HCR - Lecture 6 (Original)

Impedance

Impedance: $Z(s)$ → $F(s)=Z(s)\cdot \dot{X}(s)$

For a given force the smaller the impedance → the higher is the velocity and viceversa.

~ Ex.: Mass-Spring Problem:

$$\begin{array}{l}M\ddot{x}(t)+\beta \dot{x}(t)+kx(t)=F(t)\\ \mathcal{L}\left[\dot{x}(t)\right]=\dot{X}(s)\\ \mathcal{L}\left[\ddot{x}(t)\right]=s\dot{X}(s)\\ \mathcal{L}\left[x(t)\right]=\frac{1}{s}\dot{X}(s)\\ \\ \Rightarrow Ms\dot{X}(s)+\beta \dot{X}(s)+\frac{k}{s}\dot{X}(s)=F(s)\\ \\ \Rightarrow F(s)=\underset{Z(s)}{\underbrace{\left[Ms+\beta +\frac{k}{s}\right]}}\cdot \dot{X}(s)\end{array}$$

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Impedance:

For a given system the higher is the impedance, the lower will be its velocity.

$$Z(s):=\frac{F(s)}{\dot{X}(s)}$$

An higher impedance brings more stability to the system.

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Compliance:

For a given system the higher its compliance the higher the velocity.

$$Z^{-1}(s):=\frac{\dot{X}(s)}{F(s)}$$

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Transparency

• $Z_E$: impedance of virtual environment
• $Z_R$: impedance of the robot

I want to simulate an $F_v(s)$ so given the $x_h$ we can have:

$$F_v(s)=Z_E(s)\cdot \dot{X}(s)$$

For this particular robot the only way to give a force feedback is to apply a torque so:

$$\tau =J^{-1}\cdot F$$

where $J$ is the Jacobian Matrix. But there are 2 torques to be summed: the torque of motor ${\tau }_m$ and that of the human ${\tau }_h$

Then the torque will be:

$$\begin{array}{r}\\ \\ \\ Z_R(s):=Is+B\to \\ \end{array}\begin{array}{lc}\tau (s)& =Is\dot{\Theta }(s)+B\dot{\Theta }(s)\\ & |\\ & =(Is+B)\cdot \dot{\Theta }(s)\\ & |\\ & =Z_R(s)\cdot \dot{\Theta }(s)\end{array}$$

Also remember that:

$$\begin{array}{l}\tau =J^T\cdot F\\ \dot{X}{}_h(s)=J\cdot \dot{X}\end{array}$$

Then:

$$\begin{array}{l}\ddot{x}=J\cdot \ddot{\Theta }\\ \ddot{\Theta }=J^T\cdot \ddot{x}\end{array}$$

What is really felt by the human is the Force of the human itself $F_h$ and the velocity $\dot{X}(s)$ which is controlled by both ${\tau }_h$ and ${\tau }_m$ →

$$F=Z_p(s)\cdot \dot{X}{}_h\cdot \frac{1}{s}$$

where $Z_p(s)$ is the impedance perceived by the human.

And: