Coordinated optimization of locomotion velocity and energy consumption in vibration-driven system

Abstract

Worm-like robots have attracted more and more attention due to its excellent ability of locomotion in narrow terrains. Vibration-driven system as a discrete way for modelling locomotion of the worm-like robots has been frequently analyzed and applied. However, it is very challenging for the vibration-driven systems due to the non-smooth friction to realize efficiently high velocity and low energy consumption locomotion. Motived by such problem, this paper considers a classical vibration-driven system to optimize both the locomotion velocity and energy consumption. It is a bi-objective coordinated optimization in essence since it is impossible for these two indexes to reach the optimal values simultaneously. Then a usual Pareto front method is employed to solve such bi-objective optimization problem. Unfortunately, the obtained Pareto front only optimizes the dimensionless driving parameters for a very small friction environment. Therefore, the Pareto front method is extended and improved to reach a coordinated optimization for different friction environments. It is seen that the parameters determined by the non-Pareto solution cannot be optimized and should be excluded from the design. The results show that there are two typical Pareto region, namely, one is continuous corresponding to a small friction environment but another is discontinuous to a large friction environment. Consequently, the driving amplitude and frequency determined by combining the Pareto region with the stick–slip locomotion region are considered as the optimal parameters to achieve in the coordinated optimization of the locomotion velocity and energy consumption.

Appendix

The locomotion of the vibration-driven system is divided in four region E1-E4 in A-k plane, as shown in Fig. 2.

For region E1, the system performs reciprocating motion, and sticks twice in a cycle, as shown in Fig. 3a. The solution of Eq. (5) is expressed as

$$v_{ + }^{*} (t^{*} ) = - \cos t^{*} + \sqrt {1 - (Ak)^{2} } - Ak(t^{*} - t_{{n_{s} 1}}^{*} ),\;\;v_{ - }^{*} (t^{*} ) = - \cos t^{*} - \sqrt {1 - A^{2} } + A(t^{*} - t_{{n_{s} 3}}^{*} ),$$

(A1)

where $v_{ + }^{*}$ and $v_{ - }^{*}$ respectively describe the dimensionless velocity v^* > 0 and v^* < 0 and $t_{{n_{s} 1}}^{*}$ and $t_{{n_{s} 3}}^{*}$ are defined as Eq. (14).

For region E2, the system can perform directional motion only in this region, with one sticking motion in a cycle, as shown in Fig. 3b. The solution of Eq. (5) is expressed as

$$v_{ + }^{*} (t^{*} ) = - \cos t^{*} + \sqrt {1 - (Ak)^{2} } - Ak(t^{*} - t_{{n_{s} 1}}^{*} ).$$

(A2)

Notably, if 0 < A < 1, system is stuck when $t \in (t_{{n_{s} 4}}^{*} + 2\pi ,t_{{(n_{s} + 1)1}}^{*} + 2\pi )$, and if $A \ge 1$, it happens when $t \in (t_{{n_{s} 2}}^{*} + 2\pi ,t_{{(n_{s} + 1)1}}^{*} + 2\pi )$, where $t_{{n_{s} 2}}^{*}$ and $t_{{n_{s} 4}}^{*}$ are defined as Eq. (14).

For region E3, the system performs reciprocating motion, sticking once in a cycle, as shown in Fig. 3d. The solution of Eq. (5) is given by

$$v_{ + }^{*} (t^{*} ) = - \cos t^{*} + \sqrt {1 - (Ak)^{2} } - Ak(t^{*} - t_{{n_{s} 1}}^{*} ),\;\;\;v_{ - }^{*} (t^{*} ) = - \cos t^{*} + \cos t_{x}^{*} + A(t^{*} - t_{x}^{*} ),$$

(A3)

where $t_{x}^{*}$ satisfies $v_{ - }^{*} (t_{x}^{*} ) = 0$ and $t_{x}^{*} \in (t_{{n_{s} 3}} ,t_{{n_{s} 4}} )$,

$$- \cos t_{x}^{*} + \sqrt {1 - (Ak)^{2} } - Ak(t_{x}^{*} - t_{{n_{s} 1}} ) = 0.$$

(A4)

For region E4, the system performs reciprocating motion without sticking, as shown in Fig. 3e. The solution of Eq. (5) is

$$v_{ + }^{*} (t^{*} ) = - \cos t^{*} + \cos t_{y}^{*} - Ak(t^{*} - t_{y}^{*} ),\;\;\;v_{ - }^{*} (t^{*} ) = - \cos t^{*} + \cos t_{x}^{*} + A(t^{*} - t_{x}^{*} ),$$

(A5)

where $t_{y}^{*} \in (t_{{n_{s} 1}}^{*} ,t_{{n_{s} 2}}^{*} )$ and $t_{x}^{*} \in (t_{{n_{s} 3}} ,t_{{n_{s} 4}} )$. It is seen from Eq. (A5) that $v_{ + }^{*} (t_{y}^{*} ) = 0$ and $v_{ - }^{*} (t_{x}^{*} ) = 0$. One has $v_{ + }^{*} (t_{x}^{*} ) = 0$ and $v_{ - }^{*} (t_{y}^{*} + 2\pi ) = 0$ since $v_{ + }^{*}$ and $v_{ - }^{*}$ are continuous at $t_{x}^{*}$ and $t_{y}^{*}$, and they are periodic in $2\pi$. Thus, the Eq. (A5) yields

$$t_{x}^{*} = t_{y}^{*} + \frac{2\pi }{{1 + k}},\;\;\; - \cos (t_{y}^{*} + \frac{2\pi }{{1 + k}}) + \cos t_{y}^{*} - Ak(\frac{2\pi }{{1 + k}}) = 0,$$

(A6)