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

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: c1. 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

[0001] 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 or spin-stabilized 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, without the requirement for on-board 3-axis propulsion or mechanisms, and sensors for attitude control and pointing. 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 cannot be used to terminate the fast rotation of a spacecraft. 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 of different nature. The conservation of angular momentum ^ = ^^^ in this case can be written as: • the ones redistributing the conserving angular momentum between the main body and its special rotating mechanical parts: ^^^= −∑^^^^^^(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 (TOI) of the spinning body: ^^^ = −^^^ (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 A1 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. Symp. 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. RU 2 685 948 C1 discloses a method of controlling the orientation in space of a microsatellite. To this end, levers deflected to certain angles and with weights to be moved along them are installed on three axes of the microsatellite. To turn the microsatellite in a specified direction, levers are turned in opposite direction by moving the weights to ends of the levers to create their maximum moment of inertia. Then, the levers are reversed into their initial position by moving the weights to the levers base to create their minimum moment of inertia. As a result, the microsatellite is turned in a preset direction. A number of such cycles can be repeated relative to different axes of the microsatellite, achieving the required turn of the microsatellite in space. This is essentially a variant of the inertial systems or reaction wheels discussed above. US 2014 / 263845 A1 discloses a space vehicle having a frame with a configuration of fiber optic gyroscopes (FOGs) and control moment gyroscopes (CMGs) at the outer perimeter of the frame. The FOGs and CMGs provide guidance and control for the space vehicle. This is another example of an inertial system based on reaction wheels as discussed above. US 5255878 A discloses a method for controlling reorientation of a spacecraft's spin from a minor axis spin bias to a desired major axis spin after spin transition. A control system monitors rotational rates about the principal axes to detect a separatrix crossing of a polhode path therein. Controlled thruster firings resulting from spin rate information successively decrease and increase a characteristic parameter and capture the spacecraft during a spin transition to a desired major bias orientation. This is an example of thrust-based angular positioning as discussed above. 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: c1. 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 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 torques 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 c1) may comprise the substeps of: c1a. 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; c1b. stabilizing the misaligned orientation by restoring the spherical tensor of inertia; and c1c. 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 c2) 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 c2) 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 c) 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 c) 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 from 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 b1) of moving, prior to step c), 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 b1) may be performed as an alternative to steps c1a, 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 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 configured to act on the spinning body 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 in which 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, - through an intermediate unstable angular orientation in which the three principal moments of inertia no longer have identical values and the tensor of inertia is non-spherical, - to a selected second stable angular orientation, which may be any angular orientation of the spinning body, and in which the three principal moments of inertia have identical values again and the tensor of inertia is spherical again. 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 illustrating a system for which ^^= 4^^^^, 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 mass distribution scaling factors ^^and their derivatives ^̇^, ^ ̈^during maneuvers 1-4 as shown in Fig.2, Fig. 3E-H show the evolution over time of principal moments of inertia during the same maneuvers 1-4, Fig.3I-L show the evolution over time of kinetic energies during the same maneuvers 1-4, Fig. 4A-D show the evolution over time of mass distribution scaling factors ^^ and their derivatives ^̇^, ^ ̈^during maneuvers 6-9 as shown in Fig.2, Fig. 4E-H show the evolution over time of principal moments of inertia during the same maneuvers 6-9, Fig. 4I-4L 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 how a payload may be positioned by re-orienting a spinning body, which may be a spin-stabilized spacecraft, Fig. 6B 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.6C schematically illustrates the introduction of an additional axis of possible control of moments of inertia, Fig. 6D schematically illustrates both translating masses M and rotating masses m for changing the tensor of inertia, and Fig. 7 is a schematic representation of a spin-stabilized spacecraft comprising an embodiment of the angular orientation control system. 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 TOI. 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 TOI – 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 1^^principal direction affects both ^^and ^^. It, therefore, makes sense to choose the control parameters as mass distribution scaling factors: It is also easy to see that without loss of generality it can be accepted that ^^(^) = 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: Here ^^(^) and ^^(^) are independent control parameters that can be manipulated within a certain range between ^!^"< 1 and ^!$^> 1, to achieve the desirable maneuvering; ^^is the baseline spherical tensor of inertia. Note that in case when ^^(^) = 1, ^^(^) = 1, the TOI is spherical. Fig. 1A offers a simple mechanical interpretation of the coefficients ^^(^) and ^^(^), 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 ^^(^) and ^^(^) 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 TOI’s principal components ^^, ^^, ^^in the order of their decrease. In the case of changing principal components, this convention is 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 ^^(^) = 1, ^^(^) = 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 of ^^(^), ^^(^). As mentioned above, the choice of limits for ^^(^) and ^^(^) 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. Here and below we will define the “orientation” ((, )) as the two angles determining the direction of angular velocity * inthe own spherical coordinate system of the body (defined such that corresponds to (, / 2,0), +^ –(, / 2, , / 2, ), +^ = +^ × +^). It worth noting here that the orientation defined in this way can beinterpreted as the angles of “latitude” and “rotation” of a camera, directed along / ^, while the “azimuth” of a camera is given by *^^ + 0 (see Fig. 6A and the corresponding discussion below). The maneuver starts at a certain state with the spherical TOI ^^, angular velocity *^and the orientation ((1^2, )1^2). 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 ^^again, characterized by the angular velocity *^"3and second stable orientation ((^"3, )^"3). The conservation of angular momentum ensures that *^"3= *^(during the maneuver, however, the angular velocity varies). The sequence of TOI changes is found by an optimization procedure that ensures the desired ((^"3, )^"3). 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 ^^(^), ^^(^) cannot 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 TOI 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 0^continuous evolution of TOI’s 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: ^^= 0 (6) The angular momentum ^ is given in the local coordinate frame instantaneously aligned with the body’s principal axes as: ^4= ^4(^)^4(^) (7) In such a local coordinate frame, rotating with the angular velocity ^4(^) around its origin, the time derivative of a vector ^ is given by: (^^ 4)56^= ^^ 4(^) + ^4(^) × ^4(^) (8) The equation (7) can therefore be written in a local (rotating frame) as: This equation can be re-written in inertial Cartesian frame component-wise using indicial notation as: The non-spherical TOI ^^7(^), its time derivatives ^^^7(^), the angular velocity *^(^) and its derivative *^^(^) are found by proper rotations of the corresponding components in the local frame: The TOI components in local coordinate system ^^47 (^) is given by and A(^) is the rotation matrix defined as where C^are the orths of global Cartesian coordinate system, and / ^(^) 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 *7at mid-intervals ^ + Δ^ / 2, and all other dynamic quantities at primary intervals ^ + Δ^. The orientation is tracked in the shape of rotation matrix :(^) that is obtained by the incremental rotation of principal directions by the angle ^(^)Δ^. The equation (10) can be re-written in the matrix form as

[0002] Here the regular sign convention for tensor of inertia components is implied. Using three equations (17), six unknowns *^(^ + Δ^ / 2), have to be evaluated. Following the approach suggested in the PFC 4.0 software, the iterative algorithm is used to find these unknowns: • Set / = 0 • Set *[^]^ to the initial angular velocity. • (*) Solve (17) for *^^• Determine a new (intermediate) angular velocity: • Revise the estimate of *^as: *["M^] [^] ["^L]^ = 0.5(*^+*^) • Set / := / + 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 / 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 ^^= ^^= 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 ^^, and angular velocities in *^. This naturally introduces units of time (^^= 2, / *^), angular momentum ( ^^*^) and energy ( ^^*^^ / 2 ). The remaining quantities,characterizing the system (O, ^^, ^^) are dimensionless. The dimensionless duration of the maneuverP = (^^"3 − ^1^2) / ^^, number of reference points O and the span [^!^", ^!$^] define the parameterspace 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 ^^(^), ^^(^) which provide the maneuver highlighted in Fig.1B. In order to guide the body toward the desired final orientation ((, )) , the following definition of the functional is used: i.e., 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 ((, )) precisely matches the goal orientation ((26$4, )26$4). The time evolution of coefficients ^^(^), ^^(^) is given by cubic splines. The initial and final nodal values ^^(^1^2), ^^(^^"3) are fixed to 1 (spherical tensor of inertia), initial and final time derivatives ^^^(^), ^^^(^) are fixed to zero. The remaining 2O nodal values ^ = (^^^.. ^^Z, ^^^.. ^^Z) are varied in an unbounded and unconstrained multidimensional optimization procedure. The optimizer seeks for a vector of unknowns ∈ ℛ, which are mapped to _: ^^∈ [^!^", ^!$^] in the following way: 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 parametersevolution. The duration of the maneuver P, the number of reference values O and the range[^!^", ^!$^] are chosen empirically outside of the optimization cycle.Benchmark maneuver computations As a brief illustration of the suggested approach, the optimization of TOI control parameters to perform arbitrary angular re-orientations of a spinning body is considered. The dimensionless system of units described above is used. The control parameters are varied between 0.5 and 1.5, leading to the ranges for principal moments of inertia: The kinetic energy can therefore vary in the range h ∈ (0.222,2.0), while the angular momentum is constant and equal to 1. Every maneuver is the transition between orientation ^: ((^, )^) and orientation i: ((7, )7). As was discussed above, the initial and final points of the maneuver are characterized by unit spherical TOI, while during the maneuver TOI varies. Initial and final orientations can be visualized as points on a unit sphere - see Fig.2. Table 1 gives the set of points that was chosen to demonstrate the capabilities of the method. Point ( ) W 1 , / 2 , / 4 2f^ / ^(1,1,0) 2, / 4 , / 22f^ / ^(0,1,1)3 , / 4 0 2f^ / ^(1,0,1) 4 arccos(3f^ / ^), / 43f^ / ^(1,1,1) 5 0 0 (0,0,1) 6 , / 2 0 (1,0,0) 7 , / 2 , / 2 (0,1,0) 8 , 0 (0,0,−1) Table 1: Reference orientations, given in terms of angles ((, )) and unit direction vectors W. Table 2 summarizes the list of maneuvers considered in this section of the application. The first column gives maneuver number. The second column gives the maneuver path in terms of reference orientations specified in Table 1. The third column details whether the maneuver is the result of optimization(“O”) or a run with the prescribed control parameters (“P”). For each maneuver the control parameters are within the range [^!^", ^!$^] = (0.5,1.5). The fourth column gives the number of control reference points O (the number of optimization parameters is 2O). The fifth column gives the dimensionless duration of the maneuver P. 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 mass distribution scaling factors ^^ and their derivatives ^̇^, ^ ̈^, principal moments of inertia I and rotational kinetic energy E during the maneuvers 1-4. Figs. 3A, 3E and 3I relate to maneuver 1, Figs.3B, 3F and 3J to maneuver 2, etc. All of these maneuvers were easily achievable with P = 16 (time corresponding to 16 periods of rotation for ^^= ^^= 1), and just one reference point O = 1. The final value of the functional indicates precise convergence up to the relative tolerance levels of the optimization procedure (10f^) 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 10fj), meaning that the total work of the internal forces to accomplish the maneuver is always zero. Fig.4 details the evolution of mass distribution scaling factors ^^ and their derivatives ^̇^, ^ ̈^, principal moments of inertia I and rotational kinetic energy E during the maneuvers 6-9, whose start and / or end orientations are in close vicinity of the principal axes. Figs. 4A, 4E and 4I relate to maneuver 6, Figs. 4B, 4F and 4J to maneuver 7, etc. For these maneuvers, the algorithm also performed beyond expectations. The maneuver cannot be started if the rotation is perfectly aligned with the principal axis - any changes of ^^, ^^will not induce misalignment. However, if the initial offset from the principal axis is 10f^, rad, it achieves to the goal orientation with reasonably good precision (Table 2). It is worth noting that the slow development of misalignment and convergence to the orientation close to a principal axis lead to longer maneuvers – the dimensionless duration of the maneuvers 6-9 was increased five times compared to maneuvers 1-4. Maneuver Path O / P O P ℒ((^"3, )^"3)Oo^p1 1-2 O 1 16 0 434 2 2-3 O 1 16 0 322 3 3-1 O 1 16 0 392 4 1-4 O 1 16 0 771 5 5-8 P - 7.2 3.340 × 10f^- 6 5-8 O 5 80 5.227 × 10fj685 7 5-6 O 5 80 3.769 × 10f^2197 8 5-7 O 5 80 9.661 × 10f^4312 9 4-5 O 5 806.864 × 10f^2510 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 (21) Once the first , rad flip is completed, values of ^^and ^^are set to ^^= 1, ^^= 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, in this case 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 (maneuver 6) also resorts to development of tennis racket instability, but does it in the relaxed form, with more complex program of changes of TOI. 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. The same 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 tennisracket instability, resulting in the rotation (((l , )l)). In a second step, the transition ((l, )l)^→ ((26$4 , )26$4) is achieved by solving a proper (well-posed) optimization problem. The advantageof such an approach is that the first step should not precisely define intermediate orientation ((l, )l), it is only important to achieve a certain rotation axis 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 maneuvers 1-4 and maneuvers 6-9. Note that the oscillations of the functional with iteration number is the specific feature of Powell’s optimization algorithm, which finds the local minima by bidirectional search in a parameter space along certain directions that are explored sequentially. It can clearly be seen that in the latter case the convergence to a local minimum is much slower, and final values are far from zero. Still, these maneuvers converged with a reasonably good precision (see Table 2). An alternative approach to performing maneuvers 6-9 is illustrated in Fig. 6C. 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 / jof possible moment of inertia control may be introduced. This axis n4is not orthogonal to any of the principal axes / ^, / ^, / ^. 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 stable and allows to achieve any re-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 cannot become a deadlock, since tennis racket 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 10fjof 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, a system 1 for controlling an angular orientation of a spinning body 2, e.g. a spin-stabilized spacecraft, with respect to its axis of rotation can be arranged in the spinning body (Fig.7). The angular orientation control system 2 can comprise controllable variation means 3 configured to act on the spinning body 2 for varying a value of at least one of the principal moments of inertia in accordance with a predetermined schedule. The angular orientation control system 2 can further comprise a controller 4 for controlling the variation means 3. The controller 4 may comprise a programmable processor 7, a memory 8 and / or a receiver 9. The variation means 3 may comprise a pair of masses 5 arranged oppositely with respect to principal axis about which the principal moment of inertia is varied. The variation means 3 may further comprise moving means 6 for moving the masses in opposite directions with respect to the principal axis. In this embodiment, the calculations, similar to the ones discussed above in the context of simulations, could be performed by processor 7. Alternatively, the results of such calculations could be stored as predetermined schedules in memory 8 before launch of the spacecraft 2 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 receiver 9 on board the spacecraft 2. It is interesting to note that the orientation parameters ((, )) discussed above admit a simple interpretation. The orientation of a camera or other similar payload (telescope, dipole antenna) directed outward from a center of mass, requires specifying three angular parameters in the static case. However, for a body spinning about the fixed axis, one can alter only two angular parameters, while the “azimuth” angle is prescribed as *^^ + 0. Simple geometric considerations establish the identity of the orientation angles ((, )) with the above-mentioned angular parameters, for the case of directed payload aligned with the axis / ^. Fig.6A illustrates how the payload may be positioned by re-orienting the spacecraft through angles Δθ = θgoal– θ and Δϕ = ϕgoal- ϕ. 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. 6B. 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. 6D. 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.6D. In this particular embodiment, the masses m can additionally all rotate in the same direction, as shown in the left-hand image of Fig.6D, 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 pre-computed 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 dimensionless duration P and complexity (number of reference control points O). 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 P, O and [^!^", ^!$^]. 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 ((1^2, )1^2) ^ ((^"3, )^"3). 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 I.V. Oseledets, E.E. Tyrtyshnikov, “Breaking the curse of dimensionality, or how to use SVD in many dimensions”, SIAM 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 10f^) 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

Claims 1. 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: c1. 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.

2. Method according to claim 1, wherein the predetermined schedule includes only variations in the moment(s) of inertia which are within a capability envelope of the spinning body.

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

4. Method according to any one of the preceding claims, wherein step c) comprises varying the value of multiple principal moments of inertia to change the angular orientation of the spinning body.

5. Method according to any one of the preceding claims, wherein the first angular orientation is substantially parallel to one of the principal axes and wherein step c1) comprises the substeps of: c1a. 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; c1b. stabilizing the misaligned orientation by restoring the spherical tensor of inertia; andc1c. 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.

6. Method according to any one of the preceding claims, wherein step c2) comprises resetting the values of the three principal moments of inertia to the first value.

7. Method according to any one of the preceding claims, 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. Method according to claim 7, wherein in step c) net rotation about the principal axis is avoided when varying the mass distribution.

9. Method according to claim 7 or 8, wherein step c) comprises moving a pair of opposite masses in opposite directions with respect to the principal axis.

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

11. Method according to claim 9, wherein the opposite masses include at least two masses at each side of the principal axis, which are pivoted in opposite directions towards or away from the principal axis.

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

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

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

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

16. Method according to any one of claims 9-15, wherein energy which is consumed or generated in one part of a control cycle is at least partially recuperated or consumed, respectively, in another part of the control cycle.

17. Method according to any one of claims 9-16, wherein the masses are moved at a motion rate which is proportional to a rotation rate of the spinning body.

18. System for controlling an angular orientation of a spinning body with respect to an axis of rotation of the spinning body, 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, the system comprising controllable variation means configured to act on the spinning body 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 in which 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, - through an intermediate unstable angular orientation in which the three principal moments of inertia no longer have identical values and the tensor of inertia is non-spherical, - to a selected second stable angular orientation, which may be any angular orientation of the spinning body, and in which the three principal moments of inertia have identical values again and the tensor of inertia is spherical again.

19. System according to claim 18, further comprising a controller for controlling the variation means, wherein the controller comprises a programmable processor for calculating the schedule, a memory for storing the predetermined schedule, and / or a receiver for receiving a transmission including the predetermined schedule.

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

21. System according to any one of claims 18-20, wherein the variation means are configured to first vary the value of the at least one principal moment of inertia in accordance with a first partial schedule to develop Dzhanibekov instability, and then further vary 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 reach the second stable angular orientation.

22. System according to any one of claims 18-21, wherein the variation means are configured 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. System according to claim 22, wherein the variation means comprise a pair of masses arranged oppositely with respect to the principal axis, and means for moving the masses in opposite directions with respect to the principal axis.

24. System according to claim 23, wherein the moving means comprise an opposed pair of translating mechanisms.

25. System according to claim 23, wherein the variation means comprise at least two masses at each side of the principal axis, and the moving means comprise two opposed pairs of pivoting mechanisms.

26. System according to claim 25, wherein the pivoting mechanisms are further configured to all pivot in the same direction.

27. System according to any one of claims 23-26, wherein the variation means further comprise a pair of masses that is movable in opposite directions along an auxiliary axis which is not orthogonal to any one of the principal axes.

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

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

30. System according to any one of claims 23-29, further comprising energy storage means for supplying energy to and recuperating energy from the moving means.

31. System according to claim 30, wherein the energy storage means comprise a battery.

32. System according to any one of claims 23-31, wherein the controller is configured to control the moving means to move the masses at a motion rate which is proportional to a rotation rate of the spinning body.

33. Spinning body comprising a system according to any one of claims 18-32.

34. Spinning body according to claim 33, which is a spacecraft.

35. Spinning body according to claim 34, comprising a system according to claim 28, wherein the movable masses comprise parts of a payload of the spacecraft.

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