---
title: "Rolling Wheel & Geometric Phase Interactive Lab | Sandbox Physics"
description: "Undo every turn. Still end up somewhere else. Can the angles close while position stays shifted?"
canonical_url: "https://sandboxphysics.com/labs/rolling-holonomy/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/rolling-holonomy/"
last_modified: "2026-10-18"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

Physics tutorial

## A closed control loop can leave an open path

**Background** An upright wheel must move in the direction it faces. Rolling and steering are independent controls, but their order changes the resulting position.

**Why it matters** Connect a visible contact rule to the complete trajectory and quantitative account.

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 misconception** Returning a control or reducing yaw must undo motion or remove energy.

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

### Run the experiment

1. 01

   #### Experiment 1

   Go to the end: both angles return, but position does not.

   **What to observe:** Compare the synchronized path and numerical readouts.

2. 02

   #### Experiment 2

   Swap the command order and set steering to zero.

   **What to observe:** Compare the synchronized path and numerical readouts.

3. 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.

Physical conclusion

A velocity constraint need not integrate into a relation among positions and angles. This is one concrete nonholonomic effect.

Primary theory: [Simoes & Colombo · rolling disk](https://arxiv.org/abs/2303.17920) · [Chang & Perlmutter · Chaplygin sleigh](https://arxiv.org/abs/1811.03757)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Measure gravity, split motion into components, add contact forces and collisions, then carry the same laws into orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)