HCR - System of Ordinary Differential Equations

Differential Equations → ODE (Ordinary Differential Equation)

$$\dot{x}=f(x,t)$$

Need to keep the computation faster as possible, the allowed delays are up to some milliseconds, in case of haptic.

• Euler’s Method (1st order approximation)
• Mid Point Method (2nd order approximation)

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NOTE: 1 diff. eq. of order 2 can be splitted to 2 diff. eq.s of order 1:

$$\begin{array}{l}I\ddot{\Theta }(t)+\beta \dot{\Theta }(t)=\tau (t)\\ \\ \Rightarrow \left(\begin{array}{l}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{l}\Theta \\ \dot{\Theta }\end{array}\right)\\ \\ \Rightarrow \{\begin{array}{l}{\dot{x}}_1(t)=x_2(t)\\ I{\dot{x}}_2(t)+\beta x_2(t)=\tau (t)\end{array}\\ \\ \Rightarrow \left[\begin{array}{l}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& -\frac{\beta }{I}\end{array}\right]\left[\begin{array}{l}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ \frac{1}{I}\end{array}\right]\tau (t)\end{array}$$

System of 2 differential eq.s of order 1

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So given a differential equation of order 2 we can write in Matrix form:

$$\dot{\overline{x}}(t)=\overline{\overline{A}}\overline{x}(t)+\overline{B}\tau (t)$$