Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 437979, 14 pages

http://dx.doi.org/10.1155/2015/437979

## 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.

#### Abstract

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.

#### 1. Introduction

Humanoid robotics has become a highly important subject for the academic community in recent years due to its potential use in domestic and medical applications. Several sophisticated humanoid robots have been developed, for example, the ASIMO robot [1], created by the Honda Motor Company; the QRIO robot [2], manufactured by Sony; and the HUBO robot [3], proposed by the KAIST.

In addition to this, there is an increasing trend for the development of small-sized humanoid robots, for example, the NAO robot [4], created by Aldebaran Robotics; or the DARwIn-OP [5], manufactured by Robotis. This type of robots has found a commercial market niche in education and entertainment, offering an accessible platform for students, researchers, and hobbyists. Although commercial robots provide a simpler mechanical actuation and information processing than their noncommercial counterparts, there is an opportunity for the development of simple, analytical, and explicit kinematic models for gait kinematic control.

It is well known in the literature that legged locomotion provides several advantages when compared to wheeled locomotion [6]; for example, legs can step over obstacles and achieve a smooth ride on uneven surfaces by varying the effective length of the legs in order to match the surface geometry. Nevertheless, the design of complex dynamic motions for humanoids is only achievable through the full understanding of kinematics [7], namely, the forward and inverse kinematics. The first concept concerns the determination of the position and orientation of the end-effector, when the active joint configurations of the robot are given while the second concept deals with determination of the joint variables for a particular position and orientation of the end-effector [8].

Furthermore, the complexity of the inverse kinematics problem for open-kinematic chains has been exhaustively discussed in [9], particularly due to the fact that the nonlinear mapping of the joint and Cartesian spaces resulted in multiple solutions. Although closed-form solutions can be obtained for systems that provide six or less degrees-of-freedom (DOF), some studies prefer the determination of analytical solutions for real-time applications, since the computation time of the numerical solutions may vary significantly [10]. The computation of closed-form solutions requires the performance of complex algebraic and geometric tasks, where the challenge consists in finding the configurations in which a reduced number of unknowns can be used to express the position and orientation of the end-effector [11].

Relevant work has been conducted towards the determination of simplified kinematic models in humanoid robots. Pieper has decoupled a six-DOF robot with three intersecting axes into sets of equations for the position and orientation in [12]; Graf et al. have solved a DH chain for a robot leg using a triangle-based geometric approach in [13]; Park et al. have used a forward and reverse decoupling method to solve the kinematics of a humanoid leg using the inverse transform method in [3]; Hernández-Santos et al. have divided the walking gait into the Sagittal and Frontal planes in order to obtain the closed-form solutions for the inverse and forward kinematics of a 16-DOF humanoid robot in [14]; and Kofinas et al. have manipulated both sides of the kinematic matrix equations in order to express the translation of one foot using only three variables in [15].

This paper presents a comprehensive mathematical model that applies some of the methods listed above in addition to some new procedures, in order to determine the lower limb kinematics of the small-sized commercial humanoid NAO robot. This work approaches the position analysis by using a geometrical procedure based on triangular arrangements. Furthermore, the joint activation sequence resulting from this geometry is inserted in the analysis of the DH chain. This practice allows the conciliation of both analytical and geometrical equations in order to be solved simultaneously. Furthermore, a local coordinate frame is placed at the end-effector (either the hip or the foot). This allows the orientation functions to be obtained by equating the corresponding elements of the position and orientation matrix concatenations in a straightforward manner. The resulting orientation equations allow the feet to remain parallel to the ground, while the torso remains upright.

Furthermore, an adjustment of the DH transformation matrices is performed in order to solve the kinematics of the joint that constrains the motion of the two legs and the torso, which presents a mechanical dependency. This is of particular interest for the turn-in-place motion [16], where some joints are constrained so that the mathematical complexity of the kinematic coupling is decreased. The result of this procedure is a set of functions that allows the robot to rotate about its own vertical axis while compensating the unwanted torso rotation. Finally, a simple geometric approach is proposed in order to avoid singularities in the gait postures. The equations obtained provide the total vertical displacement that the robot needs to experience in order to reach a given step distance, resulting in a gait workspace. This approach bypasses the use of the Jacobian matrix [17], reducing the mathematical complexity of this analysis. The gait workspaces with and without joint limitations are displayed and discussed.

The remainder of this paper is organized as follows: Section 2 describes the nomenclature of the humanoid joints. Also, the DH convention is presented. Section 3 is a position analysis of the humanoid robot leg, where the forward and reversed kinematics problems are analytically solved. In Section 4 the mechanically shared joint between the two legs is analyzed, where closed-form solutions are determined for the turn-in-place motion. Section 5 presents the analysis of the workspace of a footstep; in this section the workspace of a humanoid gait is geometrically deduced and plotted. In Section 6 numerical data is presented along with experimental robot positions to corroborate the feasibility of the results obtained. Finally, Section 7 presents important conclusions about the mathematical procedures presented in this work.

#### 2. Humanoid Robot Description and Notation

This paper uses the humanoid robot NAO model H21 that is manufactured by Aldebaran Robotics [18] as an experimental platform. This robot has 21 rotational DOF and is actuated by servomotors. The wrists and hands of this model are not actuated; hence they are not considered in this work. Figure 1 shows the humanoid robot, including the navigation reference frame and the notation used to identify the joints that comprise the kinematic chains. Regarding the navigation coordinate frame of the robot, the -axis points to the forward walking direction; the -axis points to the left side of the robot; and the -axis points upwards. Each leg of the robot has 5 DOF (namely, the AnkleRoll, AnklePitch, KneePitch, HipRoll, and HipPitch) and one special joint located between the hips (comprised of the RHipYawPitch and the LHipYawPitch), coupled by a gearbox that connects the two legs. This special joint is rotated and mirrored over the -axis at each hip.