concepts of tensor analysis in which equations and variables are expressed in such ..... T.Yang and F.Freudenstein, Application of Dual Number Quarternion ...
Proceedings Intern. AM SE Conference "Signals & Systems", Geneva (Switzerland), June 17-19, 1992, AM SE Press, VoL 2, pp. 63-T8
GENERAL TENSOR METHOD FOR THE KINEMATICAL ANALYSIS OF MECHANISMS Y.PALA *
Abstract: In this study, a new analytical method including all coordinate systems is developed for the kinematical analysis of planar and spatial mechanisms. To do this, differential geometry has been used and accelerations of mechanism links have been reduced to the physical components and covariant derivative of a tensor, respectively.
Results have been applied to the three-link mechanism and Stanford
manipulator.
1. INTRODUCTION
Although there exist many different methods used for the kinematical analysis of mechanisms including rigid links connected to each other, the development of tensor method has yet been considered, except some contributions not publicly known [1 0 ]. The reason which makes preferred one method to another especially occurs when the Coriolis accelerations are available or three dimensional mechanisms are studied. For example, in the case of plane kinematics, It has been shown, for the class of problems including Coriolis acceleration, that the complex number method is easily applicable and more effective than other methods. Equivalent method of this method in three dimensions is the quarternion method [1,2]. In spite of the impressive effectiveness of these methods, application of each of these methods Is limited to certain regions, and time consuming. With the purpose of sorting out this difficulty, Ho [3] went into the way of developing a method which he called tensor method. The work of Ho [3] leans upon the work of Kalitsin [10]. On the same line, Angeles [4] proposed tensorial approachment for the kinematical analysis of rigid body systems. Recently, Casey and Lam [4] have extended the development of [3] to the kinematics
* Assistant Professor, Department of Mechanical Engineering, University of Uludag, Bursa, Turkey.
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of rigid body systems including spatial kinematical chains. However, the very important point which must be reminded here is that the term tensor passes in all these works, although the basic concepts of tensor analysis have not been used and equations found have not been put into forms which are independent of coordinate systems used. In stead of these, matrices ( Second Order Tensors ) have been used and the results have been given in the forms written according to the summation convention firstly proposed by Einstein. Whereas, in general tensor analysis, equations are so expressed that they all be valid in all coordinate systems. In the present paper, therefore, the emphasis is placed on using the basic concepts of tensor analysis in which equations and variables are expressed in such a way that they all be valid in all coordinate systems.
For this purpose, a general
tensor method has been developed for any arbitrary coordinate systems (Polar, cylindrical, spherical, parabolic, hyperbolic and In the most general case, Gaussian coordinates). By taking the geometric properties of coordinate systems employed, Riemann metric, ds2= g jjdx'dxJ, and the concept of geodesic curves representing this property have been imported. With the aid of covariant derivative, accelerations of link points have been found. Displacement analysis in this study is not different than the previously established methods since the covariant derivative and physical components are not used. Two illustrative examples using polar coordinates system are included.
2. DISPLACEMENT ANALYSIS OF LINK SYSTEMS
Let the transformation equations which provide the connection between the cartesian coordinate system of the three dimensional Euclides space and curvilinear coordinate system of the same space be given by the equality y ' = y ' ( x V 2^ 3),
(/ =1,2,3)
This, for example, may be the transformation x=rcos0, y=rsin0, z=z between the coordinates r, 0, z and x, y, z. According to this, the transformation between the base vectors gj of the curvilinear coordinate system and e( cartesian coordinate system will
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be given in the following form [6]:
(2)
However, the expression (2) implies the rotation of base vectors of curvilinear coordinate systems and does not include the relation between curvilinear coordinate systems referred to different links. This relation may be found from the connection between the positions of the tangent vectors T1, j i .
d r / dx 1
(3)
| dr/dx ' | at the initial points of the curvilinear coordinate systems.
Indeed, if S*k is a matrix
representing this rotation, then it can be written (4)
Here, when the quantities g'm are known, the quantities g™ can be obtained by means of contraction operator [6]. In order to make use of equalities given above in a mechanism, for example, a mechanism having a closed chain, it is sufficient to write
E r/ 9 ' k = 0’ k
(5)
where r ^ r ^ x 1^ 2....xn) and g'k are the position vectors and base vectors of the coordinate systems attached to different links, respectively.
3. VELOCITY ANALYSIS In this section, we will analyze the velocity cases of the parts that belong to a mechanism. The physical components of the velocity vector of a point moving along a path whose equation is given by x'=x'(t) can be written as (gaa)1/2dxa/dt [8]. Taking this
66
equation into account, for a mechanism having closed chain, we can readily write
a
/'
dt
Where a is the number for the links. However, although the equation (6), for example, for a planar mechanism, includes two equations, the number of unknown variables is four. For that reason, base vectors g “ referred to different links must be eliminated among themselves. This base transformation is a a =S °/mia a m
(7)'
and (6) takes the form I— 'dx 1 E
j
E
i
a
9 / “ -
/
^
s
j
V
=
o
(8)
We again note that if the coordinate system chosen is not ortogonal.the transformation of the base vectors is prescribed by
T 1■ dr/dx | dr/dx 1 |
4. ANALYSIS OF ACCELERATION In this section, we wish to obtain the expressions of acceleration for the mechanism constructed from links which are connected to one another. By taking the physical components of the acceleration into account, we obtain the acceleration field in the form of
a g kk a=i
12
d x
^
ot
dt2
(is ) d xap d xaq \ k dt
dt
g„ =o
by means of the equation (a) which is found in appendix.
(10)
Where a is not a
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summation index and represent the link’s number. Again,in order to obtain scalar quantities, base vectors gkm are related to one another in the form of n a -eja 9 k k m
nm » j
(11 )
By inserting the above equation into (10), we obtain for a three-dimensional mechanism that
3
,2 k
d x
n -3
EE
/=1 a=1
*kk
^2
, p , q I w \ d x dx
a I « I
a
vPÇlJ dt
a
dt
d *n-2 Po dXn-2 n-2 * 2v 2 $Hk IpqJ dt dt dt2
IM
(12)
p. , d xn~\ 1 d xnn-1 d 2xn-1 1 v $kk p q j dt dt . dt2
4M
2~nk , \( K k ] [ d xnp d xnq \
*kk
dt
qj dt
g k =o,
{k =1,2,3)
dt
We remind that in the above equation there is no restriction in choosing the base frame. 5. POLAR COORDINATE SYSTEM
The most appropriate curvilinear coordinate system for the mechanisms is cylindrical coordinate system and, in particular, polar coordinate system for planar mechanisms. In fact, this coordinate system gives us the simplest form of motion because of its great conformity which provides between the geometrical and physical symmetry of mechanism. For such a system, the loop closure equation is
r ^ g ^ 2g^....+rng^n =0
.
(13)
Transformation tensor S'k is a tensor which consists of three tensors S ^ , Smn, Snk, where
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cos(0r 0m) c/ ^ m
sin(0y-0m)
0
(14)
-sin(0^-0m) cos(Qj-Qm) 0 0
0
1
1
0
0
0
cos(0mtpn)
s\n(m-n)
0
-sin(
dxx
Putting (b) into (a), we have
where
6 A “ _ d A “ .i “
T t
dF \ i
Li j J
(e)
dt
78
is called Intrinsic derivative. Now, remembering that the velocity components v are given by v“ =dx“ /dt, we obtain that
Thus, acceleration vector a can be written in the form of
REFERENCES
1.
A.T.Yang, Application of Quarternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms,"Doctoral Dissertation", Columbia University, New York, N.Y.,1963.
2.
A..T.Yang and F.Freudenstein, Application of Dual Number Quarternion Algebra to the Analysis of Spatial Mechanisms, J. Appl. Mech., pp.300-308, 1964.
3.
C.Y.Ho, Tensor Analysis of Spatial Mechanisms, IBM, J. Res. Develop.10, pp207-212, 1966.
4.
J.Angeles, Spatial Kinematic Chains, Springer-Verlag, New York (1982).
5.
G.N.Sandor,
A.Raghavacharyulu
and A.G.Erdman,
Coriolis Acceleration
Analysis of Planar Mechanisms by Complex Number Algebra, 6.
I.S.Sokolnikoff, Tensor Analysis, Second Edition, John Wiley, New York, pp.5083, 1964.
7.
J.Casey and V.C.Lam, A Tensor Method For the Kinematical Analysis of Systems of Rigid Bodies, Mechanism and Machine Theory, Vol: 21 pp.87-97, 1986.
8.
M.R.Spiegel, Vector Analysis, Me Graw Hill Inc., New York, pp.196-225, 1986.
9.
P.R.Paul, Robot Manipulator: Mathematics, Programming and Control, The MIT Press, Massachusetts, pp.26, 56-59, 1981.
10.
S.G.Kalitsin, General Tensor Methods in the Theory of Spatial Mechanisms, Trudy Sem. Teor. Mash. Mech. Vol:14, Part:5, 1954.
