Qinchuan Li

Theory of parallel mechanisms

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Basics of Screw Theory

Screw theory is a powerful mathematical tool for the analysis of spatial mechanisms. A screw consists of two three-dimensional vectors. A screw can be used to denote the position and orientation of a spatial vector, the linear velocity and angular velocity of a rigid body, or a force and a couple, respectively. Therefore, the concept of a screw is convenient in kinematics and dynamics, while the transformation between the screw-based method and vector and matrix methods is straightforward. When applied in mechanism analysis, screw theory has the advantages of clear geometrical concepts, explicit physical meaning, simple expression and convenient algebraic calculation. It is worth noting that the preliminary requirements for screw theory are only linear algebra and basic dynamics in undergraduate level. Thus, screw theory has been widely applied and researchers have used screw theory to make great contribution to many frontier problems in mechanism theory.

Special Configuration of Mechanisms

This chapter resumptively introduces our studies on the singularity of parallel mechanisms for the Stewart manipulator and the 3-RPS mechanism. It analyzes the singular kinematic principle and the singularity classification based on the kinematic status of the machinery and the linear-complex and focuses on discussing the structure and property of the singularity loci of 3/6- and 6/6-Stewart platform for special and general orientations. The singularity of the 3-RPS mechanism is also discussed in the latter parts. Many interesting properties, such as the remarkable intersection of all six segments of the six legs of the 6/6-Stewart platform with one common line, are discovered.

Full-Scale Feasible Instantaneous Screw Motion

This chapter presents a study on the full-scale instant twist motions of parallel manipulators. The study aims to understand and correctly apply a mechanism and is based on the principal screws of the screw system. The key problem is to derive three principal screws from a given 3-DOF mechanism, and then set the relation between the pitches of the principal screws and the three linear inputs of the mechanism.

Dependency and Reciprocity of Screws

This chapter introduces the concept of a screw system and focuses on second-order and third-order screw systems, especially their various special forms. The linear dependency of screws is discussed. The principle of Grassmann line geometry and screw dependency for different geometrical spaces are also introduced. Near the end of the chapter, the concept and analysis of the reciprocal screw and constraint motion are presented.

Dynamic Problems of Parallel Mechanisms

This chapter introduces our research on some dynamics problems of parallel mechanisms. The first problem is about the over-determinate inputs. This is quite an interesting issue. In practice, there are many machines and animals that work with over-determinate input, i.e., their input-number is much bigger than their mobility number. How to set the inputs to be accordance and optimum distribute and to obtain the expectant motion acceleration is a challenge. For the second part of this chapter, we focus on the dynamic analysis, i.e., the kinetostatic analysis of parallel mechanisms. For a link with two revolute pairs, based on its free-body diagram, its unknown value is 10 for the force analysis, and each link has only six equilibrium equations in the spatial mechanism. As it is, this is insolvable directly. Some time more unknown values may appear; and even up to 130 and it needs to set a 130-order matrix for the 5-5R parallel mechanism. This is extremely difficult. To resolve this issue, we propose a new method based on the screw theory. This method will only require the setting of a six-order matrix each time the dynamics problem can be readily solved. Moreover, in the following examples we can find the screws, their reciprocal screws, and their corresponding transformations each other, these are very interesting.

Digital Topology Theory of Kinematic Chains and Atlas Database

This chapter introduces an original theory of loop algebra and its application in isomorphism identification, rigid sub-chain detection, and atlas database of kinematic chains. Introduced first is the unified topology modeling of planar kinematic chains with simple joints, multiple joints, and geared (cam) joints. Based on the array representation of loops in the topological graphs of kinematic chains, basic loop operations are introduced, and the loop algebra is established. The most important problem of isomorphism identification in the automatic structural synthesis of kinematic chains is presented by finding a unique representation of topological graphs. Finally, the digital atlas database for the topological graphs of kinematic chains is also provided.

Kinematic Influence Coefficient and Kinematics Analysis

The concept of kinematic influence coefficient (KIC) of mechanism was proposed by Tesar et al. [1–4]. Benedict and Tesar [1, 2] proposed a completely general model formulation using first- and second-order KIC. This theory has been extensively applied to both open-loop and closed-loop planar mechanisms [3]. Thomas and Tesar [4] further developed this theory into a spatial serial manipulator. Huang [5, 6] has further developed to modern parallel mechanisms.

Mobility Analysis Part-2

In this chapter we will introduce the mobility analysis by using our mobility principle based on reciprocal screw theory. First, we discuss some simple mechanisms, and then focus on the mobility-open-issue mechanisms including the classical mechanisms and modern parallel mechanisms with interesting characteristics. Besides, more complex mechanisms, such as the Multi-loop-coupling mechanisms, are also discussed.

Computer-Aided Edge Loop Theory of Kinematic Structure of Mechanisms and Its Applications

Based on the edge-based array representation of loops in the topological graphs of kinematic chains, this paper first proposes three arithmetic operations of loops. Then the concept of the independent loop set as well as it determination rules is introduced, and a new structure decomposition algorithm of kinematic chains is presented. Based on the algorithm, an automatic and efficient method for rigid subchain detection and driving pair selection of kinematic chains is proposed. Finally, an index is proposed to assess computation complexity of kinematic analysis with respect to different driving pair selections.Copyright