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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