## 4 Msa  
The MSA method is a common technical method in the machinery industry. When analysed from the perspective of structural mechanics, a link of a robot can be regarded as a single beam structural unit (Figure 4). It assumes that the materials and structures of the links have perfectly linear behaviour, and dynamical effects are negligible. The deformation \(\Delta t\) at the free end under an external load (force/torque) \(\mathbf{W}\) can be defined as (1). \[\mathbf{W}=\mathbf{K}\cdot\Delta t \tag{1}\]  
where \(\mathbf{K}\) is the \(6\times 6\) stiffness matrix. \(\Delta t\) is a 6-dimensional deformation vector including three translational displacements \(\Delta\mathbf{p}=\left[\Delta\mathbf{p}_{x},~{}\Delta\mathbf{p}_{y},\Delta\mathbf{p}_{z} \right]^{T}\) and three rotational angles \(\Delta\mathbf{\varphi}=\left[\Delta\mathbf{\varphi}_{x},~{}\Delta\mathbf{\varphi}_{y}, \Delta\mathbf{\varphi}_{z}\right]^{T}\). Correspondingly, the external load W is also a six-dimensional vector containing the force \(\mathbf{F}=\left[\mathbf{F}_{x},~{}\mathbf{F}_{y},\mathbf{F}_{z}\right]^{T}\) and torque \(\mathbf{\mathrm{M}}=\left[\mathbf{M}_{x},~{}\mathbf{M}_{y},\mathbf{M}_{z}\right]^{T}\). For typical beams with regular cross-sections, the stiffness matrix \(\mathbf{K}\) can be calculated analytically with information on the beam properties (beam cross-section area and Young's modulus and shear modulus of the beam material). When beam units are combined by joints to build a robot structure, the stiffness matrix \(\mathbf{K}\) can be obtained based on the static equilibrium of the structure. Furthermore, joint link inertia and transverse and axial deformations must be considered [62]. Subsequently, considering the rotation matrix and boundary condition constraints between each beam unit, the local stiffness matrices can be assembled into a global stiffness matrix for the robot [90; 91; 92]. Based on the existing research results and the characteristics of the MSA method, it has been widely used in the stiffness modelling of parallel robots. [PERSON] et al. used the MSA method to model the stiffness of the [PERSON] parallel robot for the first time. Two assumptions were made in the experiments. First, the connecting links of the robot were only subjected to tensile deformation without bending. The second was that the stiffness of the robot frame was much stronger than that of the connecting link. The experimental results demonstrated the effectiveness of the MSA method for stiffness modelling [93]. For further study, [PERSON] et al. first obtained the stiffness matrix of a 6-RSS parallel robot by considering the stiffness characteristics of elements such as the connecting links, actuators, and joints. Additionally, the stiffness of the joints and brakes were ignored by this simplified model [21]. To investigate the influence of joint stiffness, [PERSON] et al. established two PKM stiffness models based on MSA. The first model assumed that the robot's joints were rigid and only considered the deformation of the connecting links, while the second model considered the deformation of both the robot's connecting links and joints. Load experiments showed that the second model was closer to the true elastic deformation of the robot [63]. [PERSON] et al. also considered the flexibility of the robot's connecting links and joints. The MSA method was applied to derive a high-precision static stiffness model of a 9-DOF redundant and reconfigurable 3 \(\times\) PPPRS parallel mechanism [91]. Based on the assumption of 12 DOFs for each link (Figure 5), [PERSON] et al. proposed an MSA model of a complex EAST Articulated Maintenance Arm (EAMA). The model matrix was simplified based on boundary constraints. Compared with the FEA method, the prediction error of this method was less than 5%, which is within the acceptable prediction error range, and the computational efficiency was improved [87]. Figure 4: Deflections and wrenches of beam structural units [63]: (**a**) cantilever beam and (**b**) unsupported beam.