Neural Networks: An Overview - Neural Networks: Methodology and Applications - page 68

Fig. 1.43. Scatter plots for the prediction of the diameters of welding spots

1.4.9 An Application in Robotics: The Modeling of the Hydraulic
Actuator of a Robot Arm

The previous applications involved feedforward neural networks only. We now
turn to dynamic modeling, with recurrent neural networks.
We consider a hydraulic actuator that controls the position of a robot
arm; therefore, the position of the arm depends on the hydraulic pressure
in the actuator, which in turn depends of the angular position of a valve. A
dynamic model of the relation between the hydraulic pressure and the opening
of the valve was sought, in the framework of an informal competition between
research groups involved in nonlinear modeling. A control sequence

,
i.e., a sequence of angles of the valve, and the corresponding sequence of
the quantity to be modeled

{ u ( k ) }

y p ( k ) } , i.e., the hydraulic pressure, are shown
on Fig. 1.44. That sequence contains 1,024 samples, the first half of which,
according to the rule of the game, was to be used as a training set and the
second one as the validation set. Since no prior information was available from
the physics of the process, a black-box model must be designed.
A cursory look at the data shows that a linear model of the process would
certainly not be appropriate; the oscillations observed as responses to control
variations that are almost steps suggests that the model is at least of second
order. The training and validation sequences are approximately of the same
type and same amplitude, but the amplitude of the control signal is larger in
the validation set (around times 600 and 850) than in the training set. Thus,
the conditions are not very satisfactory.
The example is analyzed in detail in Chap. 2. The best results [Oussar
1998] were obtained with a second-order state-space model, one state variable
of which is the model output, of the form

{

y ( k +1)= x 1 ( k +1)= ψ 1 ( x 1 ( k ) ,x 2 ( k ) ,u ( k ))
x 2 ( k +1)= ψ 2 ( x 1 ( k ) ,x 2 ( k ) ,u ( k )) ,

with two hidden neurons. It is shown on Fig. 1.45.