# Analysis and Design of Infrared Search and Track System with Afocal Zoom Telescope
## 2 Homogeneous Coordinate Transformation in IRST
### Establishment of Coordinate System  
As shown in Figure 1, a typical IRST is taken as an example, which has an afocal zoom telescope with an external exit pupil, an imaging objective with an external entrance pupil, and an FSM set at the location of the inner pupil. When working in searching mode, the imaging system is mounted on the gyro-stabilized platform and rotates at a certain angular velocity \(\omega_{scan}\). In the meantime, FSM rotates accordingly at an angular velocity \(\omega_{FSM}\) to compensate for the transverse image wandering and helps obtain a high-resolution target image.  
The dynamic coordinate system of each optical element in the image system and the target should be established first, and the homogeneous coordinate transformation matrix of adjacent coordinate systems should be worked out successively. The establishment rules of each coordinate system are as follows (along the direction of light propagation):  
Figure 1: Reference frames of the general IRST model.  
C\({}_{\text{img}}\): Image plane coordinate system. The center point of the image plane is taken as the origin, the outside normal line of the image plane is Z-axis forward, and the direction parallel to the rotation axis of the scanning platform is Y-axis forward.  
C\({}_{\text{obj}}\): The imaging objective coordinate system is established by taking the main point of the image side of the lens set as the origin, the direction pointing to the object along the optical axis is Z-axis forward, and the direction parallel to the rotation axis of the scanning platform is Y-axis forward.  
C\({}_{\text{FM}}\): The folding mirror coordinate system. With the intersection point of the optical axis and the folding mirror as the origin, when it is in its initial 45\({}^{\circ}\) tilted state, the outer normal line of the mirror is Z-axis forward, and the direction parallel to the rotation axis of the scanning platform is Y-axis forward.  
C\({}_{\text{PSM}}\): The FSM coordinate system. Taking the intersection point of the optical axis and the fast steering mirror (FSM) as the origin, when it is tilted by 45\({}^{\circ}\), the outer normal line of the mirror is Z-axis forward, and the direction parallel to the rotation axis of the scanning platform is Y-axis forward.  
C\({}_{\text{tele}}\): The coordinate system of the afocal telescope system. The right-handed coordinates are set up on the image principal point of the telescope, the direction pointing to the object along the optical axis as the Z-axis forward direction and the direction parallel to the rotation axis of the scanning platform as the Y-axis forward direction.  
C\({}_{\text{s}}\): The platform coordinate system. Taking the intersection point of the platform rotation axis and the optical axis of the HST as the origin, the direction of the optical axis is Z-axis forward, and the direction parallel to the scanning platform rotation axis and upward is Y-axis forward.  
C\({}_{\text{f}}\): The target coordinate system. The intersection of the HST optical axis and the target surface is token as the origin, the direction pointing to the optical axis is Z-axis positive, and the direction parallel to the rotation axis of the scanning platform is Y-axis positive.