Some mechanical systems, that are modeled to have inelastic collisions, nonetheless possess energy-conserving intermittent-contact solutions, known as collisionless solutions. Such a solution, representing a persistent hopping or walking across a level ground, may be important for understanding animal locomotion or for designing efficient walking machines. So far, collisionless motion has been analytically studied in simple two degrees of freedom (DOF) systems, or in a system that decouples into 2-DOF subsystems in the harmonic approximation. In this paper we extend the consideration to a N-DOF system, recovering the known solutions as a special N = 2 case of the general formulation. We show that in the harmonic approximation the collisionless solution is determined by the spectrum of the system. We formulate a solution existence condition, which requires the presence of at least one oscillating normal mode in the most constrained phase of the motion. An application of the developed general framework is illustrated by finding a collisionless solution for a rocking motion of a biped with an armed standing torso.

翻译：一些被建模为具有非弹性碰撞的机械系统，却存在守恒能量的间歇接触解，即无碰撞解。这种解表现为在水平地面上的持续跳跃或行走，可能对理解动物运动或设计高效行走机器具有重要意义。迄今为止，无碰撞运动仅在简单的二自由度系统或谐波近似下解耦为二自由度子系统的系统中得到了解析研究。本文将该研究推广至N自由度系统，并将已知解作为一般公式中N=2的特例进行还原。我们表明，在谐波近似下，无碰撞解由系统的频谱决定。我们提出解的存在条件，该条件要求在运动的最强约束阶段至少存在一个振荡简正模态。通过寻找一个带有武装站立躯干的双足机器人摇摆运动中的无碰撞解，展示了所开发通用框架的应用。