Kinematics, Dynamics, and Controls (KDC)
Homework 2

General Comments:

• You need to pay attention to the relationship between your simulation and your math. Many people used continuous linearized dynamics and used the lqr() function in matlab to find the K matrix. However, if the integrator in your simulation is really discrete (e.g. simple euler), you need to discretize your dynamics, c2d(), and use dlqr().
  
  
• For determining a balance controller, it is best to attach the feet to the ground and “weld” the calf and thigh together. This turns the biped into a triple inverted pendulum. As long as the ankle torque limits we prescribed were not exceeded, this model will behave well. This ensures that your state space is just joint angles and velocities and ground contact models don’t play a role.
  
  
• Ground contact was a problem for most people. The ground can be modeled as a sort of spring-damper, but should not allow the robot to bounce or sink into the floor. Most importantly, it should not inject energy into the system. Use higher damping if you are having problems. Also it is important to minimize the effect of ground contact inaccuracies from the identification/linearization process. This relates to bolting the ankles to the floor as discussed above.
  
  
• Report quality varied. Some got better, some got worse. Overall there needs to be better presentation of results all around. Few people included the A,B, or K matrices that we asked them to find.
  
  
• There was a lot of trouble actually calculating A and B, here is a summary of the simplest method:

The continuous dynamic equations, , are the “forward dynamics” found by either deriving equations of motion or using a dynamics package that does it for you (ODE).

We can get the discrete dynamics using a simple euler approximation:

Now linearizing the system involves finding the matrices in the equation:

Here is the pseudo-matlab-code that finds the two matrices. Note that you are linearizing about , where these values may not be zero, depending on your model.

N = # of states;

M = # of inputes;

X₀ = state you are linearizing about;

U₀ = torques required to produce (inverse dynamics);

delta = small number;

%% Find A matrix

for i=1:N

X=X0;

X(i)=X(i)+delta; %% perturb state i by delta

X1 = f_D(X,U₀);

A(:,i) = (X₁-X₀)/delta;

end

%% Find B matrix

for i=1:M

U=U₀;

U(i)=U(i)+delta; %% perturb action i by delta

X₁ = f_D(X₀,U);

B(:,i) = (X₁-X₀)/delta;

end

Grading Comments:

There was a wider range of grades for this homework. Generally, either people got it or they didn’t. There 3 parts to assignment: 1) simulator, 2) LQR balance, 3) walking. If you were unable to accomplish one of these tasks, your grade reflected it.

I really want to emphasize that if you are lost or having significant trouble with the homework, please contact me or Chris and we can guide you in a constructive direction.

Highlights:

• http://gs7315.sp.cs.cmu.edu/pmwiki/pmwiki.php/Main/TeamAwesome
  
  This group had a very impressive simulator that was not based on ODE. It was strictly 2D, allowed manipulation with the mouse, and produced excellent videos.
  
  
• http://gs7315.sp.cs.cmu.edu/pmwiki/pmwiki.php/Courses/Assignment2
  
  This was probably the best all around report. Well organized and presented. They also performed a thorough investigation of every task in the assignment and had very good results. Also their movie is great.
  
  
• http://www.contrib.andrew.cmu.edu/~tgoff
  
  I want to point out that this group made a walking controller that adapted its walking cycle to keep a steady speed and maintain stability. This is very advanced work!