Today's task list:
- [x] Work on parsing the walking data
-  Work on BMD papers
-  Book hotel for BMD
-  Post update about BMD copyright
- [?] Finish reading the van der Kooij paper
-  See if our controller can drive an OpenSim model or Ton's 2D model
- [x] Wrap the HBM C code
-  Duplicate website backups on a S3 bucket
-  Work on the website theme
-  Make generic settings on the lab website
-  Review the TODO items on the Yeadon paper
-  Do CITI course
-  Do FERPA course
-  Write up database proposal
-  Try out CSympy with some mechanics problems
-  Email Mounir about teaching
I read about SWIG, Cython, and ctypes last night for wrapping the HBM code in Python. SWIG is cool because you can wrap C code by writing a seemingly simple *.i file that declares function calls in the C header and then SWIG it to create code for a variety of languages including Python, R, and Octave. Cython seems similar. You have to create a *.pyx file declaring some mappings to your C header file function calls. Then you run Cython and it generates some C code and compiles it. Both systems integrate with docutils for auto building when you run python setup.py install. Cython also supports NumPy and most numeric folks use it because of this. Finally, you can also write a python file and use ctypes to connect to functions in your header files, which seems ideal but there were many warnings about using it because it isn't flexible in the long run.
Walking System Identification
I updated the simple control solver to convert the dense A matrix to a scipy.sparse.csr_matrix and then to use scipy.sparse.linalg.lsqr to solve the sparse system (A is mostly zeros which has a thick diagonal). I thought it would be faster that solving the dense linear least square problem, but it wasn't. And I'm probably not saving any memory because I'm constructing the matrix as a dense matrix and then converting it. I need to change the code to construct a lil_matrix manually and then maybe get some memory reduction. It gave the same solution as the dense solver, so that is good.
From Ljung 1999
There are three basic methods of identifying a plant in a closed loop system, according to Ljung 1999 (page 423):
1. The Direct Approach: apply prediction error method to the output of the plant, \(y\), and the input to the plant, \(u\). Ignore feedback and the reference signal.
2. The Indirect Approach: Identify the full closed loop system from reference input, \(r\), to plant output, \(y\). If the controller is known, and you have the closed loop transfer function, you can then compute the plant.
3. The Joint Input-Output Approach: Consider the plant outputs and inputs, \(y\) and \(u\) as the outputs to a system that is driven by the reference signal \(r\).
Our human walking control system identification problem can be framed to match Ljung's framework like so:
m r + u ------------ d |+ y ---->o--->|controller|--->o--> -| ------------ +| | -------------- | -----|human |--- c |biomechanics| --------------
Here \(r\) is the reference signal, i.e. the ideal walking motion (angles, rates). \(u\) is the error between \(c\) and \(r\). \(c\) are the biomechanic model outputs, i.e. joint angles, joint rates, etc. \(d\) is the controller output, i.e. the joint torque additions. \(m\) is the nominal joint torques to keep the standard motion \(r\) and \(y\) is the sum of \(d\) and \(m\) which is fed back through the regulator, i.e. human bio model.
We measure \(c\) from motion capture. These are pretty much direct measurements. We can also use inverse dynamics with a biomechanical model to get good estimates of \(y\).
With that in mind, the direct approach doesn't seem feasible because we don't have a measurement or estimate of \(u\). The direct approach for closed loop systems also requires that you have a noise model which is close to the true noise model, otherwise bias is introduced in the identification.
For the indirect approach we do have a regulator model (i.e. the human biomechanical model) which is required, and we "measure" \(y\) but we don't measure or have an estimate of \(r\).
And thirdly, the Joint Input-Output approach requires \(y\), \(u\), and \(r\), of which only \(y\) is available.
So all three of these methods don't seem applicable as stated. But the "direct" measurements of \(c\) must be able to be used in the identification. This doesn't seem to be regarded in the framework presented in Ljung's book.