Quadrupedal jumping has been intensively investigated in recent years. Still, realizing controlled jumping with soft landings remains an open challenge due to the complexity of the jump dynamics and the need to perform complex computations during the short time. This work tackles this challenge by leveraging trajectory optimization and behavior cloning. We generate an optimal jumping motion by utilizing dual-layered coarse-to-refine trajectory optimization. We combine this with a variable impedance control approach to achieve soft landing. Finally, we distill this computationally heavy jumping and landing policy into an efficient neural network via behavior cloning. Extensive simulation experiments demonstrate that, compared to classic model predictive control, the variable impedance control ensures compliance and reduces the stress on the motors during the landing phase. Furthermore, the neural network can reproduce jumping and landing behavior, achieving at least a 97.4% success rate. Hardware experiments confirm the findings, showcasing explosive jumping with soft landings and on-the-fly evaluation of the control actions.

Introducing parallel elasticity in the hardware design endows quadrupedal robots with the ability to perform explosive and efficient motions. However, for this kind of articulated soft quadruped, realizing dynamic jumping with robustness against system uncertainties remains a challenging problem. To achieve this, we propose an impact-aware jumping planning and control approach. Specifically, an offline kino-dynamic-type trajectory optimizer is first formulated to achieve compliant 3D jumping motions, using a novel actuated spring-loaded inverted pendulum (SLIP) model. Then, an optimization-based online landing strategy, including pre-impact leg motion modulation and post-impact landing recovery, is designed. The actuated SLIP model, with the capability of explicitly characterizing parallel elasticity, captures the jumping and landing dynamics, making the problem of motion generation/regulation more tractable. Finally, a hybrid torque control consisting of a feedback tracking loop and a feedforward compensation loop is employed for motion control. Experiments demonstrate the ability to accomplish robust 3D jumping motions with stable landing and recovery. Besides, our approach can be applied to quadrupedal robots with or without additional parallel compliance.