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Nonlinear Dynamic Analysis of Planar Flexible Underactuated Manipulators
V ol. 18
No. 1
CH INES E JO U RN A L O F AER ON A U TICS
February 2005
Nonlinear Dynamic Analysis of Planar Flexible Underactuated Manipulators 1
2
HE Guang ping , LU Zhen ( 1 . School of M echanical and Electr ical Engineer ing , North Chi na Univer si ty of T echnology , Beij ing
100041 , Chi na)
( 2 . School of A utomati on Science and Electr ical Engineer ing , Beij ing Uni versity of A eronaut ics and A st ronaut ics, Beij ing Abstract:
100083 , Chi na)
T he influence of passive joint on the dynamics of planar flexible under actuated manipulators
is studied. A vibr atio n reduction method based on the internal resonance phenomenon of multi degree nonlinear dynamic system is proposed. T he dynamic simulation results rev eal that the har mo nic input for t he actuated jo int will induce the passiv e joint dev iating from its equilibrium position, the ex cursion speed and direction depend on t he amplitude of the input . A passive jo int position control scheme making use of t he vibration of the flexible structure is sugg ested, and the numerical simulation results of a model of pla nar two link flexible underactuated manipulator is shown. Key words:
dynamics and vibration; manipulator; underactuated; nonlinear; internal resonance
平面柔性欠驱动机械臂的非线性动力学分析. 何广平 , 陆
震. 中国航空学报( 英文版) , 2005, 18
( 1) : 78- 82. 摘
要: 研究了平面柔性欠驱动机械 臂中被 动关节 对系统动 态特性 的影响, 在 动力学分 析的基 础
上, 提出了一种基于内共振原理的柔性欠驱动机械臂 振动控制方法。在 对系统的稳 态周期运 动进 行分析研 究的基础上, 发现处于自由摆动状态的被动关节的平 衡位置随 系统结构的 振动而发 生漂 移, 并且被动关节平衡位置的漂移速度和方向与周期 输入的振幅有关。 提出利用结 构柔性产 生的 振动实现被动关节位置控制的方法, 通过平面二连杆 柔性欠驱动机械臂进行了仿真计算。 关键词: 动力学与振动; 机械臂; 欠驱动; 非线性; 内共振 文章编号: 1000 9361( 2005) 01 0078 05
中图分类号: T P242
T he motion cont rol of underactuated mecha
文献标识码: A
t ics of the original system. Met amorphic mecha
nism or underact uated manipulat or is proposed in
nisms have some passive joints or flex ible com po
the f ault tolerance technology of space robot system first ly [ 1] . An underactuated m anipulator also can
nents for economical sake and make the best of t he abilit y of reconf igurat ion of the mechanism. T here
be designed as a kind of assist ant robot syst em, w hich is said t o be a COBOT [ 2] . T he COBOT sys
f ore, the underactuated mechanisms and underac t uated manipulat ors have some propert ies w hich
t em can not work independent ly for t he passive
can not be provided by t he f ull actuated mecha
joint is unable t o provide t he force/ torque for ma nipulat ion under gravity. H owever, the COBOT
nism . M ost of the studies on t he underact uated ma
can cooperate wit h human being and implement some crit ical manipulation such as applicat ions in
nipulat ors assumed t hat the structure w as rigid and
bioengineering, medical t reat ment and micro elec [ 3]
[ 4, 5]
the research w as limited in position cont rol of passive joint or mot ion plan of syst em mainly.
t ronics. A met amorphic mechanism can reconf ig ure itself , and always result s in t he change of de
How ever, t he underact uat ed m anipulator alw ays is flex ible in structure for its application background
g ree of f reedoms ( DOF s) or const raint charact eris
obviously. A machine or manipulator is designed so
R eceived dat e: 2004 02 26; R evision received dat e: 2004 07 05 Foundation it em: N at ional N at ural S cience Foundat ion of China ( 50375007, 50475177)
February 2005
% 79 %
N onlinear Dynamic Analysis of Planar Flexible U nderact uat ed M anipulators
rigid or st rong t hat it has enoug h accuracy in ma
of t he links are same, E is Yong s modulus, t he
nipulat ion generally. Nevertheless, for the sake of
mass of unit lengt h of t he link is
economy of material or energy, t he dynamic behav
2,
ior of the actual m anipulator considering t he struc t ural flex ibility is crit ical and has to be investigat ed
that t he link is a Euler Bernouli beam. T he deformat ion of link described by t he as
thoroughly. On the other hand, the flexible ma
sum ed mode met hod can be writt en as
, n ) is the payload of each link end. Assume
yi ( x , t ) =
( x)
( i- 1)j
( t) +
m
ij
import ant developing direct ion of t he robot ics cur
iors of t he flex ible underactuated manipulator are
( i- 1) j j= 1
sum ing. T herefore, t he flex ible syst em is anot her [ 6]
and m i ( i = 1,
m
nipulat or has some merits such as the characterist ic in st ruct ural com pliancy or econom y in energ y con
rent ly . In t his paper, t he nonlinear vibrat ion behav
i,
( x)
ij
( i = 1, 2,
( t)
, n)
( 1)
j= 1
where
ij (
x ) is t he j th assumed mode f unction of
link i , and 0j ( x ) = 0, ij ( t ) are t he mode ampli t udes. T he mode shape funct ion ij
invest igated based on the int ernal resonance proper
( x ) = sh( !ij x ) + ∀ij sin( !ij x )
( 2)
t y of mult i deg ree nonlinear dynamic system, and a v ibrat ion reduct ion met hod is proposed. Moreover,
sat isf ies t he boundary condit ions for a simple sup
the inf luence of t he st ructural flex ibilit y on t he mo
paramet er
port ed beam w ith one end free. In Eq. ( 2) , t he ∀ij = sh( !ij L i ) / sin( !ij L i )
t ion behavior of t he passive joint is explored, and a posit ion control method for passive joint t hrough the st ruct ural vibrat ion is suggested.
1
!ij L i !
1 j + 4 # ( j = 1, 2)
( 4)
By Lag rang e equation, d ∃L ∃L = %i dt ∃ i ∃ i
Considering an n DOFs open chain planar flex ible manipulat or, t he vibration of t he syst em is
( i = 1, 2,
, n)
( 5)
T he dynamic equat ion in matrix vect or form is
induced f rom the t ransverse bending of the link and
given by M∀ &+ K&+ c = %
underact uated manipulator have no elast ic com po nents, so t he stiff ness of the passive joint is zero.
where, & # R
Assume t hat t he planar manipulat or is horizontal
nate; M = [ m ij ] # R
( Fig 1) and t he passive joint s have brakes, w hich
( 3)
and
Dynam ic Form ulation
the elast icit y of t he joint. T he passive joint s in t he
( j = 1, 2)
( n+ nm ) ∃ 1
( 6)
is t he generalized coordi
( n+ nm) ∃ ( n+ nm )
is the inert ia
( n+ nm ) ∃ ( n+ nm )
mat rix ; K = [ k ij ] # R
( n+ nm ) ∃ 1
ness mat rix ; c # R
is the st iff
contains the Coriolis,
centrifug al, elast ic and cent rif ugal stiff ening ef fects; % # R( n + nm ) ∃ 1 denot es the generalized ex t ernal t orque.
2 2. 1
Dynam ic Analysis
The vibration reduction effect with passive joints T he multi degree nonlinear dynamic system
Fig . 1
Planar tw o link flex ible under actuated manipulator model
cont rol t he passive joint to freely swing or to be
such as Eq. ( 6) can be analy zed by many approx i mat ion met hods such as averaging method, pro
locked. T he lengt h of link i is L i , t he moment of
g ression approximat ing t echnique, harmonic equi librium and so on. It has been shown t hat the sys
inert ia of the cross section area is I i , t he m at erials
t em has a plenty of dynam ic phenomena. H owev
% 80 %
HE G uang ping, LU Zhen
CJA
er, t here are a few invest ig at ions on opt imizing t he
prov ides a new vibrat ion reduction method, such as
performance of a syst em by the dynamic proper
by opt imizing the st ruct ural paramet ers.
t ies. A simple ex ample for this is show n in F ig. 2, in w hich t he vibrat ion of t he slender beam can be reduced by changing t he paramet ers of t he spring mass damp system.
F ig. 3
Fig . 2
T orque of actuating joint ( T = 0 1s)
Vibration absorber of a flexible beam
In cont rast w ith t he above scheme, t here is no need of addit ional component s for vibrat ion reduc t ion of t he underactuat ed manipulators, so that t his scheme brings some convenience in act ual applica t ion. F or bring forw ard t he benefit , a planar t wo link f lexible underact uat ed manipulat or is consid ered, w hose model is show n in Fig. 1. Assume t hat the joint near t he base is t he act uated joint , and t he
( a) b ∃ h = 3mm ∃ 50mm
joint betw een the t wo links is t he passive one, w hich is equipped w ith arrest er and is free current ly.
T he st ruct ural parameters are show n in
T able 1. Table 1
Parameters of link
Paramet er
V alue
D ensity / ( kg%m - 3)
7800
Cross section b ∃ h / ( mm ∃ mm)
3 ∃ 50
Y ong s modulus E / ( N%m - 2 ) Link Lengt h ( L 1 = L 2 ) / m Cent ralized mass ( m 1= m 2 ) / kg
2 06 ∃ 1010 0. 5
( b) b ∃ h = 2mm ∃ 50mm
1. 0
T he joint t orque is show n in F ig 3. Consider ing t he first t wo order vibrat ion modes, t he dy
Fig . 4
Vibration of tw o link flex ible underactuated manipulator under impulse input
namic response of the t wo link f lexible underactu ated manipulator is ex hibited in F ig 4( a) . When
2. 2
the cross sect ion area of the link 2 is changed to b
the foundation of m anipulat ing the underact uated
∃ h = 2mm ∃ 50mm and ot her conditions are not readjust ed, the new response is show n in Fig 4
mechanical syst em def tly. M ost of t he methods
( b) . It is obvious t hat t he vibrat ion is reduced ef
or digit al control technolog y for t he reason of sec ond order nonholonomic constraints of t he syst em,
f ect ively. T his phenomenon indicat es that the pas sive joint has the ability of reconfigurat ion of t he mechanical characterist ics of the syst em by chang ing t he st ate of t he brake. T his phenomenon also
Dynamic analysis under harmonic inputs T he posit ion cont rol problem of passive joint is
proposed before are based on t he nonlinear control
and assume t he system is rigid in st ruct ure. For prov iding some helpful instructions t o design a mo t ion controller, t he st ructural f lexibilit y is consid
February 2005
N onlinear Dynamic Analysis of Planar Flexible U nderact uat ed M anipulators
ered and t he dynamic behavior under harmonic in
% 81 %
T he tw o aspect s of dynamic analysis result s
puts is analyzed. Without lost of any generality,
manif est t hat the f lexible underact uated manipula
the t wo link flex ible underact uated manipulator is taken as exam ple. Assume that the input law of t he
t or is a spring mass damp system w hen there is ap
act uated joint is &1 = A cos( ∋t )
( 7)
w here, A and ∋ are the am plit ude and ang ular fre quency, respectively. T he angular speed and accel erat ion are g iven by &1 = - A ∋sin( ∋t )
( 8)
∀ &1 = - A ∋ cos( ∋t )
( 9)
2
Subst itut ing Eqs. ( 7) ( 9) into Eq. ( 6) , and
( a) A = 0. 001, ∋= 2#
g iving t he needed init ial condit ions, the numerical simulation results are obtained and given in F ig. 5. T he f igure shows the phase relat ion betw een t he posit ion and the angular speed of the passive joint. According to t he simulat ion results, it can be seen that the passive joint appears in harmonic oscilla t ion ( Fig 5( a) ) w hen t he am plit ude is very sm all. If t he amplit ude is enlarged, t he harmonic oscilla t ion chang es to a kind of & screw mot ion∋, a period ic motion wit h t he equilibrium posit ion of t he pas
( b) A = 0. 1, ∋= 2#
sive joint shif ts, and t he moving speed depends on the amplitude of t he harmonic input ( Fig. 5( b) ) . If the input amplitude is enlarged more t he mot ion of t he passive joint becomes a complex v ibration or chaos mot ion ( F ig. 5( c) ) . T he & screw mot ion∋ appeared in t he flex ible underact uated manipulator syst em m anif ests t hat the posit ion of the passive joint can be cont rolled by harmonic input . Nevertheless, the hig h frequency oscillat ion of the act uat ed jo int m ay arise a poor feasibility for t he lim it s of act uator performance. In fact, the vibrat ion result ed from the flex ibilit y of the st ruct ure also can induce a sim ilar eff ect on t he mot ion of passive joint. If the torque of t he act ua
( c) A = 0. 25, ∋= 2#
Fig . 5
Dynamic response of two link flexible underactu ated manipulator under harmonic input
t or can be expressed as Fig. 3 and t here is frict ion in t he passive joint , t he motion t rajectory of t he joint are given in t he Fig. 6. T he simulat ion result s show that t he st ruct ure vibrat ion w ill lead the pas sive joint to move. T his is a more economical ap proach f or position cont rolling of t he passive joint than driving t he act uated joint w ith harmonic in put.
( a) %= - 1. 0N%m
% 82 %
HE G uang ping, LU Zhen [ 2]
CJA
M oore C A , Pesh kin M A , Colgat e J E. D esign of 3R robot using continuous variable t ransmissions [ A ] . IEEE Int erna t ional Conf erence on Robot ics and Aut omation [ C ] . 1999, 3249- 3254.
[ 3]
D ai J S, Zhang Q X. M etamorphic mechanisms and t heir con f iguration models [ J ] . Chinese J of M echanical Engineering, 2000, 13( 3) : 212- 218.
[ 4]
A rai H. Posit ion control of a manipulat or wit h passive joint s using dynamic coupling [ J] . IEEE Trans on R obotics and A u t omation, 1997, 13( 4) : 528- 534.
[ 5]
A rai H. Time scaling cont rol of an underactuat ed man ipulat or
[ 6]
[ J ] . J of Robot ic Syst ems, 1998, 15( 9) : 525- 536. Nayf eh A H, Balachandran B. M odal int eractions in dynamical
( b) %= 1. 0N%m
Fig . 6
Dynamic response of two link flexible underactu ated manipulator under impulse input
propriate frict ion in t he free sw ing passive joint. T he flexibilit y of the manipulator can be utilized t o implement som e intractable operat ions.
3
Conclusions
T he passive joint of t he flex ible underact uat ed manipulator can bring the vibrat ion modes to cou pling , and t he modes coupling act ion can be used t o reduce t he st ruct ure vibration. When t he input am plitude sat isf ies t he small vibrat ion condition, t he passive joint reveals a & screw motion∋, w hich can be ut ilized to cont rolling the posit ion of t he passive joint . T he vibrat ion induced by st ructural flex ibili t y also drives t he passive joint to show a similar mot ion behavior. References [ 1]
M uk herjee R, Ch en D G . Cont rol of f ree f lying underactuated space manipulators t o equilibrium man if olds [ J] . IEEE Trans on Robot ics and Aut omat ion, 1993, 9 ( 5) : 561- 570.
and st ructural syst ems [ J ] . A SM E A pplied M echanics Re view s, 1989, 42: 175- 210.
Biographies: HE Guang Ping Born in 1972, he re ceived B. S. , M . S. and D. S. fr om Bei jing U niversity of A eronautics and Astro nautics in 1994, 1997 and 2002 respec tively. He is an associate professor of N orth China U niversity of T echnology cur rently. His main study interest in cludes robotics and no nlinear dynamics and control. T el: ( 010) 88802835; E mail: guangping he@ sohu. com LU Zhen Born in 1942, professor at Department of Electrical Eng ineering, Beijing University of Aeronautics and As tronautics. He graduated from Beijing U niversity o f T echnology in 1966, and g ot his doctorial deg ree in 1987 from Beijing Instituteof A er onautics and Astr onautics. His resear ch interests are mechanism, robotics and bio nics. He is the senior member of Chinese M echanism Engineer ing Society and Chinese Aeronautics Society , the vice chair of Per manente Commission of HM M IFT oM M , member of the Committee of Chinese Intelligent Robots Society and the Committee of robotics IFT oM M . He has superv ised 17 docto r ial candidates and published 90 papers and works.
No. 1
CH INES E JO U RN A L O F AER ON A U TICS
February 2005
Nonlinear Dynamic Analysis of Planar Flexible Underactuated Manipulators 1
2
HE Guang ping , LU Zhen ( 1 . School of M echanical and Electr ical Engineer ing , North Chi na Univer si ty of T echnology , Beij ing
100041 , Chi na)
( 2 . School of A utomati on Science and Electr ical Engineer ing , Beij ing Uni versity of A eronaut ics and A st ronaut ics, Beij ing Abstract:
100083 , Chi na)
T he influence of passive joint on the dynamics of planar flexible under actuated manipulators
is studied. A vibr atio n reduction method based on the internal resonance phenomenon of multi degree nonlinear dynamic system is proposed. T he dynamic simulation results rev eal that the har mo nic input for t he actuated jo int will induce the passiv e joint dev iating from its equilibrium position, the ex cursion speed and direction depend on t he amplitude of the input . A passive jo int position control scheme making use of t he vibration of the flexible structure is sugg ested, and the numerical simulation results of a model of pla nar two link flexible underactuated manipulator is shown. Key words:
dynamics and vibration; manipulator; underactuated; nonlinear; internal resonance
平面柔性欠驱动机械臂的非线性动力学分析. 何广平 , 陆
震. 中国航空学报( 英文版) , 2005, 18
( 1) : 78- 82. 摘
要: 研究了平面柔性欠驱动机械 臂中被 动关节 对系统动 态特性 的影响, 在 动力学分 析的基 础
上, 提出了一种基于内共振原理的柔性欠驱动机械臂 振动控制方法。在 对系统的稳 态周期运 动进 行分析研 究的基础上, 发现处于自由摆动状态的被动关节的平 衡位置随 系统结构的 振动而发 生漂 移, 并且被动关节平衡位置的漂移速度和方向与周期 输入的振幅有关。 提出利用结 构柔性产 生的 振动实现被动关节位置控制的方法, 通过平面二连杆 柔性欠驱动机械臂进行了仿真计算。 关键词: 动力学与振动; 机械臂; 欠驱动; 非线性; 内共振 文章编号: 1000 9361( 2005) 01 0078 05
中图分类号: T P242
T he motion cont rol of underactuated mecha
文献标识码: A
t ics of the original system. Met amorphic mecha
nism or underact uated manipulat or is proposed in
nisms have some passive joints or flex ible com po
the f ault tolerance technology of space robot system first ly [ 1] . An underactuated m anipulator also can
nents for economical sake and make the best of t he abilit y of reconf igurat ion of the mechanism. T here
be designed as a kind of assist ant robot syst em, w hich is said t o be a COBOT [ 2] . T he COBOT sys
f ore, the underactuated mechanisms and underac t uated manipulat ors have some propert ies w hich
t em can not work independent ly for t he passive
can not be provided by t he f ull actuated mecha
joint is unable t o provide t he force/ torque for ma nipulat ion under gravity. H owever, the COBOT
nism . M ost of the studies on t he underact uated ma
can cooperate wit h human being and implement some crit ical manipulation such as applicat ions in
nipulat ors assumed t hat the structure w as rigid and
bioengineering, medical t reat ment and micro elec [ 3]
[ 4, 5]
the research w as limited in position cont rol of passive joint or mot ion plan of syst em mainly.
t ronics. A met amorphic mechanism can reconf ig ure itself , and always result s in t he change of de
How ever, t he underact uat ed m anipulator alw ays is flex ible in structure for its application background
g ree of f reedoms ( DOF s) or const raint charact eris
obviously. A machine or manipulator is designed so
R eceived dat e: 2004 02 26; R evision received dat e: 2004 07 05 Foundation it em: N at ional N at ural S cience Foundat ion of China ( 50375007, 50475177)
February 2005
% 79 %
N onlinear Dynamic Analysis of Planar Flexible U nderact uat ed M anipulators
rigid or st rong t hat it has enoug h accuracy in ma
of t he links are same, E is Yong s modulus, t he
nipulat ion generally. Nevertheless, for the sake of
mass of unit lengt h of t he link is
economy of material or energy, t he dynamic behav
2,
ior of the actual m anipulator considering t he struc t ural flex ibility is crit ical and has to be investigat ed
that t he link is a Euler Bernouli beam. T he deformat ion of link described by t he as
thoroughly. On the other hand, the flexible ma
sum ed mode met hod can be writt en as
, n ) is the payload of each link end. Assume
yi ( x , t ) =
( x)
( i- 1)j
( t) +
m
ij
import ant developing direct ion of t he robot ics cur
iors of t he flex ible underactuated manipulator are
( i- 1) j j= 1
sum ing. T herefore, t he flex ible syst em is anot her [ 6]
and m i ( i = 1,
m
nipulat or has some merits such as the characterist ic in st ruct ural com pliancy or econom y in energ y con
rent ly . In t his paper, t he nonlinear vibrat ion behav
i,
( x)
ij
( i = 1, 2,
( t)
, n)
( 1)
j= 1
where
ij (
x ) is t he j th assumed mode f unction of
link i , and 0j ( x ) = 0, ij ( t ) are t he mode ampli t udes. T he mode shape funct ion ij
invest igated based on the int ernal resonance proper
( x ) = sh( !ij x ) + ∀ij sin( !ij x )
( 2)
t y of mult i deg ree nonlinear dynamic system, and a v ibrat ion reduct ion met hod is proposed. Moreover,
sat isf ies t he boundary condit ions for a simple sup
the inf luence of t he st ructural flex ibilit y on t he mo
paramet er
port ed beam w ith one end free. In Eq. ( 2) , t he ∀ij = sh( !ij L i ) / sin( !ij L i )
t ion behavior of t he passive joint is explored, and a posit ion control method for passive joint t hrough the st ruct ural vibrat ion is suggested.
1
!ij L i !
1 j + 4 # ( j = 1, 2)
( 4)
By Lag rang e equation, d ∃L ∃L = %i dt ∃ i ∃ i
Considering an n DOFs open chain planar flex ible manipulat or, t he vibration of t he syst em is
( i = 1, 2,
, n)
( 5)
T he dynamic equat ion in matrix vect or form is
induced f rom the t ransverse bending of the link and
given by M∀ &+ K&+ c = %
underact uated manipulator have no elast ic com po nents, so t he stiff ness of the passive joint is zero.
where, & # R
Assume t hat t he planar manipulat or is horizontal
nate; M = [ m ij ] # R
( Fig 1) and t he passive joint s have brakes, w hich
( 3)
and
Dynam ic Form ulation
the elast icit y of t he joint. T he passive joint s in t he
( j = 1, 2)
( n+ nm ) ∃ 1
( 6)
is t he generalized coordi
( n+ nm) ∃ ( n+ nm )
is the inert ia
( n+ nm ) ∃ ( n+ nm )
mat rix ; K = [ k ij ] # R
( n+ nm ) ∃ 1
ness mat rix ; c # R
is the st iff
contains the Coriolis,
centrifug al, elast ic and cent rif ugal stiff ening ef fects; % # R( n + nm ) ∃ 1 denot es the generalized ex t ernal t orque.
2 2. 1
Dynam ic Analysis
The vibration reduction effect with passive joints T he multi degree nonlinear dynamic system
Fig . 1
Planar tw o link flex ible under actuated manipulator model
cont rol t he passive joint to freely swing or to be
such as Eq. ( 6) can be analy zed by many approx i mat ion met hods such as averaging method, pro
locked. T he lengt h of link i is L i , t he moment of
g ression approximat ing t echnique, harmonic equi librium and so on. It has been shown t hat the sys
inert ia of the cross section area is I i , t he m at erials
t em has a plenty of dynam ic phenomena. H owev
% 80 %
HE G uang ping, LU Zhen
CJA
er, t here are a few invest ig at ions on opt imizing t he
prov ides a new vibrat ion reduction method, such as
performance of a syst em by the dynamic proper
by opt imizing the st ruct ural paramet ers.
t ies. A simple ex ample for this is show n in F ig. 2, in w hich t he vibrat ion of t he slender beam can be reduced by changing t he paramet ers of t he spring mass damp system.
F ig. 3
Fig . 2
T orque of actuating joint ( T = 0 1s)
Vibration absorber of a flexible beam
In cont rast w ith t he above scheme, t here is no need of addit ional component s for vibrat ion reduc t ion of t he underactuat ed manipulators, so that t his scheme brings some convenience in act ual applica t ion. F or bring forw ard t he benefit , a planar t wo link f lexible underact uat ed manipulat or is consid ered, w hose model is show n in Fig. 1. Assume t hat the joint near t he base is t he act uated joint , and t he
( a) b ∃ h = 3mm ∃ 50mm
joint betw een the t wo links is t he passive one, w hich is equipped w ith arrest er and is free current ly.
T he st ruct ural parameters are show n in
T able 1. Table 1
Parameters of link
Paramet er
V alue
D ensity / ( kg%m - 3)
7800
Cross section b ∃ h / ( mm ∃ mm)
3 ∃ 50
Y ong s modulus E / ( N%m - 2 ) Link Lengt h ( L 1 = L 2 ) / m Cent ralized mass ( m 1= m 2 ) / kg
2 06 ∃ 1010 0. 5
( b) b ∃ h = 2mm ∃ 50mm
1. 0
T he joint t orque is show n in F ig 3. Consider ing t he first t wo order vibrat ion modes, t he dy
Fig . 4
Vibration of tw o link flex ible underactuated manipulator under impulse input
namic response of the t wo link f lexible underactu ated manipulator is ex hibited in F ig 4( a) . When
2. 2
the cross sect ion area of the link 2 is changed to b
the foundation of m anipulat ing the underact uated
∃ h = 2mm ∃ 50mm and ot her conditions are not readjust ed, the new response is show n in Fig 4
mechanical syst em def tly. M ost of t he methods
( b) . It is obvious t hat t he vibrat ion is reduced ef
or digit al control technolog y for t he reason of sec ond order nonholonomic constraints of t he syst em,
f ect ively. T his phenomenon indicat es that the pas sive joint has the ability of reconfigurat ion of t he mechanical characterist ics of the syst em by chang ing t he st ate of t he brake. T his phenomenon also
Dynamic analysis under harmonic inputs T he posit ion cont rol problem of passive joint is
proposed before are based on t he nonlinear control
and assume t he system is rigid in st ruct ure. For prov iding some helpful instructions t o design a mo t ion controller, t he st ructural f lexibilit y is consid
February 2005
N onlinear Dynamic Analysis of Planar Flexible U nderact uat ed M anipulators
ered and t he dynamic behavior under harmonic in
% 81 %
T he tw o aspect s of dynamic analysis result s
puts is analyzed. Without lost of any generality,
manif est t hat the f lexible underact uated manipula
the t wo link flex ible underact uated manipulator is taken as exam ple. Assume that the input law of t he
t or is a spring mass damp system w hen there is ap
act uated joint is &1 = A cos( ∋t )
( 7)
w here, A and ∋ are the am plit ude and ang ular fre quency, respectively. T he angular speed and accel erat ion are g iven by &1 = - A ∋sin( ∋t )
( 8)
∀ &1 = - A ∋ cos( ∋t )
( 9)
2
Subst itut ing Eqs. ( 7) ( 9) into Eq. ( 6) , and
( a) A = 0. 001, ∋= 2#
g iving t he needed init ial condit ions, the numerical simulation results are obtained and given in F ig. 5. T he f igure shows the phase relat ion betw een t he posit ion and the angular speed of the passive joint. According to t he simulat ion results, it can be seen that the passive joint appears in harmonic oscilla t ion ( Fig 5( a) ) w hen t he am plit ude is very sm all. If t he amplit ude is enlarged, t he harmonic oscilla t ion chang es to a kind of & screw mot ion∋, a period ic motion wit h t he equilibrium posit ion of t he pas
( b) A = 0. 1, ∋= 2#
sive joint shif ts, and t he moving speed depends on the amplitude of t he harmonic input ( Fig. 5( b) ) . If the input amplitude is enlarged more t he mot ion of t he passive joint becomes a complex v ibration or chaos mot ion ( F ig. 5( c) ) . T he & screw mot ion∋ appeared in t he flex ible underact uated manipulator syst em m anif ests t hat the posit ion of the passive joint can be cont rolled by harmonic input . Nevertheless, the hig h frequency oscillat ion of the act uat ed jo int m ay arise a poor feasibility for t he lim it s of act uator performance. In fact, the vibrat ion result ed from the flex ibilit y of the st ruct ure also can induce a sim ilar eff ect on t he mot ion of passive joint. If the torque of t he act ua
( c) A = 0. 25, ∋= 2#
Fig . 5
Dynamic response of two link flexible underactu ated manipulator under harmonic input
t or can be expressed as Fig. 3 and t here is frict ion in t he passive joint , t he motion t rajectory of t he joint are given in t he Fig. 6. T he simulat ion result s show that t he st ruct ure vibrat ion w ill lead the pas sive joint to move. T his is a more economical ap proach f or position cont rolling of t he passive joint than driving t he act uated joint w ith harmonic in put.
( a) %= - 1. 0N%m
% 82 %
HE G uang ping, LU Zhen [ 2]
CJA
M oore C A , Pesh kin M A , Colgat e J E. D esign of 3R robot using continuous variable t ransmissions [ A ] . IEEE Int erna t ional Conf erence on Robot ics and Aut omation [ C ] . 1999, 3249- 3254.
[ 3]
D ai J S, Zhang Q X. M etamorphic mechanisms and t heir con f iguration models [ J ] . Chinese J of M echanical Engineering, 2000, 13( 3) : 212- 218.
[ 4]
A rai H. Posit ion control of a manipulat or wit h passive joint s using dynamic coupling [ J] . IEEE Trans on R obotics and A u t omation, 1997, 13( 4) : 528- 534.
[ 5]
A rai H. Time scaling cont rol of an underactuat ed man ipulat or
[ 6]
[ J ] . J of Robot ic Syst ems, 1998, 15( 9) : 525- 536. Nayf eh A H, Balachandran B. M odal int eractions in dynamical
( b) %= 1. 0N%m
Fig . 6
Dynamic response of two link flexible underactu ated manipulator under impulse input
propriate frict ion in t he free sw ing passive joint. T he flexibilit y of the manipulator can be utilized t o implement som e intractable operat ions.
3
Conclusions
T he passive joint of t he flex ible underact uat ed manipulator can bring the vibrat ion modes to cou pling , and t he modes coupling act ion can be used t o reduce t he st ruct ure vibration. When t he input am plitude sat isf ies t he small vibrat ion condition, t he passive joint reveals a & screw motion∋, w hich can be ut ilized to cont rolling the posit ion of t he passive joint . T he vibrat ion induced by st ructural flex ibili t y also drives t he passive joint to show a similar mot ion behavior. References [ 1]
M uk herjee R, Ch en D G . Cont rol of f ree f lying underactuated space manipulators t o equilibrium man if olds [ J] . IEEE Trans on Robot ics and Aut omat ion, 1993, 9 ( 5) : 561- 570.
and st ructural syst ems [ J ] . A SM E A pplied M echanics Re view s, 1989, 42: 175- 210.
Biographies: HE Guang Ping Born in 1972, he re ceived B. S. , M . S. and D. S. fr om Bei jing U niversity of A eronautics and Astro nautics in 1994, 1997 and 2002 respec tively. He is an associate professor of N orth China U niversity of T echnology cur rently. His main study interest in cludes robotics and no nlinear dynamics and control. T el: ( 010) 88802835; E mail: guangping he@ sohu. com LU Zhen Born in 1942, professor at Department of Electrical Eng ineering, Beijing University of Aeronautics and As tronautics. He graduated from Beijing U niversity o f T echnology in 1966, and g ot his doctorial deg ree in 1987 from Beijing Instituteof A er onautics and Astr onautics. His resear ch interests are mechanism, robotics and bio nics. He is the senior member of Chinese M echanism Engineer ing Society and Chinese Aeronautics Society , the vice chair of Per manente Commission of HM M IFT oM M , member of the Committee of Chinese Intelligent Robots Society and the Committee of robotics IFT oM M . He has superv ised 17 docto r ial candidates and published 90 papers and works.