By Parviz E. Nikravesh

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Example text

51. Solution The product lib was found to be [-2,5, 3f. Therefore, ill) =[ ~ -~ ~] -5 -2 0 The right-hand side of Eq. }, -2,4] [-! -~ -~] [-~ -~ -:] -8 =[ 4 12 -3 6 -12 ~ -~ ~] -5 -2 0 which verifies the validity of Eq. 51. 2 Show that axax~+~x~x~+~x~x~ 5 Solution Using algebraic vector notation, we write the left-hand side of this expression as libc + bca + clib Employing Eq. 50 for lib, be, and ca, the above terms become (baT aTbJ)c + (cbT - bTcI)a + (acT - cTaI)b Sec. 1 Vector Terms in Geometric and Algebraic Forms Algebraic Geometric a a+b aa a a+b aa aTb Q'b ax b Q .

Defined as q == [

6. It will be shown that instead of three rotational coordinates, four rotational coordinates with one equation relating these four coordinates can be used to avoid singularity problems. In this case, the coordinates for body i become qi == [x,y,z,eo,e l ,e Z,e3];' The advantage of presenting the angular orientation of a body with four coordinates instead of three is also discussed in Chap. 6. If a mechanism with b bodies is considered, the number of coordinates required is n 3 x b if the system is planar, and n 6 x b (or 7 x b) if the system is spatiaL rqi, q;', ...

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