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We investigate the performance of pose measuring systems which determine an object’s pose from measurement of a few fiducial markers attached to the object. Such systems use point-based, rigid body registration to get the orientation matrix. Uncertainty in the fiducials’ measurement propagates to the uncertainty of the orientation matrix. This orientation uncertainty then propagates to points on the object’s surface. This propagation is anisotropic, and the direction along which the uncertainty is the smallest is determined by the eigenvector associated with the largest eigenvalue of the orientation data’s covariance matrix. This eigenvector in the coordinate frame defined by the fiducials remains almost fixed for any rotation of the object. However, the remaining two eigenvectors vary widely and the direction along which the propagated uncertainty is the largest cannot be determined from the object’s pose. Conditions that result in such a behavior and practical consequences of it are presented.

The pose of a rigid object is defined by six degrees of freedom (6DOF): three angles describing the object’s orientation matrix

In this paper, our focus is on a different aspect of pose uncertainty. We are interested in how uncertainty of a single static measurement of a rigid body pose propagates to any Point of Interest (POI) associated with the object (e.g., a point on its surface). When the Computer Aided Design (CAD) model of an object is known, the location of any POI can be calculated using 6DOF data acquired by pose measuring systems [

In most practical applications, the six components of pose are not directly measured but are derived from other raw measurements. Many pose measuring systems report 6DOF data of an object based on the measurement of the 3D positions of a few points. These points, also known as fiducial markers, are rigidly attached to (or around) the measured object. Some systems may not require the use of markers as they may be trained to use some characteristic features of the measured object (e.g., well defined corner points). For systems which use 3D points, a homogenous transformation

Noise present in the measurement of the fiducials propagates to the transformation

For the class of pose measuring systems which use point-based registration to track the pose of a rigid object, propagation of orientation uncertainty to a given POI is equivalent to the propagation of the fiducials’ uncertainty to a target point and, therefore, should inherit the above-mentioned characteristics of

In this paper, we expanded the study in [

The location of any POI is fixed relative to the locations of fiducials. Therefore, for the class of pose measuring systems discussed in this paper, if a vector pointing to a POI is aligned with

In the next section, some background information and relevant equations are reviewed, followed by a brief description of the experimental setup and data postprocessing. This is followed by a presentation of the results, discussion, and conclusions.

In this section a brief review of the theoretical work relevant to our experiments is presented. Section

Given two sets of

However, noise in the measured fiducials

While noise does not affect the moment of inertia, it has a great impact on the Target Registration Error (

We note that the orientation of the noise matrix

Let vector

Repeated measurements of the orientation matrix

A commercially available, large-scale tracking system (iGPS) was used to collect 6DOF data [

Four different local frames were created and used to obtain measurements for four configurations of the vector bars; see Figure

Four configurations (a, b, c, d) of the vector bars used in the experiments to create four different local frames.

The system outputs 6DOF data

For each

Histograms of the angular deviations

The evaluation of the angular uncertainty

To show a link between the propagation of an object’s pose uncertainty to a POI and the propagation of uncertainty from fiducials to the target in registration problem, the distribution of the angular uncertainty

As mentioned earlier, the orientation of matrix

Figure

Elements of the covariance matrix

Spatial distribution of the three eigenvectors

The anisotropy of noise perturbing the locations of fiducials was checked by evaluating the ratio of eigenvalues

The distributions of the angular uncertainty ^{−2}] for

Directional distribution of the angular uncertainty

Histogram of angles

Figure

Histogram of angles

Finally, the plots in Figure ^{2}], diagonal noise matrix (variances) in (^{2}], and the axis of rotation

Distribution of uncertainty

The average orientations

The theoretical FBK distribution

The theoretical equations (

The surprising stability of eigenvector

Many pose measuring systems derive pose from the measurement of fiducial markers attached to an object. The uncertainty of the fiducials’ locations propagates to the uncertainty of the object’s orientation which, in turn, propagates in an anisotropic way to individual points on the object’s surface. The angular distribution of the orientation uncertainty propagated to a POI depends generally on the object’s orientation. However, the orientation uncertainty in the regions close to the poles defined by the eigenvector corresponding to the smallest eigenvalue of the moment of inertia matrix of fiducials is almost independent of the object’s orientation. These regions are also characterized by the smallest propagated orientation uncertainty. Thus, strategic placement of fiducials around an object ensures that the orientation uncertainty propagated to a given POI is the smallest, regardless of the object’s orientation.

The problem of finding optimal locations of fiducials for a given POI does not have a unique solution. In practical applications, the geometry of the rotated object and type of markers tracked by the instrument may impose additional constraints on possible fiducial locations. The general strategy for optimal placement is schematically shown in Figure

Optimal placement of fiducials for a given Point of Interest

In this general configuration, the principal axis of the smallest moment of inertia is denoted by

To illustrate this procedure, numerical simulations were performed. Six different randomly selected configurations of

Figure

Results from computer simulations. Six random configurations of fiducials, lines (1–3) correspond to the worst-case selection of Point of Interest

As seen in Figures

Certain trade names and company products are mentioned in the text to adequately describe the experimental procedure. In no case does it imply NIST recommendation or endorsement, nor does it imply that the products are necessarily the best available for the purpose.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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