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 four-dimensional space) by an arbitrary function of the angle of Euler rotation. They belong to the sets of non-Hamiltonian quaternions of the solid body “full” rotation.
The paper examines the new (previously published in [6]) unnormalized quaternions of rotation forming a set of non-Hamiltonian quaternions of the solid body “half-rotation”.
Non-Hamiltonian unnormalized quaternions of half-rotation are exceptional by virtue of their properties, in particular, the heterogeneity of systems of four kinematic linear differential equations corresponding to these quaternions.
Non-Hamiltonian quaternions of the half-rotation
We are considering two types of non-Hamiltonian, quaternions of the half-rotation 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 three-dimensional Euclidean vector space [1; 3; 9]; φ is the Euler final rotation angle.
Parameter λ0 and coordinates λn (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 non-Hamiltonian quaternions of half-rotation of solid body and turn out as a result of multiplication of non-traditional new normalized quaternion of half-rotation
respectively on the modules
(i.e. 
). This normalized “Hamiltonian” quaternions of half-rotation Р, М are regarded as vectors in the real four-dimensional vector space.
The different sets of the half-rotation quaternions determined by the generalized non-Hamiltonian quaternions of the half-rotation 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 non-Hamiltonian 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 λ0 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 high-precision 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 half-rotation
The multiplication formulas (rules, laws) [9, p. 109] of proper non-Hamiltonian 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 non-Hamiltonian 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 (non-Hamiltonian) multiplication [6; 10] of any non-normalized 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 non-Hamiltonian quaternions of the half-rotation
The quaternion sets Λ, U, V, form a four-dimensional quaternions representations of three-dimensional rotations classical groups [3; 4; 6] – a groups of non-Hamiltonian quaternions of three-dimensional rotations and half-rotation of the solid body or of quaternion groups of three-dimensional rotations and half-rotation with the above group multiplication formulas (3).
Multiplication formulas (3) quaternions U, V determines their name “non-Hamiltonian quaternions of the half-rotation”.
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 
Non-Hamiltonian quaternion algebra of the half-rotation
Unnormalized quaternions space U, V, together with their multiplication formulas (3) (the algebraic operations), determined the actual new, associative, non-commutative and unnormalized group [11, p. 259] of quaternions algebras of half-rotation with single-valued 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 four-dimensional 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 half-rotation are further endowed [4, p. 103–104] the structures of:
1) the commutative group under addition;
2) the non-commutative, associative four-dimensional algebra of division over the real. Thus the operations of addition and multiplication group (3) are distributive [3, p. 32].
Application of non-Hamiltonian quaternions of the half-rotation 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 αu, βu, γu > 0 and αv, βv, γv > 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 u0 > 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 non-Hamiltonian quaternions of the half-rotation 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 λ0 on the parameters u0 and v0, respectively.
Based quaternion U, V may also be prepared by new biquaternions SINS algorithms [1].
The one-step 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 four-step 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 four-step algorithms are more effective for use in SINS than the one-step, two-step and three-step algorithms. These algorithms are used intermediate orientation parameters [9, p. 144] – the coordinates φN+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 four-step 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 quasi-coordinates (gyro signal) qα (α = –2, –1, 1, 2) generated in the on-board computer SIOS or SINS on four successive “small” steps h poll gyroscopes; Q–2, Q–1, Q1, Q2 are the corresponding skew-symmetric 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 so-called conical algorithm [15] (the actual sixth-order 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 Rodrigues-Hamilton) has been implemented [2, p. 316] in the laser system “SINS-85” in serial production [16; 19; 20] since 2002 and is designed for use on aircraft Il-96-300, Tu-204, Tu-334. Modification of “SINS-85” (“SINS-77”, “SIMS-T”, “SINS SP-1”, “SINS SP-2”) are used on the aircraft An-70, Tu-95, Tu-160, Tu-214, Su-35, T-50, Yak-130 [21].
Of particular interest is the possibility of using adaptive conical algorithms [18] for the calculation of the parameters non-Hamiltonian quaternions of half-rotation in SINS. There is the only one optimal among the four-step 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 drift-order terms O(h6) 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 four-step 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 four-step 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 four-step 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 five-step 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 four-step algorithm [9, p. 218, p. 249–255].
Conclusion
The possibility of using non-Hamiltonian quaternions of half-rotation in strapdown inertial guidance and control is shown. In contrast to the classical Hamiltonian normalized quaternions of rotations the considered non-Hamiltonian half-rotation quaternions can be zero and their modules and norms depend on the corner of the end Euler rotation.
The parameters of the non-Hamiltonian quaternions of half-rotation 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.