Reference Frames and Rotations Basilio Bona DAUIN – Politecnico di Torino Semester 1, 2016-2017 B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 1 / 31
Introduction In robotics we are mainly dealing with ideal rigid bodies Rigid Body (purely kinematic definition) A rigid body is a set of finite or infinite number of points whose mutual distances are invariant in time and space. A rigid body is completely defined by a reference frame R b ( O b , i b , j b , k b ), called “ body frame ”, attached to the body. Rigid bodies move in space according to rigid motions, that include only two types of motions: 1 rigid translations 2 rigid rotations These can be combined, obtaining rigid roto-translations . B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 2 / 31
Rigid Body In particular, rigid rotations are related to several concepts 1 the physical action that makes an object rotating around an axis in space 2 the mathematical representation of this physical action, and its properties 3 the abstract concept of isometry and its properties We are also interested in the following questions: if two identical rigid bodies have different orientations in space, what is their relationship? how can we mathematically represent the transformation that acts on a vector and transforms it into another vector of equal norm, i.e., a rigid transformation? B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 3 / 31
Orthogonal reference frame Definition An Orthogonal Reference Frame (ORF) in 3D space is defined by its origin O and three mutually orthogonal unit vectors ( i , j , k ): R ( O , i , j , k ) ORF’s may be right-handed or left-handed . Since the cross product obeys the rigth-hand rule: right-hand ORF if i × j = k left-hand ORF if j × i = k We will use only right-hand ORF’s and from now on we will refer to them simply as reference frames RF . B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 4 / 31
Orthogonal reference frame There is a conceptual difference between the term Reference Frame (RF) and the term Coordinate System representing it: indeed, given a unique RF, different coordinate systems may be defined, e.g., cartesian, spherical or cylindrical coordinates, left-hand or right-hand frames, etc.; in the present context, the two terms are used as synonyms. B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 5 / 31
Rigid body Each geometric k -th point of the rigid body B is defined by a geometrical vector p B k or [ p k ] B , whose coordinates in R B are � T � p B p B p B p B k = k , 1 k , 2 k , 3 � T � p k , 1 [ p k ] B = p k , 2 p k , 3 B In a rigid body the distance between each couple of points remains constant under any rigid transformation or external action on the body. � = d ij ≥ 0; � � � p i − p j ∀ P i , P j ∈ A Therefore, the rigid body B is completely characterized by its associated body frame R B , and when we speak of a rigid body we implicitly associate it with its body frame. B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 6 / 31
Two rigid bodies, as those shown in Figure, may be interpreted as two identical objects at the same time instant, or as the same object at two different time instants. In both cases we are interested in studying the mutual relation between their orientations. Figure: Two rigid bodies with different orientations. B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 7 / 31
Relations between reference frames Given two RFs R a and R b , with a common origin O , but with different orientation, their reciprocal relation can be expressed in two ways how R b is represented in R a how R a is represented in R b The choice depends on the physical meaning associated to them: one RF can be fixed, the other moving; one RF can be given at time t 0 , the second one at a different time t 1 > t 0 ; one can be more interesting than the other, etc. B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 8 / 31
Figure: Two RF’s R a e R a with a common origin but different orientation. B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 9 / 31
Rotation matrices Representation of R b in R a : we represent the three unit vectors ( i b , j b , k b ) in R a , = R a i b j b k b b a a a Representation of R a in R b : we represent the three unit vectors ( i a , j a , k a ) in R b , i a = R b j a k a a b b b These are rotation matrices satisfying the relation b ) T a ) T R b a = ( R a R a b = ( R b B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 10 / 31
Example Figure: The same vector v represented in two rotated RFs. B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 11 / 31
Example The two previous reference frames R A and R B are rotated one with respect to the other. They are represented by two different rotation matrices: 1 0 0 R A R B represented in R A B = 0 0 − 1 0 1 0 1 0 0 R A 0 0 1 R A represented in R B B = 0 − 1 0 One matrix is the transpose of the other. Observe that R B is obtained rotating R A of α = π / 2 around i A and that R A is obtained rotating R B of α = − π / 2 around i B . B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 12 / 31
Rotation matrices Rotation matrices, denoted by R , belong to the orthonormal matrix group, i.e., RR T = R T R = I R − 1 = R T → det( R ) = +1 Given a rotation matrix r 11 r 12 r 13 R = r 21 r 22 r 23 r 31 r 32 r 33 the following constraints hold r 2 11 + r 2 12 + r 2 13 = 1 r 2 21 + r 2 22 + r 2 23 = 1 r 2 31 + r 2 32 + r 2 33 = 1 r 22 r 33 − r 23 r 32 = r 11 r 23 r 31 − r 21 r 33 = r 12 r 21 r 32 − r 22 r 31 = r 13 B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 13 / 31
Rotation matrices Euler’s theorem states that the composition of any number of rotations is equivalent to a single rotation of an angle θ around an axis represented by a unit vector u . The matrix associated to this generic rotation is called an axis-angle representation and is given by Rot( u , θ ) = u 2 1 (1 − c θ )+c θ u 1 u 2 (1 − c θ ) − u 3 s θ u 1 u 3 (1 − c θ )+ u 2 s θ u 2 u 1 u 2 (1 − c θ )+ u 3 s θ 2 (1 − c θ )+c θ u 2 u 3 (1 − c θ ) − u 1 s θ u 2 u 1 u 3 (1 − c θ ) − u 2 s θ u 2 u 3 (1 − c θ )+ u 1 s θ 3 (1 − c θ )+c θ where c θ ≡ cos θ s θ ≡ sin θ We can see that this matrix satisfies the previous constraints. B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 14 / 31
Elementary rotation matrices We call elementary rotation matrices the rotations around the unit vectors ( i , j , k ) of a generic RF. 1 0 0 1 0 0 = 0 cos α − sin α 0 c α − s α Rot( x , α ) = Rot( i , α ) = 0 sin α cos α 0 s α c α cos β 0 sin β c β 0 s β = Rot( y , β ) = Rot( j , β ) = 0 1 0 0 1 0 − sin β 0 cos β − s β 0 c β cos γ − sin γ 0 c γ − s γ 0 = Rot( z , γ ) = Rot( k , γ ) = sin γ cos γ 0 s γ c γ 0 0 0 1 0 0 1 Notice the position of 1 on the main diagonals. B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 15 / 31
Composition of rotations Every rotation can be obtained combining these three elementary rotations. Rules are simple: to obtain a generic rotation, composed by a number of elementary rotations, it is necessary to a) define which elementary rotation is applied, i.e., define the rotation axis b) define the rotation angle c) establish if the rotation is performed around fixed or mobile RF d) if the rotation takes place wrt the fixed RF, then it is necessary to pre-multiply the previous rotation matrix by the new rotation e) if the rotation takes place wrt the mobile RF, then it is necessary to post-multiply the previous rotation matrix by the new rotation. Product rule for the rotation composition Pre-fixed Post-mobile The terms “fixed” and “mobile” indicate respectively the RF assumed to be static and the RF resulting from the previous rotation. B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 16 / 31
Vector rotation Let’s assume we have a generic vector v b (both a physical or a geometrical vector) represented in a generic RF R b and we want to represent it in another RF R a , whose mutual representation is given by R a b (remember, R b in R a ). The transformation is a linear application given by v a = R a v b = R b b v b , or a v a where R b a = ( R a b ) T Notice the position of the indexes a and b . B. Bona (DAUIN) Rotations & Reference Frames Semester 1, 2016-2017 17 / 31
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