Basic Walking Simulations and Gravitational Stability Analysis for a Hexapod Robot Using Matlab

Transcription

1 Basc Walkng Smulatons and Gravtatonal Stablty Analyss for a Hexapod Robot Usng Matlab Sorn Mănou-Olaru*, Mrcea Nţulescu ** *Department of Automaton, Electroncs and Mechatroncs, Unversty of Craova, Romana (e-mal:manousorn2006@yahoo.com) ** Department of Automaton, Electroncs and Mechatroncs, Unversty of Craova, Romana (e-mal:ntulescu@robotcs.ucv.ro) Abstract: In ths paper the authors present a software program to smulate hexapod robot stablty n gravtatonal feld for a certan confguraton of legs and some basc walkng smulatons usng Matlab software package. Frst the complete knematcal model of the leg s calculated and the drect dynamcal model s presented. The knematcal model of the leg was obtaned usng Denavt Hartenberg algorthm. A workspace analyss of the leg s made n order to analyze collson durng walkng and mpose the necessary constrans. Then a hexapod robot structure usng ths knd of leg s presented and smulated usng Matlab software package. A vrtual smulaton platform was created for the robot n order to smulate robot statc stablty and basc movement algorthm. The analyss of the robot statc stablty s made for dfferent cases of locomoton on horzontal surface and for dfferent leg confguraton. Also the smulaton platform allows connecton of a physcal leg to the computer through Arduno Duemlanove development board n order to smulate dfferent movement algorthms and test the functonalty of the structure. Ths part s an expermental one and wll be mprove n the future. Movng the leg tp from one pont to another s made n two phases: stance phase and swng phase. The paper ncludes some expermental results related to the statc gravtatonal stablty dependng on the support polygon, sngle leg control and some basc walkng smulatons. Keywords: Matlab, gravtatonal stablty, Denavt-Hartenberg representaton, model, hexapod.. INTRODUCTION The nature nvented the leg and humans nvented the wheel. In nature, most arthropods have sx legs to easly mantan statc stablty, and t has been observed that a larger number of legs do not ncrease walkng speed. Moreover, hexapod robots show robustness n case of leg faults. For these reasons, hexapod robots have attracted consderable attenton n recent decades (Bensalem et. al. 2009). Most of the earth s surface s naccessble to regular vehcles. Today s robots are mostly desgned for travelng over relatvely smooth, leveled or nclned surfaces. The terran n queston s ether outdoor envronments, that s generally consdered dffcult for moble robots, or ndoor envronments lke starcases, doorsteps or tght corners can cause dffcultes. An mportant drawback of legged machnes s the complexty of the control requred to acheve walkng even on completely flat and horzontal surface n whch much smpler wheeled machnes work perfectly well (Carbone 2005). The dffculty s not movng the ndvdual legs, but n coordnaton of the movement of the legs and the body. The legged robot control system must generate a sequence of leg and body motons, a gat, whch wll propel t along desred trajectory. Gat generaton s the formulaton and selecton of a sequence of coordnated leg and body motons that propel the robot along the desred path. The most studed problem for mult-legged robots concerns how to determne the best sequence for lftng off and placng the feet. Hexapod gats have been wdely studed as a functon of robot characterstcs. The large dversty of the exstng walkng anmals offers nnumerable examples of the possbltes of ths type of locomoton. The gat analyss and selecton requres an apprecable modelng effort for the mprovement of moblty wth legs n structure/unstructured envronments. Nowadays studes are focused prmarly on usng artfcal neural networks, fuzzy logc or central pattern generators for both leg coordnaton and leg control. It s well known that to mantan a structure s poston n a three dmensonal space requres three pont of support. Machnes wth three or more legs contnuously n contact wth the ground are sad to be statcally balanced f they mantan ther projecton of the centre of gravty wthn the polygon determned by the legs on the support plane. The polygon s known as the support polygon (Fg. ). One of the most studed problem for multlegged robots concerns the stablty analyze durng lftng off and placng the legs. The moton of legged robots can be dvded nto statcally and dynamcally stable. Statc sta

2 blty means that the robot s stable at all tmes durng ts gat cycle. Dynamc stablty means that the robot s only stable when t s movng. For legged robots, statc stablty demands that the robot has at least three legs on the ground at all tmes and the robot s centre of mass s nsde the support polygon,.e. the convex polygon formed by the feet supportng the robot (Fg. ). On the left sde four legs provde support and the centre of mass s located nsde the support polygon so the robot s statcally stable. On the mddle the bottom left leg has been lfted, puttng the centre of mass outsde the support polygon whch made the robot unstable. On the rght sde three legs provde support and the centre of mass s located on one sde of the support polygon. Ths case s called crtcal stablty. The coordnate frames for the robot legs are assgned as n fg. 2. The assgnment of lnk frames follows the Denavt- Hartenberg drect geometrcal modelng algorthm. d q R z L q R 2 R x 0 L L 2 x q 2 x 2 z 2 y 0 Y G Z G X G x z Fg. 2. Model and coordnates frame for leg knematcs. Statcally stable Statcally unstable Fg.. Stablty cases for a hexapod robot: stable, unstable crtcally. Arduno Duemlanove s a development board equpped wth an AVR ATMEGA28 mcrocontroller. Today all the mcrocontrollers are made wth CMOS technology because of the large densty of ntegraton at a lower cost AVR uses Harvard archtecture, wth separate memores and buses for program and data. Some advantages of usng Arduno Duemlanove are: -easy to program. Programmng envronment s easyto-use for both begnners and advanced programmers -open source. Lots of code lbrares are avalable for free for a wde range of external components (sensors, actuator, LCD). 2. ROBOTIC LEG MODEL The successful desgn of a legged robot depends mostly on the desgn of the chosen leg. Snce all aspects of walkng are ultmately governed by the physcal lmtatons of the leg, t s mportant to select a leg that wll allow a maxmum range of moton and that wll not mpose unnecessary constrants on the walkng. 2. Drect Knematcs Crtcally stable A three-revolute knematcal chan has been chosen for each leg mechansm n order to mmc the leg structure (Fg. 2). A drect geometrcal model for each leg mechansm s formulated between the movng frame O (x,y,z ) of the leg base, where =, and the fxed frame O G (X G,Y G, Z G ). The robot leg frame starts wth lnk 0 whch s the pont on the robot where the leg s attached; lnk s the coxa, lnk 2 s the femur and lnk s the tba. Legs are dstrbuted symmetrcally about an axs n the drecton of moton (Y n ths case). The general form for the transformaton matrx from lnk to lnk - usng Denavt Hartenberg parameters [Schllng 990] s gven n (): T cosθ sn θ cosα snθ snα a cos θ snθ cos θ cosα cos θ snα a sn θ = 0 snα cosα d The transformaton matrx s a seres of transformatons:. Translate d along z - axs, 2. Rotate θ about z - axs,. Translate a along x - axs, 4. Rotate α about x - axs. The overall transformaton s obtaned as a product between three transformaton matrxes: base femur tba base Tcoxa = Tcoxa TfemurT (2) tba Consderng fg. 2 and usng (2) the coordnates of the leg tp are: x=cos θ (L + L2 cos θ2 + L cos( θ2 - θ)) y=sn θ (L + L2 cos θ2 + L cos( θ2 - θ)) () z = d + L sn θ + L sn( θ - θ ) where: d s the dstance from the ground to coxa jont. L are the lengths of the leg lnks. The centre of mass of each lnk s postoned relatve to the lnk frame by a poston vector p = [x, y, z, ] T. To fnd the poston of the centre of mass of each lnk relatve to leg frame, the coordnates p are multpled wth the D- H transformaton gvng the centre of mass postons: ()

3 p = T0 p, =... (4) CoM The poston of centre of mass of the leg s calculated usng the followng equatons: x m y m z m x y z m m m l l l = = = g =,, g = g = l l l = = = where m l s the mass of lnk. 2.2 Inverse Knematcs The geometrcal model descrbed above establshes a connecton between the jont varable and the poston and orentaton of the end frame. The nverse knematcs problem conssts of determnng the jont angles from a gven poston and orentaton of the end frame. The soluton of ths problem s mportant n order to transform the moton assgned to the end frame nto the jont angle motons correspondng to the desred end frame moton. The goal s to fnd the three jont varables θ, θ 2, and θ correspondng to the desred end frame poston. The end frames orentaton s not an ssue, snce we are only nterested n ts poston. y Fg.. Illustratons for solvng nverse knematcs. Usng () and consderng the followng constrants: all jonts allow rotaton only about one axs, femur and tba always rotate on parallel axes, and the physcal lmtaton of each jont we can determne the jont angle. The coxa jont angle can be found usng atan2(y,x) functon as can be seen from fg..a. (5) θ = atan 2( yl, xl) (6) In order to determne the other two angles a geometrcal approach was consdered. To further smplfy the approach the leg tp coordnates were transformed to coxa frame usng the transformaton matrx below: T A Leg tp x l,y l θ x y Femur jont Coxa jont T ( ) ( ) Tba jont T coxa coxa coxa R R d = 0 femur femur femur femur coxa L 2 φ 2 φ L a B φ θ x,y x (7) The angle φ 2 whch s the angle relatng to the femur servo poston, can be derved drectly from the trangle (fg..b): θ = ϕ (8) 2 2 The angle φ s the angle between the x-axs and lne a and can be calculated wth atan2 functon: ϕ = atan 2( y, x ) (9) where x and y are the leg tp coordnates n coxa frame. If we consder φ t to be the entre femur span and apply the law of cosnes results: where: L2 + a L ϕt = acos 2 L2 a 2 2 (0) a = x + y () Next the femur angle can be found from: L2 + a L θ2 = acos + atan 2( y, x) (2) 2 L2 a Agan, applyng the law of cosnes we fnd the φ angle: L2 + L a ϕ = acos () 2 L2 L Consderng fg. B we see that θ can be found as follows: 2. Dynamc Model θ = π ϕ (4) The purpose of robot dynamcs s to determne the generalzed forces requred to not only overcome the selfnertal loads (due to the weght of the lnks nsde the robot) but also to produce the desred nput motons to the robot s mechansm (Fahm 2008). In order to obtan a more precse model we dvded the mass of each lnk n two (M - servomotor mass, m -lnk mass, M > m ) (Fg. 4). Consderng the generalzed coordnates vector q= [q, q 2, q ] T the generalzed vector forces can be computed usng the below equaton: d L L τ = dt q& q where: Lqq (,&) = E( qq,&) E( q). c p (5) Consderng that all three servomotors have the same mass M =M 2 =M =M and the last two lnks have the same

4 m,l,i M 2,I 2 M,I θ Ground Fg. 4. Illustratons for developng dynamc model of the leg. mass m 2 =m =m 2 but dfferent lengths the expressons for generalzed forces are: && θ ( I I M ( l R ) m ( r r )) ( M R + m ( r + r )) && ( I I M l 2 m r 2 ) m r& 2 & τ = & θ & & & 2 2 r 2 R r r 4 τ = θ θ m g l θ + θ M + m m2 l2 cos( θ2) (2 M + m+ )]} 2 τ = && θ ( I + I ) g l cos( θ2 + θ) m2 M + m { [ cos( 2 ) ( ) (6) (7) (8) where: I -are the moments of nerta assocated wth the servomotors; I -are the moments of nerta assocated wth the lnks; r -radus of nstantaneous crcle of rotaton of the centre of mass assocated wth the lnk of the leg, =2...4; R -radus of nstantaneous crcle of rotaton of the servomotor.. WORKSPACE ANALYSIS of the LEG There are several thngs that are mportant when defnng the leg workspace. A basc one s that the workspace should not be larger than the reach of the leg. Workspace must be defne n order to maxmze moton but n the same tme to mnmze sngulartes or any other ptfalls (Mak 2007). Workspace s also useful for walkng algorthm to prevent collson of adjacent legs by mposng restrctons for drect or nverse knematcs. The actuators used on the legs are capable of rotatng 80 degrees. In order to mnmze leg collsons the coxa angle was lmted between π/4 and π/4. Ths restrcton was θ 2 z c θ m 2,l 2,I 2 M,I m,l,i ntroduced due to mechancal constructon of the leg. To get the most out of moton the operaton space of varable θ 2 was set from -π/2 to π/4 and θ from 0 to *π/4. In our case there s a sngularty when the x and y coordnates of the leg tp are zero. In ths confguraton the value for θ s arbtrary. To avod ths scenaro we can defne the x coordnate greater than zero. Physcal lmtatons due to mechancal constructon for each jont angle are: q coxa =[-π/4,π/4], q femur =[-π/4,π/2], q tba =[0,*π/4]. The defned workspace wth the above lmtaton s presented n fg. 5, 6 and 7. Fg. 5. Leg workspace (XY vew). Fg. 6. Leg workspace (XZ vew).

5 The robot structure consdered has 6 dentcal legs and each leg has degree of freedom (RRR) (Fg. 9). All the relevant ponts have been put on the model as can be seen from fg. 9: coordnates of the centre of mass of each leg G, =...6; leg numberng for easy understandng ( to 6), coordnates of the centre of mass of the robot G, projecton of the centre of mass nto the support polygon G, robot s centre of symmetry wth the attached frame O R (X R,Y R, Z R ), the global frame O w (X w,y w, Z w ). The global frame s the frame that all other frames wll be defned relatve to. The global frame s rgdly attached to the lower left corner of the world so that the z-axs s vertcal and the xy-plane s algned wth the floor surface. The orgn of the robot coordnates s attached n the centre of symmetry wth the z-axs pontng up, the x-axs pontng left and the y-axs pontng forward. Fg. 7. Leg workspace (YZ vew). The leg workspace wth the mportant pont s presented n fg. 8 and syntheszed n table. Fg. 9. Hexapod robot structure Fg. 8. Leg workspace. Extreme nterest ponts. Tabel. Leg workspace Coordnates Mn Max X -7 47,2 Y -9,6 9,6 Z -22,2 49,8 Vald for l =5, l 2 =25.2, l =7 and for coxa jont coordnates (0, 5, 20) 4. Hexapod Robot Model The legged locomoton on natural terran presents a set of complex problems (foot placement, obstacle avodance, load dstrbuton, general stablty) (Krzysztof et al 2008) that must be taken nto account both n mechancal constructon of vehcles and n development of control strateges. One way to handle these ssues s usng models that mathematcally descrbe the dfferent stuatons. Therefore modelng becomes a useful tool n understandng systems complexty and for testng and smulatng dfferent control approaches (Barreto et al, Fahm 2008). The overall transformaton from global frame to each leg tp s obtaned as follows: where: T T T T coxa = (9) W W R tp R coxa tp W TR s the transformaton matrx between the global frame and the robot frame and defnes the rotaton about y,x,z (roll, ptch, yaw matrx). R Tcoxa s the transformaton matrx from the robot centre to each leg. The poston of robot s centre of mass s calculated usng the followng equatons: x m y m z m X Y Z m m m where: m L g L g L g L = = = G =,, 6 G = 6 G = 6 L L L = = = = m, m l - mass of each leg = l (20)

6 Fg. 0. Hexapod robot control nterface. 5. SIMULATION PLATFORM The man purpose of ths smulaton platform s to show how the support polygon modfes when legs lose contact wth ground and f the projecton of the centre of mass s wthn the support polygon. The smulaton program was made usng MatLab GUIDE (Bran et al 200). The developed software platform can be used to analyze what happens wth the hexapod robot n gravtatonal feld and t also allows communcaton wth the leg n real world. The analyss of the hexapod robot n gravtatonal feld can be group n two modes: - free fall mode. In ths mode the mechancal confguraton of the legs s defned by ther jonts values, no addtonal move s allowed. - transtory analyss. Ths mode s used to analyss what happens wth the robot between two statc regmes. The communcaton between the physcal leg and Matlab envronment s made usng Arduno Duemlanove development board. Gvng a set of values for legs jonts yelds a statonary mechancal confguraton whch generates a certan support polygon n relaton wth whch we analyze the gravtatonal stablty of the hexapod robot. For a certan confguraton of legs, the robot s statcally stable f the projecton of the centre of mass s nsde the support polygon; t s at stablty lmt f the projecton of the centre of mass s on one sde of the support polygon and t s statcally unstable f the projecton of the centre of mass s outsde the support polygon. In ths last case the robot s shftng gradually ts support polygon by lftng/touchng the ground wth ts legs, due to gravty, untl the condton for statc stablty s accomplshed. The shape of the support polygon needed for mnmum statc stablty s the trangle. 5. Program Interface When the nterface s launched the robot s frst drawn n a stable confguraton as shown n fg. 0. The nterface s dvded bascally nto 2 areas: n the upper left the robot s dsplayed (plotted) accordng to the values set by the user and n the upper rght and bottom we can fnd the controls for the robot. The controls for the robot are structured manly n parts: jonts control, poston control, walkng control. Jonts control and poston control conssts of a lst where the leg number t put on and a panel where the user can set the values for each jont or poston (usng drect and nverse knematcs). Walkng control consst of choosng the terran on whch the robot wll move. The terran can be choosng usng the mmedate (controls) checkboxes. The controls for each leg are encapsulated nto a panel dentfed by leg number. Every slder has an edtable textbox where the value s dsplayed and controls a certan lnk of the robot. If we use a slder the assocated edtable textbox value s updated and vce versa. Also the robot leg poston s updated wth the data from the slder

7 or from the textbox. The controls for lnk length or mass affect all the legs because they are consdered dentcal. The button label Leg Control from fg. 0 launches a second nterface lke n fg.. The second nterface allows both hardware and smulated control for a sngle leg. Ths nterface can be used both n offlne mode or onlne mode. - determne the convex polygon - determne the area of the convex polygon (A) - form the n trangle usng 2 consecutve sdes of the support polygon and the projecton of centre of mass, G, (e.g. ABG, BCG etc.) - determne the areas of the n trangles formed (A ) - f (ΣA = A) then condton=true - else condton=false 5. Hardware Leg Control Fg.. Leg control usng Arduno Duemlanove The graphcal representaton of the robot also allows seeng the shape of the support polygon whch s updated accordng to the legs on the ground. The support polygon s drawn only f the dstance from the tp of the leg to the ground s smaller than a threshold. Ths threshold s modfable (but not present n the nterface) and was ntroduced as a way to compensate certan poston errors that may occur due to real servomotors. The smulaton program shows all the stages the robot goes through for a better understandng. For smplcty and better understandng of the robot stages the model s drawn n a smpler way Smulaton Algorthm The authors have elaborated an algorthm n order to acheve the goal of analyzng the statc stablty of a hexapod robot n gravtatonal feld. The algorthm s structured n 5 steps as followng: - settng the jonts values - determne the mechancal confguraton - determne whch legs are on the ground - evaluaton of the statc stablty condton - whle (condton of statc stablty = false) - determne the rotaton lne usng the mnmum dstance from G to support polygon s sdes - rotate the robot about the lne found - determne whch legs are on the ground - evaluaton of the statc stablty condton The determnaton of statc stablty condton s resumed at fndng f the projecton of robot s centre of mass s nsde the support polygon. For ths the authors used the followng algorthm: The system proposed by the authors (Fg. 2) s smlar wth the system xpc-target component of Matlab. The software that make possble the communcaton between Arduno board and Matlab has been released wth the last verson of Matlab. Mathworks has also developed support for Arduno n Smulnk. A part of the communcaton software s uploaded on the Arduno board and plays the server role. There are two programs that can be uploaded on the board: - adosrv.pde wth whch all the nput and output of the board can be manpulated. - motorsrv.pde desgned exclusvely for motor control va a motor sheld. The major drawback of usng the orgnal motorsrv.pde was that ths fle was developed as support for a specfc motor sheld that can only control 2 servomotors. So we modfy the fle n a way that now allows usng all 6 PWM channels avalable. Arduno AtMega28 Matlab USB P W M Fg. 2. Leg control system Leg The other part s a Matlab class (arduno.m) and plays the role of the clent. Once the class s nstantated t makes possble sendng commands over USB port. The drecton of moton of the servo s automatcally determned. If we set a rotaton wth 90 0 and then a 0 0 rotaton the servo wll rotate 60 0 antclockwse. 5.4 Walkng Algorthm Durng walkng or runnng the leg move cyclcally and n order to facltate analyss or control, the moton of the leg s often parttoned n two parts: - support phase or stance, when the robot uses the leg to support and propel. - transfer phase or swng, when the leg s moved from one foothold to the next.

8 The stance part of the walkng algorthm s supposed to move the leg n a straght lne. The swng part of the algorthm must lft the leg off the ground, move t back to the startng poston and lower t down to the ground agan. The walkng algorthm s based on the knematcal model of the leg. Speed Parameters Step length Leg lft force that acts upon the robot s the gravtatonal force. For a gven set of jont values the robot passes through many transtory stages untl t becomes statcally stable (Fg. 4. a, b, and c). Legs that have contact wth the ground determne the shape of the support polygon (trangle, quadrlateral, pentagon or hexagon). In order to know f the robot acheves statc stablty the projecton of G (G ) must be nsde the support polygon. To solve ths problem the above algorthm s appled. Algorthm Walkng control Stance Swng Leg jonts Fg.. Walkng algorthm dagram Parameters ntroduce n the algorthm (Fg. ) are: - speed; defne the speed of the leg tp, - step length; defne the length of the step, - leg lft; ths defnes how hgh the leg s lfted when t s n the swng-cycle. For a smooth straght moton, the swng tme must be equal to the stance tme for each leg. In the stance part of a step, the leg only moves n a straght lne. At tme t from the start of the step, the coordnates (x, y, z) for the trajectory wll be: x= leglength steplength y= t* speed 2 z = legheght (2) In the swng part of the step, the leg frst has to be lfted of the ground, and then moved back to where the next step s supposed to start and then lowered to the ground. To fnd out where n the swng-cycle the leg s at tme t from the start of the cycle we frst have to calculate the total length the leg has to travel: (2 * leglft steplength)* t dst = swng _ cycle (22) One step conssts of one stance cycle and one swng cycle but to mantan a smooth moton of the leg, t s mportant that the swng cycle contnues where the stance cycle ended, and that the swng cycle ends where the new stance cycle starts. Ths does not only hold for the poston, but also for the speed and the drecton of the speed. 6. EXPERIMENTAL RESULTS 6. Free fall analyss The free fall analyss represents what happens wth the robot left to fall on the ground from a heght greater than the extenson of the legs. Keepng n mnd that the jonts are locked by the values prescrbed by the user, no extra movements are allowed (no actve stablty). The only Fg. 4.a: Phase one of fallng. In fg. 4.b t can be seen that even n ths confguraton G s not nsde the support polygon and the robot contnues t s fallng and rotates about the lne determned by leg 2 and leg 5 untl the frst leg touches the ground, whch n ths case, s leg. Fg. 4.b: Phase two of fallng. In fg. 4.c a new confguraton s formed and f the algorthm descrbed above t s appled, pont G s nsde the support polygon and the robot becomes statcally stable and the fallng stops. Fg. 4.c: Phase three statcally stable. 6.2 Transtory Analyss In ths mode of analyss the robot passes between two statc regmes. A statc regme s dentfed by the condton of stablty. The user can alter a statc regme usng the controls for jont values or poston of the leg tp. Ths analyss can also be nterpreted as a contnuaton

9 of free fall analyss case f the condton of stablty has been met. In fg 5.a the robot has statc stablty, the support polygon descrbed by the legs on the ground s a quadrlateral (formed by legs, 2, 5 and 6) and the projecton of the centre of mass s nsde the support polygon. 6.. Hardware Leg Control The algorthm for controllng the leg jonts, ncludng drect knematcs and nverse knematcs, was mplemented n Matlab n order to analyze leg performance. The results of the tests can be vew n the chart below. Fg. 5.a: Phase one statcally stable. Next, lftng leg 2 the support polygon changes ts shape becomng a trangle. Usng the algorthm descrbed above, the projecton of G s not nsde the support polygon and the robot becomes statcally unstable and starts fallng (Fg. 5.b). Fg. 6. Hardware leg control results. The man errors occurred due to the fact that we can only send nteger numbers for jont values. Other errors occurred due to the fact that the jont are assembled usng bolts and nuts. Normal usage of the robot causes the nuts and bolts to loosen causng a lot of clearance n the jonts. 6.4 Inclned Plane Walkng Approach The movement on nclned plane s realzed by movng one leg from one sde at the tme as can be seen from fg. 7, 8 and 9. As mpose restrctons for the robot we can enumerate the followng: the robot body must be leveled wth respect to the ground, coxa angle s set for π/4. The angle for the plane s 0 0. Fg. 5.b: Phase two: fallng Followng the algorthm the next leg closest to the ground s leg number 4. In fg. 5.c the projecton of G s nsde the newly support polygon, the robot stops fallng and becomes statcally stable. Fg. 7. Movng leg Fg. 5.c: Phase three statcally stable Fg. 8. Movng leg.

10 After ths cycle of movement the body s dragged forward agan and the algorthm resumes. Fnally the robot wth all the legs are on the nclned plane wth ts body leveled as can be seen from fg. 24. Fg. 9. Movng leg 5. Once all the legs from one sde have been moved to a new poston the robot body wll be dragged usng all the coxa jonts so that the robot can move forward (Fg. 20). Fg. 24. Fnal poston of the robot on the nclned plane. Also t can be seen the trajectory of the robot on the nclned plane. Fg. 20. Robot body drag. After the body has been dragged to the new poston the legs 2, 4 and 6 from the other sde must be moved forward (Fg. 2, 22 and 2). Fg. 25. Trajectory of the robot durng walkng Fg. 2. Movng leg Star Step Walkng Approach As wth the prevous case of movement, ths case respects the algorthm descrbe n the prevous chapter. The movement on star step s realzed by movng one leg from one sde at the tme as can be seen from fg. 26, 27 and 28. As mpose restrctons for the robot we can enumerate the followng: the robot body must be leveled wth respect to the ground, coxa angle s set for π/4. The heght of the star step s / of the heght of the robot. Fg. 22. Movng leg 4. Fg. 2. Movng leg 6. Fg. 26. Movng leg.

11 Fg. 27. Movng the leg. Fg.. Movng leg 4. Fg. 28. Movng the leg 5. Once all the legs from one sde have been moved to a new poston the robot body wll be dragged usng all the coxa jonts so that the robot can move forward (Fg. 29). Fg. 2. Movng leg 6. After ths cycle of movement the body s dragged forward agan and the algorthm resumes. Fnally the robot wth all the legs are on the star step wth ts body leveled as can be seen from fg. Fg.. Fnal poston of the robot. Fg. 29. Robot body drag Also t can be seen the trajectory of the robot on durng walkng. After the body has been dragged to the new poston the legs 2, 4 and 6 from the other sde must be moved forward (Fg. 0, and 2). Fg. 0. Movng leg 2. Fg. 4. Trajectory of the robot durng walkng

12 7. CONCLUSION In ths paper a smulaton platform for legged moble robots was presented, that allows stablty analyss and full control of the robot. Free fall analyss s useful for nvestgatng what happens wth the robot on uneven terran or for accdents that may happen due to lose of contact, slppery surface, servomotor falure, power supply falure. Transtory analyss represents a more mportant case for locomoton, gat generaton. When startng to develop a gat cycle we can have a bg pcture of what happens when the robot starts to move, how the support polygon changes and what actons (what leg should be actuated) must be appled to the robot n order to meet condton of stablty. The nterface wll be mbung wth addtonal walkng algorthms and controls. The nterface was desgned to be smple and ntutve and to offer the user a smple and effcent way to control every aspects of the robot (angles, masses, lengths, leg control). The two analyses made n ths artcle represent the bases for next actvtes on statc stablty. Hardware leg control wll be mbung to better precson. In the future the program wll be upgraded permttng addtonal controls and functons for stablty analyss (ncludng dynamc stablty) on uneven ground and mplementng collson detecton algorthms. Also the results of these studes represent the bases for dfferent strateges of locomoton on dfferent terrans. The expermental results wll become a standard for a real hexapod robot. ACKNOWLEDGMENT Ths work was partally supported by strategc grant POSDRU/88/.5/S/5078, Project ID 5078 (2009), cofnanced by the European Socal Fund Investng n People, wthn the Sectoral Operatonal Programme Human Resource Development REFERENCES Barreto J., Trgo A.,Menezes P., Das J., Knematc and dynamc modelng of a sx legged robot. Bensalem, S.,Gallen, M., Ingrand, F., Kahloul, I., Nguyen Thanh-Hung (2009), Desgnng autonomous robots, IEEE Robotcs & Automaton Magazne, Vol. 6, March Bran R., Hunt R., Lpsman L., Rosenberg J. M.,(200), A gude to MATLAB for begnners and experenced users, ISBN-I: , Cambrdge Unversty Press Carbone G., Ceccarell M., (2005), Legged robotc systems, Cuttng Edge Robotcs, ARS Scentfc Book, pp , Wen. Fahm, F. (2008), Autonomous robots: modelng, path plannng, and control, Sprnger. Krzysztof W., Domnk B., Andrzej K.,(2008) Control and envronment sensng system for a sx-legged robot, Journal of Automaton, Moble Robotcs & Intellgent Systems, Volume 2 N o. Mak K. H.(2007) Bonspraton and Robotcs:Walkng and Clmbng Robots, ISBN , I- Tech Educaton and Publshng, Croata Schllng R. J. (990), Fundamentals of robotcs: analyss and control, ISBN: , Prentce Hall, New Jersey, USA. 7S.pdf