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.

Bubble Representation

• Definition: A local free-space region around configuration $b$:

  $$B(b)=\{q:\Vert q-b\Vert <\rho (b)\},$$

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

Force-Based Deformation

The elastic band deforms under artificial forces:

Internal Contraction Force

• Purpose: Reduces path slack and length.

• Formula: For node $b_i$:

  $$f_c(b_i)=k_c(\frac{b_{i-1}-b_i}{\Vert b_{i-1}-b_i\Vert }+\frac{b_{i+1}-b_i}{\Vert b_{i+1}-b_i\Vert })$$

  where $k_c$ is the contraction gain.

External Repulsion Force

• Purpose: Pushes the path away from obstacles.

• Formula: For node $b_i$:

  $$\begin{array}{r}{f}_r(b_i)=\{\begin{array}{ll}{k}_r({\rho }_0-\rho (b_i))\mathrm{\nabla }\rho (b_i)& \text{if }\rho (b_i)<{\rho }_0,\\ 0& \text{otherwise}.\end{array}\end{array}$$

  where $k_r$ is the repulsion gain, ${\rho }_0$ is the maximum distance for applying repulsion force, and $\mathrm{\nabla }\rho (b_i)$ is approximated via finite differences:

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

Dynamic Path Maintenance

1. 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

References

• Elastic Bands: Connecting Path Planning and Control