71 Basilio Bona - Dynamic Modelling If we use the quaternions, the Euler parameters, the Cayley-Klein parameters or the Rodrigues vectors, it in unclear what are the parameters that take the place of the α i in (2.65). Using these alternative representations, we cannot say that the α i are “angles”, but it is necessary to speak in more general terms of “angular parameters”, whose knowledge nevertheless allows to compute the chosen (Euler or RPY or Cardan) angles. For instance, using the quaternions we can define: α 1 = u 1 sin θ 2 , α 2 = u 2 sin θ 2 , α 3 = u 3 sin θ (2.106) 2 and implicitly assume the unit norm. However it is a very common practice to associate to the α i the Euler or the RPY angles, although in the last few years quaternion representation has gained much attention, mainly in the aerospace and computer graphics applications. 2.13 Point Kinematics Having introduced in the previous Sections the different representations of a geomet- rical point or of a vector, both polar or axial, in a given reference frame, and having also characterized the various types of rigid displacements in the Euclidean space, we are now ready to describe the motion of geometrical points, assumed massless. This description takes the name of kinematics and is distinct from dynamics since the former studies the motion of points or rigid bodies establishing the relations among positions, velocities and accelerations from a pure geometrical point of view, while the latter studies the influence of the forces and torques on the bodies on these quantities. In few words, kinematics study the motion without considering its causes and its effects, while dynamics studies how the external actions on the given bodies are related to their motion. Suppose to have a geometrical point P moving in the 3D Euclidean space with a time law specified by a function P ( t ); we can assume a vector representation of P in a given reference frame x 1 P ( t ) x P ( t ) = x 2 P ( t ) x 2 P ( t ) or, to keep notation simple x 1 ( t ) x ( t ) = x 2 ( t ) x 2 ( t )
72 Basilio Bona - Dynamic Modelling Recalling the vector derivative in A.5, we write d d tx 1 ( t ) x 1 ( t ) ˙ d d ≡ v ( t ) d t x ( t ) = d tx 2 ( t ) ≡ ˙ x ( t ) = x 2 ( t ) ˙ x 3 ( t ) ˙ d d tx 2 ( t ) Since the velocity is the limit − − → d P ( t ) ∆ P = lim d t ∆ T ∆ T → 0 the signed segment − − → ∆ P can be different in different reference frames. [ − ] − → If the point P is fixed in a local reference frame R m , we have ∆ P R m = 0 ; but if [ − ] − → R m moves with respect to a world frame R 0 , then ∆ P R 0 ̸ = 0 . In the first case we speak on “local” velocity, in the second case we speak of “global” or absolute velocity or total velocity . Usually when we do not specify otherwise, we will consider always the total velocity of a point. When the vector x ( t ) is subject to a time-varying rotation R ( t ) x ( t ), its derivate is computed according to the normal derivative product rule, i.e., [ d ] [ d ] d d t [ R ( t ) x ( t )] = d t R ( t ) x ( t ) + R ( t ) d t x ( t ) therefore it is necessary to derive the rotation matrix R ( t ). To understand the consequences of this operation, we have to study its properties. 2.13.1 Rotation Matrix Derivative and Angular Velocity To start we assume that the generic rotation matrix R is function of a generic variable x , that is R = R ( x ). We recall that the orthogonality of the matrix implies R ( x ) R T ( x ) = I We derive this relation with respect to a generic variable x obtaining: R T ( x ) + R ( x )d R T ( x ) d R ( x ) = O (2.107) d x d x It is evident that the first term is the transpose of the second term, therefore, recalling the definition of skew-symmetric matrix in Appendix B.6, we can define the following skew-symmetric matrix = d R ( x ) def R T ( x ) , S ( u ( x )) (2.108) d x
73 Basilio Bona - Dynamic Modelling we have introduced a generic vector u ( x ) because we know that a skew-symmetric matrix in R 3 embeds in its structure the components of a vector. We will later discuss its precise meaning. Taking both terms of (2.107) and post-multiplying them by R ( x ) we obtain d R ( x ) = S ( u ( x )) R ( x ) (2.109) d x and it results that the derivative of an orthonormal matrix is the matrix itself pre- multiplied by a suitable anti-symmetric matrix. S ( u ) in (2.108) is function of a generic vector u ( x ), itself a function of a generic scala variable x . In particular, when the rotation matrix R depends on an angle θ ( t ) around an axis given by the unit vector u , we can write: d R ( u , θ ( t )) ≡ ˙ def R ( u , θ ( t )) = S ( ω ( t )) R ( u , θ ( t )) (2.110) d t where ω ( t ) is the instantaneous total angular velocity vector of the reference frame represented by the matrix R ( t ). The angular velocity is represented in the world reference frame, but we omit to write it as ω 0 for notational simplicity. Now we take into consideration some simple cases of vector transformation R ( t ) r ( t ): we start with r ( t ) = r constant in time in a “mobile” reference frame R m a we transform it n the world reference frame R 0 r 0 ( t ) = R 0 m ( t ) r m the time derivative is computed as: 0 r 0 ( t ) = ˙ m ( t ) r m ˙ R (2.111) the vectors in this expression are represented in two different reference frames, so it is necessary to represent both in R 0 ; considering that r m = ( R 0 m ) T r 0 we have at the end, omitting for notational convenience the dependence of R on t , 0 r 0 ( t ) = ˙ 1 ( R 0 1 ) T r 0 ( t ) ˙ (2.112) R In mechanics textbooks one often encounters the following notation 0 Ω ( t ) ≡ ˙ 1 ( R 0 1 ) T R that coincide with our previous skew-symmetric matrix S ( ω ); indeed 0 1 ) T = S ( ω ) R 0 1 ) T = S ( ω ) Ω = ˙ 1 ( R 0 1 ( R 0 R
74 Basilio Bona - Dynamic Modelling This matrix is often calle the angular velocity matrix ; in some textbooks the skew-symmetric matrix Ω ( t ) = S ( ω ( t ) is denoted with the symbol � ω ( t ). Considering again the identity (2.111), we observe that it expresses the well known formula that gives the linear velocity of a point P fixed in a reference frame that rotates with an angular velocity ω : r 0 ( t ) = Ω ( t ) r 0 ( t ) = S ( ω ( t )) r 0 ( t ) = ω ( t ) × r 0 ( t ) ˙ (2.113) Other interesting properties of the skew-symmetric matrices are related to the rigid body representation in 3D space. As shown in (2.40), it is possible to compute the rotation matrix R from the rotation axis represented by the non-unit vector v and the rotation angle θ = ∥ v ∥ , considering the skew-symmetric matrix S ( v ). This equality is a theoretical consequence of the following property, in general valid for Lie groups and algebras (refer to [34] for details): ∑ ∞ 1 R ( v ) ≡ R ( u , θ ) = e S ( v ) ≡ e S ( θ, u ) = k ! S k ( θ u ) (2.114) k =0 that relates the rotation matrix R to the exponential of the skew-symmetric matrix S ( v ) = S ( u , θ ), function of the unit axis u and angle θ . Taking the Taylor series of the matrix exponential and considering that the rotations are elements of a cyclic group, and this reflects on the strusture of S i.e., S 3 = − S , S 4 = − S 2 we obtain: R = I + sin θ S ( v ) + 1 − cos θ S 2 ( v ) (2.115) θ 2 θ from which one gets the identities (2.40) and (2.41). Without entering into details, that can be found in the cited textbook, we write here the fundamental identity that relates the angular velocity vector ω ( t ), the unit vector u ( t ) and the angle θ ( t ): ω ( t ) = ˙ θ ( t ) u ( t ) + sin θ ( t ) ˙ u ( t ) + (1 − cos θ ( t )) S ( u ( t )) ˙ u ( t ) (2.116) From (2.116) we see that the vector ω ( t ) is not the formal derivative of another vector , except from the simple case that u is constant, i.e., u ( t ) = c . In such a case we have: ω ( t ) = ˙ θ ( t ) c = ˙ r ( t ) (2.117) that represents the total time derivative of the vector r ( t ) = θ ( t ) c . [ ] Another useful formula to get ω ( t ) from a rotation matrix R = is the r g b following (see [17]): ω ( t ) = 1 g + b × ˙ 2( r × ˙ r + g × ˙ b ) (2.118) Notice the similarity of this equality with (2.105).
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