CN109426147B - Adaptive gain adjustment control method for combined spacecraft after satellite acquisition - Google Patents

Adaptive gain adjustment control method for combined spacecraft after satellite acquisition Download PDF

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CN109426147B
CN109426147B CN201710727607.3A CN201710727607A CN109426147B CN 109426147 B CN109426147 B CN 109426147B CN 201710727607 A CN201710727607 A CN 201710727607A CN 109426147 B CN109426147 B CN 109426147B
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target satellite
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梁捷
李树民
刘烽
徐海航
梁武林
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Computational Aerodynamics Institute of China Aerodynamics Research and Development Center
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Abstract

The invention discloses a self-adaptive gain adjustment control method of a combined spacecraft after satellite capturing, which comprises the steps of firstly, establishing a combined spacecraft mathematical model suitable for a floating base space mechanical arm system to capture a target satellite control system design on the basis of transfer of momentum and impulse in the operation process of capturing a target satellite by a coupling space mechanical arm system, and calculating the movement speed of a joint of the combined spacecraft after the capturing operation is completed on the basis. Then aiming at the complex condition that the inertial parameters of a target satellite and a space mechanical arm system are unknown, the model, the fuzzy control theory, the sliding mode control theory and the Lyapunov stability theory are applied, and the fuzzy self-adaptive gain adjustment sliding mode control method for the stable motion of the combined spacecraft under the impact influence of collision in the capturing process is invented, so that the effective control of the captured satellite is achieved.

Description

Adaptive gain adjustment control method for combined spacecraft after satellite acquisition
Technical Field
The invention belongs to the technical field of spacecraft control. In particular to a self-adaptive gain adjustment control method of a combined spacecraft after satellite acquisition.
Background
The space mechanical arm is originally arranged on American space shuttle, international space station and complex spacecraft[1]In the above use, mainly the satellite release or the on-orbit assembly work of the space station assembly is undertaken, so that the problems of kinematic planning, dynamics and control of the non-capture process space manipulator to complete specific task operations are mainly involved. With the continuous development and maturity of the space manipulator technology, the space manipulator has the functions of on-orbit capturing, service and maintenance of satellitesEqual handling capacity is a necessary trend in the development of space manipulator technology. However, currently, related researches are not carried out much, mainly take kinematic planning for reducing grabbing impact and kinetic analysis of a grabbing process as main points, and research on control problems is less. It is worth noting that the space manipulator has completely different dynamics characteristics and limiting conditions from the ground fixed base manipulator system due to the complex space weightless environment of the space manipulator, and the system structure presents nonlinearity and strong coupling due to the free floating state of the carrier of the space manipulator system, so that the control method which is conventionally used for the ground fixed manipulator cannot be directly popularized and applied to the space manipulator control system, and the problem is particularly prominent when unknown parameters exist in the system. Meanwhile, a dynamic model involved in calm control of the combined spacecraft formed by the space mechanical arm system and the target satellite after the capturing operation has the difficulties of the space mechanical arm system model, and is also coupled with the transfer problem of momentum and impulse in the operation process of capturing the target satellite by the space mechanical arm system, and is a combined model superposed with the dynamic problems of the space mechanical arm system and the target satellite; the complexity and the correlation degree are larger than those of a single-space manipulator model, so that the research difficulty of the design problem of the related stabilization control system is larger, and the challenge is higher.
The problem of trapping in relation to space robot systems has been disclosed in the related patents CN104537151 and CN 104526695. However, most of these control methods only relate to collision dynamics modeling or trajectory planning of the capturing problem in the capturing problem of the space manipulator system, and the stabilizing control method for the combined spacecraft after the space manipulator system captures the satellite rarely relates to collision force generated by unavoidable collision due to a complex space weightless environment and when a claw at the tail end of the space manipulator contacts with the captured target satellite, so that the control problem of the combined spacecraft after the space manipulator system captures the satellite in orbit is far more complex than the problem of the space manipulator system which does not relate to the capturing process operation, and is a problem to be solved in the existing control technology.
Disclosure of Invention
The invention aims to solve the technical problem of disclosing a self-adaptive gain adjustment control method of a combined spacecraft after a satellite is captured, so as to obtain the tracking control effect that a control system asymptotically stabilizes an expected track under the condition that the combined spacecraft has uncertain factors.
The technical solution principle of the invention is as follows: firstly, on the basis of transfer of momentum and impulse in the operation process of capturing a target satellite by a coupling space mechanical arm system, a combined spacecraft mathematical model suitable for design of a control system for capturing the target satellite by a floating base space mechanical arm system is established, and on the basis, the motion speed of a joint of the combined spacecraft after the capturing operation is completed is calculated. Then aiming at the complex condition that the inertial parameters of a target satellite and a space mechanical arm system are unknown, the model, the fuzzy control theory, the sliding mode control theory and the Lyapunov stability theory are applied, and the fuzzy self-adaptive gain adjustment sliding mode control method for the stable motion of the combined spacecraft under the impact influence of collision in the capturing process is invented, so that the effective control of the captured satellite is achieved. The control method of the invention does not require the system dynamics equation to have a linear function relation with the inertia parameter, and does not need to predict the inertia parameter of the system; the control method has the obvious advantage of not needing to measure and feed back the position, the speed and the acceleration of the carrier because the position, the speed and the acceleration of the space manipulator system space shuttle carrier are eliminated in the derivation of the dynamic equation of the combined spacecraft. The method has important practical significance for on-orbit service such as capturing a failed satellite for maintenance and device replacement.
The invention discloses a self-adaptive gain adjustment control method of a combined spacecraft after satellite acquisition, which comprises the following steps:
step A: establishing a dynamic equation of a space mechanical arm system;
the space mechanical arm system is set to have a mass mPTensor of central inertia of IPInitial moving speed vx、vyInitial rotational angular velocity of ωPThe target satellite P performs on-orbit acquisition operation; establishing the following dynamic equation of the space manipulator system during the on-orbit capture by a Lagrange method:
Figure GDA0003130027280000021
wherein q is (x y theta)0 θ1 θ2)T∈R5Is a generalized coordinate vector of the system; m (q) epsilon R5×5A positive definite inertia matrix of the space mechanical arm system;
Figure GDA0003130027280000022
is a column vector containing Coriolis force and centrifugal force; fB＝(Fx Fy)T∈R2A column vector composed of the position control force of the carrier of the space shuttle; τ ═ t (τ)0 τ1 τ2)T∈R3Is composed of a joint O0、O1And O2Output torque tau of motor0、τ1And τ2A column vector of components; j is a Jacobian matrix connecting the space manipulator and the contact point; fIA contact collision force vector acting on a tail end point of a mechanical arm for a target satellite;
Figure GDA0003130027280000023
and
Figure GDA0003130027280000024
the first derivative and the second derivative of q, respectively;
Figure GDA0003130027280000025
is composed of
Figure GDA0003130027280000026
The second derivative of (a);
and B: performing collision dynamics analysis on the process of capturing the target satellite;
the following equations of the dynamics of the target satellite during in-orbit acquisition are established:
Figure GDA0003130027280000027
in the formula, MP(q)∈R3×3A positive definite inertia matrix of the target satellite;
Figure GDA0003130027280000028
is a column vector containing Coriolis force and centrifugal force; j. the design is a squarePA Jacobian matrix for connecting the target satellite with the contact point; f'IA contact collision force vector acting on the target satellite for the end point of the mechanical arm;
force and reaction relationship F 'between captured target satellite and space manipulator system in consideration of collision'I＝-FISubstituting formula (2) for formula (1) to obtain:
Figure GDA0003130027280000031
wherein,
Figure GDA0003130027280000032
is composed of
Figure GDA0003130027280000033
Moore-Penrose pseudoinverse of (1);
when the space mechanical arm system collides with the target satellite, the contact force is large and the time is short, the generalized coordinate vector does not change, and the generalized speed changes; at the same time, let us say that during a collision the system has no control input, i.e. F B0, τ is 0; defining the collision time as Δ t → 0, and integrating the collision time Δ t by equation (3) to obtain:
Figure GDA0003130027280000034
in the formula,
Figure GDA0003130027280000035
Δ t ═ O (epsilon), epsilon < 1; subscripts f, i denote the value of the vector before and after the collision, respectively; m is belonged to R5×5Is a positive of a space manipulator systemDetermining an inertia matrix; q ═ q (x y θ)0 θ1 θ2)T∈R5Is a generalized coordinate vector of the system, qfAnd q isiRespectively the generalized coordinate vector of the system before the collision and the generalized coordinate vector of the system after the collision,
Figure GDA0003130027280000036
and
Figure GDA0003130027280000037
are each qfAnd q isiThe first derivative of (a); mP(q)∈R3×3Is a positive fixed inertia matrix, M, of the target satelliteP∈R3×3Is a positive fixed inertia matrix, M, of the target satelliteP(q) and MPIs a concept, MPIs MPShorthand for (q);
Figure GDA0003130027280000038
and
Figure GDA0003130027280000039
generalized velocities before and after a target satellite collision; t is t0The time before collision is delta t;
Figure GDA00031300272800000310
is a column vector containing Coriolis force and centrifugal force,
Figure GDA00031300272800000311
and C is a concept, C is
Figure GDA00031300272800000312
The abbreviation of (1); mP∈R3×3Column vectors containing Coriolis force and centrifugal force of the target satellite; obviously, the left-hand value in the above equation is O (1), and the right-hand value in the integral term is O (1), but the integrated value is O (1)
Figure GDA00031300272800000313
Will be negligible compared to the left-hand equation, and, therefore,formula (4) may be represented as:
Figure GDA00031300272800000314
after contact collision, the tail end point of the space manipulator system and the contact point of the target satellite have the same speed
Figure GDA00031300272800000315
The generalized velocity of the target satellite at this time can be obtained from equation (5):
Figure GDA00031300272800000316
wherein,
Figure GDA00031300272800000317
is JPMoore-Penrose pseudoinverse of (1); by replacing the formula (6) with the formula (5), the speeds of the rotating hinges of the carrier and the mechanical arm of the space shuttle after contact collision can be obtained as follows:
Figure GDA00031300272800000318
in the formula,
Figure GDA00031300272800000319
and C: performing dynamic modeling on the combined spacecraft after the target satellite is captured:
after the target satellite is successfully captured, the claw at the tail end of the mechanical arm of the combined spacecraft does not generate relative displacement any more, namely,
Figure GDA0003130027280000041
and performing time derivation on the obtained product to obtain:
Figure GDA0003130027280000042
by substituting the formula (8) for the formula (2)
Figure GDA0003130027280000043
Is provided with
Figure GDA0003130027280000044
Wherein J is a Jacobian matrix connecting the space manipulator and the contact point,
Figure GDA0003130027280000045
is the first derivative of J; j. the design is a squarePTo contact the target satellite with the Jacobian matrix of contact points,
Figure GDA0003130027280000046
is JPThe first derivative of (a);
Figure GDA0003130027280000047
are the target satellite's independent generalized coordinates,
Figure GDA0003130027280000048
is composed of
Figure GDA0003130027280000049
The first derivative of (a);
combining the formula (9) and the formula (1) to obtain the kinetic equation of the combined spacecraft represented by the formula (10):
Figure GDA00031300272800000410
wherein,
Figure GDA00031300272800000411
m' (q) is a positive definite inertia matrix of a combined spacecraft composed of a space mechanical arm system and a target satellite;
Figure GDA00031300272800000412
the combined spacecraft comprises the column vectors of Coriolis force and centrifugal force,c' is
Figure GDA00031300272800000413
In short, M 'is M' (q);
to save control fuel consumption, equation (10) can be written as an under-actuated form of the kinetic equation
Figure GDA00031300272800000414
In the formula, M b2 × 2 sub-matrices of MbmIs a 2 × 3 sub-matrix, MmIs a 3 × 3 sub-matrix; cbAre the first two items of C, CmThe latter three terms; 0 is a zero column vector of order 2; theta ═ theta0 θ1 θ2)T，X＝(x y)T，
Figure GDA00031300272800000415
And
Figure GDA00031300272800000416
second derivatives of X and θ, respectively; at the same time, eliminate
Figure GDA00031300272800000417
The obtained full-drive form kinetic equation of the combined spacecraft is as follows:
Figure GDA00031300272800000418
in the formula,
Figure GDA00031300272800000419
Figure GDA00031300272800000420
is the first derivative of θ; meanwhile, the formula (12) is quasi-linearized by:
Figure GDA00031300272800000421
wherein,
Figure GDA00031300272800000422
a 3 × 3 matrix; the quasi-linearization processing only changes the expression form of the formula and does not generate any model precision loss;
step D: conventional sliding mode control of combined spacecraft
The control problem of the coordination movement of the carrier attitude of the space shuttle and the joints of the mechanical arm in the combined spacecraft after the target satellite is captured is solved by determining the control input rules of a carrier attitude control system of the space shuttle and the joints hinge drivers of the mechanical arm so as to realize the accurate tracking control of the coordination movement of the carrier and the joints hinge of the mechanical arm of the space shuttle.
A combined spacecraft is a complex system with high non-linearity, high time-variation and high coupling; meanwhile, the combined spacecraft has the characteristics of external disturbance, uncertain parameters and the like; therefore, the modeling error of the combined spacecraft dynamics model formula (13) is as follows:
Figure GDA0003130027280000051
wherein,
Figure GDA0003130027280000052
and
Figure GDA0003130027280000053
are respectively a matrix MnAnd hnEstimated value of, Δ MnFor combining M in spacecraft dynamics model formulanModeling error of,. DELTA.hnFor combining the spacecraft dynamics model formula hnThe modeling error of (2);
let θd＝[θ0d θ1d θ2d]To combine the desired output vector of the spacecraft with the actual output vector θ ═ θ0θ1θ2]The error vector between is: e-thetad(ii) a The velocity error vector is:
Figure GDA0003130027280000054
the acceleration error vector is:
Figure GDA0003130027280000055
thus, the error sliding mode switching function is defined as:
Figure GDA0003130027280000056
wherein λ ═ diag (λ)1,λ2,λ3) Is a coefficient matrix; lambda [ alpha ]i＞0(i＝1,2,3)；
The sliding mode control law is designed as follows:
Figure GDA0003130027280000057
Figure GDA0003130027280000058
Figure GDA0003130027280000059
Figure GDA00031300272800000510
in formula (16) to formula (19), the fixed gain K is diag [ K ═ K11,K12,K13]，Kii＞0(ii＝11,22,33)；A＝diag[A1,A2,A3]，ai＞0(i＝1,2,3)；
Figure GDA00031300272800000511
And
Figure GDA00031300272800000512
are all temporary variables, in particular
Figure GDA00031300272800000513
Figure GDA00031300272800000514
Figure GDA00031300272800000515
For combining M in spacecraft dynamics model formulanIs determined by the estimated value of (c),
Figure GDA00031300272800000516
for combining the spacecraft dynamics model formula hnAn estimated value of (d); k11，K12，K13，A1，A2，A3Is a constant value;
by substituting formula (16) to formula (19) for formula (13):
Mns+(hn+A)s＝Δf-K sgn s (20)
in the formula,
Figure GDA00031300272800000517
Δ f is
Figure GDA00031300272800000518
The abbreviation of (1);
as can be seen from equation 20, in the conventional sliding mode control design method, when the control input torque changes, the fixed switching gain k is large enough to ensure the system stability, which makes the control input torque buffeting obvious. In order to ensure that the combined spacecraft can accurately track expectation after a target satellite is captured, a fuzzy adaptive gain adjustment controller is added on the basis of a conventional sliding mode controller, switching items are converted into a continuous fuzzy system, and switching gains can adapt to tracking errors in real time, so that the precision of track tracking is effectively ensured, and buffeting of control input torque is also inhibited.
Step E: fuzzy adaptive gain adjustment sliding mode control of combined spacecraft
In order to enable the switching gain k to be adaptively adjusted, a control law based on fuzzy adaptive gain adjustment is designed on the basis of a conventional sliding mode control law:
Figure GDA0003130027280000061
in the formula, the adaptive gain k of the bluriInstead of the control gain Ksgns, k in formula (16)i＝(k1,k2,k3)，ki(i is 1,2,3) is the output of the ith fuzzy system;
design of fuzzy rules
If with siAs the input of the rule, a product inference engine, a single-value fuzzifier and a central mean deblurring are adopted to design a fuzzy system, and then the fuzzy rule can adopt the following form
IF siNegative for is, THEN kiis with great negative
IF siis negative, THEN kiis in negative middle
IF siNegative for is, THEN kiis with small negative
IF siis zero, THEN kiis zero
IF siis plus or minus, THEN kiis of great smallness
IF siin the middle of is, THEN kiis in the middle of the middle
IF siis greater, THEN kiis great at the positive aspect
Design the following input and output variables siAnd kiThe membership function of (a) is:
Figure GDA0003130027280000062
wherein α and σ are constants of the membership functions;
the output of the fuzzy system is:
Figure GDA0003130027280000063
Figure GDA0003130027280000064
Figure GDA0003130027280000065
Figure GDA0003130027280000066
in the formula, N is the number of fuzzy rules;
Figure GDA0003130027280000067
is an adjustable parameter vector;
Figure GDA0003130027280000068
is a fuzzy base vector;
substituting formula (21) for formula (13) to obtain:
Figure GDA0003130027280000069
get
Figure GDA0003130027280000071
Approximating the constant Δ f for the optimumiThe above-mentioned
Figure GDA0003130027280000072
Is the first derivative of the switching function S;
Figure GDA0003130027280000073
approximating a constant for the optimum;
Figure GDA0003130027280000074
in order to adjust the parameter vector,
Figure GDA0003130027280000075
is the value of the desired value thereof,
Figure GDA0003130027280000076
is an estimate of the value of the error,
Figure GDA0003130027280000077
according to the universal approximation theorem, i.e. for a given arbitrarily small constant ωi> 0, there are:
Figure GDA0003130027280000078
definition of
Figure GDA0003130027280000079
Then there is
Figure GDA00031300272800000710
The parameter adaptive control law is designed as follows:
Figure GDA00031300272800000711
step F: carrying out global stability verification on the combined spacecraft fuzzy self-adaptive gain adjustment sliding mode control closed-loop system;
theorem 1, aiming at a dynamic formula (13) of a combined spacecraft after a target satellite is captured, if a control formula (21) and a corresponding parameter self-adaptive formula (31) are designed, a motion track of the system after the target satellite is captured can be enabled to asymptotically and stably track an expected track;
and (3) proving that: designing the Lyapuloff function as
Figure GDA00031300272800000712
In the formula, MnA matrix is positively determined for a diagonal, an
Figure GDA00031300272800000713
L is positive, n is an integer from 1 to n; by deriving formula (32), the result is obtained
Figure GDA00031300272800000714
From the formula (30)
Figure GDA0003130027280000081
By substituting the parameter adaptive control law equation (31) for the equation (34), it is possible to obtain:
Figure GDA0003130027280000082
from the equation (28), there is a constant ωi> 0, assume:
Figure GDA0003130027280000083
wherein, 0 < gammai＜1；
The second term on the right side of the equal sign of formula (35) satisfies
Figure GDA0003130027280000084
Thus, there are:
Figure GDA0003130027280000085
wherein γ is diag [ γ ]1,...,γi,...,γn]；aiSelecting a for a constanti＞γiThen (A- γ) is a positive definite matrix, so there is:
Figure GDA0003130027280000086
as can be seen from equation (39), (a- γ) is a positive definite matrix, and therefore, only when s is 0,
Figure GDA0003130027280000087
adaptive control law 31 converges asymptotically; namely:
Figure GDA0003130027280000088
provided with a mechanical arm Bi(i ═ 1,2) along xiLength of shaft 3m, joint O1With the carrier centroid O of the space plane0Is 1.5m, and the mechanical arm B1Center of mass and joint O1Is 2 m; mechanical arm B2Center of mass and joint O2Are all 1.5m, capture the centroid and the joint O of the satellite P2Is 1.5 m; the mass and the inertia moment of each split are respectively as follows: m is0＝35kg,m1＝3kg,m2＝1.5kg；I0＝30kg·m2,I1＝2.7kg·m2,I2＝1.2kg·m2(ii) a The target satellite has a mass mP2kg, center inertia tensor IP＝0.8kg·m2
In simulation, the velocity of the target satellite before the acquisition operation is assumed to be vx＝1m/s、vy1m/s and ωP1rad/s and the spacearm tip position has reached the capture position; after the capturing operation is finished, assuming that the tensors of the mass and the central inertia of the target satellite are unknown during control, and assuming that the initial values of the tensors are zero;
assuming that the expected trajectory of the motion corner of the combined spacecraft system is, the unit is: rad;
θ0d＝π/3+sin(πt/5)/2；θ1d＝-π/6+3sin(πt/5)/2；θ2d＝π/3+3sin(πt/5)/2
the initial value of the movement is theta (0)＝[1.200 0.306 1.217]T(rad), simulation time from completion of the capture operation: t is 10 s; the control process ends.
The fuzzy adaptive gain adjustment sliding mode stabilization control method for the combined spacecraft has the advantages that the fuzzy adaptive gain adjustment controller is added on the basis of the conventional sliding mode controller, switching items are converted into a continuous fuzzy system, and switching gains can adapt to tracking errors in real time, so that the precision of track tracking is effectively guaranteed, and buffeting of control input torque is also restrained.
Drawings
FIG. 1 is a schematic view of a combined spacecraft after acquisition of a target satellite;
FIG. 2 is a schematic diagram of an adaptive fuzzy system for gain adjustment;
FIG. 3 is a schematic diagram of the control of the spacecraft assembly after the target satellite is captured in accordance with the present invention;
FIG. 4 is a combined spacecraft control process after acquisition of a target satellite according to the present invention;
FIG. 5 shows the variation of the spacecraft carrier position (X, Y) after acquisition of the target satellite (no calm control after acquisition);
FIG. 6 shows the carrier attitude angle θ of the space shuttle after capturing the target satellite0Variation of (3) (no stabilization control after capturing);
FIG. 7 shows a joint angle θ of a robotic arm of the combined spacecraft after capturing a target satellite1Track following situation (no stabilization control after capture);
FIG. 8 shows a joint angle θ of a robotic arm of the combined spacecraft after capturing a target satellite2Track following situation (no stabilization control after capture);
fig. 9 shows the variation of the combined spacecraft carrier position (X, Y) after acquisition of the target satellite (calm control after acquisition);
FIG. 10 is a diagram of a spacecraft carrier attitude angle θ after target satellite acquisition0Change of (3) (stabilization control after capturing);
FIG. 11 shows a joint angle θ of a robotic arm of a combined spacecraft after capturing a target satellite1Tracking situation (performing stabilization control after capturing); drawing (A)12 is a combined spacecraft mechanical arm joint angle theta after capturing a target satellite2Tracking (performing stabilization control after capturing).
Detailed Description
The invention will be further explained in detail with reference to the drawings and technical solutions.
FIG. 1 is a schematic view of a combined spacecraft after acquisition of a target satellite; FIG. 1 is a free-floating spacecraft carrier B0And a mechanical arm B1And B2The space mechanical arm system is composed by the example, wherein (P) is the target satellite to be captured, and P is the captured target satellite. Establishing a system inertial coordinate system (O-xy) and a split BiPrincipal axis coordinate system (O)i-xiyi) (i ═ 0,1, 2). The mass and central inertia tensor of each split are mi(I ═ 0,1,2) and Ii(i＝0,1,2)。l0Is O0To O1A distance of li(i ═ 1,2) is the link length of the robot arm. Definition of ri(i is 0,1,2) is each constituent BiCenter of mass OCiRadius of vector relative to O, rCIs the radial dimension of the total system centroid C relative to O. e.g. of the typeiIs along an axis xiAnd (i is 0,1,2) direction base vector.
FIG. 2 is a schematic diagram of an adaptive fuzzy system for gain adjustment; FIG. 2 is a schematic diagram of an adaptive fuzzy system for gain adjustment, where siAs input to the rule, ki(i ═ 1,2,3,. n) is the output of the i-th blur system.
FIG. 3 is a schematic diagram of the control of the combined spacecraft after acquisition of a target satellite according to the present invention. As shown in fig. 3, a self-adaptive fuzzy system 110 for gain adjustment is introduced to achieve self-adaptive approximation of switching gain in system sliding mode control 120, so that the problem of jitter in sliding mode control of the combined spacecraft after the target satellite is captured is solved, system errors are reduced, and high-precision stabilization control of the combined spacecraft after the target satellite is captured is achieved.
FIG. 4 is a control flow of the combined spacecraft after the target satellite is acquired according to the present invention.
Fig. 5 shows the change in the position (X, Y) of the space shuttle carrier after acquisition of the target satellite (no calm control after acquisition).
FIG. 6 shows the carrier attitude angle θ of the space shuttle after capturing the target satellite0The broken line represents an actual trajectory and the solid line represents a desired trajectory.
FIG. 7 shows a joint angle θ of a robotic arm of the combined spacecraft after capturing a target satellite1The broken line indicates an actual trajectory and the solid line indicates a desired trajectory in the case of trajectory tracking (no stabilization control after capturing).
FIG. 8 shows a joint angle θ of a robotic arm of the combined spacecraft after capturing a target satellite2The broken line indicates an actual trajectory and the solid line indicates a desired trajectory in the case of trajectory tracking (no stabilization control after capturing).
Fig. 9 shows the variation of the combined spacecraft carrier position (X, Y) after acquisition of the target satellite (calm control after acquisition).
FIG. 10 is a diagram of a spacecraft carrier attitude angle θ after target satellite acquisition0The broken line represents an actual trajectory and the solid line represents a desired trajectory.
FIG. 11 shows a joint angle θ of a robotic arm of a combined spacecraft after capturing a target satellite1The broken line indicates an actual trajectory and the solid line indicates a desired trajectory.
FIG. 12 shows a joint angle θ of a robotic arm of a combined spacecraft after capturing a target satellite2The broken line indicates an actual trajectory and the solid line indicates a desired trajectory.
Fig. 4 is a control flow of the combined spacecraft after capturing the target satellite according to the present invention, which includes the following specific contents:
step 210: establishing a kinetic equation of a spatial manipulator system
The space mechanical arm system is set to have a mass mPTensor of central inertia of IPInitial moving speed vx、vyInitial rotational angular velocity of ωPThe target satellite P performs on-orbit acquisition operation; establishing the following dynamic equation of the space manipulator system during the on-orbit capture by a Lagrange method:
Figure GDA0003130027280000111
wherein q is (x y theta)0 θ1 θ2)T∈R5Is a generalized coordinate vector of the system; m (q) epsilon R5×5A positive definite inertia matrix of the space mechanical arm system;
Figure GDA0003130027280000112
is a column vector containing Coriolis force and centrifugal force; fB＝(Fx Fy)T∈R2A column vector composed of the position control force of the carrier of the space shuttle; τ ═ t (τ)0 τ1 τ2)T∈R3Is composed of a joint O0、O1And O2Output torque tau of motor0、τ1And τ2A column vector of components; j is a Jacobian matrix connecting the space manipulator and the contact point; fIA contact collision force vector acting on a tail end point of a mechanical arm for a target satellite;
Figure GDA0003130027280000113
and
Figure GDA0003130027280000114
the first derivative and the second derivative of q, respectively;
Figure GDA0003130027280000115
is composed of
Figure GDA0003130027280000116
The second derivative of (a);
step 220: collision dynamics analysis of target satellite acquisition process
The following equations of the dynamics of the target satellite during in-orbit acquisition are established:
Figure GDA0003130027280000117
in the formula, MP(q)∈R3×3A positive definite inertia matrix of the target satellite;
Figure GDA0003130027280000118
is a column vector containing Coriolis force and centrifugal force; j. the design is a squarePA Jacobian matrix for connecting the target satellite with the contact point; f'IA contact collision force vector acting on the target satellite for the end point of the mechanical arm;
force and reaction relationship F 'between captured target satellite and space manipulator system in consideration of collision'I＝-FISubstituting formula (2) for formula (1) to obtain:
Figure GDA0003130027280000119
wherein,
Figure GDA00031300272800001110
is composed of
Figure GDA00031300272800001111
Moore-Penrose pseudoinverse of (1);
when the space mechanical arm system collides with the target satellite, the contact force is large and the time is short, the generalized coordinate vector does not change, and the generalized speed changes; at the same time, let us say that during a collision the system has no control input, i.e. F B0, τ is 0; defining the collision time as Δ t → 0, and integrating the collision time Δ t by equation (3) to obtain:
Figure GDA00031300272800001112
in the formula,
Figure GDA00031300272800001113
Δ t ═ O (epsilon), epsilon < 1; subscripts f, i denote the value of the vector before and after the collision, respectively; m is belonged to R5×5A positive definite inertia matrix of the space mechanical arm system; q ═ q (x y θ)0 θ1 θ2)T∈R5Is a generalized coordinate vector of the system, qfAnd q isiRespectively the generalized coordinate vector of the system before the collision and the generalized coordinate vector of the system after the collision,
Figure GDA00031300272800001114
and
Figure GDA00031300272800001115
are each qfAnd q isiThe first derivative of (a); mP(q)∈R3×3Is a positive fixed inertia matrix, M, of the target satelliteP∈R3×3Is a positive fixed inertia matrix, M, of the target satelliteP(q) and MPIs a concept, MPIs MPShorthand for (q);
Figure GDA00031300272800001116
and
Figure GDA00031300272800001117
generalized velocities before and after a target satellite collision; t is t0The time before collision is delta t;
Figure GDA0003130027280000121
is a column vector containing Coriolis force and centrifugal force,
Figure GDA0003130027280000122
and C is a concept, C is
Figure GDA0003130027280000123
The abbreviation of (1); mP∈R3×3Column vectors containing Coriolis force and centrifugal force of the target satellite; obviously, the left-hand value in the above equation is O (1), and the right-hand value in the integral term is O (1), but the integrated value is O (1)
Figure GDA0003130027280000124
Will be negligible compared to the left equation, and therefore equation (4) can be expressed as:
Figure GDA0003130027280000125
after contact collision, the tail end point of the space manipulator system and the contact point of the target satellite have the same speed
Figure GDA0003130027280000126
The generalized velocity of the target satellite at this time can be obtained from equation (5):
Figure GDA0003130027280000127
wherein,
Figure GDA0003130027280000128
is JPMoore-Penrose pseudoinverse of (1); by replacing the formula (6) with the formula (5), the speeds of the rotating hinges of the carrier and the mechanical arm of the space shuttle after contact collision can be obtained as follows:
Figure GDA0003130027280000129
in the formula,
Figure GDA00031300272800001210
and C: performing dynamic modeling on the combined spacecraft after the target satellite is captured:
after the target satellite is successfully captured, the claw at the tail end of the mechanical arm of the combined spacecraft does not generate relative displacement any more, namely,
Figure GDA00031300272800001211
and performing time derivation on the obtained product to obtain:
Figure GDA00031300272800001212
by substituting the formula (8) for the formula (2)
Figure GDA00031300272800001213
Is provided with
Figure GDA00031300272800001214
Wherein J is a Jacobian matrix connecting the space manipulator and the contact point,
Figure GDA00031300272800001215
is the first derivative of J; j. the design is a squarePTo contact the target satellite with the Jacobian matrix of contact points,
Figure GDA00031300272800001216
is JPThe first derivative of (a);
Figure GDA00031300272800001217
are the target satellite's independent generalized coordinates,
Figure GDA00031300272800001218
is composed of
Figure GDA00031300272800001219
The first derivative of (a);
combining the formula (9) and the formula (1) to obtain the kinetic equation of the combined spacecraft represented by the formula (10):
Figure GDA00031300272800001220
wherein,
Figure GDA00031300272800001221
m' (q) is a positive definite inertia matrix of a combined spacecraft composed of a space mechanical arm system and a target satellite;
Figure GDA00031300272800001223
for assembly of spacecraft containing the column vectors of Coriolis force, centrifugal force, C' is
Figure GDA00031300272800001224
In short, M 'is M' (q);
to save control fuel consumption, equation (10) can be written as an under-actuated form of the kinetic equation
Figure GDA00031300272800001222
In the formula, M b2 × 2 sub-matrices of MbmIs a 2 × 3 sub-matrix, MmIs a 3 × 3 sub-matrix; cbAre the first two items of C, CmThe latter three terms; 0 is a zero column vector of order 2; theta ═ theta0 θ1 θ2)T，X＝(x y)T，
Figure GDA0003130027280000131
And
Figure GDA0003130027280000132
second derivatives of X and θ, respectively; at the same time, eliminate
Figure GDA0003130027280000133
The obtained full-drive form kinetic equation of the combined spacecraft is as follows:
Figure GDA0003130027280000134
in the formula,
Figure GDA0003130027280000135
Figure GDA0003130027280000136
is the first derivative of θ; meanwhile, the formula (12) is quasi-linearized by:
Figure GDA0003130027280000137
wherein,
Figure GDA0003130027280000138
a 3 × 3 matrix; the quasi-linearization processing only changes the expression form of the formula and does not generate any model precision loss;
step 240: design of conventional sliding mode stabilizing control method for combined spacecraft
The control problem of the coordination movement of the carrier attitude of the space shuttle and the joints of the mechanical arm in the combined spacecraft after the target satellite is captured is solved by determining the control input rules of a carrier attitude control system of the space shuttle and the joints hinge drivers of the mechanical arm so as to realize the accurate tracking control of the coordination movement of the carrier and the joints hinge of the mechanical arm of the space shuttle.
A combined spacecraft is a complex system with high non-linearity, high time-variation and high coupling; meanwhile, the combined spacecraft has the characteristics of external disturbance, uncertain parameters and the like; therefore, the modeling error of the combined spacecraft dynamics model formula (13) is as follows:
Figure GDA0003130027280000139
wherein,
Figure GDA00031300272800001310
and
Figure GDA00031300272800001311
are respectively a matrix MnAnd hnEstimated value of, Δ MnFor combining M in spacecraft dynamics model formulanModeling error of,. DELTA.hnFor combining the spacecraft dynamics model formula hnThe modeling error of (2);
let θd＝[θ0d θ1d θ2d]To combine the desired output vector of the spacecraft with the actual output vector θ ═ θ0 θ1θ2]The error vector between is: e-thetad(ii) a The velocity error vector is:
Figure GDA00031300272800001312
the acceleration error vector is:
Figure GDA00031300272800001313
the sliding mode control law is designed as follows:
Figure GDA00031300272800001314
Figure GDA00031300272800001315
Figure GDA00031300272800001316
Figure GDA00031300272800001317
in formula (16) to formula (19), the fixed gain K is diag [ K ═ K11,K12,K13]，Kii＞0(ii＝11,22,33)；A＝diag[A1,A2,A3]，ai＞0(i＝1,2,3)；
Figure GDA00031300272800001318
And
Figure GDA00031300272800001319
are all temporary variables, in particular
Figure GDA00031300272800001320
Figure GDA00031300272800001321
Figure GDA00031300272800001322
Into a groupSynthetic spacecraft dynamics model type medium MnIs determined by the estimated value of (c),
Figure GDA00031300272800001323
for combining the spacecraft dynamics model formula hnAn estimated value of (d); k11，K12，K13，A1，A2，A3Is a constant value;
by substituting formula (16) to formula (19) for formula (13):
Mns+(hn+A)s＝Δf-K sgn s (20)
in the formula,
Figure GDA0003130027280000141
Δ f is
Figure GDA0003130027280000142
The abbreviation of (1);
as can be seen from equation 20, in the conventional sliding mode control design method, when the control input torque changes, the fixed switching gain k is large enough to ensure the system stability, which makes the control input torque buffeting obvious. In order to ensure that the combined spacecraft can accurately track expectation after a target satellite is captured, a fuzzy adaptive gain adjustment controller is added on the basis of a conventional sliding mode controller, switching items are converted into a continuous fuzzy system, and switching gains can adapt to tracking errors in real time, so that the precision of track tracking is effectively ensured, and buffeting of control input torque is also inhibited. The method can adapt to the tracking error in real time, thus effectively ensuring the precision of track tracking and inhibiting buffeting of the control input torque.
Step 250: design of fuzzy self-adaptive gain adjustment sliding mode stabilization control method for combined spacecraft
In order to enable the switching gain k to be adaptively adjusted, a control law based on fuzzy adaptive gain adjustment is designed on the basis of a conventional sliding mode control law:
Figure GDA0003130027280000143
in the formula, the adaptive gain k of the bluriInstead of the control gain Ksgns, k in formula (16)i＝(k1,k2,k3)，ki(i is 1,2,3) is the output of the ith fuzzy system;
design of fuzzy rules
If with siAs the input of the rule, a product inference engine, a single-value fuzzifier and a central mean deblurring are adopted to design a fuzzy system, and then the fuzzy rule can adopt the following form
IF siNegative for is, THEN kiis with great negative
IF siis negative, THEN kiis in negative middle
IF siNegative for is, THEN kiis with small negative
IF siis zero, THEN kiis zero
IF siis plus or minus, THEN kiis of great smallness
IF siin the middle of is, THEN kiis in the middle of the middle
IF siis greater, THEN kiis great at the positive aspect
Design the following input and output variables siAnd kiThe membership function of (a) is:
Figure GDA0003130027280000144
wherein α and σ are constants of the membership functions;
the output of the fuzzy system is:
Figure GDA0003130027280000151
Figure GDA0003130027280000152
Figure GDA0003130027280000153
Figure GDA0003130027280000154
in the formula, N is the number of fuzzy rules;
Figure GDA0003130027280000155
is an adjustable parameter vector;
Figure GDA0003130027280000156
is a fuzzy base vector;
substituting formula (21) for formula (13) to obtain:
Figure GDA0003130027280000157
get
Figure GDA0003130027280000158
Approximating the constant Δ f for the optimumiThe above-mentioned
Figure GDA0003130027280000159
Is the first derivative of the switching function S;
Figure GDA00031300272800001510
approximating a constant for the optimum;
Figure GDA00031300272800001511
in order to adjust the parameter vector,
Figure GDA00031300272800001512
is the value of the desired value thereof,
Figure GDA00031300272800001513
is an estimate of the value of the error,
Figure GDA00031300272800001514
according to the universal approximation theorem, i.e. for a given arbitrarily small constant ωi> 0, there are:
Figure GDA00031300272800001515
definition of
Figure GDA00031300272800001516
Then there is
Figure GDA00031300272800001517
The parameter adaptive control law is designed as follows:
Figure GDA00031300272800001518
step F: carrying out global stability verification on the combined spacecraft fuzzy self-adaptive gain adjustment sliding mode control closed-loop system;
theorem 1, aiming at a dynamic formula (13) of a combined spacecraft after a target satellite is captured, if a control formula (21) and a corresponding parameter self-adaptive formula (31) are designed, a motion track of the system after the target satellite is captured can be enabled to asymptotically and stably track an expected track;
and (3) proving that: designing the Lyapuloff function as
Figure GDA0003130027280000161
In the formula, MnA matrix is positively determined for a diagonal, an
Figure GDA0003130027280000162
L is positive, n is an integer from 1 to n; by deriving formula (32), the result is obtained
Figure GDA0003130027280000163
From the formula (30)
Figure GDA0003130027280000164
By substituting the parameter adaptive control law equation (31) for the equation (34), it is possible to obtain:
Figure GDA0003130027280000165
from the equation (28), there is a constant ωi> 0, assume:
Figure GDA0003130027280000166
wherein, 0 < gammai＜1；
The second term on the right side of the equal sign of formula (35) satisfies
Figure GDA0003130027280000167
Thus, there are:
Figure GDA0003130027280000171
wherein γ is diag [ γ ]1,...,γi,...,γn]；aiSelecting a for a constanti＞γiThen (A- γ) is a positive definite matrix, so there is:
Figure GDA0003130027280000172
as can be seen from equation (39), (a- γ) is a positive definite matrix, and therefore, only when s is 0,
Figure GDA0003130027280000173
adaptive control law 31 converges asymptotically; namely:
Figure GDA0003130027280000174
provided with a mechanical arm Bi(i ═ 1,2) along xiLength of shaft 3m, joint O1With the carrier centroid O of the space plane0Is 1.5m, and the mechanical arm B1Center of mass and joint O1Is 2 m; mechanical arm B2Center of mass and joint O2Are all 1.5m, capture the centroid and the joint O of the satellite P2Is 1.5 m; the mass and the inertia moment of each split are respectively as follows: m is0＝35kg,m1＝3kg,m2＝1.5kg；I0＝30kg·m2,I1＝2.7kg·m2,I2＝1.2kg·m2(ii) a The target satellite has a mass mP2kg, center inertia tensor IP＝0.8kg·m2
In simulation, the velocity of the target satellite before the acquisition operation is assumed to be vx＝1m/s、vy1m/s and ωP1rad/s and the spacearm tip position has reached the capture position; after the capturing operation is finished, assuming that the tensors of the mass and the central inertia of the target satellite are unknown during control, and assuming that the initial values of the tensors are zero;
assuming that the expected trajectory of the motion corner of the combined spacecraft system is, the unit is: rad;
θ0d＝π/3+sin(πt/5)/2；θ1d＝-π/6+3sin(πt/5)/2；θ2d＝π/3+3sin(πt/5)/2
the initial value of motion is θ (0) ═ 1.2000.3061.217]T(rad), simulation time from completion of the capture operation: t is 10 s; the control process ends. .
θ0(0)＝1.20,θ1(0)＝0.306,θ2(0)＝1.217。
Step 270: the control process ends.

Claims (1)

1. A self-adaptive gain adjustment control method for a combined spacecraft after satellite acquisition is characterized by comprising the following steps:
step A: establishing a dynamic equation of a space mechanical arm system;
the space mechanical arm system is set to have a mass mPTensor of central inertia of IPInitial moving speed vx、vyInitial rotational angular velocity of ωPThe target satellite P performs on-orbit acquisition operation; establishing the following dynamic equation of the space manipulator system during the on-orbit capture by a Lagrange method:
Figure FDA0003316909600000011
wherein q is (x y theta)0 θ1 θ2)T∈R5Is a generalized coordinate vector of the system; m (q) epsilon R5×5A positive definite inertia matrix of the space mechanical arm system;
Figure FDA0003316909600000012
is a column vector containing Coriolis force and centrifugal force; fB＝(Fx Fy)T∈R2A column vector composed of the position control force of the carrier of the space shuttle; τ ═ t (τ)0 τ1 τ2)T∈R3Is composed of a joint O0、O1And O2Output torque tau of motor0、τ1And τ2A column vector of components; j is a Jacobian matrix connecting the space manipulator and the contact point; fIA contact collision force vector acting on a tail end point of a mechanical arm for a target satellite;
Figure FDA0003316909600000013
and
Figure FDA0003316909600000014
the first derivative and the second derivative of q, respectively;
Figure FDA0003316909600000015
independent generalized coordinates for target satellites
Figure FDA0003316909600000016
The second derivative of (a);
and B: performing collision dynamics analysis on the process of acquiring the target satellite, and establishing the following dynamic equation of the target satellite during in-orbit acquisition:
Figure FDA0003316909600000017
in the formula, MP(q)∈R3×3A positive definite inertia matrix of the target satellite;
Figure FDA0003316909600000018
column vectors containing Coriolis force and centrifugal force for the target satellite; j. the design is a squarePA Jacobian matrix for connecting the target satellite with the contact point; fI' is the contact collision force vector of the mechanical arm end point acting on the target satellite;
force and reaction relationship F between captured target satellite and space manipulator system when considering collisionI′＝-FISubstituting formula (2) for formula (1) to obtain:
Figure FDA0003316909600000019
wherein,
Figure FDA00033169096000000110
is composed of
Figure FDA00033169096000000111
Moore-Penrose pseudoinverse of (1);
when the space mechanical arm system collides with the target satellite, the contact force is large and the time is short, the generalized coordinate vector does not change, and the generalized speed changes; at the same time, let us say that during a collision the system has no control input, i.e. FB0, τ is 0; defining the collision time as Δ t → 0, and integrating the collision time Δ t by equation (3) to obtain:
Figure FDA00033169096000000112
in the formula,
Figure FDA0003316909600000021
o (epsilon), epsilon < -1; subscripts f, i denote the value of the vector before and after the collision, respectively; m is belonged to R5×5Is a positive definite inertia matrix of the space mechanical arm system, and M is a shorthand of M (q); q ═ q (x y θ)0 θ1 θ2)T∈R5Is a generalized coordinate vector of the system, qfAnd q isiRespectively the generalized coordinate vector of the system before the collision and the generalized coordinate vector of the system after the collision,
Figure FDA0003316909600000022
and
Figure FDA0003316909600000023
are each qfAnd q isiThe first derivative of (a); mP∈R3×3Is a positive fixed inertia matrix, M, of the target satellitePIs MPShorthand for (q);
Figure FDA0003316909600000024
and
Figure FDA0003316909600000025
generalized velocities before and after a target satellite collision; t is t0The time before collision is delta t; c is
Figure FDA0003316909600000026
Abbreviation of (C)pIs composed of
Figure FDA0003316909600000027
The abbreviation of (1); it is obvious that the left-hand value in the above equation is O (1) and the right-hand value in the integral term is O (1), but the value after integration is O (1)
Figure FDA0003316909600000028
Will be negligible compared to the left equation, and therefore equation (4) can be expressed as:
Figure FDA0003316909600000029
after contact collision, the tail end point of the space manipulator system and the contact point of the target satellite have the same speed
Figure FDA00033169096000000210
The generalized velocity of the target satellite at this time can be obtained from equation (5):
Figure FDA00033169096000000211
wherein,
Figure FDA00033169096000000212
is JPMoore-Penrose pseudoinverse of (1); by replacing the formula (6) with the formula (5), the speeds of the rotating hinges of the carrier and the mechanical arm of the space shuttle after contact collision can be obtained as follows:
Figure FDA00033169096000000213
in the formula,
Figure FDA00033169096000000214
and C: dynamic modeling of combined spacecraft after target satellite capture
After the target satellite is successfully captured, the claw at the tail end of the mechanical arm of the combined spacecraft does not generate relative displacement any more, namely,
Figure FDA00033169096000000215
and performing time derivation on the obtained product to obtain:
Figure FDA00033169096000000216
by substituting the formula (8) for the formula (2)
Figure FDA00033169096000000217
Comprises the following steps:
Figure FDA00033169096000000218
where J is a Jacobian matrix linking the space manipulator and the point of contact,
Figure FDA00033169096000000219
is the first derivative of J; j. the design is a squarePTo contact the target satellite with the Jacobian matrix of contact points,
Figure FDA00033169096000000220
is JPThe first derivative of (a);
Figure FDA00033169096000000221
are the target satellite's independent generalized coordinates,
Figure FDA00033169096000000222
is composed of
Figure FDA00033169096000000223
The first derivative of (a);
combining the formula (9) and the formula (1) to obtain the dynamic equation of the combined spacecraft represented by the formula (10)
Figure FDA00033169096000000224
Wherein,
Figure FDA00033169096000000225
m' (q) is a positive definite inertia matrix of a combined spacecraft composed of a space mechanical arm system and a target satellite;
Figure FDA0003316909600000031
for assembly of spacecraft containing the column vectors of Coriolis force, centrifugal force, C' is
Figure FDA0003316909600000032
In short, M 'is M' (q);
to save control fuel consumption, equation (10) can be written as an under-actuated form of the kinetic equation:
Figure FDA0003316909600000033
in the formula, Mb2 × 2 sub-matrices of MbmIs a 2 × 3 sub-matrix, MmIs a 3 × 3 sub-matrix; cbAre the first two items of C, CmThe latter three terms; 0 is a zero column vector of order 2; theta ═ theta0 θ1 θ2)TWherein theta0Is the attitude angle of the carrier, theta1And theta2The relative angles of the 1 st and 2 nd joint hinges of the space manipulator respectively, X ═ X yT，
Figure FDA0003316909600000034
And
Figure FDA0003316909600000035
second derivatives of X and theta, respectively, X ═ X yTPosition coordinates of the system centroid; at the same time, eliminate
Figure FDA0003316909600000036
The obtained full-drive form kinetic equation of the combined spacecraft is as follows:
Figure FDA0003316909600000037
in the formula,
Figure FDA0003316909600000038
Figure FDA0003316909600000039
is the first derivative of θ; meanwhile, the formula (12) is quasi-linearized by:
Figure FDA00033169096000000310
wherein,
Figure FDA00033169096000000311
a 3 × 3 matrix; the quasi-linearization processing only changes the expression form of the formula and does not generate any model precision loss;
step D: conventional sliding mode control of combined spacecraft
A combined spacecraft is a complex system with high non-linearity, high time-variation and high coupling; meanwhile, the combined spacecraft has the characteristics of external disturbance, uncertain parameters and the like; therefore, the modeling error of the combined spacecraft dynamics model formula (13) is as follows:
Figure FDA00033169096000000312
wherein,
Figure FDA00033169096000000313
and
Figure FDA00033169096000000314
are respectively a matrix MnAnd hnEstimated value of hnIs composed of
Figure FDA00033169096000000315
Abbreviation of (1), Δ MnFor combining M in spacecraft dynamics model formulanModeling error of,. DELTA.hnFor combining the spacecraft dynamics model formula hnThe modeling error of (2);
let θd＝[θ0d θ1d θ2d]To combine the desired output vector of the spacecraft with the actual output vector θ ═ θ0 θ1 θ2]The error vector between is: e-thetad(ii) a The velocity error vector is:
Figure FDA00033169096000000316
the acceleration error vector is:
Figure FDA00033169096000000317
thus, an error sliding mode switching function is defined as
Figure FDA00033169096000000318
Wherein λ ═ diag (λ)1,λ2,λ3) Is a coefficient matrix; lambda [ alpha ]i＞0(i＝1,2,3)；
The sliding mode control law is designed as
Figure FDA0003316909600000041
Figure FDA0003316909600000042
Figure FDA0003316909600000043
Figure FDA0003316909600000044
In formula (16) to formula (19), the fixed gain K is diag [ K ═ K11,K12,K13]，Kii＞0(ii＝11,22,33)；A＝diag[A1,A2,A3]，
Figure FDA0003316909600000045
And
Figure FDA0003316909600000046
are all temporary variables; in particular to
Figure FDA0003316909600000047
K11，K12，K13，A1，A2，A3Is a constant;
formula (16) -formula (19) can be substituted for formula (13):
Mns+(hn+A)s＝Δf-Ksgn(s) (20)
in the formula,
Figure FDA0003316909600000048
Δ f is
Figure FDA0003316909600000049
The abbreviation of (1);
in order to ensure that the combined spacecraft can accurately track after a target satellite is captured, a fuzzy adaptive gain adjustment controller is added on the basis of a conventional sliding mode controller, and a switching item is converted into a continuous fuzzy system, so that the switching gain can adapt to a tracking error in real time, the precision of track tracking is effectively ensured, and buffeting of control input torque is also inhibited;
step E: combined spacecraft fuzzy adaptive gain adjustment sliding mode control
In order to enable the switching gain k to be adaptively adjusted, a sliding mode control law based on fuzzy adaptive gain adjustment is designed on the basis of a conventional sliding mode control law:
Figure FDA00033169096000000410
in the formula, the switching gain k is taken as the fuzzy adaptive gain kiAdaptive gain k of the ambiguityiInstead of the control gain Ksgn(s), k in formula (16)i＝(k1,k2,k3)，ki(i is 1,2,3) is the output of the ith fuzzy system;
design of fuzzy rules
If with siAs the input of the rule, a product inference engine, a single-value fuzzifier and a central mean deblurring are adopted to design a fuzzy system, and then the fuzzy rule can adopt the following form
IF siNegative for is, THEN kiis with great negative
IF siis negative, THEN kiis in negative middle
IF siNegative for is, THEN kiis with small negative
IF siis zero, THEN kiis zero
IF siis plus or minus, THEN kiis of great smallness
IF siin the middle of is, THEN kiis in the middle of the middle
IF siis greater, THEN kiis great at the positive aspect
Design the following input and output variables siAnd kiThe membership function of (a) is:
Figure FDA0003316909600000051
wherein α and σ are constants of the membership functions; the output of the fuzzy system is then:
Figure FDA0003316909600000052
Figure FDA0003316909600000053
Figure FDA0003316909600000054
Figure FDA0003316909600000055
in the formula, N is the number of fuzzy rules;
Figure FDA0003316909600000056
is an adjustable parameter vector;
Figure FDA0003316909600000057
is a fuzzy base vector; wherein
Figure FDA0003316909600000058
Is a membership function variable;
substituting formula (21) for formula (13) to obtain:
Figure FDA0003316909600000059
get
Figure FDA00033169096000000510
Approximating the constant Δ f for the optimumiThe above-mentioned
Figure FDA00033169096000000511
Is the first derivative of the error sliding mode switching function s;
Figure FDA00033169096000000512
in order to optimally approximate the constant value,
Figure FDA00033169096000000513
in order to adjust the parameter vector,
Figure FDA00033169096000000514
is the value of the desired value thereof,
Figure FDA00033169096000000515
is an estimate of the value of the error,
Figure FDA00033169096000000516
Figure FDA00033169096000000517
according to the universal approximation theorem, i.e. for a given arbitrarily small constant ωi> 0, there are:
Figure FDA00033169096000000518
definition of
Figure FDA00033169096000000519
Then there is
Figure FDA00033169096000000520
The parameter adaptive control law is designed as follows:
Figure FDA0003316909600000061
wherein s isiIs an input of a rule;
step F: aiming at a dynamic formula (13) of the combined spacecraft after a target satellite is captured, if a control law formula (21) and a corresponding parameter self-adaptive law formula (31) are designed, a motion track of the combined spacecraft after the target satellite is captured by the system can be enabled to asymptotically and stably track an expected track;
and (3) proving that: the Lyapunov function was designed as:
Figure FDA0003316909600000062
in the formula, MnA matrix is positively determined for a diagonal, an
Figure FDA0003316909600000063
L is positive, n is an integer from 1 to n; by taking the derivative of equation (32) and combining equation (30), the following can be obtained:
Figure FDA0003316909600000064
the parameter adaptive control law (31) is substituted for the expression (33) to obtain
Figure FDA0003316909600000065
From the formula (28), the number ω is as small as possiblei> 0, assume:
Figure FDA0003316909600000066
wherein, 0 < gammai＜1；
The second term on the right side of the equal sign of formula (34) satisfies
Figure FDA0003316909600000067
Thus, there are
Figure FDA0003316909600000068
Wherein γ is diag [ γ ]1,...,γi,...,γn]；aiIs constant, select ai＞γiThen (A- γ) is a positive definite matrix, so there is:
Figure FDA0003316909600000069
as can be seen from equation (38), (a- γ) is a positive definite matrix, and therefore, only when s is 0,
Figure FDA00033169096000000610
the adaptive control law (31) converges asymptotically, i.e.
Figure FDA0003316909600000071
Provided with a mechanical arm Bi(i ═ 1,2) along xiLength of shaft 3m, joint O1With the carrier centroid O of the space plane0Is 1.5m, and the mechanical arm B1Center of mass and joint O1Is 2 m; mechanical arm B2Center of mass and joint O2Is 1.5m, captures the centroid and the joint O of the satellite P2Is 1.5 m; each of which is divided intoThe mass and moment of inertia are respectively: m is0＝35kg,m1＝3kg,m2＝1.5kg；I0＝30kg·m2,I1＝2.7kg·m2,I2＝1.2kg·m2(ii) a The target satellite has a mass mP2kg, center inertia tensor IP＝0.8kg·m2；
In simulation, the velocity of the target satellite before the acquisition operation is assumed to be vx＝1m/s、vy1m/s and ωP1rad/s and the spacearm tip position has reached the capture position; after the capturing operation is finished, assuming that the tensors of the mass and the central inertia of the target satellite are unknown during control, and assuming that the initial values of the tensors are zero;
the expected track of the motion corner of the combined spacecraft system is assumed to be (unit: rad)
θ0d＝π/3+sin(πt/5)/2；θ1d＝-π/6+3sin(πt/5)/2；θ2d＝π/3+3sin(πt/5)/2
The initial value of motion is θ (0) ═ 1.2000.3061.217]T(rad), simulation time from completion of the capture operation: t is 10 s;
the control process ends.
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