# Kinematic Analysis of Continuum Robot Consisted of Driven Flexible Rods

Authors: Yingzhong Tian; Mingxuan Luan; Xu Gao; Wenbin Wang; Long Li
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper presents the kinematic analysis of a continuum bionic robot with three flexible actuation rods. Since the motion of the end-effector is actuated by the deformation of the rods, the robot structure is with high elasticity and good compliance and the kinematic analysis of the robot requires special treatment. We propose a kinematic model based on the geometry with constant curvature. The analysis consists of two independent mappings: a general mapping for the kinematics of all robots and a specific mapping for this kind of robots. Both of those mappings are developed for the single section and for the multisections. We aim at providing a guide for kinematic analysis of the similar manipulators through this paper.


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## Body

## 1. Introduction

Nowadays, people have already found the incredible ability of adaptation, excellent locomotion, and capability of being dexterous to move in complex environment presented by snakes [1], elephant’s trunk [2], and octopus tentacles [3] from nature. These outstanding performances have inspired researchers to explore further in the fields of flexible bionic robots.Robinson and Davies [4] proposed that the classification of robots can be divided into continuum robots, discrete robots, and serpentine robots. Continuum robots [5] can bend into different shapes because of the unique mechanical structure and the way of actuation in order to generate the desired movement path. This new type of robots possesses good ability of bending and dexterity; therefore we can apply these robots in narrow and unstructured environment.Currently, researchers have become increasingly interested in the design and control of continuum robots. Sun invented a cable-driven continuum robot [6] for colonoscopy, which consists of a flexible backbone, several disks, and driven cables. This flexible robot has two bending joints, and each joint has two degrees of freedom. The end-effector can move in all directions of the workspace due to the actuated cables. Sun used a method of geometry analysis to develop the kinematic model of the cable-driven robot.Walker and his team invented continuum robots such as Oct-Arm [7] and Air-Octor [8] with coupled tendons; Jones and Walker [9] designed a bionic elephant’s trunk consisting of springs, damping device, and metal disks; Festo designed Bionic Tripod 2.0 and Bionic Tripod 3.0, and a bionic robot seems like elephant’s trunk based on pneumatic bellows called Bionic Assistant [10]. Simaan [11] invented a kind of bionic robot which seems like snake with flexible backbone and cable-driven, with the diameter of 0.0042 m, and this robot was proposed for throat surgery; Gravagne et al. designed a kind of continuum robot called Clemson tentacle manipulator [12].Researchers developed different methods to establish the kinematic models of continuum robots with different structures. Jones and Walker [7, 9] used a modified homogenous transformation matrix in terms of Denavit-Hartenberg (D-H) type approach to develop and analyze the kinematic model of the continuum robot. They also simulated the parameters of the continuum robot’s trunk. This method gave the transformation of a section of the continuum robot between the base coordinate frame and the coordinate frame of the end-effector by multiplying five D-H transformation matrices. In the end, they obtained the homogenous transformation matrix of the robot. Their method reflected a transformation of robot kinematics model from rigid to flexible. Simaan et al. [13] used a differential method to divide the robot into several small units and then computed the position of the end-effector of a single manipulator by using integration method and obtained the Jacobian matrix.In this paper, we design a bionic continuum robot inspired by fins, which uses flexible rods made of glass fiber to replace the rigid structure existing in the traditional robot. We can control the position and orientation of the end-effector by changing the length of the rods, and we use continuous curvature method based on geometry to develop kinematic model of the continuum robot. We obtain the mapping among the configuration space, manipulation space, and task space to analyze the kinematics.


## 2. Design and Analysis of the Prototype Machine

### 2.1. Design of the Prototype Machine

The prototype robot under study is shown in Figure1, with its height of 1.2 m and its overall frame (③) made of aluminum alloy. The components of the prototype machine contain three modules (①), some parts of 3D printing (②), six fiber glass rods (④ and ⑥), three star configurations (⑤), four rubber base plates (⑧), twelve rollers (⑨), and any other fasteners. We assume that ⑩ is the base face.Figure 1
Prototype machine.The prototype robot has three passive rods (in gray as shown in Figure1). The modules are fixed on the aluminum frame and distributed by 120 degrees. They are actuated by step motors. Three passive rods are auxiliary and installed through every four rollers, which are fixed on the corresponding aluminum alloy rods. Three active rods are covered by black like ⑥ as shown in Figure 1, with the length of 1.6 m. Each active rod is fixed on the slider of the corresponding module, and its length can be changed by moving the slider; the variation range of the length is from 1.2 m to 1.5 m. Six flexible rods are made of fiber glass, which not only guarantees the ability of bending and makes the manipulator lighter and thus safer, but also makes it have a certain strength to ensure its stable movement.


### 2.2. Analysis of the Prototype Machine

The biomimetic fish-fin continuum robot mechanism diagram is shown in Figure2, three black lines represent three active rods (⑥) of the prototype robot, the base face is fixed, three rods are connected with the end-effector, and the prismatic pair represents corresponding slider of the module. Due to the movement of the slider, it can change the length of each rod, respectively, which can also change the position and orientation of the end-effector. We establish the coordinate frames x0-y0-z0 and x1-y1-z1 on the base face (⑩) and face of end-effector (⑦), respectively.Figure 2
The mechanic diagram of the trunk of manipulator.As shown in Figure2, we approximate the entire structure as a straight prism since the length of rod is much longer than the lengths of the sides of both faces, and we establish and analyze the kinematic model under this approximation.


## 2.1. Design of the Prototype Machine

The prototype robot under study is shown in Figure1, with its height of 1.2 m and its overall frame (③) made of aluminum alloy. The components of the prototype machine contain three modules (①), some parts of 3D printing (②), six fiber glass rods (④ and ⑥), three star configurations (⑤), four rubber base plates (⑧), twelve rollers (⑨), and any other fasteners. We assume that ⑩ is the base face.Figure 1
Prototype machine.The prototype robot has three passive rods (in gray as shown in Figure1). The modules are fixed on the aluminum frame and distributed by 120 degrees. They are actuated by step motors. Three passive rods are auxiliary and installed through every four rollers, which are fixed on the corresponding aluminum alloy rods. Three active rods are covered by black like ⑥ as shown in Figure 1, with the length of 1.6 m. Each active rod is fixed on the slider of the corresponding module, and its length can be changed by moving the slider; the variation range of the length is from 1.2 m to 1.5 m. Six flexible rods are made of fiber glass, which not only guarantees the ability of bending and makes the manipulator lighter and thus safer, but also makes it have a certain strength to ensure its stable movement.


## 2.2. Analysis of the Prototype Machine

The biomimetic fish-fin continuum robot mechanism diagram is shown in Figure2, three black lines represent three active rods (⑥) of the prototype robot, the base face is fixed, three rods are connected with the end-effector, and the prismatic pair represents corresponding slider of the module. Due to the movement of the slider, it can change the length of each rod, respectively, which can also change the position and orientation of the end-effector. We establish the coordinate frames x0-y0-z0 and x1-y1-z1 on the base face (⑩) and face of end-effector (⑦), respectively.Figure 2
The mechanic diagram of the trunk of manipulator.As shown in Figure2, we approximate the entire structure as a straight prism since the length of rod is much longer than the lengths of the sides of both faces, and we establish and analyze the kinematic model under this approximation.


## 3. Constant Curvature Kinematic Model

We used a kinematic model of continuous curvature based on geometry. Compared with the method of modified D-H and finite element analysis [14], we find that this method of modeling is more suitable for continuum robot and easier to implement. This kinematic model is based on two points:(1)
when the manipulator is bending, we can approximate as a circular arc;(2)
we ignore the gravity in the kinematic modeling.The assumption of the continuous curvature kinematic model can be divided into two mappings, as visualized in Figure3: a specific mapping is from actuation space to configuration space parameters and the other is from configuration space to task space.Figure 3
Three spaces and two mappings in the continuous kinematic model.


### 3.1. The General Mapping between Configuration Space and Task Space

The mapping between configuration space and task space can be described by homogeneous transformation matrix from the base to the end-effector. We establish the corresponding coordinate frame as shown in Figure1. At the same time, in Figure 4, we can see the kinematic model and coordinate frame of the manipulator.Figure 4
Kinematic model of the bionic robot.We consider the +z0-axis of the base coordinate frame to be tangent to the base of the robot, and the pedal is in the center of the base, the direction of the z-axis points to the top of the robot in the initial state, and the variable ϕ describes the angle of the rotation of the trunk about the z0-axis. When ϕ=0, the direction of the robot trunk bending defines the +x0-axis, and +y0-axis is defined by right-hand rule about +x0-axis and +z0-axis.We establish the terminal coordinate frame on center of the end-effector in the prototype robot; when the main body of robot rotates about +z0-axis by ϕ, the bending direction defines +x1-axis. At the same time, we consider the +z1-axis of the terminal coordinate frame to be tangent to the top of the main body of the robot. The direction of +y1-axis is defined by right-hand rule about +x1-axis and +z1-axis.In Figure4, when ϕ=0, the manipulator lies in the x-z plane, its main body becomes a circular arc with radius of r and its center at r00T, and then we can obtain the position of the end-effector as(1)p=r1-cos⁡θ0rsin⁡θT.If we rotate the arc about +z0-axis by angle of ϕ, the homogenous transformation can be obtained as(2)T=Rzϕ001Ryθp01.We note that, in some applications such as when a gripper is attached on the top of the robot, it is useful to orient the coordinate frame such that it aligns with the base frame when “sliding” along the arc to the base without rotation about the localz-axis, that is, use of Bishop’s frame [15, 16]. It is equivalent to postmultiplying T by a homogeneous transformation with rotation Rz(-ϕ), and we can obtain the following equation:(3)Tw=cos2⁡ϕcos⁡ks-1+1-sin⁡ϕcos⁡ϕcos⁡ks-1cos⁡ϕsin⁡kscos⁡ϕ1-cos⁡ksksin⁡ϕcos⁡ϕcos⁡ks-1cos2⁡ϕ1-cos⁡ks+cos⁡kssin⁡ϕsin⁡kssin⁡ϕ1-cos⁡ksk-cos⁡ϕsin⁡ks-sin⁡ϕsin⁡kscos⁡kssin⁡ksk0001.From (3), we obtain that the position of the end-effector is(4)px=cos⁡ϕ1-cos⁡θk,py=sin⁡ϕ1-cos⁡θk,pz=sin⁡θk.We can derive the inverse kinematic equation from (4) as follows:(5)a=py2pz2sin2⁡ϕ-1,θ=arccos⁡1-aa+1,ϕ=arctan⁡pypx.


### 3.2. The Specific Mapping between Actuated Space and Configuration Space

The intention of deriving the mapping between actuation space and the configuration space is to obtain the relationship between the actuation states,q=l1l2l3, and the parameters ϕ(q), k(q), and l(q) of the constant curvature model, in which variable ϕ describes the rotation angle of the manipulator, variable k describes the curvature, and variable l describes the length of the arc. We can obtain the variable θ from θ=kl.From Figures5 and 6, we can derive the relationship between bending radius and the bending radius of each flexible rod in the following form: (6)ri=r-dcos⁡ϕi.Figure 5
An illustration of the arc parameters is defined in the trunk.Figure 6
The diagram of the base section which is seen from top.The variabled describes the distance from the center of the base face to the center of the flexible rod as shown in Figure 6. The variable ϕi describes the angle between bending direction of the manipulator and the location of the ith rod. Recall that l=θr and li=θri, and then we can obtain the relationship between the length of arc (l) and the length of the ith rod (li) as follows:(7)li=l-θdcos⁡ϕi.We can obtain the relationship betweenϕi and ϕ from Figure 6 (in degree):(8)ϕi=90+120i-1-ϕ.From (7), we have(9)lq=l1+l2+l33.Substitutingi=1,2 and i=2,3 into (7), respectively, and combining them with (8), we obtain the relationship between ϕ and the actuation state q:(10)ϕq=tan-1⁡3l2+l3-2l13l2-l3.Recall thatθ=kl=li/ri, then we have ri=li/kl, and simultaneous formulas with it and (6) give(11)k=l-lildcos⁡ϕi.Substitutingϕ1=90-ϕ into (9) gives(12)k=l2+l3-2l1l1+l2+l3dsin⁡ϕ.From (12) and (10), we can derive the relationship between the curvature k and the actuated state q; that is,(13)kq=2l12+l22+l32-l1l2-l1l3-l2l3dl1+l2+l3.Equations (9), (10), and (13) describe the specific mapping of a single section of the continuum robot, and substituting them into (4), we can obtain (15), which describes the relationship between the position of the end-effector and the length of three active rods. (14)px=31.25a1-cos⁡0.0106667bb1+c2/3d2,py=18.04ca1-cos⁡0.0106667bdb1+c/3d2,pz=18.04cadb1+c2/3d2.In (14), a, b, c, and d can be computed as follows:(15)a=l1+l2+l3,b=l12-l1l2+l22-l1l3-l2l3+l32,c=-2l1+l2+l3,d=l2-l3.


## 3.1. The General Mapping between Configuration Space and Task Space

The mapping between configuration space and task space can be described by homogeneous transformation matrix from the base to the end-effector. We establish the corresponding coordinate frame as shown in Figure1. At the same time, in Figure 4, we can see the kinematic model and coordinate frame of the manipulator.Figure 4
Kinematic model of the bionic robot.We consider the +z0-axis of the base coordinate frame to be tangent to the base of the robot, and the pedal is in the center of the base, the direction of the z-axis points to the top of the robot in the initial state, and the variable ϕ describes the angle of the rotation of the trunk about the z0-axis. When ϕ=0, the direction of the robot trunk bending defines the +x0-axis, and +y0-axis is defined by right-hand rule about +x0-axis and +z0-axis.We establish the terminal coordinate frame on center of the end-effector in the prototype robot; when the main body of robot rotates about +z0-axis by ϕ, the bending direction defines +x1-axis. At the same time, we consider the +z1-axis of the terminal coordinate frame to be tangent to the top of the main body of the robot. The direction of +y1-axis is defined by right-hand rule about +x1-axis and +z1-axis.In Figure4, when ϕ=0, the manipulator lies in the x-z plane, its main body becomes a circular arc with radius of r and its center at r00T, and then we can obtain the position of the end-effector as(1)p=r1-cos⁡θ0rsin⁡θT.If we rotate the arc about +z0-axis by angle of ϕ, the homogenous transformation can be obtained as(2)T=Rzϕ001Ryθp01.We note that, in some applications such as when a gripper is attached on the top of the robot, it is useful to orient the coordinate frame such that it aligns with the base frame when “sliding” along the arc to the base without rotation about the localz-axis, that is, use of Bishop’s frame [15, 16]. It is equivalent to postmultiplying T by a homogeneous transformation with rotation Rz(-ϕ), and we can obtain the following equation:(3)Tw=cos2⁡ϕcos⁡ks-1+1-sin⁡ϕcos⁡ϕcos⁡ks-1cos⁡ϕsin⁡kscos⁡ϕ1-cos⁡ksksin⁡ϕcos⁡ϕcos⁡ks-1cos2⁡ϕ1-cos⁡ks+cos⁡kssin⁡ϕsin⁡kssin⁡ϕ1-cos⁡ksk-cos⁡ϕsin⁡ks-sin⁡ϕsin⁡kscos⁡kssin⁡ksk0001.From (3), we obtain that the position of the end-effector is(4)px=cos⁡ϕ1-cos⁡θk,py=sin⁡ϕ1-cos⁡θk,pz=sin⁡θk.We can derive the inverse kinematic equation from (4) as follows:(5)a=py2pz2sin2⁡ϕ-1,θ=arccos⁡1-aa+1,ϕ=arctan⁡pypx.


## 3.2. The Specific Mapping between Actuated Space and Configuration Space

The intention of deriving the mapping between actuation space and the configuration space is to obtain the relationship between the actuation states,q=l1l2l3, and the parameters ϕ(q), k(q), and l(q) of the constant curvature model, in which variable ϕ describes the rotation angle of the manipulator, variable k describes the curvature, and variable l describes the length of the arc. We can obtain the variable θ from θ=kl.From Figures5 and 6, we can derive the relationship between bending radius and the bending radius of each flexible rod in the following form: (6)ri=r-dcos⁡ϕi.Figure 5
An illustration of the arc parameters is defined in the trunk.Figure 6
The diagram of the base section which is seen from top.The variabled describes the distance from the center of the base face to the center of the flexible rod as shown in Figure 6. The variable ϕi describes the angle between bending direction of the manipulator and the location of the ith rod. Recall that l=θr and li=θri, and then we can obtain the relationship between the length of arc (l) and the length of the ith rod (li) as follows:(7)li=l-θdcos⁡ϕi.We can obtain the relationship betweenϕi and ϕ from Figure 6 (in degree):(8)ϕi=90+120i-1-ϕ.From (7), we have(9)lq=l1+l2+l33.Substitutingi=1,2 and i=2,3 into (7), respectively, and combining them with (8), we obtain the relationship between ϕ and the actuation state q:(10)ϕq=tan-1⁡3l2+l3-2l13l2-l3.Recall thatθ=kl=li/ri, then we have ri=li/kl, and simultaneous formulas with it and (6) give(11)k=l-lildcos⁡ϕi.Substitutingϕ1=90-ϕ into (9) gives(12)k=l2+l3-2l1l1+l2+l3dsin⁡ϕ.From (12) and (10), we can derive the relationship between the curvature k and the actuated state q; that is,(13)kq=2l12+l22+l32-l1l2-l1l3-l2l3dl1+l2+l3.Equations (9), (10), and (13) describe the specific mapping of a single section of the continuum robot, and substituting them into (4), we can obtain (15), which describes the relationship between the position of the end-effector and the length of three active rods. (14)px=31.25a1-cos⁡0.0106667bb1+c2/3d2,py=18.04ca1-cos⁡0.0106667bdb1+c/3d2,pz=18.04cadb1+c2/3d2.In (14), a, b, c, and d can be computed as follows:(15)a=l1+l2+l3,b=l12-l1l2+l22-l1l3-l2l3+l32,c=-2l1+l2+l3,d=l2-l3.


## 4. Velocity Kinematics

There are different ways to compute the forward kinematics, as shown in Table1, which represent the D-H parameters of the constant model, and (3) can be decomposed into several D-H transforms.Table 1
D-H parameters for the continuum robot.
Link θ D a α 1 ϕ 0 0 - π / 2 2 k l / 2 0 0 π / 2 3 0 d 3 0 - π / 2 4 k l / 2 0 0 π / 2 5 - ϕ 0 0 —The relationship between actuate stateq and the end-effector state can be described as(16)x=FD-HF2F1q,where x is a vector which contains the information of the position and orientation about the end-effector. We can see that there are a set of functions including FD-H, F2, and F1. These equations represent the modified D-H approach, the relationship between the D-H parameters θ1θ2d3θ4θ5T and the curve parameters ϕksT, and the relationship between three aforementioned parameters and the length of actuated flexible rods, which is represented with the vector of q=l1l2l3T.Then velocity kinematics are derived by differentiating (16) with respect to time. That is,(17)x˙=Jq˙,where J is the Jacobian between the velocity of the end-effector and the time rate of length change.From Table1, we can use the general method to compute JD-H. Then we can compute JF2 from Table 1 and JF1 from the function F1.Finally, the JacobianJ can be computed by the following equation:(18)J=JD-H·JF2·JF1.These detailed expressions of each term on the right side of (18) are shown in Appendix. Also, this approach of computing the Jacobian can be used for other continuum robots, such as the ones actuated pneumatically.The Jacobian can help us with modeling of the dynamics of this kind of robot, and it is useful to control the robot with PID or other methods of implementation. These will be reported in future papers.


## 5. The Analysis of the Manipulator Task Space

The manipulator has three degrees of freedom, there are two types of special bending and one extension alongz-axis, and we bend the manipulator by controlling the position of three slides to move the end-effector. We can simulate the task space of the manipulator through (4). We make the bending angle of the backbone change between 0 and π/2 due to the material of fiber glass, and length of each active rod changes between 1.2 meters and 1.5 meters. According to this above-mentioned rule, the task space of the manipulator is described as shown in Figure 7 by Matlab.Figure 7
The simulation of the manipulator task space (unit: m).Figure7 shows to us the positions which the end-effector can reach in the workspace.


## 6. The Analysis of Kinematic Simulation

### 6.1. The Simulation of Forward Kinematic

We can obtain the motion condition of each actuation rod. We define that the variabled is 0.0625 m, the bending angle is between π/6 and π/2, and the angle ϕ is between 0 and π, when the number of sampling points is 25. Using Matlab, the variable curve of each active rod is shown in Figure 8.Figure 8
The simulation of the manipulator task space (unit: m).Under the same conditions, if we set that the initial positon of the end-effector is (0, 0, 1.3) and when the length of the main body of the manipulator is 1.3 meters, we can simulate the position of end-effector inx-, y-, and z-axis as shown in Figure 9.Figure 9
The variable curve of the position of the end-effector which is divided intox-, y-, and z-axis (unit: m).


### 6.2. The Simulation of Inverse Kinematic

In this section, we assume a condition of the manipulator as the variableϕ equals π/4, the bending angle of the manipulator is between 0 and π/3, such as π/36,π/18,π/12,…,5π/18,11π/36,π/3, and the length of the manipulator is 1.3 meters, then we compute the variable length of each active rod, l1,l2,l3 correspondingly, and the simulation of inverse kinematic is shown as in Figure 10.Figure 10
The simulation of the inverse kinematic (unit: m).We can derive the length of each active rod with knowing the position and orientation of the end-effector by analyzing the inverse kinematics, and it can help us to control the end-effector directly.


## 6.1. The Simulation of Forward Kinematic

We can obtain the motion condition of each actuation rod. We define that the variabled is 0.0625 m, the bending angle is between π/6 and π/2, and the angle ϕ is between 0 and π, when the number of sampling points is 25. Using Matlab, the variable curve of each active rod is shown in Figure 8.Figure 8
The simulation of the manipulator task space (unit: m).Under the same conditions, if we set that the initial positon of the end-effector is (0, 0, 1.3) and when the length of the main body of the manipulator is 1.3 meters, we can simulate the position of end-effector inx-, y-, and z-axis as shown in Figure 9.Figure 9
The variable curve of the position of the end-effector which is divided intox-, y-, and z-axis (unit: m).


## 6.2. The Simulation of Inverse Kinematic

In this section, we assume a condition of the manipulator as the variableϕ equals π/4, the bending angle of the manipulator is between 0 and π/3, such as π/36,π/18,π/12,…,5π/18,11π/36,π/3, and the length of the manipulator is 1.3 meters, then we compute the variable length of each active rod, l1,l2,l3 correspondingly, and the simulation of inverse kinematic is shown as in Figure 10.Figure 10
The simulation of the inverse kinematic (unit: m).We can derive the length of each active rod with knowing the position and orientation of the end-effector by analyzing the inverse kinematics, and it can help us to control the end-effector directly.


## 7. Conclusion

(1)
Flexible continuum robots have been applied to a wide range of areas. Comparing them with traditional rigid robot, they are safer and more flexible. In this paper, we use constant curvature kinematic model based on geometry, and it is easy to implement and also suitable for robots with different kinds of actuation methods. (2)
We proposed a prototype robot based on the actuation of flexible rods, used the above-mentioned method to establish the kinematic model, and derive the inverse kinematic equations. (3)
In this paper, we analyze the kinematic model, derive the forward and inverse kinematic model, and the relationship between the length of each active rod and the position of the end-effector. (4)
The flexible continuum robot offers great potential for the robot application, and the manipulation is also an important issue. We will study the positioning error compensation method in future research to improve the accuracy.



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