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 | $m_t/m_{bnt}$ |
Type:
Form:
Model order:
4
Time domain:
Linearity:
Attachment:
Publication details:
| Title | Design, modeling and control of a hopping robot |
| Publication Type | Conference Paper |
| Year of Publication | 1993 |
| Authors | Rad, H., Gregorio P., and Buehler M. |
| Conference Name | Proceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems '93, IROS '93. |
| Date Published | 06/1993 |
| Publisher | IEEE |
| Conference Location | Yokohama |
| ISBN Number | 0-7803-0823-9 |
| Accession Number | 5050001 |
| Keywords | legged locomotion |
| Abstract | The authors report progress towards model based, dynamically stable legged locomotion with energy efficient, electrically actuated robots. The present the mechanical design of a prismatic robot leg which is optimized for electrical actuation. A dynamical model of the robot and the actuator as well as the interaction with ground is derived and validated by demonstrating close correspondence between simulations and experiments. A new continuous, and exactly implementable open loop torque control algorithm is introduced which stabilizes a limit cycle of the underlying fourth order intermittent robot/actuator/environment dynamics |
| DOI | 10.1109/IROS.1993.583877 |