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4 Model Estimation

  An estimate to the image Jacobian can be obtained by physically executing a set of calibration movements along the basis direction of motor space ([3, 8, 19]) tex2html_wrap_inline850 and approximate the Jacobian with finite differences:

(2)  displaymath903

where tex2html_wrap_inline852 . However to keep the estimate current for the typically nonlinearly changing environment the agent would need to perform these movements rather frequently. This is not appropriate in most manipulation settings, where the calibration movements would interfere with the task. Partial modeling of the viewing geometry using an ARMAX model and estimating only one or a few parameters (e.g. depth) has also been tried [2, 6]. This however restricts the camera-robot configurations and environments to structured, easy to model settings.

We seek instead, an online method, which estimates the Jacobian by simply observing the process, without introducing any extra ``calibration'' movements. In observing the process we obtain the changes in visual appearance tex2html_wrap_inline854 corresponding to a particular controller command tex2html_wrap_inline792 . This is essentially a secant approximation of the derivative of f along the direction tex2html_wrap_inline792 . We want to update the Jacobian in such a way as to satisfy our most recent observation (secant condition): tex2html_wrap_inline862

The condition above is under-determined; thus a family of updating formulas, called the Broyden hierarchy, is defined as follows:

(3) displaymath905

Where tex2html_wrap_inline864 are different rank 1 matrices so the rank of the correction term is equal to the number of non-zero tex2html_wrap_inline866 gif. We choose an unsymmetric correction term:

(4)  displaymath907

This is a first order updating formula in the above hierarchy, and it converges to the Jacobian after n orthogonal moves tex2html_wrap_inline878 . It trivially satisfies the secant condition, but to see how it works let's consider a change of basis transformation P to a coordinate system O' aligned with the last movement so tex2html_wrap_inline884 . In this basis the correction term in eq. 4 is zero except for the first column. Thus our updating schema does the minimum change necessary to fulfill the secant condition (minimum change criterion). Let tex2html_wrap_inline886 be n orthogonal moves, and tex2html_wrap_inline890 the transformation matrices as above. Then tex2html_wrap_inline892 are just circular permutations of tex2html_wrap_inline894 , and the updating schema is identical to the finite differences in eq. 2, in the tex2html_wrap_inline896 frame.

Note that the estimation in eq. 4 accepts movements along arbitrary directions tex2html_wrap_inline792 and thus needs no additional data other than what is available as a part of the manipulation task we want to solve. In the more general case of a set of non-orthogonal moves tex2html_wrap_inline900 the Jacobian gets updated along the dimensions spanned by tex2html_wrap_inline900 .


next up previous
Next: 5 Use of Estimated Up: Estimation of Visual-Motor Models Previous: 3 Visual Measures

jag@cs.rochester.edu