Dynamic analysis of a hyper-redundant space manipulator with a complex rope network

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Dynamic analysis of a hyper-redundant space manipulator with a complex rope network

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Shuguang Ma , Bin Liang , Tianshu Wang

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School of Aerospace Engineering, Tsinghua University, Beijing 100084, China b Shenzhen Key Lab of Space Robotic Technology and Telescience, Shenzhen Graduate School, Tsinghua University, Shenzhen, China

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Article history: Received 17 September 2019 Received in revised form 25 December 2019 Accepted 4 February 2020 Available online xxxx Communicated by Chaoyong Li Keywords: On-orbit servicing Space manipulator Rope-driven Dynamic modeling Vibration analysis Computational efficiency

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This paper studies the dynamics of a space manipulator. The space manipulator is designed for precise on-orbit servicing missions in a highly constrained environment. The concerned manipulator has hyperredundant degrees of freedom and moves with a piecewise constant curvature, which enhances the flexibility and controllability. Such manipulator consists of a large number of links and a complex rope network. When the manipulator is driven, the interacting forces between the links and ropes introduce complexity into the dynamic behavior. In terms of dynamic modeling, the manipulator is a very complex system. This paper proposes a dynamic model of the manipulator based on methods of multibody dynamics. The ropes are assumed to be massless and linear elastic. The equations of motion are derived using space operator algebra. The vibration of the manipulator is investigated. The governing equations of the vibration are derived by applying the perturbation method to the proposed dynamic model. The values of the natural frequencies are investigated for the elasticities of the ropes. The proposed dynamic model is also applied in numerical simulation. The explicit fourth-order Runge-Kutta method is utilized to solve the equations of motion numerically. In numerical simulation, an upper bound of the time step is encountered. The value of upper bound is found to be related to the elasticities of the ropes. Such phenomena are studied by analyzing the stability region of the Runge-Kutta methods. Besides, the computational efficiency of the numerical simulation is also limited by the value of upper bound. Two modifications of the dynamic model are introduced to relax the upper bound of the time step. The effects of the modifications are demonstrated by numerical results. © 2020 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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Recently, on-orbit servicing (OOS) has been among the most important topics in space activities [1,2]. OOS missions such as refueling, repairing, transporting and rescuing can increase the lifetime of space vehicles, improve the performance of space vehicles, and decrease launch costs. In some complex or risky OOS missions, space robots can replace astronauts to accomplish the missions. Space robots usually use manipulators to perform operations. In the first decade of this century, several demonstrations of space robots have been achieved [3,4]. Currently, with advances in robotic technologies, space robots have been designed for more precise or more complex OOS missions [5–8]. Meanwhile,

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E-mail address: [email protected] (T. Wang). https://doi.org/10.1016/j.ast.2020.105768 1270-9638/© 2020 Elsevier Masson SAS. All rights reserved.

the structures of space robots and space manipulators are becoming increasingly complicated. Generally, a space robot consists of a floating platform with one or multiple manipulators. The platform is often modeled as a single rigid body. Many of the previous space manipulators, such as the space station remote manipulator system (SSRMS) [9], the European robotic arm (ERA) [10] and the Japanese experimental module remote manipulator system (JERMS) [11], were similar to the ground-based industrial manipulators. They usually consisted of serial rigid links and were driven by electrical motors placed in the joints. The flexible space manipulators were also investigated [12,13]. Typically, the flexible space manipulators consist of two or more flexible links to increase their agility. Besides, tether-net space robot systems were also applied to capture the non-cooperative space debris [14,15]. The previous space manipulators have worked well in previous unmanned OOS missions [3–8]. However, such manipulators may not be able to perform precise operations in highly constrained environments.

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CDPR

cable-driven parallel manipulator

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DAE

differential algebra equation

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DEM

dynamically equivalent manipulator

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DOF

degree of freedom

ERA

European robotic arm

FEM

finite element method

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Nomenclature

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JERMS ODE OOS SOA SSRMS VM

Japanese experimental module remote manipulator system ordinary differential equation on-orbit servicing spatial operator algebra space station remote manipulator system virtual manipulator

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Let us consider a satellite which has failed to deploy its solar panels. Without the solar panels deploying successfully, the satellite will suffer a shortage of electricity and may not be able to work normally. To repair the satellite, we need a space robot to approach the satellite and release the pressing devices of the solar panels. In such a mission, the manipulator should be very thin and flexible to move forward and avoid obstacles in the narrow gap. Given the size limit and lack of flexibility, previous space manipulators may fail to pass through the gap. In such case, rope-driven manipulators may be a better choice. Rope-driven manipulators or cable-driven manipulators are manipulators which are driven by ropes. In rope-driven manipulators, the electrical motors of the ropes are usually placed in the base instead of the joints. The segments of rope-driven manipulators can be relatively smaller than those of industrial manipulators. Moreover, rope-driven manipulators are usually redundant or hyper-redundant, which means they can be quite flexible. Therefore, rope-driven manipulators may be more promising in the panel- deploying mission. The current rope-driven manipulators can be classified into two kinds of manipulators: rope-driven serial manipulators [16–20] and cable-driven parallel manipulators (CDPRs) [21,22]. In this paper, we mainly consider serial manipulators. Among the ropedriven serial manipulators, the so-called “continuum manipulators” or “soft manipulators” [16–18] have been widely studied recently. Soft manipulators usually consist of serial elastic backbones and several driving ropes. The bones are made up of a soft material that can be very flexible. Soft manipulators are often used to perform precise operations in highly constrained environments, for example, minimally invasive surgery. However, soft manipulators may not work well in a space environment. The temperature in space can vary considerably, leading to the failure of a soft material [23]. Soft manipulators may fail to work in space. Although soft manipulators may not be able to work well in space, we can combine the advantages of soft manipulators and previous space manipulators. In this paper, we present a multiplerope-driven space manipulator. The presented space manipulator consists of serial rigid links and a complex rope network where driving ropes and connecting ropes surround the links and joints. The number of links is high, so the space manipulator has hyperredundant degree of freedom (DOF), which enhance its flexibility like soft manipulators. In addition, the space manipulator is designed to have a piecewise constant curvature, which enhances its controllability. Such a space manipulator can be effective in a panel-deploying mission. The details of the structure of the multiple-rope-driven space manipulator are shown in Sec. 2. As the structures of space manipulators become increasingly complicated, the dynamic modeling and analysis of space manipulators become increasingly important. Several methods have been applied to model manipulators [24–28]. The previous space manipulators consist of serial rigid links and are driven in the joints. These manipulators could be studied using approximate models such as the virtual manipulator (VM) [25,26] or the dynamically equivalent manipulator (DEM) [27,28]. For soft manipulators, Cosserat rod theory [17,29] and the piecewise constant curvature

assumption [17,18] are often applied to model the backbones, and the ropes are modeled as linear springs. For CDPRs, the payload is often modeled as a six-DOF rigid body, and the ropes can be modeled using the distributed mass method [21] or the finite element method (FEM) [22]. Given the multiple-rope-driven space manipulator presented in this paper, the model of the manipulator can be divided into two parts: the model of the links and the model of the rope network. The number of links is very high. The previous approximate models [25–28] such as the VM and DEM may not be able to obtain the dynamic characteristics of the links precisely. To model the links more accurately, the methods of multibody dynamics [30–35] can be used. There are several methods of multibody dynamics, such as the Newton-Euler method, the Lagrange method, the R-W method [30], the Kane’s equation [31] and recursive method [32–35]. Among the methods mentioned above, the recursive method is relatively more computationally efficient with a computational complexity of O(N) (The time complexity of recursive method can be even reduced to O(log(N)) with the divide-andconquer algorithm [35]). Since the concerned manipulator has a large number of DOF, the computational efficiency should be considered and the recursive method may be more proper than other methods. Space operator algebra (SOA) [33,34] is one of the common algorithms of recursive method. In this paper, we use SOA to derive equations of motion of the space manipulator. Details of the deriving of the equations can be seen in Sec. 2.1. Meanwhile, the number of ropes in the rope network is also very high. Interacting forces between the links and the ropes may occur in many places. Precise models of the ropes such as the FEM may require too much computational effort. Therefore, a more computationally efficient model of the ropes is required. In this paper, the ropes are assumed to be massless and linear elastic. With this assumption, the driving ropes are modeled as serial linear springs. The connecting ropes are modeled as individual linear torsion springs. The details of the model of the rope network are shown in Sec. 2.2 and Sec. 2.3. Along with the dynamic modeling of the space manipulator, the vibration analysis [36,37] also needs to be taken into consideration. When manipulators are performing operations, extreme precision is required. While, vibration of the manipulator can be caused by orbital perturbations of the platform or the elasticity of the manipulator. The caused vibration may affect the accuracy of the operation and reduce the effectiveness of the manipulator. Several methods have been used to study the vibration of manipulators, such as the assumed mode method [38]. The concerned multiplerope-driven manipulator contains a large number of ropes, and the elasticities of the ropes are the dominant factor of the vibration characteristics. In this paper, we mainly investigate the vibration caused by the rope network. The perturbation method [39] is utilized to derive the mass matrix and stiffness matrix of the governing equations. The derived governing equations can be used to calculate the natural frequencies. The details of vibration analysis are shown in Sec. 3.

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Fig. 1. Structure of the space manipulator.

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Numerical simulation is a common way to study the dynamic behavior of multibody systems. Applying the proposed dynamic model, the derived equations of motion can be rewritten as a group of ordinary differential equations (ODEs) and solved by numerical methods of ODEs [40]. Explicit single-step methods, such as the explicit Runge-Kutta methods, are often used in numerical simulations because less computation is needed in each time step. Using explicit single-step methods, the value of a time step will be constrained in the stability region. One example of the stability region is the so-called “stiff equations”. In this paper, we use the explicit fourth-order Runge-Kutta method in the numerical simulation. An upper bound of the time step is encountered. We find that the upper bound is related to the elasticities of the ropes. Such phenomena are investigated by analyzing the stability region along with the governing equations of vibration. With the upper bound of the time step, the speed of the simulation will be limited. If we want to improve the computational efficiency, modifications need to be made to relax the upper bound of the time step. The details of the analysis of the time step and the improvement of the computational efficiency are shown in Sec. 4. The rest of this paper is organized as follows. In Sec. 2, a dynamic model of the space manipulator is proposed. In Sec. 2.1, the equations of motion for the serial links are derived based on SOA. In Sec. 2.2 and Sec. 2.3, models of the connecting ropes and driving ropes are presented, respectively. In Sec. 2.4, the scheme of the numerical simulation is presented. A numerical demonstration of the proposed dynamic model is also shown. In Sec. 3, a vibration analysis of the space manipulator is performed. In Sec. 3.1, the governing equations of the vibration are derived by applying the perturbation method to the proposed dynamic model. In Sec. 3.2, the values of the natural frequencies are investigated for the elasticities of the connecting ropes and diving ropes. In Sec. 4, the limit and improvement of the computational efficiency are discussed. In Sec. 4.1, the encountered upper bound of the time step is investigated with respect to the stability region of the RungeKutta method. In Sec. 4.2, two modifications are introduced into the proposed dynamic model to improve the computational efficiency. Finally, conclusions are presented in Sec. 5.

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2. Dynamic model of the space manipulator

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As shown in Fig. 1, the space manipulator can be divided into three sections: the base section, the middle section and the tip section. The base section has two main functions. First, the base section connects the manipulator and the platform of the space robot in the structure. Second, the base section contains the driving control box, where 15 driving ropes are pulled by electrical motors. The middle section is the main section of the space manipulator. It consists of 5 groups of links, and each group has 6 rigid links separately. The middle section has 6*5 = 30 segments in total. (In Fig. 1, only two groups of links are shown.) All 30 links are serially connected by rigid universal joints. Each link can rotate in two orthogonal directions. In the middle section, the driving

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Fig. 2. Driving ropes of the space manipulator.

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Fig. 3. Connecting ropes of the space manipulator.

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ropes pass through holes in the links, as shown in Fig. 2. When the driving ropes are pulled by the electrical motors, the links will be driven by the driving ropes. In addition to the driving ropes, there are many connecting ropes tightened along the joints, as shown in Fig. 3. The connecting ropes are used to connect the joints in the same group to make the angles of the joints approximately equal. Then, the whole group will move with a piecewise constant curve, like a continuum robot. The last section of the manipulator is the tip section. The tip section is fixed at the end of the middle section. The payloads of the manipulator, such as the laser cutter and the micro-camera, are placed in the tip section. In terms of dynamic modeling, the manipulator has two features. First, the number of DOF of the manipulator is quite high. The manipulator has 30 links connected by universal joints. The number of DOF of the manipulator is 30*2 = 60. With such a high DOF, the computational efficiency of the dynamic model should be taken into consideration. Second, the rope network in the manipulator must be treated well. There are 50 connecting ropes and 15 driving ropes in total. Interacting forces between the links and the ropes may occur in many places. Therefore, the dynamic model of the rope network should also be efficient. In this section, a dynamic model of the space manipulator is presented. In Sec. 2.1, the equations of motion for the space manipulator are derived based on SOA. The derived equations are written in the form of ODEs. The equations can be solved by an iterative algorithm with a computational complexity of O(N). In Sec. 2.2, the model of the connecting ropes is presented. The connecting ropes are assumed to be linear elastic. A specific connecting rope is modeled as a linear torsion spring. In Sec. 2.3, the model of the driving ropes is presented. The driving ropes are also assumed to be lin-

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Fig. 4. The model of serial links as a rigid multibody system.

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ear elastic. A specific driving rope is modeled as a chain of linear springs, with tension weakening because of friction. In Sec. 2.4, the scheme of the numerical simulation is presented. A numerical example is performed to verify the proposed dynamic model. The validity of the model is demonstrated at a primary level.

(1)

In Eq. (1), M denotes the mass matrix of the system, q denotes the vector of the generalized coordinates of the system, and the terms q¨ and q˙ are defined as q¨ = d2 q/dt 2 and q˙ = dq/dt. In addition, C denotes the coupling term of q and q˙ , and Q denotes the vector of the external forces. Eq. (1) presents the equations of motion for the space manipulator. Actually, Eq. (1) is also suitable for a space robot. Let us assume that the space robot consists of the space manipulator and a 6-DOF rigid platform. Since the mass and inertia of ropes are relatively much lower than those of the links and joints, all the ropes are assumed to be massless. Then, the generalized coordinates in the space robot can be defined as:

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qTplat , qTjoint

T

(2)

In Eq. (2), q plat denotes the generalized coordinates that describe the position and attitude of the platform, and q joint denotes the generalized coordinates that describe the angles of the universal joints. If q in Eq. (1) is defined as Eq. (2), the effects of the ropes can be included in the external force terms Q . In Eq. (1), Q is the sum of the effect of the ropes and the “real” external forces exerted on the space robot:

Q = Qc + Qd + Qe

(3)

In Eq. (3), Q c denotes the effect of the connecting ropes, Q d denotes the effect of the driving ropes, and Q e denotes the effect of the real external forces. The details about Q c and Q d will be shown in Sec. 2.2 and Sec. 2.3. Applying Eq. (3) to Eq. (1), the equations of motion for the space robot can be written as:

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M (q) q¨ + C (˙q, q) = Q c + Q d + Q e

q¨ = M (q)

( Q c + Q d + Q e − C (˙q, q))



As shown in Fig. 4, the serial links of the manipulator can be modeled as a multibody system. Given a multibody system, the equations of motion are often written as:



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(5)

In Eq. (5), the inverse of M (q) can be factorized as:

2.1. The equations of motion for the space manipulator

M (q) q¨ + C (˙q, q) = Q (˙q, q, t )

If Eq. (4) is solved directly, the computational complexity will be O(N3 ). In this paper, the terms in Eq. (4) are derived using SOA. Then Eq. (4) can be transformed as:

(4)

M −1 = I − H T  K

T

L

 −1

I − H TK



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(6)

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Then, Eq. (5) can be solved by a recursive method:

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f or k = N, · · · · · · , 1 P k = φk,k+1 P k++1 φkT,k+1

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+ Mk

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L k = H kT P k H k

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G k = P k H k Lk

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K k−1,k = φk−1,k G k

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G k H kT

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end loop

(7)

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ξ0 = 0

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f or k = N, · · · · · · , 1



ηk = K k,k+1 Q ck + Q dk + Q ek − C (˙q, q)k

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ξk = k,k+1 ξk−1 + ηk

εk = σk − H kT ξk −1

ζk = Lk εk end loop

λ 0 = 0 , χ0 = 0 , f or k = 1, · · · · · · , N

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2.3. The model of the driving ropes

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Let us consider a driving rope passing through holes in the links, as shown in Fig. 2. The contact regions in the holes are simplified as points. The driving rope is divided into a chain of divisions by the contact points. For two neighboring divisions with an angle θ j , the relation between the tensions in the two divisions (T j and T j +1 ) can be written as:

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Fig. 5. The model of connecting ropes.

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Besides, it should be noted that the analytical form of the terms in Eq. (4)-Eq. (6) can be quite complex, especially when the multibody system has a high number of DOF. In this paper, the process of deriving the terms in Eq. (4)-Eq. (6) is omitted here. If readers are interested in the details of the process, the references [33,34] are quite helpful.

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2.2. The model of connecting ropes

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(14)

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  l j = T j / E j · A j · l j

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ε1 ≈ ε0 + rc · θ/lc ε2 ≈ ε0 − rc · θ/lc

l = l + ldrive − l0

(15)

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In Eq. (15), l j is the elongation of the division, E j is the Young’s modulus of the material, A j is the cross-sectional area of the rope and l j is the current length of the division. The current length of the rope l is the sum of l j :



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lj

(16)

Similarly, the total elongation of the rope l is the sum of l j :

l =

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l j

(17)

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The relation between l and l can be written as:

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(18)

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In Eq. (18), l0 is the length of the rope when it is not stretched, and ldrive is the driving law that depends on the corresponding electric motor. Applying Eq. (14)-Eq. (17) to Eq. (18), the forces of all the divisions of the rope can be solved. Then, the forces exerted by the driving ropes can be calculated. Since ldrive is a function of time t, the force T j is a function of ldrive , q and q˙ . The effect of the driving ropes Q d can be written as a function of q, q˙ and t:

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Eq. (10) implies that the connecting rope can be modeled as a linear torsion spring:

Q d = Q d (q, q˙ , t )

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T = k c · θ kc = 2 A c E c rc2 /lc

From Eq. (11), we can see that the elastic coefficient kc is only related to the parameters of the space manipulator. Since kc is not determined by the initial state of the connecting rope, kc will remain constant as the manipulator moves. If the friction of the connecting ropes is taken into consideration, we can add a damping term to Eq. (11):

Eq. (14)-Eq. (19) show that a driving rope is modeled as a chain of linear springs. The tensions of the springs may weaken with the friction in the holes. Similar to the model of the connecting ropes, the presented model of the driving ropes is also relatively more concise. In Sec. 2.1, Sec. 2.2 and Sec. 2.3, the models of the links, the connecting ropes and the driving ropes are presented separately. Applying Eq. (13) and Eq. (19) to Eq. (4), we can obtain the equations of motion for the space manipulator (and the space robot):

T = kc · θ + cc · kc · θ˙

M (q) q¨ + C (˙q, q) = Q c (˙q, q) + Q d (˙q, q, t ) + Q e (t )

The torques in the joint T will become nonzero:

T = A c · E c · ε1 · rc − A c · E c · ε2 · rc

(9)

Applying Eq. (8) to Eq. (9), the relationship between θ and T can be written as:

T = 2 A c E c rc2 θ/lc

(10)

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In Eq. (14), μ j is the damping coefficient and v j is the relative velocity of the rope passing through the hole. The driving ropes are assumed to be linear elastic. The elongation of the jth division depends on the tension T j :

l=

(8)

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T j +1 = T j e   s j = −μ j · sgn v j

Fig. 5 shows the model of a tightened connecting rope between two neighboring joints. Let us assume that the cross-sectional area of the rope is A c , Young’s modulus of the rope is E c , the radius of the joint is rc , the length between the joints is lc , and rc is much smaller than lc . The connecting rope can be divided into two parts by the contact points. The initial state is assumed such that the two neighboring joints rotate with the same angles. The initial strains of both parts of the rope are ε0 (ε0 > 0). Apparently, the initial state is in equilibrium. The torques of the joints are zero. If the two joints have a difference in angles of θ , the equilibrium will be broken. The strains of the two parts of the connecting rope will change as follows:

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s j ·θ j

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(11)

(19)

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(12)

In Eq. (12), the coefficient c c is determined by experimental results or experience. The angles of the joints are included in the generalized coordinates q, so θ can be written as a function of q. Then, the effects of the connecting ropes can also be written as a function of q and q˙ :

(20)

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Eq. (20) is the basis of the dynamic analysis of the space manipulator.

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2.4. The scheme of the numerical simulation

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(13)

Eq. (20) represents the overall dynamic model of the space manipulator. In this subsection, we use a numerical example to verify the validity of the proposed model.

Eq. (12) shows that a connecting rope is modeled as a linear torsion spring.

Let us define x as x = q˙ T , qT . Eq. (20) can be written as ODEs about x:

Q c = Q c (q, q˙ )

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T

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q¨ M (q)−1 ( Q c (˙q, q) + Q d (˙q, q, t ) + Q e (t ) − C (˙q, q)) = q˙ q˙

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the space manipulator can be simulated by applying the scheme shown in Fig. 6.

Eq. (21) presents a first-order differential system. A numerical simulation of the space manipulator can be conducted by solving Eq. (21). In this paper, we use the explicit fourthorder Runge-Kutta method to solve Eq. (21). The scheme of the numerical simulation is shown in Fig. 6. Numerical Example 1. This numerical example considers the space robot as a conservative system and verifies the conservation law of the system. The initial condition is set as follows. The floating platform moves with a nonzero initial velocity (0.1 m/s, 0.2 m/s, 0.3 m/s) and a nonzero initial angular velocity (0.1 rad/s, 0.2 rad/s, 0.3 rad/s). The initial values of q joint are all zero, while the initial velocities of q joint are all 0.01 rad/s. All the connecting ropes and driving ropes are tightened initially. The connecting ropes are assumed to remain tightened in the numerical simulation. No external forces are exerted to the space manipulator. All friction effects are neglected. With such boundary conditions, the space robot can be seen as a conservative system. The translational momentum, angular momentum and mechanical energy (the sum of the kinetic energy and elastic energy) of the system should all remain constant. The parameters of the simulation are set as follows. The time step is 5 × 10−3 s, and the total simulation time is 1 s. To run the numerical example, the total CPU time is approximately 1000 s. The details of the numerical results are shown below. From Fig. 7, we can see that the translational momentum, angular momentum and mechanical energy of the system remain almost constant. More precisely, the maximum relative errors of the translational momentum and translational momentum are both less than 1 × 10 −18 , and the maximum relative error of the mechanical energy is less than 1 × 10 −6 . The conservation law is obeyed. The validity of the proposed dynamic model is verified at a primary level. It should be noted that the simulation time is only one second. This is too short to investigate the proposed model properly. A longer simulation time requires a longer CPU time. For a long-term simulation, the computational efficiency needs to be considered. In Sec. 4, we will discuss the computational efficiency of the proposed model. Numerical Example 1 shows that the proposed model can be applied in a numerical simulation conveniently. More cases of

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(21)

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Fig. 6. The scheme of the numerical simulation.

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3. Vibration analysis of the space manipulator

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This section mainly concerns the vibration analysis of the space manipulator. In the dynamic model presented in Sec. 2, both the connecting ropes and driving ropes are assumed to be elastic. The elasticities of the ropes may increase the vibration of the space manipulator. The increased vibration may affect the accuracy of the operation and reduce the effectiveness of the manipulator. In this paper, the amplitude of the vibration is assumed to be very small compared with the overall motion of the space manipulator. The perturbation method can be used to derive the governing equations of the vibration. This section is divided into two subsections. In Sec. 3.1, the mass matrix and stiff matrix of the governing equation are derived by applying the perturbation method to the proposed model. The governing equations are written as a group of linear ODEs. In Sec. 3.2, the derived governing equations are used to calculate the natural frequencies of the space manipulator. The value of the highest natural frequency is discussed for the elasticities of the ropes.

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3.1. The governing equations of the vibration

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Considering a multibody system with generalized coordinates q, the governing equations of the vibration are often written as:

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M q¨ + K q = 0 v

v

(22)

In Eq. (22), M v is the mass matrix and K v is the stiffness matrix. With the assumption of a small vibration, M v and K v can be derived using the perturbation method. Let us assume that the initial state is in equilibrium. If q has a difference q and q˙ has a difference q˙ , M v and K v can be defined as:

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2 1

v

E = qT K v q 2

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T = q˙ M q˙ T

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(23)

In Eq. (23), T denotes the total kinetic energy, and T is the difference of T . E denotes the total elastic energy, and E is the difference of E. From Eq. (23), we can derive:

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Fig. 7. Results of Numerical Example 1.

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K cv = J θT K c J θ K dv

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∂ E ∂ q2

Kv =

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∂2T M = ∂ q2 v

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(24)

In the space manipulator, the ropes are assumed to be massless. The total kinetic energy of the space robot T is the sum of kinetic energies of the platform and the links. Using Eq. (1), it is easy to verify:

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T=

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M v = M (q)

E = Ec + Ed =

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2

k c · θ +

1 2

kd · l

2

(27)

In Eq. (27), θ and kc are defined in Eq. (11), l is defined in Eq. (17), and kd is the equivalent elastic coefficient of the driving ropes, which can be defined as:

kd =

E d Ad ld

(28)

Applying Eq. (27) to Eq. (24), the stiffness matrix can be written

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as:

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(26)

Eq. (26) shows the derived mass matrix of the vibration. Similar to T , the total elastic energy E is the sum of the elastic energies of all connecting ropes and driving ropes:

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(25)

Applying Eq. (25) to Eq. (24), we can see that the mass matrix in Eq. (22) and the mass matrix in Eq. (4) are the same:

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K v = K cv + K dv

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=

(29)

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J lT K d J l

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In Eq. (29), K cv is the stiffness matrix for the connecting ropes, K dv is the stiffness matrix for the driving ropes, J θ = ∂ θ/∂ q is the Jacobian matrix of θ and q, K c is the diagonal matrix for the vector of kc , J l = ∂ l/∂ q is the Jacobian matrix of l and q, and K d is a diagonal matrix for the vector of kd . Eq. (29) shows the derived mass matrix of the vibration. Applying Eq. (26) and Eq. (29) to Eq. (22), the governing equations of the vibration can be derived. The derived equations present the vibration model of the space manipulator (actually the entire space robot). The vibration model can be used for vibration analysis.

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3.2. Natural frequencies of the space manipulator

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Applying Eq. (26) and Eq. (29), the mass matrix and stiffness matrix can be calculated. Then, the values of the natural frequencies of the space manipulator can be obtained by solving the equation:

   v   K − (2π f )2 M v  = 0

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(30)

In Table 1, we present the values of the lowest ten natural frequencies and the highest ten natural frequencies. From Table 1, we can see that the values of the highest natural frequencies are much higher than those of the lowest nonzero natural frequencies. This is similar to the stiff equations. In stiff equations, the largest eigenvalue of the coefficient matrix mostly affects the computation efficiency. A discussion about the computation efficiency can be found in Sec. 4. In this paper, we mainly investigate the highest natural frequency of the space manipulator. Given a constant mass matrix

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Table 1 Values of the natural frequencies of the space manipulator.

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Lowest ten frequencies (Hz)

0.0

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M v , the value of the highest natural frequency depends on the stiffness matrix K v . K v is the sum of K cv and K dv . If K cv is multiplied by ten while K dv remains the same, the highest natural frequency will increase from 1.22 kHz to 3.84 kHz. If K cv remains the same while K dv is multiplied by ten, the highest natural frequency will increase to just 1.27 kHz. Such phenomena show that the highest natural frequency mainly depends on K cv , i.e., the elasticities of the connecting ropes. From the above discussion, we can see that the natural frequencies of the space manipulator are distributed quite widely. Moreover, the highest natural frequency mainly depends on the elasticities of the connecting ropes. Such conclusions will be used in the next section.

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4. Discussion and improvement of the computational efficiency

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Let us reconsider the numerical example presented in Sec. 2.4. The simulation time is only one second, while the CPU time of the simulation is over 1000 s. The computational efficiency is too low for a real-time simulation. In addition, more than 2/3 of the CPU time is used for calculating the model of the ropes. Since the terms of the ropes need to be calculated at each time step, the only way to speed up the simulation is to increase the value of the time step. A limit of the time step is encountered when we want to increase its value. For the numerical example in Sec. 2.4, the limit is approximately 4 × 10−4 s. If the value of the time step exceeds the limit, the convergence of the numerical results will not be maintained. Moreover, this limit is related to the elasticity of the ropes. If we decrease the elasticity of the ropes, for example, by letting the Young’s modulus be 1/100 of its initial value, the limit of the time step will become 10 times its initial value correspondingly. This section discussed the computational efficiency of the proposed dynamic model. In Sec. 4.1, the limit of the time step is investigated by analyzing the stability region of the explicit RungeKutta methods. The limit of the time step is determined by the highest natural frequency. In Sec. 4.2, two modifications of the dynamic model are introduced to improve the computational efficiency. The limit of the time step is relaxed to less than 1% of its initial value.

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4.1. The stability region

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Fig. 8. Stability regions for the explicit Runge-Kutta methods.

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With the stability region of λh, we need to find the proper value of the coefficient λ to calculate the value of h. For linear ODEs x˙ = Ax, the largest eigenvalue of the matrix A is often chosen as λ [40]. Considering the space manipulator, we can apply the perturbation method to Eq. (21), and the equations of motion can be transferred into ODEs:







v −1

q¨ 0 −M Kv = q˙ I 0



q˙ q



(32)

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When using explicit single-step methods to solve ODEs, the value of the time step will be constrained in the stability region. For an ODE x˙ = f (x, t ), a “test equation” x˙ = λx is often used to analyze the stability of the algorithm. Numerical solutions of the two neighboring steps can be written as xn+1 = E (λh) xn . The region of λh where | E (λh)| < 1 is called the stability region. As mentioned in Sec. 2.4, we use the fourth-order explicit Runge-Kutta method to solve the derived equations of motion. The stability region for the explicit Runge-Kutta methods is shown in Fig. 8 [40]. In Fig. 8, the axes of plot are real axis and imaginary axis of λh. The parameter p denotes the order of method and the curve shows the corresponding stability boundary. We can see in imaginary axis, the boundary of λh for the fourth-order explicit Runge-Kutta method (p = 4) is about:

In Eq. (36), f is the natural frequency of the space robot, and i is the imaginary unit. From Eq. (36) and Eq. (34), the value of λ can be written as:

λh < 2.8i

λ = 2π f max i

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(34)

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Applying Eq. (33) to Eq. (34), Eq. (34) can be rewritten as:

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(35)

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Comparing Eq. (35) with Eq. (30), the eigenvalue of A is related to the natural frequency of the space robot:

α = 2π f i

(36)

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(33)

Then, the coefficient λ should be the largest eigenvalue of A:

   2 v  α M + K v  = 0

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|α I − A | = 0 λ = max {α }

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Eq. (32) can be rewritten as:



−1 q˙ 0 −M v K v ˙x = Ax, x = ,A= q I 0

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(31)

(37)

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Applying Eq. (37) to Eq. (31), the region of time step can be derived:

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f max

(38)

The upper bound of the time step is written as:

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≈

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hmax =

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Eq. (39) shows that the upper bound of the time step depends on the value of the highest natural frequency. Applying Eq. (39), we can explain the phenomena discussed in the beginning of this section. If the elasticity of the system is decreased, the value of the highest natural frequency will decrease correspondingly. The limit of the time step will increase. In addition, the value of the limit of the time step can also be explained. From Table 1, we see that the highest natural frequency of the space robot is 1217 Hz. Applying Eq. (39), we can calculate that the upper bound of the time step is approximately 3.8 × 10 −4 s. The theoretical result of Eq. (39) is consistent with the experimental result of the numerical simulation (4.0 × 10 −4 s). 4.2. Improvement of the computational efficiency

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In Sec. 4.1, we found that the value of the time step will be limited by the highest natural frequency of the space manipulator. If we want to speed up the numerical simulation, the limit of the time step must be relaxed. The highest natural frequency is determined by the mass matrix and stiffness matrix. The governing equations of the vibration are derived from the proposed dynamic model. Modifications of the dynamic model need to be introduced to improve the computational efficiency. The first modification model the connecting ropes as rigid constraints. In Sec. 3.2, we found that the highest natural frequency mainly depends on the elasticities of the connecting ropes. If the connecting ropes are modeled as rigid constraints, the elasticities of the connecting ropes will disappear. The highest natural frequency will decrease. The upper bound of the time step will be relaxed. With rigid constraints for the connecting ropes, the piecewise constant curvature of the space manipulator will be maintained perfectly. The equations of motion can be written as:

∂ R (q) T M (q) q¨ + C (˙q, q) + λ = Qd + Qe ∂q R (q) = 0

will become a constrained rigid multibody system. The limit of the stability region will also disappear. In such case, the value of the time step can increase to 0.1 s, which is enough for a real-time simulation. The effectiveness of the two modifications is presented in the following numerical example.

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(39)

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(40)

Eq. (40) is a group of DAEs. It can be solved by numerical methods for DAE, such as the expansion method or the contraction method [34]. In Eq. (40), the corresponding stiffness matrix is K v = K dv . The value of the highest natural frequency decreases to 223 Hz. The upper bound of the time step increases to 2 × 10 −3 s, which is approximately five times its initial value. The above modification relaxes the limit of the time step by approximately five times. However, such an improvement is still not sufficient. From Numerical Example 1 in Sec. 2.4, we can see that simulation needs to be sped up by at least one thousand times to realize a real-time simulation. The second modification model the driving ropes as rigid ropes. If we assume that the driving ropes are rigid, we can measure the forces of the driving ropes at the base section of the manipulator. Then, we can apply Eq. (14) to obtain the distribution of forces of the driving ropes. The equations of motion can still be written as Eq. (40). With both rigid assumptions of connecting ropes and driving ropes, the elasticities of the ropes will disappear. The space robot

Numerical Example 2. In this numerical example, we assume that both the connecting ropes and the driving ropes are rigid. The initial condition is similar to Numerical Example 1. The boundary condition is different, with constant forces of the driving ropes. We use the expansion method to solve Eq. (40). The simulation time is 100 s, and the time step is 0.1 s. The details of the numerical results are shown below. From Fig. 9, we can see that the translational momentum and angular momentum remain almost constant (Fig. 9a and Fig. 9b). The conservation law of the system is still obeyed, as in Numerical Example 1. Second, the differences of the angles in the same group are very small (Fig. 9c). The piecewise constant curvature is maintained perfectively. A violation of the constraints [41] does not occur in the numerical example. The CPU time of the numerical simulation is 59 s, which shows that a real-time simulation is realized. The performance of the modified dynamic model shows that it is much more efficient than the original model shown in Sec. 2. The speed of the numerical simulation is increased by over one thousand times. The above discussion shows that the modified model is more computationally efficient. However, the original model is more accurate. If the CPU time is not highly constrained, applying the original model can obtain more accurate results. In addition to when the elasticity must be considered (for example, the vibration analysis), the original model is more suitable. Thus, the original model and the modified model have advantages. We can apply the proper model in different cases.

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5. Conclusion

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This paper studies the dynamics of a hyper-redundant space manipulator with a complex rope network. Three main contributions are provided in this paper: First, this paper proposes a dynamic model of the space manipulator. The proposed model has two advantages. 1) The equations of motion are derived by applying the methods of multibody dynamics rather than approximate models, which makes the model more accurate. 2) The ropes are simplified as serial linear springs or linear torsion springs, which makes the model of the rope network more computationally efficient. The validity of the proposed dynamic model is verified in the numerical example presented in Sec. 2.4. Second, this paper conducts a vibration analysis of the space manipulator. The governing equations of the vibration are derived by applying the perturbation method to the dynamic model. The natural frequencies are calculated based on the derived mass matrix and stiff matrix. The highest natural frequency mainly depends on the elasticities of the connecting ropes. Such a conclusion is applied in the discussion regarding the upper bound of the time step. Third, this paper discusses the computational efficiency of the dynamic model. The encountered upper bound of the time step is explained by analyzing the stability region of the Runge-Kutta method. The value of the bound is determined by the highest natural frequency. Modifications are introduced into the dynamic model to relax the limit time step. The connecting ropes and driving ropes are assumed to be rigid rather than elastic ropes. The CPU time of the modified model is decreased to less than 0.1% of the original CPU time. Applying the modified model, a real-time simulation is realized.

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From the discussion in this paper, we can see that the key point in the dynamic modeling of the space manipulator is addressing the ropes. Similar conclusions may be suitable for other manipulators or multibody systems with complex rope networks. The presented approaches of dynamic modeling and vibration analysis may be extended in such cases.

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Declaration of competing interest

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The authors declare that no conflicts of interest exist in this manuscript.

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Acknowledgements

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This work was supported by the National Natural Science Foundation of China (Grant No. 11672145).

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