#### Abstract

To meet the different requirements in the industrial area, a novel reconfigurable parallel mechanism is proposed based on the spatial multiloop overconstrained mechanism. The configurations can be changed by driving the low-DOF (degree-of-freedom) overconstrained mechanism. The mobility of this mechanism is investigated. And the kinematic model and Jacobian matrix are both established. Based on the Jacobian matrix, the workspace, stiffness, and conditional number are all analyzed. To focus on the application in the industrial area, this paper proposes a method to establish the relationship between the performance and the structural parameters by using the modified BP neural network. Based on this method, the structural parameters can be chosen by the requirements of the special task in the industrial area. Finally, some numerical examples are presented to verify the method.

#### 1. Introduction

Recently, parallel mechanisms have gained more and more interests in the research community, due to their advantages caused by their structures, like high stiffness, good dynamic performance, and high load [1–4]. However, in the modern industrial area, the parallel mechanisms with a single performance cannot meet the requirements of the flexible manufacturing systems. In this case, the reconfigurable parallel mechanisms have been the hot area in the parallel mechanism field. In order to obtain a novel reconfigurable parallel mechanism, researchers presented many ways, like modularity [5–7], metamorphic link or joint [8–10], and reconfigurable link [11, 12] or mechanism [2]. The initial reconfigurable parallel mechanisms are proposed using the modularization, which can change their configuration by adding or locking some modular components. To change the based platform, the Bicept robot has different configurations, which was presented by Li et al. [5]. These reconfigurable parallel mechanisms cannot change their configurations by themselves, and the reconfiguration processes need manual intervention. Thus, some metamorphic structures have been proposed to make mechanisms change their configuration. By using the Bennett plano-spherical mechanism, Zhang et al. [8] presented a novel reconfigurable parallel mechanism. The mechanism has a different configuration by altering some metamorphic joints or linkages into different working phases. Similar to this way, another metamorphic parallel mechanism has been proposed [13]. At the same time, some lockable joints have been proposed to change the mechanisms’ configuration [9, 14–16]. In this way, the mechanisms should go through the singularity position when the mechanisms change their configurations. So, they cannot change their configurations, continuously. Thus, a new way to design the reconfigurable parallel mechanism attracted more and more researchers, which is using the reconfigurable link or mechanism.

Performance evaluation for the parallel mechanism is a significant issue in real industrial application [17]; thus, some indices have been proposed to evaluate the workspace [18–20], stiffness [21], and other aspects [22–24] of the mechanism. In this case, the performance analysis of reconfigurable parallel mechanisms has also gained interests. Coppola et al. [25] presented a reconfigurable hybrid robot called ReSI-Bot, and some performance like workspace, singularity, and stiffness were all discussed. Huang et al. [26] proposed a 5-DOF reconfigurable hybrid robot named TriVariant, and minimizing a global and comprehensive conditioning index was used to optimize its structural parameter. Based on the virtual coefficient, Wang et al. [27] proposed a generalized transmission index that can evaluate the motion/force transmissibility of fully parallel manipulators, while, in this paper, the overconstrained mechanism is applied to build a novel reconfigurable parallel mechanism with different configurations [28]. Compared with the existing method, the proposed mechanism can continuously change its configuration without human intervention. For the workspace of a reconfigurable parallel mechanism, Brisan and Csiszar [29] proposed some novel workspace indices to investigate the relationship between the shape, dimensions of the workspace, and the configurations of the mechanism. Viegas et al. [30] proposed three strategies including drive range extension, base translation, and dynamic joint reconfiguration to research the workspace of the reconfigurable spatial parallel mechanism. Chen [31] proposed a novel performance index to evaluate the maximum dynamic load-carrying capacity (DLCC) by using Boltzmann-Hamel-d’Alembert formulism.

However, the above performance indices are obtained after the mechanisms are determined. Well, for the reconfigurable parallel mechanisms, the most significant characteristic is that they have many different performances. In order to choose a certain configuration based on the requirement, combining artificial intelligence and parallel mechanism is a considerable method to establish the relationship between the performance and configuration.

With the intersection of mechanism and artificial intelligence, the intelligent algorithms are widely used to optimize the mechanism structural parameters [32–34], solve the kinematic solution [35, 36], plan the trajectory of the end effector [37, 38], and so on. Based on the genetic algorithm, Kelaiaia et al. [39] focused on the multiobjective optimization of a linear Delta parallel robot, regarding workspace, stiffness, kinematic, and dynamic performances as the criteria. Jamwal et al. [40] proposed a novel ankle rehabilitation robot, based on the modified genetic algorithm; the kinematic design optimization for the parallel robot was proposed. Based on the neural network, the analytical solutions of system stiffness and dexterity for a spatial parallel mechanism were calculated [41]. According to different machining requirements, single-objective optimization and multiobjective optimization algorithms were both used to optimize a serial-parallel hybrid polishing machine tool [42]. Zhang and Lei [35] used the artificial neural network to solve the forward kinematic problem of the parallel mechanism. To optimize the serial-parallel hybrid polishing machine tool, Wang et al. proposed a method to balance the machining quality and efficiency by using a neural network.

To solve the above problem that task oriented configuration determination, this paper proposes a novel method to establish the relationship between the performance indices and structural parameters by using the modified BP neural network. Based on this neural network, import the performance requirements and the structural parameters can be exported. In this way, the configuration can be determined by the special task.

In this paper, a novel reconfigurable parallel mechanism is presented, based on the spatial multiloop overconstrained mechanism. Next, the kinematic and dynamic models are both established. Then, the performance analysis for the proposed mechanism is processed. Next, according to the industrial requirements, a novel BP neural network is established to obtain the relationship between the configurations and performance indices. Finally, some numerical examples are presented to verify the correctness of these analyses.

#### 2. Design Description

##### 2.1. Model Description

The schematic of the proposed reconfigurable parallel mechanism is shown in Figure 1. The reconfigurable mechanism consists of a spatial overconstrained mechanism and a parallel mechanism. The spatial overconstrained mechanism has three identical limbs, and each limb has one horizontal joint and one revolution joint. At the same time, the parallel mechanism also has three identical limbs and each limb has three joints which are the vertical prismatic joint and two universal joints. Each limb is connected to the based platform by a horizontal joint, so the limb can move along the axis of the horizontal joint, while a connecting link connected the limbs 1 and 2, and the two sides of the link are both revolute joints, so is the relationship between limbs 2 and 3. In this case, the based platform, three vertical links, and two connecting links comprise the spatial multiloop overconstrained mechanism, which has one degree of freedom (DOF). And its DOF will be proved in the following section.

The parallel mechanism consists of three identical limbs arranged by a vertical prismatic joint and two universal joints. When the spatial multiloop overconstrained mechanism is drove, the structural parameter of the parallel mechanism will be changed; then, the whole manipulator owns different configurations, which means the reconfigurable parallel mechanism has different performance. For this manipulator, the horizontal prismatic joints and the vertical prismatic joints are regarded as the actuators.

##### 2.2. Mobility Analysis

In this section, the mobility of the manipulator is analyzed, especially for the spatial multiloop overconstrained mechanism. The initial position of the manipulator is shown in Figure 1. The fixed coordinate system is located at the central point of the based platform, which is expressed by . The coordinates of each point belonged to the spatial multiloop mechanism can be written as , , , , , and , separately. Based on the *screw theory*, in the limb , the direction of the horizontal prismatic joint, , is along the , which can be written as . The axial direction of the revolute joint, , is ; thus, the twist screw can be written as

Based on Equation (1), the wrench system of the limb can be written as

Similarly, for the limb , the axial direction of the horizontal prismatic joint, , is and the axial direction of the revolute joint, , is along with the , which can be written as ; thus, the twist system of the limb is

Based on Equation (3), the wrench system of the limb can be written as

Based on Equations (2) and (4), the twist system of the spatial single-loop overconstrained mechanism can be obtained as

According to Eqution (5), the single-loop overconstrained mechanism has one DOF, and the mobility of the other single-loop mechanism including limbs and can be calculated in the same way.

The proposed spatial multiloop overconstrained mechanism consists of the two single-loop mechanisms, and the limb is the common limb, which is attached to the driving joint. In this case, the spatial multiloop overconstrained mechanism has one DOF, which is moving along the axis of the horizontal prismatic joint.

For the other section of the manipulator, namely, the parallel mechanism is a 3-PUU parallel mechanism having three translations with the axes of , , and . So, the reconfigurable parallel manipulator needs four actuators to ensure it is working well.

##### 2.3. Kinematic and Dynamic Analysis

###### 2.3.1. The Kinematic of the Overconstrained Mechanism

The schematic of the spatial multiloop overconstrained mechanism is shown in Figure 2. The angles between any two links are all . While for this mechanism, the input parameter is , and the output parameter is . The distance between any two links equals to . Thus, based on the *cosine law*, the relationship between the input and output parameters can be written as

According to the actual application, the minus solution should be abandoned, so the result can be written as

###### 2.3.2. Inverse Kinematic for the Parallel Mechanism

The schematic of the parallel mechanism is shown in Figure 3. The point, , is the initial point attached to the ground and is represented by . The local coordinates, , is located at the central point of the moving platform. Their axial directions are shown in the following picture. As shown in Figure 3, is the driving parameter and the output parameters are the coordinates of the . And is the length of the fixed-length link, and and are the structural parameters obtained from Equation (7). The circumcircle radius of the moving platform is .

For the inverse kinematic, it means the coordinates of point are known to solve the lengths of the driving joints, . In this paper, the proposed mechanism is symmetry, so the vector equation for each limb is similar. In this case, the closed-loop vector equation of the mechanism can be given as

The parallel mechanism only has three translations, so the rotation matrix from the local coordinate system to the fixed coordinate system is the unit matrix and the coordinates of the joints in each limb can be given as

The coordinates of the central point of the moving platform located in the fixed coordinate system is . Based on the structural constraints, the lengths of the fixed-length link are known, so the equation can be obtained as

Thus, the variables of the driving joints can be calculated by Equation (10).

##### 2.4. Jacobian Matrix

For the parallel mechanisms, the Jacobian matrix is the mathematical basis to evaluate the performance of the mechanisms. In this paper, when the reconfigurable parallel mechanism is working, the configuration is determined, which means the reconfigurable mechanism is locked. The input parameters of the manipulator are only the lengths of three vertical prismatic joints, and the output parameters are the coordinates values of the central point located at the moving platform. Taking the time derivative of Equation (8) yieldswhere are the velocity of the driving joints, means the direction of the vertical prismatic joint, is the angle velocity of the fixed-length link, is the vector of the fixed-length link, is the velocity of the moving platform, is the angle velocity of the moving platform, and means the vector of in the local coordinate system.

Based on mathematical knowledge, simplifying Equation (11) yields

Thus, the velocities of the three driving joints can be obtained, when the velocity of the moving platform is given.

##### 2.5. Dynamic Analysis

In the industrial application, a dynamic model is an important section for the parallel mechanism. In this section, the inverse dynamic model is calculated for the parallel mechanism, based on the *Lagrangian Theory*. In general, there are several methods to obtain the dynamic model of the parallel mechanism, like *Lagrangian Theory*, *Newton-Euler Formulation*, and *Principle of Virtual Work*. In this paper, the proposed manipulator only has three translations, and the *Lagrangian Theory* is the best way for this manipulator to calculate this model.

For the proposed parallel mechanism, , , , , , and are regarded as the generalized coordinates. The first type of Lagrangian equations can be written aswhere is the constraint function, is the number of constraint functions, and denotes the Lagrangian multiplier. represent the , , and components of an external force exerted at the moving platform and means the actuator force. represents the total energy of the parallel mechanism.

The total energy includes kinetic energy and potential energy, and the kinetic energy can be given aswhere is the kinetic energy of the moving platform, is the kinetic energy of the vertical prismatic slider, and is the kinetic energy of the fixed-length link . In this paper, in order to simplify the dynamic model, assuming that the mass of each connecting rod, , in the fixed-length link assembly is divided evenly and concentrated at the two endpoints and . In this case, the kinetic energy of each part can be written aswhere is the mass of the vertical prismatic slider.

The potential energy of the parallel mechanism can be written aswhere is the gravitational acceleration.

Based on Equations (14) and (16), the total energy of the parallel mechanism can be written as

Substituting the given variate into Equation (13), the can be calculated.

The second type of Lagrangian equations can be written as

Thus, the input force can be calculated.

#### 3. Performance Analysis

##### 3.1. Workspace

Workspace is the shape and volume that the manipulator can be reached in the global coordinate system. The constraints of the workspace include the structural parameters and the joint angle. In order to determine the workspace in the different configurations, the range of the driving joint is given as

The range of the angle between the fixed-length link and the vertical prismatic joint is given as

Thus, the figure of the workspace in different configurations is shown in Figure 4. Figure 4(a) shows the workspace when , which means the mechanism is central symmetry. And the workspace is also central symmetry seen from Figure 4(a). When , the workspace is shown in Figure 4(b). When is minimum, compared with Figure 4(a), the range of the workspace along is declining while the range along is increasing. The other limited situation is shown in Figure 4(c), namely, . When is maximum, the range of workspace along is declining while the range along is increasing. Thus, the proposed manipulator has a different workspace in different configurations, namely, this manipulator owns different performances in the aspect of workspace.

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##### 3.2. Stiffness Analysis

Stiffness is the deformation between the actual and desired position of the end effector, when the manipulator is given a task. To simplify the stiffness model, all the joints and links of this manipulator are regarded as rigid.

The stiffness of the manipulator is related to a wrench acting on the moving platform including the forces and moment. In this system, is the wrench vector showing the forces and torques acting on the platform. In this vector, is the forces and represents the moments acting on the moving platform. For this parallel mechanism, both the constraint of structural configuration and the constraint of driving joints act on the manipulator. To balance the constraint forces or moments acting on the moving platform, the stiffness modeling of the manipulator can be written aswhere

In Equation (22), and are representing the deformation of the actuators and structural configuration. And and .

According to the *virtual work principle*, the equation of the manipulator can be given aswhere is the deformation of the moving platform.

Combining Equations (21), (22), and (24), one can havewhere is the generalized stiffness matrix which can be given as

Based on Equation (27), the stiffness can be expressed as

Due to mobility analysis, the motions in , , and are constrained. Thus, the stiffness matrix can be rewritten as

In this paper, assuming and , the stiffness of different configurations is shown in Figure 5. Figure 5(a) shows the stiffness of the manipulator when . In this configuration, the mechanism is central symmetry and the mapping of the stiffness is symmetry, as well. When the end effector is closed to the central area, the stiffness is maximum, which is about , while its stiffness is declining rapidly when the end effector is near the boundaries of the workspace. And the stiffness is declining more rapidly along the positive direction than the negative direction of . When , which means is the minimum, the stiffness of the manipulator is shown in Figure 5(b). The distribution of stiffness is similar to Figure 5(a), but the central area is moving with negative , and the whole stiffness mapping figure moves about 0.05 m along the negative . The stiffness mapping along the is as symmetrical as when . The stiffness of the manipulator is shown in Figure 5(c), when . Same as the above, near the central area, the manipulator has the highest stiffness and near the boundaries, the stiffness of the manipulator is declining. The acceptable stiffness range along the decreases, as well. However, the range along the becomes larger than and . And the shape of the mapping is the same as the mechanism’s structure in this configuration. According to these figures, the manipulator has different performance in stiffness when the mechanism locates at different configurations, which means the manipulator owns reconfigurable ability.

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#### 4. Condition Number

Condition number is a performance to evaluate the dexterity of the manipulator, which is calculated from the Jacobian matrix. It can be given as

Based on Equation (30), the condition number is determined by the position of the end effector and . When is closing to 1, the manipulator has the best dexterity. Figure 6 shows the condition number when in different configurations. When , the condition number is shown in Figure 6(a). The shaded area is the projection of the workspace. The condition number range is from 1.45 to 1.85, which means the manipulator owns the balance performance in the workspace. Figure 6(b) shows the condition number when . In this configuration, the condition number is from 1.2 to 2. When the mechanism is locating at the boundary area, the mechanism has the better performance. When , the condition number is shown in Figure 6(c). The range is from 4 to 10, which means the mechanism has a significant difference in the workspace. And in this configuration, the mechanism owns worse performance than and in the aspect of dexterity. Although the condition number is not better, the range along the is increasing. Based on Figure 6, when , the maximum condition number is 1.85, which is larger than the other two configurations. Thus, compared with another two cases, the manipulator has the best performance in condition number when . According to the results, the manipulator has different performance in different configurations.

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#### 5. Neural Network and Its Application

##### 5.1. Global Index

For the manipulator, condition number and stiffness are the performance located at a special point, not the global indices. In order to evaluate the performance in different configurations, the global indices about the stiffness and condition number should be proposed.

The global stiffness index (GSI) is given aswhere .

Similar to the GSI, the global condition number index (GCNI) can be given aswhere .

Besides the stiffness and condition number, the workspace is a global index. In this case, the performance for the manipulator in different configurations can be calculated. However, the units of the stiffness, condition number, and workspace are not identical; a method is proposed to solve this problem. This method can be written as

In this paper, the range of the reconfigurable parameter, , is . Thus, the trend of these global indices is shown in Figure 7. As shown in Figure 7, when , the workspace is increasing with , and when , the workspace is declining with . The stiffness and condition number are both increasing with . Thus, the manipulator has a different performance, when it is in a different configuration. Based on reconfigurable ability, the manipulator can be chosen as a suitable configuration for a certain requirement.

##### 5.2. Performance Choice Based on a Neural Network

Traditionally, the performance of a special mechanism will be determined when the structural parameters are given. However, in the modern industrial area, based on the special task, determining the structural parameters becomes the tendency, especially for the reconfigurable parallel mechanism. In this paper, the modified *BP neural network* is used to solve this problem. Neural networks are noted for the ability of nonlinear and complex functions, which led to their extensive applications including pattern classification, function approximation, and optimization. The neural network includes BP neural network, RBF neural network, GRNN neural network, and Hopfield neural network.

BP neural network is a multilayer neural network, including an input layer, several hidden layers, and an output layer. Its remarkable character is that it is a feedback network and it can adjust its threshold and weight based on error backpropagation algorithm. Thus, in this paper, the BP neural network is the best choice to solve the above problem. According to the complex of these performance indices, a four-layer BP neural network is designed which means it has two hidden layers. The topology of the BP neural network is shown in Figure 8. As shown in Figure 8, the input parameters are and the output parameter is the reconfigurable structural parameter, . In order to obtain the accurate neural network, the hidden layers include two sections, and the functions of both sections are given as

In this case, the output of the note of the 1^{st} hidden layer is given aswhere is the weight between the input and the 1^{st} hidden layers and is the threshold of the 1^{st} hidden layer.

Similarly to Equation (35), the function between the 2^{nd} hidden layer and the output layer is given aswhere is the weight between 1^{st} and 2^{nd} hidden layers and is the threshold of the 2^{nd} hidden layer.

The output of the output layer is given aswhere is the weight between 2^{nd} hidden and output layers and is the threshold of the output layer.

The error between actual output, , and desired output, is given as

In order to obtain the best weight and threshold faster, the additional momentum method is applied, which can be given aswhere , , and are the weight at time , , and , separately.

At the same time, the variable learning method is used to make the training process faster, which is given aswhere is the learning rate. Generally, and .

In this paper, part of all training samples is listed in Table 1. There are 24 samples in the table; however, 24 samples are too little to train the neural network. In this paper, 241 samples are used to train the BP neural network. Then, the trained neural network is obtained. Figure 9 shows the result of the trained neural network. Figure 9(a) is the error of the neural network, and in 257^{th} epoch, the neural network is trained well, which has the best performance. Figure 9(b) shows the gradient, validation, and learning rate in the training process, separately. Figure 9(c) shows the regression between the actual training samples and output parameters.

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Some random tasks are presented to verify the effectiveness of the neural network. Firstly, these desired indices are put into the input layer of the neural network; then, the reconfigurable parameter, , can be calculated by the neural network. Next, the calculated parameters are substituted into the kinematic model, while the actual indices can be obtained. Finally, comparing the desired and actual indices, the effectiveness of the neural network can be verified. These indices are listed in Table 2. Figure 10 shows the comparison between the desired and actual indices. Figures 10(a)–10(c) show the , , and between the actual and desired indices, separately. Figure 10(d) shows these indices of error between actual and desired indices. Compared with the desired indices, these errors can be neglected. Thus, the trained neural network can obtain the reconfigurable parameter well and work well. Based on the trained neural network, the configuration can be determined, according to the requirement of a certain task.

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#### 6. Numerical Example

In this section, a special task is presented to verify the correctness of kinematic and dynamic models. In this sample, the external forces are all zero and the initial position of the central point is . The structural and mass parameters are listed in Table 3.

The equation of the end effector’s trajectory is given aswhere is the time, is the radius of the curve, and is the value along the . In this sample, , , , and . And the result is shown in Figure 11.

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As shown in Figure 11, Figure 11(a) is the actual trend of driving joints, which is obtained from the simulation soft. (b) is the desired length of the joints, which is calculated by the kinematic model. (c) shows the errors between actual and desired joints and the error is very small. (d) is the error along the . (e) is the error between the actual and desired trajectory in the plane. From these figures, the error exists, but it is very small, and it can be neglectful. Thus, the inverse kinematic model can be verified.

The dynamic result is shown in Figure 12. Figure 12(a) shows the accelerations of three joints when the trajectory of the end effector is the Archimedes spiral. (b) is the driving force of these joints. According to Figures 12(a) and 12(b), the tendency of accelerations and driving forces is similar, which confirms to reality. Figures 12(c)–12(e) are the mapping of the driving forces along the , separately. Along different directions, the different driving forces are increasing, monotonously.

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#### 7. Conclusion

This paper presents a novel reconfigurable parallel mechanism based on the spatial multiloop overconstrained mechanism. According to the screw theory, the mobility of the reconfigurable is analyzed. The reconfigurable mechanism has one DOF, and it can change the parallel mechanism’s structural parameter. Then, the performance of the reconfigurable parallel mechanism will be changed. The inverse kinematic and dynamic model of the reconfigurable parallel mechanism are both established. Thus, based on the Jacobian Matrix, some performance indices are proposed. And the different performances in different configurations are analyzed. And it shows that the novel reconfigurable parallel mechanism has some configurations and the reconfigurable parallel mechanism has different performance indices. In order to choose the best configuration based on the special task requirement, by using the modified BP neural network, an algorithm is designed to solve this problem. And some samples are used to train the algorithm. Then, based on the trained neural network, some test samples are proposed to verify the correctness and accuracy of the designed neural network. Thus, a suitable configuration can be determined by the special requirement in the industrial area. Finally, based on a certain task, the numerical example is proposed to verify the correctness of the inverse kinematic and dynamic models.

#### Data Availability

The MATLAB code and the data used to support the findings of this study are available from the first author upon request.

#### Conflicts of Interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

#### Acknowledgments

The 1^{st} author would like to thank the financial support from the Youth Foundation of Zhejiang Lab (Grant No. 2020NB0AA02) and the Leading Innovation and Entrepreneurship Team of Zhejiang Province of China (Grant No. 2018R01006). The 1^{st}, 2^{nd}, and 3^{rd} authors would like to thank the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and gratefully acknowledge the financial support from the York Research Chairs (YRC) program.