The goal of this paper is to demonstrate the capacity of Model Predictive Control to generate stable walking motions without the use of predefined foot steps. Building up on well-known Model Predictive Control schemes for walking motion generation, we show that a minimal modification of these schemes allows designing an online walking motion generator which can track a given reference speed of the robot and decide automatically the foot step placement. Simulation results are proposed on the HRP-2 humanoid robot, showing a significant improvement over previous approaches.

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.

Abstract-Building on previous propositions to generate walking gaits online through the use of Linear Model Predictive Control, the goal of this paper is to show that it is possible to allow on top of that a continuous adaptation of the positions of the foot steps, allowing the generation of stable walking gaits even in the presence of strong perturbations, and that this additional adaptation requires only a minimal modification of the previous schemes, especially maintaining the same Linear Model Predictive form. Simulation results are proposed then on the HRP-2 humanoid robot, showing a significant improvement over the previous schemes.

Abstract-This article addresses the fast solution of a Quadratic Program underlying a Linear Model Predictive Control scheme that generates walking motions. We introduce an algorithm which is tailored to the particular requirements of this problem, and therefore able to solve it efficiently. Different aspects of the algorithm are examined, its computational complexity is presented, and a numerical comparison with an existing state of the art solver is made. The approach presented here, extends to other general problems in a straightforward way.

Abstract. We present a problem class of mixed-integer nonlinear programs (MINLPs) with nonconvex continuous relaxations which stem from economic test scenarios that are used in the analysis of human complex problem solving. In a round-based scenario participants hold an executive function. A posteriori a performance indicator is calculated and correlated to personal measures such as intelligence, working memory, or emotion regulation. Altogether, we investigate 2088 optimization problems that differ in size and initial conditions, based on real-world experimental data from 12 rounds of 174 participants. The goals are twofold. First, from the optimal solutions we gain additional insight into a complex system, which facilitates the analysis of a participant's performance in the test. Second, we propose a methodology to automatize this process by providing a new criterion based on the solution of a series of optimization problems. By providing a mathematical optimization model and this methodology, we disprove the assumption that the "fruit fly of complex problem solving," the Tailorshop scenario that has been used for dozens of published studies, is not mathematically accessible-although it turns out to be extremely challenging even for advanced state-of-the-art global optimization algorithms and we were not able to solve all instances to global optimality in reasonable time in this study. The publicly available computational tool Tobago [TOBAGO web site https://sourceforge.net/ projects/tobago] can be used to automatically generate problem instances of various complexity, contains interfaces to AMPL and GAMS, and is hence ideally suited as a testbed for different kinds of algorithms and solvers. Computational practice is reported with respect to the influence of integer variables, problem dimension, and local versus global optimization with different optimization codes.