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In last decades, great technology advances have been done related to the automotive sector, especially in Advanced Driver Assistance Systems (ADAS) developed to improve mobility in terms of comfort and safety during driving process; hence, automated driving is presented as an evolution of those systems in the present and upcoming years. The aim of this work is to present a complete framework of motion planning for automated vehicles, considering different constraints with parametric curves for lateral and longitudinal planners. Parametric Bézier curves are used as the core approach for trajectory design in intersections, roundabouts, and lane change maneuvers. Additionally, a speed planner algorithm is presented using the same parametric curve approach, considering comfort and safety. A simulation environment is used for testing the planning method in urban conditions. Finally, tests with the real platform in automated mode have been performed showing goods results.

Every year 1.2 million people die in road related accidents. The NHTSA conducted a study concluding that 94% of accidents are due to human errors, where a great amount of these crashes (33%) is related to wrong decisions during driving process [

In last decades, a considerable amount of institutions, research centers, and companies is seeking to improve this technology faster [

There are some other topics; those have received less attention than the ones named before, i.e., vehicle decision algorithm. This area is one of the most challenging topics in automated driving because it must deal with a great number of task, as generation of smooth trajectories [

Some methods explored in automated vehicle decision are in [

Other authors, as [

In [

In such a way, this work presents a unified motion planning framework (trajectory planning and speed profile) based on 4th and 5th degree parametric Bézier curves. It specially focuses on urban scenarios and their geometrical design. It leans on the construction of two-dimensional curves for both (lateral and longitudinal) planning methods, simplifying the possible designing parameters to the most considerable ones without degrading the curve capacities. The work specially focuses on giving a tool-set for trajectory designing under typical scenarios as intersections and lane changes and to a more specific and complex scenario as roundabouts. This, last scenario is especially interesting for its set of possible parameters and conditions; it is relevant to consider that the roundabout complexity has not been sufficiently studied for other authors in the past [

Finally, the contribution will be organize as follows: Section

This section explains all the information related to the trajectory planning approach based on real-time parametric 4th and 5th degree Bézier curves. The approach’s core will be focused on using these curves in path planning generation for urban scenarios (intersections, roundabouts, and lane changing) and speed planning for comfortable and safe behaviors. This approach is not limited to urban scenarios, although it is the main target of this work.

The planning framework explanation will be divided into three parts: (i) global planning using a global map (based on a point to define the structure of intersections, lane change or roundabouts) describing the route with reduced amount of points, (ii) local planning approach based on Bézier and specially focus on intersections, lane changes, and roundabouts (the straight and arc segment can be resolved as sequences of points), and (iii) the speed planning approach based on the same type of curves.

Bézier is a type of parametric curve, which has been commonly used for computer graphics, animations, and path generation in robotics [

These curves have several properties relevant for the purpose of the current work:

The starting point of the Bézier curve corresponds with control point

The first point tangent vector (at

The curve will lie into the convex hull formed by the control points.

Bézier curves are continuous geometrically and in curvature

Bézier curves are symmetric, and the generated curve for

This work uses 4th and 5th grade Bézier curves (some considerations with 3rd degree will be done). Higher order curves are not considered because they do not have additional benefits within current approach but increase the complexity [

Rewriting the general Bézier equation (see (

Compact Bézier polynomial Coefficients “C”.

C | Bézier curve order | ||
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3rd Order | 4th Order | 5th Order | |

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In general terms the curvature in

The previous equation explicitly shows that if the three starting points in a curve are colinear, generated Bézier curve will have zero curvature at its starting point (

In this work, global planning is used as base route to apply the local planning approach using Bézier. A simple map containing a point to define intersections, roundabouts, or lane changes (point with “x” in Figures

Trajectory planning on intersection.

Trajectory planning on lane change.

Trajectory planning on roundabouts.

Figure

In this case, the approach considers straight segments during the maneuver (Figure

Additionally, in this type of curve a new point

Figure

Roundabout entrance and exit points are defined by

In addition to these two points,

The main contribution of the current paper is in the generation of smooth and continued curvature trajectories by definition in automatic vehicles. The principal goal is urban scenarios, but it is not limited to them; hence the section will be divided into three scenarios: intersections, roundabout, and lane changes.

Figure

As it is shown in (

Using this definition is inferred that the minimum Bézier order that can be used in an intersection is 4th (5 control points), but in this case control point

Bézier control points for lateral and longitudinal approach for intersections “Int”, roundabout entrance “RE”, roundabout exit “Rex”, lane change maneuver “LC”, and speed planning.

Planning | Control points n-order Bézier | ||||||
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Lateral | Int | | | | | | |

RE | | | | | | - | |

REx | | | | | | - | |

LC | | | | | | | |

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Longitudinal | Speed | | | | | | |

The distance of control points has been set to ensure that maximum curvature of Bézier curve is reached in middle point of the trajectory, in order to know if peak curvature is under the vehicle's mechanical limits. Figure

Curvature analysis for different intersection angles.

With the purpose of finding the position of maximum curvature, it is applied derivative to (

Figure

The curvature at entrance in

Generated segments at the entrance and exit of roundabout must fit its inner part with a curvature equal to the inverse of the radius.

Bézier joining point angle must be the same of the circle arc angle (continuous trajectory).

In order to follow the aforementioned criteria, 3 colinear control points are designed to ensure the first consideration; another 2 points are selected to form a tangent segment to the circle, which assures curvature and direction continuity (second and third criteria) along the roundabout (three colineal points will generate curvature 0 and not

Table

The point

Figure

Examples of roundabout trajectories for different D values.

Generated trajectories

Curvature of the different trajectories

The approach is done using similar criteria as intersections, but control points are aligned with lane axis and equidistant by a distance

The location of control points is given in Table

The overtaking is considered as a special case composed by two lane changes; a first lane change is using the proposed method and when the first lane change is finished, this lane will be kept until the overtaking process is done. The returning will be done applying symmetry criteria and same propositions used in first lane change.

The speed control is directly affected by sudden changes in reference speed, and if these changes are smooth and continuous the reference speed tracking will be done with a better performance. In such a way a speed planner is proposed based on Bézier curves that permits:

Anticipate future conditions in the speed of the road.

Applying physical constraints in vehicle acceleration and deceleration.

Keeping safely the vehicle velocity under speed limits.

Incorporating comfort of passengers.

The speed planner approach uses 5th order Bézier curve to keep advantages of a higher degree and symmetry. Figure

Define the separation between consecutive control points

The physical constraints of the vehicle (acceleration and deceleration process) will have a direct relation with the generated curve. In this case, it is considered the general equation that relates speed to acceleration and doing a variable change

Applying derivative to (

From (

The first step in the speed planning method is applying the comfort speed limitation (reference speed) presented in [

Figure

Stages for speed planning approach.

Speed planning with overlapped intersections

Speed planning without overlapped parts

Speed planning designing method.

Finally, Figure

The architecture presented in [

Figure

Control architecture of automated vehicles.

The decision module is the main target of the current work (grey coloured module in Figure

The current section explains the approach with a numerical example using the simulation tool Dynacar [

Simulated results based on real urban scenario.

Trajectory generated with the approach

Comparison of the longitudinal acceleration

Comparison of the path curvature and simulated vehicle one

The total amount of points used in the map definition will be divided among: 1 starting point, 1 ending point, 9 intersections, and 2 roundabouts, resulting in 13 points defining all the route. Table

Point description of the map used for global planning.

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From this map a global plan is generated, which gives a better description of roundabouts in the bend segment (completion of the circle arc). In case of intersections, points in global planner coincide with those used to define the route in the scenario. All the global planner points are depicted as circles in Figure

The soft and continuous black line represents the trajectory generated using the Bézier approach. The values of

An interesting example of the entrance/exit angle in the roundabouts is the one located in the absolute coordinates

The speed planning defined for that trajectory is shown in Figure

Figure

The current section shows the trajectory planning approach used with the real vehicle. In this sense, two possible definitions of the route were implemented (Figure

Real life validation of the Bézier planning approach.

Comparison in trajectory definition

Comparison in trajectory definition

Comparison in curvature for the trajectory definition case 2

The first case is shown in the left side of Figure

The trajectory in the right side was tested with 5 different speed profiles and the behavior of the vehicle was analyzed with the curvature. Figure

Figure

Maximum lateral acceleration obtained in the real experiment using speed profile.

Uncomfortable levels | Not | A little | Fairly | Very | Extreme |

Uncomfortable magnitudes | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 |

Filtered max. values | 0.45 | 1.03 | 1.54 | 2.00 | 3.02 |

This work presents a trajectory and speed planning approach based on Bézier curves used in the decision module of an automated vehicle control architecture. The approach considers urban scenarios with intersections, roundabouts, and lane changes but it is not limited to those; the approach could also be used in scenarios such as highways. In the case of the speed planner, a parametric curve approach to adapt future changes in the speed limit considering physical constraints of the vehicle and traffic rules is presented (i.e., considering the maximum speed limit of the road).

Simplified mathematical models for trajectory planning using Bézier curves on intersections, roundabouts, and lane changes are presented reducing the time of calculation and allowing the use of these definitions for other applications, i.e., trajectory optimization process with computational efficiency. Additionally, the approach is evaluated using a simulated urban scenario based on a location of Basque Country showing that a great variety of urban scenario can be managed using this approach.

The trajectory planner was tested for automated driving, using a comparison of the trajectory with the Bézier curves and the one generated using a manual GPS recording in manual driving. Additionally the generated curvature in the vehicle for different conditions (variations in the speed profile) was compared with the curvature of the trajectory planned showing that it is feasible to be tracked by the vehicle.

The speed planning approach shows the capability of using parametric curves to generate a profile to be tracked correctly by the vehicle, giving a prediction of future behaviors. Consideration of longitudinal acceleration in curve segments is one of the future improvements related to the speed planner of this work.

This approach is considered better than others due to the fact that it includes comfort and safety criteria during trajectory and speed planning. For comfort the ISO2631-1 standard to generate speed profiles based on geometric information given by the trajectory planning was considered. Comfortable speed profiles are related to trajectories and this can be associated with safety during driving (vehicle lateral forces are related to possible dangerous situations), and additionally a continuous curvature of the trajectory planned can reduce possible unexpected situations given by vehicle' automatic control (increasing safety).

Some of the future works are related to using the presented method to generate possible optimal solutions in terms of the parameter

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

The authors declare that they have no conflicts of interest.

This work was partly supported by ECSEL Project ENABLE-S3 with Grant Agreement no. 692455-2 and the AutoDrive ECSEL Project with Grant Agreement no. 737469.