Anxin Liu

Position and Orientation Characteristic Equation for Topological Design of Robot Mechanisms

This paper presents the explicit mapping relations between topological structure and position and orientation characteristic (POC) of mechanism motion output. It deals with (I) the symbolic representation and the invariant property of the topological structure of the mechanism, (2) the matrix representation of POC of mechanism motion output, and (3) the POC equations of serial and parallel mechanisms and the corresponding symbolic operation rules. The symbolic operation involves simple mathematic tools and fewer operation rules and has clear geometrical meaning, so it is easy to use. The POC equations cannot only be used for structural analysis of the mechanism (such as determining POC of the relative motion between any two links of a mechanism and the rank of single-loop kinematic chain and calculating the full-cycle DOF of a mechanism, etc.) but can be used for structural synthesis of the mechanism as well (e.g., structural synthesis of the rank-degenerated serial mechanism, the over constrained single-loop mechanism, and the rank-degenerated parallel mechanism, etc.).

Topological Structure Synthesis of 3T1R Parallel Mechanism Based on POC Equations

A systematic method for topological structure synthesis of PM based on POC equations is introduced. The complete synthesis process includes the following several steps: (a) synthesize all candidate SOC and HSOC branches based on the POC equation for serial mechanisms, (b) determine branch combination schemes and geometrical conditions for branches to be assembled on two platforms based on POC equation for PMs, (c) check the obtained PMs for design requirements and obtain all usable PMs. Topological structure synthesis of 3T1R PM which can be used in SCARA robot is discussed in detail to illustrate to procedure of this method. 18 types of 3T1R PMs containing no prismatic pair are obtained.

Comparison Study on Three Approaches for Type Synthesis of Robot Mechanisms

Basic principles and main characteristics of three approaches for structure synthesis of robot mechanisms (the screw theory-based approach, the displacement subgroup-based approach and the approach based on position and orientation characteristic (in short, POC) ) are studied and compared in this paper. The comparison deals with the mathematical tools, the symbolic representation of mechanism topological structure, the mathematical representation of POC of the output motion link with respect to the frame link, the basic equations for structure synthesis of serial and parallel mechanisms and relevant operation rules, and the characteristics of their synthesized mechanisms, etc. This comparative study shows that the POC-based approach is totally different from the other two approaches: (1) the POC-based approach requires only simple mathematical tools (such as vector algebra, set theory, etc), (2) the POC-based approach is conceptually simpler and therefore easier to understand and to use, and (3) the POC-based approach is more general.Copyright

Structure Synthesis of Rank-Degenerated Serial Mechanisms and Over-Constrained Single-Loop Kinematic Chains

Based on the Position and orientation characteristic (POC) equation of serial mechanisms proposed by the author, this paper presents a novel systematic approach for structure synthesis of rank-degenerated serial mechanisms and over-constrained single-loop kinematic chains (KCs) (excluding the Bennett mechanism etc). This approach is totally different from the approaches based on the screw theory and based on the displacement subgroup, and only simple mathematical tools (vector algebra, etc.) are used. Using this approach, the structure types of serial mechanisms with the specified ranks and the specified degree of freedom (DOF) are synthesized firstly. After that, using the structure types of the obtained serial mechanisms, structure types of over-constrained single-loop KCs with the specified ranks and the specified DOF can be generated in a straightforward way. The structure types of the obtained serial mechanisms can be used as branches of parallel mechanisms. The structure types and the ranks of the obtained over-constrained single-loop KCs can be used to calculate the DOF of multi-loop mechanisms. In fact, the systematic approach proposed in this paper is a key component of the systematic approach for structure synthesis of parallel mechanisms.Copyright

Ordered Structure and the Coupling Degree of Planar Mechanism Based on Single-Open Chain and Its Application

A new viewpoint on structural composition of mechanism is proposed in this paper: any mechanism can be decomposed into a group of ordered single-open-chains (SOC). Based on the ordered SOC and its constraint property, the following concepts or principles are proposed or established: (1) the criteria for determination of basic kinematic chain (BKC). (2) the coupling degree of BKC, (3) the criteria for determination of types of degree-of-freedom, (4) the formula for calculation of number of mechanism configurations, and (5) a new unified modular method for structural, kinematic and dynamic analysis of mechanism.Copyright

Topology Design of (3T-1R) Parallel Mechanisms

Topology design of (3T-1R) parallel mechanism (PM) based on POC equations is discussed in detail as an example to illustrate the systematic method for topology design of PMs proposed in Chap. 9. The design process includes two stages. The first stage involves traditional structure synthesis, which results in 18 structure types of (3R-1R) PMs that contain no P pairs. The second stage includes topological performance analysis and classification of the obtained PM structure types which may help designer to select the most suitable structure type. Its core contents include: (1) Key steps of topology design for (3R-1R) PMs include: (a) Determine structure types of the SOC branch containing only R and P pairs according to the method for structure synthesis of SOC branch described in Chap.8. Then obtain corresponding HSOC branches through topologically equivalent principle in Chap.9. (b) Determine the geometrical condition for assembling branches between two platforms according to corresponding basic formulas in Chap. 9. Then determine different assembling schemes for the same geometrical condition. (c) Performance analysis and classification of the obtained PM structures are conducted based on topological characteristics of PMs in Chap. 9. The result can be used for assessment and optimization of (3R-1R) PM structure types. (2) 18 types of (3T-1R) PM structures containing no P pair are obtained, two of which have already been used in existing SCARA parallel robots. The other structure types may potentially be used in new SCARA parallel robots. (3) These 18 structure types of (3R-1R) PMs are classified into four types: Type-1 include four PMs containing only SOC branches; Type-2 include six PMs containing only HSOC branches; Type-3 include seven PMs containing both HSOC and SOC branches; Type-4 include three PMs with partial motion decoupling property. (4) According to topological performance analysis and classification of these PMs based on topological characteristics, the type-3 PM has a higher rigidity than a type-1 PM and a larger workspace than a type-2 PM. So, special attentions shall be paid to type-3 PMs.

Topological Structure of Mechanisms and Its Symbolic Representation

Topological structure of mechanisms and the symbolic representation are introduced in this chapter. The content deals with: (1) A new element for describing topological structure - geometric constraint type (the type of geometrical constraints to pair axes imposed by links, is proposed. This new element and the other two traditional elements (type of kinematic pair, connection relation between links) formed the three basic elements of topological structure. (2) The mechanism topological structure described using these three basic elements and the corresponding symbolic representation are independent of motion position of the mechanism and the fixed coordinate system. This feature is called the topological structure invariance during motion process. (3) The symbolic representation of topological structure and the topological structure invariance could be used for establishing the position and orientation characteristics (POC) equation for serial mechanisms, the POC equation for parallel mechanisms (PMs) and their operation rules (refer to Chaps. 4– 5). (4) Any serial mechanisms or multi-loop spatial mechanisms can be generated by connecting single-open-chain (SOC) in parallels or in series. The SOC unit is a new structure unit of mechanisms proposed by authors, which could be used for establishing the POC equation for serial mechanisms (refer to Chap. 4), the POC equation for PMs (refer to Chap. 5), the DOF formula (refer to Chap. 6) and the coupling degree formula (refer to Chap. 7).

Position and Orientation Characteristics Set

The position and orientation characteristics (POC) set is proposed by authors in this chapter, which is used for describing the POCs of the mechanism. The content deals with: (1) Based on the velocity analysis and the unit vector set of velocity of kinematic pairs, this unit vector set can be rewritten into the form of velocity characteristic (VC). Hence, the VC set are defined, which is used to express the velocity characteristic of the kinematic pairs. Subsequently, based on the topological structure invariance, the POC set are defined, which is used to describe the POCs of the kinematic pairs. (2) Based on the velocity analysis, the unit vector set of velocity of mechanism link is used for describing its velocity characteristics. Since this unit vector set can be rewritten into the form of velocity characteristic, the VC set is defined. Subsequently, based on one-to-one correspondence between elements of the POC set and elements of the VC set, the POC set of the mechanism is defined. Both VC set and POC set depend only on topological structure of the mechanism. (3) The dimension of the POC set is not greater than the DOF of a mechanism. There are only 15 basic types of the POC sets containing only independent elements. (4) The POC set could be used for establishing the POC equation for serial mechanisms (refer to Chap. 4) and the POC equation for PMs (refer to Chap. 5).

Mechanism Composition Principle Based on Single-Open-Chain Unit

The mechanism composition principle based on Single-Open-Chain (SOC) unit is introduced in this chapter. The content deals with: (1) The mechanism composition principle based on SOC unit is proposed. Constraint degree of SOC is defined, which is used to describe constraint features of SOC unit to DOF of the mechanism. (2) A general method for decomposing an AKC into ordered SOCs is proposed. The coupling degree of an AKC is defined and is used to represent complexity of the AKC. (3) A general method for decomposing mechanism into ordered SOCs is proposed. Criteria for AKC determination is defined and is used to determine AKCs contained in the mechanism. (4) A general method for determination of DOF types is proposed based on the AKCs in the mechanism. And a general method for motion decoupling design based on partial DOF is proposed. (5) This mechanism composition principle could be used to establish a systematic theory for unified modeling of mechanism topology, kinematics and dynamics based on SOC unit (refer to Sect. 7.7).

General Formula of Degrees of Freedom for Spatial Mechanisms

The general DOF formula which is of great importance to mechanism topology, kinematics and dynamics is introduced in this chapter. The content deals with: (1) The general DOF formula for PMs is proposed based on the topologically equivalent principle, POC equation of serial mechanisms in Chap. 4 and POC equation of PMs in Chap. 5. This DOF formula is independent of motion position of a PM and it is not necessary to establish the fixed coordinate system. Hence, it can guarantee that full-cycle DOF is obtained. (2) Based on the DOF formula, the criteria for determination of inactive pairs and the criteria for selection of driving pairs are proposed. (3) With replacement method for topological equivalent kinematic chains, DOF of general multi-loop spatial mechanisms can be calculated with this DOF formula very easily. (4) This DOF formula can be used for calculating DOF of PMs whose POC set contains non-independent element. (5) Based on the DOF formula, formulas for determining number of independent displacement equations, number of over-constraints and redundancy of mechanisms are derived. In brief , this DOF formula reveals the internal relations among topological structure, DOF and POC set of spatial mechanisms and will be used for topology design of PMs (refer to Chap. 9).