Biped Robot Gait Planning Based on 3D Linear Inverted Pendulum Model

Transcription

1 IOP Conerene Series: Materials Siene and Engineering PAPER OPEN ACCESS Biped Robot Gait Planning Based on 3D Linear Inverted Pendulum Model o ite this artile: Guohen Yu et al 218 IOP Con. Ser.: Mater. Si. Eng View the artile online or updates and enhanements. Related ontent - Gait Generation or a Small Biped Robot using Approimated Optimiation Method inh Nguen, Linh ao and Hiroshi Hasegawa - Design o a biped loomotion ontroller based on adaptive neuro-u inerene sstems M-Y Shieh, K-H Chang and Y-S Lia - Pendulums are magnetiall oupled Yaakov Kratmakher his ontent was downloaded rom IP address on 1/3/218 at 23:4

2 MSEE217 IOP Con. Series: Materials Siene and Engineering (218) 1298 doi:1.188/ x/31/1/1298 Biped Robot Gait Planning Based on 3D Linear Inverted Pendulum Model Guohen Yu, Jiapeng Zhang and Wu Bo (College o Eletroni Inormation Engineering Inner Mongolia Universit, Huhhot) Abstrat. In order to optimie the biped robot's gait, the biped robot's walking motion is simpli to the 3D linear inverted pendulum motion mode. he Center o Mass (CoM) lous is determined rom the relationship between CoM and the Zero Moment Point (ZMP) lous. he ZMP lous is planned in advane. hen, the orward gait and lateral gait are simpliied as onneting rod struture. Swing leg trajetor using B-spline interpolation. And the stabilit o the walking proess is disussed in onjuntion with the ZMP equation. Finall the sstem simulation is arried out under the given onditions to veri the validit o the proposed planning method. 1. Foreword Biped robot is a multivariable, nonlinear, strongl oupled and variable struture omple dnamis sstem. And it is a omprehensive appliation o new tehnologies in man ields suh as omputer tehnolog, eletronis, mehanis and artiiial intelligene. Gait generation is a relativel basi and ke tehnique in biped robot researh. We must stud pratial and eetive gait planning methods to ahieve and improve the robot's walking perormane. People want to use ewer variables to get the gait walk that is as real humans among man tehniques. Based on this idea, the three-dimensional linear inverted pendulum model is applied to the generation o robot gait model [1, 2]. Inverted pendulum ontrol sstem is a omple, unstable, nonlinear sstem. It is an ideal eperimental platorm or ontrol theor teahing and various ontrol eperimenting. Among man researh methods o robots, the inverted pendulum sstem is used to simulate biped robot walk in reent ears due to the great similarities between inverted pendulum sstem and biped robot walking, and it has great signiiane in the researh o biped robot gait planning and stabilit ontrol. Beome a researh diretion o biped robot [3]. his paper sstematiall anales the robot's motion sstem, establishes a genealogial data struture ] and determines the robot's motion transmission path. hen, the robot with single-leg support period is simpliied to 3D inverted pendulum movement, and the movement trajetor o CoM is obtained aording to the relationship between ZMP and CoM and the initial onditions. Using link onstraints to ind the desired angle o eah joint. Combined with ZMP theor [4, 5], the sstem simulation was arried out to veri the eetiveness o the proposed method. 2. Biped Robot Sstem Analsis and Modeling he objet o this paper is a biped robot with 21 degrees o reedom. A miniature servomotor is used as the driving element to simulate a joint. Among them, the hip o the leg has 3 mutuall perpendiular degrees o reedom, the knee has 1 degree o reedom, ankle joints has 2 mutuall perpendiular degrees o reedom, the upper etremit shoulder has 2 mutuall perpendiular degrees o reedom and the elbow has 1 degree o reedom, the head has 2 degrees o reedom, the waist has 1 degree o reedom. he sstem o biped robot is irst analed beause o its ompleit. Aording to the Content rom this work ma be used under the terms o the Creative Commons Attribution 3. liene. An urther distribution o this work must maintain attribution to the author(s) and the title o the work, journal itation and DOI. Published under liene b Ltd 1

3 MSEE217 IOP Con. Series: Materials Siene and Engineering (218) 1298 doi:1.188/ x/31/1/1298 number o joints, the robot is divided into 21 masses. Eah mass ontains a joint. he torso has no joint. When the model newl joins a joint, it an be added to the orresponding position. When programming, ou an use a uniied untion to handle programming issues or eah mass. Eah mass has two branhes, the bottom let is the parent-hild relationship, and the bottom right is the brother-sister relationship. hen these masses are onneted in a ertain order to orm the robot's motion database, whih is onvenient or reording and updating the value o eah individual. At the same time, it shows the power transmission route. For eample, when the robot supports the let oot swinging, he 13th individual's movement aets the 12th individual's movement, that is the route is: , ou an also understand the robot motion database as a genealogial tree Struture shown in Figure 1. he genealogial tree struture is onverted into a data list shown in able 1, starting rom the root o the tree marked as 1, giving eah link a number (ID), and showing the onnetion between the links in able 1. BODY RARM(Y) RARM(X) LARM(Y) RHAND LARM(X) RLEG(Z) LHAND RLEG(X) LLEG(Z) RLEG(Y) LLEG(X) HEAD(Z) RKNEE LLEG(Y) HEAD(Y) RFOO(Y) LKNEE RFOO(X) LFOO(Y) LFOO(X) Figure 1. he amil tree struture o robot 2

4 MSEE217 IOP Con. Series: Materials Siene and Engineering (218) 1298 doi:1.188/ x/31/1/1298 able 1. Robot data list ID NAME SISER CHILD 1 BODY 2 2 RARM(Y) RARM(X) 4 4 RHAND 5 LRAM(Y) LRAM(X) 7 7 LHAND 8 RLEG(Z) RLEG(X) 1 1 RLEG(Y) RKNEE RFOO(Y) RFOO(X) 14 LLEG(Z) LLEG(X) LLEG(Y) RKNEE LFOO(Y) LFOO(X) 2 HEAD(Z) HEAD(Y) 3. 3D Linear Inverted Pendulum Model Walking Parameters Setting Similar to human walking, the robot's gait le an be simpl divided into two-legged and one-legged periods. he proess o shiting the enter o gravit rom the rear oot support area to the ront oot support area during the dual oot support period. During the single leg support period, the enter o gravit o the robot is moved orward b the support leg, whih is pivoted rom the rear o the support leg to the ront o the support leg. Relative to the single-leg support period, the ontrol o the two-legged support period is less diiult. In this paper, the walking le is regarded as the le o single-legged support period, and the ehange o the two legs is ompleted in an instant [6]. In the single-leg support period, reerene [7] simpliied the robot into a 3D linear inverted pendulum model as shown in Figure 2. he torso o the robot is simpliied as a partile o all qualit. he legs o the robot are simpliied into a massless, telesopi onneting rod that onnets the torso to the oot. he enter o mass is moved orward b bending and strething the legs. And telesopi ore an be deomposed into three omponents o the oordinate sstem. As ollows, o Figure 2. 3D linear inverted pendulum 3

5 MSEE217 IOP Con. Series: Materials Siene and Engineering (218) 1298 doi:1.188/ x/31/1/1298 ( / r) ( / r) ( / r) (1) Where r is the distane between the support point and the enter o mass. he entroid is onl aeted b the ores o telesoping and gravit,it an be obtained through kineti analsis. M ( / r) M ( / r) M ( / r) Mg Considering that the entroid onstraint moves in a plane, the onstraint plane is deined as ollows Among them, k, k determine the plane tilt, k k (3) determines its height. Due to the movement o the enter o mass on the restraining surae, the resultant ore should be in the restraining surae o the enter o mass, So the onstraint surae normal vetor and resultant vetor vertial.as ollows, k ( / r) ( / r) ( / r) Mg k 1 From the above solution, and aording to equation (3), the ollowing epression is obtained. Mgr (5) Substituting equation (5) into equation (2), the equation o motion o the enter o mass on the onstraint surae is obtained as ollows: g g Solving the dierential equation b Laplae transorm, and the trajetor o, an be got. hen ombined with the onstraint surae equation, the trajetor o the entroid an be determined. (2) (6) (4) ( t) ()osh( t ) ()sinh( t ) ( t) ()osh( t ) ()sinh( t ) k k (7) Where g is a onstant o the ratio o gravitational aeleration to entroid height. () and &() are the -diretion displaements and veloities o the entroid at time ero, respetivel, () and &() are the -diretion displaements and veloities o the entroid at time ero, respetivel, the initial onditions. Ater the initial onditions have been established, the trajetor o the mass enter an be determined. Change the initial onditions, ou an get dierent urves. Sine the robot's 4

6 /m /m MSEE217 IOP Con. Series: Materials Siene and Engineering (218) 1298 doi:1.188/ x/31/1/1298 enter o mass is alwas orward in the orward diretion and swings let and right in the lateral diretion, the initial values o and are set as ollows. Simulation results shown in Figure 3: 时间 /s 时间 /s Figure 3. Inverted pendulum motion in and diretions he igure shows that in the -diretion, the entroid an ontinue to move orward beond the support point, and the entroid an not ross the support point in the -diretion. Ater reahing a ertain point, the entroid moves in the opposite diretion. 4. Generation o hree-dimensional Walk Pattern In walking movement, the three-dimensional linear inverted pendulum an be regarded as a ombination o two-dimensional linear inverted pendulum in two planes o o and o [8] Gait Unit In the single leg support period, the entroid onorms to the two-dimensional linear inverted pendulum motion in the and diretions. From [9] we an see that the trajetor o the entroid is a hperbola in the onstrained plane. Based on its smmetr, the end position o the walking unit an be determined. Re-use o the smmetr o the walking unit, we an see the initial onditions and termination onditions are smmetrial. In the -diretion, the orresponding termination ondition is ( v, ), i the initial ondition is ( v, ), sine the entroid an ross the support point. In the -diretion, the orresponding termination ondition is (, v ), i the initial ondition is ( v, ), sine the entroid an not ross the support point. he relationship between the two is: v v C ( 1) ( S ) C ( 1) ( S ) (8) p p C osh S sinh Ater the initial onditions are determined, the motion trajetor an be determined based on the dierential equation o two-dimensional linear inverted pendulum motion. he trajetor is generated b the pedestrian unit, and the same pedestrian units are onneted aording to the periodiit o the pedestrian and the initial onditions are alternatel hanged Gait Parameters In the proess o robot walking, landing point (support point) hanges in turn, we use the step sie and step width to represent and desribe the hanges in the loation, see able 2 below and Figure 4 below. 5

7 MSEE217 IOP Con. Series: Materials Siene and Engineering (218) 1298 doi:1.188/ x/31/1/1298 able 2. Step and step width o the value n ( ) S S ( n ) o ( n) Here, s is a step in the orward diretion, Figure 4. he loation o the robot landing site ( n) s is a step width in the let-right diretion, and (n) is a step in the supersript. With the let oot support point as the origin o oordinates, the n-th ( n) ( n) oothold position ( p, p ) an be epressed as ollows: () () Where (, ) p p s p p s ( n ) n( 1 ) n ( ) ( n ) n( 1 ) n n( ) ( 1) p p is the position o the irst supporting oot, the walking parameters o the n-th step are shown as ollows, /2 ( n) ( n 1 ) ( n) n1 ( n1) ( 1) s / 2 s he above equation is the oordinate o the reerene oordinate sstem, that is, the support point as the origin. At the same time, the n-th walking parameter is determined b the n+1 step and step width. his ondition is used in man preontrols. Aording to the smmetr o the gait unit we an know the walking speed o the walking unit [7]. (9) (1) ( n) ( n) v ( C 1) / ( S) ( n) ( n) v ( C 1) / ( S) (11) 6

8 MSEE217 IOP Con. Series: Materials Siene and Engineering (218) 1298 doi:1.188/ x/31/1/ he Relationship between CoM and ZMP o p C Figure 5. he relationship o CoP and ZMP he robot is in the proess o walking, support legs ontinue to alternate orward, with respet to the world oordinate sstem, the oordinates o the support leg onstantl hanges, the orresponding CoM oordinates also hange, see Figure 5. A swing le, the robot do three-dimensional inverted g pendulum motion, ZMP and CoM satis the dierential equation [1], in the ormula C P, the equation o motion or a linear inverted pendulum relative to the world oordinates is: P C Ca (12) g 5. Swing Leg Ankle rajetor Planning In the single leg support period, the swing leg swings orward rom the bak o the support point to the support point, and when it reahes the ground, it is onverted into a support leg. In the swing leg swinging proess, must go through three ke points, namel the initial point, the middle point and the end point. Using the three dierene method [11], ou an ind the swing leg trajetor. he onstraint equation o interpolation points: s t k X( t) t k 2 s t k hoot t k Z() t H h oot t k 2 hoot t k (13) (14) Where k is an integer, h oot is the height o the ankle, H is the maimum height o the ankle, and s is the step sie. Roboti Swing Legs Ankle trajetor planning, whih primaril onsiders the impat and speed between the oot and the ground. I there is aeleration when the swinging leg alls, it will hit the ground and the swinging leg beomes a supporting leg at a speed o ero. o onsider the maimum height o eet an be ahieved, this an ensure that the robot has a ertain obstale avoidane apabilit. Due to the periodiit o the robot motion, the initial veloit and aeleration o the osillating leg at time k are both ero, and the veloit and aeleration o the osillating leg at the instant o arrival (k+1) are also ero, this minimies the impat between the oot and the ground. hereore, suppose that the irst derivative and the seond derivative o swinging legs (t) and (t) at moment k and (k+1) are both ero, that is: 7

9 MSEE217 IOP Con. Series: Materials Siene and Engineering (218) 1298 doi:1.188/ x/31/1/1298 Xt () t k t k (15) Xt () t k t k (16) Zt () t k t k (17) Zt () t k t k Aording to the robot's own perormane requirements, planning a single step s=.6m, single-step le =.2s, the inal swing leg ankle X-diretion trajetor: (18) ( t).6 15* t 1125* t * t t [ p] (19) In order to eetivel avoid the obstale, planning the swinging leg bottom in the middle o the swing that is 2 t to reah the high point H.1m, he ankle height h.18 movement Z-diretion: oot m, the inal swing leg ( t) * t 216* t * t 18* t t [ p] (2) 6. Conlusion his paper anales the robot's motion arhiteture and determines the relationship between the various joints. hen, the robot's motion is redued to a three-dimensional linear inverted pendulum model, aording to the initial test onditions and the relationship between CoM and ZMP, the trajetor o CoM is obtained. In the orward gait and lateral gait, Aording to the ZMP equation, the stabilit o the gait planning is veriied, the eetiveness o the method is veriied b simulation platorm. o urther improve the speed and stabilit o walking requires more in-depth stud on planning methods and ontrol methods. 7. Aknowledgement Fund Projet: he Inner Mongolia Autonomous Region universit sientii researh projet (NJZY11) 8. Reerenes [1] S. Kajita, and K. ani, Stud o dnami biped loomotion on rugged terrain-derivation and appliation o the linear inverted pendulum mode, in Pro. IEEE Int. Con. Robotis and Automation 1991, pp (1991). [2] S. Kajita and K. ani.dnami Biped Walking Control on Rugged errain Using the Linear Inverted Pendulum Mode.he Soiet o Instrument and Control Engineers,Vol.31,No.1,1995, pp [3] Wisse M, Keliksdal G, van Frankenhen J, Moer B. Passive-based walking robot. IEEE ransations onrobotis and Automation Magaine, 27, 14(2): [4] M Vukobratovi. Biped Loomotion: Dnamis, Stabilit, Control and Appliation [M]. Spring Verlag, Berlin, 199 [5] M Vukobratovi, B Borova. Zero-Moment Point hirt Five Years o Its Lie [J]. International 8

10 MSEE217 IOP Con. Series: Materials Siene and Engineering (218) 1298 doi:1.188/ x/31/1/1298 Journal o Humanoid Robotis (S ), 24, 1(1): [6] Liu L M, ian Y. Swith ontrol between dierent speeds or a passive dnami walker. International Journal o Ad-vaned Roboti Sstems, 212, 9: 241 [7] S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Yokoi, and H.Hirukawa, Biped Walking Pattern Generation b a Simple hree-dimensional Inverted Pendulum Model, in Advaned Robotis, 17, (23). [8] M. Morisawa, S. Kajita, K. Harada, K. Fujiwara, F. Kanehiro, K. Kaneko and H. Hirukawa, Emergen Stop Algorithm or Walking Humanoid Robots, in Proeding o IEEE/RSJ International Conerene on Intelligent Robotis and Sstem, 25. [9] Q.Huang, S.Sugano and K.ani.A High Stabilit Smooth Walking Pattern or biped robot.ieee[c].con.on Robotis and Automation,22,Detroit: