Mode-Scheduling Steering Law of VSCMGs for Multi-Target Pointing and Agile Maneuver of a Spacecraft

This study proposes a method of selecting a set of gimbal angles in the final state and applies the method to the mode-scheduling steering law of variable-speed control moment gyros intended for multi-target pointing manoeuvres in the threeaxis attitude control of a spacecraft. The proposed method selects reference final gimbal angles, considering the condition numbers of the Jacobian matrix of the reaction wheel mode in the final state of a single manoeuvre and that of the constant-speed control moment gyro mode at the start of the upcoming manoeuvre to keep away from the singularities. To improve the reachability of reference final gimbal angles, the nearest set of gimbal angles among nominated sets according to the Euclidean norm is selected as the reference final set at the middle of the single manoeuvre, and then realised by adopting gimbal angle feedback steering logic using null motion. In addition, the manoeuvre profile is designed such that the second half of the single manoeuvre is more gradual and takes longer than the first. Numerical simulation confirms the validity of the proposed method in consecutive manoeuvres. Keywords—Variable-Speed Control Moment Gyros; Attitude Control; Singularity; Steering Law; Spacecraft


INTRODUCTION
A remote sensing satellite used in an emergency response requires high observation frequency as shown in Figure 1, which is achieved by making highly accurate and large-angle agile manoeuvres consecutively [1,2,3].To satisfy this requirement, the application of variable-speed control moment gyros (VSCMGs) to an attitude control actuator of a spacecraft and the steering law of the gyros have been studied [4,5].The authors previously proposed a mode-scheduling steering law for VSCMGs, where the suitable set of initial gimbal angles is selected in the constant-speed control moment gyro (CSCMG) mode considering singularity avoidance during an agile attitude manoeuvre, and the steering mode is then transited to the reaction wheel (RW) mode smoothly according to the attitude error of the spacecraft during highly accurate pointing control in the final state of the manoeuvre [6].Kasai and Kojima proposed the gain-scheduled steering law of VSCMGs [7], focusing on the condition numbers of the Jacobian matrix in an inverse matrix calculation in addition to the consideration of the attitude error.As shown in Figure 2 [7], there is a trade-off relationship between the condition numbers of the Jacobian for the CSCMG mode and that for the RW mode.The method of Kasai and Kojima changes to the gimbal angles for which the condition number of the RW mode is most well-conditioned in the final state of a single manoeuvre, and its validity has been confirmed by numerical simulation.
The present study proposes a method of selecting reference final gimbal angles considering the condition numbers of both the CSCMG mode and RW mode during multi-target pointing manoeuvres, and applies the method to the mode-scheduling steering law of VSCMGs.Through numerical analysis, it is confirmed that the condition number of the wheel Jacobian does not always need to take a minimum value, and a certain value retains the torque generation capability.The proposed method selects the reference final gimbal angles from the predefined nomination to keep not only the condition number of the wheel but also that of the gimbal at a certain level so that the CSCMG mode smoothly ends in the present manoeuvre and drives effectively in the upcoming manoeuvre.Figure 3 shows the sequence of the manoeuvre and steering of VSCMGs adopting the proposed method.Selected reference final gimbal angles are set by gimbal angle feedback steering logic [8] adopting null motion before transition to the RW mode.Since this logic does not guarantee the complete reachability of the reference final angles [9,10], two approaches are proposed to increase the reachability of the gimbal angles.Firstly, when the boundary point is defined as the start of deceleration in a rate profile of a single rest-to-rest manoeuvre, the reference final gimbal angles are selected on the condition that the Euclidean norm between gimbal angles at the boundary point and at each nomination is a minimum.Secondly, a rate profile is designed to make the gimbal angular acceleration more gradual with the intention to having a longer period of gimbal angular feedback after the boundary point.This paper presents the design of the proposed method and carries out numerical simulations to verify the feasibility of the method.

A. Attitude control of a spacecraft equipped with VSCMGs
CSCMGs consist of a wheel motor and gimbal motor that constantly rotate a wheel, whereas the VSCMGs control the spin rate of the wheel.Ford and Hall [11] proposed VSCMGs and various singularity avoidance techniques have been developed because the number of degrees of freedom is greater than that of CSCMGs [12,13].The pyramid configuration of N VSCMGs is generally used for the three-axis attitude control of a spacecraft from the viewpoint of hardware redundancy, as shown in Figure 4.The present study deals with the case N = 4 and a skew angle β = 54.7 deg.In this paper δ and Ω represent the gimbal angles and wheel rotational speed respectively.Firstly, unit direction vectors are defined as where, ˆˆ,, si ti gi g g g is the i-th unit direction vector of the spin axis, transverse axis, gimbal axis, respectively.
The inertia matrix of the spacecraft equipped with VSCMGs where, s I is the inertia matrix of the spacecraft excluding VSCMGs and i J is the inertia matrix of the i-th VSCMG.
It is necessary to solve the weighted pseudo inverse of ( 6) to calculate the desired η , introducing the weighted diagonal . This inverse kinematics solution is called a steering law and is defined as where, s W denote the weighting function for the wheel angular acceleration, and g W denote that for the gimbal angular velocity.s W and g W are therefore the weights associated with whether the VSCMGs are to perform like RWs or CSCMGs.Introducing these weights allows the control designer to distribute δ and Ω , thus realise the required r T In ( 7) and ( 8), the transition of the steering mode is then implemented as a sigmoid function of the total sum of the absolute error value error  in the spacecraft three-axis attitude control against the reference attitude, defined by where, ref θ is the reference attitude angle of the spacecraft after the rest-to-rest manoeuvre and θ is the Euler angle of the spacecraft at present.a, b, c are arbitrary positive constant values and set as a=1, b=1808, c=1.5 in this paper through numerical simulation, with the intention of not being oscillated in the transition of the steering mode and not being saturated in the wheel angular acceleration capacity when settling in the final state of a single manoeuvre.Figure 5 shows the relationship between error  and W .According to this function, the actuator drives in CSCMG mode for large-angle manoeuvring and then in VSCMG mode for the intermediate band to prevent the radical hold of the gimbals, and finally, the actuator stops the gimbals and drives in RW mode to settle on the reference attitude angle through the acceleration and deceleration of the wheels.

2) Singularity and condition numbers
A singularity should be considered when the actuator drives in CSCMG or RW mode.The singular condition occurs when all individual torque vectors are perpendicular to the required torque direction for the specific combination of gimbal angles [15].This situation means a "singularity".In the case of the mode-scheduling steering law defined in (7) distinguishing between the CSCMG mode and RW mode, the singularity of the wheel Jacobian matrix 0 D and that of the gimbal Jacobian matrix 1 D should be considered individually [16].A singularity of the RW mode and CSCMG mode occur when matrices 0 D and 1 D meet the conditions respectively


The singular index, also defined as condition number, represents the distance from singularity which is obtained through the singular value decomposition of the Jacobian matrix.The condition number of 0 D and 1 D is respectively defined as where,   ), and Larger values of s  and g  indicate that the singularity is closer for each Jacobian matrix.Provided that the spacecraft system keeps a zero-momentum status ( paper) when manoeuvring, the relationships between δ and s  , g  are those shown in Figure 2. www.ijacsa.thesai.orgTo avoid a singularity while manoeuvring in the CSCMG mode, the mode-scheduling steering law in (7) applies the singular-direction avoidance (SDA) steering law [17].By decomposing and adding a singularity avoidance parameter against 1 D , the modified matrix 1SDA D is introduced as where, are unitary matrices and 0  is a design parameter and set as a positive constant.From ( 7) and ( 12), the mode-scheduling steering law implementing SDA logic is then expressed as PROPOSED METHOD

A. Analysis of the selection of reference final gimbal angles
Considering the steering of the VSCMGs in consecutive attitude manoeuvres, the reference final gimbal angles of a single attitude manoeuvre are analysed.In the range 0 30   , which is intentionally limited from the viewpoint of symmetry in Fig. 2 From this singular value decomposition, the wheel maximum output torque is expressed as where, each element of is the maximum output torque around the roll, pitch, or yaw axis respectively.
Figure 6 shows the relationship between  and smax U .Note that each element of smax U takes an absolute value.Figure 6 reveals the possibility that the output torque around the yaw axis becomes small as  approaches zero.In general, a remote sensing satellite not only often manoeuvres around its roll and pitch axes to orient its mission sensor but also compensates for the attitude errors around the yaw axes to settle using the RW mode.Therefore, 10   is preferable from the viewpoint of acquiring the condition number of 0 D .This discussion reveals that, for the reference final gimbal angles, the domain 10 20 where, m is the positive integer and the following numerical simulation adopts m=1.

B. Applying gimbal angle feedback steering logic to the VSCMG mode-scheduling steering law
An overall design of the proposed method along with a time series of a single manoeuvre is shown in Figure 3.In a rate profile of the spacecraft profile ω , it is supposed that 0 t is the time at which the manoeuvre begins, 1 t is the time at which there is a change from acceleration to constant slew, 2 t is the time at which deceleration begins, and 3 t is the time at which the required attitude is realised.is triangular rather than having a trapezoidal shape, then 12 tt  .At 0 tt  for large-angle manoeuvring, the actuator drives in CSCMG mode and outputs torque by gimbaling.When settling in the final state of a single manoeuvre, the actuator drives in RW mode and compensates for small attitude fluctuations through the acceleration and deceleration of the wheels.To transit to the RW mode with the reference final gimbal angles of a single attitude manoeuvre, gimbal angle feedback steering logic using null motion [8] is applied.This logic allows gimbal angles to be guided to the desired angles without effecting the output torques using null motion.Gimbal angle feedback steering logic is expressed by

 
where, N K is a positive null motion gain and F δ denotes the reference final gimbal angles.When ( 17) is applied to (7) and ( 13), the proposed method is expressed by tt  is defined as the boundary point and null motion should activate after that.N K is set as


The reference final gimbal angles F δ are selected on the condition that the Euclidean norm between the boundary point and each nomination expressed in ( 16) is a minimum.F δ is set at 2 tt  so as to satisfy the condition

C. Design of the rate profile of an attitude maneuver
In terms of the gimbal angle feedback steering logic, the possibility of not converging to the reference angles by the intended time remains because null motion is not able to escape all types of singularity [10].profile ω is then designed to make the gimbal angular acceleration more gradual with the intention of having a longer period of gimbal angular feedback after the boundary point, as shown in Figure 7.The feedback controller of the attitude control system is a proportional derivative (PD) controller.In addition, the slew rate of the spacecraft and gimbal angular velocity of the VSCMGs when the attitude manoeuvre begins are supposed to be zero.To verify the validity of the rate profile design and gimbal angle feedback logic, the performances of three methods described in Table 3 are compared for the same feedback control system and the same mission.In Table 3, methods 1 and 2 are the comparative methods and method 3 is the proposed method.In method 1, the reference final gimbal angles are set so that s  has a minimum value.In method 2, profile ω is not designed and a normal rest-to-rest manoeuvre is adopted.

B. Simulation results
Figures 8 and 9 show the results of the attitude manoeuvre and weighting value for each of the three methods, respectively.In terms of the inner state of the VSCMGs, Figures 10, 11, and 12 show the results of the gimbal angles, singular index (condition number), and wheel rotational speed respectively.Figures 9 and 12 reveal that all methods can switch to the RW mode without oscillating the weighting values and without exceeding the range of the rated wheel rotational speed due to the design of weighting function in (9)  Firstly, methods 1 and 3 are compared in terms of the difference in the reference gimbal angles.Figures 8(a) (c) and 9(a)(c) reveal that methods 1 and 3 realise the reference attitude angles with approximately the same performance, switching the steering mode smoothly.By contrast, in the second manoeuvre of 80 t  , Figures 10(a) and 11(a) reveal that gimbal angles in the final state of method 1 are in a singular state for 1 D and gimbal angles of control moment gyros 1 and 2 are not able to converge although the gimbal angle feedback steering logic is active.Therefore, according to Figure 12(a), the wheel rotational speed of method 1 continuously changes even in the final state, which is not desirable for the saturation of the wheel rotational speed.By contrast in the case of method 3 from Figures 10(c because the gimbal angle feedback steering logic does not work well.Whereas in the case of method 3, Figure 10(c) shows that the gimbal angle feedback steering logic works effectively and the proposed method reaches the reference angles owing to the design that the more gradual and longer latter manoeuvre.As a result, Figure 11(c) shows that method 3 succeeds in maintaining proper values of s  and g  of the final state in both first and second manoeuvres.The numerical simulation thus confirmed that the proposed method can realise the required attitude manoeuvre of the spacecraft while reaching the intended gimbal angles of the VSCMGs considering the distance from singularity in each final state of a single manoeuvre.www.ijacsa.thesai.orgFor the multi-target pointing and agile manoeuvring of a spacecraft with VSCMGs, the present study proposed a method of selecting reference final gimbal angles of a single manoeuvre and its application to the mode-scheduling steering law of VSCMGs.Firstly, the need to consider the condition numbers of both the CSCMG mode and RW mode in the final state of each single manoeuvre during consecutive manoeuvres was presented.In addition, the desired reference final gimbal angles of the single manoeuvre were introduced.Secondly, gimbal angle feedback steering logic using null motion was applied to the mode-scheduling steering law of VSCMGs to realise the reference final gimbal angles.To improve the reachability of the reference final gimbal angles, the boundary point is defined as the start time of deceleration in a rate profile of the spacecraft, and the nearest set of gimbal angles among nominated sets according to the Euclidean norm from the boundary point were selected as the reference final set at the middle of the manoeuvre.In addition, a rate profile was designed to make the gimbal angular acceleration more gradual with the intention of having a longer duration of gimbal angular feedback after the boundary point.The numerical simulation of consecutive rest-to-rest manoeuvres confirmed that the proposed method can realise the required attitude manoeuvre of the spacecraft while reaching the intended gimbal angles of the VSCMGs in each final state of a single manoeuvre.
As gimbal angle feedback steering logic using null motion in this study does not guarantee the complete reachability to the reference final gimbal angles, further verification and improvement through Monte Carlo simulation etc. assuming various manoeuvre cases are future work.

Fig. 5 .
Fig. 5. Weighting functions s W and g W of the steering mode gimbals could be unstable and it could be difficult to transit to the RW mode smoothly because 1 D is near singularity.In addition, this state makes gimbals difficult to begin steering in the upcoming manoeuvre.Therefore, 20   is preferable from the viewpoint of acquiring the condition number of 1 D .Second, in the domain 0 10   , the output torque of the RW mode in the final state of manoeuvre could be limited depending on the direction because 0 D is near singularity.In more detail, 0 D can be decomposed as W g (Gimbal) www.ijacsa.thesai.org

Fig. 7 .
Fig. 7. Designs of the slew rate profiles profile ω


www.ijacsa.thesai.org ) and 12(c), gimbal angles and wheel rotational speed can converge to the intended values of the final state in both first and second manoeuvres.Secondly, methods 2 and 3 are compared in terms of gimbal behaviours.Figure10(b)shows that method 2 fails to reach