Design and Analysis of Kinematic Couplings for Modular Machine ...

y Fball= ( b 3 – b 2 ) × z Fball. (3.7)Hence, the coordinates of ball 2 in F ball are (x, y, z) = (0,0,0) and of ball 3 are:b 3 = ( 0, b 1 – b 2 , 0). (3.8)To find the coordinates of ball 1 in F ball , use dot products:x 1 Fball, = ( b 2 – b 1 ) ⋅ ( b 3 – b 2 )and (3.9)y 1 Fball, = ( b 2 – b 1 ) ⋅ y Fball. (3.10)Next, the origin of F ball is shifted to the location of the coupling centroid. Specifically, the couplingcentroid is the point of intersection between two lines starting at the coupling locations and bisecting therespective angles of the coupling triangle. Hence, the coupling centroid is the solution of a system of twolinear equations in the x-y plane of F ball , each derived from applying the law of cosines to the trianglegeometry. The included angles of the coupling triangle are:⎛ ( b 2 – b 1 ) 2 + ( b 3 – b 2 ) 2 – ( b 3 – b 1 ) 2⎞θ 13 = acos⎜-------------------------------------------------------------------------------------------------⎟⎝ 2 ( b 3 – b 2 ) ( b 2 – b 1 ) ⎠and (3.11)⎛ ( b 3 – b 1 ) 2 + ( b 3 – b 2 ) 2 – ( b 2 – b 1 ) 2⎞θ 12 = acos⎜-------------------------------------------------------------------------------------------------⎟. (3.12)⎝ 2 ( b 3 – b 2 ) ( b 2 – b 1 ) ⎠And the pair of linear equations (suitably between any two balls), expressed relative to F ball , is:θy 13C–ball =tan⎛ ------⎞ xC . (3.13)⎝ 2 ⎠ – ballθy 12C–ball = tan ------⎛ ⎞( xC⎝ 2 ⎠ – ball – ( b 3 – b 2 ) ). (3.14)From this solution, the location of the coupling centroid in F ball as initially placed at b 2 is denoted as(x C-ball , y C-ball, 0). Note that even under perturbations in the vertex locations, which make the triangle nolonger equilateral or isosceles, the angle bisectors of a triangle always intersect at a point; hence the intersectionof the bisectors originating at balls 2 and 3 defines the in-plane location of the coupling centroid.Now, sufficient information has been derived to calculate the HTM between the measurement systemand the coupling interface. When the axis vectors of F ball expressed in F MS are normalized to unit vectors,50