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4.5.2 Results
The abstract RNN could cope with the redundant arm postures for a given endeffector position; the MLP could not (table 4.3). The local PCA mixture approximated the training data also better then Neural Gas (table 4.3). The results from the different mixture models NGPCA, NGPCAconstV, and MPPCAext were almost equal (table 4.3). Compared to NGPCA, NGPCAconstV was slightly worse on the inverse direction. Over five different training cycles (retraining of the mixture of local PCA), the average position errors varied only slightly (for NGPCA, the maximum deviation was 2 mm).
Table 4.3:
Position and collision errors for an abstract RNN using NGPCA, NGPCAconstV, and MPPCAext for training, compared to a variant using Neural Gas for training and to a multilayer perceptron (MLP). Results are shown for two different directions of recall: forward and inverse. The inverse model takes the desired collision state as an additional input variable (third column). Position errors are averaged over all test
patterns, and are given with standard deviations. In the inverse case, the collision error is the percentage of trials
deviating from the collision input value. In the forward case, it is the
erroneous number of collision state predictions.
method 
direction 
input 
position error (mm) 
collision error (%) 
NGPCA 
inverse 
no collision 
27 ± 15 
5 
NGPCA 
inverse 
collision 
23 ± 13 
8 
NGPCA 
forward 
 
44 ± 27 
11 
NGPCAconstV 
inverse 
no collision 
31 ± 17 
5 
NGPCAconstV 
inverse 
collision 
28 ± 14 
11 
NGPCAconstV 
forward 
 
43 ± 29 
11 
MPPCAext 
inverse 
no collision 
29 ± 15 
5 
MPPCAext 
inverse 
collision 
25 ± 14 
6 
MPPCAext 
forward 
 
45 ± 29 
14 
Neural Gas 
inverse 
no collision 
58 ± 26 
2 
Neural Gas 
inverse 
collision 
56 ± 27 
4 
Neural Gas 
forward 
 
160 ± 74 
18 
MLP 
inverse 
no collision 
310 ± 111 
30 
MLP 
forward 
 
93 ± 48 
13


The mixture models distribute the training patterns among the units of the mixture. For NGPCA and NGPCAconstV, every pattern is assigned to one unit (at the end of the training). The number of patterns assigned to a unit is a measure for the weight of the unit; for MPPCAext, the weights are the prior probabilities. These weights had roughly a bellshaped distribution among the units (figure 4.10). Different from MPPCAext, the distributions for NGPCA and NGPCAconstV showed a second peak for units having few assigned patterns (around 50, the average is 250). A single peak seems to be favorable. However, the distribution of assigned patterns also depends on the structure of the data set (which is largely unknown). Apparently, in this experiment, the effect on the performance was negligible (table 4.3).
Figure 4.10:
Histogram of assigned patterns, respective prior probabilities. n is the number of units for each interval.

The remaining tests were carried out only with NGPCA. The distribution of the individual errors shows regions corresponding to different ellipsoids selected during the recall (figure 4.11). At the transition between two regions, the error as a function of the input is discontinuous.
Figure:
Position errors of the inverse model with input `collision'.
(Left) Horizontal plane (approximately 70mm above the table). (Right) Vertical plane through the origin (z = 0).

The performance depends on the number of units m and principal components q. The position and the collision errors decreased with increasing m (table 4.4). Furthermore, the position error was smallest at q = 6 (figure 4.12, right). This q value matches the local dimensionality of the distribution (figure 4.12, left).
The abstract RNN could also cope with additional noise dimensions if the number of principal components was adjusted accordingly (table 4.5). With three noise dimensions and q = 6 principal components, the position errors of the abstract RNN were more than double . However, with q = 9, the position errors were again at the nonoise level.
Table 4.4:
Dependence on the number m of units.
direction 
input 
error 
m = 50 
m = 100 
m = 200 
inverse 
no collision 
position (mm) 
48 
38 
27 
inverse 
no collision 
collision (%) 
5 
5 
5 
inverse 
collision 
position (mm) 
47 
35 
23 
inverse 
collision 
collision (%) 
8 
9 
8 
forward 
 
position (mm) 
74 
56 
44 
forward 
 
collision (%) 
16 
14 
11 

Figure 4.12:
(Left) Ratio of successive averaged eigenvalues
and
(see methods). (Right) Dependence of the position error E (here for the direction: inverse, no collision) on the number of principal components q.

Table 4.5:
Compensation of noise. The first column of numbers shows the result without noise dimensions (as in table 4.3), the second with three noise dimensions and six principal components, and the third with noise and nine principal components.
direction 
input 
error 
q = 6 (no noise) 
q = 6 
q = 9 
inverse 
no collision 
position (mm) 
27 
57 
30 
inverse 
no collision 
collision (%) 
5 
6 
6 
inverse 
collision 
position (mm) 
23 
64 
24 
inverse 
collision 
collision (%) 
8 
6 
11 
forward 
 
position (mm) 
44 
101 
45 
forward 
 
collision (%) 
11 
15 
13 

The errors for the forward direction were consistently higher than for the inverse direction (table 4.3 and 4.4). The major difference seems to be that the forward direction has six input dimensions; the inverse direction has only four. This is consistent with the finding that the square error per output dimension increased with the number of input dimensions (figure 4.13). For intermediate numbers r, the increase was even exponential. In the following section, this finding is investigated theoretically.
Figure 4.13:
Mean square error (SE) as a function of the number r of input dimensions. The dashed line is the function
a exp(br) fitted to point 2 to 8; b was
0.59±0.03.

Next: 4.6 Dependence on the
Up: 4.5 Kinematic arm model
Previous: 4.5.1 Methods
Heiko Hoffmann
20050322