After these two rotations the new x axis points toward the satellite The two

# After these two rotations the new x axis points

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x-axis points toward the satellite. The tworotations are respectively described by the two quaternions below:p=cosα2+ sinα2ˆk,q=cosβ2+ sinβ2ˆj.Since we are rotating the coordinate frame, the two operatorsLp*andLq*are applied sequentially. Thecomposite rotation operator isL(pq)*, which transforms coordinates in the station frame to those in thetracking frame.And the quaternion describing the composition rotation is the productpqwhich is asfollows.pq=parenleftBigcosα2+ sinα2ˆkparenrightBigparenleftbiggcosβ2+ sinβ2ˆjparenrightbigg=cosα2cosβ2+ cosα2sinβ2ˆj+ sinα2cosβ2ˆk+ sinα2sinβ2(ˆk×ˆj)=cosα2cosβ2sinα2sinβ2ˆi+ cosα2sinβ2ˆj+ sinα2cosβ2ˆk.The axis of the composite rotation is defined by the vectorv=parenleftbiggsinα2sinβ2,cosα2sinβ2,sinα2cosβ2parenrightbigg.(14)And the angle of rotationθsatisfiescosθ2=cosα2cosβ2,sinθ2=bardblvbardbl.The cosine is same as obtained in Section 3 of the handouts titled “Rotation in the Space” for we havecosθ=2 cos2θ21=2 cos2α2cos2β21=2cosα+ 12·cosβ+ 121=cosαcosβ+ cosα+ cosβ12.Note that the rotation axis and angle in that section transforms coordinates in the tracking frame to thosein the station frame. This explains why the axisvin (14) is opposite to the one obtained in that sectionwhile the angle is the same.5Application: 3-D Shape RegistrationAn important problem in model-based recognition is to find the transformation of a set of datapoints that yields the best match of these points against a shape model.The process is oftenreferred to asdata registration. The data points are typically measured on a real object by rangesensors, touch sensors, etc., and given in Cartesian coordinates. The quality of a match is often9
ModelrotationtranslationDatap5p2p3q2q7q4p1p4p7p6q6q3q5q1Figure 3: Matching two point setspiandqj.described as the total squared distance from the data points to the model. When multiple shapemodels are possible, the one that results in the least total distance is then recognized as the shapeof the object.Quaternions are very effective in solving the above least-squares-based registration problem. Letus begin with a formulation of the problem in 3D. Let{p1,p2, . . . ,pn}be a set of data points. Weassume thatp1, . . . ,pnare to be matched against the pointsq1, . . . ,qnon a shape model. Namely,thecorrespondencesbetween the data points and those on the model have been predetermined.Then the problem is to find a rotation, represented by an orthogonal matrixRwith det(R) = 1,and a translationbas the solution to the following minimization:minR,bnsummationdisplayi=1bardblRpi+bqibardbl2.(15)We begin by computing the centroids of the two sets of points:¯p=1nnsummationdisplayi=1pi;¯q=1nnsummationdisplayi=1qi.