Scholarly article on topic 'Validation of an Extended Approach to Multi-robot Cell Design and Motion Planning'

Validation of an Extended Approach to Multi-robot Cell Design and Motion Planning Academic research paper on "Mechanical engineering"

CC BY-NC-ND
0
0
Share paper
Academic journal
Procedia CIRP
OECD Field of science
Keywords
{"Multi-robot cells" / "Design optimization" / "Motion planning"}

Abstract of research paper on Mechanical engineering, author of scientific article — Stefania Pellegrinelli, Nicola Pedrocchi, Lorenzo Molinari Tosatti, Anath Fischer, Tullio Tolio

Abstract According to both industrial practice and literature, multi-robot cell design and robot motion planning for vehicle spot welding are two sequential activities, managed by different functional units through different software tools. Due to this sequential computation, the whole process suffers from inherent inefficiency. In this work, a new methodology is proposed, that overcomes the above inefficiency through the simultaneous resolution of design and motion planning problems. Specifically, three mathematical models were introduced that (i) select and positions the resources, (ii) allocate the tasks to the resources and (iii) identify a coordinated robot motion plan. Based on the proposed methodology, we built three ad-hoc cases with the goal to highlight the relations between design, motion planning and environment complexity. These cases could be taken as reference cases so on. Moreover, results on an industrial case are presented.

Academic research paper on topic "Validation of an Extended Approach to Multi-robot Cell Design and Motion Planning"

Available online at www.sciencedirect.com

ScienceDirect

Procedía CIRP 36 (2015) 6- 11

CIRP 25th Design Conference Innovative Product Creation

Validation of an extended approach to multi-robot cell design and motion planning

Stefania Pellegrinelliabc*, Nicola Pedrocchia, Lorenzo Molinari Tosattia, Anath Fischerc, Tullio Tolioab

"Institute of Industrial Technologies and Automation, National Research Council, ITIA-CNR, ViaBassini 15, Milan, 20133, Italy b Department of Mechanical Engineering, Politécnico di Milano, Milan, Italy c Faculty of Mechanical Engineering, Tehcnion, Haifa, Israely

* Corresponding author. Tel.: +390223699954; fax: +390223699925. E-mail address: stefanaia.pellegrinelli@itia.cnr.it

Abstract

According to both industrial practice and literature, multi-robot cell design and robot motion planning for vehicle spot welding are two sequential activities, managed by different functional units through different software tools. Due to this sequential computation, the whole process suffers from inherent inefficiency. In this work, a new methodology is proposed, that overcomes the above inefficiency through the simultaneous resolution of design and motion planning problems. Specifically, three mathematical models were introduced that (i) select and positions the resources, (ii) allocate the tasks to the resources and (iii) identify a coordinated robot motion plan. Based on the proposed methodology, we built three ad-hoc cases with the goal to highlight the relations between design, motion planning and environment complexity. These cases could be taken as reference cases so on. Moreover, results on an industrial case are presented.

© 2015 The Authors.PublishedbyElsevierB.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the scientific committee of the CIRP 25th Design Conference Innovative Product Creation Keywords: Multi-robot cells; Design optimization; Motion planning

1. Introduction

The assembly of the vehicle metal panels and vehicle body-in-white through multi-robot spot-welding cells is generally outsourced by automotive companies to original equipment manufacturers (OEMs). OEMs need to provide the best offer in terms of price per produced unit, while coping with the requests of the clients. These requests include the required production volumes which in turn define the cell cycle time for the execution of a set of welding points and the employment of a predefined body-in-white fixturing systems and transportation device which introduces a set of geometrical constraints. In such a contest, cell design and motion planning are two relevant time-consuming critical activities. Even if the mutual-influence of the multi-robot cell design and motion planning cannot be ignored, current industrial practice is based on the division of these activities and the employment of several methodologies and software tools.

In order to support OEMs to reach these goals, the conceived research focuses on the analysis of design and motion planning problems for multi-robot body-in-white

assembly cells. Specifically, this research has led to the development of a methodology able to simultaneously and automatically solve both the problems.

The paper is structured as following: Section 2 presents the state of the art; the approach is described in Section 3 highlighting the innovative aspects in comparison to previous work; Section 4 validates the approach through 3 ad-hoc cases and an industrial case; finally, conclusions and future work are given in Section 5.

2. Literary review

Although multi-robot cell design and off-line motion planning have been investigated for more than two decades, many issues are still open since (i) the complexity of the design and motion planning that represent a barrier for straightforward optimal solution, and (ii) multi-disciplinary activities and research fields are required. Specifically, the integration between the two activities has not been adequately investigated in literature so far. In [1], a 3D optimized layout for assembly

2212-8271 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the scientific committee of the CIRP 25th Design Conference Innovative Product Creation doi: 10.1016/j .procir.2015.01.062

cells is proposed, when resources, tasks and product geometry are given. A similar approach in terms of the sequential execution of the design and motion planning can be found in [2]. This paper proposed an approach for the optimization of the layout of a cell consists of two conveyor belts for part feeding, two manipulators and an assembly station. Once a possible layout is generated, robot trajectories are calculated taking into account the pre-allocated tasks. Similarly, [3] proposed a method for the selection of the most appropriate manipulator systems (combination of a robot arm and positioning table) from a set of candidate systems within the desired calculation time. Location optimization and motion coordination are integrated to derive the task completion time but robot tasks are pre-allocated. A more extended approach for the design of a cooperating robot cell can be found in [4]. Starting from an initial and rough solution, the approach leads to the definition of a final collision-free solution with optimized cycle time. However, the motion planning and collision problems are partially taken into account. A complete off-line programming toolbox for remote laser welding was proposed in [5]. The approach can provide an automated method for computing close-to-optimal robot programs. The approach has been positively tested on real industrial cases. However, the problem of robot positing is not managed.

3. Approach

The approach hereafter proposed and validated is an extension of [6,7]. The approach is based on 3 stages dealing with the motion planning for single robots, the design of the cell and the robot coordination. The simultaneous resolution of cell design and robot coordination is granted by the possibility to iteratively solve the problem till a feasible solution is found. Specifically the provided design is optimal in terms of cell investment costs and feasible in terms of robot motion plans.

The input, output and the three stages of the approach are hereafter briefly described. The differences respect to [6,7] are highlighted and are in each of the stages:

• A different trajectory generation method for stage 2 for a

better exploration of the configuration space

• A new objective function for Stage 2 to better condition the

identification

• A new mathematical model for the Stage 3

Table 1. Model inputs.

Fig. 1. The approach

3.1. Input & Output

Input Range Unit Description

BIW - Body-In-White or metal sheets that have to be welded.

WPw 1..NWP Welding Points WPs. Position, mm, and orientation deg in the cell system of the points that have the be welded. Nm denotes the number of possible WPs plus a fictitious point that represents the robot initial and ending configuration.

BF ■ Body-in-white Fixturing system of the BIW during the welding process.

BTD - BIW transportation device. It transport the BIW in and out of the cell.

KM - Robot Model. Type of robot employed.

RSM Robot Support structure Model. System on which the robot are mounted. This system influences the position and the orientation of the robots in the cell.

RPOrpa 1..Ntm - Possible Robot Position and

Orientation RPOs in the cell. denotes the number of possible RPOs.

WGMwgm 1 fywgm Welding gun models WGMs to be allocated to the robots. NWGM denotes the number of possible WGMs.

RCT R s Required cycle time. Imposed by the client, it represent the maximum cycle time of the cell.

ncrm N - Number of already aCquired RM.

^^rsm N - Number of already aCquired RSM.

m^wgm wgm N - Number of already aCquired WGMWigm.

cosj*m R € Cost of RM.

costrsm R € Cost of RSM.

cosjwgm R € Cost of WGMwgm.

WTwc R s Welding time for each WPwP

Table 2. Model outputs.

The inputs and the outputs required of the approach have been detailed in Table 1 and Table 2.

Output Range Unit Description

COST R+ € Cell investment cost

jftrm N - Total Number of required RM.

j^rsm {0,1} - Total Number of required RSM.

rr\]wgm wgm N - Total Number of required WGMy^m.

narm N - Number of RM to be Acquired.

NARSM {0,1} - Number of RSM to be Acquired.

NAwaM i isi wgm N - Number of required WGMwgm to be Acquired.

RGPwgm,rpo {0,1} Allocation of the welding guns to the robots - Equal to 1 if robot mounting WGMwgm is in RPOypa.

WPA lf.1 Slrpo.wp {0,1} - Equal to 1 if the welding point WPwp is allocated to RPO^p0.

MP-^gm, rp0i wpl, wpi {0,1} Motion plan for robot in RPOipo with WGMwgm from WPwp! to WP^ - Equal to 1 if robot in RPO?po processes WPw2 immediately after WPwl.

■MT. 1 'wgm, rpo, wpl, wpi R+ s Time necessary to robot in RPOipo mounting WGMwgm to move from WPvpi to WPwp2 and weld WPvp2.

Cwgm, rpo, wpl, wpi R+ s Completion time for robot in KPOipo mounting WGMwgm to move from WPwl to WPwp2 and weld WPwp2[^].

^wgm, rpo, ~wpl, ~wp2 R+ s Starting time for robot in RPOrpa mounting WGMwgm to move from WPvpi to WPwp2 and weld WPvp2.

Dwgm, rpo, wpl, wpi R+ s Temporal delay for robot in RPOipo mounting WGMwgm to move from WPwpl to WPwp2 and weld WP^.

OCTrpo R+ s Obtained cycle time for robot in

RPOrpo-

M^OCT R+ s Obtained cell cycle time.

3.2. Approach stages

During single robot motion planning (Stage 1), a motion plan is defined for each robot position and orientation and each welding gun, i.e. for each couple {RPOrpo,WGMwgm}. The Stage has been carefully described in [6,7] in terms of strategy, employed motion planner [8] and collision detection algorithms [9]. Respect to [6,7], trajectory generation exploits probabilistic roadmap techniques with lazy collision [10,11]. The employment of lazy collision allow the generation of extended roadmap and the simultaneous reduction of the computational time. Moreover, a criterion based on the minimization of the joint movements is selected. The idea is to evaluate the distance between the joints in the joint space. This criterion limits the unnecessary movements of the robots in the workspace.

During multi-robot cell design (Stage 2), the design of the cell is identified through the selection of the necessary resources in terms of robot, their position/orientation in the cell, allocation of the welding guns to the robots. Together with the cell design, a first motion plan solution is generated for each robot. Thus, welding points are allocated to the robots and a welding sequence is generated. This motion plan do not take into account the possible collision among the robots and will be revised during the Stage 3 of the approach. Moreover, the allocation of the welding points to the robot is not unique: a welding point can be allocated to more than one robot. Multirobot cell design is based on a mathematical mixed-integer linear mathematical model aiming at the minimization of the cell investment costs. In comparison to [6,7], the objective function has been modified eliminating the penalties for the obtainment of a cycle time greater than the required cycle time is not present. Indeed, RCT is a specific client request that have to be necessarily and correctly answered.

Minimize:

RA^COST™ + £ {NAwgm-'- COSTWGM- )+! (1)

COST = \ >

[NA^COST^ J

Subject to:

9 Resource constraints 13 Motion plan constraints 2 Cycle time constraints

Stage 3 of the approach aims at coordinating the robots on the basis of the cell design produced by the Stage 2. Robot coordination is actually based on 3 sub-stages, making the here-proposed approach a decoupled approach [10,12]. Stage 3.1 takes into account the cell design proposed by Stage 2 and provides the final allocation of the welding points to the robots and a final motion plan for each robot. Then, Stage 3.2 evaluates for each couple of trajectories belonging to the identified motion plan possible collisions. Potential collisions will be avoided though the cell motion plan scheduling in Stage

3.3. Specifically, Stage 3.1 and 3.3 are based on two mixed-integer mathematical models aiming at minimizing the cell cycle time (Eq. 2 and Eq. 3). In comparison to [6,7], the mathematical model of Stage 3.3 was modified eliminating from Eq. 3 unnecessary terms and reformulating several

constraints. On the contrary, Stage 3.2 is based on a volume-swept-like algorithm, not present in [6,7].

Minimize:

MAXOCT = max\OCTJvo}

Subject to:

13 Motion plan constraints 2 Cycle time constraints

Minimize:

MAXOCT = maxlocr^ }

Subject to:

8 Motion plan constraints 2 Cycle time constraints

4. Approach Validation

The proposed approach has been validated on 3 ad-hoc cases. These cases are hereafter described in order to be easily replicated and employed as reference cases. Finally, an industrial test case is shortly presented.

4.1. Case 1

Case 1 (Table 3) aims to solve the cell design and motion planning for the welding of 8 WPs (Table A1) with 2 possible WGMs (Table A2, Fig. A1) and with the robot "COMAU Smart NJ4-175-2.2" to be placed in 6 possible RPOs (Table A4, Fig. 2-3). The robot D-H parameters are described in Table A3. BIW, BF, BTD and RSMare represented by 13 simplified obstacles: 8 cubic obstacles and 5 parallelepipeds (Fig. 2). The obstacles position (randomly generated) and orientation are described in Table A5 through rototranslation matrices referring to the cell system. The number and cost of available resources are depicted in Table 3.

Table 3. Input - Case 1.

Input Description

BIW, BF, BTD, RSM Represented by 13 obstacles

^RPO 6

flWGM 2

RCT 30s

-\j/-<WGM wgm 0, 0

COSTRM COSTRSM 2568061 15000061

COSTraM 9255€, 965561

WTwp 0.8s

Fig. 2. Possible RPOs in cell environment

Fig. 3. Robot position and orientations (not considering robot initial orientation) - Case 1

The resolution of the problem took 48 hours, mainly for Stage 1. The final solution (Fig. 4) is characterized by the selection of 3 robots in KPO2, RPOs, KPO4. The welding gun model WGM2 is allocated to the robot in RPO2 and KPO4, while WGM] is allocated to the robot in RPO3. Moreover, RPO2 is responsible for the welding of WPs, WP7, and WPs (sequence 8^7^5); RPO3 is responsible for the welding of WPi and WP3 (sequence 1^3); RP04 is responsible for the welding of WP2, WP4 and WP,5 (sequence The cell cycle time is equal

to 20.8 s. thus coping with the RCT. Finally, the cell cost is equal to 255605 €. From the robot selection point of view, this cost is minimized since the minimum number of necessary robots is found (none combination of only 2 RPO/WGM grants the machinability of all the WPs - Table 4). From the welding gun point of view, WGM2 is selected twice even if it is more expensive than WGMj. This result can be easily explained observing Table 4: WGMj independently from the robot position present is able to reach a limited number of WPs (5 out of 8 against the 8 out of 8 of WGM2).

Table 4. WP reachability - Case 1.

WP, RPO1/WGM2; RPO2/WGM2; RPO3/WGM1; RPO4/WGM2

WP2 RPO4/WGM2; RPO5/WGM2; RPO6/WGM2

WP3 RPO3/WGM1; RPO5/WGM1; RPO6/WGM1

WP4 RPO1/WGM2; RPO2/WGM2; RPO4/WGM2

WPS RPO1/WGM1; RPO1/WGM2; RPO2/WGM2; RPO3/WGM1;

RPO4/WGM2 WP6 RPO4/WGM2; RPO6/WGM2

WP, RPO1/WGM2; RPO2/WGM2; RPO3/WGM1; RPO5/WGM1;

RPO5/WGM2; RPO6/WGM1 WPS RPO1/WGM2; RPO2/WGM2; RPO3/WGM2;_

4.2. Case 2

Case 2 is defined as an extension of Case 1. The considered set of input is unchanged apart from the number and position of obstacles in the cell. Specifically, the number of obstacles is doubled (26 obstacles). New obstacles positions are presented in Table A6 and in Fig. 5.

Results (Fig. 6) partially confirm the cell design identified in Case 1, since robot in RPOi is selected instead of robot in RPO2. Moreover, a different motion plan is generated. Specifically, RPOi visits WPi, WP4, WP5, WP?, and WPs (sequence RPO3 welds WP3; RPO4 is

responsible for the welding of WP2 and WPi (sequence 2^6). The cell cycle time is equal to 23.36 s. Since the WP reachability in Table 4 is still valid for Case 2, the different WP allocation is due to the presence of obstacles that lead to the definition of complex path.

Fig. 5. Final solution - Case 2

4.3. Case 3

As for Case 2, Case 3 present an increased complexity of the environment. The number of obstacles reaches 38. Table A7 presents the positions of the new obstacles.

Fig. 6. Final solution - Case 3

Once again, the cell design solution is confirmed (Fig. 6) as well as the welding point allocation and sequence. However, because of the increasing number of obstacles, the trajectories generated for Case 3 by Stage 1 are mainly different from the trajectories generated in Case 2. Moreover, the final cycle time requires 2 seconds more (25.72 s). Thus, it seems that the time for the final motion plan increases with the complexity of the environment.

Fig. 4. Final solution - Case 1

RPOipo/WGMwgm able to reach WP

4.4. Industrial case

The presented approach have been tested on a real industrial case. The case was provided by an Italian overall equipment manufacturer. The multi-robot cell is composed by 5 robots SMART-5 NJ4-175-2.2 (Fig. 7) mounted on a bridge support structure, 3 welding gun models (Fig. 8), a fixturing systems composed by 34 elements. The bridge structure presents 6 possible positions for the robots and 3 possible orientations for each position, for a total of 18 possible RPOs. The first robot (R01) mounts the WGMi and welds 5 points in 25.9s, imposing the cell cycle time. The second robot (R02) mounts the same welding gun as in R01 robot but is responsible for 4 welding points with a cycle time of 24.74s. The third robot (R03) mounts the WGM2. Its cycle time for execution of 4 welding points is 22.98s. Finally, fourth (R04) and fifth (R05) robots have a cycle time of 22.33s and 20.99s, respectively, for the execution of 4 WPs with the WGM3. WGMi, WGM2 and WGM3 present a different spatial occupancy and costs and grant a different accessibility.

Fig. 8. Welding guns - Industrial case

Up to now, the industrial case have been employed for the analysis of Stage 1 and 3, i.e. for the definition of the motion plan. A coordinated motion plan was successfully obtained. However, the current obtained cycle time is equal to 42 s. Indeed, some automatically path presents unnecessary motion that make complex the subsequent coordination of the robots. Then, current studies are focusing on the improvement of the techniques exploited for the definition of single-robot motion planning.

5. Conclusions and future work

The proposed approach is able to simultaneously solve the design and motion planning problems for multi-robot spot-welding cells for body-in-white assembly. The approach represents a precious software tool to support human operators during the resolution of these problems. The paper presents the novelties respect to previous works and demonstrate the approach feasibly on three ad-hoc cases to be employed as reference cases. Moreover, the results on an industrial test case are shown.

Appendix A. Test case data

Hereafter the detailed data of case 1, 2 and 3 are presented.

Table A1.Welding points -

WPw Position mm in cell Orientation deg in cell Rot range

reference system reference system - deg along

Z'Y''Z'' ZGi sys

WP, -689.36 -784.45 +257.2 -166.36 92.54 77.26 -10 15 5

WP2 1201.99 -1098.56 195.64 -59.71 108.49 -60.62 0 20 5

WP3 601.99 -1098.56 1165.64 -139.71 108.49 -80.62 -30 20 5

WP4 -659.36 -784.45 +157.2 -156.36 62.54 37.26 0 0 0

WPs -389.36 -284.45 +457.2 -166.36 92.54 77.26 0 20 5

WP„ 701.99 -798.56 395.64 -59.71 108.49 -60.62 -20 0 5

WP, -998.07 427.73 566.75 -87.68 96.95 27.24 -15 45 5

WPS 0.0 0.0 2400.0 0.0 0.0 0.0 -20 0 5

Table A2.Welding guns - Case 1.

gun system (Gi) to robot tool system (R0iT)_

gun system control point (GiCP) to gun system (Gi)

WGM, [0 -1 0 0; -1 0 0 0; 0 0 -1 0; 0 0 0 1]

WGM2 [0 0 -1 0; 1 0 0 0; 0 -1 0 0; 0 0 0 1]

[0 -1 0 0; 0 0 1 -1277; -1 0 0 -435; 0 0 0 1]

[0 1 0 0.89; -0.707 0 -0.707 -1180.16; -0.707 0 0.707 -557.84; 0 0 0 1]_

Table A3. D-H for COMAU Smart NJ4-175-2.2 - Case 1.

Input Description

a mm 350 750 250 0 0 0

a rad ft/2 ft -ft/2 2ft/3 -2ft/3 0

d mm -830 0 0 -1097 150 -198

9 rad 01 02-ft/2 63 04+ft/2 6s 06

base frame [1 0 0 0; 0 -1 0 0; 0 0 -1 0; 0 0 0 1]

tool frame [0 1 0 0; 1 0 0 0; 0 0 -1 0; 0 0 0 1]

Fig. 7. Welding guns - Industrial case

Table A4. Robot position and orientation - Case 1.

RPOrpo Rototranslation matrix in Rotation Robot initial joint

cell reference system. deg along confrad

Position in mm robot Z dir

RPO, [0 1 0 -1000; -1 0 0 1800; -45 1.3472 -0.3491

0 0 1 0; 0 0 0 1] -1.9199 0 0.3491 0

KP02 [0 1 0 -1000; -1 0 0 1800; 0 1.3472 -0.3491

0 0 1 0; 0 0 0 1] -1.9199 0 0.3491 0

RPO3 [0 -1 0 -1400; 1 0 0 45 -1.3472 -0.3491

-1800; 0 0 1 0; 0 0 0 1] -1.9199 0 0.3491 0

KPO4 [0 1 0 1400; -1 0 0 1800; -30 -1.3472 -0.3491

0 0 1 0; 0 0 0 1] -1.9199 0 0.3491 0

RPO, [0 -1 0 1000; 1 0 0 -1800; 0 1.3472 -0.3491

0 0 1 0; 0 0 0 1] -1.9199 0 0.3491 0

RPO, [0 -1 0 1000; 1 0 0 -1800; 30 1.3472 -0.3491

0 0 1 0; 0 0 0 1] -1.9199 0 0.3491 0

Table A5. Obstacles position and orientation - Case 1.

Obstacles type

Rototranslation matrix in cell reference system. Position in mm

1 0 0 -800; 0 1 0 -700; 0 0 1 1200; 0 0 0 1] 1 0 0 1000; 0 1 0 50; 0 0 1 600; 0 0 0 1] 1 0 0 -900; 0 1 0 300; 0 0 1 0; 0 0 0 1] 1 0 0 -1000; 0 1 0 150; 0 0 1 1500; 0 0 0 1] 1 0 0 -1500; 0 1 0 -300; 0 0 1 900; 0 0 0 1] 1 0 0 -100; 0 1 0 -1200; 0 0 1 0; 0 0 0 1] 1 0 0 200; 0 1 0 0; 0 0 1 100; 0 0 0 1] 1 0 0 -200; 0 1 0 -400 0 0 1 600; 0 0 0 1] 1 0 0 0; 0 1 0 0; 0 0 1 600; 0 0 0 1] 1 0 0 -2100; 0 1 0 700; 0 0 1 400; 0 0 0 1] 1 0 0 1000; 0 1 0 -400; 0 0 1 600; 0 0 0 1] 1 0 0 200; 0 1 0 400; 0 0 1 620; 0 0 0 1] 1 0 0 -300; 0 1 0 -600; 0 0 1 2400; 0 0 0 1]

Fig. A1. Welding gun position and orientation - Case 1: (a) WGMi, (b) WGM2 Table A6. Obstacles position and orientation - Case 2.

Obstacles type

Rototranslation matrix in cell reference system. Position in mm_

[1 0 0 400; 0 1 0 -300; 0 0 1 1000; 0 0 0 1] [1 0 0 500; 0 1 0 200; 0 0 1 1700; 0 0 0 1] [1 0 0 400; 0 1 0 400; 0 0 1 200; 0 0 0 1]

[1 0 0 -700; 0 1 0 -2000; 0 0 1 200; 0 0 0 1]

[1 0 0 0; 0 1 0 1000; 0 0 1 0; 0 0 0 1]

[1 0 0 -1600; 0 1 0 400; 0 0 1 0; 0 0 0 1]

[1 0 0 -2600; 0 1 0 -200; 0 0 1 0; 0 0 0 1]

[1 0 0 -2700; 0 1 0 -500; 0 0 1 1000; 0 0 0 1] [1 0 0 -3000; 0 1 0 -100; 0 0 1 300; 0 0 0 1] [1 0 0 -3000; 0 1 0 100; 0 0 1 500; 0 0 0 1] [1 0 0 -2400; 0 1 0 -100; 0 0 1 1000; 0 0 0 1]

Table A7. Obstacles position and orientation - Case 3.

Obstacles type

Rototranslation matrix in cell reference system. Position in mm_

[1 0 0 [1 0 0 [1 0 0 [1 0 0 [1 0 0 [1 0 0 [1 0 0 [1 0 0 [1 0 0 [1 0 0 [1 0 0

600; 0 1 0 0; 0 0 1 1000; 0 0 0 1] 2000; 0 1 0 -600; 0 0 1 0; 0 0 0 1] 1100; 0 1 0 -200; 0 0 1 100; 0 0 0 1] 1700; 0 1 0 800; 0 0 1 100; 0 0 0 1] 0; 0 1 0 1700; 0 0 1 700; 0 0 0 1] 2800; 0 1 0 -200; 0 0 1 1500; 0 0 0 1] 2900; 0 1 0 1100; 0 0 1 900; 0 0 0 1] -2200; 0 1 0 600; 0 0 1 500; 0 0 0 1]

-2200; 0 1 0 -600; 0 0 1 1900 -2600; 0 1 0 -200; 0 0 1 1400 -2400; 0 1 0 -100; 0 0 1 1000

0 0 0 1] 0 0 0 1] 0 0 0 1]

[1 0 0 -2400; 0 1 0 -100; 0 0 1 1000; 0 0 0 1] 1 0 0 -1000; 0 1 0 -1300; 0 0 1 1600; 0 0 0 1 ] [1 0 0 500; 0 1 0 500; 0 0 1 500; 0 0 0 1]

References

[1] Rossgoderer, U. and Woenckhaus, C. A concept for automatical layout generation. In IEEE International Conference on Robotics and Automation, 1995, Vol. 1, pp. 800-805.

[2] Hammond, F. L. and Shimada, K. Improvement of Manufacturing Workcell Layout Design Using Weighted Isotropy Metrics. In IEEE International Conference on Mechatronics and Automation, 2009, pp. 3408-3414, Changchun, China.

[3] Wang Y, Li D, Liu P, Zhang JP. An integrated accurate collision detection algorithm and its applications in construction. Applied Mechanics and Materials 2013, 353-354: 3673-3682.

[4] Papakostas, N., Alexopoulos, K., and Kopanakis, A. Integrating digital manufacturing and simulation tools in the assembly design process: A cooperating robots cell case. CIRP Journal of Manufacturing Science and Technology, 2011;4:96-100.

[5] Erdos, G., Kemeny, Z., Kovacs, A., and Vancza, J. Planning of remote laser welding processes. Forty Sixth CIRP Conference on Manufacturing Systems 2013.

[6] Pellegrinelli, S., Pedrocchi, N., Molinari Tosatti, L., Fischer, A., Tolio, T. Multi-robot spot-welding cells: an integrated approach to cell design and motion planning, CIRP Annals - Manufacturing Technology, 63(1):17-20.

[7] Pellegrinelli, S., Pedrocchi, N., Molinari Tosatti, L., Fischer, A., & Tolio, T. Design and motion planning of body-in-white assembly cells. In IEEE/RSJ Int. Conf. on Intelligent Robots and Systems. 2014. Chicago.

[8] ORL, http://www.comau.com, visited on 12/2013.

[9] Gottschalk, S., Lin, M. C., & Manocha, D. OBBTree: A Hierarchical Structure for Rapid Interference Detection. SIGGRAPH '96 Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, 1996, pp. 171-180.

[10] Choset, H., Lynch, K. M., Hutchinson, S., Kantor, G., Burgard, W., Kavraki, L. E., and Thrun, S. (2005). Principles of Robot Motion. The MIT Press, London, England.

[11] Geraerts, R. and Overmars, M. (2004). A Comparative Study of Probabilistic Roadmap Planners. In Boissonnat Joel and Goldberg, Ken and Hutchinson, Seth, J.-D. and Burdick, editors, Algorithmic Foundations of Robotics V, volume 7, pages 4358. Springer Berlin Heidelberg.

[12] Chiddarwar, S. S. and Babu, N. R. (2011). Conflict free coordinated path planning for multiple robots using a dynamic path modification sequence. Journal Robotics and Autonomous Systems, 59(7-8):508-518.