Alternating Minimization Based Trajectory Generation for Quadrotor Aggressive Flight (2002.10629v1)

Abstract: With much research has been conducted into trajectory planning for quadrotors, planning with spatial and temporal optimal trajectories in real-time is still challenging. In this paper, we propose a framework for generating large-scale piecewise polynomial trajectories for aggressive autonomous flights, with highlights on its superior computational efficiency and simultaneous spatial-temporal optimality. Exploiting the implicitly decoupled structure of the planning problem, we conduct alternating minimization between boundary conditions and time durations of trajectory pieces. In each minimization phase, we leverage the algebraic convenience of the sub-problem to escape poor local minima and achieve the lowest time consumption. Theoretical analysis for the global/local convergence rate of our proposed method is provided. Moreover, based on polynomial theory, an extremely fast feasibility check method is designed for various kinds of constraints. By incorporating the method into our alternating structure, a constrained minimization algorithm is constructed to optimize trajectories on the premise of feasibility. Benchmark evaluation shows that our algorithm outperforms state-of-the-art methods regarding efficiency, optimality, and scalability. Aggressive flight experiments in a limited space with dense obstacles are presented to demonstrate the performance of the proposed algorithm. We release our implementation as an open-source ros-package.

• The paper introduces an alternating minimization framework that decouples spatial and temporal trajectory optimization for aggressive quadrotor flight.
• It achieves efficient flight planning with up to 150Hz processing rates by leveraging iterative minimization and sparse LU factorization.
• Real-world experiments validate the approach, demonstrating agile performance under tight safety constraints and dense obstacle conditions.

Analyzing Alternating Minimization for Quadrotor Trajectory Generation

The paper "Alternating Minimization Based Trajectory Generation for Quadrotor Aggressive Flight" presents a novel framework designed to address the challenges associated with real-time trajectory planning for quadrotors, particularly in the context of aggressive flight maneuvers. The researchers focus on generating large-scale piecewise polynomial trajectories with optimal spatial and temporal properties, emphasizing computational efficiency and feasibility. This exposition provides a detailed analysis of the proposed framework, its methodology, and implications in both theoretical and practical domains.

Methodology Overview

The framework introduced employs an Alternating Minimization (AM) approach to decouple and optimize the spatial and temporal aspects of quadrotor trajectory planning. The authors leverage the inherently separable structure of trajectory planning problems, conducting minimization iteratively across trajectory boundary conditions and their respective time durations. In each phase, algebraic transformations facilitate rapid convergence and escape from suboptimal local minima.

1. Unconstrained Optimization:
   • The optimization strategy begins by partitioning the trajectory problem into spatial and temporal components.
   • The framework solves the optimization problem by iteratively determining the time allocation and trajectory boundary conditions, using methods like Sparse LU Factorization for efficient linear system solving.
2. Constrained Optimization:
   • Constraints such as maximum speed and safe distance are integrated using a feasibility check based on Sturm's Theory for multivariate polynomials.
   • The AM framework is adapted to account for these constraints, ensuring that the optimized trajectories meet feasibility requirements while minimizing the objective function.

Numerical Results and Findings

The evaluation demonstrates that the proposed framework significantly outperforms existing methods in both unconstrained and constrained settings. The benchmarks reveal an outstanding capability of handling trajectory problems involving numerous pieces with minimal computation time, achieving up to 150Hz processing rates for 60-piece trajectories. The experiments conducted show that the designed algorithm maintains trajectory feasibility under dense obstacle conditions, even at computationally inexpensive rates.

1. Global and Local Convergence Rates:
   • Theoretical analysis verifies that the AM approach achieves a global convergence rate of $O(1/\sqrt{K})$, with local convergence reaching $O(1/K)$, highlighting efficiency without relying on bespoke step-size determination.
2. Practical Implementation:
   • Real-world experiments conducted with a quadrotor validate the practical application of the proposed methodology, demonstrating agile performance without external positioning systems.

Implications and Future Directions

The contributions of this paper hold significant implications in the domain of robotic trajectory planning, particularly for autonomous aerial vehicles. The fusion of computational efficiency and robust feasibility checks propels the framework into realistic applications where both speed and safety constraints are critical. The work opens avenues for future exploration into integrating environmental interactions more intricately, addressing dynamic constraints, and extending the framework to apply to multi-agent systems.

From a theoretical perspective, the polynomial-based representation and feasible solution constraint handling direct future research towards more complex spatial geometric constraints and higher-dimensional optimization. Speculatively, with advances in computational capabilities and further refinements in algorithmic design, the techniques presented could be pivotal in orchestrating collaborative navigation in swarms of autonomous vehicles, thereby pushing the boundaries of current autonomous systems.

Thus, this research paves a foundational path toward highly efficient and feasible trajectory generation, perfectly suited for real-time applications in robotics and beyond.