Efficient trajectory planning method under multi-layer structured environment

Abstract

This invention provides an efficient trajectory planning method for multi-layered structured environments, used for autonomous navigation of ground robots in large and complex multi-layered building spaces. The problem is simplified to navigation between complex intersecting planes. Trajectory optimization is constructed based on motion state trajectories by pre-extracting trajectories from 3D point clouds and fully considering special constraints when traversing multi-layered planes to maximize the robot's maneuverability.

Description

Technical Field

[0001] This invention belongs to the field of robot trajectory planning, and in particular relates to an efficient trajectory planning method in a multi-layered structured environment. Background Technology

[0002] Robots are expected to assist in all aspects of human life, especially in the built spaces where people spend a significant amount of time. Although the mobility and precise positioning of tracked or legged robots have advanced rapidly and appear to be well-solved, the challenge of generating efficient, smooth, and high-quality trajectories within built spaces remains.

[0003] Most current planning methods only focus on single-layer environments and cannot be applied to complex, multi-layered structured environments.

[0004] Current planning methods struggle to quickly generate feasible trajectories.

[0005] Current planning methods lack consideration of semantic information, so the robot cannot know which plane it is on, resulting in dangerous operations when the planned trajectory is going up or down stairs. Summary of the Invention

[0006] To address the problems of existing technologies, this invention provides an efficient trajectory planning method for multi-layered structured environments, used for autonomous navigation of ground robots in large and complex multi-layered building spaces. The problem is simplified to navigation between complex intersecting planes. Trajectory optimization is constructed based on motion state trajectories by pre-extracting trajectories from 3D point clouds and fully considering special constraints when traversing multi-layered planes to maximize the robot's maneuverability.

[0007] The technical solution of the present invention is as follows:

[0008] An efficient trajectory planning method for multi-layered structured environments includes the following steps:

[0009] Obtain the global point cloud of a multi-layered structured environment and use it to calculate the Euclidean signed distance field;

[0010] Obtain the starting and ending coordinates, and use the multi-layered accessible plane undirected cost graph to obtain a feasible path.

[0011] in,

[0012] The feasible path adopts a motion state trajectory (MS trajectory), which is a yaw angle relative to time. w A polynomial function of θ and the forward arc length s;

[0013] The m-th trajectory i A segment can be represented as:

[0014]

[0015] s mi (t)=β T (t)c s,mi (2)

[0016] Where c mi =[c θ,mi ,c s,mi ] are the coefficients of the polynomial, and β(t) is the natural basis;

[0017] For simplicity, use σ = [w θ,s ] T .

[0018] To represent a trajectory spanning multiple planes, the entire trajectory is divided into T parts, where the t-th part is... i The segment represents the tth i Trajectories on a plane; each trajectory part consists of M ti It consists of polynomial segments, therefore the entire trajectory includes Segment; each polynomial segment is restricted to a single plane;

[0019] Since each plane has its own coordinate system, the trajectory must lie on the plane; therefore, the trajectory should be transformed into the local coordinate system for planning; at time t... i Within a plane, the trajectory should start from the starting point. w p0 or a point on the intersection line Initially, its projection on the plane is determined by... ti p0 = { ti x0, ti y0} is given;

[0020] For simplicity, w p0 and equivalence;

[0021] In Ψ ti The trajectory in can be represented as:

[0022]

[0023] in It is Ψ ti Relative to Ψ w Yaw angle offset;

[0024] Use Simpson's rule to approximate { ti x(t), ti y(t)} allows the robot to calculate Ψ at any time. ti The state in;

[0025] Based on the trajectory representation proposed in Section IVB, the algorithm sets the objective function J as:

[0026]

[0027] Where W∈R 2×2 It is a diagonal matrix used to penalize control effort. It is the duration of the trajectory, ∈ T The weights are used; the objective is to minimize the jerk σ to reduce energy loss, while introducing a time term to balance the trajectory duration; a penalty function method is employed, ensuring that the constraint d is included in the optimization function. Where w d This represents the corresponding weight;

[0028] In cross-plane programming, the trajectory starts from... It begins at the global final position or at a point on the intersection line. w p t f i ,in Due to the intersection line It was obtained through search rather than optimization, in order to ensure the optimality of the trajectory. It should be part of the optimization variables; assuming the endpoints of the intersection are The algorithm introduces the parameter η ti As:

[0029]

[0030] in From arrive Positional proportions; due to Introducing unconstrained variable η ti As an optimization variable to simplify the problem; since the MS trajectory requires integration to calculate the position, ALM is introduced to ensure that the trajectory on the plane reaches the given final position; assuming that the integration is performed on the t-th... i The final position calculated on each plane is The expected final position is On the last plane, the expected final position is the global final position. Hereinafter, algorithms are collectively referred to as Taking the x-axis on the ti-th plane as an example, the final position constraint can be expressed as:

[0031]

[0032] Therefore, the new optimization problem can be formulated as:

[0033]

[0034] in These are dual variables, and ρ>0 represents the weight of the augmented term;

[0035] Set convergence conditions so that

[0036]

[0037] In practice, algorithms typically set e max =1cm, to ensure that the final position error has the least impact on subsequent control.

[0038] Preferably, the efficient trajectory planning method in the multi-layered structured environment further includes speed constraints, direction constraints, and safety constraints on feasible paths, wherein;

[0039] Considering the influence of gravity, the velocity on the slope varies with direction, and the maximum velocity on the slope is represented by the robot's velocity in the planar coordinate system Ψ. p Yaw angle in p A function of θ;

[0040] As the robot moves forward:

[0041]

[0042] Where v max ψ is the maximum forward speed on horizontal ground, and ψ is the inclination angle of the slope; r r (ψ) and r d (ψ) is a function of the velocity ratio when ascending and descending along a plane, which is related to a specific robot;

[0043] Equation (10) uses two semi-ellipses to approximate the maximum velocity, ensuring that v max ( p θ) relative to p θ is continuous and differentiable;

[0044] Considering that differential drive robots require different wheel speeds to achieve rotation, the maximum angular velocity should depend on the current speed; the algorithm can obtain the corresponding constraints as follows:

[0045]

[0046] Where κ∈{-1,1}, ω M It is the maximum angular velocity when rotating in place. These are angular velocity and linear velocity; when the robot moves backward, the constraint C can also be obtained from equation (11). m- .

[0047] Orientation constraints: When the robot moves on stairs, the number of contact points is significantly less than on a plane; if the orientation... pIf θ deviates too much from the direction of ascent of the stairs, it may cause a side slip or even a fall.

[0048] To ensure safety, the algorithm needs to constrain the angles between them:

[0049]

[0050] Where k is an integer, θ s It is the maximum expected angle;

[0051] Because the direction of ascent of the stairs is Ψ p The x-axis, equation (12) will Ψ p The yaw angle constraint is close to 0 or π.

[0052] Safety constraints: A distance field is used to ensure safety; the constraint requires the current position {p} x ,p y} in Ψ p The ESDF value in the data is greater than the safe distance d. s :

[0053] C s ( p x, p y)=d s -E p ( p x, p y), (13)

[0054] Where E p (p x ,p y The ESDF value is obtained through bilinear interpolation in plane p.

[0055] Preferably, in the aforementioned efficient trajectory planning method for multi-layered structured environments,

[0056] After sampling and filtering the global point cloud, a normal map is obtained by estimating the normals; all planes are extracted using the normal map through a region growing method; all planes are divided into traversable planes and perpendicular planes based on the covariance matrix, and coplanar and adjacent traversable planes are merged to obtain a planar structure map; the points of each traversable plane are projected onto the plane, and the minimum convex polygon containing these points is calculated to obtain a connected multi-layer plane;

[0057] The connected multi-layered planes are meshed, and the meshes are marked as unknown meshes, safe meshes, overlapping meshes, boundary meshes, or occupied meshes. The unknown meshes, occupied meshes, and boundary meshes are used as obstacles to update the Euclidean signed distance field, obtaining vertex sets, paths, and corresponding weights, which are stored as a multi-layered passable plane undirected cost graph. If two parallel planes of the same size and close to each other exist, the lower plane is removed. The mesh graph is initialized to an unknown state; if a mesh has at least one projection point, it is set to a safe state; meshes corresponding to intersections with planes are marked as connected, while overlapping, traversable regions are marked as overlapping; overlapping meshes adjacent to safe meshes are defined as boundary meshes; vertical planes may be located above as obstacles, and Alpha Shapes are used to find the boundary points of vertical planes; if the distance to a boundary point is less than the safe distance, the corresponding mesh will be set to an occupied state.

[0058] Preferably, in the aforementioned efficient trajectory planning method for multi-layered structured environments,

[0059] These planes are described using an undirected cost graph G = (V, E, W);

[0060] The vertex set V is defined as the intersection line l i，j The point on;

[0061] Assuming it is located at l i，j Find the midpoint; use ESDF to check if the point is feasible in two adjacent planes; if not, the algorithm checks the points on both sides until the first feasible point is found;

[0062] After obtaining all V, the algorithm traverses all planes and uses A* to determine whether there exists a path connecting two vertices. The path is recorded in the plane as an edge E, at the cost of the corresponding weight W.

[0063] The present invention has the following beneficial effects:

[0064] An efficient trajectory generation planner for complex 3D architectural spaces is proposed, which simplifies the navigation problem between planes by utilizing traversable planes;

[0065] An optimization-based trajectory generation method is proposed, which effectively generates safe and kinematically appropriate trajectories.

[0066] Other advantages, objectives and features of the present invention will become apparent in part from the following description, and in part from those skilled in the art through study and practice of the invention. Attached Figure Description

[0067] Figure 1A process diagram illustrating the extraction of a traversable plane in an embodiment of the efficient trajectory planning method for multi-layered structured environments provided by the present invention.

[0068] Figure 2 The trajectory diagrams for going up and down stairs are shown in one embodiment of the efficient trajectory planning method for multi-layered structured environments provided by the present invention.

[0069] Figure 3 A trajectory comparison diagram with and without directional constraints is provided in one embodiment of the efficient trajectory planning method in a multi-layered structured environment for the present invention.

[0070] Figure 4 This is a process diagram of an embodiment of the efficient trajectory planning method in a multi-layered structured environment provided by the present invention. Detailed Implementation

[0071] The present invention will now be described in further detail with reference to the accompanying drawings, so that those skilled in the art can implement it based on the description.

[0072] This invention provides an efficient trajectory planning method for multi-layered structured environments, comprising the following steps:

[0073] Based on global point clouds, planar structures in a map can be extracted and their intersection information obtained. The ESDF of the plane can then be calculated using point clouds. Simultaneously, the algorithm utilizes the point cloud to obtain the local coordinate system and distance field of the planar structures.

[0074] Obtain the starting position w After p0, the algorithm first projects the point onto the nearest plane P1 and uses its projection point. 1 p0 is the starting node. The algorithm starts from... 1 Search for all vertices from p0 to p1. Add feasible paths and their costs to G. Final position. w p f The final node is treated similarly. A breadth-first search is performed within G to connect edges and find the path from the starting node to the final node.

[0075] Because differentially driven robots exhibit superior maneuverability in confined environments, the algorithm uses the MS trajectory, which is suitable for these robots. Intuitively, a trajectory on a plane represents the incremental position from a starting position on the plane. The MS trajectory is the yaw angle w relative to time. θ A polynomial function of the advancing arc length s. The m-th digit of the trajectory. i A segment can be represented as:

[0076]

[0077] s mi (t)=βT (t)c s,mi (2)

[0078] Where c mi =[c θ,mi ,c s,mi ] are the coefficients of the polynomial, and β(t) is the natural basis. For simplicity, the algorithm uses σ = [ w θ,s] T The higher-order continuity of the MS trajectory is inherently satisfied, and a smooth bijection is used for the nonnegativity constraint of the trajectory duration. To represent a trajectory spanning multiple planes, the algorithm divides the entire trajectory into T parts, the t-th part being... i The segment represents the tth i Trajectories on a plane. Each trajectory part consists of M. ti It consists of polynomial segments, therefore the entire trajectory includes M ti Segment. Each polynomial segment is restricted to a single plane. Unlike [1], since each plane has its own coordinate system, the trajectory must lie on the plane. Therefore, the trajectory should be transformed into the local coordinate system for planning. At the t-th i Within a plane, the trajectory should start from the starting point. w p0 or a point on the intersection line Initially, its projection on the plane is determined by... ti p0 = { ti x0, ti y0} is given. For simplicity, w p0 and Equivalent. In Ψ ti The trajectory in can be represented as:

[0079]

[0080] Where Δθ ti It is Ψ ti Relative to Ψ w The yaw angle offset. The algorithm uses Simpson's rule to approximate it.

[0081] { ti x(t), ti y(t)} allows the robot to calculate Ψ at any time. ti The state in.

[0082] Based on the trajectory representation proposed in Section IVB, the algorithm sets the objective function J as:

[0083]

[0084] Where W∈R 2×2 It is a diagonal matrix used to penalize control effort. It is the duration of the trajectory.

[0085] T These are weights. The objective is to minimize the jerk σ to reduce energy loss, while introducing a time term to balance the trajectory duration. The algorithm employs a penalty function approach, ensuring that the constraint d is included in the optimization function. ≤0, where w d This represents the corresponding weight. In cross-plane programming, the trajectory starts from... It begins at the global final position or at a point on the intersection line. in Due to the intersection line It was obtained through search rather than optimization. To ensure the optimality of the trajectory, It should be part of the optimization variables. Assume the endpoints of the intersection are... The algorithm introduces the parameter η ti As:

[0086]

[0087] in From arrive The positional proportion. Due to The algorithm introduces an unconstrained variable η ti As an optimization variable to simplify the problem. Since the MS trajectory requires integration to calculate the position, the algorithm introduces ALM

[28] to ensure that the trajectory on the plane reaches the given final position. Assume that by integration at the t-th i The final position calculated on each plane is The expected final position is On the last plane, the expected final position is the global final position. the following,

[0088] general term for algorithms Taking the x-axis on the ti-th plane as an example, the final position constraint can be expressed as:

[0089]

[0090] Therefore, the new optimization problem can be formulated as:

[0091]

[0092] in The dual variable is ρ>0, which is the weight of the augmented term. The algorithm can solve the optimization problem (8) and update λ to iteratively obtain the solution. For gradient propagation and solving optimization problems, the reader can refer to

[13] . The algorithm sets convergence conditions such that

[0093]

[0094] In practice, algorithms typically set e max =1cm, to ensure that the final position error has the least impact on subsequent control.

[0095] Velocity Constraint: Considering the influence of gravity, the robot's velocity on the slope varies with direction. To describe this variation, the algorithm represents the maximum velocity on the slope as the robot's velocity in the planar coordinate system Ψ. p Yaw angle in p A function of θ. For example, when the robot moves forward:

[0096]

[0097] Where v max ψ is the maximum forward speed on horizontal ground, and ψ is the inclination angle of the slope. r (ψ) and r d (ψ) is a function of the velocity ratio during ascent and descent along the plane, which is specific to the particular robot. Equation (10) approximates the maximum velocity using two semi-ellipses, ensuring that v max ( p θ) relative to p θ is continuous and differentiable.

[0098] Similar to

[13] , considering that differential drive robots require different wheel speeds to achieve rotation, the maximum angular velocity should depend on the current speed. The algorithm can obtain the corresponding constraints as follows:

[0099]

[0100] Where κ∈{-1,1}, ω M It is the maximum angular velocity when rotating in place. These are angular velocity and linear velocity. When the robot moves backward, constraint C can also be obtained from equation (11). m- .

[0101] Orientation constraints: When the robot moves on stairs, the number of contact points is significantly less than on a plane. If the orientation... p If θ deviates too much from the direction of ascent of the stairs, it could lead to slipping or even tipping over. To ensure safety, the algorithm needs to constrain the angle between them:

[0102]

[0103] Where k is an integer, θ s That is the maximum expected angle. Since the direction of ascent of the stairs is Ψ. p The x-axis, equation (12) will Ψp The yaw angle constraint is close to 0 or π.

[0104] Safety Constraints: The algorithm uses an ESDF (Extended Range Field) to ensure safety. The constraint requires the current position {p} to be within the specified range. x ,p y} in Ψ p The ESDF value in the data is greater than the safe distance d. s :

[0105]

[0106] Where E p ( p x, p y) is the ESDF value obtained through bilinear interpolation in plane p.

[0107] like Figure 2 As shown, the trajectory exhibits higher speeds when moving in a straight line on a horizontal plane or descending stairs, and lower speeds when ascending stairs. The trajectory switches to the next plane at the intersection, ensuring velocity continuity during the transition. The application of the Augmented Lagrange Method (ALM) minimizes trajectory connection errors between planes, while the continuity of the polynomial trajectory ensures smooth higher-order kinematics. We present the trajectory and velocity profiles for ascending and descending stairs. The proposed method effectively limits the velocity on inclined planes to a reasonable range, based on the plane's tilt angle. Due to the characteristics of traversable planes, the proposed method can constrain the robot's orientation to ensure safety on hazardous planes. Figure 3 As shown in (a), without the directional constraints in equation (12), the robot might approach the stairs at a large angle, risking obstruction or tipping over. By constraining the direction, the robot can climb the stairs effectively, as shown in (a). Figure 2 As shown in (b), the initial trajectory obtained from the search path can be approximately parallel to the x-axis of the stairs. Orientation constraints effectively limit the robot's direction on the stairs, ensuring its safety.

[0108] To demonstrate the effectiveness of the proposed trajectory generation method, we constructed four complex 3D structural environments to validate the proposed approach:

[0109] like Figure 4 As shown, the first row displays the point cloud of the environment: multi-level (c1) and buildings (d1). The second row displays the traversable planes, where white lines are intersections between planes, red lines are the plan view, and green lines are the searched paths. The third row displays the ESDF and the generated trajectory. The table shows the trajectory length L and trajectory optimization time To. The proposed method is able to generate feasible and safe trajectories in complex built spaces.

[0110] Multi-layer: A multi-level structure with complex routes that require robots to choose the correct path between ramps and stairs to reach higher floors.

[0111] Building: A common building structure with stairs, which robots need to use to move between floors.

[0112] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. They can be applied to various fields suitable for the present invention. For those skilled in the art, other modifications can be easily made. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details.

Claims

1. An efficient trajectory planning method in a multi-layered structured environment, characterized in that, Includes the following steps: Obtain the global point cloud of a multi-layered structured environment and use it to calculate the Euclidean signed distance field; Obtain the starting and ending coordinates, and use the multi-layered accessible plane undirected cost graph to obtain a feasible path. in, The feasible path uses the motion state trajectory, which is the yaw angle relative to time. w A polynomial function of θ and the forward arc length s; The m-th trajectory i A segment can be represented as: Where c mi =[c θ,mi ,c s,mi ] are the coefficients of the polynomial, and β(t) is the natural basis; For simplicity, use σ = [ w θ,s] T ; To represent a trajectory spanning multiple planes, the entire trajectory is divided into T parts, where the t-th part is... i The segment represents the tth i Trajectories on a plane; each trajectory part consists of M ti It consists of polynomial segments, therefore the entire trajectory includes Segment; each polynomial segment is restricted to a single plane; Since each plane has its own coordinate system, the trajectory must lie on the plane; therefore, the trajectory should be transformed into the local coordinate system for planning; at time t... i Within a plane, the trajectory should start from the starting point. w p0 or a point on the intersection line Initially, its projection on the plane is determined by... ti p0 = { ti x0, ti y0} is given; For simplicity, w p0 and equivalence; In Ψ ti The trajectory in can be represented as: Where Δθ ti It is Ψ ti Relative to Ψ w Yaw angle offset; Use Simpson's rule to approximate { ti x(t), ti y(t)} allows the robot to calculate Ψ at any time. ti The state in; Based on the trajectory representation proposed in Section IVB, the algorithm sets the objective function J as: Where W∈R 2×2 It is a diagonal matrix used to penalize control effort. It is the duration of the trajectory, ∈ T The weights are used; the objective is to minimize the jerk σ to reduce energy loss, while introducing a time term to balance the trajectory duration; a penalty function method is employed, ensuring that the constraint d is included in the optimization function. Where w d This represents the corresponding weight; In cross-plane programming, the trajectory starts from... It begins at the global final position or at a point on the intersection line. in Due to the intersection line It was obtained through search rather than optimization, in order to ensure the optimality of the trajectory. It should be part of the optimization variables; assuming the endpoints of the intersection are The algorithm introduces the parameter η ti As: in From arrive Positional proportions; due to Introducing unconstrained variable η ti As an optimization variable to simplify the problem; since the motion trajectory requires integration to calculate the position, ALM is introduced to ensure that the trajectory on the plane reaches the given final position; assuming that the integration is performed on the t-th... i The final position calculated on each plane is The expected final position is ti∈{1,…,T-1}; on the last plane, the expected final position is the global final position. Hereinafter, algorithms are collectively referred to as ti∈{1,…,T}; taking the x-axis on the ti-th plane as an example, the final position constraint can be expressed as: Therefore, the new optimization problem can be formulated as: in These are dual variables, and ρ>0 represents the weight of the augmented term; Set convergence conditions so that In practice, algorithms typically set e max =1cm, to ensure that the final position error has the least impact on subsequent control.

2. The efficient trajectory planning method in a multi-layered structured environment as described in claim 1, characterized in that, It also includes speed constraints, direction constraints, and safety constraints on feasible paths, among which; Considering the influence of gravity, the velocity on the slope varies with direction, and the maximum velocity on the slope is represented by the robot's velocity in the planar coordinate system Ψ. p A function of the yaw angle pθ; As the robot moves forward: Where v max ψ is the maximum forward speed on horizontal ground, and ψ is the inclination angle of the slope; r r (ψ) and r d (ψ) is a function of the velocity ratio when ascending and descending along a plane, which is related to a specific robot; Equation (10) uses two semi-ellipses to approximate the maximum velocity, ensuring that v max (pθ) is continuous and differentiable with respect to pθ; Considering that differential drive robots require different wheel speeds to achieve rotation, the maximum angular velocity should depend on the current speed; The algorithm can obtain the following constraints: Where κ∈{-1,1}, ω M It is the maximum angular velocity when rotating in place. These are angular velocity and linear velocity; When the robot moves backward, constraint C can also be obtained from equation (11). m- ; Orientation constraints: When the robot moves on stairs, the number of contact points is significantly less than that on a plane; if the direction pθ deviates too much from the upward direction of the stairs, it may cause sideslip or even tip over. To ensure safety, the algorithm needs to constrain the angles between them: Where k is an integer, θ s It is the maximum expected angle; Because the direction of ascent of the stairs is Ψ p The x-axis, equation (12) will Ψ p The yaw angle constraint is close to 0 or π. Safety constraints: Use distance fields to ensure safety; The constraint requires the current position {p x ,p y } in Ψ p The ESDF value in the data is greater than the safe distance d. s : Where E p (p x ,p y The ESDF value is obtained through bilinear interpolation in plane p.

3. The efficient trajectory planning method in a multi-layered structured environment as described in claim 2, characterized in that, After sampling and filtering the global point cloud, a normal map is obtained by estimating the normals; all planes are extracted using the normal map through a region growing method; all planes are divided into traversable planes and perpendicular planes based on the covariance matrix, and coplanar and adjacent traversable planes are merged to obtain a planar structure map; the points of each traversable plane are projected onto the plane, and the minimum convex polygon containing these points is calculated to obtain a connected multi-layer plane; The connected multi-layered planes are meshed, and the meshes are marked as unknown meshes, safe meshes, overlapping meshes, boundary meshes, or occupied meshes, respectively. The unknown meshes, occupied meshes, and boundary meshes are used as obstacles to update the Euclidean signed distance field, obtaining vertex sets, paths, and corresponding weights, which are stored as an undirected cost graph of the multi-layered passable planes. If there are two parallel planes of the same size and close to each other, the lower plane is removed. The mesh graph is initialized to an unknown state. If there is at least one projection point in the mesh, it is set to a safe state. Meshes corresponding to the intersection lines between planes are marked as connected, while overlapping traversable regions are marked as overlapping. Overlapping meshes adjacent to safe meshes are defined as boundary meshes. Vertical planes may be located above as obstacles, and Alpha Shapes is used to find the boundary points of vertical planes. If the distance from the boundary point is less than the safe distance, the corresponding mesh will be set to an occupied state.

4. The efficient trajectory planning method in a multi-layered structured environment as described in claim 3, characterized in that, These planes are described using an undirected cost graph G = (V, E, W); The vertex set V is defined as the intersection line l i，j The point on; Assuming it is located at l i，j Find the midpoint; use ESDF to check if the point is feasible in two adjacent planes; if not, the algorithm checks the points on both sides until the first feasible point is found; After obtaining all V, the algorithm traverses all planes and uses A* to determine whether there exists a path connecting two vertices. The path is recorded in the plane as an edge E, at the cost of the corresponding weight W.

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