Differential flatness-based straight-helical rotational joint motion planning method for drag system

Abstract

The application discloses a kind of linear-helical rotation interface motion planning method of tow system based on differential flatness.The application proposes a novel motion planning method for the transition motion problem of tow system from linear flight state to helical flight state, which can quickly plan the flight trajectory, speed and acceleration of the mother plane in linear-helical rotation stage, thereby indirectly controlling the stable transition flight of tow cable-towed body in linear-helical rotation stage. The method comprises the following steps: step one: constructing a multi-body dynamics model of the mother plane-cable-towed body of the tow system; step two: based on the tow system model established in step one, according to the differential flatness theory, the state variables of the mother plane and the tow cable-towed body are converted into the position of the towed body and its derivative; step three: combining the position, speed and other constraints of the mother plane and the tow cable-towed body at the start and end of the linear-helical rotation interface segment, a polynomial is used to establish the motion trajectory of the towed body in the linear-helical rotation interface segment, and the position, speed and acceleration of the mother plane are calculated according to step two.

Description

Technical Field

[0001] This invention relates to the field of aircraft trajectory planning technology, and mainly to a linear-disk rotational motion planning method for a towed system based on differential flatness. Background Technology

[0002] Aerial towing systems primarily consist of an aircraft, a towing cable, and a towed body. They have wide-ranging engineering applications in both military and civilian fields, such as aerial towed decoys, aerial towed targets, hose-and-drogue aerial refueling, soft-shell UAV recovery, and sling transport. With increasingly diversified mission requirements and rapid advancements in aviation technology, aerial towing systems have evolved from their original single straight-line flight application scenarios to hovering flight applications, such as aircraft-to-submarine communication, hovering towed UAV aerial recovery, and air-to-ground fixed-point payload recovery and deployment. To meet the mission requirements in hovering flight scenarios, ensuring the safe, stable, and efficient transition of the aircraft from straight-line flight to hovering flight has become one of the key technical challenges that urgently needs to be overcome.

[0003] It is noteworthy that current research on towed systems mainly focuses on trajectory control of towed objects in straight-line and hovering flight. However, a few methods address the transition from straight-line to hovering in towed systems by employing optimal control theory to plan the transition trajectory, which presents computationally complex problems. Therefore, there is an urgent need to develop a rapid motion planning method for towed systems transitioning from straight-line to hovering flight, thereby meeting the transition requirements of towed systems in different scenarios. Summary of the Invention

[0004] Purpose of the invention: This invention provides a motion planning method for a towed system in a straight-line to disk rotation transition. Based on the principle of differential flatness, the state variables of each node of the towed system are transformed into the positions of the towed body and their derivatives, reducing the spatial dimension of the planning. Furthermore, the method constructs the transition flight position equations of the towed body in a straight-line to disk rotation transition using polynomial functions, thereby quickly planning the transition flight position, velocity, and acceleration of the towed system.

[0005] Therefore, the present invention is achieved through the following technical solution: The linear-disk rotational connection motion planning method for towing systems based on differential flatness provided in this application includes the following steps:

[0006] Step S1: Construct a multibody dynamics model of the towing system (mother machine-cable-towing body) using the lumped mass method and Newton's second law.

[0007]

[0008] In the formula, m i =m l Let m be the mass of node i. Ndr =ml +m dr Let m be the mass of node N. l For the mass of the cable segment, m dr G is the mass of the towed body; i =m i ge z Let G be the gravitational force acting on node i. Ndr =m Ndr ge z e is the gravitational force acting on node N. z =[0,0,1] T T is the unit direction vector of gravity; i F represents the tension exerted on node i by the i-th segment of the cable; i a The aerodynamic force acting on node i, The aerodynamic force acting on the towed body.

[0009] Among them, the tension T on cable node i i It can be represented as:

[0010]

[0011] In the formula: l i =P i -P i+1 is the position vector of the i-th cable segment, l0 is the initial length of the i-th cable segment before it is stretched and deformed, E is the elastic Young's modulus of the cable, and A is the cross-sectional area of ​​the cable.

[0012] Aerodynamic force F on cable node i i a It can be represented as:

[0013]

[0014] In the formula: D i and L i Let represent the aerodynamic drag and aerodynamic lift experienced by the i-th segment of the cable, respectively; and Let represent the aerodynamic drag coefficient and aerodynamic lift coefficient of the i-th cable segment, respectively; and Let ρ represent the unit direction vector of aerodynamic drag and the unit direction vector of aerodynamic lift of the i-th cable segment, respectively; i d is the air density around the i-th cable segment; l V is the diameter of the cable; i a Let be the airspeed vector of the i-th cable segment.

[0015] Aerodynamic forces on the towed body It can be represented as:

[0016]

[0017] In the formula: D dr and L dr These represent the aerodynamic drag and aerodynamic lift experienced by the towed body, respectively. and ρ represents the aerodynamic drag coefficient and aerodynamic lift coefficient of the towed body, respectively; dr S is the air density around the towed body. dr The aerodynamic area of ​​the towed body; The airspeed vector of the towed body.

[0018] Step S2: Based on the multibody dynamics model of the towing system established in Step S1, and according to the differential flatness theory, the position, velocity, and acceleration of the mother machine are converted into the position and derivative representation of the towing body.

[0019] The towed system is differentially flat, meaning the system state and input can be written as functions of the system's flat output and its derivative. Let the three-axis positions of the towed body represent the system's flat output: δ = [p Nx ,p Ny ,p Nz ] T Based on the force balance at node N where the towing body is located, the tension T of the cable at node N can be obtained. N for:

[0020]

[0021] In the formula: e x ,e y ,e z These are the unit direction vectors in the inertial frame; T Nx ,T Ny ,T Nz These are the components of the tension at node N in the inertial frame; m dr The mass of node N; A N G Ndr , These are the acceleration vector, gravity vector, and aerodynamic vector of node N, respectively.

[0022] Meanwhile, considering that the cable tension originates from the cable's expansion and contraction, therefore, given the known segmental tension vector T of the cable... N and node N position P N At that time, the position vector P of cable node N-1 N-1 It can be represented as:

[0023]

[0024] In the formula: l0 is the initial length of the cable segment; E is the elastic modulus of the cable; A is the cross-sectional area of ​​the cable.

[0025] Subsequently, given the position vector P of the known cable node N-1 N-1 Velocity vector and acceleration vector At that time, the aerodynamic vector of cable node N-1 can be calculated. Combined with the gravitational vector G of the cable node N-1 The tension at the node can be obtained as follows:

[0026]

[0027] Further combining the position P of cable node N-1 N-1 The position P of cable node N-2 can be determined. N-2 The vector is:

[0028]

[0029] Similarly, the position vector P of node i i for:

[0030]

[0031] Among them, the cable tension T on node i i It can be represented as:

[0032]

[0033] Where: m i For the quality of node i, G i ,F i a Let represent the gravitational force vector and aerodynamic force vector acting on node i, respectively.

[0034] Finally, the position vector P0 and velocity vector of the mother machine can be obtained. and acceleration vector

[0035] Step S3: Considering the state constraints at the start and end points of the linear-disc rotary connection of the towing system, the motion position equation of the linear-disc rotary connection of the towing body is established using a polynomial, and the position, velocity and acceleration of the mother machine at the corresponding moment are calculated based on the differential flatness characteristics of the system.

[0036] Based on the towed body's states during the straight-line and hovering flight phases, and considering that the towed body maintains a constant angular velocity during the transition from straight-line to hovering flight, then the position P of the towed body is... N Speed ​​V N and acceleration A N It can be represented as:

[0037]

[0038]

[0039]

[0040] In the formula: t0 is the start time of the transition between the towed bodies, t f The moment when the transition between towed bodies ends; x0, y0, z0 are the initial positions of the towed bodies; w = V0 / R0 = V f / R f V0 is the angular velocity of the towed body during transition flight, and V0 is the linear flight speed of the towed body. f R is the towed vehicle's hovering flight speed, R0 is the radius at the initial moment of the towed vehicle's transition flight, and R f θ is the radius at the end of the towed body transition flight; θ = θ0 + wt is the transition flight angle of the towed body, and θ0 is the flight angle at the beginning of the towed body transition flight.

[0041] To determine the towed body's position equations for the straight-line to disk rotation transition flight, a polynomial is used to construct the towed body's transition flight radius as follows:

[0042] R(t) = a0 + a1(t-t0) + ... + a k (t-t0) k ,t0≤t≤t f

[0043] Meanwhile, to meet the position, velocity, and other constraints at the start and end of the towing system transition, the following towed body flight radius constraints are introduced:

[0044]

[0045]

[0046] In the formula: Let be the k-th derivative of R with respect to t.

[0047] In summary, let the transition flight time of the towed body be T = t f -t0, then the coefficient matrix K of the towed body transition flight radius R(t) can be obtained as [a0,a1,...,a k ] T for:

[0048]

[0049] Beneficial effects:

[0050] The trajectory planning method for the linear-disc rotary junction of the towing system designed in this invention, based on differential flatness, has the advantages of simple calculation and high efficiency compared with the optimal control method. It can quickly plan the linear-disc rotary junction motion position, velocity and acceleration of the towing system under calm atmosphere and constant wind disturbance, thus providing strong support for the transition motion of the towing system. Attached Figure Description

[0051] Figure 1 The flowchart of the motion planning method for the linear-disk rotating segment of the towing system based on differential flatness provided by the present invention is shown below.

[0052] Figure 2 This is a schematic diagram of the linear-disc rotary joint motion of the towing system provided by the present invention;

[0053] Figure 3 This invention provides a three-dimensional trajectory view of the motion planning of the linear-disk rotation segment of a towing system under calm atmospheric conditions.

[0054] Figure 4 This invention provides a three-dimensional trajectory view of the motion trajectory of the linear-disc rotary joint of the towing system under constant wind conditions.

[0055] Figure 5 The machine speed under calm atmosphere and constant wind conditions provided by this invention;

[0056] Figure 6 The magnitude of the mother machine acceleration under calm atmosphere and constant wind conditions is provided for this invention. Detailed Implementation

[0057] The invention will now be further described with reference to the accompanying drawings.

[0058] This embodiment provides a motion planning method for a linear-disc rotating segment of a towing system based on differential flatness. The relevant physical parameters of the towing system are: cable segment length l0 = 400m, density ρ l =970kg / m 3 , diameter d l =2×10 - 3 m, elastic modulus E = 1.2 × 10 11 pa; mass of the towed body m dr =30kg, S dr =0.785m 2 , The relevant flight parameters for the towing system are: towed body flight altitude H = 1000m, straight flight speed V0 = 80m / s, and hovering flight speed V f =20m / s, initial radius R0 = 1200m, final radius R f=300m, transition start time t0=10s, transition end time t f =210s, initial yaw angle θ0 = 0°. The specific implementation includes the following steps:

[0059] Step S1: As Figure 2 As shown, the cable is divided into N discrete units in the inertial coordinate system O. g -x g y g z g The following multibody dynamics model of the towing system, consisting of the mother machine, cable, and towing body, is constructed using the lumped mass method and Newton's second law:

[0060]

[0061] In the formula, m i =m l Let m be the mass of node i. Ndr =m l +m dr Let m be the mass of node N. l For the mass of the cable segment, m dr G is the mass of the towed body; i =m i ge z Let G be the gravitational force acting on node i. Ndr =m Ndr ge z e is the gravitational force acting on node N. z =[0,0,1] T T is the unit direction vector of gravity; i F represents the tension exerted on node i by the i-th segment of the cable; i a The aerodynamic force acting on node i, The aerodynamic force acting on the towed body.

[0062] Among them, the tension T on cable node i i It can be represented as:

[0063]

[0064] In the formula: l i =P i -P i+1 is the position vector of the i-th cable segment, l0 is the initial length of the i-th cable segment before it is stretched and deformed, E is the elastic Young's modulus of the cable, and A is the cross-sectional area of ​​the cable.

[0065] Aerodynamic force F on cable node i i a It can be represented as:

[0066]

[0067] In the formula: D i and L i Let represent the aerodynamic drag and aerodynamic lift experienced by the i-th segment of the cable, respectively; and Let represent the aerodynamic drag coefficient and aerodynamic lift coefficient of the i-th cable segment, respectively; and Let ρ represent the unit direction vector of aerodynamic drag and the unit direction vector of aerodynamic lift of the i-th cable segment, respectively; i d is the air density around the i-th cable segment; l V is the diameter of the cable; i a Let be the airspeed vector of the i-th cable segment.

[0068] Aerodynamic forces on the towed body It can be represented as:

[0069]

[0070] In the formula: D dr and L dr These represent the aerodynamic drag and aerodynamic lift experienced by the towed body, respectively. and ρ represents the aerodynamic drag coefficient and aerodynamic lift coefficient of the towed body, respectively; dr S is the air density around the towed body. dr The aerodynamic area of ​​the towed body; The airspeed vector of the towed body.

[0071] Step S2: Based on the multibody dynamics model of the towing system established in Step S1, and according to the differential flatness theory, the position, velocity, and acceleration of the mother machine are converted into the position and derivative representation of the towing body.

[0072] The towed system is differentially flat, meaning the system state and input can be written as functions of the system's flat output and its derivative. Let the three-axis positions of the towed body represent the system's flat output: δ = [p Nx ,p Ny ,p Nz ] T Based on the force balance at node N where the towing body is located, the tension in the cable at node N can be calculated as follows:

[0073]

[0074] Meanwhile, considering that the cable tension originates from the cable's expansion and contraction, therefore, given the known segmental tension vector T of the cable... N When node N is at position N, the position vector of cable node N-1 can be expressed as:

[0075]

[0076] Subsequently, given the position vector P of the known cable node N-1 N-1 Velocity vector and acceleration vector At that time, the aerodynamic vector of cable node N-1 can be calculated. Combined with the gravitational vector G of the cable node N-1 The tension at the node can be obtained as follows:

[0077]

[0078] Further combining the position P of cable node N-1 N-1 The position P of cable node N-2 can be determined. N-2 The vector is:

[0079]

[0080] Similarly, the position vector of node i is:

[0081]

[0082] Among them, the cable tension T on node i i It can be represented as:

[0083]

[0084] Finally, the position vector P0 and velocity vector of the mother machine can be obtained. and acceleration vector

[0085] Step S3: From the perspective of trajectory planning, analyze the towed body's straight-line flight and hovering flight states. Use a polynomial to construct the motion position equations for the towed body's straight-line-hovering transition segment. Based on the system's differential flatness characteristics, calculate the mother machine's position and acceleration at the corresponding moments. The calculation process is as follows: Figure 1 As shown:

[0086] Based on the towed body's states during the straight-line and hovering flight phases, and considering that the towed body maintains a constant angular velocity during the transition from straight-line to hovering flight, then the position P of the towed body is... N Speed ​​V N and acceleration A N It can be represented as:

[0087]

[0088]

[0089]

[0090] In the formula: t0 is the start time of the transition between the towed bodies, t f The moment when the transition between towed bodies ends; x0, y0, z0 are the initial positions of the towed bodies; w = V0 / R0 = V f / R f V0 is the angular velocity of the towed body during transition flight, and V0 is the linear flight speed of the towed body. f R is the towed vehicle's hovering flight speed, R0 is the radius at the initial moment of the towed vehicle's transition flight, and R f θ is the radius at the end of the towed body transition flight; θ = θ0 + wt is the transition flight angle of the towed body, and θ0 is the flight angle at the beginning of the towed body transition flight.

[0091] To determine the towed body's position equations for the straight-line to disk rotation transition flight, a polynomial is used to construct the towed body's transition flight radius as follows:

[0092] R(t) = a0 + a1(t-t0) + ... + a k (t-t0) k t0≤t≤t f

[0093] Meanwhile, to meet the position, velocity, and other constraints at the start and end of the towing system transition, the following towed body flight radius constraints are introduced:

[0094]

[0095]

[0096] In summary, the coefficient matrix K of the towed body transition flight radius R(t) can be obtained as [a0, a1, ..., a k ] T for:

[0097]

[0098] Based on the above parameters, simulations provide the motion trajectory, mother aircraft speed, and acceleration magnitude of the towed system during the straight-line to disk rotation transition phase under calm atmospheric conditions and constant wind disturbance. Figure 3-6 As shown.

[0099] The above description is only a preferred embodiment of the present invention and is not intended to limit the scope of the present invention. Several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A linear-disk rotary connection motion planning method for a towing system based on differential flatness, characterized in that, Includes the following steps: Step S1: Construct a multibody dynamics model of the towing system consisting of the mother machine, cable, and towing body; Step S2: Based on the differential flatness theory, convert the state of each node in the towing system into the three-axis position of the towing body and its derivative representation; Step S2 specifically involves: Step S21, set the three-axis position of the towed body to a flat output of the system: According to the node where the towed body is located The force balance can be obtained at the nodes The tension of the cable for: ； In the formula: These are the unit direction vectors in the inertial frame of reference; These are nodes The component of the tension applied in an inertial frame of reference; For nodes The quality; These are nodes The acceleration vector, gravity vector, and aerodynamic vector; Step S22, at the known node The tension of the cable and nodes Location At that time, the cable node position vector for: ； In the formula: The initial length of the cable segment; The elastic modulus of the cable; This represents the cross-sectional area of ​​the cable. Step S23, at a known cable node position vector Velocity vector and acceleration vector At that time, the cable node was calculated. Aerodynamic vector Combined with the gravity vector of the cable node The tension at the node can be obtained as follows: ； Further integration of cable nodes Location The cable nodes can be identified. Location The vector is: ； Similarly, we can obtain the node position vector for: ； Among them, nodes The tension of the cable Represented as: ； In the formula: For nodes quality Representing nodes respectively The vectors of gravity and aerodynamic forces acting on the object; Finally, the position vector of the mother machine is obtained. Velocity vector and acceleration vector ; Step S3: Considering the state constraints at the start and end points of the towing system's linear-disc rotation, a polynomial is used to establish the motion position equation of the towing body's linear-disc rotation. The three-axis positions of the towing body and their derivatives are further calculated. Based on step S2, the flight position, speed, and acceleration of the mother machine during the linear-disc rotation phase are planned.

2. The linear-disc rotary connection motion planning method for a towing system based on differential flatness according to claim 1, characterized in that, Step S3 is as follows: Step S31: Based on the towed system's straight-line flight and hovering flight phases, and considering that the towed body maintains a constant angular velocity during the transition from straight-line to hovering flight, the position of the towed body is determined. ,speed and acceleration , is represented as: ； ； ； In the formula: This marks the start of the transition between the towed and the main body. This is the end point of the transition between the towed bodies; This is the initial position of the towed body; For the towed body to transfer flight angular velocity, The linear flight speed of the towed body. The towed vehicle's hovering flight speed, The radius at the initial moment of the towed body's transition flight. The radius at the end of the towed body's transition flight; To facilitate the transition of flight angles for the towed vehicle. The flight angle at the initial moment of the towed body's transition flight.

3. The linear-disc rotary connection motion planning method for a towing system based on differential flatness according to claim 2, characterized in that, To determine the towed body's position equations for the straight-line to disk-rotation transition flight, a polynomial is used to construct the towed body's transition flight radius as follows: ； Meanwhile, to meet the position and velocity constraints at the start and end of the towed system transition, the following towed body flight radius constraints are introduced: ； ； In the formula: for right of Derivative order; The transition flight time of the towed body is Then the towed body transition flight radius coefficient matrix for: 。

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