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Analytic solutions to two quaternion attitude estimation problems

Yujendra Mitikiri, Kamran Mohseni

TL;DR

This work delivers analytic, geometry-based quaternion attitude estimators for two canonical problems: (i) two-vector measurements and (ii) rate measurement with a single vector. By formulating both as optimization problems in the quaternion geometry, the authors derive closed-form solutions that are instantaneous and exact, while naturally enforcing geometric constraints via feasibility cones. The approach unifies and extends classic methods such as TRIAD and Wahba, and provides connections to EKF and ECF through projection-based filtering and bias observability, validated through simulations and real experiments. The resulting framework supports frugal sensor suites, zero-latency estimates, and robust handling of noise and biases, with a consistent geometric interpretation throughout.

Abstract

This paper presents solutions to the following two common quaternion attitude estimation problems: (i) estimation of attitude using measurement of two reference vectors, and (ii) estimation of attitude using rate measurement and measurement of a single reference vector. Both these problems yield to a direct geometric analysis and solution. The former problem already has a well established analytic solution in literature using linear algebraic methods. This note shows how the solution may also be obtained using geometric methods, which are not only more intuitive, but also amenable to unconventional extensions. With respect to the latter problem, existing solutions typically involve filters and observers and use a mix of differential-geometric and control systems methods. However, no analytic solution has yet been reported to this problem. In this note, both the problems are formulated as optimization problems, which can be solved analytically to obtain a unique closed-form solution. The analytic attitude estimates are (i) instantaneous with respect to the measurements, thus overcoming the latency inherent in solutions based upon negative feedback upon an error, which can at best show asymptotic convergence, (ii) exact, thus overcoming errors in solutions based upon linear methods, and (iii) geometry-based, thus enabling imposition of geometric inequality constraints.

Analytic solutions to two quaternion attitude estimation problems

TL;DR

This work delivers analytic, geometry-based quaternion attitude estimators for two canonical problems: (i) two-vector measurements and (ii) rate measurement with a single vector. By formulating both as optimization problems in the quaternion geometry, the authors derive closed-form solutions that are instantaneous and exact, while naturally enforcing geometric constraints via feasibility cones. The approach unifies and extends classic methods such as TRIAD and Wahba, and provides connections to EKF and ECF through projection-based filtering and bias observability, validated through simulations and real experiments. The resulting framework supports frugal sensor suites, zero-latency estimates, and robust handling of noise and biases, with a consistent geometric interpretation throughout.

Abstract

This paper presents solutions to the following two common quaternion attitude estimation problems: (i) estimation of attitude using measurement of two reference vectors, and (ii) estimation of attitude using rate measurement and measurement of a single reference vector. Both these problems yield to a direct geometric analysis and solution. The former problem already has a well established analytic solution in literature using linear algebraic methods. This note shows how the solution may also be obtained using geometric methods, which are not only more intuitive, but also amenable to unconventional extensions. With respect to the latter problem, existing solutions typically involve filters and observers and use a mix of differential-geometric and control systems methods. However, no analytic solution has yet been reported to this problem. In this note, both the problems are formulated as optimization problems, which can be solved analytically to obtain a unique closed-form solution. The analytic attitude estimates are (i) instantaneous with respect to the measurements, thus overcoming the latency inherent in solutions based upon negative feedback upon an error, which can at best show asymptotic convergence, (ii) exact, thus overcoming errors in solutions based upon linear methods, and (iii) geometry-based, thus enabling imposition of geometric inequality constraints.

Paper Structure

This paper contains 14 sections, 102 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notation, definitions, and problem formulation
  3. Problem 1. Estimation using measurements of two reference vectors
  4. Problem 2. Estimation using rate measurement and measurement of single vector
  5. Nature of measurements of reference vector and angular velocity
  6. Attitude quaternion estimation
  7. Attitude estimation using two vector measurements
  8. Attitude estimation using single vector measurement and rate measurement
  9. Effect of noise in measurements on the estimation
  10. Observability and estimation of gyroscopic bias
  11. Simulation results
  12. Experimental validation of geometric attitude estimation using rate and single vector measurement
  13. Conclusion
  14. Acknowledgements

Figures (10)

  • Figure 1: Possible attitudes of a minimal rigid body formed out of three non collinear points (represented by the triangular patch) consistent with a measurement of a single vector $\mathbf{h}$. The subspace is a cone of revolution about the vector being measured.
  • Figure 2: Left: Possible axes to rotate the rigid body about, in order to measure reference vector $h$ as $b$ in the body axes. The rotation axes lie in the unit great circle spanned by $n_1 = b\times h/\|b\times h\|$, $n_2 = (b + h)/\|b + h\|$, $n_3 = -n_1$, $n_4 = -n_2$. Right: A visual depiction of the covering of the 2-sphere by the body $x$-axis using all rotations on the feasibility cone, $Q_b$. The rigid body is being rotated so as to measure the reference vector $h$ as $b$ in the body frame. In order to obtain this measurement, the body may be rotated (by differing amounts) about the set of unit vectors spanned by $n_1$ and $n_2$. As the rotation axis varies over the unit great circle spanned by these basis elements, the body $x$-axis sweeps great arcs over the 2-sphere that eventually cover all of it. Simultaneously, the yaw angle of the second rotation of the decomposition of a rotation goes from $-2\pi$ to $2\pi$. The color of the great arc is gradually varied from blue to red as the rotation axis begins at $n_1$ and goes through $n_2$, $n_3$, $n_4$, back to $n_1$.
  • Figure 3: A visual depiction of the solutions presented in Theorems \ref{['thm:twovecsoln']} and \ref{['thm:snglvecratesoln']}. The image on the left shows the two solutions $\check{r}_1 \otimes \check{q}$ (dotted triangle) and $\check{r}_2 \otimes \check{p}$ (dashed triangle) of theorem 3. The figure on the right shows the solution $\check{q}$ (solid triangle) of theorem 4 obtained by projecting the integrated attitude $\check{p}$ (dashed triangle) onto the feasibility cone of vector measurement $b$.
  • Figure 4: Matlab simulations of full attitude estimation using two vector measurements. The left figure shows the results of applying the TRIAD solution and using the geometric method of Theorem \ref{['thm:twovecsoln']}. The figure on the right shows that the two solutions are equal upto machine precision.
  • Figure 5: Matlab simulations of full attitude estimation using two vector measurements. The left figure shows the results of applying Davenport's $q$-method and an appropriate geometric filter using (\ref{['eqn:interp1']}). The figure on the right shows that the two solutions are equal upto machine precision.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Remark 10