Quintic Polynomial Trajectory

Problem Statement

Quintic trajectories are a standard choice for smooth motion generation with guaranteed continuity up to acceleration. They provide practical C2 motion profiles for UAV transitions and corridor following.

Model and Formulation

Use a fifth-order polynomial:

$$ p(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3 + a_4 t^4 + a_5 t^5 $$

Boundary conditions at initial and final times constrain position, velocity, and acceleration:

$$ p(0),\dot{p}(0),\ddot{p}(0),p(T),\dot{p}(T),\ddot{p}(T) $$

forming a linear system for a_0..a_5.

Algorithm Procedure

1. Select endpoint state and maneuver duration T.
2. Construct 6x6 boundary-condition system.
3. Solve coefficients and evaluate trajectory over time.
4. Feed resulting reference to the tracking controller.

Tuning Guidance

• Duration T controls aggressiveness directly.
• Use axis-specific duration scaling for asymmetric constraints.
• Validate peak acceleration and jerk against actuator limits.

Failure Modes and Diagnostics

• Very small T values produce infeasible accelerations.
• Discontinuous waypoint chaining can violate C2 assumptions.
• Numerical conditioning worsens when time scales vary widely.

Implementation and Execution

python -m uav_sim.simulations.trajectory_planning.quintic_polynomial_demo

Evidence

References

• Werling et al., Optimal Trajectory Generation for Dynamic Street Scenarios in a Frenet Frame (2010)
• Kelly and Nagy, Reactive Nonholonomic Trajectory Generation via Parametric Optimal Control

Related Algorithms

• Polynomial Trajectory
• Frenet Optimal