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 RHO 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.
