a fast path planning method for single and dual crane erections

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A fast path planning method for single and dual crane erections Yu-Cheng Chang, Wei-Han Hung, Shih-Chung Kang Department of Civil Engineering, National Taiwan University, Taiwan abstract article info Article history: Accepted 27 November 2011 Available online 2 January 2012 Keywords: Crane Erection Dual crane Path planning Robotic This research aims to develop a method for planning the erection path automatically and efciently. The pro- posed method is comprised of two steps. The rst step is to convert the scene of the crane erection into a con- guration space, in which the crane's load capacity and the obstacles in the environment have been included. The second step is to nd a collision-free path in the conguration space by using the probabilistic road map (PRM) method. Three tests were conducted to validate the action, crane placproposed method in this re- search. The results show that the proposed method is efcient, and can generate effective erection paths for operating in near real-time scenarios. The method is appropriate for both single and dual crane erection, and can help engineers plan more easily, and verify erection-planning decisions such as crane seleement, and logistcs. © 2011 Elsevier B.V. All rights reserved. 1. Introduction In addition to the usual single crane erection, dual-crane coopera- tive erection has become more common in modern construction pro- jects in recent years. Particularly, in industrial construction, it is often necessary to transport large facilities; this requires cranes with a ca- pacity of 5001000 t. Although available large cranes can move weights of up to 1300 t, their operation may be restricted by the lim- ited space on site, while the construction costs will increase if a larger and more expensive crane is rented. A useful, often used alternative is to utilize two less expensive cranes to perform a cooperative crane erection [10,17]. However, in the cooperative dual crane erection pro- cess, the two cranes need to work together to maintain the equilibri- um of heavy loads. The complexity of cooperative dual crane erection is far higher than that of single crane operation, and this can lead to high risk situations during construction [26]. If a feasible and safer erection path can be pre-planned for cooperative dual crane erection, then such high risk situations can be reduced to a minimum; further justifying the importance of erection path planning. Erection path planning is a complex topic, and there are three major difculties involved. Firstly, the load of the crane should be within its lift- ing capacity during the erection process; the upper limit of the capacity varies with the angle of the boom, which makes planning more difcult. Secondly, collisions among the crane, the lifting object, and any obstacle should be avoided. This makes the planning difcult when there are nu- merous obstacles on site, and the volume of the lifting object is large. Thirdly, the cable of the crane must be kept vertical plumbed during a co- operative dual crane erection in order to avoid increased tension from when the cable is inclined, as increased tension would increase the load on the crane. Erection path planning in the past was limited by computational efciency, and was applied only to path planning for moving in a sim- ple environment [19] or motion planning with low degrees of free- dom [15]. However, due to the rapid development in computing technology, computational performance has improved, and path planning, which was not possible in the past is becoming feasible now. Path planning includes planning for motions of high degrees of freedom [12], instantaneous avoidance of obstacles [13], and cooper- ative dual crane erection [22]. Due to advances in the path planning method, many studies have focused on erection path planning. Computers are used to create 3D models of construction sites and cranes; check collision in a virtual construction site; and estimate the lifting capacity of the crane. Thus, a feasible collision free erection path can be planned [25,2,11]. The method proposed in Refs. [11] and [25] effectively planned the erection path, but it can only be used for single crane operations. Wang et al. [31] proposed a sampling-based method for real-time motion planning of cranes in order to improve construction safety. Zhang et al. [28] utilized the Ant Colony algorithm to nd a collision-free-path for a mobile crane with consideration for both ef- ciency and safety. These studies have demonstrated the potential and value of utilizing path-planning technique for single crane erec- tion. The method mentioned in Ref. [2] is applicable to cooperative dual crane erection planning. It is computationally the most effective method at present for cooperative dual crane erection planning, but it still takes between 3 and 12 min to calculate each plan. Therefore, this research aims at developing a near real-time and automated method for erection path planning. The method can be ap- plied for both single crane and cooperative dual crane erection oper- ations, and can be used to nd the most suitable erection path in the Automation in Construction 22 (2012) 468480 Corresponding author. E-mail address: [email protected] (S.-C. Kang). 0926-5805/$ see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.autcon.2011.11.006 Contents lists available at SciVerse ScienceDirect Automation in Construction journal homepage: www.elsevier.com/locate/autcon

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Page 1: A fast path planning method for single and dual crane erections

Automation in Construction 22 (2012) 468–480

Contents lists available at SciVerse ScienceDirect

Automation in Construction

j ourna l homepage: www.e lsev ie r .com/ locate /autcon

A fast path planning method for single and dual crane erections

Yu-Cheng Chang, Wei-Han Hung, Shih-Chung Kang ⁎Department of Civil Engineering, National Taiwan University, Taiwan

⁎ Corresponding author.E-mail address: [email protected] (S.-C. Kang).

0926-5805/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.autcon.2011.11.006

a b s t r a c t

a r t i c l e i n f o

Article history:Accepted 27 November 2011Available online 2 January 2012

Keywords:CraneErectionDual cranePath planningRobotic

This research aims to develop a method for planning the erection path automatically and efficiently. The pro-posed method is comprised of two steps. The first step is to convert the scene of the crane erection into a con-figuration space, in which the crane's load capacity and the obstacles in the environment have been included.The second step is to find a collision-free path in the configuration space by using the probabilistic road map(PRM) method. Three tests were conducted to validate the action, crane placproposed method in this re-search. The results show that the proposed method is efficient, and can generate effective erection pathsfor operating in near real-time scenarios. The method is appropriate for both single and dual crane erection,and can help engineers plan more easily, and verify erection-planning decisions such as crane seleement, andlogistcs.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

In addition to the usual single crane erection, dual-crane coopera-tive erection has become more common in modern construction pro-jects in recent years. Particularly, in industrial construction, it is oftennecessary to transport large facilities; this requires cranes with a ca-pacity of 500–1000 t. Although available large cranes can moveweights of up to 1300 t, their operation may be restricted by the lim-ited space on site, while the construction costs will increase if a largerand more expensive crane is rented. A useful, often used alternative isto utilize two less expensive cranes to perform a cooperative craneerection [10,17]. However, in the cooperative dual crane erection pro-cess, the two cranes need to work together to maintain the equilibri-um of heavy loads. The complexity of cooperative dual crane erectionis far higher than that of single crane operation, and this can lead tohigh risk situations during construction [26]. If a feasible and safererection path can be pre-planned for cooperative dual crane erection,then such high risk situations can be reduced to a minimum; furtherjustifying the importance of erection path planning.

Erection path planning is a complex topic, and there are three majordifficulties involved. Firstly, the load of the crane should bewithin its lift-ing capacity during the erection process; the upper limit of the capacityvaries with the angle of the boom, which makes planning more difficult.Secondly, collisions among the crane, the lifting object, and any obstacleshould be avoided. This makes the planning difficult when there are nu-merous obstacles on site, and the volume of the lifting object is large.Thirdly, the cable of the cranemust be kept vertical plumbedduring a co-operative dual crane erection in order to avoid increased tension from

rights reserved.

when the cable is inclined, as increased tension would increase theload on the crane.

Erection path planning in the past was limited by computationalefficiency, and was applied only to path planning for moving in a sim-ple environment [19] or motion planning with low degrees of free-dom [15]. However, due to the rapid development in computingtechnology, computational performance has improved, and pathplanning, which was not possible in the past is becoming feasiblenow. Path planning includes planning for motions of high degrees offreedom [12], instantaneous avoidance of obstacles [13], and cooper-ative dual crane erection [22].

Due to advances in the path planning method, many studies havefocused on erection path planning. Computers are used to create 3Dmodels of construction sites and cranes; check collision in a virtualconstruction site; and estimate the lifting capacity of the crane.Thus, a feasible collision free erection path can be planned [25,2,11].The method proposed in Refs. [11] and [25] effectively planned theerection path, but it can only be used for single crane operations.Wang et al. [31] proposed a sampling-based method for real-timemotion planning of cranes in order to improve construction safety.Zhang et al. [28] utilized the Ant Colony algorithm to find acollision-free-path for a mobile crane with consideration for both ef-ficiency and safety. These studies have demonstrated the potentialand value of utilizing path-planning technique for single crane erec-tion. The method mentioned in Ref. [2] is applicable to cooperativedual crane erection planning. It is computationally the most effectivemethod at present for cooperative dual crane erection planning, but itstill takes between 3 and 12 min to calculate each plan.

Therefore, this research aims at developing a near real-time andautomatedmethod for erection path planning. The method can be ap-plied for both single crane and cooperative dual crane erection oper-ations, and can be used to find the most suitable erection path in the

Page 2: A fast path planning method for single and dual crane erections

469Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

given environment. This method allows crane operators to completethe job with higher efficiency, thereby increasing the value of auto-mation in the erection path planning process.

2. Method for path planning

In the study of path panning, the most commonly used methodsare Potential Field, Cell Decomposition, Genetic Algorithm, VisibilityGraph, and Probabilistic Roadmap.

In the Potential Field method [9], a force field is constructed aroundthe workspace of the robot, and its planned path to be planned. Attrac-tive and repulsive forces exist in this force field. The final position of theload is the center of the attractive forces, and the obstacles in space arethe center of the repulsive forces. By using this method, the robot canmove to the targeted position. Although it is easy to find the pathusing this method and it is also applicable to paths with narrow spaces,a disadvantage is that this method can fall into a local-optimum solu-tion. Applications of the Potential Field method include path planningof multiple robots [27], danger avoidance of robots [13], and path plan-ning for high-speed vehicles [6].

The Cell Decomposition method [4] divides the space into severalsubspaces, and finds the boundary conditions among the subspaces toobtain a connectivity map. The method then looks for subspaces anddecides the order in which it will proceed from the starting point tothe goal. The machine then follows this order to change from one sub-space to the next in order to arrive safely at the destination. For thesolution of non-polygonal obstacles and 3D space problems, MorseDecomposition was proposed [1]. Another method is Visibility-Based Decomposition [8]; it is designed especially for solving the pur-suit–evasion problem.

The Genetic Algorithm method [7,21] is mainly based on Darwin'stheory of evolution; based on the law of nature i.e. the principle of“the survival of the fittest”. The method considers the path of the geneand uses the biological procedure of evolution to obtain a solution foran optimal path. The algorithm comprises three steps: reproduction,crossover, and mutation. The good genes can be retained by reproduc-tion, which is followed by crossover and probabilistic mutation. Thesethree steps are repeated until the termination condition is satisfied.The disadvantages of thismethod include the large volume of computa-tion required, the fact that the termination condition may be satisfiedbefore an optimal solution is found. However, it is easier to apply theGenetic Algorithm method to solve other optimization problems thanfor path planning. Examples of such problems are circuit design [29],control system design [16], and time-history design.

The Visibility Graph method [24,18] considers the vertex of theobstacle as a path node, and considers all possible paths betweenthe starting and end point. In other words, it considers all possiblepaths that do not go through the obstacles, and connects the startingpoint, the nodes, and the end point to find the shortest path. To avoidthe problem of having too many nodes, once an obstacle is not on thepath with the shortest length, it is excluded from consideration. How-ever, a drawback of this method is that it is hard to decide whether touse the vertex of that obstacle as a node if the path with shortestlength was not calculated. In addition, this method cannot workwith smooth obstacles.

Probabilistic RoadmapMethods (PRM) [12,5] are a widely used setof methods for robotic action design and path design. The main ap-proach is to sample within the space in order to make a node, andto connect all nodes without obstacles in between in order to createthe path; thus producing a roadmap. Then, from the roadmap, wemust find all possible paths from the starting point to the end point,and the shortest path is then selected as the path for the robot. Ifsuch a path does not exist, more nodes are sampled until a path canbe identified. PRM has been proven to be relatively complete [14],as long as the probability of finding a path between the starting and

end point is not nil. If the computational time is not limited, PRMcan certainly be used to find a feasible path.

The PRM method is based on probability, where the basic processis to repeatedly guess the collision-free points and try to link theminto a collision-free path. The computation time for PRM dependson the number of path nodes. The fewer nodes there are, the shorteris the computation time. However, the probability of find a feasiblepath is also lower, and vice-versa, having more nodes leads to longercomputation time and a higher probability of finding a feasible path.The PRM method is suitable for erection path planning. Erectionpath planning is different from the maze problem of robotics, whichinvolves figuring out where the corner is and how to navigatethrough narrow paths. In construction practice, we usually maximizethe working space for cranes. This means PRM can compute (orguess) a collision-free path in a very short amount of time, and onlya few nodes are sufficient to plan an erection path. For a more compli-cated erection operation, more nodes can be used to find a feasiblepath. Furthermore, regarding efficiency, a crane operator may bemore interested in obtaining a feasible path quickly rather an optimalpath slowly. This is consistent with the main concept of the PRMmethod. Therefore, this research makes use of the PRM method forerection path planning, making it possible to find a feasible erectionpath in near real-time automatically.

3. Erection path planning for single crane erection

In this sectionwewill introduce how a Configuration Space (C-Space)is built for the single crane erection procedure, and finding a collision freepath from the C-space as the erection path for single crane operations.

3.1. Assumptions

The method for erection path planning presented in this researchis valid only under the following assumptions:

1. For the calculation of path planning, all the obstacles are assumedto be static during the erection operation, except for the crane andthe lifting object.

2. There is no change in the location where the crane is set up duringthe erection operation. In almost all erection operations, the craneis set up at a fixed location and the crane cannot move during theerection operation.

3. The sway of the lifting object during the erection process isaccounted for using a larger boundary for the lifting objectmodel. We can measure the possible sway range and set an outerboundary. Then we use the boundary to computer the collision-free path. This avoids virtually all possible collisions due to thesway. Crane operators are asked to minimize the cable sway to re-duce risks. They usually stop moving the lifting object when it isswinging until the object becomes static. However, the problemof whether the dimensions of the lifting object model are enoughis beyond the scope of this research.

3.2. Procedure of erection path planning

For the erection path planning method presented here, the flow isdivided into three parts as shown in Fig. 1. Before planning an erec-tion path, the user must provide information about the selectedcrane including its lifting capacity and initial location. A C-space isthen constructed for the selected crane, and path planning is thenconducted in the C-space. Finally, operation planning is performedfor the hoisting, and a collision free and feasible erection path isplanned. If a feasible erection path cannot be found, users then can al-ternate the crane with other cranes that have better capacity or adjustthe initial location of the selected crane until they can find a feasibleerection path.

Page 3: A fast path planning method for single and dual crane erections

Fig. 1. Flow chart for crane path planning.

Fig. 2. The range of hoist height within hmin and hmax.

470 Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

3.3. Building the C-space for single crane

We consider the crane's base-swing angle θ and its boom-luffangle φ as coordinates in order to construct a 2D C-space for thecrane. Construction of the C-space comprises three steps: first, therange of the space has to be defined; then the C-obstacle region is de-fined; and finally, we record the limits of hoist height and the heightsof obstacles for all collision free configurations in the space.

To define the region of C-space, we need to first find the extent ofpossible changes in the boom-luff angle φ. According to crane's tableof loading capabilities and considering the weightW of the lifting ob-ject, the maximum rotation angle φmax(W) and the minimum rota-tion angle φmin(W) can be identified; and thus the range of the C-space coordinates can be determined. For safety considerations, theload in the boom hold not exceed S% of the upper safety limit duringthe erection operation, the weight of the object is considered as W/S,and the table of loading capabilities should be used to find φmax(W/S)and φmin(W/S). in the planning process.

After the C-space is defined, we investigate the collision problem be-tween the crane, the lifted object, and the obstacles for all configurations[θ,φ] to establish a C-obstacle region. We propose the cObstacleCheckmethod to check whether the configuration is a C-obstacle. This methoddetermineswhether there is a collision between the crane and the obsta-cle when base-swing is θ and boom-luff angle is φ. If there is a collision,then the configuration [θ,φ] is within the C-obstacle region. If there is nocollision, then proceed to the getHoistHeightRange method for calcula-tion, which separately examines the collisions between the lifting objectand the crane, and between the lifting object and obstacles. From this,we find the range of hoist height hmin and hmax. When the hoist heightincreases (with increment Δh) and reaches a state where the lifting ob-ject and the boom are not in contact with each other, the hoist height ishmin.When the hoist height increases (byΔh) and reaches a statewherethe lifting object collides with any one of the three (Building, floor, orthe body of the crane), the hoist height at the time is hmax, as shownin Fig. 2. If hmin≥hmax, it indicates that the obstacle is too high and thelifting object cannot pass through, or that the luff angle is too largeand the lifting object is too close to the crane. In this case, the configu-ration [θ,φ] is within the C-obstacle region. When hminbhmax, the con-figuration [θ,φ] is a collision free configuration. The cObstacleCheckand getHoistHeightRange algorithm are shown in Tables 1 and 2,respectively.

After defining the C-obstacle region, the values for hoist height re-gion hmin, hmax, and the heights hob of obstacles are recorded. This in-formation is subsequently used for the planning of the erection path.

After completing the above steps, a 2D C-space can be constructedfor the crane, as shown in Fig. 3.

3.4. Path planning for single crane

In this study, we perform path planning by using the PRMmethodof path planning. First, we randomly sample a sufficient amount of

nodes in C-space, and then connect all possible nodes to form differ-ent paths. Finally, we select the optimal path from all possible pathsconnecting the starting point to the end point, which forms the erec-tion path for the crane.

To calculate the optimal path from found paths, we use the varia-tion in angle of the crane base-swing, the variation in angle of thecrane boom-luff, and variation in length of the hoist height as thebasis for evaluation, and for developing a Pcost function to calculatethe costs for the erection path In, as shown in Eq. (1). The reasonwe do not use the length of the path for evaluation is that the positionof the lifting object is the end-point result of the three degrees of free-dom (base-swing angle, boom-luff angle, and the hoist height). Theshortest path (e.g. a straight path in the 3-dimension space) is usuallymore difficult to follow for the crane operators because they have tosimultaneously control these three degrees of freedomwith changingvelocities. For a human being, it is easier, safer, and more stable tocontrol only one or two degrees of freedom with the same velocity.Therefore, we use the variation of these three degrees of freedomfor the cost measurement. Fig. 4 shows an example of cost estimationby the variation of base-swing angle and boom-luff angle.

In order to have the same evaluation criterion for angle (base-swing angle and boom-luff angle) and length (hoist height), the de-veloped cost function uses the time of operation to reach the vibra-tion as shown in Eq. (2), where tθ represents the total time ofoperation needed for the total variation of base-swing angle θ, tφ rep-resents the total time of operation needed for the total variation ofboom-luff angle φ, and th represents the total time of operation neededfor the variation of hoist heightHob _ max. The reasonwe useHob _ max in-stead of Σi=1

n−1|Δhi| will be discussed in the next section. Therefore,Eq. (2) can be transformed into Eq. (3), where ωθ is the angular speedof the base-swing, ωφ is the angular speed of the boom-luff, and Vh isthe speed of hoisting. Then we can use Eq. (3) as the cost function to

Page 4: A fast path planning method for single and dual crane erections

Table 1Algorithm of cObstacleCheck.

Algorithm cObstacleCheck(θ,φ): determine the configuration(θ,φ) is C-obstacle or notθ: base-swing angle.φ: boom-luff angle.IF obstacle collided with boom(θ,φ) or base(θ) THEN

the configuration(θ,φ) is C-obstacleELSE

getHoistHeightRange(θ,φ)IF hmin≥hmax THENthe configuration(θ,φ) is C-obstacleELSE the configuration(θ,φ) is not C-obstacle

Table 2Algorithm of getHoistHeightRange.

Algorithm getHoistHeightRange(θ,φ): find hoist height range hmin and hmax

θ: base-swing angle.φ: boom-luff angle.h: hoist heightΔh: hoist height increase in each interactionhmin: minimal hoist height for (θ,φ)hmax: maximal hoist height for (θ,φ)LET h=0REPEAT:IF hmin not found THEN

IF object(θ,φ) does not collided with boom(θ,φ) THENhmin=hELSEh=h+Δh

ELSEIF object(θ,φ) collided with obstacle, ground, or base(θ) THENhmax=hELSEh=h+Δh

UNTIL hmax foundRETURN hmin and hmax

Fig. 4. Cost estimation of a feasible path.

471Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

measure the path. The larger the calculated result of the Pcost function,the longer it will take to execute the path, and the harder it will be tooperate the crane. The path with the smallest calculated result of thePcost function is the optimal path.

However in practice, sometimes we may multiply the time cost ofeach degree-of-freedom by a weight coefficient (Wθ,Wφ, andWh) be-fore summing together as shown in Eq. (4). These coefficients can sig-nificantly influence the erection path chosen. For example if we lowerthe weight of the crane base-swing angle, the method tends to find asolution that needs fewer operations on the other parts. The erectionpath found then is easier and faster to execute for crane operators.However, it may increase the working area of the crane and thereforeraise the risk when lifting, making it unsuitable for a narrow workingspace. Conversely, if we lower the weight of hoist height variation,

Fig. 3. Layout of a crane's 2D C-space.

the erection path may tend to cross the obstacle from the top. Howev-er the retract/release action of the cable costs more time and makethe erection more inefficient. Therefore the cost function can be de-fined as Eq. (1), where the Wθ represents α ⋅ωθ

−1, the Wφ representsβ⋅ωφ

−1, and Wh represents a⋅Vh−1.

Pcost Inð Þ ¼ Wθ·Σn−1i¼1 jΔθij þWφ·Σ

n−1i¼1 jΔφij þWh·H obmax

ð1Þ

Pcost Inð Þ ¼ tθ þ tφ þ th ð2Þ

Pcost Inð Þ ¼ ωθ−1·Σn−1

i¼1 jΔθij þωφ−1·Σn−1

i¼1 jΔφij þ Vh−1·H obmax

ð3Þ

Pcost Inð Þ ¼ α·ωθ−1·Σn−1

i¼1 jΔθij þ β·ωφ−1·Σn−1

i¼1 jΔφijþ γ·Vh

−1·H obmaxð4Þ

In Erection path represented by connecting n node configura-tions where the 1-st node is the starting configuration andthe n-th node is the end configuration;

Δθi Variation in angle of the crane base-swing angle θ duringthe path section connecting the i-th node to the (i+1)-thnode in degrees;

Δφi Variation in angle of the crane boom-luff angle φ during thepath section connecting the i-th node to the (i+1)-th nodein degrees;

n Number of nodes for the path;Hob _ max The highest obstacle among configurations along the path; in

meters (m). Fig. 5 shows an example of a path with fournodes, and where Hob _ max is the maximum value of the Hob

in the Fig. 5. Since the 2DC-spacemethod in this investigationdoes not include a parameter for the hoist height, we use themaximum height of obstacle that the lifting object needs tomove past as a rough estimate for the change in hoist height.

ωθ Angular speed of base-swing, in (degrees/s).ωφ Angular speed of boom-luff, in (degrees/s).Vh Speed of hoisting, in (meter/s).α The weight of the crane base-swing angle change in the cost

estimation for the path.β The weight of the crane boom-luff angle change in the cost

estimation for the path.γ The weight of the hoist height change in the cost estimation

for the path.Wθ The factor of the crane base-swing change in the cost esti-

mation for the path.Wφ The factor of the crane boom-luff change in the cost estima-

tion for the path.Wh The factor of hoist height change in the cost estimation for

the path.

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472 Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

Fig. 5. The maximum obstacle height along the erection path.

3.5. Hoisting planning for single crane

Since the 2D crane C-space in this investigation does not includehoist height as a coordinate in space, the obtained path does not con-tain information regarding the hoisting operation. Thus, we need toperform hoisting planning on the path obtained in Section 3.4.

Initially, wemay take the maximum hoist height hmax of all config-urations along the path as the hoisting parameter. However, thisleads to unnecessary hoist operation during the erection operation,as can be seen from the solid black line in Fig. 6. To avoid this situa-tion, an additional optimization process is required on the hoistingparameter, starting from the initial configuration and before thecable elongation reaches the minimum value of hmax. If the presenthoist height is greater than hmax, then we reduce the elongation tohmax; and if the hoist height is less than hmax, then the hoist height re-mains unchanged. After the hoist height has become the smallestvalue of hmax, the hoist height is not adjusted until the goal is reachedand the lifting object is set down. The resulting hoisting parameterafter optimization is shown in Fig. 6 in gray.

Even though unnecessary change in hoist height can be avoidedeffectively after adjusting the hoisting parameters, it is possible forthe load and boom to collide with each other because the hoist heightwill no longer be adjusted after the hoist height has reached the min-imum hmax along the path, and the boom-luff angle may keep increas-ing. Therefore, we need to inspect the hoisting parameter after thehoist height has reached its minimum value. If the hoist height isless than hmin in the current configurations, then the hoist heightshould be increased to hmin, and the adjusted hoist height for thepath is shown as a dashed line in Fig. 6. After the adjustment, thepath configuration and the hoisting parameter form the optimal erec-tion path.

Fig. 6. Optimization of the cable operations.

3.6. Features of the proposed erection path planning

In the existing research on erection path planning [26], the opera-tion of crane is expressed by a 3D C-space under the condition thatthe crane does notmove during the erection operation. The coordinatesof C-space are the crane base-swing angle θ, the boom-luff φ, and thehoist height h. All crane operations are expressed in the C-space by[θ,φ,h].

In this research, we have found that it is not necessary to considerthe hoist height as one of the coordinates in the C-space. This is be-cause in the erection operation, we change the hoist height to liftthe load higher in order to avoid collisions between the load andthe ground or obstacles. In fact, we only need to know the extent ofhoist height reduction during the erection operation to enable theload to clear the obstacles. After the load is lifted during the erectionoperation, the load must be raised to a height above the obstacles inthe planned path to ensure that the load does not collide with eitherthe ground or the obstacles. Therefore, in this investigation, we havefound the minimum hoist height hmin, the maximum hoist heighthmax, and the height of obstacle Hob corresponding to each configura-tion in the C-space, we have also simplified the 3D crane C-space to a2D configuration space [θ,φ]. This method significantly reduces thecomputational time and complexity of path planning, and thus weare able to achieve a faster erection path planning procedure.

4. Erection path planning for dual cooperative crane

This study extends the aforementioned method for single craneerection path planning to a method for a dual, cooperative crane.The path planning flow chart is still the same as in Fig. 1, exceptthat in order to model the dual crane system, two 2D C-spaces areused. A method was developed to connect the two C-spaces con-structed for each crane, and find all possible path nodes so that erec-tion path planning can be conducted for dual cooperative cranes.

4.1. Building the C-space for dual cranes

In order to describe the dual crane system, we built the individualC-space for each of the two cranes, and address them as two cranes,crane A and crane B. The C-space for crane A is CA and that for craneB is CB.

We first define the coordinate ranges for CA and CB. The weight ofthe object is W. For the object, since the dual crane cooperative oper-ation is evenly divided between the two cranes, the weight of W/2 isseparately substituted into the weight tables of crane A and crane B tofind crane booms largest luff angle φmax(W) and its smallest luff angleφmin(W). Configuration CA represents a possible connection point be-tween crane A and the object, and Configuration CB represents a pos-sible connection point between crane B and the object.

We then use the method given in Section 3.3 to separately deter-mine the regions for CA-obstacle and CB-obstacle. However, unlike thecase of single crane operation, the inspection of object collision is per-formed by the use of the object and a partial cranemodel at the end ofeach connection. Since the purpose of the dual crane C-space is to findthe connectable configurations for the crane and the object, collisioninspection between the object and obstacles is carried out at thestage of path planning. After completing the C-obstacle region deter-mination, we record the cable extension range and heights of obsta-cles for each configuration.

4.2. Path planning for dual cranes

This research extends the method of single crane path planningpresented in Section 3.4 to dual crane planning. We begin by selectingone of the two cranes to be the reference crane, and then we randomlysample the configurations from the C-space to serve as the

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Fig. 8. Example of cooperative dual crane operation.

473Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

connection point between the reference crane and the load. Next, wesearch from the other crane's corresponding connection point, anduse the connection configurations between the two cranes and objectas a set of path nodes. This process continues until an adequate num-ber of nodes have been obtained and then the nodes are connected toform the path. Finally, we select the optimal solution from all possiblepaths that connect the starting point to the goal as the erection pathfor dual cooperative crane operation. The flow chart for dual cooper-ative crane path planning is shown in Fig. 7.

To explain the method of the dual crane path planning proposed inthis research further, we present a simple case of dual cooperative cranepath planning as an example, as shown in Fig. 8. First, we enter the road-map into a path node taken from the connecting point (CA _ s,CB _ s) be-tween the object and the two cranes at the starting position. Then, weenter the roadmap into a path node taken from the connecting point(CA _g,CB _g) between the object and the two cranes at the goal position.We then advance into the process of path node sampling.

During the path node sampling process, cranes A and B take turnsto serve as the reference node for sampling. When crane A is the ref-erence node, a sample is arbitrarily taken from CA (indicated as CA _n)that is not within the region of C-obstacle. CA _n then act as the con-nection point between the load and crane A. The next step is to finda possible connection point CB _n, corresponding to CA _n, which con-nects crane B and the object. Consider CA _n as the center of a circleand use the height of obstacle Hob corresponding to CA _n as the height

Fig. 7. Flow chart for dual cooperative crane path planning.

to be lifted. Then, on the XY plane of the workspace with XYZ coordi-nate system, rotate the object so that it is parallel to the line CACB ,which connects the centers of rotation of the booms of the two cranesas shown in Fig. 8. If at this time, the object in the XYZ coordinate sys-tem does not collide with either the obstacle or the cranes and CB _n

and CB are not in the region of C-obstacle, then we use (CA _n,CB _n)as a node and insert it into the roadmap.

However, if the object collides with the obstacle, then subsequent-ly we take CA _n as the center of the circle and rotate the object in theXY plane both clockwise and counterclockwise until there is no fur-ther collision. If the rotation angle of the load is θ1b π

2 or θ2b π2 and

CB _n is not C-obstacle, then we use (CA _n,CB _n) as a node and insertit to the roadmap as shown in Fig. 9.

The reason for alternatively selecting cranes A and B as the refer-ence node is to provide a variety of paths. In theory, the more diversethe sampling points, the higher the possibility of finding collision-freepaths. Since the PRM method is based on probability where we ran-domly sample the nodes in the configuration space and try to arriveat a feasible solution instead of checking all possible solutions andchoose the optimum one. Therefore, alternative selection of the refer-ence point can increase the randomness without adding to computa-tional cost. Since this study uses the connection point of the load andthe end of the crane as the center of the circle, the boom is turned inorder to avoid the obstacle so that when a different crane is used asthe reference-sampling node, various nodes will be possible.

When connecting the collision-free path, we connect the nodesalong with the connection point between the object and crane usingstraight lines to form the path in the working space with the XYZ co-ordinate system, as shown in Fig. 10. To judge whether the path be-tween the nodes is a collision-free path, the following three-stepprocess is needed. First, we find the height Hobject to lift the objectover the obstacle without collision for the section of erection path be-tween the two nodes. Secondly, we inspect the connection points be-tween the object and crane in the trajectory of path to see whetherthey will go past the C-obstacle in the spaces between CA and CB. If

2

1

CB_n

CA_n

CB_n

θ

θ

Fig. 9. Node sampling in cooperative dual crane path planning, when an object collideswith an obstacle.

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Fig. 11. Taking required height that ensure the object avoids an obstacle as the erectionpath.

474 Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

they do pass the C-obstacle, then this shows that collision will occurat the loading point of the path so that this section between nodeswill not be a part of the path. Finally, we inspect the amount ofcable extension needed to raise the height to Hobject and see if thelength exceeds what is permitted. If it exceeds the length limit, itmeans that it is not possible to lift the object over the obstacle byusing the path, and therefore the section between these nodes cannotbe a part of the path. If the section between two nodes is judged to bea collision-free path, then record the height Hobject, which will be usedin the calculation for optimal path and in planning for hoist height.

After completing the collision-free path connection, those pathsthat successfully go from the starting point to the end point throughthe nodes are designated as allowable erection paths. If no suchpath can be found, then further sampling of nodes should be per-formed and they should be added to the roadmap. The process con-tinues until a path can be found that connects the starting point tothe end point.

For calculation for the optimal path, we make use of the Pcost func-tion proposed in Section 3.4 to calculate the costs for the erectionpath. The costs for the paths of crane A and crane B are then separate-ly estimated by using the Pcost function in which results are then com-bined. The larger the resulting value is, the more time it will take toexecute the path and the harder it will be to operate the cranes.

4.3. Hoisting planning for dual cranes

In the cooperative dual crane operation, due to differences in theobstacle heights encountered by the connection point between thecrane and the load end of the two cranes, dual crane operation cannotuse the largest hoist height hmax as we do for the hoisting parameterin the case of single crane erection. From the recorded value of Hobject

during the stage of collision-free path connection, we then obtain therequired height for the object in order to avoid an obstacle, as shownin Fig. 11. Using this height, we can calculate crane A's hoisting pa-rameter hA and modify it through the method of parameter modifica-tion used in Section 3.5. This will minimize the hoist height changefor crane A and make it fall within the safety limits hminA≤hA≤hmaxA

as shown in Fig. 12. Finally, based on the corrected parameter hA, wefind the hoisting parameter hB of crane B for the same load height,and then a complete erection path can be found. After we calculatethe hoisting parameter for crane B based on the height of object andmake corrections to the hoisting parameter, we can find the hoistingparameter hA of crane A for the same object height, and obtain a dif-ferent erection path. On comparing hoist height changes in the afore-mentioned two paths, we found that the path with the smaller hoistheight change is the optimal path for cooperative dual crane opera-tion as obtained using the planning method of this research.

5. Implementation

We introduce the computer software Erection Planner in this sec-tion and its architecture and development environment for erection

Fig. 10. Trajectory for cooperative dual crane operation.

path planning. The main function of Erection Planner is to provide avisualization of a virtual construction site. Using the software, theuser can plan the erection path by providing the starting and endpoints for the object to be lifted and by selecting the type of erectionoperation. The system will show the planned erection path and theerection process in the virtual environment.

5.1. The development environment

Erection Planner was developed based on Microsoft's XNA Frame-work [20]. It renders the virtual construction site by using the graphicssoftware DirectX. In addition, PhysX developed by NVIDIA is used to de-tect collisions and to help build the C-space for the crane [23].

The following hardware configuration was used for the tests: CPUIntel Core2 Duo E7300 2.66 GHz, 3 GB RAM, NVIDIA GeForce8600GTscreen. The operation system used was Windows XP SP3 32 bits.

5.2. The software architecture

Fig. 13 shows the software architecture of the Erection Planner. It in-cludes five parts: Erection Project Information Input, C-Space Builder,Path Planner, Hoisting Planner, and Scene Visualization.

5.2.1. Erection project information inputIn order to provide a visualization of a virtual erection scene and

to check for any collisions that occur, we first use the Autodesk 3dsMax 2009 3D software [3] to build 3D models for the cranee3DSMAX, the object, and the obstacles and import them into theprogram. In order to work in the XNA Framework, we convert themodel files in the FBX format and import them into the programwhich other necessary information for erection path planning, such

Fig. 12. Correction method for the cable operation parameter.

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as the load capacity table of the crane, the weight of the object, andcoordinates of starting and end points for the object.

5.2.2. C-space builderThe purpose of this part is to build the C-space. The C-space is built

based on the method for building the C-space presented in Sections 3.3and 4.1, and is built according to the type of erection operation. To imple-ment the collision detection needed in the C-space building procedure,we use PhysX to build simple physical models for the crane, the object,and the obstacles. In the PhysX physical action simulation, the base-swing angle, the boom-luff angle, and the position of object are changedif any collisions occur.

5.2.3. Path plannerThe purpose of this part is to implement the method of path plan-

ning proposed in Sections 3.4 and 4.2, and essentially carry out erec-tion planning to obtain path information including the base-swingangle and the boom-luff angle.

5.2.4. Hoisting plannerThis part plans the hoisting operation for the erection path by the

method of hoisting planning described in Sections 3.5 and 4.3, andsends out the completed erection path information.

5.2.5. Scene visualizationIn this part, a virtual erection scene is built usingXNA, and the trajec-

tory of the object is shown in the erection scene. Fig. 5.4 shows theresulting erection path using the current method. The visualizationpart can be integrated with other 3D simulation tools for more ad-vanced and graphically detailed visualization e.g. Maya, 3ds Max andBlender. Users can retrieve the project information from the ErectionProject Information Input and combine it with the planned output ofthe Hoisting Planner (which contains the crane configurations in thepath) and then adapt it to their visualization or simulation tool.

6. Experiment result and discussion

In order to verify the method of erection path planning proposedin this research, we have conducted a series of erection scene exper-iments to test if the proposed method can be applied to both singlecrane erection operation, and dual cooperative erection operation.

Fig. 13. Software architecture of Erection Planner.

6.1. Experiment 1: Validation test of path planning for single crane

Experiment 1 uses the example of a single crane erection path plan-ning that took place in an oil refinery (CPC Dalin), and its erection sceneis shown in Fig. 14. The cranewas required to lift a cylindrical oil storagetank (radius=4.8 m, length=30m) and put it on top of a 37 m highsupporting frame. In the erection scene, the obstacles around the oilstorage tank were an eleven-story high structure (height=61 m), andtwo oil storage tanks already located on the supporting frame(height=42m). The body of the crane was 20 m long and 12 m wide,with a boom length of 90 m. It had a base-swing angular velocity of4°/s, a boom-luff angular velocity of 1°/s, and a hoist height changespeed of 1 m/s. By using the planning method presented in this re-search, an erection path was successfully identified. The object did notcollide with the aforementioned obstacles and the supporting frameat the end point. The path is shown in Fig. 15.

6.2. Experiment 2: Validation test of path planning for dual cooperativecranes

In Experiment 2, we use no.2 of dual cooperative erection opera-tion from [2] as an example. The erection scene is shown in Fig. 16.The object was a horizontal beam (length=10 m, width=1 m,height=1 m). Each crane was connected to one end of the beam.There were three obstacles between the starting and end points(the height of obstacle A was 5 m, the height of obstacle B was12 m, and the height of obstacle C was 5 m). Collisions with obstacleshad to be avoided whenmoving the object from one side of the obsta-cles to the other. The body of the crane had a length of 10 m, andwidth of 6 m. The boom length was 45 m, with a base-swing angularspeed of 4°/s, boom-luff angular speed of 1°/s, and hoist heightchange speed of 1 m/s. By using the planning method presented inthis research, an erection path was successfully identified and the ob-ject did not collide with the aforementioned obstacles. The path isshown in Fig. 17.

In addition to the problems of collision and managing the weightof object during the erection path planning, the interference and in-fluence of the erection operation on other engineering activitiesshould be also be considered. The result of erection path planning,shown in Fig. 17, required additional planning if there were workersaround the obstacles even though the plan was able to successfullylift the object over the obstacle without any collision. To avoid inter-ference with the object, the possibility of parts falling and hurting theworkers on the ground, and for the purpose of engineering safety, amore time consuming and harder to operate substitute erectionpath was required. The strategy was to move the object around theobstacles. Therefore, we tested the substituting erection paths undervarious constrained conditions. First, we hoped to find an erectionpath that did not pass through obstacle B in the middle. The test re-sults are shown in Figs. 18 and 19 (The yellow part shows the starting

Fig. 14. Erection scene of Experiment 1.

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Fig. 15. Erection path of Experiment 1 (The goal position of the object is visualized using translucent yellow color).

476 Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

position of the object and the red part shows the end position). Twodifferent paths were found: one went around obstacle B from theleft side, while the other from the right side. Afterwards, we hopedto find paths that did not go past any obstacles but passed throughthe narrow space between the cranes and obstacles. The results areshown in Figs. 20 and 21.

6.3. Experiment 3: Efficiency test

In Experiment 3, we tested using the same two erection scenesfrom Ref. [2]. We imitated the method of building the 3D C-spaceand also made use of Genetic Algorithm based method of erectionplanning, both reported in Ref. [2], and compared it with the methodpresented in this research. In addition, we also compared resultsobtained by using the proposed PRM and the planning conducted ei-ther in both 3D and 2D C-spaces.

6.3.1. Erection Scene 1Erection Scene 1 used the scene 1 from the dual cooperative crane

erection of [2] as shown in Fig. 22. The object was a beam (10 m inlength, and with height and width equal to 1 m), and the two craneswere connected to either end of the beam. The two cranes were re-quired to lift the object together from the ground and place it onthe platform on top of an obstacle 6 m high. The body of the cranewas 10 m long, 6 m wide, and the length of the boom was 45 m,with base-swing angular speed of 4°/s, boom-luff angular speed of1°/s, and hoist height change speed of 1 m/s. Fig. 23(a) shows the

Fig. 16. Erection scene of Experiment 2.

path planned using 3D C-space with the Genetic Algorithm.Fig. 23(b) shows the erection path planned using PRM in the 3D C-space, and Fig. 23(c) shows the erection path planned using PRM inthe 2D C-space.

6.3.2. Erection Scene 2Erection Scene 2 uses scene 2 from the dual cooperative crane erec-

tion of [2] as an example, and its erection scene was the same as thatin Experiment 2. The site arrangement is shown in Fig. 16, please referto Experiment 2 for details about the site. Fig. 24(a) shows the pathplanned using 3D C-space with the Genetic Algorithm. Fig. 24(b)shows the erection path planned using PRM in the 3D C-space, andFig. 24(c) shows the erection path planned using PRM in the 2D C-space.

6.4. Discussion of experimental results

The path planning result of Experiment 1 for single crane erectionobtained using the proposed method, shown in Fig. 15, successfullyavoided the obstacles and found feasible collision-free erectionpaths. The computation time for planning was 0.27 s. Thus, it took avery small amount of time to complete the planning.

The path planning result obtained in Experiment 2 for cooperativedual crane erection using the proposed method, shown in Fig. 17, suc-cessfully avoided the obstacles and found feasible collision-free erec-tion paths. The computation time for the planning was 0.52 s. Again,the time taken to complete the planning was very short. In additionto finding an erection path based on the strategy of lifting the objecthigh enough to avoid collision with obstacles, a feasible path canalso be found based on the strategy of moving the object around theobstacle. Thus, feasible paths can be found under different conditionsof constraints, as shown in Figs. 18–21.

In Experiment 3, we planned the erection path for different siteconditions using different methods of erection path planning. Weused 3D C-space with the Genetic Algorithm, the proposed PRM inthe 3D C-space, and the proposed PRM in the 2D C-space. The resultsare shown in Figs. 23 and 24. The above methods successfully foundfeasible collision-free erection paths for cooperative dual crane erec-tion. Tables 3 and 4 compare the results of path planning, wheredata was obtained from an average of 100 path-planning operations.The values compared are the planning time, change in angle of thebase-swing, change in angle of the boom-luff, and the change in thehoist height. By comparing the results, we find that the average plan-ning time using the proposed PRM method in the 2D C-space wasmuch less than the other two methods. In the comparison of craneoperation parameter, no apparent differences could be found in the

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Fig. 17. Experiment 2 — Erection path with no constrained conditions.

477Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

base-swing angle change among the three methods. This is becauseduring an erection operation, the base-swing angle usually startedfrom the initial angle and gradually rotated to the final angle; therewas not much change in angle during the operation. In the compari-son of boom-luff angle change, the PRM in the 2D C-space managedthe least changes and obtained the easiest operate erection path. 3DC-space with the Genetic Algorithm could find the approximated op-timal result. However, it was harder to use PRM in the 3D C-space tofind the optimal path compared to PRM in the 2D C-space as it wasdifficult to have path nodes evenly distributed in the C-space. ThePRM method in this investigation used probabilistic sampling to ob-tain path nodes and tried to calculate collision-free paths among allnodes. If the PRM method is improved, it should be possible to obtainbetter results. In addition, a planning method for hoisting operationhas been presented in this study for the 2D C-space, reducing unnec-essary hoist height changes.

7. Benefits and contributions

In this research, a method of automated erection path planninghas been developed. The main contributions of this research are:

1. Development of a method of 2D C-space, which reduces the com-plexity of planning and reduced the path planning time.

2. Development of a flexible method of erection path planning, whichcould be applied to both single crane operation and cooperativedouble crane operation.

3. Development of a near real-time method of erection path plan-ning. This could be used to quickly alter the path during the erec-tion operation and plan a new feasible path if, just before orduring operation, it is found that the actual site environment wasdiffers from the conditions originally planned for.

4. Development of a costs estimation function with the weight coef-ficients for each degree of freedom of crane operation. Planners

Fig. 18. Experiment 2 — Substituting path 1 when th

can adjust the weight values to generate feasible and suitable erec-tion paths for different specifications and site environments.

The developed method can potentially benefit many aspects ofcurrent construction practice. The possible benefits of this researchare summarized in the following paragraphs:

Crane selection: Rather than having a qualitative crane selection, themethods and tools developed in this research can be used to providequantitative information about the differences from using the vari-ous candidate cranes. Similarly, for large or high-rise constructionprojects, this research developed tools that provide data to enableproper evaluation of the compromises and benefits of using of mul-tiple cranes at the site as opposed to using a single crane.Crane placement: The methods developed in this research enablecomputers to simulate the erection processes automatically byutilizing the planned erection path. They can also be extended toobtain erection times produced by placing the crane at differentlocations and searching for an optimum location within the sitethat will minimize erection times.Logistics for scheduling material deliveries to the site: This study canbe used to improve various aspects of the logistics at constructionsites. In particular, the generation of a precise and detailed erec-tion plan prior to construction can minimize onsite storage re-quirements by delivering only the materials that will be erectedsoon to the site. For example, structural elements can be deliveredonly one day prior to their erection, or for a site where no onsitetemporary storage is available, they can be delivered just a fewhours prior to erection. Furthermore, elements to be lifted by thecrane can be delivered to the site and even placed in the deliverytruck in accordance with the plan for lifting them using the crane.Planning and visualization of erection virtually prior to actual erectionactivities: This research provides tools for planning and visualizing

e path did not go through the middle obstacle.

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Fig. 19. Experiment 2 — Substituting path 2 when the path did not go through the middle obstacle.

Fig. 20. Experiment 2 — Substituting path 1 when the path did not go through any obstacle.

Fig. 21. Experiment 2 — Substituting path 2 when the path did not go through any obstacle.

Fig. 22. Erection scene of Experiment 3 — Site arrangement of the example.

478 Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

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Fig. 23. Scene 1 of Experiment 3 — Erection path planned: (a) Genetic Algorithm in 3D C-space; (b) by PRM in 3D C-space; and (c) by PRM in 2D C-space.

Fig. 24. Scene 2 of Experiment 3 — Erection path planned: (a) Genetic Algorithm in 3D C-space; (b) by PRM in 3D C-space; and (c) by PRM in 2D C-space.

TabSce

P

AVVV

TabSce

P

AVVV

479Y.-C. Chang et al. / Automation in Construction 22 (2012) 468–480

erection processes on the computer before actual construction. Thisprovides an excellent planning and management tool for superinten-dents, construction managers, erection crews and crane operators toallow them to fully understand the planned erections next week,next day, next three days, next hour, etc. Being able to visualizethese erection activities prior to actual construction can help identifyand eliminate many otherwise unforeseen problems ahead of time.Computer-assisted cranes erections: Methods and tools developed inthis researchdo not just provide benefits to conventional cranes, theyprovide the basis for developing computer-assisted cranes. In the

le 3ne 1 of Experiment 3. Efficiency comparison for different methods of path planning.

lanning method Problem 1

GA(3D C-space)

PRM(3D C-space)

PRM(2D C-space)

verage planning time (sec) 36.33 11.26 0.51ariation of base-swing angle (deg) 71.47 69.81 67.64ariation of boom-luff angle (deg) 12.29 13.15 10.32ariation of hoist height (m) 31.51 32.77 26.69

le 4ne 2 of Experiment 3. Efficiency comparison for different methods of path planning.

lanning method Problem 2

GA(3D C-space)

PRM(3D C-space)

PRM(2D C-space)

verage planning time (sec) 59.42 11.32 0.52ariation of base-swing angle (deg) 144.96 143.55 142.93ariation of boom-luff angle (deg) 24.81 23.22 19.14ariation of hoist height (m) 106.11 109.78 93.06

near future, this research could be used to provide information foroperators in real time to assist them in manipulating constructioncranes more efficiently and more safely. For example, a computer-assisted crane could prevent operators from coming in contact withpower lines or in preventing the crane from colliding with obstaclesor with other cranes. They may integrate the Augmented Realityand tele-operation to visualize computer-generated collision-free(safe) and time-optimized paths to be followed by crane operators,and even “auto-pilot” capabilities for portions of the erection cycleor for entire erection cycles.Autonomous robotic crane erections: Many of the methods and al-gorithms developed in this research, together with new sensortechnology, and the development of new types of structural con-nections, can provide the basis for autonomous robotic cranes.These robotic cranes could first be used for construction in dan-gerous environments (e.g., erection of containment structures inenvironments contaminated by hazardous chemicals or radioac-tivity, erection of a bridge during armed conflict, etc.), but couldeventually be used in more standard constructions.

8. Conclusion

In this research, we developed a fast path-planning method forsingle and dual crane erections. This method replaces the use ofhoist height as a coordinate of the C-space, which can significantly re-duce computation time needed during the planning process. Thehoist height is determined after the path-planning processes. Thiscan ensure minimal changes in the hoist height, fulfilling the needsof construction practices. To validate the usability of the proposedmethod, we conducted three experiments: (1) path planning for sin-gle crane, (2) path-planning for dual crane and (3) efficiency test. The

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results of first two tests show that the proposed method can help finda collision-free erection path that satisfies safety requirements, andcan assist engineers in solving the path-planning problem. From thethird experiment, we found that the proposed method of erectionpath planning is time efficient when used to find a feasible erectionpath compared to existing methods. The obtained path by the pro-posed method is also easier to operate. The method is also flexiblein being able to find suitable erection paths under different condi-tions, as required on the erection site.

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