# From RTN reference frame to ECI and orbital elements

• GNCstudent
In summary, the conversation discusses a problem with deriving the representation of a chaser spacecraft in ECi coordinates and/or orbital elements from its position and velocity in RTN coordinates. The solution to this problem requires additional information such as mass, altitude, and gravitational constants. The target's orbit is circular and the chaser has no initial velocity. The conversation also mentions using a controller that only uses RTN coordinates and incorporating a more complex simulator that requires ECi coordinates and orbital elements.
GNCstudent
Hi all,
I'm curently meeting the following problem and can't figure it out by myself.
Assume we have to vehicles in space, a target and a chaser spacecraft . The target spacecraft defines the RTN frame of reference.
Given the position and velocity of the chaser in RTN-coordinates, how can I derive its representation in ECi coordinates and/or orbital elements ?
Is it even possible ?

The background of my problem is, that I'm using a controller, that only uses RTN coordinates. However, I'm trying to include another more commplex simulator as reference solution, that requires ECI coordinates as inital condition for the target and the relative difference coordinates in orbital elements from chaser to target.
I am aware that I need additional information, like the mass, the altitude, R_earth, mu and gravitational constant, which I could provide.
But is that enough?

A very first idea, was to assume, that the initial ECI-position of the target doesn't matter for the simple regulator. So I could just set them to a random value.. So the only thing that matters would be the distance expressed in orbital or ECI-coordinates from target to chaser. But I can't figure out , how to do that...

Some data and simplifications, that can be used here :

- the target's orbit is circular
- the RTN position of the chaser is (30;10;10) m
- the mass of the chaser is 1000kg and the mass of the target is 8000kg
- the target has an altitude of 800e3 m

GNCstudent said:
Some data and simplifications, that can be used here :

- the target's orbit is circular
- the RTN position of the chaser is (30;10;10) m
- the mass of the chaser is 1000kg and the mass of the target is 8000kg
- the target has an altitude of 800e3 m
- Also the chaser has no initial velocity (it's zero)

## 1. What is the RTN reference frame?

The RTN reference frame, also known as Radial, Transverse, and Normal frame, is a coordinate system commonly used in orbital mechanics. It is a local reference frame centered at the spacecraft, with its origin at the center of mass and its axes aligned with the radial, transverse, and normal directions with respect to the spacecraft's orbit.

## 2. How is the RTN reference frame related to the ECI reference frame?

The ECI (Earth-Centered Inertial) reference frame is a global coordinate system fixed to the Earth's center of mass and aligned with the Earth's rotational axis. The RTN reference frame is related to the ECI frame through a series of mathematical transformations, which take into account the spacecraft's position and velocity relative to the Earth's center of mass.

## 3. What are orbital elements and why are they important?

Orbital elements are six parameters that define the shape and orientation of an orbit around a celestial body. These elements include the semi-major axis, eccentricity, inclination, argument of periapsis, longitude of the ascending node, and true anomaly. They are important because they provide a complete mathematical description of an orbit and are used to predict the position and motion of spacecraft in space.

## 4. How are orbital elements calculated from the RTN reference frame?

To calculate orbital elements from the RTN reference frame, a series of transformations must be performed. First, the position and velocity vectors of the spacecraft in the RTN frame are converted to the ECI frame. Then, the classical orbital elements are calculated using the converted vectors. Finally, the elements are transformed into their final form, known as Keplerian elements, which are used to describe the shape and orientation of an orbit.

## 5. What are some applications of converting from the RTN reference frame to ECI and orbital elements?

Converting from the RTN reference frame to ECI and orbital elements has many practical applications in space missions. For example, it can be used to track and predict the position and motion of satellites and spacecraft in orbit, to plan and execute maneuvers, and to analyze and optimize orbital trajectories. It is also used in the development and operation of space-based assets, such as satellites, space probes, and spacecraft, to ensure accurate and efficient operations in space.

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