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By Jadran Lenarcic, Tadej Bajd, Michael M. Stanišić (auth.)
This ebook offers a entire advent to the realm of robotic mechanisms, essentially contemplating commercial manipulators and humanoid palms. The booklet is meant for either educating and self-study. Emphasis is given to the basics of kinematic research and the layout of robotic mechanisms. The insurance of issues is untypical. the point of interest is on robotic kinematics. The booklet creates a stability among theoretical and sensible elements within the improvement and alertness of robotic mechanisms, and comprises the newest achievements and traits in robotic technology and technology.
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E. when sin θ = 0. 4 Angular Acceleration and Time Derivatives of the YPR Angles The relation between the angular velocity and the time derivatives of the YPR orientation angles was explained in Sect. 4. The transformation is given by the W matrix, which depends on the angles ψ and θ . 8 Homogeneous Transformations 51 where ˙ = ∂W ψ˙ + ∂W θ˙ W ∂ψ ∂θ and ⎡ 0 ∂W ⎣ = 0 ∂θ 0 ⎡ 0 ∂W ⎣ = 0 ∂ψ 0 ⎤ − cos ψ cos θ − sin ψ cos θ ⎦ , 0 ⎤ 0 sin ψ sin θ 0 − cos ψ sin θ ⎦ . 0 cos θ − sin ψ cos ψ 0 The original equation can be rewritten in a more compact form ⎤ ⎡ ⎤ ⎡ ψ¨ ψ˙ θ˙ u = W0 ⎣ ψ˙ φ˙ ⎦ + W ⎣ θ¨ ⎦ , φ¨ θ˙ φ˙ where ⎡ − sin W0 = ⎣cos ψ 0 − sin ψ cos θ − sin ψ cos θ 0 ⎤ sin ψ sin θ − cos ψ sin θ ⎦ .
73) (i) Here, ej u is the corresponding element of the rotation vector ej . The above equations are valid when sin qj = 0. When sin qj = 0, there is the trivial case of (i) qj = 0 ± kπ , k = 0, 1, . . , and the rotation vector ej cannot be uniquely determined. At first it appears that we need four parameters for representing the spatial orientation of a body, the rotation angle qj and three elements of the rotation vector ej . Only three of these four parameters are independent, as there is the following constraint among the elements of the rotation vector ej21 + ej22 + ej23 = 1.
The above equations are valid when sin qj = 0. When sin qj = 0, there is the trivial case of (i) qj = 0 ± kπ , k = 0, 1, . . , and the rotation vector ej cannot be uniquely determined. At first it appears that we need four parameters for representing the spatial orientation of a body, the rotation angle qj and three elements of the rotation vector ej . Only three of these four parameters are independent, as there is the following constraint among the elements of the rotation vector ej21 + ej22 + ej23 = 1.