# Analysis of kinematic dexterity and stiffness performance based on Spring’s wire-driven 4-SPS/U rigid‒flexible parallel trunk joint mechanism

## Abstract

### Purpose

The purpose of this paper is to propose a two-degrees-of-freedom wire-driven 4SPS/U rigid‒flexible parallel trunk joint mechanism based on spring, in order to improve the robot’s athletic ability, load capacity and rigidity, and to ensure the coordination of multi-modal motion.

### Design/methodology/approach

First, based on the rotation transformation matrix and closed-loop constraint equation of the parallel trunk joint mechanism, the mathematical model of its inverse position solution is constructed. Then, the Jacobian matrix of velocity and acceleration is derived by time derivative method. On this basis, the stiffness matrix of the parallel trunk joint mechanism is derived on the basis of the principle of virtual work and combined with the deformation effect of the rope driving pair and the spring elastic restraint pair. Then, the eigenvalue distribution of the stiffness matrix and the global stiffness performance index are used as the stiffness evaluation index of the mechanism. In addition, the performance index of athletic dexterity is analyzed. Finally, the distribution map of kinematic dexterity and stiffness is drawn in the workspace by numerical simulation, and the influence of the introduced spring on the stiffness distribution of the parallel trunk joint mechanism is compared and analyzed. It is concluded that the stiffness in the specific direction of the parallel trunk joint mechanism can be improved, and the stiffness distribution can be improved by adjusting the spring elastic structure parameters of the rope-driven branch chain.

### Findings

Studies have shown that the wire-driven 4SPS/U rigid‒flexible parallel trunk joint mechanism based on spring has a great kinematic dexterity, load-carrying capacity and stiffness performance.

### Research limitations/implications

The soft-mixed structure is not mature, and there are few new materials for the soft-mixed mixture; the rope and the rigid structure are driven together with a large amount of friction and hindrance factors, etc.

### Practical implications

It ensures that the multi-motion mode hexapod mobile robot can meet the requirement of sufficient different stiffness for different motion postures through the parallel trunk joint mechanism, and it ensures that the multi-motion mode hexapod mobile robot in multi-motion mode can meet the performance requirement of global stiffness change at different pose points of different motion postures through the parallel trunk joint mechanism.

### Social implications

The trunk structure is a very critical mechanism for animals. Animals in the movement to achieve smooth climbing, overturning and other different postures, such as centipede, starfish, giant salamander and other multi-legged animals, not only rely on the unique leg mechanism, but also must have a unique trunk joint mechanism. Based on the cooperation of these two mechanisms, the animal can achieve a stable, flexible and flexible variety of motion characteristics. Therefore, the trunk joint mechanism has an important significance for the coordinated movement of the whole body of the multi-sport mode mobile robot (Huang Hu-lin, 2016).

### Originality/value

In this paper, based on the idea of combining rigid parallel mechanism with wire-driven mechanism, a trunk mechanism is designed, which is composed of four spring-based wire-driven 4SPS/U rigid‒flexible parallel trunk joint mechanism in series. Its spring-based wire-driven 4SPS/U rigid‒flexible parallel trunk joint mechanism can make the multi-motion mode mobile robot have better load capacity, mobility and stiffness performance (Qi-zhi *et al.*, 2018; Cong-hao *et al.*, 2018), thus improving the environmental adaptability and reliability of the multi-motion mode mobile robot.

## Keywords

#### Citation

Pan, Y., Chen, Y. and Li, L. (2019), "Analysis of kinematic dexterity and stiffness performance based on Spring’s wire-driven 4-SPS/U rigid‒flexible parallel trunk joint mechanism", *International Journal of Structural Integrity*, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/IJSI-02-2019-0010

### Publisher

:Emerald Publishing Limited

Copyright © 2019, Yitao Pan, Yuan Chen and Lin Li

#### License

Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode

## 1. Introduction

The trunk structure is a very critical mechanism for animals. Animals in the movement to achieve smooth climbing, overturning and other different postures, such as centipede, starfish, giant salamander and other multi-legged animals, not only rely on the unique leg mechanism, but also must have a unique trunk joint mechanism. Based on the cooperation of these two mechanisms, the animal can achieve a stable, flexible and flexible variety of motion characteristics. Therefore, the trunk joint mechanism has an important significance for the coordinated movement of the whole body of the multi-sport mode mobile robot (Huang Hu-lin, 2016).

The WABIAN-2R robot developed by Waseda University in Japan has a trunk driven in series and consists of two parts: hip and waist. Its trunk design with more degrees of freedom enables the robot to have better motion performance, such as the realization of more human-like gait movement. It rotates the pelvis, so that the legs are fully extended as the robot steps forward, unlike other robots that need to bend the knees to maintain balance. In addition, it can also achieve the complex movement of outward skimming legs and keep balance while standing on the cushion layer. But its waist mechanism also has the structure complexity, the mass overweight and so on (Kanehiro *et al.*, 2003; Ogura *et al.*, 2006). The I Struct ape-like space exploration robot jointly developed by the German artificial intelligence research center and the university of Bremen (Metta *et al.*, 2008) has a 6-dof parallel mechanism for its trunk, which makes the robot have good bearing capacity, improves the rigidity of the robot, reduces error accumulation and improves the motion accuracy. Because of the light mass and low inertia of moving parts, the dynamic performance of the system is improved. However, this kind of mechanism also has the problem of small range of motion, which limits the robot to carry out large and flexible motion. The i Cub robot (Beira *et al.*, 2006) developed by the European Association of Innovation Universities, in which the two degrees of freedom of pitch and deflection are based on a wire rope-driven differential mechanism; the degree of freedom of the roll is achieved by servo motor drive. This kind of flexible wire drive makes the humanoid robot flexible at the waist, but this kind of drive also has the problem of small driving force/torque, which makes it difficult to maintain sufficient stiffness under the circumstance of large driving load. To sum up, the early trunk joint mechanism has advantages and disadvantages: the tandem mechanism is complex and the quality is too heavy; the rigid parallel mechanism has less range of activities and low flexibility; and low bearing capacity and low stiffness of wire-driven mechanism.

In the face of complex ground environment and diverse operational requirements, higher requirements are put forward for the environmental adaptability and the whole body coordination movement ability of multi-motion mode mobile robot. Therefore, the trunk joint mechanism not only can achieve multi-attitude adjustment, but it also needs large stiffness and dexterity (Jian-feng *et al.*, 2016) to maintain the posture accuracy (Portman *et al.*, 2012) of the wooden end and large bearing capacity. In this paper, based on the idea of combining rigid parallel mechanism with wire-driven mechanism, a trunk mechanism is designed, which is composed of four spring-based wire-driven 4SPS/U rigid‒flexible parallel trunk joint mechanism in series. Its spring-based wire-driven 4SPS/U rigid‒flexible parallel trunk joint mechanism can make the multi-motion mode mobile robot have better load capacity, mobility and stiffness performance (Qi-zhi *et al.*, 2018; Cong-hao *et al.*, 2018), thus improving the environmental adaptability and reliability of the multi-motion mode mobile robot.

First, the coordinates of the spring-based wire-driven 4SPS/U rigid‒flexible parallel trunk joint mechanism are established, and the inverse kinematic position solution model of the mechanism is established by using the closed vector method, and the Jacobian matrix of velocity and acceleration is obtained. Then, combining the influence of wire of the driving part (Hai-wei *et al.*, 2013; Run-tian, 2016) and the spring elastic deformation of the restraint part (Zhe-dong *et al.*, 2018; Fan, 2014), the performance indexes of the kinematic dexterity and mechanism stiffness are established. Finally, the distribution map of the kinematic performance index of the configuration mechanism in the workspace is drawn by numerical simulation, and the kinematic dexterity and stiffness of the mechanism are analyzed.

## 2. Innovative design of the wire-driven 4-SPS/U rigid‒flexible parallel trunk joint mechanism based on spring

Similar to Figure 1, the multi-movement mode mobile robot consists of a head structure, a leg mechanism, a trunk mechanism and a propeller propulsion structure. The leg mechanism is made up of forward, middle and rear leg devices. The trunk mechanism is the critical shifting part of the multi-movement mode mobile robot, which is composed of four spring-based rope-driven 4-SPS/U rigid‒flexible parallel body joint mechanisms. As shown by Figure 1(a), the multi-movement mode mobile robot can perform different posture adjustments through the trunk joint mechanism, thereby enabling the variable direction bending trajectory crawling between the front, middle, and rear legs. As shown by Figure 1(b), after the six legs are contracted, the vector propulsion between the head structure and the propeller propulsion structure at the tail can be realized through the posture diversity adjustment of the trunk joint mechanism. As shown by Figure 1(c), when climbing obstacles, such as stairs and ditches, the robot can adjust the bending changes of different parts of the body through the trunk joint mechanism to adjust the position of the center of gravity between the middle, front and middle hind legs, so as to improve the climbing ability. As shown by Figure 1(d), the multi-movement mode mobile robot contracts the six legs at the smooth and gentle slopes, and adjusts the bending and twisting between different parts of the torso through the trunk joint mechanism to achieve a snake-like crawling. It is also possible to perform an overall rolling motion.

As shown by Figure 2(a), the wire-driven 4-SPS/U rigid‒flexible parallel trunk joint mechanism is composed of the dynamic platform and the static platform, as well as the constraint branch chain and the drive branch chain connecting the dynamic platform and the static platform. As shown in the middle of Figure 2(a), the drive branch chain includes four wire-driven SPS rigid‒flexible branches with the same structure. As shown in the upper right and lower right part of Figure 2(a), the upper and lower spherical hinges are, respectively, assembled in the spherical hinges of the dynamic and static platforms, and the spring seat is set for the upper spherical hinge and the spring seat for the lower spherical hinge. As shown in the upper left part Figure 2(a), embedding the spring in the spring base between the upper and lower spherical hingers can improve the rigidity. As shown by the picture of the lower right of Figure 2(a), piercing hole is set at the bottom of each spherical hinge seat on the static platform and also at the center of the lower spherical hinge. Then one end of the rope goes through the hole in the spherical hinge seat of the static platform, through the spring between the upper and lower spherical hinge springs, and is finally fixed on the spring seat of the upper ball joint, and the other end is fixed on the wire fixing frame of the stranding wheel at the bottom of the static platform. Thus, a spring-based wire-driven SPS rigid‒flexible branching chain is assembled. As shown by the picture of lower left in Figure 2(a), the constraint branch chain assembles the shaft end of the Hooke hinge in the same direction between the two sides of the dynamic and static platform through the sliding bearing, forming an intermediate passive constraint branched chain *U*. A corresponding drive motor for each drive branch chain is set at the bottom of the static platform, and each drive motor is provided with a stranded wheel. By driving the motor to drive the winch wheel, the driving rope on the driving branch chain is wound around, so that the length of the rope is shortened. When the winch wheel releases the driving rope line on its driving branch chain, the rope line is elongated under the spring elastic action line. Thereby, the telescopic movement of the rope drive branches realized (Qing-huan *et al.*, 2017; Trevisani *et al.*, 2006; Von Zitzewitz *et al.*, 2013; Jun, 2016; Dong-tao *et al.*, 2013), and the rotational torque of the motor output is converted into linear thrust and tension.

As shown by Figure 2(b), the spherical pair connected to the static platform is set as the first kinematic pair, the shifting pair is set as the second kinematic pair, and the spherical pair connected to the dynamic platform is set to be the third kinematic pair. *O*_{A}*-X*_{A}*Y*_{A}*Z*_{A} and *O*_{B}-*X*_{B}*Y*_{B}*Z*_{B} are the static coordinate system and moving coordinate system at the circle center of the static platform and moving platform, respectively, *O*_{A} point is at the circle center of the static platform with radius *r*, *X*_{A} axis is along with *A*_{3}*A*_{1}, *Y*_{A} axis is along with *A*_{4}*A*_{2}, Z_{A} axis accordance with right-hand rule. *O*_{B} is at the circle center of the mobile platform with radius *r*, *X*_{B} axis is along with *B*_{3}*B*_{1}, *Y*_{B} axis is along with *B*_{4}*B*_{2} and *Z*_{B} axis is in accordance with the right hand rule.

## 3. The kinematics modeling of the wire-driven 4-SPS/U rigid‒flexible parallel trunk joint mechanism

### 3.1 Inverse solution model of position

As shown in Figure 3, coordinate systems *O*_{A}*-X*_{A}*Y*_{A}*Z*_{A}, *O*_{C}*-X*_{C}*Y*_{C}*Z*_{C} and *O*_{B}*-X*_{B}*Y*_{B}*Z*_{B} are, respectively, established at the center of the static platform, the moving platform, and the hook hinge. The spherical hinge points at both ends of the four driving SPS branch chains are located at the points *A*_{1}, *A*_{2}, *A*_{3} and *A*_{4} on the static platform, and at the points *B*_{1}, *B*_{2}, *B*_{3} and *B*_{4} on the moving platform, and are evenly and symmetrically distributed. The distance between the static platform and the moving platform spherical hinge point to the center of the circle is *r*. The intermediate constraint branch *U* is located between the static platform and the moving platform, and the distance from the center of the Hook hinge to the center of the upper and lower platforms is l.

As shown by Figure 3, the coordinate system *O*_{C}-*X*_{C}*Y*_{C}*Z*_{C} at the center of the Hooke hinge rotates around *Y*_{C} axis at the angle of *θ*_{2}, and the dynamic coordinate system *O*_{B}-*X*_{B}*Y*_{B}*Z*_{B} rotates around *X*_{C} axis at the angle of *θ*_{1}. Then, the homogeneous transformation matrix (Craig, 2009) of the moving coordinate system *O*_{B}-*X*_{B}*Y*_{B}*Z*_{B} relative to the static system *O*_{A}-*X*_{A}*Y*_{A}*Z*_{A} can be obtained as follows:

*θ*

_{1}means cos

*θ*

_{2}and s

*θ*

_{1}means sin

*θ*

_{2}.

As shown by Figure 3, the vector closed quadrilateral (Wei *et al.*, 2018) *O*_{A}*A*_{1}*B*_{1}*O*_{B} formed by the black solid line arrow can obtain the variation equation of the length *A*_{i}*B*_{i} of each wire-driven branch:

In the equation, one end of the intermediate constraint branched chain *U* is connected to the fixed platform and the other end is connected to the mobile platform, so vector *O*_{B}*O*_{A} = [l·*Sθ*_{2}*Cθ*_{1}, −l·*Sθ*_{1}, l+l·*Cθ*_{2}*Cθ*_{1}]. In the moving coordinate system *O*_{B}*-X*_{B}*Y*_{B}*Z*_{B}, the coordinate of *B*_{i} can be expressed as *B*_{i}= *r*(cos *θ*_{i}, sin *θ*_{i}, 0)^{T}. In the static coordinate system *O*_{A}-*X*_{A}*Y*_{A}*Z*_{A}, the coordinate of *A*_{i} can be expressed as *O*_{A}*A*_{i}= *A*_{i}= *r*(cos *θ*_{i}, sin *θ*_{i}, 0)^{T} and *O*_{B}*B*_{i}=
*θ*_{i}= (2*π*/3)×(*i*-1). By substituting *O*_{B}*B*_{i} and *O*_{A}*A*_{i} into Equation(2), the inverse solution of the driving branch chain vector length *l*_{i} position of the parallel trunk joint mechanism can be obtained as follows:

### 3.2 Velocity and acceleration models

The velocity Jacobian matrix of the trunk joint mechanism expresses the velocity relationship between the velocity of the flexible wire-driven branched chain and that of the moving platform. It can be obtained by Equation (3):

Deriving the two ends of Equation (4) separately for time *t* can be obtained as follows:

In the equation,
*V* is the linear velocity of the mobile platform,
*W* × *O*_{B}*B*_{i,} *w* is the angular velocity of the mobile platform. Assuming *O*_{A}*O*_{B} + *O*_{B}*B*_{i} − *O*_{A}*A*_{i} = *q*_{i}, from Equation (5), we would know the following:

So, the velocity Jacobian matrix of the wire-driven 4-SPS/*U* rigid‒flexible trunk joint mechanism can be expressed as follows:

As the mobile platform is connected by Hooke joint between the point at the center *O*_{B} and the constraint branched chain mechanism, so the linear velocity of the mobile platform is zero, that is
*V* = 0. The result of substituting linear velocity *V* into Equation (6) is as follows:

In the equation,
*B*_{i}. Let *J*_{i} = *q*^{T}_{i}·

The acceleration equation of the wire-driven branch chain can be solved by taking the derivation of Equation (9) on both sides with respect to time *t*:

In the equation,

## 4. Kinematic performance analysis of the wire-driven 4-SPS/U rigid‒flexible parallel trunk joint mechanism based on spring

### 4.1 stiffness performance of parallel trunk joint mechanism

The stiffness of the wire-driven 4-SPS/U rigid‒flexible parallel trunk joint mechanism mainly depends on the slight displacement of the moving platform caused by the external force of the rope and the stiffness function of the elastic spring elastic deformation. Therefore, this paper uses the stiffness superposition method to calculate the stiffness of the rigid‒flexible parallel trunk joint mechanism. It can be assumed that the wire force vector of wire-driven branching chain is *f*_{i}=[*f*_{1}, *f*_{2}, *f*_{3}, *f*_{4}]^{T}, and that the force vector applied on the terminal moving platform is *F*_{i}=[*F*_{1}, *F*_{2}, *F*_{3}, *F*_{4}]^{T}. The virtual displacement generated by the wire of wire-driven branching chain and moving platform is, respectively, △*S*_{i}=[△*S*_{1}, △*S*_{2}, △*S*_{3}, △*S*_{4}] and △*h*_{i}=[△*h*_{1}, △*h*_{2}, △*h*_{3}, △*h*_{4}]. Then, the virtual work (Gosselin, 1990) of the force vector on the wire end and actuator is expressed as follows:

The input‒output relationship between the velocity of each drive joint and the terminal execution velocity is △*S*_{i}=*J*·△*h*_{i}. Assuming that the stiffness coefficient of each wire-driven branch chain is *K*_{i}, *f*_{i}=*k*_{i}·△*S*_{i} can be obtained. By substituting the above formula into equation (12), the following can be obtained:

In the equation, *K*_{s} is the stiffness Jacobian matrix of the mechanism. Each diagonal element in *k*_{i}=dig (*k*_{1}, *k*_{2}, *k*_{3}, *k*_{4}) represents the stiffness of the *i*th wire.

Let *V*= [*v w*]^{T} be the pose spiral vector of the moving platform, *l*=[*l*_{1}, *l*_{2}, *l*_{3}, *l*_{4}]^{T} be the length vector of the wire-driven branching chain, *δ*=[*δ*_{x}, *δ*_{y}, *δ*_{z}]^{T} represents the differential position vector along the coordinate axis direction, *n*=[*n*_{x}, *n*_{y}, *n*_{z}]^{T} represents the differential attitude vector around the coordinate axis, then the differential pose vector can be expressed as *S*=[*δ*, *n*]^{T}. Suppose the spring stiffness of the *i*th wire-driven branching chain is *H*_{i}, then *T*_{i}=*H*_{i}**dl*_{i}, where *T*_{i} represents the spring tension on the wire-driven branch chain, and *dl*_{i} represents the spring deformation of the *i* th wire-driven branch chain.

With the combination of *dl*_{i}=*J***S* and *T*_{i}=*H*_{i}**dl*_{i}, *T*_{i}=*H*_{i}**J***S* can be obtained. Both sides of this equation are multiplied by *J*^{T} to get *J*^{T}**T*_{i}=*J*^{T}**H*_{i}**J***S*. Let *F*”=*J*^{T}**T*_{i}, then *F*”=*J*^{T}**H*_{i}**J***S* is obtained further, and finally the stiffness matrix of the mechanism can be expressed as *M*_{t}=*J*^{T}**H*_{i}**J*. From the stiffness matrix expression, it can be seen that the stiffness is affected by the spring deformation configuration of the wire-driven branch chain. From the spring stiffness formula *H*_{i}=(*G***d*^{4}/8**n***D*^{3}), the stiffness matrix *M*_{t} of the spring in the wire-driven branch chain can be obtained as follows:

In the equation, *G* is the shear modulus of the spring; *d* is the wire diameter of the spring; *n* is the effective number of turns of the spring; *D* is the outside diameter of the spring.

From Equations (13) and (14), the stiffness *K*_{t} of the wire-driven 4-SPS/*U* rigid‒flexible parallel trunk joint mechanism based on spring can be obtained as follows:

### 4.2 Kinematic dexterity of parallel trunk joint mechanism

The dexterity of the mechanism expresses the movement ability of the mechanism along the specified direction under the orientation. The condition number of the Jacobian matrix can be used to evaluate the dexterity of the mechanism. Defined by the universal norm of the matrix (Li-jie, 2006):

In the equation, *α*_{max} (*J*^{T}*J*) and *α*_{min} (*J*^{T}*J*), respectively, represent the maximum eigenvalue and the minimum eigenvalue of the matrix; *β*_{max}=½ (*α*_{max} (*J*^{T}*J*)) and *β*_{min}=½ (*α*_{min} (*J*^{T}*J*)), respectively, represent the maximum and minimum singular values of the Jacobian matrix.

Generally, the reciprocal of condition number is used as the judgment index of dexterity. Assuming *Q*_{k}=(1/(ǁ*J*ǁǁ*J*^{−1}ǁ)), substituting Equation (16) into expression *Q*_{k} gives dexterity:

According to Equation (17), *Q*_{k} is the reciprocal of the number of Jacobian matrix. Its value range is 0 ⩽ *Q*_{k} ⩽ 1. The smaller the *Q*_{k} value, the worse will be the flexibility of the mechanism and the larger will be the motion deviation. At this time, the change of input joint velocity of the mechanism will have a greater impact on the change of output velocity. When *Q*_{k}=1, the mechanism has the best flexibility and is called isotropic mechanism. However, since *Q*_{K} only represents the local dexterity of the mechanism, it cannot reflect the overall flexibility of the mechanism. In this paper, the following global dexterity coefficient (Wei-fang, 2016) is adopted to comprehensively evaluate the flexibility of the rigid‒flexible parallel trunk joint mechanism:

In the equation, the greater the *C* value, the higher will be the overall dexterity of the mechanism.

## 5. Application example and mechanism performance analysis

The spring-based wire-driven 4-SPS/U rigid‒flexible parallel trunk joint mechanism parameters are set as follows: *r* = 178.5 mm, *n* = 200 mm, *l*_{imin} = 222.3 mm, *l*_{imax} = 515 mm, *θ*_{Bimax} = 90°, *θ*_{Aimax} = 90°, −45° ⩽ *θ*_{i} ⩽ 45°. The stiffness of wire *K*_{i} is 2000N·m/rad. *D*_{i} = *g*_{i}·*d*_{i}(0 ⩽ *g*_{i} ⩽ +∞) relation exists in the spring parameter, and *G* is 7.2 × 10^{10} N/m.

### 5.1 Simulation analysis of stiffness performance of the parallel trunk joint mechanism

From Equation (13), through MATLAB simulation, the maximum and minimum eigenvalue distributions of the stiffness Jacobian matrix of the parallel trunk joint mechanism only affected by the wire can be obtained. As shown in Figure 4, the branch chain driven by two adjacent ropes varies with *θ*_{1} and *θ*_{2}, and it is symmetrically decreasing distribution in the four angular directions. The stiffness value near the middle is larger; the minimum stiffness in the direction of the minimum eigenvalue is 63 (N·m/rad), and the maximum stiffness in the direction of the maximum eigenvalue is 125 (N·m/rad). On this basis, a flexible spring is added to each wire-driven branch chain, and its elastic mechanism parameters are set as *d*_{i}=0.004 m, *K*_{i}=6, and *n*_{i}=4 turns. From Equation (15), through MATLAB simulation, the stiffness matrix eigenvalue distribution of the parallel trunk joint mechanism can be obtained. As shown in Figure 5, the distribution variation of stiffness eigenvalues is the same as that in Figure 4. However, the minimum stiffness value in the direction of the minimum eigenvalue is 1,380 (N·m/rad), and the maximum stiffness value in the direction of the maximum eigenvalue is 2,750 (N·m/rad). Compared with the minimum maximum eigenvalue direction stiffness in Figure 4, which is only affected by the rope, the stiffness is increased by about 20 times. Therefore, adding a flexible spring to the wire-driven branch can improve the stiffness of the wire-driven 4SPS/U flexible rigid parallel joint mechanism.

According to Equation (14), the stiffness of the mechanism is affected by the spring elastic structure parameter ni, di, Di. In order to obtain the influence of degree of spring elastic structure parameters on the stiffness value of the mechanism, this paper obtained the maximum and minimum eigenvalue distributions of the Jacobian matrix of stiffness through the simulation based on Equation (15) under the condition of changing the parameters of spring elastic structure. Figure 6 shows the eigenvalue distribution of stiffness matrix of the parallel trunk joint mechanism when the number of turns in the spring structure of each wire-driven branching chain is 1, 2, 3, 4, and the wire diameter and outside diameter ratio are 4 mm, 6, respectively. Figure 7 shows the eigenvalue distribution of stiffness matrix of the parallel trunk joint mechanism when the linear diameter of the spring structure of each wire-driven branch chain is 10 mm, 20 mm, 30 mm and 4 mm, respectively, and the number of turns and outside diameter ratio are 4 and 6, respectively.

Figure 8 shows the eigenvalue distribution of stiffness matrix of the parallel trunk joint mechanism when the outer diameter ratio of the spring structure of each wire-driven branch is taken as 3, 4, 6, and 9, respectively, and the number of turns and the wire diameter are 4 and 4 mm, respectively. It can be seen from the distribution of the contour values of the stiffness values between the adjacent two wire-driven branches in each of the attached diagrams in Figures 6‒8 in the shape of a ring that is different in size and sparse. When the spring elastic structure parameters on the adjacent two wire-driven branches take different values, the change in the stiffness values in the direction of the maximum and minimum characteristic values is: the distribution is reduced in different sizes in the direction of the four corners with the change of *θ*_{1} and *θ*_{2} in the four angular directions ‒ the closer to the middle, the greater will be the stiffness value. The smaller the number of the spring elastic structure turns of each adjacent two wire-driven branches in Figure 6, the greater will be the contour value of stiffness between the adjacent two wire-driven branch chains. The larger the linear diameter of the spring elastic structure of each adjacent two wire-driven branches in Figure 7, the greater will be the contour value of stiffnes between the adjacent two wire-driven branches. The smaller the outer diameter of the spring elastic structure of each adjacent two wire-driven branches in Figure 8, the larger will be the contour value of stiffness between the adjacent two rope-driven branches.

Therefore, by changing the spring elastic structural parameters of each wire-driven branch chain, the following can be concluded: the fewer the number of turns, the larger will be the wire diameter and the smaller will be the outer diameter. It can make the parallel trunk joint mechanism simplify and reduce the complexity of the structure, combined with the characteristics of wire-driven and rigid parallel mechanism that can effectively enhance and improve the parallel trunk joint mechanism of the specific direction of the maximum and minimum eigenvalue direction stiffness distribution. It ensures that the multi-motion mode hexapod mobile robot can meet the requirement of sufficient different stiffness for different motion postures through the parallel trunk joint mechanism.

As can be seen from the above description of Figures 4‒8, it can only be explained that the spring elastic structure parameters have different influence on the stiffness value of the mechanism in the direction of the maximum and minimum eigenvalues, because the eigenvalues of stiffness along different eigenvectors are different, which cannot reflect that the elastic structural parameters of the spring have different absolute values of the stiffness values of the different pose positions of the mechanism. Therefore, we use the global stiffness *K*_{p} as the evaluation index. The larger the *Kp*, the better will be the global stiffness performance of the parallel trunk joint mechanism along the pose point. The global stiffness *Kp* is expressed as follows:

In the equation, *λ*_{i} represents all the eigenvalues of the stiffness matrix *K*_{t}, and *S* represents the range of the workspace.

In order to obtain the influence degree of spring elastic structure parameters on the mechanism’s global stiffness value, this paper obtained the global stiffness distribution through the simulation of Equation (19) by changing the spring elastic structure parameters. Figure 9 depicts the global stiffness distribution of the parallel trunk joint mechanism when the number of turns in the spring structure of each wire-driven branching chain is 1, 2, 3, 4, and wire diameter and outside diameter ratio are 4 mm and 6, respectively. Figure 10 depicts the global stiffness distribution of the parallel trunk joint mechanism when the linear diameter of the spring structure of each wire-driven branch chain is 10 mm, 20 mm, 30 mm and 4 mm, respectively, and the number of turns and outside diameter ratio are 4 and 6, respectively. Figure 11 depicts the global stiffness distribution of the parallel trunk joint mechanism when the outer diameter ratio of the spring structure of each wire-driven branch is taken as 2, 5, 8, and 9, respectively, and the number of turns and the wire diameter are 4 and 4 mm, respectively. According to Figures 9‒11, the global stiffness value of the contour line between each adjacent two wire-driven branches is distributed in the form of rings with different sparse and sizes in the direction of the four corners. The stiffness value of the direction of the maximum and minimum eigenvalues is decreasing with the magnitude of *θ*_{1} and *θ*_{2} in the four angular directions ‒ the closer to the middle, the greater will be the stiffness value. The smaller the number of the spring elastic structure turns of each adjacent two wire-driven branches in Figure 9, the larger will be the global stiffness value contour line value between the adjacent two wire-driven branches. The larger the linear diameter of the spring elastic structure of each adjacent two wire-driven branches in Figure 10, the larger will be the global stiffness value contour line value between the adjacent two wire-driven branches. The smaller the outer diameter of the spring elastic structure of each adjacent two wire-driven branches in Figure 11, the larger will be the global stiffness value contour line value between the adjacent two wire-driven branches.

Therefore, by changing the spring elastic structural parameters of each wire-driven branch chain, the following can be concluded: the fewer the number of turns, the larger will be the wire diameter and the smaller will be the outer diameter. It can effectively improve and adjust the global stiffness and different distributions of the parallel trunk joint mechanism by combining the characteristics of wire-driven and rigid parallel mechanism after simplifying and reducing the structural complexity. It ensures that the multi-motion mode hexapod mobile robot in multi-motion mode can meet the performance requirement of global stiffness change at different pose points of different motion postures through the parallel trunk joint mechanism.

### 5.2 Analysis of motion dexterity

According to Equation (16), the singular minimum and maximum values of dexterity, as shown in Figures 12 and 13, can be obtained through MATLAB simulation. As can be seen from the figures, the singular maximum and minimum values of the dexterity of the parallel trunk joint mechanism have similar distribution rules, both of which are symmetric about the *Y*-axis and the *X*-axis, and have a maximum value of 252.43 near the center of the circle (0, 0). Their singular maximum and minimum values decrease along both sides of the symmetry axis, with a minimum value of 168.72 at the edge. It shows that the overall transmission performance and bearing capacity of the parallel trunk joint mechanism are better.

According to Equation (17), the local dexterity of the trunk joint mechanism shown in Figure 14 can be obtained by MATLAB simulation. It can be seen from the diagram that the maximum value of dexterity is 0.971 and the minimum value is 0.679, and the local dexterity is symmetrically distributed with *Y*-axis or *X*-axis as symmetry axis. At the same time, it can be seen that the closer to the symmetry axis, the greater will be the local dexterity, and the radiation decreases to both sides. The curved surface is smooth and continuous without peak mutation, which indicates that the parallel trunk joint mechanism has good local dexterity.

According to Equation (18), the global dexterity, as shown in Figure 15, can be obtained through MATLAB simulation. Compared with the local dexterity in Figure 14, the global dexterity has the same distribution law as the local dexterity, but the maximum global dexterity is 0.999 and the minimum is 0.77266, which is higher than that of the local dexterity. It shows that the global motion performance of the parallel trunk joint mechanism is more flexible and superior.

## 6. Conclusion

Based on the spring-based wire-driven 4SPS/U rigid‒flexible parallel trunk joint mechanism and using the closed vector method to establish the position inverse equation, the Jacobian matrix of velocity and acceleration is solved. On the basis of this and combined with the flexible deformation mechanism of the wire driven pair and the spring restrained pair, the stiffness matrix of the mechanism is derived.

In this paper, the minimum and maximum eigenvalue distributions of stiffness matrix and the global stiffness index are taken as the stiffness evaluation indexes of the mechanism, and the simulation is carried out in the numerical software. The influence of the stiffness distribution on the parallel trunk joint mechanism after the introduction of the spring on the wire drive branch is compared and analyzed. It is concluded that by adjusting the spring elastic structural parameter variation, the stiffness of the mechanism in a specific direction can be improved, and the stiffness characteristic distribution of the original parallel mechanism can be improved to some extent.

Finally, the dexterity of the mechanism is analyzed by numerical simulation map, in which the maximum and minimum values of the singularity of dexterity decrease with

*X*or*Y*-axis symmetric decline, and the singular value difference from the workspace center to the boundary position is less. At the same time, the distribution maps of local dexterity and global dexterity are compared, and the results show that the parallel trunk joint mechanism has not only good bearing capacity and transmission performance, but it also has a more flexible movement performance.

## Figures

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