M.Sc Thesis
M.Sc Student Moravia Mordechai Analysis of Foot Slippage Effects on a Simple Model of Dynamic Legged Locomotion Department of Mechanical Engineering Professor Yizhar Or

Abstract

Legged locomotion is commonly used by human beings and animals on terrestrial environment. On the other hand, the common method of mechanical mobility is by wheeled vehicles. The traversability of legged locomotion is greater than of wheeled motion. Thus, in the last decades many robots were built based on the principles of legged locomotion and even inspired by legged animal in nature. Legged locomotion can be roughly divided into two categories, dynamic and quasistatic motion. Quasistatic motion is defined by moving slowly through a sequence of equilibrium postures. The robot repositions its leg while the body is in equilibrium without dynamic effects, i.e negligible inertia. Dynamic legged locomotion is defined by movement with dynamics and acceleration. The body of the robot is falling until a free leg touches the ground. Although the robot is not in equilibrium in each moment, the robot can move in a stable periodic orbit (i.e. "gait") in which the response to a small perturbation converges back to the gait.

The classical model of spring-loaded inverted pendulum (SLIP) has been widely accepted as a simple description of dynamic legged locomotion in humans, legged robots and animals at various scales. Like majority of works in the literature, the SLIP model assumes ideal sticking contact during stance phase. However, there are practical scenarios of low ground friction which causes foot slippage that can have significant influence on the dynamic behavior. Therefore, the purpose of this work is to develop an extension of the SLIP model with two masses and torque actuation, which accounts for possible slippage under Coulomb’s friction law. This model is formulated and numerically investigated in order to define the influence of stickslip transitions on the dynamic behavior of the model. In particular, the existence of periodic solutions, their stability and energetic efficiency are considered.

Periodic solutions can be found by using Poincaré map. Numerical simulations reveal several types of periodic solutions with stick↔slip transitions. Several properties of periodic solutions are plotted as a function of μ - the coefficient of friction. This is a convenient way to study the influence of the friction coefficient and slippage on the dynamics of periodic solutions of the model.  It is found, that decreasing the friction coefficient is slightly degrading stability of the periodic solutions compare to the solution under stick assumption. On the other hand, it is found that slippage due to low friction can sometimes increase the average speed and improve the energetic efficiency by significantly reducing the mechanical cost of walking. Friction is a non-conservative power, which dissipates energy from the system. Therefore, intuitively one would expect that periodic solutions with slippage are energetically inefficient. Nevertheless, this is proven wrong by our results and explained further in this work.