1. City College of New York 1 John (Jizhong) Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu Mobile Robot Control G3300: Advanced Mobile Robotics

2. City College of New York 2 Content Mobot Kinematics Kinematic Motion Control Virtual Vehicle Approach Homework

3. City College of New York 3 Kinematics of Mobile Robots Locomotion is the process of causing an autonomous robot to move. –In order to produce motion, forces must be applied to the vehicle Dynamics – the study of motion in which these forces are modeled –Includes the energies and speeds associated with these motions Kinematics – study of the mathematics of motion without considering the forces that affect the motion. –Deals with the geometric relationships that govern the system –Deals with the relationship between control parameters and the behavior of a system in state space.

4. City College of New York 4 Kinematics Model Goal:

5. City College of New York 5 Differential Drive Nonholonomic Constraint Kinematic equation Physical Meaning? Relation between the control input and speed of wheels Posture Kinematics Model: Kinematics model in world frame L

6. City College of New York 6 Differential Drive Kinematics model in robot frame ---configuration kinematics model L

7. City College of New York 7 Kinematics Differential drive VRVR VLVL L ICC R(t) robot’s turning radius  (t)  = ( v R - v L ) / L R  = L ( v R + v L ) / ( v R - v L ) v  =  R = ( v R + v L ) / 2 V(t) xbxb ybyb

8. City College of New York 8 Inverse Kinematics Key question: Given a desired position or velocity, what can we do to achieve it? V R (t) V L (t) starting position final position x y

9. City College of New York 9 Inverse Kinematics Key question: V R (t) V L (t) starting position final position x y Given a desired position or velocity, what can we do to achieve it? world information  wheel information

10. City College of New York 10 Inverse Kinematics Key question: V R (t) V L (t) starting position final position x y Given a desired position or velocity, what can we do to achieve it?

11. City College of New York 11 Inverse Kinematics Key question: V R (t) V L (t) starting position final position x y Given a desired position or velocity, what can we do to achieve it?

12. City College of New York 12 Inverse Kinematics Key question: V R (t) V L (t) starting position final position x y Need to solve these equations: for V L (t) and V R (t). x = ∫ V(t) cos(  (t)) dt y = ∫ V(t) sin(  (t)) dt  = ∫  (t) dt  = ( V R - V L ) / L V  =  R = ( V R + V L ) / 2 There are lots of solutions... Given a desired position or velocity, what can we do to achieve it?

13. City College of New York 13 Inverse Kinematics V R (t) V L (t) starting position final position x y Finding some solution is not hard, but finding the “best” solution is... Key question: It all depends on who gets to define “best”... Given a desired position or velocity, what can we do to achieve it?

14. City College of New York 14 Inverse Kinematics V R (t) V L (t) starting position final position x y Finding some solution is not hard, but finding the “best” solution is... quickest time most energy efficient smoothest velocity profiles V L (t) t Key question: It all depends on who gets to define “best”... Given a desired position or velocity, what can we do to achieve it?

15. City College of New York 15 Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time V R (t) V L (t) starting position final position x y Differential Drive

16. City College of New York 16 Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time V R (t) V L (t) starting position final position x y Differential Drive -V L (t) = V R (t) = V max (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin.

17. City College of New York 17 Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time V R (t) V L (t) starting position final position x y Differential Drive -V L (t) = V R (t) = V max V L (t) = V R (t) = V max (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. (2) drive straight until the robot’s origin coincides with the destination

18. City College of New York 18 Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time V R (t) V L (t) starting position final position x y Differential Drive (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. (2) drive straight until the robot’s origin coincides with the destination (3) rotate again in order to achieve the desired final orientation -V L (t) = V R (t) = V max V L (t) = V R (t) = V max -V L (t) = V R (t) = V max V L (t) t V R (t) only 2 settings (on and off) needed:

19. City College of New York 19 Motion Control The objective of a kinematic controller is to follow a trajectory described by its position and/or velocity profiles as function of time. Motion control is not straight forward because mobile robots are nonholonomic systems. However, it has been studied by various research groups and some adequate solutions for (kinematic) motion control of a mobile robot system are available. Most controllers are not considering the dynamics of the system

20. City College of New York 20 Open Loop Control Trajectory (path) divided in motion segments of clearly defined shape: –straight lines and segments of a circle. Control problem: –pre-compute a smooth trajectory based on line and circle segments Disadvantages: –It is not at all an easy task to pre –compute a feasible trajectory –limitations and constraints of the robots velocities and accelerations –does not adapt or correct the trajectory if dynamical changes of the environment occur. –The resulting trajectories are usually not smooth

21. City College of New York 21 Feedback Control Compute the error and change in proportion to it.

22. City College of New York 22 Motion Control Methods Virtual Vehicle Approach Curvature Steering Method Flatness Approach Dynamic Path Following

23. City College of New York 23 A Virtual Vehicle Approach Read paper: Control of mobile platforms using a virtual vehicle approach; Egerstedt, M. Hu, X. Stotsky, A., IEEE Transactions on Automatic Control, Volume: 46, Issue: 11, pp 1777-1782, 2001.Control of mobile platforms using a virtual vehicle approach

24. City College of New York 24 A Virtual Vehicle Approach The robot model: The control objective:

25. City College of New York 25 A Virtual Vehicle Approach Possible controller 1:

26. City College of New York 26 A Virtual Vehicle Approach The motion parameter:

27. City College of New York 27 A Virtual Vehicle Approach Orientation of the vehicle

28. City College of New York 28 A Virtual Vehicle Approach –The forward velocity:

29. City College of New York 29 Homework 3 Consider a differential drive mobile robot and write a short article (5~6 pages) which includes at least the following information: –The derivation of the kinematics model of the mobile robot: Include all the details, e.g., what are the assumptions for the kinematics model? What is the constraint for the kinematics model? What are the coordinate systems? –The development of a motion control algorithm for path tracking (virtual vehicle or others) –The simulation/experimental results for your algorithms with discussions: Include the simulation results for at least two paths: a circle and a sinusoidal wave. Please vary your parameters and compare your results.

30. City College of New York 30 Homework 3 The format of the short article shall follow the IEEE paper standard as follows: Title Author with affiliation Abstract Introduction Kinematics Model Path Tracking Control Simulation and Discussion Conclusions Reference

31. City College of New York 31 Thank you! Homework 3 posted Next class: Feb. 20, 2007