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Motion Planning and Control of Redundant Manipulators for Dynamical Obstacle Avoidance^{ †}

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^{†}

## Abstract

**:**

## 1. Introduction

- An off-line path planning algorithm, which plans the trajectory of the robot’s end-effector taking into account the possible presence of disturbing obstacles, modifying the path based on the positions of the obstacles before the motion starts;
- An on-line motion control algorithm, which controls in real-time the robot compensating for obstacles that are moving, or new obstacles entering the workspace;
- A redundancy control strategy that exploits the dexterity of the manipulator to avoid collisions between obstacles and the kinematic chain of the manipulator;
- A robust technique for the avoidance of singular configurations during motion.

## 2. Off-Line Path Planning

**∇**indicates the gradient operator. Norms of velocities must be set based on the type of application; as an example, values used in the simulations presented in the following of this paper are ${v}_{rep}=10\phantom{\rule{0.166667em}{0ex}}\mathrm{m}/\mathrm{s}$ and ${v}_{att}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{m}/\mathrm{s}$.

## 3. On-Line Motion Control

## 4. Results

#### 4.1. Example 1

#### 4.2. Example 2

#### 4.3. Example 3

#### 4.4. Example 4

#### 4.5. Example 5

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 3.**Comparison between position and velocity profiles of potential fields trajectory and smoothed trajectory related to the example of Figure 2.

**Figure 4.**(

**a**) Alignment condition between end-effector, obstacle and goal in a simplified 2D representation; (

**b**) example of trajectories generated by different directions of ${\mathbf{v}}_{dev}$ in the 3D space.

**Figure 6.**Linear velocity of the end-effector $\mathbf{E}$ and of the control point ${\mathbf{P}}_{r}$ closest to the obstacle ${\mathbf{P}}_{o}$.

**Figure 9.**Example 1: avoidance of a dynamic obstacle interfering with the end-effector in a fixed position.

**Figure 10.**Example 1: minimum distance robot-obstacle for the motion shown in Figure 9.

**Figure 11.**Example 1: joint rotations and speeds for the motion shown in Figure 9.

**Figure 12.**Example 2: avoidance of a dynamic obstacle interfering with an internal point of the manipulator.

**Figure 13.**Example 2: minimum distance robot-obstacle for the motion shown in Figure 12.

**Figure 14.**Example 2: joint rotations and speeds for the motion shown in Figure 12.

**Figure 16.**Example 3: minimum distance robot-obstacle for the motion shown in Figure 15.

**Figure 17.**Example 3: joint rotations and speeds for the motion shown in Figure 15.

**Figure 18.**Example 4: comparison between different methods for path generation; (

**a**) Potential Fields, (

**b**) Potential Fields with 3rd order Bézier Interpolation, (

**c**) Potential Fields with 4th order Bézier Interpolation.

**Figure 19.**Example 5: response of the control law to a moving obstacle with different values of ${k}_{v}$.

$\mathit{r}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]$ | ${\mathit{r}}_{\mathit{m}}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]$ | ${\mathit{r}}_{\mathbf{min}}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]$ | ${\mathit{v}}_{\mathbf{rep}}\phantom{\rule{0.166667em}{0ex}}[\mathbf{m}/\mathbf{s}]$ | ${\mathit{v}}_{\mathbf{att}}\phantom{\rule{0.166667em}{0ex}}[\mathbf{m}/\mathbf{s}]$ | $\mathit{T}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]$ |
---|---|---|---|---|---|

0.18 | 0.15 | 0.12 | 10 | 1 | 2 |

$\mathrm{d}t\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{s}\right]$ | ${k}_{e}$ | ${k}_{v}$ | ${\lambda}_{max}$ | $\u03f5$ | ${\dot{\theta}}_{max}\phantom{\rule{0.166667em}{0ex}}[\mathrm{rad}/\mathrm{s}]$ |

${10}^{-3}$ | 100 | 100 | ${10}^{-3}$ | ${10}^{-3}$ | $\pi $ |

$\mathit{T}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{s}\right]$ | $\mathit{L}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{mm}\right]$ | $\left|\right|\dot{\mathbf{q}}{\left|\right|}_{\mathbf{max}}\phantom{\rule{0.277778em}{0ex}}[\mathbf{rad}/\mathbf{s}]$ | $\left|\right|\ddot{\mathbf{q}}{\left|\right|}_{\mathbf{max}}\phantom{\rule{0.277778em}{0ex}}[\mathbf{rad}/{\mathbf{s}}^{2}]$ | Avg. Comp. Time [ms] | |||||
---|---|---|---|---|---|---|---|---|---|

PF | 2 | 850 | 2.23 | 134.03 | 54 | ||||

PF+3BI | 2 | 824 | (−3.1%) | 2.16 | (−3.1%) | 12.61 | (−90.6%) | 60 | (+11.1%) |

PF+4BI | 2 | 829 | (−2.5%) | 2.15 | (−3.6%) | 13.26 | (−90.1%) | 62 | (+14.8%) |

$\mathit{T}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{s}\right]$ | $\mathit{L}\phantom{\rule{0.277778em}{0ex}}\left[\mathbf{mm}\right]$ | $\left|\right|\dot{\mathbf{q}}{\left|\right|}_{\mathbf{max}}\phantom{\rule{0.277778em}{0ex}}[\mathbf{rad}/\mathbf{s}]$ | $\left|\right|\ddot{\mathbf{q}}{\left|\right|}_{\mathbf{max}}\phantom{\rule{0.277778em}{0ex}}[\mathbf{rad}/{\mathbf{s}}^{2}]$ | Avg. Comp. Time | Collision | |||||
---|---|---|---|---|---|---|---|---|---|---|

(1 Cycle) [ms] | Avoided | |||||||||

${k}_{v}=0$ | 2 | 747 | 14.87 | 402.55 | 0.86 | No | ||||

${k}_{v}=100$ | 2 | 725 | (−2.9%) | 11.89 | (−20.0%) | 273.16 | (−32.1%) | 0.85 | (−1.2%) | Yes |

${k}_{v}=500$ | 2 | 710 | (−-5.0%) | 7.54 | (−49.3%) | 79.23 | (−80.3%) | 0.87 | (+1.2%) | Yes |

${k}_{v}=1000$ | 2 | 724 | (−3.1%) | 6.48 | (−56.4%) | 187.05 | (−53.5%) | 0.90 | (+4.7%) | Yes |

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**MDPI and ACS Style**

Palmieri, G.; Scoccia, C.
Motion Planning and Control of Redundant Manipulators for Dynamical Obstacle Avoidance. *Machines* **2021**, *9*, 121.
https://doi.org/10.3390/machines9060121

**AMA Style**

Palmieri G, Scoccia C.
Motion Planning and Control of Redundant Manipulators for Dynamical Obstacle Avoidance. *Machines*. 2021; 9(6):121.
https://doi.org/10.3390/machines9060121

**Chicago/Turabian Style**

Palmieri, Giacomo, and Cecilia Scoccia.
2021. "Motion Planning and Control of Redundant Manipulators for Dynamical Obstacle Avoidance" *Machines* 9, no. 6: 121.
https://doi.org/10.3390/machines9060121