celestialTwoBodyPoint

Executive Summary

This module point one body-fixed axis towards a primary celestial object. The secondary goal is to point a second body-fixed axis towards another celestial object.

For example, the goal is to point the sensor towards the center of a planet while doing the best to keep the solar panel normal point at the sun.

The module assumes that the acceleration of the spacecraft with respect to the planets is negligible.

Message Connection Descriptions

The following table lists all the module input and output messages. The module msg connection is set by the user from python. The msg type contains a link to the message structure definition, while the description provides information on what this message is used for.

Module I/O Messages

Msg Variable Name	Msg Type	Description
attRefOutMsg	AttRefMsgPayload	attitude reference output message
transNavInMsg	NavTransMsgPayload	spacecraft translation motion input message
celBodyInMsg	EphemerisMsgPayload	primary celestial body information input message
secCelBodyInMsg	EphemerisMsgPayload	(optional) secondary celestial body information

Reference Frame Definition

This module computes a reference whose aim is to track the center of a primary target, e.g. pointing the communication antenna at the Earth, at the same time of trying to meet a secondary constraint as best as possible, e.g. a solar panel normal axis pointing the closest in the direction of the Sun. It is important to note that two pointing conditions in a three-dimensional space compose an overdetermined problem. Thus, the main constraint is always priorized over the secondary one so the former can always be met. Figure :ref:`1 <#fig:fig1>`__ shows the case where Mars is the primary celestial body and the Sun is the secondary one. Note that the origin of the desired reference frame \(\mathcal{R}\) is the position of the spacecraft.

Illustration of the restricted two-body pointing reference frame \(\mathcal{R}:\{ \hat{\mathbf r}_{1},\hat{\mathbf r}_{1}, \hat{\mathbf r}_{2} \}\) and the inertial frame \(\mathcal{N}:\{ \hat{\mathbf n}_{1},\hat{\mathbf n}_{1}, \hat{\mathbf n}_{2} \}\).

Assuming knowledge of the position of the spacecraft \(\mathbf{r}_{B/N}\) and the involved celestial bodies, \(\mathbf{R}_{P1}\) and \(\mathbf{R}_{P2}\) (all of them relative to the inertial frame \(\mathcal{N}\) and expressed in inertial frame components), the relative position of the celestial bodies with respect to the spacecraft is obtained by simple subtraction:

\[\begin{split}\begin{align} \mathbf{R}_{P1} =\mathbf{R}_{P} - \mathbf{r}_{B/N} \\ \mathbf{R}_{P2} =\mathbf{R}_{S} - \mathbf{r}_{B/N} \end{align}\end{split}\]

In analogy, the inertial derivatives of these position vectors are obtained:

\[\begin{split}\begin{align} \mathbf{v}_{P1} &=\mathbf{v}_{P} - \mathbf{v}_{B/N} \\ \mathbf{v}_{P2} &=\mathbf{v}_{S} - \mathbf{v}_{B/N} \\ \mathbf{a}_{P1} &=\mathbf{a}_{P} - \mathbf{a}_{B/N} \\ \mathbf{a}_{P2} &=\mathbf{a}_{S} - \mathbf{a}_{B/N} \end{align}\end{split}\]

Note that, while the documentation includes the full derivation including the acceleration of the spacecraft with respect to the planets, the module assumes that \(\mathbf{a}_{P1} = \mathbf{a}_{P2} = 0\).

The normal vector \(\mathbf{R}_{n}\) of the plane defined by \(\mathbf{R}_{P1}\) and \(\mathbf{R}_{P2}\) is computed through:

\[\mathbf R_{n} =\mathbf{R}_{P1} \times \mathbf{R}_{P2}\]

The inertial time derivative of \(\mathbf{R}_n\) is found using the chain differentiation rule:

\[\mathbf {v}_{n} = \mathbf{v}_{P1} \times \mathbf{R}_{P2} + \mathbf{R}_{P1} \times \mathbf{v}_{P2}\]

And the second time derivative:

\[\mathbf {a}_{n} = \mathbf{a}_{P1} \times \mathbf{R}_{P2} + \mathbf{R}_{P1} \times \mathbf{a}_{P2} + 2 \mathbf{v}_{P1} \times \mathbf{v}_{P2}\]

Special Case: No Secondary Constraint Applicable

If there is no incoming message with a secondary celestial body pointing condition or if the constrain is not valid, an artificial three-dimensional frame is defined instead. Note that a single condition pointing leaves one degree of freedom, hence standing for an underdetermined attitude problem. A secondary constrain is considered to be invalid if the angle between \(\mathbf{R}_{P1}\) and \(\mathbf{R}_{P2}\) is, in absolute value, minor than a set threshold. This could be the case where the primary and secondary celestial bodies are aligned as seen by the spacecraft. In such situation, the primary pointing axis would already satisfy both the primary and the secondary constraints.

Since the main algorithm of this module, which is developed in the following sections, assumes two conditions, the second one is arbitrarily set as following:

\[\mathbf{R}_{P2} = \mathbf{R}_{P1} \times \mathbf{v}_{P1} \equiv \mathbf{h}_{P1}\]

By setting the secondary constrain to have the direction of the angular momentum vector \(\mathbf{h}_{P1}\), it is assured that it will always be valid (\(\mathbf{R}_{P1}\) and \(\mathbf{R}_{P2}\) are now perpendicular). The first and second time derivatives are steadily computed:

\[\mathbf{v}_{P2} = \mathbf{R}_{P1} \times \mathbf{a}_{P1}\]

\[\mathbf{a}_{P2} = \mathbf{v}_{P1} \times \mathbf{a}_{P1}\]

Reference Frame Definition

As illustrated in Figure 1, the base vectors of the desired reference frame \(\mathcal{R}\) are defined as following:

\[\begin{split}\begin{align} \hat{\mathbf r}_{1} &= \frac{{\mathbf R}_{P1}} {|{\mathbf R}_{P1}|} \\ \hat{\mathbf r}_{3} &= \frac{{\mathbf R}_{n}}{|{\mathbf R}_{n}|} \\ \hat{\mathbf r}_{2} &= \hat{\mathbf r}_{3} \times \hat{\mathbf r}_{1} \end{align}\end{split}\]

Since the position vectors are given in terms of inertial \(\mathcal{N}\)-frame components, the DCM from the inertial frame \(\mathcal{N}\) to the desired reference frame \(\mathcal{R}\) is:

\[\begin{split}[RN] = \left[\begin{matrix} {}^{N}{ \hat{\mathbf r}_{1}^{T} } \\ {}^{N}{ \hat{\mathbf r}_{2}^{T} } \\ {}^{N}{ \hat{\mathbf r}_{3}^{T} } \end{matrix}\right]\end{split}\]

Base Vectors Time Derivatives

The first and second time derivatives of the base vectors that compound the reference frame \(\mathcal{R}\) are needed in the following sections to compute the reference angular velocity and acceleration. Several lines of algebra lead to the following sets:

\[\begin{split}\begin{align} \dot{\hat{\mathbf{r}}}_1 &= ([I_{3\times3}] - {\hat{\mathbf{r}}_1}{\hat{\mathbf{r}}_1}^T) \frac{{\mathbf V}_{P1}} {|{\mathbf R}_{P1}|} \\ \dot{\hat{\mathbf{r}}}_3 &= ([I_{3\times3}] - \hat{\mathbf{r}}_3 \hat{\mathbf{r}}_3^T) \frac{{\mathbf v}_{n}} {|{\mathbf R}_{n}|} \\ \dot{\hat{\mathbf{r}}}_2 &= \dot{\hat{\mathbf{r}}}_3 \times \hat{\mathbf{r}}_1 + \hat{\mathbf{r}}_3 \times \dot{\hat{\mathbf{r}}}_1 \end{align}\end{split}\]

\[\begin{split}\begin{align} \ddot{\hat{\mathbf{r}}}_1 &= \frac{1}{|{\mathbf R}_{P1}|} ( ([I_{3\times3}] - {\hat{\mathbf{r}}_1}{\hat{\mathbf{r}}_1}^T) \mathbf{a}_{P1} - 2\dot{\hat{\mathbf{r}}}_1 (\hat{\mathbf{r}}_1 \cdot \mathbf{v}_{P1}) - \hat{\mathbf{r}}_1 (\dot{\hat{\mathbf{r}}}_1 \cdot \mathbf{v}_{P1}) ) \\ \ddot{\hat{\mathbf{r}}}_3 &= \frac{1}{|{\mathbf R}_{n}|} ( ([I_{3\times3}] - {\hat{\mathbf{r}}_3}{\hat{\mathbf{r}}_3}^T) \mathbf{a}_{n} - 2\dot{\hat{\mathbf{r}}}_3 (\hat{\mathbf{r}}_3 \cdot \mathbf{v}_{n}) - \hat{\mathbf{r}}_3 (\dot{\hat{\mathbf{r}}}_3 \cdot \mathbf{v}_{n}) ) \\ \ddot{\hat{\mathbf{r}}}_2 &= \ddot{\hat{\mathbf{r}}}_3 \times \hat{\mathbf{r}}_1 + \hat{\mathbf{r}}_3 \times \ddot{\hat{\mathbf{r}}}_1 + 2 \dot{\hat{\mathbf{r}}}_3 \times \dot{\hat{\mathbf{r}}}_1 \end{align}\end{split}\]

Angular Velocity and Acceleration Descriptions

Developing some more mathematics, the following elegant expressions of \(\mathbf\omega_{R/N}\) and \(\dot{\mathbf\omega}_{R/N}\) are found:

\[\begin{split}\begin{align} \mathbf\omega_{R/N} \cdot \hat{\mathbf r}_{1} = \hat{\mathbf r}_{3} \cdot \dot{\hat{\mathbf r}}_{2} \\ \mathbf\omega_{R/N} \cdot \hat{\mathbf r}_{2} = \hat{\mathbf r}_{1} \cdot \dot{\hat{\mathbf r}}_{3}\\ \mathbf\omega_{R/N} \cdot \hat{\mathbf r}_{3} = \hat{\mathbf r}_{2} \cdot \dot{\hat{\mathbf r}}_{1} \end{align}\end{split}\]

\[\begin{split}\begin{align} \dot{\mathbf\omega}_{R/N} \cdot \hat{\mathbf r}_{1} &= \dot{\hat{\mathbf r}}_{3} \cdot \dot{\hat{\mathbf r}}_{2} + \hat{\mathbf r}_{3} \cdot \ddot{\hat{\mathbf r}}_{2} - \mathbf\omega_{R/N} \cdot \dot{\hat{\mathbf r}}_{1} \\ \dot{\mathbf\omega}_{R/N} \cdot \hat{\mathbf r}_{2} &= \dot{\hat{\mathbf r}}_{1} \cdot \dot{\hat{\mathbf r}}_{3} + \hat{\mathbf r}_{1} \cdot \ddot{\hat{\mathbf r}}_{3} - \mathbf\omega_{R/N} \cdot \dot{\hat{\mathbf r}}_{2} \\ \dot{\mathbf\omega}_{R/N} \cdot \hat{\mathbf r}_{3} &= \dot{\hat{\mathbf r}}_{2} \cdot \dot{\hat{\mathbf r}}_{1} + \hat{\mathbf r}_{2} \cdot \ddot{\hat{\mathbf r}}_{1} - \mathbf\omega_{R/N} \cdot \dot{\hat{\mathbf r}}_{3} \end{align}\end{split}\]

Note that \(\mathbf\omega_{R/N} \cdot \hat{\mathbf r}_{1}\) is the first component of the angular velocity of the reference with respect to the inertial expressed in reference frame components. Hence,

\[\begin{split}\mathbf\omega_{R/N}= {}^{R}{ \left[\begin{matrix} \mathbf\omega_{R/N} \cdot \hat{\mathbf r}_{1} \\ \mathbf\omega_{R/N} \cdot \hat{\mathbf r}_{2} \\ \mathbf\omega_{R/N} \cdot \hat{\mathbf r}_{3} \end{matrix}\right] }\end{split}\]

Similarly for the angular acceleration:

\[\begin{split}\mathbf{\dot\omega}_{R/N} = {}^{R}{ \left[\begin{matrix} \mathbf{\dot\omega}_{R/N} \cdot \hat{\mathbf r}_{1} \\ \mathbf{\dot\omega}_{R/N} \cdot \hat{\mathbf r}_{2} \\ \mathbf{\dot\omega}_{R/N} \cdot \hat{\mathbf r}_{3} \end{matrix}\right] }\end{split}\]

Eventually, in inertial frame components:

\[\begin{split}\begin{align} {}^{N} {\mathbf\omega_{R/N}} &= [RN] \textrm{ } {}^{R} {\mathbf\omega_{R/N}} \\ {}^{N} {\mathbf{\dot\omega}_{R/N}} &= [RN] \textrm{ } {}^{R} {\mathbf{\dot\omega}_{R/N}} \end{align}\end{split}\]

Class CelestialTwoBodyPoint

class CelestialTwoBodyPoint : public SysModel

Data structure for module to compute the two-body celestial pointing navigation solution.

Public Functions

void updateState(uint64_t callTime) override

This method takes the spacecraft and points a specified axis at a named celestial body specified in the configuration data. It generates the commanded attitude and assumes that the control errors are computed downstream.

Parameters:

callTime – The clock time at which the function was called (nanoseconds)

Returns:

void

void setSingularityThresh(double thresh)

Set the singularity threshold.

Parameters:

thresh – singularity threshold

double getSingularityThresh() const

Get the singularity threshold.

Returns:

double singularity threshold

Public Members

Message<AttRefMsgPayload> attRefOutMsg

The name of the output message*/.

ReadFunctor<EphemerisMsgPayload> celBodyInMsg

The name of the celestial body message*/.

ReadFunctor<EphemerisMsgPayload> secCelBodyInMsg

The name of the secondary body to constrain point*/.

ReadFunctor<NavTransMsgPayload> transNavInMsg

The name of the incoming attitude command*/.