Covariant Hamiltonian Optimization for Motion Planning (CHOMP)

Article Goal

Explain CHOMP (Covariant Hamiltonian Optimization for Motion Planning) in detail with visualizations and code implementations.

What is CHOMP?

CHOMP (Covariant Hamiltonian Optimization for Motion Planning) is a trajectory optimization algorithm that formulates motion planning as an optimization problem. It uses gradient-based optimization to find smooth, collision-free trajectories for robotic systems.

The intuitive explanation for CHOMP is that the objects can be thought of as magnets. The vehicle can be thought of as a manget with the same pole; the repulsive nature of the magnet pushes the magnet smoothly around the object.

A straight line vector is made to the target, and the trajectory is warped as necessary to equalize the repulsive efforts from the obstacles in the environment.

CHOMP: Written Walkthrough

Core Concepts

1. Trajectory Representation

CHOMP represents trajectories as a sequence of discrete waypoints in space. The trajectory is parameterized as a function of time.

2. Objective Function

The objective function in CHOMP consists of two main components:

• Smoothness term: Encourages smooth trajectories
• Obstacle term: Penalizes collisions with obstacles

3. Optimization

CHOMP uses gradient descent to minimize the objective function, iteratively improving the trajectory.

Mathematical Formulation

The objective function is given by:

Where:

• \(F_{obs}(\xi)\) is the obstacle cost
• \(F_{smooth}(\xi)\) is the smoothness cost
• \(\lambda_{obs}\) and \(\lambda_{smooth}\) are weighting parameters

Code Implementation

See Github

Advantages of CHOMP

1. Smooth trajectories: Natural smoothness through optimization
2. Collision avoidance: Explicit obstacle handling
3. Scalability: Works for high-dimensional configuration spaces
4. Flexibility: Easy to add custom cost terms

Limitations

1. Local minima: Can get stuck in local optima
2. Parameter tuning: Requires careful tuning of λ parameters
3. Computational cost: Can be expensive for complex environments
4. Initial trajectory: Quality depends on initial trajectory