Kinematics modeling of a two DOFs continuum manipulator with uniform notches

Abstract. Continuum manipulators have been widely adopted for single-port laparoscopy (SPL). A novel continuum manipulator with uniform notches which has two degrees of freedom (DOFs) is presented in this paper. The arrangement of flexible beams makes it own a higher load capacity. Its kinematic model is coupled with the mechanical model. The comprehensive elliptic integral solution (CEIS) is more practical in the actual deformation of the flexible beams. Based on that method, kinematics modeling is established from the driven space to the Cartesian space. The friction coefficient is an important factor which can affect the kinematic modeling. Therefore, an experimental platform is established to obtain the friction coefficient. The kinematic modeling is verified through the prototype. Experimental results show that the model has high precision.


Introduction
Robot design and control technology have made great progress recently. The traditional rigid manipulator can be used to complete the related tasks perfectly under the simple working environment of artificial construction. However, they cannot be used to complete relevant tasks well under the complex working environment, e.g. human body interior, complex pipelines, ruins, etc. The continuum manipulator is researched by more and more scholars to solve the problem of operations for complex space. The continuum manipulator is a kind of bionic mechanism, which is inspired by the biological structures, e.g. elephant trunk, snake and tentacles. The continuum manipulator, which has more freedom and better bending characteristics can be used to complete complex operations in a narrow space. It can be classified into discrete and continuum configurations. The discrete configuration is similar to the traditional rigid manipulator, which is formed by a series connection of multiple modular joints and supported by the elastic body. The continuum configuration can be regarded as being formed by the series connection of the flexible mechanism. The complete and uninterrupted elastic material can be taken as the skeleton of the continuum configuration directly. It not only owns no gap, high precision characteristics [1], but also has better flexibility and biocompatibility, which can help it complete more flexible movement inside the human body, and do less damage to the internal tissues of the human body.
As a typical representative of minimally invasive surgery (MIS), laparoscopic surgery generally requires opening 2-4 minimally invasive holes on the surface of the human body. SPL appeared in order to minimize the risk of such trauma and infection. The flexibility and small size of the continuum manipulator make it possible to be applied in SPL. Roh et al. have developed an SPS robot system that transmits motion through a saddle-shaped surface with end loads up to 1kgf [2]. Choset et al. designed a Cardio ARM robot system which is made up of rigid cylindrical links in series [3]. Li [8,9].
The above examples demonstrate the feasibility and effectiveness of the continuum manipulator applied in single-hole surgery. It is necessary to choose a suitable modeling method to improve the control accuracy of the manipulator. The deformation of the continuum configuration manipulator is related to the end load. Yang established a mechanical model through a cantilever beam theory which is suitable for small linear elastic deformation [8]. Yang established a mechanical model through a Timoshenko beam theory which is suitable for small linear elastic deformation [9]. The shear and rotation effects are considered in the Timoshenko beam theory. Gao et al. proposed a modeling method based on Cosserat rod theory which is suitable for elastic rods with super large deformation [10]. The shear deformation, centerline expansion and deformation, and bulk force effects are considered in Cosserat rod theory.
The main problem of continuum configuration is lower load capacity. In order to improve its practicability in SPL, more scholars have invested in research to improve its load capacity. In this paper, a uniform notch continuum with a strong load capacity is proposed. The design and characters of the structure are introduced in Section 2. The kinematic modeling method based on CEIS is presented in Section 3. The accuracy of the kinematic model can be proved by experiments in section 4.

Structural design
In this paper, a continuum manipulator with uniformed notches is obtained by the superelasticity of Nitinol. Nitinol tube's length is 142mm, outer diameter is 10mm, and inner diameter is 8.8mm. The notches are obtained by wire-cut electrical discharge machining (WEDM). The continuum manipulator proposed in this paper adopts an elliptical straight beam notch scheme, as shown in Fig. 1(a).
A and B are two fixed points on the continuum manipulator. When they are located, the ellipse can be independently adjusted for beam length l and thickness h . It is beneficial for further optimization of the structure. Besides, the ellipse can be used to reduce stress concentration.
The bending planes of adjacent flexible beams are perpendicular to each other, as shown in Fig. 1(b), (c). Another arrangement is to place flexible beams with the same bending direction together, as shown in Fig. 1(d). In Fig. 1(c), (d), flexible beam m represents the beam which is closest to the base among the beams whose bending plane is plane x-z. When the end of the continuum manipulator is subjected to a load F, the bending angle of beam m in Fig. 1(c) is larger than that in Fig. 1(d). This can increase the range of motion of the continuum manipulator in the plane x-z. Therefore, the arrangement of Fig. 1(c) is adopted in this paper. The cross-section of the flexible beam i is shown in Fig. 2. The moment of inertia of the section to the neutral axis can be obtained as follows: where Izi is the moment of inertia of the section of flexible beam i transversely facing the neutral axis z i , r 1 is the inner diameter of the nitinol tube, r 2 is the outer diameter of the nitinol tube, and h i is the thickness of flexible beam i.

Kinematics model
The deflection of some flexible beams exceeds 10% of the beam length, which is a large deflection deformation. In order to describe this deformation more accurately, the relationship between the load and deformation is described through CEIS in this paper [11]. The mechanics model is briefly introduced in Subsection 3.1. Since the kinematics modeling is inseparable from the mechanical model, the process of solving the end load of each flexible beam is mainly introduced in Subsection 3.2. Through combining the work of the first two sections, the position and pose of the continuum end are solved in Subsection 3.3.

Mechanics model
Each flexible beam can be regarded as a cantilever beam model. Load parameters, structure parameters, and deformation parameters are shown in Fig The deformation parameters of the flexible beam can be obtained by iterating the following two sets of expressions: The algorithm flow chart is shown in Fig. 4. An inflection point is a point where the curvature is zero. In the figure, m indicates the number of inflection points. When load parameters and structure parameters are known, deformation parameters can be obtained through the algorithm.

Solution for the deformation parameters of each flexible beam
The flexible beam i is placed between the rigid body i and the rigid body i+1, as shown in , which is 0 in this paper. The initial values a i , b i and θ oi are L i , 0 and 0, respectively. From Fig. 3, the homogeneous transformation matrix of {i_1} with respect to {i_2} can be observed as _ 2 _1 cos sin 0 A driven wire can be divided into many sections small driven wires. When the forces provided by driven wires outside the plane y i_1 z i_1 are known, the end-load of the flexible beam i can be solved. Only the forces provided by driven wire N i-2 M i are unknown before this iteration. The equivalent force vectors of other known forces are stored in the matrix F VD . The matrix F VD is a matrix of 3×(2i-4), and each column vector corresponds to an external force vector in the coordinate system {i-1_2}. The equivalent origins of these known forces are stored in the matrix F po . The matrix F po is a matrix of 4×(2i-4), and each column vector corresponds to a homogeneous coordinate of an origin in the coordinate system {i-1_2}. The expressions of F VD and F po in the coordinate system {i_2} are shown as Before this iteration, {i-1_2} N i-2 is a known quantity, and its expression in the coordinate system {i_2} is shown as As i>2, the friction forces between the driven wires and continuum manipulator are considered in this paper. There are three assumptions as follows: 1) The Coulomb friction model is applied, ie 2) The driven wires pass through N i-2 and M i . N i-2 is a fixed point on the rigid body i-1, and M i is a fixed point on the rigid body i; 3) The driven wires are only subjected to the force at N i-2 and M i . The direction of normal force lies in the bisector of two driven wires. The direction of the friction force is perpendicular to that of the normal force.
In order to obtain the forces provided by the driven wire N i-2 M i on the flexible beam i-1, the force analysis is carried out. Taking the driven wire N i-2 M i as the research object, the force analysis diagram is shown in Fig. 6. There are three parts in external forces applied on the driven wire N i-2 M i : 1) F T _N i-2 M i-2 is the tensile force provided by the driven wire M i-2 N i-2 , the direction is from N i-2 to M i-2 , and the magnitude is known before this iteration; 2) F T _M i N i is the tensile force provided by the driven wire M i N i , the direction is from M i to N i , and the magnitude is unknown before this iteration; 3) The friction forces and normal forces provided by the flexible beam i-1. The specific directions of friction forces are related to the deformation trend of the flexible beam i-1 [see Fig. 6(a), (b)]. Therefore, the specific directions of friction forces need to be discussed. The magnitudes of the normal forces and the friction forces are unknown.   2  2  2  2  1  2  2  2  2  2 where arccos( ), arccos( ).
The friction forces and normal forces provided by the driven wire N i-2 M i can be replaced by F T _N i-2 M i-2 and F T _M i N i equivalently. Therefore, what are actually stored in the F VD are the tensile forces provided by different driven wires. Considering that the effect of interaction forces between adjacent driven wires can be offset, the end-load of the flexible beam i(i>2) can be simplified as follows: where F T _M i-1 N i-1 is the tensile force provided by the driven wire M i-1 N i-1 , and it's known before this iteration.
The algorithm flow for solving the deformation parameters of the flexible beam i(i>2) is shown in Fig. 7. The solution process of the first two flexible beams is similar to the above process. Since one end of the first driven wire is fixed on the rigid body 1, there is no friction force between the first driven wire and the rigid body 1. Similarly, there is no friction force between the second driven wire and the rigid body 2. Therefore, friction forces don't exist during the solution process of the first two flexible beams. The end load of the flexible beam i can be simplified as follows: where F T1 is the tensile force provided by the first driven wire on the rigid body 1, and F T2 is the tensile force provided by the second driven wire on the rigid body 2.

Solving for position and pose of continuum end
After solving the deformation parameters of each flexible beam, the homogenous matrix from the tip to the base is shown as The operating space is shown in Fig. 8. As can be seen from the figure, the operating space can meet general SPL needs.

Experiments and results
The kinematics model is verified through experiments in this section. An experimental platform is established to obtain the friction coefficient in Subsection 4.1. The verification experiment of the kinematics model is introduced in Subsection 4.2.

Measurement experiment for friction coefficient
The schematic diagram of the experimental model is shown in Fig. 9(a). Before the start of each set of experiments, the angle α 1 is known. Then the angle α 2 is adjusted by the ball screw until the system reaches the critical friction state. The friction coefficient can be obtained as The physical quantities of the formula are shown in the schematic diagram. The partially enlarged view and the overall view of the experimental bench are shown in Fig. 9(b) and (c), respectively. 18 sets of measurement experiments are carried out. The mean value is taken as the friction coefficient, which is 0.3117 [see Fig. 10].

Validation of kinematics model
The experiment platform for validation of the kinematics model is shown in Fig. 11(a). The continuum manipulator is located in the fixed table. The end of the manipulator is pulled by the weight through the driven wire. The manipulator begins to deform under the tensile force of the weight. The photos of deformation results could be taken by the CCD camera, as shown in Fig. 11(b).   During the experiment, the load is increased from 0 to 1.5kg in increments of 0.1kg. Through the calibration board, the position of the tip end of the manipulator could be calculated by Matlab. The results of the experimental and theoretical values are shown in Fig.  12(a). The error of each experimental result is shown in Fig. 12(b). The results indicate that the mean tip error of all the verification experiments is 0.597 ± 0.556 mm. The max tip error is 1.153 mm, which is less than 5% of the total manipulator length. The possible sources are the gravity of the manipulator, gravity acceleration value, and image acquisition.

Summary
A two DOFs continuum manipulator with uniform notches is proposed in this paper. The kinematics model and the mechanical model are coupled. The mechanical model is established through CEIS. The tensile forces, normal forces and friction forces provided by driven wires are considered to establish the kinematics model. The friction coefficient is measured through a specific experimental platform. The results indicate that the mean tip error of all the verification experiments is 0.597 ± 0.556 mm. All the errors are less than 5% of the total manipulator length. A future study could focus on the solution of inverse kinematics, force feedback and shape perception.