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Advances in Trajectory Optimization for Space Vehicle Control

Danylo Malyuta, Yue Yu, Purnanand Elango, Behcet Acikmese

arXiv:2108.02335v2math.OCcs.LGeess.SY

TL;DR

Space vehicle missions increasingly demand control that can handle hazardous environments and tight operational constraints, while optimization tools have become practical for onboard and ground-based use. This survey synthesizes convex optimization-based methods across major space vehicle control applications, emphasizing recent progress in lossless convexification, sequential convex programming, and model predictive control. It presents a broad state-of-the-art overview while noting that specific discretization and algorithm choices retain important limitations.

  • Problem

    Future space missions require control methods that handle hazardous landing environments and numerous operational constraints while pursuing effective trajectories.

  • Method

    The paper surveys convex optimization-based space vehicle control methods across launch, landing, rendezvous, attitude, atmospheric flight, and orbital applications.

  • Results

    The survey identifies lossless convexification, sequential convex programming, and model predictive control as core technologies in recent space vehicle control advances.

  • Takeaways & Limitations

    Convex optimization offers a technology-infusion path for space vehicle control because of solver reliability and existing flight heritage.

  • Takeaways & Limitations

    Runge-Kutta discretization is computationally attractive but generally does not exactly satisfy the original continuous-time dynamics, while constrained DDP remains an active research area.

Abstract

from arXiv · show

Space mission design places a premium on cost and operational efficiency. The search for new science and life beyond Earth calls for spacecraft that can deliver scientific payloads to geologically rich yet hazardous landing sites. At the same time, the last four decades of optimization research have put a suite of powerful optimization tools at the fingertips of the controls engineer. As we enter the new decade, optimization theory, algorithms, and software tooling have reached a critical mass to start seeing serious application in space vehicle guidance and control systems. This survey paper provides a detailed overview of recent advances, successes, and promising directions for optimization-based space vehicle control. The considered applications include planetary landing, rendezvous and proximity operations, small body landing, constrained attitude reorientation, endo-atmospheric flight including ascent and reentry, and orbit transfer and injection. The primary focus is on the last ten years of progress, which have seen a veritable rise in the number of applications using three core technologies: lossless convexification, sequential convex programming, and model predictive control. The reader will come away with a well-rounded understanding of the state-of-the-art in each space vehicle control application, and will be well positioned to tackle important current open problems using convex optimization as a core technology.

1. Introduction

Optimization-based control is becoming practical for space vehicles as computing and software improve, while demanding missions increasingly require explicit handling of operational constraints. The survey focuses on convex optimization methods because they combine reliable solution methods with flight heritage across a broad range of space applications.

  • Computing hardware and software libraries have made iterative onboard computational guidance and control practical for space vehicle applications.
  • Future space missions require optimization-based control to manage hazardous landing sites, close-proximity cargo delivery, autonomous docking, and other tightly constrained operations.
  • Optimization directly imposes system constraints, which is important because optimal trajectories often operate near feasible-set boundaries.
  • Convex optimization is attractive because efficient algorithms can solve its formulations quickly with relatively low computational resource use.
  • The optimization taxonomy extends from LP, QP, and SOCP to SDP, NLP, MICP, and MINLP, with mixed-integer nonlinear programs forming the most difficult class.
  • The survey covers convex optimization-based control for launchers, landers, satellites, spacecraft, and atmospheric entry vehicles, while excluding purely atmospheric vehicles and satellite swarms.

2. Background on Optimization Methods

The survey’s algorithm overview emphasizes convexity because convex optimization offers strong convergence guarantees with low computational cost.

  • The algorithm overview focuses on methods that exploit convexity in space vehicle trajectory optimization.
  • Convex optimization is emphasized because state-of-the-art solvers provide strong convergence guarantees at relatively low computational cost.
  • The overview begins with continuous-time optimal control theory before developing methods used in spaceflight applications.

2.1. Optimal Control Theory

Optimal control formulates trajectory design as a constrained problem over continuous state and input trajectories, requiring discretization or indirect solution methods for computation. Direct methods are generally preferred for constrained onboard implementation, while indirect methods remain valuable for numerically sensitive problems and as analysis or benchmarking tools.

  • Optimal Control Problem: Optimal control seeks an input trajectory that optimizes a cost while satisfying dynamics, path constraints, and boundary conditions.
  • Optimal Control Problem: Continuous-time optimal control is semi-infinite because its input trajectory has infinitely many design parameters and cannot be implemented directly on a computer.
  • Indirect Methods: Indirect methods solve necessary optimality conditions through a two-point boundary value problem, but traditionally require advance specification of active-constraint intervals.
  • Indirect Methods: Indirect methods can be the only practical option for extreme numerical sensitivity or long-duration low-thrust trajectories.
  • Indirect Methods: Homotopy methods address slow convergence, initial-guess quality, and active-constraint specification by following solutions across a parameterized problem family.
  • Direct Methods: Direct methods discretize optimal control into parameter optimization, handle constraints effectively, and are preferred for real-time onboard implementation, whereas indirect methods provide analysis and reference solutions.

2.2. Discretization

Discretization converts continuous-time dynamics into finite-dimensional algebraic constraints on a temporal grid, using exact, approximate, or pseudospectral representations. The section compares these approaches through sparsity, trajectory recovery, and accuracy considerations.

  • Discretization overview: Discretization partitions time into N nodes and fixes a basis for state, control, or both, converting differential constraints into finite-dimensional algebraic constraints.The survey discusses zeroth-order hold, Runge-Kutta, and pseudospectral discretizations.
  • Zeroth-order hold: Zeroth-order hold represents the input as a staircase signal and exactly matches continuous-time states at grid nodes when that assumption holds.This property makes ZOH relevant to actuators that output staircase commands.
  • Zeroth-order hold: ZOH discretization propagates states across intervals using the state transition matrix, with the dynamics expressible as a stacked linear system whose sparsity can accelerate optimization.For the mass-spring-damper example, both F and G consist largely of zeros.
  • Runge-Kutta discretization: Unlike ZOH, Runge-Kutta solutions generally do not exactly satisfy the original continuous-time dynamics, despite potentially lower computational cost in some cases.The difference is especially relevant for linear time-varying dynamics, where RK4 avoids integrals required by ZOH.
  • Pseudospectral discretization: Pseudospectral collocation recovers an approximate continuous-time trajectory from state values at temporal grid nodes through interpolating polynomials.The basis coefficients correspond to trajectory values at the nodes, while equation (24) supplies values at other times.

2.3. Convex Optimization

Convex optimization provides reliable, computationally efficient tools for trajectory problems when dynamics and constraints can be represented convexly or transformed without changing the solution. Lossless convexification handles important nonconvex constraints by lifting them into a convex space while preserving optimality.

  • Convex optimization: Convex optimization is emphasized because modern solvers offer strong convergence guarantees at relatively low computational cost.The section presents convex optimization as a practical subset of broader trajectory-optimization problem classes.
  • Convex trajectory formulation: A discretized convex trajectory problem combines a convex cost, affine boundary conditions, linear time-varying dynamics, and convex path constraints.The dynamics are represented by x_k+1 = A_kx_k + B_ku_k + d_k, while path constraints use g(x_k, u_k, t_k) ≤ 0.
  • Lossless convexification: Lossless convexification exactly reformulates certain nonconvex constraints so that the solution set is neither reduced nor expanded.The survey identifies thrust lower bounds in rocket landing as a prominent example.
  • Lossless convexification: For powered descent, lifting the thrust input into an extra dimension creates a convex feasible set, and the maximum principle shows the optimum lies on its boundary.This boundary result establishes that the lifted convex problem yields an optimal solution for the original nonconvex problem.
  • Lossless convexification: Constrained spacecraft attitude reorientation admits another lossless convexification when a nonconvex quaternion constraint is combined with the unit-quaternion condition.Its proof relies on the geometry of the unit quaternion rather than the maximum principle.

2.4. Sequential Convex Programming

Sequential convex programming (SCP) iteratively solves convex approximations of nonconvex trajectory problems, updating them as new solutions are obtained. Its variants differ in approximation, update, progress-measurement, and convergence-analysis choices, while requiring only first-order information.

  • SCP is a family of nonconvex local optimization methods that iteratively solves and updates convex approximations of the trajectory problem.
  • SCP operates as a predictor-corrector loop: a forward quality check is followed by a convex solver correction when needed.The correction subproblem is a better convex approximation of the original problem.
  • Trust-region SCP formulations regulate distances from previous solutions and penalize virtual constraint violations to encourage convergence and feasibility.The trust-region sizes η_k can themselves be optimization variables in penalized trust-region methods.
  • Modern SCP variants such as SCvx, GuSTO, and PTR provide stronger convergence guarantees than earlier trajectory-generation methods.SCvx uses a discrete-time convergence proof, GuSTO uses the continuous-time maximum principle, and PTR penalizes trust-region sizes within the optimization problem.
  • Unlike SQP, SCP solves convex subproblems to full optimality and requires only first-order information, avoiding Hessian factorization and related positive-semidefiniteness concerns.This increases reverse-path iterations but can reduce forward passes because mature interior-point solvers make those iterations relatively inexpensive.
  • Related Algorithms: DDP has demonstrated successful space-trajectory applications but originally lacked natural handling of state and input constraints and is computationally more expensive as a second-order method.Constraint handling for DDP remains an active research area, with penalty, barrier, augmented-Lagrangian, active-set, and primal-dual extensions.

2.5. Mixed-integer Programming

Mixed-integer programming represents discrete events and conditional constraints with binary variables, but its worst-case runtime limits onboard use. Continuous alternatives such as STCs and homotopy methods approximate or embed the same logic while trading global optimality or generality for efficiency.

  • Mixed-integer programming uses binary variables to represent discrete events such as valve opening, thruster pulsing, and mission-phase transitions.Binary variables can also approximate nonlinear gravity, aerodynamic drag, and other trajectory features.
  • Big-M mixed-integer constraints activate conditional constraints through binary trigger variables and an activation function.When a binary trigger equals one, the corresponding trigger condition holds and the constraint is activated.
  • Traditional MIP solvers use branch-and-bound and additional presolve, cutting-plane, heuristic, parallel, branching, and symmetry methods, but worst-case runtime remains exponential in n_ζ.This computational demand hinders onboard implementation on space-vehicle hardware.
  • Continuous alternatives trade MIP global optimality for speed by approximating conditional statements without introducing integer variables.The survey presents state-triggered constraints and homotopy-based methods as two such approaches.
  • State-triggered Constraints: State-triggered constraints embed discrete if-then logic into continuous direct formulations and can handle AND and OR combinations of triggers and constraints.STCs encode forward implications rather than bidirectional statements, leaving a larger feasible space that can yield a more optimal solution.
  • Homotopy Methods: RASHS and CSC approximate binary activation with homotopy-controlled continuous sigmoids, increasing κ until the solution becomes indistinguishable from its MIP counterpart.For a fixed κ, CSC is more localized around the y-axis and therefore closer to a step signal; current RASHS and CSC formulations handle only AND combinations, while OR handling has been demonstrated in a related homotopy framework.

2.6. Model Predictive Control

Model predictive control (MPC) repeatedly solves a constrained optimization problem using the current state estimate to update feedback actions. Its constraint-handling advantages are balanced by requirements for stability, robustness, recursive feasibility, and sufficient execution frequency.

  • MPC repeatedly re-solves an optimization problem in the feedback path using the current state estimate to update the control signal.This makes the optimization problem parametric in the estimated state.
  • An MPC formulation combines dynamics, path constraints, a cost function, and terminal conditions to produce the first optimal control input.The first input is applied to the system before the problem is solved again at the next state estimate.
  • Terminal weights and constraints must support stability and recursive feasibility, meaning the problem remains feasible at the next solve.
  • MPC naturally handles system constraints in feedback control, but closed-loop stability and performance depend strongly on uncertainty robustness and execution frequency.These requirements are especially important where onboard computation is limited.
  • Explicit MPC pre-computes a lookup table for the MPC solution, exactly for quadratic programs and approximately for more general cases.

3. Applications

The applications section surveys optimization methods across major spaceflight control problems, including planetary landing, rendezvous, small body landing, attitude reorientation, atmospheric ascent and entry, and orbit transfer.

  • The survey applies optimization methods to rocket-powered planetary landing, spacecraft rendezvous, small body landing, and attitude reorientation.
  • It also covers endo-atmospheric ascent and entry and additional spaceflight control applications described later in the survey.

3.1. Powered Descent Guidance for Rocket Landing

Powered descent guidance seeks soft, precise touchdown while satisfying operational constraints, and optimization-based methods have expanded the feasible landing scenarios. Lossless convexification, sequential convex programming, and feedback-based approaches provide practical routes toward constrained and uncertainty-aware guidance.

  • Powered Descent Guidance: Powered descent guidance targets soft, precise touchdown using rocket engines during the terminal kilometers of entry, descent, and landing.The capability is important for reducing cost and reaching hazardous, scientifically rich sites.
  • Lossless Convexification: Lossless convexification solved a 3-DoF fuel-optimal powered-descent problem while handling thrust bounds and glide-slope constraints.The approach marked a major advance in solving constrained powered-descent guidance problems.
  • Sequential Convex Programming: Sequential convex programming supports tailored nonlinear formulations and real-time state-triggered constraints, including triggered angle-of-attack, line-of-sight, and collision-avoidance conditions.Compound state-triggered constraints can combine multiple trigger and constraint functions with Boolean logic.
  • Computational Performance: Pseudospectral discretization within a sequential convex programming framework yielded solutions up to 20 times faster than sequential quadratic programming.This result illustrates the computational potential of tailored discretization and optimization methods for guidance.
  • Uncertainty and Feedback: Feedback-feedforward designs incorporate a feedback gain into nominal trajectory generation to address deviations from the nominal trajectory.The surveyed approaches include simultaneous feedback-feedforward design, desensitized control, chance constraints, covariance steering, and statistical robustness analysis.

3.2. Rendezvous and Proximity Operations

RPO trajectory optimization brings an actively controlled chaser to a prescribed configuration with a target while addressing fuel use, collision avoidance, realistic thrust, robustness, and coupled position-attitude dynamics.

  • RPO brings an actively controlled chaser and passively controlled target to a prescribed relative configuration for objectives such as inspection or docking.
  • Trajectory optimization commonly minimizes fuel, while alternative objectives include control sparsity, flight-duration trade-offs, smoothness, and uncertainty sensitivity.
  • Collision avoidance is a central RPO challenge, with direct trajectory optimization offering an alternative to computationally demanding precomputed virtual nets.
  • Obstacle avoidance can use mixed-integer constraints for polytopes or conservative halfspace constraints for ellipsoids, with linearized ellipsoids requiring sequential convex programming.
  • Realistic reaction-control systems motivate nonconvex minimum-impulse constraints, while polynomial optimization, semidefinite programming, and related relaxations provide alternative solution methods.
  • Finite-duration thrust models represent pulse-width modulation, with PWPF shown to save fuel and maintain stability but potentially conflict with fixed-duration hardware.
  • Optimization-based RPO increasingly considers six-degree-of-freedom position-attitude coupling, while MPC supports online replanning under modeling errors, disturbances, and measurement noise.

3.3. Small Body Landing

Small body landing extends landing optimization to rotating, irregular celestial bodies with nonlinear gravity and uncertain parameters, requiring low-velocity touchdown and hazard avoidance.

  • Small body landing differs from planetary landing through highly nonlinear dynamics caused by irregular shape, density, and rotation.
  • The landing objective combines soft touchdown with plume-impingement and collision-avoidance requirements under rotational and gravitational nonlinearities.
  • The prototypical problem uses rotating-frame dynamics containing general nonlinear gravity and inertial forces from the small body’s angular velocity.
  • Early tube MPC work combined a robust SOCP feedforward trajectory with an SDP-optimized feedback controller while simplifying mass and dynamics.
  • A recent six-degree-of-freedom, variable-duration formulation imposed a landing-camera field-of-view constraint and was solved using sequential convex programming.
  • MPC methods address uncertainty in small-body parameters, including tube MPC and input observers for gravity-modeling errors.

3.4. Constrained Reorientation

Constrained reorientation optimizes large-angle spacecraft attitude maneuvers while keeping sensitive instruments away from bright celestial objects, especially direct sunlight or heat.

  • Scientific observation satellites require large-angle reorientation while protecting sensitive instruments from direct sunlight or heat.
  • The basic reorientation problem uses conical keep-out zones for multiple stars and prescribed initial and terminal attitude states.
  • The main computational challenge is the nonconvexity of attitude dynamics and keep-out constraints.
  • Lossless convexification can replace the star keep-out constraint with convex quadratic constraints when the quaternion remains unit norm.
  • The first greedy convex method is computationally efficient but conservative and may fail to find a feasible trajectory.
  • Alternative approaches include semidefinite programming with successive linearization, pseudospectral and nonlinear programming methods, indirect shooting, and differential evolution.

3.5. Endo-atmospheric Flight

Endo-atmospheric optimization covers ascent and entry, where aerodynamic and thermal effects create strongly nonlinear, constrained guidance problems addressed by indirect methods, sequential convex programming, and convex reformulations.

  • Ascent Flight: Ascent transfers a launch vehicle’s payload from the planetary surface to orbit while minimizing quantities such as fuel.
  • Ascent Flight: Indirect orbit-transfer methods face numerical difficulty because transversality multipliers lack physical interpretation and can vary widely in magnitude.
  • Ascent Flight: Atmospheric ascent is complicated by strong coupling between aerodynamic and thrust forces, motivating open-loop implicit guidance and specialized optimization methods.
  • Ascent Flight: Homotopy methods incorporate aerodynamics by starting from an optimal vacuum-ascent initial guess, while indirect methods solve ascent optimality conditions using finite differences.
  • Ascent Flight: Modern ascent formulations use angle of attack to direct thrust while enforcing dynamic-pressure and bending-moment constraints for structural integrity.
  • Ascent Flight: Strong nonlinearities make sequential convex programming typical for ascent, often combined with control-affine reformulations and lossless convexification.
  • Atmospheric Entry: Convex optimization contributes efficient entry-trajectory algorithms, while reformulations and filtering strategies can reduce control-signal jitter.
  • Atmospheric Entry: Entry optimization addresses jitter, a common high-frequency control oscillation associated with nonlinear coupling of state and control constraints.

3.6. Orbit Transfer and Injection

Orbit transfer and injection transport low-thrust vehicles to target orbits while minimizing objectives such as time or fuel. Recent convexification and sequential convex programming methods address nonlinear insertion constraints and support automated mission planning.

  • Orbit transfer transports a low-thrust vehicle from an initial orbit to a target orbit while minimizing quantities such as time or fuel.
  • Recent direct methods apply convex optimization technology to orbit-transfer problems that can be highly nonconvex.
  • The nonlinear orbit-insertion constraint can be linearized with a second-order correction term, yielding a sequence of SOCPs through SCP.
  • The same formulation can model optimal exo-atmospheric ascent, with the initial condition typically representing burnout from the previous launch-vehicle stage.
  • Mission planning can combine multiple orbit-transfer solutions, including planetary flybys and gravity assists, to minimize overall fuel usage or travel time.
  • The evolutionary mission trajectory generator produced a Cassini-like Saturn mission fully automatically at a fraction of the actual mission’s time and cost.

4. Outlook

The outlook emphasizes software maturity, rigorous performance guarantees, and emerging alternatives such as funnel libraries and reinforcement learning. These directions aim to extend efficient optimization-based control while addressing real-world reliability and learning challenges.

  • 4. Outlook: The survey identifies convex optimization-based techniques for next-generation space vehicle control across landing, rendezvous, attitude, orbit-transfer, ascent, and reentry applications.
  • 4. Outlook: Modern software packages support the surveyed algorithms, but the listed collection is popular and mature rather than complete.
  • 4.2.1. Guaranteed Performance: Optimization-based algorithms still require rigorous certification that convergence produces safe and robust real-world operation; general guarantees remain limited.
  • 4.2.1. Guaranteed Performance: Funnel libraries pre-compute trajectories and invariant controllers, potentially replacing onboard optimization with lookup and numerical integration for difficult nonlinearities.
  • 4.2.2. Machine Learning: Reinforcement learning is attractive because it can optimize over stochastic data streams without assuming a particular dynamic-model description.
  • 4.2.2. Machine Learning: Learning solution hyperparameters instead of high-dimensional state-input maps can make spacecraft trajectory reinforcement learning easier and require less training data.For 3-DoF powered descent guidance, 10 hyperparameters were learned instead of a 7D-state-to-3D-input map, using approximately 10^3 training trajectories.
  • 4.2.2. Machine Learning: Reinforcement-learning controllers require performance guarantees before serious onboard consideration because neural-network behavior outside training data remains difficult to characterize.
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