NL2034951B1

NL2034951B1 - Method and system for controlling an angular orientation of a spinning body

Info

Publication number: NL2034951B1
Application number: NL2034951A
Authority: NL
Inventors: Ostanin Igor
Original assignee: Univ Twente
Priority date: 2023-05-30
Filing date: 2023-05-30
Publication date: 2024-12-10

Abstract

The invention relates to a method for controlling an angular orientation of a spinning body with respect to an axis of rotation of the spinning body, comprising the steps of: a) providing the spinning body at a ﬁrst stable angular orientation with respect to its axis of rotation, the spinning body having three principal moments of inertia about three orthogonal principal axes, the three principal moments of inertia defining a tensor of inertia, wherein the three principal moments of inertia each have a first value and wherein the ﬁrst values of the three principal moments of inertia are identical so that the tensor of inertia is spherical; b) selecting a second angular orientation of the spinning body with respect to its axis of rotation, the second angular orientation being any angular orientation of the spinning body; and c) varying a value of at least one of the principal moments of inertia in accordance with a predetermined schedule, such that: Cl. initially the three principal moments of inertia no longer have identical values and the tensor of inertia is non-spherical, thus causing the angular orientation of the spinning body to gradually change from the ﬁrst angular orientation to the second angular orientation; and c2. eventually the three principal moments of inertia have identical values again and the tensor of inertia is spherical again to stabilize the spinning body in its second angular orientation. The invention further relates to a system for performing this method, and to a spinning body which comprises such a system.

Description

METHOD AND SYSTEM FOR CONTROLLING AN ANGULAR ORIENTATION OF A

SPINNING BODY

The present invention relates to a method for controlling an angular orientation of a spinning body with respect to an axis of rotation of the spinning body. Spinning is conventionally used to stabilize a body, in particular a moving body. More in particular, spinning is used to stabilize bodies in orbit, like e.g., satellites.

Exploration of the question of attitude control of a spinning body is of great practical importance for modern aerospace technology. All up-to-date satellites, spacecraft, and other systems capable of performing major orientation maneuvers, do so by introducing external torques, using small reactive thrust engines. Such a thrust-based angular positioning system can be used only a limited number of times, i.e., until the propulsion agent is fully consumed. Alternatively, existing inertial systems (reaction wheels) are capable of adjusting or stabilizing the attitude of a spacecraft very precisely, given small drift angular velocities, but fail in presence of fast rotation. These, as well as some less common systems (e.g. passive ones using the gradient of drag forces of the thin atmosphere, gravitational force gradient, yoyo de-spins, etc.), fall into two categories: » the ones using “external” moments M$ of different nature. The conservation of angular momentum L = Ig in this case can be written as:

Io = 34 MY 1) » the ones redistributing the conserving angular momentum between the main body and its special rotating mechanical parts: lo =>; La, (2)

Until recently, a third possibility has been largely neglected - altering magnitude and direction of the spin angular velocity by changing a tensor of inertia (TOL) of the spinning body:

Io = io (3)

The fundamental difference with the second category is that the change of the tensor of inertia is achieved by symmetric redistribution of weights using forces that do not create internal moments, while the reaction wheels and other similar systems imply net rotation of the masses by nonzero internal moments.

Changes in the tensor of inertia can dramatically change the behavior of a spinning body.

The presence of a deviatoric part of the tensor of inertia causes misalignment of the angular velocity and angular momentum of a spinning body, leading to chaotic motion, except in the special cases of stable periodic rotations around first or third principal axes (or quasi-periodic rotation/wobbling in the vicinity of these principal axes). Rotation in the vicinity of the second principal axis causes so- called intermediate axis instability, leading to a well-known “Dzhanibekov effect” - a series of quasi- periodic 180 degrees flips of the body orientation with respect to its spin angular velocity. Therefore,

a controlled mass redistribution leading to transformation of a first or third principal axis into a second one can be an efficient tool of orientation control.

WO 2013/002673 Al discloses a method of reorientating and controlling the thrust of a rotating spacecraft with a solar sail in which use is made of a single-axis mass distribution control to induce Dzhanibekov instability on demand. The motion of the spacecraft is stabilized once a full 180-degrees flip is accomplished. The method is disclosed in the context of re-orientation of the solar sail, to provide necessary averaged acceleration of the spacecraft.

This idea has been revisited in 2017 by P. M. Trivailo, “Utilisation of the “Dzhanibekov effect” for the possible future space missions”, Proc. of 26-th Int. Svmp. Sp. Fl. Dyn., Matsuyama,

Japan, pages 1-10, 2017, who demonstrated its feasibility in a numerical simulation. The same authors later developed and generalized the theory in P. M. Trivailo and H. Kojima, “Inertial

Morphing as a Novel Concept in Attitude Control and Design of Variable Agility Acrobatic

Autonomous Spacecraft”, Nonlinear Approaches in Engineering Application, pages 119-244,

Springer International Publishing, Cham, 2022, ISBN 978-3-030-82719-9. In this document the phase space of rigid body motion is explored, and a few different particular maneuvers are indicated that could be accomplished by altering a body’s tensor of inertia. However, these two documents only deal with particular cases of orientation control — switching between the pre-defined axes of possible stable rotation.

The invention has for its object to provide an improved method for controlling an angular orientation of a spinning body with respect to an axis of rotation of the spinning body. In accordance with the invention, such a method comprises the steps of: a) providing the spinning body at a first stable angular orientation with respect to its axis of rotation, the spinning body having three principal moments of inertia about three orthogonal principal axes, the three principal moments of inertia defining a tensor of inertia, wherein the three principal moments of inertia each have a first value and wherein the first values of the three principal moments of inertia are identical so that the tensor of inertia is spherical; b) selecting a second angular orientation of the spinning body with respect to its axis of rotation, the second angular orientation being any angular orientation of the spinning body; and c) varying a value of at least one of the principal moments of inertia in accordance with a predetermined schedule, such that: cl. initially the three principal moments of inertia no longer have identical values and the tensor of inertia is non-spherical, thus causing the angular orientation of the spinning body to gradually change from the first angular orientation to the second angular orientation; and ¢2. eventually the three principal moments of inertia have identical values again and the tensor of inertia is spherical again to stabilize the spinning body in its second angular orientation.

The method of the invention allows arbitrary angular maneuvers to be performed by changing the tensor of inertia of the spinning body. The method involves having a spherical tensor of inertia, i.e. identical moments of inertia about the three principal axes, both at the initial and the final moment of the maneuver. In this way both the initial and the final state of the spinning body are characterized by a stable periodic rotation. The desirable angular re-orientation is then achieved by optimization of the parameters of evolution of the tensor of inertia over time between the initial and the final state. The schedule may be optimized in order to minimize the time and/or the energy needed to reach the final state.

In an embodiment of the method. the predetermined schedule may include only variations in the moment(s) of inertia which are within a capability envelope of the spinning body, i.e. those done by forces, powers and torgaes that can be achieved by actuators arranged on or in the spinning body.

The schedule may be determined by calculation, may be retrieved from a database, or may be received from a remote transmitter. As such, the schedule can already be on board the spinning body from the start, or it may be calculated or retrieved at a remote location and sent to the spinning body on demand.

In order to increase the rate at which the angular orientation is varied, step c) of the method may comprise varying the value of multiple principal moments of inertia to change the angular orientation of the spinning body.

In an embodiment of the method, and when the first angular orientation is substantially parallel to one of the principal axes, step cl) may comprise the substeps of: cla. varying the value of the at least one principal moment of inertia in accordance with a first partial schedule to initiate Dzhanibekov instability, thus causing a misalignment between the first angular orientation of the spinning body and the principal axis; clb. stabilizing the misaligned orientation by restoring the spherical tensor of inertia; and cle. further varying the value of the at least one principal moment of inertia in accordance with a second partial schedule to cause the angular orientation to gradually change from the misaligned orientation to the second stable angular orientation.

In this way the schedules may be optimized, and the method may be performed in an efficient manner even when the rotation axis of the spinning body initially almost coincides with one of the principal axes of the body.

In an embodiment of the method, step ¢2) may comprise resetting the values of the three principal moments of inertia to the first value. In this way the angular velocity of the spinning body after the re-orientation maneuver will be the same as before, due to the conservation of angular momentum. Alternatively, if the value of the three principal moments of inertia at the end of step ¢2)

is not equal to the first value, the angular velocity of the spinning body after the re-orientation maneuver will be greater or smaller than before.

In another embodiment of the method, step c} may comprise varying a mass distribution of the spinning body symmetrically with respect to the at least one principal axis about which the principal moment of inertia is varied. Varying the mass distribution in a symmetrical manner will alter the moment of inertia about the principal axis.

In a further embodiment, in step ¢) net rotation about the principal axis may be avoided when varying the mass distribution. A symmetrical variation of the mass distribution may be achieved without creating internal moments.

In an embodiment of the method, step ¢) may comprise moving a pair of opposite masses in opposite directions with respect to the principal axis. By providing the spinning body with movable masses, the mass distribution may be varied easily and swiftly.

In one embodiment, the opposite masses may be translated in opposite directions towards or away from the principal axis. A translating movement may be realized in a structurally simple manner.

In another embodiment, the opposite masses may include at least two masses at each side of the principal axis, which may be pivoted in opposite directions towards or away trom the principal axis. Rotation and anti-rotation of two masses at each side cancel out any resulting moment.

In a further variant of this embodiment, the masses may further all be pivotable in a same direction. By suppressing the anti-rotation, the pivotable masses may also function in the way of a conventional reaction wheel, thus providing additional ways of controlling the angular orientation of the spinning body.

In yet another embodiment, the method may further comprise step bl) of moving, prior to step Cc), a pair of masses in opposite directions along an auxiliary axis which is not orthogonal to any one of the principal axes. In this way the direction of a principal axis may be redefined instantaneously, which may be particularly useful when the first stable angular orientation is substantially parallel to one of the principal axes of the spinning body. This step bl) may be performed as an alternative to steps cla, c1b) discussed above.

In an embodiment, the movable masses may comprise at least one functional element of the spinning body. This allows the method to be performed by simply moving existing parts of the spinning body, like e.g. the payload of a spacecraft.

Alternatively. or additionally, the movable masses may comprise at least one dead weight.

Although a dead weight adds to the total mass of the spinning body, it may allow for a simpler and lighter moving mechanism, which may offset the additional weight.

In an embodiment of the method, energy which is consumed or generated in one part of a control cycle may at least partially be recuperated or consumed, respectively, in another part of the control cycle. For instance, chemical energy from an accumulator battery may be used to power a mechanism for moving masses in one direction, and a return move of the masses may be used to regenerate the accumulator battery. In this way performing the method requires almost no energy, which is particularly advantageous if the spinning body is a spacecraft in orbit. The masses may be 5 moved at a motion rate which is proportional to a rotation rate of the spinning body. In this way, any maneuver to re-orientate the spinning body can be performed regardless of the initial rate of rotation of the body.

The invention further relates to a system for performing the method discussed above. In accordance with the invention, such a system for controlling an angular orientation of a spinning body with respect to an axis of rotation of the spinning body comprises controllable variation means for varying a value of at least one of the principal moments of inertia in accordance with a predetermined schedule to change the angular orientation of the spinning body from a first stable angular orientation to a selected second stable angular orientation, which may be any angular orientation of the spinning body.

Further embodiments of the system according to the invention for the subject matter of dependent claims 19-32.

And finally, the invention relates to a spinning body comprising such a system. Although the system is applicable to various types of spinning bodies, like smart projectiles or gyroscope- frame based devices for three-dimensional orientation, in an advantageous embodiment of the invention the spinning body may be a spacecraft. In that case, the movable masses may comprise parts of a payload of the spacecraft.

The invention will now be illustrated by way of an exemplary embodiment, with reference to the annexed drawings, in which:

Fig. 1A shows a geometric interpretation of control parameters q(t), q2(t), illustrating a system for which Ip = 2mlé ,

Fig. 1B shows the schematics of an arbitrary angular maneuver,

Fig. 2 shows schematic representations of maneuvers studied,

Fig. 3A-D show the evolution over time of principal moments of inertia during maneuvers 1-4 as shown in Fig. 2,

Fig. 3E-H show the evolution over time of kinetic energies during the same maneuvers 1-4,

Fig. 4A-D show the evolution over time of the principal moments of inertia during maneuvers 6-9 as shown in Fig. 2,

Fig. 4E-H show the evolution over time of the kinetic energies during the same maneuvers 6-9,

Fig. 5A shows the evolution of a goal functional as a function of an iteration number for maneuvers 1-4 as shown in Fig. 2,

Fig. 5B shows the evolution of the goal functional as function of the iteration number for maneuvers 6-9 as shown in Fig. 2,

Fig. 5C illustrates conservation of angular momentum during a simulation of maneuver 9,

Fig. 6A shows a possible technical implementation of a spinning body having an angular orientation control system which can be compatible with a CubeSat design specification,

Fig. 6B schematically illustrates the introduction of an additional axis of possible control of moments of inertia, and

Fig. 6C schematically illustrates both translating masses M and rotating masses m for changing the tensor of inertia.

In the context of spacecraft attitude control and maneuvering, the method and system of the invention have a number of attractive features. In contrast with existing systems, the system manipulating the spacecraft’s tensor of inertia (TOI) is capable in principle to guide the spinning body toward an arbitrarily selected orientation with respect to its axis of rotation by redistributing the energy between a chemical battery and the kinetic energy of body's own rotation. Such angular maneuvers are achieved without consuming a propulsion agent and with zero net energy losses, other than relatively small heat losses in electrical circuits and frictional mechanical contacts. Among other important features of such method of maneuvering is the possibility to change the body's TOI by displacing the useful load, and insensitivity of the maneuver to the absolute value of the spacecraft’s angular momentum — as will be discussed in-depth below.

In order to demonstrate the capabilities of the inventive approach, a special simulation- guided optimization framework was developed, based on the implementation of nonspherical particle dynamics within the open-source code MercuryDPM.

The framework convincingly demonstrates the impressive capabilities of attitude correction by optimal control of the body’s TOL Necessary manipulations with the inertia tensor of a body do have a straightforward mechanical interpretation and can be easily implemented in a real spacecraft.

Theoretical background - motion of a body changing its TOI

The equations of motion of a rigid body — Euler’s equations — admit periodic analytical solutions in only a handful of particular cases and most of the time feature chaotic solutions, that are usually approached with different schemes of numerical integration. The mathematical properties of these equations and the phase space of their solutions, as well as an extended discussion of the approaches to numerical integration. are beyond the scope of this application. Only the necessary minimum theoretical background sufficient to convey the main ideas behind the inventive way of attitude control is considered below.

Maneuvering by changing TO! — general considerations

Unlike the total mass of an isolated mass distribution (body or mechanism), its TOI can in principle be changed by altering its geometry. Hereafter the term “rigid” will be used for the motion of the body that changes its TOI, although the use of such a terminology becomes ambiguous.

By “change” of the TOI here and below is meant the change in its principal components. It is important to note, however, that simple geometric considerations show that principal components of inertia can not be changed independently. For example, scaling the mass distribution along 15¢ principal direction affects both I, and I5. It, therefore, makes sense to choose the control parameters as mass distribution scaling factors: h =q2(t)93(t)l,

I; =qi(t)q3(t)lo, (4)

I; =q (MgO

It is also easy to see that without loss of generality it can be accepted that g3(t) = 1, as it would only contribute to a scaling multiplier of the angular velocity. The scaling of principal components of TOI and their time derivatives is then given as:

L = q(®Old = 420] h =q®l I =¢ 0 (5)

I; = qq]; = (10920) + (Od)

Here q4 (t) and gq, (t) are independent control parameters that can be manipulated within a certain range between Gmin <1 and Gmax > 1, to achieve the desirable maneuvering; Jg is the baseline spherical tensor of inertia. Note that in case when q(t) = 1,q,(t) = 1, the TOI is spherical. Fig. 1A offers a simple mechanical interpretation of the coefficients q;(£}) and g,(¢), highlighting one possible way of technical implementation.

It is easy to see that certain changes in the TOI of a rotating body can be achieved with zero work of centrifugal forces. For example, the body rotating precisely around its principal axis can be arbitrarily transformed (stretched, split, etc.) along this principal axis, as long as the mass distributions around the other two axes remain the same. The other changes may be associated with positive/negative work done to move masses in the field of centrifugal forces.

Simple physical considerations lead to the conclusion that g4 (¢) and g,(¢) should be twice differentiable functions with bounded second derivatives, which ensures that the transformation of the TOI can be done using finite forces/power. Below these profiles are chosen to be cubic splines connecting equispaced reference values.

The usual convention in rigid body mechanics is the numbering of TOT's principal components [,, 1, I; in the order of their decrease. In the case of changing principal components,

this convention 1s not useful. Further below indices 1,2,3 do not imply order, the minor, major and intermediate axes are explicitly identified if necessary. Also, in case when q(t) = 1,g,(t) = 1, any axis of rotation is the body’s principal axis. However, in the present application, the term “principal axis” will only be used for the directions that remain principal directions of the body for any values ofqu(t) q(t).

As mentioned above, the choice of limits for g(t) and g, (t) ensures that the spherical TOI is available. It is therefore possible to stabilize the motion around fixed axis by making the TOI spherical. This dictates the scheme of the maneuver, depicted in Fig. 1B. The maneuver starts at a certain state with the spherical TOI I, and the first stable orientation (85eg, Doeg). defining the direction of angular velocity weg = wo in the own spherical coordinate system of the body (defined such that n, corresponds to (7/2,0), ny — (n/2,7/2,), n3 = ny X ny). In case the initial angular velocity is not aligned with one of the principal axes, the changes in the tensor of inertia initiate a complex aperiodic motion. The sequence of changes ends with the state with a spherical tensor of inertia I, again, characterized by the angular velocity weng and second stable orientation

IS (Bena, Pena). The conservation of angular momentum ensures that eng = Wo (during the maneuver, however, the angular velocity varies). The sequence of TOI changes is found by an optimization procedure that ensures the desired (Gong, Peng). The optimization technique is described below.

It is important to note that if the body rotates precisely around one of its principal axes, the changes g;{£), g(t) can not perturb the periodic motion. In such a case, the principal axis, aligned with the angular velocity, can be transformed into an intermediate axis, which causes instability, known as the Dzhanibekov effect or tennis racket theorem, and rapid development of the misalignment. Therefore, the described system of maneuvering practically does not have deadlock states.

The rigorous justification of the existence and uniqueness (non-uniqueness) of the sought maneuver trajectory is beyond the scope of this patent application. However, numerical results clearly demonstrate that the optimization algorithm, given proper search space for the maneuver parameters, always finds the maneuver leading precisely to the desired state, even for transitions between the states with close alignment of the axis of rotation with the principal axes.

Equations of motion of a body that changes its TOI

In this section the equations of rotational motion of a rigid body changing its TOL are presented. Based on the considerations above, the following set of assumptions is accepted: « a particular type of time evolution of the TOI is considered - the C2 continuous evolution of TOPs principal values, given by equation (5) as discussed above.

« the TOI and its first time derivative are prescribed precisely in the local (rotating) coordinate system at every moment of time.

The equations of motion are obtained straightforwardly by generalization of standard derivation of Euler equations. These equations are obtained from the condition of conservation of angular momentum — in case of zero total external moment acting on the body, time rate-of-change of the angular momentum in the inertial frame of reference should be zero:

L=0 (6)

The angular momentum L is given in the local coordinate frame instantaneously aligned with the body's principal axes as: =F) (7)

In such a local coordinate frame, rotating with the angular velocity ow! (£) around its origin, the time derivative of a vector L is given by: (ret = L(t) + w(t) x L(t) (8)

The equation (7) can therefore be written as:

VOODOO tw x HOael(t) =0 ©)

This equation can be re-written in inertial Cartesian frame component-wise using indicial notation as:

Oo © + I; (0); (8) + eijk (Oi (Dewy (8) = 0 (10)

The non-spherical TOI [;;(¢), its time derivatives I; (1), the angular velocity w;(£} and its derivative w;(t) are found by proper rotations of the corresponding components in the local frame: lu) = Qf (OID) Qj (2), (11) © = Qh. OO; (©), (12) w(t) = Qi; (D)w] (€). (13) @i(£) = Qi; (Oa; ©). (14)

The TOI components in local coordinate system LO is given by 12 £) 0 © 0 gilt

VOT | 0 i) 4 and Q(t) is the rotation matrix defined as egny (6) emt) esng(t)

Q(t) = [ime ean (1) 0) (16) eins(t) eznz(£) ezn3(t)

where e; are the orths of global Cartesian coordinate system, and n;(¢t) are orths of the body’s eigendirections.

Time integration of the equations of motion

Time integration of equations of motion of a rigid body is a non-trivial task, even for the case of constant tensor of inertia and absent external moments. The reason for that is the nonlinear geometric link between the angular velocity and its derivative — see the discussion below. A time integration scheme should ensure precise conservation of energy and angular momentum to ensure the reliability of the simulation results.

The time integration scheme used in this application utilizes a leap-frog algorithm of the time integration of the motion of non-spherical particle, similar to the algorithm utilized in the commercial software PFC 4.0 from Itasca Consulting Group, Inc of Minneapolis (USA). The equation (10) is solved using a finite difference procedure of the second order, computing angular velocities w; at mid-intervals t + At/2., and all other dynamic quantities at primary intervals t + At.

The orientation is tracked in the shape of rotation matrix Q(£) that is obtained by the incremental rotation of principal directions by the angle e(t) At. The equation (10) can be re-written in the matrix form as io+lIo+W =0, (Iss — Ipp)waws + 12360303 — 120202 — I3101@2 + Iwo

Ww _ [i — Iz3)wzwy +110 01 — Lizwzwz — Ipwaws + i). (22 hi): +202 121 0 — Lswzwy + 303

Iyy hz hs

I _ Ci 22 | (17)

TI liz Inf (is —Î2 hs i [a le les |, (7 Tls2 Is J

Here the regular sign convention for tensor of inertia components is implied. Using three equations (17), six unknowns w;(t + At/2), w;(t + At) have to be evaluated. Following the approach suggested in the PFC 4.0 software, the iterative algorithm is used to find these unknowns: *Setn=10 * Set ol to the initial angular velocity. e (*} Solve (17) for w; « Determine a new (intermediate) angular velocity: we! = wl" + aM A + Revise the estimate of w; as: wl = 0.5" + wh es Setn:=n + 1 and go to (*)

This algorithm gives the value of the angular velocity that is further used to update the position at the second step of leap-frog algorithm. The number of steps n necessary for the sufficient precision varies depending on the application and, based on numerical experiments, in this simulations it was set to 3.

The timestep in the present simulations have been chosen rather fine - approximately 450 timesteps per single revolution of the body, given gq; = q2 = 1. Such a timestep was chosen empirically to ensure negligible dependence of the simulation-guided optimization results on the timestep.

The described algorithm is rather simple, however, the numerical results demonstrate that in all the maneuvers described in this application it conserves the magnitude and direction of angular momentum nearly precisely, and the relative drift of the energy between same-energy initial and final state appears to be vanishingly small — as will be discussed below.

Dimensionless system of units used

It is natural to introduce the dimensionless quantities characterizing the maneuver. Moments of inertia are further measured in I, and angular velocities in wg. This naturally introduces units of time (tp = 2m /w,), angular momentum ( Iyw, ) and energy ( Iow&/2 ). The remaining quantities, characterizing the system (N, q4, q2) are dimensionless. The dimensionless duration of the maneuver

T = (tena — tpeg)/to. number of reference points N and the span [qmin, Gmax] define the parameter space where the optimal maneuver is sought.

Simulation-guided optimization procedure

The procedure to perform an optimization-based search for the optimal control parameters, providing the desired maneuver, seeks to find the control parameters q4 (t), q2{t) which provide the maneuver highlighted in Fig. 1B.

In order to guide the body toward the desired final orientation (9,¢) , the following definition of the functional is used:

£6, p, Goan Pgoar) = arccos(p (8, D)Pgoar (goan: goat) sinfcosg p(6,%) _ e= cos

SinG goat COSP goat (18)

Pgoat goats goan) = [singin \ \Cosgoa J ie, the procedure seeks to minimize the angle between the current and the desired orientation of the angular velocity direction, expressed in the local frame of a rotating body. Such functional definition does not penalize for the duration of the maneuver, complexity or rate of change of the tensor of inertia; therefore, these parameters should be prescribed as to ensure the feasibility of the maneuver. This definition also does not penalize for the energy needed to accomplish the maneuver and the number of control reference points used. This functional is strictly zero once the body's final orientation (8, ¢) precisely matches the goal orientation (8goar, $goar)-

The time evolution of coefficients q; (t), g,(t) is given by cubic splines. The initial and final nodal values q;(tpeg), qi(teng) are fixed to 1 (spherical tensor of inertia). initial and final time derivatives d; (t), d2{£) are fixed to zero. The remaining 2N nodal values q = (gt... q\, 94. 9%) are varied in an unbounded and unconstrained multidimensional optimization procedure. The optimizer seeks for a vector of unknowns X: X; € R, which are mapped to q: q; € [Gmin: Gmax] in the following

IS way: q(x) = ee _ mann cosX (19)

In the illustrated embodiment, an optimization algorithm proposed by M.J.D. Powell, “An efficient method for finding the minimum of a function of several variables without calculating derivatives.” Computer Journal, 7:155-162, 1964, is employed to vary the control parameters evolution. The duration of the maneuver T, the number of reference values N and the range [Gmin: dmax] are chosen empirically outside of the optimization cycle.

Benchmark maneuver computations

As a brief illustration of the suggested approach, the optimization of TOI control parameters to perform arbitrary angular maneuvers is considered.

The dimensionless system of units described above is used. The control parameters are varied between 0.5 and 1.5, leading to the ranges for principal moments of inertia:

I; € (0.5,1.5),1 € (0.5,1.5), 13 € (0.25,2.25). (20)

The kinetic energy can therefore vary in the range E € (0.222,2.0), while the angular momentum is constant and equal to 1.

Every maneuver is the transition between orientation i: (6;, ¢;) and orientation j: (6;, ¢;).

As was discussed above, the initial and final points of the maneuver are characterized by unit spherical TOL, while during the maneuver TOI varies. Initial and final orientations can be visualized as points on a unit sphere - see Fig. 2. Table 1 gives the set of points that was chosen to demonstrate the capabilities of the method. [ev : oo 7 | Gw

TL

Table 1: Reference orientations, given in terms of angles (6, ¢) and unit direction vectors p.

Table 2 summarizes the list of maneuvers considered in this section of the application. The first column gives maneuver number. The second columm gives the maneuver path in terms of reference orientations specified in Table 1. The third column details whether the maneuver is the result of optimization(“0") or a run with the prescribed control parameters (“P”). The fourth column gives the number of control reference points N (the number of optimization parameters is ZN). The fifth column gives the dimensionless duration of the maneuver T. The sixth column gives the value of the goal functional after convergence of the optimization procedure. The seventh column lists the number of functional evaluations during the optimization procedure (the number of simulation framework runs).

Maneuvers 1-4 illustrate the transitions between orientations that are not aligned with the principal axes of inertia.

Maneuver 5 is a single controlled Dzhanibekov flip, performed without parameter optimization - see below.

Maneuvers 6-9 are transitions between the orientations close to the principal axes.

Fig. 3. gives the evolution of principal moments of inertia and rotational kinetic energy during the maneuvers 1-4. Figs. 3A and 3E relate to maneuver 1, Figs. 3B and 3F to maneuver 2,

etc. All of these maneuvers were easily achievable with T = 16 (time corresponding to 16 periods of rotation for q; = 92 = 1). and the number of reference points N = 5. The final value of the functional indicates precise convergence up to the relative tolerance levels of the optimization procedure (1073) or higher (0 means zero up to machine precision). It is easy to see that the sought maneuvers were found by rather simple evolution of inertia tensor. Another important observation is that the kinetic energy of initial and final state matches precisely (the error does not exceed 107%), meaning that the total work of the internal forces to accomplish the maneuver is always zero.

Fig. 4 details the evolution of TOI and rotational energy for the maneuvers 6-9, whose start and/or end orientations are in close vicinity of the principal axes. Figs. 4A and 4E relate to maneuver 6, Figs. 4B and 4F to maneuver 7, etc. For these maneuvers, the algorithm also performed beyond expectations. One cannot start the maneuver if the rotation is perfectly aligned with the principal axis - any changes of q1,q2 will not induce misalignment. However, if the initial offset from the principal axis is 107377 rad, it achieves to the goal orientation with the comparable precision (Table 2). :

Ea en ps

TC CO NN ea 5p

Table 2: Benchmark maneuvers and their parameters

One interesting particular case is controlled Dzhanibekov flip - starting from the rotation around the third principal axis and spherical tensor of inertia, the axis of rotation is transformed into an intermediate axis, by setting =a, di 1a en

Once the first 7 rad flip is completed, values of q; and gq, are set to q; = 1,92 = 1 again and the rotation is stabilized. Both changes occur at the moment of alignment of rotation with the third principal axis, and therefore do not cost energy and do not induce misalignment. Numerical experiments demonstrate that the idea of such maneuver is working. However, the changes in the tensor of inertia should be instantaneous (or at least sufficiently fast) which seems impractical for some uses. The optimization routine addressing the same task also resorts to development of

Dzhanibekov instability, but does it in the relaxed form, with more complex program of changes of

TOL

The optimization problem is, however, ill-posed in this case - the final set of control parameters strongly depends on small initial misalignment of the rotation axis with the body's principal axis. This applies to all other maneuvers starting from the angular velocity close to one of the body's principal axes of inertia. Therefore, it appears to be convenient to perform such maneuvers in two steps: in a first step, a certain misalignment is produced by development of Dzhanibekov instability, resulting in the rotation ((6.,¢.)). In a second step, the transition (6.,¢.) = (Ggoar: Pgoar) is achieved by solving a proper (well-posed) optimization problem. The advantage of such an approach is that the first step should not precisely define intermediate orientation (9, fc), it is only important to achieve a certain rotation significantly misaligned with the body's principal axes.

Numerical experiments indicate that achieving the goal orientation co-oriented with one of the principal axes is rather challenging. Figs. 5A and 5B illustrate the rates of convergence of optimization algorithms for the maneuvers 1-4 and maneuvers 6-9. One can clearly see that in the latter case the convergence is much slower, and final values are much larger. Still, these maneuvers converged with a surprisingly good precision (see Table 2).

An alternative approach to performing maneuvers 6-9 is illustrated in Fig. 6B. The results above have demonstrated that the controlled development of intermediate axis instability may not be an optimal way to re-orient the spacecraft, especially if the duration of the maneuver should be minimized. Instead, an additional axis n, of possible moment of inertia control may be introduced.

This axis #2: is not orthogonal to any of the principal axes n4, ny, nz. Moving masses m along this additional or auxiliary axis allows an instantaneous re-definition of the direction of a principal axis, which in turn allows the use of a much faster pre-computed maneuver to achieve the desirable attitude.

Therefore, it is clear that the described approach to angular maneuvering is rather universal and allows to achieve any orientation of the spinning body with respect to its angular velocity. Initial alignment of the angular velocity of rotation with the principal axis slows down the maneuver, but can not become a deadlock, since Dzhanibekov instability rapidly develops even an initially tiny misalignment.

Fig. 5C illustrates the quality of angular momentum conservation in our simulations. It can be seen that the quantity that should be precisely constant, in fact, features some drift during numerical motion integration — on the order of 107% of its absolute value during the longest simulation time span. This can be considered as sufficiently good quality of time integration, which means trustworthiness of the simulations.

In a practical embodiment, the calculations, similar to the ones discussed above in the context of simulations, could be performed by a processor of a controller on board the spinning body, which could be a spacecraft. Alternatively, the results of such calculations could be stored as predetermined schedules in a memory of the controller before launch of the spacecraft and could then be retrieved whenever a maneuver would be required. It is also conceivable that the calculations would be performed or the stored schedules would be retrieved at a ground station, and the results then sent to the spacecraft.

As discussed above, varying one or more of the three moments of inertia, and thus the tensor of inertia, may be done by symmetrically changing the mass distribution around one or more of the principal axes of the spinning body. Changing the mass distribution may be done by symmetrically moving masses with respect to the principal axis, i.e. moving masses towards or away from the principal axis.

Masses M may be translated, as schematically illustrated in the context of a so-called

CubeSat in Fig. 6A. An important feature of the illustrated embodiment is that the movable masses are not dead weights, but rather parts of the payload, like e.g. a massive optical objective of an earth surveillance camera, chemical batteries and other energy storage devices, etc. Moving these masses towards the rotation axis will require force to overcome the centrifugal forces acting on these masses as a result of the rotation of the spinning body. Conversely, when the masses are allowed to move away from the rotation axis, the centrifugal forces will generate energy which may be recuperated and stored.

In order to vary the tensor of inertia of a spinning body, movement of masses m towards or away from the principal axis may also be achieved by rotation or pivoting as shown in Fig. 6C. In this embodiment a combination of translatable masses M and rotatable masses m allows the moments of inertia for all three principal axes to be varied. In order to prevent any net rotation, this embodiment of the system includes two masses m at each side of the principal axis, which may counter-rotate as shown in the right-hand image of Fig. 6C. In this particular embodiment, the masses m can additionally all rotate in the same direction, as shown in the left-hand image of Fig. 6C, which allows the system to function as a conventional reaction wheel. In this way a reaction wheel and a variable TOI may be combined in a single device, depending on the whether a co-rotation or counter- rotation of the moving parts is used. Such a design is able to align its axis of rotation of moving parts with the angular velocity by changing TOI, and then to stop the rotation of the central shaft by redistributing the angular momentum to the rotating masses acting in a “reaction wheel” mode.

As discussed above, the method and system of the present invention allow any arbitrary orientation of a body with respect to its axis of rotation to be stabilized by a certain maneuver, which can be determined by the optimization procedure. This may be achieved by starting and ending a maneuver at the state in which the body has a spherical tensor of inertia, i.e. identical moments of inertia about all three principal axes. A major advantage of the method and system of the invention is that such maneuvers cost zero energy (neglecting heat losses in electric circuits and frictional contacts).

An important feature of the invention is that it can be used in a wide range of self-spin angular velocities, since the angular velocity or rate of rotation only determines the time scale of the maneuver and does not affect its feasibility. Therefore, the same maneuver can in principle be performed by a large manned orbital station and small, rapidly spinning CubeSat.

The method of the invention can be easily implemented technologically and used in real spacecraft systems. The numerical simulations indicate that a desired maneuver can always be found, given the sufficient span of control signals [§min, Zmax]. dimensionless duration T and complexity {number of reference control points N). The maneuver is resolved with limited precision given insufficient span, duration and/or complexity, and becomes non-unique if the these parameters provided to the optimization procedure are wider than necessary minimum. Except the special case of switching between orientations close to the principal axes, the maneuver can be resolved precisely with reasonable T, N and [Gmin, Gmax].

The optimization method discussed in this application should be viewed only as a proof of concept technique, demonstrating that the maneuvering described above is possible. Using modern machine learning/optimization techniques the method can be substantially improved, producing not only precise, but shortest and the most energy-efficient maneuvers.

As could be seen above, calculation of every maneuver requires considerable computational efforts and may not be practical to be done on the fly. Therefore, for practical usage it may be necessary to pre-compute and tabulate all meaningful rotations (Gpeg, eg) — (Gena Pena). A complete pre-computed table will form a five-dimensional array, and its sufficiently dense sampling and storage may pose a problem. To address this challenge, a closer look could be taken at the structure of the phase space of the rigid motion with the changing TOI, and this array could be tabulated in adaptive manner. Alternatively, it is possible to use black-box methods, e.g. a so-called

Tensor Train cross-approximation as defined in LV. Oseledets, E.E. Tyrtyshnikov, “Breaking the curse of dimensionality, or how to use SVD in many dimensions”, SEAM Journal on Scientific

Computing 31 (5), 3744-3759, 2009, to construct a low-rank representation of the complete table of all possible rotations and the corresponding control signals. Given the symmetric structure of this array and its presumable low-rank structure, the black-box approximation should be a very efficient tool to accelerate necessary pre-computations.

One particularly interesting direction is the exploration of small corrections of the attitude.

It can be expected that small adjustments (the functional (18) is initially less than 1077) could be achieved by fast maneuvers with very few reference points. The larger maneuvers can then be represented as sequences of smaller ones.

Although the invention has been disclosed here by way of an exemplary embodiment, it will be clear that it is not limited thereto and may be varied in many ways within the scope of the following claims.

Claims

CONCLUSIONS

1. A method for controlling an angular direction of a spinning body relative to an axis of rotation of the spinning body, comprising the steps of: a) providing the spinning body in a first stable angular direction relative to its axis of rotation, the spinning body having three principal moments of inertia about three mutually perpendicular principal axes, said three principal moments of inertia defining an inertia tensor, said three principal moments of inertia each having a first value and said first values of said three principal moments of inertia being identical, such that said inertia tensor is spherical, b) selecting a second angular direction of the spinning body relative to its axis of rotation, said second angular direction being an arbitrary angular direction of the spinning body; and c) varying a value of at least one of said principal moments of inertia in accordance with a specified scheme, such that:

cl. initially the three principal moments of inertia no longer have identical values and the inertia tensor is not spherical, causing the angular direction of the spinning body to gradually change from the first angular direction to the second angular direction, and c2. finally the three principal moments of inertia again have identical values and the inertia tensor is again spherical in order to stabilize the spinning body in its second angular direction.

2. A method according to claim 1, wherein the particular scheme comprises only variations in the moment(s) of inertia that are within capacity limits of the spinning body.

3. A method according to claim 1 or 2, wherein the schedule is determined by calculation, retrieved from a database, or received from a remote transmitter.

4. A method according to any preceding claim, wherein step c) comprises varying the values of a plurality of principal moments of inertia to change the angular direction of the spinning body.

5. A method according to any preceding claim, wherein the first angular direction is substantially parallel to one of the principal axes and wherein step c1) comprises the sub-steps of!

cla. varying the value of the at least one principal moment of inertia according to a first subscheme so as to initiate Dzhanibekov instability, thereby causing an angular deviation between the first angular direction of the spinning body and the principal axis;

clb. stabilizing the deviated angular direction by restoring the spherical inertia tensor; and clc. further varying the value of the at least one principal moment of inertia in accordance with a second subscheme so as to gradually change the angular direction from the deviated angular direction to the second stable angular direction.

6. A method according to any preceding claim, wherein step c2) comprises resetting the values of the three principal moments of inertia to the first value.

7. A method according to any preceding claim, wherein step c) comprises varying a mass distribution of the spinning body symmetrically with respect to the at least one principal axis about which the principal moment of inertia is varied.

8. A method according to claim 7, wherein in step c) net rotation about the principal axis is avoided when the mass distribution is varied.

9. A method according to claim 7 or 8, wherein step c) comprises moving a pair of opposing masses in opposite directions relative to the principal axis.

10. A method according to claim 9, wherein the opposing masses are translated in opposite directions towards or away from the principal axis.

11. A method according to claim 9, wherein the opposed masses comprise at least two masses on each side of the main axis, which are pivoted in opposite directions towards or away from the main axis.

12. Method according to claim 11, wherein the masses are all further pivotable in the same direction.

13. A method according to any one of claims 9 to 12, further comprising step b1) of moving a pair of masses in opposite directions along an auxiliary axis which is not perpendicular to either of the principal axes prior to step c).

14. Method according to any one of claims 9 to 13, wherein the movable masses comprise at least one functional element of the rotating body.

15. Method according to any one of claims 9 to 14, wherein the movable masses comprise at least one ballast weight.

16. Method according to any one of claims 9 to 15, wherein energy used or generated in a part of the control cycle is at least partly recovered or used in another part of the control cycle.

17. A method according to any one of claims 9 to 16, wherein the masses are moved at a displacement speed proportional to a rotational speed of the spinning body.

18. A system for controlling an angular direction of a spinning body relative to a rotational axis of the spinning body, the spinning body having three principal moments of inertia about three mutually perpendicular principal axes, the three principal moments of inertia defining an inertia tensor, the system comprising controllable variation means for varying a value of at least one of the principal moments of inertia in accordance with a predetermined scheme so as to change the angular direction of the spinning body from a first stable angular direction to a selected second stable angular direction, which may be an arbitrary angular direction of the spinning body.

19. The system of claim 18 further comprising a controller for controlling the variation means, the controller comprising a programmable processor for calculating the schedule, a memory for storing the determined schedule, and/or a receiver for receiving a transmission containing the determined schedule.

20. A system according to claim 18 or 19, wherein the variation means are arranged to vary values of multiple principal moments of inertia.

21. A system according to claims 18-20, wherein the variation means are arranged to first vary the value of the at least one principal moment of inertia in accordance with a first subscheme so as to develop Dzhanibekov instability, and then to further vary the value of the at least one principal moment of inertia in accordance with a second subscheme so as to gradually cause the angular direction to reach the second stable angular direction.

22. A system according to any one of claims 18 to 21, wherein the variation means are arranged to vary a mass distribution of the spinning body symmetrically with respect to the at least one principal axis about which the principal moment of inertia is varied.

23. A system as claimed in claim 22, wherein the variation means comprises a pair of masses disposed opposite each other with respect to the principal axis and means for moving the masses in opposite directions with respect to the principal axis.

24. The system of claim 23, wherein the displacement means comprises an opposed pair of translation mechanisms.

25. A system as claimed in claim 23, wherein the variation means comprise at least two masses on each side of the main axis, and the displacement means comprise two opposed pairs of pivot mechanisms.

26. The system of claim 25, wherein the pivot mechanisms are further configured to all pivot in the same direction.

27. A system according to any one of claims 23 to 26, wherein the variation means further comprises a pair of masses displaceable in opposite directions along an auxiliary axis not perpendicular to either of the principal axes.

28. System according to any one of claims 23 to 27, wherein the movable masses comprise at least one functional element of the rotating body.

29. System according to any one of claims 23 to 28, wherein the movable masses comprise at least one ballast weight.

30. A system according to any one of claims 23 to 29, further comprising energy storage means for supplying energy to and recovering energy from the moving means.

31. The system of claim 30, wherein the energy storage means comprises a battery.

32. A system according to any one of claims 23 to 31, wherein the control member is arranged to actuate the displacement means to displace the masses at a displacement speed proportional to a rotational speed of the spinning body.

33. A spinning body comprising a system according to any one of claims 18 to 32.

34. A spinning body as claimed in claim 33, which is a spacecraft.

35. A spinning body as claimed in claim 34 comprising a system as claimed in claim 28, wherein the movable masses comprise portions of a payload of the spacecraft.