Strapdown Attitude Computation: Functional Iterative Integration versus Taylor Series Expansion
Yuanxin Wu, Shanghai Jiao Tong University, China; Yury A. Litmanovich, Central Scientific and Research Institute “Elektropribor”, Russia
Location: Pavilion Ballroom East
Alternate Number 4
Attitude information is vitally important for moving objects in many areas including unmanned vehicle navigation and control, virtual/augmented reality, satellite communication, robotics, and computer vision. Integrating gyroscope-measured angular velocity information is an essential and self-contained way to acquire the attitude, rotation or orientation. A number of orientation parameters has been used for attitude computation, including but not limited to the Euler angle, the rotation vector, the direction cosine matrix and quaternion. Attitude computation is in essence to numerically solve the ordinary differential equation of these attitude parameters. In the early years of strapdown inertial navigation systems, Paul Savage tried the Picard-type iterative technique to integrate the direction cosine rate equations, and the NASA technical report made the angular velocity polynomial approximation from a sequence of gyroscope outputs and then integrated the direction cosine matrix rate by the Runge-Kutta method. Shortly after in 1970s, the modern-day strapdown attitude algorithm structure was established on the Taylor series expansion approach by Jordan and Bortz, which has unexceptionally relied on the approximate rotation vector for incremental attitude update. In parallel, a number of related fields employ the quaternion to deal with attitude computation, e.g., robotics, space applications and computational mathematics, where the structure-preserving attributes of geometric integration is mostly concerned.
It has long been believed that the modern-day attitude algorithm is already good enough for applications. However, the present dynamic applications and the precision gyroscopes under way demand more accurate attitude algorithms. In principle, recent advances have shown that higher attitude accuracy can be achieved by dealing with better or exact rate forms of attitude parameters and high-order numerical integration methods. The Rodrigues vector and quaternion were used for attitude computation by way of functional iterative integration and Chebyshev polynomial approximation. The functional iterative integration approach combined with Chebyshev polynomial approximation was developed independently in the navigation community, but lately found to closely resemble the so-called Picard-Chebyshev method that was dated back to as early as 1960s. In 1980s-1990s and even quite recently, it was employed and advanced by researchers in the field of astrodynamics for orbital determination. The Picard iterative technique actually originated from the Picard–Lindelöf theorem that has been commonly used to prove the existence and uniqueness of numerical solutions of ordinary differential equations, but it has been hardly exploited in engineering applications because the repeated computation of integrals are often conceived to be inconvenient and tedious. The Taylor series method experiences a similar story in that it is conceptually easy to work with but the high-order derivatives are taken as being tedious and complicated to calculate. To avoid the need for high-order derivatives, the Runge-Kutta methods were thus devised while attempting to retain the accuracy of the Taylor series approximation. The seminal paper by Miller in 1983 used a low-order Taylor series with low-order angular velocity approximation to solve the approximate rotation vector rate equation. Very recently, the Taylor series approach is employed to directly solve the direction cosine matrix rate equation, kind of a retrospective work into the early attempt.
Another trend of intensive investigations in the strapdown attitude research was the ad-hoc algorithm optimization by means of special tuning of the algorithm coefficients to reduce the coning drift under the assumed motions such as the classical/generalized coning motion, the regular precession and the stochastic angular motion. The cost one should pay is that the optimized algorithms rank below the corresponding initial algorithms in accuracy in practical situations of irregular angular motions, say maneuvers. Needless to say it is desirable for algorithms to exhibit the same order of accuracy regardless input attitude motions.
It should be noted that the angular velocity polynomial approximation from a sequence of gyroscope outputs is an integral part of both approaches, explicitly or implicitly. Hereby in this paper we consider zero and single integral of angular velocity as gyroscope measurements as they are common in real systems, although multiple integrals could also be accounted for. The purpose of this paper is to investigate the problem of strapdown attitude computation from the general perspective of solving the differential equations involved, so as to make a comprehensive approach comparison from the accuracy standpoint. It was motivated by a long fruitful discussion of the two authors about the actual superiority of the functional iterative approach over the traditional Taylor expansion approach.
This paper compares two basic approaches to solving ordinary differential equations, which form the basis for attitude computation in strapdown inertial navigation systems, namely, the Taylor series expansion approach that was used in the low-order form for deriving all mainstream algorithms and the functional iterative integration approach developed recently. They are respectively applied to solve the kinematic equations of major attitude parameters, including the quaternion, the Rodrigues vector and rotation vector. Specifically, the mainstream algorithms, which have unexceptionally based on the simplified rotation vector, are considerably extended by the Taylor series expansion approach using the exact rotation vector and recursive calculation of high-order derivatives. The functional iterative integration approach is respectively implemented on the normal polynomial and the Chebyshev psolynomial. Numerical results under the classical coning motion are reported to assess all derived attitude algorithms. It is revealed that in the relative frequency range when the coning to sampling frequency ratio is below 0.05-0.1 (depending on the chosen polynomial truncation order), all algorithms have the same order of accuracy if the same number of samples are used to fit the angular velocity over the iteration interval; in the range of higher relative frequency, the group of Quat/Rod/RotFIter algorithms (by the functional iterative integration approach combined with Chebyshev polynomial) perform better in both accuracy and robustness to the Runge phenomenon, thanks to the excellent numerical stability and powerful functional representation capability of the Chebyshev polynomial.