introduction.md

Symbolic derivation - SymPy

"SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python."

Symbolically derived Equations of Motion

While there is an abundance of physics engines for simulating multibody dynamics (Mujoco, Drake, Dart, ...) it is still often useful to have analytically derived equations of motion. During your studies you might have come across Lagrangian mechanics, specifically Lagrange's equations of the second kind $$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \mathbf{\dot{q}}}\right) - \frac{\partial L}{\partial \mathbf{q}} = \tau ,$$ where $\mathbf{q}$ are generalized coordinates, $\mathbf{\tau}$ generalized torques, and $L$ the Lagrangian which is composed of the system's kinetic $T$ and potential $V$ energy: $L = T-V$. If you have ever been tasked with derving these equations, even for very simple systems, you know that taking the partial derivates by hand is a time-consuming process with many opportunities to make a mistake. For this reason, today we will derive them using SymPy in a slighly modified form $$\frac{\partial^2 T}{\partial \mathbf{\dot{q}}^2}\mathbf{\ddot{q}} + \frac{\partial^2 T}{\partial \mathbf{\dot{q}} \partial \mathbf{q}} \mathbf{\dot{q}} - \frac{\partial T}{\partial \mathbf{q}} + \frac{\partial V}{\partial \mathbf{q}} = \mathbf{\tau} ,$$

which also allows us to simply express $\mathbf{\ddot{q}} = \mathbf{f}(\mathbf{q},\dot{\mathbf{q}})$.

Example: cart-pole system

                              
                  \
                  /\
                 /  \
                /    \
               l      \
              /       >< m_p
             /       // 
            /       //
          \/       //
           \      //
            \    //
  |   |------\--//------|
  |   |       \//\      |
  g   |  m_c  ( ) theta |
  |   |        |_/      |
  V   |========|========|
_________OOO___|___OOO_________
               |
//|-----s----->|

Generalized coordinates

$$ \mathbf{q} = \begin{bmatrix} s & \theta \end{bmatrix}^\top . $$

Kinetic energy

$$ T = \frac{1}{2}\left( m_c \dot{s}^2 + m_p \left( \dot{x}_p^2 + \dot{y}_p^2 \right) \right) , $$

where

$$ \begin{aligned} \dot{x}_p &= \dot{s} + l \dot{\theta} \cos(\theta) \\ \dot{y}_p &= l \dot{\theta} \sin{\theta} . \end{aligned} $$

Potential energy

$$ V = g m_p (1 - l \cos(\theta) ) $$