Transforming Spatial Rigid-Body Inertias

As was discussed in Chapter 2, the transformation of a vector firom one coordinate system to an adjacent one may be accomplished using the spatial transformation X as follows  

The spatial transformation matrix, is usually written as a single genoal tiansfamation, as shown in Equation 2.5. Howevo-, as demonstrated by Feath-erstone in [9], significant savings can be obtained by using two screw transformations instead, the first on the x-axis, followed by a second on the new z-axis. Thus, the transformation, X, may be written  

Multiplication of one screw tiansfcMmation with a general spatial vector requires (10 scalar multiplications, 6 scalar additions). Thus, the complete transformation of a genoal vector requires a total of (20 multiplications, 12 additions) if two screw transformations are used. The product of a genoal transformation between adjacent coordinate systems and a general vector, on the otho- hand, requires (24 multiplications, 18 additions). 

Transforming the unit vector, f i (= %), is a special case. Since f i has only one nonzero elemoit for revolute and ixismatic joints, and this element is unity, transformation of this vector involves only the selection of a single column from X, or premultiplication of the t4)prq)riate column of by X. Thus, transformation of j i requires (0 multiplications, 0 additions) if X is available, and at worst (6 multiplications, 2 additions) if X and X are used. On-line computation of X itself, howev, requires (14 multiplications, 4 additions), while X requires only (2 multiplications, 0 additions). Note that X may be calculated off-line, since the variables involved, a,- and or,- for link i, are constants. Finally then, when genoal vectors, as well as simple vector such as 4 i, must be transformed, the most efficient method of transformation must be carefully chosen. 

The spatial inertia matrix of a rigid body may be transformed fix)m one cowdinate system to an adjacent one using Equation 2.11, rqteated here for convenience  

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The fifth section describes efficient computational methods for the transformation of spatial quantities between adjacent cowdinate frames and for the calculation of composite-rigid-body inertias. These techniques are used throughout this book to improve the computational efficiency of the algorithms. Otho-factors which simplify the equations are also discussed. 

It may be shown that this procedure for ccmiputing the spatial composite-rigid-body inertia is exactly equivalent to the procedure used by Walker and Orin in [42]. That is, if the composite mass, composite center of mass, and composite moment of inertia matrix are computed for links i through iV, they may be combined to obtain the spatial matrix K,-. After studying this equivalent approach, howev, it appears that the congruence transformation method given hoe is more efficioit... 

It can easily be shown that K,+i has the same mathematical fonn as the spatial inertia of a single rigid body. Thus, the congruence transformation of this matrix also requires (49 multiplications, 49 additions) as described above. The addition of I, requires only an extra 10 additions, if we consido the symmetry and form of Ki and I<. Note that the bottom right submatrix of is simply the diagonal matrix of the composite mass of links t through N. Thus, since this composite... 

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