Simulation is a critical step in the development of autonomous driving technologies, allowing engineers to test control algorithms, path planners, and other dynamic vehicle behaviors in a risk-free, low-cost environment. Under normal driving conditions representative of paved, most vehicle models are suitable for accurate prediction of vehicle motion. On low friction surfaces, basic vehicle models do not display the variability in vehicle response to steering input that is observed on real-world low friction surfaces such as clear ice. This work presents distribution parameters for a stochastic friction grid map for use in simulating vehicle behavior on icy surfaces. Simulation data from rapid double lane changes are compared with vehicle response to the same paths on an ice rink test course. Strong correlation between the simulation and test vehicle is achieved with validation performed using previously developed control and path planning methods.
Simulation is a key element in the development of autonomous vehicle technology. Scenarios demanding long run times, multiple agents, and repeated experiments benefit from the low-time cost of testing autonomous vehicle control, path planning, vehicle interaction with traffic, and many other dynamic behaviors [1,2]. Simulation allows for testing under specific conditions such as weather or low-frequency events . Finally, simulation provides an environment for testing which does not put passengers at risk [4,5]. This is important when environmental effects may cause undesirable vehicle responses such as vehicle instability or loss of control. One scenario where this risk is present but must be designed for to reach level 4 or 5 autonomy as defined by SAE standard J3016 , is a rapid lane change to avoid an obstacle on a low friction surface such as clear ice.
Most of the time, autonomous vehicles operate on normal paved roads, but for widespread adoption of the technology, the risk of low friction surfaces in winter weather must be accounted for. The friction coefficient in conditions with clear ice or snow-covered ice can be lower than μ = 0.1 . Previous work has demonstrated that test vehicle platforms operating on these surfaces experience high degrees of variation in their trajectory response compared to the execution of similar maneuvers on the dry pavement . While a simulation environment using a constant friction coefficient may predict a realistic vehicle response, it will not produce the variation in results observed in real-world testing. This feature of the simulation is expected due to the empirical nature of tire modeling such as the magic tire formula used in this work; the lateral and longitudinal forces generated by the tire are estimated based on the best fit of experimental data [9,10]. This work presents a stochastic method for generating the friction surface a simulated vehicle operates upon which allows for autonomous vehicle development requiring dynamic behavior on icy surfaces. The effectiveness of the method is demonstrated using rapid double lane changes for obstacle avoidance.
2 Materials and Methods
2.1 Double Lane Changes on Low Friction Surface.
Figure 1 shows a possible scenario in which an autonomous vehicle is confronted with a static or dynamic object in its intended path requiring an avoidance maneuver around the obstacle and back into its original lane of travel. An additional complication to this scenario is the presence of a low friction surface such as glare ice experienced in northern climates. This surface may be known either through detection methods [11–13] or through vehicle to vehicle/infrastructure communication. Developing control and path planning methods for navigating such a scenario will benefit from a simulation environment which allows for a high volume of testing before being implemented on a vehicle platform.
A variety of methods have been used to form lane change geometries for obstacle avoidance [14–18]. The reference geometry and target velocity of 30 mph (13.4 m/s) ISO 3888-2  standard for a double lane change maneuver are in this work as an example of a rapid lane change that would be required in an autonomous obstacle avoidance scenario. The geometry used for the ISO 3888-2 maneuver is based on the vehicle width, in this case, giving a lane width of 3.2 m center to center (Fig. 2) which is representative of a typical lane width .
2.2 Simulation Environment.
The vehicle dynamics blockset in matlab’s Simulink environment contains the 14 degrees-of-freedom (DOF) vehicle model used in this work . The vehicle model, shown in Fig. 3, consists of a 6DOF vehicle body and 2DOF wheel supported by a MacPherson independent suspension. The approximate dynamic properties of the test vehicle used in this work are used in the vehicle body and suspension with tires modeled using generic magic formula parameters for a passenger vehicle.
The vehicle model in the Simulink environment is controlled using a user defined function handling waypoint management, lateral control, and longitudinal control. Inputs required for simulation are a path matrix, a target velocity, and a grid map of friction coefficients. A diagram of the simulation environment is shown in Fig. 4.
2.3 Test Vehicle.
In order to develop improved simulation methods, baseline data from a test vehicle are required for comparison. This work uses the Chevrolet Bolt provided to Michigan Technological University through the SAE and GM sponsored AutoDrive Challenge shown in Fig. 5. Robot operating system is used to interface with the vehicle CAN bus, allowing for autonomous operation. Localization of the vehicle is performed using a NovAtel PwrPak7-E1 GNSS-INS for GPS navigation. Real-time kinematic corrections are used to achieve a positional accuracy of less than 2 cm during autonomous driving maneuvers.
2.4 Vehicle Control.
Lateral control for both the simulation and test vehicle is achieved with the commonly used pure pursuit method adapted for Ackerman steered vehicles [22–25]. Pure pursuit is a geometric steering method which uses a goal point on the path and a set look ahead distance of the vehicle to generate steering commands which steer the vehicle in a circular arc toward the goal point as shown in Fig. 6(a). Placement of the goal point on the path is achieved by checking the distance of each waypoint ahead of the vehicle starting with the current waypoint being tracked in waypoint management, linearly interpolating between waypoints to maintain the desired look ahead distance. The look ahead distance is often dependent on vehicle velocity with this work using a fixed look ahead distance of 12.5 m for the velocity of 13.4 m/s where the maneuvers are performed at.
In both the simulation environment and on the test vehicle, navigation is performed through a series of waypoints. The simulation uses a (x, y) coordinate system with the vehicle and planned path beginning at the origin of the coordinate frame. The test vehicle uses the Universal Transverse Mercator coordinate system using the vehicle’s onboard GPS and IMU to navigate. In both cases, the distance between waypoints is approximately 0.75 m. Progress along the path is required for use in lateral control and is achieved by fixing a coordinate frame to each waypoint in the series with its x-axis in line with the next waypoint in the series. When the vehicle passes the y-axis, the waypoint is achieved as shown in Fig. 6(b).
2.5 Hardware Baseline Behavior.
In order to develop a simulated friction surface conducive to the development of lateral vehicle control and path planning on icy surfaces, the baseline behavior of a test vehicle on ice must be established. Michigan Technological University’s Keweenaw Research Center is home to a winter test course which features a 300 m by 50 m ice rink test track that is of sufficient size to perform the discussed lane change maneuvers. Between Jan. 2021 and March 2021, lane change maneuvers were performed on various ice conditions, examples of which are shown in Fig. 7. Though no measurements of friction coefficient were performed during these tests, the condition of the ice was categorized qualitatively into two broad categories: sticky ice and clear ice. These categorizations are based on the length of maneuver required to meet cross track error constraints determined by concurrent work .
In addition to the ice rink test track, maneuvers were also performed on dry pavement. Figure 8 shows a strong correlation between the vehicle and simulation paths when performing both the ISO 3888-2 maneuver as well as a longer maneuver. Previous work has demonstrated a high level of repeatability for the test vehicle on dry pavement .
Figure 9 shows the vehicle behavior during sticky ice conditions compared with a simulated constant friction coefficient of μ = 0.2. While the trajectory of the simulated vehicle path is similar to that of the test vehicle, there is significant variation in vehicle response on ice with overshoot during both lane changes ranging between 0.5 m and 1.8 m at the apex of the maneuver and between 1.0 m and 3.4 m at the bottom of the maneuver. When the maneuver is complete (after the 300 m mark), the safety driver assumes manual control of the vehicle and steers off the track. When attempting a longer maneuver measuring 100 m from the beginning of the first lane change to the end of the second, the test vehicle’s behavior more closely matches the simulation.
Figure 10 shows the vehicle behavior during clear ice conditions compared with a simulated constant friction coefficient of μ = 0.1. In the case of clear ice, the trajectory of the vehicle does not match that of the simulation. Additionally, variations of vehicle path remain with undershoot/overshoot during both lane changes ranging between 0.5 m and 1.8 m at the apex of the maneuver and between 1.0 m and 3.4 m at the bottom of the maneuver. When the maneuver is complete (after the 300 m mark), the safety driver assumes manual control of the vehicle and steers off the track. When attempting a longer maneuver measuring 120 m from the beginning of the first lane change to the end of the second, the test vehicle’s behavior more closely matches the simulation.
The proposed method for improving correlation between simulation and test vehicle results is primarily for the purpose of developing lateral control and path planning strategies in simulation. It is useful to investigate other dynamic results from the simulation to ensure the simulation is giving a realistic response to model inputs. Figure 11 shows the lateral acceleration of a single test vehicle run during ISO 3888-2 maneuver on sticky ice. As is expected from a detailed vehicle model, the acceleration profile of the simulated vehicle is of a similar magnitude and shape representative of the trajectory the vehicle follows.
An additional dynamic behavior of note is the slip angle of the front tires of the vehicle. As shown in Fig. 12, the longer double lane change maneuver length of 100 m exhibits a peak slip angle of around for the test vehicle and for the simulation run at a constant friction coefficient. The more aggressive ISO 3888-2 maneuver results in a higher peak slip angle for both the test vehicle and simulated vehicle or approximately 8 deg. In this case, the simulated vehicle reaches its peak and returns to near 0 deg relatively smooth compared to the test vehicle which exhibits sustained peak slip angle as the tires of the vehicle slide across the ice. Additionally, the test vehicle does not transition smoothly near the peak slip angle suggesting an uneven tire/road interaction as it steers toward the first lane at Y = 0 m.
2.6 Methods—Friction Coefficient Map Generation.
If path planning and control methods intended for use on icy surfaces are to be developed in simulation before testing in a vehicle, it is important that the unpredictable vehicle response on ice is sufficiently represented in the simulation environment. A simple solution to this requirement is to use a stochastic environment for vehicle simulations, generating a surface for the vehicle to travel on which varies in friction coefficient according to a predetermined distribution.
During the winter of 2015, the Keweenaw Research Center gathered friction surface measurements on the winter test course . Surface friction measurements ranged from μ = 0.02 to μ = 0.22 on the ice rink test track. Though single days did not exhibit this variation in surface friction, the distribution of measurements tested throughout the season serves as a reference for possible friction values present on the test course. The distribution of friction coefficients across the season is shown in Fig. 13 along with a uniform distribution, a truncated normal distribution, and a gamma distribution which closely match the measured distribution. These three distributions are used in simulation to generate friction surfaces. The uniform distribution used is defined by minimum (μmin) and maximum (μmax) friction coefficients. The truncated normal distribution used is defined by a mean friction coefficient, a standard deviation (σ), and a truncation point (α) which eliminates the bottom tail of the distribution. The gamma distribution used is defined by a shape parameter (k), a scale parameter (θ), and a location parameter (L) which shifts the distribution to the right.
These three distributions are used as a starting point to generate the random friction surface used in the simulation. A given test uses the parameters relative to one of the three distributions to generate a friction coefficient at each (x, y) location on a grid map, an example of which is shown in Fig. 14. Along with the distribution parameters the size of each grid is given by the “patch size” (PS) parameter which specifies the length of each square grid segment on the friction map. During the simulation process, the friction coefficient fed into each wheel is determined by that wheel’s position on the friction map.
2.7 Methods—Scoring Metric.
3 Simulation Results
Determining a suitable distribution of friction coefficients across a simulated surface was performed over three simulation series shown in Table 1. Each parameter set and patch size were simulated ten times, generating a new map between each run. While this method for choosing a final distribution may not be optimal, it does deliver a method for generating a variable friction surface suitable for the development of lateral control and path planning on icy surfaces in simulation.
The first series of simulations tested each surface friction distribution shown in Fig. 13. The results from the first series of simulations are shown in Fig. 16 with each type of distribution demonstrating effectiveness at causing variation in vehicle trajectory on similar surfaces. While this series produces low simulation scores, the shape and location of the simulated trajectories do not align well with the test vehicle data.
The first simulation series serves as a starting point for a refinement of parameters used in generating the friction map. For the second series of simulations, the weighting toward the minimum friction surface measurement of μ = 0.02 was shifted to bring a higher percentage of the generated map in line with the coefficient of friction associated with ice (μ ≈ 0.1) in the literature . By increasing μmin for uniform, α for normal, and L for gamma, each distribution was shifted to the right. While this resulted in higher scores, the shape of the vehicle trajectories more closely aligns with the test vehicle data set. The results from the second simulation series are shown in Fig. 17.
All distributions in series two resulted in similar scores with the gamma distribution exhibiting the overall lowest score. Based on its low score, the gamma distribution was chosen as a focus for the third simulation series. Figure 18 shows the results from tuning the parameters of the gamma distribution, arriving at k = 1.2, θ = 0.04, L = 0.08, and PS = 3 m. A gamma distribution with these parameters leads to simulated vehicle trajectories with similar shape and variability to those of the test vehicle.
Figure 19 shows the acceleration response of a single vehicle simulation using the final gamma distribution shown in Table 1 compared with a similar trajectory from the test vehicle performing an ISO 3888-2 maneuver on ice. Both vehicle acceleration responses show similar shape and magnitude during the maneuver. This shows the simulated vehicle exhibiting expected dynamic behavior in simulation.
Figure 20 shows the front tire slip angle for both the simulated and test vehicle. Using the gamma distribution to generate the friction surface the vehicle traverses does not have a large impact on the shape of the response on the longer 100 m maneuver. For the more aggressive ISO 3888-2 maneuver, the simulated vehicle behaves similarly to the test vehicle with a longer sustained slip angle as the tires slide across the simulated surfaces.
Previous work on both optimal lane changes and lateral control methods has been done with simulation work completed using a constant friction surface. Figure 21 shows the results using the gamma distribution for a lane change which minimizes cross track error and lateral acceleration during a maneuver. Though there is still some variation in the vehicle trajectory, the length of the maneuver does not cause the extreme slip angle which increases variation in the path.
Cross track compensated (CTC) pure pursuit  has been developed to compensate for undetected low friction and has been tested using the ISO 3888-2 maneuver on the ice rink test track. The method works by increasing the pure pursuit look ahead distance as cross track error accumulates, bringing the vehicle back into the original lane without overshoot that could lead the vehicle off-road. Using this control method with the variable friction surface delivers similar results to the test vehicle performing the same maneuver (Fig. 22).
A stochastic method for generating a simulated friction surface has been developed for use in the development of control and path planning methods for autonomous vehicles operating in icy conditions. Using a gamma distribution with a shape parameter k = 1.2, a scale parameter θ = 0.04, a location parameter L = 0.08, and a patch size PS = 3 m to determine friction location at every point of a grid map in simulation causes variation in vehicle response between runs which brings simulated vehicle behavior in line with an autonomous test vehicle operating on similar surfaces. The method presented has been tested using control and path planning methods developed for use on icy surfaces showing effectiveness at predicting vehicle behavior using a variety of command paths and control strategies. Future work includes additional maneuvers and control methods in simulation and hardware to further validate this method’s effectiveness at predicting vehicle behavior on icy surfaces. Additionally, further work can be done to investigate the effects of individual parameters and distributions on vehicle response leading to an optimal set of distribution parameters and patch size.
Conflict of Interest
There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent is not applicable. This article does not include any research in which animal participants were involved.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.