A Decoupling Principle for Simultaneous Localization and Planning Under Uncertainty in Multi-Agent Dynamic Environments

Simultaneous localization and planning for nonlinear stochastic systems under process and measurement uncertainties is a challenging problem. In its most general form, it is formulated as a stochastic optimal control problem in the space of feedback policies. The Hamilton-Jacobi-Bellman equation provides the theoretical solution of the optimal problem; but, as is typical of almost all nonlinear stochastic systems, optimally solving the problem is intractable. Moreover, even if an optimal solution was obtained, it would require centralized control, while multi-agent mobile robotic systems under dynamic environments require decentralized solutions. In this study, we aim for a theoretically sound solution for various modes of this problem, including the single-agent and multi-agent variations with perfect and imperfect state information, where the underlying state, control and observation spaces are continuous with discrete-time models. We introduce a decoupling principle for planning and control of multi-agent nonlinear stochastic systems based on a small noise asymptotics. Through this decoupling principle, under small noise, the design of the real-time feedback law can be decoupled from the off-line design of the nominal trajectory of the system. Further, for a multi-agent problem, the design of the feedback laws for different agents can be decoupled from each other, reducing the centralized problem to a decentralized problem requiring no communication during execution. The resulting solution is quantifiably near-optimal. We establish this result for all the above-mentioned variations, which results in the following variants: Trajectory-optimized Linear Quadratic Regulator (T-LQR), Multi-agent T-LQR (MT-LQR), Trajectory-optimized Linear Quadratic Gaussian (T-LQG), and Multi-agent T-LQG (MT-LQG). The decoupling principle provides the conditions under which a decentralized linear Gaussian system with a quadratic approximation of the cost, obtained by linearization around an optimally designed nominal trajectory can be utilized to control the nonlinear system. The resulting decentralized feedback solution at runtime, being decoupled with respect to the mobile agents, requires no communication between the agents during the execution phase. Moreover, the complexity of the solution vis-a-vis the computation of the nominal trajectory as well as the closed-loop gains is tractable with low polynomial orders of computation. Experimental implementation of the solution shows that the results hold for moderate levels of noise with high probability. Further optimizing the performance of this approach we show how to design a special cost function for the problem with imperfect state measurement that takes advantage of the fact that the estimation covariance of a linear Gaussian system is deterministic and not dependent on the observations. This design, which corresponds in our overall design to “belief space planning”, incorporates the consequently deterministic cost of the stochastic feedback system into the deterministic design of the nominal trajectory to obtain an optimal nominal trajectory with the best estimation performance. Then, it utilizes the T-LQG approach to design an optimal feedback law to track the designed nominal trajectory. This iterative approach can be used to further tune both the open loop as well as the decentralized feedback gain portions of the overall design. We also provide the multi-agent variant of this approach based on the MT-LQG method. Based on the near-optimality guarantees of the decoupling principle and the TLQG approach, we analyze the performance and correctness of a well-known heuristic in robotic path planning. We show that optimizing measures of the observability Gramian as a surrogate for estimation performance may provide irrelevant or misleading trajectories for planning under observation uncertainty. We then consider systems with non-Gaussian perturbations. An alternative heuristic method is proposed that aims for fast planning in belief space under non- Gaussian uncertainty. We provide a special design approach based on particle filters that results in a convex planning problem implemented via a model predictive control strategy in convex environments, and a locally convex problem in non-convex environments. The environment here refers to the complement of the region in Euclidean space that contains the obstacles or “no fly zones”. For non-convex dynamic environments, where the no-go regions change dynamically with time, we design a special form of an obstacle penalty function that incorporates non-convex time-varying constraints into the cost function, so that the decoupling principle still applies to these problems. However, similar to any constrained problem, the quality of the optimal nominal trajectory is dependent on the quality of the solution obtainable for the nonlinear optimization problem. We simulate our algorithms for each of the problems on various challenging situations, including for several nonlinear robotic models and common measurement models. In particular, we consider 2D and 3D dynamic environments for heterogeneous holonomic and non-holonomic robots, and range and bearing sensing models. Future research can potentially extend the results to more general situations including continuous-time models

A Decoupling Principle for Simultaneous Localization and Planning Under Uncertainty in Multi-Agent Dynamic Environments

Simultaneous localization and planning for nonlinear stochastic systems under process and measurement uncertainties is a challenging problem. In its most general form, it is formulated as a stochastic optimal control problem in the space of feedback policies. The Hamilton-Jacobi-Bellman equation provides the theoretical solution of the optimal problem; but, as is typical of almost all nonlinear stochastic systems, optimally solving the problem is intractable. Moreover, even if an optimal solution was obtained, it would require centralized control, while multi-agent mobile robotic systems under dynamic environments require decentralized solutions. In this study, we aim for a theoretically sound solution for various modes of this problem, including the single-agent and multi-agent variations with perfect and imperfect state information, where the underlying state, control and observation spaces are continuous with discrete-time models. We introduce a decoupling principle for planning and control of multi-agent nonlinear stochastic systems based on a small noise asymptotics. Through this decoupling principle, under small noise, the design of the real-time feedback law can be decoupled from the off-line design of the nominal trajectory of the system. Further, for a multi-agent problem, the design of the feedback laws for different agents can be decoupled from each other, reducing the centralized problem to a decentralized problem requiring no communication during execution. The resulting solution is quantifiably near-optimal. We establish this result for all the above-mentioned variations, which results in the following variants: Trajectory-optimized Linear Quadratic Regulator (T-LQR), Multi-agent T-LQR (MT-LQR), Trajectory-optimized Linear Quadratic Gaussian (T-LQG), and Multi-agent T-LQG (MT-LQG). The decoupling principle provides the conditions under which a decentralized linear Gaussian system with a quadratic approximation of the cost, obtained by linearization around an optimally designed nominal trajectory can be utilized to control the nonlinear system. The resulting decentralized feedback solution at runtime, being decoupled with respect to the mobile agents, requires no communication between the agents during the execution phase. Moreover, the complexity of the solution vis-a-vis the computation of the nominal trajectory as well as the closed-loop gains is tractable with low polynomial orders of computation. Experimental implementation of the solution shows that the results hold for moderate levels of noise with high probability. Further optimizing the performance of this approach we show how to design a special cost function for the problem with imperfect state measurement that takes advantage of the fact that the estimation covariance of a linear Gaussian system is deterministic and not dependent on the observations. This design, which corresponds in our overall design to “belief space planning”, incorporates the consequently deterministic cost of the stochastic feedback system into the deterministic design of the nominal trajectory to obtain an optimal nominal trajectory with the best estimation performance. Then, it utilizes the T-LQG approach to design an optimal feedback law to track the designed nominal trajectory. This iterative approach can be used to further tune both the open loop as well as the decentralized feedback gain portions of the overall design. We also provide the multi-agent variant of this approach based on the MT-LQG method. Based on the near-optimality guarantees of the decoupling principle and the TLQG approach, we analyze the performance and correctness of a well-known heuristic in robotic path planning. We show that optimizing measures of the observability Gramian as a surrogate for estimation performance may provide irrelevant or misleading trajectories for planning under observation uncertainty. We then consider systems with non-Gaussian perturbations. An alternative heuristic method is proposed that aims for fast planning in belief space under non- Gaussian uncertainty. We provide a special design approach based on particle filters that results in a convex planning problem implemented via a model predictive control strategy in convex environments, and a locally convex problem in non-convex environments. The environment here refers to the complement of the region in Euclidean space that contains the obstacles or “no fly zones”. For non-convex dynamic environments, where the no-go regions change dynamically with time, we design a special form of an obstacle penalty function that incorporates non-convex time-varying constraints into the cost function, so that the decoupling principle still applies to these problems. However, similar to any constrained problem, the quality of the optimal nominal trajectory is dependent on the quality of the solution obtainable for the nonlinear optimization problem. We simulate our algorithms for each of the problems on various challenging situations, including for several nonlinear robotic models and common measurement models. In particular, we consider 2D and 3D dynamic environments for heterogeneous holonomic and non-holonomic robots, and range and bearing sensing models. Future research can potentially extend the results to more general situations including continuous-time models

Multi-domain Modeling and Simulation

One starting point for the analysis and design of a control system is the block diagram representation of a plant. Since it is nontrivial to convert a physical model of a plant into a blockk diagram, this can be performed manually only for small models. Based on reseach from the last 40 years, more andmore mature tools are available to achieve this transformation fully automatically. As a result, multi-domain plants, for example, systems with electrical, mechanical, thermal, and fluid parts, can be modled in a unified way and can be used directly as input-output blocks for control system design. An overview of the basic principles of this approach is given, and it is shown how to utilize nonlinear, multidomain plant models directly in a controller. Finally, the low-level "Functional Mockup Interface" standard is sketched to exchang multi-domain models between many different modeling and simulation environments

Cosmic strings and their induced non-Gaussianities in the cosmic microwave background

Motivated by the fact that cosmological perturbations of inflationary quantum origin were born Gaussian, the search for non-Gaussianities in the cosmic microwave background (CMB) anisotropies is considered as the privileged probe of non-linear physics in the early universe. Cosmic strings are active sources of gravitational perturbations and incessantly produce non-Gaussian distortions in the CMB. Even if, on the currently observed angular scales, they can only contribute a small fraction of the CMB angular power spectrum, cosmic strings could actually be the main source of its non-Gaussianities. In this article, after having reviewed the basic cosmological properties of a string network, we present the signatures Nambu-Goto cosmic strings would induce in various observables ranging from the one-point function of the temperature anisotropies to the bispectrum and trispectrum. It is shown that string imprints are significantly different than those expected from the primordial type of non-Gaussianity and could therefore be easily distinguished.Comment: 50 pages, 20 figures, uses iopart. Misprints corrected, references added, matches published versio