Optimal control of multibody systems in minimal coordinates

The present paper deals with numerical methods for optimal control of mechanical multibody systems, such as many industrial robots, whose dynamical behavior can be described in minimal coordinates by a system of semi‐explicit second order differential equations: M(q(t))q̈(t) = u(t) + h(q(t), q̇(t), t), 0 ⩽ t ⩽ t f , where q denotes the state variables, u the control variables, and M(q) the positive definite mass matrix. Numerical methods are addressed for computing an approximation of the optimal control u*(t), 0 ⩽ t ⩽ t f , which steers the system from an initial to a final position minimizing a performance index, such as time or energy, subject to bounds or nonlinear constraints on q, q̇, and u. A method is investigated in more detail which is based on piecewise polynomial approximations of state variables and utilizes the structure of the dynamical equations as well as the structure in the resulting large and sparse, nonlinearly constrained optimization problems. Results for an industrial robot with six joints demonstrate that tailored optimization methods are very well suited for fast off‐line optimization of robot trajectories.