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. ... Read more

Soft robots promise inherent safety via their material compliance for seamless interactions with humans or delicate environments. Yet, their development is challenging because it requires integrating materials, geometry, actuation, and autonomy into complex mechatronic systems. Despite progress, the field struggles to balance task-specific performance with broader factors like durability and manufacturability—a difficulty that we find is compounded by traditional sequential design processes with their lack of feedback loops. In this perspective, we review emerging co-design approaches that simultaneously optimize the body and brain, enabling the discovery of unconventional designs highly tailored to the given tasks. We then identify three key shortcomings that limit the broader adoption of such co-design methods within the soft robotics domain. First, many rely on simulation-based evaluations focusing on a single metric, while real-world designs must satisfy diverse criteria. Second, current methods emphasize computational modeling without ensuring feasible realization, risking sim-to-real performance gaps. Third, high computational demands limit the exploration of the complete design space. Finally, we propose a holistic co-design framework that addresses these challenges by incorporating a broader range of design values, integrating real-world prototyping to refine evaluations, and boosting efficiency through surrogate metrics and model-based control strategies. This holistic framework, by simultaneously optimizing functionality, durability, and manufacturability, has the potential to enhance reliability and foster broader acceptance of soft robotics, transforming human-robot interactions. ... Read more

Obtaining dynamic models of continuum soft robots is central to the analysis and control of soft robots, and researchers have devoted much attention to the challenge of proposing both data-driven and first-principle solutions. Both avenues have, however, shown their limitations; the former lacks structure and performs poorly outside training data, while the latter requires significant simplifications and extensive expert knowledge to be used in practice. This paper introduces a streamlined method for learning low-dimensional, physicsbased models that are both accurate and easy to interpret. We start with an algorithm that uses image data (i.e., shape evolutions) to determine the minimal necessary segments for describing a soft robot's movement. Following this, we apply a dynamic regression and strain sparsification algorithm to identify relevant strains and define the model's dynamics. We validate our approach through simulations with various planar soft manipulators, comparing its performance against other learning strategies, showing that our models are both computationally efficient and 25x more accurate on out-of-training distribution inputs. Finally, we demonstrate that thanks to the capability of the method of generating physically compatible models, the learned models can be straightforwardly combined with model-based control policies. ... Read more

As we increasingly strive to integrate robots into human-centric environments, safety is a top priority. Traditionally, rigid collaborative robots have relied on safety-aware computational control policies, which are susceptible to perception errors and often lead to overly cautious behavior that limits performance. In contrast, soft robotics offers a promising alternative by ensuring passive compliance throughout the robot’s structure via material softness. This mechanical compliance inherently mitigates safety issues arising from perception or control errors, although this has been paid with a substantial drop in precision. In recent years, significant advances have been seen in soft robotics, with exciting new developments in design, smart materials, actuators, sensors, models, and control strategies. However, the modeling and control of continuum soft robots continue to pose major challenges due to their infinite degrees of freedom, complex nonlinear dynamics, and time-dependent behaviors like hysteresis. As a result, soft robots often lack the necessary capability and motion precision, leading to a tradeoff where performance is sacrificed for safety. With this thesis, we argue that this tradeoff can be overcome by developing more advanced algorithms that can reason on the physics of the soft robot. More specifically, we propose combining powerful learned models with efficient and effective model-based control approaches that allow for interpretability into the actions and admit stability guarantees.

Currently, two main approaches exist for controlling soft robots. The first employs model-based control using approximated physics-based models derived from first principles. The second directly learns control policies, primarily through reinforcement learning. Both strategies face notable limitations. Existing model-based controllers are unable to fully manage and eventually exploit the dynamics of soft robots because their underlying models inadequately capture complex behaviors, particularly how actuation and external interactions affect the robot’s deformation. Moreover, deriving these models requires extensive expert knowledge. Additionally, the combined complexity and uncertainty of the dynamics between a soft robot and its environment make it currently infeasible to develop comprehensive world models from first principles alone, thereby motivating the integration of machine learning approaches that can effectively leverage data-driven insights. Conversely, directly learning the controller — such as via reinforcement learning — lacks interpretability and stability guarantees while being highly sample inefficient, a significant drawback given the time-dependent material properties and limited lifespan of current soft robots.

In this thesis, we contend that combining learned models with model-based controllers presents a promising alternative that brings together the advantages of both approaches: expressive, data-driven models that require less expert knowledge paired with controllers that are both interpretable and provably stable. Although recent years have seen increased interest in leveraging learned models for control, most work in this area depends on computationally intensive optimal control methods, such as MPC, to optimize the actuation sequence with the learned model. However, the high computational cost of solving these optimal control problems limits the maximum control frequency during deployment, preventing us from fully exploiting the dynamic capabilities of soft robots. Instead, this thesis pursues closed-form controllers that utilize the physical structure of learned models within an energy-shaping framework. The main challenge here is to develop approaches that integrate such physical structures—specifically, kinetic and potential energy terms—into the learning of dynamical models for soft robots. Before addressing this main challenge, we first had to advance physics-based models derived from first principles and identify novel techniques to leverage them for control. On one hand, this clarified which physical priors were available for learning, while on the other hand, it inspired new ways to integrate model-based controllers with learned models. The thesis addresses this topic through several interconnected key contributions.

First, we argue that quantifying the safety of soft robots is crucial for designing and controlling them to ensure that the closed-loop system meets the specific safety requirements of their intended applications. To this end, we present the first safety metric for continuum soft robots, which assesses the safety of an integrated soft robot design by accounting for both its embodied and computational intelligence.

Secondly, this thesis enhances shape sensing for soft robots by leveraging insights from kinematic models. We accomplish this by formulating and solving nonlinear optimization problems that align sensor measurements with the backbone shapes predicted by the kinematic model. We present two distinct approaches that integrate commercial sensors—namely visual and magnetic—with SLAM algorithms and a learned sensor measurement model, respectively, to accurately estimate the soft robot’s state, a key requirement for effective feedback control.

Thirdly, this thesis introduces advanced physics-based actuation models, including those for robots actuated by auxetic metamaterials - referred to as HSA robots—and models that capture the actuation dynamics of piston-driven pneumatic soft robots. We then leverage the acquired model insights to design provably stable nonlinear controllers—specifically, an integral-saturated PID combined with potential shaping and Cartesian-space impedance control for planar GSA robots, as well as a backstepping controller for pneumatic piston-driven soft robots. This contribution deepens our understanding of actuation, a critical aspect of soft robot behavior, and demonstrates how such insights can be incorporated into model-based control strategies. Moreover, experiments with HSA robots have highlighted the limitations of purely physics-based models in capturing complex phenomena like hysteresis, thereby motivating the exploration of learning-based approaches. In the future, the developed actuation models can serve as valuable physical priors for learned models.

Fourthly, the thesis presents techniques for learning soft robot models that incorporate physical structures while ensuring stability. We accomplish this by embedding physics-based dynamical models into the learning algorithm, which determines the free parameters of the dynamics and optionally optimizes a coordinate transformation—such as encoding into latent space. Two notable approaches are introduced: (1) an algorithm that extracts low-dimensional soft robot strain models from samples of the robot backbone’s shape evolution, and (2) a network of coupled harmonic oscillators for learning latent dynamics from high-dimensional observations like images. The explicit inclusion of kinematic and potential energy terms in these models allows for stability analysis using standard nonlinear system theory tools, such as Lyapunov methods. For instance, we prove that the coupled oscillator network is both globally asymptotically stable and input-to-state stable.

Fifthly, we exploit the physical structure of the learned models from contribution four to design closed-form setpoint regulators. The controller contains two key components: (1) a potential shaping feedforward term that positions the local/global minimum of the closed-loop potential energy at the setpoint by leveraging the learned model knowledge, and (2) an integral-saturated PID feedback term that rejects disturbances and compensates for modeling errors to prevent steady-state errors. The stability of the closed-loop system can then be analyzed using Lyapunov arguments.

Finally, the thesis explores methods for generating compliant motion behaviors in soft robots beyond low-level control. One approach focuses on assisting users, particularly elderly individuals, with activities of daily living by guiding the low-level controller with brain signals. This is achieved by combining motor imagery classification from wearable EEG devices with compliant impedance control in operational space. The second approach combines an orbitally stable dynamical system in latent space with a bijective neural network parametrized encoder to learn periodic motions from demonstrations. By avoiding reliance on time references, this learned motion policy enables natural and compliant tracking of demonstrated periodic motions. This contribution ensures that not just the robot structure and low-level controller are compliant, but also the high-level motion strategy. ... Read more

Robots operating alongside people, particularly in sensitive scenarios such as aiding the elderly with daily tasks or collaborating with workers in manufacturing, must guarantee safety and cultivate user trust. Continuum soft manipulators promise safety through material compliance, but as designs evolve for greater precision, payload capacity, and speed, and increasingly incorporate rigid elements, their injury risk resurfaces. In this letter, we introduce a comprehensive High-Order Control Barrier Function (HOCBF) + High-Order Control Lyapunov Function (HOCLF) framework that enforces strict contact force limits across the entire soft-robot body during environmental interactions. Our approach combines a differentiable Piecewise Cosserat-Segment (PCS) dynamics model with a convex-polygon distance approximation metric, named Differentiable Conservative Separating Axis Theorem (DCSAT), based on the soft robot geometry to enable real-time, whole-body collision detection, resolution, and enforcement of the safety constraints. By embedding HOCBFs into our optimization routine, we guarantee safety, allowing, for instance, safe navigation in operational space under HOCLF-driven motion objectives. Extensive planar simulations demonstrate that our method maintains safety-bounded contacts while achieving precise shape and task-space regulation. This work thus lays a foundation for the deployment of soft robots in human-centric environments with provable safety and performance. ... Read more

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. ... Read more

We introduce the concept of collaborative awareness as a means to enhance interoperability, resilience and self regulation in synthetic agent collectives. We discuss the theoretical, computational and engineering framework of collaborative awareness built by the EU project EMERGE, and its application to distributed robotic systems. ... Read more

Model-based strategies are a promising solution to the grand challenge of equipping continuum soft robots with motor intelligence. However, finite-dimensional models of these systems are inherently inaccurate, thus posing pressing robustness concerns. Moreover, the actuation space of soft robots is usually limited. This article aims at solving both these challenges by proposing a robust model-based strategy for the shape control of soft robots with system uncertainty and input saturation. The proposed architecture is composed of two key components. First, we propose an observer that estimates deviations between the theoretical model and the soft robot, ensuring that the estimation error converges to zero within finite time. Second, we introduce a sliding mode controller to regulate the soft robot shape while fulfilling saturation constraints. This controller uses the observer's output to compensate for the deviations between the real system and the established model. We prove the convergence of the closed-loop with theoretical analysis and the method's effectiveness with simulations and experiments. ... Read more

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. ... Read more

Electrically-actuated continuum soft robots based on Handed Shearing Auxetics (HSAs) promise rapid actuation capabilities while preserving structural compliance. However, the foundational models of these novel actuators required for precise control strategies are missing. This paper proposes two key components extending discrete Cosserat rod model (DCM) to allow for modeling HSAs. First, we propose a mechanism for incorporating the auxetic trajectory into DCM dynamical simulations. We also propose an implementation of this extension as a plugin for the Elastica simulator. Second, we introduce a Selective Piecewise Constant Strain (SPCS) kinematic parameterization that can describe an HSA segment's shape with fewer configuration variables. We verify both theoretical contributions experimentally. The simulator is used to replicate experimental data of the mechanical characterization of HSA rods. For the second component, we attach motion capture markers at various points to a parallel HSA robot and find that the shape of the HSAs can be kinematically represented with an average accuracy of 0.3 mm for positions and 0.07 rad for orientations. ... Read more

Accurate and complete terrain maps enhance the awareness of autonomous robots and enable safe and optimal path planning. Rocks and topography often create occlusions and lead to missing elevation information in the Digital Elevation Map (DEM). Currently, these occluded areas are either fully avoided during motion planning or the missing values in the elevation map are filled-in using traditional interpolation, diffusion or patch-matching techniques. These methods cannot leverage the high-level terrain characteristics and the geometric constraints of line of sight we humans use intuitively to predict occluded areas. We introduce a self-supervised learning approach capable of training on real-world data without a need for ground-truth information to reconstruct the occluded areas in the DEMs. We accomplish this by adding artificial occlusion to the incomplete elevation maps constructed on a real robot by performing ray casting. We first evaluate a supervised learning approach on synthetic data for which we have the full ground-truth available and subsequently move to several real-world datasets. These real-world datasets were recorded during exploration of both structured and unstructured terrain with a legged robot, and additionally in a planetary scenario on Lunar analogue terrain. We state a significant improvement compared to the baseline methods both on synthetic terrain and for the real-world datasets. Our neural network is able to run in real-time on both CPU and GPU with suitable sampling rates for autonomous ground robots. We motivate the applicability of reconstructing occlusion in elevation maps with preliminary motion planning experiments. ... Read more

Sensing the shape of continuum soft robots without obstructing their movements and modifying their natural softness requires innovative solutions. This letter proposes to use magnetic sensors fully integrated into the robot to achieve proprioception. Magnetic sensors are compact, sensitive, and easy to integrate into a soft robot. We also propose a neural architecture to make sense of the highly nonlinear relationship between the perceived intensity of the magnetic field and the shape of the robot. By injecting a priori knowledge from the kinematic model, we obtain an effective yet data-efficient learning strategy. We first demonstrate in simulation the value of this kinematic prior by investigating the proprioception behavior when varying the sensor configuration, which does not require us to re-train the neural network. We validate our approach in experiments involving one soft segment containing a cylindrical magnet and three magnetoresistive sensors. During the experiments, we achieve mean relative errors of 4.5%. ... Read more

The nature of continuum soft robots calls for novel perception solutions, which can provide information on the robot's shape while not substantially modifying their bodies' softness. One way to achieve this goal is to develop innovative and completely deformable sensors. However, these solutions tend to be less reliable than classic sensors for rigid robots. As an alternative, we consider here the use of monocular cameras. By admitting a small rigid component in our design, we can leverage well-established solutions from mobile robotics. We propose a shape sensing strategy that combines a SLAM algorithm with nonlinear optimization based on the robot's kinematic model. We prove the method's effectiveness in simulation and with experiments of a single-segment continuous soft robot with a camera mounted to the tip. We achieve mean relative translational errors below 9% simulations and experiments alike, and as low as 0.5% on average for some simulation conditions. ... Read more

Actuators’ dynamics have been so far mostly neglected when devising feedback controllers for continuum soft robots since the problem under the direct actuation hypothesis is already quite hard to solve. Directly considering actuation would have made the challenge too complex. However, these effects are, in practice, far from being negligible. The present work focuses on model-based control of piston-driven pneumatically-actuated soft robots. We propose a model of the relationship between the robot’s state, the acting fluidic pressure, and the piston dynamics, which is agnostic to the chosen model for the soft system dynamics. We show that backstepping is applicable even if the feedback coupling of the outer on the inner subsystem is not linear. Thus, we introduce a general model-based control strategy based on backstepping for soft robots actuated by fluidic drive. As an example, we derive a specialized version for a robot with piecewise constant curvature. ... Read more