Controller Design Method Based on Unity Quaternions and Modified Rodrigues Parameters

Abstract

The application discloses a controller design method based on unit quaternion and modified Rodrigues parameters, which comprises the following steps: S1, constructing a mathematical model of a rigid spacecraft attitude system; S2, constructing a mathematical model of a rigid spacecraft attitude error system based on modified Rodrigues parameters; S3, constructing a control target based on the attitude error; S4, constructing a disturbance observer and a Lyapunov function; S5, constructing an attitude error Lyapunov function according to an angular velocity error of the rigid spacecraft; S6, constructing an attitude system Lyapunov function according to the mathematical model of the rigid spacecraft attitude error system; and S7, constructing a moment controller according to Young inequality. The method can be used for designing a continuous controller for controlling a spacecraft, can complete asymptotic tracking of a trajectory, and can avoid occurrence of unwinding phenomenon.

Description

Technical Field

[0001] This invention relates to the field of rigid body spacecraft control technology, and in particular to a controller design method based on unit quaternions and modified Rodriguez parameters. Background Technology

[0002] Spacecraft attitude tracking control is a process designed to precisely adjust and stabilize a spacecraft from its initial stable attitude to a desired attitude. This control method has demonstrated high practical value in both civilian and military fields and has become a core attitude control mode in spacecraft operation, attracting widespread attention in the industry. Attitude dynamics are typically described using unit quaternions and modified Rodriguez parameters (MRP). However, in controller designs based on quaternions and MRP, decoupling is unavoidable. Decoupling refers to the spacecraft's initial attitude approaching the desired attitude, but instead of rotating towards it, it deviates from the desired attitude, leading to unnecessary energy consumption.

[0003] In view of this, this invention is hereby proposed. Summary of the Invention

[0004] The purpose of this invention is to address the shortcomings of existing technologies by proposing a controller design method based on unit quaternions and modified Rodriguez parameters. This method is used to design a continuous controller for controlling spacecraft, enabling asymptotic tracking of the trajectory and avoiding backtracking.

[0005] To achieve the above objectives, the present invention also employs the following technical solution:

[0006] A controller design method based on unit quaternions and modified Rodriguez parameters includes the following steps:

[0007] S1: Construct a mathematical model of the rigid body spacecraft attitude system based on unit quaternions, and obtain the equilibrium point of the mathematical model of the rigid body spacecraft attitude system.

[0008] S2: For the stable equilibrium point in the mathematical model of the rigid body spacecraft attitude system, construct a mathematical model of the rigid body spacecraft attitude error system based on the modified Rodrigues parameters.

[0009] S3: Construct a control objective based on attitude error;

[0010] S4: Construct the perturbation observer and the Lyapunov function;

[0011] S5: Construct the attitude error Lyapunov function based on the angular velocity error of the rigid body spacecraft;

[0012] S6: Construct the Lyapunov function of the attitude system based on the mathematical model of the attitude error system of the rigid body spacecraft;

[0013] S7: Construct a torque controller based on Young's inequality.

[0014] Furthermore, S1 includes the following steps:

[0015] S11: A mathematical model of the attitude system of a rigid body spacecraft is constructed based on unit quaternions. The model is as follows:

[0016]

[0017] In the formula, q represents the spacecraft's attitude, and q0 and q v Let q be the scalar and vector part, respectively; I be the identity matrix; J be the moment of inertia of the spacecraft; and ω be the angular velocity of the spacecraft in volume coordinates. It is a pure quaternion. For unit quaternion multiplication, τ is the torque of the spacecraft, d is the external disturbance experienced by the spacecraft, and S(q) is the torque of the spacecraft. v ) is q v skew-symmetric matrix;

[0018] S12: Obtain the equilibrium points of the rigid spacecraft attitude system constructed based on unit quaternions, including the unstable equilibrium point q = -1 and the stable equilibrium point q = +1.

[0019] Furthermore, S2 includes the following steps:

[0020] S21: Define the attitude error for the stable equilibrium point in the mathematical model of the attitude system of a rigid body spacecraft. Further, we can obtain Differentiating both sides of this expression, we get:

[0021]

[0022] remember ω e =ω-ω p ,in The above formula is written as in Construct the error-corrected Rodrigues vector p e as follows:

[0023]

[0024] In the formula, q e0 and q ev q e The scalar and vector parts, θ eLet n be the rotation angle error of the spacecraft. e For spacecraft Euler axes;

[0025] S22: Modified Rodriguez vector p e Taking the derivative, we obtain the error kinematics equations of the rigid body spacecraft expressed in terms of the modified Rodrigues vector, as follows:

[0026]

[0027] In the formula, ω e =ω-ω p For the angular velocity error of a rigid body spacecraft, ω p The desired angular velocity ω d In volume coordinates, p e For the attitude error of a rigid body spacecraft;

[0028] S23: The dynamic equations for the errors of a rigid body spacecraft in a volume coordinate system are as follows:

[0029]

[0030] S24: Construct a mathematical model for the attitude error system of a rigid body spacecraft. The model is as follows:

[0031]

[0032] Furthermore, step S3 includes the following steps:

[0033] S31: Construct a control objective based on the desired attitude, as follows:

[0034]

[0035] In the formula, q d The desired attitude of the rigid body spacecraft;

[0036] S32: Transform the control objective based on the desired attitude into a control objective based on the attitude error, as follows:

[0037]

[0038] Furthermore, step S4 includes the following steps:

[0039] S41: Construct the perturbation observer as follows:

[0040]

[0041] In the formula, L is the positive gain coefficient of the observer;

[0042] S42: Construct the Lyapunov function V0 and take its derivative. V0 has the following form:

[0043]

[0044] In the formula, κ is a constant, L>κ>0, where,

[0045]

[0046] Furthermore, step S5 includes the following steps:

[0047] S51: Construct the attitude error Lyapunov function V1 and take its derivative. V1 has the following form:

[0048]

[0049] S52: Constructing the angular velocity stabilization function ω α ω α The format is as follows:

[0050] ω α =-c1p e

[0051] In the formula, c1 is a constant, c1>0, and the following inequality holds.

[0052]

[0053] in,

[0054]

[0055] ω z =ω e -ω α

[0056] In the formula, S(p) e ) is p e I is a skew-symmetric matrix, where I is the identity matrix.

[0057] Furthermore, step S6 includes the following steps:

[0058] S61: Construct the Lyapunov function V2 of the attitude system and take its derivative. V2 has the following form:

[0059]

[0060] In the formula, γ is a constant, γ>0, and the following inequality holds.

[0061]

[0062] Furthermore, step S7 includes the following steps:

[0063] S71: Construct a torque controller based on Young's inequality, in the following form:

[0064]

[0065] In the formula, κ and c2 are constants, κ, c2 > 0, and the following inequality holds.

[0066]

[0067] in,

[0068]

[0069] In the formula, λ max (J) is the largest eigenvalue of J.

[0070] Compared with the prior art, the beneficial effects of this invention are as follows:

[0071] Using unit quaternions to model the spacecraft avoids singularity issues compared to the traditional Euler angles method, and can continuously and smoothly represent changes in the spacecraft's attitude. To address the decoupling phenomenon caused by controllers designed using unit quaternions, the attitude error equation is represented using modified Rodrigues parameter vectors, where the equilibrium point is located at the origin. A continuous controller is designed using modified Rodrigues parameter vectors, enabling the spacecraft to track its attitude while avoiding decoupling during tracking. Attached Figure Description

[0072] Figure 1 A flowchart illustrating the controller design method based on unit quaternions and modified Rodriguez parameters provided by this invention;

[0073] Figure 2 A coordinate system diagram of a rigid body spacecraft provided by this invention;

[0074] Figure 3 This is a schematic diagram of the spacecraft attitude dynamics model calculation module provided by the present invention;

[0075] Figure 4 The torque control module structure diagram provided by the present invention;

[0076] Figure 5 The control result experimental demonstration module provided by this invention;

[0077] Figure 6 The simulation results are for case 1.

[0078] Figure 7 The simulation results are for case 2. Detailed Implementation

[0079] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0080] Example 1:

[0081] A controller design method based on unit quaternions and modified Rodriguez parameters, such as... Figure 1-5 As shown, it includes the following steps:

[0082] S1: Construct a mathematical model of the rigid body spacecraft attitude system based on unit quaternions, and obtain the equilibrium point of the mathematical model of the rigid body spacecraft attitude system.

[0083] S2: For the stable equilibrium point in the mathematical model of the rigid body spacecraft attitude system, construct a mathematical model of the rigid body spacecraft attitude error system based on the modified Rodrigues parameters.

[0084] S3: Construct a control objective based on attitude error;

[0085] S4: Construct the perturbation observer and the Lyapunov function;

[0086] S5: Construct the attitude error Lyapunov function based on the angular velocity error of the rigid body spacecraft;

[0087] S6: Construct the Lyapunov function of the attitude system based on the mathematical model of the attitude error system of the rigid body spacecraft;

[0088] S7: Construct a torque controller based on Young's inequality.

[0089] In an optional embodiment, S1 includes the following steps:

[0090] S11: A mathematical model of the attitude system of a rigid body spacecraft is constructed based on unit quaternions. The model is as follows:

[0091]

[0092] In the formula, q represents the spacecraft's attitude, and q0 and q v Let q be the scalar and vector part, respectively; I be the identity matrix; J be the moment of inertia of the spacecraft; and ω be the angular velocity of the spacecraft in volume coordinates. It is a pure quaternion. For unit quaternion multiplication, τ is the torque of the spacecraft, d is the external disturbance experienced by the spacecraft, and S(q) is the torque of the spacecraft. v ) is q v skew-symmetric matrix;

[0093] Specifically, qv and S(q) v The details are as follows:

[0094] q v =(q v1 q v2 q v3 ) T

[0095]

[0096] S12: Obtain the equilibrium points of the rigid spacecraft attitude system constructed based on unit quaternions, including the unstable equilibrium point q = -1 and the stable equilibrium point q = +1.

[0097] In an optional embodiment, S2 includes the following steps:

[0098] S21: Define the attitude error for the stable equilibrium point in the mathematical model of the attitude system of a rigid body spacecraft. Further, we can obtain Differentiating both sides of this expression, we get:

[0099]

[0100] remember ω e =ω-ω p ,in The above formula is written as in Construct the error-corrected Rodrigues vector p e as follows:

[0101]

[0102] In the formula, q e0 and q ev q e The scalar and vector parts, θ e Let n be the rotation angle error of the spacecraft. e For spacecraft Euler axes;

[0103] S22: Modified Rodriguez vector p e Taking the derivative, we obtain the error kinematics equations of the rigid body spacecraft expressed in terms of the modified Rodrigues vector, as follows:

[0104]

[0105] In the formula, ω e =ω-ω p For the angular velocity error of a rigid body spacecraft, ω p The desired angular velocity ωd In volume coordinates, p e For the attitude error of a rigid body spacecraft;

[0106] S23: The dynamic equations for the errors of a rigid body spacecraft in a volume coordinate system are as follows:

[0107]

[0108] S24: Construct a mathematical model for the attitude error system of a rigid body spacecraft. The model is as follows:

[0109]

[0110] In an optional embodiment, step S3 includes the following steps:

[0111] S31: Construct a control objective based on the desired attitude, as follows:

[0112]

[0113] In the formula, q d The desired attitude of the rigid body spacecraft;

[0114] S32: Transform the control objective based on the desired attitude into a control objective based on the attitude error, as follows:

[0115]

[0116] In an optional embodiment, step S4 includes the following steps:

[0117] S41: Construct the perturbation observer as follows:

[0118]

[0119] In the formula, L is the positive gain coefficient of the observer;

[0120] S42: Construct the Lyapunov function V0 and take its derivative. V0 has the following form:

[0121]

[0122] In the formula, κ is a constant, L>κ>0, where,

[0123]

[0124] In an optional embodiment, step S5 includes the following steps:

[0125] S51: Construct the attitude error Lyapunov function V1 and take its derivative. V1 has the following form:

[0126]

[0127] S52: Constructing the angular velocity stabilization function ω α ω α The format is as follows:

[0128] ω α =-c1p e

[0129] In the formula, c1 is a constant, c1>0, and the following inequality holds.

[0130]

[0131] in,

[0132]

[0133] ω z =ω e -ω α

[0134] In the formula, S(p) e ) is p e skew-symmetric matrix I is the identity matrix.

[0135] Specifically, p e and S(p e The details are as follows:

[0136] p e =(p e1 p e2 p e3 ) T

[0137]

[0138] In an optional embodiment, step S6 includes the following steps:

[0139] S61: Construct the Lyapunov function V2 of the attitude system and take its derivative. V2 has the following form:

[0140]

[0141] In the formula, γ is a constant, γ>0, and the following inequality holds.

[0142]

[0143] In an optional embodiment, step S7 includes the following steps:

[0144] S71: Construct a torque controller based on Young's inequality, in the following form:

[0145]

[0146] In the formula, κ and c2 are constants, κ, c2 > 0, and the following inequality holds.

[0147]

[0148] in,

[0149]

[0150] In the formula, λ max (J) is the largest eigenvalue of J.

[0151] This embodiment presents a controller design method based on unit quaternions and modified Rodrigues parameters. It uses unit quaternions to model the spacecraft, avoiding singularity issues compared to the traditional Euler angle method, and can continuously and smoothly represent changes in spacecraft attitude. To address the decoupling phenomenon that can occur with controllers designed using unit quaternions, the attitude error equation is represented using modified Rodrigues parameter vectors, where the equilibrium point is located at the origin. A continuous controller is designed using modified Rodrigues parameter vectors, enabling the spacecraft to track attitude while avoiding decoupling during tracking.

[0152] To demonstrate the effectiveness of the controller designed using the controller design method based on unit quaternions and modified Rodriguez parameters in this embodiment, simulation verification was performed on the designed controller.

[0153] In the simulation, the moment of inertia of the rigid body spacecraft is as follows:

[0154]

[0155] The external disturbances are as follows:

[0156] d=(0.01sin(0.05t),0.005sin(0.05t),-0.01cos(0.05t)) T

[0157] Controller parameters: c1 = 0.04, c2 = 0.05, γ = 5, L = 100

[0158] The spacecraft's initial attitude is as follows:

[0159] Scenario 1:

[0160] Scenario 2:

[0161] Cases 1 and 2 correspond to two unit quaternions in the three-dimensional physical space of the initial attitude.

[0162] The initial angular velocity error of the spacecraft is as follows:

[0163] ω0=[0,0,0] T

[0164] The desired attitude of the spacecraft is as follows:

[0165]

[0166] To achieve signal ω d and Introduce the following instruction filtering:

[0167]

[0168] The superscript 'c' indicates the signal after processing by the instruction filter.

[0169] From the expression for angular velocity, we can obtain:

[0170]

[0171] Introduce the following instruction filtering:

[0172]

[0173] In the controller Can be replaced with Where k1 = k2 = 100.

[0174] Simulations were performed based on the above simulation conditions, and the simulation results for scenarios 1 and 2 are as follows: Figure 6 , 7 As shown.

[0175] Figure 6 and Figure 7 The simulation results in the upper left corner describe the change of the unit quaternion error between the spacecraft's attitude and the desired attitude over time. It can be seen that from approximately 10 seconds onwards, q tracks q... d . Figure 6 and Figure 7 The simulation results in the upper right corner show the change of the magnitude of the corrected Rodrigues parameter error vector over time, indicating that p e ≤1 indicates that the spacecraft avoids unwinding during the tracking process. Figure 6 and Figure 7 The simulation results in the lower left corner show the changes of the three components of the angular velocity error over time. The figure shows that after the 10th second, ω tracks ω... p . Figure 6 and Figure 7The simulation results in the lower right corner of the image show the change of the torque control modulus over time.

[0176] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A controller design method based on unit quaternion and modified Rodrigues parameters, characterized in that, It comprises the following steps: S1: constructing a mathematical model of a rigid spacecraft attitude system based on a unit quaternion, and obtaining an equilibrium point of the mathematical model of the rigid spacecraft attitude system; S2: for a stable equilibrium point in the mathematical model of the rigid spacecraft attitude system, constructing a mathematical model of a rigid spacecraft attitude error system based on a modified Rodrigues parameter; S3: constructing a control target based on an attitude error; S4: constructing a disturbance observer, and constructing a Lyapunov function; S5: constructing an attitude error Lyapunov function according to an angular velocity error of the rigid spacecraft; S6: constructing an attitude system Lyapunov function according to the mathematical model of the rigid spacecraft attitude error system; S7: constructing a moment controller according to a Young inequality; The S1 comprises the following steps: S11: constructing a mathematical model of a rigid spacecraft attitude system based on a unit quaternion, and the model is as follows: In the formula, q represents the spacecraft's attitude, and q0 and q v Let I be the scalar and vector part of q, respectively; let J be the identity matrix; let ω be the spacecraft's moment of inertia; and let ω be the spacecraft's angular velocity in volume coordinates. It is a pure quaternion. For unit quaternion multiplication, τ is the torque of the spacecraft, d is the external disturbance experienced by the spacecraft, and S(q) is the torque of the spacecraft. v ) is q v skew-symmetric matrix; S12: obtaining an equilibrium point of the rigid spacecraft attitude system based on the unit quaternion, including an unstable equilibrium point q=-1 and a stable equilibrium point q=+1; The S2 comprises the following steps: S21: For the stable equilibrium point in the mathematical model of the rigid spacecraft attitude system, define the attitude error Further available The derivative of both sides of the equation is available: Recall ω e = ω - ω p where The above equation is written as where The construction of the error correction Rodrigues vector p e is as follows: In the formula, q e0 and q ev are the scalar part and the vector part of q e , respectively, θ e is the rotation angle error of the spacecraft, and n e is the Euler axis of the spacecraft. The S4 comprises the following steps: S41: constructing a disturbance observer, as follows: In the formula, r is a state variable of the observer, and L is a positive gain coefficient of the observer; S42: constructing a Lyapunov function V0, and deriving it, and the form of V0 is as follows: In the formula, κ is a constant, L>κ>0, wherein, The S5 comprises the following steps: S51: constructing an attitude error Lyapunov function V1, and deriving it, and the form of V1 is as follows: S52: Construct the angular velocity stabilizing function ω α , ω α is of the form: ω α = -c1p e In the formula, c1 is a constant, c1>0, and the following inequality is established Wherein, ω z = ω e - ω α In the formula, S(p e ) is a skew-symmetric matrix of p e , and I is a unit matrix. The S6 comprises the following steps: S61: constructing an attitude system Lyapunov function V2, and deriving it, and the form of V2 is as follows: In the formula, γ is a constant, γ>0, and the following inequality is established 2. The controller design method based on unit quaternion and modified Rodrigues parameters according to claim 1, wherein, The S2 further comprises the following steps: S22: Derive the error kinematics equation of the rigid spacecraft represented by the modified Rodrigues vector p e S22: Derive the error kinematics equation of the rigid spacecraft represented by the modified Rodrigues vector p In the formula, ω e = ω - ω p is the angular velocity error of the rigid spacecraft, ω p is the desired angular velocity ω d In the body coordinate system, p e is the constructed modified Rodrigues vector; S23: constructing a dynamic equation of a rigid spacecraft error in a body coordinate system, as follows: S24: constructing a mathematical model of a rigid spacecraft attitude error system, and the model is as follows:

3. The controller design method based on unit quaternion and modified Rodrigues parameters according to claim 2, wherein, The S3 comprises the following steps: S31: constructing a control target based on an expected attitude, as follows: In the formula, q d is the desired attitude of the rigid spacecraft; S32: converting the control target based on the expected attitude to a control target based on an attitude error, as follows:

4. The controller design method based on unit quaternion and modified Rodrigues parameters according to claim 1, wherein, The S7 comprises the following steps: S71: constructing a moment controller according to a Young inequality, and the form is as follows: In the formula, κ and c2 are constants, κ,c2>0, and the following inequality is established Wherein, In the formula, λ max (J) is the largest eigenvalue of J.

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