Real-time computations of manipulator Jacobian are examined for executing on uniprocessor computers, parallel computers, and VLSI pipelines. The characteristics of the Jacobian equations are found to be in the form of the first-order linear recurrence. The time lower bound of computing the first-order linear recurrence, and hence the Jacobian, is of order O(N) on uniprocessor computers, and of order O(log2N) on parallel SIMD computers, where TV is the number of degrees-of-freedom of the manipulator. The Generalized-^ method, which achieves the time lower bound on uniprocessor computers, is derived to compute the Jacobian at any desired reference coordinate frame A; from the base coordinate frame to the end-effector coordinate frame. We find that if the reference coordinate frame k is in the range [3 , N—4], then the computational effort is the minimum. To reduce the computational complexity from the order of O (N) to O (log2N), we derive the parallel forward and backward recursive doubling algorithm to compute the Jacobian on parallel computers. Again, any reference coordinate frame k can be used, and the minimum computation occurs at k = (N—1)/2. To further reduce the Jacobian computation complexity, we design two VLSI systolic pipelined architectures. A linear VLSI pipe, which uses the least number of modular processors, takes 3N floating-point operations to compute the Jacobian, and a parallel VLSI pipe takes 3 floating-point operations. We also show that if the reference coordinate frame is selected at k — (N—1)/2, then the parallel pipe will require the least number of modular processors, and the communication paths are much shorter.
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