Analytical Mechanics: MATLAB

Transcription

1 Analytical Mechanics: MATLAB Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 1 / 47

2 Agenda Agenda 1 Vector and Matrix 2 Graph 3 Ordinary Differential Equations 4 Optimization 5 Parameter Passing 6 Random Numbers 7 Summary Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 2 / 47

3 Agenda Problem We drive a 2-DOF open loop menipulator based on joint PID control. Let us simulate the motion of the manipulator. l2 y lc2 link 2 joint 1 lc1 l1 θ1 θ2 joint 2 link 1 x Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 3 / 47

4 Agenda Problem Step 1. Step 2. Step 3. Step 4. Derive equations of motion (kinematics / dynamics) Numerically solve the derived equations of motion Describe the derived numerical solution by graphs or movies (visualization) Analyze the simulated motion Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 4 / 47

5 Agenda Problem Soloving a set of simultaneous linear equations [ ] [ ] H11 H 12 ω1 = H 12 H 22 ω 2 [ f1 f 2 ] Solving a set of ordinary differential equations [ ] [ ] ω1 α1 (θ = 1, θ 2, ω 1, ω 2 ) ω 2 α 2 (θ 1, θ 2, ω 1, ω 2 ) Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 5 / 47

6 Agenda What is MATLAB? 1 Software for numerical calculation 2 can handle vectors or matrices directly 3 Functions such as ODE solvers and optimization 4 Toolboxes for various applications 5 both programming and interactive calculation Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 6 / 47

7 Agenda What is MATLAB? MATLAB environment MATLAB Total Academic Headcount (TAH) MATLAB with all toolboxes is available April March Information matlab.html rainbow/service/software_matlab_student.html Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 7 / 47

8 Agenda What is MATLAB? Install MATLAB into your own PC or mobile Sample programs are on the web of the class Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 8 / 47

9 Vector and Matrix Vector and Matrix Column vector x = [ 2; 3; -1 ]; Row vector y = [ 2, 3, -1 ]; Matrix A = [ 4, -2, 1;... -2, 5, 2;... -2, 3, 2 ]; Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 9 / 47

10 Vector and Matrix Vector and Matrix Symbol... implies that the sentense continues. Column vector x = [ 2;... 3; ]; Column vector x = [ 2; 3; -1 ]; Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 10 / 47

11 Vector and Matrix Vector and Matrix Multiplication p = A*x; q = y*a; >> p p = >> Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 11 / 47

12 Vector and Matrix Vector and Matrix Multiplication p = A*x; q = y*a; >> q q = >> Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 12 / 47

13 Vector and Matrix Matrix operations >> A A = >> A(3,2) ans = 3 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 13 / 47

14 Vector and Matrix Matrix operations >> A A = >> A(3,2) = 6; >> A A = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 14 / 47

15 Vector and Matrix Matrix operations >> A(3,:) ans = >> A(:,2) ans = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 15 / 47

16 Vector and Matrix Matrix operations >> A A = >> A(:,2) = [ 0; 2; 1 ]; >> A A = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 16 / 47

17 Vector and Matrix Matrix operations >> A A = >> A(3,:) = [ 3, -5, -1 ]; >> A A = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 17 / 47

18 Vector and Matrix Matrix operations >> A A = >> B = A([1,3],:); >> B B = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 18 / 47

19 Vector and Matrix Matrix operations >> A A = >> C = A(:,[2,1]); >> C C = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 19 / 47

20 Vector and Matrix Basic row operations A(3,:) = 5*A(3,:); multiply the 3rd row by 5 A(1,:) = A(1,:) + 4*A(2,:); add 4-times of the 2nd row to the 1st row A([3,1],:) = A([1,3],:); exchange the 1st and the 3rd rows Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 20 / 47

21 Vector and Matrix Solving simultaneous linear equation A = [ 4, -2, 1;... -2, 5, 2;... -2, 3, 2 ]; p = [ 1; 9; 3 ]; Solve a simultaneous linear equation Ax = p >> x = A\p; >> x x = >> A*x Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 21 / 47

22 Vector and Matrix Solving simultaneous linear equation operator \ is general but less effective when coefficient matrix is positive-definite and symmetric, apply Cholesky decomposition inertia matrices are positive-definite and symmetric Cholesky decomposition positive-definite and symmetric matrix M can be decomposed as M = U T U where U is an upper trianglular matrix. Mx = b = U T Ux = b = { U T y = b Ux = y Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 22 / 47

23 Vector and Matrix Cholesky decomposition program Cholesky.m fprintf( Cholesky decomposition\n ); M = [ 4, -2, -2;... -2, 2, 0;... -2, 0, 3 ]; U = chol(m); U U *U Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 23 / 47

24 Vector and Matrix Cholesky decomposition >> Cholesky Cholesky decomposition U = ans = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 24 / 47

25 Vector and Matrix Cholesky decomposition program b = [ 4; 2; -7 ]; y = U \b; x = U\y; x result x = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 25 / 47

26 Graph Graph >> x = [0:10] x = >> f = x.*x f = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 26 / 47

27 Graph Graph >> plot(x,f) Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 27 / 47

28 Graph Graph >> t = [0:0.1:10] t = >> x = sin(t) x = Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 28 / 47

29 Graph Graph >> plot(t,x) Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 29 / 47

30 Ordinary Differential Equations Solving Ordinary Differential Equations van der Pol equation ẍ 2(1 x 2 )ẋ + x = 0 q = [ x v { ẋ = v v = 2(1 x 2 )v x ] [, q = f (t, q) = v 2(1 x 2 )v x ] Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 30 / 47

31 Ordinary Differential Equations Solving Ordinary Differential Equations File van_der_pol.m describes function f (t, q) function dotq = van_der_pol (t, q) x = q(1); v = q(2); dotx = v; dotv = 2*(1-x^2)*v - x; dotq = [dotx; dotv]; end File name van der Pol should be consistent to function name van der Pol. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 31 / 47

32 Ordinary Differential Equations Solving Ordinary Differential Equations Program solve_van_der_pol.m interval = 0.00:0.10:10.00; qinit = [ 2.00; 0.00 ]; [time, q] = ode45(@van_der_pol, interval, qinit); Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 32 / 47

33 Ordinary Differential Equations Solving Ordinary Differential Equations Draw a graph of time t and variable x plot(time, q(:,1), - ); Draw a graph of time t and variable v plot(time, q(:,2), - ); - solid line -- broken line -. dot-dash line : dotted line Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 33 / 47

34 Ordinary Differential Equations Solving Ordinary Differential Equations graph of time t and variable x Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 34 / 47

35 Ordinary Differential Equations Solving Ordinary Differential Equations graph of time t and variable v Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 35 / 47

36 Optimization Optimization Minimizing Rosenbrock function File Rosenbrock.m minimize f (x 1, x 2 ) = 100(x 2 x 2 1 ) 2 + (1 x 1 ) 2 function f = Rosenbrock( x ) x1 = x(1); x2 = x(2); f = 100*(x2 - x1^2)^2 + (1 - x1)^2; end Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 36 / 47

37 Optimization Optimization File Rosenbrock_minimize.m Result xinit = [ -1.2; 1.0 ]; [xmin, fmin] = fminsearch(@rosenbrock, xinit); xmin fmin >> Rosenbrock_minimize xmin = fmin = e-10 Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 37 / 47

38 Parameter Passing ODE with Parameter ordinary differential equation ẍ + bẋ + 9x = 0 where b is a parameter ẋ = v v = bv 9x Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 38 / 47

39 Parameter Passing Global Variable Function Program function dotq = damped_vibration (t, q) global b; x = q(1); v = q(2); dotx = v; dotv = -b*v - 9*x; dotq = [dotx; dotv]; end global b; interval = [0,10]; qinit = [2.00;0.00]; b = 1.00; [time,q] = ode45(@damped_vibration,interval,qinit); Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 39 / 47

40 Parameter Passing Nested Function Function with arguments of time, state variable vector, and parameter Program function dotq = damped_vibration_param (t, q, b) x = q(1); v = q(2); dotx = v; dotv = -b*v - 9*x; dotq = [dotx; dotv]; end interval = [0,10]; qinit = [2.00;0.00]; b = 1.00; damped_vibration damped_vibration_param (t,q,b); [time,q] = ode45(damped_vibration,interval,qinit); Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 40 / 47

41 Parameter Passing Global Variable vs Nested Function Global Variable Simple program Global variables may conflict against local variables Nested Function Somewhat complicated Must perform function defininion whenever paramerer values change Never conflict with other variables Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 41 / 47

42 Random Numbers Uniform Random Numbers Uniform Random Numbers in interval ( 0, 1 ) rng( shuffle, twister ); for k=1:10 x = rand; s = num2str(x); disp(s); end Symbol shuffle generates different random numbers whenever the program runs. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 42 / 47

43 Random Numbers Uniform Random Numbers Uniform Random Numbers in interval ( 0, 1 ) rng(0, twister ); for k=1:10 x = rand; s = num2str(x); disp(s); end specifying seed 0 generates unique random numbers whenever the program runs. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 43 / 47

44 Random Numbers dice.m function k = dice() % simulating a dice x = rand; if x < 1/6.00 k = 1; elseif x < 2/6.00 k = 2; elseif x < 3/6.00 k = 3; elseif x < 4/6.00 k = 4; elseif x < 5/6.00 k = 5; else k = 6; end end Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 44 / 47

45 Random Numbers dice run.m for i=1:10 s = []; for j=1:10 k = dice(); s = [s,, num2str(k)]; end disp(s); end Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 45 / 47

46 Random Numbers >> dice_run >> dice_run Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 46 / 47 dice run.m

47 Summary Summary Numerical calculation using MATLAB linear calculation (vectors and matrices) solving simultaneous linear equations solving ordinary differential equations numerically optimization parameter passing random numbers Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: MATLAB 47 / 47