MODEL-BASED LEARNING AND CONTROL OF ADVECTION-DIFFUSION TRANSPORT USING MOBILE ROBOTS
Mathematical models that describe different processes and phenomena are of paramount importance in many robotics applications. Nevertheless, utilization of high-fidelity models, particularly Partial Differential Equations (PDEs), has been hindered for many years due to the lack of adequate computational resources onboard mobile robots. One such problem of interest for the roboticists, that can hugely benefit from more descriptive models, is Chemical Plume Tracing (CPT). In the CPT problem, one or multiple mobile robots are equipped with chemical concentration and flow sensors and attempt to localize chemical sources in an environment of interest. This problem has important applications ranging from environmental monitoring and protection to search and rescue missions. The transport of a chemical in a fluid medium is mathematically modeled by the Advection-Diffusion (AD) Partial Differential Equation (PDE). Despite versatility, rigorous derivation, and powerful descriptive nature, the AD-PDE has seldom been used in its general form for the solution of the CPT problem due to high computational cost. Instead, often simplified scenarios that render closed-form solutions for the AD-PDE or various heuristics are used in the robotics literature.
Using the AD-PDE to model the transport phenomenon enables generalization of the CPT problem to estimate other properties of the sources, e.g., their intensity, in addition to their locations. We refer to this problem as Source Identification (SI) which we define as the problem of estimating the properties of the sources using concentration measurements that are generated under the action of those sources. We can also put one step further and consider the problem of controlling a set of sources, carried by a team of mobile robots, to generate and maintain desired concentration levels in select regions of the environment with the objective of cloaking those regions from external environmental conditions; we refer to this problem as the AD-PDE control problem that has important applications in search and rescue missions.
Both SI and AD-PDE control problems can be formulated as PDE-constrained optimization problems. Solving such optimization problems onboard mobile robots is challenging due to the following reasons: (i) the computational cost of solving the AD-PDE using traditional numerical discretization schemes, e.g., the Finite Element (FE) method, is prohibitively high, (ii) obtaining accurate knowledge of the environment and Boundary and Initial Conditions (BICs), required to solve the AD-PDE, is difficult and prone to error and finally, (iii) obtaining accurate estimates of the velocity and diffusivity fields is challenging since for typical transport mediums like air even in very small velocities, the flow is turbulent. In addition, we need to plan the actions of the mobile robots, e.g., measurement collection for SI or release rates for the AD-PDE control problem, to ensure that they accomplish their tasks optimally. This can be done by formulating a planning problem that often is solved online to take into account the latest information that becomes available to robots. Solving this planning problem by itself is a challenging task that has been the subject of heavy research in the robotics literature. The reason is that (i) the objective is often nonlinear, (ii) the planning is preferred to be done for more than the immediate action to avoid myopic, suboptimal plans, and (iii) the environment that the robots operate in is often non-convex and cluttered with obstacles.
In order to address the computational challenges that rise due to the use of numerical schemes, we propose using multiple mobile robots that decompose the high-dimensional optimization variables among themselves or using nonlinear representations of the sources. In addition we utilize Model Order Reduction (MOR) approaches that facilitate the evaluation of the AD-DPE at the expense of accuracy. In order to alleviate the loss of accuracy, we also propose a novel MOR method using Neural Networks that can straight-forwardly replace the traditional MOR methods in our formulations. To deal with uncertainty in the the PDE input-data, i.e., the geometry of environment, BICs, and the velocity and diffusivity fields, we formulate a stochastic version of the SI problem that provides posterior probabilities over all possible values of these uncertain parameters. Finally, to obtain the velocity and corresponding diffusivity fields that are required for the solution of the AD-PDE, we rely on Bayesian inference to incorporate empirical measurements, collected and analyzed by mobile robots, into the numerical solutions obtained from computational fluid dynamics models.
In order to demonstrate the applicability of our proposed model-based approaches, we have devised and constructed an experimental setup and designed a mobile robot equipped with concentration and flow sensors. To the best of our knowledge, this dissertation is the first work to use the AD-PDE, in its general form, to solve realistic problems onboard mobile robots. Note that although here we focus on the AD-PDE and particularly chemical propagation, many other transport phenomena including heat and acoustic transport can be considered and the same principles apply. Our results are a proof of concept that we hope will convince many roboticists to use more general mathematical models in their solutions.
Learning and Control
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