Research on a ø6m TBM Steel Arch Splicing Manipulator


 As a challenging task, the robotic splicing of steel arch is required to realize the grasping and docking of steel arches in a limited space. The steel arches often have a mass of over 200kg and a length of over 4m. Due to the large volume and mass of steel arches and the high requirements for the positioning accuracy of splicing, it is difficult for the general manipulator to meet its flexibility and stiffness requirements. The single-degree-of-freedom(DOF) closed-loop mechanism has a simple and reliable structure. Adding it into the manipulator can effectively improve the dynamic performance and increase the structural stiffness. In this paper, a solution model of a single-DOF closed-loop planar mechanism is presented, and alternative kinematic pairs of the mechanism with different input constraints and output requirements are derived. Based on this model, a design method of steel arch splicing manipulator with single-DOF closed-loop grasping structure is proposed. All the optional basic configurations of the manipulator are deduced, and then the optimal configuration is obtained by using the performance indexes. A prototype of the steel arch splicing manipulator is manufactured, and the reliability of the manipulator is proved by experiments.


Introduction
Generally known as a large-scale equipment, Hard Rock Tunnel Boring Machine (TBM) is specially applied to conduct the tunnel excavation inside hard rock [1][2][3][4].With the functions of excavating, slagging and supporting, TBM is also widely adopted in the field of hard rock excavation [5][6][7].The construction of TBM would unavoidably destroy the initial in-situ stress state of the surrounding rock in the original stratum, resulting in redistribution of stress in the surrounding rock which would lead to the deformation, destruction or even collapse of the surrounding rock [8,9].Especially with the unsatisfying quality of surrounding rock and the severe fragmentation of rock mass, the self-stabilization period of surrounding rock is quite short or fails to appear.Therefore, it is necessary to provide timely and effective steel arch support to control the deformation of the surrounding rock and prevent the damage of the surrounding rock [10,11].At present, the splicing of TBM steel arches is carried out manually, as shown in Fig. 1.Inside a narrow operation area of the construction personnel, the working environment is extremely harsh with strong vibration, numerous dust, low brightness as well as high humidity, where the labors are often required to stand in the stagnant water.Each steel arch is endowed with a mass of up to 200kg, requiring the manual movement, adjustment, and docking.The extremely high difficulty of the process would lower the splicing efficiency of the steel arch frame low, additionally, it would also lead to the delayed support delay or the unstable quality and finally cause collapse accident.At present, steel arch installation is mainly carried out by means of manual coordination with ring beam erector.The steel arches are delivered to ring beam erector by transport truck and TBM crane, and the workers transport the steel arches to the assembly ring of ring beam erector and fixes it.Then the assembly ring rotates at a certain angle to place the second steel arch.Workers adjust the attitude of the adjacent steel arches on the assembly ring so that the two steel arches are close to each other and the bolt holes are aligned, and then they are bolted together.As the steel arch is very heavy, it is difficult to grasp and drag, so many people are often required to carry and adjust the steel arch, and even use jacks to assist in joining the steel arch, which seriously affects the construction efficiency.In order to relieve the workload and improve the constructional quality, efficiency and safety, the concept of automatic installation of TBM steel arches is firstly proposed.A steel arch splicing manipulator is designed to realize the automatic splicing of steel arches, and improve the intelligence and assembly efficiency of TBM construction.
As the first case which investigates on the TBM steel arch splicing manipulator, related achievements have still not been publicly reported, while a series of researches could be found associated with the design of the grasping manipulator [12].Huang [13] has proposed a general design method for spatial 3-DOF isotropic manipulators, and developed 6-DOF or redundant manipulators applying 3-DOF robots as modules.Considering the operation and grasping requirements of the mechanism, Ozgür [14], Lambert [15] have analyzed the DOF and structure of the planar as well as the space manipulator, and designed a manipulator that could meet the grasping performance.Gao [16], Liang [17] considered the shape diversity of the grasped object, and designed a manipulator to realize the self-adaptive enveloping grasp.Kaluarachchi [18] has presented a design method of lightweight tendon-driving redundant manipulator.With the reduced joint torque applying a single motor, the number of actuators could get reduced and power consumption would also be minimized.Babin [19] has proposed a design method for the manipulators which could grasp small objects within the confined spaces, applying a planetary gear train to replace the traditional revolute joints.However, apart from the large mass, the steel arch is with the structure of a slender I-beam.Therefore, with the inability to guarantee the grasping stability and the docking accuracy of the steel arch, these design schemes are unsuitable for splicing steel arches.
The grasping action of steel arch needs two jaws to cooperate with each other.In order to improve the dynamic performance, increase the structural stiffness and reduce the input control, a single-DOF closed-loop planar mechanism is considered to be introduced into the grasping structure of the manipulator.At present, researches on single-DOF closed-loop mechanism are mainly focused on leg mechanism and gripper mechanism [ 20 ].Studies of closed-loop planar mechanisms such as Chebyshev Mechanism [ 21 ], Klann Mechanism [22], LARM BiPED Mechanism [23] and DQV [24] have shown that the closed-loop series structure possesses the integral rigidity, more suitable for missions under a high load, and it performs better at high stride frequency [25,26].Liu [27] designed a single DOF robot leg mechanism intended for tailed quadruped locomotion.
The design employs a four-bar planar mechanism that couples the hip and knee flexion/extension joints mechanically, which reduce the leg's DOF without violating the foot trajectory.The actuator is concentrated at a proximal location, increasing the rigidity-to-weight ratio.Hassan [28,29] introduced the crank slider mechanism into the gripper, realizing the purpose of a single driver controlling the tensioning of four fingers simultaneously.
Anwar [30] proposed a soft closed-chain modular gripper, and designed a single-DOF closed-loop plane structure on each finger, so that it can passively adapt around a curved surface, with the benefits of flexibility, low weight and controllability.However, most of the above work dedicated to reconfiguration analysis of individual mechanism and to designs combining existing mechanisms [31].The fundamental for structure synthesis of a single-DOF closed-loop mechanism has not been systematically explored.Nor has a mathematical model been derived that could serve as a basic modular foundation block for exploring similar closed-chain manipulators and grippers.
Among the innovative design theories and methods of articulated rigid body systems, screw theory [32][33][34][35] and graphical type synthesis method [36,37] present irreplaceable advantages in the design of complex manipulator due to the simple and intuitive characteristics, while they are basically only used in the design of parallel manipulators at present.The grasping action of the steel arch splicing manipulator requires the mutual cooperation between both the upper and lower jaws and also the functions of adjusting and docking the steel arch.
Regarding to this complex manipulator with multiple interrelated output tips, screw theory and graphical type synthesis method can realize the analysis and synthesis of the manipulator from the basic principle.At present, seldom research could be figured in this field.
In this paper, a solution model of single-DOF closed-loop planar mechanism is presented, which is used to analyze the law of kinematic pairs' number, arrangement and axis layout of the mechanism.Based on this model, the design scheme of single-DOF closed-loop grasping mechanism and posture adjusting mechanism for steel arch is proposed.A steel arch splicing manipulator which can realize the automatic installation of steel arch is developed, and the reliability of the manipulator is verified by experiments.

Analysis of single-DOF closed-loop mechanism
Mobility of a mechanism is generally calculated in terms of the Chebychev-Grübler-Kutzbach criterion [38] expressed in the formula 1 ( 1) where F is the mobility of a mechanism, d is the DOF of the space in which the mechanism works, n is the number of bodies connected by g joints and i f is the connectivity or DOF of the ith joint.
The single-DOF closed-loop mechanism is composed of rotating pairs and moving pairs with one . The number of bars n and the number of kinematic pairs g must be equal, meaning The screw theory is used to analyze the single-DOF closed-loop mechanism.Screw Ri $ of the ith revolute pair and screw Pj $ of the jth prismatic pair on the mechanism are respectively expressed as The sum of the spatial DOF d and the over-constrained number  of the single-DOF closed-loop mechanism is 6.If the virtual work done by the over-constraint screws in the single closed-loop mechanism on the motion-screw system is 0, then such over-constraint will not affect the motion of the mechanism, thus ensuring the correctness of the mechanism.
If the motion-screw system of the mechanism is set as   , the virtual work done by the kth constraint screw of the single closed-loop mechanism on the jth motion screw is derived as The over-constraints are divided into independent force constraints and independent couple constraints, and the kinematic pairs are divided into revolute pairs and prismatic pairs.According to Eq.( 5), it can be obtained that the virtual work should meet the following two conditions.where  and a are respectively the spatial angle and distance between the motion screw and the constraint screw.
All screws on the closed-loop mechanism must be linearly related, so that 1 0 g j j j     $ (7) where   1,2, , j jg   L is the motion velocity of the jth joint of the closed-loop mechanism, and they are not all zero.j $ is the motion screw of the jth joint.
By substituting Eq.(3) and Eq.(4) into Eq.( 7), we obtain In a single-DOF closed-loop planar mechanism, the number of overconstraints  , the constraint-screw system contains 1 independent force constraint and 2 independent couple constraints, which are described as follows (10) For revolute pairs, the condition that the virtual work is 0 can be described as All revolute pairs must be perpendicular to the two independent couple constraint screws, and intersecting or parallel to the force constraint screws.
For prismatic pairs, the condition that the virtual work is 0 can be described as 11 2 3 cos 0 0 The axes of all prismatic pairs are perpendicular to the force constraint screws, and independent of the couple constraint screws.
Where P represents prismatic joint and R represents revolute joint, and the RRRP institution is shown in Fig. 2. The constraint-screw system may contain either 1 independent force constraint and 3 independent couple constraints(denoted as F1), or 2 independent force constraints and 2 independent couple constraints(denoted as F2).
Similarly, in F1, the revolute pairs cannot be perpendicular to the 3 independent couple constraint screws, so there will be no revolute pair in the mechanism.The axis of prismatic pairs are required to be perpendicular to the force constraint, that is, the mechanism can only be PPP, as shown in Fig. 3.In F2, the revolute pairs must be perpendicular to the 2 independent couple constraint screws, and intersecting or parallel to the 2 force constraint screws.The closed-loop mechanism under this condition is triangular stable structure, which does not meet the requirements.

The solution model of planar single-degree-of-freedom closed-loop mechanism
The joint part of the single-DOF closed loop planar mechanism is divided into input joint, output joint and 11 where l and m are respectively the number of revolute pairs and prismatic pairs in the intermediate part, According to Eq.( 15), the number of revolute joints in the intermediate part under different input and output joints can be obtained( , the joint is a prismatic pair; when =1 i a , the joint is a revolute pair. The velocity relationship of the intermediate joint can be obtained by combining Eq.( 15)~(17).
When the input joint and output joint types are known, Eq.( 18) is substituted into Eq.(16) and Eq.( 17), and then the velocity relationship between input joint and output joint under different intermediate joint types can be obtained.

Design of steel arch splicing manipulator
A series of process are contained during the installation of TBM steel arch which are the transporting, grabbing, docking and bracing.The designed steel arch splicing manipulator is divided into the grasping module and the docking module, where it is required to flexibly grasp, adjust and connect two adjacent steel arches simultaneously.Moreover, in the process of steel arch transportation, the splicing manipulator would not interfere with the steel arch.

Design of the grasping module
According to the structure of the steel arch, it is envisaged to use two "L"-shaped jaws with a reverse arrangement to conduct the grasping of steel arch, as shown in Fig. 4. Directly affected by the gravity of the steel arch, the lower jaw should be settled near the manipulator base to provide better carrying capacity while designing the gripping module.In order to furtherly obtain a superior grasping effect, the lower jaws would approach to the steel arch directly below the steel arch.In addition, force sensor devices are necessarily needed by the mechanism to real-timely monitor the grasping force and ensure the grasping performance of the steel arch splicing manipulator.And in specific, the output motion form could be regarded with the form of rotation or movement.When the output joint is a revolute pair, it could be obtained from Fig. 5(a) that the output screw 1 At this point, 12 =0 =1 aa ， .According to Table 1, the number of revolute joints in the middle part is 1 or 2.
At this point, 12 = =0 aa .According to Table 1, the number of revolute joints in the middle part is 0 or 2.

Design of the docking module
Following with the grasping action, the splicing manipulator is required to adjust and dock the steel arch.
Driving the grasping module to complete the corresponding spatial rotation and lateral movement by the docking module, the grasping posture would be adjusted and the steel arch would also be docked.According to the action requirement of the docking module, the freedom space could be obtained by applying the graphical type synthesis method.In addition, based on the functional equivalence of the space line diagram, the same-dimensional subspace of the docking module is derived while the corresponding kinematics pair is also obtained, as shown in Fig. 7.

RR
Freedom space Subspace with same dimensional Kinematic pair Fig. 7 The kinematics pair corresponding to the freedom space of the grasping module

Optional configuration of steel arch splicing manipulator
Combining the grasping module and the docking module, 12 basic structure forms of steel arch splicing manipulators that can meet the requirements of single steel arch splicing motion are obtained, as shown in Fig. 8.All the mentioned 12 steel arch splicing manipulators could realize the grasping and docking actions of the steel arch through distinct working methods, additionally, the design requirements of the steel arch splicing manipulator could also be satisfied at the meanwhile.However, due to the difference in structure, each manipulator is endowed with different performance during the manufacturing and construction.In light of this, corresponding indices are needed to be proposed and evaluated.

Optimal configuration of steel arch splicing manipulator 4.1 Evaluation of motion/force transmission indices of grasping module
With a relatively complicated structure of the grasping module closed-loop mechanism, the motion/force transmission characteristics need to be analyzed to ensure the grasping requirements of the steel arch splicing manipulator could be satisfied. (

1) Local transmission index
In the closed-loop mechanism of the grasping module, the kinematic pair screws except the input screw $ i and output screw o $ are linearly independent, by which a screw system   1 n $$ L ， ， could be formed.According to the screw theory, the transmission force screw T $ could be obtained.
n $ , and independent with the linearity of the constraint-screw system of the closed-loop mechanism.
Define the input transmission coefficient  and output transmission coefficient  respectively as [37] max The local transmission index of the mechanism could be defined as

26) (2) Global transmission index
The local transmission index could only judge the effectiveness of motion/force transmission of the grasping module with a single posture.However, the grasping module operates in a continuous working space.
In order to judge the motion/force transmission in the entire working space, the global transmission index is defined as where  is the local transmission index, and W is the working space.4, where   ii xy is the position or movement direction of joint i in the coordinate system, ij l is the distance between joint i and joint j, and the angles represented by and  are shown in Fig. 6.
Defining the high-quality motion/force transmission mechanism as a mechanism with a global transmission index  which is not less than 0.7.It could be seen from Table 4 that the grasping modules PRRR_P and PRPR_P are both high-quality motion/force transmission mechanisms.

Complexity evaluation of steel arch splicing manipulator
In addition to the motion/force transmission performance, it is also necessary to analyze the mechanical complexity of the steel arch splicing manipulator to gain the best solution with low topology complexity, high performance as well as the low drive system complexity.The structural complexity evaluation indices of the steel arch splicing manipulator are proposed as follows: (1) Joint-number complexity, where N is the number of joints in the steel arch splicing manipulator, and N q is the resolution parameter defined by   max max max ln 0.1 ,for 0 = 0 , for =0.
(2) Joint-type complexity, where (3) Link diversity, B K Link diversity B K is defined to quantify the geometric constraints between neighboring joints.For a revolute joint, its axis of rotation is considered whereas for a prismatic joint, its direction.Five possible joint-constraint types between the neighboring joint axes/directions are reported in [39]: Type B2: Nonorthogonal intersection.
Type B4: Orthogonal but not intersecting.
Type B5: Skew.Link diversity B K of steel arch splicing manipulator could be expressed as where B is the entropy of the joint-constraint types and max =2.3219 B [40]. c is the number of distinct joint-constraint types and i M is the instance number of each type of joint-constraints.
(4) Actuator-number complexity, where a is the number of actuators in the robot topology at hand, while m a is the minimum number of actuators allowed.
(5) Operation mode complexity, OM K The operation complexity OM K is specifically defined for the steel arch splicing manipulator.During the docking process of the steel arch, the designed steel arch splicing manipulators are endowed with two different operation modes.The first operation mode(OM1) is that the left and right docking modules drive a prismatic pair respectively, so as to the steel arch to dock along a straight line, as shown in PR and RP structure in Fig. 7.The second operation mode(OM2) is to drive both revolute pairs at the same time, as shown in RR structure in Fig. 7.It could be seen that the motion trajectory is difficult to control with the application of the later mode, additionally, the internal interference would also be pronely caused.In this paper, the problem would be analyzed by introducing a new complexity index OM K .

=0 for OM1
(6) Total complexity, K Total complexity K∈[0, 1] is defined as a convex combination of the different complexity indices.It is defined as where Assigning equal weights to all complexity indices implies   1 5 According to the evaluation results of the motion/force transmission indices of the grasping module, it could be found that the splicing manipulators a, b, e, f, i, j present an outstanding performance.The structural complexity analysis of these 6 steel arch splicing manipulators is shown in Table 5.It could be seen that due to the different Operation modes of steel arch docking, the complexity of the latter two steel arch splicing manipulators is significantly than the first four.Among them, the complexity of manipulator j reaches the highest as 0.6911.On the contrary, the complexity of the steel arch splicing manipulator b reaches the minimum value of 0.4228, which is mainly accounted by the smaller  After a comprehensive comparison between both the motion/force transmission performance indices and complexity indices of the steel arch splicing manipulator, it could be obtained that the work performance of the steel arch splicing manipulator b is better than others.Therefore, it would be regarded as the optimal basic configuration.

Structure and working mode of steel arch splicing manipulator
The steel arch splicing manipulator is required to grasp two steel arches at the same time, hence the manipulator b is arranged symmetrically.Inside an extremely narrow working space of steel arch splicing, it is needed to conduct the transporting and splicing of steel arch.In order to avoid interference between the steel  No. 2 cylinder is the posture adjustment cylinder, which is used to adjust the posture angle of the grasped steel arch.
No. 3 cylinder is the docking cylinder, which is used to push the grasping modules to enable the docking operations by the steel arches on the two grasping modules.
No. 4 cylinder is the lower jaw control cylinder, which is used to control the up and down movement of the lower jaw and simultaneously adjust the opening angle of the grasping module.
No. 5 cylinder is the upper jaw control cylinder, which is used to control the movement of the upper jaw and realize the grasping operation of the steel arch.

In-plant tests
The steel arch splicing manipulator prototype is applied to carry out the steel arch grasping and docking experiment, as shown in Fig. 12.When the manipulator is in contact with the steel arch, it is necessary to reduce the cylinder speed to ensure the smooth operation.Furthermore, the motion control of each hydraulic cylinder during the experiment is shown in Table 6.The experimental results show that the entire splicing process of the steel arches would occupy 70s with a relatively high efficiency.During the grasping process of the steel arches, there is no steel arch sliding and manipulator interference, which proves a grasping effect.While during the docking process, the alignment effect of the pin holes of the steel arch is excellent with the distance between the center axis of the hole less than 5mm, which meets the accuracy requirements of the steel arch.
Regarding to the practical tunnel construction, the excessive weight of the steel arch would increase the difficulty of manual handling and adjustment, in light of which even jacks and other devices would be needed as assistance.The manual connection of two adjacent steel arches would take around 5-10 minutes, compared with which the splicing efficiency of the steel arch splicing manipulator is increased by 4 to 8 times.Hence, with the application of the splicing manipulator, the construction process could be greatly accelerated.

Conclusions
This paper combines screw theory and graphical type synthesis method to design the steel arch splicing manipulator.Due to the large volume and mass of steel arches, in order to ensure the splicing positioning accuracy of the manipulator and increase its structural stiffness, a single-DOF closed-loop plane mechanism is added into the grasping structure of the manipulator.Based on the basic principle of structural synthesis, a solution model of the single-DOF closed-loop mechanism is proposed.Using the screw theory, the unknown part structure of the closed-loop mechanism is deduced reversely.
The steel arch splicing manipulator is divided into two parts: grasping module and docking module.From the point of view of reducing input control, a grasping module design method with single-DOF closed-loop structure is proposed based on the derived solution model.All grasping modules that satisfy the conditions are derived in the case that only input constraints and output requirements are known.Based on the same dimensional subspace equivalence principle of graphical type synthesis method, the design method of manipulator docking module is proposed.And based on the proposed method, 12 kinds of steel arch splicing manipulator are constructed.All of these manipulators can realize the grasping and docking of steel arches and meet the design requirements of the steel arch splicing manipulator.The motion/force transmission and structural complexity of the steel arch manipulators are analyzed and discussed, and the best scheme is selected.
In order to verify the correctness of the research, a prototype of the steel arch splicing manipulator is made.
Using Adams software, the output trajectory of the end of the manipulator in space is clearly obtained.The relative spatial positions of the upper and lower jaws under different working stages are analyzed, and it is proved that the manipulator satisfies the grasping requirements.The grasping effect, docking accuracy and splicing efficiency of the manipulator are studied through the steel arch splicing experiment, which proves that the manipulator meets the design requirements.
The main contribution of this paper is to propose a manipulator which can replace the manual work to realize the steel arch splicing.The manipulator is characterized by simple structure, low manufacturing cost, simple kinematics model and so on, and has a potential application prospect.A solution model of a single-DOF closed-loop mechanism is proposed for the theoretical analysis of the steel arch splicing manipulator.This method can be used for reference for other similar closed-loop manipulator and gripper design in engineering.It has important engineering significance.In the future work, we will further study the dynamic control of the steel arch splicing manipulator.
Fig.1 Manual splicing of steel arch

2 . 2
Planar single-degree-of-freedom closed-loop mechanismOn engineering, single-DOF closed-loop planar mechanism is the most common.Due to its simple and reliable structure, it is often added into the open-chain mechanism to improve the dynamic performance and increase the structural stiffness.Given the input constraints and output requirements, how to design a closed-loop mechanism to meet the functional requirements has always been an urgent problem in engineering.

Where 1 a and 2 a
can be 1 or 0. The intermediate joint part can be divided into revolute screw position of the revolute joint in the coordinate system,

Fig. 4
Fig.4 Schematic diagram of steel arch splicing

o
Revolute output joint (b) Prismatic output joint Fig.5 Schematic diagram of grasping module A plane coordinate system is established for the single-DOF closed-loop structure of the steel arch splicingmanipulator.The lower jaw is set to move along the x-axis.When the output joint that drives the upper jaw is a revolute joint, the center of the revolute joint would be taken as the origin.When the output joint is a prismatic joint, however, the origin would be determined as the intersection between the movement directions of the input prismatic joint and the output prismatic joint.As shown in Fig.5.$ i is the input motion screw which represents the screw of the lower jaw, is the output motion screw which means the screw of the upper jaw.

2 $
joint is a prismatic joint, the output screw o could be obtained from Fig.5(b) as

6
solution of equationThe wear of the jaws and the size deviation of the steel arch would easily lead to insufficient grasping accuracy, derived by which, the grasping force would even fail to meet the grasping requirements of the steel arch.With the adjunction of a prismatic DOF to the upper jaw, a force sensor is also installed to guarantee the grasping effect.4 types of grasping module structures are shown in Fig.6.Optional structure of grasping module

Fig. 8
Fig.8 Single side basic structures of steel arch splicing manipulators

.
The motion/force transmission analysis is carried out on the single-DOF closed-loop structures of 4 kinds of grasping modules.With the purpose of better motion/force transmission, the transmission angle  adopts the most widely accepted range of [45°, 135°].As shown in Fig.6(a), regarding to the grasping module PRRR_P, 45 90     ， .In order to enable the transmission angle  to meet the requirements and ensure the transmission efficiency, it is necessary to set12 The transmission performance of the closed-loop structures of the grasping module is shown in Table

n
are the numbers of revolute and prismatic joints, respectively with RP n n n .Gx K is the geometric complexity of the pair x as introduced in [39]: Fig.10displays the grasping and docking process of the steel arch splicing manipulator.While transporting the steel arch, the manipulator would rotate and get folded as shown in Fig.10(a).In terms of the grasping process, the manipulator would reach the designated position through the revolute joint, accompanying with which the grasping module opens at the meanwhile.After the steel arch enters the grasping module, the upper jaws would close as the lower jaws move upward to complete the grasping of the steel arch, as shown in Fig.10(b).Simultaneously dragging the two grasped steel arches towards the middle, the docking operation of the steel arches would be completed as shown in Fig.10(c).

Table 1
The number of revolute joints under different input and output joints

Table 2
Closed-loop structure of grasping module for revolute output joint

Table 4
Motion/force transmission performance of the closed -loop structure of the grasping module

Table 5
Complexity indices of the manipulators