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A note on position analysis of manipulators
Mechanism and Machine Theory Vol. 19, No. 1, pp. 5 8, 1984 Printed in Great Britain.
0094-114X/84 $3.00 + .00 Pergamon Press Ltd.
A N O T E O N P O S I T I O N A N A L Y S I S OF M A N I P U L A T O R S K R I S H N A C. G U P T A Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60680, U.S.A. Abstract--The notion of zero position of a manipulator is introduced and used in conjunction with the principle of similarity to derive analysis equations. The usual link coordinate systems are not defined and the base coordinate system is the only one used in analysis. INTRODUCTION
If a picture or an engineering drawing of a manipulator is available, it is often very difficult to correctly define the special coordinate systems and associated parameters discussed in [1,2]. In this paper we present an analysis method which utilizes only the base coordinate system and is not subject to errors in interpreting manipulator geometry. This method makes use of the principle of similarity[3-5]. In view of the fundamental nature and significance of this principle, it is surprising that its potential in manipulator applications has not been exploited. ANALYSIS
Let us briefly discuss the nomenclature followed in this paper. A free vector u is represented as a 3 × 1 column matrix (vector) and a position vector (or point coordinate) p is represented as a 4 × 1 column vector. A line bound vector is indicated as ft. It is assumed that the direction u of ii and the coordinates of a point on fi are explicitly known. A formal representation of fi in terms of dual vectors is not necessary in what follows. A rotation 0 about an axis u is expressed as a 3 × 3 matrix R(O, u) such that v:= R(O, u)v~, where v~ and vf represent, respectively, the (free) vector v before and after the rotation. A displacement consisting of a rotation 0 about and translation d along the screw axis fi is expressed as a 4 × 4 matrix D(O, d, fi) such that P f = D(O, d, fi)' P~, where P~ and P: are, respectively, the coordinates of a point P before and after the displacement. The elements of matrices R and D readily follow from Rodrigues' formula and are available in the literature [5.6]. The statement of the principle of similarity can now be given. Let matrix D(O,d, fi) represent a displacement (0, d) with respect to a screw axis ft. Suppose that a displacement (~, a) with respect to a screw axis fi, whose displacement matrix is D(~, a, fi), displaces the screw axis ii to fi'. Then the displacement (0, d) with respect to the displaced screw axis fi' is represented by the matrix[3-5]
D(O,d, fi')=[D(cc, a, fi). D(O,d, fi). D '(o¢,a, fi)] (1) Let us designate an arbitrarily cohosen position of the manipulator as its reference position. As the manipulator moves from its reference position to the
current position, we define an incremental screw at a joint as the screw of relative displacement between the two adjacent links connected by the joint. F r o m the similarity principle, it follows that finite incremental joint screws do commute[4], i.e. they can be performed in any sequence or combination to achieve the same displacement of the hand and the same final position of the manipulator. F o r example, consider a 2H manipulator (H is either R or P) whose joint axes in reference position are fil~ and u2r. We follow the convention that the joint axes of the manipulator are numbered in an ascending order from the base to the hand. Let the incremental joint screws be (A0~, AdO and (A02, Ad2). A point P in the hand is displaced from its reference position P, to the final position P: as: P I = AP,. If we consider the sequence 21, then
A = [D(A0,, Ad~, fi~,).D(A02, Ad2, fi2,)].
(2)
If we consider the sequence 12, then
A' = [D(A02, Ad2, fi2)"D(AO,, Ad~, fi,,)],
(3)
where fi2 is obtained from fi2r after it is displaced by the screw (A0t, AdO with respect to the screw axis fi~. F r o m the similarity principle (1),
A' = [D (A0~, Ad,, fi,,). D(A02, Ad2, fi2,) • D -t(A01, Adl, tilt)'
D(AOI,Adt, iilr)] ---- A.
(4)
In an n H manipulator, suppose the screw axes in the reference position are ilk, k = 1, n. A point P in the hand is displaced from its reference position P, and PI when n incremental joint screws (AOk, Adk) are carried out in any sequence. Then PI = A P,, where
A = [D(AO,, aa,, r~,,)...., o ( a o . , 6clo, ~,,)1
= I1 o(a0k, aak, a~,).
(5)
k=l
Because the sequence of incremental finite joint screws is immaterial, eqn (5) has been written for the simplest sequence which is n, n - l , . . . , 2, 1. In an actual manipulator, electrical zero setting is easy to achieve and an arbitrarily chosen reference
KR]SHNAC. GUPTA
Z
..~ ~
U
50
Y
,~10~/-/-/-/-/-/-/-/-/-~z.L~...~ x/--/u~
H0
<-//'6
=Uao
Fig. 1.
position may be designated by the manufacturer as the zero position of the manipulator. Then the incremental quantities in eqn (5) become absolute quantities and subscript r is replaced by 0. All of the ~k0 are known, i.e. the directions uk0 and the coordinates of a point on kth axis are known for all k = 1, n. If the displaced position of the hand is given, then the 4 x 4 matrix A can be evaluated numerically. The governing equation for the nH manipulator then becomes
A - '. fi O(Ok, d~,,ilk0) = k=l
I.
position is neither unique nor minimal. However, it is simple to visualize and it can be obtained from (or shown in) engineering drawings in a reliable way. Fortunately, the non-uniqueness of description does not pose a significant problem because, as seen in the following, it is quite easy to reconcile the analysis done with respect to different zero positions (e.g. Figs. 1 and 2). Suppose that the first zero position is described by Uk0, k = 1, n; Ua~,Ut0 and P0. The joint variables are Ok, dk; k = 1, n. The second zero position is described by
(6)
Of the twelve equations obtained from eqn (6), only six are independent. An alternate set of equations can be obtained as follows. The manipulator hand consists of a gripper with a two fingered jaw. Let us designate a midpoint of the jaw as P, a unit vector across the jaws and at point P as u, (transverse) and a unit vector along the jaws at the point P as u, (axial). In the reference position of the manipulator, P0, Ua0 and u~ are known. In the final position of the hand, P, Ua and u, are known. Then we can write the following nine equations (an excess of 3) which can be used for analysis
p = Ik01D(Ok, dk,~iko)14×4Po
(7a)
u~=[flk=, R (0k' uk°)qd3×3u~°
(Tb)
gt : [k~I=i R(Ok, Uko)]3×3Uto.
(7C)
u
030
u50~ ~ = , For the manipulator shown in its zero position in Fig. l, the complete position analysis is given in Table I.t It should be noted that a description of manipulator geometry in terms of a conveniently chosen zero tin Figs. & Tables, ~ =- u and ~ ~ ft. Also, Table 1 uses the single valued relation 0 = 2. arctan [sin 0/(1 + cos 0)] to avoid problems with nonunique arctan function.
,~ u60 ~0~ ,li ' ao Fig. 2.
u' to
A note on position analysis of manipulators Table 1. Analysis of the manipulator shown in Fig. 1. Zero
Position
Ulo = (0,0,1) t 0 (0,0,0,1) t ,
U2o = (0,1,0) t @(0,0,0,1) t ,
U3o = (0,1,0) t@ (b,O,O,1) t ,
U4o = (1,0,0) t@ (O,a,O,1) t ,
Us@ = (0,1,0) t o (b+c,O,O,1) t ,
U6o = (1,0,0) t o ( O , a , O , 1 ) t ,
Uao = (1,0,0) t @(b+c+h,a,O,1) t ,
Uto = (0,1,0) t @ (b+c+h,a,O,1) t ,
D° = (b+e+h,a,O,1) t . Hand P o s i t i o n P' Ua'
fit
(Ua'Ua
= fit'GGt
= i)
.
Solution
~'-hfia,
H =
D = 2bZH,
e2
r~ =
x~+Y~t '
E = -2brQ
2 a r c t a n ~E-7-g} '
,
r Q = +-
¢'r~t- a2
F = a z + c 2 _ rHz _ ZH2 _ b 2 ,
e3
-e 2 + 2 a r c t a n --
G = +_¢~2 + E 2 _ F 2
-b-b sine 2+rq
f = b cos82 ÷ c cos(@ 2 + 03) fYtt - axH ) f2 ÷ f x H+ayH÷aa
e1 = 2arctan For
n =a
or
t
Vnx = UnxCOSelCOS(e 2 + e 3) ÷UnySinelcos(O 2 ÷ e 3) - UnzSin(e2 + e 3) Vny = -UnxSinel +UnyCOS01 Vnz = UnxCoselsin(02 + e 3) +UnySinSlsin(e 2 ÷ e 3) +UnzCOS(O 2 ÷ e 5) VaQ = - * v ' ~ 04
\ vaQ-vaZ/
@6 = 2 a r c t a n (v
,
0S =
Vtx ) aQ + VayVtz - VazVty
u~0, k = 1, n; u~'0,u~o and P~0.The joint variables are 0~, d~; k = 1, n. We wish to determine the relationship between Ok, dk and 0~, d~. If the final position of the hand is given, then the matrix A such that PI = A P0 and A' such that l f = A'P~ can be computed. The governing equations in the two cases are
l~ID ( ¢ ,
2arctan
4 , ~,,o) = A
(8)
k=l
f l D(O~,d~,fi~)=A"
(9)
If the second zero position corresponds to Ok = ~tk,
(10)
k=l
It is easily seen that A =A'.M,
or A ' = A
.hi, -1,
(11)
From the similarity principle,
O(O;, d;, fi~o)= [Mr-O(O;, d;, 0m)" Mr-'].
k=l
dk = ak, k = 1, n, then P6 = M. "P0, where
I
Mt = I-I D(cq, ak, ilk@), 1 < n.
(12)
After substitution of eqns (10)-(12) into eqn (9) an comparing the result with eqn (8) we obtain 0h = ak + 0~, and dk = ak + d~,, k = 1, n. Therefore, it is quite simple to reconcile the results obtained with
KRISHNA C. GUPTA
method can also be used in the analysis of instrumented bio-linkages which are used for measuring motion. However, it cannot be applied easily to closedloop kinematic chains because then it is not possible to choose an arbitrary zero position.
different zero positions. For example, the solution given in Table 1 (zero position is Fig. 1) is applicable to the analysis with respect to the zero position shown in Fig. 2 if 0, in Table 1 are replaced as: 01 --*0l, 02--,02, 0 ~ 9 0 ~ + 03, 04~ - ( 1 8 0 + 04), 05~05, 06 = _ 9 0 '~ + 06.
Acknowledgement--The financial support of the National
In terms of the notation in Refs. [1, 2] the manipulators in Figs. 1 and 2 are described as follows: at = - 90~', 72 = 0, ~3 = ~ 4 ~--- ~ 5 = 90°, $2 = a, s4 = - c, s6=h,
$1=$3=$5=0,
Science Foundation (ENG 79-.-23193) is acknowledged. REFERENCES
a2 = b, a~ = a3 = a4 = as = O.
1. R. S. Hartenberg and J. Denavit, Kinematic Synthesis o f Linkages. McGraw-Hill, New York (1964). 2. D. L. Pieper and B. Roth, Proc. 2ndlnt. Congr. on Theory o f Machines and Mechanisms, Vol. 2, pp. 159-68 (1969). 3. H. Goldstein, Classical Mechanics. Addison Wesley, New York (1950). 4. B. Roth, A S M E J. Appl. Mech. 34, 599-605 (1969). 5. O. Bottema and B. Roth, Theoretical Kinematics. North Holland, Amsterdam (1979). 6. C. H. Sue and C. W. Radcliffe, Kinematics and Mechanisms Design. Wiley, New York (1978).
Figure 2 is in fact its zero position in this notation (i.e. all 0i = 0, i = 1, 6 or all xi axes are parallel). CONCLUSION
We have presented a method of analysis which utilizes the base coordinate system and a conveniently selected zero position of the manipulator. The zero position of the manipulator can also be represented easily and reliably in engineering drawings. The
UNE N O T E
K.C.
SUR
L'ANALYSE
DE LA P O S I T I O N
DES
MANIPULATEURS
la p o s i t i o n
z~ro d ' u n
Gupta
R~sum~
-
curremment
La n o t i o n avec
t~mes
usuels
d'une
base
de
le p r i n c i p e
de c o o r d o n n ~ e s
est
le seul
de
de
la s i m i l i t u d e
d'accouplement
~ ~tre
employ~
dans
manipulateur
pour
ne s o n t
d~river pas
l'analyse.
est
les
d~finis
introduite
~quations et
et e m p l o y e e
d'analyse.
le s y s t ~ m e
Les
consys-
de c o o r d o n n ~ e s
0094-114X/84 $3.00 + .00 Pergamon Press Ltd.
A N O T E O N P O S I T I O N A N A L Y S I S OF M A N I P U L A T O R S K R I S H N A C. G U P T A Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60680, U.S.A. Abstract--The notion of zero position of a manipulator is introduced and used in conjunction with the principle of similarity to derive analysis equations. The usual link coordinate systems are not defined and the base coordinate system is the only one used in analysis. INTRODUCTION
If a picture or an engineering drawing of a manipulator is available, it is often very difficult to correctly define the special coordinate systems and associated parameters discussed in [1,2]. In this paper we present an analysis method which utilizes only the base coordinate system and is not subject to errors in interpreting manipulator geometry. This method makes use of the principle of similarity[3-5]. In view of the fundamental nature and significance of this principle, it is surprising that its potential in manipulator applications has not been exploited. ANALYSIS
Let us briefly discuss the nomenclature followed in this paper. A free vector u is represented as a 3 × 1 column matrix (vector) and a position vector (or point coordinate) p is represented as a 4 × 1 column vector. A line bound vector is indicated as ft. It is assumed that the direction u of ii and the coordinates of a point on fi are explicitly known. A formal representation of fi in terms of dual vectors is not necessary in what follows. A rotation 0 about an axis u is expressed as a 3 × 3 matrix R(O, u) such that v:= R(O, u)v~, where v~ and vf represent, respectively, the (free) vector v before and after the rotation. A displacement consisting of a rotation 0 about and translation d along the screw axis fi is expressed as a 4 × 4 matrix D(O, d, fi) such that P f = D(O, d, fi)' P~, where P~ and P: are, respectively, the coordinates of a point P before and after the displacement. The elements of matrices R and D readily follow from Rodrigues' formula and are available in the literature [5.6]. The statement of the principle of similarity can now be given. Let matrix D(O,d, fi) represent a displacement (0, d) with respect to a screw axis ft. Suppose that a displacement (~, a) with respect to a screw axis fi, whose displacement matrix is D(~, a, fi), displaces the screw axis ii to fi'. Then the displacement (0, d) with respect to the displaced screw axis fi' is represented by the matrix[3-5]
D(O,d, fi')=[D(cc, a, fi). D(O,d, fi). D '(o¢,a, fi)] (1) Let us designate an arbitrarily cohosen position of the manipulator as its reference position. As the manipulator moves from its reference position to the
current position, we define an incremental screw at a joint as the screw of relative displacement between the two adjacent links connected by the joint. F r o m the similarity principle, it follows that finite incremental joint screws do commute[4], i.e. they can be performed in any sequence or combination to achieve the same displacement of the hand and the same final position of the manipulator. F o r example, consider a 2H manipulator (H is either R or P) whose joint axes in reference position are fil~ and u2r. We follow the convention that the joint axes of the manipulator are numbered in an ascending order from the base to the hand. Let the incremental joint screws be (A0~, AdO and (A02, Ad2). A point P in the hand is displaced from its reference position P, to the final position P: as: P I = AP,. If we consider the sequence 21, then
A = [D(A0,, Ad~, fi~,).D(A02, Ad2, fi2,)].
(2)
If we consider the sequence 12, then
A' = [D(A02, Ad2, fi2)"D(AO,, Ad~, fi,,)],
(3)
where fi2 is obtained from fi2r after it is displaced by the screw (A0t, AdO with respect to the screw axis fi~. F r o m the similarity principle (1),
A' = [D (A0~, Ad,, fi,,). D(A02, Ad2, fi2,) • D -t(A01, Adl, tilt)'
D(AOI,Adt, iilr)] ---- A.
(4)
In an n H manipulator, suppose the screw axes in the reference position are ilk, k = 1, n. A point P in the hand is displaced from its reference position P, and PI when n incremental joint screws (AOk, Adk) are carried out in any sequence. Then PI = A P,, where
A = [D(AO,, aa,, r~,,)...., o ( a o . , 6clo, ~,,)1
= I1 o(a0k, aak, a~,).
(5)
k=l
Because the sequence of incremental finite joint screws is immaterial, eqn (5) has been written for the simplest sequence which is n, n - l , . . . , 2, 1. In an actual manipulator, electrical zero setting is easy to achieve and an arbitrarily chosen reference
KR]SHNAC. GUPTA
Z
..~ ~
U
50
Y
,~10~/-/-/-/-/-/-/-/-/-~z.L~...~ x/--/u~
H0
<-//'6
=Uao
Fig. 1.
position may be designated by the manufacturer as the zero position of the manipulator. Then the incremental quantities in eqn (5) become absolute quantities and subscript r is replaced by 0. All of the ~k0 are known, i.e. the directions uk0 and the coordinates of a point on kth axis are known for all k = 1, n. If the displaced position of the hand is given, then the 4 x 4 matrix A can be evaluated numerically. The governing equation for the nH manipulator then becomes
A - '. fi O(Ok, d~,,ilk0) = k=l
I.
position is neither unique nor minimal. However, it is simple to visualize and it can be obtained from (or shown in) engineering drawings in a reliable way. Fortunately, the non-uniqueness of description does not pose a significant problem because, as seen in the following, it is quite easy to reconcile the analysis done with respect to different zero positions (e.g. Figs. 1 and 2). Suppose that the first zero position is described by Uk0, k = 1, n; Ua~,Ut0 and P0. The joint variables are Ok, dk; k = 1, n. The second zero position is described by
(6)
Of the twelve equations obtained from eqn (6), only six are independent. An alternate set of equations can be obtained as follows. The manipulator hand consists of a gripper with a two fingered jaw. Let us designate a midpoint of the jaw as P, a unit vector across the jaws and at point P as u, (transverse) and a unit vector along the jaws at the point P as u, (axial). In the reference position of the manipulator, P0, Ua0 and u~ are known. In the final position of the hand, P, Ua and u, are known. Then we can write the following nine equations (an excess of 3) which can be used for analysis
p = Ik01D(Ok, dk,~iko)14×4Po
(7a)
u~=[flk=, R (0k' uk°)qd3×3u~°
(Tb)
gt : [k~I=i R(Ok, Uko)]3×3Uto.
(7C)
u
030
u50~ ~ = , For the manipulator shown in its zero position in Fig. l, the complete position analysis is given in Table I.t It should be noted that a description of manipulator geometry in terms of a conveniently chosen zero tin Figs. & Tables, ~ =- u and ~ ~ ft. Also, Table 1 uses the single valued relation 0 = 2. arctan [sin 0/(1 + cos 0)] to avoid problems with nonunique arctan function.
,~ u60 ~0~ ,li ' ao Fig. 2.
u' to
A note on position analysis of manipulators Table 1. Analysis of the manipulator shown in Fig. 1. Zero
Position
Ulo = (0,0,1) t 0 (0,0,0,1) t ,
U2o = (0,1,0) t @(0,0,0,1) t ,
U3o = (0,1,0) t@ (b,O,O,1) t ,
U4o = (1,0,0) t@ (O,a,O,1) t ,
Us@ = (0,1,0) t o (b+c,O,O,1) t ,
U6o = (1,0,0) t o ( O , a , O , 1 ) t ,
Uao = (1,0,0) t @(b+c+h,a,O,1) t ,
Uto = (0,1,0) t @ (b+c+h,a,O,1) t ,
D° = (b+e+h,a,O,1) t . Hand P o s i t i o n P' Ua'
fit
(Ua'Ua
= fit'GGt
= i)
.
Solution
~'-hfia,
H =
D = 2bZH,
e2
r~ =
x~+Y~t '
E = -2brQ
2 a r c t a n ~E-7-g} '
,
r Q = +-
¢'r~t- a2
F = a z + c 2 _ rHz _ ZH2 _ b 2 ,
e3
-e 2 + 2 a r c t a n --
G = +_¢~2 + E 2 _ F 2
-b-b sine 2+rq
f = b cos82 ÷ c cos(@ 2 + 03) fYtt - axH ) f2 ÷ f x H+ayH÷aa
e1 = 2arctan For
n =a
or
t
Vnx = UnxCOSelCOS(e 2 + e 3) ÷UnySinelcos(O 2 ÷ e 3) - UnzSin(e2 + e 3) Vny = -UnxSinel +UnyCOS01 Vnz = UnxCoselsin(02 + e 3) +UnySinSlsin(e 2 ÷ e 3) +UnzCOS(O 2 ÷ e 5) VaQ = - * v ' ~ 04
\ vaQ-vaZ/
@6 = 2 a r c t a n (v
,
0S =
Vtx ) aQ + VayVtz - VazVty
u~0, k = 1, n; u~'0,u~o and P~0.The joint variables are 0~, d~; k = 1, n. We wish to determine the relationship between Ok, dk and 0~, d~. If the final position of the hand is given, then the matrix A such that PI = A P0 and A' such that l f = A'P~ can be computed. The governing equations in the two cases are
l~ID ( ¢ ,
2arctan
4 , ~,,o) = A
(8)
k=l
f l D(O~,d~,fi~)=A"
(9)
If the second zero position corresponds to Ok = ~tk,
(10)
k=l
It is easily seen that A =A'.M,
or A ' = A
.hi, -1,
(11)
From the similarity principle,
O(O;, d;, fi~o)= [Mr-O(O;, d;, 0m)" Mr-'].
k=l
dk = ak, k = 1, n, then P6 = M. "P0, where
I
Mt = I-I D(cq, ak, ilk@), 1 < n.
(12)
After substitution of eqns (10)-(12) into eqn (9) an comparing the result with eqn (8) we obtain 0h = ak + 0~, and dk = ak + d~,, k = 1, n. Therefore, it is quite simple to reconcile the results obtained with
KRISHNA C. GUPTA
method can also be used in the analysis of instrumented bio-linkages which are used for measuring motion. However, it cannot be applied easily to closedloop kinematic chains because then it is not possible to choose an arbitrary zero position.
different zero positions. For example, the solution given in Table 1 (zero position is Fig. 1) is applicable to the analysis with respect to the zero position shown in Fig. 2 if 0, in Table 1 are replaced as: 01 --*0l, 02--,02, 0 ~ 9 0 ~ + 03, 04~ - ( 1 8 0 + 04), 05~05, 06 = _ 9 0 '~ + 06.
Acknowledgement--The financial support of the National
In terms of the notation in Refs. [1, 2] the manipulators in Figs. 1 and 2 are described as follows: at = - 90~', 72 = 0, ~3 = ~ 4 ~--- ~ 5 = 90°, $2 = a, s4 = - c, s6=h,
$1=$3=$5=0,
Science Foundation (ENG 79-.-23193) is acknowledged. REFERENCES
a2 = b, a~ = a3 = a4 = as = O.
1. R. S. Hartenberg and J. Denavit, Kinematic Synthesis o f Linkages. McGraw-Hill, New York (1964). 2. D. L. Pieper and B. Roth, Proc. 2ndlnt. Congr. on Theory o f Machines and Mechanisms, Vol. 2, pp. 159-68 (1969). 3. H. Goldstein, Classical Mechanics. Addison Wesley, New York (1950). 4. B. Roth, A S M E J. Appl. Mech. 34, 599-605 (1969). 5. O. Bottema and B. Roth, Theoretical Kinematics. North Holland, Amsterdam (1979). 6. C. H. Sue and C. W. Radcliffe, Kinematics and Mechanisms Design. Wiley, New York (1978).
Figure 2 is in fact its zero position in this notation (i.e. all 0i = 0, i = 1, 6 or all xi axes are parallel). CONCLUSION
We have presented a method of analysis which utilizes the base coordinate system and a conveniently selected zero position of the manipulator. The zero position of the manipulator can also be represented easily and reliably in engineering drawings. The
UNE N O T E
K.C.
SUR
L'ANALYSE
DE LA P O S I T I O N
DES
MANIPULATEURS
la p o s i t i o n
z~ro d ' u n
Gupta
R~sum~
-
curremment
La n o t i o n avec
t~mes
usuels
d'une
base
de
le p r i n c i p e
de c o o r d o n n ~ e s
est
le seul
de
de
la s i m i l i t u d e
d'accouplement
~ ~tre
employ~
dans
manipulateur
pour
ne s o n t
d~river pas
l'analyse.
est
les
d~finis
introduite
~quations et
et e m p l o y e e
d'analyse.
le s y s t ~ m e
Les
consys-
de c o o r d o n n ~ e s