Kinematic inversion via closed-loop schemes is at the core of robotic manipulators' success. These algorithms robustly and efficiently compute a reference for low-level controllers to position the end effector or other selected parts of the robot's body in space. This work concerns the extension of kinematic inversion algorithms beyond fully actuated systems into the realm of underactuated robotics. We revise a well-known kinematic inversion algorithm in continuum (soft) robotics, the direct task-to-actuators inversion, and generalize it beyond this field and its current strong assumptions. More precisely, in its existing formulation, the algorithm relies on geometrical assumptions that hold valid only if the robot is fully actuated. We leverage recent advancements in system dynamics of mechanical systems to generalize the method beyond soft robotics and provide a constructive strategy that does not rely on any full actuation assumption. After that, we propose a second closed-loop inversion algorithm that works in even more general conditions, not requiring the explicit derivation of end-to-end actuators to task space mapping.@en

Deep reinforcement learning (DRL) has emerged as a promising solution to mastering explosive and versatile quadrupedal jumping skills. However, current DRL-based frameworks usually rely on pre-existing reference trajectories obtained by capturing animal motions or transferring experience from existing controllers. This work aims to prove that learning dynamic jumping is possible without relying on imitating a reference trajectory by leveraging a curriculum design. Starting from a vertical in-place jump, we generalize the learned policy to forward and diagonal jumps and, finally, we learn to jump across obstacles. Conditioned on the desired landing location, orientation, and obstacle dimensions, the proposed approach yields a wide range of omnidirectional jumping motions in real-world experiments. In particular, we achieve a 90 cm forward jump, exceeding all previous records for similar robots. Additionally, the robot can reliably execute continuous jumping on soft grassy grounds, which is especially remarkable as such conditions were not included in the training stage.

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Robot feet are crucial for maintaining dynamic stability and propelling the body during walking, especially on uneven terrains. Traditionally, robot feet were mostly designed as flat and stiff pieces of metal, which meets its limitations when the robot is required to step on irregular grounds, e.g., stones. While one could think that adding compliance under such feet would solve the problem, this is not the case. To address this problem, we introduced the SoftFoot, an adaptive foot design that can enhance walking performance over irregular grounds. The proposed design is completely passive and varies its shape and stiffness based on the exerted forces, through a system of pulley, tendons, and springs opportunely placed in the structure. This article outlines the motivation behind the SoftFoot and describes the theoretical model which led to its final design. The proposed system has been experimentally tested and compared with two analogous conventional feet, a rigid one and a compliant one, with similar footprints and soles. The experimental validation focuses on the analysis of the standing performance, measured in terms of the equivalent support surface extension and the compensatory ankle angle, and the rejection of impulsive forces, which is important in events such as stepping on unforeseen obstacles. Results show that the SoftFoot has the largest equivalent support surface when standing on obstacles, and absorbs impulsive loads in a way almost as good as a compliant foot.

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We identify the nonlinear normal modes spawning from the stable equilibrium of a double pendulum under gravity, and we establish their connection to homoclinic orbits through the unstable upright position as energy increases. This result is exploited to devise an efficient swing-up strategy for a double pendulum with weak, saturating actuators. Our approach involves stabilizing the system onto periodic orbits associated with the nonlinear modes while gradually injecting energy. Since these modes are autonomous system evolutions, the required control effort for stabilization is minimal. Even with actuator limitations of less than 1% of the maximum gravitational torque, the proposed method accomplishes the swing-up of the double pendulum by allowing sufficient time.

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The control possibilities for soft robots have long been hindered by the need for reliable methods to estimate their configuration. Inertial measurement units (IMUs) can solve this challenge, but they are affected by well-known drift issues. This letter proposes a method to eliminate this limitation by leveraging the Piecewise Constant Curvature model assumption. We validate the reconstruction capabilities of the algorithm in simulation and experimentally. To this end, we also present a novel large-scale, foam-based manipulator with embedded IMU sensors. Using the filter, we bring the accuracy in IMU-based reconstruction algorithms to 93% of the soft robot's length and enable substantially longer measurements than the baseline. We also show that the proposed technique generates reliable estimations for closed-loop control of the robot's shape.

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Imitation learning (IL) facilitates intuitive robotic programming. However, ensuring the reliability of learned behaviors remains a challenge. In the context of reaching motions, a robot should consistently reach its goal, regardless of its initial conditions. To meet this requirement, IL methods often employ specialized function approximators that guarantee this property by construction. Although effective, these approaches come with some limitations: 1) they are typically restricted in the range of motions they can model, resulting in suboptimal IL capabilities, and 2) they require explicit extensions to account for the geometry of motions that consider orientations. To address these challenges, we introduce a novel stability loss function that does not constrain the function approximator's architecture and enables learning policies that yield accurate results. Furthermore, it is not restricted to a specific state space geometry; therefore, it can easily incorporate the geometry of the robot's state space. Proof of the stability properties induced by this loss is provided and the method is empirically validated in various settings. These settings include Euclidean and non-Euclidean state spaces, as well as first-order and second-order motions, both in simulation and with real robots. More details about the experimental results can be found at https://youtu.be/ZWKLGntCI6w.

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Suitable representations of dynamical systems can simplify their analysis and control. On this line of thought, this article aims to answer the following question: Can a transformation of the generalized coordinates under which the actuators directly perform work on a subset of the configuration variables be found? We not only show that the answer to this question is yes but also provide necessary and sufficient conditions. More specifically, we look for a representation of the configuration space such that the right-hand side of the dynamics in Euler-Lagrange form becomes [\boldsymbol{I}\; \boldsymbol{O}]{T}\boldsymbol{u}, being \boldsymbol{u} the system input. We identify a class of systems, called collocated, for which this problem is solvable. Under mild conditions on the input matrix, a simple test is presented to verify whether a system is collocated or not. By exploiting power invariance, we provide necessary and sufficient conditions that a change of coordinates decouples the input channels if and only if the dynamics is collocated. In addition, we use the collocated form to derive novel controllers for damped underactuated mechanical systems. To demonstrate the theoretical findings, we consider several Lagrangian systems with a focus on continuum soft robots.

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This work concerns the application of physics-informed neural networks to the modeling and control of complex robotic systems. Achieving this goal requires extending physics-informed neural networks to handle nonconservative effects. These learned models are proposed to combine with model-based controllers originally developed with first-principle models in mind. By combining standard and new techniques, precise control performance can be achieved while proving theoretical stability bounds. These validations include real-world experiments of motion prediction with a soft robot and trajectory tracking with a Franka Emika Panda manipulator.

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People suffering from conditions affecting their activities of daily living and those who do straining repetitive tasks could be assisted using supportive devices. These devices have generally been stiff in design, with more recent advances exploring soft suits, removing the need for heavier structural components. These supportive devices are often fitted with rigid actuators that lack inherent compliance and rely on feedback to regulate the assistive force. Compl iant actuators able to control stiffness and pretension have only been applied in rigid assistive devices with these devices being designed for controllable stiffness in rotation and not linear motion. This work briefly presents the results of a user study on the effects of a compliant actuator in a soft supportive device for arm flexion, the development and testing of a variable linear stiffness mechanism for a linear motion capable of controlling the stiffness and equilibrium position, and the integration of said actuator in an exosuit.

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Adopting compliant structures holds the potential to enhance the robustness and interaction capabilities of the next generation of bionic limbs. Although researchers have proficiently explored this approach in the design of artificial hands, they devoted little attention to the development of more proximal joints. This work presents a compliant prosthetic elbow prototype called BICEP. The design incorporates compliant cross-axis flexural pivots that connect the upper and lower arm without direct contact between the two links. The actuation architecture, inspired by biological mechanisms, employs one actuator and two tendons to create an agonist-antagonist mechanism. This joint enables rotation along its flexion-extension axis while maintaining flexibility in out-of-plane directions, in a system with an overall weight of 351 g. A preliminary evaluation showcases lifting capacities supporting up to 2500 g, and a maximum speed of 157^∘ per second across a 135^∘ range of motion. The soft cross-axis flexural pivots exhibit compliant behavior in both the sagittal and transversal planes, enabling a pleasant interaction with the environment and ensuring safe absorption of unintentional impacts.

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Integrating Brain-Machine Interfaces into non-clinical applications like robot motion control remains difficult - despite remarkable advancements in clinical settings. Specifically, EEG-based motor imagery systems are still error-prone, posing safety risks when rigid robots operate near humans. This work presents an alternative pathway towards safe and effective operation by combining wearable EEG with physically embodied safety in soft robots. We introduce and test a pipeline that allows a user to move a soft robot's end effector in real time via brain waves that are measured by as few as three EEG channels. A robust motor imagery algorithm interprets the user's intentions to move the position of a virtual attractor to which the end effector is attracted, thanks to a new Cartesian impedance controller. We specifically focus here on planar soft robot-based architected metamaterials, which require the development of a novel control architecture to deal with the peculiar nonlinearities - e.g., non-affinity in control. We preliminarily but quantitatively evaluate the approach on the task of setpoint regulation. We observe that the user reaches the proximity of the setpoint in 66% of steps and that for successful steps, the average response time is 21.5s. We also demonstrate the execution of simple real-world tasks involving interaction with the environment, which would be extremely hard to perform if it were not for the robot's softness.@en

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.

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Soft robots promise groundbreaking advancements across various industries. However, soft robots are susceptible to wear, fatigue, and material degradation. Their durability and long-term reliability are often overlooked, despite being critical for the successful deployment of these systems in real-world applications. This article contributes to solving this challenge by identifying metrics that reflect material wear, mechanical hysteresis, and drift occurring during long-term operations in soft architectured materials. While this same pipeline can be generalized to different soft robots, we test these metrics on the trimmed helicoid architectured materials, and we validate the improvement in performance on the Helix soft manipulator. Thanks to the proposed metrics, we demonstrate a 75% reduction in repeatability errors over long-duration experiments.

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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.

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Formulating the dynamics of continuously deformable objects and other mechanical systems analytically from first principles is an exceedingly challenging task, often impractical in real-world scenarios. What makes this challenge even harder to solve is that, usually, the object has not been observed previously, and the only information that we can get from it is a stream of RGB camera data. In this study, we explore the use of deep learning techniques to solve this nonlinear identification problem. We specifically focus on extracting dynamic models of simple deformable objects from the high-dimensional sensor input coming from an RGB camera. We investigate a two-stage approach to achieve this goal. First, we train a variational autoencoder to extract an extremely low-dimensional representation of the object configuration. Then, we learn a dynamic model that predicts the evolution of these latent space variables. The proposed architecture can accurately predict the object's state up to one second into the future.

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Parallel robots based on Handed Shearing Auxetics (HSAs) can implement complex motions using standard electric motors while maintaining the complete softness of the structure, thanks to specifically designed architected metamaterials. However, their control is especially challenging due to varying and coupled stiffness, shearing, non-affine terms in the actuation model, and underactuation. In this paper, we present a model-based control strategy for planar HSA robots enabling regulation in task space. We formulate equations of motion, show that they admit a collocated form, and design a P-satI-D feedback controller with compensation for elastic and gravitational forces. We experimentally identify and verify the proposed control strategy in closed loop.@en

Robotics is entering our daily lives. The discipline is increasingly crucial in fields such as agriculture, medicine, and rescue operations, impacting our food, health, and planet. At the same time, it is becoming evident that robotic research must embrace and reflect the diversity of human society to address these broad challenges effectively. In recent years, gender inclusivity has received increasing attention, but it still remains a distant goal. In addition, awareness is rising around other dimensions of diversity, including nationality, religion, and politics. Unfortunately, despite the efforts, empirical evidence shows that the field has still a long way to go before achieving a sufficient level of equality, diversity, and inclusion across these spectra. This study focuses on the soft robotics community?a growing and relatively recent subfield?and it outlines the present state of equality and diversity panorama in this discipline. The article argues that its high interdisciplinary and accessibility make it a particularly welcoming branch of robotics. We discuss the elements that make this subdiscipline an example for the broader robotic field. At the same time, we recognize that the field should still improve in several ways and become more inclusive and diverse. We propose concrete actions that we believe will contribute to achieving this goal, and provide metrics to monitor its evolution.@en

The compliant nature of soft robots is appealing to a wide range of applications. However, this compliant property also poses several control challenges, e.g., how to deal with infinite degrees of freedom and highly nonlinear behaviors. This paper proposes a hybrid controller for a pneumatic-actuated soft robot. To this end, a model-based feedforward controller is designed and combined with a correction torque calculated via Gaussian process regression. Then, the proposed model-based and hybrid controllers are experimentally validated, and a detailed comparison between controllers is presented. Notably, the experimental results highlight the potential benefits of adding a learning approach to a model-based controller to enhance the closed-loop performance while reducing the computational load exhibited by purely learning strategies.

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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.

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This letter concerns control-oriented and structure-preserving learning of low-dimensional approximations of high-dimensional physical systems, with a focus on mechanical systems. We investigate the integration of neural autoencoders in model order reduction, while at the same time preserving Hamiltonian or Lagrangian structures. We focus on extensively evaluating the considered methodology by performing simulation and control experiments on large mass-spring-damper networks, with hundreds of states. The empirical findings reveal that compressed latent dynamics with less than 5 degrees of freedom can accurately reconstruct the original systems' transient and steady-state behavior with a relative total error of around 4%, while simultaneously accurately reconstructing the total energy. Leveraging this system compression technique, we introduce a model-based controller that exploits the mathematical structure of the compressed model to regulate the configuration of heavily underactuated mechanical systems.

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