A matrix manipulation method using phase shifted sinewaves for real-time robot control

This work involves the theoretical development of a new realtime matrix manipulation method and the practical proof of the new theory. There is a need for such a method because it has not yet proved possible to provide non-recursive solutions to robot dynamic problems in real-time. This is because the matrix products and matrix derivatives required are, as is well known, too time consuming to numerically compute in real-time, due to the large number of additions and multiplications involved in the matrix manipulation. It is shown that the partial orthonormality of the standard Denavit/Hartenberg matrices used in the robot control equations allows them to be expressed in sinewaves such that real-time matrix manipulations are performed simply by phase shifting the sinewaves rather than by numerically calculating the matrix products. The development is introduced by initially considering two-dimensional kinematic descriptions and this is then extended to three-dimensional kinematics and later to the finding of partial derivatives used in robot dynamics. The way in which the sinewave method allows a general systematic and non-intuitive approach to robot inverse kinematic problems is also described. The practical proof of the new system is performed by Pascal simulation and also by hardware implementation. The hardware implementation involves dedicated digital hardware whose basic structure has cascade characteristics and possesses an address/data/control bus somewhat like that of a digital computer. In this hardware, sinewaves representing the various relevant matrices are stored in individual random access memories such that they can be read in a controlled manner to simulate the information about the matrix elements to be manipulated. The time it takes for the hardware, which is named the Sinusoidal Matrix Processor (SMP), to stabilize whenever the joint variables are being changed is a linear function of n compared with n4 for conventional techniques, where n is the number of degrees of freedom of the manipulator. In this time, the SMP provides all relevant first and second order partial derivatives which are involved in the inverse dynamic calculations and so a substantial- potential time saving is provided and in fact, the computing time becomes comparable to that of the recursive formulations involving the use of (4X4) matrices. Unlike recursive formulations, however, the states remain available explicitely so greatly facilitating the application of advanced control schemes