Advances in Robot Manipulators Part 17 ppt

632 Advances in Robot Manipulators Similar to the those established in (De Silva, 1976; Sooraksa & Chen, 1998), equation (16) is the fifth order TB homogeneous linear PDE with internal and external damping effects expressing the deflection w( x , t ) We have added to this equation the following initial and pinned (clamped)-mass boundary conditions (Loudini et al., 2007a, Loudini et al., 2006): w( x ,0)  w0 , Initial conditions: U U U w(0, t )  0 , Clamped end: U (18)   3 w( x , t )  w(0, t )  0 ,  M( x , t )  J h  0 xt 2  x0  Pinned end: U w( x , t )   w0 t t 0 U w( x , t ) 0 x x0 (20)   M( x , t )  2 w( x , t )   Mp   0 t 2  x   x  Free end with payload mass:   3   M( x , t )  J  w( x , t )   0   p  xt 2  x  U (19) (21) U The classical fourth order TB PDE is retrieved if the damping effects terms are suppressed: EI  4 w( x , t ) x 4  2 w( x , t ) E   4 w( x , t ) ρ 2 I  4 w( x , t )   ρI  1    ρA 0    KG  x 2 t 2 KG t 4 t 2  (22) If the effect due to the rotary inertia is neglected, we are led to the shear beam (SB) model (Morris, 1996; Han et al., 1999): EI  4 w( x , t ) x 4  ρIE  4 w( x , t ) KG x 2 t 2  ρA  2 w( x , t ) t 2 0 (23) but, if the one due to shear distortion is the neglected one, the Rayleigh beam equation (Han et al., 1999; Rayleigh, 2003) arises: EI  4 w( x , t ) x 4  ρI  4 w( x , t ) x 2 t 2  ρA  2 w( x , t ) t 2 0 (24) Moreover, if both the rotary inertia and shear deformation are neglected, then the governing equation of motion reduces to that based on the classical EBT (Meirovitch, 1986) given by Timoshenko Beam Theory based Dynamic Modeling of Lightweight Flexible Link Robotic Manipulators EI  4 w( x , t ) x 4  ρA 633  2 w( x , t ) t 2 (25) 0 If the above included damping effects are associated to the EBB, the corresponding PDE is KD I  5 w( x , t ) x t 4  EI  4 w( x , t ) x 4  ρA  2 w( x , t ) t 2  AD w( x , t ) t 0 (26) The resolution of the PDE with mixed derivative terms (16) is a complex mathematical problem Among the few methods existing in the literature, we cite the following approaches with some representative works: the finite element method (Kapur, 1966; Hoa, 1979; Kolberg 1987), the Galerkin method (Wang and Chou, 1998; Dadfarnia et al., 2005), the Rayleigh-Ritz method (Oguamanam and Heppler, 1996), the Laplace transform method resulting in an integral form solution (Boley & Chao, 1955; Wang & Guan, 1994; Ortner & Wagner, 1996), and the eigenfunction expansion method, also referred to as the series or modal expansion method (Anderson, 1953; Dolph, 1954; Huang, 1961; Ekwaro-Osire et al., 2001; Loudini et al 2006; Loudini et al 2007a; Loudini et al 2007b) In the latter one, w( x , t ) can take the following expanded separated form which consists of an infinite sum of products between the chosen transverse deflection eigenfunctions or mode shapes Wn ( x ) , that must satisfy the pinned (clamped)-free (mass) BCs, and the timedependant modal generalized coordinates δn (t ) :  w( x , t )   W (x)δ (t) n n (27) n 1 2.4 Dynamic model deriving procedure In order to obtain a set of ordinary differential equations (ODEs) of motion to adequately describe the dynamics of the flexible link manipulator, the Hamilton's or Lagrange's approach combined with the Assumed Modes Method (AMM) (Fraser & Daniel, 1991; Loudini et al 2006; Loudini et al 2007a; Loudini et al 2007b; Tokhi & Azad, 2008) can be used According to the Lagrange's method, a dynamic system completely located by n generalized coordinates qi must satisfy n differential equations of the form: d  L  dt  qi    L D    Fi , i  0,1, 2,  q q i i  (28) where L is the so-called Lagrangian given by L  T U (29) 634 Advances in Robot Manipulators T represents the kinetic energy of the modeled system and U its potential energy Also, in (28) D is the Rayleigh's dissipation function which allows dissipative effects to be included, and Fi is the generalized external force acting on the corresponding coordinate qi Theoretically there are infinite number of ODEs, but for practical considerations, such as boundedness of actuating energy and limitation of the actuators and the sensors working frequency range, it is more reasonable to truncate this number at a finite one n (Cannon & Schmitz, 1984; Kanoh & Lee, 1985; Qi & Chen, 1992) The total kinetic energy of the robot flexible link and its potential energy due to the internal bending moment and the shear force are, respectively, given by (Macchelli & Melchiorri, 2004; Loudini et al 2006; Loudini et al 2007a; Loudini et al 2007b): T 1 2   0 2  2 1  w( x , t )   γ ( x , t )  ρA  ρI   dx   dx 2 0  t   t   2   1  γ ( x , t )  2 U EI  KAG σ ( x , t ) dx  dx    2 0  x  20 1  (30)  (31) The dissipated energy due to the damping effects can be written as (Krishnan & Vidyasagar, 1988; Loudini et al 2006; Loudini et al 2007a; Loudini et al 2007b): D 2  2    3 w( x , t )  1  w( x , t )  AD  KDI   dx  dx  20 20  x 2 t   t    1   (32) Substituting these energies expressions into (28) accordingly and using the transverse deflection separated form (27), we can derive the desired dynamic equations of motion in the mass ( B ), damping ( H ), Coriolis and centrifugal forces ( N ) and stiffness ( K ) matrix familiar form: B d 2 q( t ) dt 2 H dq(t ) dt T   N  q(t ), q(t )  Kq(t )  F(t ) with q(t )  θ(t ) δ1 (t ) δ2 (t )  δn (t ) ; F(t )  τ    (33) T 0 0  0  If we disregard some high order and nonlinear terms, under reasonable assumptions, the matrix differential equation in (33) could be easily represented in a state-space form as   z(t )  Az z(t )  Bzu(t )    y(t )  C z z(t )  (34) Timoshenko Beam Theory based Dynamic Modeling of Lightweight Flexible Link Robotic Manipulators 635 T T    0  0  ; z(t )  θ(t ) δ1 (t )  δn (t ) θ(t ) δ1 (t )  δn (t )    Solving the state-space matrices gives the vector of states z (t ) , that is, the angular with u(t )  τ  displacement, the modal amplitudes and their velocities 3 A Special Case Study: Comprehensive Dynamic Modeling of a Flexible Link Manipulator Considered as a Shear Deformable Timoshenko Beam In this second part of our work, we present a novel dynamic model of a planar single-link flexible manipulator considered as a tip mass loaded pinned-free shear deformable beam Using the classical TBT described in section 2 and including the Kelvin-Voigt structural viscoelastic effect (Christensen, 2003), the lightweight robotic manipulator motion governing PDE is derived Then, based on the Lagrange's principle combined with the AMM, a dynamic model suitable for control purposes is established 3.1 System description and motion governing equation The considered physical system is shown in Fig 4 The basic deriving procedure to obtain the motion governing equation has been described in the previous section, and so only an outline giving the main steps is presented here The effect of rotary inertia being neglected in this case study, equation (10) expressing the equilibrium of the moments becomes: M( x , t ) x  S( x , t ) (35) The relation that fellows balancing forces is S( x , t ) x  ρA  2 w( x , t ) t 2 (36) Substitution of 6 and 9 into 35 and likewise 6 into 36 yields the two coupled equations of the damped SB motion: KD I  3 γ( x , t ) x t 2  EI  2 γ(x , t ) x 2  w( x , t )   kAG   γ ( x , t )  0  x    2 w( x , t ) γ ( x , t )   2 w( x , t ) kAG   0   ρA x  t 2  x 2   (37) (38) Equations 37 and 38 can be easily decoupled to obtain the fifth order SB homogeneous linear PDEs with internal damping effect expressing the deflection w( x, t ) and the slope of bending  (x, t) 636 Advances in Robot Manipulators KD I  5 w( x , t ) KD I x 4 t  5 γ( x , t ) x 4 t   ρK D I  5 w( x , t ) KG x 2 t 3 ρK D I  5 γ ( x , t ) KG x 2 t 3  EI  EI  4 w( x , t ) x 4  4 γ(x , t ) x 4   ρEI  4 w( x , t ) KG x 2 t 2 ρEI  4 γ ( x , t ) KG x 2 t 2  ρA  ρA  2 w( x , t ) t 2  2 γ( x , t ) t 2 (39) 0 (40) 0 Z0 Y0 X0 Rigid hub (Jh) Y0 Deflected link Y ( , E , I ,  ) Beam element Tip payload ( M p , J p ) y0 w( x,t ) Center of mass x X  ( x, t ) y   (t )  (t ) x0 X0 x  dx x dx S dx x M M ( x, t )  dx x S ( x, t )  w x  Y M ( x, t ) S ( x, t )  2 w( x, t ) A t 2 X Fig 4 Physical configuration and kinematics of deformation of a bending element of the studied flexible robot manipulator considered as a shear deformable beam Timoshenko Beam Theory based Dynamic Modeling of Lightweight Flexible Link Robotic Manipulators 637 We affect to the equation (39) the same initial and pinned-mass boundary conditions, given by equations 18, 19, and 21, with taking into account the result established by (Wang & Guan, 1994; Loudini et al., 2007b) about the very small influence of the tip payload inertia on the flexible manipulator dynamics:  w( x ,0)  w0 , wt ( x , t ) t0  w0 Initial conditions: U U (41) BCs at the pinned end (root of the link): U w( x , t ) x0  0 : zero average translational displacement M ( x , t ) x 0  J h  3 w( x , t ) : balance of bending moments xt 2 x0 (42) (43) BCs at the mass loaded free end: U M( x , t ) x  0 : zero average of bending moments M( x , t )  Mp x  2 w( x , t ) t 2 x  : balance of shearing forces (44) (45) x  The classical fourth order SB PDEs are retrieved if the damping effect term is suppressed: EI  4 w( x , t ) EI x 4  4γ(x , t ) x 4 ρEI  4 w( x , t )  KG x 2 t 2  ρEI  4 γ ( x , t ) KG x 2 t 2  ρA  ρA  2 w( x , t ) t 2  2 γ( x , t ) t 2 0 (46) 0 (47) Moreover, if shear deformation is neglected, then the governing equation of motion reduces to that based on the classical EBT, given by 25 If the above included damping effect is associated to the EBB, the corresponding PDE is KD I  5 w( x , t ) x 4 t  EI  4 w( x , t ) x 4  ρA  2 w( x , t ) t 2 0 (48) To solve the PDEs with mixed derivative terms (39) and (40), we have tried to apply the classical AMM which is well known as a computationally efficient scheme that separates the mode functions from the shape functions The forms of equations (39) and (40) being identical, w( x , t ) and γ ( x , t ) are assumed to 638 Advances in Robot Manipulators share the same time-dependant modal generalized coordinate δ(t ) under the following separated forms with the respective mode shape functions (eigenfuntions) Φ( x ) and Ψ( x ) that must satisfy the pinned-free (mass) BCs: w( x , t )  Φ( x )δ(t ) γ ( x , t )  Ψ( x )δ(t ) (49) Unfortunately, the application of 49 has not been possible to derive the mode shapes expressions This is due to the unseparatability of some terms of 39 and 40 To find a way to solve the problem, we have based our investigations on the result pointed out in (Gürgöze et al., 2007) In this work, it has been established that the characteristic equation of a visco-elastic EBB i.e., a Kelvin-Voigt model (given in our chapter by 48), is formally the same as the frequency equation of the cantilevered elastic beam (the EB modeled by 25) Thus, we can assume that the damping effect affects only the modal function δ(t ) So, the mode shape is that of the SB model (46, 47) Applying the AMM to the PDEs 46 and 47, we obtain EIΦiv ( x )δ(t )  EIΨ iv ( x )δ(t )  ρEI KG ρEI KG   Φ( x )δ(t )  ρAΦ( x )δ(t )  0 (50)   Ψ( x )δ(t )  ρAΨ( x )δ(t )  0 (51) Separating the functions of time, t , and space x :  δ (t ) δ(t )  Φ iv ( x ) Ψ iv ( x )   constant   λ ρA ρ ii ρA ρ ii Φ( x )  Φ (x) Ψ( x )  ψ (x) EI KG EI KG (52) The differential equation for the temporal modal generalized coordinate is  δ(t )  λδ(t )  0 (53) Its general solution is assumed to be in the following forms: δ(t )  De jωt  De  jωt  F cos(ωt  φ ) (54) λ  ω2 (55) where Timoshenko Beam Theory based Dynamic Modeling of Lightweight Flexible Link Robotic Manipulators 639 The constants D and its complex conjugate D (or F and the phase  ) are determined from the initial conditions The natural frequency ω is determined by solving the spatial problem given by Φ iv ( x )  Ψ (x)  iv ρ KG ρ KG ω2 Φ ii ( x )  ρA EI ω Ψ (x)  2 ii ω2 Φ( x )  0 ρA EI (56) ω Ψ( x )  0 2 The solutions of 56 can be written in terms of sinusoidal and hyperbolic functions Φ( x )  C 1 sin ax  C 2 cos ax  C 3 sinh bx  C 4 cosh bx (57) Ψ( x )  D1 sin ax  D2 cos ax  D3 sinh bx  D4 cosh bx where a ρ 2 ρA 2  ρ  ω   ω2   ω   2 KG EI  2 KG  ρ 2 ρA 2  ρ  ; b  ω   ω2   ω   2KG EI  2 KG  2 2 (58) The constants C k , Dk ; k  1, 4 of 57 are determined through the BCs 42-45 rewritten on the basis of 49, 53 and 55 as follows: Φ(0)  0 Ψ(0)   Jh EI ω2 Φ(0)   JΦ(0) Ψ( )  0 Φ(  )  Ψ( )  Mp KAG ω2 Φ( )  MΦ( ) (59) (60) (61) (62) By applying 59-62 to 57, we find these relations C 2  C 4 (63) aD1  bD3   aJC 1  bJC 3 (64) aD1 cos a  aD2 sin a  bD3 cosh b  bD4 sinh b  0 (65) 640 Advances in Robot Manipulators C1 a  C 2 M  D2  cos a  C 2 a  C1 M  D1  sin a  C 3b  C 4 M  D4  cosh b   C 4 b  C 3 M  D3  sinh b  0 (66) The relations between the unknown constants C k and Dk are obtained by substituting 57 into 38: D1  R  a2 a C 2 ; D2   R  a2 a C 1 ; D3  R  b2 b  R  b2 C 4 ; D4    b   C3   (67) or C1   where R  ρω 2 KG a R  a  2 D2 ; C 2  a R  a  2 D1 ; C 3  b R  b  2 D4 ; C 4  b R  b  2 D3 (68) From 63 and 67, we obtain D3    D bR  a  a R  b2 1 2 (69)  D31D1 From some combinations of 63-69, we find the relations       a R  b2   sinh b  sin a 2   b Ra  C 1  C 21C 1 C2   2 2 2 2 a b Rb   Rb cosh b  sinh b   cos a  2 2 Ra bJ R  a          (70)  R  R  R     MC 21  cos a   C 21  M  sin a  MC 21 cosh b  C 21 sinh b  a b  C  C C (71)  a  C3    31 1   1 R   M sinh b  cosh b   b   656 Advances in Robot Manipulators 5 (Horowitz et al., 1991) used the kernel of the form  K (t ,  )  f 0    f m cos  2 mt / T  cos  2 m / T   d m sin  2 mt / T  sin  2 m / T   , (12) m 1 where fi and di’s are scalar constants, which satisfy the conditions, Dm-2