Ideal 2D Unicycle Kinematics

Question

A particle is connected to a massive wheel by a rigid rod. The wheel can roll without slipping on a horizontal surface. The particle is free to rotate around the centre of the wheel.

I believe the system has two degrees of freedom: The centre of the wheel and the particle each have x- and y- positions, and the wheel has an angle of rotation. The constraints are:

• The wheel is on a horizontal surface, fixing its y-coord
• The wheel cannot slip, so its x-coord is directly proportional to its angle of rotation
• The particle is a fixed distance from the centre of the wheel

Is this correct?

This leaves two generalised coordinates, which I have taken to be the angle of rotation of the wheel and the angle between the rod and the y-axis.

After struggling (and failing) with a Newtonian approach I constructed a Lagrangian for the system and applied the Euler-Lagrange equation(s), using the angles as generalised coordinates. After much algebra out popped two second-order non-linear differential equations.

To my surprise, I could eliminate the wheel rotation completely from one equation, leaving it concerning only the rod angle and its derivatives. What, if at all, is the significance of this?

And finally, I would like to simulate the situation computationally. Is there a general way of simulating rigid constraints acting on rigid bodies/particles, or must one find and solve (numerically) the differential equations governing the system?

Answer 1

A unicycle restricted to move on a plane in three dimensions is an example of a nonholonomic mechanical system, where the nonslipping constraints cannot be integrated.

In addition, it is more interesting to give the rider degrees of freedom in order to allow him to control the unicycle. This more general case, was treated by: Zenkov Bloch and Marsden. In this more general treatment, the unicycle is not restricted to be vertical and the "rider" is modeled as a round disc, whose angle of rotation is used as the control.

The authors write the Lagrangian, explicitely, and it is not difficult to degenerate it to the restricted case of linear motion in the vertical plane, if one wishes.

In their work Zenkov Bloch and Marsden, describe a method for stabilization of the system by means of controlling the rotation angle of the rider.

The equantion of motion of mechanical systems are systems of nonlinear ODE's which rarely have exact solutions. However, a lot can be learned about the physical systems without actually solving the equations. The control rule was deduced in the above article without actually solving the system. Please see for example, the following lecture notes by Darryl Holm.

Answer 2

The significance of the separability of the differential equations is simple:

The difficulty involved in balancing a unicycle in 2D is independent of how fast you are going. (in 3D, you have to balance sideways as well, and only there does the wheel speed comes in handy).

Regarding the simulation, I'm not aware of any free, good, mechanical simulation tools, but solving the differential equations shouldn't be difficult - you could first try to see if there is an analytic solution, and if not use the many available tools for solving ODE's.