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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 437979, 14 pages
Research Article

Decoupled Closed-Form Solution for Humanoid Lower Limb Kinematics

Graduate School of Science and Engineering, Tecnológico de Monterrey, 64849 Monterrey, NL, Mexico

Received 1 January 2015; Revised 4 March 2015; Accepted 6 March 2015

Academic Editor: Francesco Franco

Copyright © 2015 Alejandro Said et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper presents an explicit, omnidirectional, analytical, and decoupled closed-form solution for the lower limb kinematics of the humanoid robot NAO. The paper starts by decoupling the position and orientation analysis from the overall Denavit-Hartenberg (DH) transformation matrices. Here, the joint activation sequence for the DH matrices is based on the geometry of a triangle. Furthermore, the implementation of a forward and a reversed kinematic analysis for the support and swing phase equations is developed to avoid matrix inversion. The allocation of constant transformations allows the position and orientation end-coordinate systems to be aligned with each other. Also, the redefinition of the DH transformations and the use of constraints allow decoupling the shared DOF between the legs and the torso. Finally, a geometric approach to avoid the singularities during the walking process is indicated. Numerical data is presented along with an experimental implementation to prove the validity of the analytical results.