# Workpiece Placement Optimization for Robot Machining Based on the Evaluation of Feasible Kinematic Directional Capabilities

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Human–Robot Collaboration

#### 2.2. Feasible Kinematic Directional Capabilities

#### 2.3. Geometric Analysis of the DTF Method for 5-Axis Robotic Machining

#### 2.4. Optimal Workpiece Placement

#### 2.5. Optimization Criterion

- -
- The joint angles of the robot should be bounded between its positional joint limits:$${\theta}_{i,\mathrm{min}}\le {\theta}_{i}\le {\theta}_{i,\mathrm{max}},i=1,\mathrm{\dots},6,$$
- -
- The workpiece should be located within a working area, which is defined relative to the robot’s base frame {B} within the range:$$\begin{array}{c}{X}_{\mathrm{min}}\le X\le {X}_{\mathrm{max}}\\ {Y}_{\mathrm{min}}\le Y\le {Y}_{\mathrm{max}}\end{array},$$
- -
- The rotation of the workpiece should be limited between:$${\phi}_{\mathrm{min}}\le \phi \le {\phi}_{\mathrm{max}},$$

## 3. Results and Discussion

#### 3.1. Experimental Setup

#### 3.2. Optimization Results

_{max}at each waypoint on the robot’s path, calculated by the DTF method (4). The results are presented in Figure 6 in the form of a swarm scatter chart for each machining path. From Figure 6, it can clearly be seen that the highest worst-case maximum achievable linear velocity ${V}_{\mathrm{max}}^{path}$ is calculated for the optimal placement of the workpiece in all six cases.

#### 3.3. Simulation Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Gajšek, B.; Stradovnik, S.; Hace, A. Sustainable Move towards Flexible, Robotic, Human-Involving Workplace. Sustainability
**2020**, 12, 6590. [Google Scholar] [CrossRef] - Costa, G.D.; Petry, M.R.; Moreira, A.P. Augmented Reality for Human-Robot Collaboration and Cooperation in Industrial Applications: A Systematic Literature Review. Sensors
**2022**, 22, 2725. [Google Scholar] [CrossRef] - Hwang, P.-J.; Hsu, C.-C.; Chou, P.-Y.; Wang, W.-Y.; Lin, C.-H. Vision-Based Learning from Demonstration System for Robot Arms. Sensors
**2022**, 22, 2678. [Google Scholar] [CrossRef] - Khawaja, F.I.; Kanazawa, A.; Kinugawa, J.; Kosuge, K. A Human-Following Motion Planning and Control Scheme for Collaborative Robots Based on Human Motion Prediction. Sensors
**2021**, 21, 8229. [Google Scholar] [CrossRef] - Kanazawa, A.; Kinugawa, J.; Kosuge, K. Adaptive Motion Planning for a Collaborative Robot Based on Prediction Uncertainty to Enhance Human Safety and Work Efficiency. IEEE Trans. Robot.
**2019**, 35, 817–832. [Google Scholar] [CrossRef] - Hähnel, S.; Pini, F.; Leali, F.; Dambon, O.; Bergs, T.; Bletek, T. Reconfigurable Robotic Solution for Effective Finishing of Complex Surfaces. In Proceedings of the 2018 IEEE 23rd International Conference on Emerging Technologies and Factory Automation 2018, Turin, Italy, 4–7 September 2018; pp. 1285–1290. [Google Scholar]
- Ye, C.; Yang, J.; Zhao, H.; Ding, H. Task-dependent workpiece placement optimization for minimizing contour errors induced by the low posture-dependent stiffness of robotic milling. Int. J. Mech. Sci.
**2021**, 205, 106601. [Google Scholar] [CrossRef] - Vosniakos, G.-C.; Matsas, E. Improving feasibility of robotic milling through robot placement optimisation. Robot. Comput.-Integr. Manuf.
**2010**, 26, 517–525. [Google Scholar] [CrossRef] - Henao, J.C.R.; Garzón, J.A.J.; Muñoz, L.D. Manipulability index study on the KUKA robot KR5 ARC HW. In Proceedings of the 2012 XVII Symposium of Image, Signal Processing, and Artificial Vision (STSIVA), Medellin, Colombia, 12–14 September 2012; pp. 72–77. [Google Scholar]
- Malhan, R.K.; Shembekar, A.V.; Kabir, A.M.; Bhatt, P.M.; Shah, B.; Zanio, S.; Nutt, S.; Gupta, S.K. Automated planning for robotic layup of composite prepreg. Robot. Comput.-Integr. Manuf.
**2021**, 67, 102020. [Google Scholar] [CrossRef] - Malhan, R.K.; Kabir, A.M.; Shah, B.; Gupta, S.K. Identifying Feasible Workpiece Placement with Respect to Redundant Manipulator for Complex Manufacturing Tasks. In Proceedings of the 2019 International Conference on Robotics and Automation (ICRA), Montreal, QC, Canada, 20–24 May 2019; pp. 5585–5591. [Google Scholar]
- Gotlih, J.; Brezocnik, M.; Balic, J.; Karner, T.; Razborsek, B.; Gotlih, K. Determination of accuracy contour and optimization of workpiece positioning for robot milling. Adv. Prod. Eng. Manag.
**2017**, 12, 233–244. [Google Scholar] [CrossRef] - Xue, Y.; Sun, Z.; Liu, S.; Gao, D.; Xu, Z. Stiffness-Oriented Placement Optimization of Machining Robots for Large Component Flexible Manufacturing System. Machines
**2022**, 10, 389. [Google Scholar] [CrossRef] - Bhatt, P.M.; Kulkarni, A.; Malhan, R.K.; Gupta, S.K. Optimizing Part Placement for Improving Accuracy of Robot-Based Additive Manufacturing. In Proceedings of the 2021 IEEE International Conference on Robotics and Automation (ICRA), Xi’an, China, 30 May–5 June 2021; pp. 859–865. [Google Scholar]
- Ur-Rehman, R.; Caro, S.; Chablat, D.; Wenger, P. Multi-objective path placement optimization of parallel kinematics machines based on energy consumption, shaking forces and maximum actuator torques: Application to the Orthoglide. Mech. Mach. Theory
**2010**, 45, 1125–1141. [Google Scholar] [CrossRef] - Santos, R.; Steffen, J.V.; Saramago, S. Optimal Task Placement of a Serial Robot Manipulator for Manipulability and Mechanical Power Optimization. Intell. Inf. Manag.
**2010**, 2, 512–525. [Google Scholar] [CrossRef] - Malhan, R.; Kabir, A.; Shah, B.; Centea, T.; Gupta, S. Determining Feasible Robot Placements in Robotic Cells for Composite Prepreg Sheet Layup. Presented at the 14th International Manufacturing Science and Engineering Conference, Erie, PA, USA, 10–14 June 2019. [Google Scholar]
- Lu, L.; Zhang, J.; Fuh, J.Y.H.; Han, J.; Wang, H. Time-optimal tool motion planning with tool-tip kinematic constraints for robotic machining of sculptured surfaces. Robot. Comput.-Integr. Manuf.
**2020**, 65, 101969. [Google Scholar] [CrossRef] - Balci, B.; Donovan, J.; Roberts, J.; Corke, P. Optimal Workpiece Placement Based on Robot Reach, Manipulability and Joint Torques. In Proceedings of the 2023 IEEE International Conference on Robotics and Automation (ICRA), London, UK, 29 May–2 June 2023; pp. 12302–12308. [Google Scholar]
- Yoshikawa, T. Manipulability of Robotic Mechanisms. Int. J. Robot. Res.
**1985**, 4, 3–9. [Google Scholar] [CrossRef] - Doan, N.; Lin, W. Optimal robot placement with consideration of redundancy problem for wrist-partitioned 6R articulated robots. Robot. Comput.-Integr. Manuf.
**2017**, 48, 233–242. [Google Scholar] [CrossRef] - Aspragathos, N.A.; Foussias, S. Optimal location of a robot path when considering velocity performance. Robotica
**2002**, 20, 139–147. [Google Scholar] [CrossRef] - Valsamos, H.; Nektarios, T.; Aspragathos, N.A. Optimal Placement of Path Following Robot Task Using Genetic Algorithms. IFAC Proc. Vol.
**2006**, 39, 132–137. [Google Scholar] [CrossRef] - Nektarios, A.; Aspragathos, N.A. Optimal location of a general position and orientation end-effector’s path relative to manipulator’s base, considering velocity performance. Robot. Comput.-Integr. Manuf.
**2010**, 26, 162–173. [Google Scholar] [CrossRef] - Stradovnik, S.; Hace, A. Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining. Sensors
**2022**, 22, 4267. [Google Scholar] [CrossRef] - ISO 10218-1/2:2011; Robots and Robotic Devices Safety Requirements for Industrial Robots Part 1: Robots/Part 2: Robot Systems and Integration. ISO: Genewa, Switzerland, 2011.
- ISO/TS 15066:2016; Robots and Robotic Devices Collaborative Robots. ISO: Genewa, Switzerland, 2016.
- Morishige, K.; Sato, Y. Optimization of Workpiece Placement in Sealing Operation Using Industrial Robot Considering Manipulability. Presented at the International Symposium on Flexible Automation, Kanazawa Japan, 15–19 July 2018. [Google Scholar]
- Zhang, L.; Guo, S.; Huang, Y.; Xiong, X. Kinematic Singularity Analysis and Simulation for 7DOF Anthropomorphic Manipulator. Int. J. Mechatron. Appl. Mech.
**2019**, 1, 157–164. [Google Scholar] - Feng, Y.; Fang, L.; Bu, W.; Kang, J. Multi-objective Optimization for Design of Redundant Serial Robots. In Proceedings of the 2020 Chinese Automation Congress (CAC), Shanghai, China, 6–8 November 2020; pp. 4987–4991. [Google Scholar]
- Chiu, S.L. Task Compatibility of Manipulator Postures. Int. J. Robot. Res.
**1988**, 7, 13–21. [Google Scholar] [CrossRef] - Bicchi, A.; Melchiorri, C.; Balluchi, D. On the mobility and manipulability of general multiple limb robots. IEEE Trans. Robot. Autom.
**1995**, 11, 215–228. [Google Scholar] [CrossRef] - Patel, S.; Sobh, T. Manipulator Performance Measures—A Comprehensive Literature Survey. J. Intell. Robot Syst.
**2014**, 77, 1–24. [Google Scholar] [CrossRef] - Boschetti, G. A Novel Kinematic Directional Index for Industrial Serial Manipulators. Appl. Sci.
**2020**, 10, 5953. [Google Scholar] [CrossRef] - Boschetti, G.; Rosa, R.; Trevisani, A. Parallel Robot Translational Performance Evaluation through Direction-Selective Index (DSI). J. Robot.
**2011**, 2011, 129506. [Google Scholar] [CrossRef] - Mansouri, I.; Ouali, M. The power manipulability—A new homogeneous performance index of robot manipulators. Robot. Comput.-Integr. Manuf.
**2011**, 27, 434–449. [Google Scholar] [CrossRef] - Yoshikawa, T. Translational and rotational manipulability of robotic manipulators. In Proceedings of the IECON ‘91: 1991 International Conference on Industrial Electronics, Control and Instrumentation, Kobe, Japan, 28 October–1 November 1991; Volume 2, pp. 1170–1175. [Google Scholar]
- Finotello, R.; Grasso, T.; Rossi, G.; Terribile, A. Computation of kinetostatic performances of robot manipulators with polytopes. In Proceedings of the 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146), Leuven, Belgium, 20 May 1998; Volume 4, pp. 3241–3246. [Google Scholar]
- Jihong, L.; Won, K.T. Inverse kinematic solution based on decomposed manipulability. In Proceedings of the 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C), Detroit, MI, USA, 10–15 May 1999; Volume 2, pp. 1514–1519. [Google Scholar]
- Long, P.; Padir, T. Constrained Manipulability for Humanoid Robots Using Velocity Polytopes. Int. J. Humanoid Robot.
**2020**, 17, 1950037. [Google Scholar] [CrossRef] - Moulianitis, V.; Katrantzis, E.; Stravopodis, N.; Aspragathos, N. A Comparative Study of Three Manipulator Performance Measures. In Proceedings of the 26th International Conference on Robotics in Alpe-Adria-Danube Region, RAAD 2017, Turin, Italy, 21–23 July 2017; pp. 19–27. [Google Scholar]
- Mansfeld, N.; Keppler, M.; Haddadin, S. Speed Gain in Elastic Joint Robots: An Energy Conversion-Based Approach. IEEE Robot. Autom. Lett.
**2021**, 6, 4600–4607. [Google Scholar] [CrossRef] - Banchoff, T.F.; Lovett, S. Differential Geometry of Curves and Surfaces; CRC Press: Boca Raton, FL, USA, 2022. [Google Scholar]
- Wang, Y.; Shoemaker, C. A General Stochastic Algorithmic Framework for Minimizing Expensive Black Box Objective Functions Based on Surrogate Models and Sensitivity Analysis. arXiv
**2014**, arXiv:1410.6271. [Google Scholar]

**Figure 1.**Intuitive definition of the robotic surface machining area through human–robot collaboration: (

**a**) the kinesthetic guidance of the robot, (

**b**) the definition of the machining area through a holographic interface, (

**c**) setting up the technological parameters through a holographic interface.

**Figure 2.**Graphical representation of the Yoshikawa manipulability index $w$; maximum feasible tool velocity ${\mu}_{2}$ (ellipsoid approach); maximum feasible tool velocity ${\mu}_{\infty}$ (polytope approach); maximum linear velocity ${v}_{a}$ (DTF method); and maximum angular velocity ${\omega}_{a}$ (DTF method) in the case of a 3-DOF planar robotic mechanism.

**Figure 3.**UR5e collaborative robot on a mobile platform and presentation of the coordinate system (red-green-blue arrows refer to x-y-z Cartesian coordinate system axes).

**Figure 4.**Three-dimensional model of a workpiece: (

**a**) Workpiece 1, (

**b**) Workpiece 2, (

**c**) Workpiece 3, and the machining paths across the surface of the workpiece: (

**d**) 1A, (

**e**) 2A, (

**f**) 3A, (

**g**) 1B, (

**h**) 2B, (

**i**) 3B; surface color is coded by the surface point height (blue—low value, yellow—high value).

**Figure 5.**Workpiece placement on the mobile platform for the machining paths: (

**a**) 1A, (

**b**) 1B, (

**c**) 2A, (

**d**) 2B, (

**e**) 3A, (

**f**) 3B. Red-colored workpiece shows its initial placement “Init”, blue- and orange-colored workpieces show its intermediate placements, respectively, and green -colored workpiece shows its optimal position “Optim”.

**Figure 6.**Representation of the calculated maximum achievable tool velocity V

_{max}at each waypoint on the robot’s path, and minimum value of V

_{max}for four placements of the workpiece in the case of the machining paths: (

**a**) 1A, (

**b**) 1B, (

**c**) 2A, (

**d**) 2B, (

**e**) 3A, (

**f**) 3B. Red, blue, orange, and green colors refer to initial workpiece placement “Init”, intermediate workpiece placements “1” and ”2” and optimal workpiece placement “Optim”, respectively.

**Figure 7.**Representation of the joint velocity infinity norm for four different placements of the workpiece in the case of the machining paths: (

**a**) 1A, (

**b**) 1B, (

**c**) 2A, (

**d**) 2B, (

**e**) 3A, (

**f**) 3B. Red, blue, orange, and green colors refer to initial workpiece placement “Init”, intermediate workpiece placements “1” and ”2” and optimal workpiece placement “Optim”, respectively.

Optimization Constraint | Value |
---|---|

Positional joint limits of the UR5e robot | $-360\xb0\le {\theta}_{i}\le 360\xb0$, $i=1,\dots ,6$ |

Working area | $\begin{array}{c}-400\mathrm{mm}\le X\le 400\mathrm{mm}\\ -200\mathrm{mm}\le Y\le -800\mathrm{mm}\end{array}$ |

Workpiece rotation | $-180\xb0\le \phi \le 180\xb0$ |

**Table 2.**Optimal workpiece placement, described by the transformation matrix for the machining paths: (

**a**) 1A, (

**b**) 1B, (

**c**) 2A, (

**d**) 2B, (

**e**) 3A, (

**f**) 3B.

(a) | (c) | (e) |

${T}_{O,Optim}^{B}=\left[\begin{array}{cccc}0.1186& -0.9929& -0.0023& 0.2461\\ 0.9929& 0.1186& 0.0030& -0.4612\\ 0.0030& -0.0059& 0.9999& -0.0662\\ 0& 0& 0& 1\end{array}\right]$ | ${T}_{O,Optim}^{B}=\left[\begin{array}{cccc}-0.9447& 0.3278& -0.0017& 0.2867\\ -0.3278& -0.9447& -0.0034& -0.3302\\ -0.0027& -0.0027& 1.0000& -0.0928\\ 0& 0& 0& 1\end{array}\right]$ | ${T}_{O,Optim}^{B}=\left[\begin{array}{cccc}0.8610& -0.5086& 0.0006& -0.0751\\ 0.5086& 0.8610& 0.0077& -0.5075\\ -0.0044& -0.0064& 1.0000& -0.0949\\ 0& 0& 0& 1\end{array}\right]$ |

(b) | (d) | (f) |

${T}_{O,Optim}^{B}=\left[\begin{array}{cccc}0.6067& -0.7949& 0.0005& -0.2991\\ 0.7949& 0.6067& 0.0104& -0.4992\\ -0.0085& -0.0059& 0.9999& -0.0662\\ 0& 0& 0& 1\end{array}\right]$ | ${T}_{O,Optim}^{B}=\left[\begin{array}{cccc}-0.9987& -0.0503& -0.0053& 0.0153\\ 0.0503& -0.9987& 0.0017& -0.5156\\ -0.0054& 0.0015& 1.0000& -0.0932\\ 0& 0& 0& 1\end{array}\right]$ | ${T}_{O,Optim}^{B}=\left[\begin{array}{cccc}0.9919& 0.1271& 0.0052& 0.1424\\ -0.1272& 0.9919& 0.0057& -0.1525\\ -0.0044& -0.0064& 1.0000& -0.0949\\ 0& 0& 0& 1\end{array}\right]$ |

Machining Parameter | Value |
---|---|

TCP speed | $50\mathrm{mm}/\mathrm{s}$ |

TCP acceleration | $500{\mathrm{mm}/\mathrm{s}}^{2}$ |

Blend radius | 1 mm |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stradovnik, S.; Hace, A.
Workpiece Placement Optimization for Robot Machining Based on the Evaluation of Feasible Kinematic Directional Capabilities. *Appl. Sci.* **2024**, *14*, 1531.
https://doi.org/10.3390/app14041531

**AMA Style**

Stradovnik S, Hace A.
Workpiece Placement Optimization for Robot Machining Based on the Evaluation of Feasible Kinematic Directional Capabilities. *Applied Sciences*. 2024; 14(4):1531.
https://doi.org/10.3390/app14041531

**Chicago/Turabian Style**

Stradovnik, Saša, and Aleš Hace.
2024. "Workpiece Placement Optimization for Robot Machining Based on the Evaluation of Feasible Kinematic Directional Capabilities" *Applied Sciences* 14, no. 4: 1531.
https://doi.org/10.3390/app14041531