Fast Direct Multiple Shooting Algorithms for Optimal Robot Control

Abstract

Summary. In this overview paper, we first survey numerical approaches to solve nonlinear optimal control problems, and second, we present our most recent algorithmic developments for real-time optimization in nonlinear model predictive control.In the survey part, we discuss three direct optimal control approaches in detail: (i) single shooting, (ii) collocation, and (iii) multiple shooting, and we specify why we believe the direct multiple shooting method to be the method of choice for nonlinear optimal control problems in robotics. We couple it with an efficient robot model generator and show the performance of the algorithm at the example of a five link robot arm. In the real-time optimization part, we outline the idea of nonlinear model predictive control and the real-time challenge it poses to numerical optimization. As one solution approach, we discuss the real-time iteration scheme.

Cost-optimal composition synthesis for modular robots

“…The general form of our optimal control problem for ordinary differential equations (ODEs) in the Lagrangian form is defined as in (4), where x(·) is the vector of state variables, defined as x = (q;q) where x ∈ R 2n , u(·) is the vector of control variables which are defined as the torques/forces acting on the joints, where u = τ and u ∈ R n , and L(x(t), u(t)) is the desired objective function as in [17]. The robot dynamics is transformed from (4) to the state space formẋ = g(x k (t), u k (t)).…”

“…When compared to other methods, direct methods have a good balance of computational efficiency and accuracy [17]. As a solution to our problem, we use the direct multiple shooting method due to its robustness, fast convergence, easy parallelizability and applicability for unstable systems [17].…”

“…Since there is a large number of different manipulators to consider, a fast trajectory planning algorithm is required to achieve the given task. There are two main approaches for optimal collision-free trajectory planning problems: i) obtaining the cost-optimal trajectory following the pre-defined path [15], [16], and ii) using optimal control methods considering the initial and the goal positions [17], [18]. The methods in the first category may fail to find the optimal path and non-collision is not guaranteed because of the possibility of collision between pre-determined positions.…”

“…Direct methods for trajectory optimization, which are in the second category, are based on the transformation of the optimal control problem into a nonlinear programming (NLP) problem. They are detailed in [17] and it is shown that the multiple direct shooting method is one of the best methods to obtain the optimal trajectory for a robot arm considering the obstacles in the environment. Another direct optimal control method is used in [18] to obtain the time-optimal trajectory for manipulators considering collision avoidance from static and dynamic obstacles as state constraints.…”

Path-Planning with Collision Avoidance in Automotive Industry

“…The fastest trajectory of a robot is the solution of an optimal control problem where the system of ordinary differential equations (ODE) are given by (1), see [3,6]. If an obstacle is present in the workspace, the collision-freeness is assured as soon as the vector w (i,j) of Proposition 1 is found at each time t and for all pairs of polyhedra.…”