Jointed Tails Enhance Control of Three-dimensional Body Rotation

TL;DR

An 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.

Abstract

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.

Figures (7)

• Figure 1: Model visualization. An example of the model structure, featuring a torso and a single-vertebra tail, showing the axes of rotation for the torso and tail.
• Figure 2: Goal torso rotation trajectories. (A, B, C) depict the 100 target trajectories for torso pitch, roll, and yaw angles, respectively. Each line represents an individual trajectory. Circles indicate the time points represented in (D). (D) depicts the torso orientation during three time points throughout one trajectory, bolded in (A, B, C). Curved colored arrows show the primary rotations that result in the differences between subsequent time steps. The asterisk indicates where the tail would attach.
• Figure 3: Each additional vertebra enhances inertial appendage performance. (A) The tracking error decreases with each additional vertebra. The single-vertebra tail had significantly higher tracking error compared to all other tail morphologies (see Tab. \ref{['tab:anova']} A for significance levels). (B) The velocity of the tail tip increases with each additional vertebra, which increases the total torque generated by the tail. These magnitudes are computed by taking the norm of the 3D vectors representing the linear velocities at the tail tip (see Tab. \ref{['tab:anova']} B for significance levels).
• Figure 4: Total control effort constraint limits performance of complex tails. Each gray line represents the value of a physical parameter for one joint degree of freedom. The red lines represent the limit imposed for that parameter. (A) represents the actuated joint angle, (B) represents the actuated joint velocity, (C) represents the joint torque, and (D) represents the total control effort. Note that only in D, the gray lines are near the limit throughout the duration of the trial, indicating that total control effort is the primary constraint on performance in this trial.
• Figure 5: Optimized vertebral lengths resemble mammal tails specialized for inertial maneuvering. (A) Optimized vertebral lengths for models with different tail configurations demonstrate a crescendo and decrescendo pattern. (B) In comparison to mammals with unspecialized mouse-like tails, mammals that use their tails for inertial maneuvering have a more pronounced crescendo and decrescendo pattern in their tail vertebral lengths.