Using fuzzy temporal logic for monitoring behavior-based mobile robots Khaled Ben Lamine and Froduald Kabanza Dept.´ de math-info Universite´ de Sherbrooke Sherbrooke, Qc J1K 2R1 Canada fbenlamin,[email protected] Abstract cal considerations make them easy to program and debug. However, their combination may cause unpredicted and undesirable results. For instance, there may be places This paper presents a model and an implementation of where the desire to reach a goal destination exactly bal- a runtime environment for specifying and monitoring ances the urge to turn away from obstacles, yielding a properties of behavior-based robot control systems. The stall (null move and turn action). In other situations, the proposed approach supports collecting events that are combination of behaviors may suggest a turn in a given recorded and examined at run-time. Temporal fuzzy situation and then a turn in the opposite direction in the logic is used as a formal language for specifying behav- next situation, so that the robot oscillates between these iors properties and new semantics are introduced to take two moves. Behaviors can also fail if the context they are into consideration environment unpredictability and un- designed for is no longer valid. For example, approach- certainty. These ideas are developed in the SAPHIRA mo- ing an object may require the object to remain visible for bile robot’s control environment, but they can also be ap- a certain period of time in order to locate it. plied to other behavior-based architectures. Experiments with two real-world robots are used to illustrate failure Detecting such failures is a non trivial problem, yet examples and the benefits of failure detection. a very important one in behavioral approaches. Without the ability to detect anomalies, a robot won’t be able to Keywords: mobile robots, behavior-based robotics, autonomously adjust its behaviors and overcome unpre- temporal reasoning, uncertainty in AI. dicted failures. Actually we need a feedback system that incorporates a “progress” criterion into behaviors, and fa- cilities to monitor this criterion. 1 Introduction Current failure detection techniques for mobile robots rely on heuristic monitoring of robot’s behaviors to detect potential failures [8, 12]. By “heuristic”, we Behavior-based approaches – see for example [1] and mean that there is no well-defined semantic behind the SAPHIRA [7]– have shown remarkable success in control- verification method. These methods rather rely on rules ling robots evolving in real world environment. Briefly, of thumbs and handle failures in an ad-hoc fashion. While in these approaches we remove the non essential assump- this effectively helps in detecting some failures, it is often tions that could prevent from an adaptation to unantici- difficult to analyze and understand the range of typical pated events. Also the decision process about the action failures covered by heuristic monitoring strategies. to take in a given situation is distributed across several simple processes. Typically, such processes, also called An another problem facing real world robot’s moni- behaviors, are implemented as direct mapping from local toring systems is uncertainty coming from the complexity sensors data to control actions. of the environment itself, from noisy sensors, or from im- precise actuators. According to [10] there is three ways to Designing concurrent reactive behaviors based on lo- 1 cope with uncertainty. over a runtime trace gathered during the system execu- tion. Another related approach was proposed by Felder and Morzenti [2]. 1. Get rid of it, by carefully engineering the robot and/or the environment; The approach we advocate here is in the same line of inquiry, but is more tailored for behavior-based robots. 2. Tolerate it, by writing robust programs able to op- First of all, we use a fuzzy temporal logic to account for erate under a wide range of situations, and recover fuzzy behaviors in the SAPHIRA mobile robot architec- from errors; or ture and noisy, uncertain information. On a more techni- 3. Reason about it, by using techniques for the repre- cal level, we use a state-based logic. Basic propositions sentation and the manipulation of uncertain informa- in our logic relates to states rather than to events. Accord- tion. ingly, our approach for checking conditions specified in that logic is different. We use an incremental method that can do the verification on the fly. This method is inspired In this paper we present a framework for monitoring from [6], where a similar approach was used to gener- behavior-based robot control systems. Along with the ate plans by verifying temporal logic goals over simulated third way above, we define a fuzzy temporal logic that traces of predicted execution sequences. is used to specify desirable system behaviors. We also provide a method for checking online the violation of Apart from validating the control system, the ulti- these behaviors. There is numerous advantages to our mate goal of monitoring, in the case of behavior-based approach including a declarative semantics for the moni- control, is to make the system more adaptive. In this set- toring knowledge and an independence of this knowledge ting, the monitoring system gives feedback to the robot’s from the implementation details of the control system. decision making processes which can then adapt their control strategies. However, the integration of monitoring In order to fix a context, these ideas are developed in and decision making is beyond the scope of the present the SAPHIRA [7] mobile robot’s control environment, but paper and hence will not be discussed. they can also be applied to other behavior-based architec- tures such as [3]. The remainder of this paper is organized as follows. 3 Temporal properties specification In the next section, we discuss related work. We then de- fine our new fuzzy temporal logic. This is followed with a description of the approach used to monitor and check Linear temporal logic formulas have been used success- the violation of behavioral properties expressed in that fully for specifying properties for the purpose of verifying logic. Finally, we present some empirical results before concurrent systems [4]. Formulas in such logics are inter- concluding. preted over models that are infinite sequences of states and temporal modalities are used to assert properties of these sequences. 2 Related Work In our case, we also use linear temporal logic formu- las, but with a fuzzy semantics. The truth of a proposition is a real value between 0 and 1. For example, the truth Monitoring is the process of recording event occurrences value of the proposition V isibleBall will be a real num- during program execution in order to gain runtime in- ber between 0 and 1 reflecting our incapacity to draw clear formation about the system states as well as information boundaries between thruthness and falseness of a propo- about the environment in which the system evolves [11, sition. This allows us to include fuzzy statements such as 5]. The work of Jahanian et al. is particularly interest- “slightly visible” or “completely visible”. On the other ing. Real-time conditions to be monitored and verified are hand, it is not wise to conclude that the ball is visible specified using a temporal logic called Real Time Logic from just one snapshot because of noise inputs. Rather, (RTL). This is a logic of events. Timing conditions are we should observe the ball on a whole period and con- specified in terms of starting time and ending time of rel- clude that it is visible based on snapshots taken during evant events. The evaluation of these conditions is made that period. To allow this, our propositions are evaluated 2 over segments of state sequences rather than over a single ² ¼(2 f; wi) = ¼(f; wi) ­ ¼(2 f; wi+1)) state. The size of the segment is determined empirically. ² ¼(f1 U f2; wi) = ¼(f2; wi) ©((¼(f1; wi) ­ ¼(f1 U f2; wi+1)) 3.1 Syntax where x ­ y is the minimum of x and y, that is, the fuzzy counter-part of and binary logic connective; x © y Our fuzzy temporal formulas are constructed from an enu- is the maximum of x and y, that is, the fuzzy counter-part merable collection of propositions; Boolean connectives of or binary logic connective. 1 ^ (and), : (not), and the temporal connectives ° (next ), 2 (always), 3 (eventually), and U ( until). The formulas The function ¼(p; wi) returns the truth value of a formation rules are: proposition p at a given state wi in a runtime trace. This truth value not only depends on the state wi, but on ² every fuzzy proposition p is a formula; in particular, a subsequence ending at wi. The length of the subse- we have built-in static propositions corresponding to quence and the interpretation mechanism are implicit in real values in [0; 1]; for the sake of clarity, the fuzzy the user-defined proposition evaluation functions. Thus, propositions corresponding to a real-value x is sim- for propositions, ¼ invokes user-defined proposition eval- ply noted x; hence 0:5 and 0:65 are fuzzy proposi- uation functions. For instance, assume p is the proposition tions; V isibleBall. We define a function that will evaluate p to a value that depends on how the vision system sees the ² if f1 and f2 are formulas, then so are :f1, f1 ^ f2, ball on each of the latest 4 states. Here, the number 4 is ° f1, 2 f1, 3 f1, and f1 U f2. set empirically. More formally, following Yager’s approach [13], we 3.2 Semantics use ordered weighted average (OWA) operators to evalu- ate the truth of propositions over histories. Formulas are interpreted over models of the form hw; ¼i, where: Definition 1 An OWA operator of dimension n is a map- ping F from [0; 1]n to [0; 1] associated with a wieghting vector W = [W1;W2;:::;Wn], such that ² w is an infinite sequence of worlds state w0; w1;:::; ² ¼ is a real-valued function that evaluates proposi- 1.