Concepts similar to Soft Walls have been described and developed
by others. Below is a (growing) list of pointers.
Please send suggested entries to
On October 15, 2001, Forbes Magazine published an
entitled "Disable the Pilots" that advocates
a Soft Walls-like scheme. Huber states that he had
described a similar idea ten years earlier, well before September 11, 2001.
(Note that following the above links will cause annoying popups to appear,
and accessing the article requires paying a nominal fee.)
Allison M. Okamura pointed out to us that the concept is related to the
idea of "virtual fixtures," which are soft or hard constraints for
surgical assistance machines. The concept is to place virtual fixtures
around delicate structures to prevent the surgeon from accidentally
contacting them during robot-assisted minimally invasive surgery and
microsurgery. Publications studying the framework and control
methodologies can be found at
Claire Tomlin, Shankar Sastry, Ian Mitchell, and others have been
working on collision avoidance between unmanned aircraft. Much of the
theory which they use applies directly to Soft Walls. Here are a few
of the papers we have found useful:
These papers, which we were not able to find online, develop the
mathematical theory of this approach:
- Application of Level Set Methods to Control and
Reachability Problems in Continuous and Hybrid Systems
Doctoral Thesis (submitted August 26, 2002)
- A Game Theoretic Approach to Controller Design for
Claire Tomlin, John Lygeros, and Shankar Sastry.
Proceedings of the IEEE, Volume 88, Number 7, July 2000.
- Conflict Resolution for Air Traffic Management: A Study
in Multi-Agent Hybrid Systems
Claire Tomlin, George J. Pappas, and Shankar Sastry.
IEEE Transactions on Automatic Control, Volume 43, Number 4,
- Some Applications of Viscosity Solutions to Optimal Control
and Differential Games.
In I. Capuzzo Dolcetta and P. L. Lions, editors, Viscosity
Solutions and Applications. Springer, 1995.
- Differential Games and Representation Formulas for Solutions of
L. C. Evans and P. E. Souganidis.
Indiana University Mathematics Journal, 33(5):773–797, 1984.