1. Data Driven Function Approximation

2. One link Given Xt what is the joint angle P Y xt P x

3. Plot1d Plot a function that maps joint angle P to tool position Xt clear all fstr=input('input a function: x.^2+cos(x) :','s'); fx=inline(fstr); range=2*pi; x=linspace(-range,range); y=fx(x); max_y=max(abs(y)); plot(x,y/max_y); hold on;

5. Black box Black box Xt P range=2*pi; linspace(-range,range); observation N=input('keyin sample size:'); x=rand(1,N)*2*range-range; n=rand(1,N)*0.1-0.05; y=fx(x)/max_y+n; figure plot(x,y,'.');

6. Paired data

7. Adaptive function Desired output Network output Input error

8. Network function Weight sum of hyper-tangent functions

9. Network b1 tanh r1 b2 1 a1 tanh r2 a2 x ……. bM rM aM tanh rM+1 1

10. Procedure 1: MLP evaluation • Input r,a,b and x,M • y=r(M+1) • Set M to the length of a • For m=1:M • Add r(i)*tanh(x*a(i)+b(i)) to y • Return y

11. Matlab code eval_MLP.m function y=eval_MLP(x,r,a,b,M) y=r(M+1); for m=1:M y=y+r(m)*tanh(x.*a(m)+b(m)); end return

12. Mean square error Given Find to minimize

13. Procedure 2: Mean square error • Input x,y,a,b,r • Set E to zero • Set n to the length of x • For i=1:n • Calculate the square error of approximating y(i) by f(x;r,a,b) • Add the error to E • Return E

14. Matlab code mean_square_error2.m function E=mean_square_error2(x,y,a,b,r) E=0; M=length(a); n=length(x); for i=1:n s_err=(y(i)-eval_MLP(x(i),r,a,b,M)).^2; E=E+s_err; end E=E/n; return

15. Matlab code

16. Desired output Network output Input error Application Black box plot1d.m Xt P observation Create paired data sampling.m Leaning learning.m

17. Matlab code Function approximation for an arbitrary target function fa1d.m

18. Y xt P x Inverse kinematics

19. Construction of inverse kinematics • Procedure • Input a 1d forward kinematics • Sampling to form paired data • Input-output swapping • Rescaling • Learning

20. Data preparation Black box plot1d.m Xt P observation Create paired data sampling.m Rescaling temp=x; x=y; y=temp; swapping max_y=max(y); y=y/max_y; plot(x,y,'.')

21. Learning inverse kinematics Desired output Network output Input error Leaning learning.m

22. Matlab code fa1d_inv.m MSE for training data 0.336461 ME for training data 0.490247 Unacceptable training errors

23. Multiple outputs • Inverse kinematics • Conflicts of I/O relation • Two distinct outcomes for the same input

24. First order derivative • Forward kinematics should be characterized by two outputs for construction of inverse kinematics Two observations Xt Black box plot1d.m P Derivative

25. Symbolic differentiation demo_diff.m function demo_diff() % input a string to specify a function % plot its derivative ss=input('function of x:','s'); fx=inline(ss); x=sym('x'); ss=['diff(' ss ')']; ss1=eval([sprintf(ss)]); fx1=inline(ss1) x=linspace(-pi,pi); plot(x,fx(x),'b');hold on; plot(x,fx1(x),'r'); return

26. Data preparation Two observations Xt Black box plot1dd.m P Dt Derivative

27. Reverse kinematics Xt Reverse Kinematics P Dt Derivative plot3(y1,y2,ix)

28. Reverse kinematics

29. Matlab code

30. Segmentation • Procedure • Input paired data,(x,y) • Sort data by y

32. Write a matlab function to divide paired data to four segments

33. Matlab code

34. multiple solutions

35. Two links (xt yt) Y P2 P1 x

36. Forward kinematics of two-link robot

37. Inverse kinematics of two-link robot

39. Reconstruction of forward Kinematics

40. Learning forward kinematics learn_MLP.m eval_MLP2.m fa2d.m

41. Forward Kinematics for planar robot

43. Inverse kinematics At the position level, the problem is stated as, "Given the desired position of the robot's hand, what must be the angles at all of the robots joints? "

44. Plot

45. Exercise • Write matlab functions to implement forward and inverse kinematics of a two-link robot

46. Example l1=1;l2=1; p1=pi/3;p2=pi/5; [x,y]=fkin(p1,p2,l1,l2) [p1,p2]=inverse_kin(x,y,l1,l2); p1/pi p2/pi

47. Nonlinear system

48. A set of nonlinear functions

49. x=linspace(-2,2); plot(x,x); hold on; plot(x,sqrt(4-x.^2)) plot(x,-sqrt(4-x.^2))

50. function F = myfun(x) F(1) = x(1).^2 + x(2)^2-4; F(2) = x(1) - x(2); return