ED7202 mds_notes

KALAIGNAR KARUNANIDHI INSTITUTE OF TECHNOLOGY
Coimbatore – 641 402
DEPARTMENT OF MECHANICAL ENGINEERING
M.E ENGINEERING DESIGN
ED7202-MECHANISMS DESIGN AND
SIMULATION
(R-2013)
YEAR: II SEM: 2

INTRODUCTION TO KINEMATICS AND MECHANISMS
kinematic pair – existing connection between two elements of a mechanism
that have a relative motion between them.
kinematic pairs was classified by reuleaux as follow:
lower pair – two links having a surface contact between them.
higher pair – two links having line or point contact between them.
• joint - guarantees the contact between two members and constrains their
relative motion.
Fig: Degrees of freedom

Degrees of freedom
 Number of independent coordinates needed to define the
position of the element/mechanism…
 ….. or number of parameters needed to determine
unambigously the geometry configuration of a system in
space….
 ….. or the number of inputs needed to obtain a predictible
output of a mechanism.
CHEVYCHEF-GRÜBLER-KUTZBACH CRITERION
Fig. kutzbatch criterion

FOUR-BAR LINCKAGE
 Very simple but very versatile. First option for design.
 Clasification depending on the task.
 Function Generator. Output rules
 Path Generator. Path rules
 Motion Generator. All important
Fig. four bar linkage

4 BAR KINEMATIC INVERSIONS
 It is the method of obtaining different mechanisms by fixing
different links of the same kinematic chain. POWERFULTOOL. See that
with the slider-crank example:
Fig 4bar kinematic inversions
GRASHOF CRITERIA
 Simple relation that describes the behavior of the kinematic inversions
of a four-bar mechanism.
S= lenght of the shortest link
L= lenght of the longest link.
P and Q are the other links.

S+L ≤ P+Q
Special Grashof mechanisms
 If s + l = p + q. Grashof Special Mechanisms.
 All inversions are double-crank or crank-rocker.
 These mechanisms suffer from the change-point
 condition.All links become collinear creating momentarily a
second DOF. OUTPUT RESPONSE IS UNDETERMINED.

Fig Grashof mechanisms
COUPLER CURVE
There are two types of path generation. One is point-to-point path generation, in which
the coupler curves only specify discrete points on the desiredpath. The other is continuous path
generation, in which the coupler curves specify the entire path, or atleast many points on it.
Because the coupler curves of linkage mechanisms are functions of their link lengths, the only
way to generate different continuous coupler curves with a single linkage mechanism is tomake
the length of at least one of its links adjustable.
One way of doing this is to replace the normal links with screw-nut links driven by
servomotors, as shown in Fig. 1. The different desired coupler curves can then be obtained by
controlling the length of the adjustable links and the angular displacement of the driving link.

Fig.1 adjustable mechanism
The investigation of new synthesis methods for path generation using linkage
mechanisms has been the subject of some research attention in recent years.
Tao and Krishnamoothy (1978) developed graphical synthesis procedures of adjustable
mechanisms for generating variable coupler curves with cusps and symmetrical coupler curves
with a double point.
McGovern and Sandor (1973) used complex number methods to synthesize adjustable
mechanisms for path generation. Kay and Haws (1975) developed a design procedure for a path
generation mechanism with a cam link, which provided accuracy over a range of motion.
Angeles et al. (1998) proposed an unconstrained nonlinear least-square optimal synthesis method
for RRRR planar path generators. Hoeltzel and Chieng (1990) proposed a pattern
matching synthesis method based on the classification of coupler curves according to moment

variants. Watanabe (1992) presented a natural equation that expressed the curvature of the path
as an equation of the arc length and was independent of the location and orientation of the path.
Ullah and Kota (1994, 1997) presented an optimal synthesis method in which the objective
function was expressed as Fourier descriptors. Shimojima et al.
(1983) developed a synthesis method for straight-line and L-shaped path generation using fixed
pivot positions as adjustable parameters. Unruh and Krishnaswami (1995) proposed a computer-
aided design technique for infinite point coupler curve synthesis of four-bar linkages. Kim and
Sodhi (1996) introduced a method of path generation that made the desired path pass exactly
through five specified points and close to other points. Chuenchom and Kota (1997) presented a
synthesis method for programmable mechanisms using adjustable dyads. Chang (2001) proposed
a synthesis method for adjustable mechanisms to trace variable arcs with prescribed velocities.
Zhou et al. (2002) proposed an optimal synthesis method with modified genetic optimization
algorithms by adjusting the position of the driven side link for continuous path generation.
Russell and Sodhi (2005) presented a design method for slider-crank mechanisms to achieve
multiphase path and function generation. In this paper, we propose a new design method for
continuous path generation by four-bar mechanisms that incorporates a screw-nut link called an
adjustable link. Different desired coupler curves can be generated by appropriately adjusting the
length of the adjustable link and controlling the angular displacement of the driving link.
Examples and experiments are provided to demonstrate this design method.
REQUIRED DRIVING LINKANGULAR DISPLACEMENT CORRESPONDING TO
THE DESIRED COUPLER CURVE
The coordinate system of a four-bar linkage is shown in Fig. 2.
The speed trajectory of the driving link, and the lengths of links 1 or 4 can be adjusted to
generate new coupler curves.

Fig. 2. The coordinate system of a four-bar linkage
Figure 2 shows that the relationship between the angular displacement of the
driving link 2 θ and the coordinate of the coupler point ( x y P , P ) can be written as
(1)
and the vector loop equation can be written as
R2 + R5 – RP = 0. (2)
Separating Eq. (2) into two scalar component equations in the x- and y-directions yields
r2 cosθ2 + r5 cos(θ3 + β) − Px = 0 and (3)
r2 sin θ2 + r5 sin(θ3 + β) − Py = 0 (4)
where ri and θi represent the length and angular displacement of the ith link, respectively.
Adding Eqs.(3) and (4) after squaring both sides gives
r5 2 = Px 2 + Py 2 − 2r2 (Px cosθ2 + Py sinθ2 ) + r2 2 , (5)
which, after rearrangement, gives
(Px cosθ2 + Py sinθ2 ) + r5 2 − Px 22−r2Py 2 −r2 2 = 0 . (6)
To reduce Eq. (6) to a form that can be solved more easily, we substitute the half angle identities
to convert the cosθ2 and sinθ2 terms to tanθ2 terms:
− tan 2(
θ
2 2 tan(
θ 2
)1 )
2cosθ = 2 ; sinθ
2
=
2 1+ tan 2(θ 2 1+ tan 2
( θ2
2
)
2
)
This results in the following simplified form, where the link lengths ( r2 and r5 ) and the
known value( Px , Py ) terms have been collected as constants A, B
and C: 2 θ2 θ2 where
A tan ( 2 ) + B tan( 2 ) +C = 0
r
2
− P
2
− P
2
− r
2
A = 5 x y 2 − P , B = 2P , and
2r2
x y
−1
P
(θ2 + μ) = tan
y
P

r 2 − P 2 − r 2
C =
5 x y 2
+ Px . The angular
2r2
displacement of the driving link can then be calculated as
θ2
2
= 2 tan
−1
− B ± B − 4AC (7)
2A
and the correspondingθ3 can be obtained from Eq. (4):
θ3 = tan
−1
(
Px − r2 cosθ2
) − β . (8)
Py − r2 sinθ2
From Figure 2, the vector loop equation can be written as
R2 + R3 – R1 – R4 =0 . (9)
If we assume that the length of link 4 can be adjusted, then we separate Eq. (9) into two scalar
component equations and rearrange as follows:
(r4 + r4 ) cosθ4 = r2 cosθ2 + r3 cosθ3 − r1 cosθ1 (10)
(r4 + r4 ) sinθ4 = r2 sinθ2 + r3 sinθ3 − r1 sinθ1
Where
By dividing Eq. (11) by Eq. (10) to
eliminate (r4 + r4 ) , the angular displacement of link 4, θ4 , can be expressed as
r4 is the length of adjustable link 4.
θ4 = tan
−1
(
r2 sinθ2 + r3 sinθ3 − r1 sinθ1
) (12)
r2 cosθ2 + r3 cosθ3 − r1 cosθ1
Then r4 can be calculated as
cosθ2 + r3 cosθ3 −r1 cosθ1
r4
r2
−r4
(13)
= cosθ4
Assuming that the length of link 1 can be adjusted, we separate Eq. 9 into two scalar
component equations and rearrange them as follows
r1 =
− B ± B
2
− 4C
− r1 (16)
2
where
B = −2r2 (cosθ1 cosθ2 + sinθ1 sinθ2 )
and− 2r (cosθ
1
cosθ
3
+ sinθ
1
sinθ
3
)
3
C = −r
2
+ r
2
+ r
2
4 2 3
. The+ 2r2 r3 (cosθ2 cosθ3 +sinθ2 sinθ3 )

corresponding θ4 is
θ
4 = tan
−1
(
r2 sinθ2 + r3 sinθ3 − (r1 + r1 ) sinθ1
)
r2 cosθ2 + r3 cosθ3 − (r1 + r1 ) cosθ1
(17)
CONDITIONS FOR GENERABLE
COUPLER CURVES
The coupler curves that can be generated must satisfy both the following conditions:
r5 − r2 ≤ rp ≤ r5 + r2 and (18)
( Px − r2 cosθ2 )2
+ (Py − r2 sinθ2 )2
= r5 2 (19)
where rp = Px
2
+ PY
2
. In Fig. 2, we assume that
r2 and r5 are not adjustable. Therefore, as long as
the desired continuous coupler curves are in the area between the two concentric circles with radii
r5 −r2 and r5 + r2 , they can be generated by
controlling the angular displacement of the driving link and adjusting the length of links l or 4.
Synthesis and analysis of coupler curves with combined planar cam
follower mechanisms
In several applications, design of mechanical systems involves the synthesis of
mechanisms in order to meet a set of kinematic requirements. Several methods of synthesis have
been proposed by various authors for selecting and scaling mechanical devices. Graphical
methods proposed by Hartenberg and Denavit (1964) and Sandor and Erdman (1984) are useful
for limited number of precision points in synthesis. Freudenstein (1954) and Erdman (1981 and
1995) developed analytical approach for synthesis of planar mechanisms. Sandor and Erdman
(1991) and Angeles and Cajin (1988) presented a computer aided numerical methods for
synthesis of planar four bar and cam mechanisms. In all above methods, objective was to guide a
rigid body through a series of specified positions (rigid body guidance) or to obtain a specific
input and output relationship (function generation), or to force a point on a linkage to move
along a prescribed trajectory (path generation), but these methods restrict the number of
precision points or coordinates. Increase in computer power has permitted the recent
development of routines that apply any number of precision points for path generation. Genetic
algorithm (GA) is one of the recent techniques of kinematic synthesis. Genetic algorithms were
first introduced by Holland (1975). Goldberg (1989) revealed that GA’s can successfully apply

to different engineering optimization problems. Laribi et al. (2004) cited a method based on
genetic algorithms and fuzzy logics for the synthesis of four bar mechanisms and combined cam
linkages. First step is based on crank and rocker mobility criterion then, mechanism is considered
with as many as degrees of freedom required by the generation task. A combined cam-linkage
mechanism synthesized for precise motion, function and path generation. Mundo et al. (2006)
proposed a genetic algorithm based method for the optimal synthesis of planar mechanisms.
Used combined cam linkage for path generation task, and also some applications of this
methodology are presented. Gabrera et al. (2000) applied genetic algorithms to synthesize the
four bar mechanisms for path generation problem. Objective function was to minimize distance
between coordinates of prescribed and designed. Lampein (2003) described the genetic
algorithms for optimize the cam mechanisms, and then this analogy is applied tosynthesize the
automotive valve trains. The work presented by Fang (1994) deals with solution methods of
optimal synthesis of planar mechanisms. A searching procedure is based on evolutionary
techniques that are genetic algorithms. Problems of four-bar mechanisms are used to test the
method. Singh and Kohli (1981) cited the existence of cam-linkage mechanisms for motion and
path generation; used the complex loop closure method and the envelope theory to define a
general approach for the synthesis of combined cam-linkage systems for exact path or motion
generation. Sadler and Yang (1990) proposed a method of dynamic optimization of cam-linkage
mechanisms. Recently Ullah and Kota (1991), Saggere and Kota (2001) and Smith and Ye
(2005) used an analytical approach to the design of mechanical systems, where planar linkages
are combined with a cam for driving an oscillating roller follower. Saxena (2005) and Kunjur
and Krishnamurthy (1997) presented genetic algorithm approach to synthesize the mechanisms,
also shown some results obtained by evolutionary techniques. These methods define a starting
population that is improved by approximations of design variables. Gen and Cheng (2000)
applied genetic algorithms to different engineering optimization problems. Also usefulness of
this approach and advantages over the conventional optimization methods are mentioned.
Different methods of kinematic analysis of planar mechanisms are described by Ghosh and
Malik (2002) and Shigley and Uicker (1980). This analysis is very essential to study the motion
of various links of a mechanism. Norton (2002) and Chen (1982) studied synthesis, analysis and
dynamics of cam mechanisms. Cam mechanism can generate complex coordinated movements,
relatively compact and easy to design. Recently Hafez and Su (2010) discussed the Synthesis of
a slider-crank fourbar linkage is presented whose coupler point traces a set of predefined task
points. There are at most 558 slider-crank four-bars in cognate pairs passing through any eight
specified task points. The problem is formulated for up to eight precision points in polynomial
equations. Umesh et al. (2010) presented Synthesis of coupler curves with combined planar cam
follower mechanisms by genetic algorithm. Synthesis and path generation by single input
combined cam mechanism using genetic algorithm was presented. Also this method applied to
the complex path generation problem.
The objective of this work is to propose a novel CCM and a method of precise path
generation with infinite number of precision points. Combine the conventional four bar and cam-

follower mechanisms to obtain a CCM and dimensional synthesis. Then estimate the error in
path generation by CCM and four bar. Finally carry out the kinematic and dynamic simulation to
verify the stability of CCM by considering the various cases of cutting profiles or shapes. The
paper is organized as follows; it presents a detailed literature survey in this field in the recent
period. From the literature survey the main objectives were formed. Profile of coupler curves
were defined, a four bar mechanism for dimension synthesized and combined with cam follower
mechanism to get CCM. Subsequently to validate CCM different curves were generated. Then
kinematic and dynamic competence of the developed CCM has been discussed.

POSITION ANALYSIS
The position analysis of planar linkages has been dominated by resultant elimination and
tangent-half-angle substitution techniques applied to sets of kinematic loop equa-tions. This
analysis is thus reduced to finding the roots of a polynomial in one variable, the characteristic
polynomial of the linkage. When this polynomial is obtained, it is said that the problem is solved
in closed form. This approach is usually preferred to numerical approaches because the degree of
the polynomial specifies the greatest possi-ble number of assembly configurations of the linkage
and modern software of personal computers provides guaranteed and fast computation of all real
roots of a polynomial equation and hence of all assembly configurations of the analyzed linkage.
A non-overconstrained linkage with zero-mobility from which an Assur group can be
obtained by removing any of its links is defined as an Assur kinematic chain, basic truss [1, 2],
or Baranov1
truss when no slider joints are considered [3]. Hence, a Baranov truss, named after
the Russian kinematician G.G. Baranov [4] who first stated it in 1952 [5], corresponds to
multiple Assur groups. The relevance of the Baranov trusses derive from the fact that, if the
position analysis of a Baranov truss is solved, the same process can be applied to solve the
position analysis of all its corresponding Assur groups. Curiously enough, despite this
importance, it is commonly accepted that the Baranov trusses with more than 9 links have not
been properly catalogued yet while all Assur groups with up to 12 links have been identified (see
Table 1) [3]. It is worth mentioning here that Yang and Yao found that the number of Baranov
trusses with 11 links is 239 using an algorithm that certainly requires further attention [6].
While the standard closed-form position analysis leads to complex systems of non-linear
equations derived from independent kinematic loop equations, the bilateration method avoids the
computation of loop equations as usually understood. It has recently been shown to be a
powerful technique by obtaining the characteristic polynomial of the three 3-loop Baranov
trusses without relying on variable eliminations nor half-angle tangent substitutions [7].
Table 1: Number of Baranov trusses as a function of the number of links (alternatively,
number of loops), and number of diﬀerent Assur groups resulting from eliminating one link from
the Baranov trusses in each class [3, 6].

Lin
ks
Loo
ps
Barano
v Resulting
trusses
Assur
groups
3 1 1 1
5 2 1 2
7 3 3 10
9 4 28 173
11 5 239 5442
13 6
unkno
wn 251638
At the end of the XIX century, it was known that there were only two six-link single-dof
planar hinged linkages. At a suggestion of Burmester [8], these two linkages were called the
Watt linkage and the Stephenson linkage. Several Stephenson linkages can be concatenated
leading to what in [9] was called a Stephenson pattern. Likewise, several Watt linkages can be
concatenated to obtain what can be called, for the same reason, a Watt pattern (see [10] for their
motion simulations). If these concatenations are circular, the results are Baranov trusses which
will be called Stepheson-Baranov and Watt-Baranov trusses, respectively (Fig. 1).
The position analysis of the Stepheson-Baranov truss of 4 loops has been solved in closed
form at least in [11, 12, 13, 14], and more recently by K. Wohlhart in [15] thus,
reaching what the author considers to be the limit of Sylvester’s elimination method. The
position analysis of the Watt-Baranov truss of 4 loops was solved in closed form by L. Han et. al.
in [16] and more recently by J. Borr`as and R. Di Gregorio [17]. Elimination methods seem to
reach their limit with the analysis of Baranov trusses with four, or five loops, depending on their
topology. Actually, the closed-form position analysis of a Baranov truss with more than five
loops has not been reported to the best of our knowledge, and only the closed-form position
analysis of one five-loop Baranov truss has been obtained [12, 18]. In this paper, we address this
challenge and we push the loop limit further by solving the closed-form position analysis of
Watt-Baranov trusses, with up to six loops, using the bilateration method.

Figure 1: Left column: The Stephenson linkage, the Stephenson pattern resulting from
concatenating four Stephenson linkages, and the Stephenson-Baranov truss resulting form the
circular concatenation of four Stephenson linkages. Right colum: The Watt linkage, the Watt
pattern resulting from concatenating four Watt linkages, and the Watt-Baranov truss resulting
form the circular concatenation of four Watt linkages.
. In Section 2, the basic formula required to apply the bilateration method is briefly reviewed.
Then, in section 3, it is shown how the bilateration method can be applied to obtain the
characteristic polynomial of a Watt-Baranov truss with an arbitrary number of kinematic loops.
To this end, it is first shown how to derive a single scalar radical equation which is satisfied if,
an only if, the truss can be assemble and, then, how the characteristic polynomial is derived by
simply clearing radicals. This last step is actually the only costly step in the whole process. Two
examples are analyzed in Section 4, including a 6-loop Watt-Baranov truss –whose characteristic
polynomial is of degree 126– with 76 assembly modes.
Synthesis of adjustable spherical four-link mechanisms for approximate
multi-path generation
In spherical mechanisms, the motion all links as well as the coupler path traced by the
mechanism lie on the surface of a sphere and, at any moment, each link of the mechanisms is
part of a great circle on the sphere. In this paper, we deal with the simplest spherical
mechanism with four links and revolute (R) joints (also known in literature as a 4R-spherical
mechanism) with all R joint axes intersecting at the centre of the sphere [1, 2]. Spherical
mechanisms have a wide variety of applications such as spherical wrists [3], surgical robots [4],
apping- wing micro airvehicle [5], grippers [6], in the swiveling fans [7], camera orienting device
[8] (Agile Eye") and space applications [9]. In all these applications, orientation of an object is
the principle requirement, and instead of using complex multi-degree-of-freedom robots, it is
often possible to use a single degreeof-freedom spherical mechanism to perform the orientation
task. Path generation is a classical problem in spherical four-link kinematics. It consists of
designing for linkage parameters such that a given point of the mechanism, usually the coupler
point, follows a prescribed path [1, 10]. There are two types of path generation problems
namely, point-to-point path generation and continuous path generation. In point-to-point path

generation the coupler path is speci_ed by small number of points and the coupler point is made
to exactly pass through all of them. For a spherical four-link mechanism the coupler point can
exactly pass through nine points on the surface of the sphere [11]. In continuous path
generation, the coupler path is specied by large number of points (much more than nine) and
the task is to design the mechanism such that the path traced by the coupler point
approximately passes through all of them. Spherical path generation is a non-linear design
problem which is generally di_cult to solve.
In this paper, we convert the non-linear design problem into a simpler optimization problem and
solve using appropriate numerical techniques. Compared to the extensive work done in
synthesis and design of planar mechanisms, a more modest amount of work has been done in
design of spherical mechanism for point-to-point and continuous
path generation. The design of 4R-spherical mechanisms using instantaneous screw axes
(ISAs) and curve matching techniques are mentioned in the work by Sodhi and co-workers [12,
13]. Synthesis of 4R-spherical path generators using the pole method was done by Tong and
Chiang [14].Spherical four-link mechanisms for _nite positions are synthesized by combining
traditional precision theory with modern approximate position synthesis in work by Bodduluri
and McCarthy [15]. Computer aided design software for 4R-spherical mechanism design based
on Burmester's theory is described in Ruth and McCarthy [16]. Four-link path generators were
synthesized using method based on numerical continuation [17] and constrained least square
optimization [18]. A triangular nomogram for symmetrical coupler curves generated by spherical
four-link crank-rocker mechanisms with special dimensions was presented in the work by Lu
[19]. The harmonic properties of coupler curves have been used to prepare an atlas of spherical
four-link generators to aid mechanism design [20, 21] and optimization based on dierential
evolution algorithm has been used for synthesis spherical 4R mechanism [22]. A computer
aided methodology for the manufacture of spherical mechanisms is discussed in reference [7]
and a review of recent advances and trends in spherical mechanisms research are listed in the
work by Liu and Yang [23]. Adjustable mechanisms are a class of mechanisms in which
different paths (orientations in case of spherical mechanisms) can be achieved by changing one
of the mechanism parameters [24].
Very little work on adjustable spherical mechanism synthesis is available in literature. Adjustable
spherical 4R linkages with _xed ground pivots and adjustable lengths for input and output links
for _ve position synthesis by the use of Burmester curves was proposed by Hong and Erdman
[25].The method can be extended to six position synthesis with adjustable ground pivot
locations. A method based on plane-to-sphere and sphere-to-plane projections was developed
by Lee [24]. Lee et al. [26] describes a least squares minimization technique to synthesize two
phase adjustable spherical mechanisms for approximate path generation and path generation
using adjustable crank-lengths of spherical four-link mechanisms is suggested in [27]. A new
chaos fractal based algorithm for path synthesis of adjustable spherical 4R mechanism is
presented in reference [28]. The synthesis of four-link adjustable mechanisms has been done in
the planar domain by an optimization based two stage process [29]. The _rst stage determines
the driving dyad and the second stage determines the driven dyad. The sequential quadratic
programming (SQP) algorithm [30] is used to search for the optimal design variables which are
the Cartesian coordinates of the joints. in a more recent work, an e_cient two stage optimization
process based on circle _tting has been proposed [31]. A similar kind of optimization based
approach is suggested for synthesis of spherical 4R mechanism in this work. In this paper, a
least squares plane _tting based formulation is suggested.
The paper deals with single adjustment, either on the driven or driving side, in one of the
spherical 4R mechanism parameters (except the crank pivot) to approximately generate multiple
paths. This paper also presents a novel technique to indirectly calculate some of the mechanism
parameters thereby reducing the number of variables required for optimization.

The SQP optimization algorithm involving minimum number of optimization variables is used in
the formulation of objective function for each type of adjustment. To the best of our knowledge
this work presents the _rst attempt in optimal design of adjustable spherical four-link
mechanisms for approximate multi-path generation. The proposed formulation is illustrated with
the help of two examples { one example deals with the generation of an oval and `8' shaped
path similar to the apping motion of a bird wing in forward motion and in the hovering mode.
The paper is organized as follows: In section 2, for the sake of completeness, all the parameters
associated with the spherical four-link mechanism are de_ned and we present a procedure for
calculating the necessary parameters. In section 3, the mechanism synthesis problem needed is
presented and the rationale behind the selection of the adjustment method is presented. In
section 4, examples illustrating our approach are presented and in section 5, conclusions are
presented.
A spherical four-link mechanism
Figure 1: Schematic of a spherical 4R mechanism

The four-link spherical mechanism OABCDP with its parameters is shown on _gure 1. The
mechanism has four revolute joints at A, B, C and D with their axes intersecting at the centerof
the sphere O. The links of the mechanism are the arcs of great circles of the sphere and the
spherical link length is the arc-length measured on the great circle between two ends of the link.
For a sphere of unit radius, the link length is same as the central angle subtended at O by the arc
on the great circle. In _gure 1, AD is the base or _xed link, AB is the crank, BC is the coupler
link, CD is the rocker link, BP is the α5-link and P is the coupler point. In spherical domain all
angles are dihedral angles, i.e., angles are measured between two great circle planes. The line of
intersection of the two circular planes is the axis about which the angle is measured. The variable
β is the coupler angle measured about the axis OB in counter-clockwise direction, ABP is the
driving dyad and DCB is the driven dyad. The crank angle ϕ and the rocker angle ψ are
measured with respect to the base link AD and about OA and OD, respectively. The center of the
sphere is O (0, 0, 0) and x2 + y2 + z2 = 1 is the equation of the sphere. The symbols A(xA, yA,
zA), α2 and α5 denote the driving side parameters and D(xD, yD, zD), α3, α4 and β are the driven
side parameters. The vector rp =OP =[rPx rPy rPz]T is the position vector of point P, and [Tn
δ ] is the rotation matrix, n = [nx ny nz]T is an unit vector corresponding to the axis of rotation
andδ is the angle of rotation about n in counter-clockwise direction. From [32], the rotation
matrix is de_ned as,
The driving crank pivot A(xA, yA, zA) remains unchanged in our approach. The
desired paths are represented by 50 to 1001 points and if lesser number of points
are prescribed then spline interpolation can be used to generate additional points on
the path. The super-script of the mechanism parameter indicates the path to which
it belongs. In planar domain, the workspace of the end-point of a dyad lies between
two concentric circles[32]. Drawing parallels from the planar case, the workspace
of the end-point of a spherical dyad lies between two coaxial spherical small
circles2, i.e., spatial circles about the same axis. All the coupler paths generated by
the mechanism must lie inside the boundaries of the workspace of the dyad where
the boundaries are dependent on the dimensions of the driving dyad ABP. The
dimensions of the driving dyad are chosen such that the workspace boundaries are
tangential to the given coupler paths. It can be seen that the small circles with
spherical radii3 αmax and αmin form the boundary of the workspace of the
driving dyad. Similar to the planar case [31], for a spherical four-link mechanism
the location of the pivot A on the sphere is outside the coupler path if α5 > α2
and inside the coupler path if α5 < α2. This fact helps in choosing _xed pivot A.
For N given points Pi (xPi , yPi , zPi) , i = 1, 2, . . . ,N, on each coupler path, we
define

where (τ, κ) are the spherical polar coordinates of A. The quantity κ is the
polar angle with respect to the +Z-axis and τ is the azimuthal angle in the XY -
plane with respect to the +X-axis. We are designing for 0 < αmax < π. The
quantities α2 and α5 are chosen such that,
For each Pi of the coupler path there is a corresponding Bi and θi. The points Bi
represent the two con_gurations of the crank to reach Pi and θi is the angle
between ABi and APmax where Pmax
is the coupler point on the spherical surface farthest from pivot A (see _gure 2).
There are two possible values of θi, as shown in _gure 2, and these can be
computed as,
θi = αi ± γi (5)
where γi determine the two con_gurations of the crank for a particular Pi. It
should be noted that the angles θ and ϕ (see _gure 1) are two di_erent quantities
although they are related to the crank.
As shown infigure 1, the angle ϕ is between the crank and the _xed link whereas
angle θ is between the crank and APmax. The angles are also with respect to two
dfferent planes { the angle ϕ is with respect to the great circular plane containing
the _xed link AD and centre of the sphere whereas θi is with respect to _xed pivot
A, the sphere centre O and the coupler point P farthest from A.

A notable fact here is that in one crank rotation, the coupler point P crosses αmax
and αmin once, thus dividing the rotation into two parts. The sign of γi in one α
max to αmin part is opposite to that of the remaining αmin to αmax part. Thus
each given coupler path will have two sets of θi and if the direction of rotation is
not speci_ed, appropriate θi must be chosen. For the reference plane,
αPi = αmax, which means that A, B and Pmax lie in the same great circle. For
the path point Pmax,
the corresponding θi = 0 and position vector of B is rBmax. Using equation (1),
we and Bi from

With the above de_nitions and kinematic equations, we next formulate the
adjustable mechanism synthesis problem.

ED7202 mds_notes

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