Towards a Comparative Measure of Legged Agility

Abstract

We introduce an agility measure enabling the comparison of two very different leaping-from-rest transitions by two comparably powered but morphologically different legged robots. We use the measure to show that a flexible spine outperforms a rigid back in the leaping-from-rest task. The agility measure also sheds light on the source of this benefit: core actuation through a sufficiently powerful parallel elastic actuated spine outperforms a similar power budget applied either only to preload the spine or only to actuate the spine during the leap, as well as a rigid backed configuration of the identical machine.

Appendices

Appendix 1: Endurance Calculations

The endurance of each leap is calculated as follows. The thermal temperature rise $\varDelta T_i$ incurred by each motor $i \in I$ during the leap is calculated via the thermal model described in Fig. 5 of [48]. Let $T_F$ denote the failure temperature of motor i and let $T_{i0}$ denote the motor i’s initial temperature before the leap. The number of times $n_i$ that motor i can perform the leap is approximated by:

$$ n_i = \frac{T_F - T_{i0}}{\varDelta T_i}. $$

The endurance of the leap is then given by the lowest individual motor endurance, or:

$$\begin{aligned} n&= \text {inf}_{i \in I} \ (n_i) \\&= \text {inf}_{i \in I} \Big (\frac{T_F - T_{i0}}{\varDelta T_i}\Big ), \end{aligned}$$

so as to extrapolate how many times the leap can be performed sequentially before thermal failure since thermal capacity represents the limiting resource for both Canid and XRL. This method allows us to sidestep the need to run repeated experiments pushing the thermal limits for each machine in order to calculate endurance which would risk motor damage.

Appendix 2: Energy and Power Density for Legged EM Actuators

Assuming that EM motors produce a magnetic field of uniform density, the motor creates force by having this field interact with permanent magnets. This interaction occurs over some area (the air gap) and so is proportional to $l^2$. Assuming that the motor does work by rotating through a fixed angle, the transformed displacement through a leg of arbitrary geometry will scale according to the characteristic length, l. The energy produced by the motor (the work done) is therefore proportional to $l^3$, so for constant density, specific energy is scale invariant.

Power density scaling is originally presented in [35], pp. 176–181, but will be reworked below with more detailed scaling analysis. Assuming energy density is mass-invariant in an actuator, the power density scaling will be considered for a hopping task. Neglecting air resistance the apex height will be constant, and so it follows that the liftoff velocity, $v_f$, will also be constant. Assuming the system starts crouched at rest, the leg must go through a fixed extension, l, and accelerate the body to $v_f$. Assuming constant acceleration, a, $v_f = a t$ and $l = \frac{1}{2} a t^2$ where t is the time the system is in contact with the ground. Substituting for a, $l = \frac{1}{2} v_f t$. Since $v_f$ is constant, t scales according to l. Given constant energy density, power density then scales according to $l^{-1}$. This means that for specific energy to remain performance limiting, specific power must scale according to $l^{-1}$. This is in sharp contrast to [49] where specific power scales according to $l^{0.5}$ in support of maintaining dynamic similarity with respect to the pendulous motion of a swinging body characteristic of certain animal climbers [50].