Precision path planning for area coverage

Abstract

A method for automatically path planning for optimizing a path of a machinery moving in an area, comprises the steps of: taking coordinates of an area contour, a working width of the machinery operating in the area, a mathematical description of the dynamics of the machinery operating in the area in form of a system of nonlinear differential equations, determines segments along the headland path which are not feasible with respect to dynamics of the machinery or whose traversal would result in area coverage gaps. A mathematical algorithm is used with a hierarchical two-step framework in combination with a special coordinate transformation from a time into a spatial domain to formulate geographic high-precision constraints. The hierarchical two-step framework addresses two objectives: a generation of headland paths and a generation of smooth transitions between headland path and mainfield lanes, whereby the first hierarchical step varies for the two objectives.

Description

[0001]TITLE PRECISION PATH PLANNING FOR AREA COVERAGE TECHNICAL FIELD The present invention relates to a method for automatically path planning for optimizing a path of a machinery moving in an area as well as a first software product for performing this method, a second software product using data obtained by this method and a device for performing the method. The invention is especially useful for agriculture, especially open- space agriculture. BACKGROUND OF THE INVENTION Path plans have to be devised for the coverage of an agricultural field or agriculturally used work area. Machinery tracking the path plans can be robots or vehicles, such as tractors and harvesters, but also airplanes, helicopters, unmanned aerial vehicles and the like. Dynamics of these machineries can mathematically be expressed in form of a system of nonlinear differential equations. Path plans enable a cost-efficient and / or a time-efficient optimized path through agricultural fields. Time is important, in particular during grain haversting due to changing weather conditions and the subsequent availiability of only a limited number of harvesting hours. Energy (typically Diesel) savings and likewise the reduction of operating hours and wear of the machinery are of interest not only for cost reduction, but for environmental reasons and regulatory exhaust emission standards. A typical agricultural field is characterized by being an usually arbitrarily and non-convexly shaped closed area with potentially sloped terrain, in general arbitrarily shaped contour boundary, and obstacle areas such as trees, stones, ponds, power pole masts or similar closed "island areas" not suited for planting and prohibited from traversal by the machinery. Underlying objectives for determining path plans for machinery, especially for large machinery, are (i) area coverage gap avoidance such that the entire work area is covered, (ii) feasibility of the path plan with respect to dynamics of the machinery, and (iii) optimization according to a criterion such as, for example, minimization of the pathlength or minimization of the number of headland turns. The computational complexity for calculating such path plans can be demanding, especially for larger fields. Methods that frequently are used for the computation of paths between two coordinates and heading directions are so-called Dubins paths (see Dubins, Lester E. "On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents." American Journal of mathematics 79.3 (1957): 497-516) andReeds-Shepp paths (see Reeds, James, and Lawrence Shepp. "Optimal paths for a carthat goes both forwards and backwards." Pacific journal of mathematics 145.2 (1990): 367- 393). Dubins paths can provide the shortest path between two coordinates and heading directions for a forward-moving vehicle with a specific turning radius. However, a Dubins path, whichis composed entirely of no more than three circular arcs of minimum turning radius orstraight lines, is derived based on a simplified mathematical model of a vehicle that in general does not represent the actual dynamics of the machinery operating in an agricultural field. A second disadvantage is that Dubins path are always only composed of straight lines and arcs of minimum turning radius, i.e., straights and maximum steering turns. For refining of path segments and generation of path segments that do not require maximum steering but instead demand focus on other constraints, such as area coverage gap avoidance, Dubins path planning is not sufficient. A Reeds-Shepp path differs from a Dubins path in that travel in the reverse direction is also allowed. While for a Dubins path only forward-motion applies, a Reeds-Shepp pathpermits combinations of forward- and backward-motion. Thus, Reeds-Shepp paths are ageneralization of Dubins paths. For Reeds-Shepp paths the same aforementioned issues as for Dubins paths apply. The following notation is used in this text. A physical area, such as an agricultural field or an agriculturally used work area, is mathematically defined by the difference (mathematical set operation) between (i) an ordered set of geographic coordinates that describe a physical area contour as a (in general non-convexly shaped) polygon, and (ii) polygons defined by ordered sets of geographic coordinates that describe the contours of any physical obstacle areas (tree islands, ponds, power pole masts, and so forth) that are not to be passed by the moving machinery within the physical field area contour. By copying the mathematically defined work area-contour and applying an erosion operation (mathematical morphological operation), an eroded working area smaller than the working area and a corresponding eroded working area contour are obtained. The eroded working area contour is herein called headland path. The mathematical erosion operation can be conducted multiple times to generate multiple headland paths. The eroded working area is herein called headland area. The eroded working area bounded by the innermost headland path is herein called mainfield area.WO 2017 / 092904 A1 discloses such a method for planning a path of a machinery movingin an area. The method considers a working width of a moving machinery. The area is eroded (mathematical morphological operation) and the eroded area is partitioned into interior lanes. The interior lanes are intersected with a contour of the eroded area, thereby generating a lane grid. A mathematical algorithm is performed for determining an optimized sequence of position coordinates on the lane grid, wherein an area covering path is generated. This method enables to find an optimized path considering obstacles such as tree islands within the area. The content of this publication is entered to this text by reference. The headland area is typically used for turning and usually planted in another direction than the mainfield area. By partitioning the eroded working area into adjacent lanes, which shall in the following be denoted as mainfield lanes, selecting an orientation of the lanes and by intersecting the mainfield lanes with the eroded area contour, a lane-grid and a corresponding connected undirected transition graph of the area is generated. Instead of straight mainfield lanes also freeform mainfield lanes, in shape similar to a path segment of the eroded area contour, can be fitted and propagated by an offset distance (the working width) to cover the eroded word area. Based on the lane-grid a path plan for field coverage can be determined. Such a path plan, in general, does (i) not yet guarantee feasibility with respect to dynamics of the machinery, and (ii) does not yet guarantee area coverage gap avoidance. SUMMARY OF THE INVENTION It is therefore an object of the invention to provide path planning which improves feasibility with respect to dynamics of a machinery and minimizes area coverage gaps. This object is solved with a method according to claim 1. This object can also be solved with a method according to the features of claim 12.The inventive methods use a transformation from a time domain into a spatial domain. Thisenables an efficient and precise path planning for big machinery, especially for agricultural machinery. By using the inventive method, the machinery can even pass difficult shapes of headlands and obstacles within the area without or with a minimum of area coverage gaps. The invention enables precision path planning for machinery moving in an area to account for detailed dynamics of the machinery given in form of a system of nonlinear differentialequations and to minimize or avoid area coverage gaps.The expressions "avoid" or "avoidance" or similar expressions using "avoid" are understood in this text as complete avoidance or best possible minimization. A first inventive method generates one or multiple precision headland paths, herein called first application. A second inventive method generates precision transitions, especiallybetween headland path and mainfield lanes, between mainfield lanes and headland, andbetween pairs of mainfield lanes, herein called second application. The first and secondmethods are preferably combined, wherein they are preferably embedded withing anoptimization loop and repeated iteratively to achieve a high-level optimization. Within this invention, both applications are addressed by the same high-level approach comprised of two hierarchical steps: -First, a first path modification, which differs for the two applications.- Second, a second path modification, which applies the same mathematical methodfor the two applications using a special coordinate transformation from a time domain into a spatial domain. The first path modification for the first application comprises the steps of(i) taking as input a headland path generated by an erosion operation(mathematical morphological operation) for a working width, (ii) determining segments along the headland path which are not feasible to betraversed due to the dynamics of the machinery or whose traversal would result in area coverage gaps for the working width of the machinery, and (iii) modifying spatial waypoint coordinates along the segments in a manner, forexample by constructing a piecewise affine or polynomial envelope function (mathematical bounding function), such that the area coverage gaps are corrected. The modifications in general do not yet guarantee feasibility of the resulting path with respect to dynamics of the machinery, i.e., there is no guarantee that the segments can actually be traversed, for example, by an autonomous tractor. Feasibility is addressed in the second hierarchical step which is described in more detail further below. The first path modification for the second application comprises the steps of (i) taking as input a path plan for area coverage given by a sequence of positioncoordinates, (ii) determining segments along the path plan for area coverage which are notfeasible with respect to dynamics of the machinery or whose traversal would result in area coverage gaps, specifically, at transitions between headland path and mainfield lanes, between mainfield lanes and headland path, and between pairs of mainfield lanes, (iii) modifying spatial waypoint coordinates along the transitions through theirreplacement with a suitably fitted (e.g., minimally deviating from the original transition) Dubins and Reeds-Shepp path segments and heuristics (e.g., the selection of interpolation and discretization spacing). The modifications in general do not yet guarantee feasibility of the resulting path with respect to dynamics of the machinery since Dubins and Reeds-Shepp paths are derived from simplified vehicle dynamics that in general may not represent the actual dynamics ofthe machinery. Feasibility is addressed in the second hierarchical step which is describednext. For the second path modification the same mathematical method is applied for both aforementioned applications. Inputs are path segments resulting from the first path modification and a mathematical description of dynamics of the machinery in form of a system of nonlinear differential equations. Then, in order to generate the second pathmodification such that the resulting path becomes feasible with respect to dynamics of themachinery, the following method is suggested. First, apply a transformation of the dynamics of the machinery from a time domain into a spatial domain. This implies replacing time with a spatial coordinate along a path as the dependent variable and transforming the differential equations for the dynamics of the machinery from a time domain into a spatial domain. The transformation entails introducing a new set of variables describing the dynamics of the machinery in a path-aligned coordinate system instead of a global coordinate system with latitude and longitude (or their equivalent in the Universal Transverse Mercator (UTM) coordinate system in order tocalculate in metric units) describing position coordinates.Second, within this new coordinate system formulate an optimization problem, preferably a convex optimization problem, subject to constraints, preferably constraints that enforce the path to avoid any area coverage gaps. The fact that these constraints are formulated in thespatial domain permits high precision results and permits to solve both of the twoapplications (i) with respect to feasibility of dynamics of the machinery and(ii) for area coverage gap avoidance.The invention uses (i) a coordinate transformation as a prerequisite for the formulation of suitableoptimization problems and (ii) embeds the coordinate transformation within a hierarchical framework that canaddress both applications, wherein the first preparatory step varies for the two applications. In detail, a first inventive computer-implemented method for automatically path planning for optimizing a path of a machinery moving in an area, the method comprising the steps of -considering at least the following parameters:- coordinates of an area contour of the area in a global coordinate system,- a working width of the machinery operating in the area,- a mathematical description of dynamics of the machinery operating in thearea in form of a system of nonlinear differential equations,- making a copy of the area contour and geometrically translating this copy in par-allel by a fraction of the working width to an interior of the area, thereby obtaining an eroded area being smaller or equal than the area and an eroded area contour of the eroded area, the eroded area contour being denoted as a headland path, -determining first segments along the headland path which are not feasible to betraversed by the machinery due to the dynamics of the machinery or whose traversal would result in area coverage gaps for the working width of the machinery, -modifying the first segments such that their traversals become feasible with re-spect to dynamics of the machinery and such that their traversals minimize area cov- erage gaps, thereby using a first mathematical algorithm, -wherein the first mathematical algorithm comprises two hierarchical steps forhandling of each of the first segments, (i) first, creating a first modification by modifying spatial waypoint coordinatesalong each of the first segments by heuristics such that the area coverage gaps are corrected, but a first resulting path is not yet feasible with respect to dynamics of the machinery, (ii) second, creating a second modification by applying a transformation of thesystem of nonlinear differential equations describing the dynamics of the machinery from a time domain into a spatial domain with space along the first resulting path of step (i) replacing time as a dependent variable in the system of nonlinear differential equations and introducing a new set of var- iables in a path-aligned coordinate system instead of the global coordinate system, formulating a first mathematical optimization problem with a set of inequality constraints based on the path resulting from the previous step (i) such that the area coverage gaps are minimized or avoided, and with a set of equality constraints such that feasibility of a second resulting path with respect to the dynamics of the machinery is addressed, solving the first mathematical optimization problem and transforming a solution of the first mathematical optimization problem from the path-aligned coordinate sys- tem back to the global coordinate system, such that after the modification of all the first segments a modified headland path is generated that is fea- sible with respect to dynamics of the machinery and minimizes area cover- age gaps. Preferably, the method comprises the further steps of selecting an orientation of straight lanes or a shape of freeform lanes within the area bounded by the modified headland path and thereby partitioning the eroded area into adjacent mainfield lanes, intersecting the mainfield lanes with the modified headland path, thereby generating a lane-grid, determining a sequence of the traversal of the mainfield lanes for area coverage, thereby generating a path plan for area coverage given by a sequence of position coordinates, determining second segments along the path plan for area coverage which are not feasible to be traversed by the machinery due to the dynamics of the machinery or whose traversal would result in area coverage gaps, specifically, at transitions be- tween headland path and mainfield lanes, between mainfield lanes and headland path, and between pairs of mainfield lanes, modifying the second segments such that their traversals become feasible with re-spect to dynamics of the machinery and such that their traversals minimize area cov-erage gaps, thereby using a second mathematical algorithm, wherein the second mathematical algorithm comprises two hierarchical steps for han- dling of each of the second segments, (i) first, creating a first modification by modifying spatial waypoint coordinatesalong each of the second segments through their replacement with asuitably fitted path segment, preferably a Dubins or Reeds-Shepp path segment, such that the second resulting path is smoothed but in gen- eral not yet feasible with respect to dynamics of the machinery, (ii) second, creating a second modification by applying a transformation ofthe system of nonlinear differential equations describing the dynamics of the machinery from a time domain into a spatial domain with space along the second resulting path of step (i) replacing time as a dependent variable in the system of nonlinear differential equations and introducing anew set of variables in a path-aligned coordinate system instead of theglobal coordinate system, formulating a second mathematical optimiza- tion problem with a set of inequality constraints based on the path result- ing from the previous step (i) such that area coverage gaps are avoided, and with a set of equality constraints such that feasibility of a second re- sulting path with respect to the dynamics of the machinery is addressed,solving the second mathematical optimization problem and transforming a solution of the second mathematical optimiziation problem from the path-aligned coordinate system back to the global coordinate system, such that after the modification of all the second segments a modified path plan for area coverage is generated that is feasible with respect to dynamics of the machinery and minimizes area coverage gaps.Preferably, a first cascade of headland paths are generated by an erosion operation (math- ematical morphological operation) for a given working width and a given contour of the field area, wherein the first mathematical algorithm is iteratively applied to each of the first cas- cade of headland paths thereby generating multiple modified headland paths each being feasible with respect to dynamics of the machinery and minimizing area coverage gaps . Preferably, the considered parameters additionally include coordinates of a contour of any obstacle being present within the area, wherein the obstacle is not to be passed by the moving machinery, and wherein the method comprises the further steps of: making a copy of the contour of any obstacle and geometrically translating this copy in parallel by a fraction of the working width away from the obstacle interior, thereby obtaining a corrected eroded area and an eroded obstacle contour of any obstacle being present within the area and performing the first mathematical algorithm for each of the eroded obstacle contours, thereby generating an obstacle headland path for each obstacle being present within the area and the path being feasible with respect to dynamics of the machinery and minimizing area coverage gaps. Preferably, a second cascade of headland paths are generated by an erosion operation for a given working width and a given contour of any obstacle present within the field area and wherein the first mathematical algorithm is iteratively applied to each of the second cascade of headland paths, thereby generating multiple obstacle headland paths each being feasible with respect to dynamics of the machinery and minimizing area coverage gaps. Preferably, the method comprises the step of determining a sequence of mainfield lanes to generate a path plan for area coverage whose traversal does not yet account for feasibility with respect to the dynamics of the machinery and for area coverage gap avoidance, and performing the method within a global optimization scheme, thereby generating an optimal path plan for area coverage that is feasible with respect to dynamics of the machinery, minimizing area coverage gaps and is optimized ac- cording to one of the criteria of: minimization of total field coverage pathlength,minimization of accumulated length of mainfield lanes, minimization of accumulatedturning for field coverage, minimization of accumulated turning when traversing be- tween mainfield lanes and headland path, minimization of maximal turning when trav-ersing between mainfield lanes and headland path, minimization of accumulated ab-solute vehicle roll angles above a user-defined threshold resulting from field slopes in hilly terrain along mainfield lanes, minimization of maximal absolute vehicle roll anglesresulting from field slopes in hilly terrain along mainfield lanes, avoidance of absolutevehicle roll angles above a user-defined threshold resulting from field slopes in hilly terrain, connections to roads and logistical harvesting organization, minimization of completion time for field coverage, as well as a weighted trade-off between at least some of the aforementioned criteria. Preferably, heuristics are applied for increased computational efficiency to discard path plan candidates that cannot attain the optimal path plan solution.Preferably the method provides as output at least a sequence of position coordinates, thesequence giving a path to be followed by the machinery. An inventive first software program for performing the inventive method comprises an algo- rithm for computing a path plan feasible with respect to dynamics of the machinery and minimizing area coverage gaps.An inventive second software program comprising storing means for storing data identifyinga path based on position coordinates of a machinery according to the inventive method gives guidance to the machinery and / or a user of the machinery in order to enable the machinery to follow the path by usage of information of at least one location sensor.An inventive device for performing the inventive method comprises input means for provid-ing parameters, means for reading of geographic location measurements returned from a location or positioning sensor and for choosing heuristics and wherein the device comprises display means for displaying data enabling a machinery and / or a user of the machin- ery to follow a path and / or output means for delivering data enabling the machinery and / or the user of the ma- chinery to follow the path.A second inventive computer-implemented method for automatically path planning for opti-mizing a path of a machinery moving in an area, the method comprising the steps of -considering at least the following parameters:- coordinates of an area contour of the area in a global coordinate system,- a working width of the machinery operating in the area,- a mathematical description of dynamics of the machinery operating in thearea in form of a system of nonlinear differential equations, -making a copy of the area contour and geometrically translating this copy in par-allel by a fraction of the working width to an interior of the area, thereby obtaining an eroded area being smaller or equal than the area and an eroded area contour of the eroded area, the eroded area contour being denoted as a headland path, -selecting an orientation of straight lanes or a shape of freeform lanes within thearea bounded by the modified headland path and thereby partitioning the eroded area into adjacent mainfield lanes, -intersecting the mainfield lanes with the headland path, thereby generating alane-grid, -determining a sequence for the traversal of the mainfield lanes for area coverage,thereby generating a path plan for area coverage given by a sequence of position coordinates, -determining second segments along the path plan for area coverage which arenot feasible to be traversed by the machinery due to the dynamics of the machin- ery or whose traversal would result in area coverage gaps, specifically, at transi- tions between headland path and mainfield lanes, between mainfield lanes and headland path, and between pairs of mainfield lanes, -modifying the second segments such that their traversals become feasible withrespect to dynamics of the machinery and such that their traversals minimize area coverage gaps, thereby using a second mathematical algorithm, -wherein the second mathematical algorithm comprises two hierarchical steps forhandling of each of the second segments, (i) first, creating a first modification by modifying spatial waypoint coordinatesalong each of the second segments through their replacement with a suit- ably fitted path segment, preferably a Dubins or Reeds-Shepp path seg- ment, such that the second resulting path is smoothed but in general not yet feasible with respect to dynamics of the machinery, (ii) second, creating a second modification by applying a transformation of thesystem of nonlinear differential equations describing the dynamics of the machinery from a time domain into a spatial domain with space along the second resulting path of step (i) replacing time as a dependent variable in the system of nonlinear differential equations and introducing a new set of variables in a path-aligned coordinate system instead of the global coordi- nate system, formulating a second mathematical optimization problem with a set of inequality constraints based on the path resulting from the previous step (i) such that area coverage gaps are avoided, and with a set of equality constraints such that feasibility of a second resulting path with respect to the dynamics of the machinery is addressed, solving the second mathe- matical optimization problem and transforming a solution of the second mathematical optimiziation problem from the path-aligned coordinate sys-tem back to the global coordinate system, such that after the modification of all the second segments a modified path plan for area coverage is gen- erated that is feasible with respect to dynamics of the machinery and mini- mizes area coverage gaps. The features of the inventive methods and the software and devices mentioned above can also be combined with the above mentioned second method without using all the steps of the above mentioned first inventive method. BRIEF DESCRIPTION OF THE DRAWINGS Preferred embodiments of the invention are described in the following with reference to the drawings, which are for the purpose of illustrating the present preferred embodiments of the invention and not for the purpose of limiting the same. In the drawings, FIG.1 is a block diagram for the flow of actions from data input comprising field and obstacle area contour coordinates, to offline computation generating an optimized field coverage path plan feasible with respect to dynamics of the machinery and minimizing area coverage gaps, to online tracking of the complete field coverage path plan according to the invention;FIG.2 is an exemplary agricultural field to illustrate data input comprising field and obstacle area contour coordinates; FIG.3 shows for an exemplary agricultural field three exemplary headland paths generated when accounting for feasibility with respect to dynamics of the machinery and seeking area coverage gap avoidance;FIG.4 illustrates for an exemplary agricultural field an area coverage gap, a headlandpath and transitions between mainfield lanes and headland path not feasible with respect to dynamics of the machinery of typical nonholonomic (mathematical system characteristics) vehicles with a limited turning radius;FIG.5 illustrates for an exemplary agricultural field a modified headland path feasiblewith respect to typical dynamics of the machinery and resulting in area coverage gap avoidance, and transitions between mainfield lanes and headland path not yet modified and feasible with respect to typical dynamics of the machinery;FIG.6 illustrates for an exemplary agricultural field and a smaller turning radius incomparison to the working width transitions between mainfield lanes and headland path when accounting for dynamics of the machinery and seeking area coverage gap avoidance;FIG.7 illustrates for an exemplary agricultural field and a larger turning radius incomparison to the working width transitions between mainfield lanes and headland path when accounting for dynamics of the machinery and seeking area coverage gap avoidance;FIG.8 illustrates for an exemplary agricultural field transitions between headland pathand mainfield lanes that are not feasible with respect to typical dynamics of the machinery;FIG.9 illustrates for an exemplary agricultural field and a smaller turning radius incomparison to the working width transitions between mainfield lanes and headland path when accounting for dynamics of the machinery and seeking area coverage gap avoidance;FIG.10 illustrates for an exemplary agricultural field and a larger turning radius incomparison to the working width transitions between mainfield lanes and headland path when accounting for dynamics of the machinery and seeking area coverage gap avoidance;FIG.11 illustrates for an exemplary agricultural field transitions between headland pathand mainfield lanes that are not feasible with respect to typical dynamics of the machinery;FIG.12 illustrates for an exemplary agricultural field and a larger turning radius incomparison to the working width transitions between mainfield lanes when accounting for dynamics of the machinery and seeking area coverage gap avoidance;FIG.13 illustrates for an exemplary agricultural field a path involving both forward andbackward motion, that is typical for Reeds-Shepp path planning, along theheadland path;FIG.14 illustrates for an exemplary agricultural field the transition along a headland pathwhen accounting for dynamics of the machinery, specifically targetting area coverage gap avoidance, and specifically desiring forward motion only in order to save pathlength and minimize tractor traces in the work area. The figures 1 to 14 show the invention. Same elements are marked with the same reference numbers. DESCRIPTION OF PREFERRED EMBODIMENTSFig. 1 displays a block diagram for the flow of actions comprising three main stepsperformed by the inventive method. The three main steps S1, S2, S3 follow each other.The first step S1 is a data input comprising field contour. If any obstacles such as treeislands, ponds, power pole masts, and so forth are present within the field area, the firststep S1 also comprises data input of obstacle area contours coordinates.The second step S2 is offline computation generating an optimized field coverage path planfeasible with respect to dynamics of the machinery and minimizing area coverage gaps.The third step S3 is online tracking of a complete field coverage path plan. Tracking can beperformed either autonomously, semi-autonomously or manually by a human-being steering the machinery operating in the agricultural field or work area. Dynamics of the machinery are described mathematically in form of a system of nonlinear differential equations. Their selection is preferably a user-choice trading off mathematicalcomplexity and accuracy in their mathematical modeling of the machinery’s mobility in thereal world. Area coverage refers to any process in which the working area is covered by a machineryfor a specific purpose. Within an agricultural context machinery may for example be a tractoror harvester. The specific purpose is for example spraying, mowing, fertilizing, seeding, harvesting and so forth for a given working width.Offline computation step S2 is divided into a first, a second and a third subroutine SR1, SR2and SR3. The second and third subroutine, SR2 and SR3, are further embedded within anoptimization loop and repeated iteratively to optimize a specific optimization criterion OL.The first subroutine SR1 is abbreviated as Module M1. This first subroutine SR1 comprisesfor a working width of the machinery the generation of one or multiple precision headland paths that are (i) feasible with respect to dynamics of the machinery and (ii) minimize area coverage gaps. Headland path generation is applicable to both the field contour as well as the contours of any obstacles present within the field area. Obstacle headland paths are generated by making a copy of the contour of any obstacle and geometrically translatingthis copy in parallel by a fraction of the working width away from the obstacle interior,thereby obtaining a corrected eroded area and an eroded obstacle contour of any obstaclebeing present within the area and performing subroutine SR1 for each of the erodedobstacle contours, thereby generating an obstacle headland path for each obstacle being present within the area and the path being feasible with respect to dynamics of themachinery and minimizing area coverage gaps.The third subroutine SR3 is abbreviated as Module M2. The third subroutine SR3 comprisesthe smoothing of transitions between any combination of headland path and mainfield lanes, e.g., transitions between headland path and mainfield lanes, between mainfield lanes and headland path, and between pairs of mainfield lanes, such that the resulting paths are (i) feasible with respect to dynamics of the machinery and (ii) minimize area coverage gaps. Modules M1 and M2 can be considered as subroutines refining the solution of the secondsubroutine SR2 such that resulting paths account (i) for feasibility with respect to dynamicsof the machinery while (ii) minimizing area coverage gaps. For example, for computational efficiency or generality, a path plan may first be computedaccording to the second subroutine SR2 assuming only a working width as input parameterand not yet specifying dynamics of the machinery, fitting a lane-grid to the mainfield area,and calculating a sequence of lane-traversals to generate a field coverage path plan, butwithout accounting yet for feasibility of the resulting path plan with respect to the dynamics and area coverage gap avoidance. Then, an extension by Modules M1 and M2 within theframework of the first subroutine S2 can refine the path plan such that the resulting pathplan is (i) feasible with respect to dynamics of the machinery and (ii) minimizing area coverage gaps.When further iterating the second and third subroutines SR2 and SR3 according to a desiredoptimization criterion in an optimization loop OL, optimal path planning results can be attained. A desired optimization criterion can represent at least one of the following: minimization of total field coverage pathlength, minimization of accumulated length ofmainfield lanes, minimization of accumulated turning for field coverage, minimization ofaccumulated turning when traversing between mainfield lanes and headland path, minimization of maximal turning when traversing between mainfield lanes and headland path, minimization of accumulated absolute vehicle roll angles above a user-defined threshold resulting from field slopes in hilly terrain along mainfield lanes, minimization of maximal absolute vehicle roll angles resulting from field slopes in hilly terrain along mainfield lanes, avoidance of absolute vehicle roll angles above a user-defined threshold resulting from field slopes in hilly terrain, connections to roads and logistical harvesting organization, minimization of completion time for field coverage, as well as a weighted trade-off between at least some of the aforementioned criteria.Fig.2 illustrates an exemplary data input comprising a field contour 1 and two obstacle areacontours 2.Fig. 3 illustrates the effect of Module M1 for the generation of headland paths. Module M1 comprises two hierarchical steps, whereby both are applied for the generation of each headland path. In the first hierarchical step, a headland path is generated by an erosion operation (mathematical morphological operation) for a working width. In the second hierarchical step, the headland path is refined and modified to account for (i) feasibility of the path with respect to dynamics of the machinery and (ii) area coverage gap avoidance. The headland path resulting from the second hierarchical step of Module M1 then serves as reference for the generation of a next headland path following the same procedure and so forth, until a desired number of headland paths is generated. In the following, both these hierarchical steps of Module M1 are discussed in more detail.In the first hierarchical step of Module M1, by copying the field contour 1 and applying anerosion operation (mathematical mophological operation) for a working width, an eroded working area smaller than the working area and a corresponding eroded working area contour are obtained. This eroded working area contour is called headland path 3, see Fig. 3. The erosion operation generates a headland path according to the first hierarchical step of Module M1. The path (i) does not guarantee feasibility with respect to dynamics of the machinery, and (ii) does in general not ensure area coverage gap avoidance. In the second hierarchical step of Module M1, as input a mathematical description in form of a system of nonlinear differential equations modeling the dynamics of the machinery andits mobility inside the work area is assumed. The headland path is modified to account forfeasibility of the path with respect to dynamics of the machinery and to minimize area coverage gaps. Therefore, (i) segments along the headland path are identified which are not feasible with respect to dynamics of the machinery, for example, with limited turning radius and similar constraints, or whose traversal would result in area coverage gaps occurring especially for highly curved field area contours. Then, (ii) spatial waypoint coordinates along the segments are modified in a manner, such that the area coverage gaps are corrected, for example by constructing a piecewise affine or polynomial envelope function (mathematical bounding function). Furthermore, (iii) the envelope function is used as a constraint in an optimization problem. The optimization problem is formulated after aspecial transformation of the mathematical description of the dynamics of the machineryfrom a time domain into a spatial domain. This special transformation from a time domain into a spatial domain is described in detail further below. The mathematical description of the dynamics of the machinery in the spatial domain is added as constraints to aforementioned optimization problem. As a result of this opimization problem formulationmethod the original objective of generating a headland path feasible with respect todynamics of the machinery and area coverage gaps avoidance is addressed. A section ofthis headland path obtained by the inventive method is marked with reference number 7 inFig.3. After the solution of aforementioned optimization problem, the corresponding modified headland path is constructed, which can then serve as the reference for the generation of a next headland path. Thus, the erosion and refinement operation of the two hierarchical steps of Module M1 can be repeated iteratively and multiple times to generate multiple headland paths, seereference numbers 3, 4 and 5 in Fig. 3, and in particular the critical path segments 8, 9, 10and 11 that require significant steering. Headland paths can be connected by transitionpaths, see reference number 6 in Fig.3. The headland area is typically used for turning andusually planted in another direction than the mainfield area 12, which is the eroded working area bounded by the innermost headland path. The insight of the usefulness of the special transformation of the mathematical description of the dynamics of the machinery from a time domain into a spatial domain for the achievement of high-precision path plans in the area coverage context is a central step of this invention and therefore discussed in more detail next. The fact that constraints can be formulated in the spatial domain permits generation of geographic high-precision paths. For the aforementioned formulation of an optimization problem a transformation of the dynamics of the machinery from a time domain into a spatial domain is applied. This implies replacing time with a spatial coordinate along a path as the dependent variable andtransforming the differential equations for the dynamics of the machinery from a timedomain into a spatial domain. The transformation entails introducing a new set of variables describing the dynamics of the machinery in a path-aligned coordinate system instead of a global coordinate system with latitude and longitude (or their equivalent in the Universal Transverse Mercator (UTM) coordinate system in order to calculate in metric units) describing position coordinates. In detail, denoting N geographic path coordinates as {xi,yi} for i=0,…,N (e.g., representing position coordinates in the UTM-coordinate system), they are transformed to {si, ρs,i, ψs,i, ^̇^^,^} for i=0,…,N, denoting the spatial coordinates along the path, the radius of curvature, heading and change of heading along that path, respecitvely, and where the dot represents the derivative with respect to time. Instead of expressing machinery position coordinates in a global coordinate system spanned by {x,y}, the machinery can now be expressed in apath-aligned coordinate frame {eψ,i, ey,i} for i=0,...,N, denoting the deviation of the machineryheading and lateral deviation of the machinery with respect to the path. The change of thespatial coordinate along the path can be expressed as the differential equation, ^̇^(^^) =^^^^^^(^^)^ ⋅ By geometric arguments, for the velocity of themachinery projected along the path, which shall be denoted by ^^^^^^(^^)^, it holds ^^^^^^(^^)^ = Combining the two equations and further partitioning theprojected velocity into a path-aligned and a path-orthogonal velocity component, which shallbe denoted by and ^^^^^^(^^)^, the following equation is derived, ^̇^(^^) = . Denoting machinery state dynamics by ^̇^(^^) = ^^^^^(^^), ^^(^^)^ for general machinery statez(t) and control variables u(t), then (i) by transformation of machinery location into a path-aligned coordinate system, which shall be denoted by ^̇^^^^^(^^)^ = ^^ ^^^^^^^(^^)^, ^^(^^)^, and(ii) elimination of time as the dependent variable and replacing it with space along the pathas the dependent variable through the elementwise transformation ^^′^(^^) =^̇^^ ^^^(^^)^⁄ ^̇^ (^^), dynamics of the machinery in the space domain are obtained, which shallbe denoted the apostrophe / inverted comma nowrepresents the derivative with respect to space. Then, for the path segment for which a path refinement is demanded such that the area coverage gap can be corrected a mathematicaloptimization problem can be formulated, in general form, {minimize ^^^^^^(^^), ^^^(^^)^ suchthat → ^^ denotes the objective function describing any desiredperformance metric (e.g. seeking minimal deviation from the path) with state and controlvector dimensions denoting H constraint functions. While in general arbitrary many constraint functions are possible, and typically designed based on heuristics with the purpose of shaping the solution with a desired result in mind, according to the method of this invention the set ofconstraint functions preferably includes the condition ≤ 0, or altenratively,generality and path shaping purpose, ^^^,^(^^) − ^^ ^^^^,^ (^^) ≤ 0 for a desired reference shapingpath ^^. The reason for the proposal of this inequality constraint is tounderstood in connection with the first hierarchical step mentioned above and the path correction. As discussed above, the path must be constructed as an envelope function such that area coverage gaps are avoided. This path correction step does standalone and per se not yet guarantee feasibility with respect to dynamics of the machinery. However, remedycan be provided through the formulation of above mentioned general optimization problemin the spatial domain which (i) includes dynamics of the machinery as equality constraints,and (ii) the path envelope is accounted for via aforementioned inequality constraints. As aresult, a path both feasible with respect to dynamics of the machinery and also minimizing area coverage gaps can be generated. In practice, inequality constraints can also be relaxed by the introduction of slack variables or the terms on the left-hand side of theinequality constraints can be included into the objective function via Lagrangian multipliers(mathematical optimization concept). Above general optimization problem formulationassumes continuous spatial coordinate ^^ ≥ 0. In practice, a linearization and discretizationaround the path may be carried out to make the optimization problem more computationally tractable. After solving the optimization problem the resulting path is obtained in the path- aligned coordinate system. After a coordinate transformation back to the global coordinate system, such as the UTM-coordinate system (to calculate in metric units), the path plan as a sequence of geographic waypoint coordinates is obtained, (i) accounting for dynamics of the machinery and (ii) minimizing area coverage gaps.Fig. 4 illustrates an area coverage gap 16 and a headland path segment 15 not feasiblewith respect to dynamics of the machinery for a typical nonholonomic (mathematical system characteristics) machinery operating with a limited turning radius. Such paths result when generating headland paths without explicitly accounting for dynamics of the machinery andarea coverage gap avoidance. Fig.4 further illustrates an exemplary mainfield lane 13 anda transition between mainfield lane and headland path at point 14 that is not feasible withrespect to dynamics of the machinery when accounting for a typical nonholonomic machinery operating with a limited turning radius.Fig.5 illustrates how the issue of the area coverage gap 16 in Fig.4 can be solved accordingto Module M1 in block diagram of Fig.1. The previous area coverage gap is now avoidedwhile the modified headland path 17 is simultaneously feasible with respect to dynamics ofthe machinery with a limited turning radius. Fig. 5 further illustrates that as a result of the headland path modification the design of the lane-grid and coordinates of mainfield lanesare also affected, compare the path 18 in Fig.5 and path 13 in Fig.4. This is in accordancewith the sequential flow of actions SR1 to SR3 in the block diagram of Fig.1.Fig. 6 illustrates a transition between headland path and mainfield lane accounting fordynamics of the machinery according to Module M2 and the flow of actions in Fig. 1, resulting in a smooth transition 19, whereby in this illustration, dynamics of the machinery with a turning radius less than half the working width are assumed. A second exemplarytransition 20 between headland path and mainfield lane is illustrated in Fig.6 as well.Fig. 7 illustrates exemplary transitions 21 and 22 between headland path and mainfieldlanes accounting for dynamics of the machinery according to Module M2 and the flow of actions in Fig.1, whereby in this illustration dynamics of the machinery with a larger turning radius more than half the working width are assumed, which results in the specific transition as shown, while ensuring area coverage gap avoidance. In the following, Module M2 is discussed in detail. Module M2 addresses the generation of precision paths for the transitions between headland path and mainfield lanes, between mainfield lanes and headland path, and between pairs of mainfield lanes. Module M2 comprises two hierarchical steps. In the first hierarchical step, a transition path is generated. In the second hierarchical step, the transition path is modified to account for (i) feasibility of the path with respect to dynamics of the machinery, and (ii) area coverage gap avoidance. In the following, both these hierarchical steps of Module M2 are discussed in more detail. The first hierarchical step of Module M2 comprises the steps of (i) taking as input a path plan consisting of a sequence of transitions between headland path and mainfield lanes, between mainfield lanes and headland path or between pairs of mainfield lanes, and (ii) modifying spatial waypoint coordinates along the transitions through their replacement with suitably fitted path segments (e.g., minimally deviating from the original transition) derived from Dubins and Reeds-Shepp path planning. These modifications in general do not yet guarantee feasibility with respect to dynamics of the machinery. There is no guarantee that the segments can actually be traversed, for example, by an autonomous tractor or harvester, since Dubins and Reeds-Shepp paths are derived from simplified vehicle dynamics that may not represent the actual dynamics of the machinery. Feasibility with respect to a mathematical description of desired dynamics of the machinery is addressed in the second hierarchical step of Module M2. Both steps are discussed in more detail next. For the path modification it is differentiated between transitions (i) from a mainfield lane to a headland path,(ii) from a headland path to a mainfield lane,(iii) from a headland path to another headland path,(iv) between two different mainfield lanes when the turning radius of a typicalnonholonomic machinery operating in the work area is larger then half the working width of the machinery such that especially significant steering is required, and finally (v) special transitions.Special transitions are for example transitions between two different headland paths, wherein this aspect is only relevant if there are multiple headland paths in the field. Other examples are transitions onto the field and the final transition to the field exit after completion of field work. Then, based on aforementioned transition types a Dubins path is fitted to smooth spatial waypoint coordinates along the transitions. A Dubins path provides the shortest pathbetween two coordinates and heading directions for a forward-moving vehicle with a specificturning radius. A shortest path is preferably desired since it minimizes compacted tractor trace area and thereby simultaneously maximizes residual field area that can be used for growing of agricultural crops. However, a Dubins path, which is composed entirely of no more than three circular arcs of minimum turning radius or straight lines, is derived based on a simplified mathematical model of a vehicle. In detail, the nonlinear differential equations describing the dynamics oflocation coordinates and heading direction in the two-dimensional plane are {^̇^(^^) =^^^^^^^^^(^^)^, ^̇^(^^) = ^^^^^^^^^(^^)^, ^̇^(^^) = ^^(^^) with ^^(^^) ∈ [−1⁄ ^^^^^ , 1⁄ ^^^^^ ] and time ^^ ≥0}, where ^^^^^ > 0 denotes the minimum turning radius. This simplified mathematicalmodel, in general, does not accurately represent the dynamics of an agricultural machinery operating in a field, which typically are more complex. Thus, in general the Dubins path does not guarantee feasibility with respect to actual dynamics of the machinery, which isrequired for high-precision path planning, especially for autonomous robots in agriculturalapplications. Therefore, according to the method of this invention an additional path plan refinement is proposed. The second path modification takes as input (i) aforementioned path plan resulting from the Dubins path fitting step, and (ii) a mathematical description of desired dynamics of the machinery in form of a system of nonlinear differential equations. Then, to generate the second path modification such that the resulting path becomes feasible with respect to dynamics of the machinery, for the second hierarchical step of Module M2 the same mathematical method as in the second hierarchical step of Module M1 is applied. Thus, first a transformation of the dynamics of the machinery from a time domain into a spatial domain is applied. This implies replacing time with a spatial coordinate along the transition path resulting from the first hierarchical step of Module M2 as the dependent variable and transforming the nonlinear differential equations for the dynamics of the machinery from a time domain into a spatial domain. The transformation entails introducing a new set of variables describing the dynamics of the machinery in a path-aligned coordinate system instead of a global coordinate system, e.g., the UTM-coordinate system. The same mathematical derivation for the transformation from a time domain into a spatial domain as described above for the discussion of Module M1 applies, and likewise the formulation of a corresponding optimization problem. After solving the optimization problem a resulting path is obtained in the path-aligned coordinate system. After a coordinate transformation back to a global coordinate system, such as the UTM-coordinate system, the path plan as a sequence of geographic waypoint coordinates is obtained, (i) accounting for dynamics of the machinery and (ii) minimizing area coverage gaps. The insight of the usefulness of using a hierarchical two-step framework in combination with a special coordinate transformation from a time domain into a spatial domain to formulate geographic high-precision constraints and the usefulness of this hierarchical two-stepframework to address both objectives, the generation of headland paths as well as thegeneration of smooth transitions between headland path and mainfield lanes, whereby the first hierarchical step varies for the two objectives, is a central step of this invention.Fig. 8 illustrates transitions 23 and 24 between headland path and mainfield lanes that arenot feasible with respect to dynamics of the machinery of typical nonholonomic vehicles with a limited turning radius.Fig. 9 illustrates transition 25 when, in contrast to Fig. 8, now accounting for dynamics ofthe machinery resulting in a steering maneuver, whereby in this illustration dynamics of the machinery with a turning radius less than half the working width are assumed. The specific path results according to Module M2 such that area coverage gaps are avoided.Fig. 10 illustrates transition 26 when, in contrast to Fig. 8, now accounting for dynamics ofthe machinery resulting in a steering maneuver, whereby in this illustration dynamics of the machinery with a larger turning radius more than half the working width are assumed.Fig. 11 illustrates transitions between mainfield lanes and headland path, in particular, asequence of transitions from headland path 27 up to mainfield lane 28, before then returningback to the headland path via mainfield lane 29, when not yet accounting for dynamics ofthe machinery with a typically limited turning radius.Fig. 12 illustrates the same transition as in Fig. 11 when now accounting for dynamics ofthe machinery resulting in a steering maneuver 30 and 31 according to Module M2, wherebyin this illustration dynamics of the machinery with a larger turning radius more than half the working width are assumed. In detail, these particular steering maneuvers result asindicated ensuring area coverage gap avoidance and, since the headland path 27 accordingto Fig. 12 must be covered as part of a high-level field coverage path plan, in this specificcase requiring in sequence the coverage of mainfield lanes 28 and 29.The aforementioned two examples are given to underline the role of modules M1 and M2in supporting a high-level path planner for field coverage and refining its path plan logic to achieve (i) feasibility with respect to dynamics of the machinery and (ii) minimization of areacoverage gaps within the framework and block diagram for the flow of actions of Fig.1.Fig. 13 illustrates the usage of a Reeds-Shepp path, which involves both forward motion,33 and 35, and backward motion 34, for the transition from headland path segment 32 toheadland path segment 36. A Reeds-Shepp path differs from a Dubins path in that travel in the reverse direction is also allowed. While for a Dubins path only forward-motion applies, a Reeds-Shepp path permitscombinations of forward- and backward-motion. Thus, Reeds-Shepp paths are ageneralization of Dubins paths. Reeds-Shepp paths can be refined to acount for dynamics of the machinery just as described above. Regarding an implementation detail, to make thecomputational problem more tractable forward- and backward-motion segments can betreated separatedly and refined separately. Reeds-Shepp paths may be of interest particularly along tight turns along headland paths and to avoid area coverage gaps along headland paths, for example, for field mowing or harvesting applications. However, they can also be replaced by Module M1 for area coverage gaps avoidance as described above. This can have two core benefits: forward motion only and typically less pathlength, which consequently also implies less undesired compacted area from tractor traces, and thereby simultaneously maximizes residual field area that can be used for growing of agricultural crops, and less traveltime or working timein the field by avoiding combined backward- and forward-motion.Fig. 14 illustrates a section of the resulting path 37 when replacing the Reeds-Shepp pathsegment of Fig. 13 with an alternative path resulting from the application of Module M1 ofthis invention when accounting for dynamics of the machinery, simultaneously targetting area coverage gap avoidance and desiring forward motion only to save pathlength. Asshown, pathlength in Fig. 14 is much shorter in contrast to the Reeds-Shepp path segmentof Fig.13, while not compromising area coverage gap avoidance.

Claims

CLAIMS1. A computer-implemented method for automatically path planning for optimizing a pathof a machinery moving in an area, the method comprising the steps of -considering at least the following parameters:- coordinates of an area contour of the area in a global coordinate system,- a working width of the machinery operating in the area,- a mathematical description of dynamics of the machinery operating in thearea in form of a system of nonlinear differential equations, -making a copy of the area contour and geometrically translating this copy in par-allel by a fraction of the working width to an interior of the area, thereby obtaining an eroded area being smaller or equal than the area and an eroded area contour of the eroded area, the eroded area contour being denoted as a headland path, -determining first segments along the headland path which are not feasible to betraversed by the machinery due to the dynamics of the machinery or whose tra- versal would result in area coverage gaps for the working width of the machinery, -modifying the first segments such that their traversals become feasible with re-spect to dynamics of the machinery and such that their traversals minimize area coverage gaps, thereby using a first mathematical algorithm, -wherein the first mathematical algorithm comprises two hierarchical steps forhandling of each of the first segments, (i) first, creating a first modification by modifying spatial waypoint coordinatesalong each of the first segments by heuristics such that the area coverage gaps are corrected, but a first resulting path is not yet feasible with respect to dynamics of the machinery, (ii) second, creating a second modification by applying a transformation of thesystem of nonlinear differential equations describing the dynamics of the machinery from a time domain into a spatial domain with space along the first resulting path of step (i) replacing time as a dependent variable in the system of nonlinear differential equations and introducing a new set of var- iables in a path-aligned coordinate system instead of the global coordinate system, formulating a first mathematical optimization problem with a set of inequality constraints based on the path resulting from the previous step (i) such that the area coverage gaps are minimized or avoided, and with a set of equality constraints such that feasibility of a second resulting path with respect to the dynamics of the machinery is addressed, solving the firstmathematical optimization problem and transforming a solution of the first mathematical optimization problem from the path-aligned coordinate sys- tem back to the global coordinate system, such that after the modification of all the first segments a modified headland path is generated that is fea-sible with respect to dynamics of the machinery and minimizes area cover- age gaps.

2. The method of claim 1 wherein the method further comprises the steps of- selecting an orientation of straight lanes or a shape of freeform lanes within thearea bounded by the modified headland path and thereby partitioning the eroded area into adjacent mainfield lanes, -intersecting the mainfield lanes with the modified headland path, thereby gener-ating a lane-grid, -determining a sequence for the traversal of the mainfield lanes for area coverage,thereby generating a path plan for area coverage given by a sequence of position coordinates, -determining second segments along the path plan for area coverage which arenot feasible to be traversed by the machinery due to the dynamics of the machin- ery or whose traversal would result in area coverage gaps, specifically, at transi- tions between headland path and mainfield lanes, between mainfield lanes and headland path, and between pairs of mainfield lanes, -modifying the second segments such that their traversals become feasible withrespect to dynamics of the machinery and such that their traversals minimize area coverage gaps, thereby using a second mathematical algorithm, -wherein the second mathematical algorithm comprises two hierarchical steps forhandling of each of the second segments,(i) first, creating a first modification by modifying spatial waypoint coordinatesalong each of the second segments through their replacement with a suit- ably fitted path segment, preferably a Dubins or Reeds-Shepp path seg- ment, such that the second resulting path is smoothed but in general not yet feasible with respect to dynamics of the machinery, (ii) second, creating a second modification by applying a transformation of thesystem of nonlinear differential equations describing the dynamics of the machinery from a time domain into a spatial domain with space along thesecond resulting path of step (i) replacing time as a dependent variable in the system of nonlinear differential equations and introducing a new set ofvariables in a path-aligned coordinate system instead of the global coordi- nate system, formulating a second mathematical optimization problem with a set of inequality constraints based on the path resulting from the previous step (i) such that area coverage gaps are avoided, and with a set of equality constraints such that feasibility of a second resulting path with respect to the dynamics of the machinery is addressed, solving the second mathe- matical optimization problem and transforming a solution of the second mathematical optimiziation problem from the path-aligned coordinate sys- tem back to the global coordinate system, such that after the modification of all the second segments a modified path plan for area coverage is gen-erated that is feasible with respect to dynamics of the machinery and mini-mizes area coverage gaps.

3. The method of any one of claims 1 or 2 wherein a first cascade of headland paths aregenerated by an erosion operation (mathematical morphological operation) for a given working width and a given contour of the field area, and wherein the first mathematical algorithm is iteratively applied to each of the first cascade of headland paths thereby generating multiple modified headland paths each being feasible with respect to dy- namics of the machinery and minimizing area coverage gaps .

4. The method of any one of claims 1 to 3 wherein the considered parameters addition-ally include coordinates of a contour of any obstacle being present within the area, wherein the obstacle is not to be passed by the moving machinery, and wherein the method comprises the further steps of: -making a copy of the contour of any obstacle and geometrically translating thiscopy in parallel by a fraction of the working width away from the obstacle interior, thereby obtaining a corrected eroded area and an eroded obstacle contour of any obstacle being present within the area and -performing the first mathematical algorithm for each of the eroded obstacle con-tours, thereby generating an obstacle headland path for each obstacle being pre- sent within the area and the path being feasible with respect to dynamics of the machinery and minimizing area coverage gaps.

5. The method of claim 4 wherein a second cascade of headland paths are generatedby an erosion operation for a given working width and a given contour of any obstacle present within the field area and wherein the first mathematical algorithm is iterativelyapplied to each of the second cascade of headland paths, thereby generating multiple obstacle headland paths each being feasible with respect to dynamics of the machin- ery and minimizing area coverage gaps.

6. The method of any one of claims 2 to 5, if referring to claim 2, wherein the methodcomprises the step of determining a sequence of mainfield lanes to generate a path plan for area coverage whose traversal does not yet account for feasibility with respect to the dynamics of the machinery and for area coverage gap avoidance, and perform- ing the method within a global optimization scheme,thereby generating an optimal path plan for area coverage that is feasible with respect to dynamics of the machinery, minimizing area coverage gaps and is optimized ac- cording to one of the criteria of: minimization of total field coverage pathlength, mini- mization of accumulated length of mainfield lanes, minimization of accumulated turn-ing for field coverage, minimization of accumulated turning when traversing between mainfield lanes and headland path, minimization of maximal turning when traversingbetween mainfield lanes and headland path, minimization of accumulated absolutevehicle roll angles above a user-defined threshold resulting from field slopes in hilly terrain along mainfield lanes, minimization of maximal absolute vehicle roll anglesresulting from field slopes in hilly terrain along mainfield lanes, avoidance of absolutevehicle roll angles above a user-defined threshold resulting from field slopes in hilly terrain, connections to roads and logistical harvesting organization, minimization of completion time for field coverage, as well as a weighted trade-off between at least some of the aforementioned criteria.

7. The method of claim 6 wherein heuristics are applied for increased computationalefficiency to discard path plan candidates that cannot attain the optimal path plan solution.

8. The method of any one of claims 1 to 7 wherein the method provides as output atleast a sequence of position coordinates, the sequence giving a path to be followed by the machinery.

9. A first software program for performing the method of any one of claims 1 to 8, thefirst software program comprising an algorithm for computing a path plan feasible withrespect to dynamics of the machinery and minimizing area coverage gaps.

10. A second software program comprising storing means for storing data identifying apath based on position coordinates of a machinery according to the method of any one of claims 1 to 8 wherein the second software program gives guidance to the ma-chinery and / or a user of the machinery in order to enable the machinery to follow the path by usage of information of at least one location sensor.

11. A device for performing the method according to any one of claims 1 to 8 wherein thedevice comprises input means for providing parameters, means for reading of geo- graphic location measurements returned from a location or positioning sensor and for choosing heuristics and wherein the device comprises -display means for displaying data enabling a machinery and / or a user of the ma-chinery to follow a path and / or -output means for delivering data enabling the machinery and / or the user of themachinery to follow the path.

12. A computer-implemented method for automatically path planning for optimizing a pathof a machinery moving in an area, the method comprising the steps of -considering at least the following parameters:- coordinates of an area contour of the area in a global coordinate system,- a working width of the machinery operating in the area,- a mathematical description of dynamics of the machinery operating in thearea in form of a system of nonlinear differential equations, -making a copy of the area contour and geometrically translating this copy in par-allel by a fraction of the working width to an interior of the area, thereby obtaining an eroded area being smaller or equal than the area and an eroded area contour of the eroded area, the eroded area contour being denoted as a headland path, -selecting an orientation of straight lanes or a shape of freeform lanes within thearea bounded by the modified headland path and thereby partitioning the eroded area into adjacent mainfield lanes, -intersecting the mainfield lanes with the headland path, thereby generating alane-grid, -determining a sequence for the traversal of the mainfield lanes for area coverage,thereby generating a path plan for area coverage given by a sequence of position coordinates,- determining second segments along the path plan for area coverage which arenot feasible to be traversed by the machinery due to the dynamics of the machin- ery or whose traversal would result in area coverage gaps, specifically, at transi- tions between headland path and mainfield lanes, between mainfield lanes and headland path, and between pairs of mainfield lanes, -modifying the second segments such that their traversals become feasible withrespect to dynamics of the machinery and such that their traversals minimize area coverage gaps, thereby using a second mathematical algorithm, -wherein the second mathematical algorithm comprises two hierarchical steps forhandling of each of the second segments, (i) first, creating a first modification by modifying spatial waypoint coordinatesalong each of the second segments through their replacement with a suit- ably fitted path segment, preferably a Dubins or Reeds-Shepp path seg- ment, such that the second resulting path is smoothed but in general not yet feasible with respect to dynamics of the machinery, (ii) second, creating a second modification by applying a transformation of thesystem of nonlinear differential equations describing the dynamics of the machinery from a time domain into a spatial domain with space along the second resulting path of step (i) replacing time as a dependent variable in the system of nonlinear differential equations and introducing a new set of variables in a path-aligned coordinate system instead of the global coordi- nate system, formulating a second mathematical optimization problem with a set of inequality constraints based on the path resulting from the previous step (i) such that area coverage gaps are avoided, and with a set of equality constraints such that feasibility of a second resulting path with respect to the dynamics of the machinery is addressed, solving the second mathe- matical optimization problem and transforming a solution of the second mathematical optimiziation problem from the path-aligned coordinate sys- tem back to the global coordinate system, such that after the modification of all the second segments a modified path plan for area coverage is gen- erated that is feasible with respect to dynamics of the machinery and mini- mizes area coverage gaps.

13. The method of claim 12 wherein the method comprises the step of determining asequence of mainfield lanes to generate a path plan for area coverage whose tra- versal does not yet account for feasibility with respect to the dynamics of themachinery and for area coverage gap avoidance, and performing the method within a global optimization scheme, thereby generating an optimal path plan for area coverage that is feasible with respect to dynamics of the machinery, minimizing area coverage gaps and is optimized ac- cording to one of the criteria of: minimization of total field coverage pathlength, mini- mization of accumulated length of mainfield lanes, minimization of accumulated turn-ing for field coverage, minimization of accumulated turning when traversing between mainfield lanes and headland path, minimization of maximal turning when traversingbetween mainfield lanes and headland path, minimization of accumulated absolutevehicle roll angles above a user-defined threshold resulting from field slopes in hilly terrain along mainfield lanes, minimization of maximal absolute vehicle roll anglesresulting from field slopes in hilly terrain along mainfield lanes, avoidance of absolutevehicle roll angles above a user-defined threshold resulting from field slopes in hilly terrain, connections to roads and logistical harvesting organization, minimization of completion time for field coverage, as well as a weighted trade-off between at least some of the aforementioned criteria.

14. The method of claim 13 wherein heuristics are applied for increased computational efficiency to discard path plan candidates that cannot attain the optimal path plan solution.

15. The method of any one of claims 12 to 14 wherein the method provides as output at least a sequence of position coordinates, the sequence giving a path to be followed by the machinery.

16. A first software program for performing the method of any one of claims 12 to 15, thefirst software program comprising an algorithm for computing a path plan feasible with respect to dynamics of the machinery and minimizing area coverage gaps.

17. A second software program comprising storing means for storing data identifying apath based on position coordinates of a machinery according to the method of any one of claims 12 to 15 wherein the second software program gives guidance to the machinery and / or a user of the machinery in order to enable the machinery to follow the path by usage of information of at least one location sensor.

18. A device for performing the method according to any one of claims 12 to 15 whereinthe device comprises input means for providing parameters, means for reading of geographic location measurements returned from a location or positioning sensor and for choosing heuristics and wherein the device comprises- display means for displaying data enabling a machinery and / or a user of the ma-chinery to follow a path and / or- output means for delivering data enabling the machinery and / or the user of themachinery to follow the path.

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