Stable Gait Synthesis and Analysis of a 12-degree of Freedom Biped Robot in Sagittal and Frontal Planes

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1 Stable Gait Sythesis ad Aalysis of a 12-degree of Freedom Biped Robot i Sagittal ad Frotal Plaes Submitted: 1 th March 2012, accepted: 22 d August 2012 A.P. Sudheer, R. Vijayakumar, K.P. Mohada Abstract: Legged machies have ot bee offered biologically realistic movemet patters ad behaviours due to the limitatios i kiematic, dyamics ad cotrol techique. Whe the degrees of freedom (DOF) icreases, the robot becomes complex ad it affects the postural stability. A loss of postural stability of biped may have potetially serious cosequeces ad this demads thorough aalysis for the better predictio ad elimiatio of the possibility of fall. This work presets the modellig ad simulatio of twelve degrees of freedom (DOF) biped robot, walkig alog a pre-defied trajectory after cosiderig the stability i sagittal ad frotal plaes based upo zero momet poit (ZMP) criterio. Kiematic modellig ad dyamic modellig of the robot are doe usig Deavit-Harteberg (DH) parameters ad Newto-Euler algorithm respectively. This paper also proposes Leveberg-Marquardt method for fidig iverse kiematic solutios ad determies the size of the foot based o ZMP for the stable motio of biped. Biped robot locomotio is simulated, kiematic ad dyamic parameters are plotted usig MATLAB. Cycloidal gait trajectory is experimetally validated for a particular step legth of the biped. Keywords: Deavit-Harteberg parameters, sagittal ad frotal plae, zero momet poit (ZMP), Leveberg- Marquardt algorithm, Cycloidal gait trajectory. Nomeclature θ i i th Joit agle a i Legth of i th lik d i offset distace of i th lik Twist agle about x i axis m ir ; m il Masses of i th right ad left leg q i i th joit parameter B, L Breadth ad legth of the foot Foot pose fuctio Hessia matrix Jacobia λ Dampig factor Iertial/mass Matrix Coriolis ad cetrifugal matrix Gravity matrix D Coefficiet matrix of torque τ R Orietatio matrix Momets at the i th joit X zmp ; Y ; Z X, Y ad Z coordiates of ZMP zmp zmp Force at the i th joit F g ; F r ; F i Total iertial ad gravity force; Reactio force; iertial force M g ; M r ; M i Total iertial momet; Reactio momet; Iertial momet. g Acceleratio due to gravity Rollig agle of circle S L Step legth v i Istataeous velocity of i th lik I i Iertial matrix of i th lik, ω i Agular velocity of i th lik ẋ, ẏ, ż x,y ad z acceleratios 1. Itroductio Robots of curret geeratio have bee used i various fields isolated from the huma society. They suffer major shortcomigs because of their limited abilities for the maipulatio ad iteractio with humas. Humaoid/biped robots are better suited for workig i huma eviromet ad have a better degree of mobility, especially i eviromet with obstacles. The mai motive behid the developmet of bipeds is its adaptability to huma eviromet, so that there is o eed to make special workig eviromet for bipeds. Early studies o bipeds were mostly o its locomotio ad ot o its real idustrial applicatios. Now it has reached the level of desigig customized bipeds for specific applicatios. Still, there are issues yet to be addressed, amog them the most basic beig stable dyamic locomotio ad gait sythesis. Bipeds ca perform both Static ad dyamic walkig. I static walkig, the complete system stays balaced by always keepig the cetre of mass (COM) of the system vertically over the support polygo formed by the feet durig locomotio [1]. I dyamic balace or walkig the vertical projectio of COM does ot stay withi the support polygo durig the motio, i.e. durig motio the COM may leave the support polygo for certai periods of time. Therefore some complicated ad coordiated movemet of other body parts oly ca balace the biped. This makes the dyamic walkig more difficult from a desig poit of view. Locatio of COM ad ZMP are the two importat issues i biped locomotio. The cocept of ZMP was put forward by M. Vukobratovic et al. [2] which revolutioized ad accelerated the studies i dyamic walkig of bipeds. ZMP is termed as the poit o the groud about which the robot s resultat momets at the groud is zero. This is used as a stability criterio for dyamic walkig i this work. If the ZMP is iside the support area, the walkig is cosidered dyamically stable, be- 36

2 cause the foot ca cotrol the robots posture. The ZMP criterio caot be applied to biped robots that do ot cotiuously keep at least oe foot o the groud or to those which do ot have active akle joits. The motio of a humaoid comprises of time-fuctios of agular positios ad velocities of the joit agles of the robot. The straight forward approach is to geerate the joit time trajectories by solvig iverse kiematics, to maitai the physical stability of the humaoid. It becomes icreasigly difficult to compute the iverse kiematics as the DOF of the biped icreases. However, such a approach is suitable for off-lie geeratio of joit trajectories. Geeratio of low-eergy gait is a ope ad otrivial issue over a cosiderable period [3] durig the motio of robot. This paper maily cocetrates o iverse kiematics ad dyamics by usig Leveberg-Marquardt algorithm (LMA) ad Newto-Euler algorithm (NEA) respectively to aalyze the stability of biped locomotio durig dyamic walkig. It also proposes a methodology to fid the foot size for the smooth motio of joits. I almost all previous works related to humaoid walkig aalysis ad sythesis, stability i sagittal ad frotal plaes are aalyzed separately with the kiematic modellig based o the geometrical approach. DH parameters are used i the preset work for the kiematic modellig ad ZMP cocept is used for the stability aalysis i sagittal ad frotal plaes.to the best kowledge of the authors, o work based o LMA ad DH modellig for aalysis ad sythesis has bee reported i the area of humaoid or biped robots with the miimizatio of foot size. This paper is structured as follows. I sectio 2, kiematic, LM algorithm ad dyamic modellig are described. The fudametal theory of the cetre of mass (COM) ad ZMP, i sigle ad double support phases are give i sectio 3. Sectio 4 deals with the simulatio of the gait trajectory for steppig motio with stability i both sagittal ad frotal plaes. Results ad discussio are show i sectio 5. Sectio 6 presets the cocludig remarks with outlook. 2. Modellig of biped robot Fig. 1 shows a 12 DOF biped robot modelled i SOL- ID WORKS which is havig six DOF per leg, two at the akle, oe kee ad three at the hip. The akle joit of both legs have yaw ad pitch motios, the kee is havig oly pitch motio ad hip joits of both the legs have roll, pitch ad yaw motios. The proposed model cosists of seve liks i order to approximate the locomotio characteristics similar to those of the lower extremities of the huma body. The complete walkig cycle cosists of three sigle support phases (SSP) i which oly oe leg is o the groud while the other swigs forward ad four double support phases (DSP) i which both legs are o the groud. The stace leg i cotact with the groud carries the whole weight of the robot. Durig the trasitio from sigle support phase to double support phase, swig leg decelerates to zero velocity. As a result of this, huge impact forces are developed at the cotact phase for a short period of time. At the ed of the DSP the swig leg accelerates which creates jerk i the joits ad liks of the robot. I DSP, Table 1. Biped mass ad dimesios Mass m 0R m 1R m 2R m 6 m 2L m 1L m 0L Kg Lik legth dd dd dd dd dd dd dd m Fig.1. Twelve DOF biped robot model the robot will be stable whe the projectio of COM stays withi the support polygo. As a result of trasitio betwee the sigle support ad double support phases, the problem of istability of the humaoid arises. The cotact phase i walkig is almost 20% of the total gait period [4]. As it is difficult to fid out the reactio forces accurately, it is assumed that the impact of swig leg is perfectly ielastic while esurig that o slippage occurs. Aother importat assumptio made is that durig the SSP, stace foot remais i flat cotact with the groud. I SSP, the robot will be stable whe the ZMP stays withi the support foot polygo Kiematic modellig Kiematic diagram of the 12 DOF biped is give i Fig. 2. Biped robots ca be described kiematically by usig joit-lik DH parameters amely joit agle (q i ), lik legth (a i ), offset distace (d i ) ad lik twist (a i ). Table 1 shows lik dimesios of the biped robot. Four DH parameters correspods to each lik of biped are give i table II based o the frame assigmet as show i Fig. 3. Forward kiematics determies the pose of robot ed effector as a fuctio of its joit ad lik parameters where as the iverse kiematics gives the values of the joit variable correspodig to ed effector or foot pose Iverse kiematics A suitable step legth is assumed for the biped walkig aalysis from the idea that the step legth for miimum eergy cosumptio is about 60% of hip height [5]. Cartesia space trajectory is plaed to get the trajectory of the swig foot which follows a cycloidal trajectory profile durig the motio of robot [6]. As this profile is made by superpositio of liear ad siusoidal fuctio, 37

3 2.2. Leveberg-Marquardt algorithm (LMA) The Leveberg-Marquardt method is a search method which gives the advatages of both Gauss-Newto directio ad the steepest descet directio methods because it uses a search directio that is a cross betwee the Gauss- Newto directio ad the steepest descet directio. Table 2. D-h parameter table Lik(i) θθ θθ θθ θθ θθ θθ ππ 22 θθ dd aa 00 dd dd dd ii ππ / ππ /2 ππ /2 0 Lik(i) θθ θθ θθ ππ 22 θθ θθ θθ θθ dd aa dd dd 00 dd Fig. 2. Kiematic diagram of biped robot ii ππ /2 ππ / ππ /2 00 it has a property of slow start, fast movig, ad slow stop. This reduces the jerk durig the start ad ed of walkig. This characteristics ca reduce the over burde at istataeous high speed motio of the actuator. Poits o the swig gait trajectory are take as poses of the foot for gettig the joit variables of each leg of the robot. Fial pose matrix of the biped robot model 0 T 12 is equal to the pose of the foot o the swig gait trajectory. Fig. 3. D-H modellig-frame assigmet for 12-dof biped LMA is used for fidig iverse kiematic solutio i this work. Solvig iverse kiematics ivolves solutio of twelve oliear equatios with trigoometric fuctios. Six idepedet equatios, three for orietatio ad three for positio are to be solved. Sice it is a bit laborious to do the iverse kiematics of the 12 DOF robot maually for the whole iterpolated poits, iverse kiematics is carried out i MATLAB ad optimized results satisfyig the boudary coditios are obtaied. Leveberg-Marquardt iterative method is used for this purpose. It is a modificatio of Newto-Euler algorithm ad gradiet descet method. It is also called damped Gauss-Newto method, as it uses a dampig factor to decide the accuracy level of solutios whe the search approaches the miima. For startig, a iitial guess is to be provided. A advatage of this method is that, the search directio is idepedet of the iitial solutio set give ad it gives the actual miima eve if the iitial assumptios are far from the global miima. LM algorithm for the preset cotext is explaied below. Mi F(q)=[F 1 (q) F 12 (q)]; Sub to π/4 q π/4, Where, q = [q1 q2...q12] T. The coefficiet of the quadratic term of local Taylor series expasio of a fuctio is, Y = f (q + dq) f (q)+ J(q)dq+dq T H(q)dq. The covergece criteria is f (q + dq ) f (q ); Therefore, J(q)dq + dq T H(q)dq = 0 dq = H(q) 1 J(q). Modified Hessia is H(q, l) = 2J T J + li 1. Set dampig factor l= Solve dq = H(q, l) 1 g 3. If (q + dq ) > f (q ), icrease l (x 10 say) ad go to Otherwise, decrease λ(x 0.1 say), let q +1 = q + dq, ad go to 2. Whe q the algorithm has coverged set l = 0 ad compute the fial solutio Where q is the iitial vector assumed ad Hessia ad Jacobia matrices, H(q)&J(q) are give i equatio 38

4 1. a & 1.b respectively. ( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) (1.a) (1.b) 2.3. Dyamic modellig The mai challeges of gait plaig are learig which icludes the selectio of specific iitial coditios, costraits ad their associated gait parameters [4]. I this sectio, differet methodologies are adopted for dyamically costraied locomotio of biped robot i sagittal ad frotal plaes. From a cotrol poit of view, the iverse dyamics problem is of solvig the joit torques from the joit agles alog with their first ad secod order derivatives. I this work, we have used the Newto-Euler recursive algorithm for dyamic aalysis. Sice stace foot is assumed to be i flat cotact, resultat groud reactio force/momet ad torques ca be computed usig Newto-Euler algorithm [10]. The process cosists of 2 iteratios, (i) forward iteratio to compute lik velocities ad acceleratios ad (ii) backward iteratio to get the torque variatio at joits. Iitially velocity ad acceleratio of base frame is take to be zero. While the stace leg is i motio o exteral forces are actig o it, except gravity loadig. It is also assumed that the cetroid of the lik ad the cetre of mass of lik coicide. A geeral dyamic model for biped walkig related to the joit coordiates vector ad joit torque vector without cosiderig the frictio ad other disturbaces is give below. ( ) ( ) M θ θ + C θ, θ θ + G( θ ) = Dτ (2) Where, is the 12 x 12 iertia matrix, is the 12 x 12 coriolis ad cetrifugal matrix ad is the 12 x 1gravity vector, is the 12 x 12 coefficiet matrix of joit torques. Joit torques at differet joits are determied through backward iteratio by usig the set of equatio 3 ad 4. T i t = ih R zˆ (3) i i-1 0 i i+1 η i = R i+1 η i i-1 i+1 + R 0 D i x i i+1 R i+1 f i+1 + ( i i-1 R 0 D i + i i R (4) 0 r ) x if + i N i i i Where, i = 12,11,1; ih i is the momet exerted o lik i by lik i 1 ad i R i-1 is the orietatio matrix, if i is the total exteral force actig at the cetre of mass of the lik, in i is the total exteral momet actig o lik at its cetre of mass, is if i the force exerted o lik i by lik i R 0 i 1 D i is the coordiates of the i th joit whe referred to frame i ad i r is the cetre of mass of lik referred to frame i. 3. Zero momet poit ad gait trajectory Static walkig stability coditio is sufficiet to esure locomotio for very slow motio of biped robot. Some of the drawbacks of this techique of motio plaig are the discrete ature of the motio of the robot ad the time required for takig a sigle step beig uusually log. It is ot always ecessary for the cetre of mass (COM) of the robot to lie vertically above the base polygo. Aother method of aalyzig stability is based o the Zero momet Poit criterio [6]. Zero Momet Poit is defied as the poit about which the momet of all the active forces actig o the robot tur out to be zero. I static gait plaig problems, biped robot is stable if the projectio of the cetre of mass (COM) falls withi the covex hull of the foot support polygo. I dyamic locomotio, lik acceleratio, iertial forces, ad groud reactio force are also to be cosidered ad the ZMP should be withi the covex hull of the foot polygo for satisfyig the stability criterio. The dyamic locomotio is highly oliear ad difficult to aalyze i real eviromet. The coditio i the static ad dyamic stability of the biped durig the sigle support phase is the locatio of the ZMP must be iside the covex hull of the supportig foot. I double support phase ZMP or projectio of COM should lie withi the covex hull of support polygo formed by left ad right foot. Accordig to D Alembert s priciple, if all forces are balaced, the motio of the biped is physically realizable. By D Alembert s priciple the total forces ad momets actig o the biped must be zero. This is give by: F r + F e = 0 : M r + M e = 0 (5) Where is the groud reactio force, F i is the total iertial ad gravity force actig o biped, M r is the reactio momet ad M e is the iertial momet actig o biped. Let F i be the iertial force, M i be the iertial momet, ad m i be the i th mass of the th segmet (i = 1...). We have: F = ( F m g) = m ( v + g) e i i i i i = 1 i = 1 (6) d M = M = ( I ) e i i i i = 1 i = 1 dt w (7) Where, v i is the istataeous velocity, I i is the iertial matrix, w i is the istataeous absolute agular velocity of ith lik at its COM, terms are relative to the fixed referece coordiate, say O as i Fig. 4 [10]. The balacig problem of the biped system ca be reduced at a assiged groud poit (x = 0, y, z ) called the ZMP, where the resultat momet (M) at the groud plae is zero (Mx = My = 0).From the relatio of the equivalet force momet, oe obtais: Fig. 4. Referece co-ordiate system for foot base 39

5 Fe = mi( v i + g) (8) i = 1 (9) (10) Fig. 5. Cycloid curve (11) Where F ad M are the resultat force ad momet at the ZMP (0, y, z ) respectively, ad (x i, y i, z i ) is the vector from the origi O of the fixed referece coordiate O to the COM of lik cosidered. x, y, z, are the correspodig compoets of acceleratios i respective directios. From the above equatios, oe obtais: X = 0 (12) zmp (13) 4. Simulatio Numerical simulatio of 12 DOF biped walkig is doe usig MATLAB.3-D kiematic patter of the biped for a sigle step is show i Fig. 6. Variatios of kiematic ad dyamic parameters ad ZMP are plotted. Iitially both legs of biped are i stace positio the the left leg is steppig a legth of 10 cm i 1 s. Simulatios are carried o a biped robot havig hip height of 25 cm ad mass of 1.7 kg. Kiematic ad dyamic modelligs help to sythesis ad aalyze the biped robot at differet scaled dimesios based o stability. (14) The costrait o the dyamic motio of the biped durig the sigle-support phase is the locatio of the ZMP (0, Y zmp, Z zmp ) must be iside the covex hull of the supportig foot. I the sigle-support phase the stable covex hull is same as the area occupied by the supportig leg o the groud. Therefore, Z mi < Z zmp < Z max ad Y mi < Y zmp < Y max where, we assume that the supportig foot is rectagular, parallel to the fixed referece coordiate O, ad betwee poits (0, Y mi,y max ) ad (0, Z mi,z max ). Mathematical iterpolatio is oe of the simplest methods for providig suitable gait trajectory i accordace to the give boudary coditios. Cartesia space trajectory plaig is carried out to get the trajectory of the swig foot. Geerally huma s akle joit motio trajectory is a cycloidal profile i ormal walkig (Kurematsu, Kitamura & Kodo, 1988) Cycloidal profile reduces effects of sudde acceleratio at the begiig ad deceleratio at the ed durig the gait geeratio. Hece the cycloidal profile is used for the trajectories of the swig foot. As this profile is made by superpositio of liear ad siusoidal fuctio, it has a property of slow start, fast movig, ad slow stop. This avoids the jerk that ca happe durig the start ad ed of walkig. This characteristics ca reduce the over burde at istataeous high speed motio of the actuator. Equatio of a cycloid i parametric form for selectig break poits o the trajectory is give i equatios 15 ad 16. Gait trajectory patter is show i Fig. 5. x i = S L (j i si j i )/2p (15) z i = S L (1 cos j i )/2p (16) Where i = 0,1...N, the umber of poses of foot o the trajectory ad S L is the step legth. Fig D Biped walkig patter 5. Results ad discussio Stable gait geeratio of a 12 DOF biped robot is demostrated i this paper. Variatios i parameters like joit agles, lik velocities ad lik acceleratio are plotted durig the stable motio of biped. Torque ad ZMP variatios are also aalyzed here. The variatio of joit agles at akle, kee ad hip for right ad left leg are varyig smoothly ad cotiuously for a sigle step as show i Fig. 7 ad this assures a smooth trasitio of the robots motio. Rollig agular variatios at the akle joit of the left leg ad hip joit pitch agular variatios (4 th ad 9 th joits)of both legs are high compared to other joit agle variatios. Because these two joits plays vital role i the stability of biped motio i this aalysis. Iitially, whe the left leg is about to lift, both hip ad kee joits should have some agular variatios for brigig the COM withi the support foot polygo. Lik velocities ad acceleratios at the COM are give i Fig. 8 ad Fig. 9, respectively. First lik is fixed at the groud durig the walkig so the velocity ad acceleratios are zero. Velocity is maximum for the swig foot (lik 12) ad miimum for the lower part (lik 2) of the stace leg. All other liks the velocities are varyig approximately i betwee 0 ad 25 cm/s. 40

6 Fig.7. Joit agle variatios of left ad right leg Fig. 8. Lik liear velocity Variatios i acceleratios are smooth but there are up ad dows because the biped is movig i high speeds with step legth of 10 cm. There are some values of acceleratios at the begiig ad ed of the gait trajectory so that the jerk will be the miimum at these two locatios. Up ad dows of velocities result i irregular variatios i the acceleratios as show i Fig. 9. This creates jerk at the joits ad liks of the robot at itermediate positios ad biped ca mostly be suited at slow speeds for small step legths ad moderate speed at higher step legths. Jerk will be reduced for higher step legths i moderate speeds but stability will be achieved with larger foot size. This will be clear from the ZMP variatios plotted i Fig. 11. It is clear from the velocity ad acceleratio diagrams that the velocity ad acceleratio variatios are same for lik 3, 4 ad 5. Similarly velocity ad acceleratio variatios are same for liks 6, 7 ad 8. This is because of the assumptio that the joits 4 th, 5 th ad 6 th are at the same origi ad also the joits 7 th, 8 th ad 9 th are at the same origi i modellig. Fig.10 shows the cotiuous variatios of torque for all joits. Startig torque for the first joit is high because this joit is oly makig the robot walk by swigig the whole system i the frotal plae. Geared motor ca be used for gettig high torque at joit 1. Torque is smallest for the akle joit of the swig foot. By chagig various kiematic ad dyamic parameters it is possible to brig the ZMP withi the limited size of Y-Z plae for attaiig stable walkig, ad variatios are plotted agaist the steppig time as show i Fig. 11. Variatio is more i Z directio compared to because the Z compoet of acceleratio has a more effect o shiftig of ZMP. The iertia compoets are small here due to the small size of biped. However those iertial 41

7 Fig. 9. Lik liear acceleratio Fig. 10. Joit torque Fig. 11. ZMP variatio i first walkig phase 42

8 Joural of Automatio, Mobile Robotics & Itelliget Systems VOLUME 6, N terms will ot be egligible i case of fast bipedal activities like ruig ad jumpig, or whe the lik masses ad dimesios are comparable to those of the actuators. There will ot be ay differece if we eglect the iertial effects i slow motio. ZMP moves i Y-Z plae approximately i a parabolic path withi the foot base. The maximum approximate rage of Yzmp ad Zzmp are -2.4 cm to 0.7 cm ad -2 cm to 3 cm respectively. Fig.1 1 shows the movemet of ZMP o the foot of the stace leg. This plot gives the feasible size of foot of stace leg for a particular step legth. The resultat values of ZMP variatios are represeted graphically for step legth of 0 to 20 cm i Fig. 12. This helps to decide the foot size for a rage of step legth based o kiematic ad dyamic costraits. As per the Fig. 12, foot size of 10 cm x 10 cm is required for biped walkig through a cycloidal gait for a step legth of 20 cm. Fig. 14. Sapshots of biped walkig Fig. 12. ZMP Vs Step legth 6. Experimetatio Walkig gait geeratio is simulated ad the results ivolvig the relevat variables are aalysed i the previous sectio. I this sectio, gait geerated for a step legth of 10 cm is experimetally validated i a 12 DOF biped. The experimetal validatio is doe by matchig simulated cycloidal trajectory with real time gait trajectory. Validatio ca also be doe experimetally by evaluatig ad comparig ZMP variatios alog with the gait trajectory. Computer / Processor is iterfaced with the biped robot through a mii maestro 12 chael servocotroller for cotrolllig actuators for the required cycloidal trajectory. ProcessorJoit Agles Servo Cotro ller Biped Robo t ated real time gait trajectory is compared with the cycloidal gait trajectory determied from the simulatio result. Robot stable motio ad Real time gait trajectory are show i Fig. 15 ad Fig. 16,respectively. Joit agles are fed to the biped for gettig the cycloidal trajectory with a fixed step legth. Oe of the experimetal real time cycloidal trajectory is give i Fig. 16. Experimet is coducted five times for the same joit agles ad steplegth. Average step legth obtaied i the real time gait gearatio is approximately 10.3 cm istead of 10 cm. A cycloid is costructed correspodig to the step legth of 10 cm ad its calibrated image is superimposed o the plot of 19 istataeous poses of swig foot. The dots i Fig. 15 are the istataeous poses obtaied durig the experimetatio. Cycloidal Gait Fig. 13. Data Flow diagram The block diagram show i Fig. 13 depicts the details of data flow for testig ad validatio of bipedal gait. Joit agles correspodig to a sigle step swig foot trajectory is determied usig MATLAB ad the sigals are set to the biped for the required motio. Fig 14 shows the sapshots of 12 DOF biped robot walkig for a step legth of 10cm. Istat motios are captured for the gait aalysis durig the foot step movemet. Evalu- Fig. 15. Poits o the stable gait trajectory 43

9 form. This research was supported by Natioal Istitute of Techology Calicut, Kerala uder the provisio of Faculty Research Grat (FRGph02/07/03-04). This support is gratefully ackowledged. AUTHORS Sudheer.A.P* Assistat Professor i the Departmet of Mechaical Egieerig, Natioal Istitute of Techology Calicut, Kerala, Idia, s: apsudheer@itc. ac.i,apsudheer@rediffmail.com Fig. 16 Real time gait trajectory Fig. 16 depicts the variatios of poits o the real time stable gait trajectory with the theoretical cycloidal gait. I this particular real time gait trajectory the step legth obtaied is approximately 10.1 cm istead of 10 cm. This Aalysis shows the correctess of modellig ad gait trajectory of the 12 DOF biped robot. 7. Coclusio ad outlook Stability aalysis of a twelve dof biped robot i the sagittal ad frotal plae for a cycloidal gait is preseted i this paper. Geeratio of a gait for the stable walkig based o zero momet poit withi a particular foot size is attempted. The iverse kiematic solutios are foud by usig Leveberg-Marquardt iterative method. Motio of the robot is costraied because of the limited umber of DOF. The preset sythesis ad aalysis gives idea of foot size for stable biped walkig. Experimetatio for determiig the real time gait trajectory is attempted ad trajectory is compared with theoretical trajectory. This experimetal result autheticates the suitability of the model for the sythesis ad aalysis of biped robot. At preset researchers are tryig to miimize the foot size to avoid self collisio ad flexibility with higher level of stability durig walkig. This work gives a clear directio for geeratig a optimum gait trajectory based o ZMP with miimum foot size. It is expected that this will lead to optimizatio of the foot size by cosiderig all the kiematic ad dyamic costraits for achievig better stability. It may be cosidered that the sythesis ad aalysis procedure ca be refied i may ways: some of them beig, optimum smooth gait plaig with miimum eergy cosumptio usig traditioal ad soft computig techiques by chagig the walkig parameters like step legths, steppig time ad height of the trajectory. Simulatio ca be exteded to various structured ad ustructured eviromets. Authors are attemptig to verify experimetal results by evaluatig ad comparig ZMP variatios alog with the gait trajectory i a complete walkig cycle. Ackowledgemets The authors are thakful to the referees ad the editor for their costructive suggestios ad commets which have immesely helped to brig this paper to the preset R.Vijayakumar Professor i the Departmet of Mechaical Egieerig, Natioal Istitute of Techology Calicut, Kerala, Idia. K.P. Mohadas retired Professor, Electrical Egieerig, Natioal Istitute of Techology Calicut, Kerala, Idia. *Correspodig author Refereces [1] M. Raibert et al., Legged robots that balace. MIT press Cambridge, MA, [2] M. Vukobratovic ad B. Borovac, Zero-momet poit-thirty five years of its life, Iteratioal Joural of Humaoid Robotics, vol. 1, o. 1, 2004, pp [3] P. Vadakkepat ad D. Goswami, Biped locomotio: stability, aalysis ad cotrol, Robotica, vol. 27, o. 1, 2009, pp [4] T. Zieliska, C. Chew, P. Kryczka, ad T. Jargilo, Robot gait sythesis usig the scheme of huma motios skills developmet, Mechaism ad Machie Theory, vol. 44, o. 3, 2009, pp [5] F. Silva, T. Machado et al., Eergy aalysis durig biped walkig. I: Robotics ad Automatio, Proceedigs of IEEE Iteratioal Coferece, vol. 1. IEEE, 1999, pp [6] Z. Tag, C. Zhou, ad Z. Su, Trajectory plaig for smooth trasitio of a biped robot. I: Proceedigs of IEEE Iteratioal Coferece o Robotics ad Automatio, ICRA 2003, vol. 2, IEEE, 2003, pp [7] M. Vukobratovic, D. Adric; B. Borovac, How to achieve various gait patters from sigle omial, Iteratioal Joural of Advaced Robotic Systems, vol. 1, o. 3, 2004, pp [8] A. Takaishi, M. Ishida, Y. Yamazaki, ad I. Kato, The realizatio of dyamic walkig by the biped walkig robot WL-10RD. I ICAR 85, 1985, vol.1, pp [9] H. Miura ad I. Shimoyama, Dyamic walk of a biped, Iteratioal Joural of Robotics Research, vol. 3, o. 2, 1984, pp [10] W. Spog, M. Vidyasagar, Robot dyamics ad cotrol, Joh Wiley& Sos, New York,