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DING Xilun, et al: Dynamics and Wheels Slip Ratio of a Wheel-legged Robot in Wheeled Motion Considering the Change of Height 2

impact of robot height on the dynamics and wheels slip.Many researches are focusing on the dynamics of

WMR in planar motion [1119] but few of them haveconsidered the factor of the height and its impact to thedynamics and wheel slip. A general dynamical model wasderived for wheeled mobile robots with nonholonomic

constraints by D'ANDREA - NOVEL, et al [12] . CAMPION ,et al, described five classes of wheel robot structures

pointing out the mobility restriction inherent to each classand derived the dynamic configuration model usingLagrange formalism in Ref. [13]. The internal dynamicsof the mobile robot under the look -ahead control wascharacterized by YUN, et al [[14] . YOSHIDA, et al [17],analyzed the slip of wheel on soft ground. The slip - basedtraction model was derived and verified by experimentson Rover Test Bed. WARD, et al [18], proposed amodel - based approach for estimating longitudinal slip of

wheel and detecting immobilized conditions ofautonomous mobile robots operating on outdoor terrain.TIAN, et al [19], established the dynamic model with lateraland longitudinal slip of wheel.

WMRs may slip in wheeled motion when it acceleratesor brakes suddenly on smooth flat ground. Slip ratio asthe measurement of wheels slip is an indispensable factorfor the traction control system which is designed to

prevent the wheeled robot from skidding. The slippage ofwheel has been analyzed in detail in Ref. [20]. The bestslip ratio is between 0.1 and 0.2. Once slip ratio becomeslarger than the boundary value of 0.2, the traction controlsystem will decrease the slip ratio value by reducing therotational velocity of wheel. However, the traditionalformula of slip ratio cannot be applied directly to therobot if the height is changing, because it is affected bythe relative velocity between the body and the wheel.Influenced by the change of height, the slip ratio of

NOROS - should be treated in a special way and theequation should be rebuilt as well.

This paper of the test is organized as follows. In section2 the wheel -legged robot NOROS - and itscharacteristics in wheeled motion are described. In section

3 the dynamics of NOROS - in wheeled motion withvariables of height is presented. With alterable heighttaken into account, the slip ratios of fore wheels and hindwheels are analyzed in section 4. Finally, the simulationresults are presented in section 5 and the conclusion insection 6.

2 NOROS - Robot and Its Wheeled Motion

NOROS - robot is a symmetric hexapod robot whichmainly consists of the body, leg stucture and lifting head.

The hemispheric body with 600mm diameter has three parts inside. The bottom part is equipped with motors thatenable the hip to rotate around the axis which is

perpendicular to the ground. The middle part contains thehardwares of control system such as CPU board, motion

drive, control cards and video capture card, etc. Binocularcamera that can be stretched out of the shell forenvironmental detection is placed at the top. Based on theresearch of bionics, the leg structure was designed like aninsect that consists of hip, thigh, knee, calf, foot andwheel. There are three joints in each leg: hip yawing joint,

hip pitching joint and knee joint. All of those joints aredriven by DC motors through the bevel gear. The angle

between the adjacent hips is 60 , and the rotation rangeof the hip joints is limited between 90 . The wheelsare placed at the knees separately. Three wheels(wheel 1,3, 5) of NOROS - are driven by motors, and the othersare passive wheels. The robot can change the motiontypes between the legged motion and the wheeled motionthrough the configuration transformation. Fig. 1(a) andFig. 1(b) show the status of legged motion and wheeledmotion of NOROS - respectively.

(a) In legged motion (b) In wheeled motionFig. 1. NOROS - Robot Developed at BUAA

When the robot moves straightly in the wheeled motion,every leg parallels to the moving direction and therotation speed of every wheel are the same. When therobot turns, three forelegs rotate b around the axis of Z i (i=1, 2, 6) in respective Cartesian coordinates P i X iY i Z i when the hind legs still maintain their postures unchanged,see Fig. 2. The rotation angle b is small (less than 30 )in this case because the rotation range of the hip is limited.When the robot has to turn sharply, it would stop andchange its direction through a series of actions such aslegs lifting, rotating and putting down. Fig. 3 shows thesituation of the two different directions. Since there aresix directions that can be chosen, the robot is able to moveto anywhere on the plane ground.

The robot height can be changed by the hip pitchingmotor when it is moving as shown in Fig. 4. In order toavoid obstacles or adjust posture for exploration tasks, therobot can change the heights between r and L+r , r and L are the radius of wheel and the length of thighrespectively. The distance x D of fore wheel is equal to

x ' D of the hind wheel but the direction is opposite.When the height of the robot increases, the relative

motion between the fore (or hind) wheel and the bodygenerates velocity in the negative (or positive) directionof x-axis. On the contrary, the moving directionexchanges when the height decreases.

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CHINESE JOURNAL OF MECHANICAL ENGINEERING 3

2

1

65

4

3

1c

1'

2'

6'

3

b

b

b

b

5

1

3

3

55c

3

5

1

c c

1 1

Fig. 2. Turning in wheeled motion

2

1

65

4

3

Direction 1

60

60

60 120

120

60

1'

2'3 '

4'

5' 6'

Fig. 3. Change of the moving direction of robot

x D x ' D

h D

Fig. 4. Change of robot height

3 Dynamics of NOROS - in WheeledMotion

The dynamics of NOROS - robot in wheeled motionon the flat ground is established in this subsection. In thewheeled motion, the supporting wheels play an importantrole to ensure the locomotion stability, and the supportingwheels produce significant effect on the dynamics ofwheeled motion on rough terrain. However, the effect ofthe supporting wheels on the dynamics is little if the robotis on the flat ground. So the robot can be regarded as threewheels robot according to the number of active wheels.Its different with the planar motion of WMR [12] , the bodyof the robot can lift in the direction of z -axis by hip

pitching motor, which means the robot has four degrees ofmobility(three translation degrees and one rotationdegree). Assume that the robot moves on the plane ground,

its center of gravity is at the center of body O b and thewheels are nondeformable. Set the body always parallel tothe ground, i.e., the angle between each leg and the bodyequals to each other d =d i, (i=1,2, ,6); assume the massof wheel is neglected compared with that of its leg, andthe mass center of each leg locates at the midpoint P ic of

thigh.A reference frame X bY b Z b is established at O b, thus the

position of robot can be specified by the following vector:

T( ) x y z q =

where, x, y, z are the coordinates of O b and q is theorientation of reference frame in the inertial coordinate OXYZ .

The positions of wheel 1,3,5 can be described by 4variables a , b , b iO P , d in the plane of OXY as shown in Fig.

2. b iO P , that is equal to R b the radius of robot body, is thelength from O b to P i , d restricted by d is the distance from

P i to the center of thigh in OXY plane. The geometriccharacteristics of the three wheels are listed in Table 1.

Table 1. Geometric characteristics of wheels

Wheel No.

Fixedangle a i / ()

Rotationalangle b i/()

Length from O b to P i

b iO P /mm

Distance from P i to P ic in OXY plane

d/ mm

1 0 b R b Lcosd /2 3 2p/3 p/3 R b Lcosd /2 5 4p/3 -p /3 R b Lcosd /2

The wheeled motion of the robot can be completelydescribed by the following 9 generalized coordinates:

1 3 5( ) x y z q b d f f f = q . (1)

3.1 Analysis on constraint and degrees of freedom The wheels are subjected to three constraint conditions:

non sideslip, pure rolling and relative speed to the body invertical axis. Non sideslip means the velocity component

perpendicular to the wheel plane is zero, thus the first

group of constraint equations is as follows:

1

1

( ) cos 0 , 1,

( ) 0 , 3,5,

L i

i

q b d

q + = =

= =

D R

D R

& &

&

x

x (2)

where 1 b( sin( ) cos( ) cos cos 0) i i i i i L R a b a b d b = - + + + D , R(q ) is a homogeneous rigid transformation matrix:

cos sin 0 0

sin cos 0 0( )

0 0 1 00 0 0 1

q q

q qq

- =

R .

The pure rolling conditions are deduced from Eq. (3).

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The velocity component in the plane of wheel, which isdetermined by the rotational speed of wheel and thevelocity of d , is equal to the speed of robot. The secondgroup of constraint equations is as follows:

2

2

( ) sin 0, 1,

( ) sin 0, 3,5, i

i

L r i

L r i

q d d f

q d d f + + = = - + = =

D R

D R

& & &

& & &

x

x (3)

where 2 b( cos( ) sin( ) cos 0) i i i i i R a b a b b = - + - + - D .Angle d i of all legs are equal, thus the vertical speeds of

all wheels are the same, the relative speed of body towheels in vertical axis is determined just by d and length ofthe leg, thus the third group of constraint equation is asfollows:

(0 0 0 1) ( ) cos 0 L q d d + = R & &g x . (4)

The constraints (Eqs. (2)(4)) can be rewritten as

( ) 0 = A q q&g , (5)

with

1 3 1 3 3

7 9 1 3 1 2 3

1 1 1 3

( ) ( )

( ) ( )

( ) 0

q q

q

q

=

C R C R 0 0

A q J R 0 J J

P R P 0

1

,

b

1 b

b

sin cos cos cos 0

0 1 cos / 2 0

0 1 cos / 2 0

L R

L R

L R

b b d b

d

d

+ = - + - +

C ,

T2 ( cos 0 0) L d = C ,

b

1 b

b

cos sin sin 0

1 0 3 / 2 0

1 0 3 / 2 0

R

R

R

b b b - - -

= -

J ,

T2 ( sin sin sin ) L L L d d d = - - J ,

3

0 0

0 0

0 0

r

r

r

=

J , 1 (0 0 0 1) = P , 2 cos L d = P .

The number of degrees of freedom is the difference between number of generalized coordinates and the rankof constraints matrix. Obviously, the rank of A(q) is 6;therefore degrees of freedom of the robot is 3.

For two possible cases of actuators implementation:three motors provide the torque of b , d , 1 f or d , 3 f ,

5 f . Here we select the first case, and torque of wheel 3, 5can be determined by the velocity around instantaneous

center of rotation.

3.2 Dynamic modeling The Lagrange formulation is used to establish equations

of motion for the robot. The total kinetic energy is the sumof kinetic energy of body T b, legs T l and wheels T w:

6 6

b l_ w_1 1

i ii i

T T T T = =

= + + , (6)

with 2 2 2 2 b b b1 1

( )2 2

T m x y z I q = + + + && & & ,

l_ leg b

2leg b

2 2

leg

2 2

1( ( sin sin ) sin cos

2 21

sin ) ( ( cos cos )2

1cos sin sin ) ( cos )2 2 2

1 1( ) , 1, 2, 6,

2 2

i i

i

z y

LT m x R d

d m y R d

L Ld m z

I I i

a b q d d b

b b a b q

b b d d b d d

q b d

= - + - -

+ + + +

- + - ++ + =

& &&

& &&

& & &&

& & &

2 2l_ leg b

2 2leg b

2leg

1 1( sin sin )

2 2 21 1

( ( cos ) )2 21

( cos ) , 3, 4,5,2 2

i i z

i y

LT m x R I

m y R d I

Lm z i

q a d d q

a q d

d d

= - + + +

- - + + +

- =

& & &&

& &&

&&

2w_ w

1, 1, 2, 62 i iT I i f = =

& L

,

where m b and mleg are the mass of body and legrespectively, R b is the radius of body, I b is the inertiamoment of body with legs around axis Z b. I y, I z are inertiamoment of leg around the axis Y ic and Z ic passing through

P ic. I w is inertia moment of wheel around the axis Y iw.The total potential energy of robot is the sum of

potential energy of body and six legs:

b l6 P P P = + , (7)

where b b ( sin ) P m g r L d = + ,

l leg ( sin / 2) P m g r L d = + .

The Lagrange formulation with Lagrange multipliersT

1 2 6( ) l l l = L and constraint equations f is asfollows:

d ( ) ( ), 1, 2, , 9

d i

i i i

T P T P fQ i

t q q q l

- - - = + =

L&

.

Based on the Lagrange formulation, the equation of

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motion for the robot in its wheeled motion is presented as

Eq. (8):

T( ) ( , ) ( ) ( ) + = + M q q V q q E q A q&& & t , (8)

where M (q) is the 9 9 inertia matrix, ( , ) V q q& is thevector of force determined by position and velocity,

T T T T4 3 3 3 2 3( ) ( ) = E q 0 I 0 ,

T( ) b d ft t t 1= , 3 3 I is the unitmatrix.

S (q) is a 9 3 matrix in the null space of A(q), i.e.,

( ) ( ) 0 = A q S qg , (9)

withT T T T

4 3 3 3 2 3( ) ( ) = S q S I S .The constraint velocities are in the null space of A(q),

denote T1( ) ( ) t b d f = v& & & , and one gets

( ) ( ) t = q S q v& . (10)

The multipliers will be cancelled after substitutingthe expression && q into Eq. (8) by differentiating Eq. (10),and pre -multiplying by T S . Finally the dynamic equationof NOROS - robot in wheeled motion is

T ( ( ) ( ) ( ) ( ) ( , )) t t + + = S M q Sv M q Sv V q q&& & t . (11)

4 Slip Ratio

The slippage of wheel may lead the robot stuck or losecontrol. To deal with this problem, traction control systemshould be designed to adjust the driving torque based onthe slip ratio which is the measure of the degree ofslippage, as seen in Fig. 5. If the slip ratio becomes largerthan the optimal slip ratio in the acceleration phase ofwheels, the drive torque should be decreased. On thecontrary, the driving torque should be increased indeceleration if slip ratio becomes smaller.

f&

Fig. 5. Traction control system based on the slip ratio

For WMR, the slip ratio is a prerequisite parameter ofthe system and it is usually defined as [17]

R R

R R R

( ) / , ( ),

( ) / , ( ),

r v r r v s

r v v r v

f f f

f f

- >= - + = - + + < +

& & &

& & (14)

Similarly, the slip ratio of the hind wheel turns

R h R h

R h R h R h

( ) , ( ),

( ) ( ) , ( ).

r v v r r v v s

r v v v v r v v

f f f

f f

- - > - = - - - < -

& & &

& & (15)

Therefore, the slip ratios of its fore wheels and hindwheels are derived respectively through Eqs. (14), (15),which contain a new variable vh.

5 Simulation and Analysis5.1 Dynamics in the different heights

The impact of vertical movement on torque of hip

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pitching motor of NOROS - on flat ground is discussedas follows. The robot body is parallel to the ground, andthus each leg bears the same burden. The torque of the hip

pitching motor is calculated in three randomly different paths of H i .

Set the robot moving along x-axis without any

displacement in the y-axis. First path is that the robotremains the same height 0.21 m during traveling. Theother two paths are sinusoids with different amplitudeand frequency. The paths is defined randomly in Eq. (16)with the variable motion time t and shown in Fig. 6:

( ) 12

3

,

0 ,

, , 0.21,

0.21 0.03sin ,

0.21 0.02sin 0.5 .

i i

x t

y

H x y z z

z t

z t

= == = = = - = -

(16)

Fig. 6. Three paths of different height

According to the notation introduced before: m body =30 kg; mleg =5 kg; L=300 mm; R b =300 mm; r =60mm; t =20 s; qini=(0 0 0 0.21 0 -0.5236 0 0 0); I y=0.0375kgm 2.

The t d for three paths of different height are calculated based on the Eq. (11) and the results are shown in Fig. 7.All the initial values of three curves are 0. Tor -z1 is thevalue of t d in the situation that the height remains at0.21m and t d at 19.1N m. When the alterable heightfollows z 2, Tor -z2 changes periodically. The cycle time is6.28 s, the maximum and minimum torques are 20.4 N mand 17.5 N m respectively. It means that when the heightof the robot increases by 0.03 m the torque decreases by1.6 N m. Similarly Tor -z3 states t d for the third path z 3.This cycle time is twice of the second one, and the peakvalue of Tor -z3 is smaller than that of Tor -z2, themaximum is 19.8 N m and minimum is 18.2 N m. Theresult demonstrates that the output torque is inversely

proportional to the height.The validity of t d for these paths is verified by

ADAMS simulation. According to the notation introduced before, the robot model is established as shown in Fig. 8.

The functions of hip pitching joint motion in those three paths are respectively: 1 0 F = ; 2 arcsin(sin / 10) F t = ;

3 arcsin(sin(0.5 ) /15) F t = . Torques of hip pitching motorsfor those three paths calculated by ADAMS are shown inFig. 9. Compared with our results (Fig. 7) the variationtrend and values of t d are almost the same. The maximum

difference is less than 0.4 N m at the minimum value ofTor -z2 which means the relative error rate is less than 2%.Therefore, it proves that the result calculated by ourdynamic model is correct.

T o r q u e o f h i p p i t c h i n g m o t o r

/ N m

d t

Fig. 7. Torques under different paths

Fig. 8. ADAMS model

d t

Fig. 9. Result of ADAMS simulation

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5.2 Slip ratio in the different heights

To illustrate the influence of robot height to the slipratio, simulations are performed in this subsection.Assume the robot still drives along x-axis and there is nodisplacement in y-axis; the revolving speed of wheel is

100 rad/min and the velocity of body in the movedirection is 4.8 m/min when the robot height remain at

Z 1 = 0.21 m, then the slip ratio is SR1=0.2 determined by

Eq. (12).When the height of robot follows the second path

'2 0.21 0.05sin Z t = + , the slip ratio of fore wheel (SR2f)

and hind wheel (SR2h) are calculated based on Eq. (13),(15), (16) and the results are shown in Fig. 10. At the

beginning the slip ratio is the default value 0.2. There is amutation value of slip ratio once the robot starts to changeheight. Then SR2f and SR2h change smoothly with varied

height. They are symmetric with SR1. The maximumvalue is 0.23 and the minimum is 0.17.The third path is '3 0.21 0.14sin0.5 Z t = + , the highest

value of Z 3 is 0.35 which nearly reaches to the

theoretical maximum height of 0.36 m(length of thigh is0.3 m and radius of wheel is 0.06 m). Slip ratio of foreand hind wheel are SR3f and SR3h respectively. Becausethe amplitude of height variation is greater and thefrequency is smaller than that of the second path, changestake place at larger scale and more slowly than SR2. Themaximum and minimum values are about 0.28 and 0.12respectively.

Fig. 11 and Fig. 12 present the relation between slipratio and alterable height in Z 2

and Z 3 respectively. The

characteristics of both situations are in common. Theresults demonstrate that the slip ratio of fore wheel islarger than SR1 but the slip ratio of hind wheel is smallerthan SR1 as the height increases. At the inflection point of

Z 2 or Z 3

, the slip ratios of both wheels become 0.2. Thenthe situation of these two wheels exchange, i.e., the slipratio of hind wheel becomes larger than SR1 while thefore wheels slip ratio decreases. Feeding the real -timeslip ratio of each wheel to the traction control system is

vital important to the safety of the robot.

Fig. 10. Slip ratio according to the change of height

SR1 SR2h SR2f

Z2'

Time /s

Fig. 11. Relation between slip ratio and height

change in path 2

Fig. 12. Relation between slip ratio and heightchange in path 3

6 Conclusions

(1) Considering the change of height, the dynamicmodel of NOROS - robot in wheeled motion is built

based on the Lagrange equation. The torques of hip pitching motors are calculated for NOROS - robot s three different paths. The result demonstrates that thetorque of hip pitching joints are inversely proportional tothe height of robot. Compared with the ADAMS results,the validity of our dynamic model is verified.

(2) New slip ratio equations with the parameter of bodys vertical velocity are proposed for fore wheels andhind wheels respectively. The impact of robot height onslip ratio is investigated. Simulation results show that theslip ratio of fore wheel is larger than that of hind wheel asthe height increases. On the contrary, the slip ratio of forewheel becomes smaller than that of hind wheel as theheight decreases.

(3) For the safety of the robot, it is very important tohave the real -time slip ratio of each wheel to be providedto the traction control system.

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Biographical notes DING Xilun, born in 1967, is currently an professor at BeihangUniversity, China . He received his PhD degree from Harbin

Institute of Technology, China , in 1997. His research interestsinclude mechanism and robotics.Tel: +86-10-82338005; E-mail: [email protected]

LI Kejia, born in 1982, is currently a PhD candidate at School ofmechanical engineering and automation, Beihang University,China . He received his bachelor degree and master degree from

Northeastern University, China , in 2004 and 2007 respectively.His research interests include mechanical design and robot

dynamics.Tel: +86-10-82339055; E-mail: [email protected]

XU Kun, born in 1981, is currently a PhD candidate at school ofmechanical engineering and automation, Beihang University,China . He received his bachelor degree and master degree alsofrom Beihang University, China , in 2005 and 2008 respectively.His research interests include robot design and kinematics.Tel: +86-10-82339055; E-mail: [email protected]

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