M083 · Controlled rolling

Rolling Wheel & Geometric Phase

If a wheel's heading and roll angle both return to their start, is the wheel back where it began? Not necessarily. Roll forward, turn, roll back the same distance, then turn back: the heading and the wheel's own angle both return, but the wheel ends up somewhere else. It can only roll the way it faces, so rolling back after a turn does not retrace the first leg. Doing the turns first sends it the opposite way; with no turning at all, the path closes.

Roll forward, turn, roll back and turn back. The upright three-dimensional wheel returns to its original orientation but ends up shifted sideways. Swap the order of the commands and the shift reverses.

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 and see how the motion changes

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

An external mechanism holds the disk upright. Each roll or turn command lasts one second, and the commanded rates switch abruptly. Ideal point contact lets the disk turn in place while stopped. This kinematic model does not include actuator work or the scrubbing of a finite contact patch.

Cyan follows the chosen command order, and dashed coral follows the other order from the same starting state. The bright trail stops at the current time. The contact point moves across the ground, yet the material point of the wheel touching the ground at that instant has zero velocity.

An exact sequence of four commands shows that the outcome depends on the path taken. The disk does not fall or lean freely, and this is not a bicycle model or a tyre-force simulation. Changing the radius changes how far the wheel turns, but not the prescribed distance along the ground.

The wheel is an ideal upright disk, controlled from outside, that rolls without slipping. The kinematics are exact, and each command is prescribed and lasts one second. Lean dynamics, actuator forces and microslip are not modeled.

Start with the essentials

Focus question: If both angles return to their initial values, must the wheel return too?
One-sentence intuition: Which way the wheel may move depends on its heading. Undoing the same rolling and steering amounts after the heading has changed does not undo the change in 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.

A roll-first loop starting from zero heading

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

Running the commands in the other order reverses this displacement. Each command lasts one second, and no dynamical forces or work are computed.

Common difficulties

The contact limits velocity, not position

Typical misconceptionReturning a control to its start must undo the motion, and slower turning must mean lost energy.

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

Run the experiment

01
Experiment 1
Go to the end of the run. Both angles return to their starting values, but the position does not.
What to observe: Compare the path with the numerical readouts, which update together.
02
Experiment 2
Swap the command order and set steering to zero.
What to observe: Compare the path with the numerical readouts, which update together.
03
Experiment 3
Change the radius while keeping the travel distance fixed. Watch the marker on the wheel turn through a different angle.
What to observe: Compare the path with the numerical readouts, which update together.

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