Elastic Bands

This is a path planning with Elastic Bands.

Core Concept

Elastic Band: A dynamically deformable collision-free path initialized by a global planner.
Objective:
- Shorten and smooth the path.
- Maximize obstacle clearance.
- Maintain global path connectivity.

Definition: A local free-space region around configuration \(b\):

\[B(b) = \{ q: \|q - b\| < \rho(b) \},\]

where \(\rho(b)\) is the radius of the bubble.

The elastic band deforms under artificial forces:

Purpose: Reduces path slack and length.
Formula: For node \(b_i\):

\[f_c(b_i) = k_c \left( \frac{b_{i-1} - b_i}{\|b_{i-1} - b_i\|} + \frac{b_{i+1} - b_i}{\|b_{i+1} - b_i\|} \right)\]

where \(k_c\) is the contraction gain.

Purpose: Pushes the path away from obstacles.
Formula: For node \(b_i\):

\[\begin{split}f_r(b_i) = \begin{cases} k_r (\rho_0 - \rho(b_i)) \nabla \rho(b_i) & \text{if } \rho(b_i) < \rho_0, \\ 0 & \text{otherwise}. \end{cases}\end{split}\]

where \(k_r\) is the repulsion gain, \(\rho_0\) is the maximum distance for applying repulsion force, and \(\nabla \rho(b_i)\) is approximated via finite differences:

\[\frac{\partial \rho}{\partial x} \approx \frac{\rho(b_i + h) - \rho(b_i - h)}{2h}.\]

Node Update:

\[b_i^{\text{new}} = b_i^{\text{old}} + \alpha (f_c + f_r),\]

where \(\alpha\) is a step-size parameter, which often proportional to \(\rho(b_i^{\text{old}})\)

2. Overlap Enforcement: - Insert new nodes if adjacent nodes are too far apart - Remove redundant nodes if adjacent nodes are too close