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The Final Exam

The Final Exam. During the last week of classes In your lab period May 5-9 Tell your friends. Servo Motor Control. Basic computer control of motor position and velocity. Reading. Chapter 11. watch the motor movement.

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The Final Exam

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  1. The Final Exam • During the last week of classes • In your lab period • May 5-9 • Tell your friends

  2. Servo Motor Control Basic computer control of motor position and velocity

  3. Reading • Chapter 11

  4. watch the motor movement http://www.youtube.com/watch?v=7coUcEHxnYA - notice the choppy movement between taught points http://www.youtube.com/user/CompassAutomation?v=lpY2P4lDYwg - notice that the movements have been smoothed out

  5. smooth movement starts slow, speeds up, finishes slow Position vs. time Velocity vs. time

  6. How do you teach a robot? • manually stepping through specific points • the results is often "choppy", because of the selection of just a few points of travel

  7. the problem You know "basically" where the robot arm should be, with a few points. Very choppy. To run smoothly in real time, you need thousands, actually a descriptive formula, p= f(t), and v= f(t)

  8. we must smooth out the curve to this taught manually derived from a formula

  9. how we solve it • Program 11 pts. Capture the (x,y) value of each point. x = [ 0 10 25 30 40 50 60 70 75 90 100 ] y = [ 0 7.5 20 30 45 50 45 30 20 7.5 0 ] remember that x represents time, and y represents velocity

  10. Find a polynomial equation that generates the same y's, from those x's (that is, an equation that matches the x,y curve). y = ax2 + bx + c • Generate 100s of x,y points using the formula to smooth out the curve

  11. polynomials • 2nd order: y = ax2 + bx + c • 3rd order: y = ax3 + bx2 + cx + d • 4th order: y = ax4 + bx3 + cx2 + dx + e number of coefficients = order+1

  12. step 1 y = [0 7.5 20 30 45 50 45 30 20 7.5 0 ] x = [0 10 25 30 40 50 60 70 75 90 100 ] figure(1) plot( x, y) reflect on this.... it's very poor vel vs time, and results in choppy movement

  13. coeff = polyfit(x, y, n) • find a polynomial of the nth order that fits the x vs y vectors • coeff will be a column vector of n+1 values • if n = 2, then MATLAB will return coeff(1), coeff(2), and coeff(3) so that • vel = coeff(1) * x2 + coeff(2) * x + coeff(3) • remember : x = [0 10 25 30 40 50 60 70 75 90 100 ] vel = [ desired 11 new pts ]

  14. order = 2 coeff = polyfit(x,y,2) % x.^2 means x2 vel = (coeff(1)*x.^2) +(coeff(2)*x) + coeff(3) %coeff: -0.0182 1.8182 -5.0000 figure(2) plot(x, vel)

  15. with n = 2 (3 Coeffs)

  16. order = 3 coeff = polyfit(x,y,3) vel = (coeff(1)*x.^3)+(coeff(2)*x.^2)+(coeff(3)*x) + coeff(4) figure(3) plot(x, vel)

  17. order = 4 (5 coeffs) coeff = polyfit(x,y,4) vel =(coeff(1)*x.^4)+(coeff(2)*x.^3)+(coeff(3)*x.^2) + (coeff(4)*x) +coeff(5) figure(4) plot(x, vel)

  18. order = 5 (6 coeffs) coeff = polyfit(x,y,5) vel = (coeff(1)*x.^5)+(coeff(2)*x.^4)+(coeff(3)*x.^3)+(coeff(4)*x.^2)+(coeff(5)*x) + coeff(6) figure(5) plot(x, vel)

  19. order = 6 (7 coeffs) coeff = polyfit(x,y,6) vel = (coeff(1)*x.^6)+(coeff(2)*x.^5)+(coeff(3)*x.^4)+(coeff(4)*x.^3)+(coeff(5)*x.^2)+ (coeff(6)*x)+coeff(7) figure(6) plot(x, vel)

  20. pretty close

  21. now plot for 1000 values of X x = [1: .1 :100] vel = (coeff(1)*x.^6)+(coeff(2)*x.^5)+(coeff(3)*x.^4)+(coeff(4)*x.^3)+(coeff(5)*x.^2)+ (coeff(6)*x)+coeff(7) figure(7) plot(x, vel)

  22. practice for the exam: %define an x row vector of 5 pts: x = [1:20:100] %get any y function of 5 pts y = 40*( 1-sin(x*pi/180) ) figure(10) plot( x, y)

  23. polyfit to order = 2 coeff = polyfit(x,y,2) vel = (coeff(1)*x.^2)+(coeff(2)*x)+coeff(3) figure(11) plot(x, vel)

  24. test that curve with 100 pts. x = [1:100] y = (coeff(1)*x.^2)+(coeff(2)*x)+coeff(3) figure(12) plot(x, y)

  25. try 6th order %define an x row vector of 5 pts: x = [1:20:100] %get a y function of 5 pts y = 40*(1-sin(x*pi/180)) coeff = polyfit(x,y,6) vel = (coeff(1)*x.^6) + (coeff(2)*x.^5) +(coeff(3)*x.^4) + (coeff(4)*x.^3) + (coeff(5)*x.^2)+(coeff(6)*x)+coeff(7) figure(13) plot(x, vel)

  26. warning Warning: Polynomial is not unique; degree >= number of data points. NOTE: the polynomial order must be less than the number of data points that is, no more coefficients allowed than the number of data points. 5 pts. - 5 coefficients - 4th order

  27. try 4th order %define an x row vector of 5 pts: x = [1:20:100] %get a y function of 5 pts y = 40*(1-sin(x*pi/180)) coeff = polyfit(x,y,4) vel = (coeff(1)*x.^4) + (coeff(2)*x.^3) + (coeff(3)*x.^2)+(coeff(4)*x)+coeff(5) figure(14) plot(x, vel)

  28. smooth it out x = [1:100] y = 40*(1-sin(x*pi/180)) coeff = polyfit(x,y,4) y = (coeff(1)*x.^4) + (coeff(2)*x.^3) + (coeff(3)*x.^2)+(coeff(4)*x)+coeff(5) figure(15) plot(x, y)

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