Jointed tails enhance control of three-dimensional body rotation

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 and 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 matched the pattern found in the tail bones of mammals specialized for inertial manoeuvring. 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 a moment of inertia in three-dimensional space, predict inertial capabilities from skeletal data and inform the design of robotic inertial appendages.