6.4.1 Simulation Results
The proposed method is validated under various noise levels and various numbers of samples. It is compared to the method that uses a single point laser introduced in Chapter 3. The relationship between the camera and the robot is given by CBθ=-90°,
CBφ =-10°, CBϕ =-170° and CBt=[510,151,430]T (mm). The line laser is at ΛEa
=[8.73,17.48,43.65]T (mm) relative to the end-effector. The three working plane poses are ΠBa1=[48.68, 31.47, -153.81]T (mm), ΠBa2=[152.41, 138.34, -163.92]T (mm), and
B 3
Πa =[ -4.0, 8.40, 29.80]T (mm). The intrinsic parameters of the camera are chosen as f =1000, u f =1000, v αc =0, u =320, 0 v =240. The single point laser is at 0 [0,-19.90,60.14]T (mm) and pointing in direction [0.98,0.02,-0.20]T , which is on the laser plane ΛEa.
In the following tests, the relative errors of one focal length are calculated as
, , / ,
u estimated u true u true
f − f f (percentage). The hand-eye rotation and translation errors are the Frobenius norms of the difference between the true and estimated CBtand CBR. Additionally, the relative percentage errors are calculated as testimated −ttrue F ttrue F and Restimated −Rtrue F Rtrue F . The relative percentage errors of a plane pose are calculated as ΠBaestimated − ΠBatrue F ΠBatrue F . For each condition, 100 independent trials are performed and the average of the errors is taken.
The first test is conducted to evaluate the performances under perturbations of different noise levels. Three types of noise are added at a particular time in each trial.
They are image position error (0~1 pixel), hand position error (0~1 mm), and hand orientation error (0~0.05 degree).
Figure 6-16 Relative errors of closed-form solution with respect to the noise level.
Focal length and camera position errors (Top). Working plane pose and camera rotation errors (Bottom).
Figure 6-17 Relative errors of nonlinear optimization with respect to the noise level.
Focal length and camera position errors (Top). Working plane pose and camera rotation errors (Bottom).
Figure 6-16 summarizes the performance of the closed-form solution and Figure 6-17 shows the results of optimization. For each of the three working plane poses, three
hand orientations, each associated with ten pure translating points, yielding a total of 90 poses, are generated independently for each trial. The results verify that the proposed method can be adopted to calibrate simultaneously hand-eye-workspace geometrical relationships and the intrinsic parameters of the camera. Furthermore, the accuracy is increased after the optimization procedure. The results reveal that the proposed method that uses a line laser outperforms that uses a point laser for values of all parameters, except the working plane poses of the nonlinear optimization solution.
Figure 6-18 Relative errors with respect to the number of samples.
Closed-form solution(Left) and Nonlinear optimization(Right).
The number of data also influences the accuracy of the proposed method. The performance of the proposed method with different numbers of samples is compared to that of the method using a single point laser. Zero mean Gaussian noise with standard deviations of 1-pixel, 1-mm, and 0.05-degree is added to each image position of point, each position of the hand, and each orientation of the hand, respectively.
Different numbers of hand orientations produce different numbers of samples with measurements of projections in images. 21 to 49 hand poses for each of the three working plane poses are generated; therefore, the number of samples varies from 63 to 147. For each number of samples, 100 independent trials are performed and the average of the relative errors is determined. Figure 6-18 summarizes the results for one focal length, fu , and the camera-to-robot position, CBt. The errors decrease as the number of samples increases. Since information of data is limited, the error of using single point laser is larger than that generated using the line laser. Notably, a comparison between the results of the closed-form solution obtained using a line laser
and those obtained using a point laser shows that the former yields an error in the focal length that is about one third and an error in the camera position that is about one fifth those generated by the latter. After optimization, these error ratios become 3/4 and 3/5.
6.4.2 Experimental Results
Figure 6-19(a) shows the overview of the experimental setup. A planar plate, of dimensions 50×50cm, is placed roughly 100cm away from the manipulator. In this configuration, the camera cannot see any part of the manipulator. The line laser module is a laser diode with an adjustable lens and is attached on the robot tool (Figure 6-19(b)).
(a) (b) Figure 6-19 Experimental setup.
(a) Overview (b) The line laser module attached to the end-effector.
The working plane is orientated at three poses. For each plane pose, the end-effector has three orientations and in each orientation, it starts from one particular position and moves sequentially in three directions. In each direction, the end-effector stops at three positions. A total of three working plane poses and 50 hand poses per plane were generated. All movements were designed to project the laser onto the working plane, and corresponding images were saved.
Laser
Table 6-5 Results with real data of different number of samples
Parameter Test 1: 80 samples
(2 working plane poses, 40 samples per plane pose) Unit
Closed-form solution Optimization σ
(f ,u f ) v (1279.98,957.97) (1121.24,1085.71) (12.75,11.16) pixel
CB Tt [858.05,200.94,-103.41] [919.32,215.41,-113.62
] [2.05,4.10,6.21] mm
E TΛa [-1.63,3.60,18.77] [-2.77,6.07,31.92] [0.04,0.10,0.45] mm
B T1
Πa [3.05,-2.26,-327.70] [-3.35,0.82,-299.68] [0.50,0.46,1.79] mm
B 2T
Πa [136.75,-20.76,-484.97] [87.45,-9.44,-435.97] [1.65,0.98,2.20] mm Distance
Error 0.6113 0.0019 1
Parameter
Test 2: 150 samples
(3 working plane poses, 50 samples per plane pose) Unit
Optimization σ
Two tests were performed in this experiment using different numbers of samples.
Data were applied to the proposed closed-form solutions to obtain initial values of parameters of the system. Next, the proposed nonlinear optimization method was used to refine these values. Table 6-5 summarizes the results. Test 1 uses data of two planes and 40 samples per plane. The first column of Table 6-5 in Test 1 presents estimates of the closed-form solution that is described. The second column presents the estimates
of nonlinear optimization in Test 1. The third column presents the standard deviations of the estimates of the nonlinear optimization, indicating the uncertainty of the refined result. The right-hand columns present results of nonlinear optimization of Test 2 using all data are in the right hand column. The last row, labeled Distance Error, presents the average distance errors which are the costs introduced in (4-23). Figure 6-20 summarizes the calibrated geometrical relationships obtained from Test 2. The results are consistent with the simulation results, which revealed that the uncertainty in the estimates declines as the number of data increases.
Figure 6-20 Calibrated geometric relationships.
6.5 Robot Kinematic Calibration using a Laser Pointer, a