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Tight Continuous-Time Reachtubes for Lagrangian Reachability

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Tight Continuous-Time Reachtubes for Lagrangian Reachability CONFIDENTIAL. Limited circulation. For review only. Tight Continuous-Time Reachtubes for Lagrangian Reachability Jacek Cyranka1, Md. Ariful Islam2, Scott A. Smolka3, Sicun Gao1, Radu Grosu4 Abstract— We introduce continuous Lagrangian reachability nonconvex optimization, and δ-satisfiability. Computing an (CLRT), a new algorithm for the computation of a tight, as tight-as-possible continuous-reachtube overestimate helps conservative and continuous-time reachtube for the solution avoid false positives when checking if a set of unsafe states flows of a nonlinear, time-variant dynamical system. CLRT employs finite strain theory to determine the deformation of can be reached from a set of initial states. the solution set from time ti to time ti+1. We have developed The class of continuous dynamical systems in which simple explicit analytic formulas for the optimal metric for we are interested is described by nonlinear, time-variant, this deformation; this is superior to prior work, which used ordinary differential equations (ODEs): semi-definite programming. CLRT also uses infinitesimal strain 0 theory to derive an optimal time increment hi between ti and x (t) = f(t; x(t)); (1a) t , nonlinear optimization to minimally bloat (i.e., using a i+1 x(t ) = x ; minimal radius) the state set at time ti such that it includes all 0 0 (1b) the states of the solution flow in the interval [ti; ti+1]. We use n δ-satisfiability to ensure the correctness of the bloating. Our where x: R ! R . We assume f is a smooth function, which results on a series of benchmarks show that CLRT performs guarantees short-term existence of solutions. The class of favorably compared to state-of-the-art tools such as CAPD in time-variant systems includes the class of time-invariant sys- terms of the continuous reachtube volumes they compute. tems. Time-variant equations may contain additional terms, I. INTRODUCTION e.g., excitation variables and periodic forcing terms. Given an initial time t , set of initial states X ⊂ n, and Recent work introduced Lagrangian ReachTube algorithm 0 R time bound T > t , CLRT computes a conservative reachtube (LRT), a new approach for the reachability analysis of con- 0 of (1), that is, a sequence of time-stamped sets of states tinuous, nonlinear, dynamical systems [4]. LRT constructs a (R ; t ); : : :; (R ; t = T ) satisfying: discrete reachtube (or flowpipe) that tightly overestimates at 1 1 k k each discrete time point the set of states reached at that time Reach ((t0; X ) ; [ti−1; ti]) ⊂ Ri for i = 1; : : : ; k; point by a dynamical system. The main idea of LRT was to construct a ball-overestimate where Reach ((t0; X ) ; [ti−1; ti]) denotes the set of all reach- in a metric space that minimizes the Cauchy-Green stretching able states of ODE system (1) in the time interval [ti−1; ti]. factor at every discrete time instant. LRT was shown to The time steps are not necessarily uniformly spaced, and are compare favorably to other reachability analysis tools, such chosen using infinitesimal strain theory (IST). as CAPD [1], [17], [18] and Flow* [2], [3] in terms of the In contrast to LRT [4], which only computes the set discrete reachtube volumes they compute on a set of well- of states reachable at discrete and uniformly spaced time known benchmarks. steps ti, for i 2 f1; : : :; kg, CLRT computes a conserva- This paper proposes a continuous-time-reachtube exten- tive overestimate for the set of states reachable in non- sion of LRT, the motivation for which is two-fold. First, LRT, uniformly spaced continuous time intervals [ti−1; ti]. Hence, while being optimal in the discrete setting, is not sound in CLRT computes space-time cylinders overestimating the the continuous setting: it is not obvious how to find a ball continuous-time reachtube. tightly overestimating the dynamics between two discrete We also note that the LRT approach, as in prior work points. Second, LRT is not directly applicable to the analysis on reachability [8], [13], employed semi-definite program- of hybrid systems, as the dynamics of a hybrid system may ming to compute a tight deformation metric minimizing the change dramatically between two discrete time points due to Cauchy-Green stretching factor. We show that this approach a mode switch. is inferior and should be avoided, as it increases the total The main goal of our algorithm, which we call continuous running time of the algorithm significantly, and can result Lagrangian ReachTube algorithm (CLRT), is to efficiently in numerical instabilities (refer to the discussion in [4]). We construct an ellipsoidal continuous-reachtube overestimate instead derive a very simple analytic formula for the tightest that is tighter than those constructed by available state-of- deformation metric. Thus, there is no need to invoke an the-art tools such as CAPD. CLRT combines a number of optimization procedure to find a tight deformation metric, techniques to achieve its goal, including infinitesimal strain as the formula for the tightest one is now available. Also, theory, analytic formulas for the tightest deformation metric, we provide a very concise proof of this fact. Let Reach ((t0; X ) ; ti−1) ⊂ BMi−1 (xi−1; δi−1), where Jacek Cyranka and Md. Ariful Islam contributted equally to this work. BM (xi−1; δi−1) is the ball computed by LRT for time 1University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 92093, i−1 2Carnegie Mellon University, 3Stony Brook University, 4Vienna University ti−1. To construct a conservative continuous reachtube of Technology overestimate for the interval [ti−1; ti], we bloat the Preprint submitted to 57th IEEE Conference on Decision and Control. Received March 20, 2018. CONFIDENTIAL. Limited circulation.98 For review only. Chapter 3. Kinematics radius of this ball to ∆i−1 > δi−1, until it becomes a conservative over-estimate for the entire interval; i.e., Reach ((t0; X ) ; [ti−1; ti]) ⊂ BMi−1 (xi−1; ∆i−1). To ensure that the bloating is as tight as possible, we proceed as follows. First, we find the largest time ti such that the displacement gradient tensor of the solutions originating in BMi−1 (xi−1; δi−1) becomes sufficiently close to linear. Second, we assume that f in (1) is linear in the interval FIGURE 3.1. The reference and current configurations of a body. [ti−1; ti], and solve a convex optimization problem to obtain an estimate ∆^ i−1. Third, we compute a sound estimate of ∆i−1 by checking the δ-satisfiability over the reals of a will sometimes be given the same notation, 9( for example. logical formula with initial estimate ∆^ i−1. A prescribed reference configuration of a body B, occupying the region We implemented prototype CLRT in C++, and thoroughly 9( with boundary 89(, is defined against which other configurations of the investigated its performance on a set of benchmarks, includ- body are to be compared. The current configuration of the body, occupying the region 9(t with boundary 89(t, is the configuration of B at time t. It is ing those used in [4]. Our results show that compared to notFig.required 1: (Top)that Thethe body referenceever actually (or initial)be in the configurationreference configuration.R and CAPD, CLRT performs favorably in terms of the continuous- However, an initial configuration of B, at some time t = to say, is often the reachtube volume they compute. Also note that contrary to naturalthe currentchoice configurationof reference configurationRt of ain bodyelasticityB subjectedproblems. toIn defor-this case, LRT, CLRT is fully implemented in C++, which greatly themationaction [16].of some Aexternal materialagents, pointthePnaturehas referenceof which are coordinatesnot of concern improves the scalability of our algorithm. At present CLRT atXthis(P )juncture,in R, andcause currentthe material coordinatespoints ofxthe(P;body t) intoRmovet, ifuntil, one at time t, they are in new positions which define the current configuration of uses externally CAPD to compute gradients of the flow, uses the same system of coordinates. The displacement vec- the body.u P but we know how to achieve this independently, and we torA bodyshowsB can howbe thought the positionof as a ofset aof materialmaterial pointpoints, sochangesthat each from R to R . (Bottom) The larger ball B (x ; ∆ ) currently work on implementing/distributing CLRT as a material point P intB is an element of the set, which is indicatedM0 0by writing0 software library written in C++. PdepictedE B. When inB blue,is in its isreference a conservativeconfiguration, over-estimatelet the position forvector the of a material point P E B relative to some prescribed origin 0 be denoted reachtube continuous segment Reach ([t0; t0 + h]; X ), that The rest of the paper is organized as follows. Section II by X(P). This will be referred to as the reference position of P. Then, is, it is such that χ[t0;t0+h] (B (x ; δ )) ⊂ B (x ; ∆ ). provides background on infinitesimal strain theory, LRT, and the region 9( occupied byt0 B when it isMin0 its0 reference0 configurationM0 0 0can be convex optimization. Sections IV and V describe the bloating thought of as a set of reference positions: factor and optimization steps that we use. Section VI presents reference (undeformed) R are called Lagrangian, whereas 9( = {X(P) I P E B} , (3.1.1) the CLRT algorithm. Section VII contains our experimental the ones of the current (deformed) Rt are called Eulerian. results. Section VIII offers our concluding remarks and the Theset ofdisplacementall X(P) such uthatofP a materialis in R The pointmaterialP frompoints its positionof B occupy directions for future work. newin Rpositionsto its positionin the current in Rconfiguration.t is defined byWhen theB followingis in its current vectorcon figuration,equation:that is, at time t, let the position vector of a material point P E B relative to the same origin 0 be denoted by x(P, t) (see Figure 3.1).
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