시뮬레이션의 기초

1. 시뮬레이션의 기초 서울대학교 컴퓨터공학부 운동연구실 이윤상

2. What is the “Simulation”? • Simulation is the imitation of some real thing, state of affairs, or process. - Wikipedia • Simulation is the process of designing a model of a real system and conducting experiments with this model … – R. E. Shannon

3. Simulation in various areas • Economics simulation • Military simulation • Urban planning • Flight simulation • Physics (dynamics) simulation • …

4. Physics Simulation • Simulate “Laws of Physics” Soft body Fluid Rigid body

5. Equations of Motion • Equations that describe the behavior of a system as a function of time

6. Solving Equations of Motion ? … t0 t1 t2 t3

7. Solving Equations of Motion • State : • Given , we want to know ? … t0 t1 t2 t3

8. Solving Equations of Motion • Integration (h : time-step)

9. Solving Equations of Motion • Integration (h : time-step)

10. Solving Equations of Motion • Integration (Euler method) • Integrate method • Euler, Runge-Kutta, backward Euler(implicit) (h : time-step)

11. Simple system : Particle System m (mass) • Particle : x (position) v (velocity)

12. Simple system : Particle System m (mass) • Particle : x (position) v (velocity) (only applied force : F = -mg) t:0.00 x:1.00 v:-0.33 t:0.03 x:0.99 v:-0.65 t:0.07 x:0.97 v:-0.98 t:0.10 x:0.93 v:-1.31 t:0.13 x:0.89 v:-1.63 t:0.17 x:0.84 v:-1.96 t:0.20 x:0.77 v:-2.29 t:0.23 x:0.70 v:-2.61 t:0.27 x:0.61 v:-2.94 t:0.30 x:0.51 v:-3.27 t:0.33 x:0.40 v:-3.59 …

13. Particle System • Spring force

14. Particle System • Spring force

15. Particle System • Spring force

16. Particle System

17. Stability & Time-step Size • Too big step size -> simulation diverges • Too small step size -> simulation goes very slow • Depends on system stiffness & integration methods timestep = 0.003 timestep = 0.033

18. Rigid Body Dynamics • Rigid body : R (orientation) ω (angular velocity) I (moment of inertia) m (mass) x (position) v (velocity)

19. Rigid Body Dynamics • State : • Integration ... • Usually used in simulation of physical systems where rotational motion is important, but material deformation does not have a significant effect.

20. Joint Constraint • Degrees of freedom (DOF) : number of independent variables that specify position (or orientation) of the system ball joint 3 DOF floating body 6 DOF universal joint 2 DOF hinge joint 1 DOF

21. Articulated Body 3 DOF 6 DOF Total DOF of system = 6 + 3*12 = 42

22. Articulated Body

23. Controller • A controller is a device which monitors and affects the operational conditions of a given dynamical system ?

24. Biped Control • Under-actuated system • We can only use internal joint torques

25. PD (Proportional Derivative) Control • Compute joint torques directly • PD gain tuning is difficult (may required to set different gains for each joint) generated torque desired pose current pose

26. Tracking Control with Inverse Dynamics • Use inverse dynamics of under-actuated system • input : desired joint accelerations, ground reaction force • output : torques at actuated joints • Little need of gain tuning

27. simulation reference motion

28. Thank you