Constraint Estimation
8.1 Constraint Estimation and Residuals
8.1.1 Fast Solution for Linear Constraint in < k
8.1.1.2 Applying Constraint Estimation in < k
Both stiness-based and constraint-based estimation were applied to test data. The PHANToM with the ngertip sensor was moved by hand to produce four groups of four experimental runs. The rst group is motion in free space, the second group is motion along a single at plane made of wood, the third group is motion along a line formed by the plane of wood and an aluminum block, the last group is a xed contact in a corner formed by the wood and two aluminum blocks. Sixteen hundred samples of the low-passed force and position data were sampled at 250 Hz.2
To investigate the eect of sample size on measurement accuracy, each group of four experimental runs was lumped into a single block of data. The measurement accuracy for the position and force measurements were then used to normalize the units of each measurement. The eects of friction where then removed from every sample using the following lter
w
~i =Id
;x
_ix
_Tix
_Tix
_i!
w
mi: (8:29)This lter computes the power being expended by the motion, and normalizes it to get a frictional force. The velocity then gives the direction of the frictional force.
This lter improves performance for the stiness approach.
Samples were then randomly drawn from the block of measurements to create an example for regression. Ten examples were drawn for each sample size. The stiness regression was applied to the data and the estimated frame was returned in ^
P
. This was compared to an assumed trueP
by computing the singular value decomposi-tion of ^P
TP
=UV
. If the two projection matrices were equivalent, the singular values would all be one. Since the singular values are the direction cosines of the misalignment, any deviation from one can be mapped to a positive angular error.We assumed a log-normal distribution for the deviation of the minimum singular value from 1. Table 8.1 shows the mean and standard deviation (timespn) of this log-normal distribution for each contact case, and the equivalent average angular error. Figure 8.1 graphically shows the eect of sample size on the estimated error.
2Chapter 7 describes the signal conditioning on the force signal
148
Chapter 8: Constraint EstimationNumber of Constraints 0 1 2 3
Mean of Log-Normal -10.9 -5.09 -4.00 -10.0 Std of Log-Normal pn 1.3 11.0 23.2 2.2 Equivalent Mean Angular Err. (deg) 0.34 6.34 11.0 0.53 Table 8.1: Error statistics for stiness based estimation in <3 using position and force measurements and friction compensation.
101
102
103 0
10 20 30 40
Angular Alignment Error and 1 Sigma Bounds (1 Contact)
Sample Size
Angular Error (deg)
Figure 8.1: Angular alignment accuracy of the stiness estimation pro-cedure with 1 contact as a function of the number of samples. Mea-surements of position and force compensated for friction were used in the estimation.
8.1: Constraint Estimation and Residuals
149
Number of Constraints 0 1 2 3
Mean of Log-Normal -10.6 -.228 -1.78 -10.0 Std of Log-Normal pn 1.54 6.37 5.89 1.9 Equivalent Mean Angular Err. (deg) 0.40 78.2 33.7 0.55 Table 8.2: Error statistics for stiness based estimation in <3 using position and force measurements without friction compensation.
The data shows, as expected, that both the full contact and no contact case produce very accurate results with little deviation. The contact cases have some bias and both have signicant variance. A plot of the position data shows that the wood surface was not at. For some data sets an 11 slope can be observered. How much of this is kinematic error in the PHANToM and how much is actual alignment error would require more careful measurements. It is very possible that the kinematic chain from the block, to the table, to the PHANToM, and nally to the ngertip has 1 ;2 of angular error.
Secondly, the data from the experiment with two contacts is not perfectly straight.
There is enough kinematic error in the robot to produce a 1 mm deviation from a straight-line t to the x and y position data over a maximum of 80 mm. In addition, the signicantly greater standard deviation in the measurements is probably due to friction and vibration.
For comparison the same experiments were run without rst preprocessing the data to account for friction. Table 8.2 shows the results. As the table indicates, sti-ness based estimation without friction compensation is terrible. Friction completely contaminates the measurements and for one contact produced an estimate that is almost 90 degrees opposed to the correct normal.
The constraint based estimator was also tested. The same sampling procedure was used to generate examples for the constraint based estimator. The results are signif-icantly improved by adding the information on the number of constraints. Table 8.3 summarizes the results. Figure 8.2 shows the performance of this estimator as a function of sample size. Even the results without friction compensation, and esti-mation based on velocities are quite accurate. Tables 8.4 and 8.5 show the results for estimation without friction compensation, and for velocity based estimation with friction compensation.
150
Chapter 8: Constraint EstimationNumber of Constraints 0 1 2 3
Mean of Log-Normal NA -8.5 -10.2 NA
Std of Log-Normal pn NA 7.3 0.38 NA
Equivalent Mean Angular Err. (deg) 0.0 1.2 0.5 0.0 Table 8.3: Error statistics for constraint based estimation in <3 using position and force measurements with friction compensation.
Number of Constraints 0 1 2 3
Mean of Log-Normal NA -8.19 -10.2 NA
Std of Log-Normal pn NA 7.2 0.36 NA
Equivalent Mean Angular Err. (deg) 0.0 1.34 0.5 0.0 Table 8.4: Error statistics for constraint based estimation in <3 using position and force measurements without friction compensation.
Number of Constraints 0 1 2 3
Mean of Log-Normal NA -10.5 -9.9 NA
Std of Log-Normalpn NA 10.6 8.7 NA
Equivalent Mean Angular Err. (deg) 0.0 0.43 0.59 0.0 Table 8.5: Error statistics for constraint based estimation in <3 using velocity and force measurements with friction compensation.
8.1: Constraint Estimation and Residuals
151
101
102
103 0
2 4
Angular Alignment Error and 1 Sigma Bounds (1 Contact)
Sample Size
Angular Error (deg)
Figure 8.2: Angular alignment accuracy of the constraint estimation procedure with 1 contact as a function of the number of samples. Mea-surements where positions and forces compensated for friction. Note the signicant improvement in performance over the stiness approach.
The completely free and xed case are t without any error (indicated by NA), since there is nothing for the estimator to estimate. The partial constraint cases are also signicantly improved.
If accurate estimates are required for assembly or other mating tasks, it may be worth the additional computational cost to use the constraint based approach. The eigenvalue computations are easier in the constraint approach since two k k sys-tems have to be solved, versus one 2k 2k system. However there is additional computational cost in the numerical iterations. In the test cases, this computation generally converged in two to three steps from the initial estimate, but each step requires searching for a minimum in three dierent directions.