Resource coordination and allocation strategies are proposed to reduce the probability of failure by aiming at the problem that the robot cannot continue to work after joint failure. Firstly, the principal component analysis method under unsupervised branches in machine learning is used to analyze the reliability function and various indexes of the robot to obtain the comprehensive evaluation function. Then, based on the fault-tolerant-control inverse-kinematics optimal algorithm, each joint can be scheduled by weighted processing. Finally, the comprehensive evaluation function is used as an index to evaluate the probability of fault occurrence, and the weight is defined to realize the coordinated resource allocation of redundant robots. Taking the planar four revolute joints (4R) redundant robot as an example, the algorithm control is compared. Based on reasonable modeling and physical verification, the results show that the method of optimal resource coordination and allocation is effective.

As the cross-discipline of robotics has flourished, it has become a major player in the agricultural sector. However, robot components can age over time or have sudden changes in load, leading to robot failure, and according to robot maintenance staff, when robots fail, they are shut down for repair, which undoubtedly increases time costs. In actual agricultural production, some fruits have a very short picking cycle, for example, cherries, which are picked in half a month, and if they are not picked in time, they will cause economic losses. It is common for crops with a short picking cycle to be picked a little early and then ripened, but this method requires good ventilation to avoid accumulating too much moisture and mold, and the ripened fruit is not as good in taste and flavor as tree-ripened fruit. In order to ensure that the robot picks the fruit at the optimum time, it needs to be able to operate even if it experiences a joint malfunction, avoiding financial losses due to maintenance and missing the optimum picking time. To meet these requirements, the robot must be highly reliable and fault tolerant. Fault-tolerant operation reduces its downtime and can also be used to extend its service life or to detect subsystem failures early and speed up the repair process. Fault tolerance in redundant robots has therefore become a major focus of attention.

The most systematic and in-depth research on fault-tolerant control of redundant robotic arms is conducted by the group of Ahmad A. Almarkhi and Khaled M. Ben-Gharbia and Anthony A. Maciejewski, whose research ranges from fault-tolerant control of planar three-joint redundant machines to fault-tolerant control of spatially redundant robotic arms, proposing the design of optimal fault-tolerant Jacobi matrices and maximizing the size of a self-moving human face to improve the fault tolerance of the robot and for optimal fault tolerance for different joint failure probabilities. Many original contributions to the design of robot kinematics have been made (Ben-Gharbia et al., 2014; Xie and Maciejewski, 2017, 2018; Almarkhi and MacIejewski, 2019; Almarkhi et al., 2020). Abdi and Nahavandi (2012) proposed the theory of optimal fault-tolerant configurations using condition numbers as optimization terms. Many domestic scholars have also researched fault-tolerant control of redundant manipulators. Jing Zhao's Beijing University of Technology team has conducted in-depth research on this aspect. They have proposed a series of control algorithms for redundant robotic arms, flexible redundant robotic arms, and robot coordination and verified the effectiveness of these algorithms through experiments (Xie and Zhao, 2010a; Zhao et al., 2014, 2012; Xie and Zhao, 2010b). Jia Qingxuan's Beijing University of Posts and Telecommunications team has researched multi-joint and multi-type fault-tolerant control, global fault-tolerant trajectory optimization methods, and joint velocity mutation suppression during force and/or position fault tolerance and verified the accuracy of the algorithms through examples. Changchun Liang from the General Design Department of Beijing Space Vehicles studied the space robotic arm using a degenerate Jacobi-matrix-based mutation suppression method and made control simulations for three operating conditions: pre-failure, operation, and post-failure suppression (Liang et al., 2016). Zhaohui Zhu from the Beijing University of Technology analyzed the seven rotating joints (7R) redundant robotic arm, from the working space of fault-tolerant performance to the screening of fault-tolerant dislocations by comprehensive indicators (Zhu, 2016). Yu Cao, Southeast University, conducted a fault tolerance analysis of an anthropomorphic redundant robot arm with joint bias and made a prototype for the algorithm's effectiveness (Cao, 2019). Yanhui Wei, Harbin Engineering University, conducted a study on the fault tolerance of reconfigurable-robot working configurations and verified the effectiveness of the fault tolerance analysis method and the feasibility of the fault tolerance control method through example validation (Wei et al., 2010). Yu She, Harbin Institute of Technology, conducted a study on the kinematics and control of a redundant robot with a single joint. The kinematics and dynamics of a single-joint robot are re-modeled, and the trajectory planning under single-joint failure is investigated (She, 2013), Bo Zhao of Jilin University conducts an active decentralized fault-tolerant control method for reconfigurable robots with multiple concurrent faults and conducts an in-depth study from the field of control (Zhao, 2014). Minghao Li of Southwest University of Science and Technology adopts a deep reinforcement learning approach to fault-tolerant control of robotic arms and does experiments to verify the algorithm's effectiveness (Li, 2019; Li and Zhang, 2020). Nie Fu Jie of Changchun University of Technology conducted a decentralized optimal active fault-tolerant control for reconfigurable robotic arms based on an adaptive dynamic programming approach (Nie, 2021). Bing Ma of Changchun University of Technology obtained a decentralized near-optimal fault-tolerant controller with an observation–compensation–evaluation network structure (Ma, 2021).

From the above, it can be seen that, at present, most of the research in the field of fault-tolerant control is directed toward fault-tolerant control after a fault has occurred, such as suppressing sudden changes in joint speed and suppressing sudden changes in torque. Although the end position and end speed are guaranteed not to change abruptly, the joint speed will change abruptly, which leads to great instantaneous acceleration of the joint and easy damage to the motor. Based on a real-time control method for the optimal total performance of a planar redundant robot (Rong et al., 2022), this paper proposes a control method for optimal coordinated resource allocation of a redundant robot and is structured as follows: this paper introduces the concept of coordinated resource allocation, derives a comprehensive evaluation function based on the robot indicators and reliability functions through principal component analysis, and thus carries out coordinated resource allocation control based on this comprehensive evaluation function. The core of the algorithm is based on the comprehensive evaluation function for all tracking points in the trajectory. The comprehensive evaluation function is used to define the weight of the joint that affects the performance the most, to reduce the degree of impact on the overall performance after the failure of the joint, to reduce the sudden change in the angle of each joint when a failure occurs, and to achieve a smooth transition between the speed of the joint before the failure and the speed of the joint after the failure.

The most fundamental analytical methods in reliability studies are probability distributions, of which four are considered to be useful in the study of the reliability and safety of robotic systems, namely (i) exponential distribution, (ii) Rayleigh distribution, (iii) Weibull distribution, and (iii) bathtub hazard rate curve distribution (Dhillon, 2015). It has been shown that the failure rate of most robots is a function of time, with the most commonly used being the bathtub hazard rate curve. The distribution is like the shape of a bathtub.

The bathtub hazard rate curve function equation is

The hazard rate function can also be expressed as

The fault density function is related to the hazard rate function as follows:

Principal component analysis (PCA) is an important evaluation method for quantitative calculations, which allows instantaneous planar data to be replaced by a small number of integrated variables with minimal loss of information from the original data, making the data structure much simpler. The data structure can be greatly simplified.

These attempts to integrate the reliability function and multiple evaluation indicators of the robot using the PCA method, from which a comprehensive evaluation function is derived, provide the basis for an optimal control method for the coordinated allocation of resources.

The principal component analysis method is divided into two main aspects in
this paper: firstly, the establishment of a comprehensive evaluation index
consisting of a reliability function and the fusion of several evaluation
indicators of the robot, and secondly, the determination of the relevance of
every single indicator according to the correlation coefficient matrix. The
main steps are outlined below.

The data are standardized for all indicators to resolve the original problem of inconsistencies in the data scale, thus enabling standardized data for further evaluation.

Let the matrix of indicators be composed of

The correlation coefficient

The eigen roots and eigenvectors of the correlation coefficient matrix

The eigenvector corresponding to the

The number of principal components is determined. In this paper, the number of principal components of the comprehensive evaluation indexes is determined by the cumulative variance contribution of each index and the reliability function, i.e., according to the proportion of variance in relation to the total variance.

We can get the expression for the principal component

The comprehensive evaluation function is determined.

Commonly used single performance indicators for robotic arms consist of
kinematic condition number (

Common performance indicators.

Note:

Let us assume we have a redundant robot with

In the above formula, the variable of angular velocity subscript

For the jumping of robot joint speed, its essence is that the structure of the Jacobian matrix in the normal working environment is different from that in the fault environment, which leads to the jumping of robot joint speed. Although the end position of the robot does not change, the speed of the robot joint will also jump. Therefore, just avoiding the singularity of the Jacobian matrix after robot failure cannot really reduce the speed jump of robot locking joints.

In this paper, the robot joint failure is defined as the locking-joint
failure so the redundant robot will face the following three problems when carrying out
the fault-tolerant operation: (1) maintaining the maximum operability of the
robot to ensure that, when the robot joint fails, it will try to avoid the
singular position; (2) ensuring that the task is as close to the center as
possible to ensure that the robot end effector can always carry out work
tasks in the fault-tolerant workspace; (3) when the robot joint fails to
lock, the joint speed jump should be reduced as much as possible – that is,
the speed change before and after the failure should be reduced as much as
possible. From the above three problems, we can express motion planning as
optimizing the joint speed of the robot and ensuring that the square of the
speed jump is minimized; the formula can be expressed as follows:

From Eq. (26), we can get that the optimal joint speed of the robot after a
joint failure of the robot can also be expressed by the gradient projection
method, which includes the general solution and the special solution, that
is

Most of the current research has focused on fault-tolerant control after a robot joint has failed, and many algorithms have been proposed, such as suppression of velocity surges and post-fault fault tolerance based on neural network methods. However, few studies have investigated a real-time resource coordination allocation strategy for a certain robot executing a certain trajectory. In this paper, based on the extensive literature, we propose a comprehensive evaluation of the robot based on a reliability function derived mathematically from a hazard function combined with the robot's performance metrics using a principal component analysis (PCA) method under the unsupervised-algorithm branch of machine learning algorithms.

The principal component analysis method described above is used here to analyze the indicators corresponding to all state points in this task state, from which the principal components and the total evaluation function are analyzed.

As shown in Table 2, the robot arm tracking trajectory is interpolated into 10 interpolation points, each with its corresponding performance index.

Performance index data of each interpolation point.

The four-sport flexibility indicators

Standardized performance index data of the manipulator.

From the above principal component analysis method equations, the correlation coefficient matrix

Correlation coefficient matrix

From Table 5, the cumulative variance contribution of the first principal component is 95.223 %, which is greater than 85 %, indicating that the first principal component responds to 95.223 % of the information we get from the original variable and basically reflects the information contained in all indicators. So the principal component is 1.

Characteristic values, principal component contribution rate, and cumulative contribution rate.

From the loadings of each indicator in Table 6,

Factor load matrix.

As can be seen from Fig. 1, the first principal component accounts for the largest proportion eigenvectors are obtained by dividing the data of the factor-loading
matrix by the square root of its corresponding eigen root. Using the
equation for principal component analysis, we multiply the resulting
eigenvectors with the normalized data to obtain the principal component
expression.

Gravel figure.

In order to ensure that the end effector can complete its expected tasks after the mechanical arm fails, this paper uses the value of the integrated evaluation function as a criterion. According to the analysis, the larger the value, the better the performance and the smaller the impact on the overall performance when the fault occurs; conversely, the worse the performance, the greater the impact on the overall performance when the fault occurs.

The derivation is as follows: the Jacobi matrix is a linear transformation of the mapping of the joint
space velocity

The results are weighted on the basis of the optimal inverse-solution
algorithm under fault-tolerant control in Sect. 4 to facilitate the allocation of individual rod speeds in the context of a coordinated resource
allocation strategy. Namely,

The analysis process is shown in Fig. 2.

Flow chart of suppression algorithm based on fault prediction model.

This paper validates the resource coordination allocation algorithm on a previously studied redundancy-degree robot object based on the concept of centralized drive arrangement. The model is shown in Fig. 3.

Three-dimensional diagram of 4R redundant robot: 1. servo motor, 2. planetary reducer, 3. axis I, 4. rod I, 5. axis II, 6. rod II, 7. axis III, 8. rod III, 9. axis IV, 10. rod IV, 11. small pulley, 12. large pulley, 13. long timing belt, 14. motor-connecting plate, 15. short timing belt, 16. stiffener, 17. wallboard.

In the Cartesian coordinate system, each linkage is defined according to the

The

In this paper, the main research of joint failure after the stuck fault forms. The optimal algorithm in Sect. 7 is used to control the arm, tracking a straight line, assuming that the fault is accurately detected and that the fault occurs in 5 s. The simulation results are shown below.

Diagram of joint angle changes without failure.

Diagram of joint velocity variation without failure.

Diagram of angle changes of each joint when joint 1 fails.

Abrupt changes in the velocity of each joint during the failure of joint 1.

Diagram of angle changes of each joint when joint 2 fails.

Abrupt changes in the velocity of each joint during the failure of joint 2.

Diagram of angle changes of joints when joint 3 fails.

Abrupt changes in the velocity of each joint during the failure of joint 3.

Diagram of angle changes of joints when joint 4 fails.

Abrupt changes in the velocity of each joint during the failure of joint 4.

The data from the moment before and the moment after the moment of failure were calculated to collate the data in Table 8.

Index data affecting the performance of different joint failures.

The analysis in Table 8 shows that the overall performance is most affected by a failure in joint 4; thus, the resource coordination allocation algorithm is controlled for joint 4.

From Sect. 8.1, it is clear that a failure of joint 4 has the greatest impact on overall performance; thus, the method in Sect. 6 is used to control joint 4. The graph below shows the change in speed and angle of each joint following a failure of joint four after control using the coordinated resource allocation strategy.

The velocity changes of each joint after the failure of joint 4 under the prediction model.

The angle changes of each joint when joint 4 fails under the prediction model.

According to Fig. 15, it can be seen that the velocity of joint 4 had reduced to approximately near zero before the 5 s failure moment occurred. As can be seen from Fig 16, the angular fluctuation is negligible; thus, the sudden change in joint velocity after the failure occurred was minimal and had very little effect on the robot trajectory tracking. It can be seen that there is almost no mutation in joint 1 at the time of failure. The mutation value of joint 2 is 0.003, that of joint 3 is 0.005, that of joint 4 is 0.004, and the total mutation value is 0.012, which greatly reduces the impact of joint failure on the overall performance. Figures 18 and 20 show the physical verification images of the tracking-line trajectory. The morphology diagram shows that the attitude of the fourth joint was mediated before the failure occurred, as can be seen in the completely different attitude of the tracking-line simulation diagrams in Figs. 17 and 19.

End tracking-status diagram without failure.

State diagram of redundant robot after 1–9 s each time a fault occurs.

Terminal tracking-trajectory-state diagram when joint 4 fails.

State diagram of redundant robot after 1–9 s each time joint 4 fails.

Since our main concern is the fault-tolerant control strategy after the fault occurs, which is to ensure the minimum instantaneous impact of the equipment failure and to ensure the stability of the job processing, we assume that the mode failure will occur at a certain time so we did not predict the fault in advance in the study. We will study robot failure prediction in future research work.

The main innovation of this paper is to integrate the bathtub hazard rate curve with various performance indicators of the robot; to propose the obtainment of the comprehensive evaluation function through the principal component analysis method; and to take the function as the mathematical model of resource coordination allocation strategy, to take the planar redundant robot as the research object based on the inverse kinematics of the fault-tolerant
control of the optimal redundancy, and to realize the adjustment of the rod that
has the greatest impact on the performance after failure. This reduces the
probability of failure and the impact on the overall performance after a
fault occurs. Through the above theoretical derivation and verification of
simulation results and experimental results, the following conclusions can
be drawn:

The sudden change in angle after the failure of each joint shows that the sudden change in the speed of the joints close to the failed joint is greater than that of the joints far from the failed joint; i.e., the performance of the joints close to the failed joint is more affected after a failure.

It can be seen from the simulation verification in Figs. 17 and 19 and the physical verification in Figs. 18 and 20 that the fault-tolerant control method based on coordinated resource allocation can ensure that the end trajectory coincides with the target trajectory after a failure.

A comparison of Figs. 14 and 15 shows that the fault-tolerant control method based on coordinated resource allocation can effectively reduce the impact on the overall performance after a failure.

A synthesis of the figures in the text shows that the fault-tolerant control method based on the coordinated allocation of resources can ensure that the sudden change in joint speed is greatly reduced without a sudden change in end speed at the moment of failure.

Code has a certain confidentiality: if there is a need for code, please contact the author, and after evaluation this can be provided.

No data sets were used in this article.

YR and TD designed the experiments, and TD carried them out. XZ developed the model code and performed the simulations.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the Natural Science Foundation of Hebei Province (grant no. E2021203018).

This paper was edited by Zi Bin and reviewed by five anonymous referees.