Successive Convexification Research Articles

Landing Guidance Control Combining Powered Descending with Nonlinear Optimization and Vertical Descending with Model Prediction

In this study, we propose a landing guidance control method for the Moon using two control methods. The landing of a spacecraft on the Moon is divided into two phases: the powered descending phase and the vertical descending phase. The powered descending phase aims to optimize the entire trajectory to satisfy the termination constraint, instead of allowing for a relatively long control period. This phase performs feedback control using trajectory updates via nonlinear optimization. The vertical-descending phase aims to achieve accurate control over a short control period. This phase applies nonlinear model predictive control for fast optimization on finite time intervals. First, the landing of a spacecraft on the Moon is modeled as a two-body problem of the Moon and spacecraft, and equations of motion are derived. Subsequently, we formulated an optimization problem for each phase and developed the proposed landing guidance method by combining the two control methods. Furthermore, numerical simulations based on the derived equations of motion were performed to confirm the effectiveness of the proposed method and to compare its performance with another optimization method, the successive convexification (SCvx) algorithm.

Sequential Convex Programming for Reentry Trajectory Optimization Utilizing Modified hp-Adaptive Mesh Refinement and Variable Quadratic Penalty

Due to the strong nonlinearity in the reentry trajectory planning problem for reusable launch vehicles (RLVs), the scale of the problem after high-precision discretization can become significantly large, and the non-convex path constraints are prone to exceed limits. Meanwhile, the objective function oscillation phenomenon may occur due to successive convexification, which results in poor convergence. To address these issues, a novel sequential convex programming (SCP) method utilizing modified hp-adaptive mesh refinement and variable quadratic penalty is proposed in this paper. Firstly, a local mesh refinement algorithm based on constraint violation is proposed. Additional mesh intervals and mesh points are added in the vicinity of the constraint violation points, which improves the satisfaction of non-convex path constraints. Secondly, a sliding window-based mesh reduction algorithm is designed and introduced into the hp-adaptive pseudospectral (PS) method. Unnecessary mesh intervals are merged to reduce the scale of the problem. Thirdly, a variable quadratic penalty-based SCP method is proposed. The quadratic penalty term related to the iteration direction and the weight coefficient updating strategy is designed to eliminate the oscillation. Numerical simulation results show that the proposed method can strictly satisfy path constraints while the computational efficiency and convergence of SCP are improved.

A novel successive convexification framework with symplectic pseudospectral solutions for nonlinear optimal control problems of constrained pantograph delay systems

This paper proposes a novel symplectic pseudospectral iteration framework for the nonlinear optimal control problem (OCP) of a pantograph delay system with inequality constraints. Traditional numerical algorithms for OCPs with pantograph delay systems have predominantly focused on unconstrained scenarios, neglecting constrained problems. Therefore, we propose a novel symplectic pseudospectral iteration framework for constrained problems. First, we employ the successive convexification (SCvx) method to transform the original problem into a series of linear quadratic (LQ) problems, addressing nonlinearity. Then, leveraging time-scale transformations and parametric variational principles, we derive the first-order necessary conditions (FONCs) for these convexified problems, including a Hamiltonian two-point boundary value problem (HTBVP) with stretching terms and a linear complementarity problem. To handle pantograph time delays effectively, we employ a local Legendre-Gauss-Lobatto (LGL) pseudospectral method with proportional grid discretization. Finally, employing a symplectic indirect algorithm, we develop a symplectic pseudospectral method (SPM) for solving the transformed LQ problems, converting them into a system of sparse linear algebraic equations and a linear complementarity problem. Validation of the framework is performed using diverse illustrative examples, demonstrating strong agreement between SCvx and SPM in computational efficiency. Numerical experiments confirm the sparsity of the core matrix and robustness to initial guesses.

Real-time fault-tolerant guidance for launch vehicle ascending flight under thrust drop failure

This paper presents a real-time fault-tolerant guidance method, which solves the autonomous rescue problem in the event of thrust drop failure during the final stage flight of launch vehicles. The core of the method is the online and optimal reconstruction of both the degraded target orbit and the subsequent flight trajectory. Aiming at achieving satisfied onboard computing performance and making the best use of the launcher’s residual energy, a unique relaxation-penalization model transformation method for the nonlinear orbit injection conditions is proposed, and a successive convexification algorithm is developed to solve the problem. Furthermore, both the two classic thrust drop failure situations, that is the mass flow rate drop and the specific impulse drop problems, can be resolved by the proposed algorithm. The average run-time of this orbit-trajectory joint optimization algorithm is below 150ms in an embedded ARM processor @1.5 GHz, which is adequate for onboard use. Using the above algorithm as a central infrastructure, a receding-horizon-strategy-based fault-tolerant guidance method is proposed. By performing the optimization repeatedly at a high frequency, the effects of parameter deviations and external disturbances can be absorbed for the most part, for example, the guidance algorithm is verified to be capable of tolerating at least ±5% thrust deviations. Comprehensive numerical and hardware-in-the-loop simulations are performed to demonstrate the computational performance, accuracy, and reliability of the proposed algorithms.

Endoatmospheric powered descent guidance

This paper addresses the problem of guiding a rocket-powered vehicle to land using aerodynamic and thrust controls. The vehicle initially glides towards the landing site, flying at an unprecedentedly high angle-of-attack to decelerate, thereby reducing the propellant needed for a later retro-thrust landing. The high angle-of-attack pure aerodynamic flight leads to highly nonlinear dynamics that have not been addressed by existing powered descent guidance methods. In particular, real-time optimization of the vehicle’s landing trajectory has been very challenging. In this paper, a closed-loop guidance method is proposed that optimizes a restricted 6-degrees-of-freedom (6-DoF) trajectory in real-time. To remedy the difficulty caused by the highly nonlinear dynamics, the general propellant-optimal problem is regularized with two modifications. The first is to augment the angular velocity to the performance index, removing possible singularity arcs that cause numerical difficulties. The second is to parameterize the thrust magnitude with a piecewise-constant form, reducing the number of control variables to be optimized. The regularized problem is then solved with successive convexification to update the restricted 6-DoF landing trajectory in each guidance cycle. Monte Carlo experiments with aerodynamic and thrust dispersions are conducted to examine the proposed guidance method.

Small Solid-Model Rocket Design and Soft Landing Trajectory Planning

A recoverable rocket is one of the keys to reusable launch technology. Currently, most recoverable rockets are powered by liquid engines, which are costly and complex compared to solid engines. In this paper, we perform soft landing trajectory planning for the designed solid rocket. First, a small model rocket equipped with a landing engine is innovatively designed. Second, a six-degree-of-freedom motion model of the rocket is established, and a minimum-landing-error problem is formed based on the landing engine. Third, successive convexification algorithm is applied to solve it. In the algorithm, a method of quaternion constraint correction is proposed to avoid the infeasibility of the problem. Lastly, the correctness and robustness of the algorithm are verified via three simulations. The possibility of rapid trajectory generation is verified by code optimization on a high-performance digital signal processor.

Successive Convexification Algorithms for Optimizing Power Systems With Energy Storage Models

Successive Convexification Algorithms for Optimizing Power Systems With Energy Storage Models

Scalable Computation of Robust Control Invariant Sets of Nonlinear Systems

Ensuring robust constraint satisfaction for an infinite time horizon is a challenging, yet crucial task when deploying safety-critical systems. We address this issue by synthesizing robust control invariant sets of perturbed nonlinear sampled-data systems. This task can be encoded as a non-convex program that we approximate by a tailored, computationally efficient successive convexification algorithm. Based on the zonotopic representation of invariant sets, we obtain an updated candidate for the invariant set and the invariance-enforcing controller by solving a single convex program. To obtain a possibly large region of safe operation, our algorithm is designed so that the sequence of candidate invariant sets is volume-wise monotonically increasing. We demonstrate the efficacy and scalability of our approach using a broad range of nonlinear control systems from the literature with up to 20 dimensions.

Successive Convexification for Nonlinear Model Predictive Control with Continuous-Time Constraint Satisfaction

We propose a nonlinear model predictive control (NMPC) framework based on a direct optimal control method that ensures continuous-time constraint satisfaction and accurate evaluation of the running cost, without compromising computational efficiency. We leverage the recently proposed successive convexification framework for trajectory optimization, where: (1) the path constraints and running cost are equivalently reformulated by augmenting the system dynamics, (2) multiple shooting is used for exact discretization, and (3) a convergence-guaranteed sequential convex programming (SCP) algorithm, the prox-linear method, is used to solve the discretized receding-horizon optimal control problems. The resulting NMPC framework is computationally efficient, owing to its support for warm-starting and premature termination of SCP, and its reliance on first-order information only. We demonstrate the effectiveness of the proposed NMPC framework by means of a numerical example with reference-tracking and obstacle avoidance. The implementation is available at https://github.com/UW-ACL/nmpc-ctcs.

Detection and estimation of spacecraft maneuvers for catalog maintenance

Building and maintaining a catalog of resident space objects involves several tasks, ranging from observations to data analysis. Once acquired, the knowledge of a space object needs to be updated following a dedicated observing schedule. Dynamics mismodeling and unknown maneuvers can alter the catalog’s accuracy, resulting in uncorrelated observations originating from the same object. Starting from two independent orbits, this work presents a convex-optimization based approach to find admissible maneuvers to correlate the two orbits. In case of a feasible maneuver detection, the time profile of the maneuver is estimated, exploiting the conjugate unscented transform to map the orbits’ covariance into the maneuver uncertainty. It is also shown that a careful choice of the coordinate system is pivotal to obtain the solution in one iteration, without successive convexification.

Convexification approaches for regional route guidance and demand management with generalized MFDs

Traffic congestion is one of the main concerns in big cities with many adverse socioeconomic effects. A promising solution is to simultaneously regulate the admission of vehicles (i.e., demand management) and redistribute traffic flows within the network (i.e., route guidance). In this work, we integrate demand management with route guidance within a Model Predictive Control framework using regional traffic dynamics with generalized Macroscopic Fundamental Diagram (MFD) shapes. Dealing with generalized MFD shapes is challenging due to the resulting nonlinear and non-convex optimization formulation. To tackle this challenge, we develop two real-time solution approaches: (i) a successive convexification approach that constructs convex bounding sets for all nonlinear terms, and (ii) a linear approximation approach that solves the problem using triangular macroscopic fundamental diagram approximations. The proposed approaches offer a trade-off between execution speed and solution quality, as the linear approximation approach runs faster while the successive convexification approach yields better quality and accuracy solutions. Macroscopic simulation results illustrate the efficiency of the successive convexification and linear approximation approaches yielding an optimality gap of less than 3.5% and 10% in all considered cases, respectively. Furthermore, both approaches outperform a state-of-practice nonlinear solver in terms of solution quality and execution time. Finally, substantial gains are also obtained regarding travel time and traffic flow efficiency in a realistic microsimulation environment.

MGV Obstacle Avoidance Trajectory Generation Considering Vehicle Shape

This study investigates the application of obstacle avoidance trajectory generation considering the vehicle shape of a micro ground vehicle by successive convexification and state-triggered constraints. The avoidance trajectory is generated by numerical computation and path-following experiments are conducted to assess the generated trajectory. The numerical computation results indicate that the trajectory obtained by the algorithm successfully avoids obstacles considering the vehicle shape and satisfies the constraints. The experiment includes the model predictive control to follow the generated trajectory. Numerical computations and experiments confirm the usefulness of the trajectory generation algorithm.

Convex optimization for post-fault ascent trajectory replanning using auxiliary phases

For ascent trajectory replanning after non-fatal dynamic faults, the terminal orbital constraints and optimization indexes under different replanning strategies usually have very strong nonconvexity, which brings significant challenges to real-time guidance. This paper introduces auxiliary phases to convexify the terminal constraints and objective functions losslessly, significantly improving the convergence and injection accuracy. After lossless and successive convexification, the original problems are converted into a series of convex subproblems. The replanning algorithm is then presented, including the fault detection module, the residual carrying capacity determination module, and the task degradation module. Two typical launch missions are simulated to verify the proposed method's pmethoderformance. Numerical results show that the method can always converge efficiently and reliably in different scenarios and has great potential for real-time onboard applications.

A joint guidance and control framework for autonomous obstacle avoidance in quadrotor formations under model uncertainty

This paper presents a novel joint guidance and control framework for autonomous obstacle avoidance and swift maneuvering of quadrotor formations under model uncertainty. In this framework, the quadrotor formation uses a primary-secondary mode, where only the primary aircraft performs online planning of obstacle avoidance trajectory to form a safe flight corridor; the secondary aircraft follows the primary aircraft for safe flight through control algorithms. Firstly, based on a known and certain 3-DOF dynamics model, a successive convexification online guidance algorithm is proposed to compute a feasible and safe flight corridor through the nonconvex flight space. The planned trajectory contains velocity and attitude angle reference commands with temporal information, which can be used for the design of control algorithms for the 6-DOF dynamics mode. Subsequently, a global controller based on robust adaptive control of the RBF neural network is proposed for compensating external disturbances and model uncertainties. Finally, simulation experiments verify the effectiveness and superiority of this guidance and control framework for quadrotor formation autonomous obstacle avoidance applications under model uncertainty. This paper provides open-source ANSI-C code, which can be deployed directly to the embedded computer to carry out experimental verification. The code link is https://github.com/mazhenwei160/Uav-formation-obstacle-avoidance.git.

Autonomous Landing Using Model Prediction via Successive Convexification

With the goal of exploring the origin of the Earth, lunar exploration has been attracting much attention. In order to achieve more flexible missions, landing with less position error is required. In addition, space missions always call for severe minimum fuel consumption. In this research, the model predictive control (MPC) is implemented to the guidance and control system for the vertical landing phase of a lunar lander. Moreover, for the purpose of reducing the computational load of MPC, successive convexification (SCvx) is also applied. The results of simulations show high feasibility in terms of trackability and robustness for landing.

A Fast Algorithm for Onboard Atmospheric Powered Descent Guidance

Atmospheric powered descent guidance (APDG) can be solved by successive convexification; however, its onboard application is impeded by high computational cost. When aerodynamic forces are ignored, powered descent guidance (PDG) can be converted to a single convex problem. In contrast, APDG has to be converted into a sequence of convex subproblems, each of which is significantly more complicated. Consequently, the computation increases sharply. A fast real-time interior point method was presented to solve the correlated convex subproblems efficiently onboard in the work. The main contributions are as follows: Firstly, an algorithm was proposed to accelerate the solution of linear systems that cost most of the computation in each iterative step by exploiting the specific problem structure. Secondly, a warm-starting scheme was introduced to refine the initial value of a subproblem with a rough approximate solution of the former subproblem, which lessened the iterative steps required for each subproblem. The method proposed reduced the run time by a factor of 9 compared with the fastest publicly available solver tested in Monte Carlo simulations to evaluate the efficiency of solvers. Runtimes on the order of 0.6 s are achieved on a radiation-hardened flight processor, which demonstrated the potential of the real-time onboard application.

Distributed MPC of Uncertain Multi-Agent Systems Considering Formations and Obstacles

This paper proposes a method to control a class of multi-agent systems with uncertainties modeled as stochastic processes with arbitrary probability distributions. The considered control problem is to lead a formation of agents through a space which is partially obstructed by obstacles. The proposed solution is to use a hierarchically structured approach of distributed stochastic model predictive control (DSMPC). The approach combines elements of formation reference structures, leader-follower concepts, and successive convexification (SC) for collision avoidance. To consider the stochastic uncertainties, over-approximated probabilistic reachable sets (PRS) are computed based on Chebyshev's inequality. The nominal (expected) agent behavior is optimized within the DSMPC such that (probabilistic) constraints are satisfied in a distributed way. For the overall approach, closed-loop stability of the distributed control concept is investigated and an illustrating example is provided.

Convex Optimization for Trajectory Generation: A Tutorial on Generating Dynamically Feasible Trajectories Reliably and Efficiently

Reliable and efficient trajectory generation methods are a fundamental need for autonomous dynamical systems. 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, successive convexification (SCvx) and guaranteed sequential trajectory optimization (GuSTO). Trajectory generation is defined as 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 difficult to solve efficiently and reliably 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 using a convex optimizer. The theoretical guarantees and computational speed offered by convex optimization have made the algorithms popular in both research and industry circles. The growing 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 such as NASA, Masten Space Systems, SpaceX, and Blue Origin. This article equips the reader with the tools and understanding necessary to work with each algorithm and know their advantages and limitations. An open source tool called the SCP Toolbox accompanies the article and provides a practical implementation of every numerical example. By the end of the article, the reader will not only be ready to use the lossless convexification and sequential convex programming algorithms, but also to extend them and to contribute to their many exciting modern applications.

Enhanced Successive Convexification Based on Error-Feedback Index and Line Search Filter

This paper introduces convexification error as an error-feedback optimization index into successive convexification optimization, widely used in aerospace engineering applications, such as reusable rocket landing. The error-feedback optimization index is not only a measure of the solution accuracy but also a criterion of convergence. The original single-objective successive convexification optimization problem is transformed into a bi-objective optimization problem, which optimizes both the convexification error and the objective. An efficient and robust convergence strategy is proposed based on a line search filter for the newly obtained optimization problem. The proposed convergence strategy improves the accuracy of the obtained solution and alleviates the potential sensitivity of the successive convexification optimization convergence to the initial guess. The proposed convergence strategy is verified using a universal controllability set approximation method. The numerical experimental results show that the proposed algorithms have good practical potential in aerospace engineering.

An iterative framework to solve nonlinear optimal control with proportional delay using successive convexification and symplectic multi-interval pseudospectral scheme

In this paper, we propose an iterative framework to solve optimal control for nonlinear proportional state-delay systems. The successive convexification technique is first implemented to convert the original nonlinear problem into a sequence of linear-quadratic problems. And a symplectic pseudospectral method, where the multi-interval pseudospectral scheme is applied with a proportional mesh, to solve the transformed problems is then developed based on the first-order necessary conditions. Each linear-quadratic problem is finally transformed into a system of linear algebraic equations with a sparse coefficient matrix. Due to the benefit of the successive convexification technique and the multi-interval pseudospectral method, initial guess on costate variables is avoided and converged solutions can be obtained with an exponential convergent rate. The proposed iterative framework is validated by four examples with distinct features, highlighting its numerical precision and efficiency. And either exponential or linear convergence property can be exhibited by tuning the approximation degree or the mesh number.