Non-Square Inverse Function Jacobians in Controlled Multibody Systems: Numerical and Application Examples

Abstract

When control algorithms of robots are constructed, the joint coordinates and the coordinates describing the dynamics might be different and the transformation between them might be necessary in both directions. The back-and-forth transformations are related to the inverse function theorem, which is well understood for single and multivariable continuous functions: the conditions are described under which the inverse function exist, furthermore the method is provided to calculate the Jacobian of the inverse function. A generalization of the theorem is necessary, when there are fewer dependent variables than independent ones, and furthermore there are constraint equations for the independent variables. It is exactly the case for model-based inverse dynamics control of multibody systems, when the dynamic model is given in terms of a redundant coordinate set, but the controller is formulated for minimum set coordinates. The widely used so-called natural coordinates are a typical redundant set. Minimum-coordinates come in the picture when the control is formulated for the joint coordinates. Clearly, when the natural coordinates are transformed to joint coordinates, there is information loss. The inverse transformation is however still possible, since there are constraint equations for the redundant set. This paper demonstrates a method for the transformation from minimum to redundant coordinates and vice versa with the help of the generalized inverse of the non-square constraint Jacobian and the projection matrices related to the constrained and admissible subspace of the redundant set. An illustrative numerical example and a robotic application demonstrate the theory. The results are relevant in the model-based control of complex-structure parallel kinematic chain robots.