# Halt Optimization Strategy for a Space Manipulator with a Joint-Locked Failure

Authors: Gang Chen; Lanpu Li; Yingzhuo Fu; Bonan Yuan; Junting Fei
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

Aiming at a space manipulator with a joint-locked failure, a halt optimization strategy is proposed in this paper. Firstly, a halt configuration optimization model (HCOM) is constructed, to select an optimal configuration where the kinematic ability of the manipulator is the best. Secondly, considering the constraint of joint running parameters and the disturbance torque of the base, we construct and solve the halt motion optimization model (HMOM), which can achieve a steady halt and ensure the safety of the manipulator. The correctness and effectiveness of the proposed strategy in this paper are verified with a 7-DOF space manipulator. This strategy firstly puts forward the idea of halt configuration optimization and realizes the minimum global disturbance torque of the base in the halt process.


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## Body

## 1. Introduction

As a high-end space equipment with large span, high flexibility, and strong operation capability, space manipulators have been widely used in the field of space exploration [1–5]. However, due to the complex joint structure and hostile working environment (high-temperature difference, high radiation, etc.), a joint-locked failure is prone to occur [6–9]. The faulty joint is locked at the failure moment and loses the ability to move. At this time, if the manipulator executes the original task without halt, its running parameters (joint velocity, acceleration, torque, etc.) will jump at the failure moment [10–12], bringing a deformation or more damage to the manipulator [13–15]. Therefore, to ensure a safe and stable operation, the faulty manipulator needs to be halted after the joint-locked failure occurs.The halt of the space manipulator with a joint-locked failure aims at designing a subsequent motion to halt the manipulator in a certain configuration. Aiming at the halt of a faulty manipulator, some scholars have carried out relevant researches. Kamata et al. [16] achieved an emergency stop of the faulty manipulator by utilizing the joint braking based on the joint limit. Dupac and Noroozi [17] proposed a halt strategy by limiting the impact of the end of the manipulator, realizing the halt of the manipulator. To avoid the joint overload, Nozawa et al. [18] achieved the halt of the manipulator by limiting the joint running parameters. In the above study, considering the parameter constraints of the manipulator, the corresponding motion planning strategy is constructed to realize the shutdown of the manipulator. But the above halt methods only consider the constraints of joint parameters, ignoring the characteristic that the running parameters of the manipulator are likely to mutate. That is to say, these halt methods belong to an emergency stop, which will seriously affect the safety of the manipulator in the halt process [19, 20]. Therefore, within relevant constraint conditions, the running parameters need to be optimized during the halt process for the faulty manipulator.Aiming at the halt optimization of a manipulator with a joint-locked failure, Xu et al. [21] used a sequence quadratic program (SQP) to obtain the halt trajectory with optimal torque by considering the torque limitation. Gao et al. [22] optimized the joint velocity based on the linear and cosine curve with the joint torque as the constraint condition. To optimize the energy consumption in the halt process, Dalla and Pathak [23] proposed a halt optimization method by limiting the joint velocity, which achieved a halt with the minimum power consumption. Although the above researches realized the halt optimization of the manipulator with a joint-locked failure, they do not consider the halt configuration, which means that the halt configuration of the manipulator is arbitrary. The space manipulator has different kinematic ability under different configurations. Considering that on-orbit operation tasks have high requirements on the kinematic ability of the space manipulator [24, 25], if the manipulator is stopped at any configuration (such as the singular configuration), its kinematic ability in the anchor configuration may not meet task requirements, which seriously restricts the reliable execution of the subsequent operation task. Therefore, the manipulator should be stopped in an optimal configuration according to task requirements, so that it can reliably perform the subsequent operation task with the best kinematic ability.In addition, the existing researches on the halt optimization of the manipulator are only aimed at the ground manipulator. But for the space manipulator, its base usually floats freely [26, 27]. If the above methods for ground manipulators are directly applied to the space manipulator, it will generate a disturbance torque on the base in the halt process because of the motion coupling between the manipulator and the base [28], and the attitude deflection of the base will be caused. However, if the attitude deflection of the base is too large, the spacecraft’s communication quality and energy acquisition efficiency will be reduced, and even the manipulator will be out of control, seriously threatening the safety of the space manipulator [29]. For this reason, to solve the problem of the halt optimization of the manipulator with a joint-locked failure, the disturbance torque of base should be limited within a small range.In summary, this paper proposes a halt optimization strategy for the space manipulator with a joint-locked failure, which is made up of two parts: (1) A comprehensive characterization of the kinematic ability, covering the manipulator itself and task requirements, is obtained. Then, a halt configuration optimization model (HCOM) is constructed with the best kinematic ability under the halt configuration as the objective. By solving the HCOM, the optimal halt configuration is obtained. (2) The halt motion is planned by using a six-order polynomial interpolation method which contains an optimization coefficient. Considering the motion coupling between the base and the manipulator, a halt motion optimization model (HMOM) is constructed with the minimum disturbance torque of the base as the objective and joint parameters as the constraint. With the HMOM solved, the halt motion optimization is achieved which can realize the steady halt and ensure the safety of the manipulator in the halt process.Aiming at the space manipulator with a joint-locked failure, the innovation of this paper mainly is shown as follows(1)
The halt configuration is optimized, ensuring that the manipulator in the halt configuration owns the best kinematic ability, and the kinematic ability in the optimal halt configure is236.8% higher than that in the suboptimal halt configuration(2)
Each unilateral kinematic ability index is analyzed quantitatively, making the comprehensive representation of the kinematic ability more accurate, which can guarantee the execution of subsequent task(3)
The disturbance torque of the base is optimized in the halt process, and its global disturbance torque is reduced by 30% at least after the optimizationThe rest of this paper is organized as follows: In Section2, we analyze the halt optimization problem and describe it as a halt configuration optimization problem (HCOP) and a halt motion optimization problem (HMOP). In Section 3, aiming at the HCOP, a HCOM is constructed and solved to obtain the optimal halt configuration. In Section 4, aiming at the HMOP, a HMOM is constructed and solved to achieve the halt optimization. In Section 5, the simulation is carried on to verify the effectiveness of the proposed strategy. In Section 6, the conclusions are summarized.


## 2. Materials and Methods

### 2.1. Problem Description

In this section, a halt optimization problem of a space manipulator with a joint-locked failure is analyzed. The halt optimization problem is described as the HCOP and the HMOP by considering task requirements and the characteristic of the base.To express easily, the symbols involved in this section are annotated in Table1. And the halt optimization problem can be described as considering the constraint lim and the optimization objective Var, take S0 as input, and optimize the halt process to make the manipulator come to Sf.Table 1
Symbols annotations of the halt.
SymbolsAnnotationst0The initial time of the halttfThe end time of the halttThe time during the haltqtThe configuration when the time istq̇tThe joint velocity when the time istq¨tThe joint acceleration when the time istτbThe disturbance torque of the base in the halt processS0The initial state of the halt,S0=q0,q̇0,q¨0⋯SfThe end state of the halt,Sf=qf,q̇f,q¨f⋯ and q̇f=0, q¨f=0VarThe parameters of the manipulator in the halt process,Var=qt,q̇t,q¨t,τb⋯limThe constraint of the halt,lim=qmin,qmax,q̇min,q̇max,q¨min,q¨max,τmin,τmax⋯Considering the high requirements of the subsequent operation tasks, it is necessary to make the kinematic ability of the manipulator under the halt configuration as high as possible. In other words, the halt configuration should be optimized with the best kinematic ability as the objective. Let the kinematic ability under the configurationq be CPq, and the HCOP can be described as
(1)findqbest,maxCPqbest,whereqbest is the optimal halt configuration.Considering the motion coupling between the manipulator and the base, we take the minimum base disturbance torque as the objective and limit the joint parameters (angle, velocity, and acceleration) within a range. On the basis, the manipulator is stably halted toqbest. The HMOP can be described as
(2)minτbs.t.limqf=qbest.At this point, the halt optimization problem is transformed into the HCOP and the HMOP, which is shown in Figure1.Figure 1
The description of the halt optimization problem.


## 2.1. Problem Description

In this section, a halt optimization problem of a space manipulator with a joint-locked failure is analyzed. The halt optimization problem is described as the HCOP and the HMOP by considering task requirements and the characteristic of the base.To express easily, the symbols involved in this section are annotated in Table1. And the halt optimization problem can be described as considering the constraint lim and the optimization objective Var, take S0 as input, and optimize the halt process to make the manipulator come to Sf.Table 1
Symbols annotations of the halt.
SymbolsAnnotationst0The initial time of the halttfThe end time of the halttThe time during the haltqtThe configuration when the time istq̇tThe joint velocity when the time istq¨tThe joint acceleration when the time istτbThe disturbance torque of the base in the halt processS0The initial state of the halt,S0=q0,q̇0,q¨0⋯SfThe end state of the halt,Sf=qf,q̇f,q¨f⋯ and q̇f=0, q¨f=0VarThe parameters of the manipulator in the halt process,Var=qt,q̇t,q¨t,τb⋯limThe constraint of the halt,lim=qmin,qmax,q̇min,q̇max,q¨min,q¨max,τmin,τmax⋯Considering the high requirements of the subsequent operation tasks, it is necessary to make the kinematic ability of the manipulator under the halt configuration as high as possible. In other words, the halt configuration should be optimized with the best kinematic ability as the objective. Let the kinematic ability under the configurationq be CPq, and the HCOP can be described as
(1)findqbest,maxCPqbest,whereqbest is the optimal halt configuration.Considering the motion coupling between the manipulator and the base, we take the minimum base disturbance torque as the objective and limit the joint parameters (angle, velocity, and acceleration) within a range. On the basis, the manipulator is stably halted toqbest. The HMOP can be described as
(2)minτbs.t.limqf=qbest.At this point, the halt optimization problem is transformed into the HCOP and the HMOP, which is shown in Figure1.Figure 1
The description of the halt optimization problem.


## 3. The Halt Configuration Optimization

The halt configuration optimization of the space manipulator with a joint-locked failure is to select the optimal configuration with the best kinematic ability, which can ensure that the manipulator under the halt configuration can meet task requirements. In this section, aiming at the HCOP, the kinematic ability of the manipulator is comprehensively characterized, and then the HCOM is constructed and solved based on the Monte Carlo method to obtain the optimal halt configuration.


### 3.1. Comprehensive Characterization of the Kinematic Ability

The representation of the kinematic ability of the space manipulator depends on its kinematic model. However, the original kinematic model of the space manipulator is no longer applicable due to the locked joint. For this reason, this section firstly reconstructs the kinematic model and then analyzes the unilateral kinematic ability indexes. Finally, the kinematic ability of the manipulator is comprehensively characterized based on the traditional entropy method and analytic hierarchy process (AHP) method.


#### 3.1.1. The Reconstruction of the Kinematic Model

For then-DOF space manipulator without a joint-locked failure, its kinematic model is shown in Figure 2. To express easily, the symbols involved in this section are annotated in Table 2.Figure 2
The kinematic model ofn-DOF space manipulator without a joint-locked failure.Table 2
Symbols annotations ofn-DOF space manipulator without a joint-locked failure.
SymbolsAnnotationsΣIInertial coordinate frameΣbBase coordinate systemΣeEnd-effector coordinate systemΣiThe coordinate system of thei‐th link (i=1,2,⋯,n)CiCentroid of thei‐th link (i=1,2,⋯,n)JiThe joint connectedi−1‐th link and i‐th link (i=1,2,⋯,n)liThe vector fromJi to Ji+1, and li=ai+biaiThe vector fromJi to Ci (i=1,2,⋯,n)biThe vector fromCi to Ji+1 (i=1,2,⋯,n−1)bn−1The vector fromCn−1 to the end-effectorzIiThe unit vector in the axis direction inΣimiThe mass of thei‐th link (i=1,2,⋯,n)mbThe mass of the baseMThe total mass of the systemIiThe inertia tensor ofi‐th link (i=1,2,⋯,n)IbThe inertia tensor of baseθiThe joint angle ofi‐th joint (i=1,2,⋯,n)qθThe position vector of joint angle in joint space,qθ=θ1,⋯,θnT∈Rn×1rIbThe position vector of centroid of base in inertial framerIiThe position vector of centroid ofi‐th in inertial frame (i=1,2,⋯,n)rIeThe position vector of the end-effectorpIiThe position vector ofJi (i=1,2,⋯,n)qbThe pose vector of the base,qb=rb,ϕbϕbThe orientation vector of the baseq̇bThe velocity of the basexeThe pose vector of the end-effector,xe=re,ϕeϕeThe orientation vector of the end-effectorvbThe velocity of the centroid of baseviThe velocity of the centroid ofi‐th link (i=1,2,⋯,n)veThe velocity of the end-effectorωbThe angular velocity of the baseωiThe angular velocity of thei‐th linkωeThe angular velocity of the end-effectorAssuming that theJl is locked, and the link Ll−1 and link Ll can be connected as a new linkL~l−1. Specifying the coordinate system of the link and joint after Jl is locked as follows: If the location of the joint or link is in front of theJl, it is consistent with the kinematic model without a joint-locked failure; otherwise, add “~” above the symbols of the kinematic model without a joint-locked failure, and the new number of the joint or link is subtracted 1 from the original basis.The dynamics parameters ofL~l−1 can be obtained according to the following equation
(1)
The mass ofL~l−1: m~l−1=ml+ml−1(2)
The centroid ofL~l−1: r~Il−1=ml−1rIl−1+mlrIl/ml−1+ml(3)
The inertia tensor ofL~l−1: I~l−1=Il−1+ml−1rIl−1TrIl−1E3−rIl−1rIl−1T, where E3∈R3×3 represents the third order identity matrixThus, the reconstruction of the kinematic model is shown in Figure3.Figure 3
The reconstructed kinematic model ofn-DOF space manipulator with a joint-locked failure.The velocity of the end-effector is represented as(3)x~̇e=J~bq~̇b+J~θq~̇θ,whereJ~b represents the base Jacobian matrix; J~θ represents the joint Jacobian matrix.Since the base of the space manipulator is in a free-floating state and is not influenced by external forces, the momentum of the system is conserved. Assuming that the momentum zero in the initial state, the velocity relation between the base and the end-effector can be expressed as(4)q~̇b=J~bθq~̇θ,whereJ~bθ represents the base-joint Jacobian matrix.Substituting Equation (3) into Equation (4), we can obtain
(5)x~̇e=J̇θ+J̇bJ~bθq~̇θ=J~Fq~̇θ,whereJ~F represents the generalized Jacobian matrix.


#### 3.1.2. The Analysis of the Unilateral Kinematic Ability

The kinematic ability of the space manipulator generally includes the minimum singular value, condition number, and manipulability, which are related to the singular value of the Jacobian matrix [30]. Next, the kinematic ability will be analyzed based on J~F.According to the singular value decomposition theory,J~F can be written as
(6)J~F=UΣV,whereU∈Rm×m and V∈Rn−1×n−1 are orthogonal matrices; Σ∈Rm×n−1 is diagonal and nonnegative real rectangular diagonal matrix, and Σ can be written as
(7)Σ=σ10⋯000σ2⋯00⋮⋮⋱⋮000⋯σm0,whereσ1 is the maximum singular value; σm is the minimum singular value. When the space manipulator is in a singular configuration, σm=0.According to the singular values obtained above, the unilateral kinematic ability of the space manipulator can be obtained, as shown in Table3.Table 3
The unilateral kinematic ability of the space manipulator with a joint-locked failure.
Minimum singular valueCondition numberManipulabilityExpressions=σmk=σ1/σmw=σ1σ2⋯σmMeaningThe degree to which the manipulator approaches the singular configurationThe condition number reflects the isotropy of the manipulator’s kinematic ability in all directions.The measure of the end-effector’s kinematic ability when manipulator moves in different directionsValue ranges∈0,+∞k∈1,+∞w∈0,+∞Value analysisThe smaller the minimum singular value is, the closer the configuration is to the singular configuration.The closer the condition number is to 1, the higher the dexterity of the manipulator is.The bigger degenerate manipulability is, the greater kinematic ability of the end-effector in all directions is.According to Table3, the bigger minimum singular value and manipulability is, the better the kinematic ability is. And the smaller condition number is, the better the kinematic ability is. As a consequence, it is necessary to carry out a standardized treatment for the above unilateral kinematic ability.The minimum singular value and the manipulability are standardized as(8)s^=standards=s−ab−a,w^=standardw=w−ab−a,wherestandardx=x−a/b−a, x∈a,b represents the unilateral kinematic ability; s^ represents the minimum singular value with standardization, and w^ represents the manipulability with standardization.Aiming at the condition number, it can be standardized as(9)k^=1k,wherek^ represents the condition number with standardization.At this point, each unilateral kinematic ability is standardized, and then the kinematic ability will be comprehensively characterized.


#### 3.1.3. The Comprehensive Characterization of the Kinematic Ability

The key to the comprehensive characterization of the kinematic ability is to determine the weight of each unilateral kinematic ability. The traditional entropy method ignores the qualitative factor that different tasks have different requirements on each unilateral kinematic ability. For example, assuming that a task needs the higher isotropy, the weight solved by the traditional entropy method only considers the unilateral kinematic ability and cannot reflect the higher isotropy, leading to an inaccuracy for the comprehensive characterization of kinematic ability. Considering that the AHP method can be used for the qualitative analysis [31], we choose the AHP method for the comprehensive characterization. Therefore, to cover the manipulator and requirements of the task, we combine and improve the AHP method and the traditional entropy method.Firstly, we calculate the subjective weight by utilizing the AHP method.(1)
Construct the hierarchical substructure model. Considering that different task requires different kinematic ability, a hierarchical structure from top to bottom is constructed as Figure4, including target layer (comprehensive characterization of kinematic ability), criterion layer (task requirements), and subcriterion layer (each unilateral kinematic ability). The target layer is the highest layer in the hierarchy, and the criterion layer and subcriterion layer are used to evaluate the subjective weight(2)
Construct judgment matrix. To judge the influence of factors at the current level on that at the next level, 1-7 scale method as shown in Table4 is used to define the importance degree of each element [32], and we construct the judgment matrix of Target-Criterion R=rijM×NFigure 4
The hierarchical substructure model.Table 4
Scale definitions of 1-7 scale method.
ScaleDefinition1i is just as important as j3i is slightly more important than j5i is more important than j7i is even more important than j2,4,6The importance lies between adjacent scalesReciprocalIf the ratio ofi and j is rij, the ratio of j and i is rji=1/rijBased on Table4, the judgment matrix R=rijM×N can be obtained by comparing the importance of each kinematic ability.
(3)
Do consistency test on the judgment matrix. By utilizing the maximum eigenvalue method, we can get the maximum eigenvalue ofR=rijM×N, and then the consistency test is used to judge whether the matrix is reasonable or notSolve the consistency indexCI(10)CI=λmax−NN−1,Whereλmax is the maximum eigenvalue of R=rijM×N; N is the order of R=rijM×N.According to the AHP theory, ifN is known, the standard consistency index RI can be obtained as shown in Table 5.Table 5
The standard consistency index.
NRI102030.5240.8951.1261.24Calculate the consistency ratioCR(11)CR=CIRI.IfCR<0.10, the judgment matrix is acceptable, if not, the judgment matrix should be modified appropriately by changing rij.
(4)
The solution of the subjective weight. For the judgment matrix satisfying the consistency test, the subjective weight of the unilateral kinematic ability can be obtained according toλmax. Then, we calculate the eigenvector of λmax, and take it as ω1,⋯,ωNT, which corresponds to the subjective weight of each unilateral kinematic ability ω1⋯ωN, respectivelyBased on the above steps, the subjective weight ofs^, w^, k^ can be obtained, which is denoted as
(12)ωCR=ωCR_1,ωCR_2,ωCR_3.Next, the entropy method is used to solve the objective weight of the unilateral kinematic ability. The configuration of the space manipulator with a joint-locked failure in the halt process can be expressed asqλ=θ1θ2⋯θn−1, where λ (λ=1,2,⋯K) represents the λ‐th configuration and K is the total number of configurations.When the manipulator is in theλ‐th configuration, the proportion of the h‐th unilateral kinematic ability in the total value of the kinematic ability is Pλ_h, which can be calculated by the following formula
(13)Pλ_h=Xλ_h∑λ=1KXλ_hh=1,2,3,whereXλ_h represents the value of the h‐th unilateral kinematic ability when the manipulator is in the λ‐th configuration.And then the entropy value of the characterization of theh‐th unilateral kinematic ability can be obtained as
(14)eh=−∑λ=1KPλ_hlogPλ_hln∑h,where0≤eh≤1.The difference coefficientgh can be calculated as
(15)gh=1−eh.The weight of each unilateral kinematic ability can be obtained as(16)ωCP_h=gh∑h=13gh,WhereωCP_hh=1,2,3 represents the weight of h‐th unilateral kinematic ability, which can be written as
(17)ωCP=ωCP_1,ωCP_2,ωCP_3.According to Equation (12) and Equation (17), the final weight of each unilateral kinematic ability can be obtained as
(18)ω=ωCR+ωCP=ωCR_1+ωCP_1,ωR_2+ωCP_2,ωCR_3+ωCP_3=ωC_1,ωC_2,ωC_3.According to the above analysis, the comprehensive characterization of the kinematic ability can be expressed as(19)CPqλ=ωC_1×s^qλ+ωC_2×k^qλ+ωC_3×w^qλλ=1,2,⋯,K,whereCPqλ is the comprehensive characterization of the kinematic ability when the manipulator is in the λ‐th configuration.At this point, the comprehensive characterization of the kinematic ability of the manipulator is completed.


### 3.2. The Construction and Solution of the HCOM

In this section, the HCOM is constructed based on the objective of maximizing the comprehensive characterization of the kinematic ability, and the optimal halt configuration is obtained by solving the HCOM with the Monte Carlo method.According to Equation (1) and Equation (19), the HCOM can be expressed as
(20)findqbest,maxCPqbest,whereqbest is the optimal halt configuration.Next, the HCOM is solved by utilizing the Monte Carlo method. The specific method is shown asStep 1.
For each jointJ~i, the random numbers of each joint angle can be generated by using the random number method: Ni=Ni_1Ni_2⋯Ni_K∈R1×K, which are between 0 and 1, and then the pseudorandom number of each joint angle can be obtained as
(21)θ~i=θ~imin+Δθ~i×Ni=θ~imin+Δθ~i⋅Ni_1θ~imin+Δθ~i⋅Ni_2⋯θ~imin+Δθ~i⋅Ni_K,
whereθ~imin=θ~iminθ~imin⋯θ~imin∈R1×K; Δθ~i=θ~imax−θ~imin; i=1,⋯,n−1.Step 2.
The pseudorandom number of the joint angles of each joint can be expressed as(22)CS=θ~1min+Δθ~1⋅N1_1θ~1min+Δθ~1⋅N1_2⋯θ~1min+Δθ~1⋅N1_Kθ~2min+Δθ~2⋅N2_1θ~2min+Δθ~2⋅N2_2⋯θ~2min+Δθ~2⋅N2_K⋮⋮⋱⋮θ~n−1min+Δθ~n−1⋅Nn−1_1θ~n−1min+Δθ~n−1⋅Nn−1_2⋯θ~n−1min+Δθ~n−1⋅Nn−1_K=θ11θ12⋯θ1Kθ21θ22⋯θ2K⋮⋮⋱⋮θn−11θn−12⋯θn−1K∈Rn−1×K.
Each column corresponds to a certain configuration, and Equation (22) can be written as
(23)CS=q1q2⋯qK∈R1×K.
And the configuration space of the manipulator can be obtained as(24)S=q1,q2,⋯,qλ,⋯,qKλ=1,2,⋯,K.Step 3.
Select a configurationqbest=θ1bestθ2best⋯θn−1best, which maximums CPqbest in Equation (19).So far, by solving the HCOM, the optimal halt configuration of the space manipulator is obtained.


## 3.1. Comprehensive Characterization of the Kinematic Ability

The representation of the kinematic ability of the space manipulator depends on its kinematic model. However, the original kinematic model of the space manipulator is no longer applicable due to the locked joint. For this reason, this section firstly reconstructs the kinematic model and then analyzes the unilateral kinematic ability indexes. Finally, the kinematic ability of the manipulator is comprehensively characterized based on the traditional entropy method and analytic hierarchy process (AHP) method.


### 3.1.1. The Reconstruction of the Kinematic Model

For then-DOF space manipulator without a joint-locked failure, its kinematic model is shown in Figure 2. To express easily, the symbols involved in this section are annotated in Table 2.Figure 2
The kinematic model ofn-DOF space manipulator without a joint-locked failure.Table 2
Symbols annotations ofn-DOF space manipulator without a joint-locked failure.
SymbolsAnnotationsΣIInertial coordinate frameΣbBase coordinate systemΣeEnd-effector coordinate systemΣiThe coordinate system of thei‐th link (i=1,2,⋯,n)CiCentroid of thei‐th link (i=1,2,⋯,n)JiThe joint connectedi−1‐th link and i‐th link (i=1,2,⋯,n)liThe vector fromJi to Ji+1, and li=ai+biaiThe vector fromJi to Ci (i=1,2,⋯,n)biThe vector fromCi to Ji+1 (i=1,2,⋯,n−1)bn−1The vector fromCn−1 to the end-effectorzIiThe unit vector in the axis direction inΣimiThe mass of thei‐th link (i=1,2,⋯,n)mbThe mass of the baseMThe total mass of the systemIiThe inertia tensor ofi‐th link (i=1,2,⋯,n)IbThe inertia tensor of baseθiThe joint angle ofi‐th joint (i=1,2,⋯,n)qθThe position vector of joint angle in joint space,qθ=θ1,⋯,θnT∈Rn×1rIbThe position vector of centroid of base in inertial framerIiThe position vector of centroid ofi‐th in inertial frame (i=1,2,⋯,n)rIeThe position vector of the end-effectorpIiThe position vector ofJi (i=1,2,⋯,n)qbThe pose vector of the base,qb=rb,ϕbϕbThe orientation vector of the baseq̇bThe velocity of the basexeThe pose vector of the end-effector,xe=re,ϕeϕeThe orientation vector of the end-effectorvbThe velocity of the centroid of baseviThe velocity of the centroid ofi‐th link (i=1,2,⋯,n)veThe velocity of the end-effectorωbThe angular velocity of the baseωiThe angular velocity of thei‐th linkωeThe angular velocity of the end-effectorAssuming that theJl is locked, and the link Ll−1 and link Ll can be connected as a new linkL~l−1. Specifying the coordinate system of the link and joint after Jl is locked as follows: If the location of the joint or link is in front of theJl, it is consistent with the kinematic model without a joint-locked failure; otherwise, add “~” above the symbols of the kinematic model without a joint-locked failure, and the new number of the joint or link is subtracted 1 from the original basis.The dynamics parameters ofL~l−1 can be obtained according to the following equation
(1)
The mass ofL~l−1: m~l−1=ml+ml−1(2)
The centroid ofL~l−1: r~Il−1=ml−1rIl−1+mlrIl/ml−1+ml(3)
The inertia tensor ofL~l−1: I~l−1=Il−1+ml−1rIl−1TrIl−1E3−rIl−1rIl−1T, where E3∈R3×3 represents the third order identity matrixThus, the reconstruction of the kinematic model is shown in Figure3.Figure 3
The reconstructed kinematic model ofn-DOF space manipulator with a joint-locked failure.The velocity of the end-effector is represented as(3)x~̇e=J~bq~̇b+J~θq~̇θ,whereJ~b represents the base Jacobian matrix; J~θ represents the joint Jacobian matrix.Since the base of the space manipulator is in a free-floating state and is not influenced by external forces, the momentum of the system is conserved. Assuming that the momentum zero in the initial state, the velocity relation between the base and the end-effector can be expressed as(4)q~̇b=J~bθq~̇θ,whereJ~bθ represents the base-joint Jacobian matrix.Substituting Equation (3) into Equation (4), we can obtain
(5)x~̇e=J̇θ+J̇bJ~bθq~̇θ=J~Fq~̇θ,whereJ~F represents the generalized Jacobian matrix.


### 3.1.2. The Analysis of the Unilateral Kinematic Ability

The kinematic ability of the space manipulator generally includes the minimum singular value, condition number, and manipulability, which are related to the singular value of the Jacobian matrix [30]. Next, the kinematic ability will be analyzed based on J~F.According to the singular value decomposition theory,J~F can be written as
(6)J~F=UΣV,whereU∈Rm×m and V∈Rn−1×n−1 are orthogonal matrices; Σ∈Rm×n−1 is diagonal and nonnegative real rectangular diagonal matrix, and Σ can be written as
(7)Σ=σ10⋯000σ2⋯00⋮⋮⋱⋮000⋯σm0,whereσ1 is the maximum singular value; σm is the minimum singular value. When the space manipulator is in a singular configuration, σm=0.According to the singular values obtained above, the unilateral kinematic ability of the space manipulator can be obtained, as shown in Table3.Table 3
The unilateral kinematic ability of the space manipulator with a joint-locked failure.
Minimum singular valueCondition numberManipulabilityExpressions=σmk=σ1/σmw=σ1σ2⋯σmMeaningThe degree to which the manipulator approaches the singular configurationThe condition number reflects the isotropy of the manipulator’s kinematic ability in all directions.The measure of the end-effector’s kinematic ability when manipulator moves in different directionsValue ranges∈0,+∞k∈1,+∞w∈0,+∞Value analysisThe smaller the minimum singular value is, the closer the configuration is to the singular configuration.The closer the condition number is to 1, the higher the dexterity of the manipulator is.The bigger degenerate manipulability is, the greater kinematic ability of the end-effector in all directions is.According to Table3, the bigger minimum singular value and manipulability is, the better the kinematic ability is. And the smaller condition number is, the better the kinematic ability is. As a consequence, it is necessary to carry out a standardized treatment for the above unilateral kinematic ability.The minimum singular value and the manipulability are standardized as(8)s^=standards=s−ab−a,w^=standardw=w−ab−a,wherestandardx=x−a/b−a, x∈a,b represents the unilateral kinematic ability; s^ represents the minimum singular value with standardization, and w^ represents the manipulability with standardization.Aiming at the condition number, it can be standardized as(9)k^=1k,wherek^ represents the condition number with standardization.At this point, each unilateral kinematic ability is standardized, and then the kinematic ability will be comprehensively characterized.


### 3.1.3. The Comprehensive Characterization of the Kinematic Ability

The key to the comprehensive characterization of the kinematic ability is to determine the weight of each unilateral kinematic ability. The traditional entropy method ignores the qualitative factor that different tasks have different requirements on each unilateral kinematic ability. For example, assuming that a task needs the higher isotropy, the weight solved by the traditional entropy method only considers the unilateral kinematic ability and cannot reflect the higher isotropy, leading to an inaccuracy for the comprehensive characterization of kinematic ability. Considering that the AHP method can be used for the qualitative analysis [31], we choose the AHP method for the comprehensive characterization. Therefore, to cover the manipulator and requirements of the task, we combine and improve the AHP method and the traditional entropy method.Firstly, we calculate the subjective weight by utilizing the AHP method.(1)
Construct the hierarchical substructure model. Considering that different task requires different kinematic ability, a hierarchical structure from top to bottom is constructed as Figure4, including target layer (comprehensive characterization of kinematic ability), criterion layer (task requirements), and subcriterion layer (each unilateral kinematic ability). The target layer is the highest layer in the hierarchy, and the criterion layer and subcriterion layer are used to evaluate the subjective weight(2)
Construct judgment matrix. To judge the influence of factors at the current level on that at the next level, 1-7 scale method as shown in Table4 is used to define the importance degree of each element [32], and we construct the judgment matrix of Target-Criterion R=rijM×NFigure 4
The hierarchical substructure model.Table 4
Scale definitions of 1-7 scale method.
ScaleDefinition1i is just as important as j3i is slightly more important than j5i is more important than j7i is even more important than j2,4,6The importance lies between adjacent scalesReciprocalIf the ratio ofi and j is rij, the ratio of j and i is rji=1/rijBased on Table4, the judgment matrix R=rijM×N can be obtained by comparing the importance of each kinematic ability.
(3)
Do consistency test on the judgment matrix. By utilizing the maximum eigenvalue method, we can get the maximum eigenvalue ofR=rijM×N, and then the consistency test is used to judge whether the matrix is reasonable or notSolve the consistency indexCI(10)CI=λmax−NN−1,Whereλmax is the maximum eigenvalue of R=rijM×N; N is the order of R=rijM×N.According to the AHP theory, ifN is known, the standard consistency index RI can be obtained as shown in Table 5.Table 5
The standard consistency index.
NRI102030.5240.8951.1261.24Calculate the consistency ratioCR(11)CR=CIRI.IfCR<0.10, the judgment matrix is acceptable, if not, the judgment matrix should be modified appropriately by changing rij.
(4)
The solution of the subjective weight. For the judgment matrix satisfying the consistency test, the subjective weight of the unilateral kinematic ability can be obtained according toλmax. Then, we calculate the eigenvector of λmax, and take it as ω1,⋯,ωNT, which corresponds to the subjective weight of each unilateral kinematic ability ω1⋯ωN, respectivelyBased on the above steps, the subjective weight ofs^, w^, k^ can be obtained, which is denoted as
(12)ωCR=ωCR_1,ωCR_2,ωCR_3.Next, the entropy method is used to solve the objective weight of the unilateral kinematic ability. The configuration of the space manipulator with a joint-locked failure in the halt process can be expressed asqλ=θ1θ2⋯θn−1, where λ (λ=1,2,⋯K) represents the λ‐th configuration and K is the total number of configurations.When the manipulator is in theλ‐th configuration, the proportion of the h‐th unilateral kinematic ability in the total value of the kinematic ability is Pλ_h, which can be calculated by the following formula
(13)Pλ_h=Xλ_h∑λ=1KXλ_hh=1,2,3,whereXλ_h represents the value of the h‐th unilateral kinematic ability when the manipulator is in the λ‐th configuration.And then the entropy value of the characterization of theh‐th unilateral kinematic ability can be obtained as
(14)eh=−∑λ=1KPλ_hlogPλ_hln∑h,where0≤eh≤1.The difference coefficientgh can be calculated as
(15)gh=1−eh.The weight of each unilateral kinematic ability can be obtained as(16)ωCP_h=gh∑h=13gh,WhereωCP_hh=1,2,3 represents the weight of h‐th unilateral kinematic ability, which can be written as
(17)ωCP=ωCP_1,ωCP_2,ωCP_3.According to Equation (12) and Equation (17), the final weight of each unilateral kinematic ability can be obtained as
(18)ω=ωCR+ωCP=ωCR_1+ωCP_1,ωR_2+ωCP_2,ωCR_3+ωCP_3=ωC_1,ωC_2,ωC_3.According to the above analysis, the comprehensive characterization of the kinematic ability can be expressed as(19)CPqλ=ωC_1×s^qλ+ωC_2×k^qλ+ωC_3×w^qλλ=1,2,⋯,K,whereCPqλ is the comprehensive characterization of the kinematic ability when the manipulator is in the λ‐th configuration.At this point, the comprehensive characterization of the kinematic ability of the manipulator is completed.


## 3.1.1. The Reconstruction of the Kinematic Model

For then-DOF space manipulator without a joint-locked failure, its kinematic model is shown in Figure 2. To express easily, the symbols involved in this section are annotated in Table 2.Figure 2
The kinematic model ofn-DOF space manipulator without a joint-locked failure.Table 2
Symbols annotations ofn-DOF space manipulator without a joint-locked failure.
SymbolsAnnotationsΣIInertial coordinate frameΣbBase coordinate systemΣeEnd-effector coordinate systemΣiThe coordinate system of thei‐th link (i=1,2,⋯,n)CiCentroid of thei‐th link (i=1,2,⋯,n)JiThe joint connectedi−1‐th link and i‐th link (i=1,2,⋯,n)liThe vector fromJi to Ji+1, and li=ai+biaiThe vector fromJi to Ci (i=1,2,⋯,n)biThe vector fromCi to Ji+1 (i=1,2,⋯,n−1)bn−1The vector fromCn−1 to the end-effectorzIiThe unit vector in the axis direction inΣimiThe mass of thei‐th link (i=1,2,⋯,n)mbThe mass of the baseMThe total mass of the systemIiThe inertia tensor ofi‐th link (i=1,2,⋯,n)IbThe inertia tensor of baseθiThe joint angle ofi‐th joint (i=1,2,⋯,n)qθThe position vector of joint angle in joint space,qθ=θ1,⋯,θnT∈Rn×1rIbThe position vector of centroid of base in inertial framerIiThe position vector of centroid ofi‐th in inertial frame (i=1,2,⋯,n)rIeThe position vector of the end-effectorpIiThe position vector ofJi (i=1,2,⋯,n)qbThe pose vector of the base,qb=rb,ϕbϕbThe orientation vector of the baseq̇bThe velocity of the basexeThe pose vector of the end-effector,xe=re,ϕeϕeThe orientation vector of the end-effectorvbThe velocity of the centroid of baseviThe velocity of the centroid ofi‐th link (i=1,2,⋯,n)veThe velocity of the end-effectorωbThe angular velocity of the baseωiThe angular velocity of thei‐th linkωeThe angular velocity of the end-effectorAssuming that theJl is locked, and the link Ll−1 and link Ll can be connected as a new linkL~l−1. Specifying the coordinate system of the link and joint after Jl is locked as follows: If the location of the joint or link is in front of theJl, it is consistent with the kinematic model without a joint-locked failure; otherwise, add “~” above the symbols of the kinematic model without a joint-locked failure, and the new number of the joint or link is subtracted 1 from the original basis.The dynamics parameters ofL~l−1 can be obtained according to the following equation
(1)
The mass ofL~l−1: m~l−1=ml+ml−1(2)
The centroid ofL~l−1: r~Il−1=ml−1rIl−1+mlrIl/ml−1+ml(3)
The inertia tensor ofL~l−1: I~l−1=Il−1+ml−1rIl−1TrIl−1E3−rIl−1rIl−1T, where E3∈R3×3 represents the third order identity matrixThus, the reconstruction of the kinematic model is shown in Figure3.Figure 3
The reconstructed kinematic model ofn-DOF space manipulator with a joint-locked failure.The velocity of the end-effector is represented as(3)x~̇e=J~bq~̇b+J~θq~̇θ,whereJ~b represents the base Jacobian matrix; J~θ represents the joint Jacobian matrix.Since the base of the space manipulator is in a free-floating state and is not influenced by external forces, the momentum of the system is conserved. Assuming that the momentum zero in the initial state, the velocity relation between the base and the end-effector can be expressed as(4)q~̇b=J~bθq~̇θ,whereJ~bθ represents the base-joint Jacobian matrix.Substituting Equation (3) into Equation (4), we can obtain
(5)x~̇e=J̇θ+J̇bJ~bθq~̇θ=J~Fq~̇θ,whereJ~F represents the generalized Jacobian matrix.


## 3.1.2. The Analysis of the Unilateral Kinematic Ability

The kinematic ability of the space manipulator generally includes the minimum singular value, condition number, and manipulability, which are related to the singular value of the Jacobian matrix [30]. Next, the kinematic ability will be analyzed based on J~F.According to the singular value decomposition theory,J~F can be written as
(6)J~F=UΣV,whereU∈Rm×m and V∈Rn−1×n−1 are orthogonal matrices; Σ∈Rm×n−1 is diagonal and nonnegative real rectangular diagonal matrix, and Σ can be written as
(7)Σ=σ10⋯000σ2⋯00⋮⋮⋱⋮000⋯σm0,whereσ1 is the maximum singular value; σm is the minimum singular value. When the space manipulator is in a singular configuration, σm=0.According to the singular values obtained above, the unilateral kinematic ability of the space manipulator can be obtained, as shown in Table3.Table 3
The unilateral kinematic ability of the space manipulator with a joint-locked failure.
Minimum singular valueCondition numberManipulabilityExpressions=σmk=σ1/σmw=σ1σ2⋯σmMeaningThe degree to which the manipulator approaches the singular configurationThe condition number reflects the isotropy of the manipulator’s kinematic ability in all directions.The measure of the end-effector’s kinematic ability when manipulator moves in different directionsValue ranges∈0,+∞k∈1,+∞w∈0,+∞Value analysisThe smaller the minimum singular value is, the closer the configuration is to the singular configuration.The closer the condition number is to 1, the higher the dexterity of the manipulator is.The bigger degenerate manipulability is, the greater kinematic ability of the end-effector in all directions is.According to Table3, the bigger minimum singular value and manipulability is, the better the kinematic ability is. And the smaller condition number is, the better the kinematic ability is. As a consequence, it is necessary to carry out a standardized treatment for the above unilateral kinematic ability.The minimum singular value and the manipulability are standardized as(8)s^=standards=s−ab−a,w^=standardw=w−ab−a,wherestandardx=x−a/b−a, x∈a,b represents the unilateral kinematic ability; s^ represents the minimum singular value with standardization, and w^ represents the manipulability with standardization.Aiming at the condition number, it can be standardized as(9)k^=1k,wherek^ represents the condition number with standardization.At this point, each unilateral kinematic ability is standardized, and then the kinematic ability will be comprehensively characterized.


## 3.1.3. The Comprehensive Characterization of the Kinematic Ability

The key to the comprehensive characterization of the kinematic ability is to determine the weight of each unilateral kinematic ability. The traditional entropy method ignores the qualitative factor that different tasks have different requirements on each unilateral kinematic ability. For example, assuming that a task needs the higher isotropy, the weight solved by the traditional entropy method only considers the unilateral kinematic ability and cannot reflect the higher isotropy, leading to an inaccuracy for the comprehensive characterization of kinematic ability. Considering that the AHP method can be used for the qualitative analysis [31], we choose the AHP method for the comprehensive characterization. Therefore, to cover the manipulator and requirements of the task, we combine and improve the AHP method and the traditional entropy method.Firstly, we calculate the subjective weight by utilizing the AHP method.(1)
Construct the hierarchical substructure model. Considering that different task requires different kinematic ability, a hierarchical structure from top to bottom is constructed as Figure4, including target layer (comprehensive characterization of kinematic ability), criterion layer (task requirements), and subcriterion layer (each unilateral kinematic ability). The target layer is the highest layer in the hierarchy, and the criterion layer and subcriterion layer are used to evaluate the subjective weight(2)
Construct judgment matrix. To judge the influence of factors at the current level on that at the next level, 1-7 scale method as shown in Table4 is used to define the importance degree of each element [32], and we construct the judgment matrix of Target-Criterion R=rijM×NFigure 4
The hierarchical substructure model.Table 4
Scale definitions of 1-7 scale method.
ScaleDefinition1i is just as important as j3i is slightly more important than j5i is more important than j7i is even more important than j2,4,6The importance lies between adjacent scalesReciprocalIf the ratio ofi and j is rij, the ratio of j and i is rji=1/rijBased on Table4, the judgment matrix R=rijM×N can be obtained by comparing the importance of each kinematic ability.
(3)
Do consistency test on the judgment matrix. By utilizing the maximum eigenvalue method, we can get the maximum eigenvalue ofR=rijM×N, and then the consistency test is used to judge whether the matrix is reasonable or notSolve the consistency indexCI(10)CI=λmax−NN−1,Whereλmax is the maximum eigenvalue of R=rijM×N; N is the order of R=rijM×N.According to the AHP theory, ifN is known, the standard consistency index RI can be obtained as shown in Table 5.Table 5
The standard consistency index.
NRI102030.5240.8951.1261.24Calculate the consistency ratioCR(11)CR=CIRI.IfCR<0.10, the judgment matrix is acceptable, if not, the judgment matrix should be modified appropriately by changing rij.
(4)
The solution of the subjective weight. For the judgment matrix satisfying the consistency test, the subjective weight of the unilateral kinematic ability can be obtained according toλmax. Then, we calculate the eigenvector of λmax, and take it as ω1,⋯,ωNT, which corresponds to the subjective weight of each unilateral kinematic ability ω1⋯ωN, respectivelyBased on the above steps, the subjective weight ofs^, w^, k^ can be obtained, which is denoted as
(12)ωCR=ωCR_1,ωCR_2,ωCR_3.Next, the entropy method is used to solve the objective weight of the unilateral kinematic ability. The configuration of the space manipulator with a joint-locked failure in the halt process can be expressed asqλ=θ1θ2⋯θn−1, where λ (λ=1,2,⋯K) represents the λ‐th configuration and K is the total number of configurations.When the manipulator is in theλ‐th configuration, the proportion of the h‐th unilateral kinematic ability in the total value of the kinematic ability is Pλ_h, which can be calculated by the following formula
(13)Pλ_h=Xλ_h∑λ=1KXλ_hh=1,2,3,whereXλ_h represents the value of the h‐th unilateral kinematic ability when the manipulator is in the λ‐th configuration.And then the entropy value of the characterization of theh‐th unilateral kinematic ability can be obtained as
(14)eh=−∑λ=1KPλ_hlogPλ_hln∑h,where0≤eh≤1.The difference coefficientgh can be calculated as
(15)gh=1−eh.The weight of each unilateral kinematic ability can be obtained as(16)ωCP_h=gh∑h=13gh,WhereωCP_hh=1,2,3 represents the weight of h‐th unilateral kinematic ability, which can be written as
(17)ωCP=ωCP_1,ωCP_2,ωCP_3.According to Equation (12) and Equation (17), the final weight of each unilateral kinematic ability can be obtained as
(18)ω=ωCR+ωCP=ωCR_1+ωCP_1,ωR_2+ωCP_2,ωCR_3+ωCP_3=ωC_1,ωC_2,ωC_3.According to the above analysis, the comprehensive characterization of the kinematic ability can be expressed as(19)CPqλ=ωC_1×s^qλ+ωC_2×k^qλ+ωC_3×w^qλλ=1,2,⋯,K,whereCPqλ is the comprehensive characterization of the kinematic ability when the manipulator is in the λ‐th configuration.At this point, the comprehensive characterization of the kinematic ability of the manipulator is completed.


## 3.2. The Construction and Solution of the HCOM

In this section, the HCOM is constructed based on the objective of maximizing the comprehensive characterization of the kinematic ability, and the optimal halt configuration is obtained by solving the HCOM with the Monte Carlo method.According to Equation (1) and Equation (19), the HCOM can be expressed as
(20)findqbest,maxCPqbest,whereqbest is the optimal halt configuration.Next, the HCOM is solved by utilizing the Monte Carlo method. The specific method is shown asStep 1.
For each jointJ~i, the random numbers of each joint angle can be generated by using the random number method: Ni=Ni_1Ni_2⋯Ni_K∈R1×K, which are between 0 and 1, and then the pseudorandom number of each joint angle can be obtained as
(21)θ~i=θ~imin+Δθ~i×Ni=θ~imin+Δθ~i⋅Ni_1θ~imin+Δθ~i⋅Ni_2⋯θ~imin+Δθ~i⋅Ni_K,
whereθ~imin=θ~iminθ~imin⋯θ~imin∈R1×K; Δθ~i=θ~imax−θ~imin; i=1,⋯,n−1.Step 2.
The pseudorandom number of the joint angles of each joint can be expressed as(22)CS=θ~1min+Δθ~1⋅N1_1θ~1min+Δθ~1⋅N1_2⋯θ~1min+Δθ~1⋅N1_Kθ~2min+Δθ~2⋅N2_1θ~2min+Δθ~2⋅N2_2⋯θ~2min+Δθ~2⋅N2_K⋮⋮⋱⋮θ~n−1min+Δθ~n−1⋅Nn−1_1θ~n−1min+Δθ~n−1⋅Nn−1_2⋯θ~n−1min+Δθ~n−1⋅Nn−1_K=θ11θ12⋯θ1Kθ21θ22⋯θ2K⋮⋮⋱⋮θn−11θn−12⋯θn−1K∈Rn−1×K.
Each column corresponds to a certain configuration, and Equation (22) can be written as
(23)CS=q1q2⋯qK∈R1×K.
And the configuration space of the manipulator can be obtained as(24)S=q1,q2,⋯,qλ,⋯,qKλ=1,2,⋯,K.Step 3.
Select a configurationqbest=θ1bestθ2best⋯θn−1best, which maximums CPqbest in Equation (19).So far, by solving the HCOM, the optimal halt configuration of the space manipulator is obtained.


## 4. The Halt Motion Optimization

To make the manipulator move continuously and stably in the halt process, the halt motion will be planned with a six-order polynomial interpolation method which contains an optimal coefficient. By taking the limitation of the joint running parameters as constraint conditions, and the minimum disturbance torque of the base as the objective, the HMOM containing the optimal coefficient is constructed. And with the HMOM solved, the halt motion of the faulty manipulator is optimized.


### 4.1. The Motion Planning in Joint Space Based on Six-Order Polynomial Interpolation

According to the characteristics of the halt (S0and Sf are known), when tf is determined, the halt motion of the manipulator is a function interpolation problem that satisfies six constraints, which includes θt0, θtf, θ̇t0, θ̇tf, θ¨t0, and θ¨tf. Therefore, the halt motion planning of the manipulator is carried out by using a six-order polynomial interpolation method, and the HMOP is described by introducing the optimization coefficient.The joint variable of the manipulator can be described as(25)θt=a0+a1t+a2t2+a3t3+a4t4+a5t5+a6t6,wherea0~a6 are the undetermined coefficients.According to the characteristics of halt motion, Equation (25) can be written as
(26)θt0=θ0=a0,θtf=θf=a0+a1tf+a2tf2+a3tf3+a4tf4+a5tf5+a6tf6,θ̇t0=θ̇0=a1,θ̇tf=a1+2a2tf+3a3tf2+4a4tf3+5a5tf4+6a6tf5,θ¨t0=θ¨0=2a2,θ¨tf=θ¨f=2a2+6a3tf+12a4tf2+20a5tf3+30a6tf4.Set the optimization coefficientk. According to Equation (26), let a6=k and we can obtain
(27)a0=θ0,a1=θ̇0,a2=θ¨02,a3=−tf3k+20θf−θ0−tf23θ¨0−θ¨f−tf8θ̇f+12θ̇02tf3,a4=3tf2k+tf23θ¨0−2θ¨f+tf14θ̇f+16θ̇0−30θf−θ02tf4,a5=−3tfk+θ¨f−θ¨0tf2−6θ̇0+θ̇ftf+12θf−θ02tf5,a6=k.According to Equation (27), the joint running parameters Var=qt,q̇t,q¨t,τt⋯ can be obtained at every moment in the halt process, which is the function with k as the variable, so it can be written as Var=Vark. Therefore, the halt motion can be optimized by optimizing the coefficient.


### 4.2. The Construction and Solution of the HMOM

In this section, considering the characteristics of the base and the safety of the halt, we construct the HMOM, where the minimum disturbance torque of the base is taken as the objective and the limitation of the joint running parameters as the constraint condition.According to the Newton-Euler dynamics equations, the resultant forcefi and moment ni of each joint can be expressed as finiT=fixfiyfiznixniynizT.The inertia force and moment of inertia of each link can be obtained as(28)F~Ii=m~iv~̇Ii,N~Ii=RIiI~iω~̇Ii+ω~Ii×RIiI~iω~Ii.The relationship of the force and torque between the joints are(29)f~Ii=F~Ii+f~Ii+1,n~Ii=N~Ii+RIia~i+b~i×f~Ii+1+n~Ii+1+RIia~i×F~Ii.For the space manipulator, the 1-th joint is directly connected to the free-floating base. Therefore, according to the model diagram shown in Figure5, the disturbance torque of the base n~b can be calculated by f~I1n~I1T.Figure 5
Solving model diagram of the disturbance torque.According to the equivalence principle of space force/torque, the disturbance force and torque of the base can be expressed as(30)f~Id=Rb1f~I1,n~Id=Rb1n~I1+Pb1×f~I1,wheref~Id=f~dxf~dyf~dzT, n~Id=n~dxn~dyn~dzT.According to Equation (30), the disturbance torque of the base in the halt process can be written as
(31)τbk=n~dxn~dyn~dzT.To reflect the overall change level of the disturbance torque during the whole halt process, the disturbance torque of the base is globalized as(32)τ¯bk=∫t0tfτbdttf−t0,whereτb represents the 2-norm of τb.Considering the safety and stability of the halt, the joint running parameters need to be constrained within a range as follows(33)limk=qtk∈qmin,qmax,q̇tk∈q̇min,q̇max,q¨tk∈q¨min,q¨max.According to the above analysis, the HMOM can be expressed as(34)findkminτ¯bks.t.limk.So far, the construction of the HMOM has been completed, and the next step is to solve the HMOM.A genetic algorithm is a method to search for the optimal solution by simulating the natural evolution process. It has a good ability to search for the optimal global solution and is widely used in solving optimization problems. The basic idea is to use the population search technology to take the population as a set of solutions and generate a new generation of population by applying a series of genetic operations to the current population, such as selection, crossover, and variation, and gradually optimize the population to the state containing the approximate optimal solution. We solve the HMOM by using a genetic algorithm, which can achieve the selection of the optimization coefficientk:Specific steps for solving the HMOM based on genetic algorithm are shown as followsStep 1.
Initialization: according to the accuracy of the solution and the solution space ofk, the binary coding method is selected to encode the population, and the initial population in the coding space is obtained. Meanwhile, the current evolution algebra is set as G=0, and the maximum evolution algebra is set as Gmax.Step 2.
Individual evaluation: set the fitness function to calculate the fitness ofk in the population. The reciprocal of τ¯bk can be selected as the fitness function
(35)f=τ¯bk−1.
According to the fitness function, the coefficient which meetsmaxf in the current population is obtained.Step 3.
Genetic operator: selection, crossover, and variation.(1)
Select Operation. The superior coefficientk in STEP 2 is passed to the next generation directly or by pairing crossover. In this paper, we use the elite mechanism for selection, that is, the coefficient corresponding to the highest fitness must be selected, and the probability of each coefficient being selected is proportional to the fitness value. Calculate the probability of each coefficient which is selected(36)pd=fkd∑d=1ξfkd,
whereξ is the number of individuals in the initial population.
(2)
Crossover Operations. The crossover probabilitypc is calculated and the crossover operation between two individuals is performed according to the crossover probability(37)pc=g1fmax−ffmax−favg,f≥favg,g2,f<favg,
wherefmax is the maximum fitness value, favg is the average fitness value, f is the larger fitness value of the two individuals, and g1 and g2 are constant.
(3)
Mutation Operation. The mutation probability is calculated and the individual is calculated according to the mutation probability(38)pm=g3fmax−f′fmax−favg,f′≥favg,g4,f′<favg,
wheref′ is the fitness value of the individual which is going to perform the mutation operation, and g3 and g4 are constant.Step 4.
Termination condition. IfG=Gmax, the coefficient k corresponding to the maximum fitness value is taken as the optimal solution; if not, repeat Step 2 to Step 4.Based on the above process, the solution of HMOM is completed, which can ensure the steady halt of the manipulator to the optimal halt configuration.


## 4.1. The Motion Planning in Joint Space Based on Six-Order Polynomial Interpolation

According to the characteristics of the halt (S0and Sf are known), when tf is determined, the halt motion of the manipulator is a function interpolation problem that satisfies six constraints, which includes θt0, θtf, θ̇t0, θ̇tf, θ¨t0, and θ¨tf. Therefore, the halt motion planning of the manipulator is carried out by using a six-order polynomial interpolation method, and the HMOP is described by introducing the optimization coefficient.The joint variable of the manipulator can be described as(25)θt=a0+a1t+a2t2+a3t3+a4t4+a5t5+a6t6,wherea0~a6 are the undetermined coefficients.According to the characteristics of halt motion, Equation (25) can be written as
(26)θt0=θ0=a0,θtf=θf=a0+a1tf+a2tf2+a3tf3+a4tf4+a5tf5+a6tf6,θ̇t0=θ̇0=a1,θ̇tf=a1+2a2tf+3a3tf2+4a4tf3+5a5tf4+6a6tf5,θ¨t0=θ¨0=2a2,θ¨tf=θ¨f=2a2+6a3tf+12a4tf2+20a5tf3+30a6tf4.Set the optimization coefficientk. According to Equation (26), let a6=k and we can obtain
(27)a0=θ0,a1=θ̇0,a2=θ¨02,a3=−tf3k+20θf−θ0−tf23θ¨0−θ¨f−tf8θ̇f+12θ̇02tf3,a4=3tf2k+tf23θ¨0−2θ¨f+tf14θ̇f+16θ̇0−30θf−θ02tf4,a5=−3tfk+θ¨f−θ¨0tf2−6θ̇0+θ̇ftf+12θf−θ02tf5,a6=k.According to Equation (27), the joint running parameters Var=qt,q̇t,q¨t,τt⋯ can be obtained at every moment in the halt process, which is the function with k as the variable, so it can be written as Var=Vark. Therefore, the halt motion can be optimized by optimizing the coefficient.


## 4.2. The Construction and Solution of the HMOM

In this section, considering the characteristics of the base and the safety of the halt, we construct the HMOM, where the minimum disturbance torque of the base is taken as the objective and the limitation of the joint running parameters as the constraint condition.According to the Newton-Euler dynamics equations, the resultant forcefi and moment ni of each joint can be expressed as finiT=fixfiyfiznixniynizT.The inertia force and moment of inertia of each link can be obtained as(28)F~Ii=m~iv~̇Ii,N~Ii=RIiI~iω~̇Ii+ω~Ii×RIiI~iω~Ii.The relationship of the force and torque between the joints are(29)f~Ii=F~Ii+f~Ii+1,n~Ii=N~Ii+RIia~i+b~i×f~Ii+1+n~Ii+1+RIia~i×F~Ii.For the space manipulator, the 1-th joint is directly connected to the free-floating base. Therefore, according to the model diagram shown in Figure5, the disturbance torque of the base n~b can be calculated by f~I1n~I1T.Figure 5
Solving model diagram of the disturbance torque.According to the equivalence principle of space force/torque, the disturbance force and torque of the base can be expressed as(30)f~Id=Rb1f~I1,n~Id=Rb1n~I1+Pb1×f~I1,wheref~Id=f~dxf~dyf~dzT, n~Id=n~dxn~dyn~dzT.According to Equation (30), the disturbance torque of the base in the halt process can be written as
(31)τbk=n~dxn~dyn~dzT.To reflect the overall change level of the disturbance torque during the whole halt process, the disturbance torque of the base is globalized as(32)τ¯bk=∫t0tfτbdttf−t0,whereτb represents the 2-norm of τb.Considering the safety and stability of the halt, the joint running parameters need to be constrained within a range as follows(33)limk=qtk∈qmin,qmax,q̇tk∈q̇min,q̇max,q¨tk∈q¨min,q¨max.According to the above analysis, the HMOM can be expressed as(34)findkminτ¯bks.t.limk.So far, the construction of the HMOM has been completed, and the next step is to solve the HMOM.A genetic algorithm is a method to search for the optimal solution by simulating the natural evolution process. It has a good ability to search for the optimal global solution and is widely used in solving optimization problems. The basic idea is to use the population search technology to take the population as a set of solutions and generate a new generation of population by applying a series of genetic operations to the current population, such as selection, crossover, and variation, and gradually optimize the population to the state containing the approximate optimal solution. We solve the HMOM by using a genetic algorithm, which can achieve the selection of the optimization coefficientk:Specific steps for solving the HMOM based on genetic algorithm are shown as followsStep 1.
Initialization: according to the accuracy of the solution and the solution space ofk, the binary coding method is selected to encode the population, and the initial population in the coding space is obtained. Meanwhile, the current evolution algebra is set as G=0, and the maximum evolution algebra is set as Gmax.Step 2.
Individual evaluation: set the fitness function to calculate the fitness ofk in the population. The reciprocal of τ¯bk can be selected as the fitness function
(35)f=τ¯bk−1.
According to the fitness function, the coefficient which meetsmaxf in the current population is obtained.Step 3.
Genetic operator: selection, crossover, and variation.(1)
Select Operation. The superior coefficientk in STEP 2 is passed to the next generation directly or by pairing crossover. In this paper, we use the elite mechanism for selection, that is, the coefficient corresponding to the highest fitness must be selected, and the probability of each coefficient being selected is proportional to the fitness value. Calculate the probability of each coefficient which is selected(36)pd=fkd∑d=1ξfkd,
whereξ is the number of individuals in the initial population.
(2)
Crossover Operations. The crossover probabilitypc is calculated and the crossover operation between two individuals is performed according to the crossover probability(37)pc=g1fmax−ffmax−favg,f≥favg,g2,f<favg,
wherefmax is the maximum fitness value, favg is the average fitness value, f is the larger fitness value of the two individuals, and g1 and g2 are constant.
(3)
Mutation Operation. The mutation probability is calculated and the individual is calculated according to the mutation probability(38)pm=g3fmax−f′fmax−favg,f′≥favg,g4,f′<favg,
wheref′ is the fitness value of the individual which is going to perform the mutation operation, and g3 and g4 are constant.Step 4.
Termination condition. IfG=Gmax, the coefficient k corresponding to the maximum fitness value is taken as the optimal solution; if not, repeat Step 2 to Step 4.Based on the above process, the solution of HMOM is completed, which can ensure the steady halt of the manipulator to the optimal halt configuration.


## 5. Simulation

In this paper, a 7-DOF space manipulator is used as the simulate object to verify the halt optimization method, which is shown in Figure6. The DH and dynamic parameters of the space manipulator are shown in Tables 6 and 7, respectively.Figure 6
The kinematic model of the 7-DOF space manipulator.Table 6
Initial DH parameters of the 7-DOF space manipulator.
iθi°dimaimαi°0\\00100.60902900.50-90300.550400.550500.50906-900.50-90700.6\\Table 7
The dynamic parameters of the 7-DOF space manipulator.
imikgIxxi,Iyyi,Izzikg⋅m2Pcim142.5[0.89, 0.05, 0.89][0, 0.25, 0]T242.5[0.05, 0.89, 0.89][0, 0.25, 0]T370[0.088, 145.83, 145.83][2.5, 0, 0]T470[0.088, 145.83, 145.83][2.5, 0, 0]T542.5[0.89, 0.89, 0.05][0, 0, -0.25]T642.5[0.89, 0.89, 0.05][0, 0, -0.25]T742.5[1.28, 1.28, 0.05][0, 0, -0.3]T


### 5.1. The Halt Configuration Optimization

Firstly, the AHP method is used to solve the subjective weight of each unilateral kinematic ability, and the judgment matrix of Target-Criterion is constructed according to Table4, which is shown as
(39)R=11/2121211/21The eigenvalue of the judgment matrix isλ=3, and the eigenvector of λ=3 is 1/4,1/2,1/4T. The result of the consistency test is CR=0<0.10, so the consistency of the judgment matrix is acceptable.According to the eigenvector ofλ=3, the subjective weights of s^, w^, and k^ are 1/4, 1/2, and 1/4, respectively. The comprehensive kinematic ability of the manipulator is simulated by combining with the traditional entropy method.Suppose there is a locked failure on the 4-th joint, the initial halt configuration except for the locked joint isq0=−180°−180°−100°−10°20°20°, the angle range of each joint is −180°,180°, and the number of random numbers is K=36000. According to Equation (19), it can be seen that the CP is a set of the 7-dimensional array, that is, the angle of six joints correspond to a CP value. Therefore, when the Monte Carlo method is used to solve the problem, the simulation results should be a 7-dimensional space diagram. To show the conclusion more conveniently and intuitively, the 7-dimensional space diagram can be converted into six 2-dimensional planar graphs in the form of a section diagram. The simulation results are shown in Figure 7.Figure 7
The simulation results of CP which are in the form of a 2-dimensional planar graph. (a) The angle of joint 1—CP. (b) The angle of joint 2—CP. (c) The angle of joint 3—CP. (d) The angle of joint 5—CP. (e) The angle of joint 6—CP. (f) The angle of joint 7—CP.
(a)(b)(c)(d)(e)(f)As we can see from Figure7, the optimal halt configuration of the manipulator is qbest=−167°−90°40°−158°90°−62°. At the same time, it can be found that the suboptimal halt configuration is qsub=−52°90°84°148°−90°157°. The kinematic ability value of the manipulator in the suboptimal halt configuration is 0.2969. By the comparison, we can see that the value in the optimal halt configuration is much higher than that in the suboptimal halt configuration: 1−0.2969/0.2969=236.8%.


### 5.2. The Halt Motion Optimization

The optimal halt configuration of the manipulator with a joint-locked failure isqf=−167°−90°40°−158°90°−62°, and the initial configuration is set to beq0=−180°−180°−100°−10°20°20°. Suppose the velocity of healthy joints isq̇0=203015254035°/s, the acceleration of healthy joints is q¨0=8101210−86°/s2, the velocity constraint is q̇min,q̇max=−40,40°/s, the acceleration constraint is q¨min,q¨max=−20,20°/s2, and the time of the entire halt process is T=tf−t0=10s.By solving the HMOM, the global variation of the base disturbance torque which changes with the coefficientk in the halt process is shown in Figure 8. According to Figures 8(a)–8(c), it can be seen that when k=0.0018, the minimum global disturbance torque is 15.55N⋅m, and when k=0, the global disturbance torque is22.25N⋅m. So the global disturbance torque of the base is reduced by 22.25−15.55/22.25=30.1% after the optimization.Figure 8
The variations of disturbance torque corresponding to differentk. (a) The variations of disturbance torque when k∈0,0.1. (b) The variations of disturbance torque when k∈0,0.01. (c) The comparison of the disturbance torque between optimization and nonoptimization.
(a)(b)(c)Whenk=0.0018, the variations of joint parameters are shown in Figure 9. At this time, the global disturbance torque of the base is minimal, and the constraint of joint parameters is also satisfied.Figure 9
The variations of joint parameters. (a) The parameters of joint 1. (b) The parameters of joint 2. (c) The parameters of joint 3. (d) The parameters of joint 5. (e) The parameters of joint 6. (f) The parameters of joint 7.
(a)(b)(c)(d)(e)(f)


## 5.1. The Halt Configuration Optimization

Firstly, the AHP method is used to solve the subjective weight of each unilateral kinematic ability, and the judgment matrix of Target-Criterion is constructed according to Table4, which is shown as
(39)R=11/2121211/21The eigenvalue of the judgment matrix isλ=3, and the eigenvector of λ=3 is 1/4,1/2,1/4T. The result of the consistency test is CR=0<0.10, so the consistency of the judgment matrix is acceptable.According to the eigenvector ofλ=3, the subjective weights of s^, w^, and k^ are 1/4, 1/2, and 1/4, respectively. The comprehensive kinematic ability of the manipulator is simulated by combining with the traditional entropy method.Suppose there is a locked failure on the 4-th joint, the initial halt configuration except for the locked joint isq0=−180°−180°−100°−10°20°20°, the angle range of each joint is −180°,180°, and the number of random numbers is K=36000. According to Equation (19), it can be seen that the CP is a set of the 7-dimensional array, that is, the angle of six joints correspond to a CP value. Therefore, when the Monte Carlo method is used to solve the problem, the simulation results should be a 7-dimensional space diagram. To show the conclusion more conveniently and intuitively, the 7-dimensional space diagram can be converted into six 2-dimensional planar graphs in the form of a section diagram. The simulation results are shown in Figure 7.Figure 7
The simulation results of CP which are in the form of a 2-dimensional planar graph. (a) The angle of joint 1—CP. (b) The angle of joint 2—CP. (c) The angle of joint 3—CP. (d) The angle of joint 5—CP. (e) The angle of joint 6—CP. (f) The angle of joint 7—CP.
(a)(b)(c)(d)(e)(f)As we can see from Figure7, the optimal halt configuration of the manipulator is qbest=−167°−90°40°−158°90°−62°. At the same time, it can be found that the suboptimal halt configuration is qsub=−52°90°84°148°−90°157°. The kinematic ability value of the manipulator in the suboptimal halt configuration is 0.2969. By the comparison, we can see that the value in the optimal halt configuration is much higher than that in the suboptimal halt configuration: 1−0.2969/0.2969=236.8%.


## 5.2. The Halt Motion Optimization

The optimal halt configuration of the manipulator with a joint-locked failure isqf=−167°−90°40°−158°90°−62°, and the initial configuration is set to beq0=−180°−180°−100°−10°20°20°. Suppose the velocity of healthy joints isq̇0=203015254035°/s, the acceleration of healthy joints is q¨0=8101210−86°/s2, the velocity constraint is q̇min,q̇max=−40,40°/s, the acceleration constraint is q¨min,q¨max=−20,20°/s2, and the time of the entire halt process is T=tf−t0=10s.By solving the HMOM, the global variation of the base disturbance torque which changes with the coefficientk in the halt process is shown in Figure 8. According to Figures 8(a)–8(c), it can be seen that when k=0.0018, the minimum global disturbance torque is 15.55N⋅m, and when k=0, the global disturbance torque is22.25N⋅m. So the global disturbance torque of the base is reduced by 22.25−15.55/22.25=30.1% after the optimization.Figure 8
The variations of disturbance torque corresponding to differentk. (a) The variations of disturbance torque when k∈0,0.1. (b) The variations of disturbance torque when k∈0,0.01. (c) The comparison of the disturbance torque between optimization and nonoptimization.
(a)(b)(c)Whenk=0.0018, the variations of joint parameters are shown in Figure 9. At this time, the global disturbance torque of the base is minimal, and the constraint of joint parameters is also satisfied.Figure 9
The variations of joint parameters. (a) The parameters of joint 1. (b) The parameters of joint 2. (c) The parameters of joint 3. (d) The parameters of joint 5. (e) The parameters of joint 6. (f) The parameters of joint 7.
(a)(b)(c)(d)(e)(f)


## 6. Conclusions

In this paper, a halt optimization strategy for a space manipulator with a joint-locked failure is proposed. By describing the halt optimization problem, we creatively put forward the halt configuration optimization method. Considering the requirements of the operation task, the HCOM is constructed and solved to obtain the optimal halt configuration. According to the constraint of joint running parameters and the disturbance torque of the base, the HMOM is constructed, and by utilizing the joint space motion planning with the six-order polynomial interpolation method, we solve it to optimize the halt of the manipulator. Finally, the simulation is carried out to verify the effectiveness of the proposed strategy. The innovation of this paper mainly includes (1) the halt configuration optimization ensures that the manipulator in the halt configuration owns the best kinematic ability, and the kinematic ability in the optimal halt configure is236.8% higher than that in the suboptimal halt configuration. (2) Each unilateral kinematic ability index is analyzed quantitatively, making the comprehensive representation of the kinematic ability more accurate. (3) The disturbance torque of the base is optimized during the halt process, and its global disturbance torque is reduced by 30% at least after the optimization.



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*Source: 8811433-2020-09-07.xml*