Geometric Fundamentals in Robotics Quaternions Basilio Bona DAUIN-Politecnico di Torino July 2009 Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 1 / 47
Introduction Quaternions were “discovered” in 1843 by Sir William Hamilton, who was looking for the extension to 3D space of the complex numbers as rotation operators. Figure: Sir William Rowland Hamilton (1805-1865) and the plaque on Broom Bridge, where the quaternions were discovered. Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 2 / 47
Definitions The generic quaternion will be indicated as q . Quaternions are elements of a 4D linear space H ( R ), defined on the real numbers field F = R , with base { 1 i j k } . i , j and k are ipercomplex numbers that satisfy the following anticommutative multiplication rules: i 2 = j 2 = k 2 = ijk = − 1 ij = − ji = k jk = − kj = i ki = − ik = j Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 3 / 47
Definitions A quaternion q ∈ H is defined as a linear combination expressed in the base { 1 i j k } : q = q 0 1 + q 1 i + q 2 j + q 3 k where the coefficients { q i } 3 i =0 are real. Another way to represent a quaternion is to define it as a quadruple of reals, ( q 0 , q 1 , q 2 , q 3 ) q = ( q 0 , q 1 , q 2 , q 3 ) in analogy with complex numbers c = a + j b , where c is represented by a couple of reals, c = ( a , b ), Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 4 / 47
Definitions Quaternions are also defined as hypercomplex numbers, i.e., those “complex numbers” having complex coefficients: q = c 1 + j c 2 , where c 1 = q 0 + k q 3 e c 2 = q 2 + k q 1 . Therefore, considering multiplication rules, it results: q = c 1 + j c 2 = q 0 + k q 3 + j q 2 + jk q 1 = q 0 1 + q 1 i + q 2 j + q 3 k Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 5 / 47
Definitions In analogy with complex numbers that are the sum of a real part and an imaginary part, quaternions are the sum of a real part and a vectorial part. The real part q r is defined as q r = q 0 , and the vectorial part q v is defined as q v = q 1 i + q 2 j + q 3 k . We write q = ( q r , q v ) or q = q r + q v ; note that the vectorial part is not “transposed” since the conventional definition for the vectorial part of a quaternion assumes a row representation. Using the convention that defines vectors as “column” vectors, we can write q = ( q r , q T v ). Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 6 / 47
Definitions Quaternions are general mathematical objects, that include real numbers r = ( r , 0 , 0 , 0) , r ∈ R complex numbers a + i b = ( a , b , 0 , 0) , a , b ∈ R real vectors in R 3 (with some caution, since not all vectorial parts represents vectors) v = (0 , v 1 , v 2 , v 3 ) , v i ∈ R . � i k � In this last case, elements are to be understood as unit vectors j � i k � forming an orthonormal base in a cartesian right-handed j reference frame. Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 7 / 47
Definitions Multiplication rules between elements i , j , k have the same properties of the cross product between unit vectors i , j , k : ij = k ⇔ i × j = k ji = − k ⇔ − j × i = − k etc. In the following we will use all the possible alternative notations to indicate quaternions q = q 0 1 + q 1 i + q 2 j + q 3 k = ( q r , q v ) = q r + q v = ( q 0 , q 1 , q 2 , q 3 ) i.e., a) a hypercomplex number; b) the sum of a real part and a vectorial part; c) a quadruple of reals. Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 8 / 47
Definitions An alternative way to write a quaternion is the following q = q 0 1 + q 1 i + q 2 j + q 3 k where now � � � � � � � � 1 0 i 0 0 1 0 i 1 = ; i = ; j = ; k = ; 0 1 0 − i − 1 0 0 i and i 2 = − 1. Hence every matrix is of the form � � c d ; − d ∗ c ∗ These matrices are called Cayley matrices [see Rotations - Pauli spin matrices ]. Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 9 / 47
Quaternions algebra Given a quaternion q = q 0 1 + q 1 i + q 2 j + q 3 k = q r + q v = ( q 0 , q 1 , q 2 , q 3 ), the following properties hold: a null or zero 0 quaternion exists 0 = 0 1 +0 i +0 j +0 k = (0 , 0 ) = 0+ 0 = (0 , 0 , 0 , 0) a conjugate quaternion q ∗ exists, having the same real part and the opposite vectorial part: q ∗ = q 0 − ( q 1 i + q 2 j + q 3 k ) = ( q r , − q v ) = q r − q v = ( q 0 , − q 1 , − q 2 , − q 3 ) Conjugate quaternions satisfy ( q ∗ ) ∗ = q Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 10 / 47
Quaternions algebra a non-negative function, called quaternion norm exists � q � , defined as 3 � q � 2 = qq ∗ = q ∗ q = ∑ q 2 ℓ = q 2 0 + q T v q v ℓ =0 A quaternion with unit norm � q � = 1 is called unit quaternion . Quaternion q and its conjugate q ∗ have the same norm � q � = � q ∗ � The quaternion q v = 0 1 + q 1 i + q 2 j + q 3 k = (0 , q v ) = 0+ q v = (0 , q 1 , q 2 , q 3 ) , that has a zero real part is called pure quaternion or vector . The conjugate of a pure quaternion q v is the opposite of the original pure quaternion q ∗ v = − q v Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 11 / 47
Quaternions algebra Given two quaternions h = h 0 1 + h 1 i + h 2 j + h 3 k = ( h r , h v ) = h r + h v = ( h 0 , h 1 , h 2 , h 3 ) and g = g 0 1 + g 1 i + g 2 j + g 3 k = ( g r , g v ) = g r + g v = ( g 0 , g 1 , g 2 , g 3 ) the following operations are defined Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 12 / 47
Sum Sum or addition h + g h + g = ( h 0 + g 0 ) 1 +( h 1 + g 1 ) i +( h 2 + g 2 ) j +( h 3 + g 3 ) k = (( h r + g r ) , ( h v + g v )) = ( h r + g r )+( h v + g v ) = ( h 0 + g 0 , h 1 + g 1 , h 2 + g 2 , h 3 + g 3 ) Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 13 / 47
Difference Difference or subtraction h − g = ( h 0 − g 0 ) 1 +( h 1 − g 1 ) i +( h 2 − g 2 ) j +( h 3 − g 3 ) k = (( h r − g r ) , ( h v − g v )) = ( h r − g r )+( h v − g v ) = ( h 0 − g 0 , h 1 − g 1 , h 2 − g 2 , h 3 − g 3 ) Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 14 / 47
Product Product = ( h 0 g 0 − h 1 g 1 − h 2 g 2 − h 3 g 3 ) 1 + hg ( h 1 g 0 + h 0 g 1 − h 3 g 2 + h 2 g 3 ) i + ( h 2 g 0 + h 3 g 1 + h 0 g 2 − h 1 g 3 ) j + ( h 3 g 0 − h 2 g 1 + h 1 g 2 + h 0 g 3 ) k = ( h r g r − h v · g v , h r g v + g r h v + h v × g v ) where h v · g v is the scalar product h v · g v = ∑ h vi g vi = h T v g v = g T v h v i defined in R n , and h v × g v is the cross product (defined only in R 3 ) h 2 g 3 − h 3 g 2 = S ( h v ) g v h v × g v = h 3 g 1 − h 1 g 3 h 1 g 2 − h 2 g 1 Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 15 / 47
Product The quaternion product is anti-commutative, since, being g v × h v = − h v × g v it follows gh = ( h r g r − h v · g v , h r g v + g r h v − h v × g v ) � = hg ; Notice that the real part remains the same, while the vectorial part changes. Product commutes only if h v × g v = 0 , i.e., when the vectorial parts are parallel. The conjugate of a quaternion product satisfies ( gh ) ∗ = h ∗ g ∗ . The product norm satisfies � hg � = � h �� g � . Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 16 / 47
Product properties associative ( gh ) p = g ( hp ) multiplication by the unit scalar 1q = q1 = (1 , 0 )( q r , q v ) = (1 q r , 1 q v ) = ( q r , q v ) multiplication by the real λ λ q = ( λ , 0 )( q r , q v ) = ( λ q r , λ q v ) bilinearity, with real λ 1 , λ 2 g ( λ 1 h 1 + λ 2 h 2 ) λ 1 gh 1 + λ 2 gh 2 = ( λ 1 g 1 + λ 2 g 2 ) h λ 1 g 1 h + λ 2 g 2 h = Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 17 / 47
Product Alternative forms Quaternion product may be written as matrix product forms: h 0 − h 1 − h 2 − h 3 g 0 g 0 − h T � � h 1 h 0 − h 3 h 2 g 1 g 1 h 0 hg = = v − h 1 h 2 h 3 h 0 g 2 h v h 0 I + S ( h v ) g 2 − h 2 h 3 h 1 h 0 g 3 g 3 = F L ( h ) g or − g 1 − g 2 − g 3 g 0 h 0 h 0 � − g T � − g 2 g 1 g 0 g 3 h 1 g 0 h 1 v hg = = g 2 − g 3 g 0 g 1 h 2 g v g 0 I + S ( g v ) h 2 g 3 g 2 − g 1 g 0 h 3 h 3 = F R ( g ) h Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 18 / 47
Quotient Since the quaternion product is anti-commutative we must distinguish between the left and the right quotient or division. Given two quaternions h e p , we define the left quotient of p by h the quaternion q ℓ that satisfies hq ℓ = p while we define the right quotient of p by h the quaternion q r that satisfies q r h = p Hence h ∗ q r = p h ∗ q ℓ = � h � 2 p ; � h � 2 Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 19 / 47
Recommend
More recommend
