Rimless Wheel Locomotion  The existence of a self-stabilizing gait for the rimless wheel without the need of control or intelligence hints at the the possibility that a good body design has a major contribution to agility in legged animal locomotion. The rimless wheel rotates about the stance foot without slipping or detaching. During a step, the motion is essentially the same as that of an inverted pendulum: where  is the angle between the stance leg and the slope normal,  is the slope angle,  is the mass,  is the leg length,  is the rotation inertia about the center of mass, and  is gravitational acceleration.When a foot reaches the ground, the assumption is that there is a plastic collision, such that this foot becomes the new stance foot and holds onto the ground. The old stance foot lifts off and the system rotates without slipping about the new stance foot. Plastic collision result in a discontinuity of the velocity. The velocity after the impact and the velocity at the end of a step, when a foot reaches the ground but before impact, have the following relationship: where  is the number of spikes, superscript "+" means post-impact while "-" means pre-impact. Note that equation (2) is consistent with the fact that a plastic collision causes a loss in kinetic energy.This expression can be derived by using conservation of generalized momentum, subject to the constraint that the new stance foot must be fixed to the ground. You can then show that  . Due to the geometric constraint, the pre-impact velocity must be  . Hence you can show that equation (2) holds.One might realize that with a point-mass model, the matrix  is singular. To derive equation (2), first let  , calculate  and then take the limit  . (In this case, you can just substitute  into equation (2).)In the Demonstration, the problem is simplified to a point-mass model, and the parameters are made dimensionless, so  ,  . Substituting these parameters into the equation of motion (equation (1)), we have:Substituting the same set of parameters into the collision equation (equation (2)), it simplifies to: On a downhill slope, the rimless wheel gains further kinetic energy from gravitational potential energy, but a proportion of the total kinetic energy is lost at the start of the next step due to the plastic collision. Steady-state walking is achieved when the potential energy gain equals the impact loss.  "Rimless Wheel Locomotion" from The Wolfram Demonstrations Project http://demonstrations.wolfram.com/RimlessWheelLocomotion/ Contributed by: Te-yuan Chyou |
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