Co-state Normalization (Co-state Normalization)

Author: Tianjiang Shuo

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

Co-state Normalization is the technique of dividing co-state variables by the Euclidean norm of their initial values, mapping infinitely many co-state solutions onto the unit sphere. In the shooting method solution of optimal control problems, co-state boundaries are typically unknown, leading to an overly large search space. Normalization effectively reduces the search space and improves the probability of finding convergent initial guesses.

Mathematical Description

Normalization Formula

For the co-state vector $\boldsymbol{\lambda}(t) = [\lambda_0; \boldsymbol{\lambda}_r(t); \boldsymbol{\lambda}_v(t); \lambda_m(t)]$, normalization is defined as:

After normalization:

Physical Significance

The physical significance of co-state normalization:

• Constrains co-state variables to the unit sphere
• Eliminates solution non-uniqueness (co-state multiplied by any non-zero constant still satisfies the equation)
• Reduces the search space from infinite to the unit sphere

Application in Two-Point Boundary Value Problems

Problem Description

In the indirect method for spacecraft orbit optimization, the optimal control problem is transformed into a Two-Point Boundary Value Problem (TPBVP):

• Initial state: Known $\mathbf{r}(t_0), \mathbf{v}(t_0), m(t_0)$
• Terminal state: Must satisfy $\mathbf{r}(t_f) = \mathbf{r}_{target}, \mathbf{v}(t_f) = \mathbf{v}_{target}$
• Unknown: Initial co-state $\boldsymbol{\lambda}(t_0)$

Complexity of Co-state Boundaries

Co-state boundaries are free (determined by transversality conditions), with each co-state component ranging in $[-\infty, +\infty]$. This leads to an almost infinite solution space for the shooting function.

Advantages of Normalization

Through co-state normalization:

1. Initial co-state is constrained to the unit sphere (7-dimensional manifold in 8-dimensional space)
2. Seven angular variables $\chi_\vartheta (\vartheta = 1, 2, \cdots, 7)$ are defined
3. Angular variables are mapped to optimization variables $X_\vartheta$ in $[0, 1]$ interval

Angular Variable Mapping

Application by Zhao Haihan et al. (2026)

In the RLEPSO-Homotopy method:

1. RLEPSO optimizes normalized co-states (7 angular variables + terminal time = 8 dimensional optimization variables)
2. Normalized co-states serve as initial guesses for homotopy shooting
3. Homotopy parameter decreases to zero to obtain fuel-optimal co-states

Related Concepts

• Shooting Method
• Co-state Variables
• Two-Point Boundary Value Problem (TPBVP)
• Homotopy Method
• Pontryagin's Maximum Principle

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.
• Betts J T. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming[M]. SIAM, 2010.