Abstract
Shape restoration is defined as the problem of constructing a desired, or goal, solid shape Sg by growing an initial solid Si, which is a subset of the goal but is otherwise unknown. This definition attempts to capture abstractly a situation that often arises in the physical world when a solid object loses its desired shape due to wear and tear, corrosion or other phenomena. For example, if the top of the femur becomes distorted, the hip joint no longer functions properly and may have to be replaced surgically. Growing it in place back to its original shape would be an attractive alternative to replacement. This paper presents a solution to the shape restoration problem by using autonomous assembly agents (robots) that self-assemble to fill the volume between Sg and Si. If the robots have very small dimension (micro or nano), the desired shape is approximated with high accuracy. The assembly agents initially execute a random walk. When two robots meet, they may exchange a small number of messages. The robot behavior is controlled by a finite state machine with a small number of states. Communication contact models chemical communication, which is likely to be the medium of choice for robots at the nanoscale, while small state and small messages are limitations that also are expected of nanorobots. Simulations presented here show that swarms of such robots organize themselves to achieve shape restoration by using distributed algorithms. This is one more example of an interesting geometric problem that can be solved by the Active Self-Assembly paradigm introduced in previous papers by the authors.