Characterising the take-off dynamics and energy efficiency in spring-driven jumping robots

Abstract

Previous design methodologies for spring-driven jumping robots focused on jump height optimization for specific tasks. In doing so, numerous designs have been proposed including using nonlinear spring-linkages to increase the elastic energy storage and jump height. However, these systems can never achieve their theoretical maximum jump height due to taking off before the spring energy is fully released, resulting in an incomplete transfer of stored elastic energy to gravitational potential energy. This paper presents low-order models aimed at characterising the energy conversion during the acceleration phase of jumping. It also proposes practical solutions for increasing the energy efficiency of jumping robots. A dynamic analysis is conducted on a multibody system comprised of rotational links, which is experimentally validated using a physical demonstrator. The analysis reveals that inefficient energy conversion is attributed to inertial effects caused by rotational and unsprung masses. Since these masses cannot be entirely eliminated from a physical linkage, a practical approach to improving energy efficiency involves structural redesign to reduce structural mass and moments of inertia while maintaining compliance with structural strength and stiffness requirements.

Figures (10)

• Figure 1: An example of (a) a nonlinear spring-driven jumping robot with a rotational linkage mechanism that exhibits a nonlinear force-length relationship (e.g. JPL_Hale_2000) and (b) its entire jumping process.
• Figure 2: (a) A prismatic jumping model driven by a massless linear spring in the standing posture. (b-c) The velocity and (d-e) acceleration of the centre of mass and (f-g) ground reaction force of the prismatic model (a) with two different body mass fractions: 1 (b,d,f) and 0.1 (c,e,h), both at a given force-to-weight ratio of 10. Without the unsprung mass (b) will take off as the spring returns to its natural length and, thus, achieves an idealised take-off. With an unsprung mass at the foot (c) undergoes a delayed take-off. Note that the greyed area is the inflight phase after the model takes off, which is out of the scope of this study. The spring displacement is normalised by the natural spring length as $y/d$; the velocity of the centre of mass is normalised by the gravitational acceleration and characteristic length as, $\dot{y}_{CG}/\sqrt{gd}$; the acceleration of the centre of mass is $\ddot{y}_{CG}/g$. The force is normalised by the peak spring charging force, $F/F_{max}$.
• Figure 3: (a) An inverted baton model driven by a massless rotational spring, and (b) its free-body diagram during the acceleration phase. (c) The spring-charging phase of the model with a natural angle of 30$\degree$. (d,e,f) The angular velocity and (g,h,i) the ground reaction force of the rotational model as a function of normalised angular displacement. The model in (d-i) has three different normalised rotational spring stiffness, $k_r/mgd$: (d,g) 4.5, (e,h) 5.5, (f,i) 10, all at a given natural angle of 30$\degree$. Note that the given natural spring angle is selected arbitrarily. For a given normalised rotational spring stiffness, an increase in natural spring angle exacerbates the premature take-off. This is because an increase in the natural spring angle increases the spring restoring torque, which increases the centripetal force and subsequently causes more premature take-off. The greyed areas in (e-f) and (h-i) are the inflight phase after the model takes off, which is out of the scope of this study. The angular displacement is normalised by the natural spring angle, $\theta/\theta_{ini}$ and the angular velocity is normalised by the gravitational acceleration and the characteristic length, $\dot{\theta}=\sqrt{d/g}$; the force is normalised by the weight of the rotational system, $F/m_Bg$.
• Figure 4: Examples of multibody systems formed by the rhomboidal linkage with (a) rotational springs, (b) horizontally placed translational spring and (c) vertically placed translational spring LO_Statics_2023.
• Figure 5: (a) The experimental model and (b) the CAD model of the main body. Working schematic of the reduction gearbox with the active latch in (c) the spring-charging and (d) the acceleration phases.