Agile and stable running locomotion control for an untethered and one-legged hopping robot

Abstract

This paper is aimed at presenting a locomotion control framework to realize agile and robust locomotion behaviors on conventional stiff-by-nature legged robots. First, a trajectory generator that is capable of characterizing angular momentum is utilized to synthesize reference CoM trajectories and associated force inputs, in accordance with the target locomotion profile. Second, the controller evaluates both force and position errors in the joint level, using a servo controller and an admittance control block. The trade-off between the position and force errors is naturally adjusted via admittance control coefficients. Implementing the controller on a 4-link, 3-jointed one-legged robot, we conducted several balancing and running experiments under challenging conditions; e.g., balancing on a moving cart, balancing on a surface with varying orientation, running on a flat surface, running on an inclined surface. The experimental study results indicated that the locomotion controller enabled the robot to perform untethered one-legged running and to maintain its balance when subject to disturbances.

Appendix: Derivation of the Lyapunov function

As previously explained, the admittance controller was expressed as $T_{ref}-T = k q_c + b \dot{q_c}$. This expression can be rearranged as follows:

$$\begin{aligned} \dot{q}_c= & {} \mu (T_{ref} - T) - \nu q_c \end{aligned}$$

(31)

$$\begin{aligned} K_v \dot{q}_c + K_p q_c= & {} \xi T_{ref} - \xi T - (k \xi - K_p)q_c \end{aligned}$$

(32)

In (31), $\mu$ and $\nu$ are diagonal matrices that are defined as $\mu _{[i,i]} = \frac{1}{b_i}$ and $\nu _{[i,i]} = \frac{k_i}{b_i}$. Using the torque command signal given in (23), the dynamics of the system may be reduced to the following form:

$$\begin{aligned} M(q)\ddot{\tilde{q}}= & {} -C(q,\dot{q})\dot{\tilde{q}} - K_v \dot{\tilde{q}} - K_p \tilde{q} - K_v \dot{q}_c - K_p q_c \nonumber \\&+\, \xi T_{ref} + T, \end{aligned}$$

(33)

where $\tilde{q} = q-q_{ref}$. Combining (32) with (33) yields the following:

$$\begin{aligned} M(q)\ddot{\tilde{q}} = -C(q,\dot{q})\dot{\tilde{q}} - K_v \dot{\tilde{q}} - K_p \tilde{q} + \beta T - \gamma q_c. \end{aligned}$$

(34)

In (34), $\beta = I+\xi$ and $\gamma = K_p - k \xi$. Recall that $T= -D_e \dot{q} - K_e (q_e - q)$, due to environmental interaction. If we utilize this information, (34) can take the final form stated below.

$$\begin{aligned} M(q)\ddot{\tilde{q}}= & {} -C(q,\dot{q})\dot{\tilde{q}} - K_v \dot{\tilde{q}} - K_p \tilde{q} - \beta D_e \dot{q} \nonumber \\- & {} \beta K_e (q_e-q) - \gamma q_c. \end{aligned}$$

(35)

First, we use the following Lyapunov function candidate:

$$\begin{aligned} V_1 = \frac{1}{2} \dot{\tilde{q}}^\intercal M(q)\dot{\tilde{q}} + \frac{1}{2} \tilde{q}^\intercal K_p\tilde{q} \end{aligned}$$

(36)

The time derivative of (36) can be expressed as below:

$$\begin{aligned} \dot{V}_1= & {} - \dot{\tilde{q}}^\intercal K_v\dot{\tilde{q}} - \dot{\tilde{q}}^\intercal \beta D_e\dot{q} - \dot{\tilde{q}}^\intercal \beta K_e (q - q_e) - \dot{\tilde{q}}^\intercal \gamma q_c\nonumber \\ \end{aligned}$$

(37)

$$\begin{aligned}= & {} - \dot{\tilde{q}}^\intercal K_v\dot{\tilde{q}} - \dot{q}^\intercal \beta D_e\dot{q} - \dot{q}^\intercal \beta K_e (q - q_e) - \dot{\tilde{q}}^\intercal \gamma q_c \nonumber \\&+\, \dot{q}_{ref}^\intercal \beta D_e \dot{q} + \dot{q}_{ref}^\intercal \beta K_e (q - q_e) \end{aligned}$$

(38)

$$\begin{aligned}= & {} - \dot{\tilde{q}}^\intercal K_v\dot{\tilde{q}} - \dot{q}^\intercal \beta D_e\dot{q} - \dot{q}^\intercal \beta K_e q + \dot{q}^\intercal \beta K_e q_e \nonumber \\&-\, \dot{\tilde{q}}^\intercal \gamma q_c + \dot{q}_{ref}^\intercal \beta T. \end{aligned}$$

(39)

As the next step, we introduce the following Lyapunov function:

$$\begin{aligned} V_2 = V_1 + \frac{1}{2} q^\intercal \beta K_e q. \end{aligned}$$

(40)

We can differentiate (40) as below.

$$\begin{aligned} \dot{V}_2= & {} \dot{q}_{ref}^\intercal \beta T \!-\! \dot{\tilde{q}}^\intercal K_v\dot{\tilde{q}} \!-\! \dot{q}^\intercal \beta D_e\dot{q} \!+\! \dot{q}^\intercal \beta K_e q_e \!-\! \dot{\tilde{q}}^\intercal \gamma q_c\nonumber \\ \end{aligned}$$

(41)

Finally, we introduce the Lyapunov function given in (24), which can be expressed in terms of $V_2$.

$$\begin{aligned} V = V_2 + \frac{1}{2} \hat{q}^\intercal P \hat{q} \end{aligned}$$

(42)

As stated previously, $\hat{q} = [q\ q_e]^\intercal$ and P was given in (25), which included a skew-symmetric matrix A and a positive semi-definite matrix C, i.e., $C \ge 0$. Its time derivative is computed as follows.

$$\begin{aligned} \dot{V} = \dot{V}_2 + \frac{d}{dt} \left( \frac{1}{2} \hat{q}^\intercal P \hat{q}\right) \end{aligned}$$

(43)

Before computing the derivation of (27) based on (43), we begin by proving the positive definiteness of $\frac{1}{2} \hat{q}^\intercal P \hat{q}$.

$$\begin{aligned} \frac{1}{2} \hat{q}^\intercal P \hat{q}= & {} \frac{1}{2} \left[ \begin{array}{c} q \\ q_e \end{array} \right] ^\intercal \left[ \begin{array}{c c} 2A &{} -\beta K_e \\ -K_e^\intercal \beta ^\intercal &{} 2C \end{array} \right] \left[ \begin{array}{c} q \\ q_e \end{array} \right] \end{aligned}$$

(44)

$$\begin{aligned}= & {} q_e^\intercal C q_e - q^\intercal \beta K_e q_e \end{aligned}$$

(45)

C is a free design parameter such that the inequality $q_e^\intercal C q_e \ge q^\intercal \beta K_e q_e$ holds true. Therefore, the Lyapunov function V given in (24) is positive definite.

As the next step, we derive (27) by showing that $\frac{d}{dt}\left( \frac{1}{2} \hat{q}^\intercal P \hat{q}\right) = - \dot{q}^\intercal \beta K_e q_e$. To do so, the first time derivative of (45) is yielded.

$$\begin{aligned} \frac{d}{dt}\left( \frac{1}{2} \hat{q}^\intercal P \hat{q}\right) \!=\! \dot{q}_e^\intercal C q_e \!+\! q_e^\intercal C \dot{q}_e \!-\! \dot{q}^\intercal \beta K_e q_e \!-\! q^\intercal \beta K_e \dot{q}_e\nonumber \\ \end{aligned}$$

(46)

Since the environment is stationary, i.e., $\dot{q}_e=0$, $\frac{d}{dt}\left( \frac{1}{2} \hat{q}^\intercal P \hat{q}\right) = - \dot{q}^\intercal \beta K_e q_e$ holds true. Therefore, Lyapunov function V and its time derivative are obtained as stated in (24) and (26). When the controller gains are assigned as $\frac{K_{vi}}{b_i} = \frac{K_{pi}}{k_i}$ for each joint, (26) is reduced to (27).