1. עקיבה אחר מטרה נעה Stable tracking control method for a mobile robot מנחה : ולדיסלב זסלבסקי מציגים : רונן ניסים מרק גרינברג

2. 1.1 Project objective  Kinematical analysis of the system in order to determine it's stability and controllability properties.  Design of a non-linear stabilizing controller by applying direct Lyaponov method.  Modification of the controller to suit the robot’s dynamic constraints.

3. 1.2 Project outline.  This Project is based on the work of Y.Kanayama, Y.Kimura, F.Miyazaki and T.Noguchi "A stable tracking control method for an autonomous mobile robot"  This project deals with the control law of a vehicle that’s tracking a moving target.  The controller manufactures the linear and angular speed outputs in order to place the robot near the target  The nature of this control system is nonlinear. As in Many control problems of this kind, the controller will be designed using a Lyaponov function.  The controller parameters, efficiency and error characteristics will be determined through simulations.

4. 1.2 Project outline. – Cont’  A linearized version of the controlled system will be studied in order to obtain the relation between the controller parameters and characteristics like overshoot and settling time  In order to examine our controller model in more realistic environment, velocity limiters and dynamics will be applied do the model. Appropriate simulations will demonstrate the updated system performance

5. 1.3 Theoretical background Lyapunov Stability Theorem Lyapunov Function

6. 1.3 Theoretical background Non-holonomic systems A non-holonomic system is a system whose motion is restricted by non- holonomic constraints. Definition: A non-holonomic constraint is a limitation on the allowable velocities of an object. For example, two wheeled robots: the robot can move in some directions (forwards and backwards), but not others (side to side). We write a constraint equation for this kinematical system: What does this equation tell us? It tells us the direction we can’t move in So if θ=0, then the velocity in y = 0 ; if θ =90, then the velocity in x = 0.

7. 2.1 Problem definition  Consider the following 2D representation of the target and controlled vehicle:

8. 2.1 Problem definition – Cont’ In the robot’s frame of reference, the error coordinates Pe are : In the robot’s frame of reference, the error coordinates Pe are :

9. 2.2 Nonlinear controller design.  Our aim is to minimize Pe. For that purpose we shall examine the following proposed Liapunov equation:  In order for this equation to be definite semi-negative, we can simply determine: Which is our control rule

10. 2.2 Nonlinear controller design.  Block diagram of the basic layout of the tracking system:

11. 2.2 Nonlinear controller design.  A simulation of this controller :

12. 2.3 Controller parameter analysis  For Larger Values of K, the tracking vehicle performs better.  Each K value has different impact on the controller ’ s velocity accommodation to the reference trajectory. Larger values of K will close the gap more rapidly, enabling the system to cope with higher frequency inputs.

13. 2.3 Controller parameter analysis  To illustrate this, let us now view the vehicle's reaction to high reference speed with varying gain parameters:

14. 2.3 Linearization Motivation Linearizing the system will benefit us in a number of ways : :  We can use the linearization method of Liapunov in order to prove that the controlled system is indeed stable :  We would like to examine the performance of the linearized system in comparison to that of the original system to show how accurately the linearized system represents the original one  Linearization enables us to use simple tools from the theory of linear system in order to characterize the system by overshoot, settling time and bandwidth, which of course isn't possible in the nonlinear system as these concepts lose their meaning.

15. 2.3 Linearization System linearization the state-space representation of the controlled closed loop system is given by : Linearizing the system around the origin:

16. 2.3 Linearization System linearization By comparing the linear and nonlinear error vector around the equilibrium, we extract the new : These changes perform the linearization in practice. (Using Routh-Horowitz algorithm, we get that, for all K values, all the eigenvalues of the matrix A are negative i.e. A is negative definite), thus, According to Lyaponov linearization method, the origin is a globally asymptotic equilibrium point. )

17. 2.3 Linearization A simulation illustrating a comparison between the linear and nonlinear controller performance :

18. 2.3 Linearization A simulation illustrating the limited convergence region of the linearized system :

19. 2.3 Linearization A simulation illustrating a comparison between the linear and nonlinear controller performance : The simulation demonstrates the limited convergence region of the linearized system.

20. 2.3 Linearization We can determine the location of each pole by Applying the following : By approximating the linearized system to a second order LTI system, we can link the controller parameters with bandwidth, overshoot and settling time requirements.

21.   We can determine the location of each pole by Applying the following : By approximating the linearized system to a second order LTI system, we can link the controller parameters with bandwidth, overshoot and settling time requirements. 2.3 Linearization

22. 2.4 Additional features intended for the robot application Distance between wheels The considered robotic vehicle has two wheels (left and right) each of which is spinning in a different velocity. Creating the new state variables:,. The dynamics of such vehicle are given by: Where V is the current velocity and Vcommand is the requested (by the controller) velocity

23. 2.4 Additional features intended for the robot aplication Applying the Dynamics to each of the wheels: We get the Following dependency on Tao

24. 2.4 Additional features intended for the robot aplication Now, adding limitors to each wheel’s velocity:

25. 2.4 Additional features intended for the robot application This dynamic response means that there are two additional state variables. Therefore an adjustment of the Lyapunov function and of the controller is needed to stabilize the new system: The above is one of the possible Lyapunov functions leading to a possible controller.

26. 2.5 Concludion  The controller was developed using classic design methods.  Our controller is simple and stabilizes the system globally-asymptotically once the target is in motion.  We have explored the limits of our controller in face of more realistic constraints such as velocity limiters and dynamic inhibition.  Possible direction for future projects: Implementation on a real lab robot.

27. 2.6 Thank you! We would like to thank our instructor Vladislav for his help and guidance and also to all the control lab staff. We would like to thank our instructor Vladislav for his help and guidance and also to all the control lab staff.