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Preliminary Design and Analysis the Mobile Robot Open WHEEL i3R Belhassen Chedli BOUZGARROU, Frédéric CHAPELLE, Jean-Christophe FAUROUX Laboratoire de Mécanique et Ingénierie Institut Français de Mécanique Avancée et Université Blaise Pascal - PowerPoint PPT Presentation
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3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
Preliminary Design and Analysis the Mobile Robot Open WHEEL i3RBelhassen Chedli BOUZGARROU, Frédéric CHAPELLE, Jean-Christophe FAUROUX
Laboratoire de Mécanique et IngénierieInstitut Français de Mécanique Avancée et Université Blaise PascalCampus Universitaire de Clermont-Ferrand – les Cézeaux BP 26563175, Aubière Cedex, France.
[email protected] , [email protected] , [email protected]
1
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
23ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
1. Introduction1. Introduction
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
1. Introduction1. Introduction
3
Existing mobile robots
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
1. Introduction1. Introduction
Existing mobile robots
4
5
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
1. Introduction1. Introduction
6
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
2. Open Wheel i3R2. Open Wheel i3R
7
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
2. Open Wheel i3R2. Open Wheel i3R
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
Kinematic diagram : 3 or 4 contact points
Mechanism graph : complex chain (3 loops).
Objective: verifying the adequacy between required mobility for the task, robot mobility and the number of actuated joints.
8
0
1
2
3
4
5
6
7
8
R
R
R
R
R
R
R
Ct
Ct
Ct
Ct
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3. Mobility analysis3. Mobility analysis
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
Mobility analysis of a single module (two contact points)
Case 1 : arbitrary wheel axes connectivity
Case 2 :parallel wheel axes connectivity
9
0
1
2
3
4
5
6
7
8
R
R
Ct
Ct
26)1232(1 M
35)1232(1 M
11
1
n
iifM
61
51
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3. Mobility analysis3. Mobility analysis
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
Two modules : introduction of a “complex joint” X simple mechanism chain (1 loop)
Mobility associated to X = 3
10
k
iifM
1
36)1332( M
0
1
2
3
4
5
6
7
8
R
R
R
X
X
Mobility analysis of the mechanism (4 contact points)
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3. Mobility analysis3. Mobility analysis
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
11
Single module in climbing phase : open mechanism chain
Two modules in climbing phase :
0
1
2
3
4
6
7
8
R
Ct
l
iifM
12
4312 M
4613431
k
iifM
0
1
2
3
4
6
7
8
R
Ct
R
R
R
X
Mobility analysis of the mechanism in climbing phase (3 contact points)
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3. Mobility analysis3. Mobility analysis
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
),(),( 1111 baba xxzz
),(),( 1111 zzyy bb
12
kinematics analysis is performed when the robot involves on a plane surface. The four wheels are in contact with the ground.13 geometric parameters :
),(),( 1010 aa yyxx
pitch angle
yaw angle
roll angle
Reference position TRzyx01O1O1O10OO
),(),(
),(),(
),(),(
424224
313113
212112
yyxx
yyxx
zzyy
Axle angles
).,(),(;),(),(
;),(),(;),(),(
848448747447
636336535335
zzxxzzxx
zzxxzzxx
wheel angles
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
4. 2D Kinematics and 4. 2D Kinematics and dynamicsdynamics
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
1313
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
12
21 xx
24
42 zz
1331 zz
35
47
3653 yy
74 yy
63 yy
1
2
3
4
5
6
7
8
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
4. 2D Kinematics and 4. 2D Kinematics and dynamicsdynamics
14
Compatibility relations (rigid body motion)
Non holonomic constraint
Wheel center velocities (rolling contacts)
3350/55)( xrRV RRO
3360/66)( xrRV RRO
4470/77)( xrRV RRO
4480/88)( xrRV RRO
)()( 36352135
e
rR
)()( 48472245
e
rR
))(sin())(sin( 363513484724)43(2
ll
rR
0))(cos())(cos( 363513484724
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
4. 2D Kinematics and 4. 2D Kinematics and dynamicsdynamics
Acceleration relations
15
313363533635
/
))(()(2033
yx
rRΓ RRO
)()( 36352135
e
rR
424484744847
/
))(()(2044
yx
rRΓ RRO
)()( 48472245
e
rR
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
4. 2D Kinematics and 4. 2D Kinematics and dynamicsdynamics
Wheel contact forces: 8 unknowns (normal reactions aren’t considered in 2D analysis)
4 Input motor torques
16
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
Dynamic analysis : deriving the vehicle equations of motion submitted to four input wheel torques.
Evaluation of acceleration capacities for given masses and body inertias, motor and components dimensioning, robot control and trajectory planning…
Newton-Euler formulation is used with appropriate projections and by isolating well chosen subsystems.
;;;; 4848474736363535 yCCyCCyCCyCC
TRT
R
TR
TR
ZYXFZYXF
ZYXFZYXF
44
33
0808080807070707
0606060605050505
;
;
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
4. 2D Kinematics and 4. 2D Kinematics and dynamicsdynamics
Contact forces elimination by using acceleration relations
17
Wheel dynamic equations (moment equations projected on rotation axes)
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
rJXrRC /))(( 05535
rJXrRC /))(( 06636
rJXrRC /))(( 07747
rJXrRC /))(( 08848
Dynamic equations of axles
mJXXe /)( 0605513
mJXXe /)( 0807524
)()()(
)()()(
87)(
)(2487)(
)(0807
65)(
)(1365)(
)(0605
225
522
5
225
522
5
CCCCXX
CCCCXX
mrmr
m
mrmr
m
JrRJe
rRe
JrRJe
JrR
JrRJe
rRe
JrRJe
JrR
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
4. 2D Kinematics and 4. 2D Kinematics and dynamicsdynamics
System of Differential Algebraic Equations
Elimination of contact forces by solving a linear system.
Equations of motion are obtained from wheel dynamic equations (accelerations of the 4 wheels) since the robot trajectory is completely determined from wheel velocities and kinematic constraints.
Subsystem dynamic equations
18
(3+5+6) rmmm
XXXrRRRO xΓ
2
)cos(353623/
13310605
033)(
(4+7+8) rmmm
XXXrRRRO xΓ
2
)cos(484724/
24310807
044)(
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
4. 2D Kinematics and 4. 2D Kinematics and dynamicsdynamics
19
5. 3D static analysis5. 3D static analysis
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
Static analysis aims to : determine the inter axle joint torque needed to lift up a
wheel as well as wheel torques that maintain vehicle equilibrium.
verify and control vehicle static stability characterized by wheel ground normal contact forces.
13 static unknowns:• 9 wheel contact forces (3 contact points)• 3 wheel input torques• 1 inter axle joint torque
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
2020
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
• Isolation of subsystem S1 = (1+2+3…+8) 6 scalar equations
0)(
0
5/
/
1
1
IM
F
Sext
Sext
• Moment equilibrium of 3 wheels 3 scalar equations00535 RXC00636 RXC 0)cos( 0747 XrRC
• Isolation of subsystem S2 = (3+5+6) 1 scalar equations0)O( 33/ 2
zM Sext
0)O( 11/ 3 xM Sext
• Isolation of subsystem S3 = (1+3+5+6) 1 scalar equations
• Isolation of subsystem S4 = (2+4+7+8) 1 scalar equations0)O( 11/ 4
xM Sext
• Isolation of subsystem S4 = (4+7+8) 1 scalar equations0)O( 44/ 5
zM Sext
Linear system of 13 equations / 13 unknowns
5. 3D static analysis5. 3D static analysis
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
21
Inter axle torque needed for climbing / inter axle angle
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
5. 3D static analysis5. 3D static analysis
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
22
Pitch angle and center of mass altitude / inter axle angle
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
5. 3D static analysis5. 3D static analysis
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
23
Contact forces / inter axle angle
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
5. 3D static analysis5. 3D static analysis
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
24
1. Introduction
2. Open Wheel i3R
3. Mobility analysis
4. 2D Kinematics and dynamics
5. 3D Static analysis
6. Conclusion
6. Conclusion6. Conclusion
3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09
A new concept of mobile robot evolving in unstructured environment is presented.
The OpenWHEEL i3R uses a serial inter-axle mechanism.
This concept was approved in term of adequacy between required mobilities, mechanism mobilities and actuated joints.
2D kinematic and dynamic modelling gives the basic relations to perform mechanical design, trajectory planning and robot control.
Static analysis allows verifying stability conditions and determining inter axle torque needed to lift one wheel in climbing manoeuvres.
Future work will focus on the implementation of the presented models on the robot control system and 3D dynamic modelling.