Towards a Modular Lower Body Robotic Model Using the Product of Exponentials

A high degree of joint synergy on the lower body is considered beneficial because it is linked to finer control of the center of mass. This is motivation to examine the stability through the prism of synergistic joint control. To identify the synergy index, the uncontrolled manifold (UCM) method can be used. This approach separates joint motion that translates to Cartesian motion from joint motion that is used for stabilization. However, research in joint synergy using the UCM staggers on the mathematical model that is used to describe the body. The human body is a non-linear multi-variable system, which makes it extremely difficult to express mathematically. As a result, certain assumptions and simplifications are needed. This has led to multiple approaches to calculate the UCM synergy index but their outcomes do not always translate to different scopes despite the theoretical background being valid. The core issue is that most of the mathematical frameworks used for joint modeling are too rigid to adapt to a different context. To address this, the product of exponentials (PoE) method to create a model of the lower body is proposed. This method allows each joint to be modeled independently as a vector and the body model can be created by simply organizing the vectors in the appropriate order. More importantly, the Jacobian matrix of the resulting model can be generated without additional differentiation. Since the UCM examines the rank and the null space of the Jacobian matrix of the system, this work is an important stepping stone towards UCM analyses for stability. Results show that the model of the right leg that was created was able to follow the hip joint center (JC) from motion capture therefore it was an accurate representation.