CN113859585A - Fixed-time unwinding-free attitude control method for spacecraft - Google Patents

Fixed-time unwinding-free attitude control method for spacecraft Download PDF

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CN113859585A
CN113859585A CN202111068565.XA CN202111068565A CN113859585A CN 113859585 A CN113859585 A CN 113859585A CN 202111068565 A CN202111068565 A CN 202111068565A CN 113859585 A CN113859585 A CN 113859585A
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CN113859585B (en
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王增
陶玙
王春阳
刘长杰
王子硕
梁书宁
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Xian Technological University
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
    • B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
    • B64G1/00—Cosmonautic vehicles
    • B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
    • B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
    • B64G1/244—Spacecraft control systems
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
    • B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
    • B64G1/00—Cosmonautic vehicles
    • B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
    • B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
    • B64G1/244—Spacecraft control systems
    • B64G1/245—Attitude control algorithms for spacecraft attitude control
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Abstract

The invention discloses a fixed-time non-unwinding attitude control method for a spacecraft, which overcomes the unwinding problem of the spacecraft in the prior art, provides a fixed-time robust attitude control method without the unwinding problem, reduces the energy consumption of attitude control, and improves the control precision of attitude control. The invention comprises the following steps: step 1, based on a unit quaternion method, considering external interference torque, and establishing an attitude tracking control model of the rigid-body spacecraft:
Figure DDA0003259300050000011
Figure DDA0003259300050000012
wherein (q)v,q4)∈R3The x R represents the orientation of the spacecraft body described by the unit quaternion relative to an inertial coordinate system, and the x R satisfy the requirement
Figure DDA0003259300050000013
And 2, designing a fixed-time robust attitude controller without an unwinding problem, and carrying out stability analysis.

Description

Fixed-time unwinding-free attitude control method for spacecraft
The technical field is as follows:
the invention belongs to the technical field of advanced control, and relates to a space vehicle fixed-time non-unwinding attitude control method based on a hybrid system.
Background art:
a spacecraft described by a quaternion method belongs to a multi-input multi-output nonlinear equation system, and has two balance points, namely (0,0,0 +/-1)T. When the initial state of the spacecraft approaches the balance point (0,0,0, -1)TIn time, the traditional attitude control technology can cause the spacecraft to converge to the balance point (0,0,0, -1) in a longer pathTCausing waste of excess energy consumption, i.e. the problem of unwinding of the spacecraft. The existing fixed-time robust attitude control technology does not consider the problem of backing-off of a spacecraft, so that energy consumption of attitude control of the spacecraft is wasted, and the fixed-time attitude control technology based on the traditional sliding mode technology has the problem of flutter, so that the system is resonated when the system is serious, and the accuracy of attitude control is reduced.
The invention content is as follows:
the invention aims to provide a fixed-time non-unwinding attitude control method for a spacecraft, which overcomes the unwinding problem of the spacecraft in the prior art, provides a fixed-time robust attitude control method without the unwinding problem, reduces the energy consumption of attitude control, and improves the control precision of attitude control.
In order to achieve the purpose, the invention adopts the technical scheme that:
a fixed-time unwinding-free attitude control method for a spacecraft is characterized by comprising the following steps: the method comprises the following steps:
step 1, based on a unit quaternion method, considering external disturbance torque, and establishing a kinematic and dynamic model of the rigid-body spacecraft:
Figure BDA0003259300030000021
Figure BDA0003259300030000022
wherein (q)v,q4)∈R3The x R represents the orientation of the spacecraft body described by the unit quaternion relative to an inertial coordinate system, and the x R satisfy the requirement
Figure BDA0003259300030000023
ω∈R3Representing the angular velocity of the spacecraft, I3∈R3×3Represents an identity matrix, J ∈ R3×3Representing a positively-defined symmetric inertial matrix, u ∈ R3And d ∈ R3Respectively representing control moment and external disturbance (.)×∈R3×3The representative antisymmetric matrix can be expressed in the form:
Figure BDA0003259300030000024
and 2, designing a fixed-time robust attitude controller without an unwinding problem, and carrying out stability analysis.
The spacecraft attitude tracking control model in the step 1 is based on a unit quaternion method, and the kinematics and dynamics model thereof is
Figure BDA0003259300030000031
Figure BDA0003259300030000032
ωe＝ω-Cωd (6)
Wherein, ω isd∈R3And ωe∈R3Other than the desired angular velocity of the spacecraft and the tracking error of the angular velocity,
Figure BDA0003259300030000033
for attitude tracking errors, where the vector ev＝[e1,e2,e3]∈R3Scalar e4E R, is the relative pose between the actual pose and the desired pose, and a corresponding rotation matrix C e R3×3Is defined as
Figure BDA0003259300030000034
Wherein the rotation matrix satisfies C1 and
Figure BDA0003259300030000035
the fixed time robust attitude controller in the step 2 is
u＝u0+u1 (7)
Figure BDA0003259300030000036
Figure BDA0003259300030000037
S＝ωe+γhsigα(ev)+λhf(ev) (10)
f(ev)＝[s(e1),s(e2),s(e3)]T (11)
Wherein λ is1,λ2And λ3Are all normal numbers, alpha > 1, s (-) can be expressed as
Figure BDA0003259300030000038
Wherein y belongs to R, and R is more than 0 and less than 1 and less than R0≤2,
Figure BDA0003259300030000039
Figure BDA00032593000300000310
d is a very small normal number, sign (·) stands for sign function, H ∈ H { -1,1} is an auxiliary variable, H { -1,1} is a sign function, H ∈ H { -1,1 { -H } is a sign function, H { -H } is a sign function, and H { -1 } is a sign function of a sign function+Continuous and hopping sets are defined as
C＝{x∈S3×R3×H:he4＞-h} (13)
D＝{x∈S3×R3×H:he4≤-h} (14)
Wherein x is { q ═ qe,ωeH, η ∈ (0,1) represents the delay gap;
Figure BDA0003259300030000041
F(ev)＝diag(l(e1),l(e2),l(e3)) (16)
Figure BDA0003259300030000042
sig(ζ)＝[sign(ζ1),sign(ζ2),sign(ζ3)]T,ζ∈R3 (18)
Figure BDA0003259300030000043
where diag (·) represents a diagonal matrix.
In step 2, the stability analysis comprises the following steps:
firstly, proving that the fixed time of the attitude error reaches the sliding mode surface;
and secondly, proving that the attitude tracking error converges to any small neighborhood of the balance point along the fixed time of the sliding mode surface, and then gradually converges to the balance point.
Compared with the prior art, the invention has the advantages and effects that:
1. compared with the existing fixed-time robust attitude control technology, the method can avoid the attitude control unwinding problem of the spacecraft described by the quaternion, ensure the rapid convergence of the attitude control, can predict the convergence time, can greatly reduce the energy consumption of the attitude control, improves the control precision of the attitude control, and has strong robustness on external interference.
2. According to the invention, by adopting a hybrid system and an advanced control theory with stable fixed time, and designing the fixed-time unwinding-free attitude controller, the unwinding problem of attitude control can be effectively avoided, the energy consumption of attitude control is reduced, the convergence time can be estimated, the external interference is strong in robustness, and the attitude control precision is high.
Description of the drawings:
FIG. 1 is a fixed time attitude control map with unwind problems, where a represents attitude tracking error convergence, b represents angular velocity tracking error convergence, c represents control torque, and d represents an enlarged view of attitude tracking error convergence with unwind problems;
FIG. 2 is a fixed time attitude control map without unwinding problems, where a represents attitude tracking error convergence, b represents angular velocity tracking error convergence, c represents control torque, and d represents an enlarged view of attitude tracking error convergence without unwinding problems;
FIG. 3 is a graph comparing attitude control energy consumption with and without unwinding problems;
FIG. 4 is a schematic flow diagram of the method of the present invention.
The specific implementation mode is as follows:
in order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example (b):
in conjunction with the method flow diagram shown in fig. 4, the proposed non-unwinding problem fixed-time robust attitude control includes the following steps:
step 1, establishing an attitude control model of a spacecraft;
step 2, designing a fixed-time robust attitude controller without an unwinding problem;
the specific process of the step 1 is as follows:
based on a unit quaternion method, considering external interference torque, establishing an attitude control model of the rigid-body space vehicle:
Figure BDA0003259300030000061
Figure BDA0003259300030000062
wherein (q)v,q4)∈R3The x R represents the orientation of the spacecraft body described by the unit quaternion relative to an inertial coordinate system, and the x R satisfy the requirement
Figure BDA0003259300030000063
ω∈R3Representing the angular velocity of the spacecraft, I3∈R3×3Represents an identity matrix, J ∈ R3×3Representing a positively-defined symmetric inertial matrix, u ∈ R3And d ∈ R3Respectively representing control moment and external disturbance (.)×∈R3×3The representative antisymmetric matrix can be expressed in the form:
Figure BDA0003259300030000064
the desired pose is described by:
Figure BDA0003259300030000065
the unit quaternion described space vehicle kinematics and dynamics model has no singular point and can describe the 360-degree space vehicle attitude. However, the spacecraft described by the quaternion has two balance points, and the existing fixed time control algorithm does not consider the unwinding problem of attitude control, so that the energy consumption of the attitude control is wasted.
Specifically, the spacecraft attitude tracking control model in the step 1 is based on a unit quaternion method, and the kinematics and dynamics model thereof is as follows:
Figure BDA0003259300030000071
Figure BDA0003259300030000072
ωe＝ω-Cωd (6)
wherein ω ∈ R3、ωd∈R3And ωe∈R3Representing the actual angular velocity of the spacecraft, the desired angular velocity and the tracking error of the angular velocity,
Figure BDA0003259300030000073
for attitude tracking errors, where the vector ev＝[e1,e2,e3]∈R3Scalar e4E R, is the relative pose between the actual pose and the desired pose. The corresponding rotation matrix C ∈ R3×3Is defined as
Figure BDA0003259300030000074
Wherein the rotation matrix satisfies C1 and
Figure BDA0003259300030000075
I3∈R3×3represents an identity matrix, J ∈ R3×3Representing a positively-defined symmetric inertial matrix, u ∈ R3And d ∈ R3Respectively representing control moment and external disturbance (.)×∈R3×3The representative antisymmetric matrix can be expressed in the form:
Figure BDA0003259300030000076
the fixed time robust attitude controller in the step 2 is
u＝u0+u1 (7)
Figure BDA0003259300030000077
Figure BDA0003259300030000078
S＝ωe+γhsigα(ev)+λhf(ev) (10)
f(ev)＝[s(e1),s(e2),s(e3)]T (11)
Wherein λ is1,λ2And λ3Are all normal numbers, alpha > 1, s (-) can be expressed as
Figure BDA0003259300030000081
Wherein y belongs to R, and R is more than 0 and less than 1 and less than R0≤2,
Figure BDA0003259300030000082
Figure BDA0003259300030000083
d is a very small normal number, sign (·) stands for sign function, H ∈ H { -1,1} is an auxiliary variable, H { -1,1} is a sign function, H ∈ H { -1,1 { -H } is a sign function, H { -H } is a sign function, and H { -1 } is a sign function of a sign function+Continuous and hopping sets are defined as
C＝{x∈S3×R3×H:he4＞-η} (13)
D＝{x∈S3×R3×H:he4≤-η} (14)
Wherein x is { q ═ qe,ωeH, η ∈ (0,1) represents the delay gap.
Figure BDA0003259300030000084
F(ev)＝diag(l(e1),l(e2),l(e3)) (16)
Figure BDA0003259300030000085
sig(ζ)＝[sign(ζ1),sign(ζ2),sign(ζ3)]T,ζ∈R3 (18)
Figure BDA0003259300030000086
Where diag (·) represents a diagonal matrix.
In order to ensure the stability of the system, the stability analysis of the designed attitude controller is required, which is mainly divided into two steps. Firstly, proving that the fixed time of the attitude error reaches the sliding mode surface. And secondly, proving that the attitude tracking error converges to any small neighborhood of the balance point along the fixed time of the sliding mode surface, and then gradually converges to the balance point.
The first step is as follows: the fixed time is proved to reach the sliding mode surface.
By derivation of the formula (8), one can obtain
Figure BDA0003259300030000091
The equation (21) can be obtained by multiplying the inertia matrix J at both ends
Figure BDA0003259300030000092
By substituting formula (2) for formula (22), the compound
Figure BDA0003259300030000093
Defining the Lyapunov candidate function as:
Figure BDA0003259300030000094
by derivation of the above formula, the result is obtained
Figure BDA0003259300030000095
By substituting formula (23) for formula (25) and using formulae (5) to (7)
Figure BDA0003259300030000096
Since the external interference is bounded and is a known constant, the consistent fixed time of the sliding mode surface is known to converge to the origin according to the law of supercoiling fixed time, and the convergence time is
Figure BDA0003259300030000097
Where e > 0, M ═ α + L, M ═ α -L, h (λ1)＝1/λ1+(2e/mλ1)1/3E is a natural constant, alpha > L, lambda1h-1(λ1)＞M，
Figure BDA0003259300030000098
When, T (x)0) Taking the minimum value.
The second step is that: and (5) proving that the system state reaches an equilibrium point along the sliding mode surface at a fixed time.
After the system state reaches the sliding mode surface, S is 0, which can be obtained from equation (8)
Figure BDA0003259300030000101
Can be obtained by combining formula (1)
Figure BDA0003259300030000102
For the system (29), defining the Lyapunov candidate function as:
Figure BDA0003259300030000103
by derivation of time, one obtains
Figure BDA0003259300030000104
When | eiIf | is > d, use h 21, obtainable from formula (31)
Figure BDA0003259300030000105
By means of e4When the ratio of (t) > 0,
Figure BDA0003259300030000106
can obtain the product
Figure BDA0003259300030000107
Wherein
Figure BDA0003259300030000108
Figure BDA0003259300030000109
Figure BDA00032593000300001010
And
Figure BDA00032593000300001011
when | eiWhen | < d, use h 21, obtainable from formula (31)
Figure BDA0003259300030000111
Application 1 < r0When the content is less than or equal to 2,
Figure BDA0003259300030000112
can obtain the product
Figure BDA0003259300030000113
When x is equal to D, V2Jump occurs to obtain
V2(x+)-V2(x)≤2he4≤-2η≤0 (36)
It can be known that the designed controller can ensure that the system is stabilized to any small neighborhood near the equilibrium point for a fixed time, and then asymptotically stabilizes to the equilibrium point.
Fig. 1-3 show that, in the presence of external disturbance torque, the attitude control method provided by the invention can ensure that the spacecraft described by the quaternion can be quickly converged to a balance point (0,0,0, ± 1)TI.e. initial attitude q (0) ═ 0.3, -0.2, -0.3,0.8832]TNear the balance point (0,0,0, -1)TIn time, the spacecraft will converge to the equilibrium point (0,0,0, -1) with the shortest pathT(shown as d in FIG. 2), the average power consumption is 3.984NM, rather than rotating a large half-turn, to converge to the equilibrium point (0,0,0,1) with a longer pathT(as shown in d in figure 1) and the average energy consumption is 14.22NM, the proposed control method greatly reduces the energy consumption of attitude control of the spacecraft (as shown in comparison with the energy consumption in figure 3), namely the problem of unwinding of the spacecraft is solved.
Experimental example:
in order to verify the effectiveness of the fixed-time robust attitude stabilization controller designed by the patent, the attitude control is performed on the space vehicle under the condition that external interference exists, the effectiveness of the unwinding problem is verified, and whether the energy consumption of the controller is reduced or not is verified. The section mainly carries out validity verification through numerical simulation, and explains the validity of a specific implementation mode and the proposed control algorithm. Nominal inertia of hypothetical rigid body spacecraftThe property matrix is J ═ 201.20.9; 1.2171.4, respectively; 0.91.415]kg·m2. Initial desired attitude and angular velocity are set q, respectivelyd(0)＝[0,0,0,1]TAnd ωd(t)＝0.05[sin(πt/100),sin(2πt/100),sin(3πt/100)]rad/s, external interference d (t) ([ 0.1sin (t)), 0.2sin (1.2t),0.3sin (1.5t)]N.m. The initial attitude and angular velocity are set to q (0) — [0.3, -0.2, -0.3,0.8832, respectively]TAnd ω (0) ═ 0.06, -0.04,0.05]Trad/s。
The above embodiments are merely illustrative of the principles and effects of the present invention, and it will be apparent to those skilled in the art that various changes and modifications can be made without departing from the inventive concept of the present invention, and the scope of the present invention is defined by the appended claims.

Claims (4)

1. A fixed-time unwinding-free attitude control method for a spacecraft is characterized by comprising the following steps: the method comprises the following steps:
step 1, based on a unit quaternion method, considering external disturbance torque, and establishing a kinematic and dynamic model of the rigid-body spacecraft:
Figure FDA0003259300020000011
Figure FDA0003259300020000012
wherein (q)v,q4)∈R3The x R represents the orientation of the spacecraft body described by the unit quaternion relative to an inertial coordinate system, and the x R satisfy the requirement
Figure FDA0003259300020000013
ω∈R3Representing the angular velocity of the spacecraft, I3∈R3×3Represents an identity matrix, J ∈ R3×3Representing a positively-defined symmetric inertial matrix, u ∈ R3And d ∈ R3Respectively representing control moment and externalInterference, (×)×∈R3×3The representative antisymmetric matrix can be expressed in the form:
Figure FDA0003259300020000014
and 2, designing a fixed-time robust attitude controller without an unwinding problem, and carrying out stability analysis.
2. The fixed-time unwinding-free attitude control method for a spacecraft of claim 1, wherein: the spacecraft attitude tracking control model in the step 1 is based on a unit quaternion method, and the kinematics and dynamics model thereof is
Figure FDA0003259300020000021
Figure FDA0003259300020000028
ωe＝ω-Cωd (6)
Wherein, ω isd∈R3And ωe∈R3Other than the desired angular velocity of the spacecraft and the tracking error of the angular velocity,
Figure FDA0003259300020000022
for attitude tracking errors, where the vector ev＝[e1,e2,e3]∈R3Scalar e4E R, is the relative pose between the actual pose and the desired pose, and a corresponding rotation matrix C e R3×3Is defined as
Figure FDA0003259300020000023
Wherein the rotation matrix satisfies C1 and
Figure FDA0003259300020000024
3. the fixed-time unwinding-free attitude control method for a spacecraft of claim 1, wherein: the fixed time robust attitude controller in the step 2 is
u＝u0+u1 (7)
Figure FDA0003259300020000025
Figure FDA0003259300020000026
S＝ωe+γhsigα(ev)+lhf(ev) (10)
f(ev)＝[s(e1),s(e2),s(e3)]T (11)
Wherein l1,l2And l3Are all normal numbers, α>1, s (x) can be represented as
Figure FDA0003259300020000027
Wherein y ∈ R,0<r<1<r0≤2,
Figure FDA0003259300020000031
δ is a very small normal number, sign (x) stands for sign function, H ∈ H { -1,1} is an auxiliary variable, H { -1,1} is a sign function, H ∈ H { -1,1} is a sign function, H { -1, H { -1 { -H { -1 { -H } is a { -H } is a { -1 { -H } is a { -H { -1 { -H+Continuous and hopping sets are defined as
C＝{x∈S3×R3×H:he4>-h} (13)
D＝{x∈S3×R3×H:he4≤-h} (14)
Wherein x is { q ═ qe,ωeH, h ∈ (0,1) represents a delay gap;
Figure FDA0003259300020000032
F(ev)＝diag(l(e1),l(e2),l(e3)) (16)
Figure FDA0003259300020000033
sig(ζ)＝[sign(ζ1),sign(ζ2),sign(ζ3)]T,ζ∈R3 (18)
Figure FDA0003259300020000034
wherein diag (x) represents a diagonal matrix.
4. The fixed-time unwinding-free attitude control method for a spacecraft of claim 1, wherein: in step 2, the stability analysis comprises the following steps:
firstly, proving that the fixed time of the attitude error reaches the sliding mode surface;
and secondly, proving that the attitude tracking error converges to any small neighborhood of the balance point along the fixed time of the sliding mode surface, and then gradually converges to the balance point.
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Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114879708A (en) * 2022-04-19 2022-08-09 四川大学 Spacecraft attitude tracking unwinding-resistant control method with fixed time convergence
CN114879708B (en) * 2022-04-19 2023-03-14 四川大学 Spacecraft attitude tracking unwinding-resistant control method with fixed time convergence
CN116804853A (en) * 2023-08-25 2023-09-26 季华实验室 Flexible spacecraft attitude control method and device, electronic equipment and storage medium
CN116804853B (en) * 2023-08-25 2023-11-07 季华实验室 Flexible spacecraft attitude control method and device, electronic equipment and storage medium

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