In this paper we solve the problem of finding a trajectory that shows that a given hybrid dynamical system with deterministic evolution leaves a given set of states considered to be safe. The algorithm combines local with global search for achieving both efficiency and global convergence. In local search, it exploits derivatives for efficient computation. Unlike other methods for falsification of hybrid systems with deterministic evolution, we do not restrict our search to trajectories of a certain bounded length but search for error trajectories of arbitrary length.
Zuthsi and co-authors @cite_27 present a method for falsification of hybrid systems that also uses multiple shooting based local search. However, the method assumes a given upper bound on the length of the error trajectory the method searches for. Moreover, their local search method always follows a given sequence of modes and transitions. They propose to search for such a sequence using tools that compute abstractions of hybrid systems, or by random search. The form of the used trajectory segments is more restricted than in our method since trajectory segments always stay in one mode, and end in the guard leading to another mode.
@cite_27
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This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations. In this paper, we address the problem of local search for the falsification of hybrid automata with affine dynamics. Namely, if we are given a sequence of locations and a maximum simulation time, we return the trajectory that comes the closest to the unsafe set. In order to solve this problem, we formulate it as a differentiable optimization problem which we solve using Sequential Quadratic Programming. The purpose of developing such a local search method is to combine it with high level stochastic optimization algorithms in order to falsify hybrid systems with complex discrete dynamics and high dimensional continuous spaces. Experimental results indicate that indeed the local search procedure improves upon the results of pure stochastic optimization algorithms. We present an algorithm for falsifying safety properties of hybrid systems, i.e., for finding a trajectory to an unsafe state. The approach is to approximate how close a point is to being an initial point of an error trajectory using a real-valued quality function, and then to use numerical optimisation to search for an optimum of this function. The function is computed by running simulations, where information coming from abstractions computed by a verification algorithm is exploited to determine whether a simulation looks promising and should be continued or cancelled. This information becomes more reliable as the abstraction becomes more refined. We thus interleave falsification and verification attempts. We present a scalable reachability algorithm for hybrid systems with piecewise affine, non-deterministic dynamics. It combines polyhedra and support function representations of continuous sets to compute an over-approximation of the reachable states. The algorithm improves over previous work by using variable time steps to guarantee a given local error bound. In addition, we propose an improved approximation model, which drastically improves the accuracy of the algorithm. The algorithm is implemented as part of SpaceEx, a new verification platform for hybrid systems, available at spaceex.imag.fr. Experimental results of full fixed-point computations with hybrid systems with more than 100 variables illustrate the scalability of the approach. We present a Monte-Carlo optimization technique for finding inputs to a system that falsify a given Metric Temporal Logic (MTL) property. Our approach performs a random walk over the space of inputs guided by a robustness metric defined by the MTL property. Robustness can be used to guide our search for a falsifying trajectory by exploring trajectories with smaller robustness values. We show that the notion of robustness can be generalized to consider hybrid system trajectories. The resulting testing framework can be applied to non-linear hybrid systems with external inputs. We show through numerous experiments on complex systems that using our framework can help automatically falsify properties with more consistency as compared to other means such as uniform sampling. We consider the problem of 20 questions with noise for multiple players under the minimum entropy criterion in the setting of stochastic search, with application to target localization. Each player yields a noisy response to a binary query governed by a certain error probability. First, we propose a sequential policy for constructing questions that queries each player in sequence and refines the posterior of the target location. Second, we consider a joint policy that asks all players questions in parallel at each time instant and characterize the structure of the optimal policy for constructing the sequence of questions. This generalizes the single player probabilistic bisection method for stochastic search problems. Third, we prove an equivalence between the two schemes showing that, despite the fact that the sequential scheme has access to a more refined filtration, the joint scheme performs just as well on average. Fourth, we establish convergence rates of the mean-square error and derive error exponents. Finally, we obtain an extension to the case of unknown error probabilities. This framework provides a mathematical model for incorporating a human in the loop for active machine learning systems. The state of the art in articulated hand tracking has been greatly advanced by hybrid methods that fit a generative hand model to depth data, leveraging both temporally and discriminatively predicted starting poses. In this paradigm, the generative model is used to define an energy function and a local iterative optimization is performed from these starting poses in order to find a "good local minimum" (i.e. a local minimum close to the true pose). Performing this optimization quickly is key to exploring more starting poses, performing more iterations and, crucially, exploiting high frame rates that ensure that temporally predicted starting poses are in the basin of convergence of a good local minimum. At the same time, a detailed and accurate generative model tends to deepen the good local minima and widen their basins of convergence. Recent work, however, has largely had to trade-off such a detailed hand model with one that facilitates such rapid optimization. We present a new implicit model of hand geometry that mostly avoids this compromise and leverage it to build an ultra-fast hybrid hand tracking system. Specifically, we construct an articulated signed distance function that, for any pose, yields a closed form calculation of both the distance to the detailed surface geometry and the necessary derivatives to perform gradient based optimization. There is no need to introduce or update any explicit "correspondences" yielding a simple algorithm that maps well to parallel hardware such as GPUs. As a result, our system can run at extremely high frame rates (e.g. up to 1000fps). Furthermore, we demonstrate how to detect, segment and optimize for two strongly interacting hands, recovering complex interactions at extremely high framerates. In the absence of publicly available datasets of sufficiently high frame rate, we leverage a multiview capture system to create a new 180fps dataset of one and two hands interacting together or with objects. This paper describes a probabilistic multiple-hypothesis framework for tracking highly articulated objects. In this framework, the probability density of the tracker state is represented as a set of modes with piecewise Gaussians characterizing the neighborhood around these modes. The temporal evolution of the probability density is achieved through sampling from the prior distribution, followed by local optimization of the sample positions to obtain updated modes. This method of generating hypotheses from state-space search does not require the use of discrete features unlike classical multiple-hypothesis tracking. The parametric form of the model is suited for high dimensional state-spaces which cannot be efficiently modeled using non-parametric approaches. Results are shown for tracking Fred Astaire in a movie dance sequence. Learning and then recognizing a route, whether travelled during the day or at night, in clear or inclement weather, and in summer or winter is a challenging task for state of the art algorithms in computer vision and robotics. In this paper, we present a new approach to visual navigation under changing conditions dubbed SeqSLAM. Instead of calculating the single location most likely given a current image, our approach calculates the best candidate matching location within every local navigation sequence. Localization is then achieved by recognizing coherent sequences of these “local best matches”. This approach removes the need for global matching performance by the vision front-end - instead it must only pick the best match within any short sequence of images. The approach is applicable over environment changes that render traditional feature-based techniques ineffective. Using two car-mounted camera datasets we demonstrate the effectiveness of the algorithm and compare it to one of the most successful feature-based SLAM algorithms, FAB-MAP. The perceptual change in the datasets is extreme; repeated traverses through environments during the day and then in the middle of the night, at times separated by months or years and in opposite seasons, and in clear weather and extremely heavy rain. While the feature-based method fails, the sequence-based algorithm is able to match trajectory segments at 100 precision with recall rates of up to 60 . While the most accurate solution to off-line structure from motion (SFM) problems is undoubtedly to extract as much correspondence information as possible and perform global optimisation, sequential methods suitable for live video streams must approximate this to fit within fixed computational bounds. Two quite different approaches to real-time SFM — also called monocular SLAM (Simultaneous Localisation and Mapping) — have proven successful, but they sparsify the problem in different ways. Filtering methods marginalise out past poses and summarise the information gained over time with a probability distribution. Keyframe methods retain the optimisation approach of global bundle adjustment, but computationally must select only a small number of past frames to process. In this paper we perform the first rigorous analysis of the relative advantages of filtering and sparse optimisation for sequential monocular SLAM. A series of experiments in simulation as well using a real image SLAM system were performed by means of covariance propagation and Monte Carlo methods, and comparisons made using a combined cost accuracy measure. With some well-discussed reservations, we conclude that while filtering may have a niche in systems with low processing resources, in most modern applications keyframe optimisation gives the most accuracy per unit of computing time. In this paper we describe a new tool, SReach, which solves probabilistic bounded reachability problems for two classes of stochastic hybrid systems. The first one is (nonlinear) hybrid automata with parametric uncertainty. The second one is probabilistic hybrid automata with additional randomness for both transition probabilities and variable resets. Standard approaches to reachability problems for linear hybrid systems require numerical solutions for large optimization problems, and become infeasible for systems involving both nonlinear dynamics over the reals and stochasticity. Our approach encodes stochastic information by using random variables, and combines the randomized sampling, a @math -complete decision procedure, and statistical tests. SReach utilizes the @math -complete decision procedure to solve reachability problems in a sound manner, i.e., it always decides correctly if, for a given assignment to all random variables, the system actually reaches the unsafe region. The statistical tests adapted guarantee arbitrary small error bounds between probabilities estimated by SReach and real ones. Compared to standard simulation-based methods, our approach supports non-deterministic branching, increases the coverage of simulation, and avoids the zero-crossing problem. We demonstrate our method's feasibility by applying SReach to three representative biological models and to additional benchmarks for nonlinear hybrid systems with multiple probabilistic system parameters. We present a policy search method that uses iteratively refitted local linear models to optimize trajectory distributions for large, continuous problems. These trajectory distributions can be used within the framework of guided policy search to learn policies with an arbitrary parameterization. Our method fits time-varying linear dynamics models to speed up learning, but does not rely on learning a global model, which can be difficult when the dynamics are complex and discontinuous. We show that this hybrid approach requires many fewer samples than model-free methods, and can handle complex, nonsmooth dynamics that can pose a challenge for model-based techniques. We present experiments showing that our method can be used to learn complex neural network policies that successfully execute simulated robotic manipulation tasks in partially observed environments with numerous contact discontinuities and underactuation. In this paper, we present a novel, general, and efficient architecture for addressing computer vision problems that are approached from an 'Analysis by Synthesis' standpoint. Analysis by synthesis involves the minimization of reconstruction error, which is typically a non-convex function of the latent target variables. State-of-the-art methods adopt a hybrid scheme where discriminatively trained predictors like Random Forests or Convolutional Neural Networks are used to initialize local search algorithms. While these hybrid methods have been shown to produce promising results, they often get stuck in local optima. Our method goes beyond the conventional hybrid architecture by not only proposing multiple accurate initial solutions but by also defining a navigational structure over the solution space that can be used for extremely efficient gradient-free local search. We demonstrate the efficacy and generalizability of our approach on tasks as diverse as Hand Pose Estimation, RGB Camera Relocalization, and Image Retrieval. Modern automated systems evolve both continuously and discretely, and hence require estimation techniques that go well beyond the capability of a typical Kalman filter. Multiple model (MM) estimation schemes track these system evolutions by applying a bank of filters, one for each discrete system mode. Modern systems, however, are often composed of many interconnected components that exhibit rich behaviors, due to complex, system-wide interactions. Modeling these systems leads to complex stochastic hybrid models that capture the large number of operational and failure modes. This large number of modes makes a typical MM estimation approach infeasible for online estimation. This paper analyzes the shortcomings of MM estimation, and then introduces an alternative hybrid estimation scheme that can efficiently estimate complex systems with large number of modes. It utilizes search techniques from the toolkit of model-based reasoning in order to focus the estimation on the set of most likely modes, without missing symptoms that might be hidden amongst the system noise. In addition, we present a novel approach to hybrid estimation in the presence of unknown behavioral modes. This leads to an overall hybrid estimation scheme for complex systems that robustly copes with unforeseen situations in a degraded, but fail-safe manner. Recent work in sampling-based motion planning has yielded several different approaches for computing good quality paths in high degree of freedom systems: path shortcutting methods that attempt to shorten a single solution path by connecting non-consecutive configurations, a path hybridization technique that combines portions of two or more solutions to form a shorter path, and asymptotically optimal algorithms that converge to the shortest path over time. This paper presents an extensible meta-algorithm that incorporates a traditional sampling-based planning algorithm with offline path shortening techniques to form an anytime algorithm which exhibits competitive solution lengths to the best known methods and optimizers. A series of experiments involving rigid motion and complex manipulation are performed as well as a comparison with asymptotically optimal methods which show the efficacy of the proposed scheme, particularly in high-dimensional spaces. In the present work, fault detection in industrial automation processes is investigated. A fault detection method for observable process variables is extended for application cases, where the observations of process variables are noisy. The principle of this method consists in building a probability distribution model and evaluating the likelihood of observations under that model. The probability distribution model is based on a hybrid automaton which takes into account several system modes, i.e. phases with continuous system behaviour. Transitions between the modes are attributed to discrete control events such as on off signals. The discrete event system composed of system modes and transitions is modeled as finite state machine. Continuous process behaviour in the particular system modes is modeled with stochastic state space models, which incorporate neural networks. Fault detection is accomplished by evaluation of the underlying probability distribution model with a particle filter. In doing so both the hybrid system model and a linear observation model for noisy observations are taken into account. Experimental results show superior fault detection performance compared to the baseline method for observable process variables. The runtime of the proposed fault detection method has been significantly reduced by parallel implementation on a GPU. In machine learning and statistics, probabilistic inference involving multimodal distributions is quite difficult. This is especially true in high dimensional problems, where most existing algorithms cannot easily move from one mode to another. To address this issue, we propose a novel Bayesian inference approach based on Markov Chain Monte Carlo. Our method can effectively sample from multimodal distributions, especially when the dimension is high and the modes are isolated. To this end, it exploits and modifies the Riemannian geometric properties of the target distribution to create wormholes connecting modes in order to facilitate moving between them. Further, our proposed method uses the regeneration technique in order to adapt the algorithm by identifying new modes and updating the network of wormholes without affecting the stationary distribution. To find new modes, as opposed to rediscovering those previously identified, we employ a novel mode searching algorithm that explores a residual energy function obtained by subtracting an approximate Gaussian mixture density (based on previously discovered modes) from the target density function. We propose a constructive control design for stabilization of non-periodic trajectories of underactuated robots. An important example of such a system is an underactuated âdynamic walkingâ biped robot traversing rough or uneven terrain. The stabilization problem is inherently challenging due to the nonlinearity, open-loop instability, hybrid (impact) dynamics, and target motions which are not known in advance. The proposed technique is to compute a transverse linearization about the desired motion: a linear impulsive system which locally represents âtransversalâ dynamics about a target trajectory. This system is then exponentially stabilized using a modified receding-horizon control design, providing exponential orbital stability of the target trajectory of the original nonlinear system. The proposed method is experimentally verified using a compass-gait walker: a two-degree-of-freedom biped with hip actuation but pointed stilt-like feet. The technique is, however, very general and can be applied to a wide variety of hybrid nonlinear systems. The main objective of this paper is to develop an efficient method for learning and reproduction of complex trajectories for robot programming by demonstration. Encoding of the demonstrated trajectories is performed with hidden Markov model, and generation of a generalized trajectory is achieved by using the concept of key points. Identification of the key points is based on significant changes in position and velocity in the demonstrated trajectories. The resulting sequences of trajectory key points are temporally aligned using the multidimensional dynamic time warping algorithm, and a generalized trajectory is obtained by smoothing spline interpolation of the clustered key points. The principal advantage of our proposed approach is utilization of the trajectory key points from all demonstrations for generation of a generalized trajectory. In addition, variability of the key points' clusters across the demonstrated set is employed for assigning weighting coefficients, resulting in a generalization procedure which accounts for the relevance of reproduction of different parts of the trajectories. The approach is verified experimentally for trajectories with two different levels of complexity. We present a novel approach for multi-object tracking which considers object detection and spacetime trajectory estimation as a coupled optimization problem. Our approach is formulated in a minimum description length hypothesis selection framework, which allows our system to recover from mismatches and temporarily lost tracks. Building upon a state-of-the-art object detector, it performs multiview multicategory object recognition to detect cars and pedestrians in the input images. The 2D object detections are checked for their consistency with (automatically estimated) scene geometry and are converted to 3D observations which are accumulated in a world coordinate frame. A subsequent trajectory estimation module analyzes the resulting 3D observations to find physically plausible spacetime trajectories. Tracking is achieved by performing model selection after every frame. At each time instant, our approach searches for the globally optimal set of spacetime trajectories which provides the best explanation for the current image and for all evidence collected so far while satisfying the constraints that no two objects may occupy the same physical space nor explain the same image pixels at any point in time. Successful trajectory hypotheses are then fed back to guide object detection in future frames. The optimization procedure is kept efficient through incremental computation and conservative hypothesis pruning. We evaluate our approach on several challenging video sequences and demonstrate its performance on both a surveillance-type scenario and a scenario where the input videos are taken from inside a moving vehicle passing through crowded city areas.