on-line stable gait generation of a two-legged robot using a genetic–fuzzy system

Robotics and Autonomous Systems 53 (2005) 15–35

On-line stable gait generation of a two-legged robotusing a genetic–fuzzy system

Rahul Kumar Jha, Balvinder Singh, Dilip Kumar Pratihar∗

Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India

Received 23 June 2004; received in revised form 22 June 2005; accepted 28 June 2005Available online 10 August 2005

Abstract

Gait generation for legged vehicles has since long been considered as an area of keen interest by the researchers. Soft computingis an emerging technique, whose utility is more stressed, when the problems are ill-defined, difficult to model and exhibit largescale solution spaces. Gait generation for legged vehicles is a complex task. Therefore, soft computing can be applied to solveit. In this work, gait generation problem of a two-legged robot is modeled using a fuzzy logic controller (FLC), whose rule baseis optimized offline, using a genetic algorithm (GA). Two different GA-based approaches (to improve the performance of FLC)are developed and their performances are compared to that of manually constructed FLC. Once optimized, the FLCs will be ableto generate dynamically stable gait of the biped. As the CPU-time of the algorithm is found to be only 0.002 s in a P-III PC, thealgorithm is suitable for on-line (real-time) implementations.© 2005 Elsevier B.V. All rights reserved.

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eywords: Two-legged robot; Dynamic stability; On-line; Staircase ascending gait; Genetic–fuzzy system

. Introduction

During locomotion, a legged robot plans both itsath as well as gait (the sequence of leg movement)imultaneously. To ensure a statically stable gait of a

Abbreviations: DSM, dynamic stability margin; FLC, fuzzy logicontroller; GA, genetic algorithm; NL, negative large; NS, negativemall; PS, positive small; PL, positive large; ZMP, zero moment point∗ Corresponding author. Tel.: +91 3222 282992;

ax: +91 3222 282278.E-mail addresses: rkjha [email protected]

R.K. Jha), [email protected] (B. Singh),[email protected] (D.K. Pratihar).

multi-legged robot, its projected center of gravity (Cshould lie within its support region, which is a convhull passing through its supporting feet[1]. It is to benoted that a statically stable gait is generally considfor a robot having four or more legs.

A two-legged robot should be dynamically banced, during which the projected CG of the vehmay not even lie on the support region. To ensdynamic stability, a parameter called dynamic stabmargin (DSM) is calculated based on the concepzero moment point (ZMP)[2,3]. The ZMP is a poinlying on the ground about which the summationall moments becomes equal to zero. It represen

921-8890/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.robot.2005.06.006

16 R.K. Jha et al. / Robotics and Autonomous Systems 53 (2005) 15–35

Nomenclature

dZMP distance of zero moment point from theankle joint

F ground reaction forceFN normal reaction forceFT friction forceg acceleration due to gravityH heightL lengthL1, . . . , L7 link lengthst time stepVmax maximum velocity of the swing legsh height of the staircasesw width of the staircasefs length of the foot

Greek lettersθ joint angleτ torque

point, at which the ground reaction force is beingapplied. Mitobe et al.[4] presented an algorithmfor controlling the angular momentum of walkingrobots through the manipulation of zero momentpoint (ZMP). To achieve a stable motion, the ZMPhas to follow a certain trajectory. But, the ZMP maydeviate from the prescribed trajectory, due to thedisturbances or tracking errors which influence thestability of the robot. Therefore, the deviation of ZMPfrom the prescribed trajectory has to be compensatedThus, the ZMP was considered as an actuating signaof the controller and a feedback control law wasproposed to control the angular momentum of walkingrobots.

Seo and Yoon[5] proposed the design of a robustdynamic gait of the biped using the concept of dy-namic stability margin. According to them, gait failureoccurred because of the discrepancy between the designed and the actual gait motions, which contributedto the changes in body forces. Thus, a gait can be considered to be robust, if it can sustain a fair magnitude oflinear impulse applied at the mass center of trunk, in thehorizontal direction, for a certain amount of time. So,the minimum magnitude of that linear impulse was de-

fined as the dynamic stability margin and the dynamicgait for a five link planar biped robot was designed bymaximizing the dynamic stability margin. A parametercalled foot strike time margin, representing the readi-ness of the foot strike, was also defined by them, whichwas supposed to have a close positive correlation withthe dynamic stability margin. A robust gait with respectto the external disturbances was obtained by maximiz-ing the foot strike time margin.

Goswami[6] solved the problem of robot stabilityusing the concept of foot rotation indicator (FRI) point,which is a point on the foot/ground contact surface,where the net ground reaction force would have to actto keep the foot stationary. To ensure no foot rotation,the FRI point must remain within the convex hull offoot support area. The concept of FRI point was usedduring single support phase and was generally treatedfor generating dynamically stable gait.

Although the above methods lay down the founda-tion of the study of dynamically balanced two-leggedrobot, they may not be suitable for on-line (real-time)implementations, due to their high computational com-plexity. Thus, suitable locomotion algorithms (adaptivein nature) are to be developed, which can negotiate un-known terrain also, on-line.

Soft computing is an emerging technique, whichconsists of fuzzy logic, neural network, genetic algo-rithm, etc. and their different combinations, can handlereal-world complex problems. It has also been used

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by various researchers, for control of humanoid robo[7]. Some of the works related to real-time control otwo-legged robot using soft computing are discussebelow.

Capi et al.[8] developed a method based on genetalgorithm to generate a human-like motion. Humanorobot gait was generated using two different cost funtions: minimum consumed energy (CE) and minimumtorque change (TC). In real-time situations, the robhas to change its gait according to the conditionsthe terrain. But, as genetic algorithm is a time consuming tool, it was used to generate feasible optimal gaitwhich were used to teach a radial basis function neunetwork. The neural network after getting trained waused for real-time gait generation.

Salatian and Zheng[9] carried out gait synthesis fora biped robot climbing sloping surfaces by using a neral network, in which an on-line learning was adoptedThey were successful in developing the gait pattern b

R.K. Jha et al. / Robotics and Autonomous Systems 53 (2005) 15–35 17

only the static stability of the robot was considered intheir work.

Park[10] developed a fuzzy logic-based ZMP tra-jectory generator, in which the leg trajectory was usedas the input. The effectiveness of his algorithm wastested through computer simulations. The ZMP trajec-tory generated by his algorithm was able to increasethe stability of the locomotion. The main drawback ofhis algorithm lies in the fact that the fuzzy logic-basedcontroller may not be optimal, as no optimizer, suchas a genetic algorithm or other tool, is used along withit.

Zhou and Meng[11] proposed a fuzzy reinforce-ment learning architecture for dynamic balance of atwo-legged robot. Through simulation analysis, theydemonstrated that it is possible for the robot to startwith some heuristic knowledge and gradually improvethe learning rate. Thus, the robot will learn to improveits performance.

Miller [12] used an NN-based on-line learning tech-nique, to tackle dynamic balance of a two-legged robot.

In the present work, an attempt is made for on-linegait generation of a two-legged robot, while going upthrough the staircases, by maintaining its dynamic sta-bility. In this paper, a genetic–fuzzy system, in whichthe performance of a fuzzy logic controller is improvedby using a GA-based tuning (off-line)[1], is developedto solve the said problem.

The rest of the manuscript is organized as follows:

in

he

for

do

rr

ofts

is

Fig. 1. A schematic diagram of a two-legged robot having 7 d.f.

assumed to be equal to 0.05 m. The stability is checkedonly along the direction of movement, but not towardsits left or right sides. The trajectory of the swing footof the robot is assumed to be a cubic spline as shownin Fig. 2, which can be expressed by the followingequation:

z = c0 + c1x+ c2x2 + c3x

3, (1)

wherez is the height of the foot of the swing leg fromthe ground at a distancex from the starting point ofeach step andc0, c1, c2 and c3 are the coefficients,whose values are found out using the followingboundary conditions:

• atx = 0, z = 0,• atx = sw − x1 − fs/2, z = sh + fs/2,

Section2 explains the mathematical formulation of theproblem, our proposed algorithm has been describedSection3. Results are presented in Section4. A com-parative study between the proposed algorithm and tpreviously developed algorithm is made in Section5.Some concluding remarks are made and the scopefuture work is discussed in Section6.

2. Mathematical formulation

In the present work, the bipedal robot is considereto have seven degrees of freedom (three at hip, twat the knees and two at the ankles).Fig. 1 showsthe schematic diagram of the bipedal robot undeconsideration. The robot consists of a trunk, two uppelegs, two lower legs and two ankles. The massesthe limbs are assumed to be lumped at some poinon the corresponding limbs. The length of each foot


Fig. 2. A schematic diagram showing cubic polynomial trajectory of the swing leg and the forces acting on the robot at the ZMP.

• atx = 2sw − x1, z = 2sh + fs/2,• atx = 2sw − x1 + x3, z = 2sh.

It is important to note that in a cycle, six intermediatepositions are considered in-between two double sup-port phases.

The variation of velocity of the swing leg withtime over each step is shown inFig. 3. The maximumduration of each time step is assumed to be equal to5.0 s and the maximum velocity of the swing leg (Vmax)is set equal to 0.056 m/s. Each time step consists of anaccelerating part during the first 1 s, and a deceleratingpart for the last 1 s and a duration for constant velocitymovement of the swing leg during the remainingperiod. The duration of the constant velocity partvaries depending upon initial and final positions of theankle joints from the edges of the stairs (refer toFig. 2).

The motion of the biped robot is consideredto be composed of a single support phase and aninstantaneous double support phase[4]. Fig. 4 showsthe posture of the robot during a single support phase.The lumped masses of different limbs are shown in thecorresponding limbs, in this figure. The gait generatedduring locomotion must be dynamically stable. Formaintaining dynamic stability, the zero moment point(ZMP) [2,3] should not lie outside the safe zone orthe sole of the supporting foot. The friction betweenground and foot is assumed to be sufficient enough toprevent the slipping.Fig. 2shows the forces acting onthe robot and the ZMP. The distance of ZMP from theankle joint can be determined as follows:

dZMP = A

B, (2)


Fig. 3. Velocity distribution of the foot of the swing leg in a step.

whereA =∑7i=1(Iiωi +mixi(g− zi) −mixizi), B =∑7

i=1mi(zi − g), ωi is the angular acceleration of theith link, xi indicates the acceleration ofith link in thex-direction, ¨zi is the acceleration ofith link in z-direction,(xi, zi) is the coordinate ofith lumped mass,Ii denotesthe moment of inertia ofith link, andmi is the mass ofith link.

During single support phase, the zero moment pointmust lie within the safe length of the foot, so that thecontact between the foot and the ground will be main-tained. But, if the ZMP does not lie within the safezone, the torque at the hip, knee and ankle joints ofthe supporting leg must be updated to move the ZMPto the safe zone, in order to achieve the dynamic sta-bility. Dynamic stability margin is defined as the dis-tance of the ZMP from the boundaries of the sup-porting polygon (of the ground leg).Fig. 2 shows theschematic diagram used for calculating the dynamicstability margin. The dynamic stability margin is deter-mined by using the expression (L1/2 − |dZMP|). Thus,its value will lie between 0.0 and 0.025 m for sta-ble gait. ConsideringFN andFT, to be the foot pres-sure and friction force at the ZMP, respectively (re-fer to Fig. 2), we can write the torque equations asfollows:

τhip = FThhip + FNδhip,

τknee = FThknee+ FNδknee,

τankle = FThankle+ FNδankle, (3)

wherehhip = l6 cosθ6 + l5 cosθ5 is the height of hipjoint from the ground, andδhip = dZMP − l6 sinθ6 −l5 sinθ5 is the distance of hip joint from the zero mo-ment point. Similarly, we can also write the follow-ing expressions as follows:hknee= l6 cosθ6, δknee=dZMP − l6 sinθ6, hankle = 0, δankle = dZMP.

AssumingFT = 0.5FN, where 0.5 is the coefficientof friction between the foot and the ground, the updatedtorque at the hip, knee and ankle joints required to shiftthe ZMP to the safe zone can be calculated by usingEq.3.

The problem considered in this paper, may be statedas follows.

A two-legged robot will have to ascend the staircasesduring its locomotion, after maintaining the maximumdynamic stability. Thus, it can be formulated as an op-timization problem as follows:

MaximizeF = L1

2−∣∣∣∣AB∣∣∣∣ , (4)


Fig. 4. A schematic diagram showing posture of the robot during asingle support phase.

whereL1 is the length of foot of the ground leg, andthe termsA andB have been defined earlier.

3. Proposed algorithm

Staircase ascending gait generation problem of thetwo-legged robot has been solved by using the fuzzylogic controllers[13]. The flowchart of the proposedalgorithm is shown inFig. 5. This problem has beensolved by using two FLCs—the first one helps to calcu-late the joint angles in double support phase and the second FLC determines the necessary variations in jointangles in a single support phase. Thus, both the fuzzylogic controllers take part in maintaining the dynamic

stability of the robot. The gait of the biped robot de-pends upon the initial and final distances of the anklejoint of the swing foot from the edges of staircases (dis-tancesx1, x2 andx3 as shown inFig. 2). Based on inputvalues,x1 andx2, the first fuzzy logic controller deter-mines the horizontal distanceL of the hip joint fromthe ankle joint of the would-be swing foot and heightH of the hip joint from the stair at which the said swingleg is resting (refer toFig. 6) for the time-being. Valuesof L and H thus determined are then used to calcu-late the joint angles during the double support phase.The mathematical expressions used to determine thejoint angles during the double support phase can beexpressed as follows:

θ2 = arcsin

(1

L2(L− L3 sinθ3)

), (5)

θ3 = arccos

(L2

3 + L2 +H2 − L22

L3√L2 +H2

)+ φ, (6)

whereφ = arctan(L/H). The values ofθ2 andθ3 canbe determined from Eqs.(5) and (6), respectively. Sim-ilarly, θ5 andθ6 can also be determined from the ge-ometry.

During the single support phase, the changed valuesof the joint angles of the swing leg are fed as inputs tothe second fuzzy logic controller. The joint angles ofthe swing leg are calculated using following equations:

d-

ills ofof.

-

θ2 = arcsin

(c1L3 sinψ + c2(L2 + L3 cosψ)

(L2 + L3 cosψ)2 + (L3 sinψ)2

),

(7)

where c1 = L5 cosθ5 + L6 cosθ6 + sh − z = L2 cosθ2 + L3 cosθ3, c2 = L5 sinθ5 + L6 sinθ6 + sw − x1+ x2 − x = L2 sinθ2 + L3 sinθ3, ψ = θ2 − θ3 =arccos((c2

1 + c22 − L2

2 − L23)/2L2L3).

Assumingθ5 andθ6 to be the same as those consiered during the double support phase,c1, c2 andψ canbe calculated and henceθ2. Thus, knowingθ2 from Eq.(7),θ3 can be calculated using the relationθ3 = θ2 − ψ.

The changes in angles of the swing leg, i.e.,�θ2 and�θ3 are fed as inputs to the second FLC, which wdetermine the changes to be made in the joint anglethe main body, supporting leg and in the ankle jointswing leg, to maintain dynamic stability of the vehicle


Fig. 5. Flowchart of the proposed algorithm.

All these changes in joint angles are implemented bythe motors mounted at different joints. The torques tobe realized by the motors are calculated using dynamicequation of motion, as given below

A(θ) × θ + B(θ) × h(θ) + C × g(θ) = Dτ, (8)

θ = ( θ1 θ2 θ3 θ4 θ5 θ6 θ7 )T,

τ = ( τ1 τ2 τ3 τ4 τ5 τ6 τ7 )T,

h(θ) = ( θ21 θ2

2 θ23 θ2

4 θ25 θ2

6 θ27

)T,

g(θ) = ( sinθ1 sinθ2 sinθ3 sinθ4 sinθ5 sinθ6 sinθ7 )T,

A(θ) = qij cos(θi − θj), B(θ) = qij sin(θi − θj),

C = −diag(hi),

D =

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

,

where q11 = I1 +m1r21, q21 = m2L2r1, q22 = I2 +

m2(L2 − r2)2 +m1L22, q33 = I3 +m3(L3 − r3)2 +

(m2 +m1)L23, q32 = m1L2L3 −m2L3(L2 − r2), q31

= m1L3r1, q44 = I4 +m4r24, q43 = q42 = q41 = 0,

q55 = I5 +m5r25 + (m4 +m3 +m2 +m1)L2

5, q54 =m4L5r4, q53 = −m3L5(L3 − r3) + (m2 +m1)L3L5,

q52 = m1L5L2 −m2L5(L2 − r2), q51 = m1L5r1,

q66 = I6 +m6r26 + (m5 +m4 +m3 +m2 +m1)L2

6,

q65 = m5L6r5 + (m4 +m3 +m2 +m1)L6L5, q64 =m4L6r4, q63 = −m3L6(L3 − r3) + (m2 +m1)L6L3,

q62 = m1L6L2 −m2L6(L2 − r2), q61 = m1r1L6,

q77 = I7 +m7r27, q76 = q75 = q74 = q73 = q72 =

q71 = 0, h7 = m7r7g, h6 = (m6r6 + (m5 +m4 +m3+m2 +m1)L6)g, h5 = (m5r5 + (m4 +m3 +m2 +m1)L5)g, h4 = m4r4g, h3 = (−m3(l3 − r3) + (m2 +m1)l3)g, h2 = (−m2(l2 − r2) +m1L2)g, h1 =m1r1g.

It is important to mention that if the FLCs fail toregain the dynamic stability of the robot by changingthe joint angles, the updated torques (calculated usingEq.(3)) can be applied to the hip, knee and ankle jointsto shift the ZMP to the safe zone. It is also to be notedthat if the dynamic stability is not maintained even afterupdating the joint torques, an external support might berequired to regain it (which is not studied in this work).

In the present work, three different approaches areimplemented for developing the FLC, which are ex-plained below.


Fig. 6. A schematic diagram showing inputs and outputs of the firstfuzzy logic controller.

Table 1Manually constructed rule base of the first FLC

x1 x2 L H

Low Low Mid LowLow Mid Low LowLow High High HighMid Low High LowMid Mid High HighMid High Low MidHigh Low Low LowHigh Mid High MidHigh High High High

Approach 1 (Manual design of FLC). Based on intu-ition, the authors have designed the membership func-tion distributions of the input and output variablesof the FLCs. Moreover, the rule bases of the FLCshave been developed based on the authors’ knowl-edge of the process to be controlled. The member-ship function distributions for the different variablesof the first FLC are shown inFig. 7. As there are threelinguistic terms for each input of the first fuzzy logiccontroller, there is a maximum of 3× 3 = 9 possiblerules.

Table 1shows the rule base of the first FLC, usedfor determining two outputs, namelyL andH. Thus, atypical rule (refer to first rule ofTable 1) of the firstFLC can be written as follows: IFx1 is Low AND x2is Low, THEN L is Mid, H is Low.

Similarly, membership function distributions of thevariables of the second FLC are displayed inFig. 8and

Fig. 7. Author-defined membership function distributions
for the inputs and outputs of the first FLC—Approach 1.


Fig. 8. Author-defined membership function distributions for the inputs and outputs of the second FLC—Approach 1.

its rule base (consisting of 16 rules) for determiningthree different outputs are shown inTable 2.

Thus, the first rule ofTable 2can be read as follows:IF�θ2 is NL AND �θ3 is NL, THEN�θ4 is NL,�θ5is NL,�θ6 is NS and�θ1 is PS.

In this approach, no optimization is carried out totune the membership function distributions or the rulebase of the FLCs.

Approach 2 (GA-based tuning of manually con-structed FLC). In this approach, membership functiondistributions of the variables (shown inFigs. 7 and 8)and the manually designed rule base of the FLC (shownin Tables 1 and 2) are optimized simultaneously (as theperformance of an FLC depends on the membershipfunction distributions of the variables and the rule base[1]), to improve its performance. The total number ofrules of the FLCs is 25 (9 for the first FLC and 16 for thesecond FLC). The membership function distributionsof a variable are assumed to be symmetrical triangles.Moreover, the membership function distributions forx1

andx2 have been assumed to be similar, as both of themindicate the distance. Thus, there are 9 real variablesand each real variable is represented by 5 bits. Thus,the GA-string consists of 15+ 9 + 30+ 16 = 70 bits,

Table 2Manually constructed rule base of the second FLC

�θ2 �θ3 �θ4 �θ5 �θ6 �θ1

NL NL NL NL NS PSNL NS NL NL PL NSNL PS NS NS NL NSNL PL PL NS NS PLNS NL PL PL NL NSNS NS PS PS NS PLNS PS NL PS NL PLNS PL PL PS NL NLPS NL PL NL PS NLPS NS PS NS NS PLPS PS NS NS PS PLPS PL PL NL NL PLPL NL NS NL NS NSPL NS NL PL NS NSPL PS PS NS NS NLPL PL PL NS PS NL


which will look like the following:

1110111001001. . .10︸︷︷︸DB of first FLC (15 bits)

, 01101. . .01︸︷︷︸RB of first FLC

,

0110111001011. . .01︸︷︷︸DB of second FLC (30 bits)

, 0110. . .01︸︷︷︸RB of second FLC

.

The first 15 bits will carry information of the real vari-ables of the first FLC, the next 9 bits represent thepresence or absence of its rules (1 and 0 indicate thepresence and absence of a rule, respectively). Simi-larly, the bits starting from 25th position to 54th po-sition will represent real variables of the second FLCand its rule base is indicated by the last 16 bits. A ge-netic algorithm (GA) helps in finding the optimizedmembership function distributions of the variables andgood rules, so that the robot can maximize the dy-namic stability margin. During optimization, the halfbase-widths of different triangles representing mem-bership function distributions of the inputs and outputsx1, H, L, �θ2, �θ3, �θ1, �θ4, �θ5, �θ6 are consid-ered as the real variables, whose ranges of variationare kept fixed to (0.001, 0.05), (0.001, 0.11), (0.001,0.11), (0, 20), (0, 20), (0, 20), (0, 20), (0, 14)(0, 15),respectively.

Approach 3 (Automatic design of FLC using a ge-eof

ch1

fe

-rhtc

its

1110110011. . .10︸︷︷︸DB of first FLC (15 bits)

, 0110. . .01︸︷︷︸RB of first FLC (9 bits)

,

0110111001. . .01︸︷︷︸Consequent part, first FLC (36 bits)

, 1110111010. . .01︸︷︷︸DB of second FLC (30 bits)

,

1111. . .011︸︷︷︸RB of second FLC (16 bits)

,

1111001101. . .011︸︷︷︸Consequent part, second FLC (128 bits)

.

It is to be noted that during optimization, the ranges ofvariation for different real variables have been kept thesame, as those inApproach 2.

3.1. Fitness of a GA-solution

Two hundred and sixteen training cases are createdby taking six equi-spaced values for each ofx1, x2 andx3. Each case is solved separately by using the twoFLCs. The objective function value (i.e.,F) is calcu-lated for each of the training cases. The average ob-jective function value is treated as the fitness of a GA-solution. A high penalty (equal to−100) is added to thefitness value, if the GA-string gives infeasible or un-stable solution for any training case. Thus, the fitnessis calculated as follows:

fitness=∑216i=1Fi

216. (9)

4. Results and discussion

The effectiveness of the proposed algorithm is testedthrough computer simulations. Different parametersfor the limbs are given inTable 3.

The performance of a GA depends on its parameters,such as crossover probability, mutation probability and

Table 3

netic algorithm). In this approach, the GA takes thresponsibility of designing a good knowledge basean FLC. For the first FLC, there are two outputs (L andH) and each output has three linguistic terms. Ealinguistic term is represented by two bits (00 and 0for Low, 10 for Mid and 11 for High). Thus, there arefour bits representing the two outputs for each rule othe first FLC. Similarly, for the second FLC, there arfour outputs (�θ4,�θ5,�θ6,�θ1) and each output hasfour linguistic terms. For this case also, each linguistic term is represented by two bits (00 for NL, 01 foNS, 10 for PS and 11 for PL). Thus, there are eigbits representing four outputs for each rule of the seond FLC. The number of rules for the first FLC is 9and that for the second FLC is 16. As inApproach2, there are 9 real variables represented by 5 beach. Thus, the GA string will be 15+ 9 + 9 × 4 +30+ 16+ 16× 8 = 234 bits long, which will look asfollows:

-

Parameters for different limbs of the two-legged robot

Limb m (kg) L (m) r (m) I (kg m2)

1 0.5 0.05 0.01667 0.0004172 2.0 0.34 0.20000 0.0210673 5.0 0.30 0.24000 0.0780004 14.0 0.28 0.10000 0.1138675 5.0 0.30 0.24000 0.0780006 2.0 0.34 0.20000 0.0210677 0.5 0.05 0.01667 0.000417


Table 4Optimized rule base of the first FLC—Approach 2

x1 x2 L H

Low Low – –Low Mid Low LowLow High High HighMid Low High LowMid Mid High HighMid High – –High Low Low LowHigh Mid High MidHigh High High High

population size. Moreover, it is dependent on the maxi-mum number of generations, the GA is allowed to run.A parametric study is carried out to determine the op-timal set of GA-parameters, by varying one parameteronly at a time and keeping the other parameters fixed.Results of the parametric study carried out during thetraining (considering 216 training cases) forApproach2 are shown inFig. 9. The optimal GA-parameters forApproach 2are found to be as follows:

• Crossover probability: 0.8,• Mutation probability: 0.01,• Population size: 60,• Maximum number of generations: 120.

Optimized membership function distributions for theinputs and outputs of the first and second FLCs asyielded byApproach 2are shown inFigs. 10 and 11,respectively. Optimized rule bases of the first and sec-ond FLCs, obtained by usingApproach 2are shown inTables 4 and 5.

Similarly, the parametric study is also carried out forApproach 3and the following optimal GA-parametersare obtained:

• Crossover probability: 0.8,• Mutation probability: 0.05,• Population size: 130,• Maximum number of generations: 180.

F c-t oft d byA theos

Table 5Optimized rule base of the second FLC—Approach 2

�θ2 �θ3 �θ4 �θ5 �θ6 �θ1

NL NL – – – –NL NS NL NL PL NSNL PS – – – –NL PL – – – –NS NL PL PL NL NSNS NS PS PS NS PLNS PS – – – –NS PL – – – –PS NL PL NL PS NLPS NS PS NS NS PLPS PS NS NS PS PLPS PL – – – –PL NL NS NL NS NSPL NS NL PL NS NSPL PS PS NS NS NLPL PL PL NS PS NL

has identified 7 good rules, whereas 10 rules are foundto be good for the second FLC.

For each approach, the variations ofθ values (i.e.,θ1, . . . , θ7) with respect to time (cycle time) are de-termined. This information (regarding variations ofθvalues with respect to time) is required to calculate thejoint torque values. It is to be noted that anti-clockwisetorque has been considered as negative and clockwisetorque is assumed to be positive.Fig. 14 shows thevariations ofθ values with time.

Twenty-two test cases are selected at random andfuzzy logic controllers as obtained byApproaches2 and 3are tested on them.Approach 2is able togive feasible solutions for 19 cases, whileApproach3 has yielded feasible solutions for 21 cases, out of22. Results of computer simulations obtained usingApproaches 1–3, for the test scenarios (each consistingof eight steps) are shown inTable 8.

Table 6Optimized rule base of the first FLC—Approach 3

x1 x2 L H

Low Low High HighLow Mid Low MidLow High High HighMid Low – –Mid Mid Low LowMid High – –High Low High HighHigh Mid High HighHigh High High High

igs. 12 and 13show the optimized membership funion distributions of the input and output variableshe first and second FLCs, respectively, as obtainepproach 3. Using these optimal GA-parameters,ptimal rule bases are found out forApproach 3ashown inTables 6 and 7. For the first FLC,Approach 3


Fig. 9. Results of parametric study forApproach 2: (a) fitness vs.pc, (b) fitness vs.pm, (c) fitness vs. population size, (d) fitness vs. maximumnumber of generations.


Fig. 10. Optimized membership function distributions for the inputs and outputs of the first FLC—Approach 2.

Approach 1is unable to give stable solution for allthe test cases and for each case, the swing leg of therobot is seen to hit the staircase. It has happened due tothe fact that the manually designed FLCs may not havethe appropriate KB. InApproach 2, feasible solutionsare obtained in 19 test cases out of 22 and in the re-

maining 3 cases, the robot is found to hit the staircase.Thus,Approach 2has outperformedApproach 1in 19cases and in the remaining 3 cases, their performancesare found to be almost the similar. InApproach 1, theKB of the FLCs are designed manually, which may notbe optimal in any sense, whereas a GA is used to tune

ions for
Fig. 11. Optimized membership function distribut the inputs and outputs of the second FLC—Approach 2.


Fig. 12. Optimized membership function distributions for the inputs and outputs of the first FLC—Approach 3.

the KB of the FLCs inApproach 2. Thus, inApproach2, an optimal KB for each FLC is developed and asa result of which, the FLCs are seen to perform bet-ter than those ofApproach 1. Approach 3has yielded21 feasible solutions out of 22 and it is found to failin only one test case.Approach 3is found to performbetter thanApproach 1in 21 cases. Moreover, out of

22 test cases,Approach 3has outperformedApproach2 in 12 cases (in terms of average dynamic stabilitymargin), whereas it has been defeated byApproach 2in the remaining 10 cases. A close watch on the fail-ure cases ofApproaches 2 and 3reveal the followinginformation. While usingApproach 2, in 10th, 15th,18th scenarios ofTable 8, the robot is found to hit the

ions for
Fig. 13. Optimized membership function distribut the inputs and outputs of the second FLC—Approach 3.


Table 7Optimized rule base of the second FLC—Approach 3

�θ2 �θ3 �θ4 �θ5 �θ6 �θ1

NL NL PS NS PL NSNL NS – – – –NL PS – – – –NL PL – – – –NS NL NS NL PL NSNS NS PS NS PS NLNS PS – – – –NS PL PS PS NL NLPS NL PS NS NL NSPS NS PL NS PL PLPS PS NL PL NS NSPS PL – – – –PL NL NS PS NS NLPL NS – – – –PL PS NS PL NL NSPL PL PS PL NL PS

staircase, so it is considered as a complete failure caseand there is no chance of further improvement, in termsof stability margin. Moreover, in 14th scenario (refer toTable 8), Approach 3is unable to yield the dynamicallystable gait but the stability can be regained by modify-

ing the hip, knee and ankle joint torques of the supportleg. The modified torques are calculated for the abovejoints and these are coming to be equal to 85.448369,−7.433359, 7.11225 (in N m) for the hip, knee andankle joints, respectively, which are lying within theranges of the motors. It is important to note that the ac-tual cycle time for bothApproaches 2 and 3are seen tobe equal to each other (if there exists a feasible solutionfor each approach) and it happens due to the fact thatthe cycle time depends on the values ofx1, x2 andx3.

Results of a particular test case (say, fourth test caseof Table 8), in which x1, x2 and x3 are set equal to0.074, 0.055, 0.082 (all in m), respectively.Figs. 15and 16show the optimal gait pattern generated by thetwo-legged robot by usingApproaches 2 and 3, respec-tively. In the above figures, gait patterns are shown ateight positions (i.e., double support phases at the begin-ning and end of the cycle and single support phases atsix intermediate positions). Out of these eight positions,Approach 2has given higher values of dynamic stabil-ity margin at four positions compared to those yieldedby Approach 3, whereasApproach 3has shown bet-ter performance (in terms of dynamic stability margin)

Table 8Simulation results obtained using three approaches for some test scenarios

Scenarios Approach 1 Approach 2 Approach 3

Average dynamic stabilitymargin (m)

Traveltime (s)


Traveltime (s)


Traveltime (s)

1 Unstable – 0.01637 82 Unstable – 0.01782 53 Unstable – 0.01940 84 Unstable – 0.02023 75 Unstable – 0.01759 66 Unstable – 0.01701 17 Unstable – 0.01207 88 Unstable – 0.01342 29 Unstable – 0.01822 5

10 Unstable – Unstable 411 Unstable – 0.01498 512 Unstable – 0.01614 613 Unstable – 0.01207 114 Unstable – 0.0134215 Unstable – Unstable 816 Unstable – 0.02078 217 Unstable – 0.01321 518 Unstable – Unstable 919 Unstable – 0.02083 720 Unstable – 0.01954 921 Unstable – 0.01783 322 Unstable – 0.01713 4

3.58688 0.01582 3.58684.17375 0.01912 4.17373.31548 0.01926 3.31543.74107 0.02084 3.74103.88506 0.01881 3.88503.84121 0.01948 3.84124.18808 0.01586 4.18804.09002 0.01030 4.09003.62395 0.01808 3.6239– 0.01900 3.98573.77495 0.01186 3.77493.25306 0.01474 3.25303.95511 0.01179 3.95514.19358 Unstable –– 0.01739 3.92613.57342 0.02013 3.57344.43175 0.01695 4.4317– 0.01681 3.97873.75957 0.02136 3.75953.91499 0.01737 3.91493.83703 0.02021 3.83704.12924 0.01929 4.1292


Fig. 14. Variation of joint angles with time.

compared toApproach 2, in the remaining four posi-tions. However,Approach 3has given a higher valueof average dynamic stability margin compared to thatof Approach 2. Fig. 17 shows the trajectory of ZMPin a cycle time, as obtained by usingApproaches 2and 3. Dynamic stability margin decreases with the in-crease ofdZMP. Thus,dZMP should be as minimum as

possible (in terms of numerical value). It is interest-ing to note thatdZMP lies within 20% (on both sides)of its maximum possible value during the period of2.5 s (approximately) (where the cycle time is foundto be equal to 3.74107 s) inApproach 3, whereas inApproach 2, it is found to lie within this limit for aperiod of only 1.5 s approximately. Thus,Approach 3has proved his supremacy overApproach 2. For thistest case, the joint torque values (τ1, τ2, . . . , τ7) as ob-tained usingApproaches 2 and 3are shown inTables9 and 10, respectively. In both the approaches, the dif-ferent joint torque values are found to lie within theirranges and there is no abrupt change in torque values.

Approach 3has shown a slightly better performancecompared toApproach 2. In Approach 3, the wholetask of designing the optimal KB of the FLCs is givento the GA, which has carried out an extensive search tofind the optimal KB of the FLCs. InApproach 2, theresponsibility of developing the optimal KB is sharedby the designer and the GA, whereas inApproach 3,the responsibility is given to the GA alone. The betterperformance ofApproach 3compared toApproach 2could be due to the reason that the designer is unableto foresee the physical problem completely and whichis done by the GA through its exhaustive search.

Once optimized, the FLCs can be used to solve theproblem in an optimal sense. The CPU time of the algo-rithm is coming out to be equal to 0.002 s, in a P-III PC.Thus, it might be suitable for on-line implementations.

5

gedr lgo-r ork

Table 9Joint torque values (in N m) for the fourth test case ofTable 8—Approach

Startingposition

Firstintermediateposition

Secondintermediateposition

Thirdintermediateposition

τ1 −0.069 −0.107 −0.088 −0.085τ2 0.839 0.312 0.684 0.725 18τ3 0.075 1.681 3.518 4.269 13τ4 −0.267 0.345 1.040 0.811 83τ5 54.086 55.333 57.650 59.137 .001τ6 −54.173 −52.221 −50.513 −50.554τ7 3.168 0.381 2.260 2.170

. Comparison with others’ work

Real-time gait generation problems of a two-legobot had been tackled by using a genetic aithm (GA) and a radial basis function neural netw

2

Fourthintermediateposition

Fifthintermediateposition

Sixthintermediateposition

Finishingposition

−0.083 −0.081 −0.078 −0.0830.611 0.427 0.219 0.24.488 4.401 4.052 4.30.851 1.104 1.273 1.258.625 56.988 55.793 56

−50.430 −49.361 −48.354 −48.5501.174 0.207 −0.845 −0.637


Fig. 15. Optimal gait generated by the robot usingApproach 2for fourth test case ofTable 8.


Fig. 16. Optimal gait generated by the robot usingApproach 3for fourth test case ofTable 8.


Table 10Joint torque values (in N m) for the fourth test case ofTable 8—Approach 3

Startingposition

Firstintermediateposition

Secondintermediateposition

Thirdintermediateposition

Fourthintermediateposition

Fifthintermediateposition

Sixthintermediateposition

Finishingposition

τ1 −0.069 −0.106 −0.089 −0.084 −0.083 −0.083 −0.081 −0.084τ2 0.851 0.294 0.670 0.708 0.593 0.396 0.180 0.153τ3 −0.105 1.912 3.484 4.047 4.294 4.299 4.028 4.141τ4 −0.156 0.222 −0.164 0.131 0.413 0.664 0.904 0.866τ5 53.489 55.441 55.878 54.201 52.227 50.762 49.247 48.507τ6 −54.806 −50.966 −49.016 −48.480 −45.981 −42.548 −39.465 −38.432τ7 2.653 −1.484 0.909 −0.053 −0.393 0.426 1.674 1.871

(RBFNN) for two different modules, namely walkingand going up-stairs[8]. In the present work, real-timestaircase ascending gait generation problems of a two-legged robot has been solved by using a fuzzy logiccontroller (FLC) and a GA. In actual locomotion, therobot collects information of the terrain with the help

of sensors, cameras, etc. and as a result of which, thecollected data could be imprecise in nature. As an FLCis a potential tool for dealing with imprecision and un-certainty, it could be a natural choice to solve this prob-lem. The knowledge base of an FLC is expressed in theform of (IF . . . THEN) rules, whereas an NN-basedcontroller, such as RBFNN, works like a black-box.Thus, it becomes easier for the designer to understandthe control action to be taken by the robot to handlea particular situation, while using an FLC. Moreover,training of an NN might be more involved computa-tionally compared to tuning of an FLC.

Salatian and Zheng[9] proposed a method of gaitsynthesis for a biped robot climbing sloping surfacesusing an NN but they considered only the static sta-bility of the robot. During locomotion, a two-leggedrobot should be dynamically balanced, which has beenconsidered in the present work.

The method of fuzzy logic-based ZMP-trajectorygeneration, as suggested by Park[10], might not be ableto yield the optimal trajectory because its knowledgebase (KB) had not been tuned by using any optimizer.In the present work, the optimal KB of the fuzzy logiccontroller has been evolved by using a genetic algo-rithm, off-line.

Zhou and Meng[11] developed a fuzzy reinforce-ment learning architecture for dynamic balance of abiped robot and its effectiveness was tested throughsimulations. The main drawback of their method liesin the fact that it may suffer from the local minima

ingas

onshee

problem due to the use of a gradient-based learnmethod. In the present work, a GA-based learning hbeen adopted. Therefore, the chance of its solutifor getting trapped into the local minima is less, as tGA does not require the derivative information of thobjective function.

Fig. 17. ZMP trajectories obtained usingApproaches 2 and 3for thefourth test case shown inTable 8.


6. Concluding remarks

In this paper, gait generation problem of a two-legged robot moving up through the staircases, issolved considering its dynamic stability. A genetic–fuzzy system is developed to generate the dynamicallystable gait for a two-legged robot. Three different ap-proaches are implemented on computer simulationsand their performances are compared.

The GA-tuned FLCs are found to perform betterthan the manually constructed FLCs. It is obvious be-cause the manually constructed FLCs may not be op-timal in any sense.Approach 3is found to performslightly better thanApproach 2. In Approach 2, the re-sponsibility of developing an appropriate KB of an FLCis shared by the designer and a GA, whereas the wholeresponsibility of designing an optimal KB of an FLC isgiven to the GA inApproach 3. The better performanceof Approach 3compared toApproach 2could be dueto the reason that the physical problem might be dif-ficult to foresee by the designer (as used inApproach2), but the GA with the help of its exhaustive search isable to solve it, inApproach 3. As the CPU time of thealgorithm is coming to be only 0.002 s, in a P-III PC,the developed algorithm might be suitable for on-lineimplementations.

The present work may be extended in a number ofways. In this paper, stability of the robot is consideredin the direction of its movement only but not along

s,h

enalers-

ofter

ee.eitof

ityanli-soes

on staircase ascending gait generation of a two-leggedrobot. It can be extended to design suitable gait of atwo-legged robot for staircase descending problems.

Acknowledgments

The authors are grateful to the anonymous review-ers, whose valuable suggestions have helped them tomodify the paper into its current form. Moreover, theyacknowledge the cooperation of Nirmal Baran Hui, Re-search Scholar, Mechanical Engineering Department,IIT Kharagpur, while modifying the manuscript.

References

[1] D.K. Pratihar, Path and gait generation of legged robots usingGA-fuzzy approach, Ph.D. Thesis, IIT Kanpur, India, 2000.

[2] M. Vukobratovic, A.A. Frank, D. Juricic, On the stability ofbiped locomotion, IEEE Trans. Biomed. Eng. 17 (1) (1970)25–36.

[3] M. Vukobratovic, B. Borovac, D. Surla, D. Stokic, BipedLocomotion—Dynamics, Stability, Control and Applications,Springer-Verlag, 1990.

[4] K. Mitobe, G. Capi, Y. Nasu, A new control method for walkingrobots based on angular momentum, Mechatronics 14 (2004)163–174.

[5] Y.J. Seo, Y.S. Yoon, Design of a robust dynamic gait of thebiped using the concept of dynamic stability margin, Robotica13 (1994) 461–468.

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[6] A. Goswami, Foot rotation indicator point: a new gait planntool to evaluate postural stability of biped robots, in: Procings of the IEEE International Conference on RoboticsAutomation, vol. 1, Detroit, USA, 1999, pp. 47–52.

[7] D. Katic, M. Vukobratovic, Intelligent Control of Robotic Sytems, Kluwer Academic Publishers, 2003.

[8] G. Capi, Y. Nasu, L. Barolli, K. Mitobe, Real time gait genetion for autonomous humanoid robots: a case study for walRobotics Auton. Syst. 42 (2003) 107–116.

[9] A.W. Salatian, Y.F. Zheng, Gait synthesis for a biped roclimbing sloping surfaces using neural networks. Part II.namic learning, in: Proceedings of the 1992 IEEE InternatiConference on Robotics and Automation, Nice, France, 1pp. 2607–2611.

10] J.H. Park, Fuzzy-logic zero-moment-point trajectory gention for reduced trunk motions of biped robots, Fuzzy Sets S134 (2003) 189–203.

11] C. Zhou, Q. Meng, Dynamic balance of a biped roboting fuzzy reinforcement learning agents, Fuzzy Sets Syst(2003) 169–187.

12] W.T. Miller, Real-time neural network control of a biped waing robot, IEEE Contr. Syst. (1994) 41–48.

its left and right sides. For real implementationstability of the robot has to be maintained in all sucdirections. Optimal gait planning problem has besolved using a fuzzy logic-based controller. A neurnetwork-based controller will be developed to solvthe same problem. Moreover, these two controllewill be compared in terms of their performances, computational complexity, and others. The performancethe developed algorithm has been tested on compusimulations. In future, the developed algorithm will bimplemented on a real robot to test its performancDuring locomotion, the torque values at hip, kneand ankle of the ground leg have been modified, ifbecomes necessary to maintain the dynamic stabilitythe robot. If the modified torque exceeds the capacof the motor, it might be needed to take the help ofextra force to maintain its dynamic stability. The appcation of external force to regain the stability, may albe included in the future work. This work concentrat


[13] E.H. Mamdani, S. Assilian, An experiment in linguistic syn-thesis with a fuzzy logic controller, Int. J. Man Mach. Stud. 7(1975) 1–13.

Rahul Kumar Jha has completed hisBTech in mechanical engineering from IITKharagpur, India, in this year (i.e., 2005).He has special interest in the areas likerobotics, soft computing.

Balvinder Singh received his BTech in me-chanical engineering from National Insti-tute of Technology, Hamirpur, India, in theyear 2002 and MTech in mechanical engi-neering from IIT Kharagpur, India, in theyear 2004. He is working, at present, as a de-sign engineer with GE India, Bangalore. Hehas special interest in the areas like robotics,soft computing.

Dilip Kumar Pratihar received his BE(Hons) and MTech in mechanical engineer-ing from National Institute of Technology,Durgapur, India, in the year 1988 and 1994,respectively. He got his PhD from IIT Kan-pur, India, in the year 2000. Besides severalscholarships, he received University GoldMedal for securing the highest marks inthe University in 1988, A.M. Das Memo-rial Medal in 1987, Institution of Engineers’Medal in 2002, and others. He completed his

post-doctoral studies in Japan (6 months) and Germany (1 year) underthe Alexander von Humboldt Fellowship Programme. He is working,at present, as an associate professor, in the Department of Mechan-ical Engineering, IIT Kharagpur, India. His research areas includerobotics, soft computing, manufacturing science. He has publishedabout 50 papers in different journals and conference proceedings. Heis a Member of the Institution of Engineers (India).

on-line stable gait generation of a two-legged robot using a genetic–fuzzy system

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