*5.1. Experiment 1*

The experiment consists of the measurement of minimum thickness *t* of the support sheet that allows the master robot to break free of the slave during the parking phase described in Section 3.4. Indeed, during this phase, the first prerequisite calls for magnetic force *Fm*,*r* to be sufficiently small for the robot to drive away. This can indeed be seen in Figure 14, and was discussed in Section 4. The same measurement was repeated for several masses of the robot, namely, 0.2, 0.4, 0.6, and 1.2 kg, by adding commensurate additional weights to the robot close to its center of mass *Gr*.

This type of experiment was selected because the mechanics of the parking phase include the most important complex phenomena that influence the behavior of the system, which are


Being these inherently complex phenomena, described and modeled in detail in Sections 3.1, 3.2, and 3.4, the experimental validation of the parking phase was deemed the most adequate to provide insight on the adherence to reality of the model as a whole.

The results of the comparison between the numerical implementation of the model described in Section 3 and the experiment results are shown in Figure 16. It should be noted that a normalized thickness index is used in place of simple thickness value *t* of the support sheet; normalization allows, in principle, to compare systems with different friction coefficients, although in the present case, normalization is a simple scaling by value 1/*μ*0,*r*.

**Figure 16.** Comparison between numerical and experiment results (Experiment 1). Top plot: data points for different robot masses *mr*; below plot: relative error.

It is apparent from the error plot shown on the bottom of the same figure that the error is comparatively small (from 0.5% to 3%) for large thicknesses and low masses of the master, while it becomes larger (approximately 14%) when thickness grows smaller following larger traction **R***μ*,0,*<sup>r</sup>* associated with larger masses *mr*. This is likely due to both the approximate model of the magnetic field that is described in Section 3.1 and the rigid-body approach used for the robot. In fact, large magnetic forces *Fm* arising from small thicknesses *t* tend to deform the rubber of the wheel, which decreases the distance of the master frame to the support sheet. The direct consequence of this is that the magnetic interaction deviates from its expected behavior. An additional possible source of error can be the model used for the friction; it assumes a linear relationship between force and frictional force. However, during deformation of the rubber due to increased mass *mr* and magnetic force *Fm*, the footprint of the wheel increases in its area, which can have an unexpected influence on traction **R***μ*,0,*<sup>r</sup>* and on traction margin *Tμ*.

#### *5.2. Experiment 2*

Phase B was experimentally investigated by measuring the maximum travel of the robot from the bottom across the curved portion of the surface, before the wheels lose traction. The angle at which slip occurs, *ϑslip*, is measured from the horizontal, coherently with the model presented in Section 3: the robot starts at *ϑ* = −*π*/2 and travels towards the top of the surface, which is located at *ϑ* = +*π*/2. Radius *ρ* of the curved sheet is 0.3 m. The test was performed for a set of four different masses *mr* of the robot and for a set of four different support-sheet thicknesses *t*. In particular, *mr* = [0.2, 0.3, 0.4, 0.5] kg, while *t* = [0.002, 0.003, 0.004, 0.005] m. This provides a total of 16 data points. The raw experimental measurements of *ϑslip*, which were taken for each data point, varied from 3 to 5.

Results shown in Figure 17 show a close similarity in the topology of the surface determined by the variation of parameters *mr* and *t*. Results show that the larger the mass, the smaller the angle *ϑslip* of slip becomes. This means that the robot slips early in its travel along the curved surface. Similarly, the larger thickness *t* is, the smaller angle *ϑslip* is. This is expected, because thicker sheets increase the distance of the magnets, which then provide less adhesion force, which clearly translates in less of a traction margin. In order to better measure the error between numerical and experimental results, we define the following:

$$\varepsilon\_{\theta,slip} = \frac{(\theta\_{slip,num} - \theta\_{slip,exp})}{\pi/2}. \tag{21}$$

This formulation measures the absolute error and is normalized against *π*/2, which is the travel from the horizontal to the vertical orientation of the robot. The comparison between numerical and experimental data, which is visible in both plots in Figure 17, shows that a comparatively small error is present in the measurements, with an average value of *εϑ*,*slip* = 5%.

It is worth noting that the system appears nonfunctional in this configuration, although this is caused by the tape on the wheels, which considerably reduces traction. According to the now-validated model and to experimentation, the system is functional without the tape.

**Figure 17.** Comparison between numerical results of the model and results from Experiment 2. **Left** plot: data points for different robot masses *mr* and for different thicknesses *t* for both the numerical and experimental campaigns; note that the z-axis is flipped. **Right** plot: relative error between numerical and experimental values, normalized with respect to *π*/2.

#### **6. Conclusions**

This paper discussed a type of climbing robot that takes advantage of a master–slave configuration that is connected through a magnetic interface. The master robot provides motion, while the slave component provides adhesion capabilities to the system even when the support sheet is nonmagnetic. Thanks to its design, once decoupled from the slave, the master robot is able to drive freely on horizontal ground.

In order to present a coherent framework, three conditions are defined that prove the feasibility of the specific design. The work envelope of the system is split in a set of phases, namely, driving on straight floor, vertical floor, curved surface, ceiling, and coupling and decoupling phases; each is characterized by its own challenges.

A Taylor expansion model was implemented to model the magnetic-field morphology, and a pseudostatic approach was used to model the master–slave robotic system to account for the inherently nontrivial interaction between the actors.

Comprehensive analysis was performed on the main variables of the master–slave system through numerical computation of the model in the operational envelope. Furthermore, experimental validation was performed both in the case of the parking phase, which is the most complex to model, and in the case of driving along a curved sheet. Both experiments showed that the numerical model had good adherence with reality, with errors around 3%–14% for the parking phase, and 5% for curved-surface driving.

Results clearly show the feasibility of the approach and provide some qualitative and quantitative insight. The experimental validation shows good accuracy of the model compared to reality. The design process of a working prototype can thus be successfully guided via the tools that are provided in the model and its analysis.

**Author Contributions:** Conceptualization, S.S., P.G.; methodology, S.S.; software, S.S.; validation, S.S.; formal analysis, P.G.; investigation, S.S., L.S.; resources, L.S.; data curating, S.S., L.S.; writing—original-draft preparation, S.S.; writing—review and editing, S.S., L.S., M.C., A.G.; visualization, S.S.; supervision, P.G., A.G.

**Funding:** This research received no external funding.

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

#### **References**


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