Instrumentation & Measurement Magazine 24-3 - 32

Evaluation Process
We assessed the performance of the implemented algorithms. The evaluation process was composed of two
phases: the training phase and the validation phase. In the
training phase, the methods using the AX = XB formulation were trained using the A and B obtained following the
above-mentioned process. As for the methods using the
point set matching formulation, they were trained using
70% of all sets of C P and B P . In the validation phase, the
remaining 30% of C P as well as B P were used for all implemented algorithms.
To achieve the best performance in all implemented algorithms, their necessary parameters were selected as listed in
Table 2. For the algorithms that required an initial solution
(e.g., ICP [13], GD-SE(3) [19], and HI-SO(3)R3), the initial rotation matrix R0 was randomly generated in SO(3). The learning
rates for HI-SO(3)R3 were selected as α  tr M1.5 λ and α  0.9
n
 
where M    P  P  P  P  in that P  n1  P is the center of B P ,
is the minimum eigenvalue of M,
tr(M) is the trace of M, λ M
min
and n is the number of points.
As explained earlier, the correspondence between C P and
B
P is required for point set matching algorithms for handeye calibration. To study the importance of the correspondence
between C P and B P , we provided it to the HI-SO(3)R3,

   
 
 

Fig. 3. Experimental set-up. The screw with the Intel RealSense D435 stereo
camera.

checkerboard pattern were acquired with different gripper
poses which were recorded as well. As for the data used for
the algorithms with the point set matching formulation, we
collected B P by guiding the robot to make a physical contact
with each corner of the checkerboard pattern using the end of
the screw (Fig. 2b). By exploiting the stereo camera's depth recovery ability, 100 sets of C P were attained from the captured
stereo images (Fig. 2a).

 

i1

 
p

M
min

 

Table 1 - Parameters of Intel D435 stereo camera
Size
90×25×25mm
Minimum Depth Distance
0.105 m

Connectors

Imager Resolution

Depth Field of View (FOV)

USB-C 3.1 Gen 1

1280×720

86°×57°(±3°)

RGB Sensor FOV

RGB Sensor Resolution

Depth Output Resolution

64.9°×42.5°×77°(±3°)

1920×1080

Up to 1280×720

Table 2 - Parameters used in the implemented algorithms
Algorithms

Parameter

Symbol

Value

Initial rotation matrix

Random R0 ∈SO(3)

Initial translation vector

Learning rate for rotation

αR

1.5
M
tr
M
   min

Learning rate for translation

αP

0.9
n

Tolerance

1×10−4

Maximum iteration number

1×104

Initial rotation matrix

Random R0 ∈ SO(3)

Initial translation vector

Tolerance

1×10−7

Maximum iteration number

1×106

Initial rotation matrix

Random R0 ∈ SO(3)

HI-SO(3)R3

GE-SE(3)

ICP

Wu's
32

Initial translation vector

Tolerance

1×10−4

Maximum iteration number

1×104

Scaling factor

1×1013

IEEE Instrumentation & Measurement Magazine

May 2021

Instrumentation & Measurement Magazine 24-3

Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 24-3