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Research Papers

Theoretical Analysis of the State of Balance in Bipedal Walking

[+] Author and Article Information
Flavio Firmani

Research Associate
Department of Mechanical Engineering,
University of Victoria,
Victoria, British Columbia, V8W 3P6, Canada
e-mail: ffirmani@me.uvic.ca

Edward J. Park

Associate Professor
Mechatronic Systems Engineering,
School of Engineering Science,
Simon Fraser University,
Surrey, British Columbia, V3T 0A3, Canada
e-mail: ed_park@sfu.ca

1Correponding author.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received July 20, 2012; final manuscript received January 18, 2013; accepted manuscript posted February 22, 2013; published online April 2, 2013. Assoc. Editor: Kenneth Fischer.

J Biomech Eng 135(4), 041003 (Apr 02, 2013) (13 pages) Paper No: BIO-12-1305; doi: 10.1115/1.4023698 History: Received July 20, 2012; Revised January 18, 2013; Accepted February 22, 2013

This paper presents a theoretical analysis based on classic mechanical principles of balance of forces in bipedal walking. Theories on the state of balance have been proposed in the area of humanoid robotics and although the laws of classical mechanics are equivalent to both humans and humanoid robots, the resulting motion obtained with these theories is unnatural when compared to normal human gait. Humanoid robots are commonly controlled using the zero moment point (ZMP) with the condition that the ZMP cannot exit the foot-support area. This condition is derived from a physical model in which the biped must always walk under dynamically balanced conditions, making the centre of pressure (CoP) and the ZMP always coincident. On the contrary, humans follow a different strategy characterized by a ‘controlled fall’ at the end of the swing phase. In this paper, we present a thorough theoretical analysis of the state of balance and show that the ZMP can exit the support area, and its location is representative of the imbalance state characterized by the separation between the ZMP and the CoP. Since humans exhibit this behavior, we also present proof-of-concept results of a single subject walking on an instrumented treadmill at different speeds (from slow 0.7 m/s to fast 2.0 m/s walking with increments of 0.1 m/s) with the motion recorded using an optical motion tracking system. In order to evaluate the experimental results of this model, the coefficient of determination (R2) is used to correlate the measured ground reaction forces and the resultant of inertial and gravitational forces (anteroposterior R2 = 0.93, mediolateral R2 = 0.89, and vertical R2 = 0.86) indicating that there is a high correlation between the measurements. The results suggest that the subject exhibits a complete dynamically balanced gait during slow speeds while experiencing a controlled fall (end of swing phase) with faster speeds. This is quantified with the root-mean-square deviation (RMSD) between the CoP and the ZMP, a relationship that grows exponentially, suggesting that the ZMP exits the support area earlier with faster walking speeds (relative to the stride duration). We conclude that the ZMP is a significant concept that can be exploited for the analysis of bipedal balance, but we also challenge the control strategy adopted in humanoid robotics that forces the ZMP to be contained within the support area causing the robot to follow unnatural patterns.

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Figures

Grahic Jump Location
Fig. 1

Morphologically-inspired human body model consisting of 17 body segments and 40 DOF. The gray rectangle illustrates the adjacent belts of the treadmill and the darker shaded area around the feet represents the convex hull of the foot-support area.

Grahic Jump Location
Fig. 2

(a) shows a posture where the two forces FGR and FF are collinear, i.e., the whole body is in rotational equilibrium; (b) shows a more general case, when FGR and FF are separated by a distance PG as the whole body undergoes a rotation about G. This rotation is explained by the change of rate in angular momentum, whose horizontal components are given by H·G=PG×FF.

Grahic Jump Location
Fig. 3

Dynamically Balanced State. (a) shows the general case where the whole body is undergoing a rotation about G with a separation of the forces FGR and FF. In the central figure, FF is replaced by an equivalent force passing through P and a couple PG × FF that counteracts the inertial moment MH·G. Consequently, (b) shows that both forces FGR and FF lie on the same line at P, therefore, the CoP and ZMP are two equivalent points.

Grahic Jump Location
Fig. 4

Dynamically Imbalance State. (a) shows the general case where the whole body is undergoing a rotation about G. In the central figure, FF is replaced by an equivalent force passing through Q and a couple QG × FF that counteracts the inertial moment MH·G. Note that the two vectors are still separated by a distance PQ. In the right figure, FF is replaced by an equivalent force passing through P and a couple PQ × FF, which represents the existing imbalance moment. In this case, points P and Q are not the same points, i.e., CoP ≠ ZMP.

Grahic Jump Location
Fig. 5

Linear regression of temporal and kinematic mean parameters

Grahic Jump Location
Fig. 6

Profiles of ground reference points: CoP, pCM, ZMP, and CMP through a gait cycle at 1.4 m/s

Grahic Jump Location
Fig. 7

Comparison of ground reference points by means of RMSD and exponential regression with least squares minimization

Grahic Jump Location
Fig. 8

Mean of mass-normalized angular momentum for each speed trial (m2/s)

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