Simulation with Software Tools

Transcription

1 Seminar Modeling and Simulation of Dynamical Systems Presented by the IEEE Control Systems Society Santa Clara Valley Sunnyvale, 5 February Session 5 Simulation with Software Tools Karl Mathia 1

2 Session 5: Software Tools Topics: Example: Simple Pendulum Dynamical model: Differential equation Discrete time Dynamics: numerical integration Software: Commercial Open source 3 Example: Simple Pendulum Dynamical model: differential equation Several methods available: Moment equation Force equation Lagrange s method Euler s method 4

3 Example: Simple Pendulum Moment equation Force equation 5 Example: Simple Pendulum Moment equation: With: 6 3

4 Example: Simple Pendulum Differential equation: =0 (unforced) As state equation: 1 ml () t ( k l () t m g sin( ()) t l M ()) in t k 1 g () t 0 sin( ()) t M (), in t 0 m m l l 7 Example: Simple Pendulum State equation: vector-valued, nonlinear function x, x x, u M k 1 x () t 0 sin( x1( t)) x( t) in x () t 0 x () t ut () m ml 8 4

5 Example: Simple Pendulum Simulation of dynamics: num. integration Simple approximation: 9 Example: Simple Pendulum More sophisticated: Runge-Kutta 4 th order 10 5

6 Session 5: Software Tools Good news: We don t need to worry about numerical integration. There are many software packages available to perform such computations. Commercial SW and Open Source SW 11 Session 5: Software Tools Commercial Software: LabView (presented earlier) MathCad (presented earlier) Matlab (general) Simulink (graphical programming) Mathematica (general) Maple (general) SPICE (simulates electrical circuits) 1 6

7 Session 5: Software Tools Pendulum in Matlab: function xdot = pend(t, x) xdot(1,1) = x(); xdot() = -k/m*x() - w0_*sin(x(1)) + 1/(m*l^)*u; Calling Function t = linspace (0.0, 5.0, 501); x0 = [pi-0.01; 0.0]; [tt,xa] = ode45(@pend,t,x0); % time axis in [s] % initial condition 13 Session 5: Software Tools Pendulum in Matlab: representations 14 7

8 Session 5: Software Tools Open Source Software: GNU Octave (emulates Matlab) Scilab (emulates Matlab) Maxima (symbolic processing) Python (scripting language) Bullet Physics Open Dynamics Engine and many other packages 15 Session 5: Software Tools Pendulum in GNU Octave: function xdot = pend(x, t) xdot(1,1) = x(); xdot() = -k/m*x() - w0_*sin(x(1)) + 1/(m*l^)*u; Calling Function t = linspace (0.0, 5.0, 501); x0 = [pi-0.01; 0.0]; xa = lsode("pend", x0, t); % time axis in [s] % initial condition 16 8

9 Bullet Physics (Open Source) Software Development Kit (SDK), Library: Collision detection, rigid body dynamics, cloth Open source (Zlib lic.) C++ PlayStation 3, Xbox 360, Multi-core CPUs Animation Movies Computer Games 17 Bullet Physics: SDK, Library 01 : Destroying Los Angeles Remember: this is open source software! 18 9

10 Open Dynamics Engine Open source SDK: Simulating rigid body dynamics Advanced joints, collision detection with friction Simulating vehicles, objects in virtual reality environments, virtual creatures C/C++ API 19 Open Dynamics Engine Physics-based skater animation Youri Dimitrov, Cheng Zhang, 008 ( 0 10

11 Session 3: Simulation Annotated bibliography Home page of Matlab, a "technical computing language." Application for computations. GNU Octave is a highlevel language, primarily intended for numerical computations. Home page of Scilab, free and open source software for numerical computation. Bullet Physics website with downloadable distribution, tutorials, documentation, demos. ODE is an open source, high performance library for simulating rigid body dynamics. 1 Session 5: Software Tools Q & A 11

12 Program Welcome Session 1 Session Session 3 08:45 09:10am 09:10 10:00am 10:10 11:00am 11:10 1:00am Coffee and bagels, Seminar kickoff at 9:00am Mathematical models of dynamical systems Dr. P.K. Menon, Optimal Synthesis System Identification - Theory and Practice Dr. Mark B. Tischler, Ames Research Center Visualization and Virtual Environments Dr. Hadi Aggoune, Cogswell Polytech. College Lunch Session 4 Session 5 1:00 1:40pm 1:40 01:30pm 01:40 :30pm Sandwiches, sodas, discussions and product demos Applications of Hardware-in-the-Loop Simulators Christoph Wimmer, National Instruments Simulation with Software Tools Elliot English, Dr. Martin Aalund, Dr. Karl Mathia 3 Sponsors