6.3 Results

The abstract RNN trained with MPPCA-ext did best at the grasping task, and NGPCA-constV was better then NGPCA (table 6.1). Furthermore, both NGPCA and NGPCA-constV were sensitive to the parameter set. A good performance was only achieved with fine tuned parameters different from the ones used in chapter 4. Kernel PCA could compete with the local PCA mixture models on the grasping performance, but not on the recall speed, which was about 2 000-times slower. All of the new methods presented in this thesis were better than a look-up table; the multi-layer perception failed since it cannot cope with redundant arm postures (see also section 4.5).

Table 6.1: Position error, orientation error, the rate of successful grasps, and the recall time for one trial. The values were averaged over all test trials. The indices for the NGPCA variants refer to the number of the training-parameter set.

method	pos. error	orient. error	grasp success	recall time
	(mm)	(degrees)	(%)	(sec)
MPPCA-ext	7	3.9	95	0.015
NGPCA1	9	4.0	90	0.015
NGPCA2	42	4.9	61	0.015
NGPCA-constV1	8	4.0	93	0.015
NGPCA-constV2	26	5.6	77	0.015
kernel PCA	9	4.6	93	31.000
look-up table	13	4.8	87	0.017
MLP	236	54.3	0	< 0.001

Table 6.2: Average performance over five different training cycles. Standard deviations are given. NGPCA and NGPCA-constV used parameter set 1.

method	position error	orientation error	grasp success
	(mm)	(degrees)	(%)
MPPCA-ext	7.3±0.2	3.9±0.2	95.2±0.7
NGPCA	9.1±0.8	4.0±0.4	90.0±1.4
NGPCA-constV	8.0±0.2	4.0±0.2	93.1±0.2

Over five different training cycles, the performance of the local PCA mixture models varied only slightly (table 6.2). MPPCA-ext and NGPCA-constV showed less variation compared to NGPCA. The advantage both have over NGPCA is probably also related to the difference in the distribution of assigned patterns per unit (or prior probabilities per unit in the MPPCA-ext case). MPPCA-ext and NGPCA-constV resulted in bell-shaped distributions; NGPCA resulted in a second peak with 34 units that have less than eight assigned patterns (figure 6.8).

In the presence of noise, the abstract RNN (tested with MPPCA-ext) showed a more robust performance than the look-up table (table 6.3). A second training data set was generated with noise uniformly drawn from the interval [- 0.1;0.1] and added to each component of each pattern. On this set, the number of successful grasps decreased only from 95% to 91% for the abstract RNN; for the look-up table, it decreased from 87% to 57%.

The last test, also using MPPCA-ext, demonstrates the utility of the population code (table 6.4). Two data-processing variants were used. The first had no population codes; each training pattern contained the brick's center of mass in the contrast image, the tilt angle of the main axis of the brick within the image, and the 12 joint angles. Here, despite the reduced dimensionality (15 compared to 68), the number of successful grasps decreased from 95% to 90%. The second variant used the same image processing as in section 6.2.3, but did not encode redundantly the joint angles. This variant decreased the success rate from 90% to 83%.

Figure 6.8: Histogram of assigned patterns, respective prior probabilities. n is the number of units for each interval. NGPCA and NGPCA-constV used parameter set 1.

Table 6.3: Performance of the abstract RNN (here, using MPPCA-ext) compared to a look-up table. A second training set was used with noise (10% of the standard deviation of the distribution) added to each pattern.

method	position error	orientation error	grasp success
	(mm)	(degrees)	(%)
abstract RNN	7.3	3.9	95
abstract RNN, noise	9.0	3.7	91
look-up	13.0	4.8	87
look-up, noise	17.5	5.2	57

Table 6.4: Performance of the abstract RNN with MPPCA-ext using different pattern processing modes (see text).

population code	position error	orientation error	grasp success
	(mm)	(degrees)	(%)
yes	7.3	3.9	95
no	9.1	2.3	90
only for vision	7.4	14.4	83