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page 225

-10, 20 or 40 neurons in the hidden layer

-a bias input was connected to each neuron

-the layers were all fully connected

-there were runs with one, and two hidden layers

16.4.1.4 - The Training Set

The problem is reduced using either left or right arm configurations, the solution is also constrained to elbow up or elbow down.

Discontinuities were avoided by not training the neural network in the region above the origin. The elbow straight configuration is also a minor singularity problem.

Training points were evenly distributed throughout the robot workspace

Only a quarter of the robot workspace was used because of the robot symettry.

The general protocol for training was,

-apply the desired position to the input, and train for the desired joint angles.

-When accuracy was high enough, the first correction net was trained by comparing the actual errors, and the desired values. Additional correction networks were also trained in some cases.

-the error was measured by using an RMS measure of the differences

page 226

oerror = oactual - odesired

oaverage = Σ oerror 3n

S =

Σ (o

error

)2

 

 

3n

 

 

 

 

 

 

 

oactual odesired oerror oaverage S

n

-Actual angle output of Neural Network

-Desired angle output of Neural Network

-Error of angle output of neural Network

-Average angle error for all joints

-R.M.S. error for all joints

-Number of training points

• A list of results are provided below,

page 227

10 hidden neurons

20 hidden neurons

40 hidden neurons

Network

Number of

average

standard

absolute error

deviation

Architecture

Connections

degrees per joint

degrees per joint

 

 

 

 

 

 

1 hidden layer

73

3.02

4.22

10 wide

 

 

 

1 hidden layer

143

2.22

3.39

20 wide

 

 

 

1 hidden layer

286

 

 

20 wide with

1.66

2.63

1 correction net

 

 

 

1 hidden layer

 

 

 

20 wide with

429

1.27

2.06

2 Correction

nets

 

 

 

1 hidden layer

 

 

 

20 wide with

572

1.19

1.92

3 correction

 

 

 

nets

 

 

 

1 hidden layer

 

 

 

20 wide with

715

1.04

1.61

4 correction

 

 

 

nets

 

 

 

1 hidden layer

 

 

 

20 wide with

858

1.01

1.55

5 correction

 

 

 

nets

 

 

 

1 hidden layers

283

1.72

2.59

40 wide

 

 

 

1 hidden layer

566

 

 

40 wide with

1.37

2.21

1 correction net

 

 

 

 

 

 

 

1 hidden layer

 

 

 

40 wide with

849

1.20

1.90

2 Correction

nets

 

 

 

1 hidden layer

 

 

 

40 wide with

1132

1.10

1.77

3 correction

 

 

 

nets

 

 

 

1 hidden layer

 

 

 

40 wide with

1415

1.10

1.76

4 correction

 

 

 

nets

 

 

 

1 hidden layer

 

 

 

40 wide with

1698

1.09

1.76

5 correction

 

 

 

nets

 

 

 

 

 

 

 

The results in the table were obtained for a variety of network configurations

A visual picture of the network configurations is shown below, and on subsequent pages. These are based on a set of test points that lie in a plane of the workspace.

page 228

********** Add figure of network point locations, and test conditions

As seen in the experimental results, there are distortions that occur near the origin, and the edges of the workspace, as would be expected with the singularities found there.

The errors also increased near the training boundaries

********* Add in more of the results figures

16.4.1.5 - Results

The mathematical sigularities caused by cartesian coordinates, and the +/- 180 degrees singularity could be eliminated by selecting another set of coordinates for space and the arm.

The best results were about 1 degree RMS.

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