2.2 Model estimation



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2.2 Model estimation

We have not yet addressed the issue of where the local model Jacobian comes from. One approach is to use geometric models, derive an analytical expression for the Jacobian, and calculate a numerical Jacobian at each sampled point . This approach has been used for manipulator kinematics in cases where analytic inversion is hard. An advantage is that a rough model can be used, depending on the feedback, to yield high precision positioning. Other approaches can be described as partially adaptive, being applicable only to a restricted set of visual motor transfer functions, requiring partial (analytic) modeling of the system (e.g. restricted to systems suitable for ARMAX modeling [Pap, Khos 93][Feddema, Lee 90], or using only image location visual measurements under a weak perspective assumption [Hollinghurst, Cipolla 1993]).

We want to solve manipulation problems that are unsuitable for analytic modeling at all, such as the manipulation of flexible foam in 12 DOF shown in section 5.3. We also want to place the fewest possible restrictions on the class of transfer functions we can model. This rules out systems depending on near linear transfer functions in the desired operating space (e.g., [Hager et al 93][Chong, Conkie 90a]), and methods that would be inefficient for high DOF systems (e.g. [Yoshimi Allen 9x] where calculating a Jacobian update requires measurements, with being the product of the number of sensory and controlled signals)

We use a Jacobian model estimation and updating scheme based on the information obtained more or less for free while performing the task. After making the movement and observing the change in the feature values, we wish to modify the Jacobian to bring our model into agreement with the measurement, i.e. to make . In general (for ), this problem does not have a unique solution. One choice is to make the minimal change necessary. In this case

which can be easily seen to satisfy our model agreement condition.

An appealing and intuitive way of visualizing how the updating works is to consider it in a coordinate frame aligned with the last movement. Let

be a coordinate change matrix from the original basis

for

to an ON basis

in which has the first basis vector aligned with the movement

we last made. Using

and

we can freely transform the measured change in visual appearance

and the motor-visual Jacobian

back and forth between the two coordinate bases. The formulas

and

, transform the objects into

and the inverse formulas bring them back.

In the course of a particular move, we obtain a difference approximation to the directional derivative of the transfer function along the direction of the movement. In the basis

, this difference approximation,

, is aligned with the first basis vector in

since

. In the Jacobian this derivative is represented in the first column. Thus to incorporate the new derivative information

into the Jacobian, we simply insert it into the first column, leaving the information along the other directions (in the other columns) unchanged. Looking at the general updating schema above in the

frame, it is clear that this is exactly what happens there also (just consider that the fact that

). Thus our updating schema fulfills the minimum change criterion.

In the practical applications we also perform adaptive weight filtering in the updating to suppress noise in the estimate. The new model

is obtained from the formula

, where

and

represents our current model confidence given as the maximum allowable step length. The estimation technique we have described falls into a class called Broyden methods [Fletcher 87] used in nonlinear optimization.



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Next: 2.3 Implementation issues Up: 2 Differential visual feedback Previous: 2.1 Derivation of a



jag@cs.rochester.edu