An Improvement of Pose Measurement Method Using Global Control Points Calibration

During the last decade pose measurement technologies have gained an increasing interest in the computer vision. The vision-based pose measurement method has been widely applied in complex environments. However, the pose measurement error is a problem in the measurement applications. It grows rapidly with increasing measurement range. In order to meet the demand of high accuracy in large measurement range, a measurement error reduction solution to the vision-based pose measurement method, called Global Control Point Calibration (GCPC), is proposed. GCPC is an optimized process of existing visual pose measurement methods. The core of GCPC is to divide the measurement error into two types: the control point error and the control space error. Then by creating the global control points as well as performing error calibration of object pose, the two errors are processed. The control point error can be eliminated and the control space error is minimized. GCPC is experimented on the moving target in the camera’s field of view. The results show that the RMS error is 0.175° in yaw angle, 0.189° in pitch angle, and 0.159° in roll angle, which demonstrate that GCPC works effectively and stably.


Introduction
Detecting the rigid transformation of images into known geometry, namely the pose measurement, is one of the central problems in aircraft inflight refueling, spacecraft docking, and comprehensive helmet mounted display [1][2][3]. In the aircraft control during aerial refueling, it is commonly used to provide accurate relative position measurements to the controller of unmanned air vehicle [4]. In spacecraft docking, pose measurement is central to the positioning of the docking assembly, and accomplished with the assistance of artificial markers or natural markers on the spacecraft [5]. In comprehensive helmet mounted display, it plays a significant role in combining the pose of helmet with direction of the weapon or sensor [6]. reference point and the global control point while the control space error is the measurement error between the measuring point and the corresponding global control point. The first error is the primary source of the measurement error and eliminated by using the standard reference data of moving space. The other error is reduced by decreasing the control range of global control point. According to the analysis above, the Global Control Point Calibration(GCPC) is proposed to create the global control points and calibrate the measurement error of object pose.

Description of System
The proposed schematic diagram of GCPC for pose measurement is shown in Fig 1. The system consists of (1) a target, (2) a three-axis turntable, (3) a turntable control box, (4) a camera, and (5) a computer.
The devices work in the following way: The target is fixed on the three-axis turntable which is controlled by the turntable control box; the image of rotating target is captured by the camera while the three standard Euler angles of turntable are read by the control box; both of the two sets of data are transmitted into the computer simultaneously.
The initial coordinate of feature point is calculated as the image of rotating target is transmitted into the computer. The algorithm of point coordinate is the Pose from Orthography and Scaling with Iterations(POSIT) [20]. POSIT is a classical algorithm and approved by scholars, companies, and defense [21,22].
The principle of POSIT is shown as Fig 2. The feature points c m P i have the depth c m z i while the orthographic projecting points c m P i 0 have the same depth c m z i 0 . The cost function is formed as:  (2) as ε x and ε y below a threshold value or when the run times reach the limit. where oj is the object coordinate system, is the matrix which corresponds to the minimum of ε x and ε y .

The Measurement Reference of GCPC
Through Eqs (1) and (2), the spatial coordinate of c m P i , based on the camera coordinate system, is obtained. The following step is to transform from c m P i to ms m P i . ms m P i is the spatial coordinate which is based on the measurement reference. Furthermore, both the creation of global control points and the performance of error calibration are conducted in measurement reference. The relationship of the measurement reference to the other coordinate system are shown in Fig 3. The relationship between the O c -X c Y c Z c and the O ms -X ms Y ms Z ms should be described as: The matrix ms R c and the vector ms T c are described as: where h 1 and h 2 are respectively the unit direction vector of o ms -x ms and o ms -y ms , The point sets c m P x and c m P y are selected respectively to establish o ms -x ms and o ms -y ms . Both of them only rotate around an axis. x is the number of object's position which rotates around the o ms -x ms and y is the number of object's position which rotates around the o ms -y ms . Taking the point sets into Eq (5) [23,24]: where Eq 5(a) is the equation of plane fitting, the coefficient (a, b, d) is the direction vector of plane; 5(b) is the equation of circle fitting, point (e, g) is the anchor point of axis which locates in the fitting plane, r is the circle radius. The unit vectors of o ms -x ms and o ms -y ms are determined through Eq (5). The fitting process is described in Fig 4. The two dotted circles are determined by Eq 5(a), and they locate in the planes determined by Eq 5(b) respectively. As the point sets c m P x , c m P y , and c m P z rotate around the axis o ms -x ms , o ms -y ms , and o ms -z ms respectively, they share a center of rotation theoretically. As the trajectories of them are noncoplanar arcs, a sphere fitting is adopted to describe the arcs. Taking the three point sets into the following sphere fitting equation, the sphere center is the shared center of roation [25]. (l, n, q) is the sphere center, h is the sphere radius.
As the measurement reference O ms -X ms Y ms Z ms is established, the coordinate ð ms m x i ; ms m y i ; ms m z i Þ of feature point ms m P i is obtained. The Principle of GCPC The information of object pose in the measurement reference is formally defined as: where the object pose is represented by I i and x i which are respectively the image feature and the standard pose vector. According to the expression, GCPC is organized as the following overview Fig 5. The control point error is the measurement error between the global control point FM k i and the reference point FM k 0 while the control space error is the measurement error between the measuring point FM t and the corresponding control point FM k i . The measured pose vector x t 0 is obtained in two ways: In Eq (8), y t k 0 is the directly measured value of x t , and shows both the control point error and the control space error. In Eq (9), y t k 5 is the measured value of ðx t À x k 5 Þ, which contains the control space error of FM k 5 . ðx k 5 À x k 0 Þ is the standard pose vector between the reference point FM k 0 and the control point FM k 5 . GCPC optimizes the object pose FM t by using Eq (9).

The Creation of Global Control Points
Given a set of feature point ms m P i;j;k , (i, j, k) are respectively the number of object's position in o mx -x ms , o mx -y ms , and o mx -z ms . A sparse point set M I ¼ f ms m P i;j;k g is selected as the initialized control points. As the moving space is parameterized by the angle information of feature points, the initialized control points are equally distributed in the angle space. The angle based space is divided as Fig 7. With the assistance of adjacent points, the points in M I divide the moving space into ideal subspaces. The measuring point in the ideal subspace is calibrated by the corresponding control point. But through the analysis of measurement reference, it can be concluded that a system error exists in the O ms -X ms Y ms Z m . The axes fitting of o ms -x ms and o ms -y ms is inaccurate and the three axes are incompletely perpendicular. An angle filter is introduced to eliminate the impact of inaccurate measurement reference. For each point in the moving space corresponding to a pose vector ( ms α, ms β, ms γ), the angle between the control point and the measuring point is calculated as: where t is the number of measuring point. The pose vector ð ms at i;j;k ; ms bt i;j;k ; ms gt i;j;k Þ is defined as the following: The matrix ms R i, j, k is defined as: 8 > > > > > > < > > > > > > : ms Ri; j; k t is the rotation matrix and turned into Euler angles through Eq (13): CbSaCg þ SbSg CbSaSg À SbCg CbCa

The Calibration of Measuring Point
The calibration process is separated into two steps: one is the determination of the pair of control point and measuring point, and the other is the calibration of the pose vector. According to Eq (10), the point pair ms m P I;J;K and ms m P t is determined by the minimum of y i;j;k t . Bring the standard pose vector of ms m P I;J;K into the following equation: where ð ms a 0 t ; ms b 0 t ; ms g 0 t Þ is the calibrated pose vector of measuring point ms m P t .

The Measurement Procedure
The measurement procedure of GCPC is shown in Fig 8. GCPC for pose measurement is divided into three steps: the establishment of measurement reference, the creation of global control points, and the calibration of measuring point. The first two steps run only once as the moving space is established.

Experiment system
For the experiment with real data, an infrared camera is used, and the camera's field angle is 80°. The internal camera parameters are calibrated [26,27], and the results are shown in Table 1.
The infrared LEDs are selected as positioning feature points, and the relative spatial position of the four feature points is shown in Table 2. Small holes are chosen to be drilled on the support board of target when the target is produced and the LEDs are selected to submerge in the hole. All devices are located on the experiment platform. Fig 9 shows the practical system in laboratory.

The global control points of GCPC
The range of moving space is -50°to 50°in yaw angle, -50°to 50°in pitch angle, and -30°to 30°in roll angle. The interval angle of sample is 5°at each DOF, and there are 5118 target images within the camera's field of view.   The images with single DOF rotation can be used to establish the measurement reference. The parameters of measurement reference are shown in Table 3.
M I = { ms P i, j, k }(i = -2,-1,0,1,2, j = -2,-1,0,1,2, k = -1,0,1) is selected as an initialized control point set. The interval angle is 20°at each DOF. Part of the M I is beyond the camera's field of view, and the cutoff frequency of M I is 125. Then M I is filtered, and its frequency of occurrence is shown as Fig 10.    Table 4.
The pose measurement results which respectively correspond to the four control point sets M I , M II , M III , and M IV are compared in the next section.

Pose measurement results
In order to prove the role of GCPC, the pose measurement of measuring target in the whole moving space is accomplished. The moving space has been established by the O ms -X ms Y ms Z ms . The gathered data are transmitted into POSIT and GCPC, and the pose measurement results are analyzed. The control point sets M I , M II , M III , and M IV are respectively adopted by GCPC. The root mean square(RMS) error of GCPC and POSIT are displayed in Fig 12. By comparing the results of GCPC and those of POSIT, it is obvious that the measurement accuracy of GCPC is higher than that of POSIT in the whole moving space. The comparisons of the four control point sets demonstrate that the creation of global control points is effective.
In order to test the error distribution of GCPC, the measuring points are classified into the surface of angle determined by three angles. The range of the first two angles are respectively -50°to 50°in yaw angle, -50°to 50°in pitch angle. The third angle changes from -30°to 30°. The RMS error of the surfaces of angle are shown in Fig 13.  The RMS error of GCPC is far less than that of POSIT. The former is stable and reduced to 0.2°while the latter fluctuates along the roll angle and reaches 1.2°. The steep trend of POSIT demonstrates that the measurement error mentioned earlier exits in the pose measurement process, and the gentle trend of GCPC proves that the measurement error is calibrated successfully in the whole moving space.
The above data analysis is based on the RMS error, and 100 measuring points with the maximal errors are selected. The optimization of GCPC to the measuring points is shown in Fig 14. Through analysis of Fig 14, it is evident that the measurement error is reduced by GCPC. The control point error which is the primary source of the measurement error is eliminated successfully, and the error curve which fluctuates around zero is caused by the control space error.

Conclusions
In this paper, GCPC is developed to optimize the pose measurement error. The control point error is redefined to be the primary source of measurement error, and calibrated by the Supporting Information S1 Dataset. Camera captured dataset. This excel contains the capture data used as the basis for the pose measurement solution described in the manuscript. The data is given by means of image coordinate. (XLSX)