# Vehicle Reference Generator for Collision-Free Trajectories in Hazardous Maneuvers

Authors: Cuauhtémoc Acosta Lúa; Gerardo De Jesus Díaz; Stefano Di Gennaro; Tarek Kabbani
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

This paper presents a reference generator for ground vehicles, based on potential fields adapted to the case of vehicular dynamics. The reference generator generates signals to be tracked by the vehicle, corresponding to a trajectory avoiding collisions with obstacles. This generator integrates artificial forces of potential fields of the object surrounding the vehicle. The reference generator is used with a controller to ensure the tracking of the accident-free reference. This approach can be used for vehicle autonomous driving or for active control of manned vehicles. Simulation results, presented for the autonomous driving, consider a scenario inspired by the so-called moose (or elk) test, with the presence of other collaborative vehicles.


---

## Body

## 1. Introduction

One important problem to be solved in the case of autonomous vehicle, as well as in active controlled vehicles, is the generation of proper reference signals ensuring the prevention from collisions with obstacles. The situation is rendered more articulated by the possible presence in the scenario of other collaborative and noncollaborative vehicles. In this paper, the artificial potential fields for autonomous robot guidance, first proposed in [1], is reformulated in the case of vehicular dynamics to determine a reference generator, which generates signals corresponding to a trajectory avoiding collisions with obstacles. Then a controller ensures the tracking of these reference signals.In robotics, the generation of a reference using potential fields has been largely investigated. The basic idea is simple: obstacles generate repulsive forces, whereas goals to be reached generate attractive forces. Different approaches have been considered to have appropriate field configurations (e.g., see [2–4]) and handle moving obstacles and goals. Environmental hazards can be handled with simplicity and these methods can be implemented in real time, since the computation effort is not heavy [5].Among the various techniques that can be used for reference generation, one can use the navigation function [6], the construction of an admissible trajectory space [7], or control and optimization techniques [8, 9]. The latter use the model predictive control to generate the reference trajectory, neglecting the dynamic environment, and are usually complex and difficult to implement online. Other methods consider only indoor robots [10], whereas [11] proposes an algorithm based on graph searching and configuration space discretization to guide an indoor robot, but collision avoidance is not considered.In this paper, a reference generator for ground vehicles, based on potential fields, is presented. The aim is the generation of trajectories avoiding collisions with obstacles. This is done by a reference generator, in which the forces due to the potentials associated with the obstacles and with the goals appear. These potential fields, similar to the barrier forces [12], are modified to the case of vehicular dynamics, adopting elliptic shapes of the potential fields, which adapt better to this environment. This adaptation allows having different force values for the different directions of the approaching obstacles. This makes this method more convenient in a dynamic environment such as the vehicular one. The reference generator represents a “reference vehicle” whose dynamics determine the reference to be tracked by the “real vehicle.” The trajectories generated by this reference generator can be used for vehicle autonomous driving or for vehicle active control. A controller will ensure the tracking of the reference signals: in the case of autonomous vehicles, the control action is imposed by the control system; in the case of vehicles with human drivers, the control system intervenes (via active control devices; see, e.g., [13] and references therein) to correct the driver’s action in order to correctly follow the references.In addition to the adaptation of the artificial potential fields to the vehicular environment on a road, adopting elliptic shapes, the originality of this paper resides in the fact that the forces of the potential fields are applied to the reference generator rather than to the real vehicle, as commonly done in the literature [14–17]. The advantages to proceed in this way are that the obstacle avoidance problem is divided into two independent subproblems (reference generation and tracking problem, where this latter can be solved with any desired technique) and that the virtual forces may be also discontinuous (as in the case of obstacles that appear as event-driven dynamics: traffic lights, sudden external disturbance, etc.), whereas the resulting generated references are continuous. This latter aspect implies that the controller is not discontinuous and therefore can be physically implemented.The proposed method is tested in a scenario inspired by the so-called moose (or elk) test, better standardized by the ISO 3888-2 and here freely modified to consider also evasive maneuvers that generically consider also the presence of other collaborative vehicles, communicating via vehicle-to-vehicle (V2V) communications. This test is quite compelling and significative of hazardous situations in the vehicular environment. The scenario has been implemented in Simulink as first step in the performance evaluation before testing this approach with more realistic vehicle dynamics software.The paper is organized as follows. In Section2, the mathematical model of a vehicle is recalled, and the control problem is stated. In Section 3, the reference generator with attractive and repulsive forces is introduced. In Section 4, the attractive and repulsive potential fields associated with obstacles and goals are proposed, and in Section 5 a controller ensuring reference tracking is designed. In Section 6, some simulations are provided, which show that the proposed method is effective in the scenario taken into account. Some comments and future work conclude the paper.


## 2. Mathematical Model of a Ground Vehicle

The attitude dynamics of a vehicle can be described by the dynamics of the longitudinal and lateral velocitiesvx and vy of the vehicle and by the dynamics of its yaw angular velocity ωz: (1)v˙x=vyωz+μxmFx,f+Fx,r,v˙y=-vxωz+μymFy,f+Fy,r,ω˙z=μyJzFy,flf-Fy,rlr+1JzMz.This model, also called bicycle model or single-track model, even if simplified, well captures the essence of the dynamics to be controlled [18]. In (1), m and Jz are the vehicle mass and inertia momentum, lf and lr are the front and rear vehicle length, and vx and vy are the longitudinal and lateral velocities of the vehicle center of mass. Moreover, μx and μy are the longitudinal and lateral tire-road friction coefficient and Fx,f and Fx,r and Fy,f and Fy,r are the longitudinal and lateral forces due to the front and rear tires, normalized with respect to μx and μy.The state variables are expressed in the frameRΓ(G,x,y,z), fixed with the vehicle and centered in its center of mass G, whereas the inertial frame is denoted as RC(O,X,Y,Z) (see Figure 1).Figure 1
Schematic diagram of the bicycle model.The forcesFx,f and Fx,r influence the longitudinal dynamics, and the lateral forces Fy,f and Fy,r influence the lateral dynamics (but also the yaw dynamics). Finally, the torque Mz influences the yaw dynamics. This latter torque can be realized by Rear Torque Vectoring (RTV) devices, that is, actuators capable of generating a torque on the vehicle, via braking or engine power differential distribution between the left and right vehicle sides (see, e.g., [19–23] for further details).The front/rear lateral forcesFy,f(αf) and Fy,r(αr) depend on the front/rear tire slip angles αf=δs-(vy+lfωz)/vx and αr=-(vy-lrωz)/vx, with δs being the steering angle imposed at the front wheel [24]. The front/rear longitudinal forces Fx,f(λf) and Fx,r(λr) depend on the front/rear tire slips λf=1-ωw,fRw/vx and λr=1-ωw,rRw/vx, with ωw,f and ωw,r being the front/rear wheel angular velocities and Rw being the wheel radius [24].The angleδs can be imposed by the driver or by the control system, in the case of manual or autonomous vehicle, or by both in the case of active control of the vehicle, with devices such as the Active Front Steering (AFS) and RTV, used to assist and correct the driver’s actions [13]. In this last case, δs is the sum of two terms, one due to the driver and the other due to the AFS. In the following, the angle δs is assumed to be at least continuously differentiable with respect to time, which is a very mild hypothesis, verified in practice.Finally, the dynamics of the wheels are(2)ω˙w,j=1Jw,jTe,j-Tb,j-RwFt,j+Fw,j,j=f,r,where Te,j and Tb,j are the torques imposed by the engine and applied by the brake to the wheel j=f,r. Moreover, for the wheel j=f,r, Jw,j is the wheel inertia, Fw,j=κjωw,j is the viscous friction force, κj is the viscous friction coefficient, Ft,j=μx,jN=f(λs,j)mg is the traction force, μx,j=f(λs,j) is the adhesion coefficient, N=mg is the normal load on the tire, and g is the gravity acceleration. For the sake of simplicity and consistently with the single-track model (4), we consider that the front wheels have the same velocity and consider the same for the rear wheels.In the literature, there are various models of tire (see, e.g., [24] and references therein). For the sake of simplicity, a simplified version of the Pacejka “magic formula” has been used in this work [24]: (3)Fy,jαj=Dy,jsin⁡Cy,jarctan⁡By,jαj,Fx,jλj=Dx,jsin⁡Cx,jarctan⁡Bx,jλj,where Bi,j, Ci,j, and Di,j (i=x,y; j=f,r) are experimental parameters. It is worth noting that the results of the paper can be applied also considering other tire models.Model (1) is quite simple and nonlinear just for the terms Fx,f, Fx,r, Fy,f and Fy,r. It can be used to design a reference generator and a control law to track such a reference. The control problem is to generate a collision-free path with reference velocities vx,ref, vy,ref, and ωz,ref for the longitudinal, lateral, and angular velocity of (1) and a control law to track these references. To this aim, in the next section, a collision-free generator, making use of potential fields, will be introduced, and a control law will be designed to track these references.


## 3. The Reference Generator with Attractive and Repulsive Forces

The dynamics of the reference vehicle mimic the dynamics of a real vehicle:(4)v˙x,ref=ωz,refvy,ref+μx,refmrefFx,f,ref+Fx,r,ref+1mrefFt,x,v˙y,ref=-ωz,refvx,ref+μy,refmrefFy,f,ref+Fy,r,ref+1mrefFt,y,ω˙z,ref=μy,refJz,refFy,f,reflf-Fy,r,reflr+lpJz,refFt,y,with some notable differences: (1)
The forcesFx,f,ref, Fx,r,ref, Fy,f,ref, and Fy,r,ref exerted by the tires of the reference vehicle are similar to those of the real vehicle (1) but correspond to a “nominal” tire and, in particular, Fy,r,ref is modified so that no tail-spins are possible in the reference vehicle. Usually, this is obtained ensuring that the rear tire characteristic is not decreasing after a certain value (corresponding to the maximal lateral force), as usually happens in a real tire.(2)
A virtual forceFt is applied to the vehicle: (5)Ft=∑j=1NrFr,j,ref+∑i=1NgFa,i,refwith componentsFt,x and Ft,y in the reference frame RΓref(Gref,xref,yref,zref) fixed with the reference vehicle. This force is the sum of the repulsive forces Fr,j,ref due to the obstacles to be avoided and of the attractive forces Fa,i,ref due to the goals to be reached. Among the obstacles to be avoided, there are also the path limits of the lane where the reference vehicle is travelling. These forces are applied to a point P, which in general does not coincide with the center of mass of the virtual vehicle and which may also be external to the geometric figure of the reference vehicle. P can be chosen so that the virtual forces determine greater torques on the reference vehicle by augmenting the distance of P from the center of mass in order to obtain better results in terms of steerability [25].Moreover,mref and Jz,ref are the mass and inertia with respect to the z-axis (z is oriented so that (x,y,z) is a right orthogonal frame), usually equal to the nominal values. Finally, μx,ref and μy,ref are the reference tire-road friction coefficients in the x and y directions, and lf and lr are the distances from the vehicle center of gravity Gref to the front and rear axles.In Figure2, a typical case of the reference vehicle in the presence of an obstacle and of a goal is shown, with the corresponding repulsive and attractive forces. The resulting force Ft determines further steering of the reference vehicle, which adds to the steering due to the tire forces.Figure 2
Reference vehicle, inertial and reference framesRC and RΓref, attractive force Fa,i due to the ith goal, and repulsive force Fr,i due to the jth obstacle.Once the reference signals (4) are generated, the “real vehicle” (1) will be controlled in order to follow these references.


## 4. Attractive and Repulsive Potential Fields

An attractive potential field can be associated with a goal that one wants to reach. Analogously, a repulsive potential field can be associated with an obstacle to be avoided. These potential fields determine attractive or repulsive forces acting, as already explained, on the reference vehicle. These potential fields have a maximal space in which they can act. Moreover, there is a space in which the reference vehicle cannot enter and corresponds to a security distance to be respected. In general, one can considerNg goals and Nr obstacles, which may be moving.Flags can be associated with specific maneuvers, which can be decided by a higher-level decision layer. In this work, flagsFo and Fb are associated with the overtaking and braking maneuvers with Fo=1 when the vehicle is overtaking and zero otherwise. It is likewise for the braking maneuver flag Fb. The use of these flags allows omitting in the reference vehicle (4) some forces that have not been considered in the maneuver, therefore improving the behavior of the reference vehicle during its maneuvers. Further flags can be used to omit repulsive forces of other vehicles that, via V2V communications, transmit their collaboration to avoid collisions to the moving vehicle.


### 4.1. Repulsive Potential Field due to an Obstacle

The repulsive potential field associated with an obstaclej is given by(6)Ur,j=undefinedifρr,j<sr,jηr,jvr,rel,j21sr,jln⁡ρr,j+sr,jrr,j+sr,jρr,j-sr,jrr,j-sr,jifρr,j∈sr,j,rr,j0otherwise,j=1,…,Nr, where (7)sr,jθr,j,vr,rel,j=1as,r,j2cos2θr,j+bs,r,j2sin2θr,j,rr,jθr,j=1ar,r,j2cos2θr,j+br,r,j2sin2θr,jare the (elliptic) safety distance from the jth obstacle and the (elliptic) influence range of the potential field, respectively, with θr,j being the bearing angle. For practical reasons, the ellipse axis as,r,j>bs,r,j>0 is chosen as(8)as,r,j=1νr,1ifγs,r,jvr,rel,j∈0,νr,11γs,r,jvr,rel,jifγs,r,jvr,rel,j∈νr,1,νr,21νr,2ifγs,r,jvr,rel,j>νr,2,where γs,r,j(|vr,rel,j|) is the major axis gain, which increases sr,j according to the velocity intensity |vr,rel,j|. For instance, one can take γs,r,j(|vr,rel,j|)=γs,r,j∘|vr,rel,j|, with νr,2>νr,1>1/bs,r,j>0 and γs,r,j∘>0. Moreover, the ellipse axes of the influence region are such that ar,r,j>br,r,j>0, so that ar,r,j>as,r,j and br,r,j>bs,r,j. In this way, rr,j>sr,j. Since sr,j depends on vr,rel,j, the safety and the influence regions have different shapes in general.The potential filed contains a logarithmic term to have a steeper increase of the repulsive force whenρr,j tends to the safety distance sr,j. The gains ηr,j, which can be generically different for each obstacle, are (e.g., linear or quadratic) functions of the relative velocity vr,rel,j, expressed in RΓr,j fixed with the obstacle, between the velocity of the jth obstacle and that of the reference vehicle. The dependence on vr,rel,j allows taking into account the different situations that can occur in practical situations. Moreover, (9)ρr,j=xP-xj2+yP-yj2is the distance of the point P of the reference vehicle from the jth obstacle; (xj,yj,0) are the coordinates of origin Ωr,j of the reference frame RΓr,j(Ωr,j,xr,j,yr,j,zr,j) fixed with the jth obstacle (usually Ωr,j coincides with the geometric center of the obstacle), expressed in the frame RΓref fixed with the reference vehicle. Note that, for a fixed distance ρr,j, the potential field varies with the bearing angle θr,j, so that one gets higher repulsive forces when θr,j=0, that is, when the reference vehicle is following the obstacle, and lower values when θr,j=π/2, that is, when the reference vehicle is avoiding the obstacle.The resulting repulsive force exerted on the reference vehicle, due to the potential fieldUr,j in (6), has components in RΓr,j given by(10)Fr,j=-∂Ur,j∂ρr,jcos⁡θr,jsin⁡θr,j=φjρr,j,θr,jcos⁡θr,jsin⁡θr,j,φjρr,j,θr,j=undefinedifρr,j<sr,j-ηr,jvr,rel,jρr,j-rr,jθr,jρr,j2-sr,j2θr,jifρr,j∈sr,j,rr,j0otherwise,j=1,…,Nr. This force expressed in the reference RΓref is (11)Fr,j,ref=FRref,inαz,refRr,j,inTαz,r,jFr,j,where F=diag{Fb,Fo} is a flag matrix and (12)Rref,inαz,ref=cos⁡αz,refsin⁡αz,ref-sin⁡αz,refcos⁡αz,ref,Rr,j,inαz,r,j=cos⁡αz,r,jsin⁡αz,r,j-sin⁡αz,r,jcos⁡αz,r,jare the rotation matrices that transform, respectively, vectors expressed in RC into vectors expressed in RΓref and vectors expressed in RC into vectors expressed in RΓr,j. Here αz,ref and αz,r,j are the yaw angles of the reference vehicle and of the obstacle, respectively.


### 4.2. Attractive Potential Field due to a Goal

The potential field associated with a goali is given by(13)Ua,i=undefinedifρa,i<sa,i-ηa,iva,rel,i21sa,iln⁡ρa,i+sa,ira,i+sa,iρa,i-sa,ira,i-sa,iifρa,i∈sa,i,ra,i0otherwise,where ηa,i is the potential field gain (e.g., a linear or a quadratic function of the relative velocity va,rel,i between the velocity of the ith goal and that of the reference vehicle, again to take into account the different situations that can occur in practical situations), ρa,i=(xP-xi)2+(yP-yi)2 is the distance from the goal, and (xi,yi,0) are the coordinates of the origin Ωa,i expressed in RΓref; (14)sa,iθa,i=1as,a,i2cos2θa,i+bs,a,i2sin2θa,i,ra,iθa,i=1ar,a,i2cos2θa,i+br,a,i2sin2θa,iare the (elliptic) safety distance to maintain when following the goal and the (elliptic) influence range of the goal, respectively, with θa,i being the goal bearing angle. Here, as,a,i>bs,a,i>0, and as,a,i has an expression similar to (8). Finally, ar,a,i>br,a,i>0 so that ar,a,i>as,a,i,br,a,i>bs,a,i, and ra,i>sa,i.The force exerted on the reference vehicle expressed inRΓref is given by (15)Fa,i,ref=Rref,inαz,refRa,i,inTαz,a,iFa,i,i=1,…,Ng, where(16)Fa,i=-∂Ua,i∂ρa,icos⁡θa,isin⁡θa,i=φiρa,i,θa,icos⁡θa,isin⁡θa,i,φiρa,i,θa,i=undefinedifρa,i<sa,iηa,iva,rel,iρa,i-ra,iθa,iρa,i2-sa,i2θa,iifρa,i∈sa,i,ra,i0otherwiseand Ra,i,in(αz,a,i) is the rotation matrix from RC to RΓa,i, with αz,a,i being the yaw angle of RΓa,i with respect to RC.


## 4.1. Repulsive Potential Field due to an Obstacle

The repulsive potential field associated with an obstaclej is given by(6)Ur,j=undefinedifρr,j<sr,jηr,jvr,rel,j21sr,jln⁡ρr,j+sr,jrr,j+sr,jρr,j-sr,jrr,j-sr,jifρr,j∈sr,j,rr,j0otherwise,j=1,…,Nr, where (7)sr,jθr,j,vr,rel,j=1as,r,j2cos2θr,j+bs,r,j2sin2θr,j,rr,jθr,j=1ar,r,j2cos2θr,j+br,r,j2sin2θr,jare the (elliptic) safety distance from the jth obstacle and the (elliptic) influence range of the potential field, respectively, with θr,j being the bearing angle. For practical reasons, the ellipse axis as,r,j>bs,r,j>0 is chosen as(8)as,r,j=1νr,1ifγs,r,jvr,rel,j∈0,νr,11γs,r,jvr,rel,jifγs,r,jvr,rel,j∈νr,1,νr,21νr,2ifγs,r,jvr,rel,j>νr,2,where γs,r,j(|vr,rel,j|) is the major axis gain, which increases sr,j according to the velocity intensity |vr,rel,j|. For instance, one can take γs,r,j(|vr,rel,j|)=γs,r,j∘|vr,rel,j|, with νr,2>νr,1>1/bs,r,j>0 and γs,r,j∘>0. Moreover, the ellipse axes of the influence region are such that ar,r,j>br,r,j>0, so that ar,r,j>as,r,j and br,r,j>bs,r,j. In this way, rr,j>sr,j. Since sr,j depends on vr,rel,j, the safety and the influence regions have different shapes in general.The potential filed contains a logarithmic term to have a steeper increase of the repulsive force whenρr,j tends to the safety distance sr,j. The gains ηr,j, which can be generically different for each obstacle, are (e.g., linear or quadratic) functions of the relative velocity vr,rel,j, expressed in RΓr,j fixed with the obstacle, between the velocity of the jth obstacle and that of the reference vehicle. The dependence on vr,rel,j allows taking into account the different situations that can occur in practical situations. Moreover, (9)ρr,j=xP-xj2+yP-yj2is the distance of the point P of the reference vehicle from the jth obstacle; (xj,yj,0) are the coordinates of origin Ωr,j of the reference frame RΓr,j(Ωr,j,xr,j,yr,j,zr,j) fixed with the jth obstacle (usually Ωr,j coincides with the geometric center of the obstacle), expressed in the frame RΓref fixed with the reference vehicle. Note that, for a fixed distance ρr,j, the potential field varies with the bearing angle θr,j, so that one gets higher repulsive forces when θr,j=0, that is, when the reference vehicle is following the obstacle, and lower values when θr,j=π/2, that is, when the reference vehicle is avoiding the obstacle.The resulting repulsive force exerted on the reference vehicle, due to the potential fieldUr,j in (6), has components in RΓr,j given by(10)Fr,j=-∂Ur,j∂ρr,jcos⁡θr,jsin⁡θr,j=φjρr,j,θr,jcos⁡θr,jsin⁡θr,j,φjρr,j,θr,j=undefinedifρr,j<sr,j-ηr,jvr,rel,jρr,j-rr,jθr,jρr,j2-sr,j2θr,jifρr,j∈sr,j,rr,j0otherwise,j=1,…,Nr. This force expressed in the reference RΓref is (11)Fr,j,ref=FRref,inαz,refRr,j,inTαz,r,jFr,j,where F=diag{Fb,Fo} is a flag matrix and (12)Rref,inαz,ref=cos⁡αz,refsin⁡αz,ref-sin⁡αz,refcos⁡αz,ref,Rr,j,inαz,r,j=cos⁡αz,r,jsin⁡αz,r,j-sin⁡αz,r,jcos⁡αz,r,jare the rotation matrices that transform, respectively, vectors expressed in RC into vectors expressed in RΓref and vectors expressed in RC into vectors expressed in RΓr,j. Here αz,ref and αz,r,j are the yaw angles of the reference vehicle and of the obstacle, respectively.


## 4.2. Attractive Potential Field due to a Goal

The potential field associated with a goali is given by(13)Ua,i=undefinedifρa,i<sa,i-ηa,iva,rel,i21sa,iln⁡ρa,i+sa,ira,i+sa,iρa,i-sa,ira,i-sa,iifρa,i∈sa,i,ra,i0otherwise,where ηa,i is the potential field gain (e.g., a linear or a quadratic function of the relative velocity va,rel,i between the velocity of the ith goal and that of the reference vehicle, again to take into account the different situations that can occur in practical situations), ρa,i=(xP-xi)2+(yP-yi)2 is the distance from the goal, and (xi,yi,0) are the coordinates of the origin Ωa,i expressed in RΓref; (14)sa,iθa,i=1as,a,i2cos2θa,i+bs,a,i2sin2θa,i,ra,iθa,i=1ar,a,i2cos2θa,i+br,a,i2sin2θa,iare the (elliptic) safety distance to maintain when following the goal and the (elliptic) influence range of the goal, respectively, with θa,i being the goal bearing angle. Here, as,a,i>bs,a,i>0, and as,a,i has an expression similar to (8). Finally, ar,a,i>br,a,i>0 so that ar,a,i>as,a,i,br,a,i>bs,a,i, and ra,i>sa,i.The force exerted on the reference vehicle expressed inRΓref is given by (15)Fa,i,ref=Rref,inαz,refRa,i,inTαz,a,iFa,i,i=1,…,Ng, where(16)Fa,i=-∂Ua,i∂ρa,icos⁡θa,isin⁡θa,i=φiρa,i,θa,icos⁡θa,isin⁡θa,i,φiρa,i,θa,i=undefinedifρa,i<sa,iηa,iva,rel,iρa,i-ra,iθa,iρa,i2-sa,i2θa,iifρa,i∈sa,i,ra,i0otherwiseand Ra,i,in(αz,a,i) is the rotation matrix from RC to RΓa,i, with αz,a,i being the yaw angle of RΓa,i with respect to RC.


## 5. A Tracking Controller of the Reference Vehicle

A simple controller is designed for the vehicle dynamics (1) to track the reference generated by (4), considering the following Lyapunov candidate [26]: (17)V=12evx2+12evy2+12eωz2,where evx=vx-vx,ref, evy=vy-vy,ref, and eωz=ωz-ωz,ref. Using (1), its derivative is (18)V˙=evxvyωz+μxmFx,f+Fx,r-v˙x,ref+evy-vxωz+μymFy,f+Fy,r-v˙y,ref+eωzμyJzFy,flf-Fy,rlr+1JzMz-ω˙z,ref,so that, considering(19)Fx,f+Fx,r=mμx-vyωz+v˙x,ref-k1vx-vx,ref,Fy,f=-Fy,r+mμyvxωz+v˙y,ref-k2vy-vy,ref,Mz=-μyFy,flf-Fy,rlr+Jzω˙z,ref-k3eωz,k1,k2,k3>0, one finally works out (20)V˙≤-k1evx2-k2evy2-k3eωz2.This proves the exponential tracking of the references. Note that (19) with (4) constitutes a dynamic controller.Once the control actions (19) are determined, the forces Fx,f, Fx,r, Fy,f, and Mz have to be physically implemented. In particular, regarding the lateral force Fy,f, since it is invertible with respect to αf in the interval [-αf,max,αf,max], where ±αf,max are the slip angle value corresponding to the maximum and minimum of Fy,f, one can calculate the angle δs from (19). In fact, for a fixed value F∘, the solution of Fy,f(αf)=F∘ is unique and is given by(21)
δ
s
=
v
y
+
l
f
ω
z
v
x
+
F
y
,
f
-
1
F
∘
if
α
f
∈
-
α
f
,
m
a
x
,
α
f
,
m
a
x
v
y
+
l
f
ω
z
v
x
±
α
f
,
m
a
x
o
t
h
e
r
w
i
s
e
.The angle δs is the angle at the wheel which has to be imposed. When dealing with autonomous driving, δs is imposed by the control system of the vehicle. If the vehicle has a human driver who imposes a steering angle δd, the active control system will impose an extra angle δc=δs-δd to on wheel in order to correctly follow the reference signals. The torque Mz can be physically applied to the vehicle using differential braking between the left and right wheels (usually the rear wheels in order to be less invasive) or by means of other mechanisms, such as active differentials. Finally, as already noted, the forces Fx,f and Fx,r depend on the wheel slips λf and λr. Denoting by Fx,f∘ and Fx,r∘ the values to be imposed at time t∘, let λf∘ and λr∘ be the corresponding slip values, obtained inverting the tires characteristics (3). They can be imposed by controlling the wheel slips [27]. To this aim, the values ωw,f∘=(1-λf∘)vx/Rw and ωw,r∘=(1-λr∘)vx/Rw are obtained and are imposed by means of Te,f and Tb,f and Te,r and Tb,r in (2), that is, controlling the velocities ωw,f and ωw,r of the front and rear wheels.


## 6. Simulation Results

In this section, a test has been considered to show the effectiveness of the proposed collision-free generator, inspired by the so-called moose (or elk) test, which is standardized by the ISO 3888-2. The scenario has been implemented in Simulink, for a first evaluation of the performance, before testing the dynamic controller (see (4) and (19)) in more realistic vehicle dynamics software. The scenario consists of a vehicle proceeding on a curving road, with a moose that suddenly crosses the road. A second vehicle (obstacle) proceeds in the opposite direction and can or cannot exchange data with the first vehicle in order to have or not a collaborative action. See Figures 3 and 4 for further details on the scenario.Figure 3
Braking maneuver before reaching the obstacle due to a moose. (a) Real (solid) and reference (dashed) trajectories. (b) Longitudinal velocityvx (black solid); reference vx,ref (black dashed) [m/s versus s]. (c) Repulsive force Fr,x [N versus s]. (d) Distance ρr between reference vehicle and the moose (black), influence distance rr (gray dashed), and safety distance sr (gray solid) [m versus s].
(a) (b) (c) (d)Figure 4
Evasive maneuver for avoiding the moose, with collaborating vehicle proceeding in the opposite lane. (a) Real (solid) and reference (dashed) trajectories. (b) Longitudinal velocityvx (black solid); reference vx,ref (black dashed) [m/s versus s]. (c) Lateral velocity vy (solid) and reference vy,ref (dashed) [m/s versus s]. (d) Angular velocities ωz (solid) and reference ωz,ref (dashed) [rad/s versus s].
(a) (b) (c) (d)To complete the scenario description, we consider the case of autonomous driving (whereδd=0). The parameters values for the reference and real vehicles and for the controller are given in Table 1, whereas those used for generating the potential fields are in Table 2. The repulsive gain has been set equal to ηr(vr,rel)=ηr∘|vr,rel|, with ηr∘ determined experimentally.Table 1
Parameters of the reference and real vehicles and of the controller.
m r e f = m = 1862 kg J z , r e f = J z = 2488 kg/m2 l f , r e f = l f = 1.18 m k 1 = 35 μ x , r e f = μ x = 1 μ y , r e f = μ y = 1 l r , r e f = l r , r = 1.77 m k 2 = 5 λ s , r e f a = λ s a = 0.12 λ s , r e f d = λ s d = - 0.12 g = 9.81 m/s2 k 3 = 40Table 2
Parameters of the potential fields.
η r ∘ = 6.5 × 1 0 3 a r , r = 1 / 50 b r , r = 1 / 4 b s , r = 1 / 2 ν r , 1 = 4 ν r , 2 = 50 γ s , r ∘ = 1.8 μ b = 0.9The high-level controller of the vehicle can decide to brake or to avoid the moose. In the first case, the control system has to calculate the braking distancesb=vref2/(2μbg) necessary to dissipate all vehicle kinetic energy and to stop before a collision, with μb being the dissipation coefficient, and g is the acceleration constant. The high-level controller decides that a braking maneuver has to be carried out by the reference generator, setting Fb=1, when the braking distance is greater than the distance s0 with the obstacle (set equal to 15 m in the simulations). Otherwise, overtaking, that is, an evasive maneuver, is decided in order to avoid the moose, setting Fo=1, if a vehicle-to-vehicle communication with the vehicle proceeding in the opposite direction gives its availability to brake or accelerate in order to ensure the existence of a proper temporal window for the evasive maneuver.Figure3 shows the case of a braking maneuver, where sb<s0, for avoiding the moose. The repulsive potential, centered on the moose, and the constraints due to the road allow generating a reference path (see Figure 3(a)), where the reference and real paths during the maneuver have been shown. The controller (19) follows perfectly the references generated by (4). In Figure 3(b), it is shown that the longitudinal velocity goes to zero, so that the distance ρr remains greater than the safety distance sa (see Figure 3(d)). The generated longitudinal potential force that acts on the reference vehicle during braking (when ρr is smaller than the influence range rr) is shown in Figure 3(c).In the second case, it is considered thatsb>s0, and it is assumed that the vehicle proceeding in the opposite direction collaborates in order to allow the evasive maneuver (see Figure 4). Figures 4(b), 4(c), and 4(d) show the longitudinal, lateral, and angular velocities with values almost identical to the generated references. The repulsive (Figures 5(a) and 5(b)) potential forces, acting on the reference vehicle, modify its dynamics.Figure 5
Evasive maneuver for avoiding the moose, with collaborating vehicle proceeding in the opposite lane. (a) Obstacle (moose) repulsive forceFr,y [N versus s]. (b) Distance ρr between reference and the moose (black), influence distance rr (gray dashed), and safety distance sr (gray solid) [m versus s].
(a) (b)It can be noticed from Figure3(c) that when the distance ρr between the reference vehicle and the obstacle is in the influence range rr of the repulsive potential field of the obstacle, the reference vehicle is affected by these repulsive forces. Note that ρr remains, as expected, greater than the safety distance sr. A temporal window allows the reference vehicle to perform the evasive maneuver. It should be noted that, due to the definition of the potential fields, the reference vehicle is not affected by the obstacle traveling in the opposite direction.It is also important to note that as the potential fields does not implement any logic regarding the direction of the evasive maneuver and if the moose was closer to the middle of the road instead of being on the right lane, in this particular curved road, repulsion forces could be generated to make the evasive maneuver to the right of the moose. In this case, the evasive maneuver could not be accomplished successfully, leading to a collision with either the moose or the lane edge. Therefore, a further logic seems necessary to be added to the reference generator to choose the correct direction of the evasive maneuver or to choose a braking maneuver in order to mitigate the collision. This logic has to take into consideration some aspects concerning the context of the collision to avoid, such as the road topology, the heading of the obstacle with respect to the vehicle heading, and the direction of the available trajectory to which the evasive maneuver should be made.Finally, it is worth noticing that both cases (braking and evasive maneuver) have to be intended as maneuvers to avoid the obstacle. In this sense, ifsb>s0 and the evasive maneuver is not possible since another noncollaborative vehicle is approaching in the opposite lane, a collision is inevitable and the braking can be interpreted as a way to mitigate the collision effects.A final remark deserves the implementability of the proposed method on commercial Electronic Control Units (ECUs). As first measure of implementability, we have proceeded to determine the computational time calculated at each cycle as the sumTp+Tr+Tc+Tv, where Tr, Tc, and Tv are the times required to calculate the reference, the control, and the dynamics of the real vehicle, respectively. Furthermore, Tp is the computational time of the potential forces applied on the reference vehicle, which can be parameterized in terms of the Nr obstacles and the Ng goals: Tp=NrTFr+NgTFa, where TFr and TFa are the times required to calculate the repulsive force for one obstacle and the attractive force for one goal. Table 3 shows the calculation times obtained for an Intel Core™ Duo i5-3230M CPU@2.60 GHz HP PC, with 8 G of RAM and clock speed 2601 MHz. It can be seen that, for each cycle, the reference generator and the controller require 0.18 ms if Nr=1 and Na=1. In the case considered in the simulations with Nr=3 and no goals, for each cycle, 0.2 ms is required, whereas this time becomes 0.36 ms for Nr=10 and Ng=1. Therefore, the proposed technique appears to be implementable on a commercial ECU, where typical computational times are of the order of 10 ms, as discussed in [28].Table 3
Computational times.
T r = 0.06 ms T c = 0.05 ms T v = 0.03 ms T F r = 0.02 ms T F a = 0.02 ms


## 7. Conclusions

In this paper, the approach with potential fields has been adapted to the case of vehicular dynamics to obtain a reference generator that generates signals of a collision-free path which are to be tracked by a vehicle. A simple controller has been designed on the basis of this generator, therefore resulting in a dynamic controller ensuring obstacle avoidance. This technique can be applied for active vehicle control and in the case of autonomous guidance. Simulation results have been presented in the autonomous case considering a scenario inspired by the so-called moose (or elk) test, with the presence of other collaborative vehicles.Future works will consist of developing a decision-making mechanism that will work at a higher level to assess threats and decide the best maneuver in the case of hazardous situations. Moreover, highly congested scenarios will be considered, which could cause the problem of local potential field minima, which could determine a block of the vehicle. Furthermore, the robustness of the proposed controller in the presence of measurement errors will be studied. Finally, a validation on software platforms, simulating the whole vehicle dynamics, will be performed.



---

*Source: 4604395-2018-02-27.xml*