# Force-Sensor-Free Implementation of a Hybrid Position–Force Control for Overconstrained Cable-Driven Parallel Robots

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## Abstract

**:**

## 1. Introduction

- a pure inverse kinematic controller (IKC), which (as the new controllers proposed in this paper) does not use force sensors, but it does not involve any correction to maintain cable tensions within predefined bounds.

## 2. CDPR Modeling

#### 2.1. Kinetostatic Model

#### 2.2. Friction Model

- $p(f,0)=0\phantom{\rule{2.84544pt}{0ex}}$; this condition is necessary to represent the behavior of the dynamic friction that is zero at rest (static friction is not modeled);
- continuity of the polynomial at the borders: $p(f,-{v}_{L})={\tau}_{F}(f,-{v}_{L})={\tau}_{vn}$ and $p(f,{v}_{L})={\tau}_{F}(f,{v}_{L})={\tau}_{vp}$;
- continuity of the derivative of the polynomial at the borders: $\frac{\partial p(f,-{v}_{L})}{\partial v}=\frac{\partial {\tau}_{F}(f,-{v}_{L})}{\partial v}=\partial {\tau}_{vn}$, and $\frac{\partial p(f,{v}_{L})}{\partial v}=\frac{\partial {\tau}_{F}(f,{v}_{L})}{\partial v}=\partial {\tau}_{vp}$.

`lsqnonlin`with a step tolerance and a function tolerance both equal to ${10}^{-6}$. The

`lsqnonlin`function implements a nonlinear least squares solver for curve-fitting problems, and it is used to find the coefficients ${c}_{in}$ and ${c}_{ip}$ on the right-hand side of Equation (5) that minimize the 2-norm of the error vector between the friction torques computed with the model in Equation (5), and the torques recorded during the friction tests. Figure 5 represents the friction torque model described in Equation (5) with the coefficients listed in Table 1. The black points are the results provided by the experimental tests. The model is considered to give a good estimation of the physical phenomenon since the coefficient of determination [45] is higher than 0.9 (if ${R}^{2}=1$, the function exactly interpolates the input data). It is interesting to note that the maximum value of the predicted friction torque is $0.15$ Nm, which is approximately $22\%$ of the nominal motor torque (equal to $0.69$ Nm). This suggests the importance of the friction model introduced in this section.

#### 2.3. Following-Error Model

`lsqnonlin`with a step tolerance and a function tolerance both equal to ${10}^{-6}$. Their values are ${c}_{f}=-0.511$, and ${c}_{v}=8.893\times {10}^{-5}$, with $\tau $ expressed in newtons per meter and v in meters per minute. These values lead to a coefficient of determination ${R}^{2}$ of the FE model equal to 0.999. The coefficients ${c}_{f}$ and ${c}_{v}$ differ by several orders of magnitude, but the value that multiplies $\tau $ in Equation (9) is ${c}_{f}/{K}_{p}=-2.55\times {10}^{-3}$. Working with coefficients with this order of magnitude was not a problem since double-precision variables were used in the PLC code.

## 3. Hybrid Control Strategy

## 4. Experimental Results and Validation

#### 4.1. Comparison between the Hybrid Control Strategies

#### 4.2. Comparison between the Hybrid Controllers and Other Controllers

#### 4.3. Effect of Changing the Pair of Force-Controlled Cables

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CDPR | Cable-driven parallel robot |

DOFs | Degrees of freedom |

HC | Hybrid joint-space control |

NC | Nullspace control |

IKC | Inverse kinematic controller |

FE | Following-error |

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**Figure 1.**IPAnema 3 Mini with the laser tracker used in the experiments. This picture is taken from [4].

**Figure 3.**Scheme of the robot kinematic chain and the areas in which the friction torque ${\tau}_{F}$ and force ${f}_{F}$ are modeled.

**Figure 4.**Scheme of the cable routing during the tests executed to estimate the parameters of the friction model in the cable actuation chain.

**Figure 6.**FE model described in Equation (8).

**Figure 7.**Workspace of the IPAnema 3 Mini with ${\sigma}_{ij}<1.5$ N when the force-controlled cables are $1,2$ (

**a**) or $3,4$ (

**b**). The yellow dots represent the exit points of the cables from the frame.

**Figure 8.**Control schemes applied to force-controlled cables by using (

**a**) force sensor feedback, HC-f; (

**b**) motor-torque feedback, HC-$\tau $; and (

**c**) FE feedback, HC-e.

**Figure 9.**Robot trajectories in the experimental tests. The paths where the force-controlled cables are wires $1,2$ are plotted in red, whereas the ones in which the force-controlled cables are wires $3,4$ are plotted in blue. (

**a**) Triangular trajectories (

**b**) Circular trajectories (

**c**) Rectangular trajectory.

**Figure 10.**Examples of the evolution in time of the cable tensions in the force-controlled cables for the triangular (plots in the first row) and circular (plots in the second row) red trajectories in Figure 9. Different controllers are considered: controller HC-f, controller HC-$\tau $, controller HC-e, controller HC-$\tau $ without the friction torque model. Solid lines represent measured forces, whereas dashed lines represent the theoretical forces computed through the force distribution algorithm.

**Figure 11.**Evolution in time of the cable forces during the tests identified by the legend in Table 3. ${f}_{min}=5$ N and ${f}_{max}=35$ N are the desired bounds of cable tensions. Solid lines represent measured forces, whereas dashed lines represent the theoretical forces computed through the force distribution algorithm.

**Figure 13.**The plots show the evolution in time of cable forces (plots in the first row) and position error (plots in the second row) when force-controlled cables are changed for increasing values of $\delta $. The areas in light blue represent the parts of the trajectory in which the force-controlled cables are 1 and 2, while the areas in yellow are the ones in which the force-controlled cables are 3 and 4.

$\mathit{i}=1$ | $\mathit{i}=2$ | $\mathit{i}=3$ | $\mathit{i}=4$ | ${\mathit{R}}^{2}$ | ||
---|---|---|---|---|---|---|

Friction torque ${\tau}_{F}$ (Nm) | ${c}_{ip}$ | $-3.64\times {10}^{-3}$ | $-5.85\times {10}^{-6}$ | $-4.66\times {10}^{-4}$ | $-2.45\times {10}^{-2}$ | $0.949$ |

${c}_{in}$ | $-5.97\times {10}^{-3}$ | $-2.31\times {10}^{-4}$ | $\phantom{\rule{6.544pt}{0ex}}3.10\times {10}^{-3}$ | $\phantom{\rule{6.544pt}{0ex}}2.52\times {10}^{-2}$ | $0.995$ | |

Friction force ${f}_{F}$ (N) | ${c}_{ip}$ | $-1.73\times {10}^{-1}$ | $\phantom{\rule{6.544pt}{0ex}}1.62\times {10}^{-2}$ | $-2.51\times {10}^{-2}$ | $2.93\times {10}^{-1}$ | $0.920$ |

${c}_{in}$ | $-1.60\times {10}^{-1}$ | $-1.40\times {10}^{-2}$ | $\phantom{\rule{6.544pt}{0ex}}1.25\times {10}^{-1}$ | $1.81\times {10}^{-1}$ | $0.986$ |

**Table 2.**Parameters of the three different high-level PID controllers shown in Figure 8.

Controller | ${\mathit{K}}_{\mathit{HL}}$ | ${\mathit{T}}_{\mathit{i}}$ (ms) | ${\mathit{T}}_{\mathit{v}}$ (ms) | ${\mathit{T}}_{\mathit{d}}$ (ms) |
---|---|---|---|---|

HC-f | $3.0\times {10}^{-4}$ | 70 | 15 | 5 |

HC-$\tau $ | $2.0\times {10}^{-2}$ | 200 | 0 | 0 |

HC-e | $4.5\times {10}^{-1}$ | 7 | 0 | 0 |

Test $\mathit{ij}$ | ||||
---|---|---|---|---|

$\mathit{i}$: | Controller | $\mathit{j}$: | Trajectory Shape | Force-Controlled Cables |

1: | IKC | 1: | Triangular | $1,2$ |

2: | HC-e | 2: | Circular | $1,2$ |

3: | HC-f | 3: | Triangular | $3,4$ |

4: | NC | 4: | Circular | $3,4$ |

5: | Rectangular |

**Table 4.**Robot accuracy with several controllers applied to different trajectories. The errors in the positioning of the marker are evaluated for a given test through the maximum (${\u03f5}_{max}$) and mean ($\overline{\u03f5}$) value over an entire motion law.

Test | ${\mathit{\u03f5}}_{\mathit{max}}$ (mm) | $\overline{\mathit{\u03f5}}$ (mm) | Test | ${\mathit{\u03f5}}_{\mathit{max}}$ (mm) | $\overline{\mathit{\u03f5}}$ (mm) |
---|---|---|---|---|---|

11 | $3.48$ | $2.59$ | 31 | $3.78$ | $2.73$ |

12 | $3.46$ | $1.85$ | 32 | $3.36$ | $2.30$ |

13 | $4.32$ | $2.65$ | 33 | $3.78$ | $2.34$ |

14 | $4.13$ | $2.31$ | 34 | $3.71$ | $2.56$ |

15 | $3.92$ | $2.44$ | 35 | $3.77$ | $1.97$ |

21 | $3.68$ | $2.70$ | 41 | $3.10$ | $2.22$ |

22 | $3.76$ | $2.40$ | 42 | $3.49$ | $1.74$ |

23 | $3.85$ | $2.44$ | 43 | $3.04$ | $2.01$ |

24 | $4.02$ | $2.61$ | 44 | $2.66$ | $1.72$ |

25 | $3.68$ | $2.04$ | 45 | $5.53$ | $3.24$ |

51 | $23.85$ | $8.03$ |

Test | ${\mathit{\rho}}_{\mathit{max}}$ (mm) | $\overline{\mathit{\rho}}$ (mm) | Test | ${\mathit{\rho}}_{\mathit{max}}$ (mm) | $\overline{\mathit{\rho}}$ (mm) |
---|---|---|---|---|---|

11 | $0.152$ | $0.047$ | 31 | $0.143$ | $0.044$ |

21 | $0.151$ | $0.051$ | 41 | $0.076$ | $0.033$ |

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**MDPI and ACS Style**

Guagliumi, L.; Berti, A.; Monti, E.; Fabritius, M.; Martin, C.; Carricato, M.
Force-Sensor-Free Implementation of a Hybrid Position–Force Control for Overconstrained Cable-Driven Parallel Robots. *Robotics* **2024**, *13*, 25.
https://doi.org/10.3390/robotics13020025

**AMA Style**

Guagliumi L, Berti A, Monti E, Fabritius M, Martin C, Carricato M.
Force-Sensor-Free Implementation of a Hybrid Position–Force Control for Overconstrained Cable-Driven Parallel Robots. *Robotics*. 2024; 13(2):25.
https://doi.org/10.3390/robotics13020025

**Chicago/Turabian Style**

Guagliumi, Luca, Alessandro Berti, Eros Monti, Marc Fabritius, Christoph Martin, and Marco Carricato.
2024. "Force-Sensor-Free Implementation of a Hybrid Position–Force Control for Overconstrained Cable-Driven Parallel Robots" *Robotics* 13, no. 2: 25.
https://doi.org/10.3390/robotics13020025