CONSTRUCTION OF CONFIDENCE REGIONS FOR UNCERTAIN SPATIAL DISPLACEMENTS WITH DUAL RODRIGUES PARAMETERS

Abstract

This paper follows our recent work on the computation of kinematic confidence regions from a given set of uncertain spatial displacements with specified confidence levels. Dual quaternion algebra is used to compute the mean displacement as well as relative displacements from the mean. In constructing a 6D confidence ellipsoid, however, we use dual Rodrigue parameters resulting from dual quaternions. The advantages of using dual quaternions and dual Rodrigues parameters are discussed in comparison with those of three translation parameters and three Euler angles, which were used for the development of the socalled the Rotational and Translational Confidence Limit (RTCL) method. The set of six dual Rodrigue parameters are used to define a parametric space in which a 6 × 6 covariance matrix and a 6D confidence ellipsoid are obtained. An inverse operation is then applied to first obtain dual quaternions and then to recover the rotation matrix and translation vector for each point on the 6D ellipsoid. Through examples, we demonstrate the efficacy of our approach by comparing it with the RTCL method known in literature. Our findings indicate that our method, based on the dual-Rodrigues formulation, yields more compact and effective swept volumes than the RTCL method, particularly in cases involving screw displacements.

1. Introduction

This paper deals with the problem of computing kinematic confidence regions from a given set of uncertain spatial displacements for a specified level of confidence. The study of such problems is motivated by the need for constructing the Planning Target Volume (PTV) to account for kinematic uncertainties in image-guided radiotherapy [1, 2]. The existing approach uses a set of three translational parameters and three Euler angles for computing confidence regions and leads to a method known as the Rotational and Translational Confidence Limit (RTCL) method [3].

Our recent research explored the use of planar quaternions and dual quaternions for constructing confidence regions from planar displacement data and spatial displacement data with uncertainties [4, 5]. However, the results shown by Ge et al. [5] were generated based on separating real and dual parts to get two 3 × 3 covariance matrices and did not illustrate the advantage of preserving kinematic properties such as the screw axis of spatial displacements. This paper investigates the utilization of dual Rodrigues parameters for developing kinematic confidence regions using one 6 × 6 covariance matrices for spatial displacements, aiming to capture the kinematic properties in $SE$(3).

The organization of the paper is as follows: Section 2 presents the kinematic preliminaries of spatial displacements, covering screw displacements, dual quaternions and dual Rodrigues parameters. The relative displacements are introduced in the same section. In Section 3, we summarize the existing Rotational and Translational Confidence Limit (RTCL) method and investigate kinematic covariance and confidence regions using a relative dual-Rodrigues formulation. Section 4 provides three examples comparing the results of the two methods under different scenarios.

2. Dual Quaternions and Dual Rodrigues Parameters

A general spatial displacement can be conveniently represented by a set of six parameters $(a,b,c;\alpha ,\beta ,\gamma )$, where $\mathbf{d}=(a,b,c)$ is the vector of translation while $(\alpha ,\beta ,\gamma )$ are Euler angles representing the rotational component. In the field of image-guided radiotherapy, such kinematic parameters can be extracted directly from medical images for statistical analysis. For the RTCL method [3], the resulting composite probability space is formed as a 6D ellipsoid with axes given by the distribution of each of the parameters $(a,b,c;\alpha ,\beta ,\gamma )$ taken independently.

However, as is well known in theoretical kinematics, spatial displacements belong to the group of $SE$(3) and its rotational components belong to the group of $SO$(3). In addition, the actual movement of an object is often subject to the constraints exist in its surrounding. For example, the movement of a tumor within a human body is constrained by human anatomy by surrounding bones and tissues. Consequently, when selecting a representation of spatial displacements, it will be advantageous to take into account these factors.

This section provides a summary of two representations of spatial displacements, namely dual quaternions and dual Rodrigues parameters, that will be used in the subsequent sections for statiscal analyses that lead to the construction of confidence regions for uncertain spatial displacements. As eloquently stated in [6], dual quaternions and dual Rodrigues parameters are isomorphic and both are defined directly in terms of the invariants of spatial displacements. When dealing with operations such as composition of two displacements or computing a relative displacement between two given displacements, dual quaternion algebra is easier to use than dual Rodrigues formula [7,8,9]. When it comes to statistical analysis and especially the construction of confidence regions, however, it is easier to use dual Rodrigues parameters as they consist of six independent parameters while dual quaternions consist of eight parameters with two nonlinear constraints.

2.1. The Invariants

According to Chasles’ theorem, every rigid transformation can be expressed as a screw displacement that captures all the invariants of the transformation [10,11]. The geometric invariant in vector form is the screw axis and the scalar invariants are the angle of rotation $\theta$ about and the distance $l$ of translation along the screw axis.

The scalar quantities $\theta$ and $l$ are immutable under a change of the coordinate frame. They can be combined to form a dual number called the dual angle [10, 8]:

$$\hat{\theta }=\theta +\epsilon l,$$ (1)

where $\epsilon$ is the dual unit with the property ${\epsilon }^2=0$.

The direction and location of a screw axis $S$ is defined by a pair of vectors ($\mathbf{s},{\mathbf{s}}^0$), known as Plücker vectors, where $\mathbf{s}$ is a unit vector indicating its direction while ${\mathbf{s}}^0$ describes its location. Let $\mathbf{c}$ be a point on $S$, then ${\mathbf{s}}^0=\mathbf{c}\times \mathbf{s}$. With the aid of the dual unit $\epsilon$, the pair of Plücker vectors can be rewritten in a more compact form as a dual vector:

$$\stackrel{\mathbf{ˆ}}{\mathbf{s}}=\mathbf{s}+\epsilon {\mathbf{s}}^0.$$ (2)

As a vector invariant, the change of a screw axis follows the rule of similarity transformation, and the corresponding dual vector induces a dual orthogonal transformation [12].

The aforementioned set of six parameters $(a,b,c;\alpha ,\beta ,\gamma )$ are not directly associated with these invariants. In contrast, dual quaternions and dual Rodrigues parameters are resulted from the above representations of the invariants.

2.2. Dual Quaternions

A dual quaternion is obtained by directly combining the dual angle (1) and the dual vector (2) [8, 10]:

$$\stackrel{\mathbf{ˆ}}{\mathbf{Q}}=\stackrel{\mathbf{ˆ}}{\mathbf{s}}\text{sin}\frac{\hat{\theta }}{2}+\text{cos}\frac{\hat{\theta }}{2},$$ (3)

where

$$\begin{gathered} \text{sin}\frac{\hat{\theta }}{2}=\text{sin}\frac{\theta }{2}+\epsilon \frac{l}{2}\text{cos}\frac{\theta }{2}, \\ \text{cos}\frac{\hat{\theta }}{2}=\text{cos}\frac{\theta }{2}-\epsilon \frac{l}{2}\text{sin}\frac{\theta }{2}. \end{gathered}$$ (4)

Let $\stackrel{\mathbf{ˆ}}{\mathbf{Q}}=\left({\hat{Q}}_1,{\hat{Q}}_2,{\hat{Q}}_3,{\hat{Q}}_4\right)$ and $\stackrel{\mathbf{ˆ}}{\mathbf{s}}=\left({\hat{s}}_1,{\hat{s}}_2,{\hat{s}}_3\right)$, then it follows from Eq. (3) that

$${\hat{Q}}_1={\hat{s}}_1\text{sin}\frac{\hat{\theta }}{2},{\hat{Q}}_2={\hat{s}}_2\text{sin}\frac{\hat{\theta }}{2},{\hat{Q}}_3={\hat{s}}_3\text{sin}\frac{\hat{\theta }}{2},{\hat{Q}}_4=\text{cos}\frac{\hat{\theta }}{2}.$$ (5)

They are called dual Euler parameters.

The conjugate of $\stackrel{\mathbf{ˆ}}{\mathbf{Q}}$ is given by

$${\stackrel{\mathbf{ˆ}}{\mathbf{Q}}}^{*}=-\stackrel{\mathbf{ˆ}}{\mathbf{s}}\text{sin}\frac{\hat{\theta }}{2}+\text{cos}\frac{\hat{\theta }}{2}.$$ (6)

It is essential in the algebra of dual quaternions such as the computation of the norm squared of a dual quaternion:

$$|\stackrel{\mathbf{ˆ}}{\mathbf{Q}}{|}^2={\stackrel{\mathbf{ˆ}}{\mathbf{Q}}}^{*}\stackrel{\mathbf{ˆ}}{\mathbf{Q}}=\stackrel{\mathbf{ˆ}}{\mathbf{Q}}{\stackrel{\mathbf{ˆ}}{\mathbf{Q}}}^{*}={\hat{Q}}_1^2+{\hat{Q}}_2^2+{\hat{Q}}_3^2+{\hat{Q}}_4^2.$$ (7)

When $\left|\stackrel{\mathbf{ˆ}}{\mathbf{Q}}\right|=1$, which is equivalent to:

$$Q_1^2+Q_2^2+Q_3^2+Q_4^2=1,$$ (8)

and

$$Q_1{Q}_1^0+Q_2{Q}_2^0+Q_3{Q}_3^0+Q_4{Q}_4^0=0,$$ (9)

the resulting dual quaternion is called a unit dual quaternion. It is said to define a unit hypersphere in dual dimensions [8].

Another form of a dual quaternion is $\stackrel{\mathbf{ˆ}}{\mathbf{Q}}=\mathbf{Q}+\epsilon {\mathbf{Q}}^0$ where $\mathbf{Q}$ is a unit quaternion of rotation and

$${\mathbf{Q}}^0=\frac{1}{2}\mathbf{d}\mathbf{Q}$$ (10)

where $\mathbf{d}$ is a vector quaternion of translation, i.e., $\mathbf{d}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$ with $\mathbf{i},\mathbf{j},\mathbf{k}$ being quaternion units.

A general formula for recovering $\mathbf{d}$ from a dual quatearnion is given in [13]:

$$\mathbf{d}=\frac{{\mathbf{Q}}^0{\mathbf{Q}}^{*}-{\mathbf{Q}\mathbf{Q}}^{0*}}{\mathbf{Q}{\mathbf{Q}}^{*}}.$$ (11)

This formula works even when $\stackrel{\mathbf{ˆ}}{\mathbf{Q}}$ is not a unit quaternion, i.e., none of the conditions (8) and (9) is satisfied.

2.3. Dual Rodrigues Parameters

A slightly different way of combining (1) and (2) results in the following:

$$\stackrel{\mathbf{ˆ}}{\mathbf{R}}=\stackrel{\mathbf{ˆ}}{\mathbf{s}}\text{tan}\frac{\hat{\theta }}{2}.$$ (12)

Three dual-number components, $\stackrel{\mathbf{ˆ}}{\mathbf{R}}=\left({\hat{R}}_1,{\hat{R}}_2,{\hat{R}}_3\right)$, are called the dual Rodrigues parameters [8, 9, 14, 15, 6, 16]. They are simply the following dual-number ratios of dual Euler parameters:

$${\hat{R}}_1=\frac{{\hat{Q}}_1}{{\hat{Q}}_4},{\hat{R}}_2=\frac{{\hat{Q}}_2}{{\hat{Q}}_4},{\hat{R}}_3=\frac{{\hat{Q}}_3}{{\hat{Q}}_4}.$$ (13)

As ${\hat{R}}_4=1+\epsilon \cdot 0$, the dual Rodrigues parameters are said to define a dual hyperplane tangent to the unit dual hypersphere.

Let ${\hat{R}}_i=R_i+\epsilon R_i^0(i=\mathrm{1,\; 2},3)$. After applying the dual number division and separating the real and dual part, equation (13) can be then expanded as:

$$\left[\begin{array}{c}{R}_1\\ R_2\\ R_3\\ 1\\ R_1^0\\ R_2^0\\ R_3^0\\ 0\end{array}\right]=\left[\begin{array}{c}{r}_1\\ r_2\\ r_3\\ 1\\ r_4\\ r_5\\ r_6\\ 0\end{array}\right]=\left[\begin{array}{c}{Q}_1/Q_4\\ Q_2/Q_4\\ Q_3/Q_4\\ 1\\ \left(Q_1^0{Q}_4-Q_1{Q}_4^0\right)/Q_4^2\\ \left(Q_2^0{Q}_4-Q_2{Q}_4^0\right)/Q_4^2\\ \left(Q_3^0{Q}_4-Q_3{Q}_4^0\right)/Q_4^2\\ 0\end{array}\right]$$ (14)

The covariance matrix which will be introduced later in this paper is based on this projection (14). It bears similarity to the stereographic projection introduced in [17].

3. Kinematic Mean, Covariance and Confidence Ellipsoids

Given a set of uncertain spatial displacements, the RTCL method [3] computes the arithmetic mean by treating each of the six displacement parameters $(a,b,c,\alpha ,\beta ,\gamma )$ separately and independently to obtain the mean displacement $\left(a_m,b_m,c_m,{\alpha }_m,{\beta }_m,{\gamma }_m\right)$ and standard deviations $\left({\sigma }_a,{\sigma }_b,{\sigma }_c,{\sigma }_{\alpha },{\sigma }_{\beta },{\sigma }_{\gamma }\right)$. The resulting 6-dimensional confidence ellipsoid is given by:

$$\frac{{\left(a-a_m\right)}^2}{{\sigma }_a^2}+\frac{{\left(b-b_m\right)}^2}{{\sigma }_b^2}+\frac{{\left(c-c_m\right)}^2}{{\sigma }_c^2}+\frac{{\left(\alpha -{\alpha }_m\right)}^2}{{\sigma }_{\alpha }^2}+\frac{{\left(\beta -{\beta }_m\right)}^2}{{\sigma }_{\beta }^2}+\frac{{\left(\gamma -{\gamma }_m\right)}^2}{{\sigma }_{\gamma }^2}\le k_{95}^2.$$ (15)

where $k_{95}$ is the scaling coefficient based on a ${\chi }^2$ distribution [3] [18]. The coefficient is determined by the level of confidence, which is 95% in this case, as well as the number of standard normal random variables or degrees of freedom in a given data set.

This process neglects the joint probability and does not take into account the geometry of $SE$(3).

3.1. Kinematic Mean

Ge et al. in [19], instead, used the dual-quaternion representation to obtain two different methods for computing relative displacements for determining the mean and variance of uncertain displacements. In this paper, we use the method that preserves the screw axis when averaging a set of screw displacements. Let ${\stackrel{\mathbf{ˆ}}{\mathbf{Q}}}_i={\mathbf{Q}}_i+\epsilon {\mathbf{Q}}_i^0(i=\mathrm{1,2},\cdots ,n)$ denote a set of unit dual quaternions, the method goes as follows:

1. First, compute the arithmetic averages of ${\mathbf{Q}}_i$ and ${\mathbf{Q}}_i^0$ to obtain

   $${\mathbf{V}}_m=\frac{1}{n}\sum _{i=1}^n{\mathbf{Q}}_i,{\mathbf{V}}_m^0=\frac{1}{n}\sum _{i=1}^n{\mathbf{Q}}_i^0.$$ (16)
2. Then normalize ${\mathbf{V}}_m$ to obtain a unit quaternion:

   $$\mathbf{V}=\frac{{\mathbf{V}}_m}{\left|{\mathbf{V}}_m\right|}.$$ (17)
3. Next, recover the translation vector quaternion using (11):

   $$\mathbf{d}={\mathbf{V}}_m^0{\mathbf{V}}^{*}-\mathbf{V}{\mathbf{V}}_m^{0*}.$$ (18)
4. Finally, use (10) to obtain

   $${\mathbf{V}}^0=\frac{1}{2}\mathbf{d}\mathbf{V}$$ (19)

   so that $\stackrel{\mathbf{ˆ}}{\mathbf{V}}=\mathbf{V}+\epsilon {\mathbf{V}}^0$ becomes a unit dual quaternion.

3.2. Kinematic Covariance Matrices

Let ${\stackrel{\mathbf{ˆ}}{\mathbf{W}}}_i$ denote the relative displacements in dual quaternion form. Then ${\stackrel{\mathbf{ˆ}}{\mathbf{W}}}_i$ are related to the mean displacement $\stackrel{\mathbf{ˆ}}{\mathbf{V}}$ and the given ${\stackrel{\mathbf{ˆ}}{\mathbf{Q}}}_i$ by

$${\stackrel{\mathbf{ˆ}}{\mathbf{Q}}}_i=\stackrel{\mathbf{ˆ}}{\mathbf{V}}{\stackrel{\mathbf{ˆ}}{\mathbf{W}}}_i.$$ (20)

Thus one can compute ${\stackrel{\mathbf{ˆ}}{\mathbf{W}}}_i$ using

$${\stackrel{\mathbf{ˆ}}{\mathbf{W}}}_i={\stackrel{\mathbf{ˆ}}{\mathbf{V}}}^{*}{\stackrel{\mathbf{ˆ}}{\mathbf{Q}}}_i,$$ (21)

where ${\stackrel{\mathbf{ˆ}}{\mathbf{V}}}^{*}$ is the conjugate of $\stackrel{\mathbf{ˆ}}{\mathbf{V}}$.

Let ${\mathbf{r}}_i=[r_{i1}{r}_{i2}{r}_{i3}{r}_{i4}{r}_{i5}{r}_{i6}{]}^T$ be components of the dual Rodrigues parameters as given by (13) and (14). Then the the covariance matrix can be established in terms of ${\mathbf{r}}_i$ as

$$\left[{\Sigma }_r\right]=\frac{1}{n}\left[\begin{array}{llll}{\mathbf{r}}_1& {\mathbf{r}}_2& \cdots & {\mathbf{r}}_n\end{array}\right]\left[\begin{array}{c}{\mathbf{r}}_1^T\\ {\mathbf{r}}_2^T\\ ⋮\\ {\mathbf{r}}_n^T\end{array}\right]$$ (22)

Hence the covariance matrix is a 6 × 6 matrix. Let ${\lambda }_i(i=\mathrm{1,\; 2},\mathrm{3,\; 4},\mathrm{5,\; 6})$ denote the eigenvalues of the covariance matrix (22) and let $\left[E\right]$ denotes a 6 × 6 matrix whose columns are the eigenvectors. The resulting confidence region is a 6D ellipsoid defined by the following equation with respect to its principal coordinate frame.

$$\frac{r_1^2}{{\lambda }_1}+\frac{r_2^2}{{\lambda }_2}+\frac{r_3^2}{{\lambda }_3}+\frac{r_4^2}{{\lambda }_4}+\frac{r_5^2}{{\lambda }_5}+\frac{r_6^2}{{\lambda }_6}\le k_{95}^2,$$ (23)

where $\mathbf{r}=\left(r_1,r_2,r_3,r_4,r_5,r_6\right)$ is an arbitrary point of the ellipsoid and is corresponding to the projection equation (14). The principal directions are defined by the eigenvectors of he covariance matrix. The relative dual quaternion ${\stackrel{\mathbf{ˆ}}{\mathbf{W}}}_{\mathbf{i}}$ associated with ${\stackrel{\mathbf{ˆ}}{\mathbf{R}}}_{\mathbf{i}}$ can be solved by combining equation (10) and (11). Spatial displacements ${\stackrel{\mathbf{ˆ}}{\mathbf{Q}}}_{\mathbf{i}}$ obtained from the resulted 6D ellipsoid with respect to the mean $\stackrel{\mathbf{ˆ}}{\mathbf{V}}$ can be found by using equation (20).

4. Examples

This section presents three examples to illustrate the dual quaternion based method for computing the kinematic confidence regions. The units for distances and angles are in millimeters and degrees, respectively.

4.1. Example 1: A Set of Screw Displacements

Let us consider first a set of screw displacements about the same axis (Figure 1). They are given in terms of $\left(a_i,b_i,c_i,{\alpha }_i,{\beta }_i,{\gamma }_i\right)(i=\mathrm{1,2},\dots ,11)$ as shown in Table 1. The set of dual Rodrigues parameters ${\mathbf{r}}_i=[r_{i1}{r}_{i2}{r}_{i3}{r}_{i4}{r}_{i5}{r}_{i6}{]}^T$, as shown in Table 2, are obtained by first computing the unit dual quaternions and then using (22). The corresponding 6 × 6 covariance matrix is then obtained as:

$$\left[{\Sigma }_r\right]=\left[\begin{array}{cccccc}0.011& 0.019& 0.030& 0.383& 0.192& 0.818\\ 0.019& 0.031& 0.050& 0.638& 0.320& 1.363\\ 0.030& 0.050& 0.080& 1.020& 0.513& 2.181\\ 0.383& 0.638& 1.020& 13.078& 6.586& 27.958\\ 0.192& 0.320& 0.513& 6.586& 3.334& 14.088\\ 0.818& 1.363& 2.181& 27.958& 14.088& 59.772\end{array}\right]$$

FIGURE 1.

Example 1: A set of screw displacements about a fixed screw axis indicated by the line in light blue.

TABLE 1.

A set of displacements that belongs to a pure screw motion.

$M_i$	$a_i$	$b_i$	$c_i$	${\alpha }_i$	${\beta }_i$	${\gamma }_i$
1	−5.129	−6.400	−12.638	−2.672	−5.246	−7.966
2	−0.640	−4.578	−4.323	5.055	6.729	11.874
3	−7.900	−7.731	−20.048	−5.819	−16.726	−23.787
4	−6.351	−6.933	−15.559	−4.303	−9.732	−14.303
5	3.859	−2.545	2.000	14.561	14.078	30.022
6	9.755	0.837	8.812	28.731	18.303	53.734
7	7.760	−0.420	6.633	23.786	17.528	45.662
8	12.744	2.978	11.921	36.257	18.203	65.926
9	5.787	−1.544	4.363	19.031	16.107	37.736
10	−7.422	−7.462	−18.539	−5.441	−14.368	−20.614
11	−8.741	−8.290	−23.095	−6.167	−21.483	−30.222

TABLE 2.

The set of dual Rodrigues parameters ${\mathbf{r}}_i$ for Example 1

$r_i$	$r_{i1}$	$r_{i2}$	$r_{i3}$	$r_{i4}$	$r_{i5}$	$r_{i6}$
1	−0.063	−0.106	−0.169	−1.999	−0.809	−4.184
2	0.002	0.003	0.004	0.144	0.174	0.354
3	−0.122	−0.204	−0.326	−4.193	−2.135	−8.975
4	−0.086	−0.144	−0.230	−2.810	−1.256	−5.934
5	0.056	0.093	0.149	1.861	0.881	3.952
6	0.126	0.210	0.336	4.245	2.061	9.040
7	0.102	0.169	0.271	3.379	1.591	7.174
8	0.166	0.276	0.442	5.761	3.011	12.365
9	0.078	0.131	0.209	2.593	1.207	5.498
10	−0.110	−0.183	−0.293	−3.705	−1.808	−7.894
11	−0.148	−0.246	−0.394	−5.275	−2.917	−11.395

The square roots of the eigenvalues of the covariance matrix are given by

$$\sqrt{{\lambda }_1}=\sqrt{{\lambda }_2}=\sqrt{{\lambda }_3}=\sqrt{{\lambda }_4}=\sqrt{{\lambda }_5}=0,\sqrt{{\lambda }_6}=8.73449.$$ (24)

Thus the confidence ellipsoid obtained using dual Rodrigues parameters reduces to a single line segment. In this case, the number of degrees of freedom is reduced to one and the corresponding scaling factor is given by $k_{95}=1.960$. Figure 2 shows the screw motion resulting from the confidence line-segment.

FIGURE 2.

Example 1: A screw motion generated from the dual Rodrigues confidence line segment. Frames in black indicate the positions of given displacement data.

If the RTCL method is used, the standard deviations obtained directly using Table 1 are given by:

$$\begin{gathered} {\sigma }_a=7.91108,{\sigma }_b=3.89537,{\sigma }_c=12.86021, \\ {\sigma }_{\alpha }={15.71}^{\circ },{\sigma }_{\beta }={15.80}^{\circ },{\sigma }_{\gamma }={34.59}^{\circ }. \end{gathered}$$ (25)

This results in a six dimensional confidence ellipsoid and the corresponding scaling factor is given by $k_{95}=3.549$. Figure 3 shows the swept volume resulting from the six dimensional confidence ellipsoid.

FIGURE 3.

Example 1: The union of displaced positions generated from the RTCL confidence ellipsoid. Frames in black indicate the positions of given displacement data.

Example 2: Screw Displacement Data with Uncertainties

For the second example, the displacement data is generated from the screw displacement data of Example 1 but with 10% uncertainties. Table 3 lists the data and Figure 4 shows the set of displacements. The corresponding dual Rodriques parameters, $\mathbf{r}$, are shown in Table 4.

TABLE 3.

A set of displacements close to a screw motion with uncertainties.

$M_i$	$a_i$	$b_i$	$c_i$	${\alpha }_i$	${\beta }_i$	${\gamma }_i$
1	−5.129	−6.400	−12.638	−2.672	−5.246	−7.966
2	−0.640	−4.578	−4.323	5.055	6.729	11.874
3	−7.900	−7.731	−20.048	−5.819	−16.726	−23.787
4	−6.351	−6.933	−15.559	−4.303	−9.732	−14.303
5	3.859	−2.545	2.000	14.561	14.078	30.022
6	9.755	0.837	8.812	28.731	18.303	53.734
7	7.760	−0.420	6.633	23.786	17.528	45.662
8	12.744	2.978	11.921	36.257	18.203	65.926
9	5.787	−1.544	4.363	19.031	16.107	37.736
10	−7.422	−7.462	−18.539	−5.441	−14.368	−20.614
11	−8.741	−8.290	−23.095	−6.167	−21.483	−30.222

FIGURE 4.

Example 2: A set of screw displacements with 10% uncertainties about a fixed screw axis indicated by the line in light blue.

TABLE 4.

Example 2: The set of dual Rodrigues parameters ${\mathbf{r}}_i$

${\mathbf{r}}_i$	$r_{i1}$	$r_{i2}$	$r_{i3}$	$r_{i4}$	$r_{i5}$	$r_{i6}$
1	−0.064	−0.105	−0.173	−1.935	−0.704	−4.080
2	0.001	0.010	−0.003	0.220	−0.022	0.163
3	−0.128	−0.201	−0.343	−4.648	−1.904	−8.970
4	−0.088	−0.142	−0.225	−2.866	−1.178	−5.312
5	0.063	0.100	0.160	2.110	0.830	4.070
6	0.138	0.215	0.351	4.659	1.835	9.390
7	0.096	0.158	0.246	3.330	1.629	7.003
8	0.159	0.274	0.484	5.588	2.862	13.204
9	0.087	0.115	0.178	3.038	1.206	5.672
10	−0.108	−0.173	−0.280	−3.694	−1.623	−8.564
11	−0.157	−0.253	−0.394	−5.803	−2.930	−12.575

The corresponding covariance matrix is given by

$$\left[{\Sigma }_r\right]=\left[\begin{array}{cccccc}0.012& 0.019& 0.031& 0.412& 0.186& 0.873\\ 0.019& 0.030& 0.050& 0.661& 0.300& 1.405\\ 0.031& 0.050& 0.082& 1.085& 0.493& 2.309\\ 0.412& 0.661& 1.085& 14.421& 6.542& 30.557\\ 0.186& 0.300& 0.493& 6.542& 3.007& 13.963\\ 0.873& 1.405& 2.309& 30.557& 13.963& 65.193\end{array}\right]$$

The square roots of the eigenvalues of the covariance matrix are given by

$$\begin{gathered} \sqrt{{\lambda }_1}=0.00297,\sqrt{{\lambda }_2}=0.00574,\sqrt{{\lambda }_3}=0.02466, \\ \sqrt{{\lambda }_4}=0.12658,\sqrt{{\lambda }_5}=0.28703,\sqrt{{\lambda }_6}=9.09105. \end{gathered}$$ (26)

The first five square roots of the eigenvalues are no longer zero but still much smaller than the six one. The resulting confidence ellipsoid is a very thin one that approximates a line segment. In this case, for the purpose of selecting $k_{95}$, we may consider the degrees of freedom to be a number between 1 and 6. We found that $k_{95}=2.795$ is a reasonable value, which corresponds to 3 DOF data.

If the RTCL method is used, the standard deviations are similar to those for Example 1:

$$\begin{gathered} {\sigma }_a=8.07276,{\sigma }_b=3.81308,{\sigma }_c=13.42286, \\ {\sigma }_{\alpha }={15.58}^{\circ },{\sigma }_{\beta }={15.710}^{\circ },{\sigma }_{\gamma }={34.809}^{\circ }. \end{gathered}$$ (27)

This results in a six dimensional confidence ellipsoid with the scaling factor $k_{95}=3.549$. Figure 5 indicates that dual Rodrigues parameter based confidence region resulting a swept volume which is very similar to the exact screw displacement case. Figure 6 is similar to Figure 3 which gives an overestimated ellipsoid.

FIGURE 5.

Example 2: A motion generated from the dual Rodrigues confidence ellipsoid which is close to a screw motion suggested by given displacements. Frames in black indicate the positions of given displacement data.

FIGURE 6.

Example 2: An ellipsoid-like region generated from the RTCL confidence ellipsoid. Frames in black indicate the positions of given displacement data.

Example 3: A Set of Spatial Displacement

For a set of general displacements shown in Figure 7, the displacement data are given in Table 5. The dual Rodriques parameters are shown in Table 6.

FIGURE 7.

Example 3: A set of general spatial displacements given in table 5.

TABLE 5.

A set of general spatial displacements.

$M_i$	$a_i$	$b_i$	$c_i$	${\alpha }_i$	${\beta }_i$	${\gamma }_i$
1	7.358	−15.910	10.657	12.186	−8.746	−18.125
2	1.455	−2.806	30.185	54.004	−1.009	−3.069
3	−2.755	1.040	9.003	13.468	−4.594	5.929
4	−10.527	10.461	5.736	7.372	−4.707	23.960
5	−16.565	16.436	1.459	7.735	5.694	34.552
6	15.653	−13.756	11.773	19.652	−3.781	−31.594
7	11.127	−10.101	9.710	18.033	−1.028	−22.526
8	−2.444	4.159	3.022	4.500	−0.047	9.000
9	0.892	−0.042	4.801	7.500	−1.500	−0.013
10	−3.156	3.635	−4.363	12.214	23.892	7.837
11	24.162	−32.416	15.900	39.576	−5.973	−66.342

TABLE 6.

Example 3: The set of dual Rodrigues parameters ${\mathbf{r}}_i$ corresponding to the dual quaternions.

${\mathbf{r}}_i$	$r_{i1}$	$r_{i2}$	$r_{i3}$	$r_{i4}$	$r_{i5}$	$r_{i6}$
1	−0.054	−0.084	−0.091	2.616	−7.324	2.231
2	0.334	0.021	0.015	−1.778	6.296	9.541
3	−0.027	0.006	0.104	−4.234	0.111	0.024
4	−0.071	0.033	0.263	−7.437	5.245	−2.334
5	−0.094	0.145	0.344	−9.478	9.251	−5.646
6	0.010	−0.086	−0.214	7.053	−3.920	2.384
7	0.004	−0.036	−0.144	3.719	−3.705	1.367
8	−0.109	0.042	0.124	−4.246	0.831	−3.160
9	−0.083	0.016	0.047	−2.580	−1.100	−1.960
10	−0.065	0.254	0.058	−4.148	−0.325	−7.260
11	0.156	−0.311	−0.505	20.513	−5.362	4.816

The resuling covariance matrix is given by.

$$\left[{\Sigma }_r\right]=\left[\begin{array}{cccccc}0.016& -0.007& -0.013& 0.457& 0.035& 0.502\\ -0.007& 0.018& 0.024& -0.937& 0.394& -0.423\\ -0.013& 0.024& 0.050& -1.747& 0.859& -0.605\\ 0.457& -0.937& -1.747& 63.576& -28.025& 20.812\\ 0.035& 0.394& 0.859& -28.025& 24.204& -5.372\\ 0.502& -0.423& -0.605& 20.812& -5.372& 20.965\end{array}\right]$$

The square roots of the eigenvalues of the covariance matrix are given by

$$\begin{gathered} \sqrt{{\lambda }_1}=0.01233,\sqrt{{\lambda }_2}=0.01684,\sqrt{{\lambda }_3}=0.07903, \\ \sqrt{{\lambda }_4}=2.59827,\sqrt{{\lambda }_5}=4.12128,\sqrt{{\lambda }_6}=9.22423. \end{gathered}$$ (28)

If the RTCL method is used, the standard deviations are obtained as

$$\begin{gathered} {\sigma }_a=11.66500,{\sigma }_b=13.68864,{\sigma }_c=8.98173, \\ {\sigma }_{\alpha }={15.37}^{\circ },{\sigma }_{\beta }={8.81}^{\circ },{\sigma }_{\gamma }={28.03}^{\circ }. \end{gathered}$$ (29)

Comparing figure 8 and 9, it can be stated that the union of displaced positions generated from dual Rodrigues method confidence ellipsoid captures the given spatial displacements more effectively than the RTCL method.

FIGURE 8.

Example 3: A swept volume generated from the dual Rodrigues confidence ellipsoid which in general captures the given displacements. Frames in black indicate the positions of given displacement data.

FIGURE 9.

Example 3: The swept volume generated from the RTCL confidence ellipsoid, which is overestimated. Frames in black indicate the positions of given displacement data.

Figure 10 shows the comparison between the confidence regions obtained from dual Rodrigues parameter method and the RTCL method, respectively, in $r_1,r_2,r_3$ space. As can be seen, the dual Rodriques formulation yields a much smaller confidence region than that of the RTCL method.

FIGURE 10.

Example 3: Confidence region shown in $r_1,r_2,r_3$ space. The red region is corresponding to dual Rodrigues method and the blue region is corresponding to the RTCL method. Black dots are the projection of given displacements in the space.

5. Conclusion

In this paper we studied the construction of confidence regions for a set of uncertain spatial displacements using dual quaternion algebra and dual Rodrigues parameters. The most commonly used coordinates $(a,b,c,\alpha ,\beta ,\theta )$ and corresponding Rotational and Translational Confidence Limit (RTCL) method of constructing confidence region tends to grossly overestimate the confidence region. The dual quaternion based projection by using Rodrigues parameters has been proved to be able to capture the screw axis of a spatial displacement and thus the resulting confidence regions preserve the kinematic properties underlying the kinematic data. Three examples have been used for comparing the resulting swept volume based on these two methods. Our examples show that the confidence ellipsoids defined using dual Rodrigues parameters captures the kinematic structure of spatial displacements more effectively especially in cases involving screw displacements.