2.1. Elastic deformation of hybrid robot end caused by parallel module

2.1.1 The contribution to the robot end deformation of the parallel module

Based on the rigidity principle, the contribution of the parallel module to end deformation can be obtained while the serial module is considered to be rigid.

According to the wrench transfer formula, the wrench acting on the point o of the parallel module can be expressed as follows:

(1) \begin{equation} \textbf{W}_{o}\textbf{=}\left[\begin{array}{l@{\quad}l} \textbf{E}_{3} & \textbf{0}_{3}\\[5pt] \left[\textbf{S}_{oT}\times \right] & \textbf{E}_{3} \end{array}\right]\textbf{W}_{T}=\textbf{D}_{To}\textbf{W}_{T}\textbf{=}\left[\textbf{f}_{o}\textbf{;}\;\textbf{m}_{o}\right] \end{equation}

where D _To denotes the mapping matrix from W _T to W _o . [S _oT ×] denotes the skew-symmetric matrix of vector S _oT .

The DCEs between limb end and the attachment point o of the moving platform of the parallel module can be obtained [Reference Li, Chen and Xu19] according to the principle of virtual work:

(2) \begin{equation} \boldsymbol{\Delta }_{i}=\textbf{J}_{i}^{T}\boldsymbol{\Delta }_{o}\left(i = 1,2,\cdot \cdot \cdot,n\right) \end{equation}

where $\boldsymbol\Delta$ _i denote the elastic deformation of the i ^th limb; J _i denote the unit constraint wrench system of the i ^th limb; and $\boldsymbol\Delta$ _o denotes the elastic deflection of the attachment point o of the moving platform.

The following equation can be obtained according to DCEs and the equilibrium equation of the spatial force system [Reference Li, Chen and Xu19].

(3) \begin{equation} \textbf{W}_{o}=\sum _{i=1}^{n}\textbf{J}_{i}\textbf{K}_{i}\textbf{J}_{i}^{T}\boldsymbol{\Delta }_{o}=\textbf{K}_{p}\boldsymbol{\Delta }_{o} \end{equation}

where K _p denotes the overall stiffness matrix corresponding to the attachment point o of the moving platform of the parallel module.

Therefore, the elastic deflection of the attachment point of the moving platform under W _o is

(4) \begin{equation} \boldsymbol{\Delta }_{o}=\textbf{K}_{p}^{-1}\textbf{W}_{o}=\textbf{C}_{p}\textbf{W}_{o} \end{equation}

where C _p denotes the overall flexibility matrix corresponding to attachment point o of the moving platform of the parallel module.

The hybrid-robot terminal elastic deflection caused by parallel module under the assumption that the serial module to be rigid can be obtained as follows.

(5) \begin{equation} \boldsymbol{\Delta }_{Tp}=\left[\begin{array}{l@{\quad}l} \textbf{E}_{3} & \left[\textbf{S}_{To}\times \right]\\[5pt] \textbf{0}_{3} & \textbf{E}_{3} \end{array}\right]\boldsymbol{\Delta }_{o}=\textbf{T}_{To}\boldsymbol{\Delta }_{o} \end{equation}

where E ₃ and 0 ₃ denote the 3 × 3 identity matrix and zero matrix, respectively, [S _To ×] denotes the skew-symmetric matrix of vector S _{To, and} T _To denotes the mapping matrix from $\boldsymbol\Delta$ _o to $\boldsymbol\Delta$ _Tp .

From the analysis above, it is evident that to determine the contribution of the parallel module to the elastic deformation of the hybrid robot end, it is necessary to first calculate the overall stiffness matrix of the parallel module. The process for calculating the limb stiffness matrix and overall stiffness matrix of the parallel module can refer to [Reference Li, Chen and Xu19].

Figure 3 presents a general model of the limb. o _ij -x _ij y _ij z _ij denotes the coordinate frame of j ^th member of i ^th limb, and axis x _ij is aligned with the direction of the limb member here.

Figure 3. General model of limb.

Unit constraint wrench system $\textbf{J}_{i}=[\textbf{\$ }_{i1}^{r},\textbf{\$ }_{i2}^{r},\textbf{\$ }_{i3}^{r}]$ of the i ^th limb can be established based on the reciprocal screw theory, where $\textbf{\$ }_{ij}^{r}=[\textbf{r}_{ij}^{r},\textbf{r}_{oij}^{r}\times \textbf{r}_{ij}^{r}]$ is the j ^th unit constraint wrench of the i ^th limb, $\textbf{r}_{ij}^{r}$ is a unit vector pointing along the direction of the screw axis, and $\textbf{r}_{oij}^{r}$ is the position of any point on the screw axis, with a magnitude of W _i = [f _i ; m _i ]. f _i and m _i denote constraint force and moment, respectively.

Based on screw theory and the principle of virtual work, the overall stiffness matrix of the parallel module can be established:

(6) \begin{equation} \textbf{K}_{p}=\sum _{i=1}^{n}\textbf{J}_{i}\textbf{K}_{i}\textbf{J}_{i}^{T} \end{equation}

where K _i denotes the stiffness matrix of the i ^th limb corresponding to W _i .

2.1.2 The contribution to the robot end deformation of each component

Elastic deformation $\boldsymbol\Delta$ _ij of the i ^th limb’s j ^th member corresponding to W _i can be obtained as follows according to Castigliano’s second theorem:

(7) \begin{equation} \boldsymbol{\Delta }_{ij}=\frac{\partial U_{ij}}{\partial \textbf{W}_{i}}=\textbf{C}_{ij}\textbf{W}_{i} \end{equation}

where U _ij denotes the strain energy of the i ^th limb’s j ^th member; C _ij denotes the flexibility matrix of the i ^th limb’s j ^th member corresponding to W _i .

The strain energy of the i ^th limb’s j ^th member is given as

(8) \begin{equation} U_{ij}=\int _{0}^{L_{ij}}\left(\frac{f_{ijx}^{2}}{2E_{ij}A_{ij}}+\frac{f_{ijy}^{2}}{2G_{ij}A_{ijy}}+\frac{f_{ijz}^{2}}{2G_{ij}A_{ijz}}+\frac{m_{ijx}^{2}}{2G_{ij}I_{ijp}}+\frac{m_{ijy}^{2}}{2E_{ij}I_{ijy}}+\frac{m_{ijz}^{2}}{2E_{ij}I_{ijz}}\right)dv_{ij} \end{equation}

where f _ij = [f _ijx f _ijy f _ijz ]^T, m _ij = [m _ijx m _ijy m _ijz ]^T denotes the internal force of the i ^th limb’s j ^th member; E _ij , G _ij , and L _ij denote the elastic modulus, shear modulus, and length of the i ^th limb’s j ^th member, respectively; A _ij , A _ijy , and A _ijz denote the area and the effective shear area of the cross-section along the y _ij and z _ij axes, respectively; I _ijy and I _ijz denote the area moment of cross-section inertia about the y _ij and z _ij axes, respectively. Finally, I _ijp denotes the polar moment of cross-section inertia.

The strain energy of the i ^th limb is as follows:

(9) \begin{equation} U_{i}=\sum _{j=1}^{c}U_{ij}=\frac{1}{2}\textbf{W}_{i}^{T}\left(\sum _{j = 1}^{c}\textbf{C}_{ij}\right)\textbf{W}_{i}\textbf{=}\frac{1}{2}\textbf{W}_{i}^{T}\textbf{C}_{i}\textbf{W}_{i} \end{equation}

where C _i denotes the flexibility matrix of the i ^th limb corresponding to W _i ; c denotes the number of the member in the i ^th limb.

Elastic deformation $\boldsymbol\Delta$ _i of the i ^th limb corresponding to W _i is as follows according to Castigliano’s second theorem:

(10) \begin{equation} \boldsymbol{\Delta }_{i}=\frac{\partial U_{i}}{\partial \textbf{W}_{i}}=\textbf{C}_{i}\textbf{W}_{i} \end{equation}

The stiffness matrix of the i ^th limb constraint wrenches can be obtained as

(11) \begin{equation} \textbf{K}_{i}=\textbf{C}_{i}^{\mathit{-}1} \end{equation}

Equations (1), (2), (3) and (10) lead to

(12) \begin{equation} \textbf{W}_{i}=\textbf{K}_{i}\boldsymbol{\Delta }_{i}=\textbf{K}_{i}\textbf{J}_{i}^{T}\boldsymbol{\Delta }_{o}=\textbf{K}_{i}\textbf{J}_{i}^{T}\textbf{C}_{p}\textbf{W}_{o}=\textbf{K}_{i}\textbf{J}_{i}^{T}\textbf{C}_{p}\textbf{D}_{To}\textbf{W}_{T} \end{equation}

Therefore, the strain energy of the i ^th limb’s j ^th member of parallel modules is as follows.

(13) \begin{equation} \begin{array}{l} U_{ij}=\dfrac{1}{2}\textbf{W}_{i}^{\mathrm{T}}\boldsymbol{\Delta }_{ij}=\dfrac{1}{2}\textbf{W}_{i}^{\mathrm{T}}\textbf{C}_{ij}\textbf{W}_{i}=\dfrac{1}{2}\left(\textbf{K}_{i}\textbf{J}_{i}^{\mathrm{T}}\textbf{C}_{p}\textbf{D}_{To}\textbf{W}_{T}\right)^{\mathrm{T}}\textbf{C}_{ij}\left(\textbf{K}_{i}\textbf{J}_{i}^{\mathrm{T}}\textbf{C}_{p}\textbf{D}_{To}\textbf{W}_{T}\right)\\[12pt] =\dfrac{1}{2}\textbf{W}_{T}^{\mathrm{T}}\left(\textbf{D}_{To}^{\mathrm{T}}\textbf{C}_{p}^{\mathrm{T}}\textbf{J}_{i}\textbf{K}_{i}^{\mathrm{T}}\textbf{C}_{ij}\textbf{K}_{i}\textbf{J}_{i}^{\mathrm{T}}\textbf{C}_{p}\textbf{D}_{To}\right)\textbf{W}_{T}=\dfrac{1}{2}\textbf{W}_{T}^{\mathrm{T}}\overline{\textbf{C}}_{ij}\textbf{W}_{T} \end{array} \end{equation}

where $\overline{\textbf{C}}_{ij}$ denotes the overall flexibility contribution matrix caused by the i ^th limb’s j ^th member of parallel modules for the hybrid robot.

The elastic deformation of the hybrid robot terminal caused by the i ^th limb’s j ^th member of parallel modules can be obtained based on the Castigliano’s second theorem.

(14) \begin{equation} \overline{\boldsymbol{\Delta }}_{ij}=\frac{\partial U_{ij}}{\partial \textbf{W}_{T}}=\overline{\textbf{C}}_{ij}\textbf{W}_{T} \end{equation}

Now, we have decoupled the elastic deformation of the hybrid robot end and the CCM caused by the i ^th limb’s j ^th member of parallel module of the hybrid robot through equation (14).

2.2. Elastic deformation of hybrid robot end caused by serial module

Elastic deformation $\boldsymbol\Delta$ _si of the j ^th member of serial module corresponding to W _T is as follows according to the strain energy and Castigliano’s second theorem.

(15) \begin{equation} \boldsymbol{\Delta }_{sj}=\frac{\partial U_{sj}}{\partial \textbf{W}_{T}}=\textbf{C}_{sj}\textbf{W}_{T} \end{equation}

where U _sj denotes the strain energy of j ^th member of the serial module; C _sj denotes the flexibility matrix corresponding to the W _T of the j ^th member of serial module.

The strain energy of the serial module U _s is as follows:

(16) \begin{equation} U_{s}=\sum _{j=1}^{g}U_{sj}=\frac{1}{2}\textbf{W}_{T}^{T}\left(\sum _{j = 1}^{g}\textbf{C}_{sj}\right)\textbf{W}_{T}\textbf{=}\frac{1}{2}\textbf{W}_{T}^{T}\textbf{C}_{s}\textbf{W}_{T} \end{equation}

where C _s denotes the flexibility matrix of the serial module corresponding to W _T ; g denotes the number of members of the serial module.

The hybrid robot end elastic deformation caused by serial modules is as follows according to Castigliano’s second theorem:

(17) \begin{equation} \boldsymbol{\Delta }_{Ts}=\frac{\partial U_{s}}{\partial \textbf{W}_{T}}=\textbf{C}_{s}\textbf{W}_{T} \end{equation}

2.3. Elastic deformation of hybrid robot end

Finally, the elastic deformation of the hybrid robot terminal is equal to the sum of elastic deformation caused by parallel and serial modules according to the rigidity principle.

(18) \begin{equation} \boldsymbol{\Delta }_{T}=\boldsymbol{\Delta }_{Tp}+\boldsymbol{\Delta }_{Ts}=\textbf{T}_{To}\textbf{C}_{p}\textbf{W}_{o}+\textbf{C}_{Ts}\textbf{W}_{T}=\left(\textbf{T}_{To}\textbf{C}_{p}\textbf{D}_{To}+\textbf{C}_{Ts}\right)\textbf{W}_{T}=\textbf{C}_{H}\textbf{W}_{T} \end{equation}

where $\boldsymbol\Delta$ _T denotes the elastic deformation of the hybrid robot terminal; C _H denotes the overall flexibility of the hybrid robot corresponding to external screw W _T .

Reference [Reference Yang, Li and Chen21] decouples the contribution of each elastic element in the parallel module using the rigidity principle. In this approach, the contribution matrix of each elastic element to the stiffness of the mechanism becomes singular in some configurations, requiring the use of the generalized inverse operation to obtain the contribution of each elastic element to the end deformation of the mechanism. In contrast, in this paper, the parallel module operates entirely within the framework of strain energy and Castigliano’s second theorem, which has a clear physical interpretation and avoids the occurrence of singularity issues that can arise when using the rigidity principle in parallel module.

4.1. Introduction of 4SRRR + 6R

The hybrid robot composed of a four-legged wall-climbing robot and serial robotic arms is developed based on the on-site work requirements of large and complex structural parts (Fig. 6). The parallel module (four-legged wall-climbing robot) consists of a moving platform, a fixed platform (the fixed foot and the working wall form a fixed platform when the four-legged wall climbing robot reaches the working position), and four identical SRRR limbs. The limb legs are symmetrical and are composed of three active rotating joints (R) and one passive ball hinge joint (S). The serial module (serial robotic arm) is composed of six active rotating joints (R), and the axes of the last three rotating joints intersect at one point. There are a total of 18 active rotating joints in the parallel and serial module as the input of the mechanism. The mobile platform and joints of hybrid robot are assumed to be rigid, and the elastic rod components are made of aluminum alloy 7075-T6, with their physical and structural parameters shown in Table 2.

Figure 5. Contours of linear and angular displacement vectors.

Figure 6. 4SRRR + 6R hybrid robot.

Table II. Physical and structure parameter values of the 4SRRR + 6R hybrid robot.

The origin of parallel module moving coordinate system o-xyz is located at the geometric center of the moving platform; the z-axis is perpendicular to the plane where the moving platform is located; the x-axis is parallel to line A ₃ A ₄; and the y-axis is determined by the right-hand rule. The origin of fixed coordinate system O-XYZ is located at the geometric center of the fixed platform. The Z-axis is vertically upward; the X-axis is parallel to the line of D ₃ D ₄; and the Y-axis is determined by the right-hand rule.

The parallel module limb is connected to the moving platform through R-joint at A _i (the axis is perpendicular to the surface of the moving platform) and connected to the fixed platform through S-joint at D _i (Fig. 7a). z _i1, z _i2, and z _i3 of the local coordinate systems A _i -x _i1 y _i1 z _i1, B _i -x _i2 y _i2 z _i2, and C _i -x _i3 y _i3 z _i3 are along the axis of A _i , B _i , and C _i , respectively. x _i1, x _i2, and x _i3 point to the directions of A _i B _i , B _i C _i , and C _i D _i , respectively. The serial module establishes a coordinate system according to the DH method.

Figure 7. Coordinate system of parallel module limb and serial module.

Figure 8. Inverse kinematics of 4SRRR wall climbing robot.

The foot must be perpendicular to the work surface due to the magnetic suction when the 4SRRR-6R hybrid robot is adsorbed on the work surface. Figure 8 illustrates the inverse kinematics calculation diagram of the limb of the 4SRRR-6R hybrid robot.

The coordinate of A _i in the moving coordinate system is ( ^o x _Ai , ^o y _Ai , ^o z _Ai ), and the coordinates of D _i are ( ^o x _Di , ^o y _Di , ^o z _Di ). The following equation according to the geometric relationships is

(24) \begin{equation} \left\{\begin{array}{l} \eta _{i1}=atan2\left({}^{o}{y}{_{Di}^{}}-{}^{o}{y}{_{Ai}^{}},\,{}^{o}{x}{_{Di}^{}}-{}^{o}{x}{_{Ai}^{}}\right)\\[5pt] L_{AiDi}=\mathrm{abs}\left[\left({}^{o}{x}{_{Di}^{}}-{}^{o}{x}{_{Ai}^{}}\right)/\cos \left(\eta _{i1}\right)\right]\\[5pt] H_{i}=\mathrm{abs}\left({}^{o}{z}{_{Ai}^{}}-{}^{o}{z}{_{Di}^{}}\right)\\[5pt] L_{BiDi}=\sqrt{\left(L_{AiDi}-L_{1}\right)^{2}+H_{i}^{2}} \end{array}\right. \end{equation}

According to the cosine theorem,

(25) \begin{equation} \left\{\begin{array}{l} \eta _{3i}=\pi -acos\left(\dfrac{L_{2}^{2}+L_{3}^{2}-L_{BiDi}^{2}}{2L_{2}L_{3}}\right)\\[12pt] \beta _{i}=acos\left(\dfrac{L_{2}^{2}+L_{BiDi}^{2}-L_{3}^{2}}{2L_{2}L_{BiDi}}\right) \end{array}\right. \end{equation}

Then,

(26) \begin{equation} \eta _{i2}+\beta _{i}=atan2\left(H_{i},L_{i}-L_{1}\right) \end{equation}

We can obtain

(27) \begin{equation} \left\{\begin{array}{l} \eta _{i2}=atan2\left(H_{i},L_{i}-L_{1}\right)-\beta _{i}\\[9pt] \eta _{i4}=\dfrac{\pi }{2}-\eta _{i2}-\eta _{i3} \end{array}\right. \end{equation}

4.2. Static stiffness modeling of hybrid robots

Figure 9 shows the constraint wrench system imposed by the limb on the mobile platform. Utilizing the screw and reciprocal screw theory, the constraint wrench system of the limb can be obtained based on local coordinate systems A _i -x _i1 y _i1 z _i1:

Figure 9. Constraint wrench system applied by the i ^th limb on the moving platform.

(28) \begin{equation} \begin{array}{l} \$ _{i1}^{r}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} 1 & 0 & 0 & 0 & z_{Di} & -y_{Di} \end{array}\right]\\[5pt] \$ _{i1}^{r}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} 0 & 1 & 0 & -z_{Di} & 0 & x_{Di} \end{array}\right]\\[5pt] \$ _{i1}^{r}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} 0 & 0 & 1 & y_{Di} & -x_{Di} & 0 \end{array}\right] \end{array} \end{equation}

The limb leg exerts three constraint forces along the x _i1, y _i1, and z _i1 axes on the robot mobile platform at point D _i when the three active joints of the limb leg are all locked according to screw and reciprocal screw theory. The amplitudes are f _i1, f _i2, and f _i3 (Fig. 9), that is, W _i = [f _i1 f _i2 f _i3 0 0 0]^T = [f _i 0]^T. The four limb legs exert 12 constraint force on the robot’s moving platform. The solving requires simultaneous equations involving additional supplementary equations and static equilibrium equations. Therefore, the four-legged climbing robot is the overconstrained mechanism.

The force screw acting on the A _i point can be expressed as follows under branch coordinate system A _i -x _i1 y _i1 z _i1 according to the translation principle of force in theoretical mechanics:

(29) \begin{equation} \textbf{W}_{Ai}=\left[\textbf{f}_{Ai};\;\textbf{t}_{Ai}\right] \end{equation}

where f _Ai represents the force at point A _i , and f _Ai = f _i ; t _Ai represents the torque at point A _i , and t _Ai = A _i D _i ×f _i .

Figure 10. Force analysis of each component of the limb.

Figure 10a, 10b and 10c illustrate the force analysis of the A _i B _i , B _i C _i and C _i D _i rods within their respective branches based on the strain energy in material mechanics, and the strain energy of the A _i B _i , B _i C _i and C _i D _i rod in Branch i can be expressed as

(30) \begin{equation} U_{ik}=a_{ik1}f_{i1}^{2}+a_{ik2}f_{i2}^{2}+a_{ik3}f_{i3}^{2}+a_{ik4}f_{i1}f_{i2}+a_{ik5}f_{i2}f_{i3}+a_{ik6}f_{i3}f_{i1} \end{equation}

where a _ik1, a _ik2 , …, and a _ik6 (k = 1,2,3) are the coefficients obtained based on the equation with W _i as the variable. C _i1 can be solved according to the equation.

The strain energy of the i ^th limb is equal to the sum of the strain energy of each member.

(31) \begin{equation} U_{i}=U_{i1}+U_{i2}+U_{i3}=a_{i1}f_{i1}^{2}+a_{i2}f_{i2}^{2}+a_{i3}f_{i3}^{2}+a_{i4}f_{i1}f_{i2}+a_{i5}f_{i2}f_{i3}+a_{i6}f_{i3}f_{i1} \end{equation}

where a _ij = a _i1j + a _i2j + a _i3j (j = 1, 2, …, 6).

The elastic deformation of the end of limb i along the axis of the constrained screw is as follows according to equation (7).

(32) \begin{equation} \begin{array}{l} \Delta _{i1}=\dfrac{\partial U_{i}}{\partial f_{i1}}=2a_{i1}f_{i1}+a_{i4}f_{i2}+a_{i6}f_{i3}\\[12pt] \Delta _{i2}=\dfrac{\partial U_{i}}{\partial f_{i2}}=a_{i4}f_{i1}+2a_{i2}f_{i2}+a_{i5}f_{i3}\\[12pt] \Delta _{i3}=\dfrac{\partial U_{i}}{\partial f_{i3}}=a_{i6}f_{i1}+a_{i5}f_{i2}+2a_{i3}f_{i3} \end{array} \end{equation}

The flexibility matrix of limb i can be obtained as

(33) \begin{equation} \textbf{C}_{i}=\left[\begin{array}{l@{\quad}l@{\quad}l} 2a_{i1} & a_{i4} & a_{i6}\\[5pt] a_{i4} & 2a_{i2} & a_{i5}\\[5pt] a_{i6} & a_{i5} & 2a_{i3} \end{array}\right] \end{equation}

Stiffness matrix K _i and overall stiffness matrix K _p can be obtained by equations (6) and (11). The contribution of the parallel module to the elastic deformation of the hybrid robot end can be obtained by equation (5), and the contribution of each member of the parallel module to the elastic deformation of the hybrid robot end can be obtained by equation (14), where $\textbf{J}_{i}=[\boldsymbol{\$ }_{i1}^{r},\$ _{i2}^{r},\$ _{i3}^{r}]$ represents the constrained screw system of limb i.

The decoupling flexibility matrix corresponding to serial module rod j and W _T , as well as the contribution to the elastic deformation of the hybrid robot end, can be obtained according to the equation (15). The overall flexibility matrix corresponding to the serial module and W _T , as well as the contribution to the elastic deformation of the hybrid robot end, can be obtained according to the equations (16) and (17).

The elastic deformation of the hybrid robot end is obtained according to the equation (18) after solving the contribution of the parallel and serial modules to the elastic deformation of the hybrid robot end, respectively.

4.3. Comparison between the analytical model and finite element model

The finite element commercial software ANSYS was used to verify the correctness and feasibility of the proposed method in the work. The material and structural parameters are given based on the parameters used in the theoretical model in the ANSYS model. The flexible limb is modeled using Beam 188 units based on the theory of Timoshenko.

Two typical working conditions of mobile welding are considered. Working pose 1: Four-legged wall-climbing robot (parallel module) limbs: θ _i1 = 0°, θ _i2 = 10°, and θ _i3 = 70° (i = 1, 2, 3, 4). Serial module: joint angle θ _s1 = 90°, θ _s2 = −45°, θ _s3 = 0°, θ _s4 = 0°, θ _s5 = −50°, and θ _s6 = 0. Working pose 2: Four-legged wall-climbing robot (parallel module) limbs: θ _i1 = 45°, θ _i2 = 10°, and θ _i3 = 70° (i = 1, 2, 3, 4). Serial module: joint angle θ _s1 = 120°, θ _s2 = −45°, θ _s3 = 0°, θ _s4 = 0°, θ _s5 = −50°, and θ _s6 = 0°. Terminal external load W _T = [50N, 80N, −100N, 0, 0, 0]^T.

Table 3 lists the comparison of tool terminal deformation results of the hybrid robot between the proposed analytical model and the finite element model (FEM), where the maximum error is 2.35%. The discrepancies between the theoretical model and the FEM mainly arise from differences in mesh partitioning and calculation errors. The correctness and accuracy of the proposed model are verified. Figure 11 shows the FEM results of the two poses of the hybrid robot.

4.4. Stiffness performance evaluation

The linear and angular displacements generated when the terminal of the hybrid robot works along the specified trajectory are used to evaluate the stiffness performance of the robot.

(34) \begin{equation} \left\{\begin{array}{l} {\unicode{x1D6E5}}_{d}=\sqrt{{\unicode{x1D6E5}}_{\mathit{d}x}^{2}+{\unicode{x1D6E5}}_{dy}^{2}+{\unicode{x1D6E5}}_{dz}^{2}}\\[5pt] {\unicode{x1D6E5}}_{r}=\sqrt{{\unicode{x1D6E5}}_{rx}^{2}+{\unicode{x1D6E5}}_{ry}^{2}+{\unicode{x1D6E5}}_{rz}^{2}} \end{array}\right. \end{equation}

The robot performs welding or polishing work along a specified trajectory. The four-legged wall-climbing robot (parallel module) limbs: θ _i1 = 45°, θ _i2 = 10°, and θ _i3 = 70° (i = 1, 2, 3, 4). The terminal tool of the robot works along the path of the sinusoidal curve (x = −0.3:0.02:0.3, y = 0.005sin(8πx) + 0.5, and z = 0.1) in a specific posture, and the external load is W _T = [50N, 80N, −100N, 0, 0, 0]^T. The linear and angular displacement reach their maximum values at the same time, which are 8.551e-5 m and 1.243e-4 rad, respectively, when the robot end is close to (−0.3 m, 0.495 m, 0.1 m) (Fig. 12).

Table III. Elastic deflection of the hybrid robot terminal.

Figure 11. Contours of linear and angular displacement vectors.

Figure 12. Linear and angular deflections of the hybrid robot terminal in the prescribed trajectory.

The contribution of each elastic element to the elastic deformation of the hybrid robot end is expressed by the equation as follows when the robot works along the specified trajectory.

(35) \begin{equation} \left\{\begin{array}{l} \overline{{\Delta} }_{ijd}=\dfrac{\sum {\Delta} _{ijd}}{N}\\[12pt] \overline{{\Delta} }_{ijr}=\dfrac{\sum {\Delta} _{ijr}}{N} \end{array}\right. \end{equation}

where ${\Delta} _{ijd}$ and ${\Delta} _{ijr}$ represent the linear and angular displacement of the elastic deformation of the robot end caused by the elastic deformation of limb i’s member j, respectively; $\overline{\unicode{x1D6E5}}_{ijd}$ and $\overline{\unicode{x1D6E5}}_{ijr}$ represent the average values of the linear and angular displacement caused by the elastic deformation of limb i’s member j when the hybrid robot end is working along a specified trajectory. N represents the number of discrete points in the work trajectory.

The average linear displacement of the hybrid robot end is 5.037 × 10⁻⁵m when it works along the sinusoidal curve path mentioned above (Fig. 12a). Table 4 shows the average linear displacement of each elastic element (the linear displacement along the x-, y-, and z-axes directions can be listed separately if necessary). The linear displacement at the tool end of the hybrid robot is mainly caused by the serial robot arm rod FG, followed by rods B ₁ C ₁, oE, and GH. Therefore, the serial robotic arm rod FG should be improved first to improve the linear stiffness of the hybrid robot.

The following five situations are considered to quantitatively evaluate the effect of improving the stiffness performance of each elastic element on the stiffness of the terminal line of the hybrid robot: (1) use the original physical parameters (Table 2), (2) the outer diameter of the FG of the serial robotic arm rod is increased by 20%, and the remaining parameters remain unchanged, (3) the outer diameter of parallel-module rod B ₁ C ₁ is increased by 20%, and the remaining parameters remain unchanged, (4) the outer diameter of the oE of the serial robotic arm rod is increased by 20%, and the remaining parameters remain unchanged, and (5) the outer diameter of the GH of the serial robotic arm rod is increased by 20%, and the remaining parameters remain unchanged.

Table 5 lists the influence of the above five conditions on the linear displacement of the terminal of the hybrid robot working along the specified trajectory. The linear displacement in working conditions (2)–(5) is reduced by 12.71%, 9.61%, 7.27%, and 6.06%, respectively, compared with that in working conditions (1). Improving the FG stiffness performance of the serial robotic arm rod can most improve the linear stiffness of the hybrid robot. In other words, the method proposed in this paper allows for effective control of the stiffness performance of the robot when the external wrench is known. This enables the maximization of robot stiffness performance or lightweight design. However, a limitation of this method is that it cannot pinpoint the components that have the greatest impact on the robot’s stiffness performance when the external wrench is unknown.

Similarly, the average angular displacement of the hybrid robot end when working along the sinusoidal curve path is 6.382 × 10⁻⁵ m (Fig. 12b). Table 6 shows the angular displacement of each elastic element (the angular displacement generated around the x-, y-, and z-axes can be listed separately if necessary). The angular displacement of the tool terminal of the hybrid robot is mainly caused by serial robot arm rod FG, followed by rod GH, oE, and A ₂ B ₂. Therefore, serial robotic arm rod FG should be improved first to improve the terminal line stiffness of the hybrid robot.

Table IV. Average linear elastic deflection contribution of each component of the 4SRRR + 6R.

Note: $\overline{\Delta }_{ijd}$ and $\overline{\Delta }_{sjd}$ denote the average linear elastic deflection contribution of the i ^th limb’s j ^th member of parallel modules and the j ^th member of serial module, respectively, its unit is m.

Table V. Comparison of the effects of each elastic component on the resulting linear deflection of the 4SRRR + 6R.

Table VI. Average angular elastic deflection contribution of each component of the 4SRRR + 6R.

Note: $\overline{\Delta }_{ijr}$ and $\overline{\Delta }_{sjr}$ denote the average angular elastic deflection contribution of the i ^th limb’s j ^th member of parallel modules and the j ^th member of serial module, respectively, its unit is rad.

Table VII. Comparison of the effects of each elastic component on the resulting angular deflection of the 4SRRR + 6R.

The following five situations are considered to quantitatively evaluate the influence of each elastic element on the stiffness of the terminal angular of the hybrid robot: (1) use the original physical parameters (Table 2), (2) the outer diameter of the FG of the serial robotic arm rod is increased by 20%, and the remaining parameters remain unchanged, (3) the outer diameter of the parallel module rod GH increased by 20%, and the remaining parameters remained unchanged, (4) the outer diameter of the oE of the serial robotic arm rod is increased by 20%, and the remaining parameters remain unchanged, and (5) the outer diameter of the serial robotic arm rod A ₂ B ₂ increased by 20%, and the remaining parameters remained unchanged.

Table 7 shows the influence of the above five conditions on the angular displacement of the terminal of the hybrid robot working along the specified trajectory. The angular displacements of cases (2)–(5) are reduced by 23.69%, 7.87%, 5.26%, and 3.92%, respectively, compared with case (1). Improving the FG stiffness performance of the serial robotic arm rod can most improve the linear stiffness of the hybrid robot.Footnote 1