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# Notebook Entry

Some thoughts on system identification of planar walking control with direct collocation.

The system's equations:

\begin{equation*} -gm_t - m_t\dot{v}_t - F_c + F_p = 0 \end{equation*}
\begin{equation*} -c\Delta y - gm_f - k\Delta y - m_f\dot{v}_f - F_a - F_o + F_c = 0 \end{equation*}
\begin{equation*} c\Delta y - gm_s + k\Delta y - m_s\dot{v}_s + F_a + F_o = 0 \end{equation*}

Now thinking about the second equation we can measure, or estimate reasonably well, these quantities: $$\Delta y, g, m_f, \dot{v}_f, F_c$$. So inverse dynamics says we can solve for $$F$$ given our measurements:

Second order Equations of motion for open loop planar walker:

\begin{equation*} 0 = F_r + F_r^* \end{equation*}
\begin{equation*} 0 = f(x(t)) + u(t) + w(t) \end{equation*}

$$u(t)$$ are the joint torques $$x(t)$$ are the states: joint angles and rates (9 joint DoFs + 3 DoFs for global location and orientation)

$$w(t)$$ is the external input which should be a force that corresponds to the random longitudinal change in belt velocity. This will need to be modeled as a specified speed input so the measured belt speed velocity can be used directly.

We also can have some output equations that spit out marker location and ground reaction forces (these are the things we actually measure).

\begin{equation*} y = g(x(t)) \end{equation*}

We then assume a controller of the form:

\begin{equation*} u(t) = m^*(\varphi) - K(\varphi) x(t) \end{equation*}

$$\varphi$$ is the proportion of the gait cycle time (percent heel strike to heel strike)

This then creates our closed loop system:

\begin{equation*} 0 = f(x(t)) + m^*(\varphi) - K(\varphi) x(t) + w(t) \end{equation*}

Then we discretize $$x(t)$$, $$u(t)$$, and $$w(t)$$. At any instant in time the equations of motion must be true.

Question, how will $$\varphi$$ be known during any given interation of the solver. Will heel strikes have to be determined for each interation? Or is that something that comes from the data, i.e. $$\varphi$$ is fixed (i.e. for a given time $$t$$ in the data recording, $$\varphi$$ can be computed).

The cost function will be designed to minimize energy (sum of square of joint torques) and to minimize the difference in measured ground reaction forces and the marker positions.

The free parameters (unknowns) in the optimization problem will be the discretized states, the discretized $$w(t)$$ input, and the controller parameters.

There are 12 markers with two coordinates each, 4 ground reaction forces, 2 ground reaction moments, and one measured belt speed. We measure each of these at 100 hz over 8 minutes for each trial. So 48000 samples.

The closed loop model has 9 DoF, 18 states, and 1 input.