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:

$$J={\int }_0^T{\Vert \frac{d^{4}p(t)}{d{t}^4}\Vert }^{2}dt$$

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

• A min-snap trajectory through sparse waypoints can loop back close to itself. That is fine for the trajectory and hostile to a naive tracker: the look-ahead sphere keeps intersecting an earlier segment and the vehicle circles behind its own carrot forever. See pure pursuit for the arc-length progress window that fixes it.

• Segment times that are too short for the distance demand accelerations the vehicle cannot produce; the polynomial is still optimal, just infeasible.

• 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 flybots.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