elastic_bands_main.rst

Elastic Bands

This is a path planning with Elastic Bands.

Code Link

.. autoclass:: PathPlanning.ElasticBands.elastic_bands.ElasticBands

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: \|q - b\| < \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 \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.

External Repulsion Force

• Purpose: Pushes the path away from obstacles.

• Formula: For node b_i:

  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}
  

  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}.
  

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