Rotation Matrix

Definition

Consider an arbitrary vector \(\mathbf{x} \in \mathbb{R}^3\) and two frames \(F_\alpha\) and \(F_\beta\) with a common origin. \(\mathbf{R}^{\beta}_{\alpha} \in SO(3)\) represents a rotation matrix from \(F_\alpha\) to \(F_\beta\).

Representation	Attitude	Rotation
Rotation Function	Passive, i.e., the function of the rotation operator is on the frames	Active, i.e., the function of the rotation operator is on the vectors
Rotation Operator	\(\mathbf{x}^\beta = \mathbf{R}^\beta_\alpha \mathbf{x}^\alpha\)	\(\mathbf{x}' = \mathbf{R} \mathbf{x}\)
Cross Relationship	\(\mathbf{R}_{\text{active}} = \mathbf{R}^T_{\text{passive}}\)
Inverse Transformation	\(\mathbf{R}^{\alpha}_{\beta} = \left(\mathbf{R}^{\beta}_{\alpha} \right)^T = \left(\mathbf{R}^{\beta}_{\alpha} \right)^{-1}\)
Composition	\(\mathbf{R}^{\gamma}_{\alpha} = \mathbf{R}^{\gamma}_{\beta} \mathbf{R}^{\beta}_{\alpha}\)
Resolving Axes Transformation	Given a linear transformation matrix \(\mathbf{M}\), \(\mathbf{M}^{\beta} = \mathbf{R}^{\beta}_{\alpha} \mathbf{M}^{\alpha} \mathbf{R}^{\alpha}_{\beta}\)
Principal Rotation Matrices	Given an angle \(\left( \cdot \right) = \left( \cdot \right)_{\beta \alpha}\) which is the angle from \(F_\beta\) to \(F_\alpha\) around a specific principle axis, the principal rotation matrices describing the rotation \(\mathbf{R}^\alpha_\beta\) are \(\begin{align} \mathbf{R}_3(\psi) &= \left[ \begin{array}{ccc} \text{cos}(\psi) & \text{sin}(\psi) & 0 \\ -\text{sin}(\psi) & \text{cos}(\psi) & 0 \\ 0 & 0 & 1 \end{array} \right] \\ \mathbf{R}_2(\theta) &= \left[ \begin{array}{ccc} \text{cos}(\theta) & 0 & -\text{sin}(\theta) \\ 0 & 1 & 0 \\ \text{sin}(\theta) & 0 & \text{cos}(\theta) \end{array} \right] \\ \mathbf{R}_1(\phi) &= \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \text{cos}(\phi) & \text{sin}(\phi) \\ 0 & -\text{sin}(\phi) & \text{cos}(\phi) \end{array} \right] \\ \end{align}\)
Time Derivative	\(\begin{align} \dot{\mathbf{R}}^\alpha_\beta(t) &= - \mathbf{R}^\alpha_\beta \boldsymbol{\Omega}^\beta_{\beta \alpha} = \mathbf{R}^\alpha_\beta \boldsymbol{\Omega}^\beta_{\alpha \beta} \\ &= -\boldsymbol{\Omega}^\alpha_{\beta \alpha} \mathbf{R}^\alpha_\beta = \boldsymbol{\Omega}^\alpha_{\alpha \beta} \mathbf{R}^\alpha_\beta \end{align}\)

Covariance Transformation

Consider a state vector \(\mathbf{x} \in \mathbb{R}^6\) with a covariance matrix \(\mathbf{P} \in \mathbb{R}^{6 \times 6}\). The covariance is defined as:

\[ \mathbf{P} = \mathbb{E}\left[ \mathbf{x} \mathbf{x}^T \right] - \mathbb{E}\left[ \mathbf{x} \right] \mathbb{E}\left[ \mathbf{x}^T \right]. \]

If there has been a state transformation such that:

\[ \mathbf{x}' = \mathbf{M} \mathbf{x}, \]

with \(\mathbf{M} \in \mathbb{R}^{6 \times 6}\), then the transformed covariance becomes:

\[ \begin{align} \mathbf{M}' &= \mathbb{E}\left[ \mathbf{x}' \left(\mathbf{x}' \right)^T \right] - \mathbb{E}\left[ \mathbf{x}' \right] \mathbb{E}\left[\left(\mathbf{x}' \right)^T \right] \\ &= \mathbb{E} \left[ \mathbf{M} \mathbf{x} \mathbf{x}^T \mathbf{M}^T \right] - \mathbb{E}\left[ \mathbf{M} \mathbf{x} \right] \mathbb{E}\left[ \mathbf{x}^T \mathbf{M}^T \right] \\ &= \mathbf{M} \mathbb{E}\left[ \mathbf{x} \mathbf{x}^T \right] \mathbf{M}^T - \mathbf{M} \mathbb{E}\left[ \mathbf{x} \right] \mathbb{E}\left[ \mathbf{x}^T \right] \mathbf{M}^T \\ &= \mathbf{M} \left( \mathbb{E}\left[ \mathbf{x} \mathbf{x}^T \right] - \mathbb{E}\left[ \mathbf{x} \right] \mathbb{E} \left[ \mathbf{x}^T \right] \right) \mathbf{M}^T \\ &= \mathbf{M} \mathbf{P} \mathbf{M}^T. \end{align} \]

Proofs

Proof - Principle Rotations

Let the coordinates of the vector \(\mathbf{p}\) be \(\left[\begin{array}{ccc} x^\beta & y^\beta & z^\beta \end{array} \right]^T\) in \(F_\beta\) and \(\left[\begin{array}{ccc} x^\alpha & y^\alpha & z^\alpha \end{array} \right]^T\) in \(F_\alpha\). Consider a case when \(F_\alpha\) is rotated by \(\psi\) about the principle 3-axis and a point \(P\) as shown in Figure 1.

Rotation about \(z^\beta\) (Groves, p31) — Figure 1 Active rotation about 3-axis

In \(F_\beta\) and \(F_\alpha\), the coordinates of \(P\) are:

\[ \begin{align} x^\beta &= r \cos \gamma \\ y^\beta &= r \sin \gamma \\ x^\alpha &= r \cos (\gamma - \psi) \\ &= r \left(\cos \gamma \cos \psi + \sin \gamma \sin \psi \right) \\ &= x^\beta \cos \psi + y^\beta \sin \psi \\ y^\alpha &= r \sin (\gamma - \psi) \\ &= r \left( \sin \gamma \cos \psi - \cos \gamma \sin \psi \right) \\ &= y^\beta \cos \psi - x^\beta \sin \psi. \end{align} \]

Hence, the relationship between the two coordinate frames is:

\[ \begin{align} \left[ \begin{array}{c} x^\beta \\ y^\beta \\ z^\beta \\ \end{array} \right] &= \left[ \begin{array}{ccc} \cos \psi & -\sin \psi & 0 \\ \sin \psi & \cos \psi & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} x^\alpha \\ y^\alpha \\ z^\alpha \\ \end{array} \right]. \end{align} \]

Proof - Poisson

\[ \dot{\mathbf{R}}^{\alpha}_{\beta}(t) = \lim_{\delta t \rightarrow 0} \left( \frac{\mathbf{R}^{\alpha}_{\beta}(t + \delta t) - \mathbf{R}^{\alpha}_{\beta} (t)}{\delta t} \right). \label{3.6} \]

If \(F_{\alpha}\) is rotating with respect to a stationary reference frame, \(F_\beta\), then:

\[ \mathbf{R}^{\alpha}_{\beta}(t + \delta t) = \mathbf{R}^{\alpha(t + \delta t)}_{\alpha(t)} \mathbf{R}^{\alpha}_{\beta}(t). \label{3.7} \]

Since the rotation of \(F_{\alpha}\) from \(t\) to \(t + \delta t\) is infinitesimal, using small angle approximation and assuming \(\boldsymbol{\omega}^{\alpha}_{\beta \alpha}\) is constant:

\[ \begin{align} \mathbf{R}^{\alpha}_{\beta} (t + \delta t) &= (\mathbf{I}_3 - \left[ \boldsymbol{\rho}_{\alpha(t) \alpha(t + \delta t)} \right]_\times) \mathbf{R}^{\alpha}_{\beta}(t) \\ &= (\mathbf{I}_3 - \delta t \left[\boldsymbol{\omega}^{\alpha}_{\beta \alpha} \right]_\times) \mathbf{R}^{\alpha}_{\beta}(t) \\ &= \left(\mathbf{I}_3 - \delta t \boldsymbol{\Omega}^{\alpha}_{\beta \alpha} \right) \mathbf{R}^{\alpha}_{\beta}(t). \label{3.8} \end{align} \]

Substituting equation (\(\ref{3.8}\)) into equation (\(\ref{3.6}\)), we get:

\[ \begin{align} \dot{\mathbf{R}}^{\alpha}_{\beta}(t) &= \lim_{\delta t \rightarrow 0} \left( \frac{ \left(\mathbf{I}_3 - \delta t \boldsymbol{\Omega}^{\alpha}_{\beta \alpha} \right) \mathbf{R}^{\alpha}_{\beta}(t) - \mathbf{R}^{\alpha}_{\beta}(t)}{\delta t} \right) \\ &= - \boldsymbol{\Omega}^{\alpha}_{\beta \alpha} \mathbf{R}^{\alpha}_{\beta} (t) \\ &= -\mathbf{R}^{\alpha}_{\beta} (t) \boldsymbol{\Omega}^{\alpha}_{\beta \alpha}, \end{align} \label{3.9} \]

which is the Poisson's equation.

Note that if the above steps are repeated under the assumption that \(F_\beta\) is rotating and \(F_\alpha\) is stionary, the result \(\dot{\mathbf{R}}^\alpha_\beta = -\mathbf{R}^\alpha_\beta \boldsymbol{\Omega}^\beta_{\beta \alpha}\) is obtained. However, the results are equivalent.

References

Groves, P., Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, Second Edition