A method for predicting the flight trajectory of an aircraft based on lateral maneuvering characteristics and no-fly zone constraints

Abstract

A method for predicting the flight trajectory of an aircraft based on lateral maneuvering characteristics and no-fly zone constraints. The method is as follows: Before prediction, first obtain the estimated values of the state variables of the aircraft in a previous period of time and the current moment through trajectory tracking; analyze the lateral maneuvering characteristics of the hypersonic aircraft during the glide phase and design a corresponding trajectory fitting model; design a method for solving the parameters of the nonlinear model, solve the fitting parameters of the fitting model, and predict the lateral trajectory of the aircraft; analyze the influence of no-fly zone constraints on the lateral trajectory of the aircraft. After introducing the constraints of the target and the bypass area, the trajectory of the enemy aircraft around the no-fly zone is divided into type C and type S, and then analyze its influence on the ballistic prediction stop time and the setting of the preset interception point; reasonably set the prediction end time and the position of the preset interception point according to the no-fly zone constraints. The present invention designs a trajectory fitting model according to the lateral maneuvering characteristics of the hypersonic aircraft, and can preferably predict the lateral ballistic trajectory.

Description

Technical Field

[0001] The present invention relates to an aircraft trajectory prediction method based on lateral maneuvering characteristics and no-fly zone constraints, which can predict subsequent lateral trajectories according to aircraft trajectory tracking results and analyze the influence of no-fly zone constraints on preset interception points. Background Art

[0002] The main motion modes of lateral motion of hypersonic vehicles in the gliding phase are swing maneuvers and turning maneuvers. The lateral trajectory prediction of the vehicle needs to be combined with its lateral maneuvering characteristics. The lateral maneuvering of the vehicle in the terminal trajectory depends on the no-fly zone constraints. It is urgent to develop a lateral trajectory prediction method and analyze the impact of no-fly zone constraints on the preset interception point. Summary of the invention

[0003] In order to overcome the shortcomings of the prior art, the present invention analyzes the influence of no-fly zone constraints on preset interception points and provides an aircraft trajectory prediction method based on lateral maneuvering characteristics and no-fly zone constraints.

[0004] The objective of the present invention is achieved through the following technical solutions:

[0005] A method for predicting aircraft trajectory based on lateral maneuvering characteristics and no-fly zone constraints, the method steps are as follows:

[0006] Step 1: Before forecasting, first obtain the estimated value of the aircraft's state at a certain time in the past and at the current moment through trajectory tracking [x 1 ,y 1 ,z 1 ,V x1 ,V y1 ,V z1 ]···[x k ,y k ,z k ,V xk ,V yk ,V zk ];

[0007] Step 2: Analyze the lateral maneuvering characteristics of the hypersonic vehicle during the gliding phase. The main lateral motion modes include swing maneuvers and turning maneuvers, and design the corresponding trajectory fitting model.

[0008] Step 3: Design a method to solve the nonlinear model parameters. The parameters that are established are used to solve the fitting parameters θ of the fitting model. k , predict the lateral trajectory of the aircraft, where S represents the fitting value at each moment With the actual value y i The minimum sum of squares;

[0009] Step 4: Analyze the influence of no-fly zone constraints on the lateral trajectory of the aircraft. After introducing the constraints of the target and the bypass area, the trajectory of the enemy aircraft around the no-fly zone is divided into type C and type S, and then analyze its influence on the ballistic prediction stop time t end and the setting of the preset interception point;

[0010] Step 5: Reasonably set the prediction end time t end and the position of the preset interception point according to the no-fly zone constraints.

[0011] Furthermore, the specific content of the said Step 2 is as follows:

[0012] Step 2-1: For the oscillating maneuver, its trajectory curve is approximately described by a straight line and a sine curve, that is, it is fitted by the DPL+LA optimal maneuver model function, and the fitting function is expressed as:

[0013] a(t) = C 2 f 2 (t)sin(f 1 (t)x(t)+η 0 )+C 1 t+C 0

[0014] Among them, a(t) represents the function of the fitting acceleration changing with time, C 2 f 2 (t) is the maneuver amplitude, f 1 (t) is the maneuver frequency, x(t) is the range on the ox axis from the start time of the maneuver to the current time, η 0 is the phase angle at the start time of the maneuver, C 1 t+C 0 is the distance at the start time of the maneuver;

[0015] Step 2-2: For the turning maneuver, its lateral trajectory is a monotonic curve, either monotonically increasing or monotonically decreasing, and it can be approximated by an Nth-degree polynomial. The fitting function is expressed as:

[0016]

[0017] Among them, a i is the coefficient of the Nth-degree polynomial, x(t) i is the range on the ox axis from the start time of the maneuver to the current time, and a(t) represents the function of the fitting acceleration changing with time;

[0018] Step 2-3: When predicting the lateral trajectory, the following prior information can also be considered. The combination of prior information can be considered in actual combat: ① The target maneuverability (the magnitude of the maneuver acceleration) is known; ② The target landing point is known; ③ The target evasion object is known.

[0019] Further, to improve the maneuvering penetration ability of the trajectory aircraft, the maneuvering amplitude and frequency are time-varying. Therefore, when predicting the lateral trajectory, let f 1 (t), f 2 (t) be linear functions of time, and we have where a, b, c, and d are constants.

[0020] Further, in step three, the steps of solving the fitting parameters θ k of the fitting model are specifically as follows:

[0021] Step 3-1: Use the nonlinear least squares method to estimate the nonlinear model parameters. Fit a nonlinear model with n (m > n) parameters using m observation values to minimize the sum of the squared residuals. The basis is to approximate the model through a linearization approximation method, and then extract the parameters through continuous iteration;

[0022] Step 3-2: Assume there are observed data (x i , y i ), i = 1, 2,..., m, where x i represents the displacement of the aircraft along the x-axis at the i-th moment, and y i represents the displacement of the aircraft along the y-axis at the i-th moment. The function form to be fitted is y = f(x, θ), θ = [θ 1 , θ 2 ,..., θ n T are the parameters to be estimated. Solve to make where is the fitted value of θ;

[0023] Step 3-3: Since the residual e j = y j - f(x j , θ), rewrite the minimum sum of squared residuals formula as where S is the sum of squared residuals, θ j is the j-th parameter to be estimated, e i is the i-th residual, and m is the number of observation values. This system of equations has no closed-form solution and requires an initial value to be given, and then the final estimated value is obtained through continuous iteration;

[0024] Step 3-4: Use the Gauss-Newton iteration algorithm to solve the estimated value. Assume θ k is the solution obtained in the k-th iteration. Take the first-order Taylor expansion of y = f(x, θ) with respect to θ k to get where J ij is the element of the Jacobian matrix, which changes with the number of iterations. The calculation method is:

[0025] ​Step 35: Further represent the residual as Δy i =y i -f(x i ,θ k ), where y i is the i-th observed value, Δy i is the difference of the i-th observed value, Δθ j is the difference of the j-th estimated parameter. Substitute them into the iterative calculation and after arrangement, we get Written in matrix form:

[0026] J T JΔθ=J T Δy

[0027] According to the above formula, the correction value Δθ of the parameter to be solved can be obtained, and the parameter value is updated to θ k+1 =θ k +Δθ, and continue the iteration until the iteration end condition is met.

[0028] Furthermore, the specific content of Step 4 is as follows:

[0029] Step 41: After introducing the constraints of the target and the no-fly zone, the trajectory of the enemy aircraft around the no-fly zone is divided into C type and S type. Several forms of the aircraft flying around the no-fly zone can be predicted for the initial maneuver, but the end time of the C-type maneuver, the start time of the second half of the S-type maneuver, and the end time of the maneuver cannot be predicted; assume that the maneuver degree is unknown and the maneuver form is known (i.e., assume that S or C type is known through the no-fly zone and the target to be flown), but the start and end times of the maneuver are unknown (the start time is judged through maneuver detection, and the end time is restricted by assuming that the aircraft will return within 50 km of the initial trajectory). For different working conditions, analyze the stop time of the ballistic prediction and the setting of the interception point;

[0030] Step 42: The interception point is in the latter part of flying around the no-fly zone. This is the most unfavorable situation. Since the end time of flying around is unknown, without other settings (such as the target aircraft needs to return to the vicinity of the original trajectory), the stop time of the ballistic prediction cannot be determined. At the same time, the preset interception point will happen to be in the stage with large prediction errors. Therefore, it is necessary to avoid presetting the interception point in this stage;

[0031] Step 43: The interception point is in the former part of flying around the no-fly zone. At this time, since the flying around has started, the prediction needs to be carried out all the time. Adopt the turning maneuver fitting model, and the trajectory of this section can be effectively predicted according to the previous section of the flight trajectory, and the influence on the interception is small;

[0032] Step 44: The interception point is after flying around the no-fly zone. Since the flying around has ended, the end time of the aircraft flying around can be confirmed through the means of maneuver mutation detection. After that, the prediction needs to be carried out all the time. Adopt the common fitting model, and it can be considered that there is no influence on the interception;

[0033] Step 4 and Step 5: Before the interception point flies around the no-fly zone, since the fly-around has not started, it can be confirmed through maneuver mutation detection means that the aircraft has not entered the fly-around moment, and the prediction needs to be carried out all the time. Using the common fitting model, it can be considered that there is no impact on the interception.

[0034] The beneficial effects of the present invention compared with the prior art are as follows:

[0035] (1) Design a trajectory fitting model according to the lateral maneuver characteristics of the hypersonic vehicle, which can better predict the lateral ballistic trajectory.

[0036] (2) Analyze the influence of different no-fly zone constraints on the aircraft trajectory, and then analyze the corresponding prediction stop moment and interception point setting, which can improve the accuracy of interception. Brief Description of the Drawings

[0037] Figure 1 It is a schematic diagram of the C-shaped and S-shaped trajectories of the aircraft around the no-fly zone and the interception point position of the present invention. Detailed Embodiment

[0038] The technical solutions of the present invention will be further described below in conjunction with the drawings and embodiments, but it is not limited thereto. Any modification or equivalent replacement of the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention shall be covered by the protection scope of the present invention.

[0039] Embodiment 1:

[0040] An aircraft ballistic prediction method based on lateral maneuver characteristics and no-fly zone constraints, and the specific steps of its ballistic prediction and analysis are as follows:

[0041] Step 1: Before prediction, first obtain the estimated values of the state variables of the aircraft in a previous period of time and the current moment through ballistic tracking [x 1 , y 1 , z 1 , V x1 , V y1 , V z1 ··· [x k , y k , z k , V xk , V yk , V zk , where x 1 represents the displacement of the aircraft on the x-axis at the initial moment, y 1 represents the displacement of the aircraft on the y-axis at the initial moment, z 1 represents the displacement of the aircraft on the z-axis at the initial moment, V x1 represents the velocity of the aircraft on the x-axis at the initial moment, V y1 represents the velocity of the aircraft on the y-axis at the initial moment, V z1Represents the z-axis velocity of the aircraft at the initial moment, x k Represents the x-axis displacement of the aircraft at the k-th moment, y k Represents the y-axis displacement of the aircraft at the k-th moment, z k Represents the z-axis displacement of the aircraft at the k-th moment, V xk Represents the x-axis velocity of the aircraft at the k-th moment, V yk Represents the y-axis velocity of the aircraft at the k-th moment, V zk Represents the z-axis velocity of the aircraft at the k-th moment;

[0042] Step 2: Analyze the lateral maneuvering characteristics of the hypersonic aircraft during the glide phase. The main motion modes of its lateral motion are wagging maneuver and turning maneuver. Design the corresponding trajectory fitting model y = F(θ 1 , θ 2 … θ k , t);

[0043] Step 3: Design a method for solving the parameters of the nonlinear model (parameters that make hold), and solve the fitting parameters θ k of the fitting model to predict the lateral trajectory of the aircraft;

[0044] Step 4: Analyze the influence of no-fly zone constraints on the lateral trajectory of the aircraft. After introducing the constraints of the target and the fly-around area, the trajectories of the enemy aircraft around the no-fly zone can be divided into C-type and S-type. Then analyze its influence on the ballistic prediction stop time t end and the setting of the preset interception point;

[0045] Step 5: Reasonably set the prediction end time t end and the position of the preset interception point according to the no-fly zone constraints.

[0046] Furthermore, the trajectory fitting model designed in Step 2 is as follows:

[0047] Step 2.1: For the wagging maneuver, its trajectory curve can be approximately described by a straight line and a sine curve, that is, use the DPL+LA optimal maneuver model function for fitting. The fitting function is expressed as:

[0048] a(t) = C 2 f 2 (t)sin(f 1 (t)x(t) + η 0 ) + C 1 t + C 0

[0049] Where: C 1 t + C 0 is the distance from the start time of the maneuver; C 2 f 2 (t) is the maneuver amplitude; f 1(t) is the maneuvering frequency; η 0 is the phase angle at the start time of maneuver; x(t) is the range on the ox axis from the start time of maneuver to the current time. Since the trajectory vehicle increases its maneuvering penetration ability by using time-varying maneuvering amplitude and frequency, when predicting the lateral trajectory, let f 1 (t), f 2 (t) be a first-order function of time, and there is where a, b, c, d are constants;

[0050] Step 2.2: For the turning maneuver, its lateral trajectory is a monotonic curve, either monotonically increasing or decreasing, and can be approximated by an Nth-degree polynomial, described as follows:

[0051]

[0052] where a i is the coefficient of the Nth-degree polynomial. To simplify the fitting function and meet the requirements of trajectory prediction at the same time, let N = 2. At this time, the fitting function is expressed as a quadratic curve model, that is, the QA model is selected as the best maneuver model for predicting the lateral trajectory of the trajectory vehicle during the maneuvering turn, and the expression is as follows:

[0053] a(t) = C 2 t 2 + C 1 t + C 0

[0054] When predicting the lateral trajectory, the following prior information can also be considered. The combination of prior information can be considered in actual combat: ① The target maneuvering ability (the magnitude of the maneuvering acceleration) is known; ② The target landing point is known; ③ The target evasion object is known.

[0055] Furthermore, the steps to solve the fitting parameters of the fitting model in Step 3 are as follows:

[0056] Step 3.1: Use the nonlinear least squares method to estimate the parameters of the nonlinear model. Use m observed values to fit a nonlinear model with n (m > n) parameters to minimize the sum of the squares of the fitting residuals. Its basis is to approximate the model through a linearization approximation method, and then extract the parameters through continuous iteration;

[0057] Step 3.2: Assume there are observed data (x i , y i ), i = 1, 2,..., m, and the function form to be fitted is y = f(x, θ), θ = [θ 1 , θ 2 , …, θ n T is the parameter to be estimated, and solve to make ​

[0058] Step 3.3: Since e j = y j - f(x j , θ), rewrite the sum of squared residuals formula as This system of equations has no closed-form solution. An initial value needs to be given, and then the final estimated value can be obtained through continuous iteration;

[0059] Step 3.4: Use the Gauss-Newton iteration algorithm to solve for the estimated value. Let θ k be the solution obtained in the k-th iteration. Take the first-order Taylor expansion of y = f(x, θ) with respect to θ k , and we have where J ij is an element of the Jacobian matrix, which changes with the number of iterations. The calculation method is:

[0060] Step 3.5: Further express the residual as Δy i = y i - f(x i , θ k ), substitute it into the iterative calculation, and after rearrangement, we get Written in matrix form:

[0061] J T JΔθ = J T Δy

[0062] Based on the above equation, Δθ can be solved, and the parameter value is updated to θ k+1 = θ k + Δθ. Continue the iteration until the iteration end condition is met.

[0063] Furthermore, the steps for analyzing the influence of the no-fly zone constraint on the lateral trajectory of the aircraft and setting the interception point in Step 4 are as follows:

[0064] Step 4.1: After introducing the constraints of the target and the bypass area, the trajectory of the enemy aircraft around the no-fly zone can be divided into C-type and S-type. For the initial maneuver of the aircraft flying around the no-fly zone, it can be predicted, but the end time of the C-type maneuver, the start time of the second half of the S-type maneuver, and the end time of the maneuver cannot be predicted. Make the following assumptions: Assume that the maneuver degree is unknown and the maneuver form is known (i.e., assume that the S or C type is known through the no-fly zone and the target to be flown), but the start and end times of the maneuver are unknown (the start time is judged through maneuver detection, and the end time is restricted by assuming that the aircraft will return within 50 km of the initial trajectory). For different working conditions, analyze the stop time of the ballistic prediction and the setting of the interception point;

[0065] Step 4.2: The interception point is in the latter stage of flying around the no-fly zone. This is the most unfavorable situation. Since the end time of flying around is unknown, without other assumptions (such as the target aircraft needs to return to the vicinity of the original trajectory), it is impossible to determine the stop time of the ballistic prediction. At the same time, the preset interception point will happen to be in the stage with a large prediction error. Therefore, it is necessary to avoid presetting the interception point in this stage;

[0066] Step 4.3: The interception point is in the former stage of flying around the no-fly zone. At this time, since the flying around has started, the prediction needs to be carried out all the time. By using the turning maneuver fitting model, this section of the trajectory can be effectively predicted according to the previous section of the flight trajectory, and the impact on the interception is small;

[0067] Step 4.4: The interception point is after flying around the no-fly zone. Since the flying around has ended, the end time of the aircraft's flying around can be confirmed through the means of maneuver mutation detection. After that, the prediction needs to be carried out all the time. By using the common fitting model, it can be considered that there is no impact on the interception;

[0068] Step 4.5: The interception point is before flying around the no-fly zone. Since the flying around has not started, the time when the aircraft has not entered the flying around can be confirmed through the means of maneuver mutation detection. The prediction needs to be carried out all the time. By using the common fitting model, it can be considered that there is no impact on the interception.

Claims

1. A method for predicting the trajectory of an aircraft based on lateral maneuvering characteristics and no-fly zone constraints, characterized in that: The steps of the method are as follows: Step 1: Before the prediction, first obtain the estimated values of the state variables of the aircraft for a previous period and the current moment through ballistic tracking [x 1 , y 1 , z 1 , V x1 , V y1 , V z1 ··· [x k , y k , z k , V xk , V yk , V zk ; Step 2: Analyze the lateral maneuvering characteristics of the hypersonic aircraft during the glide phase. The main motion modes of its lateral motion are swing maneuver and turning maneuver, and design corresponding trajectory fitting models; for the swing maneuver, its trajectory curve is approximately described by a straight line and a sine curve, that is, the DPL+LA optimal maneuver model function is used for fitting; for the turning maneuver, its lateral trajectory is a monotonic curve, either monotonically increasing or monotonically decreasing, and an Nth-degree polynomial is used to approximate it. Step 3: Design a method for solving non-linear model parameters to find the parameters that make hold, and solve the fitting parameter θ of the fitting model k , and predict the lateral trajectory of the aircraft, where S represents the fitting value at each moment and the actual value y i of the minimum sum of squares; Step 4: Analyze the influence of no-fly zone constraints on the lateral trajectory of the aircraft. After introducing the constraints between the target and the no-fly zone, the trajectory of the aircraft around the no-fly zone is divided into C-type and S-type, and then analyze its influence on the ballistic prediction stop time t end and the influence on the setting of the preset interception point; The specific content of the said Step 4 is as follows: Step 4-1: Several forms of the aircraft flying around the no-fly zone can be predicted for the initial maneuver, but the end time of the C-type maneuver, the start time of the second half of the S-type maneuver, and the end time of the maneuver cannot be predicted; assuming the maneuver degree is unknown and the maneuver form is known, that is, assuming the aircraft trajectory has been determined as C-type / S-type, but the start and end times of the maneuver are unknown. The start time is judged by maneuver detection, and the end time is judged according to the fact that the aircraft will return to within 50 km of the initial trajectory; for different working conditions, analyze the stop time of the ballistic prediction and the setting of the interception point. Step 4-2: The interception point is in the latter section of flying around the no-fly zone. This is the most unfavorable situation. Since the end time of flying around is unknown, without other settings, other settings refer to the target aircraft needs to return to the vicinity of the original trajectory, and the stop time of the ballistic prediction cannot be determined. At the same time, the preset interception point will happen to be in the stage with a large prediction error. Therefore, it is necessary to avoid presetting the interception point in this stage. Step 4-3: The interception point is in the former section of flying around the no-fly zone. At this time, since the flying around has started, the prediction needs to be carried out all the time. The turning maneuver fitting model is used, and this section of the trajectory can be effectively predicted according to the previous section of the flight trajectory, and the impact on the interception is small. Step 4-4: The interception point is after flying around the no-fly zone. Since the flying around has ended, the end time of the aircraft flying around can be confirmed by means of maneuver mutation detection. After that, the prediction needs to be carried out all the time. The common fitting model is used, and it can be considered that there is no impact on the interception. Step 4-5: The interception point is before flying around the no-fly zone. Since the flying around has not started, the moment when the aircraft has not entered the flying around can be confirmed by means of maneuver mutation detection. The prediction needs to be carried out all the time. The common fitting model is used, and it can be considered that there is no impact on the interception. Step 5: Reasonably set the forecast end time t according to the no-fly zone constraint end and the preset interception point location.

2. A method for predicting the trajectory of an aircraft based on lateral maneuvering characteristics and no-fly zone constraints according to claim 1, characterized in that: The specific steps of step 2 are as follows: Step 2-1: For the swing maneuver, its trajectory curve is approximately described by a straight line and a sine curve, that is, the DPL+LA optimal maneuver model function is used for fitting, and the fitting function is expressed as: a(t) = C 2 f 2 (t) sin(f 1 (t) x(t) + η 0 ) + C 1 t + C 0 where a(t) represents the function of the fitting acceleration varying with time, C 2 f 2 (t) is the maneuver amplitude, f 1 (t) is the maneuver frequency, x(t) is the range on the ox-axis from the start time of the maneuver to the current time, η 0 is the phase angle at the start time of the maneuver, C 1 t + C 0 is the distance at the start time of the maneuver; Step 2-2: For the turning maneuver, its lateral trajectory is a monotonic curve, either monotonically increasing or monotonically decreasing, and an Nth-degree polynomial is used to approximate it, and the fitting function is expressed as: where a i is the coefficient of the Nth-order polynomial, x(t)i is the range on the ox axis from the start time of the maneuver to the current time, and a(t) represents the function of the fitting acceleration varying with time; Step 2-3: When predicting the lateral trajectory, prior information is also considered. The prior information refers to one or more combinations of ① the target maneuvering ability, that is, the magnitude of the maneuvering acceleration is known, ② the target landing point is known, or ③ the target evasion object is known.

3. A method for predicting the trajectory of an aircraft based on lateral maneuvering characteristics and no-fly zone constraints according to claim 2, It is characterized in that: To improve the maneuvering penetration ability of the trajectory aircraft, the maneuvering amplitude and frequency are time-varying. Therefore, when predicting the lateral trajectory, let f 1 (t), f 2 (t) be a linear function of time, and we have where a, b, c, and d are constants.

4. A flight vehicle trajectory prediction method based on lateral maneuvering characteristics and no-fly zone constraints according to claim 1, It is characterized in that: In step 3, the step of solving the fitting parameter θ of the fitting model k is specifically as follows: Step 3-1: Use the nonlinear least squares method to estimate the parameters of the nonlinear model. Use m observation values to fit the nonlinear model with n parameters, where m > n, so that the sum of the squares of the fitting residuals is minimized. The basis is to approximate the model through a linearization approximation method, and then extract the parameters through continuous iteration; Step 32: Set the observed data (x i , y i ), where i = 1, 2, …, m. Here, x i represents the displacement of the aircraft along the x-axis at the i-th moment, and y i represents the displacement of the aircraft along the y-axis at the i-th moment. The function form to be fitted is y = f(x, θ), where θ = [θ 1 , θ 2 , …, θ n T are the parameters to be estimated. Solve such that where is the fitted value of θ;​ Step 3: Since the residual e j = y j - f(x j , θ), rewrite the sum of squared residuals minimization formula as where S is the sum of squared residuals, θ j is the j-th parameter to be estimated, e i is the i-th residual, m is the number of observations. This system of equations has no closed-form solution and requires an initial value to be given, and then the final estimated value is obtained through continuous iteration; Step 3 and 4: Use the Gauss-Newton iterative algorithm to solve the estimated value, and set θ k as the solution result of the k-th iteration. Take the first-order Taylor expansion of y = f(x, θ) with respect to θ k , and we have where J ij is an element of the Jacobian matrix, which changes with the number of iterations. The calculation method is: Step 35: Further represent the residual as Δy i = y i - f(x i , θ k ), where y i is the i-th observed value, Δy i is the i-th observed value difference, Δθ j is the j-th estimated parameter difference. Substitute into the iterative calculation and organize to get Write it in matrix form: J T JΔθ = J T Δy According to the above formula, the correction value Δθ of the parameter to be solved can be obtained, and the parameter value is updated to θ k+1 = θ k + Δθ, and continue the iteration until the iteration end condition is met.

Images