By Robert Seifried
Underactuated multibody structures are fascinating mechatronic platforms, as they posses fewer keep watch over inputs than levels of freedom. a few examples are sleek lightweight versatile robots and articulated manipulators with passive joints. This booklet investigates such underactuated multibody platforms from an built-in point of view. This contains all significant steps from the modeling of inflexible and versatile multibody structures, via nonlinear keep watch over thought, to optimum approach layout. The underlying theories and strategies from those diverse fields are provided utilizing a self-contained and unified procedure and notation procedure. in this case, the ebook specializes in functions to massive multibody structures with a number of levels of freedom, which require a mix of symbolical and numerical tactics. ultimately, an built-in, optimization-based layout approach is proposed, wherein either structural and regulate layout are thought of at the same time. each one bankruptcy is supplemented by means of illustrated examples.
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Additional info for Dynamics of Underactuated Multibody Systems: Modeling, Control and Optimal Design
68) which yields Hook’s material law. By rearranging the symmetric Green-Lagrange strain tensor, a strain vector can be defined as ˆ = [G 11 , G 22 , G 33 , 2G 12 , 2G 23 , 2G 31 ]T . 70) The strain vector can be computed directly from the displacements by ˆ = L L u + 1 L N (u) u. 71) Using the abbreviation ∂i = ∂/∂ Ri the operators L L , L N (u) are defined by ⎡ ∂1 ⎢0 ⎢ ⎢0 LL = ⎢ ⎢ ∂2 ⎢ ⎣0 ∂3 ⎡ 0 ∂2 0 ∂1 ∂3 0 ⎤ 0 0⎥ ⎥ ∂3 ⎥ ⎥, 0⎥ ⎥ ∂2 ⎦ ∂1 ⎤ ∂ 1 u 1 ∂1 ∂1 u 2 ∂1 ∂1 u 3 ∂1 ⎥ ⎢ ∂2 u 1 ∂2 ∂2 u 2 ∂2 ∂2 u 3 ∂2 ⎥ ⎢ ⎥ ⎢ ∂3 u 1 ∂3 ∂3 u 2 ∂3 ∂3 u 3 ∂3 ⎥.
60) Summarizing the term k¯ = J Mγγ + J k, this second order differential equation is structurally identical with the equation of motion of a multibody system in tree structure and can be solved with standard integration algorithms for ordinary differential equations. The quantities C, ctt and thus also J, γ depend on q, q˙ and, therefore, on the depended variables q d , q˙ d . One possibility is the computation of q d by solving the nonlinear loop closing constraints c(q) = c(q i , q d ) = 0. , using a Newton-Raphson method.
130) are here described in their own frame of reference K R . The use of the reference frame to describe the kinetics of an elastic body is often suitable if relative coordinates are used, which simplifies the evaluation of the equation of motion. 131) with q e = ⎣ ... ⎦ . q= qe q e, p The vector of generalized coordinates q ∈ IR f contains the coordinates q r ∈ IR fr representing the fr degrees of freedom of an equivalent rigid multibody system. The elastic coordinates q e,i of the p elastic bodies can be combined to form the global vector of elastic generalized coordinates q e ∈ IR fe .