Motion, Random Walk, and SDE, ProbMAN 3

Probabilistic Robotics Lecture 3

Linearization of E.O.M.

E.O.M means Equation of Motion.

For a physical system, the E.O.M. is usually nonlinear, with the form of $$ \begin{aligned} Let\ \vec \theta = \begin{pmatrix} \theta_1 & \theta_2 \end{pmatrix}^T \\ M(\theta)\ddot{\vec \theta} + C(\theta, \dot{\vec \theta})\dot{\vec \theta} + G(\theta) &= 0 \\ \end{aligned} $$ Where

  • M is the mass matrix
  • C is the Coriolis matrix
  • G is the gravity matrix

The general form can be written as $$ M(q)\ddot{q} + C(q, \dot{q})\dot{q} + G(q) = 0 $$

Whenever we choose generalized coordinates measured from equilibrium points, we can linearize the E.O.M. by assumptions:

  • Small angle, ||q|| « 1
  • Small velocity, ||q’|| « 1

We can throw away higher order term q’, but keep q and q". And also change cos(q) to 1, sin(q) to q.

System subject to noice level 0: Random Walk

1. Discrete time and space

Setting: Particle in 1D. At each time step, the particle moves either left or right with step 1 with equal probability.

$$ \begin{aligned} \Delta p (\Delta k, n) &\dot= \frac{1}{2}(\delta_{\Delta k,1}+\delta_{\Delta k, -1}) \\ p(k, n+1) &= (p * \Delta p)(k, n) \end{aligned} $$

Simply speaking, the probability distribution become more and more like a Gaussian distribution and wider and wider.

2. Continuous time and space

To write in SDE form, $$ \begin{aligned} dx &= dw \\ \end{aligned} $$

Level 1: Ornstein-Uhlenbeck Process

This is a forced machanical system consisting of a spring, mass and damper. $$ \begin{aligned} m\ddot{x} + c\dot{x} + kx &= f(t) \\ \end{aligned} $$ m is mass, c is damping constant, k is spring stiffness, f is external force. This second order scalar equation can be written as two first order equations. We can define x1 = x, x2 = x'. Then $$ \begin{aligned} \begin{pmatrix} dx_1 \\ dx_2 \end{pmatrix} = -\begin{pmatrix} 0 & -1 \\ k/m & c/m \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} dt + \begin{pmatrix} 0 \\ 1/m \end{pmatrix} dw \end{aligned} $$ This is essential saying that $$ \begin{aligned} \frac{dx}{dt} &= \dot x \\ m \frac{d\dot x}{dt} &= -c\dot x - kx + dw \\ \end{aligned} $$

How to get Fokker-Planck Equation from SDE