Résumé
We present a reformulation of the posture generation problem that encompasses non-Euclidean manifolds. Such a formulation allows a more elegant mathematical description of the constraints, which we exemplify through some scenarios in the simulation results section. In our previous work, the posture generation problem is formulated as a non-linear optimization program with constraints expressed only through Euclidean manifolds; we solve the latter problem using on-the-shelf solvers. Instead, we decided to implement a new SQP solver that is most suited to non-Euclidean manifolds structural objects. By doing so, we have a better mastering in the way to tune and specialize our SQP solver for robotic problems. I. INTRODUCTION Computing robot configurations to meet the requirements of a given set of tasks, within a viable state, is a recurrent problem whose complexity grows with that of the robot. In this paper, we are interested in the following generalized inverse kinematics problem: we search a configuration for which the robot fulfills tasks under constraints of joint limits, auto-collision and non-desired collision avoidance, balance, torque limits, etc. We coined it posture generation. Such a problem is encountered in both planning and control. In both cases, computation time and robustness are critical issues. We have already proposed various implementations of the humanoid posture generation problem. All of our implementations formulate the problem as a non-linear optimization program to address multi-contact planning. In [1], the multi-contact planner explores the contact space using thousands of HRP-2 humanoid posture generator (PG) queries; we used the FSQP solver [2]. In [3], the PG is extended to handle various humanoid robots and multiple agents, the solver used is IPOPT [4]. In [5] the PG is extended to various contact models and used to generate multiple related postures at once. The latter work and the DRC participation revealed that re-planning on the fly is necessary and having a robust PG is crucial in many situations. Other works also make use of PG, e.g. in [6][7]. Posture generation has been formulated as a problem over a Euclidean space. Robots variable may however be more naturally expressed over non-Euclidean manifolds. The archetypes for this are the rotation part of the root body for a humanoid robot, and ball joints, whose variables live in SO(3). Some typical tasks are also naturally formulated on