3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

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

mailto:[email protected]



1. Introduction

2. Open Wheel i3R

3. Mobility analysis

4. 2D Kinematics and dynamics

5. 3D Static analysis

6. Conclusion


1. Introduction1. Introduction

1. Introduction

2. Open Wheel i3R




6. Conclusion



3

Existing mobile robots

1. Introduction

2. Open Wheel i3R




6. Conclusion



Existing mobile robots

4

5

1. Introduction

2. Open Wheel i3R




6. Conclusion



6

1. Introduction

2. Open Wheel i3R




6. Conclusion


2. Open Wheel i3R2. Open Wheel i3R

7

1. Introduction

2. Open Wheel i3R




6. Conclusion

2. Open Wheel i3R2. Open Wheel i3R


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




6. Conclusion

3. Mobility analysis3. Mobility analysis


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




6. Conclusion



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




6. Conclusion



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




6. Conclusion



),(),( 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




6. Conclusion

4. 2D Kinematics and 4. 2D Kinematics and dynamicsdynamics


1313

1. Introduction

2. Open Wheel i3R




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



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




6. Conclusion



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




6. Conclusion



Wheel contact forces: 8 unknowns (normal reactions aren’t considered in 2D analysis)

4 Input motor torques

16

1. Introduction

2. Open Wheel i3R




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

;

;



Contact forces elimination by using acceleration relations

17

Wheel dynamic equations (moment equations projected on rotation axes)

1. Introduction

2. Open Wheel i3R




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



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




6. Conclusion



19

5. 3D static analysis5. 3D static analysis

1. Introduction

2. Open Wheel i3R




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


2020

1. Introduction

2. Open Wheel i3R




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



21

Inter axle torque needed for climbing / inter axle angle

1. Introduction

2. Open Wheel i3R




6. Conclusion



22

Pitch angle and center of mass altitude / inter axle angle

1. Introduction

2. Open Wheel i3R




6. Conclusion



23

Contact forces / inter axle angle

1. Introduction

2. Open Wheel i3R




6. Conclusion



24

1. Introduction

2. Open Wheel i3R




6. Conclusion

6. Conclusion6. Conclusion


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.

Documents

3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09