Reconstruction of Backbone Curves for 3-D Locomotion of Limbless Robots

Limbless robots composed of alternating single-axis pitch and yaw joints have many internal degrees of freedom, which make them capable of versatile three-dimensional locomotion. Typically, motions are planned kinematically by a chronological and continuous sequence of backbone curves that capture desired macroscopic shapes of the robot. However, as the geometric arrangement of single-axis rotary joints creates constraints on the rotations in the robot, the robot typically does not accurately achieve the desired shapes defined by these backbone curves. This leads to locomotor dynamics which are not well predicted by geometric models due to undesired contact with the environment. Here we propose a method for snake robots to reconstruct desired backbone curves by solving a polynomial optimization problem that exploits the robot's geometric structure. The method enables accurate curve-configuration conversions for commonly used 3-D gaits like sidewinding, sinus lifting, and helical rolling. We also demonstrated via robot experiments that the method results in smooth locomotion of the robot - in a full gait cycle, we found that the existing method resulted in an average joint angle change between time steps of 10.42 degrees, whereas the ones for our method were only 1.64 degrees.