Control Strategy of a 3-DOF Walking Robot Anca Petrişor, Nicu George Bîzdoacă, Sonia Degeratu, Adrian Drighiciu,

Gabriela Petropol Serb, Daniela Coman, Marius Constantin Popescu University of Craiova /Faculty of Electromechanical Engineering, Craiova, Romania

Abstract—The paper presents the mathematical model of a walking robot where the active pair of legs has three free joints. The mathematical model of the robot is determined considering a systemic approach of the robot as Variable Causality Dynamic Systems (VCDS). Taking into account a possible symmetrical structure, only the vertical xz-plane evolution is considered. The results can be extended to three-dimensional space. The approach is best on variable dynamic systems, considering that each leg of the pair can change the state depending of the causality order. It is considered that the first leg has both joints free and the second leg has the second joint free. The model is implemented in MATLAB environment and some evolutions examples are presented.

Keywords—robotics, variable causality, mathematical model, control strategy.

I. INTRODUCTION Behavior of walking robots is characterized by a

specific type of movement called legged locomotion [1], [2]. Legged locomotion combines continuous time differential systems and logical systems concepts, which allow describing the both fundamental aspects: leg movements and leg coordination. As a result, different types of legged movements are possible as walking gates and climbing movements. Many control algorithms implemented on the existing walking robots, [3], [4], are based on "state of the art" technologies to control the movements of articulated limbs and joint actuators.

In this present work it is presented a different approach, a systemic approach of a walking robot interpreted as Variable Causality Dynamic System (VCDS). Taking into account a possible symmetrical structure, only the vertical xz-plane evolution is considered. The mathematical model of the robot is determined considering all the points in the xz-plane as being complex numbers.

It is considered a robot leg with two joints. The angles of these joints can be free outputs or effects of other causes, or could be controls, depending on context. Because of the rigid body of the robot, these variables interact respecting some kinematics restrictions due to the finite length of the body.

The main difficulty in conceiving this mathematical model was the ununivocity of the input-output dependence of the Two Joint Arm structure in inverse kinematics.

II. ROBOT GEOMETRICAL STRUCTURE It is considered a walking robot structure, as depicted in

Fig. 1 and described in [5], [6], [7], [8], having three normal legs , ,i j pL L L and a head equivalent to another leg L0 containing the robot centre of gravity G .

The robot body RB is characterized by two position vectors O0, O1 and the leg joining points (hips) denoted iR , jR , pR .

Fig. 1. Robot geometrical structure.

The head joining point 0L is the central point O0, R0=O0, so the robot body RB is univocally characterized by the set,

0 1 0{ , , , , , }= i j pRB O O λ λ λ λ , (1)

where 0 0=λ . The geometrical structure of the robot is defined by

1 0 ⋅− = jO O e θ (2)

0 ⋅= + ⋅i i jR O e θλ (3)

0 ⋅= + ⋅j j jR O e θλ (4)

0 ⋅= + ⋅p p jR O e θλ (5)

0 0 0 0⋅= + ⋅ =jR O e Oθλ (6)

from which result,

( ) ⋅− = − ⋅i j i j jR R e θλ λ (7)

( ) ⋅− = − ⋅p j p j jR R e θλ λ (8)

( ) ⋅− = − ⋅i p i p jR R e θλ λ . (9)

The robot position in the vertical plane is defined by the pair of the position vectors O0, O1 where 1 0| | 1− =O O , or by the vector O0 and the scalar θ , the angular direction of the robot body.

u1,j u1,i

u2,i

Gi

O1O0

G0=G

Rj

u1,p

Gp

u2,j

u2,pRp

Gj

R0Ri

u1,0

u2,0

EUROCON 2007 The International Conference on “Computer as a Tool” Warsaw, September 9-12

1-4244-0813-X/07/$20.00 2007 IEEE. 2337

The robot leg, as a single element, is the Two Joint Arms (TJA) structure, as it is presented in Fig. 2.

Fig. 2. Robot leg kinematical structure in {x,z} plane.

Each of the four robot legs, 0, , ,i j pL L L L , is characterized by a Existence Relation . ( )ER L

iER( L ) : i i i iR G A B 0− − − = (10)

i 1,i 2 ,ii i j i j j u j uA a e a e e eα θ⋅ ⋅ ⋅ ⋅= ⋅ = − ⋅ ⋅ ⋅ (11)

i 2 ,ii i j i j j uB b e b e eβ θ⋅ ⋅ ⋅= ⋅ = − ⋅ ⋅ (12)

i i j iR G e AB 0θ⋅− + ⋅ = (13)

where

2 ,i 1,ii j u i i j uAB e [ b a e ]⋅ ⋅= − + ⋅ . (14)

jER( L ) : j j j jR G A B 0− − − = (15)

j 1, j 2 ,jj j j j j j u j uA a e a e e eα θ⋅ ⋅ ⋅ ⋅= ⋅ = − ⋅ ⋅ ⋅ (16)

j 2 , jj j j j j j uB b e b e eβ θ⋅ ⋅ ⋅= ⋅ = − ⋅ ⋅ (17)

0j j j jR G e AB⋅θ− + ⋅ = (18)

where

2 ,j 1, jj j u j j j uAB e [b a e ]⋅ ⋅= − + ⋅ . (19)

pER( L ) : 0p p p pR G A B− − − = (20)

p 1,p 2 ,pp p j p j j u j uA a e a e e eα θ⋅ ⋅ ⋅ ⋅= ⋅ = − ⋅ ⋅ ⋅ (21)

p 2 ,pp p j p j j uB b e b e eβ θ⋅ ⋅ ⋅= ⋅ = − ⋅ ⋅ (22)

0p p j pR G e AB⋅θ− + ⋅ = (23)

where

2 ,p 1,pp j u p p j uAB e [b a e ]⋅ ⋅= − + ⋅ . (24)

0ER( L ) : 0 0 0 0 0R G A B− − − = (25)

0 1,0 2 ,00 0 j 0 j j u j uA a e a e e eα θ⋅ ⋅ ⋅ ⋅= ⋅ = − ⋅ ⋅ ⋅ (26)

0 2 ,00 0 j 0 j j uB b e b e eβ θ⋅ ⋅ ⋅= ⋅ = − ⋅ ⋅ (27)

0 0 j 0R G e AB 0θ⋅− + ⋅ = (28)

where

2 ,0 1,00 j u 0 0 j uAB e [ b a e ]⋅ ⋅= − + ⋅ . (29)

From this point of view, the walking robot can be considered an object with five elements

0i j pR { L ,L ,L ,L ,RB }= (30)

that define an equations system with the unknown variables selection depending on the robot state.

This system is called walking robot existence relation. The mathematical model of this object is a Variable

Causality Dynamic Systems VCDS [9] and will be analyzed from this point of view.

A pair of legs { , }i jL L constitutes the Active Pair of Legs (APL) if the robot body position is the same irrespective of the feet position of all the other legs, different of iL and jL . A label is assigned to each possible APL. The APL label is expressed by a variable called Index of Activity (IA), denoted with q , which can take

aN values, numbers or strings of characters. For example, the string of characters ' '=q ij points out that the pair { , }i jL L is an APL. Instead of strings of characters, the Index of Activity can take numerical values, as for example,

12 ' '= ⇔ =q q ij 23 ' '= ⇔ =q q jp (31) 31 ' '= ⇔ =q q pi

All the other legs that at a time instant do not belong to APL are called Passive Legs (PL).

The leg in APL having a free joining point (FJP) is called slave leg, the opposite of the motor (or master) leg, whose both joining points are external controlled (EC).

III. CAUSALITY ORDERING OF AN APL WITH THREE FREE JOINTS

Let be APL= { , }i jL L that means q ' ij ' q 12= ⇔ = . Changing the indices, the below relations are available for any APL.

The kinematics structure of this APL can be followed in Fig. 1. In this structure, only one angle is external controlled (EC), so three joints are free. Considering also the pair { , }i jL L as APL, this is denoted by

' '=q ij , pc motor00, motor01, c = (32)

or ' '=q ij , pc motor00, motor02, c = . (33)

Si

Ri

Ai ai

Gi

bi

αi

βi Bi

x axis

z axis

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In this paper it is considering only the causality ordering

c=[motor00, motor 01, cp ]. In this causality ordering the angle 1, ju is EC and the

angle 2, ju is free. The block diagram of this causality structure, considering 1, ju being EC, is represented as a connection of oriented subsystems (Fig. 3).

In this structure, the position vectors iR , iG , jG can be external controlled. The vector jR depends on iR ,

jG and satisfies − =j ijiR R r . Now, the vector iG

does not affect jR . The output variables are,

j i j 1, j jjR H ( R ,G ,u ,s )= (34)

1,i i i j 1, j i1iu F ( R ,G ,G ,u ,s )= (35)

2,i i i j 1, j i2iu F ( R ,G ,G ,u ,s )= (36)

2, j i i j 1, j j2 ju F ( R ,G ,G ,u ,s )= . (37)

Fig. 3. Causality structure block diagram.

It can be observed that iα and iβ depend only on iR and iG

i j 1, j j2 jF ( R ,G ,u ,s )θθ = (38)

( , , )i i i if R G sαα = (39)

( , , )i i i if R G sββ = (40)

but the command angles 1,iu , 2,iu are referred with respect to the robot body, so they depend on the position angle θ , that means on jG and 1, ju .

The relations (39), (40) represent the mathematical model solution of a lonely leg with two free joints.

So, for the leg iL

i i i i i i i iˆ[ , ,s ] f ( R ,G ,s ,a ,b )αβα β = . (41)

The output variables jR , θ , 2, ju are calculated from the existence relations of the leg jL (18) and the kinematic restriction (7).

Making some substitutions, for ≠j iλ λ , is obtained the kinematic restriction condition moved to input,

− + = −j i j j iAB G Rλ λ . (42)

In explicitly form, the relation (42) becomes,

2, j 1, jj i ju j j ju j ie [ b a e ] G Rλ λ− + ⋅ + ⋅ = − . (43)

From (43) results the angle value

2, j 2, j 1, j i i j j2 j ˆu u F (u ,R ,G ,G ,s )= = (44)

where the state js re-establish the univocity input-output.

If it is denoted,

1, jj j ju

2 jW b a e= + ⋅ (45)

j i x z 1, j2 j 2 j 2 j 2 j 2 jQ ( ) / W q j q Q (u )λ λ= − = + ⋅ = (46)

j i2 j 2 jg (G R ) / W= − (47)

it is obtained a polynomial equation, , presented as

2,

2 2+ =jju

j jQ e g (48)

x z 2, j 2, j2 j 2 j 2 jq j q cos( u ) j sin( u ) g+ ⋅ + + ⋅ = . (49)

A trigonometric equation is obtained

2, j 2, j2 j 2 j 2 ja cos(u ) b sin(u ) c+ = (50)

having at the current moment t the following solutions

Ri

u1,j

Gi

u1,i

u2,i

Rj

u2,j

| Rj – Ri | = rij

c=[motor00, motor01, cp ]

θ

Gj

si

s j s i

u2,j

Rj

Rj

Gj

u2,j

θ θ

θ

u1,i

u2,i

Gi

Ri

αi

βi

u1,j

θ

Ri

Rj

si

s i

s j

sj

2339

2, 2, 1,2 ˆ( ) ( , , , )= =j j j i j j

ju u t F u R G s (51)

where

1,2 2 22 ( )= =x j

j j ja q a u (52)

1,2 2 22 ( )= =z j

j j jb q b u (53)

22 22 2 2 2 2 2( ) ( ) 1 1= − − − + = − − +x z z

j j j j j jc q q g Q g . (54)

The case 2 0=jb and 2 0≠ja

[ ]22

2arccos 0,

= ∈

jj

j

c

aψ π , 2 0≠ja (55)

2

2

2

02

02

>= − >

jj

j

if a

if a

π

ϕπ

(56)

( ) 22

2

= −

jj

j

u tψ

ψ

ifif

2,

2,

( ) 0

( ) 0

− >

− ≤

j

j

u t

u t

ε

ε. (57)

The case 2 0=jb and 2 0=ja , is possible only if 2 jc and it is supposed

2 0=jϕ , 2 0=jψ , 2, ( ) 0=ju t . (58)

The case 2 0≠jb It is calculated:

22

2( / 2 , / 2)= ∈ −j

jj

aarctg

bϕ π π (59)

2 22 2 22 2 2

arcsin [ / 2 , / 2]( ) ( )

= ⋅ ∈ − +

j jj

j j j

b cb a b

ψ π π

(60)

If 2, ( ) 0− ≤ju t ε

( ) 2 22,

2 2

− += − + −

j jj

j j

u tϕ ψ

ϕ ψ π

ifif

2 2

2 2

0

0

− + ≤

− + >

j j

j j

ϕ ψ

ϕ ψ. (61)

If 2, ( ) 0− >ju t ε

( ) 2 22,

2 2

− + += − +

j jj

j j

u tϕ ψ π

ϕ ψ

ifif

2 2

2 2

0

0

− + ≤

− + >

j j

j j

ϕ ψ

ϕ ψ.(62)

Knowing the angle value 2, 2,=j ju u , on calculate the jL leg input variables vector

2, 2, 1,~

22 [ ]⋅ ⋅ ⋅= ⋅ = ⋅ + ⋅j j jj j u j u j j j u

jAB e W e b a e (63)

which assumed the kinematic restriction.

In these conditions

~

2

−=− +

j ij

jj i

G ReAB

θ

λ λ (64)

hence,

1,2~

2

ˆarg ( , , , )

− = = − +

j ij i j j

jjj i

G R F u R G sAB

θθλ λ

(65)

~1,2

2~

2

( )ˆ( , , , )

− ⋅ + ⋅= =

− +

jj i j ij j i j j

jjj i

G AB RR H u R G s

AB

λ λ

λ λ(66)

Knowing θ angle value can be obtained the free angles 1,iu , 2,iu of the leg

1, = −i i iu α β (67)

2, = − −i iu β θ π . (68)

IV. EXPERIMENTAL RESULTS To allow the systematized study and to verify the

correctness of the relations regarding to different causality orderings, a package of programs was conceived using MATLAB environment and was suggestive called “Three Legs Robot”.

Fig. 4. The graphical interface of the application.

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“Three legs robot” application is a complex of MATLAB programs to analyze walking robots evolving in uncertain environments, according to a new control approach, called Stable State Transition Approach (SSTA) [10].

In the application graphical interface was provided a number of ten sliders, so that ten scalar variables can be modified. These scalar variables depend on the causality ordering. If the causality ordering cz and the activity legs index q are changed, the platform for modeling and simulation of walking robots control algorithms, in uncertain environments, allows automate change, that is automate reflector of these causality structures.

For example, in the causality structure cz=[0 1 0], presented in this paper, when as input variable is the slider 1 (that means the position of the point Ri on z axis, for the index of activity q=132), the desired variables imposed by sliders are: z

iR , xiR , 1, ju , i

zG , ixG , 1,pu , 2, pu , pa , j

zG , jxG .

These programs are adjusted to any robot parameters values, which can be in different positions when it evolves in uncertain environments and according to SSTA (Stable State Transition Approach) principle, the stable step is assured regardless of the environment shape.

The package of programs for walking robot modeling and simulation has two basis structures: test achievement and results interpretation, obtaining simple or complex figures, in Cartesian coordinates or evolution kinograms.

The physical robot can be controlled only by angles, in this case being chosen the following six angles: u1,i, u2,i, u1,j, u2,j, u1,p, u2,p. To realize a program, for instance the robot body evolution on a certain curve and with a certain angleθ , required by the planning block, some variations of these angles have to be applied. Their analytical determination is difficult considering the variation shape O0 andθ , but by the respective models are generated the angles which are then applied to the physical robot.

In the following is presented a study regarding the influence of different components on the robot values and position.

-2 0 2 4 6-6

-5

-4

-3

-2

-1

0

1

2

3RoPa1Ex2S3V1q132cz010

x abscissa (m)

y or

dina

te (

m)

Fig. 5. Robot kinematics evolution in the causality structure [0 1 0].

-4 -2 0 2 4-3

-2

-1

0

1

2

3

Ri point position on z axis

Bod

y po

sitio

n I

m(O

0), R

e(O

0), θ

Ex: RoPa1Ex2S3V1q132cz010

G1 = 0

G3 = 3

←Im(O0)

Re(O0)→

←θ

Fig. 6. Robot body position as a function of ziR point position

as input variable.

-4 -2 0 2 4-4

-3

-2

-1

0

1

2

3

4

Ri point position on z axis

Ang

les

u1,

1 , u2,

1 , u1,

3 , u2,

3

Ex: RoPa1Ex2S3V1q132cz010

G1 = 0

G3 = 3

←u1,1u2,1→

u1,3

←u2,3

Fig. 7. Controlled angles as a function of ziR point position

as input variable.

-4 -2 0 2 4-4

-3

-2

-1

0

1

2

3

4

Ri point position on z axis

Co-

ordi

nate

s Im

(R1),

Re(

R1),

Im(R

3), R

e(R

3) Ex: RoPa1Ex2S3V1q132cz010

G1 = 0

G3 = 3

←Im(R1)

Re(R1)

←Im(R3)

←Re(R3)

Fig. 8. Joints position as a function of ziR point position

as input variable.

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-4 -2 0 2 40

1

2

3

4

5

6

7

Ri point position on z axis

Ang

ular

co-

ordi

nate

s α

1 , β 1 ,

α3,

β 3

Ex: RoPa1Ex2S3V1q132cz010

G1 = 0

G3 = 3

u2,1 = 0.2302

u1,3 = 2.042

←α1

←β1

←α3

←β3

Fig. 9. Legs angular co-ordinates as a function of ziR point position

as input variable

0 0.5 1 1.5 2-3

-2

-1

0

1

2

3

Co-ordinates Real(O0)

Co-

ordi

nate

s Im

ag(O

0)

Ex: RoPa1Ex2S3V1q132cz010

G1 = 0

G3 = 3

←Imag(O0)

Fig. 10. 0zO point evolution as a function of 0

xO point, as input variable.

0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

Co-ordinates Re(R1) Re(R3) Re(R2)

Co-

ordi

nate

s Im

(R1)

Im(R

3) Im

(R2)

Ex: RoPa1Ex2S3V1q132cz010

G1 = 0

G3 = 3←Im(R1)

←Im(R3)

←Im(R2)

Fig. 11. i j pz z zR ,R ,R co-ordinates evolution as a function of i j p

x x xR ,R ,R as input variable.

V. CONCLUSIONS In this paper, a mathematical model of a walking robot

with three free joints is proposed. After the description of the robot architecture, the mathematical model is built considered the robot as variable causality dynamic system. This model is implemented in MATLAB environment. Finally, a study of the relations between the robot parameters is presented. The mathematical results are in concordance with the numerical simulations, realized using the own application, and they show that the robot reacts accordingly to the environment and it never loses its control.

“Three legs robot” application is necessary because there are a lot of causality orderings variants taking into account the causality structures and the combinations determined by the legs activity index. The program easy allows the selection of desired dependences types to be analyzed. Also, the evolution kinograms of the visualized structures can be set through graphical interfaces. These dependencies of different causality orderings are very important in practice because they are used as components of the inverse models family which allow the walking robots control achievement. This study also distinguishes the blockage area and the activity area of the robot structure, solving the extremely difficult problems regarding to the robot structure blockage, since being a rigid body structure, not every values combination is possible.

REFERENCES [1] B. Thirion and L. Thiry, “Concurent Programming for the Control

of Hexapod Walking”, 7-th International Conference on Mechatronics and Machine Vision in Practice, USA, 2001.

[2] S. Cubero, “A 6-Legged Hybrid Walking and Wheeled Vehicle“, 7-th International Conference on Mechatronics and Machine Vision in Practice, USA, 2001.

[3] WMC, The Walking Machine Catalog http://www.fzi.de/ipt/WMC/preface/walking_machines_katalog.htm, 2003.

[4] CWR, The Climbing and Walking Robots Home Page, http://www.uwe.ac.uk/clawar, 2003.

[5] A.Petrişor, C.Marin and N. G. Bizdoacă, “The mathematical model of a walking robot with one free joint“, Proceedings of International Carpathian Control Conference, ICCC2005, Miskolc-Lillafüred, Hungary, Vol.II, pag.403-408, 2005.

[6] A.Petrişor, “The mathematical model of a walking robot with four free joints“, Proceedings of 14-th International Workshop on ROBOTICS IN ALPE-ADRIA-ANUBE REGION RAAD'05, Bucharest, Romania, May 26-28, pag.473-478, 2005.

[7] A.Petrişor, C.Marin and N.G. Bizdoacă, “Variable Causality Dynamic System Approach for Robot Control”, WSEAS Transactions on Systems and Control Journal, Issue 1, Volume 1, pp.26-30, 2006.

[8] A.Petrişor and D.Coman, “Control structure of a walking robot with prescribed attitude of the body“, Proceedings of the 5th International Conference on Electromechanical and Power Systems, SIELMEN, Chişinau, Republica Moldova, 6-8 octombrie, 2005, pag.683-686.

[9] C. Marin, “On the Variable Causality Dynamic Systems”, The International Symposium on System Theory, Automation, Mechatronic Systems, SINTES 11, Romania, Craiova, Vol.I, pag.245-250, 2003.

[10] A.Petrişor, Algoritmi pentru conducerea roboţilor păşitori în condiţiile existenţei unor modele dinamice incerte, PhD Thesis, Craiova, 2004.

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