ES2994416A1 - A SYSTEM, METHOD AND COMPUTER PROGRAM PRODUCT FOR IMPROVED ATTITUDE CONTROL OF A VEHICLE UNDER TORQUE-FREE CONDITIONS (Machine-translation by Google Translate, not legally binding) - Google Patents

Abstract

A system, method and computer program product for improved attitude control of a vehicle under torque-free conditions. The system (100) comprises an attitude sensor unit (112) for measuring the angular state (113) of the vehicle (102), momentum exchange actuators (122) for exchanging momentum with the vehicle (102), a guidance module (134) for obtaining a target angular state (135). A navigation module (136) for obtaining an estimated angular state (137) and a control unit (132) configured to receive the target angular state (135), check for the occurrence of a state of rotation about the intermediate axis of inertia of the vehicle (102), DZH effect, and in that case characterize the DZH effect, obtaining at least one DZH period (232); calculate at least one activation time interval {del}t j (242) during the DZH period (232); and performing an attitude maneuver of the vehicle by actuating the pulse exchange actuators (122) for at least an activation time interval {del}t j (242). The combination of the DZH effect with the pulse exchange drives achieves high energy and time savings. (Machine-translation by Google Translate, not legally binding)

Description

DESCRIPTION

A SYSTEM, METHOD AND COMPUTER PROGRAM PRODUCT FOR A

IMPROVED ATTITUDE CONTROL OF A VEHICLE IN FREE CONDITIONS

PAIR

FIELD OF INVENTION

The present invention is within the field of systems and methods for controlling the attitude of vehicles under torque-free conditions, such as in microgravity or zero gravity conditions.

BACKGROUND OF THE INVENTION

Nowadays, regular satellites and spacecraft essentially consist of a frame that provides structural integrity and supports the rest of the subsystems. These subsystems are usually fixed to this structural frame unless they need to be specifically oriented, such as solar arrays, telescopes, etc. For space vehicles with attitude mission requirements, the attitude determination and control subsystem may also be composed of rotating parts such as reaction wheels, impulse gyroscopes, etc. By rotating these internal or external elements, the angular state (orientation and angular velocity) of the vehicle is adjusted.

Throughout the life of a satellite in orbit, different perturbations from the space environment need to be absorbed in such a way that the spacecraft is controlled. These perturbations not only come from the launch of the spacecraft from the launcher (which can cause it to “fall”), but also due to the gravitational attraction of celestial bodies, the solar wind or the impact of micro meteorites, among others. Normally, the absorption of these perturbations is achieved by means of rotary motors whose movement counteracts the perturbations in orbit, allowing the spacecraft to be oriented as desired. However, when the speed of these motors is maximum, they cannot absorb any more angular momentum and become “saturated”.

Currently, there are different systems to manage, store and deliver angular momentum, using external torques (cold gas jets, magnetic torques, gravity gradient stabilization, solar radiation pressure, etc.) or internal torques (reaction wheels, impulse wheels, control impulse gyros, etc.), hereafter referred to as momentum exchange actuators. Other mechanical solutions include fluid ring actuators and spherical reaction wheels. However, the most commonly used actuators (i.e. reaction wheels and control impulse gyros) consume non-negligible amounts of energy, in addition to requiring considerable time to place the satellite in the correct orientation.

There is a need to provide improved attitude control systems for vehicles in microgravity, zero gravity or any torque-free conditions that will reduce the energy consumption and time of attitude manoeuvres. This is particularly true for small satellites such as cubesats due to their limited inherent energy resources. In addition, the ability to incorporate more agile satellites into a constellation will reduce the number of spacecraft that need to be manufactured, launched and maintained, which will significantly reduce the overall mission cost.

BRIEF DESCRIPTION OF THE INVENTION

The present invention relates to a system, method and computer program product for improving attitude control of a vehicle in any torque-free condition (e.g., microgravity, zero gravity) that provides significant energy and time efficiency increases in attitude maneuvers.

None of the current attitude control systems take advantage of the Dzhanibekov phenomenon (DZH), which is a naturally occurring periodic 180-degree flip of a rigid body rotating around its intermediate axis of inertia. The DZH phenomenon can be controlled with an inertial transformation subsystem by setting the right inertial tensor to either turn on (180-degree periodic flip motion) or turn off (stable rotation) the phenomenon. Energy efficiency, with energy savings of up to 80%, is achieved by combining the DZH effect with current momentum exchange technology. Comparatively, maneuver time can be reduced by using it as an objective criterion instead of energy consumption. A combination of energy and time savings is also possible by compromising their respective individual gains and obtaining a trade-off between energy and time needs on a specific mission. Note that the proposed controller in charge of attitude maneuver has one more degree of freedom (control start time within the DZH period), letting the DZH effect assist the maneuvers.

The system for improving attitude control of a vehicle under torque-free conditions comprises an attitude sensor unit for measuring the angular state of the vehicle, momentum exchange actuators for exchanging momentum with the vehicle for attitude control, and an on-board computer. The on-board computer includes a control unit, a guidance module configured to obtain a target angular state, and a navigation module configured to obtain an estimated angular state from the measured angular state. The control unit is configured to, upon receipt of the target angular state, check for the occurrence of a state of rotation about the intermediate axis of inertia of the vehicle (i.e., DZH effect), and in that case: characterize the DZH effect, by obtaining at least one DZH period and the current state of the spacecraft within the DZH period; calculate at least one activation time interval during the DZH period; and execute an attitude maneuver of the vehicle by actuating one or more momentum exchange actuators of the vehicle during each time interval until the desired angular state of said time interval is reached.

The system may also include a plurality of inertia transformation actuators configured to activate or deactivate the DZH effect by changing the mass distribution within the vehicle. Thus, if the DZH effect is not active, the control unit may be configured to activate the DZH effect by changing the mass distribution within the vehicle and/or by actuating internal torque or external torque actuators.

The method for improving attitude control of a vehicle under torque-free conditions requires obtaining a target angular state; checking for the occurrence of a state of rotation about the intermediate axis of inertia of the vehicle (i.e., DZH effect), and in that case: characterizing the DZH effect, obtaining at least one DZH period; calculating at least one activation time interval during the DZH period; and executing an attitude maneuver of the vehicle by actuating a plurality of momentum exchange actuators of the vehicle during at least one activation time interval. The computer program product for improving attitude control of a vehicle under torque-free conditions comprises computer code instructions that, when executed by a processor, cause the processor to perform this method.

In addition, inertia-morph actuators allow desaturation of the momentum exchange actuators, avoiding the need for a propulsion subsystem or magnetic torques, and prolonging the lifetime of any satellite in orbit. Desaturation of the momentum exchange actuators is accomplished by controlling the rotation rate of the momentum exchange actuators and, if the rotation of the momentum exchange actuators is greater than a saturation threshold, by releasing a certain amount of mass from the inertially propelled rotating vehicle and/or by actuating the inertial transformation actuators to reallocate the total angular momentum between the primary structure and the momentum exchange actuators.

BRIEF DESCRIPTION OF THE DRAWINGS

A series of drawings that help to better understand the invention and that are expressly related to an embodiment of said invention, presented as a non-limiting example thereof, are very briefly described below.

Figure 1 shows a system for improving attitude control of a vehicle under microgravity, zero gravity, or any torque-free condition, according to one embodiment.

Figure 2 depicts a method for improving attitude control of a vehicle under microgravity, zero gravity, or any torque-free condition according to one embodiment.

Figure 3 shows in detail the steps for executing the attitude maneuver according to one embodiment.

Figure 4 shows the improved attitude control system with inertial transformation actuators.

Figure 5 shows the improved attitude control system including two movable masses as inertial transformation actuators and three reaction wheels in orthogonal configuration as momentum exchange actuators.

Figure 6 represents the implementation of a subsystem to desaturate the impulse exchange actuators.

Figure 7A shows a schematic of a spacecraft with three reaction wheels and two moving masses. Figure 7B shows the Euler angles following the x-yz rotation order.

Figure 8 shows an exemplary attitude maneuver.

Figures 9A through 9F show the energy savings (left graph) and time savings (right graph) for multi-attitude maneuvers, each graph corresponding to a fixed ^ Euler angle and as a function of the other two Euler angles (0,$).

Figure 10A shows the angular velocities during a DZH period without reaction wheel drive. Figure 10B shows the total number of maneuvers compared to maneuvers with energy gain. Figure 10C shows the percentage of maneuvers with energy gain. Figure 10D shows the average energy gain of the maneuvers.

Figure 11A shows the total number of maneuvers compared to the time-gain maneuvers. Figure 11B shows the percentage of time-gain maneuvers. Figure 11C shows the average time-gain of the maneuvers.

Figures 12A to 12C represent different examples of inertia transformation actuators: Internal moving masses (Figure 12A), external moving masses (Figure 12C), liquid base (Figure 12B).

Figures 13A to 13B show a dynamic simulation where the DZH effect is enabled by moving the masses from the initial position to the origin.

Figures 14A to 14B show the passive action of the masses, using centrifugal forces, from the origin to the final position.

DETAILED DESCRIPTION OF THE INVENTION

1 shows a system 100 for enhancing attitude control of a vehicle 102 under torque-free conditions (e.g., a spacecraft under microgravity or zero gravity conditions, a vehicle under free-fall conditions, a vehicle mounted on an air bearing installation or on a 3-axis gimbal) in accordance with one embodiment of the present disclosure. The system 100 may be divided into sensors 110, actuators 120, and an on-board computer 130.

The sensors 110 are hardware components that measure the state of the vehicle. The sensors 110 comprise an attitude sensor unit 112 comprising one or more attitude sensors (such as inertial measurement units, magnetometers, sun sensors, horizon sensors, etc.) configured to repeatedly measure the angular state of the vehicle. In one embodiment, the measured angular state 113 of the vehicle includes the attitude 0 and angular velocities w relative to a reference frame of the vehicle. The actuators 120 include a plurality of momentum exchange actuators 122 configured to exchange momentum (momentum exchange pair 104) with the vehicle 102 for attitude control. The system may employ any attitude control actuator that exchanges momentum with the main structure of the vehicle, such as momentum wheels, conventional reaction wheels, spherical reaction wheels, control momentum gyroscopes, or liquid ring actuators. Although the vehicle may include external torque actuators (magneto-reducers, thrusters, etc.), the momentum exchange actuators 122 preferably do not include such external torque actuators because the vehicle (e.g., the spacecraft) would not be in a torque-free condition. Therefore, when executing a vehicle attitude maneuver in accordance with the present invention, such maneuver preferably does not include actuating the external torque actuators, if present (instead, the internal torque actuators are actuated).

The on-board computer 130 consists of a control unit 132, a guidance module 134 and a navigation module 136.

The guidance module 134 is configured to obtain a target angular state 135 (or required angular state, the reference to be followed; for example, Euler angles and angular velocities to be reached) and provide it to the control unit 132. Depending on the mission, the target angular state 135 may be decided internally by the on-board computer 130 or commanded from a ground station. The angular state of a spacecraft is determined by the Euler angles or quaternions and the angular velocities between the vehicle's body frame and any other (usually inertial) reference frame.

The navigation module 136 is configured to repeatedly obtain an estimated angular state 137 from the measured angular state 113. The navigation module 136 is an estimator of the current angular state 106 of the vehicle 102. Given a measured angular state and a dynamic mathematical model of the system, this module provides an estimate of the state. Examples of navigation modules are Kalman filters (linear and extended) and least squares estimators.

The control unit 132 may be implemented as a microprocessor, microcontroller, or any other data processing unit. The control unit 132 processes the difference (error) between the target angular state and the estimated angular state of the vehicle 139 to generate control inputs 121 to the momentum exchange actuator. The control unit 132 executes an algorithm that improves the attitude control of the vehicle by combining traditional momentum exchange devices with the Dzhanibekov effect. The improved attitude control allows for energy and/or time savings in maneuvers. In the improved attitude control, the vehicle starts the maneuver with a state of rotation about the intermediate axis of inertia and ends the maneuver upon reaching a target angular state. The target angular state may be a fixed attitude (e.g., fixed Euler angles) and/or a state of rotation (e.g., constant rotation about the body axis X).

The control unit 132 is configured to, upon receipt of the target angular state 135, check for the occurrence of the Dzhanibekov effect (DZH) in the vehicle. The DZH effect may be assessed by checking that the vehicle is rotating about its intermediate axis of inertia. If the DZH effect is active, the control unit 132 is configured to characterize the DZH effect using a plurality of estimated angular states 137, said characterization at least including the DZH period (i.e., the period of the 180-degree flip). The characterization may also include, among other parameters, the relative timing of plateau zones in the flipping motion of the DZH effect (the plateau zones are the quasi-flat regions of the angular velocity w about the intermediate axis versus time graph), the current state of the spacecraft within the DZH period, and the angular velocity about each axis. A practical way to do this is by measuring the evolution of the angular state by successive measurements.

The control unit 132 is further configured to calculate one or more activation time slots At¡ during the DZH period in which the pulse exchange actuators 122 are to be actuated, each time slot At¡ defined by an initial activation time and a duration. The number of time slots At¡ may be a predefined value (e.g. two time slots) or a variable that the control unit 132 may also be able to evaluate. The upper limit for the number of time slots is given by the actuation frequency of the system. The initial activation times are key parameters (the duration of each time slot may be a parameter or a fixed value) that are evaluated to optimize a figure of merit (e.g. energy, maneuvering time, etc.) by including these parameters as design variables. From this evaluation a set of optimal time slots will be obtained.

The control unit 132 is also configured to execute a vehicle attitude maneuver by actuating the pulse exchange actuators 122 for at least a calculated activation time interval A t j . The combination of the pulse exchange actuators 122 with the DZH effect occurs through the actuation time bands.

Vehicle 102 in Figure 1 is a block representing the plant of the system, i.e. where the effects of the actuations occur. The inputs of this block are the drives of the actuators 120 (pulse exchange torque 104) and the dynamic disturbances 108 accounting for the external environmental torques 109 affecting the vehicle 102, and the output of this block is the new angular state 106.

In one embodiment, to execute the attitude maneuver the control unit 132 initiates an attitude maneuver to modify the angular state of the vehicle until it reaches, within a certain threshold, the target angular state 135. When the current time t is within an activation time interval At<j>, the control unit 132 repeatedly (e.g., periodically at predetermined time steps) obtains the current estimated angular state 137 at time t, e.g. (<P<t>,wt ), calculates an angular state error 139 as the difference between the target angular state 135 and the current estimated angular state 137, calculates a control input 121 to the pulse exchange actuators 122 as a function of the angular state error 139, and actuates the pulse exchange actuators 122 with the calculated control input 121. Actuation of the pulse exchange actuators 122 may be accomplished via a controller 138. The control unit 132 is configured to terminate the attitude maneuver when the angular state error 139 is less than or equal to a threshold. This may be checked at each time step iteration (e.g., before checking whether the current time t lies within an activation time interval At<j>).

2 is a flow chart of a method 200 for improving attitude control of a vehicle under torque-free conditions according to one embodiment. The method 200 comprises obtaining 210 a target angular state 135, checking for the occurrence 220 of a rotational state about the intermediate axis of inertia of the vehicle, DZH effect, and then characterizing 230 the DZH effect by obtaining at least one period of the DZH effect (DZH period 232). The DZH period 232 may be calculated using a plurality of estimated angular states 137. The method 200 further comprises calculating 240 at least one activation time interval Atj 242 during the DZH period in which a plurality of vehicle momentum exchange actuators are to be actuated. Finally, an attitude maneuver 250 is executed by actuating one or more of the vehicle's momentum swapping actuators 122 for at least one activation time interval Atj. In one embodiment, the momentum swapping actuators 122 are actuated as a function of the angular state error 139, according to a given control strategy (e.g., using a PID controller or a model predictive controller (MPC)).

The activation time intervals At<j>are calculated based on an optimization procedure including (i) an objective function dependent on one or more performance figures of merit, and (ii) a set of design variables at least including the start of each activation time interval Atj . In an example to be described later, a single activation slot is considered, the objective function involves minimizing the final attitude error, and the design variables are the initial activation time tr of the single activation slot and other control parameters, in particular, the gains (K<¡>, K<¡>, and Kd) of a PID controller used to control the pulse-reciprocated actuators 122. As a result of this exemplary optimization process, design variables (K<¡>, Kd, tr) that minimize the objective function are calculated.

3 depicts, according to one embodiment, the steps of executing the attitude maneuver 250. After the attitude maneuver 302 is initiated, a check 310 is made as to whether the current time t is within an activation time interval At<j. If so, the method repeatedly or iteratively comprises: Obtaining 320 a current estimated angular state 137 (i.e., at the current pulse t), for example, by repeatedly measuring an angular state of the vehicle and repeatedly obtaining an estimated angular state 137 from the measured angular state 113; calculating 330 an angular state error 139 as the difference between the target angular state 135 and the current estimated angular state 137; calculating 340 a control input 121 to the pulse exchange actuators 122 as a function of the angular state error 139; and actuating 350 the pulse exchange actuators 122 using the control input 121. Next, a check is made 360 as to whether the angular state error 139 is less than or equal to a predetermined threshold. In that case, the vehicle has reached the required angular state and the attitude maneuver is terminated 370; otherwise, a check is made again 310 as to whether the current time t is still within the activation time interval Atj. The calculation 340 of the control input 121 as a function of the angular state error 139 is performed by the controller 138 (e.g., a PID controller or a model predictive controller (MPC)). There are many control techniques (linear, nonlinear, based on machine learning, based on calculating an optimal path through several reference points, etc.) that can be used to control the pulse exchange actuators to reach a required angular state.

Optionally, the execution of the attitude maneuver may comprise, when the current time t is not within an activation time interval At¿, obtaining 380 a current estimated angular state 137, and recalculating 390 the activation time intervals Atj based on the evolution of the angular state of the vehicle. This operation is performed to take into account dynamic perturbations 108 (e.g., perturbations in orbit) that will cause the vehicle to deviate from the assumed behavior. These optional steps are depicted in dashed lines in Figure 3.

As shown in the flow chart of Figure 2, for the execution of the attitude maneuver the initial state of the vehicle must be a rotation at any angular speed about the intermediate axis for the DZH effect to occur naturally, and this effect can be combined with the actuation of the impulse exchange actuators 122. There are different ways to achieve this initial condition:

• By using external torque actuators such as magnetic torques, thrusters, etc. The external torque applied on the vehicle for some time will cause the vehicle to spin at the desired angular velocity. For example, external torque actuators can be used first to cancel the spin of a spacecraft just after launch and then to set the desired angular velocity.

• By using inertial transformation actuators to assign the entire angular momentum of the vehicle (e.g., an initially collapsed spacecraft) to the intermediate axis of inertia.

• By using the release conditions of the spacecraft launch systems. In this case, the initial conditions will be met and there is no need for additional hardware in the system.

The system 100 may comprise a plurality of inertial transformation actuators configured to activate or deactivate the DZH effect by changing the mass distribution within the vehicle. Similarly, the method 200 may comprise activating the DZH effect by changing the mass distribution within the vehicle. Figure 4 shows an embodiment of the system 100 including inertial transformation actuators 422 that may be used to establish the initial state in which the DZH effect is present. In this embodiment, the system 100 further comprises a plurality of inertial sensors 412 and an inertial transformation manager 432.

Inertial transformation actuators 422 are a set of actuators configured to modify the inertial tensor of the vehicle by changing its mass distribution (e.g., inertial transformation motion 404). Inertial transformation actuators 422 may be internal or external to the primary structure of the vehicle. In one embodiment shown in FIG. 5 (which depicts an improved attitude control system 100 for a spacecraft 102 that includes three orthogonally configured reaction wheels as momentum exchange actuators 122), inertial transformation actuators 422 comprise a set of at least two extendable and retractable arms 502, at least two movable masses 504 positioned along the extendable arms 502, and inertial sensors 412 configured to measure the position of the extendable arms 502.

The inertial sensors 412 are configured to measure indirect variables used to calculate the inertial characteristics of the vehicle (measured inertia tensor 413). For example, the change in inertia relative to a nominal case can be calculated by knowing the length and mass of a pair of extendable arms 502. The inertial sensors 412 may include length sensors, angular encoders, mass flow sensors, etc.

The inertia transformation manager 432 is a module configured to receive the measured inertia tensor 413 and decide the operating mode of the inertia transformation actuators 422. The inertia transformation manager 432 is responsible for controlling the inertia transformation actuators 422 according to a required inertia tensor 434.

In the embodiment of Figure 4, the inertia transformation manager 432 sends the required inertia tensor 434 to the control unit 132, which is configured to command the inertia transformation actuators 422 using a control signal (inertia control input 421). The controller 138 used to control the momentum exchange actuators 122 may also be responsible for controlling the inertia transformation actuators 422; alternatively, a different dedicated controller may be employed. In another embodiment, the inertia transformation manager 432 may directly control the inertia transformation actuators 422 (the inertia transformation manager 432 may be integrated into the control unit 132).

The inertial transformation actuators 422 may be used to activate and/or deactivate the Dzhanibekov effect. In addition, they may also be used to desaturate the momentum exchange actuators 122. In one embodiment, the system 100 includes two modes of operation of the inertial transformation actuators 422: DZH effect ON (the effect is active on the axis of rotation, i.e., the occurrence of periodic 180 degree turns); DZH effect OFF (the effect is not active on the axis of rotation). In another embodiment, the system 100 includes the following three modes of operation of the inertial transformation actuators 422: DZH effect ON; DZH effect OFF; desaturation of the momentum exchange actuators (the inertial transformation actuators throw mass outside of the vehicle to remove angular momentum that has built up due to external disturbances).

The inertia transformation actuators 422 are optional elements in the system 100 since the initial conditions (DHZ effect) and the complete operation can be achieved without them. However, their presence is very convenient especially for these three uses:

1. They can help achieve initial conditions by turning a collapsing spacecraft into a spacecraft that rotates smoothly about the desired axis. This will avoid the use of external torque actuators to detumble the spacecraft and establish the correct initial conditions.

2. They can switch roles between minor, intermediate and major axes of inertia by changing the inertia on a regular basis. This is especially interesting to make the rotation axis (one of the body axes) the intermediate or major axis depending on the operation mode. For example, to use the enhanced attitude control system, this axis must be the intermediate axis; and to achieve a constant turn, this axis must be the main axis.

3. They can desaturate momentum exchange devices, improving their operation beyond energy and time savings during attitude maneuvers, if used in combination with mass release actuators. The desaturation process is achieved by a set of inertial transformation actuators that can activate mass release actuators once they are placed in the correct location.

The system 100 may include a subsystem for desaturating the momentum exchange actuators in a vehicle. Figure 6 depicts the embodiment of a subsystem for desaturating the momentum exchange actuators 122 in a vehicle (e.g., a spacecraft) using the inertial transformation actuators 422. A series of sensors 602 monitor the rotational speed of the momentum exchange actuators 122 and send a warning signal 604 when the steady-state rotation of these actuators is reaching its limit (when it is higher than a saturation threshold). Once this is the case, it means that the momentum exchange actuators 122 are about to saturate and the desaturation procedure begins. First, the sensors 602 send a displacement command 606 to the inertial transformation actuators 422, which move from their initial position to a target position according to the amount of angular momentum to be released. The position of these actuators is adjusted with the help of the inertial sensors 412, which monitor the position of the inertial transformation actuators 422. When the inertial transformation actuators 422 are in the target position, the mass release actuators 610 are activated (for example, by a signal sent from the inertial transformation manager 432 or the control unit 132), which release mass according to the amount of angular momentum needed to be released. Physically, this will involve a reduction in the angular momentum of the spacecraft, since some of it has been released into space. The greater the released mass and its distance from the center of gravity during release, the more angular momentum will be removed from the vehicle. According to this new angular momentum, the momentum exchange actuators 122 will reduce their angular velocity, avoiding saturation. There is another way to desaturate the momentum exchange actuators 122 by using the inertia transformation actuators 422 and not by using the mass release actuators 610. For a target angular velocity of the vehicle, it is possible to transfer the angular momentum of the momentum exchange actuators 122, achieving its desaturation, to the primary structure by actuating only the inertia transformation actuators 422. The embodiment shown in Figure 6, avoiding the mass release part of the vehicle, is also valid for this alternative desaturation procedure. In this case, the total angular momentum of the vehicle contained in the primary structure and the momentum exchange actuators is preserved. However, it is managed in such a way that the momentum exchange actuators 122 are desaturated while keeping the angular velocity of the primary structure constant.

Similarly, the method 200 may further comprise desaturating the pulse exchange actuators 122 by monitoring the rotational speed of the pulse exchange actuators 122 and, if the rotation of the pulse exchange actuators 122 is greater than a saturation threshold, releasing a certain amount of mass from the vehicle.

The present invention will be explained in more detail using an example; in particular, when applied to improve the attitude control of a spacecraft under torque-free conditions. In this particular case, a single activation time interval At is used. The DZH effect occurs naturally up to the activation time of the impulse exchange actuators 122 (three reaction wheels). From that impulse, the maneuver is performed solely by the reaction wheels. The time at which the reaction wheels are activated, while the DZH effect is active, is a design variable that greatly affects the energy and time savings of this combined maneuver (DZH effect reaction wheels). The present invention achieves significant savings in energy and maneuver time during attitude maneuvers.

Vehicle 102, a spacecraft schematically represented in Figure 7A, comprises a rigid body (denoted by the subscript b) and three reaction wheels (RWS) (denoted by the subscripts fi, Í2, and fe) in orthogonal configuration, with respect to an inertial frame (denoted by the subscript /). The angular motion of the spacecraft is governed by the angular momentum equation, where the Coriolis theorem is applied to change the observer in the derivatives:

H

The total angular momentum H is the sum of the angular momentum due to the rigid angular motion of the entire spacecraft (Hb) and that of the three RWs as they rotate around their axes relative to the spacecraft (H<rw>). The inertia tensor of the spacecraft relative to the center of mass of the spacecraft is denoted by ib = isc ÍRW, which is the sum of the inertia tensor of the spacecraft without RWS (isc) and the inertia tensor of the independent RWS (iRW). Finally, the matrix i, is a diagonal matrix containing the inertia momentums of the three flywheels around their rotational axes and TE is the total external torque applied to the spacecraft. Figure 7A shows a sketch of the spacecraft, the set of 3 orthogonal RWS 122 and the 3 types of reference frames used. In this example, the mass distribution of the spacecraft can be changed (i.e., transformable spacecraft) by two movable masses 504 with mass m, which are optional elements.

Now, by including the definition of H in the equations of motion (equation (1)), we obtain:

The term — refers to the changes in inertia to which transformable spacecraft are subjected (it is a null term for other types of space vehicles whose mass distribution cannot be changed). The torque exerted by the RWS on the spacecraft is defined as:

RWS motors must generate torque whenever the inertial angular accelerations

of the flywheel body or the body are not zero. Isolating the term l in equation (3) and substituting it in equation (2), a system of three ordinary differential equations can be obtained for:

Finally, the evolution of u ,/b, derived from equation (3) and using the previous definition obtained in equation (4) is expressed as follows:

For the representation of the attitude of the body axes with respect to the inertial frame, a quaternion formulation is used with q = [qx,qy,qz,q w]T. This well-known formation avoids singularities and suffers less numerical rounding effects unlike rotation matrices. The kinematic relationship used to integrate the quaternions, and consequently the Euler angles, in time is expressed as:

Quaternions are easily transformed into Euler angles: 0= [=, 9, ? ] T. Figure 7B shows the Euler angles following the x-y-z rotation order.

The transformable subsystem chosen for this example includes two internal masses moved by a pulley-belt system in opposite directions along the x-axis and located at a distance s from the center of mass of the spacecraft, as shown in the figure. In this case, when describing the motion of one of the two masses, the location of the second mass is automatically known due to their link through the belt. To describe the motion of a mass in a rotating frame of reference, the inertial forces due to the motion of the rotating frame relative to the inertial one must be included:

where m is the mass of a moving mass, r' and v' are the position and velocity vectors with respect to the body reference frame of the mass located on the positive x^-axis, F and F, are the vector of the total applied forces and the inertial forces on the mass, respectively.

The inertial force vector is expressed as:

rf2í ___

Where TERM -^ K O b represents the acceleration of the body's reference frame as seen from the inertial one, i.e. the gravitational acceleration acting along the orbital radial direction and pointing towards the center of the Earth. Gravitational effects are neglected due to the high rotational maneuverability of the spacecraft, which does not imply a continuous effect of this gravity along the axis of moving masses, and the short simulation time.

After performing algebraic manipulations with the remaining terms considering r ' = s ib and projecting equation (7) in the direction of the x b axis, an equation of motion for the moving mass is obtained:

Refers to the belt tension as a result of half of the engine torque T<m>

about the pulley radius R<m>in charge of driving the timing belt. Half of the driving torque is used to drive only one of the two masses. The driving torque will also be included as external torque, (i.e. TE = TMj b) in the angular momentum equation (equation (4)), where the motor shaft is aligned with the ^axis direction. Thus, the coupling between the spacecraft attitude dynamics and the moving mass dynamics is enabled through T<m>and the spacecraft angular velocities (v b/iy and v b/i z).

Considering the moving masses in Figure 7A as point masses, the three impulses of change of inertia of the main axis are calculated:

with Ib xXo, Ibyyo and lb zza being the inertial impulses when the masses are at the center of mass of the spacecraft. Finally, it is possible to analytically derive the derivatives of the

inertial impulses seen from the body frame, i.e.:

In this example, a closed-loop controller implemented as a proportional-integral shunt (PID) controller is employed to steer the spacecraft using the RWS torque (TRw) as the control input in equation (4) and equation (5). At each time step of the simulation, the RWS torque is updated:

T

Where KP, K, and KD are the proportional, integral, and derivative gains arranged in 3-by-3 diagonal matrices and the Euler angle error (<t>err) is a function of the quaternion errors (qerr). The angular state error 139 (*err) can be calculated from the quaternions of the required angle qdem (target angular state 135) and the measured angle qmeas (current estimated angular state 137).

To achieve a final controlled angular state, the PID controller gains must be tuned for each different maneuver. The tuning procedure can be performed by an optimization procedure in which the final attitude error is minimized:

Being 0 err[i] the io component of the Euler angle error, t0 and tf the start and end times of the simulation, and tr the activation time to turn on the RWS, normalized with the period of the DZH effect for these simulation parameters and initial conditions. The lower limit of tr must be set to ensure that the DZH effect occurs before the RWS are turned on and Kmax must be adjusted to avoid reaching the maximum available torque. Where SF represents a safety factor:

This optimization problem needs to be solved using the kinematic and dynamic equations. These equations form a first-order dynamic system:

Where x is the state vector, u is the control vector and { is the dynamic flow of the system comprising the set of nonlinear ordinary differential equations (ODE) governing the dynamics of the system (equations (4) to (6) and (9)). Since eq. (9) is an ODE

Second-order, the kinematic relationship is used to transform the system into 2 first-order ODEs. A fourth-order Runge-Kutta numerical method was selected to solve the system.

The controller 138 calculates the discrepancy (angular state error 139) between the target angular state 135 and the estimated angular state 137 received from the guidance module 134 and the navigation module 136, respectively. In case the error is greater than a threshold, the controller 138 commands the actuators to operate. The controller 138 contains both the closed loop controller RWS indicated in equation (12) and the open loop controller that commands the motor to move the moving masses. These outputs are taken by the actuators: The inertial transformation actuators (moving masses) and the RWS that give a torque according to the controller input. These torques are commanded to the spacecraft, which changes its angular state depending on the torques and dynamic disturbances of the environment. In practice, the equations of motion given in equation (15) are contained within a block that simulates the spacecraft and are numerically integrated given a set of initial conditions and problem parameters. Finally, the measured state 113 of the spacecraft is measured by the attitude sensors 112 which are set as an input to the navigation module 136 of the system, completing the loop.

When the optimization problem is solved and thus the attitude error remains within the demanded attitude error range, the energy and time requirements for the maneuver with RWS alone and with the DZH effect in combination with RWS are compared with each other.

When the RWS closed-loop controller is combined with the DZH effect, the DZH effect acts as the “initial bump” of the spacecraft, which is then used by the RWS to place the spacecraft into a desired attitude. The time at which the RWS are activated once the DZH effect is a design variable that greatly affects the energy and time savings of this combined maneuver. Since the initial presence of the DZH effect is required, the inertial configuration of the spacecraft is one that allows for this effect. An example of an attitude maneuver where θ = 300°, θ = 120°, θ = 210° show the required Euler angles is shown in Figure 8. It can be seen that the RWS were not activated until 63% of the DZH period. After that instant, the PID closed-loop controller is able to drive the spacecraft into the desired Euler angles without exceeding either the maximum angular velocities or the maximum torque of the RWS.

A comparison is made between the energy and time taken to perform a given maneuver by combining the RWS with the DZH effect (ERW+DZH and TRW+DZH, respectively) and the values using the RWS from the beginning of the simulation (E<rw>yTrw ), i.e. the RWS without the DZH effect. Two metrics are used to quantify the comparison:

Er and Tr being the relative energy and time gains (if positive) or losses (if negative). A preliminary exploration of the energy and time gains within the 3D space of Euler angles is performed in Figure 9A.- For a selected angle ? = 60°, a grid of pairs of values is defined in the plane. For each set of Euler angles (= ,> ,?', a simulation like the one in Figure 8 is run where ERW+DZH and T<rw>+<dzh> are obtained. To analyze the scenario of using only RWS, the simulation is run with tr = 0 to activate the RWS at the start of each maneuver, thus being able to calculate ERW and<trw>. Then, the energy and time gains are calculated using equation (16) and represented with contour plots. The left plot shows that most of the attitude design space shows energy gains when combining RWS with the DZH effect compared to using the standalone RWS. As for time gains, the design space where time gains exist is smaller and more concentrated. These plots segregate the spatial regions in the Euler angle space where only energy gains, only time gains, or both energy and time gains are expected.

For further exploration of other regions using a different y angle, Figures 9B to 9F include the set of graphs for ? = (0°, 120°, 180°, 240°, 300°') as a function of the other two Euler angles (> ,= '). The entire 3D maneuver space is divided into 1728 attitude maneuvers by using 30° divisions for each Euler angle. The PID controller constants and RWS activation times for these 1728 cases are obtained through the previously explained optimization (Eq. (13)), where controllable maneuvers are obtained. Each maneuver has a different set of PID constants and RWS activation time tr given the presence of the natural DZH effect. The activation time tr of the RWSs for the simulations with independent RWSs is set to 0 to avoid the DZH effect from occurring.

Figure 10A shows the satellite angular velocities w1/i versus simulation time normalized to the DZH period during a simulation of the DZH effect without RWS actuation. The entire DZH period is divided into 11 boundaries for the RWS activation times. These 1728 cases with their summary tr are classified between the intervals defined by the boundaries. The total number of maneuvers together with those presenting energy savings are shown in Figure 10B. The fact that there is a large number of maneuvers accumulated in time steps 5 and 6 suggests that the plateau areas of the DZH effect are more suitable for RWS control. The percentage of energy-saving maneuvers, ranging from 50% to 80%, can be seen in Figure 10C. Finally, the average energy gain of the energy-saving maneuvers is shown in Figure 10D for each RWS activation time step. The combination of RWS and DZH effect not only brings a high percentage of maneuvers with energy gains, but also high energy gains (between 50% and 80%).

The same set of graphs as in Figures 10B to 10D are shown respectively in Figures 11A to 11C to analyze the time savings. As expected by looking at the right graph of Figures 9A to 9F, the percentage of maneuvers with time gains as well as the average time gains are lower (between 10% and 45%) compared to those found previously for energy. By interpreting both sets of graphs given in Figures 10 and 11, it is concluded that the combination of RWS and DZH effect for this specific case study increases the energy consumption by the attitude control system to a large extent, although it does not always reduce the maneuver time. If the space mission design is set to have a repetitive number of maneuvers with a positive energy and time gains, this strategy will be preferential.

The simulation results conclude that this strategy is faster and more energy-efficient for certain manoeuvres compared to the independent use of RWs. It is shown that between 50% and 80% of the manoeuvres can save up to 70% of energy compared to the use of independent RWs. On the other hand, time gains can be positive (in 10%-40% of the manoeuvres) or negative (losses) depending on the requested attitude.

The example illustrated in Figures 7 to 11 corresponds to a single activation time interval. However, the optimization procedure may calculate a plurality M of activation time intervals. For example, by considering two activation time intervals, the optimization procedure may calculate two initial activation times tr l tr2. The maneuver starts at t = 0 while the DZH effect is active. At t = tr l , the first activation time interval starts. The duration of the time interval may be a predefined value or a design variable. During the first activation time interval, the controller actuates the impulse exchange actuators 122 according to a predetermined control strategy (for example, following certain intermediate reference points previously calculated). The first activation time interval ends and the impulse exchange actuators 122 are not actuated, the vehicle will move with a combined movement of DZH phenomenon and an effect induced by the impulse exchange actuators. At t = tr2, the second activation time interval starts. During this time interval, the controller actuates the pulse exchange actuators 122 again according to a control strategy until the target angular state 135 is reached. The maneuver is then terminated.

As explained above, system 100 may incorporate inertial transformation actuators 422 capable of changing the inertial properties of vehicle 102 such that the DZH effect may be controlled, i.e., may be routinely activated or deactivated. Inertial transformation actuators 422 may be implemented, for example, as:

• A set of movable masses inside the vehicle. Figure 12A shows an example of a pair of masses 504 moving along an axis in opposite directions (to keep the center of mass in place). This type of system is not limited to two masses and one direction. For example, Liang et al. (“Satellite attitude maneuver using movable masses,” Acta Astronautica, vol. 176, pp. 464–475, 2020) used three internal movable masses for satellite attitude maneuvers. The limitations of this typology of morphable subsystems are imposed by the interior space of the vehicle.

• A set of mobile masses located outside the vehicle with retractable properties (Figure 12B). Although they do not normally have retractable properties, electrodynamic tethers for passive satellite deorbiting are deployed in a similar manner (García-González et al., “Attitude determination and control for the deployment preparation phase of a space tether mission,” Acta Astronautica, vol.

193, January 2022). For this configuration, there are no space limitations and the overall inertial change capabilities are much greater than with the use of internal masses. In practice, this means that a light mass placed far from the vehicle will contribute to a large modification of the inertia.

• A system of valves and pistons that moves the liquids 1204 inside the spacecraft (Figure 12C). Since the property that changes inertia is mass and not distance, it generally becomes less effective for the same mass and available space.

Inertia-morphology control refers to the property of the transformable vehicle that gradually changes the inertia of the vehicle due to the movement of these solid masses 504 or liquids 1204.

In one embodiment, the controller 138 of the control unit 132 implements an open loop inertial transformation control to activate or deactivate the DZH effect. The inertial transformation actuators 422 are actuated by the controller 138 (using the required inertia tensor 434 of the inertial transformation manager 432) to activate or deactivate the DZH phenomenon by changing the mass distribution within the vehicle, for example, by using two opposing moving masses 504. This method, alone, can serve as an improved attitude control that requires no more energy than is necessary to control the DZH effect by the movements of these masses; although it is restricted to attitude changes of 180°. For precise attitude change, the DZH effect is combined with the momentum exchange actuators.

T<m>uses an open loop controller to control the motor torque to drive the moving masses to the required position. The initial conditions for the dynamic simulations are:

On the one hand, enabling the DZH effect means switching from a stable spin of the spacecraft about the main axis of inertia to the periodic DZH tumbling motion. The masses move inwards (towards small |s| values) in the xbaxis direction to allow the main and intermediate inertia axes to exchange roles, i.e. the major axis would become the intermediate one and vice versa. Figures 13A and 13B show a dynamic simulation where the inertia transformation subsystem is activated at time t = 100 s, allowing the moving masses to move from the initial position to the origin (s = 0), enabling the DZH effect. The torque is only active for a few seconds and slightly modifies the total angular momentum of the spacecraft in the ybaxis direction. However, the increase in the yb angular velocity of the spacecraft axis (Figure 13A) after the inertial transformation actuation is mainly due to the decrease in the ybiinertial momentum of the Ibiyy axis. Conservation of angular momentum implies an increase in angular velocity when there is a decrease in inertial momentum. Figure 13A shows the DZH period 232 and the plateau zone 1302.

On the other hand, deactivation of the effect changes the periodic DZH tumbling motion of the spacecraft to a stable rotation around its main axis. The masses will move outward in the xb-axis direction (towards large |s| values). By simple inspection of equation (9), one can notice that centrifugal inertial forces felt by the masses can be used to assist the engine or even passively (i.e. without the use of external power) drive the masses outward. In Figures 14A and 14B passive actuation of the masses from the origin (with a small perturbation allowing the centrifugal force to act on the masses) to the final position is used. The activation time to allow the masses to move is set in the fifties, which coincides with the middle of the first plateau in the tumbling motion. The effect is effectively deactivated in about seven seconds. In Figure 14A, l^y l decreases, as the inertia Ibyy increases becoming the main impulse of inertia.

Claims (14)

1. A system for improving attitude control of a vehicle under torque-free conditions, wherein the system (100) comprises: an attitude sensor unit (112) configured to measure the angular state (113) of the vehicle (102); a plurality of momentum exchange actuators (122) configured to exchange momentum with the vehicle (102) for attitude control; and. an on-board computer (130) comprising a control unit (132), a guidance module (134) configured to obtain a target angular state (135) and a navigation module (136) configured to obtain an estimated angular state (137) from the measured angular state (113); characterized in that the control unit (132) is configured to, after receiving the target angular state (135), check the appearance of a state of rotation about the intermediate axis of inertia of the vehicle (102), DZH effect, and in that case: - characterize the DZH effect, obtaining at least one DZH period (232); - calculate at least one activation time interval At¡ (242) during the period DZH (232); and - execute a vehicle attitude manoeuvre by actuating the pulse exchange actuators (122) during at least the activation time interval At¡ (242). 2. The system according to claim 1, wherein to execute the attitude maneuver the control unit (132) is configured to: - if the current time is within a trigger time interval At¡ (242), repeatedly: get the current estimated angular state (137), calculate the angular state error (139) between the target angular state (135) and the current estimated angular state (137); calculating a control input (121) to the pulse exchange actuators (122) as a function of the angular state error (139); and. actuate the pulse exchange actuators (122) by the control input (121); and. - end the attitude maneuver when the angular state error (139) is less than or equal to a threshold. 3. The system according to claim 2, wherein to execute the attitude maneuver the control unit (132) is configured to: the current time is not within a trigger time interval ñt¡ (242): get a current estimated angular state (137), and. recalculate at least one Atj activation time interval (242). 4. The system according to any of the preceding claims, further comprising a plurality of inertial transformation actuators (422) configured to activate or deactivate the DZH effect by changing the mass distribution within the vehicle (102). 5. The system according to claim 4, further comprising a subsystem for desaturating the pulse exchange actuators (122) including: a plurality of sensors (602) configured to: controlling the rotation speed of the pulse exchange actuators (122); and. if the rotation of the pulse exchange actuators (122) is greater than a saturation threshold, sending a movement command (606) to the inertia transformation actuators (422) to move to a target position; and. a plurality of mass release actuators (610) configured to release mass from the inertia transformation actuators (422). 6. The system according to any of claims 4 to 5, further comprising: a plurality of inertial sensors (412) configured to measure the inertia tensor (413) of the vehicle (102) used to calculate the inertia characteristics of the vehicle (102); and. an inertia transformation manager (432) configured to receive the measured inertia tensor (413) and control the inertia transformation actuators (422) according to a required inertia tensor (434). 7. The system according to claim 6, wherein the inertial transformation actuators (422) comprise at least two extendable arms (502) and at least two movable masses (504) positioned along the extendable arms (502), and wherein the inertial sensors (412) are configured to measure the position of the extendable arms (502). 8. A method for improving attitude control of a vehicle under torque-free conditions, wherein the method (200) comprises: obtain (210) the target angular state (135); check (220) the occurrence of a state of rotation about the intermediate axis of inertia of the vehicle (102), DZH effect, and in that case: - characterize (230) the DZH effect, obtaining at least one DZH period (232); - calculate (240) at least one activation time interval Atj (242) during the period DZH (232) and - executing (250) a vehicle attitude maneuver by actuating a plurality of pulse exchange actuators (122) of the vehicle (102) during at least one activation time interval Atj (242). 9. The method according to claim 8, wherein the execution (250) of the attitude maneuver comprises: - if the current time is within an Atj activation time interval (242), repeatedly: get (320) the current estimated angular state (137), calculate (330) the angular state error (139) between the target angular state (135) and the current estimated angular state (137); calculate (340) a control input (121) to the pulse exchange actuators (122) based on the angular state error (139) and actuate (350) the pulse exchange actuators (122) by means of the control input (121) and - end (370) the attitude maneuver when the angular state error (139) is less than or equal to a threshold. 10. The method according to claim 9, wherein the execution (250) of the attitude maneuver comprises: current time is not within an Atj activation time interval (242): get (380) the current estimated angular state (137) and recalculate (390) at least one Atj activation time interval (242). 11. The method according to any one of claims 8 to 10, further comprising enabling the DZH effect by changing the mass distribution within the vehicle (102). 12. The method according to any one of claims 8 to 11, further comprising desaturating the momentum exchange actuators (122) by: controlling the rotation speed of the momentum exchange actuators (122); if the rotation of the momentum exchange actuators (122) is greater than a saturation threshold, releasing a certain amount of mass from the vehicle (102) and/or using the inertial transformation actuators (422) to desaturate the momentum exchange actuators (122). 13. A computer program product for enhanced attitude control of a vehicle under torque-free conditions, characterized by comprising computer code instructions that, when executed by a processor, cause the processor to perform the method according to any one of claims 8 to 12. 14. The computer program according to claim 13, comprising at least one computer-readable storage medium having computer code instructions recorded thereon.