An underactuated ship trajectory tracking control method

Abstract

The application discloses a kind of underactuated ship trajectory tracking control methods, comprising: ship model is constructed into the underactuated ship three degrees of freedom mathematical model containing specific environmental disturbance;Based on the ship model constructed, combined with acceleration measurement technology, design nonlinear state observer;Position error and speed relationship are obtained from nonlinear state observer, virtual speed control law is designed, and the speed error between expected speed and actual speed is obtained;Design the speed error terminal sliding surface based on hyperbolic tangent function and the external disturbance estimation value based on adaptive technology;Based on speed error terminal sliding surface, adaptive disturbance estimation value, and combined with backstepping method, calculate terminal sliding output control amount;Terminal sliding output control amount is used to design final control force and control moment with the feedforward compensation of acceleration estimation value, and ship trajectory tracking is realized.The underactuated ship trajectory can be efficiently and accurately tracked by using the application.

Description

Technical Field

[0001] This invention belongs to the field of ship trajectory tracking control, and in particular relates to an underactuated ship trajectory tracking control method. Background Technology

[0002] As sea ice coverage decreases, shipping activities in polar regions are expanding. However, due to the complex polar environment, ships must navigate along ideal, predetermined trajectories and at expected speeds in icy conditions to ensure safe navigation. Currently, underactuated ships are widely used in practical applications due to their advantages such as low cost, simple maintenance, and high adaptability. Therefore, researching the trajectory tracking capabilities of underactuated systems in complex icy environments has significant practical implications.

[0003] Many researchers have been exploring various control methods to solve trajectory tracking control problems, such as adaptive control, neural network control and model predictive control. Among them, sliding mode control has been widely studied due to its advantages of strong robustness, fast response speed and high accuracy.

[0004] For example, Chinese patent document CN119270848A discloses an adaptive sliding mode ship trajectory tracking control method based on output redefinition, including: S1: establishing an underactuated ship mathematical model and redefining its position; S2: designing the position tracking error and differentiating it; S3: designing a first linear sliding surface and a second linear sliding surface based on the position tracking error and the differentiated tracking error; S4: designing the ship's longitudinal control force and yaw control torque based on the first linear sliding surface and the second linear sliding surface and combined with adaptive technology; S5: adjusting the position tracking error through the longitudinal control force and yaw control torque to achieve convergence of the position tracking error, thereby realizing trajectory tracking.

[0005] Chinese patent document CN118051055A discloses a method and system for trajectory tracking control of underactuated ships based on identification modeling. The method includes designing an identification model, an improved extended state observer, and a trajectory tracking controller. The system comprises: S1. Determining the three-degree-of-freedom model structure through mechanistic analysis based on the motion characteristics of the underactuated ship, and identifying the model parameters using an extended Kalman filter algorithm based on ship navigation data to obtain the identification model; S2. Treating the model identification error and environmental disturbance error as the total disturbance, designing an improved extended state observer to estimate the total system disturbance; S3. Designing the trajectory tracking controller based on sliding mode control theory, employing an exponential reaching law and a linear sliding surface according to the complexity of the ship control task, and incorporating disturbance compensation into the controller design to obtain the trajectory tracking controller.

[0006] Currently, traditional sliding mode algorithms in ship motion control are widely used due to their fast performance. Sliding mode control introduces a sliding surface to guide the system state onto the desired trajectory. Further research has found that terminal sliding mode control algorithms have fast convergence speed and strong robustness, but they often suffer from problems such as chattering and singularities. Meanwhile, due to environmental interference and limitations of measuring instruments, high-frequency loads and measurement noise entering the feedback loop can cause wear on the propeller. Summary of the Invention

[0007] This invention provides an underactuated ship trajectory tracking control method, which designs an observer based on acceleration measurement to estimate the ship's state variables; combines sliding mode technology, adaptive technology, backstepping technology and acceleration feedforward to design a controller, outputs a resultant control force, and achieves efficient and accurate tracking of the underactuated ship trajectory.

[0008] A method for tracking and controlling the trajectory of an underactuated ship, characterized in that it includes:

[0009] (1) Construct the ship model into a three-degree-of-freedom mathematical model of an underactuated ship that includes specific environmental disturbances;

[0010] (2) Based on the constructed ship model, a nonlinear state observer is designed using acceleration measurement technology;

[0011] (3) Obtain the relationship between position error and velocity from the nonlinear state observer, design a virtual velocity control law, and obtain the velocity error between the desired velocity and the actual velocity;

[0012] (4) Design the velocity error terminal sliding surface based on the hyperbolic tangent function and the external disturbance estimation based on adaptive technology;

[0013] (5) Calculate the terminal sliding control quantity based on the speed error terminal sliding surface and the adaptive disturbance estimation value, combined with the backstepping method;

[0014] (6) The final control resultant force and control resultant torque are designed by using the feedforward compensation of the acceleration estimate for the terminal slid output control quantity, so as to realize ship trajectory tracking.

[0015] In step (1), the mathematical model of the underactuated ship with three degrees of freedom, including specific environmental disturbances, is expressed as follows:

[0016]

[0017] τ E =τ wind +τ current +τ ice

[0018] Where η = [x, y, ψ] TThis represents the ship's position and yaw angle in a fixed coordinate system; V = [u, v, r] T u represents the longitudinal velocity of the ship in body coordinates, v represents the sway velocity of the ship in body coordinates, and r represents the bow roll velocity of the ship in body coordinates; τ = [τ u ,0,τ r ] T τ represents the underactuated ship control input. u τ represents the oscillation control force. r τ represents the bow roll torque. E ∈R 3 This represents time-varying environmental disturbances, specifically including wind load τ. wind Flow load τ current and ice load τ ice R(η) is the transformation matrix between coordinate systems and satisfies R -1 (η)=R T (η); M represents the inertial parameter matrix; C(V) represents the Coriolis and centripetal force matrix; D represents the damping parameter matrix.

[0019] The specific expressions for R(η), M, C(V), and D are:

[0020]

[0021] Where ψ is the yaw angle of the ship in a fixed coordinate system; m is the mass of the ship; I z For the z-axis of the ship o Moment of inertia; X u ,Y v N r The linear hydrodynamic damping coefficient; A mass coefficient is added to the linear fluid.

[0022] Wind load τ wind The expression is:

[0023]

[0024] In the formula, ρ a C is the density of air. X (r rω ) represents the longitudinal wind pressure coefficient; C Y (r rω ) represents the transverse wind pressure coefficient; C N (r rω A is the wind pressure moment coefficient about the vertical axis; Fω A is the projected area of ​​the ship above the waterline. Lω L is the side projection area of ​​the ship above the waterline. oa V is the horizontal distance from the center of the ship's orthographic projection above the waterline to the origin of the coordinate system.rω It is relative wind speed; r rω It is relative to the angle of attack;

[0025] Flow load τ current The expression is:

[0026] τ current =C(V) bc V bc +DV bc

[0027]

[0028] In the formula, V bc =[u c ,v c ,0] T u c ,v c V represents the longitudinal sway velocity and transverse sway velocity of the ocean current in the volume coordinate system, respectively; c ,β c These represent the ocean current speed and direction in a fixed coordinate system; ψ represents the yaw angle.

[0029] Ice load τ ice The expression is as follows:

[0030]

[0031] In the formula, ω ice Zero-mean white noise; Ω and ∑ are positive gain matrices; τ ice For ice load.

[0032] In step (2), the nonlinear state observer is represented as:

[0033]

[0034] In the formula, This indicates the estimated position and course of the vessel; This represents the estimated values ​​of linear velocity and angular velocity; Indicates position and heading tracking error; Represents the observed acceleration; K1,x1,χ2∈R 3 Here is the diagonal matrix to be designed; fal is a continuous power function, specifically expressed as:

[0035]

[0036] In the formula, ι is a defined variable, α∈(0,1) is a nonlinear parameter, and δ>0 is the length of the linear segment interval.

[0037] Observational acceleration The expression is:

[0038]

[0039] In the formula, a is the measured acceleration, and χ3∈R 3 It is a diagonal matrix.

[0040] In step (3), the relationship between position error and velocity is obtained by the nonlinear state observer, specifically as follows:

[0041]

[0042] In the formula, x e ,y e These represent the lateral position error and the longitudinal position error, respectively; u g ,v g The observed values ​​of u and v are respectively; x d ,y d These represent the horizontal and vertical positions of the desired trajectory, respectively.

[0043] In step (3), the expression for the virtual speed control law is designed as follows:

[0044]

[0045] In the formula, c1, c2, c3, c4, C, e i ,β 1i ,β 2i ,t αi i = 1, 2 are the parameters that need to be designed; u d ,v d These represent the designed virtual velocities; if u d ≠u g ,v d ≠v g The speed error is:

[0046]

[0047] In the formula, u e v represents the error between the observed oscillation velocity and the designed virtual oscillation velocity. e This represents the error between the observed sway velocity and the designed virtual sway velocity.

[0048] In step (4), the velocity error terminal sliding surface based on the hyperbolic tangent function is expressed as:

[0049]

[0050] In the formula, s u The terminal sliding surface represents the oscillation velocity error, s vThe terminal sliding surface represents the sway velocity error, k1,k3,k4>0, p,q are design values, and 1 <p<2,1<q<2。

[0051] In step (4), the estimated external disturbance based on adaptive technology is expressed as follows:

[0052]

[0053] In the formula, η1,η2,σ1,σ2,d1,d2,κ are the design values. This represents an estimated value for the disturbance limit in the oscillation direction. This represents an estimated value of the disturbance limit in the roll direction.

[0054] In step (6), the final control resultant force and control resultant torque are designed by using feedforward compensation of the acceleration estimate for the terminal sliding diaphragm output control quantity. The expression is:

[0055]

[0056] In the formula, τ u ,τ r These are the resultant control force and resultant torque obtained by combining acceleration feedforward, τ and τ', respectively. u_sm ,τ r_sm This represents the control force and control torque obtained by the backstepping method. and For low-frequency acceleration estimation The amount.

[0057] Compared with the prior art, the present invention has the following beneficial effects:

[0058] 1. The technology of this invention enables underactuated vessels to track a preset trajectory under external disturbance conditions. By introducing an integral terminal sliding control method, the system can respond quickly to external changes and reach and maintain the preset trajectory more rapidly.

[0059] 2. Compared with the traditional terminal sliding mode method, the present invention can reduce the shortcomings of the sign function and improve the system chattering problem by using the hyperbolic tangent function.

[0060] 3. Compared with traditional sliding mode control methods, the technology of this invention is more robust and can effectively handle system nonlinearity and uncertainty.

[0061] 4. The technology of this invention is applicable to different types of ships and various environmental conditions, and has broad application potential in various fields. Attached Figure Description

[0062] Figure 1 This is a flowchart of an underactuated ship trajectory tracking control method according to an embodiment of the present invention.

[0063] Figure 2 This is a mathematical model diagram of an underactuated ship with three degrees of freedom according to the present invention.

[0064] Figure 3 This is a simulated wind load diagram of the present invention.

[0065] Figure 4 This is the simulated flow load diagram of the present invention.

[0066] Figure 5 This is a position curve variation diagram of the present invention.

[0067] Figure 6 This is a graph showing the change in the position error curve of the present invention.

[0068] Figure 7 This is a graph showing the speed curve variation of the present invention.

[0069] Figure 8 This is a control input variation diagram of the present invention. Detailed Implementation

[0070] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be noted that the embodiments described below are intended to facilitate the understanding of the present invention and do not constitute any limitation thereof.

[0071] like Figure 1 As shown, an underactuated ship trajectory tracking control method mainly includes: establishing a ship dynamics model and disturbances, designing a state observer based on acceleration measurement, designing a virtual control law, designing a sliding surface, and calculating the control input.

[0072] Specifically, it includes the following steps:

[0073] S1, construct the ship model into a three-degree-of-freedom mathematical model of an underactuated ship that includes specific environmental disturbances.

[0074] like Figure 2 As shown, the mathematical model of the three-degree-of-freedom underactuated ship in this invention is described as follows:

[0075]

[0076] In the formula, η = [x, y, ψ] T This represents the ship's position and yaw angle in a fixed coordinate system; V = [u, v, r] T Represents linear velocity and angular velocity in body coordinates; τ = [τ u ,0,τ r ] T Indicates the underactuated ship control input; τ E ∈R 3 This represents time-varying environmental disturbances, specifically including wind load τ. wind Flow load τcurrent and ice load τ ice R(η) is the transformation matrix between coordinate systems and satisfies R -1 (η)=R T (η); M represents the inertial parameter matrix; C(V) represents the Coriolis and centripetal force matrices; D(V) represents the damping parameter matrix, and their specific expressions are shown below:

[0077]

[0078] The time-varying disturbance model in this invention is described as follows:

[0079]

[0080] In the formula, ρ a C is the density of air. X (r rω ) represents the longitudinal wind pressure coefficient; C Y (r rω ) represents the transverse wind pressure coefficient; C N (r rω A is the wind pressure moment coefficient about the vertical axis; Fω A is the projected area of ​​the ship above the waterline. Lω L is the side projection area of ​​the ship above the waterline. oa V is the horizontal distance from the center of the ship's orthographic projection above the waterline to the origin of the coordinate system. rω It is relative wind speed; r rω It is the relative angle of attack; τ wind It is wind load. For example... Figure 3 The diagram shown is a simulated wind load diagram of the present invention.

[0081]

[0082] τ current =C(V) bc V bc +DV bc (9)

[0083] In the formula, V c ,β c It refers to the ocean current velocity and direction in a fixed coordinate system; V bc =[u c ,v c ,0] T It is the component of velocity in the volume coordinate system; τ current This is due to ocean current loads. For example... Figure 4 The diagram shown is a simulated flow load diagram of the present invention.

[0084]

[0085] In the formula, ω ice Zero-mean white noise; Ω and ∑ are positive gain matrices; τ ice For ice load.

[0086] S2, based on the constructed ship model, combines acceleration measurement technology to design a nonlinear state observer.

[0087] The ship nonlinear observer dynamics model in this invention is described as follows:

[0088]

[0089] In the formula, This indicates the estimated position and course of the vessel; This represents the estimated values ​​of linear velocity and angular velocity; Indicates position and heading tracking error; Represents the observed acceleration; χ1,χ2∈R 3 It is a diagonal matrix; fal is a continuous power function, specifically expressed as:

[0090]

[0091] Where α∈(0,1) is a nonlinear parameter; δ>0 is the length of the linear segment interval.

[0092] In this invention, the low-frequency estimated acceleration based on acceleration measurement is described as follows:

[0093]

[0094] In the formula, χ3∈R 3 is a diagonal matrix, where a is the measured acceleration.

[0095] S3 obtains the relationship between position error and velocity from a nonlinear state observer, designs a virtual velocity control law, and obtains the velocity error between the desired velocity and the actual velocity.

[0096] In this invention, the relationship between the defined position error variable and the velocity observation value is as follows:

[0097]

[0098] In the formula, x e ,y e These represent the lateral position error and the longitudinal position error, respectively; x d ,y d These represent the horizontal and vertical positions of the desired trajectory, respectively.

[0099] The virtual speed control law designed in this invention is described as follows:

[0100]

[0101] In the formula

[0102]

[0103] In the formula

[0104]

[0105] c1,c2,c3,c4,C, e i ,β 1i ,β 2i ,t αi i = 1, 2 are the parameters that need to be designed.

[0106] Taking the derivative of the virtual velocity control law, we get:

[0107]

[0108] If u d ≠u g ,v d ≠v g The speed error can be obtained as:

[0109]

[0110] Simultaneously, the first-order differential of the longitudinal velocity error and the second-order differential of the lateral velocity can be obtained as follows:

[0111]

[0112] In the formula, f is represented as:

[0113]

[0114] S4, designing a velocity error terminal sliding surface based on the hyperbolic tangent function and an external disturbance estimate based on adaptive technology.

[0115] In this invention, the velocity error terminal sliding surface s u With s v They are designed as follows:

[0116]

[0117] In the formula, k1,k3,k4>0, p,q are design values ​​and 1 <p<2,1<q<2。

[0118] For the speed error terminal sliding surface s u With s v Differentiating each, we get:

[0119]

[0120] in

[0121]

[0122] make We can obtain:

[0123]

[0124] In this invention, the longitudinal estimate of external disturbances is... Designed as follows:

[0125]

[0126] Where η1,σ1,d1,κ are the parameters that need to be designed.

[0127] S5 calculates the terminal sliding diaphragm output control quantity based on the speed error terminal sliding diaphragm surface and the adaptive disturbance estimation value, combined with the backstepping method.

[0128] In this invention, the longitudinal switching control law is designed as follows:

[0129]

[0130] In this invention, the longitudinal thrust in the terminal sliding membrane control is designed as follows:

[0131]

[0132] make We can obtain:

[0133]

[0134] In this invention, the estimated value of the bow-to-external disturbance is... Designed as follows:

[0135]

[0136] Among them, η2, σ2, d2 are the parameters that need to be designed.

[0137] In this invention, the steering switching control law is designed as follows:

[0138]

[0139] In this invention, the steering torque in the terminal sliding control is designed as follows:

[0140]

[0141] S6 uses feedforward compensation of acceleration estimates to design the final control resultant force and control resultant torque for the terminal slid output control quantity, thereby achieving ship trajectory tracking.

[0142] In this invention, the resultant force τ is controlled by combining acceleration feedforward design. u With resultant torque τ r for:

[0143]

[0144] In the formula, and For low-frequency acceleration estimation The amount.

[0145] The following two theorems will be proved:

[0146] Theorem 1: For any given real number, the following inequality holds:

[0147]

[0148] Where ε is the design value.

[0149] Theorem 2: For any scalar, the Lyapunov condition for practical finite-time stability is:

[0150]

[0151] Wherein, the settling time T r Bounded and satisfying:

[0152]

[0153] Constructing Lyapunov functions:

[0154]

[0155] Taking the derivative with respect to V1, we get:

[0156]

[0157] In the formula

[0158]

[0159] Then, by applying the stability criterion, we can conclude that the position tracking error tends to be stable.

[0160] make

[0161]

[0162] Constructing Lyapunov functions:

[0163]

[0164] Taking the derivative of V2, we get:

[0165]

[0166]

[0167] In the formula

[0168]

[0169] Therefore, based on the stability criterion, the velocity error and disturbance error are ultimately bounded. To verify the effectiveness of the composite controller designed in this invention, simulation experiments were conducted. The selected ship model parameters are:

[0170] m 11 =1.2×10 5 m 22 =1.779×10 5 m 33 =6.36×10 7

[0171] d 11 =2.15×10 4 d 22 =1.47×10 5 d 33 =8.02×10 6

[0172] The selected environmental interference-related parameters are:

[0173] V w =20, β w =30°, V c =0.3, β w =20°

[0174] To track a circular trajectory, the circular trajectory is defined as follows:

[0175]

[0176] The relevant parameters for the observer and controller in this invention are designed as follows:

[0177] X1=[15,15,15], χ2=[20,20,20], e i =1.2, β 1i =0.5,β 2i =0.5, i=1,2,η1=3×10 4 η2=3×10 4 σ1=3×10-9 σ² = 3 × 10 -9 , κ=0.2785, d1=0.2, d2=0.02, ω1=1×10 4 ω2=1×10 4 ρ1=2×10 4 ρ2=2×10 4 .

[0178] Simulation results are as follows Figures 5-8 As shown, Figure 5 This is a position curve change diagram of the present invention. It can be seen from the diagram that the actual position of the ship gradually approaches the target trajectory and remains stable, and the steady-state error is relatively ideal. Figure 6 This is a position error curve variation diagram of the present invention. It can be seen from the figure that the ship's position tracking error gradually approaches 0, accompanied by a slight fluctuation around 380s, but remains stable overall. Figure 5 and Figure 6 This demonstrates that despite uncertainties and interference, the ship's trajectory gradually tends to the preset trajectory, the tracking error converges to zero, and trajectory tracking control is achieved, thus proving the effectiveness of the invention. Figure 7 This is a graph showing the speed curve variation of the present invention. Figure 8 This is a control input variation diagram of the present invention. The control signal is calculated by an algorithm to obtain the corresponding control force and control torque.

[0179] The embodiments described above provide a detailed explanation of the technical solutions and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, additions, and equivalent substitutions made within the scope of the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for tracking and controlling the trajectory of an underactuated ship, characterized in that, include: (1) The ship model is constructed as a three-degree-of-freedom mathematical model of an underactuated ship that includes specific environmental disturbances, and is expressed as: in, This indicates the ship's position and yaw angle in a fixed coordinate system; , This represents the longitudinal velocity of the ship in body coordinates. This represents the sway linear velocity of the ship in body coordinates. This represents the bow roll rate of the ship in body coordinates. Indicates the control input for underactuated ships. Indicates oscillation control. Indicates the bow roll torque; This indicates time-varying environmental disturbances, specifically including wind loads. Flow load and ice load ; Let be the transformation matrix between coordinate systems and satisfy... M represents the inertial parameter matrix; Represents the Coriolis and centripetal force matrices; Represents the damping parameter matrix; The specific expression is: in, This is the yaw angle of the ship in a fixed coordinate system; For the mass of the hull; For the axis of the ship The moment of inertia; The linear hydrodynamic damping coefficient; Add a mass coefficient to the linear fluid; (2) Based on the constructed ship model, a nonlinear state observer is designed using acceleration measurement technology; the nonlinear state observer is expressed as: In the formula, This indicates the estimated position and course of the vessel; This represents the estimated values ​​of linear velocity and angular velocity; Indicates position and heading tracking error; Indicates the observed acceleration; The diagonal matrix to be designed; It is a continuous power function, specifically represented as: In the formula, To define variables, It is a nonlinear parameter; It is the length of the linear segment interval; (3) Obtain the relationship between position error and velocity from the nonlinear state observer, design a virtual velocity control law, and obtain the velocity error between the desired velocity and the actual velocity; (4) Design the velocity error terminal sliding surface based on the hyperbolic tangent function and the external disturbance estimation value based on adaptive technology; (5) Calculate the terminal sliding control quantity based on the speed error terminal sliding surface and the adaptive disturbance estimate, combined with the backstepping method; (6) The final control resultant force and control resultant torque are designed by using the feedforward compensation of the acceleration estimate for the terminal slid output control quantity, so as to realize ship trajectory tracking.

2. The underactuated ship trajectory tracking control method according to claim 1, characterized in that, Wind load The expression is: In the formula, air density; This refers to the longitudinal wind pressure coefficient. This refers to the transverse wind pressure coefficient; The wind pressure moment coefficient about the vertical axis; The projected area of ​​the ship above the waterline; The side projection area of ​​the ship above the waterline; The horizontal distance from the origin of the coordinate system to the center of the orthographic projection of the ship above the waterline. It is a relative wind speed; It is relative to the angle of attack; Flow load The expression is: In the formula, , These represent the longitudinal sway velocity and the transverse sway velocity of the ocean current in the volume coordinate system, respectively. These are the ocean current speed and direction in a fixed coordinate system; Indicates the yaw angle. Ice load The expression is as follows: In the formula, Zero-mean white noise; It is a positive gain matrix; For ice load.

3. The underactuated ship trajectory tracking control method according to claim 1, characterized in that, Observational acceleration The expression is: In the formula, To measure acceleration, It is a diagonal matrix.

4. The underactuated ship trajectory tracking control method according to claim 1, characterized in that, In step (3), the relationship between position error and velocity is obtained by the nonlinear state observer, specifically as follows: In the formula, These represent the lateral position error and the longitudinal position error, respectively. They are respectively Observed values; These represent the horizontal and vertical positions of the desired trajectory, respectively.

5. The underactuated ship trajectory tracking control method according to claim 4, characterized in that, In step (3), the expression for the virtual speed control law is designed as follows: In the formula, These are parameters that need to be designed; These represent the designed virtual speeds, if The speed error is: In the formula, This represents the error between the observed sway velocity and the designed virtual sway velocity. This represents the error between the observed sway velocity and the designed virtual sway velocity.

6. The underactuated ship trajectory tracking control method according to claim 5, characterized in that, In step (4), the velocity error terminal sliding surface based on the hyperbolic tangent function is expressed as: In the formula, The terminal sliding surface represents the oscillation velocity error. Indicates the sway velocity error at the terminal sliding surface. This is the design value, and .

7. The underactuated ship trajectory tracking control method according to claim 6, characterized in that, In step (4), the estimated external disturbance based on adaptive technology is expressed as follows: In the formula, For design values, This represents an estimated value for the disturbance limit in the oscillation direction. This represents an estimated value of the disturbance limit in the roll direction.

8. The underactuated ship trajectory tracking control method according to claim 7, characterized in that, In step (6), the final control resultant force and control resultant torque are designed by using feedforward compensation of the acceleration estimate for the terminal sliding diaphragm output control quantity. The expression is: In the formula, These are the resultant control force and resultant torque obtained by combining acceleration feedforward, respectively. This represents the control force and control torque obtained by the backstepping method. and For low-frequency acceleration estimation The amount.

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