
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00047784/00001
Material Information
 Title:
 Dynamic analysis and parametric identification for flexible serial manipulators
 Creator:
 Behi, Fariborz, 1954
 Publication Date:
 1985
 Language:
 English
 Physical Description:
 vii, 207 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Coordinate systems ( jstor )
Equations of motion ( jstor ) Inertia ( jstor ) Kinetics ( jstor ) Matrices ( jstor ) Natural frequencies ( jstor ) Stiffness ( jstor ) Structural deflection ( jstor ) Transfer functions ( jstor ) Vibration mode ( jstor ) Dissertations, Academic  Mechanical Engineering  UF Machinery, Dynamics of ( lcsh ) Manipulators (Mechanism) ( lcsh ) Mechanical Engineering thesis Ph. D Vibration ( lcsh )
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis(Ph. D.)University of Florida, 1985.
 Bibliography:
 Bibliography: leaves 204206.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Fariborz Behi.
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 University of Florida
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 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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DYNAMIC ANALYSIS AND PARAMETRIC
IDENTIFICATION FOR FLEXIBLE
SERIAL MANIPULATORS
BY
FARIBORZ BEHI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
ACKNOWLEDGEMENTS
The author wishes to thank his committee chairman,
Professor Delbert Tesar, for his support and guidance. He is also grateful to the members of his graduate commitee for their guidance and encouragement.
Special thanks go to his family and friends for their encouragement and patience. He expresses his sincere appreciation to his mother, Badri Behi, for providing financial and moral support throughout his education. Finally, thanks to Suzan Hunter for her excellent typing.
This work was funded by the office of Naval Research precision modeling grant No. N0001483K0592.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ......................................
ABSTRACT ..............................................
CHAPTER
INTRODUCTION ................................
DYNAMIC MODEL OF FLEXIBLE MANIPULATOR ARMS..
2.1 Kinematic Representation of Serial
Manipulators ...........................
2.2 Rate of Change of Orientation and
Position ...............................
2.3 Basis for the Mathematical Model .......
2.4 Dynamic Model of Serial Manipulators
With Flexible Joints ...................
2.5 The Secondary Flexible Components of a
Serial Manipulator .....................
2.6 Dynamic Model of Serial Manipulators
With Flexible Links ....................
2.7 Improvement of System Design ...........
ANALYSIS OF MANIPULATOR DYNAMIC RESPONSE .... 3.1 Forced Vibration .......................
3.2 Decoupled Equations of Motion ..........
3.3 Dynamic Response in the Frequency
Domain .................................
3.4 System Response Due to Dynamic Shocks..
3.5 Nonlinearities in the Equations of
Motion .................................
3.6 Order Analysis .........................
iii
I II
Page
ii vi
1 10 11
16 20 22 32 33 35
42 43 46 51 54 56 65
III
VI
V
VI VII APPENDICI]
A
ARM DEFLECTION DUE TO THE GROSS MOTION ...... 4.1 Complete Dynamic Model for Manipulator
Arms ...................................
4.2 Variation of System Natural Frequencies
Due to the Nominal Motion of the Arm... 4.3 Compensation for Deformations ..........
PARAMETRIC IDENTIFICATION ........................
5.1 Experimental Modal Analysis ............
5.2 Parametric Identification for
Manipulator Arms .......................
5.2a Identification of Local
Stiffnesses ......................
5.2b Identification of Local Masses...
5.2c Identification of Local Damping
Coefficients ......................
5.3 Verification of the Identified
Parameters .............................
5.4 Instrumentations .......................
5.5 Improvement of System Design ...........
ILLUSTRATIVE EXAMPLES .......................
6.1 Description of Example Manipulator ..... 6.2 System Natural Frequencies and Transfer
Functions ..............................
6.3 Manipulator's Dynamic Response ......... 6.4 Deformation Due to the Gross Motion .... CONCLUSION ..................................
ES
ARM ACCELERATION AND SECONDORDER INFLUENCE COEFFICIENTS ................................
?age 72 73 78 83 98 99 106 109 114 117 118
120 122 146
146 148 149 152
188
192
Page
B EXPRESSIONS FOR INERTIA POWER MATRIX.......... 197 C VARIATION OF THE INFLUENCE COEFFICIENTS
DUE TO MANIPULATOR'S DEFORMATION ............ 201
REFERENCES ............................................ 204
BIOGRAPHICAL SKETCH ......................... ........ 207
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy DYNAMIC ANALYSIS AND PARAMETRIC IDENTIFICATION FOR
FLEXIBLE SERIAL MANIPULATORS
By
Fariborz Behi
December 1985
Chairman: Dr. Delbert Tesar Major Department: Mechanical Engineering Vibrations in the small are low magnitude, oscillatory deformations about the mean motion equilibrium as distinct from the deformations in the large which result from external loads and large but slowly varying inertia loads. Vibrations in the small are caused by transient phenomena such as system shocks introduced when starting or stopping a machine such as a manipulator arm. The lowest natural frequency of vibration is one of the best indicators of the overall structural integrity of the manipulator system and is an indicator of the maximum acceptable speed for performing precise arm motions.
This dissertation presents a new formulation for the
dynamic model of flexible manipulators. A lumpedparameter arm model is used for tractability. This allows the dynamic equations of motion to be written as a set of coupled ordinary differential equations instead of partial
differential equations. For many manipulator designs, most of the deformation energy appears in the actuator and drive train components. Therefore, an initial model considers flexibility at the joints in the direction of relative joint motion only. The natural frequencies and their corresponding natural modes are determined from an eigenvalue problem obtained from the equations of motion. This formulation is then expanded to include other system flexibilities by introducing pseudoinputs to represent their motion.
Based on analytical modal analysis a solution algorithm is presented to estimate the dynamic response of the arm due to deterministic exciting forces. This enables the designer to obtain the response of the arm in frequency domain by considering a harmonic exciting force applied to the system.
Although the inertia loads due to the gross motion of the arm have small effect on the natural frequency of the system, they cause the system to deform due to the existence of flexible components. Therefore, the mathematical formulation is extended to include both small and gross motion of the manipulator.
Finally, an experimental procedure is introduced to determine the mass and stiffness parameters of the manipulator in order to represent a true model of the actual system.
vii
CHAPTER I
INTRODUCTION
The combination of computer control and sophisticated multidegree of freedom mechanical systems has evolved into a new discipline referred to as robotics. The potential for this discipline to greatly increase production and improve its quality has brought our social structure to the verge of a modern industrial revolution. Achieving this goal demands enormous effort to develop robot technology.
Although manipulator arms have been employed in a wide range of industrial applications, exploitation of these mechanisms to their maximum capacity is yet to come. The ability of existing manipulator arms is limited because of technological barriers in (a) "real time" computational control, and (b) design procedures [1I.* In parallel with the need for "real time" computations, mathematical formulations are necessary to incorporate complex geometries and dynamic characteristics in the design process. Current manipulator designs maintain
* The numbers in brackets refer to references at the end of this document.
simple geometries and are developed based on trial and error procedure, which is both time consuming and costly. Alternative geometries are much needed to extend manipulator capability for performing more sophisticated tasks, such as obstacle avoidance en route to a destination. In addition, system dynamic properties must be considered in the design procedure to improve the manipulator's operational capacity and precision.
An issue of extreme importance in many manipulator
tasks is the precise positioning of the arm's endeffector during movement. As speed of operation increases and the manipulator stiffness decreases, due to a need to decrease the mass of the moving parts, the flexibility of the arm generates inaccuracy in the arm motion. There are two distinct types of structural deformations which cause the arm to deflect about its equilibrium position. These are referred to as deformations in the large, and vibrations in the small [1]. Deformations in the large are nonoscillatory deflections, which are caused by external loads and inertia loads due to the gross motion of the arm. In operation, the links of the manipulator must be constantly accelerated and decelerated, leading to large inertia loads and corresponding deformations. Superimposed on this gross motion and deformation will be small scale oscillations about the equilibrium. These are the vibrations in the small. This dissertation presents a
development of mathematical formulation for analysis of manipulator dynamic behavior, considering its flexible components.
In general, structural integrity is a primary concern in manipulator design. The most direct way to reduce structural deformations is to increase system vibrational frequencies; specifically the fundamental frequency (lowest natural frequency), which is an excellent indicator of the manipulator's structural integrity. Chapter II formulates the vibratory motion of the arm in its stationary configurations from which the natural frequencies and corresponding mode shapes are deduced. In addition, based on Rayleigh's principle, the fundamental frequency is described explicitly in terms of local parameters to demonstrate the effect of each parameter on the magnitude of fundamental frequency. Not only is this frequency an indicator of system rigidity, it also provides a basis to classify the mechanism's operating speed. A manipulator's gross motion is described by a parameter, w, which indicates the rate of moving a load from one position to another and back. This parameter itself does not indicate whether a system is high speed or not, since what is considered as high speed for a particular mechanism may represent a low speed for another system. Yet, the comparison of w with the system's fundamental frequency provides a criterion for dynamic
classification of mechanisms. This criterion for single degree of freedom systems has been discussed in detail by D. Tesar and G. K. Matthew [2].
Although the source and magnitude of excitation
phenomena are extremely difficult to accurately quantify in general, the system response due to deterministic excitations, such as impulsive forces, offers valuable information about system reaction to dynamic shocks in the motion of manipulator arms. A linear formulation is presented in Chapter III to analyze system dynamic response under applied exciting loads. Also, the effect of nonlinearities, such as Coriolis and centripetal accelerations on the system response, is discussed in this chapter.
As mentioned previously, increasing the speed of a manipulator generates larger inertia forces which cause the arm to deform about its equilibrium position. To predict arm deflection due to inertia loads, the equations of motion are derived in Chapter IV, considering both gross motion and arm deformation, along with the coupling between the two motions. Ultimately, these deformations may be compensated for by modifying the nominal actuator commands.
Accurate analysis of flexible arm behavior requires
precise knowledge of system mass and stiffness properties. These parameters are commonly determined from the
continuously distributed model of the link components, or finite element methods using the mechanical properties of system elements. Manipulator link components, in general, are not uniform beams; therefore, the continuous model falls short of being practical. This problem is overcome by dividing the link elements into many small segments using finite element methods; however, this approach increases the system's representative degrees of freedom, leading to complication in analysis and numerical computations. An experimental procedure is proposed in Chapter V to obtain system mass and stiffness parameters based on the modal analysis concept. This approach offers an accurate representation of the arm since the dynamic parameters are identified so that the model behavior will match those of the actual system. Furthermore, these parameters are obtained from the prototype mechanism in its assembled form. Therefore, the effects of joint link connections on the component flexibility are included in the identification process.
In order to illustrate the application of these
formulations, Chapter VI contains analysis of an example involving a flexible manipulator arm.
During the past few decades, considerable attention has been devoted to the dynamics of highspeed flexible mechanical systems. Two remarkably complete literature surveys [3,41 present a review of research conducted on
this subject prior to 1972. Imam, Sandor, and Kramer [5] present a deflection analysis of planar linkages, where the system mass and stiffness matrices are constructed, utilizing the permutation vector method of structural analysis, and frequent recalculation of eigenvalues is avoided by using the "Rate of Change of Eigenvalues Method." Several publications [610] utilized the lumpedparameter approach for dynamic analysis of flexible planar linkages. Kohli and Sandor [11] extended this approach to investigate dynamic behavior of spatial mechanisms with elastic components.
The coupled, nonlinear, partial differential
equations of motion for a slider crank were developed by Chu and Pan [12] to include the transverse and longitudinal flexibility of the coupler link. In 1978, Sanders and Tesar [131, based on their analytical and experimental research, argued that deflection of mechanisms with elastic members can be predicted by a quasistatic method (higher time derivatives of small motion are considered to be negligible) if the mechanism operating speed is less than five percent of the system fundamental frequency. Hains [141, through an independent investigation, confirmed the conclusions drawn by Sanders and Tesar.
An analytical and experimental investigation was conducted [15] to study the dynamic behavior of a
highspeed planar four bar with elastic members. The equations of motion are derived using the assumed mode method by modeling the four bar as a cantilever beam (crank), connected with two simply supported beams representing the coupler and follower links. Dubowsky and Maatuk [16] present a formulation for the dynamic equations of motion for flexible manipulators. The manipulator's links are modeled as a series of pointconnected, slender beams, using Euler beam theory. System equations of motion are then derived by application of Lagrange's formula. As mentioned previously, this formulation is limited to mechanisms with uniform link components. Furthermore, the significance of system natural frequencies in the analysis of flexible mechanisms has not been addressed.
The transfer matrix method is utilized [17] to
develop a formulation for the analysis of flexible planar manipulators. The dynamic behavior of system components is described by constructing matrix elements which relate the motion and internal forces at one end of each component to those of the other end. The system transfer matrix is then built up while traversing the sequence of attached BernoulliEuler beams and bodies. The manipulator's natural frequencies and frequency response are obtained from the system transfer matrix and mechanism boundary conditions. This method neglects Coriolis and
centripetal accelerations for determination of system response. The transfer matrix approach is extended to spatial manipulators in [18] using a lumped mass model. By this procedure a CRL masterslave manipulator is investigated. The effects of configuration and payload on frequency response are studied both analytically and experimentally.
Tesar and Thomas [19] developed a lumped parameter model for planar three degree of freedom manipulators to determine system natural frequencies. Equations of motion are derived in terms of absolute deflection of mass elements using a Newtonian formulation.
An analytical method is presented [20,21] for the
dynamics of spatial mechanisms containing complexshaped, flexible links. The mass and stiffness properties of links are represented by a set of matrices deduced from finite element methods. These parameters are defined in the moving coordinate systems attached to the links. Therefore, they are invariant with respect to variation of the mechanism's configuration. Equations of motion are derived using Lagrange's formula and the 4 x 4 matrix dynamic analysis technique. System motion is defined in terms of absolute perturbed motion of finite element nodal points. This paper provides a significant and beneficial analytical formulation for dynamics of flexible mechanisms; however, it is not a practical procedure for
considering system flexibility in the design process. Although component mode synthesis coordinate reduction is introduced to decrease the system's degree of freedom, numerical integrations are cumbersome.
This document presents a lumped parameter model for dynamic analysis of flexible serial manipulators. The equations of motion are defined in terms of relative perturbed motion of mass components using Lagrange's formulation. Not only is this formulation relatively efficient for numerical integration, it also provides an explicit representation of system fundamental frequency in terms of local mass and stiffness parameters. Therefore, the effect of component flexibility and inertia on the system's vibration can easily be investigated through this formulation.
CHAPTER II
DYNAMIC MODEL OF FLEXIBLE MANIPULATOR ARMS
The main objective of this dissertation is to formulate the vibration behavior of manipulators, therefore providing a basis to include the system's flexibilities in the design procedure. Precision of the manipulator's operation must be assured in order to acquire a high quality product. Thus, it is required to design the system so that vibratory motions are diminished under operating conditions. The most direct way to achieve this goal is to keep the frequency content of exciting phenomena, produced by the system's operational function, far below the system's fundamental frequency. Therefore, the fundamental frequency can be utilized as a design criteria to increase system rigidity, leading to precision at higher operational speeds.
This chapter presents a mathematical model for
dynamic analysis of flexible serial manipulator arms. Of particular interest is computing the mechanism's natural frequencies, specifically the lowest one (fundamental frequency), and the corresponding mode shapes. This formulation is simple to implement on digital computers
and allows the designer to consider only the dominant, flexible components; enabling him to select the simplest, yet most accurate, representation of the mechanism to eliminate unnecessary computations. The present work is based on the formulation developed by Thomas and Tesar [22], for dynamic analysis of manipulators with rigid elements. The kinematic representation of serial manipulators developed in [22], is briefly reviewed here.
2.1 Kinematic Representation of Serial Manipulators
A serial manipulator is an assembly of links and
revolute or prismatic joints (most of the lower kinematic pairs are a combination of these joints), as shown in Fig. 2.1. The motion of each joint is defined by a parameter, Dn, representing a relative angular displacement, On, about the joint axis, %n, (revolute joint), or a relative axial motion, Sn, along Sn axis (prismatic joint). Two successive joint axes, Sj and S_, are connected by a link member along their common perpendicular vector, ajk Parameters, ajk and ajk, which are the distance along ajk, and the angle of twist about ajk between Sj and S_, define the link elements, as shown in Fig. 2.1. To describe the manipulator's motion, a global Cartesian coordinate system, along with a set of local coordinate frames, are
are chosen. The global system is a fixed frame in space located at the first joint with axes X*, Y*, and Z*, where Z* is selected along joint axis, S1. The local frames , (), Z ), are a set of Cartesian reference coordinates fixed to the link elements with X(j) lying along 2jk and Z(j) along Sj. Thus, the direction cosines of ajk and Sj in terms of the jth coordinate system, are ajk (1,0,0) and Si (0,0,1).
A body in space is completely defined by its
orientation and the position of a point on the body. The orientation of link jk is described by the direction cosines of X(j), Y(J), and Z(j) in terms of the global frame. The direction cosines of an arbitrary vector Q (Xq*, Yq*, Zq*) can be obtained by a rotational transfer of its direction cosines Q (J) (Xq (j), yq (J), Zq (j) )
defined in the jth local reference frame. Let Tj designate a 3 x 3 rotation matrix which relates the jth frame to a coordinate system with common origin to the jth frame and with axes parallel to the global coordinates. We can write
S= Tj (2)
(2.1)
Therefore, the direction cosines of ajk (Xjk*, Yjk*, Zjk*) and Sj (Xj*, Yj*, Zj*) in terms of the fixed reference frame are
ajk = Tj ajk(j) (2.2) Sj = Tj Sj(J) (2.3)
Substituting the values of ajk(j) and Sj(J) into Eqn. (2.2) and (2.3),
ajk = Tj r (2.4)
Sj = Tj K (2.5)
Equations (2.4) and (2.5) imply that the first and third columns of Tj must be formed of vectors, ajk and Sj, respectively. Furthermore, Y(J), in a right hand coordinate system, is formed by a vector crossproduct of Z(j) and X(j)
Y(J) = Z(j) x X(j) = .j x ajk (2.6)
The vector crossproduct, Sj x ajk, forms the second column of matrix Tj, since direction cosines of Y(J) in terms of the jth local reference frame are (0,1,0). The rotation matrix Tj can then be represented as
Tj = [ajk, Sj x ajk, Sj]
(2.7)
The direction cosines of Sj and ajk can be
obtained recursively from the initial direction cosines
a1 (X1*, YI*, ZI*) = (0,0,1)
(2.8)
112 (X12*, Y12*, Z12") = (Col, Sol, 0)
(2.9)
and the following expressions
Sj = T .j
ajk = T.
JI
0
Sa
(j 1)j
Caj
(j1)j
3 j
Ca Sej
(j1 )j
S(jii)jSj
where S and C represent the trigonometric functions sine and cosine.
(2.10) (2.11)
Now, let vector _j define the position of the
origin of the local frame in the fixed coordinate system. This vector is evaluated by a vector summation of the joint offsets and link lengths as shown here.
J
= + (a a + SZuSz) (2.12)
 2=2 (ii) (Zl)
Double subscripts for joint offsets represent a fixed displacement, however, it must be replaced by Sz where this value is not constant.
The position of a point, P, fixed in the jth
reference frame is defined in terms of local coordinates by a vector P(J) ( Yp(J), Zp j). Vector (Xp*,Yp*,Zp*; designates the position vector of point P in terms of global coordinates, which can be determined by transforming P(J) to the global frame. This transformation involves a translational transfer of the local frame's origin, and a rotational transfer of local coordinate axes.
P = Rj + Tj P(i)
(2. 13)
2.2 Rate of Change of Orientation and Position
The rate of change of orientation of a rigid element is defined by a vector, w (c, wY, wz), representing the angular velocity of the element about the axes of a fixed coordinate system. The absolute angular velocity of link jk in a serial manipulator can be determined simply by summing the relative angular velocities between successive links preceding link jk,
jk = 0112 + 1 23 + "' + 1jk, (2.14)
where '3wjk represents the angular velocity of link jk relative to link ij. The relative angular motion between links ij and jk is the angle Oj, which is constrained to rotate about the Sj axis; therefore, the relative angular velocities may be written as
ijjk = Oj~j (2.15)
where the dot represents the derivative with respect to time. Substituting Eqn. (2.15) into (2.14) results
Wjk = 31I + (212 + "'" + Ej~j
(2.16)
Now, Eqn. (2.15) can be written in a matrix form in terms of generalized motions, ï¿½n, and first order rotational geometric coefficients [Gjk]
wjk = (Gjk]P (2.17)
where [Gjk] is a 3 x N (N denotes the number of degrees of freedom in the system) matrix and its column vectors are
_ jk
[Gjkln = al (2.18) a n
Notice that the subscript n designates the nth column of matrix [Gjk] and notation [Gjkln is chosen in place of the common notation [Gjk];n. From Eqns. (2.16) and (2.18) the column vectors of [Gjk] are found to be
[ Sn n < j; nrevolute [Gjkln =(2.19)
0 , otherwise
The rate of change of position or translational velocity of a point, P, fixed in link jk, is the time derivative of position vector P
Vp = dP = + d (TJ)) = SS + (a Z (1)+
d't dt d 9,=2
(2.20)
SLSL + SXSï¿½) + d (TjP(j)
The time rate of change of a vector Q fixed in a rigid body jk is defined [23] as
dQ = X dt
(2.21)
Unit vectors, a(l) and Sz, are fixed in link (Zi) and vector TjP(j) is fixed in link jk. Considering this definition, we can rewrite Eqn. (2.20) in the form
p = Z SnSn + G nSn x a[ ' a)z a(z_1)z
n=1 Z=n+I
+ Sqz Z) + TjP(j)](2.22)
Notice that the terms inside the second summation of Eqn. (2.22) represent a vector from the origin of nth local reference frame to point P, since
n
Rn = S1111 + E (a(,_), a Z=2
P = $111 + Z (a a + S
2 (ï¿½1) ( I)
j
Z (a a Z=n+ I
+ Sï¿½ï¿½Sï¿½) + TjP(J).(2.25)
Eqn. (2.22) is then reduced to
J
p E Z [SnSn + n.n x (P Rn)
n= 1
(2.26)
Introducing the first order translational geometric influence coefficients [G.] and generalized motions Dn, the velocity of point P can be written in the matrix form
Vp = [Gp] D .
(2.27)
Recalling that in is equal to Gn, if joint n is revolute, and it is equal to Sn if joint n is. prismatic. By inspection of Eqn. (2.26), first order translational coefficients can be defined as
+
(2.23)
2,) + TjP(J))(2.24)
S x (P  Rn) , n < j; nrevolute
[Gpn = _n Sn , n < j; nprismatic (2.28)
0 otherwise.
Rotational and translational velocities for parallel mechanisms can also be defined in terms of first order geometric coefficients; however, the definitions of these coefficients will be more complex than definitions introduced for serial mechanisms. The analysis of parallel mechanisms is beyond the scope of this paper, although the present work may be extended to include these systems.
2.3 Basis for the Mathematical Model
The motion of manipulators with rigid components is described by highly nonlinear, coupled, differential equations in terms of the actuator's motion. However, the vibratory motions of robot arms are low magnitude, oscillatory deformations about the mean motion equilibrium caused by small relative deflections of the elastic elements. Furthermore, the representative frequency of operation of the arm, w, is kept significantly lower than the system's fundamental natural frequency. This prevents the dynamic coupling between the small and gross motion of the arm since many vibration cycles will occur about the
mean motion before any significant change in the arm configuration occurs (the detailed discussion of coupling between small and gross motion is presented in Chapter IV). In addition, the small motion assumption causes the nonlinear terms in the equations of motion to become insignificant. It is then sufficient to describe the vibratory motion of the manipulator by a set of linear, coupled, differential equations in terms of the perturbed motion of the arm about its stationary configurations.
A lumped mass arm model is used for tractability.
Not only does this produce a set of ordinary differential equations of motion instead of partial differential equations, it also provides an accurate description of system energy content. The distributed mass model is not chosen since the manipulator's links, in general, have complex geometric shapes. Because of this complexity in geometry, the distributed mass theory for beams fails to describe the mass content of the link accurately. Furthermore, the kinetic energy contribution of elements, such as electrical motors, hydraulic actuators, and drive trains, which are very significant, must be included in the total energy content of the system. This requires a mixed distributed lumped formulation, leading to a set of partial and ordinary differential equations describing the system's motion. Dynamic analysis of flexible manipulators is a complex problem; such formulation creates even more complication in this analysis.
The finite element method can be utilized to describe mass distribution of a complex shaped element by representing the element as a series of small uniform elements connected to each other. A system's degree of freedom will increase rapidly by using this method, which will lead to a large system of equations describing the manipulator's motion.
In conclusion, the lumped mass model provides an
accurate representation for most precision manipulators since the arms do not generally have long, flexible links, such as those found in the shuttle manipulator. An experimental procedure will be introduced in Chapter V to identify the concentrated mass and stiffness parameters, providing an accurate description of the manipulator's mass and flexibility content.
2.4 Dynamic Model of Serial Manipulators with Flexible Joints
For many manipulator designs, most of the deformation energy appears in the actuator and drive train components. Therefore, an initial model considers flexibility at the joints in the direction of relative joint motion only. The corresponding mass of link jk is represented by an inertia ellipsoid [23] centered at point Cj with principal axes parallel to the jth local reference frame
as shown in Fig. 2.2. Vector Cj defines the undeformed position of mass center in terms of the global reference frame.
Consider a serial manipulator with rigid components. The time derivative of vector Cj is obtained from Eqn. (2.27)
6j = [Gcj] I (2.29)
Using the mathematical definition of the time derivative, Eqn. (2.29) can be rewritten as
Aj = [Gcj] A D (2.30) At At
Canceling At from both sides leads to
ACj = [Gcj]IA (2.31)
This equation describes the variation in the position of mass center Cj, due to the small relative motion of the joints. Vector A@ portrays the elastic deformation of the
joints about their axes, and may be replaced by a set of relative perturbed motions, an, as shown in Fig. 2.2. The deformed position of point Cj is defined by a vector C , (the hat throughout this dissertation designates a vector which is displaced from its original line of action due to the deformation of the flexible components). Considering Fig. 2.2, it is evident that
Cj = Cj + &Cj = 2j + [Gcjla . (2.32)
The equations of motion for a mechanical system with N degree of freedom can be obtained from Lagrangian equations
d KE  3KE + PE = Qn (2.33)
a( q
where KE and PE are the total kinetic and potential energies of the mechanism. The generalized position coordinates are designated by qn and their corresponding generalized forces are Qn.
The velocity of mass center Cj is found by differentiating Eqn. (2.32) with respect to time
A = dc dCj d
j = _ +( [Gcj] ) S + [Gcj] . (2.34)
dt dt dt
Recalling that the manipulator arm is in a fixed configuration, and also assuming the variation of the first order coefficients due to the deflection of the arm is negligible, we can rewrite Eqn. (2.34) in a simplified form as
Cj = [Gcj]S. (2.35)
The absolute angular velocity of the link jk can be obtained from Eqn. (2.17),
wjk = [Gjk] $ (2.36)
The manipulator's kinetic energy is obtained from the sum of the kinetic energies produced by the motion of inertia mass ellipsoids
Nï¿½ ï¿½
N AA
KE = 1 I{Mj 2jTCj + jkT [ jk] Ijk} (2.37)
j=1 2
where Mj is the total mass of link jk concentrated at point Cj, and [Hjk] designates the inertia tensor about the centroid for link jk in terms of the global coordinates. The values in this inertia tensor depend on the orientation of link jk. However, the inertia tensor about the centroid, defined in terms of the local coordinate axes, [njk(J)], is a constant diagonal matrix for each link, which depends only on mass distribution of the link. To derive an expression for [fjk], recognize that the kinetic energy contribution is independent of the coordinate system so
^(T)^ ^ (J))T ( ).^3(J
! jkT [Kjk] Wjk = (jk ) [1jk ] Ijk (2.38)
The absolute angular velocity of link jk in terms of the global coordinates can be found from
' '(j)
jk = Tj jjk  (2.39)
Since Tj is an orthogonal matrix, the inverse of this matrix is equal to its transpose. So we can write
j = TT jk (2.40)
Substituting Eqn. (2.40) into (2.38) results in
A A =I
wjkT [fljk] wjk = wjkT Tj [fjk'j)] TjT _jk
(2.41)
From which, the inertia tensor, [fjkl, is defined as
[fjk] = Tj [Hjk (j)] TjT.
(2.42)
To express the system's kinetic energy in terms of the first order influence coefficients, Eqn. (2.37) is rewritten using Eqns. (2.35) and (2.36)
N
KE = Z MjST [Gcj]T [Gcj]a + _T [Gjk]T [jljk][G]jkl
2 j=1
(2.43)
= 1 ST I*]a 27
The equivalent inertia matrix of the system is represented by the N x N matrix [I*], which defines the effective mass of the system as seen at each joint. The components of this matrix are obtained from the general equation
N
[I n = Mj [Gcj]mT [Gcjln + [Gjk]rmT [ jk] [Gjk]n
m;n j=1
(2.44)
where [I*] designates the scalar component of the
m;n
inertia matrix located at mth row and nth column.
The potential energy of a manipulator in a deflected configuration is the sum of the strain energies of the springs describing the arm's flexible components
N N
PE = 1 Z Kij~i~j = 1 T [K*]a (2.45)
2 j=1 i=1 2
where Kij are the stiffness coefficients and [K*] is the equivalent stiffness matrix. Recalling that the manipulator's deflection is due to the relative deformation of joints in the direction of joint motion. The system's flexible components can be represented by a set of torsional springs attached to the revolute joint axes, and translational springs attached to the prismatic joint axes (Fig. 2.2). The energy content of each spring is due to the displacement of the corresponding joint only; therefore, there is no coupling in the system's potential energy, or
Ku = Kjj i=j (2.46) 0i i4 j.
Notice that the gravitation energy will not be included since the effect of this energy can be analyzed statically.
Selecting 5n as the generalized coordinates, the Lagrange equations for free vibration of an undamped mechanical system can be rewritten in the form
d DKE ;KE 3PE (2.47)
 + n= 0d n = 1,2,...,N.
Substituting the final expressions for the kinetic and potential energies into Eqn. (2.47), the equation of motion can be written as
d 1T+rgi =0 (2.48) 3t ( [I*]n;  T (7n [I*] ) + [K*]n;a ( . 8
The subscript, n;, designates the nth row of a matrix. The first term on the right side of Eqn. (2.48) can be expanded as shown here
d " d +""2.9
dt ( [I*]n; ) = (L [I*]n;) + [I*]n; . (2.49)
Recalling that [I*] is a function of the local mass parameters and first order geometric coefficients. Also, these coefficients are assumed to be invariant with respect to small displacements Sn, the second term on the left side of Eqn. (2.48) and the first term on the right side of Eqn. (2.49) can be neglected. The nth equation of motion is then reduced to
[I*]n; + [K*]n;_ = 0.
Next, the complete system's equations of motion can be written in the matrix form
(I*]8 + [K*] = 0.
(2.51)
Vector a describes the oscillatory motion of the manipulator's joint angles, which can be defined by a harmonic equation as
iwrt
8=B e
(2.52)
where vector B is the magnitude and wr is the frequency of the joints' vibration. Substituting Eqn. (2.52) into the equations of motion gives
{ wr2 [1*] + [K*I Be
= 0.
(2.53)
For nontrivial B, the coefficient matrix must be singular so its determinant must be zero.
wr2 [1*] + [K*1 = 0
(2.50)
(2.54)
Eqn. (2.54) can be rewritten in the form
[K*]l [1*  1 [I] = 0 (2.55) Wr2
where [I] is an N x N identity matrix. This is a standard eigenvalue problem, which can be solved to determine the system's natural frequencies, wr, and the corresponding eigenvectors r). Notice that [K*] is a diagonal matrix and simple to invert. In general, there is no closed form solution to Eqn. (2.55) and it must be solved numerically. There are several numerical techniques which can be utilized to evaluate the eigenvalues and eigenvectors of a system of equations. A matrix iteration scheme, usinq the power method in conjunction with the sweeping technique [2427], is found to be most practical to determine the first few eigenvalues and eigenvectors of a large system of equations. Also, an iteration scheme based on the QR method [2728] is best suited to evaluate all of the system's natural frequencies and the corresponding mode shapes.
2.5 The Secondary Flexible Components of A Serial Manipulator
The dynamic model of a manipulator with flexible joints may not describe the system's deformations adequately if the deformation energies contributed by other flexible components are significant. The most prominent deflection modes of a jointlink combination (Fig. 2.3a) are a) the motion about the joint axis, as discussed previously; b) the deformation of the joint bearings about two orthogonal axes perpendicular to the joint axis; and c) the torsional twisting of the link about its major axis. The joint deflection is described by an input, S1, about joint axis, SI. Two pseudoinputs, 82 and 3, are introduced to describe the bearing deflections and the torsional deformation of the link (psuedoinputs are imaginary inputs that produce small motions only). Parameter, 2, represents the combined link deflection and joint bearing deflection perpendicular to S1 about S2 (Fig. 2.3b). The combined bearings' deflection and link's torsional deformation about an axis, S3, perpendicular to S1 and S2, are described by the pseudoinput 3. The mass content of this jointlink combination is lumped at the imaginary link, a34, so that the motion of S1, a2, and 53 will contribute to the kinetic energy of this
component (Fig. 2.3b). Now, the system's equations of motion can be obtained by following the same procedure as described in Section 2.4. The length of vector B will increase to at most 3N (often, fewer than three inputs are needed for a sufficiently accurate representation). The components of inertia matrix, [I*], can be obtained from Eqn. (2.44), but the size of this matrix will increase by a factor of three. Notice that the 3N x 3N stiffness matrix [K*] will remain diagonal.
2.6 Dynamic Model of Serial Manipulators with Flexible Links
This section expands the formulation of Section 2.4
to include the flexibilities of the manipulator's links in addition to those of joint components. A general jointlink combination is shown in Fig. 2.4a. This component is shown in its rigidbody position (solid lines), as well as in its elastically deformed configuration (dashed lines). The elastic deformations of the jointlink combination may be completely described by the input, B1 (joint deflection), and six pseudoinputs, a2 through 87 (link deformations), as shown in Fig.
2.4b. The flexibility of each jointlink element can be represented by a 7 x 7 stiffness matrix, [Kjk], where subscript, jk, designates link jk. The length of vector
, will increase by a factor of seven. Considering the strains associated with displacements, a, the system's potential energy is
PE = T [K*]a (2.56)
where [K*] is a 7N x 7N matrix of the form
(K231
[K*] = K]0
K1(34] (2.57)
0 [K N(N I )]
Recalling that the size of the inertia matrix will increase by a factor of seven, its scalar component can be obtained from Eqn. (2.44). Finally, the manipulator's equations of motion are derived using the formulation of Section 2.4.
For a general six degree of freedom flexible
manipulator arm, the motion is described by fortytwo differential equations. Thus, it may be difficult to compute all the system's natural frequencies. However, higher natural frequencies have little or no effect on system vibration, since they are higher and smaller in magnitude than the fundamental frequency. It is then
adequate to evaluate the first few eigenvalues using the power method.
2.7 Improvement of System Design
The performance and operating speed of a system can be improved by increasing its fundamental natural frequency. This will reduce the robot's flexibility, leading to less deflection of the arm due to the applied loads and inertia forces.
An approximation for the lowest eigenvalue of a system of equations is given by Rayleigh's quotient
W12 = (u)T [K*] (u) (2.58)
(u)T [,*] (u)
An estimate for the fundamental frequency can be obtained from Eqn. (2.58) if the arbitrary vector u resembles the first mode of vibration. If the vector u coincides with the system fundamental mode I , we can write
W12 TK*] (2.59)
(1) )T [I*] (q (1)) M
Parameters, Ij and M1, designate the modal stiffness and modal mass of the system. Clearly, the magnitude of wl, can be increased by decreasing M1 and/or increasing Ki. To demonstrate the effect of the local stiffness and mass parameters on the system fundamental frequency, Ki and MI must be defined explicitly in terms of these parameters, as follows:
N N
K1 = Z Z Kij ' i ) 3( (2.60) i=1 j=1
N
MI = E MjDj (2.61) 3=1
where denotes the ith component of
(1) and
N N
Oj [Fj]i;y. (2.62) j=1 Z=1
[Fj] is an N x N matrix, which is constructed from
[Fj] = [G ,j]T [GPcj] + [G'jk]T [Ejk] [Gjk].
(2.63)
The column vectors of [Gp] matrices are defined as
[GicjIn n (1) [Gcj]n (2.64) [G'jkln  n(1) [Gjk]n" (2.65)
The 3 x 3 matrix, [Ejk], is obtained from
[Ejk] = [Tj] [ejk] [Tj]T (2.66)
where ejk is a 3 x 3 diagonal matrix with constant components. This matrix is defined in terms of the local mass, Mj. Assuming that the local inertia tensor of each link is proportional to its mass, we can write
1 (j) . (2.67) [ejk] Mj [L jk ]
In general, the scalar components of [ejk] fluctuate as the value of Mj varies. However, to express M1 in terms of the local masses, Mj, only, we assume that [ejk] remains constant. Now, by evaluating the coefficients of Eqns. (2.60) and (2.61), it is very simple to investigate the effect of each local mass and stiffness on the system fundamental frequency.
I p~j
e.2Z
77 1 SI, Z
Kinematic Representation of Manipulator
Figure 2. 1.
1 0
A
/777 Z
Figure 2.2. Manipulator with Elastic Joints Displaced
from their RigidBody Position.
"00,
40
s2
a2 .
1 7
(a)
S2
B2
SS
3
b
Figure 2.3. Secondary Deformations of JointLink
Combination.
,(Jj)
x i
ï¿½ m I I
I
(a)
S2 ,S3 2 S6 7
~6 I 4 4S5 M0
c1.2
( b )
Figure 2.4. General Deformation of JointLink Combination.
CHAPTER III
ANALYSIS OF MANIPULATOR
DYNAMIC RESPONSE
The response of a manipulator arm is the behavior of the system (the motion of a point in the system at any time) when subjected to a certain excitation. This excitation is caused by transient phenomena such as dynamic shocks associated with the arm motion. In addition, there is a certain harmonic content related to the loading condition of the manipulator's terminal device which may excite one or more vibrational modes of the arm.
Using the mathematical model presented in Chapter II, it is possible to formulate the forced vibration of a manipulator arm under a general exciting condition. The response of the arm can be represented by N coupled, linear, ordinary differential equations in terms of the generalized coordinates n These coordinates can be obtained from a solution of the coupled equations, but this procedure becomes progressively cumbersome as the degree of freedom of the system increases. A more suitable approach is to decouple the system of equations using the concept of modal analysis.
The transfer function is a general concept relating the system response to the excitation in the frequency domain which contains all the information concerning the system vibration characteristics. To represent the manipulator's characteristics, its transfer functions are obtained by exciting the mechanism with a harmonic force of variable frequency.
The equations of motion describing the system oscillation have been linearized by neglecting the variation of the first order geometric coefficients due to small displacements an. It is appropriate to investigate the significance of the nonlinear terms by extending the previous formulation to include the Coriolis and centripetal accelerations in the equations of motion.
3.1 Forced Vibration
In the most general case, several time varying forces and torques can be applied to each link of the manipulator arm. These applied loads can be represented by N generalized forces acting over the generalized coordinates n. Representing the generalized forces as Tn, the equations of motion can be obtained from Lagrange's equations as follows
d (3KE)  3KE + DWD + 9PE = Tn (3.1)
dt (7n) 76n n 9n
where WD is the work done by a special type of friction force arising from motion in a viscous medium. The generalized forces can be defined in terms of the first order coefficients and applied loads from the principle of virtual work, as described here.
Consider the resultant of the applied loads on link jk of the manipulator as a force, 1p, applied to point P in the link and a moment, mjk, applied to the link. The virtual work due to the applied loads on link jk can be defined as
SW = fpT Vp 6t + Mjk T wjk 6t (3.2)
The virtual displacements of point P and link jk are represented by ]p 6t and wjk 6t, where 6t is a virtual time increment. However, the virtual work can also be regarded as the product of N generalized forces Tn(jk) acting over the virtual displacements n 5t.
N (jk)
6W = E Tn Dn 6t = (T (k))T 5t (3.3)
n=1
Notice that Tn(jk) designates the equivalent force acting over the nth input due to the applied loads on link
jk. Now, Eqn. (3.2) can be rewritten in terms of the geometric coefficients using Eqns. (2.35) and (2.36)
= [Gp] (t + mjkT [Gjk]! 6t (3.4)
so that, comparing Eqns. (3.3) and (3.4), we can conclude that the generalized forces have the form
(T(Jk))T = fpT [Gp] + mjkT (Gjki . (3.5)
The total equivalent forces T are found by summing T(jk) for each link with applied loads, or
N (jk)
T . Z T (3.6) 3=1
The friction forces are nonconservative forces and are assumed to be proportional to the velocities. Therefore, the work done by these forces can be obtained from Rayleigh's dissipation function
wD I N N
2 N N Cij 8i j (3.7)
2 iZj~
1=1 j1
where Cij represents the damping coefficients. This equation can be rewritten in a matrix form as
WD  1 'T [C*] 8 (3.8)
The equivalent damping matrix is defined by [C*]. Introducing Eqns. (2.43), (2.45) and (3.8) in (3.1), we obtain a set of N coupled, ordinary, differential equations describing the response of the manipulator arm. The equations of motion can be written in the matrix form
[I*] + [C*] + [K*] = T . (3.9)
The system of equations (3.9) can be solved for the generalized coordinates Bn which represents the oscillation of the arm at any time t. But, this is possible only if the excitation is given in a deterministic form as a time dependent function.
3.2 Decoupled Equations of Motion
It is not practical to solve the coupled equations of motion simultaneously when the system has many degrees of freedom. In this case, a different approach can be used to give the equations of motion a simpler form by expressing the equations in a new set of coordinates. This is accomplished by decoupling the equations of motion using a linear coordinate transformation. The linear
transformation is obtained by assuming that the response is a superposition of the normal modes of the system multiplied by the corresponding time dependent modal coordinate. To use this procedure which is known as modal analysis, it is necessary first to obtain the system eigenvalues and eigenvectors from the eigenvalue problem described by Eqn. (2.55). Using the superposition theorem, the response can be described by the linear transformation
S = [ ] (3.10)
where n is a column matrix consisting of a set of coordinates referred to as modal coordinates. The modal matrix ['] is an N x N matrix and its column vectors are defined by the equation
= *4= (r) (3.11)
The components of [T] are independent of time, so the derivatives of the generalized coordinates with respect to time can be represented by the equations
S= [']n (3.12) I ] I ï¿½ (3.13)
Introducing Eqns. (3.10), (3.12) and (3.13) into (3.9), we obtain
(I*] [ ]n + [C*] [ ]n + (K*] [ (]TT =T .
(3.14)
Premultiplying both sides of this equation by [y]T, we can write
[x]T[I*] (]n + []T(C*] [u] T_ + [ ]T[K*] [I]T_ = [T]T T.(3.15) But, from the orthogonality condition of the normal modes we have
(4_r))T (I*] (4_(s))T =
((r))T [K*] ( _(S))T=
I M r=s ROr 'r~s
r , r=s
0 r3s
where Mr and Kr are the modal mass and modal stiffness of the system associated with the rth mode of vibration. Therefore, the system inertia and stiffness matrices corresponding to the modal coordinates are diagonal and can be defined by the equations
M ] = [i]T [I*] ['] K ] = [uf]T [K*] [T]
(3.18)
(3.19)
(3.16)
(3.17)
In general, the damping matrix associated with the modal coordinates is not diagonal. However, the damping factors are generally small and matrix [C*] can be assumed to be proportional to the system inertia and stiffness matrices, or
[C*] v 1j [K*] + v2 [I*] . (3.20)
The error introduced by this approximation for the robotic manipulator is expected to be insignificant. Hence, the damping matrix in modal coordinates can be defined by a diagonal matrix as
[ ] = [y]T [C*] ['] . (3.21)
Now, using the definitions for modal parameters we can obtain N independent equations in terms of the modal coordinates
Mr r + Cr nr + Kr nr = Nr (3.22)
where Nr represent the generalized forces associated with the modal coordinates, which are given by the equation
Nr ( r))T T
(3.23)
Dividing both sides of Eqn. (3.22) by 9r, we obtain
"" ï¿½2 Nr ( 4
flr + 2 r wr flr + rnr =r (3.24)
where by definition
= (2 "r Mr)1/2 (3.25) Wr = (Kr/Mr) I/2 (3.26)
The solution of Eqn. (3.24) can be readily obtained by means of the Laplace transformation. This solution can be expressed as
ft
1 1r rrtT l~)
 r (T)erwr(tT) sinwdr(tT)dT + nr(0)
cor d+Mr Nr (1r2) I2 cos(wdrt  Yr)+ dr sin Wdrt . (3.27)
Parameters, flr(O) and ;r(O), represent the initial displacement and velocity corresponding to the rth modal coordinate; Yr, and wdr are defined by the equations
Y r = tan1 2)1/2 (3.28)
wdr ( r2)1/2
=d (  ~rV2Wr .(3.29)
Clearly, the modal analysis is a convenient procedure for computing the manipulator's response. However, determination of all system eigenvectors becomes complicated as the degree of freedom of the system increases. This difficulty can be overcome by approximating the system response by a superposition of the first few normal modes, which requires the calculation of the first few eigenvectors only. The error introduced by this approximation is negligible, since the higher natural frequencies are generally much higher than the fundamental frequency, and the contribution of their associated modes of vibration to the system response is very small.
3.3 Dynamic Response in the Frequency Domain
The characteristics of a vibratory system are fully described by the concept of a transfer function. This function represents the magnitude of oscillation at any frequency, when the system is subjected to a periodic force. Therefore the behavior of the system can be predicted if the frequency content of the time dependent exciting function is known. For instance, consider an impulsive force and its Fourier transformation as shown in Fig. 3.1. The Fourier transform of this function suggests that an impulsive force contains an infinite range of frequencies. Consequently, this force will excite every
vibration mode of a system when applied to the system. But, the forcing function of Fig. 3.2 will excite only those vibration modes which are within the frequency content of this force.
To obtain the manipulator's transfer functions, a harmonic force F Heit is applied to a point H located on the manipulator's endeffector. The corresponding generalized forces can then be acquired from Eqn. (3.5)
TT = F T [GH] e it. (3.30) Substituting Eqn. (3.30) into (3.23), we obtain
N r = i Nri ei (3.31)
where
I RrI = u) (r))T [GH]T fH . (3.32) Introducing Eqn. (3.31) in (3.24), we can write
rr + 2 r Wr r + Wr2 Tr = INr e~t. (3.33) A solution of this differential equation is
T) = Inrl e
(3.34)
Substituting Eqn. (3.34) into (3.33) and rearranging the resultant equation, we obtain L = hrr(w) (3.35)
where hrr(w) represents the direct transfer function of the arm associated with the rth modal coordinate and is defined by the equation
1
hrr(w) = Kr [1  (w/wr)2 + 2i~r (w/wr)] (3.36)
The magnitude of the generalized coordinates B can be obtained from Eqn. (3.10) B = [,Y] HI ï¿½ (3.37) We can also find the direct transfer function, hnn(w), corresponding to the nth generalized coordinate of the arm nn(w)  I Tn I (3.38)
3.4 System Response Due to Dynamic Shocks
Any discontinuity in a time derivative of the motion of a mechanical system will create a shock in the dynamic operation of the system [2]. Such shocks may excite one or more modes of vibration of the system causing unwanted noise, fatigue, and reduced precision of operation. These shocks are most undesirable when they occur in the first and second time derivative of the system's motion (velocity and acceleration) and must be avoided if possible. A discussion of the manipulator response due to discontinuity in the velocity and acceleration curves is presented in this section.
Velocity shocks are commonly generated when starting or stopping the arm motion. Suppose that the velocity of the endeffector produced by the input commands is constant throughout the motion interval (Fig. 3.3a). The acceleration associated with this velocity (Fig. 3.3b) will be very high at the boundaries of the motion interval due to the velocity shocks at these points. Consequently, impulsive forces are generated at the boundaries by the inertia of the arm associated with this acceleration. Now, consider an impulsive force, _H6(t), applied to a point H on the manipulator's terminal device. Using Eqns. (3.5) and (3.23), the effective force corresponding to the rth modal coordinate of the arm can be obtained
where
Nr r I 6 (t)
IN = ((r))T [GH]T f
(3.39)
(3.40)
Substituting Eqn. (3.40) into (3.27) and neglecting the initial conditions, we can write
(3.41)
TrI r NrL e rwrt sinwdrt
This equation describes the behavior of the arm in the modal coordinates, from which the system response in the generalized coordinates can be obtained
3 = .I] n
(3.42)
Furthermore, the behavior of any point P on the system is described by the equation
A P = [Gp] _ ï¿½
(3.43)
Discontinuities in the acceleration curve generate similar discontinuities in the correspondinq inertia force. The exciting force produced by an acceleration
shock can be represented by a step function fHi(t) applied to point H, where
, t > 0
The modal coordinates are then obtained from Eqn. (3.27)
r=  [1  e rt (cos wdrt + sin Wdrt)].
Kr (1  Cr
(3.45)
Notice that INrI is defined by Eqn. (3.40). The generalized coordinates an and the motion of any point P on the arm can be found from equations (3.42) and (3.43).
3.5 Nonlinearities in the Equations of Motion
The response of a manipulator arm was described in Section 3.1 by a set of linear differential equations. This was accomplished by neglecting the variation of the first order geometric coefficients due to small displacements an. Further refinement of the vibration formulation is needed to investigate the influence of nonlinear terms on the response of the arm.
Consider the instantaneous configuration of the
manipulator arm shown in Fig. 3.4 (dashed lines) as it
vibrates about its rigid body position (solid lines). The total relative motion of each joint is defined by a parameter Dn(t), where
Dn(t) = Dn + Sn(t)  (3.46)
Each of the joint axes and link axes is displaced from its original line of action due to the deformation of the flexible components. These axes are defined by vectors ,j and ajk, respectively. Furthermore, the local
cordnaes ((j) Y (J) Z(j)
coordinates ( , , ) corresponding to each link X^(j) ^(i
are chosen such that x and Z lie along ajk and Sj. Thus, the rotation matrix Tj describing the orientation of the jth local frame with respect to the global coordinates can be represented in the new form
Tj = [.ajk, Sj x 2,jk, Sj]  (3.47)
Also, the direction cosines of ^j and ajk can be obtained recursively from the initial direction cosines
S1 = (0, 0, 1) (3.48)
. 12 = (C(01 + 61), S(ï¿½1 + 1), 0) (3.49)
and the following expressions
j = T j1
C(
AA
ajk = Tj1
SC
0
S
(j1)j C C (3I)3
(j + Bj)
(j1)j S(Gj + aj) (j1 S(Gj + j)
Notice that ctjk is the angle of twist about the link axis ajk The rate of change of orientation of link jk about the global coordinates is given by the sum of the relative angular velocities about the revolute joint axes, which can be written in terms of the first order rotational coefficients in the form
_jk = [Gjk]_ D = [Gjk]_
(3.52)
Recall that the arm is in a stationary configuration and parameters n are invariant with respect to time. The first order geometric coefficients [Gjk] are distinct from those defined by the matrix [Gjk], since their column vectors are given in terms of the deflected joint axes as follows
[Gjk]n =
n < j , nrevolute otherwise .
(3.53Y
(3.50) (3.51)
The deflection position of the mass center Cj in the global reference frame is described by the equation
Cj = j + [Tj] Cj() (3.54)
where, Cj(J) defines the position of Cj in the jth local frame, and vector j locates the origin of this frame in the global coordinate system. From Fig. 3.4, Rj can be obtained as
J
2j = $11_I + E (a( a + Sz Sn). (3.55)
Following the procedure described in Section 2.2, we can obtain the rate of change of mass center position
2j = [Gcj] = [Gcj] . (3.56)
Matrix [Gcj] represents the first order translational influence coefficients of the deflected arm. The nth component of [Gcjl is defined as
in x (P  Rn) , n < J ; nrevolute
(Gcj]n = , n < J ; nprismatic (3.57)
0 , otherwise .
Next, the total kinetic energy of the manipulator arm is obtained from Eqn. (2.43)
KE = ( 1*] (3.58)
where [I*] represents the inertia matrix of the manipulator in its deflected configuration. The scalar component in row m, column n of this matrix is given by the equation
^ N ^ [^j~ [^j m I^k) [I*]m;n Z (Mj [Gcj]mT [Gcjln + Gjk Tjk] [Gjkn)m n j=1 (3.59)
Matrix [fjk'j)] represents the inertia tensor about the centroid for link jk in terms of the global coordinate system. This matrix is defined by the equation (as shown in Chapter II)
[Hjk] = Tj [ jk j) TjT (3.60)
The 3 x 3 matrix [HjkI(j)] denotes the inertia tensor of link jk in terms of the jth reference frame.
The potential and dissipation energies of the mechanism are not defined in terms of the geometric
coefficients; therefore their mathematical definitions remain the same as those given by Eqns. (2.45) and (3.8). Now, substituting the final expressions for the kinetic, potential, and dissipation energies into Lagrange's equations of motion, we can write (3.61)
d 1 A
([*] _   (n [*])_ + [C*In; + [K*]n;._ Tn.
Parameter Tn is the nth component of the vector T representing the generalized forces of the arm. This vector is defined by the equation
N
T N Z (jk) (3.62)
j=1
where
(jk) = [Gp jkT [Gjk] (3.63)
The first term on the left side of Eqn. (3.61) can be expanded as follows
d d ( ^
t ([I*]n;;) = [I*]n;$ + (dt [I*]n;) (3.64)
where
d N . 3 a d ([I*]n;) = Z m  ( (I*]n; = T [i*]n; . (3.65)
m=1
Notice that 7[I*]n; represent the standard convention
differentiation with respect to a vector. The result of this differentiation is an N x N matrix, where the mth row of this matrix is obtained from the derivative of [I*]n; with respect to the mth component of vector S, or
_ ([I*n) =
a 3T (I*]n;
3.2 I*]n; .2
3B I*]n;
N
(3.66)
Using Eqns. (3.64) and (3.65) in (3.61) gives the nth equation of motion as
[i,]n;ï¿½ + ;T [( [i,]n)
n;A
+ [C*]n;A + [K*]n;a = Tn (
(3.67)
This equation can be rewritten in the form
A A *
[I*]n; + ;T [Pn] _ + [C*n;_ + [K*]n;_ = Tn (3.
where the inertia power matrix, [Pn], is defined by the equation
A  L (A 1 n (
[Pn] = ( *l*) [12*]~ (3.
68)
69)
An alternative matrix, [Pn*], is presented in [291 which can replace [Pn] (this is shown in Appendix B), such that Eqn. (3.68) becomes
[I*]n; + ;T [Pn*] + [C*]n; + [K*]n;8 = Tn ï¿½ (3.70) The scalar component of (Pn*l associated with the lth row and mth column is defined by the equation
N A A
n Mj [Hcj ] T [Gcj]n + [HjkT
[Pn ] Z;m =j=1 Z;m ;m
+ [Gjk] T [rljk] [Gjkln [Gjklm
[fljk] [Gjkln
(3.71)
The second order geometric influence coefficients are defined by [Hjk] and [Hcj], the expressions for these
coefficients are shown in Appendix A. The 3 x 3 matrix [Gjk]n is constructed from the scalar components of the vector [Gjkln as follows
0 [GjkZ]n [GjkYin
AAA
[Gjkin [GjkZ]n 0 [GjkX]n (3.72)
[GjkY]n [GjkX]n 0
This matrix is introduced in the above form so that a vector cross product can be expressed in terms of matrix multiplication
[Gjk]n x [Gjklm = [Gjk] [Gjk]m
Recognizing that displacements 3n are small, Appendix C shows that the inertia and inertia power of the manipulator arm in its deflected configuration can be approximated as
[*] & [I*] (3.74) [Pn] [Pn*] (3.75)
where matrices [I*] and [Pn*] are the inertia and inertia power of the manipulator obtained from the
undeflected configuration of the arm. It is also shown in the Appendix C that
Tn Tn . (3.76)
Considering the above approximations, Eqn. (3.70) can be simplified as
(I*]n;a + aT [Pn*] + [C*]n; + [K*In; = Tn ï¿½ (3.77)
This equation can be solved numerically to obtain the system response.
3.6 Order Analysis
In order to predict the effect of the nonlinear terms on the system response, it is necessary to determine the significance of these terms in the equations of motion. This can be accomplished by estimating the order of the nonlinear terms relative to those of the linear terms. To simplify the procedure, we consider a six degree of freedom serial manipulator arm with flexible joints only. Hence, Eqn. (3.77) can be written in the form
N N N
S[I*] ,i + E [Pn*], ,Bisj + =I n;i i=1 j=1 1;3
N
[ [C*] ,$i + Kn~n = Tn (3.78) i=1 ~
By comparing the definitions for system inertia and inertia power, it is evident that the magnitudes of their scalar components are of the same order, so it may be assumed that
I [I*]Z;m jI [Pn*]Z;mn ï¿½(3.79)
Furthermore, let us assume that the damping matrix is proportional to the stiffness matrix only, or
[C*] = v[K*] = 10d[K*] (3.80)
where d is a constant. Now, Eqn. (3.78) can be reduced to
(3.81)
N N N
E [I*] .i + E E [I*] ;i~ j + 10dKnn + Knan = Tni=I n;i i=I 3=I i;3
Since an represents the oscillation of joint n about its axis, it can be approximated by a periodic motion consisting of N harmonic functions. Considering only one mode of vibration at a time, we can write
I an I = Bn (3.82) I ;n I = Bnw (3.83) nI = Bnw2 ï¿½ (3.84)
Parameter Bn defines the magnitude of the vibratory motion. To further simplify the equations of motion, assume that the magnitude of all joint displacements is equal to a maximum displacement B, or
Bn Bm = B .
(3.85)
Introducing Eqns. (3.82) to (3.85) in (3.81), we obtain
+Bw2 N N + BW2 N N[I*]
E 3 i=1 j=1
(3.86)
+ 10dwKn + Kn} = Tn.
The order of the scalar components associated with the nth row of [I*] are greater than those corresponding to the (n+1)th row of this matrix. It is then possible to assume a worst condition for the inertia matrix, which is
N N i=I n;i
N N
i=1 j=1
[I*]
N
i=1
(3.87)
(n+1 ) ;i
[I*1
n;i .
(3.88)
N
B W2 z (I*]n;
i=1 ~
Substituting Eqn. (3.88) into (3.86) results in
N
Bw2 ( Z [I*]n )(l + NB) + BKn(10dw + 1) = Tn . (3.89)
i=1 ;
The product NB represents the significance of the nonlinear terms in the equations of motion. Assuming that the vibrational deformations are less than + .02 radians (a large motion for a precision manipulator), we can conclude that
;T [Pn*] l ! 0.12 I [I*n; ï¿½ (3.90)
Notice that the significance of the damping factor depends on the value of w.
All of the proposed approximations corr.espond to
worst conditions of the arm; therefore, the significance of the nonlinear terms is grossly exaqerated. By making more realistic assumptions it is possible to draw the same conclusion for a manipulator arm with a large number of freedoms.
f (t) F (w)
I
iw
1,t w
(a) (b)
Figure 3.1. Fourier Transform of Unit Impulse.
f(t) F(W)
(t w
(a) b
Figure 3.2. Fourier Transform of Unit Step.
( a )
( b )
Figure 3.3. Constant Velocity Curve.
/ a. ,,
1CiK ,
 A
//x
zrs,
Figure 3.4. Kinematic Representation of Oscillating Manipulator.
g A(j)
z
CHAPTER IV
ARM DEFLECTION DUE TO
THE GROSS MOTION
In all previous considerations in the vibration
analysis of flexible manipulators, it is assumed that the manipulator arms are in a fixed configuration. As the operational speed increases, the inertia loads become significantly large. At lower speeds the frequency content of these forces are much lower than the arm's fundamental frequency. Under these circumstances, the inertia loads act as statically applied loads varying throughout the cycle. Thus, the system deformations can be obtained by application of quasistatic analysis. At certain speeds, the frequency content of inertia forces becomes so large that they may excite one or more modes of vibration of the mechanism. The manipulator's deformation will then become timedependent displacements which are coupled with the mechanism's gross motion.
For precision control of a manipulator, a complete
dynamic formulation can be used to predict, and therefore compensate for, the system deformations under load
compensate for, the system deformations under load (inertia and external loads). The objective is to obtain compensation signals which, when added to the control signals, correct for hand positioning errors due to the flexible components. Knowing the active loads and manipulator's dynamic parameters (local masses and stiffnesses), the change in the terminal device position, PH, can be calculated offline, prior to operation. The compensation values are then obtained in order to modify the nominal actuator commands.
4.1 Complete Dynamic Model for Manipulator Arms
A general approach based on the formulation of
Section 3.5 is described here for deriving the equations of motion of flexible serial manipulator arms. This approach considers the kinetic energy contribution of the system inertia produced by both nominal and perturbed motion of the arm. The perturbed coordinates are regarded as the unknowns, but the nominal motion of the arm is considered to be known. Hence, the equations of motion are expressed in terms of the relative perturbed displacements of the arm.
Consider the flexible manipulator arm of Fig. 4.1.
The geometric parameters of this robot arm are defined in Section 3.5. However, the motion of each joint is described by the sum of two time dependent displacements
associated with the nominal motion and deformation of the joint. So, the relative motion of joint n can be defined by the equation
Dn(t) = On(t) + Sn(t) . (4.1)
The absolute angular velocity of link jk about the global coordinates is defined as a product of the corresponding first order rotational coefficients and the joint velocities
A A A A
jjk = [Gjk] = [Gjk] (( + ) . (4.2)
It must be borne in mind that n is identically zero if joint n represents a pseudoinput. Furthermore, the translational velocity of mass center Cj can be described in terms of the joint velocities and translational influence coefficients
Cj = [Gcj] = [Gcj] ( D + ) (4.3)
Introducing Eqns. (4.2) and (4.3) in (2.37), the kinetic energy of this mechanism can be expressed in the form
KE 1 + T I*] ( ( + ) (4.4)
Recall that gross motion of the arm has no effect on the system's potential and dissipation energies, since the mechanism is assumed to vibrate about its instantaneous equilibrium configuration. The equations of motion can now be derived from Lagrange's equations
d 3KE 3KE 9WD 3PE
d  + + =an Tn (4.5) Nominal displacements, On, do not represent the independent generalized coordinates of the system, since they are a set of given parameters which describe the time varying characteristics of the manipulator arm. Therefore, only perturbed displacements are considered as generalized coordinates in the Lagrangian formulation. Now, substitute the expressions for the system kinetic, potential, and dissipation energies into Eqn. (4.5) to obtain
d 1 + I " dt { n; )f}  (( +3 )T ( [I*] ) (( +
+ [C*]n;S + [K*]n; Tn. (4.6)
The first term on the left hand side of Eqn. (4.6) can be expanded as follows
d  .I],
n;(D + f )} + +
+ a)T(_ I*])( + ) 4.7)
Introducing Eqn. (4.7) in (4.6) results in
[I*In;(_ +) + (_ +_)T[Pn*](_ +_) + [C*In;$ + [K*In; = Tn . (4.8)
The scalar components of [Pn*] are defined by Eqn. (3.71). Expanding Eqn. (4.8) and rearranging the resultant equation gives a nonlinear differential equation in terms of the perturbed coordinates rn,
[I*]n;_ + T(Pn*] + T[R*]S + [C*]n;._ + [K*]n;=
Tn  [I*n;_  T[Pn*]_ . (4.9)
Notice that the first time derivatives of the nominal and perturbed coordinates are coupled in this equation. The coupling matrix [Rn*] is an N x N matrix defined by the equation
[Rn*] = [Pn*]T + [Pn*] ï¿½ (4.10)
Matrices [I*], [Pn*], and [Rn*] are functions of the deformation displacements $n. However, recalling Eqns. (3.74) to (3.76), the coefficients of differential equations of motion can be obtained from the nominal displacements only. Thus, Eqn. (4.9) can be rewritten in the form
[I*]n; + T[Pn*] B + T[Rn*] B + [C*In; B + [K*]n; 3
Tn  [I*]n; _T[Pn*] . (4. 11)
The last two terms on the right hand side of Eqn. (4.11) represent the equivalent load acting over the nth input due to the nominal motion of the system inertia. This load acts as an externally applied load on the system which causes the arm to deform about its rigidbody position.
In order to obtain the perturbed motion of the arm, Eqn. (4.11) must be integrated numerically. Deformation displacements can be computed at a series of time steps At, to describe the position of the arm at any time t. The accuracy of any numerical integration depends on the size of time steps, At, which becomes even more significant for Eqn. (4.11) since the coefficients of this differential equation are time varying functions. To minimize the cumulative errors in stepbystep integration and avoid numerical instability, numerical integration must be performed at small time steps At. An adequately small time step can be selected as a fraction of the time period associated with the maximum excited mode of vibration. For example, if only the first mode of the arm is excited by an applied load, the time step should be a fraction of l/wI. Clearly small time steps require long computational times for numerical integration. The
computational time may be reduced to some extent by using a suitable method such as fourthorder RungeKutta method combined with Haming predictor corrector [28]. There are several other numerical methods [30] which can be used for solving a dynamic equation such as Eqn. (4.11).
As mentioned previously, if the frequency content of the inertia loads is significantly lower than the fundamental frequency, the variation of deformation displacements with respect to time can be neqlected. Therefore, Eqn. (4.11) can be readily simplified by neglecting the time derivatives of perturbed coordinates 5n, or
[K*]n;. = Tn  [I*]n; !_T[Pn*] . (4.12)
This equation represents the quasistatic approximation of system deformations. In most cases, such analysis is adequate for describing the manipulator's deflection due to inertia loads.
4.2 Variation of System Natural Frequencies Due to the Nominal Motion of the Arm
The effect of manipulator's gross motion on the
natural frequencies may best be illustrated by an example problem. Figure 4.2 shows the example manipulator and the
kinematic notation describing it. The manipulator is an open loop spatial RRRRRR mechanism, where R stands for revolute joint and RRR indicates that the three successive joint axes are parallel. The joint axes are described by vectors Sj, and ajk is the unit vector along the common perpendicular between Sj and ï¿½.k. The nominal motion of each joint is represented by the angle ej rotating about Si axis (this angle is measured in a righthand sense between aij and ajk) Parameters Sjj and ajk represent the offset distance between links aand ajk, and the link length between axes Si and ï¿½.k, respectively. The angle of twist between Sj and S.k is designated by angle ajk. The geometric parameters for this robot arm are shown in Table 4.1. In order to represent an illustrative formulation, only one flexible component at a time is considered. Figure 4.2 shows that the manipulator arm deflects about joint axis SI by an angle al. The flexibility of the joint is represented by a torsional spring KI. Therefore, this system has one deformational degree of freedom only. Figure 4.3 illustrates the instantaneous position of the arm which lies in the X*  Z* plane. The direction cosines of the joint and link axes for arm position are listed in Table
4.2. Furthermore, the associated mass of link jk is lumped at a point Cj (Fig. 4.3) on the link, the mass moments of inertia corresponding to the link are neglected in order to avoid computational complications.
The homogeneous equations of motion for this robot
arm can be obtained from Eqn. (4.11). Recalling that the nonlinear terms, sT[Pn*]A, are insignificant and ignoring the damping forces, we can write
[I*11;181 + (Gi 62 03 64 65 66) [R1*I 0 + K1a1 = 0.
0
0
0 (4.13)
Notice that Eqn. (4.13) describes the motion of a one degree of freedom system. Assuming that the nominal angular velocities of all joints are equal to a maximum frequency, Q, Eqn. (4.13) is reduced to
[I*]1;l I + Q2Q* l + KIl 0 . (4.14)
where
6
Q1* Z [R1*]i;1 (4.15) i=1
It is necessary to obtain the first and second order translational coefficients for mass centers Cj in order to evaluate [I*]1;1 and QI*. These coefficients are listed in Tables 4.3 to 4.9. Rotational geometric coefficients are not needed since the mass moments of inertia have been neglected. Now, using Eqn. (2.44), we obtain
[I*11;1 = 1600M2 + 6403M3 + 8103M4 + 8103M5 + 12104M6 .
(4.16)
Also, from Eqns. (4.10) and (3.71) we can evaluate the scalar components of [Rl*]
[RI*]I;j = 0
[RI*]2;1 = 2400M2  11.2M3  16M4  16M5  27M6 [Rl*13;1 = 4794M3 + 5390M4 + 5390M5 + 6580M6 [Rl*]4;1 = 3.2M4  3.2M5  12M6 [Rl*15;I = [RI*16;l = 0
(4.17) (4.18)
(4.19) (4.20) (4.21)
from which the value of Q1* is determined.
QI* = 240GM2 + 4782.8M3 + 5370M4 + 5370M5 + 6541M6 (4.22)
Comparing Eqns. (4.16) and (4.22), it is evident that the value of Q1* is much smaller than [I*11;. Therefore, we can consider the worst value of QI* which is
(4.23)
QI* Z [I*]I;I
The equation of motion can now be rewritten in the form
a, + QB1 + W1281 = 0
(4.24)
where wI is the natural frequency of the arm corresponding to its fixed configuration and it is defined by the equation
I i (4.25)
The nominal angular velocity of each joint can be defined as
= ï¿½W1 (4.26)
This equation implies that Q is proportional to the natural frequency by a factor E. The eigenvalue of Eqn. (4.24) is given by the equation
wdl = I2  22 = W12i  p2 (4.27)
Thus, when the arm is operating with input velocities of ï¿½EW, its natural frequency will drop fromw, to Wdl. Selecting an exaggerated value of 0.1 for 6 results in
Wdl = 0.995 w(
(4.28)
It is apparent from this equation that the effect of the manipulator's gross motion on the system natural frequency is insignificant. This procedure can be repeated considering any or all other flexible components for any configuration of the arm to arrive at the same conclusion.
4.3 Comensation for Deformations
To compensate for the manipulator's deformations, the overall change in the spatial position of the hand can be evaluated from the known deflection in each flexible component. On the basis of this known positional error, corrections can be made to the input command signals which move the hand to its desired position. This procedure to enhance the positioning accuracy of the hand can be said to make the manipulator "electronically rigid". In this way, system inputs are tuned to the operating conditions to enhance positioning precision. The concept of compensation for deformations by tuning the system inputs is presented in [21 for highspeed cam systems.
The overall change in the spatial position of the endeffector can be represented by a vector, Ajj, consisting of six components. This vector describes the deflection of a point on the endeffector by its first three components and the change in the orientation of the
hand by its last three components. Vector, AP can be obtained from the equation
G G = [J]S (4.29) N(N+1)
where [J) is known as the Jacobian matrix. Recall that an represents the deflections in the manipulator's inputs and pseudoinputs. However, only the actual inputs of the arm can be tuned to correct for the error in the position of the endeffector. Therefore, it is necessary to predict the required change in the nominal motion of the arm to produce a deflection, APj_, in the spatial position of manipulator's hand. Considering the nominal motion of the arm, we can write
AL1 = ([Rj]4.30)
P_ = N(N+l)
Matrix [Rj] represents the Jacobian of the rigidbody mechanism and vector AD is the required correction vector. For a six degree of freedom manipulator arm, the correction values can be obtained from
L = [RJ]1 APH
(4. 31)
This should give good results when the deformations are small and the Jacobian matrix, [RJI, is not near a singularity. For manipulators with redundant degrees of freedom, the pseudoinverse of the Jacobian can be used, or
([Rj]T [Rj] )I (RjT 6PH (4.32)
Next, the tuned values for the system inputs, TD, can be obtained from
T = (4.33)
This is a relatively simple type of compensation which shows that, it is possible to electronically correct for flexibility in a manipulator system if the system loads and dynamic parameters are known.
Figure 4.1. Kinematic Representation of Oscillating Manipulator.
 vector directed into the page
 vector directed out of the page
a12
S66
Figure 4.2. Kinematic Representation of the Example Manipulator.
y(3) /(3) k.S",z
(40,0,60)
y(6)
(80,0,30) '
(90,0,30)
Figure 4.3. Arm Position for the Example Analysis.
(0,0,30)
P6z(6)
89
Table 4.1. Geometrical Parameters for the
Example Manipulator.
SJJ(in.) ajk(in.) jk(deg.)
1 30 0 90 2 0 50 0 3 0 50 0 4 0 10 90 5 0 0 90 6 20 0 
Table 4.2. Manipulator Position for the Example Analysis.
. ajk Rjk (in.) a (deg.)
1 (0,0,1) (1,0,0,) (0,0,30) 0.0 2 (0,i,0) (.8,0,.6) (0,0,30,) 36.9 3 (0,1,0) (.8,0,.6) (40,0,60) 73.7 4 (0,i,0,) (1,0,0) (80,0,30) 36.9 5 (0,0,l) (0,1,0) (90,0,30) 90.0 6 (1,0,0) (0,1,0) (110,0,30) 180.0
Table 4.3. First Order Translational Influence Coefficients
for Mass Centers C
F0 0 0 0 0 01
G] 0 0 0 0 0
0 0 0 0 0 0
L0 30 0 0 0 0
[G 2] = 40 0 0 0 0 0
0 40 0 0 0 0
ro 0 30 0 0 0]
0 80 40 0 0 0
o 0 30 0 0 0
[G C4] L90 0 0 0
0 90 50 10 0 0
[0 0 30 0 0 0
[G C5] 90 0 0 0 0 0 90 50 10 0 0
0 0 30 0 0 0
[G C6] L110 11 0 0 20 0
0 10 70 30 0 0
Table 4.4. Second Order Translational Influence Coefficients
for Mass Center C1
0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0
o o 0 0 0
o0 0 0
0 0 0 0
0 0
0 0 0 0 0 0 0 HC0 0 0 I 0
0 0 0 0 0 0
0 0
0 0
Symmetric 0 0 0 0
10
0
0
Table 4.5. Second Order Translational Influence Coefficients
for Mass Center C2
40 0 0 0 0
0 30 0 0 0
0 0 0 0 0
40 0j 0 0
0 0 0 0
30 0 0 0
0 0 0 0 0 0
_o 0 0

Symmetric
0
0
0
0 I 0 0 0 0
0 0 0 0 0 0
FHC6 C =

Full Text 
PAGE 1
DYNAMIC ANALYSIS AND PARAMETRIC IDENTIFICATION FOR FLEXIBLE SERIAL MANIPULATORS BY FARIBORZ BEHI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA Â«/ 1985
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ACKNOWLEDGEMENTS The author wishes to thank his committee chairman, Professor Delbert Tesar, for his support and guidance. He is also grateful to the members of his graduate commitee for their guidance and encouragement. Special thanks go to his family and friends for their encouragement and patience. He expresses his sincere appreciation to his mother, Badri Behi, for providing financial and moral support throughout his education. Finally, thanks to Suzan Hunter for her excellent typing. This work was funded by the office of Naval Research precision modeling grant No. N0001483K0592. 11
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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii ABSTRACT vi CHAPTER I INTRODUCTION ' 1 II DYNAMIC MODEL OF FLEXIBLE MANIPULATOR ARMS.. 10 2.1 Kinematic Representation of Serial Manipulators 11 2.2 Rate of Change of Orientation and Position 16 2.3 Basis for the Mathematical Model 20 2.4 Dynamic Model of Serial Manipulators With Flexible Joints 22 2.5 The Secondary Flexible Components of a Serial Manipulator 32 2.6 Dynamic Model of Serial Manipulators With Flexible Links 33 2.7 Improvement of System Design 35 III ANALYSIS OF MANIPULATOR DYNAMIC RESPONSE.... 42 3.1 Forced Vibration 43 3.2 Decoupled Equations of Motion 46 3.3 Dynamic Response in the Frequency Domain 51 3.4 System Response Due to Dynamic Shocks.. 54 3.5 Nonlinearities in the Equations of Motion 5 6 3.6 Order Analysis 65 iii
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Page VI ARM DEFLECTION DUE TO THE GROSS MOTION 7 2 4.1 Complete Dynamic Model for Manipulator Arms 7 3 4.2 Variation of System Natural Frequencies Due to the Nominal Motion of the Arm. . . 78 4.3 Compensation for Deformations 83 V PARAMETRIC IDENTIFICATION .' 98 5.1 Experimental Modal Analysis 99 5.2 Parametric Identification for Manipulator Arms 106 5.2a Identification of Local Stiffnesses 109 5.2b Â• Identification of Local Masses... 114 5.2c Identification of Local Damping Coefficients 117 5.3 Verification of the Identified Parameters 113 5.4 Instrumentations 120 5.5 Improvement of System Design 122 VI ILLUSTRATIVE EXAMPLES 146 6.1 Description of Example Manipulator 146 6.2 System Natural Frequencies and Transfer Functions 148 6.3 Manipulator's Dynamic Response 149 6.4 Deformation Due to the Gross Motion.... 152 VII CONCLUSION 138 APPENDICIES A ARM ACCELERATION AND SECONDORDER INFLUENCE COEFFICIENTS 192 IV
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Page B EXPRESSIONS FOR INERTIA POWER MATRIX... 197 C VARIATION OF THE INFLUENCE COEFFICIENTS DUE TO MANIPULATOR'S DEFORMATION 201 REFERENCES 204 BIOGRAPHICAL SKETCH 207 V
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DYNAMIC ANALYSIS AND PARAMETRIC IDENTIFICATION FOR FLEXIBLE SERIAL MANIPULATORS By Fariborz Behi December 1985 Chairman: Dr. Delbert Tesar Major Department: Mechanical Engineering Vibrations in the small are low magnitude, oscillatory deformations about the mean motion equilibrium as distinct from the deformations in the large which result from external loads and large but slowly varying inertia loads. Vibrations in the small arecaused by transient phenomena such as system shocks introduced when starting or stopping a machine such as a manipulator arm. The lowest natural frequency of vibration is one of the best indicators of the overall structural integrity of the manipulator system and is an indicator of the maximum acceptable speed for performing precise arm motions . This dissertation presents a new formulation for the dynamic model of flexible manipulators. A lumpedparameter arm model is used for tractability . This allows the dynamic equations of motion to be written as a set of coupled ordinary differential equations instead of partial VI
PAGE 7
differential equations. For many manipulator designs, most of the deformation energy appears in the actuator and drive train components. Therefore, an initial model considers flexibility at the joints in the direction of relative joint motion only. The natural frequencies and their corresponding natural modes are determined from an eigenvalue problem obtained from the equations of motion. This formulation is then expanded to include other system flexibilities by introducing pseudoinputs to represent their motion. Based on analytical modal analysis a solution algorithm is presented to estimate the dynamic response of the arm due to deterministic exciting forces. This enables the designer to obtain the response of the arm in frequency domain by considering a harmonic exciting force applied to the system. Although the inertia loads due to the gross motion of the arm have small effect on the natural frequency of the system., they cause the system to deform due to the existence of flexible components. Therefore, the mathematical formulation is extended to include both small and gross motion of the manipulator. Finally, an experimental procedure is introduced to determine the mass and stiffness parameters of the manipulator in order to represent a true model of the actual system. vii
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CHAPTER I INTRODUCTION The combination of computer control and sophisticated multidegree of freedom mechanical systems has evolved into a new discipline referred to as robotics. The potential for this discipline to greatly increase production and improve its quality has brought our social structure to the verge of a modern industrial revolution. Achieving this goal demands enormous effort to develop robot technology. Although manipulator arms have been employed in a wide range of industrial applications, exploitation of these mechanisms to their maximum capacity is yet to come. The ability of existing manipulator arms is limited because of technological barriers in (a) "real time" computational control, and (b) design procedures [1].* In parallel with the need for "real time" computations, mathematical formulations are necessary to incorporate complex geometries and dynamic characteristics in the design process. Current manipulator designs maintain * The numbers in brackets refer to references at the end of this document. 1
PAGE 9
2 simple geometries and are developed based on trial and error procedure, which is both time consuming and costly. Alternative geometries are much needed to extend manipulator capability for performing more sophisticated tasks, such as obstacle avoidance en route to a destination. In addition, system dynamic properties must be considered in the design procedure to improve the manipulator's operational capacity and precision. An issue of extreme importance in many manipulator tasks is the precise positioning of the arm's endeffector during movement. As speed of operation increases and the manipulator stiffness decreases, due to a need to decrease the mass of the moving parts, the flexibility of the arm generates inaccuracy in the arm motion. There are two distinct types of structural deformations which cause the arm to deflect about its equilibrium position. These are referred to as def orm.ations in the large, and vibrations in the small [1]. Deformations in the large are nonoscillatory deflections, which are caused by external loads and inertia loads due to the gross motion of the arm. In operation, the links of the manipulator must be constantly accelerated and decelerated, leading to large inertia loads and corresponding deformations. Superimposed on this gross motion and deformation will be small scale oscillations about the equilibrium. These are the vibrations in the small. This dissertation presents a
PAGE 10
3 development of mathematical formulation for analysis of manipulator dynamic behavior, considering its flexible components . In general, structural integrity is a primary concern in manipulator design. The most direct way to reduce structural deformations is to increase system vibrational frequencies; specifically the fundamental frequency (lowest natural frequency), which is an excellent indicator of the manipulator's structural integrity. Chapter II formulates the vibratory motion of the arm in its stationary configurations from which the natural frequencies and corresponding mode shapes are deduced. In addition, based on Rayleigh's principle, the fundamental frequency is described explicitly in terms of local parameters to demonstrate the effect of each parameter on the magnitude of fundamental frequency. Not only is this frequency an indicator of system rigidity, it also provides a basis to classify the mechanism's operating speed. A manipulator's gross motion is described by a parameter, uj, which indicates the rate of moving a load from one position to another and back. This parameter itself does not indicate whether a system is high speed or not, since what is considered as high speed for a particular mechanism may represent a low speed for another system. Yet, the comparison of u with the system's fundamental frequency provides a criterion for dynamic
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4 classification of mechanisms. This criterion for single degree of freedom systems has been discussed in detail by D. Tesar and G. K. Matthew [2]. Although the source and magnitude of excitation phenomena are extremely difficult to accurately quantify in general, the system response due to deterministic excitations, such as impulsive forces, offers valuable information about system reaction to dynamic shocks in the motion of manipulator arms. A linear formulation is presented in Chapter III to analyze system dynamic response under applied exciting loads. Also, the effect of nonlinearities, such as Coriolis and centripetal accelerations on the system response, is discussed in this chapter. As mentioned previously, increasing the speed of a manipulator generates larger inertia forces which cause the arm to deform about its equilibrium position. To predict arm deflection due to inertia loads, the equations of motion are derived in Chapter IV, considering both gross motion and arm deformation, along with the coupling between the two motions. Ultimately, these deformations may be compensated for by modifying the nominal actuator commands . Accurate analysis of flexible arm behavior requires precise knowledge of system mass and stiffness properties. These parameters are commonly determined from the
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5 continuously distributed model of the link components, or finite element methods using the mechanical properties of system elements. Manipulator link components, in general, are not uniform beams; therefore, the continuous model falls short of being practical. This problem is overcome by dividing the link elements into many small segments using finite element methods; however, this approach increases the system's representative degrees of freedom, leading to complication in analysis and numerical computations. An experimental procedure is proposed in Chapter V to obtain system mass and stiffness parameters based on the modal analysis concept. This approach offers an accurate representation of the arm since the dynamic parameters are identified so that the model behavior will match those of the actual system. Furthermore, these parameters are obtained from the prototype mechanism in its assembled form. Therefore, the effects of joint link connections on the component flexibility are included in the identification process. In order to illustrate the application of these formulations. Chapter VI contains analysis of an example involving a flexible manipulator arm. During the past few decades, considerable attention has been devoted to the dynamics of highspeed flexible mechanical systems. Two remarkably complete literature surveys [3,4] present a review of research conducted on
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6 this subject prior to 1972. Imam, Sandor, and Kramer [5] present a deflection analysis of planar linkages, where the system mass and stiffness matrices are constructed, utilizing the permutation vector method of structural analysis, and frequent recalculation of eigenvalues is avoided by using the "Rate of Change of Eigenvalues Method." Several publications [610] utilized the lumpedparameter approach for dynamic analysis of flexible planar linkages. Kohli and Sandor [11] extended this approach to investigate dynamic behavior of spatial mechanisms with elastic components. The coupled, nonlinear, partial differential equations of motion for a slider crank were developed by Chu and Pan [12] to include the transverse and longitudinal flexibility of the coupler link. In 1978, Sanders and Tesar [13], based on their analytical and experimental research, argued that deflection of mechanisms with elastic members can be predicted by a quasistatic method (higher time derivatives of small motion are considered to be negligible) if the mechanism operating speed is less than five percent of the system fundamental frequency. Hains [14], through an independent investigation, confirmed the conclusions drawn by Sanders and Tesar. An analytical and experimental investigation was conducted [15] to study the dynamic behavior of a
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7 highspeed planar four bar with elastic members. The equations of motion are derived using the assumed mode method by modeling the four bar as a cantilever beam (crank), connected with two simply supported beams representing the coupler and follower links. Dubowsky and Maatuk [16] present a formulation for the dynamic equations of motion for flexible manipulators. The manipulator's links are modeled as a series of pointconnected, slender beams, using Euler beam theory. System equations of motion are then derived by application of Lagrange's formula. As mentioned previously, this formulation is limited to mechanisms with uniform link components. Furthermore, the significance of system natural frequencies in the analysis of flexible mechanisms has not been addressed. The transfer m.atrix method is utilized [17] to develop a formulation for the analysis of flexible planar manipulators. The dynamic behavior of system components is described by constructing matrix elements which relate the motion and internal forces at one end of each component to those of the other end. The system transfer matrix is then built up while traversing the sequence of attached Bernoull iEuler beams and bodies. The manipulator's natural frequencies and frequency response are obtained from the system transfer matrix and mechanism boundary conditions. This method neglects Coriolis and
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8 centripetal accelerations for determination of system response. The transfer matrix approach is extended to spatial manipulators in [18] using a lumped mass model. By this procedure a CRL masterslave manipulator is investigated. The effects of configuration and payload on frequency response are studied both analytically and experimentally. Tesar and Thomas [19] developed a lumped parameter model for planar three degree of freedom manipulators to determine system natural frequencies. Equations of motion are derived in terms of absolute deflection of mass elements using a Newtonian formulation. An analytical method is presented [20,21] for the dynamics of spatial mechanisms containing complexshaped, flexible links. The mass and stiffness properties of links are represented by a set of matrices deduced from finite element methods. These parameters are defined in the moving coordinate systems attached to the links. Therefore, they are invariant with respect to variation of the mechanism's configuration. Equations of motion are derived using Lagrange's formula and the 4x4 matrix dynamic analysis technique. System motion is defined in terms of absolute perturbed motion of finite element nodal points. This paper provides a significant and beneficial analytical formulation for dynamics of flexible mechanisms; however, it is not a practical procedure for
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9 considering system flexibility in the design process. Although component mode synthesis coordinate reduction is introduced to decrease the system's degree of freedom, numerical integrations are cumbersome. This document presents a lumped parameter model for dynamic analysis of flexible serial manipulators. The equations of motion are defined in terms of relative perturbed motion of mass components using Lagrange's formulation. Not only is this formulation relatively efficient for numerical integration, it also provides an explicit representation of system fundamental frequency in terms of local mass and stiffness parameters. Therefore, the effect of component flexibility and inertia on the system's vibration can easily be investigated through this formulation .
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CHAPTER II DYNAMIC MODEL OF FLEXIBLE MANIPULATOR ARMS The main objective of this dissertation is to formulate the vibration behavior of manipulators, therefore providing a basis to include the system's flexibilities in the design procedure. Precision of the manipulator's operation must be assured in order to acquire a high quality product. Thus, it is required to design the system so that vibratory motions are diminished under operating conditions. The most direct way to achieve this goal is to keep the frequency content of exciting phenomena, produced by the system's operational function, far below the system's fundamental frequency. Therefore, the fundamental frequency can be utilized as a design criteria to increase system rigidity, leading to precision at higher operational speeds. This chapter presents a mathematical model for dynamic analysis of flexible serial manipulator arms. Of particular interest is computing the mechanism's natural frequencies, specifically the lowest one (fundamental frequency), and the corresponding mode shapes. This formulation is simple to implement on digital computers 10
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11 and allows the designer to consider only the dominant, flexible components; enabling him to select the simplest, yet most accurate, representation of the mechanism to eliminate unnecessary computations. The present work is based on the formulation developed by Thomas and Tesar [22] , for dynamic analysis of manipulators with rigid elements. The kinematic representation of serial manipulators developed in [22], is briefly reviewed here. 2 . 1 Kinematic Representation of Serial Manipulators A serial manipulator is an assembly of links and revolute or prismatic joints (most of the lower kinematic pairs are a combination of these joints), as shown in Fig. 2.1. The motion of each joint is defined by a parameter, representing a relative angular displacement, 0,^, about the joint axis, Sj^, (revolute joint), or a relative axial motion, S^, along Â£n axis (prismatic joint). Two successive joint axes, and are connected by a link member along their comm.on perpendicular vector, ^j]<. Parameters, aj}j and which are the distance along and the angle of twist about between and define the link elements, as shown in Fig. 2.1. To describe the manipulator's motion, a global Cartesian coordinate system, along with a set of local coordinate frames, are
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12 are chosen. The global system is a fixed frame in space located at the first joint with axes X*, Y* , and Z* , where Z* is selected along joint axis, S_i . The local frames are a set of Cartesian reference coordinates fixed to the link elements with X^^^ lying along and Z^^^ along S_ j . Thus, the direction cosines of and Â£j in terms of the jth coordinate system, are Â£j]^ (1,0,0) and Â£j (0,0,1). A body in space is completely defined by its orientation and the position of a point on the body. The orientation of link jk is described by the direction cosines of X^^^, Y^^^, and Z^^^ in terms of the global frame. The direction cosines of an arbitrary vector Q (Xq*, Yq* , Zq*) can be obtained by a rotational transfer of its direction cosines (Xq^^^ defined in the jth local reference frame. Let Tj designate a 3 x 3 rotation matrix which relates the jth frame to a coordinate system with common origin to the jth frame and with axes parallel to the global coordinates. We can write Q = Tj ^ ^ ( 2 . 1 )
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13 Therefore, the direction cosines of Â£jk (Xj]<*, Yj)r*, Zj]^*) and Sj (Xj*, Yj*, Zj*) in terms of the fixed reference frame are ( 2 . 2 ) (2.3) Substituting the values of Â£ j ^ ^ ^ into Eqn. (2.2) and (2.3), (j) Â£jk Â“ '^j Â£jk S j = T j S j ^ ^ ^ Â£jk = T1 0 0 Sj = Tj (2.4) (2.5) Equations (2.4) and (2.5) imply that the first and third columns of Tj must be formed of vectors, Â£jk and S_j , ( j ) respectively. Furthermore, Y in a right hand coordinate system, is formed by a vector crossproduct of Z ^ ^ ^ and ^ ^ y(3 ) ( j) X X (j) j ^ Â£ j k ( 2 . 6 )
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14 The vector crossproduct, Â£j x aj>, forms the second column of matrix T j , since direction cosines of in terms of the jth local reference frame are (0,1,0). The rotation matrix Tj can then be represented as Tj = [Â£j]^, S_j X Â£jk, Â£ j ] (2.7 The direction cosines of S_j and can be obtained recursively from the initial direction cosines Si (Xi*, Yi*, Zi*) = (0,0,1) ( 2 . 8 ) Â£12 (^12*^ ^i2*r Zi2*) = (C01, S01, 0) (2.9) and the following expressions ij = ^j1 0 Sa Ca ( j1 ) j ( 2.10 " ^j1 C0j Ca Sa SQa (jl)j ^ . . ^0 i (D1): ^ ( 2 . 11 ) where S and C represent the trigonometric functions sine and cosine.
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15 Now, let vector Rj define the position of the origin of the local frame in the fixed coordinate system. This vector is evaluated by a vector summation of the joint offsets and link lengths as shown here. Double subscripts for joint offsets represent a fixed displacement, however, it must be replaced by where this value is not constant. The position of a point, P, fixed in the jth reference frame is defined in terms of local coordinates P (Xp* ,Yp* , Zp* ) designates the position vector of point P in terms of global coordinates, which can be determined transformation involves a translational transfer of the local frame's origin, and a rotational transfer of local coordinate axes. ( 2 . 12 ) by a vector P ^ ^ ^ (Xp^^^, Yp^^\ Vector by transforming p ^ ^ ^ to the global frame. This P = R j + T j P ^ ^ ^ ( 2 . 13 )
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16 2 . 2 Rate of Change of Orientation and Position The rate of change of orientation of a rigid element is defined by a vector, oi wY, ai^), representing the angular velocity of the element about the axes of a fixed coordinate system. The absolute angular velocity of link jk in a serial manipulator can be determined simply by summing the relative angular velocities between successive links preceding link jk. link jk relative to link ij. The relative angular motion between links ij and jk is the angle 0 j , which is constrained to rotate about the Sj axis,' therefore, the relative angular velocities may be written as where the dot represents the derivative with respect to time. Substituting Eqn. (2.15) into (2.14) results (2.14) where represents the angular velocity of (2.15) iiijk " 0lÂ£l Â©2^2 + Â•Â•Â• + 0jSj (2.16)
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17 Now, Eqn. (2.15) can be written in a matrix form in terms of generalized motions, 0^/ and first order rotational geometric coefficients [Gj]^] Â“jk = tGjkii (2.17) where [Gj]^] is a 3 x N (N denotes the number of degrees of freedom in the system) matrix and its column vectors are n awjk 3$n (2.18) Notice that the subscript n designates the nth column of matrix [Gjj^] and notation [Gj]^]j^ is chosen in place of the common notation [Gj)^].^* From Eqns. (2.16) and (2.18) the column vectors of [Gjj^] are found to be n Sn 0 n Â£ j ; nrevolute otherwise (2.19) The rate of change of position or translational velocity of a point, P, fixed in link jk, is the time derivative of position vector P
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18 Xp g = =s,s, + Â£_ (TjP dt (j) ( 2 . 20 ) The time rate of change of a vector Q fixed in a rigid body jk is defined [23] as <3Q = X Q (2.21 ) dt Unit vectors, and vector is fixed in link jk. Considering this definition, we can rewrite Eqn. (2.20) in the form j . Vp = E 3n^ n=1 + i Â®nin^ )j,( j,_i ) 2, + TjP( j ) ] ( 2 . 22 ) Notice that the terms inside the second summation of Eqn. (2.22) represent a vector from the origin of nth loOal reference frame to point P, since
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19 n Sn = Sllil + <"u1)Â£ um ^ (2.23) ^ " J 2 '"(t1)Â£ H^U. " * ^dÂ£'^Â’)(2.24) i Sn = ,X,<"(21)Â£ ^Â£l)Â£ " Â’'jiÂ‘Â’'(2.25) Eqn. (2.22) is then reduced to Xp 3 = I n= 1 t2aÂ§.n * Â®nÂ§ji X (P Rn)] (2.26) Introducing the first order translational geometric influence coefficients [Gp] and generalized motions $n/ the velocity of point P can be written in the matrix form Vp = [Gp] $ . (2.27) Recalling that is equal to Q^, if joint n is revolute, and it is equal to Sp if joint n is. prismatic. By inspection of Eqn. (2.26), first order translational coefficients can be defined as
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20 , n <_ j ; nrevolute , n <_ j ; nprismatic (2.28) otherwise . Rotational and translational velocities for parallel mechanisms can also be defined in terms of first order geometric coefficients; however, the definitions of these coefficients will be more complex than definitions introduced for serial mechanisms. The analysis of parallel mechanisms is beyond the scope of this paper, although the present work may be extended to include these systems . 2.3 Basis for the Mathematical Model The motion of manipulators with rigid components is described by highly nonlinear, coupled, differential equations in terms of the actuator's motion. However, the vibratory motions of robot arms are low magnitude, oscillatory deformations about the mean motion equilibrium caused by small relative deflections of the elastic elements. Furthermore, the representative frequency of operation of the arm, co, is kept significantly lower than the system's fundamental natural frequency. This prevents the dynamic coupling between the small and gross motion of the arm since many vibration cycles will occur about the
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21 mean motion before any significant change in the arm configuration occurs (the detailed discussion of coupling between small and gross motion is presented in Chapter IV) . In addition, the small motion assumption causes the nonlinear terms in the equations of motion to become insignificant. It is then sufficient to describe the vibratory motion of the manipulator by a set of linear, coupled, differential equations in terms of the perturbed motion of the arm about its stationary configurations. A lumped mass arm model is used for tractability . Not only does this produce a set of ordinary differential equations of motion instead of partial differential equations, it also provides an accurate description of system energy content. The distributed mass model is not chosen since the manipulator's links, in general, have complex geometric shapes. Because of this complexity in geometry, the distributed mass theory for beams fails to describe the mass content of the link accurately. Furthermore, the kinetic energy contribution of elements, such as electrical motors, hydraulic actuators, and drive trains, which are very significant, must be included in the total energy content of the system. This requires a mixed distributed lumped formulation, leading to a set of P^J^tial and ordinary differential equations describing the system's motion. Dynamic analysis of flexible manipulators is a complex problem; such formulation creates even more complication in this analysis.
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22 The finite element method can be utilized to describe mass distribution of a complex shaped element by representing the element as a series of small uniform elements connected to each other. A system's degree of freedom will increase rapidly by using this method, which will lead to a large system of equations describing the manipulator's motion. In conclusion, the lumped mass model provides an accurate representation for most precision manipulators since the arms do not generally have long, flexible links, such as those found in the shuttle manipulator. An experimental procedure will be introduced in Chapter V to identify the concentrated mass and stiffness parameters, providing an accurate description of the manipulator's mass and flexibility content. 2 . 4 Dynamic Model of Serial Manipulators with Flexible Joints For many manipulator designs, most of the deformation energy appears in the actuator and drive train components. Therefore, an initial model considers flexibility at the joints in the direction of relative joint motion only. The corresponding mass of link jk is represented by an inertia ellipsoid [23] centered at point Cj with principal axes parallel to the j th local reference frame
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23 as shown in Fig. 2.2. Vector Cj defines the undeformed position of mass center in terms of the global reference frame . Consider a serial manipulator with rigid components. The time derivative of vector is obtained from Eqn. (2.27) j ] ^ . (2.29) Using the mathematical definition of the time derivative, Eqn. (2.29) can be rewritten as ^ = [Gcj] ^ (2.30) At At Canceling At from both sides leads to A^j ~ [^^c j ^ A^ Â• (2.31) This equation describes the variation in the position of mass center C j , due to the small relative motion of the joints. Vector A$ portrays the elastic deformation of the
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24 joints about their axes, and may be replaced by a set of relative perturbed motions, 6^, as shown in Fig. 2.2. The deformed position of point Cj is defined by a vector C j , (the hat throughout this dissertation designates a vector which is displaced from its original line of action due to the deformation of the flexible components). Considering Fig. 2.2, it is evident that The equations of motion for a mechanical system with N degree of freedom can be obtained from Lagrangian equations where KE and PE are the total kinetic and potential energies of the mechanism. The generalized position coordinates are designated by q^ and their corresponding generalized forces are Q^* The velocity of mass center Cj is found by differentiating Eqn. (2.32) with respect to time C j = C j + /VCj = C j + [GcjlB (2.32) (2.33)
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25 dCi dCi d Â£j = _ = _ + (__ [Gcj] ) 3 + [Gci] B . (2.34) dt dt dt Recalling that the manipulator arm is in a fixed configuration, and also assuming the variation of the first order coefficients due to the deflection of the arm is negligible, we can rewrite Eqn. (2.34) in a simplified form as Cj = [GcjlB(2.35) The absolute angular velocity of the link jk can be obtained from Eqn. (2.17), A, Â• Â“jk = [GjklB. (2.36) The manipulator's kinetic energy is obtained from the sum of the kinetic energies produced by the motion of inertia mass ellipsoids N ; A ^ ^ l{Â«j Â£jÂ£j + Â“jk^ [njk: j = 1 2 ^ KE (2.37)
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26 where Mj is the total mass of link jk concentrated at point Cj, and [Hjk^ designates the inertia tensor about the centroid for link jk in terms of the global coordinates. The values in this inertia tensor depend on the orientation of link jk. However, the inertia tensor about the centroid, defined in terms of the local coordinate axes, is a constant diagonal matrix for each link, which depends only on mass distribution of the link. To derive an expression for [lljkl , recognize that the kinetic energy contribution is independent of the coordinate system so ( j) (j) iiijk^ wjk ~ ^Â“jk [njk ] wjk (j) (2.38) The absolute angular velocity of link jk in terms of the global coordinates can be found from liijk ~ 'i'j (iijk ^ ^ ^ Â• (2.39) Since Tj is an orthogonal matrix, the inverse of this matrix is equal to its transpose. So we can write fiijk ^ 2 ) _ m . T r, . , Â“ ^3 (Â£lk (2.40) Substituting Eqn. (2.40) into (2.38) results in
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27 wjk'^ tlljk] wjk ~ '^j'^ wjk* (2.41) From which, the inertia tensor, [Iljkl / is defined as [Kjk] = Tj [Hjk^^ TjT. (2.42) To express the system's kinetic energy in terms of the first order influence coefficients, Eqn. (2.37) is rewritten using Eqns. (2.35) and (2.36) KE = 1 E Mj3^ [Gcj]^ [^cj^i + [Gj]^]T [Jlj]^] [Gj^Jg 2 j = 1 = 1 [I*]6 (2.43) 2 The equivalent inertia matrix of the system is represented by the N x N matrix [I*], which defines the effective mass of the system as seen at each joint. The components of this matrix are obtained from the general equation N [I*] Â“ E f'^cj^m'^ f*^cj]n t jk^ [*^jkln (2.44) where (1*1 designates the scalar component of the m; n inertia matrix located at mth row and nth column.
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28 The potential energy of a manipulator in a deflected configuration is the sum of the strain energies of the springs describing the arm's flexible components N N PE = j_ E E KijSiei = 1 6 T [K*]6 (2.45) 2 j=1 i=1 2 where Kj_j are the stiffness coefficients and [K*] is the equivalent stiffness matrix. Recalling that the manipulator's deflection is due to the relative deformation of joints in the direction of joint motion. The system's flexible components can be represented by a set of torsional springs attached to the revolute joint axes, and translational springs attached to the prismatic joint axes (Fig. 2.2). The energy content of each spring is due to the displacement of the corresponding joint only; therefore, there is no coupling in the system's potential energy, or K K 3J 13 1 = 3 i?^j (2.46) Notice that the gravitation energy will not be included since the effect of this energy can be analyzed statically.
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Selecting the generalized coordinates, the Lagrange equations for free vibration of an undamped mechanical system can be rewritten in the form. d 3KE 3 k: ^ 3KE 3PE (2.47) N. Substituting the final expressions for the kinetic and potential energies into Eqn. (2.47), the equation of motion can be written as The subscript, n;, designates the nth row of a matrix. The first term on the right side of Eqn. (2.48) can be expanded as shown here Recalling that [I*] is a function of the local mass parameters and first order geometric coefficients. Also, these coefficients are assumed to be invariant with respect to small displacements 8n' second term on the left side of Eqn. (2.48) and the first term on the right side of Eqn. (2.49) can be neglected. The nth equation of motion is then reduced to (2.48) (2.49)
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30 [I*]n; B + [K*ln; 6 = 0(2.50) Next, the complete system's equations of motion can be written in the matrix form [I*] g + [K*] 3 = 0. (2.51 ) Vector ^ describes the oscillatory motion of the manipulator's joint angles, which can be defined by a harmonic equation as ioj^t B = B e (2.52) where vector ^ is the magnitude and is the frequency of the joints' vibration. Substituting Eqn. (2.52) into the equations of motion gives ioOj^t {tl*] + [K*]}Be = 0. (2.53) For nontrivial B, the coefficient matrix must be singular so its determinant must be zero. + tK*] = 0 (2.54)
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31 Eqn. (2.54) can be rewritten in the form [K*]'' [I*] 1 [I] = 0 (2.55) where [I] is an N x N identity matrix. This is a standard eigenvalue problem, which can be solved to determine the system's natural frequencies, (jj^, and the corresponding eigenvectors Notice that [K*] is a diagonal matrix and simple to invert. In general, there is no closed form solution to Eqn. (2.55) and it must be solved numerically. There are several numerical techniques which can be utilized to evaluate the eigenvalues and eigenvectors of a system of equations. A matrix iteration scheme, using the power method in conjunction with the sweeping technique [2427], is found to be most practical to determine the first few eigenvalues and eigenvectors of a large system of equations. Also, an iteration scheme based on the QR method [2728] is best suited to evaluate all of the system's natural frequencies and the corresponding mode shapes.
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32 2 . 5 The Secondary^ Flexible Components of A Serial Manipulator The dynamic model of a manipulator with flexible joints may not describe the system's deformations adequately if the deformation energies contributed by other flexible components are significant. The most prominent deflection modes of a jointlink combination (Fig. 2.3a) are a) the motion about the joint axis, as discussed previously; b) the deformation of the joint bearings about two orthogonal axes perpendicular to the joint axis; and c) the torsional twisting of the link about its major axis. The joint deflection is described by an input, 6 i, about joint axis, S  . Two pseudoinputs, 32 63 ' introduced to describe the bearing deflections and the torsional deformation of the link (psuedoinputs are imaginary inputs that produce small motions only). Parameter, 32' represents the combined link deflection and joint bearing deflection perpendicular to about S_2 2.3b). The combined bearings' deflection and link's torsional deformation about an axis, ^ 3 , perpendicular to ^ and S2r ace described by the pseudoinput 33 . The mass content of this jointlink combination is lumped at the imaginary link, a 34 , so that the motion of 3 i , 33 ^ and 3 3 will contribute to the kinetic energy of this
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33 component (Fig. 2.3b). Now, the system's equations of motion can be obtained by following the same procedure as described in Section 2.4. The length of vector ^ will increase to at most 3N (often, fewer than three inputs are needed for a sufficiently accurate representation). The components of inertia matrix, [I*] , can be obtained from Eqn. (2.44), but the size of this matrix will increase by a factor of three. Notice that the 3N x 3N stiffness matrix [K*] will remain diagonal. 2 . 6 Dynamic Model of Serial Manipulators with Flexible Links This section expands the formulation of Section 2.4 to include the flexibilities of the manipulator's links in addition to those of joint components. A general jointlink combination is shown in Fig. 2.4a. This component is shown in its rigidbody position (solid lines), as well as in its elastically deformed configuration (dashed lines). The elastic deformations of the jointlink combination may be completely described by the input, 6 i (joint deflection), and six pseudoinputs , &2 through 67 (link deformations), as shown in Fig. 2.4b. The flexibility of each jointlink element can be represented by a 7 x 7 stiffness matrix, [Kjj^] , where subscript, jk, designates link jk. The length of vector
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34 _3r will increase by a factor of seven. Considering the strains associated with displacements, , the system's potential energy is PE [K*] B (2.56) where [K*] is a 7N x 7N matrix of the form [K*] = [K23] CK23] [K 34 ] 0 ^^N(N+1 ) ^ (2.57) Recalling that the size of the inertia matrix will increase by a factor of seven, its scalar component can be obtained from Eqn. (2.44). Finally, the manipulator's equations of motion are derived using the formulation of Section 2.4. For a general six degree of freedom flexible manipulator arm, the motion is described by fortytwo differential equations. Thus, it may be difficult to compute all the system's natural frequencies. However, higher natural frequencies have little or no effect on system vibration, since they are higher and smaller in magnitude than the fundamental frequency. It is then
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35 adequate to evaluate the first few eigenvalues using the power method. 2 . 7 Improvement of System Design The performance and operating speed of a system can be improved by increasing its fundamental natural frequency. This will reduce the robot's flexibility, leading to less deflection of the arm due to the applied loads and inertia forces. An approximation for the lowest eigenvalue of a system of equations is given by Rayleigh's quotient 0)1 2 ~ (u)T [K*] (u) (u)T [I*] (u) (2.58 An estimate for the fundamental frequency can be obtained from Eqn. (2.58) if the arbitrary vector Â£ resembles the first mode of vibration. If the vector u coincides with the system fundamental mode (1 ) we can write . _ (1( ^ [K*] (1( ^ ) Ki 0 )Â— ^ (^(1))T [ 1*1 (.^(1)) Ml (2.59)
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36 Parameters, K and M] , designate the modal stiffness and modal mass of the system. Clearly, the magnitude of (jjj , can be increased by decreasing M] and/or increasing K . To demonstrate the effect of the local stiffness and mass parameters on the system fundamental frequency, K and Mt must be defined explicitly in terms of these parameters, as follows: Kl N N = Z Z i=1 j=1 K ID (1 ) (2.60) _ N Ml = Z MjDj (2.6 1) j = 1 where 'P ( 1 ) and denotes the ith component of N N Dj = Z Z (2.62) j=1 Â£=1 [Fj] is an N X N matrix, which is constructed from [Fj] = [G'i'.jjT + [GÂ’4^j]^]T [Ejk] [G'^^j^]. (2.63)
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37 The column vectors of [Gi; ] matrices are defined as [ cj^n~'i^n C^cj^n (2.64) ['^Â’l^jk^n = ^ [GjkJn(2.65) The 3x3 matrix, [Ejj^] , is obtained from [Ej]^] = [Tj] [ejk] [Tj]T (2.66) where ejj^ is a 3 x 3 diagonal matrix with constant components. This matrix is defined in terms of the local mass, M j . Assuming that the local inertia tensor of each link is proportional to its mass, we can write [ejkl (2.67) In general, the scalar components of [ej^] fluctuate as the value of Mj varies. However, to express Mj in terms of the local masses, M j , only, we assume that [ej]^] remains constant. Now, by evaluating the coefficients of Eqns. (2.60) and (2.61), it is very simple to investigate the effect of each local mass and stiffness on the system fundamental frequency.
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38 Figure 2.1. Kinematic Representation of Manipulator
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39 Figure 2.2. Manipulator with Elastic Joints Displaced from their RigidBody Position.
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40 Figure 2.3. Secondary Deformations of JointLink Combination. OQ
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41 Figure 2.4. General Deformation of JointLink Combination.
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CHAPTER III ANALYSIS OF MANIPULATOR DYNAMIC RESPONSE The response of a manipulator arm is the behavior of the system (the motion of a point in the system at any time) when subjected to a certain excitation. This excitation is caused by transient phenomena such as dynamic shocks associated with the arm motion. In addition, there is a certain harmonic content related to the loading condition of the manipulator's terminal device which may excite one or more vibrational modes of the arm. Using the mathematical model presented in Chapter II, it is possible to formulate the forced vibration of a manipulator arm under a general exciting condition. The response of the arm can be represented by N coupled, linear, ordinary differential equations in terms of the generalized coordinates These coordinates can be obtained from a solution of the coupled equations, but this procedure becomes progressively cumbersome as the degree of freedom of the system increases. A more suitable approach is to decouple the system of equations using the concept of modal analysis. 42
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43 The transfer function is a general concept relating the system response to the excitation in the frequency domain which contains all the information concerning the system vibration characteristics. To represent the manipulator's characteristics, its transfer functions are obtained by exciting the mechanism with a harmonic force of variable frequency. The equations of motion describing the system oscillation have been linearized by neglecting the variation of the first order geometric coefficients due to small displacements Bn* It is appropriate to investigate the significance of the nonlinear terms by extending the previous formulation to include the Coriolis and centripetal accelerations in the equations of motion. 3.1 Forced Vibration In the most general case, several time varying forces and torques can be applied to each link of the manipulator arm. These applied loads can be represented by N generalized forces acting over the generalized coordinates Bn* Representing the generalized forces as T^, the equations of motion can be obtained from Lagrange's equations as follows d QKE ) + 3^ + 3PE = TÂ„ dt (sSn) 9Bn SSn 3Sn (3.1 )
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44 where is the work done by a special type of friction force arising from motion in a viscous medium. The generalized forces can be defined in terms of the first order coefficients and applied loads from the principle of virtual work, as described here. Consider the resultant of the applied loads on link jk of the manipulator as a force, _fp, applied to point P in the link and a moment, mjk^ applied to the link. The virtual work due to the applied loads on link jk can be defined as The virtual displacements of point P and link jk are represented by Vp 5 t and 6t, where 6 t is a virtual time increment. However, the virtual work can also be regarded as the product of N generalized forces 5W = ^p"^ Vp 6t + 6t ( 3 . 2 ) ( n ) Tn acting over the virtual displacements f 5 t ( 3 . 3 ) n= 1 Notice that Tpi designates the equivalent force acting over the nth input due to the applied loads on link
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45 jk. Now, Eqn. (3.2) can be rewritten in terms of the geometric coefficients using Eqns. (2.35) and (2.36) 6W = [Gp]Â£ 6t + mj]^T [Gj)^]Â£ 6t (3.4) so that, comparing Eqns. (3.3) and (3.4), we can conclude that the generalized forces have the form (T(jk)jT = [Gp] + mjk^ [Gjk] . (3.5) The total equivalent forces T are found by summing ( n k ) T for each link with applied loads, or T = y . (3.6) j = l The friction forces are nonconservative forces and are assumed to be proportional to the velocities. Therefore, the work done by these forces can be obtained from Rayleigh's dissipation function (3.7) where Cj_j represents the damping coefficients. This equation can be rewritten in a matrix form as
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46 ^ 6 [C*] 6 . (3.8) The equivalent damping matrix is defined by [C*]. Introducing Eqns. (2.43), (2.45) and (3.8) in (3.1), obtain a set of N coupled, ordinary, differential equations describing the response of the manipulator The equations of motion can be written in the matrix [I*] 6 + [C*] Q + [K*] 3 = T . The system of equations (3.9) can be solved for the generalized coordinates 3^ which represents the oscillation of the arm at any time t. But, this is possible only if the excitation is given in a deterministic form as a time dependent function. 3 . 2 Decoupled Equations of Motion It is not practical to solve the coupled equations of motion simultaneously when the system has many degrees of freedom. In this case, a different approach can be used to give the equations of motion a simpler form by expressing the equations in a new set of coordinates. This is accomplished by decoupling the equations of motion using a linear coordinate transformation. The linear we arm. form (3.9)
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47 transformation is obtained by assuming that the response is a superposition of the normal modes of the system multiplied by the corresponding time dependent modal coordinate. To use this procedure which is known as modal analysis, it is necessary first to obtain the system eigenvalues and eigenvectors from the eigenvalue problem described by Eqn. (2.55). Using the superposition theorem, the response can be described by the linear transformation where n is a column matrix consisting of a set of coordinates referred to as modal coordinates. The modal matrix ['i'] is an N x N matrix and its column vectors are defined by the equation The components of [ '!'] are independent of time, so the derivatives of the generalized coordinates with respect to time can be represented by the equations 3 = [H'ln (3.10) ;r = t (r) (3.11) 3 = ['!'] n (3.12) 3 = [ 'F] n . (3.13)
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48 Introducing Eqns. (3.10), (3.12) and (3.13) into (3.9), we obtain [I*] ['t'jri + [C*] ['i'ln + [K*] [â€¢]n = T . (3.14) Premultiplying both sides of this equation by we can write [4'j n + [f]T[C*] [f] n + [4']T[K*] [f] n = [4']^ T.(3. 15) But, from the orthogonality condition of the normal modes we have (r) [I*] r=s râ€¢s (3.16) (c) )T [K*] Â’ Kr , Lo . r=s r?^s (3.17) where Mjand Kjare the modal mass and modal stiffness of the system associated with the rth mode of vibration. Therefore, the system inertia and stiffness matrices corresponding to the modal coordinates are diagonal and can be defined by the equations [ M ] = [f]T [I*] [f] (3.18) [ K ] = [4']T [K*] [>f] . (3.19)
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49 In general, the damping matrix associated with the modal coordinates is not diagonal. However, the damping factors are generally small and matrix [C*] can be assumed to be proportional to the system inertia and stiffness matrices, or [C*] = v [K*] + V 2 [I*] . (3.20) The error introduced by this approximation for the robotic manipulator is expected to be insignificant. Hence, the damping matrix in modal coordinates can be defined by a diagonal matrix as [ C ] = [H']T [c*] ['}'] . (3.21 ) Now, using the definitions for modal parameters we can obtain N independent equations in terms of the modal coordinates Nr (3.22) where Njrepresent the generalized forces associated with the modal coordinates, which are given by the equation Nr = (4^^ T . (3.23)
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50 Dividing both sides of Eqn. (3.22) by Mj, we obtain n + 2 5 c 0 f + (3.24) where by definition (2 M ^) 1/2 uJi= (Kr/Mr)^"^^ . (3.25) (3.26) The solution of Eqn. (3.24) can be readily obtained by means of the Laplace transformation. This solution can be expressed as " ^drMr Nr(T)e t) 3 ( tx ) dx + ( Â•, _J^"2 Jl72 nr(0) cos(oj(j]t ~ Y ) t jjj sin dr (3.27) Parameters, nr(^) represent the initial displacement and velocity corresponding to the rth modal coordinate; and ixi are defined by the equations Yr = tan1 Â—2)Â‘W2 (328) <^dr = n Cr^) <^r Â• (3.29)
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51 Clearly, the modal analysis is a convenient procedure for computing the manipulator's response. However, determination of all system eigenvectors becomes complicated as the degree of freedom of the system increases. This difficulty can be overcome by approximating the system response by a superposition of the first few normal modes, which requires the calculation of the first few eigenvectors only. The error introduced by this approximation is negligible, since the higher natural frequencies are generally much higher than the fundamental frequency, and the contribution of their associated modes of vibration to the system response is very small. 3 . 3 Dynamic Response in the Frequency Domain The characteristics of a vibratory system are fully described by the concept of a transfer function. This function represents the magnitude of oscillation at any frequency, when the system is subjected to a periodic force. Therefore the behavior of the system can be predicted if the frequency content of the time dependent exciting function is known. For instance, consider an impulsive force and its Fourier transformation as shown in Fig. 3.1. The Fourier transform of this function suggests that an impulsive force contains an infinite range of frequencies. Consequently, this force will excite every
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52 vibration mode of a system when applied to the system. But, the forcing function of Fig. 3.2 will excite only those vibration modes which are within the frequency content of this force. To obtain the manipulator's transfer functions, a harmonic force is applied to a point H located on the manipulator's endeffector. The corresponding generalized forces can then be acquired from Eqn. (3.5) tT . FhT [GhI (3.30) Substituting Eqn. (3.30) into (3.23), we obtain (3.31 ) where 1 Nj = )'^ F_h (3.32) Introducing Eqn. (3.31) in (3.24), we can write r\^ + + ojj2nj. = i tot (3.33) A solution of this differential equation is I I 1 tot h r = I n rl e (3.34)
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53 Substituting Eqn. (3.34) into (3.33) and rearranging the resultant equation, we obtain Llri. _ I Nrl h ) (3.35) where hcr((jj) represents the direct transfer function of the arm associated with the rth modal coordinate and is defined by the equation 1 "["l' Wa3r)"2 "+ 2iVr 'u/ur^f . (3.36) The magnitude of the generalized coordinates B can be obtained from Eqn. (3.10) B = [â€¢] Ini . (3.37) We can also find the direct transfer function, hnn(co), corresponding to the nth generalized coordinate of the arm Bn hnn ( ^^ ) ~ I Tn (3.38)
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54 3 . 4 System Response Due to Dynamic Shocks Any discontinuity in a time derivative of the motion of a mechanical system will create a shock in the dynamic operation of the system [2] . Such shocks may excite one or more modes of vibration of the system causing unwanted noise, fatigue, and reduced precision of operation. These shocks are most undesirable when they occur in the first and second time derivative of the system's motion (velocity and acceleration) and must be avoided if possible. A discussion of the manipulator response due to discontinuity in the velocity and acceleration curves is presented in this section. Velocity shocks are commonly generated when starting or stopping the arm motion. Suppose that the velocity of the endeffector produced by the input commands is constant throughout the motion interval (Fig. 3.3a). The acceleration associated with this velocity (Fig. 3.3b) will be very high at the boundaries of the motion interval due to the velocity shocks at these points. Consequently, impulsive forces are generated at the boundaries by the inertia of the arm associated with this acceleration. Now, consider an impulsive force, fjj5(t), applied to a point H on the manipulator's terminal device. Using Eqns. (3.5) and (3.23), the effective force corresponding to the rth modal coordinate of the arm can be obtained
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55 Nr = I Nr I 6 (t) (3.39) where I I = [GhIT fH . (3.40) Substituting Eqn. (3.40) into (3.27) and neglecting the initial conditions, we can write _Ue 1_ e^r<^rt sinoaart (3^4^) Â“dr Mr This equation describes the behavior of the arm in the modal coordinates, from which the system response in the generalized coordinates can be obtained 6. = ['F] n . (3.42) Furthermore, the behavior of any point P on the system is described by the equation A P = [Gp] B . (3.43) Discontinuities in the acceleration curve generate similar discontinuities in the corresponding inertia force. The exciting force produced by an acceleration
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56 shock can be represented by a step function fHU(t) applied to point H, where JO , t < 0 [ 1 , t ^ 0 (3.44) The modal coordinates are then obtained from Eqn. I Nr I . CrWi1hr = Â— [1 e (cos Â‘^dr*^ (1 Cr2)V2) SI (3.27) n Udrt) ] Â• (3.45) Notice that  Nj is defined by Eqn. (3.40). The generalized coordinates motion of any point P on the arm can be found from equations (3.42) and (3.43). 3.5 Nonlinearities in the Eauations of Motion The response of a manipulator arm was described in Section 3.1 by a set of linear differential equations. This was accomplished by neglecting the variation of the first order geometric coefficients due to small displacements Bn* Further refinement of the vibration formulation is needed to investigate the influence of nonlinear terms on the response of the arm. Consider the instantaneous configuration of the manipulator arm shown in Fig. 3.4 (dashed lines) as it
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57 vibrates about its rigid body position (solid lines). The total relative motion of each joint is defined by a /% parameter $n(t)/ where
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58 0 S a ^ Â“(jl)j (3.50) Ijk " ^j1 C(0j + 6j) Ca S(0i + 6i) SOj . Bj) (3.51 ) Notice that aj}j is the angle of twist about the link axis Â£jk* The rate of change of orientation of link jk about the global coordinates is given by the sum of the relative angular velocities about the revolute joint axeS/ which can be written in terms of the first order rotational coefficients in the form ^ j k~l^jk]^ Â“[Gj]^]^ . (3.52) Recall that the arm is in a stationary configuration and parameters are invariant with respect to time. The /N first order geometric coefficients [Gj)^] are distinct from those defined by the matrix [Gj)^] , since their column vectors are given in terms of the deflected joint axes as follows n < j nrevolute j k] n otherwise . (3.53>
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59 The deflection position of the mass center Cj in the global reference frame is described by the equation Cj = Rj + [Tj] (3.54) where, Â£ j ^ ^ ^ defines the position of Cj in the jth local frame, and vector Rj locates the origin of this frame in the global coordinate system. From Fig. 3.4, Rj can be obtained as S 1 1^1 + 2 1=2 ^( l\ )l ( ) 2, ^ (3.55) Following the procedure described in Section 2.2, we can obtain the rate of change of mass center position Â£j f ^c j ^ (3.56) Matrix [G(;j] represents influence coefficients component of [G^j] is the first order of the deflected defined as n Â£ j n Â£ j ; otherwise translational arm. The nth nrevolute nprismatic (3.57) /s
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60 Next, the total kinetic energy of the manipulator arm is obtained from Eqn. (2.43) KE = 2 3 T [I*] 6 (3.58) where [I*] represents the inertia matrix of the manipulator in its deflected configuration. The scalar component in row m, column n of this matrix is given by the equation [I*] Â“ Z f^cj^m'^ f^cjln tlljk] f'^jk^n)* j = 1 (3.59) Matrix [iijk ] represents the inertia tensor about the centroid for link jk in terms of the global coordinate system. This matrix is defined by the equation (as shown in Chapter II) [Hjkl Â— Tj Tj'^ . (3.60) The 3x3 matrix denotes the inertia tensor of link jk in terms of the jth reference frame. The potential and dissipation energies of the mechanism are not defined in terms of the geometric
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61 coefficients; therefore their mathematical definitions remain the same as those given by Eqns. (2.45) and (3.8). Now, substituting the final expressions for the kinetic, potential, and dissipation energies into Lagrange's equations of motion, we can write [I*]n;i^ 2 fl*] )i + fC*]n;i + [K*]n; 6 = Parameter T^i is the nth component of the vector T representing the generalized forces of the arm. This vector is defined by the equation T N I j = 1 jk) (3.62) where [Gjkl . (3.63) The first term on the left side of Eqn. (3.61) can be expanded as follows (3.64) where < M
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62 dt N . 3 m= 1 ^ /N [! (3.65) 3 ^ Notice that r^[I*]n> represent the standard convention op ' differentiation with respect to a vector. The result of this differentiation is an N x N matrix, where the mth row of this matrix is obtained from the derivative of [I*]n; with respect to the mth component of vector or _3_ 36 ( [I*ln; tl*]n; (3.66) Using Eqns. (3.64) and (3.65) in (3.61) gives the nth equation of motion as ^ 6^ [([I*In,) 2 ^ ([1*1)1 6 + [C*]n;4 (K*]n;^ Â“ '^n (3.67)
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63 This equation can be rewritten in the form [I*] + BT [Pnli + [C*]n;i + ] n (3.68) where the inertia power matrix, [P^] , is defined by the equation ( [I*ln;) 1 9 2 3Bn ( [I*] ) . (3.69) An alternative matrix, [Pq*], is presented in [29] which can replace [P^I (this is shown in Appendix B), such that Eqn. (3.68) becomes [I*]n;6 + B^ [Pn*]i + [C*]n;6 + [K*]n;B The scalar component of [Pn*] associated with row and mth column is defined by the equation Tn . (3.70) the 1th N j=1 T 1; m Â’ ; m [njkl jk^ n (3.71 ) The second order geometric influence coefficients are /N /S defined by and the expressions for these
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64 coefficients are shown in Appendix A. The 3x3 matrix /N [Gj},]n is constructed from the scalar components of the vector [Gj)^]n as follows 0 [Gjk^ln [Gjk^ln 0 [Gjk^ln CGjk^Jn [Gjk^ln 0 ( 3 . 72 ) This matrix is introduced in the above form so that a vector cross product can be expressed in terms of matrix multiplication fGjk^n ^ fGji^lni [Gjkl fGjj^lj^ . Recognizing that displacements small. Appendix C shows that the inertia and inertia power of the manipulator arm in its deflected configuration can be approximated as ( 3 . 74 ) ( 3 . 75 ) where matrices [I*] and [Pn*] are the inertia and inertia power of the manipulator obtained from the
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65 undeflected configuration of the arm. It is also shown in the Appendix C that Tn = Tn . (3.76) Considering the above approximations, Eqn. (3.70) can be simplified as = T n (3.77 This equation can be solved numerically to obtain the system response. 3 . 6 Order Analysis In order to predict the effect of the nonlinear terms on the system response, it is necessary to determine the significance of these terms in the equations of motion. This can be accomplished by estimating the order of the nonlinear terms relative to those of the linear terms. To simplify the procedure, we consider a six degree of freedom serial manipulator arm with flexible joints only. Hence, Eqn. (3.77) can be written in the form N Â•Â• N N E [I*] . Si + E Z [Pn*] . . SiSj + i=l i=1 j=1 N E [C*] .Si + ^n Sn ~ i 1 n ; 1 (3.78)
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66 By comparing the definitions for system inertia and inertia power, it is evident that the magnitudes of their scalar components are of the same order, so it may be assumed that Furthermore, let us assume that the damping matrix is proportional to the stiffness matrix only, or where d is a constant. Now, Eqn. (3.78) can be reduced to Since represents the oscillation of joint n about its axis, it can be approximated by a periodic motion consisting of N harmonic functions. Considering only one mode of vibration at a time, we can write [C*] = v[K*] = 10"d[K*] (3.80) I I = Sn I ^n I Â“ Â®n<^ I Sn I ~ Â®n^^ Â• (3.82) (3.83) (3.84)
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67 Parameter 8^ defines the magnitude of the vibratory motion. To further simplify the equations of motion, assume that the magnitude of all joint displacements is equal to a maximum displacement B, or B n ^m = B . (3.85) Introducing Eqns. (3.82) to (3.85) in (3.81), we obtain B (3.86) {0)2 2 [I*] . + z z . + 10do)Kn + Kn } = Tn. i=l i=1 j = 1 The order of the scalar components associated with the nth row of [I*] are greater than those corresponding to the (n+1)th row of this matrix. It is then possible to assume a worst condition for the inertia matrix, which is N Z i=1 [I*] n; 1 N Z i=1 [I*] (n+1 ) ; i (3.87) or N N Z Z i=1 j=1 N N ^ [I*] i=1 n ; 1 . (3.88)
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68 Substituting Eqn. (3.88) into (3.86) results in ( 2 [I*] JCl + NB) + BKndO^ca + 1 ) = Tn . (3.89) i=1 The product NB represents the significance of the nonlinear terms in the equations of motion. Assuming that the vibrational deformations are less than + .02 radians (a large motion for a precision manipulator), we can conclude that I [Pn*U I < 0. 12 1 I Â• (3.90) Notice that the significance of the damping factor depends on the value of co. All of the proposed approximations correspond to worst conditions of the arm; therefore, the significance of the nonlinear terms is grossly exaggerated. By making more realistic assumptions it is possible to draw the same conclusion for a manipulator arm with a large number of freedoms .
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69 f(t) F(w) ( a ) ( b ) Figure 3.1. Fourier Transform of Unit Impulse. f(t) F{o)) Figure 3.2. Fourier Transform of Unit Step.
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t ( a ) ( b ) Figure 3.3. Constant Velocity Curve.
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M> Figure 3.4. Kinematic Representation of Oscillating Manipulator.
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CHAPTER IV ARM DEFLECTION DUE TO THE GROSS MOTION In all previous considerations in the vibration analysis of flexible manipulators, it is assumed that the manipulator arms are in a fixed configuration. As the operational speed increases, the inertia loads become significantly large. At lower speeds the frequency content of these forces are much lower than the arm's fundamental frequency. Under these circumstances, the inertia loads act as statically applied loads varying throughout the cycle. Thus, the system deformations can be obtained by application of quasistatic analysis. At certain speeds, the frequency content of inertia forces becomes so large that they may excite one or more modes of vibration of the mechanism. The manipulator's deformation will then become timedependent displacements which are coupled with the mechanism's gross motion. For precision control of a manipulator, a complete dynamic formulation can be used to predict, and therefore compensate for, the system deformations under load 72
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73 compensate for, the system deformations under load (inertia and external loads). The objective is to obtain compensation signals which, when added to the control signals, correct for hand positioning errors due to the flexible components. Knowing the active loads and manipulator's dynamic parameters (local masses and stiffnesses), the change in the terminal device position, APjj, can be calculated offline, prior to operation. The compensation values are then obtained in order to modify the nominal actuator commands. 4.1 Complete Dynamic Model for Manipulator Arms A general approach based on the formulation of Section 3.5 is described here for deriving the equations of motion of flexible serial manipulator arms. This approach considers the kinetic energy contribution of the system inertia produced by both nominal and perturbed motion of the arm. The perturbed coordinates are regarded as the unknowns, but the nominal motion of the arm is considered to be known. Hence, the equations of motion are expressed in terms of the relative perturbed displacements of the arm. Consider the flexible manipulator arm of Fig. 4.1. The geometric parameters of this robot arm are defined in Section 3.5. However, the motion of each joint is described by the sum of two time dependent displacements
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74 associated with the nominal motion and deformation of the joint. So, the relative motion of joint n can be defined by the equation Â• (4.1) The absolute angular velocity of link jk about the global coordinates is defined as a product of the corresponding first order rotational coefficients and the joint velocities ^ /N /S yS Â• Â• j^jk Â“ [Gjk]_$ ~ f*^jk^ (_^ 6 ) Â• (4.2) It must be borne in mind that is identically joint n represents a pseudoinput. Furthermore, translational velocity of mass center Cj can be described in terms of the joint velocities and translational influence coefficients ^j ~ [Gcj]^ = [G(^j] (^ + ^) . (4.3) Introducing Eqns. (4.2) and (4.3) in (2.37), the kinetic energy of this mechanism can be expressed in the form KE =1 ( $ + B )T [I*] ( $ + 3 ) . zero if the (4.4)
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Recall that gross motion of the arm has no effect on the system's potential and dissipation energies, since the mechanism is assumed to vibrate about its instantaneous equilibrium configuration. The equations of motion can now be derived from Lagrange's equations d 8KE 3KE 3 wD 3PE dT Â“ 3Bn 36n ^ 36n " ^ Â‘ ^ ^ Nominal displacements, do not represent the independent generalized coordinates of the system, since they are a set of given parameters which describe the time varying characteristics of the manipulator arm. Therefore, only perturbed displacements are considered as generalized coordinates in the Lagrangian formulation. Now, substitute the expressions for the system kinetic, potential, and dissipation energies into Eqn. (4.5) to obtain + [C*]n;i + [K*]n; 6 = Tn . (4.6) The first term on the left hand side of Eqn. (4.6) can be expanded as follows
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76 Introducing Eqn. (4.7) in (4.6) results in [I*]n;(i + 6) + (i + 6 )Â’^[Pn*ni + 3 ) + [C*ln; 6 + [K*ln; 3 = Â• (4.8) The scalar components of [Pn*] are defined by Eqn. (3.71). Expanding Eqn. (4.8) and rearranging the resultant equation gives a nonlinear differential equation in terms of the perturbed coordinates 3^' [I*] n;l + [Pn*]3 + l^[Rn*]6 + [C*] n;Â£ ^ n;Â£ Â“ Tn " l^[Pn*]i Â• (4.9) Notice that the first perturbed coordinates coupling matrix [Rn*l equation time derivatives of the nominal and are coupled in this equation. The is an N X N matrix defined by the [Rn*] = [Pn*]^ + [Pn*] Â• (4.10) Matrices [I*], [Pn*l ^ and [Rn*l are functions of the deformation displacements 3 ^. However, recalling Eqns. (3.74) to (3.76), the coefficients of differential equations of motion can be obtained from the nominal displacements only. Thus, Eqn. (4.9) can be rewritten in the form
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77 [I*]n;6+ i^[Pn*li+ [C*]n;6+ [K*]n;6 = Tn i^[Pn*] i Â• (4.11 The last two terms on the right hand side of Eqn. (4.11) represent the equivalent load acting over the nth input due to the nominal motion of the system inertia. This load acts as an externally applied load on the system which causes the arm to deform about its rigidbody position . In order to obtain the perturbed motion of the arm, Eqn. (4.11) must be integrated numerically. Deformation displacements can be computed at a series of time steps At, to describe the position of the arm at any time t. The accuracy of any numerical integration depends on the size of time steps. At, which becomes even more significant for Eqn. (4.11) since the coefficients of this differential equation are time varying functions. To minimize the cumulative errors in stepbystep integration and avoid numerical instability, numerical integration must be performed at small time steps At. An adequately small time step can be selected as a fraction of the time period associated with the maximum excited mode of vibration. For example, if only the first mode of the arm is excited by an applied load, the time step should be a fraction of l/aj]. Clearly small time steps require long computational times for numerical integration. The
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78 computational time may be reduced to some extent by using a suitable method such as fourthorder RungeKutta method combined with Haming predictor corrector [28]. There are several other numerical methods [30] which can be used for solving a dynamic equation such as Eqn. (4.11). As mentioned previously, if the frequency content of the inertia loads is significantly lower than the fundamental frequency, the variation of deformation displacements with respect to time can be neglected. Therefore, Eqn. (4.11) can be readily simplified by neglecting the time derivatives of perturbed coordinates Bn' [K*]n; 6 = Tn [I*]n;i ~ iT[Pn*li Â• (4.12) This equation represents the quasistatic approximation of system deformations. In most cases, such analysis is adequate for describing the manipulator's deflection due to inertia loads. 4 . 2 Variation of System Natural Frequencies Due to the Nominal Motion of the Arm The effect of manipulator's gross motion on the natural frequencies may best be illustrated by an example problem. Figure 4.2 shows the example manipulator and the
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79 kinematic notation describing it. The manipulator is an open loop spatial RRRRRR mechanism, where R stands for revolute joint and RRR indicates that the three successive joint axes are parallel. The joint axes are described by vectors Â£j , and is the unit vector along the common perpendicular between Â£j and The nominal motion of each joint is represented by the angle 0 j rotating about Â£j axis (this angle is measured in a righthand sense between Â£ij and Â£j]<). Parameters Sjj and aj]^ represent the offset distance between links Â£ij and ajk, and the link length between axes Â£j and Sjj, respectively. The angle of twist between Â£j and S}j is designated by angle aj}j. The geometric parameters for this robot arm are shown in Table 4.1. In order to represent ah illustrative formulation, only one flexible component at a time is considered. Figure 4.2 shows that the manipulator arm deflects abou't joint axis S_i by an angle 3 j . The flexibility of the joint is represented by a torsional spring K . Therefore, this system has one deformational degree of freedom only. Figure 4.3 illustrates the instantaneous position of the arm which lies in the X* Z* plane. The direction cosines of the joint and link axes for arm position are listed in Table 4.2. Furthermore, the associated mass of link jk is lumped at a point Cj (Fig. 4.3) on the link, the mass moments of inertia corresponding to the link are neglected in order to avoid computational complications.
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80 The homogeneous equations of motion for this robot arm can be obtained from Eqn. (4.11). Recalling that the Notice that Eqn. (4.13) describes the motion of a one degree of freedom system. Assuming that the nominal angular velocities of all joints are equal to a maximum frequency, Eqn. (4.13) is reduced to It is necessary to obtain the first and second order translational coefficients for mass centers Cj in order to evaluate and Qi*. These coefficients are listed in Tables 4.3 to 4.9. Rotational geometric coefficients are not needed since the mass moments of inertia have been neglected. Now, using Eqn. (2.44), we nonlinear terms, insignificant and ignoring the damping forces, we can write 0 0 . 0 0 (4.13) [I*] T + aQi*Bi + Ki8l = 0 . (4.14) where 6 Ql* = S . (4.15) i=1 obtain
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81 = I 6 OOM 2 + 6403M3 + 8 IO 3 M 4 + 8 IO 3 M 5 + 12104M6 . (4.16) Also, from Eqns. (4,10) and (3.71) we can evaluate the scalar components of [Ri*] = 0 ( 4 . 17 ) [Ri*32;1 = 2400M2 1 1 . 2 M 3 1 6 M 4 I 6 M 5 27Ms (4.18) [R 1 *] 3;1 = 4794 M 3 + 539 OM 4 + 539 OM 5 + 6580M6 (4.19) [R*] 4.1 = 3 . 2 M 4 3 . 2 M 5 12Mg (4.20) [Ri*] 5;1 = C^^1*l6;1 = 0 (4.21) from which the value of Qi* is determined. Ql* = 2400M2 + 4782. 8 M 3 + 5370M4 + 5370M5 + 6541M6 (4.22) Comparing Eqns. (4.16) and (4.22), it is evident that the value of Qi* is much smaller than [I*]!.!. Therefore, we can consider the worst value of Q]* which is Ql*5[I*]. (4.23) The equation of motion can now be rewritten in the form 0] + (101 + COINS'! = 0 (4.24)
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82 where tu Â•] is the natural frequency of the arm corresponding to its fixed configuration and it is defined by the equation a>i (4.25) The nominal angular velocity of each joint can be defined as Â£oji (4.26) This equation implies that is proportional to the natural frequency by a factor e. The eigenvalue of Eqn. (4.24) is given by the equation (^d1 (4.27) Thus, when the arm is operating with input velocities of eoj], its natural frequency will drop from go j to G 0 (j]. Selecting an exaggerated value of 0.1 for e results in ojd 1 = 0.995 G0 1 (4.28)
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83 It is apparent from this equation that the effect of the manipulator's gross motion on the system natural frequency is insignificant. This procedure can be repeated considering any or all other flexible components for any configuration of the arm to arrive at the same conclusion . 4 . 3 Compensation for Deformations To compensate for the manipulator's deformations, the overall change in the spatial position of the hand can be evaluated from the known deflection in each flexible component. On the basis of this known positional error, corrections can be made to the input command signals which move the hand to its desired position. This procedure to enhance the positioning accuracy of the hand can be said to make the manipulator "electronically rigid". In this way, system inputs are tuned to the operating conditions to enhance positioning precision. The concept of compensation for deformations by tuning the system inputs is presented in [2] for highspeed cam systems. The overall change in the spatial position of the endeffector can be represented by a vector, AP^ , consisting of six components. This vector describes the deflection of a point on the endeffector by its first three components and the change in the orientation of the
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84 hand by its last three components. Vector, AP^i/ can be obtained from the equation AFfi = ^N(N+1 ) a = [j] e (4.29) where [J] is known as the Jacobian matrix. Recall that represents the deflections in the manipulator's inputs and pseudoinputs. However, only the actual inputs of the arm can be tuned to correct for the error in the position of the endeffector. Therefore, it is necessary to predict the required change in the nominal motion of the arm to produce a deflection, A^, in the spatial position of manipulator's hand. Considering the nominal motion of the arm, we can write AÂ£h = Rg. N{N+1 ) A4> ] A^ (4.30) Matrix [^J] represents the Jacobian of the rigidbody mechanism and vector A4> is the required correction vector. For a six degree of freedom manipulator arm, the correction values can be obtained from A$ = APh (4.31 )
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85 This should give good results when the deformations are small and the Jacobian matrix, [RJ] , is not near a singularity. For manipulators with redundant degrees of freedom, the pseudoinverse of the Jacobian can be used, or = ([^J]"^ [^J] )Â“^ [^J]T APh (4.32) Next, the tuned values for the system inputs, T can be obtained from (4.33) This is a relatively simple type of compensation wtiich shows that, it is possible to electronically correct for flexibility in a manipulator system if the system loads and dynamic parameters are known.
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M> 86 Figure 4.1. Kinematic Representation of Oscillating Manipulator.
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vector directed into the page Figure 4.2. Kinematic Representation of the Example Manipulator.
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88 VO N cif > Figure 4.3. Arm Position for the Example Analysis.
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89 Table 4.1. Geometrical Parameters for the Example Manipulator. j S j j ( in . ) a ( in . ) ajj^Cdeg 1 30 0 90 2 0 50 0 3 0 50 0 4 0 10 90 5 0 0 90 6 20 0
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90 Table 4.2. Manipulator Position for the Example Analysis. j S . 3 a . , 3 k CD 1 1 Â— 1 o o O o 1 Â— 1 (0,0,30) 0.0 2 o I Â— 1 1 o Â• o 00 Â• (0,0,30,) 36.9 3 (0,1, 0) (.8,0, .6) (40,0,60) 73.7 4 (0,l,0,) (1,0,0) (80,0,30) 36.9 5 (0,0,l) (0,l,0) (90,0,30) 90.0 6 o o 1 Â— 1 (0,1,0) (110,0,30) 180.0
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91 Table 4.3. First Order Translational Influence Coefficients for Mass Centers Cj 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 80 0 0 0 30 40 0 0 0 0 0 0 0 0 0 0 90 0 0 0 90 30 0 50 0 0 10 0 0 0 0 0 0 Pc] = 0 0 90 0 0 90 30 0 50 0 0 10 0 0 0 0 0 0 30 0 70 0 0 0 20 30 0 0 0 0
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92 Table 4.4. Second Order Translational Influence Coefficients for Mass Center C^^ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Symmetric 0 0 0 0 0 0 0 _
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93 Table 4.5. Second Order Translational Influence Coefficients for Mass Center C 2 40 0 0 0 0 0 ~ 0 30 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Â— 0 0 Symmetric 0 0 0 0 0 0 0
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94 Table 4.6. Second Order Translational Influence Coefficients for Mass Center 80 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 80 40 0 0 0 0 0 0 0 0 0 30 0 0 0 40 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Symmetric 0 0 0 0 0 0 0
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95 Table 4. [Â«C4J = Second Order Translational Influenoe Coefficients for Mass Center C^ 90 0 0 0 0 0 Â” 0 0 30 0 0 0 0 0 0 0 0 0 90 50 10 0 0 0 0 0 0 0 0 30 0 0 0 50 10 0 0 0 0 0 0 30 0 0 0 10 0 0 0 0 0 0 0 0 0 0 Symmetric 0 0 0 0 0 0 __ 0
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96 Table 4.8. Second Order Translational Influenoe Coefficients for Mass Center C_ 0 0 90 50 0 0 0 I 30 0 0 0 10 0 0 10 0 0 10 0 0 Symmetric 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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97 Table 4.9. Second Order Translational Influence Coefficients for Mass Center C, D ~110 0 0 0 20 0 Â“ 0 0 30 0 0 0 0 0 0 0 0 0 110 70 30 0 0 0 0 0 0 0 0 30 0 0 0 70 30 0 0 0 0 0 0 30 0 0 0 30 0 0 0 0 0 0 0 0 20 0 Symmetric 0 0 0 0 0 0 Â° J
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CHAPTER V PARAMETRIC IDENTIFICATION Establishing the dynamic model of a mechanical system such as a robot manipulator is an essential first step in creating a modern intelligent manufacturing system. To establish the complete dynamic model of a manipulator arm, it is necessary to evaluate the system's structural properties (i.e. mass, stiffness, and damping). The structural properties of a mechanical system are commonly determined from either the continuously distributed model of the system components, or the finite element methods using the mechanical properties of system elements. In general, manipulator link components are not uniform beams, therefore, the continuous model will not represent the system accurately. , This problem is overcome by dividing the link elements into many small segments using finite element methods. Nevertheless, neither of the above methods are capable of providing a true representation of the actual mechanism. There are many factors including, the mass content of the actuators and drive trains, the deformational energy associated with the joint components, and the bearing flexibilities which 98
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99 contribute significantly to the overall structural behavior of a manipulator arm. It is not possible to predict the effect of these factors on the structural behavior of the arm using the aformentioned methods. A reliable procedure is required to identify the system's dynamic parameters which considers all significant contributing factors. An experimental procedure is proposed to obtain system stiffness and mass parameters based on the modal analysis concept. This approach develops a true representation of the arm because the dynamic model parameters are found from the identification of the actual system. 5 . 1 Experimental Modal Analysis Experimental modal analysis is a powerful tool for investigating the vibrational behavior of a flexible system. This tool has been widely used to describe the vibration characteristics of aircraft, spacecraft, automobile frames, machine tools, and other structures such as bridges. This chapter presents a technique for identifying the dynamic parameters of a robot manipulator utilizing experimental modal analysis. Before introducing this technique, experimental modal analysis is briefly discussed in order to provide a basic background for this concept .
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100 Consider a system which can be described by the two degrees of freedom model shown in Fig. 5.1. The vibratory motion of this system is fully defined by the coordinates XT(t) and X 2 (t) which respectively represent the displacements of mi and m 2 from their static equilibrium position at an arbitrary time t. The system's differential equations of motion can be obtained from Lagrange's formulation d 3KE 3KE 3WÂ° aPE dt ('3Xn) ~ 3Xn 3Xn 3Xn '^n Â• (5.1) The total kinetic energy of the system is defined by the equation KE (5.2) The system's potential energy is due to the elongation of springs ki and k 2 > which can be described by the following equation PE = ^ ki xi^+^k2 (xi X2) (5.3) Rayleigh's dissipation function is obtained from ,^D = ^ V.2 + I Â• Â• ^ Cl xi^ ^ ^ (^1 Â“ ^ 2 ) (5.4)
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101 By taking the appropriate derivatives in Eqns. (5.2), (5.3), and (5.4), and inserting the results into Eqn. (5.1), we can write the equations of motion for this two degree of freedom system in the matrix form mi 0 0 m2 (5.5) ki + k2 k 2 XI T, J<2 J<2 X 2 T2 1 . _ Cl + C2 C2 C2 C2 XI X2 T] and T 2 are the generalized forces acting over the generalized coordinates x and X 2 ^ respectively. Assuming that the system is excited by a timedependent forcing function f 2 (t) acting over the coordinate X 2 , the generalized forces Ti and T 2 can be determined from the principle of virtual work 6 W = f 2 (t) 6 x 2 = T 5x + T 2 5 X 2 . (5.6) By inspection of this equation we conclude Ti 0 T2 f2(t) The equations of motion can now be transformed into frequency domain using the Fourier transformations
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102 Pxn ( t ) = X]j ( to ) (5.8) ^Xn ( t ) = i (jjXjj ( (ij) (5.9) Pxn(t) = (5.10) Ff 2 ( t) = F2 ( 0)) (5.11) Substituting Eqns. 0)2 [M] X(oj) + (5.7) through (5.11) into (5.5) io) [C] X(to) + [K] X{^) 0 F2 ( ca) gives (5.12) In order to decouple the equations of motion it is necessary to obtain the natural frequencies and the associated mode shapes of the system from the free vibration of the undamped system. The differential equations of motion for free vibration of the system reduce to [M]x + [K]x = 0 . (5. 13) This leads to a standard eigenvalue problem described by the equation [KlI [M] [I] = 0 . (5.14) Next, the natural frequencies, cot and a>2f along with the eigenvectors, ^ ^ and are obtained from Eqn. (5.14). Furthermore, the eigenvectors can be normalized so that
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103 (1 ) , ( 2 ), _ ip ] 1 ( 1 ) ip 1 { 2 ) 1 1 {5.15) Using the superpostion theorem, the response can be described by a linear transformation of the X coordinates into the modal coordinates X( u)) ~ [ 'i'] _n ( w) (5.16) Assuming the damping matrix [C] to be proportional to the stiffness matrix [K] , the equations of motion can be decoupled by introducing Eqn. (5.16) into (5.12) and premultiplying the resultant equation by [ ~ii)^ n (w) iw [C] n (oj) t [K] n ((jj) ~ * (5.17) Matrices [M] , [C] , and [K] designate the modal parameters of the system, and N represents the generalized forces acting on the modal coordinates, which can be defined by the equation (w) 0 F2 ( w) F2(w) F2(^) (5.18) The transfer functions in the modal coordinates can then be deduced from Eqn. (5.17) as shown below
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104 Nr nr F2 Kv [1 (7) 0^ + 2i 5ji~) (5.19) where _ _ Cr . (5.20) 2 (K^Mi) V2 These transfer functions are complex functions which define the vibration characteristics of the system. The real parts of these functions are obtained from Eqn. (5.19) 1 (~) '' liir ' Re ( h rr ^00 Kr [(1 ( 7Â“)^2 + 4^^2 (5.21 ) Equation (5.21) can be graphically expressed as functions of to as shown in Fig. 5.2. By inspection of Eqn. (5.21), it is evident that the magnitude of Re ( hjj( to) ) is zero when to is equal to to^* Furthermore, the maximum and minimum of Re ( ( to) ) are obtained by differentiating Eqn. (5.21) with respect to to and letting the result be zero. If this is done, we conclude that the maximum and minimum occur at
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105 Substituting Eqn. 1 Re(hir(w)) U) COj( 1 ^ ^ J) . (5.22) into (5.21) we can write + I Re(hrr( 03) )l . = . max min 2Kr?r (5.22) (5.23) The system transfer functions in the generalized coordinates can be obtained from Eqn. (5.16). The natural modes of the system are then obtained from these transfer functions Qjx Q2r 1 (5.24) Parameters, Q\jand Q2r ' J^epresent the peak to peak values of the transfer functions as shown in Fig. 5.3. Therefore, the system characteristics are completely defined by the corresponding transfer functions. Experimental modal analysis involves the extraction of resonance frequencies o}^, modal parameters (damping, mass, and stiffness), and mode shape coefficients ( c ) from the frequency response data hnm((jj)Modal extraction involves a number of curve fitting operations. A particular type of model is fit to the data and the best estimate of associated parameters is made.
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106 These parameters can then be utilized to determine the dynamic parameters of the system from the equations Notice that the components of the mass, stiffness, and damping matrices are defined in terms of the local parameters of the system. Thus all dynamic parameters can be identified once the system matrices are calculated. 5 . 2 Parametric Identification for Manip u lator Arms This section presents a technique to identify the dynamic parameters of a particular industrial arm (Cincinnati T3776 robot shown in Fig. 5.4) based on the concept described in Section 5.1. In order to simplify the identification procedure the system's degrees of freedom are minimized by excluding the motion of the manipulator's wrist and the deflection of the arm in the X* Y* plane. The compliance of the T3776 robot associated with the servo drives is found to be insignificant relative to the structural compliance of the arm. This was determined by a comparison of the transfer functions measured with the servo brakes on and off. [M] = [f] T [>{Â»]'' [K] = [K] [^]1 [C] = [C] [^]1 . (5.27) (5.26) (5.25)
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107 Therefore, servo flexibilities are excluded by activating the servo brakes during the measurements. Although these limitations simplify the analysis substantially, they do not impose any restriction on the identification method. To describe the natural motion of the T3776 robot at a selected configuration, eleven points along the arm are chosen as shown in Fig. 5.5. The coordinates of these points in terms of the global reference frame are listed in Table 5.1. All the system modes are excited at an appropriate reference point by means of an instrumented impact hammer. The absolute vibrational response of each point is measured sequentially by an accelerometer. The excitation is obtained as an analog signal by a load cell mounted between the hammer's striking head and body. The analog signal is then transformed into the frequency domain by an AD conversion followed by the Fast Fourier transformation of the resultant digital signal. A preliminary set of measurements showed that the dominant vibrational modes of this system are concentrated in a frequency range of zero to sixty Hertz. Therefore, the frequency content of the exciting force is lowered to include only the frequency range of interest by enlarging the time duration of the impulsive force. The time duration can be increased by using a soft striking head during the measurements. Alternatively, the impulse duration can be further increased by the addition of mass
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108 to the hammer body. The exciting force and its Fourier transform under the aforementioned conditions are shown in Fig. 5.6. The vibrational motion of each point in the X* and Z* directions are measured by a large seismic accelerometer since the manipulator arm oscillates at low frequencies. The response of each point is then transformed from the time domain into frequency domain by a Fourier analyzer. An example time domain response of point 1 in the Z* direction produced by an impulsive excitation is shown in Fig. 5.7. The transfer functions of this system are obtained dividing the Fourier transform of the measured response by the Fourier transform of exciting force. Figure 5.8a represents the magnitude of the direct transfer function"' associated with point 4 in the Z* direction. The vibrational mode which occurs at 18 Hz is not included in the identification process since this mode of vibration corresponds to the motion of the arm in the X* Y* plane and it is the result of noncentral excitation at point 4. The modal parameters and mode shapes of the arm are obtained by curve fitting the transfer functions using the ' A direct transfer function is the ratio of the component of the response in the direction of the excitation and at the point of application of the excitation, to the excitation.
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109 routines of the modal analysis software, "MODAL PLUS", provided by SDRC. The modal parameters and mode shapes of this system obtained from the curve fitted transfer function (Fig. 5.8b) are listed in Tables 5.2 and 5.3, respectively. Notice that four modes of vibration are identified by this curve fitting technique and the natural motions of the arm about its rigid body configuration associated with each mode of vibration are illustrated by Fig. 5.9. 5.2a Identification of Local Stiffnesses Inspection of the arm's natural motions indicates that the oscillation of the system is produced by the deformations of the shoulder and elbow joints about their axes of rotation and the deflection of the base plate relative to the foundation of the manipulator arm. Therefore, a four degree of freedom model is chosen to represent the T3776 robot. The relative flexibility of the base plate and foundation can be represented by two concentrated springs, K and K 2 , as shown in Fig. 5.10. The deflections of the shoulder and elbow joints are due to the flexibilities of their corresponding drive assembly which can be described by two springs, K 3 and K 4 , illustrated by Fig. 5.10. Four coordinates are sufficient to fully describe the motion of a four degree
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110 of freedom system. Hence, the coordinates 1Z (the motion of point 1 in the Z* direction), 4Z, 9Z, and 11Z are selected to represent the motion of the manipulator arm. The modal matrix associated with these coordinates is deduced from Table 5.3 which can be written in the form 1 .0 1 .0 1 .0 1.0 0.399 1 o Â• 4.51 2.34 0.029 Â• 0.115 0.429 2.99 0.021 0.075 2.753 1.54 Now, we can determine the stiffness matrix associated with ^ (1Z, 4Z, 9Z, 11Z) coordinates using the modal matrix, [^] , and modal stiffnesses, K^, given in Table 5.2. The stiffness matrix is obtained from the equation [KZ] = [4<]T [k] [^]1 (5.29) It is convenient to transform the stiffness matrix, [KZ] , into a new coordinate system referred to as 7 ^ coordinates. The ^ coordinates include, gg the rotational motion of the shoulder joint about its axis, 34 the rotational motion of the elbow joint about its axes, 9Z, and 11Z as shown in Fig. 5.10. This transformation simplifies the identification process since
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Ill the stiffness matrix in terms of ^ coordinates must be a diagonal matrix of the form [K2Â®] KB3 0 0 0 0 K34 0 0 0 0 Ki 0 0 0 0 K2 (5.30) Parameter K3j_ designates the effective stiffness associated with 3j_ coordinates. It is evident that once [rz^] is obtained, the local stiffnesses are readily identified. The coordinates _Z can be transformed into the coordinates Z3 using the equation Â” Â— 1 z S3 4Z 64 9Z = [TR] 9Z 1 1Z 1 1Z , Matrix [TR] is a transformation matrix and its scalar components are functions of the robot geometry. This matrix is defined by the equation
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112 [TR] [Si ^ 1 3 [Gi 2 ] 4 "[GiZji [Gi2]2~ [Gg2] 1 [Gg2] 2 [= 4^13 [G42] 4 JG4^]i [G4^]2_ JGn^li [Gl 1 2 0 0 1 (5.32) Parameter [GjZ]j_ designates the z component of [Gj] in the ith column. Recalling that the potential energy contribution is independent of the coordinate system, we can write PE = [kZS] (^6) = \ [rZ] (z) . (5.33) The substitution of Eqn. (5.31) into (5.33) results in j (ZB)^ [kZ^] (^S) =1 (Z3)^ [TR]T [rZ] [tr] (^6). (5.34) 3y inspection of Eqn. (5.34) we conclude [rZB] = [tR]T [RZ] [TR] . (5.35)
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113 Therefore, the stiffness matrix associated with ^ coordinates can be obtained using the modal parameters deduced from the experimental measurements. The stiffness parameters, K 3 and K 4 , defining the flexibility of tne drive assembly are related to the springs, K ^3 and K 34 , by the perpendicular distances, R 3 and R 4 , between the joint axes and the axes of corresponding drive assembly Ki = ^2 K 6i (5.36) To estimate the values of local stiffnesses more accurately, the identification process is repeated for several different configurations of the arm. The final value of each local stiffness is then obtained by taking the average of the local parameters deduced from the selected configurations of the arm. The identified stiffness parameters of the T3776 robot are listed in Table 5.4.
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114 5.2b Identification of Local Masses The mass content of the manipulator arm is represented by three mass elements associated with the base housing, the upper arm, and the lower arm (Fig. 5.11a). The mass of the base housing contains, a mass M2 concentrated at a point C2 with coordinates (15,0,0) in terms of the local reference frame, and a mass moment of inertia I22 ^bout the local Z coordinate axis. Parameter M3 designates the point mass associated with the upper arm which is lumped at the point C3 (17.2,0,0) in the corresponding local reference frame, and I23 denotes the mass moment of inertia of this link about the local Z axis. The mass content of the lower arm is represented by the point mass M4 concentrated at the point C4 (24,0,0) in the associated local coordinate system, and the mass moment of inertia I24 about the local Z axis. Furthermore, the motion of base plate is defined by the psuedoprosmatic input 31 and psuedorevolute input 32 (Fig. 5.11a). The kinematic representation of this system is shown in Fig. 5.11b and its geometric parameters are listed in Table 5.5. The mass matrix in the Z coordinate system is obtained from the equation [mZ] = ['FlT [M] [')']"' (5.37)
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115 We can then determine the system inertia matrix, [I*], by transforming the _Z coordinates into the ^ ( Si ^ Szr 63' 64 ) coordinates. This transformation can be defined by the equation ^ = [TR1 ] ^ . (5.38) where [TR1] = G9Z Gi iZ .(5.39) Parameter Gj_Z designates the Z components of the translational geometric coefficients associated with point i. Recalling that the kinetic energy contribution is independent of the coordinate system, we can write KE = ( 6 )T [I*] ( 3 ) = 1 (^)T [mZ] (i) = 2 2 4 ( 8 )^ [TR1]T [mZ] [TRI] (g) . ^ (5.40) By inspection of Eqn. (5.4) we conclude [I*] = [TRI] [mZ] [TRI] . (5.41 )
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116 The system inertia matrix [I*] is a 4 x 4 symmetric matrix and its scalar components can be obtained from Eqn. (5.41). To determine the local mass parameters, each independent component of [I*] must be expressed in terms of the unknown parameters (M 2 Â» M 3 , M 4 , Iz2' ^Z3' ^Z4)* These expressions can be written in the matrix form The scalar components of coefficient matrix [MM] are obtained from Eqn. (2.44). Equation (5.42) represents a system of ten equations with six unknowns which can be solved by a least squares method to obtain the local mass parameters of the system. This process can also be performed for several configurations of the arm to provide a more accurate model. The local mass parameters of the T3776 robot identified by the above procedure are listed [MM] Iz2 M2 M 3 M4 IZ3 IZ4 111 * I12* Il 3 * Il4* I22* I23* I24* I33* I34* I44* (5.42) in Table 5.4.
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117 5.2c Identification of Local Damping Coefficients The viscous damping factors, are determined through the curve fitting routine of the measured transfer functions (listed in Table 5.2). Maintaining the basic assumption of proportional damping, we must be able to satisfy the following condition Thus, by estimating an approximate value for v we can determine the damping matrix associated with the modal coordinates. The damping factors, are defined by the equation The final value of v can be obtained from the average of this coefficient associated with each mode of vibration, or Cj= vKj. (5.43) (5.44) V (5.45)
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118 Next, the damping matrix [C] is evaluated using Eqn. (5.43). Finally, the local damping coefficients are determined by following the procedure of Section 5.2a (these values are listed in Table 5.4). 5.3 Verification of the Identified Parameters The quality of parametric identification depends on the extent to which the vibrational characteristics of the proposed model match the experimentally determined behavior of the mechanism. The suggested model verification should include, (a) the comparison of the system natural frequencies, deduced from the identified model and experimental data, throughout the arm's workspace, (b) the comparison of the static deflection of the proposed model and the actual system under applied loads, and (c) the comparison of the measured transfer functions with analytically computed transfer functions. The system natural frequencies are obtained from Eqn. (2.55) using the identified parameters. These frequencies along with the measured resonance frequencies for two geometric configurations of the arm are listed in Tables 5.6 and 5.7. Comparison of the resonance frequencies provides a basis to define the quality of the model based on the overall structural properties of the system.
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119 Static measurements were performed for several configurations in order to determine the total flexibility of the robot arm. A static load was applied to point 1 in the Z* direction and the deflection of this point was measured by means of a high resolution dial indicator. Furthermore, the total static flexibility of the robot model is determined by computing the joint deflections due to an applied load e = [K^]1 [Gi]T Fi . (5.46) Vector Â£i designates the magnitude and direction of the load applied to point 1. The total flexibility at point 1 in the Z direction can then be obtained from 1Z = [GiZ] e . (5.47) The accuracy of the total static flexibility defines the quality of the identified stiffness parameters regardless of the mass parameters. Table 5.8 presents a list of measured and computed static flexibilities at one of the selected robot configurations. A comparison of measured and modeled transfer functions provides a basis to verify the identified parameters based on the complete vibration characteristics of the system. The modeled transfer function of the robot
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120 for point 4 in the Z* direction is obtained by following the procedure introduced in Section 3.3. Figure 5.12 illustrates the measured and modeled transfer functions at a selected robot configuration. Further verification is made by adding external weight to the system and repeating the above procedure (Fig. 5.13). These comparisons indicate that the proposed model provides a relatively accurate representation for the actual system. The model can be further refined by ' performing more accurate measurements and considering other significant flexible components such as bending flexibility of the link elements. 5.4 Instrumentation In recent years, the advent of high performance, low cost minicomputers, and computing techniques such as the fast Fourier transform have given birth to powerful new "instruments" known as digital Fourier analyzers. The ability of these machines to quickly and accurately provide the frequency spectrum of a timedomain signal has opended a new era in structural dynamics testing. It is now relatively simple to obtain fast, accurate, and complete measurements of the dynamic behavior of mechanical structures, via transfer function measurements and modal analysis . Techniques have been developed which now allow the modes of vibration of an elastic structure to be identified from measured transfer function data. Once a set of transfer (frequency response) functions relating points of interest on the structure have been measured and stored, they may be operated on to obtain
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121 i.e., the natural frequency, damping factor, and characteristic mode shape for the predominant modes of vibration of the structure. Most importantly, the modal responses of many modes can be measured simultaneously and complex mode shapes can be directly identified. [31] The instrumentation package selected to conduct the test was the GEN RAD 2515 computer aided test system. This system was selected because it would meet the requirements of high baseband resolution, bias protection, interactive dataacquisition, large CPU memory, and sophisticated curvefitting software compatibility. A summary of the specifications for the test equipments are presented here. Modal Analyzer Genrad 25159715 Computer Aided Test System Eight input A/D channels with programmable antibiasing filters LSI 11/23 controller with RT1 1 operating system 512 Kilobytes of CPU memory MiniWinchester rigid disk drive with 10 megabyte total capacity MiniFloppy disk (51/4) with .5 megabyte total
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122 Sensitivity: 8 ranges (62.5 mV, .125 V, .25 V, .5 V, 1 V, 2 V, 4 V, 8 V) Bandwidths: Selectable in powers of 2 times 1.25 Hz to 20 KHZ, plus 25 KHZ Frequency Lines; 320 aliasfree lines for every 1024 input points in frame Frame Size: Selectable in powers of 2 from 526 to 8192 points/channel SDRC ModalPlus software Impactor PCB model 086B20 short sledge hammer, S.No. 625 Sensitivity: 1.04 mV/lb Range: 0500 lb PCB model 084A60 Impact super soft tip Accelerometer PCB model 393C sesmic accelerometer, S.No. 412 Sensitivity: 1.007 V/g PCB model 430 D06 power supply, S.No. 4083 5 . 5 Improvement of System Design It is possible to improve the structural integrity of the T3776 robot by increasing its fundamental natural frequency as discussed in Section 2.7. The simplest way to improve the resonance frequency of this system is to increase the modal stiffness associated with the first
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123 vibrational mode. The system's fundamental frequency is defined by Rayleigh's quotient 2 = ^ 1 M] (5.48) In order to increase the value of to a desired value 0 ) 1 ^, the modal stiffness Kj must be increased to K^, or (cjld ( 5 . 49 ) Suppose, it is desired to increase the resonance frequency by ten percent. We can write ( 1.1 )2 (5.50) Substituting Eqn. (5.48) into (5.50) results in = 1 . 21 Ki . (5.51) The stiffness matrix associated with the B coordinates is defined by the equation
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124 [K^] = 1745079 2380463 0 0 5.8300865E3 0 0 335431 X R 32 0 SyiTunetric 471 433 x R 42 (5.52 The eigenvector corresponding to the first mode of vibration is found to be 1.0 0.33 2.7 1.9 (5.53) The modal stiffness K^ is defined by the summation 4 4 Ki^ = E E i =1 j =1 [K^ (1 ) in b , n ) 3 (5.54) Therefore, the desired modal stiffness can be obtained by increasing the scalar components of [K^] . By inspecting the coefficients of in Eqn. (5.54), it is clear 173 that the stiffness parameters associated with the shoulder and elbow joints are the prominent factors in increasing the modal stiffness. We can then select these two parameters to improve the system design. Furthermore, the
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125 effective stiffness at the shoulder and elbow joints are geometrically dependent parameters. Thus, the simplest way to increase these parameters is to improve the effective geometric parameters. This can be achieved by increasing the magnitude of R^and . Rearranging Eqn. (5.54) results in (335431) (2.7)2 ^^2 + (471433) (1.9)2 R42 = Kid E E [K^] . . ^ . (5.55) i=1 j=1 Parameters R 3 and R 4 can be determined by choosing an appropriate value for either R 3 , or R 4 and computing R 4 or R 3 using Eqn. (5.55). In this case, R 4 , at a selected configuration, is chosen to be 7.572 (an increase of 1.0 in. from the previous value) and R 2 is computed to be 12.53 (an increase of 1.16 in.). Using these parameters, the eigenvalues of the improved system are computed and compared with the original system design (Tables 5.9 and 5.10). It is possible to further improve the system design by decreasing the flexibility of drive assemblies or improving the mass distribution of the arm.
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126 ////////// JJ 1 Â— 1 u i 1 1 ^1 1 Â— 1 B ^2 t d 1 1 ""2 > Â„ iojt X2(t) Fq6 Figure 5.1. The Two DegreeofFreedom System.
PAGE 134
12 ? Re ( Â— ) (in. /lb) Re ( ) (in. /lb) Figure 5.2. System Transfer Functions in the Modal Coordinates .
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128 Re ( Â— ) (in. /lb) ^ 0 ( a ) System Transfer Functions in the Generalized Coordinates . Figure 5.3.
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129 Figure 5.4. Cinncinati Milicron T3776 Robot Manipulator.
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130 Figure 5.5. Measured Point Locations (scale 1/18).
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131 FORCE (lb) . 005 t . 004 . 003 03 (KZ) ( b ) Figure 5.6. Display of Impulse and its Fourier Transform
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132 FORCE (lb) 2.0 1.0 0.0 1.0 2.0 0.0 0.1 t (sec) 0.2 0.3 1 1 i 1 1 1 ACCELERATION (in/sec ) . ^ ^ i _ 1 Â• i 1 0 .0 oÂ‘l > 0. t (sec) 2 ' ' Â’ 0. 3 ( b ) Figure 5.7. Manipulator's Response to an Impulsive Exciting Force.
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133 MAG (in/sec^lb) ( b ) Figure 5.8. Curve Fitting of System Transfer Function.
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134 / ( a ) ( b ) 40.64 Hz ( d ) Figure 5.9. The Manipulator's Mode Shapes.
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135 Figure 5.10. The Representative Four DegreeofFreedom Model.
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136 Figure 5.11
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137 MAG h^. (in/sec^lb) u (HZ) ( a ) Measured Transfer Function MAG (in/sec^lb) 0) (HZ) ( b ) Modeled Transfer Function Figure 5.12. Direct Transfer Function at 4Z.
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138 7 . 2 UlIAG. (in/sec lb) CO (HZ) ( a ) Measured Transfer Function 7 IMA.G (in/ sec lb) CO (HZ) ( b ) Modeled Transfer Function Figure 5.13. Transfer Function at 2Z with Additional Mass.
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139 Table 5.1. Coordinates of the Measured Points. Point X*(.in.) 1 74.80 2 34.85 3 20.85 4 34.85 5 22.70 6 4.85 7 11.00 8 15.00 9 15.00 10 23.00 11 23.00 Y*Cin.) z*(in.) 0.00 64.25 0.00 62.40 0.00 61.75 0.00 62.40 0.00 53.60 0.00 40.75 0.00 47.25 0.00 25.00 0.00 0.00 0.00 25.00 0.00 0.00
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140 Table 5.2. Modal Parameters. u^(HZ) K^( Ib/in) Â— 2 Mj,(lbsec /in) ^r 10.09 5.435x10^ 1.35 0.0138 24.83 7.497x10"^ 3.08 0.031 34.24 5.896x10^ 127.38 0.06 40.64 2.005x10"^ 307.45 0.055
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141 Table 5.3. Mode Shapes of the Example Manipulator u r Poi nt AX* AY* AZ* 1 4.331E1 0.00 1 .OOOEO 2 4.281E1 0.00 3 .900E1 3 6 .265E1 0.00 9 .259E2 4 3.381E1 0.00 3.995E1 10.09 Hz 5 3 .247E1 0.00 2 .384E1 6 9.732E2 0.00 7.083E2 7 8 .231E2 0.00 1 .074E2 3 3 .OOOE2 0.00 2.484E2 9 4.975E3 0.00 2.369E2 10 2.913E2 0.00 1 .563E2 11 6.714E3 0.00 2.142E2 1 1.963E1 0.00 1 .OOOEO 2 3 .438E1 0.00 4.315E1 3 1.202E1 0.00 1 .091E0 4 5 .040E1 0.00 4 .400E1 24.83 Hz 5 5 .131E1 0.00 2.390E1 6 2.759E1 0.00 l.OlOE1 7 2.611E1 0.00 1.874E2 8 1.410E1 0.00 9 .297E2 9 4 .344E2 0.00 1 . 155E1 10 1 .219E1 0.00 6.804E2 11 3 .705E2 0.00 7.469E2
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142 Table 5.3. Continued Â“r Point AX* AY* AZ* 1 2.213E0 0.00 1 .OOOEO 2 2.629E0 0.00 4 .398E0 3 3.552E0 0.00 6 .732E0 4 1.347E0 0.00 4 .309E0 34.24 Hz 5 1.992E0 0.00 2.876E0 6 3 .494E0 0.00 1.556E0 7 3.679E0 0.00 2.032E0 8 2.083E0 0.00 5 .229E1 9 7.773E1 0.00 4 .294E1 10 1 .796E0 0.00 2.725E0 11 6.361E1 0.00 2.753E0 1 8 .585E1 0.00 1 .OOOEO 2 2 .972E0 0.00 2.748E0 3 6 .299E0 0.00 2.690E0 4 1 .304E0 0.00 2.339E0 40.65 Hz 5 8 .651E2 0.00 6 .515E1 6 2 .048E0 0.00 1 .944E0 7 2.275E0 0.00 2.153E0 8 1 .602E0 0.00 2.711E0 9 8 .239E1 0.00 2.990E0 10 1.444E0 0.00 1.283E0 11 7.649E1 0.00 1 .539E0
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143 Table 5.4. Identified Parameters i (Ib/in) M. (Ibsec^/in) 2 I . (lbsec in) Zl 1 335431 0 0 2 471433 8.04 911.72 3 1118876 9.68 171.31 4 626203 0.78 1003.87 Table 5.5. Geometrical Parameters j S . . (in . ) 33 a.^dn.) (deg. ) 0 j (deg. ) 1 0 0 270 0 2 0 37.25 0 270 3 0 44 0 54. 89 4 0 40 Â— 33. 34
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144 Table 5.6. Comparison of Predicted and Measured Natural Frequencies for Configuration A (Jj Predicted Measured Error 1 10.28 10.09 1.8% 2 23.52 24.83 5 . 3% 3 36.85 34 . 24 7.6% 4 74.46 Â— Â— Table 5.7. Comparison of Predicted and Measured Natural Frequencies for Configuration B Oj Predicted Measured Error 1 11.67 11.22 3 . 9% 2 26 . 42 28 . 30 6.6% 3 44 . 08 39 . 67 11.1% 4 78 . 40 Â— Table 5.8. Comparison of Flexibilities Predicted and Measured Static for Configuration A Predicted (in/lb) Measured (in/lb) Error 2 .07x10Â“"^ 2 . 06xl0~Â‘^ 0 . 48%
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145 Table 5.9. Comparison of Original and Improved Natural Freauencies for Configuration A 0) Original Improved Improvement 1 10.28 11.23 9.2% 2 23.52 25.76 9.5% 3 36.85 38.22 3.7% 4 74.46 76.26 2.4% Table 5.10. Comparison of Original and Improved Natural Frequencies for Configuration B to Original Improved Improvement 1 11.67 12.43 6.5% 2 26.42 28.44 7.6% 3 44.08 45.05 2.2% 4 78.40 81.46 3.9%
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CHAPTER VI ILLUSTRATIVE EXAMPLES This chapter details the application of the mathematical model to a common industrial robot by computer simulation. Additionally, the system deformation due to inertia loads is determined using both the quasistatic formulation and the complete dynamic equations of motion. The comparison of the system deformation deduced from the two methods at several operating speeds illustrates the effect of the manipulator's operational frequency on the system deflection. 6 . 1 Description of Example Manipulator The Cincinnati Milacron manipulator represents a common type of manipulator in industrial use today. A drawing of its physical form and its six controllable rotational freedoms is shown in Fig. 6.1. It is an RRRRRR manipulator, where the R denotes a revolute joint and RRR indicates that these three successive joint axes are parallel, and therefore allows only planar motion. The def ormational freedoms of the arm are represented by twelve pseudoinputs, through 3^_2' shown in Fig. 6.2. The 146
PAGE 154
147 ]T6volut6 ps Â© udOÂ“ i npu t s / ^ ' ^ 2 ' ^6^ ^10^ snd ^^ 2 ^ describe the deformat ional motion of the manipulator's joints about their axes of rotation. The motion of the base housing relative to the foundation and flexibility of the housing components are assumed to be negligible. The bending deformation of the upper arm in the local XZ plane is described by the prismatic psuedoinput , and revolute pseudo~input The torsional deformation of this component is represented by 3^. The prismatic and revolute pseudoi nputs , 3^ and 3g, describe the bending deformation of the lower arm in the corresponding local XZ plane. Parameter, represents the torsional deformation of the lower arm. The bending deformation of the upper and lower arms in their corresponding XY plane and the axial deformation of these components are neglected. Furthermore, the link components associated with the manipulator's wrist are assumed to be rigid. Figure 6.2 illustrates the kinematic representation of the example manipulator and Table 6.1 lists the kinematic parameters for this arm. The inertia content of the example manipulator is described by the mass, rotational inertia, and location of the mass for each link. Table 6.2 presents the list of these inertia properties. The stiffness content of the manipulator arm is described by the stiffness matrix for each jointlink combination. These stiffness matrices are listed in Table 6.3.
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148 6 . 2 System Natural Frequencies and Transfer Functions In this section, the natural frequencies and the corresponding mode shapes of the arm at a selected configuration are determined. The system inertia matrix is obtained from Eqn . (2.44) and the stiffness matrix is constructed using the submatrices associated with each jointlink combination as described by Eqn. (2.57). The resonance frequencies and eigenvectors are then computed using the QR method. Tables 6.4 and 6.5 list the system eigenvalues and eigenvectors, respectively. The first six mode shapes of the manipulator arm are shown in Fig. 6.3. The 5.6 HZ mode is the motion of the arm in the X*Z* plane produced by the in phase vibration of the shoulder and elbow joints. The 9.5 HZ mode is the motion of the arm in the X*Y* plane caused mainly by the oscillation of the first, third, and seventh psuedoinputs . The 22 HZ mode is a higher mode to the 5.6 HZ with the elbow joint vibrating out of phase to the motion of the shoulder joint. The 24 HZ mode is the motion of the arm in the X*Y* plane and is the result of the flexure and torsional vibration of the upper and lower arms. The 34 HZ mode is a higher mode to the 9.5 HZ, the 37 HZ mode is the vibration of the arm in the X*Z* plane dominated by the oscillation of the elbow and wrist joints. In order to determine the transfer functions for this
PAGE 156
149 manipulator, the system is excited by a harmonic force, _fj^, applied to a point H located on the arm's end effector. The magnitude and direction of the exciting force are defined by the equation fj, = (6.1) The damping matrix is assumed to be proportional to the stiffness matrix by a factor of .0005. Using the procedure described in Section 3.3, the direct transfer function associated with point H is determined. The components of this transfer function in terms of the global coordinate system are shown in Fig. 6.4 (real parts) and Fig. 6.5 (imaginary parts). This transfer function indicates that' the magnitude of system vibration associated with the sixth mode and above is less than two percent of the system vibration associated with the first five modes. Therefore, only five modes of vibration contribute to the system oscillation and the other vibrational modes are insignificant . 6 . 3 Manipulator's Dynamic Response As mentioned previously, any discontinuty in a time derivative of the motion of the manipulator arm will excite one or more vibrational modes of the system. In order to
PAGE 157
150 investigate the reaction of the example manipulator to a rapid change in the velocity of the endeffector, an impulsive force, ( t ) , is applied to the point H located on the manipulator's terminal device. The exciting force is described by the equation ( t ) = ( 1 , 1 , 1 ) ^5 ( t ) (6.2) The motion of point H, produced by this exciting force can then be obtained by following the procedure discussed in Section 3.4. To examine the effect of each mode of vibration on the system response, the dynamic response is computed by considering the superposition of several consecutive normal modes. Figure 6.6a represents the X component of the response produced by the first four modes of vibration. The dynamic response in the X direction produced by the first six modes of vibration is shown in Fig. 6.6b. A comparison of these results with the response of the system produced by all twelve modes of vibration (Fig. 6.6c) indicates that the dynamic response can be adequately described by at most six modes of vibration. The components of the response in Y and Z directions are shown in Figs. 6.7 and 6.8, respectively. In order to determine the system response produced by a rapid change in the acceleration of the arm's endeffector, the exciting force, is represented by a step function.
PAGE 158
151 where f H = ( t ) = ( 1 , 1 , 1 ) ( t ) (6.3) Figures 6.9 through 6.11 respectively illustrate the dynamic response of point H in the X*, Y* , and Z* directions, produced by the first two, first four, and all twelve modes of vibration. By comparing these results, we can conclude that only the first two modes of vibration are sufficient to define the response of this system due to a step function exciting force. The system response due to the impulsive exciting force, FÂ„^(t), is determined using the nonlinear Â— ti differential equations of motion defined by Eqn. (3.77). A comparison of this response and the one deduced from the linear equations of motion illustrates the effect of Coriolis and centrepital accelerations on the manipulator's dynamic response (Figs. 6.12 through 6.14). As seen in these Figures, the effect of the nonlinear terms on the system response is insignificant. In conclusion, the vibration behavior of a manipulator arm can adequately be discribed by a linear superposition of the first few modes of vibration. Therefore, it is sufficient to determine the first few system eigenvalues and eigenvectors to investigate the behavior of the arm due to an exciting phenomena.
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152 6 . 4 Deformation Due to the Gross Motion The inertia loads resulting from the nominal motion of a manipulator arm cause the arm to deform about its rigid body position. The magnitude and behavior of the deformational motion depend on the operational speed of the arm. A manipulator's operational speed is described by a parameter, to, which indicates the rate of moving a load from one position to another and back. This section presents an investigation of the deformation of the example manipulator due to its gross motion at several operational speeds. Assume that the manipulator task is to transfer an object from one work station to another along a straight line in a plane parallel to the X*Y* plane. The initial position of the hand is described by a point P^(70,0,59) and the final position is specified by P2(77,7,59). The path length function relative to time is defined by the 345 polynomial shown in Fig. 6.15. Three operational speeds are considered for obtaining the arm's deformation. In the first case, the representative frequency of operation, is chosen to be 20 percent of the fundamental frequency. The displacement of manipulator's endÂ— effector from its specified path function is computed using the quasistatic analysis. This deformation is then compared with the resultant deformation obtained from the complete dynamic equations of motion (Eqn.
PAGE 160
153 (4.11)). Figures 6.18 through 6.20 respectively show the hand deformations in the X*, Y* , and Z* directions obtained from the two methods. In the second case, the operational frequency is assumed to be 10 percent of the manipulator's fundamental frequency (Fig. 6.16). The hand deformations for this case are shown in Figs. 6.21 through 6.23. Finally, the hand deformations produced by the gross motion of the arm at a frequency of . 022 oJj^ (Fig. 6.17) are presented in Figs. 6.24 through 6.26. At the operational frequency of 0.2a)^, the resultant deformations obtained from the quasistatic analysis are off from the actual deformations by an average of seventy percent (Table 6.6). This error will reduce to forty percent as the operational frequency is decreased to (Table 6.7). Table 6.8 shows that the error at the operational frequency of 0.022w^ is only eight percent. Thus, the comparison of the resultant deformations obtained from the quasistatic method and the complete dynamic equations of motion indicates that as the operating speed increases with respect to the fundamental frequency, the manipulator will oscillate about its static equilibrium position. However, the system deflections can be predicted by the quasistatic method, if the mechanism operating speed is less than five percent of the system fundamental frequency .
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154 Figure 6.1. Representative 6R Manipulator.
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155 CN c/ll Figure 6.2. Kinematic Representation of the Example Manipulator.
PAGE 163
156 Figure 6.3. Display of the Mode Shapes.
PAGE 164
157 Re (in/sec 2 lb) a 23 IS Z7S 271 H6H0 ) (rad/sec) Figure 6.4 Direct Transfer Function at H
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158 0 ) (rad/sec) I , Â“3 "13, IMAG (in/sec^lb) nn r~ "ISt =L Â•281. "3SI~37L ~4pl z S3 las ^ ;rad/sec 271 to (rad/sec) Figure 6.5. Direct Transfer Function at He 'te+
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159 (in.) t (sec) ..X , . , ( a ) Four Modes of Vibration h ( in . ) t (sec) ( c ) Twelve Modes of Vibration Figure 6.6. Manipulator's Response to a Unit Inpulse.
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t (sec) y , . , ( b ) Six Modes of Vibration H ( in . ) ( c ) Twelve Modes of Vibration Figure 6.7. Manipulator's Response to a Unit Impulse.
PAGE 168
161 (in.) ( b ) Six Modes of Vibration (in.) ( c ) Twelve Modes of Vibration Figure 6.8. Manipulator's Response to a Unit Impulse.
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162 (in.) (in.) ( a ) Two Modes of Vibration (in.) ( b ) Four Modes of Vibration t (sec) ( c ) Twelve Modes of Vibration Figure 6.9. Manipulator's Response to a unit step,
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( in . ) 163 ( a ) Two Modes of Vibration (in.) Â„v , . . ( b ) Four Modes of Vibration bV (in.) t (sec) ( c ) Twelve Modes of Vibration Figure 6.10. Manipulator's Response to a Unit Step.
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164 (in.) ( a ) Two Modes of Vibration (in.) ( b ) Four Modes of Vibration (in.) ( c ) Twelve Modes of Vibration Figure 6.11. Manipulator's Response to a Unit Step.
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165 (in.) 0.33 0.00 0.13 0.10 0.23 0.29 0.3M0.29 0.â€¢* 3.30 0.S 0.S0 t (sec) ( a ) Linear Equations h"" (in.) 0.33 0.00 0.13 0.10 0.23 0.29 0.3f 0.29 0.44 0.30 0.S 0.S0 t (sec) ( b ) Nonlinear Equations Figure 6.12. Manipulator's Response to a Unit Impulse.
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166 (in.) 0.22 0.3& 0.13 a.ia 0.22 0.2S 0.3Â‘f 0.23 0.44 0.50 0.S 0.S0 t (sec) ( a ) Linear Equations f 7 (in.) 0.03 0.0B 0.13 0.13 0.23 0.23 0.34 0.33 0.44 0.S0 0.S 0.S0 t (sec) ( b ) Nonlinear Equations Figure 6.13. Manipulator's Response to a Unit Impulse.
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167 (in.) 0.33 0.0B 0,13 a.ia 3.23 0.29 0.34> 0.33 3. 44 3.S0 0.S 0.S0 t (sec) ( a ) Linear Equations (in.) 0.33 0.^ 0.13 3.13 3.23 0.29 0.34 0.39 3.44 3.^ 0.S 0.53 t (sec) ( b ) Nonlinear Equations Figure 6.14. Manipulator's Response to a Unit Impulse.
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168 s (in.) 0.3G 9.1H 0.23 B.3S. 0.H1 ( a ) Position s (in/sec) 0.3S 3, It 0.E3 3.3a fl.Hl ( b ) Velocity s (in/sec ) 0.3B 0. It 0.23 3.33 0.HI ( c ) Acceleration Figure 6.15. The Path Function for Operational Speed of (sec) (sec) (sec) 0 . .
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fa p >^ 169 s (in.) ( a ) Position s (in/sec) ( b ) Velocity s (in/sec^) ( c ) Acceleration Figure 6.16. The Path Function for Operational Speed ot O.ltiJ^.
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170 s (in.) ( a ) Position ( b ) Velocity s (in/sec ) ( c ) Acceleration Figure 6.17. The Path Function for Operational Speed of 0.022oj^.
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171 (in.) ( a ) QuasiStatic Analysis (in.) ( in . ) ( c ) The Difference Between ( a ) and ( b ) Figure 6.18. Hand Deflecton for Operational Speed of 0.2ojj^.
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172 ( in . ) 0.SS 0.x 0.11 0.15 0.13 0.23 0.Z7 0.32 0.35 0.H0 (sec) ( a ) QuasiStatic Analysis (in.) ( b ) Dynamic Analysis AH^ (in.) ( c ) The Difference Between ( a ) and ( b ) Figure 6.19. Rand Def lectio for Operational Speed of O.lo)^.
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173 (in.) ( a ) QuasiStatic Analysis (in.) ( b ) Dynamic Analysis AH^ (in.) (sec) (sec) (sec) Figure 6.20. Hand Deflection for Operational Speed of 0.2uj^.
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174 (in.) ( a ) QuasiStatic Analysis a.23 3.23 a. IS a.2H 0.25 0.25 a.41 a.H7 J.5+ 3. S3 ( b ) Dynamic Analysis AH^ (in.) ( c ) The Difference Between ( a ) and ( b ) Figure 6.21. Hand Deflection for Operational Speed of (sec) (sec) (sec)
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175 (in.) ( a ) QuasiStatic Analysis ( in . ) ( b ) Dynanic Analysis (in.) ( c ) The Difference Between ( a ) and ( b ) Figure 6.22. Hand Deflection for Operational Speed of 0.10)^^. (sec) (sec) (sec)
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176 (in.) ( a ) QuasiStatic Analysis (in.) Figure 6.23. Hand Deflection for Operational Speed of O.lco^^. (sec) (sec) (sec)
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177 d. Â“3.00007 0.00015 0.000H2 0,00031 (in.) 0. 00032 i_ 0. 000^7 i i 0.000SI 0.000631 0.0^Â’[ 1 M ; 1 i ; ! M i ; M ' I ^ I , i i ! ! ' M ; ^ ; M I I I i I > i I i : ; i I i ! l i ! ' ; . i I ! i ! I I i i M : i ^ i i I I I ! I ; i i Â’ I I I I : i i ! i i I I I i i i ! I : i ! I i ! i I i i I i ^ ^ i ^ j ! ! I i i ^ ^ i ! ! 3.00 0.12 0.2S 0.38 0.SI 3. S3 0.75 0.S8 1.31 I.IH0.06 0.19 0.32 0.4H 0.S7 0.69 0.S2 0.95 1.07 1.20 ( a ) QuasiStatic Analysis ( in . ) 0. 00001 0 . 00007 !. 0.030151 0 , r 0.000321 0.303H1! 0. 000^^9 L 0. 000571 0.000661 Â“0.00^7HL ^ ! ; j Â‘ I 1 ; ! i 1 i i ! ! I i \ ! : I ; I ; I ' I ! I ^ ! I ! : ' ; : \J : I ' i i i ^ I ^ I i ! i i I i' i i i ! i i : i I i ! 1 1 : . \ * i I ' i i ! I i i i ' ! ^ i ! i ! i I I i M I 1 ; ; ; I : i\! i i i I ! ! ! I ! J i i ; i \ ! g i I I 1 : I 1 i 1 ! ' Â‘ i i 1 i 0.00 0.13 0.25 0.38 0.51 0.S3 0.75 0.38 3.06 0.19 3.32 3.4H 0.57 0.53 3.32 3.; ( b ) Dynamic Analysis .01 l.lb 1.37 1.20 AH^ (in.) 0.36 0.19 3.32 0.4Â“ 0.37 0.S9 3.32 0.3S 1.37 1.23 ( c ) The Difference Between ( a ) and ( b ) Figure 6.24. Hand Deflection for Operational Speed of 0.022oj^. (sec) (sec) (sec)
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178 ( in . ) ' ' 1 ' 1 : ' = i ' 1Â”^ 0.00i0t ^ i ! ! ^ 'i ! i ! : 1 i i ! 1 : i i ! ' 1 r 1 ! 1 i I 1 = 1 ! i 1 0 . 0007Â‘ ^ ^ L=^_J 1 1 ! 1 ! i 1 ; j : i ! 0.00Â»;1 i ! ! r / I ! i i 1 I 1 i ! 1 ^ i I j ! i ! ! i f i 0. 0003L_^_J:L1__ I 1 i i 1 1 i 1 1 0 cwia\\ y i j . ..J. ! 1 1 1 1 1 i _ 0.0fl(wLr' I i ^L ! i \ ' 1 1 1 1 1 ! 1 , f f i i1 i 0.30 0.13 0.25 Z.3S 0.51 0.S3 0.7S 0.8S 1.01 1,140.06 0.12 0.32 0.HH 0.37 0.33 0.S2 0.9S. 1,07 1.20 ( a ) QuasiStatic Analysis ( in . ) ( b ) Dynanic Analysis AH^ (in.) ( c ) The Difference Between ( a ) and ( b ) (sec) (sec) (sec) Figure 6.25. Hand Deflection for Operational Speed of 0.022co^.
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179 (in.) ( a ) QuasiStatic Analysis (in.) ( b ) Dynamic Analysis AK^ (in.) ( c ) The Difference Between ( a ) and ( b ) (sec) (sec) (sec) Figure 6.26. Hand Deflection for Operational Speed of 0.022oj^.
PAGE 187
j 1 2 3 4 5 6 7 8 9 10 11 12 180 Table 6.1. Kinematic Parameters of the Example Manipulator Sjj(in.) a^j^dn.) ajj^(deg.) e^Cdeg.) 59 0 0 0 0 0 0 0 0 0 0 13 0 40 0 0 0 40 0 0 0 8 0 0 90 0 270 90 270 0 270 90 270 90 90 0 0 0 90 90 240 0 90 90 300 90 0
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Table 6.2. Inertia Properties for the Example Manipulator 181 Â•m N M 800 700 700 40 40 40 in] I . y] 2 sec 0 700 700 40 40 40 XI rH X M 0 300 300 20 20 20 'c Â•H CN Â•1^ 0 2 QJ 01 1 5.7 0.67 0.62 0.31 0.21 0.13 (lbÂ•rÂ— 1 tsi o o o CN m 1 I T3 Â•H 0 Â— . ii tÂ— 1 Â• +J c C >H H 0) Â— u o o o o o o Â•n ?< Â•1Â—1 1 0 5 0 9 0 10 3 11 0 12 0
PAGE 189
182 Table 6 [*^23] ."34] = 3. Stiffness Matrices [^5x10Â® J 20x10^ 0 0.55x10Â® Symmetric 20x10Â® 0 0.55x10Â® Symmetric r 6 Â“1 Ul 5x10 1 1 = 5x10Â® 0 0 10.85x10Â® 0 2.89x10Â® 0 48x10Â® 0 0 10.85x10Â® 0 2.89x10Â® 0 48x10Â®
PAGE 190
183 Table 6.4. System Eigenvalues r o)^(HZ) r u^(KZ) 1 5.62 7 45.22 2 9.50 8 54.97 3 22.19 9 64 . 80 4 24.15 10 72.99 5 34.42 11 257.44 6 37.05 12 485.50
PAGE 191
Table 6 . 5 . System Eigenvectors 184
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185 Table 6.6. Hand deformation for operational speed of 0.2ca^ t (sec) dynamic analysis 1 AH 1 xlO ^ (in. ) quasistatic 1 AH 1 xlO ^ (in. ) percent error .000 .000 . 000 .000 . 025 3133.517 16055.681 Â“412.385 .050 18892 , 943 26143 . 998 33.406 .075 39582.852 31456 . 987 20.528 . 100 51405.873 32480 . 618 36.815 .125 46868. 065 30230.267 35 . 499 . 150 2 30 83 . 078 24323.010 11.591 . 175 12489.363 17295.326 Â“38.480 .200 9778 . 729 8250 .398 15.629 .225 9201.661 2543 . 651 72.357 .250 8218 . 405 12522.078 52.366 .275 9366 . 332 21920.976 134. 040 .300 23233.425 23773.706 *23.368 . 325 43292.683 31745.485 26.672 . 350 51218 . 246 28530 .997 44.198 .375 39471 . 156 1 843 0 . 902 53.305 . 400 1070 9 . 320 1719.814 33.942
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186 Table 6.7. Hand deformation for operational speed of O.lu^ t (sec) dynamic analysis 1 Ah xlO"^ (in. ) quasistatic 1 AH 1 xlO ^ (in. ) percent error .000 .000 . 000 .000 . 040 710.825 1624.717 "128 . 568 . 030 3422.217 3054 . 532 10 . 744 . 120 5288. 867 4082.293 22.813 . 160 4887 .03 1 4744.360 2.909 . 200 4214. 801 5093 . 020 "20.837 . 240 5258.330 5216.552 . 793 . 280 6254 . 241 5053.317 19.202 . 320 4690 . 716 4779 . 259 1.888 . 360 3338.311 4222.004 26 . 471 , 400 3 581 . 51+3 3513.464 1.901 . 440 3695 . 945 2767.252 25.127 . 480 1960 . 470 1313.481 7.243 . 520 325.449 849.641 151.067 . 560 439.289 238.423 34.343 . 600 730.696 1309.819 79.256
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187 Table 6.8. Hand deformation for operational speed of 0 . 022 ( 1 )^ t (sec) dynamic analysis quasistatic AHx 10 ^(in.) 1 Ahx 10 ^(in.) percent error .000 .000 .000 . 000 .080 49 . 642 46.325 6.683 . 160 114.233 101.545 11.107 . 240 ' 147.353 149.613 1.191 . 320 201.276 190.903 5.151 . 400 225 . 750 225. 319 .030 . 430 265.164 255 . 144 3.779 . 560 275.438 273.478 1.104 . 640 302.985 296.554 2 . 122 . 720 306.459 309 . 722 " 1.065 . 800 32 5 . 191 318.314 2.115 . 830 323.377 325.673 .711 . 960 328.918 326.041 .875 1 . 040 318.727 322.672 1.233 1 . 120 313 . 482 315.332 .832 1 .200 301.956 305.768 1.263
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CHAPTER VII CONCLUSION It has been demonstrated that a lumpedparameter model can provide a reliable representation of flexible manipulator arms. This model drastically simplifies the dynamic analysis of these complex mechanisms. A generic formulation has been presented using the geometric influence coefficients to describe the vibrational behavior of the system. The def ormational freedoms of the arm are represented by a set of imaginary inputs referred to as psuedoinputs . The geometric coefficients for these imaginary inputs are then generated. This allows the formulation of the equations of motion describing oscillatory motion of the system. This formulation is simple to implement on a digital computer. The designer need only consider the dominant, flexible components which allows him to select a simple, yet sufficient representation of the mechanism. The lowest natural frequencies of vibration are a good indicator of the overall structural integrity of the manipulator system and determine the maximum cycle speed for performing precise arm motion. Therefore, the fundamental 188
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189 frequency is utilized as a design criteria to increase the system rigidity, leading to precision at higher operational speeds. This frequency is described explicitly in terms of local parameters to demonstrate the effect of each parameter on the magnitude of fundamental frequency. System flexibility can then be decreased by improving the most significant local parameters. However, drastic changes of local parameters may cause a change in the overall vibration behavior of the system; thus the improvement of the local parameters must be limited in order not to alter the vibrational behavior of the mechanism. The small motion response of manipulator arms can be described by a set of linear, differential equations by neglecting the Coriolis and centripital accelerations. The investigation of the effect upon the system response of these nonlinear terms relative to the linear terms, indicated that the nonlinear terms comprise less than five percent of the total small motion response of these mechanisms. Furthermore, it was demostrated that the system response can be adequately described by the superposition of the first few modes of vibration. Although a few modes of vibration can be utilized to determine the small motion response, all significant deformat ional freedoms must be considered for developing the equations of motion. For precision control of a manipulator, it is necessary to predict and compensate for the system deformation due to
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190 the inertia loads produced by the nominal motion of the arm. As shown in Chapter IV the nominal motion of the arm has no significant effect on the manipulator's natural frequencies. Thus it is sufficient to determine the resonance frequencies in a series of stationary configurations. The system deflection can be predicted by a quasistatic method if the mechanism operating cycle speed is less than five percent of the system fundamental frequency. However, the complete dynamic equations of motion must be used for higher operational speeds. A numerical technique was attempted for solving the complete dynamic equations of motion. This was unsatisfactory, since it required long computational times. In order to overcome this difficulty a more practical technique should be developed for solving these equations. It may be convenient to assume that the dynamic equations represent a vibratory system with nonproportional damping and then apply the modal analysis method to solve these equations. Further investigation is required to demonstrate the effect of various path functions (i.e. constant acceleration curve, trapezoidal curve, etc.) on the system deformation . Experimental modal analysis is a reliable, and effective method for identifying the local, dynamic parameters of a manipulator arm. The application of this method has been demonstrated by defining the local parameters of an industrial manipulator arm. A four degree
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191 of freedom model was chosen to represent the planar motion of T3776 industrial robot. The comparison of the vibrational behavior of the representative model and actual system showed that this model can be utilized to describe the dynamic characteristics of T3776 robot. Although the final results were quite satisfactory, there was a great deal of difficulty associated with numerical evaluation of the local parameters. It is suspected that the main reason for this difficulty is the inadequacy of the proposed model. A model which includes the bending flexibilities of the upper and lower arms may prove to be an adequate representation for this manipulator arm. Furthermore, the vibration measurements were performed using a unidirectional acclerometer which makes it impossible to measure the motion of a specific point in three directions. Therfore, to avoid errors associated with the description of the motion of a specific point in space, a tridirectional accelerometer should be used.
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APPENDIX A ARM ACCELERATION AND SECONDORDER INFLUENCE COEFFICIENTS The angular acceleration of a body in space is defined by the time derivative of its angular velocity. For serial manipulators, the angular acceleration of link jk can then be obtained from ~ dT Â— Â” ^dT . (A.1) Differentiating the d dt j kl n nth component of [Gj)^] results ] Â£n ' n _< j ; nrevolute 1 0 , otherwise in (A. 2) where Sr, = Â— ( n1 ) n n1 X Sn = Z 0ra Â§m ^ Â§Ji (A. 3) m= 1 Equation (A. 2) can be rewritten in a matrix form by introducing the secondorder rotational influence coefficient 192
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193 where, is an N x N array and each component of this array is a vector of length three. The components of [Hj]^] are defined by the equation ^ 'It ^ tSjkIn Â• (A. 5) Substituting Eqn. (A. 4) into (A.1), we can write a jk = [Hjk]l (A. 6) From Eqns. (A.2), (A.3), and (A.5), the components of the secondorder rotational influence coefficients are evaluated as [Hjkl m; n m < n <_ j ; m,nrevolute otherwise . (A. 7) The components of the secondorder influence coefficients, [Hjkl r for a flexible manipulator which is deflected from its rigid body position are simply defined by the equation [Hjkl m ; n X Sji m < n <_ j ; m,nrevolute otherwise . (A. 8)
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194 Analogously, the transitional acceleration of a point, P, can be defined by the equation ^ = iT [Hpli + [Gp]i . (A. 9) The components of the secondorder translational influence coefficients, [Hp] , are defined by the equation ^^P^m;n " ^ ^It C^pln . (A. 10) These terms can be readily evaluated by differentiating the expressions in Eqn. (2.28) with respect to time and rewriting the time derivatives in terms of the joint velocity. Table A.1 represents the expressions for the secondorder influence coefficients for rigid manipulators which depend on the joint types and sequence. These expressions for flexible manipulators, which are deflected from their rigid body position, are shown in Table A. 2.
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Table A.l. SecondOrder Influence Coefficients 195 0) d rH Â«T3 > tn 0 p (0 tÂ— I d a *H c d 2 P 0 p o tn C O Â•H P o *H P 41 cn 0) Pi c 41 Â•Â• C 0) Â•H Qj O >1 C) 41 g 41 Â•Â• C 0) r4 Qj O >1 C) 41 o g >1 cn 03 41 c 0) H o *H Ml 44 0 o u o 0 c 03 d c M I Â— i d c o Â•H 41 d 41 0 Pi cnf X cnf v c V g A c p 0 c a' c c g" g" rÂ— I IÂ— H r4 fH d d Pi Pi Pi Pi C g n 03 41 C 0) *H u *p 41 44 03 O U 03 U C 03 d c M d c 0 41 d r4 03 G d p E4 "^G Pil piT 1 il X rH X cni cnf G cnT cnI X X o X X o g G G cnT cni cnI cnT V I C a' Pi 1' V C (X Pi o o V I c V Pi Pi a' V C Â•tÂ— I Vl c V g A G G Pi Pi Pi G g T 04 C >ui>u
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Table A. 2. SecondOrder Influence Coefficients for Flexible Serial Manipulators 196 0) 3 fH tt 3 > Cfl C O H +J O Â•H Sl P UJ 0) PS 4J .. c 1 h) P P Â” c o Â•H P O >1 Id p o X! e >1 w m p c 0 ) 41 44 O O u (U u c o 3 3 C O Â‘H 4> 3 P O PS c < sol X g < col V I c V e PS A 3 P O c c c e e iH IÂ— I r rH 3 3 PS PS p p p c e n 3S to P c
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APPENDIX B EXPRESSIONS FOR INERTIA POWER MATRIX This appendix presents the calculation of the inertia power matrix, derived by Thomas [29]. This matrix is defined by the equation [Pn] [I*]n;) 1 _L_ 2 [I*] . (B. 1 ) An alternative inertia power modeling matrix [Pn*l is introduced which is equivalent to [Pn^ i^ sense that 1'^ [Pn*]$ ~ [Pn^'jÂ’ (B.2) The components of this new matrix have a simplified form expressed in terms of the geometric influence coefficients and the inertia properties of the link. The component in row I, column m of [P^l is [Pn] = 3 Z ;m 3$ m [ 1*1 _ J_ 2 3$, I*' Z ;m (B.3 197
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198 Substituting the expression in Eqn. (2.44) for the scalar components of [I*] , and recognizing that 3 34>m (TjT[Gjk]n) = TjÂ”^ { [H j)^] m; n jkl n ^ Tj [Gjk]ni Tj TjTjT = [I] [Hni ] m;n [Hcj ] n ;m (B.4) (B.5) (B.6) (B.7) We can express the scalar components of [Pnl in the expanded form [P ] Z {Mj ([H^j] [Gcjln ^ Â£ ;m mto (B.8) m;Â£ 1 T 2 m; n [Gc j ] _ 1 + ~ t^jkl t^jkln + [Gjklm [Gjkln) j ([Hjk]^.^ tSjkln t^jkl^)^ C^j]^] [Gjkln, j [Gjkl^T [nj^] [Gjkln [Gjklm))
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199 Since [P^l is used in a quadratic form, it is possible to substitute the new matrix [Pnl defined by [Pnl = i ( [Pn] + tPn]^) (^.9) such that ^ [Pn^l Â“ [Pn]^ (B.10) The secondorder rotational influence coefficients for serial manipulators have the characteristic J^m;n n;m (B.1 1 ) This result and Eqn. (B.8) give the general component of [Pn] as = 1 + [Pnl J (B. 1 2) N ^ = Z {Mi [Gcjln j = 1 + J [Gjklm'^ fnjkl "2 f^jkl ['^jk^n t'^jk^m}
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200 Finally, it is convenient to return to a nonsymmetric form and define the inertia power modeling matrix [Pn*l as follows: + [Gjkl [^jkl t*^jk^m ^ Â• The scalar components of the inertia power matrix for flexible manipulators can then be expressed by the equation A. [Pn*] Â£; m N E {Mj j = 1 [He j ] T 1 ) m [Gc j ] n + [Hikl'^ tnjk] [Gjkln ^ 2 ,;m [Gjk^ [^jk] [Gjk^n } . (B.14)
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APPENDIX C VARIATION OF THE INFLUENCE COEFFICIENTS DUE TO THE MANIPULATOR'S DEFORMATION The vector components defining the geometry of flexible serial manipulators will displace from their original line of action due to the deformation of the /s flexible elements. Consider a vector Q fixed in link jk which is displaced from its original line of action Q. Since the displacements due to the manipulator's deformation are small, we can write Q = Q + AQ . (C . 1 ) The time rate of change of a vector Q fixed in a rigid body jk is defined as Q Â— ^ j k^Q.Â“ ([Gj]^]^) xQ_. (C.2) Using the mathematical definition of time derivative, Eqn. (C.2) can be rewritten as 201
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202 H ' it> a (=3 Canceling At from both sides and recognizing that vector A$ portrays the elastic deformation of the arm, _3, we can write AQ Â“ ( ) X Q_ . (C.4) Substituting Eqn. (C.4) into (C.1) results in Q = Q + ( [Gjk] B ) X Q . (C.5 ) The second term on the right hand side of Eqn. (C.5) is negligible since the components of 6. much smaller than those of vector Q. Thus, Eqn. (C.5) is reduced to Q = Q . (C.6) The components of the influence coefficients are defined in terms of the direction cosines of the manipulator's vector components. Therefore, the result of Eqn. (C.6) leads to the conclusion that fG j k] n = [Gjk] n [Gp] n = [Gp] n (C.8)
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203 [Hiv] = J ^ m ; n m ; n [Hn] P m ; n m ; n . (C.9) (C.10) Furthermore, the scalar components of the system inertia /s ^ ^ ^ matrix [I*] and inertia power matrix [?n*] defined in terms of local mass parameters which are invariant with respect to the motion of the arm, and the geometric influence coefficients. Considering Egns. (C.7) through (C.10), we can conclude that [I*] s m;n m;n rp *] s [Pn*l ^,;m Â£;m . (C.1 1 ) (C.12) These results indicate that the inertia matrix [I*] and inertia power matrix [Pn*l associated with rigid body representation of a manipulator arm can sufficiently describe the inertia properties of the corresponding flexible arm.
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REFERENCES 1. Tesar, D. , and Thomas, M. , "Assessement for the Mathematical Formulation for the Design and Digital Control of Programmable Manipulator System," NSF Grant ENG7820112 and DOE Contract ER785056102, Center of Intelligent Machines and Robotics, Gainesville, FL, 1979. 2. Tesar, D. , and Matthew, G.K., The Dynamic Synthesis, Analysis and Design of Modeled Cam Systems , Lexington Books, Lexington, MA, 1976. 3. Erdman, A.G., and Sandor, G.N., "KintoElastody namics a Review of the State of the Art and Trends," Mechanism and Machine Theory, Vol. 7, November 1972, pp. 1933. 4. Lowen, G.G. , and Jandrasits, W.G. , "Survey of Investigations into the Dynamic Behavior of Mechanisms Containing Links with Distributed Mass and Elasticity," Mechanism and Machine Theory, Vol. 7, November 1972, pp. 317. 5. Imam, I., Sandor, G.N., and Kramer, S.N., "Deflection and Stress Analysis in High Speed Planar Mechanisms with Elastic Links," Transactions of the ASME , Journal of Engineering for Industry , Vol. 95, May 1973, pp. 541548. 6. Sadler, J.P., and Sandor, G.N., "A Lumped Parameter Approach to Vibration and Stress Analysis of Elastic Linkages," Transactions of the ASME , ournal of Engineering for Industry , Vol . 9 5 , May 1973, pp. 549557 . 7. Sadler, J.P., and Sandor, G.N., "Nonlinear Vibration Analysis of Elastic FourBar Linkages," Transactions of the ASME , Journal of Engineering for Industry , Vol. 96, May 1974, pp. 411419. 8. Kohli, D., and Sandor, G.N., " Elastody namics of Planar Linkages Including Torsional Vibration of Input and Output Shafts and Elastic Deflections at Supports," Proceedings of the Fourth World Congress on Machines and Mechanisms , Univers i ty of NewcastleuponTy ne , England , 1975. 9. Imam, I., and Sandor, G.N., "A General Method of KietoElastody namic Design of High Speed Mechanisms," Mechanism and Machine Theory, Vol. 8, April 1973, pp. 497516. 204
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205 10. Sadler, J.P., "On the Analytical LumpedMass Model of an Elastic FourBar Mechanism," Transactions of the ASME , Journal of Engineering for Industry , Vol. 97, May 1975, pp. 561565. 11. Kohli, D. , and Sandor, G.N., "Lumped Parameter Approach for KinetoElastody namic Analysis of Elastic Spatial Mechanisms," Proceedings of the Fourth World Congre ss on Machines and Mechanisms , University of NewcasteluponTyne , England, 1975. 12. Chu, S.C., and Pan, K.C., "Dynamic Response of a HighSpeed SliderCrank Mechanism with an Elastic Connecting Rod," Transactions 'of the ASME , Journal of Engineering for Industry , Vol. 97, May 1975, pp. 542550. 13. Sanders, J.R., and Tesar, D. , "The Analytical and Experimental Evaluation of Vibratory Oscillations in Realistically Proportioned Mechanisms," Transactions oÂ£ the ASME, Jou rnal of Mechanical Design , Vol. 100, October 1978 , 762768. 14. Haines, R.S., "On Predicting Vibrations in Realistically Proportioned Linkage Mechanisms" Transactions of th_e ASME, Jo urnal of Mechanical Design , ol. 103, October 1981, pp. 706710. 15. Sutherland, G.H., "Analytical and Experimental Investigation of a HighSpeed Elast icMembered Linkage, Transactions of the ASME , Journal of En gineering Â— for Industry , Vol. 98, August 1976, pp. 788794. 16. Dubowsky, S. and Maatuk, J., "The Dynamic Analysis of Elastic Soatial Mechanisms," Proceedings of the Four l^ World Congress on the Theory of Machines and Mecnanisn^ , University of NewcastleuponTyne, England, 1975 . 17. Book, W.G., "Modeling, Design, and Control of Flexible Manipulator Arms," Doctoral Dissertation, Massachusetts Institute of Technology, Cambridge, 1974. 18. Beazley, W.G., "The Small Motion Dynamics of Bilaterally Coupled Kiematic Chains with Flexible Links," Doctoral Thesis, University of Texas, Austin, 1978. 19. Tesar, D., and Thomas, M. , "A Review of Modeling, Control, and Design of Manipulators in Terms of the Planar 3 DOF System," Workshop on the Required Research Activity for Generalized Robotic Manipulators, Sponsored by NSF, University of Florida, February 1978.
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206 20. Sunada, W. , and Dubowsky, S., "The Application of Finite Element Methods to the Dynamic Analysis of Flexible Spatial and CoPlanar Linkage Systems," Transactions of the ASME, Journa l of Mechanical Design , Vol. 103, July 1981 , 643651. 21. Sunada, W. , and Dubowsky, S., "On the Dynamic Analysis and Behavior of Industrial Robotic Manipulators with Elastic Members," Transactions of the ASME , Journal of Mechanical Design , Vol. 105, March 1983, pp. 4251. 22. Thomas, M. , and Tesar, D. , "Dynamic Modeling of Serial Manipulator Arms," Proceedings of the 1981 Joint Automatic Controls Conference , University of Virginia, 1981. 23. Kane, T.R., Dynamics , Holt, Rinehart and Winston, New York, NY, 1968. 24. Meirovitch, L. , Elements of Vibration Analysis , McGrawHill Book Company, New York, NY, 1975. 25. Meirovitch, L. , Analytical Methods in Vibrations , Macmillan, New York, NY, 1967. 26. Hurty , W.C., Rubinstein, M.F., Dynamics of Structures , PrenticeÂ— Hall , Englewood Cliffs, NJ, 1964. 27. Wilkinson, J.H., The Algebraic Eigenvalue Problem , Clarendon Press, Oxford, England, 1965. 28. James, M.L., Smith, G.M., Wolford, J.C., Applied Numerical Methods for Digital Computation With Fortran and CSMP, Harper and Row, Publishers , New York, NYi 1977 . 29. Thomas, M. , "Dynamic Model Formulation for RigidLink Serial Manipulator eith Revolute Joints," Master's Thesis, University of Florida, Gainesville, FL, 1981. 30. Sandor, G.N., and Erdman, A.G., Advanced Mechanism Design, PrenticeHall, Englewood Cliffs, NJ, 1984. 31. Ramsey, K.A., "Effective Measurments for Structural Dynamics Testing, Part I," Sound and Vibration, Vol. 36, November 1975, pp. 2435.
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BIOGRAPHICAL SKETCH Fariborz Behi , was born December 13, 1954, to Mr. and Mrs. N. Behi in Shiraz, Iran. After graduating from high school, he left Iran for higher education in the United States. He attended the University of Florida in 1975 and graduated with a Bachelor of Science degree in aerospace engineering in 1978. For the next two years he worked as an equipment engineer for "Iran Helicopter Industry." Graduate studies began at the University of Florida in 1980 in the Department of Mechanical Engineering. He received the Master of Engineering degree in August, 1982. 207
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Delbeiht Tesar, Chairman Graduate Research Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. tv . Josep/h Duffy Professor of'I'lec ahical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Mechanical Engineering
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. "Gary ^ Matthew Associate Professor of Mechanical Engiheering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. T^U Self ridge Ralpl G. Professor of' Computer Information Sciences and This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1985 /li^AiAs/ Gr Dean, College of Engineering Dean, Graduate School

