obtained. It allows direct calculation of the joint rates corresponding to desired velosity of the end effector without real-time calculation and invertion of the Jacobian matrix, which is time-consuming process. In general, the joint rates are obtained as solution of linear simultaneous equations. The coefficient matrix of the equations is known as the Jacobian matrix. It is difficult to solve them directly because of their complexity. In this paper, to simplify the equations, we introduce an intermediate coordinate frame which is a Cartesian coordinate frame fixed to the middle link of the manipulator. Then the equations with 6 unknowns of joint rates are simplified to two sets of linear equations with 3 unknowns. The resultant equations can be easily solved analytically. The amount of real-time computation is reduced by the analitical expression of the solution. For a PUMA robot, for instance, the calculation requires 28 additions, 46 multiplications and 4 divisions, giving computation time of about 2.5ms using a 5MHz 8086 micro processor and an 8087 coprocessor.

High Speed Computation of the Inverse Jacobian Matrix for Robot Arm Control Using Intermediate Coordinates