，，，3*

1.School of Automation Science and Electrical Engineering，Beihang University，Beijing 100191，P.R.China；2.Xi’an Flight Automatic Control Research Institute，Aviation Industry Corporation of China，Xi’an 710076，P.R.China；3.Beijing Advanced Innovation Center for Big?Data Based Precision Medicine，Beihang University，Beijing 100191，P.R.China

Abstract: This paper investigates the optimal control problem of spacecraft reorientation subject to attitude forbidden constraints，angular velocity saturation and actuator saturation simultaneously. A second-order cone programming（SOCP）technology is developed to solve the strong nonlinear and non-convex control problem in real time.Specifically，the nonlinear attitude kinematic and dynamic are transformed and relaxed to a standard affine system，and linearization and L1 penalty technique are adopted to convexify non-convex inequality constraints. With the proposed quadratic performance index of angular velocity，the optimal control solution is obtained with high accuracy using the successive SOCP algorithm.Finally，the effectiveness of the algorithm is validated by numerical simulation.

Key words：spacecraft reorientation；attitude forbidden constraints；actuator saturation；velocity saturation；secondorder cone programming（SOCP）

0 Introduction

The constrained attitude reorientation of rigid spacecraft has gained immense popularity in recent years. Some light-sensitive payloads，such as infra?red telescope and interferometers，should not be di?rectly exposed to some bright objects，leading to at?titude forbidden zones during the maneuvering. Due to the measurement range limit of the equipped sen?sors，the spacecraft can only maneuver in a lower angular velocity. Moreover，the actuators，such as flywheels and moment gyroscopes，cannot provide any requested control torques due to its physical lim?itations，which may lead to actuators’saturation. In a realistic scenario，all these issues may cause con?siderable difficulties in the design of attitude control algorithm for meeting high precision pointing re?quirement and desired control performance during the missions，especially when all these constraints are considered simultaneously.

Several nonlinear methods have been proposed to handle the spacecraft reorientation problem with attitude constraints，such as artificial potential func?tion（APF）［1-4］，path planning［5-7］，and model pre?dictive control（MPC）methods［8］. However，most of the above methods can only guarantee the feasibil?ity rather than optimality，so the optimal control problem is supposed to be considered. As a class of convex optimization，second-order cone program?ming（SOCP）can avoid this defect effectively［9-10］.Kim et al.［11-12］aimed to settle the attitude con?straints during the spacecraft maneuvering. Where?in，the Schur supplementary formula was applied to solve the dynamic constraints. It should be noted that most practical problems do not naturally satisfy the requirements of the SOCP. Therefore，several relevant attempts have been proposed. In Refs.［13-14］，the constraints in trajectory planning problems were disposed by lossless convexification to satisfy the requirements of standard SCOP framework，and a proper performance index was designed to keep the equivalence. In order to solve the optimal attitude control problem for spacecraft，a semidefi?nite relaxation method was introduced in Refs.［15-16］，and the convergence of the problem was ana?lyzed. The work in Ref.［17］proposed a SOCP method to solve a class of non-convex optimal prob?lems，where the non-convexity was caused by con?cave state constraints and nonlinear equality con?straints. Refs.［18-19］focused on the optimal prob?lems with linear or quadratic state constraints and non-convex control constraints. The original prob?lem was relaxed by slack variables，and the trans?formed problem could have the same solution with the original problem.

Other practical problems during the spacecraft reorientation maneuvering are angular velocity satu?ration and actuator saturation. The occurrence of these saturations can lead to substantial performance deterioration. As such，several control schemes have been proposed to deal with saturation con?straints. Ref.［20］designed a control algorithm that considered both angular velocity and actuator satura?tion. The adaptive control algorithm in Ref.［21］al?so focused on the spacecraft control problem with ac?tuator and velocity constraints. In Ref.［22］，the an?gular velocity constraint was addressed by using the barrier Lyapunov function. The authors in Refs.［23-25］devoted to the nonlinear MPC method based on SO（3）. The designed controllers could ensure that the angular velocity was limited within the set bound while completing the reorientation task.

Motivated by the above discussions，this paper intends to solve the optimal attitude control for spacecraft reorientation with attitude forbidden con?straints and saturation constraints based on the SOCP algorithm. The contributions are highlighted as follows：

（1）The great challenge are the strongly nonlin?ear attitude dynamics and concave constraints.Thus，great efforts are devoted to transform the original into SOCP framework by relaxation and convexification. Specially，the nonlinear attitude dy?namics are transformed and relaxed to a standard control affine system，and linearization and L1 penal?ty technique are adopted to convexify non-convex in?equality constraints.

（2）A specific quadric form performance index relative to angular velocity is provided to ensure the accuracy of the transformation.

（3）The proposed algorithm is extensible，indi?cating that the range and the number of constraints can be easily adjusted.

The remainder of this paper is organized as fol?lows：Section 1 demonstrates the mathematical de?scription of the constrained spacecraft reorientation problem. Section 2 provides the convexification and relaxation of the problem， then the successive SOCP algorithm is proposed. In section 3，the nu?merical simulation is provided，and the conclusions is detailed in section 4.

1 Optimal Control Problem State?ment

In this section，the spacecraft orientation in the body frameBrelative to the inertial frameIis repre?sented by Euler angle in a 1-2-3 sequence. Then，the mathematical descriptions of the constraints of the spacecraft reorientation problem are formulated.Finally，the formulation of the original optimal atti?tude reorientation problem is given.

1.1 Kinematic and dynamic model

Consider the attitude kinematics of a rigid spacecraft described by Euler angle［8］

And the dynamic model can be described as

whereφ，θ，ψdenote the roll，the pitch and the yaw angles of the spacecraft；ωx，ωy，ωzthe angular ve?locities around the body axes；Mx，My，Mzthe con?trol torque；andIx，Iy，Izthe diagonal values of the inertial matrix. And it makes no difference for the optimal problem given in this paper even if the ma?trix is not diagonal.

Definey=[φ θ ψ ωx ωy ωz] as the state vec?tor，and the control vector is represented byu=[Mx My Mz]. Thus，the nonlinear state equation can be governed as

1.2 Constraints of the problem

The mathematical descriptions of the initial and the terminal constrains，the attitude forbidden con?straints，the angular velocity saturation and the actu?ator saturation are given bellow.

Firstly，the initial and the terminal constrains are provided

wheret0andtfrepresent the initial and the terminal moments，respectively.

The saturation constraints，including the angu?lar velocity saturation and the actuator saturation，can be formulated as

whereωmaxrepresents the maximum value of the an?gular velocity， andMmaxthe maximum control torque that actuators can provide.

Subsequently，the mathematical descriptions of attitude forbidden constraints are given as follow?ing. The transformation matrix under 1-2-3 rotation of Euler angle is given by

It is assumed that the sensing instrument is align withZaxis of the body frame，i. e.[0，0，1]. Given any boresight vector[X0，Y0，Z0]，the attitude forbidden zone constraint can be rewritten as

In order to avoid the singularity in the kinemat?ics Eq.（1），the following constraint is introduced

whereσis a small positive constant.

1.3 Optimal control problem formulation

The optimal control problem considered in this paper is to obtain the solution subject to attitude dy?namics and physical constraints，so that the trajecto?ry of the spacecraft will start from the initial condi?tion and finally arrive at the desired terminal condi?tion，while minimizing the proposed quadric form performance index relative to the angular velocity

Therefore，the optimal control problem can be stated

2 Convexification and Relaxation

Due to the strong nonlinearity and non-convexi?ty of the original optimal control problem，convexifi?cation and relaxation are adopted to cast the original optimal problem into the SOCP framework. In par?ticular，the equivalence of the transformations is cer?tified. Finally，a successive SOCP algorithm is pro?vided to solve the problem iteratively.

2.1 Transformation of state equation

Since the SOCP frame requires linear state equation，the mathematical model（3）is supposed to be transformed by variable substitution

whereu11=sinφ，u12=cosφ，u21=sinθ，u22=cosθ，u31=sinψ，u32=cosψ.

These variables are not independent，then a new constraint is introduced

In the transformed state equation，the new con?trol vector is defined asν=[u11u21u31Mx My Mz]，and the new state vector isx=[u12u22u32ωx ωy ωz]. And the nonlinear state equation can be rewrit?ten as

Let(xk，νk)denotes thekth solution in the itera?tions，then the equation is linearized by standard Taylor series expansion

Thus，the non-convex and nonlinear state equa?tion is converted into the linear one，which meets the SCOP requirement. And it has the following form after rearrangement

To guarantee the validity of the above lineariza?tion，a trust-region constraint is considered

where?is a constant vector.

2.2 Convexification of non?convex constraints

After the above control augmentation，there have been some changes in the constraints described in the previous section. Firstly，the initial and the terminal constraints in Eq.（4）are transformed as

And the constraint in Eq.（10）becomes

The attitude forbidden zone constraint in Eq.（9）can be rewritten as

whereσis a small positive constant.

However，the SOCP framework can only deal with the linear equality constraints and the secondorder cone inequality constraints. Obviously，the constraint in Eq.（20）is supposed to be further lin?earized

To make sure that the spacecraft could avoid the attitude forbidden zone，one can transform Eq.（21）into

whereδis a small positive constant.

Besides，constraint in Eq.（13） is obviously non-convex，which cannot meet SOCP requirement certainly. But the result will not be accuracy if the same convex approximation that is applied to the at?titude forbidden constraint is employed. However，another convexification technique is utilized to relax the constraint in Eq.（13）to expand its feasible set，so that it becomes convex. Concretely，it is relaxed to be a second-order inequality constraint directly

Although the transformation does not seem equivalent，the objective functionFcan guarantee that the constraint in Eq.（23）is active almost every?where，which means that Eq.（13）can be satisfied.And the detailed demonstration will be provided in the following section.

Now the optimization problem has been put in?to to the SOCP framework，described as

2.3 L1?penalized relaxation

Since the inappropriate guess of the initial path may cause the violation of attitude forbidden con?straints，L1 penalty method is utilized to solve this issue.

Non-negative slack variables are introduced to Eq.（22）

And the objective function can be modified as

whereε4is a large constant.

With the above transformation，the optimal control problem becomes

Subject to Eqs.(5,6,16—19,23—25)

Then the assurance of the active constraint in Eq.（23）will be demonstrated as follows.

Assumption 1The constraint in Eq.（23）is inactive，which means that | (x，ν)-(xk，νk)| ＜?always holds.

Remark 1In fact，Assumption 1 is almost satisfied as the problem is not divergent and a prop?er?is chosen.

Theorem 1Let (x*，ν*) be the optimal solu?tion ofP2over a fixed interval [t0，tf]. then the con?straint in Eq.（23）will be active almost everywhere.

ProofSee the Appendix.

Remark 2If there are more than one attitude forbidden constraints in the problem，Theorem 1 will still be hold with the similar proof.

2.4 Successive SOCP algorithm

The constrained attitude reorientation problem of rigid spacecraft has been cast into an SOCP framework，then it is supposed to be discretized to several iteratively solved problems［27-29］.N+1 rep?resents the number of the discrete time points，and the time step size is defined as

where the number of iterations is depicted by the su?perscriptk，and the subscriptidenotes theith dis?crete point.

Then the successive SOCP algorithm will be given.

Algorithm

Here?2is a sufficiently small constant vector to guarantee constraint in Eq.（23）is active approxi?mately.

3 Numerical Simulation

In this section，numerical simulations are con?ducted to validate the effectiveness of the proposed spacecraft reorientation scheme. The algorithm is solved in MATLAB using YALMI［30］and MOSEK［31］，which could solve standard SOCP problems rapidly.

The spacecraft parameters in the simulation are provided as：Ix=300 kg ?m2，Iy=200 kg ?m2，Iz=190 kg ?m2. The desired attitude is (0°，0°，0°)，and the initial attitude is（-72.364 6°，0.034 4°，27.381 7°）.The maneuver time is set as 150 s. The saturation constraints of the angular velocity and control torque are restricted as 0.1and 3 N ?m，respectively.

Since the iteration algorithm requires initial val?ues，the initial maneuver path is supposed to be pro?vided. Various method can be employed to obtain the initial path. For simplicity，a PD controller in Ref.［32］is applied in the simulation，which does not consider saturation constraints and attitude for?bidden constraints.

3.1 Reorientation with single attitude forbid?den zone

In scenario 1，only one attitude forbidden zone is considered. The forbidden vector is assumed as[0.113 8，0.350 1，0.929 8]，and the keep-out angle is 11°.

Fig.1 shows the 3-D path of the spacecraft，in which the cone represents the attitude forbidden zone. Although the initial path provided by PD con?troller violates the forbidden zone，the optimal path obtained by iteration can reach the desired attitude successfully while avoiding the attitude forbidden zone. And the projection（2-D path）is presented in Fig.2.

The state and control variables of the space?craft are shown in Figs.3—5. It can be seen in Fig.3 that the Euler angles are driven to their desired ter?minal values. The angular velocity and the control torque are depicted in Fig.4 and Fig.5，respectively.As shown in Fig.6 the constraint in Eq.（23）is ac?tive during maneuvering.

Fig.1 3-D path of the spacecraft in scenario 1

Fig.2 2-D path of the spacecraft in scenario 1

Fig.3 Euler angles in scenario 1

Fig.4 Angular velocities in scenario 1

Fig.5 Control torques in scenario 1

Fig.6 Constraint in Eq.(23)in scenario 1

3.2 Reorientation with multiple attitude forbid?den zones

In scenario 2，two attitude forbidden zones are considered. Specifically，the forbidden vectors are set as [-0.068 98，0.361 6，0.929 8] and[0.291 5，0.619 4，0.729 0]，the keep-out angles are 11°and 15°，respectively.

Also，to demonstrates the superiority of the de?signed algorithm，the artificial potential functionbased controller in Ref.［1］is simulated for compari?son. The comparison results under the two algo?rithms are shown in Fig.7. Although both paths can avoid two attitude forbidden zones and then reach the desired attitude. The path generated by APF is much longer than the path obtained by the proposed control scheme，which means the proposed control scheme requires less consumption. And the corre?sponding 2-D path of the spacecraft is provided by Fig.8. The saturation constraints and the constraint in Eq.（23） are also guaranteed，as shown in Figs.9—12.

Fig.7 3-D path of the spacecraft in scenario 2

Fig.8 2-D path of the spacecraft in scenario2

Fig.9 Euler angles in scenario 2

Fig.10 Angular velocities in scenario 2

Fig.11 Control torques in scenario 2

Fig.12 Constraint in Eq.(23)in scenario 2

4 Conclusions

A successive SOCP algorithm is conducted to address the reorientation problem of rigid spacecraft in the presence of saturation constraints and attitude forbidden constraints. The core part of this paper consists of two parts：（1）The original nonlinear and non-convex constrained reorientation problem is transformed into a standard SOCP problem by con?vexification and relaxation；（2）the specific quadrat?ic-form performance index relative to angular veloci?ty can guarantee the equivalence of transformations.Finally，the algorithm is verified by a numerical sim?ulation. Further research will focus on the conver?gence analysis of the iteration.

Appendix：Proof of Proposition 1

The Hamiltonian ofP2is defined

The Lagrangian ofP2is derived as

Thus

（1）The nontriviality condition

（2）The costate differential equations

（3）The stationary conditions

（4）The Karush-Kuhn-Tucker conditions

If there exists a finite interval [ta，tb]∈[t0，tf]where the constraint in Eq.（23）is inactive，it will lead toλ7=λ8=λ9=0.will be ob?tained by Eqs.（A16—A18），andwill also be received by Eqs.（A19—A21）. And as the introduction of the L1 penalty method，λ12=0 can be guaranteed.

Then， substituting the above results into the costate differential Eqs. （A4—A9） and the stationary conditions in Eqs.（A10—A15），[P0P1P2P3P4P5P6] =0 can be received，which contradicts to the nontriviality condition.

In conclusion，there do not exist a finite inter?val [ta，tb] where the constraint in Eq.（23）is inac?tive.