imulation research of a passive dynamic walker with round feet based on non-smooth method

Passive dynamic walkers can steadily walk down gentleslope under only action of gravity without any drivers and controls, andthey have human-like characteristic of walk and high energy efficiency. Thestudy on them has a key role in designing robots and medical prosthetics.Hence, it has attracted considerable public concern in recent years.The contact configuration of the biped walkers also varies correspondinglydue to the transitions between feet and ground from slip to stick or fromcontact to detachment. For the non-smooth and nonlinear model with variablestructure, it is generally to simply regard the model as an invertedpendulum fixed at each foot in turn by means of McGeer' Step-to-Step method.Here are the basic assumptions of the traditional method (Step-to-Stepmethod): There is no slip between feet and ground, and plastic collisionoccurs at heelstrick while conserving angular momentum. In addition, it isassumed that stance leg and swing leg change instantaneously and no impulseacts on the swing foot. Although the traditional method is widely employed,the shortcomings of the method can not be neglected as follows: Firstly, slipphenomenon can not be dealt with; secondly, no direct simulation or experimentresults support the assumption of angular momentum conservation of passivedynamic walker about heelstrike; thirdly, it is not suitable for analyzinggeneral movements of the biped walker.Planar straight-legged passive dynamic walker with round foot can beconsidered as a multibody system with unilateral constrains. Based ondifferential inclusions and complementary methods, the theory of non-smoothmultibody dynamics with unilateral constraints has been well established, bywhich the multi-contact problems with friction and impact could beefficiently solved. In this paper, this passive walking phenomenon issimulated by using the non-smooth dynamic method. We mainly analyze theCoulomb friction and Newton impact phenomenon between feet and ground. Dueto the collisions, the whole dynamical equation and contact laws are set upon base of impulse-velocity level. After establishing the complementaryrelationships of all contact laws, a linear complementary problem instandard form is given. And then, we choose a one-step mid-pointtime-stepping method for numerical simulation. The programming with thenon-smooth method is easier than that of the traditional method.The following conclusions are obtained from numerical simulations in thispaper: (1) If there is no slip and the collisions are plastic, thesimulation results agree with those of the traditional methods, whichindicate that the assumptions of the Step-to-Step method are reasonable. (2)In the case of slip and plastic impact, stable periodic gaits could also befound with some suitable parameters. For this case, the friction coefficienthas a significant effect on the average walking velocity in tangentialdirection. (3) If elastic impact occurs without slipping, passive dynamicwalkers could also steadily walk down slopes with several collisions at thetransition from swing leg to stance leg. Small normal coefficient ofrestitution has little effect on average walking speed. (4) The simulationresults show that for a small normal coefficient of restitution and a largefriction coefficient, it is easy for passive dynamic walkers to realizestable periodic gaits.