[MLS94] Nonholonomic Behavior in Robotic System

Introduction

No slippage constraints:

Let system configuration $\qb = (x, y, \phi, \theta) \in \mathcal{C}$, we can rewrite the above equations as

which means $\dot{\qb} \in \mathcal{N}({\color{blue}A_q})$.

  • Q1: Is it possible to move between any two points in $\mathcal{C}$ while satisfying the constraints?

  • Q2: Is it possible to find two constraint functions $h_i (q) = 0, i = 1, 2$ such that $\mathcal{N}(A_q) = \mathcal{N}(\nabla_\qb \hb_q)$ ?

    Proof: Suppose such constraint $\hb$ exists. We have

    which is in general impossible.

Now assume the two holonomic constraints are independent and then the null space of $A_q$ has 2 degree-of-freedoms. Let $\fb(\qb)$ and $\gb(\qb)$ be the two basis of the null space, for example

then $\dot{\qb} = \fb(\qb) u_1 + \gb(\qb) u_2$. Here $\fb(\qb)$ and $\gb(\qb)$ are both vector fields.

Vector Field

A vector field on $\mathbb{R}^n$ is a smooth map which assigns to each point $\qb \in \mathbb{R}^n$ a tangent vector $\fb(\qb) \in T_q \mathbb{R}^n$. It can be thought as right-hand sides of differential equations:

$$\dot{\qb} = \fb(\qb)$$

Flow of Vector Field

A flow of vector field represents the solution of the differential equation above. Specifically, $\phi^f_t(\qb)$ represents the state of the differential equation at time $t$ starting from $\qb$ at time $0$.

Obviously, we have

and by the existence and uniqueness theorem of ODE, it satisfies the following group property:

for all $t$ and $s$.


As shown in the disk rolling example, if a system is associated with two vector fields, the direction of motion at a certain point is not unique. Now consider a infinitesimal square motion around $\qb_0$:

Motivated by this calculation, we define the Lie bracket of two vector fields $\fb$ and $\gb$

Lie Bracket

$$[\fb, \gb](\qb) = \frac{\partial \gb}{\partial \qb} \fb(\qb) - \frac{\partial \fb}{\partial \qb}\gb(\qb).$$

$\fb$ and $\gb$ are said to commute if $[\fb, \gb] = 0$.

Example: Lie brackets of linear vector fields $\fb(\qb) = A\qb$ and $\gb(\qb) = B\qb$ is

that is the commutator of the two matrices $A, B$.

Properties of Lie brackets
  1. Skew-symmetry

    $$ [f, g] = -[g, f] $$

  2. Jacobi identity

    $$ [f, [g, h]] + [h, [f, g]] + [g, [h, f]] = 0 $$

  3. Chain rule

    $$ [\alpha f, \beta g] = \alpha \beta [f, g] + \alpha(L_f \beta)g - \beta (L_g \alpha)f, $$

    where $L_f\beta$ and $L_g\alpha$ stand for the Lie derivatives of $\beta$ and $\alpha$ along the vector fields $f$ and $g$ respectively.

Controllability and Frobenius Theorem

Examples of Nonholonomic Systems

Nonholonomic Motion Planning