## 4 Msa
Figure 5: 12-DOF beam element in MSA method [87]. For different types of connections, the stiffness matrix can be different. In Figure 6, rigid and elastic connections are used in the two-link serial system. A \(12\times 12\) Cartesian stiffness matrix \(K^{r}_{\mathbb{C}}\) for rigid connection can be deduced. This matrix \(K^{r}_{\mathbb{C}}\) is non-singular and invertible. When the connection is elastic, the Cartesian stiffness matrix \(K^{r}_{\mathbb{C}}\) is rank-deficient and singular. To efficiently apply the MSA method to serial and parallel robots, [PERSON] et al. [63] developed an approach to model robot stiffness with a set of two groups of matrix equations describing the elasticity of separate links and connections between links. The general components and structures in a robot were modelled with this approach. A three-degrees-of-freedom planar parallel manipulator was selected for demonstration. The proposed MSA-based approach was approved to be in good agreement with FEA modelling. A level of 10 ms of computational efficiency can be achieved for stiffness modelling per manipulator configuration. [PERSON] et al. viewed the original MSA elements as having a limitation that could be entirely represented by beam theory. Therefore, an extended program was developed to automatically calculate the global stiffness matrix and deformation in a given structure composed of different elements under external loads. In this method, the nonlinear stiffness of the complex joints, links, and rolling contact bearings were all considered. The nonlinear stiffness characteristics were explained through several iterations of the linear model. After conducting experiments on the reconfigurable modular multiarm robot system PARAGRIP, their model demonstrated effective deformation prediction in the X, Y, and Z directions. The Z-direction had the best prediction effect, which can be used to compensate for robot errors due to gravity [65]. [PERSON] et al. introduced a dynamic external load and inertial external wrench into the overall stiffness model of a parallel robot [94]. Moreover, some scholars have applied the MSA method to model the stiffness of serial robots. [PERSON] et al. proposed nonlinear flexible stiffness modelling of 6-DOF serial robots based on the MSA method, which achieved comparable accuracy to the FEA method. The model considered the flexible connecting link and the nonlinear characteristics of the joints as the posture changed. Finally, an energy index \(\delta D\) (2) was introduced to evaluate the precision of the model, with \(\mathbf{f}\) and \(\mathbf{u}\) representing the induced force vector or wrench and the corresponding displacement vector, respectively. Compared with the FEA method, the deformation energy index was less than 0.03% in the whole workspace. In addition, the calculation time was greatly reduced. However, actual measurement experiments were not performed, and the model was only applicable for robot error predictions in stationary states [62]. \[\delta D=\mathbf{f}^{\mathbf{T}}\mathbf{u} \tag{2}\]  
Compared with the FEA method, the MSA method has clear advantages when used with large flexible components. Many component elements are simplified to some degree, which effectively reduces the calculation cost and achieves a reasonable balance between  
Figure 6: Two types of connections in a two-link serial system [63]: (**a**) a rigid connection and (**b**) an elastic connection. calculation time and accuracy [83; 84]. Unlike the VJM method, the MSA method does not use differential equations. Therefore, when describing the complex structure of a robot with a large number of closed loops and crosslinks, the MSA method reduces the complexity of the calculation [95]. However, more details need to be considered in MSA models, such as the nontrivial actions of the passive and elastic joints [63].