Mechanism and Machine Theory 37 (2002) 61±73
www.elsevier.com/locate/mechmt
Synthesis of a single-loop, overconstrained six revolute
joint spatial mechanism for two-position cylindrical
rigid body guidance
Albert J. Shih
a
a,*
, Hong-Sen Yan
b
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA
b
Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC
Received 13 February 2001; accepted 6 August 2001
Abstract
The synthesis and analysis of a single-loop, overconstrained spatial mechanism with six binary links and
six revolute joints for the guidance of a cylindrical rigid body between two positions are presented. The
geometric constraints that make the spatial 6R mechanism movable are ®rst introduced. Four features of
the geometrical constraints of this mechanism are summarized to demonstrate its mobility and used for
dimensional synthesis. Steps for mechanism dimensional synthesis are developed based on the descriptive
and analytical geometry. The mechanism analysis based on analytical geometry is also presented. One of
the advantages of this spatial mechanism is that the toggle position can be integrated into the synthesis.
This is especially attractive in the synthesis of the wheel retract and twist mechanism for aircraft landing
gears and the automated grinding wheel or tool changer for machine tools. A detailed example is used to
illustrate the mathematical models for mechanism synthesis and analysis. Ó 2002 Elsevier Science Ltd.
All rights reserved.
1. Introduction
Based on Kutzbach's mobility criterion, a single-loop spatial linkage with six binary links and
six revolute pairs, denoted as 6R, is an overconstrained mechanism with zero degrees of freedom.
A number of movable spatial 6R mechanisms have been reported. Mavroidis and Roth [1,2],
Barker [3], Lee and Yan [4], and Wohlhart [5] have reviewed these mechanisms and summarized
their con®gurations, proof of mobility, and input±output equations. The goal of this paper is to
*
Corresponding author. Tel.: +1-919-515-2365; fax: +1-919-515-7968.
E-mail address: ajshih@eos.ncsu.edu (A.J. Shih).
0094-114X/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 9 4 - 1 1 4 X ( 0 1 ) 0 0 0 5 5 - 6
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A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
present a dimensional synthesis method for a particular type of overconstrained spatial 6R
mechanism with three adjacent parallel axes. This mechanism, as shown in Fig. 1, is a special case
of the Type II movable 6R mechanisms reported in [4]. The mobility of this mechanism has been
demonstrated by solving the kinematic equations [4,6]. This mechanism is also similar, but not
identical, to the orthogonal Bricard linkage [5], which has three non-adjacent axes of revolute
joints parallel to each other. This research was initiated because of the need to design an aircraft
landing gear wheel retract and twist mechanism, which is a typical application of the two-position
cylindrical rigid body guidance. Other applications of the two-position cylindrical rigid body
guidance include automated grinding wheel or tool changers for machine tools, barrel movers for
automatic factory storage systems, etc. In this paper, mathematical models are developed to
synthesize and analyze an overcontstrained spatial 6R mechanism to solve the two-position cylindrical rigid body guidance problem.
The con®guration and Denavit and Hartenberg's notation [7] of the spatial 6R mechanism to
be used in this study is shown in Fig. 1. This mechanism has the following four features, which
make it movable and can be applied for dimensional synthesis:
Feature 1: Axes of three adjacent revolute joints, Z1 , Z2 , and Z6 for joints 1, 2, and 6, intersect at
a common point, O.
Feature 2: Axes of the other three adjacent revolute joints, Z3 , Z4 , and Z5 for joints 3, 4, and 5,
are parallel to each other.
Feature 3: Rotational axes of the two revolute joints that connect the two intersected and
parallel sets of joints are perpendicular to each other, i.e., Z2 ? Z3 and Z5 ? Z6 .
Feature 4: The rotational axis of joint 6, Z6 , intersects the origin of the X1 Y1 Z1 and X5 Y5 Z5
coordinate systems.
Only the ®rst two features are sucient to ensure the mobility of this overconstrained
spatial 6R mechanism. Features 3 and 4 are added to make the mechanism synthesis, based
on the combination of analytical and descriptive geometry methods, possible. Since the descriptive geometry [8] is used for the dimensional synthesis, the position of joints needs to be
Fig. 1. The con®guration and D±H notation of the spatial 6R mechanism and the global XYZ coordinate system.
A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
63
represented by points. The connection of two points in di€erent projective planes represents
the position of a link in descriptive geometry. Six points, O, A, B, C, D, and E, are used to
represent the positions of joints 1±6, respectively. As shown in Fig. 1, point O, representing
joint 1, is at the intersection of the X1 and Z1 axes. Point A, representing joint 2, is at the
intersection of the X3 and Z2 axes. Point B, representing joint 3, is at the intersection of the X3
and Z3 axes. Point C, representing joint 4, is at the intersection of the X4 and Z4 axes. Point D,
representing joint 5, is at the intersection of the X5 and Z5 axes. And joint 6, represented by E,
is on the extension of line OD. Points O, D, and E are collinear. The distance between points
D and E is not a design parameter listed in the D±H notation in Fig. 1 and does not in¯uence
the kinematic motion of the mechanism. The six links of this 6R mechanism are links OA,
AB, BC, CD, DE, and EO.
This spatial 6R mechanism has another feature: the six points, O, A, B, C, D, and E, always
remain in a plane. As the link OA rotates to a new position, the four points, O, A, D, and E,
determine a new plane. The motion of links moves points B and C to this plane. The three adjacent parallel axes, Z3 , Z4 , and Z5 at points B, C, and D, are perpendicular to this plane. This is
illustrated in Fig. 2. Fig. 2(a) shows the mechanism in Position 1 with positions of points A, B, and
C represented by A1 , B1 , and C1 , respectively. The plane P1 is de®ned by the three ®xed, collinear
points O, D, and E, and the point A1 . The axes at points B1 , C1 , and D (joints 3, 4, and 5) are all
perpendicular to P1 .
Assume two links, AB and BC, are separated at point C (joint 4). The link OA is then rotated to
Position 2 as shown in Fig. 2(b). The new position of point A is A2 . A2 , O, D, and E de®ne the
plane P2 . Line AB, which is perpendicular to line OA, is rotated around point A (joint 2) to the
position where B2 and link AB are both on plane P2 . Since Z2 ? Z3 (Feature 3), AB is on P2 , and Z2
is parallel to P2 , Z3 is perpendicular to P2 . This de®nes the position and orientation of joint 3 or
point B2 at Position 2. Similarly, link DE is rotated to orient Z5 at point D perpendicular to P2 .
This is possible because Z5 ? Z6 (Feature 3) and Z6 is parallel to P2 . Since Z3 , Z4 , and Z5 are parallel
to each other (Feature 1) and Z3 and Z5 are both perpendicular to P2 , Z4 at point C2 is also
perpendicular to P2 . The position of point C2 (joint 4) is determined by given lengths of links BC
and CD and known positions of B2 and D.
(a)
(b)
Fig. 2. The six joints at (a) Position 1 and (b) Position 2 located on planes P1 and P2 , respectively.
64
A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
For every incremental movement of the link OA, the mechanism can be reassembled at C. This
is another way, other than using kinematic equations [4,6], to demonstrate the mobility of this
overconstrained spatial 6R mechanism. This way of explanation may seem awkward; however, it
does provide the essential steps, to be discussed in the next section, for the graphical synthesis
based on descriptive geometry. In descriptive geometry, the side view of a plane is a straight line in
the projective plane. This particular characteristic is used for mechanism synthesis. Mechanism
design using the descriptive geometry is basically a graphical method. Several researchers have
applied this method for mechanism analysis and synthesis in the past. Herrera and Shoup [9]
demonstrated the application of descriptive geometry to analyze the RSSR and RSCP mechanisms. Lakshminarayana and Rao [10] studied the graphical synthesis of RSSR crank-rocker
mechanism. Beard [11] used the descriptive geometry to determine the position and orientation of
a revolute joint for spatial motion of two points in a rigid body.
In this paper, four steps to synthesize the spatial 6R mechanism based on the given positions
and orientations of a cylindrical rigid body are ®rst introduced. The motion of the mechanism is
analyzed using the analytical geometry. A detailed example is used to illustrate and verify the
mathematical models developed for mechanism synthesis and analysis.
2. Mechanism synthesis
The objective of mechanism synthesis is to determine locations and orientations of all six revolute joints at Position 1. The cylindrical rigid body is ®xed on link AB. Mathematically, the
cylindrical rigid body is represented by the position of a point on the center axis and the directional vector of the center axis. As shown in Fig. 3(a), the cylindrical rigid body at Positions 1 and
2 are speci®ed by points A1 and A2 on the center axes and by directional vectors of the center axes,
v1 and v2 . A1 and A2 are also the location of point A (joint 2) of the spatial mechanism at Positions
1 and 2, respectively. Fig. 2(a) shows a temporary X 0 Y 0 Z 0 coordinate system used to specify A1 , A2 ,
v1 and v2 , before the point O and X1 Y1 Z1 coordinate system are de®ned. Another XYZ coordinate
(a)
(b)
Fig. 3. Synthesis of the location of point O for two-position cylindrical rigid body guidance.
A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
65
system with the origin at O is de®ned in Fig. 1. Positions and axes of all joints in the spatial 6R
mechanism are referenced to this XYZ coordinate system in the mechanism synthesis and analysis.
The following four steps summarize the procedure for dimensional synthesis and inputs for the
mechanism synthesis are listed in Table 1.
2.1. Step 1. Location of point O
The location of point O has two constraints. First, distances from point O to A1 and A2 are
equal. Second, the angle between line OA1 and v1 (h in Fig. 3) is the same as the angle between line
OA2 and v2 . Mathematically, these two constraints can be written as:
OA1 ˆ OA2 ;
1†
OA1 v1 ˆ OA2 v2 ;
2†
where OA1 is the distance from O to A1 , OA2 is the distance from O to A2 , OA1 is the vector from O
to A1 , and OA2 is the vector from O to A2 . In the X 0 Y 0 Z 0 coordinate system, the components of
point O are …Ox0 ; Oy0 ; Oz0 †, of point A1 are …A1x0 ; A1y 0 ; A1z0 †, of vector v1 are …v1x0 ; v1y 0 ; v1z0 †, etc. Eqs.
(1) and (2) are rewritten as a set of two linear equations with three unknowns Ox0 , Oy 0 , and Oz0 :
a1 Ox0 ‡ b1 Oy 0 ‡ c1 Oz0 ˆ d1 ;
3†
a2 Ox0 ‡ b2 Oy 0 ‡ c2 Oz0 ˆ d2 ;
4†
where
a1 ˆ 2…A2x0
A1x0 †;
b1 ˆ 2…A2y 0
d1 ˆ A21x0
A22x0 ‡ A21y 0
a2 ˆ v2x0
v1x0 ;
d2 ˆ A2x0 v2x0
A1y0 †;
A22y 0 ‡ A21z0
b2 ˆ v2y 0
v1y0 ;
A1x0 v1x0 ‡ A2y0 v2y 0
c1 ˆ 2…A2z0
A1z0 †;
A22z0 ;
c2 ˆ v2z0
v1z0 ;
A1y 0 v1y0 ‡ A2z0 v2z0
A1z0 v1z0 :
By assigning a value to either Ox0 , Oy 0 , or Oz0 , Eqs. (3) and (4) can be used as a set of two linear
simultaneous equations to solve for the other two unknown components of point O. For example,
if the designer picks a value for Oz0 , then Eqs. (3) and (4) can be rearranged as Eqs. (5) and (6) to
solve for Ox0 and Oy 0 :
a1 Ox0 ‡ b1 Oy 0 ˆ d1
c 1 Oz 0 ;
5†
a2 Ox0 ‡ b2 Oy 0 ˆ d2
c 2 Oz 0 :
6†
Table 1
Summary of the inputs for mechanism synthesis
In
In
In
In
In
Step
Step
Step
Step
Step
1:
1:
3:
3:
4:
Positions of A1 and A2 and directional vectors of v1 and v2
One of the coordinates of point O in X 0 Y 0 Z 0 coordinate system, either Ox0 , Oy 0 , or Oz0
Length of link AB
Angle c and lengths of Oh Dh and Oh Eh
Ratio of the length of links BC and CD
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A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
2.2. Step 2. Angles a and b and the XYZ coordinate system at point O
As shown in Fig. 3(a), the angle between lines OA1 and OA2 is a
Ox0 A1x0 †…Ox0 A2x0 † ‡ …Oy 0 A1y 0 †…Oy0 A2y 0 † ‡ …Oz0
a ˆ arccos
OA2
A1z0 †…Oz0
A2z0 †
;
7†
where OA …ˆ OA1 ˆ OA2 † is the length of link OA.
The XYZ coordinate system with origin at point O and the unit vectors i, j, and k, as shown in
Fig. 3(a), is set up at point O. This XYZ coordinate system is also illustrated in Fig. 1. The Z-axis,
or vector k, is perpendicular to the plane OA1 A2 . The Y-axis, or vector j, is in the direction from A1
to O. The three unit directional vectors i, j, and k are:
OA2 OA1
;
OA2
OA1
jˆ
;
OA
8†
kˆ
9†
i ˆ j k:
10†
The components of O, A1 , and A2 in XYZ coordinate system are: O ˆ …0; 0; 0†, A1 ˆ …0; OA; 0†,
and A2 ˆ … OA sin a; OA cos a; 0†.
As shown in Fig. 3(b), when the link OA rotates from OA1 to OA2 and assuming that the relative
position between v1 and OA1 remains unchanged, v1 is rotated to vt . The angle between vt and OA2
is still h. b is the angle to rotate vt to v2 in the direction of w (unit vector in the direction from A2 to
O)
wˆ
OA2
;
OA
11†
b is also the angle of rotation of the rigid body around point A. To calculate vt , v1 is rotated along
the vector k by angle a. The axis rotation matrix ‰Ra;k Š is used to calculate vt [12]
8 9
8
9
< vtx0 =
< v1x0 =
12†
v 0 ˆ ‰Ra;k Š v1y0 ;
: ty ;
:
;
0
0
vtz
v1z
where
2
k 20 V a ‡ Ca
6 k 0 kx 0 V a ‡ k 0 Sa
‰Ra;k Š ˆ 4 x y
z
kx0 kz0 V a ky 0 Sa
kx0 ky 0 V a kz0 Sa
ky20 V a ‡ Ca
ky 0 kz0 V a ‡ kx0 Sa
Ca ˆ cos a;
and V a ˆ 1
Sa ˆ sin a;
3
kx0 kz0 V a ‡ ky 0 Sa
ky0 kz0 V a kx0 Sa 7
5;
kz20 V a ‡ Ca
13†
cos a;
where v1x0 , v1y 0 , and v1z0 are the components of vector v1 , vtx0 , vty 0 , and vtz0 are the components of
vector vt , and kx0 , ky 0 , and kz0 are the components of vector k in the X 0 Y 0 Z 0 coordinate system.
Fig. 3(b) illustrates how the angle b is calculated. The term l is the length of projection of v1 and
vt on w.
A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
l ˆ v1 w ˆ vt w;
"
#
v2 lw† …vt lw†
b ˆ arccos
;
2
jv2 lwj
67
14†
15†
where jv2 lwj is the length of vector v2 lw.
Eqs. (7) and (15) solve the angles a and b but fail to provide the rotational direction. It is
recommended that the designer verify the results graphically during the mechanism analysis,
which will be discussed in Section 3.
2.3. Step 3. Positions of points D and E
The positions of points D and E are determined using the descriptive geometry as the design
tool. As shown in Fig. 4, two projective planes H and F are set up. The horizontal projective
plane, H, is perpendicular to link OA1 and parallel to X and Z axes. H is the side view of plane P1 .
As discussed in Fig. 1, the line DE intersects point O, and the plane A1 ODE is called plane P1 . On
H, plane P1 is a straight line because H is the side view of P1 . The six points at Position 1 (Oh , A1h ,
B1h , C1h , Dh , and Eh ) are all located on this line. Oh and A1h overlap each other in line P1 on H.
The frontal projective plane, F, is parallel to line OA1 and perpendicular to the Z-axis and the
rotational axis at O. F is parallel to the X and Y axes. HF is the folding line of the horizontal and
frontal projection planes. On F, distances from Of to A1f and A2f are equal to the length of link
OA. The angle between lines Of A1f and Of A2f is a, as calculated in Eq. (7).
Fig. 4. Use the three projective views, H, F, and I in perspective geometry for mechanism synthesis.
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A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
The angle between line P1 on H and the folding line HF is c. c is an angle selected by the designer. The designer also needs to choose the positions of Dh and Eh , or lengths Oh Dh and Oh Eh , on
line P1 .
The third projective view, I, is the side view of plane P2 where all six points at Position 2 (OI , A2I ,
B2I , C2I , DI , and EI ) are located on. Plane P2 is a straight line on I. I is perpendicular to link OA at
Position 2, i.e., the folding line FI is perpendicular to line Of A2f , as shown in Fig. 4. The position of
OI can be found on the extension of line A2f Of . Based on descriptive geometry, as shown in Fig. 4,
the distance, d, from OI to the folding line FI is equal to the distance from Oh to the folding line HF
[8]. This de®nes the location of OI on the extension of line Of A2f . On I, A2I overlaps with OI .
A circle with radius equals to the length of link AB is drawn around OI . This circle represents
the possible trajectory of point B2 on I. The position of point B1 on I, designated as B1I , is also
located on this circle, and the angle between OI B1I and folding line FI is c, as shown in Fig. 4.
Because link AB rotates through an angle b as the cylindrical rigid body moves from Positions 1
to 2, point B2I is determined by rotating B1I relative to OI by an angle b. OI B2I de®nes the side view
of plane P2 . P2 is a straight line on I. Points DI and EI are both located on line P2 . The distance, m,
from DI to the folding line FI is equal to the known distance from Dh to the folding line HF. This
determines the position of DI . Similarly, the distance from EI to FI (n in Fig. 4) equals the distance
from Eh to HF. The position of EI on line P2 is set.
After the positions of DI and EI are found, Df and Ef can be solved. It is known that Df Dh is
perpendicular to the folding line HF and Df DI is perpendicular to the folding line FI. The intersection of two lines, one line passing through Dh and perpendicular to HF and another line
passing through DI and perpendicular to FI, intersects at Df . Similarly, the intersection of two
lines, one line passing through Eh and perpendicular to HF and another line passing through EI
and perpendicular to FI, determines the location of Ef .
Mathematically, when Dh and Eh are selected on line P1 in H, the X and Z components of points
D and E are: Dx ˆ …Oh Dh † cos c, Dz ˆ …Oh Dh † sin c, Ex ˆ …Oh Eh † cos c, and Ez ˆ …Oh Eh † sin c.
Based on the geometrical relationship shown in Fig. 4, the Y component of Dy and Ey are:
Dx
‡
tan a tan…p
Ex
Ey ˆ
‡
tan a tan…p
Dy ˆ
Dz
;
b c† sin a
Ez
:
b c† sin a
16†
17†
2.4. Step. 4. Positions of points B and C
The length of link AB is selected by the designer. Since B1h is located on line P1 , and since the
distance from A1h to B1h is equal to the length of link AB, the position of B1h can be solved. B1f is
located at the intersection of the line passing through B1h and perpendicular to HF and another
line passing through A1f and perpendicular to Of A1f .
C1h is selected on line P1 between B1h and Dh . C1f is located anywhere on the line that passing C1h
and perpendicular to HF. If C1f is selected at the point where this line intersects B1f Df , as shown
in Fig. 4, it is the toggle position for the mechanism at Position 1. This feature is especially attractive for the design of aircraft landing gear and grinding wheel loader because a locking device
can be added on joint 4 (point C) to turn the mechanism into a structure.
A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
69
The X, Y, and Z components of C1 are designated as C1x , C1y and C1z . C1x ˆ …Oh C1h † cos c and
C1z ˆ …Oh C1h † sin c. Since the selection of C1f is arbitrary on the line passing through C1h and
perpendicular to HF, C1y depends on where the C1f is located. If the toggle con®guration at
Position 1 is desired (as shown in Fig. 4), then
C1y ˆ B1y ‡ …D1y
B1y †
C1x
D1x
Bx
:
B1x
18†
Following Steps 1±4, the X, Y, and Z components of six points at Position 1, O, A1 , B1 , C1 , D
and E, are solved.
3. Mechanism analysis
A mathematical model is developed to analyze the locations of points B and C when the link
OA is rotated by an angle k from Position 1, as shown in Fig. 5. The resulting con®guration of the
mechanism is called Position 3. Positions of points A, B and C are designated as A3 , B3 and C3 .
The X, Y, and Z components of A3 are: A3x ˆ OA sin k, A3y ˆ OA cos k, and A3z ˆ 0. A point,
A3 , and a line, ODE, determine the plane P3 where B3 and C3 both located on it. As shown in Fig.
5, the unit vector p is perpendicular to P3 .
pˆ
DE OA3
;
jDE OA3 j
19†
where OA3 is the vector from O to A3 and DE is the vector from D to E.
Another unit vector q is along the direction from A3 to O, q ˆ OA3 =OA. r is the unit vector
from A3 to B3 and r ˆ p q. The position of B3 is
B3 ˆ A3 ‡ ABr;
20†
where AB is the length of link AB.
Fig. 5. Mechanism analysis: solving for positions B3 and C3 .
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A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
The location of C3 is solved based on the known lengths of links BC and CD and the locations
of points B3 and D. The angle between line B3 D and B3 C3 is designated as g, as shown in Fig. 5. g
can be solved using the law of cosines
"
#
BC†2 ‡ …B3 D†2 …CD†2
;
21†
g ˆ arccos
2…BC†…B3 D†
where BC and CD are the length of links BC and CD and B3 D is the distance from B3 to D. The
unit vector s ˆ B 3 D=B3 D. To ®nd the location of B3 , the unit vector t pointed in the direction from
B3 to C3 is used. t is obtained by rotating s along p by an angle g, as shown in Fig. 5. Applying the
same concept in Eqs. (12) and (13), t can be found by using the axis rotation matrix ‰Rg;p Š
8 9
8 9
< tx =
< sx =
ˆ ‰Rg;p Š sy :
22†
t
: y;
: ;
tz
sz
There are two possible positions of point C3 , one has the angle of rotation equals g and another
has the angle of rotation equals g. Position of C3 is
C 3 ˆ B 3 ‡ BCt;
23†
where BC is the length of link BC.
4. Numerical example
An example is presented to illustrate the procedures used to design a spatial 6R mechanism to
guide a cylindrical rigid body between two positions. First, the location of cylindrical rigid body at
Positions 1 and 2 in the X 0 Y 0 Z 0 coordinate system are speci®ed as: A1 ˆ ‰1; 1; 0Š, v1 ˆ ‰0; 0; 1Š,
A2 ˆ ‰0; 1; 1Š, and v2 ˆ ‰0; 1; 0Š. The lengths and coordinates are measured in unit length, that
could be m, cm, or inch.
The four steps described in Section 2 are used for mechanism synthesize:
Step 1. Select Oz0 ˆ 1:5. Using Eqs. (5) and (6), Ox0 ˆ 1:500 and Oy 0 ˆ 2:500. The length of link
OA is calculated as 2.179.
Step 2. Using Eq. (7), a ˆ 37:864°. Eqs. (8)±(10) calculate the X 0 , Y 0 , and Z 0 components of vectors i, j, and k as: i ˆ ‰0:826; 0:236; 0:511Š, j ˆ ‰0:229; 0:688; 0:688Š, and
k ˆ ‰0:514; 0:686; 0:514Š. Eqs. (12) and (13) are used to rotate vector v1 to
vt ˆ ‰0:477; 0:241; 0:845Š. Using Eqs. (14) and (15), angle b ˆ 116:18°. In the XYZ coordinate
system, O ˆ ‰0; 0; 0Š and A1 ˆ ‰0; 2:179; 0Š.
Step 3. Select c ˆ 30°, Oh Dh ˆ 2:5 and Oh Eh ˆ 2:7. The X, Y, and Z coordinates of D and E are
D ˆ ‰ 2:165; 5:825; 1:250Š and E ˆ ‰ 2:338; 6:290; 1:350Š.
Step 4. Select the length of link AB ˆ 0.5. Assume links BC and CD have equal length, i.e.,
point C is in the middle of B1 and D in Position 1, as illustrated in Fig. 4. Position 1 is designed
to have toggle con®guration at joint 4 (point C). The length of links BC and CD are the same,
BC ˆ CD ˆ 2:079. Positions of points B and C at Position 1 are: B1 ˆ ‰ 0:433; 2:179; 0:250Š
and C1 ˆ ‰ 1:299; 4:002; 0:750Š.
A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
71
Procedures in Section 3 are used to ®nd the positions of the three moving points A, B, and C.
Analysis results, including the X, Y, and Z components of A3 , B3 , and C3 , unit vectors p, q, r, s,
and t, and angle g, are listed in Table 2. In the ®rst column, angle k ˆ 0:0° represents Position 1 of
the mechanism. These results should match the known conditions obtained in the mechanism
synthesis. This is the ®rst check of the validity of the analysis. The ®fth column, g ˆ 37:864°,
represents the Position 2 of the mechanism. Three intermediate positions are analyzed. Graphically, the analysis results of the locations of joints and links are illustrated in Fig. 6.
The analysis results can be further veri®ed by calculating angle b from the analysis and comparing it to the known value from mechanism synthesis. Assuming the relative position between
links AB and OA remains the same while the link OA is rotated from Position 1 to 2, point B is
moved from B1 to Bt , where Bt ˆ ‰ 1:680; 1:455; 0:250Š in the XYZ coordinate system. The
angle between A2 Bt and A2 B2 , b, is calculated as 116.18°. This matches the results from Step 2 of
the mechanism synthesis and further veri®es the validity of the analysis.
Table 2
Results of the analysis of the 6R mechanism at ®ve di€erent positions
Angle k (deg)
0.000
9.466
18.932
28.398
37.864
A3x
A3y
A3z
0.000
)2.179
0.000
)0.358
)2.150
0.000
)0.707
)2.062
0.000
)1.037
)1.917
0.000
)1.338
)1.721
0.000
px
py
pz
)0.500
0.000
0.866
)0.718
0.120
0.686
)0.938
0.322
0.126
)0.723
0.391
)0.569
)0.439
0.342
)0.831
qx
qy
qz
0.000
1.000
0.000
0.164
0.986
0.000
0.324
0.946
0.000
0.476
0.880
0.000
0.614
0.789
0.000
rx
ry
rz
)0.866
0.000
)0.500
)0.676
0.113
)0.728
)0.119
0.041
)0.992
0.501
)0.271
)0.822
0.656
)0.510
)0.557
B3x
B3y
B3z
)0.433
)2.179
)0.250
)0.697
)2.093
)0.364
)0.766
)2.041
)0.496
)0.786
)2.053
)0.411
)1.010
)1.976
)0.278
sx
sy
sz
)0.417
)0.877
)0.241
)0.358
)0.909
)0.216
)0.341
)0.922
)0.184
)0.336
)0.919
)0.204
)0.279
)0.931
)0.235
Angle Z (deg)
Angle g (deg)
tx
ty
tz
0.000
0.000
)0.417
)0.877
)0.241
9.008
9.008
)0.260
)0.960
)0.104
9.269
9.269
)0.327
)0.945
)0.024
9.323
9.323
)0.429
)0.900
)0.073
6.070
6.070
)0.368
)0.912
)0.180
C3x
C3y
C3z
)1.299
)4.002
)0.750
)1.236
)4.089
)0.581
)1.447
)4.005
)0.547
)1.679
)3.924
)0.562
)1.775
)3.872
)0.653
72
A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
Fig. 6. Two projective views of six joints and links at ®ve di€erent positions.
The analysis was also carried out using the ADAMS simulation software. Hinges were used to
represent the revolute joints. The coordinates and orientations of the six joints at Position 1 were
used as the input. Link OA was rotated 37.864°. The analysis results of the position of points B
and C matched to that of Table 2. It further veri®ed the model.
5. Concluding remarks
In this paper, a single-loop, spatial movable mechanism with six binary links and six revolute joints was used to guide a cylindrical rigid body between two speci®ed positions. This
overconstrained mechanism was able to move due to special geometric constraints. The
characteristic that all six joints of the mechanism remain in a plane was applied in mechanism
synthesis using descriptive geometry as the tool. A detailed example was used to illustrate the
developed synthesis and analysis procedures. The analysis results were veri®ed by di€erent
approaches.
The research to use descriptive geometry and graphical methods for mechanism synthesis has
decreased since the advancement of computers and numerical-based methods. This paper illustrates that descriptive geometry, which gives insight to the motion of the mechanism, is still a
useful tool in mechanism design, especially in some industrial applications.
A.J. Shih, H.-S. Yan / Mechanism and Machine Theory 37 (2002) 61±73
73
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