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The humanoid bipedal locomotion requires computationally efficient solutions of the navigation and inverse kinematics problems. This paper presents analytic methods, using tools from computational geometry and techniques from the theory of Lie groups, to develop new geometric algorithms for the navigation path planning, locomotion movement, and kinematics modeling of humanoid robots. To solve the global navigation problem, we introduce the new fast marching method modified (FM3) algorithm, based on the fast marching methods (FMM) used to study interface motion, that gives a close-form solution for the humanoid collision-free whole body trajectory (WBT) calculation. For the bipedal locomotion, we build the new geometric algorithm one step to goal (OSG), to produce a general solution for the body and footstep planning which make the humanoid to move a single step towards a defined objective. We develop the new approach called sagittal kinematics division (SKD), for the humanoid modeling analysis, to allow us to solve the humanoid inverse kinematics problem using the mathematical techniques of Lie groups, like the product of exponentials (POE). The works are presented along with computed examples of the humanoid robot RH0 at the University Carlos III of Madrid. We remark that this paper introduces only closed-form solutions, numerically stable and geometrically meaningful, suitable for real-time applications.

ABSTRACT.This paper presents new analytical methods, using techniques from the theory of Lie Groups and tools from Computational Geometry, to develop algorithms for kinematics modeling and path-planning of humanoid robots. We introduce deterministic solutions, geometrically meaningful and numerically stable, based on the PoE (Product of Exponentials formula), which allows us to develop algorithms to solve the humanoid mechanical problem, even for real-time applications. Besides, to solve the path-planning problem, we introduce a close-form solution based on the FMM (Fast Marching Methods) used to study interface motion. The first development of this paper is a new approach for the analysis of the humanoid model; this is the humanoid SKD (Sagital Kinematics Division). The second development is the construction of the new FM3 (Fast Marching Method Modified) algorithm for the humanoid collision-free WBT (Whole Body Trajectory) calculation. In addition, the third main goal of this work is to build a general and analytical solution for the humanoid kinematics problem (direct and inverse); this is the new humanoid OSG (One Step to Goal) algorithm. The works are presented along with computed examples using both simulated and experimental results with the humanoid robot RH0 at the University Carlos III of Madrid.