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Recovery of Bennu’s orientation for the OSIRIS-REx mission: implications for the spin state accuracy and geolocation errors

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Abstract

The goal of the OSIRIS-REx mission is to return a sample of asteroid material from near-Earth asteroid (101955) Bennu. The role of the navigation and flight dynamics team is critical for the spacecraft to execute a precisely planned sampling maneuver over a specifically selected landing site. In particular, the orientation of Bennu needs to be recovered with good accuracy during orbital operations to contribute as small an error as possible to the landing error budget. Although Bennu is well characterized from Earth-based radar observations, its orientation dynamics are not sufficiently known to exclude the presence of a small wobble. To better understand this contingency and evaluate how well the orientation can be recovered in the presence of a large 1\(^{\circ }\) wobble, we conduct a comprehensive simulation with the NASA GSFC GEODYN orbit determination and geodetic parameter estimation software. We describe the dynamic orientation modeling implemented in GEODYN in support of OSIRIS-REx operations and show how both altimetry and imagery data can be used as either undifferenced (landmark, direct altimetry) or differenced (image crossover, altimetry crossover) measurements. We find that these two different types of data contribute differently to the recovery of instrument pointing or planetary orientation. When upweighted, the absolute measurements help reduce the geolocation errors, despite poorer astrometric (inertial) performance. We find that with no wobble present, all the geolocation requirements are met. While the presence of a large wobble is detrimental, the recovery is still reliable thanks to the combined use of altimetry and imagery data.

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Acknowledgements

The authors acknowledge support from the NASA New Frontiers OSIRIS-REx project. This work was supported by NASA contract NNM10AA11C. We thank two anonymous reviewers and the Editor-in-Chief for their comments which helped improve the manuscript.

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Correspondence to Erwan Mazarico.

Appendix

Appendix

1.1 Computing the planetary orientation parameters through dynamics

Clearly we need to be able to compute the orientation angles of Sect. 3.1 at epoch t in the dynamic case. However, it is more natural to first compute related quantities at t that are defined explicitly in the rotational equations of motion rather than the angles and their rates. This essentially involves the numerical integration of the body-fixed axes in the J2000 frame. This in turn involves the numerical integration of the angular velocities about each body-fixed axis. From the axes, we may compute the orientation angles at any time. We may compute the initial state of the axes from the initial orientation angles.

With \(\alpha \) the right ascension and \(\delta \) the declination, the rotation from the body-fixed frame to the J2000 frame is given by

$$\begin{aligned} \mathbf {R}= \left( { \begin{array}{ccc} -S_{\alpha } C_W-C_{\alpha } S_{\delta } S_W &{} S_{\alpha } S_W-C_{\alpha } S_{\delta } C_W &{} C_{\alpha } C_{\delta } \\ C_{\alpha } C_W-S_{\alpha } S_{\delta } S_W &{} -C_{\alpha } S_W-S_{\alpha } S_{\delta } C_W &{} S_{\alpha } C_{\delta } \\ C_{\delta } S_W &{} C_{\delta } C_W &{} S_{\delta } \end{array}}\right) , \end{aligned}$$
(5)

where \(C_{[\cdot ]}\) and \(S_{[\cdot ]}\) denote \(\cos {(\cdot )}\) and \(\sin {(\cdot )}\), respectively, of an angular argument. If the cross-product between two generic vectors \(\mathbf {u}\) and \(\mathbf {v}\) is expressed as \(\mathbf {u}\times \mathbf {v}=\mathbf {E}_{u}\mathbf {v}\), where

$$\begin{aligned} \mathbf {E}_{u}= \left( { \begin{array}{ccc} 0 &{} -u_z &{} u_y \\ u_z &{} 0 &{} -u_x \\ -u_y &{} u_x &{} 0 \end{array}}\right) \end{aligned}$$
(6)

is a traceless, skew-symmetric matrix and \(\mathbf {u}=\left[ {u_x\; u_y\; u_z}\right] ^\mathrm{T}\), then this rotation is a function of time and can be interpreted in one of two ways: (i) its orthogonal columns define the body-fixed axes of the rigid object in the J2000 frame such that the rate of change of this rotation is given by

$$\begin{aligned} \dot{\mathbf {R}}(t)=\mathbf {E}_{\tilde{\omega }}(t)\mathbf {R}(t), \end{aligned}$$
(7)

where \(\varvec{\tilde{\omega }}(t)\) is the angular velocity of the object in the J2000 frame, or (ii) its orthogonal rows define the J2000 axes in the body-fixed frame of the rigid object such that the rate of change is given by

$$\begin{aligned} \dot{\mathbf {R}}^\mathrm{T}(t)=\mathbf {E}_{\omega }^\mathrm{T}(t)\mathbf {R}^\mathrm{T}(t), \end{aligned}$$
(8)

where \(\varvec{\omega }(t)\) is the angular velocity of the object in the body-fixed frame. Equation 8 leads to the following relationship

$$\begin{aligned} \varvec{\omega }(t)= \left( { \begin{array}{c@{\quad }c@{\quad }c} C_{\delta } S_W &{} -C_W &{} 0 \\ C_{\delta } C_W &{} S_W &{} 0 \\ S_{\delta } &{} 0 &{} 1 \end{array}}\right) \left( { \begin{array}{c} \dot{\alpha }(t) \\ \dot{\delta }(t) \\ \dot{{W}}(t) \end{array}}\right) . \end{aligned}$$
(9)

The Euler equations of motion for a rigid body with one point fixed (Goldstein 1965) relate this angular velocity to its angular acceleration in the body-fixed frame under the affects of an applied torque, \(\varvec{\tilde{\tau }}(t)\), in the J2000 frame (here considered to be due only to the sun) such that

$$\begin{aligned} \varvec{\dot{\omega }}(t)=\mathbf {I}_c^{-1}\left[ {\mathbf {R}^\mathrm{T}(t) \varvec{\tilde{\tau }}(t)-\varvec{\omega }(t)\times \mathbf {I}_c \varvec{\omega }(t)}\right] , \end{aligned}$$
(10)

where \(\mathbf {I}_c\) is the moment of inertia matrix in the body-fixed cartesian frame. Equations 8 and 10 are the differential equations that we numerically integrate. Although Eq. 8 is of primary interest, it cannot be integrated independently from Eq. 10. Furthermore, the numerical integration of Eq. 10 provides the vector corresponding to the instantaneous spin axis. We may compute the initial conditions of the spin axis, \(\varvec{\omega }(t)\), from the initial orientation angles and their rates using Eq. 9.

If we define \(\varvec{\rho }(t)=\mathrm{vec}\left( {\mathbf {R}^\mathrm{T}(t)}\right) \), where the \(vec\left( {\cdot }\right) \) operator stacks the columns of its matrix argument, then together, Eqs. 8 and 10 and initial conditions describe the time evolution of 12 variables, i.e., the three elements of \(\varvec{\omega }(t)\) and the nine elements of \(\varvec{\rho }(t)\), and can be represented by the augmented system

$$\begin{aligned} \varvec{\dot{\zeta }}(t)=\mathbf {f}(t,\varvec{\zeta }(t),\mathbf {s}),\quad \varvec{\zeta }(t_0)=\varvec{\zeta }_0 \end{aligned}$$
(11)

where

$$\begin{aligned} \varvec{\zeta }(t)= \left( { \begin{array}{c} \varvec{\omega }(t) \\ \varvec{\rho }(t) \end{array}}\right) , \end{aligned}$$
(12)

and \(\mathbf {s}\) is a vector of length 6 containing the upper-triangular elements of the symmetric matrix \(\mathbf {I}_c\), i.e., \(\left( I_{xx}, I_{yy}, I_{zz}, I_{xy},\right. \left. I_{xz}, I_{yz}\right) \). Thus, Eq. 11 may be integrated from an initial epoch \(t_0\) to time t using Eqs. 5 and 9 as a link between \(\mathbf {R}(t)\), \(\varvec{\omega }(t)\), and the orientation angles and their rates.

In the case where \(\mathbf {R}^\mathrm{T}(t)=\left[ {\mathbf {e}_x(t)\;\mathbf {e}_y(t)\; \mathbf {e}_z(t)}\right] \) represents the J2000 axes \(\left( {\mathbf {e}_x,\mathbf {e}_y,\mathbf {e}_z}\right) \) in the principal axis frame of the body, then \(\mathbf {I}_c\) is diagonalized and Eq. 11 can be written in the more familiar form

$$\begin{aligned} \left( { \begin{array}{c} \dot{\omega }_x(t) \\ \dot{\omega }_y(t) \\ \dot{\omega }_z(t) \\ \dot{\mathbf {e}}_x(t) \\ \dot{\mathbf {e}}_y(t) \\ \dot{\mathbf {e}}_z(t) \end{array}}\right) = \left( { \begin{array}{c} \left[ {\tau _x(t)-\left( {I_{zz}-I_{yy}}\right) \omega _y(t) \omega _z(t)}\right] /I_{xx} \\ \left[ {\tau _y(t)-\left( {I_{xx}-I_{zz}}\right) \omega _x(t) \omega _z(t)}\right] /I_{yy} \\ \left[ {\tau _z(t)-\left( {I_{yy}-I_{xx}}\right) \omega _x(t) \omega _y(t)}\right] /I_{zz} \\ -\varvec{\omega }(t)\times \mathbf {e}_x(t) \\ -\varvec{\omega }(t)\times \mathbf {e}_y(t) \\ -\varvec{\omega }(t)\times \mathbf {e}_z(t) \end{array}}\right) , \end{aligned}$$
(13)

where \(I_{xx}\), \(I_{yy}\), and \(I_{zz}\) are the diagonal elements of \(\mathbf {I}_c\) and \(\tau _x\), \(\tau _y\), and \(\tau _z\) are now the torques in the principal axis frame.

When working in the principal axes frame, in the special case when \(I_{zz}\ge I_{xx}=I_{yy}\) and the torque is negligible, Eq. 13 becomes

$$\begin{aligned} \left( { \begin{array}{c} \dot{\omega }_x(t) \\ \dot{\omega }_y(t) \\ \dot{\omega }_z(t) \end{array}}\right) = \left( { \begin{array}{c} -\beta \omega _y(t) \omega _z(t) \\ \beta \omega _x(t) \omega _z(t) \\ 0 \end{array}}\right) , \end{aligned}$$
(14)

where \(\beta ={\left( {I_{zz}-I_{xx}}\right) \over I_{xx}}\), from which it is clear that \(\omega _z(t)\) is a constant, say \(\omega _{z0}\). This leads to

$$\begin{aligned} \left( { \begin{array}{c} \dot{\omega }_x(t) \\ \dot{\omega }_y(t) \end{array}}\right) = \left( { \begin{array}{cc} 0 &{}-\beta \omega _{z0}\\ \beta \omega _{z0} &{} 0 \end{array}}\right) \left( { \begin{array}{c} \omega _x(t) \\ \omega _y(t) \end{array}}\right) . \end{aligned}$$
(15)

The solution is

$$\begin{aligned} \left( { \begin{array}{c} \omega _x(t) \\ \omega _y(t) \\ \omega _z(t) \end{array}}\right) = \left( { \begin{array}{ccc} \cos (\beta \omega _{z0} t) &{} ~-\sin (\beta \omega _{z0} t) &{}~~ 0 \\ \sin (\beta \omega _{z0} t) &{} ~\cos (\beta \omega _{z0} t) &{}~~ 0 \\ 0 &{} ~~0 &{}~~ 1 \end{array}}\right) \left( { \begin{array}{c} \omega _x(0) \\ \omega _y(0) \\ \omega _z(0) \end{array}}\right) , \end{aligned}$$
(16)

and as a result, \(T=2\pi \beta ^{-1}\omega _{z0}^{-1}\) is the period of the wobble.

1.2 Estimating the planetary orientation parameters and moments of inertia in the dynamic case

The rotational dynamical equations discussed in the previous section have a direct parallel with the orbital dynamical equations used in software like GEODYN. The orbital dynamical differential equations have associated variational differential equations. The orbital variational equations are derived by differentiating the orbital dynamical equations by each force model parameter (including the initial states). They yield partial derivatives of the orbital state at any time with respect to the initial state and also with respect to force model parameters, e.g., gravitational coefficients. Likewise, the rotational dynamical equations have associated variational equations yielding partial derivatives of the orientation angles at any time with respect to the initial angular state (including angular rates) and also with respect to the other force model parameters, e.g., the moments of inertia. As discussed in Sects. 3.2 and 4.2.2, this double set of variational equations needs to be linked in order to include the orientation and moment of inertia parameters as force parameters when considering tracking measurements of the OSIRIX-REx spacecraft.

As with the computations in the previous section, we cast the rotational variational equations in terms of \(\mathbf {R}(t)\) and \(\varvec{\omega }(t)\), or \(\varvec{\zeta }(t)\), and not directly in terms of angles. Thus, from Eq. 11, the partial derivative of \(\varvec{\zeta }(t)\) with respect to \(\varvec{\zeta }_0\) (the initial rotational state) at time t, denoted as \(\varvec{\varPhi }(t)\), satisfies

$$\begin{aligned} \varvec{\dot{\varPhi }}(t)={\partial \mathbf {f}\over \partial \varvec{\zeta }}(t) \varvec{\varPhi }(t),\quad \varvec{\varPhi }(t_0)=\mathbf {I}, \end{aligned}$$
(17)

where \(\mathbf {I}\) is a \(12\times 12\) identity matrix. Likewise, the partial derivative of \(\varvec{\zeta }(t)\) with respect to \(\mathbf {s}\) (the independent moments of inertia) at time t, denoted as \(\varvec{\varPsi }(t)\), satisfies

$$\begin{aligned} \varvec{\dot{\varPsi }}(t)={\partial \mathbf {f}\over \partial \varvec{\zeta }}(t) \varvec{\varPsi }(t)+{\partial \mathbf {f}\over \partial \mathbf {s}}(t),\quad \varvec{\varPsi }(t_0)=\mathbf {0}, \end{aligned}$$
(18)

where \(\mathbf {0}\) is a \(12\times 6\) zero matrix. Note that Eqs. 17 and 18 differ in two aspects: the initial conditions and the use of explicit partial derivatives in the latter. Equations 11, 17, and 18 can be integrated together to yield \(\varvec{\zeta }(t)\) and its changes with respect to \(\varvec{\zeta }_0\) and \(\mathbf {s}\) and can be chained with partial derivatives based on Eqs. 5 and 9 to provide partial derivatives of \(\varvec{\zeta }_0\) with respect to the orientation angles and their rates at the initial time \(t_0\).

Finally, it should be mentioned that Bennu is located in a low-torque region of space such that \(\varvec{\tilde{\tau }}(t)\) is likely to be very small or negligible. In this case, Eq. 10 becomes

$$\begin{aligned} \varvec{\dot{\omega }}(t)=-\mathbf {I}_c^{-1} \left[ {\varvec{\omega }(t)\times \mathbf {I}_c \varvec{\omega }(t)}\right] , \end{aligned}$$
(19)

which means that \(\varvec{\dot{\omega }}(t)\) is invariant to a scale factor on \(\mathbf {I}_c\). Thus, any attempt to independently estimate all elements of \(\mathbf {s}\) will fail. In that case, additional constraints must be placed on \(\mathbf {s}\), such as fixing the value of one of the elements of \(\mathbf {s}\) and solving for the remainder, or fixing \(I_{xx}^2+I_{yy}^2+I_{zz}^2\) to a given value, as suggested in Sect. 5.1.

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Mazarico, E., Rowlands, D.D., Sabaka, T.J. et al. Recovery of Bennu’s orientation for the OSIRIS-REx mission: implications for the spin state accuracy and geolocation errors. J Geod 91, 1141–1161 (2017). https://doi.org/10.1007/s00190-017-1058-2

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