Kinematic and Dynamic Models of Hyper -Redundant Manipulator Based on Link Eigenvectors

Hyper-redundant manipulator has a slim geometric shape and super high degrees of freedom (DOF), which promises high motion efficiency and flexible obstacle avoidance ability in a confined space. Nevertheless, it also challenges the more complicated motion control strategy. In this article, the concepts of “link eigenvectors” and “manipulator's eigenpoints,” which apply to a widely used geometric kinematics method, are proposed. Hence, the kinematic and dynamic models can be constructed intuitively. And since the indirect joint space and local frames are replaced by the link eigenvectors and unified global frame, the kinematic and dynamic problems, especially the link's position, velocity, and acceleration, can be solved more directly. Further, based on the new models, mechanical analysis and Newton–Euler dynamic equations are effectively simplified, so as to realize the fast and precise dynamic resolution. Experiment results show that the kinematic and dynamic theoretical models are correct. And the driving forces of motors are successfully optimized, while the kinematic precision is improved simultaneously.

obstacle avoidance motion in a narrow space.As a result, it is increasingly difficult to satisfy the practical application with the development in productivity.Thus, hyper-redundant manipulator [6] with shape similar to snake [7], or elephant trunk [8], [9], or octopus tentacle [10] has received much more attention over the past decades.And, their motion control technology has been developed unprecedentedly.
Traditional forward kinematics refers to solving the end effector pose in the reference frame by the known joint parameters.And inverse kinematics refers to solving the joint parameters by the given end effector pose.Since traditional robots have low DOFs, and their joint space, configuration space, and local coordinate systems are related closely, their forward and inverse kinematics can be resolved quickly by the algebraic or numerical method.Yet, due to the super high DOFs of hyper-redundant manipulator, the algebraic or numerical inverse kinematics is too complex.
Thus, the geometric method, which refers to the specific tracking geometric curve method, is particularly widely used, such as the methods of tracking the reference curves [11], [12], follow-the-leader [13], FABRIK [14], and segmented geometry [15].These geometric methods normally generate the reference curves that the manipulator can track in the task space, and then force the manipulator to fit the curves.That optimizes the inverse resolution by reducing the computation cost.By fitting the reference curve, the method would get the positions of the link ends directly, and then the links' poses can be expressed by the link ends conveniently.Thus, in the scope of this method, the reference curve can be treated as the forward solution, while the links' poses can be treated as the inverse solution.That means the regular joint space and configuration space are substituted by the links' poses, and the local coordinate systems are not needed anymore.
As early as the 1970s, various inverse dynamics methods had already been studied [16], [17], [18].Rodriguez [19] even figured out a way to solve the inverse and forward dynamic problems with O(N ) solutions.Since then, quantities of dynamic researches on cable-driven manipulators have been done to extend the early studies, such as Euler-Lagrange formalism [20], Cosserat rod theory [21], [22], Newton-Euler dynamics [23], [24], [25], and principle of virtual work [26], [27].To improve the computation efficiency, Amouri et al. [28] proposed a dynamic model that disregards the friction effects, which might cause a bit of model error.Yuan et al. [29] presented a simpler way to analyze and verify the influence of cable friction.Although these studies successfully achieve dynamic analyses, they all still build dynamic models with joint parameters and local coordinate systems.That means the traditional methods haven't taken full use of the geometric method.
In order to solve all the global kinematic variables through the positions of the link ends directly and intuitively, the concepts of manipulator's eigenpoints and link eigenvectors are proposed in the kinematic field of the tracking geometric curve method.Then the kinematic and dynamic models are also reconstructed.As the new models are defined in a global coordinate system, some acceleration components that would be generated with the utilization of a rotating reference frame are avoided.Therefore, the Newton-Euler equations are simplified and the manipulator's dynamics are solved efficiently.The main contributions of this article are following.
1) The traditional local coordinate systems are abandoned and the simpler and more intuitive kinematic model of the manipulator is reconstructed.2) Based on several assumptions, the mechanics of the cable-driven snake-like manipulator (CSM) is analyzed accurately and globally.3) Based on the kinematic model and mechanical analysis, the dynamic model is constructed and the Newton-Euler method is simplified.The dynamics method is intuitive and can improve the static and dynamic performances of CSM.The rest of the article is organized as follows.Section II is the kinematics part.It introduces the mechanical design of the CSM and the new kinematic model.Section III is the dynamics part.First, the forces on cables, links, etc., are analyzed in detail, and then the inverse dynamics are studied and demonstrated.Section IV is the experiment part.Both static and dynamic experiments show the effectiveness of the proposed method.Finally, Section V concludes this article.

A. Mechanical Design of CSM
The mechanical structure of CSM in the article is shown in Fig. 1.It has three parts, sliding table, base box, and arm span.
The sliding table provides the translation DOF for CSM.The base box including the frame and the driving modules supplies the driving forces for the arm span's motion.The driving modules transmit the forces, i.e., transmitting the motor torques into the cable driving forces.
The arm span is composed of connecting the fixed link, six universal joints, five normal links, and the end link in series.And the fixed link is indexed as Link 0 , the end link is Link 6 .The normal links are indexed as Link 1 , Link 2 , . . .Link 5 .The distal end of the fixed link, both ends of the normal link, and the proximal end of the end link all have discs with holes for cables passing through.The discs' centers are indexed as O a1 , O b1 , O a2 , O b2 , . . .O a6 , O b6 from the fixed link to the end link.The joints are indexed as Joint 1 , Joint 2 , . . .Joint 6 , and Joint 0 is the fixed joint.Fig. 2(b) shows the view from the distal end of Link i .And it sets the center of Joint i as the origin, the axis of the revolute pair between Link i and Joint i+1 as u iy , whose direction is radial outward, the axis of the revolute pair between Link i and Joint i as u iz , whose direction is radial outward too, the central axis of Link i as u ix , whose direction is from the proximal end to the distal end.The specific u ix , u iy , u iz will be calculated in Section II-B.The holes on discs are evenly circumferentially distributed.The hole located on the proximal disc Disc 2,i of Link i and u iy is indexed as P i,1 , and the rest holes are indexed counterclockwise as P i,2 , P i,3 , . . .P i,j . . .P i, 18 .Similarly, the hole located on the distal disc Disc 1,i+1 and u iy is indexed as D (i+1,1) , and the rest holes are indexed counterclockwise as D i+ (1,2) , . . .D (i+1,j) . . .D (i+1,18) .Disc 1,i and Disc 2,i belong to the same Joint i .Ψ j is the included angle between and also the included angle between The cables passing through the corresponding holes are indexed as C 1 , C 2 , . . .C j . . .C 18 .As is shown in Fig. 2, each> universal joint with two DOFs is driven by three cables, i.e., Joint i is driven by C i , C i+6, C i+12 .The three cables would pass through the two discs of Link 0 , Link 1 , . . .Link i−1 and the proximal disc of Link i .The base box can alter the lengths of C i , C i+6, C i+12 in Joint i by controlling the movement of C i , C i+6, C i+12 , so as to alter the distances between D i,i and P i,i , D i,i+6 and P i,i+6 , D i,i+12 and P i,i+12 .And finally, Joint i rotates as required.
The structure puts the bulky power unit rear, making the arm span lighter and motion more flexible.

B. Construction of Kinematic Model
Definition: Manipulator's eigenpoints are the set of proximal and distal end points of all links.
Since there is no torsional motion between the adjacent links, the manipulator's eigenpoints can fully determine the position and attitude of the manipulator.Thus, it can be taken as the kinematic model.In traditional kinematics, the pose information of the manipulator is described with many local coordinate systems.These local coordinate systems can be transformed into each other with transformation matrices.However, the tracking geometric curve method gets the manipulator's eigenpoints first but not the joint parameters, so it is indirect and troublesome to solve the transformation matrices utilizing joint parameters.Therefore, although the traditional kinematic model is universal, it's not as simple and intuitive as the reconstructed kinematic model utilizing manipulator's eigenpoints directly.Additionally, the reconstructed dynamic model based on the new kinematic model can avoid the discussion of some acceleration components and is more efficient and intuitive.
Actually, manipulator's eigenpoints are enough to describe the motion state, but it is necessary to introduce the more important concept of link eigenvectors.
Definition: Link i 's independent link eigenvector is a unit vector pointing from the proximal end to the distal end of Link i .It is u ix : Independent eigenvector.O mi : Proximal end of Link i .
Link i also has two derived link eigenvectors, u iy andu iz , which is shown in Fig. 2(b). Let Let u diry = u dirz × u ix , then u diry is collinear with u iy .Therefore It can improve the accuracy of u iy andu iz by determining their directions first and then normalizing them respectively.
As is shown in Fig. 1(b), the global coordinate system can be defined as: The view is from the distal end of Link 0 .Set the proximal end of Link 0 as the origin.Set −−−−−→ O m0 O m1 as x 0 -axis.Set y 0 -axis horizontally to the right.Set z 0 -axis vertically upward.Then, vectors u 0x , u 0y , u 0z is defined to coincide with the global coordinate system.If the manipulator's eigenpoints are known, u ix (i = 1, 2 • • • 6) can be calculated by (2), u iy , u iz (i = 1, 2 • • • 6) can be calculated by (3).Thus, all link eigenvectors can be calculated.Conversely, manipulator's eigenpoints can be calculated by the recursive formula, It is worth noting that the link eigenvectors are highly collinear with the link's inertial principal axes.
Actually, link eigenvectors and O mi can be transformed into transformation matrix.Let 0 i T refers to the transformation matrix from the global coordinate system to the local coordinate system fixed on Link i , then, As is shown in Fig. 2(a), there is a hollow ring numbered i (HR i ) in the universal joint.Analogously, HR i also has its own eigenvectors, v ix , v iy , v iz .Due to HR i 's symmetrical structure, its eigenvectors are collinear with the inertial principal axes and their values are

C. Mapping From Link Eigenvectors to Cables' Lengths
The cable C j within the range of arm span can be divided into the part between P i−1,j and D i,j , and the part between D i,j and P i,j , i.e., the link's part and the joint's part.After the cable is tensioned, the length of the link's part is fixed, while the length of the joint's part, i.e., the cable's length of C j between the two discs in Joint i , is approximately equal to the distance between D i,j and P i,j .Therefore, the following will only consider the joint's part.The symbols are defined as follows: Radius of the distribution circle of the hole on the disc.s i,j : Distance between D i,j and P i,j .C sj : Pseudo total length of C j , calculated by the adding of related s i,j .
Then (O ai and O bi have been defined in Section II-A), Let the initial values of O b0 and P 0,j be O b0 (0, 0, 0) and Thus, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Let n j be the remainder of dividing j by 6, then, If the cable is tensioned, C sj can be taken as the real cable's total length.Thus, the mapping is solved.

III. DYNAMICS
As is shown in Fig. 3, the traditional dynamic methods [23], [24], [25] solve problems with the local coordinate systems, i.e., the kinematic and dynamic variables are analyzed in the local frames or transforming the joint parameters into global variables or some other ways.Since classical mechanics is only effective in the inertial frame, the process of utilizing the local frames is complicated and indirect.Actually, for the tracking geometric curve method, all parameters can be solved in a unified inertial global frame, so the local frames and joint space can be abandoned.In this article, the Newton-Euler method is adopted and the dynamic equations are simplified with some techniques.

A. Analysis of Cable Forces
Cable forces are the driving forces for the manipulator's joints.Due to the small mass and low speed of the cables, the paper ignores the dynamics of the cables themselves.
As the lubrication between the cables and the holes is good and there is no stick-slip phenomenon, the Coulomb friction model with continuous saturation function is adopted.The friction coefficient is, μ 0 : Coefficient of sliding friction.Set μ 0 = 0.14.
it is the velocity of C j relative to P i,j .And if the cable is moved to the distal end relative to the hole, v i,j,k is defined as positive.sat(): Saturation function, which is defined below, As is shown in Fig. 4(c), the force analysis of a cable passing through the hole is simplified as the case of a cable twining around a cylinder.Arc AB is the contact line between the cable and D i,j , and its contact angle is θ i,j,1 .The cable is subjected to the distribution pressure N i,j,1 , friction force f i,j,1 , and tension forces T i,j,1 , T i,j,2 .Then, though Link i−1 is subjected to −N i,j,1 and −f i,j,1 at D i,j , the resultant force is, Similarly, the resultant force on Link i−1 from C j at P i−1,j is Therefore, the total acting force on Link i−1 from C j is The infinitesimal arc element ds is cut from Arc AB, and its contact angle is dθ.T , T are the tension forces on both ends of ds, dN is the pressure force on ds from the hole, and df is the friction force.Then, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
And, dN , df can be decomposed as follows: The variables above satisfy the following expressions: o() refers to the higher order infinitesimal.Then, if ( 16) is projected along the tangent and normal directions, we can get, T, T are the value of T , T .And as dθ is the infinitesimal Substitute the above equations into (21) and ignore the higher order infinitesimal, we can get, Then, we can get, Therefore, we can get, Similarly, let θ i,j,2 be the contact angle between C j and P i,j , According to the positions of P i−1,j , D i,j , P i,j , and D i+1,j , we can further get the unit direction vectors of T i,j,1 , . . .T i,j, 4 , Then, As a result, as long as the tension force at one end of the cable is known, every tension force at each section of the cable can be calculated.And then all cable forces acting on the links can be calculated through (15).

B. Analysis of Hollow Ring and Link Forces
1) Analysis of Hollow Ring Forces: HR i is the key part of Joint i , which connects Link i−1 and Link i .There are three kinds of forces on HR i .The first are the acting forces from Link i−1 and Link i .The second is the gravity force.The third is the inertia force.
When the bearings are distributed on two ends of a shaft, the bending moment on the shaft could be balanced by the supporting forces of the bearings at both ends.Therefore, the bearing at each end of HR i would only have supporting forces on the shaft, but no bending moment.Additionally, since the friction force at each revolute pair is negligible with good lubrication, the torque moments can also be ignored.Thus, the first kind of forces at each bearing of HR i have no moment, which are shown in Fig. 5.And the forces from Link i are F Ni,1 , F Ni,2 , the forces from Link i−1 are F Ni,3 , F Ni, 4 .Furtherly, 2z act on the same line, they can be merged as one, F Ni,zc .Similarly, F Ni,3y , F Ni,4y can be merged as F Ni,yc .Then, the acting forces on HR i from adjacent links are as follows: The gravity of HR i is m H : Mass of HR i , and the mass of every hollow ring is equal.g : Gravity acceleration.HR i 's inertia force's solving needs the kinematic variables, which are calculated through HR i 's eigenvectors and O mi 's position.In this article, the discrete control is adopted.And since the cycle time is short enough, the discrete kinematic Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
variables, including velocity, acceleration, angular velocity and angular acceleration, are approximately equal to the continuous kinematic variables.Thus, the inertia force can be calculated with the discrete variables.
The velocity of HR i 's centroid at current period is O mi : Position of HR i 's centroid at the current period, which is also the proximal end of Link i .O mi : Position of HR i 's centroid at last period.ΔT : Cycle time.Set ΔT = 10 ms.
Then, the acceleration of HR i 's centroid at current period is v cH,i : Velocity of HR i 's centroid at last period.
Then, the inertia force on HR i is In order to calculate the angular velocity ω Hi and the angular acceleration α Hi , let v iy , v iz : Two eigenvectors of HR i at last period.
If viz × viy = 0, the angular velocity of HR i is If viz × viy = 0, let Then, the angular velocity of HR i is Then, the angular acceleration of HR i is ω Hi : Angular velocity of HR i at last period.
Let I cH be the inertia tensor of HR i relative to the center of mass, which is calculated in the local coordinate system.The local coordinate system is defined as: Set HR i 's eigenvectors as x-, y-, and z-axes, and O mi as the origin.Then I cH is constant and can be solved in advance according to the mass distribution of the hollow ring.I cH is defined as dm: Mass element of HR i .r: Vector from mass element to the centroid of HR i in the local coordinate system.I 3 : Identity matrix with 3-order.
Let r 0 be the vector from mass element to the centroid of HR i in the global coordinate system, then then the inertia tensor of HR i in the global coordinate system is In the global coordinate system, the inertia moment on HR i about the centroid is Substitute ( 41) into (42), we get Let ω Hi = [ω Hi,x , ω Hi,y , ω Hi,z ] T , then its corresponding skew symmetric cross product matrix is Then, the time derivative of the rotation matrix is Then, (43) can be transformed into Then, Then, substitute (47) into (42), the inertia moment is As the hollow ring's mass and inertia tensor are small, its dynamics sometimes can be ignored for calculation efficiency.
2) Analysis of Link Forces: There are five kinds of forces on Link i .The first are the cable forces.The second are the acting forces from adjacent hollow rings.The third are the environment forces.The fourth is the gravity force.The fifth is the inertia force.Fig. 6 shows the first and second kind of forces on Link 4 .
The first kind of forces acting on Link i can be divided into the known cable forces and the driving forces.The known cable forces are from the cables passing through Link i , which are fixed on Link m (m > i).Since Link m 's force analysis is done before Link i 's, T m,m,3 , T m,m+6,3 , T m,m+12,3 are all known parameters.Thus, the tension forces on every section of C m , C m+6 , C m+12 can be solved according to (25) and (26).Then, the known cable forces can be calculated as (T i+1,j,2 + T i,j, 3 ) . (49) The driving forces are unknown and to be solved.They are from the cables C j (j = i, j = i + 6, j = i + 12), which are fixed on Link i .Since T i+1,j,2 actually is the internal force of Link i , the resultant force from C j on Link i is only T i,j,3 .Then, the driving forces are as follows: The acting forces on Link i from HR i and HR i+1 are When i = 6, Link i is the end link.Then, F N (i+1),3x + F N (i+1),4x + F N (i+1),3z + F N (i+1),4z + F N (i+1),yc can be treated as the environment forces.In addition, the middle links are also subjected to the environment forces, which can be from the outside and inside of the manipulator.Although the environment influence is not analyzed here, the corresponding forces can still be added into dynamics equations if necessary.
The gravity force on Link i not only comes from the link itself, but also the water pipe, gas pipe, cables, etc.For the end link, the operation tools even need to be considered.Link i 's total gravity is m i : Total mass of Link i .
On the analogy of HR i , the inertia force on Link i is a cc,i : Acceleration of Link i 's centroid at current period.
v c,i : Velocity of Link i 's centroid at current period.v c,i : Velocity of Link i 's centroid at last period.
Let P ci be the position of Link i 's centroid at current period and its value is, a i , b i , c i : Position coefficients of Link i 's centroid, which are related to the mass distribution.
P ci : Position of Link i 's centroid at last period.
The inertia moment on Link i about the centroid is, I c,i : Inertia tensor of Link i in the global coordinate system.ω i : Angular velocity of Link i .α i : Angular acceleration of Link i .
The calculations of I c,i , ω i , α i are similar to the calculation of I cH,i , ω Hi , α Hi .

C. Inverse Dynamics Method
The traditional Newton-Euler method includes outward and inward iterations.Outward iteration solves the kinematic parameters, while the inward iteration solves the driving forces.In the tracking geometric curve method, the traditional dynamics would figure out the joint parameters and their first and second derivatives first based on the known manipulator's eigenpoints, and then the kinematic parameters of the links.That not only wastes the convenience of the geometric method to solve the kinematic parameters in the global coordinate system but also makes the dynamic description cumbersome and complicated.Thus, the dynamic model in this article makes the calculation from manipulator's eigenpoints to kinematic parameters more direct.As is shown in Section III-B, the inertia forces of Link i and HR i are already calculated from the global kinematic parameters, so the desired solution of the outward iteration has already been done.Additionally, compared to the joint parameters, link eigenvectors are more intuitive and easier to be obtained.The following text will demonstrate the inverse dynamics in detail.
1) Method for Setting the Optimal Driving Forces: The driving forces of the CSM is from the cables.Since each universal joint with 2 DOFs is driven by three cables, it belongs to the redundant actuation.The ith recursive Newton-Euler equations Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
have 12 independent equations involving HR i and Link i .And the unknown vectors in the equations include 3 driving forces, T i,i,3 , T i,i+6,3 , T i,i+12,3 , 5 interaction forces between HR i and Link i , F Ni,1x , F Ni,2x , F Ni,zc , F Ni,1y , F Ni,2y , and 5 interaction forces between Link i−1 and HR i , Since the directions of these vectors are known, the 12 independent equations have 13 unknown variables, which also means the actuation is redundant.Thus, for a better dynamic effect, it is meaningful to set the optimal driving forces.The goals are as follows.
1) Driving forces are greater than the pretensioning force T pre , which promises the accuracy and stiffness of the CSM.
2) The cable forces, not only the maximum cable force but every cable force, should be as small as possible.Although the optimal driving forces can be obtained by the setting and solving of the optimal objective function, it only needs to set the minimum of Joint i 's 3 driving forces equal to T pre .The proof is as follows.
Let S MT 1,i be the resultant moment of the known cable forces on Link i about O mi , S MT 2,i be the resultant moment of the driving forces on Link i about O mi , M G i be the moment of G i about O mi , and M F I,i be the moment of 4 , the moment equations about O mi of Link i , HR i are as follows: The left side of (58) includes the unknown variables, while the right side is with the known variables.Except for M I,i and M IH,i whose expressions are already known, the other variables in (58) are expressed as follows: The values of d i+1,j and p i,j are as follows: S MT 2,i contains 3 unknown variables, T i,i,3 , T i,i+6,3 , and T i,i+12,3 .It can be expressed as follows: The expression of M F Ni,k needs to be transformed for reducing the number of unknown variables of (58).Let R k be the vector from O mi to the acting point of F Ni,k , then, Similarly, Then, Similarly, And the value of R k : Vector from O mi to the acting point of F N (i+1),k .

The value of M G i is
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The value of M F I,i is (Note the difference between M F I,i and M I,i .) (71) Finally, T i,i,3 , T i,i+6,3 , T i,i+12,3 , x i,1 , x i,2 , x i,3 , x i,4 are the 7 unknown variables of (58).Let and move X i , one term of S MT 2,i in the first equation of (58), to the right side, (58) can be transformed to The right part of the equations can be expressed as (73) And let D be the coefficient matrix of (72), then Coefficients k 1 , c 1 are defined for convenient description Similarly, k 2 , c 2 can also be defined.Thus, Next, we will prove k 1 is positive in any reasonable case.Set P ix as the projection of u ix in the plane determined by u (i−1)y and u (i−1)z .Let ϕ be the included angle between P ix and u (i−1)y , γ be the included angle between u ix and u (i−1)x .Actually, γ reflects the bending degree of the CSM and γ ≤ 0.2π.P i,i is the hole on Disc 2,i , which is passed through by the cable C i .Let Ψ be the included angle between When Ψ traverses from 0 to 2π, ϕ traverses from 0 to 2π and γ traverses from 0 to 0.2π, the value of k 1 has been calculated for all cases.Fig. 7(a) shows the relation among ϕ, γ and k 1 when Ψ = 2π.Fig. 7(b) shows the relation among ϕ, γ and k 2 when Ψ = 0.5π.The result shows k 1 is always greater than 0, i.e., T i,i,3 , T i,i+6,3 are positively correlated.And since T i,i,3 , T i,i+6,3 , T i,i+12,3 are rotationally symmetric, they are all positively correlated, which means all increase when one increases, and all decrease when one decreases.
As a result, when the minimum of T i,i,3 , T i,i+6,3 , T i,i+12,3 (i = 1, 2 • • • 6) is equal to T pre , the two goals of setting the optimal driving forces can be satisfied.Furthermore, since the variation of the cable force is continuous, the cable force, which is minimum at the last period can be preset as T pre during the Newton-Euler equations' solution, which can improve the efficiency of the iteration.After calculation, if the cable force, which is minimum at the last period is not minimum this time, the minimum cable force at this time should be set as T pre and then the solution that meets the requirements can be figured out by again solving the Newton-Euler equations.
It has been verified that when the manipulator is in the reachable space, the coefficient matrix of (72) is nonsingular.Therefore, (72) must have a unique solution, i.e., theoretically proving that the moment equilibrium of Link i and HR i can be achieved by setting the driving forces.In fact, if the values of T i,i,3 , T i,i+6,3 , T i,i+12,3 can be positive and negative, the moment equilibrium of Link i and HR i could be achieved by 2 driving forces, and the 2 driving forces are unique.However, the driving force is the cable's tension force, i.e., the value is always positive, so 3 driving forces should drive the joint together to drive the Link i successfully and scheme the driving forces uniformly.
It is also worth noting that if γ closes to the joint limit, the values of k 1 and k 2 would increase significantly or close to 0, which would lead to too large value difference among the driving forces and the balance would be hard to maintain.
2) Newton-Euler Equations: After one of the 3 driving forces on Link i is set as T pre , the ith Newton-Euler equations have 12 linear equations and 12 unknown variables.Equation (72) shows a way how to solve the Euler equations and Newton equations separately, which would reduce the calculation cost.Let 4 , then Newton equations are as follows: (77) Then, all related forces including the driving forces on Link i and HR i can be solved by ( 72) and (77) with much less calculation.And then all related forces of the manipulator can be solved through the inward iteration.Afterward, the driving torques of all motors can be solved.Finally, the inverse dynamics is completed.

IV. EXPERIMENT
The experiments have been presented on a CSM physical prototype to verify the effectiveness of the kinematic and dynamic models.There are two types of experiments.The first is about the static tests, which study the cable forces when the manipulator stays in the homing position.The second experiment is about the dynamic tests, which study the cable forces when the manipulator stays in motion.Parameters of the CSM have been listed in Table I.
There are two special sensors on the manipulator.First, a force transducer is set on the driving module's end for each cable, which is for the cable force's monitoring.Second, two joint angle sensors are set on each joint, which are for monitoring the joint parameters and then deriving the end-effector pose.Fig. 8 shows how the cable force transducers and the joint angle sensors are assembled.

A. Static Experiment
Experiment purpose: Verify the accuracy of the driving force predicted by the models, when the CSM is static.
Experiment procedures: 1) Loosen all cables and let the manipulator naturally lie on the sliding table.2) Tension all cables and let the manipulator in an uncertain posture.caused by complex coupling effects among the cable forces.For instance, when the manipulator remains in the homing position, if the cable forces of C 6 , C 12 , C 18 increase, the cable forces of C 5 , C 11 , C 17 would decrease.The other is the systematic errors caused by the force transducers, such as zero-drift and linearity errors.Furthermore, the inadequate accuracy of the mass and geometric dimension would also lead to some small errors.
The above experiment has been repeated to reduce the random errors.The 15 groups' data (∅ A ) of each experiment are averaged again, which are recorded as ∅ B .As is shown in Fig. 9(b), the slightly fluctuating polylines refer to the new averaged data ∅ B , and the straight lines refer to the theoretically calculated cable forces.It can be found that the fluctuations of the cable forces have been significantly reduced and the errors are within a reasonable range.
Experiment results: The static test results show that the measured cable forces fluctuate each time, indicating that the dynamic system of the manipulator is complex and the coupling effect among the cables cannot be ignored.However, if multiple groups of the data are averaged, the random errors would be sharply reduced and the theoretical driving forces are basically the same as the feedback values.That verifies the accuracy and reliability of the proposed models when dealing with the static problems of the manipulator.

B. Dynamic Experiment
Experiment purpose: Verify the accuracy of the driving force predicted by the models, when the CSM is dynamic.
Experiment procedures: 1) Let the manipulator in the homing position state.
2) Set the driving torques of the motors the same as the calculation result of the dynamic method, when the manipulator does the tip-following motion.Fig. 10 illustrates how the manipulator moves.3) Read the feedback value of each force transducer every 0.1 seconds and record the data.Fig. 11 shows the variation of the cable forces, andC 1 , C 2 , • • • C 18 refer to the theoretical values, while C (1) , C (2) , • • • C (18) refer to the feedback values.Fig. 12(a) is the variation of joint parameters, and α 4 , α 5 , α 6 refer to the theoretical values, while α (4) , α (5) , α (6) refer to the feedback values.It shows that when the joint angle is relatively large, insufficient rotation may occur.That is because the shortest cable on the joint would bend more severely, which results in a bending resistance effect.Fig. 12(b) shows the errors of the manipulator's end, including three-axis errors.The experiment  proves that the theoretical driving forces are basically consistent with the real values during the motion, and simultaneously the dynamic accuracy is good.
Experiment results: The experiment data show that the coupling effect of the cable forces seems to be weakened in the dynamic tests, and the cable forces and the joint angles both are close to the theoretical values.Additionally, the algorithm of the kinematic and dynamic models works efficiently and reliably, which means it is applicable to real-time control.Therefore, the proposed method is practical.

C. Comparison Experiment
The comparison experiment with the conventional methods has been conducted to verify the advantages of the proposed method.Compared to traditional Newton-Euler methods [23], [24], [25], the paper constructs more direct and intuitive models skipping the intermediate joint parameters.Thus, according to the experiment, the computation efficiency of the kinematic parameters is improved by 23.9%.The result of the comparison experiment is shown in Table II.
In addition, since the traditional methods omit the analysis of hollow ring force, the modeling error is inevitable."Theoretical driving force error" in Table II refers to the relative driving force difference between the traditional method and this article's method.Despite the tiny experiment difference, it Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE II RESULT OF COMPARISON EXPERIMENT
would increase with the increasing speed of CSM.And although CSM's motion is still slow at present, it will be faster with the development of technology.And when CSM's end acceleration is more than 10m/s 2 in some cases, the theoretical driving force error could be more than 10%.
Moreover, to confirm the reliability and efficiency of the method, we have tested various trajectories on the CSM.

V. CONCLUSION
While the sufficient DOFs of hyper-redundant manipulator bring extremely flexible motion performance, it also challenges the more complex kinematic and dynamic algorithms.The excellent motion control should be not only efficient and stable, but also draw out the super dexterity of the hyper-redundant manipulator.Based on the widely used tracking geometric curve method, the paper reconstructs the kinematic and dynamic models in the global coordinate system with the link eigenvectors.And based on the new models, the manipulator's mechanics and the Newton-Euler equations are optimized and simplified.Thus, the dynamic solution can be obtained precisely and quickly with a reliable solving process.The experiments prove that the cable driving forces are optimized and the motion accuracy and smoothness are also improved.And the comparison with the traditional methods shows the improvements in efficiency and accuracy.
However, as the research went by, we also found a series of issues that have not been resolved but are worthy of further study.When the universal joint angle is close to the limit, the relations among the driving forces may change significantly, and some cables may even have severe bending effect.That would affect the driving of the joint.Later, we will study the ways of how to weaken these effects, such as setting T pre dynamically, optimizing the mechanical structure.In addition, the complex coupling effects of the cables would cause random errors.Next, we will research and try to grasp the coupling laws.And that could not only furtherly verify the correctness of the dynamic model, but also is conducive to achieving the more precise dynamic close-loop control.Furtherly, we will also study the forward dynamics of the CSM according to the proposed method, so as to facilitate the dynamics optimization and the simulation of the manipulator.

Fig. 4 .
Fig. 4. Cable forces' analysis on Joint i .(a) Joint i .(b) C j s force on Joint i .(c) Force analysis.
Fig. 4(a) shows Joint i , and (b) shows the force analysis of C j in Joint i .There are 4 tension forces in the figure.And T i,j,2 , T i,j,3 are equal with opposite directions, while T i,j,4 , T i+1,j,1 are equal with opposite directions.As is shown in Fig.4(c), the force analysis of a cable passing through the hole is simplified as the case of a cable twining around a cylinder.Arc AB is the contact line between the cable and D i,j , and its contact angle is θ i,j,1 .The cable is subjected to the distribution pressure N i,j,1 , friction force f i,j,1 , and tension forces T i,j,1 , T i,j,2 .Then, though Link i−1 is subjected to −N i,j,1 and −f i,j,1 at D i,j , the resultant force is,

D 1 :
Matrix obtained by replacing the first column of D with B. D X : Matrix obtained by replacing the first column of D with −[p i,j × u vi,i,3 000] T .D Y : Matrix obtained by replacing the first column of D with [M Omi (−M IH,i )] T .

Fig. 8 .
Fig. 8. Sensors' configuration.(a) Way of how the cable force transducer is assembled.(b) Way of how the joint angle sensors are assembled.

Fig. 9 .
Fig. 9. Cable forces' recordings in static state.(a) Cable forces' recording in one experiment.(b) Cable forces' recording with the new averaged data.

3 )
Drive the motors, and let the manipulator return to and remain in the homing position from the uncertain posture.Keep the error of each universal joint angle less than 0.01 • .4) Read the feedback value of each force transducer every second.Read a total of 100 times and average the feedback values.The average data are recorded as ∅ A . 5) Repeat the above steps for 15 times.Fig. 9(a) shows all feedback cable forces in one experiment.Every single figure shows 3 driving forces that drive the same joint.Every single cable force in the figures fluctuates with errors.There are 2 main reasons for that.One is the random errors

TABLE I PARAMETERS
OF THE CSM