=====================================================================
Potential Issues/Advanced Topics/Future Features in Physics/Mechanics
=====================================================================
This document will describe some of the more advanced functionality that this
module offers but which is not part of the "official" interface. Here, some of
the features that will be implemented in the future will also be covered, along
with unanswered questions about proper functionality. Also, common problems
will be discussed, along with some solutions.
Common Issues
=============
Here issues with numerically integrating code, choice of ``dynamicsymbols`` for
coordinate and speed representation, printing, differentiating, and
substitution will occur.
Numerically Integrating Code
----------------------------
See Future Features: Code Output
Differentiating
---------------
Differentiation of very large expressions can take some time in SymPy; it is
possible for large expressions to take minutes for the derivative to be
evaluated. This will most commonly come up in linearization.
Choice of Coordinates and Speeds
--------------------------------
The Kane object is set up with the assumption that the generalized speeds are
not the same symbol as the time derivatives of the generalized coordinates.
This isn't to say that they can't be the same, just that they have to have a
different symbol. If you did this: ::
>> KM.coords([q1, q2, q3])
>> KM.speeds([q1d, q2d, q3d])
Your code would not work. Currently, kinematic differential equations are
required to be provided. It is at this point that we hope the user will
discover they should not attempt the behavior shown in the code above.
This behavior might not be true for other methods of forming the equations of
motion though.
Printing
--------
The default printing options are to use sorting for ``Vector`` and ``Dyad``
measure numbers, and have unsorted output from the ``mprint``, ``mpprint``, and
``mlatex`` functions. If you are printing something large, please use one of
those functions, as the sorting can increase printing time from seconds to
minutes.
Differentiating
---------------
Differentiation of very large expressions can take some time in SymPy; it is
possible for large expressions to take minutes for the derivative to be
evaluated. This will most commonly come up in linearization.
Substitution
------------
There are two common issues with substitution in mechanics:
- When subbing in expressions for ``dynamicsymbols``, sympy's normal ``subs``
will substitute in for derivatives of the dynamic symbol as well: ::
>>> from sympy.physics.mechanics import dynamicsymbols
>>> x = dynamicsymbols('x')
>>> expr = x.diff() + x
>>> sub_dict = {x: 1}
>>> expr.subs(sub_dict)
Derivative(1, t) + 1
In this case, ``x`` was replaced with 1 inside the ``Derivative`` as well,
which is undesired.
- Substitution into large expressions can be slow.
If your substitution is simple (direct replacement of expressions with other
expressions, such as when evaluating at an operating point) it is recommended
to use the provided ``msubs`` function, as it is significantly faster, and
handles the derivative issue appropriately: ::
>>> from sympy.physics.mechanics import msubs
>>> msubs(expr, sub_dict)
Derivative(x(t), t) + 1
Linearization
-------------
Currently, the linearization methods don't support cases where there are
non-coordinate, non-speed dynamic symbols outside of the "dynamic equations".
It also does not support cases where time derivatives of these types of dynamic
symbols show up. This means if you have kinematic differential equations which
have a non-coordinate, non-speed dynamic symbol, it will not work. It also
means if you have defined a system parameter (say a length or distance or mass)
as a dynamic symbol, its time derivative is likely to show up in the dynamic
equations, and this will prevent linearization.
Acceleration of Points
----------------------
At a minimum, points need to have their velocities defined, as the acceleration
can be calculated by taking the time derivative of the velocity in the same
frame. If the 1 point or 2 point theorems were used to compute the velocity,
the time derivative of the velocity expression will most likely be more complex
than if you were to use the acceleration level 1 point and 2 point theorems.
Using the acceleration level methods can result in shorted expressions at this
point, which will result in shorter expressions later (such as when forming
Kane's equations).
Advanced Interfaces
===================
Advanced Functionality
----------------------
Remember that the ``Kane`` object supports bodies which have time-varying
masses and inertias, although this functionality isn't completely compatible
with the linearization method.
Operators were discussed earlier as a potential way to do mathematical
operations on ``Vector`` and ``Dyad`` objects. The majority of the code in this
module is actually coded with them, as it can (subjectively) result in cleaner,
shorter, more readable code. If using this interface in your code, remember to
take care and use parentheses; the default order of operations in Python
results in addition occurring before some of the vector products, so use
parentheses liberally.
Future Features
===============
This will cover the planned features to be added to this submodule.
Code Output
-----------
A function for generating code output for numerical integration is the highest
priority feature to implement next. There are a number of considerations here.
Code output for C (using the GSL libraries), Fortran 90 (using LSODA), MATLAB,
and SciPy is the goal. Things to be considered include: use of ``cse`` on large
expressions for MATLAB and SciPy, which are interpretive. It is currently unclear
whether compiled languages will benefit from common subexpression elimination,
especially considering that it is a common part of compiler optimization, and
there can be a significant time penalty when calling ``cse``.
Care needs to be taken when constructing the strings for these expressions, as
well as handling of input parameters, and other dynamic symbols. How to deal
with output quantities when integrating also needs to be decided, with the
potential for multiple options being considered.