CN109835200B

CN109835200B - Path planning method for single-row wireless charging sensor network of bridge

Info

Publication number: CN109835200B
Application number: CN201910001921.2A
Authority: CN
Inventors: 刘贵云; 彭百豪; 蒋文俊; 向建化; 张杰钊; 唐冬
Original assignee: Guangzhou University
Current assignee: Guangzhou University
Priority date: 2019-01-02
Filing date: 2019-01-02
Publication date: 2022-02-01
Anticipated expiration: 2039-01-02
Also published as: CN109835200A

Abstract

The invention discloses a path planning method for a single-file wireless charging sensor network of a bridge, which comprises the following specific steps: (1) obtaining bridge nodes required to be set based on a linear topological structure; (2) solving to obtain the optimal node number according to the coverage of 2 and the condition of the minimum line number; (3) according to the optimal spiral point and the optimal path distribution under the condition that the two nodes are linearly and uniformly distributed; (4) according to the optimal spiral point and the optimal path distribution under the condition that the three nodes are linearly and uniformly distributed; (5) and obtaining the optimal spiral point and the optimal path for charging the unmanned aerial vehicle under the condition of n nodes according to the optimal disc point selection condition of the two nodes and the three nodes. The invention provides a planning method for an optimal charging path of an unmanned aerial vehicle, which can enable a sensor network in a bridge parameter monitoring system to operate for a long time.

Description

Path planning method for single-row wireless charging sensor network of bridge

Technical Field

The invention relates to the technical field of wireless energy transmission, in particular to a path planning method for a single-row wireless charging sensor network of a bridge.

Background

With the rapid development of domestic economy, the construction level of domestic bridges is continuously improved, various river-crossing and sea-crossing bridges and highway-railway dual-purpose bridges are opened out, and as the bridges are longer in construction and larger in size, the structures are more complex, and the maintenance of the bridge construction becomes particularly important. And the damage of internal structures of many domestic bridges is difficult to detect after long-term exposure to the sun and rain, and once the structural damage cannot be timely detected and repaired, irreparable great loss can be caused.

Therefore, it is necessary to monitor the bridge periodically. However, the traditional manual maintenance is not only high in labor cost, but also may cause insufficient measurement accuracy due to subjective reasons. But with the popularization of wireless sensor networks, the problem is well solved. However, with the continuous increase of the scale of the wireless network and the continuous increase of the data transmission rate, the consumption of the wireless terminal device in the wireless network to energy increases explosively, and the battery capacity and the battery life of the wireless device in the wireless network greatly limit the performance of the wireless network, so that the limited battery life is the bottleneck of designing and developing the wireless network. In recent years, wireless charging technology has become prevalent, which can well solve the problem of extending the life cycle of wireless networks.

Disclosure of Invention

The invention aims to provide a path planning method of a single-row wireless charging sensor network of a bridge, which aims at a wireless charging sensor network based on a common bridge structure, adopts an optimal node arrangement scheme, prolongs the life cycle of the wireless charging sensor network and provides a method for planning paths of the single-row wireless charging sensor network of the bridge.

The purpose of the invention can be realized by the following technical scheme:

a path planning method for a single-row wireless charging sensor network of a bridge comprises the following specific steps:

(1) obtaining bridge nodes required to be set based on a linear topological structure;

(2) solving to obtain the optimal node number according to the coverage of 2 and the condition of the minimum line number;

(3) obtaining an optimal spiral point and optimal path distribution according to the linear uniform distribution condition of the two nodes;

(4) obtaining an optimal spiral point and optimal path distribution according to the linear uniform distribution condition of the three nodes;

(5) and obtaining the optimal hovering point and the optimal path for charging the unmanned aerial vehicle under the condition of n nodes according to the optimal hovering point conditions of the two nodes and the three nodes.

Specifically, in step (3), for the case that the energy of two nodes is maximized, theorem 1, theorem 2 and theorem 3 are proposed, and based on the theorem 1, the theorem 2 and the theorem 3, theorem 1 and theorem 2 are proposed for the unmanned aerial vehicle charging path with the maximized energy in the case of two nodes.

Further, the method can be used for preparing a novel materialAnd 2, the scenes based on theory 1, theory 2 and theory 3 are the same, and the assumed scenes are as follows: fixing two nodes on a bridge, wherein the two nodes are parallel to one side of the bridge, using a Cartesian space rectangular coordinate system as a reference, symmetrically placing the two nodes at two ends of a zero point, assuming that the linear distance between the two nodes is D, placing the nodes on an x axis, and the coordinates of the left node are

And the coordinates of the right node are

The base station is positioned on the same side of the bridge with the two nodes and is at a certain distance from the node on the right side, and the coordinate of the base station is assumed to be (xi, 0,0), wherein

When the charging period starts, the unmanned aerial vehicle drives away from the base station to start flying; the fastest flying speed of the unmanned aerial vehicle is V, and the unmanned aerial vehicle flies in parallel to the connecting line of the nodes; the distance between the unmanned aerial vehicle and the x axis is constant h, namely the flying height of the unmanned aerial vehicle is constant h, and the coordinates of the flying point of the unmanned aerial vehicle are (xi-Vt, 0, h); the unmanned aerial vehicle starts to charge the node from leaving the base station, and the charging process comprises a flight process and a hovering process;

the theory 1 specifically comprises the following steps:

there is and only one optimal hover point when the following conditions are met:

condition 1: the lagrangian multipliers corresponding to the nodes are equal;

condition 2: the ratio of the distance D between the nodes to the flying height H of the unmanned aerial vehicle is equal to or less than

The optimal hover point is (0,0, h);

the theory 2 specifically comprises the following steps:

two optimal hover points exist when the following conditions are met:

condition 1: the lagrangian multipliers corresponding to the nodes are equal;

condition 2: the ratio of the distance D between the nodes to the flying height H of the unmanned aerial vehicle is larger than

The coordinates of the optimal hover point are then: (x)₁0, h) and (x)₂0, h) in which x₁And x₂Satisfies the following conditions:

the theory 3 specifically comprises the following steps:

there is only one optimal hover point when the following conditions are met:

condition 1: the lagrangian multipliers corresponding to the nodes are unequal;

the corresponding optimal hover point coordinate at this time is (x, 0, h). Where x is determined by the particular numerical value, there is no definite algebraic form.

Further, theorem 1 specifically includes:

let the coordinates of the left node be

And the coordinates of the right node are

The coordinates of the base station are (xi, 0, 0); the fastest flight speed of the unmanned aerial vehicle is V; the charging period of the unmanned aerial vehicle is T, and the numerical value of T is larger than the time of a roundtrip spiral point; the flying height of the unmanned aerial vehicle is assumed to be h; y in the corresponding coordinates of the nodes on the two sides, the base station and the unmanned aerial vehicle in the flight path is 0;

in the case of theorem 1 or 3 being satisfied, that is, when only one optimal hover point is included, the description of the optimal path of the drone is:

the unmanned aerial vehicle drives away from the base station from 0 moment

In the time of (1), no one is presentThe machine directly flies to the optimal spiral point at the speed V and the spiral time is required at the optimal spiral point

Finally the remaining time is

The drone needs to fly back at speed V immediately and return to the base station.

During the flight of the unmanned aerial vehicle, the total energy captured by the two nodes is E1, during the hovering of the unmanned aerial vehicle, the total energy captured by the two nodes is E2, and then based on the theorem 1 or the theorem 3, the maximum energy captured by the unmanned aerial vehicle is: E-E1 + E2.

The theorem 2 specifically includes:

considering the charging path of the unmanned aerial vehicle under the condition of energy maximization of two nodes, firstly, it is assumed that: the coordinates of the left node are

And the coordinates of the right node are

The coordinates of the base station are (xi, 0, 0); the fastest flight speed of the unmanned aerial vehicle is V; the charging period of the unmanned aerial vehicle is T, and the numerical value of T is larger than the time required by the roundtrip circling point; the flying height of the unmanned aerial vehicle is h; the flight period of the unmanned aerial vehicle is T, and T must be greater than

Y in the corresponding coordinates of the nodes on the two sides, the base station and the unmanned aerial vehicle in the flight path is 0;

in the case of theorem 2 being satisfied, i.e. when there are two optimal hover points, the description of the optimal path is:

unmanned aerial vehicle drives away from base station from 0 moment in time

In the interior, the drone must fly at speed V to the first optimal hover point on the right, at which hover point the time is coiled

Then needs to return to the base station at the speed V.

During the flight process of the unmanned aerial vehicle, the total energy captured by the two nodes is E1, during the hovering process of the unmanned aerial vehicle, the total energy captured by the two nodes is E2, and based on the theorem 2, the maximum energy captured by the unmanned aerial vehicle is as follows: E-E1 + E2.

Specifically, in step (4), for the case where the energy of three nodes is maximized, theorem 4 and theorem 5 are proposed, and based on theorem 4 and theorem 5, theorem 3 and theorem 4 are proposed for the charging path of the unmanned aerial vehicle where the energy is maximized in the case of three nodes.

Further, the scenes corresponding to the theorem 4 and the theorem 5 are consistent, and the assumed scene is: the nodes are symmetrically distributed and divided into a left point, a middle point and a right point, the linear topological structure is adopted, the base station is positioned on one side of the right node, the coordinates of the left node are (-d,0,0), the coordinates of the middle node are (0,0,0), the position corresponding to the origin point is (d,0, 0). The base station x-axis distance is xi, and the corresponding coordinate is xi, 0, 0. The starting point of the unmanned aerial vehicle is located at the base station; the fastest flight speed of the unmanned aerial vehicle is V;

the theory 4 specifically comprises the following steps:

when the following three conditions are satisfied:

condition 1:

wherein

Condition 2:

wherein Δ₁＝720h⁸+2304h⁴d⁴+576h²d⁶，

Δ₂＝10368h¹²+82944h⁸d⁴+165888h⁴d⁸；

Condition 3:

when the following condition 4 is not satisfied, there is only one optimal hover point corresponding to the node capture energy maximization, and the coordinates are (0,0, h):

condition 4:

when the above condition is satisfied, but the following condition 5 is not satisfied:

condition 5:

at the moment, two optimal spiral points corresponding to the maximum node capture energy exist, and the coordinates are (x) respectively₁，0，h)，(x₂0, h) in which x₁，x₂As follows:

when the node satisfies the condition 4, the condition 5, but does not satisfy the condition 6:

condition 6:

the number of the spiral points corresponding to the maximum node capture energy is 3, and the corresponding coordinates are (x)₁，0，h)，(x₂0, h) and (0,0, h), wherein the corresponding hover point on the left and right sides is the maximum hover point, the corresponding hover point on the origin is the maximum hover point, wherein x₁And x₂The coordinates of (c) correspond to:

when the node meets the conditions 4, 5 and 6, 4 hover points exist corresponding to the maximum node capture energy, and the corresponding hover point coordinate is (x)₁，0，h)，(x₂，0，h)，(x₃0, h) and (x)₄，0，h)，x₁，x₂，x₃And x₄The corresponding coordinates are:

the theory 5 specifically comprises the following steps:

when any one of condition 1, condition 2 or condition 3 in lemma 4 is not satisfied, there is and only one optimal hover point:

only one optimal hover point corresponds to coordinates (0,0, h) at this time.

According to a formula of the point-to-point charging power consumption of the unmanned aerial vehicle, after the coordinates of the left node, the middle node and the right node are respectively substituted, Q is respectively obtained₁，Q₂And Q₃. Corresponding to

And order

ψ＝Q₁+Q₂+Q₃。

Further, the theorem 3 specifically includes:

the unmanned aerial vehicle's of three node energy maximization condition charging path assumes: the three nodes are linearly distributed; the coordinates of the left node are (-d,0, 0); the coordinates of the intermediate node are (0,0,0), corresponding to the position of the origin; the coordinates of the right node are (d,0, 0); the coordinates of the base station are (xi, 0, 0); the flying height of the unmanned aerial vehicle is h; the charging period of the unmanned aerial vehicle is T; the fastest flight speed of the unmanned aerial vehicle is V; and the y values corresponding to the three nodes, the base station and the flight path coordinates of the unmanned aerial vehicle are 0.

According to the different conditions in the theorem 4, the following 4 cases can be classified:

case 1: on the premise of meeting the conditions 1, 2 and 3 in the conditions of the theorem 4, the condition 4 is not met, and meanwhile, the charging period of the unmanned aerial vehicle is larger than that of the unmanned aerial vehicle

The moving path description considering the maximum charging energy of the unmanned aerial vehicle at this time should be:

starting from the base station at zero time, the unmanned aerial vehicle directly flies to the optimal circle point corresponding to the origin at the speed V and the circle time

Then, the signal flies back to the base station at the same speed V.

Case 2: when the conditions 1, 2, 3 and 4 in the conditions of the theorem 4 are met, the condition 5 is not met, and the charging period of the unmanned aerial vehicle is larger than that of the unmanned aerial vehicle

The description of the moving path considering the maximum charging energy of the unmanned aerial vehicle is as follows:

the unmanned aerial vehicle starts from the base station at zero time and directly flies to x at speed V₁Corresponding optimum hover point, hover time

And then flies back to the base station at speed V.

Case 3: when condition 1, condition 2, condition 3, condition 4 and condition 5 in the condition of theory 4 are satisfied, but do not satisfy condition 6, guarantee that the charge cycle of unmanned aerial vehicle is greater than this moment

Consider unmanned aerial vehicle charging abilityThe moving path description for maximizing the amount should be:

Then, the signal flies back to the base station at the same speed V.

Case 4: when all the conditions in lemma 4 are met, there are four maximum hover points, if x is₁Corresponding maximum greater than x₂The corresponding maximum value ensures that the charging period of the unmanned aerial vehicle is greater than

The moving path description considering the maximum charging energy of the drone should be:

And then flies back to the base station at speed V.

If x₁Corresponding maximum value less than x₂The corresponding maximum value ensures that the charging period of the unmanned aerial vehicle is greater than

the unmanned aerial vehicle drives away from the base station from zero time and directly flies to x at a speed V₂Corresponding optimum hover point, hover time

And then flies back to the base station at the speed V.

Assuming that the total energy captured by the three nodes is E1 when the drone flies back and forth, and the total energy captured by the three nodes when the drone is hovering is E2, then, based on the relevant condition of lemma 4, the maximum energy captured by the nodes, that is, the maximum energy charged by the drone is E1+ E2.

The theorem 4 specifically includes:

the unmanned aerial vehicle's of three node energy maximization condition charging path assumes: the three nodes are linearly distributed; the coordinates of the left node are (-d,0, 0); the coordinates of the intermediate nodes are (0,0, 0); the coordinates of the right node corresponding to the position of the origin are (d,0, 0); the coordinates of the base station are (xi, 0, 0); the flying height of the unmanned aerial vehicle is h; the flying speed of the unmanned aerial vehicle is V; the flight period of the drone is T, where T must be greater than

The y values corresponding to the coordinates of the three nodes, the base station and the flight path of the unmanned aerial vehicle are all 0;

when the condition of lemma 5 is satisfied, there are only 1 optimal hover point. At this time, to maximize the energy captured by the node, the description of the charging path of the unmanned aerial vehicle should be:

starting from the 0 moment, after the unmanned aerial vehicle drives away from the base station, the unmanned aerial vehicle must drive to the optimal circle point at the highest speed V, and the rotation time is set at the optimal circle point

And then immediately returns to the base station at the fastest speed V.

Specifically, in step (5), theorem 5, theorem 6 and theorem 7 are obtained according to the optimal hovering point conditions of the two nodes and the three nodes, so that the optimal hovering point and the optimal path for charging the unmanned aerial vehicle under the condition of n nodes can be obtained.

Further, the theorem 5 specifically includes:

in the case that n nodes are symmetrically and uniformly distributed about the origin, the derivatives of the total energy captured by the nodes all contain only odd terms.

The theorem 6 specifically includes:

under the condition that n nodes are symmetrically and uniformly distributed about the origin, when the number of the optimal spiral points is even, the origin must be a minimum value point.

The theorem 7 specifically includes:

under the condition that n nodes are symmetrically and uniformly distributed about the origin, when the number of the optimal spiral points is odd, the origin is necessarily the maximum value point.

Compared with the prior art, the invention has the following beneficial effects:

the invention provides a linear topology optimal node arrangement scheme based on a bridge, namely, the number of nodes with the coverage degree of 2 is ensured to be minimum, the optimal charging path of an unmanned aerial vehicle under the conditions that the number of the nodes is 2 and the number of the nodes is 3 is obtained by solving, and then rules of n nodes are summarized, so that the optimal charging path is obtained.

Drawings

FIG. 1 is a flow chart showing the steps of the present invention.

Detailed Description

The present invention will be described in further detail with reference to examples and drawings, but the present invention is not limited thereto.

Examples

(3) according to the optimal spiral point and the optimal path distribution under the condition that the two nodes are linearly and uniformly distributed;

(4) according to the optimal spiral point and the optimal path distribution under the condition that the three nodes are linearly and uniformly distributed;

Specifically, in the step (2), the bridge is divided into three regions according to the symmetry in the x direction and the symmetry in the y direction, and only the case where the coverage is 2 is considered based on reducing the number of nodes as much as possible.

The bridge width is 24m, the search radius of the sensor node is 10m, the sensor node is away from one end of the bridge, the length of the closest end is defined as D, and the half of the connecting line of the centers of the two sensor nodes in the y direction is defined as H.

As the positions of the nodes of the sensors are close to the center of the bridge, namely, the distance between the nodes on two sides of the bridge is reduced, the value of D is uniformly an integer for the convenience of operation.

The first condition is as follows: d is 3m, H is 9 m. Under the condition that the coverage is 2, the distance of the node in the x direction is obviously reduced, namely the intersection point of the circle on the right side in the x direction and the upper end bridge passes through the intersection point of the circle center of the node on the left side and the vertical distance of the upper end bridge, and the condition is as follows:

the distance between the two nodes can be solved through the two nodes and a right triangle formed by the intersection points of the right side circle and the upper end of the bridge, and the distance between the two nodes is as follows:

the value is 9.5m, and it can be found that the distance between the two nodes is smaller than the distance between the two nodes corresponding to the coverage of 1, and the condition does not satisfy the verification condition when the coverage is 2.

Case two: d is 4m, H is 8 m. In the case of coverage 2, the distance between two nodes is found to be 9.2m by the method of case one, and at this time, the only area where coverage 1 is possible is that a certain node, the nodes on the left and right sides, have substantially completely covered the node in the x direction, and the only area is a sector area where the node is covered by the left and right nodes, and if a circle can cover the area in the y direction, the area of the node on the bridge can be completely covered. It is verified that the circle in the y direction can cover the sector area. At this time, the total number of required nodes is 43.

Case three: d is 5m, H is 7 m. Under the condition that the coverage is 2, the center distance is 7.1m, and verification shows that the node circle in the y direction can completely cover the sector area with the coverage of 1, and the required number of nodes is 47 at this time.

Case four: d is 6m, H is 6 m. Under the condition that the coverage degree is 2, the distance between two nodes is 8, and the verification finds that the circle in the y direction can just cover the sector area, and the number of the required nodes is 50.

Case five: d is 8m, H is 4 m. At this time, the distance between the intersection point of the circle center of the node perpendicular to the upper end of the bridge and the intersection point of the circle center of the node and the upper end of the bridge is larger than the distance between the circle centers of the two nodes, which means that the intersection point of the two circle centers of the node is positioned at the lower side of the upper end of the bridge and an uncovered area exists. So in order to cover the area, the right node must be translated to the left until just covering the area, and in the case of coverage of 2, the number of nodes required for the distance between the two nodes to be 12m is 66.

In summary, as the nodes move toward the center line of the bridge, the number of nodes required for the coverage of 2 increases. With a coverage of 2, the minimum required number of nodes is 43.

Specifically, in step (3), the scenes on which lemma 1, lemma 2 and lemma 3 are based are the same, assuming that the scenes are: fixing two nodes on a bridge, wherein the two nodes are parallel to one side of the bridge, using a Cartesian space rectangular coordinate system as a reference, symmetrically placing the two nodes at two ends of a zero point, assuming that the linear distance between the two nodes is D, placing the nodes on an x axis, and the coordinates of the left node are

And the coordinates of the right node are

At the beginning of the charging cycle,the unmanned aerial vehicle drives away from the base station and starts flying; the fastest flying speed of the unmanned aerial vehicle is V, and the unmanned aerial vehicle flies in parallel to the connecting line of the nodes; the distance between the unmanned aerial vehicle and the x axis is constant h, namely the flying height of the unmanned aerial vehicle is constant h, and the coordinates of the flying point of the unmanned aerial vehicle are (xi-Vt, 0, h); the unmanned aerial vehicle starts to charge the node from leaving the base station, and the charging process comprises a flight process and a hovering process;

assuming that the drone has a continuously stable output energy consumption P, the energy power obtained by each node can be described by the following formula:

where η represents the energy conversion efficiency of each node, and the value of η is greater than 0 and less than 1, and it is generally assumed that the value of η is always 1 in an ideal case. β represents the channel power gain obtained at a reference distance, when the distance is 1 meter. h represents the vertical distance of two parallel lines when the flight path of the unmanned aerial vehicle is parallel to the connecting line of the two nodes. x is the number of_k，y_kRespectively representing x-axis coordinates and y-axis coordinates corresponding to the nodes. And x and y represent x and y coordinates corresponding to the flying track of the unmanned aerial vehicle.

Further, the theorem 1 specifically includes:

condition 1: the lagrangian multipliers corresponding to the nodes are equal;

The optimal hover point is (0,0, h).

Further, the theorem 2 specifically includes:

two optimal hover points exist when the following conditions are met:

condition 1: the lagrangian multipliers corresponding to the nodes are equal;

condition2: the ratio of the distance D between the nodes to the flying height H of the unmanned aerial vehicle is larger than

further, the theorem 3 specifically includes:

there is only one optimal hover point when the following conditions are met:

condition 1: the lagrangian multipliers corresponding to the nodes are unequal;

Specifically, when two nodes capture the maximum energy, the following conditions must be satisfied:

condition 1: establishing a constraint equation, and constraining two nodes, wherein the sum of the energy of the nodes is represented as E, and the percentage of the left node in the energy is a₁The percentage of energy occupied by the right node is a₂Thus having a₁+a₂＝1。

Condition 2: the node on the left side is constrained by

Similarly, the right node obtains an energy value greater than or equal to a₂E. The MaxE can be solved by solving lagrangian functions.

According to conditions 1 and 2, to obtain the maximum energy E, the following three formulas must be satisfied:

in order to obtain Max L, a lagrange dual method is adopted to analyze the problem, L must be assumed first according to the principle of the lagrange dual method, and Max L is assumed to be bounded and can be taken as the maximum value, that is, E can be taken as the maximum value.

Based on this assumption, the following formula can be derived:

since when E tends to infinity, the polynomial for E will be

If it is

Then E will have a decisive influence on L, i.e. L will also go to infinity, so in order to guarantee the presence of MaxL it must be assumed that equation (5) holds.

In the original problem, MaxE is the objective function, and the constraint conditions are formula (3) and formula (4), where α₁+α₂With the lagrange dual method, the lagrange multiplier is first fixed, where λ corresponds to 1₁And λ₂Wherein λ₁And λ₂Are all more than or equal to 0, then the optimal unknown number is solved, the x coordinate of the flight path of the unmanned aerial vehicle is corresponded to, and finally the optimal lambda is solved₁And λ₂So a new equation f (λ) is established using the Lagrangian dual method₁,λ₂) The following are:

according to the dual method, the Minf (λ) can be solved₁,λ₂) The method of (3) to obtain MaxE.

When solving the MaxL, the L function can be split into two parts, and the two parts are both maximized, so that the L can certainly obtain the maximum value. L is split into the following two parenthesis:

in the first bracket due to

Therefore, the value is constant 0, so MaxL in this case can only be determined by the content of the second bracket, that is, in the second bracket, the unmanned aerial vehicle can be firstly determined assuming that the unmanned aerial vehicle is at the optimal hover point in the whole time period, and the optimal hover point is firstly determined

After the optimal hovering point is solved, the unmanned aerial vehicle flies to the optimal hovering point at the highest speed, the fact that the unmanned aerial vehicle can return to the base station at the highest speed in the last period is guaranteed, and the fact that the unmanned aerial vehicle has the time on the optimal hovering point as much as possible in the period is guaranteed.

Order to

To maximize ψ, the sign of the derivative in the field of the point is verified to determine whether the point is an extreme point.

Because the left and right sides node all is located the x axle, so corresponding to its y axle, its coordinate abscissa is 0, so unmanned aerial vehicle can be the deformation of identity to the formula of node charging power:

in correspondence with the node on the left side,

and for the node on the right

Substituting it into ψ yields the following equation:

then derivative the psi with the variable x, i.e. solve

Where λ needs to be discussed₁λ₂Two cases are: lambda [ alpha ]₁＝λ₂And λ₁≠λ₂；

The proof of lemma 1 and lemma 2 is as follows:

when lambda is₁＝λ₂When this is true, the constant terms that appear after the reduction of the numerator will be completely cancelled out, and the remaining terms are all polynomial components containing x, as follows:

order to

The numeralization is the simplest equation of the fourth power containing x, and

after, the molecular simplification is:

the root formula of the quadratic equation is utilized and substituted into the solution to obtain:

because of the fact that

So if a<0, x will not have a solution, so the content of a needs to be classified and discussed, where a is 0 and the solution is obtained

So when

When x has two unequal or multiple roots, when

There is one and only one zero solution at this time. When in use

In the case of (1), since the solution of a is that if the sign is negative, a will not have a solution, so the sign can only be positive, and a is guaranteed to be greater than zero, according to equation (13), the solution of the equation is:

the zero solution is also when

One solution at the time, verified, is in the range (- ∞, x)₁) The function is monotonically increasing, in the interval (x)₁0) time function is monotonically decreasing in the interval (0, x)₂) While the corresponding function is monotonically increasing, in zonesM (x)₂, + ∞) is monotonically decreasing, so psi corresponds to x₁Origin and x₂The position of (a) gets an extremum. The extreme value is the maximum value at the moment, and psi (x) is found after the numerical values of the extreme values are compared₁)＝ψ(x₂) Phi psi (0), corresponding to zero, is a minimum value, so when

There are two optimal hover points. Thus, introduction 2 is complete.

When in use

When there is one and only one zero solution, the verification is obtained when x<At 0, the function increases monotonically, when x>When 0, the function is monotonically decreasing, so that when x is 0, the maximum is obtained, i.e. when x is 0, the optimal hover point is obtained. Thus, introduction 1 is done.

The proof of lemma 3 is as follows:

when lambda is₁≠λ₂When this is true, the numerator will become a 5 th order equation, comprising a polynomial with x from 0 to 5 times-24 total terms, with only one optimal hover point due to no corresponding derivation method, as follows:

1. when in use

And lambda₁＞λ₂When the optimal solution is in the interval

2. When in use

And lambda₁＜λ₂When the optimal solution interval is

3. When in use

And lambda₁＞λ₂When the optimal solution interval is

4. When in use

And lambda₁＜λ₂When the optimal solution interval is

According to and λ₁＝λ₂When the base station is closer to the node on the right side in the comparative analysis of (2), it is obvious that lambda is₁＞λ₂The optimal hovering point is closer to the left node, and in the process that the unmanned aerial vehicle drives to the optimal hovering point from the right node, the total receiving energy of the sensor node is subjected to a reduction process. Selection of lambda₁＜λ₂When the base station is closer to the left node, λ is selected₁＞λ₂The situation of (2) is then optimal. Thus, the introduction of 3 is completed.

Through the above discussion proof procedure, one can conclude that:

under the condition of two nodes, when lambda is equal, the ratio of the distance D between the nodes to the flying height h of the unmanned aerial vehicle is larger than

When there are two optimal hover points, when the ratio is less than

There is only one optimal hover point. When lambda is not equal, there is only one maximum anywayThe spiral point is preferred.

Further, theorem 1 specifically includes:

And the coordinates of the right node are

the unmanned aerial vehicle drives away from the base station from 0 moment

In the time of (3), the unmanned aerial vehicle directly flies to the optimal circle point at the speed V and circles for time at the optimal circle point

Finally the remaining time is

Note that the total energy captured by the two nodes in the flight process of the unmanned aerial vehicle is E1, and the total energy captured by the two nodes in the hovering process of the unmanned aerial vehicle is E2, so based on theorem 1 or theorem 3, the maximum energy captured by the unmanned aerial vehicle is: E-E1 + E2.

Further, the theorem 2 specifically includes: considering the charging path of the unmanned aerial vehicle under the condition of energy maximization of two nodes, firstly, it is assumed that: the coordinates of the left node are

And the coordinates of the right node are

unmanned aerial vehicle drives away from base station from 0 moment in time

Then needs to return to the base station at the speed V.

Note that the total energy captured by the two nodes in the flight process of the unmanned aerial vehicle is E1, and the total energy captured by the two nodes in the hovering process of the unmanned aerial vehicle is E2, so based on the theorem 2, the maximum energy captured by the unmanned aerial vehicle is: E-E1 + E2.

The proof process of theorem 1 is as follows:

considering the case of only one optimal hover point, the distance from the base station to the origin is xi, when lambda is₁＝λ₂And is

At the moment, the optimal hover point is just at the origin.

When in use

When the unmanned aerial vehicle drives away from the base station, the unmanned aerial vehicle directly flies to the optimal circle point corresponding to the origin at the speed V. The energy obtained for the time period for the left and right nodes is:

in addition, the

In the time quantum, the unmanned aerial vehicle is just above the origin, and the energy obtained by the left node and the right node is the same as that obtained by the following two nodes:

during the last remaining time of the day, the user may,

in, unmanned aerial vehicle finishes to two node charging, returns the basic station and takes down, and the energy that the left side node obtained will be:

the right node gets the energy:

the sum of the total energy of the unmanned aerial vehicle charging the two nodes in the whole period, namely the maximum energy captured by the two nodes under the condition of the theorem 1 or the theorem 3 is as follows:

when lambda is₁≠λ₂The analysis is the same as the above-mentioned proving process, and will not be described in detail here. Thus, the certification of theorem 1 is completed.

The proof process of theorem 2 is as follows:

when two optimum hover points are involved, i.e.

The fastest flight speed of the unmanned aerial vehicle is assumed to be V, and other assumed situations are the same as those when only one optimal hover point exists. The proof idea at this time is consistent with that of theorem 1, that is, in short, within a certain period T, the flight time is shortest and the hover time is longest.

At this time, because the coordinates of the base station are (xi, 0,0), the coordinates of the two optimal spiral points are (x)₁，0，h)，(x₂，0，h)，x₁，x₂The two optimal solutions respectively corresponding to the lemma 2 only need to hover at one of the hover points, the problem of energy balance is not considered, the problem of energy maximization is only considered, and the hover at the hover point closest to the base station is obvious, so that the node can capture the most energy.

In that

Within the time, the unmanned aerial vehicle drives away from the base station, flies directly to the optimal spiral point on the right side with speed V, and the energy supplemented to the node on the left side at this moment is:

the right node receives energy as follows:

when time is

The drone is located at the right optimal hover point. At this time, the two nodes receive different energy because of different distances, and at this time, for the node on the left side:

and for the right node, this time:

at the right spiral point x₁After a while of hover, it needs to return to the base station at speed V immediately.

The energy received by the node during the return process is the same as the energy received by the node during the arrival of the hover point from the base station, at this time:

so when there are two nodes, the drone also only needs to go through 3 periods. First, starting from the base station and reaching the first optimal hover point x to the right₁. Second, both nodes are charged at the hover point. And finally, under the condition that the remaining time is enough, directly flying back to the base station from the first optimal spiral point on the right side to complete one round of charging, wherein the sum of the total energy received by the two nodes, namely the maximum energy captured by the two nodes based on the theorem 2 is as follows:

thus, the certification of theorem 2 is completed.

In summary, the following conclusions can be drawn:

1. when the unmanned aerial vehicle has only one optimal hovering point, the assumed condition is based on theorem 1, after the unmanned aerial vehicle drives away from the base station, the unmanned aerial vehicle needs to fly directly to the hovering point, and when the unmanned aerial vehicle returns, the unmanned aerial vehicle also needs to fly directly back, and the maximum energy captured by the two nodes is shown in the formula (20).

2. When the unmanned aerial vehicle has two hover points, the assumption condition is based on theorem 2, and the unmanned aerial vehicle must fly at the fastest speed when flying, including flying at the optimal hover point and between the hover point and the base station, but only needs to hover at the hover point closest to the base station, and does not need to hover at both hover points. The maximum energy captured by the two nodes at this time is shown as equation 27.

further, the theorem 4 specifically includes:

when the following three conditions are satisfied:

condition 1:

wherein

Condition 2:

wherein Δ₁＝720h⁸+2304h⁴d⁴+576h²d⁶，

Δ₂＝10368h¹²+82944h⁸d⁴+165888h⁴d⁸；

Condition 3:

condition 4:

condition 5:

condition 6:

the number of the spiral points corresponding to the maximum node capture energy is 3, and the corresponding coordinates are (x)₁，0，h)，(x₂0, h) ofAnd (0,0, h), wherein the corresponding spiral point of the left side and the right side is the maximum value spiral point, the corresponding spiral point of the original point is the maximum value spiral point, and x₁And x₂The coordinates of (c) correspond to:

further, the theorem 5 specifically includes:

only one optimal hover point corresponds to coordinates (0,0, h) at this time.

Respectively substituting the coordinates of the left, middle and right nodes into the formula (1) of the point-to-point charging power consumption of the unmanned aerial vehicle to respectively obtain Q₁，Q₂And Q₃. Corresponding to

And order

ψ＝Q₁+Q₂+Q₃ (28)

And 4, proving the theorem 4 and the theorem 5 by using a derivation method.

In summary, the three node case can be concluded as follows:

1. when condition 1, condition 2 and condition 3 in lemma 4 are not satisfied, there is only one and only one optimal hover point of the drone to maximize the capture energy in the case of linear arrangement of three nodes.

2. When condition 1, condition 2, and condition 3 in lemma 4 are satisfied, if the captured energy is the maximum in the case where three nodes are linearly arranged, there may be 1, 2, 3, and 4 optimal hover points depending on the conditions.

Further, the theorem 3 specifically includes:

On the premise of meeting the conditions 1, 2 and 3 in the conditions of the theorem 4, the condition 4 is not met, and meanwhile, the charging period of the unmanned aerial vehicle is larger than that of the unmanned aerial vehicle

Then, the signal flies back to the base station at the same speed V.

Assuming that the total energy captured by the three nodes is E1 when the unmanned aerial vehicle flies back and forth, and the total energy captured by the three nodes is E2 when the unmanned aerial vehicle hovers, then based on the relevant condition of the theorem 4, the maximum energy captured by the nodes, that is, the maximum energy charged by the unmanned aerial vehicle is E1+ E2;

further, the theorem 4 specifically includes:

considering the problem of maximizing energy of three nodes, it is first assumed that: the three nodes are linearly distributed; the coordinates of the left node are (-d,0, 0); the coordinates of the intermediate node are (0,0,0), corresponding to the position of the origin; the coordinates of the right node are (d,0, 0); the flying height of the unmanned aerial vehicle is assumed to be h; y values corresponding to the coordinates of the three nodes, the base station and the unmanned aerial vehicle are all 0;

And then immediately returns to the base station at the fastest speed V.

the proof process of theorem 4 is as follows:

unmanned aerial vehicle need directly fly to first spiral point with speed V after taking off from the basic station, and in this process, the energy that the left side node that corresponds received is:

the received energy of the intermediate node is then:

and the received energy of the right node is:

same time of day

During the time, unmanned aerial vehicle is located directly over the initial point, is in the optimum point of circling, and at this moment, the left and right sides node is because the symmetry, so the energy of the receipt that corresponds is:

the energy obtained by the intermediate node is maximum, because the connection distance between the unmanned aerial vehicle and the intermediate node is the closest at the moment, the corresponding received energy is

After ensuring sufficient remaining time, in

In the time, the unmanned aerial vehicle must fly back to the base station at the fastest flying speed, and the energy received by the corresponding left, middle and right nodes at the moment is respectively

The sum of the energy transmitted by the drone to the three nodes at this time, i.e. the maximum energy captured based on the lemma 5 in the case of the three nodes, is:

at this point, theorem 4 proves complete.

The proof process of theorem 3 is as follows:

unmanned aerial vehicle starts from the basic station, directly drives to this optimum spiral point with speed V footpath, and the spiral point that the original point corresponds promptly moves the in-process, and the energy that the left side node received is:

the received energy of the intermediate node is then:

and the received energy of the right node is:

same time of day

After ensuring sufficient remaining time, in

thus, theorem 3 case 1 proves complete.

In summary, in the case of three nodes, the following conclusions can be drawn:

1. when only one hover point exists, in order to maximize the energy captured by the node, the unmanned aerial vehicle always flies at the fastest speed when driving away from the base station to the optimal hover point, and flies at the fastest speed in the process of returning to the base station, wherein the total energy captured by the node is shown as a formula (32);

2. when there is not only one point of hovering, that is, three points of hovering, the drone moves at the fastest speed when moving between nodes except when hovering. At this time, a total of 7 cases can be classified according to various satisfying conditions satisfying the theorem 4.

Specifically, in step (5), theorems 5 and 6 are obtained according to the optimal hovering point conditions of the two nodes and the three nodes, so that the optimal hovering point and the optimal charging path of the unmanned aerial vehicle under the condition of n nodes can be obtained.

Further, the theorem 5 specifically includes:

In the proving process of two, three and four nodes, it can be known that there are at most two maximum spiral points between two adjacent nodes, and for the adjacent nodes, the number of peripheral nodes on the line can only make the maximum spiral point position move but not change the number of the maximum spiral points to the maximum spiral points, so corresponding to n nodes, theoretically, according to the relationship between the number of unknown times and the solution, that is, the number of the solution is equal to the number of unknown times, and according to the derivation formula of the above 2, 3 and 4 nodes, the maximum times satisfies the following formula:

the highest degree term is 1+4(n-1) (47)

Therefore, when n nodes are included, 4n-3 maximum spiral points should be included theoretically, and the origin is a polynomial solution anyway, because even terms (including 0 degree) are mutually cancelled out due to the linear symmetrical distribution of the nodes found in the process of proving operation of 2, 3 and 4 nodes.

Because the nodes are distributed in a linear and symmetrical mode, even terms are cancelled out in a positive mode and a negative mode, namely if even nodes exist, the nodes are distributed symmetrically and uniformly around the origin, the number of the nodes is 2n +2, the nodes are distributed linearly and uniformly, and in the derivative of the received total energy, the even terms are cancelled out in a left-right mode because the nodes are distributed symmetrically and uniformly around the origin.

The mathematical expression proves that when the number of the nodes is even, the denominators are the same after the derivatives of the corresponding nodes on the left side and the right side are reduced, the numerators are only different in two terms, and the numerators after the derivatives of the left side and the right side are respectively shown as the following formula:

(x+d)[x⁴-4x³d+6x²d²+2x²h²-4xd³-4xdh²+d⁴+2d²h²+h⁴]......

(48)

(x-d)[x⁴+4x³d+6x²d²+2x²h²+4xd³+4xdh²+d⁴+2d²h²+h⁴]......

(49)

wherein d is the position of the left and right side nodes from the origin, the derivative of the corresponding nodes on the left and right sides of the ellipsis part is the same part after the approximate division integration, and different parts are also written. After expanding the equations (48) and (49), it is found that the even terms have the same value but opposite signs, and the odd terms have the same value and the same signs. Therefore, because the ellipses are identical, the polynomial degree has odd numbers and even numbers, after multiplication, the sign is unchanged, that is, the value of the even number term of the corresponding formula (48) multiplied by the ellipses is equal to the value of the even number term of the formula (49) multiplied by the ellipses, the sign is opposite, and the values are finally cancelled out, the value of the odd number term of the remaining corresponding formula (48) multiplied by the ellipses is equal to the value of the odd number term of the formula (49) multiplied by the ellipses, the sign is identical, and finally the two formulas are added to become twice of the original value. The other pairwise symmetric node analysis is the same as the above formula, and will not be described herein.

When the number of the nodes is odd, the analysis of the corresponding nodes on the left side and the right side is the same as the above formula. The denominator after the derivative reduction integration of the energy at the origin node is the same as the polynomial corresponding to other nodes, and the numerator is shown as the following formula:

x[(x-d)²+h²]²[(x+d)²+h²]²....[(x-(2n+1)d)²+h²]²[(x+(2n+1)d)²+h²]² (50)

the omitted part of the equation (50) is the same as the left and right polynomials, and the part except the x part of the above equation can be understood as the product of the square product of the denominators of the symmetric node energy capture equation. One pair of polynomials is assumed to be [ (x-d)²+h²]²[(x+d)²+h²]²After deployment, the following formula is shown:

x⁸-4x⁶d²+4x⁶h²-4x⁴h²d²+6x⁴d⁴+6x⁴h⁴-4x²d⁴h²-4x²d⁶+4x²d²h⁴4x²h⁶+6h⁴d⁴+4h²d⁶+d⁸+h⁸ (51)

from equation (51), it is found that all odd terms are eliminated and the rest are even terms. Except for this pair, the polynomial products and equations (51) of the remaining symmetric nodes are similar, with only the corresponding coefficients being different. Therefore, after the multiplication of the n-half pairs, i.e., the multiplication of innumerable even terms, the result must also be even terms, and all the multiplication by x in the expression (50) becomes odd terms.

Up to this point, the derivatives for the n nodes case contain only odd terms to justify.

It is also easy to prove that the min term must be 1, i.e., only 0 solution at this time.

Correspondingly, because the solution must contain the maximum and minimum values, i.e. the number of the maximum values plus the number of the minimum values must be 1+4(n-1), the total energy received by the node must be monotonically increased when reaching the first maximum spiral point from infinity, and the total energy received by the node must be monotonically decreased when going from the last spiral point to infinity. The left side and the right side tend to be minus infinity, and the maximum value point is less than the minimum value point by one. Theorem 5 proves that the process is finished.

Theorem 6: under the condition that n nodes are symmetrically and uniformly distributed about the origin, when the number of the optimal spiral points is even, the origin must be a minimum value point.

And (3) proving that: this rule is certainly true for 2, 3, and 4 nodes, extending to the case of multiple nodes. The minimum value point means that the left area of the derivative of the point is negative and the right area is positive.

The same is proved by the same theorem 6, because the spiral points on the left side and the right side are symmetrical pairwise, only one side cannot be present, but the other side does not exist, when the number of the spiral points is even, the maximum spiral point can only exist on the left side and the right side of the original point, and the original point can only be a minimum point. Theorem 6 proves that the process is finished.

Theorem 7: under the condition that n nodes are symmetrically and uniformly distributed about the origin, when the number of the optimal spiral points is odd, the origin is necessarily the maximum value point.

And (3) proving that: the proof method is consistent with the theorem 7 analysis method when four nodes exist, and the method is named as a symmetric analysis method. And the symmetrical analysis method continuously performs superposition comparison aiming at the two conditions, and finally determines that the origin is the maximum point.

Firstly, because the maximum spiral points on the left side and the right side of the origin point are pairwise appeared, the number of the spiral points can be changed from an even number to an odd number only by the appearance of the maximum spiral point corresponding to the origin point, and the existence certification is finished.

The first condition that uses the symmetry analysis method is that there is the condition of two maximum values spiral points in node both sides, uses arbitrary node to do the perpendicular line of perpendicular to x axle as the intersect this moment, assumes that this node right-hand member still has x nodes, the first maximum value of perpendicular line right-hand member spirals the point, must have a point a at the perpendicular line left end, to unmanned aerial vehicle at the first maximum value spiral point of perpendicular line right-hand member with spiral at a point, to using this node as the center, the x node in right side, the energy that the x node in left side received must equal. In contrast, a node to the left of the origin receives more energy than before because point a is closer to the origin than the node, and therefore more energy is received at point a than at the first maximum hover point to the right of the vertical line, and at this time, the maximum hover point to the left of the vertical line may only coincide with or be to the left of point a. The energy received at the maximum hover point on the side of the node closer to the origin is necessarily much greater than the energy received at the maximum hover point on the side of the node further from the origin.

The second scenario can be applied to a scenario where two maximum circle points exist between adjacent nodes, a vertical straight line is drawn at the center point of the adjacent node, and if x nodes are located on the right side of the vertical line, the maximum circle point located at the right end of the vertical line inevitably has a point b which is the same as the maximum circle point at the right end of the vertical line in distance from the vertical line, but the point b is located on the left side of the vertical line. At point b or the maximum hover point to the right of the perpendicular, the energy received by the x nodes to the right of the perpendicular and the x nodes to the left of the perpendicular must be the same, while the remaining n-2x nodes remain, and it is apparent that since point b is closer to the origin, the remaining n-2x nodes receive more energy than when hovering at the first maximum hover point to the right of the perpendicular. The maximum hover point between adjacent nodes is hovering, and the more energy a node captures as the maximum hover closer to the origin.

The linear and uniform distribution of the n nodes is formed by the first situation and the second situation, and the two situations are continuously compared and superposed, because the nodes are symmetrical about the original point, the maximum spiral points can be the same in pairs, only one side of the maximum spiral points needs to be analyzed, and finally the maximum spiral point at the original point is much larger than all the maximum spiral points at the left side and the right side, namely, the original point corresponds to the maximum spiral point which is the maximum spiral point, namely, the optimal spiral point.

Therefore, when the number of the nodes is odd, the maximum point corresponding to the origin must be the maximum point. Theorem 7 proves that the process is finished.

According to the above theorem, an exhaustive search method is adopted to obtain the optimal path for charging the unmanned aerial vehicle based on the optimal hover point in this embodiment: the distance between the nodes on one side of the bridge is 9.2 meters, and the distance between the two rows of nodes of the bridge is 16 meters; the coordinates of the upper sides of the nodes are (Xi, 8, 0), wherein Xi is the abscissa of the corresponding node, and the coordinates of the lower sides of the nodes are (Xi, -8, 0), wherein Xi is the abscissa of the corresponding node; the coordinates of the base station are (ξ, -12,0), where ξ is greater than either Xi; the nodes are distributed symmetrically on the x axis and the y axis; unmanned planeThe fastest flying speed of the unmanned aerial vehicle is V, and the lower limit of the flying height h of the unmanned aerial vehicle is 5 meters; the flight period of the unmanned aerial vehicle is T, and T must be greater than T at the moment

When the flying height h of the unmanned aerial vehicle is lower than 11 meters and higher than 5 meters, two optimal hovering points exist, corresponding coordinates are [0, Y, h ] and [0, -Y, h ], and the value range of Y is between 0 and 8. Then, to enable the nodes on the two sides of the bridge to capture the maximum energy, the description of the charging path of the unmanned aerial vehicle should be as follows:

the unmanned aerial vehicle drives away from the base station from zero moment and directly flies to the optimal circle point nearest to the base station at a speed V, and the coordinate of the circle point nearest to the base station is [0, -Y, h ]]So that the disk rotation time is at the disk rotation point

And then directly returning to the base station at the speed V. The energy captured by the node is maximum at this time.

When the flying height of the unmanned aerial vehicle is larger than 11 meters, only one optimal hovering point exists, and the corresponding coordinate is [0, 0, h ], so that the nodes on two sides of the bridge capture the maximum energy, the description of the charging path of the unmanned aerial vehicle is as follows:

the unmanned aerial vehicle drives away from the base station from zero time and drives to the optimal circle point at the maximum speed V, and the unmanned aerial vehicle circles

Returning at speed V immediately after time. The energy captured by the node is maximum at this time.

The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.

Claims

1. A path planning method for a single-row wireless charging sensor network of a bridge is characterized by comprising the following specific steps:

(3) obtaining an optimal hover point and optimal path distribution according to the linear and uniform distribution condition of the two nodes, wherein the optimal hover point is obtained based on the maximization of energy captured by the two nodes, and the optimal path distribution is obtained based on the optimal hover point and the total energy captured by the two nodes during the hovering and flying of the unmanned aerial vehicle;

(4) obtaining an optimal hover point and optimal path distribution according to the linear and uniform distribution condition of the three nodes, wherein the optimal hover point is obtained based on the maximization of energy captured by the three nodes, and the optimal path distribution is obtained based on the optimal hover point and the total energy captured by the three nodes during the hovering and flying of the unmanned aerial vehicle;

(5) obtaining an optimal hovering point and an optimal path for charging the unmanned aerial vehicle under the condition of n nodes according to the optimal hovering point conditions of the two nodes and the three nodes, wherein the optimal hovering point and the optimal path for charging the unmanned aerial vehicle are obtained based on the total energy captured by the nodes, and the optimal path is obtained based on the optimal hovering point by adopting an exhaustive search method in combination with the step (3) and the step (4);

in the step (5), theorems 5, 6 and 7 are obtained according to the optimal hovering point conditions of the two nodes and the three nodes, so that the optimal hovering point and the optimal charging path of the unmanned aerial vehicle under the condition of n nodes can be obtained;

the theorem 5 specifically includes:

under the condition that n nodes are symmetrically and uniformly distributed about an origin, the derivatives of the total energy captured by the nodes only contain odd terms;

the theorem 6 specifically includes:

under the condition that n nodes are symmetrically and uniformly distributed about the origin, when the number of the optimal spiral points is an even number, the origin must be a minimum value point;

the theorem 7 specifically includes:

2. The path planning method for the single-file wireless charging sensor network of the bridge according to claim 1, wherein in the step (3), for the case that the energy of two nodes is maximized, theorem 1, theorem 2 and theorem 3 are proposed, and based on the theorem 1, the theorem 2 and the theorem 3, theorem 1 and theorem 2 are proposed for the charging path of the unmanned aerial vehicle with the maximized energy in the case of two nodes;

the theorem 1, the theorem 2 and the theorem 3 are based on the same scene, and the scene is as follows: fixing two nodes on a bridge, wherein the two nodes are parallel to one side of the bridge, using a Cartesian space rectangular coordinate system as a reference, symmetrically placing the two nodes at two ends of a zero point, the linear distance between the two nodes is D, placing the nodes on an x axis, and the coordinate of the node on the left side is

And the coordinates of the right node are

The base station is positioned at the same side of the two nodes of the bridge and is at a certain distance from the node at the right side, and the coordinate of the base station is ([ xi ], 0 and 0), wherein

the theory 1 specifically comprises the following steps:

condition 1: the lagrangian multipliers corresponding to the nodes are equal;

The optimal hover point is (0,0, h);

the theory 2 specifically comprises the following steps:

two optimal hover points exist when the following conditions are met:

condition 1: the lagrangian multipliers corresponding to the nodes are equal;

the theory 3 specifically comprises the following steps:

there is only one optimal hover point when the following conditions are met:

condition 1: the lagrangian multipliers corresponding to the nodes are unequal;

the corresponding optimal circle point coordinate is (x, 0, h); wherein x is determined according to specific numerical values and has no definite algebraic form;

the theorem 1 specifically includes:

the coordinates of the left node are

And the coordinates of the right node are

The coordinates of the base station are (xi, 0, 0); the fastest flight speed of the unmanned aerial vehicle is V; unmanned aerial vehicleThe charging period is T, and the numerical value of T is larger than the time of the roundtrip spiral point; the flying height of the unmanned aerial vehicle is h; y in the corresponding coordinates of the nodes on the two sides, the base station and the unmanned aerial vehicle in the flight path is 0;

in the case of theorem 1, that is, when only one optimal hover point is included, the description of the optimal path of the drone is:

the unmanned aerial vehicle drives away from the base station from 0 moment

Finally the remaining time is

The unmanned aerial vehicle needs to return to the base station at the speed V immediately;

the sum of the total energy of the unmanned aerial vehicle charging the two nodes, namely the maximum energy captured by the two nodes under the condition of lemma 1 is as follows:

wherein E is_{Left 1}And E_{Right 1}Respectively show when

When the unmanned aerial vehicle drives away from the base station, the unmanned aerial vehicle directly flies to the optimal circle point corresponding to the origin at the speed V, and the energy obtained by the left node and the right node is obtained; e_{Left 2}And E_{Right 2}Are respectively shown in

In the time period, the unmanned aerial vehicle is just above the optimal hovering point, and the energy obtained by the left node and the right node is obtained; e_{Left 3}And E_{Right 3}Respectively represent

In the method, after the unmanned aerial vehicle finishes charging the two nodes, the unmanned aerial vehicle returns to the base station to be rested, and energy obtained by the left node and the right node is obtained;

the theorem 2 specifically includes:

considering the unmanned aerial vehicle charging path under the condition of energy maximization of two nodes, firstly, the coordinate of the left node is

And the coordinates of the right node are

unmanned aerial vehicle drives away from base station from 0 moment in time

In, the drone flies at speed V to the first optimal hover point on the right, at which hover point the time is convoluted

Then, the mobile terminal returns to the base station at the speed V;

the sum of the total energy charged by the unmanned aerial vehicle to the two nodes, namely the maximum energy captured by the two nodes based on the lemma 2, is:

wherein E is_{Left 1}And E_{Right 1}Respectively show when

When the unmanned aerial vehicle drives away from the base station, the unmanned aerial vehicle directly flies to the right optimal circle point at the speed V, and the energy obtained by the left node and the right node is obtained; e_{Left 2}And E_{Right 2}Respectively represent the time

The unmanned aerial vehicle is positioned at the optimal circle point on the right side, and the energy obtained by the left node and the right node is obtained; e_{Left 3}And E_{Right 3}Respectively represent

In, unmanned aerial vehicle finishes to two node charges, returns the basic station and takes down, to the energy that left and right node obtained.

3. The path planning method for the single-file wireless charging sensor network of the bridge according to claim 1, wherein in the step (4), for the case that the energy of three nodes is maximized, theorem 4 and theorem 5 are proposed, and based on the theorem 4 and the theorem 5, theorem 3 and theorem 4 are proposed for the charging path of the unmanned aerial vehicle with the maximized energy of three nodes;

the scenes corresponding to the theorem 4 and the theorem 5 are consistent, and the scenes are as follows: the nodes are symmetrically distributed and divided into a left point, a middle point and a right point, the linear topological structure is adopted, the base station is positioned on one side of the right node, the coordinates of the left node are (-d,0,0), the coordinates of the middle node are (0,0,0), the position corresponding to the origin point is (d,0, 0); the distance of the x axis of the base station is xi, and the corresponding coordinate is (xi, 0, 0); the starting point of the unmanned aerial vehicle is located at the base station; the fastest flight speed of the unmanned aerial vehicle is V;

the theory 4 specifically comprises the following steps:

when the following three conditions are satisfied:

condition 1:

wherein

Condition 2:

wherein Δ₁＝720h⁸+2304h⁴d⁴+576h²d⁶，

Δ₂＝10368h¹²+82944h⁸d⁴+165888h⁴d⁸；

Condition 3:

condition 4:

when the condition 4 is satisfied, but the following condition 5 is not satisfied:

condition 5:

condition 6:

the theory 5 specifically comprises the following steps:

at this time, the corresponding coordinates of only one optimal spiral point are (0,0, h);

the theorem 3 specifically includes:

the three nodes are linearly distributed in a charging path of the unmanned aerial vehicle under the condition that the energy of the three nodes is maximized; the coordinates of the left node are (-d,0, 0); the coordinates of the intermediate node are (0,0,0), corresponding to the position of the origin; the coordinates of the right node are (d,0, 0); the coordinates of the base station are (xi, 0, 0); the flying height of the unmanned aerial vehicle is h; the charging period of the unmanned aerial vehicle is T; the fastest flight speed of the unmanned aerial vehicle is V; the y values corresponding to the three nodes, the base station and the flight path coordinates of the unmanned aerial vehicle are 0;

The moving path of the unmanned aerial vehicle with the maximum charging energy is described as follows:

Then, directly flying back to the base station at the same speed V;

the unmanned aerial vehicle starts from the base station at zero time and directly flies to x at speed V₁Corresponding optimumHover point, hover time

Then, directly flying back to the base station at the speed V;

The movement path for maximum charging energy of the drone is described as:

Then, directly flying back to the base station at the same speed V;

case 4: when all of the conditions in lemma 4 are met, there are four maximum hover points, when x is₁Corresponding maximum greater than x₂The corresponding maximum value ensures that the charging period of the unmanned aerial vehicle is greater than

The movement path for maximum charging energy of the drone is described as:

Then, directly flying back to the base station at the speed V;

when x is₁Corresponding maximum value less than x₂The corresponding maximum value ensures that the charging period of the unmanned aerial vehicle is greater than

The moving path of the unmanned aerial vehicle for charging energy maximization is described as：

Then, directly flying back to the base station at the speed V;

when the unmanned aerial vehicle flies back and forth, the total energy captured by the three nodes is E1, and when the unmanned aerial vehicle is hovering, the total energy captured by the three nodes is E2, so that the maximum energy captured by the nodes, namely the maximum energy charged by the unmanned aerial vehicle is E1+ E2 based on the relevant condition of the lemma 4;

the theorem 4 specifically includes:

the three nodes are linearly distributed in a charging path of the unmanned aerial vehicle under the condition that the energy of the three nodes is maximized; the coordinates of the left node are (-d,0, 0); the coordinates of the intermediate nodes are (0,0, 0); the coordinates of the right node corresponding to the position of the origin are (d,0, 0); the coordinates of the base station are (xi, 0, 0); the flying height of the unmanned aerial vehicle is h; the flying speed of the unmanned aerial vehicle is V; the flight period of the drone is T, where T must be greater than

when the condition of lemma 5 is satisfied, only 1 optimal hover point exists; at this time, to maximize the energy captured by the node, the description of the charging path of the unmanned aerial vehicle should be:

Then returning to the base station at the fastest speed V;

when the unmanned aerial vehicle flies back and forth, the total energy captured by the three nodes is E1, and when the unmanned aerial vehicle hovers, the total energy captured by the three nodes is E2, so that the maximum energy captured by the nodes, namely the maximum energy charged by the unmanned aerial vehicle is E1+ E2 based on the relevant condition of the lemma 5.