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APPENDIX D QUATERNIONS

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The quaternion has four parameters b = (b1 , b2 , b3 , b4 ) R4 and can be represented by a generalized (four component) complex number: b = b1 + b2 i + b3 j + b4 k (D1)

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where 1, i, j, k are the quaternion basis The symbol will be used to denote the quaternion product The product of two quaternions yields a third quaternion The quaternion product has the following properties: i i = 1, i j = k, i k = j, j j = 1, j k = i, j i = k, k k = 1, k i = j, k j = i The conjugate or adjoint of b is b = b1 b2 i b3 j b4 k (D2)

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Addition or subtraction of quaternions is de ned as the addition and subtraction of the corresponding components of the quaternions By the distributive properties of multiplication and the above properties, the product of quaternions b and c is b c (b1 c1 b2 c2 b3 c3 b4 c4 ) + (b1 c2 + b2 c1 + b3 c4 b4 c3 ) i + (b1 c3 b2 c4 + b4 c2 + b3 c1 ) j + (b1 c4 + b2 c3 b3 c2 + b4 c1 ) k c1 b1 b2 b3 b4 b b1 b4 b3 c2 (D3) = 2 b3 b4 b1 b2 c3 b4 b3 b2 b1 c4 b1 c1 c2 c3 c4 c c1 c4 c3 b2 (D4) = 2 c3 c4 c1 c2 b3 c4 c3 c2 c1 b4 =

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It is important to note that quaternion multiplication is not commutative: b c = c b; but is associative: a (b c) = (a b) c The norm of a quaternion is (D5) b = b b = b2 + b2 + b2 + b2

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1 2 3 4

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This inverse of quaternion b is b 1 = b b

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The quaternion b can also be expressed in the vector form b = b1 + b (D6)

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D2 ROTATIONS

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where b = [b2 , b3 , b4 ] The vector form allows compact representation of quaternion operations For example, the quaternion conjugate is b = b1 b

The quaternion product can be written as b c = b1 c1 b c + b1 c + c1 b + b c (D7)

Based on the quaternion b we can form the matrices Qb = b1 b b (b1 I + [b ]) and Qb = b1 b b (b1 I [b ]) (D8)

From which it is clear that Qb = Qb and Qb = Qb Using the matrices Qb and Qb , the quaternion product can be expressed as b c = Qb c = Qc b (D9) (D10)

which are the same matrices written in component form in eqns (D3 D4) It can also be shown, by direct multiplication, that Qb and Qc commute, c = Qc Qb Qb Q

Rotations

Let frame a be aligned with frame b by rotating frame a by radians about unit vector E The quaternion b that represents the rotational transformation from frame a to frame b is b= cos( /2) E sin( /2)

Note that b has the normality property that b = 1 Therefore, for the representation of rotational transformations, the quaternion b has only three degrees of freedom Let z = Rb v where v is coordinatized in frame a and z is the represena tation of v when coordinatized in frame b Each vector can be expressed as the quaternion form as qv = 0 v and qz = 0 z

Using quaternions, the transformation of the vector quantity v from frame a to frame b is qz = b qv b 1 = b qv b (D11)