Qualitative Spatial Calculi with Assymetric Granularity

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1 Qualitative Spatial Calculi with Assymetric Granularity Sree Harsha R November 13, 2007 Abstract Qualitative Spatial/Direction Calculi gives the orientation of any two points to eachother in a 2-D space as a binary relation. Previously proposed calculi were built having either fixed or arbitrary granularity. But all these calculi are symmetric, that is they have the same granularity at the front(right) and the back(left). In this report we propose a new Qualitative Spatial/Direction Calculi called MST AR m with an assymetric granularity between the front and the back. The aim of this work is to produce a comprehensive calculi where the granularity of the front is independent of that of the back. This calculi will be very useful when representing the behavior of human beings or rules that are made keeping human beings in mind (Eg. Traffic rules). 1 Qualitative Spatial Reasoning Qualitative Spatial Reasoning(QSR) is an abstraction that summarizes similar quantiative states into one qualitative characterization. From a cognitive perspective, the qualitative method compares features of the domain rather than measuring them in terms of some artificial external scale. QSR basically investigates qualitative representations of space that can be abstracted from the metrical values recieved from the physical world together with reasoning techniques that allow predictions about spatial relations. Spatial reasoning, mainly for humans, is driven by qualitative abstractions rather than complete quantitative knowledge. Human beings in their everyday tasks rarely come across quantitative metrical data, we make decisions based on 1

2 qualitative symbols or representations. But for computers this is compeletely different. Looking at an image, a computer can not decide whether a certain object is smaller or just farther away. [1]. QSR is typically realized in form of a Qualitative Spatial Calculi, over sets of spatial relations (like north-of, ahead-of) along with a set of operations on these relations. In this report we have tried to define a new calculi, MST AR m. Qualitative spatial calculi, like MST AR m, is used to abstract the quantitative metrical data of the scene that is observable by an agent and a representation of rule conguration. From these qualitative spatial calculi, symbolical reasoning processes can be applied by an agent to decide rule-compliant or desirable actions. A qualitative substrate allows us to decouple low-level agent behavior from high-level reasoning. [2] To predict the MST AR m relation between two points after a certain action(s) is(are) committed we need the idea of conceptual neighborhoods. The notion of conceptual neighborhood has been introduced by Freksa [3]. Two spatial relations of a qualitative spatial calculus are conceptually neighbored if they can be continuously transformed into each other without resulting in a third relation in between. To test our calculus, we create a stable and rule-compliant decision agent in Golog to control individual vessels in the Sailaway simulator. Each individual vessel s position and orientation information is converted into binary relation using a qualitative spatial calculi, MST AR m. This binary relation between two vessels is mapped to a rule from the International Regulations for Preventing Collissions at Sea, COLREGS (Collission Regulations) and corresponding action is taken. 2 MSTAR Calculus Definition MSTAR Calculus: Given a 2D plane P and a global reference direction in P. For each point p P, the MStar Calculus MST AR m [δ 1,..., δ m ](δ 1 ) where 0 δ 1 < δ 2... < δ m < 180 specifies 2m lines which intersect at p. Line j forms an angle δ j with the global reference direction if 1 j m and δ j m if m + 1 j 2m. For each point p P these 2m lines partition P into 4m + 1 disjoint zones with respect to the reference direction. 2m lines, 2m sectors each bound by two lines. A unique identifier is assigned to each zone as follows: 2

3 First m lines are make an angle δ j with the global reference direction if 1 j m and the next m lines form an angle δ j m if m + 1 j 2m. They are numbered in clockwise direction 0 p, 2 p,..., (4m 2) p 2m lines and point p bind the 2m sectors. We assign uniquie identifiers 1 p, 3 p,..., (4m 1) p in clockwise order starting with sector bound by p and the lines 0 p and 2 p Figure 1: MST AR m [15, 30, 45, 120](15) Using these zones 4m + 1 basic MStar relations can be defined as follows: the identity relatioin id {(p, p) p P } and the relations I {(p, q) p, q P, q I p }, I {0, 1,..., 4m 1} we denote the relations {1, 3, 5,...} as odd relations, and {0, 2, 4,...} as even relations 3 Customization for MStar Calculus To prove that any new Spatial Calculus is valid for qualitative reasoning algorithms, the set of relations produced by this calculus is closed under operations like Union( ), Intersection( ), Converse( ) 3

4 and Composition(o) etc. But it is difficult to define the requisite operations for MSTAR Calculus. So a simpler way is to just customize an existing calculus to MStar Calculus. Before defining any of the operations, we need to understand the idea of customizing a calculus. In our case, we have customized the Star Calculus [4] to the MStar Calculus. Initially we wanted to use the Star Calculus in Sailaway. But it is mandatory in Star Calculus that granularity of the front is same as that of the back. The MStar Calculus, as indicated in Section 2, can have a very fine granularity on the front and but coarse granularity in the back. It has been proven that, it is possible to customize an existing calculus to a new calculus which inherits computational properties from the existing calculus. [5] 3.1 Star Calculus Definition STAR Calculus: Given a 2D plane P and a global reference direction in P. For each point p P, the MStar Calculus MST AR m [δ 1,..., δ m ](δ 1 ) where 0 δ 1 < δ 2... < δ m < 180 specifies m lines which intersect at p. Line j forms an angle δ j with the global reference direction. For each point p P these 2m lines partition P into 4m + 1 disjoint zones with respect to the reference direction. 2m lines, 2m sectors each bound by two lines. See Fig Figure 2: ST AR m [15, 30, 45, 60, 120, 135, 150, 165](15) 4

5 3.2 Customization of Star Calculus to MStar Calculus Here we present and algorithm to find an MStar Calculus for every given Star Calculus. That is, we customize this Star Calculus to the given MStar Calculus. The operations of the Star Calculus should be used to create the operations for MStar Calculus. [5] Customization of calculus has three types Customization by Macro Relations: Merging some of the base relations to form a new set of base relations. Customization by Unused Relations: Excluding some of the base relations to form a smaller set of base relations. Customization by Reduced Domain: Reducing the base relations by restricting the domain. However the new customized calculus should still be closed under the operators (,,,o) Algorithm to transform a Star Calculus into MStar Given notation MST AR m [δ 1,..., δ m ](δ 1 ) where 0 δ 1 < δ 2... < δ m < Create a Star Calculus S ST AR 2m [δ 1,..., δ m, 180 δ 1,..., 180 δ m ](δ 1 ). 2. Consider each line\region in the new calculus S. If it is required in M, keep it and assign a number accordingly.else, merge it with the preceeding line\region. 3. Maintain a mapping(s2m) between line\region of the calculi. For example given the MStar Calculus M from [prev section]. M has 8 half-lines and 8 regions. Now according to the above mentioned algorithm, we first define a Star Calculus with the same parameters. This will create 16 lines and 16 regions [figure]. Now we will look at each half-line\region of S and see if it is relevant in M. For example line\region 0 4 are the same in both S and M. But region 5 in M is equivalent to 5, 6, 7 in S. Similarly region 7 in M is equivalent to 9 21 in S. Every line/region in M is mapped to a set of line\region in S by a mapping M2S. This is depicted in Fig. 3 5

6 Figure 3: Conversion of ST AR 8 [15, 30, 45, 60, 120, 135, 150, 165](15) to ST AR 4 [15, 30, 45, 120](15) given by S2M. 7 Mapping from Star to MStar S2M(0) {0} S2M(1) {1} S2M(2) {2} S2M(3) {3} S2M(4) {4} S2M(5) {5,6,7} S2M(6) {8} S2M(7) {9,10,11,12,13,14,15,16,17,18,19,20,21} S2M(8) {22} S2M(9) {23,24,25} S2M(10) {26} S2M(11) {27} S2M(12) {28} S2M(13) {29} S2M(14) {30} S2M(15) {31} s 6

7 3.3 Operators If R, R 1, R 2 bas(m), M MST AR m and S = S2M(M). M is the MST AR m calculus and S is its corresponding ST AR 2m calculus. 1. Converse Operator: If R = id then R = id If 0 R 4m 1 then R S = S2M(R) and R = {A 1 A 2 A 3...}, where A 1, A 2, A3,... R S 2. Composition Operator : If R 1 = id then R 1 R 2 = R 2 If R 1 = R 2 then R 1 R 2 = R 1 If R 1 = R2 and R odd, then R 1 R 2 = { } If R 1 = R2 and R even, then R 1 R 2 = {R, S, id} If 0 R 1, R 2 4m 1 then R 1S = S2M(R 1 ), R 2S = S2M(R 2 ) and R 1 R 2 = R 1S R 2S 3.4 Uniqueness of MSTAR Previously proposed calculi were built having fixed or arbitrary granularity. But all these calculi are symmetric, that is they have the same granularity at the front(right) and the back(left). MST AR m gives the option to introduce an assymetry between front and back. This is makes it the perfect vehicle to capture the perception of humans or humanoids. We feel that rules defined in natural language for human usage, like traffic rules, are best expressed by an assymetric calculus. Assymetry is important because humans have fine grained perception in the front(through eyes) and little or no perception of what is behind them. Such an assymetry is best expressed in MST AR m calculus. Here we use the MST AR m calculus to model and control navigational tasks of vessels according to the COLREGS. The COLREGS were originally written for human usage. Hence we felt that the best possible way to simulate automated navigation is through this assymetric calculus which best represents the human way of percieving spaces. 7

8 4 Sailaway The Sailaway project examines individual agents/vessels which solve navigational tasks in sea. This goes far beyond single agent goal directed deliberation, that is, an agent s action in a particular situation often effects the other agents actions at the same time. The environment for these agents is dynamic or constantly changing. This means they have to constantly review the situation or worldmodel and make decisions in order to avoid collissions. Decision made by each agent is governed by a set of rules and laws. Traffic rules at sea are specified by the International Regulations for Preventing Collissions at Sea, COLREGS. [6] These rules define behavior and actions with respect to spatial configurations and properties of the participating agents. Since rules are mostly expressed in natural language and use qualitative terms. These qualitative terms can be translated, to a certain extent, into a qualitative spatial calculi. Qualitative Spatial Reasoning(QSR) abstracts similar metrical values to qualitative representations. MST AR m formalisms abstract from metric data by summarizing similar quantitative states into a single qualitative description. For this reason, such formalisms are suited as a basis for representing rules in a formal way. We use conceptual neighborhoods to reason about the transitions in spatial situations. Articial cognitive agents that interact with humans should be able to process such rule sets. Sensors deliver numerical values which should be mapped to qualitative terms. An agent must be able to identify itself with the current spatial situation, recognise the rules that will be applicable to this situation and with respect to the rule set decide which action gives reasonable rule-compliant behavior. Hence we expect rule-compliant behavior from an agent which can only perceive its surroundings in terms of numerical values. Spatial relations must be deduced from these numerical values. [7] For example a rule that governs the head-on situation could be, When two power driven vessels are meeting head-on so as to involve risk of collision each shall alter her course to starboard so that each ship will pass on the port side of the other. To decide whether a certain vessel or agent is moving towards another agent or heading for collission, we need a qualitative calculi. Qualitative calculi reveal the relative nature of spatial information, that is, properties of objects are compared to one another rather than 8

9 comparing the properties to some external scale [7]. The objective of using qualitative caculi is to find the orientation of a vessel with respect another vessel. Here we use the MST AR m family of calculi which describe relations between oriented points (points in the same plane with directional parameter). [7] Our focus is mainly to translate rules from natural language descriptions to a qualitative formalization for agent control. We intend to translate the rules in natural language to logic programming constructs in Golog. That is the agents will make independent decisions based on the Golog based constructs [6]. 4.1 Software Architecture See Fig. 4 for the basic architechture. In this project we have used three main components Sailaway simulator:- Supplies the exact positional information of the vessel in the simulator to Golog Decision Agent. The simulator also recieves the commands from the Decision Agent. Golog Decision Agent :- Each vessel has its own Decision Agent. Using the MST AR m binary relation the current spatial situation is mapped to a certain rule in the rule set and the corresponding decision is made. The decision is then supplied to the Sailaway simulator. SPARQ :- Converts the positional information into an MST AR m binary relation and this information is sent back to Decision Agent. 5 Adapting MST AR m to Sailaway For effective decision making in Sailaway, it is not enough to have MST AR m calculus in its purest form. That is, if there are two vessels A and B in a potential collission situation, we need the orientation of A to B (AR 1 B) and the orientation of B to A (BR 2 A) to make an effective decision. The problem with MST AR m and ST AR m is that they provide only the orientation of a certain point to another point. So we need to tweak MST AR m to get this relation. We need to change the (MST AR m ) calculus so that it will encode both R 1 and R 2. This can be done simply by concatenation of both the 9

10 Sailaway Simulator Positional Info Navigation command Positional Info High Level Interface Golog Decision Agent Vessel A Positional Info MSTAR Relation SparQ MSTAR Navigation command High Level Interface Golog Decision Agent Vessel B Positional Info MSTAR Relation SparQ MSTAR Figure 4: Software Architecture relations into one single relation. That is, in MST AR m the relation between A and B will be A R 2 R 1 B and the relation between B and A will be A R 1 R 2 B. 5.1 Example Rule Figure 5: Rule 1 - Head On 10

11 Figure 6: Rule 2 - On the side The Rule 1 in Fig. 5 shows that when two vessels are heading into each other (when two vessels are HeadOn ). The International Regulations for Avoiding Collisions at Sea(COLREGS), specify that both the vessels should turn starboard or should turn to the right to avoid a collission. 5.2 Using MST AR m The Golog Decision Agent uses MST AR m to map each situation to a certain rule and then make the corresponding decision. In a scenario like HeadOn shown in Fig. 5, the MST AR m relation between the two vessels is This situation has to be mapped to the rule in Fig. 5. Since region 15 is front (around the reference direction), both the vessels know that there is potential collission on the front. The corresponding rule should be invoked and both the vessels should turn starboard. The whole decision process is coded in Golog [8]. The requisite code is 1 proc(decision_motor(v1), [ [V2,Rel,Rel_P1,Rel_P2] = epf_collissionwith(getname), if( belongsto(to(rel_p1),front) = true, turn_starboard( V1 ) 5 if( and([belongsto(to(rel_p1),right), ecf_vesseltype(v2) = 2, approaching(rel_p1,rel_p2) = true]), turn_starboard( V1), keep_midships(v1)))]). There are various types of vessels like Sport vessel, Motor vessel, Sail vessel etc. Rule compliant vessel behavior is dependent on its 11

12 type. This code basically uses the MST AR m to decide whether the potential collission is on its front or right and takes the corresponding decision depending on its colliders vessel type. Here we recieve the information about the MST AR m as an exogenous fluent in line 3. An exogenous fluent is a sensor value which is taken from the outside or from the environment. The MST AR m relation and the name of the colliding vessel is considered as an exogenous fluent. In line 3 we decide whether the opponent is to my front using the to and belongst o functions and the corresponding action is sent to the simulator. 5.3 Conceptual Neighborhoods Two spatial relations of a qualitative spatial calculus are conceptually neighbored if they can be continuously transformed into each other without resulting in a third relation in between. Conceptual neighborhood is given by the function [3] [9]. cn( j i ) = ( j i+1, j i 1, j+1 i, j 1 i, j 1 i 1, j 1 i+1, j+1 i+1, j 1 i+1, ) (1) For example - When two vessels are in the relation 3 13 and vessel A turns starboard (slightly to the right), the expected relation is According to this we can deduce the successor of each MST AR m relation given the action being taken. Using this we can predict how a vessel can go from a collission prone situation to a collission free situation. That is, if two vehicles are in HeadOn situation( 15 15), both of them will have to turn starboard, and it will make the MST AR m relation But this is also a collision prone situation, so there should be another turn starboard which which makes the relation In this way we chart the course of a certain a vessels navigation from a collission prone situation to a collission free situation in terms of the MST AR m relation between them. This is represented in the transition graph of Fig. 7. [10] In this graph the rectangular boxes represent severly collission 15 prone relations of MST AR 4 ( 15 and 14 14). The circles represent 13 moderately collission prone relations( , 12, 11 and ). The 9 concentric circles represent collission free relations ( 9, 8 8, 7 7 ). Actions are reprsented in the brackets. 12

13 6 Future Work Future work will focus on implenting MStar Calculus in SparQ software package and creating an effective Golog Decision Agent to handle the relations produced by the calculus. Also introducing the concept of planning into the Decision Agent will be important. Planning will enable autonomous vessels in the simulator to plan their path both offline and online when encountered with a potential collission situation. References [1] Jan Oliver Wallgrn Frank Dylla. Qualitative spatial reasoning with conceptual neighborhoods for agent control. Journal of Intelligent and Robotic Systems, 48:55, [2] Reinhard Moratz. Representing relative direction as binary relation of oriented points. Proceedings of the 17th European Conference on AI, [3] Christian Freksa. Conceptual neighborhood and its role in temporal and spatial reasoning. Proceedings of the IMACS Workshop on Decision Support Systems and Qualitative Reasoning, page 181, [4] Debasis Mitra Jochen Renz. Qualitative direction calculi with arbitrary granularity. Pacific Rim International Conference on Artificial Intelligence, [5] Falko Schmid Jochen Renz. Customizing qualitative spatial and temporal calculi. Australian Joint Conference on Artificial Intelligence, [6] F. Dylla, L. Frommberger, J.O. Wallgrun, D. Wolter, B. Nebel, and S. Wol. Sailaway: Formalizing navigation rules. Proceedings of the AISB 07 Artificial and Ambient Intelligence Symposium on Spatial Reasoning and Communication, [7] F. Dylla, L. Frommberger, J.O. Wallgrun, D. Wolter, B. Nebel, and S. Wol. Qualitative spatial reasoning for rule compliant agent navigation. American Association of Artificial Intelligence, [8] Henrik Grosskreutz and Gerhard Lakemeyer. On-line execution of cc-golog plans. In IJCAI-01,

14 [9] Christian Freksa. Using orientation information for qualitative spatial reasoning. Proceedings of the International Conference GIS - From Space to Territory: Theories and Methods of Spatio- Temporal Reasoning on Theories and Methods of Spatio-Temporal Reasoning in Geographic Space, pages , [10] Jan Oliver Wallgrun Frank Dylla. On generalizing orientation information in opram. 29th German conference on Artificial Intelligence,

15 15_15 (S,S) 14_14 (S,S) 13_13 (S,S) 12_12 (S,S) 11_11 (S,S) 10_10 (M,M) 9_9 (M,M) (P,P) 8_8 7_7 Figure 7: Transition Graph 15