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A COMPLETE DYNAMIC ANALYSIS OF STEWART PLATFORM INCLUDING
SINGULARUTY DETECTION
by
Burcu GÜNERİ
October, 2007
İZMİR
A COMPLETE DYNAMIC ANALYSIS OF STEWART PLATFORM INCLUDING
SINGULARITY DETECTION
ABSTRACT
In recent years, the field of parallel manipulators has expanded significantly in applications
requiring high precision, loading capacity, accuracy, rigidity and high velocity. Stewart
platform (hexapod) is the most common type of parallel manipulators. The limited workspace,
complex kinematic / kinetic solutions and singularities inside the workspace are the most
important problems encountered with regard to this type of robots. In this study, Stewart
platform is investigated in details by modeling and simulation, dynamic analysis and
singularity detection. Design and simulation stage, force analysis (including limb weights)
were performed by VisualNastran software. Workspace and singularity situation ere searched
by MATLAB program. In this manner, a complete package of analysis focused on Stewart
platform is presented.
Keywords: Stewart platform, hexapod, workspace, singularity, trajectory planning.
STEWART PLATFORMUNUN TEKİLLİK ARAMAYI İÇEREN
TAM DİNAMİK ANALİZİ
ÖZ
Son yıllarda, paralel manipülatörlerin kullanımı yüksek hassasiyet, taşıma kapasitesi,
doğruluk, rijitlik ve yüksek hız gerektiren uygulamalarda önemli ölçüde artmıştır. Stewart
platformu (hegzapod) paralel manipülatörlerin en yaygın tipidir. Kısıtlı çalışma uzayı,
karmaşık kinematik / kinetik çözümler ve çalışma uzayı içerisindeki tekillikler bu tip
robotlarda karşılaşılan en önemli problemlerdir. Bu çalışmada, Stewart platformu modelleme
ve benzetim, dinamik analiz ve tekillik arama yolu ile ayrıntılı bir şekilde incelenmiştir.
Tasarım ve benzetim aşaması, kuvvet analizi (bacak ağırlıkları dahil edilerek) VisualNastran
yazılımı ile gerçekleştirilmiştir. MATLAB programı ile çalışma uzayı ve tekillik durumu
araştırılmıştır. Bu şekilde, Stewart platformu odaklı tam bir analiz paketi sunulmuştur.
Anahtar sözcükler: Stewart platformu, hegzapod, çalışma uzayı, tekillik, yörünge planlama.
1. Introduction
Manipulators have become indispensable parts of human lives because of their wide range
of applications. Studies firstly started with serial manipulators continued rapidly with parallel
ones and combination of both, hybrid ones. As the technology has been developing gradually
day by day, using manipulators in applications that require micro-positioning has become an
important issue. As a result, studies on parallel manipulators have been greatly increased
because of high precision and accuracy of parallel manipulators (Merlet, 2006). After the
innovation of airplane simulator in 1965, parallel manipulators have become very popular and
great interest has been devoted by the researchers to this subject because of technological
requirements.
This study has focused on Stewart platform which is a parallel manipulator with six
degrees of freedom. The objective of this study is to examine dynamic characteristics of
Stewart platform with the singularity-free path planning. Actually, this is an integrated
research project supported by TÜBİTAK (Karagülle, H., Sarıgül, S., Kıral, Z., Varol, K., &
Malgaca, L., 2006, 2007a, 2007b).
There are many valuable studies published in the literature on different aspects or types of
parallel manipulators. The first important study was presented by Fichter (1986) on a general
theory and practical construction of Stewart platform. In this paper, kinetic, kinematic and
singularity analysis is included and a summary of the work done in Oregon State University
over the past several years is given. Gosselin (1990) introduced a method for determination of
workspace of 6-dof parallel manipulators. Masory & Wang (1995) calculated the workspace
volume and dexterity of Stewart platform by considering the effects of some parameters such
as link lengths, joint locations, and design dimensions on the workspace. Merlet, Gosselin &
Mouly (1998) classified workspace of parallel manipulators and tried to determine the
boundaries of the workspace with some geometrical computer algorithms. Dasgupta &
Mruthyunjaya (1998) investigated singularity-free path planning of Stewart platform. With
given two end-poses of the manipulator, they tried to find singularity-free path between these
points with the algorithm constructed based on the condition number. Tanev (2000) showed
kinematic analysis of a new type of hybrid (parallel-serial) robot manipulator; and presented
closed-form solutions to the forward and inverse kinematic problems. Tsai & Joshi (2000)
presented a paper on kinematics and optimization of a spatial 3-UPU parallel manipulator
incuding singularity. Merlet (2001) developed a generic trajectory verifier for the motion of
parallel robots. Lee & Shim (2001) introduced forward kinematics of Stewart-Gough platform
using algebraic elimination. Xi (2001) presented a study on hexapods with fixed-length legs;
and compared three types of hexapods: Hexaglide, sliding in the horizontal direction;
Linapod, sliding in the vertical direction and HexaM, sliding in a slanted angle. Wang, Wang,
Liu & Lei (2001) developed an algorithm for the determination of the workspace of a parallel
machine tool based on the cutter point; and preliminary solutions are given to the problem of
positioning the workpiece. Geike & McPhee (2003) studied inverse dynamic analysis of
parallel manipulators with full mobility. Sen, Dasgupta & Mallik (2003) worked on
singularity-free path planning of parallel manipulators using Lagrangian equation composed
of kinetic energy term. Gallordo, Rico, Frisoli, Checcacci & Bergamasco (2003) used another
approach to the dynamic analysis of parallel manipulators by using screw theory. Harib &
Srinivasan (2003) performed kinematic and dynamic analysis of Stewart platform based
machine structures with inverse and forward kinematics, singularity, inverse and forward
dynamics including joint friction and actuator dynamics. Hiller, Fang, Mielczarek, Verhoeven
& Franitza (2005) presented a study on tendon-based parallel manipulators. Ider (2005)
studied drive singularity condition at which actuators can not influence the end-effector
accelerations instantaneously and they lose the control of one or more degrees of freedom.
In this study, a Stewart platform (hexapod), having SPS (spherical-prismatic-spherical)
joints in each leg is considered. Workspace analysis of Stewart platform constitutes the core
of the study. The effects of geometric and kinematic constraints on the workspace were
examined. An effort was devoted to kinetic considerations, which is believed to form the
originality of this study. As far as it is known, there is no other study in the literature on the
triple relationship among “position – workspace – force” of Stewart platform. On the other
hand, inclusion of limb masses via solid modeling approaches the simulations to reality. All
these analyses and simulations provide better design and manufacturing facilities. The
stability and reliable motion ability of Stewart platform are also satisfied by detailed
singularity analysis codes.
Complete dynamic analysis of the robot refers to kinetic and kinematic studies with
analytical and simulation solutions. Solid model of Stewart platform was performed by
VisualNastran software. The parts of the robot were drawn by I-DEAS software and
transferred to VisualNastran for assembly. Some computer codes were developed in
MATLAB program using inverse kinematic equations in order to find the workplane and
workspace boundaries under the effect of geometric and kinematic parameters. The
orientation effect of the moving platform was also included in the analysis. Joint position -
workspace - actuator force relationship was examined by using the workplane boundary
curves found from MATLAB program as the inputs of the prescribed motion data in
VisualNastran. Then the actuator force data was recorded during the simulation. Completing
both kinematic and kinetic analyses, singularity analysis of Stewart platform was performed
by developed MATLAB codes and singularity-free trajectory planning was carried and a safe
motion of the robot may be satisfied.
2. Modeling and Inverse Kinematic Analysis of Stewart Platform
Figure 1. A 6-dof, 6SPS Stewart platform
Stewart platform considered in this study has 6 degrees of freedom and 6 limbs each
having spherical-prismatic-spherical joints as shown in Figure 1. Let u, v, and w are three unit
vectors defined along the x, y, and z axes of the moving coordinate system on the moving
platform, respectively. Rotation matrix which defines moving platform relative to the fixed
base can be written as:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
zzz
yyy
xxx
BA
wvuwvuwvu
R (1)
The elements of the rotation matrix should satisfy the orthogonal conditions:
1222 =++ zyx uuu
1222 =++ zyx vvv
1222 =++ zyx www (2)
0=++ zzyyxx vuvuvu
0=++ zzyyxx wuwuwu
0=++ zzyyxx wvwvwv
[ ]Tiziyix aaa=ia and [ ]TBiwiviu bbb=ib being the position vectors of a point Ai on
the fixed frame and of another point Bi on the moving frame respectively, in Figure 1, the
loop closure equation can be written as:
iii abRpBA −+= iB
BA (3)
where p is the position vector stating the location of moving platform relative to the fixed
base. To be able to calculate the length di of thi limb, the dot product of ii BA is taken with
itself. After doing that operation, one can obtain:
[ ] [ ] 6,1,2, ifor 2 …=−+−+= iiB
iiB abRpabRp B
ATB
Aid (4)
For the inverse kinematic problem, the position vector p and rotation matrix BA R are
known and the limb lengths are the values to be calculated. By expanding equation (4) and
taking the square root of the result, a general expression for the limb lengths is found as
follows:
[ ] iiB
iT
iBT
iT
iiBT
iBT abRa2pbR2paabbpp
TB
AB
Aid 2−−+++±= (5)
Equation (5) is written 6 times for each limb, to be able to calculate limb lengths
corresponding to each given location. As it is seen, two possible solutions exist; however
negative limb length is not feasible so only positive one is considered. The computer program
developed in MATLAB utilizes equation (3) for the inverse kinematic analysis in the
computation of the workplane of the robot.
Drawing the parts by using I-DEAS program in *.iges* format and transferring them to
VisualNASTRAN 4D 2004, solid model of the hexapod was prepared as in Figure 2 after
combining parts with the required joint constraints.
Figure 2. Solid model of the hexapod by VisualNastran 4D 2004
After completing the complex solid model, performing some calculations and required
simulations, finally the prototype hexapod whose photograph is shown in Figure 3 was
produced.
Figure 3. Photograph of Stewart platform produced at Dokuz Eylül
University Laboratories.
3. Workspace Analysis
Parallel manipulators have complex kinematic equations so determination of their
workspace is a challenging problem. The workspace of Stewart platform can be defined as the
reachable region of the coordinate system attached to the center of the mobile platform. There
are some geometrical or analytical methods in literature in order to calculate the workspace.
In this study, a workspace algorithm was developed in MATLAB to define the workspace of
Stewart platform. The method depends on slicing the space by parallel planes. The reachable
region of the hexapod is calculated on a defined plane and this procedure is applied for
different horizontal planes for different levels of the z- coordinate. The boundary curves of the
regions are calculated, drawn and then curves at different “z” levels are combined. As a result,
the boundary surface of the workspace volume appears (Masory & Wang, 1995).
Figure 4 shows the simple model of Stewart platform and the required notations for the
formulae used in developed MATLAB code. Spherical joints are located on the base platform
with αb angle from the axes which divide the platform into three pieces by 120º intervals.
Similarly, αp is the angle of the spherical joints on the moving platform from the axes which
Limbs
Base Platform
Moving Platform
divide the platform into three pieces by 120º intervals. Rb and Rp are radii of the circles on
which spherical joints are located on the base and moving platforms, respectively. Bi and Pi
denote the points on which spherical joints are located on the base and moving platforms,
respectively (i=1, 2,..., 6).
Figure 4. A simple model of the Stewart platform.
The program developed in MATLAB utilizes the inverse kinematic analysis in the
computation of the workplane of the robot. In order to check the actuator strokes whether they
exceed the link length limits or not, the loop closure equation (3) is used with rotation matrix
in equation (1). The constraints which are used in calculating the workplane can be
formulized as following:
)(maxmin aLLLL
i
i
≤≤
=il
)(.cos
)6()(.
cos
max1
max1
c
bR
bbi
pP
B
pi
θθ
θθ
≤=
≤=
−
−
i
bii
i
pii
lnl
lnl
Here il is the limb vector and θ is the rotation angle of the joints as shown in Figure 5. Lmin
and Lmax denote the minimum and maximum link length limitations, respectively. θbmax and
θpmax are the maximum joint angles for the joints mounted on the base and moving platforms,
respectively. Besides, nbi and npi are unit vectors in the spherical joint directions in the base
and moving platform coordinate systems, respectively.
Figure 5. Rotation angles of the joints and unit vectors in the
joint directions
In following three sections, three parameters are investigated by considering their effects
on the workplane for different orientations of the moving platform. Cross-sectional areas on a
definite plane are calculated by changing the parameter examined and keeping the other
parameters constant in all analyses. These analyses were performed for z= 0.235m plane.
3.1 Effect of Link Length Limitation
Link length limitations for defining the workspace include maximum and minimum
lengths of the links. Computer codes developed in MATLAB calculate the boundary of the
curves which define the maximum area reached by the hexapod for different motion range of
the actuators on a definite plane. Analyses were performed for different link length limits
using the formulations presented in equations (3) and (6) in order to observe the change of the
working area. Figure 6 shows the shape of workplane boundary curves for parallel orientation
of the moving platform. The workplane area becomes smaller as the link length limit
decreases. The same procedure was applied for different orientations of the moving platform
and similar curves were obtained. As a general summary of these curves, Figure 7 is
generated showing the working areas for different orientations of the moving platform.
biθpiθ
pin
bin
(a)
(b) (c)
(d) (e)
Figure 6. Workplane boundaries for βx=0o, βy=0o, βz=0o (a) Lmin=0.18m, Lmax=0.29m
(b) Lmin=0.20m, Lmax=0.275m (c) Lmin=0.218464m, Lmax=0.268928m (d) Lmin=0.22m,
Lmax=0.26m (e) Lmin=0.235m, Lmax = 0.245m
According to Figure 7, the increase in the moving range of actuators enlarges the
workplane for all orientations. It can also be stated that as the rotation of the platform
increases, the workplane area decreases. The largest working area can be obtained for the
horizontal orientation of the moving platform whereas the smallest one is obtained for the
combination of the rotations about both axes where the manipulator has very restricted
motion.
Figure 7. Working area for different link length limits and orientations.
3.2 Effect of Maximum Joint Angle Limitation
Joint angle is the tilt angle of the spherical joints. MATLAB computations give the
boundary curves which define the maximum area reached by the hexapod for different joint
angle limitations of the spherical joints on a definite plane. Similar to the previous one, this
analysis is repeated for different orientations of the moving platform. Therefore, the
relationships of workplane area, joint angle, and orientation of the moving platform can be
examined. Figure 8 shows the workplane area boundaries for six different maximum joint
angle limitations for the parallel orientation of the moving platform. As it is seen from that
figure, the decrease in maximum joint angle limit causes a decrease in the workplane area.
The same procedure was also applied for the other orientations and Figure 9 presents
workplane-joint angle limit relationship in terms of the moving platform orientation. This
curve makes a peak at 30° joint angle after which the workplane is almost constant since
beyond that angle, link length is the controlling parameter and robot can not go further.
However; for other orientations, the workplane becomes constant after 40° joint angle.
Therefore; joints can be chosen by taking link length limitations into account in order not to
have needless rotational range of the spherical joints.
(a) (b)
(c) (d)
Figure 8. Workplane boundaries for βx=0o, βy=0o, βz=0o (a) φp=45o, 40o, 30o (b) φp=25o
(c) φp=20o (d) φp=15o
Figure 9. Working area for different maximum joint angles and orientations
3.3 Effect of Joint Location
Joint location effect is considered by changing locations of the joints on the moving
platform. The base platform joint positions are kept constant as 15° for all analyses, only for
convenience. Figure 10 shows the workplane boundary curves for different joint angles for
the parallel orientation of the moving platform. When that figure is examined, it is seen that
the closer the joint location angles for the base and moving platform, the larger the
workspace. Figure 11 summarizes all the analysis by presenting the joint location angle -
workplane area relationship including the orientation of the moving platform. Workplane has
maximum area when the joint location angle is 15o, which is equal to the joint location angle
on the base platform. However, such a configuration is not recommended since in that case,
Jacobian matrix of the moving platform is singular (Masory and Wang, 1995). In the design
and assembly of the moving platform 30o joint locations were applied. This is a good
selection for reaching large workspace without the risk of singularity due to joint locations.
(a) (b)
(c) (d)
(e) (f)
Figure 10. Workplane boundaries for βx=0o, βy=0o, βz=0o (a) αp=10o (b) αp=15o
(c) αp=25o (d) αp=35o (e) αp=50o (f) αp=70o
Figure 11. Working area for different joint location angles and orientations.
3.4 Joint position – Workspace - Actuator Force Relationship
This section presents the results of kinetic analysis subjected to Stewart platform. Joint
position – workspace – actuator force relationship is investigated by considering five different
designs simulated according to the joint location angles of the moving platform. This
investigation includes analyses performed by both VisualNastran and MATLAB.
For MATLAB analyses, below procedure are followed:
• In order to see the workplane - force relationships in terms of joint locations, five
different joint location angles were used for the moving platform. The procedure of
obtaining the workplane boundary curves and finding the actuator forces in pursuing
this trajectory was applied for αp= 20°, 25°, 30°, 36°, 44° on z=0.25763m plane which
lies approximately on one-third of the platform’s moving range.
• Base platform’s joint location angles were kept constant as αb= 15° in each analysis.
• Cross-sectional area inside boundary curves of the workspace was calculated
numerically by Green’s theorem for each joint location angle. The coordinates of the
points on those workplane boundary curves were recorded while the developed
MATLAB code was running.
For VisualNastran simulations, below procedure are followed:
• The recorded coordinates of the points on boundary curve of the computed workplane
were transferred to the VisualNastran 4D 2004 program as inputs of the prescribed
motion. It was adjusted the robot to complete the motion on this trajectory in 360
seconds with 0.5 second intervals. It should be noted that in this first set of simulations,
force constraint was chosen and set to zero for all of the actuators. During the
simulation, actuator length data was saved for all actuators.
• Then, before starting the second set of simulation, the prescribed motion input for the
moving platform was disabled. For the actuators, actuator length constraint was chosen
instead of the force constraint.
• Second simulation set was started with the previously recorded actuator length inputs
giving the same trajectory; and forces on the actuators were measured.
• Force values with respect to time were recorded and force- time graphs obtained are
presented in Figure 12 for each actuator with the joint location angle of 30o.
• These analyses were performed in the same manner four more times for the other joint
location angles which are 20°, 25°, 36° and 44°.
Figure 12. Variation of actuator forces on the pursuit of boundary trajectory
3.5 Combined Results of Both Analyses
It can be said that trajectories and the variation of forces are similar in each analysis. When
the force-time graphs in five analyses are compared, it is seen that larger workplane areas
result in larger forces on the linear actuators. In Figure 13, the MATLAB results for
workspace area values of five joint location angles are shown. In Figure 14, VisualNastran
results for maximum forces among all of the actuators are shown with respect to joint angles.
As shown in Figure 13, as the value of the joint location angle on the moving platform
becomes closer to 15o, the workplane area increases and reaches its maximum value.
Generally, actuator forces decrease with closer joint location angles for the same trajectories,
since the limbs stand more vertically. However, in Figure 14, they become greater since the
robot follows workplane boundary trajectories in order to compare the worst conditions
among all of the actuators in terms of the maximum force. As the robot reaches further points
with increasing workplane boundaries, the forces become larger. Comparison of Figures 13
and 14 shows that, when joint location angles become closer to each other, the gain in the
surface area is very small whereas the change in the force value is really significant. This fact
shows that an optimization study is required in order to determine the best joint location
suitable for the desired application.
Figure 13. Workplane areas for αp= 20°, 25°, 30°, 36°, 44°.
Figure 14. Maximum actuator forces for αp= 20°, 25°, 30°, 36°, 44° .
3.6 3-D Workspace Analysis
A closed 3-D workspace of the hexapod is constructed by using the procedure of obtaining
2-D workplane boundary curves several times for each z level and then combining these
curves. The workspace of the produced Stewart platform is established for the four previously
analyzed orientations of the moving platform. The “z” plane interval is [0.23m- 0.27m] with
Δz= 0.001m steps. It should be noted that “z” level indicates the center point of the plane
where the centers of the spherical joints lie. The computations with regard to workspace led to
Figure 15. In Figure 15, isometric views of workspace for four orientations of the moving
platform are shown. The workspace gets smaller with the orientation angle. According to
direction of rotation, the symmetry of the graph changes as shown in Figures 15 (b)-(d). As
the height of the moving platform increases, linear actuators elongate both horizontally and
vertically and so the workspace becomes narrower.
(a) (b)
(c) (d)
Figure 15. Isometric views of workspace of produced Stewart platform for (a) βx=0o, βy=0o, βz=0o
(b) βx=5o, βy=0o, βz=0o (c) βx=0o, βy=5o, βz=0o (d) βx=5o, βy=5o, βz=0o.
4. Singularity Analysis
Singularity is an important phenomenon in order to provide success in the motion of the
robots in terms of accomplishing the desired task. Singularity analysis of robots should be
performed before giving a task to the robot including the desired motion.
4.1 Jacobian Analysis of Stewart Platform
Derivation of the velocity loop-closure equations and evaluation of the Jacobian matrix
should be performed in order to describe the singular or ill-conditioned configurations of the
Stewart platform. It is important to identify the singular positions and orientations since in
such configurations, manipulator may gain extra degrees of freedom; becomes uncontrollable;
or the forces on the actuators may diverge to infinity, which results in the breakage of the
actuators. Jacobian matrix provides a relation between input limb velocities and output linear
and angular velocities of the moving platform.
For the manipulator in Figure 4, input vector is Tddd ],...,,[.
6
.
2
.
1=.ρ , which defines the
velocities of six limbs and output vector is ],,,,,[ pzpypxpzpypx vvv ωωω=.x , which defines the
linear and angular velocities of the moving platform. Loop closure equation for the ith limb is
given as,
ObOP + OPPi = ObBi + Bi Pi (7)
Differentiating equation (7) with respect to time yields,
iiiiPp ssωpωv.
ii dd +×=×+ (8)
where pi and si denote the vector ObOP and a unit vector along Bi Pi , respectively. ωi denotes
angular velocity of the ith limb with respect to the global coordinate frame having origin Ob.
If both sides of equation (8) are multiplied by si as dot-product multiplication; ωi is eliminated
and .
)( ip d=⋅×+⋅ ωspvs iipi (9)
is obtained. By writing equation (9) for each limb and combining these six scalar equations in
matrix form,
.
ρ
.
x ρJxJ = (10)
is obtained. Here,
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
×
××
=
TT
TT
TT
sps
spssps
)(...
)()(
666
222
111
xJ and IJ ρ = ( 6 x 6 identity matrix).
4.2 Condition Number
Condition number is a term which is used as a measure of problem’s amenability to be
solved; that is, how numerically well-posed a problem is. It is also used in singularity analysis
of manipulators since this analysis helps classifying the configurations as singular, ill-
conditioned or well-conditioned by examining the condition number of the Jacobian matrix.
Condition number of Jacobian matrix shows the stability or sensitivity of that matrix to
numerical operations. It can be defined as the ratio of the largest eigenvalue to the smallest
eigenvalue of the related matrix. Minimum possible value of the condition number is “1” and
small values indicate well-conditioning. Large values of the condition number indicate ill-
conditioning and at a singular configuration, condition number becomes infinite. Condition
number κ (J) of the Jacobian matrix may be formulated as (Tsai & Joshi, 2000):
1JJJ −=)(κ (11)
where . denotes the norm of the matrix inside it.
In order to have non-singular configuration of the robot, condition number must be lower
than a limit value, κlim. (Sen, Dasgupta & Mallik, 2003). This limit may change for different
design parameters. When condition number is evaluated, it can be seen that it generally lies in
an interval; therefore, the limit for the condition number can be chosen by multiplying the
upper interval bound with a safety factor. For example, this safety factor may be assumed
between 1.5 and 4 but it should be remembered that taking large condition number limits
causes more rough results and may lead researcher to lose some of the singular points.
5. Trajectory Planning
Defining the path to be followed by a manipulator is required in order to perform a desired
task. The most important step in this procedure is checking whether the given trajectory is
valid or not; and if it is not valid, arranging singularity-free path accordingly. This kind of
study is called “motion or trajectory planning”. For parallel manipulators, singularities exist
both on the workspace boundaries and inside the workspace. The validation criteria for
parallel manipulators are given as following:
• The trajectory must lie inside the workspace of the robot; which means that link
length and maximum joint angle limitations can not be exceeded and link
interference should be avoided.
• The trajectory must not have any singular points on it.
• If the singularity is present on the trajectory, the path should be designed again
without passing the singular points found.
In this section, singularity analysis is performed on the basis of trajectory planning. By
examining singularity requirements, it is possible to determine whether the given trajectory is
valid or not.
5.1 Reason for Condition Number Based Analysis
At the beginning of the study, some computer codes which check the singularity of the
Jacobian matrix, Jx, had been developed. By these codes, the Jacobian matrix had been
calculated both inside the workplane and on the workplane boundaries. The points where
determinant of Jacobian matrix equals to zero had been investigated. Actually, it is very
difficult to find the points for which the determinant of Jacobian matrix is exactly zero.
Therefore, the points for which the algebraic sign of the determinant changed had been
investigated as the first attempt. Many analyses had been performed for different paths and
configurations of the robot in order to find a sign change. However, computed Jacobians had
very low values and they did not have a sign change. For this reason, another method based
on condition number was used. In this method, stability of Jacobian matrix was investigated
with the calculation of the condition numbers on the points that the robot passes.
5.2 Singularity Analysis in the Workspace
Singularity analysis was performed within the workspace and on the workspace boundaries
for which singularity is more probable. In order to perform the analysis, the Stewart
platform’s workplane boundary data was used at different heights for different orientations of
the moving platform. It should be noted that singularity analysis was performed for the
continuous workplanes introduced. That is, the height of the moving platform was so chosen
that on these planes there was no unreachable region for the manipulator.
Condition numbers were calculated for so many regularly placed points within the
workspace. Four of the condition numbers computed for each workplane boundary are
presented in Table 1 as suggested by Sen, Dasgupta and Mallik (2003). These are first point,
last point, minimum and maximum condition numbers. First point and last point correspond
to the beginning and end of the trajectory. These four condition numbers are sufficient to
detect the variation of their values. The upper limit of the non-singular condition number can
be taken as 1.5 times of the higher condition number for the first point or last point on the
trajectory. It should be noted that these analyses were performed for the link length limit of
0.050464m and maximum rotational joint angle limit of 45o on different z planes. The
procedure was also repeated for different orientations of the moving platform about the main
axes.
In Table 1, condition numbers for the parallel orientation of the moving platform at
different heights are shown. The empty row indicates that the workplane is not continuous at
that height. When the results are examined, it can be said that condition numbers found are
small enough implying no singularity.
Figure 16 shows the nature of the condition numbers on the same graph with different
orientations. When Figure 16 is examined, it is seen that lowest condition numbers are
reached for the parallel orientation since the motion ability of the robot is maximum for that
configuration and the robot moves without extra force. On the other hand, rotations about
only one axis give larger results than the rotation about both axes. The reason for this
phenomenon may be the dimension of the workspace. It is smaller for the combined
orientation so condition numbers for this case are lower than those for single axis rotation.
Lastly, the decrease in condition numbers due to the height is another observation for this
analysis. As shown in Figure 16, the robot has maximum condition numbers for the lowest
“z” values in all orientations since the workplane area is maximum for these heights.
Condition numbers increase with increasing workspace in the same orientation.
Table 1. Condition numbers for moving platform orientation: βx = 0o, βy = 0o, βz = 0o.
Plane level z (m) First point condition number
Last point condition number
Minimum condition number
Maximum condition number
0.210 - - - -
0.220 57.0480 70.4531 57.0480 70.5771
0.230 59.6411 68.9893 59.6411 69.0905
0.240 62.2341 67.9158 62.2341 67.9930
0.250 64.8272 67.2726 64.8272 67.2726
0.260 67.4203 67.6089 67.4203 67.6089
67
67,5
68
68,5
69
69,5
70
70,5
71
71,5
72
0,22 0,225 0,23 0,235 0,24 0,245 0,25 0,255 0,26 0,265
z (m)
max
imum
con
ditio
n nu
mbe
rs
βx = 0 βy = 0 βz = 0
βx = 5 βy = 0 βz = 0
βx = 0 βy = 5 βz = 0
βx = 5 βy = 5 βz = 0
Figure 16. Maximum condition numbers for different orientations.
Conclusion
In the area of robotics, mathematical design strategy based on the theory and simulation of
the theoretical model is an important issue before the production process. Such studies
guarantee the working ability of the produced robot. Therefore, the possible failure of the
robot due to choosing wrong design parameters can be prevented by real time simulations.
In this study, Stewart platform was investigated in details in terms of inverse kinematics,
kinetics, workspace and singularity by modeling, simulation and running generated computer
codes. The methodology considered in this study is general and applicable to any Stewart
platform. As a result, possible errors of design can be eliminated before the production stage
and robot’s motion space can be determined.
In this study, some important geometric and kinematic parameters affecting the workspace
of Stewart platform were examined based on inverse kinematic analysis. These analyses were
performed by the computer codes developed in MATLAB. The parameters are link length
limit, maximum joint rotation angle limit and joint location angle. The orientation angle of the
moving platform is another factor considered.
Findings related to workspace:
• Increase in the link length and joint rotation angles limits results in the increase of the
workspace.
• Choosing a moving platform joint location angle which is close to the value of the
base platform joint location angle increases the workspace. However, same joint
location angles for the base and moving platforms cause singularity or ill-
conditioning of the robot.
• Maximum workspace volume is obtained for the parallel orientation whereas
minimum workspace volume is provided by orientation about both axes. In addition,
the change in the shape of workspace occurs due to the different orientations of the
moving platform.
The actuator forces were determined via VisualNastran 4D software by including the limb
weights. Combined relationship of joint location angle – workspace – actuator force was
investigated.
Findings related to actuator forces:
• Closer joint location angles for moving and base platforms decrease actuator forces if
the same trajectory is followed in each case. However, for the motion on the
boundaries of the workspace, the actuator forces increase as joint location angles get
closer.
• When the decreasing effect of close joint location angles and increasing effect of the
motion on the boundary curves in terms of actuator forces are compared, it is seen that
the increasing effect of the motion on the workspace boundaries is much more
dominant.
• Therefore, the necessity of an optimization study arises for deciding joint location
angles according to the task.
Singularity analysis was performed based on the computation of the Jacobian matrix and
the condition number by running the developed MATLAB codes.
Findings related to singularity:
• As the robot moves further in its workspace, condition number of the corresponding
position increases.
• The orientation angle of the moving platform affects the condition numbers and
increasing of orientation angle causes a slight increase in condition numbers.
• The analysis on the effect of limb connections puts forward that cross type connection
for a known singular configuration prevents the robot from having singularity. Cross
type connection results in reasonable condition numbers.
In this study, all the developed algorithms and simulations were applied to prototype
hexapod designed and produced in Mechanical Engineering Department of Dokuz Eylül
University as the TÜBİTAK Research Project. All the foregoing analysis reveals that design
parameters chosen for the prototype model are really good and the mechanism is very stable
with reasonable results. Singularity analysis can be also performed for real paths as soon as
the trajectory of the robot in practical application is decided. By this way, any undesired
characteristics of motion can be prevented by MATLAB analysis without causing the failure
of the real model.
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