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12thIFToMM World Congress, Besanon (France), June18-21, 2007 CK-xxx

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Special new configuration of wheeled robot to work over uneven terrain: sand,

mud and snow

T. Akinfiev*

M. Armada

A. Ramirez

Industrial Automation Institute

Madrid, Spain

AbstractIn the paper, a wheeled robot with changeable

structure is presented. The robot is able to work under two

modes: under continuous movement mode over sufficiently

solid and smooth surfaces and under discontinuous movement

mode over soft surfaces. Under discontinuous movement mode

the wheels of the robot alternately act as supporting legs

(blocked wheels) and as freely rolling wheels. Specific dynamic

properties of the robot have been discovered using analytical methods. The problem of time optimal control is solved.

Experimental results obtained using a prototype confirmed

theoretical considerations. Field of application of the robot is

connected with disabled people transportation.

Keywords: mobile robot, design, changeable structure,

dynamical analysis.

I. Introduction

Principally, two main kinds of mobile robots are known:

legged robots and wheeled robots. Legged robots can

travel over uneven terrain; however, they have a very

complex construction. Among some of their features are:usually they are working with on-board computer, they

have numerous DOF, they have low efficiency, very highenergy expenses and low autonomy; in general, they are

very expensive and act with low velocity [1].

On the other hand, wheeled robots have comparatively

simple construction, they are not expensive, they have

high efficiency, they can work with high velocity and

usually they have a simple control system. However, theycan move only over solid and smooth surfaces and not

able to cross big obstacles [2-7].

There exist special devices with wheels of big diameter

and big width [12, 13]. These devices allow for

transportation of disabled people along soft surface;however, because of the huge dimensions they cannot gothrough standard door so that they cannot be used inside

buildings.

The authors of several studies [8-10] attempted to mix

positive properties of legged and wheeled robots.

Normally, it is done on the bases of legged robot

configuration adding wheels at the end of the legs. This

kind of robots is called hybrid robots. Hybrid robots can

have high velocity over smooth surfaces and use legs to

cross obstacles.

However, usually they have a very complex design, even

more complex than legged robots; they also have

complicate control system, they require a big number of

motors, etc.

In this paper it is considered a new and simple

configuration of mobile robot which takes the advantages

of positive properties of both: wheeled and legged robots.The construction of the robot is based on a new concept

of design of wheeled mobile robot, which can work in two

different regimes. Like wheeled robot it can work on even

(solid and smooth) surfaces and it can also move as hybrid

robot on an uneven rugged terrain (like sand, mud and

snow) [11].

Figure 1 shows the robots configuration. The robot isformed by a body of the robot, by several axis connected

to the wheels and to the body of the robot, and by wheels

which are able to rotate around the axis. Due to the newdesign concept, the axis of the rear wheels is directly

connected to the body of robot and the axis of the front

wheels is located on a special mobile element, which isconnected to the body of robot and can slide forward and

back. Motor 1 is fixed on the body and is kinematically

connected to the mobile element (through a transmission

belt-pulley or screw-nut). The rear wheels are fed from

motors 2, which rotate these wheels and determine the

direction of the movement of the robot. Each wheel has aguided stopper. With two-side stoppers, rotation of wheels

can be prevented.

Motor 1

ReducerPassive pulley

BeltPulley

Mobile element

Motor 2 Front wheel

Stopper

Stopper

Fig. 1. Configuration of the robot

*E-mail: [email protected]

E-mail: [email protected]

E-mail: [email protected]

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III. Robots working regime

If the surface is sufficiently plain, smooth and solid, the

robot is working under the first regime of work. In this

regime, the motor 1 is disconnected, the stoppers are

disconnected and the robot is working helped by the

motors 2 like any other traditional wheeled robot with all

advantages for such kind of machines.

If the robot starts to work under the same regime on a

sandy surface, the torque on the powered wheel (or

wheels) starts to move the under-wheel sand, the wheel

would be caved in the sand and the robot would not beable to continue moving forward.

For this case the robot works under its second regime

(the motor 2 is disconnected to prevent self-excavation of

wheels). The second regime of work consists of:

(1) The stopper of front wheel is disconnected, the

stoppers of rear wheels are connected; motor 1 starts tomove mobile element forward together with front axis and

front wheel; in this step the front wheel makes passive

forward rotation; the rear wheels do not move because of

the applied stoppers.

(2) When mobile element arrives to the switching point,

motor 1 starts to apply force to the mobile element inopposite direction. The motor stops when mobile element

arrives to the extreme position.

(3) At this moment, the stoppers of rear wheels are

switched off and are now disconnected; the stopper of

front wheel is connected. Motor 1 tries to move the

mobile element back but the mobile element does notexecute any movement because of the force of friction

between the front wheel and the ground, and because front

wheel is fixed. Consequently, the body of robot starts to

be moved forward with positive acceleration; in this

step the rear wheels make passive forward rotation;

(4) When the body of robot arrives to the switching point,

motor 1 starts to apply force to the mobile element in

opposite direction. The body of robot is moved forward

with negative acceleration; motor 1 stops when mobile

element arrives to the extreme position.

After that the whole process starts again.

Figure 2 shows the above described steps of the second

regime.

Thanks to the stoppers applied on rear wheels, duringsteps 1 - 2 these wheels behave together like legs in

walking robot, which can hold a given position on the

ground. During these steps the front wheel acts as

conventional passive wheel.

In the same way, in steps 3-4, thanks to the stopper on

the front wheel, it together with the mobile element,

behave like a leg in walking robot, which can hold the

given position on the ground. During the same steps, the

rear wheels act as conventional passive wheels.

Along one full cycle of movement, front and rear wheels

are playing consecutively two roles: wheel and leg,

conferring to the robot a hybrid robots typical

performing.

IV. Robots dynamical propertiesDynamical properties of the robot working under the

first regime of movement are out of consideration in this

paper because they have been discussed by other authors

and their results are very well known [2-7]. Only

dynamical properties of the robot working under the

second regime of movement will be considered.

One full cycle of movement under the second regime of

work consists of 4 steps.

A. Step 1To analyze the dynamical properties it is necessary to

make an independent description of the forces that affect

the body of the robot (including rear wheels, fig. 3) and

the mobile element (including front wheel, fig. 4).

Figure 3 describes all forces applied to the body of the

FP1L1

d

N21

FFR11

MgFT1

FFR21

O

X

Y

Fig. 3. Forces applied to the body of the robot

Initial position

Va Step 1

V

a Step 2

V

a

Step 3

Va

Step 4

Fig. 2. Full cycle of movement of robot

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robot, so that the following system of equations can be

written:

0121 =+ MgFN P (1)

012111 =+TFRFR FFF (2)

02 21111 = dFLFMgL FRP (3)

21221 NKFFR (4)where:

M- mass of body of robot.

g- acceleration of gravity.K2 - coefficient of static friction between wheels of

robot and the ground.

O - pivot point (for calculation of moments of forces).L1 - distance between the pivot point O and the centre of

gravity of the body of robot.

d - distance between the pivot point O and the ground.

N21 - normal force applied to the rear wheels of robotfrom the ground.

FP1 - force applied between the mobile element and thebody of robot.

FFR11 force of sliding friction applied between thebody of robot and the mobile element.

FFR21 force of static friction applied between blockedwheels of robot and the ground.

FT1 - force applied to the body of robot or to the mobileelement from the transmission system.

N11 - normal force applied to the front wheel of themobile element from the ground.

The equations (1) and (2) correspond to the projection

of forces onXand Ycoordinates. Because of the stoppersapplied on rear wheels and the force of friction between

the rear wheels of robot and the ground, the robots body

does not move and the acceleration of the body of robot is

equal to zero.

The equation (3) corresponds to the rotational

equilibrium of the body of robot.

The stoppers of rear wheels are connected; in

consequence there is no rotation of the rear wheels.

Besides, it is assumed that rear wheels do not slide as

well. In this case, the force of friction applied between the

rear wheels and the ground, when there is no motion, is

static friction force. The magnitude of the static friction

force is given by the inequality (4).Figure 4 describes all forces applied to the mobile

element (together with front wheel). The following system

of equations can be established:

0111 = mgFN P (5)

1111 maFF FRT = (6)

1111 PFR FKF = (7)where:

m - mass of the mobile element including front wheel.

1a - acceleration of mobile element.

K1 - coefficient of sliding friction between the mobile

element and the body of robot.

Equations (5) and (6) correspond to the projection of all

forces onXand Ycoordinates.

During step 1 the mobile element is in motion; it means

that the force of friction applied between the mobile

element and the body of robot is a sliding friction. The

magnitude of the sliding friction force is given by the

equation (7).

In this point an additional remark has been done. The

front wheel is not fixed by the stoppers, and it can rotate

freely. The inertia of front wheel is very small and in

consequence the magnitude of the force of friction applied

between the front wheel and the ground is assumed to be

very small, so in the present analysis this force of friction

is not taken into account.An important condition of the considered system

consists of the normal force applied to the wheels of robot

from the ground has to be positive (N11>0) in order tokeep the front wheel in contact to the ground all the time.

This limitation is especially important when the robot is

working over inclined surfaces.

From the system of equations (1) (7) it is easy to find

the maximum possible value of acceleration ( max1a ) for

the mobile element. It is interesting to note, that the

magnitude of acceleration is a robots self-property, whichis connected to the forces of friction and the mass of

mobile element. This means, if motor 1 is not powerful

enough, then the mobile element would not be able toreach the maximum possible acceleration; however, even

if motor 1 is very powerful, then it would be impossible to

reach acceleration higher than max1a .

Fig. 5 shows the maximum possible velocity of mobile

element and body of robot versus time. According to this

figure, the mobile elements movement takes place with

maximum possible acceleration max1a and max2a (step 1

and step 2, accordingly) and the bodys movement takes

place with maximum possible acceleration max3a and

max4a (step 3 and step 4, accordingly).

mg

FT1

FP1

N11

FFR11

1a

X

Y

Fig. 4. Forces applied to the mobile element

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0

0,5

1

1,5

2

0 0,5 1

t (s)

Velocity(m/s)

Step 1 & 2 Step 3 & 4

Fig. 5. Maximum possible velocity of the robots body and mobile

element.

The maximum possible magnitude of acceleration(a1max) during step 1 can be achieved when the static force

of friction applied between the rear wheels of robot and

the ground (FFR21) is equal to its limit value before sliding,i.e., whenFFR21 = K2N21

In this case:

max1

21

12

2a

dKL

MgL

m

K=

(8)

The force applied to the body of robot or to the mobile

element from the transmission system corresponding to

the maximum possible value of acceleration (a1max) is:

( )2111

1max1max12 dKL

MgLKMgKmaFT

+= . (9)

It is important to mention that during step 1 the value of

FT1 has to be less or equal than FT1max; it has to be morethan some minimal value (FFR11), i.e.,FFR11

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max3

21

12

2

2a

dKL

mM

M

gLK=

+

+ (28)

where FT3max can be described by:

+

++= mg

dKL

mMgLKMaFT

21

11max3max32

2(29)

D. Step 4

For step 4 it is possible to write the following system ofequations:

0424 =+ MgFN P (30)

4144 MaFF FRT = (31)

4114 PFR FKF = (32)

0414 = mgFN P (33)

024144 =+ FRFRT FFF (34)

14224 NKFFR (35)

022 2411141 =++ FRdFLmgLNMgL (36)From the above system of equations follows:

4

11

1414

2a

LdK

MgLdF

M

K

M

F TT =

++ (37)

Then we obtain:

max4

21

12

2

2a

dKL

mM

M

gLK=

+ (38)

where the value ofFT4max corresponding to a4max can bedescribed by:

+= mg

dKL

mMgLKMaFT

21

11max4max42

2(39)

As it has been shown above, for each step it is possible to

obtain the maximum possible value of acceleration aimax,

i=1,,4. This takes place when the static force of friction,applied between the wheels of the robot connected to the

stoppers (rear wheels during steps 1-2, and front wheels

during steps 3-4) and the ground, is equal to its limit value

before sliding. The values of aimax are not related to the

properties of the motor 1. Those values are obtainedconsidering only self properties of robot and the workingsurface. Even if the motor 1 is very powerful, it is

impossible to support values of acceleration higher than

aimax.

V. Time optimal control

In order to improve the system performance, time

optimal control has to be included. Time optimal control

for one full cycle of movement must be found, according

to the values of maximum possible acceleration calculated

in each step. The optimization has to be made

independently for steps 1-2 and 3-4.

To solve the problem of time optimal control the

distance of each step Xi is calculated. This allows finding

switching points for motor control.

The figure 5 illustrates the relationship between velocity

and time during steps 1-4. In the initial moment mobile

element starts the movement with zero velocity. Themobile element moves with maximum possible

acceleration, and when it is at a switching point, step 1 is

over and step 2 begins. During step 2 the mobile element

moves with maximum possible negative acceleration. Step

2 is completed when velocity of mobile element is equal

zero. Steps 3 and 4 are performed in the same manner, the

body of the robot is in motion instead of mobile element.

Solving the problem of optimization for step 1, it is

possible to find:

1max11 2 XaVS = (40)

2max21

2 XaVS

= (41)

X1 + X2 =l (42)

max1

11

2

a

Xt = . (43)

max2

22

2

a

Xt = (44)

where l is a full distance of the displacement of themobile element.

From the above system of equations it is easy to findX1,

X2, t1, t2.

For one full cycle of movement, the average robotsmovement velocity is:

0

14

~

T

lV = (45)

where the total time required to execute one full cycle of

movement is described by:

43210 ttttT +++= (46)

VI. Experiments

To confirm the effectiveness of the use of new designed

robot, which can work over uneven terrain, a special

prototype was constructed, built and tested. A bottomview of prototype is presented in fig. 6.

The tests made with the prototype confirmed the

dynamical properties described above. During

experiments a new property of the robot was found. It was

observed that if any of the moving wheel (for example,

front wheel) contacts to an obstacle, then fixed wheels (in

this case, rear wheels) start to slide back and move back a

top layer of sand. Sand is elevated in front of the sliding

rear wheels, and a support is created preventing further

sliding of the rear wheels. At this moment the front wheel

begins to cross the obstacle. This effect is shown in fig. 7.

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The same effect helps robot to continue movement even in

a situation, when K1 < K2. Self creation of a special sandsupport is equivalent to increasing static force of friction.

Of course, in this case average velocity of movement

decreases because during some period of the wholeworking time robot is moving back.

VII. Conclusion

New design of a special configuration of wheeled robot

for displacements over uneven terrain (sand, mud and

snow) has been developed. Several working regimes of

the robot have been described. Dynamical properties of

robot have been calculated. The problem of time optimal

control has been solved. Experimental results with a

prototype confirmed theoretical considerations. Such

robot can be used for transportation of disabled people

along soft or hard surfaces.

References

[1] Walking Machine Catalogue: http://www.walking-machines.org/.[2] Fierro, R. and Lewis, F. L. Control of a nonholonomic mobile

robot: Backstepping kinematics into dynammics. In 34th IEEEConference on Decision and Control, 1999, pages 3805-3810, New

Orleans, USA.

[3] Akinfev, T., Armada, M., and Fernandez, R. Vehicle controlmethod. Patent No2187375, Spain.

[4] Astolfi, A. Exponential stabilization of a wheleed mobile robot viadiscontinuos control. Journal of Dynamic Systems, Measurements,

and Control, 1999, 121, 121-126.

[5] Everett, H. R. Sensors for Mobile Robots: Theory and Application.AK Peters Ltd., 1995.

[6] Fukao, T., Nakagawa, H., and Adachi, N. Adaptive tracking controlof a nonholonomic mobile robot. In IEEE Transactions onRobotics and Automation, 2000, 16(5), 609-615.

[7] Guldner, J. and Utkin, V. I. Stabilization of nonholonomic mobilerobots using Lyapunov functions for navigation and sliding mode

control. In 33rd IEEE Conference on Decision and Control, 1994,pages 2967-2972, Orlando, Florida, USA.

[8] Leppanen I., Salmi S. and Halme A. Workpartner HUTAutomations new hybrid walking machine. CLAWAR 98, Brussels.

[9] Matsumoto O., Kajita S., Saigo M. and Tani K. Biped type leg wheeled robot.Advanced Robotics , vol. 13, No 3, pp. 235 236.

[10] Adachi H., Koyachi N., Arai T., Shimuzu A. and Nogami Y.Mechanism and control of leg wheel hybrid mobile robot.

IEEE/RSJ International Conference on Robotics and Automation,

pp. 17921797, 1999.

[11] Akinfiev T., Fernandez R., Armada M. and A. Ramirez. Device forpeople or payload transportation and its control method. Patent

application P200503060, Spain.[12] http://www-robot.mes.titech.ac.jp/robot/wheeled/helios3/helios

3_e.html

[13] http://www.ncaonline.org/products/beach-wheelchairs/index.shtml#skipnav

Fig. 7. The illustration of the sand support creation.

Fig. 6. A bottom view of the robots prototype

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