Motion Planning and Trajectory Based Controller Synthesis for Optimal and Correct System Behavior

This dissertation is primarily focused on synthesizing control and motion plan that guarantee correct system behaviors while optimizing a performance objective. In the approaches presented here, the correct system behaviors are defined in terms of safety/reachability specifications and temporal logic specifications. Safety/reachability specifications involve requiring the system to avoid a set of unsafe states at all times and reach a set of goal states within a specific amount of time. Temporal logic specifications extend these system behaviors to more complex specifications such as the system is required to visit different sets of states in a specific order and at specific times.;To solve the safety/reachability control problems we propose to use the concept of trajectory robustness that provides a robust neighborhood around any nominal trajectory of the system satisfying the given safety/reachability specification. If the system is initialized within the robust neighborhood of the nominal trajectory, we can apply the same control input as the nominal trajectory and the resulting system trajectory is guaranteed to always stay within the robust tube defined by that nominal trajectory and its neighborhood. As a result, the new trajectory also satisfies the safety/reachability specifications.;We then develop a motion planner based on Mixed-Integer Linear Programming (MILP) approach such that given a system whose dynamics can be represented with linear equations and desired system specifications in terms of Metric Temporal Logic (MTL) formula or simpler reach-avoid specifications, the motion planner finds a system trajectory that satisfies the specifications, even in dynamically changing environments. In this work, the end goal is to be able to find such trajectories in real-time such that if the environment of the system is changing dynamically in the run-time, the system can still satisfy its desired task specifications given by the MTL formula.;To account for any general nonlinear systems, we also propose a gradient-search based optimization algorithm for motion planning to satisfy MTL specifications. Both the MILP method and this one are capable of minimizing (maximizing) a desired performance criteria. We showcase the usefulness of these approaches in real-world applications, by solving a task and motion planning problem for a manipulator arm for an object rearrangement task. We use a hierarchical framework, where at the high-level the MILP method is used to find proper sequences of movement of the manipulator arm. At the low-level the gradient-based method finds the actual control inputs to realize the arm movement sequence.;Finally, we extend the MILP motion planning approach from a single-agent system to a multi-agent system, where multiple individual agents need to cooperate to realize a group MTL task. We solve this problem by proposing a semi-decentralized approach, where each agent solves its own MILP problem locally, in parallel to all other agents. This leads to a speed-up in the execution time for finding the individual trajectories for each agent, so that the motion planning approach can be implemented in real-time, in a dynamically changing environment.