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Mission-based online generation of probabilistic monitoring models for mobile robot navigation using Petri nets
Robotics and Autonomous Systems (
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Mission-based online generation of probabilistic monitoring models for mobile robot navigation using Petri nets L. Zouaghi ∗ , A. Alexopoulos, A. Wagner, E. Badreddin Institute for Computer Engineering, Automation Laboratory, University of Heidelberg, Germany
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Article history: Available online xxxx Keywords: Hybrid monitoring Mobile robots Navigation Environment modelling
abstract This paper presents a generic hybrid monitoring approach, which allows the detection of inconsistencies in the navigation of autonomous mobile robots using online-generated models. A mission on the context of the navigation corresponds to an autonomous navigation from a start to an end mission point. The operator defines this mission by selecting a final goal point. Based on this selection the monitoring models for the current mission must be generated online. The originalities of this work are (i) the association of classic state estimation based on a particle filter with a special class of Petri net in order to deliver an estimation of the next robot state (position) as well as the environment state (graph nodes) and to use both pieces of information to distinguish between external noise influences, internal component faults and global behaviour inconsistency (ii) the integration of the geometrical and the logical environment representation into the monitor model (iii) the online generation of the corresponding monitoring model for the present mission trajectory while the system is running. The model takes simultaneously into account the uncertainty of the robot and of the environment through a unified modelling of both. To show the feasibility of the approach we apply it to an intelligent wheelchair (IWC) as a special type of autonomous mobile robot. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Monitoring, fault detection and diagnosis play an important role for autonomous and intelligent robotic systems due to the increasing demands on dependable and fault tolerant real-time applications. This fault tolerant behaviour in mobile robots refers to the possibility to autonomously detect and identify faults early before they result in catastrophic failures as well as to the capability to continue operating after a fault has occurred by switching to a safe state. However, fault detection and identification for robots is a complex problem because of the large space of possible faults (e.g. robot sensors, actuators, the uncertainty of the environment-models). Mobile robots are best modelled as hybrid systems since their behaviours result from the interaction between continuous and discrete dynamics. Several methods have been developed to deal with the monitoring of such systems. The most important approaches are these that combine the basic model of continuous systems, which are differential equations, with the basic model of discrete event systems, which
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Corresponding author. Tel.: +49 0 621 1813924; fax: +49 0 621 181 2739. E-mail addresses: [email protected] (L. Zouaghi), [email protected] (A. Alexopoulos), [email protected] (A. Wagner), [email protected] (E. Badreddin). 0921-8890/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.robot.2012.07.012
are automata or Petri nets. Using Petri nets to model hybrid systems offers advantages over finite automata when concurrency and complexity issues are of concern. Multiple comparisons from the literature between Petri nets and automata show that the former model is powerful and it has several advantages over the latter, not only because it is more general but also because it offers a better structure for modelling. The combination of such discrete models with numerical filters eases the modelling of the interaction of discrete and continuous dynamics and enables the extension of these approaches to other aspects such as the probabilistic framework to cover uncertainties. Some approaches take account of uncertainties of the robot and its environment and detect faults using numerical filters such as a particle filter, which estimates the most probable state of the robot and detect faults by comparison with the robot model. Also a mixture of Kalman filters [1,2] was used for tracking multiple hypotheses about the state of the system. In these approaches the estimate is then compared with the measured value to generate a residual, which declares the occurrence of a fault when a certain threshold value is exceeded. However fault detection based on residual interpretation is not enough, because exceeding a threshold does not necessarily always correspond to a fault. Hybrid monitoring based on Petri nets has been developed, for example, in [3]. However, this kind of behaviour prediction does not really consider the hybrid system’s nature consisting of a combination of discrete-event and continuous state evolution. All
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these restrictions make the approaches linking a discrete model to a numerical filter [4,5] more interesting. The approach, which seems to be the most interesting in the hybrid modelling and dealing with uncertainty is the particle Petri net [6,7]. It was used for the analysis of flight procedures and deals with situation monitoring. However this monitoring approach is not real-time critical and suffers from some restrictions that make it not quite suitable for the monitoring of real-time and safety-critical applications. In the classical particle Petri nets the numerical part is similar to differential Petri nets, where token colours are solutions to differential equations associated with numerical places. The symbolic part is a possibilistic Petri net [8], in which a token in a place is associated with a possibility value denoting the possibility that the token is really in this place. It estimates the system state using the particle filter for the numerical state and the possibilistic formalism for the symbolic state. However it doesn’t considerate the interaction between both parts of the model and in addition to this restriction particle Petri nets were used to describe concrete flight procedures which means that corresponding models must be developed offline. For mobile robots, which navigate arbitrarily in the environment or for autonomous systems that perform various tasks this technique is unsuitable since the model would have to describe all possible paths. To make the approach suitable for the monitoring of such systems under uncertain conditions several modifications and extensions were required. For this purpose we defined in [9] a new class of Petri nets called modified particle Petri nets (MPPN). They use the basics of the classical particle Petri nets but with an additional transition type called a hybrid transition which has specific firing rules. The main concept of the MPPN is the consideration of the influence of the numerical state when estimating and correcting the symbolic state. This dependency enables a symbolic estimation in a probabilistic framework unlike the classical particle Petri nets, which use the possibilistic formalism (by ranking the configurations according to a partial preorder) for the symbolic state without taking into account the interaction between both parts of the model. Since the user can select a different mission each time, a generic xmldescription of the model allows the online generation of the models. This is very useful in the mission monitoring because it allows an online adaptation of the monitor model to each change in the mission or to a re-planning step. The concept is shown on an autonomous wheelchair on the navigation level, where the position as well as the mission-execution has been monitored. 2. Problem formulation and solution approach The navigation process in the context of mobile robots [10] includes finding a safe path given a start and goal positions, guiding the robot safely along this path and updating the robot’s position from time to time. The problem we are interested in concerns the online monitoring of such navigation process by considering the uncertainty of the sensors and the environment models. Therefore, our formulation of the navigation monitoring problem requires:
• Estimating robot and environmental states, as they change over time, from sensor measurements that provide noisy, partial information about the states and considering the close coupling of robot–environment interaction. • Online generation of the corresponding monitoring models for the computed paths using a textual description of the Petri net [11], which eases the online generation of the corresponding monitoring model for the present mission trajectory while the system is running. • Fault monitoring and detection mechanism enabling the distinction between external noise influences, internal component faults and global behaviour inconsistency.
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The solution for the problem statement will be based on the following steps:
• The integration of the geometrical and topological paradigms in a hybrid environment model and solving the find-path problem using a graph-search algorithm. • The combination of a Monte Carlo method, such as particle filters with a discrete model, such as a Petri net in order to monitor the robot together with its operating environment under consideration of the uncertainty and to provide the probability of the robot being in each of the states given by the Petri net model. • Development of a hybrid monitoring and fault detection mechanism, which uses both the continuous information (the residual) and the discrete information (the Petri net marking) in order to detect inconsistencies in the navigation process. The paper is organized as follows: In Section 3, we introduce our probabilistic hybrid monitoring approach and we define a new form of Petri net called the modified particle Petri net (MPPN). In Section 4, we apply the method in the mobile robotic area to an intelligent wheelchair in order to monitor its position and path execution. Experimental results are presented in Section 5. Some concluding remarks are offered in the final section. 3. Probabilistic hybrid monitoring using modified particle Petri net 3.1. Modified particle Petri net (MPPN) We define the modified particle Petri net (MPPN) as a 9-tuple
⟨P , T , Pre, Post, X , F , γ , Ω , M0 ⟩, where:
• P: the set of places partitioned into numerical places PN and symbolic places Ps .
• T : the set of transitions (numerical, symbolic or hybrid). • Pre and Post are the pre-incidence and the post-incidence matrices, respectively, of dimension |P | × |T |. The incidence matrix of the net is defined as: ∀ p ∈ P and ∀ t ∈ T C (p, t ) = Post(p, t ) − Pr e(p, t ). • X ⊂ ℜn : is the state space of the numerical state vector. • F : difference equation system associated with numerical places and representing the continuous state evolution.
• γ (p): is the application that associates a configuration with each symbolic place p ∈ PS . • Ω : a set of conditions associated with the transitions. • M0 : is the initial marking of the net. A numerical place pn ∈ P N is associated with a differential equation representing the continuous evolution of the system state (e.g. position, temperature, . . .). Numerical places are marked by a set of particles πki = [qik , wki ] defined by their corresponding continuous valued state qk ∈ X and weights wk ∈ [0, 1] at time k and representing the uncertainty distribution over the value of the numerical state vector (Fig. 1). A state change from one numerical place to another occurs when the process state exceeds the boundary of its state space. After the state change, the process follows the solution of the differential equation that may be different from the previous one, until the next jump occurs. A symbolic place ps ∈ P S is marked by configurations (black (j) token) δk+1|k representing the symbolic state of the system itself resulting from external events or the symbolic state of another system, which is in interaction with it (a human, an environment, a technical system,. . .). The marking of the Petri net Mk = {Πk , ∆k } is the marking of the MPPN at time k consisting of the set of configurations ∆k in symbolic places and the set of particles Πk in numerical places.
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Fig. 1. Numerical (a) symbolic (b) and hybrid (c) transitions linking numerical places (blue circles) marked by particles and symbolic places (black circles) marked by configurations.
Fig. 3. Estimation of the continuous-valued state using the Particle Filter principle.
density function Pr = (qk , ONk , µOk ), which depends on the noisy observation (sensory data) ONk and a random noise µOk :
w ˜ ki +1 = Pr(qik+1 , ONk , µOk ),
(3)
where w ˜ ki +1 is the likelihood value of the i-th particle. The updated particle weights are then normalized by the sum over all particle weights:
wki +1 =
w ˜ ki +1 , J i w ˜ k +1
(4)
i=1
Fig. 2. Hybrid state estimation process of MPPN.
3.2. Concept of hybrid state estimation using MPPN In the MPPN the hybrid behaviour of the system is represented by the evolution of its hybrid state vector, which is composed of numerical and symbolic variables. The state estimation using MPPN consists of a prediction and a correction. The prediction step is based on the evolution of the Petri net marking and the particle values in order to determine all possible next states of the system considering both numerical and symbolic uncertainties. The correction step is carried out by updating this prediction according to a new observation made by the system (Fig. 2). The prediction step computes the next state estimate from the current state. It consists of the estimation of the discrete state given by the Petri net marking at time k + 1 and the estimation of the continuous state by estimating the particle values. The evolution of a given marking Mk at time k to a reachable marking Mk+1 at the instant k + 1 is obtained from: Mk+1 = Mk + C · Tfired
(1)
where C is the incidence matrix and Tfired is the firing vector, which describes the transitions that are fired at a firing instant. The continuous-valued state to be used for residual computation is then estimated by estimating the particle values as follows:
∀π
i k+1|k
ˆ k+1|k (p), ∈Π
qik+1
= f( )+ν qik
k+1 p
(2)
with p ∈ PN , f (·) is the difference equation associated with the place p and νk is the process noise. The correction step is a hybrid process, which compares the actual hybrid observation with the prediction and corrects the predicted state using the observation model of the system. All particles πki +1|k = [qik+1|k , wki +1|k ] are weighted by a probability
J is the number of particles and wki +1 the normalized weight (Fig. 3). Resampling is the last step in this estimation process. The aim of this step is to delete the particles with low weights and to keep all samples with a high probability to represent the real continuous system state. After resampling the initial number N of particles and the initial particle weight, N −1 is restored. The inputs for ˆ k+1|k and the outputs ˆ k+1|k ⊆ M resampling are the particles Π
ˆ k+1|k+1 representing the estimated are the resampled particles Π numerical state of the system at the time step k + 1. The result of the numerical correction step is a probability distribution PrDN over the numerical state vector computed as follows: ∀p ∈ P N ,
ˆ k+1|k (p) = PrDN Π
mk+1|k (p)
wki +1 ,
(5)
i=1
mk+1|k (p) is the number of particles at place p at time k+1 and wki +1 the corresponding weights. The most probable numerical state is the one with the highest probability PrDN . The symbolic marking of the Petri net is updated from the observation of the configuration. The first step in this process is to calculate the probability distribution PrDS over the symbolic states in dependence on the numerical distribution PrDN . This dependency of both parts is another feature that characterizes the MPPN. In order to consider the influence of the numerical part, we define a state graph GS : ∀pn ∈ P N , ∀ps ∈ P S , GS
S1,1
.
GS = .. S N ,1
··· .. . ···
S1,K
.. , .
Si,j =
1 0
(pin , pjs ) ∈ RG otherwise
(6)
SN ,K j
where RG is the reachability graph, (pin , ps ) is the hybrid state containing a numerical state, which corresponds to the i-th numerical place and a symbolic state corresponding to the j-th symbolic place of the Petri net and S indicates if that hybrid state is
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Fig. 5. Experimental environment for the wheelchair navigation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 4a. The intelligent wheelchair.
4.2. Kinematics model of the wheelchair The wheelchair’s pose at time k represents its state and is de⃗ (k) = (xpos (k), ypos (k), φpos (k)) correspondscribed by the vector X ing to two position coordinates xpos (k), ypos (k) and the orientation φpos (k) transformed from the wheelchair coordinate to the world coordinate system. The numerical part of the Petri net tracks the ⃗k+1 = f (X⃗k , uk ). The wheelchair state using the motion model X model defines the relationship between the new pose with respect to the current pose and to the input given by the odometry sensors on the wheelchair as follows:
⃗k+1 = f (X⃗k , uk ) + vk X xpos,k + ∆sk · cos(φpos,k ) ypos,k + ∆sk · sin(φpos,k ) + vk
=
reachable (S = 1) or not (S = 0). The probability distribution PrDS over the symbolic state vector is then calculated as follows: K
(8)
φpos,k + ∆φk where uk = (∆s, ∆φ) is the control input, ∆s and ∆φ are the dis-
Fig. 4b. And the control structure.
ˆ k+1|k (ps )) = PrDS (∆
ˆ k+1|k (pn )), GS (i, j) PrDN (Π
placement and the rotation of the wheelchair respectively during the same sample interval and vk is a noise assumed to be zero mean Gaussian with covariance Qk modelling the position uncertainty of the wheelchair. 4.3. Environment modelling and planning a safe trajectory
n =1
∀pn ∈ P N , ∀ps ∈ P S .
(7)
4. Hybrid monitoring within robot control architecture 4.1. Wheelchair system description The intelligent wheelchair (IWC) depicted in Fig. 4a is a conventional powered wheelchair (OttoBock GmbH, B600) with two motorized rear wheels and castors in front [12]. It is equipped with different sensors: ultrasonic sensors arranged around the wheelchair providing collision avoidance functionality, two incremental encoders attached on the two rear wheels in order to measure the wheelchair displacement, a gyro measuring the angular rate of the wheelchair orientation and an RFID sensor system detecting the RFID-tags, which are used as landmarks in the environment to correct odometry errors. The wheelchair’s behaviour is produced using a hierarchical control architecture based on the RNBC approach [13], which decomposes the system into behavioural subsystems in a nested manner (Fig. 4b). Each behavioural level in this structure possesses its own monitor, which communicates with the monitor of the next lower level and sends status to the monitor of the next higher level. In this paper we present only the monitor on the path planning, which monitors both the path and the position of the wheelchair (as a part of the next lower monitor ‘‘position control’’) to assure that it remains on the desired path within a tolerance.
The environment in which the wheelchair navigates is assumed to be known and contains known convex polygon-shaped obstacles. Unknown obstacles are treated by the collision-avoidance behaviour. The wheelchair has a rectangular shape and three degrees of freedom that define its possible motions. Planning a trajectory means to plan the motion of every point of the wheelchair shape according to its degrees of freedom, but that remains a hard problem: the wheelchair would need to make sure, at every point in its computed trajectory, that it does not intersect obstacles. So the solution is to treat the wheelchair as a moving point and to enlarge all object boundaries by the wheelchair size. The problem of computing collision-free path of a rectangle is now reduced to that of a point in the free navigation space (the white space in Fig. 5). Using the approach of the analogue mapping developed in [14] and generating a logical model directly from the geometrical model, the path-planning problem is solved by inflating the obstacle boundaries and by simultaneously shrinking the boundaries of the navigation space until the objects come into contact with each other or with the boundaries of the navigation space. We use the obtained logical graph as a topological map: the green points are the nodes, each node contains a landmark (RFID-Tag) and the lines represent the links between the nodes which are obstacle free paths from one landmark to the next. The blue point represents the start position and the red point is the goal position. The grey circle shows the current location of the robot, the red line is the desired trajectory and the blue line is the estimated trajectory. The placement of the landmarks depends on the topology of the environment and is used for constructing the symbolic Petri net of the monitor.
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Fig. 7. Petri net model for navigation monitoring. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
4.5. Consistency analysis and fault detection
Fig. 6. Textual description of the monitor.
4.4. Model generation Through a graphical interface the user can define a target to be reached by the robot. A path planner based on artificial intelligence uses Dijkstra’s algorithm [15] to produce a path of waypoints (green points on Fig. 5). Taking under account the location uncertainty given by the system specification [14] as maximal allowed deviations εx and εy of the position, the waypoints will be expanded automatically to waypoint-areas. The wheelchair does not have to go exactly through the i-th waypoint WPi = (xref ,i , yref ,i ), which represents the centre point of the i-th area but it suffices to enter its correct area and to begin to adjust its orientation to the next waypoint. The areas are defined as follows: Areai = {xref ,i − εx ≤ x ≤ xref ,i + εx ; yref ,i
− εy ≤ y ≤ yref ,i + εy }.
(9)
Based on the user’s goal selection, the monitor model is generated automatically. This textual description corresponds to the graphical model (for visualization) in Fig. 7. The robot navigation depends significantly on the environmental topology, that’s why it needs to be monitored in due consideration of the robot–environment interaction (Fig. 6). For this purpose the Petri net model is composed of two parts in interaction: (i) a symbolic Petri net (in blue), which contains the waypoint-areas (symbolic states) lying in the planned path (symbolic state vector) and indicates when navigating in an area and when going towards the next area (ii) a numerical Petri net (in black), which tracks the x- and y-positions (numerical state vector) of the wheelchair using a particle filter and estimates with respect to the robot position if an area is entered (the numerical state ‘‘reached’’) or not (‘‘not reached’’). On the hybrid transitions intervals are defined, indicating when the waypoint must be reached as well as conditions indicating the occurrence of events.
In contrast to the classical monitoring and diagnosis methods for hybrid systems, our proposed fault detection mechanism is based not only on the estimated value for the continuousvalued state, and thus only on residual computation but also on a consistency analysis of the symbolic state with respect to the numerical state. This joint consideration of both parts of the model (numerical and symbolic) allows the distinction between the causes of deviation in system behaviour. This deviation can be caused by a stochastic disturbance (external noise influences) in the normal operation or an error (internal component faults) or a global behaviour inconsistency (reaction to an unexpected situation). The fault detection mechanism works as follows: the particles (in the entire network, regardless of their position) are sorted according to their weighting and grouped in classes κi . To one class belong only particles, which belong to the same numerical place. A numerical place pn_ max ∈ PN containing the class with the highest weight is tested in combination with the most likely symbolic place qs_ max ∈ PS (resulting from the symbolic correction) for reachability. This check is necessary because the class with the highest weighting does not necessarily belong to the most likely numerical state (with the highest PDN ) and therefore does not represent necessarily in combination with the symbolic state, a reachable marking. If the result of this special reachability test is positive (i.e. (pmax , qmax ) ∈ GR ), where GR is the reachability graph of the Petri net, then the symbolic state is considered as consistent otherwise it is inconsistent. Considering such consistency analysis and the residual, we are able to attribute the discrepancy in predictions and observations to noise or to a fault or to global behaviour inconsistency. Such different fault cases are discussed in the next session. 5. Experimental results The estimation results of the y-position corresponding to the trajectory of Fig. 5, where the predicted system behaviour (blue trajectory) fits well with the desired one (red trajectory), are shown in Figs. 8 and 9. In Fig. 8 only the estimation results of the numerical part are depicted: the red trajectory represents the observations and the blue crosses are the particles, representing the estimate of the continuous-valued state (y-position). The probabilities of the numerical states ‘‘reached’’ and ‘‘not reached’’ are shown in the same plot (right axis): when the wheelchair enters an area (red histogram) the most probable discrete state is then ‘‘reached’’ (in blue) otherwise ‘‘not reached’’ (in green). The ID number of the driven area corresponds to its height on the right axis (1-3-4-7-8-9). As shown the estimation
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Fig. 10. Deviation of the wheelchair trajectory.
Fig. 8. Estimation results of the numerical Petri net. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 11. Detection of a global behaviour fault.
Fig. 9. Estimation results of the symbolic Petri net. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
smoothly fits the observations and corresponds to the correct discrete state. While the estimation of the numerical state uses the odometry observation and refers to the geometrical information of the environment, the estimation of the symbolic state (‘‘Area’’) uses the symbolic observation (RFID-sensor) and refers to the logical information of the environment. The result of such symbolic estimation is a marking of the symbolic Petri net, which estimates entering or leaving a certain area and when navigating between areas (red segments of crosses in Fig. 9). Inconsistencies in the navigation behaviour: In order to study the consistency of the wheelchair behaviour, three fault cases are considered. Fig. 10 shows the case when the wheelchair deviates from the planned trajectory and the last area no. 9 goes undetected. The results of the numerical as well as the symbolic estimation process are illustrated on the same plot (Fig. 11). We can notice that the consistency analysis of the symbolic state with respect to the numerical shows a transition from consistent to inconsistent state (from time step 778) and the residual exceeds the maximal allowed deviation indicating a fault. Such faults (residual + inconsistency) are classified as a global behaviour inconsistency and need feedback from another behavioural level (e.g. the collision-avoidance or the axis-control level) to know if this is an error (e.g. problem at the wheels) or a reaction to unexpected situation (e.g. deviation in order to avoid an unknown obstacle). Fig. 12 shows the case of the trajectory (nodes: 1-3-4-78-10-12-13) when the wheelchair behaves correctly according to
Fig. 12. Detection of component fault.
the numerical estimation (entering the area is correct estimated) and the continuous state shows that the wheelchair does not deviate from its intended behaviour. But the symbolic Petri net does not detect the landmark and shows an inconsistency with respect to the numerical part (from step 790 to 836), which indicates a component fault (no residual and inconsistency), namely a sensor failure. Fig. 13 shows the case of an external noise, which leads to a lack of symbolic observation from step 104 to 116 (e.g. landmark undetected because of floor material or sensor noise). This case looks similar to the sensor failure case but thanks to consistency analysis it was possible to distinguish between a component fault and a noise: The symbolic state stays consistent all the time.
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[10] [11]
[12]
[13]
[14] [15] [16]
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uncertainties, IEEE/SICE International Symposium on System Integration, Kyoto University, Kyoto, Japan December 20–22. E. Badreddin, Control and system design of wheeled mobile robots, Habilitation Script, ETH Zürich, 1997. L. Zouaghi, A. Wagner, E. Badreddin, Monitoring of hybrid systems using behavioural system specification, in: On the 18th IFAC World Congress, IFAC 2011, 2011. C. Bartolein, A. Wagner, M. Jipp, E. Badreddin, Dependable system design for assistance systems for electrically powered wheelchairs, in: International Workshop on the Design of Dependable Critical Systems, 2009. E. Badreddin, Recursive control structure for mobile robots, in: International Conf. on Intelligent Autonomous Systems 2 (IAS.2), Amsterdam, 1989, pp. 11–14. E. Badreddin, Associative memory implementation in path-planning for mobile robots, in: IEEE Conf. on Robotics and Automation, 1990. Edsger W. Dijkstra, A note on two problems in connexion with graphs, Numerische Mathematik. 1 (1959) S.269–271. A. Alexopoulos, L. Zouaghi, E. Badreddin, Associative memory for modified petri-net based monitoring of mobile robot navigation, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, Vilamoura, Algarve, Portugal October 7–12, 2012, (submitted for publication).
Fig. 13. Detection of a noise.
6. Conclusion The paper presents a method for hybrid monitoring enabling the on online generation of the models, which estimate and predict the states of mobile robots by coupling continuous and discrete information in a unified framework. It was proven that the monitor is robust against disturbances in the landmark detection process and that it enables the distinction between noise, component faults and global behaviour inconsistency. The disadvantages are that if an expected landmark stays undetected during the navigation process one can only assume that a fault in the landmark detection system has occurred without taking for example modelling and/or software errors into account. For this reason an error alarm causes the fault-handling mechanism to stop the navigation process immediately because everything else would be careless for the dependability of the AMR. To overcome this problem a long-term memory for the successfully traversed paths on the topological map would be suitable. This long-term memory is realized with associative memory and it is presented in [16]. References [1] S. Roumeliotis, G. Sukhatme, G. Bekey, Fault detection and identification in a mobile robot using multiple-model estimation, in: IEEE Int. Conf. on Robotics and Automation,, Leuven, Belgium, May, 1998 2223–2228. [2] U. Lerner, R. Parr, D. Koller, G. Biswas, Bayesian fault detection and diagnosis in dynamic systems, in: Proc. 17th Nat. Conf. on Artificial Intelligence, 2000. [3] X.D. Koutsoukos, K.X. He, M.D. Lemmon, P.J. Antsaklis, Timed Petri nets in hybrid systems: Stability and supervisory control, Journal of Discrete Event Dynamic Systems: Theory and Applications 8 (2) (1998) 137–173. [4] S. McIlraith, Diagnosing hybrid systems: a bayesian model selection problem, in: Proceedings of the 11th International Workshop on Principles of Diagnosis, DX’2000, 2000. [5] U. Lerner, R. Parr, D. Koller, G. Biswas, Bayesian fault detection and diagnosis in dynamic systems, in: Proc. 17th Nat. Conf. on Artificial Intelligence, 2000. [6] C. Lesire, C. Tessier, Particle Petri nets for aircraft procedure monitoring under uncertainty, in: ATPN, Miami, FL, USA, 2005. [7] F. Dehais, A. Goudou, C. Lesire, C. Tessier, Towards an anticipatory agent to help pilots. in: AAAI Fall Symposium. From Reactive to Anticipatory Cognitive Embodied Systems. Arlington, USA, 2005. [8] J. Cardoso, R. Valette, D. Dubois, Possibilistic Petri nets’, IEEE Transactions on Systems, Man, and Cybernetics, Part B (1999) 573–582. [9] L. Zouaghi, A. Alexopoulos, A. Wagner, E. Badreddin, 2011 Modified particle Petri nets hybrid dynamical systems monitoring under environmental
Leila Zouaghi was born in Cologne, Germany. She received her Diploma in electrical engineering from the Technical University Kaiserslautern, Germany, in 2005. She is currently working toward the Ph.D. degree in computer engineering at the Institute of Computer Engineering, University of Mannheim (now Heidelberg University). Her research interests include hybrid monitoring, fault detection, mobile robots, Petri nets and nonlinear state estimation.
Alexander Alexopoulos was born in Heidelberg, Germany. He received his Diploma in computer science from the University of Mannheim, Germany, in 2011. He is currently working toward the Ph.D. degree in computer engineering at the Institute of Computer Engineering, University of Mannheim (now Heidelberg University). His research interests include game theory, fuzzy control and neuronal networks.
Achim Wagner obtained his Diploma in Information Technology from the Saarland University, Saarbrücken, Germany, in 1996. Also at the Saarland University he received his doctoral degree in the area of electrical materials science in 2002. Afterwards he worked as an associate researcher in medical robotics projects at the Laboratory for Biomechanics and Experimental Orthopedics, OUZ, Heidelberg University and at the Automation Laboratory, Institute of Computer Engineering, University of Mannheim (now Heidelberg University). Since 2004 he has led the robotic research group. His current scientific interests are dependable autonomous and assisting robotic systems, fault-tolerant system design and control, and human–machine interaction. Essam Badreddin is currently the head of the Automation laboratory at the Heidelberg University, Germany. He earned his Swiss Diploma in Electrical Engineering from the Swiss Federal Institute of Technology, Zurich, Switzerland (ETHZ), the Doctor of Technical Sciences in Control Theory (ETHZ) and Habilitation in Mechanical Engineering (ETHZ). There, he founded and led the first robotics research group. Prof. Badreddin served as Monbushu Professor in Japan before he joining the University of Mannheim and later Heidelberg University. He served in industry at Contraves-Zurich as an R&D System Engineer in the air-defence sector, where he also holds several international patents (Silver Dollar awarded).
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Robotics and Autonomous Systems journal homepage: www.elsevier.com/locate/robot
Mission-based online generation of probabilistic monitoring models for mobile robot navigation using Petri nets L. Zouaghi ∗ , A. Alexopoulos, A. Wagner, E. Badreddin Institute for Computer Engineering, Automation Laboratory, University of Heidelberg, Germany
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Article history: Available online xxxx Keywords: Hybrid monitoring Mobile robots Navigation Environment modelling
abstract This paper presents a generic hybrid monitoring approach, which allows the detection of inconsistencies in the navigation of autonomous mobile robots using online-generated models. A mission on the context of the navigation corresponds to an autonomous navigation from a start to an end mission point. The operator defines this mission by selecting a final goal point. Based on this selection the monitoring models for the current mission must be generated online. The originalities of this work are (i) the association of classic state estimation based on a particle filter with a special class of Petri net in order to deliver an estimation of the next robot state (position) as well as the environment state (graph nodes) and to use both pieces of information to distinguish between external noise influences, internal component faults and global behaviour inconsistency (ii) the integration of the geometrical and the logical environment representation into the monitor model (iii) the online generation of the corresponding monitoring model for the present mission trajectory while the system is running. The model takes simultaneously into account the uncertainty of the robot and of the environment through a unified modelling of both. To show the feasibility of the approach we apply it to an intelligent wheelchair (IWC) as a special type of autonomous mobile robot. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Monitoring, fault detection and diagnosis play an important role for autonomous and intelligent robotic systems due to the increasing demands on dependable and fault tolerant real-time applications. This fault tolerant behaviour in mobile robots refers to the possibility to autonomously detect and identify faults early before they result in catastrophic failures as well as to the capability to continue operating after a fault has occurred by switching to a safe state. However, fault detection and identification for robots is a complex problem because of the large space of possible faults (e.g. robot sensors, actuators, the uncertainty of the environment-models). Mobile robots are best modelled as hybrid systems since their behaviours result from the interaction between continuous and discrete dynamics. Several methods have been developed to deal with the monitoring of such systems. The most important approaches are these that combine the basic model of continuous systems, which are differential equations, with the basic model of discrete event systems, which
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Corresponding author. Tel.: +49 0 621 1813924; fax: +49 0 621 181 2739. E-mail addresses: [email protected] (L. Zouaghi), [email protected] (A. Alexopoulos), [email protected] (A. Wagner), [email protected] (E. Badreddin). 0921-8890/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.robot.2012.07.012
are automata or Petri nets. Using Petri nets to model hybrid systems offers advantages over finite automata when concurrency and complexity issues are of concern. Multiple comparisons from the literature between Petri nets and automata show that the former model is powerful and it has several advantages over the latter, not only because it is more general but also because it offers a better structure for modelling. The combination of such discrete models with numerical filters eases the modelling of the interaction of discrete and continuous dynamics and enables the extension of these approaches to other aspects such as the probabilistic framework to cover uncertainties. Some approaches take account of uncertainties of the robot and its environment and detect faults using numerical filters such as a particle filter, which estimates the most probable state of the robot and detect faults by comparison with the robot model. Also a mixture of Kalman filters [1,2] was used for tracking multiple hypotheses about the state of the system. In these approaches the estimate is then compared with the measured value to generate a residual, which declares the occurrence of a fault when a certain threshold value is exceeded. However fault detection based on residual interpretation is not enough, because exceeding a threshold does not necessarily always correspond to a fault. Hybrid monitoring based on Petri nets has been developed, for example, in [3]. However, this kind of behaviour prediction does not really consider the hybrid system’s nature consisting of a combination of discrete-event and continuous state evolution. All
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these restrictions make the approaches linking a discrete model to a numerical filter [4,5] more interesting. The approach, which seems to be the most interesting in the hybrid modelling and dealing with uncertainty is the particle Petri net [6,7]. It was used for the analysis of flight procedures and deals with situation monitoring. However this monitoring approach is not real-time critical and suffers from some restrictions that make it not quite suitable for the monitoring of real-time and safety-critical applications. In the classical particle Petri nets the numerical part is similar to differential Petri nets, where token colours are solutions to differential equations associated with numerical places. The symbolic part is a possibilistic Petri net [8], in which a token in a place is associated with a possibility value denoting the possibility that the token is really in this place. It estimates the system state using the particle filter for the numerical state and the possibilistic formalism for the symbolic state. However it doesn’t considerate the interaction between both parts of the model and in addition to this restriction particle Petri nets were used to describe concrete flight procedures which means that corresponding models must be developed offline. For mobile robots, which navigate arbitrarily in the environment or for autonomous systems that perform various tasks this technique is unsuitable since the model would have to describe all possible paths. To make the approach suitable for the monitoring of such systems under uncertain conditions several modifications and extensions were required. For this purpose we defined in [9] a new class of Petri nets called modified particle Petri nets (MPPN). They use the basics of the classical particle Petri nets but with an additional transition type called a hybrid transition which has specific firing rules. The main concept of the MPPN is the consideration of the influence of the numerical state when estimating and correcting the symbolic state. This dependency enables a symbolic estimation in a probabilistic framework unlike the classical particle Petri nets, which use the possibilistic formalism (by ranking the configurations according to a partial preorder) for the symbolic state without taking into account the interaction between both parts of the model. Since the user can select a different mission each time, a generic xmldescription of the model allows the online generation of the models. This is very useful in the mission monitoring because it allows an online adaptation of the monitor model to each change in the mission or to a re-planning step. The concept is shown on an autonomous wheelchair on the navigation level, where the position as well as the mission-execution has been monitored. 2. Problem formulation and solution approach The navigation process in the context of mobile robots [10] includes finding a safe path given a start and goal positions, guiding the robot safely along this path and updating the robot’s position from time to time. The problem we are interested in concerns the online monitoring of such navigation process by considering the uncertainty of the sensors and the environment models. Therefore, our formulation of the navigation monitoring problem requires:
• Estimating robot and environmental states, as they change over time, from sensor measurements that provide noisy, partial information about the states and considering the close coupling of robot–environment interaction. • Online generation of the corresponding monitoring models for the computed paths using a textual description of the Petri net [11], which eases the online generation of the corresponding monitoring model for the present mission trajectory while the system is running. • Fault monitoring and detection mechanism enabling the distinction between external noise influences, internal component faults and global behaviour inconsistency.
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The solution for the problem statement will be based on the following steps:
• The integration of the geometrical and topological paradigms in a hybrid environment model and solving the find-path problem using a graph-search algorithm. • The combination of a Monte Carlo method, such as particle filters with a discrete model, such as a Petri net in order to monitor the robot together with its operating environment under consideration of the uncertainty and to provide the probability of the robot being in each of the states given by the Petri net model. • Development of a hybrid monitoring and fault detection mechanism, which uses both the continuous information (the residual) and the discrete information (the Petri net marking) in order to detect inconsistencies in the navigation process. The paper is organized as follows: In Section 3, we introduce our probabilistic hybrid monitoring approach and we define a new form of Petri net called the modified particle Petri net (MPPN). In Section 4, we apply the method in the mobile robotic area to an intelligent wheelchair in order to monitor its position and path execution. Experimental results are presented in Section 5. Some concluding remarks are offered in the final section. 3. Probabilistic hybrid monitoring using modified particle Petri net 3.1. Modified particle Petri net (MPPN) We define the modified particle Petri net (MPPN) as a 9-tuple
⟨P , T , Pre, Post, X , F , γ , Ω , M0 ⟩, where:
• P: the set of places partitioned into numerical places PN and symbolic places Ps .
• T : the set of transitions (numerical, symbolic or hybrid). • Pre and Post are the pre-incidence and the post-incidence matrices, respectively, of dimension |P | × |T |. The incidence matrix of the net is defined as: ∀ p ∈ P and ∀ t ∈ T C (p, t ) = Post(p, t ) − Pr e(p, t ). • X ⊂ ℜn : is the state space of the numerical state vector. • F : difference equation system associated with numerical places and representing the continuous state evolution.
• γ (p): is the application that associates a configuration with each symbolic place p ∈ PS . • Ω : a set of conditions associated with the transitions. • M0 : is the initial marking of the net. A numerical place pn ∈ P N is associated with a differential equation representing the continuous evolution of the system state (e.g. position, temperature, . . .). Numerical places are marked by a set of particles πki = [qik , wki ] defined by their corresponding continuous valued state qk ∈ X and weights wk ∈ [0, 1] at time k and representing the uncertainty distribution over the value of the numerical state vector (Fig. 1). A state change from one numerical place to another occurs when the process state exceeds the boundary of its state space. After the state change, the process follows the solution of the differential equation that may be different from the previous one, until the next jump occurs. A symbolic place ps ∈ P S is marked by configurations (black (j) token) δk+1|k representing the symbolic state of the system itself resulting from external events or the symbolic state of another system, which is in interaction with it (a human, an environment, a technical system,. . .). The marking of the Petri net Mk = {Πk , ∆k } is the marking of the MPPN at time k consisting of the set of configurations ∆k in symbolic places and the set of particles Πk in numerical places.
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Fig. 1. Numerical (a) symbolic (b) and hybrid (c) transitions linking numerical places (blue circles) marked by particles and symbolic places (black circles) marked by configurations.
Fig. 3. Estimation of the continuous-valued state using the Particle Filter principle.
density function Pr = (qk , ONk , µOk ), which depends on the noisy observation (sensory data) ONk and a random noise µOk :
w ˜ ki +1 = Pr(qik+1 , ONk , µOk ),
(3)
where w ˜ ki +1 is the likelihood value of the i-th particle. The updated particle weights are then normalized by the sum over all particle weights:
wki +1 =
w ˜ ki +1 , J i w ˜ k +1
(4)
i=1
Fig. 2. Hybrid state estimation process of MPPN.
3.2. Concept of hybrid state estimation using MPPN In the MPPN the hybrid behaviour of the system is represented by the evolution of its hybrid state vector, which is composed of numerical and symbolic variables. The state estimation using MPPN consists of a prediction and a correction. The prediction step is based on the evolution of the Petri net marking and the particle values in order to determine all possible next states of the system considering both numerical and symbolic uncertainties. The correction step is carried out by updating this prediction according to a new observation made by the system (Fig. 2). The prediction step computes the next state estimate from the current state. It consists of the estimation of the discrete state given by the Petri net marking at time k + 1 and the estimation of the continuous state by estimating the particle values. The evolution of a given marking Mk at time k to a reachable marking Mk+1 at the instant k + 1 is obtained from: Mk+1 = Mk + C · Tfired
(1)
where C is the incidence matrix and Tfired is the firing vector, which describes the transitions that are fired at a firing instant. The continuous-valued state to be used for residual computation is then estimated by estimating the particle values as follows:
∀π
i k+1|k
ˆ k+1|k (p), ∈Π
qik+1
= f( )+ν qik
k+1 p
(2)
with p ∈ PN , f (·) is the difference equation associated with the place p and νk is the process noise. The correction step is a hybrid process, which compares the actual hybrid observation with the prediction and corrects the predicted state using the observation model of the system. All particles πki +1|k = [qik+1|k , wki +1|k ] are weighted by a probability
J is the number of particles and wki +1 the normalized weight (Fig. 3). Resampling is the last step in this estimation process. The aim of this step is to delete the particles with low weights and to keep all samples with a high probability to represent the real continuous system state. After resampling the initial number N of particles and the initial particle weight, N −1 is restored. The inputs for ˆ k+1|k and the outputs ˆ k+1|k ⊆ M resampling are the particles Π
ˆ k+1|k+1 representing the estimated are the resampled particles Π numerical state of the system at the time step k + 1. The result of the numerical correction step is a probability distribution PrDN over the numerical state vector computed as follows: ∀p ∈ P N ,
ˆ k+1|k (p) = PrDN Π
mk+1|k (p)
wki +1 ,
(5)
i=1
mk+1|k (p) is the number of particles at place p at time k+1 and wki +1 the corresponding weights. The most probable numerical state is the one with the highest probability PrDN . The symbolic marking of the Petri net is updated from the observation of the configuration. The first step in this process is to calculate the probability distribution PrDS over the symbolic states in dependence on the numerical distribution PrDN . This dependency of both parts is another feature that characterizes the MPPN. In order to consider the influence of the numerical part, we define a state graph GS : ∀pn ∈ P N , ∀ps ∈ P S , GS
S1,1
.
GS = .. S N ,1
··· .. . ···
S1,K
.. , .
Si,j =
1 0
(pin , pjs ) ∈ RG otherwise
(6)
SN ,K j
where RG is the reachability graph, (pin , ps ) is the hybrid state containing a numerical state, which corresponds to the i-th numerical place and a symbolic state corresponding to the j-th symbolic place of the Petri net and S indicates if that hybrid state is
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Fig. 5. Experimental environment for the wheelchair navigation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 4a. The intelligent wheelchair.
4.2. Kinematics model of the wheelchair The wheelchair’s pose at time k represents its state and is de⃗ (k) = (xpos (k), ypos (k), φpos (k)) correspondscribed by the vector X ing to two position coordinates xpos (k), ypos (k) and the orientation φpos (k) transformed from the wheelchair coordinate to the world coordinate system. The numerical part of the Petri net tracks the ⃗k+1 = f (X⃗k , uk ). The wheelchair state using the motion model X model defines the relationship between the new pose with respect to the current pose and to the input given by the odometry sensors on the wheelchair as follows:
⃗k+1 = f (X⃗k , uk ) + vk X xpos,k + ∆sk · cos(φpos,k ) ypos,k + ∆sk · sin(φpos,k ) + vk
=
reachable (S = 1) or not (S = 0). The probability distribution PrDS over the symbolic state vector is then calculated as follows: K
(8)
φpos,k + ∆φk where uk = (∆s, ∆φ) is the control input, ∆s and ∆φ are the dis-
Fig. 4b. And the control structure.
ˆ k+1|k (ps )) = PrDS (∆
ˆ k+1|k (pn )), GS (i, j) PrDN (Π
placement and the rotation of the wheelchair respectively during the same sample interval and vk is a noise assumed to be zero mean Gaussian with covariance Qk modelling the position uncertainty of the wheelchair. 4.3. Environment modelling and planning a safe trajectory
n =1
∀pn ∈ P N , ∀ps ∈ P S .
(7)
4. Hybrid monitoring within robot control architecture 4.1. Wheelchair system description The intelligent wheelchair (IWC) depicted in Fig. 4a is a conventional powered wheelchair (OttoBock GmbH, B600) with two motorized rear wheels and castors in front [12]. It is equipped with different sensors: ultrasonic sensors arranged around the wheelchair providing collision avoidance functionality, two incremental encoders attached on the two rear wheels in order to measure the wheelchair displacement, a gyro measuring the angular rate of the wheelchair orientation and an RFID sensor system detecting the RFID-tags, which are used as landmarks in the environment to correct odometry errors. The wheelchair’s behaviour is produced using a hierarchical control architecture based on the RNBC approach [13], which decomposes the system into behavioural subsystems in a nested manner (Fig. 4b). Each behavioural level in this structure possesses its own monitor, which communicates with the monitor of the next lower level and sends status to the monitor of the next higher level. In this paper we present only the monitor on the path planning, which monitors both the path and the position of the wheelchair (as a part of the next lower monitor ‘‘position control’’) to assure that it remains on the desired path within a tolerance.
The environment in which the wheelchair navigates is assumed to be known and contains known convex polygon-shaped obstacles. Unknown obstacles are treated by the collision-avoidance behaviour. The wheelchair has a rectangular shape and three degrees of freedom that define its possible motions. Planning a trajectory means to plan the motion of every point of the wheelchair shape according to its degrees of freedom, but that remains a hard problem: the wheelchair would need to make sure, at every point in its computed trajectory, that it does not intersect obstacles. So the solution is to treat the wheelchair as a moving point and to enlarge all object boundaries by the wheelchair size. The problem of computing collision-free path of a rectangle is now reduced to that of a point in the free navigation space (the white space in Fig. 5). Using the approach of the analogue mapping developed in [14] and generating a logical model directly from the geometrical model, the path-planning problem is solved by inflating the obstacle boundaries and by simultaneously shrinking the boundaries of the navigation space until the objects come into contact with each other or with the boundaries of the navigation space. We use the obtained logical graph as a topological map: the green points are the nodes, each node contains a landmark (RFID-Tag) and the lines represent the links between the nodes which are obstacle free paths from one landmark to the next. The blue point represents the start position and the red point is the goal position. The grey circle shows the current location of the robot, the red line is the desired trajectory and the blue line is the estimated trajectory. The placement of the landmarks depends on the topology of the environment and is used for constructing the symbolic Petri net of the monitor.
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Fig. 7. Petri net model for navigation monitoring. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
4.5. Consistency analysis and fault detection
Fig. 6. Textual description of the monitor.
4.4. Model generation Through a graphical interface the user can define a target to be reached by the robot. A path planner based on artificial intelligence uses Dijkstra’s algorithm [15] to produce a path of waypoints (green points on Fig. 5). Taking under account the location uncertainty given by the system specification [14] as maximal allowed deviations εx and εy of the position, the waypoints will be expanded automatically to waypoint-areas. The wheelchair does not have to go exactly through the i-th waypoint WPi = (xref ,i , yref ,i ), which represents the centre point of the i-th area but it suffices to enter its correct area and to begin to adjust its orientation to the next waypoint. The areas are defined as follows: Areai = {xref ,i − εx ≤ x ≤ xref ,i + εx ; yref ,i
− εy ≤ y ≤ yref ,i + εy }.
(9)
Based on the user’s goal selection, the monitor model is generated automatically. This textual description corresponds to the graphical model (for visualization) in Fig. 7. The robot navigation depends significantly on the environmental topology, that’s why it needs to be monitored in due consideration of the robot–environment interaction (Fig. 6). For this purpose the Petri net model is composed of two parts in interaction: (i) a symbolic Petri net (in blue), which contains the waypoint-areas (symbolic states) lying in the planned path (symbolic state vector) and indicates when navigating in an area and when going towards the next area (ii) a numerical Petri net (in black), which tracks the x- and y-positions (numerical state vector) of the wheelchair using a particle filter and estimates with respect to the robot position if an area is entered (the numerical state ‘‘reached’’) or not (‘‘not reached’’). On the hybrid transitions intervals are defined, indicating when the waypoint must be reached as well as conditions indicating the occurrence of events.
In contrast to the classical monitoring and diagnosis methods for hybrid systems, our proposed fault detection mechanism is based not only on the estimated value for the continuousvalued state, and thus only on residual computation but also on a consistency analysis of the symbolic state with respect to the numerical state. This joint consideration of both parts of the model (numerical and symbolic) allows the distinction between the causes of deviation in system behaviour. This deviation can be caused by a stochastic disturbance (external noise influences) in the normal operation or an error (internal component faults) or a global behaviour inconsistency (reaction to an unexpected situation). The fault detection mechanism works as follows: the particles (in the entire network, regardless of their position) are sorted according to their weighting and grouped in classes κi . To one class belong only particles, which belong to the same numerical place. A numerical place pn_ max ∈ PN containing the class with the highest weight is tested in combination with the most likely symbolic place qs_ max ∈ PS (resulting from the symbolic correction) for reachability. This check is necessary because the class with the highest weighting does not necessarily belong to the most likely numerical state (with the highest PDN ) and therefore does not represent necessarily in combination with the symbolic state, a reachable marking. If the result of this special reachability test is positive (i.e. (pmax , qmax ) ∈ GR ), where GR is the reachability graph of the Petri net, then the symbolic state is considered as consistent otherwise it is inconsistent. Considering such consistency analysis and the residual, we are able to attribute the discrepancy in predictions and observations to noise or to a fault or to global behaviour inconsistency. Such different fault cases are discussed in the next session. 5. Experimental results The estimation results of the y-position corresponding to the trajectory of Fig. 5, where the predicted system behaviour (blue trajectory) fits well with the desired one (red trajectory), are shown in Figs. 8 and 9. In Fig. 8 only the estimation results of the numerical part are depicted: the red trajectory represents the observations and the blue crosses are the particles, representing the estimate of the continuous-valued state (y-position). The probabilities of the numerical states ‘‘reached’’ and ‘‘not reached’’ are shown in the same plot (right axis): when the wheelchair enters an area (red histogram) the most probable discrete state is then ‘‘reached’’ (in blue) otherwise ‘‘not reached’’ (in green). The ID number of the driven area corresponds to its height on the right axis (1-3-4-7-8-9). As shown the estimation
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Fig. 10. Deviation of the wheelchair trajectory.
Fig. 8. Estimation results of the numerical Petri net. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 11. Detection of a global behaviour fault.
Fig. 9. Estimation results of the symbolic Petri net. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
smoothly fits the observations and corresponds to the correct discrete state. While the estimation of the numerical state uses the odometry observation and refers to the geometrical information of the environment, the estimation of the symbolic state (‘‘Area’’) uses the symbolic observation (RFID-sensor) and refers to the logical information of the environment. The result of such symbolic estimation is a marking of the symbolic Petri net, which estimates entering or leaving a certain area and when navigating between areas (red segments of crosses in Fig. 9). Inconsistencies in the navigation behaviour: In order to study the consistency of the wheelchair behaviour, three fault cases are considered. Fig. 10 shows the case when the wheelchair deviates from the planned trajectory and the last area no. 9 goes undetected. The results of the numerical as well as the symbolic estimation process are illustrated on the same plot (Fig. 11). We can notice that the consistency analysis of the symbolic state with respect to the numerical shows a transition from consistent to inconsistent state (from time step 778) and the residual exceeds the maximal allowed deviation indicating a fault. Such faults (residual + inconsistency) are classified as a global behaviour inconsistency and need feedback from another behavioural level (e.g. the collision-avoidance or the axis-control level) to know if this is an error (e.g. problem at the wheels) or a reaction to unexpected situation (e.g. deviation in order to avoid an unknown obstacle). Fig. 12 shows the case of the trajectory (nodes: 1-3-4-78-10-12-13) when the wheelchair behaves correctly according to
Fig. 12. Detection of component fault.
the numerical estimation (entering the area is correct estimated) and the continuous state shows that the wheelchair does not deviate from its intended behaviour. But the symbolic Petri net does not detect the landmark and shows an inconsistency with respect to the numerical part (from step 790 to 836), which indicates a component fault (no residual and inconsistency), namely a sensor failure. Fig. 13 shows the case of an external noise, which leads to a lack of symbolic observation from step 104 to 116 (e.g. landmark undetected because of floor material or sensor noise). This case looks similar to the sensor failure case but thanks to consistency analysis it was possible to distinguish between a component fault and a noise: The symbolic state stays consistent all the time.
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[12]
[13]
[14] [15] [16]
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uncertainties, IEEE/SICE International Symposium on System Integration, Kyoto University, Kyoto, Japan December 20–22. E. Badreddin, Control and system design of wheeled mobile robots, Habilitation Script, ETH Zürich, 1997. L. Zouaghi, A. Wagner, E. Badreddin, Monitoring of hybrid systems using behavioural system specification, in: On the 18th IFAC World Congress, IFAC 2011, 2011. C. Bartolein, A. Wagner, M. Jipp, E. Badreddin, Dependable system design for assistance systems for electrically powered wheelchairs, in: International Workshop on the Design of Dependable Critical Systems, 2009. E. Badreddin, Recursive control structure for mobile robots, in: International Conf. on Intelligent Autonomous Systems 2 (IAS.2), Amsterdam, 1989, pp. 11–14. E. Badreddin, Associative memory implementation in path-planning for mobile robots, in: IEEE Conf. on Robotics and Automation, 1990. Edsger W. Dijkstra, A note on two problems in connexion with graphs, Numerische Mathematik. 1 (1959) S.269–271. A. Alexopoulos, L. Zouaghi, E. Badreddin, Associative memory for modified petri-net based monitoring of mobile robot navigation, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, Vilamoura, Algarve, Portugal October 7–12, 2012, (submitted for publication).
Fig. 13. Detection of a noise.
6. Conclusion The paper presents a method for hybrid monitoring enabling the on online generation of the models, which estimate and predict the states of mobile robots by coupling continuous and discrete information in a unified framework. It was proven that the monitor is robust against disturbances in the landmark detection process and that it enables the distinction between noise, component faults and global behaviour inconsistency. The disadvantages are that if an expected landmark stays undetected during the navigation process one can only assume that a fault in the landmark detection system has occurred without taking for example modelling and/or software errors into account. For this reason an error alarm causes the fault-handling mechanism to stop the navigation process immediately because everything else would be careless for the dependability of the AMR. To overcome this problem a long-term memory for the successfully traversed paths on the topological map would be suitable. This long-term memory is realized with associative memory and it is presented in [16]. References [1] S. Roumeliotis, G. Sukhatme, G. Bekey, Fault detection and identification in a mobile robot using multiple-model estimation, in: IEEE Int. Conf. on Robotics and Automation,, Leuven, Belgium, May, 1998 2223–2228. [2] U. Lerner, R. Parr, D. Koller, G. Biswas, Bayesian fault detection and diagnosis in dynamic systems, in: Proc. 17th Nat. Conf. on Artificial Intelligence, 2000. [3] X.D. Koutsoukos, K.X. He, M.D. Lemmon, P.J. Antsaklis, Timed Petri nets in hybrid systems: Stability and supervisory control, Journal of Discrete Event Dynamic Systems: Theory and Applications 8 (2) (1998) 137–173. [4] S. McIlraith, Diagnosing hybrid systems: a bayesian model selection problem, in: Proceedings of the 11th International Workshop on Principles of Diagnosis, DX’2000, 2000. [5] U. Lerner, R. Parr, D. Koller, G. Biswas, Bayesian fault detection and diagnosis in dynamic systems, in: Proc. 17th Nat. Conf. on Artificial Intelligence, 2000. [6] C. Lesire, C. Tessier, Particle Petri nets for aircraft procedure monitoring under uncertainty, in: ATPN, Miami, FL, USA, 2005. [7] F. Dehais, A. Goudou, C. Lesire, C. Tessier, Towards an anticipatory agent to help pilots. in: AAAI Fall Symposium. From Reactive to Anticipatory Cognitive Embodied Systems. Arlington, USA, 2005. [8] J. Cardoso, R. Valette, D. Dubois, Possibilistic Petri nets’, IEEE Transactions on Systems, Man, and Cybernetics, Part B (1999) 573–582. [9] L. Zouaghi, A. Alexopoulos, A. Wagner, E. Badreddin, 2011 Modified particle Petri nets hybrid dynamical systems monitoring under environmental
Leila Zouaghi was born in Cologne, Germany. She received her Diploma in electrical engineering from the Technical University Kaiserslautern, Germany, in 2005. She is currently working toward the Ph.D. degree in computer engineering at the Institute of Computer Engineering, University of Mannheim (now Heidelberg University). Her research interests include hybrid monitoring, fault detection, mobile robots, Petri nets and nonlinear state estimation.
Alexander Alexopoulos was born in Heidelberg, Germany. He received his Diploma in computer science from the University of Mannheim, Germany, in 2011. He is currently working toward the Ph.D. degree in computer engineering at the Institute of Computer Engineering, University of Mannheim (now Heidelberg University). His research interests include game theory, fuzzy control and neuronal networks.
Achim Wagner obtained his Diploma in Information Technology from the Saarland University, Saarbrücken, Germany, in 1996. Also at the Saarland University he received his doctoral degree in the area of electrical materials science in 2002. Afterwards he worked as an associate researcher in medical robotics projects at the Laboratory for Biomechanics and Experimental Orthopedics, OUZ, Heidelberg University and at the Automation Laboratory, Institute of Computer Engineering, University of Mannheim (now Heidelberg University). Since 2004 he has led the robotic research group. His current scientific interests are dependable autonomous and assisting robotic systems, fault-tolerant system design and control, and human–machine interaction. Essam Badreddin is currently the head of the Automation laboratory at the Heidelberg University, Germany. He earned his Swiss Diploma in Electrical Engineering from the Swiss Federal Institute of Technology, Zurich, Switzerland (ETHZ), the Doctor of Technical Sciences in Control Theory (ETHZ) and Habilitation in Mechanical Engineering (ETHZ). There, he founded and led the first robotics research group. Prof. Badreddin served as Monbushu Professor in Japan before he joining the University of Mannheim and later Heidelberg University. He served in industry at Contraves-Zurich as an R&D System Engineer in the air-defence sector, where he also holds several international patents (Silver Dollar awarded).