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In this project, I make different use of the optimization process in CMC in order to optimize the control input force for active actuators.

• CMC procedure is static optimization process, and it minimizes the cost function J which can be represented

  as 

  as  

  Mathblock
  J = \sum_{i=1}^{n_x} x_i^2 + \sum_{j=1}^{n_q} w_j \left( \ddot{q}_j\,^* - \ddot{q}_j \right)^2

• When we add active actuators to an OpenSim model, the activation term in the cost

  function becomes

  function becomes 

  Mathblock
  x = \begin{bmatrix} x_{\mathrm{muscle}} \\ x_{\mathrm{actuator}} \end{bmatrix}

where X_muscle is muscle control and X_actuator is actuator control. As X_actuator is part of activation states, it is also adjusted after the optimization process.

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1. 1. My initial guess was to saturate the optimal input force that I found earlier at 400 N. I generated an new input force which is identical to the optimal input force up to 400 N, and saturated once the optimal input force exceeded 400 N.
   2. The second input force I tried was a new result from my CMC simulations. The new CMC result was acquired by assigning 4000 N to the maximum actuation force and bounding the control input between 0 and 0.1. In other words,

F_{\mathrm{actuator}}^{\mathrm{max}} = 400 N, \quad 0 \leq x_{\mathrm{actuator}} \leq 0.1

When we compare the saturated optimal input and the result from the new CMC procedure, we find similarity. Now, let's compare the metabolic cost reduction when each control input is applied to ankle actuators.

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