Dynamics of Tree-Type Robotic Systems
As reviewed in Chap. 2, Newton-Euler (NE) equations of motion are found to be popular in dynamic formulations. Several methods were also proposed by various researchers to obtain the Euler-Langrage’s form of NE equations of motion. One of these methods is
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Dynamics of Tree-Type Robotic Systems
As reviewed in Chap. 2, Newton-Euler (NE) equations of motion are found to be popular in dynamic formulations. Several methods were also proposed by various researchers to obtain the Euler-Langrage’s form of NE equations of motion. One of these methods is based on velocity transformation of the kinematic constraints, e.g., the Natural Orthogonal Complement (NOC) or the Decoupled NOC (DeNOC), as obtained in Chap. 4. The DeNOC matrices of Eq. (4.28) are used in this chapter to obtain the minimal order dynamic equations of motion that have several benefits.
5.1 Dynamic Formulation Using the DeNOC Matrices The DeNOC-based methodology for dynamic modeling of a general multibody system, be it serial, tree-type or closed-loop, begins with the uncoupled NewtonEuler (NE) equations of motion of all the links constituting the system. The NE equations of motion are first formulated in a matrix-vector form for a serial kinematic module followed by a tree-type system consisting of the modules.
5.1.1 NE Equations of Motion for a Serial Module Let us consider the ith module of Fig. 4.6. The ith module contains i serially connected links. The NE equations of motion for the kth link of the ith module can be written as (Greenwood 1988) Ick ¨ P k C ¨k Ick ¨k D nck mk cR k D fck
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 5, © Springer ScienceCBusiness Media Dordrecht 2013
(5.1)
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5 Dynamics of Tree-Type Robotic Systems
Fig. 5.1 Motion of link ki in module Mi
Ck: Center of mass of the kth link Ok: Origin of the kth link
Ck dk
. ok
a #k
k Ok fk
rk
nk
wk
where nCk and fCk are the resultant moment about and force applied at the Centre of Mass (COM), Ck , whereas, ICk is the inertia tensor about Ck , and mk is the mass of kth link. Moreover, cR k is the linear acceleration of Ck and ¨k is the angular velocity of the link. It is worth mentioning that the main objective of dynamic analysis is to calculate either joint torques or joint motions. Hence, if the origin of a link, which lies on the joint axis, is selected as a reference point, instead of the COM, efficient recursive inverse and forward dynamics algorithms can be obtained. This was also shown by Stelzle et al. (1995). In order to represent Eq. (5.1) with respect to the origin, Ok , of the kth link, Fig. 5.1, the entities cR k ; ICk ; fCk and nCk , respectively, are represented in terms of the linear acceleration of origin Ok , i.e., oR k , the inertia tensor about Ok , namely, Ik , the resultant force applied at Ok , fk , and the resultant moment about Ok , nk , as cR k D oR k dk ¨ P k ¨k .dk ¨k / ICk D Ik C mi .dk 1/.dk 1/ fck D fk
(5.2)
nck D nk dk fk where dk is the 3-dimensonal vector from the origin of the kth link to its COM, as shown in Fig. 5.1, whereas dk 1 is the 3 3 cross-product tensor associated with dk [d1 d2 d3 ]T . The product of dk 1 with any vector x gives the
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