Galloping is a common high-speed gait in both animals and quadrupedal robots, yet its energetic characteristics remain insufficiently explored. This study systematically analyzes a large number of possible galloping gaits by categorizing them based on the number of flight phases per stride and the phase relationships between the front and rear legs, following Hildebrand's framework for asymmetrical gaits. Using the A1 quadrupedal robot from Unitree, we model galloping dynamics as a hybrid dynamical system and employ trajectory optimization (TO) to minimize the cost of transport (CoT) across a range of speeds. Our results reveal that rotary and transverse gallop footfall sequences exhibit no fundamental energetic difference, despite variations in body yaw and roll motion. However, the number of flight phases significantly impacts energy efficiency: galloping with no flight phases is optimal at lower speeds, whereas galloping with two flight phases minimizes energy consumption at higher speeds. We validate these findings using a quadratic programming (QP)-based controller, developed in our previous work, in Gazebo simulations. These insights advance the understanding of quadrupedal locomotion energetics and may inform future legged robot designs for adaptive, energy-efficient gait transitions.

Ship roll motion in high sea states has large amplitudes and nonlinear dynamics, and its prediction is significant for operability, safety, and survivability. This paper presents a novel data-driven methodology to provide a multi-step prediction of ship roll motions in high sea states. A hybrid neural network is proposed that combines long short-term memory (LSTM) and convolutional neural network (CNN) in parallel. The motivation is to extract the nonlinear dynamic characteristics and the hydrodynamic memory information through the advantage of CNN and LSTM, respectively. For the feature selection, the time histories of motion states and wave heights are selected to involve sufficient information. Taken a scaled KCS as the study object, the ship motions in sea state 7 irregular long-crested waves are simulated and used for the validation. The results show that at least one period of roll motion can be accurately predicted. Compared with the single LSTM and CNN methods, the proposed method has better performance in predicting the amplitude of roll angles. Besides, the comparison results also demonstrate that selecting motion states and wave heights as feature space improves the prediction accuracy, verifying the effectiveness of the proposed method.