A novel approach of overtaking maneuvering using modified RG method.

Intelligent and safe overtaking maneuvering is always a challenging task for autonomous vehicles. This paper proposes and experimentally implements a novel approach of overtaking maneuvering using modified form of Rendezvous Guidance (RG) algorithm for trajectory planning and obstacle avoidance, considering driver safety and comfort during autonomous overtaking. The simulations for all possible scenarios are conducted to ensure the effectiveness of proposed modified RG algorithm. These scenarios involved presence and absence of obstacle vehicle in overtaking lane alongside leading vehicle in driving lane. In addition, the enhanced performance of modified RG algorithm is established over conventional RG algorithm by comparative analysis. The results indicate that overtaking maneuvering period could be decreased by 10% using a modified RG algorithm and vehicle will cover less distance to complete overtaking. The efficacy of proposed method is justified by performing experiments using mobile robots. The experimental results and simulation results of modified RG algorithm are compared, and their plots are almost identical.

Introduction

The sales of Autonomous vehicles have been a center of attention for researchers nowadays as it enables vehicles to perform regular driving tasks automatically. Automatic driving has numerous advancements within the domain of lane-keeping, distance maintenance, cruise control, and lane departure, etc. Such improvements were vital in ensuring safety and comfort but unfortunately, several challenges are still associated with autonomous vehicles [1-4]. The most significant amongst them is decision making. Overtaking is one of the arduous tasks that come under the umbrella of decision making. It involves lateral and longitudinal motion to avoid collisions and includes various other tasks like lane-changing, lane-keeping, and returning to driving lane [5].

Previously proposed approaches were quite promising in terms of handling moving obstacle avoidance problems. Incremental search algorithm and sampling-based trajectory planning method like RRT is proposed for ensuring safe trajectories while overtaking [6, 7]. The Model Predictive Control (MPC) has been utilized in [8] by having non-convex avoidance constraints in the optimization framework that limits the uniqueness and feasibility of an obtained solution. Authors in [9, 10] employed potential field method for trajectory generation by compromising on vehicle user comfort, emergency situations, and reaction of other (obstacle and vehicle to be overtaken) vehicles. Similarly, receding horizon control in [11] modeled surrounding vehicles as Markov Gaussian Processes and multi-policy decision making in [12] via geometric partition through Voronoi cells deals with lane changing problem but had same aforementioned drawbacks of user comfort, emergency situations, etc. In [13], authors proposed a practical approach for driver assistance during overtaking by analyzing the acceleration and lane of next vehicle from opposite direction. Radar and video camera sensors are used in sensors fusion concept. But it failed when multiple vehicles have same acceleration and moving in different lane. In [14], follow the gap algorithm was proposed for vehicle overtaking during tense situations i.e., multiple vehicles in same line and having same speed. Results were compared with sin-X algorithm [15] and were found comparatively better. But follow the gap algorithm is not efficient when next vehicle varies speed regularly or changes lane due to heavy traffic density. X-sin function uses mathematical function for analyzing the scenario for comfortable overtaking maneuvering. It calculates longitude and lateral position of the ego vehicle. Technology independent sensor (TIS) is implemented via MATLAB and Pre-Scan software’s by [16], which scan geometries of the vehicles to assists driver in safe overtaking maneuvering. Acceleration and vehicle direction were not considered which reduced the accuracy. Hierarchical overtaking technique for driver assistance is proposed in [17] which uses acceleration and speed signals from the surrounding vehicles and uses them in clustering methods to achieve high probability density function, which results in predicting expected motion. Computed reference is then tracked using the Linear Parameter Varying (LPV) control design method to guarantee safe motion. This proposed technique is very complex and requires lot of computational power. It can handle high density traffic interactions. [18] proposed deep learning-based reinforcement framework method for generating vehicle motion model, but huge data set is required for training safe overtaking safe maneuvering strategy. In [19] trajectory optimization for safe overtaking is performed via behavior and trajectory planning algorithm. This technique minimizes intrusion onto the adjacent lane. The proposed technique is very slow in performance therefore it is not adopted for further implementation. But this technique is more accurate than all mentioned techniques. [20] proposed brix model for overtaking maneuvering of two tankers. Moment acting on the ship hulls and maneuvering motion were analyzed for safe overtaking. Special function was established for calculating minimum distance, but this technique is only valid for two vehicles. Behavior of 3010 vehicles were analyzed in [21] during overtaking in China. Nonparametric survival analysis was performed to model the overtaking time before conducting log-rank test. It was found that conventional vehicle is faster in overtaking than electrical vehicles, as well as men are also safer than women during overtaking. Dynamic trajectory planning algorithm is prosed in [22], in which overtaking maneuvering trajectory is divided into short time trajectories to manipulate safe overtaking. This method becomes complex for three lane road. In [23] overtaking maneuvering is taken as an optimization problem, pontryagain’s minimum principle is used to reduce the fuel consumption by 18% during overtaking period. However, constraints of the said scenario do not fit for every vehicle and high-density traffic. In [24] optimal and shortest path selection for overtaking maneuver is done using optimization algorithms, but this is only valid when a single vehicle has many choices for overtaking and have very low traffic density on road.

Therefore, it is the need of an hour to devise a solution that shall address the safety and user comfort problems while overtaking.

The article proposes a motion planning method to enable pursuer vehicle to perform optimal decision for overtaking maneuver based upon Rendezvous Guidance (RG) technique. Leading vehicle changes speed and lane regularly, which is modeled in expected six scenarios. Initially, the scenario of an absence of any other vehicle in overtaking lane is considered. Once the distance between two vehicles gets 2s, RG algorithm creates imaginary target points ‘M’, ‘M’, and ‘M’ to ensure successful overtaking of leading vehicle ‘L’ by chaser vehicle ‘C’. The second scenario involves blocking vehicle ‘OB’ in overtaking lane which inhibits ‘C’ to overtake ‘L’. Therefore, proposed algorithm comes into effect and allows ‘OB’ to overtake ‘L’ first to create a room for ‘C’ to overtake ‘L’. The third scenario involves cancellation of overtaking decision after ‘C’ gets into overtaking lane. This situation arises when ‘L’ starts accelerating and the velocity of ‘L’ becomes greater than ‘C’ and eventually, ‘C’ needs to get back in driving lane. The final situation is complex in comparison with previous situations as it involves multiple vehicles ‘L’ and ‘L’ in driving lane and also the blocking vehicle ‘OB’ in overtaking lane. In this scenario, RG algorithm will not allow ‘C’ to cross both vehicles straight away as it waits for a particular distance between ‘OB’ and ‘L’- ‘L’ to ensure the ‘C’ gets safely back into a driving lane after overtaking ‘L’ and ‘L’.

The aforementioned scenarios were simulated on RG technique and its modified form. The movement in highway is highly predictable therefore, instead of using Rendezvous line (RL), velocity line (VL) is used for efficient lane changing and overtaking decision making. A comparative analysis has been performed for RG technique and its modified form via simulations. The results demonstrate that modified form of algorithm allows 10% less time for chasing vehicle to overtake a leading vehicle and thus less distance is covered by chasing vehicle while overtaking. The experimental setup for comparison of both approaches are also elaborated by a specification of hardware equipment involved. Two different experiments are performed for justification of simulated results of modified RG algorithm.

The major contribution of the article are as follows.

Devising a methodology based upon RG technique that not only performs well by treating vehicle in driving lane as an obstacle but also yields effective results in presence of blocking vehicles in overtaking lane alongside leading vehicles in driving lane.

A novel modified form of RG technique is proposed and its comparative analysis with conventional RG technique is also conducted to validate the effectiveness of the proposed approach.

Modified RG technique is also compared with conventional offline technique.

Four different scenarios are modeled, which caters almost maximum possible situations.

The uniqueness of manuscript is portrayed by an attempt to address the scenario of overtaking which involves multiple vehicles (driving and overtaking lane) through RG technique and proposed modified form of RG technique. The proposed technique yields better results as it allows 10% less time for overtaking. In addition, the decisions in these complex overtaking scenarios have always been made considering the safety and comfort of chasing vehicle user.

The manuscript comprises of six sections. The research scenario is introduced in first section while the second section elaborates the problem description and proposed approach to tackle the discussed problem. The overtaking scenarios and decision making are highlighted in third section. Section IV covers the simulation results for all scenarios using modified RG algorithm and comparative analysis of modified and conventional RG algorithm. The experimental results and simulation results are compared in fifth section to verify the applicability of proposed algorithm alongside the description of the experimental setup. Finally, the manuscript is concluded in sixth section.

Problem formulation

As mentioned earlier, the primary objective of our research work is to devise a methodology which provides safety and comfort to vehicle users while overtaking. Therefore, it is deemed necessary to discuss user comfort in detail while overtaking as we will be incorporating parameters in our proposed mathematical model to ensure the comfort of vehicle owners. The ride comfort is evaluated upon acceleration and angular motion. The sudden motion causes intense comfort disturbance and high lateral acceleration will result in indirect comfort disturbance [25]. To ensure a comfortable ride, the value of lateral acceleration and axial acceleration shall not exceed 1.25 m/s2 and 5 m/s2 respectively [26, 27]. The time interval of 2s is considered an ideal value for a safe distance between two vehicles to ensure braking operation without collision [28-30]. The values of ride comfort and braking distance are incorporated as constraints in a mathematical model. The model is focused upon the development of safe and optimal trajectory for lane changing to ensure seamless overtaking process.

Three and four phase overtaking maneuver

To understand the mathematical model in detail, the description of terminologies and potential scenarios involved is required. We begin with a simplest lane changing scenario in which we have labeled the chasing vehicle as ‘C’ and leading vehicle with ‘L’. If ‘C’ wants to overtake ‘L’ then the velocity of chasing vehicle must be greater than a leading vehicle. The velocity of ‘C’ is indicated by and velocity of ‘L’ is given by . It is assumed that there is no blocking vehicle ‘OB’ in overtaking lane and thus the process of overtaking is reduced to three stages only. These three stages are overtaking or lane changing maneuver i.e. maneuver from driving lane to overtaking lane, travelling in overtaking lane, and ultimately returning to the driving lane.

If the blocking vehicle ‘OB’ is present in overtaking lane, it will not allow ‘C’ to overtake. Therefore, ‘C’ has to adjust its velocity in a manner so that once overtaking lane is cleared by ‘OB’, only then ‘C’ proceeds to overtake ‘L’. In case of presence of a blocking vehicle in overtaking lane, the stages of overtaking maneuver are increased by 1 as fourth stage involves velocity adjustment. The whole scenario is depicted in Fig 1. Upper lane in Fig 1 is overtaking lane while lower lane is travelling lane. Same is the case in all scenarios figure.

Fig 1

Stages of overtaking in complex scenario.

Rendezvous Guidance technique

The optimal overtaking maneuver for chasing vehicle can be planned via Rendezvous Guidance (RG) technique. RG technique is originally introduced for spaceships rendezvous missions with space stations and asteroids [31, 32]. RG technique has delivered promising results for non-maneuvering targets interception which corresponds to vehicles moving in a straight path in our case. Therefore, this technique is adopted in our research article, in which six different scenarios are analyzed. However, there are two major limitations associated with RG technique. The first limitation is RG technique is designed for rendezvous missions with a particular target and it does not include overtaking the target. Secondly, numerous constraints are attached to chasing vehicle which are not associated with spaceships, so an amendment is required in technique to make it useful for optimal overtaking maneuver.

The first limitation is addressed by the introduction of a shadow target. Shadow target will guide ‘C’ through all the phases of overtaking to ensure user comfort and safety. The location of shadow target will be defined according to the location of vehicle which is to be overtaken. The second limitation is addressed by adopting RG technique proposed in [33-37] for robotic autonomous vehicle interception. The technique mentioned in [33-37] enabled us to gather relevant information of ‘C’, ‘OB’, and ‘L’ for generation of single acceleration command to be used for ‘C’. The acceleration command would enable ‘C’ to overtake ‘OB’ and simultaneously avoiding ‘L’ in a time-optimal manner. The acceleration command is calculated via velocity-matching with shadow target considering constraints applied due to chasing vehicle dynamics and user comfort. The shadow target will also avoid vehicle collision in overtaking lane by avoiding obstacle. When RG is compared to other overtaking techniques, it results in higher accuracy, less complexity and can model maximum scenarios.

Rendezvous Guidance law based trajectory

The understanding of RG law-based trajectory requires discussion of relevant parameters in two-dimensional geometry to help readers grasp the functionality of RG technique. Consider two-dimensional geometry which involves ‘C’ and shadow target having a velocity of and respectively. The imaginary line connecting ‘C’ and shadow target is referred as Line-of-Sight (LOS). The angle created by LOS with fixed reference (x-axis) as given in Fig 2 is given by λ which is calculated as given in (1).

Fig 2

Construction of Rendezvous set.

Where h is the distance between chasing vehicle and shadow target in a lateral direction and l is the distance between them in an axial direction. The length of LOS is defined as range ‘’. As per the parallel navigation law, the LOS direction shall remain constant with respect to non-rotating frame while chaser approaches the target. Therefore, the relative velocity indicated by ‘‘ should remain parallel to LOS. If the particular rule stays intact throughout the motion of chasing vehicle, the distance between chaser and target would decrease till zero. The parallel navigation law is expressed by (2) and (3).

Eq (2) ensures that and remains collinear while (3) ensures that C is not receding from the target. By solving (2) and (3) in parametric form would result in (4).

In (4), a is a positive real number. The instantaneous relative velocity can now be written in the form of chasing vehicle and shadow target velocities which is given in (5).

By substituting (4) into (5), the resulting expression is given in (6).

The primary objective of trajectory planner is to obtain optimal chaser velocity command as per parallel navigation law for upcoming instant command. The value of is obtained through proximity sensors installed on a vehicle. By substituting vector in (6) would result in locus for the chasing vehicle velocity vectors that lie in semi-line parameterized by a. This semi-line is known as Rendezvous line. The endpoints of velocity vectors and in indicate the positions of chaser and shadow target after a unit time. If chasing vehicle consistently follows velocity command that lies on RL, the direction of LOS remains constant which guarantees positional matching of chasing vehicle and shadow target. To find the value of a given that velocity matching is realized, we assume that the acceleration of chaser in a given direction is indicated by A. The simultaneous reduction of velocity and position difference in the direction of LOS could be written in the form of (7).

Where is the magnitude of the maximum allowable closing velocity, and t is the remaining time-to-intercept from the current instant. The maximum instantaneous allowable closing velocity is obtained by solving (7) which is given in (8).

The maximum closing velocity, as imposed by the frequency of velocity command generation by the trajectory planner for a fast asymptotic interception, is given in (9).

The value of n in (9) is determined empirically. The final value of permissible closing velocity component for velocity command is obtained by considering (8) and (9) which results in (10).

The endpoints of all velocity command vectors on RL that have a closing velocity component smaller than constitute a line segment extending from = to which is given as red line in Fig 2. This set of points is referred to herein as the Rendezvous Set (RS). The velocity may not be achieved by chasing vehicle within time interval Δt due to constraints of user comfort and vehicle dynamics. Therefore, we need to determine a feasible region which includes a set of velocities that can be achieved by chasing vehicle within Δt considering kinematic, dynamic, and passenger comfort constraints. This region is determined by imposing a limitation on lateral acceleration of chasing vehicle. We assume that maximum value for lateral acceleration of chaser vehicle is defined by (11).

Where a is maximum lateral acceleration, , h is the width of the lane, and ϑ is the maximum angle turning angle of chaser vehicle.

We assume that δ is current heading angle of chasing vehicle and by taking into consideration kinematic and dynamic vehicle constraints alongside user comfort constraints, feasible velocity region (FVR) is realized which is given in Fig 2. The velocity selected by vehicle for an interval Δt is the component of RS that lies within FVR. The maximum velocity with FVR is (t + Δt) as given in Fig 2. If chaser vehicle obeys the velocity commands having maximum velocity within FVR and velocity is also a component of RS then time-efficient interception can be realized.

Modified Rendezvous Guidance algorithm

The RG algorithm may not achieve predictability when it comes to robots but as the vehicle movements on highways is predictable so we can increase the velocity of ‘C’ to reduce lane changing and overtaking time. However, increase in velocity of ‘C’ is limited by lateral acceleration and user comfort constraints for the generation of trajectory command. The predictability provides us with an advantage to define a velocity line (VL). VL originates from a starting point of RL and makes an angle of ϑ with x-axis. Therefore, utilization of VL instead of RL for velocity command of upcoming instant ensures enhanced efficient lane changing and overtaking.

Overtaking maneuver decisions for multiple scenarios

As we have mentioned earlier that under the description of RG technique, shadow target will guide ‘C’ through all phases of overtaking. Therefore, a marker target (‘M’) which corresponds to shadow target is utilized for guiding ‘C’. The location of ‘M’ depends upon the instant of lane changing and location of ‘OB’. Initially, vision module gathers position and velocities of vehicles lies within the specific range. The information of vehicle velocity and position enables algorithm to compute RS and eventually yields a closing velocity component . The desired velocity to be realized by ‘C’ in upcoming instant via modified RG method is represented by . To obtain desired value of acceleration, the acceleration command ( ) is fed to ‘C’ for next instance. must lie within the FVR and if that’s not the case then optimal velocity is selected from RS for the upcoming instance. A new feasible velocity region (NFVR) is formed by the intersection of VL with FVR which contains velocities to be generated for upcoming instant commands. 1 and 2 are two intersection points where VL intersects FVR as given in Fig 3 and selection of any point within NFVR ensures a rendezvous with ‘M’. The selection of maximum velocity from NFVR which is 1 will enable ‘C’ to get nearest to ‘M’. Therefore, for upcoming instant, the required velocity for ‘C’ could be written in the form of (12).

Fig 3

Generation of command for chasing vehicle.

The corresponding lateral acceleration command could be written in terms of (13).

Overtaking scenario 1: Absence of vehicle in overtaking lane

The absence of any vehicle in overtaking lane makes overtaking maneuver quite simple for ‘C’. As the distance between ‘C’ and ‘L’ gets to 2.5s, the proposed algorithm will check if there is any vehicle in overtaking lane or not? If an overtaking lane is free of any obstacle vehicle, the vehicles will continue to move forward till the distance between them gets 2s. The RG algorithm will come to effect at a distance of 2s by the creation of imaginary marker targets ‘M1’, ‘M2’, and ‘M3’ for guiding ‘C’ throughout the overtaking maneuver. During overtaking, the velocity of ‘C’ remains constant within each particular phase. The pictorial representation of overtaking scenario 1 is given in Fig 4. Upper lane is overtaking lane while lower lane is travelling lane.

Fig 4

Position of marker targets for scenario 1.

Overtaking scenario 2: Presence of vehicle in overtaking lane

The presence of any vehicle in overtaking lane makes overtaking maneuver a bit tricky for ‘C’. As the distance between ‘C’ and ‘L’ gets to 2.5s, the proposed algorithm will check if there is any vehicle in overtaking lane or not? If overtaking lane has any obstacle vehicle ‘OB’, then ‘C’ will wait for ‘OB’ to overtake ‘L’ first so that it can perform overtaking maneuver with safety. In this scenario, ‘M’ is created 2s behind ‘L’ and velocity of ‘M’ is equal to ‘L’. Once overtaking lane is cleared of ‘OB’, ‘C’ can start overtaking maneuvering. However, in this scenario, velocities of ‘M’ and ‘M’ can either be set to initial velocity of ‘C’ which is = or initial velocity of ‘OB’ which is = . The selection between the two velocities will be done by utilization of velocity with less magnitude. The pictorial representation of overtaking scenario 2 is given in Fig 5.

Fig 5

Position of marker targets for scenario 2.

Overtaking scenario 3: Quitting overtaking maneuver

The overtaking scenario may get complicated when ‘C’ starts overtaking maneuver and suddenly ‘L’ accelerates to increase its velocity such that its velocity gets greater than ‘C’ ( > ). Therefore, ‘C’ cannot overtake ‘L’ in this scenario so algorithm will decide to abort the overtaking process and allows ‘C’ to travel with the same velocity in overtaking lane till the distance between them gets 3s. Now, the algorithm creates a marker target ‘M’ in driving lane to allow ‘C’ to get back into the driving lane. The position of ‘M’ is stationary and a pictorial representation of overtaking scenario 3 is given in Fig 6.

Fig 6

Position of marker targets for scenario 3.

Overtaking scenario 4: Overtaking maneuvering with multiple vehicles

The overtaking scenario gets complex with the involvement of overtaking multiple vehicles and presence of ‘OB’ in overtaking lane. By assuming a scenario in which there are two vehicles ‘L’ and ‘L’ in driving lane that ‘C’ needs to takeover and ‘L’ is ahead of ‘L’. The velocities of ‘L’ and ‘L’ are and respectively. ‘OB’ is also moving in overtaking lane so due to safety considerations; RG algorithm will not permit ‘C’ to perform overtaking straightaway. In addition, the distance between ‘L’ and ‘L’ is lower enough so that ‘C’ cannot return to driving lane before overtaking both vehicles. Therefore, under such circumstances, a new location of marker is added 1s ahead of ‘L’ which is given as ‘M’ in Fig 7. Therefore, the given set of markers in Fig 7 will allow ‘C’ to overtake ‘L’ and ‘L’ alongside avoiding collision with ‘OB’ in overtaking lane.

Fig 7

Position of marker targets for scenario 4.

Simulation results

To demonstrate the effectiveness of the proposed algorithm, several simulations were conducted to confirm the accuracy of their response to the location of leading and obstacle vehicles. Various combinations of ‘C’, ‘L’, and ‘OB’ parameters have been tested to check the efficacy of proposed algorithm. The obtained results indicate that proposed algorithm is effective in guiding chasing vehicle to perform overtaking maneuver comfortably without any sort of accidents. The comparative analysis of proposed algorithm and RG algorithm is also performed to highlight the benefits of modified RG algorithm over the conventional RG algorithm.

Modified RG algorithm results

The simulation results for all four overtaking scenarios with modified form of RG algorithm are discussed separately.

Overtaking scenario 1. In this scenario, chasing vehicle needs to overtake leading vehicle and there is no blocking vehicle obstructing the path of chasing vehicle in overtaking lane. The preliminary velocity of ‘C’ is taken as 30 m/s and that of ‘L’ as 20 m/s. ‘L’ is accelerating to achieve a velocity of 22 m/s. The velocity and path of ‘C’ are given in Fig 8. The lateral and axial velocities of chasing vehicle are given in Fig 9. The obtained values of time taken for overtaking, maximum velocity achieved, maximum lateral acceleration, and maximum axial acceleration are listed in Table 1.

Fig 8

Profiling–velocity and path for C in scenario 1.

Fig 9

Lateral and axial velocities of C in scenario 1.

Table 1

Summarized results for scenario 1.

S.No.	Parameters	Values
1	Total time taken for overtaking (s)	16.3
2	Maximum velocity achieved (m/s)	34
3	Maximum lateral acceleration (m/s2)	1.04
4	Maximum axial acceleration (m/s2)	2.5

Overtaking scenario 2. In this scenario, chasing vehicle needs to overtake leading vehicle and blocking vehicle is also obstructing the path of chasing vehicle in overtaking lane. The velocities of ‘L’ and ‘OB’ are varying sinusoidally and had an initial velocity of 20 m/s and 27.5 m/s respectively. The velocity and path of ‘C’ are given in Fig 10. The lateral and axial velocities of chasing vehicle are given in Fig 11. The driving lane and passing lane obstacle velocities for chasing vehicle are given in Fig 12. The obtained values of time taken for overtaking, maximum velocity achieved, maximum lateral acceleration, and maximum axial acceleration are listed in Table 2.

Fig 10

Profiling–velocity and path for C in scenario 2.

Fig 11

Lateral and axial velocities of C in scenario 2.

Fig 12

Driving lane and passing lane obstacle velocities.

Table 2

Summarized results for scenario 2.

S.No.	Parameters	Values
1	Total time taken for overtaking (s)	26.8
2	Maximum velocity achieved (m/s)	34
3	Maximum lateral acceleration (m/s2)	0.88
4	Maximum axial acceleration (m/s2)	2.35

Overtaking scenario 3. In this scenario, once chasing vehicle enters overtaking lane and starts to overtake leading vehicle, the leading vehicle accelerates, and its velocity gets greater than the chasing vehicle. Therefore, chasing vehicle needs to quit overtaking maneuver and get back in the driving lane. As ‘C’ starts to overtake ‘L’, ‘L’ starts accelerating to a velocity of 34 m/s which leads ‘C’ to abort overtaking maneuver. The velocity and path of ‘C’ are given in Fig 13.

Fig 13

Profiling–velocity and path for C in scenario 3.

Overtaking scenario 4. In this scenario, chasing vehicle needs to overtake multiple (two) leading vehicles in driving lane and blocking vehicle is also obstructing the path of chasing vehicle in overtaking lane. The velocities of ‘L’ and ‘L’ are 20 m/s and the preliminary velocity of ‘C’ is 30 m/s. The velocity and path of ‘C’ are given in Fig 14.

Fig 14

Profiling–velocity and path for C in scenario 4.

Comparative analysis of modified and conventional RG algorithm

The comparative analysis required that simulations which were performed for modified RG algorithm should be repetitively performed for the conventional RG method. All four scenarios are utilized for comparison of both methods. Tabular representation of the parameter values of the total time taken for vehicle to perform overtaking maneuver and distance travelled during overtaking maneuvering are done for each scenario for readers to grasp which method performs better.

Overtaking scenario 1. In this scenario, chasing vehicle needs to overtake leading vehicle and there is no blocking vehicle obstructing the path of chasing vehicle in overtaking lane. ‘L’ is deaccelerating to drop its velocity from 22 m/s to 18 m/s. The parameter values for both methods are listed in Table 3. In this case, time taken for overtaking maneuver in modified RG method is less which means that chasing vehicle has to cover less distance to complete overtaking maneuvering.

Table 3

Comparison of time and distance in scenario 1.

Parameters	Modified RG Method	Conventional RG Method
Total time (s)	11.5	12.7
Distance travelled (m)	368	390

Overtaking scenario 2. In this scenario, chasing vehicle needs to overtake leading vehicle and blocking vehicle is obstructing the path of chasing vehicle in overtaking lane. ‘L’ is moving with constant velocity and ‘C’ is accelerating which increases its velocity from 25 m/s to 30 m/s. The parameter values for both methods are listed in Table 4. In this case, time taken for overtaking maneuver in modified RG method is less which means that chasing vehicle has to cover less distance to complete overtaking maneuvering.

Table 4

Comparison of time and distance in scenario 2.

Parameters	Modified RG Method	Conventional RG Method
Total time (s)	28.4	28.8
Distance travelled (m)	728	736

Overtaking scenario 3. In this scenario, chasing vehicle wants to overtake leading vehicle but speed of leading vehicle is more than chasing vehicle so chasing vehicle comes down to lower lane without overtaking. Initially starting speed of chasing vehicle is 20 m/s while speed of vehicle to overtaken is 25 m/s. during overtaking speed of vehicle to be overtaken increases to 30 m/s. The parameter values for both methods are listed in Table 5. In this case, time taken for returning to travelling in modified RG method is less which means that chasing vehicle has to cover less distance to complete overtaking maneuvering.

Table 5

Comparison of time and distance in scenario 3.

Parameters	Modified RG Method	Conventional RG Method
Total time (s)	24	26
Distance travelled (m)	730	740

Overtaking scenario 4. In this scenario, chasing vehicle needs to overtake two leading vehicle and there is also blocking vehicle obstructing the path of chasing vehicle in overtaking lane. Initial velocity of chasing vehicle is 25m/s while vehicles to overtaken are having velocity of 20m/s. The parameter values for both methods are listed in Table 6. In this case, time taken for overtaking maneuver in modified RG method is less which means that chasing vehicle has to cover less distance to complete overtaking maneuvering.

Table 6

Comparison of time and distance in scenario 3.

Parameters	Modified RG Method	Conventional RG Method
Total time (s)	22	25
Distance travelled (m)	690	725

Comparative analysis of RG algorithm with conventional off-line overtaking method

The comparison of the proposed on-line methodology was also carried with an off-line method presented in [38]. Since the technique proposed cannot cope with variations in obstacle velocity, for this comparison, both vehicles are moving with constant velocities. Simulations are performed for all four scenarios in this comparison. The results of the simulations for the comparison are presented via table. The table shows the time taken and the distance travelled using both methodologies.

Scenario 1. There is no obstacle vehicle present which could delay/restrict the overtaking manoeuvre. In this scenario, vehicle to be overtaken, is moving with a constant velocity of 20 m/s and the velocity of C is 30 m/s. Table 7 shows basic overtaking parameters of both methods for scenario 1.

Table 7

Basic overtaking parameters for scenario 1 –a comparison.

	Modified RG Method	Off-Line Method
Total time (s)	13.5	15.5
Distance travelled (m)	430	466

Scenario 2. There is another vehicle present in overtaking lane which could delay/restrict the overtaking manoeuvre. In this scenario, both vehicles are moving with constant velocities of 20 m/s and 25 m/s, respectively. The starting velocity of C is taken as 30 m/s. Table 8 shows basic overtaking parameters of both methods for scenario 2.

Table 8

Basic overtaking parameters for scenario 2 –a comparison.

	Modified RG Method	Off-Line Method
Total time (s)	34.2	39.5
Distance travelled (m)	844	966

Scenario 3. Chasing vehicle is having slow speed as compared to vehicle to be overtaking, so overtaking does not happen. In this scenario, chasing vehicle and vehicle to be overtaken are moving with initial velocities of 25 m/s and 20 m/s, respectively. Velocity of vehicle to be overtaken increases to 30m/s, so the chasing vehicle comes back to travelling lane. Table 9 shows basic overtaking parameters of both methods for scenario 2.

Table 9

Basic overtaking parameters for scenario 3 –a comparison.

	Modified RG Method	Off-Line Method
Total time (s)	23	27
Distance travelled (m)	720	770

Scenario 4. Chasing vehicle is to overtake vehicle with double lanes and in the presence of obstructing vehicle in overtaking lane. Table 10 shows basic overtaking parameters of both methods for scenario 2.

Table 10

Basic overtaking parameters for scenario 4 –a comparison.

	Modified RG Method	Off-Line Method
Total time (s)	22	25
Distance travelled (m)	690	725

Experimental setup

Numerous experiments were conducted by having different numbers of chasing and leading vehicles. With the help of these experimental trials, the comparative analysis of modified and conventional RG method is performed. The experimental results clearly indicate a 10% decrease in overtaking maneuver period by modified RG method. The experimental trials are conducted using robots and it illustrates almost similar behavior as its simulations. The image of workspace is captured and processed to gather information of all objects within workspace. The information gathering from workspace is performed with the help of image acquisition and processing module of software. The extracted information is then used for trajectory planning which is used for the calculation of command to be sent to chasing vehicle.

The specifications of hardware used in the experiment are given in Table 11. The software used for the experiment comprises of three primary modules which include image acquisition and processing, trajectory planning, and communication modules. At first stage, analog CCD camera captures image of workspace and second stage involves extraction of positional information by vision algorithm. Third stage is about forwarding extracted information to trajectory planner for computation of acceleration command (real-time) for chasing vehicle.

Table 11

Experimental hardware specifications.

Component	Characteristics
Pursuer and Obstacle Vehicles	Miabot PRO BT v2 Differential-Drive mobile Robots with Bluetooth Communication
CCD Camera	Resolution: 640 × 480 pixels
	Lens Focal Length: 6 mm
	Vertical Distance from Floor: 3000 mm
Floor Workspace	2740 × 1500 mm

First experiment

In this experiment, the overtaking lane is free of blocking vehicle. The width of each lane is 160 mm, initial velocity of ‘C’ is 8 mm/s and initial velocity of ‘L’ is 6 mm/s. The image of marker targets ‘M’ are given in Fig 15 for each phase during overtaking maneuvering. The comparison between experimental and simulated results is demonstrated with the help of simulation results in Fig 16 and experimental results in Fig 17. Three experimental trials are conducted with similar parameters under identical conditions. The results demonstrate that modified RG method has less time period for completing overtaking maneuvering. The conventional RG method yields overtaking distance of 2200 mm and modified RG method results in a distance of 2000 mm. Thus, a decrease of 9.8% is witnessed using modified RG method.

Fig 15

Marker target position for situation 1.

Fig 16

Experiment 1- simulated path (a) Original RG method (b) Modified RG method.

Fig 17

Experiment 1- experimental path (a) Original RG method (b) Modified RG method.

Second experiment

In this experiment, the overtaking lane has a presence of blocking vehicle. The width of each lane is 160 mm, initial velocity of ‘C’ is 8 mm/s, initial velocity of ‘L’ is 6 mm/s, and initial velocity of ‘OB’ is 8 mm/s. The results demonstrate that modified RG method has less time period for completing overtaking maneuvering. The conventional RG method yields overtaking distance of 2227 mm and modified RG method results in a distance of 2075 mm. Thus, a decrease of 9% is witnessed using modified RG method.

The takeaways from experimental results are as follows:

In a noisy environment, RG method performs overtaking maneuvering free of accidents which illustrates the robustness of approach.

The experimental results bolster the simulation results by demonstrating similar paths with and without the presence of blocking vehicle in overtaking lane. In addition, it indicates that the proposed modified approach performs better in comparison with the conventional RG method.

Conclusion

This article presents robust and time-optimal guidance-based algorithm for trajectory planning. The proposed approach not only adapts to environmental changes rather reacts to them in an appropriate manner that is particular to a given situation. The proposed modified RG algorithm is simulated for four different scenarios which involve an absence of ‘OB’ in overtaking lane, presence of ‘OB’ in overtaking lane, single leading vehicle, and multiple leading vehicles in driving lane. The simulation results in all four scenarios reflect accident-free overtaking maneuvering completion. In addition, comparative analysis for simulation results of conventional and modified RG method is performed using two of the four scenarios discussed. The modified RG approach enables overtaking maneuvering time to decrease by 10%. After establishing the significance of modified RG method over the conventional RG method, it is required to check if the simulated results of modified method are comparable to its experimental results or not? Therefore, for similar two situations, the experimental results are compared with simulated results and experimental results support simulated results as they are almost identical. The presented modified RG method ensures accident-free overtaking in all scenarios which makes it better than off-line solutions suggested by previous works. However, future work may concentrate on comparative analysis of experimental and simulation results of both approaches with complex scenarios having multiple blocking and leading vehicles in overtaking and driving lane respectively.

(RAR)

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(ZIP)

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(ZIP)

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(RAR)

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13 Sep 2021

PONE-D-21-28089A Novel Approach of Overtaking Maneuvering using Modified RG MethodPLOS ONE

Dear Dr. Usman Ghumman,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

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Reviewer #1: Partly

Reviewer #2: Yes

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: I Don't Know

Reviewer #2: Yes

**********

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The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

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Reviewer #2: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: 1、The overtaking manoeuvre is a 2-dimensional movement. The literature review is poor. No relevant studies are introduced. At least, the following two references should be cited:

(1) Zhao, J., Knoop, V.L., Wang, M., 2020. Two-dimensional vehicular movement modelling at intersections based on optimal control. Transportation Research Part B: Methodological 138, 1-22.

(2) Bichiou, Y., Rakha, H.A., 2018. Developing an optimal intersection control system for automated connected vehicles. IEEE Trans. Intell. Transp. Syst. 20, 1908–1916.

2、RG technique is originally introduced for spaceships rendezvous missions with space stations and asteroids. Why is RG technology used in overtaking in this paper? What are the advantages of this method over traditional overtaking methods?

3、The scope of application of this method should be explained. leading vehicle may change speed, lane, etc. is this method applicable? In addition, the influence of lane changing behavior of rear vehicles on overtaking cannot be ignored.

4、The improved RG technology is better than that before the improvement. This conclusion is easy to draw. But will the improved RG model be better than the traditional classical overtaking models? The improved method should be compared with the classical overtaking models.

Reviewer #2: This paper focuses on the overtaking maneuvering for trajectory planning and obstacle avoidance. The topic is interesting, but the treatment of the methodology is weak. The authors should make great efforts to make this paper published.

The layout of this paper should obey the format.

The figures are unclear and ambiguous for readers. And it is undesirable to include so many figures in this paper.

Many relative scientific papers are missing in the Introduction section.

The contributions of this paper is unclear.

The paper contains many typos and errors of units.

The Rendezvous Guidance technique should be expressed clearly in section 2.2.

How does this algorithm distinguish the overtaking scenarios? Sometimes two or more scenarios may occur at the same time. For instance, there are multiple vehicles but the subject vehicle achieves the unsuccessful overtaking. Will this algorithm handle this situation?

In figure 24, it seems the two vehicles collide with each other, which is unreasonable.

**********

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Reviewer #1: No

Reviewer #2: No

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Submitted filename: Recommendation.docx

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9 Oct 2021

All data sets of experiments, scenarios and simulation codes along with simulation videos are uploaded upon request of editor as a confirmation of data avalibility. All the concerns and recommendations of editors and reviewers are addressed upto the best effort

Submitted filename: Response to Reviewers.docx

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1 Nov 2021

PONE-D-21-28089R1A Novel Approach of Overtaking Maneuvering using Modified RG MethodPLOS ONE

Dear Dr. Usman Ghumman,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript by Dec 16 2021 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

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If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

Jing Zhao, Ph.D.

Academic Editor

PLOS ONE

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Additional Editor Comments (if provided):

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Comments to the Author

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Reviewer #1: All comments have been addressed

Reviewer #2: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: No

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: (No Response)

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: (No Response)

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: A novel modified form of RG technique is proposed and its comparative analysis with conventional RG technique is also conducted to validate the effectiveness of the proposed approach. The proposed technique yields better results as it allows 10% less time for overtaking. Modified RG technique is also compared with conventional offline technique. The proposed technique is technically sound, and the data support the conclusions.

Reviewer #2: Although there are so many figures in the revised manuscript (which however are not shown), the presented figures convey little information and the explanation of the figures are quite poor. For instance, where is the overtaking lane and OB in Figure 7 to Figure 10?

Many figures in the compressed file are still unclear. In addition, I still hold the idea that too many figures without sufficient information may distract readers, so summarizing and combining these figures are necessary. I hesitantly appreciate the authors' action of concern 3.

I did not see 6 scenarios in this paper. There are actually 4 overtaking scenarios. Be careful when using "case", "scenario" and "situation".

As to Figure 11, the y-distance path subfigure fails to show the time in second, so relationship between the speeds and the lateral position is unclear. The same holds for the other similar figures.

The comparison between the modified and conventional RG algorithms under scenario 3 and 4 should also be included.

More comparison cases with the off-line overtaking method are encouraged, since scenario 1 is quite simple.

Why the indicator values of the modified RG method in Table 3, 5 and 6 are different? The simulation settings when comparing with the conventional RG and the off-line overtaking method are supposed to be identical.

There are still many grammar and typo errors throughout the paper. Please check.

**********

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Reviewer #1: No

Reviewer #2: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

4 Nov 2021

Minor revison and concerns of reviewer#2 are adressed.

Submitted filename: Response to Reviewers v1.docx

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10 Nov 2021

A Novel Approach of Overtaking Maneuvering using Modified RG Method

PONE-D-21-28089R2

Dear Dr. Usman Ghumman,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

Kind regards,

Jing Zhao, Ph.D.

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #2: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: (No Response)

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: (No Response)

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: (No Response)

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: (No Response)

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: A novel modified form of RG technique is proposed and its comparative analysis with conventional RG technique is also conducted to validate the effectiveness of the proposed approach. The proposed technique yields better results as it allows 10% less time for overtaking.

Reviewer #2: (No Response)

**********

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Reviewer #1: No

Reviewer #2: No

10 Dec 2021

PONE-D-21-28089R2

A novel approach of overtaking maneuvering using modified RG method

Dear Dr. Ghumman:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

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on behalf of

Dr. Jing Zhao

Academic Editor

PLOS ONE