Conference Paper

Towards Time-Optimal Race Car Driving Using Nonlinear MPC in Real-Time

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Conference Paper

Towards Time-Optimal Race Car Driving Using Nonlinear MPC in Real-Time

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Abstract

This paper addresses the real-time control of autonomous vehicles under a minimum traveling time objective. Control inputs for the vehicle are computed from a nonlinear model predictive control (MPC) scheme. The time-optimal objective is reformulated such that it can be tackled by existing efficient algorithms for real-time nonlinear MPC that build on the generalized Gauss-Newton method. We numerically validate our approach in simulations and present a real-world hardware setup of miniature race cars that is used for an experimental comparison of different approaches.

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... The simulation shows that under the condition of real-time computation, MPC can only obtain suboptimal solutions [4]. Verschueren et al. (2014) built on the generalized Gauss-Newton method to handle the real-time nonlinear MPC and got calculation time by an experiment on a Real-time Debian system [5]. Borrelli et al. (2014) presented the robust invariant set to satisfy robust MPC constraints of states and inputs, and did experiment on dSPACE [6]. ...
... The simulation shows that under the condition of real-time computation, MPC can only obtain suboptimal solutions [4]. Verschueren et al. (2014) built on the generalized Gauss-Newton method to handle the real-time nonlinear MPC and got calculation time by an experiment on a Real-time Debian system [5]. Borrelli et al. (2014) presented the robust invariant set to satisfy robust MPC constraints of states and inputs, and did experiment on dSPACE [6]. ...
... Existing MPC optimizers usually adopt the online calculation, which can not satisfy policy accuracy and the millisecond-level time requirements of onboard standard controllers, simultaneously. Taking the computing power of the Audi A8 L3 level autopilot controller, zFAS, as the benchmark, the calculation time of MPC algorithms mentioned above ( [6], [10], [8], [9], [5] ) is 203.4 ms, 103.6 ms, 98.9 ms, 98.4 ms, and 41.6 ms respectively. In general, the computing time assigned to the automated vehicle motion control task is less than 10ms due to the limited computing capability. ...
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The design of an automated vehicle controller can be generally formulated into an optimal control problem. This paper proposes a continuous-time finite-horizon approximate dynamicprogramming (ADP) method, which can synthesis off-line near-optimal control policy with analytical vehicle dynamics. Lying on the general Policy Iteration framework, it employs value andpolicy neural networks to approximate the mappings from thesystem states to value function and control inputs, respectively. The proposed method can converge to the near-optimal solutionof the finite-horizon Hamilton-Jacobi-Bellman (HJB) equation. We further applied our algorithm to the simulation of automated vehicle control for the path tracking maneuver. The results suggest that the proposed ADP method can obtain the near-optimal policy with 1% error and less calculation time. What is more, the proposed ADP algorithm is also suitable for nonlinear control systems, where ADP is almost 500 times faster than the nonlinear MPC ipopt solver.
... Applications of this spatial parameterization to planar vehicles demonstrated its ability to balance reference tracking and obstacle avoidance [2], [3]. Combining the spatial pathparameterization with advances in embedded-optimization allowed for real-time and near time-optimal Nonlinear Model Predictive Control (NMPC) applicable to miniature racing cars [4]. Subsequently, online obstacle avoidance was achieved in [5] by formulating a singularity-free parameterization of the system dynamics. ...
... where A x = A(x(t)) and b x = b(x(t)) in constraint (22e) stand for the half-space matrixes of the polyhedron associated to the location of the point-mass p W , which can be computed from (4). With this constraint, it is guaranteed that the motion planning takes place within the free space. ...
Preprint
This paper presents a two-stage prediction-based control scheme for embedding the environment's geometric properties into a collision-free Pythagorean Hodograph spline, and subsequently finding the optimal path within the parameterized free space. The ingredients of this approach are twofold: First, we present a novel spatial path parameterization applicable to any arbitrary curve without prior assumptions in its adapted frame. Second, we identify the appropriateness of Pythagorean Hodograph curves for a compact and continuous definition of the path-parametric functions required by the presented spatial model. This dual-stage formulation results in a motion planning approach, where the geometric properties of the environment arise as states of the prediction model. Thus, the presented method is attractive for motion planning in dense environments. The efficacy of the approach is evaluated according to an illustrative example.
... In [6] a bicycle model for a minimum-time maneuvering NMPC is used to enable real-time feasibility on the embedded hardware of small-scale racecars. First, the authors reformulated the economic optimization problem to a tracking NMPC, which was solved via an approximation scheme. ...
... Hence, for the low level model the algebraic loop, as described in Subsection II.B, is relaxed with a first-order lag using the sampling time as time constant τ. So we introduce the states a x,AL and a y,AL with the following dynamics to be used in (6). ...
... In Law, Dalal, and Shearrow (2018), a MPC approach for controlling front steering of an autonomous vehicle is presented using a successive online linearization of a non-linear vehicle model. In Verschueren, De Bruyne, Zanon, Frasch, and Diehl (2014), a real-time MPC control scheme was presented that solve the racing problem and test it in miniature race cars. Also in Rosolia, Carvalho, and Borrelli (2017), Learning MPC was introduced to solve the racing problem. ...
... Some works solve this problem using an optimal control technique, as e.g. Rosolia et al. (2017) and Verschueren et al. (2014). Unlike them, we propose a planning layer where the complete optimal time trajectory is calculated, thus allowing to reduce the complexity inside the motion control layer. ...
Article
This article presents an innovative control approach for autonomous racing vehicles. Linear Parameter Varying (LPV) theory is used to model the dynamics of the vehicle and implement an LPV-Model Predictive Controller (LPV-MPC) that can be computed online with reduced computational cost. The optimal time problem is solved by an optimal off-line trajectory planner that calculates the best trajectory under the constraints of the circuit. An identification of the system model based on optimization is also carried out. The planning and control scheme is validated in simulation and experimentally in a real platform where the effectiveness of the proposed LPV-MPC is demonstrated.
... Extensions of this reformulation to planar vehicles proved its ability to trade-off between reference tracking and obstacle avoidance [22], [23]. Leveraging this reparameterization and exploiting advances in embedded optimization solvers, [24] developed a real-time NMPC for miniature racing cars that could approximate time-optimal performance. In [25], further modifications in the spatial representation of the plant model allowed including online obstacle avoidance, while still being near-time-optimal. ...
... To approach online time-optimal navigation, we approximate the OCP (8) by a Nonlinear Program (NLP) according to the multiple-shooting approach [29] in which the optimization horizon T is split into N sections with constant decision variables. Similarly to [24] and [25] To leverage the computational advantages of the Gauss-Newton Hessian approximation, the cost function is implemented in a quadratic least-squares fashion. When doing so, the tractable NLP that is solved at every NMPC iteration and approximates OCP (8) is given by: ...
Preprint
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Minimum-time navigation within constrained and dynamic environments is of special relevance in robotics. Seeking time-optimality, while guaranteeing the integrity of time-varying spatial bounds, is an appealing trade-off for agile vehicles, such as quadrotors. State of the art approaches, either assume bounds to be static and generate time-optimal trajectories offline, or compromise time-optimality for constraint satisfaction. Leveraging nonlinear model predictive control and a path parametric reformulation of the quadrotor model, we present a real-time control that approximates time-optimal behavior and remains within dynamic corridors. The efficacy of the approach is evaluated according to simulated results, showing itself capable of performing extremely aggressive maneuvers as well as stop-and-go and backward motions.
... In [12] the authors reformulated the autonomous racing control task as a non-convex optimization problem and then proposed a linearization strategy to compute an approximate solution. The authors in [13] proposed a Nonlinear Model Predictive Control (NMPC) strategy which exploits a Pacejka tire model identified form experimental data. The NMPC is implemented on an experimental setup using an exact Hessian SQP-type optimization algorithm. ...
... Notice that the optimizer in (13) can be used to approximate the evolution of vehicle's velocities, ...
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We present a learning model predictive controller (LMPC) for autonomous racing. We model the autonomous racing problem as a minimum time iterative control task, where an iteration corresponds to a lap. The system trajectory and input sequence of each lap are stored and used to systematically update the controller for the next lap. In the proposed approach, the race time does not increase at each iteration. The first contribution is to propose a local LMPC which reduces the computational burden associated with existing LMPC strategies. In particular, we show how to construct a local safe set and approximation to the value function, using a subset of the stored data. The second contribution is to present a system identification strategy for the autonomous racing iterative control task. We use data from previous iterations and the vehicle's kinematic equations of motion to build an affine time-varying prediction model. The effectiveness of the proposed strategy is demonstrated by experimental results on the Berkeley Autonomous Race Car (BARC) platform.
... On the higher level, a dynamic optimization routine find the optimal geometric trajectory and the lower level tracking controller uses the resulting solution as a time-dependent reference [7]. In [8], the authors proposed a real-time NMPC scheme based on a multiple shooting discretization. A direct solution with a nonlinear economic MPC problem is presented in [9], taking into account nonlinear vehicle dynamics, path constraints and control bounds. ...
Conference Paper
Real-time planning of feasible motion, and the accurate tracking of that motion are key aspects of autonomous vehicles. Various planning and lateral control algorithms have been proposed in the literature. However, no experimental comparison has been carried out to confront the performance of different algorithms. In this work some of the most popular algorithms have been experimentally tested to drive a scale vehicle model along a track. A new algorithm for path tracking based on clothoid curves is here proposed and experimentally validated. The same curves are also used in a new path planning algorithm proposed. This planning is compared to the off-line solution of an optimal control problem. The results of the comparison and the validations are then reported and discussed.
... Tires become highly saturated during sudden path changes, which require "large actuator inputs" within a limited timeframe [143]. Lastly, due to their large computational costs, most control algorithms have been tested only in simulations rather than in actual AVs [37,51,144] and they have only been validated under conditions of minimal parameter variations and unexpected environmental changes [138]. To ensure controllers' applicability to real-world AV implementations, studies have proposed using "hardware-in-the-loop" (HIL) simulations that includes a physical actuator in the simulation tests, developing a V2X system that utilises environmental data to update the AV's control parameters as driving conditions change and that is robust to wireless network disruptions, and developing controllers that integrate the steering, braking, and suspension controls during various road conditions [51,138]. ...
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Autonomous Vehicles (AVs) are increasingly embraced around the world to advance smart mobility and more broadly, smart, and sustainable cities. Algorithms form the basis of decision-making in AVs, allowing them to perform driving tasks autonomously, efficiently, and more safely than human drivers and offering various economic, social, and environmental benefits. However, algorithmic decision-making in AVs can also introduce new issues that create new safety risks and perpetuate discrimination. We identify bias, ethics, and perverse incentives as key ethical issues in the AV algorithms’ decision-making that can create new safety risks and discriminatory outcomes. Technical issues in the AVs’ perception, decision-making and control algorithms, limitations of existing AV testing and verification methods, and cybersecurity vulnerabilities can also undermine the performance of the AV system. This article investigates the ethical and technical concerns surrounding algorithmic decision-making in AVs by exploring how driving decisions can perpetuate discrimination and create new safety risks for the public. We discuss steps taken to address these issues, highlight the existing research gaps and the need to mitigate these issues through the design of AV’s algorithms and of policies and regulations to fully realise AVs’ benefits for smart and sustainable cities.
... Tires become highly saturated during sudden path changes, which require "large actuator inputs" within a limited timeframe [143]. Lastly, due to their large computational costs, most control algorithms have been tested only in simulations rather than in actual AVs [37,51,144] and they have only been validated under conditions of minimal parameter variations and unexpected environmental changes [138]. To ensure controllers' applicability to real-world AV implementations, studies have proposed using "hardware-in-the-loop" (HIL) simulations that includes a physical actuator in the simulation tests, developing a V2X system that utilises environmental data to update the AV's control parameters as driving conditions change and that is robust to wireless network disruptions, and developing controllers that integrate the steering, braking, and suspension controls during various road conditions [51,138]. ...
Preprint
Full-text available
Autonomous Vehicles (AVs) are increasingly embraced around the world to advance smart mobility and more broadly, smart, and sustainable cities. Algorithms form the basis of decision-making in AVs, allowing them to perform driving tasks autonomously, efficiently, and more safely than human drivers and offering various economic, social, and environmental benefits. However, algorithmic decision-making in AVs can also introduce new issues that create new safety risks and perpetuate discrimination. We identify bias, ethics, and perverse incentives as key ethical issues in the AV algorithms' decision-making that can create new safety risks and discriminatory outcomes. Technical issues in the AVs' perception, decision-making and control algorithms, limitations of existing AV testing and verification methods, and cybersecurity vulnerabilities can also undermine the performance of the AV system. This article investigates the ethical and technical concerns surrounding algorithmic decision-making in AVs by exploring how driving decisions can perpetuate discrimination and create new safety risks for the public. We discuss steps taken to address these issues, highlight the existing research gaps and the need to mitigate these issues through the design of AV's algorithms and of policies and regulations to fully realise AVs' benefits for smart and sustainable cities.
... In contrast to the PID control method, which is traditional and highly error-prone, the other one that is highly desirable and efficient is the MPC 1 method [14]. Using the kinematic model of the car and measuring the parameters at any given moment, this method predicts the amount of time needed to accelerate the vehicle and the amount of rotation of the steering wheel to achieve the desired state. ...
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Due to increasing use of vehicles and road traffic, drive safety has become an important issue. Therefore, an Advanced Driving Assistant System (ADAS) can be a convenient option to increase driving safety. This paper presents a model of an ADAS that is capable of driving from one place to another in different paths such as curved, straight and straight line followed by curved lines. This area of research is divided into several sub-domains, such as deep learning, hardware platform, computer vision, and control. Every self-driving car must be aware of its surroundings and act accordingly. Advances in neural networks and deep learning made it possible to extract information from cameras easily and robustly. In this paper, a neural network is trained to identify various objects such as traffic lights and pedestrians. In addition, image processing is used to detect road lines. due to high computational costs of image processing operations, a network of embedded systems are utilized. Furthermore, MPC control method is used for automated and intelligent steering that makes the right decisions in a timely manner with a small error.
... In contrast to the PID control method, which is traditional and highly error-prone, the other one that is highly desirable and efficient is the MPC 1 method [14]. Using the kinematic model of the car and measuring the parameters at any given moment, this method predicts the amount of time needed to accelerate the vehicle and the amount of rotation of the steering wheel to achieve the desired state. ...
... Proposition 1: Consider system (1) with initial state x s ∈ X, final state X f = {x f }, optimal control problem (5) and control law (10). Choose N min ≥ 1 and define ∆t min ≥ 0 and N ≥ N min ensuring ∆t min < ∆t * (x s , N ). ...
Preprint
This paper deals with the development and analysis of novel time-optimal point-to-point model predictive control concepts for nonlinear systems. Recent approaches in the literature apply a time transformation, however, which do not maintain recursive feasibility for piecewise constant control parameterization. The key idea in this paper is to introduce uniform grids with variable discretization. A shrinking-horizon grid adaptation scheme ensures convergence to a specific region around the target state and recursive feasibility. The size of the region is configurable by design parameters. This facilitates the systematic dual-mode design for quasi-time-optimal control to restore asymptotic stability and establish a smooth stabilization. Two nonlinear program formulations with different sparsity patterns are introduced to realize and implement the underlying optimal control problem. For a class of numerical integration schemes, even nominal asymptotic stability and true time-optimality are achieved without dual-mode. A comparative analysis as well as experimental results demonstrate the effectiveness of the proposed techniques.
... The latter seeks the optimal control for an OCP with fixed final time, which is considered at and given by the upper level. This approach avoids any time or spatial transformation, which are commonly adopted for this class of problems, especially those arising, e.g., in automotive and robotic applications [3], [16], [19], [20]. Hence, the dynamics do not change and no local optima are introduced by the transformation. ...
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This paper discusses a bilevel optimization approach for free finite final time optimal control problems and addresses a numerical method for their approximate solution. The core idea is to decouple the final time optimization from the optimal control and state trajectory. This is rigorously formulated as an equivalent bilevel problem seeking, at the upper level, the optimal final time and optimal control and corresponding state at the lower level. Standard solvers for nonlinear optimal control can deal with the latter, while the former is a box-constrained optimization problem with one scalar decision variable. The interface between the two levels is based on the Hamilton function associated to the problem and its relationship with the cost function. A method for solving the upper level problem is developed, that combines a tailored fast first-order method with a robust and guaranteed root-finding algorithm. Finally, numerical results demonstrate the robustness of the method and show its limitations.
... In the literature, several control techniques for autonomous racing are known. They can be roughly categorized into two categories, predictive (Liniger et al., 2015;Verschueren et al., 2014;Funke et al., 2017;Rosolia et al., 2017;Liniger and Lygeros, 2019) and non-predictive controllers (Zhang et al., 2001;Kritayakirana and Gerdes, 2012;Klomp et al., 2014). Since non-predictive racing controllers are not able to plan the motion of the car, it is necessary to compute the racing line beforehand. ...
Preprint
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This paper presents the algorithms and system architecture of an autonomous racecar. The introduced vehicle is powered by a software stack designed for robustness, reliability, and extensibility. In order to autonomously race around a previously unknown track, the proposed solution combines state of the art techniques from different fields of robotics. Specifically, perception, estimation, and control are incorporated into one high-performance autonomous racecar. This complex robotic system, developed by AMZ Driverless and ETH Zurich, finished 1st overall at each competition we attended: Formula Student Germany 2017, Formula Student Italy 2018 and Formula Student Germany 2018. We discuss the findings and learnings from these competitions and present an experimental evaluation of each module of our solution.
... We proceed to implement MPC-CBF in a more complex scenario: competitive car racing. In some previous car racing control work [24], [25], they only consider static obstacles on the track while we deal with dynamic obstacles such as other cars using MPC-CBF. ...
Preprint
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The optimal performance of robotic systems is usually achieved near the limit of state and input bounds. Model predictive control (MPC) is a prevalent strategy to handle these operational constraints, however, safety still remains an open challenge for MPC as it needs to guarantee that the system stays within an invariant set. In order to obtain safe optimal performance in the context of set invariance, we present a safety-critical model predictive control strategy utilizing discrete-time control barrier functions (CBFs), which guarantees system safety and accomplishes optimal performance via model predictive control. We analyze the stability and the feasibility properties of our control design. We verify the properties of our method on a 2D double integrator model for obstacle avoidance. We also validate the algorithm numerically using a competitive car racing example, where the ego car is able to overtake other racing cars.
... In related literature, several control techniques for autonomous racing are known. They can be roughly categorized into two categories: predictive (Liniger et al., 2015;Verschueren et al., 2014;Funke et al., 2017;Rosolia et al., 2017;Liniger and Lygeros, 2019) and non-predictive controllers (Zhang et al., 2001;Kritayakirana and Gerdes, 2012;Klomp et al., 2014;Kapania and Gerdes, 2015). Since non-predictive racing controllers are not able to plan the motion of the car, it is necessary to compute the racing line beforehand. ...
Article
Full-text available
This paper presents the algorithms and system architecture of an autonomous racecar. The introduced vehicle is powered by a software stack designed for robustness, reliability, and extensibility. To autonomously race around a previously unknown track, the proposed solution combines state of the art techniques from different fields of robotics. Specifically, perception, estimation, and control are incorporated into one high‐performance autonomous racecar. This complex robotic system, developed by AMZ Driverless and ETH Zürich, finished first overall at each competition we attended: Formula Student Germany 2017, Formula Student Italy 2018 and Formula Student Germany 2018. We discuss the findings and learnings from these competitions and present an experimental evaluation of each module of our solution.
... Classical Approaches Classical approaches to autonomous car racing approach the problem by separating it in a chain of submodules consisting of perception, trajectory planning, and control. In particular, model predictive control (MPC) [2], [11]- [15] is a promising approach for controlling the vehicle at high speed. In [16], an MPC controller is combined with learned system dynamics based on Gaussian Processes for the task of autonomous car racing. ...
Preprint
Full-text available
Autonomous car racing raises fundamental robotics challenges such as planning minimum-time trajectories under uncertain dynamics and controlling the car at its friction limits. In this project, we consider the task of autonomous car racing in the top-selling car racing game Gran Turismo Sport. Gran Turismo Sport is known for its detailed physics simulation of various cars and tracks. Our approach makes use of maximum-entropy deep reinforcement learning and a new reward design to train a sensorimotor policy to complete a given race track as fast as possible. We evaluate our approach in three different time trial settings with different cars and tracks. Our results show that the obtained controllers not only beat the built-in non-player character of Gran Turismo Sport, but also outperform the fastest known times in a dataset of personal best lap times of over 50,000 human drivers.
... Several authors have validated the reliability of the ACADO Toolkit C/C ++ code in PC-based simulations with sampling time in microsecond range [13]. The previous works [14], [15], [16] shows the implementation of NMPC with computational time results obtained from PC-based simulations. The author [17] implemented C/C ++ code from ACADO Toolkit on RaspberryPi which is an OS-based embedded platform. ...
Preprint
Full-text available
Today, various nonlinear programming problem (NLP) solvers and C/C ++ code generation frameworks are available as open source for solving nonlinear model predictive control (NMPC). Almost all the solvers are written in C/C ++ code which are more compatible for the PC-based simulation environment. These codes are not directly compatible for embedded implementation and real-time control. An attempt has been made to address this shortcoming by creating a customized framework on top of the C code generated from ACADO Toolkit to make it directly compatible with ARM based embedded platforms. The study also analyzes the embedded implementation aspects using C code generated for qpOASES, qpDUNES, and HPMPC solvers from ACADO Toolkit. We show the results of hardware-in-loop (HIL) simulations with detailed analysis and comparison of memory requirement and achievable sampling time for three benchmark dynamical systems on different embedded platforms viz ARM Cortex M3, PYNQ FPGA and Raspberry Pi. The results show that qpOASES outperforms as compared to the other two solvers when the computational time is of prime importance for small prediction horizon. Similarly, when there are limited on-chip memory resources, qpDUNES can prove beneficial.
... Other solutions aim at combining trajectory generation, subject to obstacle avoidance constraints, and trajectory tracking, subject to actuation limits and track boundaries, in a single optimization problem, as shown in [29,17,30,31,32,33]. ...
Thesis
A Nonlinear Model Predictive Control (NMPC) strategy aimed at controlling a scale race car model for autonomous racing competitions has been discussed in this thesis. The proposed approach combines motion planning and control and is concerned with minimizing the lap time by avoiding obstacles and keeping the vehicle within the track boundaries. The optimization problem set up considers both the vehicle’s actuation limits and the lateral and longitudinal forces acting on the car described by a simplified version of the Pacejka’s Magic Formula and a drivetrain model. The resultant optimization problem has been designed and implemented through Optimization Engine (OpEn), a framework with which it is possible to achieve real-time performance such as to make the algorithm potentially suitable to be integrated into an embedded system. The feasibility and effectiveness of the proposed approach are demonstrated through closed-loop simulations achieved in Gazebo. Specifically, the F1/10 Autonomous Racing Simulator has been used to test the algorithm in an environment quite close to real implementations. The simulations obtained by considering three different racing tracks show how the proposed control strategy allows the car model to safely race on a circuit, even when obstacles are placed at critical points of the track.
... At the same time, the policy has to control the vehicle at the limit of handling, where the behavior of the model is highly nonlinear. Currently, the most advanced methods use modelbased predictive control techniques [1], [2], [3], [4]. Several groups showed that the performance of such methods can be increased by the use of data to improve the prediction model [5], [6]. ...
Preprint
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We present a reinforcement learning-based solution to autonomously race on a miniature race car platform. We show that a policy that is trained purely in simulation using a relatively simple vehicle model, including model randomization, can be successfully transferred to the real robotic setup. We achieve this by using novel policy output regularization approach and a lifted action space which enables smooth actions but still aggressive race car driving. We show that this regularized policy does outperform the Soft Actor Critic (SAC) baseline method, both in simulation and on the real car, but it is still outperformed by a Model Predictive Controller (MPC) state of the art method. The refinement of the policy with three hours of real-world interaction data allows the reinforcement learning policy to achieve lap times similar to the MPC controller while reducing track constraint violations by 50%.
... Classical Approaches Classical approaches to autonomous car racing approach the problem by separating it in a chain of submodules consisting of perception, trajectory planning, and control. In particular, model predictive control (MPC) [2], [11]- [15] is a promising approach for controlling the vehicle at high speed. In [16], an MPC controller is combined with learned system dynamics based on Gaussian Processes for the task of autonomous car racing. ...
Article
Full-text available
Autonomous car racing is a major challenge in robotics. It raises fundamental problems for classical approaches such as planning minimum-time trajectories under uncertain dynamics and controlling the car at its limits of handling. Besides, the requirement of minimizing the lap time, which is a sparse objective, and the difficulty of collecting training data from human experts have also hindered researchers from directly applying learning-based approaches to solve the problem. In the present work, we propose a learning-based system for autonomous car racing by leveraging high-fidelity physical car simulation, a course-progress-proxy reward, and deep reinforcement learning. We deploy our system in Gran Turismo Sport, a world-leading car simulator known for its realistic physics simulation of different race cars and tracks, which is even used to recruit human race car drivers. Our trained policy achieves autonomous racing performance that goes beyond what had been achieved so far by the built-in AI, and, at the same time, outperforms the fastest driver in a dataset of over 50,000 human players.
... We proceed to implement MPC-CBF in a more complex scenario: competitive car racing. In some previous car racing control work [24], [25], they only consider static obstacles on the track while we deal with dynamic obstacles such as other cars using MPC-CBF. ...
... We proceed to implement MPC-CBF in a CHAPTER 4. MODEL PREDICTIVE CONTROL WITH CONTROL BARRIER FUNCTIONS 44 more complex scenario: competitive car racing. In some previous car racing control work [117,193], they only consider static obstacles on the track while we deal with dynamic obstacles such as other cars using MPC-CBF. ...
Thesis
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Control barrier functions (CBFs) are one of the many used approaches for achieving safety in robot autonomy. This thesis tackles several challenges present in control barrier functions in different aspects, including optimization, control, planning and navigation. This thesis is composed of three parts. In Part I, we point out the optimization infeasibility between CBF constraint and input constraint, and address the feasibility problem in optimal control for quadratic programming using control barrier functions under input constraints. We also notice that the potential conflict between input constraints and safety constraints also exists in the discrete-time domain together with model predictive control. This conflict is formally identified in reachability analysis and reachability is later enhanced in proposed formulations. In the Part II, we focus on obstacle avoidance for control and trajectory generation in a tight environment, which requires polytopic obstacle avoidance. We analyze the optimization problem for obstacle avoidance between polytopes. The novel optimizations for obstacle avoidance in continuous domain and discrete-time domain are proposed to solve this challenge. In Part III of the thesis, we discuss several applications of motion planning and navigation using control barrier functions. We firstly address the motion planning problem within a finite state machine which unifies a mid-level planner and a low-level safety-critical controller with application to autonomous driving. Next, we propose parallelism for motion planning with control barrier functions with application to autonomous racing. Finally, we explore the possibility of safety-critical motion planning for high dimensional systems such as Cassie, a life-size bipedal robot.
... 1) The Racing Problem: We first consider a single-player racing problem, in which the goal is to drive a race car on a given track in minimum time. Instead of minimizing the time directly, the time-optimal objective is normally reformulated as minimizing the path of least curvature or the shortest path in order to use numerical optimization methods [19]. In [6], the authors propose a course-progressproxy reward formulation, which closely represents the lap time and can be maximized using reinforcement learning. ...
... Two common approaches to the racing problem are time-optimal control [26] and contouring control [27], i.e., where the progress is maximized along the track. Both approaches have been implemented successfully on hardware platforms akin to ours [28], [29]. In order to account for modeling errors, several learning approaches have been proposed in the context of race cars. ...
Preprint
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Model predictive control has been widely used in the field of autonomous racing and many data-driven approaches have been proposed to improve the closed-loop performance and to minimize lap time. However, it is often overlooked that a change in the environmental conditions, e.g., when it starts raining, it is not only required to adapt the predictive model but also the controller parameters need to be adjusted. In this paper, we address this challenge with the goal of requiring only few data. The key novelty of the proposed approach is that we leverage the learned dynamics model to encode the environmental condition as context. This insight allows us to employ contextual Bayesian optimization, thus accelerating the controller tuning problem when the environment changes and to transfer knowledge across different cars. The proposed framework is validated on an experimental platform with 1:28 scale RC race cars. We perform an extensive evaluation with more than 2'000 driven laps demonstrating that our approach successfully optimizes the lap time across different contexts faster compared to standard Bayesian optimization.
... The free end-time version of the optimal control problems (3.6) and (3.7) are solved through a bilevel approach, which aims at decoupling the determination of the end-time from the solution of the optimal control problems. This allows avoiding time transformation techniques, commonly used in automotive and robotics applications [15,63,93,95], which usually require solving a problem very sensitive to the initial guess, see [64]. ...
Thesis
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This thesis is devoted to a theoretical and numerical investigation of methods to solve open-loop non zero-sum differential Nash games. These problems arise in many applications, e.g., biology, economics, physics, where competition between different agents appears. In this case, the goal of each agent is in contrast with those of the others, and a competition game can be interpreted as a coupled optimization problem for which, in general, an optimal solution does not exist. In fact, an optimal strategy for one player may be unsatisfactory for the others. For this reason, a solution of a game is sought as an equilibrium and among the solutions concepts proposed in the literature, that of Nash equilibrium (NE) is the focus of this thesis. The building blocks of the resulting differential Nash games are a dynamical model with different control functions associated with different players that pursue non-cooperative objectives. In particular, the aim of this thesis is on differential models having linear or bilinear state-strategy structures. In this framework, in the first chapter, some well-known results are recalled, especially for non-cooperative linear-quadratic differential Nash games. Then, a bilinear Nash game is formulated and analysed. The main achievement in this chapter is Theorem 1.4.2 concerning existence of Nash equilibria for non-cooperative differential bilinear games. This result is obtained assuming a sufficiently small time horizon T, and an estimate of T is provided in Lemma 1.4.8 using specific properties of the regularized Nikaido-Isoda function. In Chapter 2, in order to solve a bilinear Nash game, a semi-smooth Newton (SSN) scheme combined with a relaxation method is investigated, where the choice of a SSN scheme is motivated by the presence of constraints on the players’ actions that make the problem non-smooth. The resulting method is proved to be locally convergent in Theorem 2.1, and an estimate on the relaxation parameter is also obtained that relates the relaxation factor to the time horizon of a Nash equilibrium and to the other parameters of the game. For the bilinear Nash game, a Nash bargaining problem is also introduced and discussed, aiming at determining an improvement of all players’ objectives with respect to the Nash equilibrium. A characterization of a bargaining solution is given in Theorem 2.2.1 and a numerical scheme based on this result is presented that allows to compute this solution on the Pareto frontier. Results of numerical experiments based on a quantum model of two spin-particles and on a population dynamics model with two competing species are presented that successfully validate the proposed algorithms. In Chapter 3 a functional formulation of the classical homicidal chauffeur (HC) Nash game is introduced and a new numerical framework for its solution in a time-optimal formulation is discussed. This methodology combines a Hamiltonian based scheme, with proximal penalty to determine the time horizon where the game takes place, with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed end-time. The resulting numerical optimization scheme has a bilevel structure, which aims at decoupling the computation of the end-time from the solution of the pursuit-evader game. Several numerical experiments are performed to show the ability of the proposed algorithm to solve the HC game. Focusing on the case where a collision may occur, the time for this event is determined. The last part of this thesis deals with the analysis of a novel sequential quadratic Hamiltonian (SQH) scheme for solving open-loop differential Nash games. This method is formulated in the framework of Pontryagin’s maximum principle and represents an efficient and robust extension of the successive approximations strategy in the realm of Nash games. In the SQH method, the Hamilton-Pontryagin functions are augmented by a quadratic penalty term and the Nikaido-Isoda function is used as a selection criterion. Based on this fact, the key idea of this SQH scheme is that the PMP characterization of Nash games leads to a finite-dimensional Nash game for any fixed time. A class of problems for which this finite-dimensional game admits a unique solution is identified and for this class of games theoretical results are presented that prove the well-posedness of the proposed scheme. In particular, Proposition 4.2.1 is proved to show that the selection criterion on the Nikaido-Isoda function is fulfilled. A comparison of the computational performances of the SQH scheme and the SSN-relaxation method previously discussed is shown. Applications to linear-quadratic Nash games and variants with control constraints, weighted L1 costs of the players’ actions and tracking objectives are presented that corroborate the theoretical statements.
... To ensure a trajectory tracking able to reduce the lap time, the state-of-the-art offers different approaches both from the architecture and methodology point of view. From the architecture point of view, the research is divided between the ones that used a real-time single algorithm to plan and track the trajectory [2,3], others that preferred to separate the planner from the tracker by having the first offline and the second work in real-time [4], and others that divided trajectory planning and tracking but implemented them both in realtime [5][6][7]. The first architecture type, as said in [8], gives the advantage of incorporating tire force constraints in a straight-forward way. ...
Article
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The aim of this study was to develop trajectory planning that would allow an autonomous racing car to be driven as close as possible to what a driver would do, defining the most appropriate inputs for the current scenario. The search for the optimal trajectory in terms of lap time reduction involves the modeling of all the non-linearities of the vehicle dynamics with the disadvantage of being a time-consuming problem and not being able to be implemented in real-time. However, to improve the vehicle performances, the trajectory needs to be optimized online with the knowledge of the actual vehicle dynamics and path conditions. Therefore, this study involved the development of an architecture that allows an autonomous racing car to have an optimal online trajectory planning and path tracking ensuring professional driver performances. The real-time trajectory optimization can also ensure a possible future implementation in the urban area where obstacles and dynamic scenarios could be faced. It was chosen to implement a local trajectory planning based on the Model Predictive Control(MPC) logic and solved as Linear Programming (LP) by Sequential Convex Programming (SCP). The idea was to achieve a computational cost, 0.1 s, using a point mass vehicle model constrained by experimental definition and approximation of the car’s GG-V, and developing an optimum model-based path tracking to define the driver model that allows A car to follow the trajectory defined by the planner ensuring a signal input every 0.001 s. To validate the algorithm, two types of tests were carried out: a Matlab-Simulink, Vi-Grade co-simulation test, comparing the proposed algorithm with the performance of an offline motion planning, and a real-time simulator test, comparing the proposed algorithm with the performance of a professional driver. The results obtained showed that the computational cost of the optimization algorithm developed is below the limit of 0.1 s, and the architecture showed a reduction of the lap time of about 1 s compared to the offline optimizer and reproducibility of the performance obtained by the driver.
... In particular, rapid, simultaneous optimization of the driven path and speed via MPC has been used successfully in the control of miniature cars. An early example is given by Verschueren et al. in [12]. Work by Liniger et al. with a vehicle model that accounted for tire slip appeared in [13] and was similarly demonstrated through the racing of 1:43 scale RC cars. ...
Article
In emergency situations, autonomous vehicles will be forced to operate at their friction limits in order to avoid collisions. In these scenarios, coordinating the planning of the vehicle's path and speed gives the vehicle the best chance of avoiding an obstacle. Fast reaction time is also important in an emergency, but approaches to the trajectory planning problem based on nonlinear optimization are computationally expensive. This paper presents a new scheme that simultaneously modifies the desired path and speed profile for a vehicle in response to the appearance of an obstacle, significant tracking error, or other environmental change. By formulating the trajectory optimization problem as a quadratically constrained quadratic program, solution times of less than 20 milliseconds are possible even with a 10 second planning horizon. A simplified point mass model is used to describe the vehicle's motion, but the incorporation of longitudinal weight transfer and road topography mean that the vehicle's acceleration limits are modeled more accurately than in comparable approaches. Experimental data from on an autonomous vehicle in two scenarios demonstrate how the trajectory planner enables the vehicle to avoid an obstacle even when the obstacle appears suddenly and the vehicle is already operating near the friction limits.
... 1) The Racing Problem: We first consider a single-player racing problem, in which the goal is to drive a race car on a given track in minimum time. Instead of minimizing the time directly, the time-optimal objective is normally reformulated as minimizing the path of least curvature or the shortest path in order to use numerical optimization methods [19]. In [6], the authors propose a course-progressproxy reward formulation, which closely represents the lap time and can be maximized using reinforcement learning. ...
Preprint
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Professional race car drivers can execute extreme overtaking maneuvers. However, conventional systems for autonomous overtaking rely on either simplified assumptions about the vehicle dynamics or solving expensive trajectory optimization problems online. When the vehicle is approaching its physical limits, existing model-based controllers struggled to handle highly nonlinear dynamics and cannot leverage the large volume of data generated by simulation or real-world driving. To circumvent these limitations, this work proposes a new learning-based method to tackle the autonomous overtaking problem. We evaluate our approach using Gran Turismo Sport -- a world-leading car racing simulator known for its detailed dynamic modeling of various cars and tracks. By leveraging curriculum learning, our approach leads to faster convergence as well as increased performance compared to vanilla reinforcement learning. As a result, the trained controller outperforms the built-in model-based game AI and achieves comparable overtaking performance with an experienced human driver.
... Even if recent contributions to theory, algorithms and efficient implementations [25,26] have enlarged their application spectrum and have allowed to consider nonlinear firstprinciple models, the need of real-time performances calls for simplified models with mildly nonlinear tire models, approximate load transfers and steady-state or negligible roll and pitch motions of the vehicle body. Moreover, actual implementations, to the best of authors' knowledge, are limited to RC model cars travelling in controlled indoor environments [27] and [28]. In [28], real-time feasibility at 50 Hz sampling rate has been demonstrated in impressive experimental results with fast 1:43 scale models enabling obstacle avoidance in dynamic racing situations. ...
Article
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This paper addresses the problem of the link between the driving style of an ideal driver, modelled as an optimal controller, and fundamental set-up parameters of a vehicle in the GP2 motorsport class. The aim is to evaluate quantitatively how set-up parameters, like distribution of aerodynamic loads, weight and roll stiffness between front and rear axles, affect the driving style, encoded in the shape of the optimal trajectory and in the acceleration, brake and steer inputs. To this aim, we develop an optimization code that includes a double-track vehicle model capable of solving the minimum lap-time problem (MLTP) on a given track. The track is represented via NURBS curves and the MLTP is framed and solved as an optimal control problem by transcription into a nonlinear program using direct collocation. To assess the accuracy of the vehicle model and the optimization pipeline, we also validate our results against real telemetry data. The developed software framework lends itself to easily perform both sensitivity analysis and concurrent trajectory planning and set-up parameter optimization: this is obtained by simple promotion of static parameters of interests to variables in the optimal control problem. Some results along these lines are also included.
Article
In this study, an experimental 1:43-scale car was developed to verify a vehicle control strategy for autonomous race cars. The power slide driving technique was employed as a maneuver to overcome sharp corners at high speed. To develop a proper control strategy for autonomous race cars, the experimental scale car trajectories when driven by a human driver were analyzed. The side slip angle and the turn rate during sharp cornering were used as reference values for the developed autonomous control method that has a hierarchical architecture. The slip-based control method was employed as a high-level controller to stabilize the vehicle state when turning sharp corners. The performance of the developed control strategy was experimentally verified using a real-time environmental 1:43-scale car.
Article
Full-text available
If the driving behavior of an autonomous vehicle is similar to that of a skilled driver, the human driver can extricate himself from fatigue operation and the comfort of passengers can also be guaranteed. Therefore, this paper studies the human-like lane-changing model of an autonomous vehicle. The lane-changing characteristic data of skilled drivers are collected and analyzed through a real vehicle test. Then, comparing the MPC-based driver model with the steering wheel angle of human drivers, we found that the MPC-based model could hardly reflect the maneuvering characteristics of human drivers, so we proposed a driver model with steering wheel angle continuity for human drivers. This paper uses four neural network models to compare the prediction on the test set, then uses different input types to compare the prediction accuracy of the model, and finally verifies the generalization ability of the model on the verification set. These three test results show that the prediction results of the human-like lane-changing driving model based on Bi-LSTM are closest to the real steering wheel angle sequence of skilled drivers. The test results demonstrate that the Bi-LSTM-based human-like lane-changing driving model achieves 9.8% RMSE and 6.8% MAE, which improves 10.8% RMSE and 10.3% MAE over LSTM. The model can generate the steering wheel angle sequence in the process of lane changing like a human, so as to realize the human simulation control of an autonomous vehicle for lane-changing conditions.
Thesis
In this thesis, we discuss several techniques for solving nonlinear optimization problems arising in nonlinear model predictive control (NMPC). They share two things in common: they all approximate some non-convex functions with convex ones, and they are all useful in a real-time, embedded context. First, a convexification method for structured indefinite quadratic programming problems is presented. Such problems are found in sequential quadratic programming (SQP) methods for NMPC problems. Its main advantage is a convergence speedup, i.e. local quadratic convergence, under some assumptions compared to a quasi-Newton or a generalized Gauss-Newton approach. A second new technique, called sequential convex quadratic programming (SCQP), is useful in the presence of constraint and objective functions that are ‘convex-over-nonlinear’. SCQP is a generalization of the GGN method, similar to sequential convex programming (SCP) but at a lower computational cost. For example, it exploits the convexity in ellipsoidal constraints, often encountered in practice. Two additional convex approximation methods are presented for time-optimal problems. The first one is an approximation of the time-optimal problem with convex l 1 penalties. The second method reformulates nonlinear, non-convex path constraints as simple bounds, under some conditions. As a final contribution, we present a novel software framework, acados, in which the above techniques are implemented. It is a modular framework for embedded optimization, meant to facilitate rapid prototyping of new algorithms. New features with respect to existing software like ACADO are that it is built on the optimized linear algebra library BLASFEO and on the automatic differentiation library CasADi. Interfaces to higher-level languages such as Python and Matlab are available. Some numerical examples show the ease-of-use and a significant speedup with respect to ACADO.
Article
This paper deals with the development and analysis of novel time-optimal point-to-point model predictive control concepts for nonlinear systems. Recent approaches in the literature apply a time transformation, however, which do not maintain recursive feasibility for a piecewise constant control parameterization. The key idea in this paper is to introduce uniform grids with variable discretization. A shrinking-horizon grid adaptation scheme ensures convergence to a specific region around the target state and recursive feasibility. The size of the region is configurable by design parameters. This facilitates the systematic dual-mode design for quasi-time-optimal control to restore asymptotic stability and establish a smooth stabilization. Two nonlinear program formulations with different sparsity patterns are introduced to realize and implement the underlying optimal control problem. For a class of numerical integration schemes, even nominal asymptotic stability and true time-optimality are achieved without dual-mode. A comparative analysis as well as experimental results demonstrate the effectiveness of the proposed techniques.
Article
Recent road trials have shown that guaranteeing the safety of driving decisions is essential for the wider adoption of autonomous vehicle technology. One promising direction is to pose safety requirements as planning constraints in nonlinear, nonconvex optimization problems of motion synthesis. However, many implementations of this approach are limited by uncertain convergence and local optimality of the solutions achieved, affecting overall robustness. To improve upon these issues, we propose a novel two-stage optimization framework: In the first stage, we find a solution to a mixed-integer linear programming (MILP) formulation of the motion synthesis problem, the output of which initializes a second nonlinear programming (NLP) stage. The MILP stage enforces hard constraints of safety and road rule compliance generating a solution in the right subspace, while the NLP stage refines the solution within the safety bounds for feasibility and smoothness. We demonstrate the effectiveness of our framework via simulated experiments of complex urban driving scenarios, outperforming a state-of-the-art baseline in metrics of convergence, comfort, and progress.
Article
Time optimal model predictive control (MPC) strategies are investigated in this paper. We first derive a series of controllable sets described by ellipsoids or convex polytopes offline, then obtain the control input that can steer initial state into terminal set in the shortest time steps by solving an online optimization problem with lower computational burden. The constraints of online optimization problem are determined by the controllable sets. We give the methods to calculate the ellipsoidal controllable sets for linear time-invariant systems and linear time-varying systems, respectively, and propose a new method to calculate the polytopic controllable sets for linear time-varying systems. The recursive feasibility of online optimization problem and the stability of closed-loop system can be guaranteed. Numerical examples show that the algorithms work well.
Article
In this paper, we present an effective online planning solution for autonomous vehicles that aims at improving the computational load while preserving high levels of performance in racing scenarios. The method follows the structure of the model predictive (MP) optimal strategy where the main objective is to maximize the velocity while smoothing the dynamic behavior and fulfilling varying constraints. We focus on reformulating the non-linear original problem into a pseudo-linear problem by convexifying the objective function and reformulating the non-linear vehicle equations to be expressed in a Linear Parameter Varying (LPV) form. In addition, the ability of avoiding obstacles is introduced in a simple way and with reduced computational cost. We test and compare the performance of the proposed strategy against its non-linear approach through simulations. We focus on testing the performance of the trajectory planning approach in a racing scenario. First, the case of free obstacles track and afterwards a scenario including static obstacles. Simulation results show the effectiveness of the proposed strategy by reducing the algorithm elapsed time while finding appropriate trajectories under several input/state constraints.
Thesis
This thesis deals with the development and analysis of novel time-optimal model predictive control concepts for nonlinear systems. Common realizations of model predictive controllers apply direct transcription methods to first discretize and then optimize the subordinate optimal control problems. The key idea of the proposed concepts is to introduce discretization grids in which the underlying discretization is explicitly treated as temporally variable during optimization. A single optimization parameter for all grid intervals leads to the global uniform grid, while the definition of an individual parameter for each interval results in the local uniform and quasi-uniform grid representations. The proposed grids are well-suited for established direct transcription methods such as multiple shooting and collocation. In addition, a proposed non-uniform grid with extended multiple shooting is highly beneficial for bang-singular-bang control systems with simple constraint sets. Minimizing the local time information in all grids leads to the overall time-optimal transition. Integration with state feedback does not immediately guarantee asymptotic stability and recursive feasibility. To this end, the thesis provides a grid adaptation scheme capable of ensuring practical stability and, under more restricted conditions, also asymptotic stability while maintaining feasibility. The practical stability results facilitate the systematic dual-mode control design that restores asymptotic stability and establishes smooth stabilization. The secondary objective of this thesis is the computationally efficient realization of time-optimal model predictive control by exploiting the inherent sparse structures in the optimal control problems. In particular, the efficient computation of first- and second-order derivatives required for iterative optimization is facilitated by a so-called hypergraph. The hypergraph captures the structure of the transcribed optimal control problems and leads to an almost linear relation between computation time and grid size. In addition, the hypergraph shows negligible computation times for each reconfiguration that is essential for grid adaptation. Numerous examples in simulation and with a real experimental system demonstrate the capabilities and potentials of the proposed concepts. Extensive benchmarks in C++ compare the proposed methods with each other and the current state of the art. The methods based on variable discretization outperform the current time-optimal model predictive control methods in the literature, especially with regard to computation time.
Article
This article focuses on the design, test, and validation of a deep neural network (DNN)-based control scheme capable of predicting optimal motion commands for autonomous ground vehicles (AGVs) during the parking maneuver process. The proposed design utilizes a multilayer structure. In the first layer, a desensitized trajectory optimization method is iteratively performed to establish a set of time-optimal parking trajectories with the consideration of noise-perturbed initial configurations. Subsequently, by using the preplanned optimal parking trajectory data set, several DNNs are trained in order to learn the functional relationship between the system state-control actions in the second layer. To obtain further improvements regarding the DNN performances, a simple yet effective data aggregation approach is designed and applied. These trained DNNs are then utilized as the motion controllers to generate feedback actions in real time. Numerical results were executed to demonstrate the effectiveness and the real-time applicability of using the proposed control scheme to plan and steer the AGV parking maneuver. Experimental results were also provided to justify the algorithm performance in real-world implementations.
Article
This work presents the real-time control of 1:43 scale autonomous race cars using nonlinear model predictive control based on a singularity-free prediction model. This model allows the car to drive at both low and high speeds and in stop-and-go maneuvers. Additional constraints are imposed in the optimal control problem to ensure the validity of the model assumptions. Moreover, the control scheme is capable of avoiding obstacles online. The experimental results show that the proposed method converges to nearly time-optimal behavior by maximizing the progress on the track and achieves competitive lap time results.
Article
This study develops an autonomous vehicle control method that enables it to perform a drift maneuver which is an expert driving technique consisting of sliding the rear wheel intentionally for fast cornering. By developing an autonomous control algorithm for such an agile maneuver, the safety of the future autonomous vehicle on extreme conditions such as slippery road, will be increased. Drift equilibrium states are derived to find the suitable feedforward control input for the scale car to enter the drifting region. In addition, a feedback controller is designed based on the linear quadratic regulator method in order to track the circular trajectory and maintain drift equilibrium states. To validate the performance of the developed control algorithm a 1:10 scale car experimental platform is developed with on-board control and sensor system. The feasibility of the developed method for the autonomous vehicle is confirmed through both simulation and experiments following circular trajectories while maintaining the desired equilibrium states.
Chapter
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The control of autonomous vehicles is a challenging task that requires advanced control schemes. Nonlinear Model Predictive Control (NMPC) and Moving Horizon Estimation (MHE) are optimization-based control and estimation techniques that are able to deal with highly nonlinear, constrained, unstable and fast dynamic systems. In this chapter, these techniques are detailed, a descriptive nonlinear model is derived and the performance of the proposed control scheme is demonstrated in simulations of an obstacle avoidance scenario on a low-fricion icy road.
Conference Paper
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In this paper we address the minimum lap-time problem for a single-track rigid car which includes tire models and load transfer. Given a planar track including lane boundaries, our goal is to find a trajectory of the car minimizing the lap time subject to tire and steering limits. By using a new set of coordinates, the time-dependent system is transformed into a “space-dependent” (and space-variant) system. The choice of a suitable set of coordinates and the partition of the dynamics into a “longitudinal” one and a “transverse” one, allows us to convert the minimum time problem into a fixed horizon constrained optimal control problem. Based on a projection operator nonlinear optimal control technique, we propose a minimum lap-time strategy to push the rigid car to the limit of its handling capabilities. Finally, we provide numerical computations that: (i) show the effectiveness of the proposed strategy, and (ii) allow us to highlight important features of minimum lap-time trajectories.
Conference Paper
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We address the problem of real-time obstacle avoidance on low-friction road surfaces using spatial Nonlinear Model Predictive Control (NMPC). We use a nonlinear four-wheel vehicle dynamics model that includes load transfer. To overcome the computational difficulties we propose to use the ACADO Code Generation tool which generates NMPC algorithms based on the real-time iteration scheme for dynamic optimization. The exported plain C code is tailored to the model dynamics, resulting in faster run-times in effort for real-time feasibility. The advantages of the proposed method are shown through simulation.
Article
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This paper focuses on time-optimal path tracking, a subproblem in time-optimal motion planning of robot systems. Through a nonlinear change of variables, the time-optimal path tracking problem is transformed here into a convex optimal control problem with a single state. Various convexity-preserving extension are introduced, resulting in a versatile approach for optimal path tracking. A direct transcription method is presented that reduces finding the globally optimal trajectory to solving a second-order cone program using robust numerical algorithms that are freely available. Validation against known examples and application to a more complex example illustrate the versatility and practicality of the new method.
Article
Full-text available
Optimization problems in chemical engineering often involve complex systems of nonlinear DAE as the model equations. The direct multiple shooting method has been known for a while as a fast off-line method for optimization problems in ODE and later in DAE. Some factors crucial for its fast performance are briefly reviewed. The direct multiple shooting approach has been successfully adapted to the specific requirements of real-time optimization. Special strategies have been developed to effectively minimize the on-line computational effort, in which the progress of the optimization iterations is nested with the progress of the process. They use precalculated information as far as possible (e.g. Hessians, gradients and QP presolves for iterated reference trajectories) to minimize response time in case of perturbations. In typical real-time problems they have proven much faster than fast off-line strategies. Compared with an optimal feedback control computable upper bounds for the loss of optimality can be established that are small in practice. Numerical results for the Nonlinear Model Predictive Control (NMPC) of a high-purity distillation column subject to parameter disturbances are presented.
Conference Paper
Full-text available
In this paper we present a Model Predictive Control (MPC) approach for combined braking and steering systems in autonomous vehicles. We start from the result presented in F. Borrelli et al. (2005) and P. Falcone et al. (2006), where a Model Predictive Controller (MPC) for autonomous steering systems has been presented. We formulate a predictive control problem in order to best follow a given path by controlling the front steering angle and the brakes at the four wheels independently, while fulfilling various physical and design constraints.
Article
A large class of algorithms for nonlinear model predictive control (MPC) and moving horizon estimation (MHE) is based on sequential quadratic programming and thus requires the solution of a sparse structured quadratic program (QP) at each sampling time. We propose a novel algorithm based on a dual two-level approach involving a nonsmooth version of Newton's method that aims at combining sparsity exploitation features of an interior point method with warm-starting capabilities of an active-set method. We address algorithmic details and present the open-source implementation qpDUNES. The effectiveness of the solver in combination with the ACADO Code Generation tool for nonlinear MPC is demonstrated based on set of benchmark problems, showing significant performance increases compared to the established condensing-based approach, particularly for problems with long prediction horizons.
Chapter
The vertical motions of road and rail vehicles can be treated jointly again. The investigations of the vertical motion provide a basis for the optimization of the driving comfort, and the driving safety as far as road vehicles are concerned. The contact forces in wheel contact patches most important for the longitudinal and lateral motions are directly affected by the vertical motion according to the rolling contact laws. The fundamental principles of a vehicle suspension can be completely studied by a model with two vertical degrees of freedom. However, for a more detailed analysis more complex models can be used, too. For magnetically levitated vehicles the active suspension control has to be considered, too. For wheeled vehicles actively controlled suspensions are also of increasing interest.
Conference Paper
In this paper we present a Model Predictive Control (MPC) approach for combined braking and steering systems in autonomous vehicles. We start from the result presented in (Borrelli et al. (2005)) and (Falcone et al. (2007a)), where a Model Predictive Controller (MPC) for autonomous steering systems has been presented. As in (Borrelli et al. (2005)) and (Falcone et al. (2007a)) we formulate an MPC control problem in order to stabilize a vehicle along a desired path. In the present paper, the control objective is to best follow a given path by controlling the front steering angle and the brakes at the four wheels independently, while fulfilling various physical and design constraints.
Conference Paper
Two frameworks based on Model Predictive Control (MPC) for obstacle avoidance with autonomous vehicles are presented. A given trajectory represents the driver intent. An MPC has to safely avoid obstacles on the road while trying to track the desired trajectory by controlling front steering angle and differential braking. We present two different approaches to this problem. The first approach solves a single nonlinear MPC problem. The second approach uses a hierarchical scheme. At the high-level, a trajectory is computed on-line, in a receding horizon fashion, based on a simplified point-mass vehicle model in order to avoid an obstacle. At the low-level an MPC controller computes the vehicle inputs in order to best follow the high level trajectory based on a nonlinear vehicle model. This article presents the design and comparison of both approaches, the method for implementing them, and successful experimental results on icy roads.
Conference Paper
This paper presents a hierarchical control framework for the obstacle avoidance of autonomous and semi-autonomous ground vehicles. The high-level planner is based on motion primitives created from a four-wheel nonlinear dynamic model. Parameterized clothoids and drifting maneuvers are used to improve vehicle agility. The low-level tracks the planned trajectory with a nonlinear Model Predictive Controller. The first part of the paper describes the proposed control architecture and methodology. The second part presents simulative and experimental results with an autonomous and semi-autonomous ground vehicle traveling at high speed on an icy surface.
Conference Paper
Receding horizon control requires the solution of an optimization problem at every sampling instant. We present efficient interior point methods tailored to convex multistage problems, a problem class which most relevant MPC problems with linear dynamics can be cast in, and specify important algorithmic details required for a high speed implementation with superior numerical stability. In particular, the presented approach allows for quadratic constraints, which is not sup-ported by existing fast MPC solvers. A categorization of widely used MPC problem formulations into classes of different complexity is given, and we show how the computational burden of certain quadratic or linear constraints can be decreased by a low rank matrix forward substitution scheme. Implementation details are provided that are crucial to obtain high speed solvers. We present extensive numerical studies for the proposed methods and compare our solver to three well-known solver packages, outperforming the fastest of these by a factor 2-5 in speed and 3-70 in code size. Moreover, our solver is shown to be very efficient for large problem sizes and for quadratically constrained QPs, extending the set of systems amenable to advanced MPC formulations on low-cost embedded hardware.
Article
The best race driver is the one that, with a given vehicle, is able to drive on a given track in the shortest possible time. Thus, the only target is the lap time. A race driver model has to do the same.The first step towards this target is to decide which trajectory to follow. In fact, the optimal trajectory is the best compromise between the shortest track and the track that allows to achieve the highest speeds (least curvature track). Thus, the problem of trajectory planning is a bounded optimisation problem that has to take into account not only the geometry of the circuit but also the dynamics of the vehicle. A simplified vehicle dynamic model is used for this purpose. Due to the fact that the vehicle will be driven at its limit performances, although simplified, the model has to correctly reproduce the maximum possible acceleration, a function of the vehicle speed, the maximum possible deceleration, again a function of the vehicle speed, and the maximum lateral acceleration, a function of both the vehicle speed and the steering angle. Knowing the trajectory, the vehicle model allows to determine the lap time. Through an optimisation algorithm it is therefore possible to determine the best compromise between shortest track and track with the minimum curvature, i.e. the trajectory (in terms of track and speed profile) that allows to minimize the time lap.Once the best trajectory has been determined (both in terms of best track and best speed profile), it is necessary to identify the driver’s inputs to follow the given trajectory. This task is carried out by considering the driver as a controller that acts on a nonlinear plant (the vehicle) in order to achieve the desired results. Thus, the driver converts the best trajectory into vehicle’s inputs. The mutual interaction between plant and controller (the driver’s inputs are not only a function of the best trajectory but also of the driver’s reactions due to the vehicle’s dynamics) is not taken into account in this paper.
Article
In this paper, a novel approach to autonomous steering systems is presented. A model predictive control (MPC) scheme is designed in order to stabilise a vehicle along a desired path while fulfilling its physical constraints. Simulation results show the benefits of the systematic control methodology used. In particular we show how very effective steering manoeuvres are obtained as a result of the MPC feedback policy. Moreover, we highlight the trade off between the vehicle speed and the required preview on the desired path in order to stabilise the vehicle. The paper concludes with highlights on future research and on the necessary steps for experimental validation of the approach.
Article
In this paper, we present an automatic C-code generation strategy for real-time nonlinear model predictive control (NMPC), which is designed for applications with kilohertz sample rates. The corresponding code export module has been implemented within the software package ACADO Toolkit. It is capable of exporting fixed step-size integrators together with their sensitivities as well as a real-time Gauss–Newton method. Here, we employ the symbolic representation of optimal control problems in ACADO in order to auto-generate plain C-code which is optimized for final production. The exported code has been tested for model predictive control scenarios comprising constrained nonlinear dynamic systems with four states and a control horizon of ten samples. The numerical simulations show a promising performance of the exported code being able to provide feedback in much less than a millisecond.
Article
Nearly all algorithms for linear model predictive control (MPC) either rely on the solution of convex quadratic programs (QPs) in real time, or on an explicit precalculation of this solution for all possible problem instances. In this paper, we present an online active set strategy for the fast solution of parametric QPs arising in MPC. This strategy exploits solution information of the previous QP under the assumption that the active set does not change much from one QP to the next. Furthermore, we present a modification where the CPU time is limited in order to make it suitable for strict real-time applications. Its performance is demonstrated with a challenging test example comprising 240 variables and 1191 inequalities, which depends on 57 parameters and is prohibitive for explicit MPC approaches. In this example, our strategy allows CPU times of well below 100 ms per QP and was about one order of magnitude faster than a standard active set QP solver. Copyright © 2007 John Wiley & Sons, Ltd.
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P. Spengler and C. Gammeter. Modeling of 1:43 scale race cars. Master's thesis, ETH Zrich, 2010.
Low complexity MPC schemes for integrated vehicle dynamics control problems
  • P Falcone
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  • H E Tseng
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P. Falcone, F. Borrelli, J. Asgari, H.E. Tseng, and D. Hrovat. Low complexity MPC schemes for integrated vehicle dynamics control problems. 9 th International Symposium on Advanced Vehicle Control, 2008.
A Condensing Algorithm for Nonlinear MPC with a Quadratic Runtime in Horizon Length
  • A E Joel
  • Janick V Andersson
  • Milan Frasch
  • Moritz Vukov
  • Diehl
Joel A. E. Andersson, Janick V. Frasch, Milan Vukov, and Moritz Diehl. A Condensing Algorithm for Nonlinear MPC with a Quadratic Runtime in Horizon Length. Automatica, 2013. Submitted.
Model Predictive Control of Autonomous Vehicles
  • M Zanon
  • J V Frasch
  • M Vukov
  • S Sager
  • M Diehl
M. Zanon, J. V. Frasch, M. Vukov, S. Sager, and M. Diehl. Model Predictive Control of Autonomous Vehicles. In Proceedings of the Workshop on Optimization and Optimal Control of Automotive Systems, pages 41-57. 2014.