Modeling of continuum robots: A review

Continuum robots offer good environmental adaptability and superior safety in human-robot interaction. This paper describes the state of the art in continuum robot modeling and summarizes and evaluates the mainstream continuum robot modeling approaches in the current community through three different classifications: continuum models, geometric models, and data-driven models. Finally, the paper provides a summary of existing research approaches and provides future research opportunities in continuum robot modeling.


Introduction
Continuum robots are robots with flexible structures.Typically, they do not consist of discrete joints but have a flexible central skeleton.This curved structure enables the robot to adapt to complex curved terrain and to pass through body cavities or surgical incisions [1].Therefore, it has great potential in the field of medical surgery and there are many commercial applications [2].Although the application of continuum robots has been socially accepted, the development is facing great challenges.Modeling and control of continuum robots are often considered difficult and complex because of their lack of discrete structure.In relative, modeling of the kinematics and dynamics of rigid robots can be defined by coordinates and is now well established [3].However, the infinite degrees of freedom and flexible couplings of continuum robots are challenging to model using a similar approach to rigid robots.Currently, the study of reasonable models is the main principle that must be considered in order to model continuum robots with a low computational effort and sufficient computational accuracy.Over the past decade, continuum robotics has evolved at a high rate, and related modeling techniques have continued to iterate [4].
In order to cope with the ever-changing technological developments, this paper analyzes and evaluates the currently available research on continuum robot modeling and numerical theory.In this paper, the modeling approach is divided into the kinematic-based geometric model [5], the mechanicsbased continuum mechanics model [6], and the data-driven model [7].This paper provides a deeper understanding of continuum robotics through a comparison of continuum robot modeling and an evaluation and analysis of modeling approaches.Thus, promoting the development of continuum robotics.Overview of this paper, the continuum model is introduced in II-A, while the geometric model-based continuum machine is in II-B.Data-driven models combined with neural Networks and other data-driven models are introduced in III.Finally, the technical challenges facing the development of continuum robot modeling nowadays will be given in IV, and V is the concluding section.

Modeling classification
There are several modeling methods that are currently used to represent the behavior of continuum robots.These methods range from analytical models, which are based on mathematical equations, to computational models, which are based on numerical simulations.The basic components of the continuum robot model are the kinematic frameworks.Traditional rigid discrete linkage robotic models use the Denavit Hartenberg (D-H) method.An approximate representation of the continuous structure is obtained by performing a series of homogeneous transformations on the discrete model.For example, the early snake bionic robots [8] and the hyper redundant robots [9] are based on rigid discrete linkage.Initially, this approach was used for continuum robots, but nowadays the more mainstream approaches are constant curvature theory and Cosserat rod theory.The model structure is shown in Fig. 1.

Continuum mechanics models
Continuum mechanics is now the mainstay in continuum robotics research, this framework is universal and is emerging as an industry standard.The reason is that they are computationally efficient and accurate in statics systems.The classical continuum mechanics system contains the Kirchhoff rod and Cosserat rod.In the beginning, Kirchhoff and Clebsch introduced Kirchhoff rods to solve the problem of deformation of non-planar elastic materials, they envisioned the rod as a short one-dimensional linear segment with dynamics as three-dimensional contact forces and moments.The Cosserat brothers, on the other hand, extended the assumption of extensibility on the rod and the orthogonality of the rod center line to the cross-section.This establishes a 6-dimensional mathematical framework based on Euclidean three-dimensional spatial position vectors and a three-dimensional angular vector composition.Cosserat rods are flexible and versatile in the analysis of various systems.Important to note that the Cosserat rod theory is only applicable to robots with slender structures since one of their lengths is much larger than the cross-sectional area.In this case, the dynamics of each cross-section of the rod can be approximated by the law of averaging balance.The Cosserat rod consists of two partial differential equations(PDE) sets.Equation ( 1 (1) In general, these two sets of PDE do not have analytic solutions.The main solutions for the continuum robotics community are as follows: 1)Direction-based numerical method: The shooting method is a numerically based method of solving the boundary value problem of a system of differential equations by converting it into an initial value problem.The community proposed to discretize the time inverse in the PDE and then transform the PDE into Boundary Value Problem (BVP) for the Ordinary Differential Equation (ODE) [12].This is a numerical method for the forward dynamics of the Cosserat rod model [13].Lagrangian dynamics is also the most used mechanical model in the mechanical modeling of continuum robots [14].The Lagrangian method is more applicable to complex systems interacting with multiple rods than the boundary value problem based on the shooting method.In the latest study, the two Cosserat rod-based methods were shown to be mathematically equivalent, with both methods yielding approximate solution answers [15].The finite difference method can also be used for rod theory solutions, this method can be chosen as either space finite difference or time finite difference, and this method makes the integration of BVP by the Runge-Kutta method feasible.
2)Finite element method: In addition to the boundary elements, the finite element method (FEM) can also create space discretization.The finite element method is a common method for finding approximate partial differential solutions.FEM is a numerical method that is widely used in the field of mechanical engineering to analyze and simulate the behavior of structures.In the context of continuum robots, FEM has been used to model the mechanical behavior of flexible structures, such as cables, tubes, or sheets, and to optimize the design and control parameters for a given task.In FEM, a flexible structure is represented as a collection of elements, such as beams or shells, that are connected at discrete nodes.The behavior of each element is described by a set of equations that consider the material properties, the geometric shape, and the loads and constraints acting on the structure.The solution to these equations is then used to predict the response of the flexible structure to external loads, such as tension or bending.The finite element simulation for a single cable continuum robot is shown in Fig. 2. The introduction of the FEM method can transform PDEs into a static system of discrete equations or a dynamic system of ODE equations [16].

Geometrical models
Geometric methods do not rely on partial differential equations, rather, it relies on geometric shapes.The constant curvature (cc) model and piecewise-constant-curvature (PCC) (See Fig. 1b) model in the community is the geometrical model.The constant curvature kinematic framework consists of a finite number of curve segments.These curve segments are tangent and have a constant curvature along their length.In this kinematic frame, the curvature, angle, and length of each curve segment form a set of coordinates.This coordinate called the arc parameters, describes the shape of the robot.In PCC modal piecewise means that the curvature of the robot structure is constant over short segments.
The CC method has been successfully applied to several continuum robots in the early research of the last 10 years in continuum robotics [18].This model is a very significant simplification, so its use is often limited.But this simplification can make kinematic calculations and control simpler and save computing power.
For complex shapes and deformations, more advanced models, such as finite element models or Cosserat rod models, can require to accurately capture the behavior of the robot.

Data-driven models
The data-based approach essentially does not use any theoretical model, they use data to make predictions or to perform tasks, rather than relying on a priori knowledge or assumptions.The datadriven model based on the koopman operator, and the neural network model are described below:

Koopman operator
Koopman operator theory is a mathematical framework for studying the dynamics of nonlinear systems.It is named after the mathematician Samuel Koopman, who introduced the theory in the early 20th century.In Koopman operator theory, the behavior of a nonlinear system is studied by constructing a linear operator, called the Koopman operator, that describes how the system evolves over time.The Koopman operator is defined on functions of the state of the system, rather than on the state itself, which allows the dynamics of the system to be studied in a linear, rather than a nonlinear, framework.As shown in Fig. 3, the Koopman operator serves to extract the value of the downstream measurement function, Even though the flow may be nonlinear in its phase space M, the dynamics are linear.One of the main advantages of Koopman operator theory is that it provides a way to study the long-term behavior of nonlinear systems, such as their stability and attractors, without having to solve the nonlinear differential equations that describe the system.This makes it useful for analyzing complex systems, such as chaotic systems, that are difficult to study using traditional methods.Koopman operators are used to model the behavior of high-dimensional and nonlinear systems, and large amounts of data are usually required to capture the complex behavior of these systems and train the Koopman operators.In the study by Bruder et al. [20], a datadriven modeling and control approach based on the theory of Koopman operators achieved as good results as nonlinear model predictions, demonstrating that the transformation of complex nonlinear problems in continuum robots into linear problems by the Koopman operator does not affect the performance of the predictions.However, a limitation of this study is that the model lacks external loads and contact forces for simulating the interaction of a continuum robot with its environment.

Neural network
Neural network (NN) models are a class of machine learning models that are inspired by the structure and function of the biological brain.These models are used in a wide range of applications, including image recognition, natural language processing, and control systems.In continuum robotics systems, neural networks can also be used to predict the robot's kinematics and enhance control accuracy.Consider a neural network with an input layer, one or more hidden layers, and an output layer.Let X be the input vector, Wi be the weight matrix connecting layer i to layer i +1, bi be the bias vector for layer i, and f be the activation function.The output of the network, Y, can then be computed using Equation 3.
Where n is the number of hidden layers in the network.The activation function f can be any nonlinear function.This formula shows how the inputs are processed through the network, and how the weights and biases of the connections between neurons determine the output of the network.The weights and biases are learned during the training process, and the goal of training is to find the values of W and b that minimize the error between the predicted output and the actual output.
Feed-forward neural networks have been used to describe the static and kinematic models of continuum robots.Further, feed-forward neural networks can also be used to compensate for the uncertain non-linear dynamics of CRs [21].This approach does not require an accurate dynamic model but provides performance improvements and is easily accessible to other continuum robot designs.
Time zeroing neural network in continuum robot control has been validated initially as well, while in 2022 research the discrete zeroing neural networks can also be applied to CRs kinematic control problems [22].This modality-free control scheme obtains the positive discretization formulation, furthermore, with high control accuracy.However, the portability of the control scheme based on the neural network algorithm needs further study and the scheme only considers the kinematics of the continuum robot and does not consider the dynamical system.
Overall, data-driven models offer a powerful approach to modeling and predicting complex systems.But the mainstream is still providing performance improvements for continuum robots.

Challenges of continuum robot modeling
Faster, more accurate modeling and mathematical reasoning to solve forward and inverse mechanics of continuum robots is the goal of the community.However, an increase in computational accuracy means a decrease in speed, so numerous modeling methods aim to give designers more options.On the other hand, The continuum robot will face a strong friction environment, as well as an external loading environment.Because the continuum robot lacks a rigid structure, small external loads can change the shape of the continuum robot However, the kinematic problems in these scenarios have not yet been solved.The impact of friction and localized losses on the control of continuum robots is also the focus of current research.Possible future modeling approaches are listed below:

Geometrically exact FEM
At present, no modeling method can guarantee both accuracy and computational efficiency.Data-driven models are now the mainstream models with good generalization capability.Save researchers time in learning complex dynamics and kinematic systems because the principles are generalized.However, for continuum robots, this means that each robot requires a huge database and a long learning process.Further, neural network models cannot explain the inherent relationships of modeled systems.Finite elements seem to be a more common and reliable choice, and commercial finite element software based on geometrically exact FEM (GE-FEM) is considered a strong contender [23].However, the form of its generated models is based on nonlinear structural mechanics, which cannot be easily used in robotics.

Energetic approaches
A modal Ritz method such as geometric variable-strain (GVS) provides a highly simplified dynamic model of the usual explicit Lagrangian form of a rigid robot and, therefore, facilitates the transfer of the method from rigid to soft robots.This modeling approach is also applicable to robots with variable stiffness.However, the GVS method still has shortcomings in simulating the interaction with the environment.

Conclusion
This paper summarises the modeling methods that have been widely used.Continuum robotics is an interdisciplinary field that involves traditional mechanical mechanics dynamics as well as geometry.As it is often composed of soft and flexible materials, materials science is also essential in the characterization of these materials.Machine learning, computer vision, and other disciplines are also important developments in continuum robotics.These intersections of continuum robotics are critical to the development and advancement of the field, and their integration and cooperation are key to overcoming the technical challenges and limitations of continuum robotics and realizing its full potential for applications in various fields.
)[10] [11] represents linear momentum, and Equation (2) (3) (4) expresses the angular momentum.The dynamics of the Cosseart rod are based on the balance of linear and angular momentum in the cross-section.