Multiple constrained rivaling actuators in the optimal control of miniaturized manipulators
The joints of a miniaturized multi-link manipulator, used in high-precision telesurgery with artificial fingers and hands, are driven by weak, but fast, and strong, but slow actuators acting in parallel. The respective actuator is characterized by constraints of its torque and by the rate of change of its torque. This leads to a multiple-constrained problem of rivaling controls. Defining the derivatives of the controls as new control variables, the transformation of the original controls into additional state variables eliminates the time derivatives of the torques and leads to a system of differential-algebraic equations (DAE) for the optimal control problem of differential index up to 4, for which a stable numerical solution is difficult to obtain (driftoff effect). In addition, the structure of this DAE system is very complicated. The traditional approach yields a very sensitive DAE control problem. The need to derive complicated interior point conditions in every individual case further restricts the application of the classical approach. Here the authors develop new approach. The original control problem is being reformulated as a series of piecewise defined nonlinear multipoint boundary value problems for ordinary differential equations with non-constant dimensions. The dimensions are locally adapted to the actual demand. The control structure is significantly simplified, and a stable evaluation of the controls becomes possible. Based on the minimum principle, the mathematical analysis of coupling of single parts of the boundary value problem leads to new and standardized interior point conditions at the junction points. The numerical stability and efficiency of the new approach is demonstrated on the example of a two-finger manipulator with six degrees of freedom. Very accurate numerical solutions are obtained.
scientific article; zbMATH DE number 5277389