The Development of gearless reducers with rolling balls

Transcription

1 Journl of Mechnicl Science nd Technology 4 () 89~95 DOI.7/s [Invited Pper] The Development of gerless reducers with rolling blls Hidetsugu Terd * Grdute school of Medicl nd Engineering, Deprtment of Mechnicl System Engineering, University of Ymnshi, Tked 4-3-, Kofu, Ymnshi 4-85, Jpn (Mnuscript Received My, 9; Revised November, 9; Accepted November 6, 9) Abstrct To stisfy the vrious demnds such s no-bcklsh chrcteristic nd so on, we hve newly developed three types of no-bcklsh reducers: the cycloid bll reducer, the precession bll reducer nd the reciprocting motion input type bll reducer. Especilly, these reducers re developed for the robot joints. Also, these reducers re kind of the constnt velocity cm mechnisms with rolling blls; their inputs use eccentric motion, precession motion nd reciprocting motion. The motion principle nd the profile clcultion method of ech reducer re proposed, using the vector nlysis. Keywords: Bll reducer; No-bcklsh; Eccentric motion; Precession motion; Reciprocting motion Introduction The chrcteristic of no-bcklsh trnsmission is the most importnt performnce for industril robots. To stisfy tht performnce, mny vrieties of the no-bcklsh reducers hve been developed [-3]. However, the conventionl no-bcklsh reducers hve limited reduction rtio. So, we cnnot select n rbitrry reduction rtio. Also, these reducers cnnot often be used to the robot wrist joint; tht size is limited strictly. And the conventionl orthogonl xis output type reducers cnnot eliminte the bcklsh. To solve these problems, it is difficult to develop only one type of reducer. Therefore, we hve pid ttention s follows: () The plnetry motion cn generte the reduced rottion with high reduction rtio. () The cm mechnism is designed using motion loci; tht is, no-bcklsh in theoreticl clcultion. (3) The simplest rolling element is the bll shpe. Considering these points, we hve developed three input motion type reducers which re kind of the constnt velocity cm mechnisms with rolling blls; they hve eccentric motion, precession motion nd reciprocting motion. The eccentric input type is useful for the co-xil output type reducer. And the precession motion input type is useful to generte the reduced orthogonl xis output rottion. Also, the reciprocting motion input type is suitble for the hollow shpe nd the lrge dimeter of the reducer elements. This pper ws presented t the ICMDT 9, Jeju, Kore, June 9. This pper ws recommended for publiction in revised form by Guest Editors Sung-Lim Ko, Keiichi Wtnuki. * Corresponding uthor. Tel.: , Fx.: E-mil ddress: terd@ymnshi.c.jp KSME & Springer In this report, n overview of the developed reducers is shown. And the motion principle nd the profile clcultion method re shown, using the some developed reducers.. Development of the eccentric motion input type bll reducer. Single stge type cycloid bll reducer The single stge type cycloid bll reducer [4] consists of two sections: the reduction section nd the motion trnsmitting section with mny circulr grooves, shown in Fig.. The reduction section is driven by the eccentric input shft shown in Fig. s. To eliminte the excittion which is cused by eccentric motion, counter weight is needed s. The reduction section consists of fixed disc 4 with n epi-trochoidl Fig.. Fundmentl structure of single stge type cycloid bll reducer.

2 9 H. Terd / Journl of Mechnicl Science nd Technology 4 () 89~95 Mny circulr grooves Tble. Terminology of two stge type trochoidl wve bll reducer. Epi-trochoidl wve on n eccentric motion disc Z Z Z 6 Z 5 Hypo-trochoidl wve on n eccentric motion disc Input rottion Output rottion Epi-trochoidl Hypo-trochoidl wve on wve on fixed Output disc disc First stge Second stge reduction section reduction section Motion trnsmitting section Fig.. Equivlent model of two stge type cycloid bll reducer..3 Kinemtic nlysis of reduction section In generl, the epi/hypo-trochoidl is well known s the locus of the rbitrry point on the rolling circle without slipping round the bse circle. To decide the rbitrry position on the rolling circle, we hve newly defined the trochoid fctor. So, using the stndrd rdius, the trochoid fctor nd the number of trochoidl wve on ech bse circle, epitrochoidl P 3 is defined s Eq. () nd the hypotrochoidl P 4 is defined s Eq. (). P = ( n + ) r e α r e () 3 jθ3 j( + n) θ3 P = ( n ) r e + α r e () 4 jθ 4 j( n) θ4 Fig. 3. Fundmentl structure of two stge type cycloid bll reducer. wve groove 5, n eccentric motion disc 6 with hypotrochoidl wve groove 7, nd mny rolling blls 8. To eliminte the eccentric rottion, the motion trnsmitting section consists of mny rolling blls, the circulr grooves of the eccentric motion disc 8 which re rrnged on the opposite side of trochoidl wve groove, nd the output disc 9 with circulr grooves. Also, the generted reduced motion rottes the output shft.. Two stge type cycloid bll reducer At the single stge type reducer, to eliminte the excittion which is cused by eccentric motion, counterweight is needed. Furthermore, the improvement of reducer configurtion is necessry to stisfy the light weight nd smll size. So, we hve developed two-stge type cycloid bll reducer [5] tht consists of two sections, the reduction section nd the motion trnsmitting section with mny circulr grooves, shown in Fig.. Ech fundmentl element is similr to the single stge type reducer, s shown in Fig. 3. Ech terminology is defined in Tble. The trochoid fctor α defines the shpe of the trochoidl. In cses in which tht fctor is., tht hs cycloidl shpe, nd when tht is higher thn., tht hs wve shpe with loop. Therefore, the trochoid fctor should be selected s Eq. (3). < α. (3) And the eccentricity e is defined s shown in Eq. (4). e = α r (4) The center locus of the wve groove on ech disc conforms to the trochoidl, so we nlyze the center loci. And when the fixed disc hs hypo-trochoidl wve groove, nd the eccentric motion disc hs n epi-trochoidl wve groove, the geometry of reduction section is shown in Fig. 4. The motion of the epi-trochoidl wve locus P on the eccentric motion disc is defined by Eq. (5). jθin jδ P = α re + e P3 jθin = α re + ( n + ) r e α r e j( θ3+ δ) j{( + n) θ3+ δ} (5)

3 H. Terd / Journl of Mechnicl Science nd Technology 4 () 89~95 9 Pitch circle of rolling blls lloction Bse circle of hypo-trochoidl Meshed points with blls Eccentricity Epi-trochoidl Reduction rtio i f: f:3 f:4 f:5 O 4 O 3 P 4 6 Number of n epi-trochoidl wve n 5 Bse circle of n epi-trochoidl Hypo-trochoidl Rolling circle of n epi-trochoidl Rolling circle of hypo-trochoidl Fig. 4. Geometry of n epi-trochoidl wve nd hypo-trochoidl wve meshing. In this eqution, the δ shows the reduced rottion ngle round the first stge eccentric shft. On the other hnd, the hypo-trochoidl wve groove on the fixed disc conforms to Eq. (). When these grooves re meshed cross mny blls, s shown in Fig. 4, the first stge design conditions re needed s follows. n n = δ = θin n n + θ = θ + θ4 = θin n + 3 in n( n ) The motion of the epi-trochoidl wve locus P is shown in Eq. (7). This eqution is stisfied on the rbitrry rottion. P j ( θin + mπ ) n + jθin α = ( n + ) r e + r e ( m=,,, L, n ) (7) In cses in which ll meshed points with blls conform to the points on tngents of the epi-trochoidl nd the hypo-trochoidl, ll blls re rrnged to the equivlent pitch. The number of blls is n +. At the second stge section, the motion nlysis is similr to the first stge section. However, to rotte with constnt velocity, the f must be n integer which shows the design condition prmeter of wve numbers, s Eq. (8). f n + ( n + ) 5 = (8) n So the reltion between the number of first stge epitrochoidl wve n 5 nd the reduction rtio i is needed s shown in Fig. 5. (6) Fig. 5. Reltion between the numbers of stge epi-trochoidl wve. Second stge side pitch circle of circulr grooves Circulr grooves on the second stge side disc O 9 O 8 θ in δ θ 8 θ 9 Fig. 6. Geometry of circulr groove wve meshing. Meshed points with blls δ Circulr grooves on the first stge side disc First stge side pitch circle of circulr grooves.4 Kinemtic nlysis of motion trnsmitting section At the two stge type reducer, the two reduction sections re driven by two eccentricities which re not equl. So, the trnsmitting section is needed to eliminte the different rottionl phse nd the different eccentricities. The circulr grooves of the eccentric motion discs re rrnged on the opposite side of ech trochoidl wve groove. The geometricl model is shown in Fig. 6. The eccentricity of the second stge e is defined s shown in Eq. (9). e = α r (9) b b The second stge eccentricity is rrnged on n opposite phse to ech other. The initil position of circulr grooves is locted on n opposite phse, too. Using Eq. (7), the stndrd rdius r 8, the circulr groove rdius r 7, nd rottion ngle of ech stge θ 8, θ 7, the circulr locus P 8 on the first stge side re defined s Eq. (), nd the circulr locus P 7 on the second stge side is defined s Eq. (). P = α + + ( α + α ) j () 8 in 8 re jθ re jδ 8 r b rb e θ in 7 P = αb re jθ b + re jγ 7 + ( α r + αb rb) e jθ 7 () In this eqution, the γ shows the reduced rottion ngle round the second stge eccentric shft. When these grooves re

4 9 H. Terd / Journl of Mechnicl Science nd Technology 4 () 89~95 meshed cross mny blls, the design conditions re needed s Eq. (). r8 = r7 δ = γ = θin n θ7 = θin θ8 = θin + π () 3. Development of the precession motion input type bll reducer 3. Co-xil output type precession bll reducer Fig. 7. Fundmentl structure of co-xil type precession bll reducer. The co-xil type precession bll reducer [6] consists of n inclined input shft, precession motion rotor which rottes round the input shft freely, mny rolling blls on the rotor 3, fixed ring 4 with sptil circulr grooves 5 of n inside surfce nd n output ring 6 with the trochoidl wve groove 7 of n inside surfce, s shown Fig. 7. The output ring hs coxil output to the input shft. Mny blls which re rrnged on the rotor re meshing with the sptil circulr grooves 5, to generte precession motion of pre-cession rotor. The sptil cycloidl or trochoidl wve groove 7, which is bsed on precession motion, genertes the reduced rotting motion. Tht sptil wve groove is the locus of precession motion with the reduced rotting motion, which genertes using the reltive motion of precession rotor nd n output ring. On both sides of the pre-cession rotor, mny rotting blls re rrnged in uniform ngulr pitch. For ech bll, the rotting phse which is cused by n inclined input shft is different. 3. Orthogonl-xis output type precession bll reducer The orthogonl-xis output type precession bll reducer [7] consists of mny prts s shown in Fig. 8. The generting section of precession motion consists of n inclined input shft with the inclined shft which rottes freely, precession motion rotor 3 which rottes round the input shft freely, mny rolling blls 4 for precession motion nd fixed ring 5 with the sptil circulr grooves 6 of n inside surfce. These sptil circulr grooves mke the precession motion. The converting section of the precession motion direction consists of converting rotor 7 nd mny rolling blls 9, nd 3 on tht rotor. The precession rotor hs mny grooves 8 for converting tht motion direction. And mny circulr grooves re ssigned round tht rotor, to generte precession motion round the output xis. Also, the section of reduced output motion consists of n output rotor with the sptil trochoidl-wve groove of n inside surfce. This groove is generted from the rbitrry required reduction rtio. The generted reduced motion rottes the output shft 4 which is the orthogonl direction of the input shft. Fig. 8. Fundmentl structure of n orthogonl xil type precession bll reducer. 3.3 Kinemtic nlysis of sptil circulr grooves on the fixed ring In generl, the precession motion is well known s the rotting motion of cone without slipping. This motion cn be modeled to the motion of disc rotting round the equivlent circle in which the rotting xis coincides with the i-xis on the Crtesin spce. When precession motion is generted, it is necessry to limit the rbitrry sptil motion. So, sptil circulr grooves on fixed ring re needed. Also, the reduced rotting motion is generted from the precession motion for the precession rotor. The rection torque for reduced rotting motion cts on the precession rotor. So the mechnism which holds rection torque is needed s well s the generl plnetry gering system, t this proposed reducer. Therefore, the circulr sptil grooves on fixed ring re investigted.

5 H. Terd / Journl of Mechnicl Science nd Technology 4 () 89~95 93 The section generting the precession motion is shown in Fig. 9. The xis of the input shft intersects with the inclined shft t the origin point O p. Then the pitch rdius of blls locted is defined s r. All blls re locted on the rbitrry position on precession rotor, so the number of locted blls is defined s n. Ech bll is locted round the rotting xis w t the initil ngle λ n. Tht is seprted t the initil ngle τ n round j-xis. Also the inclined ngle is defined s α tht rotor which rottes freely is ssembled to the inclined shft. When the inclined input shft rottes s θ, the precession rotor rottes without slipping on the equivlent disc s φ. So the position of the bll center P is defined s Eq. (3) [3]. w φ i θ wλn jα jτn w φ iθ = = P Ε Ε E E E R Ε Ε P n = (,,,, Z ) (3) bn In this eqution, ech bll is numbered s the n-th. Also, the initil position is defined s Eq. (4). R = (,, r ) T (4) The rotting mtrix round the w xis is defined s Eq. (5). cos φ + λ ( cos φ ) w φ Ε = λµ ( cos φ) + υ sinφ υλ( cos φ ) µ sinφ λµ ( cos φ) υsin φ υλ( cos φ ) + µ sinφ cos φ + µ ( cos φ ) µυ( cos φ) λsinφ µυ( cos φ) + λ sinφ cos φ+ υ ( cos φ) (5) At this eqution, the λ, µ nd ν re defined s the directionl cosine of unit vector t the w -xis. Bsed on these vector nlyses, the end-mill cutter genertes circulr grooves the dimeter of which is the sme s the bll. When these grooves re generted, tht cutter pose hs to conform to the direction of P. 3.4 Kinemtic nlysis of the specil circulr grooves on converting rotor The precession motion generted from n input rotting is shown s Eq. (3). However, the xis of output rotting is locted on orthogonl direction to the xis of input rotting. So, it is different from the required precession motion which rottes round output xis s shown Eq. (6). wφ kθ =Ε Ε P P (6) Ech bll on the converting rotor rottes round xis w. Tht is seprted t the initil ngle τ round j-xis. Also the inclined ngle of output side is defined s α. The pitch rdius of blls on the converting rotor is defined s Eq. (7). R = ( r,,) T (7) The position of ech bll is defined s Eq. (8). P k ε w λ n jα jτ =Ε Ε Ε Ε R n= (,,,, Z ) (8) bn In this eqution, ech bll is numbered s the n-th. Also, the rbitrry initil pose of converting rotor is defined s the offset ngle ε. In cses in which motion is converted on the co-ordinte frme of tht rotor, the grooves on the converting rotor re generted with considertion of the reltive motion. So, the position of the bll center is clculted, which is similr to the sptil circulr grooves on the fixed ring. The exmple of the specil circulr shpe loci nd grooves is shown in Fig.. These loci hve the individul non-symmetricl circulr shpes. 3.5 Kinemtic nlysis of the sptil trochoidl wve groove on n output rotor The input rottion t 36 degrees genertes one cycle of precession motion for tht precession rotor round the i-xis. This motion is limited using the sptil circulr grooves on fixed ring. Then, this motion is converted to the precession motion round the k-xis using the converting rotor. Also, the reduced rotting motion is generted from the converting rotor Fig. 9. Geometry of precession motion nd circulr groove vector. Fig.. Specil circulr shpe grooves on the precession motion converting rotor.

6 94 H. Terd / Journl of Mechnicl Science nd Technology 4 () 89~95 Output ring θ k Wve groove w φ Fixed ring Precession motion converting rotor Circulr Grooves O p α Q τ 3 τ P w R R i Fig.. Geometry of reduced output motion. Fig. 3. Structure of reciprocting motion bll reducer. Fig.. Trochoidl wve groove on n output ring. motion. The rection torque of reduced rotting motion round the k-xis cts on tht converting rotor. So, this section consists of the prt which limits sptil motion round the k- xis, nd the generting prt of reduced motion. This generting section of reduced output motion is shown in Fig.. The sptil trochoidl wve groove is shown in Fig.. With this type of reducer, the rolling blls re rrnged in uniform ngulr pitch on the rotor converting the precession motion. Therefore, ech bll does not seprte from tht locus t these loop points nd rottes smoothly. In other words, the loops on the trochoidl wve groove cn be used. 4. Development of the precession motion input type bll reducer The reciprocting motion type bll reducer [8] consists of the reciprocting motion generting section nd the reduced motion generting section. Ech element is n input rotor, fixed rotor, reduced motion rotor 3, mny reciprocting motion sliders 4 nd mny rolling blls 5 nd 6, s shown in Fig. 3. These sliders re locted between the fixed rotor nd ech input/reduced rotor to hold the rolling blls. Ech rotor meshes with blls by use of the outside groove on themselves. Also, the fixed rotor hs internl liner grooves 7, which generte the reciprocting motion by bll rolling. At the reciprocting motion generting section, to generte the doubling reciprocting cycle motion using the constnt velocity input rottion, the input rotor hs specil circulte groove 8, just like "8" chrcter, which is on the rotor outside surfce. The meshing with the bll locted on tht slider nd the groove genertes the hlf speed reciprocting motion. Tht bll meshes with fixed rotor simultneously. The motion direction is trnslted by use of the component of the reciprocting motion direction. It is similr to grooved cm mechnism which hs converting follower. Also, t this section, mny reciprocting motion sliders which hve different motion phse ech other re ssigned on the uniform ngulr pitch. On the other hnd, t the reduced motion generting section, the reduced rottion is generted by use of the meshing with blls nd the reduced groove. These blls re locted on the reciprocting motion slider. Tht groove hs wve groove just like "trochoidl-wve" 9 on tht rotor outside surfce. At this section, these reciprocting sliders drive reduced motion rotor which is different from the generl cm mechnism. 5. Conclusions In this report, n overview of reducers which were developed to stisfy the vrious demnds is given. Using the two stge type cycloid bll reducer nd the orthogonl-xis output type precession bll reducer, the motion principle nd the profile clcultion method re shown. In future work, to relize higher performnces for the robot joint, we'll improve the structures of reducers. References [] W. G. Molyneux, The follower tooth reduction ger, Mechnisms 97, I. Mech. E., London, G.B., (973) 5-3. [] Cyclo Reducer ctlog, Sumitomo Hevy Industry Ltd., Tokyo, Jpn, (999) F-. [3] Hrmonic Drive CSF-3 Series Technicl Dt, Hrmonic Drive systems Inc., Tokyo, Jpn, (8) 7-R-TCSF3. [4] H. Terd, K. Imse nd H. Mkino, Fundmentl nlysis of cycloid bll reducer, (st report) Motion principle, J. Jpn Soc. for Precision Engineering 54() (988) - 6. [5] H. Terd nd K. Imse, Motion nlysis of multi-stge type trochoidl wve bll reducer, Proc. of the th Int. Conf.

7 H. Terd / Journl of Mechnicl Science nd Technology 4 () 89~95 on the Theory of Mchines nd Mechnisms, Liberec, Czech Republic, (8) [6] H. Terd, T. Fukzw nd R. Irie, Motion nlysis of coxil output type precession motion bll reducer, Proc. of the th World Congress in Mechnism nd Mchine Science, Tinjin, Chin, (4) [7] H. Terd nd R. Irie, Motion nlysis of n orthogonl output type precession motion bll reducer, Proc. of the 9th Int. Conf. on the Theory of Mchines nd Mechnisms, Liberec, Czech Republic, (4) [8] H. Terd, T. Msud nd S. Yoshid, Motion nlysis of reciprocting motion type bll reducer, Proc. of the th World Congress in Mechnism nd Mchine Science, Besncon, Frnce, (7) Lc Hidetsugu Terd received his Bchelor nd Mster degrees in Mechnicl Engineering from the University of Ymnshi, Jpn, in 986 nd 988, respectively. He then received his Doctorte from the Tokyo Institute of Technology in 993. Dr. Terd is currently n Associte Professor t the Grdute School of Medicl nd Engineering Science t the University of Ymnshi in Kofu, Jpn. He serves s the IFToMM Technicl Committee member of the Linkge nd cms. His reserch interests include ger-less reducers, robotics, nd micro-mchining.