Symmetric systems

Finally, one property of systems that is important in some planning algorithms is symmetry.^14.1 A system ${\dot x}= f(x,u)$ is symmetric if the following condition holds. If there exists an action trajectory that brings the system from some ${x_{I}}$ to some ${x_{G}}$, then there exists another action trajectory that brings the system from ${x_{G}}$ to ${x_{I}}$ by visiting the same points in $X$, but in reverse time. At each point along the path, this means that the velocity can be negated by a different choice of action. Thus, it is possible for a symmetric system to reverse any motions. This is usually not possible for systems with drift. An example of a symmetric system is the differential drive of Section 13.1.2. For the simple car, the Reeds-Shepp version is symmetric, but the Dubins version is not because the car cannot travel in reverse.