Indirectly controlled limit cycle walking of combined rimless wheel based on entrainment to active wobbling motion

Abstract

It has been clarified that a passive combined rimless wheel (CRW) that consists of two identical eight-legged rimless wheels can increase the walking speed either by adjusting the phase difference between the fore and rear legs or by using a passive wobbling mass that vibrates up-and-down in the body frame. Toward a further speeding-up of the CRW, this paper investigates the effect of an active wobbling mass driven by an actuator and the effect of an indirect excitation control. First, we develop the mathematical model and numerically show that the CRW generates a walking motion, which is entrained to the up-and-down motion of the active wobbling mass at frequencies higher than the natural frequency of the CRW. We discuss the gait properties mainly from the viewpoints of frequency and phase relationships. Second, we conduct verification experiments using our prototype CRW machine and describe the results.

Appendix A

The details of the matrix M(q) and the vector $\boldsymbol{h}( \boldsymbol{q}, \dot{\boldsymbol{q}} )$ in (1) are as follows:

$$\begin{aligned} \boldsymbol{M}(\boldsymbol{q}) =& \left [ \begin{array}{c@{\quad}c@{\quad}c} \boldsymbol{M}_1 (\boldsymbol{q}) & \mathbf{0}_{3 \times 3} & \mathbf{0}_{3 \times 4} \\ \mathbf{0}_{3 \times 3} & \boldsymbol{M}_2 (\boldsymbol{q}) & \mathbf{0}_{3 \times 4} \\ \mathbf{0}_{4 \times 3} & \mathbf{0}_{4 \times 3} & \boldsymbol{M}_3 (\boldsymbol{q}) \end{array} \right ], \end{aligned}$$

(28)

$$\begin{aligned} \boldsymbol{M}_1 (\boldsymbol{q}) =& \left [ \begin{array}{c@{\quad}c@{\quad}c} m_1 & 0 & m_1 L_1 \cos \theta_1 \\ 0 & m_1 & - m_1 L_1 \sin \theta_1 \\ m_1 L_1 \cos \theta_1& - m_1 L_1 \sin \theta_1 & m_1 L_1^2 + I_1 \\ \end{array} \right ], \end{aligned}$$

(29)

$$\begin{aligned} \boldsymbol{M}_2 (\boldsymbol{q}) =& \left [ \begin{array}{c@{\quad}c@{\quad}c} m_2 & 0 & m_2 L_2 \cos \theta_2 \\ 0 & m_2 & - m_2 L_2 \sin \theta_2 \\ m_2 L_2 \cos \theta_2& - m_2 L_2 \sin \theta_2 & m_2 L_2^2 + I_2 \\ \end{array} \right ], \end{aligned}$$

(30)

$$\begin{aligned} \boldsymbol{M}_3 (\boldsymbol{q}) =& \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} m_3 + m_c & 0 & - m_c L_c \cos \theta_3 & - m_c \sin \theta_3 \\ 0 & m_3 + m_c & m_c L_c \sin \theta_3 & - m_c \cos \theta_3 \\ - m_c L_c \cos \theta_3 & m_c L_c \sin \theta_3 & m_c L_c^2 & 0 \\ - m_c \sin \theta_1& - m_c \cos \theta_3 & 0 & m_c \end{array} \right ], \end{aligned}$$

(31)

$$\begin{aligned} \boldsymbol{h}( \boldsymbol{q},\dot{\boldsymbol{q}} ) =& \left [ \begin{array}{c} \boldsymbol{h}_1 ( \boldsymbol{q},\dot{\boldsymbol{q}} ) \\ \boldsymbol{h}_2 ( \boldsymbol{q},\dot{\boldsymbol{q}} ) \\ \boldsymbol{h}_3 ( \boldsymbol{q},\dot{\boldsymbol{q}} ) \end{array} \right ], \end{aligned}$$

(32)

$$\begin{aligned} \boldsymbol{h}_1 ( \boldsymbol{q},\dot{\boldsymbol{q}} ) =& \left [ \begin{array}{c} -m_1 L_1 \mbox{$\dot{\theta}$}_1^2 \sin \theta_1 \\ m_1 ( g - L_1 \mbox{$\dot{\theta}$}_1^2 \cos \theta_1 ) \\ - m_1 g L_1 \sin \theta_1 \end{array} \right ], \end{aligned}$$

(33)

$$\begin{aligned} \boldsymbol{h}_2 ( \boldsymbol{q},\dot{\boldsymbol{q}} ) =& \left [ \begin{array}{c} -m_2 L_2 \mbox{$\dot{\theta}$}_2^2 \sin \theta_2 \\ m_2 ( g - L_2 \mbox{$\dot{\theta}$}_2^2 \cos \theta_2 ) \\ - m_2 g L_2 \sin \theta_2 \end{array} \right ], \end{aligned}$$

(34)

$$\begin{aligned} \boldsymbol{h}_3 ( \boldsymbol{q},\dot{\boldsymbol{q}} ) =& \left [ \begin{array}{c} m_c \mbox{$\dot{\theta}$}_3^2 ( L_c \mbox{$\dot{\theta}$}_3 \sin \theta_3 - 2 \dot{L}_c \cos \theta_3 ) \\ (m_3 + m_c) g + m_c \mbox{$\dot{\theta}$}_3 ( L_c \mbox{$\dot{\theta}$}_3 \cos \theta_3 + 2 \dot{L}_c \sin \theta_3 ) \\ m_c L_c ( g \sin \theta_3 + 2 \dot{L}_c \mbox{$\dot{\theta}$}_3 ) \\ -m_c ( g \cos \theta_3 + L_c \mbox{$\dot{\theta}$}_3^2 ) \end{array} \right ]. \end{aligned}$$

(35)

Note that we added inertia moments for the fore and rear RWs, I ₁ and I ₂, as indicated in (29) and (30). This is necessary to calculate M(q)⁻¹ in the derivations of λ and $\ddot{\boldsymbol{q}}$. After that, for deriving (9), we finally took the limits as I ₁→0 and I ₂→0.