Scientific journal
European Journal of Natural History
ISSN 2073-4972

ON THE NEW NON-HAMILTONIAN QUATERNIONS OF HALF-ROTATION AND THEIR APPLICATION TO PROBLEMS OF ORIENTATION

Panov A. 1 Ponomarenko S. 2
1 International Academy of Navigation and Motion Control
2 State Research Institute of Aviation
The application new (non-traditional) groups, algebras of the half-rotation solid body and their application in the orientation tasks for the strapdown inertial navigation system and orientation systems. Non-Hamiltonian quaternions of the half-rotation can be zero in contrast to the classical Hamiltonian normalized quaternions of the full-rotation with the parameters of the Euler (Rodrigues – Hamilton), their rates are not constant and depend on the Euler angles of final rotation.
non-Hamiltonian quaternion of rotation
half-rotation
groups and algebras of quaternions
strapdown inertial systems
guidance
navigation
control
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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:

Panov01.wmf Panov02.wmf

where u0 = 1 – λ0; v0 = 1 + λ0; Panov03.wmf Panov04.wmf Panov05.wmf Panov06.wmf 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 Panov07.wmf (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]:

Panov08.wmf

with unit norm

Panov09.wmf Panov10.wmf

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

Panov11.wmf Panov12.wmf Panov13.wmf

Panov14.wmf Panov15.wmf

Panov16.wmf

respectively on the modules

Panov17.wmf Panov18.wmf

(i.e. Panov19.wmf Panov20.wmf). 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 Panov21.wmf and their norms are equal to doubled scalar parts:

Panov22.wmf

Panov23.wmf (1)

where Panov24.wmf Panov25.wmf are conjugate quaternions.

In addition, the following equalities hold: Panov26.wmf and Panov27.wmf Panov28.wmf unlike inequality Panov29.wmf 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 Panov30.wmf for quaternion Λ by substitution of variable λ0 with variables u0, v0 and are as follows:

Panov31.wmf Panov32.wmf (2)

where Panov33.wmf is the angular velocity quaternion; Panov34.wmf is the vector of absolute rotational velocity of the solid body; Panov35.wmf Panov36.wmf Panov37.wmf 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

Panov38.wmf

where Panov39.wmf is a scalar unit quaternion; Panov40.wmf 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:

Panov41.wmf Panov42.wmf

Panov43.wmf

as well as:

Panov44.wmf

Panov45.wmf (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:

Panov46.wmf

Panov47.wmf (4)

where Panov48.wmf are zero quaternion; Panov49.wmf 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 Panov50.wmf Panov51.wmf

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]:

Panov52.wmf

Panov53.wmf (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; Panov54.wmf 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 Panov55.wmf 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]

Panov56.wmf (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.