Design of a Ball Handling Mechanism for RoboCup

Transcription

1 Design of a Ba Handing Mechanism for RoboCup T. Bouens, DCT Bacheor Fina Project Superisor: Coach: Prof.dr.ir. M. Steinbuch J.J.T.H. de Best Eindhoen Uniersity of Technoogy Department of Mechanica Engineering Dynamics and Contro Technoogy group Eindhoen, June, 009

2 Abstract This report describes the design, reaization and testing of a ne ba handing mechanism for Tech United, the Robocup team of the Eindhoen Uniersity of Technoogy. The goa of this project is to enabe the robot to receie the ba and dribbe ith the ba. The report i describe in detai the preious concepts that ere impemented in the different ersions of the robots. It i describe the improements made on the ne concept. The ne concept inoes motorized hees attached to eers mounted under an ange of 90 degrees ith respect to each other and optica sensors measuring the ange of the eer in respect to the robot. These hees exert forces on the ba in order to contro it. A theoretica mode of the ne concept has been inferred, using the Lagrange equations of motion, resuting in a inearized mode around a desired equiibrium point. This mode is erified by measuring frequency response on the experimenta setup. From this comparison a controer has been deised. This controer uses a cascaded structure, meaning it impements both a position and a eocity contro oop to contro the ange of the eer and therefore the position of the ba. The eocity contro oop is added to add the option of introducing feed forard information of the moements of the robot itsef. These moements are the biggest knon disturbances on the ba handing mechanism. This cascading resuts in a cosed oop system ith a bandidth of approximatey 1 Hz.

3 Contents 1. Introduction 4. Design criteria 5 Rues and reguations 5 Tech United criteria 5 3. Preious concepts 6 Static or passie contro: 6 Actie controed mechanism ithout ba position feedback 7 Actie controed mechanism ith ba position feedback 8 4. Current concept 10 Layout 10 Sensors Identification and Contro 11 Theoretica mode 11 Practica Impementation 15 Identification Resuts 1 7. Concusion and Recommendations 3 Concusion 3 Recommendations 3 8. References 4 9. Appendix 5 M-fies used in FRF measurements 5 3

4 1. Introduction RoboCup 1 is an internationa project that promotes A.I. (Artificia Inteigence), robotics, and reated fieds. It is an attempt to foster A.I. and inteigent robotics research by proiding a standard probem here a ide range of technoogies can be integrated and examined. RoboCup chose to use the soccer game as a centra topic of research, aiming at innoations to be appied for sociay significant probems and industries. The utimate goa of the RoboCup project is to deeop a team of fuy autonomous humanoid robots that can in against the human ord soccer champion by 050. Robocup consists of arious eagues. There are separate soccer eagues for arious heeed, egged and humanoid robots, but there are aso simuation and rescue eagues. The Robocup team of the Eindhoen Uniersity of Technoogy, Tech United, competes in the Midde Size League, in hich heeed robots autonomousy pay soccer. This report describes the design, reaization and testing of a ba dribbing mechanism for the Tech United TURTLE (Tech United Robot Limited Edition) robots. Various efforts hae been made to buid a mechanism to contro the ba, but this sti is an ongoing process. The goa of this project is to describe the preious concepts and eauate and contro the current concept. The outine of the report is as foos. First, the design criteria i be presented in chapter. Chapter 3 i discuss the preious concepts of stopping and dribbing ith the ba. This is fooed by a description of the current concept in chapter 4 and the system identification and controer design in chapter 5. Finay, the resuts of these tests and the design itsef i be eauated in chapter 6. 4

5 . Design criteria This section i describe the design criteria to hich the ba handing mechanism has to compy ith. Firsty the rues and reguation of the Robocup competition and secondy the criteria set by the Tech United Robocup team are described. Rues and reguations This section describes the Rues and Reguations of the Mid-League soccer game 3 regarding ba handing mechanisms. These rues are as cose as possibe to the FIFA rues but aso contain extra parts that are reeant for ba handing. These rues are as fooed: During a game, the ba must not enter the conex hu of a robot by more than a third of its diameter except hen the robot is stopping the ba. The ba must not enter the conex hu of a robot by more than haf of its diameter hen stopping the ba. The robot may exert a force onto the ba ony by direct physica contact beteen robot and ba. Forces exerted onto the ba that hinder the ba from rotating in its natura direction of rotation are aoed for no more than 4 seconds and a maxima distance of moement of one meter. Exerting this kind of forces repeatedy is aoed ony after a aiting time of at east 4 seconds. Natura direction of rotation means that the ba is rotating in the direction of its moement. Vioating any of the aboe rues is considered ba hoding. Tech United criteria The goa of the ba handing mechanism for the Tech United TURTLE (Tech United Robot Limited Edition) is to catch the ba and keep it during a dribbing motion of the robot hie obeying the rues mentioned aboe. Another important criteria is the actie ba handing. Most competitors use a static construction, consisting of points and/or edges to guide the ba, reying on the robot itsef to contro and hande the ba. The designed ba handing mechanism for the TURTLE has to consist of parts that are actiey drien to exert extra forces onto the ba in order to change eocity and position in reation to the robot. 5

6 3. Preious concepts This section i describe the preious concepts that ere impemented in the Tech United Turtes. They can be subdiided into 3 different categories. Firsty, the static or passie contro mechanism. Secondy, the actie controed mechanism ithout ba position feedback and thirdy the actie controed mechanism ith ba position feedback. Static or passie contro: Static or passie contro means that the ba handing mechanism is not an actiey actuated mechanism, but consists of points and/or edges, as seen in Fig 3.1. These guide the ba in the desired direction ithout preenting the ba to ro in its natura direction of rotation. The first probem ith this concept is that the robot has to moe around the center of the ba to change direction, as seen in Fig 3., hich eads to more compex trajectory panning. Catching the ba is the second probem. Whie the robot is catching the ba the eocity of both robots shoud be synchronized to preent the ba from bouncing aay from the robot. Fig 3.1 Passie contro. Desired ba trajectory Robots position in reation to the ba Fig 3. Robot trajectory panning for static contro. 6

7 Actie controed mechanism ithout ba position feedback Actie contro means that the ba handing mechanism consists of parts that are actiey drien to exert extra forces onto the ba in order to change eocity and position in reation to the robot. The first ersion of this concept had motorized rubber bets, shon in Fig 3.3, to push or pu the ba according to the moement of the turte. This eocity is estimated by using the onboard encoders mounted on each of the Omni hees. Changing the bas direction sti inoed moing the robot therefore a compex trajectory as sti necessary. The contro of the ba mechanism hoeer as imited to going forards and backards because the bets ere paced parae to each other. A simpified schematic can be seen in Fig 3.5a. The parae bets are represented by a singe actuator, here the red arro is the appied torque and the hite arro is the direction of rotation. Fig 3.3 Actie by controed bets. The next ersion had to 4-bar inkage systems ith motorized hees. The eers are paced under an ange of 90 degrees ith respect to each other as seen in Fig 3.4. This as an improement oer the bets because they did not get cut that easiy and they contacted the ba at an ange reatie to each other, thus enabing a more directiona contro of the ba. A simpified schematic can aso be seen in Fig 3.5b. The 4-bar inkages are represented by to actuators, here the red arro is the appied torque and the hite arro is the direction of rotation. Because of the different rotation directions some sideays moement of the ba is possibe. Fig 3.4 Actie by controed hees. Possibe ba moement Actuator Fig 3.5a Bets. Fig 3.5b Whees. 7

8 Actie controed mechanism ith ba position feedback In the preious concept the bas position ith respect to the robot as unknon and it as assumed that the ba as in the correct position to be abe to moe the ba forards or backards. In order to get the desired position feedback a simpe potentiometer as mounted at a hinge point of the 4-bar inkage such that the configuration can be measured. The signa of this sensor as used to contro/adjust the eocity of the hees. Adding a sensor that gae the position of the ba ith respect to the robot added the feedback necessary for controing the ba ithout haing to moe the robot itsef. Desired ba moement Robots position in reation to the ba Fig 3.6 Robot trajectory panning for actie contro. First of a, it creates the opportunity to drie sideays and turning around the center of the ba instead of the center of the ba, hie sti possessing the ba. Secondy, the position of the ba ith respect to the robot can be controed ithin the range of the ba handing mechanism. This property can be expoited during a kick, here the ba can be pued against the kicker or can be paced to the eft or to the right of the kicker, hich enabes the robot to aim during a shot, ithout rotating the robot itsef. Furthermore, during a dribbe the ba i rotate in its natura direction at a times, independent of the texture of the fied. Finay, the trajectory panning probem becomes much simper since it is not necessary to rotate around the ertica axis of the robot in order to change the direction during a dribbing motion, as seen in Fig

9 The 4-bar inkage turned out to be too big for the ne TURTLE robots and as repaced ith a singe eer, as seen in Fig 3.7. During testing this system orked great, making sure the ba as kept at the correct position during moing and enabing more compex moements to aoid opponents hie keeping the ba in possession. It turned out to be the opponents and other robots that ere the next big probem. The mechanism as not robust enough to cope ith coisions ith other robots and oud break or, resuting in improper functioning. Fig 3.7 Actie controed eer. The next ersion of the concept as designed to hande these coisions, hie sti being abe to contro the ba. A spring as attached to the eer to absorb the impacts, see Fig 3.8. The spring as stiff enough to preent infuencing the ba handing behaior, but compiant enough to fex on impact. The fex proed to be insufficient to preent damage but a softer spring oud infuence the ba handing, so a different concept as desired. Fig 3.8 Actie controed eer ith spring. 9

10 4. Current concept This section i describe the current concept of the ba handing. This concept as produced using the proen ba handing principa used by the preious TURTLEs but improing the robustness. Firsty, the ne ayout ith the improements of the ba handing mechanism i be described and finay the ne sensors that are used are discussed. Layout The preious concept had the eers mounted hafay up the robot hich needed extra bracing on the robot, making the construction more spacious and compex. The ne design has the eers of the ba handing mechanism mounted on a rigid base pate, making a ery compact construction possibe. The eers are repaced by stronger A-arms and more protection is added to the hees, as shon in Fig 4.1. The A-arms are mounted on the base pate using rubber-cushioned ba bearings to sti be abe to absorb energy during coisions. Sensors Fig 4.1 The ne concept. Motor/Whee Optica sensor A-arm Rubbercushioned ba bearing The potentiometers are repaced ith optica sensors because the attachment of the eer has changed from a fixed hinge to the rubber-cushioned bearings, (see fig 4.1) This preents the potentiometer from being mounted at the hinge-point. The optica sensors no detect the distance beteen the robot and the hee directy, instead of the ange of the eer. 10

11 5. Identification and Contro Theoretica mode This section describes the theoretica mode for the ne ba handing mechanism. The - dimensiona simpified mode of the ba handing mechanism is schematicay depicted in Fig In this figure, the cart represents the soccer robot. Fig 5.1: The theoretica mode. A eer is mounted on the robot that can freey rotate around a fixed axis on the robot by means of a hinge. The height of this point is ocated at a distance h beo the center of the ba as can be seen in Fig The eer has a ength, a mass m and a rotationa inertia J. At the end of the eer a hee ith mass m and rotationa inertia J is mounted. The radius of the hee is denoted by R. A torque T can be appied beteen the hee and the eer by means of a motor to contro the ange of the eer and to counteract the iscous damping d that is present beteen the eer and the hee. During dribbing the hee is assumed to be in contact ith the ba at a times. The ba has a radius R b, a mass m b and a rotationa inertia J b. Furthermore, it is assumed that there is no sip beteen the hee and the ba nor beteen the ba and the foor. The motion of the robot is prescribed and is denoted by s(t). The idea is to measure the ange of the eer φ and maintain this ange at a preferred ange, hich corresponds ith a desired distance beteen the ba and the robot. 11

12 Lagrange equation of motion Using Lagrange equation of motion 4 to describe the system, the position ectors of the bodies center of mass, i.e. r, r, r b are deried as a function of the generaized coordinate φ and the prescribed coordinate s(t) and are gien by r r r b s = 1 () t cos( φ ) 1 sin( φ ) () t cos( φ ) sin( φ ) s + = s + =,, () t + cos( φ ) + ( R + R ) ( sin( φ ) h) b h. [1] Aso the rotation ector of each body, i.e. θ, θ, θ can be cacuated as a function of the generaized coordinate φ and the prescribed coordinate s(t), gien by b θ = φ, s θ = b s θ = () t + cos( φ ) + ( R + R ) sin( φ ) b R sinφ h ( h) arcsin( ) () t + cos( φ ) + ( Rb + R ) ( sin( φ ) h). R b R R + R b, [] Using the position ectors the eocity ectors of the bodies can be cacuated as b dr dφ dr = + dφ dt ds dr dφ dr = + dφ dt ds dr b dφ dr b = + dφ dt ds ds dt ds dt ds dt dr dr = & φ + s&, dφ ds dr dr = & φ + dφ ds dr b dr = & φ + dφ ds b s&. s&, [3] 1

13 The rotationa eocities can be cacuated simiary using the rotation ectors, gien by dθ dφ dθ ω = + dφ dt ds dθ dφ dθ ω = + dφ dt ds dθ b dφ dθ b ω b = + dφ dt ds ds dt ds dt ds dt dθ & dθ = φ + dφ ds dθ b & dθ b = φ + dφ ds s&, dθ & dθ = φ + dφ ds s&. s&, [4] With these resuts the kinetic energy of the system becomes K = i {,, b} 1 m i T i i + 1 J ω i T i ω. i [5] The potentia energy can be ritten as V ( φ ) + m g sin( ) m gh 1 = m g sin φ, [6] + b here g is the graity constant. With the potentia and kinetic energy knon the Lagrange s equations of motion can be cacuated as d dt dk dk dv T Qnc d + = &, [7] φ dφ dφ here Q nc are the non-conseratie forces. These forces consist of the appied torque T and the torque caused by the damping d T = d & φ & φ beteen the eer and the hee modeed as ( ) These non-conseratie forces are formuated as d Q nc dθ = dφ dθ ( Td T ). φ [8] Finay, a non-inear mode can be constructed hich has the form ( φ, & φ, s& () t,& s () t, T ). & φ = f [9] 13

14 Linearization In the preious section the non-inear mode of the ba handing mechanism as deried. For the design of a stabiizing feedback controer of the feedback controer the non-inear mode is inearized around the fooing desired equiibrium point π φ = & φ s & & 0, T [10], eq,, eq = 0, eq = 0, s 4 eq = eq In the choice of this equiibrium point it is checked hether or not the chosen equiibrium point satisfies the preferred distance beteen the ba and the robot. Other equiibrium points may be chosen as e, since this does not infuence the presented contro method. The torquet eq, associated ith the chosen equiibrium point, can be cacuated by substituting the aues of [10] into [9] and soing for T eq from ( φ, & φ, s&,& s T ) 0 &, eq, eq eq eq = f, [11] eq Linearization around the point of operating resuts in the fooing second order inear mode x& = Ax + Bu G :, [1] y = C x here the state ector is gien by [ ] T output y is defined as φ φ, gien by eq x = φ φ & φ, the input u is gien by T Teq and the, eq. The system matrix A, input matrix B and output matrix C are A = df df, B = df, C = dφ d & φ dt [ 1 0], [13] here A and B are both eauated at the equiibrium point, [10]. 14

15 Practica Impementation This section i describe the controer structure used for the ba handing system. The structure i consists of to contro oops, as seen in Fig 5.. One contro oop for the ange of the eer and the other for the rotationa eocity of the hee. The optica sensor attached to the eer gies the position feedback for the controer. This feedback is then used to contro the motor/hee ith the eocity contro oop. The motor has a tachometer attached to the shaft for rotationa feedback. ϕ&,r () t + - T( t) H C 1,1 1 H 1, ϕ ( t) ϕ& ( t) () t ϕ,r + - ϕ& ( ) ( t),r t C H ϕ Fig 5.: The cascaded controer diagram. The main reason for this cascading is the robot itsef. The moement of the robot is the biggest knon disturbance on the ba handing mechanism and using a eocity contro oop this moement can be anticipated, making the handing more accurate. Furthermore it adds fexibiity to the controer. Een if the robot does not possess the ba, the eocity of the hees can be controed, for exampe for catching the ba. ϕ&,r () t T () t ϕ& ( t) + - C 1 H 1, Figure 5.3: Veocity contro oop consisting of controer C 1 and pant H 1,. The rotationa eocity contro oop gies an input torque T ( t) to the pant H 1,, as seen in Fig 5.3, based on the error beteen the ϕ& t and reference output eocity ( ) eocity ϕ& ( t).,r The position contro oop gies an input reference eocity ϕ&,r () t to the pant H, based on the error beteen the output eer ange ϕ () t and reference ange ϕ () t, as seen in Fig 5.4.,r ϕ ( t) ( t) ( t),r + ϕ&,r - C H Figure 5.4: Position contro oop consisting of controer C and pant H. ϕ 15

16 Identification This section i describe the identification of the three pants H 1,1, H 1, and H in order to determine the controers for an effectie ba handing system. These pants represent transfer functions. In order to determine these transfer functions or the transfer function estimates (tfe) a Fourier anaysis of the input and output signas can be used. This anaysis orks on the basis that eery signa consists of mutipe trigonometric functions. Using a hite noise signa as a disturbance on the system, the response of the system can be measured. White noise is a random signa ith a fat poer spectra density. In other ords, the signa contains equa poer ithin a fixed bandidth at any center frequency. Using the Fourier anaysis on both the noise signa and the input signa, the response of the system to a specific input frequency can be cacuated. This produces a Frequency Response Function (FRF) pot of the pant. The FRF shos the gain and phase potted ersus the frequencies. ϕ&,r () t e() t () ϕ& ( t) () t T t C 1 H 1, () t ϕ,r () t e() t () ϕ ( t) ϕ& C,r t H Fig 5.5: Measurement diagram for both eocity and position contro oop In a Fourier anaysis the input signa u(t), a reference signa r(t) and a hite noise signa (t) hae to be measured. According to Fig 5.5 these input signas are T(t) and ϕ&,r ( t) for the eocity an position contro oop. The hite noise signa (t) is the same in both cases. The reference signa is 0 for both oops. The Fourier anaysis uses a Fourier transformation on the signas, hich has the form of F () s f () t here st = e dt s = σ + iω. [14] Using [14] e get the fooing transformed signas; T () t U ( s), () t W ( s), () t Y ( s), () t E () s Y () s ϕ& e ϕ, r () t U p ( s), () t W p ( s), () t Yp ( s), () t E () s Y () s ϕ e p p for the eocity oop and for the position oop. 16

17 Using the fooing formuas the transfer function estimate of the pant H 1, () s can be determined for each frequency. Firsty the Sensitiity function of the eocity contro oop, the transfer of noise W () s to input U () s, the measured signas. () s () s U S () s =. [15] W Rearranging the components, [15] can be ritten as U U U () s = W () s + C1() s E () s, () s = W () s C1() s Y () s, () s = W () s C () s H () s U (). s 1 1, [16] This eads to 1 S () s = [17] 1+ C () s H (). 1 1, s Secondy the Process Sensitiity of the eocity contro oop, the transfer of error E ( s) to noise W () s, is deried. E E E () s = Y () s, () s = H1,() s U() s, () s = H () s C () s E () s + W () s 1, ( ). 1 [18] This eads to 1, () s () s H () s H PS () s =. [19] 1+ C1 1, Combining [17] and [19] the transfer function estimate of ( s) () s () s H, 1 has the fooing form, PS H1,() s =. [0] S 17

18 Pant H 1,, as described in the preious section, is the transfer function of input motor torque () t ϕ& t of the hee. T to the rotationa eocity ( ) Using [0] and measuring the input T ( t) and the hite noise signa ( t), the transfer function H 1, can be determined and potted as an FRF, as seen in Fig 5.6. This is done in MatLab and Simuink, using m-fies to do the cacuations; these can be found in the Appendix Pant Ba Handing Veocity Left Right Ampitude [db] Phase [deg] Frequency [Hz] Fig 5.6: Measured frequency response function for eocity contro oop From this FRF the infuence of the inertia of the system can ceary be seen in the - sope of the ampitude ersus frequency pot. 18

19 Pant H coud be determined simiary, measuring the input eocity ϕ&,r () t and the output eer ange ϕ () t, to determine the transfer beteen them. The robot hoeer uses to eers in order to contro the robot in a -dimensiona enironment and grab and hande the ba. This eads to a muti-input muti-output (MIMO) system, since there are no to actuators that can infuence the motion of the ba. Furthermore, there are to sensors to measure the ba position. In Fig 5.7 the FRF of each eer and the cross-couping beteen eers can be seen. 0 Transfer of right input to right output 0 Transfer of right input to eft output 0 0 Ampitude [db] Ampitude [db] Frequency [Hz] Frequency [Hz] 0 Transfer of eft input to right output 0 Transfer of eft input to eft output 0 0 Ampitude [db] Ampitude [db] Frequency [Hz] Frequency [Hz] Fig 5.7: Measured frequency response function for the position contro oop in a MIMO system. By pacing the eers under an ange of 90 degrees ith respect to each other, seen from the top ie, the cross couping of the muti-input muti-output system can be minimized. This minimization is aidated ia the measured reatie gain array 5 (RGA). The reatie gain array proides a measure of interaction hich is independent of input and output scaing. ~ ~ ~ 1 The RGA is defined as Λ ( H ) = H ( H ) T, here H ~ is the to-input to-output pant and denotes eement-by-eement mutipication (the Schur product). In order to appy decentraized contro it is preferred that this matrix is cose to the unity matrix I for a frequencies. 19

20 From Fig. 5.8 it can be seen that the system can be considered to be decouped up to approximatey 60 Hz. For this reason, the aimed bandidth of the feedback controer i hae to be ess than 60 Hz in order to use a singeinput singe-output (SISO) controer. The measured FRF for H is then reduced to ony the diagona terms. Fig. 5.8: The measured reatie gain array ϕ, In order to get H 1,1, the transfer of input motor torque T ( t) to the output eer ange ( t) the same kind of measurements coud be done. Hoeer after some simpifying of the origina bock diagram (Fig 5.9), it can be concuded that H C 1 = H1,1, 1+ C1H1, here the eocity contro oop is depicted by C1 1+ C H 1 1,. This eads to H 1,1 1 = + H1, H C. 1 So H 1,1 is a combination of the measured pants H 1, and H. The ony parameter in the equation is controer C 1, hich has been chosen 1 for the measurements. This eads to H ( + H1, ) =. 1,1 1 H 0

21 6. Resuts This section describes the resuts obtained from the theoretica mode compared ith the frequency response measurements. The parameters of the theoretica setup are gien in Tabe 6.1: Parameter Description Vaue Unit Length of the eer 0. [m] m Mass of the eer 0.4 [kg] J Inertia of the eer 1 m [kgm ] r Radius of the hee 0.08 [m] m Mass of the hee 0.1 [kg] J Inertia of the hee m r 0.89 r b Radius of the ba π [kgm ] m b Mass of the ba 0.43 [kg] J b Inertia of the ba m b r b 3 [m] [kgm ] d Damping beteen eer and hee [Nmrad/s] h Height of rotation point of ba 0.1 [m] g Graity constant 9.81 [m/s ] Tabe 6.1: Parameters of the ba handing setup With these parameters the inearized mode G [1] becomes 0 x& = 6.43 y = [ 1 0] x 1 0 x + u [0] The poes of this system are ocated at 0.93 and -8.9, hich indicates that the considered system is unstabe. The inearized system gies the theoretica transfer beteen input torque T(t) and the output ange ϕ () t hich is the same transfer as the pant H 1,1. The inearized system can be erified by comparing the FRF of the system ith the measured FRF of pant H 1,1. This comparison can be seen in Fig

22 It can be seen that the mode matches the measured frequency response measurement for o frequencies. Hoeer, beyond 40 Hz, the mode does not match the measured FRF s. Unmodeed artifacts of the ba handing mechanism sho up in the measured frequency response function. One of these artifacts might represent the finite stiffness of the tires of the ba handing hees. Moreoer, since the contro agorithm is impemented in discrete time the system aso inherenty suffers from time deay, hich is aso not taken into account in the modeing of dynamics of the ba handing mechanism. 50 Pant Ba Handing Left Right Mode Ampitude [db] Phase [deg] Frequency [Hz] Fig 6.1: Cacuated frequency response function for the pant H and the theoretica mode. By using oop shaping techniques 6, an output feedback controer ith a singe proportiona gain can be used to stabiize the system gien in [0]. This feedback controer C is gien by u = e = r y. Then, Fig. 6.1 can be interpreted as the open oop of the system, ith a bandidth of approximatey 1 Hz. The cosed oop poes are gien by 11.3 ± 4.7i, hence a stabe cosed oop system is obtained.

23 7. Concusion and Recommendations Concusion Preious concepts ere eauated for eak and strong points. A ne setup as deised that incorporated the strong points and improed on the eak points. A non-inear mode as formuated for the setup, hich after inearization is used for the feedback contro design, eading to a stabe cosed oop system ith a bandidth of approximatey 13 Hz. The mode is compared to experimenta resuts and matches these resuts up to a frequency of 40 Hz. The difference can be expained by unmodeed artifacts in the system and time deay due to discrete contro agorithms. Recommendations The fooing improements of the ba handing mechanism for the Tech United TURTLEs can be made: 1. Eauating the step response of the controer, further improing the handing. This i decrease the setting time of the mechanism on arge disturbances, ike catching the ba.. Impement feed forard into the contro strategy, improing the dynamics of the system during moing. The motion of the robot itsef can be anticipated resuting in a more accurate handing. Furthermore it adds fexibiity to the controer. Een if the robot does not possess the ba, the eocity of the hees can be controed. 3

24 8. References [1] RoboCup, RoboCup Officia site, [] Tech United. Eindhoen Uniersity of Technoogy: (009). [3] RoboCup Rues and Reguations [4] A. de Kraker, D.H. an Campen, Mechanica Vibrations, Shaker Pubishing BV, (001) [5] S. Skogestadt and I. Postethaite, Mutiariabe Feedback Contro, Anaysis and, Wie, (007) [6] G.F. Frankinn, J.D Poe, A Enami-Naeihi, Feedback Contro of Dynamic Systems, Addison-Wesey Pubishing Company (1994) 4

25 9. Appendix M-fies used in FRF measurements Bahanding_tota.m cose a cear a cc oad G_bapos1.mat oad G_bapos.mat oad G_bae1.mat oad G_bae.mat G_ba1 = (1+15*G_bae1).*G_bapos1; G_ba = (1+15*G_bae).*G_bapos; F=[0 1; ]; G=[0;757.47]; H=[1 0]; J=[0]; sys=ss(f,g,h,j); frfsys = 10*squeeze(freqresp(sys,*pi*f)); figure subpot(11) semiogx(f,db(abs(g_ba)),'coor','r','linewidth',1) yabe(['ampitude [db]']) tite(['pant Ba Handing']) grid on hod on semiogx(f,db(abs(g_ba1)),'coor','c','linewidth',1) hod on semiogx(f,db(abs(frfsys)),'coor','k','inestye','--','linewidth',) egend('left','right','mode') subpot(1) semiogx(f,ange(g_ba)./pi*180,'coor','r','linewidth',1) xabe(['frequency [Hz]']) yabe(['phase [deg]']) grid on hod on semiogx(f,ange(g_ba1)./pi*180,'coor','c','linewidth',1) hod on semiogx(f,ange(frfsys)./pi.*180,'coor','k','inestye','-- ','LineWidth',) 5

26 Bae.m cear a cose a cc Npoints = 048; Noerap = 1/*Npoints; Fs = 1000; oad Tim_e.mat % Measured error ba handing 1 e1 = rt_baedata.signas.aues(:,1); % Measured error ba handing e = rt_baedata.signas.aues(:,); % Injected noise = rt_baedata.signas.aues(:,3); % Measured pant input ba handing 1 u1 = rt_baedata.signas.aues(:,4); % Measured pant input ba handing u = rt_baedata.signas.aues(:,5); [c_s u1,f] [c_s u,f] [c_ps e1,f] [c_ps e,f] = mscohere(,u1,hanning(npoints),noerap,npoints,fs); = mscohere(,u,hanning(npoints),noerap,npoints,fs); = mscohere(,e1,hanning(npoints),noerap,npoints,fs); = mscohere(,e,hanning(npoints),noerap,npoints,fs); [S u1,f] = tfestimate(,u1,hanning(npoints),noerap,npoints,fs); [S u,f] = tfestimate(,u,hanning(npoints),noerap,npoints,fs); [PS e1,f] = tfestimate(,e1,hanning(npoints),noerap,npoints,fs); [PS e,f] = tfestimate(,e,hanning(npoints),noerap,npoints,fs); % S11 % S1 % PS11 % PS1 PS e1 = -PS e1; PS e = -PS e; G_bae1 = PS e1./s u1; G_bae = PS e./s u; C_bae1 = (1./S u1-1)./g_bae1; C_bae = (1./S u-1)./g_bae; % Pot resuts figure subpot(11) 6

27 semiogx(f,db(abs(g_bae1))) yabe(['ampitude [db]']) tite(['pant Ba Handing Veocity 1']) grid on hod on semiogx(f,db(abs(g_bae)),'r') subpot(1) semiogx(f,ange(g_bae1)./pi*180) xabe(['frequency [Hz]']) yabe(['phase [deg]']) grid on hod on semiogx(f,ange(g_bae)./pi*180,'r') sae G_bae1 G_bae1 f sae G_bae G_bae f 7

28 Bapos.m function frf_bapos() Npoints = 048; Noerap = 1/*Npoints; Fs = 1000; oad Tim_pos.mat % Measured error ba handing 1 e1 = rt_baposdata.signas.aues(:,1); % Measured error ba handing e = rt_baposdata.signas.aues(:,); % Injected noisefrf_bapos1 1 = rt_baposdata.signas.aues(:,3); % Injected noise = rt_baposdata.signas.aues(:,4); % Measured pant input ba handing 1 u1 = rt_baposdata.signas.aues(:,5); % Measured pant input ba handing u = rt_baposdata.signas.aues(:,6); if (~isequa(1,zeros(size(1)))) & (isequa(,zeros(size()))) % noise injected in oop 1 [c_s_1_u1,f] = mscohere(1,u1,hanning(npoints),noerap,npoints,fs); [c_s_1_u,f] = mscohere(1,u,hanning(npoints),noerap,npoints,fs); [c_ps_1_e1,f] = mscohere(1,e1,hanning(npoints),noerap,npoints,fs); [c_ps_1_e,f] = mscohere(1,e,hanning(npoints),noerap,npoints,fs); [S_1_u1,f] = tfestimate(1,u1,hanning(npoints),noerap,npoints,fs); [S_1_u,f] = tfestimate(1,u,hanning(npoints),noerap,npoints,fs); [PS_1_e1,f] = tfestimate(1,e1,hanning(npoints),noerap,npoints,fs); [PS_1_e,f] = tfestimate(1,e,hanning(npoints),noerap,npoints,fs); % S11 % S1 % PS11 % PS1 PS_1_e1 = -PS_1_e1; PS_1_e = -PS_1_e; time_of_experiment1 = cock; sae frf_bapos1 f c_s_1_u1 c_s_1_u c_ps_1_e1 c_ps_1_e S_1_u1 S_1_u PS_1_e1 PS_1_e time_of_experiment1 8

29 eseif (isequa(1,zeros(size(1)))) & (~isequa(,zeros(size()))) [c_s u1,f] = mscohere(,u1,hanning(npoints),noerap,npoints,fs); [c_s u,f] = mscohere(,u,hanning(npoints),noerap,npoints,fs); [c_ps e1,f] = mscohere(,e1,hanning(npoints),noerap,npoints,fs); [c_ps e,f] = mscohere(,e,hanning(npoints),noerap,npoints,fs); [S u1,f] = tfestimate(,u1,hanning(npoints),noerap,npoints,fs); [S u,f] = tfestimate(,u,hanning(npoints),noerap,npoints,fs); [PS e1,f] = tfestimate(,e1,hanning(npoints),noerap,npoints,fs); [PS e,f] = tfestimate(,e,hanning(npoints),noerap,npoints,fs); % S11 % S1 % PS11 % PS1 PS e1 = -PS e1; PS e = -PS e; time_of_experiment = cock; sae frf_bapos f c_s u1 c_s u c_ps e1 c_ps e S u1 S u PS e1 PS e time_of_experiment ese error('ony injected noise in one of the to oops') end 9

30 Bapos_tot.m cear a cose a cc try oad frf_bapos1.mat oad frf_bapos.mat catch error('first perform the to experiments') end S(1,1,:) = S_1_u1; S(1,,:) = S u1; S(,1,:) = S_1_u; S(,,:) = S u; S_bapos = frd(s,f,'units','hz'); PS(1,1,:) = PS_1_e1; PS(1,,:) = PS e1; PS(,1,:) = PS_1_e; PS(,,:) = PS e; PS_bapos = frd(ps,f,'units','hz'); G_bapos = PS_bapos*in(S_bapos); [G,Hz] = FRDATA(G_bapos); figure; subpot(11) semiogx(hz,db(abs(squeeze(g(1,1,:))))) hod on semiogx(hz,db(abs(squeeze(g(,,:)))),'r') subpot(1) semiogx(hz,ange(squeeze(g(1,1,:)))./pi.*180) hod on semiogx(hz,ange(squeeze(g(,,:)))./pi.*180,'r') % Pot resuts % Pot measured sensitiity (ampitude) figure; bodemag(s_bapos) grid on % Pot measured process sensitiity (ampitude) figure; bodemag(ps_bapos) grid on % Pot measured pant (ampitude) figure; bodemag(g_bapos) grid on 30

31 % Cacuate reatie gain array (RGA) RGA = []; [G_bapos_frf,f] = frdata(g_bapos); for i=1:ength(f) RGA(:,:,i)=G_bapos_frf(:,:,i).*(pin(G_bapos_frf(:,:,i))).'; end RGA = frd(rga,f,'units','hz'); % Pot reatie gain array (RGA) figure; ymin = 0; ymax = 5; xmin = min(f); xmax = max(f); f=f(3:end); RGA=RGA(:,:,3:end); subpot(1) semiogx(f,abs(squeeze(rga(1,1,:))),'coor','k','ineidth',) xim([xmin xmax]) yim([ymin ymax]) grid on yabe(['rga [-]']) subpot() semiogx(f,abs(squeeze(rga(1,,:))),'coor','k','ineidth',) xim([xmin xmax]) yim([ymin ymax]) grid on subpot(3) semiogx(f,abs(squeeze(rga(,1,:))),'coor','k','ineidth',) xim([xmin xmax]) yim([ymin ymax]) grid on yabe(['rga [-]']) xabe(['frequency [Hz]']) subpot(4) semiogx(f,abs(squeeze(rga(,,:))),'coor','k','ineidth',) xim([xmin xmax]) yim([ymin ymax]) grid on xabe(['frequency [Hz]']) def_fontsize = 10; def_tex_ineidth_cm = 1.0*3.5146e-; subpot(1); set(gca,'ytick',[ ]) subpot(); set(gca,'ytick',[ ]) subpot(3); set(gca,'ytick',[ ]) subpot(4); set(gca,'ytick',[ ]) G_bapos1 = squeeze(g_bapos_frf(1,1,:)); G_bapos = squeeze(g_bapos_frf(,,:)); sae G_bapos1 G_bapos1 f sae G_bapos G_bapos f 31