modelization of a 3-psp 3-dof parallel manipulator used as
TRANSCRIPT
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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