1. Lecture 5: Basic Dynamical Systems CS 344R: Robotics Benjamin Kuipers

2. Dynamical Systems A dynamical system changes continuously (almost always) according to A controller is defined to change the coupled robot and environment into a desired dynamical system.

3. In One Dimension Simple linear system Fixed point Solution –Stable if k < 0 –Unstable if k > 0

4. In Two Dimensions Often, position and velocity: If actions are forces, causing acceleration:

5. The Damped Spring The spring is defined by Hooke’s Law: Include damping friction Rearrange and redefine constants

6. The Linear Spring Model Solutions are: Where r 1, r 2 are roots of the characteristic equation

7. Qualitative Behaviors Re(r 1 ), Re(r 2 ) < 0 means stable. Re(r 1 ), Re(r 2 ) > 0 means unstable. b 2 -4c < 0 means complex roots, means oscillations. b c unstable stable spiral nodal unstable

8. Generalize to Higher Dimensions The characteristic equation for generalizes to –This means that there is a vector v such that The solutions are called eigenvalues. The related vectors v are eigenvectors.

9. Qualitative Behavior, Again For a dynamical system to be stable: –The real parts of all eigenvalues must be negative. –All eigenvalues lie in the left half complex plane. Terminology: –Underdamped = spiral (some complex eigenvalue) –Overdamped = nodal (all eigenvalues real) –Critically damped = the boundary between.

10. Node Behavior

11. Focus Behavior

12. Saddle Behavior

13. Spiral Behavior (stable attractor)

14. Center Behavior (undamped oscillator)

15. The Wall Follower (x,y)

16. The Wall Follower Our robot model: u = (v  ) T y=(y  ) T   0. We set the control law u = (v  ) T = H i (y)

17. The Wall Follower Assume constant forward velocity v = v 0 –approximately parallel to the wall:   0. Desired distance from wall defines error: We set the control law u = (v  ) T = H i (y) –We want e to act like a “damped spring”

18. The Wall Follower We want For small values of  Assume v=v 0 is constant. Solve for  –This makes the wall-follower a PD controller.

19. Tuning the Wall Follower The system is Critically damped is Slightly underdamped performs better. –Set k 2 by experience. –Set k 1 a bit less than

20. An Observer for Distance to Wall Short sonar returns are reliable. –They are likely to be perpendicular reflections.

21. Experiment with Alternatives The wall follower is a PD control law. A target seeker should probably be a PI control law, to adapt to motion. Try different tuning values for parameters. –This is a simple model. –Unmodeled effects might be significant.

22. Ziegler-Nichols Tuning Open-loop response to a step increase. dT K

23. Ziegler-Nichols Parameters K is the process gain. T is the process time constant. d is the deadtime.

24. Tuning the PID Controller We have described it as: Another standard form is: Ziegler-Nichols says:

25. Ziegler-Nichols Closed Loop 1.Disable D and I action (pure P control). 2.Make a step change to the setpoint. 3.Repeat, adjusting controller gain until achieving a stable oscillation. This gain is the “ultimate gain” K u. The period is the “ultimate period” P u.

26. Closed-Loop Z-N PID Tuning A standard form of PID is: For a PI controller: For a PID controller: