Rotation Matrices

Rotation matrices are used extensively in skplatform and although the subject is conceptually straight-forward it is very easy to get confused with the subject.

Intrinsic and Extrinsic

The purpose of a rotation matrix is to rotate a vector to produce a new vector. This immediately opens up two common options:

1. Extrinsic: hold the coordinate axes constant and rotate the given vector. The final vector is expressed in terms of the fixed axes.

2. Intrinsic: hold the vector constant in the original coordinate system and rotate the coordinate axes to a new set of axes. The final vector is the original vector expressed in the new rotated corodinate system.

Wikipedia Davenport Chained Rotations and Wikipedia Euler Angles describe the two rotations in much more detail.

Both types of rotation have application within skplatform.

For example, specifying instrument look vectors in the Instrument Control Frame (ICF) are easily visualized using extrinsic rotations while orientian of the Platform Control Frame (PCF) using yaw, pitch and roll is best suited for intrinsic rotations.

Extrinsic Rotation Matrix

Rotation matrices are used to (i) rotate vectors within one coordinate system or (ii) to transform the coordinates of a vector in one coordinate system to coordinates in a rotated coordinate system.

Extrinsic rotations execute all the rotations in a sequence of rotations around the axes in the starting coordinate system. This starting coordinate system is a global system that does not rotate.

The standard approach for a 3-axis cartesion coordinate system is to consider the effect of rotation around each of the three axes upon each of the primary unit vectors: i.e. rotate the \(\hat{x}\), \(\hat{y}\) and \(\hat{z}\) unit vectors around the \(\hat{z}\) axis and use this to express the new, primed unit vectors in terms of the original unit vector.

For example, rotation around the \(\hat{z}\) axis,

\[\begin{split}\hat{x}' &= &\cos\theta\:\hat{x} &+ \sin{\theta}\:\hat{y} &+ 0\:\hat{z} \\ \hat{y}' &= &-\sin\theta\:\hat{x} &+ \cos{\theta}\:\hat{y} &+ 0\:\hat{z} \\ \hat{z}' &= &0\:\hat{x} &+ 0\:\hat{y} &+ 1\:\hat{z} \\\end{split}\]

We can then immediately write the extrinsic rotation matrix, \(R_z\), for rotation around the Z axis as,

\[\begin{split}R_{z} = \begin{bmatrix} \cos\theta & -\sin{\theta} & 0 \\ \sin\theta & \cos{\theta} & 0 \\ 0 & 0 & 1 \end{bmatrix}\end{split}\]

and similary for Y and X gives,

\[\begin{split}R_{y} = \begin{bmatrix} \cos\theta & 0 & \sin{\theta} \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos{\theta} \end{bmatrix}\end{split}\]

\[\begin{split}R_{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin{\theta} \\ 0 & \sin\theta & \cos{\theta} \end{bmatrix}\end{split}\]

This rotation matrix will rotate any unit vector around the Z axis by angle \(\theta\) and present the new unit vector in terms of the original unit vectors.

For example: extrinsic rotation of the \(\hat{x}\) unit vector is given by:

\[\hat{x}' = R_{z}\:\hat{x}\]

or more explicitly,

\[\begin{split}\hat{x}' = \begin{bmatrix} \cos\theta & -\sin{\theta} & 0 \\ \sin\theta & \cos{\theta} & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\end{split}\]

which upon evaluation gives

\[\hat{x}' = \cos\theta\:\hat{x} + \sin{\theta}\:\hat{y} + 0\:\hat{z}\]

Intrinsic Rotation Matrix

Intrinsic rotations execute rotations in a sequence of rotations around the axes of a coordinate system after the axes have been rotated by the previous rotation matrix. The rotating coordinate system is a local system that has a new orientation after application of each rotation matrix. This rotating local system is easy to visualize and describe, especially for platforms like aircraft, gondola, and spacecraft, where yaw, pitch and roll are often used.

Theorem

Consider a sequence of rotations, zyx, applied to a vector \(\bar{v}\), where we first rotate around the Z axis, then the Y axis and finally the X axis.

First consider the extrinsic case. In the extrinsic rotation the final vector, \(\bar{v}'_{ext}\), is given by the equation,

\[\bar{v}'_{ext} = \boldsymbol{R_x}\:\boldsymbol{R_y}\:\boldsymbol{R_z}\:\bar{v}\]

where we note that the order of rotations must be expressed mathematically from right to left so \(R_z\) is first appled to \(\bar{v}\) then \(R_y\) and finally \(R_x\) and all rotations are around the fixed, starting, global coordinate system.

Now consider the intrinsic case. This is trickier to write down as the axes of rotation are changing between rotations, but a proof is given below that proves the intrinsic rotation is equivalent to an extrinsic rotation with the order of rotation matrices reversed:

\[\boxed{\bar{v}'_{int} = \boldsymbol{R_z}\:\boldsymbol{R_y}\:\boldsymbol{R_x}\:\bar{v}}\]

Proof

Wikipedia states

Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice-versa. For instance, the intrinsic rotations x−y′−z′′ by angles α,β,γ are equivalent to the extrinsic rotations z−y−x by angles γ,β,α.

The rotation matrix, \(R\), can be considered to be a coordinate transform that expresses the unit vector \(\hat{x}'\) in terms of the unprimed unit vectors, \(\hat{x}\), \(\hat{y}\) and \(\hat{z}\). Conversely, the inverse of the rotation matrix, \(R^{-1}\), expresses the unprimed unit vectors, \(\hat{x}\) in terms of the primed unit vectors \(\hat{x}'\).

Intrinsic rotations rely upon rotation around axes that are rotated along with the target vector, \(\bar{v}\). This presents an issue as simply applying a rotation matrix to an unprimed vector:

\[\bar{v}'_g = \boldsymbol{R_z} \: \bar{v}_g\]

creates a new rotated vector, \(\bar{v}'_g\), where we haved used the g subscript to indicate the vector is expressed in terms of the original unprimed, global, unit vectors. It is not appropriate to leave the rotated vector expressed in global unprimed components if we are going to perform a second (or third) rotation around a local, primed axis. The global rotated vector must be written in terms of the new local unit vectors. This is achieved with a coordinate transform using the inverse of the rotation matrix,

\[\bar{v}'_l = \boldsymbol{R_z^{-1}} \: \bar{v}'_g\]

Having transformed the primed vector to the local, rotated coordinate system we can now apply the second rotation, \(R_y\) to produce a double primed rotated vector,

\[\bar{v}''_l = \boldsymbol{R_y} \: \bar{v}'_l\]

This second,double primed, rotated vector is expressed in terms of the first set of local coordinates. To perform the third rotation around the thrid axis we must transform the second, double primed from the first local system to the second local system,

\[\bar{v}''_{ll} = \boldsymbol{R_y^{-1}} \: \bar{v}''_l\]

We have now expressed the second rotation vector in the second local coordinate system and we can now apply the third and final rotation,

\[\bar{v}'''_{ll} =\boldsymbol{R_x} \: \bar{v}''_{ll}\]

We now have the fully rotated vector but it is expressed in the second local system and we wish to get it back to the original unprimed global system. This is straight forward as we can easily transform the vector from the second, local system to the first local system using the second rotation matrix,

\[\bar{v}'''_{l} =\boldsymbol{R_y} \: \bar{v}'''_{ll}\]

and from the first local system to the global system using the first rotation matrix,

\[\bar{v}'''_{g} =\boldsymbol{R_z} \: \bar{v}'''_{l}\]

Putting all the above equations together we get

\[\bar{v}'''_{g} = \boldsymbol{R_z} \: \boldsymbol{R_y} \: \boldsymbol{R_x} \: \boldsymbol{R_y^{-1}} \: \boldsymbol{R_y} \: \boldsymbol{R_z^{-1}} \: \boldsymbol{R_z}\: \bar{v}_{g}\]

or more succinctly

\[\boxed{ \bar{v}'''_{g} = \boldsymbol{R_z} \: \boldsymbol{R_y} \: \boldsymbol{R_x} \: \bar{v}_{g} }\]

And we have the proof of the standard, intrinsic rotation formula that rotation around the axes in the order z, y, x is implemented by applying the (extrinsic like) rotation matrices in reverse order (from right to left), i.e. \(R_x\) is applied first, \(R_y\) second and \(R_z\) third.