State Transition Matrix

Author: Tianjiang Says

Website: https://cislunarspace.cn

Definition

The State Transition Matrix (STM), denoted Φ(t,t0)\Phi(t, t_0), is a matrix that relates small changes in the initial state of a dynamical system to changes at a later time. It describes how perturbations propagate along a reference orbit.

δx(t)=Φ(t,t0)δx(t0)\delta \mathbf{x}(t) = \Phi(t, t_0) \delta \mathbf{x}(t_0)

Computation

The STM is computed by integrating the variational equations simultaneously with the equations of motion:

Φ˙(t,t0)=A(t)Φ(t,t0),Φ(t0,t0)=I\dot{\Phi}(t, t_0) = A(t) \Phi(t, t_0), \quad \Phi(t_0, t_0) = I

where A(t)A(t) is the Jacobian matrix of the dynamics and II is the identity matrix.

For the CR3BP, the state vector is 6-dimensional (position and velocity), so Φ\Phi is a 6×66 \times 6 matrix.

Applications

ApplicationDescription
Stability analysisEigenvalues of the monodromy matrix determine orbital stability
Differential correctionUsed to solve boundary value problems for periodic orbits
Orbit designSensitivity analysis for trajectory optimization
Station-keepingControl law design based on linearized dynamics

References

  • Chen Yuju. DRO Orbit Design and Control Research for Cislunar Space Situation Awareness[D]. 2024.