Scale synthesis refers to selecting appropriate performance indicators (kinematic, static stiffness, or dynamic performance indicators, etc.) after selecting a parallel mechanism configuration, and optimizing the size parameters of the mechanism members. Today, we will mainly elaborate on the scale synthesis based on kinematic performance indicators and briefly introduce the dynamic performance indicators.

The kinematic performance indicators of parallel mechanisms are mainly divided into two categories: one is based on the algebraic characteristics of Jacobian matrices, and the other is based on the motion/force transfer performance indicators of screw theory. For a long time, using the algebraic features of Jacobian matrices to construct performance indicators (such as condition numbers, operability, etc.) and conducting size synthesis has been a common method for evaluating and optimizing the design of parallel mechanisms. However, Merlet pointed out that for mixed degree of freedom parallel mechanisms, there is an inconsistency in the physical dimensions of each item in the Jacobian matrix. Therefore, the performance indicators constructed from this have defects such as unclear physical meanings, and cannot be directly used to evaluate the performance of the mechanism.

To solve the problem of inconsistent physical dimensions, scholars have proposed various methods to achieve the homogenization of Jacobian matrices.

• Angeles et al. used the concept of feature length to normalize the Jacobian matrix of the mechanism, but the limitation of this method is that the feature length of the mechanism is not constant throughout the entire motion process.
  
• Kim et al. established a Jacobian matrix with the same dimension between the linear velocity and driving velocity at each point on the moving platform, but the singular values based on this matrix still have unclear physical meanings.
  
• Pond et al. proposed a new homogeneous Jacobian matrix and compared and analyzed 3-PRS parallel mechanisms with different layouts using condition number indicators.
  
• In the case of a 3-PRS mechanism used for friction stir welding, Shi et al. analyzed the performance of the mechanism using the global dexterity index of a homogeneous Jacobian matrix, and combined spatial graphing method for size synthesis.
  

The performance index of motion/force transfer refers to the efficiency of energy transfer from the input end to the output end of the mechanism, and can also be used to evaluate the resistance of the driver to external loads. In previous studies, Yuan et al. and Sutherland et al. explored the transmission performance of mechanisms, however, they only considered the output transmission index (OTI) of the mechanism. Based on this, Liu et al. proposed the input transmission index (ITI), which comprehensively considers the impact of input/output transmission performance and defines the local transmission index (LTI) to describe the performance of a certain configuration of the mechanism.

Figure 1. LTI Performance Indicator Chart

The performance indicators of motion/force transfer have been widely applied in the performance analysis and optimization design of 2R1T parallel mechanisms.

• Xie et al. used these indicators to conduct detailed analysis and optimization design on some 2R1T parallel mechanisms, such as 2PUR-PRS and PSS-PRU-PU.
  
• For redundant drive parallel mechanisms, Li et al. used the method of gradually locking the drive branch for single degree of freedom mechanism combination, analyzed the mapping relationship between multiple drives and single output, proposed indicators to describe the motion/force transfer performance of redundant drive parallel mechanisms, and successfully applied them to 2R1T redundant drive parallel mechanisms.
  
• Recently, Xu et al. conducted performance evaluation and scale synthesis on a large number of 2R1T non redundant and redundant parallel mechanisms using motion/force transfer performance indicators.
  

Compared to the kinematic performance indicators, there is relatively less research on the dynamic performance indicators of parallel mechanisms. This is mainly because the dynamic modeling of parallel mechanisms is more complex, and early designs focused more on the evaluation of kinematic performance. Asada proposed the generalized inertial ellipsoid method in, which can describe the distribution of inertial characteristics of mechanisms in different postures by analyzing the relationship between kinetic energy and generalized velocity. By 1985, Yoshikawa introduced the Dynamic Manipulation Ellipsoid (DME) by deriving the mapping relationship between the generalized acceleration and generalized driving force of the mechanism, which was used to evaluate the ability of the mechanism's end effector to change its pose under a certain driving force. Subsequently, Yoshikawa subdivided the dynamic operability ellipsoid index into mobile dynamic operability and rotational dynamic operability, which is suitable for evaluating the dynamic performance of mixed degree of freedom mechanisms.

On the basis of DME indicators, researchers have carried out many application works. For example, Wu et al. used the DME index to analyze the dynamic performance of a two degree of freedom parallel mechanism, while Chen et al. used this index to evaluate the dynamic performance of a 3-PRRU parallel mechanism. Unlike the dynamic operability ellipsoid, Graettinger et al. proposed an acceleration radius to measure the acceleration performance of mechanisms. Wang et al. proposed a performance index for describing the time-varying coupling dynamics of mechanisms, which can be used to evaluate changes in dynamic loads. Recently, Liu et al. proposed a new indicator for evaluating the acceleration performance of high-speed mobile robots from the perspective of energy transfer.

Figure 2. Dynamic Operability Ellipsoid Index

(Left: rotational operability; Right: mobile operability)

The current dynamic performance evaluation methods mainly focus on considering the inertia matrix of the mechanism, but lack consideration for the comprehensiveness and accuracy of the mechanism. In addition, many proposed evaluation methods lack global indicators with reasonable explanations, making it difficult to determine the dynamic performance of different mechanisms.