Time-Optimal Planning for Long-Range Quadrotor Flights: An Automatic Optimal Synthesis Approach

TL;DR

This work tackles the challenge of time-optimal planning for long-range quadrotor flights by introducing Automatic Optimal Synthesis (AOS), a polynomial-based framework that represents trajectories with a minimal yet sufficient number of polynomial pieces. By leveraging differential flatness and Pontryagin Maximum Principle–guided structure (bang-bang and bang-singular controls), AOS determines an appropriate piece count $N$ and solves a multi-piece optimality problem that accommodates arbitrary starts, ends, and waypoints while respecting full quadrotor dynamics. The approach achieves near time-optimal performance with orders-of-magnitude faster computation than discretization-based methods and scales well to long ranges and multi-waypoint tasks, as validated by extensive simulations and real-world experiments showing aggressive maneuvering with peak speeds up to about $8.86$ m/s. The results demonstrate significant practical impact for drone racing, large-area missions, and waypoint-laden routes, and establish a robust, generalizable pathway for extending AOS to other dynamical systems. Future work aims to further reduce tracking errors by integrating data-driven quadrotor models into the AOS pipeline.

Abstract

Time-critical tasks such as drone racing typically cover large operation areas. However, it is difficult and computationally intensive for current time-optimal motion planners to accommodate long flight distances since a large yet unknown number of knot points is required to represent the trajectory. We present a polynomial-based automatic optimal synthesis (AOS) approach that can address this challenge. Our method not only achieves superior time optimality but also maintains a consistently low computational cost across different ranges while considering the full quadrotor dynamics. First, we analyze the properties of time-optimal quadrotor maneuvers to determine the minimal number of polynomial pieces required to capture the dominant structure of time-optimal trajectories. This enables us to represent substantially long minimum-time trajectories with a minimal set of variables. Then, a robust optimization scheme is developed to handle arbitrary start and end conditions as well as intermediate waypoints. Extensive comparisons show that our approach is faster than the state-of-the-art approach by orders of magnitude with comparable time optimality. Real-world experiments further validate the quality of the resulting trajectories, demonstrating aggressive time-optimal maneuvers with a peak velocity of 8.86 m/s.

Figures (19)

• Figure 1: Time-optimal trajectory generated by the proposed AOS approach to pass 19 waypoints and executed by a quadrotor platform in a motion capture room. Our approach achieves near-optimal racing performance and can handle long-range time-critical tasks with superior efficiency.
• Figure 2: Illustration of the AOS approach. Step 1): understand the bang-bang or bang-singular structure of the optimal collective thrust and rotational rate trajectories. Step 2): leverage the obtained information to determine an appropriate number of polynomials for trajectory representation. Step 3): solve the time-optimal planning problem parameterized by the selected polynomials. It is expected that the polynomial can accommodate all rapid state or input changes with reasonable order and thereby generate close-to-optimal results.
• Figure 3: Illustration of the non-dimensional quadrotor model to facilitate the analysis of time-optimal maneuvers.
• Figure 4: Demonstration of singular flows (orange lines) in a time-optimal maneuver of a pure horizontal translation. Note that the optimal rotation trajectory $\theta^{*}$ consists of three phases, represented by two yellow lines and one green curve in the middle. The corresponding optimal rotational rate input $u_{R}^{*}$ is depicted at the bottom trajectory (black line). In the first phase, the vehicle pitches forward with a maximal value of $u_{R}^{*}\!=\!1$, which corresponds to a bang arc (yellow line). Once it intersects the singular flow, we have $\theta^{*}\!=\!\varTheta$ for a long time where $\Phi_{R}\!=\!\dot{\Phi}_{R}\!=\!0$ holds, implying that $u_{R}^{*}$ is singular. This phase entails a smooth transition from acceleration to deceleration of the vehicle. In the last phase, the vehicle pitches forward again with $u_{R}\!=\!1$ to return a hover state. We see that by drawing singular flows, we know when $u_{R}^{*}$ can be singular and what the structure of optimal maneuvers looks like.
• Figure 5: Demonstration of how to use singular flows (orange lines) to determine the optimal thrust input, $u_{T}^{*}$ (black line), for a purely vertical translation. It is known that the time-optimal maneuver entails a flip. By checking the intersections between the optimal rotation trajectory (green and yellow lines) and the singular flows shifted by $\pi/2$ as per Corollary \ref{['cor:ut_switch']}, we get the exact switching times of $u_{T}^{*}$. After taking into account one additional switch caused by $p_{2}(\hat{\tau})=p_{4}(\hat{\tau})=0$, we obtain the full structure of $u_{T}^{*}$ as shown in the bottom trajectory (black).

Theorems & Definitions (46)

• Lemma 1
• proof
• Lemma 2
• proof
• Definition 1: Singular Flow
• Lemma 3
• proof
• Remark
• Definition 2: Flat Singular Flow
• Lemma 4