coordinates for the next pose. In tasks where speed and precision are important, the real-time requirement puts heavy constraints on the computation time of the inverse kinematic algorithm.

 The forward kinematics problem, the conversion from joint space to Cartesian space, is a much simpler problem that has a unique closed-form solution. In most cases a robot manipulator can achieve a desired end-effector pose in more than one configuration. The question of just how many distinct solutions there are to the inverse kinematics problem of general six-degree-of-freedom (DOF) robot manipulators has interested a few researchers. Roth, Rastegar, and Scheinman (1973) put an upper bound of 32 on the degree of a polynomial equation (in one joint variable) that can be derived from the inverse kinematics problem of six-DOF manipulators. A similar result was obtained by Duffy and Crane (1980), using the equivalence between an open 6-revolute-DOF kinematic chain and the 7-revolute

single-loop spatial mechanism. Therefore, the number of inverse kinematic solutions for 6-revolute-DOF manipulators could be at most 32. More recently, Lee and Liang (in

press), using Duffy's method, were able to reduce the degree of the inverse kinematic polynomial equation to 16, thereby reducing the upper bound on the number of inverse kinematic solutions to 16. Tsai and Morgan (1984), illustrating a new inverse kinematic method capable-of producing all soldtions,

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