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Restraint analysis of assembly work carriers
Robotics & Computer-Integrated Manufacturing, Vol. 10, No. 4, pp. 257-265, 1993 Printed in Great Britain.
•
0736-5845/93 $6.00+0.00 © 1993 Pergamon Press Ltd
Paper i
RESTRAINT ANALYSIS OF ASSEMBLY WORK CARRIERS EDWARD C. DE METER Department of Industrial & Management Systems Engineering, College of Engineering, Pennsylvania State University, 207 Hammond Building, University Park, PA 16802, U.S.A.
Work carrier design is critical to the performance and economic feasibility of automatic assembly systems. Unless parts are securely held throughout transport, insertion, and fastening operations, precision assembly is impossible to achieve. This paper presents analytical techniques for determining whether carrier locators provide sufficient kinematic restraint. It presents techniques for determining whether carrier clamps and locators have the potential to resist arbitrary external forces. Finally, it presents a model for determining the minimum clamp actuator intensities necessary to restrain parts throughout transport and assembly.
Ohwovoriole, 19 and Salisbury and Roth 21 have used restraint models to characterize the conditions for which a set of point contacts provide complete restraint. Chou 7 and Chou et al. s'9 applied these models to the restraint analysis of machining fixtures. De Meter 11 developed methods for modeling the wrench systems of frictionless and frictional surface contact regions. Kerr and Roth x2 formulated a linear program to determine the set of gripper actuator intensities necessary to resist an external load with the least reliance on frictional contact. Assada and Kitagawa x developed a linear program for determining whether a set of frictionless point contacts provide total restraint. Other related works include Assada and Andre, 2 Li and Sastry, x5 Montana, ~7 and Sturges. 22 These models were restricted to an individual part. With respect to multiple parts in contact, Blum et al. 5 formulated a linear program to determine the stability of multiple blocks in frictionless and frictional point contact. Bonesehanscher et al.6 modified the work of Blum et al. to include friction cones. This paper develops models for analysing the kinematic restraint imposed by carrier locators. It develops models for determining whether locators and clamps can prevent relative part motion during manual handling. Finally it develops a linear program for determining the set of minimum clamp actuator intensities necessary to prevent relative part motion throughout all transport and assembly operations. An illustrative example is provided.
INTRODUCTION Work carriers are transportable fixtures which carry partially complete assemblies to and from work stations within an automatic assembly system. The carriers are transported between work stations by an automated material handling system. Upon arrival at a station, the carrier is clamped into position, which serves to locate and immobilize all critical parts within the assembly. Additional parts are then added to the assembly by a robot or insertion machine. Upon completion of fastening operations, the carrier is released and transported to the next work station. Parts held by a carrier are located with respect to a reference frame attached to the carrier. To facilitate their loading, parts are placed in contact with carrier locators and clamped. The part-locator and part-part contact regions serve to kinematically restrain the assembled parts. In addition, they provide reactive forces to resist part movement. Clamps are pressed against part surfaces to establish preload forces at the part-clamp, part-locator, and part-part contact regions. The preload forces permit the existence of frictional forces which prevent slippage. Three questions encountered during work carrier design are: (1) do the locators provide the proper kinematic restraint, (2) can the locators and clamps prevent relative part motion during manual handling, and (3) are clamp actuator intensities sufficient to prevent relative part motion during high-speed transport and assembly operations? This paper presents new methods of restraint analysis to answer these questions. Restraint models have been developed to solve many theoretical and practical problems in gripper and fixture design. Bausch and Youcef-Toumi.4 Lakshminarayana, ~4 Markenscoff et al., ~6 Nguyen,1 a
TWISTS, WRENCHES, AND VIRTUAL POWER The instantaneous velocity of a rigid body is minimally described by the rotational velocity, 8, of the body about a point and the translational velocity, v, of the point. Associated with a body in motion is a 257
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Robotics & Computer-Integrated Manufacturing • Volume 10, Number 4, 1993
unique, instantaneous screw axis, qJ. For each point on q', ,9 and v share the same direction as q?. Defining o~ and v as the intensities (signed magnitudes) of,9 and v, respectively, the ratio A (A = v/~) is called the pitch of q~. In combination, '9 and v define a twist, t. Analogously, a system of forces and couples acting on a rigid body can be resolved into a single force, f, and couple, e, directed along a unique screw axis, q'. Defining )L and p to be the intensities of f and e, respectively, A = p/2. In combination, f and e define a wrench, w. In R ~, t and w are represented as t = ['grvor] r
(1)
w = [fTmo~] r
(2)
vo=(p × ' 9 ) + v
(3)
+/
~
with mo=(p×f)+e
(4)
.....)
art t
tor B
where p = an arbitrary point on ~P, vo = translational velocity of the reference origin, and m o = m o m e n t about the reference origin.
0
ca,o°
3
The virtual power, W, of a body undergoing a twist, t, while being acted upon by a wrench, w, is the permuted dot product, *, between the two vectors such that Fig. 1. Work carrier example.
W=f.vo+mo.,9.
(5)
If W = 0, then t and w are reciprocal? If W > 0, then t and w are repelling, z° If W < 0 , then t and w are contrary. L O C A T I O N ANALYSIS Before a work carrier begins its route through an automatic assembly system it is manually loaded with n parts. Each part must be located with respect to the carrier reference frame. To facilitate loading, the n parts are placed in contact with carrier locators and with each other• If properly loaded, contact regions S 1 . . . S,, are created• If contact is maintained, the desired locations of the n parts are established. Chou et al. 8 have stated that parts loaded into a fixture must be deterministicaUy located. This is not true. In m a n y circumstances parts can displace and still maintain a desired geometric relationship with respect to the carrier reference frame. An example is provided in Fig. 1. Part 2 is a cylinder. After being loaded into the work carrier, part 2 can rotate about its X-axis and still maintain its geometric relationship to the carrier as well as to parts 1 and 3. Thus locator analysis should not investigate deterministic location. Instead it should investigate whether the m contact regions provide the proper kinematic restraint. This is accomplished by determining the twists of the n parts which satisfy the kinematic constraints imposed by S 1 . . . S,,. Assume that parts i and 1 contact at region S t. Let t~ and t~ be the twists of parts i and 1 with respect to the carrier reference frame. If the relative
motion between parts i and 1 is constrained kinematically, then all relative twists are dictated by the joint at Sj. Joint constraints can be modeled explicitly or implicitly. The explicit model is presented first• It is known 3 that a joint constrains all twists of a rigid body to lie within the linear hull of t j, . . . tj where xj < 6. t ~ , . . , tj are hnearly independent twtsts whose screw axes are the j o m t axes. As a result, the relative motion between bodies i and 1 must satisfy •
Kj
.
.
.
xj
.
t i - t 1 = It j, . .. tj~j]oj
(6)
where toj~R ~j. Note that if body 1 is a locator, then t 1 = [0].
Let A be a vector which describes the simultaneous twists of the n parts. Then A must satisfy the explicit joint constraints at the m contact regions. These linear constraints are written as H A = [0]
(7)
where A = [ t [ • . . t,r ta T r T, and 1 , . . ~o,,] H = 6m x [6n + ~ s:j] j=l
explicit kinematic constraint coefficient matrix. All A which satisfy (7) exist within the null space of H and are spanned linearly by {A l, . . . , Aim }, where fH = [6n + ~ j ~ 1 ~cj]--rank (H). fn equals the number of degrees of kinematic freedom of the n parts. To
Restraint analysisof assemblywork carriers. E. C. DE METER determine whether the locators provide the proper kinematic restraint, A~, . . . , Ay,, are examined to determine whether they span only permissible twists. The joints are modeled implicitly using reciprocality constraints• It is known 3 that all relative twists permitted by joint j are reciprocal to a system of wrenches linearly spanned by w:J 1 w:J,oJ where p J, = 6 - - x J. w . . w, are linearly independent wrenches " J t " ' " ~'aj , • • associated with pj points of frlcnonless contact m St. To satisfy the implicit joint constraint, the relative motion between bodies i and 1 must satisfy "
"
"
[w~ . . . %3 T(t~ - t~)= [o3
(8)
where w'.J t = r m rO j i ffj l iJ t , for i = 1 L
pj. "
"
"
Let B be a vector which describes the simultaneous twists of the n parts. Then B must satisfy the m sets of reciprocality constraints. These constraints are written as
VB =
[0]
(9)
where B = [t~ • . . t ~r] r , a n d V= [ 6 m - ~ x j] x 6n j=l
implicit kinematic constraint coefficient matrix• All B which satisfy (9) exist within the null space of V, and are spanned linearly by { B I , . . . , Bf~}, where f~ = 6 n - r a n k (V). f~ equals the number of degrees of kinematic freedom of the n parts, and by necessity equals fn. To determine whether the locators provide the proper kinematic restraint, B~ . . . . . By~ are examined to determine whether they span only permissible twists. Finding a basis for the null space of either H or V requires the application of Gaussian elimination and back substitution. However, V is always a more compact matrix than H. As a result (9) is a more efficient formulation than (7)•
wrenches which can act upon bodies i and 1. Let E~ consist of all possible wrenches which can be exerted by Sj on body i. The screw axes of all wrenches within Ej are constrained by the geometry, location, and frictional coefficient of Sj. It has been proven 11 that, for restricted cases of frictionless planar, spherical, and cylindrical contact, E~ is the nonnegative linear hull of a finite set of wrenches which act through the bounding vertices of the contact region. For a planar region, the restriction is that it is convex• For a spherical region, the restriction is that it is bounded by n vertices and n circular arcs and that the included angle between any two surface normals is less than 7~. For a cylindrical region, the restriction is that it is bounded by four vertices, two circular arcs, and two lines, and that the included angle between any two surface normals is less than re. For frictional point contact, E~ is approximated well if modeled by the extreme directions of a polyhedral cone inscribed within the friction cone• Likewise for frictional planar contact, excellent approximations are achieved if E i is modeled with the extreme directions of polyhedral cones inscribed within the friction cones at the bounding vertices. Ohwovoriole 19 showed that a part is completely restrained by a gripper against small arbitrary forces if it can not displace without resulting in negative virtual power at the gripper fingers. This is equivalent to stating that the part can not experience a nonzero twist which will result in zero or positive virtual power relative to the system of wrenches exerted by the gripper fingers. This concept is extendible to the relative motion between rigid bodies• Assume that E i is spanned by w j , . . , wj . Note that this is a different set ~J . . than wj . . . w , J P J . For no negatwe wrtual power to . . occur at S i, the relatwe motion between bodms i and 1 must satisfy t
I
T
[ws, . . . ws, fl ( t , - t x ) >
[0]
(10)
where W'.
.ll
P O T E N T I A L RESTRAINT ANALYSIS Upon loading, a work carrier is manually transferred to the automatic assembly system. During transport, the carrier locators and clamps must restrain the n parts against gravity and small, random inertia forces. Due to their nature the inertia forces are difficult to model. However their magnitudes are much smaller than those encountered during processing. As a result, the designer is primarily concerned as to whether the carrier can resist small, arbitrary wrenches acting on the n parts. This is determined through potential restraint analysis. 14 Assume that a carrier consists of L clamps, and that the total number of clamp-part, locator-part, and part-part contact regions is M. The first step is to model the wrench systems associated with the M contact regions. Contact region S t provides a system of
259
=
I-m
t..
r
Oji
f.T-I T, ji-J
for i = 1 •
• •
zj.
For the n parts to displace without resulting in negative virtual power, the following linear constraints must be satisfied:
OB_~fO]
(11)
where M
G= ~
~jx6n
j=l
virtual power constraint coefficient matrix. All B which satisfy (ll) exist within a closed polyhedral cone and are nonnegatively spanned by M {B1, . . ., BI~ ), w h e r e f a >_ ~ j = 1 z~. B 1. . . . . BI~ are positively independent, and are called the extreme directions of the cone. Note that every B within the cone is a ray. Hence any positive multiple of it will also exist within the cone•
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Robotics & Computer-Integrated Manufacturing • Volume 10, Number 4. 1993
A linear program may be formulated to determine whether a nonzero solution exists to (11). To do so, (11 ) is transformed into standard format by replacing B with the difference between two nonnegative vectors, B ' t - B ~ , and by subtracting a nonnegative slack vector, B~, from the inequality constraints to change them to equality constraints. This results in GB'~ - GB~ - B~ = [0] B', > [03, B~ > E0], B, > [03.
(12)
To search for a solution Assada and Kitagawa ~ advocate sequentially constraining each element within the resultant of B'~-B~ to be a positive or negative constant. An alternative approach is to sequentially maximize and minimize each element within B'I - B E . The resulting formulation is For i = 1 . . . 6n, Maximize B'I,-B~, subject to (12) Minimize B'~ - B ~ , subject to (12).
(13)
I f a nonzero solution to a subproblem of (13) is found, it will be unbounded, since all solutions to (11 ) are rays. Note that both (13) and Assada and Kitagawa's formulation require 12n iterations. A much more efficient method of finding a nonzero solution is to first determine whether any nonzero B satisfy the reciprocality constraints at the M contact regions. This is equivalent to attempting to find a nonzero solution to GB = [0].
(14)
If the rank of G is 6n, then the only solution to (14) is the zero vector. This indicates that it is impossible for the n parts to displace with generating virtual power at at least one spanning wrench. Next it is determined whether a nonzero B can result in positive virtual power at at least one spanning wrench. This is accomplished by creating a linear program which attempts to drive elements of B s positive. In standard form this is Maximize
j=l
~ B~,
(15)
i=1
subject to (12). If the solution to (15) is zero, no positive virtual power can occur. Assume that parts i and 1 belong to a mechanism such that contact regions 1. . . . , q form close a joint, r. Then the size of (14) and (15) can be reduced by replacing the q sets of virtual power constraints with one set of implicit joint constraints. This can be seen by recognizing that t~-t~ must be reciprocal to w~, . . . . w~,. However, it is known 13 that ~q = x r j-> P~ + 1. As a consequence, upon reformulation, the number of constraints in (14) and (15) will be reduced by ~ = 1 zj-p~, while the number of slack variables in (15) will be reduced by ~ = 1 zj. C L A M P A C T U A T O R I N T E N S I T Y ANALYSIS Clamps are mechanisms which rely on force and torque actuation to exert contact forces. Critical to
work carrier design is knowledge of the set of minimum actuator intensities necessary to insure restraint throughout all transport and assembly operations. The set is defined as the one in which the maximum intensity is minimized. Actuator intensities designed in excess of these values are undesirable since they needlessly increase carrier costs. Intensities belov, these values are insufficient for restraint. The following optimization model is derived to determine this minimum set. Assume that the L clamps contact the parts at regions 1. . . . . L. Further assume that clamp./consist of ~j actuators and that a~ is comprised of their intensities. By design aj is nonnegative. Assume that clamp j exerts a wrench, w. If clamp link masses and moments of inertia are small relative to those of the assembled parts, then the effects of gravity and inertia forces on the clamp links can be ignored. In this case. t~ direct linear relationship ~° can be derived between zt~ and w. This relationship is expressed as ~j =/iiW
i I (3 }
a~_>[0] where F j = a ~i × 6 actuator intensity constraint matrix. However, w must exist within E~ and is thus spanned by wj, . . . wj.. As a result (16) may be rewritten for the L clamps as For j = 1 . . . L
(171
~ - F~[wj,... wj~.~ = [0] ~.j> [0], %_> [0] where k j = the intensity vector for wj, . . . wi~. Assume that during transport to a work station, the carrier holds c unfastened parts. In this case, unfastened means that, if removed from the carrier, a part can move relative to those parts that it is in contact with. c changes throughout the assembly process as parts are added and fastened to the initial n. During transport, the c parts are subject to gravitational forces, inertia forces, and contact forces. Assume that the travel time is t* s. Let ~,i(t) and Xi(t) be the instantaneous inertia force and inertia moment acting on part i. Let ai(t) be the acceleration of the center of mass of part i, and 9i(t) and O~(t) be the angular velocity and acceleration of part i. If ~i(t), Xi(t), ai(t), 81(t), and Oi(t) are defined relative to a reference frame at the center of mass of part i, then the following relationships exist: ~i(t) = - m#i(t) X~(t) = - ~ i O ~ ( t ) - a ~ ( t )
(18) × ~.ia~(t)
O<_t
Restraint analysis of assembly work carriers. E. C. DE METER
must equal [ [ - ~ i ( t ) - g O r - x r ( t ) ] T, where gi equals the gravitation force acting on part i. Again assume that part i and part 1 contact at Sj. Let w i be a wrench exerted by S 1 on part i, and let w I be wi defined with respect to the part i reference frame. Define w 1 and w~ in a similar fashion, w I and w] must satisfy the following constraints: w i, - [
w ji , . . .
w~.]Xj
- - [wj, ...
wi = [wj, . . . wj.]kj [w j, • • • wj,j]~j ~j__ [0].
~l(t)+gl Xx(t)
:
= [0]
(20)
Let N(s) be a multi-variate function 23 which describes the external contact wrenches exerted on the c parts during insertion and fastening. In addition, let gs be a vector which describes the gravitational wrenches acting on the c parts. Using the same logic applied to the development of (21), the simultaneous static equilibrium constraints may be written as r~ + gs + N(s) = [03 ~_> [0], s~<_s<_s~
~(t)+g~ xAt)
(22)
w~ = -
w j1, ] X j
The dynamic equilibrium constraints for the c parts maybe written as
D~ +
parts are subject to gravitational forces and contact forces. Since relative motion is not permitted, each part is subject to static equilibrium constraints. Simply put, the sum of the wrenches acting on each part must be the zero vector. Thus at Sj, w i and wl must satisfy
(19)
~.j_> [o].
261
(23)
where
~_> [0], O<_t<_t*
M
F=6cx
where
~ vj j=l
= [ L T . . . k ~ ] r , and
static equilibrium contact wrench coefficient matrix.
u D = 6c x ~ rj dynamic equilibrium contact wrench j=l
coefficient matrix.
Equations (17) and (23) must be satisfied if the c parts are not to experience relative displacement during assembly. Equations (17) and (23) maybe combined to form
Equations (17) and (20) must be satisfied if the c parts are not to experience relative displacement during transport. Let a t = [ a t r , . . , , etr'l r r • Then (17) and (20) maybe combined to form:
Q ~at=P(t)
(21)
~> [o], ~> [o], o < t < t *
X ~ = E(s)
(24)
; > [0], ¢~___[03, s';
E,s,=
£I gs
where
[o] -~l(t)-gl P(t) =
-Xl(t)
combined constraint coefficient matrix. , and
-~(t)-g~
-x~(t) Q = I 6 c + i~1 l j ] × Ij=~ z i + j ~ I lj] combined constraint coefficient matrix. Equation (21) will always be satisfied if P(t) exists within the nonnegative column space of Q for 0 < t < t*. Upon arrival at the work station, the work carrier is held motionless. A new part is subsequently inserted and fastened to the c parts. During assembly, the c
Equation (24) will always be satisfied if E(s) exists within the nonnegative column space of A for s* < s < s~. Our objective is to determine the set of minimum actuator intensities which will satisfy (21) and (24). In general P(t) and E(s) are not linear. As such they make model formulation difficult. However this problem is overcome if P(t) and E(s) are replaced by polygonal and polyhedral approximations. For example, assume that P(t) is replaced by a polygon with r/vertices such that P1 = P ( t l ) • • • P , = P ( t , ) . If P 1 . . . P~ exist within the nonnegative column space of Q, then each point within the polygon is guaranteed to exist since it can be represented as a convex combination of two vertice. A similar argument can be made if E(s) is replaced with a polyhedral surface with vertices, E 1 . . . Er.
262
Robotics & Computer-Integrated Manufacturing • Volume 10, Number 4, 1993
U s i n g these a p p r o x i m a t i o n s , the set of m i n i m u m 6t must guarantee that (21) is satisfied for P ( t ) = P1 - • • P , , and that (24) is satisfied for E(s) = E 1 . . . E~.
This minimum set maybe found using the following linear program: Minimize d
W14 W15
W12
/
?'P:I
(25) L
subject to: d>e~ for i = 1 . . . ~
b
j-I
Q~=Pi
forj=l..q W8
A
SLs-P3 W22230..'%
~"a+j=Ej f o r j = l . . 7 P3//~
a _>[0], ~j>_[0] f o r j = l . . r / + 7. To find the minimum a for the entire assembly process, (25) must include the combined constraints (21) and (24) of all transport and assembly operations. If the effects of gravitational and inertia forces on the clamp links are significant, then (16) and (17) no longer apply. Instead the system of clamp links and c parts are modeled using (21) and (24). For those clamp links connected by an activated joint, Ej is spanned by r/ passive wrenches and ~cj actuated wrenches whose screw axes are those of tj, . . . tj j. As before % will be defined as the intensities of thes~ wrenches. WORK CARRIER EXAMPLE Refer to the work carrier in Fig. 1. It consists of four locators and three linearly actuated clamps. The contact regions and their spanning wrenches are illustrated in Fig. 2. Wrench data with respect to the carrier reference fram are provided in Table 1.0~i, 3(2, and % are the intensities of the linear clamp actuators. Note that the origins of the center of mass reference systems for parts 1, 2, and 3 lie on the Z-axis of the carrier reference frame. Their distances from the carrier reference frame origin are 0.688, 2, and 3.312 in., respectively. If properly located, parts 1 and 3 should have no kinematic freedom while part 2 should be able to rotate about its X-axis. To check this let B~
T
(26)
, and
tl
! W2
t W3
t W 4.
i W5
! W6
[o]
[o]
[o]
[o3 [o]
[o]
[o3 [o]
[o]
[o]
[o]
W
Vr=
TTT [tat2t3]
[0]
E0]
E0]
E0]
! Wll
[0]
E0]
E0]
[o] t W7
! W8
t W9
r WIO
t ! ! --W 7 --W 8 --W 9 --WIo
[o3 w~ w~ w;~ [o] [o3 [o]
[o3
[o3
[o3
[o3
[o] I
Wi2
W'13
W14
WI5
-w~
-wi~ - w l ,
-wi~
Upon checking the basis of the null space of V, it is
/"
/°-"~'~Wl W34 W22 W5 SL'p1W Wt9
SL7 W16// SL4 P I ~ tW',-~
'
21~;~ S
W2~// ,,, ~" /,W 6 W!'J
W4 Sl: PI SI/bP1
S lw:~
l 3 P[
W3o Sc3 P2' ~ W33w29W27 ~'~W W W3132 W28~ SC3_P;' S,L6 p2 H W26 Fig. 2. Contact regions and spanning wrenches.
Table 1. Spanning wrench data (carrier reference frame) fx (lb)
fy (lb)
f= (lb)
mo~ lib-in.)
mo, (lb-in.)
m,= lib-in.)
0.000 0.000
0.000 0.000
1.000 1.000
-- 1.000
2
1.000
1.000 - 1.000
0.000 0.000
3
0.000
0.000
1.000
1.000
0.000
0.000
4 5 6 7 8 9 10
0.000 0.000 - 1.000 0.000 0.000 0.000 0.000
1.000 1.000 0.000 -0.866 0.866 0.866 0.866 0.000 0.866 0.866 0.866 0.866 1.000 1.000 - 1.000 -- 1.000 - 1.000 - 1.000 -- 1.000 - 1.000 - 1.000 - 1.000 1.000 -- 1.000 - 1.000 - 1.000 1.000 1.000 1.000 1.000 0.000
0.000 0.000 0.000 0.500 0.500 -0.500 -0.500 0.000 0.500 0.500 0.500 0.500 0.000 0.000 0.500 -0.500 0.000 0.000 0.500 -0.500 0.000 0.000 0.500 -0.500 0.0t30 0.000 0.500 - 0.500 0.000 0.000 0.000
0.000 0.000 0.000 1.732 - 1.732 1.732 1.732
0.000 0.000 -- 1.000 0.500 0.500 -0.500 - 0.500
1.000 1.000 0.000 0.866 0.866 0.866 I).866
ff
1
I1
1.000
12 13 14 15 t6 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.000 0.577 -0.577 0.000 0.000 0.000 0.000 0.000 0.500 --0.500 0.000 0.000 0.500 0.500 0.000 0.000 0.500 -0.500 0.000 0.000 0.500 -0.500 1.000
0.000
- 2.000
0.000
1.732 - 1.732 1.732 - 1.732 3.000 3.000 0.750 -0.750 0.000 0.000 3.750 2.250 3.000 3.000 2.500 1.500 2.000 2.000 -- 2.500 1.500 -- 2.000 .... 2.000 0.000
- 0.500 - 0.500 0.500 0.500 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.500 - 1.500 -- 1.000 1.000 1.000 - 1.000 - 1.000 1.000 1.000 -- 1.000 4.000
0.866 0.866 0.866 0.866 1.000 - 1.000 0.000 0.000 - 0.750 0.750 0.000 0.000 -0.750 0.750 2.000 2.000 - 2.500 1.500 2.000 2.000 2.500 1.500 0.000
Restraint analysis of assembly work carriers • E. C. DE METER Table 2. Robot link parameters
found to contain only one vector, [0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0] r, which confirms that part 2 can only rotate about its X-axis. To determine if the carrier can restrain the parts during manual handling, let
Link i
1 2 3
w'~ w'~ W~o w~ Gr= v r [o] EO] [o] EO]
[03 [0] [0] Eo] Eo] [o] [o]
[o] [o] [o]
01
)'i - 1 (rad)
a~_ 1 (in.)
bi (in.)
(rad)
0.000 0.000 0.000
0.000 20.000 20.000
0.000 0.000 0.000
02(t ) 0.000
Or(t)
Consequently at(t ) and Or(t) are defined by
[0]
iJ1 sin 02 - 0 2 cos 02 - ( 0 1 + 02) 2
Eo] Eo] [o]
01 cos tJ2+O~ sin 02 + (1~1+02)
at(01 (t), 02(t ) =
t l t l t l t ! W26 W27 W28 W29 W30 W31 W32 W33
[o] [o]
263
0 for i = 1, 2, 3
[o] [o] [o] [o]
[o]
[o] [o]
[o]
[0]
[0]
[0]
[0]
t
i
t
t
0
(27)
Ot(01(t), 0tit))=
01 +02
Upon checking the null space of G it is found to contain only the zero vector. Thus it is impossible for the parts to move without generating virtual power at at least one spanning wrench. Applying (15) results in an objective value of zero. This indicates that it is impossible for parts to move without generating negative virtual power at at least one spanning wrench. Thus the carrier can set up a system of wrenches to resist any small wrench acting on the three parts. The carrier is transported to a work station by the two-link manipulator shown in Fig. 3. Its DenavitHartenberg parameters are defined in Table 2. To determine the set of minimum actuator intensities necessary to restrain the parts during this move, the trajectories of the three parts are first defined. The gripper reference frame and carrier reference frame are coincident. Due to part location and the planar motion of the manipulator, at(t ) and Ot(t ) for the three parts are equal to the linear and angular acceleration of the gripper reference frame, t* equals 1 s, and the trajectories of 01(t) and 02( 0 are defined by the functions 01 (t) = 02(t ) =
6.283t--
{)l(t) = 02(t ) =
6.283 - 12.566t
6.283t 2
for i=1, 2, 3
0
W22 W23 W24 W25
01(t)=O2(t)=3.142t2-2.094t 3
(29)
0 _ < t < l sec. Due to symmetry, the inertial matrix of each part is defined by Itxx ~i "~- 0
0
0
fiyy
0
0
It."
0
for i= 1, 2, 3.
(30)
However due to the zero terms in Or(t), only It," appears in the application of (19). Values of mt , gt, and It,, are presented in Table 3. Values of ~l(t) and Xt(t ) for t = 0, 0.2, 0.5, 0.8, and 1 s are presented in Tables 4 Table 3. Part inertia and gravitational force data m~
1~
g~
Part
(lb)
(lb-~i~.2)
(lb)
~1~~)
gi, (lb)
1 2 3
6.097 5.334 6.097
17.182 17.336 17.182
0.000 0.000 0.000
0.000 0.000 0.000
- 6.097 - 5.334 - 6.097
(28) Table 4. Inertia force data for parts 1 and 3
t
~
~
~,
(sec)
(lb)
(lb)
(Ib)
X~ (Ib-in.)
Xy (Ib-in.)
X~ (Ib-in.)
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.559 0.335 0.000 -0.335 -0.559
~, (Ib)
X~ (Ib-in.)
Xy (Ib-in.)
X, (Ib-in.)
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.564 0.338 0.000 -0.338 -0.564
O < t < l sec. Work Carrier at Final Location = 1 see, 01 =e 2 = 60")
0 0.2 0.5 0.8 1
0.000 5.948 - 1.463 3.596 - 3.788 0.389 -2.423-2.826 -1.717 -4.957
Table 5. Inertia force data for part 2
(see)
'Jd', ///// Xo' Xl
X2
Manipulator
7
x3 t _ Work Carrier at Initial Location (t = O, 01 = e2 = o')
Fig. 3. Work carrier transport.
0 0.2 0.5 0.8 1
(lb)
(lb)
0.000 5.204 - 1.280 3.147 --3.150 0.340 --2.120 -2.472 --1.502-4.337
264
Q=
Robotics & Computer-Integrated Manufacturing • Volume 10, Number 4, 1993 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-1
-I
-1
-I
0
0
0
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
-~
-1
-~ -~
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 w~ w~ w~ ~t/1 w~ w I w I w l w I wll0 [0] [0] [0] [0] [0] [0] [OJ Wll Wll w~O W~l [0] [0] [0] [0] [0] [0] [0]-W~ " ~ "~9 "WI20"W~I'WI22"W~3"~II,'W~I, [0] [0] [0] [0] [0] [0] [0]
0 0 [0] [0] [0] [0]
0
0
0
0
0
0
0
0
0
o
o
o
o
o
o
o
o
o
o
0 -I -1 -1 -1 -I 0 0 0 0 [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] W26,2W~7 ~ $ W29 W~0 W~I ~ 2 W~3 [0]
0
0 0 0
1
0
0
0
1
o
o i
0
0
[01 [Ol [01 [0] [0] [0] [Ol [01 [Ol [oI [01 w~2 w~3 w~4 w~ w~6 w~7 [0] [Ol [01 [01 w~7 w~3 w~ w~5 f0l [01 [01 fo] [Ol [01 [0] [01 w~4 0 Fig. 4. Q-matrix.
and 5. These are referred to as {~, . . . {~, and X~, . . . X~, respectively. The elements of Q are defined in Fig. 4. Data for the spanning wrenches with respect to the reference frames of parts 1, 2, and 3 are provided in Tables 6, 7, and 8 respectively. Equation (25) is formulated by defining = [ k j, . . . ).j~] r, = [ ~ 1 1 ~ 2 ~ 3 ] T,
(31 }
and
[o] ~lj--gl X2j
P j = ~2~-g2
forj=l ...5.
Table 7. Spanning wrench data (part 2 reference frame)
w
f~ {lb)
f~. (lb)
f. (lb)
m,,~ lib-in.)
m,, (lb-ii~.)
7 8 9 10 !1 12 13 14 15 26 27 28 29 30 31 32 33
0.000 0.000 0.000 0.000 - 1.000 0.000 0.577 -0.577 0.000 0.000 0.000 0.500 0.500 0.000 0.000 0.500 0.500
-0.866 0.866 -0.866 0.866 0.000 -0.866 0.855 .....0.866 0.866 1.000 - 1.000 - 1.000 1.000 1.000 1.000 1.000 1.000
-0.500 -0.500 -0.500 --0.500 0.000 0.500 0.500 0.500 0.500 0.500 -0.500 0.000 0.000 0.500 - 0.500 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0,500 0.500 0.000 0.000 -0.500 0.500 0.000 0.000
0.500 0.500 -0.500 --0.500 0.000 0.500 --0.500 0.500 0.500 1.000 1.000 0.000 0.000 - 1.000 1.000 0.000 0.000
m,,~ (lb-m) 0.866 0866 0.866 0.866 0.0(X) 0.866 {}.866 0.866 0.866 2.000 2.0(X) 2.51/1! 1.500 2.000 2.000 2.500 1.500
X2j
~3~ - g3 X3~
Upon solving (25), the set of minimum actuator i n t e n s i t i e s is f o u n d to b e ~1 = ~2 = {~3 = 9 . 7 5 8 lbf.
CONCLUSIONS The models developed in this paper aid the design of assembly work carriers. They are used to evaluate locator functionality, the potential restraint offered by locators and clamps, and the set of minimum actuator intensities necessary for restraint throughout all transport and assembly operations. The models discussed in this paper are also applicable to other workholding devices such as grippers and machining fixtures. Future work will investigate formulations for optimizing work carrier and gripper configurations.
Table 6. S p a n n i n g wrench data (part 1 reference flame)
,6,
f~ (lb)
fy (lb)
f~ (lb)
mo~ (lb-in.)
mo~ (lb-in.)
mo~ (lb-in.)
1 2 3 4 5 6 7 8 9 tO 18 19 20 21
0.000 0.000 0.000 0.000 0.000 - 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.500 --0.500
0.000 0.000 0.000 1.000 1.000 0.000 --0.8.66 0.866 --0.866 0.866 - 1.000 -- 1.000 -- 1.000 - 1.000
1.000 1.000 1.000 0.000 0.000 0.000 -0.500 --0.500 --0.500 --0.500 0.500 -0.500 0.000 0.000
- 1.000 - 1.000 1.000 0.688 0.688 0.000 1.136 -- 1.136 1.136 -- 1.136 0.062 - 1.438 -0.688 -0.688
1.000 - 1.000 0.000 0.000 0.000 --0.312 0.500 0.500 --0.500 --0.500 0.000 0.000 -0.344 0.344
0.000 0.000 0.000 - 1.000 1.000 0.000 --0.866 0.866 0.866 -0.866 0.000 0.000 -0.750 0.750
Table 8. Spanning wrench data (part 3 reference frame)
12 13 14 15 16 17 22 23 24 25
f,
f~.
(lb)
fib)
0.000 0.577 -0.577 0.000 0.000 0.000 0.000 0.000 0.500 --0.500
-0.866 0.866 -0.866 0.866 1.000 1.000 - 1.000 - 1.000 - 1.000 - 1.000
f. (11~) 0.500 0.500 0.500 0.500 0.000 0.000 0.500 -0.500 0.000 0.000
m,,~
m,,
m,,~
lib-in.)
(lb-in.)
(lb-in.~
0.500 -0.500 0.500 0.500 0.000 0.000 0.000 0.000 0.156 0.156
0.866 11.866 1/.866 - 1/.866 1.000 1.000 0.000 0.(X10 0.750 0.750
1.136 1.136 1.136 1.136 0.312 0.312 0.438 1.062 0.312 0.312
REFERENCES 1. Assada, H., Kitagawa, M.: Kinematic analysis and planning for form closure grasps by robotic hands. Robotics Computer-lntegr. Mfg. 5: 293-299, 1989. 2. Assada, H., Andre, B.: Kinematic analysis of workpart fixturing for flexible assembly with automatically recon3. 4.
5.
6.
7.
f i g u r a b l e fixtures. IEEE J. Robotics Automn R A - I : 86-94, 1985. Ball, R. S., A Treatise on the Theory oj Screws. C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e . 1900. B a u s c h , J. J., Y o u c e f - T o u m i , K.: C o m p u t e r p l a n n i n g m e t h o d s f o r a u t o m a t e d f i x t u r e l a y o u t s y n t h e s i s . Proceedings of Manufacturing International 90 Conference 1990, Vol. 1, pp. 225-232. Blum, M., Griflith, A., N e u m a n n , B.: A stability test for c o n f i g u r a t i o n s of blocks. Artificial Intelligence Memo No. 188, M I T Artificial Intelligence L a b o r a t o r y , C a m bridge, M A . , 1970, pp. 1-24. B o n e s c h a n s c h e r , N., v a n d e r D r i f t , H., Buckly, S., T a y l o r , R.: S u b a s s e m b l y Stability. In Proceedings ,1 1988 A A A I Conf., St. Paul, M N . , 1988, pp. 7 8 0 785. C h o u , Y. C.: A m e t h o d o l o g y f o r a u t o m a t i c l a y o u t of
Restraint analysis of assemblywork carriers • E. C. DE METER
8.
9.
10. 11. 12. 13.
14.
fixture elements based on machining force considerations. Proceedings of ASME Winter Annual Meeting, 1990, pp. 181-189. Chou, Y. C., Chandru, V., Barash, M. M.: A mathematical approach to automatic configuration of machining fixtures: analysis and synthesis. ASME J. Engng Ind. 111: 299-306, 1989. Chou, Y. C., Chandru, V., Barash, M. M.: A mathematical approach to automatic design of machining fixtures: analysis and synthesis. Proceedings of ASME Winter Annual Meeting, 1987, pp. 11-27. Craig, J. J.: Introduction to Robotics Mechanics and Control. Addison-Wesley, Reading, MA, 1986. De Meter, E. C.: Restraint analysis of rigid bodies constrained by surface contact. IMSE Working Paper 91-133, pp. 1-33, 1991. Kerr, J., Roth, B.: Analysis ofmultifingered hands. Int. J. Robotics 4(4): 3-17, 1986. Kerr, D. R., Sanger, D. J.: The analysis of kinematic restraint. In Proceedings of Sixth World Congressfor the Theory of Mechanisms and Machines, New Delhi. 1983, pp. 299-303. Lakshminarayana, K.: Mechanics of form closure. ASME Technical Paper, 78-DET-32, pp. 1-8, 1979.
265
15. Li, Z., Sastry, S.: Task-oriented optimal grasping by multi-fingered robot hands. IEEE J. Robotics Automn 4: 32-44, 1988. 16. Markenscoff, X., Ni, L., Papadimitriou, C. H.: The geometry of grasping. Int. J. Robotics Res. 9: 61-74, 1990. 17. Montana, D. J.: The kinematics of contact and grasp. Int. J. Robotics Res. 7(3): 17-32, 1988. 18. Nguyen, V. Constructing force-closure grasps. Int. J. Robotics Res. 7(3): 3-16, 1988. 19. Ohwovoriole, E. N.: Kinematics and friction in grasping by robotic hands. ASME J. Mech. Transm. Automn Design 109: 398-404, 1987. 20. Ohwovoriole, E. N., Roth, B.: An extension of screw theory. ASME J. mech. Design 103: 725-735, 1981. 21. Salisbury, J. K., Roth, B.: Kinematic and force analysis of articulated mechanical hands. ASME J. Mech. Transm. Automn Design 105: 35-41, 1983. 22. Sturges, R. H.: A three-dimensional assembly task quantification with application to machine dexterity. Int. J. Robotics Res. 7(4): 34-78, 1988. 23. Whitney, D. E.: Quasi-static assembly of compliantly supported rigid parts. ASME J. dyn. Systems Meas. Control 104: 65-77, 1982.
•
0736-5845/93 $6.00+0.00 © 1993 Pergamon Press Ltd
Paper i
RESTRAINT ANALYSIS OF ASSEMBLY WORK CARRIERS EDWARD C. DE METER Department of Industrial & Management Systems Engineering, College of Engineering, Pennsylvania State University, 207 Hammond Building, University Park, PA 16802, U.S.A.
Work carrier design is critical to the performance and economic feasibility of automatic assembly systems. Unless parts are securely held throughout transport, insertion, and fastening operations, precision assembly is impossible to achieve. This paper presents analytical techniques for determining whether carrier locators provide sufficient kinematic restraint. It presents techniques for determining whether carrier clamps and locators have the potential to resist arbitrary external forces. Finally, it presents a model for determining the minimum clamp actuator intensities necessary to restrain parts throughout transport and assembly.
Ohwovoriole, 19 and Salisbury and Roth 21 have used restraint models to characterize the conditions for which a set of point contacts provide complete restraint. Chou 7 and Chou et al. s'9 applied these models to the restraint analysis of machining fixtures. De Meter 11 developed methods for modeling the wrench systems of frictionless and frictional surface contact regions. Kerr and Roth x2 formulated a linear program to determine the set of gripper actuator intensities necessary to resist an external load with the least reliance on frictional contact. Assada and Kitagawa x developed a linear program for determining whether a set of frictionless point contacts provide total restraint. Other related works include Assada and Andre, 2 Li and Sastry, x5 Montana, ~7 and Sturges. 22 These models were restricted to an individual part. With respect to multiple parts in contact, Blum et al. 5 formulated a linear program to determine the stability of multiple blocks in frictionless and frictional point contact. Bonesehanscher et al.6 modified the work of Blum et al. to include friction cones. This paper develops models for analysing the kinematic restraint imposed by carrier locators. It develops models for determining whether locators and clamps can prevent relative part motion during manual handling. Finally it develops a linear program for determining the set of minimum clamp actuator intensities necessary to prevent relative part motion throughout all transport and assembly operations. An illustrative example is provided.
INTRODUCTION Work carriers are transportable fixtures which carry partially complete assemblies to and from work stations within an automatic assembly system. The carriers are transported between work stations by an automated material handling system. Upon arrival at a station, the carrier is clamped into position, which serves to locate and immobilize all critical parts within the assembly. Additional parts are then added to the assembly by a robot or insertion machine. Upon completion of fastening operations, the carrier is released and transported to the next work station. Parts held by a carrier are located with respect to a reference frame attached to the carrier. To facilitate their loading, parts are placed in contact with carrier locators and clamped. The part-locator and part-part contact regions serve to kinematically restrain the assembled parts. In addition, they provide reactive forces to resist part movement. Clamps are pressed against part surfaces to establish preload forces at the part-clamp, part-locator, and part-part contact regions. The preload forces permit the existence of frictional forces which prevent slippage. Three questions encountered during work carrier design are: (1) do the locators provide the proper kinematic restraint, (2) can the locators and clamps prevent relative part motion during manual handling, and (3) are clamp actuator intensities sufficient to prevent relative part motion during high-speed transport and assembly operations? This paper presents new methods of restraint analysis to answer these questions. Restraint models have been developed to solve many theoretical and practical problems in gripper and fixture design. Bausch and Youcef-Toumi.4 Lakshminarayana, ~4 Markenscoff et al., ~6 Nguyen,1 a
TWISTS, WRENCHES, AND VIRTUAL POWER The instantaneous velocity of a rigid body is minimally described by the rotational velocity, 8, of the body about a point and the translational velocity, v, of the point. Associated with a body in motion is a 257
258
Robotics & Computer-Integrated Manufacturing • Volume 10, Number 4, 1993
unique, instantaneous screw axis, qJ. For each point on q', ,9 and v share the same direction as q?. Defining o~ and v as the intensities (signed magnitudes) of,9 and v, respectively, the ratio A (A = v/~) is called the pitch of q~. In combination, '9 and v define a twist, t. Analogously, a system of forces and couples acting on a rigid body can be resolved into a single force, f, and couple, e, directed along a unique screw axis, q'. Defining )L and p to be the intensities of f and e, respectively, A = p/2. In combination, f and e define a wrench, w. In R ~, t and w are represented as t = ['grvor] r
(1)
w = [fTmo~] r
(2)
vo=(p × ' 9 ) + v
(3)
+/
~
with mo=(p×f)+e
(4)
.....)
art t
tor B
where p = an arbitrary point on ~P, vo = translational velocity of the reference origin, and m o = m o m e n t about the reference origin.
0
ca,o°
3
The virtual power, W, of a body undergoing a twist, t, while being acted upon by a wrench, w, is the permuted dot product, *, between the two vectors such that Fig. 1. Work carrier example.
W=f.vo+mo.,9.
(5)
If W = 0, then t and w are reciprocal? If W > 0, then t and w are repelling, z° If W < 0 , then t and w are contrary. L O C A T I O N ANALYSIS Before a work carrier begins its route through an automatic assembly system it is manually loaded with n parts. Each part must be located with respect to the carrier reference frame. To facilitate loading, the n parts are placed in contact with carrier locators and with each other• If properly loaded, contact regions S 1 . . . S,, are created• If contact is maintained, the desired locations of the n parts are established. Chou et al. 8 have stated that parts loaded into a fixture must be deterministicaUy located. This is not true. In m a n y circumstances parts can displace and still maintain a desired geometric relationship with respect to the carrier reference frame. An example is provided in Fig. 1. Part 2 is a cylinder. After being loaded into the work carrier, part 2 can rotate about its X-axis and still maintain its geometric relationship to the carrier as well as to parts 1 and 3. Thus locator analysis should not investigate deterministic location. Instead it should investigate whether the m contact regions provide the proper kinematic restraint. This is accomplished by determining the twists of the n parts which satisfy the kinematic constraints imposed by S 1 . . . S,,. Assume that parts i and 1 contact at region S t. Let t~ and t~ be the twists of parts i and 1 with respect to the carrier reference frame. If the relative
motion between parts i and 1 is constrained kinematically, then all relative twists are dictated by the joint at Sj. Joint constraints can be modeled explicitly or implicitly. The explicit model is presented first• It is known 3 that a joint constrains all twists of a rigid body to lie within the linear hull of t j, . . . tj where xj < 6. t ~ , . . , tj are hnearly independent twtsts whose screw axes are the j o m t axes. As a result, the relative motion between bodies i and 1 must satisfy •
Kj
.
.
.
xj
.
t i - t 1 = It j, . .. tj~j]oj
(6)
where toj~R ~j. Note that if body 1 is a locator, then t 1 = [0].
Let A be a vector which describes the simultaneous twists of the n parts. Then A must satisfy the explicit joint constraints at the m contact regions. These linear constraints are written as H A = [0]
(7)
where A = [ t [ • . . t,r ta T r T, and 1 , . . ~o,,] H = 6m x [6n + ~ s:j] j=l
explicit kinematic constraint coefficient matrix. All A which satisfy (7) exist within the null space of H and are spanned linearly by {A l, . . . , Aim }, where fH = [6n + ~ j ~ 1 ~cj]--rank (H). fn equals the number of degrees of kinematic freedom of the n parts. To
Restraint analysisof assemblywork carriers. E. C. DE METER determine whether the locators provide the proper kinematic restraint, A~, . . . , Ay,, are examined to determine whether they span only permissible twists. The joints are modeled implicitly using reciprocality constraints• It is known 3 that all relative twists permitted by joint j are reciprocal to a system of wrenches linearly spanned by w:J 1 w:J,oJ where p J, = 6 - - x J. w . . w, are linearly independent wrenches " J t " ' " ~'aj , • • associated with pj points of frlcnonless contact m St. To satisfy the implicit joint constraint, the relative motion between bodies i and 1 must satisfy "
"
"
[w~ . . . %3 T(t~ - t~)= [o3
(8)
where w'.J t = r m rO j i ffj l iJ t , for i = 1 L
pj. "
"
"
Let B be a vector which describes the simultaneous twists of the n parts. Then B must satisfy the m sets of reciprocality constraints. These constraints are written as
VB =
[0]
(9)
where B = [t~ • . . t ~r] r , a n d V= [ 6 m - ~ x j] x 6n j=l
implicit kinematic constraint coefficient matrix• All B which satisfy (9) exist within the null space of V, and are spanned linearly by { B I , . . . , Bf~}, where f~ = 6 n - r a n k (V). f~ equals the number of degrees of kinematic freedom of the n parts, and by necessity equals fn. To determine whether the locators provide the proper kinematic restraint, B~ . . . . . By~ are examined to determine whether they span only permissible twists. Finding a basis for the null space of either H or V requires the application of Gaussian elimination and back substitution. However, V is always a more compact matrix than H. As a result (9) is a more efficient formulation than (7)•
wrenches which can act upon bodies i and 1. Let E~ consist of all possible wrenches which can be exerted by Sj on body i. The screw axes of all wrenches within Ej are constrained by the geometry, location, and frictional coefficient of Sj. It has been proven 11 that, for restricted cases of frictionless planar, spherical, and cylindrical contact, E~ is the nonnegative linear hull of a finite set of wrenches which act through the bounding vertices of the contact region. For a planar region, the restriction is that it is convex• For a spherical region, the restriction is that it is bounded by n vertices and n circular arcs and that the included angle between any two surface normals is less than 7~. For a cylindrical region, the restriction is that it is bounded by four vertices, two circular arcs, and two lines, and that the included angle between any two surface normals is less than re. For frictional point contact, E~ is approximated well if modeled by the extreme directions of a polyhedral cone inscribed within the friction cone• Likewise for frictional planar contact, excellent approximations are achieved if E i is modeled with the extreme directions of polyhedral cones inscribed within the friction cones at the bounding vertices. Ohwovoriole 19 showed that a part is completely restrained by a gripper against small arbitrary forces if it can not displace without resulting in negative virtual power at the gripper fingers. This is equivalent to stating that the part can not experience a nonzero twist which will result in zero or positive virtual power relative to the system of wrenches exerted by the gripper fingers. This concept is extendible to the relative motion between rigid bodies• Assume that E i is spanned by w j , . . , wj . Note that this is a different set ~J . . than wj . . . w , J P J . For no negatwe wrtual power to . . occur at S i, the relatwe motion between bodms i and 1 must satisfy t
I
T
[ws, . . . ws, fl ( t , - t x ) >
[0]
(10)
where W'.
.ll
P O T E N T I A L RESTRAINT ANALYSIS Upon loading, a work carrier is manually transferred to the automatic assembly system. During transport, the carrier locators and clamps must restrain the n parts against gravity and small, random inertia forces. Due to their nature the inertia forces are difficult to model. However their magnitudes are much smaller than those encountered during processing. As a result, the designer is primarily concerned as to whether the carrier can resist small, arbitrary wrenches acting on the n parts. This is determined through potential restraint analysis. 14 Assume that a carrier consists of L clamps, and that the total number of clamp-part, locator-part, and part-part contact regions is M. The first step is to model the wrench systems associated with the M contact regions. Contact region S t provides a system of
259
=
I-m
t..
r
Oji
f.T-I T, ji-J
for i = 1 •
• •
zj.
For the n parts to displace without resulting in negative virtual power, the following linear constraints must be satisfied:
OB_~fO]
(11)
where M
G= ~
~jx6n
j=l
virtual power constraint coefficient matrix. All B which satisfy (ll) exist within a closed polyhedral cone and are nonnegatively spanned by M {B1, . . ., BI~ ), w h e r e f a >_ ~ j = 1 z~. B 1. . . . . BI~ are positively independent, and are called the extreme directions of the cone. Note that every B within the cone is a ray. Hence any positive multiple of it will also exist within the cone•
260
Robotics & Computer-Integrated Manufacturing • Volume 10, Number 4. 1993
A linear program may be formulated to determine whether a nonzero solution exists to (11). To do so, (11 ) is transformed into standard format by replacing B with the difference between two nonnegative vectors, B ' t - B ~ , and by subtracting a nonnegative slack vector, B~, from the inequality constraints to change them to equality constraints. This results in GB'~ - GB~ - B~ = [0] B', > [03, B~ > E0], B, > [03.
(12)
To search for a solution Assada and Kitagawa ~ advocate sequentially constraining each element within the resultant of B'~-B~ to be a positive or negative constant. An alternative approach is to sequentially maximize and minimize each element within B'I - B E . The resulting formulation is For i = 1 . . . 6n, Maximize B'I,-B~, subject to (12) Minimize B'~ - B ~ , subject to (12).
(13)
I f a nonzero solution to a subproblem of (13) is found, it will be unbounded, since all solutions to (11 ) are rays. Note that both (13) and Assada and Kitagawa's formulation require 12n iterations. A much more efficient method of finding a nonzero solution is to first determine whether any nonzero B satisfy the reciprocality constraints at the M contact regions. This is equivalent to attempting to find a nonzero solution to GB = [0].
(14)
If the rank of G is 6n, then the only solution to (14) is the zero vector. This indicates that it is impossible for the n parts to displace with generating virtual power at at least one spanning wrench. Next it is determined whether a nonzero B can result in positive virtual power at at least one spanning wrench. This is accomplished by creating a linear program which attempts to drive elements of B s positive. In standard form this is Maximize
j=l
~ B~,
(15)
i=1
subject to (12). If the solution to (15) is zero, no positive virtual power can occur. Assume that parts i and 1 belong to a mechanism such that contact regions 1. . . . , q form close a joint, r. Then the size of (14) and (15) can be reduced by replacing the q sets of virtual power constraints with one set of implicit joint constraints. This can be seen by recognizing that t~-t~ must be reciprocal to w~, . . . . w~,. However, it is known 13 that ~q = x r j-> P~ + 1. As a consequence, upon reformulation, the number of constraints in (14) and (15) will be reduced by ~ = 1 zj-p~, while the number of slack variables in (15) will be reduced by ~ = 1 zj. C L A M P A C T U A T O R I N T E N S I T Y ANALYSIS Clamps are mechanisms which rely on force and torque actuation to exert contact forces. Critical to
work carrier design is knowledge of the set of minimum actuator intensities necessary to insure restraint throughout all transport and assembly operations. The set is defined as the one in which the maximum intensity is minimized. Actuator intensities designed in excess of these values are undesirable since they needlessly increase carrier costs. Intensities belov, these values are insufficient for restraint. The following optimization model is derived to determine this minimum set. Assume that the L clamps contact the parts at regions 1. . . . . L. Further assume that clamp./consist of ~j actuators and that a~ is comprised of their intensities. By design aj is nonnegative. Assume that clamp j exerts a wrench, w. If clamp link masses and moments of inertia are small relative to those of the assembled parts, then the effects of gravity and inertia forces on the clamp links can be ignored. In this case. t~ direct linear relationship ~° can be derived between zt~ and w. This relationship is expressed as ~j =/iiW
i I (3 }
a~_>[0] where F j = a ~i × 6 actuator intensity constraint matrix. However, w must exist within E~ and is thus spanned by wj, . . . wj.. As a result (16) may be rewritten for the L clamps as For j = 1 . . . L
(171
~ - F~[wj,... wj~.~ = [0] ~.j> [0], %_> [0] where k j = the intensity vector for wj, . . . wi~. Assume that during transport to a work station, the carrier holds c unfastened parts. In this case, unfastened means that, if removed from the carrier, a part can move relative to those parts that it is in contact with. c changes throughout the assembly process as parts are added and fastened to the initial n. During transport, the c parts are subject to gravitational forces, inertia forces, and contact forces. Assume that the travel time is t* s. Let ~,i(t) and Xi(t) be the instantaneous inertia force and inertia moment acting on part i. Let ai(t) be the acceleration of the center of mass of part i, and 9i(t) and O~(t) be the angular velocity and acceleration of part i. If ~i(t), Xi(t), ai(t), 81(t), and Oi(t) are defined relative to a reference frame at the center of mass of part i, then the following relationships exist: ~i(t) = - m#i(t) X~(t) = - ~ i O ~ ( t ) - a ~ ( t )
(18) × ~.ia~(t)
O<_t
Restraint analysis of assembly work carriers. E. C. DE METER
must equal [ [ - ~ i ( t ) - g O r - x r ( t ) ] T, where gi equals the gravitation force acting on part i. Again assume that part i and part 1 contact at Sj. Let w i be a wrench exerted by S 1 on part i, and let w I be wi defined with respect to the part i reference frame. Define w 1 and w~ in a similar fashion, w I and w] must satisfy the following constraints: w i, - [
w ji , . . .
w~.]Xj
- - [wj, ...
wi = [wj, . . . wj.]kj [w j, • • • wj,j]~j ~j__ [0].
~l(t)+gl Xx(t)
:
= [0]
(20)
Let N(s) be a multi-variate function 23 which describes the external contact wrenches exerted on the c parts during insertion and fastening. In addition, let gs be a vector which describes the gravitational wrenches acting on the c parts. Using the same logic applied to the development of (21), the simultaneous static equilibrium constraints may be written as r~ + gs + N(s) = [03 ~_> [0], s~<_s<_s~
~(t)+g~ xAt)
(22)
w~ = -
w j1, ] X j
The dynamic equilibrium constraints for the c parts maybe written as
D~ +
parts are subject to gravitational forces and contact forces. Since relative motion is not permitted, each part is subject to static equilibrium constraints. Simply put, the sum of the wrenches acting on each part must be the zero vector. Thus at Sj, w i and wl must satisfy
(19)
~.j_> [o].
261
(23)
where
~_> [0], O<_t<_t*
M
F=6cx
where
~ vj j=l
= [ L T . . . k ~ ] r , and
static equilibrium contact wrench coefficient matrix.
u D = 6c x ~ rj dynamic equilibrium contact wrench j=l
coefficient matrix.
Equations (17) and (23) must be satisfied if the c parts are not to experience relative displacement during assembly. Equations (17) and (23) maybe combined to form
Equations (17) and (20) must be satisfied if the c parts are not to experience relative displacement during transport. Let a t = [ a t r , . . , , etr'l r r • Then (17) and (20) maybe combined to form:
Q ~at=P(t)
(21)
~> [o], ~> [o], o < t < t *
X ~ = E(s)
(24)
; > [0], ¢~___[03, s';
E,s,=
£I gs
where
[o] -~l(t)-gl P(t) =
-Xl(t)
combined constraint coefficient matrix. , and
-~(t)-g~
-x~(t) Q = I 6 c + i~1 l j ] × Ij=~ z i + j ~ I lj] combined constraint coefficient matrix. Equation (21) will always be satisfied if P(t) exists within the nonnegative column space of Q for 0 < t < t*. Upon arrival at the work station, the work carrier is held motionless. A new part is subsequently inserted and fastened to the c parts. During assembly, the c
Equation (24) will always be satisfied if E(s) exists within the nonnegative column space of A for s* < s < s~. Our objective is to determine the set of minimum actuator intensities which will satisfy (21) and (24). In general P(t) and E(s) are not linear. As such they make model formulation difficult. However this problem is overcome if P(t) and E(s) are replaced by polygonal and polyhedral approximations. For example, assume that P(t) is replaced by a polygon with r/vertices such that P1 = P ( t l ) • • • P , = P ( t , ) . If P 1 . . . P~ exist within the nonnegative column space of Q, then each point within the polygon is guaranteed to exist since it can be represented as a convex combination of two vertice. A similar argument can be made if E(s) is replaced with a polyhedral surface with vertices, E 1 . . . Er.
262
Robotics & Computer-Integrated Manufacturing • Volume 10, Number 4, 1993
U s i n g these a p p r o x i m a t i o n s , the set of m i n i m u m 6t must guarantee that (21) is satisfied for P ( t ) = P1 - • • P , , and that (24) is satisfied for E(s) = E 1 . . . E~.
This minimum set maybe found using the following linear program: Minimize d
W14 W15
W12
/
?'P:I
(25) L
subject to: d>e~ for i = 1 . . . ~
b
j-I
Q~=Pi
forj=l..q W8
A
SLs-P3 W22230..'%
~"a+j=Ej f o r j = l . . 7 P3//~
a _>[0], ~j>_[0] f o r j = l . . r / + 7. To find the minimum a for the entire assembly process, (25) must include the combined constraints (21) and (24) of all transport and assembly operations. If the effects of gravitational and inertia forces on the clamp links are significant, then (16) and (17) no longer apply. Instead the system of clamp links and c parts are modeled using (21) and (24). For those clamp links connected by an activated joint, Ej is spanned by r/ passive wrenches and ~cj actuated wrenches whose screw axes are those of tj, . . . tj j. As before % will be defined as the intensities of thes~ wrenches. WORK CARRIER EXAMPLE Refer to the work carrier in Fig. 1. It consists of four locators and three linearly actuated clamps. The contact regions and their spanning wrenches are illustrated in Fig. 2. Wrench data with respect to the carrier reference fram are provided in Table 1.0~i, 3(2, and % are the intensities of the linear clamp actuators. Note that the origins of the center of mass reference systems for parts 1, 2, and 3 lie on the Z-axis of the carrier reference frame. Their distances from the carrier reference frame origin are 0.688, 2, and 3.312 in., respectively. If properly located, parts 1 and 3 should have no kinematic freedom while part 2 should be able to rotate about its X-axis. To check this let B~
T
(26)
, and
tl
! W2
t W3
t W 4.
i W5
! W6
[o]
[o]
[o]
[o3 [o]
[o]
[o3 [o]
[o]
[o]
[o]
W
Vr=
TTT [tat2t3]
[0]
E0]
E0]
E0]
! Wll
[0]
E0]
E0]
[o] t W7
! W8
t W9
r WIO
t ! ! --W 7 --W 8 --W 9 --WIo
[o3 w~ w~ w;~ [o] [o3 [o]
[o3
[o3
[o3
[o3
[o] I
Wi2
W'13
W14
WI5
-w~
-wi~ - w l ,
-wi~
Upon checking the basis of the null space of V, it is
/"
/°-"~'~Wl W34 W22 W5 SL'p1W Wt9
SL7 W16// SL4 P I ~ tW',-~
'
21~;~ S
W2~// ,,, ~" /,W 6 W!'J
W4 Sl: PI SI/bP1
S lw:~
l 3 P[
W3o Sc3 P2' ~ W33w29W27 ~'~W W W3132 W28~ SC3_P;' S,L6 p2 H W26 Fig. 2. Contact regions and spanning wrenches.
Table 1. Spanning wrench data (carrier reference frame) fx (lb)
fy (lb)
f= (lb)
mo~ lib-in.)
mo, (lb-in.)
m,= lib-in.)
0.000 0.000
0.000 0.000
1.000 1.000
-- 1.000
2
1.000
1.000 - 1.000
0.000 0.000
3
0.000
0.000
1.000
1.000
0.000
0.000
4 5 6 7 8 9 10
0.000 0.000 - 1.000 0.000 0.000 0.000 0.000
1.000 1.000 0.000 -0.866 0.866 0.866 0.866 0.000 0.866 0.866 0.866 0.866 1.000 1.000 - 1.000 -- 1.000 - 1.000 - 1.000 -- 1.000 - 1.000 - 1.000 - 1.000 1.000 -- 1.000 - 1.000 - 1.000 1.000 1.000 1.000 1.000 0.000
0.000 0.000 0.000 0.500 0.500 -0.500 -0.500 0.000 0.500 0.500 0.500 0.500 0.000 0.000 0.500 -0.500 0.000 0.000 0.500 -0.500 0.000 0.000 0.500 -0.500 0.0t30 0.000 0.500 - 0.500 0.000 0.000 0.000
0.000 0.000 0.000 1.732 - 1.732 1.732 1.732
0.000 0.000 -- 1.000 0.500 0.500 -0.500 - 0.500
1.000 1.000 0.000 0.866 0.866 0.866 I).866
ff
1
I1
1.000
12 13 14 15 t6 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.000 0.577 -0.577 0.000 0.000 0.000 0.000 0.000 0.500 --0.500 0.000 0.000 0.500 0.500 0.000 0.000 0.500 -0.500 0.000 0.000 0.500 -0.500 1.000
0.000
- 2.000
0.000
1.732 - 1.732 1.732 - 1.732 3.000 3.000 0.750 -0.750 0.000 0.000 3.750 2.250 3.000 3.000 2.500 1.500 2.000 2.000 -- 2.500 1.500 -- 2.000 .... 2.000 0.000
- 0.500 - 0.500 0.500 0.500 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.500 - 1.500 -- 1.000 1.000 1.000 - 1.000 - 1.000 1.000 1.000 -- 1.000 4.000
0.866 0.866 0.866 0.866 1.000 - 1.000 0.000 0.000 - 0.750 0.750 0.000 0.000 -0.750 0.750 2.000 2.000 - 2.500 1.500 2.000 2.000 2.500 1.500 0.000
Restraint analysis of assembly work carriers • E. C. DE METER Table 2. Robot link parameters
found to contain only one vector, [0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0] r, which confirms that part 2 can only rotate about its X-axis. To determine if the carrier can restrain the parts during manual handling, let
Link i
1 2 3
w'~ w'~ W~o w~ Gr= v r [o] EO] [o] EO]
[03 [0] [0] Eo] Eo] [o] [o]
[o] [o] [o]
01
)'i - 1 (rad)
a~_ 1 (in.)
bi (in.)
(rad)
0.000 0.000 0.000
0.000 20.000 20.000
0.000 0.000 0.000
02(t ) 0.000
Or(t)
Consequently at(t ) and Or(t) are defined by
[0]
iJ1 sin 02 - 0 2 cos 02 - ( 0 1 + 02) 2
Eo] Eo] [o]
01 cos tJ2+O~ sin 02 + (1~1+02)
at(01 (t), 02(t ) =
t l t l t l t ! W26 W27 W28 W29 W30 W31 W32 W33
[o] [o]
263
0 for i = 1, 2, 3
[o] [o] [o] [o]
[o]
[o] [o]
[o]
[0]
[0]
[0]
[0]
t
i
t
t
0
(27)
Ot(01(t), 0tit))=
01 +02
Upon checking the null space of G it is found to contain only the zero vector. Thus it is impossible for the parts to move without generating virtual power at at least one spanning wrench. Applying (15) results in an objective value of zero. This indicates that it is impossible for parts to move without generating negative virtual power at at least one spanning wrench. Thus the carrier can set up a system of wrenches to resist any small wrench acting on the three parts. The carrier is transported to a work station by the two-link manipulator shown in Fig. 3. Its DenavitHartenberg parameters are defined in Table 2. To determine the set of minimum actuator intensities necessary to restrain the parts during this move, the trajectories of the three parts are first defined. The gripper reference frame and carrier reference frame are coincident. Due to part location and the planar motion of the manipulator, at(t ) and Ot(t ) for the three parts are equal to the linear and angular acceleration of the gripper reference frame, t* equals 1 s, and the trajectories of 01(t) and 02( 0 are defined by the functions 01 (t) = 02(t ) =
6.283t--
{)l(t) = 02(t ) =
6.283 - 12.566t
6.283t 2
for i=1, 2, 3
0
W22 W23 W24 W25
01(t)=O2(t)=3.142t2-2.094t 3
(29)
0 _ < t < l sec. Due to symmetry, the inertial matrix of each part is defined by Itxx ~i "~- 0
0
0
fiyy
0
0
It."
0
for i= 1, 2, 3.
(30)
However due to the zero terms in Or(t), only It," appears in the application of (19). Values of mt , gt, and It,, are presented in Table 3. Values of ~l(t) and Xt(t ) for t = 0, 0.2, 0.5, 0.8, and 1 s are presented in Tables 4 Table 3. Part inertia and gravitational force data m~
1~
g~
Part
(lb)
(lb-~i~.2)
(lb)
~1~~)
gi, (lb)
1 2 3
6.097 5.334 6.097
17.182 17.336 17.182
0.000 0.000 0.000
0.000 0.000 0.000
- 6.097 - 5.334 - 6.097
(28) Table 4. Inertia force data for parts 1 and 3
t
~
~
~,
(sec)
(lb)
(lb)
(Ib)
X~ (Ib-in.)
Xy (Ib-in.)
X~ (Ib-in.)
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.559 0.335 0.000 -0.335 -0.559
~, (Ib)
X~ (Ib-in.)
Xy (Ib-in.)
X, (Ib-in.)
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.564 0.338 0.000 -0.338 -0.564
O < t < l sec. Work Carrier at Final Location = 1 see, 01 =e 2 = 60")
0 0.2 0.5 0.8 1
0.000 5.948 - 1.463 3.596 - 3.788 0.389 -2.423-2.826 -1.717 -4.957
Table 5. Inertia force data for part 2
(see)
'Jd', ///// Xo' Xl
X2
Manipulator
7
x3 t _ Work Carrier at Initial Location (t = O, 01 = e2 = o')
Fig. 3. Work carrier transport.
0 0.2 0.5 0.8 1
(lb)
(lb)
0.000 5.204 - 1.280 3.147 --3.150 0.340 --2.120 -2.472 --1.502-4.337
264
Q=
Robotics & Computer-Integrated Manufacturing • Volume 10, Number 4, 1993 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-1
-I
-1
-I
0
0
0
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
-~
-1
-~ -~
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 w~ w~ w~ ~t/1 w~ w I w I w l w I wll0 [0] [0] [0] [0] [0] [0] [OJ Wll Wll w~O W~l [0] [0] [0] [0] [0] [0] [0]-W~ " ~ "~9 "WI20"W~I'WI22"W~3"~II,'W~I, [0] [0] [0] [0] [0] [0] [0]
0 0 [0] [0] [0] [0]
0
0
0
0
0
0
0
0
0
o
o
o
o
o
o
o
o
o
o
0 -I -1 -1 -1 -I 0 0 0 0 [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] W26,2W~7 ~ $ W29 W~0 W~I ~ 2 W~3 [0]
0
0 0 0
1
0
0
0
1
o
o i
0
0
[01 [Ol [01 [0] [0] [0] [Ol [01 [Ol [oI [01 w~2 w~3 w~4 w~ w~6 w~7 [0] [Ol [01 [01 w~7 w~3 w~ w~5 f0l [01 [01 fo] [Ol [01 [0] [01 w~4 0 Fig. 4. Q-matrix.
and 5. These are referred to as {~, . . . {~, and X~, . . . X~, respectively. The elements of Q are defined in Fig. 4. Data for the spanning wrenches with respect to the reference frames of parts 1, 2, and 3 are provided in Tables 6, 7, and 8 respectively. Equation (25) is formulated by defining = [ k j, . . . ).j~] r, = [ ~ 1 1 ~ 2 ~ 3 ] T,
(31 }
and
[o] ~lj--gl X2j
P j = ~2~-g2
forj=l ...5.
Table 7. Spanning wrench data (part 2 reference frame)
w
f~ {lb)
f~. (lb)
f. (lb)
m,,~ lib-in.)
m,, (lb-ii~.)
7 8 9 10 !1 12 13 14 15 26 27 28 29 30 31 32 33
0.000 0.000 0.000 0.000 - 1.000 0.000 0.577 -0.577 0.000 0.000 0.000 0.500 0.500 0.000 0.000 0.500 0.500
-0.866 0.866 -0.866 0.866 0.000 -0.866 0.855 .....0.866 0.866 1.000 - 1.000 - 1.000 1.000 1.000 1.000 1.000 1.000
-0.500 -0.500 -0.500 --0.500 0.000 0.500 0.500 0.500 0.500 0.500 -0.500 0.000 0.000 0.500 - 0.500 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0,500 0.500 0.000 0.000 -0.500 0.500 0.000 0.000
0.500 0.500 -0.500 --0.500 0.000 0.500 --0.500 0.500 0.500 1.000 1.000 0.000 0.000 - 1.000 1.000 0.000 0.000
m,,~ (lb-m) 0.866 0866 0.866 0.866 0.0(X) 0.866 {}.866 0.866 0.866 2.000 2.0(X) 2.51/1! 1.500 2.000 2.000 2.500 1.500
X2j
~3~ - g3 X3~
Upon solving (25), the set of minimum actuator i n t e n s i t i e s is f o u n d to b e ~1 = ~2 = {~3 = 9 . 7 5 8 lbf.
CONCLUSIONS The models developed in this paper aid the design of assembly work carriers. They are used to evaluate locator functionality, the potential restraint offered by locators and clamps, and the set of minimum actuator intensities necessary for restraint throughout all transport and assembly operations. The models discussed in this paper are also applicable to other workholding devices such as grippers and machining fixtures. Future work will investigate formulations for optimizing work carrier and gripper configurations.
Table 6. S p a n n i n g wrench data (part 1 reference flame)
,6,
f~ (lb)
fy (lb)
f~ (lb)
mo~ (lb-in.)
mo~ (lb-in.)
mo~ (lb-in.)
1 2 3 4 5 6 7 8 9 tO 18 19 20 21
0.000 0.000 0.000 0.000 0.000 - 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.500 --0.500
0.000 0.000 0.000 1.000 1.000 0.000 --0.8.66 0.866 --0.866 0.866 - 1.000 -- 1.000 -- 1.000 - 1.000
1.000 1.000 1.000 0.000 0.000 0.000 -0.500 --0.500 --0.500 --0.500 0.500 -0.500 0.000 0.000
- 1.000 - 1.000 1.000 0.688 0.688 0.000 1.136 -- 1.136 1.136 -- 1.136 0.062 - 1.438 -0.688 -0.688
1.000 - 1.000 0.000 0.000 0.000 --0.312 0.500 0.500 --0.500 --0.500 0.000 0.000 -0.344 0.344
0.000 0.000 0.000 - 1.000 1.000 0.000 --0.866 0.866 0.866 -0.866 0.000 0.000 -0.750 0.750
Table 8. Spanning wrench data (part 3 reference frame)
12 13 14 15 16 17 22 23 24 25
f,
f~.
(lb)
fib)
0.000 0.577 -0.577 0.000 0.000 0.000 0.000 0.000 0.500 --0.500
-0.866 0.866 -0.866 0.866 1.000 1.000 - 1.000 - 1.000 - 1.000 - 1.000
f. (11~) 0.500 0.500 0.500 0.500 0.000 0.000 0.500 -0.500 0.000 0.000
m,,~
m,,
m,,~
lib-in.)
(lb-in.)
(lb-in.~
0.500 -0.500 0.500 0.500 0.000 0.000 0.000 0.000 0.156 0.156
0.866 11.866 1/.866 - 1/.866 1.000 1.000 0.000 0.(X10 0.750 0.750
1.136 1.136 1.136 1.136 0.312 0.312 0.438 1.062 0.312 0.312
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6.
7.
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Restraint analysis of assemblywork carriers • E. C. DE METER
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9.
10. 11. 12. 13.
14.
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