Homework 1

• Assignment 1 Handout
• For this first homework we are to make a humanoid robot touch target points in its “world”. For my solution, I used Matlab's fmincon function and provided it with several constraints

hw1_submission.zip

• Just run main.m, and change computeGradient if you would like to have it compute the analytical gradient (is not accurate nor fast, but the foundation is there)

• Due to a limited timeframe (for various reasons) in completing the assignment, I went with the provided sample Matlab code for a 3-link arm and modified the criterion and constraints code to work for a full humanoid robot.

• Both of the humanoid's feet cannot move (they can bend though).
  • I had a tough time implementing this, as fmincon didn't like having the x,y, and z coordinate errors combined into one constraint for some reason. It worked great when they were separated though! A snippet of constraints.m is shown below for illustration:

%Was not able to optimize using this code
eq_violations(1) = (tree2.j(17).position - [-0.0484 -0.0890 0.0811])*(tree2.j(17).position - [-0.0484 -0.0890 0.0811])';
 
%But it worked just fine with this code
eq_violations(1:3) = (tree2.j(17).position - [-0.0484 -0.0890 0.0811]);

• The center of mass of the humanoid cannot go outside its base of support (or else it would tip over!)

• For my criterion, I used the euclidean distance between the humanoid robot's right wrist and the target point, which worked well.

score = (tree1.j(29).position - target)*(tree1.j(29).position - target)';

• For part 2 of the assignment, we were asked to analytically compute the gradient (instead of letting Matlab compute it numerically).

• I was able to use Matlab's Symbolic Math Toolbox to compute the partial derivatives of the forward kinematics equations analytically. This seemed to be a pretty simple operation, as all I had to do was pass in a symbol called angles(i) instead of a numerical angle. See below inside of drc_forward_kinematics_symbolic.m:

angles = sym('angles',[29,1]);
for i = 2:29
 parent = tree.l(i).parent;
 tree.j(i).rotation = generate_axis_angle_rotation_matrix_symbolic( tree.j(i).axis, angles(i) );
.
.
.

• However, the equations became complicated very quickly and took around 20 seconds to evaluate them for a given set of angles (inside of criterion.m).
  • Also, the robot seems to crumple on the floor unnecessarily, smells fishy…

• Is there much of a difference between evaluating analytic derivatives at a point and doing numerical differentiation ((x+dx - x)/ dy) at the same point? Much to my chagrin, I saw fmincon requested a numerical gradient and not an analytic one, which was my cause for surprise :)
  • Taking the effort to do the 2nd derivative would make more sense, as you get even more of a picture of where global minimas might be
  • On that note, why not 3rd or 4th derivatives if you can?
• The analytic result is definitely not right yet, as the humanoid seems to crumple up on the floor unnecessarily. I need to do some more testing with simple examples!