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

1. 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 Campus Universitaire de Clermont-Ferrand – les Cézeaux BP 265 63175, Aubière Cedex, France. bouzgarrou@ifma.fr , chapelle@ifma.fr , fauroux@ifma.fr 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

2. 1. Introduction • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

3. 1. Introduction Existing mobile robots • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

4. 1. Introduction Existing mobile robots • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 4 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

5. 1. Introduction • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

6. 2. Open Wheel i3R • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

7. 2. Open Wheel i3R • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

8. 0 1 2 3 4 5 6 7 8 3. Mobility analysis Objective: verifying the adequacy between required mobility for the task, robot mobility and the number of actuated joints. • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion Kinematic diagram : 3 or 4 contact points R R Ct Ct Mechanism graph : complex chain (3 loops). R R R Ct Ct R R 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

9. 6 8 7 0 1 2 3 4 5 3. Mobility analysis Mobility analysis of a single module (two contact points) • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion R Case 1 : arbitrary wheel axes  connectivity Ct Ct R Case 2 : parallel wheel axes  connectivity 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

10. 8 0 1 2 3 4 5 6 7 3. Mobility analysis Mobility analysis of the mechanism (4 contact points) • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion Two modules : introduction of a “complex joint” X  simple mechanism chain (1 loop) Mobility associated to X = 3 R X R X R 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

11. 2 7 6 1 2 3 4 5 7 0 8 5 8 0 1 6 3 4 3. Mobility analysis Mobility analysis of the mechanism in climbing phase (3 contact points) Single module in climbing phase : open mechanism chain Ct • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion R R Two modules in climbing phase : Ct R R X R 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

12. 4. 2D Kinematics and dynamics kinematics analysis is performed when the robot involves on a plane surface. The four wheels are in contact with the ground. 13 geometric parameters : • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion yaw angle pitch angle roll angle Reference position Axle angles wheel angles 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

13. 4. 2D Kinematics and dynamics • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 2 4 8 1 3 7 6 5 13 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

14. 4. 2D Kinematics and dynamics Wheel center velocities (rolling contacts) • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion Compatibility relations (rigid body motion) Non holonomic constraint 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

15. 4. 2D Kinematics and dynamics Acceleration relations • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

16. 4. 2D Kinematics and dynamics • 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. • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 4 Input motor torques Wheel contact forces: 8 unknowns (normal reactions aren’t considered in 2D analysis) 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

17. 4. 2D Kinematics and dynamics Wheel dynamic equations (moment equations projected on rotation axes) • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion Dynamic equations of axles Contact forces elimination by using acceleration relations 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

18. 4. 2D Kinematics and dynamics Subsystem dynamic equations (3+5+6)  • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion (4+7+8)  • 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. 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

19. 5. 3D static analysis • 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. • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion • 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

20. 5. 3D static analysis • Isolation of subsystem S1 = (1+2+3…+8)  6 scalar equations • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion • Moment equilibrium of 3 wheels  3 scalar equations • Isolation of subsystem S2 = (3+5+6)  1 scalar equations • Isolation of subsystem S3 = (1+3+5+6)  1 scalar equations • Isolation of subsystem S4 = (2+4+7+8)  1 scalar equations • Isolation of subsystem S4 = (4+7+8)  1 scalar equations  Linear system of 13 equations / 13 unknowns 20 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

21. 5. 3D static analysis Inter axle torque needed for climbing / inter axle angle • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

22. 5. 3D static analysis Pitch angle and center of mass altitude / inter axle angle • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

23. 5. 3D static analysis Contact forces / inter axle angle • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09

24. 6. 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. • Introduction • Open Wheel i3R • Mobility analysis • 2D Kinematics and dynamics • 3D Static analysis • Conclusion 3ème Congrès International Modélisation et Conception des Systèmes Mécaniques CMSM’09