Rigid Body Transformations

The rotmat module in c4dynamics provides a collection of functions for defining and exploring the rotation of bodies from one frame of reference to another.

Background Material

A rotation matrix is a mathematical representation of a rotation in three-dimensional space.

It’s a 3x3 matrix that, when multiplied with a vector, transforms the vector to represent a new orientation.

Each element of the matrix corresponds to a directional cosine, capturing the rotation’s effect on the x, y, and z axes.

Euler Angles Order

Frame based vectors are related through a Direction Cosine Matrix (DCM). [HS]

When Euler angles are employed in the transformation of a vector expressed in one reference frame to the expression of the vector in a different reference frame, any order of the three Euler rotations is possible, but the resulting transformation equations depend on the order selected. [MIs]

In aerospace applications for example, the common order is that the first Euler rotation is about the z-axis, the second is about the y-axis, and the third is about the Xaxis. Such a transformation order is called z-y-x, or 3-2-1. With reference to a body orientation, the resulting order is yaw, pitch, and roll. With reference to geographical orientation, the resulting order is azimuth (heading), elevation (pitch), and roll (bank angle).

Right Hand Frame

The positive directions of coordinate system axes and the directions of positive rotations about the axes are arbitrary. In right-handed systems:

i x j = k
j x k = i
k x i = j

where i is the unit vector in the direction of the x-axis, j is the unit vector in the direction of the y-axis, k is the unit vector in the direction of the z-axis.

Positive rotations are clockwise when viewed from the origin, looking out along the positive direction of the axis.

These conventions are illustrated in Fig-1.

Fig-1: Coordinate System Conventions

References

[HS]

Hanspeter Schaub, “Spacecraft Dynamics and Control” lecture notes, module 2: rigidbody kinematics.

[MIs]

Ch 4 in “Missile Flight Simulation Part One Surface-to-Air Missiles”, Military Handbook, 1995, MIL-HDBK-1211(MI).

Examples

For examples, see the various functions.

See Also

rotx	Direction Cosine Matrix for a rotation about the x
roty	Direction Cosine Matrix for a rotation about the y
rotz	Direction Cosine Matrix for a rotation about the z
dcm321	Direction Cosine Matrix for a for a sequence of rotations in following order: \(z\), then \(y\), then \(x\)
dcm321euler	Extract Euler angles (roll, pitch, yaw) from a Direction Cosine Matrix (DCM) of 3-2-1 order
animate	Animates the rigid body’s motion using a 3D model according to the 3-2-1 Euler angles histories.