Walking Locomotion System for Quadruped Mobile Platform with Adaptive Hoofs

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Copyright © IF AC Intelligent Assembly and Disassembly, Bucharest, Romania, 2003

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WALKING LOCOMOTION SYSTEM FOR QUADRUPED MOBILE PLATFORM WITH ADAPTIVE HOOFS Comeliu Riidulescu and Stefan Varga

Politechnical University ofTimi!joara. Department of Mechatronics. Mechanical Engineering Faculty Dui. Mihai Viteazu. 1. 1900 Timi!joara. Romania E-mail: [email protected]. Abstract: The mobile platforms are used to perform programmed or remote controlled movement in a very large working space. In the terresnial locomotion two systems are used: based on rolling (wheeled and tracked), or based on walking. This paper presents a walking locomotion system with four legs, soine identical jointed kinematics chains, where every kinematics pair is actuated. The legs have the platform as the base element; the other end of the leg is jointed with an adaptive hoof The hoofs are actuated by synchronous transmission. Together, the legs and the hoofs, define a quadruped walking locomotion system. Copyright () 2003IFAC Keywords: Robotics, Mobile Robots, Walking Robot, Mobile Platform and Platform Stability.

I . INTRODUCTION

joint is the frame of the adaptive hoofing mobile platform (AHMP) and the leaded element is called thighbone.

The mobile platforms are equipped with a locomotion system, in order to be able to perform programmed or remote controlled movement in a very large working space (Kovacs, 2(00). In terrestrial locomotion the following systems are used: based on rolling (wheeled and tracked), with or without air cushion lifting technique, or based on walking (Manko, 1992). The wheeled locomotion systems need a rolling track. They travel, with relatively great speed, on various routes between start and target stations, dynamically selected during driving (Riidulescu, 2(00). The walking locomotion systems can access start and target stations on various routes defined on natural terrain with obstacles, but the speed is not so fast. This paper presents a quadruped locomotion system with legs that have one end jointed with an adaptive hoof The mobile platform equipped with wheeled-walking locomotion system (Varga and Radulescu, 200la) has greater moving capability, because it permits a rapid movement using wheeled locomotion and slow movement using walking locomotion, thus having the capability to move by walking, creeping, climbing and jumping. The adaptive hoofs oriented according to the terrain relief, improve the platform stability, in the moment of passing from the support role oflegs to stepping role.

Actuator system for the hip joint A The brake of the ankle joint C

Fig. I. Scheme of one hoofed leg For the B joint, the thighbone is the base element and the leaded element is called tibia. For the C joint, the tibia is the base element, and the adaptive hoof represents the leaded element. The A and B joints actuator systems, are placed on the frame, and have DC servomotors with rotor disk and harmonic drive gear. The movements from the motor shafts arrive to the three joints through the synchronous transmissions. The A and B joints servomotors are equipped with a

2. DESCRIPTION OF THE ADAPTIVE HOOFED LEG The adaptive hoofed leg mechanisms are a RRR structure (R-Rotation joint), where every kinematics pair is actuated as shown in Fig. I. The A kinematics pair is called hip j oint, the B is called knee joint. and the C is called the ankle joint. The base element of the A

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memory, in the order required by the movement strategy, which is automatically elaborated depending on the signal values received from sensors just before sending the command. The movement strategy depends on the number of the legs of the walking locomotion system. This paper presents a locomotion system with four legs having adaptive hoofs.

incremental rotation encoders, that send the reject signal needed to achieve the programmed movements for walking, by a follow-up control system.

3. THE STABLE MOVEMENT CONDITION OF THE AHMP QUADRUPED The most frequently displacement mode of the AHMP is the walking. In the walking mode, the legs fulfill the movements with various movement rules of the ankle point C, called characteristic point. On smooth terrain surface, the legs movement rules are periodical repeated, after every 4 steps, executed in the stepping order succession. While one of the legs steps, the other three play the support role and ensure the platform advance (Kovacs, 2000). Every support legs combination determines, between the afferent C points on the terrain a support triangle. In order to ensure the stability while walking, the triangle transformation must be made in the area under the center of gravity (Frik and Amendt, 1995). The number of the support triangle changed during the platform advance with one step, is given by C~ = 4 . They are shown in Fig. 2, for the stepping order ~ - ®

- CD - @ . During stepping with one leg with a step length of L forward, the platform frame advances with T SApas = U4. The movement of the C extremity of support legs in respect of the frame is in opposite sense to the frame movement in respect of natural terrain, so consequently for one step duration they are equal to ASc sup = - U4. TSc pas =

t;

.

~ .5

'"

~

.5

4

0. fr 3

V;

E.s E ;;

2

~~

Fig. 2. AHMP - step-by-step movement schemas for o - @) - (j) - Q) stepping order: a) - lateral view; b) - top view

o

E

1 E 0 <.s

o~
-1

- .c ~.:: -2

The C joint is controlled by one torque sensor that is mounted in the hoof body. This sensor disconnect the brake, mounted on the motor wheel in the A joint of the ankle transmission, when the sole hoof is in contact with an inclined relief of the terrain. When the torque from C joint is canceled, the brake is connecting again, because the sole is in full contact with the terrain relief The connection of the brake authorizes the control system to continue the movement program to advance theAHMP. The electronic control system is placed on the frame and has an on board computer as central unit The control software is based on the mathematical algorithms, structured by specific movement sequence routines. The program's routines are elaborated off-line and loaded into the ROM memory. In order to execute the program routines, they are loaded into the RAM

f- 0

~ep

1

~ep

2

~ep

3

~ep4

Fig. 3. The legs extremity movement in respect of the frame of AHMP at four consecutive steps, with stepping order 0 - @) - (j) - Q) The composition of transport and relative movements are used to calculate the AScpas movement in respect of the frame, during the advance step of the C point of the walking leg: (I) ASc pas

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3U4 is obtained from relation (I). The

movements of the 4 legs extremities in respect of the frame, during the AHMP advance with L distance is shown in Fig. 3 histograms. In this figure the base unit is U4. It can be observed, that after one step sequence made by one of the legs with 3 units ahead of the frame, 3 support sequences follow and for each of the leg extremity C moves in opposite sense of the advancing direction only one unit The histograms emphasize the duration rule of the AHMP quadruped walking. According to this rule, between the stepping duration and the support duration of every leg, there must be a 113 ratio. The total duration to make one complete cycle is 4 time units. The duration rule ensures the closing of the C point trajectory in respect of frame, absolute necessary condition to maintain the legs united with the platfonn during walking. The algorithms for walking locomotion control program use the trajectory segmentation operation, which is based on this rule and is presented in next paragraph.

(3)

and the coefficients with the relations:

(4)

x 4. ALGORITHMS FOR THE LOCOMOTION SYSTEM CONFIGURATION This paragraph describes that a support polygon corresponds to every AHMP quadruped movement sequence. The quadruped cannot make another step, without bringing the support leg hoofs in the support polygon corners. The support polygon creation is called locomotion system configuration. To pass from stopped configuration to the turning configuration or to start of walking configuration or inverse, the movement of the involved legs into the configuration operation is similar to the movement that perfonns one step, but the support legs are maintained at relative rest. The programmed routines elaborated to implement the configuration operations into the control software are based on the leg's inverse geometrical model (Varga and Riidulescu, 200Ia). The Axz reference system is chosen with the origin into the hip joint, with the sense of abscise axis oriented into the frame moving direction and with Az axis oriented after the vertical line in A point towards the terrain (Fig. 4). The inverse geometrical model supposes that the support polygon's corner points P (xp, zp) are defined. The unknown variables are the orientation angles a and ~ of the thighbone and tibia elements. They are detennined for the C(x, z) leg's extremity, situated on the vertical line that crosses through the P support point, at r - hoof height from 0 reference level where the leg 's ankle joint is. It is specified that the level 0 is the reference level, which will be chosen at H distance from the hip joint. According to the algorithm (Varga and Riidulescu 200 I b) the angles are detennined with the relations: .

u

a=arcsln-

lAB

Il=arcsin~

H

Fig. 4. The leg's configuration based on the inverse geometrical model Knowing the lAB and lee lengths, the H height and the hoof height r, after detennining the C point coordinates with the relations: x = xp

z =zp -r = H-r

(5)

the algorithm is run in (4), (3), (2) relations order, to obtain the orientation angles a and ~ unique values. During the configuration, the leg's characteristic point goes through a fragmented trajectory (Fig. 5).

9-············································0 · . ···· ... ~ .~ ····· ... o·

(2)

loc

Fig. 5. A fragmented trajectory

the u and v function with the relations:

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h

This trajectory has the C I as the start point and the Cl as the stop point, in successive positions of the ankles joint, before and after the configuration operation. The fragmented trajectory contains the first rising segment CIQh the second horizontal movement segment QI02, and the third descending segment 02Cl . The start and stop points coordinates are detennined using the (5) relations, knowing the coordinates of the support points PI and Pl between which the transition is made. To obtain the QI and 02 coordinates, from Cl and Cl points if is translated the axis of the points with the h height as in Fig. 5. This algorithm associates to each coordinate pair, one angle pair. With the four pair of angles (ao, 13C1 ), (Cla, 130), (CIoh 1301), (002, ~) the angular range, needed in the hip joint and the knee joint is defined: ~aCIQI = aQI -acl

~aQIQ2 = aQ2 - aQI

~l3clQI = I3QI -l3cl

~I3QIQ2 = I3Q2 -I3QI

~aQ2C2 = aCl - aQ2

routine, is broke up the trajectory of the characteristic point in respect of the frame, into 2 branches. In the first branch is kept the stepping phase related part, and in the second the support phase related part. This operation is called brute segmentation and is presented in Fig. 6a. The branch passed in the stepping phase, contains the same segments CIQh QIQl, QlCl like the configuration trajectory that was discussed for the immobile supports. Therefore the afferent movements for this branch can be controlled also by routines based on the described algorithm. The branch passed during the support phase (ClC I segment), is passed in three sequences, which on duration rule has same lengths equal with U4. In the support phase of the leg's 8, the first sequence correspond to the ClC interval - when the leg ® is stepping, the second sequence correspond to the interval CC" - when the leg is stepping. After the brute segmentation a fine segmentation is made (Fig.6b). This operation divides each support segment in subintervals, i.e. CCQIo ~I~' ~lC" having the lengths proportional with the other segments CIQIo QIQl, Q l C 2 from the stepping phase branch (because they have constant speeds). According to the two segmentations, for the sub interval lengths, the following relations are obtained:

(6)

~I3Q2c2 = I3c2 -I3Q2

in order to reconfigure the system. Every configuration routines, used by the control software to change the walking method, are based to the same relations like (6).

3L14 r~-----A.

QI

h ·L ICCQI = IQ2c• = - - 8h+3L L h ·L 2 . 8h+3L ICQlcQ2

"'I

tolQ2

Q2

="4-

r········~~~···············Q: : : h ~----.,.

~t: '" 0 P-Po P-

;;

~ ~

<.> 0) '" c£..c:

EooP-

.0 0)

C

·c

t:.a

j l

Leg
; tclQl ~ : 0

Cl

:

C'

C"

These relations permit to calculate the co-ordination of the intermediary points ~Io ~, and with the inverse geometrical algorithm to deduce the joints ranges, needed to correlate each support leg movement with the other stepping leg movement These calculations must be repeated for each support interval and for each involved leg, in both the support phase and in the stepping phase. They use the same algorithm and software routine, only with changed dates (the coordinates of the trajectory points). Because, at the walking locomotion there are cyclical repeated movements, is predetermined off-line the joints ranges for the whole duration of the cycle and store these results in a database. In this case, the commands to generate the stepping movement are reduced to repeat readings the database and sending this movement information to the axis regulator to be executed.

to2C2

Cl

r"p" 'T"'(=l"'r"p"'j 80+<» 80+
80+®

a)

~ Ll4

U4

Ll4

talQ2

~

be2

telal

It

M

CQl

CQI

0 . . .. . . . . .0

C"

(7)

b)

C'

5. THE SOLE ROLE TO ASSURE THE AHMP STABILITY

Fig. 6. Breaking up the walking trajectory: a) - brute segmentation b) - fine segmentation

The sole hoof is necessary to increase the mobile platform stability. The stability can be examined if the stepping of the legs with sole hoof is compared with the stepping of the legs with claw hoof. In case of legs with claw hoof the P support point is

In the walking locomotion, the system is permanently reconfigured. This process depends on both the stepping leg' s movement and on support leg's movement. In order to formulate the algorithm, that permits to include these influences into the control

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found at the intersection between the vertical line, crossing by the characteristic point C, and the terrain (Fig. 4). The great pressure in this point, increase the claw adherence to the terrain. This is an advantage because it generates the greatest propulsion force, but is a disadvantage if the terrain is soft.

support triangle afferent to the start of stepping with the leg@. If the coordinates of the points P" G and P3 (Fig. 7a and 7b) are: A

XPI

a)

P4

~8

A x

PI

L

B

=2"-8

YPI

xG = 0

u~

X P3 = {

i

=+"2

YG = 0

~ - i)

Y P3 = -

(8)

~

these three points are situated on same line:

~B2 +(L/2~

~B2 +(A -

L/4)2

XPI

YPI

XG

YG

X P3

YP3

(9)

1=0

Also, if the coordinates of the points P2 , G and P4 (Fig. 7c and 7d) are:

b)

.-~-~--.Y

xp2

A L =---

YP2 = - -

xG

=0

YG

2

8

B

2

(10)

=0

B

YP4 = - -

2

these three points are situated on same line:

c)

(I I)

...--".......- - - - . Y In these four particular moments having the limits of instability and any perturbation in balance of outside forces can overturn the platform with claw hoofs. To avoid the overturn of platform the hoof with sole are used. The sole hoofs have a greater support surface than the claw hoofs, resulting in a reduced pressure on the terrain and in consequence reducing the propulsion force generated by friction. The stability of platform increases, because the soles of the support legs expand the support triangle, and the center of gravity G is made into the support triangles, in relation with line P IP3 or P2 P4 .

d)

0, Hip joint

Fig. 7 The support triangle positions of the AHMP with claw in instability moments An other disadvantage of the claw hoof consists in the

I, Thighbone

fact that, in the moment that it goes out from the support phase or goes in the new support phase, the diagonal support claws points and the AHMP center of gravity, G, arrive at same line. Fig. 7a represents the support triangle in the end moment of stepping with the leg Q) and Fig.7b represents the support triangle afferent to the start of stepping with the leg @). Fig. 7c represents the support triangle in the end moment of stepping with the leg ill and Fig. 7d represents the

2, Knee joint Fig. 8. Planetary transmission with timing belt Because the hoofs represent the last element of story

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mechanism of the leg, the sole orientation depends of the thighbone and tibia movement. Regarding the sole horizontal position (if the frame is also horizontal), in Fig. I is proposed a mechanical solution, based on the story synchronous transmission with timing belts and the hip wheel fixed on the frame through the brake. This is a story planetary transmission, where the hip wheel represents the central element and the wheels from knee joint and ankle joint represent the two satellites. Fig. 8 presents the portion between the hip joint 0 and the knee joint 2, where the thighbone is the satellite ann I. Using the Willis relation for this story planetary transmission the self-ratio is:

if the R support reaction, from the terrain on the hoof, is symmetrical or asymmetrical. The symmetrical reactions appear when stepping on a smooth terrain and do not cause torque around the knee joint. The reaction signal from the torque sensor has the value r = 0 and the relay breaks the current i of the brake. As a result, the brake blocks the transmission, and maintains invariably the hoof orientation. The asymmetrical reactions appear when stepping on the irregular terrain and cause torque around the knee joint. The reaction signal from the torque sensor has the value r #: 0 and the relay connects the brake on the current i. As a result the brake unblocks the transmission, and gives the possibility to modify the hoof orientation, till the sole is in full contact with the terrain relief.

and the relation is obtained:

6. CONCLUSION

. (1) .

102

Z2

iOl2 -I

z2 -zo

112=~=--­

• To assure the stability of the AHMP in the stepping phase, a lot of rules need to be respected. The most important rules are: the stepping order, the duration rule, the not coupled movement between the hoof, the thighbone and tibia and the adaptive put of the hoof on the terrain relief. • Some advantages of the mechanical solution of the not coupled movement between the hoof, the thighbone and tibia are the design simplicity and the reduced weight of the timing belt. • The adaptive hoof control with electromagnetic brake, controlled by torque sensor, represents a simple solution, easy to implement.

(13)

If the number of teeth Zo=Z2, i 12=oo. In this situation the satellite 2 orientations is invariable, indifferently to the thighbone movement. By making Z2=Z3, into transmission between knee and ankle, also, is obtained that the wheel 3 orientation does not depend to the tibia movement. As the element 2 and element 3 do not have a couple movement with the hoof, the sole keeps the initial orientation, there horizontally must be ensured relative to the hoof assembly, if the frame is in horizontal position. This solution gives to the mobile platfonn a good stability, but just on a smooth terrain.

REFERENCES Frik M., Amendt O. (1995) Neural Control of a Walking Robot in Variable Terrain, Proceedings ofl)'h World Congress on the Theory of Machine and Mechanisms , Milano, Volume nr. 3, page 2297-2301. Kovacs Fr., Varga St., Pau V. (2000). Introducere in Roboticii, page 85-101. Printech Publisher, Bucure~ti, ISBN 973-632-230-X. Manko D.J. (1992). A general model of legged locomotion on natural terrain, page 39-49. Kluwer Academic Publishers, Boston, ISBN 07923-9247-7 Radulescu C. (2000). Robocare ~i Sisteme de robocare, Mirton Publisher, Timi~oara, ISBN 973-585-258-6. Varga St., Radulescu C. (2001 a). Mobile Platfonn with Wheeled - Walking Locomotion System. In: The J(jh International Workshop on Robotics in Alpe-Adria-Danube Region, RAAD'OJ. paper Rd OIl, Vienna University of Technology Vienna. Varga St., Radulescu C. (2001 b). Un sistem de locomotie pe rotile pentru platfonne mobile. In: The 2'h National Workshop on mobile Robots. WMRC'OJ. Craiova, Romania, ISBN 973-804339-5.

r

g 5

L - - - - - - I 8" ~ o Cl)

E-tn

Fig. 9 Adaptive hoof remote control If the mobile platfonn move on the irregular terrain, it is inevitable to meet relief with varied inclination. In this case adaptive hoofs must ensure the mobile platfonn stability. The control scheme of the adaptive hoof position is presented in Fig.9. There are 2 situations, depending

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