Arc-length Continuation

Author: 天疆说

Site: https://cislunarspace.cn

Definition

Arc-length continuation is a numerical method for tracing solution curves along a parameter branch. Rather than treating the parameter as a free variable to solve directly, an arc-length constraint is introduced in parameter space so that each iteration step advances along the arc direction, bypassing singular points (e.g., limit points) and enabling global tracing of solution branches.

Principles

Arc-length continuation is typically used together with the shooting method. Let λ\lambda be the parameter, x\mathbf{x} the state vector, and F(x,λ)=0\mathbf{F}(\mathbf{x}, \lambda) = 0 the shooting equation. The arc-length constraint is:

G(x,λ,s)=∥x−x0∥2+(λ−λ0)2−Δs2=0G(\mathbf{x}, \lambda, s) = \|\mathbf{x} - \mathbf{x}_0\|^2 + (\lambda - \lambda_0)^2 - \Delta s^2 = 0

where Δs\Delta s is the prescribed arc-length step and (x0,λ0)(\mathbf{x}_0, \lambda_0) is the current known solution.

Predictor-corrector steps:

  1. Predict: advance one step along the tangent vector to get (xp,λp)(\mathbf{x}_p, \lambda_p)
  2. Correct: use (xp,λp)(\mathbf{x}_p, \lambda_p) as initial guess and solve the coupled system of shooting equation and arc-length constraint via Newton iteration
  3. Step size control: adaptively adjust Δs\Delta s based on corrector convergence

Applications in Cislunar Space

  • Halo orbit family continuation: starting from planar Lyapunov orbits (Az=0A_z = 0), gradually increase AzA_z amplitude, solving each new orbit with shooting at each step
  • DRO orbit branch tracing: trace DRO configurations along the AxA_x parameter to construct the complete periodic orbit branch diagram
  • NRHO family analysis: trace the evolution of Earth-Moon L1/L2 NRHO branches as the mass parameter varies

Arc-length continuation extends the shooting method from single-orbit solving to systematic generation of entire orbit families, making it a core tool in libration point orbit design.

References

  • Zimovan E M. Characteristics and design strategies for near rectilinear halo orbits within the Earth-Moon system[D]. Purdue University, 2017.
  • Seydel R. Practical bifurcation and stability analysis[M]. Springer, 2010.