References

Concepts similar to Soft Walls have been described and developed by others. Below is a (growing) list of pointers. Please send suggested entries to eal@eecs.berkeley.edu.

• On October 15, 2001, Forbes Magazine published an article by Peter Huber 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 http://www.me.jhu.edu/~allisono/publications.html.

• 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:

  1. Application of Level Set Methods to Control and Reachability Problems in Continuous and Hybrid Systems
     Ian Mitchell
     Doctoral Thesis (submitted August 26, 2002)
  2. A Game Theoretic Approach to Controller Design for Hybrid Systems
     Claire Tomlin, John Lygeros, and Shankar Sastry.
     Proceedings of the IEEE, Volume 88, Number 7, July 2000.
  3. 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, April 1998.
  These papers, which we were not able to find online, develop the mathematical theory of this approach:
  1. Some Applications of Viscosity Solutions to Optimal Control and Differential Games.
     Martino Bardi.
     In I. Capuzzo Dolcetta and P. L. Lions, editors, Viscosity Solutions and Applications. Springer, 1995.
  2. Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations.
     L. C. Evans and P. E. Souganidis.
     Indiana University Mathematics Journal, 33(5):773–797, 1984.