Question

MATLAB EXERCISE 5 This exercise focuses on the Jacobian matrix and determinant, simulated resolved-rate control, and inverse statics for the planar 3-DOF, 3R robot. (See Figures 3.6 and 3.7; the DH parameters are given in Figure 3.8.) The resolved-rate control method [9] is based on the manipulator velocity equation x = kve, where ky is the Jacobian matrix, is the vector of relative joint rates, X is the vector of commanded Cartesian velocities (both translational and rotational), and k is the frame of expression for the Jacobian matrix and Cartesian velocities. This figure shows a block diagram for simulating the resolved-rate control algorithm: Xc éc өс Robot Resolved-Rate-Algorithm Block Diagram As is seen in the figure, the resolved-rate algorithm calculates the required commanded joint rates c to provide the commanded Cartesian velocities Xc; this diagram must be calculated at every simulated time step. The Jacobian matrix changes with configuration A. For simulation purposes, assume that the commanded joint angles c are always identical to the actual joint angles achieved, A (a result rarely true in the real world). For the planar 3-DOF, 3R robot assigned, the velocity equations X=kJė for k=0 are i y = [-L1s1 - L2312 - L3$123 -L2912 – L3s123 -L3$123] |L1C1 +L2012 + L3C123 L2012+ L3C123 L3C123 {12} 1 16 where $123 = sin(@1 + 2 +83), C123 = cos(@ +62 +63), and so on. Note that ºx gives the Cartesian velocities of the origin of the hand frame (at the center of the grippers in Figure 3.6) with respect to the origin of the base frame {0}, expressed in (0) coordinates. Now, most industrial robots cannot commandé directly, so we must first integrate these commanded relative joint rates to commanded joint angles , which can be commanded to the robot at every time step. In practice, the simplest possible integration scheme works well, assuming a small control time step Ar: new = @old + Ar. In your MATLAB resolved-rate simulation, assume that the commanded @cey can be achieved perfectly by the virtual robot. (Chapters 6 and 9 present dynamics and control material for which we do not have to make this simplifying assumption.) Be sure to update the

Answer 1

Code function x = my_jacobi(A, b, tot_it) Function Input for the function SA - Matrix input $b - Vector input $tot_it - Number of iterations Output of the function 8x - Solution after tot it number of iterations Length of matrix lenofMatrix = length (A); % Compute x x = zeros (lenofMatrix, 1); Variable to store total total = 0; Update x Old x Old = x For loop() for idx3 = 1:tot_it For loop() for idxl = 1:lenOfMatrix For loop() for idx2 = 1:lenofMatrix Condition check if (idx2 -= idx1) Update total total = total + x_old(idx2); (A(idxl, idx2)/A(idxl, idx1)) # Otherwise else $Continue continue; End of condition end End of for loop() end x(idx1) = -total + b(idxl) /A(idxl, idxl); total = 0; End of for loop() end

x old = x; &End of for loop() end End of function end Sample input for the function A=[4 -2 0; -2.6 -2; 0 -2 4]; b=[2;8; -10]; tot_it=100; XJacobi = my jacobi (A,b, tot it) XT = Alb errJacobi = norm (xT - xJacobi) Sample output Default Term - Browser octave: X11 DISPLAY environment variable not set octave: disabling GUI features warning: function ./demo.m shadows a core library function x_old = xJacobi = 1.00000 1.eeeee -2.00000 6.6613e-16 errJacobi = sh-4.3$
Executable Code
function x = my_jacobi(A, b, tot_it) %Function
  
   %Input for the function
   %A - Matrix input
   %b - Vector input
   %tot_it - Number of iterations
  
   %Output of the function
   %x - Solution after tot_it number of iterations
  
   %Length of matrix
   lenOfMatrix = length(A);
  
   %Compute x
   x = zeros(lenOfMatrix,1);
  
   %Variable to store total
   total = 0;
  
   %Update x_Old
   x_Old = x
  
   %For loop()
   for idx3 = 1:tot_it
      
       %For loop()
       for idx1 = 1:lenOfMatrix
          
           %For loop()
           for idx2 = 1:lenOfMatrix
              
               %Condition check
               if (idx2 ~= idx1)
              
                   %Update total
                   total = total + (A(idx1,idx2)/A(idx1,idx1)) * x_Old(idx2);
              
               %Otherwise
               else
              
                   %Continue
                   continue;
              
               %End of condition
               end
              
           %End of for loop()
           end
           x(idx1) = -total + b(idx1)/A(idx1,idx1);
           total = 0;
      
       %End of for loop()
       end
       x_Old = x;
  
   %End of for loop()
   end
%End of function
end

%Inputs for funtion
A=[4 -2 0; -2 6 -2; 0 -2 4];
b=[2;8;-10];
tot_it=100;
xJacobi = my_jacobi(A,b,tot_it)
xT = A\b

errJacobi = norm(xT - xJacobi)