On Symbolic Solution of the Matrix Algebraic and Differential Riccati Equations

In the specification of compliant motion for robots one meets the use of the vector DOT and CROSS products between the system state and a given, or measured vector, as part of the required performance index, e.g. fox f·x = 0 and f×x = 0. This performance index form allows one to obtain a symbolic solution to the algebraic Riccati equation arising from an LQ formulation of the optimal control problem involved in a special but common case. Of special interest is the fact that the symbolic solutions obtained are very simple and can be evaluated in real time. Also, the simple analytic form of the solution shows clearly the behavior of the resulting control system and gives insight into the control of more complex control systems which are not of this very special form.