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International Journal of Electrical Engineering and Computing
Vol. 6, No. 1 (2022)
1
Original research paper
UDC 004.45:004.388]:621.9
DOI 10.7251/IJEEC2206001N
Workspace and kinematic structure analysis of a
6-DOF Lambda parallel kinematic machine
Ljubomir Nešovanović1 and Saša Živanović2
1 LOLA Institute, Belgrade, Serbia
2University of Belgrade, Faculty of Mechanical Engineering, Belgrade, Serbia
E-mail address: ljubomir.nesovanovic@li.rs, szivanovic@mas.bg.ac.rs
Abstract— This paper presents workspace and kinematic analysis of a parallel kinematic machine based on the Lambda mechanism.
The considered parallel kinematic machine has six degrees of freedom (DOF), achievable with six actuated translation joints. The
kinematic analysis includes the definition of every active part of the machine, as well as the definition of every active or passive joint
used to connect machine parts. The mathematical model of the machine is created for the better understanding of the machine's
operation. The proposed mathematical model of the machine includes inverse kinematic equations, whose solving presents the first
step in workspace analysis. In this case, the offered parallel kinematic machine has six active-joint variables, and every active-joint
variable is the result of one inverse kinematic equation. Verification of the inverse kinematic equations has been done analytically,
using MatLab software and a CAD/CAM system. Workspace analysis, as one of the most important parameters of the parallel
kinematic machine, presents the main topic of this paper. In this case two approaches to the workspace analysis are given. The first
considered analysis is used to determine the achievable workspace of the machine, and the second analysis is used to determine the
total orientation workspace of the machine. Polar coordinates are used to simplify the process of the workspace analysis.
Keywords- Lambda mechanism; inverse kinematic equations; workspace analysis; CAD/CAM.
I. INTRODUCTION
This paper is based on one part of the research presented in
[1] and describes a parallel kinematic machine with six DOF.
Parallel kinematic machines have many advantages over serial
ones, but one of the biggest problems is a deficient ratio between
machine space and workspace [2]. This paper presents one way
to achieve the workspace of a parallel kinematic machine.
Before analyzing the workspace, it is essential to form the proper
kinematic model of the machine and prepare the initial kinematic
parameters of the mechanisms, for further analysis.
The main characteristic of the analyzed machine is the
Lambda mechanism, on which the machine is based. The first
one to use the Lambda configuration was Stewart [3], and the
interpreted machine is established on this research. Many ideas
of parallel kinematic machines are based on the Lambda
mechanism [4-6] and actuated, constant length translation joints
are characteristic for all of them, including the machine
proposed in this paper. Parallel kinematic machines based on the
Lambda mechanism may have two to six DOF. The machine
based just on one Lambda mechanism has two DOF and can be
upgraded to the hybrid mechanism with four DOF [4]. Same as
the machine presented in [5], the parallel kinematic machine
proposed in this paper has three Lambda mechanisms and can
acquire 6-DOF. The machine shown in [6] has four DOF
enabled using three Lambda mechanisms with some limitations.
The machine presented in [7], as the machine proposed in this
paper, has six DOF achieved using six kinematic chains, whilst
two chains are connected to the same shaft using the translation
joints. Differences between the proposed machine and the
machine presented in [7] are in length and connection of
kinematic chains.
The concept of a parallel kinematic machine based on
Lambda mechanism with actuated translation joints, the
kinematic structure analysis, and inverse kinematic equations
are presented below. This paper also presents the verification of
inverse kinematic equations and workspace analysis.
II. THE CONCEPT OF A PARALLEL KINEMATIC MACHINE
BASED ON THE LAMBDA MECHANISM
The machine analyzed in this paper consists of a stationary
base and a moving platform connected with three independent
Lambda mechanisms (Fig. 1). Each Lambda mechanism is
defined by two kinematic chains, one longer than another,
connected with one rotary joint. Therefore, the machine has six
kinematic chains.
This paper is a revised and expanded version of the paper presented at
the XXI International Symposium INFOTEH-JAHORINA 2022
Ljubomir Nešovanović et al.
2
Figure 1. The kinematic model of the machine
Before mathematical analysis, it is necessary to define the
design of the Lambda mechanism correctly. The specific
position of kinematic chains, their structure, and the type of used
joints of the Lambda mechanism are shown in Fig. 2.
Actuated translation joints are positioned on the stationary
base. One of the characteristics of the analyzed machine are two
translation joints positioned on the same shaft on the stationary
base. Each Lambda mechanism is connected with two
translation joints on the same shaft on the stationary base, using
the spherical joint. The connection between each Lambda
mechanism and the moving platform is provided by one
spherical joint. As previously said, the Lambda mechanism is
defined by two kinematic chains, one longer than another. The
rotary connection between the two chains is provided by
connecting one side of the smaller kinematic chain to the body
of the longer one.
It is crucial to say that the presented machine has three types
of joints. The spherical joints offer three DOF, and every DOF
provides rotation around one of the three perpendicular axes.
The rotary joints provide one DOF for rotation around defined
axes. The translation joints provide one DOF for translation in a
direction of the required axis. The only actuated joints of the
machine are translation joints, and the others are passive joints.
For better understanding of the presented machine, graph
diagram is shown in Fig. 2.
III. THE KINEMATIC STRUCTURE ANALYSIS OF THE
PROPOSED MACHINE
Before analyzing the workspace and optimizing the machine
parameters, it is necessary to form a proper mathematical model
of the kinematic structure. The proper representation of the
kinematic structure of the machine are inverse kinematic
equations. For achieving inverse kinematic equations, it is
necessary to solve the inverse kinematic problem. Solving the
inverse kinematic problem (IKP) means transforming the
moving platform’s position and orientation vector into the
active-joint variables [8].
Figure 2. Graph diagram
The machine analyzed in this paper has six independent
kinematic chains (Fig. 3), and every chain is connected to the
actuated translation joint. The position of every translation joint
on the shaft directs the position and orientation of the moving
platform. Consequently, the active-joint variable of this machine
is the position of the translation joint on the shaft. The vector of
every active-joint variable is:
l =l1
l2
l6 (1)
The first three elements of the vector are the variables that
describe the position of translation joints connected to the longer
kinematic chains of the Lambda mechanism. The moving
platform’s position and orientation vector
xe=[px, py, pz, Ψ, θ, Φ]T (2)
Figure 3. The kinematic model with active-joint positions
International Journal of Electrical Engineering and Computing
Vol. 6, No. 1 (2022)
3
is given for inverse kinematic equation solving [8]. The value of
the machine parameters is also given. The machine parameters
are the stationary base dimension (C), the moving platform
dimension (D), the dimension of longer kinematic chains (ci),
the dimension of smaller kinematic chains (ui), and the
dimension between kinematic chain connection (of Lambda
mechanism) and platform (ri).
In this case, geometric methods solve the inverse kinematic
problem. It is necessary for the machine with six active-joints to
create six equations. The starting point was creating a vector
equation that could connect the zero-position point of the active-
joint on the stationary base (R) and the corresponding point on
the moving platform (N). Vectors used in mathematical
derivations are platform position vector ( p
B
OP), joint position on
the platform vector ( p
P
Ni ), joint position on the base vector
(p
B
Ri), the direction of the actuated joint vector ( a
Bi), and unit
vectors of orientation ( w
Biz
Biq
B
i). Fig. 4 shows that the
connection between the N point on the stationary base and the R
point on the moving platform (ki) can be described with three
vector equations [9]:
kiw
Bi= p
B
OP + R∙
P
Bp
P
Ni− p
B
Ri (i=1,2,…,6) (3)
kiw
Bi=lia
Bi +ciz
Bi (i=1,2,3) (4)
kiw
Bi=lia
Bi +uiq
B
i +riz
Bi (i=4,5,6) (5)
The equation (3) can be used for solving all six equations.
Equation (4) is reserved for finding the result of the first three
active-joint variables. Equation (5) can solve equations
connected to the last three active-joint variables. This equation
presents a starting point for solving the inverse kinematic
problem. After mathematical derivation, the solution of the
inverse kinematic problem is presented with six equations for
every active-joint variable [1]:
Figure 4. The kinematic model with required vectors
l1=−zp+D∙sθ− −zp+D∙sθ2−(k1
2− c12), (6)
l2=−zp−1
2D∙sθ+3
2D∙cθ∙sΨ−
−zp− 1
2D∙sθ+3
2D∙cθ∙sΨ2
−(k2
2− c22), (7)
l3=−zp− 1
2D∙sθ −3
2D∙cθ∙sΨ−
−zp− 1
2D∙sθ −3
2D∙cθ∙sΨ2
−k3
2− c32, (8)
l4
=l1+ a
B
4 ∙ (c1 −r4)∙ z
B
4 +
(a
B4 ∙ (c1 − r4)∙ z
B
4)2 −((c1 − r4)2−u42), (9)
l5
=l2+ a
B
5 ∙ (c2 −r5)∙ z
B
5 +
(a
B
5 ∙ (c2 − r5)∙ z
B
5)2 −((c2 − r5)2−u52), (10)
l6
=l3+ a
B
6 ∙ (c3 −r6)∙ z
B
6 +
(a
B
6 ∙ (c3 − r6)∙ z
B
6)2 −((c3 − r6)2−u62). (11)
IV. VERIFICATION OF THE INVERSE KINEMATIC EQUATIONS
Verifying the inverse kinematic equations on a virtual
prototype is vital before using equations (6)-(11) in workspace
analysis. The inverse kinematic equations have been verified
using two software, MatLab and PTC Creo Parametric. Usage
of PTC Creo Parametric has created a simplified CAD model of
the machine, and MatLab software has been used to find the
most effective solution of the inverse kinematic equations.
The important measurements are done on a virtual model
using PTC Creo software (Figs. 5 and 6). Dimensions required
from the model were parameters of the machine and the moving
platform position and orientation vector. The starting point for
all measurements is defining the proper coordinate systems of
the stationary base (IKP) and the moving platform (TP). The
stationary base coordinate system is presented as a zero vector
and presents a starting point for every analysis. The coordinate
system of the platform represents the platform’s position and
orientation. Software PTC Creo Parametric generated a
measurement tool to provide the transformation matrix that has
all the necessary information about the platform's position and
orientation (Fig. 5). After measurement, the required dimensions
are imported into the MatLab program. The product of the
created program are the active-joint variables. The active-joint
variable can be measured on the virtual prototype of the machine
(Fig. 5). The measured dimension between every translation
joint and XY plane of the stationary base’s coordinate system
presents an active-joint variable.
Ljubomir Nešovanović et al.
4
Figure 5. The measurement’s on CAD model
Figure 6. Second measuring experiment
Evaluation of inverse kinematic equations is done by
comparing the active-joint variable measured on a virtual model
and generated using the MatLab program. This comparison is
made for two different positions of the moving platform. The
first position of the platform shown in Fig. 5 is used for
experiment 1, while the platform position shown in Fig. 6 is used
for experiment 2. Comparing both ways generated active-joint
variables confirms inverse kinematic equations (Tab. 1).
TABLE I. COMPARISON OF BOTH WAYS GENERATED ACTIVE-JOINT
VARIABLES
Experiment 1
Experiment 2
Active-joint
variable
CAD
model
[mm]:
MatLab
code
[mm]:
CAD
model
[mm]:
MatLab
code
[mm]:
398.89
398.88
164.59
164.60
299.68
299.67
223.16
223.16
414.67
414.66
467.37
467.37
1758.75
1758.73
1587.58
1587.59
1714.02
1714.01
1565.02
1565.03
1723.08
1723.07
1514.00
1514.00
The difference between the active-joint variable measured
on a CAD model and generated using the computer program
shown in Tab. 1 results from multiple conversions and
measurement errors. Proposed inverse kinematic equations
accuracy is proven, and offered equations can be used in future
machine analysis.
V. WORKSPACE ANALYSIS
Workspace is one of the most important parameters in
machine tool design. For parallel kinematic machine tools, the
workspace is usually a weak point of the machine design and
presents a vital parameter. Because of this characteristic,
workspace analysis is often a starting point for parallel kinematic
machine designing. Some of the main dimensions of the
machine can be generated from the workspace analysis, and the
workspace can have different shapes and sizes depending on the
machine's design.
There are multiple approaches to workspace analysis
because of many possible usages of parallel kinematic machines.
This paper presents the two approaches to workspace analysis.
The first approach is based on the achievable workspace, and the
second is based on the total orientation workspace.
A. Achievable workspace
Achievable workspace is the machine's workspace whose
end-effector can reach every workspace point in any orientation
[10]. The provided machine parameters are the stationary base's
dimension, the moving platform's dimension, and the
dimensions of the kinematic chains. The machine parameters
used for workspace analysis are adopted for regular size machine
tools and implemented in a simplified CAD model of the
machine. The exact value of every parameter is shown in [1].
Because of the specific machine design, it is difficult or even
impossible to generate the machine's workspace geometrically.
In this case, the analytical method based on the computer code
programmed in MatLab software is used for obtaining the
achievable workspace, and created code is acquired on the
algorithm shown in Fig. 7.
As previously said, the analyzed machine has 6-DOF and,
intuitively, both types of the workspace are tridimensional.
Because of the specific kinematics, the achievable workspace of
the machine has no traditional design. In this case, polar
coordinates (z, ρ, β) are used for achievable workspace analysis.
For the better understanding, the procedure of the workspace
analysis is shown graphically in Fig. 8. The first step in finding
the achievable workspace design is to divide the achievable
workspace into the planes perpendicular to height (z). After
splitting, all the analyses are done on each plane of the divided
workspace. The starting position for the analysis of each plain is
zero value of the polar coordinates (ρ and β). The first polar
coordinate (ρ) presents the axial distance between the z-axis and
the desired point, and the second polar coordinate (β) is the angle
from the positive x-axis to the first polar coordinate.
After defining all values of polar coordinates, it is necessary
to convert polar coordinates to the Cartesian coordinates (x, y,
z). This conversion uses the inverse kinematics equations and
the required limits to verify the defined point possible to achieve
in any end-effector orientation. The first step in this procedure is
to find the first height that can be reached with the end-effector.
International Journal of Electrical Engineering and Computing
Vol. 6, No. 1 (2022)
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Figure 7. Block diagram for achievable workspace analysis
The second step is to raise the value of polar coordinate ρ
iteratively and, after every raise, convert polar coordinates to
Cartesian coordinates and verify them. The first point that does
not come through verification is dismissed, and the polar
coordinate β rises value, while the polar coordinate ρ is set to
value zero. The variable used for raising the value of polar
coordinates is a given constant.
The shape and size of the achievable workspace analyzed on
a plain defined by height (z) is completed, after achieving the
total circle value with polar coordinate β, by connecting the
points with a high value of polar coordinate ρ for every polar
coordinate β (Fig. 9). The process is iteratively repeated for
every plain defined by height. The machine's achievable
workspace is shown in Fig. 10, and it presents the analyzed
planes while every plain is connected.
Figure 8. The workspace analysis
The presented achievable workspace has a specific shape
because of the required limits of the active-joint variables. The
value of the defined limits is set using the geometry to avoid the
collision of active machine parts. Required limits are used to
determine the connections between active machine parts as well
as to define the proposed mechanism itself. Importing the
needed limits in the mathematical model of the machine shows
a more realistic presentation of the achievable workspace.
Figure 9. Workspace analysis on a single plain
Ljubomir Nešovanović et al.
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Figure 10. The achievable workspace
The first limitation of the proposed mechanism is the
maximal and minimal length that the translation joint can reach
on the shaft. The total length of all three shafts is the limitation
that limits the height of the achievable workspace. Fig. 11 a)
shows the design of an achievable workspace generated with
previously defined limits.
The second limitation is needed because the proposed
mechanism has six translation joints and three shafts, where
every shaft has two translation joints positioned on it. Because
of this characteristic, a collision between two translation joints
positioned on the same shaft is possible. This type of collision is
avoided by including limitations in the mathematical model,
which defines minimal length between two translation joints on
the same shaft. Design of the achievable workspace generated
with previously described limitations is shown in Fig. 11 b).
The final design of achievable workspace is demonstrated in
Fig. 11 c) and includes one more limitation imported into the
mathematical model. The last limitation presents the maximal
length between two translation joints positioned on the same
shaft. This limitation prevents collision between two kinematic
chains of the same Lambda mechanism.
Figure 11. The TOW workspace
B. Total orientation workspace
Total orientation workspace (TOW) presents the workspace
achievable with the end-effector in every orientation for
determined Euler angles [11]. In the TOW analysis for the
proposed machine, Euler angles (Ψ, θ and Φ) are set on a value
between -15 and 15 degrees. Intuitively the TOW is part of the
achievable workspace, and TOW can also be defined as a set of
points reachable with an end-effector in every orientation for the
specified Euler angles.
As the part of the achievable workspace, the process of
gaining the TOW is similar to the previously explained one, for
gaining the achievable workspace. As in the case of the analysis
of achievable workspace, the design of TOW is generated using
the computer code programmed in MatLab. The algorithm used
International Journal of Electrical Engineering and Computing
Vol. 6, No. 1 (2022)
7
for acquiring this computer code is shown in Fig. 12. Because of
the similarity in the procedure of gaining the workspace, the
presented algorithm is similar to the one used in achievable
workspace analysis.
In the achievable workspace analysis, every point which can
be reached with the end-effector and does not cross any defined
limitation is imported into the database. This is not the case for
TOW analysis, and the coordinates of the considered point must
fulfill one more condition. The necessary condition is that the
considered point must be reachable with the end-effector in
every orientation for defined Euler angles values.
Figure 12. Block diagram for TOW workspace analysis
If this condition is fulfilled, the coordinates of the considered
point can be imported into the database. This means that the
active joint variables can pass defined limits for every end-
effector orientation. The explained condition is the crucial
difference between achievable workspace analysis and TOW
analysis. It is important acknowledging that if any of the defined
limitations is crossed, considered point coordinates are erased,
and the process of TOW analysis is continued by increasing the
value of the polar coordinate (β).
Furthermore, it is important emphasizing the fact that the
limitations used in TOW analysis are the same as the ones used
for achievable workspace. Defined limitations have the same
effect on the TOW design as they have on the design of
achievable workspace shown in Fig. 11.
After comparing the generated achievable workspace and the
TOW, the conclusion is that TOW has much smaller dimensions
than the achievable workspace (Fig. 13). This conclusion is
implied because of the conditions included in the TOW analysis
but not in the achievable workspace analysis. It is important to
state that the design of the TOW and the achievable workspace
is symmetrical around the X-axis, and the defined symmetry is
caused by the specific kinematics design of the proposed
machine.
Figure 13. The TOW workspace
Ljubomir Nešovanović et al.
8
VI. CONCLUSION
The main results established in this paper are inverse
kinematic equations generated from the geometric model of the
machine and workspace analysis. These shown results are
helpful for better understanding of the proposed machine.
Derived inverse kinematic equations with given input
parameters can generate the active joint variables – the positions
of the active translation joints on the shaft. With inverse
kinematic equations, it is possible to develop the proper
workspace of the parallel kinematic machine. Before using the
inverse kinematic equations in the workspace analysis, it is
crucial to verify them. The verification is done by using PTC
Creo Parametric and MatLab software. Comparison of active-
joint variables generated with these two software packages has
confirmed the accuracy of the inverse equations.
The established workspace has a complex tridimensional
shape, and the characteristic structure of the machine can explain
the complex design of the workspace. Presented workspace
analysis is based on the polar coordinates, because polar
coordinates may offer many advantages in the geometrical
analysis of the tridimensional workspace of the machine.
The workspace analysis can help define the dimensions of
the machine's active elements and the dimension of passive
components. This characteristic can help optimize the basic
parameters and whole size of the machine.
Future research may possess the optimization of workspace
based on changing the parameters of the machine. Optimized
workspaces can show proper directions for the usage of the
machine.
ACKNOWLEDGMENT
The presented research was supported by the Ministry of
Education, Science and Technological Development of the
Republic of Serbia by contract no. 451-03-68/2022-14/200105
dated 4 February 2022 and by contract 451-03-68/2022-14/
200066 dated 2022.
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Ljubomir Nešovanović (1997) received
the B.Sc. and M.Sc. degree in Mechanical
Engineering from the University of
Belgrade, Republic of Serbia. He is
working as a junior researcher at the
LOLA Institute. His field of research
interest include parallel kinematic
machines and manufacturing.
Saša Živanović (1969) is a Full Professor
at Faculty of Mechanical Engineering,
University of Belgrade, Serbia. His current
research interests are machine tools,
parallel kinematic machine tools,
reconfigurable machine tools, STEP-NC,
robots for machining, CAD/CAM, Wire
EDM, and rapid prototyping.
