CSE 579 Project Milestone 4 – Individual Project Report Automated Warehouse Scenario.

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CSE 579 Project Milestone 4 – Individual Project Report Automated Warehouse Scenario Sang-Hun Sim School of Computing, Informatics, and Decision Systems Engineering ssim6@asu.edu

Background

In my individual project report, I suggested a solution to the problem of an automated warehouse. My approach involved creating a transition system affected by robots' actions, and a methodology to reach the desired state. I used Knowledge Representation and Reasoning (KRR) to define all the hard constraints for the automated warehouse, and aimed to find the minimum time for the robots to fulfill the order. I established a scenario as close to reality as possible, by adding rules and conditions based on the given input data. Each step in my approach and the analysis of the results are described. Ultimately, I found satisfactory results for all five inputs..

I chose Automated Warehouse as my project topic, inspired

by Module 5 of CSE 579. In that module, I learned about the need to reason about actions and the concept of Fluents. With Answer Set Programming (ASP) as my key tool, I was able to define each element with Atoms and express actions and their effects in a timely manner. I needed to consider the complexity of the algorithm, logical contradictions, and optimization potential while creating the necessary constraints. To gain an understanding of the project, I referred to the CSE 579 lecture notes and Vladimir’s book, Answer Set Programming. I also turned to Artificial Intelligence, A Modern Approach to learn about reasoning procedures

Problem Statement

Approaches to Solve

Abstract

In the automated warehouse scenario, robots are tasked with delivering products to picking stations. This requires them to horizontally and vertically move through grid-shaped cells and transfer shelves containing the products to the picking stations. Robots can move beneath the shelves and pick them up due to their flat body shape. My goal was to deliver all products specified in the order state to the picking stations. I was given five instances as initial inputs, each of which had a different format in the 'init' function. The required output was the robots' actions, expressed in the 'occur' function, and rules or constraints were added to ensure the scenario was as close to reality as possible. For instance, a robot could not pass under the shelves when carrying another shelf, and had to be moved out of the way to put the shelf down. The time was counted in steps, with each step corresponding to a single action - or none at all - and there was to be no collision between robots as they moved, picked shelves up, put them down, delivered products or remained idle. The highways on the grid cells were also to remain free from shelves. These rules were outlined in the original project description, but I also added additional constraints during programming.

Sketching the Program Before programming, I looked into the instance file for the types of objects and Fluents, assessing what needed to be included in the parameters. I visualized how the Automated Warehouse should look, with a 4x4 grid and Fluents at their indicated coordinates. Following the instance file, I determined the initial statuses of the Fluents, as well as the four actions the robot could make, and I made a list of required constraints that could easily be added in the code.

Fluents and Objects I next dealt with the domain-independent axioms, realizing that the Fluents were not initially exogenous and thus had to be converted from the initial instance input. I declared them as instances and Fluents, defined the Atoms depicting their statuses, and incorporated object parameters, such as coordination and corresponding objects. For each object, like “node”, I had to include a unique value, like “Nid”, and applicable parameters, such as “pair(X,Y)” for location. Most of the other objects were on the same basis, but I also had to construct counting functions for the objects to define constraints and other rules. For example, information regarding the total number of nodes was essential in specifying the goal states.

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all objects that have a certain time value in their state function at the end.

Transitions I then specified the transition statements to define the four actions of the robots, outlining the casual relationship between their action and the effected status. As a precaution, I made sure that the actions and their effects were made in a timely manner, and that the changes made by the actions were applied without contradiction.

Constraints

I have created extra rules to link the suggested output format to a format I have created that includes more information. In the output for picking up, for example, it does not have the identification of the shelf, so I added new rules to fill in this missing information. I made sure that the time of quick changes and being applied without defect.

Goal Statement The final quantity of products in the order statement must be zero and it is also specified that there are some number of products in the initial state. This ensures that the program will only consider the order statement specified in the instance file and eliminate any cases with incorrect locations of pick up-stations..

The Law of Inertia I also had to create a law of inertia for objects without any robot action that keeps the value the same as the immediate preceding timestamp. Therefore, I introduced this rule for

The constraints have a very important role, as they make the program more practical and useful for the real world.

There are two types: action constraints and status constraints. As there are four robot actions, there are four action constraint categories. For the robot's movement, I have enforced that they can't travel beyond one slot and must stay within the boundary of the grid.

Regarding picking up, a shelf cannot be handled by both node and robot at the same time and a robot cannot take a shelf away from another robot. Additionally, a robot can only handle one shelf at a time. When a robot is putting down the shelf, they must travel to the picking station and they are not allowed to drop it on the highway. The delivery works similarly - a robot can create a delivery action when they are at the picking station and they possess a shelf with any products. Plus, it is not possible to deliver more items than are contained in the shelf.

I concluded that state constraints are only necessary for robots and shelves. For robots, I prevented collisions by ensuring that they don't take each other's positions or occupy the same coordinates at once. I enforced that a shelf can only ever be located in one node or robot at any one time.

Main Results and Analysis Types of Result Encountered

Result of instance#2: occurs(object(robot,1),move(0,-1),0) occurs(object(ro-

I encountered three types of results, UNSATISFIABLE, bot,2),move(0,1),0) occurs(object(robot,1),move(-1,0),2) ocSATISFIABLE with incorrect behavior of fluent, and curs(object(robot,2),move(-1,0),4) occurs(object(roSATISFIABLE with correct actions. UNSATISFIABLE bot,1),move(1,0),5) occurs(object(robot,1),move(0,1),6) occame to me with errors making code logically wrong. This curs(object(robot,2),move(0,-1),6) occurs(object(rohappened especially when the naming concept were in inbot,1),move(0,1),7) occurs(object(robot,2),move(1,0),8) occurs(object(ro-curs(object(robot,1),move(-1,0),8) occonsistency. SATISFIABLE with inappropriate output bot,1),move(1,0),9) occurs(object(robot,1),move(-1,0),10) happened when there were not enough constraints. Thereoccurs(object(robot,2),move(1,0),10) occurs(object(rofore, I had to keep aware of errors and carefully analyze bot,1),move(0,-1),12) occurs(object(robot,2),move(0,-1),12) the results although I got SATISFIABLE result. occurs(object(robot,2),pickup,2) occurs(object(robot,1),pickup,4) occurs(object(robot,2),pickup,9) occurs(ob Analyze of Satisfiable Results ject(robot,2),putdown,7) occurs(object(robot,2),de-liver(1,1,1),5) occurs(object(robot,1),deliver(1,3,2),13) ocIt seems like instance#3 and #5 have lower number of n curs(object(robot,2),deliver(2,2,1),13) since they have only two orders to deliver at the same pick- Optimization: 11 ing station it requires smaller n values, while #1 has the OPTIMUM FOUND

largest number of deliveries.

Result of instance#3: N-values for each instance: Instance Number #1

Minimum Time Taken 13

occurs(object(robot,1),move(-1,0),2) occurs(object(robot,2),move(1,0),2) occurs(object(robot,1),move(0,-1),4) occurs(object(robot,2),move(-1,0),4) occurs(object(robot,1),move(0,-

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1),9) occurs(object(robot,2),move(1,0),9) occurs(object(robot,1),move(1,0),12) occurs(object(robot,2),move(0,1),12) occurs(object(robot,1),pickup,8) occurs(object(robot,2),pickup,8) occurs(object(robot,1),deliver(2,4,1),10) occurs(object(robot,2),deliver(1,2,1),13) When I implement the program with such the command Optimization: 7 clingo test.txt simpleInstances\inst1.asp -c n=15, I get the OPTIMUM FOUND #2 #3 #4 #5

11 7 10 6

output as specified in Result of each instance Result of instance#4: I confirmed that constraints are in good compliance. Every moves one slot per one turn and keep their movement ocwithin the range of grid. They pick up shelf after they

occurs(object(robot,1),move(-1,0),0) occurs(object(rorobot bot,2),move(0,-1),0) occurs(object(robot,1),move(-1,0),1) curs(object(robot,1),move(0,-1),2) occurs(object(ro-

bot,2),move(-1,0),2) occurs(object(robot,1),move(0,-1),4) ocreach to the shelf’s location and do not take it away from curs(object(robot,2),move(0,1),4) occurs(object(roother robots.

Most importantly, they bring the product to bot,1),move(1,0),5) occurs(object(robot,2),move(0,1),5) octhe pickup station as order statement specifies. curs(object(robot,2),move(1,0),6) occurs(object(robot,2),move(-1,0),10) occurs(object(robot,1),move(-1,0),11)

Result of instance#1:

occurs(object(robot,2),pickup,1) occurs(object(ro-

occurs(object(robot,1),move(-1,0),0) occurs(object(rooccurs(object(robot,1),move(-1,0),1) oc-

bot,1),pickup,3) occurs(object(robot,2),pickup,7) occurs(obbot,2),move(-1,0),0)

ject(robot,2),putdown,3) occurs(object(robot,1),decurs(object(robot,2),move(0,1),2)

occurs(object(ro- liver(2,2,1),6) occurs(object(robot,1),deliver(3,2,2),10) ocbot,1),move(-1,0),4) occurs(object(robot,2),move(0,-1),4) occurs(object(robot,2),deliver(1,1,1),11) curs(object(robot,2),move(1,0),6) occurs(object(ro- Optimization: 10 bot,1),move(0,-1),7) occurs(object(robot,2),move(1,0),7) oc- OPTIMUM FOUND curs(object(robot,1),move(1,0),8) occurs(object(ro- bot,2),move(0,-1),10) occurs(object(robot,1),move(1,0),11) Result of

instance#5: occurs(object(robot,1),move(0,-1),12) occurs(object(ro- occurs(object(robot,1),move(-1,0),2) occurs(object(robot,2),move(1,0),12) occurs(object(robot,2),pickup,1) oc- bot,2),move(-1,0),2) occurs(object(robot,1),move(-1,0),3) occurs(object(robot,1),pickup,2) occurs(object(ro- curs(object(robot,2),move(0,1),4) occurs(object(robot,2),pickup,8) occurs(object(robot,1),pickup,10) occurs(obbot,1),move(-1,0),7) occurs(object(robot,2),move(0,-1),7) ocject(robot,2),putdown,5) occurs(object(robot,1),putdown,6) curs(object(robot,2),pickup,3) occurs(object(rooccurs(object(robot,2),deliver(1,3,4),3) occurs(object(robot,1),pickup,4) occurs(object(robot,2),putdown,8) ocbot,1),deliver(1,1,1),5) occurs(object(robot,2),de- curs(object(robot,2),deliver(1,3,4),6) occurs(object(roliver(3,4,1),11) occurs(object(robot,1),deliver(2,2,1),13) bot,1),deliver(1,1,1),8) Optimization: 13 Optimization: 6 OPTIMUM FOUND

OPTIMUM FOUND

Conclusion (Self-Assessment) I found that this project was an invaluable extension of the principles learned in CSE 579 module 5, as it required me to use my knowledge of solving complex reasoning problems in a hands-on setting. I was able to devise state constraints to ensure that robot collisions and shelf locations were prevented and maintained. Even though I was not able to directly control the grippers' movement, I still found this to be an excellent learning opportunity..

This project enabled me to apply the concepts from my CSE 579 Module 5 course, as I was able to solve complex reasoning problems and see the outcome of my work in practice. My successful completion of the mission and my

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results satisfying all constraints confirmed to me that I had a good grasp of the material...

I had difficulty finding errors resulting in an UNSATISFIABLE response. This normally occurred when the code I wrote was incorrect, and I had to examine the entire section. For cases with SATISFIABLE responses but conflicting answers, I gradually added constraints until I arrived at a reasonable outcome. Unfortunately, I admit that I could have omitted a few constraints in the definition..

I acknowledge that there may be more efficient approaches to solving this issue. For now, I manually defined the individual elements, but there could be generic methods of carrying out this task. If I had additional time, I would look into more innovative ways of defining the functions.

Opportunities for Future Work I can confidently state that ASP is highly effective in resolving complex issues in our world, since its applications have numerous practical uses. It would be truly meaningful if I could craft a program that operates a warehouse automation system as expressed in the project's narrative..

It would be favorable to proceed with this project alongside other courses related to Robotics Engineering, thus allowing me to join in constructing a real automated warehouse model. I am confident that I could make a substantial contribution to such a project if I am familiar with the effective logics.

References 1) Vladimir Lifschitz. Answer Set Programming. 2009 2) Stuart Russel, Peter Norvig. Artificial Intelligence, A Modern Approach, Third Edition. 2016

Appendix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% GENERAL DEFINITION %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% NODE %%%

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state(Nid,pair(X,Y)):- init(object(node,Nid),value(at,pair(X,Y))). node(Nid):- init(object(node,Nid),value(at,pair(X,Y))). node_Num(N):N=#count{Nid:init(object(node,Nid),value(at,pair(X,Y)))}.

%%% GRID %%% state(object(highway,Nid)):- init(object(highway,Nid),value(at,pair(X,Y))). num_Col(num_cols):num_cols = #count{X:init(object(node,Nid),value(at,pair(X,Y)))}. num_Row(num_rows):num_rows = #count{Y:init(object(node,Nid),value(at,pair(X,Y)))}.

%%% ROBOT %%% state(object(robot,Rid),on(node,Nid),0):- state(Nid,pair(X,Y)), init(object(robot,Rid),value(at,pair(X,Y))). robot(Rid):- init(object(robot,Rid),value(at,pair(X,Y))). move(1,0;-1,0;0,1;0,-1). numRobots(Num):- Num = #count{Rid:init(object(robot,Rid),value(at,pair(X,Y)))}. {occurs(object(robot,Rid),move(X1,Y1),T):move(X1,Y1)}1:- numRobots(Num), Rid=1..Num, T=0..n-1. state(object(robot,Rid),on(node, Moved_Nid), T+1):- state(Nid,pair(X,Y)), state(object(robot,Rid),on(node,Nid),T), occurs(object(robot,Rid),move(X1,Y1),T), state(Moved_Nid, pair(X+X1,Y+Y1)).

%%% ORDER %% state(Oid,object(node,Nid),product_info(Pid,PQ),0):init(object(order,Oid),value(pickingStation,PKid)), state(object(pickingStation,PKid),object(node,Nid)), init(object(order,Oid),value(line,pair(Pid,PQ))).

%%% PICKING STATION %%% state(object(pickingStation,PKid),object(node,Nid)) init(object(node,Nid),value(at,pair(X,Y))).

:-

init(object(pickingStation,PKid),value(at,pair(X,Y))),

%%% SHELF %%% shelf(Sid):- init(object(shelf,Sid),value(at,pair(X,Y))). state(object(shelf,Sid),on(node,Nid),0) :init(object(shelf,Sid),value(at,pair(X,Y))), state(Nid,pair(X,Y)).

%%% PRODUCT %%% product(Pid):- init(object(product,Pid),value(on,pair(Sid,EA))). state(object(product,Pid),object(shelf,Sid),with(quantity,EA),0):init(object(product,Pid),value(on,pair(Sid,EA))). numProducts(Num):Num=#count{Pid:init(object(product,Pid),value(on,pair(X,Y)))}.

%%% PICKUP %%% % Need to specifiy the shelf

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{occurs(pickup(Rid,Sid),T):shelf(Sid)}1:- Rid=1..NR, numRobots(NR), T=0..n-1. occurs(object(robot,Rid),pickup,T):occurs(pickup(Rid,_),T). % Effect of picking up a shelf state(object(shelf,Sid),on(robot,Rid),T+1) :occurs(object(robot,Rid),pickup,T), state(object(shelf,Sid),on(node,Nid),T), state(object(robot,Rid),on(node,Nid),T), R=1..NR, numRobots(NR).

%%% PUTDOWN %%% {occurs(putdown(Rid,Sid),T):shelf(Sid)}1:- Rid=1..NR, numRobots(NR), T=0..TN,TN=n-1. occurs(object(robot,Rid),putdown,T):occurs(putdown(Rid,_),T). % Effect of putting down a shelf state(object(shelf,Sid),on(node,Nid),T+1) :occurs(putdown(Rid,Sid),T), state(object(shelf,Sid),on(robot,Rid),T), state(object(robot,Rid),on(node,Nid),T).

%%% DELIVERY %%% {occurs(object(robot,Rid),for(order,Oid),deliver(Sid,Pid,EA),T):state(Oid,object(node,Nid),product_info(Pid,EA),T), state(object(product,Pid),object(shelf,Sid),with(quantity,PQ),T), EA=1..PQ}1:- Rid=1..NR, numRobots(NR), T=0..TN,TN=n-1. occurs(object(robot,Rid),deliver(Oid,Pid,EA),T):-occurs(object(robot,Rid),for(order,Oid),deliver(Sid,Pid,EA),T). state(Oid,object(node,Nid),product_info(Pid,EA-QQ),T+1):- occurs(object(robot,Rid),for(order,Oid),deliver(Sid,Pid,QQ),T), state(Oid,object(node,Nid),product_info(Pid,EA),T). state(object(product,Pid),object(shelf,Sid),with(quantity,PQ-QQ),T+1):occurs(object(robot,R),for(order,OI),deliver(Sid,Pid,QQ),T), state(object(product,Pid),object(shelf,Sid),with(quantity,PQ),T).

%%%%%%%%%%%%%%%%%%%%% GOAL %%%%%%%%%%%%%%%%%%%%% :- not state(Oid,object(node,Nid),product_info(Pid,0),T), state(Oid,object(node,Nid),product_info(Pid,_),0), Nid=1..NN, node_Num(NN), T=n.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% LAW OF INERTIA %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% state(object(robot,Rid),on(node,Nid),T+1):- state(object(robot,Rid),on(node,Nid),T), not occurs(object(robot,Rid),move(_,_),T), T<n. state(Oid,object(node,Nid),product_info(Pid,OQ),T+1):- state(Oid,object(node,Nid),product_info(Pid,OQ),T), state(object(product,Pid),object(shelf,Sid),with(quantity,PQ),T), not occurs(object(robot,_),for(order,Oid),deliver(Sid,Pid,_),T), T<n. state(object(product,Pid),object(shelf,Sid),with(quantity,PQ),T+1):- state(object(product,Pid),object(shelf,Sid),with(quantity,PQ),T), not occurs(object(robot,_),for(order,_),deliver(Sid,Pid,_),T), T<n. state(object(shelf,Sid),on(robot,Rid),T+1):state(object(shelf,Sid),on(robot,Rid),T), not occurs(putdown(Rid,Sid),T), T<n. state(object(shelf,Sid),on(node,Nid),T+1):state(object(shelf,Sid),on(node,Nid),T), not occurs(pickup(_,Sid),T), T<n.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% CONSTRAINTS %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% ROBOT %%%%%%%%%%%%%%%%%%%%% % Cannot move to more than 1 slot apart :not move(1,0;-1,0;0,1;0,-1). :- occurs(object(robot,R),move(DX,DY),T), DX>1, DX<-1, DY>1, DY<-1. % Within the range of grid :- occurs(object(robot,Rid),move(X1,Y1),T), state(object(robot,Rid),on(node,Nid),T), state(Nid,pair(X,Y)), X+X1>NC, num_Col(NC).

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:- occurs(object(robot,Rid),move(X1,Y1),T), state(object(robot,Rid),on(node,Nid),T), state(Nid,pair(X,Y)), Y+Y1>NR, num_Row(NR). :- occurs(object(robot,Rid),move(X1,Y1),T), state(object(robot,Rid),on(node,Nid),T), state(Nid,pair(X,Y)), X+X1<1. :- occurs(object(robot,Rid),move(X1,Y1),T), state(object(robot,Rid),on(node,Nid),T), state(Nid,pair(X,Y)), Y+Y1<1. % Cannot do pickup/putdown/deliver and move concurrently :- occurs(object(robot,Rid),move(X1,Y1),T), occurs(pickup(Rid,_),T). :- occurs(object(robot,Rid),move(X1,Y1),T), occurs(putdown(Rid,_),T). :- occurs(object(robot,Rid),move(X1,Y1),T), occurs(object(robot,Rid),for(order,_),deliver(_,_,_),T). % Cannot move under other shelves if a robot holds a shelf already :state(object(robot,Rid),on(node,Nid),T), occurs(object(robot,Rid),move(DX,DY),T), state(Nid,pair(X,Y)), state(object(shelf,Sid),on(robot,Rid),T), state(object(shelf,Sid_other),on(node,Nid_other),T), state(Nid_other,pair(X+DX,Y+DY)), Sid!=Sid_other, Nid!=Nid_other. % No two actions concurrently :- occurs(object(robot,Rid),Move,T), occurs(object(robot,Rid),Move_other,T), Move!=Move_other.

%%%%%%%%%%%%%%%%%%%%% PICKUP %%%%%%%%%%%%%%%%%%%%% % No 2 shelves on the same location :- state(object(shelf,Sid),on(node,Nid),T), state(object(shelf,Sid2),on(node,Nid),T), Sid!=Sid2. :- state(object(shelf,Sid),on(robot,Rid),T), state(object(shelf,Sid2),on(robot,Rid),T), Sid!=Sid2. % Cannot pick up the shelve more than 1 :- occurs(pickup(Rid,Sid),T), state(object(shelf,Sid_other),on(robot,Rid),T), Sid!=Sid_other. % Cannot take away the shelf from other robot :- occurs(pickup(Rid,Sid),T), state(object(shelf,Sid),on(robot,Rid_other),T), Rid!=Rid_other. % No 2 robots pick up the same shelf :- 2{occurs(pickup(Rid,Sid),T): robot(Rid)}, shelf(Sid). % Can pick a shelf only on the node where the shelf reside :- occurs(pickup(Rid,Sid),T), state(object(shelf,Sid),on(node,Nid),T), not state(object(robot,Rid),on(node,Nid),T).

%%%%%%%%%%%%%%%%%%%%% PUTDOWN %%%%%%%%%%%%%%%%%%%%% % Robot can putdown only if they holds a shelf. :- occurs(putdown(Rid,Sid),T), not state(object(shelf,Sid),on(robot,Rid),T). % No putdown a shelf on the highway :- occurs(putdown(Rid,Sid),T), state(object(robot,Rid),on(node,Nid),T), state(object(highway,Nid)).

%%%%%%%%%%%%%%%%%%%%% DELIVERY %%%%%%%%%%%%%%%%%%%%% % Robot can do deliver only when it holds a shelves containing product :- occurs(object(robot,Rid),for(order,Oid),deliver(Sid,Pid,_),T), state(object(product,Pid),object(shelf,Sid),with(quantity,_),T), not state(object(shelf,Sid),on(robot,Rid),T). % Robot can do deliver only when it is at corresponding pickingstation :- occurs(object(robot,Rid),for(order,Oid),deliver(_,Pid,_),T), state(Oid,object(node,Nid),product_info(Pid,_),T), not state(object(robot,Rid),on(node, Nid),T), state(object(pickingStation,PKid),object(node,Nid)). % Robot cannot do deliver more than product amount :- occurs(object(robot,Rid),for(order,Oid),deliver(Sid,Pid,QQ),T), state(object(product,Pid),object(shelf,Sid),with(quantity,PQ),T), QQ>PQ. % Robot cannot do deliver if order amount is larger than product amount :- occurs(object(robot,Rid),for(order,Oid),deliver(Sid,Pid,QQ),T), state(Oid,object(node,Nid),product_info(Pid,PQ),T), QQ>PQ.

%%%%%%%%%%%%%%%%%% ROBOT STATES % Cannot swap the location = no collision

%%%%%%%%%%%%%%%%%%

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:- state(object(robot,Rid1),on(node,Nid),T), state(object(robot,Rid1),on(node,Nid2),T+1), state(object(robot,Rid2),on(node,Nid2),T), state(object(robot,Rid2),on(node,Nid),T+1), Rid1!=Rid2. % No 2 robots are on same location :- 2{state(object(robot,Rid),on(node,Nid),T):node(Nid)}, robot(Rid), T=0..n. :2{state(object(robot,Rid),on(node,Nid),T):robot(Rid)}, node(Nid), T=0..n.

%%%%%%%%%%%%%%%%%% SHELF STATES %%%%%%%%%%%%%%%%%% % Shlef cannot be at both robot and node concurrently :- state(object(shelf,Sid),on(node,_),T), state(object(shelf,Sid),on(robot,_),T). :- state(object(shelf,Sid),on(node,Nid),T), state(object(shelf,Sid),on(node,Nid2),T), Nid!=Nid2. :state(object(shelf,Sid),on(robot,Rid),T), state(object(shelf,Sid),on(robot,Rid2),T), Rid!=Rid2. timestamp(N):-N=#count{T:occurs(A,B,T)}. #minimize{N: timestamp(N)}. #show occurs/3.

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