automated path planning of cooperative crane lifts using heuristic search

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Automated Path Planning of Cooperative Crane Lifts UsingHeuristic Search

PL. Sivakumar1; Koshy Varghese2; and N. Ramesh Babu3

Abstract: The use of cooperative cranes can improve the cost effectiveness of heavy lift operations. However, the compdeveloping a reliable lift plan prevents the widespread use of cooperative crane lifts. The availability of a computer-aided plannincan improve planning efficiency and reliability. Path planning is an important subtask of the lift planning process. This paperwork done to develop a computer aided path planner for two crane lifts. Two heuristic search methods, hill climbing and A*, wereimplemented for automating the path-planning task. Search space was represented using the concept of configuration space.tiveness of the search methods was evaluated by solving three problems with increasing levels of complexity. The formulationproblems was based on the type of movement of cooperative cranes~in synchronous or asynchronous manner! and the presence otrapping space. It was found that while the hill climbing approach found feasible paths in a few seconds or minutes, these pathsfrom optimal in situations containing trapping space. In contrast, the A* search resulted in near optimal paths, but the execution timeof the order of hours.

DOI: 10.1061/~ASCE!0887-3801~2003!17:3~197!

CE Database subject headings: Construction; Robotics; Construction equipment; Cranes; Project planning; Heuristics.

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using suitable control systems to ensure coordination amcranes.

Among the several tasks required to plan a crane lift, pplanning is a key task. Path planning involves finding the paththe object from a pick to a place configuration subjected to spaand nonspatial constraints among cooperative cranes, objectsite obstacles. The present work focuses on developing an amated path-planning system for cooperative crane lifts. This stem will assist the planner to prepare reliable lift plans moefficiently.

In attempting to automate the path-planning task, the authrealize that partial automation would be more practical for currneeds. However, in the future, complete automation wouldrequired, especially when working in harsh environments. Fther, elements of the fully automated approach can be used fopartial automated solution. Hence, the main objective of thevestigation was focused on attempting complete automationthe path planning of cooperative cranes.

The subobjectives required to achieve the main objecwere: ~1! identify key factors influencing the path-planning taof cooperative crane lifts;~2! select appropriate space represention and search techniques; and~3! formulate, implement, and tesalternative path-planning approaches.

Literature Review

Past works that form the basis for this study are in the followareas: heavy-lift planning, path planning of construction manilators, and robot motion planning.

Heavy-lift planning consists of several iterative steps. Amothem, eight steps have been identified as critical~Varghese et al.1997!. Attempts were made toward the development of comperized heavy-lift planning by both academic institutions and cstruction organizations~Brown and Root 1991; Hornaday et a1993; Lin and Haas 1996; Varghese et al. 1997!. These computer

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1Research Scholar, Dept. of Civil Engineering, I.I.T. Madras, Chen600 036, India.

2Associate Professor, Dept. of Civil Engineering, I.I.T. MadraChennai 600 036, India.

3Professor, Dept. of Mechanical Engineering, I.I.T. Madras, Chen600 036, India.

Note. Discussion open until December 1, 2003. Separate discusmust be submitted for individual papers. To extend the closing dateone month, a written request must be filed with the ASCE ManagEditor. The manuscript for this paper was submitted for review and psible publication on February 26, 2002; approved on December 30, 2This paper is part of theJournal of Computing in Civil Engineering,Vol. 17, No. 3, July 1, 2003. ©ASCE, ISSN 0887-3801/2003/197–207/$18.00.

Introduction

Cranes have become essential for lifting large prefabricated pcomponents in industrial construction and maintenance options. When the object is too heavy or too large for a sinconventional crane to handle, options such as the use of speassembled large capacity cranes or jacking systems are usedficulties associated with the use of specialized lift equipmentclude their limited availability, difficulty in transportation, anhigh utilization cost. In addition, these resources must be sculed months in advance. In such circumstances, the applicatiomultiple medium capacity cranes that are commonly availablconstruction sites to cooperatively carry out a lift task is a suitaalternative.

When two or more cranes are cooperatively used to liftobject, the risk is much higher than that of a single crane lift dto the interaction among cooperative cranes~Shapiro et al. 1991!.The risks associated with cooperative crane lifts can be catrized by the risk of human error in preparing lift plans and the rof uncoordinated motion during lift plan execution. These riscan be overcome by automating the planning tasks using cputer aided planning tools and automating lift plan execut

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aided planning tools contain a database of crane librariesprovide powerful built-in features such as visualization, interfence detection, capacity checking, and the selection of suitcrane/rigging assembly. However, none of these tools providerequired features for automated path planning of cooperacrane lifts.

A tool for automating path planning of single crane lifts wdeveloped using the configuration space~C-Space! concept andheuristic search procedures~Reddy 1997!. In addition to cranesattempts were also made to automate the planning task of oconstruction manipulators. A path planner based on dynamicgramming concepts was developed for a six degree of freepipe manipulator~Alciatore 1989!. A system called Path-Findewas developed based on a heuristic search method to findeffective paths for a generic construction manipulator~Moradet al. 1992!. The need and the applicability of large-scale manilators for piping erection, elevated concrete placement, painstructural-steel erection and demolition were evaluated~Hsiehand Haas 1993!. However, the scope of the previous studies wlimited to single manipulators.

Motion planning is a broad area that includes sensing, pning, and actuation~Latombe 1991!. Path planning, a subset omotion planning, is an active area of research in the fieldsrobotics and artificial intelligence. Schwartz and Sharir~1988!presented a detailed review of research on algorithmic andmetric aspects of motion planning. Fundamental issues on moplanning and different problem solving approaches weresented~Latombe 1999!. Hwang and Ahuja~1992! presented acomprehensive review on past work and various issues conered in motion planning. The different factors limiting the praccal applications of motion-planning concepts and approacheovercome them have also been addressed~Gupta and del Pobi1998!.

Among the research groups working on path planning inbotics, only a few are focusing on path planning of cooperamanipulators. Different types of cooperation and kinematic prlems when using cooperative robots for various assembly thave been analyzed~Dauchez et al. 1984!. A simple geneticalgorithm-based trajectory planning method was developedmultiple coordinating robots~Rana and Zalzala 1996; Sun et a1996!. An algorithm based on potential function was formulatfor path planning of two cooperative manipulators~Mohri et al.1996!.

From the above, it can be seen that path planning is an aresearch area in robotics. However, only few studies have focon path planning of cooperative robot manipulators. A few uniqcharacteristics that make a cooperative crane lift problem difent from conventional robotics include the following:~1! cranescan have telescoping capability, which is not common in rob~2! heavy loads are freely suspended from the hooks of the crthrough steel wire ropes; whereas, relatively lesser weight obare gripped by the end effector of robot manipulators;~3! hoistingis a unique operation in the case of crane manipulators; and~4!for cranes that are capable of traveling while loaded, the degof freedom~DOF! are based both on manipulating capability amobility. In most robotic situations, the DOF is based eitherthe manipulating capability or mobility. Further, path planningcooperative crane lifts is a computationally challenging probbecause of the high DOF of the system and the need to mcooperation among the cranes. No attempt has been made ttomate the path planning of cooperative crane lifts.

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Problem Formulation

In order to develop a path planner for cooperative crane lseveral problem-specific aspects such as DOF, nature of mment, slope of the load line~hoisting rope!, and the criteria toidentify a best lift path need to be considered. In addition, apppriate assumptions need to be made to simplify the problemmulation and thereby to minimize the complexity of paplanning algorithms based on the constraints on computatitime.

Degrees of Freedom

Different DOF of a real world lattice boom crane are shownFig. 1. These DOF include:~1! location and orientation of thebase in plan, i.e.,Cx, Cy, andRb , respectively;~2! swinging ofthe cab@f#; ~3! luffing of the boom@u#; ~4! telescoping action othe boom@bL#; ~5! hoisting @hL#; and ~6! slope of the load lineabout thex andy axes@huX ,huY#. A configuration set represening a unique configuration of the crane is expressed@Cx,Cy,Rb ,f,u,bL ,hL ,huX ,huY#.

The scope of the present work is limited to fixed base latboom cranes. In addition, both base and cab are modeledsingle entity. Configuration variables@huX ,huY#, which representthe slope of the load line are not modeled as DOF becauseare kept within the permissible limits in order to keep to a mimum the additional load transferred to each crane~Shapiro et al.1991!. Hence, the configuration set of a single crane is simpliinto @f,u,hL#. This work focuses on path planning of a coopetive crane lift using two cranes. Therefore, the corresponding cfiguration set is expressed as$@f,u,hL#C1 ,@f,u,hL#C2% whereC1 and C2 represent crane 1 and crane 2, respectively. Tparticular configuration is referred to as crane 233 manipulator,because it consists of two cranes, each with three DOF.

Nature of Movement

During the lift, the movement of cooperative cranes can be eisynchronous or asynchronous. Synchronous movement refean identical movement of cooperative cranes between pickplace locations. For example, the two successive configura$@90,45,2#C1 ,@90,45,2#C2% and$@90,45,5#C1 ,@90,45,5#C2% repre-sent a synchronous hoisting operation from two to five units.

In the case of asynchronous movement, cooperative crwill not move identically. For example, consider the two succsive configuration sets $@140,55,2#C1 ,@30,40,2#C2% and

Fig. 1. Degrees of freedom of lattice boom crane

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$@150,55,2#C1 ,@30,45,2#C2%. Here, the first crane performsswing operation from 140° to 150°; whereas, the second crperforms a luffing operation from 40° to 45°. The present woconsiders both synchronous and asynchronous movementplanning the path.

Slope of Load Line

The slope of the load line is a common occurrence in casesmultiple crane lifts due to cooperation constraints. When the lolines are not vertical, additional load is transferred to each cr~Shapiro et al. 1991!. Hence, this slope and the resultant addtional load must be kept within permissible limits. This has betaken into account implicitly in modeling cooperation among tcranes.

Criteria for Best Lift Path

A lift path must have a minimum number of swing, luff, and hoioperations~Shapiro et al. 1991! to minimize the work done by thecrane. Apart from this, several other factors influence the feasity and optimality of the path such as~1! priority among swing,luff, and hoist operations based on the ease of lift executionthe crane operator;~2! permissible limit for the capacity utiliza-tion of each crane;~3! minimum clearances needed betweecranes, load, and site obstacles; and~4! avoiding lifts over dan-gerous operating areas. In reality, capacity utilization and cleances determine the feasibility and optimality of a lift path whother factors primarily influence the optimality of the path. In thstudy, path planning is based on the work done by the maniptor and the clearances required between the cranes, obstaclesobject. A check for capacity is not included in the present woHowever, it can be incorporated into the path-planning algoritwithout any major change at a later stage.

Assumptions

The assumptions that were made in formulating the cooperacrane lift problem include:• Both cooperating cranes are identical;• Both cranes are assumed to be located at the same level;• The object is symmetric, and the load is distributed to ea

crane equally; and• All obstacles in the construction site are static.In addition, certain simplifications that were made in representthe geometry of cooperative cranes include:• Both base and cab are integrated into a solid of rectang

shape;• Both boom and obstacles in the construction site are solid

rectangular shape; and• The geometry of the object lifted by cooperative cranes

assumed to be a line.Fig. 2 shows the simplified geometry of cooperative cran

based on the previous assumptions.In summary there are five key attributes of the problem.

1. Crane: two identical cranes working cooperatively; threegrees of freedom for each crane—swinging, luffing ahoisting; both synchronous and asynchronous motion is csidered; and limits of slope of load line can be specified.

2. Load: symmetric and load are distributed equally to eacrane and lifted at the ends.

3. Site: contains static obstructions to cranes and load.4. Feasibility criteria: collision free path.

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5. Optimality criteria: minimize work done.Even with these simplifications, the cooperative crane problecomplex. The assumptions were required to permit the authofocus on issues such as degrees of freedom and spacerepresentation. These assumptions limit the practical applicaof the current prototype system. However, they do not compmise the investigation of the proof-of-concept of the solutionproach.

Solution Methodology

The problem defined has a total of six degrees of freedomgeneral, path planning for manipulators having more than fDOF is considered to be complex~Hwang and Ahuja 1992!.

Although a wide variety of solution approaches are availafor simple path-planning problems, few have been successapplied to solve complex problems. To deal with the complexof the current problem in a systematic manner, three benchmmanipulator systems of increasing complexity were formulaThese problems are planar 132 manipulator, planar 232 manipu-lator, and planar 233 manipulator and are shown in Fig. 3.planarN3M manipulator represents a system that consists oNmanipulators, each withM DOF. Each solution strategy was intially evaluated on the benchmark systems. This enabledevaluation of the scalability of different approaches and fituning of selected approaches to solve the more complexcrane system.

Solution Concepts

There are three key concepts in formulating a path planninggorithm:~1! representation of search space;~2! selection of searchtechnique; and~3! selection of search option.

Representation of Search Space

Search space can be represented either in real spaceC-Space. Real space is the cartesian space withX, Y andZ axes.In C-Space, each axis represents a DOF of the manipulHence, each configuration of the manipulator is representedpoint in C-Space. Udupa~1977! first identified this idea of shrinking the configuration of the manipulator to a point. Later, it wsystematically analyzed and popularized with the term C-Sp~Lozano-Perez and Wesley 1979; Lozano-Perez 1983!. The maxi-mum number of dimensions in real space is three; whereas,equal to the DOF of the manipulator in C-Space.

Key advantages of C-Space over real space representatioclude: ~1! Each point refers to a unique configuration of the m

Fig. 2. Simplified geometry of two cooperative cranes


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Fig. 3. Benchmark Problems:~a! Planar 132 manipulator;~b! Planar232 manipulator;~c! Planar 233 manipulator

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nipulator because it is expressed in terms of configuration vables. Hence, there is no redundancy in representingconfiguration;~2! Constraints on the manipulator movement cbe expressed specifically; and~3! There is no need to solve inverse kinematic problems because the path is in terms of conration variables. Hence, C-Space was selected to represensearch space.

Fig. 4~a! shows a planar manipulator with two angular degreof freedom. As this manipulator is planar, the Cartesian spaclimited to two dimensions. Independently, because there aretwo degrees of freedom, the configuration space is representtwo dimensions. Figs. 4~a and b! show two configurations—@135,45# and@45,135# of the manipulator for the same end effetor positionP(x,y). Fig. 4~c! shows these configurations reprsented as points in C-Space. A configuration in the C-Spacebe either feasible or infeasible. When there is interference orvalues of DOF are out of their respective ranges, the configurais said to be infeasible.

The dimension of the C-Space for the cooperative crane plem is six, because there are six DOF. For a cooperative cconfiguration to be feasible the slope of the load line shouldwithin the permissible limit; there should be no interferen

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among cranes, object, and obstacles; and the range for eachshould be within the permissible limits.

Search Techniques

Three types of artificial intelligence based search techniquesclude exhaustive search~systematic exploration!, heuristic search~guided exploration!, and probabilistic search~Johnson and Pic-ton 1995!.

Exhaustive search techniques explore the search space syatically without any guidance. Depth first and breadth fisearches belong to this category. This search is inefficient becit does not evaluate the proximity of the goal node at each stepcontrast, a heuristic search attempts to find the most promisdirection at each step based on certain rules. Hill climbing and*

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are two popular heuristic search techniques. Probabilistic setechniques such as simulated annealing and genetic algorithmlize randomness in addition to certain guidelines. One of themerits of these search techniques includes the ability to esfrom local minima. Among the search techniques, heuristicprobabilistic searches have the potential to consistently fingood solution within a short time.

Attempts were made to apply genetic algorithms for benmark problems such as planar 132 manipulator and planar 232manipulator~Sivakumar et al. 1999, 2000!. Although the initialresults were good for simple problems, they were not satisfacfor complex problems. In addition, the application of these teniques to complex problems needs fine-tuning of certain pareters based on experimental data and observations. Hencepresent work focuses on investigating the applications of heursearch techniques. A detailed description of various heurisearch techniques has been presented by Rich and Knight~1991!.Among the various heuristic search techniques, hill climbsearch and A* search were selected for this work.

Search Options

For the selected search space representation and search techa search can be carried out either as a tree search in open spas a graph search in feasible space. Search in the open srefers to a search in a space containing both feasible and insible configurations. Hence, feasibility checks are performed ding the search for only those configurations that are exploduring the search. In the case of a search in the feasible spfeasible configurations are first identified by exhaustively testall configurations of cooperative cranes. Next, the connectirelationship is established among the feasible configurations,the path is found by searching the graph containing feasible cfigurations. As precomputation of feasible C-Space can be cputationally expensive for higher dimensions of C-Space; a tbased search in open C-Space was taken as a search option

Formulation and Implementation of Path PlanningAlgorithms

Based on the concepts selected in the previous section, two pplanning algorithms were formulated and implemented: trbased hill climbing search in open C-Space and tree based*search in open C-Space. The path planner developed usingtwo algorithms consists of three key modules: input, path seaand path display.

Module I: Input

Input data consist of location and dimensions of each cralength of the object, number of obstacles, location and dimensof each obstacle, pick and place configurations, step size forDOF, number and the DOF that vary concurrently betweensuccessive steps, and tolerance used to limit the slope of theline. These data are entered into an ASCII text file, and it is rby the system using a C11 code.

Module II: Path Search

Hill climbing and A* search were used to find the feasible/neoptimal path. Because hill climbing is a local search, it keeps othe information about the current node and its neighbors attime during the search. In contrast, A* is a global search. It retain

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information of all the configurations that are explored and tconnectivity relationship among them until reaching the placonfiguration. Linked lists were used to retain the connectivrelationship during the implementation of the A* search.

Hill Climbing Search

The logic for a hill climbing search is as follows:1. Set the current crane configuration to the pick configurati2. Generate neighbors for the current configuration;3. Perform feasibility check for the neighbors generated;4. Identify the feasible neighbors;5. Compute the heuristic search function for the feasible nei

bors;6. Choose the best neighbor based on the heuristic func

value;7. Set the current configuration to the neighbor chosen;8. If the current configuration is equal to the place configu

tion, stop, and go to step 9 or else go to step 2; and9. Print the path.

A* Search

The logic for A* search is as follows:1. Declare open list and closed list to store the nodes that

expanded and ready for expansion;2. Compute the heuristic function value of the pick config

ration and add it to the open list;3. Identify the best configuration in the open list;4. Remove that configuration from the open list and add it

the closed list;5. If the best configuration is equal to the place configurati

go to step 9;6. Generate neighbors for the best configuration and perfor

feasibility check for all;7. Compute the heuristic function value for each feasib

neighbor and add it to the open list;8. Point these neighbors toward the best configuration to

tain the parent-child relationship, and go to step 3;9. Trace the path from place to pick configuration; and

10. Print the pathWhen a configuration is added to the open or closed lists ofA* search, there are chances for a copy of that node to be in thlists. In such cases, the best between the two is retained, andchanges in the parent-child relationship are done if necessardetailed note on this has been presented by Rich and Kn~1991!.

Details of Steps Common to Hill Climbing and A *Search

Three important steps common to hill climbing and A* searchesinclude the generation of neighbors, the feasibility check, andcomputation of the heuristic search function. Among these,implementation of the first two steps are the same for theclimbing and the A* searches; whereas, the third step is differe

Generation of Neighbors

In order to find the most promising direction at each step, a fixnumber of neighbors are generated for each configuration.total number of neighbors generated for each configurationpends on the number and the type of DOF that vary between


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order to ensure cooperation among the cranes. Hence, ipresent work, only two DOF are allowed to vary between sucsive steps—one DOF for each crane. Each crane can havone of the following six movements:

$f1Df,u,hL%, $f2Df,u,hL%, $f,u1Du,hL%

$f,u2Du,hL%, $f,u,hL1DhL%, $f,u,hL2DhL%

In synchronous movement, both cranes vary the sameand move in the same direction by the same magnitude.results in six neighbors for each configuration, as follows:

$@f1Df,u,hL#C1 ,@f1Df,u,hL#C2%

$@f2Df,u,hL#C1 ,@f2Df,u,hL#C2%

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Table 1. Input Data for Crane 233 Manipulator Problems

Input variables

Problem Ibased on synchronized

movement

Problem IIbased on asynchronized

movement

Problem IIIbased on asynchronize

movement

Location @X,Y,Z#

Crane-1 @65,60,0# @5,20,0# @5,20,0#Crane-2 @95,60,0# @45,20,0# @45,20,0#

Dimensions@L,B,H#

Base @10,8,2.5# @10,8,2.5# @10,8,2.5#Boom @20,3,3# @20,3,3# @20,3,3#

Object length 30 10 10Number of obstacles 1 1 3Obstacles details:Coordinates of opposite corners A@~17,12,0!,~27,15,5!#

@~55,78,0!,~105,80,6!# @~15,16,0!,~30,24,6!# B@~20,18,3!,~24,22,12!#@(X1 ,Y1 ,Z1),(X2 ,Y2 ,Z2)# C@~17,25,0!,~27,28,5!#Pick configuration @~90,60,1!,~90,60,1!# @~140,55,2!,~30,40,2!# @~140,55,2!,~30,40,2!#Place configuration @~90,30,1!,~90,30,1!# @~180,65,8!,~0,55,8!# @~220,55,2!,~330,40,2!#

successive steps. This is based on what combinations of Dvariations are possible for cooperative cranes during its moment from one step to another step.

Consider@X,Y#, a two-dimensional node~configuration! forwhich neighbors have to be generated. LetDX, DY be the step-size values forX and Y, respectively. If only one DOF variesbetween two successive steps, the number of neighbors willfour, and they are$@X1DX,Y#, @X2DX,Y#, @X,Y1DY# and@X,Y2DY#%. If only two DOF vary, the number of neighbors willbe four, and they are$@X1DX,Y1DY#, @X1DX,Y2DY#, @X2DX,Y1DY#, @X2DX,Y2DY#%. If either one or two DOFvary, there can be eight neighbors, which will be the combinatiof the above two sets of neighbors.

In general, for a multiple crane lift, the number of DOF varying between two successive steps should be kept to a minimum

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Check for InterferenceThere can be four possible combinations of interference in tcooperative crane system. These include crane component to locrane component to obstacles in the construction site, loadobstacles, and crane boom to crane boom. In order to identinterference in the cooperative crane system due to any one ofabove four combinations, two basic interference detection rotines were developed using C11 based on computational geom-etry concepts. These two routines can find the interference btween a line-to-plane and a line-to-box. Each component of tcrane, object, and obstacles was modelled in such a way that thtwo basic routines are used to check for the collision in the coperative crane system.

Check for Limits of DOFDuring crane movement, each DOF should be kept within therespective permissible ranges. In general, a crane can swthrough 360°. The limits for luffing and hoisting basically depenon the crane’s specifications. These limits are interdependent aare also governed by the interference between load-to-boom aload-to-ground. A procedure to monitor the limits of the DOF ialso incorporated.

Computation of Heuristic Function

The path planner uses a heuristic function to find the most proising direction in order to move from one step to another. Amonthe several criteria considered for arriving at the best lift path, thpresent work utilizes only the work done by the manipulato

Fig. 7. Path found using A* search for Problem I:~a! Isometric view;~b! plan

$@f,u1Du,hL#C1 ,@f,u1Du,hL#C2%

$@f,u2Du,hL#C1 ,@f,u2Du,hL#C2%

$@f,u,hL1DhL#C1 ,@f,u,hL1DhL#C2%

$@f,u,hL2DhL#C1 ,@f,u,hL2DhL#C2%

In asynchronous movement, each crane can vary any DOF. Acombinations~636! were considered in this work. Hence, therewill be a total of 36 neighbors for each configuration.

Feasibility Check

The feasibility of cooperative crane configuration depends othree factors: cooperation among the cranes, interference, andlimits for DOF.

Check for CooperationThe two cranes must handle the object cooperatively during themovement from pick to place configuration. A perfect coordination between the two cranes refers to a situation where the dtance between the boom tips of the two cranes is always equathe length of the object carried by them. However, this is noalways possible in practice. Hence, there is a difference betwethe object length and the distance between the boom tips. Tresults in a slope of the load line and the subsequent transfer ofadditional load to the crane in proportion to the slope. To ensuacceptable cooperation, the slope of the load line and the adtional load transferred due to this slope should be kept withpermissible limits.


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Fig. 8. Path found using hill climbing search for Problem II:~a! Isometric view;~b! plan

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Results and Discussion

The performance of path-planning approaches is illustrated uthree test problems. For each problem, both hill climbing and*are applied, and the results are compared. Of the three probthe first one is based on synchronous movement; whereassecond and third are based on asynchronous movement. Tapresents the summary of input data for these problems.

The step values used for swinging, luffing, and hoisting wset at 10°, 5°, and 2 units respectively. Ideally, the step-size vfor each DOF are based on the required least count of hook mment. A high step resolution should be used for paths thaexecuted by automatic control systems; a low resolution caused for manual execution. In general, very high resolutionrequire computational time in terms of hours and even dwhereas, very low resolution may result in poor coordinaamong the cranes. Because the present work focuses on opath planning where the planning is automated and the execis manual, the lower resolution is acceptable. However, oneiting assumption in utilizing the lower resolution is that all ostacles on site are adequately large to be captured in theresolution C-Space.

Problem I: Synchronous Movement

For this problem, simple pick and place configurations werelected because the path does not exist for complex situausing synchronous movement. The problem requires the cranlift the load from one side of a wall~obstacle! to the other sideFigs. 6 and 7 show the details of the load path generated bclimbing and A*, respectively.

The hill climbing search took 37 steps to reach the placefiguration, and 32 steps were required to climb over the obst

Fig. 9. Path found using A* search for Problem II:~b! Isometric View;~b! Plan

Work done is estimated as the Euclidean distance travelled by thobject in real space corresponding to the movement of coopertive cranes between pick and place configurations. Euclidean ditance is commonly used to estimate the path cost because ofsimplicity and scope for generating better solutions for a widerange of problems. However, it may be noted that this functiononly gives an approximate measurement on the quality of a paand does not include any problem specific heuristics.

In a hill climbing search, the value of the heuristic function isequal to the Euclidean distance between the current configuratioand the place configuration. In contrast, in the A* search, it isequal to the sum of the actual distance from pick to current configuration and the Euclidean distance between current configurtion to place configuration.

Module III: Path Display

Path refers to a step-by-step movement of cooperative cranes. Tpath for the crane 233 manipulator is expressed as follows:

@C1,C2#PICK , @C1,C2#1 , @C1,C2#2 ,

@C1,C2#3 , @C1,C2#N21 , @C1,C2#N , @C1,C2#PLACE

In the above expression,N represents the number of stepsdenoting the path and@C1,C2# represents a unique configurationof cooperative cranes, which is$@f,u,hL#C1 ,@f,u,hL#C2%. Tosimulate the path generated, a 3D visualization platform was developed in AutoCAD. As a part of this platform, customized tool-bars were developed to control each DOF of cooperative craneusing Object ARX~Autodesk 1999!. Fig. 5 shows a screen snap-shot of this platform. In addition to path simulation, this platformcan also be used to visually check cooperation and interferencamong cranes.

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Fig. 10. Pick and place locations for Problem III:~a! Pick configuration@~140,55,2!#, @~30,40,2!#; ~b! place configuration@~220,55,2!#,@~330,40,2!#

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Problem III: Asynchronous Movement

The pick and place configuration for this problem is shown in Fi10. It requires the cranes to pick the load from the ground at olocation and move it to the place location while avoiding threobstacles: A, B, and C. It should be noted that there is free spbelow obstacle B. Figs. 11 and 12 show the details of the lopath generated by hill climbing and A*, respectively.

Hill climbing took 31 steps to reach the place configuration.can be seen in Fig. 11 that the hill climbing search was misdrected by the free space below obstacle B. Although the pagenerated by hill climbing is feasible in this case, it is not optmal.

The A* search took 25 steps to reach the place configuratioIt can be seen from Fig. 12 that the A* search avoids the freespace below obstacle B, and it proceeds toward the place confiration along the side of the obstacle.

Comparison of Search Approaches

From the solutions of Problem III shown in the previous sectioit can be seen that the performance of the A* search is better thanhill climbing when there is trapping space in the search spaWhen there is no trapping space, as in Problem II, there is nmuch difference between the paths generated by hill climbing aA*. The performance of the hill climbing and A* searches can becompared based on factors such as search time, path cost,search complexity. Table 2 presents details of comparison forthree test problems based on these factors.

The time required for finding the path depends on the seatechnique used, number of neighbors generated for each noand the presence of trapping space. It can be seen from Tab

Fig. 11. Path found using hill climbing search for Problem III:~a!Isometric view;~b! plan

Fig. 12. Path found using A* search for Problem III:~a! Isometricview; ~b! plan

This is because hill climbing is guided by a heuristic that selecthe next configuration based on the Euclidean distance betwethe next potential configuration and the place configuration. Thheuristic is not always correct because, in some cases, a seastep might have to be directed away from the place configuratioto avoid obstacles in an optimal manner.

As this problem illustrates, to reach the place location the oject must be lifted over the obstacle. However, instead of directmoving up, the hill-climbing search explores all the configurations near the face of the obstacle that are closer to the placonfiguration at each step upward. Thus, numerous steps arequired to climb the obstacle. Situations such as this, which milead and trap the search, are said to contain trapping spaceswas noted that the presence and size of trapping spaces depenthe location of the pick and place configuration as well as thlocation and orientation of the obstacle.

It can be seen from Fig. 7 that the A* search did not gettrapped and took only eight steps to reach the place configuratio

Problem II: Asynchronous Movement

For this problem, the cranes that are located on either side ofobject must pick the load from the ground and place it on top othe object. Figs. 8 and 9 show the details of load path generatby hill climbing and A* respectively. Both hill climbing and A*took 10 steps to reach the place configuration. In this case, tperformance of hill climbing is almost equal to the A* searchbecause there is no trapping space to mislead the hill climbinsearch.

L OF COMPUTING IN CIVIL ENGINEERING © ASCE / JULY 2003 / 205

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Table 2. Comparison of Search Approaches for Crane 233 Manipulator Problems

Input variables

Problem I Problem II Problem III

Hillclimbing A*

Hillclimbing A*

Hillclimbing A*

Number of steps representing the path~excluding pick and place location!

37 11 8 8 31 25

Path cost 149.1 38.6 38.5 37.8 46.9 41.5Time 42 s 1 min

29 s52 s 1 h

4 min2 min 2 h

24 minNumber of nodes tested:@Hill climbing: ~Number of steps11!3Numberof neighbors; A* : Number of nodes in theclosed list3Number of Neighbors#

228@3836#

450@7536#

324@9*36#

19,476@541336#

1,152@32*36#

99,972@2777336#

Number of directions that A* wassimultaneously looking for when itreached the place configuration@Number of nodes in the open list#

— 73 — 605 — 509

Note: Computations were carried out on a HP Kayak Work Station with P-II 400 MHz processor and 128 MB RAM. All search approaches are tresearch in open C-Space. Number of search directions is not given for hill climbing because it is a sequential search.

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In this paper, two methods, a tree-based hill climbing searchopen C-Space and a tree-based A* search in open C-Space, wereimplemented and tested for automating the path planning task fa two-crane lift. The effectiveness of these search methods wassessed by solving problems that required the synchronous aasynchronous movements of cooperative cranes with single amultiple obstacles.

Of the two heuristic search methods evaluated, hill climbinwas found to be effective in generating feasible solutions verquickly when the environment does not contain any trappinspace. In contrast, the A* search was found to generate near optimal paths in an environment that contains trapping space. However, the A* search took a considerably longer time, even withcoarse resolution for each degree of freedom. Further, if addtional cranes are added to the system or if addition degreesfreedom such as telescoping are included, the dimension of tC-Space will increase, and there will be an exponential increain computational time.

Although the current implementation has established the proof concept in using the proposed approaches for automating tpath-planning task of cooperative crane lifts, it cannot be usedirectly for automated planning in practical cases because of tfollowing limitations: ~1! The geometry of crane and load wassimplified; ~2! The heuristic search is only based on distance; an~3! Issues such as unsymmetrical loads and its distribution on twcooperative cranes are not included.

In its current form, the prototype system can be used to amanual planning. By using the system, alternative macro paoptions can be generated, and these can be detailed manuallythis mode, it would reduce conventional planning time as it caeliminate exploration of infeasible paths at a preliminary stage.

Further work in this area is focused on overcoming the limitations mentioned previously. In addition, investigations are underway to improve the speed of execution when performingsearch in a finer resolution. Higher dimensions of C-Space thconsider telescoping and mobility of crane components are alunderway. This includes experiments with powerful hardwareimproved heuristics based on clustering techniques, parallel prcessing, and alternate search methods such as genetic algorith

The current work is a preliminary step toward developing aautomated path planner for complex systems. It is likely that

that the time spent by the hill climbing search is less than the timspent by A* for all the test cases. This is because hill climbingconsiders only the neighbors of the current configuration to movfrom each step; whereas, A* retains and compares informationabout all the configurations that it tests until reaching the placconfiguration. Although the A* search takes more time than thehill climbing search, the path it generates will be optimal andhence has a lower cost.

The complexity of a search is based on the total number oconfigurations tested as well as total number of alternative diretions that the search is simultaneously evaluating. In the caseProblem III, the hill climbing search tested 1,152 configurationswhereas, the A* search tested a total of 99, 972 configuration~Table 2!. The A* search was evaluating 509 alternate directionin the search space when it reached the place configuration. Tnumber of alternative search directions is not presented for hclimbing because it does not search in multiple directions.

It can be seen from the figures that there are a few jerks in thpath generated by the A* search for the asynchronous case. Thiarises due to a combination of grid size and the specificationthe search heuristic. To avoid this, the heuristic can be formulateto penalize any jerks in the motion, and the search can be coducted using a finer grid resolution.

Conclusions

The use of cooperative cranes can improve the cost-effectiveneof heavy lifts. However, the complexity in developing a reliablelift plan prevents the widespread use of multiple crane lifts. Thavailability of a computer aided planning system can improve thefficiency and reliability of the lift plan. The work presented inthis paper is a preliminary step toward the development of suchsystem.

Path planning is an important subtask of the lift-planning process. The automation of the path-planning task will assist the lplanner to develop a reliable lift plan. Further, the use of cooperative manipulators is of interest in other areas, such as manfacturing. The approaches used for path planning of cooperaticranes can also be applied for cooperative manipulators in othareas.

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will take a number of years of research and development bethe planner can be used autonomously in practical applicatiHowever, once developed, it will have widespread applicationa number of domains.

Acknowledgments

The writers would like to acknowledge the financial support pvided by the Department of Science and Technology, New DeIndia, @No. III. 5~110!/97-ET and III. 5~23!/2000-ET# for this re-search work.

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