Reduced-order models are central to motion planning and control of quadruped robots, yet existing templates are often hand-crafted for a specific locomotion modality. This motivates the need for automatic methods that extract task-specific, interpretable low-dimensional dynamics directly from data. We propose a methodology that combines a linear autoencoder with symbolic regression to derive such models. The linear autoencoder provides a consistent latent embedding for configurations, velocities, accelerations, and inputs, enabling the sparse identification of nonlinear dynamics (SINDy) to operate in a compact, physics-aligned space. A multi-phase, hybrid-aware training scheme ensures coherent latent coordinates across contact transitions. We focus our validation on quadruped jumping—a representative, challenging, yet contained scenario in which a principled template model is especially valuable. The resulting symbolic dynamics outperform the state-of-the-art handcrafted actuated spring-loaded inverted pendulum (aSLIP) baseline in simulation and hardware across multiple robots and jumping modalities.

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.

Stepping strategy, including step time and step location modulation, and hip strategy, i.e., upper-body movement, play crucial roles in achieving robust humanoid locomotion. However, exploiting these balance strategies in a unified and flexible manner has not been well addressed. In this article, we propose a sequential convex optimization approach. Based on the linear inverted pendulum model, we modulate step parameters, including step location and step time, using quadratically constrained quadratic programming in real time. Then, based on the nonlinear inverted pendulum plus flywheel model, we regulate angular momentum using the linear model predictive control. To accommodate for scenarios with height variation, we consider nonlinear 3-D locomotion dynamics explicitly. The proposed approach is validated via comparison studies and extensive experiments on the humanoid with planar and linear feet. The results demonstrate enhanced robustness against dynamic disturbances and adaptability to real-world scenarios. On average, the enhanced stepping strategy rejects 135% larger external forces than our previous article. Also, robust locomotion across height-varying stepping stones is realized, which is rarely reported for a humanoid robot with planar feet.

Current quadrupedal robot control methods face critical challenges: model-based approaches require extensive manual parameter tuning, while reinforcement learning (RL) methods demand prohibitive training time. To promote quadrupedal robot development, simplify motion controller design, and facilitate robot deployment, this study proposes a novel bio-inspired control scheme. Specifically, inspired by the differentiated modalities of the animal's proximal and distal joints, a multi-model fusion scheme is constructed. First, the hip movement in joint space is obtained by a central pattern generator(CPG), whereby motion gaits, including trotting and galloping, are generated by a coupling network. Then, to generate the knee motion, a CPG-driven finite state machine is first proposed to determine the gait state. On top of this, the spring-loaded inverted pendulum model is utilized to regulate the knee joint's torque command. To enhance forward stability and speed tracking accuracy, this study incorporates online feedback regulation that adjusts both the CPG frequency and joint oscillation amplitude based on attitude angle and forward velocity information. And, a virtual model control strategy is designed to modify the torque profile of the knee torque. To verify the proposed methodology, hardware experiments are conducted on a newly developed quadrupedal robot. Results demonstrate that (i) the small-sized robot can reach 0.8 m/s (2.0 BL/s), with minimal tracking errors and relatively stable robot postures; (ii) compared with the traditional case where CPG generates both hip and knee trajectory directly, the energy consumption is reduced by 11.2% with our method; (iii) the robot can realize smooth trot-gallop-trot gait transition on flat ground and robust walking across uneven terrains.

The usage of parallel elastic actuators (PEA) in legged robots could potentially enhance the joints and increase energy efficiency by providing extra torques. However, the current design that adopts tension springs or spiral springs usually requires additional working space for PEA add-ons and enlarges size and mass too much. Besides, they often tune the spring parameters especially the spring constant by hand, failing to achieve optimal performance when considering multiple objectives. To tackle these issues, this article designs a compact dual-slide PEA (DS-PEA) leg that adopts a compression spring structure. Through integrating with a dual-slide mechanism, the PEA elements are attached tightly to the transmission, resulting in a small-size and light-weighted design. Furthermore, we adopt a multiobjective optimization method, i.e., multi-Pareto fronts quantify, to automatically choose the proper spring constant. Simulation and hardware experiments demonstrate that peak torque, motor power, and cost of transport for motion tracking are all largely reduced, even when working at multiple trajectories. Extensive hopping experiments further validate the dynamic motion capability and the energy efficiency of the delicate design. The compact DS-PEA leg will be used in a quadruped robot shortly.

Controlled execution of dynamic motions in quadrupedal robots, especially those with articulated soft bodies, presents a unique set of challenges that traditional methods struggle to address efficiently. In this study, we tackle these issues by relying on a simple yet effective two-stage learning framework to generate dynamic motions for quadrupedal robots. First, a gradient-free evolution strategy is employed to discover simply represented control policies, eliminating the need for a predefined reference motion. Then, we refine these policies using deep reinforcement learning. Our approach enables the acquisition of complex motions like pronking and back-flipping, effectively from scratch. Additionally, our method simplifies the traditionally labour-intensive task of reward shaping, boosting the efficiency of the learning process. Importantly, our framework proves particularly effective for articulated soft quadrupeds, whose inherent compliance and adaptability make them ideal for dynamic tasks but also introduce unique control challenges.

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.

To promote the research in compliant quadrupedal locomotion, especially with parallel elasticity, we present Delft E-Go, which is an easily accessible quadruped that combines the Unitree Go1 with open-source mechanical add-ons and control architecture. Implementing this novel system required a combination of technical work and scientific innovation. First, a dedicated parallel spring with adjustable rest length is designed to strengthen each actuated joint. Then, a novel 3-D dual spring-loaded inverted pendulum model is proposed to characterize the compliant locomotion dynamics, decoupling the actuation with parallel compliance. Based on this template model, trajectory optimization is employed to generate optimal explosive motion without requiring reference defined in advance. To complete the system, a torque controller with anticipatory compensation is adopted for motion tracking. Extensive hardware experiments in multiple scenarios, such as trotting across uneven terrains, efficient walking, and explosive pronking, demonstrate the system’s reliability, energy benefits of parallel compliance, and enhanced locomotion capability. Particularly, we demonstrate for the first time the controlled pronking of a quadruped with asymmetric legs.

Bipedal locomotion has been widely studied in recent years, where passive safety (i.e., a biped rapidly brakes without falling) is deemed to be a pivotal problem. To realize safe 3-D walking, existing works resort to nonlinear optimization techniques based on simplified dynamics models, requiring hand-tuned reference trajectories. In this article, we propose to integrate safety constraints into constrained task-space imitation learning, endowing a humanoid robot with adaptive walking capability. Specifically, unlike previous work using nonlinear and coupled capturability dynamics, we first linearize the 3-D capture conditions using appropriate extreme values and then seamlessly incorporate them into constrained imitation learning. Furthermore, we propose novel heuristic rules to define control points, enabling adaptive locomotion learning. The resulting framework allows robots to learn locomotion skills from a few demonstrations efficiently and apply the learned skills to unseen 3-D scenarios while satisfying the constraints for passive safety. Unlike deep enforcement learning, our framework avoids the need of a large number of iterations or sim-to-real transfer. By virtue of the task-space adaptability, the proposed imitation learning framework can reuse collected demonstrations in a new robot platform. We validate our method by hardware experiments on Walker2 robot and simulations on COMAN robot.

Quadrupeds deployed in real-world scenarios need to be robust to unmodelled dynamic effects. In this work, we aim to increase the robustness of quadrupedal periodic forward jumping (i.e., pronking) by unifying cutting-edge model-based trajectory optimization and iterative learning control. Using a reduced-order soft anchor model, the optimization-based motion planner generates the periodic reference trajectory. The controller then iteratively learns the feedforward control signal in a repetition process, without requiring an accurate full-body model. When enhanced by a continuous learning mechanism, the proposed controller can learn the control inputs without resetting the system at the end of each iteration. Simulations and experiments on a quadruped with parallel springs demonstrate that continuous jumping can be learned in a matter of minutes, with high robustness against various types of terrain.

Robust locomotion is a challenging task for humanoid robots, especially when considering dynamic disturbances. This article proposes a disturbance observer-based cascaded model predictive control (MPC) approach for bipedal locomotion, with the capability of exploiting ankle, stepping, hip and height variation strategies. Specifically, based on the variable-height inverted pendulum model, a nonlinear MPC that is run at a low frequency is built for 3-D locomotion (i.e., with height variation) while accounting for the footstep modulation as well. Differing from previous works, the nonlinear MPC is formulated as a convex optimization problem by semidefinite relaxation. Subsequently, assuming a flywheel at the pelvis center, a linear MPC that is run at a high frequency is proposed to regulate angular momentum (e.g., through rotating the upper body), which is solved by convex quadratic programming. To run the cascaded MPC in a closed-loop manner, a high order sliding mode observer is designed to estimate system states and dynamic disturbances simultaneously. Simulation and hardware experiments demonstrate the walking robustness in real-world scenarios, including 3-D walking with varying speeds, walking across non-coplanar terrains and push recovery.