Efficient Kinematic and Algorithmic Singularity Resolution for Multi-Contact and Multi-Level Constrained Dynamic Robot Control
Résolution efficace de singularité cinématique et algorithmique pour le contrôle de robot dynamique contraint à plusieurs contacts et à plusieurs niveaux
Résumé
The introduction of the latest solvers for least-squares programs allows very fast solving of multi-level constrained robotic control problems. However, seldom are all these constraints perfectly achievable at the same time. This leads to kinematic and algorithmic singularities, an issue that has concerned the research of many roboticists.
With this thesis, we build on this already available knowledge and introduce new methods for singularity resolution tailored to such hierarchically constrained least-squares programs for kinematic structures like humanoid robots.
In previous works the connection between the Gauss-Newton algorithm, Newton's method and the Levenberg-Marquardt algorithm has been shown. All have their origin in the second order Taylor expansion of the non-linear function of the quadratic error norm. It is expressible in least-squares form and suitable for our hierarchical least-squares solvers. The Gauss-Newton algorithm, which neglects Taylor second order terms, corresponds to the standard control algorithm exhibiting numerical instabilities when close to singularities. The Levenberg-Marquardt algorithm, which can be considered a classical measure of singularity resolution, approximates the Taylor second-order terms with a weighted identity matrix. Newton's method uses the analytic expression. Based on this circumstance, we assume that Newton's method is a valid approach for singularity resolution. In a first step, we formulate the Lagrangian gradient and Hessian of the constrained optimization problem. The former enables positive definite updates of the Hessian by the Broyden-Fletcher-Goldfarb-Shanno algorithm which then can be used in a Quasi-Newton method. The latter one can be directly used for Newton's method of multi-level constrained optimization. It also inspires another approximation method based on the Symmetric Rank 1 method. We introduce a method to switch from the Gauss-Newton algorithm to Newton's method when in the vicinity of singularities. We validate our methods in simulation on a set of different kinematic problems going in and out of singularities while being numerically stable and exceeding Levenberg-Marquardt based methods in terms of error reduction.
Next, we make the step from pure optimization to robot control which requires the introduction of a suitable control time-step. This allows the formulation of controllers which imitate acceleration-based control in the velocity domain as it is required for Newton's method. By doing so, we are able to incorporate the robot dynamics in the form of the equation of motion into our control framework. Validation is conducted with real robot experiments on the HRP-2Kai.
We conclude this work by giving some hints on how to leverage bound constraints in hierarchical least-squares solvers. Our implementation is tailored to the LexLSI solver and allows computational speed-ups for problems dominated by bound constraints.
L’usage en robotique des tous derniers solveurs de problèmes de moindres carrés linéaires hiérarchiques permet de résoudre très rapidement des problèmes de contrôle multiniveaux. Cependant, il est rare que les contraintes puissent être toutes satisfaites en même temps. Cela introduit des singularités cinématiques et algorithmiques, un problème qui a fait l’objet de nombreuses études.
Dans cette thèse, nous nous basons sur l’état de l’art et introduisons de nouvelles méthodes de résolution des singularités dédiées à ces problèmes de moindres carrés hiérarchiques pour des structures cinématiques telles que des robots humanoïdes. Au voisinage des deux types de singularités, la linéarisation du modèle géométrique non-linéaire qui décrit des structures cinématiques comme les robots, typiquement utilisée en contrôle, n’approxime pas suffisamment bien la fonction originale. Cela se traduit par un mauvais conditionnement de la matrice Jacobienne, ce qui entraîne des changements rapides de la configuration du robot. Comme cela peut amener à des comportements dangereux pour le robot ou son environnement, une telle situation doit être évitée.
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