Theoretical Analysis of the State of Balance in Bipedal Walking

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.