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A unified state model of orbital trajectory and attitude dynamics

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Abstract

The trajectory and attitude dynamics of an orbital spacecraft are defined by a unified state model, which enables efficient and rapid machine computation for mission analysis, orbit determination and prediction, satellite geodesy and reentry analysis. The state variables are momenta — a general form for attitude, and a parametric form for orbital motion. The orbital parameters are the velocity state characteristics of the orbital hodograph. The coordinate variables are sets of four Euler parameters, which define the rotation transformation by the quaternion algebra.

The unified state model possesses many analytical properties which are invaluable for dynamical system synthesis, numerical analysis and machine solution: regularization, unified matrix algebra, state graphs and transforms. The analytic partials of position and velocity with the state and coordinate variables are presented, as well as representative perturbation functions such as air drag, gravitational potential harmonics, and propulsion thrust.

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Abbreviations

A :

wind cross-section of orbital body

Az :

direction of the wind velocityq in the local horizon plane, referred to the projection of the Z-axis on that plane

C, R :

parametric variables of orbital velocity state

C D :

drag coefficient of orbital body

C 22,S 22 :

coefficients of the cos and sin terms (respectively) of the tesseral harmonics for the gravitational potential function (m=n=2)

[E]:

transformation matrix of body attitude rotation, with elementsɛ ij

e 1,e 2,e 3 :

rotating polar set of unit vectors defining the point-mass motion about the planetocentric origin 0

e 1,e 2,e 3,e 4 :

Euler parameters defining rotation of a coordinate set

e a1,e a2,e a3,e a4 :

Euler parameters defining attitude rotation of the orbital body about its center-of-mass

e 01,e 02,e 03,e 04 :

Euler parameters defining rotation of the orbital trajectory frame about the planetocentric origin 0

f 1,f 2,f 3 :

intermediate set of unit vectors defining coordinates in the instantaneous orbital plane, referred to axisX′

g 1,g 2,g 3 :

planetocentric inertial set of unit vectors fixed in inertial space about the planetocentric origin 0

[H]:

angular momentum matrix of the body attitude dynamics

[J]:

inertia matrix of the orbital body, with elementsJ ij

J 2,J 3,J 4 :

coefficients of the zonal harmonics for the gravitational potential function (n=2, 3, 4 respectively)

Lo, La :

topocentric coordinates of longitude and latitude, respectively

m :

mass of the orbital body

g :

a unitary quaternion

q :

atmospheric wind velocity in the local horizon plane

R e :

spherical Earth's radius

r, v, α,J :

position, velocity, acceleration and jerk (i.e., time-rate of acceleration change) vectors of orbital motion, respectively

T :

torque vector of attitude perturbation

u :

angle of rotation of a coordinate set about a directed axis whose points are invariant

v r :

relative velocity

w x, wy, wz :

angular velocity of the body attitude rotation about its principal axes

w 1,w 3 :

angular velocity of the rotating polar coordinates of the orbital body motion about the planetocentric origin 0

α, β, γ:

direction angles defining space orientation of a directed line

δ:

angle of rotation of the axisX about the line-of-nodesLN, to axisX′ in the instantaneous orbital plane

ζ:

square of the Mach number for the orbital velocity at the given altitude

η:

constant (ζ/r) assumed in the exponential model of atmospheric density as a function of altitude

θ:

flight path angle (i.e., angle between the orbital velocity vector and the local horizon in the instantaneous orbital plane)

ι:

angle of incidence between the instantaneous orbital plane and the equatorial planeXY

λ:

angle of rotation from the axisX′ to the orbital position vectorr, in the instantaneous orbital plane

μ:

planetary gravitational constant

ν≡ui:

angle from the line-of-nodesLN to the orbital position vectorr

ϱ:

atmospheric density

ϱ0:

atmospheric density at sea level

ϕ:

true anomaly

σ:

angle of rotation from the axisX′ to the perigee apsis

Ω:

angle from the inertial axisX to the line-of-nodesLN

ω e :

planetary rate of rotation about its spin axisZ

B :

orbital body axes

b :

ballistic

e1,e2,e3:

rotating polar coordinate components of orbital motion

f1,f2:

intermediate coordinate components of orbital motion

I :

inertial axes

T1,T2,T3,T4:

Euler parameter components of the topocentric coordinates

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Authors and Affiliations

  1. System Development Corporation, Santa Monica, Calif., U.S.A.

    Samuel P. Altman

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  1. Samuel P. Altman
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Altman, S.P. A unified state model of orbital trajectory and attitude dynamics. Celestial Mechanics 6, 425–446 (1972). https://doi.org/10.1007/BF01227757

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