# Elasto-Geometrical Model-Based Control of Industrial Manipulators Using Force Feedback: Application to Incremental Sheet Forming

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## Abstract

**:**

## 1. Introduction

- Robot category upgrade;
- Absolute pose feedback control;
- Force control;
- Model-based compensation.

- MSA models are based on the Euler–Bernoulli beam theory. They are well-suited for simple and slender geometrical structures used for parallel manipulators [26,27]. This method allows the description of the behavior of the joints using an appropriately located stiffness matrix setting up the radial, axial, radial rotational and axial rotational stiffnesses of each joint.
- A special case of MSA is the VJM, also called lumped-stiffness modeling, where the elastic deformations are only localized at the joints [28]. Indeed, several research works have demonstrated that for industrial anthropomorphic robots, deflection errors are mainly due to joint elasticity [29]. Furthermore, for the sake of simplicity, the elasticity of each joint is usually modeled by a single torsional spring located along its motorized axis to integrate the elastic deformations of the structure, the joint and the actuator [30].

- The first feature is the development of an efficient test-model approach to identify the model structure and calibrate the elastic parameters of an industrial serial robot. Using rigorous iterations, the elasto-geometrical model is identified and enhanced, aiming at the best compromise between complexity and accuracy before being validated during experimental tests.
- The second feature is the implementation of an elasto-geometrical model-based position control loop with force feedback to elastically correct the Tool Center Point (TCP) pose of any serial robot.
- The third feature is the validation of both the identification approach and the elastic correction strategy on a real ISF application.

## 2. Force-Feedback Position Control Based on Elasto-Geometrical Modeling

#### 2.1. Position Control Strategy of Industrial Manipulators Using a Force-Feedback Loop

- ${R}_{0}=({\mathbf{O}}_{0},{\mathbf{x}}_{0},{\mathbf{y}}_{0},{\mathbf{z}}_{0})$ is the base frame of the robot;
- ${R}_{t}=({\mathbf{O}}_{t},{\mathbf{x}}_{t},{\mathbf{y}}_{t},{\mathbf{z}}_{t})$ is the robot tool frame.

#### 2.2. Elasto-Geometrical Modeling

## 3. Elasto-Geometrical Model Identification and Calibration

#### 3.1. Refinement of the Elasto-Geometrical Model of the Stäubli TX200

#### 3.2. Validation of the TX200 Elasto-Geometrical Model

## 4. Application to Forming Processes: ISF Experiment

#### 4.1. Experimental Setup

#### 4.2. Hardware Implementation of the Experimental Tests

#### 4.3. Results

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CAM | Computer-Aided Manufacturing |

CAD | Computer-Aided Design |

CNC | Computer Numerical Control |

FSW | Friction-Stir Welding |

ISF | Incremental Sheet Forming |

RISF | Robotized Incremental Sheet Forming |

VJM | Virtual-Joint Method |

MSA | Matrix Structural Analysis |

FEA | Finite Element Analysis |

TCP | Tool center Point |

EE | End-Effector |

DoF | Degree of Freedom |

mDoF | Motorized Degree of Freedom |

eDoF | Elastic Degree of Freedom |

mDH | Modified Denavit–Hartenberg |

PID | Proportional-Integral-Derivative |

RMS | Root-Mean-Square |

## Appendix A

Step 1 : Identification of${k}_{\theta 1,z}$and${k}_{\theta 3,x}$ $\mathbf{q}={\left[0\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ $\mathbf{w}={\left[0\phantom{\rule{0.222222em}{0ex}}{f}_{y}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ |

Step 2 : Identification of ${k}_{\theta 2,z}$ $\mathbf{q}={\left[0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ $\mathbf{w}={\left[{f}_{x}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ |

Step 3 : Identification of ${k}_{\theta 3,z}$ $\mathbf{q}={\left[0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ $\mathbf{w}={\left[0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}{f}_{z}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ |

Step 4 : Identification of${k}_{\theta 5,z}$$\mathbf{q}={[0\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}0-{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0]}^{T}$ $\mathbf{w}={\left[{f}_{x}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ |

Step 5 : Identification of ${k}_{\theta 4,z}$ $\mathbf{q}={[0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{3\pi}{4}}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}-{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0]}^{T}$ $\mathbf{w}={\left[0\phantom{\rule{0.222222em}{0ex}}{f}_{y}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ |

**Figure A1.**Deviation from nominal trajectory in mm along the ${\mathbf{x}}_{p}$-axis (

**a**); the ${\mathbf{y}}_{p}$-axis (

**b**); the ${\mathbf{z}}_{p}$-axis (

**c**); and in norm (

**d**); without online correction (left) and with online correction (right).

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**Figure 3.**Deflection of the TCP due to the external wrench $\mathbf{w}$ applied on the tool: desired (

**a**); real (

**b**); configurations.

**Figure 6.**Flow chart of the coupled test-model approach used for identifying the elasto-geometrical model.

**Figure 7.**Specific configuration for stiffness model structure identification and calibration of the Stäubli TX200 (top view): articular configuration $\mathbf{q}={\left[0\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$; external wrench $\mathbf{w}={\left[0\phantom{\rule{0.222222em}{0ex}}{f}_{y}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ with the maximum value being ${f}_{y}=1400$ N.

**Figure 8.**Elastic displacement $\delta {p}_{x}$ measured for several points along the ${\mathbf{x}}_{0}$-axis and estimated by the one-eDoF-per-joint model and the presented model: articular configuration $\mathbf{q}={\left[0\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$; external wrench $\mathbf{w}={\left[0\phantom{\rule{0.222222em}{0ex}}{f}_{y}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ with ${f}_{y}=1400$ N.

**Figure 9.**Elastic displacement $\delta {p}_{y}$ measured for several points along the ${\mathbf{x}}_{0}$-axis and estimated by the one-eDoF-per-joint model and the presented model: articular configuration $\mathbf{q}={\left[0\phantom{\rule{0.222222em}{0ex}}{\textstyle \frac{\pi}{2}}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$; external wrench $\mathbf{w}={\left[0\phantom{\rule{0.222222em}{0ex}}{f}_{y}\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\phantom{\rule{0.222222em}{0ex}}0\right]}^{T}$ with ${f}_{y}=1400$ N.

**Figure 10.**Comparison of ${}^{0}\delta {p}_{z}$ error for the presented model predictions (blue), one-eDoF-per-joint elastic model predictions (red) and experimental measurements (green) in a 0.6-by-0.6 meter square workspace plane with a vertical applied force of 600 N.

**Figure 11.**Heatmaps of the deviation between the one-eDoF-per-joint model predictions and experimental measurements (

**a**); and between the presented model predictions and experimental measurements (

**b**); both for a vertical applied force of 600 N.

**Figure 12.**Comparison of ${}^{0}\delta {p}_{z}$ error for the presented model predictions (blue), one-eDoF-per-joint elastic model predictions (red) and experimental measurements (green) in a 0.6-by-0.6 meter square workspace plane with a vertical applied force of 1100 N.

**Figure 13.**Heatmaps of the deviation between the one-eDoF-per-joint model predictions and experimental measurements (

**a**); and between the presented model predictions and experimental measurements (

**b**); both for a vertical applied force of 1100 N.

**Figure 14.**(

**a**) Desired spiral trajectory; (

**b**) Comparison of ${}^{0}\delta {p}_{z}$ error for the presented model predictions (continuous blue) and experimental measurements (dashed orange) along a spiral trajectory with a constant vertical load of 600 N.

**Figure 15.**Position of the ISF setup inside the robotic cell (

**left**); desired tool path and CAD model of the clamping system with a formed metal sheet (

**right**).

**Figure 17.**Deviation from desired trajectory measured at the TCP during the cone-forming experiment without and with online correction.

**Figure 18.**External force along the ${\mathbf{x}}_{p}$-, ${\mathbf{y}}_{p}$- and ${\mathbf{z}}_{p}$-axes measured at the robot TCP during the cone-forming experiment without and with online correction.

**Figure 19.**Trajectory-tracking error norm as a function of the external force norm (both measured at the robot TCP during the cone forming experiment without and with online correction).

**Table 1.**Torsional stiffness parameters ${k}_{\theta i,j}$ of the Stäubli TX200 (${10}^{6}\mathrm{Nm}\xb7{\mathrm{rad}}^{-1}$).

Joint 1 | Joint 2 | Joint 3 | Joint 4 | Joint 5 | Joint 6 | |
---|---|---|---|---|---|---|

${k}_{\theta i,x}$ | - | - | 1.45 | - | - | - |

${k}_{\theta i,y}$ | - | - | - | - | - | - |

${k}_{\theta i,z}$ | 2.32 | 1.76 | 2.04 | 0.09 | 0.02 | - |

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## Share and Cite

**MDPI and ACS Style**

Johra, M.; Courteille, E.; Deblaise, D.; Guégan, S.
Elasto-Geometrical Model-Based Control of Industrial Manipulators Using Force Feedback: Application to Incremental Sheet Forming. *Robotics* **2022**, *11*, 48.
https://doi.org/10.3390/robotics11020048

**AMA Style**

Johra M, Courteille E, Deblaise D, Guégan S.
Elasto-Geometrical Model-Based Control of Industrial Manipulators Using Force Feedback: Application to Incremental Sheet Forming. *Robotics*. 2022; 11(2):48.
https://doi.org/10.3390/robotics11020048

**Chicago/Turabian Style**

Johra, Marwan, Eric Courteille, Dominique Deblaise, and Sylvain Guégan.
2022. "Elasto-Geometrical Model-Based Control of Industrial Manipulators Using Force Feedback: Application to Incremental Sheet Forming" *Robotics* 11, no. 2: 48.
https://doi.org/10.3390/robotics11020048