Energy Shaping Nonlinear Acceleration Control for a Mobile Inverted Pendulum with a Slider Mechanism Utilizing Instability

Transcription

1 Proceeding of the World Congre on Engineering and Coputer Science Vol I WCECS, October 4-6,, San Francico, USA Energy Shaping Nonlinear Acceleration Control for a obile Inverted Pendulu ith a Slider echani Utilizing Intability K. Yokoyaa and. akahahi Abtract A nonlinear controller for accelerating a obile inverted pendulu (IP) ith a lider echani i propoed. he controller hape the total energy of the yte and utilize intability of the IP for acceleration. he body angle and the diplaceent are controlled to keep tate here the IP i tatically untable, hich lead to tranlational acceleration due to intability of the yte. he total energy of the yte i haped to have the iniu at given deired tate and the yte i controlled to converge to the. he propoed controller can achieve variou propertie through the energy haping procedure. Epecially an energy function that ill lead to afe operation of the IP i propoed. he function enure that otion of the IP i retricted ithin predefined region and converge to the deired tate. he controller alo return the yte back to the deired tate ith tate-dependent gain that becoe large if the yte coe cloe to fall over. Effectivene of the propoed controller and utilization of intability for the IP ith the lider echani are verified through iulation. Index er Energy Shaping, obile Inverted Pendulu, Intability, Slider echani x n l l t l r 3 Fig.. A diagra of a obile inverted pendulu ith a lider Reference Diplaceent f iniu Energy for Diplaceent Reference Angle iniu Energy for Angle Intability A I. INRODUCION obile inverted pendulu (IP) ha a all footprint and can turn in a all radiu. It obility i energy-efficient due to it being lighter than ordinary car. It i applied to peronal obility vehicle that are expected to be ued near huan living pace. here are to type of IP. he one i the tanding-type, for exaple, Segay. he other i the itting-type, for exaple, OBIRO and en-v, hich have a lider echani. A diagra of the yte i hon in Fig.. he lider hift the center of a (CO) of the IP hile a rider i itting till. hi echani enable the IP to accelerate ithout the rider hifting hi/her CO back and forth. herefore variou people including the elderly can ue the IP ith the lider eaily. In thi tudy, thi type of the IP i focued on, and a afe controller to accelerate the obility i propoed. We conider the afety in ter of the anucript received uly 3, ; revied Augut 7,. hi ork a upported in part by apan Society for the Prootion of Science. K. Yokoyaa i ith School of Science for Open and Environental Syte, Keio Univerity, 3-4- iyohi, Kohoku-ku, Yokohaa 3-85, apan (correponding author to provide phone: ; fax: ; e-ail: kazuto_yokoyaa@8.jukuin.keio.ac.jp).. akahahi. i ith Departent of Syte Deign Engineering, Keio Univerity, 3-4- iyohi, Kohoku-ku, Yokohaa 3-85, apan (e-ail: takahahi@d.keio.ac.jp). ISBN: ISSN: (Print); ISSN: (Online) Acceleration heoretical Retriction of Angle and Diplaceent Fig.. he concept of the propoed controller body angle and the lider diplaceent. he controller utilize intability of the IP. he body angle and the lider diplaceent are controlled to keep tatically untable but dynaically table tate hile theoretically retricting otion of the IP in a predefined range. he intentional detabilization lead to indirect control of tranlational acceleration. ypical controller for the IP ith the lider in previou tudie are linear one baed on linearly approxiated euation of otion[]-[3]. hey control the yte to converge to tatically table tate. he reference body angle i et in the vertical direction, and the lider diplaceent at the poition here the IP can tand till. oever, the tability i only guaranteed in the neighborhood of the euilibriu. hu, if the IP i largely inclined due to diturbance or the ay a rider operate it, it ay fall over. Driving control i baed on tracking a given reference tranlational velocity. Although the IP ut be detabilized traniently becaue of the dynaic hen it accelerate or WCECS

2 ddx [/] Proceeding of the World Congre on Engineering and Coputer Science Vol I WCECS, October 4-6,, San Francico, USA able I PARAEERS OF E OBIE INVERED PENDUU Paraeter Unit Value kg 5 kg kg 3 kg kg kg kg l. l.5 lt.5 r.75 n - 3 decelerate, otion of the body i not directly conidered in the previou controller. herefore, the IP ay fall over becaue of an inappropriate reference velocity. hu, the author have propoed controller for the IP ithout the lider that retrict otion of the body in a predefined range[4][5]. Epecially, the acceleration controller utilize intability of the yte[5]. A reference body angle i directly given to the yte, and the body i controlled to incline it intentionally, hich lead to the indirect control of tranlational acceleration. he ethod i afe becaue the otion of the body i directly conidered and eaier to predict than that of previou controller. Baed on the background above, thi tudy extend our acceleration controller for the IP ith the lider that utilize intability. he body angle and the lider diplaceent are controlled to keep tatically untable but dynaically table tate. he tranlational acceleration i controlled indirectly through the direct control of the to tate. he propoed control ethod provide the rider of a itting IP ith a feeling of aneuvering hile inclining back and forth, a can experienced ith the tanding IP. In our previou tudy, there a a one-one relationhip beteen the body angle and the tranlational acceleration at teady tate. When a large acceleration need to be achieved, the IP ha to be inclined largely. he large body angle can be uppreed uing the ne degree of freedo of the lider. he cobination of the to tate can odify otion of the IP, leading to afe operation. o achieve thi concept, a nonlinear controller i applied called an interconnection and daping aignent paivity-baed control (IDA-PBC)[6][7]. he ethod hape the total energy of yte to tabilize the. he controller can achieve variou propertie that are difficult to achieve ith a linear controller through the energy haping procedure. In thi tudy, an energy function i propoed that enure that the body angle and the lider diplaceent are retricted ithin predefined region and converge to reference tate. he controller alo help to return the IP to the deired tate uing tate-dependent gain that becoe large if the IP coe cloe to falling over. Effectivene of the controller i verified in iulation. he conceptual diagra of the propoed controller i hon in Fig ref [] ref ref [rad] Fig. 3. Relationhip beteen intability and tranlational acceleration II. EQUAIONS OF OION AND ANAYSIS OF DYNAICS A. Euation of otion A diagra of the IP i hon in Fig.. In thi tudy, the hifting body ean a part of the body on the lider and bae body i one under it.,, and repreent ae of the bae body, heel, and hifting body repectively.,,, and are inertia of the bae body, the heel, the hifting body, and the rotor of the otor repectively. l, l, and l t repreent iniu ditance beteen the center of the heel and the CO of the bae body, hifting body, and the lider repectively. r i heel radiu and n i reduction ratio of the actuator. Phyical paraeter of the yte are hon in able I. he value are elected baed on an experiental etup of the IP hich i being developed in our laboratory. i the body angle ith repect to the vertical direction, i the heel angle relative to the body, and 3 i the diplaceent of the lider. It origin i et at the poition here the IP can keep the upright poture tatically hen rad. x r i the travel ditance. 3 i the generalized poition vector and g i the acceleration of gravity. he euation of otion of the IP are decribed a follo: S CS GS GIu () r l l r l co l co in S 3 3 r l co l co in S 3 S3 r l r co (4) r n (5) S r co (6) S3 S33 h (7) C C x [/ ] 4 - r in lr in S r in lr in S 3 r co l r in 3 r co l r in 3 () (3) (8) (9) ISBN: ISSN: (Print); ISSN: (Online) WCECS

3 Proceeding of the World Congre on Engineering and Coputer Science Vol I WCECS, October 4-6,, San Francico, USA C () S3 3 G g co l in l in () S 3 G () S G g in (3) S3 G I (4) u f. (5) Sij denote the i - j ter of S, and C Si and G Si denote the i -th eleent of C S and G S repectively. he potential energy of the yte are co V gr l l g g in. 3 B. Relationhip beteen Intability and Acceleration (6) Conider the cae in hich the body angle and the diplaceent of the hifting body converge to reference angle ref and diplaceent 3, repectively. he folloing relation are obtained. g l l in co ref ref S S (7) (8) f S 3 g in ref (9) here,, and f f are contant. he relationhip i graphically repreented in Fig. 3. he teady tranlational acceleration x r can be controlled by applying the reference tate ref and to the yte. In our previou tudy[5], the lider echani a not introduced to the IP and the body angle and the tranlational acceleration had a one-one relationhip. When a large tranlational acceleration need to be achieved, the body angle ut be large, hich i undeirable in ter of afe operation of the IP. he proble can be olved by introducing the lider echani. A deired tranlational acceleration can be achieved ith all ref cobined ith the degree of freedo of. III. DERIVAION OF POR-AIONIAN SYSE o achieve the concept, a nonlinear control ethod called IDA-PBC[6][7] i applied. he ethod hape the total energy preerving port-ailtonian (P) tructure[8] of the yte. Stabilization i achieved uing paivity of the P yte. Paivity i an eential energetic property of phyical yte. In general, control ethod that utilize paivity are expected to be robut[9][]. In addition, the IDA-PBC ha been hon to have poerful tabilizing perforance[6]. We ho that the IP ith the lider can be appropriately decribed a a P yte and derive the IDA-PBC controller. Feedback linearization[][] hich ill be fragile to odeling error, and linear approxiation of the euation otion[3][4] hich lead to guaranteeing tability at leat in the neighborhood of an euilibriu are not ued. herefore, robutne and large doain of attraction are expected in our controller. he folloing euation can be derived by eliinating fro the euation of otion (). C G GIu () S S S 3 S S3 S S S S 3 S 3 S3 S33 S S S S3 C C C C3 C S S S S3 G G G G3 G S S l l g in g3 co g in () () (3) G I S S S 3 S 3 here (4). We aue that the potential energy of the above yte that i related to G a co 3 in V l l g g. (5) We alo conider a the inertia atrix of the yte () and calculate Euler-agrange euation of otion V (6) d Q. (7) dt We can check Q correpond to the left-hand ide of (). herefore, the yte can be repreented a a P yte becaue Euler-agrange yte are contained in P yte[8]. I u p I p GI (8) p p V (9) here p. IV. IDA-PBC CONROER DESIGN A. Derivation of Controller In thi tudy, the range of the body angle i conidered in rad and the diplaceent of the lider in 3. ISBN: ISSN: (Print); ISSN: (Online) WCECS

4 [rad] 3 [] 3 [] 3 [] [rad] 3 [] 3 [] Proceeding of the World Congre on Engineering and Coputer Science Vol I WCECS, October 4-6,, San Francico, USA. 5 x -3 5 x -3 [rad].5 ith Slider Reference ref x -3 3 [] (a) Body angle (b) Diplaceent of center of gravity Fig. 4. Coparion of IDA-PBC controller for yte ith or ithout the lider echani x Reference Reference baed on the experiental etup. in the above range can be checked nuerically ith the phyical paraeter of the IP in able I. We define the deired inertia atrix, the deired potential energy V, and the deired total energy, hich are going to be aigned to the P yte ith the IDA-PBC controller. It can be repreented a belo uing arbitrary and a ke-yetric atrix becaue the open-loop P det G [6]. yte i full-actuated and u u u (3) e di u G p (3) e I udi Kiy c (3) here Kdi K di i a paraeter of the controller and y c i the paive output of the cloed-loop P yte. u e i the energy haping control input hich hape the total energy of the open-loop P yte and guarantee tability. u di i called daping injection that i ued to achieve ayptotic tability. he IDA-PBC controller (3) change the open-loop P yte into the cloed-loop one. p [rad] 3 - Reference Reference ref I GIK ig I p (33) p p V (34) y G G p. (35) c I p I 3 [] (a) Body angle (b) Diplaceent of center of gravity Fig. 5. racking for contant reference tate fro the edge of the doain of attraction. function at leat in the neighborhood of, p, are a follo: ref ref (36) arg inv (37) ref here 3 interpreted a ref ref ref ref. he condition (37) can be V (38) V. (39) ref - Under all the condition above, e have y K y. (4) c i c Ayptotic tability i guaranteed hen zero-tate detectability i atified. he IDA-PBC controller ha degree of freedo in deigning and V. Variou propertie can be achieved that are difficult to realize ith a linear controller. C. Deign of Energy Function In thi tudy, the kinetic energy i not going to be haped, that i,, to verify baic propertie of the propoed controller. he deired potential energy i deigned a K p V l ref K p3 3l Reference Reference. 3 (4) B. Condition for Stability Condition to guarantee tability ith the yapunov l and 3l are paraeter hich have to be et to retrict otion of the IP in l and 3 3l. K p and K p3 ISBN: ISSN: (Print); ISSN: (Online) WCECS

5 ddx [/] d [rad] 3 [] [rad] Input orue [N] Input Force [N] [rad] 3 [] [rad] [rad] Proceeding of the World Congre on Engineering and Coputer Science Vol I.. WCECS, October 4-6,, San Francico, USA r = r = 5..5 r =.5 r =.5 r ref = r ref. =. r ref =.5 r ref. =. r ref.5 =.5 Slider r ref.5 =.5 5 r = r = ithout. Slider ithout -.5 Slider tie.5[] tie.5[] are poitive contant. A long a initial condition are et atifying l, 3 3l and reference tate are alo elected fro the range, the propoed deired potential energy (4) atifie the condition (37). V ha the uniue iniu at, p ref,. Stability of the yte i guaranteed ith the yapunov function. We can alo check y, c udi GIp p, and V ref uing (33). Zero-tate detectability i atified and ayptotic tability i guaranteed. D. Propertie of Propoed Controller he deired potential energy V contribute the control input in the for of V K pv ref K pv3 3 (4) K pvi il i iref il i K pi i,3. (43) hi i a kind of a proportional control uing deviation K. i [rad] [rad] [N] x [/ ] ultiplied ith the tate-dependent gain iref (a) Body angle (b) Diplaceent of center of gravity. r r ref= =..5 r r ref= =.5 r ref=.5 r r ref= =. - r ref=.5 r ref= x (c) Actuator r torue = (d) Slider force r =.53 - refr = refr. =. xreference ref refr.5=.5 ithout Slider r = tie ithout [] Slider r ref=.5 r ref= r.5 ref= (e) ranlational acceleration (f) otal energy of cloed-loop P yte Fig. 6. Siulation reult of acceleration control pvi i 3 [] [N] f he gain property i deigned baed on our previou tudy[5]. If the body angle coe cloe to falling over beyond the reference angle, the gain increae and the controller actively return the IP fro the dangerou tate. A iilar property i alo achieved for the diplaceent of the lider. hey are deirable in ter of afe operation of the IP. V. SIUAION In iulation, the propoed controller i copared ith one fro our previou tudy that deal ith a IP ithout the lider[5]. Effectivene of the acceleration control uing the intentional detabilization of the body angle and the diplaceent i verified. he control input (3) i applied to the yte () in the folloing iulation. In all cae, initial value of the heel angle and it angular velocity are zero. he paraeter of the propoed controller are et a l.55 rad, 3 l., K p.5, K p3 8,, and K i = diag(.6, 35). Although the IDA-PBC controller for the IP ithout the lider ha a tructure different fro that ith the lider, e oit the detail in thi tudy. Pleae refer to our previou tudy[5]. Fig. 4 copare baic perforance of the to controller. ISBN: ISSN: (Print); ISSN: (Online) WCECS

6 Proceeding of the World Congre on Engineering and Coputer Science Vol I WCECS, October 4-6,, San Francico, USA he initial condition are all zero. he reference body angle and diplaceent are ref. rad,. he controller for the IP ithout the lider i applied to the yte () ith a contraint 3. Both controller are deigned to ho iilar repone in the body angle, a hon in Fig. 4 (a). In Fig. 4 (b), the hifting body lightly ove and converge to zero in about. heoretical retriction of l and 3 3l are checked in Fig. 5. he initial condition are et a.54 rad and 3.9, hich are on the edge of the retriction range. o ake the ituation difficult, e et the initial body angular velocity rad/ and the lider velocity 3 / hich are in the direction of coing cloer to the edge. he reference tate are ref. rad,.. In Fig. 5, although the repone are precipitou, the to tate converge to the reference ithout violating the retriction. Fig. 6 copare the to controller ith the reference tranlational acceleration xref x /. he initial condition are all zero. Fro (7) to (9), there are any cobination of ref and to achieve the given reference acceleration. hree cae of ref are verified in Fig. 6. i uniuely deterined for each ref. Fig. 6 (e) ho the tranlational acceleration converge to the reference in all cae. Fig. 6 (f) ho that i onotonically non-increaing in all cae and play the role of the yapunov function a i theoretically expected. In Fig. 6 (a), the cae ho the larget axiu body angle, herea it i uppreed in the other three cae by utilizing the lider. In Fig. 6 (b), the cae ref ho the larget axiu diplaceent, hich i about 3.9. oever, there i a argin againt the retriction of 3 3 l.. he reference acceleration i reaonably achieved by utilizing the cobination of the body angle and the diplaceent. he cae ref and ref.5 in Fig. 6 (a) ho underhoot. hi ill be undeirable for the rider of the IP. Although controlling the IP ith the reference angle a the vertical direction ha been very coon in previou tudie, the cae ref. doe not ho underhoot. he reult indicate that utilizing intability effectively accelerate the IP ith the lider. he propoed controller can odify the otion of the body angle uing the hifting echani appropriately. In Fig. 6 (c), the axiu input torue of the propoed controller are ore uppreed than previou one. hi ill prevent lipping hen the IP tart to accelerate. Fig. 6 (d) ho that the larger the reference angle, the aller the axiu lider force. he acceleration control ith the cobination of inclining and hifting the body ha the erit of reducing the reuired axiu torue and force of the actuator a ell a odifying the otion of the IP. VI. CONCUSION An energy haping nonlinear acceleration controller for a obile inverted pendulu (IP) ith a lider echani i propoed. he controller accelerate the obility utilizing intability of the yte. he body angle and the lider diplaceent are controlled to keep tate here the yte i ISBN: ISSN: (Print); ISSN: (Online) tatically untable, hich lead to tranlational acceleration due to the intability of the IP. Variou controller propertie can be achieved through the energy haping procedure. Epecially an energy function hich ill lead to afe operation of the IP i propoed. he function enure that the body angle and the lider diplaceent are retricted in predefined region and converge to a deired tate. he controller alo return the yte back to the deired tate ith tate-dependent gain that becoe large if that the yte coe cloe to fall over. Effectivene of the propoed ethod i verified in iulation. he reult have indicated that otion of the IP can be odified by the cobination of intentional inclination and hift of the body. Reuired axiu input torue and force to accelerate the IP are hon to be uppreed ith the utilization of intability. REFERENCES [] N. ookuni,. Shino, and. Kaata, Iproving tability for a to-leg-heeled inverted-pendulu-type vehicle euipped ith a lider, Proceeding of the ranportation and ogitic Conference of the apan Society of echanical Engineer, 4, pp , 8. (in apanee) [] N. ookuni,. Shino, and. Kaata, Effect of eat lider on otion control for PR, Proceeding of the Conference of the Robotic Society of apan, E-, 9. (in apanee) [3] N. iroe, K. Sukigara,. Kajia, and. Yaaoka, ode itching control for peronal obility robot -ooth tranition fro four-heel ode to heeled-inverted-pendulu ode by inirial value copenation during otion-, ranaction on Indutry Application of the Intitute of Electrical Engineer of apan, vol. 3, no. 9, pp ,. (in apanee) [4] K. Yokoyaa, and. akahahi, Stabilization of a obile inverted pendulu ith IDA-PBC and experiental verification, ournal of Syte Deign and Dynaic, vol. 5, no. 8, pp ,. [5] K. Yokoyaa, and. akahahi, Energy haping nonlinear acceleration control of a pendulu type obility, Proceeding of International Aociation for Vehicle Syte Dynaic, 38., Augut,. [6] R. Ortega,. W. Spong, F. Góez -Etern, and G. Blankentein, "Stabilization of a cla of underactuated echanical yte via interconnection and daping aignent", IEEE ranaction on Autoatic Control, vol. 47, no. 8, pp. 8-33,. [7] R. Ortega, A.. van der Schaft, B. achke, and G. Ecobar, Interconnection and daping aignent paivity-baed control of port-controlled hailtonian yte, Autoatica, vol. 38, pp ,. [8] A.. van der Schaft, -Gain and Paivity echniue in Nonlinear Control, nd edition, Springer, 999. [9] K. Ouka, and F. atuno, "On robutne of paivity of anipulator", IEEE Conference on Deciion and Control, pp , 999. [] R. Ortega, A.. van der Schaft, I. areel, and B. achke, "Putting energy back in control", IEEE Control Syte agazine, vol., no., pp. 8-33,. [] K. Pathak,. Franch, and S. K. Agraal, "Velocity and poition control of a heeled inverted pendulu by partial feedback linearization", IEEE ranaction on Robotic, vol., no. 3, pp. 553, 5. [] N. atakeyaa, and A. Shiada, oveent Control Uing Zero Dynaic of o-heeled Inverted Pendulu Robot, ranaction of the Society of Intruent and Control Engineer, vol. 44, no. 3, pp. 59, 8. (in apanee) [3] O. atuoto, S. Kajita, and K. ani, Etiation and control of the Attitude of a dynaic obile robot uing internal enor, Advanced Robotic, vol. 7, no., pp , 993. [4] F. Graer, A. D'Arrigo, S. Colobi, and A. C. Rufer, "OE: a obile, inverted pendulu", IEEE ranaction on Indutrial Electronic, vol. 49, no., pp. 7-4,. WCECS