Add additional notes after tanker control example in Ch 5 (dynamics)

[Richard Murray]

Lars Nielsen suggests the following:

After the tanker stability you start a new example Example 5.9 "Stability of an oscillation". Since I only see the bottom of page 5-17, I don't know how you continue, but we have used exactly the same equations for path tracking, where stability to the path is automatic while still leaving velocity along the path free to control. You may look at Section 2 "A conceptual example" in the attached paper. Thus the separation principle between radial and tangential control in the example has wider applications, and if it is not already mentioned I think it is worthwhile to do so. Should you be interested, feel free from our side to use material from our paper. Further, I have checked with Björn that he can be helpful if you would need assistance with text or in adapting a plot or something else.

He suggests adding the following text:

\subsection{Remark --- Separation Property Utilized in Robotics for Path Tracking} The elegant separation property shown by the differential equations in (5.15) and its inherent different stability properties has wider applications, and can, e.g., be utilized for path tracking in robotics. Let $x_1$ and $x_2$ in Example~5.9 represent coordinates of the robot position in a local coordinate system, favorably in the standard notation from differential geometry of a curve, called natural coordinates. Let $R$ be the local radius of curvature for the path to be tracked. Then by extending the dynamics with $(R^2-x_1^2-x_2^2)$ as in Example~5.9, a system is obtained where the robot is \emph{attracted} to the path with exponential convergence. On the other hand, in the tangential direction (which in Example~5.9 corresponds to the $\varphi$ direction), it is a clear advantage that it is not asymptotically stable since it leaves speed along the path free to control. This can be used to devise a speed controller that adjusts the speed along the path to cope with actuator constraints or other requirements implied by the application. \endRemark

Further, if you have a section for further reading you are of course more than welcome to use the following text.

\subsection{Further Reading} An approach using the separation property in Example~5.9 in robotics is available in \cite{olofsson_nielsen_2017}.

@article{olofsson_nielsen_2017, title = "Path-tracking velocity control for robot manipulators with actuator constraints", author = "Bj{\"o}rn Olofsson and Lars Nielsen", year = "2017", volume = "45", pages = "82--99", journal = "Mechatronics", }

[Richard Murray]

24 Jul: probably too detailed to include in the book, though perhaps worth noting that there is a common separation of “lateral” versus “longitudinal” control. Karl will have a look.

[Richard Murray]

Karl: I read the material. I think that we should leave the text as is.