DE3927299A1 - Motion path computer for optimising course, e.g. of cruise missile - has 1st computer processing topographical and optimising data and 2nd computer which improves initial optimal path - Google Patents

Abstract

The course computer for flight path optimisation contains a first computer unit (50) receiving start and target points, information, esp. topographical information, about the terrain between the start and target points and at least one optimisation parameter. The computer applies a series of supporting points between the start and finish points and determines the paths including the supporting points which are optimal for the entered parameter. A second computer unit (52) improves the optimal path from the first computer. USE/ADVANTAGE - E.g. for use in cruise missiles for reliable, rapid optimal flight path determination at low cost.

Description

The invention relates to a trajectory computer for Path optimization, especially for missiles.

For devices that are between different locations (here start and called the target point) should be moved, whereby a Many possible ways come into question, that arises Problem, like that regarding at least one Optimization parameters, such as B. minimal movement time, optimal way reliable and without too much effort can be determined. Especially with autonomous flight bodies (e.g. cruise missile) that are preferred in low The problem arises if flight altitudes are to fly Programming the trajectory is because the straight flight path with correspond to the constant rise and fall of the missile according to the elevation profile of the landscape along this path in terms of flight duration and fuel consumption in general is unfavorable. Also from a security perspective, i. H. best possible coverage of the missile during the Flight, the straight flight is usually unfavorable because these cover options are not taken into account. So should because of larger water areas and mountain peaks low coverage should be avoided if possible. Under certain circumstances the pre-programmed trajectory must be during the flight be changed because suddenly an obstacle or a Hazard area appears. Another trajectory optimization during flight taking into account these new circumstances would be very desirable. Not only for unmanned missiles, robotic vehicles or the like. The problem of movement path optimization put; even with manned devices, such as. B. aircraft, it would be conceivable for self-control of the aircraft (Auto-Pilot) to determine an optimal trajectory.

Optimization methods as such are generally known. They require a lot of equipment and computing time  wall when high demands on the reliability of the Optimization. A known problem is here that the optimization method used in each case often "sticks" at a sub-minimum, i.e. H. in converges to a minor and therefore not at all provides the global optimal value. To avoid this danger reduce, either with extremely great effort entire multidimensional search space or one leaves the optimization calculation under ongoing review run by an experienced scientist who may intervenes in the calculation.

The object of the invention is a movement train calculator to indicate which with low equipment Effort and low computing time reliably the optimal Trajectory determined.

This task is done by a trajectory calculator solved, comprehensive

• - A first computing unit, in which starting point and Target point and information, especially topogra phical information about the between starting point and target area and at least one Optimization parameters can be entered and which ones in the area establishes a number of discrete bases and those tracks among the possible ones in start and Destination point ending, running over the support points Pathways identified in terms of optimization parameters is optimal,
• - A second arithmetic unit, which of the first The computing unit determined the optimal path as the starting value in a continuous optimization process uses and with this method an improved optimal path determined.

The first arithmetic unit covers the questionable Area with a relatively rough grid and checked the  possible paths going through these grid points at least one optimization condition. So there that entire 2- or 3-dimensional space between the start point and target point is examined, can be assumed be determined that the determined optimal path (through the discrete support points) near the at continuous The calculation of the resulting global optimum. The latter is then determined in the second computing unit telt, with this as the starting value from that of the first Calculating unit determines the optimal path. Because of as expected, close to the global optimum there is a rapid convergence the continuous optimization process, also with Use of a less powerful computer relatively slow computing speed and relatively low according to memory. It has been found that in in most cases a relatively rough one Grid is sufficient to use the first arithmetic unit obtainable starting values. So it is often enough 100 × 100 grid points or even 50 × 50 grid points lay. The requirements are correspondingly low the memory requirement and the computing speed also first arithmetic unit. A movement path calculator is sufficient therefore in most cases a computer in Personal-Com computer size. Such a computer requires Determining the optimal path a computing time in minutes area that enables "real-time operation", ie a running motion path optimization calculation too changing during flight to take into account Flight conditions. You can also work with parallel computers be reduced with a corresponding reduction in computing time.

In a preferred development of the invention, one third arithmetic unit provided by the first Computing unit determined, corresponding to a polygon optimal path smoothes and then the second re unit as the starting value. Based on these  Smoothing the starting value results in an even faster Convergence of the continuous optimization process in the second arithmetic unit.

The invention further relates to a method for moving trajectory planning, in particular for missiles. This The method has the following steps:

• a) discretize an area between the starting point and Target point by defining a number of support points len;
• b) determining the with respect to a given opti optimal parameters among the possible Chen, ending in the start and finish point, over the Support points running tracks;
• c) continuous optimization calculation with optima len path step b) as a starting value.

A step d) between step b) and c) is preferred provided, in which the determined according to step b), optimal path corresponding to a polyline is, and then taken as the starting value for step c) becomes.

In a development of the invention it is proposed that area underlying steps b) and c) between Start and finish in a previous step f) Art is modeled with an optimization parameters, in particular the minimum hazard placed excessive terrain or artificial obstacles, possibly in danger zones. In this way, on the one hand automatically ensures that the missile dangers Zones overflown at a higher altitude, provided that they are also below these new conditions optimal path over this area leads. In many cases, the optimization calculation  however, result in a path that avoids this danger zone avoids in advance, for example when an opti The fuel consumption parameter is - that Optimization process will then seek out ways in which Changes in flight altitude are avoided as far as possible.

A simple attractive procedure to use in discreet space to find the global solution rather than a local one Landing Minium is the BELLMAN'SCHE procedure is preferably used in step b).

For smoothing in step d), the least Square method used, for example for the path searched for uses a Fourier series approach.

For step c) the NELDER-MEAD or the is preferred POWELL algorithm used. These standard procedures quickly deliver the closest minimum. Due to the im Step b) determined the starting value in the vicinity of the globa len minimum (optimum) is the result of the Step c) the optimum sought.

For the sake of simplicity, the orbit can function as a function be put. In cases where more complicated Webs such as B. Meander tracks come into question, the Path preferably shown in parameter form. According to the invention it is also provided that one in the Steps b) and / or c) several optionally weighted Optimization parameters, in particular minimal risk and speed or minimal danger and drift material consumption, taken into account.

Also in step b) and / or c) secondary condition conditions, in particular minimal flight altitude profile, or Flyability conditions, especially maximum acceleration inclination or minimum flight curve radius will.  

The invention is based on the drawing preferred embodiments explained. It shows:

Fig. 1 in a ground plan view of a flight path determined by discrete optimization calculation;

FIG. 2 shows a top view corresponding to FIG. 1 with the trajectory according to FIG. 1 as a starting value for a continuous optimization calculation with the optimal trajectory that then results;

Fig. 3 is a top view of an embodiment with terrain grid and elevation information and with three trajectories 1 , 2 , 3 determined according to the invention, each of which are optimal with respect to different optimization parameters (safe or fast or minimal fuel consumption);

FIG. 4 shows the trajectory 1 in Fig 3 in side view with landscaping height profile.

5 shows the trajectory 2 in Fig 3 in side view with landscaping height profile..;

Fig. 6 the trajectory 3 in Fig 3 in side view with landscaping height profile.

Figure 7 is a plan view of a labyrinth according to the invention determined path of movement.

Fig. 8 is a plan view of a labyrinth corresponding to FIG. 7 with an optimal path and starting from another starting point

Fig. 9 is a flowchart for explaining the optimization method according to the invention.

In principle, this is how it works when optimizing the trajectory before that one over all possible tracks or ways Cost function L, which depends on the respective strategy hangs, summed up and the resulting ge total cost of this route with the total cost of other routes compares and looks for the path that is minimal Has total cost. So you assign a number to each path so-called cost functional J, for:

With

The vectors i, j, and k stand for the three Cartesian coordinate directions, the numbers x, y and z for the respective Cartesian components and t for the Time. L contains both flight dynamics and Eigen shafts of the room, such as B. obstacles or actual or assumed terrain. The determination of the optimal way is therefore a problem of variation calculation or optimization calculation. By variation of J the EULER-LAGRANGE equations arise. Their associated partial differential equation is the HAMILTON-JAKOBI- Equation that can be solved numerically.

The cost function L expresses which strategy the optimization should be based on, i.e. so with regard to which optimization parameter or which Optimization parameters the path to be chosen is optimal should. Here come pure strategies, such as speed (minimal flight time), regardless of other facial points such as Fuel consumption, in question or also Mixing strategies in which several parameters are specified weighting must be taken into account when optimizing  such as security and speed or Security and fuel consumption.

The height z of the missile above the earth's surface can be as Measure of the security of the missile against foreigners influence (z should be as small as possible, because the aim is to fly as low as possible).

A cost function that takes both flight time into account as well as the security defined in this way following form:

The first bracket with the squared speed component is a measure of flight time; the second Parenthesis is a measure of security, being the Parame ter R the ratio of speed to safety the optimization regulates. For R against infinity we get for example, the limit case of flight duration minimization.

The variation problem must be under the following constraints be solved:

z (t) V (x (t), y (t)). (4)

V (x (t), y (t)) is the height of the effective landscape can consist of the following parts:

Hazardous areas are therefore taken into account in that receive high effective V-values, e.g. Wasserge offer and mountain peaks, whereas areas with good Dec low V-values. Through the Overlay of the different parts of the landscape Vi  The missile "sees" or takes into account the optimization an area that is strongly influenced by natural Terrain can vary and therefore as an effective country shaft is called.

If the equals sign in equation 4 applies, then z (t) is given (holonomic constraint for the variation problem) and can therefore be eliminated from L ₁ . The 3-dimensional movement is therefore limited to a predetermined (generally 3-dimensional) surface with the two degrees of freedom x and y. This case applies to a robot vehicle, for example, and also approximately for missiles, insofar as its flight characteristics allow it to follow the terrain or the effective landscape exactly. In general, the flying body is not able to do this, so that the holonomic constraint must be relaxed and z (t) in Equation 4 can also be greater than V. However, in order to be able to keep the deviation from the specified effective landscape as low as possible when optimizing the path, a function L _{2 of} the following form, which is referred to as a penalty function and is added to the cost function L ₁ , is introduced:

L₂ = K₁ · [z (t) - V (x (t), y (t))] ² · R (V (x (t), y (t)) - z (t)]. (6)

R is the step function, which is 1 whenever z (t) is below V (otherwise 0); thus, depending on the freely selectable parameter K ₁ (K ₁ significantly greater than 0), a deviation from the effective landscape is "punished".

Another constraint can be that the maximum acceleration in the Z direction is limited to a predeterminable maximum value a _z :

| (t) | a _z . (7)

Likewise, the degree of change of direction (curvature) of the trajectory in the area V (x, y) can be limited to a value a _{x, y} :

[((t)) 2 + ((t)) 2] ^1/2 a _xy . (8th)

These additional conditions can be taken into account by assigning further penalty functions (such as L ₂ ) and / or introduced in the subsequent continuous optimization calculation as an additional condition for the path curve to be calculated. Another such secondary condition is, for example, the maximum speed

U _max ² + ² + ² U² _max . (8a)

If the deviations in z (t) from V are small, then they are decoupled the variation problem in a first approximation in a path search on the surface V with the variables x, y and independent of which in a search for the vertical component z (t) the path x (t), y (t), which is called "terrain following" can be.

In the approximation z (t) = V (x (t), y (t)) and in the borderline case R → ∞ the problem is reduced to the search for the geodesic table lines, i.e. the shortest connecting lines of the (non-Euclidean) area V. At constant Ge speed of the missile results in the geodesics optimal ways in terms of flight duration. In the case of large ones but finite values of R can differ from the general Parameter representation x (t), y (t) for the simpler function representation y (x) can be passed over, however Uniqueness demands, so that meandering loops out are closed.

To take fuel consumption into account, a further cost function contribution L _{V can be} introduced according to the following equation:

Here is the infinitesimal energy consumption with dJ along an infinitesimal path ds or dx be draws, consisting of a friction thermal and a Therm, which is different from zero whenever the Missile must rise:

Here m is the mass, Q the cross-sectional area of the missile, u its velocity, o the air density, g the gravitational constant and c _w the drag coefficient.

The procedure according to the invention for path optimization is explained below with reference to FIG. 9.

First, the appropriate terrain that has the starting and finishing point is selected (block 10 in FIG. 9).

Next, 12 danger zones are determined in a block, in particular areas of low coverage such as water surfaces and mountain peaks, and accordingly artificial terrain elevations or terrain obstacles are stored. These obstacles affect the equation 4 on the cost function L ₁ according to equation 3, so that the danger zones are included in the cost functions and are accordingly taken into account in the optimization calculation. However, there are also conceivable cases in which block 12 can be skipped, since natural terrain can be expected (broken connecting line between block 10 and block 14 following block 12 ).

In block 14 it is determined which of the possible strategies S 1 , S 2 etc. should be used as the basis for the optimization calculation, for example whether the flight path is as safe as possible ( FIG. 4), the fastest possible flight path ( FIG. 5) or a flight path minimal fuel consumption ( Fig. 6) is to be determined. Mixed strategies are also possible, such as "as quickly and safely as possible" (Equation 3).

It then becomes the one assigned to the respective strategy Cost function based on the optimization calculation.

Depending on whether relatively simple or meandering lanes are to be expected, block 14 is followed either by a discrete route search in functional representation (block 16 ) (in the case of simple lanes) or a discrete route search in parameter representation (block 18 ).

The cost function L ₁ in the case of the functional representation y (x) has the following form:

With

This is generally mentioned at the beginning of the optimization calculation Defined cost functional J minimized, the as existent minimum value in the following with S referred to as. The following applies in the case of the functional representation

For the minimal case J = S, the 1st summand is also on the right side of the equation because of the optimality of the partial routes a minimum. For small Δx one can therefore do the following Derive recursion formula:

This is the discrete form of the HAMILTON-JAKOBI equation. It is solved iteratively (BELLMAN'SCHES method in Bellmann, R .: Dynamic Programming, New York, Priceton University Press 1957), in which one slope iterations y '(x ₁ ) tried out. This ensures that the global solution is obtained within the discrete space (true minimum) and does not end up in a local minimum. By choosing the y ′ (x ₁ ) a discrete 2-dimensional grid is spanned. The computing time is proportional to the number of grid points n.

For the parameter representation (x (t), y (t)) the search space is a 3-dimensional grid with computing time proportional to n ^3/2 .

The same procedure is also carried out with L ₂ for the z component, the secondary condition (equation 7) being taken into account when selecting the different directions y ′ (x ₁ ).

In FIG. 1, the procedure is shown in simplified form for the case of the functional representation. If one starts from target point A, for example, six different gradients y '(e.g. minus 60 °, minus 30 °, 0 °, plus 30 °, plus 60 °) and an iteration step size Δx. The x, y coordinate system of the terrain is rotated by a corresponding orthogonal transformation such that the start and finish lie on the X axis. Starting, for example, from target point A, a series of planes E 1 parallel to the Y direction is then placed in the area between target and start, each at a distance Δx.

From the target point A, the straight lines corresponding to the given five slopes are placed in the surface and their intersections P 11 , P 12 , P 13 , P 14 and P 15 are determined with the next following plane E 1 . The functional J or its minimum S is determined for each of these points. For the points in the first level E 1 , the solution is trivial, since here the direct connection between the point in question and the previous target point A is always the optimal route.

This is different in the points of the other levels, e.g. B. in points P 21 to P 29 of the next level E 2 . These points in turn result from the fact that in each of the points P 11 to P 15 of the plane E 1 the five straight lines are laid with the predetermined slopes and the intersection points of these straight lines with the plane E 2 are determined. When determining the minimum value S for each of the points P 21 to P 29 of the plane E 2 , for example for the point P 24 , one proceeds according to equation 14 in such a way that from this point P 24 to the points P 11 to P 15 the Level E 1 connects routes a 1 , a 2 , a 3 , a 4 and a 5 and calculates the route share of the cost function J for each of these routes, i.e. L (x, y, y ′) · Δx. If one adds to this the value of S of the corresponding point in level E 1 , this results in the cost functional J for the path from P 24 via the relevant point in level E 1 to the target point A. It is now an easy, by ver to determine the minimum S equal to the different cost functionals J for the tracks running between one of the points P 11 to P 15 of the level E 1 between the point P 24 and the target point A. The values S for the other points of level E 2 and for the remaining levels between the destination and the start are determined in a corresponding manner.

The result is for all meshes or grids points between goal A and start B the minimum values S known and the one assigned to the respective minimum value optimal path between the respective point and the destination A. As soon as this procedure has reached the starting point B, there is the desired result, namely this  Minimum value S assigned to the point together with the associated optimal value Train.

Depending on the strategy used, different optimal courses are obtained. In Fig. 1, a path 30 is indicated with a double line, which results when terrain elevations (area D) are to be avoided (strategy: safe). If, on the other hand, minimal fuel consumption is important, so that the path length must be kept short, the optimization calculation yields, for example, a path 32 which partially penetrates into the elevated region D, which is indicated in FIG. 1 by two height lines 20 and 22 .

According to FIG. 9, a smoothing of the determined path corresponding to a polyline is carried out in a following block 40 , for which purpose a compensation calculation using a Fourier series approach is preferably carried out. The following condition applies to the smoothed function (x):

(x) represents a complete system, which approximates each function as precisely as desired. a denotes the Fourier components. x ₀ and x ₁ denote the x coordinates of the start and finish. The summation index k runs between 1 and the limit value N. This is preferably chosen equal to 7 in order to exclude curvatures of the trajectory that are too strong and thus to meet the corresponding secondary condition. By expanding this approach, predetermined gradients in the start and finish can also be taken into account.

In Fig. 2 is the broken line of the polygon of the web 30 shown in FIG. 1 and with a solid line this web 30 'after appropriate smoothing.

Using this trajectory, the height profile defined by this trajectory is then determined in a block 42 and a discrete flight path calculation is carried out along this profile to determine the z component of the trajectory.

With y (x), inserted in the cost function L, one obtains, after integration, a multidimensional function I with N variables a _k , which are called the RITZ parameters:

I (a₁, a₂,..., A _N ) = ∫ L ((x), (x) ′, x) dx. (16)

The task now is to minimize function I, using standard methods, preferably the NELDER-MEAD algorithm (Nelder, JA and Mead, R. in Computer Journal, Vol. 7, 308 (1965)) or the POWELL-Al algorithm (Brent, RR: "Algorithms for Minimization without derivations", Erylewood Cliffs, Prentice Hall). This calculation step is symbolized by block 44 in FIG. 9.

As a result, the optimal path is obtained either in the functional representation (block 46 ) or in the functional representation (block 68 ).

Since the discrete route search in accordance with block 16 or 18 provides a solution in the discrete search area, which, as a rule, including in the case of flight path problems, is in the catchment area of the global minimum in continuous space, a solution that is in the global minimum is obtained after the continuous optimization calculation has been carried out. A direct application of the continuous optimization calculation would generally lead to a local minimum, which is more or less far from the optimum. The z component can also be included directly in block 44 , giving up the previous decoupling.

In Fig. 2 is obtained as a result of the continuous optimization calculation path 30 '' with a dash-dot-dot line.

In the Figs. 3 to 6, a further execution is illustrated, for example. It can be seen in FIG. 3 provided with a Höhenrasterung area with the starting point B and target A at the bottom and top of the screen. Some grid levels are assigned height information in FIG. 3.

The trajectory 1 was determined with the strategy of the greatest possible security (ie avoiding surveys even when taking detours).

The trajectory 2 essentially follows a geodetic one, since the flight duration is as short as possible.

The trajectory 3 minimizes fuel consumption, ie tries to climb as little as possible and to be as short as possible.

Fig. 7 shows a labyrinth. Even in such a complicated situation, the method according to the invention provides a usable path between target point A and starting point B. If starting point B is at a different location, another optimal path may result, as shown in FIG. 8. In both cases, the calculation is carried out in the parameter representation of the route sought.

Blocks 14, 16 and 18 can be assigned as completed arithmetic operations to a first arithmetic unit 50 and, accordingly, block 44 to a second arithmetic unit 52 , block 40 to a third arithmetic unit 54 and block 42 to a fourth arithmetic unit 56 (interrupted in FIG. 9) Outlines indicated).

The one addressed in relation to Equation 14 iterative solution of the HAMILTON-JAKOBI equation according to the invention a parallelization in the calculation implementation, d. H. several arithmetic steps can be dependent on each other, can be performed simultaneously. The computing time requirement is reduced accordingly. It parallel computers can be used, mostly have a variety of processors, at the same time performing the calculations.

The parallelization is possible because, in the case the functional representation, when calculating the minimum value S in one point only those cost radio tionale must be known in the previous Level E and that of the one currently to be calculated Can be reached from there (limited incline range).

If you choose the parameter representation, you get in correspondingly that one for the calculation in Time step i in a grid node (x (t = i), y (t = i)) only the information of the neighboring nodes in question required for the previous time step t = i-1.

One can tie multiple knots into a bandage for a pro processor summarize the communication effort between processors. Everyone can too Processor with local memory.

Overall, there is a drastic reduction in computing time, which is proportional to n ^2/3 in the case of parameter display with sequential calculation and is reduced to the order of magnitude in parallel processing.

Claims (13)

1. Trajectory computer for trajectory optimization, in particular for missiles, comprising

• - A first arithmetic unit ( 50 ), in which the starting point and destination as well as information, in particular special topographical information, about the area between the starting point and the destination and at least one optimization parameter can be entered and which places a number of discrete support points in the area and those tracks determined from the possible orbits that end in the start and destination points and run over the support points, which is optimal with regard to the optimization parameter,
• - A second arithmetic unit ( 52 ) that uses the optimal path determined by the first arithmetic unit ( 50 ) as a starting value in a continuous optimization process and with this method determines an improved optimal path.

2. trajectory computer according to claim 1, characterized by a third computing unit ( 40 ), which smoothes the determined from the first computing unit ( 50 ), a polygon corresponding optimal trajectory and then supplies the second computing unit ( 52 ) as a start value. 3. trajectory computer according to claim 1, characterized in that the first computing unit ( 50 ) comprises a parallel computer. 4. Procedure for planning the trajectory, in particular for missiles, with the following steps:

• a) discretize an area between the starting point and the destination by defining a number of support points;
• b) determining the path which is optimal with regard to a predetermined optimization parameter, among the possible paths which end in the start and destination and run over the support points;
• c) continuous optimization calculation with the optimal path step b) as a starting value.

5. The method according to claim 4, characterized in that a step d) between step b) and c) vorese hen is, in which the determined according to step b), optimal path smoothed according to a polyline tet, and then as the starting value for step c) is taken. 6. The method according to claim 4 or 5, characterized records that this is the basis of steps b) and c) area between start and finish in one previous step f) is artificially modeled with a corresponding optimization parameter, esp in particular the minimum risk Excessive terrain or artificial obstacles, possibly in danger zones. 7. The method according to any one of claims 4 to 6, characterized characterized in that step b) paralleli siert. 8. The method according to any one of claims 4 to 7, characterized characterized in that in step b) the Bellman ′ uses procedures. 9. The method according to any one of claims 4 to 8, characterized characterized in that in step d) the least Square method is used.   10. The method according to any one of claims 4 to 9, characterized characterized in that in step c) a discrete Minimum search algorithm, preferably the Nelder Mead algorithm or the Powell algorithm puts. 11. The method according to any one of claims 4 to 10, characterized characterized in that in step b) the web as Represents function or in parameter form. 12. The method according to any one of claims 4 to 11, characterized characterized in that in steps b) and / or c) at least one optionally weighted optimization risk parameters, in particular minimal risk and Speed or minimal danger and drift material consumption, taken into account. 13. The method according to any one of claims 4-12, characterized in that in steps b) and / or c) secondary conditions, in particular minimum flight height profile (zv) or flight conditions, in particular maximum acceleration (| (t) | a _z ) or minimum flight curve radius.