gait synthesis for quadruped robot walking up and down slope

Proceedings of the 1993 IEEmSJ International Conference on Intelligent Robots and Systems Yokohama, Japan July 26-30.1993

GAIT SYNTHESIS FOR QUADRUPED ROBOT WALKING UP AND DOWN SLOPE

JunNin Pan Junslii Cheng

Shanghai Jiao Tong University 1954 Hua shan Road, Shanghai 200030. P.R.Chiiia

ABSTBACT \

Authors of this paper have already published a paper concerned with quadruped robot walking up or down slope [l].In that paper authors studied slope recognition, gradient measuring method for quadruped robot walking from level to slope,the reason and algorithm of body 4" turn,gait adjustment in transition area.maximum gradient of the slope for the robot to climb and optimum height of the body.Three rules used to recognize a slope are proposed in that paper but they were not proved . This

c *I - paper is a continuation of that paper. V L I W bauthors study motion laws of quadruped robot in other environments which are concerned with slope) i. e. from

recisnition and gradient measuring rethod for all environments which are concerned with s1opQ)he

hich are used to recognize a slope are uthors study prediction of posture after posture adjustment before body turn, the

JTUYH-I1 a quadruped robot by Shanghai Jiao Tong Univ.

a slope to another s l o p e o p cr

length of@adruped robot in

walking up slope.

2. UNIFICATION OF SLOPE RECOGNITION AND GRADIENT HEASURING NETUOD

There ai-e tactile sensors mounted on the botiom and sides of each foot.With the sensors the quadruped robot have adaptability to rough terrain and obstacles. In walkin9 wlieri a t;ictilc sensor o i i s ide of swinging foot is touched the robot judges an obstacle is set, then the robot lifts the foot as high as possible to stride over the obstacle.When a tactile sensor on bottom of swinging foot is touched the robot judses the foot has touched ground, theii the foot stops moving and control system records the Z-coordinate of the foot. Slope also can cause changes of Z-coordinate of swinging foot.According to the information from tactile sensors on feet the robot can judge whether a slope is met and what the gradient of the slope is. Assume that intersection line of surfaces of two environments is perpendicular to the direction of walking authors of this paper propose following rules to recognize a new slope : a. The two front feet of a quadruped walking robot have

chanyes of position as colupared with normal position coritinuously two times in 7. direction when touching ground. the changes have the same direction.

b. The latter changes of position is bigger than double former changes of position.

c. The former changes of position of the two front feet are the "the latter changes of position of the two front feet also are the sase.

I f above three conditions are satified the robot judses a new slope is met. Here slope has broad sense.leve1 can be seen as c? slope which has sradient zero. When robot walks on down slope i f its front feet have positive clianyes of position , the robot judges a i~cw slope xliicli has positive differential sradjeiit is met, the differential gradient of the two enviroiimeiits can be expressed as

0-7803-0823-9/93/$3.00 (C) 1993 IEEE 5 3 2

Authorized licensed use limited to: Khajeh Nasir Toosi University of Technology. Downloaded on December 21, 2009 at 05:53 from IEEE Xplore. Restrictions apply.

\

lAhl

S a=tan-l - (1)

Assume tha t the gradient of or ig ina l slope is a o . if a h o new slope is UP slope as shown i n Fig.Z(a), i f u=ao new environment is level a s shown i n Fiy.f(b), i f a<ao new slope still is down slope but the gradient

I f its f ront fee t have negative clianyes of posit ion the robot judges a new slope which liave negative d i f fe re i i t i a l gradient is met. Calculation forwu1;i of U still is (1). The new environment still is down slope,but the gradient of new slope is la rger than or iy ina l oiie as sliown in Fig.Z(d).Yhen the robot walks on up slope i f i ts f ront f e e t have negative changes of posit ion. the robot judges a new slope which have negative d i f f e ren t i a l gradient is met. Tlie d i f f e ren t i a l gradient of two environment can be expressed as (1) .Assume tha t the gradient of or ig ina l slope is a. i f a h o new slope is down slope as shown i n 'Fig.3(a), i f o = a o new environment is leve l as shown in Fig.J(b), i f a < a D new slope s t i l l is up slope but the gradient

I f its f ront fee t have pos i t ive clianyes of posit ion the robot judyes a new slope which have pos i t ive d i f f e ren t i - a l gradient is met. calculation formula of I still is (1) ..The new eiivironment still is up slope, but the gradient of iiew slope is la rger than or ig ina l one as s l i o ~ n i n Fig.3(d).

is smaller as shown i n Fiy.2(c).

of new slope is smaller as shown i n Fig.S(c).

a. new slope is a up slope. b. new environment is level. c. new slope st i I I is a down slope,

but the gradient i s snwllsr than original one.

d. new slope still is a down slope. but the gradient is larger than origlnal one.

Fig.2. Robot walks on a down slope. and w e t s a new rlopz.

.? '< ?\ *4

a. new slope is a down slope. b. new environaent is level. C. nrw slope still is a up slope

/-

but the gradient is smaller than otiginal one.

but the grndiant i s larger than original one.

Fig.3. Robot walks on a up slope

and meits a new slope.

' d. new slope still Is a up slope

Synthesize above two cases fo1lol;iiig coilclwion uray be obtained: a . posture sensor gives the gradieiit uf the slope on

wliicli ltre robot welks.If the gradient is pos i t ive the slope is up slopc.If the gradieilt is neydtive the slope is down slope. I f Ll~c yradieiit is zero the eilviroilwent is lcvcl.

L. ulieii tlic frolit fee t of the robot have posit ive cllnriUcs of yositiori in 2 direction. differeii t iul gr.idiciiL J is calcu1;itcd with (1) .The robot slioiild turn i I s body II dcgrccs coiiiilcrclocl,visc. llic posture seiisor will g ive the irisliiiatiuii of the body.

c. wlicfi the frolit f e e t of the W b O L have negative ch,iiiycs of positioii in 2 di rec t ion d i f f e ren t i a l gradient U is still calculated with (1). The robot should turn its body II deyrees clockk'ise.The posture sensor will give the inc l ina t ion of the body.

Three ru l e s for recognition of a slope a re proved ;IS

Apparelit l y v i th three rules, obstacle, d i tcli, s top, s t a i r s a r e impossible to be confused with slope. The oidy environwent which is possible t o be confused with slope is rough t e r r a in as shown in F i u . 4 . The problciu is how large the probabi l i ty of confusion is. ,\ssuiuc that on rouoli t e r ra in the protrusions, depressions and f l a t f l oo r area a r e d is t r ibu ted a t randoru and probabili ty of each cnvironiuent is 1/3. Thc event t ha t when touching ground the r igh t f ront foot has f i r s t posit ion cliangcs as compared w i t h normal posit ion is dcnoted a s A l l , the event tha t when touching ground the r igh t froiit foot has second posit ion change as compared wi th normal pos i t ion in same direction is denoted as Bll,the event t h a t Khen touching ground the l e f t f ron t foot has f i r s t pos i t ion change as compared with norural posit ion in same d i rec t ion is denoted as d s l , the e \wl t tliallt %hen totichiits wound the the l e f t f ron t foo t has second pos i t ion chaiioe a s compared w i t h normal posit ion i n sawc di rec t ion is denoted a s BZ1. because All, Dll, A n a , Bvl are independent events,the probabili ty of event t ha t the tvo froiit fee t liave posit ion changes i i i same di rec t ion coiitinuously two times P(Al1~Bl1~AZl~BZl) is as follows

Probabili ty densit) function of a rnndour variable X which represents height of protrusions or depth of depressions is denoted as f(X) erid f i r s t posit ion chanye of f ront f e e t is deiioted a s XI the probabili ty of second posit ion chanye of f ront f ee t vhich is la rger than PX,is as follows:

f 01 1 OYS .

P(A,,~B,,~A,~~B,,~=P~~~~) s~(~,l) *P(.L,) .p(nZl) (2)

P(X > 2x1)= jz, f(x)dx (3)

In (3) X a is niaxiniuur Iieight of protrusions or waximuur depth of depressions. Assuwe tha t F(X) is d is t r ibu t ion function of a randoto variable X, from (3) following formula is got:

For calculating probabili ty of confusion X1 should be equal or s ~ l a l l e r than X J 2 . The event t ha t height of t he protrusion r igh t front foot f i r s t touches is X I and height of another protrusion r igh t front foot continuously touches is equal or la rger than 2Xl is denoted as Cll, the event theat same tliing happens t o l e f t froiit foot is denoted as Clz.Tlie cvcnt tha t depth of the depression rislit froirt foot f i r s t touches is Xl and depth of another depression r igh t f ront foot coiitinuously touches is equal 01' larger than 2X, is denoted as CZ1. the event that s u e L l i i i i v Iiappeiis to l e f t f ron t foot is dciioted as

P (X>ZYl) =l-P(2XJ ( 4 )

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Cz2.Probabi1ity of event C,, is as follows:

(5)

Assume tha t f (XI is uniform distributioi1,so f ( x 1 = l/Xz F( X 1 = X/Xz

P (c l l )= J: 2 f(Xl)[l-F(2Xi)ldXi

1

4 r -

According t o same reason probabi l i ty of event Cl,, Czl, and Czz seperately is

The event t ha t two f ront f e e t meet protrusions continuously two times and height of f i r s t protrusion is X1 and height of second protrusion is equal or larger than 2Xl is denoted as D1.Probability of D1 is

P ( D i ) = P ( A I ~ ~ B I ~ * A z ~ + B z ~ ) 'P(Cii) .P(Ciz) (6) The event t ha t two f ront f ee t meet depressions continuously two times and depth of f i r s t depression is X I and depth of second depression is equal or larger than 2X1 is denoted a s D2. Probabili ty of events D r is

(7) Total probabili ty of events D1 and Dz is denoted a s Plz

(8) Calculating P l z with values of A l l , B 1 1 , A 2 1 . B ~ 1 , C ~ 1 , C 1 Z Czl,Ca2,the probabili ty of confusion of rough t e r r a i n with slope is obtained a s follows:

Further i f uniformity t h a t two f ront f e e t have posit ion changes is considerd Plz will be grea t ly smaller than value of (9).So the method of slope recognition with three ru les is re l i ab le . I t is almost impossible t o make slope t o be confused with other environments.

3. POSITION PREDiCTION AND POSTURE R E G U d O N

In JTUWH-8 pantograph mechanism is use&. There a re limits of motion space f o r every degree of' freedom.It is necessary for body of the walking rob01 t o turn t o pa ra l l e l with sloping surface. B u t before body turn control system should predic t t he posit ions of four f e e t t o see whether they will go beyond the limits of motion space.If they will coiitrol system should appropriately regulate posture of tlie robot f i r s t then has body of the robot turn. When the robot meets a slope which i has posit ive d i f f e ren t i a l s rad ien t body of t he robot should turn counterclockwise.2-coordinatrs of f ron t f e e t reduce and Z-coordinates of back f e e t increase. I f control systeB predic t s t h a t backfeet will so beyond upper edge of motion space while front feet w i l l keep within motion space a f t e r body turn counterclockwise control system should appropriately r a i s e body f i r s t then turn it.The amount of r i s e of body of the robot should guarantee the I ront f e e t and back f e e t within motion space before and a f t e r body turn.further more the back f ee t should be i n some distance from upper edge of motion space a f t e r bofy turn t o guarantee tha t back f ee t are able t o r i s e and swing.If r i s e of body of the robot have back f ee t keep within motion space but have front

P(Ciz) = 1/4 , P(Cz1) = 1/4 , P(Czz) 2 1/4.

P(Dz)= P ( A i 1.Bi1 ~ A z i *Bni) 8 P (Czi) 'P(Czz1

P i z=P (D 1) t P (Dz 1

P i z = 1/648 (9)

BEFORE BODY TURN I

f ee t go out of lnotion space n f t e r body turn control system judves tlie slope is too s teep $01' the robot t o climb.hccording to the same reason i f control systen pi~etlict 111'11 back feet will keep tiitilia wotioti s iwc wliile f runt f e e t will go beyotrd lower edge of' motion space ; lf ter body tui- i r control system sl iuuld appropriately lowcr body of the robot f irst their t u i m i t . I n X- d i r e c t i o n ~ I I ~ I I body o r 1Iie robot turns count erclockwise X-i:oordiiist,es 01' four f e e t a l l reduce.If coutrol systen p1,edict tkit any one of four f ee t h i l l go beyolld b d edge of motion space body of the robot should move backward cer ta in distance f i r s t then turns. When the robot meets a slope which has negative d i f f e ren t i a l s rad ien t body of the robot should t u r n clockwise. Iii this case prediction is still heeded . i f there a re f ee t which will go beyond edge of notioii space a f t e r body turn clockuise posture of the body should be regulated then the body turns.The t r a i n of thouglit is the same but d i rec t ion of reyulation is opposite t o f i r s t case.

3.1 DETERHIIIATION OF REGULATIVE QUANTITY

k'hen the robot meets a slope khich has posit ive d i f f e i m t i n l g r d i e n t motion space in 2-direction of back fee t of the robot is shown on Fig.5.

/I \\ -'

Fig.4. Robot walking on F i g . 5 . Sidc view of notion space

of back feet i n 2-direction. rough terrain.

Z3 i s Z-coordinate of back f e e t before body turn, Zmhr is tipppcr edge of wotion space in Z-direction,Z,' is 2-coordiiite 01 back f e e t n f t e r body turn counterclockwise. Z8' goes beyond upper edge of motion s ince in 2-direction and it is obtained frow follouillg for mula

From (10) z y ' = (-sin*) sx.tcosu , zS ' (11) A L ' is quantity of going beyond uppcr edge A Z j ' =Z i l ' -Z~so i * (12)

Assume tha t body of the robot rise A L f i r s t then turns counteiwlockuise back f e e t reach Z,,, exactly, from (10)

Le,= (-sini) .Xytcosn (Z3-bZa) From (11) (12)

b23= (Z3'-2m*x)/COSU = A Z s ' / C O S t (13) The regulate quantity AZS can be obtained from (13). In (13) bZ3' is from predictioii with (11) and (12). Hotion space i n X-direction of f ron t f e e t of the robot is shown in Fig.6. Xi is X-coordinate of f ront f w t before body turn.Xpin is back edge of notion space in X -direction.Xl' is X-coordinate of f ront f ee t a f t e r body turn coiinterc1ockwise.Xlr goes beyond back edge of motion space i n X-directioii. From (10)

534


Xi'= cosa ~Xltsine~Z1 (14 ) AX1' is quantity of going beyond back edge

AXi'. Xmin-Xi' ( 1 5 ) Assume that body of the robot moves backward AX1 first then turns counterclockwise front feet reach X m l n exactly.Prom (IO)

Xmln= cosa 0 (Xl+AXl)+sina *Z1

Prom (141, (15) AXi= (Xmxn -Xi')/COSU = AXi'/cosa (16)

The regulative quantity AX1 can be obtained from (IG). In (16) AX,' is from prediction with (14) and (15). When the robot meets a slope which has negative differential gradient regulative quantity can be obtained with sane train of thought.

= cosa~Xltcosa~AXl+sina *Z1

3.2 EFFECTS OF BODY REGULATION TO COORDINATES OF 4 FEET AFTER BODY TURN

Body regulation in X-direction will affect foot positions in Z-direction after body turn.Body regulation in 2-direction will also affect foot positions in X-direction after body turn. When the effects are big enough those degrees of freedom which originally don't go beyond edges of motion space may go beyond edges of motion space after body turn. So prediction needs to be done repeatedly until appropriate regulative quantities in X-direction and in Z-direction are obtained. Vith these regulative quantities there is no foot going beyond edges of motion space after body turn. When the robot meets a slope which has positive differential gradient as shown in Pig.7. If body of the robot moves backward in X-direction first as regulation Z-coordinates of four feet after body turn counterclockwise will all reduce compared with corresponding ones without body regulatioii in X-direction. because body turn counterclockwise causes Z-coordinates of back feet to increase less and Z-coordinates of front feet reduce more body regulation in backward direction will be favourable to Z-coordinates of back feet but unfavourable to Z-coordinates of front feet. If body of the robot rises first as regulation X- coordinates of four feet after body turn counterclockwise will all reduce compared with corresponding ones without body rise. Because body turn counterclockwise causes X-coordinates of four feet to reduce body regulstion in up direction will be unfavourable to X-coordinates of four feet. To all environments which are concerned with slope effects of body regualtion on X-coordinates and Z-coordinates of four feet list in table 1.

Fig.6. Vertical view of notion Fig.7. Effect of body regulation in X-direction to coordinntes of 4 feet afer body turn.

space of front feet in X-direction.

4. STABILITY JUDGEHENT OF QUADRUPED WALKING ROBOT IN TRANSITION AREA

In static walking when a leg is going to suing other

I-- '

I

thrce lcgs will support the body, projection of the center of gravity of the robot 0, must be in triangle composed of the three supporting feet to maintain stability of the robot.If O1 is out of the triangle the robot must move it.s body until Ol is in the triangle before a leg is raised neglecting effect of position change of 4 legs on the center of gravity of the robot O.!4hen the robot walks on the flat floor,because the three supporting feet have the sane Z-coordinate, the method of calculating area of plane triangle can be used to judge whether the projection of the center of gravity of the robot is in the triangle[". In the transition area frow level to slope( or, from one slope to another slope) because the body of the robot is parallel with sloping furface and supporting feet have different Z- coordiiiate the method of calculating area of solid triangle should be used to judge whether O1 is in the trianglc. The formula of calculating area of solid triangle is as folloks

1 S=- y.-y, z2-z, "1-21 xy-P, xa-x, ) 'a-y, * q ys-y, za-2.1: I =,-=I xI-xl I I xs-x. y3-y, I * (i'i)

But calculatiny area of solid trianyle is more comples than calcualting area of Plane triaiigle.So tie propose a imaginary coordinate transformation nethod in Khich the body of the robot is ilnngiiicd to turn t o parallel with 1evel.In new coordiiiate system whether 0, is i n the triaiigle composed of three supporting feet is decided by calculatiiiy area of a plane triangle.As shom in Fig.8 in coordiilatc system syz leg4 is going to Le raised, suppclrliilg f e e t P1,P2,P:, couposed il solid triiigle. In or&r to siluplify calcirlating for stability judseaait

535


Fig.8. Transformation from Fig.8. Vertical view of quadruped solid triangle into plans triangle.

coordinate system xyz turns P' clockwise on y-axis t o yet new coordinate system x'y'z'.Old coordinates of P1, Pz,Ps a r e transformed in to new coordinates of P l , P Z , P s with ( 10) . In new coordinate system 2' ax i s is perpendicular.If Ol is i n or out of APIPzPy,Projection of the center of gravity of the robot O2 should be in o r out of AP1'Pz1PS' which is the projection of APIPzPs on the f l a t f loor.In AP1'Pz'Py' P I ' . Pa', Ps' have the same 2'-c0ordinate.h coordinate system x'y'z' pl' ,ep', p3' have the same x'-coordinate and y'-coordinate a s P I , PZ,Ps seperately,so method of ca lcu la t ing area of plane t r iangle can still be used i n t r ans i t i on area t o judge the s t a b i l i t y of the robot supporting by three f e e t , while x'-coordinate and y'-coordinate of P I , Pz , PS a r e used fo r calculating a rea of plane t r iangle . S t ab i l i t y margin eth hod[^^ is used for s t a b i l i t y judgement. I f OZ is i n APIJPz'Pa' and the rPininm distnnce from OZ t o s ides of AP1'P2'Pa' is bigger than s e t margin r t he robot is s tab le , otherwise the robot is unstable.The margin r should be chosen big enough to compensate the change of t he center of gravity of t he robot i n motion. See reference (31 about s t a b i l i t y margin method in de ta i l .

5. HAXHUH STEP LENGTn OF QUADRUPED WALKING

robot In transition area.

. ROBOT IN TRANSITION MEA

Step length of quadruped walking robot is concerned with i n i t i a l posit ions of its four f ee t . According t o reference [ l ] when quadruped robot walks on leve l and periodical symmetry g a i t a r e used , optimum i n i t i a l posit ion i n X-direction of four f e e t is 0.6L ,maximum s t ep i s 0.8L. L is motion space i n X-direction. When quadruped robot walks on s lope optimum init ial posit ion i n X-direction of four f ee t is (0.6-0.2Aj)L. A B is duty fac tor of AL. AB=AL/L , A L is displacement of projection of t he center of gravity on s1ope .h t h i s case Sm=0.8( 1- 2AB) L. Sm is maximum s tep length.cZ1 When quadruped robot walks i n t r ans i t i on a rea from leve l t o slope,because body of the robot tu rn t o pa ra l l e l with sloping surface i n i t i a l posit ions in X-direction of its four f ee t a r e not symmetrical any more. The ve r t i ca l view of the quadruped robot is shown in Fig.9. Footl and Foot2 move backward AL. in X-direction.Foot3 aiid Footl move backvard AL3 i n X-direction.The new X-coordinates of four f ee t may be obtained from (IO).

X'= cosa~X t s ina -Z (18)

= (l-coso).Xz - s ina*Zo (19)

= (I-cosa)-XY - sinuSZi (20)

ALz= Xz - Xz'

AL3= Xs - Xs'

Tbe swing order of 4 l e s s is 4 -1 -13 +Z. According t o

reference 111 the distance from initLa1 pos i t ion in X- di rec t ion of back f e e t t o back edye of motlon space L,' should meet followiny formula

In (21) S is s t ep length.

The distance from i n i t i a l posit ion i n X- di rec t ion of f ront f ee t to back cdye of motioii space L.' should neet

Lr' 2 1/2 s (21)

La' = I3 - A13 ' (22)

following formula Is' b 31.1 s Lz' = Lz - ALz (2.1)

When Lz'and Ls'are de f in i t e , waxiwulu s t ep lengths StU2 and Smr may be seperately o b t a i n d frolu L,' and 13'

s,, = U 3 La' (25) S m 3 = 2 L Y ' (26)

Prac t ica l utnxiuum s t ep length Sol chooses minkuu one between SmZ and SmY

S, = Hin(Smz,Sms) (27) In (19) and (20) amounts of A l a and AL3 a r e coliceriled with gradient of slope P , X-coordinates and Z- '

coordinates of four f e e t before body turn, while these values are coliceriled with envionluent in vhich the robot walks. I f quadruped robot walks in trt insit ion a rea frow slope t o level or frou one slope t o another slope, the maximuw s t ep leibgth are still calculated from (191, (201, (22),(24).(25),(26),(27),but 3 i a (19),(20) has broad sciisc i n 14iicIi u wealis d i f fe rer i t i a l wadieit t of tvo slopcs.

G. COBCLUSION

This paper and reference [I] have studied comprehensive problems of quadruped robot walking up and down slope. Solution of these problems lays a foundation fo r experimental study of quadruped robot walkiny up and dowii slope.llere up and down slope has broad sense . I t not only includes from leve l t o slope or vice versa but a l so includes from one slope t o another slope. Level is seen as a slope which has gradient zero. So the study r e su l t s of the two papers a r e su i t ab le t o various environments concerned with slope . But in the papers some assuwptions a re taken: a. one assumption is tha t surface of slope is f l a t : b. another assumption is tha t d i rec t ion of walking is perpendicular t o in te rsec t ion l i ne of surfaces of two slopes. I f surface of slope is not f l a t formula (1) is unsuitable t o uleasure gradient of the slope. I f d i rec t ion of ualkiiiy is not perpendicular t o i i i tersection l i ne of surfaces of tvo slope g a i t in trailsi t ion area got from the papers is unsuitable eitlrer.Kevcrtlieless. the two papers a r e still valuable. To couplei case fur ther study will be made 011 the bas i s of tile t vo papers.

REFERENCE

(11 Junluin roil and Junshi Chens, "Study of aundruped k'alking Robot Climbing and L'alking Doiin Slope" rroc.of In t . rorkshop on In t e l l i gen t Robots and Systews'91 PP1513-PP153-l

I 2 1 Junmin Pan and Junslri Clieng, "Study of Quadruped Robot Walkilly on Slope" Robot RO.4 1991. (111 Cliiiiese) PP22-PP26

131 Junslri Clreiig and Juiiluin Pan,"Study of Valkiny Coiitrol Algorithm fo r Quadruped Robot" Journal of Shanghai Jiao Tong Univ. NO.6 1991 (In Chinese) r 1'33- PP .I .I

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gait synthesis for quadruped robot walking up and down slope

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