# Kinematic and Dynamic Characteristics of the Free-Floating Space Manipulator with Free-Swinging Joint Failure

Authors: Qingxuan Jia; Bonan Yuan; Gang Chen; Yingzhuo Fu
Journal: International Journal of Aerospace Engineering
 (2019)

---

## Abstract

For the free-floating space manipulator with free-swinging joint failure, motions among its active joints, passive joints, free-floating base, and end-effector are coupled. It is significant to make clear all motion coupling relationships, which are defined as “kinematic coupling relationships” and “dynamic coupling relationships,” inside the system. With the help of conservation of system momentum, the kinematic model is established, and velocity mapping relation between active joints and passive joints, velocity mapping relation between active joints and base, velocity mapping relation between active joints and end-effector. We establish the dynamic model based on the Lagrange equation, and the system inertia matrix is partitioned according to the distribution of active joints, passive joints, and the base. Then, kinematic and dynamic coupling relationships are explicitly derived, and coupling indexes are defined to depict coupling degree. Motions of a space manipulator with free-swinging joint failure simultaneously satisfy the first-order nonholonomic constraint (kinematic coupling relationships) and the second-order nonholonomic constraint (dynamic coupling relationships), and the manipulator can perform tasks through motion planning and control. Finally, simulation experiments are carried out to verify the existence and correctness of the first-order and second-order nonholonomic constraints and display task execution effects of the space manipulator. This research analyzes the kinematic and dynamic characteristics of the free-floating space manipulator with free-swinging joint failure for the first time. It is the theoretical basis of free-swinging joint failure treatment for a space manipulator.


---

## Body

## 1. Introduction

The application of a space manipulator can not only improve execution efficiency of space tasks but also decrease risks of an astronaut’s EVA. Therefore, space manipulators play an important role in space exploration [1, 2]. However, space manipulators are exposed to hazardous space environment with high temperature difference and intense radiation and usually execute many heavy tasks, like carrying a load whose mass reaches up to 8-25 tons, so parts in joints suffer from severe abrasion, and joints are prone to fail during the long-term service [3]. Subjected to hazardous environment, maintaining fault joints in orbit is accompanied with high costs and risks, so we hope to design a failure treatment strategy to keep using space manipulators after joint failure [4, 5].Wu et al. [6] divided joint failure of space manipulators into two types: a fail-on type with infinite torque and a fail-off type with zero torque, according to joint failure cases in Canadarm1. Afterwards, these two failure types are explicitly defined as locked joint failure and free-swinging joint failure [7, 8]. Locked joint failure means the fault joint is rigidly locked so long as the abnormal operation is detected. Reconfiguring the kinematic and dynamic models, the space manipulator with locked joints is degraded into a new manipulator with less degrees of freedom (DOF) to be used [9, 10]. Free-swinging joints cannot support torque at all, so they swing freely. It is usually caused by motor excitation loss, gear engagement separation, or cracking of the transmission shaft in alternant speed-up and speed-down. However, there are few researches on free-swinging joint failure treatment for space manipulators [11].For the space manipulator with free-swinging joint failure, its DOF is larger than the number of actuated joints, so it is an underactuated system. Referring to the service experience of underactuated manipulators on the ground, we conclude that free-swinging joint failure treatment methods mainly include the following: (1) direct control of underactuated manipulators to execute tasks [12]; (2) with the help of the kinematic and dynamic coupling relationships between active joints and passive joints, free-swinging joints are passively regulated to target angles and are locked. Then, the manipulator with locked joints comes into use [13]. Because free-swinging joints can only be regulated passively, they are called as “passive joints,” and the healthy joints regulating passive joints are called “active joints.” The free-swinging joint failure treatment methods above either depend on motion planning or depend on the control method. No matter which method is used, it is significant to establish kinematic and dynamic models of underactuated manipulators and analyze kinematic and dynamic characteristics to make clear all coupling relationships inside the system at the beginning.Kinematic and dynamic characteristic analysis has been extensively studied around underactuated manipulators on the ground. Aiming at a 2 DOF planar manipulator whose second joint is passive, Nakamura et al. [14] established its dynamic model based on the Hamiltonian function in phase space. Introducing periodic motion into an active joint, he drew Poincaré mapping figures to show the displacement relation between an active joint and a passive joint. Arai et al. [15] derived the unintegrable acceleration coupling relationship between active joints and a passive joint of a 3 DOF underactuated planar manipulator. The acceleration coupling relationship was called as “dynamic coupling relationship,” and it was the second-order nonholonomic constraint inside system. Lai et al. analyzed the dynamic coupling relationships of 2 DOF [16, 17], 3 DOF [18], 4 DOF [19], and n DOF [20] underactuated planar manipulators with the passive first joint and pointed out that when the first joint is passive, the second-order nonholonomic constraint could be integrated into the first-order nonholonomic constraint, so velocities of active joints and a passive joint were coupled. The researches above tell us that for underactuated planar manipulators, if only the first joint is passive, the underactuated system is with the first-order nonholonomic constraint, which represents that velocities of active joints and passive joints are coupled. Otherwise, the second-order nonholonomic constraint exists, and accelerations of active joints and passive joints are coupled.The researches above used many geometric constraints of planar mechanism to simplify the analysis procedure, so they are only appropriate for underactuated planar manipulators. High DOF manipulators have three-dimensional geometric structure, so their kinematic and dynamic characteristic analysis is more complicated. Considering balance of gravity, English and Maciejewski [21] established the velocity mapping relation between joints and end-effector for n DOF underactuated manipulators. Bergerman and Xu [22] derived the second-order nonholonomic constraint which contained centrifugal force, Coriolis force, and gravitational torque and defined the local and global dynamic coupling indexes to depict acceleration transmission efficiency from active joints to passive joints, namely, dynamic coupling degree. Analysis of high DOF underactuated manipulators on the ground must consider gravity effects. However, the space manipulator is free from gravity. Mukherjee and Chen [23] established a dynamic model of the space manipulator with free-swinging joints through the Lagrange equation and derived dynamic coupling relationship between active joints and passive joints based on the Hamiltonian operator. Aiming at the case that active joints are more than passive joints, He et al. [24] pointed out that “passive redundancy” existed when regulating passive joints. Then, they designed a sliding-mode controller with end-effector position/attitude correction, to minimize end-effector disturbance when regulating passive joints. Chen et al. [25] analyzed dynamic coupling relationship between an adjacent active joint and a passive joint for a space manipulator with single free-swinging joint failure and proposed a passive joint regulation strategy based on iteration control. The researches above treated the space manipulator base as fixed. In practice, space manipulators are installed on a free-floating base and motions between joints and the base are coupled. This coupled motion will act on kinematic and dynamic modelling. If it is neglected, we cannot establish valid kinematic and dynamic models and cannot reveal kinematic and dynamic coupling relationships of the space manipulator with free-swinging joint failure.For the free-floating space manipulator, motion between the base and joints satisfies the first-order nonholonomic constraint originally because of momentum conservation without external force [26]. Moreover, according to existing researches about kinematic and dynamic modelling for underactuated manipulators, the second-order nonholonomic constraint also exists in accelerations among active joints, passive joints, and the base. Therefore, the free-floating space manipulator with free-swinging joint failure simultaneously contains the first-order nonholonomic constraint (kinematic coupling relationships) and the second-order nonholonomic constraint (dynamic coupling relationships), and they are independent. However, for underactuated manipulators on the ground, either the first-order nonholonomic constraint or the second-order nonholonomic constraint exists inside systems. Therefore, the kinematic and dynamic characteristic analysis of the free-floating space manipulator with free-swinging joint failure is more complicated than conventional underactuated manipulators, and all kinematic and dynamic coupling relationships need to be studied.Aiming at the free-floating space manipulator with free-swinging joint failure, we establish the kinematic model based on the momentum conservation. Then, the partitioned dynamic equation corresponding to the distribution of active joints, passive joints, and the base are derived based on the Lagrange equation. Then, the kinematic and dynamic coupling relationships among active joints, passive joints, the base and end-effector are derived, and local and global coupling indexes are defined to evaluate the coupling degree. Finally, we briefly introduce motion planning and a control method for the space manipulator with free-swinging joint failure, as the reference of free-swinging joint failure treatment. The traits of this paper mainly include the following:(1)
The general kinematic and dynamic modelling procedure of the free-floating space manipulator with free-swinging joint failure is proposed for the first time
(2)
We point out that the free-floating space manipulator with free-swinging joint failure simultaneously contains the first-order and second-order nonholonomic constraints, and we explicitly reveal all kinematic and dynamic coupling relationships among active joints, passive joints, free-floating base, and end-effector
(3)
We simply design motion planning and the control method based on the kinematic and dynamic coupling relationships for the free-floating space manipulator with free-swinging joint failure, and many tasks can be completed successfullyThe remainder of this paper is as follows: Section II is the kinematic modelling of the free-floating space manipulator with free-swinging joint failure, and the velocity mapping relations between active joints, passive joints, base, and end-effector are derived. Then, we establish the partitioned dynamic model based on the Lagrange equation in Section III. In Section IV, all kinematic and dynamic coupling relationships inside the system are derived, and the local and global coupling indexes are defined. Then, motion planning and control methods for the free-floating space manipulator with free-swinging joint failure to execute tasks are designed. In Section V, simulation experiments are carried out to verify the existence and correctness of the first-order and second-order nonholonomic constraints and to display the execution effects of many tasks of the free-floating space manipulator with free-swinging joint failure. The conclusion is in Section VI.


## 2. Kinematic Modelling of the Free-Floating Space Manipulator with Free-Swinging Joint Failure

In this section, the kinematic model of the free-floating space manipulator with free-swinging joint failure is established and the expression of the conservation of system momentum is derived. Then, the velocity mapping relations between active joints, passive joints, the base, and end-effector are obtained.


### 2.1. Kinematic Modelling

Ann DOF space manipulator with free-swinging joint failure is in Figure 1. In this paper, vectors and matrixes with subscript in the upper left corner are with respect to the corresponding coordinate frame and others with no subscript in the upper left corner are with respect to the inertial coordinate frame. Definitions of frequently used variables are as follow:Figure 1
Kinematic model of then-DOF space manipulator with free-swinging joint failure.C0: base centroid,Ci: centroid of the i-th link,CG: centroid of the whole system,Ji: i-th joint, i=1,2,⋯,n,∑I: inertial coordinate frame built at CG,∑0: base coordinate frame built at C0,∑i: i-th link coordinate frame built at Ci,∑E: end-effector coordinate frame,m0: base mass,mi: mass of the i-th link,M: mass of the whole system,I0∈R3×3: base inertia matrix with respect to C0,Ii∈R3×3: inertia matrix of the i-th link with respect to Ci,pi∈R3×1: position vector of Ji, i=1,2,⋯,n,pe=xe,ye,zeT∈R3×1: position vector of end-effector,Ψe=αe,βe,γeT∈R3×1: attitude vector of end-effector,ri∈R3×1: position vector of Ci, i=1,2,⋯,n,rg∈R3×1: position vector of CG (if ∑I is built at CG, rg≡0),r0∈R3×1: position vector of C0,Ψ0∈R3×1: attitude vector of base,b0∈R3×1: position vector from C0 to J1,θ=θ1,θ2,⋯,θnT∈Rn×1: joint angle vector,θ̇=θ̇1,θ̇2,⋯,θ̇nT∈Rn×1: joint velocity vector,θ¨=θ¨1,θ¨2,⋯,θ¨nT∈Rn×1: joint acceleration vector,q=r0T,Ψ0T,θTT∈Rn+6×1: state variable of system (q̇ and q¨ are velocity and acceleration, respectively),Rij∈R3×3: attitude transformation matrix from ∑j to ∑i,aii∈R3×1: position vector from Ji to Ci (ai is its norm, i=1,2,⋯,n),bii∈R3×1: position vector from Ci to Ji+1 (bi is its norm, i=1,2,⋯,n),zi∈R3×1: unit vector of the i-th joint axis, i=1,2,⋯,n,v0,ω0∈R3×1: velocity of C0/base angular velocity,vi,ωi∈R3×1: velocity of Ci/angular velocity of the i-th link,ve,ωe∈R3×1: end-effector velocity/end-effector angular velocity.For a free-floating space manipulator, linear/angular velocities ofi-th link and end-effector are
(1)viωi=E3−r0i×O3E3v0ω0+JLiJAiθ̇=Jbiv0ω0+Jmiθ̇,(2)veωe=E3−p0e×O3E3v0ω0+JLeJAeθ̇=Jbv0ω0+Jmθ̇,where JLi=z1×ri−p1,z2×ri−p2,⋯,zi×ri−pi,0,⋯,0∈R3×n, JLe=z1×pe−p1,z2×pe−p2,⋯,zi×pe−pi,⋯,zn×pe−pn∈R3×n, JAi=z1,z2,⋯,zi,0,⋯,0∈R3×n, and JAe=z1,z2,⋯,zi,⋯,zn∈R3×n.and where p0e=pe−r0 and r0i=ri−r0 and p0e× and r0i× are the antisymmetric matrixes of p0e and r0i, respectively. E3∈R3×3 is a unit matrix, and O3∈R3×3 is a zero matrix. Jb∈R6×6 denotes base Jacobian matrixes representing the transmission from v0T,ω0TT to viT,ωiTT,veT,ωeTT. Jmi∈R6×n and Jm∈R6×n denote manipulator Jacobian matrixes representing the transmission from θ̇ to viT,ωiTT,veT,ωeTT.Jm consists of n column vectors: Jm=jm1,jm2,⋯,jmn∈R6×n. So Jmθ̇ can be expressed as
(3)Jmθ̇=∑i=1njmiθ̇i.As can be seen from equation (3), jmi depicts the contribution of i-th joint velocity θ̇i to veTωeTT. If selecting the rest of the healthy joints as active joints, we can obtain a serial number set of passive joints as SP=sPi∀sPi∈1,2,⋯,n,i=1,2,⋯,p and serial number set of active joints as SA=sAj∀sAj∈1,2,⋯,n,j=1,2,⋯,a. So the passive joint velocity is θ̇P=θ̇P1,θ̇P2,⋯,θ̇PpT=θ̇sP1,θ̇sP2,⋯,θ̇sPpT, and the active joint velocity is θ̇A=θ̇A1,θ̇A2,⋯,θ̇AaT=θ̇sA1,θ̇sA2,⋯,θ̇sAaT. With the help of the commutative law of addition, we express equation (3) as
(4)Jmθ̇=∑j=1pjmsPjθ̇Pj+∑k=1ajmsAkθ̇Ak=JmPθ̇P+JmAθ̇A=JmPJmAθ̇Pθ̇A,where jmsPj,j=1,2,⋯,p, is the column vectors corresponding to passive joints, and they make up JmP∈R6×p which represents transmission from θ̇P to veTωeTT. jmsAk,k=1,2,⋯,a, is the column vectors corresponding to active joints, and they make up JmA∈R6×a which represents transmission from θ̇A to veTωeTT. If making θ̇b=v0Tω0TT∈R6×1, equation (2) becomes
(5)veωe=Jbθ̇b+JmPθ̇P+JmAθ̇A=JbJmPJmAθ̇bθ̇Pθ̇A.If we know the desiredveT,ωeTT, we can acquire θ̇b, θ̇P, and θ̇A from equation (5). However, the base and passive joints cannot actively supply θ̇b and θ̇P because of lacking actuators. All motions of passive joints, base, and end-effector can only be actuated by active joints. In the next subsection, we are going to derive specific velocity mapping relations among active joints, passive joints, base, and end-effector based on conservation of system momentum.


### 2.2. Velocity Mapping Relations between Active Joints, Passive Joints, Base, and End-Effector Based on Momentum Conservation

Linear momentum and angular momentum of the whole system are(6)P=∑i=0nmivi=Mv0−Mr0×ω0+JLMθ̇,(7)L=∑i=0nIiωi+miri×vi=I0+∑i=1nIi−miri×r0i×ω0+JAMθ̇,where JLM=∑i=1nmiJLi∈R3×n represents the contribution of θ̇ to linear momentum P and JAM=∑i=1nIiJAi+miri×JLi∈R3×n represents the contribution of θ̇ to angular momentum L. JLM and JAM can be expressed as JLM=jLM1,jLM2,⋯,jLMn and JAM=jAM1,jAM2,⋯,jAMn, respectively, and we can find column vectors corresponding to passive joints and active joints to make up matrixes
(8)JLMA=jLMsA1jLMsA2⋯jLMsAa∈R3×a,JAMA=jAMsA1jAMsA2⋯jAMsAa∈R3×a,JLMP=jLMsP1jLMsP2⋯jLMsPp∈R3×p,JAMP=jAMsP1jAMsP2⋯jAMsPp∈R3×p.Equations (6) and (7) can be expressed in matrix form
(9)PL=ME3−Mr0×O3Iωv0ω0+JLMAJAMAθ̇A+JLMPJAMPθ̇P,where Iω=I0+∑i=1nIi−miri×r0i×∈R3×3. Without the external force, system momentum is conservative. Setting the initial momentum is zero, so
(10)ME3−Mr0×O3Iωv0ω0+JTMLAJAIAAθ̇A+JTMLPJAIAPθ̇P=00.From equation (10), we know that θ̇A, θ̇P, and θ̇b are coupled with each other, but we care more about the transmission from θ̇A to θ̇P and θ̇b. From equation (10), we have
(11)JTMLAJAIAAθ̇A+ME3−Mr0×JTMLPO3IωJAIAPθ̇bθ̇P=00.Making(12)HA=JTMLAJAIAA∈R6×a,HbP=ME3−Mr0×JTMLPO3IωJAIAP∈R6×p+6,Equation (11) becomes
(13)HAθ̇A+HbPθ̇bθ̇P=00.Then, we can obtain transmission fromθ̇A to θ̇P and θ̇b:
(14)θ̇bθ̇P=JbPAθ̇A+E6+p−HbP+HbPξ.JbPA=−HbP+HA∈R6+p×a represents the Jacobian matrix from θ̇A to θ̇P and θ̇b. E6+p−HbP+HbPξ is the null space of HbP, and ξ is an arbitrary vector. Because 6<6+p, when HbP is full rank, the null space exists. In particular, making ξ=0, we can obtain transmission from θ̇A to θ̇P and from θ̇A to θ̇b:
(15)θ̇b=JbAθ̇A,θ̇P=JPAθ̇A.JbA∈R6×a is the first six rows of vectors of JbPA, and JPA∈Rp×a is the last p rows of vectors of JbPA. Substituting equation (15) into equation (5), we obtain the transmission from θ̇A to veT,ωeTT:
(16)veωe=JbPJbPA+JmAθ̇A=JeAθ̇A,where JbP=JbJmP∈R6×p+6, and JeA=JbPJbPA+JmA∈R6×a is the Jacobian matrix representing transmission from θ̇A to veT,ωeTT. Equations (15) and (16) can be reduced to the following constraints:
(17)h1θ̇P,θ̇A,t=0,h2θ̇b,θ̇A,t=0,h3ve,ωe,θ̇A,t=0.Obviously, for space manipulators, there is no definite relation betweenθA and θP,θb,pe,Ψe, which means we cannot find constraints like h10θP,θA,t=0, h20θb,θA,t=0, and h30pe,Ψe,θA,t=0 making h1=dh10/dt, h2=dh20/dt, and h3=dh30/dt, so constraints in equation (17) are nonholonomic. Because equation (17) includes constraints among the first-order derivatives of θb,θA,θP,pe,Ψe, it is called as the first-order nonholonomic constraint. The first-order nonholonomic constraint is unique for the free-floating space manipulator with free-swinging joint failure because its momentum is conservative.


## 2.1. Kinematic Modelling

Ann DOF space manipulator with free-swinging joint failure is in Figure 1. In this paper, vectors and matrixes with subscript in the upper left corner are with respect to the corresponding coordinate frame and others with no subscript in the upper left corner are with respect to the inertial coordinate frame. Definitions of frequently used variables are as follow:Figure 1
Kinematic model of then-DOF space manipulator with free-swinging joint failure.C0: base centroid,Ci: centroid of the i-th link,CG: centroid of the whole system,Ji: i-th joint, i=1,2,⋯,n,∑I: inertial coordinate frame built at CG,∑0: base coordinate frame built at C0,∑i: i-th link coordinate frame built at Ci,∑E: end-effector coordinate frame,m0: base mass,mi: mass of the i-th link,M: mass of the whole system,I0∈R3×3: base inertia matrix with respect to C0,Ii∈R3×3: inertia matrix of the i-th link with respect to Ci,pi∈R3×1: position vector of Ji, i=1,2,⋯,n,pe=xe,ye,zeT∈R3×1: position vector of end-effector,Ψe=αe,βe,γeT∈R3×1: attitude vector of end-effector,ri∈R3×1: position vector of Ci, i=1,2,⋯,n,rg∈R3×1: position vector of CG (if ∑I is built at CG, rg≡0),r0∈R3×1: position vector of C0,Ψ0∈R3×1: attitude vector of base,b0∈R3×1: position vector from C0 to J1,θ=θ1,θ2,⋯,θnT∈Rn×1: joint angle vector,θ̇=θ̇1,θ̇2,⋯,θ̇nT∈Rn×1: joint velocity vector,θ¨=θ¨1,θ¨2,⋯,θ¨nT∈Rn×1: joint acceleration vector,q=r0T,Ψ0T,θTT∈Rn+6×1: state variable of system (q̇ and q¨ are velocity and acceleration, respectively),Rij∈R3×3: attitude transformation matrix from ∑j to ∑i,aii∈R3×1: position vector from Ji to Ci (ai is its norm, i=1,2,⋯,n),bii∈R3×1: position vector from Ci to Ji+1 (bi is its norm, i=1,2,⋯,n),zi∈R3×1: unit vector of the i-th joint axis, i=1,2,⋯,n,v0,ω0∈R3×1: velocity of C0/base angular velocity,vi,ωi∈R3×1: velocity of Ci/angular velocity of the i-th link,ve,ωe∈R3×1: end-effector velocity/end-effector angular velocity.For a free-floating space manipulator, linear/angular velocities ofi-th link and end-effector are
(1)viωi=E3−r0i×O3E3v0ω0+JLiJAiθ̇=Jbiv0ω0+Jmiθ̇,(2)veωe=E3−p0e×O3E3v0ω0+JLeJAeθ̇=Jbv0ω0+Jmθ̇,where JLi=z1×ri−p1,z2×ri−p2,⋯,zi×ri−pi,0,⋯,0∈R3×n, JLe=z1×pe−p1,z2×pe−p2,⋯,zi×pe−pi,⋯,zn×pe−pn∈R3×n, JAi=z1,z2,⋯,zi,0,⋯,0∈R3×n, and JAe=z1,z2,⋯,zi,⋯,zn∈R3×n.and where p0e=pe−r0 and r0i=ri−r0 and p0e× and r0i× are the antisymmetric matrixes of p0e and r0i, respectively. E3∈R3×3 is a unit matrix, and O3∈R3×3 is a zero matrix. Jb∈R6×6 denotes base Jacobian matrixes representing the transmission from v0T,ω0TT to viT,ωiTT,veT,ωeTT. Jmi∈R6×n and Jm∈R6×n denote manipulator Jacobian matrixes representing the transmission from θ̇ to viT,ωiTT,veT,ωeTT.Jm consists of n column vectors: Jm=jm1,jm2,⋯,jmn∈R6×n. So Jmθ̇ can be expressed as
(3)Jmθ̇=∑i=1njmiθ̇i.As can be seen from equation (3), jmi depicts the contribution of i-th joint velocity θ̇i to veTωeTT. If selecting the rest of the healthy joints as active joints, we can obtain a serial number set of passive joints as SP=sPi∀sPi∈1,2,⋯,n,i=1,2,⋯,p and serial number set of active joints as SA=sAj∀sAj∈1,2,⋯,n,j=1,2,⋯,a. So the passive joint velocity is θ̇P=θ̇P1,θ̇P2,⋯,θ̇PpT=θ̇sP1,θ̇sP2,⋯,θ̇sPpT, and the active joint velocity is θ̇A=θ̇A1,θ̇A2,⋯,θ̇AaT=θ̇sA1,θ̇sA2,⋯,θ̇sAaT. With the help of the commutative law of addition, we express equation (3) as
(4)Jmθ̇=∑j=1pjmsPjθ̇Pj+∑k=1ajmsAkθ̇Ak=JmPθ̇P+JmAθ̇A=JmPJmAθ̇Pθ̇A,where jmsPj,j=1,2,⋯,p, is the column vectors corresponding to passive joints, and they make up JmP∈R6×p which represents transmission from θ̇P to veTωeTT. jmsAk,k=1,2,⋯,a, is the column vectors corresponding to active joints, and they make up JmA∈R6×a which represents transmission from θ̇A to veTωeTT. If making θ̇b=v0Tω0TT∈R6×1, equation (2) becomes
(5)veωe=Jbθ̇b+JmPθ̇P+JmAθ̇A=JbJmPJmAθ̇bθ̇Pθ̇A.If we know the desiredveT,ωeTT, we can acquire θ̇b, θ̇P, and θ̇A from equation (5). However, the base and passive joints cannot actively supply θ̇b and θ̇P because of lacking actuators. All motions of passive joints, base, and end-effector can only be actuated by active joints. In the next subsection, we are going to derive specific velocity mapping relations among active joints, passive joints, base, and end-effector based on conservation of system momentum.


## 2.2. Velocity Mapping Relations between Active Joints, Passive Joints, Base, and End-Effector Based on Momentum Conservation

Linear momentum and angular momentum of the whole system are(6)P=∑i=0nmivi=Mv0−Mr0×ω0+JLMθ̇,(7)L=∑i=0nIiωi+miri×vi=I0+∑i=1nIi−miri×r0i×ω0+JAMθ̇,where JLM=∑i=1nmiJLi∈R3×n represents the contribution of θ̇ to linear momentum P and JAM=∑i=1nIiJAi+miri×JLi∈R3×n represents the contribution of θ̇ to angular momentum L. JLM and JAM can be expressed as JLM=jLM1,jLM2,⋯,jLMn and JAM=jAM1,jAM2,⋯,jAMn, respectively, and we can find column vectors corresponding to passive joints and active joints to make up matrixes
(8)JLMA=jLMsA1jLMsA2⋯jLMsAa∈R3×a,JAMA=jAMsA1jAMsA2⋯jAMsAa∈R3×a,JLMP=jLMsP1jLMsP2⋯jLMsPp∈R3×p,JAMP=jAMsP1jAMsP2⋯jAMsPp∈R3×p.Equations (6) and (7) can be expressed in matrix form
(9)PL=ME3−Mr0×O3Iωv0ω0+JLMAJAMAθ̇A+JLMPJAMPθ̇P,where Iω=I0+∑i=1nIi−miri×r0i×∈R3×3. Without the external force, system momentum is conservative. Setting the initial momentum is zero, so
(10)ME3−Mr0×O3Iωv0ω0+JTMLAJAIAAθ̇A+JTMLPJAIAPθ̇P=00.From equation (10), we know that θ̇A, θ̇P, and θ̇b are coupled with each other, but we care more about the transmission from θ̇A to θ̇P and θ̇b. From equation (10), we have
(11)JTMLAJAIAAθ̇A+ME3−Mr0×JTMLPO3IωJAIAPθ̇bθ̇P=00.Making(12)HA=JTMLAJAIAA∈R6×a,HbP=ME3−Mr0×JTMLPO3IωJAIAP∈R6×p+6,Equation (11) becomes
(13)HAθ̇A+HbPθ̇bθ̇P=00.Then, we can obtain transmission fromθ̇A to θ̇P and θ̇b:
(14)θ̇bθ̇P=JbPAθ̇A+E6+p−HbP+HbPξ.JbPA=−HbP+HA∈R6+p×a represents the Jacobian matrix from θ̇A to θ̇P and θ̇b. E6+p−HbP+HbPξ is the null space of HbP, and ξ is an arbitrary vector. Because 6<6+p, when HbP is full rank, the null space exists. In particular, making ξ=0, we can obtain transmission from θ̇A to θ̇P and from θ̇A to θ̇b:
(15)θ̇b=JbAθ̇A,θ̇P=JPAθ̇A.JbA∈R6×a is the first six rows of vectors of JbPA, and JPA∈Rp×a is the last p rows of vectors of JbPA. Substituting equation (15) into equation (5), we obtain the transmission from θ̇A to veT,ωeTT:
(16)veωe=JbPJbPA+JmAθ̇A=JeAθ̇A,where JbP=JbJmP∈R6×p+6, and JeA=JbPJbPA+JmA∈R6×a is the Jacobian matrix representing transmission from θ̇A to veT,ωeTT. Equations (15) and (16) can be reduced to the following constraints:
(17)h1θ̇P,θ̇A,t=0,h2θ̇b,θ̇A,t=0,h3ve,ωe,θ̇A,t=0.Obviously, for space manipulators, there is no definite relation betweenθA and θP,θb,pe,Ψe, which means we cannot find constraints like h10θP,θA,t=0, h20θb,θA,t=0, and h30pe,Ψe,θA,t=0 making h1=dh10/dt, h2=dh20/dt, and h3=dh30/dt, so constraints in equation (17) are nonholonomic. Because equation (17) includes constraints among the first-order derivatives of θb,θA,θP,pe,Ψe, it is called as the first-order nonholonomic constraint. The first-order nonholonomic constraint is unique for the free-floating space manipulator with free-swinging joint failure because its momentum is conservative.


## 3. Dynamic Modelling of the Free-Floating Space Manipulator with Free-Swinging Joint Failure

Establishment of a dynamic model is the basis of dynamic coupling relationship analysis and controller design, so the dynamic model of the space manipulator with free-swinging joint failure is established in this section.


### 3.1. Kinetic Energy of System

The Lagrange function of the space manipulator with free-swinging joint failure is as follows:(18)L=K−E,where K is system kinetic energy and E is the system potential energy. The space manipulator is free from gravity, and we take no account of joint and link flexibility, so E=0. The Lagrange function equals to total kinetic energy:
(19)L=K=∑i=0n12miviTvi+12ωiTIiωi=12θ̇bTθ̇TMbMbmMmbMmθ̇bθ̇=12θ̇bTθ̇TMθ̇bθ̇,where
(20)Mb=ME3−Mr0×−Mr0×T∑i=0nmir0i×Tr0i×+Ii∈R6×6,Mbm=∑i=0nmiJLi∑i=0n−mir0i×TJLi+IiJAi∈R6×n,Mmb=MbmT=∑i=0nmiJLiT∑i=0n−miJLiTr0i×+JAiTIi∈Rn×6,Mm=∑i=0nmiJLiTJLi+JAiTIiJAi∈Rn×n,M=MbMbmMmbMm∈Rn+6×n+6.Mb denotes the base inertia matrix. Mm denotes the manipulator inertia matrix. Mbm and Mmb are the coupling inertia matrixes between base and manipulator. M is the inertia matrix of the whole system. Expanding equation (19), we have
(21)L=12θ̇bTMbθ̇b+θ̇TMmbθ̇b+θ̇bTMbmθ̇+θ̇TMmθ̇.We can know thatMbm, Mmb, and Mm are relative to joint velocity, so we can utilize commutative law of addition to divide these matrixes into
(22)θ̇TMmb=θ̇1mmb_row1+θ̇2mmb_row2+⋯+θ̇nmmb_rown,Mbmθ̇=mbm_col1θ̇1+mbm_col2θ̇2+⋯+mbm_colnθ̇n,θ̇TMmθ̇=∑i=1n∑j=1nθ̇immijθ̇j.The row vectormmb_rowi of Mmb, the column vector mbm_coli of Mbm, and the element mmij of Mm represent the contribution of the corresponding joint velocity to kinetic energy. Then, equation (21) can be expressed as
(23)L=12θ̇bTMbθ̇b+θ̇PTθ̇ATMmbPMmbAθ̇b+θ̇bTMbmPMbmAθ̇Pθ̇A+θ̇PTθ̇ATΜmPPΜmPAΜmAPΜmAAθ̇Pθ̇A,where
(24)MmbP=mmb_rowSP1mmb_rowSP2⋮mmb_rowSPp∈Rp×6,MmbA=mmb_rowSA1mmb_rowSA2⋮mmb_rowSAa∈Ra×6,MbmP=mbm_colSP1mbm_colSP2⋯mbm_colSPp∈R6×p,MbmA=mbm_colSA1mbm_colSA2⋯mbm_colSAa∈R6×a,MmPP=mmSP1SP1mmSP1SP2⋯mmSP1SPpmmSP2SP1mmSP2SP2⋯mmSP2SPp⋮⋮⋱⋮mmSPpSP1mmSPpSP2⋯mmSPpSPp∈Rp×p,MmAA=mmSA1SA1mmSA1SA2⋯mmSA1SAammSA2SA1mmSA2SA2⋯mmSA2SAa⋮⋮⋱⋮mmSAaSA1mmSAaSA2⋯mmSAaSAa∈Ra×a,MmAP=mmSA1SP1mmSA1SP2⋯mmSA1SPpmmSA2SP1mmSA2SP2⋯mmSA2SPp⋮⋮⋱⋮mmSAaSP1mmSAaSP2⋯mmSAaSPp∈Ra×p,MmPA=mmSP1SA1mmSP1SA2⋯mmSP1SAammSP2SA1mmSP2SA2⋯mmSP2SAa⋮⋮⋱⋮mmSPpSA1mmSPpSA2⋯mmSPpSAa∈Rp×a.M becomes
(25)M=MbMbmPMbmAMmbPMmPPMmPAMmbAMmAPMmAA.And the Lagrange function of the system is(26)L=12θ̇bTθ̇PTθ̇ATMbMbmPMbmAMmbPMmPPMmPAMmbAMmAPMmAAθ̇bθ̇Pθ̇A=12θ̇bTMbθ̇b+θ̇PTMmbPθ̇b+θ̇ATMmbAθ̇b+θ̇bTMbmPθ̇P+θ̇PTMmPPθ̇P+θ̇ATMmAPθ̇P+θ̇bTMbmAθ̇A+θ̇PTMmPAθ̇A+θ̇ATMmAAθ̇A.


### 3.2. Lagrange Equation of the System

The Lagrange equation is(27)ddt∂L∂θ̇b−∂L∂θb=τb,ddt∂L∂θ̇P−∂L∂θP=τP,ddt∂L∂θ̇A−∂L∂θA=τA.Substituting equation (26) into equation (27), and calculating derivatives, we have
(28)ddt∂L∂θ̇b=Mbθ¨b+MbmPθ¨P+MbmAθ¨A+∑i=16+n∂Mb∂qiq̇i⋅θ̇b+∑i=16+n∂MbmP∂qiq̇i⋅θ̇P+∑i=16+n∂MbmA∂qiq̇i⋅θ̇A,ddt∂L∂θ̇P=MmbPθ¨b+MmPPθ¨P+MmPAθ¨A+∑i=16+n∂MmbP∂qiq̇i⋅θ̇b+∑i=16+n∂MmPP∂qiq̇i⋅θ̇P+∑i=16+n∂MmPA∂qiq̇i⋅θ̇A,ddt∂L∂θ̇A=MmbAθ¨b+MmAPθ¨P+MmAAθ¨A+∑i=16+n∂MmbA∂qiq̇i⋅θ̇b+∑i=16+n∂MmAP∂qiq̇i⋅θ̇P+∑i=16+n∂MmAA∂qiq̇i⋅θ̇A.Makingcτb_1=∑i=16+n∂Mb/∂qiq̇i⋅θ̇b+∑i=16+n∂MbmP/∂qiq̇i⋅θ̇P+∑i=16+n∂MbmA/∂qiq̇i⋅θ̇A, cτP_1=∑i=16+n∂MmbP/∂qiq̇i⋅θ̇b+∑i=16+n∂MmPP/∂qiq̇i⋅θ̇P+∑i=16+n∂MmPA/∂qiq̇i⋅θ̇A, and cτA_1=∑i=16+n∂MmbA/∂qiq̇i⋅θ̇b+∑i=16+n∂MmAP/∂qiq̇i⋅θ̇P+∑i=16+n∂MmAA/∂qiq̇i⋅θ̇A. And cτb_2=∂L/∂θb, cτP_2=∂L/∂θP, and cτA_2=∂L/∂θA.cτb_1,cτb_2∈R6×1, cτP_1,cτP_2∈Rp×1, and cτA_1,cτA_2∈Ra×1are centrifugal forces and Coriolis forces acting on the base, passive joints, and active joints. So equation (28) becomes
(29)MbMbmPMbmAMmbPMmPPMmPAMmbAMmAPMmAAθ¨bθ¨Pθ¨A+cbcPcA=τbτPτA=00τA,where
(30)cbcPcA=cτb_1−cτb_2cτP_1−cτP_2cτA_1−cτA_2∈Rn+6×1.Passive joints and base cannot supply torque, soτb≡0 and τP≡0. If we know the motions of active joints, passive joints, and the base, the required active joint torque τA can be calculated to actuate the space manipulator to arrive at the target state.


## 3.1. Kinetic Energy of System

The Lagrange function of the space manipulator with free-swinging joint failure is as follows:(18)L=K−E,where K is system kinetic energy and E is the system potential energy. The space manipulator is free from gravity, and we take no account of joint and link flexibility, so E=0. The Lagrange function equals to total kinetic energy:
(19)L=K=∑i=0n12miviTvi+12ωiTIiωi=12θ̇bTθ̇TMbMbmMmbMmθ̇bθ̇=12θ̇bTθ̇TMθ̇bθ̇,where
(20)Mb=ME3−Mr0×−Mr0×T∑i=0nmir0i×Tr0i×+Ii∈R6×6,Mbm=∑i=0nmiJLi∑i=0n−mir0i×TJLi+IiJAi∈R6×n,Mmb=MbmT=∑i=0nmiJLiT∑i=0n−miJLiTr0i×+JAiTIi∈Rn×6,Mm=∑i=0nmiJLiTJLi+JAiTIiJAi∈Rn×n,M=MbMbmMmbMm∈Rn+6×n+6.Mb denotes the base inertia matrix. Mm denotes the manipulator inertia matrix. Mbm and Mmb are the coupling inertia matrixes between base and manipulator. M is the inertia matrix of the whole system. Expanding equation (19), we have
(21)L=12θ̇bTMbθ̇b+θ̇TMmbθ̇b+θ̇bTMbmθ̇+θ̇TMmθ̇.We can know thatMbm, Mmb, and Mm are relative to joint velocity, so we can utilize commutative law of addition to divide these matrixes into
(22)θ̇TMmb=θ̇1mmb_row1+θ̇2mmb_row2+⋯+θ̇nmmb_rown,Mbmθ̇=mbm_col1θ̇1+mbm_col2θ̇2+⋯+mbm_colnθ̇n,θ̇TMmθ̇=∑i=1n∑j=1nθ̇immijθ̇j.The row vectormmb_rowi of Mmb, the column vector mbm_coli of Mbm, and the element mmij of Mm represent the contribution of the corresponding joint velocity to kinetic energy. Then, equation (21) can be expressed as
(23)L=12θ̇bTMbθ̇b+θ̇PTθ̇ATMmbPMmbAθ̇b+θ̇bTMbmPMbmAθ̇Pθ̇A+θ̇PTθ̇ATΜmPPΜmPAΜmAPΜmAAθ̇Pθ̇A,where
(24)MmbP=mmb_rowSP1mmb_rowSP2⋮mmb_rowSPp∈Rp×6,MmbA=mmb_rowSA1mmb_rowSA2⋮mmb_rowSAa∈Ra×6,MbmP=mbm_colSP1mbm_colSP2⋯mbm_colSPp∈R6×p,MbmA=mbm_colSA1mbm_colSA2⋯mbm_colSAa∈R6×a,MmPP=mmSP1SP1mmSP1SP2⋯mmSP1SPpmmSP2SP1mmSP2SP2⋯mmSP2SPp⋮⋮⋱⋮mmSPpSP1mmSPpSP2⋯mmSPpSPp∈Rp×p,MmAA=mmSA1SA1mmSA1SA2⋯mmSA1SAammSA2SA1mmSA2SA2⋯mmSA2SAa⋮⋮⋱⋮mmSAaSA1mmSAaSA2⋯mmSAaSAa∈Ra×a,MmAP=mmSA1SP1mmSA1SP2⋯mmSA1SPpmmSA2SP1mmSA2SP2⋯mmSA2SPp⋮⋮⋱⋮mmSAaSP1mmSAaSP2⋯mmSAaSPp∈Ra×p,MmPA=mmSP1SA1mmSP1SA2⋯mmSP1SAammSP2SA1mmSP2SA2⋯mmSP2SAa⋮⋮⋱⋮mmSPpSA1mmSPpSA2⋯mmSPpSAa∈Rp×a.M becomes
(25)M=MbMbmPMbmAMmbPMmPPMmPAMmbAMmAPMmAA.And the Lagrange function of the system is(26)L=12θ̇bTθ̇PTθ̇ATMbMbmPMbmAMmbPMmPPMmPAMmbAMmAPMmAAθ̇bθ̇Pθ̇A=12θ̇bTMbθ̇b+θ̇PTMmbPθ̇b+θ̇ATMmbAθ̇b+θ̇bTMbmPθ̇P+θ̇PTMmPPθ̇P+θ̇ATMmAPθ̇P+θ̇bTMbmAθ̇A+θ̇PTMmPAθ̇A+θ̇ATMmAAθ̇A.


## 3.2. Lagrange Equation of the System

The Lagrange equation is(27)ddt∂L∂θ̇b−∂L∂θb=τb,ddt∂L∂θ̇P−∂L∂θP=τP,ddt∂L∂θ̇A−∂L∂θA=τA.Substituting equation (26) into equation (27), and calculating derivatives, we have
(28)ddt∂L∂θ̇b=Mbθ¨b+MbmPθ¨P+MbmAθ¨A+∑i=16+n∂Mb∂qiq̇i⋅θ̇b+∑i=16+n∂MbmP∂qiq̇i⋅θ̇P+∑i=16+n∂MbmA∂qiq̇i⋅θ̇A,ddt∂L∂θ̇P=MmbPθ¨b+MmPPθ¨P+MmPAθ¨A+∑i=16+n∂MmbP∂qiq̇i⋅θ̇b+∑i=16+n∂MmPP∂qiq̇i⋅θ̇P+∑i=16+n∂MmPA∂qiq̇i⋅θ̇A,ddt∂L∂θ̇A=MmbAθ¨b+MmAPθ¨P+MmAAθ¨A+∑i=16+n∂MmbA∂qiq̇i⋅θ̇b+∑i=16+n∂MmAP∂qiq̇i⋅θ̇P+∑i=16+n∂MmAA∂qiq̇i⋅θ̇A.Makingcτb_1=∑i=16+n∂Mb/∂qiq̇i⋅θ̇b+∑i=16+n∂MbmP/∂qiq̇i⋅θ̇P+∑i=16+n∂MbmA/∂qiq̇i⋅θ̇A, cτP_1=∑i=16+n∂MmbP/∂qiq̇i⋅θ̇b+∑i=16+n∂MmPP/∂qiq̇i⋅θ̇P+∑i=16+n∂MmPA/∂qiq̇i⋅θ̇A, and cτA_1=∑i=16+n∂MmbA/∂qiq̇i⋅θ̇b+∑i=16+n∂MmAP/∂qiq̇i⋅θ̇P+∑i=16+n∂MmAA/∂qiq̇i⋅θ̇A. And cτb_2=∂L/∂θb, cτP_2=∂L/∂θP, and cτA_2=∂L/∂θA.cτb_1,cτb_2∈R6×1, cτP_1,cτP_2∈Rp×1, and cτA_1,cτA_2∈Ra×1are centrifugal forces and Coriolis forces acting on the base, passive joints, and active joints. So equation (28) becomes
(29)MbMbmPMbmAMmbPMmPPMmPAMmbAMmAPMmAAθ¨bθ¨Pθ¨A+cbcPcA=τbτPτA=00τA,where
(30)cbcPcA=cτb_1−cτb_2cτP_1−cτP_2cτA_1−cτA_2∈Rn+6×1.Passive joints and base cannot supply torque, soτb≡0 and τP≡0. If we know the motions of active joints, passive joints, and the base, the required active joint torque τA can be calculated to actuate the space manipulator to arrive at the target state.


## 4. The Analysis of Kinematic and Dynamic Coupling Relationships

Based on kinematic and dynamic models of the free-floating space manipulator with free-swinging joint failure, the kinematic and dynamic coupling relationships inside system are derived, and coupling indexes are defined to depict the coupling degree.


### 4.1. Kinematic Coupling Relationships and Coupling Indexes

For the space manipulator with free-swinging joint failure, passive joints, base, and end-effector can only be passively actuated by active joints, so we call them as “passive units” [27]. Combining them with equations (15) and (16), we have
(31)vt=θ̇bθ̇Pveωe=JbAJPAJPgθ̇A=Jtθ̇A.vt∈R6+p+6×1 denotes the vector consisting of all passive unit velocities, and Jt∈R6+p+6×a is the total Jacobian matrix of the system. Equation (31) reflects the kinematic coupling relationships between active joints and all passive units. For a specific task, it requires no more than a part of the passive units to move as the desired velocities in vt, so vt is the velocity of the entire task space. From equation (31), we can acquire nonhomogeneous linear equations:
(32)Jtθ̇A−vt=jtr1θ̇A−vt1=0,jtr2θ̇A−vt2=0,⋮jtr12+pθ̇A−vt12+p=0.jtri,i=1,2,⋯,12+p, of Jt represents the contribution of θ̇A to vti. The necessary and sufficient condition for the existence of a solution of equation (32) is rankJt=rankJt,vt≤a, which is equivalent to
(33)a≥12+p,rankJt=12+p.It tells us that if active joints are more than passive units, andJt is full rank, we can find at least one set of solutions of Jtθ̇A−vt=0, which means active joints can actuate all passive units at the same time. For the desired vt, we can calculate θ̇A:
(34)θ̇A=Jt+vt.In particular, ifa>12+p, the system is redundant, and equation (34) has the general solution
(35)θ̇A=Jt+vt+Ea−Jt+Jtφ,where φ∈Ra×1 is an arbitrary vector. Equation (35) means there are infinite solutions of θ̇A corresponding to the same vt because of the arbitrariness of φ. Generally speaking, redundancy empowers the space manipulator with free-swinging joint failure to perform secondary tasks, such as obstacle-avoidance, singularity-avoidance, and dexterity-optimization, as long as we find an analytical expression of φ according to secondary task requirements.If conditions in equation (33) are unsatisfactory, the row vectors of Jt are linearly dependent and some velocities in vt cannot be actuated by θ̇A. In other words, the transmission efficiency from θ̇A to vt is insufficient, corresponding to insufficiency of kinematic coupling degree. The kinematic coupling degree depends on the degree of closeness of Jt to singularity, which can be represented by the minimum singular value, condition number, and manipulability. Because manipulability can reflect the general closeness to singularity, which means it can reflect the comprehensive transmission capacity from active joint motion/torque to passive units’ motion. We define the kinematic coupling index based on the manipulability of Jt to evaluate kinematic coupling degree.
(36)wt=∏i=112+pσi=detJtJtT,where σt1≥σt2≥⋯≥σt6+p+6≥0 are singular values of Jt. When wt is close to zeros, the transmission efficiency from θ̇A to vt is inefficient, which means vt is kinematically uncontrollable for θ̇A. Because wt=wtq, in motion planning, we should continually calculate wt according to current state q.However, assuming at least one joint fails, so12+p≥13. If we hope all passive units are controllable, the manipulator must have 14 joints at least. It is unrealistic for real space manipulators because of complicated structure and high difficulty in design. Fortunately, many space tasks only require active joints to actuate a part of passive units. We are going to discuss some typical tasks in the following.


#### 4.1.1. End-Effector Trajectory Tracking Task

This task requires the end-effector to move along the predefined trajectory to arrive at the desired positon/attitude, so the task velocity vector isveA=veT,ωeTT. The transmission from θ̇A to veA depends on JeA. When a≥6, the end-effector could be effectively actuated by active joints. The kinematic coupling index is defined as the manipulability of JeA:
(37)weAq=∏i=16σeAi=detJeAJeAT.IfweAq is close to zero, veA is uncontrollable for θ̇A and the end-effector trajectory tracking task is unrealizable. If weAq is big enough, we should interpolate veT,ωeTT to track the predefined trajectory and calculate θ̇A=JeA†veA. The manipulator can complete the task successfully with the obtained θ̇A.


#### 4.1.2. Passive Joint Regulating Task

This task requires active joints to regulate passive joints to target angles based on the first-order nonholonomic constraint betweenθ̇A and θ̇P. The task velocity vector is vPA=θ̇P, and the transmission from θ̇A to θ̇P depends on JPA. When a≥p, passive joints could be actuated by active joints. The kinematic coupling index is defined as the manipulability of JPA:
(38)wPAq=∏i=1pσPAi=detJPAJPAT.IfwPAq is close to zero, vPA is uncontrollable for θ̇A and the passive joint regulating task is unrealizable. If wPAq is big enough, we should interpolate θ̇P according to the task requirement and calculate θ̇A=JPA†θ̇P.


#### 4.1.3. Base Position/Attitude Controlling Task

This task requires the base to move to the desired positon/attitude relying on the first-order nonholonomic constraint betweenθ̇A and θ̇b, so the task velocity vector is vbA=v0T,ω0TT. The transmission from θ̇A to vbA depends on JbA. When a≥6, the base can be actuated. The kinematic coupling index is defined as the manipulability of JbA:
(39)wbAq=∏i=16σbAi=detJbAJbAT.IfwbAq is close to zero, vbA is uncontrollable for θ̇A so the task is unrealizable. If wbAq is big enough, we should interpolate v0T,ω0TT as the requirement and calculate θ̇A=JbA†vbA.


#### 4.1.4. The Combination Tasks

The space manipulator with free-swinging joint failure usually executes some combination tasks, like the task that requires active joints to actuate the end-effector to the desired position and passive joints to target angles at the same time. For this task, we select the task velocity vector asvvePA=veT,θ̇PTT∈R3+p×1 and the Jacobian matrix is
(40)JvePA=JeA_vJPA∈R3+p×a,where JeA_v is the first three rows of JeA. The kinematic coupling index is the manipulability of JvePA. For the task to eliminate the base attitude deflection when regulating passive joints, we select the task velocity vector as vωbP=θ̇PT,ω0TT∈Rp+3×1 and the Jacobian matrix is
(41)JωbPA=JPAJbA_ω∈Rp+3×a,where JbA_ω is the last three row vectors of JbA. The kinematic coupling index is the manipulability of JωbPA. To eliminate base attitude disturbance, we keep ω0≡0 [28].For general space tasks, we should selectvtask=vtask1,vtask2,⋯,vtaskfT∈Rf×1 according to controlled passive units for current task requirements. f is the dimensionality of vtask. Then, the corresponding row vectors from Jt need to be taken out to construct the Jacobian matrix:
(42)JvtaskA=Jtrv_task1Jtrv_task2⋮Jtrv_taskf∈Rf×a,and the kinematic coupling index is
(43)wvtaskAq=∏i=1fσvtaskA=detJvtaskAJvtaskAT.Ifa≥f and wvtaskAq>0, we can interpolate vtask as requested and calculate θ̇A as θ̇A=JvtaskA+vtask; for a>f, the space manipulator with free-swinging joint failure is redundant for the current task, and the general solution of θ̇A exists as
(44)θ̇A=JvtaskA+vtask+Ea−JvtaskA+JvtaskAφ̇.The analysis above depends on the conservation of system momentum. Space manipulators are usually installed on the base with large mass and inertia, such as a space station or space laboratory. The partition of Jacobian matrix is equivalent to move all passive joints to the base, so passive joints and the base make up a large combination, and all active joints make up a small combination. If system momentum is conservative, it is difficult to use a small combination to regulate a large combination. So when the base mass and inertia is large, as well as the kinematic coupling degree between active joints and passive joints, the base will be seriously inefficient. Passive joint regulation and base control tasks cannot be performed relying on motion planning.


### 4.2. Dynamic Coupling Relationships and Coupling Indexes

Expanding equation (29):
(45)Mbθ¨b+MbmPθ¨P+MbmAθ¨A+cb=0,(46)MmbPθ¨b+MmPPθ¨P+MmPAθ¨A+cP=0,(47)MmbAθ¨b+MmAPθ¨P+MmAAθ¨A+cA=τA.Dynamic coupling relationships inside the system can be derived from equations (45), (46), and (47).


#### 4.2.1. Dynamic Coupling Relationship between Active Joints and Passive Joints

Eliminatingθ¨b in equations (45) and (46), we have
(48)−Mb−1MbmPθ¨P−Mb−1MbmAθ¨A−Mb−1cb=−MmbP+MmPPθ¨P−MmbP+MmPAθ¨A−MmbP+cP.So(49)θ¨P=MPAθ¨A+CPA,where MPA=MmbP+MmPP−Mb−1MbmP+Mb−1MbmA−MmbP+MmPA∈Rp×a is the coupling inertia matrix between active joints and passive joints and CPA=MmbP+MmPP−Mb−1MbmP+Mb−1cb−MmbP+cP∈Rp×1 is the nonlinear item. The first term in the right of equation (49) reflects the transmission from θ¨A to θ¨P, and it is the dynamic coupling relationship between active joints and passive joints. We can obtain nonhomogeneous linear equations MPAθ¨A+CPA−θ¨P=0. If p≤a and rankMPA=p, we have the solution of θ¨A. Therefore, referring to [29], the dynamic coupling index is defined as the manipulability of MPA to evaluate the transmission efficiency from θ¨A to θ¨P.
(50)ηPAq=∏i=1pεPAi=detMPAMPAT,where εPA1≥εPA2≥⋯≥εPAp≥0 are singular values of MPA. When ηPAq is close to zero, the dynamic coupling degree between passive joints and active joints is inefficient and θ¨P is dynamically uncontrollable for θ¨A. When ηPAq is big enough, we can design the control law for θ¨P, and θ¨A which actuates the manipulator to complete the passive joint regulating task can be obtained by θ¨A=MPA†θ¨P−CPA.


#### 4.2.2. Dynamic Coupling Relationship between Active Joints and Base

Eliminatingθ¨P in equations (45) and (46), we have
(51)−MbmP+Mbθ¨b−MbmP+MbmAθ¨A−MbmP+cb=−MmPP−1MmbPθ¨b−MmPP−1MmPAθ¨A−MmPP−1cP.So(52)θ¨b=MbAθ¨A+CbA,where MbA=MmPP−1MmbP−MbmP+Mb+MbmP+MbmA−MmPP−1MmPA∈R6×a is the coupling inertia matrix between active joints and the base. CbA=MmPP+MmbP−MbmP+Mb+MbmP+cb−MmPP+cP∈R6×1 is the nonlinear item. The first term in the right of equation (52) is the dynamic coupling relationship between active joints and the base. The dynamic coupling index is defined as the manipulability of MbA to evaluate the transmission efficiency from θ¨A to θ¨b.
(53)ηbAq=∏i=16εbAi=detMbAMbAT.IfηbAq is close to zero, θ¨b is uncontrollable for θ¨A. On the contrary, we can design the control law for θ¨b to calculate θ¨A=MbA+θ¨b−CbA.In particular, we can get transmission fromθ¨A to θ¨b,θ¨P by combining equations (49) and (52):
(54)θ¨Pθ¨b=MPAMbAθ¨A+CPACbA.If we calculate the derivative of equation (22), we have
(55)θ¨Pθ¨b=JPAJbAθ¨A+J̇PAJ̇bAθ̇A.Equation (55) has a similar form with equation (54), but obviously, JPA≠MPA, JbA≠MbA, J̇PAθ̇A≠CPA, and J̇bAθ̇A≠CbA, so equations (22) and (54) are different.Equation (54) can also be derived from equations (45) and (46):
(56)θ¨bθ¨P=−MbMbmPMmbPMmPP−1MbmAMmPAθ¨A+cbcP.Substituting equation (56) into equation (47), we have
(57)τA=MτAθ¨A+CτA,where
(58)MτA=−MmbAMmAPMbMbmPMmbPMmPP−1MbmAMmPA+MmAA∈Ra×a,represents the mapping relation between θ¨A and τA and
(59)CτA=−MmbAMmAPMbMbmPMmbPMmPP−1cbcP+cA∈Ra×1,is the nonlinear item. Equation (57) establishes the direct relation between θ¨A and τA. After designing the control law of θ¨P or θ¨b, we can calculate θ¨A, and then we can obtain the torque τA that actuates the manipulator. We can further find the transmission from τA to θ¨P,θ¨b:
(60)θ¨P=MPττA+CPτ,θ¨b=MbττA+Cbτ,where MPτ=MPAMτA−1∈Rp×a, CPτ=CPA−MPAMτA−1CτA∈Rp×1, Mbτ=MbAMτA−1∈R6×a, and Cbτ=CbA−MbAMτA−1CτA∈R6×1. So we can directly calculate τA from equation (60) corresponding to the desired θ¨P or θ¨b. Dynamic coupling indexes are defined to evaluate transmission efficiency from τA to θ¨P,θ¨b:
(61)ηPτq=∏i=1pεPτi=detMPτMPτT,ηbτq=∏i=16εbτi=detMbτMbτT.


#### 4.2.3. Dynamic Coupling Relationship between Active Joints and End-Effector

Becausev̇eT,ω̇eTT does not appear in the Lagrange equation, we cannot find a mapping relation between θ¨A and v̇eT,ω̇eTT in equation (29), but we can find the dynamic coupling relationship between active joints and the end-effector from another perspective [12]. Calculating derivative of equation (16), we have
(62)v̇eω̇e=JeAθ¨A+J̇eAθ̇A.Multiplying equation (57) by JeAMτA−1, we have
(63)JeAθ¨A+JeAMτA−1CτA=JeAMτA−1τA⇒JeAθ¨A+J̇eAθ̇A=JeAMτA−1τA+J̇eAθ̇A−JeAMτA−1CτA⇒v̇eω̇e=MeττA+Ceτ,where Meτ=JeAMτA−1∈R6×a is the coupling inertia matrix representing the transmission from τA to v̇eT,ω̇eTT and Ceτ=J̇eAθ̇A−JeAMτA−1CτA∈R6×1 is the nonlinear item. Equation (63) is the dynamic coupling relationship between active joints and the end-effector. The dynamic coupling index between active joints and the end-effector is the manipulability of Meτ:
(64)ηeτq=∏i=16εeτi=detMeτMeτT.Ifηeτq is close to zero, v̇eT,ω̇eTT is uncontrollable for τA. On the contrary, we can design the control law for v̇eT,ω̇eTT to calculate
(65)τA=Meτ+v̇eω̇e−Ceτ.Equations (60) and (63) can be reduced to the following constraints:
(66)g1θ¨b,θ¨AorτA,t=0,g2θ¨b,θ¨AorτA,t=0,g3v̇e,ω̇e,τA,t=0.Theorem.
The constraints in equation (70) are also nonholonomic.Proof.
Taking equation (49) as an example, we have
(67)MPAθ¨A−θ¨P+CPA=0⇒0p×6−Ep×pMPAq¨+CPA=0,making MsPA=0p×6−Ep×pMPA∈Rp×6+n. Assuming there exists the first-order constraint pq,q̇=0 whose derivative with respect to time is
(68)dpdt=∂p∂qq̇+∂p∂q̇q¨.Assuming constraint in equation (66) is holonomic, we have
(69)∂p∂qq̇=CPA,∂p∂q̇=MsPA.Consideringq̇=0, we can obtain the general solution of pq,q̇:
(70)pq,q̇=MsPAq̇+sq+C=0.sq is relative to the state variable q. C is a constant vector. The derivative of equation (70) is
(71)dpdt=MsPAq¨+ṀsPAq̇+dsdqq̇=0.Substitutingcb and cP into equation (71), and comparing it with equation (67), we obtain
(72)M1ṀbṀbmPṀbmA−M2ṀmbPṀmPPṀmPA−ṀsPA−dsdqq̇+12q̇T∂Μ∂qP−∂Μ∂qbq̇=0.where M1=MmbP+MmPP−Mb−1MbmP+Mb−1 and M2=MmbP+MmPP−Mb−1MbmP+MmbP+. To satisfy the above equality, we have
(73)dsdq=M1ṀbṀbmPṀbmA−M2ṀmbPṀmPPṀmPA−ṀsPA,∂Μ∂qP=∂Μ∂qb.sq is only relative to q instead of q̇, so the first equality is not satisfied. The second equality is also not satisfied because θP and θb are not cyclic coordinates in Μ. Therefore, pq,q̇=0 does not exist and the constraints in equation (66) are nonholonomic. Because equation (66) is the constraints among the second-order derivatives of θb,θA,θP,pe,Ψe, they are called as the second-order nonholonomic constraint.The free-floating space manipulator with free-swinging joint failure simultaneously contains the first-order nonholonomic constraint and the second-order nonholonomic constraint, and they are independent with each other. From a mathematical viewpoint, equations (17) and (66) belong to differential equations, and they have the same solution. In fact, because null space exists in equation (14), we can find the same solution to these two differential equations. Therefore, motions of the free-floating space manipulator with free-swinging joint failure simultaneously satisfy the first-order and second-order nonholonomic constraints.The analysis above is not based on conservation of system momentum, so even if base mass and inertia are very large, the degree of dynamic coupling may be large enough to enable active joints to actuate passive units through the control method.In summary, dynamic coupling relationship between active joints and all passive units is(74)abP=θ¨bθ¨P=MbPθ¨A+CbP,orat=θ¨bθ¨Pv̇eω̇e=MtττA+Ctτ.at∈Rp+12×1 is the acceleration of the entire task space.
(75)MbP=MbAMPA∈Rp+6×a,is the coupling inertia matrix representing the transmission from θ¨A to θ¨bT,θ¨PTT, and
(76)Mtτ=MbτMPτMeτ∈Rp+12×a,is the coupling inertia matrix representing the transmission from τA to at.
(77)CbP=CbACPA∈Rp+6×1,Ctτ=CbτCPτCeτ∈Rp+12×1,are nonlinear items. Based on dynamic coupling relationships, active joints can actuate the manipulator to execute tasks as the same as motion planning. For general tasks, we should select the task acceleration vector atask∈Rf×1 according to task requirements and construct MataskA∈Rf×a and CataskA∈Rf×1. Then, we design the control law for atask to guarantee that the selected passive units will converge to the equilibrium point, and τA is obtained. If a>f, the space manipulator is redundant and the solutions of θ¨A and τA are
(78)θ¨A=MataskA+atask−CataskA+I−MataskA+MataskAς,orτA=MataskA+atask−CataskA+I−MataskA+MataskAς.ς¨∈Ra×1 is an arbitrary vector. Appropriately designing ς, the manipulator can perform primary and secondary tasks at the same time.


### 4.3. Global Kinematic and Dynamic Coupling Indexes

The kinematic and dynamic coupling indexes defined above are with respect to system state variablesq. If their values are large, it only indicates that motion transmission efficiency from active joints to passive units is high only at the current q, so the indexes above are called as local indexes for kinematic and dynamic coupling in state space. However, before structure design, motion planning, and controller design, the system state is unknown, so the local coupling indexes are not appropriate at this moment. We care more about the global coupling degree in the entire state space, and global kinematic and dynamic coupling indexes are defined as
(79)w∗g=∫qw∗2qdq∫qdq,η∗g=∫qη∗2qdq∫qdq.w∗ and w∗g correspond to local and global manipulability of a Jacobian matrix J∗, respectively. η∗ and η∗g correspond to local and global manipulability of an inertia matrix M∗, respectively. If kinematic and dynamic parameters are determined, global indexes can be calculated. The higher the values of global indexes, the higher the motion transmission efficiency from active joints to passive units whatever state the manipulator is in. If selecting different healthy joints as active joints, different values of global indexes will be obtained. There exists the case that values of global indexes are close whether we select some healthy joins. It indicates that these joints are incapable to actuate passive units, so we should not select them as active joints.


## 4.1. Kinematic Coupling Relationships and Coupling Indexes

For the space manipulator with free-swinging joint failure, passive joints, base, and end-effector can only be passively actuated by active joints, so we call them as “passive units” [27]. Combining them with equations (15) and (16), we have
(31)vt=θ̇bθ̇Pveωe=JbAJPAJPgθ̇A=Jtθ̇A.vt∈R6+p+6×1 denotes the vector consisting of all passive unit velocities, and Jt∈R6+p+6×a is the total Jacobian matrix of the system. Equation (31) reflects the kinematic coupling relationships between active joints and all passive units. For a specific task, it requires no more than a part of the passive units to move as the desired velocities in vt, so vt is the velocity of the entire task space. From equation (31), we can acquire nonhomogeneous linear equations:
(32)Jtθ̇A−vt=jtr1θ̇A−vt1=0,jtr2θ̇A−vt2=0,⋮jtr12+pθ̇A−vt12+p=0.jtri,i=1,2,⋯,12+p, of Jt represents the contribution of θ̇A to vti. The necessary and sufficient condition for the existence of a solution of equation (32) is rankJt=rankJt,vt≤a, which is equivalent to
(33)a≥12+p,rankJt=12+p.It tells us that if active joints are more than passive units, andJt is full rank, we can find at least one set of solutions of Jtθ̇A−vt=0, which means active joints can actuate all passive units at the same time. For the desired vt, we can calculate θ̇A:
(34)θ̇A=Jt+vt.In particular, ifa>12+p, the system is redundant, and equation (34) has the general solution
(35)θ̇A=Jt+vt+Ea−Jt+Jtφ,where φ∈Ra×1 is an arbitrary vector. Equation (35) means there are infinite solutions of θ̇A corresponding to the same vt because of the arbitrariness of φ. Generally speaking, redundancy empowers the space manipulator with free-swinging joint failure to perform secondary tasks, such as obstacle-avoidance, singularity-avoidance, and dexterity-optimization, as long as we find an analytical expression of φ according to secondary task requirements.If conditions in equation (33) are unsatisfactory, the row vectors of Jt are linearly dependent and some velocities in vt cannot be actuated by θ̇A. In other words, the transmission efficiency from θ̇A to vt is insufficient, corresponding to insufficiency of kinematic coupling degree. The kinematic coupling degree depends on the degree of closeness of Jt to singularity, which can be represented by the minimum singular value, condition number, and manipulability. Because manipulability can reflect the general closeness to singularity, which means it can reflect the comprehensive transmission capacity from active joint motion/torque to passive units’ motion. We define the kinematic coupling index based on the manipulability of Jt to evaluate kinematic coupling degree.
(36)wt=∏i=112+pσi=detJtJtT,where σt1≥σt2≥⋯≥σt6+p+6≥0 are singular values of Jt. When wt is close to zeros, the transmission efficiency from θ̇A to vt is inefficient, which means vt is kinematically uncontrollable for θ̇A. Because wt=wtq, in motion planning, we should continually calculate wt according to current state q.However, assuming at least one joint fails, so12+p≥13. If we hope all passive units are controllable, the manipulator must have 14 joints at least. It is unrealistic for real space manipulators because of complicated structure and high difficulty in design. Fortunately, many space tasks only require active joints to actuate a part of passive units. We are going to discuss some typical tasks in the following.


### 4.1.1. End-Effector Trajectory Tracking Task

This task requires the end-effector to move along the predefined trajectory to arrive at the desired positon/attitude, so the task velocity vector isveA=veT,ωeTT. The transmission from θ̇A to veA depends on JeA. When a≥6, the end-effector could be effectively actuated by active joints. The kinematic coupling index is defined as the manipulability of JeA:
(37)weAq=∏i=16σeAi=detJeAJeAT.IfweAq is close to zero, veA is uncontrollable for θ̇A and the end-effector trajectory tracking task is unrealizable. If weAq is big enough, we should interpolate veT,ωeTT to track the predefined trajectory and calculate θ̇A=JeA†veA. The manipulator can complete the task successfully with the obtained θ̇A.


### 4.1.2. Passive Joint Regulating Task

This task requires active joints to regulate passive joints to target angles based on the first-order nonholonomic constraint betweenθ̇A and θ̇P. The task velocity vector is vPA=θ̇P, and the transmission from θ̇A to θ̇P depends on JPA. When a≥p, passive joints could be actuated by active joints. The kinematic coupling index is defined as the manipulability of JPA:
(38)wPAq=∏i=1pσPAi=detJPAJPAT.IfwPAq is close to zero, vPA is uncontrollable for θ̇A and the passive joint regulating task is unrealizable. If wPAq is big enough, we should interpolate θ̇P according to the task requirement and calculate θ̇A=JPA†θ̇P.


### 4.1.3. Base Position/Attitude Controlling Task

This task requires the base to move to the desired positon/attitude relying on the first-order nonholonomic constraint betweenθ̇A and θ̇b, so the task velocity vector is vbA=v0T,ω0TT. The transmission from θ̇A to vbA depends on JbA. When a≥6, the base can be actuated. The kinematic coupling index is defined as the manipulability of JbA:
(39)wbAq=∏i=16σbAi=detJbAJbAT.IfwbAq is close to zero, vbA is uncontrollable for θ̇A so the task is unrealizable. If wbAq is big enough, we should interpolate v0T,ω0TT as the requirement and calculate θ̇A=JbA†vbA.


### 4.1.4. The Combination Tasks

The space manipulator with free-swinging joint failure usually executes some combination tasks, like the task that requires active joints to actuate the end-effector to the desired position and passive joints to target angles at the same time. For this task, we select the task velocity vector asvvePA=veT,θ̇PTT∈R3+p×1 and the Jacobian matrix is
(40)JvePA=JeA_vJPA∈R3+p×a,where JeA_v is the first three rows of JeA. The kinematic coupling index is the manipulability of JvePA. For the task to eliminate the base attitude deflection when regulating passive joints, we select the task velocity vector as vωbP=θ̇PT,ω0TT∈Rp+3×1 and the Jacobian matrix is
(41)JωbPA=JPAJbA_ω∈Rp+3×a,where JbA_ω is the last three row vectors of JbA. The kinematic coupling index is the manipulability of JωbPA. To eliminate base attitude disturbance, we keep ω0≡0 [28].For general space tasks, we should selectvtask=vtask1,vtask2,⋯,vtaskfT∈Rf×1 according to controlled passive units for current task requirements. f is the dimensionality of vtask. Then, the corresponding row vectors from Jt need to be taken out to construct the Jacobian matrix:
(42)JvtaskA=Jtrv_task1Jtrv_task2⋮Jtrv_taskf∈Rf×a,and the kinematic coupling index is
(43)wvtaskAq=∏i=1fσvtaskA=detJvtaskAJvtaskAT.Ifa≥f and wvtaskAq>0, we can interpolate vtask as requested and calculate θ̇A as θ̇A=JvtaskA+vtask; for a>f, the space manipulator with free-swinging joint failure is redundant for the current task, and the general solution of θ̇A exists as
(44)θ̇A=JvtaskA+vtask+Ea−JvtaskA+JvtaskAφ̇.The analysis above depends on the conservation of system momentum. Space manipulators are usually installed on the base with large mass and inertia, such as a space station or space laboratory. The partition of Jacobian matrix is equivalent to move all passive joints to the base, so passive joints and the base make up a large combination, and all active joints make up a small combination. If system momentum is conservative, it is difficult to use a small combination to regulate a large combination. So when the base mass and inertia is large, as well as the kinematic coupling degree between active joints and passive joints, the base will be seriously inefficient. Passive joint regulation and base control tasks cannot be performed relying on motion planning.


## 4.1.1. End-Effector Trajectory Tracking Task

This task requires the end-effector to move along the predefined trajectory to arrive at the desired positon/attitude, so the task velocity vector isveA=veT,ωeTT. The transmission from θ̇A to veA depends on JeA. When a≥6, the end-effector could be effectively actuated by active joints. The kinematic coupling index is defined as the manipulability of JeA:
(37)weAq=∏i=16σeAi=detJeAJeAT.IfweAq is close to zero, veA is uncontrollable for θ̇A and the end-effector trajectory tracking task is unrealizable. If weAq is big enough, we should interpolate veT,ωeTT to track the predefined trajectory and calculate θ̇A=JeA†veA. The manipulator can complete the task successfully with the obtained θ̇A.


## 4.1.2. Passive Joint Regulating Task

This task requires active joints to regulate passive joints to target angles based on the first-order nonholonomic constraint betweenθ̇A and θ̇P. The task velocity vector is vPA=θ̇P, and the transmission from θ̇A to θ̇P depends on JPA. When a≥p, passive joints could be actuated by active joints. The kinematic coupling index is defined as the manipulability of JPA:
(38)wPAq=∏i=1pσPAi=detJPAJPAT.IfwPAq is close to zero, vPA is uncontrollable for θ̇A and the passive joint regulating task is unrealizable. If wPAq is big enough, we should interpolate θ̇P according to the task requirement and calculate θ̇A=JPA†θ̇P.


## 4.1.3. Base Position/Attitude Controlling Task

This task requires the base to move to the desired positon/attitude relying on the first-order nonholonomic constraint betweenθ̇A and θ̇b, so the task velocity vector is vbA=v0T,ω0TT. The transmission from θ̇A to vbA depends on JbA. When a≥6, the base can be actuated. The kinematic coupling index is defined as the manipulability of JbA:
(39)wbAq=∏i=16σbAi=detJbAJbAT.IfwbAq is close to zero, vbA is uncontrollable for θ̇A so the task is unrealizable. If wbAq is big enough, we should interpolate v0T,ω0TT as the requirement and calculate θ̇A=JbA†vbA.


## 4.1.4. The Combination Tasks

The space manipulator with free-swinging joint failure usually executes some combination tasks, like the task that requires active joints to actuate the end-effector to the desired position and passive joints to target angles at the same time. For this task, we select the task velocity vector asvvePA=veT,θ̇PTT∈R3+p×1 and the Jacobian matrix is
(40)JvePA=JeA_vJPA∈R3+p×a,where JeA_v is the first three rows of JeA. The kinematic coupling index is the manipulability of JvePA. For the task to eliminate the base attitude deflection when regulating passive joints, we select the task velocity vector as vωbP=θ̇PT,ω0TT∈Rp+3×1 and the Jacobian matrix is
(41)JωbPA=JPAJbA_ω∈Rp+3×a,where JbA_ω is the last three row vectors of JbA. The kinematic coupling index is the manipulability of JωbPA. To eliminate base attitude disturbance, we keep ω0≡0 [28].For general space tasks, we should selectvtask=vtask1,vtask2,⋯,vtaskfT∈Rf×1 according to controlled passive units for current task requirements. f is the dimensionality of vtask. Then, the corresponding row vectors from Jt need to be taken out to construct the Jacobian matrix:
(42)JvtaskA=Jtrv_task1Jtrv_task2⋮Jtrv_taskf∈Rf×a,and the kinematic coupling index is
(43)wvtaskAq=∏i=1fσvtaskA=detJvtaskAJvtaskAT.Ifa≥f and wvtaskAq>0, we can interpolate vtask as requested and calculate θ̇A as θ̇A=JvtaskA+vtask; for a>f, the space manipulator with free-swinging joint failure is redundant for the current task, and the general solution of θ̇A exists as
(44)θ̇A=JvtaskA+vtask+Ea−JvtaskA+JvtaskAφ̇.The analysis above depends on the conservation of system momentum. Space manipulators are usually installed on the base with large mass and inertia, such as a space station or space laboratory. The partition of Jacobian matrix is equivalent to move all passive joints to the base, so passive joints and the base make up a large combination, and all active joints make up a small combination. If system momentum is conservative, it is difficult to use a small combination to regulate a large combination. So when the base mass and inertia is large, as well as the kinematic coupling degree between active joints and passive joints, the base will be seriously inefficient. Passive joint regulation and base control tasks cannot be performed relying on motion planning.


## 4.2. Dynamic Coupling Relationships and Coupling Indexes

Expanding equation (29):
(45)Mbθ¨b+MbmPθ¨P+MbmAθ¨A+cb=0,(46)MmbPθ¨b+MmPPθ¨P+MmPAθ¨A+cP=0,(47)MmbAθ¨b+MmAPθ¨P+MmAAθ¨A+cA=τA.Dynamic coupling relationships inside the system can be derived from equations (45), (46), and (47).


### 4.2.1. Dynamic Coupling Relationship between Active Joints and Passive Joints

Eliminatingθ¨b in equations (45) and (46), we have
(48)−Mb−1MbmPθ¨P−Mb−1MbmAθ¨A−Mb−1cb=−MmbP+MmPPθ¨P−MmbP+MmPAθ¨A−MmbP+cP.So(49)θ¨P=MPAθ¨A+CPA,where MPA=MmbP+MmPP−Mb−1MbmP+Mb−1MbmA−MmbP+MmPA∈Rp×a is the coupling inertia matrix between active joints and passive joints and CPA=MmbP+MmPP−Mb−1MbmP+Mb−1cb−MmbP+cP∈Rp×1 is the nonlinear item. The first term in the right of equation (49) reflects the transmission from θ¨A to θ¨P, and it is the dynamic coupling relationship between active joints and passive joints. We can obtain nonhomogeneous linear equations MPAθ¨A+CPA−θ¨P=0. If p≤a and rankMPA=p, we have the solution of θ¨A. Therefore, referring to [29], the dynamic coupling index is defined as the manipulability of MPA to evaluate the transmission efficiency from θ¨A to θ¨P.
(50)ηPAq=∏i=1pεPAi=detMPAMPAT,where εPA1≥εPA2≥⋯≥εPAp≥0 are singular values of MPA. When ηPAq is close to zero, the dynamic coupling degree between passive joints and active joints is inefficient and θ¨P is dynamically uncontrollable for θ¨A. When ηPAq is big enough, we can design the control law for θ¨P, and θ¨A which actuates the manipulator to complete the passive joint regulating task can be obtained by θ¨A=MPA†θ¨P−CPA.


### 4.2.2. Dynamic Coupling Relationship between Active Joints and Base

Eliminatingθ¨P in equations (45) and (46), we have
(51)−MbmP+Mbθ¨b−MbmP+MbmAθ¨A−MbmP+cb=−MmPP−1MmbPθ¨b−MmPP−1MmPAθ¨A−MmPP−1cP.So(52)θ¨b=MbAθ¨A+CbA,where MbA=MmPP−1MmbP−MbmP+Mb+MbmP+MbmA−MmPP−1MmPA∈R6×a is the coupling inertia matrix between active joints and the base. CbA=MmPP+MmbP−MbmP+Mb+MbmP+cb−MmPP+cP∈R6×1 is the nonlinear item. The first term in the right of equation (52) is the dynamic coupling relationship between active joints and the base. The dynamic coupling index is defined as the manipulability of MbA to evaluate the transmission efficiency from θ¨A to θ¨b.
(53)ηbAq=∏i=16εbAi=detMbAMbAT.IfηbAq is close to zero, θ¨b is uncontrollable for θ¨A. On the contrary, we can design the control law for θ¨b to calculate θ¨A=MbA+θ¨b−CbA.In particular, we can get transmission fromθ¨A to θ¨b,θ¨P by combining equations (49) and (52):
(54)θ¨Pθ¨b=MPAMbAθ¨A+CPACbA.If we calculate the derivative of equation (22), we have
(55)θ¨Pθ¨b=JPAJbAθ¨A+J̇PAJ̇bAθ̇A.Equation (55) has a similar form with equation (54), but obviously, JPA≠MPA, JbA≠MbA, J̇PAθ̇A≠CPA, and J̇bAθ̇A≠CbA, so equations (22) and (54) are different.Equation (54) can also be derived from equations (45) and (46):
(56)θ¨bθ¨P=−MbMbmPMmbPMmPP−1MbmAMmPAθ¨A+cbcP.Substituting equation (56) into equation (47), we have
(57)τA=MτAθ¨A+CτA,where
(58)MτA=−MmbAMmAPMbMbmPMmbPMmPP−1MbmAMmPA+MmAA∈Ra×a,represents the mapping relation between θ¨A and τA and
(59)CτA=−MmbAMmAPMbMbmPMmbPMmPP−1cbcP+cA∈Ra×1,is the nonlinear item. Equation (57) establishes the direct relation between θ¨A and τA. After designing the control law of θ¨P or θ¨b, we can calculate θ¨A, and then we can obtain the torque τA that actuates the manipulator. We can further find the transmission from τA to θ¨P,θ¨b:
(60)θ¨P=MPττA+CPτ,θ¨b=MbττA+Cbτ,where MPτ=MPAMτA−1∈Rp×a, CPτ=CPA−MPAMτA−1CτA∈Rp×1, Mbτ=MbAMτA−1∈R6×a, and Cbτ=CbA−MbAMτA−1CτA∈R6×1. So we can directly calculate τA from equation (60) corresponding to the desired θ¨P or θ¨b. Dynamic coupling indexes are defined to evaluate transmission efficiency from τA to θ¨P,θ¨b:
(61)ηPτq=∏i=1pεPτi=detMPτMPτT,ηbτq=∏i=16εbτi=detMbτMbτT.


### 4.2.3. Dynamic Coupling Relationship between Active Joints and End-Effector

Becausev̇eT,ω̇eTT does not appear in the Lagrange equation, we cannot find a mapping relation between θ¨A and v̇eT,ω̇eTT in equation (29), but we can find the dynamic coupling relationship between active joints and the end-effector from another perspective [12]. Calculating derivative of equation (16), we have
(62)v̇eω̇e=JeAθ¨A+J̇eAθ̇A.Multiplying equation (57) by JeAMτA−1, we have
(63)JeAθ¨A+JeAMτA−1CτA=JeAMτA−1τA⇒JeAθ¨A+J̇eAθ̇A=JeAMτA−1τA+J̇eAθ̇A−JeAMτA−1CτA⇒v̇eω̇e=MeττA+Ceτ,where Meτ=JeAMτA−1∈R6×a is the coupling inertia matrix representing the transmission from τA to v̇eT,ω̇eTT and Ceτ=J̇eAθ̇A−JeAMτA−1CτA∈R6×1 is the nonlinear item. Equation (63) is the dynamic coupling relationship between active joints and the end-effector. The dynamic coupling index between active joints and the end-effector is the manipulability of Meτ:
(64)ηeτq=∏i=16εeτi=detMeτMeτT.Ifηeτq is close to zero, v̇eT,ω̇eTT is uncontrollable for τA. On the contrary, we can design the control law for v̇eT,ω̇eTT to calculate
(65)τA=Meτ+v̇eω̇e−Ceτ.Equations (60) and (63) can be reduced to the following constraints:
(66)g1θ¨b,θ¨AorτA,t=0,g2θ¨b,θ¨AorτA,t=0,g3v̇e,ω̇e,τA,t=0.Theorem.
The constraints in equation (70) are also nonholonomic.Proof.
Taking equation (49) as an example, we have
(67)MPAθ¨A−θ¨P+CPA=0⇒0p×6−Ep×pMPAq¨+CPA=0,making MsPA=0p×6−Ep×pMPA∈Rp×6+n. Assuming there exists the first-order constraint pq,q̇=0 whose derivative with respect to time is
(68)dpdt=∂p∂qq̇+∂p∂q̇q¨.Assuming constraint in equation (66) is holonomic, we have
(69)∂p∂qq̇=CPA,∂p∂q̇=MsPA.Consideringq̇=0, we can obtain the general solution of pq,q̇:
(70)pq,q̇=MsPAq̇+sq+C=0.sq is relative to the state variable q. C is a constant vector. The derivative of equation (70) is
(71)dpdt=MsPAq¨+ṀsPAq̇+dsdqq̇=0.Substitutingcb and cP into equation (71), and comparing it with equation (67), we obtain
(72)M1ṀbṀbmPṀbmA−M2ṀmbPṀmPPṀmPA−ṀsPA−dsdqq̇+12q̇T∂Μ∂qP−∂Μ∂qbq̇=0.where M1=MmbP+MmPP−Mb−1MbmP+Mb−1 and M2=MmbP+MmPP−Mb−1MbmP+MmbP+. To satisfy the above equality, we have
(73)dsdq=M1ṀbṀbmPṀbmA−M2ṀmbPṀmPPṀmPA−ṀsPA,∂Μ∂qP=∂Μ∂qb.sq is only relative to q instead of q̇, so the first equality is not satisfied. The second equality is also not satisfied because θP and θb are not cyclic coordinates in Μ. Therefore, pq,q̇=0 does not exist and the constraints in equation (66) are nonholonomic. Because equation (66) is the constraints among the second-order derivatives of θb,θA,θP,pe,Ψe, they are called as the second-order nonholonomic constraint.The free-floating space manipulator with free-swinging joint failure simultaneously contains the first-order nonholonomic constraint and the second-order nonholonomic constraint, and they are independent with each other. From a mathematical viewpoint, equations (17) and (66) belong to differential equations, and they have the same solution. In fact, because null space exists in equation (14), we can find the same solution to these two differential equations. Therefore, motions of the free-floating space manipulator with free-swinging joint failure simultaneously satisfy the first-order and second-order nonholonomic constraints.The analysis above is not based on conservation of system momentum, so even if base mass and inertia are very large, the degree of dynamic coupling may be large enough to enable active joints to actuate passive units through the control method.In summary, dynamic coupling relationship between active joints and all passive units is(74)abP=θ¨bθ¨P=MbPθ¨A+CbP,orat=θ¨bθ¨Pv̇eω̇e=MtττA+Ctτ.at∈Rp+12×1 is the acceleration of the entire task space.
(75)MbP=MbAMPA∈Rp+6×a,is the coupling inertia matrix representing the transmission from θ¨A to θ¨bT,θ¨PTT, and
(76)Mtτ=MbτMPτMeτ∈Rp+12×a,is the coupling inertia matrix representing the transmission from τA to at.
(77)CbP=CbACPA∈Rp+6×1,Ctτ=CbτCPτCeτ∈Rp+12×1,are nonlinear items. Based on dynamic coupling relationships, active joints can actuate the manipulator to execute tasks as the same as motion planning. For general tasks, we should select the task acceleration vector atask∈Rf×1 according to task requirements and construct MataskA∈Rf×a and CataskA∈Rf×1. Then, we design the control law for atask to guarantee that the selected passive units will converge to the equilibrium point, and τA is obtained. If a>f, the space manipulator is redundant and the solutions of θ¨A and τA are
(78)θ¨A=MataskA+atask−CataskA+I−MataskA+MataskAς,orτA=MataskA+atask−CataskA+I−MataskA+MataskAς.ς¨∈Ra×1 is an arbitrary vector. Appropriately designing ς, the manipulator can perform primary and secondary tasks at the same time.


## 4.2.1. Dynamic Coupling Relationship between Active Joints and Passive Joints

Eliminatingθ¨b in equations (45) and (46), we have
(48)−Mb−1MbmPθ¨P−Mb−1MbmAθ¨A−Mb−1cb=−MmbP+MmPPθ¨P−MmbP+MmPAθ¨A−MmbP+cP.So(49)θ¨P=MPAθ¨A+CPA,where MPA=MmbP+MmPP−Mb−1MbmP+Mb−1MbmA−MmbP+MmPA∈Rp×a is the coupling inertia matrix between active joints and passive joints and CPA=MmbP+MmPP−Mb−1MbmP+Mb−1cb−MmbP+cP∈Rp×1 is the nonlinear item. The first term in the right of equation (49) reflects the transmission from θ¨A to θ¨P, and it is the dynamic coupling relationship between active joints and passive joints. We can obtain nonhomogeneous linear equations MPAθ¨A+CPA−θ¨P=0. If p≤a and rankMPA=p, we have the solution of θ¨A. Therefore, referring to [29], the dynamic coupling index is defined as the manipulability of MPA to evaluate the transmission efficiency from θ¨A to θ¨P.
(50)ηPAq=∏i=1pεPAi=detMPAMPAT,where εPA1≥εPA2≥⋯≥εPAp≥0 are singular values of MPA. When ηPAq is close to zero, the dynamic coupling degree between passive joints and active joints is inefficient and θ¨P is dynamically uncontrollable for θ¨A. When ηPAq is big enough, we can design the control law for θ¨P, and θ¨A which actuates the manipulator to complete the passive joint regulating task can be obtained by θ¨A=MPA†θ¨P−CPA.


## 4.2.2. Dynamic Coupling Relationship between Active Joints and Base

Eliminatingθ¨P in equations (45) and (46), we have
(51)−MbmP+Mbθ¨b−MbmP+MbmAθ¨A−MbmP+cb=−MmPP−1MmbPθ¨b−MmPP−1MmPAθ¨A−MmPP−1cP.So(52)θ¨b=MbAθ¨A+CbA,where MbA=MmPP−1MmbP−MbmP+Mb+MbmP+MbmA−MmPP−1MmPA∈R6×a is the coupling inertia matrix between active joints and the base. CbA=MmPP+MmbP−MbmP+Mb+MbmP+cb−MmPP+cP∈R6×1 is the nonlinear item. The first term in the right of equation (52) is the dynamic coupling relationship between active joints and the base. The dynamic coupling index is defined as the manipulability of MbA to evaluate the transmission efficiency from θ¨A to θ¨b.
(53)ηbAq=∏i=16εbAi=detMbAMbAT.IfηbAq is close to zero, θ¨b is uncontrollable for θ¨A. On the contrary, we can design the control law for θ¨b to calculate θ¨A=MbA+θ¨b−CbA.In particular, we can get transmission fromθ¨A to θ¨b,θ¨P by combining equations (49) and (52):
(54)θ¨Pθ¨b=MPAMbAθ¨A+CPACbA.If we calculate the derivative of equation (22), we have
(55)θ¨Pθ¨b=JPAJbAθ¨A+J̇PAJ̇bAθ̇A.Equation (55) has a similar form with equation (54), but obviously, JPA≠MPA, JbA≠MbA, J̇PAθ̇A≠CPA, and J̇bAθ̇A≠CbA, so equations (22) and (54) are different.Equation (54) can also be derived from equations (45) and (46):
(56)θ¨bθ¨P=−MbMbmPMmbPMmPP−1MbmAMmPAθ¨A+cbcP.Substituting equation (56) into equation (47), we have
(57)τA=MτAθ¨A+CτA,where
(58)MτA=−MmbAMmAPMbMbmPMmbPMmPP−1MbmAMmPA+MmAA∈Ra×a,represents the mapping relation between θ¨A and τA and
(59)CτA=−MmbAMmAPMbMbmPMmbPMmPP−1cbcP+cA∈Ra×1,is the nonlinear item. Equation (57) establishes the direct relation between θ¨A and τA. After designing the control law of θ¨P or θ¨b, we can calculate θ¨A, and then we can obtain the torque τA that actuates the manipulator. We can further find the transmission from τA to θ¨P,θ¨b:
(60)θ¨P=MPττA+CPτ,θ¨b=MbττA+Cbτ,where MPτ=MPAMτA−1∈Rp×a, CPτ=CPA−MPAMτA−1CτA∈Rp×1, Mbτ=MbAMτA−1∈R6×a, and Cbτ=CbA−MbAMτA−1CτA∈R6×1. So we can directly calculate τA from equation (60) corresponding to the desired θ¨P or θ¨b. Dynamic coupling indexes are defined to evaluate transmission efficiency from τA to θ¨P,θ¨b:
(61)ηPτq=∏i=1pεPτi=detMPτMPτT,ηbτq=∏i=16εbτi=detMbτMbτT.


## 4.2.3. Dynamic Coupling Relationship between Active Joints and End-Effector

Becausev̇eT,ω̇eTT does not appear in the Lagrange equation, we cannot find a mapping relation between θ¨A and v̇eT,ω̇eTT in equation (29), but we can find the dynamic coupling relationship between active joints and the end-effector from another perspective [12]. Calculating derivative of equation (16), we have
(62)v̇eω̇e=JeAθ¨A+J̇eAθ̇A.Multiplying equation (57) by JeAMτA−1, we have
(63)JeAθ¨A+JeAMτA−1CτA=JeAMτA−1τA⇒JeAθ¨A+J̇eAθ̇A=JeAMτA−1τA+J̇eAθ̇A−JeAMτA−1CτA⇒v̇eω̇e=MeττA+Ceτ,where Meτ=JeAMτA−1∈R6×a is the coupling inertia matrix representing the transmission from τA to v̇eT,ω̇eTT and Ceτ=J̇eAθ̇A−JeAMτA−1CτA∈R6×1 is the nonlinear item. Equation (63) is the dynamic coupling relationship between active joints and the end-effector. The dynamic coupling index between active joints and the end-effector is the manipulability of Meτ:
(64)ηeτq=∏i=16εeτi=detMeτMeτT.Ifηeτq is close to zero, v̇eT,ω̇eTT is uncontrollable for τA. On the contrary, we can design the control law for v̇eT,ω̇eTT to calculate
(65)τA=Meτ+v̇eω̇e−Ceτ.Equations (60) and (63) can be reduced to the following constraints:
(66)g1θ¨b,θ¨AorτA,t=0,g2θ¨b,θ¨AorτA,t=0,g3v̇e,ω̇e,τA,t=0.Theorem.
The constraints in equation (70) are also nonholonomic.Proof.
Taking equation (49) as an example, we have
(67)MPAθ¨A−θ¨P+CPA=0⇒0p×6−Ep×pMPAq¨+CPA=0,making MsPA=0p×6−Ep×pMPA∈Rp×6+n. Assuming there exists the first-order constraint pq,q̇=0 whose derivative with respect to time is
(68)dpdt=∂p∂qq̇+∂p∂q̇q¨.Assuming constraint in equation (66) is holonomic, we have
(69)∂p∂qq̇=CPA,∂p∂q̇=MsPA.Consideringq̇=0, we can obtain the general solution of pq,q̇:
(70)pq,q̇=MsPAq̇+sq+C=0.sq is relative to the state variable q. C is a constant vector. The derivative of equation (70) is
(71)dpdt=MsPAq¨+ṀsPAq̇+dsdqq̇=0.Substitutingcb and cP into equation (71), and comparing it with equation (67), we obtain
(72)M1ṀbṀbmPṀbmA−M2ṀmbPṀmPPṀmPA−ṀsPA−dsdqq̇+12q̇T∂Μ∂qP−∂Μ∂qbq̇=0.where M1=MmbP+MmPP−Mb−1MbmP+Mb−1 and M2=MmbP+MmPP−Mb−1MbmP+MmbP+. To satisfy the above equality, we have
(73)dsdq=M1ṀbṀbmPṀbmA−M2ṀmbPṀmPPṀmPA−ṀsPA,∂Μ∂qP=∂Μ∂qb.sq is only relative to q instead of q̇, so the first equality is not satisfied. The second equality is also not satisfied because θP and θb are not cyclic coordinates in Μ. Therefore, pq,q̇=0 does not exist and the constraints in equation (66) are nonholonomic. Because equation (66) is the constraints among the second-order derivatives of θb,θA,θP,pe,Ψe, they are called as the second-order nonholonomic constraint.The free-floating space manipulator with free-swinging joint failure simultaneously contains the first-order nonholonomic constraint and the second-order nonholonomic constraint, and they are independent with each other. From a mathematical viewpoint, equations (17) and (66) belong to differential equations, and they have the same solution. In fact, because null space exists in equation (14), we can find the same solution to these two differential equations. Therefore, motions of the free-floating space manipulator with free-swinging joint failure simultaneously satisfy the first-order and second-order nonholonomic constraints.The analysis above is not based on conservation of system momentum, so even if base mass and inertia are very large, the degree of dynamic coupling may be large enough to enable active joints to actuate passive units through the control method.In summary, dynamic coupling relationship between active joints and all passive units is(74)abP=θ¨bθ¨P=MbPθ¨A+CbP,orat=θ¨bθ¨Pv̇eω̇e=MtττA+Ctτ.at∈Rp+12×1 is the acceleration of the entire task space.
(75)MbP=MbAMPA∈Rp+6×a,is the coupling inertia matrix representing the transmission from θ¨A to θ¨bT,θ¨PTT, and
(76)Mtτ=MbτMPτMeτ∈Rp+12×a,is the coupling inertia matrix representing the transmission from τA to at.
(77)CbP=CbACPA∈Rp+6×1,Ctτ=CbτCPτCeτ∈Rp+12×1,are nonlinear items. Based on dynamic coupling relationships, active joints can actuate the manipulator to execute tasks as the same as motion planning. For general tasks, we should select the task acceleration vector atask∈Rf×1 according to task requirements and construct MataskA∈Rf×a and CataskA∈Rf×1. Then, we design the control law for atask to guarantee that the selected passive units will converge to the equilibrium point, and τA is obtained. If a>f, the space manipulator is redundant and the solutions of θ¨A and τA are
(78)θ¨A=MataskA+atask−CataskA+I−MataskA+MataskAς,orτA=MataskA+atask−CataskA+I−MataskA+MataskAς.ς¨∈Ra×1 is an arbitrary vector. Appropriately designing ς, the manipulator can perform primary and secondary tasks at the same time.


## 4.3. Global Kinematic and Dynamic Coupling Indexes

The kinematic and dynamic coupling indexes defined above are with respect to system state variablesq. If their values are large, it only indicates that motion transmission efficiency from active joints to passive units is high only at the current q, so the indexes above are called as local indexes for kinematic and dynamic coupling in state space. However, before structure design, motion planning, and controller design, the system state is unknown, so the local coupling indexes are not appropriate at this moment. We care more about the global coupling degree in the entire state space, and global kinematic and dynamic coupling indexes are defined as
(79)w∗g=∫qw∗2qdq∫qdq,η∗g=∫qη∗2qdq∫qdq.w∗ and w∗g correspond to local and global manipulability of a Jacobian matrix J∗, respectively. η∗ and η∗g correspond to local and global manipulability of an inertia matrix M∗, respectively. If kinematic and dynamic parameters are determined, global indexes can be calculated. The higher the values of global indexes, the higher the motion transmission efficiency from active joints to passive units whatever state the manipulator is in. If selecting different healthy joints as active joints, different values of global indexes will be obtained. There exists the case that values of global indexes are close whether we select some healthy joins. It indicates that these joints are incapable to actuate passive units, so we should not select them as active joints.


## 5. Simulation Experiments

In simulation, we plan motions for the space manipulator with free-swing joint failure to perform some tasks to verify the existence and correctness of the first-order nonholonomic constraint. Then, we give a study case for calculating kinematic and dynamic indexes and discuss the variation ofwPAg and ηPAg with the increasing base mass and inertia. In particular, aiming at the case where a passive regulating task cannot be completed through motion planning with large base mass and inertia, we design the control law based on PD controller for θ¨P, and the task is completed based on the second-order nonholonomic constraint. The practicability of the coupling index and correctness of the second-order nonholonomic constraint are verified.Simulation experiments are carried out surrounding SSRMS-type 7 DOF space manipulator in Figure2. Its kinematic and dynamic parameters are in Tables 1 and 2.Figure 2
SSRMS-type 7 DOF space manipulator.Table 1
Kinematic parameters of SSRMS-type 7 DOF space manipulator.
Joint no.
Torsion angleαi−1 (°)
Link lengthai−1 (m)
Joint angleθi (°)
Link offsetdi (m)
1
0
0
180
1.1
2
−90
0
−90
0.5
3
−90
0
0
0.5
4
0
5
0
0.5
5
0
5
0
0.5
6
90
0
90
0.5
7
90
0
180
1.1Table 2
Dynamic parameters of SSRMS-type 7 DOF space manipulator.
Joint no.
Link massmi (kg)
Link inertiamdiagIxxi,Iyyi,Izzi (kg·m2)
Link centroid positionPcii (m)
0
106
Diag[106, 106, 106]
[0, 0, 0]T
1
42.5
Diag[0.8854, 0.0531, 0.8854]
[0, 0.25, 0]T
2
42.5
Diag[0.8854, 0.0531, 0.8854]
[0, 0.25, 0]T
3
70
Diag[0.0875, 145.8333, 145.8333]
[2.5, 0, 0]T
4
70
Diag[0.0875, 145.8333, 145.8333]
[2.5, 0, 0]T
5
42.5
Diag[0.8854, 0.8854, 0.0531]
[0, 0, 0.25]T
6
42.5
Diag[0.8854, 0.8854, 0.0531]
[0, 0, 0.25]T
7
42.5
Diag[1.2750, 1.2750, 0.0531]
[0, 0, 0.55]T


### 5.1. Motion Planning for the Space Manipulator with Free-Swinging Joint Failure

Aiming at the end-effector trajectory tracking and passive joint regulating tasks and the combination tasks that track end-effector trajectory and limit base attitude deflection when regulating passive joints, we plan motions for the space manipulator based on the first-order nonholonomic constraint. The existence of the first-order nonholonomic constraint is verified, and the motion planning based on kinematic coupling relationships is applied.


#### 5.1.1. End-Effector Trajectory Tracking Task

For the SSRMS-type space manipulator with the single joint failure, it has 6 healthy joints. Selecting all healthy joints as active joints, the end-effector trajectory tracking task can be performed. Assuming the first joint fails, the setting initial manipulator configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and the initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires the end-effector to move along an arc trajectory to position/attitude xe_desT,Ψe_desTT=5m,2m,6m,60°,−30°,−90°T and pass the middle point xe_mid=4m,3m,5.5mT. Task execution time is 40 s. We interpolate the velocity of the end-effector with the quartic polynomial
(80)vt=1.0734e−6t4−8.5874e−5t3+1.7175e−3t2.The variation ofxeT,ΨeTT=xe,ye,ze,αe,βe,γeT is in Figure 3, and the comparison between the actual and the desired end-effector trajectory is in Figure 4.Figure 3
The variation ofxeT,ΨeTT with execution of the trajectory tracking task.
(a)
End-effector position
(b)
End-effector attitudeFigure 4
Comparison between the actual and desired end-effector trajectory.The joint motion is shown in Figure5.Figure 5
The variations of joint angle and velocity with execution of the trajectory tracking task.
(a)
Joint angle
(b)
Joint velocityAs can be seen from Figures3–5, end-effector moves along the desired trajectory to arrive at the desired position/attitude, and the joint motion is smooth, so the end-effector trajectory tracking task is accomplished based on motion planning. Velocity of the passive first joint is calculated from equation (15), and its angle is obtained by integral.


#### 5.1.2. Passive Joint Regulating Task

If the number of passive joints is less than or equal to 3, there will be at least 4 healthy joints and the manipulator can perform passive joint regulating task. Assuming the first and fourth joints fail, the initial configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and the initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires all passive joints to be regulated to θP_des=−30°,−30°T in 40 s. However, for the base mass and inertia in Table 2, active joints cannot regulate passive joints because the kinematic coupling degree is seriously insufficient. Therefore, we reset the base mass as m0=100kg and inertia as I0=diag100kg⋅m2,100kg⋅m2,100kg⋅m2. Interpolating θ̇P with the quartic polynomial, variations of θP and θ̇P are shown in Figure 6.Figure 6
The variations of passive joint angle and velocity with execution of the passive joint regulating task.
(a)
Passive joint angle
(b)
Passive joint velocityVariations ofθA and θ̇A are shown in Figure 7.Figure 7
The variations of active joint angle and velocity with execution of the passive joint regulating task.
(a)
Active joint angle
(b)
Active joint velocityIt can be seen that two passive joints are regulated to −30° in 40 s and the motion of active joints is smooth, so the passive joint regulating task is accomplished.


#### 5.1.3. Tracking End-Effector Trajectory When Regulating Passive Joints

If the number of passive joints is less than or equal to 2, there will be at least 5 healthy joints, and active joints can actuate the end-effector to the desired position and regulate passive joints to target angles simultaneously. Assuming the first joint fails, the initial configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires the end-effector to track a straight trajectory to the desired position xe_des=4m,2m,3mT and the passive joints to be regulated to θP_des=0° in 40 s. Base mass and inertia are the same with the passive regulating task. Variations of xe and θP are in Figure 8. The comparison between the actual and the desired trajectory is in Figure 9.Figure 8
The variations of the end-effector position and passive joint angle with execution of the combination task.
(a)
End-effector position
(b)
Passive joint angleFigure 9
Comparison between the actual and the desired end-effector trajectory for the combination task.The end-effector arrives at the desired position, and the passive joint is regulated to the target angle in 40 s, so the combination task is accomplished.


#### 5.1.4. Limiting Base Attitude Deflection When Regulating Passive Joints

If the number of passive joints is less than or equal to 2, active joints can eliminate base attitude deflection when regulating passive joints. Assuming the first joint fails, the initial configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and the initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires the passive joint to be regulated to θP_des=0°, and base attitude is not disturbed. Base mass and inertia are also the same with the passive regulating task. Variations of θP and θb are in Figure 10.Figure 10
The variations of passive joint angle and base attitude with execution of the combination task.
(a)
Passive joint angle
(b)
Base attitudeFigure10 shows that the passive joint converges to 0° in 40 s and the order of magnitudes of base attitude deflection is only 10−11°, so the combination task is accomplished.During task execution, the values of kinematic coupling indexes are large enough, or else the tasks cannot be accomplished. The simulation results above tell us the space manipulator can execute many typical tasks based on motion planning, which verifies the existence and correctness of the first-order nonholonomic constraint inside the system.


### 5.2. Calculation and Application Example for Kinematic and Dynamic Coupling Indexes

Aiming at the space manipulator in Figure2 with free-swinging first joint failure, we give an example for calculating local and global kinematic and dynamic coupling indexes and discuss the differences of values of coupling indexes when selecting different healthy joints as active joints. Then, we analyze the variations of wPAg and ηPAg with the increasing base mass and inertia in particular and point out that when the base mass and inertia are large, passive joint regulation and base deflection limitation cannot be realized through motion planning. In particular, aiming at the passive joint regulating task, we design the control law for θ¨P based on PD controller and calculate τA to actuate the manipulator to complete the task.


#### 5.2.1. Calculation of Coupling Indexes

Setting the configuration asθ=−50°,−170°,150°,−60°,130°,170°,0°T, the base position/attitude is θb=0,0,0,0,0,0T, the base mass is m0=100kg, and the base inertia is I0=diag100kg⋅m2,100kg⋅m2,100kg⋅m2. Assuming that the rotation range of every joint is θi_min,θi_max=−180°,180°,i=1,2,⋯,7, the base position deflection limitation is x0j_min,x0j_max=−1m,1m,j=1,2,3 and the attitude deflection limitation is Ψ0k_min,Ψ0k_max=−20°,20°,k=1,2,3. For the space manipulator with free-swinging first joint failure in Figure 2, selecting all healthy joints as active joints, we calculate the kinematic and dynamic coupling indexes in Table 3.Table 3
Kinematic and dynamic coupling indexes.
Local value
Global value
Single
Maximum
Minimum
weA
0.0016
1.9130
1.0320×10−7
0.0085
wPA
0.1475
1.6794
0.0082
0.1962
wbA
8.8645×10−10
8.8645×10−6
7.4366×10−16
3.4419×10−16
ηPA
0.6014
210.9814
0.0165
9.8958
ηbA
0
2.1145×10−35
0
0
ηPτ
0.0079
1.6818×103
2.1660×10−4
15.9581
ηbτ
0
2.1145×10−31
0
0
ηeτ
6.8770×10−11
1.7706×10−5
2.5938×10−16
4.2568×10−15In Table3, we make the value which is less than 10−40 equal to zero. Thus, the dynamic coupling degree between active joints and the base is insufficient. In particular, if we lock the fourth and seventh joints, and select other healthy joints as active joints, kinematic and dynamic coupling indexes turn into what is found in Table 4.Table 4
Kinematic and dynamic coupling indexes with the fourth and seventh joints locked.
Local value
Global value
Single
Maximum
Minimum
weA4,7lock
3.2168×10−18
8.4015×10−16
0
7.4488×10−35
wPA4,7lock
0.1338
2.0813
0.0025
0.1471
wbA4,7lock
6.4487×10−21
7.3665×10−18
0
3.7389×10−40
ηPA4,7lock
0.5825
145.4495
0.0072
8.0662
ηbA4,7lock
0
1.2862×10−36
0
0
ηPτ4,7lock
0.0078
968.7484
1.2646×10−4
13.3688
ηbτ4,7lock
0
1.0415×10−31
0
0
ηeτ4,7lock
3.9886×10−26
1.1401×10−14
5.7742×10−30
6.0911×10−34As can be seen from Table4, active joints without the fourth and seventh joints are incapable of actuating the end-effector or controlling the base because the number of active joints is less than the DOF of the end-effector and base. However, values of wPA, ηPA, and ηPτ in Tables 3 and 4 are similar, especially for global indexes, so active joints without the fourth and seventh joints can still regulate passive joints.The global indexes can not only depict the motion transmission efficiency from active joints to passive units but also serve as the reference to reducing the number of active joints. Users can select the fewest active joints with enough coupling degree to actuate the manipulator.


#### 5.2.2. Effect of Base Mass and Inertia onwPAg and ηPAg

Making base mass and inertia are(81)m0=mc,I0=mc000mc000mc.mc is the equivalent mass of the base [30]. We set the values of mc as mc∈101,102,103,104,105,106,107kg. Assuming the first joint fails and selecting all healthy joints as active joints and taking lgmc as the x-axis, we draw the figures to exhibit the variation of wPAg and ηPAg with mc in Figure 11.Figure 11
The variations ofwPAg and ηPAg with mc.
(a)
Variation ofwPAg
(b)
Variation ofηPAgWith increase of the base mass and inertia,wPAg and ηPAg decrease continually. When mc>106, the value of wPAg reduces to 10−8 and the magnitudes of change in wPAg reach up to 107. But value of ηPAg stabilizes around 1.4, and the magnitudes of change in ηPAg are only 101. Thus, active joints cannot effectively regulate passive joints based on motion planning, but can regulate passive joints based on the control method at this moment.We design the control method to regulate passive joints based on the second-order nonholonomic constraint. We makeθP_des=−30°, θ̇P_des=0°/s, and θ¨P_des=0°/s2 as the desired states of passive joints and design the PD controller for θ¨P:
(82)θ¨P=θ¨P_des+KDθ̇P_des−θ̇P+KPθP_des−θP.Makinge=θP_des−θP, we have e¨+KDė+KPe=0. According to the characteristic of the second-order homogeneous differential equation, e exponentially converges to zero. Taking KD=0.5 and KP=0.1, θ¨P and θP are in Figure 12 and θA, θ¨A, and τA are in Figure 13.Figure 12
The motion of passive joints with the PD controller.
(a)
Passive joint acceleration
(b)
Passive joint angleFigure 13
The variations of active joint motion and torque.
(a)
Active joint angle
(b)
Active joint acceleration
(c)
Active joint torqueFigure12 shows that the passive joint is regulated to the desired angle at 28.1 s, and the active joint’s motion and torque are smooth in Figure 13. Thus, the passive joint regulating task is accomplished based on the control method. The PD controller above is only appropriate to the space manipulator with an accurate dynamic model. Considering model uncertainty and external disturbance, a robust control system needs to be designed. But in this paper, we mainly analyze the kinematic and dynamic characteristics of the space manipulator with free-swinging joint failure, instead of a robust controller design. In the future study, we will combine the sliding mode and fuzzy control method to design a controller robust to the bounded unknown disturbance and unmodeled uncertainty for the space manipulator with free-swinging joint failure.The simulation above verifies the following theories:(1)
The first-order nonholonomic constraint and the second-order nonholonomic constraint do simultaneously exist inside the system of the space manipulator with free-swinging joint failure
(2)
The kinematic and dynamic coupling indexes proposed in this paper can objectively evaluate kinematic and dynamic coupling degrees from active joints to passive units
(3)
Many space tasks are either based on motion planning with the first-order nonholonomic constraint or based on a controller design with the second-order nonholonomic constraint


## 5.1. Motion Planning for the Space Manipulator with Free-Swinging Joint Failure

Aiming at the end-effector trajectory tracking and passive joint regulating tasks and the combination tasks that track end-effector trajectory and limit base attitude deflection when regulating passive joints, we plan motions for the space manipulator based on the first-order nonholonomic constraint. The existence of the first-order nonholonomic constraint is verified, and the motion planning based on kinematic coupling relationships is applied.


### 5.1.1. End-Effector Trajectory Tracking Task

For the SSRMS-type space manipulator with the single joint failure, it has 6 healthy joints. Selecting all healthy joints as active joints, the end-effector trajectory tracking task can be performed. Assuming the first joint fails, the setting initial manipulator configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and the initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires the end-effector to move along an arc trajectory to position/attitude xe_desT,Ψe_desTT=5m,2m,6m,60°,−30°,−90°T and pass the middle point xe_mid=4m,3m,5.5mT. Task execution time is 40 s. We interpolate the velocity of the end-effector with the quartic polynomial
(80)vt=1.0734e−6t4−8.5874e−5t3+1.7175e−3t2.The variation ofxeT,ΨeTT=xe,ye,ze,αe,βe,γeT is in Figure 3, and the comparison between the actual and the desired end-effector trajectory is in Figure 4.Figure 3
The variation ofxeT,ΨeTT with execution of the trajectory tracking task.
(a)
End-effector position
(b)
End-effector attitudeFigure 4
Comparison between the actual and desired end-effector trajectory.The joint motion is shown in Figure5.Figure 5
The variations of joint angle and velocity with execution of the trajectory tracking task.
(a)
Joint angle
(b)
Joint velocityAs can be seen from Figures3–5, end-effector moves along the desired trajectory to arrive at the desired position/attitude, and the joint motion is smooth, so the end-effector trajectory tracking task is accomplished based on motion planning. Velocity of the passive first joint is calculated from equation (15), and its angle is obtained by integral.


### 5.1.2. Passive Joint Regulating Task

If the number of passive joints is less than or equal to 3, there will be at least 4 healthy joints and the manipulator can perform passive joint regulating task. Assuming the first and fourth joints fail, the initial configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and the initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires all passive joints to be regulated to θP_des=−30°,−30°T in 40 s. However, for the base mass and inertia in Table 2, active joints cannot regulate passive joints because the kinematic coupling degree is seriously insufficient. Therefore, we reset the base mass as m0=100kg and inertia as I0=diag100kg⋅m2,100kg⋅m2,100kg⋅m2. Interpolating θ̇P with the quartic polynomial, variations of θP and θ̇P are shown in Figure 6.Figure 6
The variations of passive joint angle and velocity with execution of the passive joint regulating task.
(a)
Passive joint angle
(b)
Passive joint velocityVariations ofθA and θ̇A are shown in Figure 7.Figure 7
The variations of active joint angle and velocity with execution of the passive joint regulating task.
(a)
Active joint angle
(b)
Active joint velocityIt can be seen that two passive joints are regulated to −30° in 40 s and the motion of active joints is smooth, so the passive joint regulating task is accomplished.


### 5.1.3. Tracking End-Effector Trajectory When Regulating Passive Joints

If the number of passive joints is less than or equal to 2, there will be at least 5 healthy joints, and active joints can actuate the end-effector to the desired position and regulate passive joints to target angles simultaneously. Assuming the first joint fails, the initial configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires the end-effector to track a straight trajectory to the desired position xe_des=4m,2m,3mT and the passive joints to be regulated to θP_des=0° in 40 s. Base mass and inertia are the same with the passive regulating task. Variations of xe and θP are in Figure 8. The comparison between the actual and the desired trajectory is in Figure 9.Figure 8
The variations of the end-effector position and passive joint angle with execution of the combination task.
(a)
End-effector position
(b)
Passive joint angleFigure 9
Comparison between the actual and the desired end-effector trajectory for the combination task.The end-effector arrives at the desired position, and the passive joint is regulated to the target angle in 40 s, so the combination task is accomplished.


### 5.1.4. Limiting Base Attitude Deflection When Regulating Passive Joints

If the number of passive joints is less than or equal to 2, active joints can eliminate base attitude deflection when regulating passive joints. Assuming the first joint fails, the initial configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and the initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires the passive joint to be regulated to θP_des=0°, and base attitude is not disturbed. Base mass and inertia are also the same with the passive regulating task. Variations of θP and θb are in Figure 10.Figure 10
The variations of passive joint angle and base attitude with execution of the combination task.
(a)
Passive joint angle
(b)
Base attitudeFigure10 shows that the passive joint converges to 0° in 40 s and the order of magnitudes of base attitude deflection is only 10−11°, so the combination task is accomplished.During task execution, the values of kinematic coupling indexes are large enough, or else the tasks cannot be accomplished. The simulation results above tell us the space manipulator can execute many typical tasks based on motion planning, which verifies the existence and correctness of the first-order nonholonomic constraint inside the system.


## 5.1.1. End-Effector Trajectory Tracking Task

For the SSRMS-type space manipulator with the single joint failure, it has 6 healthy joints. Selecting all healthy joints as active joints, the end-effector trajectory tracking task can be performed. Assuming the first joint fails, the setting initial manipulator configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and the initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires the end-effector to move along an arc trajectory to position/attitude xe_desT,Ψe_desTT=5m,2m,6m,60°,−30°,−90°T and pass the middle point xe_mid=4m,3m,5.5mT. Task execution time is 40 s. We interpolate the velocity of the end-effector with the quartic polynomial
(80)vt=1.0734e−6t4−8.5874e−5t3+1.7175e−3t2.The variation ofxeT,ΨeTT=xe,ye,ze,αe,βe,γeT is in Figure 3, and the comparison between the actual and the desired end-effector trajectory is in Figure 4.Figure 3
The variation ofxeT,ΨeTT with execution of the trajectory tracking task.
(a)
End-effector position
(b)
End-effector attitudeFigure 4
Comparison between the actual and desired end-effector trajectory.The joint motion is shown in Figure5.Figure 5
The variations of joint angle and velocity with execution of the trajectory tracking task.
(a)
Joint angle
(b)
Joint velocityAs can be seen from Figures3–5, end-effector moves along the desired trajectory to arrive at the desired position/attitude, and the joint motion is smooth, so the end-effector trajectory tracking task is accomplished based on motion planning. Velocity of the passive first joint is calculated from equation (15), and its angle is obtained by integral.


## 5.1.2. Passive Joint Regulating Task

If the number of passive joints is less than or equal to 3, there will be at least 4 healthy joints and the manipulator can perform passive joint regulating task. Assuming the first and fourth joints fail, the initial configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and the initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires all passive joints to be regulated to θP_des=−30°,−30°T in 40 s. However, for the base mass and inertia in Table 2, active joints cannot regulate passive joints because the kinematic coupling degree is seriously insufficient. Therefore, we reset the base mass as m0=100kg and inertia as I0=diag100kg⋅m2,100kg⋅m2,100kg⋅m2. Interpolating θ̇P with the quartic polynomial, variations of θP and θ̇P are shown in Figure 6.Figure 6
The variations of passive joint angle and velocity with execution of the passive joint regulating task.
(a)
Passive joint angle
(b)
Passive joint velocityVariations ofθA and θ̇A are shown in Figure 7.Figure 7
The variations of active joint angle and velocity with execution of the passive joint regulating task.
(a)
Active joint angle
(b)
Active joint velocityIt can be seen that two passive joints are regulated to −30° in 40 s and the motion of active joints is smooth, so the passive joint regulating task is accomplished.


## 5.1.3. Tracking End-Effector Trajectory When Regulating Passive Joints

If the number of passive joints is less than or equal to 2, there will be at least 5 healthy joints, and active joints can actuate the end-effector to the desired position and regulate passive joints to target angles simultaneously. Assuming the first joint fails, the initial configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires the end-effector to track a straight trajectory to the desired position xe_des=4m,2m,3mT and the passive joints to be regulated to θP_des=0° in 40 s. Base mass and inertia are the same with the passive regulating task. Variations of xe and θP are in Figure 8. The comparison between the actual and the desired trajectory is in Figure 9.Figure 8
The variations of the end-effector position and passive joint angle with execution of the combination task.
(a)
End-effector position
(b)
Passive joint angleFigure 9
Comparison between the actual and the desired end-effector trajectory for the combination task.The end-effector arrives at the desired position, and the passive joint is regulated to the target angle in 40 s, so the combination task is accomplished.


## 5.1.4. Limiting Base Attitude Deflection When Regulating Passive Joints

If the number of passive joints is less than or equal to 2, active joints can eliminate base attitude deflection when regulating passive joints. Assuming the first joint fails, the initial configuration isθini=−50°,−170°,150°,−60°,130°,170°,0°T and the initial base position/attitude is θb_ini=0,0,0,0,0,0T. The task requires the passive joint to be regulated to θP_des=0°, and base attitude is not disturbed. Base mass and inertia are also the same with the passive regulating task. Variations of θP and θb are in Figure 10.Figure 10
The variations of passive joint angle and base attitude with execution of the combination task.
(a)
Passive joint angle
(b)
Base attitudeFigure10 shows that the passive joint converges to 0° in 40 s and the order of magnitudes of base attitude deflection is only 10−11°, so the combination task is accomplished.During task execution, the values of kinematic coupling indexes are large enough, or else the tasks cannot be accomplished. The simulation results above tell us the space manipulator can execute many typical tasks based on motion planning, which verifies the existence and correctness of the first-order nonholonomic constraint inside the system.


## 5.2. Calculation and Application Example for Kinematic and Dynamic Coupling Indexes

Aiming at the space manipulator in Figure2 with free-swinging first joint failure, we give an example for calculating local and global kinematic and dynamic coupling indexes and discuss the differences of values of coupling indexes when selecting different healthy joints as active joints. Then, we analyze the variations of wPAg and ηPAg with the increasing base mass and inertia in particular and point out that when the base mass and inertia are large, passive joint regulation and base deflection limitation cannot be realized through motion planning. In particular, aiming at the passive joint regulating task, we design the control law for θ¨P based on PD controller and calculate τA to actuate the manipulator to complete the task.


### 5.2.1. Calculation of Coupling Indexes

Setting the configuration asθ=−50°,−170°,150°,−60°,130°,170°,0°T, the base position/attitude is θb=0,0,0,0,0,0T, the base mass is m0=100kg, and the base inertia is I0=diag100kg⋅m2,100kg⋅m2,100kg⋅m2. Assuming that the rotation range of every joint is θi_min,θi_max=−180°,180°,i=1,2,⋯,7, the base position deflection limitation is x0j_min,x0j_max=−1m,1m,j=1,2,3 and the attitude deflection limitation is Ψ0k_min,Ψ0k_max=−20°,20°,k=1,2,3. For the space manipulator with free-swinging first joint failure in Figure 2, selecting all healthy joints as active joints, we calculate the kinematic and dynamic coupling indexes in Table 3.Table 3
Kinematic and dynamic coupling indexes.
Local value
Global value
Single
Maximum
Minimum
weA
0.0016
1.9130
1.0320×10−7
0.0085
wPA
0.1475
1.6794
0.0082
0.1962
wbA
8.8645×10−10
8.8645×10−6
7.4366×10−16
3.4419×10−16
ηPA
0.6014
210.9814
0.0165
9.8958
ηbA
0
2.1145×10−35
0
0
ηPτ
0.0079
1.6818×103
2.1660×10−4
15.9581
ηbτ
0
2.1145×10−31
0
0
ηeτ
6.8770×10−11
1.7706×10−5
2.5938×10−16
4.2568×10−15In Table3, we make the value which is less than 10−40 equal to zero. Thus, the dynamic coupling degree between active joints and the base is insufficient. In particular, if we lock the fourth and seventh joints, and select other healthy joints as active joints, kinematic and dynamic coupling indexes turn into what is found in Table 4.Table 4
Kinematic and dynamic coupling indexes with the fourth and seventh joints locked.
Local value
Global value
Single
Maximum
Minimum
weA4,7lock
3.2168×10−18
8.4015×10−16
0
7.4488×10−35
wPA4,7lock
0.1338
2.0813
0.0025
0.1471
wbA4,7lock
6.4487×10−21
7.3665×10−18
0
3.7389×10−40
ηPA4,7lock
0.5825
145.4495
0.0072
8.0662
ηbA4,7lock
0
1.2862×10−36
0
0
ηPτ4,7lock
0.0078
968.7484
1.2646×10−4
13.3688
ηbτ4,7lock
0
1.0415×10−31
0
0
ηeτ4,7lock
3.9886×10−26
1.1401×10−14
5.7742×10−30
6.0911×10−34As can be seen from Table4, active joints without the fourth and seventh joints are incapable of actuating the end-effector or controlling the base because the number of active joints is less than the DOF of the end-effector and base. However, values of wPA, ηPA, and ηPτ in Tables 3 and 4 are similar, especially for global indexes, so active joints without the fourth and seventh joints can still regulate passive joints.The global indexes can not only depict the motion transmission efficiency from active joints to passive units but also serve as the reference to reducing the number of active joints. Users can select the fewest active joints with enough coupling degree to actuate the manipulator.


### 5.2.2. Effect of Base Mass and Inertia onwPAg and ηPAg

Making base mass and inertia are(81)m0=mc,I0=mc000mc000mc.mc is the equivalent mass of the base [30]. We set the values of mc as mc∈101,102,103,104,105,106,107kg. Assuming the first joint fails and selecting all healthy joints as active joints and taking lgmc as the x-axis, we draw the figures to exhibit the variation of wPAg and ηPAg with mc in Figure 11.Figure 11
The variations ofwPAg and ηPAg with mc.
(a)
Variation ofwPAg
(b)
Variation ofηPAgWith increase of the base mass and inertia,wPAg and ηPAg decrease continually. When mc>106, the value of wPAg reduces to 10−8 and the magnitudes of change in wPAg reach up to 107. But value of ηPAg stabilizes around 1.4, and the magnitudes of change in ηPAg are only 101. Thus, active joints cannot effectively regulate passive joints based on motion planning, but can regulate passive joints based on the control method at this moment.We design the control method to regulate passive joints based on the second-order nonholonomic constraint. We makeθP_des=−30°, θ̇P_des=0°/s, and θ¨P_des=0°/s2 as the desired states of passive joints and design the PD controller for θ¨P:
(82)θ¨P=θ¨P_des+KDθ̇P_des−θ̇P+KPθP_des−θP.Makinge=θP_des−θP, we have e¨+KDė+KPe=0. According to the characteristic of the second-order homogeneous differential equation, e exponentially converges to zero. Taking KD=0.5 and KP=0.1, θ¨P and θP are in Figure 12 and θA, θ¨A, and τA are in Figure 13.Figure 12
The motion of passive joints with the PD controller.
(a)
Passive joint acceleration
(b)
Passive joint angleFigure 13
The variations of active joint motion and torque.
(a)
Active joint angle
(b)
Active joint acceleration
(c)
Active joint torqueFigure12 shows that the passive joint is regulated to the desired angle at 28.1 s, and the active joint’s motion and torque are smooth in Figure 13. Thus, the passive joint regulating task is accomplished based on the control method. The PD controller above is only appropriate to the space manipulator with an accurate dynamic model. Considering model uncertainty and external disturbance, a robust control system needs to be designed. But in this paper, we mainly analyze the kinematic and dynamic characteristics of the space manipulator with free-swinging joint failure, instead of a robust controller design. In the future study, we will combine the sliding mode and fuzzy control method to design a controller robust to the bounded unknown disturbance and unmodeled uncertainty for the space manipulator with free-swinging joint failure.The simulation above verifies the following theories:(1)
The first-order nonholonomic constraint and the second-order nonholonomic constraint do simultaneously exist inside the system of the space manipulator with free-swinging joint failure
(2)
The kinematic and dynamic coupling indexes proposed in this paper can objectively evaluate kinematic and dynamic coupling degrees from active joints to passive units
(3)
Many space tasks are either based on motion planning with the first-order nonholonomic constraint or based on a controller design with the second-order nonholonomic constraint


## 5.2.1. Calculation of Coupling Indexes

Setting the configuration asθ=−50°,−170°,150°,−60°,130°,170°,0°T, the base position/attitude is θb=0,0,0,0,0,0T, the base mass is m0=100kg, and the base inertia is I0=diag100kg⋅m2,100kg⋅m2,100kg⋅m2. Assuming that the rotation range of every joint is θi_min,θi_max=−180°,180°,i=1,2,⋯,7, the base position deflection limitation is x0j_min,x0j_max=−1m,1m,j=1,2,3 and the attitude deflection limitation is Ψ0k_min,Ψ0k_max=−20°,20°,k=1,2,3. For the space manipulator with free-swinging first joint failure in Figure 2, selecting all healthy joints as active joints, we calculate the kinematic and dynamic coupling indexes in Table 3.Table 3
Kinematic and dynamic coupling indexes.
Local value
Global value
Single
Maximum
Minimum
weA
0.0016
1.9130
1.0320×10−7
0.0085
wPA
0.1475
1.6794
0.0082
0.1962
wbA
8.8645×10−10
8.8645×10−6
7.4366×10−16
3.4419×10−16
ηPA
0.6014
210.9814
0.0165
9.8958
ηbA
0
2.1145×10−35
0
0
ηPτ
0.0079
1.6818×103
2.1660×10−4
15.9581
ηbτ
0
2.1145×10−31
0
0
ηeτ
6.8770×10−11
1.7706×10−5
2.5938×10−16
4.2568×10−15In Table3, we make the value which is less than 10−40 equal to zero. Thus, the dynamic coupling degree between active joints and the base is insufficient. In particular, if we lock the fourth and seventh joints, and select other healthy joints as active joints, kinematic and dynamic coupling indexes turn into what is found in Table 4.Table 4
Kinematic and dynamic coupling indexes with the fourth and seventh joints locked.
Local value
Global value
Single
Maximum
Minimum
weA4,7lock
3.2168×10−18
8.4015×10−16
0
7.4488×10−35
wPA4,7lock
0.1338
2.0813
0.0025
0.1471
wbA4,7lock
6.4487×10−21
7.3665×10−18
0
3.7389×10−40
ηPA4,7lock
0.5825
145.4495
0.0072
8.0662
ηbA4,7lock
0
1.2862×10−36
0
0
ηPτ4,7lock
0.0078
968.7484
1.2646×10−4
13.3688
ηbτ4,7lock
0
1.0415×10−31
0
0
ηeτ4,7lock
3.9886×10−26
1.1401×10−14
5.7742×10−30
6.0911×10−34As can be seen from Table4, active joints without the fourth and seventh joints are incapable of actuating the end-effector or controlling the base because the number of active joints is less than the DOF of the end-effector and base. However, values of wPA, ηPA, and ηPτ in Tables 3 and 4 are similar, especially for global indexes, so active joints without the fourth and seventh joints can still regulate passive joints.The global indexes can not only depict the motion transmission efficiency from active joints to passive units but also serve as the reference to reducing the number of active joints. Users can select the fewest active joints with enough coupling degree to actuate the manipulator.


## 5.2.2. Effect of Base Mass and Inertia onwPAg and ηPAg

Making base mass and inertia are(81)m0=mc,I0=mc000mc000mc.mc is the equivalent mass of the base [30]. We set the values of mc as mc∈101,102,103,104,105,106,107kg. Assuming the first joint fails and selecting all healthy joints as active joints and taking lgmc as the x-axis, we draw the figures to exhibit the variation of wPAg and ηPAg with mc in Figure 11.Figure 11
The variations ofwPAg and ηPAg with mc.
(a)
Variation ofwPAg
(b)
Variation ofηPAgWith increase of the base mass and inertia,wPAg and ηPAg decrease continually. When mc>106, the value of wPAg reduces to 10−8 and the magnitudes of change in wPAg reach up to 107. But value of ηPAg stabilizes around 1.4, and the magnitudes of change in ηPAg are only 101. Thus, active joints cannot effectively regulate passive joints based on motion planning, but can regulate passive joints based on the control method at this moment.We design the control method to regulate passive joints based on the second-order nonholonomic constraint. We makeθP_des=−30°, θ̇P_des=0°/s, and θ¨P_des=0°/s2 as the desired states of passive joints and design the PD controller for θ¨P:
(82)θ¨P=θ¨P_des+KDθ̇P_des−θ̇P+KPθP_des−θP.Makinge=θP_des−θP, we have e¨+KDė+KPe=0. According to the characteristic of the second-order homogeneous differential equation, e exponentially converges to zero. Taking KD=0.5 and KP=0.1, θ¨P and θP are in Figure 12 and θA, θ¨A, and τA are in Figure 13.Figure 12
The motion of passive joints with the PD controller.
(a)
Passive joint acceleration
(b)
Passive joint angleFigure 13
The variations of active joint motion and torque.
(a)
Active joint angle
(b)
Active joint acceleration
(c)
Active joint torqueFigure12 shows that the passive joint is regulated to the desired angle at 28.1 s, and the active joint’s motion and torque are smooth in Figure 13. Thus, the passive joint regulating task is accomplished based on the control method. The PD controller above is only appropriate to the space manipulator with an accurate dynamic model. Considering model uncertainty and external disturbance, a robust control system needs to be designed. But in this paper, we mainly analyze the kinematic and dynamic characteristics of the space manipulator with free-swinging joint failure, instead of a robust controller design. In the future study, we will combine the sliding mode and fuzzy control method to design a controller robust to the bounded unknown disturbance and unmodeled uncertainty for the space manipulator with free-swinging joint failure.The simulation above verifies the following theories:(1)
The first-order nonholonomic constraint and the second-order nonholonomic constraint do simultaneously exist inside the system of the space manipulator with free-swinging joint failure
(2)
The kinematic and dynamic coupling indexes proposed in this paper can objectively evaluate kinematic and dynamic coupling degrees from active joints to passive units
(3)
Many space tasks are either based on motion planning with the first-order nonholonomic constraint or based on a controller design with the second-order nonholonomic constraint


## 6. Conclusions

We analyze kinematic and dynamic characteristics of the free-floating space manipulators with free-swinging joint failure in this paper. Based on momentum conservation and the Lagrange equation, the kinematic and dynamic models are established. On this basis, the kinematic and dynamic coupling relationships are derived and we point out that the first-order nonholonomic constraint and the second-order nonholonomic constraint simultaneously exist in the system. Local and global kinematic and dynamic coupling indexes are correspondingly defined. Finally, simulation experiments surrounding the SSRMS-type space manipulator are designed to verify the existence and correctness of the first-order and second-order nonholonomic constraints and display the execution effect of many tasks based on motion planning and the control method. The main contributions in this paper include the following:(1)
We explicitly propose kinematic and dynamic modelling methods for the space manipulator with free-swinging joint failure for the first time
(2)
All kinematic coupling relationships (the first-order nonholonomic constraint) and dynamic coupling relationships (the second-order nonholonomic constraint) inside the system of the space manipulator with free-swinging joint failure are derived
(3)
For general tasks, we propose the motion planning method based on the first-order nonholonomic constraint and design a control strategy based on the second-order nonholonomic constraint. This is the basis of treatment of free-swinging joint failureIn future research, we will pay more attention to task optimization to make the space manipulator with free-swinging joint failure perform primary and secondary tasks in the meantime. And we further design the control system to be robust to model uncertainty and external disturbance.



---

*Source: 2679152-2019-09-12.xml*