Theoretical and experimental investigation of viscoelastic serial ...

Original Article

Theoretical and experimental investigation of viscoelastic serial robotic manipulators with motors at the joints using Timoshenko beam theory and Gibbs–Appell formulation

Proc IMechE Part K: J Multi-body Dynamics 2016, Vol. 230(1) 37–51 ! IMechE 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1464419315574406 pik.sagepub.com

MH Korayem, AM Shafei, M Doosthoseini, F Absalan and B Kadkhodaei

Abstract In this article, the mathematical modeling of n-viscoelastic-link robotic manipulators based on the Gibbs–Appell formulation and the assumed mode method is developed. The elastic properties of the links are modeled according to the Timoshenko beam theory and its associated mode shapes. Also the dynamic effects of the motors at the joints are fully taken into consideration. In the mathematical modeling, the effects of torsion, extensional deformation, bending in two directions, structural damping, and viscous air as well as the gravity effects have also been considered. A recursive algorithm based on 4 4 transformation matrices has been developed to ease the derivation of motion equations. The mismatch between the high-speed vibrations of flexible robotic arms and the low-speed operation of the software and hardware interface in the computer and the experimental setup is a serious obstacle for the measuring and controlling of such systems. So, to verify the simulation results and the results obtained from the experimental test bed, high-speed digital-to-analog converter components on an electronic interface board are used in conjunction with the National Instrument’s LABVIEW software package. Keywords Kelvin–Voigt, Timoshenko, recursive, Gibbs–Appell, experimental setup Date received: 8 October 2014; accepted: 22 January 2015

Introduction Nowadays, there is a growing interest in the use of robotic manipulators with flexible links in many situations, such as in the aerospace industry, and for the maintenance of power plants. Compared to the conventionally heavy and bulky rigid-arms manipulators, the elastic-link robotic manipulators enjoy unique benefits such as lower energy consumption, higher operation speed, better maneuverability, and better transportability. However, the accuracy of operation in these kinds of robotic manipulators is drastically limited by their end-effector deflections and vibrations which persist for a moment after a move is completed. Achieving a balance between high operational speed and high desirable accuracy in these robotic manipulators has made it a challenging issue for the researchers. The purpose of this paper is not to review all the existing works in the field of elastic robotic manipulators. Instead, this paper focuses only on those

research works that are important and relevant to the present research topic. Finite element method (FEM) and assumed mode method (AMM) are two basic approaches for the mathematical modeling of elastic robotic manipulators. FEM is a general technique and it can be extended to manipulators with complexly shaped links. This method is not the subject of this paper, and for more detailed studies one may be referred to Geradin and Cardona1 in which the mathematical modeling of flexible multibody systems has been Department of Mechanical Engineering, Center of Excellence in Experimental Solid Mechanics and Dynamics, Iran University of Science and Technology, Tehran, Islamic Republic of Iran Corresponding author: MH Korayem, Iran University of Science and Technology, Narmak, Tehran 18846, Islamic Republic of Iran. Email: [email protected]

38 presented. As was mentioned earlier, the other method is the AMM in which the flexibility of the links is represented by a truncated finite modal series in terms of spatial mode shapes and timedependent mode amplitudes. Although this technique has been exploited extensively, in most of these works the Euler–Bernoulli beam theory (EBBT) and its related mode Eigenfunctions have been employed.2 For a more precise modeling of the links’ elastic properties, the assumption of Timoshenko beam theory (TBT), which considers the effects of rotational inertia and shear deformation, can be used. Compared to a large number of investigations on the mathematical modeling of elastic robotic manipulators based on the FEM and TBT,3 there are only a few works in which the combination of AMM and TBT has been applied to model these kinds of robotic systems. The most complete and comprehensive work in this regard belongs to Loudini,4 who has presented the governing equations of a single flexible link on the basis of the Lagrangian formulation. But because of the complexity of the presented system, the equations of motion have been derived and solved only for a single flexible link with one mode shape. The dissipative forces of a vibrating beam can arise from various internal factors such as beam material or from different external factors such as air drag. To include the effects of damping in the governing equations of transversely vibrating beams, Sezawa5 suggested the use of the Voigt viscoelastic model for the mathematical modeling of solid material. Gu¨rgo¨ze et al.6 derived the characteristic equations of a cantilevered, viscoelastic bending beam (Kelvin–Voigt model); however, he simplified this continuous system by an equivalent spring–damper–mass system. The Timoshenko beam subjected to different blends of damping effects has been investigated by Banks et al.7 In the field of robotics, Sooraksa and Chen8 and Loudini et al.9 investigated the combinations of TBT and internal and external damping (as distributed damping). But, again, all these studies are confined to single flexible links. To properly design and control elastic robotic manipulators, it is crucial to develop an accurate mathematical model. However, since the governing equations of these robotic systems are highly coupled and nonlinear, most investigations in this field are restricted to a single flexible link. So, if the benefits associated with the accurate modeling of multiflexible links are not to be sacrificed, it will be inevitable to develop a recursive algorithm that derives the governing equations systematically. A recursive formulation based on the Lagrangian methodology for elastic-link manipulators was suggested by Book,10 in which the combination of AMM and EBBT was applied to model the elastic properties of the links; but no numerical example was presented to show the applicability of the proposed algorithm. Jin and Sankar11 presented a symbolic algorithm for flexible arms by

Proc IMechE Part K: J Multi-body Dynamics 230(1) using the AMM and EBBT on the basis of the Lagrangian principle. After presenting too many formulations with long terms, the governing equations were ultimately solved for a robot with only one elastic link. Mohan and Saha12 exploited the benefits of combining the Euler–Lagrange and Newton–Euler formulations for serial flexible multibody systems. Although they simulated a six-link Space Shuttle Remote Manipulator System whose second and third links are assumed flexible, they ignored the effect of motor mass and also the effects of internal and external damping in their modeling. The Gibbs–Appell (G–A) equations are one of the least used dynamic principles for deriving the dynamic equations of manipulator robots. This method makes use of a scalar function (Gibbs function or acceleration energy (S)) to derive the governing equations of motion. Although this is a powerful method for systems with nonholonomic constraints,13 it can be applied as effectively to holonomic systems. In the field of robotics, Vukobratovic and Potkonjak14 used the concept of ‘‘acceleration energy’’ for the first time to develop an algorithm and a computer program based on that for the automatic generation of motion equations for a chain of N-rigid links. However, in their work, the Gibbs function of the system was not developed explicitly. Another approach was suggested by Mata et al.15 in which a new form of the G–A equations was developed. In their work, the Gibbs function of a series of N-rigid links was presented. Then by taking the necessary derivatives of the Gibbs function with respect to quasi-accelerations, the inverse dynamic equations were obtained. Finally, a recursive algorithm was presented for the automatic construction of the governing equations. A more recent investigation on this subject can be found in Korayem and Shafei,16 where a methodology has been implemented based on 33 rotational matrices to model and solve the inverse and forward dynamic problems of elastic robotic manipulators. However, the derived formulations are too long and too hard to be computed in real time, the results are not validated by experimental data, and the effects of motors at the joints are not considered in the mathematical modeling of these kinds of robotic systems. Despite a large number of theoretical works on elastic robotic manipulators, only a few experimental investigations can be found in the literature. Regarding the single-link experimental test beds, the paper most relevant to this study belongs to Martins et al.,17 in which the simulation and experimental exercises are compared with the consideration of the bang-bang input torque. The main drawback of this work is the absence of structural damping in the presented mathematical modeling. This has caused significant discrepancy in the theoretical and experimental results. Khorrami et al.18 have conducted experiments on two-link elastic robotic

Korayem et al. manipulators by attaching an accelerometer at the end point and have presented the experimental results for the end-point positioning of these manipulators. Yim et al.19 focused on the implementation of a dual-mode controller for the maneuver of a two-axis flexible robotic arm. Through the control system proposed in this investigation, accurate joint angle tracking and elastic mode stabilization can be accomplished. Also Nagaraj et al.20 presented a mathematical model along with an experiment for a two-link flexible system which undergoes locking during motion. As previously mentioned, by using the recursive G–A principle and applying the TBT and AMM, this paper presents the mathematical modeling of N-viscoelastic-link. The rest of the paper is organized as follows: In the next section, the kinematics of the system is described. In the subsequent section, the dynamics of the system including the Gibbs’ function, the potential energy, and the Rayleigh’s dissipation function and their derivatives, and also the governing equations of the system are successively presented. In the following section, the forward dynamic equations of motion including the inertia matrix and the remaining dynamics and external forcing terms are evaluated. In ‘‘Algorithm for forming dynamic model’’ section, a recursive algorithm based on the G–A formulation and 44 transformation matrices is developed to generate the motion equations of n-viscoelastic-link robotic manipulators symbolically. The description of the experimental setup is given in the following section. In ‘‘Implementation and results’’ section, the comparison between the

Figure 1. Manipulator with elastic links and motors at the joints.

39 theoretical and experimental results is illustrated and in the final section, the conclusions of the present work are summarized and the merits of the proposed method are highlighted.

Kinematics of flexible links Here, the kinematic of n-viscoelastic-link robotic manipulators will be considered. The coordinate system of every link (xi yi zi and x^ i y^ i zî ) is attached exactly according to the rules developed in Korayem and Shafei.16 Figure 1 shows the arbitrary differential element Q. Let i r~Q=Oi be the position vector of this element with respect to the origin of the xi yi zi coordinate system. To include the small displacement of the link, the method of modal analysis is exploited. So i

r~Q=Oi ðÞ ¼ ~ þ

mi X

ij ðtÞ~rij ðÞ

ð1Þ

j¼1

where ~ ¼ f 0 0 1 gT is the position vector of differential element Q with respect to Oi when link i has no deformation, r~ij ¼ f xij yij zij 0 gT is the Eigenfunction vector whose components (xij , yij , zij ) are the jth extensional and lateral mode shapes of link i, ij is the jth time-varying modal generalized coordinate of link i, and mi is the number of mode shapes used to express the deformation of link i. The rotation of differential element Q in the Oi xi , Oi yi , and Oi zi directions can be considered as xi , yi ,

40

Proc IMechE Part K: J Multi-body Dynamics 230(1)

and zi , respectively. By using the truncated modal expansions, these small angles can be represented as i~ i

¼ xi

yi

zi

T

¼

mi X

ij ðtÞ~ij ðÞ

ð2Þ

j¼1

where ~ij ¼ f xij yij zij gT is the Eigenfunction vector whose components (xij , yij , zij ) are the jth rotational mode shapes of link i in the Oi xi , Oi yi , and Oi zi directions, respectively. The position of differential element Q on ith link can be represented in any coordinate system j if the transformation matrix j Ti is known 3 xj component of Oi yj component of Oi 7 7 zj component of Oi 5 1

2 j

6 j Ri Ti ¼ 6 4 0~ T

r~Q=O0 ¼ 0 Ti i r~Q=Oi ¼ Ti i r~Q=Oi

ð5Þ

where Ai is defined as the joint’s transformation matrix from the xi yi zi to x^ i1 y^ i1 zî1 coordinate system and Ei is the transformation matrix of the ith link, which shows the orientation and transformation of the x^ i y^ i zî coordinate system with respect to the xi yi zi . This matrix can be presented as Ei ¼ Hi þ

mi X

# ij ðtÞMij ðli Þ

0

r~€Oiþ1 =O0 ¼ T€ i i r~Oiþ1 =Oi þ 2T_ i i r~_Oiþ1 =Oi þ Ti i r~€Oiþ1 =Oi

ð9Þ

ð10Þ where i r~Oiþ1 =Oi can be expressed as r~Oiþ1 =Oi

¼ l~i þ

mj X

ij ðtÞ~rij ðli Þ

ð11Þ

ð3Þ

To simplify the formulations, Ti can be divided into the following two parts

"

r~€Q=O0 ¼ T€ i i r~Q=Oi þ 2T_ i i r~_Q=Oi þ Ti i r~€Q=Oi

i¼1

ð4Þ

Ti ¼ Ti1 Ei1 Ai ¼ T^ i1 Ai

0

i

Here, j Ri is the rotation matrix that demonstrates the orientation of the ith body’s local reference system with respect to that of the jth body and 0~ ¼ f 0 0 0 gT . Now, the position of Q with respect to the base coordinate system can be specified as 0

The absolute acceleration of differential element Q and also the absolute acceleration of the (i þ 1)th body’s local reference system Oiþ1 , which is expressed in the X0 Y0 Z0 global coordinate system, can be represented as

Also l~i ¼ f li 0 0 1 gT is the position vector of Oiþ1 with respect to Oi when link i has no deformation. T_ i and T€ i will be, respectively, obtained by taking the first and second derivatives of equation (5) with respect to time as _ T_ i ¼ T^ i1 Ai þ T^ i1 A_ i

ð12Þ

€ _ T€ i ¼ T^ i1 Ai þ 2 T^ i1 A_ i þ T^ i1 A€ i

ð13Þ

The first and second time derivatives of T^ i appear in equations (12) and (13), respectively. By considering equation (5), these two terms can be represented as _ T^ i ¼ T_ i Ei þ Ti E_ i

ð14Þ

€ T^ i ¼ T€ i Ei þ 2 T_ i E_ i þ Ti E€ i

ð15Þ

And finally, the time derivatives of the joints’ transformation matrices appearing in the above equations can be written as

ð6Þ

j¼1

@Ai A_ i ¼ q_i ¼ Ui q_ i @qi

ð16Þ

where 2

1 60 Hi ¼ 6 40 0 2

0 1 0 0

0 6 zij Mij ¼ 6 4 yij 0

0 0 1 0

2

@ Ai @Ai A€ i ¼ 2 q_ 2i þ q€ i ¼ U2i q_2i þ Ui q€i @qi @qi

3

li 07 7 05 1

zij 0 xij 0

ð7Þ

yij xij 0 0

3 xij yij 7 7 zij 5 0

ð8Þ

It should be noted that all the angles in Ei are evaluated at ¼ li , where li is the length of the ith link.

ð17Þ

In the next section, equations (9) and (10) will be used to obtain the acceleration energies due to links’ motions.

Dynamic of flexible robots This section includes three subsections. At first, the system’s acceleration energy (Gibbs function) is computed. Then, the system’s potential energy and the Rayleigh’s dissipation function for modeling the

Korayem et al.

41

damping effects are presented. And at last, the governing equations of motion are derived.

mi X mi X ij C2ij þ CT2ij þ ik ij C1ikj

j¼1

B6i ¼

The acceleration energy of the system

mi X

B5i ¼ C3i þ

mi X mi X

k¼1 j¼1

€ik €ij C4ikj

ð2126Þ

k¼1 j¼1

The G–A method makes use of a scalar function in terms of accelerations to derive the equations of motion. The acceleration energy of this robotic system can come from two sources: (1) links’ motion and (2) motors’ motion. The Gibbs’ function of the links based on the TBT can be represented as 1 T i ðÞTr 0 r~€Q=O0 0 r~€Q=O0 S1 ¼ 2 i¼1 0 T 1 € € þ i ~i Ji ðÞi ~i d 2 n Z X

li

ð18Þ

In equation (20) there is a term called the ‘‘irrelevant terms.’’ In fact, in the Gibbs’ function the terms that are not functions of q€ j and €jf can be ignored, because the partial derivatives of these terms with respect to quasi-accelerations are equal to zero. All the other variables appearing in equations (21) to (26) can be represented as

C1ikj

1 ¼ 2

C2ij ¼ In the above equation, i ðÞ is the mass per unit length of the ith link, while Ji ðÞ is the mass moment of € inertia per unit length of this link. Also, i ~i is the angular acceleration of Q and it will be obtained by twice differentiating equation (2) with respect to time. Now, the acceleration energy due to motors’ motion is considered. The Gibbs’ function of the motors’ motion can be expressed as n X 1

S2 ¼

i¼1

2

T Mmi Tr 0 r~€Oiþ1 0 r~€Oiþ1

ð19Þ

where Mmi is the mass of the ith motor which is fixed at the origin of the (i þ 1)th body’s local reference system. The summation of equations (18) and (19) leads to the acceleration energy of the entire system S¼ S¼

n Z X

C3i ¼ C4ikj

1 2 1 2

1 ¼ 2

Z Z Z

0 li 0

0 li

Z 0

þ

Mmi r~ik ðli Þ~rTij ðli Þ

i ~r~Tij d þ Mmi l~i r~Tij ðli Þ i ~~T d þ Mmi l~i l~Ti

T ~ik Ji ~ij d

ð2730Þ

Using the G–A formulation, the motion equations of the system will be obtained by differentiating the Gibbs’s function with respect to quasi-accelera tions @S=@q€ j , @S=@€jf . These two terms can be evaluated as € n X @S @Ti ¼2 Tr B5i T€ Ti þ 2B4i T_ Ti þ B3i TTi @q€ j @q€ j i¼j

( ) n X @S @T€ i T T T € _ ¼2 Tr B5i Ti þ 2B4i Ti þ B3i Ti @€jf @€jf i¼jþ1 (" mj X _jk C1jkf þ 2Tr T€ j Njf þ 2T_ j

dsi

Tr T€ i B5i T€ Ti þ 4T€ i B4i T_ Ti þ 2T€ i B3i TTi

i¼1

þ4T_ i B2i TTi þ Ti B1i TTi

þ B6i þ irrelevant terms

ð20Þ þ Tj

mj X

#

mi X mi X

mj X

€jk C4jkf

ð32Þ

k¼1

where

€ik €ij C1ikj

k¼1 j¼1 mi X mi X

k¼1

)

€jk C1jkf TTj þ 2

k¼1

where

B2i ¼

li

i r~ik r~Tij d

ð31Þ li

i¼1 0 n X

B1i ¼

li

Njf ¼ C2jf þ

_ik €ij C1ikj

mj X

jk C1jkf

ð33Þ

k¼1

k¼1 j¼1

B3i ¼

mi X

€ij C2ij þ

j¼1

B4i ¼

mi X j¼1

_ij C2ij þ

mi X mi X k¼1 j¼1 mi X mi X k¼1 j¼1

ik €ij C1ikj ik _ij C1ikj

Potential energy and Rayleigh’s dissipation function of the system The potential energy of the system can arise from two sources: (1) gravity and (2) elastic deformations.

42


The gravitational potential energy of the whole system including the gravitational potential energies of the links and motors can be represented as gT Vg ¼ ~

n X

Ti i r~i

ð34Þ

i¼1

where g~ is the vector of gravity field and i r~i can be written as i

r~i ¼ C~5i þ

mi X

ik C~6ik

taking the partial derivatives of the Rayleigh’s dissipation function with respect to quasi-velocities. These two terms can be represented as @D ¼0 @q_ j

ð42Þ

m

j @D X _jk ðtÞDjkf ¼ @_jf k¼1

ð43Þ

ð35Þ

k¼1

Governing equations of the system

where C~5i ¼ Mmi l~i þ

Z

li

i ~d

ð36Þ

0

C~6ik ¼ Mmi r~ik ðli Þ þ

Z

1. The joint equations of motion

li

i r~ik d

ð37Þ

0

The strain potential energy function for n-viscoelastic-link robotic manipulators based on TBT has been expressed in Korayem and Shafei16 as Ve ¼

Now, by using the G–A formulation, the dynamic equations of viscoelastic robotic manipulators will be finalized as

mi X mi n X 1X ij ðtÞik ðtÞKijk 2 i¼1 j¼1 k¼1

ð38Þ

@S @D @V þ þ ¼ j @q€j @q_j @qj

j ¼ 1, 2, . . . , n

2. The deflection equations of motion @S @D @V þ þ ¼ 0 j ¼ 1, 2, . . . , n; @€jf @_jf @jf f ¼ 1, 2, . . . , mj

The summation of equations (34) and (38) leads to the potential energy of the whole system. For deriving the conservative generalized forces caused by gravity and elastic deformations, the partial derivatives of the potential energy function with respect to quasicoordinates (i.e. qj and jf ) should be evaluated. These two terms can be represented as n X @V @Ti i ¼ ~ gT r~i @qj @qj i¼j

ð39Þ

mj n X X @V @Ti i T ¼ ~ g jk ðtÞKjkf r~i g~T Tj C~6jf þ @jf @jf i¼jþ1 k¼1 ð40Þ The viscous damping forces can be appropriately considered in the Rayleigh’s dissipation function. This function has been presented in Korayem and Shafei16 for viscous air damping (as external damping) and for structural viscoelasticity effect (as internal damping), in the following form D¼

mi X mi n X 1X _ij ðtÞ_ik ðtÞDijk 2 i¼1 j¼1 k¼1

ð41Þ

The nonconservative generalized forces of the internal and external damping will be obtained by

ð44Þ

ð45Þ

where j is the external torque applied to the jth joint. The above equations are in the inverse dynamic form. In this form of dynamics, the exerted forces and the applied torques by the actuators will be evaluated algebraically by knowing the configurations (position, velocity, and acceleration) of the system.21

Motion equations in forward dynamic form In this section, the motion equations of viscoelastic robotic manipulators will be arranged in the forward dynamic form as ~€ ¼ ! IðÞ Re

ð46Þ

In the above equation, IðÞ is the inertia matrix of ~ is the vector of quasi-coordinates, and the system, ! Re is the vector of the remaining dynamics including the strain, gravitational, Coriolis, and centrifugal forces as well as the external torques applied to the joint variables. These two vectors can be written as ~ ¼ q1

11

...

1m1

. . . qj

j1

. . . jmj

q2

. . . 2m2 T . . . nmn ð47Þ 21

Korayem et al.

43

! Re ¼ Req1 Re11 ... Re1m1 Req2 Re21 ... Re2m2 oT ... Reqj Rej1 ... Rejmj ... Renmn ð48Þ where Reqj and Rejf will be obtained from equations (44) and (45), respectively, after all the terms with quasi-accelerations are excluded from those equations. To achieve the objectives of this section, the sum€ mation form of T€ i and T^ i must be presented. Having the summation form of these terms makes it possible to evaluate the partial derivatives of Ti and T€ i with respect to qj , q€ j , jf , and €jf , which appeared in equations (31), (32), (39), and (40). Let us express T^ i and Ti in two different ways T^ i ¼ A1 E1 A2 E2 . . . . . . Ah Eh . . . . . . Ai Ei ¼ T^ h1 Ah h T i ¼ Th Eh h T^ i

ð49Þ

Ti ¼ A1 E1 A2 E2 . . . . . . Ah Eh . . . . . . Ei1 Ai ¼ T^ h1 Ah h T~ i ¼ Th Eh h Ti

ð50Þ

Equations (49) and (50), respectively, represent T^ i and Ti in the recursive form. Taking the second derivative of these two terms with respect to time and changing them to the summation form yields € € € T^ i ¼ T^ s,i þ T^ v,i

ð51Þ

ð52Þ T€ i ¼ T€ s,i þ T€ v,i € € T^ s,i and T€ s,i represent those constructive terms of T^ i € and T€ i that include q€ j and €jf , whereas T^ v,i and T€ v,i € denote those components of T^ i and T€ i that do not have q€ j and €jf as quasi-accelerations. These terms can be expressed as X X € T^ s,i ¼ T^ h1 Uh h T i q€ h þ Th Mhk h T^ i €hk i

mh

h¼1

k¼1

þ

mh i X X h¼1 i X

T€ s,i ¼

ð53Þ

_ T_ h Mhk h T^ i þTh Mhk h T^ i _hk

ð54Þ

mh i1 X X

Th Mhk h Ti €hk

h¼1 k¼1

i X _ _ T^ h1 Uh h T~ i þ T^ h1 U2h h T~ i q_ h þ T^ h1 Uh h T~ i q_ h

þ

mh i1 X X h¼1

k¼1

ð58Þ

Inertia coefficients For assembling the coefficients of the inertia matrix, equations (52), (57), and (58) should be substituted into equations (44) and (45). Keeping all the terms that have quasi-accelerations (q€ h and €hk ) as their coefficients on the left-hand side, and arranging them in the matrix form, leads to the inertia matrix of the system. The details of the above procedure will be explained in the following sections. The coefficients of quasi-accelerations in the joint equations. In equation (44), if all the terms that have q€h and €hk as their coefficients are grouped, the following expression will be obtained 2

n X

n o Tr T^ j1 Uj j F~h UTh T^ Th1 q€ h

h¼1

þ 2

mh n1 X X

n o Tr T^ j1 Uj j Fh MThk TTh

h¼1 k¼1

þ2

mh n X X

n

Tr T^ j1 Uj j T~ h Nhk TTh

! o

€hk

ðIÞ

where

j

F~h ¼

n X

j

T~ i B5i h T~ Ti

j

Fh ¼

n X

j

T~ i B5i h TTi

ð5960Þ

i¼maxðhþ1,j Þ

T^ h1 Uh h T~ i q€h þ

h¼1

@T€ i @Ti ¼ ¼ Tj Mjf j Ti @€jf @jf

i¼maxðh,j Þ

!

ð55Þ T€ v,i ¼

ð57Þ

h¼j k¼1

k¼1

h¼1

@T€ i @Ti ¼ ¼ T^ j1 Uj j T~ i @q€ j @qj

!

i X € _ T^ v,i ¼ T^ h1 Uh h T i þ T^ h1 U2h h T i q_ h þ T^ h1 Uh h T_ i q_ h h¼1

Now, the partial derivatives of Ti and T€ i with respect to quasi-coordinates and quasi-accelerations can be written as

! T_ h Mhk h Ti þTh Mhk h T_ i _hk

ð56Þ

By writing expression (I) in matrix form, an obviously symmetric matrix is obtained. In the next section, this important fact is exploited to reduce the volume of computations. Also, in proposed algorithm in Section 5, equations (59) and (60) are evaluated in the recursive form to improve the computational complexity. The coefficients of quasi-accelerations in deflection equations. As was mentioned earlier, the inertia matrix of the whole system is symmetric. So, in the deflection equations, the coefficient of q€ h is the same

44


as the coefficient of €hk in the joint equations. But, the coefficient of €hk can be expressed as mh n1 X X

2

Tr Tj Mjf j h MThk TTh

h¼1 k¼1 n X

þ2

mh X Tj Mjf j Th Nhk TTh

j1 X mh X

Tr Tj C1jkf TTj

þ2

! o

€hk

ðIIÞ

ð66Þ

" þ T€ v,j Njf þ 2T_ j

mj X

# ) _jk C1jkf TT j

k¼1

h i þ g~ Tj Mjf Ajþ1 P~jþ1 þ C~6jf

where h ¼

Rejf ¼ 2Tr Tj Mjf Ajþ1 Qjþ1

k¼1

j

jk ðtÞKjkf þ _jk ðtÞDjkf

Substituting equation (58) into equation (66) and changing the obtained equation to a recursive form yields (

n o Tr Th Mhk h Tj Njf TTj

h¼1 k¼1 mj n X

mj X k¼1

h¼jþ1 k¼1

þ2

!

n X @T i i r~i þ Tj C~6jf þ g~T @jf i¼jþ1

T

n X

j

Ti B5i h TTi

ð61Þ

i¼maxð jþ1,hþ1Þ

mj X jk ðtÞKjkf þ _jk ðtÞDjkf

ð67Þ

k¼1

In the next section, expression (II) will be arranged in matrix form. Also, equation (61) will be evaluated recursively in the following section.

Equations (63) and (67) show the recursive form of the right-hand side of the governing equations.

Algorithm for forming dynamic model Right-hand side of motion equations Here, the right-hand side of the governing equations will be obtained in recursive form. To achieve the aim of this section, all the terms in equation (44) that do not have q€ j and €jf should be transferred to the righthand side as n X @Ti

i r~i @qj € h n i X @Ti T T € _ 2 Tr B5i Tv,i þ 2B4i Ti @q€j i¼j

Reqj ¼j þ g~T

i¼j

ð62Þ

Substituting equation (57) into equation (62) and converting the obtained equation to a recursive form yields n o Reqj ¼ j þ g~T T^ j1 Uj P~j 2Tr T^ j1 Uj Qj ð63Þ where P~j ¼ j r~j þ Ej Ajþ1 P~jþ1

ð64Þ

Qj ¼ B5j T€ Tv,j þ 2B4j T_ Tj þ Ej Ajþ1 Qjþ1

ð65Þ

Now, let us consider Rejf . If all the terms in equation (45) that do not have quasi-accelerations are transferred to the right-hand side, the following expression will be achieved ( ) n i X @T€ i h T T € _ Rejf ¼ 2 Tr B5i Tv,i þ 2B4i Ti @€jf i¼jþ1 (" # ) mj X _jk C1jkf TT 2Tr T€ v,j Njf þ 2T_ j j

k¼1

In this section, an algorithm is developed for deriving the motion equations of viscoelastic robotic manipulators, recursively. In this algorithm, all the mathematical operations are performed by only 4 4 and 4 1 matrices. The successive procedures of this algorithm are outlined in the following steps: Step 1. Calculating the compound transformation matrix for i ¼ 1 : 1 : n; j ¼ i : 1 : n if ði ¼ j Þ i Ti ¼ I44 ; elseif ð j ¼ i þ 1Þ i Tj ¼ Aiþ1 ; else i Tj ¼ i Tj1 Ej1 Aj ; By knowing i Tj , this algorithm can evaluate i T~ j . for i ¼ 1 : 1 : n; j ¼ i : 1 : n if ði ¼ j Þ i ~ Ti ¼ I44 ; else i ~ Tj ¼ Ei i Tj ; Ti is calculated by using i T~ j Initialize: T1 ¼ A1 ; for i ¼ 2 : 1 : n Ti ¼ T1 1 T~ i ; And finally, T^ i is presented as for i ¼ 1 : 1 : n 1 T^ i ¼ Ti Ei ; _ Step 2. Calculating T_ i and T^ i in recursive form Initialize: T_ 1 ¼ U1 q_ 1 ; for i ¼ 2 : 1 : n _ T_ i ¼ T^ i Ai þ T^ i1 Ui q_ i ;

_ T^ 1 ¼ T_ 1 E1 þ T1 E_ 1 ;

Korayem et al.

45

_ T^ i ¼ T_ i Ei þ Ti E_ i ;

i

€ Step 3. Calculating T€ v,i and T^ v,i in recursive form € Initialize: T€ v,1 ¼ U21 q_ 21 ; T^ v,1 ¼ T€ v,1 E1 þ 2T_ 1 E_ 1 ; for i ¼ 2 : 1 : n if ði 6¼ nÞ € _ T€ v,i ¼ T^ v,i1 Ai þ 2T^ i1 Ui q_ i þ T^ i1 U2i q_ 2 ; i

€ T^ v,i ¼ T€ v,i Ei þ 2T_ i E_ i ;

F~j ¼ B5j þ j F~jþ1 ðEi Aiþ1 ÞT ;

else i ~ Fj ¼ Ei Aiþ1 iþ1 F~j ; j F~i ¼ i F~Tj ; . Calculating i Fj for i ¼ 1 : 1 : n; j ¼ 1 : 1 : n 1 i Fj ¼ i F~jþ1 ATjþ1 ; . Calculating i j

else € _ T€ v,n ¼ T^ v,n1 An þ 2T^ n1 Un q_ n þ T^ n1 U2n q_ 2n ; Step 4. Calculating P~j and Qj in recursive form Initialize: P~n ¼ n r~n ; for j ¼ n 1 : 1 : 1 P~j ¼ j r~j þ Ej Ajþ1 P~jþ1 ; Initialize: Qn ¼ B5n T€ Tv,n þ 2B4n T_ Tn ; for i ¼ n 1 : 1 : 1 Qj ¼ B5j T€ Tv,j þ 2B4j T_ Tj þ Ej Ajþ1 Qjþ1 ; Step 5. Assembling the right-hand sides of motion equations Initialize: Re10 ¼ 1 þ g~T U1 P~1 2Tr U1 Q1 ; for j ¼ 2 : 1 : n n o Rej0 ¼ j þ g~T T^ j1 Uj P~j 2Tr T^ j1 Uj Qj ; for f ¼ 1 : 1 : mn

for i ¼ 1 : 1 : n 1; j ¼ 1 : 1 : n 1 i j ¼ Aiþ1 iþ1 F~jþ1 ATjþ1 ; Step 7. Assembling the coefficients of the inertia matrix . The inertia coefficients of joint variables in the joint equations for j ¼ 1 : 1 : n; h ¼ j : 1 : n if ðj ¼ 1 && h ¼ 1Þ Jjh ¼ 2Tr Uj j F~h UTh ; elseif ðj ¼ 1 && h 6¼ 1Þ Jjh ¼ 2TrfUj j F~h UTh T^ Th1 g; Jhj ¼ JTjh ; else Jjh ¼ 2TrfT^ j1 Uj j F~h UT T^ T g; Jhj ¼ JT ; h

(" T€ v,n Nnf þ 2T_ n

Renf ¼ 2Tr

mn X

#

)

_nk C1nkf TTn

h1

jh

. The inertia coefficients of deflection variables in the joint equations

k¼1 mn X þ g~T Tn C~6nf nk ðtÞKnkf þ _nk ðtÞDnkf ; k¼1

for j ¼ 1 : 1 : n 1; f ¼ 1 : 1 : mj ( Rejf ¼ 2Tr Tj Mjf Ajþ1 Qjþ1 " þ T€ v,j Njf þ 2T_ j

mj X

# ) _jk C1jkf TT j

k¼1

þ g~ Tj Mjf Ajþ1 P~jþ1 þ C~6jf T

mj X

jk ðtÞKjkf þ _jk ðtÞDjkf ;

for j ¼ 2 : 1 : n; h ¼ 1 : 1 : j 1; k ¼ 1 : 1 : mh Jjhk ¼ 2TrfT^ j1 Uj j Fh MThk TTh g; for j ¼ 1 : 1 : n 1; h ¼ j : 1 : n 1; k ¼ 1 : 1 : mh if ðj ¼ 1Þ Jjhk ¼ 2Tr Uj j T~ h Nhk þ j Fh MThk TTh ; else Jjhk ¼ 2TrfT^ j1 Uj ½j T~ h Nhk þ j Fh MThk TTh g; for j ¼ 1 : 1 : n; k ¼ 1 : 1 : mn if ðj ¼ 1Þ Jjnk ¼ 2Tr Uj j T~ n Nnk TTn ; else Jjnk ¼ 2TrfT^ j1 Uj j T~ n Nnk TTn g; . The inertia coefficients of deflection variables in the deflection equations

k¼1

for j ¼ 1 : 1 : n 1; f ¼ 1 : 1 : mj ; k ¼ 1 : 1 : mj Jjfjk ¼ 2TrfMjf j j MTjk þ C1jkf g; Step 6. Preparing the recursive forms of summation variables in the inertia matrix . Calculating i F~j for j ¼ n : 1 : 1; i ¼ j : 1 : 1 if ðj ¼ iÞ if ðj ¼ nÞ n ~ Fn ¼ B5n ; else

for j ¼ 1 : 1 : n 2; h ¼ j þ 1 : 1 : n 1; f ¼ 1 : 1 : mj ; k ¼ 1 : 1 : mh

Jjfhk ¼ 2Tr Tj Mjf j Th Nhk þ j h MThk TTh ; for j ¼ 1 : 1 : n 1; f ¼ 1 : 1 : mj ; k ¼ 1 : 1 : mn Jjfnk ¼ 2Tr Tj Mjf j Tn Nnk TTn ; for f ¼ 1 : 1 : mn ; k ¼ 1 : 1 : mn Jnfnk ¼ 2TrfC1nkf g;

46


Figure 2. The IUST two-link elastic robotic manipulator.

In a general comparison, the proposed recursive algorithm used in the mathematical modeling of multiple flexible links is more complete and comprehensive than the previous counterparts because it considers the effects of internal and external damping, shear and rotational inertia, mass of the motors, gravity, torsion, and longitudinal elongation. Moreover, using the G–A formulation based on 44 transformation matrices helps combine the necessary expressions for deriving the governing equations. In fact, saving even a few mathematical operations may have a significant effect on the efficiency of the algorithm when these computations are performed over thousands and millions of discretized steps.

Experimental setup The experimental setup depicted in Figure 2 is a twolink flexible robot which was built at the Department of Mechanical Engineering of the Iran University of Science and Technology (IUST). This experimental test bed consists of two flexible links and two rigid joints (the first joint is driven by a servo-tech AC servo motor and the second joint is driven by a DC motor). The 400 W AC servo motor can be driven in position, speed, and toque modes without needing a controller. Also, an incremental encoder with a resolution of 2500 pulse per revolution is used to measure the rotational position of this motor. However, to control the second motor a digital PID controller based on the AVR microcontrollers has been used. This controller uses the current as a feedback to control the output torque of the motor. In order to measure the current, a Hall effect-based linear current sensor (Model: ACS712) is used. Also, an H-bridge IC (L298) is used to amplify the signal that drives the DC motor. The data of the current are read by a microcontroller with a resolution of 10 bit. The rotational position of this motor is measured by a small absolute magnetic encoder. The deFection and vibration of every point of each link are estimated by the data collected from the three strain gauge bridges mounted on

both sides of each link. The analog signals of the strain gauge bridges are amplified through a signal conditioner IC (AD7190) and sent to a computer. A 64-bit personal computer is used for control motor output via an Advantech PCI-1710HG data acquisition card. PCI-1710HG is a multifunction card with 12 bits resolution and 100 kS=s sampling rate. The physical parameters of the flexible links are shown in Table 1. The flexible manipulator has a planar motion; therefore the gravitational effects can be ignored. The measured data from the two shaft encoders and strain gauge bridges are used to graphically represent the flexible links’ position and vibration in the LabView software. The schematic diagram of the experimental setup is shown in Figure 3.

Implementation and results In this section, the simulation and experimental results pertaining to the responses of flexible manipulators are presented and compared in time domain. The torque-driven movements of the gearless, low friction actuation mechanism are directly applied to the flexible links. In this modeling, the AMM with the first three clamped-tip mass mode shapes for the first link and the first three clamped-free mode shapes for the second link is applied. These modes can be represented as yij ¼ C1ij sin aij : þ C2ij cos aij : þ C3ij sinh bij : þ C4ij cosh bij : ð68Þ zij ¼ ij :C2ij sin aij : ij :C1ij cos aij : þ ij :C4ij sinh bij : þ ij :C3ij cosh bij : ð69Þ All the necessary coefficients of the mode shapes can be found in Table 2. Due to the stiffness of the mentioned mechanism, a smooth input torque causes a very small

Korayem et al.

47

Table 1. Parameters of the two flexible links. Parameters

Value

Unit

Length of the links Height of the links Thickness of the links Bending stiffness Shear correction factor Mass per unit length

L1 ¼ L2 ¼ 0:5 h1 ¼ 4; h2 ¼ 5:17 t1 ¼ 4; t2 ¼ 1:5 EIz1 ¼ 14:93; EIz2 ¼ 1:017 k ¼ 5=6 1 ¼ 0:504; 2 ¼ 0:2442 2 0:584 0 0 6 J1 ¼ 4 0 0:578 0

m cm mm N m2 – kg m1

Mass moment of inertia per unit length

Mass of the motor (DC) Shear modulus Kelvin–Voigt damping coefficient Air damping coefficient

2

0

0:4685 6 J2 ¼ 4 0 0

0

0:00578

3 7 5 104

3 0 0 7 0:4681 0 5 104 0 0:0004

Mm1 ¼ 0:495 G1 ¼ G2 ¼ 27 109 Kv1 ¼ Kv2 ¼ 1100 ¼ 0:2

kg m1

kg N m2 kg m1 s kg m1 s

Figure 3. Schematic diagram of the experimental setup.

displacement, which is not sufficient to demonstrate the desired effects. So, the input torque should be interrupted suddenly. Also, the shape of the input torque should be such that it allows the manipulator to accelerate and then decelerate and finally to stop at a target location. Examples of these input torques are shown in Figures 4 and 5. In order to compare the simulation and experimental results, four system responses, namely, hub angles (Figures 6 and 8), hub velocities (Figures 7 and 9), end point deflections (Figures 10 and 12), and end point velocities (Figures 11 and 13) of both joints and links

are investigated. The system responses are measured at a sampling rate of 200 Hz. It should be noted that the angular velocities of the motors and the end point velocities of the links are, respectively, obtained by taking the time derivatives of motors’ angular positions and the links’ end point deflections. Although a quick look at the above figures may cause one to conclude that there are huge differences in the simulation and experimental results, a quantitative analysis proves the opposite. For this purpose, and for a quantitative analysis of the obtained results, the mean values and the standard

48


Table 2. Necessary coefficients of the mode shapes.

1j 1j

C12j C22j C32j C42j a2j b2j 2j

1:00002719 1:00436397 1 1:00436397 2:08745814 2:08743922 2:08742031 2:08747706 1:00001234 1:36222328 1 1:36222328 3:75020565 3:75019022 3:75017479 3:75022108

1:00379293 1 1:00339706 1 7:95141011 7:95036469 7:94931940 7:95245567 1:01854889 1 1:01847012 1 9:38812524 9:38788322 9:38764121 9:38836726

1:00125768 1:00006865 1 1:00006865 14:1957043 14:1897581 14:1838143 14:2016531 1:00021656 1:00077660 1 1:00077660 15:7093715 15:7082376 15:7071039 15:7105054

0.5

25 20 15 10 5 0

0

0

0.2

0.4

0.6 t (sec)

0.8

1

1.2

Figure 6. Hub angular position for the first joint.

SIM EXP

100

50

0

-50 0

0.2

0.4

0.6 t (sec)

0.8

1

1.2

Figure 7. Hub angular velocity for the first joint.

-0.5 0

0.2

0.4

0.6 t (sec)

0.8

1

1.2

Figure 4. Input torque for the first joint.

Applied Torque to the Second Joint (N.m)

First Joint Angular Position (deg)

j¼3

0.1

Second Joint Angular Position (deg)

Applied Torque to the First Joint (N.m)

2j

j¼2

First Joint Angular Velocity (deg/sec)

C11j C21j C31j C41j a1j b1j

30

j¼1

80

60 EXP SIM

40

20

0 0

0.2

0.4

0.6 t (sec)

0.8

1

1.2

0.05

Figure 8. Hub angular position for the second joint. 0

-0.05

-0.1 0

0.2

0.4

0.6 t (sec)

0.8

1

1.2

Figure 5. Input torque for the second joint.

deviations of these results are calculated and presented in Table 3. Contrary to what was believed, the mean values and the standard deviations of the obtained results indicate that the simulation and experimental results

match very well. Thus, the proposed model is valid and it can be reliably used in the development of effective control algorithms for manipulator with flexible links of varying lengths.

Conclusion In this paper a mathematical model for a multilink viscoelastic robotic system is presented. The contributions of this work can be characterized as follows: . The G–A methodology, which is in contrast to the Lagrangian dynamic modeling (wherein the

EXP SIM 200

100

0

-100 0

0.2

0.4

0.6 t (sec)

0.8

1

1.2

End Point Deflection of the First Link (mm)

Figure 9. Hub angular velocity for the second joint.

2 0 -2 -4 -6

0

0.2

0.4

0.6 t (sec)

0.8

1

1.2

End Point Velocity of the First Link (m/s)

Figure 10. End point deflection for the first link.

0.4 0.2 0 -0.2 -0.4 -0.6 0.2

0.4

0.6 t (sec)

0.8

1

1.2

Figure 11. End point velocity for the first link.

mathematical complexities and the computational requirements limit the design of suitable controllers), has been applied to generate the governing motion equations of elastic robotic manipulators. In dynamic systems with holonomic constraints, the quasi-velocities can be the same as the generalized velocities. So, one can show that

@S d @T @T ¼ @q€j dt @q_ j @qj

10 5 0 -5 -10 -15 -20

0

0.2

0.4

0.6 t (sec)

0.8

1

1.2

2

SIM EXP

1

0

-1

-2

0

0.2

0.4

0.6 t (sec)

0.8

1

1.2

Figure 13. End point velocity for the second link.

SIM EXP

0.6

15

Figure 12. End point deflection for the second link.

SIM EXP

4

End Point Deflection of the 2nd Link (mm)

49

End Point Velocity of the 2nd Link (m/s)

Second Joint Angular Velocity (deg/s)

Korayem et al.

j ¼ 1, 2, . . . , n

In above equation when the degree of freedom of system (j) increases, the implementation of the G–A

method becomes more efficient, because it needs fewer differentiations relative to the Lagrangian formulation. . Deriving the motion equations of a chain of nelastic-link robotic manipulators by hand is likely to produce errors. So, a recursive algorithm based on 44 transformation matrices has been developed to form the governing equations, automatically. The main advantage of using 44 transformation matrices (where the rotations and translations are combined) instead of 33 rotational matrices is the compact form of the derived symbolic expressions, which reduces the mathematical operations needed to generate motion equations. . Almost all the important parameters that affect the vibration of links in different direction (lateral bending in two directions, torsional vibration, and extensional deformation) including the internal and external damping, shear and rotational inertias of the links, masses of motors, and also gravity have been considered in this paper. Although previous works have studied the above-mentioned effects, none of them has combined these effects in multiple elastic links in a recursive form. . In the mathematical modeling of elastic robotic manipulators by the AMM, the proper selection

50


Table 3. Mean values and standard deviations of the achieved results. Mean Value

Hub angular position for the first joint Hub angular velocity for the first joint Hub angular position for the second joint Hub angular velocity for the second join End point deflection for the first link End point velocity for the first link End point deflection for the second link End point velocity for the second link

Standard Deviation

SIM

EXP

SIM

EXP

14.6423 22.1755 45.1767 45.0623 –0.0328 –0.0013 0.0001 –0.0020

14.7819 22.1814 46.9478 45.2899 –0.0269 –0.0015 –0.0099 –0.0059

10.3249 30.6958 27.8593 84.3205 2.5425 0.2411 0.6721 0.3936

10.2495 28.0345 27.1956 80.3166 2.8058 0.1914 0.5827 0.3084

of mode shapes is an important step for improving the results. In this paper, for the first and intermediate links, the mode shapes with clamped-tip mass boundary conditions and for the last link, the eigenfunctions with clamped-free boundary conditions based on the TBT (in which the mode shapes of deflections and rotations are independently obtained) have been adopted. In all the previous works that have considered multiple flexible links, the eigenfunctions are obtained on the basis of the EBBT in which the rotational mode shapes are acquired by simply differentiating the bending eigenfunctions. . Most of the experimental works in the field of elastic robotic manipulators are confined to single elastic links. Here, a two-link elastic robotic manipulator has been prepared to study the interactions between the links, the coupling effects due to the simultaneous large motions and small deflections and the effects of driving motors at intermediate joints. . The mismatch between the high-speed vibrations of a flexible robotic arm and the low-speed operation of the software and hardware interface between the computer and experimental setup is a problematic obstacle for the measuring and controlling of such systems. So, the combined use of high-speed digital-to-analog convertor (DAC) components on an electronic interface board and the National Instrument’s LABVIEW software package has been proposed as a solution for this problem. In this paper, to avoid the modeling of joint flexibility and friction, the flexible manipulators are directly attached to motor shafts. As a consequence, the experimental tests are restricted to torque-controlled modes to avoid the large position changes. For future work, by using gears and modeling the joints in multiple flexible links, more tests can be performed on the position-controlled modes of motors. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Acknowledgements The authors are grateful to our robotic team which this work was benefited from them: Arastoo Azimi, Amir Kabir University of Technology; Amin Younesi, Pars Artiman Javid Company; Ali Reza Pashazad, Azad University, Research and Science Branch.

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