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3.1 Derivation of a Control Method

  In general, an unknown visual-motor function tex2html_wrap_inline3516 can be written as a multidimensional Taylor expansion. In particular we are interested in the linear approximation:

(1) displaymath3573

where for a sufficiently ``smooth'' f the residual term tex2html_wrap_inline3520 is small for tex2html_wrap_inline3522 . J is called the visual motor Jacobian and is defined as

(2) displaymath3591

The visual motor Jacobian relates small changes in vision space to small changes in motor (control) space, e.g., for tex2html_wrap_inline3466 , tex2html_wrap_inline3528 , tex2html_wrap_inline3530 and tex2html_wrap_inline3532 we have:

(3) displaymath3577

Thus the visual motor Jacobian acts as a local model of the function behavior around tex2html_wrap_inline3534 (and tex2html_wrap_inline3536 ).

Given that we observe an initial perceptual state represented in the feature vector tex2html_wrap_inline3536 and that we wish to bring about the (goal) state tex2html_wrap_inline3540 , the best prediction using our linear model J, of the required change tex2html_wrap_inline3542 in robot control signals, is given by solving the system of equations:

(4) displaymath3579

Executing this move, i.e. setting tex2html_wrap_inline3544 changes the perceived state to tex2html_wrap_inline3546 . Unless f is a linear model, tex2html_wrap_inline3546 is typically not equal to tex2html_wrap_inline3540 . However, if tex2html_wrap_inline3554 is small, and f smooth, tex2html_wrap_inline3558 is likely to be smaller than tex2html_wrap_inline3554 , and the process can be repeated yielding a sequence tex2html_wrap_inline3562 successively closer to tex2html_wrap_inline3540 . What we have is a quasi Newton method for solving a set of nonlinear equations in tex2html_wrap_inline3462 (find tex2html_wrap_inline3568 such that tex2html_wrap_inline3570 ). In a continuous setting we have a similar control law:

(5) displaymath3581

where K is a gain matrix.

These two control laws implement the image based look-and-move and the image based visual servoing.



Martin Jägersand