There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Filtering and smoothing methods are used to produce an accurate estimate of the state of a time-varying system based on multiple observational inputs (data). Interest in these methods has exploded in recent years, with numerous applications emerging in fields such as navigation, aerospace engineering, telecommunications, and medicine. This compact, informal introduction for graduate students and advanced undergraduates presents the current state-of-the-art filtering and smoothing methods in a unified Bayesian framework. Readers learn what non-linear Kalman filters and particle filters are, how they are related, and their relative advantages and disadvantages. They also discover how state-of-the-art Bayesian parameter estimation methods can be combined with state-of-the-art filtering and smoothing algorithms. The book’s practical and algorithmic approach assumes only modest mathematical prerequisites. Examples include MATLAB computations, and the numerous end-of-chapter exercises include computational assignments. MATLAB/GNU Octave source code is available for download at www.cambridge.org/sarkka, promoting hands-on work with the methods.

贝叶斯滤波与平滑 (Bayesian filtering and smoothing)

The Architecture Of Cognition

We propose a cooperative motion and task planning scheme for multi-agent systems where the agents have independently assigned local tasks, specified as linear temporal logic formulas. These tasks contain hard and soft sub-specifications. A least-violating initial plan is synthesized first for the potentially infeasible task and the partially-known workspace. This discrete plan is then implemented by the potential-field-based navigation controllers. While the system runs, each agent updates its knowledge about the workspace via its sensing capability and shares this knowledge with its neighbouring agents. Based on the knowledge update, each agent verifies and revises its motion plan in real time. It is ensured that the hard specification is always fulfilled for safety and the satisfaction for the soft specification is improved gradually. The design is distributed as only local interactions are assumed. The overall framework is demonstrated by a case study and an experiment.

Multi-agent plan reconfiguration under local LTL specifications

We describe a computational framework for automatic deployment of a robot with sensor and actuator noise from a temporal logic specification over a set of properties that are satisfied by the regions of a partitioned environment. We model the motion of the robot in the environment as a Markov decision process (MDP) and translate the motion specification to a formula of probabilistic computation tree logic (PCTL). As a result, the robot control problem is mapped to that of generating an MDP control policy from a PCTL formula. We present algorithms for the synthesis of such policies for different classes of PCTL formulas. We illustrate our method with simulation and experimental results.

Temporal Logic Motion Planning and Control With Probabilistic Satisfaction Guarantees

We present a computational framework for automatic deployment of a robot from a temporal logic specification over a set of properties of interest satisfied at the regions of a partitioned environment. We assume that, during the motion of the robot in the environment, the current region can be precisely determined, while due to sensor and actuation noise, the outcome of a control action can only be predicted probabilistically. Under these assumptions, the deployment problem translates to generating a control strategy for a Markov Decision Process (MDP) from a temporal logic formula.We propose an algorithm inspired from probabilistic Computation Tree Logic (PCTL) model checking to find a control strategy that maximizes the probability of satisfying the specification. We illustrate our method with simulation and experimental results.

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Motion planning and control from temporal logic specifications with probabilistic satisfaction guarantees

Robot control for tasks such as moving around obstacles or grasping objects has advanced significantly in the last few decades. However, controlling robots to perform complex tasks is still accomplished largely by highly trained programmers in a manual, time-consuming, and error-prone process that is typically validated only through extensive testing. Formal methods are mathematical techniques for reasoning about systems, their requirements, and their guarantees. Formal synthesis for robotics refers to frameworks for specifying tasks in a mathematically precise language and automatically transforming these specifications into correct-by-construction robot controllers or into a proof that the task cannot be done. Synthesis allows users to reason about the task specification rather than its implementation, reduces implementation error, and provides behavioral guarantees for the resulting controller. This article reviews the current state of formal synthesis for robotics and surveys the landscape of abstract...

Synthesis for Robots: Guarantees and Feedback for Robot Behavior

Formal methods are increasingly being used for control and verification of dynamic systems against complex specifications. In general, these methods rely on a relatively simple system model, such as a transition graph, Markov chain, or Markov decision process, and require abstraction of the original continuous-state dynamics. It can be difficult or impossible, however, to find a perfectly equivalent abstraction, particularly when the original system is stochastic. Here we develop an abstraction procedure that maps a discrete-time stochastic system to an Interval-valued Markov Chain ( IMC ) and a switched discrete-time stochastic system to a Bounded-parameter Markov Decision Process ( BMDP ). We construct model checking algorithms for these models against Probabilistic Computation Tree Logic ( PCTL ) formulas and a synthesis procedure for BMDP s. Finally, we develop an efficient refinement algorithm that reduces the uncertainty in the abstraction. The technique is illustrated through simulation.

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Formal Verification and Synthesis for Discrete-Time Stochastic Systems

This paper considers the problem of motion planning for a hybrid robotic system with complex and nonlinear dynamics in a partially unknown environment given a temporal logic specification. We employ a multi-layered synergistic framework that can deal with general robot dynamics and combine it with an iterative planning strategy. Our work allows us to deal with the unknown environmental restrictions only when they are discovered and without the need to repeat the computation that is related to the temporal logic specification. In addition, we define a metric for satisfaction of a specification. We use this metric to plan a trajectory that satisfies the specification as closely as possible in cases in which the discovered constraint in the environment renders the specification unsatisfiable. We demonstrate the efficacy of our framework on a simulation of a hybrid second-order car-like robot moving in an office environment with unknown obstacles. The results show that our framework is successful in generating a trajectory whose satisfaction measure of the specification is optimal. They also show that, when new obstacles are discovered, the reinitialization of our framework is computationally inexpensive.

/pdf/iterative-temporal-motion-planning-for-hybrid-systems-in-7rttp6og5q.pdf

Morteza Lahijanian

Papers

Temporal Logic Motion Planning and Control With Probabilistic Satisfaction Guarantees

Motion planning and control from temporal logic specifications with probabilistic satisfaction guarantees

Synthesis for Robots: Guarantees and Feedback for Robot Behavior

Formal Verification and Synthesis for Discrete-Time Stochastic Systems

Iterative temporal motion planning for hybrid systems in partially unknown environments