Sensitivity of the propagated two-body state to the initial state.
Definition
- State transition matrix (STM): the Jacobian matrix (matrix of partial derivatives) of the orbital state at the final time with respect to the state at the initial time
- Notation: initial position and velocity subscript 1, final subscript 2
- , , , are blocks
- Propagation itself uses the Lagrange coefficients; the STM needs their derivatives too
Battin’s Method
- References:
- Battin’s method
- Arora, Russell, Strange, Ottesen, “Partial derivatives of the solution to the Lambert boundary value problem”, JGCD, 2015, doi: 10.2514/1.G001030
- Universal formulation: Battin’s technique formulates the STM blocks , , , in universal variables, so one set of formulas covers every conic
Universal Formulation
- Inverse semimajor axis from the initial state:
- Radial-velocity-like terms at both ends:
- Universal anomaly from the time of flight (TOF):
- Auxiliary quantities (with the Lagrange coefficients , ,
over the transfer):
- And
STM Blocks
- is the identity matrix; bold , are vectors, , magnitudes
Position-to-Position Submatrix
Velocity-to-Position Submatrix
Position-to-Velocity Submatrix
Unclear in notes
- In the term inside the brackets is printed without a transpose, which does not give a matrix; presumably it is the outer product . Check against Battin / Arora et al.
Velocity-to-Velocity Submatrix
- The identity terms (, , , ) are the propagation matrix; the rest accounts for , changing with the initial state
- Other STMs in the course: the HCW STM for relative motion about a circular orbit, and the Yamanaka-Ankersen STM for elliptical reference orbits