Métodos algébricos aplicados ao estudo do movimento de robôs humanoides

Abstract

Humanoid robots are multibody systems with appearance of humans, with two legs, two arms and a head. There are different applications where humanoid robots can be used, as in domestic’s activities, rescue of victims in natural disasters and to support the human rehabilitation. The kinematic model of humanoid robots has been extensively studied and there are different approaches to solve it. The interest is to obtain such models that represent the humanoid gait as similar as possible to the human gait. Despite the solutions presented in the literature, the robots still walking with behavior different of human’s, even on flat surfaces and without external disturbances. Possible reasons for these differences are the limitations of the inverse kinematics model used for the robot design. In general, the motions are split on two independent planes, keeping constant the torso and feet orientation, that does not give a general solution to describe a spatial trajectory of the humanoid robot. A common approach to obtain these models applies the conventional geometric methods to model the inverse kinematics, with matrix transformation and trigonometric expressions, where the multibody orientation is, in general, given by Euler angles. This approach presents parametrization singularities given by Gimbal Lock problem. The complexity to interpolate the motion described by Euler angles is another problem. Then, the objective of this work is to present a new method to solve the humanoid robot’s inverse kinematics maintaining its dynamic equilibrium. The method is based on algebraic geometry and dual quaternions approach to obtain the inverse kinematics model, which is able to describe the spatial motion of humanoid robots without restrictions of orientation and required displacement. The proposed methodology enable to interpolate the joint angles, velocities and accelerations, resulting in polynomial functions, allowing to obtain the acceleration of the center of mass. The dynamic stability analysis is done to the proposed motion, by computational simulations and experimental tests, based on the method of Zero Moment Point (ZMP). The influence of dynamic effects is also evaluated by the variation of the robot center of mass acceleration. The method was applied to design the gait of two robots of different sizes. The tests have shown the dynamic stability of the proposed trajectory and validated the methodology to design and simulate the humanoid robot gait.