High speed computation of the inverse Jacobian matrix and of servo inputs for robot arm control

For controlling in real time robot arms having six degrees of freedom, the most efficient way is to use an incremental method which consists in deriving the variations of the six variables si associated to the degrees of freedom from those of the six variables xj representing the position and the orientation of the end effector, using the inverse jacobian matrix J-1. We have developed a method avoiding the inversion of the 6×6 jacobian matrix J. We show that in all cases it is possible to derive, and often very easily, the mathematical expressions of the 36 terms of J-1 from those of the terms of J and from the inverse of a 3×3 sub-matrix whose choice is arbitrary and usually directed by simplicity of inversion. The computations involve only 3×3 matrices and consist in multiplications and only 2 inversions. Moreover it is sufficient to determine only the 9 terms of a 3×3 sub-matrix of J-1 for computing directly the sis, then the computing time is significantly reduced. We demonstrate the use of this method for controlling our hydraulic telethesis: the time needed for computing the inputs of the servos associated to the actuators, from the true positions of the joints delivered by transducers, is less than 8 milliseconds using a Texas Instruments TMS 9900 microprocessor.