2023 17th International Conference on Engineering of Modern Electric Systems (EMES) | 979-8-3503-1063-4/23/$31.00 ©2023 IEEE | DOI: 10.1109/EMES58375.2023.10171704 Tracking Trajectory by Using The Polynomial Method for ACC System Based on Smart Car Platform Duc Lich Luu Nguyen Viet Hung Ciprian Lupu Faculty of Transportation Mechanical Engineering University of Science and Technology The University of Danang Da Nang 550000, Vietnam ldlich@dut.udn.vn Faculty of Information Technology East Asia University of Technology Bac Ninh, Viet Nam hungnv@eaut.edu.vn Department of Automatic Control and Systems Engineering University Politehnica of Bucharest Bucharest, Romania ciprian.lupu@upb.ro Abstract—A well-known advanced driver assistance technology that can be employed for that is the Adaptive Cruise Control System (ACC) Cars equipped with ACC system are to control the car speed to follow a driver’s set speed closely when there is no leading car When a slower leading car is present based on distance sensor, the ACC controlled car are able to keep desired distance with respect to the position and velocity of the car in front This paper presents a design methodology of Adaptive Cruise Control (ACC) system for two smart car platform consist of the leading smart car and a host smart car move on a single lane on a laboratory installation RST algorithm in the multi algorithms structure is proposed to design in realtime architecture for ACC system RST algorithm is one of the most effective solutions for the real-time control of nonlinear systems or working regimes The results ofthe experiment have been conducted to illustrate the effectiveness of the proposed algorithm Keywords—RST controller, Adaptive cruise control, Automotive electronics, Tracking trajectory, Real-time systems I I NTRODUCTION As travel demand has increased over the years, the expressway network has also been expanded to accommodate the demand To overcome these problems, the concept of the Advanced Driver Assistant System is one of the approaches which are under considerable research in the autonomous vehicle technology It ensure safety measures and comfort level for drivers, passengers in urban areas [1] and improves the road capacity [2] This claim is possible due to the development of mechatronics technology [3] The car started to be driven by hardware and software systems, creating the concept of autonomous cars [4] The ACC system is one of the subsystem of the Advanced Driver Assistant System which is one application of keeping small distance with the preceding vehicle The ACC system have two modes of steady state operation: velocity control and vehicle following i.e distance control based on the measured signals of sensors [5], [6] In autonomous robot area, the smart car platform has been studied in recent years, some researchers of [7], [8] have Fig Two smart cars following each other on a single lane been proposed many robot in the platooning to applied for the ACC system with different controllers including predictive controller, sliding mode controller, controller and review into account for the noise from hardware (devices and motors) Some other papers [6], [9], [10], they have been made and tested smart car platform to stay a formation same time avoiding collisions, spotting people, fire or a tracking lane system However, one of the main problems of employing these algorithms is that the computation is large, complex and difficult to implement in a real-time application or in a discretetime system To the best of our knowledge, there are very few scientific works on multi algorithms structure consist of RST algorithms applied to autonomous cars, especially ACC systems Some researchers have introduced the RST control approach as in [9], [11], [12], [13] In this paper, RST algorithm in the multi algorithms structure is proposed to design for ACC system Digital Control Design by using the polynomial method such as RST algorithm with two outstanding advantages such as: simplicity and realtime applicability We focus on implementing digital control design by the polynomial method in the discrete time system for ACC system The leading car is controlled by the ACC system with the reference of the closed-loop (CL) system is the desired velocity, based on a RST controller 979-8-3503-1063-4/23/$31.00 ©2023 IEEE Authorized licensed use limited to: Soongsil University Downloaded on July 12,2023 at 00:00:03 UTC from IEEE Xplore Restrictions apply Fig RST algorithm for ACC system The host car is controlled by the ACC system with the reference of the CL system is the desired distance, based on a RST controller, it is to follow the leading car at the desired distance based on the constant spacing policy and the testing results is to verify the performances of two smart car platform consist of the leading smart car and a host smart car as in Fig.1 This paper is structured as follows: In Section 2, we first consider the mathematical modeling of cars Then ACC Control Structure for host car is presented in Section The Real-Time Implementation and Results will be described in Section and in Section 5, conclusions will close this paper II M ATHEMATICAL MODELING OF CARS For simplicity, the longitudinal dynamics of car assume be described by the following differential equations [14], [15]: = v(t) p(t) (1) pă(t) = a(t) p (t) + pă(t) = u(t) Where, the position, velocity, acceleration in longitudinal axis of the car at time instant t are indicated by p(t), v(t), a(t), respectively As a result, the longitudinal dynamics of the car can be represented by: H(s) = P (s) = U (s) s (ζs + 1) (2) III ACC C ONTROL S TRUCTURE FOR LEADING CAR AND HOST CAR Two smart cars move on a single lane in as Fig.1 The ACC system may also be consider as an autonomous control system that is made to operate with good performances when there is uncertainty in the system and in the environment for a long time and must be able to compensate for system failures without any outside interference This is employing radar sensors, an electronic control unit and an appropriate software which is processing the sensor data and providing the necessary output to track the car ahead in safe conditions The leading car is designed by the ACC system which is to adjust the velocity of car, the desired for the CL system is the velocity of car Utilizing the ACC system of host car is to keep the same speed with the lead car while keeping the value of desired distance with respect to leading car The transfer function (2) of the longitudinal dynamics used for digital controller design, expressed by the irreducible fraction: A(z −1 ) (3) H(z −1 ) = B(z −1 ) Where, A(z −1 ), B(z −1 ) polynomials are: mP a =3 A(z −1 ) = + aj z −j j=1 = + a1 z −1 + a2 z −2 + a3 z −3 mP b =3 bj z −j B(z −1 ) = j=1 = b0 + b1 z −1 + b2 z −2 + b3 z −3 Various techniques, such as the RST controller have been applied in recent years The acronym RST stands for (R)Regulation, (S)-Sensitivity , and (T)-Tracking control It is also linked to R(z −1 ), S(z −1 ), T (z −1 ) polynomials that are employed in the two-degree of freedom controller structure [11] The RST control algorithm in multi algorithms structure with a closed-loop system was used with the ACC system structure (Fig.2): R(z−1 ) T (z−1 ) y(k) u(k) ≡ − S(z−1 ) S(z−1 ) yref (k) (4) −1 R(z ) T (z −1 ) ≡− y(k) + yref (k) S(z −1 ) S(z −1 ) yref (k) - trajectory or filtered set point, in which: dref (k) is the desired distace defined by the constant spacing policy for host car or vref (k) is the desired velocity set by the human for leading car; u(k) - algorithm output, the desired acceleration for host car (control signal) dref (k) = l0 (k)(m) (5) y(k) - process output, in which: d(k) is actual distance for host car or v(k) is velocity for leading car, L is the car length, and it is defined as: Authorized licensed use limited to: Soongsil University Downloaded on July 12,2023 at 00:00:03 UTC from IEEE Xplore Restrictions apply IV R EAL -T IME I MPLEMENTATION AND R ESULTS In the demonstrative section, a small laboratory installation that is used by a real-time software application for two smart car platform travelling single lane as in Fig.3 that allows practical verification of the proposed theoretical elements is presented The cost of components of each electric car can be seen in table.1 The smart car platform is described as in [15], consisting of a leading car and a host car The cars is moved with the distance which is actually very small, so employing the infrared device is measured the distance between the leading car and the host car The longitudinal velocity is measured from encoder sensor mounted on the rear wheels RST Controller Parameters Calculation: the choice of the RST controller parameters allows to solve as well the problem of regulation as well as of tracking, and using a sampling period of Ts = 0.1 second, the system constraints as in [16] are −2.5(m/s2 ) ≤ u(k) ≤ 2.5(m/s2 ) and these polynomials are given by: Fig The smart car platform in the laboratory [15] TABLE I C OMPONENTS OF A ELECTRIC CAR [15] Component Qty Electric Car Module MEGA 2560 microcontroller L9110S driver module Distance Sensor Lines Sensor Speed Sensor Battery WeMos D1 Mini ESP8266 1 2 Unit price (USD) 20 9.0 2.0 7.2 7.5 2.0 7.5 7.0 Total Total price (USD) 20 9.0 4.0 7.2 15.0 8.0 15.0 7.0 85.2 R(z −1 ) = 249.4258 − 387.5863z −1 + 148.0126z −2 T (z −1 ) = 9.8521 S(z −1 ) = + 0.7841z −1 + 0.1404z −2 d(k) = plead (k) − phost (k) − L −1 −1 (6) −1 R(z ), T (z ) and S(z ) polynomials of the proposed RST digital feedback controller, the corresponding parameters are defined as below: m r =2 X rj z −j = r0 + r1 z −1 + r2 z −2 R(z −1 ) = j=0 T (z −1 ) = m t =0 X tj z −j = t0 (7) j=0 1 = m =2 = −1 + s z −2 s P S(z −1 ) s + s z sj z −j j=0 ms , mr , and mt illustrate the respective polynomial degrees as well as the memory dimension for the algorithm’s software implementation For example, if mr = 2, three memory places should be saved for the process’s output: y(k), y(k −1), y(k − 2) The same method applies to u(k), respectively To apply ms = mr = mt = 2, the control law computation is as the following: u(k) = [T (z −1 )yref (k) − r0 y(k) − r1 y(k − 1) s0 (8) − r2 y(k − 2) − s1 u(k − 1) − s2 u(k − 2)] and formula (8) gives the algorithm’s memory actualization for the next iteration: u(k − 1) = u(k), y(k − 1) = y(k), yref (k − 1) = yref (k) The constraint condition are considered as follows: umin ≤ u(k) ≤ umax where umin < and umax > are bounds of control input This sector only focus for the host car using ACC system based on the constant spacing policy which is a common type of strategy that shows real-time applicability RST algorithm is embedded in two smart cars (leading car and host car as in Fig.3), in which the leading car equipped with ACC system, maintains at reference velocity as in Fig.4 and the host car equipped with ACC system to keep the same speed with the leading car as in Fig.4 while keeping the value of desired distance with respect to leading car i.e,0.35m as in Fig.5 Adaptive Cruise control for the host car, i.e following the reference trajectory with the desired distance dref (k) at a discrete time k The constant spacing policy, the ACC system maintains at a fixed constant between leading smart car and host smart car in Fig.3), with the initial distance and the fixed desired distance is set to l0 = 0.35m The lead car stays the velocity at 0.5m/s in during time interval [0, 38.0s], and then decelerates from 0.5m/s to 0.4cm/s during interval [38.1s, 42.0s], and then stays the speed at 0.4m/s during interval [42.1s, 70.0s] Their velocities, the distances between leading smart car and host smart car are indicated in Figs.4, respectively From these figures, the signals resulted after testing can be observed that velocity tracking operates well Clearly, the distance of the host smart car converges to the desired value, i.e 0.35m The host car exist a little large distance at time at 0s (start) and a little overshoot after that, mainly caused by a higher starting voltage than the minimum working voltage From testing results, we observe that RST algorithm results for host cars satisfied with the request for real-time implementation The time delay has existed as the micro-controler Authorized licensed use limited to: Soongsil University Downloaded on July 12,2023 at 00:00:03 UTC from IEEE Xplore Restrictions apply errors due to the influence of noise, disturbance, environment from hardware devices However, does not seriously affect the results In general, the accurate sensors will bring better performance Next step in future, mathematical models of cars in the continuous time system convert to discrete time system ACC systems using RST algorithm will be implemented and simulated, thereby comparing simulation and testing ACKNOWLEDGMENT This paper was performed in the PRECIS Research Center, Laboratory 10 - Advanced Control Systems for Real - Time Applications The authors would like to thank the support of this institutions Fig The velocities of two mart car platform based on the constant spacing policy Fig Inter-car distance of the host smart car platform based on the constant spacing policy of cars calculate the control command for the motor driver; the devices are affected by brightness, disturbance or noise which leads to unstable measurement results However, the result error is not large In general, an accurate sensor will bring better performance to the smart car platform V C ONCLUSIONS In this paper, RST algorithm in the multi algorithms structure for the ACC system was studied The polynomial method in the discrete time system for ACC system will eliminate the need complex mathematical derivatives, model uncertainties and linearization In real-time applications, two smart car platform equipped with the ACC system is tested and implemented on a laboratory control structure, each smart car platform based on sensors Its purpose is to make the efficiency of RST algorithm Testing results show the leading car maintains at reference speed and the host car to keep the same speed with the leading car while keeping the desired distance with respect to leading car However, there was the time delay and some R EFERENCES [1] J Guanetti, Y Kim, and F Borrelli, “Control of connected and automated vehicles: State of the art and future challenges,” Annual reviews in control, vol 45, pp 18–40, 2018 [2] M Nagai, “The perspectives of research for enhancing active safety based on advanced control technology,” Vehicle System Dynamics, vol 45, no 5, pp 413–431, 2007 [3] K Reif, Automotive Mechatronics Konrad Reif Ed Springer Vieweg, 2019 [4] H Yu, X Li, R M Murray, S Ramesh, and C J Tomlin, Safe, autonomous and intelligent vehicles Springer, 2018 [5] D L Luu and C Lupu, “Dynamics model and design for adaptive cruise control vehicles,” in 2019 22nd International Conference on Control Systems and Computer Science (CSCS) IEEE, 2019, pp 12–17 [6] D L Luu, C Lupu, I Cristian et al., “Speed control and spacing control for autonomous mobile robot platform equipped with infrared sensors,” in 2021 16th international conference on engineering of modern electric systems (EMES) IEEE, 2021, pp 1–4 [7] G Guo and W Yue, “Sampled-data cooperative adaptive cruise control of vehicles with sensor failures,” IEEE Transactions on Intelligent Transportation Systems, vol 15, no 6, pp 2404–2418, 2014 [8] M Trudgen, R Miller, and J M Velni, “Robust cooperative adaptive cruise control design and implementation for connected vehicles,” International Journal of Automation and Control, vol 12, no 4, pp 469–494, 2018 [9] C Lupu, C.-C Mihai, F.-D Secuianu, and C Petrescu, “Fast disturbance rejection in mimo process based on algorithms switching,” in 2018 22nd International Conference on System Theory, Control and Computing (ICSTCC) IEEE, 2018, pp 469–473 [10] D L Luu and C Lupu, “Experimenta1evaluation of smart cars model for a platoon of vehicles,” in 2019 23rd International Conference on System Theory, Control and Computing (ICSTCC) IEEE, 2019, pp 815–820 [11] I D Landau and G Zito, Digital control systems: design, identification and implementation Springer, 2006, vol 130 [12] C Lupu, D Popescu, and A Udrea, “Real-time control applications for nonlinear processes based on adaptive control and the static characteristic,” WSEAS Transactions on Systems and Control, vol 3, no 6, pp 607–616, 2008 [13] D Popescu, A Gharbi, D Stefanoiu, and P Borne, Process Control Design for Industrial Applications John Wiley & Sons, 2017 [14] R Rajamani, Vehicle dynamics and control London: Springer Science & Business Media, 2011 [15] D L Luu, C Lupu, and T Van Nguyen, “Design and simulation implementation for adaptive cruise control systems of vehicles,” in 2019 22nd International Conference on Control Systems and Computer Science (CSCS) IEEE, 2019, pp 1–6 [16] D L Luu, H T Pham, C Lupu, T B Nguyen, and S T Ha, “Research on cooperative adaptive cruise control system for autonomous vehicles based on distributed model predictive control,” in 2021 International Conference on System Science and Engineering (ICSSE) IEEE, 2021, pp 13–18 Authorized licensed use limited to: Soongsil University Downloaded on July 12,2023 at 00:00:03 UTC from IEEE Xplore Restrictions apply