Fast, Accurate Models for Predicting the Compliance of Elastic Flexure-jointed Robots

Robot manipulators having elastic links or flexure joints have a number of advantages, especially in simplifying the control of contact with other objects. However, current simplified parametric models of flexure motion do not accurately predict the behavior of these mechanisms under large deflections. This paper presents a “smooth curvature model” of flexure behavior that describes the curvature of a highly flexible member such as a flexure joint using a basis of three orthogonal polynomials. Using this model, we show that it is possible to predict the planar stiffness these mechanisms, even in cases where the deformation of the hinge is too large for the linear Euler-Bernoulli beam bending model. Using both finite element methods and the much less computationally expensive proposed model, numerical results will demonstrate that it is possible to accurately predict the in-plane compliance of a highly flexible mechanism in the presence of an external load. The results of this work are significant because they demonstrate that the behavior of flexure-based robotic mechanisms can be modeled quickly and with few parameters, enabling their use in closedloop control for situations where collision safety is a concern, and rigorous model-based path planning for obstacle avoidance, among other applications. INTRODUCTION For many years, there has been a general recognition that physical or simulated compliance can be used to improve the ability of a robot to perform certain tasks, specifically tasks involving contact with the environment, such as grasping, manipulation, and locomotion [1-5]. Mechanical compliance in a robot hand, such as the SDM Hand shown in Fig. 1, simplifies control of a robot because contact does not need to be exactly predicted or modeled. The robot can simply be commanded to collide with an object, and the soft contact will ensure that the load is limited by compliance, or balanced between multiple contacts through elastic averaging. Another key advantage of planned-in compliance is robustness to unforeseen collisions with objects in an uncertain environment [6]. One obstacle to progress in this field is the lack of good constitutive models for large-deformation flexure joints. These elastic elements are often modeled as Euler-Bernoulli beams connecting two rigid bodies. Because the beam bending equations are nonlinear for large deformation, approximated beam profiles are often used in place of exact solutions [7]. For example, a flexure is sometimes modeled as an arc of constant curvature [8], or as a single rotational degree of freedom at the center of the joint [9]. These methods are capable of estimating the gross kinematic behavior, in the sense that the angles between the manipulator’s rigid bodies are described; as a consequence, these models are useful for inverse kinematics or motion planning. However, the obvious drawback to these single-parameter models is their inability to predict the compliance of a flexure. A one-degree of freedom model will, by definition, admit motion in one direction locally, and it will be infinitely stiff in any other direction. This problem is shared by other, more accurate models such as the family of pseudorigid body models [10]. The remaining alternatives are finite Figure 1. Left: the SDM Hand, a flexure-based, underactuated robotic hand designed by the authors. Right: a computational model of the SDM Hand created to study compliant finger behavior