Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics

Abstract

As can be seen by the results obtained by calculations as well by experiments on the prototypes, four-bar mechanisms can be rearranged in rhombus and kite configurations which lead to very performant prismatic pairs allowing to design very fast linear actuators. Moreover, to improve performance and to reduce emcumbrance, networking the rhombus four-bars can lead to very good results. In the author’s knowledge, this is the first time that four-bars were envisaged to be applied as linear actuators. The next step will be to analyze their dynamics design integrating force analysis. Then, to design a large scale parallel robot prototype will help investigate their worthiness towards the design a very high speed milling machine. Finaly, several optimization problems may arise to determine proper linkage sizing.

Full text

6
Kinematics Synthesis of a New Generation of
Rapid Linear Actuators for High Velocity
Robotics with Improved Performance
Based on Parallel Architecture
Luc Rolland
Ecole Normale Supérieure des Arts et Métiers, Metz
France
1. Introduction
This article studies several classes of linear actuators based on parallel topology featuring
lower mobility.
Translation actuator design represents a very important issue in manipulator design in areas
like machine tools for example and more recently hexapods. Actual designs are usually
limited to low accelerations actually limited to 2 g. Moreover, alignment problems are
difficult to circumvent and usually lead to non-uniform friction in the translation motion
refered as hard spots. Despite important breakthroughs, linear motors are still limited to
accelerations of 5g and they are plagued by problems such as surrounding magnetisation
and limited torque. As for any parallel mechanisms, the proposed architectures do provide
for a more rigid linkage. Their rigidity advantage leads to larger actuator bandwidth,
thereby allowing for increased accelerations which result in larger forces being applicable to
the extremity while keeping overall mass very low. The main disadvantage will be their
transverse emcumbrance which will be minimized through mechanism networking.
Two diamond and one rhombus configurations have been designed, analyzed, constructed
and compared verifying their ability for very fast accelerations. Their kinematics are
investigated allowing to write the forward and inverse problems for position, velocity and
accelerations where closed-form solutions are determined. Motion limitations and
singularity analysis are also provided from which configuration recommendations can be
derived. These actuators will then be easily controllable despite their non-linear nature.
In parallel manipulators, the prismatic pairs are usually encountered as the linear actuators
for several architectures such as the planar 3RPR, the general Gough platform, (Gough &
Whitehall, 1962; Fichter, Kerr and Rees-Jones) and the Kanuk (Rolland, 1999) for examples.
These prismatic actuators, may they be guided or not, do play a very important role in
robotics design and their performance has been an issue. According to the author’s
obsevration on several high speed milling projects, these actuators have been hampering the
advent of high speed milling by being unable to provide for adequate accelerations in low
inertia and high rigidity packages.

Advanced Strategies for Robot Manipulators
108
(a) Unimate (b) Versatran (c) Stanford Arm
Fig. 1. Cylindrical robots
Typically, contemporary linear actuators have generally evolved in devices which can be
classified under the nine following categories:
• Piston in a Cylinder or diaphragm being driven by a fluid
• Linear motors
• machine screw and nut
• A worm gear and screw
• Rack and pinion
• Belts and pulleys
• Cam and plunger
• Crank-slider
• Linkages
Lets consider Euclid’s definition of a straight line: ”A straight line is a line which lies evenly
with the points on itself”, (Euclid, 2002).
A straight line mechanism is defined as a mechanism that generates a straight-line output
motion from an input actuator which rotates, oscillates, or moves in a straight line.
Inventing a straight line mechanism, referred as SLM hereafter, has been the concern of
many researchers and engineers long before the industrial revolution. The use of linkages as
SLM can be traced as far back as in the XIII century when sawmill drawings showed
mechanisms for changing circular motion to straight-line motion. Even Da Vinci himself has
drawn one mechanism to convert rotation to translation having slides acting as guides
(DaVinci, 1493). Door locking mechanisms are other old examples where the rotation of the
key was converted into translation motion of the lock element.
In 1603, Christopher Scheiner invented the pantograph, (Scheiner, 1631). It may be regarded
as the first example of the four-bar linkage. The pantograph is a device for copying and
enlarging drawings. Knowing that the actuator is located to one end, this device can be
made to move on a straight-line providing that the input follows a straight-line, therefore
becoming a pure amplification linkage.
In the late seventeenth century, it was extremely difficult to machine straight line and flat
surfaces. Knowing that prismatic pair construction without backlash had become an
important and difficult challenge, much effort was then diverted towards the coupler curve
of a linkage comprising only revolute joints which were much easier to produce.

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
with Improved Performance Based on Parallel Architecture
109
Fig. 2. Typical linear actuator
Later, James Watt proposed a four-bar mechanism generating motion approximating
roughly a straight-line. We would have to wait until 1864 when Peaucellier introduced the
first planar linkage capable of transforming rotary motion into exact straight-line motion.
Until this invention, no planar mechanism existed for producing straight-line motion
without reference guideways which could not be made very straight themselves.
It was soon followed by the grasshopper linkage which also provided for an exact straight-
line. These mechanisms were essential in the development of steam engines and machine
tools. Then, Hart’s Linkage and A-frame both reduce the link number to only five.
The Kmoddl library from Cornell University presents 39 linkages imagined to produce
linear motion which come from Franz Reuleaux Collection of Kinematic Mechanisms
(Reuleaux, 1876; Moon, 2007). Most of them feature relatively complex architectures where
linkages cannot easily be practically applied in systems such as robots or milling machines.
Several proposals were patented trying to simplify the linkage producing straight-line
motion. In the class of nearly straight line linkages, one can identify several linkages by
inventors such as Hoekens, Chebyshev, Evans, Roberts and Burmester. With appropriate
linkage dimensions, part of the motion can be a straigth line. Hoekens linkage can be
considered a Cognate linkage of the Chebyshev linkage since it produces a similar motion
pattern. These simpler designs always applied the properties of special points on one of the
links of a four-bar linkage. They could often produce straight-lines over some limited range
of their motion. The commonality of all these ingenious mechanisms is in the fact that they
feature linkages based on closed loops or so-called parallel topology.
Very early on, the designers were faced with the fact that a prismatic pair or joint is much
more difficult to build than a revolute joint. This is even more the case when trying to have a
linear actuator, (Soylemez, 1999). This observation holds on even today.
Practically, parallel mechanism architectures have been able to provide solutions to
industrial problems and needs with improved performance manipulators.
Theoritically, they may even improve accuracy but this is still an open problem at the
moment, especially when control in concerned.
When the proposed configurations allow to bring the actuators fixed on or jointed to the
base, the inertia of mobile elements can be significantly reduced so the extremity or end-
effector can move at higher accelerations resulting in the deliverance of larger forces.

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110
Fig. 3. Scheiner pantograph
The principal drawback which shall be studied is the construction of kinematics models
characterized by non-linear equations where an implicit relationship is produced between
the manipulator configuration parameters, actuator joint positions and the end-effector
position and orientation.
The simplest forms of parallel manipulators are the ones producing one degree-of-freedom.
Performance evaluation for these single DOF mechanisms includes the four following
criterias:
• workspace
• singularity avoidance
• linkage ecumbrance
• linearity in motion transmission
In a one DOF problem, the workspace criteria is then reduced to a simple range with two
extremum values te be determined: the minimal and maximal positions.
Moreover, the design of machine-tools based on parallel robots have been concerned by
problems related to inherent difficulties with prismatic actuator designs which have
hampered their successful implementation.
This paper original contribution is in the design of a new generation of linear actuators for
increased performance where planar parallel linkages are applied. For example, if
implemented to replace classical linear actuators on Gough platform or even planar 3RPR
manipulators, they allow to bring the motors in positions directly jointed to the base.
The introduction is followed by chapter on kinematic topology synthesis consisting of a
review of various kinds of mechanisms to provide straight-line motion where mobility is

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
with Improved Performance Based on Parallel Architecture
111
analyzed. Chapter 3 is dedicated to the kinematics analysis of several promising alternatives
based on the four-bar mechanism. Then, chapter four investigates the selected performance
criterias. This paper closes on a design chapter where prototypes are shown with motion
analysis in terms of position, velocity and acceleration.
2. Kinematics topology synthesis
Firstly, in this section, we shall make a review of some interesting planar mechanisms which
can perform the specified set of functionnal requirements. In this case the tasks shall be to
achieve straight-line motion.
2.1 Background study
We need two definitions related to degree-of-freedoms.
The DOF of the space is defined as the number of independant parameters to define the
position of a rigid body in that space, identified as
λ
.
The DOF of a kinematic pair is defined as the number of independant parameters that is
required to determine the relative position of one rigid body with respect to the other
connected rigid body through the kinematic pair.
The term mechanism is defined as a group of rigid bodies or links connected together to
transmit force and motion.
Mobility and kinematics analyses are possible under some assumptions:
• Ideal mechanisms with rigid bodies reducing the mechanism motion to the geometric
domain.
• Elastic deformations are neglected
• Joint clearance and backlash are insignificant
2.2 Functionnal requirements
Historically, the need for straight-line motion has resulted on linkages based on closed loops
or so-called parallel topology. The idea is to convert rotation motion into translations or
straight-line motions. It is usually considered that prismatic pairs are much harder to build
than revolute joints, (Soylemez, 1999).
Prismatic actuators as well as slides have the following problems:
• the side reactions of prismatic pairs produce friction leading to wear
• these wears are uneven, non-uniform and unpredictable along the path of the slide
since the flat surfaces in contact are not well defined due to construction imperfections.
Some mechanisms are designed to generate a straight-line output motion from an input
element which rotates, oscillates or moves also in a straight line.
The kinematic pair DOF is defined as the number of independent parameters necessary to
determine the relative position of one rigid body with respect to the other connected to the
pair, (Soylemez, 1999).
The linkages are designed to generate motion in the plane and are then limited to three
DOFs, therefore the only available joints are either with one or 2 dofs only.
The actual problem is addressed from a robotics or even machine-tool point of view. It can
be summarized by this question: how can you draw a straight line without a reference edge?
Most robotics manipulators or machine tools are applying referenced linear motions with
guiding rails and even now linear motors. In design of parallel manipulators such as 3RPR

Advanced Strategies for Robot Manipulators
112
or Gough platforms, the actuators have especially to generate straight lines without any
guiding rails.
This question is not new and it actually comes from the title taken from the book written by
Kempe, where he describes plane linkages which were designed to constrain mechanical
linkages to move in a straight line (Kempe, 1877).
2.3 Mobility analysis of linkages
Here is the mobility formula that is applied for topology investigation, (Rolland, 1998):
m = Σji −
λ
n (1)
where Σji is the sum of all degree-of-freedoms introduced by joints and
λ
= 3 is the available
DOF of the planar space in which the actuator is evolving.
Finally, the number of closed loops in the system is n. This number can be multiplied and
shall be a natural number n ∈ {1,2,3, . . .}
2.4 Four-bar mechanisms
If n = 1 and only revolute joints are selected, then the mechanisms can be selected in the
large variety of four-bar mechanisms. These linkages feature one closed-loop or one
mechanical circuit. According to Grashof’s law, the sum of the shortest and longest link
cannot exceed the sum of the remaining two links if there is to be continuous relative motion
between the links. Hence, they can be classified as four types as shown in figure 4.
Fig. 4. Four-bar mechanism classification (from Wikipedia)
Three four-bar mechanisms can produce partial straight-line motion. They are characterized
by two joints connected to the fixed base.
The Chebyshev linkage is the epitome of the four-bar mechanical linkage that converts
rotational motion to approximate straight-line motion. It was invented by the 19th century
mathematician Pafnuty Chebyshev. It is a four-bar linkage therefore it includes 4 revolute
joints such that Σji = 4 ∗ 1 where n = 1 since there is only one closed loop. The resulting
mobility: m = 4−3 ∗ 1=1. Hoekens linkage happens to be a Cognate linkage of the Chebyshev
linkage. It produces a similar motion pattern. With appropriate linkage dimensions, part of
the motion can be an exact straigth line.
Robert’s linkage can have the extremity P set at any distance providing it is layed out on
that line perpendicular to the coupler, i-e link between A and B. This means that P can be
positionned on top of the coupler curve instead of below.

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
with Improved Performance Based on Parallel Architecture
113
This mobility calculation holds fo any four-bar mechanism including the free ones, i-e not
being attached to the base.
If properly designed and dimensionned, four-bar linkages can become straight-line motion
generators as will be seen in the next section on kinematics. This is one of the contribution of
this work.
2.5 True straight-line mechanisms
If n > 1 and only revolute joints are selected, then the mechanisms become more complex
and will integrate two closed loops or two mechanical circuits.
Three mechanisms can produce exact straight-line motion: the Peaucelier linkage, the
Grasshoper mechanism and a third one which has no name.
This linkage contains nine revolute joints such that Σji = 9 ∗ 1 = 9. Please note that where
three links meet at one point, two revolute joints are effectively existing. Three closed loops
can be counted for n = 3. The resulting mobility: m = 9 − 3 ∗ 3 = 0. The linkage designed by
Peaucelier is one of those mechanisms which cannot meet the mobility criterion but do
provide the required mobility. Very recently, Gogu has reviewed the limitations of mobility
analysis, (Gogu, 2004).
P
P
P
P
Fig. 5. Four-bar mechanisms: Chebyshev, Hoekens and Robert linkages
(a) Peaucelier
P
(b) The grasshoper
P
(c) An alternate case
Fig. 6. Exact straigth-line mechanisms
The two other linkages do provide for seven revolute joints for Σji = 7 ∗ 1 = 7 and two closed
loops for n = 2. The resulting mobility: m = 7 − 3 ∗ 2 = 1 which is verified by experiments.

Advanced Strategies for Robot Manipulators
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These three mechanisms do provide for straight-line motion at the cost of complex linkages
which do occupy very valuable space. This makes them less likely to be applied on robots.
3. Kinematics analysis
A mechanism is defined as a group of rigid bodies connected to each other by rigid
kinematics pairs to transmit force and motion. (Soylemez, 1999).
Kinematics synthesis is defined as the design of a mechanism to yield a predetermined set of
motion with specific characteristics.
We shall favor dimensional synthesis of function generation implementing an analytical
method. The function is simply a linear function describing a straight-line positioned
parallel to one reference frame axis.
The method will implement a loop-closure equation particularily expressed for the general
four bar linkage at first. The first step consists in establishing the fixed base coordinate system.
3.1 Four-bar mechanism
O2O4
r2
r3
r4
A
B
(a) Fixed four-bar
r2
r3
r4
O4
O2
A
B
(b) Semi-free four-bar
Fig. 7. General four-bar linkages
Lets define the position vectors and write the vector equation. Taking O2 and O4 as the link
connecting points to the fixed base located at the revolute joint center, taking A and B as the
remainder mobile revolute joint centers, the general vectorial formulation is the following,
(Uicker, Pennock and Shigley, 2003):
(r1 + r2 + r3 + r4 = 0) (2)
This last equation is rewritten using the complex algebra formulation which is available in
the textbooks, (Uicker, Pennock and Shigley):
3
12 4
1234
=0
j
jj j
re re re re
θ
θθ θ
++− (3)
where
θ
1,
θ
2,
θ
3 and
θ
4 are respectively the fixed base, crank, coupler and follower angles
respective to the horizontal X axis.
If we set the x axis to be colinear with O2O4, if we wish to isolate point B under study, then
the equation system becomes:

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
with Improved Performance Based on Parallel Architecture
115
342
3412
=
j
jj
re re r re
θ
θ
θ
−− (4)
Complex algebra contains two parts directly related to 2D geometry. We project to the x and
y coordinate axes, in order to obtain the two algebraic equations. The real part corresponds
to the X coordinates and the imaginary part to the Y coordinates. Thus, the equation system
can be converted into two distinct equations in trigonometric format.
For the real or horizontal part:
(
)
(
)
(
)
3344122
cos = cos cosrrrr
θ
θθ
−− (5)
For the imaginary or vertical part:
(
)
(
)
(
)
334422
sin = sin sinrrr
θ
θθ
− (6)
When O2O4 is made colinear with the X axis, as far as r1 is concerned, there remains only one
real part leading to some useful simplification.
The general four bar linkage can be configured in floating format where the O4 joint is
detached from the fixed base, leaving one joint attached through a pivot connected to the
base. Then, a relative moving reference frame can be attached on O2 and pointing towards
O4. This change results in the same kinematic equations.
Since, the same equation holds and we can solve the system:
()
222
4= 2arctan BABC
AC
θ
⎛⎞
+− + −
⎜⎟
⎜⎟
+
⎝⎠
(7)
where the A,B,C parameters are:
()
()
()
1
2
2
2
2222
12 1234
424
=cos
=sin
cos 1
=2
r
Ar
B
rrrrr
Crrr
θ
θ
θ
−
−
+−+
−
(8)
To determine the position of joint center B in terms of the relative reference frame O:
(
)
(
)
214 44 4
=[ cos , sin ]
t
OB r r r
θθ
+ (9)
Then, the norm of the vector OB gives the distance between O and B:
()
()
()
()
22
2
214444
== cos sinxOB rr r
θθ
++ (10)
This explicit equation gives the solution to the forward kinematics problem. An expression
spanning several lines if expanded and which cannot be shown here when the expression of
θ
4, equation 7, is substitued in it. This last equation gives the distance between O and B, the
output of the system in relation to the angle
θ
2, the input of the system as produced by the
rotary motor. The problem can be defined as: Given the angle
θ
2, calculate the distance x
between O and B.

Advanced Strategies for Robot Manipulators
116
The four-bar can be referred as one of the simplest parallel manipulator forms, featuring one
DOF in the planar space (
λ
= 3). One family of the lowest mobility parallel mechanisms.
The important issue is the one of the path obtained by point B which is described by a
coupler curve not being a straight line in the four-bar general case.
However, in the floating case, if applied as an actuator, the general four-bar can be made to
react like a linear actuator. The drawbacks are in its complex algebraic formulation and non-
regular shape making it prone for collisions.
3.2 Specific four bar linkages
We have two questions if we want to apply them as linear actuators:
• Can we have the four-bar linkage to be made to move in a straight-line between point
O2, the input, where the motor is located and B, the output, where the extremity or end-
effector is positionned?
• Can simplification of resulting equations lead to their inversions?
As we have seen earlier, specific four bar linkages can be made to produce straight-line
paths if they use appropriate dimensions and their coupler curves are considered on link
extensions. In this case, we still wish to study the motion of B with the link lengths made
equal in specific formats to produce specific shapes with interesting properties. Three
solutions can be derived:
• the parallelogram configuration,
• the rhombus configuration,
• the kite or diamond shape configuration, (Kempe, 1877).
3.2.1 The parallelogram configuration
Parallelograms are characterized by their opposite sides of equal lengths and they can have
any angle. They even include the rectangle when angles are set to 90 degrees. They have
been applied for motion transmission in the CaPaMan robot, (Ceccarelli, 1997).
The parallelogram four-bars are characterized by one long and one short link length. They
can be configured into two different formats as shown in figure 8.
O2
r2
r3
O4
r4
O4
r3
O2
r2
r4
AB
B
A
Fig. 8. The two parallelogram four-bar cases

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
with Improved Performance Based on Parallel Architecture
117
The follower follows exactly the crank. This results in the equivalence of the input and
following angles:
θ
4 =
θ
2.
If we set R and r as the link lengths respectively, then to determine the position of joint
center B in terms of the relative reference frame O2; an simple expression is derived from the
general four-bar one:
(
)
(
)
2=[ cos , sin ]OB R r r
θ
θ
+ (11)
Then, the norm of the vector O2B gives the distance between O2 and B:
22
2
=| |= 2 cos( )xOB Rr Rr
θ
++ (12)
This last equation is the result of the forward kinematics problem.
Isolation of the
θ
variable will lead to the inverse kinematics problem formulation:
222
=arccos 2
xRr
Rr
θ
⎛⎞
−−
⎜⎟
⎝⎠
(13)
Detaching joint O4 from the fixed base, the parallelogram becomes a semi-free linkage which
can be considered as one prismatic actuator.
3.2.2 The rhombus configuration
The rhombus configuration can be considered a special case of the parallelogram one. All
sides of a Rhombus are congruent and they can have any angle. Therefore, r1 = r2 = r3 = r4 or
even one can write r = R as for the parallelogram parameters. The mechanism configuration
even includes the square when angles are set to 90 degrees.
The forward kinematics problem becomes:
=2 cos 2
xr
θ
⎛⎞
⎜⎟
⎝⎠
(14)
The Inverse kinematics problem is expressed as:
O
A
C
B
Fig. 9. The Rhombus detailed configuration

Advanced Strategies for Robot Manipulators
118
=2arccos 2
x
r
θ
⎛⎞
⎜⎟
⎝⎠
(15)
Simple derivation will lead to differential kinematics.
The forward differential kinematics is expressed by the following equation:
=sin()
2
vr
θ
ω
− (16)
where =d
dt
θ
ω
We take the following geometric property:
cos =
22
x
θ
⎛⎞
⎜⎟
⎝⎠ (17)
We apply Pythagore’s theorem:
2
2
1
sin = 4
22
x
r
θ
⎛⎞ −
⎜⎟
⎝⎠ (18)
Then, the FDP can be rewritten in terms of the length x:
2
2
=4
2
rx
vr
ω
−− (19)
Inversion of equation 19 lead to the inverse differential kinematics problem being expressed
as:
2
=sin( )
v
r
θ
ω
− (20)
Substituting equation 17 and equation 18 into the former, we obtain:
=1
2
vx
rr
ω
−− (21)
Further derivation will give the extremity acceleration where the FDDP can be expressed as:
2
1
=sin cos
22 2
ar r
θ
θ
αω
⎛⎞ ⎛⎞
−−
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
(22)
Substituting equation 17 into the former lead to the following expression of the FDDP:
2
2
2
11
=4
24
x
ar x
r
α
ω
−−− (23)
The IDDP:

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
with Improved Performance Based on Parallel Architecture
119
2
1cos
22
=
sin 2
ar
r
θ
ω
αθ
⎛⎞
−− ⎜⎟
⎝⎠
⎛⎞
⎜⎟
⎝⎠
(24)
Substituting equation 21, equation 17 and equation 18 into the former, we obtain:
3/2
14
2
2
11
=2 2
2
4
x
ar vx r r
x
r
α
−
−−
⎛⎞
−−
⎜⎟
⎝⎠
−
(25)
O
A
B
C
Fig. 10. The diamond shape four-bar
3.2.3 The kite or diamond shape configuration
The kite configuration is characterized by two pairs of adjacent sides of equal lengths,
namely R and r.
Then, two configurations into space depending on which joint the motor is attached. The
motor is also located on the joint attached on the fixed base.
To obtain the first configuration, the first pair is located at O2, the crank joint center where
the motor is located, as its articulation center and the second pair at B, the extremity joint, as
its center.
The second configuration integrates the actuator on O4. However, the actuator x output is
defined as the linear distance between O2 and B making this actuator moving sideways. The
problem will be that the change of four-bar width is going to introduce parasitic transverse
motion which will in turn prevent real linear motion due to the pivot effect caused by the
motor joint. This approach is thus rejected.
To obtain the second disposition, one can mount the driven joint between two unequal links
and have the output on the opposite joint also mounted between two unequal links. This
results in sideways motion. However, this would also result in parasitic transverse motion
which would mean that the final motion would not be linear being their combination.
Therefore, this last configuration will not be retained further.
Lets R be the longest link length, the links next to B, and r be the smallest link one, the links
next to O2.
Since this configuration is symmetric around the axis going through O2 and B, it is thus
possible to solve the problem geometrically by cutting the quadrilateral shape into two
mirror triangles where the Pythagorean theorem will be applied to determine the distance
between O2 and B giving:

Advanced Strategies for Robot Manipulators
120
()
()
2
22
sin
=1sin 1
2
r
xr R R
θ
θ
⎛⎞
⎛⎞
−+−
⎜⎟
⎜⎟
⎝⎠
⎝⎠ (26)
This equation expresses then the forward kinematics problem.
Using the law of cosinuses on the general triangle where the longest side is that line between
O2 and B, it is possible to write a more compact version for the FKP:
(
)
22 1
2
2cosxRr r
θ
=−− (27)
The inverse kinematics problem requires the distance or position x as input which completes
the two triangle lengths into the diamond shape. Hence, the cosinus laws on general
triangles can be applied to solve the IKP:
22 2
1
=2arccos 2
Rrx
r
θ
⎛⎞
−−
⎜⎟
⎝⎠
(28)
To obtain the differential kinematics models, the kinematics models are differentiated.
FDP:
(
)
()
2
22
1
2
1
2
sin
1
=22cos
r
v
Rr r
θω
θ
−− (29)
Differentiation of the IKP leads to the following IDP expression:
22
2
1
2
1
2
42cos(
=44cos(
)
)
Rr r
vr
ω
θ
θ
∗−−
− (30)
After testing several approach for obtaining the differential model leading to accelerations,
it was observed that starting with the inverse problem leads to more compact expressions:
The IDDP is obtained by differentiating the IDP:
22 2 2
3/2
22 22 22 22
3
22
48()
=( 1 2) where 1 , 2
() ()
44
xR r x
aA A A A
Rrx Rr x
rr
rr
α
−−
+= =−
−− ⎛⎞
−−
−−
⎜⎟
⎝⎠
(31)
Inverting the IDDP produces the FDDP but it cannot be shown in the most compact form.
The Kite configuration models are definitely more elaborate and complex than for the
rhombus configuration without necessarily leading to any kinematics advantages.
3.2.4 The rhombus configuration repetition or networking
The rhombus four-bar linkage can be multiplied as it can be seen in platform lifting devices.
The repetition of these identical linkages helps reduce the encumbrance and this will be
studied in this section in the context of linear actuator design.

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
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(a) Single rhombus (b) Double rhombus (c) Triple rhombus
Fig. 11. Rhombus networking
The distance traveled by the first moving central joint (FKP) is:
(
)
12
=2 cosxr
θ
(32)
This problem can be solved just like solving the original single rhombus FKP.
The distance traveled by the second moving central joint (FKP) is:
(
)
21 2
=2 =4 cosxxr
θ
(33)
The impact of adding the second rhombus is doubling the distance or position reach.
The distance traveled by the third moving central joint or the solution of the FKP of a triple
rhombus is:
(
)
31 2
=3 =6 cosxxr
θ
(34)
This trend can be generalized to a repetition of n identical rhombuses.
(
)
12
==2cos
n
xnxnr
θ
(35)
The result of the four-bar rhombus repetition is the linear motion amplification by that
repetition number.
To obtain the inverse kinematics problem, one can proceed with inversion of the FKP.
The double rhombus angular position of the actuator can then be deduced:

Advanced Strategies for Robot Manipulators
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1
=2arccos 4
x
r
θ
⎛⎞
⎜⎟
⎝⎠
(36)
This equation can then also be extrapolated to a repetition of n identical rhombuses.
1
=2arccos 2
x
nr
θ
⎛⎞
⎜⎟
⎝⎠
(37)
The forward differential model is obtained by derivation of the forward kinematics model.
For a double rhombus configuration, the relative speed of the second central joint is equal to
the absolute speed of the first central joint:
2/ 1
=
r
vv (38)
212/
=r
vvv+ (39)
21
=2vv
(40)
where
(
)
12
=sinvr
θ
ω
− (41)
Hence, the actual speed of the second extremity or the final end-effector becomes:
(
)
22
=2 sinvr
θ
ω
− (42)
The impact of adding the second rhombus is doubling the end-effector velocity.
The same result would be obtained by derivation of the equation for x2.
We now calculate the velocity of the third moving central joint which corresponds to the
solution of the FDP of a triple rhombus.
(
)
31 2
=3 = 3 sinvv r
θ
ω
− (43)
This trend can be generalized to a repetition of n identical rhombuses:
(
)
12
== sin
n
vnv nr
θ
ω
− (44)
The inverse differential model can be obtained in two ways, either by derivation of the
inverse kinematics model or inversion of the forward differential model.
By inversion of the FDP, the double rhombus angular position of the actuator can then be
deduced:
1
2
2
1
=
4
vr x
r
ω
−
−
(45)
For the triple rhombus, we extrapolate:

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
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2
2
21
=34
vrx
r
ω
−
(46)
For a linkage with the repetition of n rhombuses, we obtain the following equation:
2
2
21
=
4
vnr x
r
ω
−
(47)
To determine the accelerations, we will again differentiate the former differential models.
We calculate derivation of the equation for v2 for the second rhombus; it results in doubling
the end-effector acceleration.
The FDDP for the case where we are doubling the rhombus leads to:
(
)
(
)
2
21 22
1
=2 = 2 sin cos
2
aa r r
θ
θ
αω
−− (48)
For the triple rhombus, we can determine that:
(
)
(
)
2
31 22
1
=3 = 3 sin cos
2
aa r r
θ
θ
αω
−− (49)
For n rhombuses, it is possible to extrapolate:
(
)
(
)
2
122
1
== sin cos
2
n
ana nr r
θ
θ
αω
−− (50)
Multying n times the rhombus linkage results in multiplying the acceleration likewise.
The IDDP, inverse model for a double rhombus, through derivation of the IDP or inversion
of the FDDP, the calculation returns this equation:
2
124
2
2
2
3
2
11 1
=2 1
816
16
x
ar v xr r
x
r
α
−−
−
⎛⎞
−−
⎜⎟
⎝⎠
−
(51)
For three rhombuses, the angular acceleration can then be determined:
2
124
2
2
2
3
2
11 1
=2 7 1
236
36
x
ar v xr r
x
r
α
−−
−
⎛⎞
−−
⎜⎟
⎝⎠
−
(52)
We have then extrapolated for a linear actuator constructed with n rhombuses:
2
234
22
2
22
3
2
11
=2 1 4
4
ax
vxn r
nr n r
x
nr
α
−−
−
⎛⎞
−−
⎜⎟
⎝⎠
−
(53)

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3.2.5 The kite configuration repetition or networking
There seems to be no advantage to gain from networking the kite configuration. This will
even add complexity to the kinematics models. Therefore, this prospect has not been
investigated further.
4. Kinematics performance
4.1 Singularity analysis
4.1.1 General four bar linkage
For the general four bar linkage, singularities can be found when A + C = 0 using the values
of equation 8. The solution to this equation results in:
()
2222
141234
2
241
2
1
=arccos 2
rr r r r r
rrr
θ
⎛⎞
−++−+
⎜⎟
⎜⎟
−+
⎝⎠
(54)
4.1.2 The parallelogram configuration
Singularities could be found only when Rr = 0 which is impossible since all links obviously
have lengths larger then zero.
From the kinematics point of view, no limitations apply on the application of parallelograms
since the rocker can follow the crank in any position allowing full rotation capability,
therefore having no kinematics singularity whatesoever.
This mechanism could be considered somewhat similar or equivalent to the belt and pulley
one where the two pulleys are of equal lengths if the belt is considered without elasticity.
4.1.3 The rhombus configuration
For the IDP, singularities exist and they can be determined by cancelling the denominator in
the equations 20 and 21 leading to the two following equations.
The first one calculates the singularity in terms of the input angle
θ
:
(
)
2
sin = 0
θ
(55)
Hence, we find a singularity at
θ
= 0 and its conterpart
θ
= 360 degrees.
For the second one determines the singularity in terms of the extremity position x:
2
2
4=0
x
r
− (56)
Hence, the singular position x = 2r corresponds to the same posture as
θ
= 0.
From a geometric point of view, links have no material existence (no mass) and they can
occupy the same position in space. In reality, the masses do not allow such cases and
therefore the singularity will be alleviated by bar width as will be explained later in the
design section. The IDDP models bring singularities. Observation of the denominator allows
us to determine that the singular configurations are just the same as the one studied for the
IDP since the equations feature the same denominators under the power.
4.1.4 The kite or diamond shape configuration
If R > r, then this results then into an amplified motion without any singularity with full 360
degrees rotation of the input crank. This configuration has an advantage over the other

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
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125
types of four-bars. This would surely represent one reason to apply this mechanism as a
linear actuator.
If R < r, then the mechanism cannot reach an input angle of 180 degrees since this would
mean 2R > 2r in contradiction with stated configurations. Hence, the system will block into
position θmax < 180° unable to go further. The angular range will be limited to [0,
θ
max] where:
(
)
=2arcsin
max R
r
θ
(57)
This posture also yield a singularity which can also enforce mechanism blockage. Hence,
this type will not be retained.
4.1.5 The rhombus configuration repetition or networking
In terms of singularities, finding the roots of the FDP and IDP will lead to the same
singularities as for a single rhombus as it would seem logical. In terms of singularities,
finding the roots of the FDDP and IDDP is equivalent to finding the same singularities
solving the roots of only the IDDP as for a single rhombus.
Therefore, networking rhombuses will not introduce any singularity.
4.2 Workspace
The second important performance criterion for robotic design is usually the workspace. In the
case of single DOF device, this narrows down to a simple range which we wish to maximize.
4.2.1 The general four-bar linkage
The mechanism can reach the following maximum length where two links are aligned,
either r1 and r4 or r2 and r3. Then, the mechanism reach will be xmax and is calculated by the
length of the extension of the two shortest links going from O2 and leading to the extremity B:
(
)
1423
=min ,
max
xrrrr++ (58)
The mechanism can also reach a minimum length which is a far more difficult problem to
determine depending upon the configuration and relative link lengths. This is where
Graschoff’s formulas could help solve this problem. Despite the fact that link lengths value
could be found leading to a coupler curve being a straight line, this constitutes another
reason to avoid the general four-bar mechanisms.
4.2.2 The parallelogram configuration
The maximum and minimum actuator values of x can be determined by looking for the
roots of the x(
θ
) function derivative or by geometric reasoning. Hence, using the simplest, i-e
the second approach, we can determine that the extremas are found at
θ
= k
π
where k ∈
{0,1,2,3, . . .}. With n = 0, the maximum value is found xmax = R + r and with n = 1, the
minimum value is xmin = |R − r|. We do not need to go further because of the repetitive
nature of the trigonometric signal. These correspond to the posture where the four-bar is
folded on itself: one fold to the left and one to the right.
4.2.3 The rhombus configuration
To determine the maximum and minimum values, several methods lead to the same results.

Advanced Strategies for Robot Manipulators
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Taking the FKP, equation 14, the maximum value is obtained when cos( 2
θ
) = 1 and the
minimum value will be when cos(2
θ
) = −1. Hence, xmax = 2r and the related angle is then
θ
=
0. Moroever, xmin = −2r and the related angle is then
θ
= −2
π
. These values imply that the
pure geometric nature of the kinematics analysis allows the mechanism to reverse by going
unto itself. Hence, the minimum can be seen on the left or negative side of the reference
frame and the maximum is located on the right or positive side.
With considerations of the linkage dimensions, the geometric analysis can be augmented by
taking into account the linkage width.
Firstly, two linkages cannot occupy the same space, therefore, the rhombus linkage
configuration will have a pair of opposite linkages below and one pair above. This lead to
physical constraints equations. This property can also be translated into geometric
information.
These opposite links are parallel pairs which will eventually touch each other alongside at
two mechanism rotations. These postures could be considered as folded ones. The first
corresponds to the minimum rotation and the second to the maximum rotation.
Let the rhombus linkage be constructed by four bars of identical width w.
The minimal rotation angle increases to:
>0
min
θ
(59)
=2arcsin
min
w
r
θ
⎛⎞
⎜⎟
⎝⎠
(60)
Taking into account that the kinematics chain cannot reverse by going unto itself, the
maximal rotation angle reduces to:
<
max
θ
π
(61)
=2arccos
max
w
r
θ
⎛⎞
⎜⎟
⎝⎠
(62)
The final range of the linear actuator is then the interval determined by: [2 arcsin( w
r,
2 arccos( w
r].
X
Xmax
Xmin
B
O
O
X
B
Fig. 12. The rhombus extreme positions

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127
The extreme positions can be determined from these values. Minimal length is going to
occur at maximal angular displacement.
=2 cos 2
min max
xr
θ
⎛⎞
⎜⎟
⎝⎠
(63)
Substituting
θ
max into the equation, we get:
(
)
(
)
= 2 cos 2arccos
min w
r
xr (64)
Similarily, maximal length is going to occur at minimal angular displacement.
=2 cos 2
max min
xr
θ
⎛⎞
⎜⎟
⎝⎠
(65)
Substituting
θ
min into the equation, we get:
(
)
(
)
=2 cos 2arcsin
max w
r
xr (66)
4.2.4 The kite or diamond configuration
Let the kite linkage be constructed by four bars of identical width w.
The minimal rotation angle is exactly the same as the rhombus:
>0
min
θ
(67)
(
)
= 2arcsin
min w
r
θ
(68)
Taking into account that the kinematics chain cannot reverse by going unto itself, the
maximal rotation angle reduces to the case where the long bars touch each other in the
negative sense of the reference frame. We have to take then the angle outside the triangle
formed by these long bars:
<2
max
θ
π
(69)
=2 2arcsin
max
w
R
θπ
⎛⎞
−⎜⎟
⎝⎠
(70)
The extreme positions can be determined from these values. In the case of maximal position,
the two geometric triangles formed by the long and short links add up:
Minimal length is going to occur at maximal angular displacement.
22
=()cos
22
min max
w
xR r
θ
⎛⎞
−−
⎜⎟
⎝⎠
(71)
Maximal length is going to occur at minimal angular displacement.
22
=()cos
22
max max
w
xR r
θ
⎛⎞
−+
⎜⎟
⎝⎠
(72)

Advanced Strategies for Robot Manipulators
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Emax
Emin
B
OO
B
X
Fig. 13. The rhombus encumbrance
4.3 Encumbrance
4.3.1 The rhombus configuration
The proposed linear actuators are based on four-bar linkage where encumbrance becomes
an issue considering that the mechanism spread sideways making them subject to collisions
if other actuators would be located in the vicinity such as it is often the case with parallel
manipulators.
Ecumbrance is defined as the distance from one side of the mechanism to the other side
taking into account the linkage width.
A Rhombus would have minimum encumbrance of Emin = 2w when the angle is at
θ
min. This
characteristic is relatively unimportant compared to the maximum encumbrance occuring at
the maximum input angle posture
θ
max:
22
=2
max
Erww
−
+ (73)
4.3.2 The kite configuration
A folded kite would have minimum encumbrance of Emin = 2w when the angle is at
θ
min just
like the rhombus. The maximum encumbrance is occuring at the maximum input angle
posture
θ
max = 2
π
when the smaller links are aligned:
=2
max
Erw
+
(74)
As can be observed, the kite encumbrance only really depends on the dimension of the
shortest links.
4.4 The repeated rhombus configuration
The problem of encumbrance justifies the design of a mechanism based on the repetition of
identical rhombuses as it is done for lifting platforms.
The repetition of the four-bar rhombuses is not affecting the rotation input and the
θ
min and
θ
max values are only related to the first rhombus, therefore these extrema are unchanged.
4.4.1 The double rhombus
For the double rhombus, the minimum position is determined by:
(
)
(
)
= 4 cos 2arccos
min w
r
xr (75)

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129
And the maximum position is calculated using:
(
)
(
)
=4 cos 2arcsin
max w
r
xr (76)
4.4.2 The triple rhombus
For the triple rhombus, the minimum position is determined by:
(
)
(
)
=6 cos 2arcsin
max w
r
xr (77)
And the maximum position is calculated using:
(
)
(
)
=6 cos 2arcsin
max w
r
xr (78)
4.4.3 The multiple rhombus
For the generalized case with n rhombuses, the minimum position is determined by:
(
)
(
)
=2cos2arcsin
max w
r
xnr (79)
And the maximum position is calculated using:
(
)
(
)
=2cos2arcsin
max w
r
xnr (80)
4.4.4 Encumbrance of the multiple rhombus
The networking of rhombuses is not affecting encumbrance in the sense that the values are
exactly the same. However, the main advantage is that the reach which can be defined as the
maximum position is increasing while the encumbrance remains unchanged. This could not
happen with a simple rhombus where we would need to increase the link lengths in order to
increase reach resulting in larger emcumbrance.
Lets define the encumbrance ratio.
The encumbrance ratio is defined as the ratio of reach divided by the transverse
encumbrance perpendicular to the axis of motion.
=max
max
X
eE (81)
For the repeated rhombus, this occus when
θ
=
θ
min and the encumbrance ratio becomes:
(
)
(
)
22
= 2 cos 2arcsin 2
w
r
enr r w w
−
+ (82)
Hence, the motion to encumbrance ratio is increasing proportionaly with the rhombus repetition.
5. Design examples
5.1 Initial prototypes
A first group of prototypes were constructed and tested using a Meccano set while the
author was working at the Ecole Nationale des Arts et Metiers in Metz. This resulted in the
constuction of a planar parallel manipulator as seen in figure 14. The DC motor was the
typical Meccano 36 VDC.

Advanced Strategies for Robot Manipulators
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Fig. 14. Planar parallel manipulator with four-bar actuators
They were sufficient to prove and validate the concept. In effect, one rhombus four-bar with
a Meccano motor could not be seen moving due to very high accelerations.
5.2 Actual prototypes
This work was then completed during the author stay at Middle East Technical University,
Northern Cyprus Campus.
Three typical linear actuators were constructed as seen in figure 15. One comprising one
rhombus, one with two rhombuses and one with a kite configuration using the same links
and motors whenever possible.
5.2.1 Configuration
Here are the mechanism geometric parameters. They were constructed with two
standardized bars. The short bars have length r = 10 cm and width w = 3 cm. The long bars
have length R = 20 cm with same width.
The geared electrical motors were selected to provide maximum rotation speed of 120 RPM.
Hence,
ω
max = 4
π
radians/s.
5.2.2 Extreme positions
For the rhombus, from the proposed equations, the minimum input angle is then
θ
min = 0,301
radian. The maximum input angle is
θ
max = 2,84 radian. These values are confirmed by
measurements on the prototypes.
From these angular positions, we calculate the maximum position or reach as xmax = 19,77 cm
and the minimum value is xmin = 3 cm. Again these calculated values are confirmed by
measurements.

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
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131
(a) Kite (b) Single rhombus
(c) Double rhombus
Fig. 15. The three four-bar actuator prototypes
The encumbrance can also be deduced. At
θ
max, the maximum encumbrance is calculated as
Emax = 22,77 cm. This was confirmed by measurements.
5.2.3 Motion analysis
In the curves of figure 16, t is in fact
θ
2, the angular input position in radian, and ω is the
angular velocity made to change from −4
π
to 4
π
radians/s.
The first question is about the actuator linearity and this issue can be answered by plotting
the extremity position in relation to the input angle position. The motion becomes non-linear
at the angular extremities and it becomes almost linear for a large number of angular
positions from -1 to 1 radian corresponding to the position range of -18 to 18 cm.
The second question is to determine the extremity velocity profile according to input angle
position at angular velocities going from the minimum until the maximum. This last value
comes from the geared motor specifications. End-effector velocity changes almost linearily
with the angular velocity but changes non-linearily with the angular position. It cancels out
at angular extremities and it becomes maximal at
θ
=
π
.
This velocity profile also corresponds to the accuracy profile. This means that for a
predefined encoder accuracy located on the gearmotor shaft, the resulting extremity
accuracy will be changing accordingly. One can foresee that the lowest accuracy is attained
at
θ
=
π
and the best accuracies are achieved near the angular position extremities.
The third question involves the extremity acceleration in relation to input angle position at
angular velocities going from the minimum until the maximum. We have to fix some
angular acceleration and the value 1rad/s2 has been selected arbitrarily. Extremity

Advanced Strategies for Robot Manipulators
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acceleration changes non-linearily with the angular velocity. It changes almost linearily with
the angular position. It reaches very high values when the linkage reaches very close to the
maximum position.
(a) Position
(b) Velocity
(c) Acceleration
Fig. 16. The rhombus kinematics performance

Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics
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133
6. Conclusion
As can be seen by the results obtained by calculations as well by experiments on the
prototypes, four-bar mechanisms can be rearranged in rhombus and kite configurations
which lead to very performant prismatic pairs allowing to design very fast linear actuators.
Moreover, to improve performance and to reduce emcumbrance, networking the rhombus
four-bars can lead to very good results.
In the author’s knowledge, this is the first time that four-bars were envisaged to be applied
as linear actuators.
The next step will be to analyze their dynamics design integrating force analysis.
Then, to design a large scale parallel robot prototype will help investigate their worthiness
towards the design a very high speed milling machine.
Finaly, several optimization problems may arise to determine proper linkage sizing.
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