Control for autonomous vehicles in collision avoidance maneuvers : LPV modeling and static feedback controller
Penco Dario, Davins-Valldaura Joan, Godoy Emmanuel, Kvieska Pedro, Valmorbida Giorgio
CControl for autonomous vehicles in collision avoidance maneuvers :LPV modeling and static feedback controller
Dario Penco , , Joan Davins-Valldaura , Emmanuel Godoy , Pedro Kvieska and Giorgio Valmorbida Abstract — This article presents a state feedback control designstrategy for the stabilization of a vehicle along a referencecollision avoidance maneuver. The stabilization of the vehicle isachieved through a combination of steering, acceleration andbraking. A Linear Parameter-Varying (LPV) model is obtainedfrom the linearization of a non-linear model along the referencetrajectory. A robust state feedback control law is computedfor the LPV model. Finally, simulation results illustrate thestabilization of the vehicle along the reference trajectory.
I. I
NTRODUCTION
One of the current challenges in the development ofautonomous driving is to guarantee safety for increased levelsof autonomous driving [1]. For low velocity maneuvers, suchas parking, the trajectory planning and execution can rely onsimple mathematical models. On the other hand, collisionavoidance maneuvers, carried out in high velocities requiremore sophisticated control strategies to be automated and,due to its dynamics involving suspension, brakes and tireforces call for more complex models.In the context of autonomous driving, maneuvers witha maximum vehicle’s acceleration of . g on just onevehicle axis are considered as low dynamics maneuvers.Maneuvers with an acceleration larger than . g and possiblya combination of longitudinal and lateral acceleration areconsidered high dynamics maneuvers.Literature is rich of solutions for vehicle control in lowdynamics maneuvers. In low dynamics the longitudinal andlateral dynamics of the vehicle are commonly considereduncoupled. The control of the longitudinal and lateraldynamics is hence treated separately. A common solutionemployed for longitudinal control is the Linear ProportionalDerivative (PID) control law [2], [3], [4] and [5]. Lateralcontrol is usually based on the linear bicycle model [6].Several solutions use H and H ∞ control laws [2], [7] and[8].For high dynamics maneuvers the coupling of the vehicle’slongitudinal and lateral dynamics cannot be neglected.Moreover in these conditions the combined longitudinaland lateral tire slip is an important factor related to the tireforces saturation [9].Solutions for high dynamics strategies may also considera decoupling of longitudinal and lateral dynamics [10]. Universit´e Paris-Saclay, CNRS, CentraleSup´elec,Laboratoire des signaux et syst`emes, 91190 Gif-sur-Yvette, France. { dario.penco, emmanuel.godoy,giorgio.valmorbida } @centralesupelec.fr Groupe Renault, Technocentre, 78288 Guyancourt,France. { dario.penco, joan.davins-valldaura,pedro.kvieska } @renault.com However, since the cross effects on lateral and longitudinalvariables are more significant, these strategies demand a finertuning of the controller and a better knowledge of the vehicleparameters. These strategies may thus lack robustness.The use of joint lateral and longitudinal models will thereforebe sought for high dynamics.
Model Predictive Control (MPC) is a control strategythat attacks simultaneously the path generation and itscontrol. Moreover, it allows to account for input- andstate-constraints, which is useful since the obstacles andlane limits can be easily translated into constraints ofthe underlying optimization problem. Because of that inliterature we find several solutions for vehicle controlin high dynamics maneuvers based on the MPC controlstrategy. In [12] an MPC controller is developed based ona non-linear bicycle model that takes into considerationboth longitudinal and lateral dynamics. The simulation andexperimental tests show good results, but the computationalload required by the controller is high. To solve this problemanother MPC controller based on local linearizations of thenon-linear model is proposed. The computational load isreduced, but the commands computed show chattering. In[13] an improvement is proposed. The controller is basedon a four-wheel non-linear model. Also in this case thecomputational is considerable. A similar solution is proposedin [14]. In [15] tube-based
MPC is used to address alsorobustness criteria. In this case, to reduce the computationalload by reducing the complexity of the model on which thecontroller is based, the longitudinal and lateral tire forceshave been considered as the system’s inputs. A low levelcontroller is then required to compute the steering angleand wheel torques. Despite the good results present inliterature concerning MPC, for commercial automobiles, thecertification of a computationally demanding strategy maybe more difficult. Due to the limited computational powerthe online optimization-based strategies as MPC may requiread hoc circuits.The goal of this paper is to propose a controller for vehiclestabilization along a reference high dynamics maneuver. Toachieve this goal we propose a
Linear Parameter-Varying (LPV) model for the vehicle dynamics along the maneuver.The LPV framework offers methods for the synthesis ofcontrollers that do not require a large computation load [16],[17], [18], [19].We develop the LPV model based on a nonlinear modelthat considers the coupling between longitudinal and lateraldynamics. Moreover, since in a high dynamics maneuversthe tires may operate on their physical limits, presenting a r X i v : . [ m a t h . O C ] F e b ombined tire slip, we further complexify the model byadding the tire nonlinear behavior. Clearly, other dynamicscan also influence high maneuvers, such as the suspensionand braking systems. In this paper we will not consider thesedynamics as we focus on the study of the influence of thetire forces.The linearization of the proposed nonlinear model along areference maneuver trajectory yields the LPV system we usefor the synthesis of the control law. In particular, due to analgebraic loop introduced by the saturation function of thetires, we propose a strategy to obtain an uncertain parameterrelated to the slope of sector non-linearities.The outline of the article is: • in Section II the non-linear vehicle model and theprocedure to obtain the LPV model are discussed; • the method used for the synthesis of the controller isshown in Section III; • in Section IV the results of the application of thecontroller in a simulation environment are discussed; • the conclusions and future work are discussed in SectionV. II. P RELIMINARY RESULTS
A. Vehicle model description and equations of motion
To obtain a nonlinear model of the vehicle we consider thatthe two wheels of each axle have been regrouped at the centerof the axle track. The vehicle degrees of freedom allow it tomove on the xy plane, so it can translate along the x and y axis and rotate around the z axis. The vehicle axis are shownin Figure 1.In Table I there are the model state variables and inputs. Fig. 1. Vehicle’s xy plane, with the forces listed in Table II. Table II contains the forces shown in Figures 1 and 2. Theforces applied on the front wheel have each a componenton the x and y axis of the vehicle, that depends on δ f . Wedenote the components of F xf on the xy plane: F xxf = F xf cos δ f F yxf = F xf sin δ f . The model equations of motion in the vehicle frame are: m ( ˙ v − r u ) = 2 ( F xxf + F xr − R xxf − R xr − F xyf ) − F aero (1a) Fig. 2. Wheel’s x w z w plane.TABLE IM ODEL STATE VARIABLES AND INPUTS
State variables v (cid:2) ms (cid:3) longitudinal speed u (cid:2) ms (cid:3) lateral speed r (cid:2) rads (cid:3) yaw rate ω wf (cid:2) rads (cid:3) front wheel rotational speed ω wr (cid:2) rads (cid:3) rear wheel rotational speed x [ m ] absolute longitudinal position y [ m ] absolute lateral position ψ [ deg ] yaw angleSystem’s inputs δ f [ deg ] front wheel steering angle τ wf [ Nm ] front wheel torque τ wr [ Nm ] rear wheel torque TABLE IIM
ODEL ’ S FORCES F xf front tire longitudinal force F xr rear tire longitudinal force F yf front tire lateral force F yr rear tire lateral force R xf front wheel rolling friction R xr rear wheel rolling friction F aero aerodynamic drag P weight N f front wheel normal force N r rear wheel normal force m ( ˙ u + r v ) = 2 ( F yxf − R yxf + F yyf + F yr ) (1b) I zz ˙ r = 2 ( F yxf + F yyf − R yxf ) (cid:96) f − F yr (cid:96) r (1c) I w y ˙ ω wf = τ wf − F xf r e (1d) I w y ˙ ω wr = τ wr − F xr r e . (1e)For the sake of brevity, just the tires forces and the normalforces will be detailed. For the other forces description see[6] and [20].The vehicle position and orientation on the inertial frame aregoverned by: ˙ x = v cos ψ − u sin ψ (2a) ˙ y = u cos ψ + v sin ψ (2b) ψ = r. (2c) B. Tires forces
In high dynamics maneuvers, the tires are subject to saturationand combined slip [21]. It is important then to take intoconsideration these phenomena by using a non-linear modelfor the tire forces.The tire model in [22], developed for control purposes, isa variation of the
Dugoff’s model [23]. It is simpler andrequires less parameters than other commonly used models,as the
Pacejka’s magic formula [24] and the brush model [21], [25], [26]. We rely on this model since it allows torepresent tire forces saturation and combined slip.Longitudinal and lateral forces are linear respectively on thelongitudinal and lateral slip, κ and α : ˆ F x = c ∗ κ ( µ, N, α ) κ (3a) ˆ F y = c ∗ α ( µ, N, κ ) α. (3b)Longitudinal and lateral slip κ and α are defined in [21]. Thelongitudinal slip, or slip ratio, κ is the the difference betweenwheel longitudinal speed and wheel tangent speed divided bythe wheel longitudinal speed. The lateral slip angle α , shownin Figure 3, is the difference between the wheel steeringangle δ and the wheel vector speed angle θ w . Fig. 3. Wheel lateral slip angle α . The expressions of κ and α for the front and rear wheels are: κ f = − v cos δ f + ( u + (cid:96) f r ) sin δ f − ω wf r e v cos δ f + ( u + (cid:96) f r ) sin δ f (4a) κ r = − v − ω wr r e v (4b) α f = δ f − arctan (cid:18) u + (cid:96) f rv (cid:19) (4c) α r = − arctan (cid:18) u − (cid:96) r rv (cid:19) . (4d)The coefficients c ∗ κ et c ∗ α are: c ∗ κ ( µ, N, α ) = N c κ µ (cid:18) (cid:113) c α tan( α ) + c κ κ ∗ + N µ ( κ ∗ − (cid:19) c α tan( α ) + 4 c κ κ ∗ (5a) c ∗ α ( µ, N, κ ) = N c α µ (cid:0) √ α ∗ c α + c κ κ + N µ ( κ − (cid:1) α ∗ c α + 4 c κ κ . (5b) N is the normal force on the wheel and µ the frictioncoefficient between tire and ground. The expressions of κ ∗ and α ∗ are: κ ∗ = N µ (cid:16) c κ + (cid:112) N µ + 8 c κ N µ + N µ (cid:17) c κ (6a) α ∗ = N µ c α . (6b)The parameters c κ et c α are respectively the longitudinal andlateral tire stiffness.Here we use a circle to approximate the saturation of the tireforces as in [10] and [27], thus simplifying the more generalellipse model [21], [28], [29]. The radius of the saturationcircle is given by F max = µN . ˆ F x + ˆ F y ≤ ( µN ) (7)The logistic function, shown Figure 4, has been used toexpress the saturation: σ ( x, u, (cid:96) ) = u − (cid:96) e − u − (cid:96) ( x − u − (cid:96) ) + (cid:96) (8)Where x is the force before saturation, computed as in (3), u and (cid:96) are the upper and lower saturation bounds. A diagramof the tire forces saturation is shown in Figure 5.The normal force N on the tire is present both in thecomputation of the coefficients c ∗ κ et c ∗ α , expressed in (5),and the saturation bound (7). Considering the importance of N in the computation of the tire forces, as shown in Figure6, a variable normal force model has then been developed.We consider the equations of motion of the vehicle neglectedmovements: N f + N r − P (9a) F yxf + F yyf + F yr ) h − R yxf ( h − r e ) (9b) − F xxf − F xyf + F xr ) h + 2 ( R xxf + R xr ) ( h − r e ) + N r (cid:96) r − N f (cid:96) f . (9c) Fig. 4. Logistic function.ig. 5. Tire forces saturation.
Equation (9a) describes the vehicle vertical motion.Equations(9b) and (9c) describe respectively the roll and pitch motion,i.e. the rotation about the vehicle’s x and y axis.Accordingto the model definition of Section II-A, the vertical, roll andpitch accelerations are equal to zero.Considering both (1) and (9), we solve the system of equationswith the following unknowns: N f , N r , F xf , F xr , F yf and F yr . The solution of N f and N r does not depend on the tiremodel used. The expressions obtained for N f and N r , notshown in this article for the sake of brevity, depend on thevehicle dynamics, in particular the longitudinal acceleration ˙ v . , , , , , , , , α [deg] F y [ N ] N Fig. 6. Lateral tire force with respect to N . N f and N r are saturated between and P z using the logisticfunction. C. Model linearization
The state and input vectors of the model in (1) are: x = vurω wf ω wr u = δ f τ wf τ wr (10)The model expressed by (1) is in implicit form f ( ˙ x , x , u ) = 0 .Due to the saturation functions used for the tire forces, it isnot possible to express the model in explicit form.It is instead possible to isolate the implicit part of the modelas shown in Figure 7: ˙ x = g ( x , u ) + B σ ( u ) σ ( h ) (11) With: σ ( h ) = σ ( h ( ˙ x , x , u )) σ ( h ( ˙ x , x , u )) σ ( h ( ˙ x , x , u , σ )) σ ( h ( ˙ x , x , u , σ )) σ ( h ( ˙ x , x , u , σ )) σ ( h ( ˙ x , x , u , σ )) (12) h ( ˙ x , x , u , σ ) = h ( ˙ x , x , u ) h ( ˙ x , x , u ) h ( ˙ x , x , u , σ ) h ( ˙ x , x , u , σ ) h ( ˙ x , x , u , σ ) h ( ˙ x , x , u , σ ) = ˆ N f ˆ N r ˆ F xf (cid:16) σ (cid:16) ˆ N f (cid:17)(cid:17) ˆ F xr (cid:16) σ (cid:16) ˆ N r (cid:17)(cid:17) ˆ F yf (cid:16) σ (cid:16) ˆ N f (cid:17)(cid:17) ˆ F yr (cid:16) σ (cid:16) ˆ N r (cid:17)(cid:17) . (13) Fig. 7. Isolation of the implicit part of (1).
The vector h ( ˙ x , x , u , σ ) contains the inputs of the saturationfunctions.The vector σ ( h ) contains the output of the satura-tion functions.The model non-linearities due to the saturationfunctions have then been isolated and they result as an inputof the model.The expressions of g ( x , u ) and B σ ( u ) , notshown in this article for the sake of brevity, are derived from(1) after extraction of σ ( h ) .The state dynamics expressed as in (11) can be linearizedalong a reference trajectory ( ˙ x ( t ) , x ( t ) , u ( t ) , σ ( t )) , with σ ( t ) = σ ( h ( ˙ x ( t ) , x ( t ) , u ( t ))) . For simplicity, in thefollowing the reference trajectory will be indicated with T ( t ) . The vector σ ( h ) is also non linear with respect to h ( ˙ x , x , u , σ ) . It is linearized too along the trajectory T ( t ) . ∆ ˙ x = A ( t )∆ x + B ( t )∆ u + B σ ( t )∆ σ (14a) ∆ h = C ( t )∆ x + D ( t )∆ u + D σ ( t )∆ σ (14b) A ( t ) = ∂g ( x , u ) ∂ x (cid:12)(cid:12)(cid:12)(cid:12) T ( t ) (14c) B ( t ) = ∂g ( x , u ) ∂ u (cid:12)(cid:12)(cid:12)(cid:12) T ( t ) + ∂ ( B σ ( u ) σ ( h )) ∂ u (cid:12)(cid:12)(cid:12)(cid:12) T ( t ) (14d) B σ ( t ) = B σ ( u ) | T ( t ) (14e) C ( t ) = ∂ h ( ˙ x , x , u , σ ) ∂ ˙ x (cid:12)(cid:12)(cid:12)(cid:12) T ( t ) A ( t ) + ∂ h ( ˙ x , x , u , σ ) ∂ x (cid:12)(cid:12)(cid:12)(cid:12) T ( t ) (14f) D ( t ) = ∂ h ( ˙ x , x , u , σ ) ∂ ˙ x (cid:12)(cid:12)(cid:12)(cid:12) T ( t ) B ( t ) + ∂ h ( ˙ x , x , u , σ ) ∂ u (cid:12)(cid:12)(cid:12)(cid:12) T ( t ) (14g) σ ( t ) = ∂ h ( ˙ x , x , u , σ ) ∂ ˙ x (cid:12)(cid:12)(cid:12)(cid:12) T ( t ) B σ ( t ) + ∂ h ( ˙ x , x , u , σ ) ∂σ (cid:12)(cid:12)(cid:12)(cid:12) T ( t ) (14h)In this work the reference trajectory of Figure 8 has beenconsidered. The trajectory has a steering to the left followedby an opposite steering to the right.That results in a vehicle’slateral displacement of approximately 6m.No torque is appliedto the wheels.This trajectory corresponds to a two lane changemaneuver at 70km/h.The saturation functions can be approximated as sector- . . . [s] [ m ] Lateral displacement y . . . [s] [ k m / h ] Speed v . . . − [s] [ d e g ] Steering angle δ f . . . − − . . [s] [ N m ] Torque τ wf τ wr Fig. 8. Reference trajectory. bounded non-linearities.This means that they are approxi-mated to a line, whose slope varies according to the systemworking point.Figure 9 shows the slope sector for eachelement of σ ( h ) along the reference trajectory. Since inthe reference trajectory the wheel torques are equal to zero,the longitudinal slip ratios are small, hence the longitudinaltire forces are small too.As a consequence they are far fromtheir saturation bounds and their slope sector is small.Theabsence of torque on the wheel causes a small longitudinalacceleration ˙ v .Since the wheel normal forces vary with ˙ v , asdiscussed in Section II-A, the load transfer is small. Thereforetheir slope sector is small too.On the contrary the slope sectorof the lateral forces is large.Indeed the input steering angle δ f produces large lateral forces, which move closer to theirsaturation bounds.It is then possible to consider σ ( h ) ≈ K σ h . K σ is a diagonalmatrix, with the slope of the saturation functions along thediagonal. Applying this approximation in the expression of ∆ h in (14b) we obtain: ∆ h = C ( t )∆ x + D ( t )∆ u + D σ ( t ) K σ ∆ h ∆ h = ( I − D σ ( t ) K σ ) − C ( t )∆ x + ( I − D σ ( t ) K σ ) − D ( t )∆ u . Replacing this expression of ∆ h in (14a), the linearizedstate dynamics with the sector-bounded approximation of the − , − , −
500 0 500 1 ,
000 1 , − , ,
000 ˆ N f σ (cid:16) ˆ N f (cid:17) − , − , −
500 0 500 1 ,
000 1 , − , ,
000 ˆ N r σ (cid:16) ˆ N r (cid:17) − , − , −
500 0 500 1 ,
000 1 , − , ,
000 ˆ F xf σ (cid:16) ˆ F x f (cid:17) − , − , −
500 0 500 1 ,
000 1 , − , ,
000 ˆ F xr σ (cid:16) ˆ F x r (cid:17) − , − , −
500 0 500 1 ,
000 1 , − , ,
000 ˆ F yf σ (cid:16) ˆ F y f (cid:17) − , − , −
500 0 500 1 ,
000 1 , − , ,
000 ˆ F yr σ (cid:16) ˆ F y r (cid:17) Fig. 9. Slope sector for reference trajectory of Figure 8. saturation functions is: ∆ ˙ x = (cid:101) A ( t )∆ x + (cid:101) B ( t )∆ u (15a) (cid:101) A ( t ) = A ( t ) + B σ ( t ) K σ ( I − D σ ( t ) K σ ) − C ( t ) (15b) (cid:101) B ( t ) = B ( t ) + B σ ( t ) K σ ( I − D σ ( t ) K σ ) − D ( t ) . (15c) D. LPV polytopic model for control synthesis
The matrices (cid:101) A ( t ) and (cid:101) B ( t ) in (15b) and (15c) are time-varying.In the following, the elements of the matrix K σ will beconsidered as the average slope coefficients of the slopesector shown in Figure 9.It is possible to consider the elements of (cid:101) A ( t ) and (cid:101) B ( t ) thatvary with time as varying parameters. We call θ i , i = 1 , ... , q these parameters. Θ is the vector containing all the parameters.It is then possible to express the linearized model in (15) asan LPV model, with the state matrix A and the input matrix B affinely dependent on the varying parameters.Along the reference trajectory shown in Figure 8, a total of 23elements of (cid:101) A ( t ) and (cid:101) B ( t ) vary with time. However severalof these elements have a small variation. It is possible toconsider just the 6 elements of (cid:101) A ( t ) and (cid:101) B ( t ) that vary themost to constitute the Θ vector. The range of each parameter θ i , i = 1 , ... , is between a minimum θ i and a maximum θ i . θ i and θ i are the minimum and maximum values of θ i along the reference trajectory T ( t ) .For the control synthesis we add the longitudinal, lateral andyaw angle errors with respect to T ( t ) to the state of themodel. They are defined as: x L = x − x (16a) y L = y − y (16b) ∆ ψ = ψ − ψ . (16c) x , y and ψ are the vehicle’s absolute position and attitudealong the reference trajectory T ( t ) .The vehicle’s position dynamics in (2a) and (2b) can besimplified neglecting the vehicle’s lateral speed u : ˙ x = v cos ψ (17a) y = v sin ψ. (17b)Considering that v = v +∆ v , u = u +∆ u and ψ = ψ +∆ ψ and that the ∆ ψ is small along the reference trajectory, thedynamics of the error variables are: ˙ x L = ∆ v (18a) ˙ y L = ∆ v ψ + ∆ ψ v cos ψ (18b) ∆ ˙ ψ = ∆ r. (18c)The expressions ψ and v cos ψ are added to Θ .It is finally possible to define the polytope as the hypercubeof dimension 8, each dimension being the parameter θ i , i =1 , ... , , between θ i and θ i . The polytopic model is definedat its N = 2 vertices: S i = (cid:18) A i B i C i D i (cid:19) i = 1 , ... , N. (19)III. M AIN RESULTS
A static state feedback controller has been designed startingfrom the polytopic LPV model described in Section II-D. Astabilizing state feedback for (19) can be computed solving aproblem under LMI constraints, as shown in [19]. The system(19) is stabilizable by a feedback controller K = R Q − witha specific level of contractivity β > if there exist a matrix Q = Q T > and a matrix R such that: QA Ti + A i Q + R T B Ti + B i R + 2 βQ < , i = 1 , . . . , N. (20)The LMI problem expressed in (20) allows to specify, asperformance criteria, just the level of contractivity.In order to specify other constraints on the poles of the closedloop, it is possible to define an LMI region as in [30].An LMI region D of order s is a subset of the complex planedefined as: D = (cid:8) z ∈ C : L + zM + ¯ zM T < (cid:9) (21)With L = L T ∈ R s × s and M ∈ R s × s .The system (19) is D -stabilizable by a state feedbackcontroller K = R Q − if there exist a matrix Q = Q T > and a matrix R such that: L ⊗ Q + M ⊗ ( A i Q ) + M T ⊗ (cid:16) QA Ti (cid:17) + M ⊗ ( B i L ) + M T ⊗ (cid:16) L T B Ti (cid:17) < i = 1 , . . . , N. (22)The controller presented in this paper has been developedusing this last method.The LMI region D has been defined in order to restrict thedynamics of the closed loop between λ = − and λ = − ,as shown in Figure 10. This choice guarantees a certain levelof contractivity and limits fast dynamics.IV. N UMERICAL RESULTS
In this Section some numerical results of the controller areshown.The controller has been tested on Simulink on the non-linear bicycle model described in Section II-A. The trajectoryconsidered is the reference trajectory shown in Section II-C,
Fig. 10. LMI region for control synthesis.
Figure 8.Several simulations have been run, with a set of differentinitial conditions for ∆ v and ∆ u . The purpose is to establishempirically the region of initial conditions for which thecontroller is able to stabilize the system and converge to thereference trajectory. Figure 11 shows the empirical region ofattraction of the controller for a non-zero initial conditionsfor the variables ∆ v and ∆ u .The capacity of the controller to stabilize the system andbring back the vehicle along the reference trajectory is limitedfor | ∆ v (0) | (cid:46) . m/s and | ∆ u (0) | (cid:46) . m/s.A case with initial conditions of ∆ v (0) = 0 . m/s and ∆ u (0) = 0 . m/s is shown more in detail.Figure 12 shows the commands computed by the controller.Figure 13 shows the vehicle’s position and attitude error withrespect to the reference trajectory. Finally Figure 14 showsthe vehicle’s dynamics error with respect to the referencetrajectory. Both the dynamics and position errors convergeto zero, i.e. the vehicle is brought back to the referencetrajectory. − . − . − − . − . − . − . . . . . . . − . − − . . . v [m/s] ∆ u [ m / s ] No convergenceConvergence
Fig. 11. Empirical region of attraction.
It has been observed that in the cases in which the closedloop diverges, the torques computed by the controller are toolarge, outside the limits of what could be possibly realizedon a vehicle. . . . . . . . . . . . . − . − . [s] [ d e g ] ∆ δ f . . . . . . . . . . . . − , − [s] [ N m ] ∆ τ wf . . . . . . . . . . . . − − − − [s] [ N m ] ∆ τ wr Fig. 12. Commands computed by the controller. . . . . . . . . . . . . . . [s] [ c m ] x L . . . . . . . . . . . . [s] [ c m ] y L . . . . . . . . . . . . − . − . − · − [s] [ d e g ] ∆ ψ Fig. 13. Vehicle’s position and attitude error. . . . − . [s] [ d e g/ s ] ∆ r . . . − − [s] [ r a d / s ] ∆ ω wf . . . − − [s] [ r a d / s ] ∆ ω wr . . . . . . [s] [ m / s ] ∆ v . . . . . . [s] [ m / s ] ∆ u Fig. 14. Vehicle’s dynamics error.
V. C
ONCLUSION AND PERSPECTIVES
The results of Section IV show that the LPV model developedin Section II can be successfully used to develop a controllerthat stabilizes the vehicle along a collision avoidance ma-neuver. Even if the set of initial conditions for which thecontroller is able to stabilize the system is limited, the LPV systems framework offers many possibilities to improve thecontroller. The LPV model could be used for the synthesis ofa dynamic feedback controller, as the one proposed in [16].The tire forces will also be studied in the absolute stabilitycontext, that is, considering the whole sector for the non-linearity description instead of considering a single slopeapproximation. R
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