Flight dynamics (spacecraft)


Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.
The principles of flight dynamics are used to model a spacecraft's orbital flight; maneuvers to change orbit; translunar and interplanetary flight; a vehicle's powered flight during launch from the Earth, or spacecraft launch and landing on a celestial body, with or without an atmosphere; entry through the atmosphere of the Earth or other celestial body; and attitude control. They are generally programmed into a vehicle's inertial navigation systems, and monitered on the ground by a member of the flight controller team known in NASA as the flight dynamics officer, or in the European Space Agency as the spacecraft navigator.
Flight dynamics depends on the disciplines of propulsion, aerodynamics, and astrodynamics. It cannot be reduced to simply attitude control; real spacecraft do not have steering wheels or tillers like airplanes or ships. Unlike the way fictional spaceships are portrayed, a spacecraft actually does not bank to turn in outer space, where its flight path depends strictly on the gravitational forces acting on it and the propulsive maneuvers applied.

Basic principles

A space vehicle's flight is determined by application of Newton's second law of motion:
where F is the vector sum of all forces exerted on the vehicle, m is its current mass, and a is the acceleration vector, the instantaneous rate of change of velocity, which in turn is the instantaneous rate of change of displacement. Solving for a, acceleration equals the force sum divided by mass. Acceleration is integrated over time to get velocity, and velocity is in turn integrated to get position.
Flight dynamics calculations are handled by computerized guidance systems aboard the vehicle; the status of the flight dynamics is monitored on the ground during powered maneuvers by a member of the flight controller team known in NASA's Human Spaceflight Center as the flight dynamics officer, or in the European Space Agency as the spacecraft navigator.
For powered atmospheric flight, the three main forces which act on a vehicle are propulsive force, aerodynamic force, and gravitation. Other external forces such as centrifugal force, Coriolis force, and solar radiation pressure are generally insignificant due to the relatively short time of powered flight and small size of spacecraft, and may generally be neglected in simplified performance calculations.

Propulsion

The thrust of a rocket engine, in the general case of operation in an atmosphere, is approximated by:
The effective exhaust velocity of the rocket propellant is proportional to the vacuum specific impulse and affected by the atmospheric pressure:
where:
The specific impulse relates the delta-v capacity to the quantity of propellant consumed according to the Tsiolkovsky rocket equation:
where:

Atmospheric force

Aerodynamic forces, present near a body with significant atmosphere such as Earth, Mars or Venus, are analyzed as: lift, defined as the force component perpendicular to the direction of flight ; and drag, the component parallel to, and in the opposite direction of flight. Lift and drag are modeled as the products of a coefficient times dynamic pressure acting on a reference area:
where:
The gravitational force that a celestial body exerts on a space vehicle is modeled with the body and vehicle taken as point masses; the bodies are simplified as spheres; and the mass of the vehicle is much smaller than the mass of the body so that its effect on the gravitational acceleration can be neglected. Therefore the gravitational force is calculated by:
where:

Orbital flight

Orbital mechanics are used to calculate flight in orbit about a central body. For sufficiently high orbits, aerodynamic force may be assumed to be negligible for relatively short term missions When the central body's mass is much larger than the spacecraft, and other bodies are sufficiently far away, the solution of orbital trajectories can be treated as a two-body problem.
This can be shown to result in the trajectory being ideally a conic section with the central body located at one focus. Orbital trajectories are either circles or ellipses; the parabolic trajectory represents first escape of the vehicle from the central body's gravitational field. Hyperbolic trajectories are escape trajectories with excess velocity, and will be covered under Interplanetary flight below.
Elliptical orbits are characterized by three elements. The semi-major axis a is the average of the radius at apoapsis and periapsis:
The eccentricity e can then be calculated for an ellipse, knowing the apses:
The time period for a complete orbit is dependent only on the semi-major axis, and is independent of eccentricity:
where is the standard gravitational parameter of the central body.
of a spacecraft orbiting a central body, defining orientation of the orbit in relation to its fundamental reference plane
The orientation of the orbit in space is specified by three angles:
The orbital plane is ideally constant, but is usually subject to small perturbations caused by planetary oblateness and the presence of other bodies.
The spacecraft's position in orbit is specified by the true anomaly,, an angle measured from the periapsis, or for a circular orbit, from the ascending node or reference direction. The semi-latus rectum, or radius at 90 degrees from periapsis, is:
The radius at any position in flight is:
and the velocity at that position is:

Types of orbit

Circular

For a circular orbit, ra = rp = a, and eccentricity is 0. Circular velocity at a given radius is:

Elliptical

For an elliptical orbit, e is greater than 0 but less than 1. The periapsis velocity is:
and the apoapsis velocity is:
The limiting condition is a parabolic escape orbit, when e = 1 and ra becomes infinite. Escape velocity at periapsis is then

Flight path angle

The specific angular momentum of any conic orbit, h, is constant, and is equal to the product of radius and velocity at periapsis. At any other point in the orbit, it is equal to:
where φ is the flight path angle measured from the local horizontal This allows the calculation of φ at any point in the orbit, knowing radius and velocity:
Note that flight path angle is a constant 0 degrees for a circular orbit.

True anomaly as a function of time

It can be shown that the angular momentum equation given above also relates the rate of change in true anomaly to r, v, and φ, thus the true anomaly can be found as a function of time since periapsis passage by integration:
Conversely, the time required to reach a given anomaly is:

Orbital maneuvers

Once in orbit, a spacecraft may fire rocket engines to make in-plane changes to a different altitude or type of orbit, or to change its orbital plane. These maneuvers require changes in the craft's velocity, and the classical rocket equation is used to calculate the propellant requirements for a given delta-v. A delta-v budget will add up all the propellant requirements, or determine the total delta-v available from a given amount of propellant, for the mission. Most on-orbit maneuvers can be modeled as impulsive, that is as a near-instantaneous change in velocity, with minimal loss of accuracy.

In-plane changes

Orbit circularization
An elliptical orbit is most easily converted to a circular orbit at the periapsis or apoapsis by applying a single engine burn with a delta v equal to the difference between the desired orbit's circular velocity and the current orbit's periapsis or apoapsis velocity:
To circularize at periapsis, a retrograde burn is made:
To circularize at apoapsis, a posigrade burn is made:
Altitude change by Hohmann transfer
A Hohmann transfer orbit is the simplest maneuver which can be used to move a spacecraft from one altitude to another. Two burns are required: the first to send the craft into the elliptical transfer orbit, and a second to circularize the target orbit.
To raise a circular orbit at, the first posigrade burn raises velocity to the transfer orbit's periapsis velocity:
The second posigrade burn, made at apoapsis, raises velocity to the target orbit's velocity:
A maneuver to lower the orbit is the mirror image of the raise maneuver; both burns are made retrograde.
Altitude change by bi-elliptic transfer
A slightly more complicated altitude change maneuver is the bi-elliptic transfer, which consists of two half-elliptic orbits; the first, posigrade burn sends the spacecraft into an arbitrarily high apoapsis chosen at some point away from the central body. At this point a second burn modifies the periapsis to match the radius of the final desired orbit, where a third, retrograde burn is performed to inject the spacecraft into the desired orbit. While this takes a longer transfer time, a bi-elliptic transfer can require less total propellant than the Hohmann transfer when the ratio of initial and target orbit radii is 12 or greater.
Burn 1 :
Burn 2, to match periapsis to the target orbit's altitude:
Burn 3 :

Change of plane

Plane change maneuvers can be performed alone or in conjunction with other orbit adjustments. For a pure rotation plane change maneuver, consisting only of a change in the inclination of the orbit, the specific angular momentum, h, of the initial and final orbits are equal in magnitude but not in direction. Therefore, the change in specific angular momentum can be written as:
where h is the specific angular momentum before the plane change, and Δi is the desired change in the inclination angle. From this it can be shown that the required delta-v is:
From the definition of h, this can also be written as:
where v is the magnitude of velocity before plane change and φ is the flight path angle. Using the small-angle approximation, this becomes:
The total delta-v for a combined maneuver can be calculated by a vector addition of the pure rotation delta-v and the delta-v for the other planned orbital change.

Translunar flight

Vehicles sent on lunar or planetary missions are generally not launched by direct injection to departure trajectory, but first put into a low Earth parking orbit; this allows the flexibility of a bigger launch window and more time for checking that the vehicle is in proper condition for the flight. A popular misconception is that escape velocity is required for flight to the Moon; it is not. Rather, the vehicle's apogee is raised high enough to take it to a point where it enters the Moon's gravitational sphere of influence This is defined as the distance from a satellite at which its gravitational pull on a spacecraft equals that of its central body, which is
where D is the mean distance from the satellite to the central body, and
mc and ms are the masses of the central body and satellite, respectively. This value is approximately from Earth's Moon.
A significant portion of the vehicle's flight requires accurate solution as a three-body problem, but may be preliminarily modeled as a patched conic approximation.

Translunar injection

This must be timed so that the Moon will be in position to capture the vehicle, and might be modeled to a first approximation as a Hohmann transfer. However, the rocket burn duration is usually long enough, and occurs during a sufficient change in flight path angle, that this is not very accurate. It must be modeled as a non-impulsive maneuver, requiring integration by finite element analysis of the accelerations due to propulsive thrust and gravity to obtain velocity and flight path angle:
where:
Altitude, downrange distance, and radial distance from the center of the Earth are then computed as:

Mid-course corrections

A simple lunar trajectory stays in one plane, resulting in lunar flyby or orbit within a small range of inclination to the Moon's equator. This also permits a "free return", in which the spacecraft would return to the appropriate position for reentry into the Earth's atmosphere if it were not injected into lunar orbit. Relatively small velocity changes are usually required to correct for trajectory errors. Such a trajectory was used for the Apollo 8, Apollo 10, Apollo 11, and Apollo 12 manned lunar missions.
Greater flexibility in lunar orbital or landing site coverage can be obtained by performing a plane change maneuver mid-flight; however, this takes away the free-return option, as the new plane would take the spacecraft's emergency return trajectory away from the Earth's atmospheric re-entry point, and leave the spacecraft in a high Earth orbit. This type of trajectory was used for the last five Apollo missions.

Lunar orbit insertion

In the Apollo program, the retrograde lunar orbit insertion burn was performed at an altitude of approximately on the far side of the Moon. This became the pericynthion of the initial orbits, with an apocynthion on the order of. The delta v was approximately. Two orbits later, the orbit was circularized at. For each mission, the flight dynamics officer prepares 10 lunar orbit insertion solutions so the one can be chosen with the optimum fuel burn and best meets the mission requirements; this is uploaded to the spacecraft computer and must be executed and monitored by the astronauts on the lunar far side, while they are out of radio contact with Earth.

Interplanetary flight

In order to completely leave one planet's gravitational field to reach another, a hyperbolic trajectory relative to the departure planet is necessary, with excess velocity added to the departure planet's orbital velocity around the Sun. The desired heliocentric transfer orbit to a superior planet will have its perihelion at the departure planet, requiring the hyperbolic excess velocity to be applied in the posigrade direction, when the spacecraft is away from the Sun. To an inferior planet destination, aphelion will be at the departure planet, and the excess velocity is applied in the retrograde direction when the spacecraft is toward the Sun. For accurate mission calculations, the orbital elements of the planets must be obtained from an ephemeris, such as that published by NASA's Jet Propulsion Laboratory.

Simplifying assumptions

For the purpose of preliminary mission analysis and feasibility studies, certain simplified assumptions may be made to enable delta-v calculation with very small error:
Since interplanetary spacecraft spend a large period of time in heliocentric orbit between the planets, which are at relatively large distances away from each other, the patched-conic approximation is much more accurate for interplanetary trajectories than for translunar trajectories. The patch point between the hyperbolic trajectory relative to the departure planet and the heliocentric transfer orbit occurs at the planet's sphere of influence radius relative to the Sun, as defined above in Orbital flight. Given the Sun's mass ratio of 333,432 times that of Earth and distance of, the Earth's sphere of influence radius is .

Heliocentric transfer orbit

The transfer orbit required to carry the spacecraft from the departure planet's orbit to the destination planet is chosen among several options:
The required hyperbolic excess velocity v∞ is the difference between the transfer orbit's departure speed and the departure planet's heliocentric orbital speed. Once this is determined, the injection velocity relative to the departure planet at periapsis is:
The excess velocity vector for a hyperbola is displaced from the periapsis tangent by a characteristic angle, therefore the periapsis injection burn must lead the planetary departure point by the same angle:
The geometric equation for eccentricity of an ellipse cannot be used for a hyperbola. But the eccentricity can be calculated from dynamics formulations as:
where h is the specific angular momentum as given above in the Orbital flight section, calculated at the periapsis:
and ε is the specific energy:
Also, the equations for r and v given in Orbital flight depend on the semi-major axis, and thus are unusable for an escape trajectory. But setting radius at periapsis equal to the r equation at zero
anomaly gives an alternate expression for the semi-latus rectum:
which gives a more general equation for radius versus anomaly which is usable at any eccentricity:
Substituting the alternate expression for p also gives an alternate expression for a. This gives an equation for velocity versus radius which is likewise usable at any eccentricity:
The equations for flight path angle and anomaly versus time given in Orbital flight are also usable for hyperbolic trajectories.

Launch windows

There is a great deal of variation with time of the velocity change required for a mission, because of the constantly varying relative positions of the planets. Therefore, optimum launch windows are often chosen from the results of porkchop plots that show contours of characteristic energy plotted versus departure and arrival time.

Powered flight

The equations of motion used to describe powered flight of a vehicle during launch can be as complex as six degrees of freedom for in-flight calculations, or as simple as two degrees of freedom for preliminary performance estimates. In-flight calculations will take perturbation factors into account such as the Earth's oblateness and non-uniform mass distribution; and gravitational forces of all nearby bodies, including the Moon, Sun, and other planets. Preliminary estimates can make some simplifying assumptions: a spherical, uniform planet; the vehicle can be represented as a point mass; flight path assumes a two-body patched conic approximation; and the local flight path lies in a single plane) with reasonably small loss of accuracy.
The general case of a launch from Earth must take engine thrust, aerodynamic forces, and gravity into account. The acceleration equation can be reduced from vector to scalar form by resolving it into its tangential and angular time rate-of-change components relative to the launch pad. The two equations thus become:
where:
Mass decreases as propellant is consumed and rocket stages, engines or tanks are shed.
The planet-fixed values of v and θ at any time in the flight are then determined by numerical integration of the two rate equations from time zero :
Finite element analysis can be used to integrate the equations, by breaking the flight into small time increments.
For most launch vehicles, relatively small levels of lift are generated, and a gravity turn is employed, depending mostly on the third term of the angle rate equation. At the moment of liftoff, when angle and velocity are both zero, the theta-dot equation is mathematically indeterminate and cannot be evaluated until velocity becomes non-zero shortly after liftoff. But notice at this condition, the only force which can cause the vehicle to pitch over is the engine thrust acting at a non-zero angle of attack and perhaps a slight amount of lift, until a non-zero pitch angle is attained. In the gravity turn, pitch-over is initiated by applying an increasing angle of attack, followed by a gradual decrease in angle of attack through the remainder of the flight.
Once velocity and flight path angle are known, altitude and downrange distance are computed as:
The planet-fixed values of v and θ are converted to space-fixed values with the following conversions:
where ω is the planet's rotational rate in radians per second, φ is the launch site latitude, and Az is the launch azimuth angle.
Final vs, θs and r must match the requirements of the target orbit as determined by orbital mechanics, where final vs is usually the required periapsis velocity, and final θs is 90 degrees. A powered descent analysis would use the same procedure, with reverse boundary conditions.

Atmospheric entry

Controlled entry, descent, and landing of a vehicle is achieved by shedding the excess kinetic energy through aerodynamic heating from drag, which requires some means of heat shielding, and/or retrograde thrust. Terminal descent is usually achieved by means of parachutes and/or air brakes.

Attitude control

Since spacecraft spend most of their flight time coasting unpowered through the vacuum of space, they are unlike aircraft in that their flight trajectory is not determined by their attitude, except during atmospheric flight to control the forces of lift and drag, and during powered flight to align the thrust vector. Nonetheless, attitude control is often maintained in unpowered flight to keep the spacecraft in a fixed orientation for purposes of astronomical observation, communications, or for solar power generation; or to place it into a controlled spin for passive thermal control, or to create artificial gravity inside the craft.
Attitude control is maintained with respect to an inertial frame of reference or another entity. The attitude of a craft is described by angles relative to three mutually perpendicular axes of rotation, referred to as roll, pitch, and yaw. Orientation can be determined by calibration using an external guidance system, such as determining the angles to a reference star or the Sun, then internally monitored using an inertial system of mechanical or optical gyroscopes. Orientation is a vector quantity described by three angles for the instantaneous direction, and the instantaneous rates of roll in all three axes of rotation. The aspect of control implies both awareness of the instantaneous orientation and rates of roll and the ability to change the roll rates to assume a new orientation using either a reaction control system or other means.
Newton's second law, applied to rotational rather than linear motion, becomes:
where is the net torque about an axis of rotation exerted on the vehicle, Ix is its moment of inertia about that axis, and is the angular acceleration about that axis in radians per second per second. Therefore, the acceleration rate in degrees per second per second is
Analogous to linear motion, the angular rotation rate is obtained by integrating α over time:
and the angular rotation is the time integral of the rate:
The three principal moments of inertia Ix, Iy, and Iz about the roll, pitch and yaw axes, are determined through the vehicle's center of mass.
The control torque for a launch vehicle is sometimes provided aerodynamically by movable fins, and usually by mounting the engines on gimbals to vector the thrust around the center of mass. Torque is frequently applied to spacecraft, operating absent aerodynamic forces, by a reaction control system, a set of thrusters located about the vehicle. The thrusters are fired, either manually or under automatic guidance control, in short bursts to achieve the desired rate of rotation, and then fired in the opposite direction to halt rotation at the desired position. The torque about a specific axis is:
where r is its distance from the center of mass, and F is the thrust of an individual thruster
For situations where propellant consumption may be a problem, alternative means may be used to provide the control torque, such as reaction wheels or control moment gyroscopes.