Planning in Dynamic EnvironmentsThrough Temporal Logic Monitoring

Bardh Hoxha and Georgios FainekosArizona State University660 W 6th St, STE 203

Tempe, AZ 85281{bhoxha, fainekos}@asu.edu

Abstract

We present a framework that enables online planning forrobotic systems in dynamic environments. The PLANrmframework presented in this work utilizes the theory of ro-bustness and monitoring of Metric Temporal Logic (MTL)specifications to inspect and modify available plans to bothavoid obstacles and satisfy specifications in a dynamic envi-ronment. The use of MTL allows the practitioner to set com-plex event and timing based specifications that need to besatisfied in the execution of the plan. The monitoring algo-rithm inspects the possible paths in a bounded window andselects and adjusts a path to satisfy the specifications. In thispaper, we present initial results on the framework and an ex-tended summary of the algorithmic results. The approach isillustrated using a running example of a car-like model witha number of MTL specifications.

IntroductionThe problem of motion planning has been researched exten-sively in the past few decades. As a result, robotic systemshave been successfully deployed in a variety of fields, fromdisaster response and recovery to autonomous warehousesystems, autonomous driving and industrial manufacturing.In general, the operating environment is dynamic, wherenew obstacles may appear, or the mission may change.To support planning for such environments, we proposethe PLANrm framework that incorporates a Metric Tem-poral Logic (MTL) (Koymans 1990) monitoring algorithm(Dokhanchi, Hoxha, and Fainekos 2014) that inspects possi-ble plans in a bounded time window and selects a path thatsatisfies changing MTL specifications.

MTL is a formal language which enables practitioners toexpress complex specifications that take into account boththe occurrence and timing of events through time. It enablesthe definition of specifications such as ”if you pass throughregion A within a five second period, then visit region Bwhile avoiding region C” or ”always visit the refueling re-gion R before visiting region B”.

Temporal Logic has been previously utilized to synthe-size controllers for motion-planning of dynamical systems.In (Fainekos, Kress-Gazit, and Pappas 2005; Fainekos et al.

Copyright c© 2016, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.

2009; Kloetzer and Belta 2007), the authors convert Lin-ear Temporal Logic (LTL) specifications to Buchi automataand create abstraction models for the dynamical systems.Then they utilize automata-based methods to synthesizecontrollers. One of the main challenges with this approachare the increasing computational costs due to both the lengthof the specification and the level of abstraction of the dynam-ical system. Other varieties of Temporal Logic such as MTLhave been also utilized in planning. In (Kabanza 1995), theauthor uses MTL to create synchronized plans for teams ofagents. In (Karaman and Frazzoli 2008b), the authors con-sider the vehicle routing problem with MTL specifications.In (Ding, Lazar, and Belta 2014), for online planning, theauthors present an approach for receding horizon control forfinite deterministic systems subject to LTL formulas. In (Ra-man et al. 2014), the authors present a method for develop-ing model predictive controllers for systems that can be rep-resented as a mixed integer-linear program. The controlleris developed subject to Signal Temporal Logic (Maler andNickovic 2004) specifications.

In this work, we consider the online planning problem fordynamic environments where both the environment and mis-sion may change. Furthermore, MTL enables the specifica-tion of time bounds for temporal operators and thus the defi-nition of more elaborate specifications. Our planning frame-work utilizes the notion of robustness of MTL specifications(Fainekos and Pappas 2006; 2009), with a bounded-timemonitoring algorithm (Dokhanchi, Hoxha, and Fainekos2014), to enable online planning for dynamic environments.Finally, we illustrate our framework with Matlab Simulinkusing a car-like model from the Robotics toolbox (Corke2011) with various MTL specifications.

Summary of Contributions: We present an online plan-ning problem for robotic systems where the environmentis dynamically changing and mission compliance specifi-cations may be updated. To the authors knowledge, this isthe first attempt to include specification updates in an onlineplanning problem for dynamically changing environments.In the following, we provide an overview of our solutionand the planning framework. We illustrate the approach us-ing an example of a car-like model with a number of MTLspecifications.

Acknowledgments: This work has been partially sup-ported by award NSF CNS 1446730.

The Workshops of the Thirtieth AAAI Conference on Artificial Intelligence Planning for Hybrid Systems: Technical Report WS-16-12

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PreliminariesMetric Temporal LogicMetric Temporal Logic (Koymans 1990) enables reasoningover quantitative timing properties of boolean signals. In thispaper, we use the standard fragment of MTL with boundedfuture, but also we allow the use of past time operators. Inthe following, we assume we a constant sampling rate forthe robot and thus any specification on time reduces to aspecification on sampling points.

Definition 1 (MTL<+∞+pt Syntax) Let AP be the set of atomic

propositions and I be any non-empty interval of N, and Ibe any non-empty interval of N ∪ {+∞}. The set MTL<+∞

+ptformulas is inductively defined as

ϕ ::= > | p | ¬ϕ | ψ ∨ ϕ | ψUIϕ | ψSIϕ

where p ∈ AP and > stands for true.

The propositional operators conjunction (∧) and implica-tion (→) are defined the usual way. Other temporal operatorsare defined as follows.

Future operators:

� (Next) as �ϕ ≡ >U[1,1]ϕ

^ (Eventually) as ^Iϕ ≡ >UIϕ

� (Always) as �Iϕ ≡ ¬^I¬ϕ

The intuitive meaning of the ψU[a,b]ϕ operator at samplingtime i is a follows: ψ has to hold at least until ϕ becomes truewithin the time interval of [i + a, i + b] in the future.

Past operators:

� (Previous) as �ϕ ≡ >S[1,1]ϕ

� (Eventually in the past) as �Iϕ ≡ >S

Iϕ

� (Always in the past) as �Iϕ ≡ ¬�

I¬ϕ

The intuitive meaning of the ψS[a,b]ϕ operator at samplingtime i is as follows: since ϕ became true in the past interval[i − b, i − a], ψ must hold till now (current time i).

The atomic propositions in our case label subsets of theoutput space Y. Each atomic proposition is a shorthand foran arithmetic expression of the form p ≡ g(y) ≤ c, whereg : Y → R and c ∈ R. We define an observation map O :AP → P(Y) such that for each p ∈ AP the correspondingset is O(p) = {y | g(y) ≤ c} ⊆ Y.

Robustness of MTL<+∞+pt Formulas

The robustness estimate, formally presented in (Fainekosand Pappas 2009; Abbas et al. 2013), is used to indicate howwell a signal satisfies (or falsifies) MTL specifications. Inthe rest of the paper, we will use the terms time state se-quence and (execution or simulation) trace interchangeably.We define a timed state sequence as µ = µ0µ1µ2 . . . µm whereµi = (τi, si) represents a tuple of the time stamp and the vec-tor containing the values of the state variables at samplinginstance i. Given a timed state sequence µ of a system out-put trajectory and a MTL specification ϕ, the robustness es-timate returns a value ρ ∈ R ∪ {−∞,∞}. A positive (resp.negative) robustness means that the system is satisfied (resp.

falsified) for a particular system output trajectory. In the fol-lowing, we use the notation [[ϕ]] to denote the robustness es-timate with which µ satisfies the specification ϕ. In additionto the boolean semantics, the robustness measure ρ defineshow robustly µ satisfies or falsifies ϕ.

Formally, we define the robust semantics of MTL<+∞+pt as

follows. Using a metric d (Seda and Hitzler 2008), we candefine a distance function that captures how far away a pointx ∈ X is from a set S ⊆ X.

Definition 2 (Signed Distance) Let x ∈ X be a point, S ⊆X be a set and d be a metric. Then, we define the SignedDistance from x to S to be

Distd(x, S ) :={− inf{d(x, y) | y ∈ S } if x < Sinf{d(x, y) | y ∈ X\S } if x ∈ S

where inf is the infimum.

Using Definition 2, we provide the formal definition forthe robustness semantics.

Definition 3 (MTL<+∞+pt Robustness Semantics) Let s be a

trace s : N → X, and O be an observation map O : AP →P(X), then the robust semantics of any formula ϕ ∈MTL<+∞

+ptwith respect to s is recursively defined as:

[[>]](s, i) := +∞

[[p]](s, i) := Distd(s(i),O(p))~¬ϕ�(s, i) := −~ϕ�(s, i)

~ψ ∨ ϕ�(s, i) := ~ψ�(s, i) t ~ϕ�(s, i)

~ψU[l,u]ϕ�(s, i) :=⊔i+u

j=i+l

(~ϕ�(s, j) u

� j−1

k=i~ψ�(s, k)

)~ψS[l′ ,u′〉ϕ�(s, i) :=⊔i−l′

j=max{0,i−u′}

(~ϕ�(s, j) u

�i

k= j+1~ψ�(s, k)

)wheret stands for max,u stands for min, p ∈ AP, l, u, l′ ∈ Nand u′ ∈ N ∪ {∞}. Furthermore, the symbol 〉 in S[l′,u′〉 willbe ) when u′ = +∞ and ] when u′ , +∞.

Monitoring of MTL<+∞+pt Formulas

The monitoring algorithm, formally presented in(Dokhanchi, Hoxha, and Fainekos 2014), enables theonline robustness estimation of MTL<+∞

+pt formulas duringsystem execution or simulation. Given a MTL<+∞

+pt formulaϕ and a current time state sequence step i, the algorithmdetermines the number of time steps needed in the future(resp. past) needed to compute the robustness of ϕ. We referto the number of time steps as the window size. Next, adynamic algorithm is utilized to calculate the robustness es-timate. The future (resp. past) window size for specificationϕ is called the horizon (resp. history) and denoted by hrz(ϕ)(resp. hst(p)). The finite horizon and the history are definedrecursively as follows:

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offline/rarely calledglobal planner

inputs online/runtime

Finite HorizonLocal MTL Planner

GlobalPlanner (1)

EnvironmentMap

Global LTLSpecification

Local MTLSpecification

Local RRTPlanner (2)

MTLMonitoring

(3)

PathEvaluationMetric (4)

RoboticSystem

globalmap

state

vehicleinputs

localpaths

feasiblepaths

local sensing/update map

Figure 1: The planning process using the PLANrm framework.

Horizon:hrz(p) = 0hrz(¬ψ) = hrz(ψ)hrz(ψ OP ϕ) = max{hrz(ψ), hrz(ϕ)}hrz(ψU[l,u]ϕ) = max{hrz(ψ) + u − 1, hrz(ϕ) + u}hrz(ψS[l′,u′〉ϕ) = max{hrz(ψ), hrz(ϕ)}

History:hst(¬ψ) = hst(ψ)hst(ψU[l,u]ϕ) = max{hst(ψ), hst(ϕ)}hst(ψ OP ϕ) = max{hst(ψ), hst(ϕ)}hst(ψS[l′,u′〉ϕ) ={

max{hst(ψ) + u′ − 1, hst(ϕ) + u′} if u′ , +∞max{hst(ψ) + l′ − 1, hst(ϕ) + l′} if u′ = +∞

where p ∈ AP. Here, OP is any binary operator in proposi-tional logic, and ψ, ϕ are MTL<+∞

+pt formulas.

Problem FormulationOur high level goal is to provide an online planning frame-work for dynamic environments where part of the missionspecifications may be updated. We aim to do so by utilizingthe notion of robustness and monitoring of MTL specifica-tions. Formally, we aim to solve the following problem:

Problem 1 Given a model of a robotic system Σ and a pathp that satisfies an LTL formula ϕ, design an online con-troller which computes local paths which still satisfy ϕ and,moreover, they satisfy a set {ψi} of potentially dynamicallychanging local MTL mission requirements in a dynamicallychanging environment.

The planning framework is presented in Fig. 1. Given aglobal LTL specification ϕ, a global planner is utilized togenerate a path from the initial position to the goal posi-tion that satisfies ϕ. Then, the online, finite horizon localMTL planner generates plans that take into account poten-tially changing MTL specifications until the destination isreached. In certain conditions, the local planner may notfind a valid local path that satisfies the specification. In thosecases, an updated plan from the global planner is requested.

However, at this point, we must make sure that the futureobligations from the local MTL requirements are preserved.

In the following, we review motion planning over LTLspecifications and then we provide an overview of our on-line planning framework for solving Problem 1 through anexample.

Review of LTL PlanningMotion planning over LTL specifications has a long his-tory (Loizou and Kyriakopoulos 2004; Fainekos, Kress-Gazit, and Pappas 2005; Kloetzer and Belta 2007; Kara-man and Frazzoli 2008a; Bhatia, Kavraki, and Vardi 2010;Plaku and McMahon 2013). One of the earliest works in LTLplanning in robotics appears in (Lamine and Kabanza 2002).There, the authors use LTL runtime monitoring to check forviolations of LTL specifications while the robot executes itsplan. Also, the framework includes a simulation based plan-ner with LTL goals in order to choose a behavior that wouldbest satisfy the LTL goals.

A general approach to the high-level planning problemwith LTL is as follows. First, a discretized abstraction Mof the workspace using decomposition techniques is gener-ated (Fainekos, Kress-Gazit, and Pappas 2005). Then, theLTL specification is converted into a Buchi automaton B.Next, a search algorithm over the product automaton M × Bis utilized to find a feasible plan. In the LTL planning frame-work, partially known environments can be handled as well.For example, see the recent work in (Guo, Johansson, andDimarogonas 2013), where the authors present an approachto locally update the product automaton when the environ-ment changes. Closer in spirit to our work are the works by(Plaku, Kavraki, and Vardi 2009; Bhatia, Kavraki, and Vardi2010), where the authors present an approach that combineshigh-level planners with sampling-based low-level plannersto explore for feasible paths that satisfy LTL specifications.

In general, finding a tractable method for motion plan-ning over MTL requirements is still an open problem. Theapproach utilized for LTL formulas is not feasible for MTL.MTL formulas can be represented by timed automata (Alur

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Figure 2: Left: Environment map with the initial position for the robot (yellow) at xinit = (10, 10, π) and the goal position forthe robot (green) at xgoal = (90, 90, 0). Middle: RRT generated on the configuration space considering the vehicle dynamics.Right: Best path based on the RRT from xinit to xgoal.

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Figure 3: Vehicle progression from xinit = (10, 10, π) to xgoal = (90, 90, 0) at various times in the path traversal. The blue pathrepresents the shortest path obtained by the global planner. The green path represents the path traversed by the vehicle. The redregions represent areas where the vehicle should not go through. Also, every time the vehicle is in the yellow set then withinthe next 20s the vehicle should enter set blue set.

and Dill 1994) and currently algorithms exist only for frag-ments of MTL (Maler, Nickovic, and Pnueli 2006). There-fore, in this work, we utilize LTL for the global specification.

Planning in DynamicEnvironments and Specifications

In real world applications of robotic systems, both the en-vironment and the mission can change dynamically as thesystem is running. In the latter case, we assume that suchchanges are defined as MTL specifications. We aim to utilizehistory and horizon window sizes for MTL specifications todetermine the timespan for the local plan generator in orderto generate specification compliant plans. For the local plan-ner, we utilize MTL since it can capture specifications thatinclude timing intervals for temporal operators. In many ap-plications it is often necessary to capture timing properties.

We present an overview of the planing framework in

Fig. 1. First, a global planner (1) is utilized to generate pathsconsidering the environment and system dynamics. For ex-ample, using the framework presented in (Bhatia, Kavraki,and Vardi 2010). Then, the global plan is executed. If thereare any local MTL formulas that must be satisfied, then a lo-cal RRT planning framework is executed. The Local Planner(2) generates paths from the current system location, whileconsidering a dynamically changing environment and spec-ification. The set of paths generated are then classified asfeasible and infeasible through MTL Monitoring (3). Out ofthe feasible local plans, in the Path Evaluation Metric (4)stage, the framework choses the one that most closely cor-responds to the global established path. Here, we can utilizea weighted average of the robustness metric, in conjunctionwith a similarity measure, such as the euclidean distance,to chose one path from the set of feasible paths. It is im-portant to note that the local plan needs to stay close tothe global plan, either in terms of path distance, regions or

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some other high-level abstraction. In case no feasible pathsare found, the Local Planer (2) requests an update from theglobal planner (1) and the process repeats until the destina-tion is reached.

To illustrate the approach we utilize the car-like modelfrom the Robotics Toolbox (Corke 2011). A rear wheel isfixed to the body, and a front wheel which rotates to steerthe vehicle. We assume that the vehicle only moves forward.The configuration of the vehicle is represented by general-ized coordinates q = (x, y, θ) ∈ C ⊂ S E(2). The kinematicmodel of the vehicle is presented with the following equa-tions of motion:

x = v cos θy = v sin θ

θ =vL

tan γ

where (x, y, θ) represent the (x, y) position and θ orien-tation of the robot, v represents the forward speed, L repre-sents the length of the vehicle and γ is the steering angle. Inour study, the only input to the system is the steering anglewith a constant velocity. For more details on the model see(Corke 2011). The Matlab Simulink model is also providedin the toolbox. In Fig. 2 we present the environment alongwith the initial and goal positions. We also present the globalplan generated using an RRT algorithm.

In Figure 3, we illustrate vehicle progression in a dynam-ically changing environment and mission specification. Themission goal is to go from initial position xinit = (10, 10, π)to xgoal = (90, 90, 0). At T = 15s, a specification φ = �¬ais introduced where a is the set [30, 40] × [40, 60]. Thespecification states that set a should never be visited. AtT = 60s and T = 75s, the local planner generates a paththat avoids set a. At T = 90s, the specification is updated toφ = �(¬a ∧ ¬b), where b is the set [70, 80] × [40, 60]. AtT = 120s, the local planner fails to find a path towards xgoaland therefore requests an updated global plan. The globalplanner then presents a new route to xgoal. At T = 150s,the specification is updated to φ = �(¬a ∧ ¬b) ∧ �(c →^[0,20]d), where c is the set [40, 50]× [60, 70] and d is the set[30, 40] × [70, 80]. In addition to the previous specification,the updated specification states that every time the vehicleis in set c then within the next 20s the vehicle should enterset d. At T = 120s, the local planner provides a plan thatsatisfies the specification. It should be noted that it straysaway from the plan provided by the global planner in orderto satisfy the updated specification. At T = 240s, the vehiclereaches the destination.

In practice, when an updated local MTL specification isreceived, we should verify that it does not contradict theglobal LTL specification. In certain scenarios, we can utilizethe work presented in (Fainekos 2011; Dokhanchi, Hoxha,and Fainekos 2015) to detect and debug contradictory or in-consistent specifications.

The Local Planner (2) used in this framework generatesa set of trajectories by slightly modifying the steering angleat a number of control points within the specification hori-zon/history window. Namely, given a horizon/history win-dow size H, the steering angle is modified at ti equidistant

control points such that H =∑n

i=1 ti. This method is aimed toimprove the smoothness of the trajectories generated as op-posed to general RRT methods. This local planning methodis illustrated in Figure 4. To generate the RRT, we utilize aslightly modified version of the algorithm provided in theRobotics Toolbox (Corke 2011). The spread-based methodutilized is inspired by the works presented in (Von Hun-delshausen et al. 2008; Yu et al. 2012) which may be used togenerate smooth vehicle motion plans.

The set of trajectories generated from the local plannerare then tested over the specifications using the MTL Moni-toring (3) algorithm. We use S-TaLiRo (Annapureddy et al.2011; Hoxha et al. 2014) to conduct the robustness com-putation over MTL formulas. In Figure 4, we illustrate thisprocess with the specification ϕ = �(a→ ^[0,2]b) where a isthe set [10, 12] × [9, 11] and b is the set [13, 15] × [12, 14].

Once the set of feasible trajectories is provided, we uti-lize the Path Evaluation Metic (4) to choose the best path.Here, we can utilize the robustness metric, in conjunctionwith a trajectory similarity measure, to choose a path thatcorresponds closely to the global path provided by the globalplanner. We refer to this as the selection criterion. In our ex-ample, we utilize a weighted average of the robustness met-ric and the similarity measure. For a similarity measure, weutilize a Euclidean distance measure, i.e. for two trajectoriesT1 and T2, Ed(T1,T2) =

∑ni=1 ‖(T1(i),T2(i))‖ /n, where T (i)

represents the vehicle position at time i.Determining the best selection criterion is a challenging

problem, since non-optimal choices may lead to livelock sit-uations where the vehicle moves but does not make overallprogress towards the goal. If a livelock does happen, it isimportant to detect it and mitigate the issue by possibly re-questing a new plan from the global planner. This topic willbe covered in future work.

Conclusion and Future WorkIn this work, we presented an overview of the frameworkthat enables online planning for robotic systems in dynamicenvironments where the mission specification may changeduring system operation. We illustrated the framework on acar-like model from the Robotics Toolbox (Corke 2011).

As future work, we will consider the challenges presentedin the previous section. In certain situations, when no fea-sible paths are presented by the local planner, an updatedglobal path is requested by the global planner. However, inthose cases, the future obligations from the local MTL speci-fication should be maintained. Also, we will investigate can-didates for the selection criterion and conduct experimen-tal results to evaluate them. We will also investigate meth-ods for determining livestock conditions and how to resolvethem. Finally, we plan to utilize the PLANrm framework ona more complex hybrid system and evaluate the performanceand computational overhead during this process.

ReferencesAbbas, H.; Fainekos, G. E.; Sankaranarayanan, S.; Ivancic,F.; and Gupta, A. 2013. Probabilistic temporal logic falsi-

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Figure 4: Top: Left: Local planner used in the bicycle modelexample for generating smooth and evenly distributed tra-jectories. Right: An additional method for generating localplans through RRTs. Bottom: Trajectories that satisfy (resp.falsify) specification ϕ = �(a→ ^[0,2]b) represented in blue(resp. red) where a is the set [10, 12] × [9, 11] and b is theset [13, 15] × [12, 14].

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