Flight control method and system for tilt rotorcraft in hovering or low-speed variable pitch attitude and storage medium

Abstract

The invention discloses a flight control method and system for a tilt rotorcraft in hovering or low-speed variable pitch attitude and a storage medium, and the method comprises the following steps: converting a ground coordinate system and a body coordinate system into a dynamic reference coordinate system through constructing a conversion matrix; various parameters under the ground coordinate system and the body coordinate system are converted to be under the dynamic reference coordinate system, and expected manipulated variables of the position and the attitude of the tilt rotorcraft are calculated according to the parameters; according to the invention, variable pitch attitude control of the tilt rotorcraft is realized, and variable attitude take-off and landing, hovering and low-speed flight can be realized.

Description

Technical Field

[0001] The invention relates to flight control of a tiltrotor aircraft, and in particular to a flight control method, system and storage medium for a tiltrotor aircraft when it is hovering or changing its pitch attitude at a low speed. Background Art

[0002] Tilt-rotor UAV is a rotorcraft with unique performance. It combines the vertical take-off and landing and hovering capabilities of ordinary helicopters with the high-speed cruising flight capabilities of turboprop aircraft. This performance enables tilt-rotor UAVs to have a variety of unique flight modes. Therefore, researching and designing these new flight modes can expand the flight envelope of tilt-rotor aircraft, reveal the flight mechanism, and improve the application capabilities of tilt-rotor aircraft.

[0003] The take-off and hovering process of the tilt-rotor aircraft demonstrate the unique characteristics and advantages of this type of aircraft, and are essential processes for the tilt-rotor aircraft to perform mission operations. The conventional method for implementing the above process is to take off in helicopter mode on horizontal ground, and then switch to fixed-wing mode to achieve horizontal flight. The take-off process of the conventional method is that the tilt-rotor aircraft takes off in helicopter mode, rises to a certain point and hovers; at the hovering point, the tilt-rotor aircraft begins to tilt the nacelle forward to switch modes and continue to fly; finally, the nacelle completes a 90° tilt, at which time the tilt-rotor aircraft completes the switch from helicopter mode to fixed-wing mode; but in slope take-off and landing, that is, when the nacelle mechanism and the fuselage are in a non-vertical state to perform vertical take-off and landing and switch to horizontal flight, the conventional take-off method cannot be well adapted, and the traditional method can only keep the aircraft in a fixed trim attitude when hovering and flying forward at a low speed, which limits its application flexibility. Therefore, the present invention overcomes the above shortcomings. Summary of the invention

[0004] Purpose of the invention: The purpose of the present invention is to provide a flight control method, system and storage medium for a tilt-rotor aircraft to hover or change its pitch attitude at a low speed, which can realize the tilt-rotor aircraft to change its attitude and take off and landing, hover and fly at a low speed.

[0005] Technical solution: The flight control method for a tiltrotor aircraft when it is hovering or changing its pitch attitude at a low speed according to the present invention comprises the following steps:

[0006] S1, constructing a first transformation matrix from the ground coordinate system to the body coordinate system;

[0007] S2, constructing a second transformation matrix for body coordinate system transformation according to the desired state of the body;

[0008] S3, constructing a third conversion matrix for converting the ground coordinate system into the dynamic reference coordinate system according to the first conversion matrix and the second conversion matrix;

[0009] S4. Convert various parameters in the ground coordinate system and the body coordinate system to the dynamic reference coordinate system, and calculate the expected control amount of the position and attitude of the tilt-rotor aircraft based on the dynamic reference coordinate system.

[0010] Because the attitude and position data of the tiltrotor measured by the sensors on the tiltrotor are only completed in the body coordinate system, when the tiltrotor is on a slope, the measured data can only be used in the body coordinate system, such as the pitch angle. The pitch angle measured by the sensors on the tiltrotor is the slope angle. When designing the control law, the first conversion matrix and the second conversion matrix are used to construct the third conversion matrix, and various parameters are converted to the corrected dynamic reference coordinate system. In the initial state, the three-dimensional coordinate axes of the dynamic reference coordinate system are in the same direction as the three-dimensional coordinate axes of the ground coordinate system, and the initial values ​​of the attitude angles are all zero, which meets the requirements of the new attitude angle controller, and the control law design can be performed. In this way, the tiltrotor's attitude change flight can be accurately controlled according to the various control quantities calculated in the dynamic reference coordinate system.

[0011] The flight control system for a tiltrotor aircraft when hovering or changing pitch attitude at a low speed according to the present invention comprises:

[0012] Ground and body coordinate system conversion module: used to construct the first conversion matrix from the ground coordinate system to the body coordinate system;

[0013] Aircraft desired state conversion module: used to construct a second conversion matrix according to the desired pitch attitude of the aircraft, so as to establish the relationship between the aircraft coordinate system and the dynamic reference coordinate system;

[0014] A ground and dynamic reference coordinate system conversion module: used to construct a third conversion matrix for converting the ground coordinate system into the dynamic reference coordinate system according to the first conversion matrix and the second conversion matrix;

[0015] Control variable calculation module: used to convert various parameters in the ground coordinate system and the body coordinate system into the dynamic reference coordinate system, and calculate the expected control variables of the position and attitude of the tilt-rotor aircraft based on this.

[0016] The computer-readable storage medium storing one or more programs described in the present invention includes one or more programs and instructions. When the instructions are executed by a computing device, the computing device can execute any of the above methods.

[0017] Beneficial effect: Compared with the prior art, the present invention has the following significant effects: by constructing a first conversion matrix and a second conversion matrix, and then constructing a third conversion matrix therefrom, the ground coordinate system is converted to a dynamic reference coordinate system by the third conversion matrix. The dynamic reference coordinate system can reset the initial attitude of the tiltrotor on the slope to zero, so that the subsequent control system can be designed normally, so that the control amount calculated in the dynamic reference coordinate system can accurately control the take-off and landing of the tiltrotor, thereby realizing the tiltrotor take-off and landing, hovering and low-speed flight with a variable pitch attitude. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 This is a diagram of the conventional take-off process of a tiltrotor aircraft;

[0019] Figure 2 This is a diagram of the take-off process of a tiltrotor aircraft located on a slope;

[0020] Figure 3 It is a schematic diagram of three coordinate systems and the initial state of the body;

[0021] Figure 4 Schematic diagram of the control structure of this method. DETAILED DESCRIPTION

[0022] As shown in the figure, the conventional method of taking off a tiltrotor aircraft is to take off in helicopter mode on level ground, and then transform into fixed-wing mode to achieve horizontal flight. The process is as follows Figure 1 As shown: the tiltrotor aircraft takes off from starting point A in helicopter mode, rises to point B and hovers; at point B, the tiltrotor aircraft begins to tilt the nacelle forward to switch modes, and continues to fly to the end point C; at point C, the nacelle completes a 90° tilt, and the tiltrotor aircraft completes the switch from helicopter mode to fixed-wing mode.

[0023] When the tilt-rotor aircraft is on a slope, the specific take-off process is as follows: Figure 2 As shown: At point a on the slope, the tilt-rotor aircraft fuselage remains parallel to the slope, and the nacelle is tilted to the slope angle to ensure that the nacelle is at a 90° angle to the horizontal plane; take off to point b in this attitude, and at point b, the fuselage is converted from a pitch attitude state to a horizontal state c through fixed-point hovering. 1 ; or keep the pitch attitude unchanged, with c 2 The state can realize forward, backward, left and right flight.

[0024] Because the data measured by the sensors on the tilt-rotor aircraft are relative to the aircraft itself, when its own position and attitude have not changed, because the tilt-rotor aircraft is on a slope, the actual first pitch angle is not zero. If the conventional method is used to control the flight of the tilt-rotor aircraft based on the data measured by the sensors, it is rather cumbersome. Therefore, the control method of the present invention is proposed.

[0025] Figure 3 In, θ 0 is the pitch angle between the body coordinate system and the ground coordinate system in the desired pitch state of the body, γ 0 x is the angle of rotation of the nacelle from the state parallel to the Z axis of the body coordinate system and facing upward to the state facing vertically upward; b 、z b are the X-axis and Z-axis in the body coordinate system, and the Y-axis is perpendicular to the paper surface; d 、z d are the X-axis and Z-axis in the dynamic reference coordinate system, and the Y-axis is perpendicular to the paper surface; when the tilt-rotor aircraft is in a non-slope state, that is, a plane, and in helicopter mode, γ 0 When γ is 0°, it is in fixed-wing mode. 0 is 90°.

[0026] The flight control method for a tiltrotor aircraft during hovering or low-speed pitch attitude change according to the present invention comprises the following steps:

[0027] S1, constructing a first transformation matrix from the ground coordinate system to the body coordinate system;

[0028] The dynamic reference coordinate system is denoted as [X d Y d Z d ]; the ground coordinate system is denoted as [X e Y e Z e ]; the body coordinate system is recorded as [X b Y b Z b ];

[0029] The positional relationship of one coordinate system relative to another coordinate system can be expressed by three angles: yaw angle ψ, pitch angle θ, and roll angle φ, which are called Euler angles. The first Euler angle converted from the ground coordinate system to the body coordinate system is recorded as: the first yaw angle ψ e , the first pitch angle θ e and the first roll angle φ e .

[0030] The process of gradually rotating from the ground coordinate system to the body coordinate system is as follows:

[0031] S1.1, rotate the first yaw angle ψ around the Z axis of the ground coordinate system e , get the first intermediate coordinate system;

[0032] S1.2, rotate the first intermediate coordinate system obtained by step S1.1 around the Y axis to obtain a first pitch angle θ e , and obtain the second intermediate coordinate system;

[0033] S1.3, rotate the second intermediate coordinate system obtained by step S1.2 about the X axis by a first roll angle φ e , and finally the body coordinate system is obtained.

[0034] According to the above steps, the first transformation matrix R be The calculation formula

[0035] R be =R X (φ e )·R Y (θ e )·R Z (ψ e )

[0036] in,

[0037]

[0038] Among them, ψ e ,θ e and φ e They are the first yaw angle, the first pitch angle and the first roll angle converted from the ground coordinate system to the body coordinate system respectively.

[0039] S2. constructing a second transformation matrix according to the desired pitch posture of the aircraft, thereby establishing a relationship between the aircraft coordinate system and the dynamic reference coordinate system;

[0040] In the body coordinate system [X b Y b Z b ] around Y b Axis rotation angle θ 0 , relative to the body coordinate system [X b Y b Z b The second transformation matrix R y (θ 0 ), θ 0 is the pitch angle between the body coordinate system and the ground coordinate system in the desired pitch state of the body; the second transformation matrix is ​​R y (θ 0 ) is calculated according to the following formula

[0041]

[0042] S3. Construct a third conversion matrix that converts the ground coordinate system into the dynamic reference coordinate system based on the first conversion matrix and the second conversion matrix. The calculation formula of the third conversion matrix is:

[0043] R de =R y (θ 0 )·R be

[0044] Among them, R de is the third transformation matrix.

[0045] According to the third transformation matrix, the second Euler angle ξ of the ground coordinate system to the dynamic reference coordinate system can be extracted. d =(ψ d θ d φ d ), which is the actual Euler angle of the tiltrotor aircraft. The specific extraction method is:

[0046] The second pitch angle (that is, the actual pitch angle) θ d By R de R in the matrix de (1,3) element calculation:

[0047] θ d =arcsin(-R de (1,3))

[0048] The second yaw angle (that is, the actual yaw angle) ψ d By R de R in the matrix de (1,1) and R de (1,2) element calculation:

[0049] ψ d =arctan2(R de (1,2),R de (1,1))

[0050] Second roll angle (ie actual roll angle) φ d By R de R in the matrix de (2,3) and R de (3,3) element calculation:

[0051] φ d =arctan2(R de (2,3),R de (3,3))

[0052] Through the above conversion, the Euler angle between the body coordinate system and the ground coordinate system can be converted into the Euler angle between the dynamic reference coordinate system and the ground coordinate system; except for the Euler angle, the other related position and posture variables can be calculated according to the above method. This calculation is common knowledge in this field and will not be repeated in this article.

[0053] Through such conversion calculations, the actual position and attitude changes of the tilt-rotor aircraft relative to the ground can be reflected, and then the corresponding control amount can be calculated to control the flight state of the tilt-rotor aircraft.

[0054] S4. Convert various parameters in the ground coordinate system and the body coordinate system to the dynamic reference coordinate system, and calculate the expected control amount of the position and attitude of the tiltrotor aircraft based on this. It should be noted that all variables used for calculation in this step are obtained in the dynamic reference coordinate system. In order to distinguish them from the corresponding variables in the other two coordinate systems, a subscript "d" is added to the variable without a subscript, and a superscript "d" is added to the variable with a subscript to indicate that the variable is a variable in the dynamic reference coordinate system.

[0055] The calculation formula of the expected vertical control amount in the dynamic reference coordinate system is:

[0056]

[0057] in, is the desired vertical control amount in the dynamic reference coordinate system, m is the mass of the tiltrotor, g is the gravitational acceleration, is the proportional gain of the vertical velocity error in the dynamic reference coordinate system; is the integral gain of the vertical velocity error in the dynamic reference coordinate system; is the differential gain of the vertical velocity error in the dynamic reference coordinate system;

[0058] is the vertical velocity error in the dynamic reference coordinate system, and the calculation formula is:

[0059]

[0060] in, is the expected vertical velocity in the dynamic reference coordinate system, is the actual vertical velocity in the dynamic reference coordinate system;

[0061] The calculation formula is

[0062]

[0063] in, is the proportional gain of the height error in the dynamic reference coordinate system, is the expected vertical height in the dynamic reference coordinate system; z d It is the actual vertical height in the dynamic reference coordinate system.

[0064] The calculation formula of the expected control amount of the attitude in the dynamic reference coordinate system is:

[0065]

[0066] in, is the expected operation amount of the posture in the dynamic reference coordinate system, is the proportional gain of the angular velocity error in the dynamic reference coordinate system; is the integral gain of the angular velocity error in the dynamic reference coordinate system; is the differential gain of the angular velocity error in the dynamic reference coordinate system;

[0067] is the angular velocity error in the dynamic reference coordinate system, and the calculation formula is:

[0068]

[0069] in, is the desired angular velocity in the dynamic reference coordinates, ω d is the actual angular velocity in the dynamic reference coordinate system;

[0070] The calculation formula is

[0071]

[0072] in, is the expected Euler angle between the dynamic reference coordinate system and the ground coordinate system, ξ d is the actual Euler angle between the dynamic reference coordinate system and the ground coordinate system;

[0073] is the proportional gain matrix of the Euler angle error between the dynamic reference coordinate system and the ground coordinate system, specifically:

[0074]

[0075] in, and They are respectively the proportional gain of the pitch angle error, the proportional gain of the roll angle error and the proportional gain of the yaw angle error between the dynamic reference coordinate system and the ground coordinate system.

[0076] Expected Euler angles in the dynamic reference coordinate system The expression is

[0077]

[0078] in, and are the expected roll angle, expected pitch angle and expected yaw angle between the dynamic reference coordinate system and the ground coordinate system, respectively. For the given input value, the expected roll angle and the desired pitch angle The calculation formula is

[0079]

[0080] in, is the proportional gain of the horizontal velocity error in the dynamic reference coordinate system; is the integral gain of the horizontal velocity error in the dynamic reference coordinate system; is the differential gain of the horizontal velocity error in the dynamic reference coordinate system;

[0081] is the horizontal velocity error in the dynamic reference coordinate system, and the calculation formula is:

[0082]

[0083] in, is the actual horizontal velocity in the dynamic reference coordinate system, and the specific expression is is the expected horizontal velocity in the dynamic reference coordinate system, and the calculation formula is:

[0084]

[0085] in, and are the components of the expected horizontal velocity in the X-axis and Y-axis in the dynamic reference coordinate system respectively; is the expected horizontal position in the dynamic reference coordinate system, and the specific expression is and are the X-axis and Y-axis coordinates of the desired horizontal position in the dynamic reference coordinate system, respectively; is the actual horizontal position, and the specific expression is Among them, x d and d The X-axis and Y-axis coordinates of the actual horizontal position in the dynamic reference coordinate system respectively;

[0086] is the proportional gain of the horizontal position error in the dynamic reference coordinate system, and the specific expression is

[0087]

[0088] in, and are the proportional gains of the horizontal position errors in the X-axis and Y-axis directions in the dynamic reference coordinate system, respectively.

[0089] The actual angular velocity ω in the dynamic reference coordinate system d The calculation formula is

[0090]

[0091] Among them, p d ,q d and rd are the roll angular velocity, pitch angular velocity and yaw angular velocity in the dynamic reference coordinate system, respectively, b ,q b and r b are the roll angular velocity, pitch angular velocity and yaw angular velocity in the body coordinate system respectively.

[0092] The flight control system for a tiltrotor aircraft when hovering or changing pitch attitude at a low speed according to the present invention comprises:

[0093] Ground and body coordinate system conversion module: used to construct the first conversion matrix from the ground coordinate system to the body coordinate system;

[0094] Aircraft desired state conversion module: used to construct a second conversion matrix according to the desired pitch attitude of the aircraft, so as to establish the relationship between the aircraft coordinate system and the dynamic reference coordinate system;

[0095] A ground and dynamic reference coordinate system conversion module: used to construct a third conversion matrix for converting the ground coordinate system into the dynamic reference coordinate system according to the first conversion matrix and the second conversion matrix;

[0096] Control variable calculation module: used to convert various parameters in the ground coordinate system and the body coordinate system into the dynamic reference coordinate system, and calculate the expected control variables of the position and attitude of the tilt-rotor aircraft based on this.

[0097] The computer-readable storage medium storing one or more programs described in the present invention includes one or more programs and instructions. When the instructions are executed by a computing device, the computing device executes any one of the above methods.

Claims

1. A flight control method for a tiltrotor aircraft when it is hovering or changing its pitch attitude at a low speed, characterized in that: The following steps are involved: S1, constructing a first transformation matrix from the ground coordinate system to the body coordinate system; S2. constructing a second transformation matrix according to the desired pitch posture of the aircraft, thereby establishing a relationship between the aircraft coordinate system and the dynamic reference coordinate system; S3, constructing a third conversion matrix for converting the ground coordinate system into the dynamic reference coordinate system according to the first conversion matrix and the second conversion matrix; S4. Convert various parameters in the ground coordinate system and the body coordinate system to the dynamic reference coordinate system, and calculate the expected control amount of the position and attitude of the tilt-rotor aircraft based on the dynamic reference coordinate system.

2. A flight control method for a tiltrotor aircraft when hovering or changing pitch attitude at a low speed according to claim 1, characterized in that: In step S1, the first conversion matrix is ​​calculated according to the following formula R be =R X (f e )·R Y (i e )·R Z (ψ e ) Among them, R be is the first transformation matrix; Among them, ψ e ,θ e and φ e They are the first yaw angle, the first pitch angle and the first roll angle converted from the ground coordinate system to the body coordinate system respectively.

3. The flight control method for a tiltrotor aircraft when hovering or changing pitch attitude at a low speed according to claim 1, characterized in that: In step S2, the second conversion matrix is ​​calculated according to the following formula Among them, R y (θ0) is the second transformation matrix, and θ0 is the pitch angle between the body coordinate system and the ground coordinate system in the desired pitch state of the body.

4. The flight control method for a tiltrotor aircraft when hovering or changing pitch attitude at a low speed according to claim 1, characterized in that: The third conversion matrix calculation formula in step S3 is: R de =R y (θ0)·R be Among them, R de is the third transformation matrix.

5. The flight control method for a tiltrotor aircraft when hovering or changing pitch attitude at a low speed according to claim 1, characterized in that: The calculation formula of the desired vertical control amount in the dynamic reference coordinate system in step S4 is: in, is the desired vertical control amount in the dynamic reference coordinate system, m is the mass of the tiltrotor, g is the gravitational acceleration, is the proportional gain of the vertical velocity error in the dynamic reference coordinate system; is the integral gain of the vertical velocity error in the dynamic reference coordinate system; is the differential gain of the vertical velocity error in the dynamic reference coordinate system; is the vertical velocity error in the dynamic reference coordinate system, and the calculation formula is: in, is the expected vertical velocity in the dynamic reference coordinate system, is the actual vertical velocity in the dynamic reference coordinate system; The calculation formula is in, is the proportional gain of the height error in the dynamic reference coordinate system, is the expected vertical height in the dynamic reference coordinate system; z d It is the actual vertical height in the dynamic reference coordinate system.

6. The flight control method for a tiltrotor aircraft when hovering or changing pitch attitude at a low speed according to claim 1, characterized in that: The calculation formula of the expected manipulation amount of the posture in the dynamic reference coordinate system in step S4 is: in, is the expected operation amount of the posture in the dynamic reference coordinate system, is the proportional gain of the angular velocity error in the dynamic reference coordinate system; is the integral gain of the angular velocity error in the dynamic reference coordinate system; is the differential gain of the angular velocity error in the dynamic reference coordinate system; is the angular velocity error in the dynamic reference coordinate system, and the calculation formula is: in, is the desired angular velocity in the dynamic reference coordinate system, ω d is the actual angular velocity of the dynamic reference coordinate system; The calculation formula is in, is the expected Euler angle between the dynamic reference coordinate system and the ground coordinate system, ξ d is the actual Euler angle between the dynamic reference coordinate system and the ground coordinate system; is the proportional gain matrix of the Euler angle error between the dynamic reference coordinate system and the ground coordinate system, specifically: in, and They are respectively the proportional gain of the pitch angle error, the proportional gain of the roll angle error and the proportional gain of the yaw angle error between the dynamic reference coordinate system and the ground coordinate system.

7. A flight control method for a tiltrotor aircraft when hovering or changing pitch attitude at a low speed according to claim 6, characterized in that: The expected Euler angle The expression is in, and are the expected roll angle, expected pitch angle and expected yaw angle between the dynamic reference coordinate system and the ground coordinate system, respectively. For the given input value, the expected roll angle and the desired pitch angle The calculation formula is in, is the proportional gain of the horizontal velocity error in the dynamic reference coordinate system; is the integral gain of the horizontal velocity error in the dynamic reference coordinate system; is the differential gain of the horizontal velocity error in the dynamic reference coordinate system; is the horizontal velocity error in the dynamic reference coordinate system, and the calculation formula is: in, is the actual horizontal velocity in the dynamic reference coordinate system, and the specific expression is is the expected horizontal velocity in the dynamic reference coordinate system, and the calculation formula is: in, and are the components of the expected horizontal velocity in the X-axis and Y-axis in the dynamic reference coordinate system respectively; is the expected horizontal position in the dynamic reference coordinate system, and the specific expression is and are the X-axis and Y-axis coordinates of the desired horizontal position in the dynamic reference coordinate system, respectively; is the actual horizontal position, and the specific expression is Among them, x d and d The X-axis and Y-axis coordinates of the actual horizontal position in the dynamic reference coordinate system respectively; is the proportional gain of the horizontal position error in the dynamic reference coordinate system, and the specific expression is: in, and are the proportional gains of the horizontal position errors in the X-axis and Y-axis directions in the dynamic reference coordinate system respectively; Actual Euler angle ξ d The calculation formula is x d =(ψ d i d f d ) Among them, ψ d is the actual yaw angle, and the calculation formula is ψ d =arctan2(R de (1,2),R de (1,1)) θ d is the actual pitch angle, and the calculation formula is θ d =arcsin(-R de (1,3)) φ d is the actual roll angle, and the calculation formula is φ d =arctan2(R de (2,3),R de (3,3))。 8. The flight control method for a tiltrotor aircraft when hovering or changing pitch attitude at a low speed according to claim 6, characterized in that: The actual angular velocity ω in the dynamic reference coordinate system d The calculation formula is Among them, p d ,q d and r d are the roll angular velocity, pitch angular velocity and yaw angular velocity of the dynamic reference coordinate system, respectively, b ,q b and r b are the roll angular velocity, pitch angular velocity and yaw angular velocity of the body coordinate system respectively.

9. A flight control system for a tiltrotor aircraft when it is hovering or changing pitch attitude at a low speed, characterized in that: The system comprises: Ground and body coordinate system conversion module: used to construct the first conversion matrix from the ground coordinate system to the body coordinate system; Aircraft desired state conversion module: used to construct a second conversion matrix according to the desired pitch attitude of the aircraft, so as to establish the relationship between the aircraft coordinate system and the dynamic reference coordinate system; A ground and dynamic reference coordinate system conversion module: used for constructing a third conversion matrix for converting the ground coordinate system into the dynamic reference coordinate system according to the first conversion matrix and the second conversion matrix; Control variable calculation module: used to convert various parameters in the ground coordinate system and the body coordinate system into the dynamic reference coordinate system, and calculate the expected control variables of the position and attitude of the tilt-rotor aircraft based on this.

10. A computer-readable storage medium storing one or more programs, characterized in that: The one or more programs include instructions which, when executed by a computing device, cause the computing device to perform any one of the methods according to claims 1 to 8.

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