Editing Nonholonomic system (section)

==Examples==
===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.

===Rolling sphere===
This example is an extension of the 'rolling wheel' problem considered above.

Consider a three-dimensional orthogonal Cartesian coordinate frame, for example, a level table top with a point marked on it for the origin, and the ''x'' and ''y'' axes laid out with pencil lines.  Take a sphere of unit radius, for example, a ping-pong ball, and mark one point ''B'' in blue.  Corresponding to this point is a diameter of the sphere, and the plane orthogonal to this diameter positioned at the center ''C'' of the sphere defines a great circle called the equator associated with point ''B''.  On this equator, select another point ''R'' and mark it in red.  Position the sphere on the ''z''&nbsp;=&nbsp;0 plane such that the point ''B'' is coincident with the origin, ''C'' is located at ''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0, ''z''&nbsp;=&nbsp;1, and ''R'' is located at ''x''&nbsp;=&nbsp;1, ''y''&nbsp;=&nbsp;0, and ''z''&nbsp;=&nbsp;1, i.e. ''R'' extends in the direction of the positive ''x'' axis.  This is the initial or reference orientation of the sphere.

The sphere may now be rolled along any continuous closed path in the ''z''&nbsp;=&nbsp;0 plane, not necessarily a simply connected path, in such a way that it neither slips nor twists, so that ''C'' returns to ''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0, ''z''&nbsp;=&nbsp;1.  In general, point ''B'' is no longer coincident with the origin, and point ''R'' no longer extends along the positive ''x'' axis.  In fact, by selection of a suitable path, the sphere may be re-oriented from the initial orientation to any possible orientation of the sphere with ''C'' located at ''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0, ''z''&nbsp;=&nbsp;1.<ref>The Nonholonomy of the Rolling Sphere, Brody Dylan Johnson, The American Mathematical Monthly, June–July 2007, vol. 114, pp.&nbsp;500–508.</ref>  The system is therefore nonholonomic.  The anholonomy may be represented by the doubly unique [[quaternion]] (''q'' and −''q'') which, when applied to the points that represent the sphere, carries points ''B'' and ''R'' to their new positions.

===Foucault pendulum===
An additional example of a nonholonomic system is the [[Foucault pendulum]].  In the local coordinate frame the pendulum is swinging in a vertical plane with a particular orientation with respect to geographic north at the outset of the path.  The implicit trajectory of the system is the line of latitude on the Earth where the pendulum is located.  Even though the pendulum is stationary in the Earth frame, it is moving in a frame referred to the Sun and rotating in synchrony with the Earth's rate of revolution, so that the only apparent motion of the pendulum plane is that caused by the rotation of the Earth.  This latter frame is considered to be an inertial reference frame, although it too is non-inertial in more subtle ways.  The Earth frame is well known to be non-inertial, a fact made perceivable by the apparent presence of [[centrifugal force]]s and [[Coriolis effect|Coriolis]] forces.

Motion along the line of latitude is parameterized by the passage of time, and the Foucault pendulum's plane of oscillation appears to rotate about the local vertical axis as time passes.  The angle of rotation of this plane at a time ''t'' with respect to the initial orientation is the anholonomy of the system.  The anholonomy induced by a complete circuit of latitude is proportional to the [[solid angle]] subtended by that circle of latitude.  The path need not be constrained to latitude circles.  For example, the pendulum might be mounted in an airplane.  The anholonomy is still proportional to the solid angle subtended by the path, which may now be quite irregular.  The Foucault pendulum is a physical example of [[parallel transport]].

===Linear polarized light in an optical fiber===
Take a length of optical fiber, say three meters, and lay it out in an absolutely straight line.  When a vertically polarized beam is introduced at one end, it emerges from the other end, still polarized in the vertical direction.  Mark the top of the fiber with a stripe, corresponding with the orientation of the vertical polarization.

Now, coil the fiber tightly around a cylinder ten centimeters in diameter.  The path of the fiber now describes a [[helix]] which, like the circle, has constant [[curvature]].  The helix also has the interesting property of having constant [[torsion of a curve#Properties|torsion]].  As such the result is a gradual rotation of the fiber about the fiber's axis as the fiber's centerline progresses along the helix.  Correspondingly, the stripe also twists about the axis of the helix.

When linearly polarized light is again introduced at one end, with the orientation of the polarization aligned with the stripe, it will, in general, emerge as linear polarized light aligned not with the stripe, but at some fixed angle to the stripe, dependent upon the length of the fiber, and the pitch and radius of the helix.  This system is also nonholonomic, for we can easily coil the fiber down in a second helix and align the ends, returning the light to its point of origin.  The anholonomy is therefore represented by the deviation of the angle of polarization with each circuit of the fiber.  By suitable adjustment of the parameters, it is clear that any possible angular state can be produced.

===Robotics===
In [[robotics]], nonholonomic has been particularly studied in the scope of [[motion planning]] and [[feedback linearization]] for [[mobile robot]]s.<ref>''Robot Motion Planning and Control'', Jean-Paul Laumond (Ed.), 1998, Lecture Notes in Control and Information Sciences, Volume 229, Springer, {{doi|10.1007/BFb0036069}}.</ref>