This book aims at gathering roboticists, control theorists, neuroscientists, and mathematicians, in order to promote a multidisciplinary research on movement analysis. It follows the workshop “Geometric and Numerical Foundations of Movements” held at LAAS-CNRS in Toulouse in November 2015. 1 Its objective is to lay the foundations for a mutual understanding that is essential for synergetic development in motion research. In particular, the book promotes applications to robotics—and control in general—of new optimization techniques based on recent results from real algebraic geometry.

Starting from a robotics perspective, the generation of goal oriented motion for robots obeys classically to a two-step paradigm. The first step is the planning, where the typical problem is to find a geometric path that allows the robot to reach the desired configuration starting from the current position while ensuring obstacle avoidance and enforcing the satisfaction of kinematic constraints. Motion planning lays its grounding on the decidability properties of this classic geometrical problem. Moreover, the traditional approaches that are used to find solutions rely on the global probabilistic certainty of the convergence of path construction stochastically sampled in the configuration space. The second step of motion generation is the control, where the robot has to perform the planned motion while ensuring the respect of dynamical constraints. Motion control seeks primarily for local controllability or at least the stability of the motion. The basic instances of these problems have long been tackled using local state-space control. However, the typical nonlinearity of the dynamics, together with the non-controllability of its linearization, leads more and more solutions to resort to model preview control. These methods allow to predict the outcome of a control strategy in a future horizon and to improve it accordingly, usually by using numerical optimizations which take