Soft robots' ability to safely navigate complex environments motivates the development of algorithms for accurate environmental interaction assessment, enabling greater autonomy. Specifically, strain-based shape and force estimation of continuum robots with embedded soft sensors poses an open challenge mainly owing to continuous softness, anisotropic deformation, and non-linear properties. Mathematical description of deformable soft bodies and accurate estimation of external forces are crucial for achieving controllable and intelligent behaviors of these robots. In this paper, a kinetostatic strain-based modeling for rod-driven soft robots (RDSR) with embedded stretch sensors is proposed, which incorporates local strains, actuation variables, and external interactions. The strain model enables full shape estimation of the robot and prediction of strain variations in soft bodies. Building on this, we develop a force estimator based on predicted and measured sensor and actuator lengths to evaluate 3D external forces, accounting for both orthogonal and tangential components relative to the backbone. Moreover, we introduce a methodology using a novel ellipsoid representation to handle tangential forces that may become insensitive in certain singular configurations. This estimator allows us to either disregard such forces when they do not influence deformation or estimate them when they become observable. Our simulations and experiments demonstrate how this approach can be used to analyze the robot's configuration and successfully estimate external forces. Finally, it is demonstrated that when the continuum arm follows trajectories with higher strain sensitivity, tangential force estimation is significantly improved.

Underwater monitoring is extremely complex due to the lack of a global localization system, limited communication and environmental factors such as turbidity and darkness that limit visibility, affecting control and situational awareness. Typically, monitoring relies on a single autonomous underwater vehicle (AUV) or a set of independent AUVs; techniques which are prone to failure as they rely only on onboard odometry and sensors, making missions vulnerable to malfunctions, damage, and noise. To address these challenges, we propose a Multi-Agent Reinforcement Learning (MARL) framework to enable cooperation among multiple AUVs, mitigating the limitations of the underwater environment. Our in-silico solution focuses on a group of robots learning a strategy to follow a partially hidden underwater pipe without global localization, while dealing with environmental disturbances affecting sensors and actuators. The numerosity of the agents, and most importantly their collaboration, helps overcome underwater visibility constraints. By sharing relative position information of neighboring agents with respect to the pipe, navigation is improved. By introducing quantitative measures for pipe exploration, we show that cooperation significantly enhances system performance compared to independent agents. Emerging collaboration among robots allows the swarm to complete pipe inspections faster and more efficiently than non-cooperative baseline models of non-interacting agents, even under extremely reduced visibility scenarios. Moreover, single agents also benefit from cooperation, learning effective policies more quickly and covering a longer portion of the pipe. Finally, our model guarantees explainability. We analyze learned strategies and provide a visualization method that allows the interpretation of the learned policies. • Reinforcement learning is applied to pipeline following by underwater robotic agent. • In conditions of poor visibility, a single agent is not able to complete the mission. • In contrast, multi-agent team completes this task using reinforcement learning. • Swarm collaboration enables faster and more efficient task completion. • Collaboration enables agents develop more efficient individual strategies.

Soft robots show compliance and have infinite degrees of freedom. Thanks to these properties, such robots can be leveraged for surgery, rehabilitation, biomimetics, unstructured environment exploring, and industrial grippers. In this case, they attract scholars from a variety of areas. However, nonlinearity and hysteresis effects also bring a burden to robot modeling. Moreover, following their flexibility and adaptation, soft robot control is more challenging than rigid robot control. In order to model and control soft robots, a large number of data-driven methods are utilized in pairs or separately. This review first briefly introduces two foundations for data-driven approaches, which are physical models and the Jacobian matrix, then summarizes three kinds of data-driven approaches, which are statistical method, neural network, and reinforcement learning. This review compares the modeling and controller features, e.g., model dynamics, data requirement, and target task, within and among these categories. Finally, we summarize the features of each method. A discussion about the advantages and limitations of the existing modeling and control approaches is presented, and we forecast the future of data-driven approaches in soft robots. A website (https://sites.google.com/view/23zcb) is built for this review and will be updated frequently.Note to Practitioners—This work is motivated by the need for a review introducing soft robot modeling and control methods in parallel. Modeling and control play significant roles in robot research, and they are challenging especially for soft robots. The nonlinear and complex deformation of such robots necessitates specific modeling and control approaches. We introduce the state-of-the-art data-driven methods and survey three approaches widely utilized. This review also compares the performance of these methods, considering some important features like data amount requirement, control frequency, and target task. The features of each approach are summarized, and we discuss the possible future of this area.

Soft robots offer remarkable adaptability and safety advantages over rigid robots, but modeling their complex, nonlinear dynamics remains challenging. Strain-based models have recently emerged as a promising candidate to describe such systems, however, they tend to be high-dimensional and time-consuming. This article presents a novel model order reduction approach for soft and hybrid robots by combining strain-based modeling with proper orthogonal decomposition (POD). The method identifies optimal coupled strain basis functions—or mechanical synergies—from simulation data, enabling the description of soft robot configurations with a minimal number of generalized coordinates. The reduced order model (ROM) achieves substantial dimensionality reduction in the configuration space while preserving accuracy. Rigorous testing demonstrates the interpolation and extrapolation capabilities of the ROM for soft manipulators under static and dynamic conditions. The approach is further validated on a snake-like hyper-redundant rigid manipulator and a closed-chain system with soft and rigid components, illustrating its broad applicability. Moreover, the approach is leveraged for shape estimation of a real six-actuator soft manipulator using only two position markers, showcasing its practical utility. Finally, the ROM's dynamic and static behavior is validated experimentally against a parallel hybrid soft-rigid system, highlighting its effectiveness in representing the high-order model and the real system. This POD-based ROM offers significant computational speed-ups, paving the way for real-time simulation and control of complex soft and hybrid robots.

Soft continuum robots, characterized by their inherent compliance and dexterity, are increasingly pivotal in applications requiring delicate interactions with the environment such as the medical field. Despite their advantages, challenges persist in accurately modeling and controlling their shape during interactions with surrounding objects. This is because of the difficulty in modeling the large degrees of freedom in soft-bodied objects that become more active during interactions. In this study, we present a deep visual model to predict the interaction shapes of a soft continuum robot in contact with surrounding objects. By formulating this task as a forward-statics problem, the model uses the initial state images containing the object configuration and future actuation values to predict interactive state images of the robot under this actuation condition. We developed and tested the model in both simulated and physical environments, explored the model’s predictive capabilities using monocular and binocular views, and tested the model’s generalization ability on different datasets. Our results show that deep learning methods are a promising tool for solving the complex problem of predicting the shape of a soft continuum robot interacting with the environment, requiring no prior knowledge about the system dynamics and explicit mapping of the environment. This study paves the way for future explorations in robot-environment interaction modeling and the development of more adaptable interaction shape control strategies.

Rod-driven soft robots (RDSR) with a well-balanced performance in terms of perception, precision, and intelligence have a great potential for application. Mathematical description and predicted sensing of deformable soft bodies are crucial to achieve controllable and intelligent behaviors of these robots. In this work, we propose a kinetostatic model for RDSR embedded with co-located sensors based on the Geometric Variable Strain (GVS) approach where local deformations, actuation lengths and external interactions are included. This approach allows us to estimate the shape of RDSR and predict the strain variation of soft bodies under internal and external interactions. Simulations and experimental results show that tip position errors are not greater than 1.8% with respect to the whole body length under different loads (0, 100, 200, 300 gf). The maximum error of predicted sensor length change is up to 2 mm and its percentage relative to the actual length does not exceed 4%. The results demonstrate the accuracy and effectiveness of the proposed model.

In this article, we propose a new algorithm for solving the forward dynamics of multibody systems consisting of rigid bodies connected in arbitrary topologies by localized joints and/or soft links, possibly actuated or not. The simulation is based on the implicit time integration of the Lagrangian model of these systems, where the soft links are modeled by Cosserat rods parameterized by assumed strain modes. This choice imposes a predictor–corrector structure on the approach, and requires computing both the residual vector and the Jacobian of the residual vector of the dynamics constrained by the time integrator. These additional calculations are handled here with a new Newton–Euler recursive inverse dynamics algorithm and its linearized tangent version. The approach is illustrated with numerical examples from the Cosserat rod literature and from recent robotic applications.

Suitable representations of dynamical systems can simplify their analysis and control. On this line of thought, this paper 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?Not only we show that the answer to this question isyes, but we 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 [IO]tu, being u the system input. We identify a class of systems, calledcollocated, 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.

The robotics community has seen an exponential growth in the level of complexity of the theoretical tools presented for the modeling of soft robotics devices. Different solutions have been presented to overcome the difficulties related to the modeling of soft robots, often leveraging on other scientific disciplines, such as continuum mechanics, computational mechanics, and computer graphics. These theoretical and computational foundations are often taken for granted and this leads to an intricate literature that, consequently, has rarely been the subject of a complete review. For the first time, we present here a structured overview of all the approaches proposed so far to model soft robots. The chosen classification, which is based on their theoretical and numerical grounds, allows us to provide a critical analysis about their uses and applicability. This will enable robotics researchers to learn the basics of these modeling techniques and their associated numerical methods, but also to have a critical perspective on their uses.

This article explores the relationship between optimal control and Cosserat beam theory from the perspective of solving the forward and inverse dynamics (and statics as a subcase) of continuous manipulators and snake-like bioinspired locomotors. By invoking the principle of minimum potential energy and the Gauss principle of least constraint, it is shown that the quasi-static and dynamic evolutions of these robots are the solutions of optimal control problems in the space variable, which can be solved at each step (of loading or time) of a simulation with the shooting method. In addition to offering an alternative viewpoint on several simulation approaches proposed in the recent past, the optimal control viewpoint allows us to improve some of them while providing a better understanding of their numerical properties. The approach and its properties are illustrated through a set of numerical examples validated against a reference simulator.

Prokaryotic flagellum is considered as the only known example of a biological “wheel,” a system capable of converting the action of rotatory actuator into a continuous propulsive force. For this reason, flagella are an interesting case study in soft robotics and they represent an appealing source of inspiration for the design of underwater robots. A great number of flagellum-inspired devices exists, but these are all characterized by a size ranging in the micrometer scale and mostly realized with rigid materials. Here, we present the design and development of a novel generation of macroscale underwater propellers that draw their inspiration from flagellated organisms. Through a simple rotatory actuation and exploiting the capability of the soft material to store energy when interacting with the surrounding fluid, the propellers attain different helical shapes that generate a propulsive thrust. A theoretical model is presented, accurately describing and predicting the kinematic and the propulsive capabilities of the proposed solution. Different experimental trials are presented to validate the accuracy of the model and to investigate the performance of the proposed design. Finally, an underwater robot prototype propelled by four flagellar modules is presented.

The Fin-Ray principle, inspired by the physiology of fish rays, represents the foundation of a large number of robotic devices. However, despite their popularity, there is not anyad-hoctheoretical model technique for the analysis of this family of fingers. This lack is the main motivation of the presented work, which provides the mathematical modeling, analysis, and prototyping of a closed-chain Fin-Ray finger. In this scenario, the contribution of this article is twofold. At one end, we provide a general discrete Cosserat approach for the modeling of closed-chain soft robots which shares the geometrical structure of the rigid robotics counterpart. On the other end, the approach is employed to explore the family of Fin-Ray effect fingers. Finally, an improved design, which is able to conform to contacting surfaces, while maintaining stiffness out of its grasping plane, is fabricated and its performances are compared to those of a previously proposed prototype.

In this article, we propose a new dynamic model of Cosserat beams in view of its application to continuum and soft robotics manipulation and locomotion. In contrast to usual approaches, it is based on the nonlinear parameterization of the beam shape by its strain fields and their reduction on a functional basis of strain modes. While remaining geometrically exact, the approach provides us with a minimal set of ordinary differential equations in the usual Lagrange matrix form that can be exploited for analysis and control design. Inspired from rigid robotics, the calculation of the matrices of this Lagrangian model is performed with a new reduced inverse Newton-Euler algorithm. To assess the approach, this Lagrangian model is compared against a well-validated finite element method through several benches of nonlinear structural statics and dynamics.

Flagellated micro-organism are regarded as excellent swimmers within their size scales. This, along with the simplicity of their actuation and the richness of their dynamics makes them a valuable source of inspiration to design continuum, self-propelled underwater robots. Here we introduce a soft, flagellum-inspired system which exploits the compliance of its own body to passively attain a range of geometrical configurations from the interaction with the surrounding fluid. The spontaneous formation of stable helical waves along the length of the flagellum is responsible for the generation of positive net thrust. We investigate the relationship between actuation frequency and material elasticity in determining the steady-state configuration of the system and its thrust output. This is ultimately used to perform a parameter identification procedure of an elastodynamic model aimed at investigating the scaling laws in the propulsion of flagellated robots.

Dynamic control of soft robotic manipulators is an open problem yet to be well explored and analyzed. Most of the current applications of soft robotic manipulators utilize static or quasi-dynamic controllers based on kinematic models or linearity in the joint space. However, such approaches are not truly exploiting the rich dynamics of a soft-bodied system. In this paper, we present a model-based policy learning algorithm for closed-loop predictive control of a soft robotic manipulator. The forward dynamic model is represented using a recurrent neural network. The closed-loop policy is derived using trajectory optimization and supervised learning. The approach is verified first on a simulated piecewise constant strain model of a cable driven under-actuated soft manipulator. Furthermore, we experimentally demonstrate on a soft pneumatically actuated manipulator how closed-loop control policies can be derived that can accommodate variable frequency control and unmodeled external loads.

In this paper, a geometric and unified model of soft-rigid multi-body systems is presented, based on a discrete Cosserat approach of the soft-body dynamics. The model is in fact a generalization to soft and hybrid systems of the geometric theory of rigid robotics characterized by the exponential map. A generalization of the recursive Newton-Euler algorithm is also presented, able to solve inverse and forward dynamic problems with linear O(N) complexity. The proposed model provides several improvements with respect to the existing flexible multi-body models, which make it particularly suitable to study the dynamics of modern soft robots as shown for a multi-body system inspired by motile bacteria.

Nowadays, the most adopted model for the design and control of soft robots is the piecewise constant curvature model, with its consolidated benefits and drawbacks. In this work, an alternative model for multisection soft manipulator dynamics is presented based on a discrete Cosserat approach, in which the continuous Cosserat model is discretized by assuming a piecewise constant strain along the soft arm. As a consequence, the soft manipulator state is described by a finite set of constant strains. This approach has several advantages with respect to the existing models. First, it takes into account shear and torsional deformations, which are both essential to cope with out-of-plane external loads. Furthermore, it inherits desirable geometrical and mechanical properties of the continuous Cosserat model, such as intrinsic parameterization and greater generality. Finally, this approach allows to extend to soft manipulators, the recursive composite-rigid-body and articulated-body algorithms, whose performances are compared through a cantilever beam simulation. The soundness of the model is demonstrated through extensive simulation and experimental results.

International audience

Modeling and control of soft robots is an up-to-date and exciting area of research which has been tackled with complementary approaches so far. In this paper, we modify the existing continuum Cosserat approach optimizing it for soft robot arms which can be discretized in a finite number of sections and degrees of freedom. The resulting new piece-wise constant strain model extends the existing piece-wise constant curvature model by allowing torsion and shears strains which are fundamental to cope with out-of-the-plane external forces as appearing for example during ground locomotion. A first experimental comparison has been also conducted using one fluidic actuated leg of the soft crawler FASTT.

This paper reports on the development and validation of the elastodynamics model of an innovative underwater soft-bodied robot inspired by cephalopods. The vehicle, for which the model is devised, is propelled by a discontinuous activation routine which entails the collapse of an elastic shell via cable transmission and its following passive re-inflation under the action of the elastic energy stored in the shell walls. Activation routine and thrust characterization have been determined to depend massively on the capability of the shell to elastically return to its unstrained state, hence an accurate description of the dynamics of the shell during all stages of actuation and at various degrees of deformation is essential. The model, based on a geometrically exact Cosserat theory, is validated against measurement achieved from an ad-hoc experimental apparatus, bringing evidence of its aptness at capturing the key parameters of the system. Eventually the model is employed for simulating a proper propulsion routine in water demonstrating that, upon suitable parametrization of the internal and external hydrodynamics, it can reliably be employed for the realistic quantitative characterization of the cephalopod-inspired robot.

The solution of the inverse kinematics problem of soft manipulators is essential to generate paths in the task space. The inverse kinematics problem of constant curvature or piecewise constant curvature manipulators has already been solved by using different methods, which include closed-form analytical approaches and iterative methods based on the Jacobian method. On the other hand, the inverse kinematics problem of nonconstant curvature manipulators remains unsolved. This study represents one of the first attempts in this direction. It presents both a model-based method and a supervised learning method to solve the inverse statics of nonconstant curvature soft manipulators. In particular, a Jacobian-based method and a feedforward neural network are chosen and tested experimentally. A comparative analysis has been conducted in terms of accuracy and computational time.

The new and promising field of soft robotics has many open areas of research such as the development of an exhaustive theoretical and methodological approach to dynamic modeling. To help contribute to this area of research, this paper develops a dynamic model of a continuum soft robot arm driven by cables and based upon a rigorous geometrically exact approach. The model fully investigates both dynamic interaction with a dense medium and the coupled tendon condition. The model was experimentally validated with satisfactory results, using a soft robot arm working prototype inspired by the octopus arm and capable of multibending. Experimental validation was performed for the octopus most characteristic movements: bending, reaching, and fetching. The present model can be used in the design phase as a dynamic simulation platform and to design the control strategy of a continuum robot arm moving in a dense medium.

In this work we address the inverse kinetics problem of a non-constant curvature manipulator driven by three cables. An exact geometrical model of this manipulator has been employed. The differential equations of the mechanical model are non-linear, therefore the analytical solutions are difficult to calculate. Since the exact solutions of the mechanical model are not available, the elements of the Jacobian matrix can not be calculated. To overcome intrinsic problems of the methods based on the Jacobian matrix, we propose for the first time a neural network learning the inverse kinetics of the soft manipulator moving in three-dimensional space. After the training, a feed-forward neural network (FNN) is able to represent the relation between the manipulator tip position and the forces applied to the cables. The results show that a desired tip position can be achieved with a degree of accuracy of 1.36% relative average error with respect to the total arm length.

This paper describes the design and development of a robot with six soft limbs, with the dual capability of pushing-based locomotion and grasping by wrapping around objects. Specifically, a central platform lodges six silicone limbs, radially distributed, with cables embedded. A new mechanism-specific gait, invariant regarding the number of limbs, has been implemented. Functionally, some limbs provide stability while others push and pull the robot to locomote in the desired direction. Once the robot is close to a target, one limb is elected to wrap around the object and, thanks to the particular limb structure and the soft material, a friction-based grasping is achieved. The robot is inspired by the octopus and implements the key principles of locomotion in this animal, without coping the full body structure. For this reason it works in water, but it is not restricted to this environment. The experiments show the effectiveness of the original solution in locomotion and grasping.

Control of soft robots remains nowadays a big challenge, as it does in the larger category of continuum robots. In this paper a direct and inverse kinetics models are described for a non-constant curvature structure. A major effort has been put recently in modelling and controlling constant curvature structures, such as cylindrical shaped manipulators. Manipulators with non-constant curvature, on the other hand, have been treated with a piecewise constant curvature approximation. In this work a non-constant curvature manipulator with a conical shape is built, taking inspiration from the anatomy of the octopus arm. The choice of a conical shape manipulator made of soft material is justified by its enhanced capability in grasping objects of different sizes. A different approach from the piecewise constant curvature approximation is employed for direct and inverse kinematics model. A continuum geometrically exact approach for direct kinetics model and a Jacobian method for inverse case are proposed. They are validated experimentally with a prototype soft robot arm moving in water. Results show a desired tip position in the task-space can be achieved automatically with a satisfactory degree of accuracy.

Recently, continuum manipulators have drawn a lot of interest and effort from the robotic community, nevertheless control and modeling of such manipulators are still a challenging task especially because they require a continuum approach. In this paper, a general mechanical model with a geometrically exact approach for tendon-driven continuum manipulators is presented. This model can be applied to a wide range of manipulators thanks to the generality of the parameters which can be set. The approach proposed could as well be a powerful tool for developing the control strategy. The model is also capable of properly simulating the coupled tendon drive, because it takes into account the torsion of the robot arm rather than neglecting it, as it is common practice in other existing models.