  This paper presents a novel methodology to model and optimize trajectories of
a quadrupedal robot with spinal compliance to improve standing jump performance
compared to quadrupeds with a rigid spine. We introduce an elastic model for a
prismatic robotic spine that is actively preloaded and mechanically
lock-enabled at initial and maximum length, and develop a constrained
trajectory optimization method to co-optimize the elastic parameters and motion
trajectories toward enhanced jumping distance. Results reveal that a less stiff
spring is likely to facilitate jumping performance not as a direct propelling
source but as a means to unleash more motor power for propelling by trading-off
overall energy efficiency. We also visualize the impact of spring coefficients
on the overall optimization routine from energetic perspectives to identify the
suitable parameter region.

Robotics;Systems and Control;Systems and Control