mp Co uta & tio d n e a li Brezov, J Appl Computat Math 2018, 7:3 l p M p a Journal of A t h DOI: 10.4172/2168-9679.1000410 f e o m l a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access Optimization and Gimbal Lock Control via Shifted Decomposition of Rotations Brezov D* Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, Hristo Smirnenski, Bulgaria Abstract The present paper studies the effects of introducing a fourth rotation in the generalized Euler factorization. As a first immediate consequence, it is shown to weaken the well-known Davenport orthogonally condition on the decomposition axes, thus allowing much more freedom in the geometry of attitude control mechanisms. Furthermore, the additional parameter is used for avoiding gimbal lock singularities and optimization purposes. Some of these features have been pointed out by other authors and partially studied in the case of two axes. This text, however, provides exact closed form solutions based on recent results on the generalized Euler decomposition problem. The examples considered below clearly illustrate that the approach of four factor sequences can be significantly more efficient than standard Euler type decompositions. Finally, dynamical effects, such as the inertial anisotropy, are being modelled via introducing weight coefficients in the cost function. Keywords: Rotations; Spacecraft attitude; Robot kinematics; of attitude maneuvers. Here we consider only minimization of the Optimal control rotation path, while the impact of specific asymmetries is being modelled by introducing weights in the cost function. A more detailed Introduction discussion from dynamical perspective [9,10]. Euler angles have been a major tool in the description of attitude Vector Decompositions of Rotations orientation of spacecraft and aircraft since these technologies exist. ζ The vector-parameter may be introduced as c= where the unit However, the method causes some troubles, such as the so-called ζ 0 gimbal lock problem related to the inevitable topological singularity quaternion 22 2 of the parameterization map between the three-dimensional torus and ζ=( ζ0,ζ) =+ ζζ 01i + ζ 2 j+ ζ 3 k ∈ ,|| ζ|| =+= ζζ0 1 3 the rotation group in which has the topology of projective space is a representative of the spin group SU(2) 3, so c actually inhibits 3 3 ← SO(3). This problem does not appear if we use quaternions or 3 the projective space SO(3) . Note that it may also be expressed axis-angle decomposition [1] associated with the Hopf bration , but φ 1 via the Euler’s trigonometric substitution as≅ c=τn where τ =tan ∈� ≅ 2 in practice this may be even less efficient in terms of attitude adjustment. is usually referred to as the≅ scalar parameter (ϕ being the angle of It has been [2] the gimbal lock singularity may be utilized within the 2 rotation) and n stands for the unit vector along the invariant axis. Euler angles' framework using a simple optimization procedure on This parameterization provides several major advantages. Firstly, it the excessive free parameter at the singular point. The present paper allows for expressing∊ the rotation matrix entries as rational functions suggests an analogous procedure in the regular case as well, by means of c, namely of introducing a fourth “shift" parameter. Such an additional degree of 2 t × × (1−c) I ++ 22 cc c I + c R (c) = = (1) freedom allows for optimizing and thus, saving a significant portion 1+−cc2 I × of the energy spent on spacecraft maneuvers as well as manipulating where stands for the identity, cct denotes the tensor (dyadic) the gimbal lock condition. A similar idea for optimization [3,4] for product and c× is an extension of the Hodge duality that de nes the sequences of arbitrary length in the two-axes setting. Here we allow three 3 cross product in as a×b=a×b. As the right-hand side in eqn. (1) is axes while restricting to four factors, which turns out to be sufficient in the well-known Cayley map, we have (c)=Cay(c×). many cases [5]. Hence, the cost function may be expressed explicitly in terms of α, due to the closed form solutions obtained below. The Another advantage is the compact form of the group composition derivation of these results is based on a recent study on the generalized law cccc2++× 1 21 Euler decomposition problem [6] considering arbitrarily oriented axes cc21,,= ⇔=RR(c2) ( c 1) R cc 21 1− (cc) (2) and the corresponding necessary and sufficient condition [7]. In this 21 3 1 where (ab; ) = a⋅ bstands for the scalar (dot) product in . Note extended framework we may weaken the Davenport condition γ12+γ23 ≥ π for the relative angles between the axes [8] to γ12+γ23+γ31 ≥ π, thus allowing for different geometries of attitude control mechanisms in vehicles and robots, beyond the classical orthogonal settings. The text *Corresponding author: Brezov D, Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, Hristo Smirnenski, Bulgaria, Tel:+359 is organized as follows: in the following section the quaternion based 2 963 5245; E-mail: [email protected] vector-parameter is introduced and utilized to solve the generalized three-axes decomposition problem. Section three explains in detail Received July 10, 2018; Accepted August 07, 2018; Published August 09, 2018 the shift parameter construction, while sections four and five focus on Citation: Brezov D (2018) Optimization and Gimbal Lock Control via Shifted its applications in gimbal lock control and optimization of rotational Decomposition of Rotations. J Appl Computat Math 7: 410. doi: 10.4172/2168- 9679.1000410 sequences, respectively. Several examples are given to illustrate the way variations of the shift parameter influence the energy efficiency Copyright: © 2018 Brezov D. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted 1 Here γij denotes the minimal angle between the rotation axes, so orthogonally is use, distribution, and reproduction in any medium, provided the original author and implied source are credited. J Appl Computat Math, an open access journal Volume 7 • Issue 3 • 1000410 ISSN: 2168-9679 Citation: Brezov D (2018) Optimization and Gimbal Lock Control via Shifted Decomposition of Rotations. J Appl Computat Math 7: 410. doi: 10.4172/2168-9679.1000410 Page 2 of 7 also that formula in eqn. (2) is just the quaternion multiplication rule decomposition. In the case eqn. (7) instead of eqn. (4) this procedure projected onto 3. Composing rotations in this manner appears far yields. 1 g γ more efficient in terms of computational complexity compared to the 12 31 ∆=′′gg10 ≥ (9) usual matrix multiplication [5]. 12 23 γ 31g 23′ 1 With the aid of formula in eqns. (1) and (2) one may derive [6] where we may express the α-dependent matrix entry as the explicit solutions to the generalized Euler decomposition problem g′′(α) ==++(ccˆˆ, ) gg g cosαω sin α =τ 23 2 3 12 13 1[1g2 ]3 2 (c)=(c3)(c2)(c1), where ci iicˆ stands for the vector-parameter φi 2 of the i-th rotation in the decomposition,τ i = tan , cˆ ∈ i.e., More 2 i with the standard notation a[ibj]=aibj−ajbi. On the other hand, the precisely, we have condition in eqn. (9) may be written in the equivalent form [7]. ± σ i τ= σε = −> γ 22 22 i ,,i ijk(g jk jk ) jk (3) ′ g12γ 31 −−−≤≤+−−(11 g12)( γγ 32) gg 23 12 31 (11 g12)( γ 31 ) ωi ±∆ where εijk denotes the Levi-Civita symbol. Furthermore, we make use Moreover, using a standard trigonometric substitution one may of the notation express gij= (ccˆˆ i,,cc j) γ ij= ( ˆ i ,R (c) ˆj ) g23′ (α) = Acos( αα −+0) gg 12 13 as well as with t 2 ωi =×=×=(cˆ1 c,ˆR (c) cˆ3), ωω 2( cˆ 1cc ˆˆ 23 ,, c ) 3 ((c)cˆˆ12 ,c × ˆ 3) 2 ω2 Ag=( 1[ g ]3) +=ωα 2, 0 arctan 12 g 1[12 g ]3 and the necessary and sufficient condition for the existence of real solutions This allows for writing the above inequalities in the simpler form 1 g12γ 31 cos(γ12+− γ 31) gg 12 13 ≤ Acos(αα −≤0 ) cos( γ12 −− γ 31) gg 12 13 ∆=gg2110 23 ≥ (4) γ 31g 32 1 where we make use of the notation γij =arccosgij = (ccˆˆ i , j) ,γγ ij = arccosij = (cˆ i , cˆ j ) Whenever any of the conditions γji=gji (i≠j) holds, one may decompose into a pair of rotations about the i-th and the j-th axis as Note that since we have the restriction |cos(α−α )|≤1, formula ()c= (ττ , ccˆˆ ) ( ) 0 j j ii with eqn. (15) determines a real interval for the shift angle α for arbitrary γ − jj 1 γ ii −1 compound rotation if and only if ττij=οο, = (5) ωωij cos(γγ12−−≥−+−≤ 31) gg 12 13 A, cos(γγ12 31) gg 12 13 A ο ο t ωi =ccˆˆ × , (c)c ˆ ωi =(c)ccˆˆ, ×c ˆ where the notation ( ijj ) and ( ii j) is used. With On the other hand, one has i=1,j=2 in particular, these expressions apply also to the gimbal lock ω 2=det{g } ⇒= A 1 −gg22 1 − setting 2 ij ( 12)( 13 ) ccˆˆ3 =±⇔± 1γ 31 1 (6) so the above conditions may be written also as cos(γγ12−≥ 31 ) cos( γγ12 + 31 ) in which the first equality in eqn. (5) yields the scalar parameter of ϕ1 (10) cos(γγ12+≤ 31 ) cos( γγ12 − 31 ) ± ϕ3.[2,6]. which finally yields for the unknown parameter2 The Shift Parameter Construction γγ12− 13 −≤≤ γγ 12 312 γγ 12 + 13 (11) Let us consider a specific type of decompositions involving four rotations about three initially given axes.