A Restatement Of The Optimum Synthesis Of Function Generators With Planar Four-Bar And Slider-Crank Mechanisms Examples

60 Int. J. Mechanisms and Robotic Systems, Vol. 3, No. 1, 2016
Copyright © 2016 Inderscience Enterprises Ltd.
A restatement of the optimum synthesis of function
generators with planar four-bar and slider-crank
mechanisms examples
P.A. Simionescu
School of Engineering and Computing Sciences,
College of Science and Engineering,
Texas A&M University-Corpus Christi,
Unit 5733,
6300 Ocean Drive,
Corpus Christi, TX 78412, USA
Email: pa.simionescu@tamucc.edu
Abstract: The problem of optimum synthesising of a mechanism to best
approximate a function, while simultaneously ensuring good motion
transmission characteristics is discussed. Distinction is made between the
design variables that determine the shape of the input-output (I/O) function of
the mechanism, and the design variables that affect the degree of overlap
between this I/O function and the function to be mechanised, through scaling,
mirroring and rotations in 90° increments. Examples are given of designing the
planar four-bar and slider-crank linkages of a logarithmic scale, and that of a
tangent-function generator. These are performed on modified mechanisms with
an added degree-of-freedom, which substantially simplify the synthesis
problem.
Keywords: four bar; function generation; optimisation algorithm; planar
kinematics, slider crank.
Reference to this paper should be made as follows: Simionescu, P.A. (2016)
‘A restatement of the optimum synthesis of function generators with planar
four-bar and slider-crank mechanisms examples’, Int. J. Mechanisms and
Robotic Systems, Vol. 3, No. 1, pp.60–79.
Biographical note: Petru Aurelian Simionescu is an Engineering Faculty at
Texas A&M University in Corpus Christi. He received his BSc from University
Polytechnica of Bucharest, a doctorate in Technical Sciences from the same
university, and a PhD in Mechanical Engineering from Auburn University. He
taught and performed research at several Romanian, British, and American
universities, and worked for 4 years in industry as an Automotive Engineer. His
research interests include kinematics, dynamics and design of multibody
systems, evolutionary computation, CAD, computer graphics, and information
visualisation. So far he has authored over 50 technical papers and has been
granted 7 patents.

Restatement of the optimum synthesis of function generators 61
1 Introduction
Design of function generators like those shown in Figure 1 is a classical mechanism
kinematics problem (Svoboda, 1943, 1948; Freudenstein, 1955; Hartenberg and Denavit,
1964; Tao, 1965; Suh and Radcliffe, 1978; Simionescu and Beale 2002). Depending on
application, the input and output links of such a mechanism can be imposed certain
maximum displacements, or one or both of these displacements can be adjusted during
the synthesis process. Examples from the first category are steering linkages of
automobiles, which must ensure a correlated pivoting of the steerable wheels in
accordance with the condition of correct turning (https://commons.wikimedia.org/
wiki/ File:Ackerman_Steering_Linkage.gif; https://commons. wikimedia.org/wiki/
File:Bell-Crank_Steering_Linkage.gif; https://commons.wikimedia.org/wiki/File:Rack-
And-Pinion_Steering_Linkage.gif). In this case, the input and output members, i.e., the
steering-knuckles of the left and right wheel have imposed limit turning angles, dictated
by the required minimum radius of turn, by the possible interferences of the wheels with
the car body, and in case of front-wheel drive vehicles, by the angular capabilities of the
constant velocity joints of the front axle.
Figure 1 Four-bar function generator that mechanises the function log(u) with 1≤u≤10 (a) and
slider-rocker function generator that mechanises the function tan(u) with 0≤u≤ 45° (b).
These are scale drawings of two of the numerical results discussed in Section 5 of the
paper. See also the animations (https://commons.wikimedia.org/wiki/File:
Func_Geen_Log(u).gif) and (https://commons.wikimedia.org/wiki/File:
TRRR_Func_Geen_Tan(u).gif) (see online version for colours)

62 P.A. Simionescu
The scale mechanisms in Figure 2 belong to the second category. Here, the transmission
ratio between the rack or gear sector and its pinion can be established post synthesis, once
the geometry of the base linkage has been established. This allows the maximum travels
of the input and/or output links to be included among the design variables in the synthesis
problem.
Figure 2 Logarithmic scales consisting of a linkage in series with one or two gear amplifiers.
Scale (a) employs a four-bar, scale (b) employs a slider-rocker, and scale (c) employs a
rocker-slider. Each linkage corresponds to numerical examples discussed in Section 5
In this paper, a revised formulation of the problem of synthesising four-bar and slider-
crank linkages for the generation of functions, using optimisation techniques is presented.
Distinction will be made between the design variables that influence the I/O function of
the mechanism, and the design variables that affect the relative disposition and the degree
of overlap between the graph of the function to be generated and the said I/O function,
through scaling, mirroring, and rotations in 90° increments. The invariances of the I/O
function of the base mechanism with respect to these transformations will also be
observed for a more effective problem formulation. Additional simplifications will be
achieved by defining objective functions that utilise modified mechanisms with
extensible couplers. Such objective functions are easier to formulate, take less central

processor unit (CPU) time to evaluate, and additionally provide an extended search space
to the optimisation problem (Simionescu and Beale, 2002).
The idea of using during synthesis of a modified four-bar linkage with a variable-
length coupler was first proposed by Artobolevsky, Levitskii and Cercudinov (1959).
Others researchers have used modified mechanisms with an added degrees of freedom to
synthesise four-bar and six-bar (Stephenson II, Stephenson III, and Watt II) planar
mechanisms, as well as spatial four-bar and spatial slider-rocker mechanisms (Levitskii,
Sarkissyan and Geckian, 1972; Suh and Mecklenburg, 1973; Alizade Mohan Rao and
Sandor, 1975; Avilés, Amezua and Hernandez, 1994; Simionescu and Alexandru, 1995;
Simionescu, Smith and Tempea, 2000; Simionescu and Talpasanu, 2007; Avilés et al.,
2010; Collard, Duysinx and Fisette, 1910).
2 Function generation seen as a curve fitting problem
The problem of synthesising a function generating mechanism has been tackled by many
kinematicians in the past (Svoboda, 1943, 1948; Freudenstein, 1955; Hartenberg and
Denavit, 1964; Tao, 1965; Suh and Radcliffe, 1978; Simionescu and Beale, 2002; Plecnik
and McCarthy, 2011; Mehar, Singh and Mehar, 2015). It is suggested here for the first
time to describe the approximation of a function by a linkage as a curve-fitting problem.
Figure 3(a) shows the plot of the function f(u) = log(u) with 1≤u≤10 that is supposed to
be approximated by some type of linkage mechanism, in particular a four-bar linkage.
For a given set of geometric parameters, the I/O function ψ(ϕ) of the mechanism can
have the shape in Figure 3(b). In the same diagram it is shown in dashed line the variation
of the pressure angle γ, defined as the angle between the velocity vector of the floating
joint of the output-member, and the reaction force delivered to that joint in the absence of
any gravitational or inertia forces.
Figure 3 Synthesis for the generation of functions can be described as best fitting the graph of the
prescribed function (a) with a portion the graph of the I/O function of a mechanism (b)
(see online version for colours)

64 P.A. Simionescu
In case of planar four-bar and slider-crank mechanisms, more frequently used is the
transmission angle, i.e., the complement about 90° of the pressure angle (Volmer and
Jensen, 1962; Balli and Chanda, 2002; Söylemez, 2002). The transmission angle however
does not have a direct equivalent in case of spatial linkage mechanisms (Simionescu,
1999), which can be considered a disadvantage. To avoid joint jamming, pressure angle γ
should not depart more than ±50° from the ideal value of zero, with deviations close to
±60° being sometimes considered acceptable (Hartenberg and Denavit, 1964; Suh and
Radcliffe, 1978).
With reference to Figures 3–6, the synthesis problem can be stated as follows: find a
portion of the I/O curve ψ(ϕ) of the mechanism that closely fits the graph of the function
f(u) to be mechanised. For proper motion transmission characteristics, the portion of the
I/O curve utilised should additionally be associated with favourable pressure angle
values. In the process of best fitting, the graph of the function f(u) with the I/O graph of
the mechanism ψ(ϕ), the following geometric transformations can be applied:
1 horizontal translation, i.e., modify ϕs
2 vertical translation, i.e., modify ψs
3 horizontal scaling, i.e., modify Δϕ
4 vertical scaling, i.e., modify Δψ
5 mirroring about a vertical axis (i.e., change the sign of Δϕ), which is equivalent with
driving the input link in reverse
6 mirroring about a horizontal axis (i.e., change the sign of Δψ), which is equivalent
with switching between the forward and the return strokes of the mechanism, or
changing the closure in case of single loop mechanisms
7 rotations in plane by ±90°. This is equivalent with swapping Δϕ and Δψ (i.e., making
the input link output, and vice versa), followed by changing the sign of either one as
explained at nos. 5 and 6 above.
Note that throughout the paper, index ‘s’ designates the initial position, and index ‘f’
designates the final position, while Δϕ = ϕf − ϕs is the motion range of the input link and
Δψ = ψf − ψs is the motion ranges of the output link: Evidently, in case of the crank-
slider and slider-crank mechanisms, the I/O function ψ(ϕ) will instead be S(ϕ) and ψ(S)
respectively, and the slider motion range will be ΔS = Sf − Ss.
Consequently, it can be distinguished between two types of design variables:
• Variables that describe the disposition and size of the graph of the prescribed
function f(u) relative to the I/O function of the mechanism. These are the
i initial position of the input and output links ϕs and ψs
ii motion ranges Δϕ and Δψ
iii signs of Δϕ and Δψ (positive or negative)

• Variables that determine the I/O function of the mechanism. In general, these are the
i link lengths
ii ground-joint location
iii fixed angles of ternary and polynary links
iv orientation (closure) of mechanism loop(s), which for single loop mechanisms it
is equivalent with mirroring the entire mechanism and driving it in reverse.
Motion ranges Δϕ and Δψ will determine the values of the scale factors K1 and K2
(Simionescu, 1999).
f s f s
1 2
f s f s
and
( ) ( )
K K
u u F u F u
φ φ ψ ψ
− −
= =
− −
(1)
where K1 relates a current displacement ϕ of the input member of the mechanism to the
independent variable u, and K2 relates the corresponding output members displacement
ψ(ϕ) to the function value f(u) according to the following equations:
s 1 s
( ) ( )
u K u u
φ φ
= + ⋅ − (2)
( )
i 2 s
( ) ( ) ( )
u K f u f u
ψ ψ
= + ⋅ − (3)
One can notice that for topologically symmetric mechanisms like the four-bar linkage,
swapping Δϕ and Δψ will not reveal additional solutions when a non-monotonic function
f(u) is to be mechanised - see the dashed line graphs in Figures 4(b), (c), and (e). Second,
for functions with symmetric graphs, like odd or even functions over symmetric intervals
about the origin, the transformations shown in thin line in Figures 4(c), (d), and (e) are
redundant, and will not reveal additional solutions either. For non-monotonic functions
like Figure 4(b), (c), and (e), the input-link motion range should be extended to include
both extrema of the I/O function of the mechanism.
As Figures 5 and 6 show, the I/O functions of the planar crank-rocker and crank-
slider mechanisms exhibit invariances with respect to mirroring the entire mechanism
about the OX and OY axes. These are two-by-two equivalent with changing the closure
of the mechanism loop. Versions of the same mechanisms that do not allow a full rotation
of their input links (the non-Grashof linkages) exhibit similar properties with respect to
changing the loop closure.
Since the I/O function of any mechanism with rotational input and output links
(including the four-bar linkage) is a scaling invariant, its link lengths will be normalised
with respect to the input link length, i.e., AB will be assumed equal to one. For the slider-
crank mechanism, a similar normalisation will be applied during the synthesis process,
but with respect to the slider travel range, i.e., ΔS will be assumed equal to one. In case of
the four-bar mechanisms, more common is to normalise the ground link (Freudenstein,
1955), which has the drawback of the search converging to degenerate mechanism
solutions with zero input and output link lengths (Simionescu and Beale, 2002).

66 P.A. Simionescu
Figure 4 Scale, mirror, and rotation transformations applicable in the search for a best
fit between the graph of the function f(u) and a portion of the I/O function of the
mechanism; (a) and (d) correspond to monotonic functions, (b), (c), and
(e) correspond to non-monotonic functions, (c), (d), and (e) correspond to functions
having symmetric graphs over the considered intervals, while (a) and (b) have no
symmetries

Figure 5 Invariances of the I/O function ψ(ϕ) and of the pressure angle γ(ϕ) of a crank-rocker
mechanism

68 P.A. Simionescu
Figure 6 Invariances of the transmission function ψ(s) and pressure angle γ(s) of a crank-slider
mechanism
Awareness of the properties discussed above allows one to avoid using redundant design
variables, and also allows to best select the side constraints in an optimisation problem. In
addition, it was found that it is better to specify a reference point (ϕ0,ψ0), with ψ0
calculated using Eq. (3), instead of prescribing the length of the coupler (Simionescu and
Beale, 2002). Such a reference point, which will correspond to an exact point within the
working range of the mechanism, can sometimes be explicitly imposed, for example as
the initial position of the mechanism. The reader is certainly aware that 16 or more
significant digits are used internally by the computer, and any real mechanism with
rounded-off dimensions will not satisfy precisely this reference point. Also the I/O
function of the mechanism will deviate from the one obtained through synthesis. To

eliminate such limitations, in the numerical implementations discussed in this paper,
certain design variables will be restricted right from within the optimisation algorithm to
only a small number of decimals.
3 Optimum synthesis of the planar four-bar linkage for the generation of
functions
Figure 7 depicts a planar four-bar mechanism with unit rocker length (i.e., OA = 1),
equipped with a variable-length coupler AB. This allows any correlation (ϕj,ψj) of the
input and output links to be exactly satisfied by modifying the coupler length in the
amount δABj.
Synthesising an actual four-bar mechanism for the generation of functions requires
minimising the deviation δψ between the imposed output-link motion given by Eq. (3),
and the actual I/O function ψ(ϕ(u)), where ϕ(u) is calculated with Eq. (2). At a control
point uj within the interval [ui…uf] to be mechanised, this deviation will be:
j j j
( ( )) ( ( ))
u f u
δψ ψ φ ψ
= − (4)
For the modified linkage in Figure 7, a change in length δABj = *
j
AB − ABj is
necessary in order to cancel the error δψj.
Figure 7 Variable length coupler four-bar mechanism shown in a prescribed position j.
*
j
R is the
reaction force inside joint B which is collinear with link AB, and
*
j
V is the velocity
vector of joint B
An objective function to be minimised that is equal to the maximum relative error of a
four-bar mechanism with rigid coupler is defined as follows:
j
1
( )
(...) max 100%
j n
u
F
δψ
ψ
=
= ⋅
Δ
…
. (5)
where j = 1…n are discrete control points. Objective function F(…) has the drawback
that it requires a complete displacement analysis of the mechanism for every control
point j, being in addition not defined in the positions where the mechanism cannot be
assembled.

70 P.A. Simionescu
Based on kinematic or energy-conservation considerations, it can be shown that the
coupler-length variation δABj is related to the exactly calculated output error δψj
according to the following approximate relation (Simionescu and Beale, 2002):
( )
* *
j j j j
AB AB cos
δψ δ γ δψ
≅ ⋅ = (6)
where *
j
γ is the pressure angle at the joint B for the mechanism with adjusted coupler
length. Using this approximation, an alternative objective function is defined, i.e.,
j
1 *
1
j
AB 1
(...) max 100%
AB cos
j n
F
δ
ψ
γ
=
= ⋅ ⋅
Δ
⋅
…
. (7)
which is very easier to evaluate, and in addition returns real values for any point of the
design space, less for Δψ=0. Angle *
j
γ occurring in Eqs. (6) and (7) can be determined by
subtracting 90° from the angle formed by vectors *
j
AB and *
j
CD associated to the
respective links (Figure 7):
* * * * *
j j j j j
90 arctan
γ = ° − × ⋅
AB CB AB CB (8)
Because in a synthesis problem angles γ*
are calculated for the adjusted-length coupler,
Eq. (8) provides only an approximation to the actual pressure angle. As the search
progresses towards the minimum of F1 however, this approximation of γ will be
increasingly accurate, and likewise F1 will approximate better the exactly calculated error
function F. This means that during the optimisation process γ*
can be used confidently to
verify the imposed limits upon the pressure angle of the mechanism.
The seven parameters that can be adjusted during the optimum synthesis process of
the four-bar (i.e., the variables of the objective functions F and F1) are: link lengths BC
and OC; maximum displacement of the input link Δϕ (positive only); maximum
displacement of the output link Δψ (either positive or negative); initial angles ϕs and ψs
of the input and output links respectively; a number k between 0 and 1 defining an exact
point (ϕ0,ψ0), with ϕ0=ϕs+k⋅Δϕ and ψ0 calculated with Eq. (3). This exact point (ϕ0,ψ0)
will serve to calculate the coupler length AB.
4 Optimum synthesis of planar slider-crank linkages for the generation of
functions
The above derivations valid for the four-bar linkage can be extended to the synthesis of
slider-rocker and rocker-slider function generators (Figures 8 and 9). In these cases, the
link length of the mechanism will be normalised relative to the slider travel, i.e., ΔS will
be assumed equal to one.

Figure 8 Planar slider-rocker mechanism with a variable coupler AB in an arbitrary position j
There are six parameters that can be adjusted during the optimum synthesis process of the
slider-rocker mechanism (Figure 8) as follows: slider eccentricity e; length BC; distance
OC; maximum displacement Δψ and initial angle of the rocker ψs; and a number k
between 0 and 1 serving to define an exact point (S0,ψ0), with S0=SS+k⋅ΔS and with ψ0
calculated using Eq. (3). Same as above, the exact point (S0,ψ0) will be used to determine
the coupler length AB. Of these design variables, e, OC, and Δψ can be either positive or
negative. The objective function similar to F1 employed in synthesising the slider-rocker
function generating mechanism will be:
j
2 *
1
j
AB 1
(...) max 100%
AB cos
j n
F
δ
ψ
γ
=
= ⋅ ⋅
Δ
⋅
…
. (9)
where angle *
j
γ is calculated using Eq. (8).
Figure 9 Planar rocker-slider mechanism with a variable coupler AB shown in an arbitrary
position j, where
*
j
R is the force delivered by coupler AB, and
*
j
V is the velocity
vector of the slider

72 P.A. Simionescu
The design variables of the rocker-slider function generator mechanism in Figure 9 (also
six in number) have been chosen as follows: slider eccentricity e; rocker length OA;
coordinate xBs from where the slider displacement S is measured; maximum input-link
displacement Δϕ (either positive or negative) and its initial angle ϕs; and a number k
between 0 and 1 that defines the exact point (ϕ0,S0) with ϕ0=ϕs+k⋅Δϕ and S0=Ss.
( )
s 2 s
( ) ( ) ( )
S u S K f u f u
= + ⋅ − (10)
Similarly, the exact point (ϕ0,S0) will serve to calculate the coupler length AB. The
variable-length-coupler-based objective function will be in this case:
j
3 *
1
j
AB 1
(...) max 100%
cos
j n
F
S
δ
γ
=
= ⋅ ⋅
Δ
…
. (11)
where the approximate pressure angle *
j
γ is the angle formed by vectors *
j
AB associated
to the deformed coupler AB, and a horizontal unit vector i.e.
* * *
Bj j j
90 arctan (1,0)
γ = °− ×
AB AB (12)
5 Numerical results
Objective functions F1, F2, and F3 with n = 120 control points subjected to ⎪ *
max
γ ⎪ ≤ 45°
have been minimised using a multi-start Nelder and Mead algorithm (Press et al., 2007).
Knowing that any practical function-generator will have rounded-off dimensions, all
linear design variables have been limited to only three decimals right from within the
search algorithm (Simionescu, 2014). Similarly, the angular motion ranges Δϕ and/or Δψ
have been rounded to a multiple of 0.5° by the search algorithm. A post-synthesis
analysis has been finally performed to evaluate the exactly calculated relative error, as
well as the exactly calculated pressure angle γ.
In case of objective function F1, the limits of the design variables and the optimum
solutions found through optimisation for Δϕ > 0 and Δψ > 0 and Δϕ > 0 and Δψ < 0 are
summarised in Table 1. These mechanism solutions are drawn at scale in Figures 1 and
2(a). Their kinematic and performance diagrams are shown in Figure 10.
Table 1 Search domains and solutions of the optimum four-bar generator of function log(u)
with 0 ≤ u ≤ 10 and ⎪
*
max
γ ⎪ ≤ 45°
The mechanism in Figures 1 and 10(a) The mechanism in Figures 2(a) and 10(b)
OA 1.0 OA 1.0
0.1 ≤ OC ≤ 2.5 0.451 0.1 ≤ OC ≤ 2.5 2.123
0.1 ≤ BC ≤ 2.5 0.630 0.1 ≤ BC ≤ 2.5 0.671
0 ≤ ϕs ≤ 360° 8.9625° 0 ≤ ϕs ≤ 360° 83.5930°
0 ≤ ψs ≤ 360° 96.7979° 0 ≤ ψs ≤ 360° 276.1691°
60 ≤ Δϕ ≤ 120° 115.50° 60°≤ Δϕ ≤ 120° 86.00°
60 ≤ Δψ ≤ 120° 70.50° –120°≤ Δψ ≤ 60° –60.00°

Table 1 Search domains and solutions of the optimum four-bar generator of function log(u)
with 0 ≤ u ≤ 10 and ⎪
*
max
γ ⎪ ≤ 45° (continued)
The mechanism in Figures 1 and 10(a) The mechanism in Figures 2(a) and 10(b)
0 ≤ k ≤ 1 0.02253 0 ≤ k ≤ 1 0.39563
AB 0.783 AB 2.647
Approximately maximum
error (F1)
2.162% Approximately maximum error
(F1)
3.504%
Exact maximum error 2.305% Exact maximum error 3.776%
⎪
*
max
γ ⎪ ≤ 45° 44.434° ⎪
*
max
γ ⎪ ≤ 45° 44.729°
⎪γmax⎪ 45.512° ⎪γmax⎪ 43.020°
Figure 10 Kinematic and performance diagrams of the four-bar function generator in Figures 1
and 2(a). The dashed ψ(ϕ) curves are the ideal I/O functions, while the dashed error
and pressure angle γ curves are the approximate ones (see online version for colours)

74 P.A. Simionescu
The minimisation of objective functions F2 and F3 yielded the solutions summarised in
Table 2. The corresponding rocker-slider and slider-rocker mechanisms drawn at scale
are those shown in Figures 2(b) and 2(c), while their kinematic and performance
diagrams are available for comparison in Figures 11.
Table 2 Search domains and of solutions of the optimum slider-rocker and rocker-slider
generators of function log(u) with 0 ≤ u ≤ 10 and⎪
*
max
γ ⎪ ≤ 45°
The mechanism in Figures 2(b) and 11(a) The mechanism in Figures 2(c) and 11(b)
ΔS 1.0 ΔS 1.0
Ss 0.0 Ss 0.0
−2.5 ≤ OC ≤ 2.5 0.215 0.2 ≤ OA ≤ 2.5 1.754
0.5 ≤ BC ≤ 2.5 0.707 −1.5 ≤ yB ≤ 1.5 −0.277
−1.5 ≤ yA ≤ 1.5 −0.599 0 ≤ xBs ≤ 360° −1.118
0 ≤ ψs ≤ 360° 318.095° 0 ≤ ϕs ≤ 360° 280.803°
60°≤ Δψ ≤ 120° 60.00° 60°≤ Δϕ ≤ 120° 117.50°
0 ≤ k ≤ 1 0.03400 0 ≤ k ≤ 1 0.79284
AB 0.785 AB 2.034
Approximately maximum error
(F2)
5.698%
Approximately maximum error
(F3)
1.615%
Exact maximum error (F) 6.427% Exact maximum error (F) 1.619%
⎪
*
max
γ ⎪ ≤ 45° 44.193° ⎪
*
max
γ ⎪ ≤ 45° 44.984°
⎪γmax⎪ 43.017° ⎪γmax⎪ 45.306°
The aforementioned numerical experiments have been repeated for the same objective
functions F1, F2, and F3, same number of control points n and same side constraints, but
for ⎪ *
max
γ ⎪ ≤ 60°. The results obtained are gathered in Table 3. The rocker-slider
mechanism exhibits again the best performance, being capable of generating the function
log(u) over the interval 1≤ u ≤10 with a maximum relative error of only 0.037% (Figures
12 and 13 and animation (https://commons.wikimedia.org/wiki/File:RRRT_Func_Geen_
Log(u).gif)). Interestingly, the number of precision points is only two, contrary to the
expectation that higher precision comes with an increased number of precision points
(Mehar, Singh and Mehar, 2015). Another observation is that the search domain of the
design variable k used to calculate coupler length AB can be reduced from [0…1] to only
[0…0.5] since at least two exact points of the function generating mechanism are
expected to occur in this reduced interval.

Figure 11 Kinematic and performance diagrams of the optimum slider-rocker and rocker-slider
function generators in Figures 2(b) and 2(c). The dashed ψ(ϕ) and S(ϕ) curves are the
ideal I/O functions, while the dashed error and pressure angle γ curves are the
approximately calculated ones (see online version for colours)
Table 3 Optimum four-bar, slider-rocker and rocker-slider generators of function log(u) with
0 ≤ u ≤ 10 and ⎪
*
max
γ ⎪ ≤ 60°
Four-bar mechanisms Slider-rocker mechanism
Rocker-slider mechanism
(Figures 12 and 13)
OA = 1.0 OA = 1.0 ΔS = 1.0 ΔS = 1.0
OC = 0.699 OC = 1.46100 Ss = 0.0 Ss = 0.0
BC = 1.028 BC = 0.677 OC = 1.34100 OA = 2.35400
ϕs = 32.48771° ϕs = 61.5497° BC = 0.51700 yB = –0.49700
ψs = 112.91018° ψs = 273.2961° yA = 0.53400 xBs = –1.15400
Δϕ = 118.50° Δϕ = 96.5° ψs = 178.01804 ϕs = 152.23159
Δψ = 63.00° Δψ = –74.0° Δψ = 60.00° Δϕ = 90.00
AB = 0.68500 AB = 1.85600 AB = 0.98500 AB = 1.84500
F1 = 0.5981% F1 = 1.340% F2 = 4.587% F3 = 0.037%
F = 0.5939% F = 1.383% F = 4.664% F = 0.037%
⎪ *
max
γ ⎪ = 9.86° ⎪
*
max
γ ⎪ = 9.94° ⎪
*
max
γ ⎪ = 59.93° ⎪
*
max
γ ⎪ = 59.764°
⎪γmax⎪ = 59.75° ⎪γmax⎪ = 59.29° ⎪γmax⎪ = 56.16° ⎪γmax⎪ = 59.747°

76 P.A. Simionescu
Figure 12 Kinematic and performance diagrams of the optimum slider-rocker and rocker-slider
function generators in Figure 13. Note that the ideal and the actual I/O curves, as well
as the exact and the approximately calculated error and pressure angle curves are near
identical (see online version for colours)
Figure 13 A rocker-slider function generating mechanism designed to mechanise the function
log(u) with 1≤ u ≤10. See also animation (https://commons.wikimedia.org/wiki/File:
RRRT_Func_Geen_Log(u).gif) (see online version for colours)
An additional set of numerical experiments have been performed for a comparison with
recently reported results obtained using a precision-point method (Mehar, Singh and
Mehar, 2015). These experiments consisted of synthesising planar four-bar, slide-rocker,

and rocker-slider mechanisms for the generation of the function tan(u) over the interval
0 ≤ u ≤ 45°, while imposing the pressure angle not to exceed 60°. The results obtained are
summarised in Table 4 and in Figure 14. The slider-rocker mechanism provided the
best solution, with a maximum relative error of only 0.00645%. Figure 1(b) and
animation (https://commons.wikimedia.org/wiki/File:TRRR_Func_Geen_Tan(u).gif) are
scale representations of this slider-rocker mechanism.
Table 4 Optimum four-bar, slider-rocker and rocker-slider generators of the function tan(u)
with 0 ≤ u ≤ 45° and ⎪
*
max
γ ⎪ ≤ 60°
Four-bar mechanism Slider-rocker mechanism Rocker-slider mechanism
OA = 1.0 ΔS = 1.0 ΔS = 1.0
OC = 2.063 Ss = 0.0 Ss = 0.0
BC = 0.517 OC = 1.77600 OA = 2.37600
ϕs = 207.17832° BC = 0.94600 yB = −0.76100
ψs = 162.61021° yA = −0.51000 xBs = 0.02400
Δϕ = 60° ψs = 356.00137 ϕs = −178.10006°
Δψ = −64° Δψ = 74.5° Δϕ = 61.5°
AB = 2.534 AB = 1.6000 AB = 2.49300
F1 = 0.02530% F2 = 0.006447% F3 = 0.087767%
F = 0.02532% F = 0.006449% F = 0.087789%
⎪ *
max
γ ⎪ = 58.65° ⎪
*
max
γ ⎪ = 59.93° ⎪
*
max
γ ⎪ = 33.1469°
⎪γmax⎪ = 58.66° ⎪γmax⎪ = 56.16° ⎪γmax⎪ = 33.1570°
Figure 14 Variation with input link displacement of the relative errors and pressure angles of the
four-bar (curves 1), slider-rocker (curves 2) and rocker-slider (curves 3) optimised
generators of the function tan(u) with 0 ≤ u ≤45° (see online version for colours)

78 P.A. Simionescu
6 Conclusions
The planar four-bar, slider-rocker, and rocker-slider mechanisms have been synthesised
for function generation using optimisation techniques. Distinction has been made
between the design variables that determine the shape of the input-output function of the
mechanism, and the design variables that affect the degree of overlap between this and
the function to be mechanised. The latter parameters include the motion ranges and
relative motion of the input and output links. For added simplicity, adjustable-coupler
mechanisms that have an added degree-of-freedom were assumed in the definition of the
respective objective-function. Such objective functions come with the benefit of an
extended design spaces, and are less CPU time intensive in comparison with full
kinematic-analysis-based objective functions. Also accounted for during the synthesis
process was the effect of rounding off of the linear and angular design variables, inherent
to any practical implementation of the respective mechanism solutions. A number of
examples for the generation of logarithm and tangent functions have been presented. The
optimum planar rocker-slider and slider-rocker linkages (which have been overlooked in
the past in favour of the four-bar linkage), emerged as particularly good solutions.
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