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Research Papers

# Dynamic Motion Planning of 3D Human Locomotion Using Gradient-Based Optimization

[+] Author and Article Information
Hyung Joo Kim, Qian Wang, Salam Rahmatalla, Jasbir S. Arora, Karim Abdel-Malek, Jose G. Assouline

Center for Computer Aided Design, College of Engineering, The University of Iowa, Iowa City, IA 52242

Colby C. Swan1

Center for Computer Aided Design, College of Engineering, The University of Iowa, Iowa City, IA 52242colby-swan@uiowa.edu

The ZMP is discussed in Sec. 21.

1

Corresponding author; surface mail: Department of Civil and Environmental Engineering, 4120 Seamans Center, The University of Iowa, Iowa City, IA 52242.

J Biomech Eng 130(3), 031002 (Apr 21, 2008) (14 pages) doi:10.1115/1.2898730 History: Received May 23, 2006; Revised September 04, 2007; Published April 21, 2008

## Abstract

Since humans can walk with an infinite variety of postures and limb movements, there is no unique solution to the modeling problem to predict human gait motions. Accordingly, we test herein the hypothesis that the redundancy of human walking mechanisms makes solving for human joint profiles and force time histories an indeterminate problem best solved by inverse dynamics and optimization methods. A new optimization-based human-modeling framework is thus described for predicting three-dimensional human gait motions on level and inclined planes. The basic unknowns in the framework are the joint motion time histories of a $25-degree$-of-freedom human model and its six global degrees of freedom. The joint motion histories are calculated by minimizing an objective function such as deviation of the trunk from upright posture that relates to the human model’s performance. A variety of important constraints are imposed on the optimization problem, including (1) satisfaction of dynamic equilibrium equations by requiring the model’s zero moment point (ZMP) to lie within the instantaneous geometrical base of support, (2) foot collision avoidance, (3) limits on ground-foot friction, and (4) vanishing yawing moment. Analytical forms of objective and constraint functions are presented and discussed for the proposed human-modeling framework in which the resulting optimization problems are solved using gradient-based mathematical programing techniques. When the framework is applied to the modeling of bipedal locomotion on level and inclined planes, acyclic human walking motions that are smooth and realistic as opposed to less natural robotic motions are obtained. The aspects of the modeling framework requiring further investigation and refinement, as well as potential applications of the framework in biomechanics, are discussed.

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## Figures

Figure 1

Forces on a walking biped including GRFs RC, gravity force mg, and inertial force (−mẍG) and inertial moment (−ḢG)

Figure 2

Illustration of BOS ΓBOS in (a) SSP and (b) DSP

Figure 3

The BOS polygon during RDS: (a) side view and (b) plan view. (LH1∕LH2=medial/lateral left foot heel, LB1∕LB2=medial/lateral left foot ball, LT1∕LT2=medial/lateral left foot toe, RH1∕RH2=medial/lateral right foot heel, RB1∕RB2∕medial/lateral right foot ball, RT1∕RT2∕medial/lateral right foot toe). The BOS region ΓBOS is defined in Eq. 9 and is an intersection of up to six half-planes Li defined in Eqs. 10,10, and ΓBOSC denotes its complement with L5C, the complement of L5, as shown in (c).

Figure 4

The 31DOF human linkage model with dimensions (L0–L13) and joint rotation axes (q0–q27) denoted by cylinders. Three additional degrees of freedom (q28–q30) capture the rigid body translation of the body. Specific segment lengths and mass values represent the values used in the computations of Sec. 4.

Figure 5

Aids to FCA during RDS: (a) at left the half-plane Lm1 lateral to the medial edge of the left foot and (Lm1)C its complement and (b) at right the half-plane Lm2 lateral to the medial edge of the right foot and (Lm2)C its complement

Figure 6

Locomotion of a biped that demonstrates normal walking on a level plane ((a) and (b)), on a gently inclined plane ((c) and (d)), and on a steeply inclined plane ((e) and (f)). To facilitate viewing, the posture has been offset walking direction.

Figure 7

The trajectories for the BOS, PCOM, and ZMP for normal walking on a level plane. The ZMP (black dots) and PCOM (blue open squares) traverse the BOS (red outline).

Figure 8

ZMP (black filled dots) and PCOM (blue open squares) trajectories over the BOS (gray regions) in anterior-posterior direction ((a), (c), and (e)), and in medial-lateral direction ((b), (d), and (f)) for normal walking on a level plane, walking on a plane sloped at 0.15, and walking on a plane sloped at 0.30, respectively

Figure 9

Stage 3 locomotion during DSP with RDS: (a) view from the top and (b) side view from the right. The figures in dashed red lines denote the array of body postures; the six black dots trace out the ZMP trajectory, and blue open squares denote the PCOM trajectory.

Figure 10

Locomotion during RSS Stage 4 as the left leg swings forward: (a) view from the top and (b) side view from the right. The figures in dashed red lines again denote the array of the postures, while the black filled dots show the ZMP trajectory, and the blue open squares the PCOM.

Figure 11

Computed gait stage durations in seconds for normal walking on a level plane

Figure 12

Profiles of joint angles for (a) hip flexion, (b) knee extension, and (c) ankle dorsiflexion. Both first and second time derivatives of the joint angles are also shown to illustrate their continuity characteristics. The time derivatives are scaled as noted to fit onto their angle.

Figure 13

Biped model’s ascent of an inclined slope: (a) xyz axes are attached to the slope. (b) When the system is rotated by α, the biped ascent is equivalent to normal walking with fraction of gravity force (mgsinα) that pulls the biped backward.

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