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Update the nonholonomic equation of motion chapter #155

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May 11, 2023
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Update the nonholonomic equation of motion chapter #155

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78af74e
Added learning objectives and clarified relation to equations in moti…
moorepants May 11, 2023
b8fdb3c
Added notation for all the linear equations for constraints.
moorepants May 11, 2023
b6aaf4b
Back to a longer simulation.
moorepants May 11, 2023
9384e19
Add cse true to all lambdify calls.
moorepants May 11, 2023
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51 changes: 38 additions & 13 deletions nonholonomic-eom.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -2,8 +2,6 @@
Equations of Motion with Nonholonomic Constraints
=================================================

.. warning:: This page as not yet been updated for the 2022-2023 course.

.. note::

You can download this example as a Python script:
Expand Down Expand Up @@ -42,6 +40,18 @@ Equations of Motion with Nonholonomic Constraints
**kwargs)
me.ReferenceFrame = ReferenceFrame

Learning Objectives
===================

After completing this chapter readers will be able to:

- formulate the :math:`p` dynamical differential equations for a nonholonomic system
- simulate a nonholonomic multibody system
- calculate trajectories of dependent speeds

Introduction
============

In chapters, :ref:`Holonomic Constraints` and :ref:`Nonholonomic Constraints`,
I introduced two types of constraints: holonomic (configuration) constraints
and nonholonomic (motion) constraints. Holonomic constraints are nonlinear
Expand All @@ -52,9 +62,6 @@ of motion first, as they are slightly easier to deal with.
.. [#] They can be linear in the coordinates, but then there is little reason
not to solve for the depedendent coordinates and eliminate them.

Nonholonomic Equations of Motion
================================

Nonholonomic constraint equations are linear in both the independent and
dependent generalized speeds (see Sec. :ref:`Snakeboard`). We have shown that
you can explicitly solve for the dependent generalized speeds :math:`\bar{u}_r`
Expand All @@ -73,7 +80,15 @@ and :math:`u_r` can be solved for as so:
.. math::
:label: eq-dep-speeds-solve

\bar{u}_r = -\mathbf{M}_n(\bar{q}, t)^{-1}\bar{g}_n(\bar{u}_s, \bar{q}, t) \\
\bar{u}_r = -\mathbf{M}_n(\bar{q}, t)^{-1}\bar{g}_n(\bar{u}_s, \bar{q}, t)

which is the same as Eq. :math:numref:`eq-contraint-linear-form-solve` we
originally developed:

.. math::
:label: eq-dep-speeds-repeat

\bar{u}_r = \mathbf{A}_n \bar{u}_s + \bar{b}_n\\

Using Eq. :math:numref:`eq-dep-speeds-solve` equation we can now write our
equations of motion as :math:`n` kinematical differential equations and
Expand All @@ -93,6 +108,11 @@ and these can be written in explicit form:
\dot{\bar{q}} = -\mathbf{M}_k(\bar{q}, t)^{-1} \bar{g}_k(\bar{u}_s, \bar{q}, t) \\
\dot{\bar{u}}_s = -\mathbf{M}_d(\bar{q}, t)^{-1} \bar{g}_d(\bar{u}_s, \bar{q}, t) \\

This leaves us with :math:`n+p` equations of motion, instead of :math:`2n`
equations seen in a holonomic system. Nonholonomic constraints reduce the
number of degrees of freedom and thus fewer dynamical differential equations
are necessary to fully describe the motion.

Snakeboard Equations of Motion
==============================

Expand All @@ -104,6 +124,7 @@ inertia of the three bodies are the same.

.. figure:: figures/motion-snakeboard.svg
:align: center
:width: 80%

Configuration diagram of a planar Snakeboard model.

Expand Down Expand Up @@ -150,7 +171,11 @@ expressions we create.
urd_zero = {udi: 0 for udi in urd}
usd_zero = {udi: 0 for udi in usd}

qd_zero, ur_zero, us_zero, urd_zero, usd_zero
qd_zero, ur_zero, us_zero

.. jupyter-execute::

urd_zero, usd_zero

2. Establish the kinematics
---------------------------
Expand Down Expand Up @@ -477,19 +502,19 @@ coordinates.
We now have :math:`\mathbf{M}_k, \bar{g}_k, \mathbf{M}_d` and :math:`\bar{g}_d`
and can proceed to numerical evaluation.

.. todo:: Also show how Fr and Frs can be formed using An.

Simulate the Snakeboard
=======================

We now move to numerical evaluation for the simulation. First, create a
function that evaluates the matrices of the equations of motion.

.. todo:: lambdify(cse=True) fail for this. Open an issue on SymPy.

.. todo:: sm.Matrix.count_ops() doesn't seem like it exists. Open an issue.

.. jupyter-execute::

eval_kd = sm.lambdify((q, us, p), (Mk, gk, Md, gd))
eval_kd = sm.lambdify((q, us, p), (Mk, gk, Md, gd), cse=True)

Now create a function that evaluates the right hand side of the explicit
ordinary differential equations for use with ``solve_ivp()``.
Expand Down Expand Up @@ -563,7 +588,7 @@ later use in the animation of the motion.

.. jupyter-execute::

t0, tf = 0.0, 10.0
t0, tf = 0.0, 8.0

.. jupyter-execute::

Expand Down Expand Up @@ -638,7 +663,7 @@ the points we want to plot given the generalized coordinates.
for point in [Co, Cr, Co, Ao, Bo, Bl, Br]:
coordinates = coordinates.row_join(point.pos_from(O).to_matrix(N))

eval_point_coords = sm.lambdify((q, p), coordinates)
eval_point_coords = sm.lambdify((q, p), coordinates, cse=True)
eval_point_coords(q0, p_vals)

Now create a plot of the initial configuration:
Expand Down Expand Up @@ -699,7 +724,7 @@ the constraint equations. Here I loop through time to calculate
.. jupyter-execute::

x = sm.Matrix([q1, q2, q3, q4, q5, u3, u4, u5])
eval_ur = sm.lambdify((x, p), ur_sol)
eval_ur = sm.lambdify((x, p), ur_sol, cse=True)

ur_vals = []
for xi in xs:
Expand Down
37 changes: 37 additions & 0 deletions notation.rst
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Expand Up @@ -90,6 +90,8 @@ Angular and Translational Kinematics
Constraints
===========

.. todo:: Add Y_k, kin diff eqs.

:math:`N,M,n,m,p`
:math:`N` coordinates, :math:`M` holonomic constraints, :math:`n`
generalized coordinates and generalized speeds, :math:`m` nonholonomic
Expand All @@ -101,6 +103,21 @@ Constraints
:math:`\bar{f}_n(u_1, \ldots, u_n, q_1, \ldots, q_n, t) = 0 \textrm{ where } \bar{f}_n \in \mathbb{R}^m`, ``fn``
Vector function of :math:`m` nonholonomic constraint equations among the
:math:`n` generalized speeds and generalized coordinates.
:math:`\mathbf{A}_r`
Linear coefficient matrix for :math:`\bar{u}_r` in the nonholonomic
constraint equations.
:math:`\mathbf{A}_s`
Linear coefficient matrix for :math:`\bar{u}_s` in the nonholonomic
constraint equations.
:math:`\bar{b}_{rs}`
Terms not linear in :math:`\bar{u}_s` or :math:`\bar{u}_r` in the
nonholonomic constraint equations.
:math:`\mathbf{A}_n`
Linear coefficient matrix for :math:`\bar{u}_s` in the equation for
:math:`\bar{u}_r=\mathbf{A}_n\bar{u}_s + \bar{b}_n`.
:math:`\bar{b}_n`
Terms not linear in :math:`\bar{u}_s` in the equation for
:math:`\bar{u}_r=\mathbf{A}_n\bar{u}_s + \bar{b}_n`.

Mass Distribution
=================
Expand Down Expand Up @@ -194,6 +211,26 @@ Unconstrained Equations of Motion
:math:`\bar{g}_m`
Terms not linear in :math:`\dot{\bar{x}}` in the equations of motion.

Equations of Motion with Nonholonomic Constraints
=================================================

:math:`\bar{f}_n(\bar{u}_s, \bar{u}_r, \bar{q}, t) = 0`
Nonholonomic constraint equations.
:math:`\mathbf{M}_n=\mathbf{A}_r`
Linear coefficient matrix for :math:`\bar{u}_r` in the nonholonomic
constraint equations.
:math:`\bar{g}_n=\mathbf{A}_s\bar{u}_s+\bar{b}_{rs}`
Terms not linear in :math:`\bar{u}_r` in the nonholonomic constraint
equations.
:math:`\dot{\bar{f}}_n(\dot{\bar{u}}_s, \dot{\bar{u}}_r, \bar{u}_s, \bar{u}_r, \bar{q}, t) = 0`
Time derivative of the nonholonomic constraint equations.
:math:`\mathbf{M}_{nd}`
Linear coefficient matrix for :math:`\dot{\bar{u}}_r` in the time
differentiated nonholonomic constraint equations.
:math:`\bar{g}_{nd}`
Terms not linear in :math:`\dot{\bar{u}}_r` in the time differentiated
nonholonomic constraint equations.

.. |notation-scalar| image:: figures/notation-scalar.svg
:height: 10px

Expand Down