Optimizing Spring-Damper Design in Human Like Walking that is Asymptotically Stable Without Feedback
Special purpose numerical optimal control algorithms can be used to create biped multibody systems and open loop joint torque histories that create periodic motions that are asymptotically stable without any feedback. In this context, we have produced ope
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IWR, Universit¨ at Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany [email protected] Dept. of Mechanical Engineering, Columbia University, New York, 10027, USA [email protected]
Abstract Special purpose numerical optimal control algorithms can be used to create biped multibody systems and open loop joint torque histories that create periodic motions that are asymptotically stable without any feedback. In this context, we have produced open-loop stable biped walking, running, hopping, somersaults and flip-flops. In this paper, we specifically investigate the stabilizing role of springs and dampers added to a biped walking system by including the spring and damper constants in the stability optimization. It is shown that stability and robustness to state disturbances of the asymptotically stable open loop gaits can be very substantially increased by an optimization-based selection of spring and damper components and that springs and dampers help to induce a more natural appearing solution.
1 Introduction A fundamental understanding of the principles underlying dynamic walking and running motions is important in many fields of research and industry, e.g. for the construction of better walking robots, in prosthetics and rehabilitation, or for the visualization of motions in computer graphics. In order to achieve this understanding, model-based numerical optimization and optimal control is a very helpful tool making it possible to ”look inside the dynamics” of the gait or to improve its performance. One crucial property of real life walking and running motions is stability, and the faster a motion, the harder it is to make it stable. We are especially interested in exploiting the natural stability of the system as much as possible since this should facilitate the task of any feedback control system to cope with disturbances. The idea of exploiting the natural stability has been introduced in robotics in the form of the passive-dynamic walkers (e.g. Mochon & McMahon [10], McGeer [9]), which are purely mechanical devices without
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any actuators or sensors moving down inclined slopes. More recent robots of this type are capable of free 3D dynamic walking on level ground power by very simple actuation and sensors (Collins et al. [5]) . If these ideas are to be transferred to more complex walking and running robots with general forms of actuations, this will not be possible based on intuition and experimental testing only, but instead guidelines produced by a thorough theoretical analysis should be used to support the design process of these robots. In our opinion, numerical optimization is the right tool to approach this issue. In our previous research, we have developed special stability optimization techniques and we were able to determine, for a range of robot models, limit cycle gaits that are asymptotically stable without any feedback at all [11, 12, 14]. There are different properties of a robot to be tuned in order to improve its stability, namely: • • • •
the t
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