A mobile robot to be applied in high-voltage power lines

TECHNICAL PAPER

A mobile robot to be applied in high-voltage power lines

Rogerio Sales Goncalves • Joao Carlos Mendes Carvalho

Received: 19 July 2011 / Accepted: 13 February 2013

� The Brazilian Society of Mechanical Sciences and Engineering 2014

Abstract Many theoretical and experimental studies have

been carried out in order to develop autonomous robots to travel

along telecommunication lines and power transmission lines to

perform inspection and/or repair work. These robots can

improve the efficiency, reduce labor and are expected to reduce

the risks of accident of maintenance personnel. In spite of the-

oretical researches and technological developments of robotic

systems, problems related to stability, controllability, ability to

transpose/surmounting obstacles and autonomy still exist. Thus,

this paper presents a mobile robot that moves suspended on

electric power lines which can surmount different types of

obstacles, placed at any position on cable. Firstly is presented a

review of some existing robots. After, the kinematics model of

the mobile robot and a methodology for spherical obstacle

surmounting are presented. Finally, numerical and experimental

results are presented to validate the proposed model.

Keywords Robotics � Mobile robots � Linkage

mechanism � Transmission lines

1 Introduction

The economic expansion and the development of urban cen-

ters increase the demand for electricity requiring an important

network for its transmission and distribution. Many of the

power transmission lines are installed higher than 150 m

above the ground and require a detailed inspection to identify

and repair defects in the cable, clamps, insulators, installation

or exchange signaling spheres, and for cleaning the cable.

Then, regular checks and inspections are crucial to the security

of energy supply, reliability and reduction of maintenance

costs. Workers who maintain these cables not only face the

constant risk of falling from a great height but also electric

shocks. Thus, due to the difficulties in maintaining such

transmission lines, automatic systems/machines are proposed

and developed in order to reduce the danger for workers, with

less human labor and increasing labor savings.

Many theoretical and experimental studies have been

carried out in order to develop autonomous machines to

travel along telecommunication and power transmission

lines to perform inspection and/or repair work. These

machines can improve the efficiency, reduce labor and are

expected to reduce any danger to maintenance personnel.

In spite of theoretical researches and technological devel-

opments, problems related to stability, controllability,

ability and autonomy still exist [22].

Basically one can identify two types of autonomous or

semi-autonomous systems. The first one uses insulated

telescopic boom where at its extremity is installed a

manipulator. These systems cannot be used in inaccessible

places such as valleys, rivers and mountain regions, and

lines higher than the operation range of the crane arm [3, 5,

6, 13, 14, 16]. Although they are commonly used, they

present problems related to mobility, controllability and

accessibility, and basically are applied to repair insulators.

The second type is mobile robots that can move along the

cable. The sustained system, in general, is composed of a

linkage mechanism coupled to a device that maintains the

robot on cable, such as wheels, grippers and feet. Two

locomotion methods have been studied. The first method

Technical Editor: Glauco Caurin.

R. S. Goncalves (&) � J. C. M. Carvalho

Federal University of Uberlandia School of Mechanical

Engineering, Uberlandia, MG 38400-902, Brazil

e-mail: [email protected]

J. C. M. Carvalho

e-mail: [email protected]

123

J Braz. Soc. Mech. Sci. Eng.

DOI 10.1007/s40430-014-0152-0

uses wheeled robots and the second one uses a walking

technique. Although wheels seem the simplest way for

locomotion, the wheeled robots have problems to cross

obstacles along the cable such as clamps, vibration dampers

and insulators. Examples of wheeled robots can be seen in [2,

10, 15, 19–21, 23–26]. The second method is based on

consideration that a walking robot can move in highly

unstructured environments [1] and then they can be used to

perform inspection and/or repair works on transmission/

power lines, avoiding obstacles. Thus, studies have been

made to obtain an adequate system related to locomotion and

transposition method, mechanical structure and controller.

One locomotion approach resembles the motion of a

cabbage worm, which moves with movements of expan-

sion and contraction of their own body [17].

Another robot had been studied by Sawada, named as

self-guided type, which uses wheels for its normal motion,

‘‘but when the robot encounters a tower, it unfolds an arc-

shaped arm and attaches it to the ground cable on opposite

sides of the tower. The robot then travels along the arm to

pass around to the other side of the tower. Once it is firmly

affixed to the cable on the other side of the tower it deta-

ches the arm and folds it up until it is needed again’’ [19].

Higuchi [11] proposed the balancer type that has a

unique locomotion mechanism for the robot, equipped with

four crawlers, two arms with pulleys, and four actuators

that can go over the towers with stable motion.

Campos et al. [4] proposed a semi-autonomous robot for

autonomous installation and removal of signaling spheres on

overhead cables, but this robot does not surmount obstacles.

Li and Ruan [12] developed a mobile robot capable of

clearing obstacles such as counterweights, anchor clamps,

and torsion tower. The robot has 13 degree of freedom

(dof) and is designed to imitate the monkey’s behavior.

Tsujimura and collaborators proposed a ‘‘legged robot’’ that

walks suspended on telecommunication aerial cables where its

own weight assures its equilibrium. The proposed robot has

two pairs of legs, having the same length, where each leg is

made of slider-crank mechanism with one motor coupled to the

front legs and another to the rear ones [23–26]. From the

kinematical and dynamical robot models they obtained rela-

tionships between the length links of the mechanism legs in

such way the robot can move stably at regular speed. The

experimental results confirmed the good behavior of the robot

when obstacles are placed at equal intervals and, the maximum

dimensions of the surmounted object is function of the foot

trajectory above the cable. If the obstacles are irregularly

spaced an algorithm is necessary to control the motor motion in

order to change the gait for avoiding the collision between the

foot and obstacles [24].

In this paper, we propose one alternative to provide the

necessary mobility to Tsujimura’s robot, using only one

motor to move it, in order to surmount some kinds of

existing obstacles on cable, posed in any place, and to be

applied for high-voltage power lines.

A methodology is proposed to surmount the main obsta-

cles on cable which are vibration dampers, clamps, insulators

for supporting the cable on tower, and signaling spheres. The

signaling spheres are important transmission line compo-

nents that require regular inspections, maintenance, substi-

tution and installation of new ones. They are used for aerial

security close to highways, airports, rivers, valleys and urban

centers and its use and installing are regulated by specific

norms in each country. The task is generally carried out by an

electrician suspended by a helicopter or that moves on the

cable [18], this operation that is done closed to and/or in

contact with a high-voltage line represents great risk of

accidents. Moreover, the costs involved are high.

In particular, one degree of freedom is increased to each

robot leg, allowing a variation of its length. In this way,

when the robot foot collides with an obstacle, detected by a

sensitive edge system (safety edge) placed at each foot, the

correspondent legs come back, have their lengths aug-

mented and tries to surmount the obstacle again. This

procedure is repeated for each leg until the robot over-

comes the obstacle. In any situation, when the feet are on

the wire all legs have the same length. Because the robot at

the surmounting step is in a quasi-static condition, in which

the acceleration and the velocity are approximated to zero,

the robot’s behavior can be analyzed by its kinematic.

Then, in this paper the kinematic model of the mobile

robot, a methodology for surmounting signaling sphere,

and numerical and experimental results are presented.

2 Kinematic analysis

The robot sketch is shown in Fig. 1 and its parameters are

shown in Fig. 2. It is composed of four slider-crank mech-

anisms, used as legs, sprockets, timing belts and only one

motor to transmit a synchronized movement to the legs. It has

two pairs of identical legs, one at the front and another at the

rear allowing the robot to walk suspended on cable hanging

by feet at the extremity of each leg. In Fig. 1, the feet are

represented by simple hooks. Each pair of legs is composed

of two slider-crank mechanism in which each leg has one

degree of freedom for varying its length. Thus, the robot

structure has two legs at right and two at left where the legs on

the same side have the same movement. The stability is

guaranteed by its own weight and, at each instant, by the

simultaneous contact of the feet on the same side with the

cable. In this way the leg motion can be divided into two

modes: the first one is the surmount mode where the foot

describes a trajectory above the cable and the second mode,

the contact mode, when the foot maintains contact with the

cable and supports the robot’s weight and moves the robot


123

forward. Each robot leg repeats these two modes in turn. The

proposed robot has a movement like the sloth.

Figure 2 shows a sketch of the lateral view of the mobile

robot, where the left feet are in contact with the cable. The

subscript ‘‘r’’ represents the rear legs and ‘‘f’’ the front

ones. The kinematic parameters of a pair of legs are shown

in Fig. 3, where each pair of legs is composed of two sli-

der-crank mechanism OABC and OA0BC0. The input

motion, given by angle h, is defined as the angle between

the horizontal line and link OA0. The phase between link

OA and link OA0 is p radian. As the pairs of legs at rear and

at front are equals, in order to simplify the notation, on next

sessions we will exclude the ‘‘r’’ and ‘‘f’’ subscripts that

identify the rear and frontal legs. Segments OB, OA, AC

and A0C0 are denoted by a, r, d and c, respectively.

The goal of the mobile robot is to surmount obstacles on

cable which can be done by increasing adequately the leg

length when an obstacle is detected. In order to analyze the

robot’s behavior its kinematics is studied by considering

equal and unequal leg length. Tsujimura and Yabuta [24]

obtained optimal relations for the robot parameters for which

its mass center presents little vertical oscillations allowing its

trajectory to be considered as linear. In this work the leg

length for these conditions is called as ‘‘optimal leg length’’.

Fig. 1 General configuration of

the mobile robot suspended on

cable

Fig. 2 Lateral sketch of the

mobile robot. The left feet are

on the cable


123

Considering a reference frame XYZ fixed on cable whose

origin coincides with the foot that contacts the cable, foot

C0 in Fig. 3, and using trigonometric relations the trajectory

of point O, given by coordinates xo and zo, can be obtained.

The initial analysis is done by considering the legs length

AC and A0C0 as unequal. Then the coordinates xo and zo can

be given by [7, 8]:

xo¼xoinitialþr �r 1� dffiffiffiffiffiffiffiffiffiffiffiffiffi

a2þr2p

� ��

�

�

�

þr cosh 1� dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2þr2þ2a r sinhp

� ��

�

�

�

þ ð1�rÞ rcffiffiffiffiffiffiffiffiffiffiffiffiffi

a2þr2p �1

� ��

þrcoshc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2þr2�2 a r sinp

h�1

� ��

þ2ð1�rÞ r 1� dffiffiffiffiffiffiffiffiffiffiffiffiffi

a2þr2p

� ��

�

�

�

�

�

�

�

ð1Þ

zo ¼ ð1� rÞ �r sin h� c a� r sin h½ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ r2 � 2 a rp

sin h

�

þr � d aþ r sin h½ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ r2 þ 2 a rp

sin hþ r sin h

� ð2Þ

where xoinitial represents an initial position of the mobile

robot, defined by point O, and r is a binary parameter

identifying if a shorter leg’s foot contacts the cable or not.

r ¼ 1 when the shorter leg’s foot is in the surmount mode

and r ¼ 0 if it is in contact mode.

In a similar way the trajectory of the foot that is in the

surmount mode is given by coordinates xp and zp as [7]:

Equations (1) and (2) are used for studying the trajectory

of the point O of the mobile robot and Eqs. (3) and (4) for

the analysis of the trajectory of its feet for the surmount

mode.

From Eqs. (1) to (4) simulations had been carried through

in order to analyze the robot motion and the foot trajectory

xp ¼ r

r cos h 2� cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ r2 � 2 a r sin hp � d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ r2 þ 2 a r sin hp

� �

þrc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ r2p þ d


a2 þ r2p

� �

� 2

2

6

6

6

4

3

7

7

7

5

þ 1� rð Þ r cos hc


a2 þ r2 � 2 a r sin hp þ d


a2 þ r2 þ 2 a r sin hp � 2

� ��

ð3Þ

zp ¼ ð1� rÞ d aþ r sin hð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ r2 þ 2 a r sin hp � c a� r sin hð Þ


a2 þ r2 � 2 a r sin hp � 2 r sin h

� �

þr 2 r sin h� d aþ r sin hð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ r2 þ 2 a r sin hp þ c a� r sin hð Þ


a2 þ r2 � 2 a r sin hp

� � ð4Þ

Fig. 3 Pair of legs composed of two slider-crank mechanisms. The

foot C’ is on the cable


123

when the legs have unequal length and/or when the slider-

crank mechanism does not have the optimal relations

obtained by Tsujimura [24]. In this case, the robot motion is

not uniform and when the foot leaves the cable, its trajectory

can to present a sector under the cable as shown in Fig. 4. In

Fig. 4 the trajectories of the right and left feet, in sequence,

are presented for the leg kinematic parameters a = 80 mm;

r = 50 mm, and c = d = 390 mm. In a real case the foot

cannot pass under the cable due to its mechanical design.

Thus, the mobile robot will have a non-uniform motion over

the cable. Moreover, when legs have unequal length the

obstacle that the robot can surmount is smaller than when it

has equal leg length, as shown in Goncalves [7].

Thus, the proposed robot has legs of varying lengths in

such a way that it maintains the optimal relations only for the

leg whose foot is in contact with the cable. In this case the leg

whose foot starts the trajectory above the cable has its length

increased to surmount the obstacle. As the robot motion is

defined by the legs which are in contact with the cable, for a

stable motion it is sufficient that only these legs satisfy the

optimal kinematic parameters and then, the legs on the sur-

mounting mode can vary their lengths in an adequate way to

surmount the obstacle. After the surmounting process the leg

comes back to its optimal length. In the next section, a

methodology to detect and surmount obstacles is presented.

3 Methodology for obstacle surmounting

In this section a surmount methodology is presented for

signaling spheres, and some kinds of clamps and vibration

dampers. If the obstacles are not spheres, the proposed

methodology can still be applied because it is based on the

collision between the robot foot and the obstacle which, in

this case, is detected by a sensitive edge system (safety

edge) placed at each foot. When a collision between the

robot foot and an obstacle occurs the control system ‘‘read’’

the crank angular position and, using Eqs. (3) and (4), the

corresponding foot coordinates xp and zp are obtained.

The surmount method consists of moving back the foot

when it finds an obstacle, which is done by turning back the

input crank OA, Fig. 3, of a predefined angle Dh. After the

backward movement of the input crank, the leg whose foot

had collided with the obstacle, increases by a predefined

length Dc. After the increase of the leg, the input crank

moves forward to surmount the obstacle. If a second

Fig. 4 Foot trajectory using a

non-optimal parameters for the

slider-crank mechanism

(a = 80 mm; r = 50 mm,

c = d = 390 mm)

Fig. 5 Procedure to obtain the radius of the obstacle, R, and its center

coordinate, xm, after the second collision


123

collision occurs, one can calculate the radius of the

obstacle and its center position since it is considered as a

sphere. From the angular position of the input crank both

collisions’ coordinates can be obtained using Eqs. (3) and

(4) and, applying trigonometric relations and parameters of

the robot legs, the spherical obstacle radius R, Eq. (5), its

center coordinate xm, Eq. (6), and its extremity coordinate

xf ¼ xm þ R can be obtained. In Fig. 5, the procedure to

obtain these relations is sketched where P1 and P2 repre-

sent the first and the second point of collision, respectively.

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z22 þ ðxm � x2Þ2

q

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z21 þ

z22 � z2

1 þ x22 � x2

1

2 ðx2 � x1Þ

� �2

�2 x1

z22 � z2

1 þ x22 � x2

1

2 ðx2 � x1Þ

� �

þ x21

s

ð5Þ

xm ¼z2

2 � z21 þ x2

2 � x21

2 ðx2 � x1Þð6Þ

From Fig. 5, the maximum coordinate of the sphere, zp,

is given for its center coordinate xm; i: e:; zp ¼ R. But if a

leg length is used to attain this position, given by (xm, R)

another collision can occur along the foot trajectory when it

comes back to the cable. Then, in order to avoid another

collision, an adequate leg length must be considered,

defined by dxm. From simulations one can consider a zp

coordinate as zp ¼ Rþ Dc, where Dc is the step leg length

variation. Thus, the maximum leg length dxm can be

obtained from Eq. (4) for r ¼ 0 and hm is the input crank

angle when the foot, at the surmount mode, pass by coor-

dinate xm, Fig. 6, as

dxm¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2þr2þ2 a r senhm

p

ðaþr senhmÞ

� �

:

� ðRþDcÞþ c ða�r senhmÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2þr2�2 a r senhm

p þ2 r senhm

� �

ð7Þ

Other relations can be used for the zp coordinate at the

correspondent center position of the sphere. Nevertheless,

numerical simulations have shown that the proposed value

is suitable for controlling the variation of the leg length.

However, the leg length given by Eq. (7) does not avoid a

collision in all conditions. When the leg has optimal length,

a collision can occur when the obstacle is on the position of

the cable to which the foot is returning, as shown in Fig. 7

(dashed lines).

Then, the obstacle surmounting methodology must be

analyzed for two cases. In the first case, the obstacle is not

on the return position of the foot on the cable, as sketched

in Fig. 9. In the second case, the obstacle is on the return

position of the foot, as sketched in Fig. 8.

When the obstacle is not on the return position of the

foot after the foot surmounts the obstacle and before the

foot touches the cable, the robot legs must be reduced to its

optimal length.

In the second case, in order to avoid the problem sket-

ched in Fig. 4, the robot legs must have equal lengths at

start and at the end of the surmounting process. However,

to overpass the obstacle position, avoiding the collision, the

leg length must be greater than its optimal length, i.e., it

must be long enough to put the surmounting foot on the

cable at the end of the obstacle, given by xf. At this posi-

tion, the leg length is named as dxf as sketched in Fig. 7

(solid lines—dxf), which can be obtained from Eq. (4)

where r ¼ 0; x ¼ xf ; c ¼ d ¼ dxf and h ¼ 2p, as

Fig. 6 Maximum leg length to surmount the sphere

Fig. 7 Condition when the obstacle is on the return position of the

foot for the optimal leg length


123

dxf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ r2p

2 r

!

:

� xm þ Rþ 2 r � rc


a2 þ r2p þ c


a2 þ r2p � 2

� ��

ð8Þ

When the surmounting foot touches the cable at the end

of the obstacle, the right and left legs should have the same

length dxf. For this, the leg which foot is in contact mode

must increase its length to dxf too. When all legs have the

same dxf length, all feet are on the cable and the point O is

at a lower position than for the optimal leg length, as

sketched in Fig. 8 (Adjusted position—solid line).

To continue the robot motion, the legs must return to its

optimal length, Fig. 8 (dashed lines). For that, at the beginning

of a new step, the foot that just finished the surmount mode has

its leg length reduced to its optimal length. At this moment, the

foot that starts the surmount mode leaves the cable (because its

leg length is still dxf), and the foot starts its trajectory out of the

cable as sketched in Fig. 8 (dashed lines). This condition avoids

collisions at the beginning of the new trajectory and, during the

surmount mode, the leg length returns to its optimal length.

In order to simplify the control a linear function to

describe the length variation of the legs as a function of the

input angle h has been used, which can be given as

d ¼ q � hþ s, whose coefficients q and s can be obtained

by solving this equation for initial and final conditions of

leg length and its corresponding crank angles. Using the

subscript ‘‘i’’ to identify the initial condition and ‘‘e’’ for

Fig. 8 Procedure for correction

of the legs length when the

obstacle is on the return position

of the foot on the cable

Fig. 9 Condition where the

obstacle does not cause collision

(a = 0.08 m; r = 0.05 m;

c = d = 0.33 m)


123

the final ones, the above equation can be written as

di ¼ q � hi þ s and de ¼ q � he þ s. Assigning d the length

of the right leg and c the length of left leg, then lengths can

be obtained by:

d ¼ de � di

he � hi

hþ di he � de hi

he � hi

� �

ð9Þ

c ¼ ce � ci

he � hi

hþ ci he � ce hi

he � hi

� �

ð10Þ

The analysis of the trajectory of the robot center can be

found in [7].

4 Numerical simulations

For simulations, to design a compact robot and construct a

prototype for initial experimental tests in order to validate

the theoretical considerations, the following parameters

had been used: a = 0.08 m, r = 0.05 m the optimal length

of the legs c = d = 0.33 m, and Dh ¼ 5�. Figure 9 pre-

sents a simulation where collision does not occur: the

obstacle is small and remains below the trajectory of the

surmounting foot. In this case, the length of all legs

remains unchanged during the trajectory.

Figure 10 represents the case where there are two col-

lisions and the obstacle is not on the return position of the

surmounting foot. After the first collision, at point P1, the

input crank turns back by an angle Dh, the foot going to

point M. From this point, the leg increases by Dc and the

input crank moves forward simultaneously until the foot

reaches the position corresponding to the x coordinate of

the first collision, point N. From this point, the leg shrinks

and collides again with the obstacle at point P2. After the

second collision and from Eqs. (5) to (7) one can obtain the

radius of the sphere, the coordinate of its center, xm, its

extreme, xf, and the leg length dxm. Continuing the sur-

mounting process after the second collision, the input crank

turns back by an angle Dh. Then, the leg increases until it

reaches the position corresponding to the x coordinate of

the obstacle center, xm, whose length is dxm and, simulta-

neously, the input crank moves forward, the foot going to

point Q. From this point, the leg shrinks until hit its optimal

length when it touches the cable at point T. The final tra-

jectory is given by C–P1–M–N–P2–N–Q–T.

An example where the obstacle is on the return position of

the foot on the cable and with two collisions is presented in

Fig. 11. In this example, from the first collision, at point P1,

to point Q, where the leg has its maximum length dxm, the

sequence is the same as shown in Fig. 10. After the second

collision and using the obtained parameters: the radius of the

sphere, the coordinate of its center, xm, its extreme, xf and the

leg length dxm, it can be verified that the obstacle is on the

position where the foot return to the cable. In this case the

final leg length dxf can be obtained from Eq. (8).

From position given by point Q, the input crank moves

forward at same time that the leg change its length from dxm

until reaches the length dxf and, simultaneously, the legs

whose feet are on the cable have their lengths augmented to

dxf too in such way that when foot C touches the cable at

point T all legs have the same length dxf In the region Q–T

of the trajectory, the coefficients of Eq. (9) are given by

Fig. 10 Simulation of surmounting process with two collisions when

the obstacle is not on the return position of the foot, for the leg length

variation of Dc = 0.03 m

Fig. 11 Trajectory of the right foot when occur two collisions and the

obstacle is on the return position of the foot on the cable (used

parameters: a = 0.08 m; r = 0.05 m; Dh = 5�; Dc = 0.04 m)


123

di ¼ dxm; hi ¼ hm; he ¼ p and de ¼ dxf , where dxf is given

by Eq. (8). Simultaneously the Eq. (10) coefficients for the

other leg are ci ¼ c; hi ¼ hm; he ¼ p and ce ¼ dxf .

At the beginning of the next gait, both legs must return

to their normal length. Then, while the support leg, whose

foot is C, reduces its length, the foot C0 goes over the cable

as shown in Fig. 8.

In Fig. 11 the point H represents the return position of

the foot on the cable in the absence of an obstacle. The

dotted lines P1–H corresponds to the foot trajectory in the

absence of obstacle and P2–H the foot trajectory after the

first leg augmentation if no obstacle exists.

Figure 12 presents the same example as shown in Fig. 11

but, in this case, both feet trajectories are presented. C repre-

sents the right foot, C’ the left ones and the surmounting

process starts by moving the foot C. If no collision occurs the

right foot trajectory is represented by the line C–M–H and the

left foot C0 by C0–I. Considering two collisions, the trajectory

of the right foot C is the same as shown in Fig. 11. After right

foot C had surmounted the obstacle, at point T, all legs have the

same length given by dxf. In order to guarantee a stable robot

motion its legs must return to their optimal length. After the

surmounting process the right foot C is on the cable and, for

continuing the robot motion, its leg shrinks until its optimal

length. As the other leg, the left ones, initially maintains its dxf

length, its foot C0 goes over the cable until it reaches the point

F. From this point the foot starts its normal trajectory over the

cable at same time that its leg reduces to its optimal length

until touch the cable at point G. Then the left foot trajectory is

given by line C0–F–G.

5 Experimental system

A prototype of the mobile robot was built in order to verify

the validity of the surmounting methodology. The linkage

parameters are those for describing a stable trajectory i. e.,

a:r = 1.6 and c:r = 6.6 and, a = 0.8 m, r = 0.05 m, each

leg can increase from its optimal length given by

c = d = 0.33 m to its maximum length 0.62 m, and the

distance between the front and rear legs is 0.295 m. The

control is based on a PC system using a control board to

command the four-step motors used to vary the leg length,

and the dc motor that drives all legs. To verify the contact

between feet and the obstacle, a sensitive edge system

placed at each foot was used, which is similar to a con-

tinuous limit switch. The mobile robot structure is built in

aluminum and the synchronism of the motion legs is given

by a dc motor, pulley and timing belt. The robot weight is

approximately 8 kg. Other information about the mechan-

ical design and control system of the mobile robot can be

found in [7]. It should be noted that for applications of

high-voltage power lines, the robot must have dimensions

compatible with the size of the biggest obstacle which is, in

general, the signaling spheres. Then, the distance between

the front and rear legs of the robot and between the feet that

are on the cable must be longer than the obstacle.

In Fig. 13, the built prototype surmounting an obstacle

that is not on the return position of the foot on the cable and

with one collision is presented. In Fig. 13a, all feet of the

robot are on the cable. From this condition, the robot starts

its movement with the right feet leaving the cable

(Fig. 13b). After the right feet pass through its higher

position and is come back to the cable, a collision with the

obstacle takes place (Fig. 13c, d). From this position, the

input crank turns back 5� (Fig. 13e). After the return

position, the legs increase by 30 mm and, simultaneously,

the input crank goes forward until the front right foot

reaches the corresponding collision position. At this posi-

tion, the right legs have 360 mm length. From this point,

the right feet continue their motion to the cable, shrinking

the right legs (Fig. 13f). When the right legs reach their

optimal length the right feet are on the cable (Fig. 13g) and

from this position the robot can continue its motion by

moving the left legs.

6 Conclusions

As related in several works about mobile robots that move

suspended on cable, most of them present problems related to

controllability, stability and ability to surmount obstacles.

In this paper, a four-leg robot that moves suspended on

cable with sufficient mobility to surmount obstacles inde-

pendently of their position on cable was presented.

The robot is based on the Tsujimura’s robot and their

results related to the kinematical and dynamical models. In

the proposed robot, the two pairs of legs, in which each

structure leg is made up of a slider-crank mechanism, are

Fig. 12 Example for feet trajectories for surmounting an obstacle


123

driven by only one electrical motor and one motor to vary the

length of each leg, improving a simple structure system. This

kinematic structure allows the robot to surmount obstacles

placed at any position on the cable. The proposed structure

was defined by considering characteristics such as stability,

simplicity and controllability. The own weight assuring its

equilibrium and the uniform robot motion provide its sta-

bility since the kinematic parameters are as those obtained by

the optimal relations from Tsujimura’s analysis.

A methodology has been proposed to surmount signal-

ing spheres and some kinds of clamps and vibration

dampers that depend of their dimensions and model.

Numerical simulation has been carried through proving the

validity of the proposed methodology. Using linear func-

tions to control the length variation of the legs simplifies

the mathematical model of the feet trajectories.

Finally, a prototype was built to verify the behavior of

such mobile robot during its motion on the cable and for

surmounting an obstacle.

As the stable motion of the robot can be done without a

precise control of the actuator, the robot is in a quasi-static

motion for surmounting an obstacle, and from numerical

and experimental simulations one can say that the robot is

well controllable.

Fig. 13 The built prototype surmounting an obstacle. a The initial

position with the four feet on the cable; b right feet start the trajectory

above of the cable; c right feet start to return to the cable when a

collision occurs; d right front foot collides with the obstacle; e the

input crank returns from Dh; f the right legs increase to surmount the

obstacle and after they shrink to the right feet return to the cable; g the

four feet contact the cable with the optimal leg length [7]


123

Another methodology has been proposed for the robot to

surmount insulators that sustain the cable at towers, the

results of which were published previously in [9].

Next step of the work consists in adapting the prototype

robot feet to transpose insulators that sustain the cable at

towers and develop an onboard control system.

Acknowledgments The authors are thankful to CNPq, CAPES and

FAPEMIG for the partial financing support of this research work.

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A mobile robot to be applied in high-voltage power lines

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