#### 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.

#### 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 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 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 successfully

The 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

An 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:

: base centroid,

: centroid of the -th link,

: centroid of the whole system,

: -th joint, ,

: inertial coordinate frame built at ,

: base coordinate frame built at ,

: -th link coordinate frame built at ,

: end-effector coordinate frame,

: base mass,

: mass of the -th link,

: mass of the whole system,

: base inertia matrix with respect to ,

: inertia matrix of the -th link with respect to ,

: position vector of , ,

: position vector of end-effector,

: attitude vector of end-effector,

: position vector of , ,

: position vector of (if is built at , ),

: position vector of ,

: attitude vector of base,

: position vector from to ,

: joint angle vector,

: joint velocity vector,

: joint acceleration vector,

: state variable of system ( and are velocity and acceleration, respectively),

: attitude transformation matrix from to ,

: position vector from to ( is its norm, ),

: position vector from to ( is its norm, ),

: unit vector of the -th joint axis, ,

: velocity of /base angular velocity,

: velocity of /angular velocity of the -th link,

: end-effector velocity/end-effector angular velocity.

For a free-floating space manipulator, linear/angular velocities of -th link and end-effector are where , , , and .and where and and and are the antisymmetric matrixes of and , respectively. is a unit matrix, and is a zero matrix. denotes base Jacobian matrixes representing the transmission from to . and denote manipulator Jacobian matrixes representing the transmission from to .

consists of column vectors: . So can be expressed as

As can be seen from equation (3), depicts the contribution of -th joint velocity to . If selecting the rest of the healthy joints as active joints, we can obtain a serial number set of passive joints as and serial number set of active joints as . So the passive joint velocity is , and the active joint velocity is . With the help of the commutative law of addition, we express equation (3) as where is the column vectors corresponding to passive joints, and they make up which represents transmission from to . is the column vectors corresponding to active joints, and they make up which represents transmission from to . If making , equation (2) becomes

If we know the desired , we can acquire , , and from equation (5). However, the base and passive joints cannot actively supply and 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 where represents the contribution of to linear momentum and represents the contribution of to angular momentum . and can be expressed as and , respectively, and we can find column vectors corresponding to passive joints and active joints to make up matrixes

Equations (6) and (7) can be expressed in matrix form where . Without the external force, system momentum is conservative. Setting the initial momentum is zero, so

From equation (10), we know that , , and are coupled with each other, but we care more about the transmission from to and . From equation (10), we have

Making

Equation (11) becomes

Then, we can obtain transmission from to and :

represents the Jacobian matrix from to and . is the null space of , and is an arbitrary vector. Because , when is full rank, the null space exists. In particular, making , we can obtain transmission from to and from to :

is the first six rows of vectors of , and is the last rows of vectors of . Substituting equation (15) into equation (5), we obtain the transmission from to : where , and is the Jacobian matrix representing transmission from to . Equations (15) and (16) can be reduced to the following constraints:

Obviously, for space manipulators, there is no definite relation between and , which means we cannot find constraints like , , and making , , and , so constraints in equation (17) are nonholonomic. Because equation (17) includes constraints among the first-order derivatives of , 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: where is system kinetic energy and is the system potential energy. The space manipulator is free from gravity, and we take no account of joint and link flexibility, so . The Lagrange function equals to total kinetic energy: where

denotes the base inertia matrix. denotes the manipulator inertia matrix. and are the coupling inertia matrixes between base and manipulator. is the inertia matrix of the whole system. Expanding equation (19), we have

We can know that , , and are relative to joint velocity, so we can utilize commutative law of addition to divide these matrixes into

The row vector of , the column vector of , and the element of represent the contribution of the corresponding joint velocity to kinetic energy. Then, equation (21) can be expressed as where

becomes

And the Lagrange function of the system is

##### 3.2. Lagrange Equation of the System

The Lagrange equation is

Substituting equation (26) into equation (27), and calculating derivatives, we have

Making , , and . And , , and .

, , and are centrifugal forces and Coriolis forces acting on the base, passive joints, and active joints. So equation (28) becomes where

Passive joints and base cannot supply torque, so and . If we know the motions of active joints, passive joints, and the base, the required active joint torque 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

denotes the vector consisting of all passive unit velocities, and 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 , so is the velocity of the entire task space. From equation (31), we can acquire nonhomogeneous linear equations:

, of represents the contribution of to . The necessary and sufficient condition for the existence of a solution of equation (32) is , which is equivalent to

It tells us that if active joints are more than passive units, and is full rank, we can find at least one set of solutions of , which means active joints can actuate all passive units at the same time. For the desired , we can calculate :

In particular, if , the system is redundant, and equation (34) has the general solution where is an arbitrary vector. Equation (35) means there are infinite solutions of corresponding to the same because of the arbitrariness of . 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 according to secondary task requirements.

If conditions in equation (33) are unsatisfactory, the row vectors of are linearly dependent and some velocities in cannot be actuated by . In other words, the transmission efficiency from to is insufficient, corresponding to insufficiency of kinematic coupling degree. The kinematic coupling degree depends on the degree of closeness of 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 to evaluate kinematic coupling degree. where are singular values of . When is close to zeros, the transmission efficiency from to is inefficient, which means is kinematically uncontrollable for . Because , in motion planning, we should continually calculate according to current state .

However, assuming at least one joint fails, so . 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 is . The transmission from to depends on . When , the end-effector could be effectively actuated by active joints. The kinematic coupling index is defined as the manipulability of :

If is close to zero, is uncontrollable for and the end-effector trajectory tracking task is unrealizable. If is big enough, we should interpolate to track the predefined trajectory and calculate . The manipulator can complete the task successfully with the obtained .

###### 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 between and . The task velocity vector is , and the transmission from to depends on . When , passive joints could be actuated by active joints. The kinematic coupling index is defined as the manipulability of :

If is close to zero, is uncontrollable for and the passive joint regulating task is unrealizable. If is big enough, we should interpolate according to the task requirement and calculate .

###### 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 between and , so the task velocity vector is . The transmission from to depends on . When , the base can be actuated. The kinematic coupling index is defined as the manipulability of :

If is close to zero, is uncontrollable for so the task is unrealizable. If is big enough, we should interpolate as the requirement and calculate .

###### 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 as and the Jacobian matrix is where is the first three rows of . The kinematic coupling index is the manipulability of . For the task to eliminate the base attitude deflection when regulating passive joints, we select the task velocity vector as and the Jacobian matrix is where is the last three row vectors of . The kinematic coupling index is the manipulability of . To eliminate base attitude disturbance, we keep [28].

For general space tasks, we should select according to controlled passive units for current task requirements. is the dimensionality of . Then, the corresponding row vectors from need to be taken out to construct the Jacobian matrix: and the kinematic coupling index is

If and , we can interpolate as requested and calculate as ; for , the space manipulator with free-swinging joint failure is redundant for the current task, and the general solution of exists as

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):

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

Eliminating in equations (45) and (46), we have

So where is the coupling inertia matrix between active joints and passive joints and is the nonlinear item. The first term in the right of equation (49) reflects the transmission from to , and it is the dynamic coupling relationship between active joints and passive joints. We can obtain nonhomogeneous linear equations . If and , we have the solution of . Therefore, referring to [29], the dynamic coupling index is defined as the manipulability of to evaluate the transmission efficiency from to . where are singular values of . When is close to zero, the dynamic coupling degree between passive joints and active joints is inefficient and is dynamically uncontrollable for . When is big enough, we can design the control law for , and which actuates the manipulator to complete the passive joint regulating task can be obtained by .

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

Eliminating in equations (45) and (46), we have

So where is the coupling inertia matrix between active joints and the base. 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 to evaluate the transmission efficiency from to .

If is close to zero, is uncontrollable for . On the contrary, we can design the control law for to calculate .

In particular, we can get transmission from to by combining equations (49) and (52):

If we calculate the derivative of equation (22), we have

Equation (55) has a similar form with equation (54), but obviously, , , , and , so equations (22) and (54) are different.

Equation (54) can also be derived from equations (45) and (46):

Substituting equation (56) into equation (47), we have where represents the mapping relation between and and is the nonlinear item. Equation (57) establishes the direct relation between and . After designing the control law of or , we can calculate , and then we can obtain the torque that actuates the manipulator. We can further find the transmission from to : where , , , and . So we can directly calculate from equation (60) corresponding to the desired or . Dynamic coupling indexes are defined to evaluate transmission efficiency from to :

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

Because does not appear in the Lagrange equation, we cannot find a mapping relation between and 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

Multiplying equation (57) by , we have where is the coupling inertia matrix representing the transmission from to and 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 :