GSoC, Week 8

I don’t have as much to talk about this week, as I spent two days in travel. Mostly, I worked on the sphinx documentation for mechanics. I am continuing with the previous organization of the documentation: first the mathematical descriptions, then the SymPy implementations. I’m still not convinced that this is the best way to do things, but at the moment, at least the subsubmodule’s documentation is consistent. Also, there will most likely be changes to the code, probably relating to the switch to using Brian Granger’s functional derivative branch and the different printing. I’m also trying to use the same examples in both the mathematical sections and in the code sections. I think this is the most logical approach, and if I choose to rearrange the documentation, I won’t have to rewrite much. This is all in my ‘pydy’ branch.

The other thing I am working on is forming the equations of motion for the bicycle. Right now they are being computed, but I’m not sure of the accuracy. There are already tests in place for holonomic systems, but non for non-holonomic systems (for reference: http://en.wikipedia.org/wiki/Nonholonomic_system). Unfortunately, the equations for non-holonomic systems get very, very big. In our lab, with the bicycle, we form our equations, linearize them, substitute in the system parameters, and examine the eigenvalues in order to validate our equations. I’m currently using my SymPy code to do this with the bicycle model, but am encountering some difficulties. I’ve got the mass matrix and forcing terms solved for and linearized, and the udots have a solution, in the form udot = A x, where x = [q2,q4,q5,u2,u3,u5,u1,u4,u6].T . What should happen next is that the relevant entries from this matrix are extracted, as our final state is [u2,u5,q2,q5] (this is lean rate, steer rate, lean angle, steer angle), then a smaller matrix is constructed and the eigenvalues are examined. Now, I know that the eigenvalues are off already. Examining this “A” matrix though shows that some of the elements are correct, compared to a reference “A” matrix. It would appear that the partial derivative of the udots w.r.t. the lean angle and lean rate are correct, but are not right w.r.t. the steer angle and steer rate. I’m not sure how to interpret this, as it is showing that half of the correct numbers are there (and these are correct to ~13 significant digits when evaluated numerically), and the other half are wrong. I’m hoping that I have just performed the linearization wrong, or the order in which I have arranged my matrices is incorrect; I don’t see many other options.

The full equations come out in the form: MM udot = forcing, or the mass matrix multiplied by the time derivatives of the generalized speeds equals the forcing terms. MM is 6×6, and forcing is 6×1 (as is udot). What is done next is substitute in the current state into the mass matrix, find the jacobian of the forcing vector w.r.t. the vector x (from above), and substitute in the numerical values. Next, udot is found by performing MM.inv() * forcing (where forcing is now a 6×9 matrix). Since I am getting some of the correct values, it would appear that there are elements in both the MM and the forcing matrix which are correct. I’m going to continue to play around with things, as again, I feel that I am most likely assembling these things in the incorrect order. Hopefully this will prove successful though, and will provide a good, non-trivial example of what physics.mechanics can do.

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