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38
LQRTrees: Feedback motion planning via sums of squares verification
 International Journal of Robotics Research
, 2010
"... Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses rigorously computed stability regions to build a sparse tree ..."
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Cited by 67 (23 self)
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Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses rigorously computed stability regions to build a sparse tree of LQRstabilized trajectories. The region of attraction of this nonlinear feedback policy “probabilistically covers ” the entire controllable subset of the state space, verifying that all initial conditions that are capable of reaching the goal will reach the goal. We numerically investigate the properties of this systematic nonlinear feedback design algorithm on simple nonlinear systems, prove the property of probabilistic coverage, and discuss extensions and implementation details of the basic algorithm. 1
Computational Methods for Reachability Analysis of Stochastic Hybrid
 Systems, Hybrid Systems: Computation and Control 2006 LNCS 3927
, 2006
"... Abstract. Stochastic hybrid system models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing algorithms for reachability analysis is challen ..."
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Cited by 26 (8 self)
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Abstract. Stochastic hybrid system models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing algorithms for reachability analysis is challenging because of the interaction between the discrete and continuous stochastic dynamics. In this paper, we propose a probabilistic method for reachability analysis based on discrete approximations. The contribution of the paper is twofold. First, we show that reachability can be characterized as a viscosity solution of a system of coupled HamiltonJacobiBellman equations. Second, we present a numerical method for computing the solution based on discrete approximations and we show that this solution converges to the one for the original system as the discretization becomes finer. Finally, we illustrate the approach with a navigation benchmark that has been proposed for hybrid system verification. 1
Computational Approaches to Reachability Analysis of Stochastic Hybrid Systems
"... Abstract. This work investigates some of the computational issues involved in the solution of probabilistic reachability problems for discretetime, controlled stochastic hybrid systems. It is first argued that, under rather weak continuity assumptions on the stochastic kernels that characterize the ..."
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Cited by 21 (8 self)
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Abstract. This work investigates some of the computational issues involved in the solution of probabilistic reachability problems for discretetime, controlled stochastic hybrid systems. It is first argued that, under rather weak continuity assumptions on the stochastic kernels that characterize the dynamics of the system, the numerical solution of a discretized version of the probabilistic reachability problem is guaranteed to converge to the optimal one, as the discretization level decreases. With reference to a benchmark problem, it is then discussed how some of the structural properties of the hybrid system under study can be exploited to solve the probabilistic reachability problem more efficiently. Possible techniques that can increase the scaleup potential of the proposed numerical approximation scheme are suggested. 1
Approximate reachability computation for polynomial systems
 in HSCC’06, vol. 3927 in LNCS
, 2006
"... Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time ste ..."
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Cited by 20 (11 self)
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Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial. We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2. In addition, we have successfully applied our reachability algorithm to two models of a biological system. 1
Logics of Dynamical Systems
"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded ..."
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Cited by 15 (15 self)
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We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded systems and cyberphysical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multiagent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic
Computational Methods for Verification of Stochastic Hybrid Systems
 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS  PART A
, 2008
"... Stochastic hybrid system (SHS) models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing sound computational methods for verification is ch ..."
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Cited by 15 (5 self)
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Stochastic hybrid system (SHS) models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing sound computational methods for verification is challenging because of the interaction between the discrete and the continuous stochastic dynamics. In this paper, we propose a probabilistic method for verification of SHSs based on discrete approximations focusing on reachability and safety problems. We show that reachability and safety can be characterized as a viscosity solution of a system of coupled Hamilton–Jacobi–Bellman equations. We present a numerical algorithm for computing the solution based on discrete approximations that are derived using finitedifference methods. An advantage of the method is that the solution converges to the one for the original system as the discretization becomes finer. We also prove that the algorithm is polynomial in the number of states of the discrete approximation. Finally, we illustrate the approach with two benchmarks: a navigation and a room heater example, which have been proposed for hybrid system verification.
Robust ReachAvoid Controller Synthesis for Switched Nonlinear Systems
"... Abstract — In this paper, we describe a method to automatically synthesize controllers that provide hard guarantees of safety and target reachability for sampleddata switched systems under bounded continuous disturbances. Techniques from hybrid system verification are used to perform continuous tim ..."
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Cited by 10 (3 self)
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Abstract — In this paper, we describe a method to automatically synthesize controllers that provide hard guarantees of safety and target reachability for sampleddata switched systems under bounded continuous disturbances. Techniques from hybrid system verification are used to perform continuous time differential game calculations on each sampling interval. Iterative procedures are given for computing the set of states for which there exists an admissible control policy so that the closedloop system satisfies the properties of safety and reachability over a finite time horizon. From this computation, we show how to obtain an explicit state feedback policy in the form of multiple reachable sets, and an algorithm is given for using this feedback law in closedloop control of the switched system. A simulation example of automated aerial refueling is used to illustrate the application of our approach. I.
Reachability calculations for automated aerial refueling
 in Decision and Control, 2008. CDC 2008. 47th IEEE Conference on
, 2008
"... Abstract — This paper describes HamiltonJacobi (HJ) reachability calculations for a hybrid systems formalism governing unmanned aerial vehicles (UAVs) interacting with another vehicle in a safetycritical situation. We use this problem to lay the foundations toward the goal of refining or designing ..."
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Cited by 7 (4 self)
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Abstract — This paper describes HamiltonJacobi (HJ) reachability calculations for a hybrid systems formalism governing unmanned aerial vehicles (UAVs) interacting with another vehicle in a safetycritical situation. We use this problem to lay the foundations toward the goal of refining or designing protocols for multiUAV and/or manned vehicle interaction. We describe here what mathematical foundations are necessary to formulate verification problems on reachability and safety of flight maneuvers. We finally show how this formalism can be used in the chosen application to inform UAV decisions on avoiding unsafe scenarios while achieving mission objectives. I.
DIFFEOMORPHIC SURFACE FLOWS: A NOVEL METHOD OF SURFACE EVOLUTION
, 2008
"... We describe a new class of surface flows, diffeomorphic surface flows, induced by restricting diffeomorphic flows of the ambient Euclidean space to a surface. Different from classical surface PDE flows such as mean curvature flow, diffeomorphic surface flows are solutions of integrodifferential equa ..."
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Cited by 7 (1 self)
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We describe a new class of surface flows, diffeomorphic surface flows, induced by restricting diffeomorphic flows of the ambient Euclidean space to a surface. Different from classical surface PDE flows such as mean curvature flow, diffeomorphic surface flows are solutions of integrodifferential equations in a group of diffeomorphisms. They have the potential advantage of being both topologyinvariant and singularity free, which can be useful in computational anatomy and computer graphics. We first derive the Euler–Lagrange equation of the elastic energy for general diffeomorphic surface flows, which can be regarded as a smoothed version of the corresponding classical surface flows. Then we focus on diffeomorphic mean curvature flow. We prove the shorttime existence and uniqueness of the flow, and study the longtime existence of the flow for surfaces of revolution. We present numerical experiments on synthetic and cortical surfaces from neuroimaging studies in schizophrenia and auditory disorders. Finally we discuss unresolved issues and potential applications.