Minimum-Snap Trajectory

Problem Statement

Waypoint-only paths are not directly flyable because they ignore high-order dynamic smoothness. Minimum-snap planning computes polynomial trajectories that reduce aggressive jerk/snap behavior and improve tracking performance.

Model and Formulation

Each segment is represented by a polynomial:

$$ p(t) = \sum_{i=0}^{n} c_i t^i $$

The optimization minimizes integrated snap:

$$ J = \int_{0}^{T}\left|\frac{d^4p(t)}{dt^4}\right|^2dt $$

subject to waypoint and continuity constraints for position, velocity, acceleration, and jerk.

Algorithm Procedure

1. Allocate segment times across waypoints.
2. Build quadratic cost matrix for snap objective.
3. Apply boundary and continuity equality constraints.
4. Solve constrained QP for polynomial coefficients.

Tuning Guidance

• Time allocation dominates smoothness-quality trade-offs.
• Enforce corridor constraints for cluttered environments.
• Increase continuity order for aggressive maneuvers with tight tracking budgets.

Failure Modes and Diagnostics

• Unrealistic segment times create numerically stiff trajectories.
• Sparse waypoints can violate obstacle-clearance assumptions.
• Overly smooth trajectories may become too conservative for time-critical tasks.

Implementation and Execution

python -m uav_sim.simulations.trajectory_planning.min_snap

Evidence

References

• Mellinger and Kumar, Minimum Snap Trajectory Generation and Control for Quadrotors (2011)
• Richter, Bry, Roy, Polynomial Trajectory Planning for Aggressive Quadrotor Flight (2016)

Related Algorithms

• Polynomial Trajectory
• Feedback Linearisation