# Modelling an Industrial Robot and Its Impact on Productivity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Adaptive Fuzzy Sliding Mode Control (AFSMC)

## 3. Time from the Trajectory Planner

^{j}= C

^{j}(α

_{i}

^{j}, p

_{k}

^{j}) is defined unambiguously using the Cartesian coordinates of important points in the robot α

_{i}

^{j}= (α

_{xi}

^{j}, α

_{yi}

^{j}, α

_{zi}

^{j}).

^{i}to C

^{f}is calculated by solving an optimization problem.

- (1)
- Robot dynamics$$M\left(q\left(t\right)\right)\ddot{q}\left(t\right)+C\left(q\left(t\right),\dot{q}\left(t\right)\right)\dot{q}\left(t\right)+g\left(q\left(t\right)\right)=\tau (t)$$
- (2)
- Boundary conditions (position, velocity and acceleration) for intermediate configurations$$q\left({t}_{int-1}\right)={q}_{int-1};q\left({t}_{int}\right)={q}_{int}\phantom{\rule{0ex}{0ex}}\dot{q}\left({t}_{int-1}\right)={\dot{q}}_{int-1};\dot{q}\left({t}_{int}\right)={\dot{q}}_{int-1}\phantom{\rule{0ex}{0ex}}\ddot{q}\left({t}_{int-1}\right)={\ddot{q}}_{int-1};\ddot{q}\left({t}_{int}\right)={\ddot{q}}_{int-1}$$
- (3)
- Boundary conditions for initial and final configuration$$\begin{array}{l}q\left(0\right)={q}_{o};q\left({t}_{f}\right)={q}_{f}\\ \dot{q}\left(0\right)=0;\dot{q}\left({t}_{f}\right)=0\end{array}$$
- (4)
- Collision avoidance with obstacles$${d}_{ij}\ge {r}_{j}+{w}_{i}$$
_{i}is the radius of cylinder that wraps the arm i. - (5)
- Driving torque$${\tau}_{i}^{\mathrm{min}}\le {\tau}_{i}\left(t\right)\le {\tau}_{i}^{\mathrm{max}}\forall t\in \left[0,{t}_{min}\right],i=1,\dots ,dof$$
- (6)
- Power in the driving motors,$${P}_{i}^{min}\le {\tau}_{i}\left(t\right){\dot{q}}_{i}\left(t\right)\le {P}_{i}^{max}\forall t\in \left[0,{t}_{min}\right],i=1,\dots ,dof$$
- (7)
- Jerk on the driving motors,$${\stackrel{\u20db}{q}}_{i}^{min}\le {\stackrel{\u20db}{q}}_{i}\left(t\right)\le {\stackrel{\u20db}{q}}_{i}^{max}\forall t\in \left[0,{t}_{min}\right],i=1,\dots ,dof$$
- (8)
- Energy consumed$${{\displaystyle \sum}}_{j=1}^{m-1}\left({{\displaystyle \sum}}_{i=1}^{dof}{\epsilon}_{ij}\right)\le E,$$
_{ij}is the energy used by the driving motor i to move from configuration c^{j}to c^{j+}^{1}.

^{3}. The maximum energy considered is different for each solved example.

_{j}= t

_{j}− t

_{j−}

_{1}, at each interval, so that the objective function is

## 4. Assessment of the Productivity of an Assembly Line

_{p}: is the market unitary price of the item p (€).

_{p}: is the unitary cost of the item p (€), which covers costs of raw materials, amortization, energy, maintenance, labour force, taxes to direct and indirect costs.

_{p}(t): is a function which depends on the number of items manufactured per hour:

_{k}: is the set of tasks required to manufacture an item (p), i.e., the workload, where k represents the number of tasks.

_{min}.

_{min p}to carry out a certain robot task with the aim to manufacture a product p. Other required tasks are defined as a constant value (${t}_{j}$). The lower the time the greater the number of items manufactured per hour, thus increasing the firm total profit. Therefore, the cumulative time of all needed tasks is defined as:

## 5. Experimental Case Studies for Methodology Validation

_{minp}. This function has been calibrated to reflect the market environment, for instance, the current demand of each item.

_{p}∗ N

_{p}) provides the cost to produce p items (N

_{p}). P

_{p}provides the profit as shown in Equation (22), i.e., the item price multiplied by the number of items manufactured. This exercise has been applied for product 2 and case 17. The statistics of the Gaussian function are defined as follows: the mean is that previous defined for product 2 (i.e., m

_{c}= 0.75 EUR for the cost and m

_{p}= 1.02 EUR for the price) and the standard deviation is m/4. The Gaussian functions are sampled and then these values are used in the algorithm. The profits due to fluctuations are shown in Figure 3 and Figure 4. The methodology allows translating market fluctuations directly into profits, which allows making informed decisions about production, for instance, about what items should be manufactured and, moreover, to do it with an efficient scheduling of the robot tasks. In this way, the production is significantly improved.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Pareto frontiers obtained with the optimization algorithm for case 27 for the three different products manufactured.

Driving Actuator | Maximum Torque (N·m) | Minimum Torque (N·m) | Maximum Power (W) | Minimum Power (W) |
---|---|---|---|---|

1 | 140 | −140 | 275 | −275 |

2 | 180 | −180 | 350 | −350 |

3 | 140 | −140 | 275 | −275 |

4 | 80 | −80 | 150 | −150 |

5 | 80 | −80 | 150 | −150 |

6 | 40 | −40 | 75 | −75 |

**Table 2.**Task execution time (s) obtained with the optimization algorithm for the set of cases. These times represent the desired motion and consider the physical restraints of the robot.

Case | Max. Jerk Constraint (rad/s^{3}) | Consumed Energy Constraint (J) | Execution Time (s) from the Optimization Algorithm | Consumed Energy (J) | Jerk in Third Actuator (rad/s ^{3}) | |
---|---|---|---|---|---|---|

Max. | Min. | |||||

1 | Unconstrained | Unconstrained | 3.79 | 140.1 | 1615 | −841 |

2 | Unconstrained | 75 | 22.55 | 75 | 107 | −28 |

3 | 5 | Unconstrained | 19.27 | 80.1 | 5 | −5 |

4 | 5 | 75 | 25.76 | 75 | 5 | −4 |

5 | Unconstrained | Unconstrained | 5.14 | 203.49 | 1098 | −984 |

6 | Unconstrained | 200 | 5.15 | 200 | 1096 | −980 |

7 | Unconstrained | 175 | 5.3 | 175 | 954 | −872 |

8 | Unconstrained | 150 | 5.62 | 150 | 704 | −627 |

9 | Unconstrained | 125 | 6.42 | 125 | 425 | −418 |

10 | Unconstrained | 100 | 12.25 | 100 | 167 | −145 |

11 | Unconstrained | 95 | 21.08 | 95 | 176 | −37 |

12 | 5 | Unconstrained | 23.05 | 103.3 | 5 | −5 |

13 | 5 | 95 | 26.35 | 95 | 5 | −5 |

14 | Unconstrained | Unconstrained | 2.27 | 87.5 | 897 | −944 |

15 | Unconstrained | 50 | 7.34 | 50 | 125 | −147 |

16 | 5 | Unconstrained | 14.82 | 51.5 | 5 | −5 |

17 | 5 | 50 | 17.94 | 50 | 2.8 | −3.6 |

18 | 5 | Unconstrained | 18.28 | 42.14 | 5 | −3.9 |

19 | 10 | Unconstrained | 14.51 | 42.39 | 10 | −7.7 |

20 | 25 | Unconstrained | 10.69 | 43.1 | 25 | −19.3 |

21 | 5 | Unconstrained | 18.28 | 42.14 | 5 | −3.9 |

22 | 10 | Unconstrained | 14.51 | 42.39 | 10 | −7.7 |

23 | 25 | Unconstrained | 10.69 | 43.1 | 25 | −19.3 |

24 | 50 | Unconstrained | 8.49 | 43.92 | 50 | −38.7 |

25 | 100 | Unconstrained | 6.74 | 45.36 | 100 | −77.4 |

26 | 1000 | Unconstrained | 3.21 | 63.37 | 856.8 | −776.2 |

27 | Unconstrained | Unconstrained | 2.41 | 88.46 | 1361.50 | −1802.90 |

28 | 5 | 40 | 18.65 | 40 | 5 | −4 |

29 | Unconstrained | 40 | 9.94 | 40 | 337 | −13.2 |

30 | Unconstrained | Unconstrained | 3.08 | 106.2 | 972 | −1009 |

31 | Unconstrained | 40 | 9.18 | 40 | 42.2 | −37.8 |

32 | 5 | Unconstrained | 15.91 | 40.3 | 5 | −5 |

33 | 5 | 40 | 15.93 | 40 | 5 | −5 |

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**MDPI and ACS Style**

Llopis-Albert, C.; Rubio, F.; Valero, F.
Modelling an Industrial Robot and Its Impact on Productivity. *Mathematics* **2021**, *9*, 769.
https://doi.org/10.3390/math9070769

**AMA Style**

Llopis-Albert C, Rubio F, Valero F.
Modelling an Industrial Robot and Its Impact on Productivity. *Mathematics*. 2021; 9(7):769.
https://doi.org/10.3390/math9070769

**Chicago/Turabian Style**

Llopis-Albert, Carlos, Francisco Rubio, and Francisco Valero.
2021. "Modelling an Industrial Robot and Its Impact on Productivity" *Mathematics* 9, no. 7: 769.
https://doi.org/10.3390/math9070769