CA1287029C - Method for controlling the attitude of a spinning body in orbit - Google Patents
Method for controlling the attitude of a spinning body in orbitInfo
- Publication number
- CA1287029C CA1287029C CA000539476A CA539476A CA1287029C CA 1287029 C CA1287029 C CA 1287029C CA 000539476 A CA000539476 A CA 000539476A CA 539476 A CA539476 A CA 539476A CA 1287029 C CA1287029 C CA 1287029C
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- Prior art keywords
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- spin
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- axis
- rate
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- 238000000034 method Methods 0.000 title claims abstract description 34
- 238000009987 spinning Methods 0.000 title claims abstract description 25
- 230000033001 locomotion Effects 0.000 claims abstract description 17
- 230000005484 gravity Effects 0.000 claims abstract description 11
- 230000000087 stabilizing effect Effects 0.000 claims abstract description 7
- 230000001174 ascending effect Effects 0.000 claims description 5
- 208000031872 Body Remains Diseases 0.000 claims description 3
- 238000010586 diagram Methods 0.000 description 10
- 238000005265 energy consumption Methods 0.000 description 5
- 230000006641 stabilisation Effects 0.000 description 5
- 238000011105 stabilization Methods 0.000 description 5
- 230000014509 gene expression Effects 0.000 description 3
- 238000012423 maintenance Methods 0.000 description 3
- 238000013461 design Methods 0.000 description 2
- 230000009977 dual effect Effects 0.000 description 2
- 239000000446 fuel Substances 0.000 description 2
- BHMLFPOTZYRDKA-IRXDYDNUSA-N (2s)-2-[(s)-(2-iodophenoxy)-phenylmethyl]morpholine Chemical compound IC1=CC=CC=C1O[C@@H](C=1C=CC=CC=1)[C@H]1OCCNC1 BHMLFPOTZYRDKA-IRXDYDNUSA-N 0.000 description 1
- 238000013459 approach Methods 0.000 description 1
- 238000010276 construction Methods 0.000 description 1
- 238000011161 development Methods 0.000 description 1
- 230000002349 favourable effect Effects 0.000 description 1
- 238000005286 illumination Methods 0.000 description 1
- 238000004519 manufacturing process Methods 0.000 description 1
- 230000007246 mechanism Effects 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 239000003380 propellant Substances 0.000 description 1
Classifications
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B64—AIRCRAFT; AVIATION; COSMONAUTICS
- B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
- B64G1/00—Cosmonautic vehicles
- B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
- B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
- B64G1/28—Guiding or controlling apparatus, e.g. for attitude control using inertia or gyro effect
- B64G1/281—Spin-stabilised spacecraft
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B64—AIRCRAFT; AVIATION; COSMONAUTICS
- B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
- B64G1/00—Cosmonautic vehicles
- B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
- B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
- B64G1/244—Spacecraft control systems
Abstract
METHOD FOR CONTROLLING THE ATTITUDE
OF A SPINNING BODY IN ORBIT
ABSTRACT OF THE DISCLOSURE
A method for passively stabilizing the attitude of a spinning orbiting body subject to orbital precession to maintain the spin axis orientation of the body essentially fixed and stable relative to the orbital plane. The preferred embodiment comprises selecting a body mass geometry and orbital parameters, including an orbit inclination, an orbital rate and a rate of regression of orbit line of nodes, so that the precessional motion of the body and the orbit are equal and opposite in direction and locating the spin axis of the body in a plane containing the north axis of the body and the orbit normal and between the north axis and the orbit normal, so that the gravity gradient precession of the spin axis equals the regressional motion of the orbit normal to produce a planar equilibrium configuration.
OF A SPINNING BODY IN ORBIT
ABSTRACT OF THE DISCLOSURE
A method for passively stabilizing the attitude of a spinning orbiting body subject to orbital precession to maintain the spin axis orientation of the body essentially fixed and stable relative to the orbital plane. The preferred embodiment comprises selecting a body mass geometry and orbital parameters, including an orbit inclination, an orbital rate and a rate of regression of orbit line of nodes, so that the precessional motion of the body and the orbit are equal and opposite in direction and locating the spin axis of the body in a plane containing the north axis of the body and the orbit normal and between the north axis and the orbit normal, so that the gravity gradient precession of the spin axis equals the regressional motion of the orbit normal to produce a planar equilibrium configuration.
Description
MET~OD FOR CONT~OLLING THE ATTITUDE
OF A SPIN~ING BODY IN ORBIT
sAcKGRouND OE THE INVENTION
lo Field of the Invention -This invention relates to spin axis stabilization of a spinning orbital body and, more particularly, to a method of stabilizing a spinning orbiting body without resort to active stabilizing mechanisms.
OF A SPIN~ING BODY IN ORBIT
sAcKGRouND OE THE INVENTION
lo Field of the Invention -This invention relates to spin axis stabilization of a spinning orbital body and, more particularly, to a method of stabilizing a spinning orbiting body without resort to active stabilizing mechanisms.
2. Description of Related Art In previously known methods of attempting to stabilize the orientation of a spinning orbital body, such as a satellite, various active means are generally employed.
The most common of these means is the use of thrusters using some form of mass expulsion to produce attitude control moments which interact with the subject body mass and geometry to produce the desired precession of the spin axis of the body to maintain a stable attitude orientation.
Another stabilization method found in the prior art is the use of electrical energy or magnetic ~ields to produce magnetic moments which react with the earth's magnetic field to produce control torques for precessing the spin axis of the orbitaI body.
Thus, to create and maintain the desired control torque moments, these known prior art methods require additional complexity, weight and energy consumption that lessen the useful payload and efficiency of the orbiting body for its designed mîssion.
:, . .
While these known prior art methods are acceptable for relatively short-lived orbital bodies and missions, where the necessary trade-off of weight, complexity and energy consumption may be feasible, the present goals of establishing a permanent orbiting body, such as an earth orbiting space station, present unique demands on payload weights and overall system efficiency requiring new stabilization methods and systems that will result in greater reliability and dependability of operation at reduced levels of weight, complexity and energy consumption.
Accordingly, each of these previously known methods has a number of disadvantages which are overcome in the practice of the present invention. Specifically, the desired orientation of the body may be produced without the use of thrusters or mass expulsion, thus eliminating the need to carry attitude control fuel or propellant on the orbiting body. The desired orientation of the body may also be produced without the expenditure of electrical energy or the development of magnetic fields of any kind, thus greatly reducing the energy requirements of the orbiting body. The desired orientation of the body may be produced without the need for attitude sensing devices to produce control system error signals, thus providing for reduced cost and improved reliability. Finally, as a result of the system design and configuration embodied in the method of this invention, the desired attitude o the spinning body may be passively maintained, and the maximum excursion of the sun angle from the equatorial plane of the spinning body may be kept suitably small for solar array efficiency.
SUMMARY OF THE INVENTION
The present invention provides a method to passively stabilize a spinning body in a fixed orientation designed to satisfy the aforementioned needs of reduced weight, complexity and energy consumption. This invention embodies a unique method of choosing the vehicle body mass geometry, ~i7~
spin rate, orientation and orbit in a prescribed fashion.
The orbit does not have to be equatorial, or the attitude orientation normal to the orbit plane. Furthermore, the method of the inventlon allows passive maintenance of the desired attitude of the spinning body, and the maximum excursion of the sun angle from the equatorial plane of the spinning body may be kept suitably small for solar array efficiency or antenna gain.
Accordingly, the present invention relates to a method for passively stabilizing the attitude of a spinning orbiting body having as its orbital parameters an orbit inclination i, an orbital rate QO and a rate of regression of the orbit line of nodes y , so that the spin axis orientation angle ~O of the body remains essentially fixed and stable relative to the orbital plane of the body, even when the orbit of the body is precessing. The axis orientation ~O is the angle of the spin axis from polar north in the plane formed by north and the orbit normal.
This method comprises selecting a mass geometry ~ for the body defined by the ratio of the bodyls spin inertia to the body's transverse inertia; selecting a spin rate Q for the body so that the spin orientation angle ~O ~or the body is an equllibrium solution to the following relationship:
~ ) where, for an xyz orbital coordinate system having z along the orbit normal axis and x at the ascending node, the following relations hold:
xO = O
zO sin i cos i - kz ~7 with zO being a solution to the quartic:
k2Zo4-2 k cos i zo3~ k2) zo2~2 k cos i Zo-COS2 i=0 S and k being a constant defined by the following expression:
k = l o S ( ~ ) Q~, and initially orienting the body in orbit with spin orientation angle ~O. The quantities xO, yO and zO which define the orientation angle ~ are the components in the xyz system of a unit vector along the body spin axis.
Other aspects of the invention are as follows:
A method of passively stabilizing the attitude of a spinning earth-orbiting body which is precessing, the method serving to maintain the spin axis orientation of said body essentially fixed and stable relative to the orbital plane of said body, even when the orbit of said body is also precessing, the method comprising:
selecting a body mass geometry and orbital parameters comprising an orbit inclination, an orbital rate and a rate of regression of orbit line o~ nodes so that the prece,ssional motions of said body and said orbit are equal and opposite in direction;
locating the spin axis of said body in a plane defined by the earth polar axis and the orbit normal of the body and between said polar axis and said orbit normal; and selecting the spin axis angle so that the gravity gradient precession of said spin axis equals the regressional motion of the orbit normal to produce an axis equilibrium configuration.
~ 4a An earth orbiting system in which a spinning satellite is located in orbit about the earth, the system comprising:
a spinning orbiting body having a mass geometry defined by the ratio of the body's spin inertia to the body's transverse inertia and a selected spin rate consistent with desired orbital parameters satisfying the following deEinitionso i = orbit inclination k = 1.5( ~ r where tvehicle spin inertia) a (vehicle transverse inertia) QO = orbital rate Q = spin rate Y = rate of regression of orbit line of nodes.
and wherein the spin axis of the orbiting body is at an angle ~O defined as:
= arc tan ~ ~ + i - ~/2 Yo where, for an Xy2 orbital coordinate system having z along the orbit normal axis and x at the ascending noder the following relations hold:
xO = 0 y zO sin i cos i - kz - I ,;
J;
~70~3 4b and zO is a solution to the quartic k2zo4-2 k cos i zo3~ k2) Zo2~2 k cos i ~O-cos~ i=O
BRIEF DESCRIPTION_OF ~E DRAWINGS
A ~etter understanding of the present invention may be gained from a consideration of the following detailed description, taken in conjunction with the accompanying drawings in which:
FIG. lA is a diagram illustrating the relationship of an orbiting body such as a satellite whose spin axis attitude is passively maintained in accordance with the present invention;
FIG. lB is a diagram similar to that of FIG. lA from a top plan view illustrating relative positions of the spin axis and the orbit normal at differen~ times;
2~ FIG. 2A ïs a diagram illustrating the change in satellite orientation resulting from the regressional motion of the orbit normal which is produced by the earth's oblateness;
FIG. 2B is a diagram similar to that of the diagram in FIG. 2R from a top plan view FIG. 3A is a diagram illustrating the regressional motion of the spin axis of a satellite FIG. 38 is a diagram si.milar to that of the diagram in FIG. 3A from a top plan view;
~o ~ .
~ t~ 9 FIG. ~ is a diagram illustrating the coordinate system definition employed in the description herein; and FIG. 5 is a graph of the equilibrium spin axis angle ~O as a function of spin speed for a space station application having an orbit altitude of 500 km and inclination of 28~5.
~ESCRIPTION OF THE PREFERRED EMBODIMENT
The essence of this invention lies in the design of the orbiting body mass properties (body inertia and geometry), the production of a particular spin rate, and a unique initial orientation of the body in orbit such that the ensuing inertial motion of the spin axis continuously and passiveIy follows the ~ction of the precessing orbit plane while also limiting the total excursion of the sun line from the equatorial plane of the spinning body. ~his unique arrangement of system elements accounts for the basic system diagram of FIGS. lA and lB.
As illustrated in FIGS. lA and lB, the spin axis is fixed in the plane defined by the earth polar axis and the orbit plane normalj and at a fixed angle ~O with respect to north. If the spin axis is placed in this position initially, it will remain there indefinitely, even though the orbit itself is precessing, provided the angle ~O is correctly chosen. Furthermore, the spin axis orientation is stable in that small errors in the initial placement remain small indefinitely. ~hus, the spin axis is passively maintained in a fixed position in the earth axis/orbit plane without recourse to fuel or energy expenditure, provided it is placed at the correct initial angle, ~O~ to begin with-This angle ~O is a function of two parameters i and k, as defined in the following relationship:
i - orbit inclination k = 1.5( a-l) where ~vehicle spin inertia) (vehicle transverse inertia~
~O = orbital rate Q = spin rate y = rate of regression of orbit line of nodes.
It is important in practicing the method of the present invention to select the mass geometry of the body and the spin rate ( ~ and Q , respectively) consistent with the desired orbital parameters referenced above (i, QO~
and y ) so that the angle ~O is small enough to provide adequate solar cell power over the mission life; and then to place the vehicle's spin axis in the correct orientation at the start of the mission.
In a preferred embodiment of the method of the invention, the spin axis orientation angle ~ O may be determined as follows. For a vehicle in an orbit which is inclined with respect to the equatorial plane, the oblateness of the earth causes the orbit normal to precess about the north/south axis in a retrograde sense. This regressional motion is shown in Figs. 2A and 2B. In addition, gravity gradient torques across the body cause the spin axis of the vehicle to precess about the orbit normal in a positive sense if the body is rod-shaped that is, having a roll-to-pitch ratio less than unity, and in a retrograde sense if the body is disk-shaped. The motion for the disk-shaped case is illustrated in FIGS. 3A and 3B.
~ ~3~
These two precessions will generally cause the spin axis of the vehicle to wander over large regions of the sky unless mass and/or energy are expended to counteract these forces.
An alternative approach, and the fundamental principle of the method of the present invention, Ls to size the vehicle and orbit parameters so that ~he precessional motions combine favorably to allow passive maintenance of the spin axis attitude in the useful, known orientation illustrated in FIGS. lA and ls. This favorable combination is achieved by locating the spin axis between the north axis and the orbit normal at a position such that the gravity gradient precession of the spin axis is just balanced by the regressional motion of the orbit normal to produce the planar equilibrium configuration as illustrated in FIGS. lA
and lB.
To derive the method of the invention mathematically, consider a rotating xyz orbital coordinate system as illustrated in FIG. 4 with z along the orbit normal axis and x at the ascending node. The angular rate ~ of this system with respect to the celestial XYZ system is O
~ sin i (1) cos i with the equations of motion for the vehicle angular moment vector h satisfying the relation d~
d~ ~ x h T~G (2) where ~G = gravity gradient torque.
Since the gravity disturbance torque is small and requires long time intervals to move the h~ vector appreciably, it is replaced by its average value over an orbit to yield the governing equations of motion for the o~
system: (Equation 3) /hx\ /-hy\ /hz sin i - hy cos i~
~ h ) = _ 3 0 (C-A) hz ~ hx) + Y ~hx cos i where C is the vehicle spin inertia and A is the transverse inertia about the vehicle center o gravity.
Letting ~ denote the unit vector, ~ = (Y~ = lh The above equation can be rewritten:
x' = (kz-cos i) y + z sin i y' = (cos i - kz) x (5) z' = -x sin i where: d ' = , ~ = Yt d ~ 2 k = 1.5( ~ ) ~y and .~ = C
These expressions (5) have the two Eirst integrals, x2 + y2 + z2 = l kz2 ~ z cot i t Cl (6) 2 sin i and the equilibrium solution xO = o (7) zO sin i Y cos i - kz with zO a solution to the quartic relation, k2Zo4 - 2 k cos i zo3+(1 k2~ Zo2~2 k cos i zo-cos2 i=0 (8) The equilibrium spin angle ~O is then determined from the equilibrium solution, ~O = arc tan~- ) + i - ~/2 Yo The dependence of ~ O on the parameters i and k is evident in the form of the equilibrium solution to Equations (7), (8) and (9)~ That the solution is stable follows from the first integral expressions in Equation (6) above which show the motion of the spin axis to be on the intersection of the unit sphere and a parabolic cyli.nder. For the equilibrium solution, this intersection is a single ~oint~
This closed contour behavior demonstrates the stability of the equilibrium motion described by this se.t of equations.
To illustrate the concept, consider a large space station which is to be placed in a 500 kilometer orbit inclined at an angle o 28.5. For such an orbit, the nodal regression rate is 6.72 per day and the orbital period of 94.13 minutes. Let the vehicle have a dual spinner construction consisting of a large rotor, which is spun to provide both gyroscopic stiffness and a spinning gravity environment, and a despun zero gravity section.
environment, and a despun zero gravity section.
FIG. 5 illus-trates the critical angle ~O for spin rates from one to six revolutions per minute and for values of vehicle inertia ratios a ranging from 1.2 to 1.~. For the dual spin application, a is the ratio of the rotor spin inertia to the vehicle transverse iner~ia. Since small values of ~O are desired for power purposes, higher spin rates and lower O values are preferred ~or this application.
Hence the spacecraft designer would size the rotor mass properties and spin rate to generate an acceptable spin axis angle ~O for such an objective.
The method of the invention applies to both spinning and d~al spinning spacecraft. In the latter casel however, the vehicle spin inertia must be replaced by the rotor spin inertia in determining the spin axis orientation angle ~O~
Also it will be noted from the equilibrium condition of Equations (7), ~8) and (9) that there can exist as many as four orientation angles ~O for each orbit and spacecraft.
Generally, the smallest ~O value is of most interest because of power considerations. ~owever, the other angle solutions can be used to provide similar spin axis maintenance.
Thus, there has been described a method for passively maintaining the attitude of a spinning orbiting body having as its orbital parameters an orbit inclination, an orbital rate and a rate of regression of orbit line of nodes, so that the spin axis orientation angle of the body remains essentially fixed and stable relative to the orbital plane of the body, even when the orbit of the body is precessing, to provide optimum antenna gain and optimum solar cell illumination. By using the method of the invention of passive stabilization to orient the satellite, the penalties in weight, complexity, and energy consumption existing in conventional active stabilization systems of the prior art have been minimized~
Although there have been described above specific arrangements of a method for controlling the attitude of a .. ~ , .. ........ ... .
spinning body in orbit in accordance with the .invention for the purpose of illustra~ing the manner in which the invention may be used to advantage, it will be appreciated that the invention is not limited thereto. Accordingly, any S and all modifications, variations or equivalent arrangements which may occur to those skilled in the art should be considered to be within the scope of the invention as defined in the annexed claims.
The most common of these means is the use of thrusters using some form of mass expulsion to produce attitude control moments which interact with the subject body mass and geometry to produce the desired precession of the spin axis of the body to maintain a stable attitude orientation.
Another stabilization method found in the prior art is the use of electrical energy or magnetic ~ields to produce magnetic moments which react with the earth's magnetic field to produce control torques for precessing the spin axis of the orbitaI body.
Thus, to create and maintain the desired control torque moments, these known prior art methods require additional complexity, weight and energy consumption that lessen the useful payload and efficiency of the orbiting body for its designed mîssion.
:, . .
While these known prior art methods are acceptable for relatively short-lived orbital bodies and missions, where the necessary trade-off of weight, complexity and energy consumption may be feasible, the present goals of establishing a permanent orbiting body, such as an earth orbiting space station, present unique demands on payload weights and overall system efficiency requiring new stabilization methods and systems that will result in greater reliability and dependability of operation at reduced levels of weight, complexity and energy consumption.
Accordingly, each of these previously known methods has a number of disadvantages which are overcome in the practice of the present invention. Specifically, the desired orientation of the body may be produced without the use of thrusters or mass expulsion, thus eliminating the need to carry attitude control fuel or propellant on the orbiting body. The desired orientation of the body may also be produced without the expenditure of electrical energy or the development of magnetic fields of any kind, thus greatly reducing the energy requirements of the orbiting body. The desired orientation of the body may be produced without the need for attitude sensing devices to produce control system error signals, thus providing for reduced cost and improved reliability. Finally, as a result of the system design and configuration embodied in the method of this invention, the desired attitude o the spinning body may be passively maintained, and the maximum excursion of the sun angle from the equatorial plane of the spinning body may be kept suitably small for solar array efficiency.
SUMMARY OF THE INVENTION
The present invention provides a method to passively stabilize a spinning body in a fixed orientation designed to satisfy the aforementioned needs of reduced weight, complexity and energy consumption. This invention embodies a unique method of choosing the vehicle body mass geometry, ~i7~
spin rate, orientation and orbit in a prescribed fashion.
The orbit does not have to be equatorial, or the attitude orientation normal to the orbit plane. Furthermore, the method of the inventlon allows passive maintenance of the desired attitude of the spinning body, and the maximum excursion of the sun angle from the equatorial plane of the spinning body may be kept suitably small for solar array efficiency or antenna gain.
Accordingly, the present invention relates to a method for passively stabilizing the attitude of a spinning orbiting body having as its orbital parameters an orbit inclination i, an orbital rate QO and a rate of regression of the orbit line of nodes y , so that the spin axis orientation angle ~O of the body remains essentially fixed and stable relative to the orbital plane of the body, even when the orbit of the body is precessing. The axis orientation ~O is the angle of the spin axis from polar north in the plane formed by north and the orbit normal.
This method comprises selecting a mass geometry ~ for the body defined by the ratio of the bodyls spin inertia to the body's transverse inertia; selecting a spin rate Q for the body so that the spin orientation angle ~O ~or the body is an equllibrium solution to the following relationship:
~ ) where, for an xyz orbital coordinate system having z along the orbit normal axis and x at the ascending node, the following relations hold:
xO = O
zO sin i cos i - kz ~7 with zO being a solution to the quartic:
k2Zo4-2 k cos i zo3~ k2) zo2~2 k cos i Zo-COS2 i=0 S and k being a constant defined by the following expression:
k = l o S ( ~ ) Q~, and initially orienting the body in orbit with spin orientation angle ~O. The quantities xO, yO and zO which define the orientation angle ~ are the components in the xyz system of a unit vector along the body spin axis.
Other aspects of the invention are as follows:
A method of passively stabilizing the attitude of a spinning earth-orbiting body which is precessing, the method serving to maintain the spin axis orientation of said body essentially fixed and stable relative to the orbital plane of said body, even when the orbit of said body is also precessing, the method comprising:
selecting a body mass geometry and orbital parameters comprising an orbit inclination, an orbital rate and a rate of regression of orbit line o~ nodes so that the prece,ssional motions of said body and said orbit are equal and opposite in direction;
locating the spin axis of said body in a plane defined by the earth polar axis and the orbit normal of the body and between said polar axis and said orbit normal; and selecting the spin axis angle so that the gravity gradient precession of said spin axis equals the regressional motion of the orbit normal to produce an axis equilibrium configuration.
~ 4a An earth orbiting system in which a spinning satellite is located in orbit about the earth, the system comprising:
a spinning orbiting body having a mass geometry defined by the ratio of the body's spin inertia to the body's transverse inertia and a selected spin rate consistent with desired orbital parameters satisfying the following deEinitionso i = orbit inclination k = 1.5( ~ r where tvehicle spin inertia) a (vehicle transverse inertia) QO = orbital rate Q = spin rate Y = rate of regression of orbit line of nodes.
and wherein the spin axis of the orbiting body is at an angle ~O defined as:
= arc tan ~ ~ + i - ~/2 Yo where, for an Xy2 orbital coordinate system having z along the orbit normal axis and x at the ascending noder the following relations hold:
xO = 0 y zO sin i cos i - kz - I ,;
J;
~70~3 4b and zO is a solution to the quartic k2zo4-2 k cos i zo3~ k2) Zo2~2 k cos i ~O-cos~ i=O
BRIEF DESCRIPTION_OF ~E DRAWINGS
A ~etter understanding of the present invention may be gained from a consideration of the following detailed description, taken in conjunction with the accompanying drawings in which:
FIG. lA is a diagram illustrating the relationship of an orbiting body such as a satellite whose spin axis attitude is passively maintained in accordance with the present invention;
FIG. lB is a diagram similar to that of FIG. lA from a top plan view illustrating relative positions of the spin axis and the orbit normal at differen~ times;
2~ FIG. 2A ïs a diagram illustrating the change in satellite orientation resulting from the regressional motion of the orbit normal which is produced by the earth's oblateness;
FIG. 2B is a diagram similar to that of the diagram in FIG. 2R from a top plan view FIG. 3A is a diagram illustrating the regressional motion of the spin axis of a satellite FIG. 38 is a diagram si.milar to that of the diagram in FIG. 3A from a top plan view;
~o ~ .
~ t~ 9 FIG. ~ is a diagram illustrating the coordinate system definition employed in the description herein; and FIG. 5 is a graph of the equilibrium spin axis angle ~O as a function of spin speed for a space station application having an orbit altitude of 500 km and inclination of 28~5.
~ESCRIPTION OF THE PREFERRED EMBODIMENT
The essence of this invention lies in the design of the orbiting body mass properties (body inertia and geometry), the production of a particular spin rate, and a unique initial orientation of the body in orbit such that the ensuing inertial motion of the spin axis continuously and passiveIy follows the ~ction of the precessing orbit plane while also limiting the total excursion of the sun line from the equatorial plane of the spinning body. ~his unique arrangement of system elements accounts for the basic system diagram of FIGS. lA and lB.
As illustrated in FIGS. lA and lB, the spin axis is fixed in the plane defined by the earth polar axis and the orbit plane normalj and at a fixed angle ~O with respect to north. If the spin axis is placed in this position initially, it will remain there indefinitely, even though the orbit itself is precessing, provided the angle ~O is correctly chosen. Furthermore, the spin axis orientation is stable in that small errors in the initial placement remain small indefinitely. ~hus, the spin axis is passively maintained in a fixed position in the earth axis/orbit plane without recourse to fuel or energy expenditure, provided it is placed at the correct initial angle, ~O~ to begin with-This angle ~O is a function of two parameters i and k, as defined in the following relationship:
i - orbit inclination k = 1.5( a-l) where ~vehicle spin inertia) (vehicle transverse inertia~
~O = orbital rate Q = spin rate y = rate of regression of orbit line of nodes.
It is important in practicing the method of the present invention to select the mass geometry of the body and the spin rate ( ~ and Q , respectively) consistent with the desired orbital parameters referenced above (i, QO~
and y ) so that the angle ~O is small enough to provide adequate solar cell power over the mission life; and then to place the vehicle's spin axis in the correct orientation at the start of the mission.
In a preferred embodiment of the method of the invention, the spin axis orientation angle ~ O may be determined as follows. For a vehicle in an orbit which is inclined with respect to the equatorial plane, the oblateness of the earth causes the orbit normal to precess about the north/south axis in a retrograde sense. This regressional motion is shown in Figs. 2A and 2B. In addition, gravity gradient torques across the body cause the spin axis of the vehicle to precess about the orbit normal in a positive sense if the body is rod-shaped that is, having a roll-to-pitch ratio less than unity, and in a retrograde sense if the body is disk-shaped. The motion for the disk-shaped case is illustrated in FIGS. 3A and 3B.
~ ~3~
These two precessions will generally cause the spin axis of the vehicle to wander over large regions of the sky unless mass and/or energy are expended to counteract these forces.
An alternative approach, and the fundamental principle of the method of the present invention, Ls to size the vehicle and orbit parameters so that ~he precessional motions combine favorably to allow passive maintenance of the spin axis attitude in the useful, known orientation illustrated in FIGS. lA and ls. This favorable combination is achieved by locating the spin axis between the north axis and the orbit normal at a position such that the gravity gradient precession of the spin axis is just balanced by the regressional motion of the orbit normal to produce the planar equilibrium configuration as illustrated in FIGS. lA
and lB.
To derive the method of the invention mathematically, consider a rotating xyz orbital coordinate system as illustrated in FIG. 4 with z along the orbit normal axis and x at the ascending node. The angular rate ~ of this system with respect to the celestial XYZ system is O
~ sin i (1) cos i with the equations of motion for the vehicle angular moment vector h satisfying the relation d~
d~ ~ x h T~G (2) where ~G = gravity gradient torque.
Since the gravity disturbance torque is small and requires long time intervals to move the h~ vector appreciably, it is replaced by its average value over an orbit to yield the governing equations of motion for the o~
system: (Equation 3) /hx\ /-hy\ /hz sin i - hy cos i~
~ h ) = _ 3 0 (C-A) hz ~ hx) + Y ~hx cos i where C is the vehicle spin inertia and A is the transverse inertia about the vehicle center o gravity.
Letting ~ denote the unit vector, ~ = (Y~ = lh The above equation can be rewritten:
x' = (kz-cos i) y + z sin i y' = (cos i - kz) x (5) z' = -x sin i where: d ' = , ~ = Yt d ~ 2 k = 1.5( ~ ) ~y and .~ = C
These expressions (5) have the two Eirst integrals, x2 + y2 + z2 = l kz2 ~ z cot i t Cl (6) 2 sin i and the equilibrium solution xO = o (7) zO sin i Y cos i - kz with zO a solution to the quartic relation, k2Zo4 - 2 k cos i zo3+(1 k2~ Zo2~2 k cos i zo-cos2 i=0 (8) The equilibrium spin angle ~O is then determined from the equilibrium solution, ~O = arc tan~- ) + i - ~/2 Yo The dependence of ~ O on the parameters i and k is evident in the form of the equilibrium solution to Equations (7), (8) and (9)~ That the solution is stable follows from the first integral expressions in Equation (6) above which show the motion of the spin axis to be on the intersection of the unit sphere and a parabolic cyli.nder. For the equilibrium solution, this intersection is a single ~oint~
This closed contour behavior demonstrates the stability of the equilibrium motion described by this se.t of equations.
To illustrate the concept, consider a large space station which is to be placed in a 500 kilometer orbit inclined at an angle o 28.5. For such an orbit, the nodal regression rate is 6.72 per day and the orbital period of 94.13 minutes. Let the vehicle have a dual spinner construction consisting of a large rotor, which is spun to provide both gyroscopic stiffness and a spinning gravity environment, and a despun zero gravity section.
environment, and a despun zero gravity section.
FIG. 5 illus-trates the critical angle ~O for spin rates from one to six revolutions per minute and for values of vehicle inertia ratios a ranging from 1.2 to 1.~. For the dual spin application, a is the ratio of the rotor spin inertia to the vehicle transverse iner~ia. Since small values of ~O are desired for power purposes, higher spin rates and lower O values are preferred ~or this application.
Hence the spacecraft designer would size the rotor mass properties and spin rate to generate an acceptable spin axis angle ~O for such an objective.
The method of the invention applies to both spinning and d~al spinning spacecraft. In the latter casel however, the vehicle spin inertia must be replaced by the rotor spin inertia in determining the spin axis orientation angle ~O~
Also it will be noted from the equilibrium condition of Equations (7), ~8) and (9) that there can exist as many as four orientation angles ~O for each orbit and spacecraft.
Generally, the smallest ~O value is of most interest because of power considerations. ~owever, the other angle solutions can be used to provide similar spin axis maintenance.
Thus, there has been described a method for passively maintaining the attitude of a spinning orbiting body having as its orbital parameters an orbit inclination, an orbital rate and a rate of regression of orbit line of nodes, so that the spin axis orientation angle of the body remains essentially fixed and stable relative to the orbital plane of the body, even when the orbit of the body is precessing, to provide optimum antenna gain and optimum solar cell illumination. By using the method of the invention of passive stabilization to orient the satellite, the penalties in weight, complexity, and energy consumption existing in conventional active stabilization systems of the prior art have been minimized~
Although there have been described above specific arrangements of a method for controlling the attitude of a .. ~ , .. ........ ... .
spinning body in orbit in accordance with the .invention for the purpose of illustra~ing the manner in which the invention may be used to advantage, it will be appreciated that the invention is not limited thereto. Accordingly, any S and all modifications, variations or equivalent arrangements which may occur to those skilled in the art should be considered to be within the scope of the invention as defined in the annexed claims.
Claims (6)
1. A method of passively stabilizing the attitude of a spinning earth-orbiting body having as its orbital parameters an orbit inclination i, an orbital rate .OMEGA.o and a rate of regression of orbit line of nodes ? , so that the spin axis orientation angle .PHI.o of said body remains essentially fixed and stable relative to the orbital plane of said body, even when the orbit of said body is precessing, which comprises:
selecting a mass geometry .sigma. for said body defined by the ratio of said body's spin inertia to said body's transverse inertia;
selecting a spin rate .OMEGA. for said body so that said spin axis orientation angle .PHI.0 for said body is an equilibrium solution to the following relationship .PHI.o = where for an xyz orbital coordinate system having z along the orbit normal axis and x at the ascending node Xo = 0 Yo = i = orbit inclination K = and zo is a solution to the quartic k2zo4-2 k cos i zo3+(1-k2) zo2+2 k cos i zo-cos2 i=0 and initially orienting said body in orbit with said spin axis orientation angle.
selecting a mass geometry .sigma. for said body defined by the ratio of said body's spin inertia to said body's transverse inertia;
selecting a spin rate .OMEGA. for said body so that said spin axis orientation angle .PHI.0 for said body is an equilibrium solution to the following relationship .PHI.o = where for an xyz orbital coordinate system having z along the orbit normal axis and x at the ascending node Xo = 0 Yo = i = orbit inclination K = and zo is a solution to the quartic k2zo4-2 k cos i zo3+(1-k2) zo2+2 k cos i zo-cos2 i=0 and initially orienting said body in orbit with said spin axis orientation angle.
2. The method of Claim 1 wherein the spin axis of said body is located in a plane defined by the polar axis of the earth and the orbit plane normal of the body.
3. The method of Claim 2 wherein the spin axis of the body is oriented between polar north and the orbit normal.
4. A method of passively stabilizing the attitude of a spinning earth-orbiting body which is precessing, the method serving to maintain the spin axis orientation of said body essentially fixed and stable relative to the orbital plane of said body, even when the orbit of said body is also precessing, the method comprising:
selecting a body mass geometry and orbital parameters comprising an orbit inclination, an orbital rate and a rate of regression of orbit line of nodes so that the precessional motions of said body and said orbit are equal and opposite in direction;
locating the spin axis of said body in a plane defined by the earth polar axis and the orbit normal of the body and between said polar axis and said orbit normal; and selecting the spin axis angle so that the gravity gradient precession of said spin axis equals the regressional motion of the orbit normal to produce an axis equilibrium configuration.
selecting a body mass geometry and orbital parameters comprising an orbit inclination, an orbital rate and a rate of regression of orbit line of nodes so that the precessional motions of said body and said orbit are equal and opposite in direction;
locating the spin axis of said body in a plane defined by the earth polar axis and the orbit normal of the body and between said polar axis and said orbit normal; and selecting the spin axis angle so that the gravity gradient precession of said spin axis equals the regressional motion of the orbit normal to produce an axis equilibrium configuration.
5. The method of Claim 4, further including the step of initially orienting said body in orbit with said spin axis angle.
6. An earth orbiting system in which a spinning satellite is located in orbit about the earth, the system comprising:
a spinning orbiting body having a mass geometry defined by the ratio of the body's spin inertia to the body's transverse inertia and a selected spin rate consistent with desired orbital parameters satisfying the following definitions:
i = orbit inclination k = where .OMEGA.o = orbital rate .OMEGA. = spin rate .gamma. = rate of regression of orbit line of nodes.
and wherein the spin axis of the orbiting body is at an angle ?o defined as:
?o = arc tan + i - .pi./2 where, for an xyz orbital coordinate system having z along the orbit normal axis and x at the ascending node, the following relations hold:
Xo = 0 and zo is a solution to the quartic k2zo4-2 k cos i zo3+(1-k2) zo2+2 k cos i zo-cos2 i=0
a spinning orbiting body having a mass geometry defined by the ratio of the body's spin inertia to the body's transverse inertia and a selected spin rate consistent with desired orbital parameters satisfying the following definitions:
i = orbit inclination k = where .OMEGA.o = orbital rate .OMEGA. = spin rate .gamma. = rate of regression of orbit line of nodes.
and wherein the spin axis of the orbiting body is at an angle ?o defined as:
?o = arc tan + i - .pi./2 where, for an xyz orbital coordinate system having z along the orbit normal axis and x at the ascending node, the following relations hold:
Xo = 0 and zo is a solution to the quartic k2zo4-2 k cos i zo3+(1-k2) zo2+2 k cos i zo-cos2 i=0
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| US886,481 | 1986-07-17 | ||
| US06/886,481 US4757964A (en) | 1986-07-17 | 1986-07-17 | Method for controlling the attitude of a spinning body in orbit |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA1287029C true CA1287029C (en) | 1991-07-30 |
Family
ID=25389108
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA000539476A Expired - Fee Related CA1287029C (en) | 1986-07-17 | 1987-06-11 | Method for controlling the attitude of a spinning body in orbit |
Country Status (7)
| Country | Link |
|---|---|
| US (1) | US4757964A (en) |
| EP (1) | EP0287572B1 (en) |
| JP (1) | JPH01500308A (en) |
| CN (1) | CN1010000B (en) |
| CA (1) | CA1287029C (en) |
| DE (1) | DE3787093T2 (en) |
| WO (1) | WO1988000730A1 (en) |
Families Citing this family (16)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4776540A (en) * | 1986-10-03 | 1988-10-11 | Communications Satellite Corporation | Method of orienting a synchronous satellite |
| KR960015106B1 (en) * | 1986-11-25 | 1996-10-28 | 가부시기가이샤 히다찌세이사꾸쇼 | Surface package type semiconductor package |
| US5100084A (en) * | 1990-04-16 | 1992-03-31 | Space Systems/Loral, Inc. | Method and apparatus for inclined orbit attitude control for momentum bias spacecraft |
| US5669586A (en) * | 1994-12-06 | 1997-09-23 | Space Systems/Loral, Inc. | Satellite gravity gradient compensation using on-orbit solar array reorientation |
| US5738309A (en) * | 1996-02-28 | 1998-04-14 | Hughes Electronics | Single axis correction for orbit inclination |
| US6128575A (en) * | 1998-06-10 | 2000-10-03 | Hughes Electrnoics Corporation | Methods for accurately inserting satellite constellations into common orbit planes |
| DE10342866A1 (en) * | 2003-09-15 | 2005-04-21 | Eads Astrium Gmbh | Method for determining the position of a spacecraft with the aid of a direction vector and a total twist measurement |
| JP4926962B2 (en) | 2004-05-18 | 2012-05-09 | アコーダ セラピューティクス、インク． | Purification method of chondroitinase and its stable preparation |
| CN100439204C (en) * | 2007-07-03 | 2008-12-03 | 北京控制工程研究所 | Estimating and compensating method for reducing influence of attitude controlling jetting on the orbit |
| CN103116358B (en) * | 2013-01-15 | 2015-07-29 | 南京航空航天大学 | Folding and unfolding control method stablized by a kind of spacecraft rope |
| CN103786896A (en) * | 2014-03-03 | 2014-05-14 | 邹珺 | Manned spacecraft with gravity loading device |
| CN105159309B (en) * | 2015-09-01 | 2018-01-23 | 西北工业大学 | It is a kind of to utilize the spacecraft Attitude stable control method for biasing tether |
| US10005568B2 (en) | 2015-11-13 | 2018-06-26 | The Boeing Company | Energy efficient satellite maneuvering |
| CN106052694B (en) * | 2016-07-11 | 2017-03-15 | 中南大学 | Based on the method that gravitational vectors and its gradient tensor carry out locating and tracking to single moving object |
| CN106516177B (en) * | 2016-10-18 | 2019-07-19 | 南京航空航天大学 | It is a kind of based on rope be technology space junk recycling and control method |
| CN112061424B (en) * | 2020-07-16 | 2022-04-12 | 北京控制工程研究所 | Maneuvering process energy angle dynamic tracking method based on fusion target attitude |
Family Cites Families (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US3271565A (en) * | 1962-01-16 | 1966-09-06 | Bell Telephone Labor Inc | System for determining the orbital parameters of a terrestrial satellite |
-
1986
- 1986-07-17 US US06/886,481 patent/US4757964A/en not_active Expired - Lifetime
-
1987
- 1987-06-08 DE DE87903983T patent/DE3787093T2/en not_active Expired - Fee Related
- 1987-06-08 WO PCT/US1987/001340 patent/WO1988000730A1/en active IP Right Grant
- 1987-06-08 EP EP87903983A patent/EP0287572B1/en not_active Expired - Lifetime
- 1987-06-08 JP JP62503667A patent/JPH01500308A/en active Pending
- 1987-06-11 CA CA000539476A patent/CA1287029C/en not_active Expired - Fee Related
- 1987-07-16 CN CN87104891A patent/CN1010000B/en not_active Expired
Also Published As
| Publication number | Publication date |
|---|---|
| CN87104891A (en) | 1988-03-30 |
| EP0287572A1 (en) | 1988-10-26 |
| WO1988000730A1 (en) | 1988-01-28 |
| EP0287572B1 (en) | 1993-08-18 |
| JPH01500308A (en) | 1989-02-02 |
| DE3787093D1 (en) | 1993-09-23 |
| US4757964A (en) | 1988-07-19 |
| CN1010000B (en) | 1990-10-17 |
| DE3787093T2 (en) | 1994-04-21 |
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