[Paper Review] Smooth time optimal trajectory generation for drones

Srinath Tankasala, Can Pehlivantürk+2 more|arXiv (Cornell University)|Feb 18, 2022

Robotic Path Planning Algorithms4 citations

TL;DR

This paper presents a smooth, time-optimal trajectory generation method for drones by combining Pontryagin’s Minimum Principle (PMP) for two-point boundary value problems with a nonlinear programming (NLP) formulation in discrete time to extend optimal control across multiple waypoints. The approach ensures continuous thrust input at the boundary of the control set, achieving 20% faster flight times than minimum-snap polynomial methods while enabling smoother, more trackable trajectories compared to bang-bang controls.

ABSTRACT

In this paper, we address a minimum-time steering problem for a drone modeled as point mass with bounded acceleration, across a set of desired waypoints in the presence of gravity. We first provide a method to solve for the minimum-time control input that will steer the point mass between two waypoints based on a continuous-time problem formulation which we address by using Pontryagin's Minimum Principle. Subsequently, we solve for the time-optimal trajectory across the given set of waypoints by discretizing in the time domain and formulating the minimum-time problem as a nonlinear program (NLP). The velocities at each waypoint obtained from solving the NLP in the discretized domain are then used as boundary conditions to extend our two-point solution across those multiple waypoints. We apply this planning methodology to execute a surveying task that minimizes the time taken to completely explore a target area or volume. Numerical simulations and theoretical analyses of this new planning methodology are presented. The results from our approach are also compared to traditional polynomial trajectories like minimum snap planning.

Motivation & Objective

• To develop a continuous-time, smooth time-optimal control input for a point-mass drone model with bounded acceleration and gravity.
• To extend two-point time-optimal solutions to multiple waypoints using a discrete-time nonlinear program (NLP) formulation.
• To generate trajectories that use maximum thrust at all times (boundary control) while ensuring smoothness for improved tracking on real UAVs.
• To compare the proposed method with traditional minimum-snap polynomial planning in terms of flight time and input smoothness.
• To evaluate the impact of switching points on solution quality and computational cost in the discretized NLP formulation.

Proposed method

• Formulate the minimum-time control problem between two waypoints using Pontryagin’s Minimum Principle (PMP) for a point-mass drone with bounded thrust and gravity.
• Discretize the time domain and solve the multi-waypoint problem as a nonlinear program (NLP), using waypoint velocities as decision variables.
• Use the NLP solution to define position and velocity references that are interpolated into a continuous trajectory.
• Enforce that the control input remains on the boundary of the thrust constraint set (hypersphere) at all times, ensuring maximum acceleration usage.
• Apply switching points in the input function to improve smoothness, with one or more switching points used to refine the control input shape.
• Compare the resulting trajectory with minimum-snap polynomial trajectories in terms of flight time and input smoothness.

Experimental results

Research questions

• RQ1 Can a smooth, time-optimal control input be generated for a drone across multiple waypoints while respecting bounded thrust and gravity?
• RQ2 How does the performance of the proposed NLP-based method compare to minimum-snap polynomial planning in terms of total flight time?
• RQ3 What is the impact of increasing the number of switching points in the discretized control input on flight time and input smoothness?
• RQ4 To what extent does direct interpolation of waypoint states approximate the true time-optimal control input?
• RQ5 Can the two-point PMP solution be effectively extended to multi-waypoint paths using a discrete-time NLP formulation?

Key findings

• The proposed method reduces total flight time by 20% compared to minimum-snap polynomial trajectory planning, demonstrating superior time optimality.
• The continuous-time optimal input generated via PMP is significantly smoother than bang-bang control and more trackable on real UAVs.
• Using multiple switching points (e.g., four) improves input smoothness slightly but provides no significant reduction in total flight time compared to one switching point.
• The NLP-based approach successfully generates a continuous trajectory that respects the boundary of the thrust constraint set at all times, ensuring maximum acceleration usage.
• Direct interpolation of waypoint states produces a trajectory that is close to the optimal PMP solution but not time-optimal, highlighting the need for proper optimization.
• The Hamiltonian is not constant across all waypoints in the initial solution, indicating that further refinement of waypoint velocity guesses could yield better time-optimal solutions.