Lossless Convexification Research Articles

Lossless convexification and duality

The main goal of this paper is to investigate the strong duality of non-convex semidefinite programming problems (SDPs). In the optimization community, it is well-known that a convex optimization problem satisfies strong duality if Slater’s condition holds. However, this result cannot be directly generalized to non-convex problems. In this paper, we prove that a class of non-convex SDPs with special structures satisfies strong duality under Slater’s condition. Such a class of SDPs arises in SDP-based control analysis and design approaches. Throughout the paper, several examples are given to support the proposed results. We expect that the proposed analysis can potentially deepen our understanding of non-convex SDPs arising in the control community and promote their analysis based on KKT conditions.

Convex Approach to Real-Time Multiphase Trajectory Optimization for Urban Air Mobility

Electric vertical takeoff and landing technology has emerged as a promising solution to alleviate ground transportation congestion. However, the limited capacity of onboard batteries enforces hard time constraints for vehicle operations in a complex urban environment. Therefore, a constrained, real-time trajectory optimization method is necessary to enable safe and energy-efficient vehicle flight. Unfortunately, most existing trajectory optimization methods suffer from low computational efficiency, unpredictable convergence processes, and strong dependence on a good initial trajectory. In this paper, we employ a successive convex programming approach to solve the multiphase minimum-energy-cost electric vertical takeoff and landing vehicle trajectory optimization problem for urban air mobility applications. The main contribution is the development and validation of two novel convexification methods that find approximate optimal solutions to the multiphase nonlinear trajectory optimization problem in real time by solving a sequence of convex subproblems. Specifically, the first method transforms the original nonlinear problem into a sequence of second-order cone programming problems through a convenient change of variables and lossless convexification, while the second approach achieves a similar goal without the need for control convexification. The resulting convex subproblems can be solved reliably in real time by advanced interior point methods. The proposed methods are demonstrated through numerical simulations of two different urban air mobility scenarios and compared with the solution obtained from GPOPS-II, a state-of-the-art general-purpose optimal control solver. Our proposed sequential convex programming methods can obtain near-optimal solutions with faster computational speed than GPOPS-II.

탐색기 시야각 제한을 고려한 장거리 공대공 유도탄의 중기 궤적최적화

This paper describes a midcourse trajectory optimization for long-range missiles with a seeker field-of-view limit. Trajectory optimization is performed using convex programming, which is effective for real-time applications. Field-of-view limit, minimum dynamic pressure, and angle-of-attack limit are considered as path constraints. The terminal velocity is used as a performance index to ensure maneuverability in the terminal homing phase. The discontinuity of dynamics at the end of boost time is solved by separating the flight phase into a boost phase and a glide phase. The nonlinear dynamics are converted to input affine form using new input variables and partially linearized. The admissible input set is relaxed into a convex set through lossless convexification. Slack variables are introduced in the quadratic trust region and linearized path constraint, where they are augmented into the performance index for robust convergence. The transformed dynamics are discretized through a pseudo-spectral method and optimal trajectory is obtained using sequential convex programming. The effectiveness and robustness of the proposed method are verified through numerical simulation.

Strong observability as a sufficient condition for non-singularity and lossless convexification in optimal control with mixed constraints

Strong observability as a sufficient condition for non-singularity and lossless convexification in optimal control with mixed constraints

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.

Online trajectory optimization and guidance algorithm for space interceptors with nonlinear terminal constraints via convex programming

PurposeIn this paper, an online convex optimization method for the exoatmospheric ascent trajectory of space interceptors is proposed. The purpose of this paper is to transform the original trajectory optimization problem into a sequence of convex optimization subproblems.Design/methodology/approachFor convenience in calculating accuracy and efficiency, the complex nonlinear terminal orbital elements constraints are converted into several quadratic equality constraints, which can be better computed by a two-step correction method during the iteration. First, the nonconvex thrust magnitude constraint is convexified by the lossless convexification technique. Then, discretization and successive linearization are introduced to transform the original problem into a sequence of one convex optimization subproblem, considering different flight phases. Parameters of trust-region and penalty are also applied to improve the computation performance. To correct the deviation in real time, the iterative guidance method is applied before orbit injection.FindingsNumerical experiments show that the algorithm proposed in this paper has good convergence and accuracy. The successive progress can converge in a few steps and 3–4 s of CPU time. Even under engine failure or mission change, the algorithm can yield satisfactory results.Practical implicationsThe convex optimization method presented in this paper is expected to generate a reliable optimal trajectory rapidly in different situations and has great potential for onboard applications of space interceptors.Originality/valueThe originality of this paper lies in the proposed online trajectory optimization method and guidance algorithm of the space inceptors, especially for onboard applications in emergency situations.

Aerogravity-assist capture into the three-body system: A preliminary design

Aero-gravity assist (AGA) is a low-cost maneuver for an interplanetary vehicle since it can potentially replace thrust with aerodynamic force to save fuel consumption. In this study, an AGA-maneuver capture scheme to deliver a vehicle into a three-body system is presented, and a successive approximation-based method is proposed to optimize atmospheric trajectories. Based on the analysis of the aerodynamic deceleration process, the AGA capture maneuver is first formulated as a coplanar atmospheric flight process with fixed time and strict endpoint-state constraints. Then, the lossless convexification and successive approximation techniques are applied to construct the solving process of iteratively optimizing the AGA trajectories. Finally, the relaxation strategy to the strict terminal state constraints and the heavily penalized-function method are developed to avoid the artificial infeasibility of the problem and improve the convergence. Numerical simulations, demonstrated by the AGA-capture into Sun-Mars L2 Halo orbits and Sun-Mars backward stable orbits, show the effectiveness and reliability of the proposed method. It is the first time that the AGA-maneuver is combined with the dynamic property of the three-body system, and the proposed method can potentially enrich the possibility of the low-energy interplanetary transfer.

Lossless convexification of optimal control problems with annular control constraints

This paper presents new mathematical results for lossless convexification of optimal control problems with a non-convex annular control constraint. The problem is relevant because it is representative of a rocket landing problem. It was studied previously with an assumption at the final point (e.g., free final time), and it was shown that controllability is a sufficient condition to solve the problem as a sequence of convex programs. Herein, a sufficient condition is given for certain fixed time problems to be solvable as a single convex program. The main result is that controllability is also a sufficient condition for solving the general fixed time problem as a sequence of convex programs.

A convex approach to trajectory optimization for boost back of vertical take-off/vertical landing reusable launch vehicles

In this paper, a convex optimization algorithm is proposed to solve the online trajectory optimization problem of boost back of vertical take-off/vertical landing reusable launch vehicles. To achieve high-precision landing of launch vehicles, trajectory optimization of the boost-back flight phase considering the accuracy of entry is carried out, especially in emergencies. The trajectory optimization problem is formulated as an optimal control problem with minimum fuel consumption, and then it is transformed into a series of convex optimization subproblems, which can be solved by primal-dual interior-point method accurately and rapidly. During the transformation, flip-Radau pseudospectral discretization method, lossless convexification and successive convexification technology are applied. To drive the vehicle to predetermined entry points at the expected velocity, terminal constraints are expressed as orbital constraints of the endpoint in the boost-back flight phase. Considering the influence of Earth's rotation, the right ascension of the ascending node of the target orbit is updated according to the time and true anomaly at the end of the boost-back flight phase. Furthermore, the homotopy method is applied to the situation where there is no good initial guess when emergency happens. The algorithm presented in this paper performs well upon the simulation experiments of mission change and thrust decline. With good accuracy, high computational efficiency, and excellent robustness, the convex approach proposed has a great potential for onboard application in reusable launch vehicles and other space vehicles.

An Iterative Convex Programming Method for Rocket Landing Trajectory Optimization

A rapid trajectory optimization method is proposed to solve the fuel-optimal Earth-landing problem of reusable rockets, in which the nonlinear aerodynamic drag force is non-negligible. To enable the online and autonomous operation ability, the method is designed based on convex optimization, which features rapid and deterministic convergence properties, and a homotopic-iterative strategy is proposed to convexify the nonlinear system dynamics of the rocket. In the proposed iterative algorithm, the problem is first solved based on the lossless convexification method while the drag force is considered to be zero. Then, during subsequent iterations, the drag profile is approximated by the last solution and homotopically added to the problem. Thus, the nonlinear drag is gradually included while the problem remains convex. Because the convexification of the nonlinear terms is not based on linearization, no reference trajectory or initial guess is needed, which greatly enhances the autonomy of the algorithm. Numerical experiments are provided to demonstrate the effectiveness, rapidness, and accuracy of the proposed algorithm.

Convex relaxation for optimal rendezvous of unmanned aerial and ground vehicles

In this paper, we present an optimal rendezvous approach that facilitates the coordination of unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs). Of particular interest is the problem of UAV-UGV rendezvous for autonomous aerial refueling, which is an important capability for the future use of UAVs. The main contribution of this work is the development of a promising method for the generation of optimal rendezvous trajectories in real time using convex optimization. First, the UAV-UGV rendezvous problem is formulated as a nonconvex optimal control problem using an error dynamic model, where the state and control variables are highly coupled. Then, great effort is devoted to reducing the nonconvexity and coupling of the dynamics through change of variables. Based on a lossless convexification technique, a sequential convex programming algorithm is designed and the solution is obtained by solving a sequence of convex optimization problems. Theoretical proof is provided to demonstrate the exactness of convexification. Furthermore, a simple line search technique is introduced to handle the model error resulting from the linear approximations and to improve the convergence of the designed successive process. Simulation results of two refueling rendezvous situations validate that the proposed method is capable of converging to the solutions within a second, which shows great potential for real-time applications.

Online trajectory optimization for power system fault of launch vehicles via convex programming

This paper presents an online trajectory optimization algorithm for launch vehicles based on convex programming to ensure flight safety in case of power system fault. Due to high complexity of the power system, the engine may break down during the flight, causing significant decrease of thrust or energy. In this case, the nominal trajectory will be infeasible as the dynamical model and energy state is different from the normal status, thus the online trajectory optimization and re-planning are considered. For different kinds of engine failures, different terminal orbital constraints are proposed. When the mass flow rate of fuel decreases, the energy loss is little but the dynamical model changes obviously, so the location of the injection point cannot be guaranteed. In this case, the terminal orbital elements are constrained except the true anomaly, so that the payload of launch vehicles can still settle into the nominal orbit, and the true anomaly is optimized for minimum fuel consumption. As for the energy-loss failure, the strategy to change the target orbit is proposed considering the requirement of launch mission and subsequent orbit transfer insertion. The terminal constraints are proposed analytically in this paper. In order to solve the nonconvex trajectory optimization problem accurately and rapidly, the optimization problem is transformed into convex optimization problems by various convexification techniques, including the lossless convexification and successive convexification. Finally, the high efficiency and accuracy of the proposed algorithm is verified by numerical experiments. The algorithm proposed in this paper has potential applications in onboard trajectory optimization and re-planning of launch vehicles in case of power system fault to ensure the accomplishment of the launch mission.

A Convex Programming Method for Rocket Powered Landing With Angle of Attack Constraint

Real time trajectory planning is vital in rocket precision powered landing guidance. Due to the nonconvex angle of attack (AOA) constraint and other constraints, including nonlinear dynamics and thrust constraint, the powered landing trajectory planning problem is highly nonconvex, which makes it difficult to be solved in real time via existing nonconvex optimization algorithms. As the main contribution of this paper, the AOA constraint is taken into account in the problem. A convex feasible set method is presented to handle it, in which a quadratic concave function is introduced to find a convex feasible subset in the original AOA constraint and the second order term of this function is estimated by Gersgorin disc theorem. For the remaining nonconvexities, the lossless convexification is employed to address the thrust constraint, and the successive linearization is performed to handle the nonlinear dynamics. Thus, a convex optimization problem with AOA constraint for landing trajectory planning is built. The optimal solution to the original problem is obtained by iteratively solving convex problems. Numerical simulations show that the proposed method can find a feasible landing trajectory where the AOA constraint is satisfied and has a better convergence performance compared with the traditional linearization method.

Lossless Convexification of Optimal Control Problems with Semi-continuous Inputs

This paper presents a novel one-shot convex optimization method for finding globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems with non-convex constraints that restrict the input norms to be either zero or lower-and upper-bounded. The non-convex problem is relaxed to a convex one whose optimal solution is proved to be optimal almost everywhere for the original problem, a procedure known as lossless convexification. The solution relies on second-order cone programming and demonstrates that a meaningful class of optimal control problems with binary variables can be solved reliably and in polynomial time. A rocket landing example with a coupled thrust-gimbal constraint corroborates the effectiveness of the approach.

A Convex Programming Approach to Mid-course Trajectory Optimization for Air-to-Ground Missiles

This paper addresses trajectory optimization in the mid-course phase of an air-to-ground missile, when the main objectives are (a) to ensure that the target is locked on in the center of the missile’s field-of-view at a specified flight path angle and (b) to attain maximum possible speed to allow for sufficient maneuverability in the terminal phase. The method presents as a second-order cone program (SOCP) formulation for this trajectory optimization, taking advantage of partial linearization and lossless convexification techniques that effectively handle underlying non-convex characteristics of the problem. A well-established SOCP solver can then be readily used to obtain the optimal solution to this convex program. The proposed approach is validated by (a) proving the losslessness of the convexification, and (b) numerically comparing the results with an existing pseudo-spectral method.

Uncertain parameters analysis of powered-descent guidance based on Chebyshev interval method

This paper focuses on the powered-descent guidance (PDG) problem involving uncertain-but-bounded parameters. An efficient numerical algorithm based on the high approximation accuracy of the Chebyshev series expansion is presented to solve the nonlinear optimal control problem with interval uncertain parameters. First, an infinite number of uncertain optimal control problems is translated into a series of deterministic optimal control problems by exploiting the Chebyshev interval inclusion. Subsequently, the deterministic optimal control problem is solved by a convex optimization method based on the lossless convexification of the PDG problem. Afterwards, these techniques are used to generate the enclosure of the PDG trajectory with uncertainty and the upper and lower bounds of the minimum fuel consumption during the pinpoint landing mission. Finally, the effectiveness and superiority of the proposed approach are validated by illustrative numerical simulations.

Minimum-Fuel Low-Thrust Transfers for Spacecraft: A Convex Approach

This paper presents a convex approach to the numerical solution of the minimum-fuel low-thrust orbit transfer problem. The main contribution is the transformation of the original nonlinear optimal control problem into a sequence of convex optimization problems. First, the control is decoupled from the states through a change of variables. Then, by introducing a lossless convexification technique, the control constraints are convexified, and the original problem is relaxed into a sequence of second-order cone programming problems. The resulting subproblems can be solved in real time by efficient interior-point methods. Finally, the effectiveness of the proposed methodology is demonstrated through numerical simulations of the three-dimensional minimum-fuel Earth-to-Mars low-thrust transfer problem.

Fuel-Optimal Low-Thrust Trajectory Optimization Using Indirect Method and Successive Convex Programming

Although the robustness of indirect methods is enhanced by the homotopic approach and switching detection technique when applied to fuel-optimal low-thrust trajectory optimization, the bottleneck in adjoint initialization still needs further investigation. This paper overcomes this bottleneck by the adjoint mapping between the Lagrange multipliers of direct methods to the adjoint variables. The nonconvex optimization problem deduced from direct methods is converted into a convex one by lossless convexification and successive convex programming. By combining these techniques, a framework is built to effectively solve the fuel-optimal low-thrust trajectory optimization problem.

Lossless convexification of nonconvex MINLP on the UAV path‐planning problem

SummaryMost unmanned aerial vehicle path‐planning problems have been modeled as linear optimal control problems, ie, as mixed‐integer linear programming problems. However, most constraints cannot be described accurately in linear form in practical engineering applications. In this paper, the traditional unmanned aerial vehicle path‐planning problem is modified as a nonconvex mixed‐integer nonlinear programming problem, whose continuous relaxation is a nonconvex programming problem. A lossless convexification method is introduced into the generalized Benders decomposition algorithm framework. Thus, an optimal solution can be obtained without directly solving the nonconvex programming problem. The output of the proposed algorithm has been rigorously proved to be the optimal solution to the original problem. Meanwhile, the simulation results verify the validity of the theoretical analysis and demonstrate the superior efficiency of the proposed algorithm.