Motion-planning via a forest of dynamics-based trees

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Introduction

My current research mostly focuses on control theory and nonlinear control. In general, a control function synthesis can be categorized into two classes of linear and nonlinear control. The advances in linear system theory and the available mathematical tools makes synthesis and analysis of linear controllers significantly simpler than of the nonlinear controllers. On the other hand, the nonlinear nature of real physical systems and limitations imposed on dynamic models due to linearization processes reduces the performance of the systems controlled by linear controllers. This lack of efficiency is more apparent in systems with large number of degrees of freedom and complex dynamics such as mobile robots, power plants, stock markets, human interactions etc. To date, unlike linear systems, the analytical methods that have been developed for nonlinear systems are not generalizable to be used in an algorithmic approach in synthesizing nonlinear controllers. Consequently, most of the nonlinear controllers are designed for specific platforms and are not generalizable. Additionally, the corresponding numerical methods developed for nonlinear systems are relatively slow and suffer from curse of dimensionality.

In this regard, as a mindless generalizable numerical approach, we have introduced the concept of Regionally Growing Random Trees (RGRT) as a powerful tool that synergistically combines motion planning and control tasks. RGRT is a forest of Dynamics-based Rapidly Expanding Trees (DRETs) that grow in the state-space of a dynamic system without requiring any distance function or explicit solutions of the differential equations of motion. The growth of multiple DRETs results in paths between the tree roots and forms a roadmap that is utilized in a planning algorithm to find a feasible path between a current state and a goal state. A path-tracking algorithm is then used to convert the open-loop commands of the planner into a feedback controller, which provides robustness against disturbances and modeling errors. The RGRT motion planning and control scheme allows complete utilization of system nonlinearities, which provides solutions for overcoming actuator constraints and eliminates the limitations imposed on the system by traditional feedback control approaches.

The following two videos show the response of a simple pendulum with input torque limitation to a PD and RGRT controllers. In both situations, the desired state is the upward configuration of the pendulum. In the first video, the pendulum is controlled by a simple PD controller, and due to the input torque limitations, it cannot swing pass the maximum torque point (horizontal configuration). Consequently, the pendulum continuously oscillates about an angle below horizon.

In the following video, we demonstrate the response of the same system when it uses a RGRT forest as the main controller. As illustrated in the video, the pendulum swings back and forth to gain enough momentum to pass the maximum torque point and finally reaches to the desired state.

The following video shows a solution using RGRT for an under-actuated cart-pole regulation problem with limited input where it is desired to move the cart pole system to the zero position with the pendulum standing in the upward configuration. Note that the initial condition set in this simulation will cause instability in normal linear controllers.