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Application of Local Linear Steering Models with Model Predictive Control for Collision Avoidance Maneuvers
9th IFAC Symposium on Intelligent Autonomous Vehicles 9th IFAC Intelligent Autonomous June 29 -Symposium July 1, 2016.on Messe Leipzig, Germany Vehicles 9th IFAC Symposium on Intelligent Autonomous Vehicles 9th IFAC Intelligent Autonomous Vehicles June 29 -Symposium July 1, 2016.on Messe Leipzig, Germanyonline at www.sciencedirect.com June 29 - July 1, 2016. Messe Leipzig,Available Germany June 29 - July 1, 2016. Messe Leipzig, Germany
ScienceDirect IFAC-PapersOnLine 49-15 (2016) 187–192
Application of Local Linear Steering Application of Local Linear Steering Application of Local Linear Application of Local Linear Steering Steering Models with Model Predictive Control for Models with Model Predictive Control for Models with Model Predictive Control for Models with Model Predictive Control for Collision Avoidance Maneuvers Collision Avoidance Maneuvers Collision Collision Avoidance Avoidance Maneuvers Maneuvers Boliang Boliang Boliang Boliang Adam Adam Adam Adam
Yi, Jens Ferdinand, Norbert Simm, Frank Bonarens Yi, Jens Ferdinand, Norbert Simm, Frank Bonarens Yi, Yi, Jens Jens Ferdinand, Ferdinand, Norbert Norbert Simm, Simm, Frank Frank Bonarens Bonarens Opel AG, Bahnhofsplatz 1, 65423 R¨ usselsheim, Germany Opel AG, Bahnhofsplatz 1, 65423 R¨ u sselsheim, Germany Opel AG, Bahnhofsplatz 1, u sselsheim, (e-mail: jens.ferdinand, Opel AG,{boliang.yi, Bahnhofsplatz 1, 65423 65423 R¨ R¨ unorbert.simm, sselsheim, Germany Germany (e-mail: {boliang.yi, jens.ferdinand, norbert.simm, (e-mail: jens.ferdinand, norbert.simm, frank.bonarens}@de.opel.com). (e-mail: {boliang.yi, {boliang.yi, jens.ferdinand, norbert.simm, frank.bonarens}@de.opel.com). frank.bonarens}@de.opel.com). frank.bonarens}@de.opel.com).
Abstract: Abstract: Abstract: Collision avoidance systems demand sophisticated control algorithms to ensure driving on a safe Abstract: Collision avoidance systems demand sophisticated control algorithms to ensure driving on a safe Collision systems demand sophisticated algorithms to driving on preplanned path. Model predictive (MPC) control is suitable for this task as different Collision avoidance avoidance systems demandcontrol sophisticated control algorithms to ensure ensure drivinginfluences on aa safe safe preplanned path. Model predictive control (MPC) is suitable for this task as different influences preplanned path. Model predictive control (MPC) is suitable for this task as different influences can be considered by thepredictive algorithm. However MPC a model of the plant to guarantee preplanned path. Model control (MPC) is requires suitable for this task as different influences can be considered by the algorithm. However MPC requires a model of the plant to guarantee can considered by However MPC the to good characteristics. Especially the task to requires generate aaamodel simpleof model covering can be becontrol considered by the the algorithm. algorithm. However MPC requires ofsteering the plant plant to guarantee guarantee good control characteristics. Especially the task to generate aamodel simple steering model covering good control characteristics. Especially the task to generate simple steering model covering the considered range of vehicle dynamics is challenging. In this work a concept for an adaptive goodconsidered control characteristics. Especially the task to generate a simple steering model covering the range of vehicle dynamics is challenging. In this work a concept for an adaptive the considered range of vehicle dynamics is challenging. In this work a concept for an adaptive steering model is presented. The adaptive steering model is integrated in a model predictive the considered range of vehicle dynamics is steering challenging. In this work a concept for anpredictive adaptive steering model is presented. The adaptive model is integrated in a model steering model presented. adaptive steering is in predictive control collision The avoidance maneuvers. Simulation results show improvement in steering concept model is isfor presented. The adaptive steering model model is integrated integrated in aa model model predictive control concept for collision avoidance maneuvers. Simulation results show improvement in control concept for collision avoidance maneuvers. Simulation results show improvement in prediction quality, while achieving modest improvement in control performance. control concept for collision avoidance maneuvers. Simulation results show improvement in prediction quality, while achieving modest improvement in control performance. prediction quality, while achieving modest improvement in control performance. prediction quality, while achieving modest improvement in control performance. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Automotive, collision avoidance, model predictive control, local linear model tree, Keywords: Automotive, collision collision avoidance, model model predictive control, control, local linear linear model tree, tree, Keywords: Automotive, steering system Keywords: Automotive, collision avoidance, avoidance, model predictive predictive control, local local linear model model tree, steering system steering steering system system 1. INTRODUCTION for different speeds and other arbitration modules. These 1. INTRODUCTION INTRODUCTION for different different speeds speeds and and other other arbitration arbitration modules. modules. These These 1. for features helpspeeds the driver handle complex modules. and fast chang1. INTRODUCTION for different andto other arbitration These features help the driver to handle complex and fast changfeatures help the driver to handle complex and fast changvehicle dynamics states and increase safety as well as Vehicle dynamics control for evasive maneuvers have been ing features help the driver to handle complex and fast changing vehicle vehicle dynamics dynamics states states and and increase increase safety safety as as well well as as Vehicle dynamics dynamics control control for for evasive evasive maneuvers maneuvers have have been been user ing experience in the vehicle. Influenced by these features, Vehicle the focus of different work for collision avoidance e.g. ing vehicle dynamics increaseby safety well as Vehicle dynamics control for evasive maneuvers have been user experience in the the states vehicle.and Influenced theseasfeatures, features, the focus of different work for collision avoidance e.g. user experience in vehicle. Influenced by these the model will changeby itsthese character acthe of work for e.g. Schorn (2007), as it allows distanceavoidance to the object useranalytical experiencesteering in the vehicle. Influenced features, the focus focus of different different work lower for collision collision e.g. the the analytical steering model will will change its its character character acSchorn (2007), as it it allows allows lower distanceavoidance to the the object object analytical steering model change according to different vehicle dynamics conditions. Modelling Schorn (2007), as lower distance to at higher velocities compared to braking maneuvers when the analytical steering model will change its character acSchorn (2007), as itcompared allows lower distance to the object cording to different vehicle dynamics conditions. Modelling at higher velocities to braking maneuvers when cording to different vehicle dynamics conditions. Modelling of a steering system is thus a crucial task, to provide at higher velocities compared to braking maneuvers when starting the maneuver, see Moshchuk (2015). cording to different vehicle dynamics conditions. Modelling at higher velocities compared to braking maneuvers when of a steering system is thus a crucial task, to provide starting the the maneuver, maneuver, see see Moshchuk Moshchuk (2015). (2015). of system is aa crucial task, provide predictability the overall dynamics control esstarting of aa steering steering for system is thus thusvehicle crucial task, to to provide starting maneuver, see Moshchuk (2015). predictability for the overall overall vehicle dynamics control esIn order the to assist the driver in emergency situations only, pecially predictability for the vehicle dynamics control in maneuvers with short duration. Keller et esal. In order to assist the driver in emergency situations only, predictability for the overall vehicle dynamics control especially in maneuvers with short duration. Keller et al. In order to assist the driver in emergency situations only, the principle to the react at the last possible moment is (2015) pecially in maneuvers with short duration. et al. proposed a cascaded control concept toKeller control the In order to assist driver in emergency situations only, the principle to react at the last possible moment is pecially in maneuvers with short duration. Keller et al. (2015) proposed a cascaded control concept to control the the principle to react at the last possible moment is followed, so that the system intervenes only in critical (2015) proposed aa cascaded control to control the steering wheel angle in anconcept inner loop by a linear the principle to react at theintervenes last possible moment is demanded followed, so that the system only in critical (2015) proposed cascaded control concept to control the demanded steering wheel angle in an inner loop by a linear followed, situations. demanded steering wheel angle in inner aa linear using steering as anby interface followed, so so that that the the system system intervenes intervenes only only in in critical critical controller situations. demandedthus steering wheel anglewheel in an antorque inner loop loop linear controller thus using steering wheel torque as an anbyinterface interface situations. controller thus using steering wheel torque as the vehicle. However tuning of the controller for differsituations. controller thus using steering wheel torque as an interface Model predictive control (MPC) has been one of the most for for the vehicle. However tuning of the controller for differModel predictive predictive control control (MPC) (MPC) has has been been one one of of the the most most ent for vehicle. However tuning controller for vehicle dynamics conditions is athe tough task. Moshchuk Model for the the vehicle. However tuning of of controller for differdifferpopular control methods in the last ent vehicle dynamics conditions is aathe tough task. Moshchuk Moshchuk Model predictive control (MPC) hasyears beenand one is ofpresented the most (2015) popular control methods in the last years and is presented ent vehicle dynamics conditions is tough task. provided a control concept using analytical steering popular control methods in the last years and is presented ent vehicle dynamics conditions is a tough task. Moshchuk and discussed in Borrelli (2003). The idea of the control (2015) provided a control concept using analytical steering popular controlinmethods the last years and presented and discussed Borrelliin(2003). (2003). The idea of isthe the control system aa control concept analytical steering with constant parameters, which will further be and in idea of control (2015) provided provided control concept using using analytical steering method is to use a time discrete model of the plant and (2015) system with constant constant parameters, which will further further be and discussed discussed in Borrelli Borrelli (2003). The The idea of the control method is to use a time discrete model of the plant and system with parameters, which will be used in this work. method is to use a time discrete model of the plant and system with constant parameters, which will further be predict the future behaviour of the system. According to used in this work. method is to use a time discrete model of the plant and predict the the future future behaviour behaviour of of the the system. system. According According to to used in this work. used in this work. apredict cost function and an optimization problem the control predict the future of the system. to cost function function andbehaviour an optimization optimization problemAccording the control control aaa cost an the output with the and lowest value willproblem be chosen. cost function and an cost optimization problem the control The main contributions of this paper are as follows: output with the lowest cost value will be chosen. The main main contributions contributions of of this this paper paper are are as as follows: output the lowest cost value will output with with the works lowesthave cost been valueconducted will be be chosen. chosen. The main contributions of this papercharacteristics are as follows: follows:at difSeveral research for vehicle dy- The - The problem of steering system Several research works have been conducted for vehicle dy- The The problem problem of of steering steering system system characteristics characteristics at at difdifSeveral research works have been conducted for vehicle dynamics control with MPC. Falcone et al. (2007) presented ferent conditions is described. Thus the need for approSeveral control researchwith works haveFalcone been conducted for vehicle dynamics MPC. et al. al. (2007) (2007) presented - ferent The problem of steering system characteristics at difconditions is described. Thus the need for appronamics control with MPC. Falcone et presented linear time variant MPC approaches using a single track ferent described. Thus need appropriate modelling, is considering dynamics namicstime control withMPC MPC.approaches Falcone et using al. (2007) presented linear variant single track ferent conditions conditions described.different Thus the thevehicle need for for appropriate modelling, is considering different vehicle dynamics linear time variant MPC using aaa single model and steering wheel approaches angle as command signal track compriate modelling, considering different vehicle dynamics conditions of the steering system, is deduced. linear time variant MPC approaches using single track model and steering wheel angle as command signal compriate modelling, considering different vehicle dynamics conditions of the steering system, is deduced. model and wheel angle command combined a solver for quadratic problems. of steering system, is - conditions Further, an adaptive model predictive controller for model with and steering steering wheel angle as asprogramming command signal signal combined with solver for for quadratic programming problems. of the the steering system, is deduced. deduced. - conditions Further, an an adaptive model predictive controller for for bined with aaa solver quadratic programming problems. Katriniok (2013) additionally incorporate the longitudinal Further, adaptive model predictive controller collision avoidance maneuvers is proposed. To improve bined with solver for quadratic programming problems. Katriniok (2013) additionally incorporate the longitudinal Further, an adaptive model predictive controller for collision avoidance maneuvers is proposed. To improve Katriniok (2013) additionally incorporate the longitudinal deceleration into the MPC problem for combined braking collision avoidance maneuvers proposed. To improve prediction quality at different is vehicle dynamics states, Katriniok (2013) additionally incorporate the longitudinal deceleration into the MPC problem for combined braking collision avoidance maneuvers is proposed. To improve prediction quality at different vehicle dynamics states, deceleration the MPC problem for braking and steering.into All of above works use the steering wheel prediction quality at vehicle dynamics state dependant parameters learnedstates, using deceleration thethe MPC problem for combined combined braking and steering.into All of of the above works use use the steering steering wheel prediction qualitysteering at different different vehicle are dynamics state dependant steering parameters are learnedstates, using and steering. All the above works the angle as command for the system. state dependant parameters are learned local linear modelsteering trees, described in Nelles (2001).using and steering. All of for the the above works use the steering wheel wheel angle as command system. state dependant steering parameters are learned using local linear model trees, described in Nelles (2001). angle as command for the system. linear trees, described in Nelles - local Results in amodel simulation environment with an(2001). embedded angle as command for the system. local linear model trees, described in Nelles (2001). Results in in aa simulation simulation environment environment with with an an embedded embedded Most vehicles use steering torque as intervention interface. --- algorithm Results the electricenvironment steering control for difMost vehicles vehicles use use steering steering torque torque as as intervention intervention interface. interface. Results in of a simulation withmodule an embedded algorithm of the electric steering control module for difdifMost Steering torque ensures good interaction of lane keepalgorithm of the electric steering control module for ferent trajectories show the effectiveness of the proposed Most vehicles use steering torque as intervention interface. Steering torque ensures good interaction of lane keepalgorithm of the electric steering control module for different trajectories show the effectiveness of the proposed Steering torque ensures good of keeping and lane centering with the driver. ferent the of methods comparedshow to constant steering parameters with Steering torque ensuresconcepts good interaction interaction of lane laneMature keeping and lane lane centering concepts with the the driver. driver. Mature ferent trajectories trajectories show the effectiveness effectiveness of the the proposed proposed methods compared to constant steering parameters with ing and centering concepts with Mature steering provideconcepts different with functionality like Mature friction methods compared to constant respect prediction ing and systems lane centering the driver. steering systems provide different different functionality functionality like friction friction methodsto to quality. constant steering steering parameters parameters with with respect tocompared prediction quality. steering systems provide like compensation, torque amplification curves (boost curve) respect to prediction quality. steering systems provide different functionality like friction compensation, torque amplification curves (boost curve) respect to prediction quality. compensation, compensation, torque torque amplification amplification curves curves (boost (boost curve) curve) Copyright 2016 IFAC 187 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 187 Copyright 2016 responsibility IFAC 187Control. Peer review©under of International Federation of Automatic Copyright © 2016 IFAC 187 10.1016/j.ifacol.2016.07.730
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2. VEHICLE DYNAMICS DEPENDANT STEERING SYSTEM CHARACTERISTICS Electrical steering support is provided in many vehicles to ensure comfortable and safe driving on the street. One of the functions of this support is to guarantee low actuation force for the driver in different speed ranges, which is done by using boost curves, describing the speed dependant torque amplification character. Also friction compensation, active steering wheel centering and other complex functionalities are integrated in the steering system. Parallel to these fundamental functions a steering torque interface needs to be provided for active safety driver assistance system functions. These functions need to work neatly with existing safety concepts. In this work an Opel Insignia model year 2014 is used. It is equipped with an EPS (Electrical Power Steering) steering actuator, providing a 6Nm steering wheel torque interface. In the following we will show the causal relation with the steering system and the vehicle dynamics in Fig.1, which will be considered in our control concept. The EPS Mechanical Components: Steering System
EPS Control Unit Steering Wheel Torque Command TSW
Boost Curve Friction Compensation
Total Assisted Torque
Vehicle Dynamic
Steering Wheel Angle
TASS
Apart from boost curves different functionalities are implemented in the steering system like friction compensation and active steering wheel centering. A steering model with a simple structured steering system like (1) cannot consider all these features. Collision avoidance maneuvers induce transitions through a high range of vehicle dynamics accelerations, in which the features mentioned above have different effects on the vehicle dynamics. The collision avoidance algorithm needs to consider changing behaviour due to these features as well as changing environmental parameters like self-alignment torque at different speeds. Thus the steering parameters will be adapted using local linear model tree, which will be presented in the next sections. The main focus of this work lies in an easy and adaptive concept to consider parameter changes during control of a vehicle in different vehicle dynamics operation conditions. 3. VEHICLE DYNAMICS MODEL
δ
3.1 Vehicle Model and Tire Model
Active Steering Wheel Centering
Fig. 1. Plant structure with Steering- and Vehicle Dynamics System control unit receives a steering wheel command torque. Here different function modules have an influence on the total assisted torque provided by the electrical machine. The boost curve has the main influence on the steering system behaviour. It translates the commanded torque into a resulting total assisted torque generated by the electrical device. The amplification rate is speed dependant and is illustrated in Fig. 2 schematically. Here v0 is a lower vehicle speed, therefore higher assistance torque is demanded at the same torque command than for a higher speed v1 . The applicable total torque is limited due to actuator limits of the steering system. In a next step the applied total assisted torque leads to a change in the steering wheel angle according to the mechanical dynamic of the steering system. In this work we use a simple equation Total Assisted Torque
In the equation above, δ is the steering wheel angle, JStr is the inertia of the steering wheel and steering column, TASS is the total assisted torque, αf is the slip angle at the tires of the front axis, dStr is the damping parameter, DStr is the self alignment constant consisting of the cornering stiffness and pneumatic trail length.
v0 v1
Steering Wheel Torque Command
Vehicle dynamics modeling has been discussed in various studies e.g. by Schorn (2007) and Mitschke and Wallentowitz (2004). In this section the single-track model is derived. The dynamical equations representing the lateral vehicle dynamics are given by ˙ = Fyf + Fyr , mv(β˙ + ψ) Jz ψ¨ = lf Fyf − lr Fyr .
(3) Here m is the mass of the vehicle, v is the vehicle speed, Jz is the inertia for the z-axis, lf and lr are the distances of the center of gravity to the front and rear axis respectively. β is the side slip angle, ψ is the yaw angle, ψ˙ is the yaw rate, Fyf and Fyr are the lateral forces generated by the tires at the front and rear axis. The lateral force is calculated using Pacejka’s tire formula, given by Fy (sα ) = D sin{C arctan[Bsα − E(Bsα − arctan(Bsα ))]} (4) where sα is the lateral tire slip given by sα = sin(α) and B, C, D, E are tire specific parameters. In order to simplify the equation, Pacejka’s tire formula is linearized around the operation point according to Choi and Choi (2014), such that the linear equation is given by Fyf = cαf αf + Fyf 0 ,
(5)
Fyr = cαr αr + Fyr0 . with the front and rear slip angle
(6)
Fig. 2. Boost Curve for different velocities with v0 < v1 from Moshchuk (2015) to represent the steering system behaviour. JStr δ¨ + dStr δ˙ + DStr αf = TASS (1) 188
(2)
αf =
δ lf ˙ − β − ψ, iL v lr αr = −β + ψ˙ v
(7) (8)
IFAC IAV 2016 June 29 - July 1, 2016. Messe Leipzig, Germany Boliang Yi et al. / IFAC-PapersOnLine 49-15 (2016) 187–192
x˙ = Ax + Bu + E,
(9)
z = Cx
(10)
wz
1
X
[h11,...,h1p]
wz
wz
2
X
[h21,...,h2p]
+
^ [h^1,...,h p]
...
where cαf and cαr are the cornering stiffness of the front and rear axis correspondingly. Equation (2) together with (5) and (1) can be integrated in a linear state space model (9)-(10) which can be further used for control design. The state vector consists of x = [β ψ˙ ψ y δ˙ δ]T where y is the lateral position. The output vector is chosen to be T z = [ψ y] and the command signal is u = TASS . The overall state space model is given by
189
wz
M
[hM1,...,hMp]
X
with A=
Fig. 3. Structure of modified local linear model tree cαf + cαr
cαr lr − cαf lf
−
mv cα lr − cα lf f r Jz 0 v 2DStr JStr 0
B=
−1 0 0
mv2 cαr lr2 + cαf lf2 −
Jz v 1 0 2lf DStr vJStr 0
0 0 0 0 v 0
0 0
0 0 −dStr 0 0 − JStr 0 0 1
cαf mviL cαf lf
The validity area of each local model is given by the validity function ηj (w), a multi-dimensional gaussian curve with centers cji and standard deviations σji , where i is the dimension in the z-regressor space.
J z iL , 0 0 −2DStr
The validity function can be calculated by (11) and (12). Here the membership function Ψj (w) describes the membership of the recent operating point to each of the fixed local linear models in the q-dimensional z-regressor space. We note that the sum of all membership functions equals one. � � 2 2 2
iIStr 0
Fyf0 + Fyr0 Fyr0 lrmv − Fyf0 lf � � 0 01 0 0 0 J z . ,C = 0 0 0 1 0 0 ,E = 0 0
0 0 0 0 1 JStr 0
ηj (wz ) = e
0 0
− 12
(wz1 −cj1 ) σ2 j1
+
(wz2 −cj2 ) σ2 j2
+...++
ηj (wz )
Ψj (wz ) = �M
t=1
4. LOCAL LINEAR MODEL TREE Local linear model tree (LOLIMOT) is a method for model identification and provides the opportunity to solve the problem addressed in section 2. By adapting steering parameters according to different vehicle dynamics conditions, LOLIMOT provides the opportunity to improve prediction quality and control performance of MPC. In this section the original concept of LOLIMOT will not be described in detail here, instead we propose an adaption to local linear model trees, where overlapped parameters of seperate local models are outputs of the model shown in Fig. 3. For more information on the original LOLIMOT concept, please refer to Nelles (2001). The principles of LOLIMOT presented in Nelles (2001) is based on neural networks and works by dividing the input space in different multi-dimensional subspaces (hypercubes) such that vehicle state dependant characteristics can be ensured by the model structure. The space is spanned by the z-regressors wz = [wz1 , · · · , uzq ]. Here these signals may consist of vehicle dynamics states like vehicle speed, steering wheel angle and lateral acceleration. In each hyper-cube a local model represents the desired behaviour for the specific area. In our method the jth local model is represented by the parameter vector hj = [hj1 , ..., hjp ] . 189
M �
ηt (wz )
Ψj (wz ) = 1
(wzq −cjq ) σ2 jq
(11)
(12) (13)
j=1
The standard deviation σji can be calculated by σji = kσ ∆ji (14) with kσ a tuning variable and ∆ji the width of the jth hyper-cube in the z-regressor dimension i . The output parameters of this LOLIMOT model are calculated using the parameters of the seperate local models and the validity value of each hyper-cube. The structure of our adapted LOLIMOT is shown in Fig. 3. By multiplication of the local parameter vectors with the corresponding validity value. The output parameter vector can be calculated by ˆ 1 , · · · , ˆhp ] ˆ = [h h (15) with ˆi = h
M �
hi Ψj (wz )
(16)
j=1
ˆ i are the resulting output In the equations above, h parameters from (16), q is the number of z-regressors, p the number of x-regressors and M the number of local models. The proposed concept works with q = 7 z-regressors and p = 3 x-regressors. M , the number of local models is 3.
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This value is determined by experience, ensuring a tradeoff between overfitting and underfitting. The z-regressors are vehicle speed v, longitudinal and lateral acceleration ax ˙ and ay , steering wheel angle δ, steering wheel gradient δ, ¨ steering wheel acceleration δ. The x-regressors are steering ˙ steeering wheel acceleration δ¨ and slip wheel gradient δ, angle of the front tire αf corresponding to the input signals in (1). All model parameters in LOLIMOT need to be positive to be compatible with physical effects. The identification of these has been done using normed x-regressor signals, which are steering angle gradient, steering angle acceleration and slip angle on the front wheel in accordance to (1).
∆t Qy Qφ R 0.1 s 15 50 0.001 Table I: MPC tuning parameter m 2000kg
Jz lf lr iL 3500kgm2 1.23m 1, 51m 16 Tire Parameters [D, C, B, E] [ 2.05, 0.44, 19.57, -0.70] Table II: Vehicle dynamics parameter of the model
previous control step. Using this approach the MPC can benefit by considering the operation point dependant characteristics in the prediction horizon. The overall closed loop structure is illustrated in Fig.4.
In the next section we will show how to improve the control performance of the MPC by using LOLIMOT steering parameter adaptation. 5. MODEL PREDICTIVE CONTROL In this work we use MPC to control the vehicle on a preplanned trajectory. The optimization problem demands the vehicle to follow a trajectory for yaw angle and for lateral position while the command signal is constrained. Here Q, P and R contain tuning parameters of the controller and Ak , Bk , Ck and Ek are discretized matrices of the continuous state space model from (9) and (10) with the discretization time ∆t for the time step k. J = min
wx ,wz
Fig. 4. Structure of closed control loop with LOLIMOT The control tuning parameters of the MPC and the vehicle dynamics model parameters are given in Table I and II.
N −1
(yref (k) − y(k))T Q(yref (k) − y(k)) + ...
k=1
+ u(k)Ru(k) T
+ (yref (N ) − y(N )) P (yref (N ) − y(N ))... + u(N )Ru(N ) (17) x(k + 1) =Ak x(k) + Bk u(k) + Ek y(k) =Ck x(k) Qφ 0 Q= 0 Qy ulow < u(k) < uhigh
(18) (19) (20)
To further consider changing vehicle dynamics conditions during driving, steering parameters are not assumed to be constant during the prediction horizon, but changing due to predicted vehicle dynamics condition changes. The vehicle transition equation in (18) can thus be substituted by ˆk )x(k) + Bk (h ˆk )u(k) + Ek (h ˆk ) x(k + 1) =Ak (h z(k) =Ck x(k) ˆ k =[h ˆ 1k ... ˆ h hpk ] ˆ ik = h
M
To quantify this a criteria is formulated below, which gives the average rooted square error of the MPC for the steering wheel angle prediction within the horizon. K N 1 2 1 (δ(ti + k∆t) − δpred,i (k)) ∆δpred = K i=1 N k=1
(22) ∆t is the MPC discretization time, K is the total number of sampling time steps, ti is the sampling time step and N is the number of prediction steps. In addition to the prediction criteria above, we will also investigate vehicle dynamics stability and control performance criteria to evaluate the benefit of adaptive steering models in the MPC. We choose the lateral overshoot ∆o over the end value of the lateral trajectory as well as the maximum deviation from trajectory ∆ymax as criteria for evaluating the control performance. 6. SIMULATION RESULTS
(21)
hi Ψj (wk )
j=1
Note that the discrete state space matrices are now deˆ k , which is dependant pendant on the parameter vector h on predicted vehicle dynamics of prediction step k from previous control step. In each prediction step, the relevant parameters are estimated for the relevant predicted vehicle state of the 190
In this section, the proposed MPC with an adaptive steering model will be compared with a MPC with constant steering model. This section begins by showing and describing the learned LOLIMOT model used in this work in Table III and IV. Here the model structure is represented by the center of the local linear model tree in Table III and the model characteristic is shown by the model parameters in Table III. Table III shows that the three local models distinguish from each other by the steering wheel angle position. Here LM 1 and LM 3 have high validities for
IFAC IAV 2016 June 29 - July 1, 2016. Messe Leipzig, Germany Boliang Yi et al. / IFAC-PapersOnLine 49-15 (2016) 187–192
Center of TDriv [Nm] v [m/s] δ [rad] δ˙ [-] δ¨ [-] z-Regres. LM 1 -0.1279 16.35 0.06 -2.07 -0.03 LM 2 -0.1279 16.35 0.06 0.33 -0.03 LM 3 -0.1279 16.35 0.06 1.93 -0.03 Center of 2 2 ax [m/s ] ay [m/s ] TASS [Nm] z-Regres. LM 1 -2.09 -0.69 0.04 LM 2 -2.09 -0.69 0.04 LM 3 -2.09 -0.69 0.04 Table III: Centers of local models in LOLIMOT
191
100
100 Measured Predicted
80
60
Steering Wheel Angle [deg]
Steering Wheel Angle [deg]
60 40 20 0 −20 −40
20 0 −20 −40
−80
−80
Parameters for Dst [−] d [-] JSt [-] LM 1 0.57 1.35 0.47 LM 2 2.91 1.10 0.98 LM 3 0.49 2.26 0.64 Table IV: Parameters of local models in LOLIMOT
40
−60
−60
−100
Measured Predicted
80
4
4.5
5
5.5
6
Time in [sec]
6.5
−100
7
4
4.5
5
5.5
6
Time in [sec]
6.5
7
Fig. 5. Control Performance of MPC with constant steering parameters (left) and with steering parameters adapted by LOLIMOT (right) on maneuver 1 with 2.2m lateral distance
First we evaluate the steering wheel angle prediction quality of the control algorithms by simulating two control maneuvers. The first maneuver starts with an initial velocity of 60 km h and a planned maximum lateral acceleration of 7 sm2 using a sigmoidal function. The lateral offset is 2.2 m at the end of the maneuver. The applied steering wheel angle and the predicted steering wheel angle for the MPC with constant and with adaptive steering models are shown Fig.5. It can be seen that the proposed method shows improved prediction quality. This can also be inferred for the second maneuver, where the planned maximum lateral acceleration is 9 sm2 and the lateral offset is 3 m. The figures are consistent with the defined criteria for prediction quality from (22) in Table V and VI. Here the predictive deviation ∆δpred for MPC with adaptive parameters is lower than with constant parameters for the two maneuvers. In a next step we want to show the effect of the improved steering model for the control performance of the MPC. In Fig.7 and 8 we show the results for the two maneuvers described above. The main characteristics are summarized in Table V and VI. As expected the tables show improved 191
100
100 Measured Predicted
80
Measured Predicted
80 60
Steering Wheel Angle [deg]
60
Steering Wheel Angle [deg]
high negative and positive steering angles (-2.07rad and 1.93rad) whereas LM 2 has high validity around center position. In addition to Table III, Table IV shows the model parameters of LOLIMOT, which can be explained by physical effects. Note that the values for parameters and the xregressors have no units due to normalization. The values of the self alignment constant Dst is six times bigger around center position of steering wheel angle (operation point of LM 2) than for the endpoints of steering wheel angle position (LM 1 and LM 3). This can be explained by the characteristic of self alignment torque which has a big slope at zero position and lower slope for big slip angles, see Mitschke and Wallentowitz (2004). Typically high slip angles appear at high steering wheel position, which is consistent with the learned model in Table III. Another physical effect lies in the values of JSt . It can be seen that the inertia in LM 2 is approximately twice the value of the inertia in LM 1 and 3. This can be explained by the initial friction force in the steering system at the start of the maneuver when steering wheel angle is at zero position.
40 20 0 −20 −40
40 20 0 −20 −40 −60
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Fig. 6. Control Performance of MPC with constant steering parameters (left) and with adaptive parameters by LOLIMOT (right) on maneuver 2 with 4m lateral distance MPC with ∆δpred ∆ymax ∆o Const. Prm 8.76◦ 0.70m 0.15m LOLIMOT 5.76◦ 0.66m 0.11m Table V: Results on MPC with constant parameters and predictive LOLIMOT adapted parameters for maneuver 1 MPC with ∆δpred ∆ymax ∆o Const. Prm 12.7◦ 0.91m 0.18m LOLIMOT 7.9◦ 0.90m 0.16m Table VI: Results on MPC with constant parameters and predictive LOLIMOT adapted parameters for maneuver 2
results in maximum lateral deviation, overshoot and time delay. Due to high robustness and the prediction characteristic of the MPC, the effect on the control performance is not significant. 7. CONCLUSION This paper has investigated the influence of steering parameters for a collision avoidance maneuver using a model predictive control approach. Complex steering functionalities influence the model quality of the steering system. We suggested LOLIMOT to improve prediction quality of the controller using simple steering structure. The performance of the controller is shown by comparing predicted steeering wheel angle course for the maneuver time. LOLIMOT using predictive states cannot improve the controller performance and prediction quality significantly compared to constant steering parameters. This may be explained by the robustness of the model predictive approach to model errors, which can be further investigated
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100 Constant Parameters LOLIMOT parameters
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Fig. 7. Control Performance of MPC with constant steering parameters and with steering parameters adapted by LOLIMOT on maneuver 1 with 2.2m lateral distance in a next step. To improve prediction quality of steering model, more work could be done to increase the complexity of the steering model and to use enhanced parameter learning algorithms. In a next step, vehicle dynamics characterstic will be considered for the generation of the trajectory, to ensure driveability and performance. ACKNOWLEDGEMENTS This research was conducted as part of the research project ”UR:BAN Urbaner Raum: Benutzergerechte Assistenzsysteme und Netzmanagement” funded by the German Federal Ministry of Economics and Energy (BMWi) in the frame of the third traffic research program of the German Bundestag. REFERENCES Borrelli, F. (2003). Constrained optimal control of linear and hybrid systems. Springer-Verlag, Berlin. Choi, M. and Choi, S. (2014). Model predictive control for vehicle yaw stability with practical concerns. Vehicular Technology, IEEE Transactions on, 63(8), 3539–3548. Falcone, P., Borrelli, F., Asgari, J., Tseng, H., and Hrovat, D. (2007). Predictive active steering control for au192
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−50 −100
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−5
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0
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5
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50
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Yaw Angle [deg]
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Steering Wheel Angle [deg]
100
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4
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Fig. 8. Control Performance of MPC with constant steering parameters and with steering parameters adapted by LOLIMOT on maneuver 2 with 3m lateral distance tonomous vehicle systems. Control Systems Technology, IEEE Transactions on, 15(3), 566–580. Katriniok, A. (2013). Optimal vehicle dynamics control and state estimation for a low-cost GNSS-based collision avoidance system. VDI-Verlag, D¨ usseldorf. Keller, D.I.M., Haß, I.C., Seewald, I.A., et al. (2015). A vehicle lateral control approach for collision avoidance by emergency steering maneuvers. In 6th International Munich Chassis Symposium, 175–197. Mitschke, M. and Wallentowitz, H. (2004). Dynamik der Kraftfahrzeuge. VDI-Buch. Springer-Verlag, Berlin. Moshchuk, N. (2015). Collision avoidance with steering: Towards autonomous driving: ADAS and role of steering systems. Steering Systems. Nelles, O. (2001). Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Engineering online library. Springer-Verlag, Belin. Schorn, M. (2007). Quer- und L¨ angsregelung eines Personenkraftwagens f¨ ur ein Fahrerassistenzsystem zur Unfallvermeidung. Ph.D. thesis, Technische Universit¨ at Darmstadt.
ScienceDirect IFAC-PapersOnLine 49-15 (2016) 187–192
Application of Local Linear Steering Application of Local Linear Steering Application of Local Linear Application of Local Linear Steering Steering Models with Model Predictive Control for Models with Model Predictive Control for Models with Model Predictive Control for Models with Model Predictive Control for Collision Avoidance Maneuvers Collision Avoidance Maneuvers Collision Collision Avoidance Avoidance Maneuvers Maneuvers Boliang Boliang Boliang Boliang Adam Adam Adam Adam
Yi, Jens Ferdinand, Norbert Simm, Frank Bonarens Yi, Jens Ferdinand, Norbert Simm, Frank Bonarens Yi, Yi, Jens Jens Ferdinand, Ferdinand, Norbert Norbert Simm, Simm, Frank Frank Bonarens Bonarens Opel AG, Bahnhofsplatz 1, 65423 R¨ usselsheim, Germany Opel AG, Bahnhofsplatz 1, 65423 R¨ u sselsheim, Germany Opel AG, Bahnhofsplatz 1, u sselsheim, (e-mail: jens.ferdinand, Opel AG,{boliang.yi, Bahnhofsplatz 1, 65423 65423 R¨ R¨ unorbert.simm, sselsheim, Germany Germany (e-mail: {boliang.yi, jens.ferdinand, norbert.simm, (e-mail: jens.ferdinand, norbert.simm, frank.bonarens}@de.opel.com). (e-mail: {boliang.yi, {boliang.yi, jens.ferdinand, norbert.simm, frank.bonarens}@de.opel.com). frank.bonarens}@de.opel.com). frank.bonarens}@de.opel.com).
Abstract: Abstract: Abstract: Collision avoidance systems demand sophisticated control algorithms to ensure driving on a safe Abstract: Collision avoidance systems demand sophisticated control algorithms to ensure driving on a safe Collision systems demand sophisticated algorithms to driving on preplanned path. Model predictive (MPC) control is suitable for this task as different Collision avoidance avoidance systems demandcontrol sophisticated control algorithms to ensure ensure drivinginfluences on aa safe safe preplanned path. Model predictive control (MPC) is suitable for this task as different influences preplanned path. Model predictive control (MPC) is suitable for this task as different influences can be considered by thepredictive algorithm. However MPC a model of the plant to guarantee preplanned path. Model control (MPC) is requires suitable for this task as different influences can be considered by the algorithm. However MPC requires a model of the plant to guarantee can considered by However MPC the to good characteristics. Especially the task to requires generate aaamodel simpleof model covering can be becontrol considered by the the algorithm. algorithm. However MPC requires ofsteering the plant plant to guarantee guarantee good control characteristics. Especially the task to generate aamodel simple steering model covering good control characteristics. Especially the task to generate simple steering model covering the considered range of vehicle dynamics is challenging. In this work a concept for an adaptive goodconsidered control characteristics. Especially the task to generate a simple steering model covering the range of vehicle dynamics is challenging. In this work a concept for an adaptive the considered range of vehicle dynamics is challenging. In this work a concept for an adaptive steering model is presented. The adaptive steering model is integrated in a model predictive the considered range of vehicle dynamics is steering challenging. In this work a concept for anpredictive adaptive steering model is presented. The adaptive model is integrated in a model steering model presented. adaptive steering is in predictive control collision The avoidance maneuvers. Simulation results show improvement in steering concept model is isfor presented. The adaptive steering model model is integrated integrated in aa model model predictive control concept for collision avoidance maneuvers. Simulation results show improvement in control concept for collision avoidance maneuvers. Simulation results show improvement in prediction quality, while achieving modest improvement in control performance. control concept for collision avoidance maneuvers. Simulation results show improvement in prediction quality, while achieving modest improvement in control performance. prediction quality, while achieving modest improvement in control performance. prediction quality, while achieving modest improvement in control performance. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Automotive, collision avoidance, model predictive control, local linear model tree, Keywords: Automotive, collision collision avoidance, model model predictive control, control, local linear linear model tree, tree, Keywords: Automotive, steering system Keywords: Automotive, collision avoidance, avoidance, model predictive predictive control, local local linear model model tree, steering system steering steering system system 1. INTRODUCTION for different speeds and other arbitration modules. These 1. INTRODUCTION INTRODUCTION for different different speeds speeds and and other other arbitration arbitration modules. modules. These These 1. for features helpspeeds the driver handle complex modules. and fast chang1. INTRODUCTION for different andto other arbitration These features help the driver to handle complex and fast changfeatures help the driver to handle complex and fast changvehicle dynamics states and increase safety as well as Vehicle dynamics control for evasive maneuvers have been ing features help the driver to handle complex and fast changing vehicle vehicle dynamics dynamics states states and and increase increase safety safety as as well well as as Vehicle dynamics dynamics control control for for evasive evasive maneuvers maneuvers have have been been user ing experience in the vehicle. Influenced by these features, Vehicle the focus of different work for collision avoidance e.g. ing vehicle dynamics increaseby safety well as Vehicle dynamics control for evasive maneuvers have been user experience in the the states vehicle.and Influenced theseasfeatures, features, the focus of different work for collision avoidance e.g. user experience in vehicle. Influenced by these the model will changeby itsthese character acthe of work for e.g. Schorn (2007), as it allows distanceavoidance to the object useranalytical experiencesteering in the vehicle. Influenced features, the focus focus of different different work lower for collision collision e.g. the the analytical steering model will will change its its character character acSchorn (2007), as it it allows allows lower distanceavoidance to the the object object analytical steering model change according to different vehicle dynamics conditions. Modelling Schorn (2007), as lower distance to at higher velocities compared to braking maneuvers when the analytical steering model will change its character acSchorn (2007), as itcompared allows lower distance to the object cording to different vehicle dynamics conditions. Modelling at higher velocities to braking maneuvers when cording to different vehicle dynamics conditions. Modelling of a steering system is thus a crucial task, to provide at higher velocities compared to braking maneuvers when starting the maneuver, see Moshchuk (2015). cording to different vehicle dynamics conditions. Modelling at higher velocities compared to braking maneuvers when of a steering system is thus a crucial task, to provide starting the the maneuver, maneuver, see see Moshchuk Moshchuk (2015). (2015). of system is aa crucial task, provide predictability the overall dynamics control esstarting of aa steering steering for system is thus thusvehicle crucial task, to to provide starting maneuver, see Moshchuk (2015). predictability for the overall overall vehicle dynamics control esIn order the to assist the driver in emergency situations only, pecially predictability for the vehicle dynamics control in maneuvers with short duration. Keller et esal. In order to assist the driver in emergency situations only, predictability for the overall vehicle dynamics control especially in maneuvers with short duration. Keller et al. In order to assist the driver in emergency situations only, the principle to the react at the last possible moment is (2015) pecially in maneuvers with short duration. et al. proposed a cascaded control concept toKeller control the In order to assist driver in emergency situations only, the principle to react at the last possible moment is pecially in maneuvers with short duration. Keller et al. (2015) proposed a cascaded control concept to control the the principle to react at the last possible moment is followed, so that the system intervenes only in critical (2015) proposed aa cascaded control to control the steering wheel angle in anconcept inner loop by a linear the principle to react at theintervenes last possible moment is demanded followed, so that the system only in critical (2015) proposed cascaded control concept to control the demanded steering wheel angle in an inner loop by a linear followed, situations. demanded steering wheel angle in inner aa linear using steering as anby interface followed, so so that that the the system system intervenes intervenes only only in in critical critical controller situations. demandedthus steering wheel anglewheel in an antorque inner loop loop linear controller thus using steering wheel torque as an anbyinterface interface situations. controller thus using steering wheel torque as the vehicle. However tuning of the controller for differsituations. controller thus using steering wheel torque as an interface Model predictive control (MPC) has been one of the most for for the vehicle. However tuning of the controller for differModel predictive predictive control control (MPC) (MPC) has has been been one one of of the the most most ent for vehicle. However tuning controller for vehicle dynamics conditions is athe tough task. Moshchuk Model for the the vehicle. However tuning of of controller for differdifferpopular control methods in the last ent vehicle dynamics conditions is aathe tough task. Moshchuk Moshchuk Model predictive control (MPC) hasyears beenand one is ofpresented the most (2015) popular control methods in the last years and is presented ent vehicle dynamics conditions is tough task. provided a control concept using analytical steering popular control methods in the last years and is presented ent vehicle dynamics conditions is a tough task. Moshchuk and discussed in Borrelli (2003). The idea of the control (2015) provided a control concept using analytical steering popular controlinmethods the last years and presented and discussed Borrelliin(2003). (2003). The idea of isthe the control system aa control concept analytical steering with constant parameters, which will further be and in idea of control (2015) provided provided control concept using using analytical steering method is to use a time discrete model of the plant and (2015) system with constant constant parameters, which will further further be and discussed discussed in Borrelli Borrelli (2003). The The idea of the control method is to use a time discrete model of the plant and system with parameters, which will be used in this work. method is to use a time discrete model of the plant and system with constant parameters, which will further be predict the future behaviour of the system. According to used in this work. method is to use a time discrete model of the plant and predict the the future future behaviour behaviour of of the the system. system. According According to to used in this work. used in this work. apredict cost function and an optimization problem the control predict the future of the system. to cost function function andbehaviour an optimization optimization problemAccording the control control aaa cost an the output with the and lowest value willproblem be chosen. cost function and an cost optimization problem the control The main contributions of this paper are as follows: output with the lowest cost value will be chosen. The main main contributions contributions of of this this paper paper are are as as follows: output the lowest cost value will output with with the works lowesthave cost been valueconducted will be be chosen. chosen. The main contributions of this papercharacteristics are as follows: follows:at difSeveral research for vehicle dy- The - The problem of steering system Several research works have been conducted for vehicle dy- The The problem problem of of steering steering system system characteristics characteristics at at difdifSeveral research works have been conducted for vehicle dynamics control with MPC. Falcone et al. (2007) presented ferent conditions is described. Thus the need for approSeveral control researchwith works haveFalcone been conducted for vehicle dynamics MPC. et al. al. (2007) (2007) presented - ferent The problem of steering system characteristics at difconditions is described. Thus the need for appronamics control with MPC. Falcone et presented linear time variant MPC approaches using a single track ferent described. Thus need appropriate modelling, is considering dynamics namicstime control withMPC MPC.approaches Falcone et using al. (2007) presented linear variant single track ferent conditions conditions described.different Thus the thevehicle need for for appropriate modelling, is considering different vehicle dynamics linear time variant MPC using aaa single model and steering wheel approaches angle as command signal track compriate modelling, considering different vehicle dynamics conditions of the steering system, is deduced. linear time variant MPC approaches using single track model and steering wheel angle as command signal compriate modelling, considering different vehicle dynamics conditions of the steering system, is deduced. model and wheel angle command combined a solver for quadratic problems. of steering system, is - conditions Further, an adaptive model predictive controller for model with and steering steering wheel angle as asprogramming command signal signal combined with solver for for quadratic programming problems. of the the steering system, is deduced. deduced. - conditions Further, an an adaptive model predictive controller for for bined with aaa solver quadratic programming problems. Katriniok (2013) additionally incorporate the longitudinal Further, adaptive model predictive controller collision avoidance maneuvers is proposed. To improve bined with solver for quadratic programming problems. Katriniok (2013) additionally incorporate the longitudinal Further, an adaptive model predictive controller for collision avoidance maneuvers is proposed. To improve Katriniok (2013) additionally incorporate the longitudinal deceleration into the MPC problem for combined braking collision avoidance maneuvers proposed. To improve prediction quality at different is vehicle dynamics states, Katriniok (2013) additionally incorporate the longitudinal deceleration into the MPC problem for combined braking collision avoidance maneuvers is proposed. To improve prediction quality at different vehicle dynamics states, deceleration the MPC problem for braking and steering.into All of above works use the steering wheel prediction quality at vehicle dynamics state dependant parameters learnedstates, using deceleration thethe MPC problem for combined combined braking and steering.into All of of the above works use use the steering steering wheel prediction qualitysteering at different different vehicle are dynamics state dependant steering parameters are learnedstates, using and steering. All the above works the angle as command for the system. state dependant parameters are learned local linear modelsteering trees, described in Nelles (2001).using and steering. All of for the the above works use the steering wheel wheel angle as command system. state dependant steering parameters are learned using local linear model trees, described in Nelles (2001). angle as command for the system. linear trees, described in Nelles - local Results in amodel simulation environment with an(2001). embedded angle as command for the system. local linear model trees, described in Nelles (2001). Results in in aa simulation simulation environment environment with with an an embedded embedded Most vehicles use steering torque as intervention interface. --- algorithm Results the electricenvironment steering control for difMost vehicles vehicles use use steering steering torque torque as as intervention intervention interface. interface. Results in of a simulation withmodule an embedded algorithm of the electric steering control module for difdifMost Steering torque ensures good interaction of lane keepalgorithm of the electric steering control module for ferent trajectories show the effectiveness of the proposed Most vehicles use steering torque as intervention interface. Steering torque ensures good interaction of lane keepalgorithm of the electric steering control module for different trajectories show the effectiveness of the proposed Steering torque ensures good of keeping and lane centering with the driver. ferent the of methods comparedshow to constant steering parameters with Steering torque ensuresconcepts good interaction interaction of lane laneMature keeping and lane lane centering concepts with the the driver. driver. Mature ferent trajectories trajectories show the effectiveness effectiveness of the the proposed proposed methods compared to constant steering parameters with ing and centering concepts with Mature steering provideconcepts different with functionality like Mature friction methods compared to constant respect prediction ing and systems lane centering the driver. steering systems provide different different functionality functionality like friction friction methodsto to quality. constant steering steering parameters parameters with with respect tocompared prediction quality. steering systems provide like compensation, torque amplification curves (boost curve) respect to prediction quality. steering systems provide different functionality like friction compensation, torque amplification curves (boost curve) respect to prediction quality. compensation, compensation, torque torque amplification amplification curves curves (boost (boost curve) curve) Copyright 2016 IFAC 187 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 187 Copyright 2016 responsibility IFAC 187Control. Peer review©under of International Federation of Automatic Copyright © 2016 IFAC 187 10.1016/j.ifacol.2016.07.730
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2. VEHICLE DYNAMICS DEPENDANT STEERING SYSTEM CHARACTERISTICS Electrical steering support is provided in many vehicles to ensure comfortable and safe driving on the street. One of the functions of this support is to guarantee low actuation force for the driver in different speed ranges, which is done by using boost curves, describing the speed dependant torque amplification character. Also friction compensation, active steering wheel centering and other complex functionalities are integrated in the steering system. Parallel to these fundamental functions a steering torque interface needs to be provided for active safety driver assistance system functions. These functions need to work neatly with existing safety concepts. In this work an Opel Insignia model year 2014 is used. It is equipped with an EPS (Electrical Power Steering) steering actuator, providing a 6Nm steering wheel torque interface. In the following we will show the causal relation with the steering system and the vehicle dynamics in Fig.1, which will be considered in our control concept. The EPS Mechanical Components: Steering System
EPS Control Unit Steering Wheel Torque Command TSW
Boost Curve Friction Compensation
Total Assisted Torque
Vehicle Dynamic
Steering Wheel Angle
TASS
Apart from boost curves different functionalities are implemented in the steering system like friction compensation and active steering wheel centering. A steering model with a simple structured steering system like (1) cannot consider all these features. Collision avoidance maneuvers induce transitions through a high range of vehicle dynamics accelerations, in which the features mentioned above have different effects on the vehicle dynamics. The collision avoidance algorithm needs to consider changing behaviour due to these features as well as changing environmental parameters like self-alignment torque at different speeds. Thus the steering parameters will be adapted using local linear model tree, which will be presented in the next sections. The main focus of this work lies in an easy and adaptive concept to consider parameter changes during control of a vehicle in different vehicle dynamics operation conditions. 3. VEHICLE DYNAMICS MODEL
δ
3.1 Vehicle Model and Tire Model
Active Steering Wheel Centering
Fig. 1. Plant structure with Steering- and Vehicle Dynamics System control unit receives a steering wheel command torque. Here different function modules have an influence on the total assisted torque provided by the electrical machine. The boost curve has the main influence on the steering system behaviour. It translates the commanded torque into a resulting total assisted torque generated by the electrical device. The amplification rate is speed dependant and is illustrated in Fig. 2 schematically. Here v0 is a lower vehicle speed, therefore higher assistance torque is demanded at the same torque command than for a higher speed v1 . The applicable total torque is limited due to actuator limits of the steering system. In a next step the applied total assisted torque leads to a change in the steering wheel angle according to the mechanical dynamic of the steering system. In this work we use a simple equation Total Assisted Torque
In the equation above, δ is the steering wheel angle, JStr is the inertia of the steering wheel and steering column, TASS is the total assisted torque, αf is the slip angle at the tires of the front axis, dStr is the damping parameter, DStr is the self alignment constant consisting of the cornering stiffness and pneumatic trail length.
v0 v1
Steering Wheel Torque Command
Vehicle dynamics modeling has been discussed in various studies e.g. by Schorn (2007) and Mitschke and Wallentowitz (2004). In this section the single-track model is derived. The dynamical equations representing the lateral vehicle dynamics are given by ˙ = Fyf + Fyr , mv(β˙ + ψ) Jz ψ¨ = lf Fyf − lr Fyr .
(3) Here m is the mass of the vehicle, v is the vehicle speed, Jz is the inertia for the z-axis, lf and lr are the distances of the center of gravity to the front and rear axis respectively. β is the side slip angle, ψ is the yaw angle, ψ˙ is the yaw rate, Fyf and Fyr are the lateral forces generated by the tires at the front and rear axis. The lateral force is calculated using Pacejka’s tire formula, given by Fy (sα ) = D sin{C arctan[Bsα − E(Bsα − arctan(Bsα ))]} (4) where sα is the lateral tire slip given by sα = sin(α) and B, C, D, E are tire specific parameters. In order to simplify the equation, Pacejka’s tire formula is linearized around the operation point according to Choi and Choi (2014), such that the linear equation is given by Fyf = cαf αf + Fyf 0 ,
(5)
Fyr = cαr αr + Fyr0 . with the front and rear slip angle
(6)
Fig. 2. Boost Curve for different velocities with v0 < v1 from Moshchuk (2015) to represent the steering system behaviour. JStr δ¨ + dStr δ˙ + DStr αf = TASS (1) 188
(2)
αf =
δ lf ˙ − β − ψ, iL v lr αr = −β + ψ˙ v
(7) (8)
IFAC IAV 2016 June 29 - July 1, 2016. Messe Leipzig, Germany Boliang Yi et al. / IFAC-PapersOnLine 49-15 (2016) 187–192
x˙ = Ax + Bu + E,
(9)
z = Cx
(10)
wz
1
X
[h11,...,h1p]
wz
wz
2
X
[h21,...,h2p]
+
^ [h^1,...,h p]
...
where cαf and cαr are the cornering stiffness of the front and rear axis correspondingly. Equation (2) together with (5) and (1) can be integrated in a linear state space model (9)-(10) which can be further used for control design. The state vector consists of x = [β ψ˙ ψ y δ˙ δ]T where y is the lateral position. The output vector is chosen to be T z = [ψ y] and the command signal is u = TASS . The overall state space model is given by
189
wz
M
[hM1,...,hMp]
X
with A=
Fig. 3. Structure of modified local linear model tree cαf + cαr
cαr lr − cαf lf
−
mv cα lr − cα lf f r Jz 0 v 2DStr JStr 0
B=
−1 0 0
mv2 cαr lr2 + cαf lf2 −
Jz v 1 0 2lf DStr vJStr 0
0 0 0 0 v 0
0 0
0 0 −dStr 0 0 − JStr 0 0 1
cαf mviL cαf lf
The validity area of each local model is given by the validity function ηj (w), a multi-dimensional gaussian curve with centers cji and standard deviations σji , where i is the dimension in the z-regressor space.
J z iL , 0 0 −2DStr
The validity function can be calculated by (11) and (12). Here the membership function Ψj (w) describes the membership of the recent operating point to each of the fixed local linear models in the q-dimensional z-regressor space. We note that the sum of all membership functions equals one. � � 2 2 2
iIStr 0
Fyf0 + Fyr0 Fyr0 lrmv − Fyf0 lf � � 0 01 0 0 0 J z . ,C = 0 0 0 1 0 0 ,E = 0 0
0 0 0 0 1 JStr 0
ηj (wz ) = e
0 0
− 12
(wz1 −cj1 ) σ2 j1
+
(wz2 −cj2 ) σ2 j2
+...++
ηj (wz )
Ψj (wz ) = �M
t=1
4. LOCAL LINEAR MODEL TREE Local linear model tree (LOLIMOT) is a method for model identification and provides the opportunity to solve the problem addressed in section 2. By adapting steering parameters according to different vehicle dynamics conditions, LOLIMOT provides the opportunity to improve prediction quality and control performance of MPC. In this section the original concept of LOLIMOT will not be described in detail here, instead we propose an adaption to local linear model trees, where overlapped parameters of seperate local models are outputs of the model shown in Fig. 3. For more information on the original LOLIMOT concept, please refer to Nelles (2001). The principles of LOLIMOT presented in Nelles (2001) is based on neural networks and works by dividing the input space in different multi-dimensional subspaces (hypercubes) such that vehicle state dependant characteristics can be ensured by the model structure. The space is spanned by the z-regressors wz = [wz1 , · · · , uzq ]. Here these signals may consist of vehicle dynamics states like vehicle speed, steering wheel angle and lateral acceleration. In each hyper-cube a local model represents the desired behaviour for the specific area. In our method the jth local model is represented by the parameter vector hj = [hj1 , ..., hjp ] . 189
M �
ηt (wz )
Ψj (wz ) = 1
(wzq −cjq ) σ2 jq
(11)
(12) (13)
j=1
The standard deviation σji can be calculated by σji = kσ ∆ji (14) with kσ a tuning variable and ∆ji the width of the jth hyper-cube in the z-regressor dimension i . The output parameters of this LOLIMOT model are calculated using the parameters of the seperate local models and the validity value of each hyper-cube. The structure of our adapted LOLIMOT is shown in Fig. 3. By multiplication of the local parameter vectors with the corresponding validity value. The output parameter vector can be calculated by ˆ 1 , · · · , ˆhp ] ˆ = [h h (15) with ˆi = h
M �
hi Ψj (wz )
(16)
j=1
ˆ i are the resulting output In the equations above, h parameters from (16), q is the number of z-regressors, p the number of x-regressors and M the number of local models. The proposed concept works with q = 7 z-regressors and p = 3 x-regressors. M , the number of local models is 3.
IFAC IAV 2016 190 Boliang Yi et al. / IFAC-PapersOnLine 49-15 (2016) 187–192 June 29 - July 1, 2016. Messe Leipzig, Germany
This value is determined by experience, ensuring a tradeoff between overfitting and underfitting. The z-regressors are vehicle speed v, longitudinal and lateral acceleration ax ˙ and ay , steering wheel angle δ, steering wheel gradient δ, ¨ steering wheel acceleration δ. The x-regressors are steering ˙ steeering wheel acceleration δ¨ and slip wheel gradient δ, angle of the front tire αf corresponding to the input signals in (1). All model parameters in LOLIMOT need to be positive to be compatible with physical effects. The identification of these has been done using normed x-regressor signals, which are steering angle gradient, steering angle acceleration and slip angle on the front wheel in accordance to (1).
∆t Qy Qφ R 0.1 s 15 50 0.001 Table I: MPC tuning parameter m 2000kg
Jz lf lr iL 3500kgm2 1.23m 1, 51m 16 Tire Parameters [D, C, B, E] [ 2.05, 0.44, 19.57, -0.70] Table II: Vehicle dynamics parameter of the model
previous control step. Using this approach the MPC can benefit by considering the operation point dependant characteristics in the prediction horizon. The overall closed loop structure is illustrated in Fig.4.
In the next section we will show how to improve the control performance of the MPC by using LOLIMOT steering parameter adaptation. 5. MODEL PREDICTIVE CONTROL In this work we use MPC to control the vehicle on a preplanned trajectory. The optimization problem demands the vehicle to follow a trajectory for yaw angle and for lateral position while the command signal is constrained. Here Q, P and R contain tuning parameters of the controller and Ak , Bk , Ck and Ek are discretized matrices of the continuous state space model from (9) and (10) with the discretization time ∆t for the time step k. J = min
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Fig. 4. Structure of closed control loop with LOLIMOT The control tuning parameters of the MPC and the vehicle dynamics model parameters are given in Table I and II.
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To further consider changing vehicle dynamics conditions during driving, steering parameters are not assumed to be constant during the prediction horizon, but changing due to predicted vehicle dynamics condition changes. The vehicle transition equation in (18) can thus be substituted by ˆk )x(k) + Bk (h ˆk )u(k) + Ek (h ˆk ) x(k + 1) =Ak (h z(k) =Ck x(k) ˆ k =[h ˆ 1k ... ˆ h hpk ] ˆ ik = h
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To quantify this a criteria is formulated below, which gives the average rooted square error of the MPC for the steering wheel angle prediction within the horizon. K N 1 2 1 (δ(ti + k∆t) − δpred,i (k)) ∆δpred = K i=1 N k=1
(22) ∆t is the MPC discretization time, K is the total number of sampling time steps, ti is the sampling time step and N is the number of prediction steps. In addition to the prediction criteria above, we will also investigate vehicle dynamics stability and control performance criteria to evaluate the benefit of adaptive steering models in the MPC. We choose the lateral overshoot ∆o over the end value of the lateral trajectory as well as the maximum deviation from trajectory ∆ymax as criteria for evaluating the control performance. 6. SIMULATION RESULTS
(21)
hi Ψj (wk )
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Note that the discrete state space matrices are now deˆ k , which is dependant pendant on the parameter vector h on predicted vehicle dynamics of prediction step k from previous control step. In each prediction step, the relevant parameters are estimated for the relevant predicted vehicle state of the 190
In this section, the proposed MPC with an adaptive steering model will be compared with a MPC with constant steering model. This section begins by showing and describing the learned LOLIMOT model used in this work in Table III and IV. Here the model structure is represented by the center of the local linear model tree in Table III and the model characteristic is shown by the model parameters in Table III. Table III shows that the three local models distinguish from each other by the steering wheel angle position. Here LM 1 and LM 3 have high validities for
IFAC IAV 2016 June 29 - July 1, 2016. Messe Leipzig, Germany Boliang Yi et al. / IFAC-PapersOnLine 49-15 (2016) 187–192
Center of TDriv [Nm] v [m/s] δ [rad] δ˙ [-] δ¨ [-] z-Regres. LM 1 -0.1279 16.35 0.06 -2.07 -0.03 LM 2 -0.1279 16.35 0.06 0.33 -0.03 LM 3 -0.1279 16.35 0.06 1.93 -0.03 Center of 2 2 ax [m/s ] ay [m/s ] TASS [Nm] z-Regres. LM 1 -2.09 -0.69 0.04 LM 2 -2.09 -0.69 0.04 LM 3 -2.09 -0.69 0.04 Table III: Centers of local models in LOLIMOT
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Fig. 5. Control Performance of MPC with constant steering parameters (left) and with steering parameters adapted by LOLIMOT (right) on maneuver 1 with 2.2m lateral distance
First we evaluate the steering wheel angle prediction quality of the control algorithms by simulating two control maneuvers. The first maneuver starts with an initial velocity of 60 km h and a planned maximum lateral acceleration of 7 sm2 using a sigmoidal function. The lateral offset is 2.2 m at the end of the maneuver. The applied steering wheel angle and the predicted steering wheel angle for the MPC with constant and with adaptive steering models are shown Fig.5. It can be seen that the proposed method shows improved prediction quality. This can also be inferred for the second maneuver, where the planned maximum lateral acceleration is 9 sm2 and the lateral offset is 3 m. The figures are consistent with the defined criteria for prediction quality from (22) in Table V and VI. Here the predictive deviation ∆δpred for MPC with adaptive parameters is lower than with constant parameters for the two maneuvers. In a next step we want to show the effect of the improved steering model for the control performance of the MPC. In Fig.7 and 8 we show the results for the two maneuvers described above. The main characteristics are summarized in Table V and VI. As expected the tables show improved 191
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high negative and positive steering angles (-2.07rad and 1.93rad) whereas LM 2 has high validity around center position. In addition to Table III, Table IV shows the model parameters of LOLIMOT, which can be explained by physical effects. Note that the values for parameters and the xregressors have no units due to normalization. The values of the self alignment constant Dst is six times bigger around center position of steering wheel angle (operation point of LM 2) than for the endpoints of steering wheel angle position (LM 1 and LM 3). This can be explained by the characteristic of self alignment torque which has a big slope at zero position and lower slope for big slip angles, see Mitschke and Wallentowitz (2004). Typically high slip angles appear at high steering wheel position, which is consistent with the learned model in Table III. Another physical effect lies in the values of JSt . It can be seen that the inertia in LM 2 is approximately twice the value of the inertia in LM 1 and 3. This can be explained by the initial friction force in the steering system at the start of the maneuver when steering wheel angle is at zero position.
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Fig. 6. Control Performance of MPC with constant steering parameters (left) and with adaptive parameters by LOLIMOT (right) on maneuver 2 with 4m lateral distance MPC with ∆δpred ∆ymax ∆o Const. Prm 8.76◦ 0.70m 0.15m LOLIMOT 5.76◦ 0.66m 0.11m Table V: Results on MPC with constant parameters and predictive LOLIMOT adapted parameters for maneuver 1 MPC with ∆δpred ∆ymax ∆o Const. Prm 12.7◦ 0.91m 0.18m LOLIMOT 7.9◦ 0.90m 0.16m Table VI: Results on MPC with constant parameters and predictive LOLIMOT adapted parameters for maneuver 2
results in maximum lateral deviation, overshoot and time delay. Due to high robustness and the prediction characteristic of the MPC, the effect on the control performance is not significant. 7. CONCLUSION This paper has investigated the influence of steering parameters for a collision avoidance maneuver using a model predictive control approach. Complex steering functionalities influence the model quality of the steering system. We suggested LOLIMOT to improve prediction quality of the controller using simple steering structure. The performance of the controller is shown by comparing predicted steeering wheel angle course for the maneuver time. LOLIMOT using predictive states cannot improve the controller performance and prediction quality significantly compared to constant steering parameters. This may be explained by the robustness of the model predictive approach to model errors, which can be further investigated
IFAC IAV 2016 192 Boliang Yi et al. / IFAC-PapersOnLine 49-15 (2016) 187–192 June 29 - July 1, 2016. Messe Leipzig, Germany
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Fig. 7. Control Performance of MPC with constant steering parameters and with steering parameters adapted by LOLIMOT on maneuver 1 with 2.2m lateral distance in a next step. To improve prediction quality of steering model, more work could be done to increase the complexity of the steering model and to use enhanced parameter learning algorithms. In a next step, vehicle dynamics characterstic will be considered for the generation of the trajectory, to ensure driveability and performance. ACKNOWLEDGEMENTS This research was conducted as part of the research project ”UR:BAN Urbaner Raum: Benutzergerechte Assistenzsysteme und Netzmanagement” funded by the German Federal Ministry of Economics and Energy (BMWi) in the frame of the third traffic research program of the German Bundestag. REFERENCES Borrelli, F. (2003). Constrained optimal control of linear and hybrid systems. Springer-Verlag, Berlin. Choi, M. and Choi, S. (2014). Model predictive control for vehicle yaw stability with practical concerns. Vehicular Technology, IEEE Transactions on, 63(8), 3539–3548. Falcone, P., Borrelli, F., Asgari, J., Tseng, H., and Hrovat, D. (2007). Predictive active steering control for au192
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Fig. 8. Control Performance of MPC with constant steering parameters and with steering parameters adapted by LOLIMOT on maneuver 2 with 3m lateral distance tonomous vehicle systems. Control Systems Technology, IEEE Transactions on, 15(3), 566–580. Katriniok, A. (2013). Optimal vehicle dynamics control and state estimation for a low-cost GNSS-based collision avoidance system. VDI-Verlag, D¨ usseldorf. Keller, D.I.M., Haß, I.C., Seewald, I.A., et al. (2015). A vehicle lateral control approach for collision avoidance by emergency steering maneuvers. In 6th International Munich Chassis Symposium, 175–197. Mitschke, M. and Wallentowitz, H. (2004). Dynamik der Kraftfahrzeuge. VDI-Buch. Springer-Verlag, Berlin. Moshchuk, N. (2015). Collision avoidance with steering: Towards autonomous driving: ADAS and role of steering systems. Steering Systems. Nelles, O. (2001). Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Engineering online library. Springer-Verlag, Belin. Schorn, M. (2007). Quer- und L¨ angsregelung eines Personenkraftwagens f¨ ur ein Fahrerassistenzsystem zur Unfallvermeidung. Ph.D. thesis, Technische Universit¨ at Darmstadt.