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## 33 Quaternions

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.

— Function File: [a, b, c, d] = quaternion (w)
— Function File: [vv, theta] = quaternion (w)
— Function File: w = quaternion (a, b, c, d)
— Function File: w = quaternion (vv, theta)

Construct or extract a quaternion

```          w = a*i + b*j + c*k + d
```

from given data.

— Function File: qconj (q)

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
```

— Function File: qderiv (omega)

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) * Q
```

If the passive convention is used (rotate the frame, not the vector), then

```          Q' = -qderivmat (omega) * Q
```

— Function File: qderivmat (omega)

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) * Q
```

If the passive convention is used (rotate the frame, not the vector), then

```          Q' = -qderivmat (omega) * Q.
```

— Function File: qinv (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]
```

— Function File: qmult (a, b)

Multiply two quaternions.

```          [w, x, y, z] = w*i + x*j + y*k + z
```

identities:

```          i^2 = j^2 = k^2 = -1
ij = k                 jk = i
ki = j                 kj = -i
ji = -k                ik = -j
```

— Function File: qtrans (v, q)

Transform the unit quaternion v by the unit quaternion q. Returns v` = `q`*`v`/`q.

— Function File: qtransv (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.

— Function File: qtransvmat (qib)

Construct a 3x3 transformation matrix from quaternion qib that is equivalent to rotation of th radians about axis vv, where `[`vv`, `th`] = quaternion (`qib`)`.

— Function File: qcoordinate_plot (qf, qb, qv)

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.