ROE Formation Controller Software

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Description

The ROE Formation Controller Software is a flight software module that computes translational force commands for a chaser spacecraft to follow a desired trajectory relative to a target (chief) spacecraft. The controller operates in the Local-Vertical/Local-Horizontal (LVLH) frame and supports multiple formation flying modes including station-keeping, elliptical orbits, and approach trajectories. A Proportional-Integral-Derivative (PID) controller with optional Clohessy-Wiltshire feedforward drives the position and velocity errors to zero.

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Example Use Cases

• Rendezvous Operations: Execute R-bar or V-bar approach trajectories to a target spacecraft for docking.
• Formation Flying: Maintain a stationary or walking ellipse around a chief spacecraft for distributed missions.
• Inspection Missions: Perform teardrop fly-around maneuvers for visual inspection of a target.
• Station-Keeping: Hold a fixed offset position (perch) relative to a target in the LVLH frame.

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Module Implementation

LVLH Frame

The controller computes relative motion in the LVLH (Hill) frame centered on the target spacecraft. The frame axes are defined as:

Axis	Direction
$x$ (radial)	Along the target’s position vector from the central body
$y$ (along-track)	In the direction of orbital motion
$z$ (cross-track)	Completes the right-handed system (orbit normal)

The transformation from inertial to LVLH uses the target’s position ${\mathbf{r}}_T$ and velocity ${\mathbf{v}}_T$:

$$\hat{\mathbf{x}}=\frac{{\mathbf{r}}_T}{|{\mathbf{r}}_T|},\hat{\mathbf{z}}=\frac{{\mathbf{r}}_T\times {\mathbf{v}}_T}{|{\mathbf{r}}_T\times {\mathbf{v}}_T|},\hat{\mathbf{y}}=\hat{\mathbf{z}}\times \hat{\mathbf{x}}$$

Formation Flying Modes

Mode	Description
Perch	Fixed LVLH offset with zero relative velocity
Stationary Ellipse	Closed elliptical relative orbit with no drift
Walking Ellipse	Elliptical relative orbit with along-track drift
Teardrop	Asymmetric approach-and-retreat trajectory
R-Bar Approach	Radial approach along the $+x$ axis
V-Bar Approach	Along-track approach along the $+y$ axis

Control Law

The controller computes acceleration commands using a PID control law:

$${\mathbf{a}}_c=K_p{\mathbf{e}}_r+K_d{\mathbf{e}}_v+K_i\int {\mathbf{e}}_{r}dt$$

where ${\mathbf{e}}_r={\mathbf{r}}_{ref}-\mathbf{r}$ is the position error, ${\mathbf{e}}_v={\mathbf{v}}_{ref}-\mathbf{v}$ is the velocity error, and $K_p$, $K_d$, $K_i$ are diagonal gain matrices with per-axis values.

The integral term includes anti-windup limiting and is only applied in Perch mode to eliminate steady-state error.

Clohessy-Wiltshire Feedforward

When enabled, the controller adds feedforward terms to cancel the natural drift described by the Clohessy-Wiltshire equations:

$${\mathbf{a}}_{ff}=\left[\begin{array}{c}-3n^2{x}_{ref}-2n{\dot{y}}_{ref}\\ 2n{\dot{x}}_{ref}\\ n^2{z}_{ref}\end{array}\right]$$

where $n$ is the target’s mean motion. This feedforward significantly improves tracking accuracy during station-keeping and approach maneuvers.

Approach Trajectories

For R-bar and V-bar approaches, the controller supports two profile types:

Hermite Profile: A smooth polynomial trajectory over a fixed duration when MaxApproachSpeed is zero.

Trapezoidal Profile: An accelerate-coast-decelerate profile when MaxApproachSpeed is positive. The trajectory phases are:

1. Accelerate: Ramp up to the maximum approach speed
2. Coast: Maintain constant velocity
3. Decelerate: Brake to zero velocity at the target position

The approach begins only when the chaser is within the axis position tolerance and has nulled its relative velocity.

Port-Relative Approach

When PortRelativeApproach is enabled, the controller tracks the target docking adapter’s position and orientation rather than a static LVLH offset. The reference point is computed as:

$${\mathbf{r}}_{ref}={\mathbf{r}}_{port}+d_{standoff}\cdot {\hat{\mathbf{u}}}_{port}$$

where ${\mathbf{r}}_{port}$ is the target port position, $d_{standoff}$ is the configured standoff distance, and ${\hat{\mathbf{u}}}_{port}$ is the port’s mating axis. Final closure is gated by alignment and corridor constraints.

Force Output

The acceleration command is transformed to the inertial frame and converted to force:

$${\mathbf{F}}_N=m\cdot [R_{LVLH}^N{]}^T{\mathbf{a}}_c$$

where $m$ is the chaser spacecraft mass. An optional force deadband zeros the output when the commanded force magnitude falls below a threshold.

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Assumptions/Limitations

• Requires ephemeris messages from both the chaser and target spacecraft.
• The target spacecraft is assumed to be in a near-circular orbit for accurate Clohessy-Wiltshire feedforward.
• Ellipse trajectories require a valid target mean motion; invalid values defer trajectory creation.
• The integral term is only active in Perch mode to prevent windup during dynamic maneuvers.
• Port-relative approach requires docking adapter telemetry from both spacecraft.