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Euler-Angle-Joints (EAJs)

As frequently noted in the literature on robotics (Sugihara et al. 2002; Kurazume et al. 2003; Vukobratovic et al. 2007; Kwon and Park 2009) and mechanisms (Duffy 1978; Chaudhary and Saha 2007), a higher Degrees-of-Freedom (DOF) joint, say, a universal, a cylindrical or a spherical joint, can be represented using a combination of several intersecting 1-DOF joints. For example, a universal joint also known as Hooke’s joint is a combination of two revolute joints, the axes of which intersect at a point, whereas a cylindrical joint is a combination of a revolute joint and a prismatic joint. Similarly, the kinematic behavior of a spherical joint may be simulated by the combination of three revolute joints whose axes intersect at a point. The joint axes can be represented using the popular Denavit and Hartenberg (DH) parameters (Denavit and Hartenberg 1955). For the spherical joints, an alternative approach using the Euler angles can also be adopted, as there are three variables. For spatial rotations, one may also use other minimal set representations like Bryant (or Cardan) angles, Rodriguez parameters, etc. or non-minimal set representation like Euler parameters, quaternion, etc. The non-minimal sets are not considered here due the fact that the dynamic models obtained in this book are desired in minimal sets. The minimal sets, other than Euler/Bryant angles, are discarded here as they do not have direct correlation with the axis-wise rotations. It is worth mentioning that the fundamental difference between the Euler and Bryant angles lies in a fact that the former represents a sequence of rotations about the same axis separated with a rotation about a different axis, denoted as ˛–ˇ–˛, whereas the latter represents the sequence of rotations about three different axes, denoted as ˛–ˇ–”. They are also commonly referred to as symmetric and asymmetric sets of Euler angles in the literature. For convenience, the name Euler angles will be referred to both Euler and Bryant angles, hereafter. Euler Angles are defined as rotations about three orthogonal axes, which may be identified using DH parameters. However, the only difficulty is that, under the DH parameter scheme the variable rotations always occur about Z axis, whereas the Euler Angles are defined by rotation about all three axes, i.e., X, Y and Z. In

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 3, © Springer ScienceCBusiness Media Dordrecht 2013

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3 Euler-Angle-Joints (EAJs)

this chapter, a method will be presented to correlate these rotations so that one can take the advantage of the DH notations in defining the Euler angles to simulate the kinematic behavior of spherical joints. Such correlations are termed as EulerAngle-Joints (EAJs) (Shah et al. 2012b). The EAJs have the specific advantage that they make a unified representation of multiple-DOF joints and also provide computational efficiency in formulating dynami