Convex Optimization for Trajectory Generation

Abstract: Reliable and efficient trajectory generation methods are a fundamental need for autonomous dynamical systems of tomorrow. The goal of this article is to provide a comprehensive tutorial of three major convex optimization-based trajectory generation methods: lossless convexification (LCvx), and two sequential convex programming algorithms known as SCvx and GuSTO. In this article, trajectory generation is the computation of a dynamically feasible state and control signal that satisfies a set of constraints while optimizing key mission objectives. The trajectory generation problem is almost always nonconvex, which typically means that it is not readily amenable to efficient and reliable solution onboard an autonomous vehicle. The three algorithms that we discuss use problem reformulation and a systematic algorithmic strategy to nonetheless solve nonconvex trajectory generation tasks through the use of a convex optimizer. The theoretical guarantees and computational speed offered by convex optimization have made the algorithms popular in both research and industry circles. To date, the list of applications includes rocket landing, spacecraft hypersonic reentry, spacecraft rendezvous and docking, aerial motion planning for fixed-wing and quadrotor vehicles, robot motion planning, and more. Among these applications are high-profile rocket flights conducted by organizations like NASA, Masten Space Systems, SpaceX, and Blue Origin. This article aims to give the reader the tools and understanding necessary to work with each algorithm, and to know what each method can and cannot do. A publicly available source code repository supports the provided numerical examples. By the end of the article, the reader should be ready to use the methods, to extend them, and to contribute to their many exciting modern applications.

• The paper presents convex optimization methods that reformulate nonconvex trajectory challenges into solvable convex problems for autonomous systems.
• It details lossless convexification (LCvx) and sequential convex programming techniques (SCvx and GuSTO) that iteratively linearize nonconvex constraints for precise motion planning.
• The study emphasizes theoretical guarantees and practical computational efficiency, demonstrating successful applications in aerospace and robotics.

Convex Optimization for Trajectory Generation

The article provides a comprehensive examination of convex optimization techniques, specifically, their use in trajectory generation for autonomous dynamical systems. With the increasing demands on autonomous vehicles and robotic systems to operate safely and efficiently, the focus of this tutorial is on three major methods: lossless convexification (LCvx), and two sequential convex programming algorithms, SCvx and GuSTO.

Key Methods Overview

1. Lossless Convexification (LCvx):
   • LCvx is a modeling technique designed to handle nonconvex trajectory generation problems by reformulating them into convex problems. The article details how this method can handle specific nonconvexities such as input norm constraints and pointing constraints, drawing on theoretical underpinnings from control theory.
   • The LCvx approach has been particularly effective in aerospace applications, such as rocket landing and aerial vehicle motion planning, where input and state constraints are non-negligible. Its capability to solve trajectories with a high degree of precision while maintaining computational efficiency has been demonstrated by its application in high-profile experiments by NASA and companies like SpaceX.
2. Sequential Convex Programming (SCP):
   • SCP, which includes methods like SCvx and GuSTO, approaches trajectory generation through iterative convex approximations of a nonconvex trajectory optimization problem.
   • In SCP, the problem is linearized around a reference solution and solved as a sequence of convex subproblems. This allows it to handle a wide variety of trajectory optimization problems effectively.
   • The article discusses SCP's adaptability and efficiency, making it an attractive option for real-time use in autonomous systems, especially in challenging environments where computational resources may be limited.

Numerical Results and Applications

The paper showcases various applications to illustrate the practical implications of these methods. The effectiveness of convex optimization in trajectory planning is backed by successful implementations in areas such as:

• Rocket Landing: The use of convex optimization algorithms such as LCvx has enabled significant advancements in precision and reliability in guided rocket landing, even under stringent constraints and real-time computational limits.
• Aerospace Motion Planning: SCP methods have been used extensively for spacecraft rendezvous and docking, as well as hypersonic reentry trajectory generation for spacecraft, scenarios where nonconvex problems must be tackled under considerable uncertainty and limited actuation.

Theoretical and Practical Implications

The paper highlights the theoretical robustness and computational practicality of using convex optimization in trajectory planning. The ability of these algorithms to produce globally optimal or locally optimal solutions in polynomial time is emphasized, given the complexities involved in real-world autonomous systems.

• Theoretical Implications: LCvx and SCP provide strong theoretical guarantees, enabling deterministic solutions for complex trajectory planning problems. The lossless nature of LCvx ensures that the convexified problems accurately reflect the original nonconvex intentions.
• Practical Implications: These methods are versatile across disciplines, supporting applications not just in aerospace, but also in robotics, underwater vehicles, and even vehicular networks. Their adaptability to various constraints and objectives means they can cater to many autonomous systems' needs.

Future Directions in AI and Autonomous Systems

The tutorial opens doors to further research and developments in trajectory generation, particularly in crafting algorithms that can handle even broader classes of nonconvex constraints and in promoting efficiency in higher-dimensional spaces. As AI and machine learning techniques begin to merge with traditional control methods, the methods discussed could see integration with probabilistic models that enhance their performance in uncertain and dynamic environments.

In summary, convex optimization, particularly as discussed in the realms of LCvx and SCP, provides a potent toolkit for addressing the challenges posed by trajectory generation in autonomous systems, promising reliability, efficiency, and the capacity to handle complex nonconvex problems adeptly.