1. Dynamics Kris Hauser I400/B659, Spring 2014

2. Agenda Ordinary differential equations Open and closed loop controls Integration of ordinary differential equations Dynamics of a particle under force field Rigid body dynamics

3. Dynamics How a system moves over time as a reaction to forces and torques Distinguished from kinematics, which purely describes states and geometric paths Uncontrolled dynamics: From initial conditions that include state x 0 and time t 0, the system evolves to produce a trajectory x(t). Controlled dynamics: From initial conditions x 0, time t 0, and given controls u(t), the system evolves to produce a trajectory x(t)

4. Dynamic equations

7. From second-order ODEs to first-order ODEs

9. From time-dependent to time- independent dynamics

10. Dynamic equation as a vector field Can ask: From some initial condition, on what trajectory does the state evolve? Where will states from some set of initial conditions end up? Point (convergence), a cycle (limit cycle), or infinity (divergence)?

11. Numerical integration of ODEs

12. Integration errors

13. Open vs. Closed loop Open loop control: The controls u(t) only depend on time, not x(t) E.g., a planned path, sent to the robot No ability to correct for unexpected errors Closed loop control : The controls u(x(t),t) depend both on time and x(t) Feedback control Requires the ability to measure x(t) (to some extent) In either case we have an ODE, once we have chosen the control function

14. Controlled Dynamics -> 1 st order time-independent ODE

17. DYNAMICS OF RIGID BODIES

18. Rigid Body Dynamics The following can be derived from first principles using Newton’s laws + rigidity assumption Parameters CM translation c(t) CM velocity v(t) Rotation R(t) Angular velocity  (t) Mass m, local inertia tensor H L

19. Rigid body ordinary differential equations

20. Kinetic energy for rigid body Rigid body with velocity v, angular velocity  KE = ½ (m v T v +  T H  ) World-space inertia tensor H = R H L R T vv T vv H 0 0 m I 1/2

21. Kinetic energy derivatives

22. Summary Gyroscopic “force”

23. Force off of COM x F

24. x F

25. Generalized torque f Now consider the equivalent force f, torque τ at COM

26. Generalized torque f

27. f

28. Principle of virtual work f F

29. f F fτfτ

30. Next class Feedback control Principles App J