- Email: [email protected]
A Stochastic Hybrid Structure for Predicting Disturbances in Mixed Automated and Human-Driven Vehicular Scenarios
2nd IFAC Conference on Cyber-Physical & Human-Systems 2nd IFAC on Cyber-Physical & Human-Systems Miami, FL,Conference USA, Dec. 14-15, 2018 2nd IFAC on Cyber-Physical & Human-Systems Miami, FL,Conference USA, Dec. 14-15, 2018 Available online at www.sciencedirect.com 2nd IFAC Conference on Cyber-Physical & Human-Systems Miami, FL,Conference USA, Dec. 14-15, 2018 2nd IFAC on Cyber-Physical & Human-Systems Miami, FL, USA, Dec. 14-15, 2018 Miami, FL, USA, Dec. 14-15, 2018
ScienceDirect
IFAC PapersOnLine 51-34 (2019) 400–402
A Stochastic Hybrid Structure for Predicting Disturbances in Mixed Automated A Stochastic Hybrid Structure for Predicting Disturbances in Mixed Automated A Stochastic Hybrid Structure for Predicting Disturbances Mixed Automated and Human-Driven Vehicular Scenariosin A Structure for Predicting Disturbances Mixed Automated Human-Driven Vehicular Scenariosin A Stochastic Stochastic Hybrid Hybridand Structure for Predicting Disturbances in Mixed Automated and Human-Driven Vehicular Scenarios and Human-Driven Vehicular Scenarios Hossein Nourkhiz Mahjoub*, Mohammadreza Davoodi**, Yaser P. Fallah*, Javad M. Velni** and Human-Driven Vehicular Scenarios
Hossein Nourkhiz Mahjoub*, Mohammadreza Davoodi**, Yaser P. Fallah*, Javad M. Velni** Hossein Hossein Nourkhiz Nourkhiz Mahjoub*, Mahjoub*, Mohammadreza Mohammadreza Davoodi**, Davoodi**, Yaser Yaser P. P. Fallah*, Fallah*, Javad Javad M. M. Velni** Velni** *UniversityMohammadreza of Central Florida, Orlando, Yaser FL, 32816, USA Javad M. Velni** Hossein Nourkhiz Mahjoub*, Davoodi**, P. Fallah*, *University of Central Florida, Orlando, FL, 32816, USA (e-mail: hnmahjoub @knights.ucf.edu, [email protected]). *University of Florida, Orlando, FL, 32816, USA (e-mail: hnmahjoub @knights.ucf.edu, [email protected]). *University of Central Central Florida, Orlando, 32816, **University of Georgia, Athens, GA, FL, 30602, USAUSA *University of Central Florida, Orlando, FL, 32816, USA (e-mail: hnmahjoub @knights.ucf.edu, [email protected]). (e-mail: [email protected], **University Georgia, Athens, [email protected]). GA, [email protected]) 30602, USA (e-mail:[email protected], (e-mail: hnmahjoub @knights.ucf.edu, [email protected]). **University of Georgia, Athens, GA, 30602, USA (e-mail:[email protected], **University of Georgia, Athens, GA, [email protected]) 30602, USA **University of Georgia, Athens, GA, 30602, USA (e-mail:[email protected], (e-mail:[email protected], [email protected]) [email protected]) [email protected]) Abstract: In this work,(e-mail:[email protected], we introduce a stochastic prediction method which can be utilized in applications Abstract: In this work, we introduce a stochastic prediction method which can be movements. utilized in applications such as cooperative adaptive cruise control (CACC) to predict interfering vehicles’ One of the Abstract: In this work, we introduce aa stochastic prediction method which can be utilized in such as cooperative adaptive cruise control (CACC) to predict interfering vehicles’ movements. One of the Abstract: In this work, we introduce stochastic prediction method which can be utilized in applications applications main criteria in the design of automated vehicle systems is their robustness against the disturbances resulted Abstract: In this work, we introduce a stochastic prediction method which can be utilized in applications such as cooperative adaptive cruise control (CACC) to predict interfering vehicles’ movements. One of main criteria in the design of automated vehicle systems is their robustness against the disturbances resulted such adaptive of cruise (CACC) to predictThe interfering vehicles’ movements. One to of the the from as thecooperative non-homogeneity the control vehicular environment. non-homogeneity isdisturbances mainly due the such as cooperative adaptive cruise control (CACC) to predict interfering vehicles’ movements. One of the main criteria in the design of automated vehicle systems is their robustness against the resulted from the non-homogeneity of the vehicular environment. The non-homogeneity is mainly due to the main criteria in and the design of automated vehicle systemsco-existence. is their robustness against the disturbances resulted human-driven automated/autonomous vehicles It is therefore imperative for the main criteria in the design of automated vehicleenvironment. systems is their robustness against theisdisturbances resulted from the non-homogeneity of the vehicular The non-homogeneity mainly due to the from the non-homogeneity of the vehicular environment. The non-homogeneity is mainly due to human-driven and automated/autonomous vehicles co-existence. It is therefore imperative for the automated applications to beofdesigned with the capability ofThe handling the uncertain of humanfrom the non-homogeneity the vehicular environment. non-homogeneity isbehaviors mainly due to human-driven and automated/autonomous vehicles co-existence. is therefore imperative for human-driven and automated/autonomous vehicles co-existence. Itfor isvehicle therefore imperative for the the automated applications to bemanner. designedThis withpaper the capability of handlingIt the uncertain behaviors of humandriven vehicles in a robust presents a method movements time-series human-driven and automated/autonomous vehicles co-existence. Itthe is uncertain therefore behaviors imperative for the automated applications to be designed with the capability of handling of humanautomated applications to be designed with the capability of handling the uncertain behaviors of humandriven vehicles in a robust manner. This paper presents a method for vehicle movements time-series forecasting using a powerful Bayesian inference method, Gaussian Processes. The automated applications to benon-parametric designedThis with the capability of handling thenamely uncertain behaviors of humandriven vehicles aa robust manner. presents aa method for vehicle movements time-series driven vehicles ina powerful robust manner. using This paper paper presents method for vehicle movements time-series forecasting usingin non-parametric Bayesian inference method, namely Gaussian Processes. The proposed methodology is evaluated realistic vehicle trajectory data from NGSIM dataset and is driven vehicles in a robust manner. This paper presents a method for vehicle movements time-series forecasting using powerful non-parametric Bayesian inference method, namely Gaussian Processes. forecasting using aamore powerful non-parametric Bayesian inference method, namely Gaussian Processes. The proposed methodology is evaluated using realistic vehicle trajectory data from NGSIM dataset andThe is shown to provide accurate results compared to baseline methods that use constant velocity coasting. forecasting using a powerful non-parametric Bayesian inference method, namely Gaussian Processes. The proposed methodology is using realistic data from NGSIM dataset and shown to provide more accurate results compared to vehicle baselinetrajectory methods that constant velocity coasting. proposed methodology is evaluated evaluated using realistic vehicle trajectory datause from NGSIM dataset and is is proposed methodology is evaluated using realistic vehicle trajectory data from NGSIM dataset and is Keywords: Discrete Hybrid Stochastic Automata, Model Predictive Control, Non-parametric Bayesian shown to provide more accurate results compared to baseline methods that use constant velocity coasting. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. shown to provide more accurate results compared to Model baselinePredictive methods that use constant velocity coasting. Discrete Hybrid Stochastic Automata, Control, Non-parametric Bayesian Keywords: shown to provide more accurate results compared baseline methods that use constant velocity coasting. Inference, Gaussian Processes, Vehicular Networks, to Model-Based Communication Discrete Hybrid Automata, Model Control, Keywords: Inference, Gaussian Vehicular Networks, Communication Keywords: DiscreteProcesses, Hybrid Stochastic Stochastic Automata,Model-Based Model Predictive Predictive Control, Non-parametric Non-parametric Bayesian Bayesian DiscreteProcesses, Hybrid Stochastic Automata,Model-Based Model Predictive Control, Non-parametric Bayesian Keywords: Inference, Gaussian Vehicular Networks, Communication Inference, Gaussian Processes, Vehicular Networks, Model-Based Communication Inference, Gaussian Processes, Vehicular Networks, Model-Based Communication These models could be generated either in the vehicle which is 1. INTRODUCTION and SYSTEM DESCRIPTION These models could (fully be generated either in is directly controlled or partially) bythe thevehicle humanwhich driver 1. INTRODUCTION and SYSTEM DESCRIPTION These models could be generated either in the vehicle which directly controlled (fully or partially) by the human driver models could be agents. generated either in the vehicle which is is 1. and SYSTEM DESCRIPTION itself or also in target The former case is considered Plausible disturbances from different sources such as manned These These models could be agents. generated either in the vehicle which is 1. INTRODUCTION INTRODUCTION and SYSTEM DESCRIPTION directly controlled (fully or partially) by the human driver itself or also in target The former case is considered Plausible disturbances from different sources such as manned directly controlled (fully or partially) by the human driver 1. INTRODUCTION and SYSTEM DESCRIPTION here, since it is more effective if the models are derived by the vehicles or vulnerable road users (VRUs) should be directly controlled (fully or partially) by case the human driver itself or also in target agents. The former is considered Plausible disturbances from different sources such as manned vehicles or vulnerable users (VRUs) should be here, itself since or also target agents. The case is considered it isinmore effective the models are derived the Plausible disturbances fromroad different such as manned manned vehicle and thenif be former transmitted overby the appropriately considered during thesources design phase of the itself since or also inmore target agents. The former case is considered Plausible disturbances fromroad different sources such as manned here, it is effective if the models are derived by the vehicles or vulnerable users (VRUs) should be manned vehicle and then be transmitted over the appropriately considered during the design phase of the here, since it is more effective if the models are derived by vehicles or vulnerable road users (VRUs) should be network for other vehicles which need these distributed vehicular systems, such as cooperative here, sincevehicle it is more effective if the models are derived by vehicles orautomated vulnerable road users (VRUs) should be communication manned and then be transmitted over the appropriately considered during the design phase of the manned vehicle andhas then transmitted overinthese the communication network for otherbeproposed vehicles which need appropriately considered duringsystems, the design phase of the models. distributedcruise automated vehicular such as cooperative This notion been recently the adaptive control (CACC), platooning management manned vehicle andhas then beproposed transmitted overinthese the appropriately considered duringsystems, the design phase of the models. communication network for other vehicles which need distributed automated vehicular such as cooperative This notion been recently the adaptive cruise control (CACC), platooning management communication network for other vehicles which need these distributed automated vehicular systems, such as cooperative vehicular literature as Model-Based Communication (MBC) (merging/splitting) systems, etc. Although partially automated communication network for other vehicles which need these distributed automated vehicular systems, such as cooperative models. This notion has been proposed recently in the adaptive cruise control (CACC), platooning management vehicular literature as Model-Based Communication (MBC) (merging/splitting) systems, etc. Although partially automated models. This notion has been proposed recently in the adaptive cruise control (CACC), platooning management [1], [2], [3], [4].notion Theasbetter of the MBC (MBC) against and fully cruise autonomous vehicles are expected to management have a high models. This has performance been proposed recently in the adaptive control (CACC), platooning vehicular literature Model-Based Communication (merging/splitting) systems, etc. Although partially automated [1], [2], [3], [4]. The better performance of the MBC against and fully autonomous vehicles are expected to have a high vehicular literature as Model-Based Communication (MBC) (merging/splitting) systems, etc. Although partially automated the latter case is due to the availability of more accurate raw penetration rate after being commercialized attoanhave affordable vehicular literature asto Model-Based Communication (MBC) (merging/splitting) systems, etc. Although partially automated [1], [2], [4]. The better performance the MBC against and fully autonomous vehicles are aa high case isthe duehost the availability ofof accurate raw penetration rate after being commercialized attoanhave affordable [1], latter [2], [3], [3], [4]. The better performance ofmore the imperfect MBC against and fully autonomous vehicles are expected expected high the information for vehicle rather than the raw price, they will definitely co-exist with manned (human [1], [2], [3], [4]. The better performance of the MBC against and fully autonomous vehicles are expected to have a high the latter case is due to the availability of more accurate raw penetration rate after being commercialized at an affordable information for the host vehicle rather than the imperfect raw price, they will definitely co-exist with manned (human the latter case is due to the availability of more accurate penetration rate after commercialized atThis an affordable received remote agents toimperfect the network driven)they vehicles for anbeing enduring time horizon. is(human a valid information the latter case isthe duehost tobythe availability ofdue more accurate penetration rate after being commercialized at an affordable information for vehicle rather than the raw price, will definitely co-exist with manned for the host rather than thetoimperfect raw information received byvehicle remotedrops. agents due thethat network price, they willfor definitely co-exist with manned driven) vehicles an enduring time are horizon. This atis(human such as delay and packet It is obvious more fact, even when automated vehicles deployed aaa valid large issues for the host vehicle rather than thetoimperfect raw price, they willfor definitely co-exist with manned (human information received by remote agents due the network driven) vehicles an enduring time horizon. This is valid issues such as delay and packet drops. It is obvious that more fact, even when automated vehicles are deployed at a large information received by remote agents due to the network driven) vehicles for an enduring time horizon. This is valid precise rawasreceived information results in forecasting models with scale. Invehicles addition toanthe different autonomy levels, various information by remote agents due to the network driven) for enduring time horizon. This is valid issues such delay and packet drops. It is obvious that more fact, even when automated vehicles are deployed at a large rawasinformation results in forecasting models with scale. Instrategies addition to the different autonomy levels, various issues such delay packet drops. It is obvious thatismore fact, even when automated are deployed a large precise fidelity. The and behavior modeling block, which the control utilized byvehicles automated vehicles isat issues such asinformation delay and packet drops. It is obvious thatismore fact, even when automated vehicles are deployed at another a large higher precise raw results in forecasting models with scale. In addition to the different autonomy levels, various higher fidelity. The behavior modeling block, which the control strategies utilized by automated vehicles is another precise raw information results in forecasting models with scale. In addition to the different autonomy levels, various main focus of this work, could be decomposed into two factor which might induce behavioral non-homogeneities in precise raw information results in forecasting models with scale. In addition to the different autonomy levels, various higher fidelity. The behavior modeling block, which is the control strategies utilized by automated vehicles is another main focus of this work, could be decomposed into two factor which might induce behavioral non-homogeneities in higher fidelity. The behavior modeling block, which is the control strategies utilized by automated vehicles is another hierarchical layers which arecould inmodeling charge ofblock, modeling the largethe vehicular networks. Aby possible approach this higher fidelity. The behavior which is two the control strategies utilized automated vehiclesto isface another main focus of this work, be decomposed into factor which might induce behavioral non-homogeneities in hierarchical layers which are in charge of modeling the largethe vehicular networks. A possible approach to face this main focus of this work, could be decomposed into two factor which might induce behavioral non-homogeneities in scale and small-scale behaviors, respectively. Large-scale problem and design a robust distributed control scheme for a main focus of this work, could be decomposed into two factor which might induce behavioral non-homogeneities in hierarchical layers which are of modeling the the networks. A possible approach to face this scale and are small-scale respectively. Large-scale problem and design a robust distributed control hierarchical layers whichbehaviors, are in in charge charge ofand modeling the largelargethe vehicular vehicular networks. Ascenario possible toappropriate face for thisa behaviors high-level driving actions maneuvers such non-homogenous vehicular is approach finding thescheme hierarchical layers whichbehaviors, are in charge ofand modeling the largethe vehicular networks. Ascenario possible approach toappropriate face for thisa behaviors scale and small-scale respectively. Large-scale problem and design a robust distributed control scheme are high-level driving actions maneuvers such non-homogenous vehicular is finding the scale and small-scale behaviors, respectively. Large-scale problem and design a human robust distributed control scheme for a as lane-change, take-over, joining or leaving a platoon, steady stochastic models for interventions in the system and scale and are small-scale behaviors, respectively. Large-scale problem and design a human robustscenario distributed control for a as behaviors high-level driving actions and maneuvers such non-homogenous vehicular is finding thescheme appropriate lane-change, take-over, joining or leaving a platoon, steady stochastic models for interventions in the system and behaviors are high-level driving actions and maneuvers such non-homogenous vehicular scenario is finding appropriate cruise, etc.are The set of large-scale driving behaviors could be then treating thesevehicular intervention models in in different vehicles behaviors high-level driving actions and maneuvers such non-homogenous scenario is finding appropriate as lane-change, take-over, joining or leaving a platoon, steady stochastic models for human interventions the system and cruise, etc. The set of large-scale driving behaviors could be then treating these intervention models in different vehicles as lane-change, take-over, joining or leaving a platoon, steady stochastic models for human interventions in the system and interpreted as the system discrete behavioral modes (states). (network agents) in a consistent manner. In our proposed as lane-change, take-over, joining or leaving a modes platoon, steady stochastic models for human interventions in the system and cruise, etc. The set of large-scale driving behaviors could be then treating these intervention models in different vehicles interpreted as the system discrete behavioral (states). (network agents) in a consistent manner. In our proposed cruise, etc. The set of large-scale driving behaviors could be then treating these intervention models in different vehicles Small-scale behaviors the dynamics (patterns) of the set framework inthese this work, these stochastic models are cruise, etc. as The setsystem of define large-scale driving behaviors could be then treating intervention models in different vehicles interpreted the discrete behavioral modes (states). (network agents) in a consistent manner. In our proposed interpreted as the system discrete behavioral modes (states). Small-scale behaviors define the dynamics (patterns) of the set (network agents) in a consistent manner. In our proposed framework in this work, these stochastic models are of kinematic states of the targeted agent inside different interpreted asin stochastic disturbances generated byproposed remote interpreted as the system discrete behavioral modes (states). (network agents) in a consistent manner. In our Small-scale behaviors define the dynamics (patterns) of the set framework this work, these stochastic models are kinematic states the the targeted inside different interpreted asinconsidered stochastic disturbances generated by remote Small-scale behaviors define dynamics (patterns) of the set framework this work, stochastic are of modes. Theseof kinematic statesagent which are required by vehicles and in these the design of themodels stochastic Small-scale behaviors define the dynamics (patterns) of the set framework in stochastic this work, these stochastic models are discrete of kinematic states of the targeted agent inside different interpreted as disturbances generated by remote discrete modes. These kinematic states which are required by of kinematic states of the targeted agent inside different interpreted as stochastic disturbances generated by remote vehicles and considered in the design of the stochastic the host vehicle controllers are simultaneously affected by controllers of the host vehicle. This methodology enables the of kinematic states of the targeted agent inside different interpreted as stochastic disturbances generated by remote discrete modes. These kinematic states which are required by vehicles and considered in the design of the stochastic the host vehicle controllers are simultaneously affected by controllers of the host vehicle. This methodology enables the discrete modes. These kinematic states which are requiredand vehicles and toconsidered in therobust designbehavior of the and stochastic inherent remote vehicle physical kinematic characteristics host vehicle show a more wiser discrete modes. These kinematic states which are required by vehicles and considered in the design of the stochastic the host vehicle controllers are simultaneously affected controllers of the host vehicle. This methodology enables the inherent remote vehicle physical kinematic characteristics and host vehicle to show a more robust behavior and wiser the host vehicle controllers are simultaneously affected by controllers of the host vehicle. Thisimposed methodology enables the driving style of the driver (driver behavior). Discrete Hybrid reactions against the uncertainty by unpredictable hoststyle vehicle controllers are kinematic simultaneously affected by controllers of the host vehicle. This methodology enables the the vehicle physical characteristics and inherent remote host vehicle to show aa more robust behavior and wiser driving of the driver (driver behavior). Discrete Hybrid reactions against the uncertainty imposed by unpredictable inherent remote vehicle physical kinematic characteristics and host vehicle to show more robust behavior and wiser Automata (DHSA) [5] is a well-known structure human interventions caused by the remote vehicles’ inherentstyle remote vehicle physical kinematic characteristics and host vehicle to show a more robust behavior anddrivers. wiser Stochastic driving of driver (driver behavior). Discrete Hybrid reactions against uncertainty imposed by Stochastic Automata (DHSA) [5] ismentioned a well-known structure human interventions by the remote vehicles’ drivers. drivingincorporates style of the the all driver (driver behavior). Discrete Hybrid reactions against the the caused uncertainty imposed by unpredictable unpredictable which of the above components in driving style of the driver (driver behavior). Discrete Hybrid reactions against the uncertainty imposed by unpredictable Stochastic Automata (DHSA) [5] is a well-known structure human interventions caused by the remote vehicles’ drivers. This interventions material is based caused on work supported in part byvehicles’ the Nationaldrivers. Science Stochastic Automata [5] ismentioned a well-known structure which incorporates all(DHSA) ofisthe above components in human by the remote a formal way. DHSA an extension of Discrete Hybrid This material is based on work supported in part by the National Science Stochastic Automata (DHSA) [5] ismentioned a well-known structure human interventions caused by the and remote Foundation under CAREER Grant 1664968 in part vehicles’ by the Qatardrivers. National which incorporates all of the above components in a formal way. DHSA is an extension of Discrete Hybrid which incorporates all of the above mentioned components in This material is based on work supported in part by the National Science Foundation under CAREER 1664968 part Qatar National Automata (DHA) [6], which handles theoftransition between Research Fund Project NPRP 8-1531-2-651. incorporates all of the above mentioned components in This material is based onGrant work supportedand in in part byby thethe National Science awhich formal way. DHSA is an extension Discrete Hybrid Foundation under CAREER Grant 1664968 and in part by the Qatar National a formal way. DHSA is an extension of Discrete Hybrid Automata (DHA) [6], which handles the transition between This material is based on work supported in part by the National Science Research Fund Project NPRP 8-1531-2-651. modes (discrete states DHSA terminoFoundation under CAREER Grant 1664968 and in part by the Qatar National adifferent formal behavioral way. DHSA is an extension ofin Discrete Hybrid Automata (DHA) which handles the transition between Research Fund Project NPRPGrant 8-1531-2-651. Foundation under CAREER 1664968 and in part by the Qatar National different behavioral modes (discrete states DHSA terminoAutomata (DHA) [6], [6], which handles the in transition between Research Fund Project NPRP 8-1531-2-651. Automata (DHA) [6], which handles the transition between different behavioral modes (discrete states Research Fund Project NPRP 8-1531-2-651. different behavioral modes (discrete states in in DHSA DHSA terminoterminoCopyright ©2018 456Hosting 2405-8963 © 2019, IFAC IFAC (International Federation of Automatic Control) bybehavioral Elsevier Ltd.modes All rights reserved. (discrete states in DHSA terminoCopyright ©2018 IFAC 456 different Peer review©2018 under IFAC responsibility of International Federation of Automatic Copyright 456Control. Copyright ©2018 IFAC 456 10.1016/j.ifacol.2019.01.006 Copyright ©2018 IFAC 456
IFAC CPHS 2018 Miami, FL, USA, Dec. 14-15, 2018
Hossein Nourkhiz Mahjoub et al. / IFAC PapersOnLine 51-34 (2019) 400–402
401
Fig. 1. DHSA control framework in host vehicle, which considers the GP models of the remote vehicles as its stochastic input disturbances logy) of the system in a probabilistic manner via defining the stochastic Finite State Machine (sFSM) rather than deterministic FSM of DHA. The other elements of DHSA are the mode selector (MS), which defines the patterns of continuous dynamics within each discrete state, event generator (EG) which defines the set of possible events which could be generated according to different linear constrains on the continuous state values, and the switched affine system (SAS) which defines a standard linear state space model (discrete difference equation) for the continuous states, continuous inputs, and system disturbances. The events generated by the event generator are affecting the functionalities of both sFSM and mode selector components.
presented. Different continuous state variables of the vehicle are considered as separate time series here. Our goal in this section is deriving the appropriate forecasting models which give accurate estimates for future values of these time series. In order to propose an analytically tractable procedure for modeling the joint vehicle-driver behavior, which at the same time is not limited to some certain criteria, non-parametric Bayesian inference techniques, and particularly Gaussian Processes (GPs) [7], are among the most promising methods in the literature. In the GP formulation, sequence of observed samples from a signal are treated as one instant of an N-dimensional multivariate Gaussian random vector. N here denotes the number of available observations.
In our configuration, any vehicle sets up a DHSA for itself, including its own continuous dynamics (mainly its position, velocity, and acceleration) as the continuous SAS states in addition to the received models of the remote vehicles’ continuous dynamics as the SAS stochastic disturbances. DHSA discrete states (behavioral modes) and their evolution over time are defined by our large-scale behavior modeling block. In fact, this block plays the role of sFSM in DHSA framework. Appropriate SAS dynamic selection (MS functionality) at each moment is handled according to the current discrete state of the system, in addition to the current set of generated events by EG. These events depend on the constraints have been defined on the continuous states, such as the instantaneous distance with the preceding vehicle, current speed offset from the speed limit or from the driver-specified speed through cruise control system, etc.
(1) {𝑌𝑌𝑌𝑌1 , 𝑌𝑌𝑌𝑌2 , … 𝑌𝑌𝑌𝑌𝑁𝑁𝑁𝑁 }~𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇̅ , Σ𝑁𝑁𝑁𝑁⨯𝑁𝑁𝑁𝑁 ) This N-dimensional multivariate Gaussian is considered as the joint marginalized distribution of the original function at the observation points. This is based on the assumption that all other function values, other than values at observation moments, have been integrated out. The main component of the GP formulation is its N by N correlation matrix, denoted by Σ in (1). This matrix, which is called the kernel function in the GP context, defines the trend of sample functions (sample paths) from the posterior distribution, after Bayesian inference, based on the similarity pattern among the observed values. By choosing an appropriate combination of different kernel types, different patterns could be captured within the available observations. In this work, we used a realistic vehicular dataset provided by US-DOT, known as NG-SIM (I-80) dataset [8]. We tried to derive a predictive model for the next position of a vehicle based on a history of its previous states (i.e., position, and speeds). We tried a powerful kernel type, i.e., spectral mixture kernel [9], and observed its prediction results versus the constant speed model as the baseline. We have focused on those fractions of the trajectories in which vehicles are making special lateral maneuvers such as lane-change. It has been
The overall schematic of the described system architecture is presented in Figure 1. 2. NON-PARAMETRIC BAYESIAN MODELING 2.1 Time-Series Forecasting with Gaussian Processes In this section, the details of the method used inside the previously mentioned small-scale behavior modeling block is 457
IFAC CPHS 2018 402 Miami, FL, USA, Dec. 14-15, 2018
Hossein Nourkhiz Mahjoub et al. / IFAC PapersOnLine 51-34 (2019) 400–402
[2] E. Moradi-Pari, H. N. Mahjoub, H. Kazemi, Y. P. Fallah and A. TahmasbiSarvestani, "Utilizing Model-Based Communication and Control for Cooperative Automated Vehicle Applications," in IEEE Transactions on Intelligent Vehicles, vol. 2, no. 1, pp. 38-51, March 2017.
observed from this analysis that during these special maneuver moments, GP prediction outperforms the baseline model. This is very important, since during a normal driving scenario along the longitudinal direction in a normal traffic situation, constant speed model works well enough and makes predictions with enough precision. In fact, the main bottle neck of this model is at deviation moments from this normal trend, such as lanechange maneuvers.
[3] H. N. Mahjoub, B. Toghi and Y. P. Fallah, “A Stochastic Hybrid Framework for Driver Behavior Modeling Based on Hierarchical Dirichlet Process,” 88th IEEE Vehicular Technology Conference (VTC 2018-Fall), Chicago, IL, August 2018 [4] H. N. Mahjoub, B. Toghi and Y. P. Fallah, “A Driver Behavior Modeling Structure Based on Nonparametric Bayesian Stochastic Hybrid Architecture,” 2018 IEEE Connected and Automated Vehicles Symposium (CAVS), Chicago, IL, August 2018
Figure 2 is an example of the lateral position forecast error comparison at the next sample time for the baseline and SM kernel GP models, assuming we have the last two samples of the speed for each prediction. The forecast moment has been swept during the complete lane change maneuver for each model. It is clear that almost during the whole lateral action, GP generates better predictions for the next moment lateral position.
[5] A. Bemporad and S. Di Cairano, "Model-Predictive Control of Discrete Hybrid Stochastic Automata," in IEEE Transactions on Automatic Control, vol. 56, no. 6, pp. 1307-1321, June 2011. [6] F. D. Torrisi and A. Bemporad, "HYSDEL-a tool for generating computational hybrid models for analysis and synthesis problems," in IEEE Transactions on Control Systems Technology, vol. 12, no. 2, pp. 235-249, March 2004. [7] Gaussian Processes for Machine Learning Carl Edward Rasmussen and Christopher K. I. Williams The MIT Press, 2006. ISBN 0-262-18253-X. [8] NGSIM dataset is available online at: https://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm [9] Andrew Wilson, Ryan Adams, Gaussian Process Kernels for Pattern Discovery and Extrapolation, Proceedings of the 30th International Conference on Machine Learning, PMLR 28(3):1067-1075, 2013.
Fig. 2. Lateral position tracking error comparison of SMkernel GP and baseline (constant speed) models during a lanechange maneuver from NGSIM (I-80) dataset 3. CONCLUDING REMARKS In this abstract, we have proposed a stochastic hybrid framework based on DHSA and non-parametric Bayesian notions to model the states of the remote vehicles as stochastic disturbance components, which could be utilized in order to design a more robust stochastic MPC in the host vehicle. More specifically, it has been proposed to augment the baseline constant speed forecasting model with a non-parametric Bayesian model, i.e. GPs, during the special driving moments where the baseline model accuracy deviates from its normal level. For instance, lane change maneuvers which introduce some non-linearities in the lateral position trend over time, might not be modeled with a good precision by the constant speed (linear position) model. Thus, taking a hybrid modeling approach which switches between the baseline and GP in an online adaptive manner seems a better solution than using each of these models individually. However, one challenge here is defining a criterion which enables the model to automatically detect the switching moments. It seems that using other parameters, such as heading or steering angle which are highly correlated with the lateral movements, in conjunction with the speed could be a good starting point to tackle this problem. We are currently investigating the design of such hybrid forecasting methods. REFERENCES [1] Y. P. Fallah, "A model-based communication approach for distributed and connected vehicle safety systems," 2016 Annual IEEE Systems Conference (SysCon), Orlando, FL, 2016, pp. 1-6. 458
ScienceDirect
IFAC PapersOnLine 51-34 (2019) 400–402
A Stochastic Hybrid Structure for Predicting Disturbances in Mixed Automated A Stochastic Hybrid Structure for Predicting Disturbances in Mixed Automated A Stochastic Hybrid Structure for Predicting Disturbances Mixed Automated and Human-Driven Vehicular Scenariosin A Structure for Predicting Disturbances Mixed Automated Human-Driven Vehicular Scenariosin A Stochastic Stochastic Hybrid Hybridand Structure for Predicting Disturbances in Mixed Automated and Human-Driven Vehicular Scenarios and Human-Driven Vehicular Scenarios Hossein Nourkhiz Mahjoub*, Mohammadreza Davoodi**, Yaser P. Fallah*, Javad M. Velni** and Human-Driven Vehicular Scenarios
Hossein Nourkhiz Mahjoub*, Mohammadreza Davoodi**, Yaser P. Fallah*, Javad M. Velni** Hossein Hossein Nourkhiz Nourkhiz Mahjoub*, Mahjoub*, Mohammadreza Mohammadreza Davoodi**, Davoodi**, Yaser Yaser P. P. Fallah*, Fallah*, Javad Javad M. M. Velni** Velni** *UniversityMohammadreza of Central Florida, Orlando, Yaser FL, 32816, USA Javad M. Velni** Hossein Nourkhiz Mahjoub*, Davoodi**, P. Fallah*, *University of Central Florida, Orlando, FL, 32816, USA (e-mail: hnmahjoub @knights.ucf.edu, [email protected]). *University of Florida, Orlando, FL, 32816, USA (e-mail: hnmahjoub @knights.ucf.edu, [email protected]). *University of Central Central Florida, Orlando, 32816, **University of Georgia, Athens, GA, FL, 30602, USAUSA *University of Central Florida, Orlando, FL, 32816, USA (e-mail: hnmahjoub @knights.ucf.edu, [email protected]). (e-mail: [email protected], **University Georgia, Athens, [email protected]). GA, [email protected]) 30602, USA (e-mail:[email protected], (e-mail: hnmahjoub @knights.ucf.edu, [email protected]). **University of Georgia, Athens, GA, 30602, USA (e-mail:[email protected], **University of Georgia, Athens, GA, [email protected]) 30602, USA **University of Georgia, Athens, GA, 30602, USA (e-mail:[email protected], (e-mail:[email protected], [email protected]) [email protected]) [email protected]) Abstract: In this work,(e-mail:[email protected], we introduce a stochastic prediction method which can be utilized in applications Abstract: In this work, we introduce a stochastic prediction method which can be movements. utilized in applications such as cooperative adaptive cruise control (CACC) to predict interfering vehicles’ One of the Abstract: In this work, we introduce aa stochastic prediction method which can be utilized in such as cooperative adaptive cruise control (CACC) to predict interfering vehicles’ movements. One of the Abstract: In this work, we introduce stochastic prediction method which can be utilized in applications applications main criteria in the design of automated vehicle systems is their robustness against the disturbances resulted Abstract: In this work, we introduce a stochastic prediction method which can be utilized in applications such as cooperative adaptive cruise control (CACC) to predict interfering vehicles’ movements. One of main criteria in the design of automated vehicle systems is their robustness against the disturbances resulted such adaptive of cruise (CACC) to predictThe interfering vehicles’ movements. One to of the the from as thecooperative non-homogeneity the control vehicular environment. non-homogeneity isdisturbances mainly due the such as cooperative adaptive cruise control (CACC) to predict interfering vehicles’ movements. One of the main criteria in the design of automated vehicle systems is their robustness against the resulted from the non-homogeneity of the vehicular environment. The non-homogeneity is mainly due to the main criteria in and the design of automated vehicle systemsco-existence. is their robustness against the disturbances resulted human-driven automated/autonomous vehicles It is therefore imperative for the main criteria in the design of automated vehicleenvironment. systems is their robustness against theisdisturbances resulted from the non-homogeneity of the vehicular The non-homogeneity mainly due to the from the non-homogeneity of the vehicular environment. The non-homogeneity is mainly due to human-driven and automated/autonomous vehicles co-existence. It is therefore imperative for the automated applications to beofdesigned with the capability ofThe handling the uncertain of humanfrom the non-homogeneity the vehicular environment. non-homogeneity isbehaviors mainly due to human-driven and automated/autonomous vehicles co-existence. is therefore imperative for human-driven and automated/autonomous vehicles co-existence. Itfor isvehicle therefore imperative for the the automated applications to bemanner. designedThis withpaper the capability of handlingIt the uncertain behaviors of humandriven vehicles in a robust presents a method movements time-series human-driven and automated/autonomous vehicles co-existence. Itthe is uncertain therefore behaviors imperative for the automated applications to be designed with the capability of handling of humanautomated applications to be designed with the capability of handling the uncertain behaviors of humandriven vehicles in a robust manner. This paper presents a method for vehicle movements time-series forecasting using a powerful Bayesian inference method, Gaussian Processes. The automated applications to benon-parametric designedThis with the capability of handling thenamely uncertain behaviors of humandriven vehicles aa robust manner. presents aa method for vehicle movements time-series driven vehicles ina powerful robust manner. using This paper paper presents method for vehicle movements time-series forecasting usingin non-parametric Bayesian inference method, namely Gaussian Processes. The proposed methodology is evaluated realistic vehicle trajectory data from NGSIM dataset and is driven vehicles in a robust manner. This paper presents a method for vehicle movements time-series forecasting using powerful non-parametric Bayesian inference method, namely Gaussian Processes. forecasting using aamore powerful non-parametric Bayesian inference method, namely Gaussian Processes. The proposed methodology is evaluated using realistic vehicle trajectory data from NGSIM dataset andThe is shown to provide accurate results compared to baseline methods that use constant velocity coasting. forecasting using a powerful non-parametric Bayesian inference method, namely Gaussian Processes. The proposed methodology is using realistic data from NGSIM dataset and shown to provide more accurate results compared to vehicle baselinetrajectory methods that constant velocity coasting. proposed methodology is evaluated evaluated using realistic vehicle trajectory datause from NGSIM dataset and is is proposed methodology is evaluated using realistic vehicle trajectory data from NGSIM dataset and is Keywords: Discrete Hybrid Stochastic Automata, Model Predictive Control, Non-parametric Bayesian shown to provide more accurate results compared to baseline methods that use constant velocity coasting. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. shown to provide more accurate results compared to Model baselinePredictive methods that use constant velocity coasting. Discrete Hybrid Stochastic Automata, Control, Non-parametric Bayesian Keywords: shown to provide more accurate results compared baseline methods that use constant velocity coasting. Inference, Gaussian Processes, Vehicular Networks, to Model-Based Communication Discrete Hybrid Automata, Model Control, Keywords: Inference, Gaussian Vehicular Networks, Communication Keywords: DiscreteProcesses, Hybrid Stochastic Stochastic Automata,Model-Based Model Predictive Predictive Control, Non-parametric Non-parametric Bayesian Bayesian DiscreteProcesses, Hybrid Stochastic Automata,Model-Based Model Predictive Control, Non-parametric Bayesian Keywords: Inference, Gaussian Vehicular Networks, Communication Inference, Gaussian Processes, Vehicular Networks, Model-Based Communication Inference, Gaussian Processes, Vehicular Networks, Model-Based Communication These models could be generated either in the vehicle which is 1. INTRODUCTION and SYSTEM DESCRIPTION These models could (fully be generated either in is directly controlled or partially) bythe thevehicle humanwhich driver 1. INTRODUCTION and SYSTEM DESCRIPTION These models could be generated either in the vehicle which directly controlled (fully or partially) by the human driver models could be agents. generated either in the vehicle which is is 1. and SYSTEM DESCRIPTION itself or also in target The former case is considered Plausible disturbances from different sources such as manned These These models could be agents. generated either in the vehicle which is 1. INTRODUCTION INTRODUCTION and SYSTEM DESCRIPTION directly controlled (fully or partially) by the human driver itself or also in target The former case is considered Plausible disturbances from different sources such as manned directly controlled (fully or partially) by the human driver 1. INTRODUCTION and SYSTEM DESCRIPTION here, since it is more effective if the models are derived by the vehicles or vulnerable road users (VRUs) should be directly controlled (fully or partially) by case the human driver itself or also in target agents. The former is considered Plausible disturbances from different sources such as manned vehicles or vulnerable users (VRUs) should be here, itself since or also target agents. The case is considered it isinmore effective the models are derived the Plausible disturbances fromroad different such as manned manned vehicle and thenif be former transmitted overby the appropriately considered during thesources design phase of the itself since or also inmore target agents. The former case is considered Plausible disturbances fromroad different sources such as manned here, it is effective if the models are derived by the vehicles or vulnerable users (VRUs) should be manned vehicle and then be transmitted over the appropriately considered during the design phase of the here, since it is more effective if the models are derived by vehicles or vulnerable road users (VRUs) should be network for other vehicles which need these distributed vehicular systems, such as cooperative here, sincevehicle it is more effective if the models are derived by vehicles orautomated vulnerable road users (VRUs) should be communication manned and then be transmitted over the appropriately considered during the design phase of the manned vehicle andhas then transmitted overinthese the communication network for otherbeproposed vehicles which need appropriately considered duringsystems, the design phase of the models. distributedcruise automated vehicular such as cooperative This notion been recently the adaptive control (CACC), platooning management manned vehicle andhas then beproposed transmitted overinthese the appropriately considered duringsystems, the design phase of the models. communication network for other vehicles which need distributed automated vehicular such as cooperative This notion been recently the adaptive cruise control (CACC), platooning management communication network for other vehicles which need these distributed automated vehicular systems, such as cooperative vehicular literature as Model-Based Communication (MBC) (merging/splitting) systems, etc. Although partially automated communication network for other vehicles which need these distributed automated vehicular systems, such as cooperative models. This notion has been proposed recently in the adaptive cruise control (CACC), platooning management vehicular literature as Model-Based Communication (MBC) (merging/splitting) systems, etc. Although partially automated models. This notion has been proposed recently in the adaptive cruise control (CACC), platooning management [1], [2], [3], [4].notion Theasbetter of the MBC (MBC) against and fully cruise autonomous vehicles are expected to management have a high models. This has performance been proposed recently in the adaptive control (CACC), platooning vehicular literature Model-Based Communication (merging/splitting) systems, etc. Although partially automated [1], [2], [3], [4]. The better performance of the MBC against and fully autonomous vehicles are expected to have a high vehicular literature as Model-Based Communication (MBC) (merging/splitting) systems, etc. Although partially automated the latter case is due to the availability of more accurate raw penetration rate after being commercialized attoanhave affordable vehicular literature asto Model-Based Communication (MBC) (merging/splitting) systems, etc. Although partially automated [1], [2], [4]. The better performance the MBC against and fully autonomous vehicles are aa high case isthe duehost the availability ofof accurate raw penetration rate after being commercialized attoanhave affordable [1], latter [2], [3], [3], [4]. The better performance ofmore the imperfect MBC against and fully autonomous vehicles are expected expected high the information for vehicle rather than the raw price, they will definitely co-exist with manned (human [1], [2], [3], [4]. The better performance of the MBC against and fully autonomous vehicles are expected to have a high the latter case is due to the availability of more accurate raw penetration rate after being commercialized at an affordable information for the host vehicle rather than the imperfect raw price, they will definitely co-exist with manned (human the latter case is due to the availability of more accurate penetration rate after commercialized atThis an affordable received remote agents toimperfect the network driven)they vehicles for anbeing enduring time horizon. is(human a valid information the latter case isthe duehost tobythe availability ofdue more accurate penetration rate after being commercialized at an affordable information for vehicle rather than the raw price, will definitely co-exist with manned for the host rather than thetoimperfect raw information received byvehicle remotedrops. agents due thethat network price, they willfor definitely co-exist with manned driven) vehicles an enduring time are horizon. This atis(human such as delay and packet It is obvious more fact, even when automated vehicles deployed aaa valid large issues for the host vehicle rather than thetoimperfect raw price, they willfor definitely co-exist with manned (human information received by remote agents due the network driven) vehicles an enduring time horizon. This is valid issues such as delay and packet drops. It is obvious that more fact, even when automated vehicles are deployed at a large information received by remote agents due to the network driven) vehicles for an enduring time horizon. This is valid precise rawasreceived information results in forecasting models with scale. Invehicles addition toanthe different autonomy levels, various information by remote agents due to the network driven) for enduring time horizon. This is valid issues such delay and packet drops. It is obvious that more fact, even when automated vehicles are deployed at a large rawasinformation results in forecasting models with scale. Instrategies addition to the different autonomy levels, various issues such delay packet drops. It is obvious thatismore fact, even when automated are deployed a large precise fidelity. The and behavior modeling block, which the control utilized byvehicles automated vehicles isat issues such asinformation delay and packet drops. It is obvious thatismore fact, even when automated vehicles are deployed at another a large higher precise raw results in forecasting models with scale. In addition to the different autonomy levels, various higher fidelity. The behavior modeling block, which the control strategies utilized by automated vehicles is another precise raw information results in forecasting models with scale. In addition to the different autonomy levels, various main focus of this work, could be decomposed into two factor which might induce behavioral non-homogeneities in precise raw information results in forecasting models with scale. In addition to the different autonomy levels, various higher fidelity. The behavior modeling block, which is the control strategies utilized by automated vehicles is another main focus of this work, could be decomposed into two factor which might induce behavioral non-homogeneities in higher fidelity. The behavior modeling block, which is the control strategies utilized by automated vehicles is another hierarchical layers which arecould inmodeling charge ofblock, modeling the largethe vehicular networks. Aby possible approach this higher fidelity. The behavior which is two the control strategies utilized automated vehiclesto isface another main focus of this work, be decomposed into factor which might induce behavioral non-homogeneities in hierarchical layers which are in charge of modeling the largethe vehicular networks. A possible approach to face this main focus of this work, could be decomposed into two factor which might induce behavioral non-homogeneities in scale and small-scale behaviors, respectively. Large-scale problem and design a robust distributed control scheme for a main focus of this work, could be decomposed into two factor which might induce behavioral non-homogeneities in hierarchical layers which are of modeling the the networks. A possible approach to face this scale and are small-scale respectively. Large-scale problem and design a robust distributed control hierarchical layers whichbehaviors, are in in charge charge ofand modeling the largelargethe vehicular vehicular networks. Ascenario possible toappropriate face for thisa behaviors high-level driving actions maneuvers such non-homogenous vehicular is approach finding thescheme hierarchical layers whichbehaviors, are in charge ofand modeling the largethe vehicular networks. Ascenario possible approach toappropriate face for thisa behaviors scale and small-scale respectively. Large-scale problem and design a robust distributed control scheme are high-level driving actions maneuvers such non-homogenous vehicular is finding the scale and small-scale behaviors, respectively. Large-scale problem and design a human robust distributed control scheme for a as lane-change, take-over, joining or leaving a platoon, steady stochastic models for interventions in the system and scale and are small-scale behaviors, respectively. Large-scale problem and design a human robustscenario distributed control for a as behaviors high-level driving actions and maneuvers such non-homogenous vehicular is finding thescheme appropriate lane-change, take-over, joining or leaving a platoon, steady stochastic models for interventions in the system and behaviors are high-level driving actions and maneuvers such non-homogenous vehicular scenario is finding appropriate cruise, etc.are The set of large-scale driving behaviors could be then treating thesevehicular intervention models in in different vehicles behaviors high-level driving actions and maneuvers such non-homogenous scenario is finding appropriate as lane-change, take-over, joining or leaving a platoon, steady stochastic models for human interventions the system and cruise, etc. The set of large-scale driving behaviors could be then treating these intervention models in different vehicles as lane-change, take-over, joining or leaving a platoon, steady stochastic models for human interventions in the system and interpreted as the system discrete behavioral modes (states). (network agents) in a consistent manner. In our proposed as lane-change, take-over, joining or leaving a modes platoon, steady stochastic models for human interventions in the system and cruise, etc. The set of large-scale driving behaviors could be then treating these intervention models in different vehicles interpreted as the system discrete behavioral (states). (network agents) in a consistent manner. In our proposed cruise, etc. The set of large-scale driving behaviors could be then treating these intervention models in different vehicles Small-scale behaviors the dynamics (patterns) of the set framework inthese this work, these stochastic models are cruise, etc. as The setsystem of define large-scale driving behaviors could be then treating intervention models in different vehicles interpreted the discrete behavioral modes (states). (network agents) in a consistent manner. In our proposed interpreted as the system discrete behavioral modes (states). Small-scale behaviors define the dynamics (patterns) of the set (network agents) in a consistent manner. In our proposed framework in this work, these stochastic models are of kinematic states of the targeted agent inside different interpreted asin stochastic disturbances generated byproposed remote interpreted as the system discrete behavioral modes (states). (network agents) in a consistent manner. In our Small-scale behaviors define the dynamics (patterns) of the set framework this work, these stochastic models are kinematic states the the targeted inside different interpreted asinconsidered stochastic disturbances generated by remote Small-scale behaviors define dynamics (patterns) of the set framework this work, stochastic are of modes. Theseof kinematic statesagent which are required by vehicles and in these the design of themodels stochastic Small-scale behaviors define the dynamics (patterns) of the set framework in stochastic this work, these stochastic models are discrete of kinematic states of the targeted agent inside different interpreted as disturbances generated by remote discrete modes. These kinematic states which are required by of kinematic states of the targeted agent inside different interpreted as stochastic disturbances generated by remote vehicles and considered in the design of the stochastic the host vehicle controllers are simultaneously affected by controllers of the host vehicle. This methodology enables the of kinematic states of the targeted agent inside different interpreted as stochastic disturbances generated by remote discrete modes. These kinematic states which are required by vehicles and considered in the design of the stochastic the host vehicle controllers are simultaneously affected by controllers of the host vehicle. This methodology enables the discrete modes. These kinematic states which are requiredand vehicles and toconsidered in therobust designbehavior of the and stochastic inherent remote vehicle physical kinematic characteristics host vehicle show a more wiser discrete modes. These kinematic states which are required by vehicles and considered in the design of the stochastic the host vehicle controllers are simultaneously affected controllers of the host vehicle. This methodology enables the inherent remote vehicle physical kinematic characteristics and host vehicle to show a more robust behavior and wiser the host vehicle controllers are simultaneously affected by controllers of the host vehicle. Thisimposed methodology enables the driving style of the driver (driver behavior). Discrete Hybrid reactions against the uncertainty by unpredictable hoststyle vehicle controllers are kinematic simultaneously affected by controllers of the host vehicle. This methodology enables the the vehicle physical characteristics and inherent remote host vehicle to show aa more robust behavior and wiser driving of the driver (driver behavior). Discrete Hybrid reactions against the uncertainty imposed by unpredictable inherent remote vehicle physical kinematic characteristics and host vehicle to show more robust behavior and wiser Automata (DHSA) [5] is a well-known structure human interventions caused by the remote vehicles’ inherentstyle remote vehicle physical kinematic characteristics and host vehicle to show a more robust behavior anddrivers. wiser Stochastic driving of driver (driver behavior). Discrete Hybrid reactions against uncertainty imposed by Stochastic Automata (DHSA) [5] ismentioned a well-known structure human interventions by the remote vehicles’ drivers. drivingincorporates style of the the all driver (driver behavior). Discrete Hybrid reactions against the the caused uncertainty imposed by unpredictable unpredictable which of the above components in driving style of the driver (driver behavior). Discrete Hybrid reactions against the uncertainty imposed by unpredictable Stochastic Automata (DHSA) [5] is a well-known structure human interventions caused by the remote vehicles’ drivers. This interventions material is based caused on work supported in part byvehicles’ the Nationaldrivers. Science Stochastic Automata [5] ismentioned a well-known structure which incorporates all(DHSA) ofisthe above components in human by the remote a formal way. DHSA an extension of Discrete Hybrid This material is based on work supported in part by the National Science Stochastic Automata (DHSA) [5] ismentioned a well-known structure human interventions caused by the and remote Foundation under CAREER Grant 1664968 in part vehicles’ by the Qatardrivers. National which incorporates all of the above components in a formal way. DHSA is an extension of Discrete Hybrid which incorporates all of the above mentioned components in This material is based on work supported in part by the National Science Foundation under CAREER 1664968 part Qatar National Automata (DHA) [6], which handles theoftransition between Research Fund Project NPRP 8-1531-2-651. incorporates all of the above mentioned components in This material is based onGrant work supportedand in in part byby thethe National Science awhich formal way. DHSA is an extension Discrete Hybrid Foundation under CAREER Grant 1664968 and in part by the Qatar National a formal way. DHSA is an extension of Discrete Hybrid Automata (DHA) [6], which handles the transition between This material is based on work supported in part by the National Science Research Fund Project NPRP 8-1531-2-651. modes (discrete states DHSA terminoFoundation under CAREER Grant 1664968 and in part by the Qatar National adifferent formal behavioral way. DHSA is an extension ofin Discrete Hybrid Automata (DHA) which handles the transition between Research Fund Project NPRPGrant 8-1531-2-651. Foundation under CAREER 1664968 and in part by the Qatar National different behavioral modes (discrete states DHSA terminoAutomata (DHA) [6], [6], which handles the in transition between Research Fund Project NPRP 8-1531-2-651. Automata (DHA) [6], which handles the transition between different behavioral modes (discrete states Research Fund Project NPRP 8-1531-2-651. different behavioral modes (discrete states in in DHSA DHSA terminoterminoCopyright ©2018 456Hosting 2405-8963 © 2019, IFAC IFAC (International Federation of Automatic Control) bybehavioral Elsevier Ltd.modes All rights reserved. (discrete states in DHSA terminoCopyright ©2018 IFAC 456 different Peer review©2018 under IFAC responsibility of International Federation of Automatic Copyright 456Control. Copyright ©2018 IFAC 456 10.1016/j.ifacol.2019.01.006 Copyright ©2018 IFAC 456
IFAC CPHS 2018 Miami, FL, USA, Dec. 14-15, 2018
Hossein Nourkhiz Mahjoub et al. / IFAC PapersOnLine 51-34 (2019) 400–402
401
Fig. 1. DHSA control framework in host vehicle, which considers the GP models of the remote vehicles as its stochastic input disturbances logy) of the system in a probabilistic manner via defining the stochastic Finite State Machine (sFSM) rather than deterministic FSM of DHA. The other elements of DHSA are the mode selector (MS), which defines the patterns of continuous dynamics within each discrete state, event generator (EG) which defines the set of possible events which could be generated according to different linear constrains on the continuous state values, and the switched affine system (SAS) which defines a standard linear state space model (discrete difference equation) for the continuous states, continuous inputs, and system disturbances. The events generated by the event generator are affecting the functionalities of both sFSM and mode selector components.
presented. Different continuous state variables of the vehicle are considered as separate time series here. Our goal in this section is deriving the appropriate forecasting models which give accurate estimates for future values of these time series. In order to propose an analytically tractable procedure for modeling the joint vehicle-driver behavior, which at the same time is not limited to some certain criteria, non-parametric Bayesian inference techniques, and particularly Gaussian Processes (GPs) [7], are among the most promising methods in the literature. In the GP formulation, sequence of observed samples from a signal are treated as one instant of an N-dimensional multivariate Gaussian random vector. N here denotes the number of available observations.
In our configuration, any vehicle sets up a DHSA for itself, including its own continuous dynamics (mainly its position, velocity, and acceleration) as the continuous SAS states in addition to the received models of the remote vehicles’ continuous dynamics as the SAS stochastic disturbances. DHSA discrete states (behavioral modes) and their evolution over time are defined by our large-scale behavior modeling block. In fact, this block plays the role of sFSM in DHSA framework. Appropriate SAS dynamic selection (MS functionality) at each moment is handled according to the current discrete state of the system, in addition to the current set of generated events by EG. These events depend on the constraints have been defined on the continuous states, such as the instantaneous distance with the preceding vehicle, current speed offset from the speed limit or from the driver-specified speed through cruise control system, etc.
(1) {𝑌𝑌𝑌𝑌1 , 𝑌𝑌𝑌𝑌2 , … 𝑌𝑌𝑌𝑌𝑁𝑁𝑁𝑁 }~𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝜇𝜇𝜇𝜇̅ , Σ𝑁𝑁𝑁𝑁⨯𝑁𝑁𝑁𝑁 ) This N-dimensional multivariate Gaussian is considered as the joint marginalized distribution of the original function at the observation points. This is based on the assumption that all other function values, other than values at observation moments, have been integrated out. The main component of the GP formulation is its N by N correlation matrix, denoted by Σ in (1). This matrix, which is called the kernel function in the GP context, defines the trend of sample functions (sample paths) from the posterior distribution, after Bayesian inference, based on the similarity pattern among the observed values. By choosing an appropriate combination of different kernel types, different patterns could be captured within the available observations. In this work, we used a realistic vehicular dataset provided by US-DOT, known as NG-SIM (I-80) dataset [8]. We tried to derive a predictive model for the next position of a vehicle based on a history of its previous states (i.e., position, and speeds). We tried a powerful kernel type, i.e., spectral mixture kernel [9], and observed its prediction results versus the constant speed model as the baseline. We have focused on those fractions of the trajectories in which vehicles are making special lateral maneuvers such as lane-change. It has been
The overall schematic of the described system architecture is presented in Figure 1. 2. NON-PARAMETRIC BAYESIAN MODELING 2.1 Time-Series Forecasting with Gaussian Processes In this section, the details of the method used inside the previously mentioned small-scale behavior modeling block is 457
IFAC CPHS 2018 402 Miami, FL, USA, Dec. 14-15, 2018
Hossein Nourkhiz Mahjoub et al. / IFAC PapersOnLine 51-34 (2019) 400–402
[2] E. Moradi-Pari, H. N. Mahjoub, H. Kazemi, Y. P. Fallah and A. TahmasbiSarvestani, "Utilizing Model-Based Communication and Control for Cooperative Automated Vehicle Applications," in IEEE Transactions on Intelligent Vehicles, vol. 2, no. 1, pp. 38-51, March 2017.
observed from this analysis that during these special maneuver moments, GP prediction outperforms the baseline model. This is very important, since during a normal driving scenario along the longitudinal direction in a normal traffic situation, constant speed model works well enough and makes predictions with enough precision. In fact, the main bottle neck of this model is at deviation moments from this normal trend, such as lanechange maneuvers.
[3] H. N. Mahjoub, B. Toghi and Y. P. Fallah, “A Stochastic Hybrid Framework for Driver Behavior Modeling Based on Hierarchical Dirichlet Process,” 88th IEEE Vehicular Technology Conference (VTC 2018-Fall), Chicago, IL, August 2018 [4] H. N. Mahjoub, B. Toghi and Y. P. Fallah, “A Driver Behavior Modeling Structure Based on Nonparametric Bayesian Stochastic Hybrid Architecture,” 2018 IEEE Connected and Automated Vehicles Symposium (CAVS), Chicago, IL, August 2018
Figure 2 is an example of the lateral position forecast error comparison at the next sample time for the baseline and SM kernel GP models, assuming we have the last two samples of the speed for each prediction. The forecast moment has been swept during the complete lane change maneuver for each model. It is clear that almost during the whole lateral action, GP generates better predictions for the next moment lateral position.
[5] A. Bemporad and S. Di Cairano, "Model-Predictive Control of Discrete Hybrid Stochastic Automata," in IEEE Transactions on Automatic Control, vol. 56, no. 6, pp. 1307-1321, June 2011. [6] F. D. Torrisi and A. Bemporad, "HYSDEL-a tool for generating computational hybrid models for analysis and synthesis problems," in IEEE Transactions on Control Systems Technology, vol. 12, no. 2, pp. 235-249, March 2004. [7] Gaussian Processes for Machine Learning Carl Edward Rasmussen and Christopher K. I. Williams The MIT Press, 2006. ISBN 0-262-18253-X. [8] NGSIM dataset is available online at: https://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm [9] Andrew Wilson, Ryan Adams, Gaussian Process Kernels for Pattern Discovery and Extrapolation, Proceedings of the 30th International Conference on Machine Learning, PMLR 28(3):1067-1075, 2013.
Fig. 2. Lateral position tracking error comparison of SMkernel GP and baseline (constant speed) models during a lanechange maneuver from NGSIM (I-80) dataset 3. CONCLUDING REMARKS In this abstract, we have proposed a stochastic hybrid framework based on DHSA and non-parametric Bayesian notions to model the states of the remote vehicles as stochastic disturbance components, which could be utilized in order to design a more robust stochastic MPC in the host vehicle. More specifically, it has been proposed to augment the baseline constant speed forecasting model with a non-parametric Bayesian model, i.e. GPs, during the special driving moments where the baseline model accuracy deviates from its normal level. For instance, lane change maneuvers which introduce some non-linearities in the lateral position trend over time, might not be modeled with a good precision by the constant speed (linear position) model. Thus, taking a hybrid modeling approach which switches between the baseline and GP in an online adaptive manner seems a better solution than using each of these models individually. However, one challenge here is defining a criterion which enables the model to automatically detect the switching moments. It seems that using other parameters, such as heading or steering angle which are highly correlated with the lateral movements, in conjunction with the speed could be a good starting point to tackle this problem. We are currently investigating the design of such hybrid forecasting methods. REFERENCES [1] Y. P. Fallah, "A model-based communication approach for distributed and connected vehicle safety systems," 2016 Annual IEEE Systems Conference (SysCon), Orlando, FL, 2016, pp. 1-6. 458