9711

Rimless Wheel Locomotion

The rimless wheel was proposed by McGeer in 1990. It consists of a mass with equally spaced spikes extending outwards, like a wagon wheel without an outer rim. On a downhill slope, with the right mechanical design and under the right conditions, the trajectory of the device will automatically settle into a stable limit cycle without the need of control. The self-stabilizing walking gait is due purely to the interaction between the inertial property of the wheel and the force of gravity.
You can change the number of spikes and the slope using the sliders, and select the initial state of the wheel on the state-space plot. Under the right conditions, the motion trajectory converges to a limit cycle. Click the "run" button to see how the wheel walks. For some bad choices of parameters and initial conditions, the rimless wheel slows down or cannot take even one step.

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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:
, (1)
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:
, (2)
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.
Defining as the center of mass of the system, the velocity of the next stance foot is given by:
.
Then the generalized moment of the system in and the constraint can be written as:
,
,
where is the impulsive force due to the plastic collision and
.
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.
Reference:
T. McGeer, "Passive Dynamic Walking," Intern. J. Robot. Res. 9(2), 1990 pp. 62–82.
T. McGeer, "Dynamics and Control of Bipedal Locomotion," J. Theor. Biol. 163(3), 1993, pp. 277–314.
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