Cosserat Rod Theory Research Articles

Chained Spatial Beam Adomian Decomposition Model: A Novel Model of Flexible Slender Beams for Large Spatial Deflections

Abstract The main element of compliant mechanisms and continuum robots is flexible slender beams. However, the modeling of beams can be complicated due to the geometric nonlinearity becoming significant at large elastic deflections. This paper presents an explicit nonlinear model called the spatial beam Adomian decomposition model (SBADM) for intermediate spatial deflections of a slender beam with uniform, bisymmetric sections subjected to general end-loading. Specifically, the elongation, bending, torsion, and shear deformations of the beams are modeled based on Timoshenko's assumptions and Cosserat rod theory. Then, the quaternion transformation and Adomian decomposition are used to solve the nonlinear governing differential equations for the beam by truncating the higher-order terms, yielding an explicit expression for spatially deflected beams within intermediate deflection ranges. Simulations demonstrate the accuracy and time-wise efficiency of the SBADM, as well as its advantages over the state-of-the-art. In addition, this paper also introduces a discretization-based scheme called the chained SBADM (CSBADM) for large spatial deflections of flexible beams. Real-world experiments with two different configurations have also been performed to validate the effectiveness of the CSBADM. The results indicate that the CSBADM can accurately calculate the load-displacement relations for large deformed beams.

An explicit nonlinear model for large spatial deflections of symmetric slender beams

Flexible slender beams are commonly used in compliant mechanisms and continuum robots. However, the modeling of these beams can be complicated due to the geometric nonlinearity becoming significant at large elastic deflections. This paper presents an explicit nonlinear model for large spatial deflections of a slender beam with uniform, symmetrical sections subjected to general end-loading. The elongation, bending, torsion, and shear deformations of the beams are modeled based on Timoshenko’s assumptions and Cosserat rod theory. Subsequently, the nonlinear governing differential equations for the beam are derived from the quaternion representation of the rotation matrix. The explicit load–displacement relations of the beam are obtained using the improved Adomian decomposition method. This method is superior to the classical Adomian decomposition method in terms of convergence rate and domain. The convergence and superiority of the method are also rigorously demonstrated. Simulations are provided to verify the one-, two-, and three-dimensional deflections of beams. Real-world experiments have also been performed to validate our method’s effectiveness with two different beam configurations. The results indicate that the proposed method accurately estimates large spatial deflections of flexible beams.

Simulation and parameterization of nonlinear elastic behavior of cables

AbstractThis work contributes to the simulation, modeling, and characterization of nonlinear elastic bending behavior within the framework of geometrically nonlinear rod models. These models often assume a linear constitutive bending behavior, which is not sufficient for some complex flexible slender structures. In general, nonlinear elastic behavior often coexists with inelastic behavior. In this work, we incorporate the inelastic deformation into the rod model using reference curvatures. We present an algorithmic approach for simulating the nonlinear elastic bending behavior, which is based on the theory of Cosserat rods, where the static equilibrium is calculated by minimizing the linear elastic energy. For this algorithmic approach, in each iteration the static equilibrium is obtained by minimizing the potential energy with locally constant algorithmic bending stiffness values. These constants are updated according to the given nonlinear elastic constitutive law until the state of the rod converges. To determine the nonlinear elastic constitutive bending behavior of the flexible slender structures (such as cables) from the measured values, we formulate an inverse problem. By solving it we aim to determine a curvature-dependent bending stiffness characteristic and the reference curvatures using the given measured values. We first provide examples using virtual bending measurements, followed by the application of bending measurements on real cables. Solving the inverse problem yields physically plausible results.

Exact derivative propagation method to compute the generalized compliance matrix for continuum robots: Application to concentric tubes continuum robots

This paper introduces the concept of Generalized Compliance for continuum robots, specifically for those modeled with the Cosserat Rod theory. Unlike existing models based on tip compliance, the proposed approach considers interactions along the entire body of the flexible robot. The paper also presents a novel method referred to as the Low-Level Derivative Propagation Method, which is designed for the computationally efficient derivation of the Generalized Compliance matrix. The proposed method streamlines calculations and reduces integration time. The presented method, which is general and applies to various types of continuum robot models, is demonstrated on the case of a Concentric Tubes Continuum Robot. We provide detailed derivations of the equations and computation techniques leading to the derivation of the Generalized Compliance matrix, as well as a large-scale numerical validation of the method. The code used in this paper is available on the following GitHub website: https://github.com/benoitrosa/Generalized-Compliance-Computation-for-Continuum-Robots.

Model-based trajectory tracking of a compliant continuum robot.

Introduction: Compliant mechanisms, especially continuum robots, are becoming integral to advancements in minimally invasive surgery due to their ability to autonomously navigate natural pathways, significantly reducing collision severity. A major challenge lies in developing an effective control strategy to accurately reflect their behavior for enhanced operational precision. Methods: This study examines the trajectory tracking capabilities of a tendon-driven continuum robot at its tip. We introduce a novel feedforward control methodology that leverages a mathematical model based on Cosserat rod theory. To mitigate the computational challenges inherent in such models, we implement an implicit time discretization strategy. This approach simplifies the governing equations into space-domain ordinary differential equations, facilitating real-time computational efficiency. The control strategy is devised to enable the robot tip to follow a dynamically prescribed trajectory in two dimensions. Results: The efficacy of the proposed control method was validated through experimental tests on six different demand trajectories, with a motion capture system employed to assess positional accuracy. The findings indicate that the robot can track trajectories with an accuracy within 9.5%, showcasing consistent repeatability across different runs. Discussion: The results from this study mark a significant step towards establishing an efficient and precise control methodology for compliant continuum robots. The demonstrated accuracy and repeatability of the control approach significantly enhance the potential of these robots in minimally invasive surgical applications, paving the way for further research and development in this field.

PDE-based control synthesis for a planar cable-driven continuum arm

Classical wave and beam models cannot describe the arc-like planar configuration of a continuum arm. Based on the Cosserat rod theory, a strain vector and the rotation matrix are introduced into this work. Two groups of nonlinear partial differential equations (PDEs) are obtained by analyzing kinetic and potential energy densities and virtual power done by the cable tension, which are explicit both in the spatial coordinates and in the physical parameters. Although involving algebraic–trigonometric descriptions, the obtained dynamical model is the most concise formulation but difficult to use for control design. We develop a feedforward plus proportional–derivative feedback boundary control scheme for the nonlinear PDE system to regulate the planar shape of the continuum arm and then linearize the dynamical equations about the equilibrium to assess the control performance. Some interesting results are the closed-loop stability and the well-posedness analysis of the linearized model, which have not previously been noted. Finally, numerical results validate the dynamical equations and the performance of the controller.

Haptic Localization with a Soft Whisker from Moment Readings at the Base.

This article focuses on haptic localization of very lightweight and delicate objects while applying a contact force >5000 times lower than the weight of the object. A soft whisker integrated with a Force/Moment (F/M) sensor at the base, and a novel reconstruction algorithm have been proposed for this purpose. Initially, the mathematical relationships between the deformations of the whisker and the F/M sensor outputs were used to reconstruct the shape of the whisker and the position of the touched object. The Cosserat rod theory was used under the assumption that only one contact point occurs during the exploration, and friction effects are negligible. A new methodology we called moment only reading (MOR) has been tested, verified, and compared with previous methods that employed Force and Moment Readings (FMR). Experimental investigations revealed that the spatial position estimation error of the MOR method was confined within 13 mm, when the force applied ranged between 0.001 and 0.01 N. Moreover, the comparison with FMR demonstrated that MOR is capable of retrieving the position of objects even when the force readings drop below the force resolution of the sensor. Eventually, the MOR method has been applied to demonstrate the localization and grasping, with a soft gripper, of delicate crops like tomatoes and strawberries.

Cosserat Rod-Based Dynamic Modeling of a Hybrid-Actuated Soft Robot for Robot-Assisted Cardiac Ablation

Soft robotics has emerged as a promising field due to the unique characteristics offered by compliant and flexible structures. Overcoming the challenge of precise position control is crucial in the development of such systems that require accurate modeling of soft robots. In response, a hybrid-actuated soft robot employing both air pressure and tendons was proposed, modeled, and validated using the dynamic Cosserat rod theory. This approach comprehensively addresses various aspects of deformation, including bending, torsion, shear, and extension. The designed robot was intended for robot-assisted cardiac ablation, a minimally invasive procedure that is used to treat cardiac arrhythmias. Within the framework of the Cosserat model, dynamic equations were discretized over time, and ordinary differential equations (ODEs) were solved at each time step. These equations of motion facilitated the prediction of the robot’s response to different control inputs, such as the air pressure and tension applied to the tendons. Experimental studies were conducted on a physical prototype to examine the accuracy of the model. The experiments covered a tension range of 0 to 3 N for each tendon and an air pressure range of 0 to 40 kPa for the central chamber. The results confirmed the accuracy of the model, demonstrating that the dynamic equations successfully predicted the robot’s motion in response to diverse control inputs.

A Soft Variable Stiffness Hand With an Integrated Actuating–Cooling System

In this paper, a new soft finger is presented to achieve a fast variable stiffness mechanism using polycaprolactone based on an integrated actuating–cooling system. First, the variable stiffness finger demonstrates a dramatic 15-fold stiffness variation from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10.2 \times {10}^4{\bm{\ }}$</tex-math></inline-formula> to 0.67 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \times {10}^4$</tex-math></inline-formula> Nmm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> when the temperature is changed in a range from 20 °C to 60 °C. The cycle time of the variable stiffness based on the integrated actuating–cooling hydraulic system is approximately 505 s, which is much shorter than that of 3870 s based on the pneumatic method. Second, due to the varied weight of water in the chamber, Cosserat rod theory is used to model the bending shape of the soft finger instead of the constant curvature model. A feasible method to control the position and stiffness simultaneously based on the Cosserat model is presented. Finally, a three-fingered soft hand is fabricated based on the variable stiffness and self-locking method, which can hold objects up to 3560 g successfully, which is more than 5.2 times its own weight of 680 g.

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.

Model-aided 3D shape and force estimation of continuum robots based on Cosserat rod theory and using a magnetic localization system

Model-aided 3D shape and force estimation of continuum robots based on Cosserat rod theory and using a magnetic localization system

General Kinetostatic Modeling and Deformation Analysis of a Two-Module Rod-Driven Continuum Robot with Friction Considered

Continuum robots actuated by flexible rods have large potential applications, such as detection and operation tasks in confined environments, since the push and pull actuation of flexible rods withstand tension and compressive force, and increase the structure’s rigidity. In this paper, a generalized kinetostatics model for multi-module and multi-segment continuum robots considering the effect of friction based on the Cosserat rod theory is established. Then, the model is applied to a two-module rod-driven continuum robot with winding ropes to analyze its deformation and load characteristics. Four different in-plane configurations under the external load term as S1, S2, C1, and C2 are defined. Taking a bending plane as an example, the tip deformation along the x-axis of these shapes is simulated and compared, which shows that the load capacity of C1 and C2 is generally larger than that of S1 and S2. Furthermore, the deformation experiments and simulations show that the maximum error ratio without external loads relative to the total length is no more than 3%, and it is no more than 4.7% under the external load. The established kinetostatics model is proven sufficient to accurately analyze the rod-driven continuum robot with the consideration of internal friction.

Piecewise Linear Strain Cosserat Model for Soft Slender Manipulator

Recently soft robotics has rapidly become a novel and promising area of research with many designs and applications due to their flexible and compliant structure. However, it is more difficult to derive the nonlinear dynamic model of such soft robots. The differential kinematics and dynamics of the soft manipulator can be formulated as a set of highly nonlinear partial differential equations (PDEs) via the classic Cosserat rod theory. In this work, we propose a discrete modeling technique named piecewise linear strain (PLS) to solve the PDEs of Cosserat-based models, based on which the associated analytic models are deduced. To validate the accuracy of the proposed Cosserat model, the static model of the conical cantilever rod under gravity as a simple example is simulated by using different discretization methods. Results indicate that PLS Cosserat model is comparable to the mechanical deformation behavior of a real-world soft manipulator. Finally, a parameters identification scheme for this model is established, and the simulation as well as experimental validation demonstrate that using this method can identify the model physical parameters with high accuracy.

Cosserat Rod Modeling of Continuum Robots from Newtonian and Lagrangian Perspectives

Cosserat rod theory proved efficient modeling performances in robotics, especially in the context of continuum robots, in the past decade. The implementation of such theory is far from being unique and straightforward. We consider the illustrative example of multisegment, general routing tendon actuated continuum robots in their nominal static operating regime. This article details two main approaches based on Cosserat rod modeling, namely, the Newtonian and Lagrangian approaches. We provide a walk-through guide regarding theoretical derivations and numerical implementation of both approaches, together with a proof of equivalence. This comparative study is supplemented with novel contributions and extensions of each approach and in-depth discussion of their performances and applicability, as well as highlighting their special features.

Data based constitutive modelling of rate independent inelastic effects in composite cables using Preisach hysteresis operators

AbstractThis contribution aims at introducing first steps to develop hysteresis operator type inelastic constitutive laws for Cosserat rods for the simulation of cables composed of complex interior components. Motivated by the basic elements of Cosserat rod theory, we develop a specific approach to constitutive modelling adapted for this application. Afterwards, we describe the hysteretical behaviour arising from cyclic bending experiments on cables by means of the Preisach operator. As shown in pure bending experiments, slender structures such as electric cables behave inelastically, and open hysteresis loops arise with noticeable difference between the first load cycle and the following ones. The Preisach operator plays an important role in describing the input-output relation in hysteresis behaviours, and it can be expressed as a superposition of relay operators. Hence, a mathematical formulation of the problem is introduced, and a first attempt is made to determine the hysteresis behaviour that describes the relation between curvature and bending moment. Therefore, a suitable kernel function is identified in a way that its integration over the Preisach plane results in the bending moment of the specimen, and a comparison between different kernel functions is performed.

Kinetostatic Modeling of Continuum Delta Robot With Variable Curvature Continuum Joints

Abstract Continuum robots have attracted lots of attention due to their structural compliance, manipulation dexterity, and design compactness. To extend the application scenarios, a slender continuum robot, the CurviPicker, was developed for low-load medium-speed pick-and-place tasks in a previous study. To improve the payload capacity and positioning accuracy of the CurviPicker, a novel Continuum Delta Robot (CDR) was then proposed with three dual-continuum-joint translators in a preliminary investigation. However, the initial version of the CDR did not fully utilize the bending ranges of its continuum joints. In addition, while being modeled using the constant curvature assumption for the continuum joints, the CDR shows lowered positioning accuracy for heavier objects, as the CDR’s continuum joints diverge from the assumed constant curvature shapes. In this paper, the design of the CDR was re-optimized to enable wider bending ranges of the continuum joints (&amp;gt;90 deg) to generate an enlarged workspace, taking into consideration several possible structural interferences. Furthermore, a kinetostatic model is derived based on the Cosserat rod theory to reduce the positioning errors caused by the external loads. The experimental result showed that the workspace is enlarged to approximately 9.47 × 107 mm3 compared with the volume of 6.57 × 107 mm3 of the initial version. Within this enlarged workspace, the average positioning error with a 1000-g load was reduced to 1.93 mm, compared with 4.43 mm obtained by the previous constant curvature assumption.

Curvature-based force estimation for an elastic tube

AbstractContact force is one of the most significant feedback for robots to achieve accurate control and safe interaction with environment. For continuum robots, it is possible to estimate the contact force based on the feedback of robot shapes, which can address the difficulty of mounting dedicated force sensors on the continuum robot body with strict dimension constraints. In this paper, we use local curvatures to estimate the magnitude and location of single or multiple contact forces based on Cosserat rod theory. We validate the proposed method in a thin elastic tube and calculate the curvatures via Fiber Bragg Grating (FBG) sensors or image feedback. For the curvature feedback obtained from multicore FBG sensors, the overall force magnitude estimation error is$0.062 \pm 0.068$N and the overall location estimation error is$3.51 \pm 2.60$mm. For the curvature feedback obtained from image, the overall force magnitude estimation error is$0.049 \pm 0.048$N and the overall location estimation error is$2.75 \pm 1.71$mm. The results demonstrate that the curvature-based force estimation method is able to accurately estimate the contact force.

Mechanics of fiber-reinforced soft manipulators based on inhomogeneous Cosserat rod theory

In this study, an inhomogeneous Cosserat rod theory is introduced and compared to the conventional homogeneous rod for modeling soft manipulators. The inhomogeneity is addressed by considering the pressure actuation as part of the rod’s constitutive laws, resulting in shifting the neutral axis. This shift is investigated for a soft manipulator with three parallel fiber-reinforced actuators. Furthermore, a fiber-reinforced actuator is modeled using nonlinear continuum mechanics to extract the effect of radial pressure on axial deformation and is combined with Cosserat model. Finally, several numerical methods are employed to solve the proposed model and validated by a series of experiments.

Observed Control of Magnetic Continuum Devices

This paper models an extensible catheter with an embedded magnet at its distal tip subject to an external magnetic field. We implement a control method coined observed control to perform model-based predictive control of the catheter using a Kalman smoother framework. Using this same smoother framework, we also solve for catheter shape and orientation given magnetic and insertion control using Cosserat rod theory and implement a disturbance observer for closed-loop control. We demonstrate observed control experimentally by traversing a 3D cube trajectory with the catheter tip. The catheter achieved positional accuracy of 3.3 mm average error in open-loop, while closed-loop control improved the accuracy to 0.33 mm.

Kinematics and Stiffness Modeling of Soft Robot With a Concentric Backbone

Abstract Soft robots can undergo large elastic deformations and adapt to complex shapes. However, they lack the structural strength to withstand external loads due to the intrinsic compliance of fabrication materials (silicone or rubber). In this paper, we present a novel stiffness modulation approach that controls the robot’s stiffness on-demand without permanently affecting the intrinsic compliance of the elastomeric body. Inspired by concentric tube robots, this approach uses a Nitinol tube as the backbone, which can be slid in and out of the soft robot body to achieve robot pose or stiffness modulation. To validate the proposed idea, we fabricated a tendon-driven concentric tube (TDCT) soft robot and developed the model based on Cosserat rod theory. The model is validated in different scenarios by varying the joint-space tendon input and task-space external contact force. Experimental results indicate that the model is capable of estimating the shape of the TDCT soft robot with an average root-mean-square error (RMSE) of 0.90 (0.56% of total length) mm and average tip error of 1.49 (0.93% of total length) mm. Simulation studies demonstrate that the Nitinol backbone insertion can enhance the kinematic workspace and reduce the compliance of the TDCT soft robot by 57.7%. Two case studies (object manipulation and soft laparoscopic photodynamic therapy) are presented to demonstrate the potential application of the proposed design.