In strapdown inertial navigation system (SINS) [1] and orientation systems (SIOS) [9] of the aerospace aerial vehicle the classical “Hamiltonian” quaternions of solid body rotation with the parameters of the Euler (Rodrigues – Hamilton) [1], [9] are now (from the beginning of the 70s of the last century) widely used. These quaternions are normalized (with unit norm) and they cannot be zero [1–10].
The possibility of using an unnormalized quaternion for SINS with no single the norms, depending on the angle of the final Euler rotation of solid body first is shown in [5 (2000), 10 (1999)]. Such quaternions are obtained by multiplying the normalized Hamiltonian quaternions of rotation (as unit vectors of the real fourdimensional space) by an arbitrary function of the angle of Euler rotation. They belong to the sets of nonHamiltonian quaternions of the solid body “full” rotation.
The paper examines the new (previously published in [6]) unnormalized quaternions of rotation forming a set of nonHamiltonian quaternions of the solid body “halfrotation”.
NonHamiltonian unnormalized quaternions of halfrotation are exceptional by virtue of their properties, in particular, the heterogeneity of systems of four kinematic linear differential equations corresponding to these quaternions.
NonHamiltonian quaternions of the halfrotation
We are considering two types of nonHamiltonian, quaternions of the halfrotation of solid body:
where u0 = 1 – λ0; v0 = 1 + λ0; is the unit vector of Euler’s axis of finite rotation (turn) of the solid body in threedimensional Euclidean vector space [1; 3; 9]; φ is the Euler final rotation angle.
Parameter λ0 and coordinates λ_{n} (n = 1, 2, 3) of three dimensional vector (coordinate orthonormal basis with unit vectors related to a solid body) are Euler (Rodrigues – Hamilton as a function of the angle φ) real parameters [1; 3; 9; 10]. They define the classic Hamiltonian quaternion of “full” rotation [1; 3]:
with unit norm
Quaternions U, V are considered here as nonHamiltonian quaternions of halfrotation of solid body and turn out as a result of multiplication of nontraditional new normalized quaternion of halfrotation
respectively on the modules
(i.e. ). This normalized “Hamiltonian” quaternions of halfrotation Р, М are regarded as vectors in the real fourdimensional vector space.
The different sets of the halfrotation quaternions determined by the generalized nonHamiltonian quaternions of the halfrotation UC = cUU; VC = cVV, where cU, cV are arbitrary constant coefficients. When cU = cV = 1 viewed quaternions U, V are obtained.
Unlike quaternions Λ, unnormalized quaternions U, V can be zero (at φ = 0 and φ = 2π respectively) and their modules depend on angle φ. Therefore, they are of special practical interest in solving two major problems: inertial sensing and inertial attitude control of the solid body provided that the shortest turns (at angles φ < π and φ > π) are ensured.
Quaternions U, V are exceptional (from the set of possible nonHamiltonian unnormalized quaternions of rotation [1; 5; 6; 10]) as those quaternions and their corresponding kinematic differential equations and groups, group quaternions algebras of rotation have a number of special or unique properties.
By the way for example, quaternions U, V in addition to going to zero, have a common vector and their norms are equal to doubled scalar parts:
(1)
where are conjugate quaternions.
In addition, the following equalities hold: and unlike inequality where (°) is the sign algebraic operations “Hamiltonian” quaternion multiplication [1; 3].
Quaternion differential kinematic equations
Quaternion kinematic differential equations for “proper” quaternions [1; 9, p. 109] U, V, are linear, but not homogeneous. Those equations are obtained from the known [1] linear kinematic equations for quaternion Λ by substitution of variable λ_{0} with variables u0, v0 and are as follows:
(2)
where is the angular velocity quaternion; is the vector of absolute rotational velocity of the solid body; is the relative derivatives of quaternions in time.
The equations (2) have a joint first integral u0 + v0 = 2.
These equations because of their inhomogeneity are of special interest for the solution of tasks of synthesis of highprecision conical precession computer algorithms of SIOS (the sixth or tenth order of accuracy) using Taylor’s rows [9].
The formulas for the multiplication of quaternions of the halfrotation
The multiplication formulas (rules, laws) [9, p. 109] of proper nonHamiltonian quaternions U, V, are obtained from the classic (group) [1; 3] multiplication formulas of normalized own quaternions Λ by substitution of quaternion Λ with quaternions U, V, according to the following formulas
where is a scalar unit quaternion; zero vector.
For two sequential finite rotations (turns) of the solid body, the group multiplication formulas of normalized quaternions Λ and nonHamiltonian quaternions U, V are written in symbolic form as:
as well as:
(3)
where Λ, U, V are the resulting rotation quaternions, Λ1, U1, V1 are the first rotation quaternions, Λ2, U2, V2 are the second rotation quaternions; (⊗) is a conventional sign of the group (nonHamiltonian) multiplication [6; 10] of any nonnormalized quaternions; (°) is a sign of the algebraic operation of Hamiltonian multiplication.
The formula (3) includes the operation of addition of quaternions, in contrast of the multiplication formulas of the classical Hamiltonian quaternions with the parameters of the Euler [1; 3; [4].
The group of nonHamiltonian quaternions of the halfrotation
The quaternion sets Λ, U, V, form a fourdimensional quaternions representations of threedimensional rotations classical groups [3; 4; 6] – a groups of nonHamiltonian quaternions of threedimensional rotations and halfrotation of the solid body or of quaternion groups of threedimensional rotations and halfrotation with the above group multiplication formulas (3).
Multiplication formulas (3) quaternions U, V determines their name “nonHamiltonian quaternions of the halfrotation”.
The following equalities follow from the above formulas:
(4)
where are zero quaternion; is a zero vector.
These equalities show that unit elements in groups of quaternions of U, V are respectively the zero quaternion and the doubled single quaternion 2Е4, and reverse quaternions U–1, V–1 are equal to the conjugate
NonHamiltonian quaternion algebra of the halfrotation
Unnormalized quaternions space U, V, together with their multiplication formulas (3) (the algebraic operations), determined the actual new, associative, noncommutative and unnormalized group [11, p. 259] of quaternions algebras of halfrotation with singlevalued division and without zero divisors [11; 12] (since these group algebras and group there is no zero divisors).
Multiplicity of the quaternions U, V forms a linear fourdimensional Euclidean vector space, while the Hamiltonian quaternions rotation Λ not form a vector space, since haven’t zero quaternions.
By analogy with the algebra of Hamiltonian quaternions Λ of rotation the exceptional quaternions algebras U, V of halfrotation are further endowed [4, p. 103–104] the structures of:
1) the commutative group under addition;
2) the noncommutative, associative fourdimensional algebra of division over the real. Thus the operations of addition and multiplication group (3) are distributive [3, p. 32].
Application of nonHamiltonian quaternions of the halfrotation in the< problems of the control orientation
Parameters of quaternions of U, V are used for the solution of tasks of control of orientation of the spacecraft (SC), as solid body, in positive definite quaternion functions fu and fv Lyapunov of a square look [5; 10]:
(5)
where α_{u}, β_{u}, γ_{u} > 0 and α_{v}, β_{v}, γ_{v} > 0; Аu, Av are definitely positive symmetric constant operators; is the momentum kinematics vector of the spacecraft; J is the operator (tensor) of inertia of the spacecraft; is the angular velocity vector of the spacecraft.
To ensure control shortest reversals spacecraft function is used fu when u0 < 1, v0 > 1 (0 < φ < π), or function fv when u_{0} > 0, v0 < 0 (π < φ < 2φ).
With an appropriate choice of formulas determine the vector of control points (as described, for example, in [10]) a negative definition of the derivative of Lyapunov functions in time provides. The result is the asymptotic stability of the processes controlling the orientation of the spacecraft and its shortest spreads throughout the range of variation of the angle from 0° to 360°.
Application of nonHamiltonian quaternions of the halfrotation in the algorithms of the orientation determine
Parameters – the coordinates of exceptional quaternions U, V used in control algorithms by orientation of spacecraft are calculated on computer algorithms of SIOS with are similar known algorithm for computing the classical quaternions rotation Euler (Rodrigues – Hamilton) parameters [1; 2; 7–9; 14–18]. This calculation algorithms parameters quaternions U, V easily obtained from the many known algorithms for calculating parameters of Euler (Rodrigues – Hamilton) by simply replacing the scalar parameter λ_{0} on the parameters u0 and v0, respectively.
Based quaternion U, V may also be prepared by new biquaternions SINS algorithms [1].
The onestep algorithms of the third and fourth orders of accuracy in the “scaled” [9, p. 78, 79] quaternion type 0,5U used in the “HARTRON” Corp. (Kharkov, Ukraine), in the task of determining the orientation of the spacecraft [13].
Of particular practical interest now becomes a fourstep algorithm of the fourth – sixth order accuracy [1; 2; 9; 14; 15; 17; 18], it is possible recurrence computing quaternion U, V with a time step H = 4h (h – a constant and minimum possible sample rate in the computer SINS of signals gyroscopes in time).
The article [7; 8] shows that the fourstep algorithms are more effective for use in SINS than the onestep, twostep and threestep algorithms. These algorithms are used intermediate orientation parameters [9, p. 144] – the coordinates φ_{N+4,k} (k = 1, 2, 3) small vector characterizing finite Euler rotation of the object to a small angle for a time equal to step H. The algorithms for computing these parameters may be represented by a generalized fourstep algorithm of the form [9, p. 172]
(6)
where qN+4 = q–2 + q–1 + q1 + q2 ; q–2, q–1, q1, q2 are column matrix, composed of angular increments corresponding quasicoordinates (gyro signal) q_{α} (α = –2, –1, 1, 2) generated in the onboard computer SIOS or SINS on four successive “small” steps h poll gyroscopes; Q–2, Q–1, Q1, Q2 are the corresponding skewsymmetric matrix.
The values of the constant coefficients av (v = 1…4) to (6), that determine the specific form of the case considered algorithms fourth order of accuracy [9, p. 173], are presented in Table 1 in the form of fractions. Algorithms 1, 2, 3, 5 are given in [9, p. 169; 153; 173; 157], the algorithm 4 – article [14] (“smoothing” algorithm of the fourth order obtained on the basis of Chebyshev polynomials).
Algorithm 3 was first published in 1986 [7] and was also considered in the paper [8] (1987). The monograph [9, p. 158] in the algorithm (3) under the number (3.3.45) contains a typo (instead of the coefficient a4 = 32/45 printed a4 = 32/55).
Table 2 shows for comparison the values of constant speed calculation drift of the algorithms (with conical vibrations of SIOS gyroscopes block [14] with conditions: nutation angle – 1 deg, the frequency of vibrations of tapered – 10 Hz, step with computing – 0,01 s) obtained in computer simulations by the method of the parallel accounts [9, p. 218]. As can be seen from Table 2, algorithm 3 is significantly superior in accuracy and other algorithms are substantially socalled conical algorithm [15] (the actual sixthorder of accuracy). Further analysis showed the benefits of the algorithm 3 and also in operation performance [8; 9].
The algorithm 3 (as the main part of the calculation algorithm parameters RodriguesHamilton) has been implemented [2, p. 316] in the laser system “SINS85” in serial production [16; 19; 20] since 2002 and is designed for use on aircraft Il96300, Tu204, Tu334. Modification of “SINS85” (“SINS77”, “SIMST”, “SINS SP1”, “SINS SP2”) are used on the aircraft An70, Tu95, Tu160, Tu214, Su35, T50, Yak130 [21].
Of particular interest is the possibility of using adaptive conical algorithms [18] for the calculation of the parameters nonHamiltonian quaternions of halfrotation in SINS. There is the only one optimal among the fourstep algorithms the best in terms of accuracy and operation performance adaptive algorithm conical (algorithm 6 of the 6th order from Tables 1, 2). It is obtained based on the algorithm (6) with coefficients insist on a conical motion. This configuration by choosing values of the coefficient b23 in the formulas (3.3.107) of [9, p. 173]. This algorithm is performed complete (ideal) compensation conical error due coefficients k05, k14, k23 in square terms of the asymptotic estimates (4.3.31) constant speed computing driftorder terms O(h^{6}) when ϑ > 0 (ϑ – nutation angle) [9, p. 215]. The accuracy of the algorithm, as shown by computer simulation exceeds the accuracy of the algorithm 3 a decimal (2,2·10^{–5} deg/h) under the conditions of calculation, the relevant Table 2.
Table 1
The constant coefficients of fourstep algorithms
Factors 
Number of algorithm 

1 
2 
3 
4 
5 
6 

а1 
0 
0 
22/45 
184/315 
–74/45 
534/945 
а2 
16/9 
0 
22/45 
112/315 
–9/2 
486/945 
а3 
0 
4/3 
22/45 
212/315 
86/45 
414/945 
а4 
0 
0 
32/45 
52/105 
0 
696/945 
Table 2
The constant velocity of the drift computing of fourstep algorithms
Option 
Number of algorithm 

1 
2 
3 
4 
5 
6 

The actual order of accuracy 
4 
4 
6 
6 
6 
6 
The drift velocity, deg/h 
2,5 
1,4 
3,9·10^{–4} 
9,6·10–2 
1,1·10^{–2} 
2,2·10^{–5} 
Optimum conical algorithm 6 exceeds the accuracy even of the fourstep algorithm American company Litton [15] providing for filtering signals of laser gyroscopes [21; 22]. The computational complexity of optimal algorithm 6 equal to the computational complexity of the algorithm 4 and the algorithm of the company Litton.
There is also the only one among the fivestep algorithms the optimal conical algorithm of 6th order with the ideal correction of the conical error. A method for constructing such an algorithm and computer study of its accuracy and operation performance based on asymptotic estimates of similar cases fourstep algorithm [9, p. 218, p. 249–255].
Conclusion
The possibility of using nonHamiltonian quaternions of halfrotation in strapdown inertial guidance and control is shown. In contrast to the classical Hamiltonian normalized quaternions of rotations the considered nonHamiltonian halfrotation quaternions can be zero and their modules and norms depend on the corner of the end Euler rotation.
The parameters of the nonHamiltonian quaternions of halfrotation are appropriate to use in advanced SIOS and SINS of aerospace aircrafts, along with the classic parameters of Euler (Rodrigues–Hamilton), or instead of them.