Editing Nonholonomic system (section)

===Rolling wheel===
A wheel (sometimes visualized as a unicycle or a rolling coin) is a nonholonomic system.

====Layperson's explanation====
Consider the wheel of a bicycle that is parked in a certain place (on the ground).  Initially the [[Valve stem|inflation valve]] is at a certain position on the wheel.  If the bicycle is ridden around, and then parked in ''exactly'' the same place, the valve will almost certainly not be in the same position as before. Its new position depends on the path taken. If the wheel were holonomic, then the valve stem would always end up in the same position as long as the wheel were always rolled back to the same location on the Earth. Clearly, however, this is not the case, so the system is nonholonomic.

====Mathematical explanation====
{{See also|Holonomic constraints#Terminology|Holonomic constraints#Pfaffian form|Holonomic constraints#Universal test for holonomic constraints}}
[[File:Unicycle drawing updat.png|thumb|upright=2|An individual riding a motorized unicycle. The configuration space of the unicycle, and the radius <math>r</math> of the wheel, are marked. The red and blue lines lay on the ground.]]
It is possible to model the wheel mathematically with a system of constraint equations, and then prove that that system is nonholonomic.

First, we define the configuration space. The wheel can change its state in three ways: having a different rotation about its axle, having a different steering angle, and being at a different location. We may say that <math>\phi</math> is the rotation about the axle, <math>\theta</math> is the steering angle relative to the <math>x</math>-axis, and <math>x</math> and <math>y</math> define the spatial position. Thus, the configuration space is:
<math display="block"> \mathbf{u}=\begin{bmatrix}x & y & \theta & \phi \end{bmatrix}^\mathrm{T} </math>

We must now relate these variables to each other. We notice that as the wheel changes its rotation, it changes its position. The change in rotation and position implying velocities must be present, we attempt to relate angular velocity and steering angle to linear velocities by taking simple time-derivatives of the appropriate terms:
<math display="block">\begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix} = \begin{pmatrix}r\dot{\phi} \cos\theta \\ r\dot{\phi}\sin\theta \end{pmatrix}</math>
The velocity in the <math>x</math> direction is equal to the angular velocity times the radius times the cosine of the steering angle, and the <math>y</math> velocity is similar. Now we do some algebraic manipulation to transform the equation to ''Pfaffian form'' so it is possible to test whether it is holonomic, starting with:
<math display="block">\begin{pmatrix}\dot{x}-r\dot{\phi}\cos\theta \\ \dot{y}-r\dot{\phi}\sin\theta\end{pmatrix} = \mathbf{0}</math>

Then, let's separate the variables from their coefficients (left side of equation, derived from above). We also realize that we can multiply all terms by <math>\text{d}t</math> so we end up with only the differentials (right side of equation):
<math display="block">\begin{pmatrix} 1 & 0 & 0 & -r\cos\theta \\ 0 & 1 & 0 & -r\sin\theta \end{pmatrix} \begin{pmatrix} \dot{x} \\ \dot{y} \\ \dot{\theta} \\ \dot{\phi} \end{pmatrix} = \mathbf{0} = \begin{pmatrix} 1 & 0 & 0 & -r\cos\theta \\ 0 & 1 & 0 & -r\sin\theta \end{pmatrix} \begin{pmatrix} \text{d}x \\ \text{d}y \\ \text{d}\theta \\ \text{d}\phi \end{pmatrix}</math>
The right side of the equation is now in ''Pfaffian form'':
<math display="block"> \sum_{s=1}^n A_{rs}du_s = 0;\; r = 1, 2 </math>

We now use the [[Holonomic constraints#Universal test for holonomic constraints|universal test for holonomic constraints]]. If this system were holonomic, we might have to do up to eight tests. However, we can use mathematical intuition to try our best to prove that the system is nonholonomic on the ''first'' test. Considering the test equation is:
<math display="block">A_\gamma\left(\frac{\partial A_\beta}{\partial u_\alpha}-\frac{\partial A_\alpha}{\partial u_\beta}\right)+A_\beta \left(\frac{\partial A_\alpha}{\partial u_\gamma}-\frac{\partial A_\gamma}{\partial u_\alpha}\right)+A_\alpha\left(\frac{\partial A_\gamma}{\partial u_\beta}-\frac{\partial A_\beta}{\partial u_\gamma}\right)=0</math>

we can see that if any of the terms <math>A_\alpha</math>, <math>A_\beta</math>, or <math>A_\gamma</math> were zero, then that part of the test equation would be trivial to solve and would be equal to zero. Therefore, it is often best practice to have the first test equation have as many non-zero terms as possible to maximize the chance of the sum of them not equaling zero. Therefore, we choose:
:<math>A_\alpha=1</math>
:<math>A_\beta=0</math>
:<math>A_\gamma =-r \cos \theta </math>
:<math>u_\alpha=dx</math>
:<math>u_\beta=d\theta</math>
:<math>u_\gamma=d\phi</math>

We substitute into our test equation:
<math display="block">-r \cos \theta \left(\frac{\partial}{\partial x}(0)-\frac{\partial}{\partial \theta}(1)\right) + 0 \left(\frac{\partial}{\partial \phi}(1)-\frac{\partial}{\partial x}(-r \cos \theta)\right) + 1\left(\frac{\partial}{\partial \theta}(-r \cos \theta) - \frac{\partial}{\partial \phi}(0)\right) = 0</math>

and simplify:
<math display="block">r\sin\theta=0</math>

We can easily see that this system, as described, is nonholonomic, because <math>\sin\theta</math> is not always equal to zero.

=====Additional conclusions=====
We have completed our proof that the system is nonholonomic, but our test equation gave us some insights about whether the system, if further constrained, could be holonomic. Many times test equations will return a result like <math>-1=0</math> implying the system could never be constrained to be holonomic without radically altering the system, but in our result we can see that <math>r\sin\theta</math> ''can'' be equal to zero, in two different ways:
* <math>r</math>, the radius of the wheel, can be zero. This is not helpful as the system in practice would lose all of its degrees of freedom.
* <math>\sin\theta</math> can be zero by setting <math>\theta</math> equal to zero. This implies that if the wheel were not allowed to turn and had to move only in a straight line at all times, it would be a holonomic system.
There is one thing that we have not yet considered however, that to find all such modifications for a system, one must perform ''all'' eight test equations (four from each constraint equation) and collect ''all'' the failures to gather all requirements to make the system holonomic, if possible. In this system, out of the seven additional test equations, an additional case presents itself:
<math display="block">-r\cos\theta=0</math>
This does not pose much difficulty, however, as adding the equations and dividing by <math>r</math> results in:
<math display="block">\sin\theta -\cos\theta=0</math>
which with some simple algebraic manipulation becomes:
<math display="block">\tan\theta=1</math>

which has the solution <math display="inline">\theta = \frac{\pi}{4}+n\pi;\;n\in\mathbb{Z}\;</math> (from <math>\theta = \arctan(1)</math>).

Refer back to the [[Nonholonomic system#Layman's explanation|layman's explanation]] above where it is said, "[The valve stem's] new position depends on the path taken. If the wheel were holonomic, then the valve stem would always end up in the same position as long as the wheel were always rolled back to the same location on the Earth. Clearly, however, this is not the case, so the system is nonholonomic." However it is easy to visualize that if the wheel were only allowed to roll in a perfectly straight line and back, the valve stem ''would'' end up in the same position! In fact, moving parallel to the given angle of <math>\pi/4</math> is not actually necessary in the real world as the orientation of the coordinate system itself is arbitrary. The system can become holonomic if the wheel moves only in a straight line at any fixed angle relative to a given reference. Thus, we have not only proved that the original system is nonholonomic, but we also were able to find a restriction that can be added to the system to make it holonomic.

However, there is something mathematically special about the restriction of <math>\theta = \arctan(1)</math> for the system to make it holonomic, as <math>\theta = \arctan(y/x)</math> in a Cartesian grid. Combining the two equations and eliminating <math>\theta</math>, we indeed see that <math>y = x</math> and therefore one of those two coordinates is completely redundant. We already know that the steering angle is a constant, so that means the holonomic system here needs to only have a configuration space of <math> \mathbf{u}=\begin{bmatrix} x & \phi \end{bmatrix}^\mathrm{T} </math>. As discussed [[Holonomic constraints#Configuration spaces of two or one variable|here]], a system that is modellable by a Pfaffian constraint must be holonomic if the configuration space consists of two or fewer variables. By modifying our original system to restrict it to have only two degrees of freedom and thus requiring only two variables to be described, and assuming it can be described in Pfaffian form (which in this example we already know is true), we are assured that it is holonomic.