On the use of Generalized Rodrigues Parameters for Representing Rigid Body Rotations

Abstract

This work complements the body of studies on the use of generalized Rodrigues parameters (GRPs) for describing the orientation of a rigid body. A simple decision logic is designed for handling the GRPs so that singular configurations are avoided, while enforcing the one-to-one mapping between the set of orientation parameters and the unique rotation matrix.

Appendices

Appendix

A.1 Proof: $\mathbf {R}(\mathbf {p}) \in \mathbb {SO}(3) , \forall \mathbf {p} \in \mathbb {R}^{3}$

The product R^T(p)R(p) reads, using (7)

$$ \mathbf{R}^{\scriptscriptstyle \mathrm{T}}(\mathbf{p})\mathbf{R}(\mathbf{p}) = \mathbf{I}_3 + 4\xi^2 \left( \mathbf{I}_3 + \xi^2 \boldsymbol{\Omega}_{\mathbf{p}}^2 - (a-\xi)^2\mathbf{I}_3 \right) \boldsymbol{\Omega}_{\mathbf{p}}^2 $$

(39)

By substituting $\boldsymbol {\Omega }_{\mathbf {p}}^{2} = \mathbf {p}\mathbf {p}^{\scriptscriptstyle \mathrm {T}}-\mathbf {p}^{\scriptscriptstyle \mathrm {T}}\mathbf {p}\mathbf {I}_{3}$, one obtains

$$ \mathbf{R}^{\scriptscriptstyle \mathrm{T}}(\mathbf{p})\mathbf{R}(\mathbf{p})= \mathbf{I}_3 $$

(40)

since Ω_pp = 0.

The determinant of matrix R(p) can be derived as follows:

$$ \det \left( \mathbf{R}^{\scriptscriptstyle \mathrm{T}}(\mathbf{p})\mathbf{R}(\mathbf{p})\right) = \left( \det \mathbf{R}(\mathbf{p})\right)^2 = 1 $$

(41)

Thus, det R(p) = ± 1, but since the determinant for p = 0 is + 1, and R(p) is a continuous function of the GRPs, the determinant is identically one for any real-valued vector p. This demonstrates that expression (7) always produces a proper rotation matrix.

A.2 Inverse map

The orientation matrix in terms of GRPs is explicitly given as

$$ \begin{array}{@{}rcl@{}} \mathbf{R}(\mathbf{p}) &=& \mathbf{I}_3 + 2\xi^2 \left[\frac{a-\xi}{\xi}\boldsymbol{\Omega}_{\mathbf{p}} + \boldsymbol{\Omega}_{\mathbf{p}}^2 \right]\\ &=& \left[\begin{array}{lll} 1-2\xi^2(p_{2}^2+p_{3}^2) & 2\xi^2p_{1}p_{2}+2\xi(a-\xi)p_{3} & 2\xi^2p_{1}p_{3}-2\xi(a-\xi)p_{2} \\ 2\xi^2p_{1}p_{2}-2\xi(a-\xi)p_{3} & 1-2\xi^2(p_{1}^2+p_{3}^2) & 2\xi^2p_{2}p_{3}+2\xi(a-\xi)p_{1} \\ 2\xi^2p_{1}p_{3}+2\xi(a-\xi)p_{2} & 2\xi^2p_{2}p_{3}-2\xi(a-\xi)p_{1} & 1-2\xi^2(p_{1}^2+p_{2}^2) \end{array}\right]\\ \end{array} $$

(42)

from which relations (14)-(15) are easily obtained.

Note that, should the sum r₁₂ + r₂₁ in (15) approach zero, it is sufficient to use one of the two alternatives

$$ \begin{array}{@{}rcl@{}} \underline{\xi} &=& a \pm \frac{1}{2} \sqrt{\frac{(r_{32}-r_{23})(r_{21}-r_{12})}{r_{13}+r_{31}}}\\ &=& a \pm \frac{1}{2} \sqrt{\frac{(r_{13}-r_{31})(r_{21}-r_{12})}{r_{23}+r_{32}}} \end{array} $$

(43)

The same reasoning applies to the shadow set of GRPs.

Expressions (14)–(20) are specifically worked out to analyze internal consistency properties. From a computational perspective, it is easier to use the quaternion inversion formulas to obtain the inverse mapping:

$$ q_0 = \frac{1}{2} \sqrt{1+\text{tr}(\mathbf{R})} \quad ; \quad \mathbf{q} = \frac{1}{4q_0} \left( \begin{array}{lll} r_{23}-r_{32} \\ r_{31}-r_{13} \\ r_{12}-r_{21} \end{array}\right) $$

(44)

from which

$$ \mathbf{p} = \frac{1}{\sqrt{1+\text{tr}(\mathbf{R})} \left( 2a+\sqrt{1+\text{tr}(\mathbf{R})} \right)} \left( \begin{array}{lll} r_{23}-r_{32} \\ r_{31}-r_{13} \\ r_{12}-r_{21} \end{array}\right) $$

(45)

This function diverges in two cases. The first is for rotations about any axis and angle ϕ = (2k + 1)π ($k \in \mathbb {Z}$), for which tr(R) = − 1, corresponding to a null quaternion scalar component. Obviously, this is not a degenerate point, and the issue is circumvented by taking one of the following alternative inverse functions [13]:

$$ \begin{array}{@{}rcl@{}} \gamma &=& \frac{1}{2}\sqrt{1+r_{11}-r_{22}-r_{33}}\\ \mathbf{p} &=& \frac{1}{4\gamma a+r_{23}-r_{32}} \left( \begin{array}{lll} (2\gamma)^2 \\ r_{12}+r_{21} \\ r_{13}+r_{31} \end{array}\right) \end{array} $$

(46)

or

$$ \begin{array}{@{}rcl@{}} \gamma &=& \frac{1}{2}\sqrt{1+r_{11}-r_{22}-r_{33}}\\ \mathbf{p} &=& \frac{1}{4\gamma a+r_{23}-r_{32}} \left( \begin{array}{lll} (2\gamma)^2 \\ r_{12}+r_{21} \\ r_{13}+r_{31} \end{array}\right) \end{array} $$

(47)

or

$$ \begin{array}{@{}rcl@{}} \gamma &=& \frac{1}{2}\sqrt{1+r_{11}-r_{22}-r_{33}}\\ \mathbf{p} &=& \frac{1}{4\gamma a+r_{23}-r_{32}} \left( \begin{array}{lll} (2\gamma)^2 \\ r_{12}+r_{21} \\ r_{13}+r_{31} \end{array}\right) \end{array} $$

(48)

Since both (q₀,q) and (−q₀,−q) are valid parameterizations of the same attitude matrix R, for each of the four possible inversions (45)–(48) both the direct and the shadow GRPs can be recovered by simply inverting the sign of scalar a. It is then straightforward to apply the logic reported in Table 5 to select the appropriate set that guarantees both internal and external consistency.

The second degenerate point is at a rotation for which $2a+\sqrt {1+\text {tr}(\mathbf {R})}=0$, corresponding to a quaternion scalar component q₀ = −a, which is indeed a singularity point for the GRPs direct representation. This singularity is avoided by switching to the shadow set through inversion of the sign of parameter a.

A.3 Derivation of identity (26)

Using identity (25) in (24), one obtains

(49)

Using (6), expression (49) works out as

(50)

with $\beta =\sqrt {(1-a^{2})\mathbf {p}^{\scriptscriptstyle \mathrm {T}}\mathbf {p}+1}$. The last ratio on the right-hand side in (50) only retains the sign of the scalar ap^Tp ∓ β, and identity (26) is thus obtained.

A.4 Rotations characterized by $\mathbf {p}^{\scriptscriptstyle \mathrm {T}}\mathbf {p}=\mathbf {p}_{s}^{\scriptscriptstyle \mathrm {T}}\mathbf {p}_{s}=\frac {1}{a^{2}}$

Functions (26) and (29) may take an indeterminate value when the squared norm of the GRPs approaches the value $\frac {1}{a^{2}}$. However, any of the alternative inversion functions (46)–(48) can be used to study internal consistency. Such rotations are characterized by symmetric rotation matrices (the scalar component of quaternion is null, thus ξ = a and ξ_s = −a), and R(p) = R(−p) = R(p_s), thus the direct and the shadow GRPs can be swapped without altering the result.

A.5 Derivation of identity (32)

Substituting (6) into (31) gives

$$ \frac{\xi-a}{q_0} = \frac{1}{q_0}\left( \frac{a\pm\sqrt{(1-a^2)\mathbf{p}^{\scriptscriptstyle \mathrm{T}}\mathbf{p}+1}}{1+\mathbf{p}^{\scriptscriptstyle \mathrm{T}}\mathbf{p}}-a\right) $$

(51)

From relation (4) one obtains $\mathbf {p}^{\scriptscriptstyle \mathrm {T}}\mathbf {p}=\frac {1-{q_{0}^{2}}}{(q_{0}+a)^{2}}$. Expression (51) then works out as

$$ \begin{array}{@{}rcl@{}} \frac{\xi-a}{q_0} &=& \frac{1}{q_0} \frac{\pm\sqrt{(q_0+a)^2(aq_0+1)^2}-a(1-q_0^2)}{a^2+2aq_0+1}\\ &=& \frac{1}{q_0} \frac{\pm|q_0+a||aq_0+1|+a(q_0^2-1)}{a^2+2aq_0+1} \end{array} $$

(52)

Since both a and q₀ are only defined in the interval [− 1; + 1], the term |aq₀ + 1| is always larger than, or equal to, zero. Expression (32) is then obtained.