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.

Modular robotics offers a promising approach for developing versatile and adaptive robotic systems capable of autonomous reconfiguration. This paper presents a novel modular robotic system in which each module is equipped with independent actuation, battery power, and control, enabling both individual mobility and coordinated locomotion. The system employs a hierarchical Central Pattern Generator (CPG) framework, where a low-level CPG governs the motion of individual modules, while a high-level CPG facilitates inter-module synchronization, allowing for seamless transitions between independent and collective behaviors.To validate the proposed system, we conduct both simulations in MuJoCo and hardware experiments, evaluating the system’s locomotion capabilities under various configurations. We first assess the fundamental motion of a single module, followed by two-module and four-module cooperative locomotion. The results demonstrate the effectiveness of the CPG-based control framework in achieving robust, flexible, and scalable locomotion. The proposed modular architecture has potential applications in search-and-rescue operations, environmental monitoring, and autonomous exploration, where adaptability and reconfigurability are essential for mission success.

This study explores the dynamics of asymmetrical bounding gaits in quadrupedal robots, focusing on the integration of torso pitching and hip motion to enhance speed and stability. Traditional control strategies often enforce a fixed posture, minimizing natural body movements to simplify the control problem. However, this approach may overlook the inherent dynamical advantages found in natural locomotion. By considering the robot as two interconnected segments, we concentrate on stance leg motion while allowing passive torso oscillation, drawing inspiration from natural dynamics and underactuated robotics principles. Our control scheme employs Linear Inverted Pendulum (LIP) and Spring-Loaded Inverted Pendulum (SLIP) models to govern front and rear leg movements independently. This approach has been validated through extensive simulations and hardware experiments, demonstrating successful high-speed locomotion with top speeds nearing 4 m/s and reduced ground reaction forces, indicating a more efficient gait. Furthermore, unlike conventional methods, our strategy leverages natural torso oscillations to aid leg circulation and stride length, aligning robot dynamics more closely with biological counterparts. Our findings suggest that embracing the natural dynamics of quadrupedal movement, particularly in asymmetrical gaits like bounding, can lead to more stable, efficient, and high-speed robotic locomotion. This investigation lays the groundwork for future studies on versatile and dynamic quadrupedal gaits and their potential applications in scenarios demanding rapid and effective locomotion.

Online control design using a high-fidelity, full-order model for a bipedal robot can be challenging due to the size of the state space of the model. A commonly adopted solution to overcome this challenge is to approximate the fullorder model (anchor) with a simplified, reduced-order model (template), while performing control synthesis. Unfortunately it is challenging to make formal guarantees about the safety of an anchor model using a controller designed in an online fashion using a template model. To address this problem, this paper proposes a method to generate safety-preserving controllers for anchor models by performing reachability analysis on template models by relying on functions that bound the difference between the two models. This paper describes how this reachable set can be incorporated into a Model Predictive Control framework to select controllers that result in safe walking on the anchor model in an online fashion. The method is illustrated on a 5-link RABBIT model, and is shown to allow the robot to walk safely while utilizing controllers designed in an online fashion.

In this paper, we develop theoretical foundations for a new class of rehabilitation robot: body powered devices that route power between a user's joints. By harvesting power from a healthy joint to assist an impaired joint, novel bimanual and self-assist therapies are enabled. This approach complements existing robotic therapies aimed at promoting recovery of motor function after neurological injury. We employ hydraulic transmissions for routing power, or equivalently for coupling the motions of a user's joints. Fluid power routed through flexible tubing imposes constraints within a limb or between homologous joints across the body. Variable transmissions allow constraints to be steered on the fly, and simple valve switching realizes free space and locked motion. We examine two methods for realizing variable hydraulic transmissions: using valves to switch among redundant cylinders (digital hydraulics) or using an intervening electromechanical link. For both methods, we present a rigorous mathematical framework for describing and controlling the resulting constraints. Theoretical developments are supported by experiments using a prototype fluid-power exoskeleton.

This paper uses optimal control to simultaneously optimize the motion and morphology of a realistic model of a 2D Monoped. In particular, we compare the energetics of four different actuator configurations: a parallel elastic actuator (PEA) in the hip and a series elastic actuator in the leg (SEA), series hip and parallel leg, series hip and series leg, and parallel hip and parallel leg. We use realistic models with mass in the legs and feet, damping in the springs, and detailed DC electric motor models. The comparison is carried out for the cost of transport of three energetic measures: positive motor work, electrical losses, and positive electrical work, and evaluated as a function of velocity. In our optimization we include motor parameters, stiffness, and spring pre-compression terms as free variables, ensuring that we compare the energetically optimal version of each configuration at each velocity. We show that for the positive motor work and the electrical losses costs of transport (COT), the parallel hip and series leg configuration is energetically optimal. For the electrical work, the optimal configuration is speed dependent, with series hip and parallel leg optimal at low speeds, and both series hip series leg and parallel hip series leg optimal at high speeds.

If the control authority of a running system is insufficient to reach a target state in a single step, i.e. if deadbeat control is not possible, then a stabilizing controller is faced with the decision on how to plan intermediate steps. In this work, we compare the performance of a simple greedy control policy (that computes deadbeat inputs and simply caps them) with the optimal performance found by an exhaustive search through decision space. The performance criterion used in this study is the basin of attraction: the set of all states from where the target state will be reached in a finite number of steps. Using the planar spring-loaded inverted pendulum (SLIP) as a model for a running robot, we compare the two control schemes and a fully passive behavior. To this end, we extended the passive slip model to include a controllable, yet limited variation of the touchdown angle and of the damping in the leg spring. We quantified the number of steps that it would take for the model to fall or converge from arbitrary initial states. The paper highlights how the passive stabilization, that is inherent to the SLIP model, greatly influences the dynamics of the controlled system. Furthermore, it reveals some new insights into the structure of basins of attractions of SLIP-like running models.

This paper presents a novel variable transmission system that is based on the concept of digital hydraulics. In the proposed system, sets of rolling-diaphragm cylinders are mounted via different effective lever arms to an input and output joint. A variable subset of these cylinders is connected via three-way two-position on/off valves to a common hydraulic manifold. This introduces a controllable constraint on the hydraulic flow and creates a programmable hydraulic transmission. With three single-acting cylinders, we could realize 37 different transmission ratios. We investigated the nonholonomic flow constraint analytically, in simulation, and with an experimental prototype. Using water as fluid, we show that a very stiff transmission (124.2 Nm/rad) can be achieved within the range of ±6°. Theoretical transmission ratios are tracked with R-squared values of more than 0.996 and backlash is smaller than 1.4%. Furthermore, we show the applicability of the proposed transmission in the simulation of a body-powered knee-ankle exoskeleton.

Building on our previous work on passive dynamic walking with quadrupeds, we show that a large variety of gaits can be created completely passively by a quadrupedal model with elastic legs. Similar to the well-known Spring Loaded Inverted Pendulum model for bipeds, we created a conceptual quadrupedal model with elastic massless legs. To obtain a well-defined sequence of ground contact, we defined three distinct phases for each leg: stance, swing, and wait for touch down. Since a leg cannot make contact during swing, modifying the duration of this phase allows us to prevent feet from striking the ground prematurely. Gaits were identified in a single shooting implementation, such that the contact sequence was only influenced by the starting values of the numerical integration. By varying these values, we were able to identify trotting, pacing, walking, toelting, bounding, and galloping within a single model. For each of the identified gaits, we report the footfall pattern, ground contact forces, speed, and first order limit cycle stability.