Future directions for implementation of aerial robot

aMirjana Filipovic, Member, IEEE, bLjubinko Kevac, cAna Djuric

aMihajlo Pupin Institute, The University of Belgrade, Volgina 15,

11060 Belgrade, Serbia, e-mail: mira@robot.imp.bg.ac.rs

bSchool of Electrical Engineering,The University of

Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia, e-mail:

ljubinko.kevac@gmail.com

cWayne State University , 4855 Fourth St. Detroit, MI 48202,

U.S.A., e-mail: ana.djuric2@wayne.edu

Abstract— The aim of this paper is to highlight the importance of forming a mathematical model of Cable-suspended Parallel Robot – CPR (aerial robot), since only that way CPR provides precise guidance in the area. Only highly intelligent control systems, based on high authentic system mathematical model, can provide the realization of very complex tasks. The established mathematical model provides an opportunity to modernize CPR significantly and to make its application become much wider.

Keywords- aerial robot (Cable-suspended Parallel Robot), observation; workspace; kinematics; dynamics; analysis; synthesis

I. INTRODUCTION It can be said that similar system groups, Cable-suspended

Parallel Robots (CPR), were analyzed and modelled as evidenced by numerous publications. The cable driven redundant parallel manipulator for a large spherical radio telescope (LSRT), is analyzed and modelled in papers [1],[2].

Nonlinear dynamic analysis of the suspended cable system is carried out with some sensible results presented in [1] that could be useful to the real engineering of LSRT. Integrated mechanical, electronic, optic and automatic control technologies are employed to make considerable improvement upon the same system. A multiple cable robotic crane designed in [2] to provide improved cargo handling is investigated. The equations of motion are derived for the cargo and flexible cable using Lagrange’s equations and assumed modes method. The results are compared against desired cable lengths and results achieved in previous research using a rigid cable model. This is one of a few papers dealing with flexible ropes.

This work was done for the suspension system in four points. The camera moves quietly and continuously following the observed object. Camera’s carrier moves in space freely allowing the capture of objects from above. It gives a unique feeling to the event viewer to follow smoothly from an unusual proximity, and that is very close to the action regardless of the size of observed space. Free motion in space opens up completely new and unique perspective. The modular design system for observation of workspace and light weight components and a small force in the ropes allow the successful setting up a system to almost any location, which offers wide

possibilities for its use in closed or open space, which can be small or large sizes. It may be placed in the most unlikely places, for example when filmmaking of different genres or for military or police purposes. The CPR system should fly over the audience without being able to be off and fall to the ground.

This paper has aims at forming the mathematical model for CPR system according to the original scientific principles. This paper proposes the application of theoretical methods in solving specific problems rather than the implementation of systems engineering approaches.

In Section II is given its mathematical model for the CPR. The samples of the system responses are analyzed for different conditions. In section III there are concluding remarks.

II. MATHEMATICAL MODEL OF CABLE-SUSPENDED PARALLEL ROBOT (AERIAL ROBOT)

The chosen type of CPR, marked as CPR-B, has been developed in this paper. The whole CPR-B system operates as shown, using two ropes and pulleys that are fixed for up to four corners of point’s space observation. See Fig. 1.

The whole system is very reliable in the physical sense, because there is only minimal possibility that the camera carrier falls to the shot ground. This would happen only if both ropes broke at the same time. Video from the camera is sent to the user via a separate fiber-optic cable.

Winch 4 is driven by a motor that realizes the motion 4θ winding or unwinding fiber-optic cable, depending on the position of the camera carrier in the space. The purpose of the motor motion 4θ is to ensure that the fiber-optic cable is never tight or too loose to lower too much hanging above the recorded surface and thus not to obstruct the recorded participants.

To define a dynamic model of the CPR-B system for observation of moving objects in work space, depicted in Fig. 1, it is necessary first of all to define the kinematics model of the same system. There is a strong coupling between the motion of the camera in the space of Cartesian coordinates x , y , z , and rotation angles of each motor 1θ , 2θ , 3θ .

978-1-4673-1176-2/12/$31.00 ©2012 IEEE

x, y

, z

-dire

ctio

n [m

]

time (s) b) a)time (s)

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car r

ier

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]o p0 1 2 3 4 5 6

-1

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b)

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y

O

1

pulleys

motorized winch 4of fiber-optic kable

wall an

chor

motorized winch-1

θ

θ

xO

θ

motor 3winch 3

motor 4winch 4

fiber-optic cable

x

yz

O

A

motor 1winch 1

v

contour of the workspace

y

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our o

f the

wor

kspa

ce

motorized winch-2

motorized winch-3

pulleys

wal

l anc

hor

θ 4θ

d

k

n m

h

h

mn

k

a)

b)

A

camera carrier

camera carrier

fiber-optic cable

x

yzA motor 2

winch 2sy k

n m

hcamera carrier

z..

Figure 1. CPR-B, a) in the space, b) top view

This relationship is clearly defined by the Jacobi matrix bJ , which connects the external velocity of change of

coordinates Tzyxp ][ &&&& = with the velocity of change of

internal coordinates T][ 321 θθθφ &&&& = .

For the realization of any trajectory in x , y , z space, it is necessary to provide very precise and mutually coordinated motion of all three motors 1θ , 2θ , 3θ . This relationship is:

pJb && ⋅=φ . (1)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

333231

232221

131211

bbb

bbb

bbb

b

JJJJJJJJJ

J . (2)

The elements of this matrix that are beyond the diagonal show the strong coupling between the external and internal coordinates. The mathematical model of the system has the following form:

bvvv MSLGu ⋅+⋅+⋅= φφ &&& . (3)

Vector equation (3) is given by applying Lagrange’s equation on generalized coordinates 1θ , 2θ , 3θ .

RFM bb ⋅= . (4)

( )

FR

JF

Tb

b ⋅⎟⎠⎞⎜

⎝⎛

=

−

◊

1

. (5)

pcc PapmF ++⋅= )( && . (6)

),( bb JfJ ◊=◊ . (7)

The presence of the factors ◊ is a consequence of structural systems with Fig.1. Since the CPR-B has two parallel ropes, suspending cameras in all four directions, then equation (7) takes the form:

bb JJ ⋅◊=◊ . (8)

◊ the factor which multiplies only one direction where there are two parallel ropes. In this direction the force in each rope is half of the impact forces F acting on a camera carrier. Considered CPR-B has two ropes from a camera carrier to all four-point suspension (line k , h , m , n ). The connection between the resultant forces of motors bF and forces acting on a camera carrier F in Cartesian space coordinates is given by the relation (5). This is a geometrical relationship, which is uniquely defined. To obtain the relationship between internal and external forces the virtual work principle is applied. Equation (5) is particularly important because it participates in the formation of a dynamic model of CPR-B. It is assumed that ropes are rigid. Substituting (4)-(8) in (3) gets a dynamic model of CPR-B:

( )( ) FSLGu Tbvvv J ⋅⋅+⋅+⋅=

−

◊

1φφ &&& . (9)

The matrix ( )( ) 1−◊

TbJ is characterized a strong coupling

between presented motors, [ ]Tb MMMM 321= - load moment of motor, [ ]Tb FFFF 321= - resultant force of motor,

Figure 2. The reference trajectory motion of a) position ox , oy , oz , b)

velocity op& of camera carrier (Example 1).

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igna

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[N

]u o

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op

Figure 3. Reference and real trajectories of motion position a) x , y ,

z camera carrier, b) 1θ , 2θ , 3θ motor shaft (Example 1).

R - winch radius, Tzyx FFFF ][= - external force,

Tzyxp PPPP ][= - perturbation force acting on the camera

carrier, Tuuuu ][ 321= , viv GdiagG = , viv LdiagL = ,

viv SdiagS = , m - mass of camera carrier, Tcc ga ]00[ −= -

gravity acceleration. The CPR-B system is in a process of developing at the Mihajlo Pupin Institute (see [3]-[6]). Control law is selected by the local feedback loop for position and velocity of the motor shaft in the following form:

)()(i

o

ilvii

o

ilpii KKu θθθθ && −⋅+−⋅= . (10)

Travel speed camera carrier is trapezoid form )/(417.0

maxsmpo =& , as shown on Fig. 2b). The motors are type

Heinzman SL100F and gears HFUC14-50-2A-GR+belt.

A. Example 1 The example CPR-B from Fig. 1 is analyzed. The camera

carrier has starting point )](8.02.02.0[ mpostart

−= , and the

end point )](2.02.00.2[ mpoend

−= . See Fig. 2a). The camera moves in x and z directions, while the coordinate y is constant. Fig. 3a) shows the results for the motion of the camera carrier in all three directions of Cartesian coordinates at

the real x , y , z and reference ox , oy , oz level.

Figure 4. Deviation real from the reference values of a motion trajectory) a camera carrier, b) the motor shaft (Example 1)

Figure 5. Real and reference a) control signal iu , b) resultant forces for all three motors (Example 1).

Trajectory of motor rotation angles at the real 1θ , 2θ , 3θ

and reference o

1θ , o

2θ , o

3θ level is given in Fig. 3b). The

mathematical model of the system, at the reference level, is defined by (1)-(9). The all three motors 1θ , 2θ , and 3θ must participate in the realization of this task coordinated. This is especially evident in Fig. 3b). This is a confirmation that all of these motions are mutually coupled. There was a good track of desired trajectory at the level of motor motions (in the order of about )(310 rad− , see Fig. 4b)) and at the level of motion of

the camera carrier (on the order of about )(410 m− , see Fig. 4a)). The level of control signals is given in Fig. 5a) and does not exceed the limits of )(24 V± . In Fig. 5b) there are resultant forces of motors that are of orders of magnitude up to )(17 N .

B. Example 2 CPR-B from Fig. 1 is analyzed, also. All system and

control parameters are the same as in Example 1. The mathematical model of the system, at the reference level, is defined by (1)-(9). The camera carrier has starting point )](0.12.02.0[ mpo

start−= , and the end point

)](2.02.00.2[ mpoend

−= . See Fig. 6. There is a very good tracking of the desired trajectory until the moment when the motor 3 enters saturation.

Figure 6. The reference trajectory motion of a) position ox , oy , oz , b)

velocity op& of camera carrier (Example 2).

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]u o

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F 3

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time (s) time (s)b)a)

o

θ2 θ2-

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atio

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dire

ctio

n [r

ad]

θi

x x o-

z oz-

y y o-

o

θ2 θ2-

o

θ1 θ1-

o

θ3 θ3-

Figure 7. Reference and real trajectories of motion position a) x , y ,

z camera carrier, b) 1θ , 2θ , 3θ motor shaft (Example 2).

In Fig. 9a) it is shown that the third motor enters the saturation point in )(95.0 s at the real and reference level it reaches the value of )(243min Vu −= and it leaves the time of saturation in )(35.1 s at the reference level, while for the real level the time is )(6.1 s . The first and second motors do not enter saturation at all. There are resultant forces of motors in Fig. 9b). The CPR-B is modelled and analyzed by software package AIRCAB.

III. CONCLUSION The result of this work is the form of mathematical model

of CPR-B system. Due to large demand and bright prospects of CPR-B development, the importance of establishing its highly authentic mathematical model is especially emphasized. For the purpose of analysis and selection of parameters of the CPR-B system, the software packages AIRCAMB is defined. This software package can be used to verify the choice of some CPR-B parameters; different type of trajectories as example in this paper. Any motion from the selected workspace can be simulated. The validity of the defined mathematical model can be confirmed due to the software package. Future researches go a step further in implementation of the features of elastic ropes (type of nonlinear dynamic elasticity as defined in [7]-[14] in the mathematical model of CPR-B.

ACKNOWLEDGMENT This research was supported by the Ministry of Science

and Technological Development of Serbia for project “Ambientally intelligent service robots of anthropomorphic characteristics” TR-35003 and “The dynamics of hybrid systems of complex structure“, OI-174001, and partially by the: SNSF Care-robotics project no. IZ74Z0_137361/1.

Figure 8. Deviation real from the reference values of a motion trajectory) a camera carrier, b) the motor shaft (Example 2)

Figure 9. Real and reference a) control signal iu , b) resultant forces for all three motors (Example 2).

We are grateful to Prof. Dr. Katica R. (Stevanovic) Hedrih from Mathematical Institute, Belgrade, Serbia, for helpful consultations during the implementation of this paper.

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