Tails used as inertial appendages induce body rotations of animals and robots--a phenomenon that is governed largely by the ratio of the body and tail moments of inertia. However, vertebrate tails have more degrees of freedom (e.g., number of joints, rotational axes) than most current theoretical models and robotic tails. To understand how morphology affects inertial appendage function, we developed an optimization-based approach that finds the maximally effective tail trajectory and measures error from a target trajectory. For tails of equal total length and mass, increasing the number of equal-length joints increased the complexity of maximally effective tail motions. When we optimized the relative lengths of tail bones while keeping the total tail length, mass, and number of joints the same, this optimization-based approach found that the lengths match the pattern found in the tail bones of mammals specialized for inertial maneuvering. In both experiments, adding joints enhanced the performance of the inertial appendage, but with diminishing returns, largely due to the total control effort constraint. This optimization-based simulation can compare the maximum performance of diverse inertial appendages that dynamically vary in moment of inertia in 3D space, predict inertial capabilities from skeletal data, and inform the design of robotic inertial appendages.

Biological flying, gliding, and falling creatures are capable of extraordinary forms of inertial maneuvering: free-space maneuvering based on fine control of their multibody dynamics, as typified by the self-righting reflexes of cats. However, designing inertial maneuvering capability into biomimetic robots, such as biomimetic unmanned aerial vehicles (UAVs) is challenging. Accurately simulating this maneuvering requires numerical integrators that can ensure both singularity-free integration, and momentum and energy conservation, in a strongly coupled system - properties unavailable in existing conventional integrators. In this work, we develop a pair of novel quaternion variational integrators (QVIs) showing these properties, and demonstrate their capability for simulating inertial maneuvering in a biomimetic UAV showing complex multibody-dynamics coupling. Being quaternion-valued, these QVIs are innately singularity-free; and being variational, they can show excellent energy and momentum conservation properties. We explore the effect of variational integration order (left-rectangle vs. midpoint) on the conservation properties of integrator, and conclude that, in complex coupled systems in which canonical momenta may be time-varying, the midpoint integrator is required. The resulting midpoint QVI is well-suited to the analysis of inertial maneuvering in a biomimetic UAV - a feature that we demonstrate in simulation - and of other complex dynamical systems.