I Introduction
Ia Motivation
Recently, autonomous racing is an active subtopic in the field of autonomous driving research. In autonomous racing, the ego car is required to drive along a specific track with an aggressive behavior, such that it is capable of competing with other agents on the same track. By overtaking other leading vehicles and moving ahead, the ego vehicle can finish the racing competition with a smaller lap time. While the behavior of overtaking other vehicles has been studied in autonomous driving on public roads, however, these techniques are not effective on a race track. This is because autonomous vehicles are guided by dedicated lanes on public roads to succeed in lane follow and lane change behaviors, while the racing vehicles compete in the limitedwidth tracks without guidance from welldefined lanes. Existing work focuses on a variety of algorithms for autonomous car racing, but most of them could not provide a timeoptimal behavior with high update frequency in the presence of other moving agents on the race track. In order to generate racing behaviors for the ego racing car, we propose a racing algorithm for planning and control that enables the ego vehicle to maintain timeoptimal maneuvers in the absence of local vehicles, and fast overtake maneuvers when local vehicles exist, as shown in Fig. 1.
IB Related Work
In recent years, researchers have been focusing on planning and control for autonomous driving on public roads. For competitive scenarios like autonomous lane change or lane merge, both modelbased methods [34] and learningbased methods [42] have been demonstrated to generate the ego vehicle’s desired trajectory. Similarly, control using modelbased methods [13, 26] and learningbased methods [21] have also been developed. However, the criteria to evaluate planning and control performance are different for car racing compared to autonomous driving on public roads. For car racing [25], when the ego racing car competes with other surrounding vehicles, most onroad traffic rules are not effective. Instead of maneuvers that offer a smooth and safe ride, aggressive maneuvers that push the vehicle to its dynamics limit [38] or even beyond its dynamics limit [11] are sought to win the race. In order to quickly overtake surrounding vehicles, overtake maneuvers with tiny distances between the cars and large orientation changes are needed. Moreover, due to the bigger slip angle caused by changing the steering orientation more quickly during racing, more accurate dynamical models should be used for autonomous racing planning and control design. We next enumerate the related work in several specific areas.
Approach  GP  DRL  GraphSearch  Game Theory  ModelBased  
Publication  [15]  [10]  [37]  [24]  [40]  [14]  [32]  [18]  [19]  [9]  [6]  [44]  Ours 
Lap Timing  Yes  Yes  Yes  Yes  No  Yes  Yes  Yes  Yes  No  No  No  Yes 
Static Agent  No  No  Yes  Yes  Yes  No  No  No  No  Yes  Yes  Yes  Yes 
Moving Agent  No  No  One  One  Multiple  No  No  No  No  Multiple  One  One  Multiple 
Update Frequency (Hz)  N/A  N/A  15  30  2  Offline  20  20  Offline  <1  10  <10  >25 
Planner  No  No  Yes  Yes  Yes  Yes  No  No  No  Yes  Yes  No  Yes 
Dynamics Accuracy  N/A  N/A  Yes  Yes  No  Yes  Yes  Yes  Yes  Yes  Yes  Yes  Yes 
This indicates if the lap timing performance was considered in the work.  
This indicates the optimization update frequency. Learningbased approaches like GP don’t have this update frequency.  
This indicates the dynamics model used in controller. “Yes”, “No”, “N/A” are for dynamic model, kinematic model and modelfree one. 
IB1 Planning Algorithms
For car racing, the planner is desired to generate a timeoptimal trajectory. Although some work using convex optimization problems [23, 20, 5, 7, 14, 36, 1] or Bayesian optimization (BO) [16] reduces the ego vehicle’s lap time impressively, either no obstacles [20, 5, 7, 14, 36, 1, 16] or only static obstacles [23] are assumed to be on the track. When moving vehicles exist on the track, nonlinear dynamic programming (NLP) [9], graphsearch [37] and game theory [24, 40, 17] based approaches have demonstrated their capabilities to generate collisionfree trajectories. Additionally, in order to improve the chance of overtaking, offline policies are learnt for the overtake maneuvers at different portions of a specific track [3]. However, these approaches don’t solve all challenges. For instance, work in [9, 40, 17, 3] does not take lap timing enhancement into account. In [37], the ego vehicle is assumed to compete on a straight track with one constantspeed surrounding vehicle. These assumptions are relatively simple for a real car racing competition. In [24], it is assumed that the planner knows the other vehicle’s strategy and the complexity of the planner increases excessively when multiple vehicles compete with each other on the track.
IB2 Control Algorithms
Researchers focus on enhancing performance of the ego vehicle by achieving its speed and steering limits through better control design, e.g., obtaining optimal lap times by driving fast. The majority of existing work focuses on developing controllers with no other vehicles on the track. The learningbased controllers [32, 18, 19, 41] leverage the control input bounds to achieve optimal performance in iterative tasks. Modelfree methods like Bayesian optimization (BO) [27], Gaussian processes (GPs) [15]
, deep neural networks (DNN)
[28, 39]and deep reinforcement learning (DRL)
[10, 31] have also been exploited to develop controllers that result in agile maneuvers for the ego car. To deal with other surrounding vehicles, DRL has also been used in [35] to control the ego vehicle during overtake maneuvers. Recently, model predictive based controllers (MPC) with nonlinear obstacle avoidance constraints has become popular to help the ego vehicle avoid other vehicles in the free space. A nonconvex nonlinear optimization based controller is implemented in [33] to help the ego vehicle avoid static obstacles. Researchers in [22] use mixedinteger quadratic programs (MIQP) to help the ego vehicle compete with one moving vehicle. In [6], GPs was applied to formulate the distance constraints of a stochastic MPC controller with a kinematic bicycle model. However, large slip angles under aggressive maneuvers will cause a mismatch between real dynamics model and the kinematic model used in the controller, resulting in the controller being unable to guarantee the system’s safety in some cases. In [44], a safetycritical control design by using control barrier functions is proposed to generate a collisionfree trajectory without a highlevel planner, where infeasibility could arise due to the high nonlinearity of the optimization problem. Moreover, due to the lack of a trajectory planner, deadlock could happen very often during the overtake maneuvers, such as in [44, 6]. A comparison of various approaches and their features are enumerated in TABLE I.As mentioned above, all the previous work on planning and control design for autonomous racing could not enhance the lap timing performance and simultaneously compete with multiple vehicles. Inspired by the work on iterative learningbased control and optimizationbased planning, we propose a novel racing strategy to resolve the challenges mentioned above with a steady low computational complexity.
IC Contribution
The contributions of this paper are as follows:

We present an autonomous car racing strategy that switches between a learningbased MPC trajectory planner (in the absence of surrounding vehicles) and optimizationbased homotopic trajectory planner with a lowlevel safetycritical controller (when the ego vehicle competes with surrounding vehicles).

The learningbased MPC approach guarantees timeoptimal performance in the absence of surrounding vehicles. When the ego vehicle competes with surrounding vehicles, multiple homotopic trajectories are optimized in parallel with different geometric reference paths and the best timeoptimal trajectory is selected to be tracked with an optimizationbased controller with obstacle avoidance constraints.

We validate the robust performance together with steady low computational complexity of our racing strategy in numerical simulations where randomly moving vehicles are generated on a simulated race track. It is shown that our proposed strategy allows the ego vehicle to succeed in overtaking tasks when there are multiple vehicles moving around the ego vehicle.
Ii Background
In this section, we revisit the vehicle model and learningbased MPC for iterative tasks. The learningbased MPC will be used as the trajectory planner when no surrounding vehicles exist.
Iia Vehicle Model
In this work, we use a dynamic bicycle model under Frenet coordinates. The system dynamics is described as follows,
(1) 
where and show the state and input of the vehicle, is a nonlinear dynamic bicycle model in [30]. The definition of state and input is as follows,
(2) 
where acceleration at vehicle’s center of gravity and steering angle are the system’s inputs. denotes the curvilinear distance travelled along the track’s center line, and show the deviation distance and heading angle error between vehicle and center line. , and are the longitudinal velocity, lateral velocity and yaw rate, respectively.
In this paper, this model (1
) is applied for precise numerical simulation using Euler discretization with sampling time 0.001s (1000Hz). Through linear regression from the simulated reference path, an affine timeinvariant model as below,
(3) 
will be used in the trajectory planner to avoid excessive complexity from nonlinear optimization, where represents the equilibrium point for linearized dynamics. On the other hand, an affine timevarying model as below,
(4) 
where matrices , , and are obtained at local equilibrium point on reference trajectory with iterative data which is close to . The dynamics (4) will be used on racing controller design for better tracking performance. The data collection for iterative tasks for reference trajectory will be presented in the next part.
IiB Iterative Learning Control
A learningbased MPC [32], which improves the ego vehicle’s lap timing performance through iterative tasks, will be used in this paper. This has the following components:
IiB1 Data Collection
The learningbased MPC optimizes the lap timing through historical states and inputs from iterative tasks. To collect initial data, a simple tracking controller like PID or MPC can be used for the first several laps. During the data collection process, after the
th iteration (lap), the controller will store the ego vehicle’s closedloop states and inputs as vectors. Meanwhile, through offline calculation, every point of this iteration will be associated with a cost, which describes the time to finish the lap from this point.
IiB2 Online Optimization
After the initial laps, the learningbased MPC optimizes the vehicle’s behavior based on collected data. At each time step, a convex set for terminal constraint (green convex hull in Fig. 3) is built to represent the states that can drive the ego vehicle to the finish line in the previous laps. By constructing the cost function to create a minimumtime problem, an openloop optimized trajectory can be generated. Since the cost function is based on the previous states’ timing data, the vehicle is able to drive to the finish line with time that is no greater than the time from the same position during previous laps. As a result, the ego vehicle will reach the timeoptimal performance after several laps.
More details of this method can be found in [32]. In our work, this approach will be used for trajectory planning when the ego vehicle has no surrounding vehicles. This helps with better lap timing without surrounding vehicles. Notice that the data for iterative learning control will be collected through offline simulation with no obstacles on the track, shown in Fig. 2.
Iii Racing Algorithm
After introducing the background of vehicle modeling and learningbased MPC, we will present an autonomous racing strategy that can help the ego vehicle enhance lap timing performance while overtaking other moving vehicles.
Iiia Autonomous Racing Strategy
There are two tasks in autonomous racing: enhancing the lap timing performance and competing with other vehicles. To deal these two problems, our proposed strategy will switch between two different planning strategies. When there are no surrounding vehicles, trajectory planning with learningbased MPC is used to enhance the timing performance through historical data. Once the leading vehicles are close enough, an optimizationbased trajectory planner optimizes several homotopic trajectories in parallel and the collisionfree optimal trajectory is selected with an optimaltime criteria, which will be tracked by a lowlevel MPC controller. By adding obstacle avoidance constraints to the lowlevel controller, it has the ability to guarantee the system’s safety. The racing strategy is summarized in Fig. 2.
IiiB Overtaking Planner
To determine if a leading vehicle is in the ego vehicle’s range of overtaking, following condition must be satisfied:
(5) 
where and are ego vehicle’s and th surrounding vehicle’s traveling distance, and are ego vehicle’s and th surrounding vehicle’s longitudinal speed. indicates the vehicle’s length. and are safetymargin factor and prediction factor which we can tune for different performance.
As shown in Fig. 4, when there are vehicles in the ego vehicle’s range of overtaking, there exists potential areas, each leading to paths with a different homotopy, that the ego vehicle can use to overtake these surrounding vehicles. These areas are the one below the th vehicle, the one above the 1st vehicle, and the ones between each group of adjacent vehicles. +1 groups of optimizationbased trajectory planning problems are solved in parallel, enabling steady low computational complexity even when competing with different numbers of surrounding vehicles. To reduce each optimization problem’s computational complexity through fast convergence, geometric paths with a distinct homotopy class that laterally lay between vehicles or vehicle and track boundary (black dashed curves in Fig. 4) are used as reference paths in the optimization problems. By comparing the optimization problems’ costs, the optimal trajectory is selected from optimized solutions. For example, as the case shown in Fig. 4, the dashed orange line in area 2 will be selected since it avoids surrounding vehicles and finishes overtake maneuver with smaller time. The function to minimize during the selection is shown as follows,
(6)  
where is a scalar used in metric for timing and is a nonzero penalty cost if the new potential area of overtaking is different from the area of overtaking in the last time step. A bigger value of is applied such that the ego vehicle is optimized to reach a farther point during the overtake maneuver, which results in a shorter overtaking time since the planner’s prediction horizon and sampling time are fixed. Additionally, the other terms in (6) prevents the ego vehicle from changing direction abruptly during an overtake maneuver and guarantees the ego vehicle’s safety.
Beziercurves are widely used in path planning algorithms in autonomous driving research [12, 8, 29]
because it is easy to tune and formulate. Thirdorder Beziercurves are used in this work. Each Beziercurve is interpolated from four control points, including shared start and end points with two additional intermediate points, shown in Fig.
5. Specifically, the start point for the Bezier curve is the ego vehicle’s current position and end point is on the timeoptimal trajectory generated from learningbased MPC planner. The selection of end point makes vehicle’s state as close as possible to the timeoptimal trajectory after overtake behavior. To make all curves smoother and have no or fewer conflicts with the surrounding vehicles, the other two control points will be between the track’s boundary and vehicle for Areas 1 and +1 shown in Fig. 4(a), or between two adjacent vehicles shown in Fig. 4(b). These two intermediate control points will have the same lateral deviation from the center line. The key advantage of our selection of control points is that the interpolated geometric curve won’t cross the connected lines between control points with its convexity, shown in Fig. 5. This property makes our reference paths collisionfree with respect to other surrounding vehicles in most cases, which speed up the computational time of the trajectory generation at each area.Remark 1 ().
When the ego vehicle is approaching other surrounding vehicles with big lateral difference (see Fig. 6), the Beziercurve path used in planner might be not collisionfree with other surrounding vehicles. In this case, the corresponding area is not an ideal choice for the overtake maneuver since it asks the ego vehicle to change direction abruptly. Therefore, the optimized trajectory is either infeasible or has a large overtaking time and is consequently not selected.
Remark 2 ().
In order to ultimately speed up convergence of the optimization problem, the ideal choice for reference paths in the optimization problem should be a collisionfree path for the ego vehicle. However, this comes with a high computational complexity with nonlinear optimization. Therefore, we use the curves that has no or fewer conflicts with other vehicles instead of the collisionfree curves in this paper to find an appropriate tradeoff between complexity and performance of optimized trajectory. Spline curves like cubic curvature polynomials, Dubin’s paths and Bezier curves are suitable candidate types for reference paths.
The details of the optimization formulation for trajectory generation will be illustrated in the next section.
IiiC Trajectory Generation
After illustrating the planing strategy, this subsection will show the details about the optimization problem used for trajectory generation for each potential area with different homotopic paths that the ego vehicle can use to overtake the surrounding vehicles.
The optimization problem is formulated as follows,
(7a)  
s.t.  (7b)  
(7c)  
(7d)  
(7e) 
where (7b), (7c), (7d) are constraints for system dynamics, state/input bounds and initial condition. The system dynamics constraint describes the affine linearized model described in (3). The cost function (7a) is composed with three parts, the terminal cost , the stage cost and the state/input changing rate cost . The construction of cost function and constraints in the optimization will be presented in details in the following subsections.
IiiC1 Terminal Cost
Terminal cost is about the ego vehicle’s traveling distance along the track during the overtaking process
(8) 
This compares the openloop predicted traveling distance at the th step with the ego vehicle’s current traveling distance . This works as the cost metric for timing during the overtaking process.
IiiC2 Stage Cost
The stage cost introduces the lateral position differences between the openloop predicted trajectory and other two paths along the horizon.
(9) 
and are the reference path and timeoptimal trajectory in Frenet coordinates. The timeoptimal trajectory is generated by the learningbased MPC trajectory planner used on a track without other agents, discussed in Sec. IIB. is an initial guess for the traveling distance at the th step, which is equal to , where a constant longitudinal speed is assumed along the prediction horizon.
IiiC3 State/Input Changing Rate Cost
To make the predicted trajectory smoother, the state/input changing rate cost is formulated as follow:
(10)  
IiiC4 Obstacle Avoidance Constraint
In order to generate a collisionfree trajectory, collision avoidance constraint (7e) is added in the optimization problem. To reduce computational complexity, only linear lateral position constraint will be added when the ego vehicle overlapes with other vehicles longitudinally. will be used to check if the ego vehicle is overlapping with other vehicle longitudinally along the horizon. In (7e), shows the lateral position difference, and are the vehicle’s length and width, is a safe margin.
IiiD Overtaking Controller
After introducing the algorithm for trajectory generation, a lowlevel tracking controller with model predictive control used for overtaking will be discussed in this part. The constrained optimization problem is described as follows:
(11a)  
(11b)  
(11c)  
(11d)  
(11e)  
where 
(11b), (11c), (11d) describe the constraints for system dynamics (4), input/state bounds and initial conditions, respectively. The represents the stage cost, which tracks the desired trajectory optimized by the trajectory planner. Equation (11e) with represents discretetime control barrier function constraints [43] with relaxation ratio , which could guarantee the system’s safety by guaranteeing along the horizon with forward invariance. In this project, is used to represent the distance between the ego vehicle and other vehicles. The optimization (11) allows us to find the optimal control in a manner similar to MPC.
Iv Results
Iva Simulation Setup
Having illustrated our autonomous racing strategy and algorithm, we are going to validate our approach through numerical simulations in this section. In all simulations, all vehicles are with a length of 0.4m and a width of 0.2m. The track’s width is set to 2 m. The horizon lengths for trajectory planner and controller are , and shared discretization time . Both state and input noises are considered in the simulations. The optimization problems (7) and (11) are implemented in Python with CasADi [2] used as modeling language, are solved with IPOPT [4] on Ubuntu 18.04 on a laptop with a CPU i79850 processor at a 2.6Ghz clock rate.
IvB Racing With Other Vehicles
Snapshots shown in Fig. 7 illustrate examples of overtaking behavior in both straight and curvy track segments when competing with other vehicles. When the ego vehicle competes with three surrounding vehicles, it could overtake them on one side of all vehicles (Fig. 6(a)) or between them (Fig. 6(b)). The animations of more challenging overtaking behavior can be found on https://youtu.be/41r0Vu6rf4. As shown in TABLE I, the proposed racing planner could update at 25 Hz and could help the ego vehicle overtake multiple moving vehicles. By switching to a trajectory planning based on learningbased MPC, the ego vehicle is able to reach its speed and steering limit when there are no surrounding vehicles.
Speed Range [m/s]  0  0.4  0.4  0.8  0.8  1.2  1.2  1.6 

mean [s]  1.613  2.312  3.857  13.095 
min [s]  0.8  1.2  1.8  3.5 
max [s]  3.6  5.2  21.6  36.1 
To better analyze the performance and limitations of our autonomous racing strategy in different scenarios, random tests are introduced under two groups. The first group of simulation aims to show the overtaking time for passing one leading vehicle with different speeds, and statistical results are summarized in TABLE II. We can observe that when the surrounding vehicles’ speed reaches between 1.2m/s and 1.6m/s, much more time is needed for the ego vehicle to overtake the leading vehicle. This is because as the leading vehicle’s speed increases, less space becomes available for the ego vehicle to drive safely. Especially in a curve, the ego vehicle’s speed limit decreases when less space can be used to make a turn. Since more than half of our track is with curves, the ego vehicle needs to wait for a straight segment to accelerate to pass the leading vehicle.
Speed Range [m/s]  0  0.4  0.4  0.8  0.8  1.2  1.2  1.6 

Single  100  100  96  84 
Two  100  100  98  66 
Three  100  98  84  36 
The second group of simulation shows the proposed racing strategy’s success rate to overtake multiple leading vehicles in one lap, and statistical results are summarized in TABLE III. We can find that when more than one surrounding vehicle exists, much more space would be occupied by other vehicles. As a result, the ego vehicle might not have enough space to accelerate to high speed to pass surrounding vehicles. Although in these cases, the ego vehicle can not overtake all surrounding vehicles after one lap, our proposed racing strategy can still guarantee the ego vehicle’s safety along the track.
During our simulation, the mean solver time for our planner for single, two or three surrounding vehicles is 39.21ms, 39.41ms and 40.23ms. We also notice that when the number of surrounding vehicles is larger than three, the steady complexity still holds but the track becomes too crowded for the ego vehicle to achieve high success rate of overtake maneuver. This validates the steady low computational complexity of proposed planning strategy thanks to the parallel computation for multiple trajectory optimizations.
IvC Racing Without Other Vehicles
As discussed in Sec. IIIA, when there are no other surrounding vehicles, the ego vehicle adopts the learningbased MPC formulation for trajectory generation and control. In this paper, the learningbased MPC uses historical data from two previous laps (implying in Fig. 3) and the initial data are calculated offline before the racing tasks. For the same setup as shown in Fig. 6(a), ego vehicle’s speed and deviation from track’s center line along the track is shown with learningbased MPC’s profile in Fig. 8. The overtake maneuver happens in a hairpin curve and the curve’s apex is occupied by other moving vehicles, resulting in less space being available for the ego vehicle and thus causing it to slow down to avoid a potential collision. After it passes all surrounding vehicles, the ego vehicle goes back to drive at its speed and steering limit to achieve timeoptimal behavior.
V Conclusion
In this paper, we presented an autonomous racing strategy that could help the ego vehicle enhance its lap timing performance while overtaking other moving vehicles. We verify the performance of our proposed algorithm through numerical simulation, where several surrounding vehicles are simulated to start from random position with random speed on a track. Other competing car racing strategies such as blocking will be envisaged for the future.
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