Transformation between common coordinate systems and coordinate systems

Coordinate System: What coordinate systems are there and what is the transformation matrix between them? (Mainly three coordinate systems and three corners)

Geographic coordinate system (N Series)

Take the coordinate system origin on as the aircraft center. The XN axis points to the north, the YN axis points to the day, and the Zn axis points to the east.

Simply put, XYZ is North Tiandong.

The relationship between the geographic coordinate system and the Earth coordinate system is the geographic location of the aircraft, that is, the longitude λ and latitude L.

Load (bullet) body coordinate system (Series B)

The origin 0 of the coordinate system is the center of the carrier (missile. The ox axis is the axis of symmetry (vertical axis) of the carrying (bullet) body shell, pointing to the missile head. Define an instantaneous shot as the primary symmetric plane of the missile. The oy axis is perpendicular to the ox axis (also called the vertical axis) in the primary symmetric plane of the missile ). The Oz axis (horizontal axis) is perpendicular to the primary symmetric plane and points to the right side along the transmitting direction. The oy axis points to the direction of the right-hand Cartesian coordinate system.

The source ob of the coordinate system is the aircraft center. The XB axis follows the vertical axis of the carrier, the Yb axis along the vertical axis of the carrier, and the ZB axis along the side axis of the carrier.

Simply put, X is the missile flight direction, Z points to the right, and Y is the right hand coordinate system (that is, pointing to the day)

PS: in the right-hand coordinate system, the thumb points to Z, and the finger bending direction is 90 ° from X to Y.

The relation between the carrier coordinate system and the geographic coordinate system is the carrier's posture.

Speed Coordinate System

The missile center is the origin O. the OXC axis follows the movement direction of the missile center and indicates that the forward direction is positive. The oyc axis is in the vertical symmetric plane of the missile, perpendicular to OXC, and positive upward. The ozc axis is perpendicular to the OXC and oyc planes, the direction is determined by the right-hand Cartesian coordinate system.

To put it simply, it is the same as the coordinate system on the bullet

Other coordinate systems

1. World coordinate system (W-BLH)

W is the origin, B is the longitude, L is the latitude, H is the altitude

2. The image-Space Coordinate System of the camera in the lower-view of the aircraft's fixation

The Origin O of the o_uvz Coordinate System of the image of the camera in the lower-View frame of the aircraft is the camera light center and the center of the aircraft. The camera's optical axis overlaps with the Oz axis, and the Oz axis is re-merged with the vertical axis of the aircraft and directed to the roof at the bottom. The forward and vertical axes of the ov axis are re-merged, and the head points to the tail. The ov axis and the oy axis overlap, but the forward direction of the ov axis is opposite to that of the oy axis. The ou axis is perpendicular to the vertical axis of the aircraft and points to the left wing from the right wing. The ou axis and the ox axis overlap, but the ou axis is opposite to the ox axis.

3. plane coordinate system of the image taken by an aircraft

When taking a photo, the starting point C of c_hw in the photo plane coordinate system is the starting point in the upper left corner of the image. The CW axis is adding direction to the photo column. The CW axis is parallel to the ov axis and is parallel to the vertical axis of the aircraft and points the head to the end of the camera. The ch axis is adding direction to the row of the photo. The ch axis is parallel to the ou axis and perpendicular to the Aircraft Vertical Axis and points to the left wing from the right wing.

Three important points

There are three kinds of missile attitudes: the attitude angle that rotates around the X axis (along the tangent direction of the orbit forward), which is called rolling. The attitude angle that rotates around the Y axis (perpendicular to the orbital plane) is called pitching; the attitude angle that rotates around the Z axis (perpendicular to the 0-xy plane), which is called the deviation.

Pitch Angle

The angle between the X axis and the horizontal plane of the coordinate system. When the positive and half axes of the X axis are located on the horizontal plane above the coordinate origin, the pitch angle is positive. According to habits, the range of the pitch angle θ is-π/2 ≤ θ ≤ π/2. (In short, the upper part is positive and the lower part is negative)

Pitch angle = Angle of Attack + trajectory angle (understanding)

Flight angle Yaw

The angle between the missile vertical axis projection on the horizontal plane and the ax axis of the ground coordinate system (in the horizontal plane, pointing to the target is positive). The yaw angle is positive when the ax axis is clockwise transferred to the projection line of the missile vertical axis, the opposite is negative (the yaw angle in the figure is positive ). (In short, the left is positive and the right is negative)

Roll (also called roll angle and roll angle)

The angle between the vertical line of the target object and its projection on the horizontal plane.

Side Slide (AOS, angle of sideslip)

The angle between the missile axis and the missile's flight speed in the horizontal plane.

The slide angle is the angle between the velocity vector V and the missile's longitudinal symmetric plane, and is the relationship between the velocity coordinate system and the missile coordinate system; the yaw angle is the angle between the missile vertical axis projection on the horizontal plane and the ax axis of the ground coordinate system (on the horizontal plane, pointing to the target is positive). It is the angle relationship between the ground coordinate system and the missile coordinate system. (Differentiated from the flight angle)

Major transformations between Coordinate Systems

1. Transformation between two vector Coordinate Systems

2. transition between axes in the plane coordinate system

As shown in, vector R has the following components in the coordinate system where the two origins overlap: (XP, YP), (XQ, yq), and α is the rotation angle from Q to P.

3. transition between axes of the 3D Coordinate System

The turn Matrix

PS: The following two matrices are inverse matrices.