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On generating the motion of industrial robot manipulators K. Kaltsoukalas, S. Makris, G. Chryssolouris n,1 Laboratory for Manufacturing Systems and Automation, University of Patras, Greece a r t i c l e i n f o Article history: Received 17 October 2013 Received in revised form 19 September 2014 Accepted 8 October 2014 Available online 29 October 2014 Keywords: Path planning Industrial robot motion Grid search a b s t r a c t In this study, an intelligent search algorithm is proposed to defi ne the path that leads to the desired position and orientation of an industrial robots manipulator end effector. The search algorithm gradually approaches the desired confi guration by selecting and evaluating a number of alternative robots confi gurations. A grid of the robots alternative confi gurations is constructed using a set of parameters which are reducing the search space to minimize the computational time. In the evaluation of the alternatives, multiple criteria are used in order for the different requirements to be fulfi lled. The alternative confi gurations are generated with emphasis being given to the robots joints that mainly affect the position of the end effector. Grid resolution and size parameters are set on the basis of the desired output. High resolution is used for a smooth path and lower for a rough estimation, by providing only a number of the intermediate points to the goal position. The path derived is a series of robot confi gurations. This method provides an inexperienced robot programmer with fl exibility to generate automatically a robotic path that would fulfi ll the desired criteria without having to record intermediate points to the goal position. MNA MNA N 2 From 1 and 2 pDH; MNA MNA 2k1DH 3 Therefore, in the example with the 6 DOFs robot where, the number of the alternative confi gurations was found to be N27 (for DH3) If MNA20, the probability of getting the alternative confi g- uration that is closer to the desired position is given by Eq.(2) Fig. 2. Available joint angles for each degree of freedom in the DH. Fig. 3. COMAU Smart5 Six, 6 DOF, Industrial Manipulator. Fig. 4. Alternative confi gurations using MNA3, DH3 and SR2 parameters for 6 DOFs. K. Kaltsoukalas et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 657167 Probability to get the best alternative confi guration in DH, P(DH3, MNA20) 20=27 74% Consequently, for exhaustive search in DH (P1), MNAN27 Giving sample rate (SR)2 for each alternative in the decision horizon, two samples are taken from the rest of the joints; thus, the number of complete alternative confi gurations becomes N completeMNA 27; SR 2 MNAnSR 27 ? 2 54 complete alternatives In general, the number of complete alternative confi gurations for the predefi ned MNA and SR parameters is given by the following equation: Number of complete alternative configurationsMNA;SR; Ncomplete MNAnSR4 The proposed algorithm does not have to search the entire work- space of the robot. During each iteration, only a maximum number of neighbor confi gurations are evaluated. Calculation time for a complete target path depends on the distance of the starting point to the target. Calculation time also increases when more inter- mediate points are requested for a smoother path that better fulfi lls the desired criteria. 2.2. Evaluation of the alternative confi gurations Multiple criteria are used for the evaluation of the alternative confi gurations. A decision matrix is built as shown in the following table. In the context of this study, two criteria have been taken into consideration, those of the distance due to translation and the distance due to rotation from the target position and the robots orientation. Despite the fact that the proposed algorithm could also be used just for the defi nition of the joint parameters for a given position and orientation of the robots end effector (inverse kinematics), the main purpose of this study is to plan the robots path, which better fulfi lls the multiple criteria defi ned by the user. The search algorithm is easily extensible for more criteria. (Tables 1 and 2) The utility for each of the alternatives is calculated as the weighted sum of the distance due to translation and to orientation. Ui WtjjXi?XjjWrfqi;q5 where Xi?X, is the Euclidean distance of the end effector from the target position and fqi;qtarget is the distance due to rotation (orientation of the target confi guration). The weight factors Wtand Wrare selected from the user in order to give emphasis to the desired criterion. If the user is only interested in the position of the end effector, the factors Wt1 and Wr0 should be used. The metric of the distance between rotations is the Norm of the Difference of Quaternions, described in detail in 17. fqi;qtarget min fjjqi?qtargetjj;jjqiqtargetjjg6 where, J J denotes the Euclidean norm (or 2-norm) and q the orientation of the end effector, expressed in quaternions. The metric gives values in the range 0; ffiffiffi 2 p ?. The alternative confi guration with the smaller utility function is selected at each decision point. Path search algorithm Input: Target position (X Y Z), target orientation (Euler angles ZYZ”), DH, MNA, SR, (k, d: grid size MNAnSR complete alternatives are evaluated. The alternative confi guration that provides the smaller value of the utility function is selected. 7. The resolution and the size of the grid are redefi ned. 8. Steps 17 are repeated until there is an alternative confi gura- tion that provides the target position and target orientation within the pre-defi ned distance error. 2.3. Industrial manipulator motion generation The proposed algorithm calculates the robots sequential, intermediate confi gurations in order to approach the target posi- tion while fulfi lling the predefi ned criteria for the path. Every confi guration of the robot is within its joint limits. The robot controller uses the derived path in order to generate the motion of the industrial manipulator, taking into consideration the dynamic constraints of the robot. 3. Implementation The proposed algorithm has been implemented in Matlab with the use of the Robotics Toolbox 18. The fl owchart of the algorithm is presented in the following fi gure. Fig. 5. Industrial robot motion generation. Table 1 Evaluation of the alternatives according to the distance criteria. Alternative Confi gurations Normalized criteriaUtility value Distance due to translation Distance due to rotation Ui W1Ci1 W2Ci2(where W1 and W2the criteria weights) Alternative 1C11C12U1 Alternative 2C21C22U2 Alternative 3C31C32U3 Alternative mMNAnSR Cm1Cm2Um K. Kaltsoukalas et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 657168 4. Results In Figs. 7 and 8, it is observed that the grid size and resolution parameters (k, d ) have a great infl uence on the smoothness of the path towards the desired position. Lower values of these parameters lead to better paths, however, the computational time is increased. 4.1. Search algorithm parameters correlation In order for the correlation among the search parameters MNA, DH and SR to be examined, a set of experiments was designed using the Taguchi method with the objective of process time minimization. The initial values of the grid parameters were selected to be k5 and d0.1 rad (E61). 4.1.1. Taguchi design of experiments The effect of the search parameters DH, MNA, and SR will be examined so as for the process time required for fi nding the path to be minimized to the target position. Four levels are selected for each parameter. The proposed set of experiments, according to the Taguchi method, is given in L16 table. L16 table: Fig. 7. Grid resolution effect on the on the path (a) d0.01 rad and (b) d0.1 rad. Fig. 6. Flowchart of the proposed algorithm. Table 2 Set of experiments for 4 levels of the parameters DH, MNA, and SR. Exp. no.DHMNASRTime (Sec) 122510.60 225020.57 327531.12 4210041.82 532540.72 635030.91 737520.91 8310011.17 942520.55 1045010.91 1147542.16 12410031.60 1352531.29 1455042.84 1557510.48 16510022.01 K. Kaltsoukalas et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 657169 . Analysis of means (ANOM) ?From Figs. 9 and 10, it is observed that the target position of the end effector is better approached for DH3 (fi rst three degrees of freedom of the robot). The higher values of MNA and SR are suffi cient only when the orientation is taken into consideration. In order for both the target position and orientation of the end effector to be approached, the best results (lowest computing time) are given for DH3, MNA25 and SR2. ?The interaction among the parameters DH, MNA and SR and their effect on the computing time is presented in Fig. 11. It is confi rmed that for lower DH values suffi cient SR has to be consider whilst for higher DH values the SR value should be minimum for less computing time. Fig. 8. Grid size effect on the path (a) path generated for k1 and (b) path generated for k5. Fig. 9. DH, MNA and SR vs. processing time (target position). Fig. 10. DH, MNA and SR vs. processing time (target position and orientation). Fig. 11. Interaction of DH with SR (target position). K. Kaltsoukalas et al. / Robotics and Computer-Integrated Manufacturing 32 (2015) 657170 5. Conclusions In this study, an intelligent search algorithm is proposed to defi ne the path that leads to the desired position and orientation of the end effector of an industrial robot manipulator. The grid parameters as well as the search algorithm parameters DH, MNA, SR are proven to be drastically reducing the processing time. As regards the problem of approaching the target position, it is shown that the best results are obtained when the fi rst three joints of the robot have been considered (DH3). This is consistent with the initial assumption that the fi rst three degrees of the robots freedom (joints) are responsible for the end effectors position. For the rest of the joints, only a few samples are suffi cient in order for the path towards the target position to be determined. When the orientation of the end effector is considered, a higher sample rate for the joint angles, outside the decision horizon, should be used. The criteria considered for the calculation of the distance from the target position and orientation through weight factors, are predefi ned by the user. The path is sent to the robot controller, where the motion program of the industrial manipulator is generated. The algorithm is extensible to the use of more criteria in the future. Free collision paths will be addressed in a future study via a collision detection module integrated into the algorithm. Acknowledgments This study has received funding by the project X-act/FoF-ICT- 314355, funded by the European Commission under the 7th Framework Program. References 1 Chryssolouris G. Manufacturing Systems Theory and Practice. 2nd ed. . New York: Springer-Verlag; 2006. 2 Mourtzis D, Alexopoulos K, Chryssolouris G. Flexibility consideration in the design of manufacturing systems: an industrial case study. CIRP J Manuf Sci Technol 2012;5(4):27683. 3 Karl F, Reinhart G, Zaeh MF. 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