An Evaluation of Moreau’s Time-stepping Scheme for the Simulation of a Legged Robot

A state-of-the-art simulation technique that solves the equations of motion together with the set-valued contact and impulse laws by the time-stepping scheme of Moreau is introduced to the legged robotics community. An analysis is given that shows which of the many variations of the method fits best to legged robots. Two different methods to solve the discretized normal cone inclusions are compared: the projected over-relaxed Jacobi and Gauss-Seidel iteration. The methods are evaluated for an electrically-driven quadrupedal robot in terms of robustness, accuracy, speed and ease of use. Furthermore, the dependence of the simulation speed on the choice of the generalized coordinates is examined. The proposed technique is implemented in C++ and compared to a fast and simple approach based on compliant contact models. In conclusion, the introduced method with hard contacts is very beneficial for the simulation of legged robots. INTRODUCTION Research on legged robots heavily relies on multibody simulations, which are required to safely test new locomotion control algorithms or may be used for motion planning and design optimizations. Simulators for such robots need to cope with highdimensional systems and non-smooth dynamics due to unilateral frictional contacts and impacts. At the same time they should be efficient for real-time control algorithms and very accurate Address all correspondence to this author: [email protected] to avoid any unrealistic opportunities that an optimization could possibly exploit. Most existing simulation tools used in robotics cannot meet all of these demanding requirements. Therefore, the development of powerful physics engines for legged robotics is a topic of active interest, in particular, as legged machines become more and more complex. The speed requirement is most often a knock-out criterion in robotics and therefore efficient computation methods for the forward dynamics have been developed, for instance, the Articulated-Rigid-Body algorithm or the Composite-Rigid-Body method [1]. When contacts are involved, compliant contact models are mostly preferred to increase simulation speed, since their evaluation is considered to be very cheap compared to hard contact models which need numerical iterations. This may however be a fallacy, because compliant contact models may require more simulation steps to generate accurate and robust solutions than others. To tackle the numerical issues, different compliant models, even tailored for legged robots, have been proposed and analyzed [2, 3]. The major shortcomings of such simple models however remain: their parameters have to be tuned manually, which is a tedious process as they often have both numerical and physical effects. There exists a variety of general-purpose physics engines [4, 5], often designed for games, but widely used in robotics [6, 7]. Such physics engines are able to simulate intricate virtual reality environments, but often make use of approximations of the constraints, which may result in unrealistic motions. Commonly, 1 Copyright c © 2014 by ASME FIGURE 1: QUADRUPED ROBOT STARLETH the mechanical system is described using a set of coordinates that define the pose of each body in Cartesian space while enforcing the joint constraints numerically and approximate the non-smooth dynamics, for instance, by formulating the contact dynamics as linear complementary problems [8]. These shortcomings that originate from a trade-off between speed, universal applicability and verisimilitude are not always acceptable for the simulation of legged robotics and may be far from optimal. Various simulators were developed for legged robots in recent years. Kanehiro et al. [9] developed their own simulation technique to simulate a humanoid robot. A simulator for a humanoid with compliant contacts was developed by Dallali et al. [10]. Todorov et al. [8] use a discrete-time velocity-based formulation based on the work of Stewart et al. [11], and could achieve impressive simulation speeds. Their simulator is however closed source and designed as multi-purpose engine, which may not exploit all characteristics of a legged robot. In this paper, we introduce a state-of-the-art simulation technique to the legged robotics community, which was developed during the past years in the field of mechanics and was successfully applied to various mechanical problems [12], including a robotic snake [13]. The proposed method solves the equations of motion together with the set-valued contact laws and the impact equations by the time-stepping scheme of Moreau [14]. The set-valued contact and impulse laws are modeled as discretized normal cone inclusions, which need to be solved iteratively. Two popular numeric iteration methods are thus analyzed in this work: the projected over-relaxed Jacobi (JOR) and the Gauss-Seidel (SOR) iteration. Due to the distinctive structure of legged robots (large, heavy trunk and thin, lightweight legs), a subtle choice of the coordinates to describe the multibody system may significantly improve the accuracy and speed of the simulation. Three different sets of coordinates are therefore examined in this work. We investigate which of the many variations of the method with hard contacts fits best to legged robots and provide an evaluation of the overall performance. We focus on the simulation of a medium-dog-sized electrically-driven quadrupedal robot called StarlETH [15] shown in Fig. 1. The quadruped with 18-degreesof-freedom (DOF) is capable of various dynamic motions including fast walking, trotting, and pronking. The locomotion control algorithms were so far developed with a simple simulation technique using compliant contacts in combination with a standard Runge-Kutta integrator [16]. The current paper thus compares the newly proposed and previously used method in terms of robustness, accuracy, speed and user-friendliness. NON-SMOOTH DYNAMICS We first introduce the simulation algorithm based on Moreau’s time-stepping scheme together with the hard and compliant contact model to solve the equations of motion of scleronomic dynamical systems. For a more complete introduction to non-smooth dynamics briefly covered in the next sections, the reader is referred to [17]. Equations of Motion The equations of motion of a non-smooth scleronomic dynamical system (cf. [18]) described with the generalized coordinates q and velocities u can be represented by a set of equations of the form M(q) u̇− h(q,u)−WUλU −WBλB = 0,