Convex Programming Algorithm Research Articles

Trajectory reconstruction method for spacecraft formation considering collision probability constraints

Abstract This paper introduces a trajectory reconstruction algorithm for spacecraft formation, combining collision probability and sequential convex programming algorithms. When reconstructing the trajectory of spacecraft formation, external collision avoidance constraint is added to avoid collision threats from space objects. The constraint accuracy of the external collision avoidance constraints of the BOX method and the collision probability method is analyzed. The simulation results indicate that the collision probability method has higher accuracy compared to the BOX method. It addresses the issue of inaccurate division of collision avoidance zones in the BOX method, which tends to be aggressive in the velocity direction and conservative in other directions. A reference is provided for trajectory reconstruction when formation satellites encounter non-cooperative targets.

Weight-adaptive augmented Lagrange multiplier sequential convex programming for nonlinear trajectory optimization

Sequential convex programming has become a prominent research focus in aerospace trajectory optimization due to its high computational efficiency. One critical challenge in sequential convex programming is the infeasibility of subproblems, which significantly impacts its stability. To address this issue, certain constraints must be relaxed using relaxation variables, which are then incorporated into the objective function as a penalty term. However, the choice of penalty weight greatly influences the optimality and convergence of sequential convex programming. Therefore, this paper proposes a weight-adaptive augmented Lagrange multiplier sequential convex programming algorithm to tackle the challenge. First, this paper analyzes the difficulty of determining an appropriate penalty weight from the perspective of its penalty effect. Based on this analysis, an augmented penalty term is introduced, and the update rule for the augmented Lagrange multiplier is formulated using the Karush-Kuhn-Tucker conditions. Subsequently, a method with adaptive weight adjustment is designed based on the change in the penalty term to further improve the algorithm's solving speed. Finally, the proposed method is validated through numerical simulations in a Mars-powered landing problem and a small glide vehicle reentry problem. The results show that the proposed method is 20.59 % and 25.16 % faster compared to the general sequential convex programming algorithm, respectively. Furthermore, the proposed method is demonstrated to be robust to the initial choice of penalty weight.

Trajectory optimization for aerodynamically controlled missiles by chance-constrained sequential convex programming

The flight environment of aerodynamically controlled missiles is full of complexity and uncertainty. To cope with the uncertainty more effectively and enhance the convergence performance in trajectory optimization problems for aerodynamically controlled missiles simultaneously, the chance-constrained sequential convex programming (CC-SCP) algorithm is proposed in this paper. The uncertainty is regarded as the chance constraint, and a smooth and differential approximation function is designed to transform this chance constraint into the constraint that the convex optimization method can handle. Subsequently, the originally non-convex trajectory optimization problem is reformulated into a series of convex optimization subproblems, in which an initial reference trajectory guess generation strategy is proposed, and a theoretical proof of the exact convex relaxation is given to enhance the algorithm's convergence performance and theoretical value, respectively. Numerical simulations are provided to verify the convergence and effectiveness of the CC-SCP algorithm, and the advantages of using the CC-SCP algorithm to cope with the uncertainty are illustrated. Furthermore, comparative simulation examples show that the proposed algorithm possesses a low conservatism, which means the proposed algorithm can obtain a bigger convergence region and a better solution than other current methods when handling the same chance constraints. Finally, the robustness of the algorithm is discussed.

Collision-free trajectory planning for UAVs based on sequential convex programming

This paper proposes an optimization-based approach to solve the collision-free trajectory planning problem for unmanned aerial vehicles (UAVs) with nonlinear dynamical systems. The problem is modeled as a non-convex optimization problem, and then iteratively solved by sequential convex programming algorithm which approximates non-convex constraints by convexification methods. We linearize the dynamical equations of the UAV and penalize collision-free constraints with a hinge loss. Furthermore, a variable trust region method is employed to ensure the convergence and stability of the algorithm. We apply the algorithm to three-degree-of-freedom UAV case studies. Numerical results validate that our method can provide collision-free trajectories with a 27.19% reduction in total flight time and a 93.61% reduction in computation time on average compared with the genetic algorithm method.

Rapid reconfiguration of gravitational detection spacecraft using high-accuracy successive convex programming

This article aims to address the challenge of achieving fast and high-accuracy reconfiguration of the TianQin constellation, with a focus on developing trajectory optimization algorithms to enhance both computation speed and accuracy in the reconfiguration process within TianQin Gravity Wave (GW) detection constellations. Initially, through perturbation analysis on the TianQin constellation, we formulated a dynamic model based on a virtual formation strategy, incorporating factors such as solar radiation pressure (SRP) and differences in third-body perturbation (DTBP). Subsequently, to meet the dual requirements of rapidly trajectory generation and optimal control accuracy, we proposed a new time-optimal high-accuracy successive convex programming (HASCP) algorithm and theoretically proved its convergence and optimality. Finally, we present simulations focusing on the TianQin gravitational detection spacecraft to validate the efficacy of the algorithm.

Reentry trajectory planning for hypersonic vehicles via an improved sequential convex programming method

The reentry trajectory planning problem of hypersonic vehicles is generally a continuous and nonconvex optimization problem. The solution efficiency and convergence properties of the planning algorithm become an essential point when considering the practical application scenarios. In this paper, an augmented-Lagrange-based improved sequential convex programming algorithm is proposed to solve it and achieve superior solutions with higher accuracy and efficiency. Firstly, the mapped Chebyshev pseudo-spectral method is utilized to convert the continuous optimal control problem into a highly approximately discretized one. In this framework, the discretization technique is employed to discretize the states and control variables at the time interval, which leads to the improvement of ill-conditioned differential matrix and better approximation properties of the variables. Moreover, by introducing slack variables, automatically updated penalty parameters and Lagrange multipliers during the convexification processing, the original nonconvex problem is transformed into an augmented-Lagrange-function-formed relaxed convex one. Then, the arrived convex subproblem can be solved iteratively by the matured interior-point method until the predetermined precision is satisfied. Based on this transformation, it can effectively alleviate the numerical difficulty and improve the convergence property of the proposed improved sequential convex programming algorithm. Finally, Rigorous theoretical analysis is conducted to prove that the solution sequence generated by the proposed algorithm could converge to an approximate global solution of the original problem. Numerical simulations are also presented, and it is demonstrated that the solution time of the proposed algorithm is more than 95% less than that of the traditional nonlinear programming methods without losing the merits of high precision.

DDPG-Based Convex Programming Algorithm for the Midcourse Guidance Trajectory of Interceptor

To address the problem of low accuracy and efficiency in trajectory planning algorithms for interceptors facing multiple constraints during the midcourse guidance phase, an improved trajectory convex programming method based on the lateral distance domain is proposed. This algorithm can achieve fast trajectory planning, reduce the approximation error of the planned trajectory, and improve the accuracy of trajectory guidance. First, the concept of lateral distance domain is proposed, and the motion model of the midcourse guidance segment in the interceptor is converted from the time domain to the lateral distance domain. Second, the motion model and multiple constraints are convexly and discretely transformed, and the discrete trajectory convex model is established in the lateral distance domain. Third, the deep reinforcement learning algorithm is used to learn and train the initial solution of trajectory convex programming, and a high-quality initial solution trajectory is obtained. Finally, a dynamic adjustment method based on the distribution of approximate solution errors is designed to achieve efficient dynamic adjustment of grid points in iterative solving. The simulation experiments show that the improved trajectory convex programming algorithm proposed in this paper not only improves the accuracy and efficiency of the algorithm but also has good optimization performance.

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.

A computationally efficient sequential convex programming using Chebyshev collocation method

This paper aims to develop a computationally efficient sequential convex programming algorithm for a class of nonlinear systems. A Chebyshev collocation discretization (CCD) technique is proposed that transforms the continuous convex optimization problem into a finite-dimensional discrete optimization problem. The CCD technique uses Chebyshev polynomials to construct a closed-form approximate solution to the convex dynamic equation. Moreover, it establishes a linear mapping between the control inputs and the system states. This enables the constraints of the dynamics equations to be externalized from the optimization problem and computed using efficient linear equation solving algorithms. On the one hand, this strategy significantly reduces the number of constraint equations and optimization variables, improving the speed of the optimizer. On the other hand, it preserves the path constraints on the system states in the optimization problem. A numerical simulation example of the perching maneuver for a fixed-wing unmanned aerial vehicle is presented to validate the efficiency of the algorithm. The result shows significant improvements in both speed and accuracy compared to the Euler discretization, and greatly improves the solution speed while maintaining almost the same solution accuracy compared to the Gauss pseudospectral optimization method.

Real-Time Trajectory Planning for Hypersonic Entry Using Adaptive Non-Uniform Discretization and Convex Optimization

This paper introduces an improved sequential convex programming algorithm using adaptive non-uniform discretization for the hypersonic entry problem. In order to ensure real-time performance, an inverse-free precise discretization based on first-order hold discretization is adopted to obtain a high-accuracy solution with fewer temporal nodes, which would lead to constraint violation between the temporal nodes due to the sparse time grid. To deal with this limitation, an adaptive non-uniform discretization is developed, which provides a search direction for purposeful clustering of discrete points by adding penalty terms in the problem construction process. Numerical results show that the proposed method has fast convergence with high accuracy while all the path constraints are satisfied over the time horizon, thus giving potential to real-time trajectory planning.

Mixed-integer trajectory optimization with no-fly zone constraints for a hypersonic vehicle

This paper focuses on mixed-integer trajectory optimization of no-fly zones avoidance for a hypersonic vehicle. When there are multiple no-fly zones, the trajectory optimization problem has many local optimal solutions, and the performance of the solution depends on the selection of an initial guess. Essentially, the selection of the initial guess can be abstracted as a discrete path decision. To deal with this issue, this paper first integrates the discrete path decision and continuous trajectory optimization for no-fly zones avoidance and models these as a mixed-integer optimal control problem (MIOCP). As the corner stone, a creative directed graph is established to model the path decision problem, which is then formulated by integer constraints. Combined with the minimum effort optimal control problem (Problem 1), we form the MIOCP (Problem 2), thus integrating the path decision and trajectory optimization to improve global performance. Additionally, to solve the MIOCP, an effective iterative mixed-integer convex programming (IMICP) algorithm is customized by adding two ad-hoc techniques, relaxation and a proximity term, thus remedying the artificial infeasibility. Simulation results show that this approach is practically independent of the initial guess and has better performance than existing methods.

Whole body motion generation with centroidal dynamics of legged robots using sequential bounds tightening of McCormick envelopes

In this paper we introduce a Sequential Convex Programming (SCP) algorithm for the motion generation with the centroidal dynamics of legged robots using a sequential bounds tightening of McCormick envelopes strategy to cope with the nonconvexity of the problem (related to bilinear terms). Therefore, the proposed SCP algorithm is initialized with relaxed McCormick envelopes and then their bounds are sequentially tightened around the current estimate of the solution enforcing this way convergence to a feasible point. The SCP algorithm solves a quadratic program at each iteration by an interior point method. Additionally, the proposed SCP algorithm is alternated with an inverse kinematics algorithm to achieve the whole body motion generation. Finally, extensive numerical experiments show the effectiveness of the proposed algorithm in generating highly agile motions such as trotting, bounding, stotting and running for humanoid and quadruped robots.

Joint Optimization of Configuration, Component Sizing, and Energy Management for Input-Split Hybrid Powertrains

Configuration design, component sizing, and energy management are intertwined. However, the full potential of hybrid powertrains has not yet been thoroughly investigated due to the multi-dimensional design space. Therefore, in this paper, a joint optimization method combining particle swarm optimization (PSO) and convex programming (CP) is proposed for the first time. In the outer loop, the PSO algorithm searches for the optimal design for the powertrain configuration. In contrast, the component size and power management are simultaneously optimized by the CP algorithm in the inner loop. A combined driving cycle is designed to integrate both acceleration and energy-saving performance, where an extreme acceleration driving process is intended to act as the acceleration constraint. The results show that, compared to Prius 1st, the acceleration and total cost-saving of the optimal 1-PG (planetary gear) powertrain are improved by 13.09% and 11.08%, respectively. In particular, the acceleration and total cost-saving of the optimal 2-PG powertrain (MG2-RSG) are upgraded by 34.23% and 15.18%, respectively. Finally, the effectiveness and efficiency of the proposed method are verified by comparing it with dynamic programming (DP) and multi-objective evolutionary algorithm based on the decomposition (MOEA/D) method.

Energy-Efficient Trajectory Design of a Multi-IRS Assisted Portable Access Point

In this article, we propose a framework for energy efficient trajectory design of an unmanned aerial vehicle (UAV)-based portable access point (PAP) deployed to serve a set of ground nodes (GNs). In addition to the PAP and GNs, the system consists of a set of intelligent reflecting surfaces (IRSs) mounted on man-made structures to increase the number of bits transmitted per Joule of energy consumed measured as the global energy efficiency (GEE). The GEE trajectory for the PAP is designed by considering the UAV propulsion energy consumption and the Peukert effect of the PAP battery, which represents an accurate battery discharge profile as a non-linear function of the UAV power consumption profile. The GEE trajectory design problem is solved in two phases: in the first, a path for the PAP and feasible positions for the IRS modules are found using a multi-tier circle packing method, and the required IRS phase shift values are calculated using an alternate optimization method that considers the interdependence between the amplitude and phase responses of an IRS element; in the second phase, the PAP flying velocity and user scheduling are calculated using a novel multi-lap trajectory design algorithm. Numerical evaluations show that: neglecting the Peukert effect overestimates the available flight time of the PAP; after a certain threshold, increasing the battery size reduces the available flight time of the PAP; the presence of IRS modules improves the GEE of the system compared to other baseline scenarios; the multi-lap trajectory saves more energy compared to a single-lap trajectory developed using a combination of sequential convex programming and Dinkelbach algorithm.

Distributed primal outer approximation algorithm for sparse convex programming with separable structures

This paper presents the distributed primal outer approximation (DiPOA) algorithm for solving sparse convex programming (SCP) problems with separable structures, efficiently, and in a decentralized manner. The DiPOA algorithm development consists of embedding the recently proposed relaxed hybrid alternating direction method of multipliers (RH-ADMM) algorithm into the outer approximation (OA) algorithm. We also propose two main improvements to control the quality and the number of cutting planes that approximate nonlinear functions. In particular, the RH-ADMM algorithm acts as a distributed numerical engine inside the DiPOA algorithm. DiPOA takes advantage of the multi-core architecture of modern processors to speed up optimization algorithms. The proposed distributed algorithm makes practical the solution of SCP in learning and control problems from the application side. This paper concludes with a performance analysis of DiPOA for the distributed sparse logistic regression and quadratically constrained optimization problems. Finally, the paper concludes with a numerical comparison with state-of-the-art optimization solvers.

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.

Convex Trajectory Planning [About This Issue

This <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEEE Control Systems</i> issue includes one feature and one application for control. The feature “Convex Optimization for Trajectory Generation,” by Danylo Malyuta, Taylor P. Reynolds, Michael Szmuk, Thomas Lew, Riccardo Bonalli, Marco Pavone, and Behçet Açikmes¸e, provides a tutorial on the optimization methods that have proven themselves in both theory and practice to be fast and reliable at producing dynamically feasible trajectories for nonlinear systems subject to nonconvex constraints. The authors and their colleagues are original developers of the methods discussed: lossless convexification and two sequential convex programming algorithms called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SCvx</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GuSTO</i> . At their core, all three methods leverage a reliable, low-level convex optimizer that solves nonconvex control problems through problem reformulation, linearization, or both. Over the past decade, these methods have generated growing interest across academia and industry, including from organizations such as NASA, Masten Space Systems, SpaceX, and Blue Origin. The article provides the reader with an intimate understanding of what it takes to use—or even extend—each of the methods to generate nontrivial trajectories. Open source code written in the Julia language accompanies the article and can serve as a ready-to-use toolbox for the reader’s further work on sequential convex programming.

Model predictive control for close-proximity maneuvering of spacecraft with adaptive convexification of collision avoidance constraints

This study investigated model predictive control (MPC) for close-proximity maneuvering of spacecraft. It is essential for a designed MPC to effectively handle collision avoidance between the servicer spacecraft and the client spacecraft, especially while the client is rotating. The rotating motion of the client leads to dynamic changes in the collision avoidance constraints, which increases the difficulty of optimizing the control input in the MPC framework. Therefore, this study presents a method to improve the performance and computational efficiency of MPC for rendezvous and docking with a nonrotating or rotating client. An ellipsoid is adopted to model the client’s keep-out zone (KOZ). Given the spherical KOZ of the servicer, an expanded ellipsoid is introduced to describe the KOZ for the center of mass of the servicer and modeled as a nonlinear constraint. The linearization method for reference points located at the expanded ellipsoid is adopted to convexify the nonlinear constraints. The reference points are adaptively determined according to the positions of the servicer, client, and expanded ellipsoidal KOZ. The resulting hyperplanes are then used to describe the collision avoidance constraints. By utilizing the aforementioned strategies, combined with the calculated reference points, an adaptive convex programming algorithm suitable for real-time implementation of MPC is derived. The performance of the proposed method is demonstrated through numerical simulations.

Unmanned Aerial Vehicle (UAV)-Assisted Path Planning for Unmanned Ground Vehicles (UGVs) via Disciplined Convex-Concave Programming

Thelast decade has witnessed growing research interests in the path-planning problem for unmanned ground vehicles (UGVs) equipped with RGB-D cameras or LiDARs. However, apart from their extremely high cost, these sensors may be obscured by obstacles, which makes them impractical for many complex scenarios. Besides, most of the existing works for path planning have not considered energy efficiency for UGVs that are usually constrained by limited on-board batteries. Distinct from existing methods, this work presents a vision-based unmanned aerial vehicle (UAV)-assisted cooperative system for multiple UGVs. The proposed system harnesses the broad vision of UAV and operates in both general outdoor and global positioning system (GPS)-denied indoor environments. In sharp contrast to the conventional heuristic algorithms such as the rapidly exploring random tree (RRT) algorithm and the Dijkstra methods, the proposed energy-efficient path planning for UGVs is formulated as a non-convex optimization problem by considering the collision-aware obstacle-avoidance. More specifically, the proposed path-planning scheme contains two stages, namely the semantic segmentation that localizes UGVs and obstacles in the environment by exploiting the wide-angle camera mounted on the UAV, followed by the trajectory generation in which a disciplined convex and concave programming (DCCP) algorithm is devised for the non-convex energy-minimization problem. Extensive experiments validate the effectiveness of the UAV-UGV cooperative system and the proposed DCCP-based path-planning scheme.

Fusion Localization Algorithm Based on Robust IMM Model Combined with Semi-Definite Programming

With the continuous development of wireless sensor network (WSN) technology, WSN has gradually become one of the key technologies of the Internet, and is widely used in indoor target location technology. However, the obstacles will have a great influence on the distance measurement, and it will result in a large positioning error. Therefore, how to deal with the non-line-of-sight (NLOS) error becomes an important problem. In this paper, Interacting Multiple Model (IMM) was used to identify NOLS/LOS. The NLOS probability was calculated by Markov transform probability, and the likelihood function was calculated by extended Kalman filter (EKF). The NLOS probability was compared with the LOS probability to judge whether the measurement contained the NLOS error. A robust algorithm combining IMM model with semidefinite programming (IMM-SDP) was proposed. The improved convex programming algorithm was proposed to reduce the NLOS error. Simulation and experimental results showed that the proposed algorithm can effectively reduce the influence of NLOS error and achieve higher positioning accuracy compared with the existing positioning methods.