A Geometrically Exact Model for Soft Continuum Robots: The Finite Element Deformation Space Formulation

Performance evaluation of PSO-PID and PSO-FLC for continuum robot's developed modeling and control

Continuum robots are complex structures that require sophisticated modeling and control methods to achieve accurate position and motion tracking along desired trajectories. They are highly coupled, nonlinear systems with multiple degrees of freedom that pose a significant challenge for conventional approaches. In this paper, we propose a system dynamic model based on the Euler-Lagrange formulation with the assumption of piecewise constant curvature (PCC), where we accounts for the elasticity and gravity effects of the continuum robot. We also develop and apply a particle swarm optimization (PSO) algorithm to optimize the parameters of our developed controllers: an inverse dynamic proportional integral derivative (PID) controller and an inverse dynamic fuzzy logic controller (FLC), where we use the integral time of absolute error (ITAE) as the objective function for the PSO algorithm. We validate our proposed model and optimized controllers through different designed trajectories, simulated using our developed unique animated MATLAB simulation. The results show that the PSO-PID controller improves the rise time, overshoot percentage, and settling time by 16.3%, 31.1%, and 64.9%, respectively, compared to the PID controller without PSO. The PSO-FLC controller shows the best performance among all controllers, with a settling time of 0.7 s and a rise time of 0.4 s, leading to the highest level of precision in trajectory tracking. The ITAE error for the PSO-FLC controller is 11.4% and 29.9% lower than that of the PSO-PID and FLC controllers, respectively.

Deep CNN-Based Static Modeling of Soft Robots Utilizing Absolute Nodal Coordinate Formulation

Soft continuum robots, inspired by the adaptability and agility of natural soft-bodied organisms like octopuses and elephant trunks, present a frontier in robotics research. However, exploiting their full potential necessitates precise modeling and control for specific motion and manipulation tasks. This study introduces an innovative approach using Deep Convolutional Neural Networks (CNN) for the inverse quasi-static modeling of these robots within the Absolute Nodal Coordinate Formulation (ANCF) framework. The ANCF effectively represents the complex non-linear behavior of soft continuum robots, while the CNN-based models are optimized for computational efficiency and precision. This combination is crucial for addressing the complex inverse statics problems associated with ANCF-modeled robots. Extensive numerical experiments were conducted to assess the performance of these Deep CNN-based models, demonstrating their suitability for real-time simulation and control in statics modeling. Additionally, this study includes a detailed cross-validation experiment to identify the most effective model architecture, taking into account factors such as the number of layers, activation functions, and unit configurations. The results highlight the significant benefits of integrating Deep CNN with ANCF models, paving the way for advanced statics modeling in soft continuum robotics.

Kinetostatic Modeling of Soft Robots: Energy-Minimization Approach and 99-Line MATLAB Implementation

Soft robots have received a great deal of attention from both academia and industry due to their unprecedented adaptability in unstructured environment and extreme dexterity for complicated operations. Due to the strong coupling between the material nonlinearity due to hyperelasticity and the geometric nonlinearity due to large deflections, modeling of soft robots is highly dependent on commercial finite element software packages. An approach that is accurate and fast, and whose implementation is open to designers, is in great need. Considering that the constitutive relation of the hyperelastic materials is commonly expressed by its energy density function, we present an energy-based kinetostatic modeling approach in which the deflection of a soft robot is formulated as a minimization problem of its total potential energy. A fixed Hessian matrix of strain energy is proposed and adopted in the limited memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, which significantly improves its efficiency for solving the minimization problem of soft robots without sacrificing prediction accuracy. The simplicity of the approach leads to an implementation of MATLAB with only 99-line codes, which provides an easy-to-use tool for designers who are designing and optimizing the structures of soft robots. The efficiency of the proposed approach for predicting kinetostatic behaviors of soft robots is demonstrated by seven pneumatic-driven and cable-driven soft robots. The capability of the approach for capturing buckling behaviors in soft robots is also demonstrated. The energy-minimization approach, as well as the MATLAB implementation, could be easily tailored to fulfill various tasks, including design, optimization, and control of soft robots.

Reduced order modeling and model order reduction for continuum manipulators: an overview

Soft robot's natural dynamics calls for the development of tailored modeling techniques for control. However, the high-dimensional configuration space of the geometrically exact modeling approaches for soft robots, i.e., Cosserat rod and Finite Element Methods (FEM), has been identified as a key obstacle in controller design. To address this challenge, Reduced Order Modeling (ROM), i.e., the approximation of the full-order models, and Model Order Reduction (MOR), i.e., reducing the state space dimension of a high fidelity FEM-based model, are enjoying extensive research. Although both techniques serve a similar purpose and their terms have been used interchangeably in the literature, they are different in their assumptions and implementation. This review paper provides the first in-depth survey of ROM and MOR techniques in the continuum and soft robotics landscape to aid Soft Robotics researchers in selecting computationally efficient models for their specific tasks.

Trajectory tracking control of a pneumatically actuated continuum manipulator in the presence of obstacles by using terminal sliding mode control

This paper proposes an efficient trajectory planning and dynamic tracking controller scheme for a pneumatic continuum manipulator under the effect of material hysteresis. First of all, generalized governing nonlinear dynamic equations in the form of partial differential equations for pneumatic continuum manipulator dynamics are developed by using discrete Cosserat-rod theory, where the manipulator material hysteresis is modeled by using a fractional order Bouc-Wen model. Then, the trajectory planning for the end-effector of a continuum manipulator is proposed, which accounts for the static obstacles in the workspace and the Jacobian singularity. Subsequently, an adaptive terminal sliding mode controller for the joint space control combined with a simple PI controller for task space control is proposed. The proposed controller guarantees exponential convergence of the manipulator tip positional error in finite time, even in the existence of external disturbances and model uncertainties, without any need for prior knowledge of their upper bounds. Finally, the proposed controller is applied to a two-segment continuum manipulator, the trunk of Robotino-XT, through numerical simulations and the performance gain over two controllers proposed in the literature for similar pneumatic continuum manipulators is demonstrated.

Control-Oriented Models for Hyperelastic Soft Robots Through Differential Geometry of Curves

The motion complexity and use of exotic materials in soft robotics call for accurate and computationally efficient models intended for control. To reduce the gap between material and control-oriented research, we build upon the existing piece-wise constant curvature framework by incorporating hyperelastic and viscoelastic material behavior. In this work, the continuum dynamics of the soft robot are derived through the differential geometry of spatial curves, which are then related to finite-element data to capture the intrinsic geometric and material nonlinearities. To enable fast simulations, a reduced-order integration scheme is introduced to compute the dynamic Lagrangian matrices efficiently, which in turn allows for real-time (multilink) models with sufficient numerical precision. By exploring the passivity and using the parameterization of the hyperelastic model, we propose a passivity-based adaptive controller that enhances robustness toward material uncertainty and unmodeled dynamics-slowly improving their estimates online. As a study-case, a soft robot manipulator is developed through additive manufacturing, which shows good correspondence with the dynamic model under various conditions, for example, natural oscillations, forced inputs, and under tip-loads. The solidity of the approach is demonstrated through extensive simulations, numerical benchmarks, and experimental validations.

Model Analysis of Robotic Soft Arms including External Force Effects

Because robotic soft arms have a high power-to-weight ratio, low cost, and ease of manufacturability, increasing numbers of researchers have begun to focus on their characteristics in recent years. However, many urgent problems remain to be resolved. For example, soft arms are made of hyperelastic material, making it difficult to obtain accurate model predictions of the soft arm shape. This paper proposes a new modeling method for soft arms, combining the constant curvature model with Euler–Bernoulli beam theory. By combining these two modeling methods, we can quickly solve for the soft arm deformation under the action of an external force. This paper also presents an experimental platform based on a cable-driven soft arm to verify the validity of the proposed model. We carried out model verification experiments to test for different external effects. Experimental results show that the maximum error of our proposed soft arm deformation model is between 2.86% and 8.75%, demonstrating its effectiveness.

Coupled Dynamic Modeling and Control of Aerial Continuum Manipulation Systems

Aerial continuum manipulation systems (ACMSs) were newly introduced by integrating a continuum robot (CR) into an aerial vehicle to address a few issues of conventional aerial manipulation systems such as safety, dexterity, flexibility and compatibility with objects. Despite the earlier work on decoupled dynamic modeling of ACMSs, their coupled dynamic modeling still remains intact. Nonlinearity and complexity of CR modeling make it difficult to design a coupled ACMS model suitable for practical applications. This paper presents a coupled dynamic modeling for ACMSs based on the Euler–Lagrange formulation to deal with CR and the aerial vehicle as a unified system. For this purpose, a general vertical take-off and landing vehicle equipped with a tendon-driven continuum arm is considered to increase the dexterity and compliance of interactions with the environment. The presented model is independent of the motor’s configuration and tilt angles and can be applied to model any under/fully actuated ACMS. The modeling approach is complemented with a Lyapunov-wise stable adaptive sliding mode control technique to demonstrate the validity of the proposed method for such a complex system. Simulation results in free flight motion scenarios are reported to verify the effectiveness of the proposed modeling and control techniques.

Review of the exponential and Cayley map on SE(3) as relevant for Lie group integration of the generalized Poisson equation and flexible multibody systems

The exponential and Cayley maps on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe–Kaas and generalized- α schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed-form relations along with the relevant proofs, including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized- α scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.

Visual Servoing Pushing Control of the Soft Robot with Active Pushing Force Regulation

Soft robots characterize operational safety due to inherent compliance from their soft mechanism, whereas their mechanism enhances the difficulty in accurate closed-loop control. To explore their applicability in manipulation tasks, in this article, we propose a visual servoing pushing controller considering the effect of contact. The controller is designed to simultaneously complete the positioning and manipulation tasks. To further improve the manipulation performance, an active pushing force regulation method is proposed and integrated into the controller. The proposed control algorithm is validated experimentally. The results show that the controller guarantees the convergence to the image error and meanwhile, it improves the pushing manipulation performance.

Task space adaptation via the learning of gait controllers of magnetic soft millirobots

Untethered small-scale soft robots have promising applications in minimally invasive surgery, targeted drug delivery, and bioengineering applications as they can directly and non-invasively access confined and hard-to-reach spaces in the human body. For such potential biomedical applications, the adaptivity of the robot control is essential to ensure the continuity of the operations, as task environment conditions show dynamic variations that can alter the robot’s motion and task performance. The applicability of the conventional modeling and control methods is further limited for soft robots at the small-scale owing to their kinematics with virtually infinite degrees of freedom, inherent stochastic variability during fabrication, and changing dynamics during real-world interactions. To address the controller adaptation challenge to dynamically changing task environments, we propose using a probabilistic learning approach for a millimeter-scale magnetic walking soft robot using Bayesian optimization (BO) and Gaussian processes (GPs). Our approach provides a data-efficient learning scheme by finding the gait controller parameters while optimizing the stride length of the walking soft millirobot using a small number of physical experiments. To demonstrate the controller adaptation, we test the walking gait of the robot in task environments with different surface adhesion and roughness, and medium viscosity, which aims to represent the possible conditions for future robotic tasks inside the human body. We further utilize the transfer of the learned GP parameters among different task spaces and robots and compare their efficacy on the improvement of data-efficient controller learning.

A Recurrent Neural-Network-Based Real-Time Dynamic Model for Soft Continuum Manipulators

This paper introduces and validates a real-time dynamic predictive model based on a neural network approach for soft continuum manipulators. The presented model provides a real-time prediction framework using neural-network-based strategies and continuum mechanics principles. A time-space integration scheme is employed to discretize the continuous dynamics and decouple the dynamic equations for translation and rotation for each node of a soft continuum manipulator. Then the resulting architecture is used to develop distributed prediction algorithms using recurrent neural networks. The proposed RNN-based parallel predictive scheme does not rely on computationally intensive algorithms; therefore, it is useful in real-time applications. Furthermore, simulations are shown to illustrate the approach performance on soft continuum elastica, and the approach is also validated through an experiment on a magnetically-actuated soft continuum manipulator. The results demonstrate that the presented model can outperform classical modeling approaches such as the Cosserat rod model while also shows possibilities for being used in practice.

Dynamic modeling of soft continuum manipulators using lie group variational integration

This paper presents the derivation and experimental validation of algorithms for modeling and estimation of soft continuum manipulators using Lie group variational integration. Existing approaches are generally limited to static and quasi-static analyses, and are not sufficiently validated for dynamic motion. However, in several applications, models need to consider the dynamical behavior of the continuum manipulators. The proposed modeling and estimation formulation is obtained from a discrete variational principle, and therefore grants outstanding conservation properties to the continuum mechanical model. The main contribution of this article is the experimental validation of the dynamic model of soft continuum manipulators, including external torques and forces (e.g., generated by magnetic fields, friction, and the gravity), by carrying out different experiments with metal rods and polymer-based soft rods. To consider dissipative forces in the validation process, distributed estimation filters are proposed. The experimental and numerical tests also illustrate the algorithm's performance on a magnetically-actuated soft continuum manipulator. The model demonstrates good agreement with dynamic experiments in estimating the tip position of a Polydimethylsiloxane (PDMS) rod. The experimental results show an average absolute error and maximum error in tip position estimation of 0.13 mm and 0.58 mm, respectively, for a manipulator length of 60.55 mm.

Toward Task Autonomy in Robotic Cardiac Ablation: Learning-Based Kinematic Control of Soft Tendon-Driven Catheters

The goal of this study was to propose and validate a control framework with level-2 autonomy (task autonomy) for the control of flexible ablation catheters. To this end, a kinematic model for the flexible portion of typical ablation catheters was developed and a 40-mm-long spring-loaded flexible catheter was fabricated. The feasible space of the catheter was obtained experimentally. Furthermore, a robotic catheter intervention system was prototyped for controlling the length of the catheter tendons. The proposed control framework used a support vector machine classifier to determine the tendons to be driven, and a fully connected neural network regressor to determine the length of the tendons. The classifier and regressors were trained with the data from the feasible space. The control system was implemented in parallel at user-interface and firmware and exhibited a 0.4-s lag in following the input. The validation studies were four trajectory tracking and four target reaching experiments. The system was capable of tracking trajectories with an error of 0.49 ± 0.32 and 0.62 ± 0.36 mm in slow and fast trajectories, respectively. Also, it exhibited submillimeter accuracy in reaching three preplanned targets and ruling out one nonfeasible target autonomously. The results showed improved accuracy and repeatability of the position control compared with the recent literature. The proposed learning-based approach could be used in enabling task autonomy for catheter-based ablation therapies.

Dynamic simulation of articulated soft robots

Soft robots are primarily composed of soft materials that can allow for mechanically robust maneuvers that are not typically possible with conventional rigid robotic systems. However, owing to the current limitations in simulation, design and control of soft robots often involve a painstaking trial. With the ultimate goal of a computational framework for soft robotic engineering, here we introduce a numerical simulation tool for limbed soft robots that draws inspiration from discrete differential geometry based simulation of slender structures. The simulation incorporates an implicit treatment of the elasticity of the limbs, inelastic collision between a soft body and rigid surface, and unilateral contact and Coulombic friction with an uneven surface. The computational efficiency of the numerical method enables it to run faster than real-time on a desktop processor. Our experiments and simulations show quantitative agreement and indicate the potential role of predictive simulations for soft robot design.

Energy-shaping control of soft continuum manipulators with in-plane disturbances

Soft continuum manipulators offer levels of compliance and inherent safety that can render them a superior alternative to conventional rigid robots for a variety of tasks, such as medical interventions or human–robot interaction. However, the ability of soft continuum manipulators to compensate for external disturbances needs to be further enhanced to meet the stringent requirements of many practical applications. In this paper, we investigate the control problem for soft continuum manipulators that consist of one inextensible segment of constant section, which bends under the effect of the internal pressure and is subject to unknown disturbances acting in the plane of bending. A rigid-link model of the manipulator with a single input pressure is employed for control purposes and an energy-shaping approach is proposed to derive the control law. A method for the adaptive estimation of disturbances is detailed and a disturbance compensation strategy is proposed. Finally, the effectiveness of the controller is demonstrated with simulations and with experiments on an inextensible soft continuum manipulator that employs pneumatic actuation.

Online Disturbance Estimation for Improving Kinematic Accuracy in Continuum Manipulators

Continuum manipulators are flexible robots which undergo continuous deformation as they are actuated. To describe the elastic deformation of such robots, kinematic models have been developed and successfully applied to a large variety of designs and to various levels of constitutive stiffness. Independent of the design, kinematic models need to be calibrated to best describe the deformation of the manipulator. However, even after calibration, unmodeled effects such as friction, nonlinear elastic and/or spatially varying material properties as well as manufacturing imprecision reduce the accuracy of these models. In this paper, we present a method for improving the accuracy of kinematic models of continuum manipulators through the incorporation of orientation sensor feedback. We achieve this through the use of a "disturbance wrench", which is used to compensate for these unmodeled effects, and is continuously estimated based on orientation sensor feedback as the robot moves through its workspace. The presented method is applied to the HydroJet, a waterjet-actuated soft continuum manipulator, and shows an average of 40% reduction in root mean square position and orientation error in the two most common types of kinematic models for continuum manipulators, a Cosserat rod model and a pseudo-rigid body model.

TMTDyn: A Matlab package for modeling and control of hybrid rigid–continuum robots based on discretized lumped systems and reduced-order models

A reliable, accurate, and yet simple dynamic model is important to analyzing, designing, and controlling hybrid rigid–continuum robots. Such models should be fast, as simple as possible, and user-friendly to be widely accepted by the ever-growing robotics research community. In this study, we introduce two new modeling methods for continuum manipulators: a general reduced-order model (ROM) and a discretized model with absolute states and Euler–Bernoulli beam segments (EBA). In addition, a new formulation is presented for a recently introduced discretized model based on Euler–Bernoulli beam segments and relative states (EBR). We implement these models in a Matlab software package, named TMTDyn, to develop a modeling tool for hybrid rigid–continuum systems. The package features a new high-level language (HLL) text-based interface, a CAD-file import module, automatic formation of the system equation of motion (EOM) for different modeling and control tasks, implementing Matlab C-mex functionality for improved performance, and modules for static and linear modal analysis of a hybrid system. The underlying theory and software package are validated for modeling experimental results for (i) dynamics of a continuum appendage, and (ii) general deformation of a fabric sleeve worn by a rigid link pendulum. A comparison shows higher simulation accuracy (8–14% normalized error) and numerical robustness of the ROM model for a system with a small number of states, and computational efficiency of the EBA model with near real-time performances that makes it suitable for large systems. The challenges and necessary modules to further automate the design and analysis of hybrid systems with a large number of states are briefly discussed.

The softness distribution index: towards the creation of guidelines for the modeling of soft-bodied robots

Modeling soft robots is not an easy task owing to their highly nonlinear mechanical behavior. So far, several researchers have tackled the problem using different approaches, each having advantages and drawbacks in terms of accuracy, ease of implementation, and computational burden. The soft robotics community is currently working to develop a unified framework for modeling. Our contribution in this direction consists of a novel dimensionless quantity that we call the softness distribution index (SDI). The SDI for a given soft body is computed based on the distribution of its structural properties. We show that the index can serve as a tool in the choice of a modeling technique among multiple approaches suggested in literature. At the moment, the investigation is limited to bodies performing planar bending. The aim of this work is twofold: (i) to highlight the importance of the distribution of the geometrical and material properties of a soft robotic link/body throughout its structure; and (ii) to demonstrate that a classification based on this distribution provides guidelines for the modeling.