A theory of degrees of freedom for mechanisms

doi:10.1016/j.mechmachtheory.2003.12.005

Mechanism and Machine Theory

Volume 39, Issue 6, June 2004, Pages 621-643

https://doi.org/10.1016/j.mechmachtheory.2003.12.005 Get rights and content

Abstract

In this paper, we put forward a stricter and more complete theory of degrees of freedom (DOF) for mechanisms, especially for the complex spatial mechanisms, which may not be solved correctly with traditional theories. We also point out that it is more appropriate to calculate the DOF of the mechanism with an output member rather than that of the whole mechanism. Furthermore, the new concept of configuration degrees of freedom (CDOF) in this theory can form a more complete and reasonable theoretical base for analyzing the mobility, singularity and stability of the mechanisms.

Introduction

The DOF of a mechanism is the first consideration in the study of kinematics and dynamics of mechanisms [1, p. 66]. As a matter of fact, the DOF of any mechanism should be determined before it is designed, and then the drive and control can be determined. Many scholars have brought forward various theories about the analysis and calculation of the DOF of mechanism [1], [2], [3], [4], [5], [6], [7]. However, on the part of the existing theories and formulas, all of them are right only under certain circumstances. In the case of unusual mechanisms, the existing theories may lead to a completely wrong result. For example, these theories and methods are valid in solving the problems with the planar and simple spatial mechanisms, but when they are applied to certain special mechanisms, for example, the 4-PTT spatial mechanism shown in Fig. 1 and the 3-PTT spatial mechanism shown in Fig. 2, they all lead to a wrong result, which is quite different from the reality.

In this paper, therefore, we suggest a stricter and more complete theory system of DOF that can be used to analyze and compute the DOF of all kinds of mechanisms. This new theory not only includes the right part of all kinds of the traditional theories of DOF, but also includes the analysis and calculation problems of the DOF in many complex spatial mechanisms, which may not be addressed by traditional theories. In the theory, we also point out that it is more appropriate to calculate the DOF of the mechanism with an output member rather than the DOF of the whole mechanism. Besides, the concept of configuration degree of freedom (CDOF) is put forward, which forms a more complete and reasonable theoretical base to analyze the mobility, singularity and stability of a mechanism synchronously.

Section snippets

Problems existing in the traditional theories of DOF

The universally accepted theory of DOF is that of Kutzbach Grübler [4, pp. 13 17 19]. Later, Hunt [2, pp. 34–35 376 382], Dobrovolski [5, p. 12] and etc. [7, p. 20], [1, pp. 66–94] introduced new methods to analyze the DOF of the mechanism. In the following, we will first analyze the DOF of the mechanisms shown in Fig. 1, Fig. 2 with the traditional theories of DOF.

New theory of DOF

In order to avoid the problems existing in the traditional methods, we will first introduce the conception of constraints spaces in this paper. The definition is:

The constraints spaces, S_C, of a rigid body are the linear spaces spanned by all of the constraints applied by its neighbors.

For example, the constraints spaces of a fixed rigid body can be denoted as: $S_{fixed}^{C} = T_{x} T_{y} T_{z} R_{x} R_{y} R_{z},$ where S_fixed^C––the constraints spaces of a fixed rigid body, T_x––the unitary vector of the translational

Proof of new DOF theory

In order to accurately depict the position and orientation of any spatial rigid object, only six independent variables are needed. That is, the DOF of any spatial rigid object is 6. According to the screw theory [2], [7], [8], the inverse screws of kinematic screws are a set of general forces, including three orthogonal forces and three orthogonal moments of couples. As a result, the constraints spaces that the inverse screws applied by the n kinematic chains can be spanned are obtained. And

Analyze the DOF of the spatial four-bar mechanism with link BC as the output member

The spatial four-bar mechanism, shown in Fig. 4, is made up of 2-RR (2 rotational joints) kinematic chains. To study the DOF of the mechanism with an output member––link BC, we can first decompose the spatial parallel mechanism to form two kinematic chains connecting link BC with the base and then study the inverse screws of each kinematic chain.

Because the two kinematic chains connecting the link with the base are identical in isomorphism, we can only select one of them, say, member AB to

Conclusion

By uniformly depicting the kinematic and geometrical constraints of mechanism with screw theory, we put forward a new theory system of DOF that should be widely used to study the systematic DOF of spatial parallel mechanisms with an output member and that can solve the analysis and calculation problems of the DOF in many complex spatial mechanisms, which may not be correctly solved with traditional theories. Besides, the new concept of CDOF in this theory can form a more complete and reasonable

References (8)

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There are more references available in the full text version of this article.

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