M083 · Controlled rolling

Rolling Wheel & Geometric Phase

Roll, turn, roll back and turn back. Watch an upright three-dimensional wheel recover its original orientation while retaining a sideways displacement. Change the command order to reverse the result.

Horizontal displacement $0$

Transverse displacement $0$

Heading change $0$

Wheel rotation $0$

Net position shift $0$

Material contact speed $0$

Geometry & initial state

Change the constraint. Follow the consequence.

Wheel radius

R

0.40\,\mathrm m

Signed distance in each roll command

d

2.00\,\mathrm m

Steering angle per turn

\alpha

90.00\,{}^\circ

Command order: roll first or turn first

b_{\rm order}

0.00\,

Initial heading

\theta_0

0.00\,{}^\circ

The disk is held upright by an external mechanism. Each roll or yaw command lasts one second; command rates switch discontinuously. Ideal point contact permits yaw while stopped. Actuator work and finite contact-patch scrub are outside this kinematic model.

Cyan is the selected command order; dashed coral is the alternative order with the same initial state. The bright completed trail stops at the current time. A ground contact location travels; the instantaneous material point touching it has zero velocity.

An exact four-command construction demonstrates path dependence. It is not a freely falling or leaning disk, a bicycle model, or a tire-force simulation. Radius changes wheel rotation but not the prescribed ground distance.

An upright, externally controlled ideal disk with no slip. Exact kinematics; prescribed one-second commands, no lean dynamics, actuator forces or microslip.

Start with the essentials

Focus question: If both angles return to their initial values, must the wheel return too?
One-sentence intuition: The allowed velocity directions depend on heading. Undoing the scalar controls in a new heading does not undo the position.

Core mathematical model

Pure rolling with prescribed steering

\dot x=R\dot\phi\cos\theta,\qquad\dot y=R\dot\phi\sin\theta,\qquad\mathbf v_{\rm contact}=\mathbf 0

Heading is measured counterclockwise from the horizontal axis, seen from above. Positive wheel spin rolls forward.

Roll-first loop at zero initial heading

\Delta\theta=\Delta\phi=0,\qquad\Delta x=d(1-\cos\alpha),\quad\Delta y=-d\sin\alpha

The alternative command order reverses this displacement. Each command lasts one second; there is no dynamical force or work claim.

Common difficulties

Contact is a velocity rule

Typical misconceptionReturning a control or reducing yaw must undo motion or remove energy.

Better mental modelCompare the two command orders at the end. The allowed velocity depends on the current heading.

Run the experiment

01
Experiment 1
Go to the end: both angles return, but position does not.
What to observe: Compare the synchronized path and numerical readouts.
02
Experiment 2
Swap the command order and set steering to zero.
What to observe: Compare the synchronized path and numerical readouts.
03
Experiment 3
Change radius while keeping travel fixed. Watch the wheel marker rotate by a different angle.
What to observe: Compare the synchronized path and numerical readouts.

Primary theory: Simoes & Colombo · rolling disk · Chang & Perlmutter · Chaplygin sleigh