# Hopping robot - Stance

## Model description:

$$\dot{x}=A_Sx+b_S\tau+e_S(x)$$

with

$A_S = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -K\dfrac{J}{\beta_Sr^2} & -C\dfrac{J}{\beta_Sr^2} & K\dfrac{J}{\beta_Sr^2} & C\dfrac{J}{\beta_Sr^2} \\ 0 & 0 & 0 & 1 \\ K\dfrac{m_b}{\beta_s} & C\dfrac{m_b}{\beta_s} & -K\dfrac{m_b}{\beta_s} & -C\dfrac{m_b}{\beta_s} \\ \end{bmatrix},\\ x = \begin{bmatrix} z\\ \dot{z}\\ p\\ \dot{p} \end{bmatrix}, b_S = \dfrac{\eta}{\beta_{S}r} \begin{bmatrix} 0\\ m_n\\ 0\\ m_{bn} \end{bmatrix},\\ e_S(x)=\dfrac{1}{\beta_s} \begin{bmatrix} 0\\ \alpha(ks_0 - m_{bn}g- f_{fS} + m_n(m_ng - ks_0 - f_a)\\ 0\\ 0\\ \end{bmatrix}.$

 $z$ Body Height $p$ Actuator Length $\tau$ Motor Torque $\theta$ Motor angle, $\theta = p/r$ $s$ Spring Length $m_b$ 9.5kg Upper Leg Mass $m_n$ 0.25kg Ball Nut Mass $m_t$ 0.5kg Toe Mass $k$ 400 N/m Spring Constant $F_p$ 5.0N Leg Dry Friction $F_z$ 1.5N Planarized Dry Friction $F_a$ 0N Ball Screw Dry Friction $c$ 5.5Ns/m Spring Viscous Friction $\hat{\tau}$ 1.78Nm Stall Torque $\hat{\omega}$ 2800RPM Max Speed $\eta$ 0.95 Ball Screw Efficiency $s_0$ 0.608m Spring Rest Length $l_0$ 0.595m Maximum Leg Length $J$ 2.7$\times$10$^{-4}$kgm$^2$ Motor Inertia $\alpha$ 0.34kgm $J/r^2+m_n$ $\mu$ 0.05 $m_t/m_{bnt}$

4

## Publication details:

 Title Design, modeling and control of a hopping robot Publication Type Conference Paper Authors Rad, H., Gregorio P., and Buehler M.