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.

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 [

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. [

Aiming at the halt optimization of a manipulator with a joint-locked failure, Xu et al. [

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 [

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

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 is

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

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 optimization

The rest of this paper is organized as follows: In Section

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 Table

Symbols annotations of the halt.

Symbols | Annotations |
---|---|

The initial time of the halt | |

The end time of the halt | |

The time during the halt | |

The configuration when the time is | |

The joint velocity when the time is | |

The joint acceleration when the time is | |

The disturbance torque of the base in the halt process | |

The initial state of the halt, | |

The end state of the halt, | |

The parameters of the manipulator in the halt process, | |

The constraint of the halt, |

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 configuration

where

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 to

At this point, the halt optimization problem is transformed into the HCOP and the HMOP, which is shown in Figure

The description of the halt optimization problem.

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.

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.

For the

The kinematic model of

Symbols annotations of

Symbols | Annotations |
---|---|

Inertial coordinate frame | |

Base coordinate system | |

End-effector coordinate system | |

The coordinate system of the | |

Centroid of the | |

The joint connected | |

The vector from | |

The vector from | |

The vector from | |

The vector from | |

The unit vector in the axis direction in | |

The mass of the | |

The mass of the base | |

The total mass of the system | |

The inertia tensor of | |

The inertia tensor of base | |

The joint angle of | |

The position vector of joint angle in joint space, | |

The position vector of centroid of base in inertial frame | |

The position vector of centroid of | |

The position vector of the end-effector | |

The position vector of | |

The pose vector of the base, | |

The orientation vector of the base | |

The velocity of the base | |

The pose vector of the end-effector, | |

The orientation vector of the end-effector | |

The velocity of the centroid of base | |

The velocity of the centroid of | |

The velocity of the end-effector | |

The angular velocity of the base | |

The angular velocity of the | |

The angular velocity of the end-effector |

Assuming that the

The dynamics parameters of

The mass of

The centroid of

The inertia tensor of

Thus, the reconstruction of the kinematic model is shown in Figure

The reconstructed kinematic model of

The velocity of the end-effector is represented as

where

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

where

Substituting Equation (

where

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 [

According to the singular value decomposition theory,

where

where

According to the singular values obtained above, the unilateral kinematic ability of the space manipulator can be obtained, as shown in Table

The unilateral kinematic ability of the space manipulator with a joint-locked failure.

Minimum singular value | Condition number | Manipulability | |
---|---|---|---|

Expression | |||

Meaning | The degree to which the manipulator approaches the singular configuration | The 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 directions |

Value range | |||

Value analysis | The 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 Table

The minimum singular value and the manipulability are standardized as

where

Aiming at the condition number, it can be standardized as

where

At this point, each unilateral kinematic ability is standardized, and then the kinematic ability will be comprehensively characterized.

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 [

Firstly, we calculate the subjective weight by utilizing the AHP method.

Construct the hierarchical substructure model. Considering that different task requires different kinematic ability, a hierarchical structure from top to bottom is constructed as Figure

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 Table

The hierarchical substructure model.

Scale definitions of 1-7 scale method.

Scale | Definition |
---|---|

1 | |

3 | |

5 | |

7 | |

2,4,6 | The importance lies between adjacent scales |

Reciprocal | If the ratio of |

Based on Table

Do consistency test on the judgment matrix. By utilizing the maximum eigenvalue method, we can get the maximum eigenvalue of

Solve the consistency index

Where

According to the AHP theory, if

The standard consistency index.

1 | 0 |

2 | 0 |

3 | 0.52 |

4 | 0.89 |

5 | 1.12 |

6 | 1.24 |

Calculate the consistency ratio

If

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 to

Based on the above steps, the subjective weight of

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 as

When the manipulator is in the

where

And then the entropy value of the characterization of the

where

The difference coefficient

The weight of each unilateral kinematic ability can be obtained as

Where

According to Equation (

According to the above analysis, the comprehensive characterization of the kinematic ability can be expressed as

where

At this point, the comprehensive characterization of the kinematic ability of the manipulator is completed.

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 (

where

Next, the HCOM is solved by utilizing the Monte Carlo method. The specific method is shown as

For each joint

where

The pseudorandom number of the joint angles of each joint can be expressed as

Each column corresponds to a certain configuration, and Equation (

And the configuration space of the manipulator can be obtained as

Select a configuration

So far, by solving the HCOM, the optimal halt configuration of the space manipulator is obtained.

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.

According to the characteristics of the halt (

The joint variable of the manipulator can be described as

where

According to the characteristics of halt motion, Equation (

Set the optimization coefficient

According to Equation (

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 force

The inertia force and moment of inertia of each link can be obtained as

The relationship of the force and torque between the joints are

For the space manipulator, the 1-th joint is directly connected to the free-floating base. Therefore, according to the model diagram shown in Figure

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

where

According to Equation (

To reflect the overall change level of the disturbance torque during the whole halt process, the disturbance torque of the base is globalized as

where

Considering the safety and stability of the halt, the joint running parameters need to be constrained within a range as follows

According to the above analysis, the HMOM can be expressed as

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 coefficient

Specific steps for solving the HMOM based on genetic algorithm are shown as follows

Initialization: according to the accuracy of the solution and the solution space of

Individual evaluation: set the fitness function to calculate the fitness of

According to the fitness function, the coefficient which meets

Genetic operator: selection, crossover, and variation.

Select Operation. The superior coefficient

where

Crossover Operations. The crossover probability

where

Mutation Operation. The mutation probability is calculated and the individual is calculated according to the mutation probability

where

Termination condition. If

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.

In this paper, a 7-DOF space manipulator is used as the simulate object to verify the halt optimization method, which is shown in Figure

The kinematic model of the 7-DOF space manipulator.

Initial DH parameters of the 7-DOF space manipulator.

0 | \ | \ | 0 | 0 |

1 | 0 | 0.6 | 0 | 90 |

2 | 90 | 0.5 | 0 | -90 |

3 | 0 | 0.5 | 5 | 0 |

4 | 0 | 0.5 | 5 | 0 |

5 | 0 | 0.5 | 0 | 90 |

6 | -90 | 0.5 | 0 | -90 |

7 | 0 | 0.6 | \ | \ |

The dynamic parameters of the 7-DOF space manipulator.

1 | 42.5 | [0.89, 0.05, 0.89] | [0, 0.25, 0]^{T} |

2 | 42.5 | [0.05, 0.89, 0.89] | [0, 0.25, 0]^{T} |

3 | 70 | [0.088, 145.83, 145.83] | [2.5, 0, 0]^{T} |

4 | 70 | [0.088, 145.83, 145.83] | [2.5, 0, 0]^{T} |

5 | 42.5 | [0.89, 0.89, 0.05] | [0, 0, -0.25]^{T} |

6 | 42.5 | [0.89, 0.89, 0.05] | [0, 0, -0.25]^{T} |

7 | 42.5 | [1.28, 1.28, 0.05] | [0, 0, -0.3]^{T} |

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 Table

The eigenvalue of the judgment matrix is

According to the eigenvector of

Suppose there is a locked failure on the 4-th joint, the initial halt configuration except for the locked joint is

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.

As we can see from Figure

The optimal halt configuration of the manipulator with a joint-locked failure is

By solving the HMOM, the global variation of the base disturbance torque which changes with the coefficient

The variations of disturbance torque corresponding to different

When

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.

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 is

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there is no conflict of interest regarding the publication of this paper.

All authors gratefully acknowledge the financial support by the National Natural Science Foundation of China under Grant 51975059, the Research Fund of the manned space engineering (No. 18051030101), and the Fundamental Research Funds for the Center Universities (Grant No. 2019PTB-012).