Rotational matrix operations

Rotational matrix operations#

Rotational Matrix Operations#

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.

../_images/frame_conventions.svg

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.

Functions

rotmat.rotx(phi)

Generate a 3x3 Direction Cosine Matrix for a positive rotation about the x-axis by an angle \(\phi\) in radians.

rotmat.roty(theta)

Generate a 3x3 Direction Cosine Matrix for a positive rotation about the y-axis by an angle \(\theta\) in radians.

rotmat.rotz(psi)

Generate a 3x3 Direction Cosine Matrix for a positive rotation about the z-axis by an angle \(\psi\) in radians.

rotmat.dcm321([phi, theta, psi])

Generate a 3x3 Direction Cosine Matrix (DCM) for a sequence of positive rotations around the axes in the following order: \(z\), then \(y\), then \(x\).

rotmat.dcm321euler(dcm)

Extract Euler angles (roll, pitch, yaw) from a Direction Cosine Matrix (DCM) of 3-2-1 order.

animate.animate(rb, modelpath[, angle0, ...])

Animate a rigidbody.
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