Homotopy Method (Homotopy Method)

Author: 天疆说

Contributing Institution: School of Astronautics, Harbin Institute of Technology, National Key Laboratory of Rapid Design and Intelligent Swarm of Small Spacecraft

Reference: 关宇同等. 面向航天器远距离协同交会的超参数自主调优-同伦方法[J]. 航天器环境工程, 2026.

Definition

The Homotopy Method is an iterative numerical method for solving nonlinear problems. Its basic idea is to construct a continuous deformation process from simple problems to complex problems, solving a series of intermediate problems to finally obtain the solution to the target problem. In orbit optimization, homotopy methods are often used to transition from easily solvable energy-optimal problems to difficult-to-directly-solve fuel-optimal problems.

Principles

Fuel-Optimal vs Energy-Optimal

In spacecraft rendezvous optimal control problems:

  • Fuel-optimal problem: Minimizes fuel consumption, with discontinuous bang-bang control law
  • Energy-optimal problem: Minimizes thrust energy, with continuous control law, easy to solve

Homotopy Construction

Introduce homotopy parameter ε∈[0,1]\varepsilon \in [0,1], construct performance index:

J=∑j=12λj0FjIspg0∫t0tf[uj−εuj(1−uj)]dtJ = \sum_{j=1}^{2} \frac{\lambda_{j0}F_j}{I_{sp}g_0} \int_{t_0}^{t_f} \left[u_j - \varepsilon u_j(1-u_j)\right] dt

  • When ε=1\varepsilon = 1: Degenerates to energy-optimal problem (continuous control)
  • When ε→0\varepsilon \to 0: Approaches fuel-optimal problem (bang-bang control)

Smooth Transition Mechanism

The homotopy method achieves smooth transition from energy-optimal to fuel-optimal by gradually decreasing ε\varepsilon:

εd=10−(d/15),d=1,2,⋯ ,N\varepsilon_d = 10^{-(d/15)}, \quad d = 1, 2, \cdots, N

Each step uses the co-state from the previous step as initial guess, effectively enlarging the convergence domain.

Application in Orbit Optimization

Solving Bang-bang Control Non-smooth Integration Problem

The bang-bang characteristics of fuel-optimal control cause discontinuities on the right-hand side of differential equations, preventing direct numerical integration. The homotopy method addresses this by:

  1. When ε>0\varepsilon > 0, control law is continuous and differentiable
  2. Gradually decrease ε\varepsilon to zero to obtain fuel-optimal solution
  3. Effectively avoids numerical difficulties from directly solving bang-bang control

Application by 赵海涵 et al. (2026)

赵海涵 et al. combined RLEPSO with homotopy method:

  • RLEPSO quickly obtains high-quality energy-optimal initial co-states
  • Homotopy method smoothly transitions to fuel-optimal control
  • Solved fuel-optimal problem for long-distance cooperative rendezvous under J₂ perturbation

Comparison with Continuation Method

CharacteristicHomotopy MethodContinuation Method
PurposeProblem smoothingOrbit family exploration
ParameterHomotopy parameter ε\varepsilonOrbit parameter (amplitude, period, etc.)
ApplicationOptimal control transitionPeriodic orbit family generation
Initial solution requirementLow (energy-optimal easily obtained)High (periodic solution required)

References

  • 关宇同, 高长生, 胡玉东, 赵海涵. 面向航天器远距离协同交会的超参数自主调优-同伦方法[J]. 航天器环境工程, 2026.
  • Topputo F, et al. A survey on direct optimal control via homotopy continuation[C]. AIAA/AAS Astrodynamics Specialist Conference, 2014.