Mathematical formulation of the MPC Controller

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Collision avoidance between vehicles

Platooning is a concept where multiple vehicles follow one another with short in-tervehicle distances to improve tra c e ciency and throughput, and reduce fuel consumption due to reduced aerodynamic resistance. Fuel consumption reduced by around 10% [19] because of constant speed. According to experiments, at 10 m spacing and a velocity of 80 km/h, the reduction in fuel consumption was about 21 percent [36]. Other authors also have similar results on the reduction of fuel consumption due to platooning [37, 38]. Closer the vehicles, better are the results of platooning, but higher are the risks of collision.
Japans semi-governmental New Energy and Industrial Technology Development Organization, developed a technology for large and small trucks to safely maintain a 4 m distance between vehicles in a single lane while driving 80 kmph [39]. Whereas in Germany, a project at RWTH Aachen University in Germany operated a platoon of four trucks spaced at 10m intervehicle distance [40]. A European project: Safe Road Trains for the Environment (Sartre) based on platooning has explored using cars and lorries simultaneously at 85 kmph with a gap between each vehicle of 6m[41], whereas [19] mentions the velocity range for heavy vehicle platooning is between 37 to 50 kmph with 10 meters distance between them.
Most of the work has been done on collision avoidance in platoons with vehicles moving in a single lane. However, authors in [42] focus on collision avoidance in a scenario where alternate vehicles from the platoon steer towards right or left to stop somewhere along the diagonal with the vehicle in front. There is a big assumption that the roads have atleast three lanes with platoon moving on the lane in the middle, or in case of roads with just one lane, there is free open space on both sides of the roads. In this chapter, we focus on approaches to avoid collisions in platoons based on di erent scenarios. Some of the modern day vehicles already have Adaptive Cruise Control(ACC) systems which are usually radar-based systems that maintain a particular velocity in case there are no vehicles in front, or a safe distance with the vehicle in front, but the safety gap maintained is pretty big, and it serves neither purposes of platooning. The distance between vehicles can either be constant, which is known as constant spacing policy or velocity dependent, which is also known as constant time gap policy. In the rst type of spacing policy, the distance between the vehicle remains xed irrespective of the speed of the vehicle [43, 44]. [44] states the recommended headway in Germany is 1.8 sec for manual driving in a platoon scenario. Moreover, [43] considers not just the vehicle in front but also the vehicle behind for autonomous driving.
String stability is another critical measure of the safety of ACC vehicles. If a disturbance or an error in the platoon of vehicles magni es down the platoon, the system of is said to be string unstable. If the error is absorbed, the system (or the control behavior) is termed string stable. Usually, the use of on board sensors in an ACC system restricts the vehicle only to have information of the preceding vehicle. Thus, the following vehicle only reacts based on the vehicle in front. This creates a delay in information transfer and thus a delay in reaction, which has a substantial impact on the string stability.
A system of three or more ACC vehicles is very likely to get string unstable ac-cording to Lu, Wang et al. [45] and this was proven in [46]. A minimum intervehicle distance of 1:2 m can be achieved for two identical vehicles without endangering a collision, assuming that there is no delay present in the feedback system. In case of a delay, intervehicle distance is deduced based upon the vehicles maximum deceler-ation ability [21].
Milanes et al. mention that ACC in a multi vehicle scenario is unstable because while designing ACC systems, researchers do not consider vehicle actuation and sensor data evaluation delay, whereas in practise it is non-negligible [47]. Moreover, with CACC, the last vehicle in the vehicle stream can have information of the actions of the rst vehicle, whereas, in ACC, the last vehicle will only have information of the vehicle in front. Milanes and Shladover recommended time gap for ACC to be set to 1.1 sec whereas CACC is set to 0.6 s.
[48, 49] consider CACC vehicle longitudinal dynamics to be composed of two parts: the ACC part (feedback control) and CACC part (feedforward control). The former is computed using relative distance and velocity sensed by onboard sensors whereas the latter is computed using the information received over V2V communications.
There may be di erent means of communication and information transfer in V2V, used for platooning like IEEE 802.11p, Visible Light Communications (VLC), cellular technologies, etc. Authors in [50] propose using VLC as a backup or as an o oad measure to DSRC technology as they show that it can very well handle the demands of vehicular communication within a range of say 25 meters, except for the induced time delay leading to an increased safety distance between the vehicles.

Collision avoidance in platooning

One of the main purposes of the introduction of autonomous (ACC/CACC) vehicles is to reduce the number of collisions occurring due to human errors and human factors like fatigue, drowsiness, etc. Automating driving procedure using ACC and CACC control algorithms removes the human factor in driving and can thus be considered as algorithms that reduce collisions. String stability is usually veri ed to ensure ACC and CACC control algorithms are stable. String stability ensures collision avoidance in a platoon under nominal operational conditions, and other methods are required to ensure collision avoidance in a platoon when the system of vehicles is out of the operational domain.
[51] introduces an algorithm for safe collision free platoon braking in the event of a total communication failure. They assume it takes some time (perception response time tprt) to detect a communication loss, and thus the response can begin only after tprt s. Di erent situations like hard braking by the leader, hard braking by any vehicle in the middle of the platoon, etc. are simulated with no communication loss or total communication loss. String stability is proved mathematically using transfer functions. When a vehicle in the middle of the platoon starts braking, the original platoon is split into two. The rst one with the original leader, whereas the vehicle in the middle which starts braking becomes the leader of the second platoon. As the authors consider all vehicles to have the same braking capability, the proposed system and mathematical model might not be valid if di erent vehicles have di erent braking capacities.
[52] focusses on longitudinal control for collision avoidance of ACC vehicle pla-toon in all conditions. Longitudinal collision avoidance is achieved by keeping a minimum safe distance between vehicles based on the deceleration capacity, and the velocity of these vehicles. On the same lines, [53] proposes vehicle following algorithm based on safe following distance dependent on inter vehicular distance and relative velocity. These approaches are conservative and usually require having a large intervehicle distance thus leading to a decrease in road capacity. [54] pro-poses laws using which two independent platoons can engage in activities like lane changing, merging of platoons, platoon splitting, etc.
Next, we look at approaches proposed for the ego vehicle collision avoidance with neighboring vehicles. These approaches can be extended to multiple vehicles in the platoon for collision avoidance in the platoon. Long ago, researchers from Ford [55] had a vision of inter vehicular communication for collision warning system. They had formulated a method to avoid collisions by taking into account various factors like time to a collision, time to collision avoidance, human reaction time, warning issuable time, etc.
[56, 57] focusses on rear-end collisions, either the following vehicle should be informed as to when by latest it should start braking[56] or alternately leading vehicle can accelerate at the last moment [57]. [58] focusses on longitudinal collision avoidance based on collision cone approach. The use of elastic band models to compute lateral control to ensure collision avoidance, where di erent vehicles in the vicinity asserting di erent forces on the ego-vehicle has also been proposed [59]. In [60], a xed inter vehicular distance based approach is discussed arising from the idea of the subject vehicle being towed by the vehicle in front using V2V com-munications only. The goal of the subject vehicle is to align angles using a lateral controller, and use the longitudinal controller to hard maintain a xed distance between vehicles. Achieving these goals would be di cult given communication de-lays, non-synchronisation of the information exchange of vehicles, calculation and implementation delays in real life situations with unpredictable behaviors of the leading vehicles. Shorter the turn radius, larger was the error observed.

Coordinated CAV applications

There are various applications based on V2V communications which will help reduce tra c and accidents in the future. Some of these applications which have interested researchers are platooning, coordinated intersection clearance, ramp merging and round about clearance. Platooning has already been detailed before, in this section, we look at the other applications. When multiple vehicles are approaching an intersection, ideally, they should clear the intersection one after another without any vehicle requiring to come to a halt. Moreover, the intersection should be occupied by the next vehicle as soon as the vehicle before quits the intersection. These intersection clearance algorithms can either be centralized or decentralized. Rios-Torres et al. [77] produce a well-written survey of work on centralized and decentralized coordination for collision avoidance and intersection clearance at intersections. Intersection clearance algorithms are evaluated by comparing their performance with the results on tra c light based approach.
[78] Gives an overview of the coordination between cooperative autonomous ve-hicles for collision avoidance in intersections and other cases. Three major issues to solve coordination problems namely, sensing (and signal processing), communi-cation, and control are discussed. Techniques used for vehicle coordination can be broadly split into 1. Rule-based coordination, 2. Optimization based coordination. Di erent centralized and decentralized control algorithms are discussed. [79, 80] in-troduces the concept of decentralized coordination problem solving and centralized coordination method for intersection management. The coordination strategy de-cided by a centralized server is optimal as all vehicles implement controls generated by the centralized server. The decentralized system has two steps:
1.: Decision order: It decides the sequence in which the participating vehicles are o ered to make their control choice. there are multiple options: FIFO, distance to the intersection, etc. to decide the ordering. 2.: Sequential control computation: Each independent vehicle makes a control decision such that it either enters the intersection after all vehicles (1… i-1) have left or enters the intersection before any of the vehicle (1…i-1) has entered.
They mention the complexity is lower in distributed approach and doesn’t depend on the number of agents in case of a decentralized approach as collision avoidance is enforced by local state constraints at given time stamps. [80] mentions the re-quirement of having an emergency mode if the participating vehicles which cannot nd any solution to the coordination problem.
[81] focuses on decentralized control approach for intersection crossing and pro-poses two algorithms. First, prioritizing access to the intersection based on their time of arrival and second, based on the inertia of the vehicle and time of arrival at the intersection. [82] focuses on intelligent intersection clearance and collision avoid-ance with the vehicle in front while approaching the intersection. They use a hybrid system between a centralized and distributed system for intersection clearance.
[83] focuses on a centralized intersection management issues where multiple ve-hicles need to coordinate to pass through an intersection. Vehicle dynamics and physical restrictions act as constraints to this issue and sum of local costs (at all vehicles) is optimized to get the best strategy to clear the intersection.
[84] Authors adopt a simple interesting approach to solve the issue of intersec-tion collision avoidance, they use Pontryagin’s minimum principle and Hamiltonian equations to get closed form solution. To compare results of their algorithm, they assume the base case where all vehicles coming from the main lane pass by freely whereas those coming from the side lane/merging lane need to come to a full halt and then enter. The aw is that the decision making order is based solely on dis-tance from the intersection (thus, when velocities of vehicles are not the same, their decision order fails). [67] proposes a centralized control algorithm robust to localization errors for in-tersection management in mixed tra c scenario involving conventional, connected and automated vehicles. Optimal intersection clearance strategy is found using branch and bound algorithm used in a tree search problem. Their proposed algo-rithm is compared with the scenario where: tra c signal is controlled using tra c sensors placed on the roads to estimate the tra c statistics, etc. [85] focuses on evaluating communication requirements in a centralized coordi-nated intersection management system for CACC vehicles; they provide results in terms of transmission power, the probability of reception of a packet in the uplink and in the downlink for designing centralized controllers. Whereas [86] evaluates the centralized control algorithm for autonomous vehicle intersection clearance under unreliable uplink communication.
Schildbash et al. focus on a centralized Robust MPC (RMPC) based collision avoidance (intersection clearing) system for a mixed vehicle scenario [87]. The centralized controller nds safe gaps in the crossing tra c and optimizes the longi-tudinal motion to make the CACC vehicle cross the intersection (rest of the vehicles are MDVs). Authors counter uncertain MDV behavior by assuming the worst case (minimum and maximum) values of acceleration possible to create a robust MPC based technique for intersection clearance in a mixed vehicle scenario [87]. Ro-bustness comes from the fact that despite the uncertainty of the future control of MDV, the maximum and minimum acceleration of MDV is considered to have a safe intersection clearance.

Communication error

1. Communication delays impact a centralized and a decentralized controller di er-ently. In a decentralized control system, every vehicle broadcasts their state pa-rameters and control computations are done locally by each vehicle. A central-ized controller requires transmission of vehicles’ state parameters to the centralized server (computational unit) on the uplink and computed controls back on the down-link to the vehicles. Communication impairments like packet delays, packet losses and out-of-order delivery of packets can thus manifest themselves either on the uplink or on the downlink. Out-of-order delivery of packets can be addressed by discarding the newly received packet if the time stamp of a newly received packet is older than that of the last received packet [90]. The occurrence of out-of-order delivery of packets also indirectly signi es packet delays (and/or losses). propagation delays: unequal propagation delays for di erent packets causes delayed, and disordered reception and packet losses (sometimes due to very long propagation delays) are inevitable in V2X communications [91].
2. Delay introduced at the PHY layer: Multiple nodes might need to compete for the right to access the network causing an additional time delay.
3. Delay introduced due to MAC layer algorithm: Di erent data types could have di erent priorities, (introduced in Enhanced Distributed Channel Access (EDCA) as an improvement over Distributed Coordination Function (DCF)), which can introduce delays of varying magnitude as well.This paragraph looks at the modeling of communication errors in a centralized control scenario. Communication delay on the uplink is understood as feedback delay which results into a di erence in the received state and the actual state infor-mation if V2V communication is used (or di erence in measured state and actual state if onboard sensors are used). This issue arising due to communication delay is considered as model mismatch [73]. In the case of uplink packet loss, the predicted behavior of MDVs can be used to estimate MDVs’ state parameters. Control values from the last transmission on the downlink can be used to predict future states for CACC vehicles. These estimated states can be used to compute controls [92]. A centralized intersection manager coping with communication errors on the uplink is introduced in [93]. Nazari et al. compute a closed form expression for centralized in-tersection coordination of automated vehicles valid under di erent communication conditions like no packet reception, all packet reception, and few packets recep-tion on the uplink using Bernoulli’s random variable; downlink communications are assumed to be perfect [86].
   Continuous periodic uplink data transfer is assumed, and event-based down-link communication is proposed [94] to reduce the load on the network. Delay threshold of a downlink communication channel which allows collision free control of vehicles is analyzed. At the end of the threshold, as there would not be any control information at the vehicle, emergency braking must be activated. Communication requirements based on control algorithm to ensure collision avoidance is discussed [94]. The impact of communication disturbances on centralized and de-centralized controllers impacting a centralized intersection clearance algorithms has also been surveyed [77]. Design guidelines for both uplink (whereby vehicles send intentions to the central controller) and downlink (where the controller prescribes vehicles of safe control actions) are suggested based on the communication system analysis for the centralized intersection crossing coordination [85].
   Team AnnieWAY observed that vehicles not transmitting anything or some outdated data result into unavailability of communicated information, resulting in problems during cooperation during the Grand Cooperative Driving challenge [95]. Authors assume, in case a packet is not received, it will be retransmitted, thus e ec-tively assuming, di erent communication delays for di erent vehicles and no packet losses [96]. End-to-end communication delays of the IEEE 802.11p communication system is assumed to be 0.09 sec by authors in [97]. Naus et al. implemented 802.11g for V2V communication and found the communication delay of 10 ms dur-ing a CACC vehicle platoon experiment [98], whereas Ploeg et al. found a delay of 150 ms in a V2V communication based on 801.11a [99]. In [100] Ploeg assumes a nominal value of 0.02 s as the CACC communication delay time. Xu et al. model communication delays in 3 types: 1. xed delay 2. distance dependant delay 3. random delay. Distance dependent delay is assumed to be squarely proportional to the distance between vehicles [101]. [102] assumes a communication delay of 22 ms, and they design a min-max MPC controller robust to communication delays. Dey et al. [103] review the impact of communication delays on di erent control algorithms operating ACC and CACC vehicles in a decentralized control scenario.

Perception response time

Two major human factors a ecting MDVs are the reaction time and the limited visibility. This perception response time (PRT) is added to imitate response delay of a human driver [32,141]. We assume MDV’s visibility to be limited to the vehicle in immediate front only. We de ne ti;i 1 and ti;1 as the pair of PRT of a MDV i compared to the vehicle in front and the delay vehicle i shows before it starts to brake with respect to the rst vehicle respectively. ti;1 is also thus called the e ective perception response time. Moreover, we assume MDV i would react ti;i 1 seconds after vehicle i 1 and ti;1 seconds after vehicle 1. And ti;1= ti;i 1+ti 1;i 2+…+ t2;1 if all 2; 3::i front vehicles are MDVs (refer to Fig. 2.2).
CACC vehicles (labeled CACC in the diagram) are assumed to be able to imple-ment received controls synchronously. In heterogeneous tra c where there is a mix of CACC and MDVs, the tprt of the MDV would depend on the number of vehicles between itself and the immediate CACC vehicle in the front. From the gure, we can see that for vehicle 3 (i=3), t3;1 = t3;2 + t2;1. Although the perception response time of vehicle 3 is t3;2 as it reacts t3;2 s after vehicle 2, the e ective perception response time is t3;1. The fourth vehicle is CACC and thus can begin reacting at approximately the same time as the rst CACC vehicle. Whereas for i =5, t5;1 = t5;4 as the vehicle in front is a CACC vehicle.

Obstacle detection and gathering vehicle information

There are multiple ways of gathering state information of MDVs. First: MDVs have onboard units (OBUs) which can be plugged into On-board diagnostics (OBD) port or the CAN bus to retrieve state parameters. Second: A CACC vehicle can use its sensors like radars and lidars to detect where the neighboring vehicles are. If those vehicles are not communicating, they can be assumed to be MDVs else, CACC enabled. In this way, state information of all vehicles can be obtained and transmitted to the centralized controller.
Next, we introduce two ways of detecting the scenario with a broken vehicle (obstacle) or an intersection. Case A – the leading vehicle is a CACC vehicle:
(1) There is a broken vehicle on the road which is transmitting DENMs or the RSU at the intersection is transmitting messages to the CACC vehicle approaching the intersection. (2) the CACC vehicle uses its sensors or cameras. In this case, the probability of awareness depends on the probability of receiving a message or the probability of sensing, at a particular distance. This probability of awareness thus changes with distance and depends on factors like communication channel occupancy, atmospheric conditions, etc. When two vehicles are approaching one another, or a vehicle is approaching an obstacle, the distance reduces gradually over time. The probability that the communication succeeds when they are far is small. Case B – the leading vehicle is a MDV: There is a broken vehicle or an intersection which needs to be detected by a human driver. The probability of awareness, in this case, depends on the attentiveness and visibility of the driver to perceive and understand the scenario.
This work does not focus on how the information is extracted, it focuses on processing and using the information. We consider noti cation distance as the distance at which the rst message is received (or the object is sensed) by the CACC vehicle, or the vehicle is detected by the MDV. A range of noti cation distances is used to cover di erent values of awareness at di erent distances to the obstacle or the intersection.

Table of contents :

1 Introduction 
1.1 General Introduction
1.2 Methodology
1.3 Contribution
1.4 Structure of thesis
2 State of the Art 
2.1 Collision avoidance between vehicles
2.1.1 Platooning
2.1.2 Collision avoidance in platooning
2.1.3 Vehicle Following Models
2.1.4 Coordinated CAV applications
2.1.5 Centralized vs decentralized controller
2.2 Types of errors
2.2.1 Communication error
2.2.2 Localization error
2.2.3 Control error
2.2.4 Model mismatch
2.2.5 Perception response time
2.2.6 Achievable braking capacity
2.3 Problem formulation
3 Analysis of Centralized Controller Operation 
3.1 Centralized Controller Operation
3.1.1 Obstacle detection and gathering vehicle information
3.1.2 Model predictive control basic principle
3.1.3 Mathematical formulation of the MPC Controller
3.2 Modeling Errors inuencing the centralized controller
3.2.1 Model Mismatch
3.2.2 Control Error
3.2.3 Localization Error
3.2.4 Communication Error
4 Robust Centralized Controller 
4.1 Centralized controller conguration
4.1.1 Simulation parameters
4.2 Buer
4.3 Robustness to Model Mismatch
4.3.1 Interface with Driving simulator
4.4 Impact of Control errors
4.5 Robustness to Communication error
4.5.1 Evaluation set 1
4.5.2 Evaluation set 2
4.6 Robustness to Localization error
4.6.1 Controller model robust to localization error
4.6.2 Evaluation of the robust controller
4.6.3 Warning for enhanced safety
4.7 Robustness of the controller to various errors
4.7.1 Communication Overhead
4.8 Parameter Sensitivity
5 Conclusions and Perspectives
5.1 Conclusions
5.2 Perspectives
A List of Publications and Contributions 

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