3.3 Adaptive vs. Non-adaptive control



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3.3 Adaptive vs. Non-adaptive control

  
Figure 7: Results from 50 runs of an experiment injecting varying amount of random noise into the Jacobian, thus disturbing the internal model in the controller

In this experiment we compared performance of adaptive and non-adaptive controllers in 3 DOF. The basic experimental plan was to add varying amounts of noise to the Jacobian model, and then compare trajectory and endpoint errors for the adaptive and non-adaptive cases. The disturbed Jacobian is defined by the equation

where

between

and

indicates the model accuracy and

is a random

matrix with elements drawn from a uniform distribution on the interval

. This restricts the gain, but not the directional effect of the random Jacobian. 50 trials were run with varying values of the model accuracy parameter.

In fig. 7, we see that for the non-adaptive algorithm, the percentage of the trials which converged to the correct endpoint (within

in this case) quickly falls to zero for model accuracies

below

. The adaptive algorithm has no problem, even with completely random initial Jacobians.

We define trajectory error as the average deviation in visual space from the planned straight line between the initial feature values

and the goal

. Model error is the average difference between the predicted move and the actual move for each sampling interval

. The difference between model and trajectory error is that trajectory error does not penalize errors in gain along the desired trajectory, whereas model error does. On the other hand if the robot starts with a large initial error in position, the trajectory error will accumulate until the robot is back on the desired trajectory whereas model error will only penalize for incorrect predictions.

For random models (

small) both model and trajectory error increases somewhat for the adaptive algorithm. This increase can be accounted for by a few early erroneous movements. These noisy movements quickly update the Jacobian along the corresponding directions, and the error is reduced to near zero. In theory, a number of independent moves equal to number of controlled DOF should suffice to determine the Jacobian exactly, but because the measurements are noisy we use variable gain filtering to reduce its effect. This increases the robustness of the updating procedure, but slows down the convergence somewhat. For the non-adaptive algorithm, we find a huge increase in both trajectory and model error. This is caused by divergent trials, where the robot goes off in a completely incorrect direction. The peak in both time and model error for

around

is caused by long period oscillatory behavior at the transition between convergence and divergence. An illustration of a typical oscillatory move is shown in fig. 8.

  
Figure 8: Left: Visually planned path. Right: Typical oscillatory behavior of non-adaptive controller when model error is near the convergence limit.



next up previous
Next: 3.4 6 DOF adaptive Up: 3 Experiments with the Previous: 3.2 Repeatability



jag@cs.rochester.edu