Design and Analysis of an Adaptable Wheeled-Legged Robot for Vertical Locomotion

Abstract

Most of the developed and studied service robots for vertical locomotion, as visual inspection, are made up by a rigid body with legs, wheels, or both. Thus, the robot can only displace over regular and/or flat surfaces since it is not able to adapt to the irregularities and projections of the wall. Therefore, this paper presents the design and analysis of an adaptable robot for vertical locomotion service tasks, which has a body made up of four wheeled legs that can easily adapt to the different irregularities and projections of building facades. The robot uses an Electric Ducted Fan (EDF) as the vortex adhesion system. Each leg has a rubber cover, which allows a higher mechanical adaptability of the robot over different irregularities of the wall. Theoretical backgrounds and open issues are addressed by considering some challenging problems such as mechanical adaptability modeling as well as kinematic and static analysis. Laser sensors are mounted over the robot to measure the adaptability of the robot, between the legs and body, at each time of the experimental tests for vertical locomotion.

1. Introduction

In recent years, the demands for service robotics are increasing, and service robots need to be able to perform various tasks in different surroundings [1,2,3,4]. Such service tasks require the interaction of the robot with the environment, facing obstacles, people, or rough terrains [5]. In the particular case of vertical locomotion robots, the importance of professional application has been increased, such as inspection, maintenance, or cleaning, where the robot increases the operational efficiency and protects human health and safety [5,6,7,8]. In most of these service operations, the climbing robots need to navigate manifold and vertical human-made structures [5,6,7,8], like nuclear power plants, skyscrapers, or cultural heritage buildings. Almost all the current developed climbing robots have been designed to displace over flat surfaces, like glass or metal structures.

The locomotion issues and displacement limitations of this kind of robots are mainly due to their body configuration, shape, and adhesion principle. Thus, the robot should have a shape and configuration that allows displacing over all projections and reliefs without falling, and it must have a suitable adhesion technology to not harm the vertical surface. Specifically, for stone-made buildings, the climbing robots should adapt to all the irregularities, projections, and reliefs without causing any type of damage to the structure.

Within climbing robots there are different adhesion systems that can be used on rocky surfaces, such as pneumatic adhesion systems, mechanical adhesion, electro-adhesion, and chemical adhesion [5,6,7,8,9]. In pneumatic adhesion systems the main adhesion is by negative pressure, which in turn can be performed by passive suction cups, active suction cups, and a vortex system [10]. Passive suction cups are suitable only on very flat surfaces, such as glass; it is easy to combine them with different types of locomotion as in the Mapache robot in [9] and either the vehicle presented in [10] or the Dexter robot in [11], which is equipped with two articulated feet. This principle has the great disadvantage that it is not robust against disturbances such as dust, in addition to the fact that the robot must be kept in motion since the suction cups lose the negative pressure due to small leaks. As for the active suction cups, an electric generation system by vacuum pump is used, or mechanically by venturi vacuum nozzles. These active systems have the advantage that they can work in more rugged terrain compared to passive suction cups and that they are capable of generating high attractive forces. This adhesion system is also simple to use with almost any type of locomotion, such as the Wall-Bot hexapod robot presented in [12] or the robot with wheels in [13].

The vortex system is a patented method of adhesion known as Vram (movement of regenerative vortex air), which generates adhesive forces through a vortex inside the robot. This vortex is created by a high-speed rotor and allows adhesion without additional sealing elements. Until now, only a few systems have been developed by using this principle, such as the Clarifying Climber robot by Clarifying Technologies [14], the City Climber [15], or the one presented in [16], which are driven by wheels. This principle is still being investigated to improve the adhesion principle and to analyze its aerodynamics [17].

In terms of mechanical adhesion systems, the system of claws or spines is available through a grip or clamping mechanism. These systems can be used on surfaces that are sufficiently rough so that the claws or spines find sufficient points of contact, or that provide protruding elements in a structure that can be grasped. Its main advantage lies in the energy consumption of the adhesion mechanism, its applicability to rough surfaces or structures, and its safety. The adhesion based on grip is especially very safe since even a loss of power does not necessarily lead to a fall in the system (if the mechanical configuration admits it). On the other hand, these systems are not very fast and are of limited maneuverability. In addition, its payload is low compared to robots that use magnetic attraction or a pneumatic system [8]. This adhesion system can be found in robots such as CLIBO in [18], the robot RISE in [19], or the robot LEMUR in [20].

On the other hand, electro-adhesion, a growing discipline in the field of climbing robot research, consists of electro-adhesive pads composed of conductive electrodes that generate electrostatic force or Van der Waals forces between the surface and the robot [8]. Until now there have been only some robots that use this active adhesion system such as the vehicles presented in [21,22]. This technology seems to be a very promising approach since it is safe, energetically efficient, and robust with respect to different surfaces [8].

Chemical adhesion systems include different adhesion methods, such as the adhesive-tapes-in-wheels combination presented in [23], or multi-leg systems with sticky elastomers on the feet [24]. The main advantage of this adhesion principle lies in its low energy consumption when the system is not moving, while during movement a certain force is needed to detach or separate the material from the surface. Another chemical principle of this adhesion system is the thermal glue [25] that changes its characteristics depending on the temperature.

Nowadays, a commercial solution for vertical locomotion over non-ferromagnetic surfaces, like wood or stone, are the HB1 and HB2 robots by HausBots. HB2 is the most recent version of the HB1 robot. HB1 and HB2 implement AEROGrip Technology, a developed technology by HausBots, which consists of a suction and airflow hybrid which allows for high downforce generation on many surfaces, rough and smooth, ferrous and non-ferrous [26]. This technology allows the robots to keep them added into the surface. As a locomotion system, they use the tack system that can allow overcoming small obstacles, such as wires and surface bolts [26], but not adapting to bigger irregularities.

An easy solution to solve these issues is the use of underactuated mechanisms (UMs) as extremities for the robot. UMs can easily grasp an object with high adaptability to comply to its shape with low sensor information and an easy control strategy [27,28,29]. Mechanical adaptability, as terminology, refers to the capacity of a UM to take the shape of any object and hold it with uniform pressure. It is defined as the relation between the joint state q and the external restrictions, which generates a change in the torque of the mechanism. Mechanical adaptability is measured by the relation between the joint’s displacement and the generated torque, until reaching a locking point where the mechanism can no longer move [28,29,30]. Thus, an infinity value of adaptability determines a free motion and the initial contact state, where the UM can move until it contacts the object; meanwhile, a zero value of adaptability determines the grasping convergent state, where the UMs no longer move, and the contact force is higher [31].

The aim of this paper is to present the design of an adaptable service robot for vertical locomotion, which is made up of a body with four wheeled legs. Each leg is composed by a tendon-driven underactuated mechanism with two rotational joints and a wheel in the end extremity. The legs have a rubber cover. The main function of this cover is as a spring for each joint, allowing the legs to return to their main position when the tendon is not driven, as well as allowing to improve the mechanical adaptability of the legs. An Electric Ducted Fan (EDF) is used for a vortex system as the adhesion principle. The vortex system is located at the middle of the body. Theoretical backgrounds and open issues are addressed by considering some challenging problems such as mechanical adaptability modeling as well as kinematic and static analysis. Laser sensors are mounted over the robot to measure the adaptability of the robot, between the legs and body, at each time of the experimental tests for vertical locomotion. Also, an IMU sensor is mounted to measure the angular displacements over each axis. This paper demonstrates the suitability and feasibility of the proposed service robot for vertical locomotion.

2. Conceptual Design of a Novel Wheeled-Legged Robot

A functional analysis is carried out with the state of the art.

Table 1. Functional analysis of the proposed robot.

	What to Do	How to Do
Adhesion system	Hold the robot	Produce more than minimum required force to hold the robot during the vertical displacement.
	Preserve the stone	Do not be in contact with the vertical surface to not damage the stone.
Adaptable configuration	Displacement	The robot must displace to almost any part of the vertical surface.
	Adaptation to reliefs	The robot must have legs to easily adapt to the different reliefs of the surface.
Control unit	Tele-operation	The operator must be able to manage the displacement of the robot from a wireless control command.

presents the main tasks to be achieved with a proposal of solutions to fulfill them. The attached problem can be divided into two main parts, namely an adhesion system which can hold the robot over the vertical surface without damaging the surface and an adaptable configuration which allows the robot to displace over the different surfaces and reliefs. An operator should be able to manage the operation of the robot, giving instructions for displacement. This requires a simple control unit with a user-friendly operation interface.

Conceptual design of the robot is performed by the design requirements presented in

Table 2. Technical specifications of the proposed robot.

	Value	Notes
Max size	600 × 600 × 200 (mm)	To cover the columns
Weight	2000 g	Maximum weight of the robot
Payload	1000 g	Maximum weight of payload
Materials	Lightweight	Use materials as PLA or elastomer for a lightweight body
Material surface	Stone	Like marble and quarry

. These requirements have been selected for a general-purpose service robot for vertical locomotion, considering a low-cost fabrication. A specific type-synthesis has been carried out to identify the main working principles of the proposed configuration. Figure 1 presents a flow chart of the design of the robot. The conceptual design consists of a rigid body with four wheeled legs with rubber cover (see Figure 2 and Figure 3). This cover allows improving the mechanical adaptability of the leg. The wheels are actuated by servo motors, and the legs are tendon-driven by using micromotors (see Figure 4). The main parts of the proposed legs are presented in Figure 5.

At the center of the body is located an EDF as the vortex system. The wheeled-legged configuration has three main proposes: (1) to provide faster and easier displacement of the robot by using the wheels, (2) to adapt to the different irregularities and reliefs of the surface by using the legs, and (3) to vary the distance between the shroud of the vortex system and the surface. Thus, the robot can easily adapt to almost any part of the vertical surfaces. Also, two ball casters are mounted at the ends of the body to give major support to the robot, as well as to give a constant gap between the shroud of the vortex system and the surface. Different sensors are located along the robot to validate its adaptability and performance. The main dimensions of the robot are presented in Figure 6 and Figure 7. Figure 8 presents a photograph of the built robot.

3. Analysis and Modeling of the Proposed Concept

3.1. Kinematic Analysis of Legs

In underactuated mechanisms (UMs), where the number of actuators is less than the degree of freedom (DoF), uncontrollable motions and forces occur to communicate passively with the physical world, such as complying with robotic hands in the form of grasped objects [14] and adapting with walking robots to complex road conditions [11]. The use of this kind of mechanisms for adaptative movements produces quite efficient and interesting results, since with low-quality sensory information the UM can easily adapt to the shape of any object. The proposed legs are composed by an underactuated mechanism and a wheel at its end extremity. The leg is considered a single tendon-driven articulated mechanism, where the joint values can be found by considering the principle for transmission lines and coaxiality in tendon-driven mechanisms [13,14,15]. This principle denotes that the links are labeled sequentially from 0 to n and the pulleys are labeled from j to j + n − 1. Considering the schematic diagram of the leg presented in Figure 9, the angular displacement of link j with respect to link i can be denoted by ${\theta }_{j,i}$; thus, the fundamental circuit equation, once for each pulley pair, is represented by Equation (1). The sign of each term of ${\theta }_{k,k-1}$ in Equation (1) is determined by the orientation of the routing: it is negative if it is a crossed routing; otherwise, it is positive [11]. The angular displacement of each the leg is denoted by Equation (2).

$${\theta }_{j,0}={\theta }_{\mathrm{1,0}}\pm \left({\displaystyle \frac{R_{j+1}}{R_j}}\right){\theta }_{\mathrm{2,1}}\pm \left({\displaystyle \frac{R_{j+2}}{R_j}}\right){\theta }_{\mathrm{3,2}}\dots \pm \left({\displaystyle \frac{R_{j+m-1}}{R_j}}\right){\theta }_{m,m-1}$$

(1)

$${\theta }_{j\mathrm{2,0}}={\theta }_{\mathrm{1,0}}+\left({\displaystyle \frac{R_{j2}}{R_{j1}}}\right){\theta }_{\mathrm{2,1}}$$

(2)

3.2. Mechanical Adaptability

The mechanical adaptability of a UM is defined as the ability to adapt or take the shape of an object by holding it. This capacity can be quantified as presented in Equation (3), where $\delta q$ is the change in the joint’s position, and $\delta \tau$ is the change in the generated torque. Adaptability, $Ad$, is defined as the relationship between the joint state q and the external restrictions, which generate a change in the torque of the mechanism. Therefore, adaptability is the relationship between the articular movement of the mechanism and the torque generated until reaching a point of lock where the mechanism cannot move anymore [28,32,33]. Due to the above, the mechanical adaptability of the proposed legs is studied. The schematic diagram of the leg for mechanical adaptability is presented in Figure 10, where $r_1,r_2$ are the radio of each rotational joint, $q_1,q_2$ are the angular displacements of each joint, $K$ is the resistance of the rubber acting over each joint as a spring, and $a$ is the displacement of the tendon. The relationship between the deformation of the elastic elements, $\Delta$, given by the displacement of the actuator, $a$, and the angles of the joints, $q_1,q_2$, is denoted by Equation (4) [34]. Thus, the elastic potential energy of the mechanism can be expressed as in Equation (5). The relationship of change between the joint movement and the torque, or the generated force, is expressed by the second partial derivative of Equation (4), as presented in Equation (6) [17].

$$Ad={\displaystyle \frac{‖\delta q‖}{‖\delta \tau ‖}}$$

(3)

$$\Delta =r_1{q}_1+r_2{q}_2-a$$

(4)

$$V\left(q,a\right)={\displaystyle \frac{1}{2}}K{\Delta }^2$$

(5)

$$\left[\begin{array}{c}\delta \tau \\ \delta f_a\end{array}\right]=-{\displaystyle \frac{{\partial }^{2}V}{\partial {\left(q,a\right)}^2}}\left[\begin{array}{c}\delta q\\ \delta a\end{array}\right]$$

(6)

In the first instants of movement of the mechanism, before contacting the surface, it is free and without external constraints of movements, so the torque generated to be adapted is zero. When the mechanism contacts the surface, a matrix or vector of external constraints must be considered that allows evaluating its movement. The restricted movement of the mechanism, given the joint state, can be represented as Equation (7), where $H\left(q\right)$ is the matrix of restrictions and $\delta \lambda$ is the restricted movement. Thus, consider a linear constraint in joint space, as expressed in Equation (8), where c represents the restricted joint positions of the mechanism.

$$\delta q=H\left(q\right)\delta \lambda$$

(7)

$$H\left(q\right)=\left[\begin{array}{c}\mathit{cos}\left(c\right)\\ \mathit{sin}\left(c\right)\end{array}\right]$$

(8)

Therefore, by solving Equation (6), the elements of the adaptability equation can be obtained:

$$\delta q=-H{\left(H^{T}KqH\right)}^{-1}{H}^{T}Kqa\delta a$$

(9)

$$\delta \tau =\left(KqH{\left(H^{T}KqH\right)}^{-1}{H}^T-I\right)Kqa\delta a$$

(10)

Considering the parameters of the leg, presented in

Table 3. Parameters of one leg.

Parameters	${\mathit{l}}_1,{\mathit{l}}_2$ (mm)	${\mathit{r}}_1,{\mathit{r}}_2$ (mm)	K (N/mm)
Values	50	7.5	20

, the mechanical adaptability is simulated for a circular surface with radii of 250 mm. The behavior of the adaptability allows knowing the range of movement of the mechanism; the top value of the adaptability, when the mechanism can move without producing any force over the surface; and the lower value of adaptability, when the mechanism cannot move anymore and is producing the higher value of torque over the surface [28,34,35,36,37,38,39].

3.3. Vortex System

EDFs have been traditionally popular for their use as a propulsion method for both big and small-scale aircrafts, mainly due to an increased thrust efficiency by the reduction of the ducts of tip vortices and the pressure drops at the blade tips [40,41,42]. Thus, EDFs do not require the adhesion mechanism to be in contact with the target surface, which solves the design challenges of adhesion in cases of curved, rough, or nonferromagnetic surfaces [40,41,42,43,44]. In addition, no external equipment (e.g., compressor, tubes etc.) must be used for maintaining adhesion, which produces an easy operating solution.

The experimental evaluation of this adhesion system has been constrained to investigating the effect of the gap, i.e., the distance of the EDF shroud to the target surface, on the adhesion efficiency. Negative pressure adhesion works on the principle of generating and maintaining a low-pressure zone, $P_a$, inside a cavity compared to the surrounding outside pressure, $P_b$, and $v$ is the air flow velocity into $A_{in}$. As depicted in Figure 11, the difference in pressure will induce a force, $F_s$, across the projected cavity area onto the target surface, A, by the high-pressure region [42,43,44] as follows:

$$F_S=A_{in}{P}_a$$

(11)

$$P_a={\displaystyle \frac{1}{2}}{v}^2{P}_b$$

(12)

Considering an EDF-based adhesion mechanism, the low-pressure zone lies within its shroud; hence, the active pressure area is depicted in Equation (13).

$$A_{in}=2\pi h{r}_s$$

(13)

Furthermore, air flow through the low-pressure zone, $P_a$, is introduced by the distance between the shroud’s end and the test surface, h, as displayed in Figure 11. In addition to the adhesion force produced through the negative pressure, the EDF generates a thrust as it pulls air through the duct. At small h, the air density will not be enough to create a big thrust force, but as the gap increases and airflow is less restricted, the generated thrust will increase. Different simulations have been carried out through the CFD (computational fluid dynamics) of SolidWorks 2022 SP5 to evaluate the EDF equations (see Figure 12) considering the barometric pressure of the city of Queretaro in Mexico, its mean temperature, and the parameters in

Table 4. EDF parameters for simulations.

Parameters	Value
rs (mm)	65
rd (mm)	69
RPM	53,000
h (mm)	2, 6, 10

. CFD simulations demonstrate that over a 10 mm gap, the suction force begins to fall. Table 4 presents the simulation parameters. Simulation results are presented in

Table 5. Simulation results.

Suction Pressure (KPa)
	h = 2	h = 6	h = 10
Theoretical	−0.528	−0.952	−1.062
Simulation	−0.562	−0.938	−1.115
Force (N)	−3.5	−54.96	−61.30

.

3.4. Rolling Resistance Analysis

When a wheel runs over a flexible surface, it finds out resistance to the movement, since the surface immediately ahead is deforming. This phenomenon is called rolling resistance. Figure 13 shows a wheel with a vertical load, W, and a horizontal force, P, that produces the wheel to move with a constant velocity. The reaction pressure from the surface over the wheel passes through point B at the contact area (see Figure 13). Since the speed of the wheel is constant, the three forces acting on it are in equilibrium; therefore, the surface reaction R over the wheel must pass through the center of the wheel, O. The above generates the following moments in B.

$$\sum M_B=W\times AB-P\times OA=0$$

(14)

Thus, the force P is equal to the horizontal component of the reaction R and is called rolling friction or rolling resistance; the distance a is called the coefficient of rolling resistance. The rolling friction or rolling resistance (f) for a sloping surface is given by the following:

$$f={\displaystyle \frac{Wa}{r}}$$

(15)

Considering the free body diagram of Figure 14, the following equation is obtained:

$$\sum F_y=-mg-{\displaystyle \sum _1^4}{F}_{f_i}+{\displaystyle \sum _1^4}{F}_{F_i}=0$$

(16)

Thus, the force F and the minimum required torque are defined in Equation (17) and Equation (18).

Table 6. Robot parameters for static analysis.

Mass (g)	a (mm)	r (mm)
1500	1.016	30

and

Table 7. Results for static analysis.

Friction (N)	Force (N)	Torque (Nm)
0.4983	15.22	0.4546

present the robot parameters and the static analysis results, respectively.

$$F=W+f$$

(17)

$$\tau =Fr$$

(18)

4. Experimental Evaluation

An experimental evaluation of the robot has been carried out to demonstrate its feasibility and features over a controlled environment. The experimental evaluation consists in evaluating the performance of the proposed robot during a vertical locomotion over a concrete wall. To validate the performance, ten different tests were carried out; each test was performed with a different obstacle.

Table 8. Proposed tests for experimental evaluation of the robot.

Test No.	Object	Shape	Dimensions (H × D × W) mm
1	No object	NA	NA
2	Curved		8 × 60 × 60
3	Step		15 × 100 × 60
4	Big step		24 × 150 × 60
5	OB2		2 × 20 × 60
6	OB4		4 × 20 × 60
7	OB6		6 × 20 × 60
8	OB10		10 × 20 × 60
9	OB16		16 × 20 × 60
10	OB18		18 × 20 × 60

presents the proposed experimental tests and the obstacle used in each one. Test 1 is used as reference test, with no obstacle and a free vertical locomotion, to compare it with the other tests. Figure 15, Figure 16, Figure 17 and Figure 18 present the desired behavior of the robot, and adaptation to obstacles, during the experimental tests.

A LiPo battery (6s) is used as the main power supply of the robot; a voltage regulator allows setting a voltage of 5 volts to operate an Arduino UNO and the servomotors (MG995); an ESC controller is used to control the speed of the EDF; and a servo tester regulates the PWM to the ESC controller (see Figure 19). MG995 servomotors are used to drive the wheels, with a maximum torque of 15 kg-cm. Different sensors have been mounted over the robot to test its efficiency. An IMU sensor (GY-87) is mounted over the body to obtain the behavior of the angular displacements along the tests; thus, one can obtain the rotation during a straight, forward displacement. An ultrasonic sensor (HC-SR04) is mounted in the front of the body of the robot; thus, one can obtain its linear velocity by the relation between the displaced distance and the time lapse. Four laser sensors are located over the body to measure the distance between the body and the surface. One laser sensor is mounted over the servomotor of each leg (see Figure 1); thus, the distance between the leg and the wall can be measured. The above allows relating the displacement of the leg and the displacement of the body along the experimental tests; namely, it is possible to measure the adaptability of the robot when it is carrying out a vertical locomotion over different obstacles. Unlike the mechanical adaptability of UMs, this approach allows measuring the percentage that the leg is complying to the obstacle’s shape. Figure 20 and Figure 21 present a time lapse of the robot during the experimental evaluation for tests one and four, respectively. To measure the adaptability of the robot over the obstacles Equation (19) is used, where $\delta$ is the relative error between the theoretical high and the experimental high. The theoretical high is the sum of the distance of the leg to the wall (61 mm) and the high of the obstacle. The experimental high is the sensed distance between the leg and the wall.

$$A=\left(1-\delta \right)\times 100\%$$

(19)

$$\delta ={\displaystyle \frac{TH-EH}{EH}}$$

(20)

There are four representative tests used to demonstrate the feasibility of the robot, tests 2, 4, 5, and 10. The distance between the body and the wall during these three representative tests is presented in Figure 22, where it is possible to note that the body stays always parallel to the wall with a continuous distance of 52 mm with a variation of ±1 mm. Figure 23 presents the behavior of the legs during the representative tests. It is possible to appreciate that the legs of the robot are reaching the maximum high of each obstacle, according with Table 8. Therefore, if the body of the robot keeps straight and parallel to the wall, and if the legs are taking the shape of the obstacles, the robot is adapting into the shape of the obstacles during its vertical locomotion. The angular displacements, obtained with the IMU, are presented in Figure 24. Roll and pitch are the angular displacements for the x- and y-axis, respectively. The variation in the behavior of these axes is mainly due to the vibrations of the EDF; meanwhile, yaw (z-axis) represents the turning of the robot due to the vertical locomotion and the adaptability to the obstacles. Comparing Figure 23 and Figure 24, one can note that the variation in the angular displacements coincides with the movement of the legs during the test, (see Figure 23a and Figure 24a.

Table 9. Summary of experimental results.

Test No.	TH (mm)	EH (mm)	$\mathit{\delta }$	A (%)	Roll (°)	Pitch (°)	Yaw (°)	LS (mm/s)
1	61	61	0.0000	100%	−0.6	−0.34	−1.45	38.24
2	69	69	0.0000	100%	−0.3	1.01	−2.07	38.35
3	76	75	0.0131	98.69%	−4.06	−0.71	0.27	38.42
4	85	84	0.0117	98.83%	−1.56	0.07	0.4	38.32
5	63	62	0.0158	98.42%	−0.9	−0.16	1.53	38.35
6	65	65	0.0000	100%	−2.36	0.99	2.02	37.92
7	67	66	0.0149	98.51%	−1.35	2.07	0.87	38.16
8	71	71	0.0000	100%	−0.9	−1.17	2.58	37.95
9	77	76	0.0129	98.71%	2.96	0.03	2.01	38.22
10	79	79	0.0000	100%	−3.02	2.18	2.89	38.33

TH = theoretical high, EH = experimental high, $\delta$ = relative error, A = adaptability, LS = linear speed.

presents the summary of the experimental results of the robot.

Table 10. Average measured values of the proposed robot performance.

$\mathit{\delta }$	Adaptability (%)	Roll (°)	Pitch (°)	Yaw (°)	LS (mm/s)	Weight (g)
0.00684	99.316	−1.21	0.39	0.90	37.62	1545

presents the average measured values of the robot performance. To obtain the mean power consumption, an ACS712 current sensor has been used. The maximum power consumption, considering the servomotor and dc motor, is 1.54 watts for each leg; therefore, for the four legs it is 6.16 watts. Moreover, for the vortex system, the mean power consumption of the EDF is 1900 watts. Thus, the mean total power consumption of the robot is 1906.16 watts.

Table 11. Statistical data of the experimental evaluation.

Test No.	$\mathit{\sigma }$	${\mathit{\sigma }}^2$	MSE
1	0.7454	0.5557	0.5524
2	0.7556	0.5753	0.5772
3	0.6771	0.4584	0.4583
4	0.7413	0.5576	0.5567
5	0.7486	0.5495	0.5743
6	0.7512	0.5856	0.5709
7	0.7507	0.5604	0.5833
8	0.7585	0.5643	0.5625
9	0.7467	0.5636	0.5631
10	0.7312	0.5432	0.5873

MSE = mean squared error.

presents the statistical data of the carried out experimental evaluation.

5. Conclusions

This paper presents the design of a novel wheeled-legged service robot for vertical locomotion tasks. The design of the robot allows displacing over concrete walls by using wheels, and the underactuated legs with rubber cover help the robot to adapt to almost any irregularity and relief of the surface. The robot uses an EDF as the vortex system for adhesion. This vortex system is located at the middle of the robot body. The use of the theory of mechanical adaptability for UMs gives a general perspective of the behavior of the legs during a grasping task. The use of UMs instead of fully driven mechanisms helps the robot to have legs with a higher adaptability with an easier control strategy. The EDF simulations demonstrate the feasibility of the proposed configuration since, by varying h and the EDF speed, the suction force can support the required load and weight specifications of the robot. Different sensors have been used to compare the performance of the robot on each test, mainly the adaptability of the legs in respect of the main body. The experimental tests demonstrate that the wheels follow the shape of the obstacles during vertical locomotion; meanwhile, the main body of the robot keeps straight and parallel to the climbing surface, with an average relative error of 0.00684 and an adaptability of 99.316%.