Introduction

Finds rotation matrix which rotates the x and y unit vectors at some position such that the satellites scanline "matches" a reference scanline. The optical (z) axis should be pointing towards the ground target.

Note: the across track direction (y') is the y axis projected to the plane tangential to the earth at the ground target.

Inputs

• unit position vector: np.array - N*3
• ground target: np.array - N*3
• across track direction vector: np.array - N*3

Outputs

• rotation matrix: np,array - 3*3*N

Implementation

1. The rotated y axis is in the plane shared by the z axis at the spacecraft's position and the across track direction vector at the ground target.
2. Therefore, the rotated x axis is normal to this plane and is equal to cross(z, y')
3. See 'Rotations Stage 1' for how to get rotation matrix which rotates x to desired orientation - process is the same as rotating z. This rotation matrix also orients the y axis correctly.

Introduction

Finds rotation matrix which rotates the z unit vector at some position such that it points towards the ground target.

Inputs

• unit position vector: np.array - N*3
• ground target: np.array - N*3

Outputs

• rotation matrix: np,array - N33

Implementation

• Source

• let the z unit vector before rotation be A and after rotation be B

• consider the basis vectors u, v, and w constructed by the vector projection of B on A, the vector rejection of B on A, and the normal of the plane which A and B form

• the change of basis matrix for this basis is C=[u v w]

• the rotation matrix R_z rotates vectors about the z axis

• cos(theta) and sin(theta) can be replaced with dot(A, B) and cross(A, B)

• U = C^-1 R_z C transforms A to the new basis, rotates it to the desired position, and then transforms it back to the original basis

Introduction

This module takes in a latitude and longitude on the surface of the Earth, then generates equivalent ECI positions given an input array of times

Inputs

• latitudes: np.array - N*1
• longitudes: np.array - N*1
• times: np.array - N*1

Outputs

• ECI positions: np.array - N*3

Implementation

Using Celest, we will do the following:

1. Generate a "fake" Satellite object using a code snippet similar to satellite = Satellite(position=gcrs, frame='gcrs', time=julian, offset=0)
2. This will take in an array of latitude and longitude positions corresponding to the location on the ground at each desired time
   etc.

Write requirements for all of these