ME451 Kinematics and Dynamics of Machine Systems

1. ME451 Kinematics and Dynamics of Machine Systems MATLAB Tutorial September 13, 2011 Radu Serban University of Wisconsin-Madison

2. Before getting started… • Almost entirely, this tutorial was compiled from bits of information gathered from various internet sources • Please let me know of any mistakes you find (serban@engr.wisc.edu) • The right frame of mind: • You will not be able to say at the end of workshop that you know MATLAB but rather that you have been exposed to MATLAB. • Use MATLAB’s help – this is your first stop • Second stop: search the web for examples that come close to what you need • You learn how to use MATLAB by using it, that’s why the start might be slow and at times frustrating • http://www.mathworks.com/academia/student_center/tutorials/launchpad.html

3. Contents • What is MATLAB? • MATLAB Components • MATLAB Desktop • Matrices • Importing and Exporting Data • Elementary math with MATLAB • M-file Programming • Functions vs. Scripts • Variable Type/Scope • Debugging MATLAB functions

4. What is MATLAB? • Integrated Development Environment (IDE) • Programming Language • Collection of Toolboxes • A package that excels in Linear Algebra support

5. MATLAB as an IDE • Integrated development environment • Write your own code for computation • Good visualization (plotting) tools • Easy-to-use environment • Command Window • Command History • Help Browser • Workspace Browser • Editor/Debugger

6. MATLAB as Programming Language • High-level language • Data types • Functions • Control flow statements • Input/output • Graphics • Object-oriented programming capabilities

7. Toolboxes • Collections of functions to solve problems from various application fields: • DSP (Digital Signal Processing) Toolbox • Image Toolbox • Wavelet Toolbox • Neural Network Toolbox • Fuzzy Logic Toolbox • Control Toolbox • Multibody Simulation Toolbox • And many other… • http://www.tech.plym.ac.uk/spmc/links/matlab/matlab_toolbox.html • http://www.mathworks.com/matlabcentral/fileexchange/ • amazing number of toolboxes available: if you need something, it’s out there somewhere available for download

8. Getting help • help command >> help • lookforcommand >> lookfor • Help browser >> doc • helpwincommand >> helpwin • Search engine • Printable documents • Link to MathWorks

9. General Functions • whos: List current variables and their size • clear: Clear variables and functions from memory • cd: Change current working directory • dir:List files in directory • pwd:Tells you the current directory you work in • echo:Echo commands in M-files • format: Set output format (long, short, etc.) • diary(foo): Saves all the commands you type in in a file in the current directory called “foo”

10. Calculations at the Command Line Assigning Variables MATLAB as a calculator • -5/(4.8+5.32)^2 ans = -0.0488 • (3+4i)*(3-4i) ans = 25 • cos(pi/2) ans = 6.1230e-017 • exp(acos(0.3)) ans = 3.5470 • a = 2; • b = 5; • a^b • ans = • 32 • x = 5/2*pi; • y = sin(x) • y = • 1 • z = asin(y) • z = • 1.5708 Semicolon suppresses screen output Results assigned to “ans” if name not specified () parentheses for function inputs A Note about Workspace: Numbers stored in double-precision floating point format

11. MATLAB for (linear) algebra

12. Matrices • Entering and Generating Matrices • Subscripts • Scalar Expansion • Concatenation • Deleting Rows and Columns • Array Extraction • Matrix and Array Multiplication • NOTEWe don’t have time to carefully look at all these topics. Be aware that these facilities exist in MATLAB and that you can access them when needed by first invoking helpon that command

13. Entering Numeric Arrays • a=[1 2;3 4] • a = • 1 2 • 3 4 • b=[-2.8, sqrt(-7), (3+5+6)*3/4] • b = • -2.8000 0 + 2.6458i 10.5000 • b(2,5) = 23 • b = • -2.8000 0 + 2.6458i 10.5000 0 0 • 0 0 0 0 23.0000 Use square brackets [ ] NOTE: 1) Row separator semicolon (;) 2) Column separator space OR comma (,) • Any MATLAB expression can be entered as a matrix element (internally, it is regarded as such) • In MATLAB, the arrays are always rectangular

14. Columns (n) 1 2 3 4 5 A (2,4) 1 2 Rows (m) 3 4 5 A (17) The Matrix in MATLAB 1 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 5 10 15 20 25 4 10 1 6 2 8 1.2 9 4 25 7.2 5 7 1 11 A = 0 0.5 4 5 56 23 83 13 0 10 Rectangular Matrix: Scalar: 1-by-1 array Vector: m-by-1 array 1-by-n array Matrix: m-by-n array

15. Entering Numeric Arrays Scalar expansion • w=[1 2;3 4] + 5 • w = • 6 7 • 8 9 • x = 1:5 • x = • 1 2 3 4 5 • y = 2:-0.5:0 • y = • 2.0000 1.5000 1.0000 0.5000 0 • z = rand(2,4) • z = • 0.9501 0.6068 0.8913 0.4565 • 0.2311 0.4860 0.7621 0.0185 Creating sequences: colon operator (:) Utility functions for creating matrices.

16. Numerical Array Concatenation(Tiling) Use [ ] to combine existing arrays as matrix “elements” Row separator: semicolon (;) Column separator: space / comma (,) • a=[1 2;3 4] • a = • 1 2 • 3 4 • cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a] • cat_a = • 1 2 2 4 • 3 4 6 8 • 3 6 4 8 • 9 12 12 16 • 5 10 6 12 • 15 20 18 24 Use square brackets [ ] 4*a Note: The resulting matrix must be rectangular

17. Array Subscripting / Indexing 1 2 3 4 5 A = 1 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 5 10 15 20 25 4 10 1 6 2 1 2 3 4 5 8 1.2 9 4 25 A(1:5,5) A(:,5) A(21:25) A(1:end,end) A(:,end) A(21:end)’ 7.2 5 7 1 11 0 0.5 4 5 56 A(3,1) A(3) 23 83 13 0 10 A(4:5,2:3) A([9 14;10 15])

18. Deleting Rows and Columns • A=[1 5 9;4 3 2.5; 0.1 10 3i+1] • A = • 1.0000 5.0000 9.0000 • 4.0000 3.0000 2.5000 • 0.1000 10.0000 1.0000+3.0000i • A(:,2)=[] • A = • 1.0000 9.0000 • 4.0000 2.5000 • 0.1000 1.0000 + 3.0000i • A(2,2)=[] • ??? Indexed empty matrix assignment is not allowed. “:” is a VERY important construct in MATLAB

19. c(2,4) = a(2,4)*b(2,4) Matrix Multiplication Array Multiplication [2x4] • a = [1 2 3 4; 5 6 7 8]; • b = ones(4,3); • c = a*b • c = • 10 10 10 • 26 26 26 [4x3] [2x4]*[4x3] [2x3] a(2nd row) b(3rd column) Array Multiplication (component-wise operation) • a = [1 2 3 4; 5 6 7 8]; • b = [1:4; 1:4]; • c = a.*b • c = • 1 4 9 16 • 5 12 21 32

20. Matrix Manipulation Functions • zeros: Create an array of all zeros • ones: Create an array of all ones • eye: Identity Matrix • rand: Uniformly distributed random numbers • diag: Diagonal matrices and diagonal of a matrix • size: Return array dimensions • fliplr: Flip matrices left-right • flipud: Flip matrices up and down • repmat: Replicate and tile a matrix • reshape: Reshape array

21. Matrix Manipulation Functions • transpose (’):Transpose matrix • rot90: rotate matrix 90 • tril: Lower triangular part of a matrix • triu: Upper triangular part of a matrix • cross: Vector cross product • dot: Vector dot product • det: Matrix determinant • inv: Matrix inverse • eig: Evaluate eigenvalues and eigenvectors • rank: Rank of matrix

22. Elementary Math

23. Elementary Math • Logical Operators • Math Functions • Polynomials and Interpolation

24. = = equal to > greater than < less than >= Greater or equal <= less or equal ~ not & and | or isfinite(), etc. . . . all(), any() find Logical Operations • Mass = [-2 10 NaN 30 -11 Inf 31]; • each_pos = Mass>=0 each_pos = 0 1 0 1 0 1 1 • all_pos = all(Mass>=0) all_pos = 0 • all_pos = any(Mass>=0) all_pos = 1 • pos_fin = (Mass>=0)&(isfinite(Mass)) pos_fin = 0 1 0 1 0 0 1 • Note: • 1 = TRUE • 0 = FALSE

25. Elementary Math Function • abs, sign: Absolute value and signum function • sin, cos, asin, acos, etc.: Trigonometric functions • exp, log, log10: Exponential, natural and common (base 10) logarithm • ceil, floor: Round to integer, toward +/-infinity • fix: Round to integer, toward zero

26. Elementary Math Function • round: Round to the nearest integer • gcd: Greatest common divisor • lcm: Least common multiple • sqrt: Square root function • real, imag: Real and Imaginary part of complex • rem: Remainder after division

27. Elementary Math Function Operating on Arrays • max, min: Maximum and minimum of arrays • mean, median: Average and median of arrays • std, var: Standard deviation and variance • sort: Sort elements in ascending order • sum,prod: Summation and product of elements • trapz: Trapezoidal numerical integration • cumsum, cumprod: Cumulative sum, product • diff, gradient: Differences and numerical gradient

28. Programming and Application Development

29. Topics discussed… • The concept of m-file in MATLAB • Script versus function files • The concept of workspace • Variables in MATLAB • Type of a variable • Scope of a variable • Flow control in MATLAB • The Editor/Debugger

30. Function M-file function r = ourrank(X,tol) % rank of a matrix s = svd(X); if (nargin == 1) tol = max(size(X)) * s(1)* eps; end r = sum(s > tol); Multiple input arguments, use ( ) • r=ourrank(rand(5),.1); function [mean,stdev] = ourstat(x) [m,n] = size(x); if m == 1 m = n; end mean = sum(x)/m; stdev = sqrt(sum(x.^2)/m – mean.^2); Multiple output arguments, use [ ] • [m std]=ourstat(1:9);

31. Basic Parts of a Function M-File Input Arguments Output Arguments Function Name Online Help function y = mean(x) % MEAN Average or mean value. % For vectors, MEAN(x) returns the mean value. % For matrices, MEAN(x) is a row vector % containing the mean value of each column. [m,n] = size(x); if m == 1 m = n; end y = sum(x)/m; Function Code

32. Script and Function Files • Script Files • Work as though you typed commands into MATLAB prompt • Variable are stored in MATLABworkspace • Function Files • Let you make your own MATLAB Functions • All variables within a function are local • All information must be passed to functions as parameters • Subfunctions are supported

33. The concept of Workspace • At any time in a MATLAB session, the code has a workspace associated with it • The workspace is like a sandbox in which you find yourself at a certain point of executing MATLAB • The “Base Workspace”: the workspace in which you live when you execute commands from prompt • Remarks: • Each MATLAB function has its own workspace (its own sandbox) • A function invoked from a calling function has its own and separate workspace (sandbox) • A script does not create to a new workspace (unlike a function), but rather uses the workspace from which it was invoked

34. Variable Types in MATLAB • Local Variables • In general, a variable in MATLAB has local scope, that is, it’s only available in its workspace • The variable disappears when the workspace ceases to exist • Recall that a script does not define a new workspace – be careful, otherwise you can step on variables defined at the level where the script is invoked • Since a function defines its own workspace, a variable defined in a function is local to that function • Variables defined outside the function should be passed to function as arguments. Furthermore, the arguments are passed by value • Every variable defined in the subroutine, if to be used outside the body of the function, should be returned back to the calling workspace

35. Variable Types in MATLAB • Global Variables • These are variables that are available in multiple workspaces • They have to be explicitly declared as being global • Not going to expand on this, since using global variables is a badprogramming practice • A note on returning values from a function • Since all variables are local and input arguments are passed by value, when returning from a function a variable that is modified inside that function will not appear as modified in the calling workspace unless the variable is either global, or declared a return variable for that function

36. Flow Control Statements if Statement if ((attendance >= 0.90) & (grade_average >= 60)) pass = 1; end; while Loops eps = 1; while (1+eps) > 1 eps = eps/2; end eps

37. Flow Control Statements a = zeros(k,k) % Preallocate matrix for m = 1:k for n = 1:k a(m,n) = 1/(m+n -1); end end for Loop: method = 'Bilinear'; ... (some code here)... switch lower(method) case {'linear','bilinear'} disp('Method is linear') case 'cubic' disp('Method is cubic') otherwise disp('Unknown method.') end Method is linear switchStatement:

38. Editing and Debugging M-Files • The Editor/Debugger • Debugging M-Files • Types of Errors (Syntax Error andExceptions) • Usingkeyboardand “;” statement • Setting Breakpoints • Stepping Through • Continue, Go Until Cursor, Step, Step In, Step Out • Examining Values • Selecting the Workspace • Viewing Datatips in the Editor/Debugger • Evaluating a Selection

39. Debugging Select Workspace Breakpoint Set Auto- Breakpoints Tips

40. Importing and Exporting Data • Using the Import Wizard • Using the saveand loadcommands save fname save fname x y z save fname -ascii save fname -mat load fname load fname x y z load fname -ascii load fname -mat

41. Input/Output for Text File • Read formatted data, reusing the format string N times. • Suppose the text file stars.dat contains data in the following form: • Jack Nicholson 71 No Yes 1.77 • Helen Hunt 45 No No1.73 • >> fid = fopen(‘stars.dat’); • >> C = textscan(fid, ‘%s%s%d%s%s%f’); • >> fclose(fid); • C is a 1x6 cell array, each cell containing a column of the data. • Import and export numeric data from ASCII delimited file using dlmread/dlmwrite • C = textscan(fid,’format’,N); • M = dlmread(filename,delimiter,range)

42. Input from Text File • fopen: Open a file for input/output • fgetl: Read line from file • fclose: Close one or more open files • fid= fopen('myFile.txt' , ‘r'); • str1= fgetl(fid); • fclose (fid);

43. Miscellaneous

44. The inline Function • inlinefunction • f = inline('3*sin(2*x.^2)','x') • f = • Inline function: • f(x) = 3*sin(2*x.^2) • f(2) • ans = • 2.9681 Use charfunction to convert inline object to string • Numerical Quadrature using quad • Q = quad('1./(x.^3-2*x-5)',0,2); • F = inline('1./(x.^3-2*x-5)'); • Q = quad(F,0,2); • Q = quad('myfun',0,2) Note: quadfunction uses adaptive Simpson quadrature function y = myfun(x) y = 1./(x.^3-2*x-5);

45. Symbolic Math • syms t; • func1=sym('3*cos(10*t)'); • symbolic variable • symbolic function • syms t; • func1=sym('3*cos(10*t)'); • value=subs(func1,t,2.3) • value = -1.5985 • diff(func1) • ans = (-30)*sin(10*t) • Substitution • symbolic differentiation

46. Graphics Fundamentals

47. Graphics and Plotting in MATLAB • Basic Plotting • plot, title, xlabel, grid, legend, hold, axis • Editing Plots • Property Editor • Mesh and Surface Plots • meshgrid, mesh, surf, colorbar, patch, hidden • Handle Graphics

48. 2-D Plotting marker line color Syntax: Example: plot(x1, y1, 'clm', x2, y2, 'clm', ...) x=[0:0.1:2*pi]; y=sin(x); z=cos(x); plot(x,y,x,z,'linewidth',2) title('Sample Plot','fontsize',14); xlabel('X values','fontsize',14); ylabel('Y values','fontsize',14); legend('Y data','Z data') grid on

49. Title Grid Ylabel Legend Xlabel Sample Plot

50. Displaying Multiple Plots • Nomenclature: • Figure window– the window in which MATLAB displays plots • Plot– a region of a window in which a curve (or surface) is displayed • Three typical ways to display multiple curves in MATLAB (other combinations are possible…) • One figure contains one plot that contains multiple curves • Requires the use of the command “hold” (see MATLAB help) • One figure contains multiple plots, each plot containing one curve • Requires the use of the command “subplot” • Multiple figures, each containing one or more plots, each containing one or more curves • Requires the use of the command “figure” and possibly “subplot”