  We consider the classic motion planning problem defined over a roadmap in
which a vehicle seeks to find an optimal path from a source to a destination in
presence of an attacker who can launch attacks on the vehicle over any edge of
the roadmap. The vehicle (defender) has the capability to switch on/off a
countermeasure that can detect and permanently disable the attack if it occurs
concurrently. We model the problem of traveling along en edge using the
framework of a simultaneous zero-sum dynamic game (edge-game) with a stopping
state played between an attacker and defender. We characterize the Nash
equiliria of an edge-game and provide closed form expressions for two actions
per player. We further provide an analytic and approximate expression on the
value of an edge-game and characterize conditions under which it grows
sub-linearly with the number of stages. We study the sensitivity of Nash
equilibrium to the (i) cost of using the countermeasure, (ii) cost of motion
and (iii) benefit of disabling the attack. The solution of an edge-game is used
to formulate and solve for the secure planning problem known as a meta-game. We
design an efficient heuristic by converting the problem to a shortest path
problem using the edge cost as the solution of corresponding edge-games. We
illustrate our findings through several insightful simulations.
