ExoMars 2016 Schiaparelli Module Trajectory and Atmospheric Profiles Reconstruction

Abstract

On 19^th October 2016 Schiaparelli module of the ExoMars 2016 mission flew through the Mars atmosphere. After successful entry and descent under parachute, the module failed the last part of the descent and crashed on the Mars surface. Nevertheless the data transmitted in real-time by Schiaparelli during the entry and descent, together with the entry state vector as initial condition, have been used to reconstruct both the trajectory and the profiles of atmospheric density, pressure and temperature along the traversed path.

The available data-set is only a small sub-set of the whole data acquired by Schiaparelli, with a limited data rate (8 kbps) and a large gap during the entry because of the plasma blackout on the communications.

This paper presents the work done by the AMELIA (Atmospheric Mars Entry and Landing Investigations and Analysis) team in the exploitation of the available inertial and radar data. First a reference trajectory is derived by direct integration of the inertial measurements and a strategy to overcome the entry data gap is proposed. First-order covariance analysis is used to estimate the uncertainties on all the derived parameters. Then a refined trajectory is computed incorporating the measurements provided by the on-board radar altimeter.

The derived trajectory is consistent with the events reported in the telemetry and also with the impact point identified on the high-resolution images of the landing site.

Finally, atmospheric profiles are computed tacking into account the aerodynamic properties of the module. Derived profiles result in good agreement with both atmospheric models and available remote sensing observations.

Appendices

Appendix A: Equations of Motion

Let be $M$ the inertial MMED reference frame and $G$ the GNC body frame. The EDM state is defined as follows

$$ \boldsymbol{x} = \bigl[ ^{M}\boldsymbol{v} \, ^{G} \boldsymbol{q}_{M} \, ^{M}\boldsymbol{p} \bigr] ^{T} $$

(A.1)

with the velocity and position expressed in Cartesian coordinates with respect to MMED frame and the attitude quaternion representing the transformation from the MMED frame to the GNC frame. In the following the used reference frames will be indicated only if needed.

We assume that both the acceleration $^{G}\boldsymbol{a}$ and angular rate $^{G}\boldsymbol{\omega }$ are given with respect to the GNC frame and are already compensated for offsets and biases. This results in the following set of equations

$$ \frac{d}{dt} \left [ \textstyle\begin{array}{c} \boldsymbol{v} \\ \boldsymbol{q} \\ \boldsymbol{p} \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{c} \boldsymbol{a}_{g} + \boldsymbol{C}^{T}(\boldsymbol{q}) \, \boldsymbol{a} \\ \boldsymbol{K}(\boldsymbol{q}) \, \boldsymbol{\omega } \\ \boldsymbol{v} \end{array}\displaystyle \right ] $$

(A.2)

where $\boldsymbol{C}(\boldsymbol{q})$ is the rotation matrix corresponding to the attitude quaternion, $\boldsymbol{K}(\boldsymbol{q})$ is the quaternion kinematic matrix and $\boldsymbol{a}_{g}$ is the gravitational acceleration. The quaternion is composed by a scalar and vectorial part $\boldsymbol{q} = [ \boldsymbol{q}_{v} q _{s} ] ^{T}$ and then

$$\begin{aligned} \boldsymbol{C}(\boldsymbol{q})& = \boldsymbol{I} - 2 q_{s} S(\boldsymbol{q}_{v}) + 2 S( \boldsymbol{q}_{v}) S( \boldsymbol{q}_{v}) \end{aligned}$$

(A.3)

$$\begin{aligned} \boldsymbol{K}(\boldsymbol{q}) &= \frac{1}{2} \left [ \textstyle\begin{array}{c} q_{s} \boldsymbol{I} + S(\boldsymbol{q}_{v}) \\ - \boldsymbol{q}_{v}^{T} \end{array}\displaystyle \right ] \end{aligned}$$

(A.4)

with the cross-product, skew-symmetric matrix operator $S$ defined as

$$ S(\boldsymbol{x}) = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & -x_{3} & x_{2} \\ x_{3} & 0 & -x_{1} \\ -x_{2} & x_{1} & 0 \end{array}\displaystyle \right ] $$

(A.5)

refer to Shuster (1993) for more details.

Inertial measurements are modeled using additive Gaussian white noise

$$\begin{aligned} ^{G}\boldsymbol{y}_{\omega } =& \, ^{G}\boldsymbol{w} + \boldsymbol{\epsilon } _{\omega } \end{aligned}$$

(A.6)

$$\begin{aligned} ^{G}\boldsymbol{y}_{a} =& \, ^{G}\boldsymbol{a} + \boldsymbol{\epsilon }_{a} \end{aligned}$$

(A.7)

hence the best available estimate of the acceleration and angular rate to be used in Eq. (A.2) and Eq. (A.14) are the measurements itself $\widehat{\boldsymbol{a}} = \boldsymbol{y}_{a}$ and $\widehat{\boldsymbol{\omega }} = \boldsymbol{y} _{\omega }$.

Velocity and position are modeled using additive errors

$$\begin{aligned} \boldsymbol{v} =& \widehat{\boldsymbol{v}} + \boldsymbol{\delta }_{v} \end{aligned}$$

(A.8)

$$\begin{aligned} \boldsymbol{p} =& \widehat{\boldsymbol{p}} + \boldsymbol{\delta }_{p} \end{aligned}$$

(A.9)

Since a quaternion is by definition unit length its covariance matrix is singular hence, to avoid numerical problems, a small angle multiplicative model is used (Lefferts et al. 1982; Markley 2003). The Rotation from MMED to GNC frame is represented as the composition of two consecutive rotations

$$ ^{G}\boldsymbol{q}_{M} = ^{G}\boldsymbol{q}_{\widehat{G}} \, \otimes \, ^{\widehat{G}}\boldsymbol{q}_{M} = \boldsymbol{\delta }_{q} \, \otimes \, ^{\widehat{G}}\boldsymbol{q}_{M} $$

(A.10)

the first quaternion rotates from inertial $M$ to estimated body frame $\widehat{G}$ the second quaternion rotates from the estimated body frame $\widehat{G}$ to the true body frame $G$. Since the first rotation is very small we can approximate the error quaternion $\boldsymbol{\delta } _{q}$ with a $3\times1$ error angle vector $\boldsymbol{\delta }_{\theta }$ as follow

$$ \boldsymbol{\delta }_{q} = \left [ \textstyle\begin{array}{c} \boldsymbol{u}\sin (\delta _{\theta }/2) \\ cos(\delta _{\theta }/2) \end{array}\displaystyle \right ] \approx \left [ \textstyle\begin{array}{c} \frac{1}{2} \boldsymbol{\delta }_{\theta } \\ 1 \end{array}\displaystyle \right ] $$

(A.11)

with $\boldsymbol{u}, \lVert \boldsymbol{u} \rVert =1$ the rotation axis and $\delta _{\theta }$ the rotation angle. In this hypothesis the corresponding rotation matrix results

$$ \boldsymbol{C}(\boldsymbol{\delta }_{q}) \approx \boldsymbol{I} - S(\boldsymbol{\delta }_{\theta }) $$

(A.12)

The state error is defined as $\boldsymbol{\delta }_{x} = [ \boldsymbol{\delta }_{v} \boldsymbol{\delta }_{\theta } \boldsymbol{\delta }_{p} ] ^{T}$ and evolves according to the following equation

$$ \frac{d}{dt} \boldsymbol{\delta }_{x} = \boldsymbol{A}\,\boldsymbol{\delta }_{x} + \boldsymbol{B}\, \boldsymbol{\epsilon } + \boldsymbol{\eta } $$

(A.13)

where $\boldsymbol{A}$ is the Jacobian of the state equation (A.2) with respect to the state $\boldsymbol{x}$ and $\boldsymbol{B}$ is the Jacobian with respect to the input noise $\boldsymbol{\epsilon } = [ \boldsymbol{\epsilon }_{\omega } \boldsymbol{\epsilon }_{a}]^{T}$ and $\boldsymbol{\eta } = [\boldsymbol{\eta }_{v} \boldsymbol{\eta }_{\theta }\boldsymbol{\eta }_{p}]$ is the state error modeled as Gaussian white noise. The corresponding system of equations reads as

$$ \frac{d}{dt} \left [ \textstyle\begin{array}{c} \boldsymbol{\delta }_{v} \\ \boldsymbol{\delta }_{\theta } \\ \boldsymbol{\delta }_{p} \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & -\boldsymbol{C}^{T}(\boldsymbol{q}) S(\boldsymbol{a}) & 0 \\ 0 & -S(\boldsymbol{\omega }) & 0 \\ \boldsymbol{I} & 0 & 0 \end{array}\displaystyle \right ] \,\left [ \textstyle\begin{array}{c} \boldsymbol{\delta }_{v} \\ \boldsymbol{\delta }_{\theta } \\ \boldsymbol{\delta }_{p} \end{array}\displaystyle \right ] + \left [ \textstyle\begin{array}{c@{\quad}c} 0 & \boldsymbol{C}^{T}(\boldsymbol{q}) \\ \boldsymbol{I} & 0 \\ 0 & 0 \end{array}\displaystyle \right ] \,\left [ \textstyle\begin{array}{c} \boldsymbol{\epsilon }_{\omega } \\ \boldsymbol{\epsilon }_{a} \end{array}\displaystyle \right ] + \left [ \textstyle\begin{array}{c} \boldsymbol{\eta }_{v} \\ \boldsymbol{\eta }_{\theta } \\ \boldsymbol{\eta }_{p} \end{array}\displaystyle \right ] $$

(A.14)

note that the effects of uncertainties on the gravitational acceleration are assumed to be negligible. The corresponding first order state covariance $\boldsymbol{P}$ is a $9\times 9$ matrix computed integrating the following continuous time equation

$$ \frac{d}{dt}\boldsymbol{P} = \boldsymbol{A}\boldsymbol{P} + \boldsymbol{P} \boldsymbol{A}^{T} + \boldsymbol{B}\boldsymbol{Q} \boldsymbol{B}^{T} + \boldsymbol{R} $$

(A.15)

with $\boldsymbol{Q}$ and $\boldsymbol{R}$ respectively the input and state noise covariance matrices.

Appendix B: RDA Measurements Model

RDA had four beams oriented as shown in Fig. 1, the sensing direction of each beam is represented by the constant versors ${}^{G}\boldsymbol{s}_{n},\,n=0,1,2,3$. Assuming a flat horizontal surface as described Sect. 7, the slant-range and slant-out measurements correspond to the distance in meters along the beam from the centre of GNC frame to the plane below the DM at the altitude $h$ AGL. The altitude AGL is computed from the position vector $\boldsymbol{p}$ as

$$ h = \lVert \boldsymbol{p} \rVert - r_{MOLA} - r_{GROUND} $$

(B.1)

with constant $r_{MOLA} = 3396000~\text{m}$ and $r_{GROUND} = -1440~\text{m}$. The rotation from the GNC frame to the local NED frame is ${}^{N}\boldsymbol{C}_{G} = ^{N}\boldsymbol{C}_{M}\,^{M} \boldsymbol{C}_{G}$, where the first rotation from the MMED to the local NED frame ${}^{N}\boldsymbol{C}_{M}$ depends on the DM position and the second rotation is computed using the attitude quaternion. Expressing the beam direction with respect to the local NED reference frame, the Z component of the versor reads as

$$ z_{n} = \boldsymbol{e}_{Z}\,^{N} \boldsymbol{C}_{M}\,\boldsymbol{C}^{T}( \boldsymbol{q})\,\boldsymbol{s}_{n} $$

(B.2)

with $\boldsymbol{e}_{Z}\,= [ 0\,0\,1 ]$. The ${}^{N} \boldsymbol{C}_{M}$ is computed using the estimated position but it is considered constant in the derivation of the error model. The slant-range measurement model is simply

$$ y_{s,n} = \frac{h}{z_{n}} = \frac{\lVert \boldsymbol{p}\rVert - r_{MOLA} - r_{GROUND} }{ \boldsymbol{e}_{Z} \,^{N}\boldsymbol{C}_{M}\, \boldsymbol{C}^{T}(\boldsymbol{q})\,\boldsymbol{s}_{n}} $$

(B.3)

note that Eq. (B.2) and hence this equation are invariant for rotations along the local vertical, this issue is considered in the implementation of the estimator as discussed in Sect. 8. The slant-range error model is then

$$ \delta _{y_{s,n}} = \boldsymbol{H}_{s,n}\,\boldsymbol{\delta }_{x} + \epsilon _{s,n} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \boldsymbol{0}\ &\ \frac{h}{z^{2}_{n}}\,\boldsymbol{e}_{Z}\,^{N}\boldsymbol{C} _{M}\,\boldsymbol{C}^{T}(\boldsymbol{q})\,S(\boldsymbol{s}_{n})\ &\ \frac{1}{z_{n}\, \lVert \boldsymbol{p}\rVert }\,\boldsymbol{p}^{T} \end{array}\displaystyle \right ] \, \left [ \textstyle\begin{array}{c} \boldsymbol{\delta }_{v} \\ \boldsymbol{\delta }_{\theta } \\ \boldsymbol{\delta }_{p} \end{array}\displaystyle \right ] + \epsilon _{s,n} $$

(B.4)

where $\epsilon _{s,n}$ is the uncertainty in the slant range measurement of each beam; measurement uncertainties were modeled as Gaussian white noise. To model the slant-out measurements we used only the B0 model $y_{s,0}$ while the slant-range is derived stacking the equations related to B0, B1, B2 and B3 in a vector $\boldsymbol{y}_{s} = [y_{s,0}\,\,y _{s,1}\,\,y_{s,2}\,\,y_{s,3}]^{T}$.

The terrain relative velocity was determined independently along each RDA beam (Bombaci et al. 2016) and then processed on-board to get the co-rotating velocity of the DM expressed in the GNC reference frame. The velocity-out measurements are modeled expressing the DM terrain relative velocity in the MMED frame by means of the state variables in Eq. (A.1) and then rotating this velocity to the GNC. This reads as

$$ \boldsymbol{y}_{v} = \boldsymbol{C}(\boldsymbol{q})\, ( \boldsymbol{v} - \boldsymbol{\varOmega }\,\boldsymbol{p} ) $$

(B.5)

where $\boldsymbol{\varOmega } = S(\boldsymbol{w}_{MARS})$ and $\boldsymbol{w}_{MARS}$ is the angular rate of Mars. The corresponding error model is

$$ \delta _{y_{v}} = \boldsymbol{H}_{v}\,\boldsymbol{\delta }_{x} + \boldsymbol{\epsilon }_{v} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \boldsymbol{C}(\boldsymbol{q})\ &\ \boldsymbol{C}(\boldsymbol{q})\,S(\boldsymbol{v}-\boldsymbol{\varOmega }\,\boldsymbol{p}) \boldsymbol{C}^{T}(\boldsymbol{q})\ &\ -\boldsymbol{C}(\boldsymbol{q})\,\boldsymbol{\varOmega } \end{array}\displaystyle \right ] \, \left [ \textstyle\begin{array}{c} \boldsymbol{\delta }_{v} \\ \boldsymbol{\delta }_{\theta } \\ \boldsymbol{\delta }_{p} \end{array}\displaystyle \right ] + \boldsymbol{\epsilon }_{v} $$

(B.6)

with $\boldsymbol{\epsilon }_{v}$ the uncertainty in the velocity estimate; measurement uncertainties were modeled as Gaussian white noise.