Object-oriented Dynamics Modeling for Legged Robot Trajectory Optimization and Control

o This paper discusses the development and applica- tion of object-oriented modeling and implementation techniques to achieve a representation of the mechanical model amenable to the various requirements by legged robot applications. This leads to a uniform, modular, and exible code generation while reaping the performance of efcient domain-specic articulated body algorithms. Trajectory optimization problems of bipedal and quadrupedal robots are investigated as applications. It is shown how a high-level specication of multibody dynamics models using component libraries serves as a basis for generation of a number of modules forming an ioveralli computational dynamics model. The selected examples illustrate how one copes with the emerging complexity by integrating various modules for, i. e. , equations of motion, non-linear boundary conditions, symmetry and transition conditions, for multiple phases during motion. Variations of the mechanical structure, e. g. , contact conditions can be treated either by multiple models or by reconguration of one model. Numerical results are presented for time- and energy-optimal trajectories of full three-dimensional models of a humanoid robot and a Sony four-legged robot. I. INTRODUCTION The development of e xible, dynamic and autonomous (i. e. guided by external and internal sensor data) motions for quadrupedal and humanoid robots is still a challenging and mainly unsolved task. The dynamics of legged robots is characterized by a free-oating, tree-like multibody sys- tems structure with a high number of degrees of freedom consisting of many links and many actuated joints. Due to the frequent changes of the contact situation during legged locomotion frequent changes in the dynamic model occur. This special structure can usually not be utilized for an ef- cient implementation when using general purpose multibody systems (MBS) formalisms and tools. To cope with the high complexity of the nonlinear dynamics of legged robots model- based methods for real-time actuator control, for trajectory optimization, and for controller design specically tailored to the one scenario of legged robots must be developed. For rapid and virtual prototyping a most desireable goal is not only to apply the same abstract dynamic model representations and mathematical models but also as many parts as possible of the same code for off- and on-line evaluations of legged robot dynamics during all stages of design, development, implementation and operation of a legged robot. To facilitate the investigation of new concepts of nonlinear model-based optimization and control methods also the sensitivities of the legged robot dynamics model with respect to its state variables and parameters are needed. The formalisms and tools applied at the same time have to cope with (i) the complex underlying mechanical model, characterized by many degrees of freedom, actuator dynamics, and interaction of the robot with its physical environment including time-varying contacts and collisions, (ii) the wide range of required numerical schemes, including kinematics, dynamics, sensitivity informa- tion, etc., (iii) efcient code generation which is particularly vital for on-line computations, and must rely on the power of dedicated (recursive) algorithms, and (iv) interaction of the computational model with the system it is running on, e. g. , communication with sensors and actuators.