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A Convex Optimization Approach for Finite-Thrust Time-Constrained Cooperative Rendezvous

Boris Benedikter, Alessandro Zavoli, Guido Colasurdo

arXiv:1909.09443v1math.OC

TL;DR

The paper addresses finite-thrust cooperative rendezvous, where two spacecraft maneuver together under fixed-time and fuel objectives. It combines lossless and successive convexification with filtered reference updates, and reports convergence to optimal solutions from trivial initial guesses in short computation times. The approach remains subject to challenging future extensions such as highly elliptical target-orbit boundary conditions.

  • Problem

    Cooperative rendezvous is harder to initialize with indirect methods because two maneuvering spacecraft create many possible burn structures, while direct methods have computational and optimality drawbacks.

    Source: INTRODUCTION · pp. 1–2

  • Method

    The method combines lossless and successive convexification, variable changes, constraint relaxation, successive dynamics linearization, and filtered reference-solution updates.

    Source: INTRODUCTION · p. 2

  • Results

    The convex approach converges to optimal solutions from a trivial first guess, identifies the most convenient solution among multiple locally optimal families, and converges in a short time.

    Source: CONCLUSIONS · p. 16

  • Takeaways & Limitations

    The methodology provides a convex solution procedure for cooperative rendezvous while retaining the problem’s nonlinear dynamics and improving successive-convexification robustness.

    Source: CONCLUSIONS · p. 15

    Future work must address challenging convexification of boundary conditions for highly elliptical target orbits and test the filtering method on other problems.

    Source: CONCLUSIONS · p. 16

Abstract

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This paper presents a convex approach to the optimization of a cooperative rendezvous, that is, the problem of two distant spacecraft that simultaneously operate to get closer. Convex programming guarantees convergence towards the optimal solution in a limited, short, time by using highly efficient numerical algorithms. A combination of lossless and successive convexification techniques is adopted to handle the nonconvexities of the original problem. Specifically, a convenient change of variables and a constraint relaxation are performed, while a successive linearization of the equations of motion is employed to handle the nonlinear dynamics. A filtering technique concerning the recursive update of the reference solution is proposed in order to enhance the algorithm robustness. Numerical results are presented and compared with those provided by an indirect method.

INTRODUCTION

The paper formulates a finite-thrust cooperative rendezvous in which two spacecraft maneuver simultaneously, then develops a convex approach intended to address the initialization and computational challenges of existing methods.

INTRODUCTION

  • Indirect methods are accurate but difficult to initialize when burn and coast structures are unknown, especially with two simultaneously maneuvering spacecraft.Direct methods are easier to initialize but remain sensitive to the first guess, computationally burdensome, and of questionable optimality.Source: INTRODUCTION · pp. 1–2
  • A weighted filter over recent reference solutions is proposed to improve robustness against artificial unboundness or infeasibility during successive convexification.The paper reports that this simple filtering approach was effective for the studied problem.Source: INTRODUCTION · p. 2
  • The proposed method combines lossless and successive convexification to preserve the nonlinear dynamics while converting the problem into tractable convex subproblems.A variable change, constraint relaxation, and successive equation-of-motion linearization are used in the transcription.Source: INTRODUCTION · p. 2

ORIGINAL PROBLEM FORMULATION

  • Two spacecraft start in phase opposition on the same circular orbit and must rendezvous on a larger assigned circular orbit at a fixed final time.The spacecraft share initial mass, maximum thrust, and specific impulse; the optimization seeks trajectories and controls minimizing overall propellant use.Source: ORIGINAL PROBLEM FORMULATION · p. 2

System Dynamics

  • The spacecraft are modeled as point masses using polar Earth-centered inertial position coordinates and local-vertical-local-horizontal velocity components.The dynamics use an inverse-square gravity model, thrust as the control, and propellant depletion linked to thrust through effective exhaust velocity.Source: System Dynamics · p. 3

Objective Function

The objective evaluates mission performance through total propellant consumption, equivalently maximizing the combined final mass of the two spacecraft.

Objective Function

  • The optimization maximizes the sum of the final masses of the two spacecraft, which is equivalent to minimizing their overall propellant consumption.The satellite index identifies either spacecraft in the objective expression.Source: Objective Function · p. 4

Boundary Conditions

The formulation imposes shared final position and velocity together with a common target circular orbit, then organizes the dynamics for convex transcription through variable changes and successive linearization.

Boundary Conditions

  • Rendezvous requires both spacecraft to share final position and velocity, while the unbounded right-ascension angle requires accounting for relative extra revolutions.For the studied case, the optimal relative-revolution integer is always zero.Source: Boundary Conditions · p. 4
  • Both spacecraft must finish on the same circular orbit of assigned radius, although only semi-major axis and eccentricity are specified for the target orbit.Because rendezvous already equates final position and velocity, these orbit constraints can be imposed on only one spacecraft.Source: Boundary Conditions · pp. 4–5

CONVEX TRANSCRIPTION

  • The continuous problem is converted into a second-order cone program with a linear objective, linear equalities, and second-order cone constraints.The transcription proceeds through lossless convexification, successive linearization, and finite-dimensional discretization.Source: CONVEX TRANSCRIPTION · p. 5

Change of Variables

  • A change of variables replaces nonlinear dynamic terms with linear terms in a control-affine system without introducing approximation.The new controls include thrust direction and thrust-to-mass ratio, and additional state construction completes the affine dynamics.Source: Change of Variables · p. 5

Constraint Relaxation

The method relaxes a nonconvex control equality into a convex constraint while linearizing the remaining thrust bound, enabling mixed burn and coast arcs in one phase.

Constraint Relaxation

  • The nonlinear relation among the newly defined control variables is a nonconvex equality constraint.This relation is the target of the subsequent convex relaxation.Source: Constraint Relaxation · p. 6
  • Relaxing the control equality preserves useful nonlinear structure and can be exact when the relaxed and original problems share the same optimum.The paper identifies this relaxation as particularly favorable for successive-convexification convergence.Source: Constraint Relaxation · p. 6
  • The relaxed control relation becomes a second-order cone constraint, while the maximum-thrust bound is linearized around a reference solution.The thrust-to-mass variable remains bounded by the engine’s maximum thrust.Source: Constraint Relaxation · p. 6
  • Allowing burn and coast arcs within one phase avoids requiring a priori knowledge of the burn structure, which is valuable when both spacecraft maneuver simultaneously.A uniform mesh must nevertheless be dense enough for burn arcs, increasing the number of variables, though efficient algorithms partly mitigate the cost.Source: Constraint Relaxation · p. 6

Successive Linearization

Remaining nonconvexities are handled by linearizing constraints and dynamics around an iteratively updated reference solution. The resulting dynamics depend on the reference state rather than intermediate controls, improving iterative robustness.

  • Successive linearization handles remaining nonconvexities by linearizing constraints around an iteratively updated reference solution.Source: Successive Linearization · pp. 6–7
  • The equations of motion are linearized using state and control derivative matrices, with a control-affine formulation simplifying the result.Source: Successive Linearization · pp. 6–7
  • Reference-state-dependent dynamics prevent intermediate high-frequency control jitters from affecting the next iteration, providing robustness to the iterative method.Source: Successive Linearization · p. 7

Discretization

The continuous-time optimal control problem is converted into a finite-dimensional problem through direct transcription. Time, states, controls, differential constraints, and path constraints are represented on a mesh.

  • Direct transcription converts the infinite-dimensional optimal control problem into a finite-dimensional optimization problem.Source: Discretization · p. 7
  • Time is discretized into M −1 intervals and M mesh points, whose spacing need not be uniform.Source: Discretization · p. 7
  • State and control variables are discretized at mesh nodes, replacing differential constraints with algebraic defect constraints using trapezoidal integration.Source: Discretization · p. 7
  • Path constraints are converted into algebraic constraints by imposing them at every mesh node.Source: Discretization · p. 8

Mesh Refinement

Mesh refinement evaluates the continuous-time approximation against a discretization-error tolerance and adds nodes selectively where needed. The starting mesh must balance approximation accuracy against convergence behavior.

  • The discrete solution is converted into a continuous-time approximation using cubic-spline states and linearly interpolated controls to evaluate discretization error.Source: Mesh Refinement · p. 8
  • The integral used for error estimation can be evaluated with two trapezoidal half-steps on each interval.Source: Mesh Refinement · p. 8
  • If the error exceeds tolerance, new nodes are selected by minimizing the maximum discretization error while limiting mesh growth between refinements.Source: Mesh Refinement · p. 8
  • The initial mesh must be dense enough for accurate continuous-time approximation but coarse enough to avoid convergence problems.Source: Mesh Refinement · p. 8

Successive Convexification Algorithm

Successive convexification repeatedly solves approximating SOCP problems and updates the reference solution until successive references differ by a tolerance. A filtering update improves robustness, while initialization can use a simple coasting trajectory.

Successive Convexification Algorithm

Initialization

  • Although convex subproblems have globally optimal local minima, successive convexification still requires an initial reference solution.Source: Initialization · p. 9
  • A coasting trajectory on the initial orbit provides an effective initialization without guessing controls or burn-arc sequences, making initialization non-obstructive.Source: Initialization · pp. 9–10

COPLANAR CASE

In the coplanar case, the convex approach is evaluated against an indirect method across mission durations, including two coexisting solution families with distinct trajectory structures.

COPLANAR CASE

  • The planar experiment compares convex-programming results with an indirect-method solution for a rendezvous between identical satellites starting on the same circular equatorial orbit.The target orbit has radius r̃_f = 1.2, while the initial orbit radius is r̃_0 = 1.Source: COPLANAR CASE · p. 10
  • The equatorial setting makes latitude and normal velocity zero throughout the mission, allowing the final velocity condition to enter the convex formulation without linearization.This simplification follows from both spacecraft departing from the same orbital plane.Source: COPLANAR CASE · p. 10
  • The convex approach converges to the indirect method’s solution for every mission duration and automatically selects the less expensive family when both are viable.The comparison concerns overall propellant consumption.Source: COPLANAR CASE · p. 10
  • At approximately 11.165 time units, the spacecraft roles invert as one solution family becomes more convenient than the other.Durations below this threshold are classified as family A; longer durations as family B.Source: COPLANAR CASE · p. 11
  • Family A uses about one revolution for the leading satellite and one and a half for the other, with the leading vehicle on an external orbit.The second spacecraft remains near the initial orbit while recovering phase.Source: COPLANAR CASE · p. 11
  • Family B uses nearly two revolutions by the trailing satellite on a faster internal orbit, while the other spacecraft can wait on the final orbit.These trajectory patterns distinguish the two coplanar solution families.Source: COPLANAR CASE · p. 11

Phasing duty

The paper separates rendezvous effort into transfer and phasing costs, then evaluates convergence and extends the analysis to non-coplanar missions. The convex approach matches the indirect benchmark, identifies solution families, and remains convergent with an exact relaxation.

Phasing duty

  • For times longer than 17.17, phasing duty is null and total mission cost equals the transfer cost of both spacecraft.The transfer cost is 0.083 per satellite for all considered durations exceeding the Hohmann transfer time.Source: Phasing duty · p. 12
  • Family A increases cooperation through an elevated apoapsis for the leading satellite, whereas shorter missions force the trailing satellite to use lower, costlier orbits.Family B instead places the phasing burden entirely on satellite II through an internal-orbit maneuver.Source: Phasing duty · p. 12

Convergence Behavior

  • The algorithm converges for every investigated duration, with solutions already closely resembling the final result after five iterations.A 25-iteration limit is effective despite small oscillations in a few cases.Source: Convergence Behavior · p. 12
  • The transformed control-affine dynamics prevent intermediate control oscillations from undermining successive linearization.Trajectories stabilize early while burn structures can continue changing substantially across iterations.Source: Convergence Behavior · p. 12
  • The relaxed control constraint is exact in both coplanar and non-coplanar cases, since the computed controls satisfy the original equality within tolerance.Mesh refinement adds nodes only in intervals above tolerance, chiefly burn arcs, to retain a small problem size.Source: Convergence Behavior · p. 13, NON-COPLANAR CASE · p. 15

NON-COPLANAR CASE

  • In the non-coplanar case, convex and indirect methods produce practically identical overall propellant consumption, while the convex method selects the more convenient family at each duration.The scenario includes an expensive plane-change maneuver and two families that coexist over a limited time interval.Source: NON-COPLANAR CASE · pp. 13–14, Figure 4 · p. 14
  • Non-coplanar family A retains one-versus-one-and-a-half-revolution patterns, whereas family B uses nearly two revolutions by the trailing spacecraft.The three-dimensional families closely resemble those of the coplanar problem.Source: NON-COPLANAR CASE · p. 14, Figure 5 · p. 14
  • The optimal final inclination depends on how plane rotation is distributed, with family B showing oscillations because efficient rotation timing becomes important.For family A, the spacecraft performing most maneuvers also performs the greatest plane change, aided by combining rotation with energy changes near nodes.Source: NON-COPLANAR CASE · p. 15, Figure 6 · p. 15

CONCLUSIONS

The paper’s convex methodology solves cooperative rendezvous while preserving the problem’s nonlinear dynamics, and its filtering update improves robustness. It converges from a trivial first guess, identifies convenient solutions among multiple locally optimal families, and does so quickly, while highly elliptical target-orbit requirements remain challenging.

  • The convex methodology preserves the real nonlinear dynamics while combining lossless and successive convexification for cooperative rendezvous.Source: CONCLUSIONS · p. 15
  • The filtering update is simple but effective for improving successive-convexification robustness on the studied problem.Source: CONCLUSIONS · p. 15
  • Convergence to the optimal solution is achieved from a trivial first guess, despite two spacecraft having difficult switching-control structures.Source: CONCLUSIONS · pp. 15–16
  • The convex approach recognizes the most convenient solution among multiple locally optimal families identified by the indirect method.Source: CONCLUSIONS · p. 16
  • Future work must address target-orbit requirements such as highly elliptical orbits, whose boundary-condition convexification may be challenging.Source: CONCLUSIONS · p. 16
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