ÉCOLE POLYTECHNIQUE DE MONTRÉAL

MODELIZATION OF A 3-PSP 3-DOF PARALLEL MANIPULATOR USED

AS FLIGHT SIMULATOR MOVING SEAT.

MASTER IN ENGINEERING PROJET III

MEC6903

SUBMITTED TO:

Luc Baron Ph.D.

Mechanical Engineering Department

SUBMITTED BY:

Jaime Pacheco (1562321)

April 23, 2013.

I

Table of Contents

1.0 Introduction ………………………………………………………………………………………………………………………

1

2.0 Manipulator Description …………………………………………………………………………………………

3

2.1 Inverse Kinematic Model ……………………………………………………………………………………

5

2.2 Direct Kinematic Model ………………………………………………………………………………………

9

3.0 Jacobian Matrix Analysis ………………………………………………………………………………………

11

4.0 Workspace analysis ………………………………………………………………………………………………………

14

5.0 Conclusion ……………………………………………………………………………………………………………………………

21

6.0 References …………………………………………………………………………………………………………………………… 22

II

Figure List

Fig. 1. Flight simulator moving seat using a 3-PSP parallel manipulator............. 2

Fig. 2. Kinematic diagram of the 3-PSP parallel manipulator. ........................................ 3

Fig. 3. Tilt-and-torsion angle representation [12]............................................................... 4

Fig. 4. Vectors in 3-PSP parallel manipulator. ......................................................................... 6

Fig. 5. Kinematic simulation of the 3-PSP parallel manipulator performed in

Catia V5................................................................................................................................................................... 7

Fig. 6. Parasite displacement for different values of and for a 3-PSP

parallel manipulator with . ........................................... 8

Fig. 7. Manipulator simplification used to obtain direct kinematic equations. 9

Fig. 8. Result of the numerical simulation using jacobian matrix. ........................ 13

Fig. 9. Workspace in coordinated system for a 3-PSP parallel manipulator

with . Ratio . .................................................... 15

Fig. 10. Workspace in coordinated system for a 3-PSP parallel manipulator

with and . Ratio ....................... 16

Fig. 11. Workspace in coordinated system for a 3-PSP parallel manipulator

with and . Ratio . ............................ 17

Fig. 12. Workspace for a 3-PSP parallel Manipulator in coordinated system .

..................................................................................................................................................................................... 18

Fig. 13. Workspace in coordinated system for a 3-PSP parallel manipulator

with and whit (a) and Ratio ,

(b) and Ratio ,and (c) and Ratio . ............ 19

Fig. 14. Reference point position for the flight simulator moving seat

workspace. ............................................................................................................................................................. 20

Fig. 15. Workspace in coordinated system ; (a), (b)and(c); and (d) in

coordinate system for the flight simulator moving seat. ......................................... 20

Fig. 16. Relation between the maximum reachable value and the ratio .

..................................................................................................................................................................................... 21

1

1.0 Introduction

In recent years the use of parallel manipulator has been increased due they

have a number of advantages, when compared to traditional serial arms, such

as high accuracy, high rigidity, low inertia, high load-to weight ratio and

good dynamic performance [1]. The actual research is focused over the

applications like high-speed machine tools, motion simulators, micro motion

manipulators, force /torque sensor, etc [2]. However the parallel manipulator

has some disadvantage such as small workspace, complex forward kinematics,

and a complicated set of universal and spherical joints.

Initially the attention was concentrated in parallel manipulators with six

degrees of freedom (DOF) [3]. For flight simulators, the most used mechanism

is the Steward Platform, a 6-DOF fully parallel manipulator. However this

mechanism is very expensive and complex, and it is best suited for full

trained simulators. Actually, the research increasingly focuses on parallel

manipulators with a limited-DOF because there are many applications that

don’t need all DOF, and specifically, flight simulator with a limited-DOF can

be used in the first stages of pilot training [4]. Furthermore, these

mechanisms have a lot of advantages in terms of simplicity in construction

and control, and a reduced cost [5]. Especially the 3-DOF spatial parallel

manipulators (SPM) have been considered to be the most useful SPM for hybrid

machine tool [5,6] , e.g., the 3-PRS (Prismatic-Revolute-Spherical) type.

They are used primarily as a plug and play module for numerical control

machine for the fabrication of large components in the aircraft and

automobile industries [7]. However, some parallel manipulator presents

complications due a coupled orientation and position for the end-Effector

[8].

In the literature, the most studied configuration for limited-DOF SPM is

three equal kinematical chains used as legs that join two platforms and has

3-DOF. This configuration is very useful from the point of view of design

and construction, and moreover, that permits to interchange the role of the

base and moving platform because the mechanism is topologically symmetric

[9]. One of these manipulators is the 3-PSP (prismatic-Spherical-Prismatic)

3-DOF, which is studied in this paper for an application as flight simulator

moving seat shown in Fig. 1. In this case all passive joints are kept in the

base platform in order to ensure the security of the pilot, obtaining a

configuration that is upside-down relative to those studied in the literature

[2,7,8].

The 3-DOF SPM are grouped in function of the DOF used by their end effector

as: (1) translational, (2) Rotational and (3) coupled rotational-

translational. The most famous full translational mechanism is the Delta

robot presented By Clavel [10] that have been quite studied in the

literature. There is also a lot of research in fully rotational 3-DOF

parallel manipulator like The Agile Eye presented by Gosselin & Hamel[11].

The 3-PSP is in the third group, because position is coupled with the

orientation.

2

For this report, two rotational DOF ( ) and one transnational (z) DOF are

studied, and displacements in X and Y are considered as parasitic motion

depended on and . First, in section 2, a description of the 3-PSP

manipulator and the orientation representation Tilt and Torsion [12] is done.

A geometric method is use to solve the forward kinematic problem and a

simplification of the mechanism is used to solve the direct kinematic

problem. Next, the jacobian matrix is obtained from the forward equations and

then a numerical simulation of the position, speed and accelerations is

performed. Finally in section 4, the workspace is analyzed using an iterative

technique.

Fig. 1. Flight simulator moving seat using a 3-PSP parallel manipulator.

3

2.0 Manipulator Description

The 3-PSP parallel manipulator has a Moving platform, three legs and a fixed

base. The moving platform is an equilateral triangle inscribed into a

circumference with radius r. There are three legs and each one has an

actuated prismatic joint rigidly attached to the mobile platform and normal

to it. The connection points between legs and the moving platform are the

vertex of the triangle identified as . These points define the plane

containing the mobile platform which will be named { }. The fixed base has three non-actuated prismatic joints that are coplanar and equally spaced at

120°. A spherical joint makes the connection between the active prismatic

joint of the leg and the passive one on the fixed base. The plane containing the fixed base and the three passive prismatic joints will be called and it

is defined by the points , and . These points correspond to the center

of the spherical joints. Thus, we have three kinematic chains, Prismatic-

Spherical-Prismatic, in this mechanism and 3-DOF.

A1

A2

A3

B1

B2

B3

XY

Z

O

120°

Active prismatic

joint

Passive prismatic

joint

MobilePlatform

Fixedbase

UV

WP

Fig. 2. Kinematic diagram of the 3-PSP parallel manipulator.

As shown in Fig. 2, the fixed coordinated frame denoted as { } has its

origin at the intersection point of the axis of displacement of the three

passive prismatic joints in the fixed base. The -axis is normal to the base

platform and oriented toward moving platform, and the -axis is orientated

Plane { }.

Plane { }

4

toward point . The origin of the moving coordinate frame { } is

placed in the center of the moving platform, the -Axis is oriented toward

point and the -axis is normal to the moving platform. The relation of

moving coordinate frame { } with respect to base coordinate frame { } is

defined by rotation matrix

[2].

[

] 1)

In this case an orientation representation called till-and-torsion [12] is

used, which is represented by three angles. Figure 3(a) shows The angle ,

called Tilt, that is a rotation around axis placed onto original plane.

The orientation of axis is done by angle , called azimuth, which is the

angle between the projection of axis onto the fixed plane and the fixed

axis. The third angle is shown by Fig. 3(b) and is a rotation about the

and is called Torsion. For 3-PSP the angle is always zero. In order to

avoid the singularity at , we set the ranges of azimuth and tilt angles,

respectively, and .

Fig. 3. Tilt-and-torsion angle representation [12].

5

2.1 Inverse Kinematic Model

The unitary vector, in the coordinate frame { } for the direction of passive prismatic joints can be written as:

(2)

where ⁄ .

Equally, the position vector of point in the coordinate frame { } is defined by:

(3)

where is the radius of the moving platform.

The vector expressed in the coordinate frame { } is given as:

(4)

The position vector of the origin of frame { }, i.e. point , can be expressed

in the coordinate frame { } as:

The vector from origin point of the frame { } to the point is defined by:

(5)

Let be defined as a unit vector normal to the moving platform in the

coordinate frame { } and having the same positive direction of the

displacement axis of active prismatic joints:

(6)

Thus, vector expressed in coordinate frame { } is:

(7)

6

bi

O

a

i

ki

UV

WP

Ai

Bim

i

Bip

MobilePlatform

Fixedbase

Active prismatic

joint

Passive prismatic

joint

Fig. 4. Vectors in 3-PSP parallel manipulator.

As shown in Fig. 4, vectors and are coplanar, therefore we must have:

(8)

(9)

where is the distance from point to point ,

is the distance from point to point and

is a unit vector defined in (2).

The inverse kinematic model of the 3-PSP parallel manipulator can be obtained

from (8). The displacements of the active prismatic joints are done as

function of the generalized coordinates and :

(10)

7

√

(11)

√

(12)

The 3-PSP have 1-DOF in translation, and 2-DOF in rotation and . An important feature of the 3-PSP parallel manipulator is the parasitic

displacement in and , which can be calculated from (9):

(13)

(14)

The invers kinematic equation was verified using a kinematical simulation

performed in Catia V5 as shown in Fig. 5.

Fig. 5. Kinematic simulation of the 3-PSP parallel manipulator performed in

Catia V5.

Moving

platform.

Fixed base

Plan { }.

Active prismatic

joint axis

Passive prismatic

joint axis

8

Fig. 6. Parasite displacement for different values of and for a 3-PSP

parallel manipulator with .

Figure 6 shows the parasite displacement for various values of z and . We can see how the capacity of the 3-PSP Parallel manipulator to rise different

position and and orientation change as function of z. In this case the manipulator finds its maximum capability at .

-15 -10 -5 0 5 10 15 20 25 30-25

-20

-15

-10

-5

0

5

10

15

20

25

X (mm)

Y(m

m)

Parasitic displacement for z = 80

-30 -20 -10 0 10 20 30 40 50-40

-30

-20

-10

0

10

20

30

40

X (mm)

Y(m

m)

Parasitic displacement for z = 56

-20 -15 -10 -5 0 5 10 15 20 25-25

-20

-15

-10

-5

0

5

10

15

20

25

X (mm)

Y(m

m)

Parasitic displacement for z = 40

-6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

2

3

4

5

X (mm)

Y(m

m)

Parasitic displacement for z = 20

(a) (b)

(c) (d)

Workspace limit Workspace limit

Workspace limit

Workspace limit

⁄ ⁄

⁄ ⁄

⁄ ⁄

⁄

⁄ ⁄

⁄

⁄

⁄

⁄

9

2.2 Direct Kinematic Model

In order to simplify the calculation of the direct kinematic equations is

necessary to clarify that the value of generalized coordinate is always the

minimal distance between the point and the fixed plane { }; and is the

angle between moving plane { } and the fixed plan { }. The plans { } and { } has been defined in section 2. Thus, it is possible to invert the manipulator

placing the original moving platform as base plane and introducing a new

coordinate frame { } where axis is normal to the { } plane and

axis is oriented toward point , as shown in Fig.7. Furthermore, we ignore

the passive prismatic joins for this calculation because the only information

needed is the position of the center of the spherical joints defined as point

U’V’

W’P

A2

B2

A3

B3A1

B1

n2

n1

Fig. 7. Manipulator simplification used to obtain direct kinematic equations.

The position vector of the point , in coordinated frame { } can be written as:

(15)

Plane{ }.

Plane { }

Plan { }.

10

The position vector of the origin point in the coordinate frame { }, can be written as:

(15)

To define the plane { } formed by points , and , it is necessary set two

vectors as:

(16)

(17)

To obtain the equation of the plane { } the cross product between (16) and (17) is computed:

[ √

√

]

(18)

where is a vector normal to plane { }.

Thus, the equation of plane { } is:

√

√

√

(19)

From (19) we can calculate the value of the generalized coordinate , that is

the distance from plane { } to point .

| |

√

(20)

Calculating the angle between the normal vectors to each plane, can be

obtained, as follow:

(

‖ ‖‖ ‖)

(21)

11

where is the unit vector of the -axis in the coordinate frame { }.

Developing (21), the direct kinematic equation of can be written as:

(

√

)

(22)

From (10) and (11), the direct equation of can be computed and written as:

(23)

√

(24)

(25)

3.0 Jacobian Matrix Analysis

The velocities and accelerations of the active prismatic joins of the 3-PSP

parallel manipulator can be obtained by direct differentiation of the inverse

kinematics equations (10),(11) and (12); and can be expressed as follow:

(26)

(27)

Equation (26) can be rewritten in matrix form as:

(28)

where is the serial jacobian matrix, is the parallel jacobian matrix,

and

.

12

From (26) and (28), the parallel jacobian matrix can be written as:

[

( √ )

√

( √ )

√

]

(28)

In this case the serial jacobian matrix is a 3x3 identity matrix.

The jacobian matrix was validated using the resolved motion rate (RMR)

method. The RMR can be expressed as:

[

] (28)

Using this method and knowing the initial value of and , and

applying small increments of the generalized coordinates, the final position

of the active prismatic joints can be obtained. These increments must be as

smallest as possible to ensure the accurate of the calculation.

A numerical simulation using a MatLab Script have been performed applying a

constant velocity for the coordinates and . The values used in the

simulation are:

⁄

⁄

⁄

The results obtained are shown in fig.8. As we can see, constant velocities

in and , develop a great variation in linear speed and acceleration in

the active prismatic joints.

13

Fig. 8. Result of the numerical simulation using jacobian matrix.

14

4.0 Workspace analysis

To obtain the workspace of the 3-PSP parallel manipulator, an iterative

method has been performed. The algorithm has three nested loops. In the

outermost, z value increases from zero to a maximum value, which corresponds

to the maximum value of displacement of the three active prismatic joints

( ). The second loop sweeps from – to . Finally, the

inner loop sweeps from zero to the point at which the limits of the

prismatic joints is achieved . At that point, the value of the

three generalized coordinates and , and the parasite displacements

coordinates, and are stored. With this list of points, the external

surface of the manipulator workspace can be obtained. Fig. 9 shows the

workspace in the coordinate system for a manipulator with

.

Algorithm to obtain the external point of the workspace using Invers

Kinematic solution.

for z=0 TO 10, step size= 0.1 for = - TO ; step size= 0.1 =0; for = 0 TO ; step size= 0.1 Computing Invers kinematic of the 3-PSP having as

results if Point inside of workspace. else Point outside of workspace

Save de previous point as a limit element of the workspace

end end end end

An alternative method using the direct kinematic solution is performed. In

this case three nested loop increase de value of from 0 to , keeping

the active prismatic joints displacements inside their joints limits. Thus,

all the points obtained by using the algorithm are inside the workspace.

Algorithm to obtain the workspace using Direct Kinematic solution.

for = z=0 TO 10, step size= 0.1

for = z=0 TO 10, step size= 0.1

for = z=0 TO 10, step size= 0.1

Computing direct kinematic of the 3-PSP having as results end end end

15

Fig. 9. Workspace in coordinated system for a 3-PSP parallel manipulator

with . Ratio ⁄ .

Figure 9(a) shows the volume that forms the workspace. This begins with a

point in . As can be seen in the Fig. 9(b) and 9(c), the cross sections

begin to increase until reaching approximately , at this point the

cross section begins to decrease again until a point at z = 100mm.

Additionally, the work space is symmetrical every 120°. An important factor

-40-20

020

4060

-40

-20

0

20

400

20

40

60

80

100

X (mm)Y (mm)

Z (

mm

)

-30 -20 -10 0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50

X (mm)

Y (

mm

)

-30 -20 -10 0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50

X (mm)

Y (

mm

)(a)

(b) (c)

16

that we have found is the ratio between the radius of the moving platform and

the maximum displacement of the active prismatic joints, ⁄ . When this

ratio is changed, the form of the work space is modified. As can be seen in

Fig. 10, when the ratio is reduced to 0.5, the z value where the maximal

extension of the workspace is reached increase front 56mm to 64mm.

Additionally, making a comparison between Fig. 9, 10 and 11, we can see that

when this ratio increases, the upper working space becomes convex approaching

a spherical section and the workspace volume become bigger, when the ratio is

reduced, the workspace becomes more concave.

Fig. 10. Workspace in coordinated system for a 3-PSP parallel manipulator

with and . Ratio ⁄ .

-15 -10 -5 0 5 10 15 20 25 30-25

-20

-15

-10

-5

0

5

10

15

20

25

X (mm)

Y (

mm

)

-15 -10 -5 0 5 10 15 20 25 30-25

-20

-15

-10

-5

0

5

10

15

20

25

X (mm)

Y (

mm

)

(a)

(b) (c)

-20-10

010

2030

-40

-20

0

20

400

20

40

60

80

100

X (mm)Y (mm)

Z (

mm

)

17

Fig. 11. Workspace in coordinated system for a 3-PSP parallel manipulator

with and . Ratio ⁄ .

Fig. 12 shows the workspace in the coordinate system , for a 3-PSP parallel

manipulator with . In this case, only the surface that

indicate the upper and lower boundaries is shown. The lateral limit are

determined by the range defined for and in section 2.0, i.e. ,

and . Thus, the workspace is all the points between the upper and lower

boundaries.

-40 -20 0 20 40 60 80-60

-40

-20

0

20

40

60

X (mm)

Y (

mm

)

-40 -20 0 20 40 60 80-60

-40

-20

0

20

40

60

X (mm)

Y (

mm

)

(a)

(b) (c)

mm mm

mm

-50

0

50

100

-100

-50

0

50

1000

20

40

60

80

100

X (mm)Y (mm)

Z (

mm

)

18

Fig. 12. Workspace for a 3-PSP parallel Manipulator in coordinated system .

Making a comparison between the fig. 13(a), 13(b) and 13(c) is easy to

determine that for a larger ratio ⁄ , we obtain a greater orientation

capacity of manipulator in . The Relation between the maximum attainable

value of and the ratio ⁄ is shown in Fig. 16.

So far, we have studied the case in which the displacement of the active

prismatic joints is equal to the maximum length of the leg. This case does

not apply to our flight simulator in which the actuators only have a maximum

displacement of 25 mm and the benchmark for the workspace is located in the

center of the driver's head, at a height of 1255 mm about the fixed base, as

shown in Fig. 14. To get this workspace, we use the algorithm based on the

direct kinematics and taking a displacement between 1530mm and 1555mm for the

prismatic joint and a mobile base with 382 mm radius. Figure 15 (a), (b) and

(c) show the workspace in the coordinate system , and Fig. 15 (d) shows it

in the system . As we can see, the result achieved is very different from

that obtained when considered complete displacement actuators. In addition,

analyzing Fig. 15 (d) we can observe that the orientation in theta capacity

is extremely low, reaching only a maximum of 0.044 rad. This because the

ratio between the and is very small ⁄ .

19

Fig. 13. Workspace in coordinated system for a 3-PSP parallel manipulator

with and whit (a) and Ratio ⁄ ,

(b) and Ratio ⁄ ,and (c) and Ratio ⁄ .

20

Fig. 14. Reference point position for the flight simulator moving seat

workspace.

Fig. 15. Workspace in coordinated system ; (a), (b)and(c); and (d) in

coordinate system for the flight simulator moving seat.

(a) (b)

(c) (d)

1255mm

21

Fig. 16. Relation between the maximum reachable value and the ratio ⁄ .

5.0 Conclusion

1. A new architecture of the 3-PSP parallel manipulator 3-DOF was

introduced using the orientation representation Tilt and Torsion.

2. The inverse kinematic model was obtained with a geometric method and

verified using a kinematic simulation performed in Catia V5.

3. To resolve the direct kinematic problem, a simplification of the 3-PSP

parallel manipulator mechanism with a geometric approach was used to

develop the equation for and . The Angle was obtained from

inverse kinematic equations.

4. The Parallel jacobian matrix was obtained by direct differentiation of

the inverse kinematic equations and validated using Resolved Motion

Rate method.

5. Finally, the robot reachable workspace was determinate. The variations

for different configuration were analyzed, introducing the ratio

⁄ as an index of the ability of orientation in for the 3-PSP

manipulator. The difference in the workspace between full and partial

displacement actuator for the active prismatic joints was established.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

qi max/r

(

rad

)

22

6.0 References

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Parallel Manipulator. Robotica, 2006,24:239-250.

[2] Hao Q, guan L, wang J, wang L. Dynamic feedfoward control of a

novel 3-PSP 3-DOF parallel Manipulator. Chinese Journal of

Mechanical Engineering,2009, 20: 1-8.

[3] Harib K, Ullah A, Sharif M, Hammami A. A hexapod-based machine

tool with hybrid structure: kinematic analysis and trajectory

planning. International journal of machine tools & manufacture,

2007,47:1426-1432

[4] Pouliot N, Gosselin C, Nahon M. Motion simulation capabilities of

three-degree-of-freedom flight simulators. AIAA Journal Aircraft,

1998,35:9–17. [5] Sun T, Song Y, LI Y, LIU L. Dimensional synthesis of 3-DOF

parallel manipulator based on dimensionally homogeneous Jacobian

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[6] Chen C, Huang Y, Han X, Cheng X, Guo S. Study and Analysis of the

3-PRS Parallel Mechanism. IEEE International conference on

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[7] Rezaei A, Akbarzadeh A, Aahmoodi P, Akbarzadeh M. Position,

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manipulator. Robotics and Computer-integrated manufactured, 2013,

29:158-173.

[8] HAO Q, GUAN L, LIU X, WANG L. Dynamic analysis of a novel 3-PSP

3-DOF parallel manipulator. ASME/IFToMM International Conference on

Reconfigurable Mechanisms and Robots, 2009: 309-314.

[9] Di Gregorio R, Parenti-Castelli V. Position Analysis in

analytical form of the 3-PSP Mechanism. Journal of Mechanical

Design, 2001, 123: 51-57.

[10] Clavel R. DELTA, a fast robot with parallel geometry. Proceedings of the 18th International Symposium on Industrial Robots, 1988:

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[11] Gosselin C, Hamel J. The Agile Eye - a high-performance 3-

degree-of-freedom camera-orienting device. IEEE international

conference on robotics and automation proceedings, 1994: 781-786.

[12] Bonev I, Gosselin C. Singularity locy of spherical parallel

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