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Quaternions are hypercomplex numbers used to represent spatial rotations in three dimensions. This set of routines provides a useful basis for working with quaternions in Octave. A tutorial is in the Octave source, scripts/quaternion/quaternion.ps.
These functions were written by A. S. Hodel, Associate Professor, Auburn University.
Construct or extract a quaternion
w = a*i + b*j + c*k + dfrom given data.
Conjugate of a quaternion.
q = [w, x, y, z] = w*i + x*j + y*k + z qconj (q) = -w*i -x*j -y*k + z
Derivative of a quaternion.
Let Q be a quaternion to transform a vector from a fixed frame to a rotating frame. If the rotating frame is rotating about the [x, y, z] axes at angular rates [wx, wy, wz], then the derivative of Q is given by
Q' = qderivmat (omega) * QIf the passive convention is used (rotate the frame, not the vector), then
Q' = -qderivmat (omega) * Q
Derivative of a quaternion.
Let Q be a quaternion to transform a vector from a fixed frame to a rotating frame. If the rotating frame is rotating about the [x, y, z] axes at angular rates [wx, wy, wz], then the derivative of Q is given by
Q' = qderivmat (omega) * QIf the passive convention is used (rotate the frame, not the vector), then
Q' = -qderivmat (omega) * Q.
Return the inverse of a quaternion.
q = [w, x, y, z] = w*i + x*j + y*k + z qmult (q, qinv (q)) = 1 = [0 0 0 1]
Multiply two quaternions.
[w, x, y, z] = w*i + x*j + y*k + zidentities:
i^2 = j^2 = k^2 = -1 ij = k jk = i ki = j kj = -i ji = -k ik = -j
Transform the unit quaternion v by the unit quaternion q. Returns v
=
q*
v/
q.
Transform the 3-D vector v by the unit quaternion q. Return a column vector.
vi = (2*real(q)^2 - 1)*vb + 2*imag(q)*(imag(q)'*vb) + 2*real(q)*cross(imag(q),vb)Where imag(q) is a column vector of length 3.
Construct a 3x3 transformation matrix from quaternion qib that is equivalent to rotation of th radians about axis vv, where
[
vv,
th] = quaternion (
qib)
.
Plot in the current figure a set of coordinate axes as viewed from the orientation specified by quaternion qv. Inertial axes are also plotted:
- qf
- Quaternion from reference (x,y,z) to inertial.
- qb
- Quaternion from reference to body.
- qv
- Quaternion from reference to view angle.