A wheel moving in free space has the six degrees of freedom of a rigid body. If we constrain it to be upright on a plane and to roll without slipping, how many holonomic and nonholonomic constraints is the wheel subject to?

This is a question from NWUs's course on Foundations of Robot Motion,

According to my understanding, the wheel has 2 non-holonomic constraints:

  1. wheel is rolling and not slipping and
  2. the wheel cannot slide along its axis of rotation.

but I don't understand how many holonomic constraints it would have.

Isn't, the wheel not able to slide along its axis of rotation also a holonomic constraint?

I hope someone can help me with the number of holonomic constraints on the wheel.


1 Answer 1


Holonomic constraints are ones that are based on the position only. And restrict the system to an envelope.

You've identified one stated constraint that is non-holonmic and broken it down into two components as they have constraints related to the velocity.

The remaining constraints in your list are then holonomic.

  • The wheel is upright
  • The wheel is on the plane

You've above decided that not sliding is a non-holonomic constraint. Why are you questioning that? Because it's related to the previous position of the system you're restricting the change in position from one state to the next, thus it's related to the derivative of the position not the value of the position.

If you create a state space representation of the wheel and write the constraints out you may find it more revealing that way such that you can reveal the structure of the constraints and apply the definition of a holonomic constraint to each one.

There's also a good discussion of the difference here: https://physics.stackexchange.com/questions/409951/what-are-holonomic-and-non-holonomic-constraints

  • $\begingroup$ okay..So are my 2 nonholonomic constraints correct? $\endgroup$ Nov 22, 2021 at 13:57
  • $\begingroup$ you said that I identified one velocity constraint and divided it into two. So if that is just 1 nonholonomic constraint what is the other one? $\endgroup$ Nov 22, 2021 at 14:00
  • 1
    $\begingroup$ And thank you so much for making me understand holonomic constraints. It was really very helpful. $\endgroup$ Nov 22, 2021 at 14:01
  • $\begingroup$ For the number of constraints, you are just recomposing it into individual components making more simple constraints, which is likely what your course is looking for. Think of it like writing fractions or equations in the simplest form. $\endgroup$
    – Tully
    Nov 22, 2021 at 16:40
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    $\begingroup$ okay...Got it.Thank you. $\endgroup$ Nov 23, 2021 at 12:40

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