A Selection Framework of Multiple Navigation Primitives Using Generalized Stochastic Petri Nets *Gunhee Kim, gWoojin Chung, and *Munsang Kim g
*Intelligent Robotics Research Center Korea Institute of Science and Technology 39-1 Hawolgok-dong, Sungbuk-ku, Seoul, 136-791, Korea
Department of Mechanical Engineering Korea University Anam-dong, Sungbuk-ku, Seoul, 136-713, Korea
{knir38, munsang}@kist.re.kr
[email protected]
Abstract - This paper proposes a selection framework of multiple navigation primitives for a service robot using Generalized Stochastic Petri Nets (GSPN’s). By adopting probabilistic approach, our framework helps the robot to select the most desirable navigation primitive in run time through the performance estimation according to environmental conditions. Moreover, after a mission, the robot evaluates prior navigation performance from accumulated data, and uses the results for the improvement of future operations. Modeling, analysis, and performance evaluation are conducted on firm mathematical foundation. Also, GSPN’s have several advantages over classic automata or direct use of Markov Process. We conducted simulations of the model derived from our experience of practical installations. The results showed that the framework is useful for primitive selection and performance analysis. Index Terms - Mobile robot navigation, behavior selection, a guide robot, Generalized Stochastic Petri Nets, Markov process
I. INTRODUCTION In these days, a guide robot is one of the most emerging fields of service robot application. At the KIST (Korea Institute of Science and Technology), a guide robot Jinny is under development for permanent installation at the National Science Museum of Korea as shown in Fig.1 [1]. The Jinny was developed by using a Petri net (PN) based control architecture, which was designed for multifunctional service robots in our previous work [2] [3]. Also, we implemented an integrated navigation system, which was successfully applied to real applications [4]. Through our experiences for permanent installation, we concluded it is indispensable that a guide agent uses several navigation primitives in order to operate robustly in a dynamic and unmodified environment. In other words, it is important for the robot adaptively to select its navigation primitives according to the conditions of environments. For
. Fig. 1 The guide robot Jinny at the National Science Museum of Korea.
example, in general cases, it is advisable that the robot uses a map-based navigation. However, if the robot is in a crowded or narrow region between exhibits, the sensorbased navigation, like a contour tracking, is much dependable since it is less affected by localization accuracy and uncertainties of environment. Navigation using multiple behaviors has been widely accepted in biomimetic study [9]. In general, navigation task is accomplished by the cooperation of several components such as a localizer and a path planner. Thus, the primitive selection problem is influenced by internal states of these navigation components. As the related components and navigation primitives increase, it becomes troublesome to manage the relationships between them. Also, these operations usually have nondeterministic properties. Therefore, the selection problem should be solved not by pre-defined logics but by performance estimation in run-time. A major scope of this paper is to propose a selection framework of multiple navigation primitives for a service robot. By using probabilistic approach, our framework selects the most desirable navigation primitive in run time through the performance estimation according to environmental conditions. Moreover, the robot evaluates prior navigation performance from accumulated data, and uses the results for the improvement of efficiency in future tasks. This approach is particularly useful for the most service robots which repeatedly operate in the workspace. In our approach, modeling, analysis, and performance evaluation are carried out based on the Generalized Stochastic Petri Nets (GSPN’s) [6][7][8]. GSPN’s are useful since the activities of the selection framework are eventdriven and can be divided into the possible discrete states. Owing to the formalism, our strategy has following three major advantages. First, our framework is developed on firm mathematical foundation. This advantage makes it possible to set up state equations and other mathematical models governing the behaviors of a system. Therefore, it allows not only to fully identify qualitative behavioral properties of designed logics, but also to carry out quantitative performance analysis. Second, our method supports modular and incremental designs of navigation framework since GSPN’s have powerful modeling ability. It can model concurrency, asynchronous events, logical priority relations, and structural interactions. Also, the transformation from GSPN model to the mathematical representation can be automated by several free or commercialized tools. Third, as a graphical tool, GSPN’s can represent both static and dynamic aspects of a system. They can visualize the control logic and make it easy to check the states of a system. Moreover, the dynamic changes of states can be explicitly
described since the GSPN’s exploit tokens, which clearly indicate state transitions with respect to event firings. Chapter II introduces the GSPN modeling of the proposed framework. Chapter III explains the performance estimation for navigation primitive selection from the GSPN model. Chapter IV shows the results of simulations. Finally, concluding remarks are given in Chapter V. II. GSPN MODELING OF THE PROPOSED FRAMEWORK A. Basics of GSPN’s The basics of GSPN’s are briefly reviewed in this section. The details of Petri net theory can be found in many references like [6], [7], and [8]. The concept of time is introduced to PN’s for performance evaluation and scheduling problems. Stochastic Petri Nets (SPN’s) are PN’s where each transition is associated with an exponentially distributed random variable which expresses the delay from the enabling to the firing of the transition. The firing rate λ is the sole parameter which characterizes the exponential distribution. The physical meaning of this rate is reflected by the fact that the mean of the exponential random variable is E(t) = 1/λ. The SPN models which allow for immediate transitions, i.e., with zero time delay, are called Generalized Stochastic Petri Nets (GSPN’s). The immediate transition is used to represent a logical control or an activity whose delay is negligible compared with those associated with timed transitions. Due to the memoryless property of the exponential distribution of firing delays, it has been shown that GSPN models, including extensions such as priority transitions, random switches, inhibitor arcs, and probabilistic arcs, can be converted into their equivalent Markov process (MP) representations [8]. MP’s can be obtained from the reachability graph of the PN’s. In the reachability graph, nodes indicate all reachable markings generated from the initial marking and each arc represents a transition firing. The construction algorithm can be found in [6] and [7]. The reachability graph is one of the powerful structural analysis methods for PN’s. For example, it formally identifies deadlocks, mutual exclusion, liveness, and boundedness. This structural analysis prevents developers’ design mistakes which can cause malfunctions of a robot. Through the analysis of the derived MP’s, it is possible to find various performance estimates of a system, which includes the probability of a particular condition, the expected value of the number of tokens, the mean number of firings per time unit, and the mean system throughput [6]. Our framework computes these performance estimates for the selection of navigation primitives. From the view of control logic design, there are benefits for choosing Petri net over classic automata used in other systems [10] or in UML [11]. First, automata are inadequate to describe the concurrence of activities in the system. On the other hand, tokens in PN’s can explicitly monitor the statuses of several components simultaneously. It means that automata only represent the status of the entire system at a certain instance, whereas the PN’s can describe statuses of components individually. Second, the qualitative or
quantitative analysis for systems modeled by PN’s is more powerful than ones by automata. The PN based approach is also advantageous over direct use of MP’s like Partially Observable Markov Decision Process (POMDP) models [12]. GSPN’s are isomorphic to embedded Markov Chains [8]. The main advantage of PN is that the number of places and transitions only increases slightly as the system complexity increases, while the number of states in the MP’s increases exponentially. Also, there is no need to manually enumerate all the possible states of MP’s since they are automatically generated from the GSPN model. This frees the modeler from having to painstakingly account for all possible states of the system. Thus, modular design and incremental changes to a PN models can be carried out simply by adding tokens, places, or transitions. On the other hand, minor changes in a MC model usually require redefining all the states in the model. Due to PN’s mathematical superiority, the overall processes mentioned above, for example, the derivation of the reachability graph and the equivalent MP model, the computations of steady-state probabilities, and simulations are automated and incorporated into free software packages such as Design/CPN [14] and TimeNet [15]. B. Problem Statement The Jinny’s navigation is a range sensor based scheme without modification of an environment [1]. Range sensors are used for mapping, path planning, and localization. Our localization method is a probabilistic map-matching scheme based on Monte Carlo localization [5]. The localizer considers not only position estimation but also state analysis, which helps a robot to take an appropriate action. We implemented four navigation primitives into the Jinny [1]. We learned by experience that this approach is more robust and reliable than the method using a single navigation strategy in uncertain and dynamic surroundings. In this paper, we mainly consider two types of navigation primitives, AutoMove and Contour tracking. The detailed description of these motions is summarized in Table 1. The AutoMove is our fundamental navigation strategy that uses modified Konolige’s gradient method [16]. The AutoMove computes optimal paths to the goal in real-time, avoiding local minima. The advantages of the AutoMove are generality and optimality. It is applicable in any situations, and it also makes it possible for a robot to move the desired position with shortest collision-free trajectory. On the other hand, the Contour tracking is a wall following technique which makes the robot move with a fixed distance to the wall. It generates velocity commands based on only raw laser sensing data in every sampling time. Thus, the Contour tracking has a better localization property than the AutoMove. The performance of the Contour tracking is less affected by localization accuracy since it is a sensor-based navigation. It works well even if the localizer fails. Also, the Contour tracking improves localization reliability, as known as a coastal navigation paradigm [13]. As the robot navigates along the wall, it is more likely that the robot collects accurate environmental information. Thus, it decreases the likelihood of getting lost.
TABLE I DESCRIPTION OF TWO NAVIGATION PRIMITIVES Type AutoMove Contour Tracking - Shortest path planning with - a (left, right, center) wallAlgorithm obstacle avoidance following technique using only laser scan data - Optimality (shortest path to - Reactive any points on the maps) - Rise localization reliability Merits - Generality (applicable in - Less affected by any situations) localization accuracy - Generally applicable, but - An area where there are Desirable the performance drops in a many static feature like walls environment narrow or crowded region. or exhibits
From this observation, we make one rule for the primitive selection. It is that if the localizer falls into the Warning state, Contour tracking is unconditionally selected. The main problem of this research is to select one navigation primitive out of two primitives according to internal and environmental conditions. The criterion of this selection problem is “which primitive leads the robot to a goal faster than the other with guaranteeing localization safety.” Obviously, the AutoMove is more efficient than the Contour tracking since the moving distance is shorter in most cases. However, if the area is so crowded that the localization Warning takes place too frequently, the AutoMove may not be advantageous. Since localization Warning causes motion changes to Contour tracking, using Contour tracking from the beginning to end may be more efficient that using AutoMove. Therefore, the primitive should be selected by considering both the traveling distance and estimated localization success rate. This problem is formulated from the basis of mathematical models using GSPN’s. For simplicity, we assume that the localizer is the only component which has an influence on the navigation performance. However, this simplified assumption doesn’t undermine the proposed approach since our method completely supports the modular and incremental design. Rather, more complicated the model, the more advantageous our approach. It is not cumbersome to add or revise the models of primitives or navigation components. Simply by adding relative tokens, places, or transitions to the GSPN’s, we can obtain a model without complexity explosion.
(a) GSPN model
C. Modeling Method The modeling method goes through following procedure. First, based on a given system description, navigation primitives and required components are identified. Primitives are designed as places, and the changes between them are modeled as transitions. Each component is represented as an independent GSPN’s model. Primitive models and component models are used as basic building blocks for the model of performance estimation. And then, they are linked by transition and arcs, according to the relationships between them. The primitive model and each component model have its own token in order to clearly express its internal status independently. The resultant GSPN model is shown in Fig.2. Table II describes the physical meaning of places and transitions of the model. The GSPN model has six places, seven timed transitions (drawn as white bars), and three immediate transitions (drawn as black bars). The initial marking is M0= (1 0 0 1 0 0), which is denoted as P0P3 in the reachability graph in Fig.2.(b) by specifying the places having tokens. The localizer has two internal states, Success and Warning. In the initial marking, a token is assigned to P3, i.e., it is assumed that the localizer initially knows its position. The Warning event t5 fires when the localizer fails in estimating robot’s accurate position for several steps. It doesn’t mean that the localizer completely loses its position but that it gives warning about several repetitions of matching failures. TABLE II DESCRIPTION OF THE PLACES AND THE TRANSITION Place Description P0 (P5) Navigation available (Completion) P1 (P2) Running AutoMove (Contour tracking) P3 (P4) Localization Success (Warning) Firing rate Transition Description t0 Start AutoMove (prob. p) t1 Start Contour tracking (prob. 1-p) Convert to Contour tracking (AutoMove) due t2 (t4) λ1 (λ2) to performance estimation Convert to Contour tracking due to t3 localization Warning t5 (t6) Localization Warning (Success) event fired λ3 (λ4) t7 (t8) AutoMove (Contour tracking) completed λ5 (λ6) t9 Initialization λ7
(b) Reachability graph Fig. 2 GSPN modeling for performance estimation
(c) Reduced embedded Markov chain
Two navigation primitives, AutoMove and Contour tracking, are modeled as P1, P2, respectively. Initially, the robot selects its motion by a random switch comprising the transitions t0 and t1 with corresponding probabilities p and 1-p, respectively. The transition between them takes place according to the change of localizer states. The immediate transition t3 means that the robot takes Contour tracking as soon as the localizer Warning event fires. The other transition between two primitives, t2 and t4, are modeled as timed transitions in order to express that the robot can change its current navigation primitive during the localizer Success state, if necessary. One of the most important modeling issues is how to set the firing rates Λ={λ1,…,λ7}. These parameters are based on the following sources of information shown in Table III. It will be more clearly explained in simulations to appear in Chapter IV. In order to perform the evaluation of GSPN designs, it is necessary to obtain an embedded Markov chain (EMC). Fig.2.(c) shows the EMC induced from the rechability graph of Fig.2.(b), which is derived from GSPN model of Fig.2.(a). Fig.2 clearly shows the advantages of PN’s over direct use of automata or MP’s. The PN model in Fig.2.(a) are more intuitive than the automata model in Fig.2(b), since PN model describes the statuses of primitives and the localizer independently, while the automata model represents only the status of the entire system. Also, GSPN’s don’t go through exponential state explosion and the MP model in Fig.2(c) can be automatically generated without manual enumeration of all the possible states. Rate
λ1 (λ2)
λ3 (λ4)
λ5 (λ6) λ7
TABLE III DESCRIPTION OF FIRING RATES Description λ1 and λ2 are firing rates of transitions between AutoMove and Contour tracking during the localization Success. These values are set differently according to which primitive is considered. When the performance estimation of Contour tracking is computed, λ1 is set to a large number and λ2 to a small number. Large λ1 expedites the transition from AutoMove to Contour tracking, and small λ2 restrains the reverse transition. Physically meaning is that when the localizer is in Success, Contour tracking is selected as possible. Reversely, when AutoMove is considered, λ1 is a small number, and λ2 is a large number. λ3 is the localization warning rate and λ4 is the recovery rate. Thus, λ3 means how frequently the state of the localizer is changed from Success to Warning on the average. In other words, its reciprocal indicates how long the localizer sojourns in Success state. Conversely, λ4 is the opposite meaning. These values are obtained from the robot’ s experience by iterating navigation in workspaces. λ5 (λ6) is the rates of task completion using AutoMove (Contour tracking). These are estimated from following equation; λ5(λ6) = 1/(estimated travel time to the goal) = (average speed)/(remained distance to the goal) Since λ7 has little effect for performance analysis, it is assigned to a large fixed number.
III. PERFORMANCE ESTIMATION USING GSPN ANALYSIS A. Steady-state probabilities In order to perform quantitative analysis, the steadystate probability πi of marking Mi (i=1,2,3,5,6) should be obtained from the EMC model in Fig.2.(c). The process to get the steady-state probability πi is explained in detail in
[6]. To summarize, it goes through following procedure. First, the transition probability matrix U is derived from the EMC in Fig.2.(c). U is shown in (1). Next, a vector of real numbers Y=(y1, y2, …, ys) is calculated, and it is the solution of the system of linear equations (2). Then, the steady-state probability πi of marking Mi (i=1,2,3,5,6), which means the proportion of time the marking process spends in Mi, is obtained from (3). mi is the mean sojourn time of the marking Mi, and E(Mi) is the set of transitions enabled in Mi. λ1 λ5 λ3 0 ( λ 1 + λ 3 + λ 5 ) ( λ 1 + λ 3 + λ 5 ) 0 (λ 1 + λ 3 + λ 5) λ2 λ6 λ3 0 (λ 2 + λ 3 + λ 6 ) 0 (1) λ λ 2 + λ 3 + λ 6) 2 + λ 3 + λ 6) ( ( λ7 ⋅ p λ 7 ⋅ (1 − p ) λ3 0 0 U = λ λ λ 3 + λ 7) 3 + λ 7) 3 + λ 7) ( ( ( λ4 λ7 0 0 0 (λ 4 + λ 7 ) (λ 4 + λ 7 ) λ4 λ6 0 0 0 (λ 4 + λ 6 ) (λ 4 + λ 6 ) s
∑y
Y = Y ⋅U
i =1
πi =
yi mi j =1
j
(2)
=1
where, mi =
s
∑y m
i
j
(3)
1
∑
tk ∈E ( M i )
λk
B. Framework for the selection of navigation primitives The criterion of the primitive selection issue is “which primitive leads the robot to a goal faster than the other with guaranteeing localization safety.” This selection problem can be solved by comparing the frequencies of firing a transition of t7 and t8 in Fig.2. The physical meanings of t7 and t8 are the completion of a navigation task using AutoMove and Contour tracking, respectively. Thus, the robot selects the primitive which has higher frequency than the other. In the analysis of GSPN’s, the frequency of firing a transition, i.e., the average number of transition firings in unit time, can be computed as the weighted sum of the transition firing rate shown in (4) [6]: fj =
∑
i:t j ∈E ( M i )
λ j (M ) ⋅ π i
(4)
i
where fj is the frequency of firing tj, and λj(Mi) is the firing rate of tj at Mi. Thus, in our system, the frequencies of navigation completion of each primitive can be computed like (5). Therefore, if fauto is larger than fcontour, AutoMove is selected. Otherwise, Contour tracking is chosen.
f auto = λ 5 ⋅ π 1 f cont = λ 6 ⋅ (π 2 + π 6)
(5)
The performance estimation, i.e., the comparison of (5), takes place when following occasions arise; 1) At the beginning of a navigation task: Starting a navigation task, the robot should plan which primitive are advantageous in a current situation.
2) At the localization state conversion from Warning to Success: When the localization warning occurs, the robot unconditionally executes contour tracking for improving localization accuracy. On its way, if the localizer recovers to Success state, the robot should decide whether it remains Contour tracking or changes to the AutoMove. B. Performance analysis In addition to the primitive selection, the GSPN model is also used to calculate some other performance indices of the system. Typical examples are the utilization of AutoMove and Contour tracking and the probabilities of localization Success and Warning during navigation. The steady-state probability of the event A defined through a condition which holds true for the markings Mi ∈ H is obtained as (6) [6].
P{ A} =
∑π
M i ∈H
i
(6)
Since P1 and P2 models the executions of AutoMove and Contour tracking, the utilization of each of them is the probability when P1 and P2 are marked, respectively. Thus, the unitization of each primitive is obtained from (7). Similarly, the probabilities of localization Success and Warning are derived from (8).
Uauto = P{M ( P1) = 1} = π 1 Ucont = P{M ( P 2) = 1} = π 2 + π 6 Psuccess = P{M ( P 3) = 1} = π 1 + π 2 + π 3
Pwarning = P{M ( P 4) = 1} = π 5 + π 6
(a) A typical environment in National Science Museum of Korea
(7) (8)
IV. SIMULATION RESULTS A. A simulation environment To show the feasibility of the proposed framework, simulations are conducted. In the National Science Museum of Korea, the most contours of environments are polygonal due to exhibits as presented in Fig.3.(a). Based on this fact, a simplified environment used for simulations is set like Fig.3.(b). It is assumed that the average speeds of both primitives are 0.2 m/s. Additionally, it is supposed that the distance between the center of a robot and a wall is about 1m during Contour tracking. As shown in Fig.3.(b), the moving distance of AutoMove is 14m, and the travel distance of Contour tracking is 28m. Thus, λ5 = 0.2/14 = 0.014286 and λ6 = 0.2/28 = 0.007143 (see Table III). B. Primitive selection model A quantitative model of the primitive selection is presented and analyzed. We mainly focus on the effects of localization performance on the primitive selection. Although the moving distance using contour tracking is twice as longer as that using AutoMove, sometimes it is safe to choose contour tracking when the localizer stays the Warning state for a long time. Therefore, the primitive selection varies with both the localization warning rate λ3 and the recovery rate λ4. We obtain the result of primitive selection with variable λ3 and fixed λ4 in Fig.4, and the results with fixed λ3 and variable λ4 in Fig.5.
(b) A simplified workspace for simulations Fig. 3 A simulation environment
Fig.4 shows the variation of fauto and fcont with respect to λ3, given the other λi’s are fixed. They are computed from (5). The firing rates Λ={0.001, 1000, λ3, 1/60, 1/70, 1/140, 1000} is used for the calculation of fauto, and Λ={1000, 0.001, λ3, 1/60, 1/70, 1/140, 1000} for fcont. As described in Table III, when fauto is considered, λ1 is assigned to a small number (λ1=0.001) in order to restrain the transition from AutoMove to Contour tracking during localization Success. In addition, λ2 is large (λ2=1000) in order to promote AutoMove. On the contrary, for the computation of fcont, λ1 and λ2 are assigned to 1000 and 0.001, respectively. λ7 is set to a large number (λ7=1000) since it has little practical effect on performance analysis. In both cases, the localization recovery rate λ4 is fixed to 1/60, which means the sojourn time of localization Warning is 60 sec. on the average. That is, the localizer Success is recovered after 60 sec. from falling into Warning state. As depicted in Fig. 4, fauto is larger than fcont when λ3 is smaller than 0.0117. That is, Contour tracking is selected if the localizer falls into Warning state more frequently than 85 (≈1/0.0117) sec. on the average. fcont remain constant to 0.0071428 since Contour tracking can be used regardless of state changes of localizer all the time. That is, if the robot moves by Contour tracking only from beginning to end, it takes about 140 (≈1/0.0071428)sec. This result is consistent with the fact that λ3 = 0.2/28 = 0.007143. Fig.5 is the result of changes of fauto and fcont with respect to λ4, given the other λi’s are fixed. The firing rates Λ={0.001, 1000, 1/90, λ4, 1/70, 1/140, 1000} is used for fauto, and Λ={1000, 0.001, 1/90, λ4, 1/70, 1/140, 1000} for fcont. In this example, the localization warning rate λ3 is fixed to 1/90, which means that the localization goes to Warning state every 90 sec. on the average. In this case, AutoMove is chosen if λ4 is larger than 0.0161, i.e., if the sojourn time of localization warning is smaller than about 62 sec.
V. CONCLUSION
Fig. 4 Variation of fauto and fcont according to λ3
Throughput
fauto fcont
This paper presents a selection framework of multiple navigation primitives for a service robot based on GSPN formalism. Modeling, analysis, and performance evaluation are conducted on firm mathematical foundation. Although, we consider simplified model which considers only two navigation primitives and one localization component, it doesn’t undermine the advantages of the proposed approach. As the model becomes more complex, the merit of our method increases. We conducted simulations of the model derived from experiences of practical installations. Currently, the proposed framework is under implementation to the Jinny to verify its feasibility in a real museum environment. REFERENCES
= 0.0161
Fig. 5 Variation of fauto and fcont according to λ4
Fig. 6 Variation of Psuccess and Pwarning according to λ3
C. Performance analysis model As the example of navigation performance analysis, we present the probabilities of localization Success and Warning derived from (8). The utilizations of AutoMove and Contour tracking are also calculated from (7) in the same way. Fig.6 shows the variation of Psuccess and Pwarning according to λ3, given the other λi’s are fixed. The firing rate Λ is the same with that of Fig.4. Naturally, Psuccess decreases with the increase of the localization warning rate λ3. At λ3=0.0117, both Psuccess and Pwarning equal 0.5, which implies that the sojourn times of localizer Success and Warning states are the same. This is related to the result of Fig.4. Intersection points in both graphs (Fig.4 and Fig.5) exist at λ3=0.0117. This fact implies that the AutoMove is selected when Psuccess is larger than Pwarning, i.e., when the sojourn time of localizer Success is larger than that of localizer Warning. It is caused by the fact that λ5 (= 0.2/14 = 0.014286) is exactly twice λ6 (= 0.2/28 = 0.007143).
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