Potential Function-Based String Stable Controller for Heavy Road Vehicle Platoons

Heavy Commercial Road Vehicle (HCRV) platooning is a promising solution to meet increasing freight transportation demands. This paper applies the concept of Artificial Potential Functions (APF) to design a string stable controller, viz., Potential Function-based String Stable (PFSS) controller, for establishing stable HCRV platoons under various operating conditions. A methodical approach for the design and selection of potential functions that would maintain the desired intervehicular spacing in the presence of perturbations has been presented. The proposed methodology has been evaluated on a complete vehicle dynamics model-based platoon framework, which would emulate actual vehicle operation. It provides string stable operation under different lead vehicle perturbation maneuvers on various road friction and slope conditions with homogeneous and heterogeneous loading conditions. Compared to existing string stable control approaches, the proposed PFSS controller has attributes such as minimal data requirement from neighboring vehicles, actuator feasible control actions, and simpler control structures.


I. INTRODUCTION
Heavy Commercial Road Vehicle (HCRV) platoons have the capability in ensuring fuel economic freight movement [1]. In an HCRV platoon, the vehicles travel one after the other with close inter-vehicular distance, such that the aerodynamic drag force between them is minimized, leading to reduced fuel consumption [2], [3]. Moreover, this would lead to efficient utilization of the available road infrastructure [4].
For string stable platoon operation, the spacing error between the vehicles should not propagate along the platoon in the event of a perturbation [5]. An effective string stable controller is crucial for collision-free autonomous platoon operation, and this area has received adequate research attention, as summarized in Table 1.
During string stable controller design, factors such as vehicle dynamics, including aerodynamic drag and rolling resistance, appropriate tire modeling, wheel dynamics, and brake/powertrain actuator dynamics, are significant while considering high-speed vehicle operation. Moreover, these factors are critical in HCRVs due to higher brake actuation The associate editor coordinating the review of this manuscript and approving it for publication was Nasim Ullah . delay and response time, higher mass variation during laden and unladen operating conditions, and more significant dynamic load transfer during braking. The current study has considered all these factors. Further, the variation in road friction coefficient (µ) has been considered to incorporate operation in different road conditions. Vehicles of different masses are also considered to characterize the heterogeneity in loading conditions that can be seen in HCRV platoons. The effect of road gradients on string stability has also been investigated since these are encountered in long haul operations. From Table 1, existing studies on string stable controllers have not explicitly considered all the aforementioned factors. In this context, the current study attempts to incorporate all these factors in the string stable control design for HCRV platoons using an Artificial Potential Function (APF) approach.
APF has been widely used in modelling interaction among robots and their environment [29]- [35]. Recent research applied APF for automotive applications [10], [11], [26], [36]. In [10], the APF control approach has been used for designing an automated driving strategy by integrating cooperative adaptive cruise control with collision avoidance and gap closing functionalities. APF has been used to design a longitudinal control scheme for the cooperative merging of car platoons [11]. Feng et al. used APF for generating safe collision-free paths for autonomous vehicles [36]. A hierarchical control scheme for fuel-efficient and collisionfree car platoons using dynamic programming and APF has been presented in [26].
In [28], the authors have incorporated factors such as vehicle dynamics and µ variations in the design of string stable controllers for HCRV platoons using sliding mode control. However, like other commonly used approaches, the controller demands individual vehicle's position, speed, and/or acceleration information from both the preceding and succeeding vehicles [8]. The practical implementation of such control methods would be highly sensing dependent. Motivated by these factors, this paper designs a string stable controller with lower data requirements from neighboring vehicles by utilizing the concept of APF. The Potential Function-based String Stable (PFSS) controller has been proposed such that it would require position and speed information from the preceding vehicle alone.
The proposed PFSS controller would synthesize control inputs (acceleration/deceleration commands) for each individual vehicle in the platoon based on instantaneous spacing and speed errors, such that the spacing error is attenuated along the platoon. The idea is to decelerate the vehicle when the inter-vehicular distance between the constituent HCRVs in the platoon becomes less than the desired threshold value to avoid collisions and accelerate the vehicle when the inter-vehicular distance becomes more than the desired threshold. Since the proposed approach does not incorporate the error propagation factor directly into the controller design as in conventional controllers [8], [28], the controller structure becomes simpler, making practical implementation easier.
The major attributes of the proposed PFSS controller, viz., error attenuation capability, minimal data requirement from the neighboring vehicles, and ability to synthesize actuator feasible control inputs, have been investigated by comparing its performance with the commonly used sliding mode based string stable controller. Further, the proposed PFSS controller has been tested extensively for its efficacy under different operating scenarios.

II. VEHICLE MODEL
The platoon is assumed to have N + 1 vehicles, with one leader and N followers (Fig. 1). The following assumptions were made in this study: • Only longitudinal dynamics of the HCRV has been considered.
• The tire model parameters and vehicle parameters were assumed to be known.
• Equal load distribution on the left and the right wheel of a specific axle of the vehicle was assumed.

A. HCRV MODEL
The equation that characterizes the leader vehicle motion iṡ where, x 0 (t) and v 0 (t) are the position and longitudinal speed of the leader vehicle, v 0 (t) being the speed that has to be followed by the follower vehicles. The position and longitudinal speed dynamics of the i th follower vehicle are represented aṡ where, τ ji (t) is the drive/brake input with j = f , r, indicating front and rear wheels, respectively, and (v i (t), τ ji (t)) represents a nonlinear function in v i (t) and τ ji (t), which is given by The mass of the i th follower vehicle is represented by m i . F xji (λ ji (t), τ ji (t)) represents the longitudinal force at the tireroad interface and is a function of the longitudinal wheel slip ratio, λ ji (t), the friction coefficient (µ) between the tire and the road, and the normal load on the tire (F zji (t)) [37]. These are characterized by , during braking, , during drive, where, r i is the tire radius, and ω ji represents the angular speed of the wheels, I ji represents the moment of inertia of wheels, a i (t) represents the longitudinal acceleration, h i is the height of the Center of Gravity (CG) of the vehicle, and h ai is the height of the location at which the equivalent aerodynamic force acts. The longitudinal distances of the front and rear axles from the CG of the vehicle are represented by l fi and l ri , respectively. The road gradient is represented by θ and is defined to be positive for an up-grade and negative for a down-grade. Using the Magic Formula (MF) tire model [38], the longitudinal force at the tire-road interface is represented as, where, = Bλ x (t), and λ x (t) = λ ji (t) + S H . In this study, the MF model parameters, B, C, D, E, S H , S V were obtained from the vehicle dynamic simulation software, The resistive force on the i th vehicle, F Ri (t), is given by where, f i is the rolling friction coefficient, ρ is the air density, a fi is the vehicle frontal area, and C Di is the drag coefficient. The transfer function of the air brake system developed by Sridhar etal. [39] via Hardware in Loop experimentation is where, δ dji is the time constant, and T dji is the time delay, and τ desji and τ ji represent the demanded brake torque and actual brake torque developed, respectively. The experimentally corroborated values for the model parameters as given in [39] are δ dji = 260 ms and T dji = 45 ms, and these have been utilized in this study. Literature suggests using a firstorder model with a time constant and delay for approximating the powertrain dynamics during the acceleration/drive phase [40]. Hence, a similar model as presented in Eq. (7) has been used to incorporate actuation dynamics in the drive phase. Literature supports the use of δ dji = 260 ms and T dji = 45 ms as feasible choices during acceleration/drive phase as well [40]. The first order Padé approximation has been used to represent Eq. (7) as The state space representation of Eq. (8) can be written as Now, on substituting η 1ji (t), η 2ji (t) from (10) in (4), the rotational dynamics of the wheels can be expressed aṡ Considering this complete vehicle model including actuator dynamics, each follower vehicle's dynamics in the platoon can be represented aṡ where,

III. ARTIFICIAL POTENTIAL FUNCTION BASED STRING STABLE CONTROLLER A. CONCEPT OF ARTIFICIAL POTENTIAL FUNCTION
Artificial Potential Functions (APFs) are used to represent the virtual energy level of physical systems and have been widely used for path planning in robotic systems [29]- [32].
In this approach, all entities in the configuration space are assumed to be carrying potential charges, both attractive and repulsive. By properly selecting these potentials, it would be possible to regulate vehicle movements and have a collisionfree operation. In order to achieve this, a positive definite ''artificial potential'', (S) : n → , is defined, where S ∈ n is the state of the dynamical system. The potential function, (S) could consist of both attractive ( a (S)) and repulsive ( r (S)) components, that regulate the vehicle behavior.
A significant advantage of APF-based approaches is that they could provide a simple but flexible mechanism to integrate multiple design requirements and constraints while ensuring closed-form solutions [10], [11]. For instance, it is desirable to keep the vehicles at a minimum intervehicular distance in platoon applications. In case of any perturbation, this would require continuous adjustment of the vehicle position. A potential function, (S) is an energy-like function and can be designed to have a minimum value at the desired intervehicular distance. If vehicles become close to each other (spacing less than desired value), the repulsive field should push them towards the desired intervehicular distance, avoiding collisions. Conversely, if they become far apart, the attractive potential should move them towards the desired distance. This can be achieved by using control inputs derived using the gradient of the potential function. Hence, using APF, the desired (equilibrium) intervehicular distances could be maintained, ensuring a stable platoon operation.

B. ERROR DYNAMICS
From Fig. 1, the spacing between a pair of vehicles in the platoon at each time instant is given by, Every follower vehicle in the platoon should follow the preceding vehicle with a desired inter-vehicular spacing given by where, s o and h i represent standstill spacing and the time-headway respectively. To avoid collision between two consecutive vehicles, a follower vehicle should follow its preceding vehicle's speed profile (since s d i (t) is a function of v i (t)). Hence the control objective in a string stable controller design problem is Given a perturbation, the spacing error between two consecutive vehicles is, For string stability, e i (t) → 0 by making d i (t) → s d i (t), and this could be achieved through, (17) An augmented error function ( i (t)) is defined as, where, κ is a positive real number, and i (t) considers error in both intervehicular distance and longitudinal speed. The value of i (t) would be zero for a platoon in equilibrium.

C. PROPERTIES OF THE POTENTIAL FUNCTION
For a continuous and differentiable function To maintain the follower vehicles at a desired intervehicular distance, i (t) should satisfy The goal is to maintain the intervehicular spacing d i (t) at s d i (t) ( i (t) = 0). Since the control input is computed as the gradient of the potential function, the follower vehicle would move towards the minimum energy point. If the vehicle is perturbed from this point, the vehicle would be automatically brought back by acceleration/braking. In a scenario where the follower vehicle moves closer to the preceding vehicle, such that the intervehicular spacing is below the threshold value (and could cause collisions), brakes should be applied. The speed reduction (and brake force) should be higher as the follower vehicle is closer to the vehicle in front and should decrease as it moves away towards the desired intervehicular distance. This could be ensured by choosing the potential function to satisfy When the function is monotonically decreasing as presented by equation (20), the control input (as gradient of potential function d i (t) d i (t) ) would have negative values, indicating braking, and thus avoiding collisions.

3) THE FUNCTION SHOULD BE (STRICTLY) MONOTONOUSLY INCREASING IN THE RANGE [s d i (t ), ∞)
A converse scenario exists in which the follower vehicle is further away from the preceding vehicle, with a spacing value much higher than the desired intervehicular distance. In such a condition, the follower vehicle should accelerate and move towards the preceding vehicle. The spacing error should be VOLUME 9, 2021 reduced quickly, and this is achieved by ensuring When the function is monotonically increasing as presented by equation (21), the control input has positive values, indicating acceleration, thus maintaining the platoon formation.
To achieve these properties, a parabolic potential function is chosen as a candidate function in this study.
Proposition 1: For the error function presented in Eq. (18) and for the system under study, the choice of i (t) = 1 2 σ i i (t) 2 would satisfy the above properties for σ i > 0.
Proof: The selected potential function should satisfy the aforementioned three properties.

EXISTENCE OF GLOBAL MINIMA
It can be noted that, Checking the second derivative, Thus, the function is minimum at d i (t) = s d i (t).

MONOTONICITY
To check the monotonicity, one should check for the sign of

D. PFSS CONTROLLER DESIGN FOR STRING STABLE OPERATION
The gradient of i (t) is taken with respect to i (t) and used to obtain the acceleration/deceleration control input as, Now, substituting Eqs. (14), (13), (16), and (18), in Eq. (24), Algorithm 1 presents the PFSS control methodology. It is to be noted that, for the realization of the control equation described by Eq. (25), only the position and speed information from the preceding (i − 1) th vehicle is required in addition to that of the i th vehicle. The control equation does not require information from the succeeding (i + 1) th vehicle. Moreover, this approach does not require acceleration information from the (i − 1) th vehicle. Now, the total desired torque can be obtained as, where, τ sati represents the saturation magnitude characterizing the maximum physical limits of the actuator. During acceleration (since HCRVs are typically rear wheel driven), During braking, where, β represents the torque distribution factor between front and rear wheels.

IV. RESULTS AND DISCUSSIONS A. TEST CONDITIONS
A platoon of 5 HCRVs (one leader plus four follower configuration) has been considered. The vehicle parameters used in [28] were adopted here. The platoon leader was assumed to undergo an accelerating maneuver from 10 m/s to 15 m/s and a decelerating maneuver from 10 m/s to 5 m/s. Two different rates of 1 m/s 2 and 2 m/s 2 were considered. Both dry and wet road conditions, corresponding to µ values of 0.8 and 0.4, respectively, were used in the study. Road slopes (θ) of −5 • , 0 • , and +5 • , indicating downhill, straight and uphill operating conditions, respectively, in accordance with [41], were considered. The test cases were simulated using σ i = σ for all the follower vehicles. To test the efficacy of the controller, a limiting scenario where the platoon leader  undergoes an acceleration maneuver on an uphill road is considered, and plots and analyses corresponding to the same are presented next. Results corresponding to other sets of operating conditions have been presented in Table 2. Figure 2 presents the scenario where the platoon leader undergoes an acceleration maneuver on an uphill road. The maneuver starts at t = 15 s and ends at t = 20 s, with leader having an acceleration of 1 m/s 2 . The follower vehicles were found to be following the leader, ensuring a stable platoon operation. Figure 3 presents the spacing error attenuation plots for the speed tracking profiles presented in Fig. 2. The spacing errors were found to be attenuating along the platoon length (96%, 90%, and 84% of the first vehicle error magnitude). The magnitude of the instantaneous spacing errors was found to be in the order of millimeters, which is negligible compared to the size of the vehicle.

C. COMPARISON WITH CONVENTIONAL STRING STABLE CONTROLLERS
To establish the efficacy of the proposed PFSS controller over existing string stable controllers, it has been compared with the well-established sliding mode-based string stable control design approach [8], [28]. Due to the simple control structure as presented in Eq. (25), the proposed PFSS controller requires tuning of only two parameters (σ i , κ), whereas conventional design approaches presented in [8] and [28] require a minimum of five and seven parameters to be simultaneously tuned to ensure string stable operation. Further, different aspects, viz., error attenuation capability, performance under actuator constraints, and lower data requirement, have been analyzed and compared to show the advantages of the proposed PFSS controller.

1) ERROR ATTENUATION CAPABILITY
The same scenario as presented in Section 2 has been considered using the sliding mode approach [28] for comparison, and the corresponding error attenuation profile is presented in Fig. 4. Even though string stable operation was achieved, the magnitude of spacing error values was found to be much higher (approximately three orders of magnitude higher) compared to the proposed approach (Fig. 3). In addition to this, using the sliding mode approach, the magnitude of the torque required was also found to be much higher compared to that of the proposed APF methodology (Fig. 5). The control input exceeded the actuator limit with the SMC controller. The error plots presented in Fig. 4 have been generated by relaxing the actuator constraints. The sliding mode-based approach used an augmented error propagation function as the switching function, and the controller has been designed to attenuate the error propagation. On the other hand, the proposed APF approach does not consider error propagation terms in the control design and maintains the string stability by mitigating individual intervehicular errors instantaneously. This prevents the errors from growing to higher values. As soon as a non-zero error exists, the controller would try to mitigate it by either accelerating/decelerating the vehicles.    Figure 5 presents rear torque input profiles of the (follower) vehicle 1 (for both PFSS and conventional SMC controllers) for tracking the speed profile as presented in Fig. 2. The maximum permissible torque value (obtained from the vehicle datasheet and IPG TruckMaker ® software), limited by the actuator physical constraints, is also marked. The plot shows that the torque inputs are well within the actuator's limits with the proposed controller and are practically realizable. On the other hand, if one were to use a conventional approach as presented in [28], the computed torque inputs were found to exceed the aforesaid actuator physical limit. When the actuator constraints were strictly imposed, then the corresponding torque inputs were not sufficient to track a lead vehicle perturbation (as presented in Fig. 2) and resulted in collisions.

3) DATA REQUIREMENT AND STABILITY
The control equation (Eq. (25)) for the PFSS controller is a function of the position and speed information of the (i − 1) th and the i th vehicles alone, ). However, well-established string stable control design approaches would require data from preceding and succeeding vehicles [8], [28]. In order to demonstrate the impact of reduced data requirement from neighboring vehicles, the same scenario as presented in Fig. 2 has been evaluated using the PFSS controller in the presence of a communication delay. It was assumed that a constant communication delay of 100 ms was present during data transmission from the neighboring vehicles [42]. A conventional controller as presented in [28] has also been used to track the same profile. From the results, one can observe that using the conventional control strategy, communication delay during data transmission from preceding and succeeding vehicles resulted in string instability as presented in Fig. 6 (a). However, the proposed PFSS controller could maintain string stability even with the delay in data transmission from the preceding vehicles ( Fig. 6 (b)). It is interesting to note that the conventional controller could maintain a stable operation if the same communication delay were assumed to be present only during data transmission from the preceding vehicle [28], signifying the importance of minimal data requirement.

D. PFSS CONTROLLER PERFORMANCE UNDER HETEROGENEOUS LOADING CONDITIONS
A heterogeneous condition where (follower) vehicle 1 was overloaded by 40% and vehicle 3 underloaded by 40% [43] has also been considered to evaluate for the robustness of the proposed method to vehicle mass variations. The proposed approach was found to be performing well under such conditions, and the error attenuation plot for the heterogeneously loaded platoon is presented in Fig. 7. Similar to the homogeneous case, the errors are bounded to smaller values (order of 10 −3 ) and were found to be attenuating along the length of the platoon. Table 2 presents the performance evaluation of the proposed PFSS controller for different operating scenarios. Both homogeneous and heterogeneous loading conditions (both overloaded and underloaded) were considered. For each  set, along with string stability, the magnitude of the torque inputs was also checked for possible actuator limit infringements. For profiles with 1 m/s 2 and −1 m/s 2 acceleration, the proposed PFSS controller could ensure string stable operation (with torque inputs well within the physical limits of actuator) for both homogeneous and heterogeneous loading conditions. The impact of various operating conditions is visible from Table 2. For instance, consider the case where the platoon leader is accelerating from 10 m/s to 15 m/s at 2 m/s 2 on a straight and level dry road. While the proposed PFSS controller could ensure string stable operation for a homogeneously loaded platoon, the same was not ensured for a heterogeneous platoon. For overloaded vehicles, the torque input exceeded the actuator physical limit in this case and resulted in string instability.
Remark 1: It can be observed that for µ < 0.4, the proposed approach could not ensure string stability in all the considered scenarios. A scenario where vehicle 1 in the platoon lost its stability due to an increase in wheel slip ratio is shown in Fig. 8. For µ < 0.4, the wheel slip ratio increases (Fig. 8), leading to wheel spin and potential vehicle instability. This caused the vehicles to collide, resulting in string instability. A similar situation exists in panic braking scenarios, which could be addressed by integrating with a suitable Wheel Slip Regulation (WSR) algorithm.

V. CONCLUSION
A string stable controller design methodology for HCRV platoons using artificial potential function has been proposed. This approach was found to be efficient in attenuating the spacing error propagation by controlling the individual vehicle instantaneous spacing error magnitudes. The proposed controller was seen to possess attributes such as lesser data requirement from neighboring vehicles, actuator feasible control input magnitudes, and a simpler control structure. Due to these attributes, this approach could potentially be taken towards the real-time implementation of heavy vehicle platooning for long-haul freight transportation. Integration of the proposed PFSS control algorithm with a suitable Wheel Slip Regulation (WSR) algorithm would be considered as a potential immediate future scope of this study.