# Dynamic models of parallel robots in the form of multi-mass systems with stiffness coefficients

I want to develop a dynamic model of a parallel robot for control tasks. It is known that equations of motion based on Lagrange equations can be transformed into a state space and use such a model.

I was wondering if it is possible to somehow include the stiffness matrix of the manipulator in this model and get a model like the one shown in the second figure.

• We can use a nonlinear state-space.
– dtn
Jan 8 '20 at 13:02
• True but you can't put it in a matrix form. This is what I meant. Jan 8 '20 at 13:04
• I agree. But now, please help me deal with my task. The model has not yet planned non-linear elements (friction, gaps, etc.).
– dtn
Jan 8 '20 at 13:39
• So you only want to model it ? Jan 8 '20 at 14:14
• Yes, for now, I only need a model, but in the form of a multi-mass structural diagram.
– dtn
Jan 8 '20 at 14:32

The equations of motion is simply \begin{align} J_m \ddot{\theta}_1 + K_{md}(\theta_1-\theta_2) &= \tau_e \tag{1} \\ J_d \ddot{\theta}_2 + K_{md}(\theta_2-\theta_1) &= \tau_L + \tau_s \tag{2} \end{align} Rewriting (1)&(2), we get $$\begin{bmatrix} J_m & 0 \\ 0 & J_d \end{bmatrix} \begin{bmatrix} \ddot{\theta}_1 \\ \ddot{\theta}_2 \end{bmatrix} + \begin{bmatrix} K_{md} & -K_{md} \\ -K_{md} & K_{md} \end{bmatrix} \begin{bmatrix} \theta_1 \\ \theta_2 \end{bmatrix} = \begin{bmatrix} \tau_e \\ \tau_L + \tau_s \end{bmatrix}$$

• Since this is a linear system, it is not difficult to compose its equations. I am interested in a more specific question: how to relate the stiffness matrix to the Lagrange equation and obtain on this basis a similar multi-mass structural diagram. Note the article: Stiffness Analysis and Comparison of 3-PPR Planar Parallel Manipulators With Actuation Compliance Guanglei Wu, Shaoping Bai, Jørgen A. Kepler Thanks again for the answer.
– dtn
Jan 8 '20 at 14:40
• the Lagrange equation is nothing but an approach to derive the equations of motion. I've used Newtonian approach because it is simple in this case. I don't know what you're trying to do. Jan 8 '20 at 14:44
• Also, note that the block diagram is clearly for the transfer function. Jan 8 '20 at 14:45
• I agree, this block diagram can be converted into a transfer function. CroCo, as far as I know, a special stiffness matrix is being built for parallel robots, which characterizes the relationship of the applied forces and moments with small deformations of rotation and displacement. I want to understand how to combine this matrix with the equations of motion of a parallel robot.
– dtn
Jan 8 '20 at 14:54
• If you are interested, I draw on ideas from this article: Mobile platform center shift in spherical parallel manipulators with flexible limbs See paragraph 3.1. Cartesian Stiffness Matrix
– dtn
Jan 8 '20 at 14:56

Yes it is possible.

You can look at the following paper

Stiffness Analysis and Comparison of 3-PPR Planar Parallel Manipulators With Actuation Compliance Guanglei Wu , Shaoping Bai , Jørgen A. Kepler

• Unfortunately, this work did not clarify the situation, but only created more questions. In the works of this team, the stiffness matrix includes coefficients related to rotational deformations and to displacement deformations. In the two-mass model that I want to do, the stiffness coefficient is the stiffness of the shaft connecting the two big masses.
– dtn
Jan 8 '20 at 9:01
• Is this coefficient for a parallel robot taken from the matrix? And then what is its masses? Two large masses connected by an elastic element are taken in these schemes, and a model is built on this basis. And in this article, stiffness applies to the body itself (from one of the parts of the link). And so it is not clear how the circuit for a parallel robot will look if it has large massive links and stiffnesses that are not separated from each other. Nevertheless, thank you for your help.
– dtn
Jan 8 '20 at 9:01
• P.S. Sorry for the amateurish questions, I am not in its pure form a professional robotics and mechanic.
– dtn
Jan 8 '20 at 9:01