# Kinematic-Model-Free Redundancy Resolution Using Multi-Point Tracking and Control for Robot Manipulation

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## Abstract

**:**

## 1. Introduction

#### 1.1. Contributions

#### 1.2. Paper Structure

## 2. Related Work

## 3. Mathematical Background

#### 3.1. Quaternions

**i**,

**j**,

**k**are unit vectors pointing in the x, y, z directions.

#### 3.2. Dual Quaternions

#### 3.3. Rigid Transformation

#### 3.4. Interpolation

#### 3.5. Relative Dual Quaternions

#### 3.6. Dual Quaternion Distance

#### 3.7. Dual Quaternion Regression

## 4. Proposed Approach

## 5. Simulation Results

#### 5.1. Single-Point Control

#### 5.2. Multi-Point Control

#### 5.3. Robustness

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The controller performed the task of reaching, bringing a single control point at the end effector to a desired target pose. As more joints were enabled, the performance was degraded. In particular, in the 2-degrees-of-freedom case, the robot reaches the target in 21 steps, while, in the 9-degrees-of-freedom case, the robot reaches the target in 150 steps.

**Figure 4.**The three arrangements of the control points (CP’s) along the robot’s kinematic chain. In the first arrangement, only one control point was used which was placed at the end effector. In the second and third arrangements, two and three equally spaced control points were used, respectively.

**Figure 5.**The three reaching tasks to be performed. The translucent pose is the initial configuration from which the robot is to actuate towards the opaque pose.

**Figure 6.**The controller performed three reaching tasks. In all 3 tasks, the controller converged faster when using more control points (CPs).

**Figure 7.**These joint positions correspond to reaching task 1. The larger jitters are due to exploratory behavior, which is triggered to collect more information about the robot’s local kinodynamics.

**Figure 8.**In this simulation run, the controller attempted to actuate the robot to a back arching configuration. It achieved to reach this configuration within 150 steps.

Link | ${\mathit{d}}_{\mathit{i}}$ | ${\mathit{\theta}}_{\mathit{i}}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}$ |
---|---|---|---|---|

1 | 0 | 0 | 0.05 | 1.5708 |

2 | 0 | 0 | 0.05 | 1.5708 |

3 | 0 | 0 | 0.05 | −1.5708 |

4 | 0 | 0 | 0.05 | 1.5708 |

5 | 0 | 0 | 0.05 | −1.5708 |

6 | 0 | 0 | 0.05 | 1.5708 |

7 | 0 | 0 | 0.05 | −1.5708 |

8 | 0 | 0 | 0.05 | 1.5708 |

9 | 0 | 0 | 0.05 | 0 |

Degrees of Freedom | Steps to Reach |
---|---|

2 | 21 steps |

3 | 40 steps |

4 | 42 steps |

5 | 35 steps |

6 | 41 steps |

7 | 76 steps |

8 | 127 steps |

9 | 150 steps |

Arrangement | Control Point 1 | Control Point 2 | Control Point 3 |
---|---|---|---|

1 | end effector | - | - |

2 | end of link 5 | end effector | - |

3 | end of link 3 | end of link 6 | end effector |

**Table 4.**Steps required to complete three reaching tasks using three arrangements of control points.

Task | Arrangement | ||
---|---|---|---|

1 | 2 | 3 | |

1 | 70 steps | 64 steps | 46 steps |

2 | 93 steps | 59 steps | 35 steps |

3 | 185 steps | 147 steps | 92 steps |

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**MDPI and ACS Style**

AlAttar, A.; Cursi, F.; Kormushev, P.
Kinematic-Model-Free Redundancy Resolution Using Multi-Point Tracking and Control for Robot Manipulation. *Appl. Sci.* **2021**, *11*, 4746.
https://doi.org/10.3390/app11114746

**AMA Style**

AlAttar A, Cursi F, Kormushev P.
Kinematic-Model-Free Redundancy Resolution Using Multi-Point Tracking and Control for Robot Manipulation. *Applied Sciences*. 2021; 11(11):4746.
https://doi.org/10.3390/app11114746

**Chicago/Turabian Style**

AlAttar, Ahmad, Francesco Cursi, and Petar Kormushev.
2021. "Kinematic-Model-Free Redundancy Resolution Using Multi-Point Tracking and Control for Robot Manipulation" *Applied Sciences* 11, no. 11: 4746.
https://doi.org/10.3390/app11114746