# Vision-Based Hybrid Controller to Release a 4-DOF Parallel Robot from a Type II Singularity

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. 3UPS+RPU Parallel Robot

- The position (${x}_{m}$, ${z}_{m}$) and the orientation ($\theta $, $\psi $) of the mobile platform.
- The orientation of the four universal joints: ${q}_{l1}$, ${q}_{l2}$ for limbs $l=1\dots 3$ and ${q}_{43}$, ${q}_{44}$ for limb 4.
- The length of the four linear actuators given by ${q}_{l3}$ for limbs $l=1\dots 3$ and ${q}_{42}$ for limb 4.
- The orientation of the three spherical joints represented by ${q}_{l4}$, ${q}_{l5}$, ${q}_{l6}$ for external limbs $l=1\dots 3$.
- The orientation of the revolute joint ${q}_{41}$.

#### 2.2. Type II Singularities

#### 2.3. Angle between Two Output Twist Screws

#### 2.4. 3D Tracking System

#### 2.5. Hybrid Controller Description

Algorithm 1. Initialization 3 |

INITIALIZATION$\overrightarrow{\Delta i}=\overrightarrow{0}$ ${N}_{ch}=$ number of columns of ${\mathrm{M}}_{inc}$. BEGIN${\overrightarrow{q}}_{{ind}_{d}}={\overrightarrow{q}}_{{ind}_{r}}+{\nu}_{d}\xb7{t}_{s}\xb7\overrightarrow{\Delta i}$ IF ${e}_{pin}==true$${\mathrm{min}\mathsf{\Omega}}_{c}=$ minimum element in ${\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{c}$ IF ${\mathrm{min}\mathsf{\Omega}}_{c}<{\mathsf{\Omega}}_{lim}$ OR $\Vert {J}_{D}\Vert {}_{c}<\Vert {J}_{D}\Vert {}_{lim}$IF ${\mathrm{min}\mathsf{\Omega}}_{c}=={\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{c}\left(1\right)$${\overrightarrow{i}}_{ch}=\left[\begin{array}{cc}1& 2\end{array}\right]$ $*{}^{}{\overrightarrow{i}}_{nc}=\left[\begin{array}{cc}3& 4\end{array}\right]$ ELSEIF ${\mathrm{min}\mathsf{\Omega}}_{c}=={\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{c}\left(2\right)$${\overrightarrow{i}}_{ch}=\left[\begin{array}{cc}1& 3\end{array}\right]$ $*{}^{}{\overrightarrow{i}}_{nc}=\left[\begin{array}{cc}2& 4\end{array}\right]$ ELSEIF ${\mathrm{min}\mathsf{\Omega}}_{c}=={\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{c}\left(3\right)$${\overrightarrow{i}}_{ch}=\left[\begin{array}{cc}1& 4\end{array}\right]$ $*{}^{}{\overrightarrow{i}}_{nc}=\left[\begin{array}{cc}2& 3\end{array}\right]$ ELSEIF ${\mathrm{min}\mathsf{\Omega}}_{c}=={\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{c}\left(4\right)$${\overrightarrow{i}}_{ch}=\left[\begin{array}{cc}2& 3\end{array}\right]$ $*{}^{}{\overrightarrow{i}}_{nc}=\left[\begin{array}{cc}1& 4\end{array}\right]$ ELSEIF ${\mathrm{min}\mathsf{\Omega}}_{c}=={\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{c}\left(5\right)$${\overrightarrow{i}}_{ch}=\left[\begin{array}{cc}2& 4\end{array}\right]$ $*{}^{}{\overrightarrow{i}}_{nc}=\left[\begin{array}{cc}1& 3\end{array}\right]$ ELSE ${\mathrm{min}\mathsf{\Omega}}_{c}=={\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{c}\left(6\right)$${\overrightarrow{i}}_{ch}=\left[\begin{array}{cc}3& 4\end{array}\right]$ $*{}^{}{\overrightarrow{i}}_{nc}=\left[\begin{array}{cc}1& 2\end{array}\right]$ ENDIF${\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{ch}=$ column vector of ${N}_{ch}$ zeros FOR ${c}_{1}=1:{N}_{ch}$${\overrightarrow{q}}_{ch}={\overrightarrow{q}}_{{ind}_{d}}$ ${\overrightarrow{q}}_{ch}\left({\overrightarrow{i}}_{ch}\right)={\overrightarrow{q}}_{ch}\left({\overrightarrow{i}}_{ch}\right)+{\nu}_{d}\xb7{t}_{s}\xb7{\mathrm{M}}_{inc}\left(:,{c}_{1}\right)$ $**{}^{}{\overrightarrow{q}}_{ch}\left({\overrightarrow{i}}_{nc}\right)={\overrightarrow{q}}_{ch}\left({\overrightarrow{i}}_{nc}\right)+{\nu}_{d}\xb7{t}_{s}\xb7{\mathrm{M}}_{inc}\left(:,{c}_{1}\right)$ IF ${\overrightarrow{minq}}_{ind}<{\overrightarrow{q}}_{ch}<{\overrightarrow{maxq}}_{ind}$ (element-wise comparison)${\overrightarrow{\mathrm{X}}}_{ch}=$ Solve the Forward Kinematics for ${\overrightarrow{q}}_{ch}$, using ${\overrightarrow{X}}_{c}$ as initial condition ${\overrightarrow{\mathsf{\alpha}}}_{ch}=$ Angle of spherical joints for ${\overrightarrow{\mathrm{X}}}_{ch}$ IF ${\overrightarrow{\mathsf{\alpha}}}_{ch}<{\overrightarrow{\mathsf{\alpha}}}_{lim}$ (element-wise comparison)${\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{ch}\left({c}_{1}\right)=$ Calculate the index ${\mathsf{\Omega}}_{i,j}$ for ${\overrightarrow{\mathrm{X}}}_{ch}$ with $i,j={\overrightarrow{i}}_{ch}$ ENDIFENDIFENDFOR${c}_{1}=argmax\left({\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{ch}\right)$ $\overrightarrow{\Delta i}\left({\overrightarrow{i}}_{ch}\right)=\overrightarrow{\Delta i}\left({\overrightarrow{i}}_{ch}\right)+{\mathrm{M}}_{inc}\left(:,{c}_{1}\right)$ ${\overrightarrow{q}}_{{ind}_{d}}={\overrightarrow{q}}_{{ind}_{r}}+{\nu}_{d}\xb7{t}_{s}\xb7\overrightarrow{\Delta i}$ ENDIFENDIFEND |

## 3. Results

#### 3.1. Simulation of the Vision-Based Hybrid Controller

- The mean absolute error (MAE)$$\mathrm{MAE}=\frac{1}{F}{\displaystyle \sum}_{i=1}^{F}\left(\frac{1}{n}{\displaystyle \sum}_{j=1}^{n}\left|{q}_{ind}{}_{r}\left(i,j\right)-{q}_{ind}{}_{c}\left(i,j\right)\right|\right)$$
- The mean absolute percentage error (MAPE)$$\mathrm{MAPE}=\frac{100}{F}{\displaystyle \sum}_{i=1}^{F}\left(\frac{1}{n}{\displaystyle \sum}_{j=1}^{n}\left|\frac{{q}_{ind}{}_{r}\left(i,j\right)-{q}_{ind}{}_{c}\left(i,j\right)}{{q}_{ind}{}_{r}\left(i,j\right)}\right|\right)$$
- The mean distance travelled for type II singularity release (MDSR)$$\mathrm{MDSR}=\frac{1}{F}{\displaystyle \sum}_{i=1}^{F}\left({\displaystyle \sum}_{j=1}^{n}\left|{q}_{ind}{}_{r}\left(i,k\right)-{q}_{ind}{}_{c}\left(i,j\right)\right|\right)$$

#### 3.2. Experimentation of the Vision-Based Hybrid Controller

- ${\overrightarrow{X}}_{c}$ is provided by processing the data stream from the 3DTS in real time.
- During the 15 s before the SRM is activated, an external perturbation is applied to the PR. Since in a type II Singularity the PR can vary its position and orientation without moving any actuators, the researcher can apply some forces to the PR by hand to check whether the mobile platform experiences uncontrolled motion.

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 10.**(

**a**) $\Vert {J}_{D}\Vert $ (

**b**) $\mathrm{min}\mathsf{\Omega}$ for trajectory 1 in the simulation.

**Figure 11.**(

**a**) $\Vert {J}_{D}\Vert $ (

**b**) $\mathrm{min}\mathsf{\Omega}$ for trajectory 1 in the experimentation.

${R}_{1}\left(m\right)$ | ${R}_{2}\left(m\right)$ | ${R}_{3}\left(m\right)$ | ${\beta}_{FD}\left(\xb0\right)$ | ${\beta}_{FI}\left(\xb0\right)$ | ${d}_{s}\left(m\right)$ |

0.4 | 0.4 | 0.4 | 90 | 45 | 0.15 |

${R}_{m1}\left(m\right)$ | ${R}_{m2}\left(m\right)$ | ${R}_{m3}\left(m\right)$ | ${\beta}_{MD}\left(\xb0\right)$ | ${\beta}_{MI}\left(\xb0\right)$ | |

0.3 | 0.3 | 0.3 | 50 | 90 |

Parameters | ||
---|---|---|

Variable | Description | Default |

${\nu}_{d}$ | $\mathrm{releasing}\mathrm{velocity}\mathrm{module}\mathrm{in}m/s$ | 0.01 |

${t}_{s}$ | $\mathrm{controller}\mathrm{sample}\mathrm{time}\mathrm{in}s$ | 0.01 |

$\Vert {J}_{D}\Vert {}_{lim}$ | $\mathrm{experimental}\mathrm{limit}\mathrm{for}\Vert {J}_{D}\Vert $ | 0.015 |

${\mathsf{\Omega}}_{lim}$ | $\mathrm{experimental}\mathrm{limit}\mathrm{for}{\mathsf{\Omega}}_{i,j}$ | $1.8\xb0$ |

${\overrightarrow{maxq}}_{ind}$ | maximum feasible values for the actuators’ length in $m$$,4x1$ vector | ${\left[\begin{array}{cc}\begin{array}{cc}0.93& 0.92\end{array}& \begin{array}{cc}0.93& 0.82\end{array}\end{array}\right]}^{T}$ |

${\overrightarrow{minq}}_{ind}$ | minimum feasible values for the actuators’ length in $m$$,4x1$ vector | ${\left[\begin{array}{cc}\begin{array}{cc}0.65& 0.64\end{array}& \begin{array}{cc}0.65& 0.54\end{array}\end{array}\right]}^{T}$ |

${\overrightarrow{\mathsf{\alpha}}}_{lim}$ | experimental limits for the spherical joints, $3x1$ vector | $38\xb0$ |

${\mathrm{M}}_{inc}$ | $\mathrm{possible}\mathrm{increments}/\mathrm{decrements}\mathrm{for}\overrightarrow{\Delta i}$ | See equation (10) |

$\overrightarrow{\Delta i}$ | $\mathrm{column}\mathrm{vector}4x1$, persistent variable | - |

Inputs | ||

Variable | Description | Default |

${e}_{pin}$ | enable pin | - |

$\Vert {J}_{D}\Vert {}_{c}$ | determinant of the forward Jacobian matrix, feedback signal | - |

${\overrightarrow{\mathrm{V}\mathsf{\Omega}}}_{c}$ | column vector with the six ${\mathsf{\Omega}}_{i,j}$ indices, feedback signals | - |

${\overrightarrow{X}}_{c}$ | position and orientation of the mobile platform, feedback signal | - |

${\overrightarrow{q}}_{{ind}_{r}}$ | trajectory for the actuators, reference signal | - |

Outputs | ||

Variable | Description | Default |

${\overrightarrow{q}}_{{ind}_{d}}$ | trajectory for the actuators, desired signal | - |

Trajectory | Description | Type II Singularity | |||
---|---|---|---|---|---|

${\mathit{x}}_{\mathit{m}}\left(\mathbf{m}\right)$ | ${\mathit{z}}_{\mathit{m}}\left(\mathbf{m}\right)$ | $\mathit{\theta}$ (rad) | $\mathit{\psi}$ (rad) | ||

1 | Hip flexion | 0.01 | 0.70 | 0.15 | 0.31 |

2 | Partial internal–external knee rotation | 0.01 | 0.70 | −0.02 | 0.14 |

3 | Flexion–extension of the knee combined with ankle and knee rotations | 0.05 | 0.72 | −0.01 | 0.15 |

4 | Flexion–extension of the knee combined with hip flexion | 0.12 | 0.77 | −0.06 | 0.11 |

5 | Complete internal–external knee rotation | −0.05 | 0.73 | 0.10 | 0.33 |

Trajectory | MAE (mm) | MAPE (%) | MDSR (mm) | |||
---|---|---|---|---|---|---|

SRM-V1 | SRM-V2 | SRM-V1 | SRM-V2 | SRM-V1 | SRM-V2 | |

1 | 3.87 | 10.74 | 0.53 | 1.40 | 7.01 | 18.18 |

2 | 1.09 | 2.04 | 0.14 | 0.28 | 5.05 | 2.92 |

3 | 1.77 | 6.15 | 0.24 | 0.82 | 4.78 | 6.74 |

4 | 3.00 | 10.24 | 0.38 | 1.25 | 7.48 | 10.81 |

5 | 10.74 | 10.44 | 1.43 | 1.37 | 15.47 | 35.23 |

MEAN | 4.09 | 7.92 | 0.54 | 1.02 | 7.95 | 14.77 |

Trajectory | MAE (mm) | MAPE (%) | MDSR (mm) | AVR (N) |
---|---|---|---|---|

1 | 3.26 | 0.45 | 3.64 | 0.22 |

2 | 3.02 | 0.41 | 7.61 | 0.52 |

3 | 2.05 | 0.27 | 1.60 | 0.17 |

4 | 2.14 | 0.27 | 1.90 | 0.46 |

5 | 10.66 | 1.42 | 11.82 | 1.44 |

MEAN | 4.22 | 0.56 | 5.31 | 0.56 |

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## Share and Cite

**MDPI and ACS Style**

Pulloquinga, J.L.; Escarabajal, R.J.; Ferrándiz, J.; Vallés, M.; Mata, V.; Urízar, M.
Vision-Based Hybrid Controller to Release a 4-DOF Parallel Robot from a Type II Singularity. *Sensors* **2021**, *21*, 4080.
https://doi.org/10.3390/s21124080

**AMA Style**

Pulloquinga JL, Escarabajal RJ, Ferrándiz J, Vallés M, Mata V, Urízar M.
Vision-Based Hybrid Controller to Release a 4-DOF Parallel Robot from a Type II Singularity. *Sensors*. 2021; 21(12):4080.
https://doi.org/10.3390/s21124080

**Chicago/Turabian Style**

Pulloquinga, José L., Rafael J. Escarabajal, Jesús Ferrándiz, Marina Vallés, Vicente Mata, and Mónica Urízar.
2021. "Vision-Based Hybrid Controller to Release a 4-DOF Parallel Robot from a Type II Singularity" *Sensors* 21, no. 12: 4080.
https://doi.org/10.3390/s21124080