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Discrete positions method in kinematics and control of spatial linkages
Mechanism and Machine Theory, Vol. 15, pp. 47-60
Pergamon Press Ltd., 1960. Printed in Great Britain
Discrete Positions Method in Kinematics and Control of Spatial Linkages M. S. Konstantinovt and
M. D. Markov~ Received 6 July 1977; for publication 6 July 1979
Abstract Using the screw theory in the field of infinitesimal displacements, the first and second order transfer functions of spatial kinematic chains are algebrised. Based on this, a method of finite increments of the relative displacements in the joints is offered. Finally, an algorithm for the control strategy of open kinematic chains of manipulators and robots was evolved using an imaginary closing of the kinematic chain while bearing in mind the conditions for instantaneous mobility (transitory).
Introduction THE WIDESPREADuse of spatial mechanisms in different branches of technology and the growing interest of researchers in the field of analysis and synthesis of spatial chains, has led to the enrichment and updating of a number of classical methods and the creation of some new investigation concepts. Displacement analysis, which is fundamental to spatial kinematics, is the object of quite a number of studies, as for example Zinoviev's vector method[l], Dimentberg's motor-algebra method[2], the dual numbers method of Bayer[3], the matrix method of Denavit and Hartenberg [4], the spinor method developed by Denavit [5], the method of Yang and Freudenstein using dual quaterniaus[6], the tensor[7], and vector-complex methods[8] and other modifications [9], [10]. Regardless of the method applied, the deciphering of the positions functions is associated with the solution of systems of transcedental equations and finding the roots of polynomials of very high orders. The application of iterative numerical methods usually leads to residual inaccuracies which are hard to estimate. Very few authors treat the question of the direct solution of the position functions equations. A good approach, adapting Newton's iterative method to the solution of matrix equations of the position functions, is found in[l l] and turns out to be a logical conclusion of Denavit and Hartenberg's method. Qualitatively different are the ideas for the displacement analysis by linear-geometric means exposed by Freudenstein and Woo in their fundamental study[12]. The authors have the feeling that their own ideas, presented in this paper, stay very close to the concepts stated in [12].
Kinematically Equivalent Discrete Model of a Spatial Kinematic Linkage An arbitrary spatial kinematic chain in any of its discrete configurations can be reduced kinematically to an equivalent chain containing only revolute R, prismatic P and screw H joints. It is well known, as for instance in[3], [13] that a C joint can be replaced by two consecutive coaxial R and P joints, while introducing an additional link, an S joint is equivalent to three noncoplanar R joints whose axes intersect at its center and two additional links and an tProfessor Head of the Central Laboratory of Manipulators and Robots at the Higher Institute of Mechanical and Electrical Engineering, Sofia, Bulgaria Darvenitza. ~Dipl. Engineer Research fellow at the Central Laboratory of Manipulators and Robots at the Higher Institute of Mechanical and Electrical Engineering, Sofia, Bulgaria Darvenitza. 47
48
E joint can be replaced by two P joints with nonparallel axes lying in its plane, one R joint perpendicular to the P axes and two additional links. Other joints can be similarly replaced by additional links and joints with one relative degree of freedom. For this reason, the generality of this paper will not be reduced if we restrict ourselves to the study of kinematic chains containing only R, P and H joints. In the case of nonoverconstrained mechanisms, the kinematic analysis of single loop spatial structures with one degree of freedom is reduced to the classical seven bar problem.
Algebraic Interpretation of the First Transfer Functions by Screws Lately, screw methods, founded by Ball[14], Kotelnikov[15], and further developed by a number of other authors[16], are once again in vogue among researchers due to their natural adaptivity to analytic description of spatial mechanical structures. Modern authors are intentionally not mentioned here because of the large amount of results that have been accumulated with the help of screw methods in the field of kinematic, force and dynamic research. The instantaneous helicoidal motion of a free body in space is associated to the notion of a kinematic screw, so that lfcl=(to+tzV)~,=to(ia+jfl+ky)+lz[i(Va+tol)+j(Vfl+tom)+k(Vy+ton)]
(I)
where/(4 is a kinematic screw of the velocities; to is the angular velocity modulus; V is the minimum linear velocity modulus; /z is Clifford's screw operator[17]; .i. is a unit screw determining the position of the instantaneous screw axis (ISA), a, fl, y, I, m, n - are normal Plfiker coordinates[18], defining uniquely the position and orientation of the ISA in the Euclidean space, the first three being the cosine directories of the ISA, while the last three represent the scalar components of the moment of the unit vector which lies on the ISA with respect to the origin of a ground Cartesian coordinate system, i, j, k are unit vectors of the ground coordinate system. The concepts so defined are valid when both absolute and relative infinitesimal displacements are interpreted. Of practical interest are those kinematic screws of the relative displacements of the links, which are due to the constraints introduced by the joints. For the kinematically equivalent discrete model of a spatial linkage here chosen the kinematic screw of the relative velocity of the links joined by R joints, is lf/IR = toR[(iaR + jflR + kyR) + #(iIR + jmR + knR)].
(2)
In the case of P joints Mp = lz Vp(iae + jfle +kyp)
(3)
and lastly in the case of H joints 1Vl= ¢ou{iaH + jflH + kyu +/z[i(Pua. + 1.) + j(Puflu + mn) + k(PuyH + nu)]}
(4)
where PH = (Vn/to.) is the pitch of the screw joint. It is well known[19] that for any discrete position of a single loop closed kinematic chain consisting of h links, connected by kR revolute joints, kp prismatic joints and k. screw joints, the condition for instantaneous compatibility of the infinitesimal relative displacements and the closedness of the chain is given by the commutative screw sum KR
Ke
KU
U=!
U=I
U=I
which might be further recorded in a bivector form in accordance with the substitutions (2), (3) and (4). If we denote the first transfer functions by the ratios
49 KRR. U _ OJR.U
KRP, U = Vp, u
6OR, 1 '
KpR, U _ WR, U
KRH, U = O)H.~U
O.)R,1 '
KP, U = Vp, v
(.OR. I
(6)
KpH, U = WH, U
gp, 1 '
gp, I '
gp, i
(OH.1 '
(-OH, I '
¢J)H,I
where the lower index determines the type of joint between the input link and the frame, the upper left index determines the type of joint and lastly the upper right index determines the serial number of the joint of a given type in the chain, then these first transfer functions as it is obvious from (6), might have dimensions "length", "length to the power of minus one", or have no dimension at all. The screw eqn (5), after the substitutions (2), (3) and (4) and the relations (6), is equivalent to a linear nonhomogeneous algebraic system of six scalar equations with respect to n - 1 desired first transfer functions. For instance, in the case when the joint between the first link and the frame is of the type R, the expanded matrix of the corresponding system will be OlR.2...
OtR, K R O . . .
• -- OlR, 1 --
Ot~H,1
. .
OIH, K ~
O~m
..
flH, K H
--~R,I
T R , 2 . . . TR.KR O . • • OTH, I
• .
~/H,K H
--~R.1
..
(aH, KHPH.K. + IN, K . )
--[R.I
(~H.KHPH,KH + mH, KH)I
-- mR, j
( r.,n,,P,-,,K. + nn.KH)
- nR,~._j
f 3 R , 2 . . . [3R, K ~ O .
. .
IR.z. . . IR, K.ap, l . . . ae, x p ( a n , , P n , , mR,2..,
+ lu, O
mR, K.B.,I . . . [3p, Ke([3n.,Pn.j + r a n , 3 . .
h R . 2 . • • nR,KRYP,1 • • • Y P . K p ( Y H . 1 P H , I +
nn, O
..
(7)
Similarly we can record the expanded matrix in the case of a P, or H joint between the input link and the frame.
Mobility of Spatial Kinematic Chains The question of mobility of a mechanical system of joined rigid bodies is reduced to studying its ability to be driven by one or more input links. The classical approach to the problem of mobility[20, 21] based on structural equations carrying information about the number of links and the types of joints, is in many cases inadequate. An objective estimation of the mobility can be carried out based on the metric parameters of the chain and its kinematic properties. Mobility has been thoroughly studied by Hunt[22]t on the basis of the properties of screw systems and their special forms, and transitivity and gross mobility have been separated. Back in the beginning of the present century, quite intuitively, certain overconstrained mechanisms were created, and some researchers explained their gross mobility[23, 24] by geometric and analytic methods. Lately, Savage[25] and Waldron[26] created techniques for synthesis of some overconstrained kinematic chains with gross mobility, which in itself is a difficult and in no way a satisfactorily resolved problem. A spatial kinematic chain has instantaneous mobility[27] if the nonhomogeneous linear system which is characterized by the expanded matrix (7), corresponding to the equivalent discrete kinematic model of the mechanism, is compatible. The necessary and sufficient condition for this, according to Kroneker-Kapelli's theorem, is that the rank r of the base matrix is equal to the rank r' of the expanded one. It is invariant with respect to the choice of the frame but not so with respect to the choice of the input link. Satisfying the conditions for instantaneous mobility of a given structure does not necessarily guarantee mobility in the case of larger displacements since the instantaneous mobility can be transitory if for all infinitely close neighbouring positions the condition for instantaneous tThe authors beg Professor Hunt to excuse them for quoting his unofficialpaper[23], but they were stimulated to do so because of the freshness and originality of the ideas set forth therein.
MMT Vol 15, No 1--D
50 mobility is not satisfied. The necessary condition for a non-transitory instantaneous mobility is the existence of instantaneous mobility for a set of neighbouring positions. On the other hand, even those cases are of practical interest for which the condition for instantaneous mobility is not fulfilled but the existence of suitably chosen clearances in the joints and the fact that this condition is satisfied for the infinitely close neighbouring positions, are enough to allow the mechanism to overcome that critical state and even function with full cycle gross mobility. The instantaneous mobility as outlined above is equivalent to the question of linear relation between screws and the theory of screw groups [15, 28].
Determination of Second and Higher Order Transfer Functions Without Differentiation The kinematic analysis on the level of second order derivatives of the position functions is suggested by the modern tendencies for the fast action of mechanisms in mechanical automatics. It provides for a quantitative estimation of the effect of the inertia forces, most strongly expressed in the case of open kinematic chains. Research in the field of acceleration analysis was carried out by a number of scientists as for example[29, 30], including screw-geometric methods[31] but in all cases this was accomplished through double differentiation of the displacement functions. The method described in this paper and which was already used for finding the first transfer functions can be further applied to include second transfer functions[32]. The formally differentiated screw eqn (1) written in the form (8) contains the derivative of the unit screw
(9)
,~ = toe + a + / ~ [ p + a + p x(toe xa)]
assuming the following new notations: a is the unit vector of the ISA; p is the radius vector of an arbitrary point on the ISA; tOe is instantaneous angular velocity of the ISA. Since the relations (8) and (9) hold both for absolute as well as relative displacements of the spatial kinematic chain's links, then for the joints included in the equivalent kinematic model, the bivector equation (8), for R joints, becomes 2,
MR = dR~'R + tOR,4R
(10)
tip =
(11)
for P joints +
and for H joints /Vl. = (1 +/zPH)(o3ng, n + w.g,.).
(12)
All components of eqn (9) can be found algebraically, if the assumption is made that the first order transfer functions are already known and that the chain is closed. It can be assumed for tOe, i.e. the transfer angular velocity of a ISA of the relative motion, to be the absolute angular velocity of one of the links connected by the joint, while the derivative ti we can assume equal to the velocity of a point on the joint's axis regarded as a point belonging to one of the links in its absolute motion. The formally differentiated eqn (5), in the case when the joint between the first link and the frame is of the type R, bearing in mind (10), (11) and (12), can be recorded in the form KR
Kp
+ U=2
U=I
KR U=I
KH
v,,u L,.u +
0 + U=I
Kp
KH
U=I
U=I
(13)
51 Likewise in the case of the first transfer functions the introduction of the ratios for the second transfer functions in the manner of [33] 1,RR,U - -
FpR,U
tf)R,U -
vRP,U = 7PtU '
lJRH'U =
tbz,v
FpP,U
vpU, U
d')H,U
~rP,U
vL'
v R.v - (oR.u
v
.u -
vL' f'P,u
v
vL
(14)
.u = ';'H.u
the screw eqn (13) is equivalent to the nonhomogeneous algebraic system of six scalar equations with respect to n - 1 desired second transfer functions. The expanded matrix of this system is analogous to the matrix (7) with the exception of the column comprising the free terms which are determined by the right-hand side of eqn (13). The strategy described so far can be applied for determining the highest order transfer functions without resolving to differentiation. Besides, the base matrices of the corresponding linear non-homogeneous algebraic systems of equations stay invariant but with increasing the order of the transfer functions the last column on the right-hand side of the corresponding expanded matrices is considerably complicated. The second and higher order transfer functions of the type (14) and the corresponding higher analogies are dependent on the assumed values for the input transfer functions of the respective order, as seen from eqn (13). The ratios (14) and their higher order analogies, as well as (6), can be with or without dimensions, and the study of the systems that determine them also bears relation to the problem of mobility. Computation of the Finitely Small Relative Displacements in the Joints of a Kinematic Chain For almost all spatial mechanisms of practical importance, there exist special discrete configurations of their kinematic chains, for which the computation of the numerical values of the position functions is not so labour consuming. It is therefore convenient to use them when determining the initial conditions in the displacement analysis along the lines of the strategy proposed in this paper. The finitely small relative displacements in the joints, expanded in a Taylor series [34, 43] when the input link and the frame are connected by an R joint, for the three pairs of joints figuring in the equivalent kinematic model, for a chosen input step A~0R,are as follows: A~oRR'U -~ KRR'UA~oR +~ UR ' A~OR + . . .
f
l RU. 2 A~e'v rUe'VA~R+~VRev.' A~R2 +... 1
(15)
A~off'v KRn'VA~OR+~1 VRuv."A¢t¢2 + .... In the case when a P joint is employed between the input link and the frame, the corresponding Taylor series for an input step A~Cpare A~p R'U = KpR'U A~p +~
f
1 vff.UA~ff+,, " 1
A~opn'v rpn'vA~p +~1 vpu.vA~e2+ .... And finally, when an H joint is employed, the Taylor series for an input step Aq~n are
(16)
52 A~DHR'U = KHR'U ~
1
RU--
H -+- ~ lJH • A~DH + -j-. , .
A~. e.u = K.P.t; A~H + ~1 VHV't'A ~ f +. ACHH,U =
KHH'U A c H
(17)
1
-~'~ 1jHH'U A c H 2 q"- . ..
The increments of the position functions in the joints, obtained from (15), (16) or (17), respectively, define a neighbouring position of the kinematic chain, corresponding to the chosen input step. Following step by step the proposed strategy we are in a position to find, using a computer, a numerical solution of the position functions, applying the following obvious relations _- C v j+ + a C . V l +
(18) , = C ' . v [ . + A¢',vl+
where q is an index denoting the consecutive step. It is clear that using a larger step, in order not to decrease the accuracy of the calculations, it is necessary to include higher order members of the Taylor series. On the other hand, the introduction of each next higher order transfer function is connected with a corresponding increase in the computation difficulties and additional machine time expenditures. The authors recommend the use of Taylor series up to second degree. The use of first order transfer functions only is not effective in the case of fast changing position functions, while in the case of mechanisms which pass through dead points it is not always possible.
Control of Open Spatial Kinematic Chains With the introduction of open kinematic chains in technology and medicine as basic mechanic structures of manipulators, robots, prosthetic appliances, walking machines and certain mechanimals, the question of their expedient positioning and control has become imperative. Two fundamental problems of spatial kinematics arise -given the relative displacement in the kinematic joints, i.e. given a particular configuration of the chain, find the position of the last link, -given the position of the last link find the configuration of the kinematic chain, i.e. find the relative displacements in the joints. The first problem, also known as the +'direct problem" of displacement analysis, of kinematics, respectively, has in all cases a unique solution which is the result of direct methods, and algorithms exist to this effect, like for example [35, 36]. The second problem, known as the '+inverse problem" of displacement analysis is not always uniquely determined and is of great practical importance since its solution is fundamental to the mathematical modelling of the control processes of open mechanical systems of connected rigid bodies. It is well known that the theoretical minimum of degrees of freedom which permit the free control of an object in the three-dimensional Euclidean space is six. The solution of the inverse problem by direct methods has so far proved to be impossible, as Pieper and Roth[37] have shown, since in the case of a robot with 6 degrees of freedom, the problem is reduced to finding the roots of a polynomial of degree 524288. It is only through additional restrictions on the structure of the kinematic chain, like, for instance, permanent intersection of pairs or triplets of axes defining relative motions, that some authors [37, 38, 8] were able, using direct methods, to produce original solutions of the inverse problem for certain particular structures with up to 6 degrees of freedom. The method proposed in this paper for the control of an open kinematic chain on the basis of target variation of the positions and orientation of the end link is founded on the computation technique of finite displacements in the joints. An open kinematic chain with 6 degrees of
53 freedom and only R, P and H joints, is considered. Let two orthogonal reference axes be connected to the last link, and let along these axes B and d be two unit screws. Let F be the origin of the two axes. The end position of the last link will be defined by the position of the two reference axes and the pole namely by the unit screws/3e and ~ , and the point Fe. The displacement which has to be realized by the last link, can be modelled by two noncommutative finite displacements, translation and rotation, rotation and translation, respectively. The unit screws defining the initial and finite displacement of the last link (the transported object) are expressed in a bivector form, as follows:
~=h+#pxb
Be = be +/2,pc X be
C = c + / z p xc
(2e = ce +/zpe xc~
b'e=O
(19)
be "ee =0.
First variant: translation, rotation. The screw of the finite translation displacement, according to (19), is defined as q£ = IZ(pe - p) = tS/z~-
(20)
where s~ is the modulus, and ~- is the unit vector of pe-P. The screw of the finite rotation displacement superposing B with Be and (7 with t~e, is denoted by
M=
~fi
(21)
where ¢ is the angle of finite rotation, R is the unit screw of the axis of this finite rotational displacement, while B' and C' denote the unit screws of the displacement of the axes B and (~ after performing the translation (20). The screw R in a bivector form = r +/./,joe x r
(22)
contains the unknown unit vector r which will be defined by using the ideas of Saussure [39], namely
r = V(gZq
g2 +_ q(g. q)2)
(23)
where g = be - b,
q = Ce - c
(24)
while g and q are the corresponding moduli of the vectors, defined by (24). To determine the angle ~ of finite rotation, the following relation is used:
{d. d~ '~
¢ = arc cos kd .-----~)
(25)
d = b x r,
(26)
where de = be x r
are introduced, while d and de are the corresponding moduli of the vectors defined by (26). Second variant: rotation, translation. The corresponding screws of finite displacements are determined according to the technique applied for the first variant. Every finite displacement of the described two possible variants can be modelled by an
54 imaginary introduction of a prismatic P, respectively revolute R-joint closing the chain [401. In this way the open kinematic chains can be treated in a kinematic aspect as closed onest, and this allows to realize the two necessary finite displacements according to the exposed method for forecasting the finite small relative displacements in the joints of the chain. A further improvement of this strategy, allowing to decrease the number of steps and making possible the realization of the control in real time by a processor of moderate power, is achieved by an imaginary closing of the chain through a screw H-joint. For this purpose, it is necessary to define the normal Pliiker coordinates of the axis of the finite screw displacement leading the last link (the transported object) from an arbitrary initial position to the target finite position and to find the complex angle of rotation about and translation along this axis. Dimentberg and Averianova[41], by using the principle of transmission of the operations of vector algebra into bivector (screw) algebra. The same principle, formulated almost at the same time and independently by Kotelnikov[15] and Study[161, is also used by the authors, and for determining the normal PKiker coordinates of the required axis, the screw analogue of the relation (23) is applied j~ =
GxQ x/(G2Q 2 - (0" ~,)2)
(27)
where
In the formula (27) R is the unit screw of the axis of finite screw displacement, while G and Q are the complex moduli of the screws defined by (28). Analogically, to (25), for the complex angle of the finite displacement one obtains:
D. b e cos ~b = D" De
(29)
where the following screws have been introduced:
~=~x~,
6e=~eX~
(30)
while D and De are the corresponding complex moduli of the screws defined by (30). As it is known, the kinematic sense of the complex angle cb is defined by cos ~ = cos ~, - #n sin 0
(31)
where 0 is the angle of finite rotation about the axis R, and r/ is the length of the finite translation displacement along the axis. Note that for one and the same initial and finite position of the object, the angle ~o obtained from (25) for the strategy "translation-rotation", is equal to the angle ~, obtained from (29), (3 I) for the strategy of finite screw displacement, while for the values of finite translation displacements for the two different strategies, the following dependence holds: n ~ £.
(32)
Since the screw relation (27) for some particular cases, degenerates and becomes unsuitable for practical calculations, the following technique is reccommended in order to overcome this difficulty. If b=be, c~ce,
resp. c = c e b#be
then r = b = be,
resp. r = c = c e
*Thelast link of the openkinematicchain is treated as an input link in the imaginarilyclosedchain.
55 when carrying out the strategy "translation-rotation". If b = be
and c = ce
then the rotation vanishes and the finite translation displacement is d e t e r m i n e d f r o m (20), In case, when carrying out the target p r o g r a m m e directly, or step by step, the imaginarily closed kinematic chain with a changing structure falls into a state of a partial or full jamming (loss of mobility), one has to introduce deviating impulses for displacements, and after that the system is brought back to the accepted control algorithm. Since the mathematical modelling of the kinematics of the last link of the open kinematic chain is carried out in the six-dimensional space, while the regional positioning of the pole is accomplished in the three-dimensional space, then, when constituting the target assignments it is n e c e s s a r y for them to be in a c c o r d a n c e with the volume of the serviced six-dimensional space. This involves a compatibility between the prescribed target orientation of the last link and the space angle defining the service coefficient [42] at the given point of the service space.
References 1. V. A. Zinoviev, ProstranstvenieMechanizmi s Nizshimi Parami. (in Russian). Gostehizdat, Moscow (1952). 2. F. M. Dimentberg, The Determination of Positions of Spatial Mechanisms. pp. 142. Izdat. Akad. Nauk SSSR, Moscow (1950). 3. R. Beyer, TechnischeRaumkinematik. Springer-Verlag, Berlin 0963). 4. J. Denavit and R. S. Hartenberg, A kinematic notation for lower pair mechanisms based on matrices. J. appl. Mech, 22, Trans. ASME, 77, 215-221 0955). 5. J. Denvait, Displacement analysis of mechanisms based on 2 × 2 matrices of dual numbers. VDI-Berichte, B. 29, 81-89 (1958). 6. A. T. Yang and F. Freudenstein, Application of dual-number quaternion algebra to the analysis of spatial mechanisms, Trans. ASME J. appl. Mech. 31E, 300-308 (1964). 7. D. Mangeron and C. Dr/igan, Asupra unei noi metode tensoriale de studiu a mecanismelor. Bull. Institutului politechnic, Jasi, III, No. 1-2, 151-164 (1957). 8. M. S. Konstantinov and Z. I. Zankov, A kinematical algorithm and dynamical point mass simulation in robots and manipulators. Proc. of First CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, Udine (Italy) 1972 Vol. I, Springer-Verlag Wien, New York. 9. P. A. Lebedev, Kinematika prostranstvenih mechanismov. Mashinostroenie, Moscow 0966). 10. V. Brat, Kinematic analysis of spatial mechanisms with general kinematic pairs (a matrix method). 3rd World Congress on the TMM, Proc. Vol. C, Cupari, Yugoslavia (1971). I 1. J. J. Uicker, J. Denavit and R. S. Hartenberg, An iterative method for the displacement analysis of spatial mechsnisms. J. appl. Mech. 31,309-314 (1964). 12. L Woo and F. Freudenstein, Application of line geometry to theoretical kinematics and the kinematic analysis of mechanical systems. 3. Mechanisms, 5, 417-460 (1970). 13. I. 1. Artobolevski, Teoria mehanismov i mashin (in Russian). lzdat Nauka, Moscow (1975). 14. R. S. Ball, A Treatise on the Theory of Screws. Cambridge University Press, Cambridge (1900). 15. A. P. Kotelnikov, Theory of Screws. (Russian) Kazan (1895-96). 16. E. Study, Geometrie der Dynamen. Teubner Verlagsgesellschaft GmbH. Liepzig (1903). 17. W. Clifford, Preliminary sketch of biquaternions. Proc. of the London Math. Soc., Vol. IV, (1873). 18. J. Pliiker, NeueGeometriedes Raumesgegriindet aufdieBetrachtungdergeraden Linieals Raumelement. B. G. Teubner, Leipzig (1868--69). 19. M. S. Konstantinov and M. D. Markov, Parvi predavatelni funktsii na prostranstveni kinematichni verigi. J. Theoretical and Appl. Mech. Bulg. Acad. Sci., 2, (1977). 20. K. Kutzbach, Mechanische Leitungsverzweigung. Maschinenbau, Der Betreib 8, No. 21,710-716 (1929). 21. P. O. Somov, O stepeniah svobodi kinematicheskoi tsepi. J. of the Russian Physico-chemical Soc. SBP Univ., Vol. XIX, part phys. issue 1 9 (1887). 22. K. H. Hunt, Screw Systems in Spatial Kinematics. Monash University, Clayton, Victoria, Australia (1970). 23. A. Bennet, A new mechanism, Engineering 76, 777-778 (1903). 24. R. Bricard, Lemons de Cin~matique Vol. 2, Gauthier-Villars, Paris (1927). 25. M. Savage, Four-link mechanisms with cylindric, revolute and prismatic pairs. Mechanism and Machine Theory, 7, 191-210 (1973). 26. K. J. Waldron, A study of overconstrained linkage geometry by solution of closure equation. Part I. Method of Study. Mechanism and Machine Theory, 8(1), 95-104 (1973). 27. M. D. Markov, Mobilnost na prostranstvenite kinematichni verigi. Godishnik na VMEI "V. I. Lenin", Jub. izd. (1976). 28. F. M. Dimentberg, Metod vintov v prikladnoi mehanike. Mashinostroenie, Moscow (1971). 29. M. S. Konstantinov, Inertia forces of robots and manipulators, paper presented at the Second Conference on Remotedly Manned Systems (June 9-1 I, 1975), University of Southern California, L. Angeles, U.S.A. 30. J. I)enavit, R. S. Hartenberg, R. Razi and J. J. Uicker, Velocity, acceleration and static-force analysis of spatial linkages. J. appl. Mech., 32, Trans. ASME Vol. 87, (B) 903--910(1965). 31. M. Skreiner, Acceleration analysis of spatial linkages using axodes and the instantaneous screw axis. J. of Eng. forlnd. 97 (February 1967). 32. M. S. Konstantinov and M. D. Markov, Vtori predavatelni funktsii na prostranstveni kinematichni verigi. J. of Theoretical and appl. Mech., Vol. 3, Bulg. Acad. Sci. (1977).
56 33. 1. I. Artobolevski, Dinamicheskie kriterii regima dvizenia mashini. J. Theoretical and Appl. Mech. Bulg. Acad. Sci., Sofia Year 11, No. 1 (1971). 34. M. S. Konstantinov and M. S. Markov, Pozitsiona identifikatsia na prostranstveni kinematichni verigi..1"+ Theoretical and Appl. Mech. Bulg. Acad. Sci. Sofia 4 (1977). 35. R. Seyffarth, Geometrische Konstruktionsverfarhen zur Darstellung viergliedriger r~iumlicher Kurbelgetribe. Maschinenbautechnik, 10, Heft 12, Berlin (1%1). 36. M. S. Konstantinov, Structural and kinematic analysis of robots and manipulators. Proc. Symposium on lndugtrml Robots. Tokyo (1974). 37. D. L. Pieper and B. Roth, The kinematics of manipulators under computer control. 2nd Int. Congress on the Theory o( Machines and Mechanisms, Proc.+ Zakopane, Poland (1%9). 38. A. G. Ovakimov, Kinematicheskoe isledovanie prostranstvenoi tsepi upravlenia manipulatora. Izv. vuzov, series "Mashinostroenie'++ No. 14 (1971). 39. R. Saussure, Etude de geometrie cinematique regl6e. Amer. J. of Maths. 19, (1877). 40. M. S. Konstantinov and M. D. Markov, Deshifrirane na konfiguratsiata na otvorena kinematichna veriga po polozenieto i orientatsiata na krainoto zveno. C.R. Acad. Bulg. Sci., 30, 5 (1977). 41. V. G. Averianova and F. M. Dimentberg, Opredelenie vinta peremeshtenia po nachalnomui konechnomu polozeniu tverdovo tela. Machinovedenie, No. 2 (1%5). 42. A. E. Kobrinski and J. A. Stepanenko, Nekotorie problemi teorii manipulatorov. Mehanika mashin, issue 7-8, (1%7). 43. P. I. Genova, Priblizitelen metod za opredeljane na pozitsionite funktsii na ravninite mehanizmi, Proc. of the 3rd National Congress of Theoretical and Applied Mechanics. Varna (1977).
Appendix 1 We will determine the first and second transfer functions for one discrete position of the spatial crank and connecting-rod assembly, shown in Fig. I. The following numerical data are accepted for the considered mechanism: a = 0.5, b = 0.2, c = 1, 0 = 60°. Consider the discrete position of the mechanism when the link is parallel to the plane xOy of the chosen ground coordinate system. The kinematically equivalent scheme of the mechanism in this position is described on Fig. 2, the axes of R-joints replacing the S-joint being chosen parallel to the coordinate axes. For the case in consideration, the screw equation (5), written in a bevector form, will be WR.li+ toR.2(k+/.t 0.45825 i) + ~OR.~(j- / z 0.2 i) + tOR+4[i+ #(0.2 j - 0.45825 k)] + ,OR.s(0.72842i -- 0.68512 j + /Z(0.13702 i + 0.145711 j -- 1.33392 k)] + o~R.6[k+ ~(1.18667 i -0.68512 j)] + Vp.1#(0.5 i - 0.86602 j)
0.
I33)
After separation of the scalar equations equivalent to the bivector equation (33) and introduction of the relations (6), the following values for the first transfer functions were determined: KR'~'2-- 0.10273, KRn'4=
.,+RR+2= 0.31392
1.33376, ~Rk's =0.45820
K,~'~'~- 0.10273,
134)
K,~e'f .... 0.14%6[m].
When determining the second transfer functions, we assume
~'R.t
=
IIS I],
~.,+.l =0+
For the same discrete position and dimensions of the mechanism, by using the obtained numerical values (34) for the first transfer functions, the screw eqn (13) assumes the bivector form, as follows:
,! 'r Z
/
1X ib
Figure 1. Spatial crank and connecting-rod assembly.
57
I
A"
b = a sin q~,t,,
pi~,t
?,4.
Figure 2. Kinematically equivalent scheme of the spatial crank and connecting-rod assembly from Fig. 1.
o~R.z(k+ # 0.45825 i)+ dR,3(J - # 0.2 i)+ O~R.4[i+ #(0.2 j --0.45825 k)] + o)R,5[0.72842 i - 0.68512 j + #(0.13702 i+ 0.14570 j - 1.33392 k)] + o)R.dk + #(1.18667 i - 0.68512 j)]+ Ve: #(0.5 i - 0.86602 j)
(35)
= 0.20546 j - 0.73263 k + #(-0.22573 i + 0.61889 j - 0.2 k). After separation of the scalar equations equivalent to the bivector eqn (35) and introduction of the relations (14), the following values for the second transfer functions are obtained: uR R'2= -- 0.54915,
VR R'3 = 0.06844
7,RR'4= --0.14568,
UR R'5
= - 0.2
vRR'6= - 0.18347,
vR e : =
-
(36)
0.56947 [m].
Appendix 2 Following the method described in the present paper, an analysis of the position of 6R mechanism Turbula (System Schatz, Maschinenfabrik W.A. Bachofen Switzerland) will be carried out, its kinematic scheme being shown in Fig. 3. According to the classical theory for mobility [20, 21], the kinematic chain should be a statically determinate rigid frame but the mechanism has one degree of freedom due to the suitably chosen dimensions and angles of intersection or crossing between the axes of kinematic joints. This is explained by the fact that for every position of the mechanism the joints' axes are straight lines belonging to one and the same linear complex, i.e. the unit screws defining the position of the joints' axes in the three dimensional Euclidean space belong to one and the same 5-term screw group. Fig. 4 describes the kinematic scheme of the mechanism in one particular position which is suitable to be chosen as initial. The same figure indicates the correlations between the metric chain parameters assuring its full cycle gross mobility. From Fig. 3 and Fig. 4 it is clear that for the chosen initial position, the position functions have the values, as follows: tpe,i = 90o, ¢e.2 = 900, ~oe.3= 60o, ~pe.4= 180o, ~e.~ = 150o, q~e.6= 900. For the initial position of the mechanism in consideration, the screw eqn (5) assumes the following bivector form: mR.Ik + tOR.2J+ ~0R.3(i+ #j) + tOR,4(0.5 j + 0.5 X/(3)k + # 0.51 i) - tOR,5(i -- # ' x / ( 3 ) k ) - tOR.6(k+ #\/(3)i) = 0.
(37)
After separation of the scalar equations equivalent to the bivector equation (37) and introduction of the relations (6), the following values for the first transfer functions were calculated: KRR': = 0.5 X/(3),
KRR'3 = 0,
KRR'4= -- X/(3),
KRR'5 = 0,
K R R ' 6 = --
0.5.
(38)
In order to determine the second transfer functions, it is again assumed that coR,~= I[S -~] and &R.~=0. Using the obtained numerical values for the first transfer functions (38), after some transformations, the screw eqn (13) is written in a bivector form, as follows:
+ O)R,4(0.5 j + 0.5 X/(3) k + # 0.5 i) - O)R,5(i - # X/(3) k) - o)R.6(k + # ~ / ( 3 ) i) = 0.25 ~/(3) i - #(0.5 x/(3) j - 0.75 k).
O)R,2J + O)R3(i + # j )
(39)
58 F-Z
+:U I
/ , q0~'2
A~
o-
i
.,
Y
Al, ll
Figure 3. Kinematic scheme of the "l'urbulamechanism. From the equation (39), by using the substitutions (14), the following values of the second transfer functions are obtained: VRR'2 : O,
VRR'3
= 0.5 \/(3),
VRR'4 = 0,
VRR'5 =
-- 0.25 \/{3},
VR~'6 : O.
(40)
Using the Taylor series (15), the finitely small relative displacements in the joints, corresponding to the chosen input step, are calculated to within the second derivative. For the mechanism Turbula the composed program by the method of discrete positions was applied. For the chosen initial position of the mechanism, two of the first transfer functions vanish, according to (38), whence the first transfer functions only are not always in a position to forecast the finitely small increments of the displacement
iz
~
~,,., .L ~,.2
"
r
..
Figure 4. Kinematic scheme of the Turbula mechanism in the chosen initial position.
59
Table 1. Entry IJJCPD1N 00D278 Entry IJJCPD3 00D278 CSECT IJ2E0005 00D208 00C400 1
2
3
4
5
6
1:5707%33 1:65806279 1:74532925 1:83259571 1:91986218 2:00712864 2:09439510 2:18166156 2:26892803 2:35619449 2:44346095 2:53072742 2:61799388 2:70526034 2:79252680 2:87979327 2:%705973 3:05432619 3:14159265
1:5707%33 1:64634767 1:72175438 1:7%86461 1:87151310 1:94551312 2:01864056 2:09065137 2:16120529 2:22990447 2:2%27123 2:35%1805 2:41909903 2:47363502 2:52187991 2:56222941 2:59291532 2:61223795 2:61892612
1:04719755 1:05051075 1:06052269 1:07750413 1:10192156 1:13446123 1:17606403 1:22797244 1:29178919 1:36954480 1:46371426 1:57739853 1:71414925 1:87779376 2:07187702 2:29858732 2:55710257 2:84181384 3:14160555
3:14159265 2:99049090 2:83%8035 2:68946483 2:54017513 2:39218506 2:24593135 2:10193863 1:96085207 1:82348006 1:69086564 1:56431454 1:44552043 1:33664254 1:24031305 1:15972719 1:09846817 1:05993187 1:04665731
2:61799388 2:61633535 2:61132740 2:60284356 2:59062349 2:57435096 2:55354634 2:52758817 2:49567477 2:45679043 2:40967274 2:35279263 2:28437580 2:20250281 2:10546519 1:99216115 1:86294309 1:72060892 1:57071202
1:5707%33 1:52708087 1:48286424 1:43762389 1:39079751 1:34176022 1:28979853 1:23407999 1:17361818 1:10723444 1:03352129 0:95081888 0:85723147 0:75073522 0:62945560 0:49220393 0:33930274 0:17347248 0:00018954
functions, i.e. they are not always in a position to determine the neighbouring configuration of the mechanism for an arbitrarily chosen finite step. If for any position of a given mechanism the first and second transfer functions vanish simultaneously in some of the joints, the use of higher derivatives might impose itself but it is not typical for the single loop spatial kinematic chains. With every following step the errors accumulate which makes useful to accept the computation strategy, as follows: From the initial position the displacement analysis to be carried out for one half of the interval studied clockwise, while after that from the same initial position it must be carried out counterclockwise for the other half of the interval. The Turbula mechanism as an overconstrained mechanism, is a rough test for the method since with the accumulation of impermissible errors, the equality of the ranks of the base and expanded matrices of the systems for determining the first and higher transfer functions will be violated. Under double-accuracy computation on the EC 10-40 computer, using a step 1,5', the results given in Table 1 were obtained, where the values of the angles in the joints are registered in radians in the separate columns. The number of the column corresponds to the number of the joint. The digital print was done for every 200 steps.
Appendix 3 The a~s of the complex angle of the finite screw displacement superposing the initial position of a solid object, defined by f = i, C = j and the finite position, defined by fie = j, Ce = k +/zl will be determined. According to (28), G=-i+j,
{~=-j+k+#i.
141)
Substituting in (27), for the unit screw of the axis of the required finite displacement, one obtains i+j+k-#k
1
. .
1
.
_2k)].j
(42)
By using (30), one gets 1 [ - j + k + l ( 2 j + k ) ],
•
I
D e = ~' l ~ [ |/- k -( 3 / z3( 2 i) + k ) ]
143)
After normalization of the screws D and fie, their complex moduli are obtained, as follows: D=~/
1 - Z 3--~'
De = ~/
I -/z 3X/(6)'
G)
!44)
Substituting (43) and (44) in (29), for the complex angle of the finite displacement, one finds 1 1 cos~=-~-L.
(45)
or, in accordance with (31) 2 = arc cos ( 1-~) =~', are calculated.
I ~ = x/3(3)(length ) "0 = 2 sin
(46)
6O EJ.NE }~L~THODE DER D I S K R E T ~ RAUMLI CH/~ MECHANISM]~
LAG~2~ IN DER KINE%~ATIK UND STEUERUNG DER B~LEGDNG~h?~
M.S.Konstantinov und M.D.Ma±'kov
Kurzfassun~ - Die ersten und zweiten Ableitungen der relativen Bewegungen riumlicher Ketten in jeder diskreten La~e werden bei Einf0/Lvung der Schraubentheorie der infini~esimalen Verschiebungen
bestim~it, ohne Differentiationen
im Ra~nen
anzuwenden. Die
endlich kleine Ver~nderungen der in Ta~loz-Re£hen entwickelten Positionsfunktionen werden Schritt f~r Sc~ritt integriert, und somit ist die Bewegung der Kette ermittelt Fdr die Steuerung offener kinematischer Ketten mit sechs Freiheitsgraden wird ein Algorithmus auf Grund scheinbarer EinschlieBung der Kette aufgebaut. Dabei wird die Bewegungsf/higkeit
berOcksichtigt.
bestimmte Schraubenbewegung
Eine beliebige Ziellage des Endgliedes wi~d dutch
erreicht, womit die Zahl der Schritte vermindert und die
Computerzeit wesentlich ve~ki~zt wird. Entsprechende offene Ketten werden betrachtet.
Beispiele fir geschlossene und
Pergamon Press Ltd., 1960. Printed in Great Britain
Discrete Positions Method in Kinematics and Control of Spatial Linkages M. S. Konstantinovt and
M. D. Markov~ Received 6 July 1977; for publication 6 July 1979
Abstract Using the screw theory in the field of infinitesimal displacements, the first and second order transfer functions of spatial kinematic chains are algebrised. Based on this, a method of finite increments of the relative displacements in the joints is offered. Finally, an algorithm for the control strategy of open kinematic chains of manipulators and robots was evolved using an imaginary closing of the kinematic chain while bearing in mind the conditions for instantaneous mobility (transitory).
Introduction THE WIDESPREADuse of spatial mechanisms in different branches of technology and the growing interest of researchers in the field of analysis and synthesis of spatial chains, has led to the enrichment and updating of a number of classical methods and the creation of some new investigation concepts. Displacement analysis, which is fundamental to spatial kinematics, is the object of quite a number of studies, as for example Zinoviev's vector method[l], Dimentberg's motor-algebra method[2], the dual numbers method of Bayer[3], the matrix method of Denavit and Hartenberg [4], the spinor method developed by Denavit [5], the method of Yang and Freudenstein using dual quaterniaus[6], the tensor[7], and vector-complex methods[8] and other modifications [9], [10]. Regardless of the method applied, the deciphering of the positions functions is associated with the solution of systems of transcedental equations and finding the roots of polynomials of very high orders. The application of iterative numerical methods usually leads to residual inaccuracies which are hard to estimate. Very few authors treat the question of the direct solution of the position functions equations. A good approach, adapting Newton's iterative method to the solution of matrix equations of the position functions, is found in[l l] and turns out to be a logical conclusion of Denavit and Hartenberg's method. Qualitatively different are the ideas for the displacement analysis by linear-geometric means exposed by Freudenstein and Woo in their fundamental study[12]. The authors have the feeling that their own ideas, presented in this paper, stay very close to the concepts stated in [12].
Kinematically Equivalent Discrete Model of a Spatial Kinematic Linkage An arbitrary spatial kinematic chain in any of its discrete configurations can be reduced kinematically to an equivalent chain containing only revolute R, prismatic P and screw H joints. It is well known, as for instance in[3], [13] that a C joint can be replaced by two consecutive coaxial R and P joints, while introducing an additional link, an S joint is equivalent to three noncoplanar R joints whose axes intersect at its center and two additional links and an tProfessor Head of the Central Laboratory of Manipulators and Robots at the Higher Institute of Mechanical and Electrical Engineering, Sofia, Bulgaria Darvenitza. ~Dipl. Engineer Research fellow at the Central Laboratory of Manipulators and Robots at the Higher Institute of Mechanical and Electrical Engineering, Sofia, Bulgaria Darvenitza. 47
48
E joint can be replaced by two P joints with nonparallel axes lying in its plane, one R joint perpendicular to the P axes and two additional links. Other joints can be similarly replaced by additional links and joints with one relative degree of freedom. For this reason, the generality of this paper will not be reduced if we restrict ourselves to the study of kinematic chains containing only R, P and H joints. In the case of nonoverconstrained mechanisms, the kinematic analysis of single loop spatial structures with one degree of freedom is reduced to the classical seven bar problem.
Algebraic Interpretation of the First Transfer Functions by Screws Lately, screw methods, founded by Ball[14], Kotelnikov[15], and further developed by a number of other authors[16], are once again in vogue among researchers due to their natural adaptivity to analytic description of spatial mechanical structures. Modern authors are intentionally not mentioned here because of the large amount of results that have been accumulated with the help of screw methods in the field of kinematic, force and dynamic research. The instantaneous helicoidal motion of a free body in space is associated to the notion of a kinematic screw, so that lfcl=(to+tzV)~,=to(ia+jfl+ky)+lz[i(Va+tol)+j(Vfl+tom)+k(Vy+ton)]
(I)
where/(4 is a kinematic screw of the velocities; to is the angular velocity modulus; V is the minimum linear velocity modulus; /z is Clifford's screw operator[17]; .i. is a unit screw determining the position of the instantaneous screw axis (ISA), a, fl, y, I, m, n - are normal Plfiker coordinates[18], defining uniquely the position and orientation of the ISA in the Euclidean space, the first three being the cosine directories of the ISA, while the last three represent the scalar components of the moment of the unit vector which lies on the ISA with respect to the origin of a ground Cartesian coordinate system, i, j, k are unit vectors of the ground coordinate system. The concepts so defined are valid when both absolute and relative infinitesimal displacements are interpreted. Of practical interest are those kinematic screws of the relative displacements of the links, which are due to the constraints introduced by the joints. For the kinematically equivalent discrete model of a spatial linkage here chosen the kinematic screw of the relative velocity of the links joined by R joints, is lf/IR = toR[(iaR + jflR + kyR) + #(iIR + jmR + knR)].
(2)
In the case of P joints Mp = lz Vp(iae + jfle +kyp)
(3)
and lastly in the case of H joints 1Vl= ¢ou{iaH + jflH + kyu +/z[i(Pua. + 1.) + j(Puflu + mn) + k(PuyH + nu)]}
(4)
where PH = (Vn/to.) is the pitch of the screw joint. It is well known[19] that for any discrete position of a single loop closed kinematic chain consisting of h links, connected by kR revolute joints, kp prismatic joints and k. screw joints, the condition for instantaneous compatibility of the infinitesimal relative displacements and the closedness of the chain is given by the commutative screw sum KR
Ke
KU
U=!
U=I
U=I
which might be further recorded in a bivector form in accordance with the substitutions (2), (3) and (4). If we denote the first transfer functions by the ratios
49 KRR. U _ OJR.U
KRP, U = Vp, u
6OR, 1 '
KpR, U _ WR, U
KRH, U = O)H.~U
O.)R,1 '
KP, U = Vp, v
(.OR. I
(6)
KpH, U = WH, U
gp, 1 '
gp, I '
gp, i
(OH.1 '
(-OH, I '
¢J)H,I
where the lower index determines the type of joint between the input link and the frame, the upper left index determines the type of joint and lastly the upper right index determines the serial number of the joint of a given type in the chain, then these first transfer functions as it is obvious from (6), might have dimensions "length", "length to the power of minus one", or have no dimension at all. The screw eqn (5), after the substitutions (2), (3) and (4) and the relations (6), is equivalent to a linear nonhomogeneous algebraic system of six scalar equations with respect to n - 1 desired first transfer functions. For instance, in the case when the joint between the first link and the frame is of the type R, the expanded matrix of the corresponding system will be OlR.2...
OtR, K R O . . .
• -- OlR, 1 --
Ot~H,1
. .
OIH, K ~
O~m
..
flH, K H
--~R,I
T R , 2 . . . TR.KR O . • • OTH, I
• .
~/H,K H
--~R.1
..
(aH, KHPH.K. + IN, K . )
--[R.I
(~H.KHPH,KH + mH, KH)I
-- mR, j
( r.,n,,P,-,,K. + nn.KH)
- nR,~._j
f 3 R , 2 . . . [3R, K ~ O .
. .
IR.z. . . IR, K.ap, l . . . ae, x p ( a n , , P n , , mR,2..,
+ lu, O
mR, K.B.,I . . . [3p, Ke([3n.,Pn.j + r a n , 3 . .
h R . 2 . • • nR,KRYP,1 • • • Y P . K p ( Y H . 1 P H , I +
nn, O
..
(7)
Similarly we can record the expanded matrix in the case of a P, or H joint between the input link and the frame.
Mobility of Spatial Kinematic Chains The question of mobility of a mechanical system of joined rigid bodies is reduced to studying its ability to be driven by one or more input links. The classical approach to the problem of mobility[20, 21] based on structural equations carrying information about the number of links and the types of joints, is in many cases inadequate. An objective estimation of the mobility can be carried out based on the metric parameters of the chain and its kinematic properties. Mobility has been thoroughly studied by Hunt[22]t on the basis of the properties of screw systems and their special forms, and transitivity and gross mobility have been separated. Back in the beginning of the present century, quite intuitively, certain overconstrained mechanisms were created, and some researchers explained their gross mobility[23, 24] by geometric and analytic methods. Lately, Savage[25] and Waldron[26] created techniques for synthesis of some overconstrained kinematic chains with gross mobility, which in itself is a difficult and in no way a satisfactorily resolved problem. A spatial kinematic chain has instantaneous mobility[27] if the nonhomogeneous linear system which is characterized by the expanded matrix (7), corresponding to the equivalent discrete kinematic model of the mechanism, is compatible. The necessary and sufficient condition for this, according to Kroneker-Kapelli's theorem, is that the rank r of the base matrix is equal to the rank r' of the expanded one. It is invariant with respect to the choice of the frame but not so with respect to the choice of the input link. Satisfying the conditions for instantaneous mobility of a given structure does not necessarily guarantee mobility in the case of larger displacements since the instantaneous mobility can be transitory if for all infinitely close neighbouring positions the condition for instantaneous tThe authors beg Professor Hunt to excuse them for quoting his unofficialpaper[23], but they were stimulated to do so because of the freshness and originality of the ideas set forth therein.
MMT Vol 15, No 1--D
50 mobility is not satisfied. The necessary condition for a non-transitory instantaneous mobility is the existence of instantaneous mobility for a set of neighbouring positions. On the other hand, even those cases are of practical interest for which the condition for instantaneous mobility is not fulfilled but the existence of suitably chosen clearances in the joints and the fact that this condition is satisfied for the infinitely close neighbouring positions, are enough to allow the mechanism to overcome that critical state and even function with full cycle gross mobility. The instantaneous mobility as outlined above is equivalent to the question of linear relation between screws and the theory of screw groups [15, 28].
Determination of Second and Higher Order Transfer Functions Without Differentiation The kinematic analysis on the level of second order derivatives of the position functions is suggested by the modern tendencies for the fast action of mechanisms in mechanical automatics. It provides for a quantitative estimation of the effect of the inertia forces, most strongly expressed in the case of open kinematic chains. Research in the field of acceleration analysis was carried out by a number of scientists as for example[29, 30], including screw-geometric methods[31] but in all cases this was accomplished through double differentiation of the displacement functions. The method described in this paper and which was already used for finding the first transfer functions can be further applied to include second transfer functions[32]. The formally differentiated screw eqn (1) written in the form (8) contains the derivative of the unit screw
(9)
,~ = toe + a + / ~ [ p + a + p x(toe xa)]
assuming the following new notations: a is the unit vector of the ISA; p is the radius vector of an arbitrary point on the ISA; tOe is instantaneous angular velocity of the ISA. Since the relations (8) and (9) hold both for absolute as well as relative displacements of the spatial kinematic chain's links, then for the joints included in the equivalent kinematic model, the bivector equation (8), for R joints, becomes 2,
MR = dR~'R + tOR,4R
(10)
tip =
(11)
for P joints +
and for H joints /Vl. = (1 +/zPH)(o3ng, n + w.g,.).
(12)
All components of eqn (9) can be found algebraically, if the assumption is made that the first order transfer functions are already known and that the chain is closed. It can be assumed for tOe, i.e. the transfer angular velocity of a ISA of the relative motion, to be the absolute angular velocity of one of the links connected by the joint, while the derivative ti we can assume equal to the velocity of a point on the joint's axis regarded as a point belonging to one of the links in its absolute motion. The formally differentiated eqn (5), in the case when the joint between the first link and the frame is of the type R, bearing in mind (10), (11) and (12), can be recorded in the form KR
Kp
+ U=2
U=I
KR U=I
KH
v,,u L,.u +
0 + U=I
Kp
KH
U=I
U=I
(13)
51 Likewise in the case of the first transfer functions the introduction of the ratios for the second transfer functions in the manner of [33] 1,RR,U - -
FpR,U
tf)R,U -
vRP,U = 7PtU '
lJRH'U =
tbz,v
FpP,U
vpU, U
d')H,U
~rP,U
vL'
v R.v - (oR.u
v
.u -
vL' f'P,u
v
vL
(14)
.u = ';'H.u
the screw eqn (13) is equivalent to the nonhomogeneous algebraic system of six scalar equations with respect to n - 1 desired second transfer functions. The expanded matrix of this system is analogous to the matrix (7) with the exception of the column comprising the free terms which are determined by the right-hand side of eqn (13). The strategy described so far can be applied for determining the highest order transfer functions without resolving to differentiation. Besides, the base matrices of the corresponding linear non-homogeneous algebraic systems of equations stay invariant but with increasing the order of the transfer functions the last column on the right-hand side of the corresponding expanded matrices is considerably complicated. The second and higher order transfer functions of the type (14) and the corresponding higher analogies are dependent on the assumed values for the input transfer functions of the respective order, as seen from eqn (13). The ratios (14) and their higher order analogies, as well as (6), can be with or without dimensions, and the study of the systems that determine them also bears relation to the problem of mobility. Computation of the Finitely Small Relative Displacements in the Joints of a Kinematic Chain For almost all spatial mechanisms of practical importance, there exist special discrete configurations of their kinematic chains, for which the computation of the numerical values of the position functions is not so labour consuming. It is therefore convenient to use them when determining the initial conditions in the displacement analysis along the lines of the strategy proposed in this paper. The finitely small relative displacements in the joints, expanded in a Taylor series [34, 43] when the input link and the frame are connected by an R joint, for the three pairs of joints figuring in the equivalent kinematic model, for a chosen input step A~0R,are as follows: A~oRR'U -~ KRR'UA~oR +~ UR ' A~OR + . . .
f
l RU. 2 A~e'v rUe'VA~R+~VRev.' A~R2 +... 1
(15)
A~off'v KRn'VA~OR+~1 VRuv."A¢t¢2 + .... In the case when a P joint is employed between the input link and the frame, the corresponding Taylor series for an input step A~Cpare A~p R'U = KpR'U A~p +~
f
1 vff.UA~ff+,, " 1
A~opn'v rpn'vA~p +~1 vpu.vA~e2+ .... And finally, when an H joint is employed, the Taylor series for an input step Aq~n are
(16)
52 A~DHR'U = KHR'U ~
1
RU--
H -+- ~ lJH • A~DH + -j-. , .
A~. e.u = K.P.t; A~H + ~1 VHV't'A ~ f +. ACHH,U =
KHH'U A c H
(17)
1
-~'~ 1jHH'U A c H 2 q"- . ..
The increments of the position functions in the joints, obtained from (15), (16) or (17), respectively, define a neighbouring position of the kinematic chain, corresponding to the chosen input step. Following step by step the proposed strategy we are in a position to find, using a computer, a numerical solution of the position functions, applying the following obvious relations _- C v j+ + a C . V l +
(18) , = C ' . v [ . + A¢',vl+
where q is an index denoting the consecutive step. It is clear that using a larger step, in order not to decrease the accuracy of the calculations, it is necessary to include higher order members of the Taylor series. On the other hand, the introduction of each next higher order transfer function is connected with a corresponding increase in the computation difficulties and additional machine time expenditures. The authors recommend the use of Taylor series up to second degree. The use of first order transfer functions only is not effective in the case of fast changing position functions, while in the case of mechanisms which pass through dead points it is not always possible.
Control of Open Spatial Kinematic Chains With the introduction of open kinematic chains in technology and medicine as basic mechanic structures of manipulators, robots, prosthetic appliances, walking machines and certain mechanimals, the question of their expedient positioning and control has become imperative. Two fundamental problems of spatial kinematics arise -given the relative displacement in the kinematic joints, i.e. given a particular configuration of the chain, find the position of the last link, -given the position of the last link find the configuration of the kinematic chain, i.e. find the relative displacements in the joints. The first problem, also known as the +'direct problem" of displacement analysis, of kinematics, respectively, has in all cases a unique solution which is the result of direct methods, and algorithms exist to this effect, like for example [35, 36]. The second problem, known as the '+inverse problem" of displacement analysis is not always uniquely determined and is of great practical importance since its solution is fundamental to the mathematical modelling of the control processes of open mechanical systems of connected rigid bodies. It is well known that the theoretical minimum of degrees of freedom which permit the free control of an object in the three-dimensional Euclidean space is six. The solution of the inverse problem by direct methods has so far proved to be impossible, as Pieper and Roth[37] have shown, since in the case of a robot with 6 degrees of freedom, the problem is reduced to finding the roots of a polynomial of degree 524288. It is only through additional restrictions on the structure of the kinematic chain, like, for instance, permanent intersection of pairs or triplets of axes defining relative motions, that some authors [37, 38, 8] were able, using direct methods, to produce original solutions of the inverse problem for certain particular structures with up to 6 degrees of freedom. The method proposed in this paper for the control of an open kinematic chain on the basis of target variation of the positions and orientation of the end link is founded on the computation technique of finite displacements in the joints. An open kinematic chain with 6 degrees of
53 freedom and only R, P and H joints, is considered. Let two orthogonal reference axes be connected to the last link, and let along these axes B and d be two unit screws. Let F be the origin of the two axes. The end position of the last link will be defined by the position of the two reference axes and the pole namely by the unit screws/3e and ~ , and the point Fe. The displacement which has to be realized by the last link, can be modelled by two noncommutative finite displacements, translation and rotation, rotation and translation, respectively. The unit screws defining the initial and finite displacement of the last link (the transported object) are expressed in a bivector form, as follows:
~=h+#pxb
Be = be +/2,pc X be
C = c + / z p xc
(2e = ce +/zpe xc~
b'e=O
(19)
be "ee =0.
First variant: translation, rotation. The screw of the finite translation displacement, according to (19), is defined as q£ = IZ(pe - p) = tS/z~-
(20)
where s~ is the modulus, and ~- is the unit vector of pe-P. The screw of the finite rotation displacement superposing B with Be and (7 with t~e, is denoted by
M=
~fi
(21)
where ¢ is the angle of finite rotation, R is the unit screw of the axis of this finite rotational displacement, while B' and C' denote the unit screws of the displacement of the axes B and (~ after performing the translation (20). The screw R in a bivector form = r +/./,joe x r
(22)
contains the unknown unit vector r which will be defined by using the ideas of Saussure [39], namely
r = V(gZq
g2 +_ q(g. q)2)
(23)
where g = be - b,
q = Ce - c
(24)
while g and q are the corresponding moduli of the vectors, defined by (24). To determine the angle ~ of finite rotation, the following relation is used:
{d. d~ '~
¢ = arc cos kd .-----~)
(25)
d = b x r,
(26)
where de = be x r
are introduced, while d and de are the corresponding moduli of the vectors defined by (26). Second variant: rotation, translation. The corresponding screws of finite displacements are determined according to the technique applied for the first variant. Every finite displacement of the described two possible variants can be modelled by an
54 imaginary introduction of a prismatic P, respectively revolute R-joint closing the chain [401. In this way the open kinematic chains can be treated in a kinematic aspect as closed onest, and this allows to realize the two necessary finite displacements according to the exposed method for forecasting the finite small relative displacements in the joints of the chain. A further improvement of this strategy, allowing to decrease the number of steps and making possible the realization of the control in real time by a processor of moderate power, is achieved by an imaginary closing of the chain through a screw H-joint. For this purpose, it is necessary to define the normal Pliiker coordinates of the axis of the finite screw displacement leading the last link (the transported object) from an arbitrary initial position to the target finite position and to find the complex angle of rotation about and translation along this axis. Dimentberg and Averianova[41], by using the principle of transmission of the operations of vector algebra into bivector (screw) algebra. The same principle, formulated almost at the same time and independently by Kotelnikov[15] and Study[161, is also used by the authors, and for determining the normal PKiker coordinates of the required axis, the screw analogue of the relation (23) is applied j~ =
GxQ x/(G2Q 2 - (0" ~,)2)
(27)
where
In the formula (27) R is the unit screw of the axis of finite screw displacement, while G and Q are the complex moduli of the screws defined by (28). Analogically, to (25), for the complex angle of the finite displacement one obtains:
D. b e cos ~b = D" De
(29)
where the following screws have been introduced:
~=~x~,
6e=~eX~
(30)
while D and De are the corresponding complex moduli of the screws defined by (30). As it is known, the kinematic sense of the complex angle cb is defined by cos ~ = cos ~, - #n sin 0
(31)
where 0 is the angle of finite rotation about the axis R, and r/ is the length of the finite translation displacement along the axis. Note that for one and the same initial and finite position of the object, the angle ~o obtained from (25) for the strategy "translation-rotation", is equal to the angle ~, obtained from (29), (3 I) for the strategy of finite screw displacement, while for the values of finite translation displacements for the two different strategies, the following dependence holds: n ~ £.
(32)
Since the screw relation (27) for some particular cases, degenerates and becomes unsuitable for practical calculations, the following technique is reccommended in order to overcome this difficulty. If b=be, c~ce,
resp. c = c e b#be
then r = b = be,
resp. r = c = c e
*Thelast link of the openkinematicchain is treated as an input link in the imaginarilyclosedchain.
55 when carrying out the strategy "translation-rotation". If b = be
and c = ce
then the rotation vanishes and the finite translation displacement is d e t e r m i n e d f r o m (20), In case, when carrying out the target p r o g r a m m e directly, or step by step, the imaginarily closed kinematic chain with a changing structure falls into a state of a partial or full jamming (loss of mobility), one has to introduce deviating impulses for displacements, and after that the system is brought back to the accepted control algorithm. Since the mathematical modelling of the kinematics of the last link of the open kinematic chain is carried out in the six-dimensional space, while the regional positioning of the pole is accomplished in the three-dimensional space, then, when constituting the target assignments it is n e c e s s a r y for them to be in a c c o r d a n c e with the volume of the serviced six-dimensional space. This involves a compatibility between the prescribed target orientation of the last link and the space angle defining the service coefficient [42] at the given point of the service space.
References 1. V. A. Zinoviev, ProstranstvenieMechanizmi s Nizshimi Parami. (in Russian). Gostehizdat, Moscow (1952). 2. F. M. Dimentberg, The Determination of Positions of Spatial Mechanisms. pp. 142. Izdat. Akad. Nauk SSSR, Moscow (1950). 3. R. Beyer, TechnischeRaumkinematik. Springer-Verlag, Berlin 0963). 4. J. Denavit and R. S. Hartenberg, A kinematic notation for lower pair mechanisms based on matrices. J. appl. Mech, 22, Trans. ASME, 77, 215-221 0955). 5. J. Denvait, Displacement analysis of mechanisms based on 2 × 2 matrices of dual numbers. VDI-Berichte, B. 29, 81-89 (1958). 6. A. T. Yang and F. Freudenstein, Application of dual-number quaternion algebra to the analysis of spatial mechanisms, Trans. ASME J. appl. Mech. 31E, 300-308 (1964). 7. D. Mangeron and C. Dr/igan, Asupra unei noi metode tensoriale de studiu a mecanismelor. Bull. Institutului politechnic, Jasi, III, No. 1-2, 151-164 (1957). 8. M. S. Konstantinov and Z. I. Zankov, A kinematical algorithm and dynamical point mass simulation in robots and manipulators. Proc. of First CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, Udine (Italy) 1972 Vol. I, Springer-Verlag Wien, New York. 9. P. A. Lebedev, Kinematika prostranstvenih mechanismov. Mashinostroenie, Moscow 0966). 10. V. Brat, Kinematic analysis of spatial mechanisms with general kinematic pairs (a matrix method). 3rd World Congress on the TMM, Proc. Vol. C, Cupari, Yugoslavia (1971). I 1. J. J. Uicker, J. Denavit and R. S. Hartenberg, An iterative method for the displacement analysis of spatial mechsnisms. J. appl. Mech. 31,309-314 (1964). 12. L Woo and F. Freudenstein, Application of line geometry to theoretical kinematics and the kinematic analysis of mechanical systems. 3. Mechanisms, 5, 417-460 (1970). 13. I. 1. Artobolevski, Teoria mehanismov i mashin (in Russian). lzdat Nauka, Moscow (1975). 14. R. S. Ball, A Treatise on the Theory of Screws. Cambridge University Press, Cambridge (1900). 15. A. P. Kotelnikov, Theory of Screws. (Russian) Kazan (1895-96). 16. E. Study, Geometrie der Dynamen. Teubner Verlagsgesellschaft GmbH. Liepzig (1903). 17. W. Clifford, Preliminary sketch of biquaternions. Proc. of the London Math. Soc., Vol. IV, (1873). 18. J. Pliiker, NeueGeometriedes Raumesgegriindet aufdieBetrachtungdergeraden Linieals Raumelement. B. G. Teubner, Leipzig (1868--69). 19. M. S. Konstantinov and M. D. Markov, Parvi predavatelni funktsii na prostranstveni kinematichni verigi. J. Theoretical and Appl. Mech. Bulg. Acad. Sci., 2, (1977). 20. K. Kutzbach, Mechanische Leitungsverzweigung. Maschinenbau, Der Betreib 8, No. 21,710-716 (1929). 21. P. O. Somov, O stepeniah svobodi kinematicheskoi tsepi. J. of the Russian Physico-chemical Soc. SBP Univ., Vol. XIX, part phys. issue 1 9 (1887). 22. K. H. Hunt, Screw Systems in Spatial Kinematics. Monash University, Clayton, Victoria, Australia (1970). 23. A. Bennet, A new mechanism, Engineering 76, 777-778 (1903). 24. R. Bricard, Lemons de Cin~matique Vol. 2, Gauthier-Villars, Paris (1927). 25. M. Savage, Four-link mechanisms with cylindric, revolute and prismatic pairs. Mechanism and Machine Theory, 7, 191-210 (1973). 26. K. J. Waldron, A study of overconstrained linkage geometry by solution of closure equation. Part I. Method of Study. Mechanism and Machine Theory, 8(1), 95-104 (1973). 27. M. D. Markov, Mobilnost na prostranstvenite kinematichni verigi. Godishnik na VMEI "V. I. Lenin", Jub. izd. (1976). 28. F. M. Dimentberg, Metod vintov v prikladnoi mehanike. Mashinostroenie, Moscow (1971). 29. M. S. Konstantinov, Inertia forces of robots and manipulators, paper presented at the Second Conference on Remotedly Manned Systems (June 9-1 I, 1975), University of Southern California, L. Angeles, U.S.A. 30. J. I)enavit, R. S. Hartenberg, R. Razi and J. J. Uicker, Velocity, acceleration and static-force analysis of spatial linkages. J. appl. Mech., 32, Trans. ASME Vol. 87, (B) 903--910(1965). 31. M. Skreiner, Acceleration analysis of spatial linkages using axodes and the instantaneous screw axis. J. of Eng. forlnd. 97 (February 1967). 32. M. S. Konstantinov and M. D. Markov, Vtori predavatelni funktsii na prostranstveni kinematichni verigi. J. of Theoretical and appl. Mech., Vol. 3, Bulg. Acad. Sci. (1977).
56 33. 1. I. Artobolevski, Dinamicheskie kriterii regima dvizenia mashini. J. Theoretical and Appl. Mech. Bulg. Acad. Sci., Sofia Year 11, No. 1 (1971). 34. M. S. Konstantinov and M. S. Markov, Pozitsiona identifikatsia na prostranstveni kinematichni verigi..1"+ Theoretical and Appl. Mech. Bulg. Acad. Sci. Sofia 4 (1977). 35. R. Seyffarth, Geometrische Konstruktionsverfarhen zur Darstellung viergliedriger r~iumlicher Kurbelgetribe. Maschinenbautechnik, 10, Heft 12, Berlin (1%1). 36. M. S. Konstantinov, Structural and kinematic analysis of robots and manipulators. Proc. Symposium on lndugtrml Robots. Tokyo (1974). 37. D. L. Pieper and B. Roth, The kinematics of manipulators under computer control. 2nd Int. Congress on the Theory o( Machines and Mechanisms, Proc.+ Zakopane, Poland (1%9). 38. A. G. Ovakimov, Kinematicheskoe isledovanie prostranstvenoi tsepi upravlenia manipulatora. Izv. vuzov, series "Mashinostroenie'++ No. 14 (1971). 39. R. Saussure, Etude de geometrie cinematique regl6e. Amer. J. of Maths. 19, (1877). 40. M. S. Konstantinov and M. D. Markov, Deshifrirane na konfiguratsiata na otvorena kinematichna veriga po polozenieto i orientatsiata na krainoto zveno. C.R. Acad. Bulg. Sci., 30, 5 (1977). 41. V. G. Averianova and F. M. Dimentberg, Opredelenie vinta peremeshtenia po nachalnomui konechnomu polozeniu tverdovo tela. Machinovedenie, No. 2 (1%5). 42. A. E. Kobrinski and J. A. Stepanenko, Nekotorie problemi teorii manipulatorov. Mehanika mashin, issue 7-8, (1%7). 43. P. I. Genova, Priblizitelen metod za opredeljane na pozitsionite funktsii na ravninite mehanizmi, Proc. of the 3rd National Congress of Theoretical and Applied Mechanics. Varna (1977).
Appendix 1 We will determine the first and second transfer functions for one discrete position of the spatial crank and connecting-rod assembly, shown in Fig. I. The following numerical data are accepted for the considered mechanism: a = 0.5, b = 0.2, c = 1, 0 = 60°. Consider the discrete position of the mechanism when the link is parallel to the plane xOy of the chosen ground coordinate system. The kinematically equivalent scheme of the mechanism in this position is described on Fig. 2, the axes of R-joints replacing the S-joint being chosen parallel to the coordinate axes. For the case in consideration, the screw equation (5), written in a bevector form, will be WR.li+ toR.2(k+/.t 0.45825 i) + ~OR.~(j- / z 0.2 i) + tOR+4[i+ #(0.2 j - 0.45825 k)] + ,OR.s(0.72842i -- 0.68512 j + /Z(0.13702 i + 0.145711 j -- 1.33392 k)] + o~R.6[k+ ~(1.18667 i -0.68512 j)] + Vp.1#(0.5 i - 0.86602 j)
0.
I33)
After separation of the scalar equations equivalent to the bivector equation (33) and introduction of the relations (6), the following values for the first transfer functions were determined: KR'~'2-- 0.10273, KRn'4=
.,+RR+2= 0.31392
1.33376, ~Rk's =0.45820
K,~'~'~- 0.10273,
134)
K,~e'f .... 0.14%6[m].
When determining the second transfer functions, we assume
~'R.t
=
IIS I],
~.,+.l =0+
For the same discrete position and dimensions of the mechanism, by using the obtained numerical values (34) for the first transfer functions, the screw eqn (13) assumes the bivector form, as follows:
,! 'r Z
/
1X ib
Figure 1. Spatial crank and connecting-rod assembly.
57
I
A"
b = a sin q~,t,,
pi~,t
?,4.
Figure 2. Kinematically equivalent scheme of the spatial crank and connecting-rod assembly from Fig. 1.
o~R.z(k+ # 0.45825 i)+ dR,3(J - # 0.2 i)+ O~R.4[i+ #(0.2 j --0.45825 k)] + o)R,5[0.72842 i - 0.68512 j + #(0.13702 i+ 0.14570 j - 1.33392 k)] + o)R.dk + #(1.18667 i - 0.68512 j)]+ Ve: #(0.5 i - 0.86602 j)
(35)
= 0.20546 j - 0.73263 k + #(-0.22573 i + 0.61889 j - 0.2 k). After separation of the scalar equations equivalent to the bivector eqn (35) and introduction of the relations (14), the following values for the second transfer functions are obtained: uR R'2= -- 0.54915,
VR R'3 = 0.06844
7,RR'4= --0.14568,
UR R'5
= - 0.2
vRR'6= - 0.18347,
vR e : =
-
(36)
0.56947 [m].
Appendix 2 Following the method described in the present paper, an analysis of the position of 6R mechanism Turbula (System Schatz, Maschinenfabrik W.A. Bachofen Switzerland) will be carried out, its kinematic scheme being shown in Fig. 3. According to the classical theory for mobility [20, 21], the kinematic chain should be a statically determinate rigid frame but the mechanism has one degree of freedom due to the suitably chosen dimensions and angles of intersection or crossing between the axes of kinematic joints. This is explained by the fact that for every position of the mechanism the joints' axes are straight lines belonging to one and the same linear complex, i.e. the unit screws defining the position of the joints' axes in the three dimensional Euclidean space belong to one and the same 5-term screw group. Fig. 4 describes the kinematic scheme of the mechanism in one particular position which is suitable to be chosen as initial. The same figure indicates the correlations between the metric chain parameters assuring its full cycle gross mobility. From Fig. 3 and Fig. 4 it is clear that for the chosen initial position, the position functions have the values, as follows: tpe,i = 90o, ¢e.2 = 900, ~oe.3= 60o, ~pe.4= 180o, ~e.~ = 150o, q~e.6= 900. For the initial position of the mechanism in consideration, the screw eqn (5) assumes the following bivector form: mR.Ik + tOR.2J+ ~0R.3(i+ #j) + tOR,4(0.5 j + 0.5 X/(3)k + # 0.51 i) - tOR,5(i -- # ' x / ( 3 ) k ) - tOR.6(k+ #\/(3)i) = 0.
(37)
After separation of the scalar equations equivalent to the bivector equation (37) and introduction of the relations (6), the following values for the first transfer functions were calculated: KRR': = 0.5 X/(3),
KRR'3 = 0,
KRR'4= -- X/(3),
KRR'5 = 0,
K R R ' 6 = --
0.5.
(38)
In order to determine the second transfer functions, it is again assumed that coR,~= I[S -~] and &R.~=0. Using the obtained numerical values for the first transfer functions (38), after some transformations, the screw eqn (13) is written in a bivector form, as follows:
+ O)R,4(0.5 j + 0.5 X/(3) k + # 0.5 i) - O)R,5(i - # X/(3) k) - o)R.6(k + # ~ / ( 3 ) i) = 0.25 ~/(3) i - #(0.5 x/(3) j - 0.75 k).
O)R,2J + O)R3(i + # j )
(39)
58 F-Z
+:U I
/ , q0~'2
A~
o-
i
.,
Y
Al, ll
Figure 3. Kinematic scheme of the "l'urbulamechanism. From the equation (39), by using the substitutions (14), the following values of the second transfer functions are obtained: VRR'2 : O,
VRR'3
= 0.5 \/(3),
VRR'4 = 0,
VRR'5 =
-- 0.25 \/{3},
VR~'6 : O.
(40)
Using the Taylor series (15), the finitely small relative displacements in the joints, corresponding to the chosen input step, are calculated to within the second derivative. For the mechanism Turbula the composed program by the method of discrete positions was applied. For the chosen initial position of the mechanism, two of the first transfer functions vanish, according to (38), whence the first transfer functions only are not always in a position to forecast the finitely small increments of the displacement
iz
~
~,,., .L ~,.2
"
r
..
Figure 4. Kinematic scheme of the Turbula mechanism in the chosen initial position.
59
Table 1. Entry IJJCPD1N 00D278 Entry IJJCPD3 00D278 CSECT IJ2E0005 00D208 00C400 1
2
3
4
5
6
1:5707%33 1:65806279 1:74532925 1:83259571 1:91986218 2:00712864 2:09439510 2:18166156 2:26892803 2:35619449 2:44346095 2:53072742 2:61799388 2:70526034 2:79252680 2:87979327 2:%705973 3:05432619 3:14159265
1:5707%33 1:64634767 1:72175438 1:7%86461 1:87151310 1:94551312 2:01864056 2:09065137 2:16120529 2:22990447 2:2%27123 2:35%1805 2:41909903 2:47363502 2:52187991 2:56222941 2:59291532 2:61223795 2:61892612
1:04719755 1:05051075 1:06052269 1:07750413 1:10192156 1:13446123 1:17606403 1:22797244 1:29178919 1:36954480 1:46371426 1:57739853 1:71414925 1:87779376 2:07187702 2:29858732 2:55710257 2:84181384 3:14160555
3:14159265 2:99049090 2:83%8035 2:68946483 2:54017513 2:39218506 2:24593135 2:10193863 1:96085207 1:82348006 1:69086564 1:56431454 1:44552043 1:33664254 1:24031305 1:15972719 1:09846817 1:05993187 1:04665731
2:61799388 2:61633535 2:61132740 2:60284356 2:59062349 2:57435096 2:55354634 2:52758817 2:49567477 2:45679043 2:40967274 2:35279263 2:28437580 2:20250281 2:10546519 1:99216115 1:86294309 1:72060892 1:57071202
1:5707%33 1:52708087 1:48286424 1:43762389 1:39079751 1:34176022 1:28979853 1:23407999 1:17361818 1:10723444 1:03352129 0:95081888 0:85723147 0:75073522 0:62945560 0:49220393 0:33930274 0:17347248 0:00018954
functions, i.e. they are not always in a position to determine the neighbouring configuration of the mechanism for an arbitrarily chosen finite step. If for any position of a given mechanism the first and second transfer functions vanish simultaneously in some of the joints, the use of higher derivatives might impose itself but it is not typical for the single loop spatial kinematic chains. With every following step the errors accumulate which makes useful to accept the computation strategy, as follows: From the initial position the displacement analysis to be carried out for one half of the interval studied clockwise, while after that from the same initial position it must be carried out counterclockwise for the other half of the interval. The Turbula mechanism as an overconstrained mechanism, is a rough test for the method since with the accumulation of impermissible errors, the equality of the ranks of the base and expanded matrices of the systems for determining the first and higher transfer functions will be violated. Under double-accuracy computation on the EC 10-40 computer, using a step 1,5', the results given in Table 1 were obtained, where the values of the angles in the joints are registered in radians in the separate columns. The number of the column corresponds to the number of the joint. The digital print was done for every 200 steps.
Appendix 3 The a~s of the complex angle of the finite screw displacement superposing the initial position of a solid object, defined by f = i, C = j and the finite position, defined by fie = j, Ce = k +/zl will be determined. According to (28), G=-i+j,
{~=-j+k+#i.
141)
Substituting in (27), for the unit screw of the axis of the required finite displacement, one obtains i+j+k-#k
1
. .
1
.
_2k)].j
(42)
By using (30), one gets 1 [ - j + k + l ( 2 j + k ) ],
•
I
D e = ~' l ~ [ |/- k -( 3 / z3( 2 i) + k ) ]
143)
After normalization of the screws D and fie, their complex moduli are obtained, as follows: D=~/
1 - Z 3--~'
De = ~/
I -/z 3X/(6)'
G)
!44)
Substituting (43) and (44) in (29), for the complex angle of the finite displacement, one finds 1 1 cos~=-~-L.
(45)
or, in accordance with (31) 2 = arc cos ( 1-~) =~', are calculated.
I ~ = x/3(3)(length ) "0 = 2 sin
(46)
6O EJ.NE }~L~THODE DER D I S K R E T ~ RAUMLI CH/~ MECHANISM]~
LAG~2~ IN DER KINE%~ATIK UND STEUERUNG DER B~LEGDNG~h?~
M.S.Konstantinov und M.D.Ma±'kov
Kurzfassun~ - Die ersten und zweiten Ableitungen der relativen Bewegungen riumlicher Ketten in jeder diskreten La~e werden bei Einf0/Lvung der Schraubentheorie der infini~esimalen Verschiebungen
bestim~it, ohne Differentiationen
im Ra~nen
anzuwenden. Die
endlich kleine Ver~nderungen der in Ta~loz-Re£hen entwickelten Positionsfunktionen werden Schritt f~r Sc~ritt integriert, und somit ist die Bewegung der Kette ermittelt Fdr die Steuerung offener kinematischer Ketten mit sechs Freiheitsgraden wird ein Algorithmus auf Grund scheinbarer EinschlieBung der Kette aufgebaut. Dabei wird die Bewegungsf/higkeit
berOcksichtigt.
bestimmte Schraubenbewegung
Eine beliebige Ziellage des Endgliedes wi~d dutch
erreicht, womit die Zahl der Schritte vermindert und die
Computerzeit wesentlich ve~ki~zt wird. Entsprechende offene Ketten werden betrachtet.
Beispiele fir geschlossene und