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Model Predictive Control for Autonomous Vehicles with Speed Profile Shaping
10th IFAC Symposium on Intelligent Autonomous Vehicles 10th Symposium on Autonomous 10th IFAC IFACPoland, Symposium on Intelligent Intelligent Autonomous Vehicles Vehicles Gdansk, July 3-5, 2019 10th IFAC Symposium on Intelligent Autonomous Vehicles Gdansk, Poland, July 3-5, 2019 Available online at www.sciencedirect.com Gdansk, July 3-5, 2019 10th IFACPoland, Symposium on Intelligent Autonomous Vehicles Gdansk, Poland, July 3-5, 2019 Gdansk, Poland, July 3-5, 2019
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IFAC PapersOnLine 52-8 (2019) 31–36
Model Predictive Control for Model Predictive Control for Model Predictive Control for Model Predictive Control for Shaping Autonomous Vehicles with Speed Profile Model Predictive Control for Shaping Autonomous Vehicles with Speed Profile Autonomous Vehicles with Speed Profile Autonomous Vehicles with Speed Profile Shaping Shaping Autonomous Vehicles with Speed Profile Shaping ∗ ∗∗ ∗∗∗ Y. Mizushima ∗ I. Okawa ∗∗ K. Nonaka ∗∗∗
2[m/s Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration [m/s [m/s [m/s [m/s [m/s ] 22]]22]]2]
∗∗∗ Y. Mizushima ∗∗ I. Okawa ∗∗ K. Nonaka Y. Y. Mizushima Mizushima I. I. Okawa Okawa ∗∗ K. K. Nonaka Nonaka ∗∗∗ ∗ I. Okawa ∗∗ K. Nonaka ∗∗∗ Y.∗∗ Mechanics, MizushimaTokyo City University, University, Japan Japan ∗ Mechanics, Tokyo City Tokyo City ∗ Mechanics, Mechanics, Tokyo City University, University, Japan Japan (e-mail: [email protected]). (e-mail: [email protected]). ∗ Mechanics, (e-mail: [email protected]). ∗∗ Tokyo City University, Japan ∗∗ DENSO CORPORATION, Japan (e-mail: [email protected]). ∗∗ DENSO CORPORATION, Japan DENSO CORPORATION, Japan ∗∗ (e-mail: [email protected]). (e-mail: isao [email protected]). DENSO CORPORATION, Japan (e-mail: isao [email protected]). ∗∗ DENSO isao [email protected]). ∗∗∗ Faculty(e-mail: CORPORATION, Japan (e-mail: isao [email protected]). ∗∗∗ of Engineering, Tokyo City University, Japan ∗∗∗ Faculty of Engineering, Tokyo City University, of (e-mail: Engineering, Tokyo City University, Japan Japan ∗∗∗ Faculty(e-mail: [email protected]) [email protected]). Faculty of Engineering, Tokyo City University, Japan [email protected]) ∗∗∗ Faculty of (e-mail: (e-mail: [email protected]) Engineering, Tokyo City University, Japan (e-mail: [email protected]) (e-mail: [email protected]) Abstract: In recent years, automatic driving has been widely studied which is expected to promote Abstract: recent years, driving has been studied is to Abstract:ofIn Intraffic recentaccidents. years, automatic automatic driving has control, been widely widely studied which which is expected expected to promote promote reduction Focusing on velocity it is sometimes required to decelerate greatly Abstract: In recent years, automatic driving has been widely studied which is expected to promote reduction of traffic accidents. Focusing on velocity control, it is sometimes required to decelerate greatly reduction of traffic accidents. Focusing on velocity control, it is sometimes required to decelerate greatly Abstract: In recent years, automatic driving has been widely studied which is expected to promote reduction of traffic accidents. Focusing on velocity control, it is sometimes required to decelerate greatly to prevent prevent collision collision with with other other vehicles. vehicles. On On the the other other hand, hand, excessive excessive acceleration acceleration or or deceleration deceleration is to is to prevent collision with other vehicles. On the other hand, excessive acceleration or deceleration is reduction ofcollision traffic accidents. Focusing onfor velocity control, it ispaper sometimes required tomethod decelerate greatly to prevent with other vehicles. On the other hand, excessive acceleration or deceleration is undesirable to provide comfortable ride passengers. This presents a novel to reshape undesirable to provide comfortable ride for passengers. This paper presents a novel method to reshape undesirable to provide comfortable ride for passengers. This paper presents a novel method to reshape to prevent collision with other vehicles. On the other hand, excessive acceleration or deceleration is the profile of vehicle velocity based on model predictive control. Introducing combined hard and soft undesirable to provide comfortable ride for passengers. This paper presents a novel method to reshape the profile of vehicle velocity based on model predictive control. Introducing combined hard and soft the profile of vehicle velocity based on model predictive control. Introducing combined hard and soft undesirable to provide comfortable ride for passengers. This paper presents a novel method to reshape constraints, and sudden is suppressed while aa collision with other vehicles is the profile ofexcessive vehicle velocity basedacceleration on model predictive control. Introducing combined hard and soft constraints, excessive and sudden acceleration is suppressed while collision with other vehicles is constraints, and sudden acceleration is suppressed while aa numerical collision with other vehicles is the profile The ofexcessive vehicle velocity based on model predictive control. Introducing hard and soft constraints, excessive andof sudden acceleration is is suppressed while collisioncombined with other vehicles is prevented. advantage the proposed method verified through simulation supposing prevented. The advantage of the proposed method is verified through a numerical simulation supposing prevented. The advantage of the proposed method is verified through a numerical simulation supposing constraints, excessive andof sudden acceleration isinis suppressed whileofaathe collision with other vehicles is aaprevented. practical situation where an another vehicle cuts the front space controlled vehicle. It is shown The advantage the proposed method verified through numerical simulation supposing situation where an another vehicle cuts in front space the controlled vehicle. It is a practical practical situation whereproperly anthe another vehicle cuts inisthe the front spacesoof ofthat controlled vehicle. Itsupposing is shown shown prevented. The advantage of proposed method verified through athe numerical simulation that the proposed method shapes the acceleration profile it adapts to such situations. athat practical situation where an another vehicle cuts in the front space of the controlled vehicle. It is shown the method properly shapes the acceleration profile so that it adapts the proposed proposed method shapes thecuts acceleration profile that adapts to to such such situations. situations. athat practical situation whereproperly an another vehicle in the front spaceso theit It is shown that the proposed method properly shapes the acceleration profile soofthat itcontrolled adapts to vehicle. such situations. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier All rights reserved. that the proposed method properly shapes the acceleration profile so that it adaptsLtd. to such situations. Keywords: Automatic Automatic driving, driving, Model Model predictive predictive control, control, Speed Speed control, control, Preventing Preventing collision, collision, Speed Speed Keywords: Keywords: Automatic driving, Model predictive control, Speed control, Preventing collision, Speed profile Keywords: Automatic driving, Model predictive control, Speed control, Preventing collision, Speed profile profile Keywords: Automatic driving, Model predictive control, Speed control, Preventing collision, Speed profile profile 1. INTRODUCTION Range 1 Range 2 Range Range 33 1. Range 1. INTRODUCTION INTRODUCTION Range 1 1 Range Range 2 2 Range 33 1. INTRODUCTION Range 1 Range 2 Range 1. INTRODUCTION Range 1 Range 2 Range 3 0.0 0.0 0.0 In recent recent years, years, automatic automatic driving driving has has been been widely widely studied studied 0.0 In In recent years, automatic driving has been widely studied 0.0 which is expected expected to increase increase fuel efficiency, efficiency, reduce driver’s driver’s In recent years, automatic driving has been widely studied which is to fuel reduce which is expected to increase fuel efficiency, reduce driver’s In recent years, automatic driving has been widely studied load and prevent traffic accidents, for example. Lane trackwhich is expected to increase fuel efficiency, reduce driver’s load and prevent traffic accidents, for example. Lane trackload and prevent traffic accidents, for example. Lane trackwhich isadaptive expected to increase fuel efficiency, reduce loadand and preventcruise traffic accidents, forhas example. Lanedriver’s tracking and cruise control (ACC) has already been impleing adaptive control (ACC) already been impleTime [s] ing and adaptive cruise control (ACC) has already been impleTime [s] load andon prevent traffic accidents, forhas example. Lane impletracking and adaptive cruise control (ACC) already been mented highways (Aeberhard et al., 2015; Eichelberger and Time mented on highways (Aeberhard et al., 2015; Eichelberger and Time [s] [s] mented on highways (Aeberhard et al., 2015; Eichelberger and ing and adaptive cruise control (ACC) has already been impleMcCartt, 2016). In the future, automatic driving is expected mented on highways (Aeberhard et al., 2015; Eichelberger and Time [s] manipulates the brake McCartt, 2016). In the future, automatic driving is expected Fig. 1. Referential deceleration. A driver McCartt, the future, automatic is expected Referential deceleration. A manipulates mented on2016). highways (Aeberhard et al., 2015; and Fig. Fig. 1. 1. Referential deceleration. A driver driver manipulates the the brake brake to be be available available inIn urban areas (Ziegler (Ziegler et driving al.,Eichelberger 2014). In these these McCartt, 2016). Inurban the future, automatic driving is expected to in areas et al., 2014). In pedal only in Range 1 or Range Fig. 1. Referential deceleration. A driver to be available in urban areas (Ziegler et al., 2014). In these pedal only only in in Range Range 11 or or Range Range 3. 3. manipulates the brake McCartt, 2016). Inurban the future, automatic driving isaccording expected to be available in areas (Ziegler et al., 2014). In these pedal 3. situations, it is required to change speed frequently Fig. 1. Referential deceleration. A driver manipulates the brake situations, it is required to change speed frequently according pedal only in Range 1 or Range 3. situations, it lights is required toareas change speedet frequently to be available in urban (Ziegler al., 2014). In these deceleration and cutting-in caused by surrounding vehicles to the the traffic and the the surrounding vehicles. Onaccording the other den situations, it lights is required to change speedvehicles. frequently according pedal only in Range 1 or Range 3. den deceleration and cutting-in caused by surrounding vehicles to traffic and surrounding On the other to the ittraffic and the surrounding vehicles. Onaccording the other den deceleration and cutting-in caused by surrounding vehicles situations, it lights is required to change frequently should be studied for practical application. hand, is important important to realize realize properspeed acceleration and decelerato the ittraffic lights and the surrounding vehicles. On decelerathe other den deceleration and cutting-in caused by surrounding vehicles should be studied for practical application. hand, is to proper acceleration and hand, it is important to realize proper acceleration and decelerashould be studied for practical application. to the traffic lights and the surrounding vehicles. On the other den deceleration and cutting-in caused by surrounding vehicles tion because it is directly related to riding comfort (Wand et al., hand, it is important to realize proper acceleration and decelerashould be studied for practical application. tion because it is directly related to riding comfort (Wand et al., This paper presents acceleration/deceleration trajectory genertion because it is directly related to riding comfort (Wand et al., hand, it is important to realize proper acceleration and decelerashould be studied for practical application. This paper presents acceleration/deceleration trajectory genertion because it is directly related to riding comfort (Wand et al., 2000). This paper presents acceleration/deceleration trajectory gener2000). ation which achieves both reducing load on passengers and 2000). This paper presents acceleration/deceleration trajectory genertion because it is directly related to riding comfort (Wand et al., ation which achieves both reducing load on passengers and 2000). ation which achieves both reducing load on passengers and This paper presents acceleration/deceleration trajectory generconsidering the motion of surrounding vehicles. The proposed In previous researches, speed control has been proposed (Ioannou ation which achieves both reducing load on passengers and 2000). considering the motion of surrounding vehicles. The proposed In previous researches, speed control has been proposed (Ioannou In previous researches, speed control has been proposed (Ioannou considering the motion of surrounding vehicles. The proposed ation which achieves both reducing load ondistance passengers and In previous researches, speed control has been proposed (Ioannou considering the motion of surrounding vehicles. The proposed method firstly designs the target velocity and based on and Xu, 1994; Yamamura et al., 2008) in order to suppress method firstly designs the target velocity and distance based and Xu, 1994; Yamamura et al., 2008) in order to suppress and Xu, 1994; Yamamura etACC al., with 2008) in order to suppress method firstly designs the target velocity and road distance based on on In previous researches, speed control has model been proposed (Ioannou considering the(CC) motion of surrounding vehicles. The proposed cruise control and ACC. CC considers environment both acceleration and jerk. predictive conand Xu, 1994; Yamamura et al., 2008) in order to suppress method firstly designs the target velocity and distance based on cruise control (CC) and ACC. CC considers road environment both acceleration and jerk. ACC with model predictive conboth acceleration and jerk. ACC with model predictive concruise control (CC) and ACC. CC considers road environment and Xu, 1994; Yamamura et al., 2008) in order to suppress method firstly designs the target velocity and distance based on such as deceleration at crossroad. ACC considers a motion trol (MPC) has been proposed (Luo et al., 2010; Stanger and both acceleration and jerk. ACC with model predictive concruise control (CC) and ACC. CC considers road environment such as as deceleration deceleration at at crossroad. crossroad. ACC ACC considers considers aa motion motion of of trol (MPC) has been proposed (Luo et al., 2010; Stanger and trol (MPC) has been proposed (Luo et al., 2010; Stanger and such of both acceleration and jerk. ACC with model predictive concruise control (CC) and ACC. CC considers road environment surrounding vehicles such as sudden deceleration cuttingdel Re, Re, 2013; Peng and Umit, 2015; 2015; He et al., al., 2017; Liu et et and al., such trol (MPC) hasPeng beenand proposed (Luo He et al., 2010; Stanger as deceleration at crossroad. ACC considers and a motion of surrounding vehicles such as sudden deceleration and cuttingdel 2013; Umit, et 2017; Liu al., del Re, 2013; Peng and Umit, 2015; He et al., 2017; Liu et al., surrounding vehicles such as sudden deceleration and cuttingtrol (MPC) has been proposed (Luo et al., 2010; Stanger asoptimal deceleration at crossroad. ACC considers abymotion of del Re, 2013; and Umit, 2015; He etwith al., 2017; Liuvehicle et and al., such surrounding vehicles such astrajectory sudden deceleration and cuttingin. An acceleration is generated MPC so 2017; Zhao etPeng al., 2017), where collision with another vehicle in. An optimal acceleration trajectory is generated by MPC so 2017; Zhao et al., 2017), where collision another 2017; Zhao etPeng al., 2017), where collision another vehicle An optimal acceleration is generated bytracks MPC so del Re, 2013; and Umit, 2015; He etwith al., 2017; Liu et al., in. surrounding vehicles such astrajectory sudden deceleration and cuttingthat the velocity profile is properly tuned as well as it the has been prevented by constraints on inter-vehicular distance. 2017; Zhao et al., 2017), where collision with another vehicle in. An optimal acceleration trajectory is generated by MPC so that the velocity profile tuned as as tracks the has been prevented by on distance. has been prevented by constraints constraints on inter-vehicular inter-vehicular distance. that thevelocity. velocityFigure profile1is is properly properly tuned as well well as it it the 2017; Zhao et al., 2017), where withusing another vehicle in. An optimal acceleration trajectory is generated bytracks MPCtraso target the referential acceleration Those methods suppress load oncollision passengers constraints has been prevented by constraints on inter-vehicular distance. that thevelocity. velocityFigure profile1isdepicts properly as well as it tracks the target depicts thetuned referential acceleration traThose methods suppress load on passengers using constraints Those methods suppress load on passengers using constraints target velocity. Figure 1 depicts the referential acceleration trahas been prevented by constraints on inter-vehicular distance. that the velocity profile is properly tuned as well as it tracks the jectoryvelocity. during deceleration, deceleration, where the brake pedal pedal is depressed depressed on acceleration acceleration and jerk. load However, constraintsusing on acceleration acceleration Those methods and suppress on passengers constraints target Figure 1 depicts thethe referential acceleration trajectory during where brake is on jerk. However, constraints on on acceleration and However, constraints on during deceleration, where the brake pedal is depressed Those methods suppress load onfor passengers using constraints target velocity. Figure 1 depicts the referential acceleration traon jerk acceleration and jerk. However, constraints on acceleration acceleration jectory during deceleration, where the brakein pedal is 3. depressed in Range 11 and the brake pedal is released Range Unnecor jerk contradict thejerk. constraint inter-vehicular distance; it jectory in Range and the brake pedal is released in Range 3. Unnecor contradict the constraint for inter-vehicular distance; it or jerk contradict thejerk. constraint for constraints inter-vehicular distance; it in Range 1 anddeceleration, theacceleration brake pedal is released Range Unnecon acceleration and However, on acceleration jectory during where the discomfort brakein pedal is 3. depressed essary change of causes on passenger is sometimes difficult to obtain control parameters which sator jerk contradict the constraint for inter-vehicular distance; it in Range 1 and the brake pedal is released in Range 3. Unnecessary change of acceleration causes discomfort on passenger is sometimes difficult to parameters which satis difficult to obtain obtain control control parameters which satessary change acceleration causes discomfort on passenger or jerk contradict the constraint for inter-vehicular it in Range 1 andof brake pedal is released in Range 3. Unnecand deterioration of driving performance due to aa fluctuation isfysometimes different kind of constraints constraints simultaneously. Todistance; overcome is sometimes difficult to obtain control parameters which satessary change oftheacceleration causes discomfort on passenger and deterioration of driving performance due to fluctuation isfy different kind of simultaneously. To overcome isfy different kind of constraints simultaneously. To overcome and deterioration of driving performance due to a fluctuation is sometimes difficult to obtain control parameters which satessary change of acceleration causes discomfort on passenger isfy different kind of constraints simultaneously. To overcome and deterioration of driving performance due to a fluctuation of load load distribution. distribution. Thus, Thus, the the referential referential acceleration acceleration is is conconthis difficulty, difficulty, ACC ACC with with relaxed relaxed acceleration acceleration constraints constraints has has of this this difficulty, ACC acceleration load distribution. Thus, the referential acceleration is isfy ofwith constraints simultaneously. To overcome and deterioration of driving performance due totoa realize fluctuation this different difficulty, ACC with relaxed acceleration constraints has of load distribution. Thus, the referential acceleration is conconstant excluding Range 11 and Range 3. In order the been proposedkind (Shengbo etrelaxed al., 2011; 2011; Zhang et etconstraints al., 2018).has In of stant excluding Range and Range 3. In order to realize the been proposed (Shengbo et al., Zhang al., 2018). In been proposed (Shengbo al., 2011; Zhang acceleration etconstraints al., 2018).and In stant excluding Range 1depicted and 3. In order toimpose realize the this difficulty, ACC with et relaxed acceleration has of load distribution. Thus, theRange referential acceleration is conreferential acceleration in Fig. 1, we also soft these methods, both suppression of excessive been proposed (Shengbo et al., 2011; Zhang et al., 2018). In stant excluding Range 1 and Range 3. In order to realize the referential acceleration depicted in Fig. 1, we also impose soft these methods, both suppression of excessive acceleration and these methods, both suppression of excessive acceleration and referential acceleration depicted in Fig. 1, we also impose soft been proposed (Shengbo et al., 2011; Zhang et al., 2018). In stant excluding Range 1 and Range 3. In order to realize the constraints on acceleration and jerk, where the constraints are prevention of collision are considered. But the speed profile these methods, both suppression of excessive acceleration and referential acceleration depicted in Fig. 1, we also impose soft constraints on acceleration and jerk, where the constraints are prevention of collision are considered. But the speed profile prevention of collision are considered. But the speed profile constraints on acceleration and jerk, where the constraints are these methods, both suppression of excessive acceleration and referential acceleration depicted in Fig. 1, we also impose soft prevention of collision are considered. But the speed profile constraints on acceleration and jerk, where the constraints are relaxed utilizing utilizing slack slack variables. variables. In In addition, addition, we we impose impose aa hard hard has not not been been designed designed explicitly explicitly which which possibly possibly affects affects the the relaxed has has not been designed explicitly which possibly affects the relaxed utilizing slack variables. In addition, we impose a hard prevention of designed collision are considered. But the speed profile constraints onan acceleration and jerk, where the constraints are has not been explicitly which possibly affects the relaxed utilizing slack variables. In addition, we impose a hard constraint on upper limit of running distance to avoid colcomfortability of passengers. In addition, a motion such as sudconstraint on limit distance to colcomfortability of In a motion as comfortability of passengers. passengers. In addition, addition, motion such such as sudsudconstraint on an an upper uppervariables. limit of of running running distance to avoid avoid colhas not been designed explicitly whichaapossibly affects the relaxed utilizing In addition, we impose a hard comfortability of passengers. In addition, motion such as sudconstraint on an slack upper limit of running distance to avoid colcomfortability of IFAC passengers. In addition, a motion such as Control) sud- constraint on an upper limit ofreserved. running distance to avoid col2405-8963 © 2019, (International Federation of Automatic Hosting by Elsevier Ltd. All rights
Copyright © 2019 IFAC Copyright © 2019 Copyright © under 2019 IFAC IFAC Peer review responsibility of International Federation of Automatic Control. Copyright © 2019 IFAC 10.1016/j.ifacol.2019.08.044 Copyright © 2019 IFAC
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State of leader vehicle (Δ𝑆𝑆, 𝑣𝑣p )
lision with other vehicles, so that the proposed method shapes the acceleration trajectory while a collision with other vehicle is prevented. We verify the advantage of the proposed method in a simulation where another vehicle cuts in on the front. We compare the proposed method with a method without trajectory shaping to show that the trajectory is properly shaped in the situation.
ACC
CC
Adaptive Cruise Control Cruise Control (𝑣𝑣t,acc , 𝑑𝑑acc ) (𝑣𝑣t,cc , 𝑑𝑑cc , 𝑣𝑣cc ) Combination function
2. SPEED CONTROL SYSTEM
Target state of MPC (𝑑𝑑t , 𝑣𝑣t , 𝑎𝑎t ), Constraint value (𝑑𝑑, 𝑣𝑣)
2.1 Overview Figure 2 depicts a model of the control system for the longitudinal velocity used in this study. The system adapts various situations, such as cruising, acceleration/deceleration at crossroad, deceleration of a leader vehicle and cutting-in by another vehicle. The system is divided into the upper and the lower level controller. In the upper level controller, optimal vehicle running distance d, velocity v and acceleration a are calculated. The upper level controller consists of CC, ACC, combination function of them and MPC. CC generates a target velocity and a movable distance based on the static environment of the road. ACC generates a target velocity which keeps a safe inter-vehicular distance and an upper bound distance which prevents collision with a leading vehicle. Based on the output of ACC and CC, the combination function determines the target state and the upper limit of running distance, which are used in MPC to calculate an optimal input. Details of these functions are described in later sections. This flow is repeated at each control cycle to obtain an optimal vehicle velocity. In the lower layer controller, the engine output is controlled so that the actual velocity follows the optimal velocity. This study focuses only on the upper layer controller which mainly determines the speed profile of running vehicles.
Model Predictive Control Vehicle model
Optimal vehicle state (𝑑𝑑, 𝑣𝑣, 𝑎𝑎)
Upper level controller
Lower level controller
Engine throttle Actual vehicle
Fig. 2. The speed control system consisting of the upper and lower level controller. The upper level controller generates the optimal vehicle behavior. The lower level controller tracks the optimal behavior.
Running distance along the reference path 𝑑𝑑
2.2 Vehicle Model We focus on longitudinal vehicle motion as depicted in Fig. 3, where d is the running distance along the reference path and v is the vehicle velocity, respectively. We define vehicle acceleration a and state vector x := [d, v, a]T . We also define input u of the vehicle jerk. Each state has relationships described as d˙ = v, v˙ = a and a˙ = u. A state equation of continuous-time system is described as follows: ] [ ] [ 010 0 x˙ = 0 0 1 x + 0 u. (1) 000 1
Velocity 𝑣𝑣
Fig. 3. Vehicle model considering longitudinal motion. vehicle decelerates/accelerates near crossroad, the target velocity is designed to be with a constant acceleration; this target velocity expects both avoiding inappropriate change of steering characteristics and improving ride comfort. The duration required for acceleration/deceleration ∆td is given as follows: ve − vs ∆td = , (3) ar where vs is initial velocity, ve is terminal velocity and ar is target acceleration, respectively. Start time of deceleration ts is given as follows: 1 dr + ar ∆td2 2 , (4) ts = vs where dr is running distance to the stop line. Finish time of deceleration te is calculated by te = ts + ∆td . In addition, the maximum velocity v¯cc [k] is generated to drive within the limit of velocity. In cruise section, v¯cc [k] is the maximum velocity. In crossroad, v¯cc [k] is 0.5 km/h so that the vehicle can stop immediately. Figure 4(a) depicts the target velocity and the
Using zero-order hold, the state equation is represented by discrete-time system as follows: 3 ∆t /6 1 ∆t ∆2t /2 x[k + 1] = 0 1 ∆t x[k] + ∆2t /2 u[k], (2) 0 0 1 ∆t
where ∆t is the sampling time. 2.3 Cruse control (CC)
CC generates a target velocity and a movable distance based on static environment of the road such as crossroad. When the vehicle cruises at a constant velocity, the target velocity is usually set to be equivalent to the maximum speed. When the 32
(a) The vehicle accelerate.
𝑡𝑡e
Horizon with MPC 𝑣𝑣t,ACC
Time 𝑡𝑡 [s] Target Constraint
0
(b) The vehicle decelerate.
Fig. 4. Target and constraint velocity with CC. Controlled vehicle
Δ𝑆𝑆
𝑣𝑣
Target position
33
Running distance [m]
Time 𝑡𝑡 [s] Target Constraint
𝑡𝑡s
Velocity [m/s]
Δ𝑡𝑡d
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Velocity [m/s]
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𝑣𝑣t,CC
Target of MPC 𝑣𝑣t Time 𝑡𝑡 [s]
(a) Target velocity.
𝑇𝑇H
0
Horizon with MPC
𝑑𝑑ҧCC
𝑑𝑑ҧACC
Upper limit of MPC 𝑑𝑑ҧ Time 𝑡𝑡 [s]
(b) Upper limit of distance
𝑇𝑇H
Fig. 6. Integrating with CC and ACC. Δ𝑆𝑆t
2.5 Combination of CC and ACC
Leader vehicle
The target state in MPC is generated by the velocity trajectory of CC and ACC. Figure 6(a) depicts combination the target velocity, where vt [k] is target velocity. Adopting one with the lower velocity at each time, the combine function generates vt [k]. The function integrates vt [k] with respect to time in order to obtain a target running distance dt [k]. In addition, the function differentiates vt [k] with respect to time in order to obtain a target acceleration at [k]. Since vt [k] is designed using (6) and (7), noise amplification due to differentiation is negligible.
𝑣𝑣p
Fig. 5. Relationship between controlled vehicle and leader vehicle. upper limit of velocity when the vehicle starts. Figure 4(b) depicts them when the vehicle decelerates to the stop line. In addition, the upper limit of running distance d¯cc is set at stop line to prevent running over the line.
The limit of running distance by CC and ACC are combined to generate the limit of running distance in MPC. Figure 6(b) depicts generation of the upper limit of running distance, where ¯ is the upper limit of running distance. Adopting one with d[k] the lower distance at each time, the combination function ¯ Also, the upper limit of velocity v[k] generates d[k]. ¯ is v¯cc [k].
2.4 Adaptive cruse control (ACC) In ACC, the controlled vehicle follows the leader vehicle while safety inter-vehicle distance is preserved. Figure 5 depicts the relationship between the leader vehicle and the controlled vehicle, where v is velocity of the controlled vehicle, vp is velocity of the leader vehicle and ∆S is actual inter-vehicle distance, respectively. The target inter-vehicle distance ∆St [k] is expressed as follows: ¯ (5) ∆St [k] = v[k]tTHW − (vp [k] − v[k])tTTC + ∆S,
3. MODEL PREDICTIVE SPEED CONTROL Preventing collision with other vehicles and following a limit speed are necessary. Automatic driving must prevent the acceleration which exceeds the limit of vehicle performance for guaranteeing safety. In addition, excessive acceleration and sudden acceleration should be avoided for comfortability of passengers. To satisfy these requests, we propose a velocity controller based on MPC, which calculates the optimal control input by minimizing an index function while satisfying constraints and guaranteeing the safety of the vehicle. Moreover, in order to suppress excessive acceleration that passenger feels uncomfortable, we introduce slack variables into the constraints which render constraints flexible while physical or legal limitations are satisfied by hard constraints.
where tTHW is the constant time-headway, tTTC is the constant ¯ = 5 m is inter-vehicle distance when time-to-collision and ∆S these vehicles stop. In (5), the first term shows the inter-vehicle distance in proportion to the velocity of the controlled vehicle. When the controlled vehicle is faster, the inter-vehicle distance is extended. The second term modifies the inter-vehicle distance depending on the difference in velocity of these vehicles. In this study, we set tTHW = 3.3 s so that inter-vehicle distance is 60 m when the vehicle drives at 60 km/h. In addition, we set tTTC = 0.3 s. Target acceleration at,acc [k] is expressed as follows: at,acc [k] = KS (∆S[k] − ∆St [k]) − Kv (v[k] − vp [k]),
3.1 Constraint condition
(6)
We impose various constraints on the state x and the input u to ensure safety and comfortability. To prevent running over the stop line, inequality constraint of running distance is given as: ¯ d[k] ≤ d[k], (8)
where Kd and Kv are constant gains. In addition, the target velocity vt,acc is expressed as follows: vt,acc [k + 1] = vt,acc [k] + at,acc [k]∆h , (7) where ∆h is time step of horizon in MPC. To obtain the target velocity trajectories, (6) and (7) are calculated in the horizon of the MPC. It is assumed that behavior of the leader vehicle can be predicted by map data: the vehicle decelerates to stop when a traffic light is red.
where d¯ is the upper limit of running distance. Collision with other vehicle can also be considered by (8). To follow the limit speed, inequality constraint on velocity is given as: ¯ (9) v[k] ≤ v[k] ≤ v[k],
In order to avoid collision with the leader vehicle, a limit of running distance d¯acc [k] is 5 m behind the leader vehicle.
where v¯ is the upper limit of velocity and v is the lower limit of velocity, respectively. To prevent acceleration exceeding a 33
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Controlled vehicle Another vehicle
Penalty range Max
Penalty [-]
Min
Red light on 𝑎𝑎
𝑎𝑎s
𝑎𝑎𝑠𝑠
Acceleration a [m/s2]
𝑎𝑎
Fig. 8. Simulation condition: the another vehicle cuts in the controlled vehicle while these vehicles decelerate for stop line.
Fig. 7. Penalty by slack value. The blue line is evaluation value and the orange area is penalty range.
where Q, R, Qa , Qu are weights and N is the number of prediction step. In (16), the first and second terms evaluates the deviation of the target state and the input value. The third term evaluates the deviation from the comfortable acceleration and the forth term evaluates the deviation from the comfortable jerk. Minimizing (16), excessive acceleration and jerk are suppressed while the magnitude of state tracking error and that of input are balanced.
performance limit, inequality constraint on acceleration is given as: a ≤ a[k] ≤ a, ¯ (10)
where a¯ is upper limit of acceleration and a is lower limit of acceleration, respectively. To protect engine and powertrain from sudden acceleration, inequality constraint on jerk is given as: u ≤ u[k] ≤ u, ¯ (11)
3.3 Formulation as an optimization problem
where u¯ is upper limit of jerk and u is lower limit of jerk, respectively.
The proposed MPC is formulated by optimization problem as follows: minimize J, with respect to u[k], σu [k] for (k = 0, 1, · · · , N − 1), σa [k] for (k = 1, 2, · · · , N), subject to (2), (8) ∼ (13).
In addition, we impose constraints which prevent uncomfortable acceleration. To suppress excessive and sudden acceleration, soft constraints with slack variables are given as: as − σa [k] ≤ a[k] ≤ a¯s + σa [k], (12) us − σu [k] ≤ u[k] ≤ u¯s + σu [k], (13)
When (8) or (9) cannot be achieved, slack variable σa [k] and σu [k] increase. Since these slack variables are minimized in the index function, the acceleration trajectory is approximated to the referential one as close as possible.
where σa is slack variable of acceleration, σu is slack variable of jerk, as , a¯s are the boundaries of comfortable deceleration and us , u¯s are the boundaries of comfortable jerk, respectively. It is noted that we distinguish the performance limit a¯ from boundary of comfortable acceleration a¯s . Since slack variable σa is an amount of deviation from as or a¯s , we require σa ≥ 0. The magnitude of slack variable σa is minimized in the evaluation function given later in (16) in order to suppress excessive acceleration. Figure 7 depicts the effect of (10) and (12) on the evaluation function which indicates that acceleration that passenger feels uncomfortable is suppressed while acceleration that exceeds performance limit is prevented. Slack variable σu is also amount of deviation from us or u¯s , we can give that σa ≥ 0. Owing to (10) and (13), excessive jerk can be suppressed while the physical limitation is achieved.
4. SIMULATION 4.1 Condition We conduct numerical simulations to verify advantage of the proposed controller. The simulation assumes an interference by another vehicle. Another vehicle decelerating at −1.47 m/s2 cuts in the controlled vehicle when the controlled vehicle decelerates at −0.981 m/s2 from 40 km/h as depicted in Fig. 8. We assume that the controlled vehicle cannot predict cutting-in by another vehicle.
3.2 Index function
To verify the performance, the proposed method is compared with MPC without penalty and PD controller. Input jerk of the PD controller is expressed as follows: ePD [k] − ePD [k − 1] , (17) uPD [k] = KP ePD [k] + KD ∆t
The deviation from the target state e[k] is represented as follows: (14) e[k] = x[k] − xt [k],
where
xt [k] := [dt [k] vt [k] at [k]] .
(15)
where KP is proportional gain, KD is derivative gain and ePD [k] := at [k] − a[k], respectively.
The evaluation function is represented as follows: J=
Moreover, the proposed method is compared with another method with constant soft constraint on (12) and (13). Another method has soft constraints of acceleration and jerk as follows: as − σa,cns ≤ a[k] ≤ a¯s + σa,cns , (18) us − σu,cns ≤ u[k] ≤ u¯s + σu,cns . (19)
1 N−1 1 ∑ (e[k]T Qe[k] + Ru[k]2 ) + 2e[N]T Qe[N] 2 k=0
N N 1 1 + Qa ∑ σa [k]2 + Qu ∑ σu [k]2 , 2 k=1 2 k=1
(16)
34
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Element
Symbol
Value
Maximum acceleration Minimum acceleration Maximum acceleration penalty Minimum acceleration penalty Maximum jerk penalty Minimum jerk penalty Weight of state Weight of input Weight of acceleration penalty Weight of jerk penalty Gain of inter-vehicular distance in ACC Gain of velocity in ACC Proportional gain of PD controller Derivative gain of PD controller
a¯ a a¯s as u¯s us Q R Qa Qu KS Kv KP KD
1.471 m/s2 −1.96 m/s2 0.981 m/s2 −0.981 m/s2 0.909 m/s3 −1.25 m/s3 diag(0.002, 0.05, 1.0) 0.005 600 900 0.4 0.35 4.0 0.1
Running distance d [m]
Table 1. Simulation parameters.
Time t [s]
(a) Running distance d.
2
Acceleration a [m/s ]
where σa,cns and σu,cns are slack variables which are same through the horizon. Since σa,cns and σu,cns are given in every prediction step, acceleration and jerk may deviate same amount at every prediction step. As with (16), the magnitude of slack variables σa,cns and σu,cns are minimized in index function in order to suppress excessive acceleration and jerk. In this paper, MPC which utilizes multiple slack variables in the horizon is called “Multiple slack variables”, while MPC which utilizes constant slack variables in the horizon is also called “Constant slack variables”. Table 1 shows the parameters of the simulation. Since sudden acceleration is more uncomfortable than excessive acceleration, the weight of the jerk penalty is set to be larger than the weight of acceleration penalty. To enable safety decelerate from 60 km/h, we set the horizon length 7.0 s and the number of the prediction step N = 30.
Time t [s]
(b) Acceleration a.
Jerk u [m/s ]
4.2 Results and discussion 3
Figure 9(a), (b) and (c) depicts the vehicle running distance, acceleration and jerk with the conventional methods, respectively. In Fig. 9(a), the running distance is less than the upper limit; collision with another vehicle is prevented in the conventional methods. As shown in Fig. 9(b), PID results in acceleration beyond the limit, while the acceleration of MPC without penalty is suppressed by the hard constraint. However, MPC was not feasible when acceleration was more suppressed by the hard constraint whose lower limit is a = −0.981 m/s2 since the hard constraint on acceleration contradicts the hard constraint on running distance. Thus, the conventional methods cannot shape the acceleration trajectory.
Time t [s] (c) Jerk u.
Fig. 9. Simulation result with the conventional methods.
Figure 10(a), (b) and (c) depicts the vehicle running distance, acceleration and jerk of the proposed methods, respectively. According to Fig. 10(a), the running distance is converged to the upper limit; the vehicle stops behind another vehicle in both the proposed methods. Figure 10(b) shows that the acceleration with the proposed methods exceed the penalty in order to avoid collision with another vehicle. The deviations from the referential acceleration were suppressed about 59% and 70% by the proposed methods compared with MPC without penalty. The acceleration of MPC with the multiple slack variables is changed after 14 s where it takes minimum at the beginning and increases with time. This is due to the profile of the target acceleration which takes minimum after the cut-in and then it monotonously increases later. However, the acceleration of MPC with the constant slack variables is constant since its
predicted value is same through the horizon. Moreover, the jerk with the proposed methods are suppressed by the penalty as shown in Fig. 10(c). Thus, the acceleration is formed as the shape given in Fig. 1 by the proposed method. Therefore the proposed method can reduce the load on passengers while satisfying various limitations. 5. CONCLUSION We proposed a velocity control based on MPC to suppress acceleration and jerk for passengers while guaranteeing various limitations. Collision with other vehicles was prevented by 35
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REFERENCES Running distance d [m]
Aeberhard, M., Rauch, S., Bahram, M., Tanzmeister, G., Thomas, J., Pilat, Y., Homm, F., Huber, W., and Kaempchen, N. (2015). Experience, results and lessons learned from automated driving on germany’s highways. IEEE Intelligent Transportation Systems Magazine, 7(1), 42–57. Eichelberger, A.H. and McCartt, A.T. (2016). Toyota drivers’ experiences with dynamic radar cruise control, pre-collision system, and lane-keeping assist. Journal of Safety Research, 56, 67 – 73. He, D., He, W., Song, X., and Guo, H. (2017). Predictive cruise control of vehicles with pre-planned acceleration/deceleration command. In 2017 29th Chinese Control And Decision Conference (CCDC), 7558–7563. Ioannou, P. and Xu, Z. (1994). Throttle and brake control systems for automatic vehicle following. I V H S Journal, 1(4), 345–377. Liu, K., Gong, J., Kurt, A., Chen, H., and Ozguner, U. (2017). A model predictive-based approach for longitudinal control in autonomous driving with lateral interruptions. In 2017 IEEE Intelligent Vehicles Symposium (IV), 359–364. Luo, L.h., Liu, H., Li, P., and Wang, H. (2010). Model predictive control for adaptive cruise control with multiobjectives: comfort, fuel-economy, safety and car-following. Journal of Zhejiang University SCIENCE A, 11(3), 191–201. Peng, L. and Umit, O. (2015). Predictive control of a vehicle convoy considering lane change behavior of the preceding vehicle. In American Control Conference, 4374–4379. Chicago, IL. Shengbo, L., Keqiang, L., Rajesh, R., and Jianqiang, W. (2011). Model predictive multi-objective vehicular adaptive cruise control. IEEE Transactions on Control Systems Technology, 19(3), 556–566. Stanger, T. and del Re, L. (2013). A model predictive cooperative adaptive cruise control approach. In 2013 American Control Conference, 1374–1379. Wand, F., Sagawa, K., and Inooka, H. (2000). A study of the relationship between the longitudinal acceleration/deceleration of automobiles and ride comfort. The Japanese journal of ergonomics, 36(4), 191–200. (in Japanese). Yamamura, Y., Seto, Y., Nishira, H., and Kawabe, T. (2008). An acc design method for achieving both string stability and ride comfort. Journal of System Design and Dynamics, 2(4), 979–990. Zhang, J., Li, Q., and Chen, D. (2018). Integrated adaptive cruise control with weight coefficient self-tuning strategy. Applied Sciences, 8(6). Zhao, R., Wong, P., Xie, Z., and Zhao, J. (2017). Realtime weighted multi-objective model predictive controller for adaptive cruise control systems. International Journal of Automotive Technology, 18(2), 279–292. Ziegler, J., Dang, T., Franke, U., Lategahn, H., Bender, P., Schreiber, M., Strauss, T., Appenrodt, N., Keller, C.G., Kaus, E., Stiller, C., and Herrtwich, R.G. (2014). Making bertha drive - an autonomous journey on a historic route. IEEE Intelligent Transportation Systems Magazine, 6(2), 8 – 20.
Time t [s]
2
Acceleration a [m/s ]
(a) Running distance d.
Time t [s]
3
Jerk u [m/s ]
(b) Acceleration a.
Time t [s]
(c) Jerk u.
Fig. 10. Simulation result with the proposed methods. the constraint of running distance. In this paper, we showed suppression of acceleration and jerk even when another vehicle cuts in the controlled vehicle. Hence, the vehicle can be expected to shape the acceleration trajectory by the proposed method. For the future work, we will conduct experiments using an actual vehicle. ACKNOWLEDGEMENTS The authors gratefully acknowledge the helpful comments of the reviewers. 36
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IFAC PapersOnLine 52-8 (2019) 31–36
Model Predictive Control for Model Predictive Control for Model Predictive Control for Model Predictive Control for Shaping Autonomous Vehicles with Speed Profile Model Predictive Control for Shaping Autonomous Vehicles with Speed Profile Autonomous Vehicles with Speed Profile Autonomous Vehicles with Speed Profile Shaping Shaping Autonomous Vehicles with Speed Profile Shaping ∗ ∗∗ ∗∗∗ Y. Mizushima ∗ I. Okawa ∗∗ K. Nonaka ∗∗∗
2[m/s Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration [m/s [m/s [m/s [m/s [m/s ] 22]]22]]2]
∗∗∗ Y. Mizushima ∗∗ I. Okawa ∗∗ K. Nonaka Y. Y. Mizushima Mizushima I. I. Okawa Okawa ∗∗ K. K. Nonaka Nonaka ∗∗∗ ∗ I. Okawa ∗∗ K. Nonaka ∗∗∗ Y.∗∗ Mechanics, MizushimaTokyo City University, University, Japan Japan ∗ Mechanics, Tokyo City Tokyo City ∗ Mechanics, Mechanics, Tokyo City University, University, Japan Japan (e-mail: [email protected]). (e-mail: [email protected]). ∗ Mechanics, (e-mail: [email protected]). ∗∗ Tokyo City University, Japan ∗∗ DENSO CORPORATION, Japan (e-mail: [email protected]). ∗∗ DENSO CORPORATION, Japan DENSO CORPORATION, Japan ∗∗ (e-mail: [email protected]). (e-mail: isao [email protected]). DENSO CORPORATION, Japan (e-mail: isao [email protected]). ∗∗ DENSO isao [email protected]). ∗∗∗ Faculty(e-mail: CORPORATION, Japan (e-mail: isao [email protected]). ∗∗∗ of Engineering, Tokyo City University, Japan ∗∗∗ Faculty of Engineering, Tokyo City University, of (e-mail: Engineering, Tokyo City University, Japan Japan ∗∗∗ Faculty(e-mail: [email protected]) [email protected]). Faculty of Engineering, Tokyo City University, Japan [email protected]) ∗∗∗ Faculty of (e-mail: (e-mail: [email protected]) Engineering, Tokyo City University, Japan (e-mail: [email protected]) (e-mail: [email protected]) Abstract: In recent years, automatic driving has been widely studied which is expected to promote Abstract: recent years, driving has been studied is to Abstract:ofIn Intraffic recentaccidents. years, automatic automatic driving has control, been widely widely studied which which is expected expected to promote promote reduction Focusing on velocity it is sometimes required to decelerate greatly Abstract: In recent years, automatic driving has been widely studied which is expected to promote reduction of traffic accidents. Focusing on velocity control, it is sometimes required to decelerate greatly reduction of traffic accidents. Focusing on velocity control, it is sometimes required to decelerate greatly Abstract: In recent years, automatic driving has been widely studied which is expected to promote reduction of traffic accidents. Focusing on velocity control, it is sometimes required to decelerate greatly to prevent prevent collision collision with with other other vehicles. vehicles. On On the the other other hand, hand, excessive excessive acceleration acceleration or or deceleration deceleration is to is to prevent collision with other vehicles. On the other hand, excessive acceleration or deceleration is reduction ofcollision traffic accidents. Focusing onfor velocity control, it ispaper sometimes required tomethod decelerate greatly to prevent with other vehicles. On the other hand, excessive acceleration or deceleration is undesirable to provide comfortable ride passengers. This presents a novel to reshape undesirable to provide comfortable ride for passengers. This paper presents a novel method to reshape undesirable to provide comfortable ride for passengers. This paper presents a novel method to reshape to prevent collision with other vehicles. On the other hand, excessive acceleration or deceleration is the profile of vehicle velocity based on model predictive control. Introducing combined hard and soft undesirable to provide comfortable ride for passengers. This paper presents a novel method to reshape the profile of vehicle velocity based on model predictive control. Introducing combined hard and soft the profile of vehicle velocity based on model predictive control. Introducing combined hard and soft undesirable to provide comfortable ride for passengers. This paper presents a novel method to reshape constraints, and sudden is suppressed while aa collision with other vehicles is the profile ofexcessive vehicle velocity basedacceleration on model predictive control. Introducing combined hard and soft constraints, excessive and sudden acceleration is suppressed while collision with other vehicles is constraints, and sudden acceleration is suppressed while aa numerical collision with other vehicles is the profile The ofexcessive vehicle velocity based on model predictive control. Introducing hard and soft constraints, excessive andof sudden acceleration is is suppressed while collisioncombined with other vehicles is prevented. advantage the proposed method verified through simulation supposing prevented. The advantage of the proposed method is verified through a numerical simulation supposing prevented. The advantage of the proposed method is verified through a numerical simulation supposing constraints, excessive andof sudden acceleration isinis suppressed whileofaathe collision with other vehicles is aaprevented. practical situation where an another vehicle cuts the front space controlled vehicle. It is shown The advantage the proposed method verified through numerical simulation supposing situation where an another vehicle cuts in front space the controlled vehicle. It is a practical practical situation whereproperly anthe another vehicle cuts inisthe the front spacesoof ofthat controlled vehicle. Itsupposing is shown shown prevented. The advantage of proposed method verified through athe numerical simulation that the proposed method shapes the acceleration profile it adapts to such situations. athat practical situation where an another vehicle cuts in the front space of the controlled vehicle. It is shown the method properly shapes the acceleration profile so that it adapts the proposed proposed method shapes thecuts acceleration profile that adapts to to such such situations. situations. athat practical situation whereproperly an another vehicle in the front spaceso theit It is shown that the proposed method properly shapes the acceleration profile soofthat itcontrolled adapts to vehicle. such situations. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier All rights reserved. that the proposed method properly shapes the acceleration profile so that it adaptsLtd. to such situations. Keywords: Automatic Automatic driving, driving, Model Model predictive predictive control, control, Speed Speed control, control, Preventing Preventing collision, collision, Speed Speed Keywords: Keywords: Automatic driving, Model predictive control, Speed control, Preventing collision, Speed profile Keywords: Automatic driving, Model predictive control, Speed control, Preventing collision, Speed profile profile Keywords: Automatic driving, Model predictive control, Speed control, Preventing collision, Speed profile profile 1. INTRODUCTION Range 1 Range 2 Range Range 33 1. Range 1. INTRODUCTION INTRODUCTION Range 1 1 Range Range 2 2 Range 33 1. INTRODUCTION Range 1 Range 2 Range 1. INTRODUCTION Range 1 Range 2 Range 3 0.0 0.0 0.0 In recent recent years, years, automatic automatic driving driving has has been been widely widely studied studied 0.0 In In recent years, automatic driving has been widely studied 0.0 which is expected expected to increase increase fuel efficiency, efficiency, reduce driver’s driver’s In recent years, automatic driving has been widely studied which is to fuel reduce which is expected to increase fuel efficiency, reduce driver’s In recent years, automatic driving has been widely studied load and prevent traffic accidents, for example. Lane trackwhich is expected to increase fuel efficiency, reduce driver’s load and prevent traffic accidents, for example. Lane trackload and prevent traffic accidents, for example. Lane trackwhich isadaptive expected to increase fuel efficiency, reduce loadand and preventcruise traffic accidents, forhas example. Lanedriver’s tracking and cruise control (ACC) has already been impleing adaptive control (ACC) already been impleTime [s] ing and adaptive cruise control (ACC) has already been impleTime [s] load andon prevent traffic accidents, forhas example. Lane impletracking and adaptive cruise control (ACC) already been mented highways (Aeberhard et al., 2015; Eichelberger and Time mented on highways (Aeberhard et al., 2015; Eichelberger and Time [s] [s] mented on highways (Aeberhard et al., 2015; Eichelberger and ing and adaptive cruise control (ACC) has already been impleMcCartt, 2016). In the future, automatic driving is expected mented on highways (Aeberhard et al., 2015; Eichelberger and Time [s] manipulates the brake McCartt, 2016). In the future, automatic driving is expected Fig. 1. Referential deceleration. A driver McCartt, the future, automatic is expected Referential deceleration. A manipulates mented on2016). highways (Aeberhard et al., 2015; and Fig. Fig. 1. 1. Referential deceleration. A driver driver manipulates the the brake brake to be be available available inIn urban areas (Ziegler (Ziegler et driving al.,Eichelberger 2014). In these these McCartt, 2016). Inurban the future, automatic driving is expected to in areas et al., 2014). In pedal only in Range 1 or Range Fig. 1. Referential deceleration. A driver to be available in urban areas (Ziegler et al., 2014). In these pedal only only in in Range Range 11 or or Range Range 3. 3. manipulates the brake McCartt, 2016). Inurban the future, automatic driving isaccording expected to be available in areas (Ziegler et al., 2014). In these pedal 3. situations, it is required to change speed frequently Fig. 1. Referential deceleration. A driver manipulates the brake situations, it is required to change speed frequently according pedal only in Range 1 or Range 3. situations, it lights is required toareas change speedet frequently to be available in urban (Ziegler al., 2014). In these deceleration and cutting-in caused by surrounding vehicles to the the traffic and the the surrounding vehicles. Onaccording the other den situations, it lights is required to change speedvehicles. frequently according pedal only in Range 1 or Range 3. den deceleration and cutting-in caused by surrounding vehicles to traffic and surrounding On the other to the ittraffic and the surrounding vehicles. Onaccording the other den deceleration and cutting-in caused by surrounding vehicles situations, it lights is required to change frequently should be studied for practical application. hand, is important important to realize realize properspeed acceleration and decelerato the ittraffic lights and the surrounding vehicles. On decelerathe other den deceleration and cutting-in caused by surrounding vehicles should be studied for practical application. hand, is to proper acceleration and hand, it is important to realize proper acceleration and decelerashould be studied for practical application. to the traffic lights and the surrounding vehicles. On the other den deceleration and cutting-in caused by surrounding vehicles tion because it is directly related to riding comfort (Wand et al., hand, it is important to realize proper acceleration and decelerashould be studied for practical application. tion because it is directly related to riding comfort (Wand et al., This paper presents acceleration/deceleration trajectory genertion because it is directly related to riding comfort (Wand et al., hand, it is important to realize proper acceleration and decelerashould be studied for practical application. This paper presents acceleration/deceleration trajectory genertion because it is directly related to riding comfort (Wand et al., 2000). This paper presents acceleration/deceleration trajectory gener2000). ation which achieves both reducing load on passengers and 2000). This paper presents acceleration/deceleration trajectory genertion because it is directly related to riding comfort (Wand et al., ation which achieves both reducing load on passengers and 2000). ation which achieves both reducing load on passengers and This paper presents acceleration/deceleration trajectory generconsidering the motion of surrounding vehicles. The proposed In previous researches, speed control has been proposed (Ioannou ation which achieves both reducing load on passengers and 2000). considering the motion of surrounding vehicles. The proposed In previous researches, speed control has been proposed (Ioannou In previous researches, speed control has been proposed (Ioannou considering the motion of surrounding vehicles. The proposed ation which achieves both reducing load ondistance passengers and In previous researches, speed control has been proposed (Ioannou considering the motion of surrounding vehicles. The proposed method firstly designs the target velocity and based on and Xu, 1994; Yamamura et al., 2008) in order to suppress method firstly designs the target velocity and distance based and Xu, 1994; Yamamura et al., 2008) in order to suppress and Xu, 1994; Yamamura etACC al., with 2008) in order to suppress method firstly designs the target velocity and road distance based on on In previous researches, speed control has model been proposed (Ioannou considering the(CC) motion of surrounding vehicles. The proposed cruise control and ACC. CC considers environment both acceleration and jerk. predictive conand Xu, 1994; Yamamura et al., 2008) in order to suppress method firstly designs the target velocity and distance based on cruise control (CC) and ACC. CC considers road environment both acceleration and jerk. ACC with model predictive conboth acceleration and jerk. ACC with model predictive concruise control (CC) and ACC. CC considers road environment and Xu, 1994; Yamamura et al., 2008) in order to suppress method firstly designs the target velocity and distance based on such as deceleration at crossroad. ACC considers a motion trol (MPC) has been proposed (Luo et al., 2010; Stanger and both acceleration and jerk. ACC with model predictive concruise control (CC) and ACC. CC considers road environment such as as deceleration deceleration at at crossroad. crossroad. ACC ACC considers considers aa motion motion of of trol (MPC) has been proposed (Luo et al., 2010; Stanger and trol (MPC) has been proposed (Luo et al., 2010; Stanger and such of both acceleration and jerk. ACC with model predictive concruise control (CC) and ACC. CC considers road environment surrounding vehicles such as sudden deceleration cuttingdel Re, Re, 2013; Peng and Umit, 2015; 2015; He et al., al., 2017; Liu et et and al., such trol (MPC) hasPeng beenand proposed (Luo He et al., 2010; Stanger as deceleration at crossroad. ACC considers and a motion of surrounding vehicles such as sudden deceleration and cuttingdel 2013; Umit, et 2017; Liu al., del Re, 2013; Peng and Umit, 2015; He et al., 2017; Liu et al., surrounding vehicles such as sudden deceleration and cuttingtrol (MPC) has been proposed (Luo et al., 2010; Stanger asoptimal deceleration at crossroad. ACC considers abymotion of del Re, 2013; and Umit, 2015; He etwith al., 2017; Liuvehicle et and al., such surrounding vehicles such astrajectory sudden deceleration and cuttingin. An acceleration is generated MPC so 2017; Zhao etPeng al., 2017), where collision with another vehicle in. An optimal acceleration trajectory is generated by MPC so 2017; Zhao et al., 2017), where collision another 2017; Zhao etPeng al., 2017), where collision another vehicle An optimal acceleration is generated bytracks MPC so del Re, 2013; and Umit, 2015; He etwith al., 2017; Liu et al., in. surrounding vehicles such astrajectory sudden deceleration and cuttingthat the velocity profile is properly tuned as well as it the has been prevented by constraints on inter-vehicular distance. 2017; Zhao et al., 2017), where collision with another vehicle in. An optimal acceleration trajectory is generated by MPC so that the velocity profile tuned as as tracks the has been prevented by on distance. has been prevented by constraints constraints on inter-vehicular inter-vehicular distance. that thevelocity. velocityFigure profile1is is properly properly tuned as well well as it it the 2017; Zhao et al., 2017), where withusing another vehicle in. An optimal acceleration trajectory is generated bytracks MPCtraso target the referential acceleration Those methods suppress load oncollision passengers constraints has been prevented by constraints on inter-vehicular distance. that thevelocity. velocityFigure profile1isdepicts properly as well as it tracks the target depicts thetuned referential acceleration traThose methods suppress load on passengers using constraints Those methods suppress load on passengers using constraints target velocity. Figure 1 depicts the referential acceleration trahas been prevented by constraints on inter-vehicular distance. that the velocity profile is properly tuned as well as it tracks the jectoryvelocity. during deceleration, deceleration, where the brake pedal pedal is depressed depressed on acceleration acceleration and jerk. load However, constraintsusing on acceleration acceleration Those methods and suppress on passengers constraints target Figure 1 depicts thethe referential acceleration trajectory during where brake is on jerk. However, constraints on on acceleration and However, constraints on during deceleration, where the brake pedal is depressed Those methods suppress load onfor passengers using constraints target velocity. Figure 1 depicts the referential acceleration traon jerk acceleration and jerk. However, constraints on acceleration acceleration jectory during deceleration, where the brakein pedal is 3. depressed in Range 11 and the brake pedal is released Range Unnecor jerk contradict thejerk. constraint inter-vehicular distance; it jectory in Range and the brake pedal is released in Range 3. Unnecor contradict the constraint for inter-vehicular distance; it or jerk contradict thejerk. constraint for constraints inter-vehicular distance; it in Range 1 anddeceleration, theacceleration brake pedal is released Range Unnecon acceleration and However, on acceleration jectory during where the discomfort brakein pedal is 3. depressed essary change of causes on passenger is sometimes difficult to obtain control parameters which sator jerk contradict the constraint for inter-vehicular distance; it in Range 1 and the brake pedal is released in Range 3. Unnecessary change of acceleration causes discomfort on passenger is sometimes difficult to parameters which satis difficult to obtain obtain control control parameters which satessary change acceleration causes discomfort on passenger or jerk contradict the constraint for inter-vehicular it in Range 1 andof brake pedal is released in Range 3. Unnecand deterioration of driving performance due to aa fluctuation isfysometimes different kind of constraints constraints simultaneously. Todistance; overcome is sometimes difficult to obtain control parameters which satessary change oftheacceleration causes discomfort on passenger and deterioration of driving performance due to fluctuation isfy different kind of simultaneously. To overcome isfy different kind of constraints simultaneously. To overcome and deterioration of driving performance due to a fluctuation is sometimes difficult to obtain control parameters which satessary change of acceleration causes discomfort on passenger isfy different kind of constraints simultaneously. To overcome and deterioration of driving performance due to a fluctuation of load load distribution. distribution. Thus, Thus, the the referential referential acceleration acceleration is is conconthis difficulty, difficulty, ACC ACC with with relaxed relaxed acceleration acceleration constraints constraints has has of this this difficulty, ACC acceleration load distribution. Thus, the referential acceleration is isfy ofwith constraints simultaneously. To overcome and deterioration of driving performance due totoa realize fluctuation this different difficulty, ACC with relaxed acceleration constraints has of load distribution. Thus, the referential acceleration is conconstant excluding Range 11 and Range 3. In order the been proposedkind (Shengbo etrelaxed al., 2011; 2011; Zhang et etconstraints al., 2018).has In of stant excluding Range and Range 3. In order to realize the been proposed (Shengbo et al., Zhang al., 2018). In been proposed (Shengbo al., 2011; Zhang acceleration etconstraints al., 2018).and In stant excluding Range 1depicted and 3. In order toimpose realize the this difficulty, ACC with et relaxed acceleration has of load distribution. Thus, theRange referential acceleration is conreferential acceleration in Fig. 1, we also soft these methods, both suppression of excessive been proposed (Shengbo et al., 2011; Zhang et al., 2018). In stant excluding Range 1 and Range 3. In order to realize the referential acceleration depicted in Fig. 1, we also impose soft these methods, both suppression of excessive acceleration and these methods, both suppression of excessive acceleration and referential acceleration depicted in Fig. 1, we also impose soft been proposed (Shengbo et al., 2011; Zhang et al., 2018). In stant excluding Range 1 and Range 3. In order to realize the constraints on acceleration and jerk, where the constraints are prevention of collision are considered. But the speed profile these methods, both suppression of excessive acceleration and referential acceleration depicted in Fig. 1, we also impose soft constraints on acceleration and jerk, where the constraints are prevention of collision are considered. But the speed profile prevention of collision are considered. But the speed profile constraints on acceleration and jerk, where the constraints are these methods, both suppression of excessive acceleration and referential acceleration depicted in Fig. 1, we also impose soft prevention of collision are considered. But the speed profile constraints on acceleration and jerk, where the constraints are relaxed utilizing utilizing slack slack variables. variables. In In addition, addition, we we impose impose aa hard hard has not not been been designed designed explicitly explicitly which which possibly possibly affects affects the the relaxed has has not been designed explicitly which possibly affects the relaxed utilizing slack variables. In addition, we impose a hard prevention of designed collision are considered. But the speed profile constraints onan acceleration and jerk, where the constraints are has not been explicitly which possibly affects the relaxed utilizing slack variables. In addition, we impose a hard constraint on upper limit of running distance to avoid colcomfortability of passengers. In addition, a motion such as sudconstraint on limit distance to colcomfortability of In a motion as comfortability of passengers. passengers. In addition, addition, motion such such as sudsudconstraint on an an upper uppervariables. limit of of running running distance to avoid avoid colhas not been designed explicitly whichaapossibly affects the relaxed utilizing In addition, we impose a hard comfortability of passengers. In addition, motion such as sudconstraint on an slack upper limit of running distance to avoid colcomfortability of IFAC passengers. In addition, a motion such as Control) sud- constraint on an upper limit ofreserved. running distance to avoid col2405-8963 © 2019, (International Federation of Automatic Hosting by Elsevier Ltd. All rights
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State of leader vehicle (Δ𝑆𝑆, 𝑣𝑣p )
lision with other vehicles, so that the proposed method shapes the acceleration trajectory while a collision with other vehicle is prevented. We verify the advantage of the proposed method in a simulation where another vehicle cuts in on the front. We compare the proposed method with a method without trajectory shaping to show that the trajectory is properly shaped in the situation.
ACC
CC
Adaptive Cruise Control Cruise Control (𝑣𝑣t,acc , 𝑑𝑑acc ) (𝑣𝑣t,cc , 𝑑𝑑cc , 𝑣𝑣cc ) Combination function
2. SPEED CONTROL SYSTEM
Target state of MPC (𝑑𝑑t , 𝑣𝑣t , 𝑎𝑎t ), Constraint value (𝑑𝑑, 𝑣𝑣)
2.1 Overview Figure 2 depicts a model of the control system for the longitudinal velocity used in this study. The system adapts various situations, such as cruising, acceleration/deceleration at crossroad, deceleration of a leader vehicle and cutting-in by another vehicle. The system is divided into the upper and the lower level controller. In the upper level controller, optimal vehicle running distance d, velocity v and acceleration a are calculated. The upper level controller consists of CC, ACC, combination function of them and MPC. CC generates a target velocity and a movable distance based on the static environment of the road. ACC generates a target velocity which keeps a safe inter-vehicular distance and an upper bound distance which prevents collision with a leading vehicle. Based on the output of ACC and CC, the combination function determines the target state and the upper limit of running distance, which are used in MPC to calculate an optimal input. Details of these functions are described in later sections. This flow is repeated at each control cycle to obtain an optimal vehicle velocity. In the lower layer controller, the engine output is controlled so that the actual velocity follows the optimal velocity. This study focuses only on the upper layer controller which mainly determines the speed profile of running vehicles.
Model Predictive Control Vehicle model
Optimal vehicle state (𝑑𝑑, 𝑣𝑣, 𝑎𝑎)
Upper level controller
Lower level controller
Engine throttle Actual vehicle
Fig. 2. The speed control system consisting of the upper and lower level controller. The upper level controller generates the optimal vehicle behavior. The lower level controller tracks the optimal behavior.
Running distance along the reference path 𝑑𝑑
2.2 Vehicle Model We focus on longitudinal vehicle motion as depicted in Fig. 3, where d is the running distance along the reference path and v is the vehicle velocity, respectively. We define vehicle acceleration a and state vector x := [d, v, a]T . We also define input u of the vehicle jerk. Each state has relationships described as d˙ = v, v˙ = a and a˙ = u. A state equation of continuous-time system is described as follows: ] [ ] [ 010 0 x˙ = 0 0 1 x + 0 u. (1) 000 1
Velocity 𝑣𝑣
Fig. 3. Vehicle model considering longitudinal motion. vehicle decelerates/accelerates near crossroad, the target velocity is designed to be with a constant acceleration; this target velocity expects both avoiding inappropriate change of steering characteristics and improving ride comfort. The duration required for acceleration/deceleration ∆td is given as follows: ve − vs ∆td = , (3) ar where vs is initial velocity, ve is terminal velocity and ar is target acceleration, respectively. Start time of deceleration ts is given as follows: 1 dr + ar ∆td2 2 , (4) ts = vs where dr is running distance to the stop line. Finish time of deceleration te is calculated by te = ts + ∆td . In addition, the maximum velocity v¯cc [k] is generated to drive within the limit of velocity. In cruise section, v¯cc [k] is the maximum velocity. In crossroad, v¯cc [k] is 0.5 km/h so that the vehicle can stop immediately. Figure 4(a) depicts the target velocity and the
Using zero-order hold, the state equation is represented by discrete-time system as follows: 3 ∆t /6 1 ∆t ∆2t /2 x[k + 1] = 0 1 ∆t x[k] + ∆2t /2 u[k], (2) 0 0 1 ∆t
where ∆t is the sampling time. 2.3 Cruse control (CC)
CC generates a target velocity and a movable distance based on static environment of the road such as crossroad. When the vehicle cruises at a constant velocity, the target velocity is usually set to be equivalent to the maximum speed. When the 32
(a) The vehicle accelerate.
𝑡𝑡e
Horizon with MPC 𝑣𝑣t,ACC
Time 𝑡𝑡 [s] Target Constraint
0
(b) The vehicle decelerate.
Fig. 4. Target and constraint velocity with CC. Controlled vehicle
Δ𝑆𝑆
𝑣𝑣
Target position
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Running distance [m]
Time 𝑡𝑡 [s] Target Constraint
𝑡𝑡s
Velocity [m/s]
Δ𝑡𝑡d
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Velocity [m/s]
Velocity [m/s]
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𝑣𝑣t,CC
Target of MPC 𝑣𝑣t Time 𝑡𝑡 [s]
(a) Target velocity.
𝑇𝑇H
0
Horizon with MPC
𝑑𝑑ҧCC
𝑑𝑑ҧACC
Upper limit of MPC 𝑑𝑑ҧ Time 𝑡𝑡 [s]
(b) Upper limit of distance
𝑇𝑇H
Fig. 6. Integrating with CC and ACC. Δ𝑆𝑆t
2.5 Combination of CC and ACC
Leader vehicle
The target state in MPC is generated by the velocity trajectory of CC and ACC. Figure 6(a) depicts combination the target velocity, where vt [k] is target velocity. Adopting one with the lower velocity at each time, the combine function generates vt [k]. The function integrates vt [k] with respect to time in order to obtain a target running distance dt [k]. In addition, the function differentiates vt [k] with respect to time in order to obtain a target acceleration at [k]. Since vt [k] is designed using (6) and (7), noise amplification due to differentiation is negligible.
𝑣𝑣p
Fig. 5. Relationship between controlled vehicle and leader vehicle. upper limit of velocity when the vehicle starts. Figure 4(b) depicts them when the vehicle decelerates to the stop line. In addition, the upper limit of running distance d¯cc is set at stop line to prevent running over the line.
The limit of running distance by CC and ACC are combined to generate the limit of running distance in MPC. Figure 6(b) depicts generation of the upper limit of running distance, where ¯ is the upper limit of running distance. Adopting one with d[k] the lower distance at each time, the combination function ¯ Also, the upper limit of velocity v[k] generates d[k]. ¯ is v¯cc [k].
2.4 Adaptive cruse control (ACC) In ACC, the controlled vehicle follows the leader vehicle while safety inter-vehicle distance is preserved. Figure 5 depicts the relationship between the leader vehicle and the controlled vehicle, where v is velocity of the controlled vehicle, vp is velocity of the leader vehicle and ∆S is actual inter-vehicle distance, respectively. The target inter-vehicle distance ∆St [k] is expressed as follows: ¯ (5) ∆St [k] = v[k]tTHW − (vp [k] − v[k])tTTC + ∆S,
3. MODEL PREDICTIVE SPEED CONTROL Preventing collision with other vehicles and following a limit speed are necessary. Automatic driving must prevent the acceleration which exceeds the limit of vehicle performance for guaranteeing safety. In addition, excessive acceleration and sudden acceleration should be avoided for comfortability of passengers. To satisfy these requests, we propose a velocity controller based on MPC, which calculates the optimal control input by minimizing an index function while satisfying constraints and guaranteeing the safety of the vehicle. Moreover, in order to suppress excessive acceleration that passenger feels uncomfortable, we introduce slack variables into the constraints which render constraints flexible while physical or legal limitations are satisfied by hard constraints.
where tTHW is the constant time-headway, tTTC is the constant ¯ = 5 m is inter-vehicle distance when time-to-collision and ∆S these vehicles stop. In (5), the first term shows the inter-vehicle distance in proportion to the velocity of the controlled vehicle. When the controlled vehicle is faster, the inter-vehicle distance is extended. The second term modifies the inter-vehicle distance depending on the difference in velocity of these vehicles. In this study, we set tTHW = 3.3 s so that inter-vehicle distance is 60 m when the vehicle drives at 60 km/h. In addition, we set tTTC = 0.3 s. Target acceleration at,acc [k] is expressed as follows: at,acc [k] = KS (∆S[k] − ∆St [k]) − Kv (v[k] − vp [k]),
3.1 Constraint condition
(6)
We impose various constraints on the state x and the input u to ensure safety and comfortability. To prevent running over the stop line, inequality constraint of running distance is given as: ¯ d[k] ≤ d[k], (8)
where Kd and Kv are constant gains. In addition, the target velocity vt,acc is expressed as follows: vt,acc [k + 1] = vt,acc [k] + at,acc [k]∆h , (7) where ∆h is time step of horizon in MPC. To obtain the target velocity trajectories, (6) and (7) are calculated in the horizon of the MPC. It is assumed that behavior of the leader vehicle can be predicted by map data: the vehicle decelerates to stop when a traffic light is red.
where d¯ is the upper limit of running distance. Collision with other vehicle can also be considered by (8). To follow the limit speed, inequality constraint on velocity is given as: ¯ (9) v[k] ≤ v[k] ≤ v[k],
In order to avoid collision with the leader vehicle, a limit of running distance d¯acc [k] is 5 m behind the leader vehicle.
where v¯ is the upper limit of velocity and v is the lower limit of velocity, respectively. To prevent acceleration exceeding a 33
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Controlled vehicle Another vehicle
Penalty range Max
Penalty [-]
Min
Red light on 𝑎𝑎
𝑎𝑎s
𝑎𝑎𝑠𝑠
Acceleration a [m/s2]
𝑎𝑎
Fig. 8. Simulation condition: the another vehicle cuts in the controlled vehicle while these vehicles decelerate for stop line.
Fig. 7. Penalty by slack value. The blue line is evaluation value and the orange area is penalty range.
where Q, R, Qa , Qu are weights and N is the number of prediction step. In (16), the first and second terms evaluates the deviation of the target state and the input value. The third term evaluates the deviation from the comfortable acceleration and the forth term evaluates the deviation from the comfortable jerk. Minimizing (16), excessive acceleration and jerk are suppressed while the magnitude of state tracking error and that of input are balanced.
performance limit, inequality constraint on acceleration is given as: a ≤ a[k] ≤ a, ¯ (10)
where a¯ is upper limit of acceleration and a is lower limit of acceleration, respectively. To protect engine and powertrain from sudden acceleration, inequality constraint on jerk is given as: u ≤ u[k] ≤ u, ¯ (11)
3.3 Formulation as an optimization problem
where u¯ is upper limit of jerk and u is lower limit of jerk, respectively.
The proposed MPC is formulated by optimization problem as follows: minimize J, with respect to u[k], σu [k] for (k = 0, 1, · · · , N − 1), σa [k] for (k = 1, 2, · · · , N), subject to (2), (8) ∼ (13).
In addition, we impose constraints which prevent uncomfortable acceleration. To suppress excessive and sudden acceleration, soft constraints with slack variables are given as: as − σa [k] ≤ a[k] ≤ a¯s + σa [k], (12) us − σu [k] ≤ u[k] ≤ u¯s + σu [k], (13)
When (8) or (9) cannot be achieved, slack variable σa [k] and σu [k] increase. Since these slack variables are minimized in the index function, the acceleration trajectory is approximated to the referential one as close as possible.
where σa is slack variable of acceleration, σu is slack variable of jerk, as , a¯s are the boundaries of comfortable deceleration and us , u¯s are the boundaries of comfortable jerk, respectively. It is noted that we distinguish the performance limit a¯ from boundary of comfortable acceleration a¯s . Since slack variable σa is an amount of deviation from as or a¯s , we require σa ≥ 0. The magnitude of slack variable σa is minimized in the evaluation function given later in (16) in order to suppress excessive acceleration. Figure 7 depicts the effect of (10) and (12) on the evaluation function which indicates that acceleration that passenger feels uncomfortable is suppressed while acceleration that exceeds performance limit is prevented. Slack variable σu is also amount of deviation from us or u¯s , we can give that σa ≥ 0. Owing to (10) and (13), excessive jerk can be suppressed while the physical limitation is achieved.
4. SIMULATION 4.1 Condition We conduct numerical simulations to verify advantage of the proposed controller. The simulation assumes an interference by another vehicle. Another vehicle decelerating at −1.47 m/s2 cuts in the controlled vehicle when the controlled vehicle decelerates at −0.981 m/s2 from 40 km/h as depicted in Fig. 8. We assume that the controlled vehicle cannot predict cutting-in by another vehicle.
3.2 Index function
To verify the performance, the proposed method is compared with MPC without penalty and PD controller. Input jerk of the PD controller is expressed as follows: ePD [k] − ePD [k − 1] , (17) uPD [k] = KP ePD [k] + KD ∆t
The deviation from the target state e[k] is represented as follows: (14) e[k] = x[k] − xt [k],
where
xt [k] := [dt [k] vt [k] at [k]] .
(15)
where KP is proportional gain, KD is derivative gain and ePD [k] := at [k] − a[k], respectively.
The evaluation function is represented as follows: J=
Moreover, the proposed method is compared with another method with constant soft constraint on (12) and (13). Another method has soft constraints of acceleration and jerk as follows: as − σa,cns ≤ a[k] ≤ a¯s + σa,cns , (18) us − σu,cns ≤ u[k] ≤ u¯s + σu,cns . (19)
1 N−1 1 ∑ (e[k]T Qe[k] + Ru[k]2 ) + 2e[N]T Qe[N] 2 k=0
N N 1 1 + Qa ∑ σa [k]2 + Qu ∑ σu [k]2 , 2 k=1 2 k=1
(16)
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Element
Symbol
Value
Maximum acceleration Minimum acceleration Maximum acceleration penalty Minimum acceleration penalty Maximum jerk penalty Minimum jerk penalty Weight of state Weight of input Weight of acceleration penalty Weight of jerk penalty Gain of inter-vehicular distance in ACC Gain of velocity in ACC Proportional gain of PD controller Derivative gain of PD controller
a¯ a a¯s as u¯s us Q R Qa Qu KS Kv KP KD
1.471 m/s2 −1.96 m/s2 0.981 m/s2 −0.981 m/s2 0.909 m/s3 −1.25 m/s3 diag(0.002, 0.05, 1.0) 0.005 600 900 0.4 0.35 4.0 0.1
Running distance d [m]
Table 1. Simulation parameters.
Time t [s]
(a) Running distance d.
2
Acceleration a [m/s ]
where σa,cns and σu,cns are slack variables which are same through the horizon. Since σa,cns and σu,cns are given in every prediction step, acceleration and jerk may deviate same amount at every prediction step. As with (16), the magnitude of slack variables σa,cns and σu,cns are minimized in index function in order to suppress excessive acceleration and jerk. In this paper, MPC which utilizes multiple slack variables in the horizon is called “Multiple slack variables”, while MPC which utilizes constant slack variables in the horizon is also called “Constant slack variables”. Table 1 shows the parameters of the simulation. Since sudden acceleration is more uncomfortable than excessive acceleration, the weight of the jerk penalty is set to be larger than the weight of acceleration penalty. To enable safety decelerate from 60 km/h, we set the horizon length 7.0 s and the number of the prediction step N = 30.
Time t [s]
(b) Acceleration a.
Jerk u [m/s ]
4.2 Results and discussion 3
Figure 9(a), (b) and (c) depicts the vehicle running distance, acceleration and jerk with the conventional methods, respectively. In Fig. 9(a), the running distance is less than the upper limit; collision with another vehicle is prevented in the conventional methods. As shown in Fig. 9(b), PID results in acceleration beyond the limit, while the acceleration of MPC without penalty is suppressed by the hard constraint. However, MPC was not feasible when acceleration was more suppressed by the hard constraint whose lower limit is a = −0.981 m/s2 since the hard constraint on acceleration contradicts the hard constraint on running distance. Thus, the conventional methods cannot shape the acceleration trajectory.
Time t [s] (c) Jerk u.
Fig. 9. Simulation result with the conventional methods.
Figure 10(a), (b) and (c) depicts the vehicle running distance, acceleration and jerk of the proposed methods, respectively. According to Fig. 10(a), the running distance is converged to the upper limit; the vehicle stops behind another vehicle in both the proposed methods. Figure 10(b) shows that the acceleration with the proposed methods exceed the penalty in order to avoid collision with another vehicle. The deviations from the referential acceleration were suppressed about 59% and 70% by the proposed methods compared with MPC without penalty. The acceleration of MPC with the multiple slack variables is changed after 14 s where it takes minimum at the beginning and increases with time. This is due to the profile of the target acceleration which takes minimum after the cut-in and then it monotonously increases later. However, the acceleration of MPC with the constant slack variables is constant since its
predicted value is same through the horizon. Moreover, the jerk with the proposed methods are suppressed by the penalty as shown in Fig. 10(c). Thus, the acceleration is formed as the shape given in Fig. 1 by the proposed method. Therefore the proposed method can reduce the load on passengers while satisfying various limitations. 5. CONCLUSION We proposed a velocity control based on MPC to suppress acceleration and jerk for passengers while guaranteeing various limitations. Collision with other vehicles was prevented by 35
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REFERENCES Running distance d [m]
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Time t [s]
2
Acceleration a [m/s ]
(a) Running distance d.
Time t [s]
3
Jerk u [m/s ]
(b) Acceleration a.
Time t [s]
(c) Jerk u.
Fig. 10. Simulation result with the proposed methods. the constraint of running distance. In this paper, we showed suppression of acceleration and jerk even when another vehicle cuts in the controlled vehicle. Hence, the vehicle can be expected to shape the acceleration trajectory by the proposed method. For the future work, we will conduct experiments using an actual vehicle. ACKNOWLEDGEMENTS The authors gratefully acknowledge the helpful comments of the reviewers. 36