Kinematics of Tree-Type Robotic Systems
Kinematic modeling of a tree-type robotic system is presented in this chapter. In order to obtain kinematic constraints, a tree-type topology is first divided into a set of modules. The kinematic constraints are then obtained between these modules by intr
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Kinematics of Tree-Type Robotic Systems
Kinematic modeling of a tree-type robotic system is presented in this chapter. In order to obtain kinematic constraints, a tree-type topology is first divided into a set of modules. The kinematic constraints are then obtained between these modules by introducing the concepts of module-twist, module-joint-rate, etc. This helps in obtaining the generic form of the Decoupled Natural Orthogonal Complement (DeNOC) matrices for a tree-type system with the help of module-to-module velocity transformations. Using the present derivation, link-to-link velocity transformation (Saha 1999b) turns out to be a special case of the module-to-module velocity transformation (Shah et al. 2012a) presented in this chapter.
4.1 Kinematic Modules Conventionally, a tree-type system is considered to have a set of links or bodies connected by kinematic pairs as shown in Fig. 4.1a. However, here a more generic approach is introduced, where the tree-type architecture is considered to have a set of kinematic modules (Shah et al. 2012a). Each module is defined as a set of serially connected links. This is shown in Fig. 4.1b, where the kinematic modules are depicted by dotted boundaries. For the purpose of analysis, the tree-type system is first modularized before its kinematic constraints are obtained. In order to modularize a tree-type system, a link in the tree-type system is first identified as its base, for example, link #0 in Fig. 4.1b, which may be fixed or floating. This is referred to as module M0 . Once the base module is established, the system is then modularized outward such that each module 1. contains serially connected links only; 2. emerges from the last link of its parent module. The first condition defines a module while the second one defines its connectivity with the adjoining modules. The resulting modules are indicated with M1 , M2 , etc. S.V. Shah et al., Dynamics of Tree-Type Robotic Systems, Intelligent Systems, Control and Automation: Science and Engineering 62, DOI 10.1007/978-94-007-5006-7 4, © Springer ScienceCBusiness Media Dordrecht 2013
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4 Kinematics of Tree-Type Robotic Systems
a
b Ms
A link or body
A kinematic module
Mi Mb M1
Base
0
Base
0
M0
Fig. 4.1 Tree-type architectures (a) Conventional (b) Multi-modular
It is worth noting that any tree-type system can be modularized such that it follows the above two conditions. For a given tree-type system, however, there may be several module architectures which obey the above conditions. This is evident from Fig. 4.2a, b which represent two different module architectures of the same treetype system shown in Fig. 4.1a. Moreover, if we consider each link in the tree-type system as one module then the resultant module architecture will follow the above two modularization conditions. As a result, Fig. 4.1a turns out to be a special case of the proposed architecture in Fig. 4.1b. Figure 4.2c shows another arbitrary way of modularization that satisfies only the first modularization condition, i.e., each module is
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