Time-invariant Motion Planner in discretized C-Spacetime for MRS

In this work, a new class of planners for MRS is introduced: Time-invariant Motion Planners, a class of planners that operate indifferently in forward or in backward planningtime direction. Thanks to the specular symmetry (with respect to the timeline) of the motion operators, the planning algorithm can operate both in top-down way (from the goal to the starting pose) or vice versa bottom-up (from the starting pose to the goal) addressing different types of problems. The planner underlying mechanism is an artificial field over a lattice (CAs), where the robots are shrunk to points subjected to attractive and repulsive forces (Lagrangian mechanics). Building a regular manifold of potential values and following its minimum valleys, a trajectory in the spacetime is extracted, corresponding to a robot movement (geometrization of the motion). The potential manifold is constructed on the base of the motion operators. These are the atomic (non interruptible) moves over the space and the time lattice and the set of all of them represents the entire kinematics of a robot. Every robot has its own set and there are contemporarily robots with different kinematics. The manifold emerges from the interaction of the set of operators, the world model and the representation of the robots’ shapes. Using a discretized C-Spacetime, the definition of velocity of a robot becomes an intrinsical (geometrical) property emerging from the interaction between the motion operators and the spacetime. It is fundamental for the correctness of the planning to take care of the actual robot occupancy during an atomic move to avoid collisions with other robots/obstacles. It derives the definition of Motion Silhouette, a conceptual evolution of the Sweeping Silhouette (2002), which is itself an evolution of the Obstacles Enlargement concept by Lozano-Pérez in 1983. To avoid the problem of the swapping of two robots, it is important to take in consideration of the well-known Shannon’s Theorem in the discretization phase of the C-Spacetime and consequently in the definition of Motion Silhouette.