Inverted Pendulum-type Personal Mobility Considering Human Vibration Sensitivity

Transcription

1 (IJACSA) Inernaional Journal of Advaned Compuer Siene and Appliaions, Vol. 5, No. 3, 14 Invered Pendulum-ype Personal Mobiliy Considering Human Vibraion Sensiiviy Misaki Masuda Shool of Siene for Open and Environmenal Sysems Kauo Yokoyama Shool of Siene for Open and Environmenal Sysems Takuma Suuki Graduae Shool of Siene and Tehnology Masaki Takahashi Deparmen of Sysem Design Engineering Absra An invered pendulum-ype PM (personal mobiliy) has been araing aenion as a low-arbon vehile. For many people who like o use he PM, ride omfor is imporan. However, ride omfor of PM has no been foused on in previous sudies. The vibraion is one of auses ha make riders feel unomforable. The PM is unsable sysem and horional vibraion may be aused by a sabiliing onrol. Addiionally, verial vibraion may also be aused by road disurbanes. This sudy analyes he vibraion of he rider s head in hese wo direions when he PM runs on a road wih disurbanes in numerial simulaions, and evaluaes ride omfor wih he frequeny haraerisis of he vibraion. To onsider human vibraion sensiiviy, he frequeny weighing proposed in ISO is used as an evaluaion sandard. The improvemen mehods are proposed from boh sofware and hardware, and i is onfirmed ha he proposed mehod an improve ride omfor. Keywords Invered Pendulum-Type Personal Mobiliy; Ride Comfor; Human Vibraion; Frequeny Analysis; Vibraion Conrol; Frequeny Shaped LQG I. INTRODUCTION Reenly in Japan, he number of single-person households has inreased hrough delining birhrae and a rapidly growing proporion of elderly people, and he populaion dereases [1]. Based on his urren siuaion, he PM has been araing aenion as a ne-generaion mobiliy vehile. The PM is a vehile for personal shor-disane rips, and i is an eleri vehile. The single-person household using a PM insead of a gasoline vehile is epeed o onribue o he low-arbon soiey. Sudies of dynamis and sabiliy of he PM have been ondued []. Sudies of model and onrol haraerisis of human on he PM have been reenly ondued [3][4]. Many sabiliaion onrol mehods for he PM have been proposed [5]-[8]. I is imporan o improve ride omfor so ha many people use he PM. However, he ride omfor problem has no been foused on in previous sudies. This paper aims a improving ride omfor of he PM. The model ha was sudied was an invered pendulum PM ha has many advanages and is widely used. The sysem of he invered pendulum PM is unsable. Therefore, horional vibraion ours by a sabiliing onrol. Addiionally, verial vibraion ours when raveling due o road disurbanes. This sudy analyes he vibraion in hese wo direions and evaluaes ride omfor of he oupan onsidering he human vibraion sensiiviy. This sudy proposes an invered pendulum PM wih greaer ride omfor using boh hardware and sofware approahes based on he analysis resuls. To verify is effeiveness, numerial simulaions were arried ou. II. MODELING A. Model and Parameer Figs. 1 (A), (B) and (C) show a oordinae sysem, a model for numerial simulaion, and a model for design of onrol sysem respeively. In our sudy, he models of an invered pendulum-ype PM are limied o wo-dimension, horional and verial. The following variables have been hosen o desribe he vehile (see also Fig. 1): θ: Pendulum angle wih respe o ais. φ: Tire angle wih respe o pendulum angle. u: Conrol inpu orque. d: Displaemen of road disurbane. Parameer definiions and values are olleed in TABLE I. Parameers of he vehile are referring o Segway PT i [9]. The oupan was assumed ha he heigh is 1.7 m, and weigh is 65. kg. This refers o average heigh and weigh of Japanese adul male [1]. Tire siffness and damping oeffiien are referring o he sudy of Sharp e al [11]. 189 P a g e

2 (IJACSA) Inernaional Journal of Advaned Compuer Siene and Appliaions, Vol. 5, No. 3, 14 Yaw TABLE I. MODEL PARAMETERS Parameer Symbol Uni Value Mass of he pendulum m kg 13. Mass of he ires M kg 1. Momen of ineria of he pendulum J p kgm 1.7 y Momen of ineria of he ire J kgm.9 Heigh of he oupan l m 1.7 Roll Pih Disane beween he ire ener and ener of graviy of he pendulum L m.7 Radius of he ire r m.4 Tire siffness k N/m (A) Coordinae sysem Tire damping oeffiien kg/s 1. 1 l L r k u M J m J p d M J L u m J p r By using he Lagrangian, he following moion equaions are obained: M m r mrl os ml J p J M m r mrl os J mlsin mrl mgl sin sin os M m r mrl J M m r J mrl sin u (4) (5) (B) A model for simulaion (C) A model for onrol sysem design Fig. 1. Diagrams of an invered pendulum-ype personal mobiliy B. Moion Equaions of Model for Numerial Simulaion Coordinaes of he ener of he ire (, ), and of he ener of graviy of he pendulum ( p, p ) are (, ) ( r( ), ) (1) (, ) ( Lsin, Los ) () p p In Fig. 1(B), kinei energy T, poenial energy V, and dissipaion energy D of he numerial simulaion model are ml sin M m ml os k( d) ( d) III. DESIGN OF CONTROL SYSTEM A. Moion Equaions of he Model for Conrol Sysem In Fig. 1(C), he following moion equaions are obained as M m r mrl os ml J p J M m r mrl os J mrl sin mglsin os M m r mrl J M m r J mrl sin u (6) (7) (8) T M ( ) m( ) J J ( ) 1 V mgl os k( d) mgl os k( d) 1 ( ) ( ) D d d p p p (3) B. Sae-Spae Represenaion θ is assumed o be near, herefore sin, os 1, are assumed. On his assumpion, he equaions are linearly approimaed, and represened as he following saespae. 19 P a g e

3 Frequeny weighing [db] (IJACSA) Inernaional Journal of Advaned Compuer Siene and Appliaions, Vol. 5, No. 3, 14 A B (9) u The onrol arge veor is [ ] T. C. Design of Opimal Regulaor The design of he opimal regulaor has done using hree gains. We all hem Conroller 1, Conroller, and respeively. TABLE II shows he feedbak gains, K K K, and he seling ime of θ of eah onroller. Conrol inpu orque is u K.The seling ime of θ is he ime ha θ has been less han rad. As shown in TABLE II, he onrollers have differen seling imes. Seling ime of eah onroller is relaively slow, medium, and fas. IV. CONDITIONS OF SIMULATION AND ANALYSIS A. Iniial Sae Sae variable veor for numerial simulaion model is defined as [ ] T (1) The iniial sae veor is [ ] T (11) B. Road Disurbane There are wo ypes of road disurbane. One is sinusoidal, and he oher one is random. 1) Sinusoidal road: The sinusoidal road is defined as d asinb (1) a is ampliude of he disurbane, and a =.5 m. b is a onversion faor. When he PM is raveling a desired speed (in his paper, 5.6 m/s), b =.18 so as o saisfy b. When f is subsiued o Eq. (1), Eq. (1) is assumed o be equivalen o he sinusoidal road displaemen whose frequeny is f H. In his paper, he frequeny is 7 paern,.1,.5, 1., 5., 1., 16., and. H. ) Random road: The design of random road is refer o ISO 868 [1]. In his paper, lassifiaion of road is Level A, B, C, D, and E. Eah level means ha he road is very good, good, normal, poor, and very poor. C. Targe of Analysis Coordinaes of he head of he oupan ( h, h ) is (, ) ( l sin, l os ) (13) h h Feedbak gains TABLE II. Symbol PROPERTIES OF CONTROLLERS Uni Conroller 1 3 K Nm/rad K Nm/rad Seling ime T s Fig.. Frequeny weighing urves shown in ISO Aeleraion of he head of he oupan is 1) Horional direion -3 Verial direion h l l h l l Frequeny [H] os sin sin os (14) D. Evaluaion Mehod In his paper, Power Sperum Densiy (PSD) and Roo Mean Square (RMS) are used for evaluaion. 1) PSD: PSD is evaluaed by using he frequeny weighing urves ha are proposed in ISO [13], as ) shown in Fig.. Fig. shows ha he unpleasan frequeny range is abou H in he horional direion and H in he verial direion. 3) RMS: RMS is defined [13]: 1 T w() (15) 1 aw a d T T is simulaion ime, in his paper, T = 3 s. The aeleraion of he head of oupan is subsiued o aw(). The RMS alulaed by Eq. (15) does no refle human vibraion sensiiviy. Therefore, he aeleraion daa is filered o onsider human vibraion sensiiviy. The ransfer funion represenaion of he frequeny weighing proposed in ISO631-1 is used as a filer [14]. The ransfer funions are 191 P a g e

4 (IJACSA) Inernaional Journal of Advaned Compuer Siene and Appliaions, Vol. 5, No. 3, s 163.7s 6.64s 1.79 Wd () s 4 3 s 3.77s 36.1s 69.8s s 989.s.18 Wk () s 3 s 78.9s 41s 5614 (16) W d is he ransfer funion in he horional direion, and W k is ha in he verial direion. V. EVALUATION OF RIDE COMFORT Fig. 3 shows a rider s head aeleraion when he road disurbane is sinusoidal (f = 1. H). Figs. 4 and 5 show he PSD and RMS of he head aeleraion, respeively. In he horional direion, he resul is differen due o hanging he onroller, as shown in Fig, 3(A). Figs. 4(A) and 5(A) show ha he faser he seling ime of he onroller is, he beer ride omfor beome. However when, whose seling ime is he fases of he hree onrollers, is used, he ride omfor ges worse. I is onfirmed ha if he seling ime is less han 1. s, he ride omfor ges worse. Therefore, onroller whose seling ime is oo fas, for eample,, is undesirable as a onrol sysem onsidering he ride omfor. In he verial direion, he resuls are almos he same even if he onrollers are hanged, as shown in Fig. 3(B). The same rends an be seen in Figs. 4(B) and 5(B). This suggess ha improving on he onrol sysem design is diffiul. In addiion, in Fig. 4(B), here are wo peaks of PSD. One is he frequeny of he vibraion ha is inluded in he road disurbane. The oher is 6.7 H on any roads. The frequeny, ha is he naural frequeny of his model, is in he frequeny range ha human feels unomforable in he verial direion ( H). Therefore, in he verial direion, i is required o hange he naural frequeny using a hardware approah. VI. IMPROVEMENT OF RIDE COMFORT A. Proposal for Improving Ride Comfor. Improvemen Mehod in he Horional Direion: When he opimal regulaor onrollers are used, he faser he seling ime of he onroller is, he beer ride omfor beome. However, if he seling ime is oo fas, ride omfor ges worse. Therefore we apply a frequeny shaping Linear- Quadrai-Gaussian (LQG) onroller [15] o onrol PM as an approah from he onrol sysem. Using his mehod, i isinended o redue he vibraion in he frequeny range ( H), while keeping he seling ime of θ abou 1.4 s whih is he seling ime of onroller. Weighing on he sae variable, Q ( j), and weighing on he onrol inpu, R ( j), are designed for he frequeny shaping LQG onrol. Q( j) is Q ( j) Q ( j) 1 (17) Fig. 6 shows Q ( j) and R ( j). Q ( j) is designed o ake large value in a low-frequeny band up o 1. H, and R( j) is designed o ake large value in a frequeny band greaer han 1.6 H. These weighings represened in he ransfer funions are as follows. Q R f f () s s () s s s +.5 s s s (18) The seling ime of θ is 1.37 s. This ime is equivalen o ha of he onroller. We all his LQG onroller Conroller 4. 1) Improvemen Mehod in he Verial Direion: In he verial direion, i is onfirmed ha improving on he onrol sysem design is diffiul. We propose a hange of he ires as a hardware approah. The naural frequeny of he PM s ire is f n 1 k M m (19) Fig. 7 shows he relaion beween he siffness of he ire and he naural frequeny, when he oupan weigh is hanged. The oupan weigh is kg, whih refers o a manual of Segway PT i. The urren ire siffness is N/m. The naural frequeny is always onained in he frequeny range ( H) in whih human feels unomforable. In order ha he naural frequeny is no inluded in he range, he ire siffness needs o be more han. 1 5 N/m or less han N/m. I is known ha he ire siffness is dependen on he air pressure. The air pressure is dependen on he ire oblaeness [16]. The ire oblaeness is alulaed from he ire widh and heigh. The ire oblaeness of he Segway PT i is.7. To know he ire haraerisi of Segway PT i, we refer o he ire haraerisis of auomobiles and mooryles [16][17]. Auomobile and mooryle ires whose oblaeness is near.7 are seleed. Wih respe o hose ires, he ire siffness and widh are shown in Fig. 8. In Fig. 8, i is onfirmed ha he siffness has nearly linear relaionship wih he ire widh. Consequenly, using linear inerpolaion of daa of mooryle ires, we obain k 5W 87. The ire widh an be deermined from he desired ire siffness. If he ire siffness is more han. 1 5 N/m, he ire widh needs o be more han 14 mm. In he same way, if he ire siffness is less han N/m, he ire widh needs o be less han 55 mm. When ire siffness is N/m, he RMS value is smaller han ha when N/m. Therefore, we propose a ire whose widh is 55 mm. When he ire is used, he naural frequeny beomes 3.4 H. B. Simulaion Resuls and Disussion We proposed improvemen mehods for eah direion: horional and verial. In our proposed mehod, he wo mehods are used simulaneously. 19 P a g e

5 PSD [(m/s ) /H] Naural Frequeny [H] PSD [(m/s ) /H] PSD [(m/s ) /H] Gain [db] Aeleraion [m/s ] RMS [m/s ] Aeleraion [m/s ] RMS [m/s ] (IJACSA) Inernaional Journal of Advaned Compuer Siene and Appliaions, Vol. 5, No. 3, (A) Horional direion Conroller 1 Conroller Conroller 1 Conroller (A) Horional direion 1-1 Conroller 1 - Conroller Conroller 1 Conroller Fig. 3. A rider s head aeleraion (Sinusoidal road,f =1. H) 1 Fig. 5. RMS value of a rider s head aeleraion (Sinusoidal road, f =1. H ) 1-5 Conroller 1 Conroller Frequeny [H] (A) Horional direion - -4 Q (( j j ) f R( R f ( j j) ) Frequeny [H] Fig. 6. Weighing funions for frequeny shaped LQG Conroller 1 Conroller Frequeny [H] Fig. 4. PSD of a rider s head aeleraion (Sinusoidal road, f =1. H ) Mass of Person [kg] Fig. 7. Naural Frequeny of he personal mobiliy k = 5 k = 5 k = 1 k = 193 P a g e

6 Aeleraion [m/s ] Aeleraion [m/s ] PSD [(m/s ) /H] Tire siffness [ 1-4 N/m] PSD [(m/s ) /H] (IJACSA) Inernaional Journal of Advaned Compuer Siene and Appliaions, Vol. 5, No. 3, Tire widh [mm] Fig. 8. Relaions beween he ire widh and he siffness Auomobile ires Mooryle ires Linear approimaion (Auomobile ires) Linear approimaion (Mooryle ires) Conroller 1, k = 1 Conroller, k = 1, k = 1 Conroller 4, k = Frequeny [H] (A) Horional direion (A) Horional direion Conroller 1, k = 1 Conroller, k = 1, k = 1 Conroller 4, k = 5-1 Conroller 1, k = 1 Conroller, k = 1 -, k = 1 Conroller 4, k = Fig. 9. A rider s head aeleraion (Sinusoidal road,f =1. H) Fig. 9 ompares rider s head aeleraion of Conroller 1, Conroller, and he proposed mehod. The road disurbane is sinusoidal (f = 1. H). Fig. 1 and 11 show he resuls when he road disurbane is random (Level C) whih simulae he aual road. Fig. 1 shows PSD of he head aeleraion and Fig. 11 shows RMS of i. Based on he resuls in Chaper V, only Conroller 1 and are used in simulaion for verifiaion. In Fig. 9(A), he proposed mehod does no modify he maimum value of he horional head aeleraion. In Fig. 1(A), where he road disurbane is random, he vibraion in he unomforable frequeny range ( H) is suppressed by he proposed mehod. 1-1 Conroller 1, k = 1 Conroller, k = 1, k = 1 Conroller 4, k = Frequeny [H] Fig. 1. PSD of a rider s head aeleraion (Random road, Level C) Fig. 11(A) shows ha he RMS value of Conroller IV is 13% smaller han ha of Conroller whih has he same seling ime as Conroller 4. In Fig. 9(B), he proposed mehod suppresses he verial head aeleraion and makes he frequeny of vibraion lower. In Fig, 1(B), he peak of PSD moved from 6.7 H o 3.4 H, he naural frequeny afer ire hanging. In Fig. 11(B), Conroller 4 anno suppress he RMS value. However, if he ire wih he desired siffness is ombined wih Conroller 4, he RMS value is abou 79% smaller han he value of ohers. From hese resuls, in boh direions, he effeiveness of he proposed mehod was onfirmed. VII. CONCLUSION In his paper, we evaluaed ride omfor of an invered pendulum personal mobiliy (PM) and proposed a mehod o improve i. Firs, we suggesed a model limied o wo dimensions, horional and verial, onsidering he ire siffness. We assumed ha he PM runs on a sinusoidal or random road and evaluaed he horional and verial vibraion of he head of he oupan in numerial simulaions. The evaluaion mehods are frequeny analysis using PSD and RMS. We se he evaluaion sandard in aordane wih ISO We proposed improvemen mehods for eah direion. In he horional direion, we proposed frequeny shaped LQG onrol. In he verial direion, analysis showed ha i is diffiul o improve on he onrol sysem design approah. Therefore we proposed a hange he ire as a hardware approah. Using boh mehods simulaneously, in he 194 P a g e

7 RMS [m/s ] RMS [m/s ] (IJACSA) Inernaional Journal of Advaned Compuer Siene and Appliaions, Vol. 5, No. 3, 14 Horional direion, we redued he vibraion of he frequeny range ( H) in whih human feels unomforable. A one ime, in he verial direion, he peak of PSD moved from 6.7 H, inluding wihin he unomforable frequeny range ( H), o 3.4 H, he naural frequeny afer ire hanging. The resuls show ha he proposed mehod was effeive k 1 k 5 (A) Horional direion k 1 k 5 Fig. 11. RMS value of a rider s head aeleraion (Random road, Level C) As fuure works, we ry o verify he validiy of our proposed mehod hrough he simulaions using he deailed model of oupan and he eperimens using our developed invered pendulum-ype personal mobiliy. REFERENCES [1] Minisry of Inernal Affairs and Communiaions, 11. The 1 Census. [] S. Arakawa, C. Nakagawa, A. Shinani and T. Io, Basi Sudy on he Behavior of Invered Pendulum Vehile and Driver Using Mulibody Dynamis, Transaions of he JSME, vol. 78, no.789, pp , 1 (in Japanese). [3] S. Ajisaka, T. Kuboa and H. Hashimoo, Human Balane Conrol Abiliy for Affiniive Personal Vehile, 1h Inernaional Conferene on Ubiquious Robos and Ambien Inelligene (URAI), pp , 13. [4] C. Nakagawa, K. Nakano, Y. Suda and R. Hayashi, Sabiliy of he Dynamially Sabilied Two-Wheeled Vehile Traveling on a Rough Road, Journal of Mehanial Sysems for Transporaion and Logisis, vol., no. 1, pp , 9. [5] F. Grasser, A. D.Arrigo, S. Colombi and A. C. Rufer, JOE: a Mobile, Invered Pendulum, IEEE Transaions on Indusrial Eleronis, vol. 49, no.1, pp ,. [6] G. C. Guerrero, J. J. R. Pasaye, Real-ime Conrol o Hold Uprigh Equilibrium of a Two-wheeled Auo-balaning Vehile, Power, Eleronis and Compuing (ROPEC), IEEE Inernaional Auumn Meeing on Power, Eleronis and Compuing (ROPEC), pp. 1 5, 13. [7] J. Huang, F. Ding, T. Fukuda, T. Masuno, Modeling and Veloiy Conrol for a Novel Narrow Vehile Based on Mobile Wheeled Invered Pendulum, IEEE Transaions on Conrol Sysems Tehnology, vol. 1, no. 5, pp , 13. [8] L. Zhijun, Y. Chenguang, Neural-Adapive Oupu Feedbak Conrol of a Class of Transporaion Vehiles Based on Wheeled Invered Pendulum Models, IEEE Transaions on Conrol Sysems Tehnology, vol., Issue 6, pp , 1. [9] Segway Japan, Ld., Home Segway Japan, segway-japan.ne. See also URL hp:// (aessed Jan, 14). [1] Minisry of Eduaion, Culure, Spors, Siene and Tehnology, Invesigaion of Physial Finess and Eerise Capaiy. See also URL hp:// (aessed Jan, 14) [11] R. S. Sharp, S. Evangelou, D. J. N. Limebeer, Advanes in he Modeling of Mooryle Dynamis, Mulibody Sysem Dynamis, vol. 1, no. 3, pp , 4. [1] Inernaional Organiaion for Sandardiaion ISO 868, Mehanial vibraion -Road surfae profiles - Reporing of measured daa, [13] Inernaional Organiaion for Sandardiaion ISO 631-1, Mehanial vibraion and shok evaluaion of human eposure o whole-body vibraion par 1: general requiremens, [14] L. Zuo, S. A. Nayfeh, Low order oninuous-ime filers for approimaion of he ISO human vibraion sensiiviy weighings, Journal of Sound and Vibraion, 65, pp , 3. [15] N. K. Gupa, Frequeny-Shaped Cos Funionals: Eension of Linear- Quadrai-Gaussian Design Mehods, Journal of Guidane, Conrol and Dynamis, vol. 3, no. 6, pp , 198. [16] Sakai, H., Tire Engineering, Grand Pri, 1-3 Kanda Jinbo-ho, Chiyoda-ku, Tokyo (in Japanese). [17] Kinei Charaerisis of Mooryles and Environmen Surrounding I, Soiey of Auomoive Engineers of Japan, P a g e