A modified modal method for solving the mission-oriented inverse kinematics of hyper-redundant space manipulators for on-orbit servicing

Accepted Manuscript A modified modal method for solving the mission-oriented inverse kinematics of hyper-redundant space manipulators for on-orbit servicing Wenfu Xu, Zonggao Mu, Tianliang Liu, Bin Liang PII:

S0094-5765(17)30138-8

DOI:

10.1016/j.actaastro.2017.06.015

Reference:

AA 6350

To appear in:

Acta Astronautica

Received Date: 24 January 2017 Revised Date:

19 March 2017

Accepted Date: 15 June 2017

Please cite this article as: W. Xu, Z. Mu, T. Liu, B. Liang, A modified modal method for solving the mission-oriented inverse kinematics of hyper-redundant space manipulators for on-orbit servicing, Acta Astronautica (2017), doi: 10.1016/j.actaastro.2017.06.015. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

A Modified Modal Method for Solving the Mission-oriented Inverse Kinematics of Hyper-redundant Space Manipulators for On-orbit Servicing Wenfu Xu1,2*, Zonggao Mu1, Tianliang Liu1, Bin Liang3

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1. Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China 2. State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China [email protected], [email protected], [email protected] 3. Department of Automation, School of Information Science and Technology, Tsinghua University, Beijing 100084, China [email protected]

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Abstract: A hyper-redundant space manipulator has extreme flexibility and is suitable to work in highly cluttered or multi-obstacles environment. However, its inverse kinematics is very challenging due to a large number of degrees of freedom (DOFs). In this paper, a modified modal method is

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proposed to solve the mission-oriented inverse kinematics. The spatial backbone of the manipulator is defined using a mode function, according to the mission requirement and working environment. All the universal joints are divided into M/2 groups, i.e. two adjacent universal joints comprise a group (M is the number of universal joints; it is assumed an even number. If it is an odd number, the remaining universal joint is a separated group). The whole manipulator is then segmented into M/2 sub-manipulators. Each sub-manipulator has 4-DOFs and is redundant for position or orientation. The last sub-manipulator is used to match the desired direction vector and the position of the

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end-effector with respect to the previous sub-manipulator’s end. The remaining sub-manipulators are used to control the relative position between each other with one redundant degree of freedom. The equivalent link is fitted to the backbone function. The Cartesian coordinates of each node is then determined by combining the total length of the manipulator and the mode function. Then, the joint an-

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gles are solved through the position of each node. For each 4-DOF group, a parameter called arm angle is used to denote the redundancy and optimize its local configuration. Finally, typical cases of a

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12-DOF and a 20-DOF manipulators are simulated. The results show that the method is very efficient for resolving the inverse kinematics of hyper-redundant space manipulators. Keywords: Space Manipulator; On-orbit Servicing; Hyper-redundant Manipulator; Inverse kinematics; Mode Function

1 Introduction

A space robotic system will play an important role in on-orbital servicing [1-6], such as spacecraft construction, repair, maintenance [7] and orbital debris removal [8]. A hyper-redundant manipulator has great advantages when working in narrow and multi-obstacles environments. The redundancy can be used to deal with the singularity, obstacle avoidance and joint torque optimization problems synchronously. It is also useful for minimizing the reaction torque of a space manipulator. However, the inverse kinematics of such manipulators becomes very complex and challenging,

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ACCEPTED MANUSCRIPT due to the existence of a large number of degrees of freedom. Recently, the scholars proposed some methods to solve this problem. These methods can be divided into three categories [9]: numerical, geometrical, and artificial neural network methods. For numerical methods, the pseudo-inverse of the Jacobian matrix is used to deal with the differential kinematics equations. With the increase of degrees of freedom, the computational load becomes very large, affecting the real-time control. Recently, artificial neural network [10-11] was used to solve the inverse kinematics problem of general

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manipulators. The size of the training set will change significantly with the increase of the number of degrees of freedom. As for geometrical methods, Chirikjian [12] proposed the concept of backbone curve. He used a backbone function to describe the macro-configuration of hyper-redundant manipulator. This method has lower computation cost, and is suitable for on-line path planning and real-time control. However, the determination and resolution of the backbone function rely on experience and

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skill. Another important geometrical method is the mode function method, proposed by Ref. [13]. The backbone curve was defined as a piecewise continuous curve. This curve was used to express the

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geometric characteristics of a hyper-redundant manipulator. It was composed of a series of mode functions, which could be chosen as needed. Once the backbone curve was determined according to the position of the end effector, a certain fitting algorithm was used to determine the joint angles. This method was successfully used for obstacle avoidance [14], motion planning [15], and control [16-17]. Fahimi et al. [18-19] used a different mode function to extend the working space. He applied a recursive fitting algorithm for the universal joint configuration, avoiding solving a large number of nonlinear equations. Samer et al. [20-21] proposed another geometric method, which could find a

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solution from the infinitely solutions of hyper-redundant manipulator. This method set the angles of adjacent joints to be equal. It is very effective for simple planar manipulators with equal link length. The singularity problem was also avoided. By now, the backbone curve method was considered as an effective way to solve the inverse kinematics problem. The commonly used backbone cures are listed

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as: mode function curve [22], space arc curve [23], Bezier curve [24-25], and so on. For these methods, each joint of the hyper-redundant manipulator is placed on the backbone, there are three shortcomings: 1) The position of the end-effector is assured, but the direction of the end-effector is diffi-

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cult to accurately adjust; 2) The local configuration of the hyper-redundant manipulator cannot be adjusted independently, and a lot of degrees of freedom are sacrificed; 3) Determining a reasonable backbone function is very complex. For a practical task, the backbone function is often required be re-designed on real time, hence the inverse kinematics will become much more complex and the computation load will increase largely. In this paper, a modified modal method is proposed to solve the mission-oriented inverse kinematics of a hyper-redundant space manipulator. A spatial backbone is defined using a mode function to represent the geometry of the manipulator. The redundant degrees of freedom are then transferred to some geometry parameters. The mission configuration can be easily denoted by these parameters. The desired position and direction of the end-effector can be simultaneously attained. All the joint angles are analytically solved and the computation efficiency is very high. The method has generality

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ACCEPTED MANUSCRIPT and can be applied for typical spatial hyper-redundant manipulators. The remainder of this paper is organized as follows. Section 2 introduces the working environment of a spatial hyper-redundant manipulator. The kinematics model is then analyzed. Section 3 proposes a modified inverse fitting resolution method, derived from the traditional method. The important improvements are introduced. In Section 4, the resolving of the inverse kinematics based on the modified modal function method is addressed, considering the 3D position and direction of the

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hyper-redundant manipulator. The determination of the universal joints and the resolution of all joint angles are detailed. In section 5, the simulation of a 12-DOF and a 20-DOF manipulator is presented to verify the proposed method. The last section summarizes the whole paper and gives the conclusions.

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2 A Hyper-redundant Space Manipulator for Space Station

A hyper-redundant space manipulator will play an important role in the construction and maintenance of large spacecraft. It can enter narrow space to inspect, assemble and repair the key

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devices. The application concept for the Chinese space station is shown in Fig. 1, where a hyper-redundant manipulator is used to cross the truss structure to supply inspection information and

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intervention means when needed.

Fig. 1 The hyper-redundant manipulator in working environment

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The manipulator is composed of multiple universal joints, arranged in sequence. Each universal joint has two orthogonal degrees of freedom, which are formed by Pitch axis and Yaw axis. All the joints are driven by cables. The joint layout is (Pitch-Yaw)-(Yaw-Pitch)-(Pitch-Yaw)…., i.e. PY-YP-PY…structure. The manipulator has three-dimensional motion capability and is suitable for complex task in narrow space.

For the application of the Chinese space station, the hyper-redundant manipulator has M universal joints, i.e. the manipulator has n (n=2M) DOFs. For convenience, two adjacent universal joints are defined as a 4-DOF joint group (PY-YP joint layout). So all the universal joints can be classified into N=n/4 groups. Then, the whole manipulator is segmented into N sub-manipulators (PY-YP joint layout), each has 4-DOF and redundant for positioning. If n is not exactly 4 times of N, the remaining joints are grouped separately.

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ACCEPTED MANUSCRIPT The D-H coordinate systems and the D-H parameters are respectively shown in Fig. 2 and listed in Table 1, where n (n=2M=4N) is the number of the degrees of freedom. The symbols are defined as follows: U 2i−1 ——the odd numbered ( 2i − 1) th universal joint, i = 1, 2,L, N U 2i ——the even numbered ( 2i ) th universal joint, i = 1, 2,L, N

( N = n / 4) ;

( N = n / 4) ;

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θ 4i −3 , θ 4i −2 ——the two rotation angles of the universal joint U 2i−1 , i = 1, 2,L, N ( N = n / 4 ) ; θ 4i −1 , θ4i ——the two rotation angles of the universal joint U 2i , i = 1, 2,L, N ( N = n / 4 ) . U 2i−1

U2

θ3

θ2

θ 4 i −3

θ 4i

θ 4 i −2

θ 4i −1

U 2 N −1

U2N

θ n −3

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θ4

θ1

U 2i

θ n −2

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U1

θn

θ n −1

Fig. 2 D-H coordinate systems of a n-DOF Cable-driven Hyper-Redundant Manipulator Table 1 D-H parameters of a Hyper-Redundant Manipulator

θi (°)

αi (°)

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Link i

0 0 0 0 ... 0 0 0 0

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1 2 3 4 ... n-3 n-2 n-1 n

90 0 -90 0 ... 90 0 -90 0

ai （m） 0 L 0 L ... 0 L 0 L

d i （m） 0 0 0 0 ... 0 0 0 0

3 Modified Fitting Method for Backbone Curve As pointed out above, the traditional methods place each universal joint on the backbone curve, which is a piecewise continuous curve fitting the macroscopic geometric features of the hyper-redundant manipulator. There are some shortcomings for these methods. Here, a modified method will be proposed to overcome the shortcomings. Two adjacent universal joints comprise a group, which constitutes a 4-DOF sub-manipulator (including two links). The straight-line distance between two adjacent odd-numbered universal joints is defined as the equivalent link, the length is ρi . The concept is shown in Fig. 3.

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T= ( xT , yT , zT )

U 2i

U2N

U 2i+1

<2>

<2> Improvement<2>:The position of Even numbered universal joint are parameterized by arm-angle ψ i

U 2i+1

U 2i −1

%a U 2i

U2

O0

U1

Improvement<3>:The length of equivalent link between two adjacent odd universal joints are parameterized by ρi

U1

<3>

Y0

Z0

<3>

Z0

O0

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a)Traditional fitting method

T= ( xT , yT , zT )

X0

Explanation Backbone: thin red line Link: thick black solid line Possible position of U2i : green dashed line Equivalent link: purple thick solid line

%b U 2i

U2N

U 2 i−1

U 2 i+1 %a U 2i

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Universal Joint: white solid point

Even universal joint: blue solid point

U2

Universal joint rely on arm-angle: green solid point

Transformation of direction vector: dash-dotted line <1>

U1 O0

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Transformation of arm-angle: dash-dotted line <2>

Transformation of equivalent link: dash-dotted line <3>

Y0

Z0 b)Modified fitting method

Fig. 3 The concept of the modified fitting method

For convenience of discussion, the following symbols are defined:

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Y0

U2N

%b U 2i

<1>

U 2 i−1

U2

T= ( xT , yT , zT )

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backbone

X0

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X0

Improvement<1>:Release the last universal joint to match the desired direction of the end-effector k k

ρi ——the equivalent link length of the ψ i ——the arm angle of the

i th

i th

sub-manipulator, i = 1, 2,L, N

sub-manipulator, i = 1, 2,L, N

( N = n / 4) ;

( N = n / 4) ;

k ∈ R 3 ——the desired direction vector of the end-effector; τ ∈ R 3 ——the tangential direction of the backbone curve at the end point; T ——the origin of the end-effector’s frame. Its coordinates are denoted by T ( xT , yT , zT ) ; O0 ——the origin of the base frame, i.e. frame {0};

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ACCEPTED MANUSCRIPT Firstly, the backbone curve is determined based on the position of the end-effector, and each universal joint is placed on the curve, similarly with the traditional method. Then, the even numbered universal joints, i.e. the 2nd, 4th, 6th, …, joints are released from the backbone curve. The Cartesian position of each released joint is parameterized by a scalar called arm angle, denoting the redundancy of the segmented sub-manipulators. Hence, the local configuration can be optimized by this parameter. Another improvement is that the odd numbered universal joints, i.e. the 1st, 3th, 5th, …, joints can

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move along the backbone curve, by parameterizing the equivalent link length between two adjacent odd numbered universal joints. Based on these two improvements, the new method can be used to meet additional tasks, such as obstacles avoidance, singularity avoidance, joint movement overrun avoidance and so on. The application in dynamic environment can be easily realized by adjusting the method is actually a special case of the modified method.

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above parameters, instead of re-determining the backbone curve. From this viewpoint, the traditional In contrast, the modified method has the following three aspects of improvement:

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1) The direction vector k is introduced, and the last universal joint is no longer required to be located on the backbone. It is used to match the direction vector of end-effector which can be met better.

2) The position of each even universal joint is parameterized by arm-angleψ i . Hence, the even universal joints are also no longer required to locate on the backbone. The possible position will be optimized by arm-angle.

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3) The equivalent link length between two adjacent odd joints ρ i is parameterized, so the position of each odd universal joint will move along the backbone by adjusting the equivalent link parameter.

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4 Inverse Kinematics Resolution

4.1 The main steps of the resolution method The main steps of the inverse kinematics resolution method are shown as follows:

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1) Determining the backbone curve A reasonable mode function is determined to represent the macroscopic shape according to the desired end position and local configurations. The overall shape of the backbone (i.e.①red line in Fig. 4) is then obtained by the numerical integration of the mode functions.

2) Matching the desired direction vector k The end point (i.e. T ) of the backbone is the desired position. Theoretically, the tangent direction of the backbone curve at the end point can be used as the direction of the manipulator. Generally, the desired direction vector of the end-effector is dependent on the tool and the practical mission. That is to say, when the backbone curve is determined using the traditional method, the tangential vector (i.e. τ ) at the end point of the backbone curve will not be the same as the desired direction

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ACCEPTED MANUSCRIPT uuuuuv vector (i.e. U 2 N T ) because of the length of the last link. So we make the direction vector (i.e. k ) at the end point to collide with the desired direction of the end-effector by releasing the last universal joint U 2 N , whose position can be calculated based on the position of the end point and the direction vector k .

3) Determining the Cartesian position of the universal joint U 2 N −1

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uuuuuuuuuuv The position of U 2 N −1 can be determined by backbone curve fitting using link U 2 N −1U 2 N . Because the backbone is similar to an arc, in general, the position of U 2 N is difficult to fall on the backbone. In this paper, the point U 2 N −1 is fitted to the backbone by using one-link fitting which is uuuuuuuuuuv defined in 4.3.2 (i.e. U 2 N −1U 2 N ,③).

4) Determining the Cartesian position of odd numbered universal joints (except the last

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universal joint)

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It can be determined by using the equivalent link length between two adjacent odd numbered uuuuuuuuuuv universal joints to fit the backbone curve, (i.e. U 2i −1U 2i +1 ④).

5) Determining the Cartesian positions of even numbered universal joints The Cartesian position of the even numbered universal joints (i.e. U 2 , U 4 , U 6 L U 2 N −2 , ⑤) can be determined by combining those of the odd-numbered joints and the arm-angle parameters.

6) Solving the angle of each degree of freedom

When the Cartesian position of each universal joint is determined, the angle of each degree of Pitch-Yaw type).

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freedom can be determined according to the specific joint layout (such as Yaw-Pitch type, or The concept is shown in Fig. 4. The important steps will be detailed in the following parts.

T= ( xT , yT , zT )

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τ

k

U2N

U 2i−1

U 2i+1

U 2i

U2

U1

Fig. 4 Modified inverse fitting method based on backbone

4.2 Backbone curve defined by a mode function The backbone curve is used to describe the macro-geometric characteristics of a hy-

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ACCEPTED MANUSCRIPT per-redundant manipulator. The key is to obtain reasonable mode function and its coefficients. X

e1

e2

backbone

µ ( s)

u(s) Z

ϕ ( s)

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Y

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e3

Fig. 5 The overall shape of backbone curve and the position vector

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The spatial backbone curve (shown in Fig. 5) is expressed as follows:

X ( s ) = ∫ lu (σ )dσ s

0

(1)

where s ∈ [0,1] is a normalized parameter denoting the length of the curve; u (σ ) is the unit vector which is tangent to the curve at σ ; l is the actual length of the curve. Fig. 5 also shows the definition of the position vector u( s ) . Equation (1) can be further expressed in the reference frame

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{XYZ} as the following form:

(2)

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 s l sin ϕ (σ ) cos µ (σ )d σ   ∫0   s  X ( s ) =  ∫ l cos ϕ (σ ) cos µ (σ )d σ  0   s   l sin µ ( σ ) d σ ∫0  

Function ϕ (σ ) and µ (σ ) are the linear combination of the following mode functions: (3)

µ (σ ) = a3 (1 − cos(2πσ ))

(4)

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ϕ (σ ) = a1 sin(2πσ ) + a2 (1 − cos(2πσ ))

The mode coefficients ai ( i = 1, 2,3) can be solved by numerical solution [13, 18].

4.3 Backbone curve fitting The backbone curve fitting process includes the following three steps: direction vector fitting to match the desired direction of the end-effector; one-link fitting to re-locate the ( 2 N − 1)th universal joint onto the backbone curve; equivalent dual-link fitting to place the odd-numbered universal joints, by take into account the shape of backbone curve and local configuration adjustment. 4.3.1 Direction vector fitting To achieve a specific task, the tool of the hyper-redundant manipulator must meet the posture

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ACCEPTED MANUSCRIPT requirements. If the desired direction of the end-effector is denoted by k , the position of the last universal joint can be solved as:

uuuuuuuv k T T O0 U 2 N = [ x2N , y2N , z2N ] = [ xT , yT , zT ] − L k

(5)

4.3.2 One-link fitting

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The end-effector of the hyper-redundant manipulator is set to coincide with the ends of the backbone. When the position of the universal joint U 2 N is known, the following step is to solve the position of U 2 N −1 . These points on the backbone will be recursively calculated such that the spatial

where, uuuuuuuv O0 U 2 N ——the Cartesian position of the

( 2N )

uuuuuuuuuuv U 2 N −1U 2 N ——the vector of the

universal joint, denoted as

( 2 N − 1)

th

( 2 N − 1)

th

(6)

( x2 N , y 2 N , z 2 N ) ;

( x2 N −1 , y2 N −1 , z2 N −1 ) ;

universal joint, i.e.

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uuuuuuuuv O0 U 2 N −1 ——the Cartesian position of the

th

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distance between the adjacent universal joints satisfies the link length relationship: uuuuuuuuv uuuuuuuv uuuuuuuuuuv O 0 U 2 N −1 = O0 U 2 N − U 2 N −1U 2 N

uuuuuuuuuuv link, its length is U 2 N −1U 2 N = L .

The position of the universal joint is a function of the length of the normalized backbone (i.e. s ). In order to ensure that the universal joint falls on the backbone, the stepwise scanning method and

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the bisection method is used to search in the interval s ∈ [0,1] . The equation f ( sk ) = 0 is regarded as the stop condition. Then, the position of U 2 N −1 can be solved based on the following equation:

f ( sk ) =

(x

2 N −1

− x2 N

) +(y 2

2 N −1

− y2 N

) + (z 2

2 N −1

− z2 N

)

2

−L

(7)

4.3.3 Equivalent dual-link fitting

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Different from the traditional one-link fitting, the rest of the universal joints are solved by the equivalent link fitting (called equivalent dual-link fitting) to solve the inverse solution of the hy-

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per-redundant manipulator. The traditional one-link fitting can guarantee an outstanding match to the backbone curve. However, since the backbone is a configuration determined qualitatively by the mode function, it is often difficult to satisfy the demand of the actual working condition. The equivalent dual-link fitting relates the redundancy of the hyper-redundant manipulator to the intuitive controllable parameters (i.e. the parameter of equivalent length ρ i and arm-angle ψ i ). Compared with the traditional one-link fitting, the redundancy of the hyper-redundant manipulator is further released.

4.4 Determining the Cartesian Position of Each Universal Joint 4.4.1 Solving the Cartesian positions of odd numbered universal joints

For the ith 4-DOF universal joint group (shown in Fig. 6), the length of the equivalent dual-link (i.e. ρ i ) is only dependent on the 3rd and 4th joints of this group, denoted as θ3( ) and θ 4( ) . i

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i

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T= ( xT , yT , zT )

ρi

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U2 N

%b U 2i U 2 i−1

U 2i+1

%a U 2i

U2

k

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U1

Fig. 6 The principle of solving odd universal joints (i ) 1

(i ) 2

Setting θ =θ =0 , we get:  Lc1(i ) c2(i ) − Lc4(i ) s1(i ) s3(i ) − c1(i ) c2( i )c3(i ) − Lc1(i ) s2( i ) s4(i )     0 R (i ) Ls (i )c (i ) + Lc(i ) c(i ) s (i ) + s (i ) c(i ) c(i ) − Ls (i ) s (i ) s (i )  0 (i ) 1 2 4 1 3 1 2 3 1 2 4  T4 =  4   i i i i i i Ls2( ) + Lc2( ) s4( ) + Ls2( ) c3( )c4( )    0  1 (i ) (i ) 3 4

L − Lc c     Ls3(i ) c4(i )   0 R4( i ) = Lc2( i ) s4(i )     0 1  

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  0 (i )  R = 4   0 

) )

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( (

(8)

(i ) x

p   p (yi )   p z(i )  1 

( ) ( ) = ( L − Lc ( )c ( ) ) + ( Ls ( ) c ( ) ) + ( Lc ( ) s ( ) ) =2L (1 + c ( ) c ( ) )

where, s (j ) = sin θ (j ) , c (j ) = cos θ (j ) , and i

i

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i

ρi 2

i 3

i 4

i

2

i 3

i 4

2

i 2

i 4

2

2

i 3

i 4

(9)

then,

( (

min 2L2 1 + c3(i )c4(i ) i i

() () θ3 ,θ4

)) ≤ ρ

2 i

≤ 4 L2

(10)

Supposing the limit of the joints is 0 ≤ θ3( ) ,θ 4( ) ≤ 90o , the equivalent dual-link length satisfies the i

i

following inequality (see Fig. 7): 1.42 L ≤ ρ i ≤ 2.00 L

(11)

For a n-DOF manipulator, there will be N − 1 equivalent links to be fitted. The corresponding parameters ρ i is selected to control the length of the corresponding equivalent links. The default

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ACCEPTED MANUSCRIPT value is ρ 2 = ... = ρ N −1 = 1.7 L . The length ρ1 is an undetermined parameter. In order to obtain the reasonable length of the equivalent links, the points on the backbone will be recursively computed so that the spatial distance between the adjacent odd universal joints satisfies the length relationship in (12). When the value ρ1 satisfies (11), there will exists reasonable inverse solutions of the hyper-redundant manipulator. uuuuuuuv uuuuuuuv uuuuuuuuuuv O 0 U 2i −1 = O 0 U 2i +1 − U 2i −1U 2i +1

(12)

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uuuuuuuv th where O0 U 2i +1 ——the position of the ( 2i + 1) universal joint, denoted as

( x2i+1 , y2i +1 , z2i +1 ) ;

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uuuuuuuv th O0 U 2i −1 ——the position of the ( 2i − 1) universal joint, denoted as ( x2i −1 , y2i −1 , z2i −1 ) ; uuuuuuuuuuv uuuuuuuuuuv th U 2i −1U 2i +1 ——the ( 2i − 1) equivalent dual-link, i.e. U 2i −1U 2i +1 = ρ i .

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U 2 i +1

ρi

U '2i+1

U 2i

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U 2 i −1

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Fig. 7 The relationship between the equivalent link and the joint angles

In equation (12), the position of the universal joint is a function of the length of the backbone (i.e. s ). In order to ensure that the universal joint falls on the backbone, the stepwise scanning meth-

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od and the bisection method is used to search in the interval s ∈ [0,1] .The equation f ( sk ) = 0 is regarded as the stop condition, then the position of the odd universal joints can be solved according to the following equation:

f ( sk ) =

( x2i +1 − x2i −1 )

2

+ ( y2i +1 − y2i −1 ) + ( z2i +1 − z2i −1 ) − ρ i 2

2

(13)

As the value of ρ i is set, then the position of U 2i−1 can be solved. At the last step, the reasonability of the position of U 3 should be re-judged by the parameter ρ1 : uuuuuv

+ ( y3 − y1 ) + ( z3 − z1 ) (14) uuuuuv If the parameter meets ρ1 ≥ 2 L , it means that the value of U1U 3 is too long to reach, then the

ρ1 = U1U 3 =

( x3 − x1 )

2

2

2

equivalent link should be adjusted adaptively by set ρ 2 = ρ 2 + ∆,...ρ N −1 = ρ N −1 + ∆ ( ∆ is an adjusta-

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ACCEPTED MANUSCRIPT ble parameters). A finite iteration will be performed until ρ1 − 1.7 L ≤ δ ( δ is the threshold). uuuuuv If the parameter meets ρ1 ≤ 1.42 L , it means that the value of U1U 3 is too short to reach, then the equivalent link should also be adjusted adaptively by set ρ 2 = ρ 2 + ∆,...ρ N −1 = ρ N −1 + ∆ ( ∆ is an adjustable parameters). A finite iteration will be performed until ρ1 − 1.7 L ≤ δ ( δ is a threshold). When all the parameters ρ1 are determined, all the odd universal joints can be solved easily.

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4.4.2 Solving the Cartesian positions of even numbered universal joints

When the Cartesian positions of the odd numbered universal joints are determined, the Cartesian positions of the even numbered universal joints can be solved by introducing redundancy parameters. For each 4-DOF sub-manipulator, a parameter called arm-angle is defined to represent its redundancy. For the ith sub-manipulator, the arm plane, the reference plane and arm-angle are defined as follows:

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(a) Arm plane: the plane defined by the three points U2i-1 U2i and U2i+1, denoted by plane U2i-1U2iU2i+1. When the condition (11) is satisfied, the arm plane always exists. (b) Reference plane: the plane defined by the three points U1, U2i-1 and U2i+1, denoted by

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U1U2i-1U2i+1 When these three points are collinear, the algorithm singularity happens. For this case, we can use the plane determined by line U2i-1U2i+1 and the line Z%0 or X% 0 (a line through the point U2i-1 parallel to the Z0 or X0 axis) as the reference plane.

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(c) Arm angle: the angle between the arm plane and the reference plane for the ith sub-manipulator, denoted by ψ i . The dual arm-angle parameterization method [26] can also be used to avoid the algorithm singularity. When the three points U1(x1, y1, z1), U2i-1(x2i-1, y2i-1, z2i-1) and U2i+1(x2i+1, y2i+1, z2i+1) are determined, we can solve the Cartesian position of U2i (x2i, y2i, z2i) about a given arm-angle (i.e. ψ i ) according to the following steps:

(1) Determining the function of the reference plane according to the three points U1(x1, y1, z1),

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U2i-1(x2i-1, y2i-1, z2i-1) and U2i+1(x2i+1, y2i+1, z2i+1) The plane function is as follows:

A ( x − x1 ) + B ( y − x1 ) + C ( z − x1 ) = 0

(15)

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where

 A = ( y2i −1 − y1 )( z2i +1 − z1 ) − ( z2i −1 − z1 )( y2i +1 − y1 )   B = ( z2i −1 − z1 )( x2i +1 − x1 ) − ( x2i −1 − x1 )( z2i +1 − z1 )  C = ( x2i −1 − x1 )( y2i +1 − y1 ) − ( y2i −1 − y1 )( x2i +1 − x1 )

(2) Determining the special points of U2i which is located on the reference plane and denoted as % ( x% , y% , z% ) U 2i 2i 2i 2i

There are three conditions for U% 2i . First, it lies in the reference plane, so its coordinates satisfy the plane function, i.e.(15); second, the distance between U% 2i and U2i-1 equals the link length L, i.e. uuuuuuuuv uuuuuuuuuv % U % U % U = L ; third, the distance between U and U equals L too, i.e. U = L . So, the fol2i+1 2i 2i −1 2i 2i

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2 i +1

ACCEPTED MANUSCRIPT lowing system of equations should be satisfied for U% 2i :

 A ( x%2i − x1 ) + B ( y% 2i − x1 ) + C ( z%2i − x1 ) = 0  2 2 2   ( x% 2i − x2i −1 ) + ( y% 2i − y2i −1 ) + ( z%2i − z2i −1 ) = L  2 2 2  ( x% 2i − x2i +1 ) + ( y% 2i − y2i +1 ) + ( z%2i − z2i +1 ) = L

(16)

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Then, the coordinates of U% 2i will be solved from Eq.(16). There are two solutions. They are denoted as U% a2i and U% b2i respectively. In fact, U% a2i and U% b2i are two positions of U2i for ψ i = 0 and ψ i = π respectively.

(3) Determining the desired Cartesian position of U2i for an arbitrary arm-angle ψ i

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The center point of line U2i-1U2i+1 is denoted as Oi ( x0i , y0i , z0i ) , which is determined as follows: x2i +1 + x2i −1   x0i = 2  y2i +1 + y2i −1  (17)  y0i = 2  z2i +1 + z2i −1   z0 i = 2 

For a given parameter ψ i , the Cartesian position of the universal joint U2i is denoted as Uψ2i . The trajectory of Uψ2i with ψ i ∈ [0, 2π ) is a circle whose center is Oi and radius is uuuuuuv % a = ( x% − x ) 2 + ( y% − y )2 + ( z% − z ) 2 Ri = Oi U 2i 2i 0i 2i 0i 2i 0i

(18)

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A frame {xciycizci} is defined to represent the circle. Its x-axis and z-axis are respectively defined % a and O U . The y-axis is then determined according to the right hand rule. The unit as Oi U 2i i 2 i−1 vectors of x-, y- and z-axis of {xciycizci} are denoted as nci , oci and aci respectively. They are

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calculated according to the following equations: uuuuuuv %a Oi U 1 nci = uuuuuu2vi = 2 2 2 %a Oi U ( x%2i − x0i ) + ( y% 2i − y0i ) + ( z%2i − z0i ) 2i

uuuuuuuv Oi U 2i −1 aci = uuuuuuuv = Oi U 2i −1

 x%2i − x0i   y% − y  0i   2i  z%2i − z0i 

1

( x2i −1 − x0i ) + ( y2i −1 − y0i ) + ( z2i −1 − z0i ) 2

2

oci = aci × nci

So, it is easily to get the coordinates of Uψ2i , i.e.:  xψ2i   Ri cosψ i   ψ    y2i  = [ nci oci aci ]  Ri sinψ i   zψ2i   0   

2

 x2i −1 − x0i  y − y   2i −1 0i   z2i −1 − z0i 

(19)

(20)

(21)

(22)

Now, the coordinates of the even-numbered universal joints with respect to arm-angle parameter are determined. Because the arm angle is a variable that can be adjusted, it can be used to optimize

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ACCEPTED MANUSCRIPT the local configuration. The solving process is shown in Fig. 8.

T= ( xT , yT , zT )

X0 backbone %b U 2i

U2 N

Oi

ψi

U 2i+1 U 2i −1

%a U 2i

U1 O0

x ci

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Z0

ψi

Oi

z ci

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U2

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Uψ2i

Uψ2i

U 2i−1

Y0

y ci

k

Fig. 8 The concept of calculating Uψ2i

4.5 Solving the joint angles based on relative position of universal joints When Cartesian coordinates of all the universal joints are determined, the joint angles are then solved using geometry or algebraic methods for each sub-manipulator. As pointed out in Section 2, each sub-manipulator has 4 DOFs, whose rotation axes are arranged in PY-YP layout.

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The universal joints of the ith sub-manipulator are U2i-1 (PY-type universal joint) and U2i (YP-type universal joint). The corresponding rotation angles are

(θ 4i −3 ,θ4i −2 )

and

(θ4i −1 ,θ4i )

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relationship is shown in Fig. 9, and the calculation will be given in the following sections.

T= ( xT , yT , zT )

k

U2N

U 2i −1 (θ 4i −3 , θ 4i − 2 )

frame{4i − 4},{4i − 3}

U 2i+1

frame{4i},{4i + 1}

U 2i (θ 4i −1 , θ 4i )

U 2 (θ 3 ,θ 4 )

frame{4i − 2},{4i − 1}

U1 (θ1 ,θ 2 )

Fig. 9 Relationship between joint coordinate system and joint angles

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ACCEPTED MANUSCRIPT 4.5.1 PY-type universal joints Solving

The homogeneous transformation from (4i-4)th frame to (4i-2)th frame are dependent on the two rotation angles

( i.e. (θ

4 i −3

, θ 4i −2 ) ) of U2i-1 (PY-type universal joint). According to the D-H notation,

we get

U 2i with respect to the (4i-4)th frame, i.e. Uψ2i = [ Lc4i −3c4i − 2

− Ls4i − 2 ]

T

Ls4i −3c4i − 2

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4i − 4

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 c4i −3c4i −2 − c4i −3s4i −2 − s4i −3 Lc4i −3c4i −2  s c − s4i −3s4i −2 c4i −3 Ls4i −3c4i −2  4 i −4  T4i −2 =  4i −3 4i −2 (23) − c4i −2 0 − Ls4i −2   − s4i −2   0 0 1  0  The first three numbers in the fourth column is actually the Cartesian position of the universal joint

(24)

On the other hand, the Cartesian position of the universal joint U 2i relative to the frame {0} is 0

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obtained in Section 4.4.2 and denoted as (22). It is expressed as: Uψ2i =  xψ2i , yψ2i , zψ2i 

T

(25)

For the 1st sub-manipulator, i.e. i=1, Eq. (24) is the same as Eq. (25). So we can easily solve the joints by setting the right side of Eq. (24) equal to that of Eq. (25). That is to say, θ1 and θ 2 corresponding to the 1st sub-manipulator are firstly solved. Its other two joint angles, i.e. θ3 and θ 4 , will be solved under the known values of θ1 and θ 2 (the details will be given in Section 4.5.2). st

nd

0

Uψ2i should be transformed into the frame

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For i>1, the Cartesian position

{4i − 4} . When all

th

the joint angles of the 1 , 2 , …, (i-1) sub-manipulators are solved, the transformation matrix from 0th to (4i-4)th frames (i.e. 0T4i −4 ) is then determined under the solved joint angles. Then, the

Cartesian position of Uψ2i in (4i-4)th frame can be obtained as follows: Uψ2i = ( 0T4i −4 ) ⋅ 0 Uψ2i =  4i −4 xψ2i , 4i −4 yψ2i , 4i −4 zψ2i  Combining equations (24) and (26), −1

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4 i −4

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θ 4i −2

 = arctan 2  −  

4i −4 ψ 2i

z ,± L



θ 4i −3 = arctan 2 

(

x

y

 L cos θ 4i −2 

)

,

y

  L cos (θ 4i − 2 )   4i − 4 ψ 2i

x

)

4i − 4 ψ 2 2i

L

4i −4 ψ 2i

(

) +(

4i − 4 ψ 2 2i

T

    

(26)

(27)

(28)

4.5.2 YP-type universal joints

Similarly, the homogeneous transformation from (4i-2)th frame to (4i)th frame are dependent on the two rotation angles (i.e. θ 4i −1 , θ4i ) of U2i (YP-type universal joint). According to the D-H notation, we get

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ACCEPTED MANUSCRIPT s4 i Lc4i −1c4i   c4i −1c4i − c4i −1s4i s c − s4i −1s4i − c4i −1 Ls4i −1c4i  4 i −2  T4i =  4i −1 4i (29) c4i 0 Ls4i   s4 i   0 0 1   0 The first three numbers in the fourth column is actually the Cartesian position of the universal joint

4i − 2

U 2i +1 = [ Lc4i −1c4i

Ls4i ]

T

Ls4i −1c4i

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U 2i+1 with respect to the (4i-2)th frame, i.e. (30)

On the other hand, the Cartesian position of the universal joint U 2i+1 relative to the frame {0} is obtained in Section 4.4.1 and is expressed as follows: U 2i +1 = [ x2i +1 , y2i +1 , z2i +1 ]

T

(31)

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0

Under the solved joints, 0T4i− 2 is known. So, the Cartesian position of frame can also be obtained as follows:

Uψ2i +1 = ( 0T4i −2 ) ⋅ 0 U 2i +1 =  4i −2 x2i +1 , 4i − 2 y2i +1 , 4i −2 z2i +1  −1

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4 i −2

T

4i − 2

U 2i +1 in (4i-2)th (32)

Combing (30) and (32), the two angles are solved as follows:  θ 4i −1 = arctan 2  4i − 2 z2i +1 / L, ±  

(

4i − 2

2 2  x2i +1 ) + ( 4i − 2 y2i +1 )   L  

5 Simulation study

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4i − 2  4i − 2 y2i +1 x2i +1  θ 4i = arctan 2  ,   L cos (θ 4i −1 ) L cos (θ 4i −1 ) 

(33)

(34)

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5.1 Spatial circular tracking based on a 12-DOF manipulator A space circle is defined by the center O ( xo , yo , zo ) and radius R. The end-effector of manipulator is required to move along a section of the arc of the circle. The starting angle is α 0 and the arc center angle is ζ . We divide the arc into W segment using W + 1 points (numbered form 0 to W ).

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Therefore, for any point p j ( x j , y j , z j )(1 ≤ j ≤ W ) on the circle, the coordinates with the X-Y-Z are as follows:

 x j = xo + R ⋅ sin(α 0 + j ⋅ ∆ζ )   y j = yo + R ⋅ cos(α 0 + j ⋅ ∆ζ )  z j = zo 

(35)

For a practical example, the values of the parameters of the circle are given as R = 0.1500m , ζ ∈ [ 0, π / 2 ] , and O= [ xo , yo , zo ] = [ 0.3000m,0.3000m,0.5000m ] .

Firstly, the joint angles corresponding to each node of the end-effector are solved using the inverse kinematics resolving method presented above. Then, the cubic spline functions are used for interpolating the data in Table 2 to obtain the movement values of all the joints at each sample period. The joint angles of the manipulator at any time between t0 and t f are then planned. The planned

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ACCEPTED MANUSCRIPT the joint angle is shown in Fig. 10, the variation of the position and attitude of the end-effector is shown in Fig. 11. The 3D state in the simulation system is shown in Fig. 12. The configuration varies as Fig. 13. All the results show that the proposed method effectively realize the smooth circular movement of the end-effector of the manipulator. The method is also effect for other types of Cartesian trajectory.

θ1 /o

θ 2 /o

θ3 /o

θ 4 /o

θ5 /o

θ6 /o

θ7 /o

θ8 /o

1 2 3 4 5 6

83.54 79.76 72.10 66.82 65.83 66.16

47.14 44.78 40.58 37.70 35.96 35.34

17.40 17.23 18.44 23.44 33.36 39.39

-49.27 -44.57 -35.33 -32.23 -36.62 -39.17

-19.00 -23.08 -32.40 -38.05 -36.71 -35.27

-15.97 -15.23 -13.12 -9.99 -7.32 -6.20

-72.16 -71.78 -71.43 -72.04 -73.20 -73.60

24.53 26.78 28.86 25.07 15.46 9.57

θ9 /o

θ10 /o

θ11 /o

θ12 /o time

52.14 51.75 51.05 50.99 51.61 51.97

-11.43 -10.49 -9.44 -10.58 -13.71 -15.54

71.36 71.48 71.65 71.50 70.98 70.67

29.40 28.58 27.61 28.51 31.13 32.60

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No.

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Table 2 Multiple nodes corresponding to the space circle

Fig. 11 Position and orientation of the end-effector

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Fig. 10 Joint angle curves

Fig. 12 The 3D state in the simulation system

Fig. 13 The variation of the configuration

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Unit

Φ45×1500

mm

1.5 5

m kg

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20

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Value

DOF /

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Item Envelope size (without control box) Maximum length Total mass Degree of freedom Number of Motors Power Load Accuracy

W kg mm

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≤60 3.5 ≤4.5

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Fig. 14 The 3D model of the 20-DOF manipulator

5.2.2 Simulation design and results

To verify the proposed method for the manipulator working in narrow space, the inspection task

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of the Chinese space station is simulated, i.e. the manipulator is planned to cross into the truss to perform the inspection task. The simulation system is shown in Fig. 15. The desired end-effector position and direction are defined as the movement nodes, which are given according to the mission and working environment. For the truss inspection mission, we design 4 movement nodes. The ith node includes the desired end-effector position

[ xTi , yTi , zTi ]

T

and direc-

tion ki (i=1, …, 4). The corresponding time is denoted as ti. They are given as follows: [ xT1 , yT1 , zT1 ]T = [0.7500m,0.2000m,0.1000m ]T  T   k1 = 0,1/ 2,1/ 2   t1 = 0s

18

(36)

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(37)

[ xT3 , yT3 , zT3 ]T = [ 0.6500m,0.2500m,0.1000m ]T   k3 = [0,1,0] t = 10s  3

(38)

[ xT4 , yT4 , zT4 ]T = [ 0.6500m,0.3500m,0.1000m ]T  T  k4 = [0,1, 0] t = t = 20s f  4

(39)

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[ xT2 , yT2 , zT2 ]T = [ 0.7000m,0.1500m,0.1000m ]T  T  k2 = [0,1, 0] t = 5s  2

The initial state of the simulation is shown as Fig. 16. Under each movement node, the joint an-

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gles are solved using the method proved above. Then, the 3rd spline functions are used to plan the trajectory of each joint. The joint angles of the manipulator at any time between t0 and t f are then generated and used for simulation. The planned joint angles are shown in Fig. 17, and the variation curve of the end-effector position and attitude during the simulation is shown in Fig. 18. The manipulator is controlled to move along the planned joint angles at each step. Some typical 3D state of the manipulator in the simulation system are shown in Fig. 19 and Fig. 20. These results show that the proposed method is feasible for a hyper-redundant manipulator to perform missions in confined

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ulator.

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space. It also reflects the extremely narrow space-crossing capability of the hyper-redundant manip-

Fig. 16 The initial state of the simulation ( t1 = 0s )

Fig. 15 The simulation scene

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Fig. 17 Joint angles during the simulation

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Fig. 18 End-effector position and orientation of the end-effector during the simulation

Fig. 19 Intermediate state ( t3 = 10s )

Fig. 20 Final state

(t f

= 20s )

6 Conclusion The modified fitting method based on the mode function is proposed to solve the inverse kine-

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ACCEPTED MANUSCRIPT matics of the hyper-redundant manipulator. This method can be applied to a variety of configurations, such as those configurations composed of parallel arranged joints, orthogonal arranged joints or universal joints (Only the concrete expressions for the 4-DOF sub-manipulator, i.e. the equations in Section 4.5, are different). Except for the desired position and direction requirement, the calculated joint angles can satisfy the additional tasks, such as avoiding the obstacles, avoiding the singular configuration and avoiding the joint limit. The method transforms the redundant degree of freedom into ad-

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justable parameters which can be controlled. By adjusting these parameters the hyper-redundant manipulator can move along a specific trajectory or cross through a narrow and complex environment. To verify its generality, typical cases of a 12-DOF manipulator and a 20-DOF manipulator are simulated. The results show that the method is efficient for the inverse kinematics resolution and configuration planning for spatial hyper-redundant manipulators in three dimension. The method itself does

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not depend on the movement characteristics of the target. When the hyper-redundant manipulator is mounted on a free floating platform to capture a moving target, this method can also be used, com-

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bining the position-based visual servoing concept. The concept has generality and can be applied for the spatial hyper-redundant manipulators of arbitrary structure. In the future, the relationship between the mode functions and the narrow or multi-obstacle environment will be further studied to provide the theoretical basis for the selection of backbone. New backbone functions to represent the macro-geometry of the hyper-redundant manipulator is another research interest. More, we will also focus on the control method with high stability performance.

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Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 61573116, U1613227), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China（Grant No.51521003), and the Basic Research Program of Shenzhen

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References

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(JCYJ20160427183553203, JCYJ 20150529141408781, CKFW2016033016372515).

[1] M. Shan, JianGuo and E. Gill, "Review and comparison of active space debris capturing and removal methods," Progress in Aerospace Sciences, vol.80, pp. 18-32, 2016. [2] A. Flores-Abad, O. Ma, K. Pham and S. Ulrich, "A Review of Space Robotics Technologies for On-orbit Servicing," Progress in Aerospace Sciences, vol.68, pp. 1-26, 2014. [3] P. Huang, M. Wang, Z. Meng, F. Zhang and Z. Liu, "Attitude Takeover Control for Post-capture of Target Spacecraft using Space Robot," Aerospace Science and Technology, vol.51, pp. 171-180, 2016. [4] P. Huang, D. Wang, Z. Meng, F. Zhang and Z. Liu, "Impact Dynamic Modelling and Adaptive Target Capturing Control for Tethered Space Robots with Uncertainties," IEEE/ASME Transactions on Mechatronics, vol.21, no.5, pp. 2260-2271, 2016. [5] P. Huang, D. Wang, Z. Meng, F. Zhang and J. Guo, "Adaptive Postcapture Backstepping Control for Tumbling Tethered Space Robot–Target Combination," Journal of Guidance, Control and Dynamics, vol.39, no.1, pp. 150-156, 2016.

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Highlights (for review)

(1) Solving mission-oriented inverse kinematics of hyper-redundant manipulators; (2) Segmenting the whole manipulator into 4-DOF sub-manipulators;

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(4) Analytical method assures the computational efficiency.

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(3) Arm-angles and equivalent link length for redundancy parameterization;