**6. Experimental Results**

Experiments were carried out on the cube robot to compare between the two configurations, **C**<sup>1</sup> (see Figure 21) and **C**<sup>2</sup> (see Figure 22), in the swimming pool at Montpellier University. The cube in the water and a video link for the cube's operations can be seen in Figure 23.

**Figure 21. C**<sup>1</sup> of the cube robot.

**Figure 22. C**<sup>2</sup> of the cube robot.

**Figure 23.** Cube robot in the water https://www.youtube.com/watch?v=RKiWUOxDKdw (accessed on 18 October 2019)

#### *6.1. Attainability Validation*

The incremental torques about the **u**-axis, **v**-axis, and **w**-axis were applied to cube robot respectively, and angular velocities and PWM input values were stored to evaluate these two configurations. For safety, the experiments were stopped when one thruster reached the saturation values. The experimental results are shown in Figures 24–26. For rotating about the **u**-axis (Figure 24), the attainability of configurations **C**<sup>1</sup> and **C**<sup>2</sup> was almost the same: all thrusters operated in a feasible region. Otherwise, for rotating about the **v**-axis or **w**-axis, the attainability of configuration **C**<sup>2</sup> was better than that of **C**1. In particular, with the **v**-axis experiment (Figure 25), the cube robot with **C**<sup>1</sup> stopped the mission earlier than with **C**<sup>2</sup> (at Time Step 771) because one thruster reached saturation. The same thing happened with the **w**-axis experiment (at Time Step 451) (see Figure 26).

**Figure 24.** The cube rotates about the **u**-axis for **C**<sup>1</sup> and **C**<sup>2</sup> (X-axis = **u**-axis).

**Figure 25.** The cube rotates about the **v**-axis for **C**<sup>1</sup> and **C**<sup>2</sup> (Y-axis = **v**-axis).

**Figure 26.** The cube rotates about the Z–axis for **C**<sup>1</sup> and **C**<sup>2</sup> (Z-axis = **w**-axis).

#### *6.2. Energetic Validation*

In this section, we verify the energy consumption during these experiments for the two configurations. An energy-like criterion is proposed:

$$\mathbf{E} = \sum\_{i=1}^{m} \int\_{t=0}^{T} |P\mathsf{W}\mathcal{M}^{i}(t) - 1500| dt \tag{43}$$

where *m* is the number of thrusters, *T* is the time of the experiment, and *PWM<sup>i</sup>* (*t*) is the PWM inputs of the *i*th thruster.

Table 6 shows the energy consumption of the robot during the three rotation experiments. For **u**-axis rotation, the attainability of the two configurations was the same, but the energy consumption of **C**<sup>2</sup> was lower than that of **C**1. For **v**-axis and **w**-axis rotations, the duration of the experiments of **C**<sup>2</sup> was longer than that of **C**1, and the energy consumption, therefore, was higher.


**Table 6.** Energy consumption of the two configurations.

Table 7 shows the comparison of the energy consumption of the two configurations with the same time duration. For the **v**-axis rotation, the energy value of **C**<sup>2</sup> was lower than that of **C**1. However, for the **w**-axis, the energy value of **C**<sup>2</sup> was higher. This happened because the robot dived deeper in **C**<sup>2</sup> in the experiment of the **w**-axis rotation, and the robot had to deliver more power to maintain a greater constant depth.

**Table 7.** Energy consumption of the two configurations with the same time duration.


#### *6.3. Robustness and Reactive Validation*

This section validates the robustness and reactivity of the optimal configuration (**C**2) in comparison to the normal one (**C**1). For robustness, the robot performed a mission, and one or two thrusters were turned off. For the normal configuration **C**1, the mission would fail, and for the optimal configuration **C**2, the mission would be guaranteed. Specifically, for the robustness index, we carried out the following experiments:


For the reactive index, we measured how fast the robot changed missions. The following experiments were carried out:


The experimental results for the robustness validation of **C**<sup>1</sup> and **C**<sup>2</sup> are shown in Figures 27–29. In the case of one or two motors stopped, the depth control performances of **C**<sup>1</sup> and **C**<sup>2</sup> were almost the same (see Figure 27). The differences are clear in the case of three thrusters stopped (Figure 29): the performance of **C**<sup>1</sup> was not guaranteed (Figure 28) and violations of the PWM values occurred (see Figure 29a).

**Figure 27.** Depth control for **C**<sup>1</sup> and **C**<sup>2</sup> with one and two motors stopped. (**a**) Depth control of two configurations with one motor stopped. (**b**) Depth control of two configurations with two motors stopped.

**Figure 28.** Depth control for **C**<sup>1</sup> and **C**<sup>2</sup> with three motors stopped.

**Figure 29.** PWM evaluation for **C**<sup>1</sup> and **C**<sup>2</sup> with 3 motors stopped. (**a**) PWM of **C**1. (**b**) PWM of **C**2.

The results for the reactive validation are shown in Figures 30–32. We measured the reactive time of the angular velocities when the directions of the cube's actions changed. It is clear that the reactive time of **C**<sup>2</sup> was faster than that of **C**1. Specifically, the reactive time is the region formed by the vertical dashed lines in Figures 30–32. It is obvious that the reactive time of **C**<sup>2</sup> was smaller than that of **C**<sup>2</sup> (see Figures 31 and 32).

**Figure 30.** Angular velocity evaluation for **C**<sup>1</sup> and **C**2: diving, rotating about the **u**-axis, and rotating about the diagonal-axis (**Wx** = **p**; **Wy** = **q**; **Wz** = **r**). (**a**) Angular velocities of **C**1. (**b**) Angular velocities of **C**2.

**Figure 31.** Angular velocity evaluation for **C**<sup>1</sup> and **C**2: diving, rotating about the **u**-axis, and rotating about the **v**-axis (**Wx** = **p**; **Wy** = **q**). (**a**) Angular velocities of **C**1. (**b**) Angular velocities of **C**2.

**Figure 32.** Angular velocity evaluation for **C**<sup>1</sup> and **C**2: diving, rotating about the **u**-axis, and rotating about the **v**-axis (**Wx** = **p**; **Wy** = **q**).

#### **7. Conclusions and Future Work**

In this paper, an approach for designing an optimal configuration matrix (which depends on the positions and directions of the thrusters) of overactuated underwater robots was presented. The performance indices (related to manipulability, energy, workspace, reactivity, and robustness) were proposed and analyzed. Specifically, the manipulability index shows the isotropic properties of a robot; the energetic index minimizes the energy consumption under some assumptions; the workspace index is related to the attainable spaces (i.e., the force and torque spaces) of the robot; the reactive index presents how fast the robot changes the direction of the resulting actuation force; finally, the robustness index is related to the capacity of the robot to maintain its performance in the case of failures (i.e., some thrusters are completely stopped). It was formulated as a multi-objective optimization problem. Because the different indices exhibit different magnitudes and physical meanings, the goal-attainment method was chosen to find one Pareto-optimal solution. Simulation and experimental results showed that the performances of the optimal configuration were better than a "normal" configuration, which is often used (thrusters are installed vertically or horizontally). Because of the nonconvexity of the problem, finding all Pareto-optimal solutions, the Pareto front, remains a challenging problem and will be future work. Moreover, a design problem relaxing the assumptions (i.e., perfectly known characteristics of the actuators, pseudo-inverse dispatcher) is also an interesting direction for future research.

**Author Contributions:** Conceptualization, T.D., L.L., and R.Z.; methodology, T.D., L.L., and R.Z.; software, T.D., B.R., and P.L.; validation, T.D., L.L., R.Z., B.R., and P.L.; writing—original draft preparation, T.D.; writing—review and editing, L.L.; supervision, L.L. and R.Z.; project administration, L.L.; funding acquisition, L.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This project was supported by the LabEx NUMEV (ANR-10-LABX- 0020) within the I-SITE MUSE (ANR-16-IDEX-0006) and the Region Occitanie (french FEDER funds).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank Numev Labex, MUSE, Montpellier University; Region Occitanie; and FEDER for supporting this research.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


