Geometrically Constrained Trajectory Optimization for Multicopters (2103.00190v4)

Abstract: We present an optimization-based framework for multicopter trajectory planning subject to geometrical configuration constraints and user-defined dynamic constraints. The basis of the framework is a novel trajectory representation built upon our novel optimality conditions for unconstrained control effort minimization. We design linear-complexity operations on this representation to conduct spatial-temporal deformation under various planning requirements. Smooth maps are utilized to exactly eliminate geometrical constraints in a lightweight fashion. A variety of state-input constraints are supported by the decoupling of dense constraint evaluation from sparse parameterization, and backward differentiation of flatness map. As a result, this framework transforms a generally constrained multicopter planning problem into an unconstrained optimization that can be solved reliably and efficiently. Our framework bridges the gaps among solution quality, planning efficiency, and constraint fidelity for a multicopter with limited resources and maneuvering capability. Its generality and robustness are both demonstrated by applications to different flight tasks. Extensive simulations and benchmarks are also conducted to show its capability of generating high-quality solutions while retaining the computation speed against other specialized methods by orders of magnitude. The source code of our framework is available at: https://github.com/ZJU-FAST-Lab/GCOPTER

• The paper introduces optimality conditions that enable efficient trajectory planning by balancing solution quality with computational speed.
• It presents the MINCO trajectory class which decouples dense constraint evaluations from sparse parameterizations for robust performance.
• Extensive simulations validate the framework's ability to produce smooth, adaptable trajectories under diverse dynamic conditions.

Geometrically Constrained Trajectory Optimization for Multicopters

The paper presents a sophisticated optimization framework for trajectory planning in multicopters, addressing both geometrical and user-defined dynamic constraints. The focus is on developing a mechanism that efficiently balances solution quality, planning speed, and constraint fidelity.

Framework Overview

The proposed framework leverages a novel trajectory representation that minimizes control efforts, utilizing optimality conditions that facilitate a controlled transformation of the spatial-temporal structure of trajectories. This representation is particularly valuable, offering linear-complexity operations for adapting to various planning objectives.

The framework incorporates smooth map techniques to remove geometrical constraints effectively, thus simplifying the optimization problem without sacrificing fidelity. Importantly, constraints on state and input are managed through decoupling dense constraint evaluations from sparse parameterizations, along with leveraging backward differentiation of flatness maps.

Key Contributions

1. Optimality Conditions: The paper introduces optimality conditions for multi-stage control effort minimization, which ensure the existence and uniqueness of solutions. These conditions are structured to provide both sufficiency and necessity, and they enable the direct construction of optimal trajectories with linear complexity.
2. Novel Trajectory Class: The MINCO trajectory class is designed to accommodate user-defined objectives while retaining trajectory smoothness through spatial-temporal deformation. This class focuses on an efficient balance of geometry and dynamics to satisfy constraints.
3. Constraint Elimination and Penalty Methods: The framework employs lightweight constraint elimination techniques, transforming the complex multi-constrained trajectory problem into a simpler unconstrained optimization problem. A penalty-based approach is utilized to generalize continuous-time constraints, integrating them into the optimization process efficiently.

Performance and Validations

The framework achieves notable computation speed improvements, outperforming existing methods in efficiency by orders of magnitude. It is validated through extensive simulations and benchmarks, producing high-quality trajectory solutions under various settings and validating its applicability across different flight tasks.

Implications and Future Work

The proposed optimization framework opens up several avenues for practical and theoretical advancements:

• Computational Efficiency:

By mitigating computation overhead without detracting from trajectory quality, this framework is well-suited for scenarios where time-critical decision-making is crucial.

• Versatility and Adaptability:

The flexible design accommodates a wide range of user-defined constraints, making it applicable to diverse flight conditions and multicopter designs.

• Future Developments:

The paper suggests potential for extending this methodology to other dynamic systems with different configurations and constraints. Further developments may involve integrating real-time obstacle recognition and adaptive planning capabilities, which would enhance autonomous navigation systems in continuously changing environments.

In conclusion, the framework represents a comprehensive approach to multicopter trajectory optimization, achieving an efficient synergy between computational complexity, trajectory quality, and constraint fidelity. The research contributes significantly to autonomous aerial navigation, paving the way for more resilient and adaptive flight systems.