Note
You can download this example as a Python script: :jupyter-download:script:`sympy` or Jupyter notebook: :jupyter-download📓`sympy`.
SymPy is an open source, collaboratively developed computer algebra system (CAS) written in Python. We will use it extensively for manipulating symbolic expressions and equations. All of the mathematics needed to formulate the equations of motion of multibody systems can be done with pencil and paper, but the book keeping becomes extremely tedious and error prone for systems with even a small number of bodies. SymPy allows us to let a computer handle the tedious aspects, e.g. differentiation, solving linear systems of equations, and reduce the errors one would encounter with pencil and paper. This chapter introduces SymPy and the primary core SymPy features needed we will be using.
We will consistently import SymPy as follows:
.. jupyter-execute:: import sympy as sm
Since SymPy lets you work with mathematical symbols it's nice to view SymPy
objects in a format that is similar to the math in a textbook. Executing
:external:py:func:`~sympy.interactive.printing.init_printing` at the beginning
of your Jupyter Notebook will ensure that SymPy objects render as typeset
mathematics. I use the use_latex='mathjax' argument here to disable math
image generation.
.. jupyter-execute:: sm.init_printing(use_latex='mathjax')
Symbols are created with the :external:py:func:`~sympy.core.symbol.symbols` function. A symbol a is created like so:
.. jupyter-execute::
a = sm.symbols('a')
a
This symbol object is of the :external:py:class:`~sympy.core.symbol.Symbol` type:
.. jupyter-execute:: type(a)
Multiple symbols can be created with one call to symbols() and SymPy
recognizes common Greek symbols by spelled out name.
.. jupyter-execute::
b, t, omega = sm.symbols('b, t, omega')
b, t, omega
Note that the argument provided to symbols() does not need to match the
Python variable name it is assigned to. Using more verbose Python variable
names may make code easier to read and understand, especially if there are many
mathematical variables that you forget the meaning of. Note that the subscripts
are recognized too.
.. jupyter-execute::
pivot_angle, w2 = sm.symbols('alpha1, omega2')
pivot_angle, w2
We will also work with undefined mathematical functions in addition to symbols. These will play an important role in setting up differential equations, where we typically don't know the function, but only its derivative(s). You can create arbitrary functions of variables. In this case, we make a function of t. First create the function name:
.. jupyter-execute::
f = sm.Function('f')
f
This is of a type :external:py:class:`~sympy.core.function.UndefinedFunction`.
.. jupyter-execute:: type(f)
Now we can create functions of one or more variables like so:
.. jupyter-execute:: f(t)
Due to SymPy's internal implementations, the type of this object is not defined as expected:
.. jupyter-execute:: type(f(t))
so be aware of that.
The same UndefinedFunction can be used to create multivariate functions:
.. jupyter-execute:: f(a, b, omega, t)
Now that we have mathematical variables and functions available, they can be
used to construct mathematical expressions. The most basic way to construct
expressions is with the standard Python operators +, -, *, /,
and **. For example:
.. jupyter-execute:: expr1 = a + b/omega**2 expr1
An expression will have the type Add, Mul, or Pow:
.. jupyter-execute:: type(expr1)
This is because SymPy stores expressions as a tree. You can inspect this internal representation by using the :external:py:func:`~sympy.printing.repr.srepr` function:
.. jupyter-execute:: sm.srepr(expr1)
This is a visual representation of the tree:
.. graphviz::
:align: center
digraph{
# Graph style
"ordering"="out"
"rankdir"="TD"
#########
# Nodes #
#########
"Add(Symbol('a'), Mul(Symbol('b'), Pow(Symbol('omega'), Integer(-2))))_()" ["color"="black", "label"="Add", "shape"="ellipse"];
"Symbol('a')_(0,)" ["color"="black", "label"="a", "shape"="ellipse"];
"Mul(Symbol('b'), Pow(Symbol('omega'), Integer(-2)))_(1,)" ["color"="black", "label"="Mul", "shape"="ellipse"];
"Symbol('b')_(1, 0)" ["color"="black", "label"="b", "shape"="ellipse"];
"Pow(Symbol('omega'), Integer(-2))_(1, 1)" ["color"="black", "label"="Pow", "shape"="ellipse"];
"Symbol('omega')_(1, 1, 0)" ["color"="black", "label"="omega", "shape"="ellipse"];
"Integer(-2)_(1, 1, 1)" ["color"="black", "label"="-2", "shape"="ellipse"];
#########
# Edges #
#########
"Add(Symbol('a'), Mul(Symbol('b'), Pow(Symbol('omega'), Integer(-2))))_()" -> "Symbol('a')_(0,)";
"Add(Symbol('a'), Mul(Symbol('b'), Pow(Symbol('omega'), Integer(-2))))_()" -> "Mul(Symbol('b'), Pow(Symbol('omega'), Integer(-2)))_(1,)";
"Mul(Symbol('b'), Pow(Symbol('omega'), Integer(-2)))_(1,)" -> "Symbol('b')_(1, 0)";
"Mul(Symbol('b'), Pow(Symbol('omega'), Integer(-2)))_(1,)" -> "Pow(Symbol('omega'), Integer(-2))_(1, 1)";
"Pow(Symbol('omega'), Integer(-2))_(1, 1)" -> "Symbol('omega')_(1, 1, 0)";
"Pow(Symbol('omega'), Integer(-2))_(1, 1)" -> "Integer(-2)_(1, 1, 1)";
}
This representation is SymPy's "true" representation of the symbolic
expression. SymPy can display this expression in many other representations,
for example the typeset mathematical expression you have already seen is one of
those representations. This is important to know, because sometimes the
expressions are displayed to you in a way that may be confusing and checking
the srepr() version can help clear up misunderstandings. See the
manipulation section of the SymPy tutorial for more information on this.
Undefined functions can also be used in expressions:
.. jupyter-execute:: expr2 = f(t) + a*omega expr2
SymPy has a large number of elementary and special functions. See the SymPy documentation on functions for more information. For example, here is an expression that uses :external:py:class:`~sympy.functions.elementary.trigonometric.sin`, :external:py:class:`~sympy.functions.elementary.complexes.Abs`, and :external:py:func:`~sympy.functions.elementary.miscellaneous.sqrt`:
.. jupyter-execute:: expr3 = a*sm.sin(omega) + sm.Abs(f(t))/sm.sqrt(b) expr3
Note that Python integers and floats can also be used when constructing expressions:
.. jupyter-execute:: expr4 = 5*sm.sin(12) + sm.Abs(-1001)/sm.sqrt(89.2) expr4
Warning
Be careful with numbers, as SymPy may not intepret them as expected. For example:
.. jupyter-execute:: 1/2*a
Python does the division before it is multiplied by a, thus a floating
point value is created. To fix this you can use the S() function to
"sympify" numbers:
.. jupyter-execute:: sm.S(1)/2*a
Lastly, an expression of t:
.. jupyter-execute:: expr5 = t*sm.sin(omega*f(t)) + f(t)/sm.sqrt(t) expr5
We introduced the srepr() form of SymPy expressions above and mentioned
that expressions can have different representations. For the following
srepr() form:
.. jupyter-execute:: sm.srepr(expr3)
There is also a standard representation accessed with the repr() function:
.. jupyter-execute:: repr(expr3)
This form matches what you typically would type to create the function and it
returns a string. print() will display that string:
.. jupyter-execute:: print(expr3)
SymPy also has a "pretty printer" (:external:py:func:`pprint() <sympy.printing.pretty.pretty.pretty_print>`) that makes use of unicode symbols to provide a form that more closely resembles typeset math:
.. jupyter-execute:: sm.pprint(expr3)
Lastly, the following lines show how SymPy expressions can be represented as \LaTeX code. The double backslashes are present because double backslashes represent the escape character in Python strings.
.. jupyter-execute:: sm.latex(expr3)
.. jupyter-execute:: print(sm.latex(expr3))
Warning
When you are working with long expressions, which will be the case in this course, there is no need to print them to the screen. In fact, printing them to the screen make take a long time and fill your entire notebook with an unreadable mess.
One of the most tedious tasks in formulating equations of motion is the differentiation of complex trigonometric expressions. SymPy can calculate derivatives effortlessly. The :external:py:func:`~sympy.core.function.diff` SymPy function takes an undefined function or an expression and differentiates it with respect to the symbol provided as the second argument:
.. jupyter-execute:: sm.diff(f(t), t)
All functions and expressions also have a .diff() method which can be used
like so (many SymPy functions exist as standalone functions and methods):
.. jupyter-execute:: f(t).diff(t)
expr3 is a more complicated expression:
.. jupyter-execute:: expr3
It can be differentiated, for example, with respect to b:
.. jupyter-execute:: expr3.diff(b)
You can also calculate partial derivatives with respect to successive variables, as in the following:
\frac{\partial^2 h(a, \omega, t, b)}{\partial t \partial b}
.. jupyter-execute:: expr3.diff(b, t)
Note that the answer includes real and imaginary components and the signum function.
Warning
SymPy assumes all symbols are complex valued unless told otherwise. You can attach assumptions to symbols to force them to be real, positive, negatives, etc. For example, compare these three outputs:
.. jupyter-execute::
h = sm.Function('h')
sm.Abs(h(t)).diff(t)
.. jupyter-execute::
h = sm.Function('h', real=True)
sm.Abs(h(t)).diff(t)
.. jupyter-execute::
h = sm.Function('h', real=True, positive=True)
sm.Abs(h(t)).diff(t)
Sometimes you may need to add assumptions to variables, but in general it will not be necessary.
Lastly, a typical type of derivative you may encounter:
.. jupyter-execute:: expr5
.. jupyter-execute:: expr5.diff(t)
SymPy expressions can be evaluated numerically in several ways. The :external:py:meth:`~sympy.core.basic.Basic.xreplace` method allows substitution of exact symbols or sub-expressions. First create a dictionary that maps symbols, functions or sub-expressions to the replacements:
.. jupyter-execute::
repl = {omega: sm.pi/4, a: 2, f(t): -12, b: 25}
This dictionary can then be passed to .xreplace():
.. jupyter-execute:: expr3.xreplace(repl)
Notice how the square root and fraction do not automatically reduce to their decimal equivalents. To do so, you must use the :external:py:meth:`~sympy.core.evalf.EvalfMixin.evalf` method. This method will evaluate an expression to an arbitrary number of decimal points. You provide the number of decimal places and the substitution dictionary to evaluate:
.. jupyter-execute:: expr3.evalf(n=31, subs=repl)
.. jupyter-execute:: type(expr3.evalf(n=31, subs=repl))
Note that this is a SymPy :external:py:class:`~sympy.core.numbers.Float` object, which is a special object that can have an arbitrary number of decimal places, for example here is the expression evaluated to 300 decimal places:
.. jupyter-execute:: expr3.evalf(n=300, subs=repl)
To convert this to Python floating point number, use float():
.. jupyter-execute:: float(expr3.evalf(n=300, subs=repl))
.. jupyter-execute:: type(float(expr3.evalf(n=300, subs=repl)))
This value is a machine precision floating point value and can be used with standard Python functions that operating on floating point numbers.
To obtain machine precisions floating point numbers directly, it is better to use the :external:py:func:`~sympy.utilities.lambdify.lambdify` function to convert the expression into a Python function:
.. jupyter-execute:: eval_expr3 = sm.lambdify((omega, a, f(t), b), expr3)
.. jupyter-execute:: help(eval_expr3)
Now you have a function that operates on and returns floating point values:
.. jupyter-execute:: eval_expr3(3.14/4, 2, -12, 25)
.. jupyter-execute:: type(eval_expr3(3.14/4, 2, -12, 25))
This distinction between SymPy Float objects and regular Python and NumPy
float objects is important. In this case, the Python float and the NumPy
float are equivalent. The later will compute much faster because arbitrary
precision is not required.
Note
In these materials, you will almost always want to convert SymPy expressions
into machine precision floating point numbers, so use lambdify() almost
exclusively.
SymPy supports matrices of expressions and linear algebra. Many of the operations needed in multibody dynamics are more succinctly formulated with matrices and linear algebra. Matrices can be created by passing nested lists to the :external:py:class:`Matrix() <sympy.matrices.dense.MutableDenseMatrix>` object. For example:
.. jupyter-execute:: mat1 = sm.Matrix([[a, 2*a], [b/omega, f(t)]]) mat1
.. jupyter-execute:: mat2 = sm.Matrix([[1, 2], [3, 4]]) mat2
All matrices are two dimensional and the number of rows and columns, in that
order, are stored in the .shape attribute.
.. jupyter-execute:: mat1.shape
Individual elements of the matrix can be extracted with the bracket notation taking the row and column indices (remember Python indexes from 0):
.. jupyter-execute:: mat1[0, 1]
The slice notation can extract rows or columns:
.. jupyter-execute:: mat1[0, 0:2]
.. jupyter-execute:: mat1[0:2, 1]
Matrix algebra can be performed. Matrices can be added:
.. jupyter-execute:: mat1 + mat2
Both the * and the @ operator perform matrix multiplication:
.. jupyter-execute:: mat1*mat2
.. jupyter-execute:: mat1@mat2
Element-by-element multiplication requires the :external:py:func:`~sympy.matrices.expressions.hadamard.hadamard_product` function:
.. jupyter-execute:: sm.hadamard_product(mat1, mat2)
Differentiation operates on each element of the matrix:
.. jupyter-execute:: mat3 = sm.Matrix([expr1, expr2, expr3, expr4, expr5]) mat3
.. jupyter-execute:: mat3.diff(a)
.. jupyter-execute:: mat3.diff(t)
The Jacobian matrix of vector (column matrix) can be formed with the :external:py:meth:`~sympy.matrices.matrices.DenseMatrix.jacobian` method. This calculates the partial derivatives of each element in the vector with respect to a vector (or sequence) of variables.
.. jupyter-execute:: mat4 = sm.Matrix([a, b, omega, t]) mat4
.. jupyter-execute:: mat3.jacobian(mat4)
You'll need to solve linear systems of equations often in this course. SymPy offers a number of ways to do this, but the best way to do so if you know a set of equations are linear in specific variables is the method described below. First, you should know you have equations of this form:
a_{11} x_1 + a_{12} x_2 + \ldots + a_{1n} x_n + b_1 = 0 \\
a_{21} x_1 + a_{22} x_2 + \ldots + a_{2n} x_n + b_2 = 0 \\
\ldots \\
a_{n1} x_1 + a_{n2} x_2 + \ldots + a_{nn} x_n + b_n = 0
These equations can be put into matrix form:
\mathbf{A}\bar{x} = \bar{b}
where:
\mathbf{A} =
\begin{bmatrix}
a_{11} & a_{12} & \ldots & a_{1n} \\
a_{21} & a_{22} & \ldots & a_{2n} \\
\ldots & \ldots & \ldots & \ldots \\
a_{n1} & a_{n2} & \ldots & a_{nn}
\end{bmatrix}
\bar{x} =
\begin{bmatrix}
x_1 \\
x_2 \\
\ldots \\
x_n
\end{bmatrix}
\bar{b} =
\begin{bmatrix}
-b_1 \\
-b_2 \\
\ldots \\
-b_n
\end{bmatrix}
Finally, \bar{x} is found with matrix inversion (if the matrix is invertible):
\bar{x} = \mathbf{A}^{-1}\bar{b}
Taking the inverse is not computationally efficient, so some form of Gaussian elmination should be used to solve the system.
To solve with SymPy, start with a column matrix of linear expressions:
.. jupyter-execute::
a1, a2 = sm.symbols('a1, a2')
exprs = sm.Matrix([
[a1*sm.sin(f(t))*sm.cos(2*f(t)) + a2 + omega/sm.log(f(t), t) + 100],
[a1*omega**2 + f(t)*a2 + omega + f(t)**3],
])
exprs
Since we know these two expressions are linear in the a_1 and
a_2 variables, the partial derivatives with respect to those two
variables will return the linear coefficients. The \mathbf{A} matrix
can be formed in one step with the .jacobian() method:
.. jupyter-execute:: A = exprs.jacobian([a1, a2]) A
The \bar{b} vector can be formed by setting a=b=0, leaving the terms that are not linear in a_1 and a_2.
.. jupyter-execute::
b = -exprs.xreplace({a1: 0, a2:0})
b
Lastly, the LUsolve() method performs Gaussian-Elimination to solve the
system:
.. jupyter-execute:: A.LUsolve(b)
The above result from :external:py:meth:`~sympy.matrices.matrices.MutableDenseMatrix.LUsolve` is a bit complicated. SymPy has some functionality for automatically simplifying symbolic expressions. The function :external:py:func:`~sympy.simplify.simplify.simplify` will attempt to find a simpler version:
.. jupyter-execute:: sm.simplify(A.LUsolve(b))
But you'll have the best luck at simplifying if you use specific functions that target what type of expression you may have. The :external:py:func:`~sympy.simplify.trigsimp.trigsimp` function only attempts trigonometric simplifications, for example:
.. jupyter-execute:: sm.trigsimp(sm.cos(omega)**2 + sm.sin(omega)**2)
Warning
Only attempt simplification on expressions that are several lines of text. Larger expressions become increasingly computationally intensive to simplify and there is generally no need to do so in these materials.
As mentioned earlier, SymPy represents expressions as graphs (trees). Symbolic expressions can also be represented as directed acyclic graphs that contain only one node for each unique expression (unlike SymPy's trees which may have repeated expressions in nodes). These unique expressions, or "common sub-expressions", can be found with the :external:py:func:`~sympy.simplify.cse_main.cse` function. This function will provide a simpler form of the equations that minimizes the number of operations to compute the answer.
.. jupyter-execute:: substitutions, simplified = sm.cse(A.LUsolve(b))
The substitutions variable contains a list of tuples, where each tuple has
a new intermediate variable and the sub-expression it is equal to.
.. jupyter-execute:: substitutions[0]
The :external:py:class:`Eq() <sympy.core.relational.Equality>` class with tuple
unpacking (*) can be used to display these tuples as equations:
.. jupyter-execute:: sm.Eq(*substitutions[0])
.. jupyter-execute:: sm.Eq(*substitutions[1])
.. jupyter-execute:: sm.Eq(*substitutions[2])
.. jupyter-execute:: sm.Eq(*substitutions[4])
The simplified variable contains the simplified expression, made up of the
intermediate variables.
.. jupyter-execute:: simplified[0]
This section only scratches the surface of what SymPy can do. The presented concepts are the basic ones needed for this course, but getting more familiar with SymPy and what it can do will help. I recommend doing the SymPy Tutorial. The "Gotchas" section is particularly helpful for common mistakes when using SymPy. The tutorial is part of the SymPy documentation https://docs.sympy.org, where you will find general information on SymPy.
The tutorial is also available on video:
If you want to ask a question about using SymPy (or search to see if someone else has asked your question), you can do so at the following places:
- SymPy mailing list: Ask questions via email.
- SymPy Gitter: Ask questions in a live chat.
- Stackoverflow: Ask and search questions on the most popular coding Q&A website.