The stiffness study of parallel robots is an important field in robotics. Stiffness refers to the ability of a robot to resist deformation when subjected to external forces. Stiffness is particularly important for parallel robots as it directly affects the accuracy, dynamic performance, and load capacity of the robot. Next, we will delve into the definition, classification, and modeling methods of stiffness。

1.Definition and classification of stiffness

Static stiffness: refers to the ability of a robot to resist external forces in a specific position and direction under static conditions. Usually analyzed by calculating the Jacobian matrix and the structural parameters of the robot.

Dynamic stiffness: It involves the stiffness characteristics of robots during dynamic operation, which is usually more complex as it requires consideration of inertia, damping, and other dynamic factors.

Figure 1 Large deformation flexible linkage parallel mechanism

2. Stiffness modeling method

In order to predict the stiffness characteristics of robots under different positions, postures, and load conditions, we need to establish a mathematical model of the robot system. The following are several commonly used stiffness modeling methods:

2.1 Jacobian Matrix Method

The Jacobian matrix method is used to calculate the stiffness model of parallel robots through the kinematic relationships of the robots. Firstly, by establishing a Jacobian matrix between the robot joint space and the task space, the relationship between joint motion and end effector motion is described. Then, map the stiffness matrix of the joint space to the task space and calculate the stiffness matrix of the task space through a formula.

Among them is the Jacobian matrix, which is the stiffness matrix of the joint space.

Advantages: This method is computationally simple and suitable for real-time analysis.

Disadvantage: Assuming the stiffness matrix is a constant, it cannot accurately reflect the nonlinear effects of complex structures or materials.

2.2 Elastic Rod Theory

The stiffness modeling method of elastic rod theory treats each rod in a parallel robot as an elastic body and analyzes the influence of their elastic deformation on the overall stiffness under stress. The stiffness of each member is determined by its material properties (such as elastic modulus), geometric shape (such as cross-sectional area and moment of inertia), and length, including axial, bending, and torsional stiffness. By using the principle of virtual work and elastic mechanics formulas, establish the local stiffness matrix of each member, and then transform it into the global coordinate system through coordinate transformation. Overlay the stiffness matrices of all members to form the global stiffness model of the entire robot. This model can be used to analyze the stiffness characteristics of parallel robots under different working conditions, in order to optimize design or control strategies.

Advantages: Suitable for stiffness analysis of complex structures and can consider the nonlinear behavior of materials.

Disadvantages: Complex calculation, especially in multi degree of freedom robots, with a large amount of computation.

Figure 2 Elastic Rod Model

2.3 Finite Element Analysis (FEA)

Finite Element Analysis (FEA) calculates the stiffness model of a parallel robot by discretizing the robot structure into multiple finite element elements, performing local stiffness analysis on each element, and then integrating the stiffness of these elements into the overall model. Specifically, the material properties and geometric characteristics of each unit are used to construct its stiffness matrix, and then the global stiffness matrix of the entire robot is generated by assembling the stiffness matrices of all units.

Advantages: Very precise, able to consider material heterogeneity and geometric complexity, suitable for high-precision analysis.

Disadvantage: Requires a large amount of computing resources and is not suitable for real-time applications.

Figure 3 Finite Element Analysis

The stiffness research of parallel robots is a complex and delicate field, which not only requires us to have a deep understanding of the structure and materials of the robots, but also requires us to use advanced mathematical tools and calculation methods. Through precise stiffness modeling, we can optimize the design of parallel robots, improve their performance, and enable them to play a greater role in industrial, medical, scientific research, and other fields. In the next article, we will continue to explore another important aspect of stiffness research for parallel robots - stiffness optimization design. Coming soon!

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