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46
Reachability Analysis of Nonlinear Systems with Uncertain Parameters using Conservative Linearization
"... Abstract — Given an initial set of a nonlinear system with uncertain parameters and inputs, the set of states that can possibly be reached is computed. The approach is based on local linearizations of the nonlinear system, while linearization errors are considered by Lagrange remainders. These error ..."
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Cited by 33 (15 self)
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Abstract — Given an initial set of a nonlinear system with uncertain parameters and inputs, the set of states that can possibly be reached is computed. The approach is based on local linearizations of the nonlinear system, while linearization errors are considered by Lagrange remainders. These errors are added as uncertain inputs, such that the reachable set of the locally linearized system encloses the one of the original system. The linearization error is controlled by splitting of reachable sets. Reachable sets are represented by zonotopes, allowing an efficient computation in relatively highdimensional space. I.
Continuous Collision Detection for Articulated Models using Taylor Models and Temporal Culling
"... configurations of two moving mannequin models consisting of 15 links and 20K triangles each. (b) Motion interpolation from the initial and final configurations. (c) Finding the first time of contact between the two mannequins (contact features highlighted in pink). (d)(e)(f) Selfcollision between t ..."
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Cited by 24 (3 self)
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configurations of two moving mannequin models consisting of 15 links and 20K triangles each. (b) Motion interpolation from the initial and final configurations. (c) Finding the first time of contact between the two mannequins (contact features highlighted in pink). (d)(e)(f) Selfcollision between the left and right arms of a mannequin. Our algorithm can perform such continuous collision detection in a fraction of a millisecond. We present a fast continuous collision detection (CCD) algorithm for articulated models using Taylor models and temporal culling. Our algorithm is a generalization of conservative advancement (CA) from convex models [Mirtich 1996] to articulated models with nonconvex links. Given the initial and final configurations of a moving articulated model, our algorithm creates a continuous motion with constant translational and rotational velocities for each link, and checks for interferences between the articulated model under continuous motion and other models in the environment and for selfcollisions. If collisions occur, our algorithm reports the first time of contact (TOC) as well as collision witness features. We have implemented our CCD algorithm and applied it to several challenging scenarios including locomotion generation, articulatedbody dynamics and character motion planning. Our algorithm can perform CCDs including selfcollision detection for articulated models consisting of many links and tens of thousands of triangles in 1.22 ms on average running on a 3.6 GHz Pentium 4 PC. This is an improvement on the performance of prior algorithms of more than an order of magnitude.
On Taylor model based integration of ODEs
 SIAM Journal on Numerical Analysis
, 2007
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Interval Computations and IntervalRelated Statistical Techniques: Tools for Estimating Uncertainty of the Results of Data Processing and Indirect Measurements
"... In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on ..."
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Cited by 15 (9 self)
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In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on [−∆, ∆], and to use the corresponding statistical techniques. In some situations, however, this approach underestimates the error of indirect measurements. It is therefore desirable to directly process this interval uncertainty. Such “interval computations” methods have been developed since the 1950s. In this chapter, we provide a brief overview of related algorithms, results, and remaining open problems.
Affine Arithmetic: Concepts and Applications
, 2003
"... Affine arithmetic is a model for selfvalidated numerical computation that affine arithmetic keeps track of firstorder correlations between computed and input quantities. We explain the main concepts in affine arithmetic and it handles the dependency problem in standard interval arithmetic. We also ..."
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Cited by 14 (1 self)
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Affine arithmetic is a model for selfvalidated numerical computation that affine arithmetic keeps track of firstorder correlations between computed and input quantities. We explain the main concepts in affine arithmetic and it handles the dependency problem in standard interval arithmetic. We also describe some of its applications.
Metalevel Interval Arithmetic and Verifiable Constraint Solving
, 2001
"... CLIP is an implementation of CLP(Intervals) which has been designed to be verifiably correct in the sense that the answers it returns are mathematically correct solutions to the underlying arithmetic constraints. This fundamental design criteria affects many aspects of the implementation from the in ..."
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Cited by 10 (2 self)
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CLIP is an implementation of CLP(Intervals) which has been designed to be verifiably correct in the sense that the answers it returns are mathematically correct solutions to the underlying arithmetic constraints. This fundamental design criteria affects many aspects of the implementation from the input and output of decimal constants to the design of the interval arithmetic libraries and the constraint solving algorithms. In particular, to enhance verifiability, CLIP employs the simplest model of constraint solving in which constraints are decomposed into sets of primitive constraints which are then solved using a library of primitive constraint contractors. This approach results in a simple constraint solver whose correctness is relatively straightforward to verify, but the solver is only able to solve relatively simple constraints. In this paper, we present the syntax, semantics, and implementation of CLIP, and we show how to use metalevel techniques to enhance the power of the CLIP constraint solver while preserving the simple structure of the system. In particular, we demonstrate that several of the boxnarrowing algorithms from the Newton and Numerica systems can be easily implemented in CLIP. The principal advantages of this approach are (1) the resulting solvers are relatively easy to prove correct, (2) new solvers can be rapidly prototyped since the code is more concise and declarative than for imperative languages, and (3) contractors can be implemented directly from mathematical formulae without having to first prove results about interval arithmetic operators. Finally, the source code for the system is publicly available, which is a clear prerequisite for public, independent verifiability.
Combining symbolic simulation and interval arithmetic for the verification of AMS designs
 in: IEEE International Conference on Formal Methods in ComputerAided Design, 2007
"... Abstract—Analog and mixed signal (AMS) designs are important integrated circuits that are usually needed at the interface between the electronic system and the real world. Recently, several formal techniques have been introduced for AMS verification. In this paper, we propose a difference equation ..."
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Cited by 10 (2 self)
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Abstract—Analog and mixed signal (AMS) designs are important integrated circuits that are usually needed at the interface between the electronic system and the real world. Recently, several formal techniques have been introduced for AMS verification. In this paper, we propose a difference equations based bounded model checking approach for AMS systems. We define model checking using a combined system of difference equations for both the analog and digital parts, where the state space exploration algorithm is handled with Taylor approximations over interval domains. We illustrate our approach on the verification of several AMS designs including ∆Σ modulator and oscillator circuits. I.
Computing reachable sets for uncertain nonlinear monotone systems
, 2009
"... We address nonlinear reachability computation for uncertain monotone systems, those for which flows preserve a suitable partial orderings on initial conditions. In a previous work [1], we introduced a nonlinear hybridization approach to nonlinear continuous reachability computation. By analysing the ..."
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Cited by 10 (1 self)
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We address nonlinear reachability computation for uncertain monotone systems, those for which flows preserve a suitable partial orderings on initial conditions. In a previous work [1], we introduced a nonlinear hybridization approach to nonlinear continuous reachability computation. By analysing the signs of offdiagonal elements of system’s Jacobian matrix, a hybrid automaton can be obtained, which yields componentwise bounds for the reachable sets. One shortcoming of the method is induced by the need to use whole sets for addressing mode switching. In this paper, we improve this method and show that for the broad class of monotone dynamical systems, componentwise bounds can be obtained for the reachable set in a separate manner. As a consequence, mode switching no longer needs to use whole solution sets. We give examples which show the potentials of the new approach.
Progress in the Solving of a Circuit Design Problem
 JOURNAL OF GLOBAL OPTIMIZATION
, 2001
"... A new branchandprune algorithm for globally solving nonlinear systems is proposed. The pruning technique combines a multidimensional interval Newton method with the constraint satisfaction algorithm HC4 [1]. The main contributions of this paper are the finegrained interaction between both algori ..."
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Cited by 10 (2 self)
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A new branchandprune algorithm for globally solving nonlinear systems is proposed. The pruning technique combines a multidimensional interval Newton method with the constraint satisfaction algorithm HC4 [1]. The main contributions of this paper are the finegrained interaction between both algorithms which avoids some unnecessary computation,and the description of HC4 in terms of a chain rule for constraints’ projections. Our algorithm is experimentally compared with two global methods from Ratschek and Rokne [17] and from Puget and Van Hentenryck [16] on Ebers and Moll’ circuit design problem [6]. An interval enclosure of the solution with a precision of twelve significant digits is computed in four minutes, providing an improvement factor of five on the same machine.
SetBased Computation of Vehicle Behaviors for the Online Verification of Autonomous Vehicles
"... Abstract — We compute the set of all possible behaviors of an autonomous vehicle using reachability analysis. A reachable set is the set of states a system can possibly reach for a given set of initial states, disturbances, and sensor noise values. We consider autonomous vehicles which plan trajecto ..."
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Cited by 6 (5 self)
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Abstract — We compute the set of all possible behaviors of an autonomous vehicle using reachability analysis. A reachable set is the set of states a system can possibly reach for a given set of initial states, disturbances, and sensor noise values. We consider autonomous vehicles which plan trajectories for a certain lookahead horizon which are followed using feedback control. While a perfectly followed trajectory might not violate specified safety properties (e.g. lane departures or vehicle collisions), there might exist a violating deviation from the planned trajectory. Given the mathematical model of the controlled vehicle and bounds on uncertainty, our approach detects any possible violation. In addition, the approach provides results faster than real time such that maneuvers of vehicles can be checked before they are fully executed. I.