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5.1 Trust Region

  We adopt a trust region methodgif [Dahlquist and Björck, 1995, Fletcher, 1987] which adapts the maximum step length tex2html_wrap_inline3918 automatically by taking steps as long as is possible or desirable, while maintaining convergence. The trust region method adjusts tex2html_wrap_inline3918 so that the controller never moves out of the region where the linear approximation given by the current Jacobian estimate is valid. To do this we want to solve a constrained problem for tex2html_wrap_inline3944 instead of taking the whole Newton step tex2html_wrap_inline3542 :

(19)  displaymath3993

Define a model agreement as tex2html_wrap_inline3948 and adjust tex2html_wrap_inline3918 according to:

(20)  displaymath3995

The objective of eq. 5.4 is to keep the model agreement within the two constants tex2html_wrap_inline3958 and tex2html_wrap_inline3960 . The convergence of the algorithm is quite insensitive to their exact values. However the values of the parameters affect how accurately the controller follows the trajectory, and they are selected differently for coarse transportation and fine manipulation moves, trading accuracy for time. Typical suitable values are tex2html_wrap_inline3962 and tex2html_wrap_inline3964 [Dahlquist and Björck, 1995].

The trust region method has very strong convergence properties. Convergence to an accumulation point tex2html_wrap_inline3966 satisfying the first and second order minimum conditions tex2html_wrap_inline3968 and tex2html_wrap_inline3970 is guaranteed for any function tex2html_wrap_inline3878 , rather than only for a quadratic function, as in the case of the unmodified quasi-Newton method in eq. 5.2 (See [Fletcher, 1987]). For the quasi-Newton method to work, we need the starting point tex2html_wrap_inline3534 to be much closer to the goal tex2html_wrap_inline3568 than for the trust region method. The trust region method is efficient. When far from the goal it takes steps as long as possible given the accuracy and validity region of the estimated Jacobian model. When close to the goal the constraint tex2html_wrap_inline3944 becomes inactive, and we achieve the super-linear convergence of the quasi-Newton method.

The above trust region modification to the quasi-Newton method is guaranteed to converge to a minimum. However, unless the functional tex2html_wrap_inline3878 is convex and thus only has one minimum, we cannot guarantee that tex2html_wrap_inline3966 is the desired global minimum tex2html_wrap_inline3568 . Another problem is that we have no control over which path tex2html_wrap_inline3986 the controller takes, something which is of vital importance in robotics where, unlike optimization, the numerical search corresponds to actual movements. So far we have only considered the static problem. In a real robot, unless it is moved in a look-and-move fashion [Weiss and Sanderson, 1987], we also need to consider how to control velocities and acceleration.

With the above approach we can control velocities and accelerations by placing additional constraints on tex2html_wrap_inline3988 . With a fixed sampling frequency (typically limited to 60 Hz by the video hardware), the constraint tex2html_wrap_inline3990 limits the joint velocity, and tex2html_wrap_inline3992 limits the joint acceleration.


next up previous contents
Next: 5.2 Visual Space Trajectory Up: 5 Visual Servo Control Previous: 5 Visual Servo Control

Martin Jägersand