Artificial Potential Fields

Introduction

Artificial Potential Fields (APFs) are used in path planning and trajectory planning, particularly for robotic navigation. This method involves conceptualizing a surface, denoted as \(\phi\), defined in the workspace. The robot is visualized as moving down the gradient of this surface.

Key Concepts

• Robot Position: The robot is considered to be at a position \(b \in \mathbb{R}^3\) with a certain radius \(\rho_r\).
• Goal Configuration: The goal or final configuration is treated as an attractive point in the potential field.
• Obstacle Boundaries: Obstacles are located at \(o_i\) with radii \(\rho_i > 0\). These create repelling effects in the potential field.
• Critical Points: The goal is to have one minimum (attractive point) and ensure that every other critical point is nondegenerate.

APF Definition

The potential field \(\phi\) is a function from a field \(F\) to the interval [0, 1], defined as a composition:

Where \(\hat{\phi}\) is another function from \(F\) to [0, ∞), encoding the goal point and obstacles:

• Goal Function \(\gamma\): For \(\gamma : F \rightarrow [0, \infty)\), it is defined as \(\gamma(b) = (b - g)^T (b - g)\), where \(g\) is the goal position.
• Obstacle Function \(\beta\): The obstacle function \(\beta : F \rightarrow [0, \infty)\) is defined in terms of distances to obstacles, \(\beta(b) = \Delta_{j=1}^N \beta_j(b)\), where each \(\beta_j(b) = (b - o_j)^2 - (\rho_r + \rho_j)^2\).
• Parameter \(k\): Represents the balance between goal attraction and obstacle repulsion.

Handling Freespace Boundaries

The freespace boundary \(\partial F\) is the zero-level set of \(\beta^{-1}(0)\). Since \(\hat{\phi}\) can blow up on \(\partial F\), it’s not admissible. This is remedied by squashing the function using \(\sigma : [0, \infty] \rightarrow [0, 1]\), defined as \(\sigma(x) = \frac{x}{1 + x}\).

To avoid degenerate goal points, a sharpening function \(\sigma_d : [0, 1] \rightarrow [0, 1]\) is applied, where \(\sigma_d(x) = x^{1/k}\). This results in \(\phi\) becoming admissible with a non-degenerate minimum at \(b = g\).

Here we implemented the APF by getting the obstacle positions through RGB-D cameras. Target point was selected from the initial position. The remaining robots were responsible to follow the robot in front from a distance with the same algorithm.