Exploiting Linear Structure for Precision Control of Highly Nonlinear Vehicle Dynamics

Drifting - operating a vehicle at a high sideslip angle - offers intriguing possibilities for controlling autonomous vehicles in critical situations. While drifting is a very dynamic process, occurring in a region of the state space with saturated tires and unstable equilibria, autonomous vehicles have been successfully controlled in this region. Previous control approaches to path tracking while drifting have relied on nonlinear vehicle models. In this paper, however, we demonstrate that linearized models capture the necessary dynamics for control in a large region surrounding a drift equilibrium. Using this linearized model, we develop a controller based on a linear quadratic regulator. This controller uses steering, throttle, and brakes to track both a desired path and a desired speed profile, making the system fully actuated. We demonstrate the fidelity of this linearized model and the utility of this controller by implementing this controller on MARTY, an electric DMC DeLorean, and accurately tracking equilibrium and quasi-equilibrium paths with centimeter-level accuracy that exceeds prior work.


I. INTRODUCTION
T HERE is more than one way for a car to go around a corner. Normally, steady cornering represents an equilibrium condition where drivers turn the steering wheel in the direction they wish to turn. The equilibrium is defined by linear dynamics at low steering angles, transitions to nonlinear dynamics as the steer angle increases and ultimately becomes unstable or uncontrollable when one axle reaches the friction limit. A less conventional way to turn, drifting, represents another possible equilibrium condition of the vehicle in steady cornering. Drifting involves generating high sideslip angles to keep the rear tires operating at their friction limits while the front tires countersteer to operate below their friction limits. This produces unstable but controllable dynamics.
Drifting can be a useful cornering method for low and uncertain friction surfaces. Tavernini et al. showed that for rear wheel drive and all wheel drive vehicles on low friction surfaces, drifting maneuvers were time-optimal for navigating a hairpin turn [1], while Berntorp et al. calculated that the minimum time maneuver through a hairpin turn would involve 30 degrees or more of sideslip at some points. Velenis et al. showed that two rally racing maneuvers that involve drifting (trail-braking and pendulum turn cornering) are the time optimal way to maneuver certain corners on low friction, off-road surfaces [2]. In particular, they found the trail-braking drifting maneuver allows the car to exit the corner and quickly resume straight line driving, giving the driver the ability to react to uncertain road and environmental conditions [2]. Velenis and Tsiotras found that the optimal trajectory to navigate a corner in order to maximize exit velocity includes larger sideslip angles in situations with low lateral friction, generating trajectories qualitatively similar to rally racing techniques [3]. Beyond optimizing time or exit velocity, Gray et al. found drifting maneuvers could be harnessed to plan paths for obstacle avoidance [4].
Given this applicability to obstacle and collision avoidance, several researchers have developed techniques for tracking a path while drifting using modern nonlinear control techniques. Goh et al. developed a controller that used steering and throttle to track a path while drifting [5] leaving velocity as a free variable. Goel et al. developed a fully-actuated controller that used steering, throttle, and front brakes to track a path and velocity profile while drifting [6]. Both of these controllers relied on nonlinear models of the vehicle dynamics. In experiments on full-sized vehicles, these controllers had root-mean-square (RMS) lateral path tracking errors of 18 cm [5] and 42 cm [6]. However, professional rally race drivers and drift competition drivers seem to obtain even better path tracking results. Rally drivers use drift maneuvers to precisely navigate the uncertain conditions and poor visibility of dirt rally tracks, and drift competition drivers position their vehicle door-to-door with another vehicle or their bumper only inches away from the wall-all while maintaining very high sideslip angles. Keen and Cole hypothesized that human drivers use linear models to make steering decisions [7], so the professional drivers' out-performance of controllers employing nonlinear dynamics in a region of the state space where the vehicle dynamics are highly nonlinear seems surprising.
The existence of multiple equilibria for cornering is a direct consequence of the nonlinearity of the equations of motion. Different models predict different numbers of equilibria, with Hindiyeh and Gerdes [8] showing three distinct equilibria arising from a three state single track vehicle model and Edelmann et al. finding four with a two track model [9]. Equilibria corresponding This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ to drifting are unstable [10] and their linear representations contain an unstable real eigenvalue [11]. Bárdos et al. showed experimentally that linear control based on such a linearized model could stabilize the velocity states of a vehicle around a drift equilibrium [12].
This paper demonstrates that the vehicle dynamics can be treated as linear over a large area of the state space surrounding the equilibrium. Furthermore, the eigenstructure remains consistent even as the equilibrium conditions vary. Therefore, linear techniques that exploit this structure can be leveraged to not only stabilize velocity states but also to track a trajectory. We validate this claim by developing a controller based on this linear model and implementing it on a full sized test platform, MARTY, shown in Fig. 1. This controller uses linear techniques to simultaneously maintain a drift equilibrium and track a path using steering, throttle, and front brakes. By exploiting the underlying linear structure, this linear controller achieves path tracking performance that exceeds results with nonlinear controllers previously seen in the literature. The key to harnessing the highly nonlinear dynamics of drifting for automated vehicles is, therefore, to focus on the linearity these nonlinear equations can obscure. This paper is organized as follows: section II presents the single track vehicle and tire models used for the analysis. Section III describes the linearization of the dynamics and demonstrates the similarity to the nonlinear dynamics around an equilibrium. Section IV explains the development of the equilibrium and path tracking linear quadratic regulator (LQR) controller that can leverage this linearity. Experimental results with this controller implemented on a full size test vehicle in section V demonstrate its ability to track paths of constant or varying curvature and its straightforward response to tuning parameters. Section VI shows how the tracking performance exceeds previously published results and Section VII shares conclusions and directions for future work.

A. Single Track Model
The single track model strikes a balance between model complexity and accuracy. As shown in Fig. 2 this planar model has three states: U x , velocity in the direction of the vehicle's heading, U y , velocity in the direction perpendicular to the vehicle's heading, and r, the yaw rate of the vehicle. The system has three inputs: steering, modeled as the steering angle δ, throttle, modeled as a force on the rear tires F xr , and front brakes, modeled as a force on the front tires, F xf . Lateral forces on the tires, F yf and F yr , are generated by vehicle states and inputs as explained below. Vehicle parameters include mass, m, moment of inertia, I z , and a and b, which are the distance from the center of mass to the front and rear of the vehicle, respectively. Using the single track model, the dynamics are modeled as shown in equations (1)- (3).

B. Tire Model
A modified version of the Fiala brush model [8] is used to model both the front and rear lateral tire forces. The tire forces are a function of the vehicle states, inputs, and parameters, and the tire parameters: C α , the cornering stiffness of the tire and μ, the friction coefficient of the tire. The first step in determining the lateral tire force is calculating the slip angle α: the angle between the center line of the tire and the tire's direction of travel.

C. Path Tracking Model
A curvilinear coordinate system defines the vehicle's position relative to a desired reference path, as shown in Fig. 3. The lateral error, e, is the distance from the nearest point on the path to the vehicle's center of mass. The distance along the path is called s, and Δψ is the angle between the vehicle's heading and the tangent to the path at the nearest point. While drifting, Δψ is always nonzero because the vehicle is no longer pointing the same direction it is traveling. At equilibrium, the heading error is the opposite of the sideslip angle (Δψ = −sideslip). The path tracking dynamics are described by the equations below.
Note the nonlinearities in the single track model equations, the Fiala tire model, and the path tracking model. While this system of equations represents the vehicle dynamics well, it does not allow for direct application of linear control techniques.

III. LINEARIZATION OF DYNAMICS
As shown by Bárdos et al., linearizing dynamics about drift equilibria leads to a model with sufficient accuracy for successful control and with enough simplicity for linear control techniques [12]. The research presented here takes a deeper look at the linearized dynamics. In this section, we explain the method of linearization, demonstrate in simulation why linearizing about drift equilibria is valid, and investigate the eigenvectors of various drift equilibria for further insight into the behavior of dynamics near drift equilibria.

A. Linearization Method
An equilibrium is defined by a combination of states and inputs where the state derivatives are zero. Solving (1)-(3) for zero state derivatives yields an equilibrium defined by six values:Û x ,Û y ,r,δ,F xr ,F xf . The hat accent is used to denote equilibrium values for the remainder of the paper.
The linearization method uses different approaches for the terms in the state equations involving the vehicle dynamics and the terms related to path tracking. We have chosen to linearize the vehicle dynamics equations numerically using a Taylor series expansion. While the equations of motion, (1), (2), and (3), can be analytically linearized as shown by Hindiyeh in [11], this requires additional assumptions. Since one goal of this paper is to investigate the similarity of the original and linearized equations about the equilibrium, the disadvantages of introducing additional assumptions outweighs the advantages of a closed-form expression. In contrast, analytical linearization of the path equations- (8) and (10)-does not require making assumptions beyond constant U x and κ. Therefore, analytical linearization can be applied to the path equations to provide additional insight at no cost to accuracy.
The analytical partial derivatives of the state equations for Δψ and e with respect to the state vector of interest where S = sin Δψ and C = cos Δψ (11) After calculating the analytical partial derivatives, the partials are evaluated at the equilibrium values of the states to yield (12).
Under the assumption of constant longitudinal velocity, U x , the full linearized model is: In this set of state equations, the steering angle, δ, and the rear longitudinal force, F xr , comprise the inputs available for controller design. The experiments that follow use the front brakes to enforce the assumption of constant velocity in a separate control loop. For simplicity, the numerical linearization does not consider the influence of the front brakes on the front Fig. 4. Phase portrait and trajectories plotted in the lateral velocity-yaw rate phase plane. The dynamics were linearized about the equilibrium denoted by an asterisk, located at U x = 6.6, U y = −4.6, and r = 0.8. The phase portrait, calculated using a nonlinear model, is shown in light blue arrows. Simulated trajectories starting throughout the phase plane are calculated using (blue lines) and the linearized model (dashed red lines). Note that trajectories simulated with linearized dynamics match those simulated with full nonlinear dynamics well. lateral tire force since the front lateral force is generally below its peak value in a drift equilibrium [11]. The experimental results discuss the impact of this choice in more detail.

B. Analysis in Simulation
Phase portraits provide a useful method to analyze the behavior of nonlinear systems. Voser et al. used phase portraits to establish that drift equilibria are unstable saddle points [13]. Fig. 4 shows the phase portrait of the full nonlinear drift equations, centered around a drift equilibrium. The phase portrait is overlaid with trajectories calculated using both the full nonlinear equations and the numerically linearized equations. In order to generate the linearized trajectories, the dynamics were linearized once about the drift equilibrium shown, and then the state derivatives were calculated using those dynamics for the entire trajectory. The region shown in the figure represents approximately the range of state variable variation around the equilibrium observed in path tracking experiments.
Near the drift equilibrium, the trajectories calculated with the linearized equations qualitatively match the trajectories calculated with the nonlinear equations. There are no dynamics or behaviors obviously missing. Furthermore, the trajectories match closely in a quantitative sense as well. This suggests that the linear model represents the system dynamics well enough over a sufficiently large area around the equilibrium to enable control with linear techniques. While the linearization of only a single equilibrium is presented here, Fig. 4 is representative of a large range of drifting equilibria.
In order to better understand how the linearized dynamics vary from equilibrium to equilibrium, we calculate and plot the eigenvectors for equilibria throughout the lateral velocity -yaw rate phase plane. A two state representation of drifting has one stable and one unstable eigenvector; Fig. 5 shows the eigenvectors, scaled by eigenvalue, of drift equilibria in the U y − r phase plane. Qualitatively, the stable eigenvector maintains a remarkably consistent direction for a variety of equilibria. The direction of the unstable eigenvector gradually changes as the drift equilibria changes. The eigenvalue magnitudes also change smoothly and slowly as the equilibrium values vary.
The consistency of the eigenstructure gives some additional insight into drift equilibria. While drifting is unstable, the characteristics of the instability are consistent and well predicted using a linear understanding. For a given linear controller, the consistency of the eigenstructure of the open loop dynamics leads to an expected consistency in the closed-loop dynamics as the system moves to adjacent equilibria or in a region around a single equilibrium. This gives us additional confidence that we can control the vehicle throughout this region using a controller based on the linear model.

IV. CONTROLLER DEVELOPMENT
The ability to represent the vehicle dynamics with a linear model without sacrificing model fidelity allows traditional linear control techniques to be applied to the problem of controlling a drifting vehicle.

A. Stabilizing and Path Tracking Controller
As shown in (13a), this system is represented by four states: U y −Û y , r −r, Δψ − Δψ, and e, and two inputs: δ −δ and F xr −F xr . All states and inputs are measured as the deviation from their equilibrium values such that the states and inputs will all equal zero when the system achieves the desired equilibrium (the equilibrium value of e is already zero). While any number of linear control techniques could be applied to the linearized state equations, this paper focuses on Linear Quadratic Regulator (LQR) control for its familiarity and transparency. The gain matrix of infinite-horizon LQR can be easily calculated and the Q and R matrices have intuitive interpretations. This enables an additional test for the linearized dynamics -does the system respond in an intuitive manner to changes in these matrices? Section V-D answers this question experimentally.
The Q and R matrices were initially chosen using Bryson's Rule [14], tuned in simulation, and finally adjusted after preliminary experiments. The final Q and R matrices-shown below-are diagonal, penalizing states and commands on an individual basis. Each diagonal entry is (1/maximum acceptable deviation from equilibrium value) 2 .
The inputs take the form u = −Kx where K is the solution to the infinite-horizon LQR problem. The closed loop system has the form:ẋ Note that the desired drift equilibrium-calculated using the single track model, the Fiala tire model, and the estimated parameters of the vehicle-may not be an actual equilibrium of the real system. Modeling and parameter errors abound in drifting since the tires heat up and shed rubber over the course of the test, producing changes from the static values assumed in the model. This controller does not have the means to correct for steady state error if the calculated equilibrium is not a true equilibrium of the system. Therefore, steady-state tracking errors should be expected with such an approach. The consistency of the eigenstructure, however, suggests that such parameter variations should not have a significant impact on the system's closed-loop stability. The experiments presented later investigate whether or not the eigenstructure consistency is sufficient to handle real-world variation in parameters even with a simple controller design.

B. Longitudinal Velocity Controller
We use a separate control loop with the front brake force, F xf , to regulate the longitudinal velocity of the vehicle. The equilibrium F xf serves as a feedforward term, and an additional proportional term corrects for any errors in U x , giving In simulation and in experiment, a gain of K U x = 2400 Ns/m proved sufficient for speed tracking.  Table 1 and discussed in [5], [6], and [15].
For experimental tests, equilibrium and quasi-equilibrium trajectories were prepended with an entry trajectory to initiate the drift and bring the vehicle to a starting equilibrium, using the trajectory generation and controller discussed in [5]. Data sets shown below exclude the drift entry and exit periods, focusing on the data collected with the LQR controller active. The front brake force, F xf was distributed equally between the left and right tires. The experiments produced the desired rear drive force, F xr , by closing a feedback loop around wheelspeed, as described in [5].

B. Equilibrium Trajectory
For the first experiment, the equilibrium tracked is located at U x = 6.5532 m/s, U y = −4.5886 m/s, r = 0.8 rad/s, and Δψ = 35 deg. Following this equilibrium traces out a circular trajectory. The linearized dynamics of the tracked equilibrium are shown below:    6 shows the four vehicle states using the stabilizing LQR controller. All four of the states shown in the plot have very low error for this application relative to prior art. In particular, the lateral path tracking error, e, is quite low, with RMS error of 4.2 cm and a standard deviation of only 1.4 cm. The data displays a distinct periodicity, for example the spike seen in the lateral velocity just before 41, 49, and 57 seconds in Fig. 6. This data set represents about 2.5 laps of the circular trajectory. This periodicity correlates with variations in the track surface. The controller maintains the trajectory so precisely that driving over the same variation in the track surface manifests in the same visible dynamic response with each lap. Fig. 7 visualizes the lateral velocity and yaw rate state data in the phase plane. For the duration of this experiment, the vehicle remains near the desired equilibrium. Even though the equilibrium is not perfectly modeled-the centroid of measured states is not on top of desired equilibrium-the system remains stable and the tracking errors remain small. These experiments support the hypothesis that the eigenstructure consistency enables closedloop stability in the presence of real-world uncertainty. As expected, uncertainty produces steady-state tracking errors with the magnitudes of these errors depending upon the specific gain choice. Comparing the range of the axes of Fig. 4 and Fig. 7, we see that most of the trajectory shown in Fig. 7 is well within the phase space shown in Fig. 4. The simulated trajectories in Fig. 4 indicate that the dynamic model linearized about the equilibrium adequately predicts behavior in this region. The experimental data shown in Fig. 7 confirm that this single linearized model is accurate enough to develop a controller that keep the vehicle near this equilibrium.  Fig. 6 shown in the lateral velocity-yaw rate phase plane. The reference equilibrium for this experiment is denoted by an asterisk while the measured lateral velocity and yaw rate values are shown by the blue line. Note that the states remain close to the equilibrium throughout the data set. The two stabilizing inputs commanded by the LQR controller are shown in Fig. 8. We do not have direct measurements of the tire forces available, so the forces have been estimated from inertial sensors (see Appendix). The variation in input commands is minimal: the commanded steering angle has a standard deviation of only 0.4 degrees and the commanded rear drive force has a standard deviation of 213 N. This variation is readily achievable by the actuators, as demonstrated by the similarity between the measured and commanded input signals.
Finally, the longitudinal velocity and the front braking force used to control the longitudinal velocity are shown in Fig. 9. Even with the simple control law of constant, equilibrium feedforward plus proportional feedback, the error in longitudinal velocity is acceptably low. By feeding forward the pure equilibrium value rather than calculating a feedforward based on  The LQR controller employed here does not eliminate the small steady-state tracking errors but does robustly stabilize the vehicle in the presence of uncertainty. The magnitude of this robustness can be seen in another equilibrium trajectory experiment during which the right rear tire delaminated. Despite the large disturbance in the tire parameters as the tire shreds and the tread separates in chunks, the vehicle maintained the desired states as shown in Fig. 10. While this is an extreme example, the sliding rear tires that define drifting continually generate heat and, therefore, tire parameters continually change throughout a drift. The linear eigenstructure's robustness to changes in tire parameters enables robust control even with such a simple linear controller.

C. Quasi-Equilibrium Trajectory 1) Expanding Controller to Track Quasi-Equilibria Trajectories:
To demonstrate that this linearization approach works at more than a single operating point, we employ this controller to track a quasi-equilibrium trajectory. Initially described in [15] and further developed in [5], each point along a quasiequilibrium trajectory is an equilibrium. Fig. 5 shows that the dynamics vary smoothly between nearby drift equilibria, suggesting that linear control would be appropriate for such trajectories.
We use the controller from section IV to track a quasiequilibrium trajectory by linearizing every 0.25 m along the reference trajectory and updating the linearization used based on the vehicle's progress. The entire quasi-equilibrium trajectory uses the same Q and R matrices and K U x value employed in tracking a single equilibrium. By changing the linearization point, the controller becomes a gain scheduling controller. At each point along the trajectory we update the linearization and calculate new feedback gains, but the controller structure otherwise remains the same.
2) Quasi-Equilibrium Results: Fig. 11 shows the path tracked during this quasi-equilibrium experiment. Known as a "cassette tape," the radius of this path varies between 7 and 12 m, all while maintaining equilibrium reference states.
We plot the experimental results in the U y − r phase plane in Fig. 12. Visualized in this plane, the quasi-equilibrium trajectory gently curves through the phase space. The measured states remain close to the reference trajectory at all points, staying well within a region of similar size to that shown in Fig. 4. This gives good confidence that for each operating point along this trajectory, the linearization represents the dynamic behavior well. Furthermore, these results validate the conclusions about smoothly changing dynamics drawn from Fig. 5. Even as the Fig. 12. Data from the same quasi-equilibrium experiment shown in Fig. 13 visualized in the U y − r phase plane. The black asterisks represent equilibria along the quasi-equilibrium path, while the blue line represents the measured state data. The states remain close to the equilibria even as the reference equilibria move throughout the phase plane. desired equilibrium and linearization used changes, the vehicle maintains the path. Fig. 13 shows the state tracking data from the quasiequilibrium experiment. Despite the changing drift equilibria, which causes a dramatic change in equilibrium lateral velocity and yaw rate, the vehicle tracks the states with minimal error. For this data set, the RMS lateral path tracking error is 3.7 cm. The greatest error in the states happens around 52-55 seconds, As seen in Fig. 14, the front brakes regulate longitudinal velocity as it increases and decreases. This plot exposes a limitation of this controller. The front brakes control the longitudinal velocity, not the throttle, so the only way for the controller to increase longitudinal velocity is to ease off the brakes. While this control strategy works overall-the RMS velocity error for this data set is only 0.34 m/s-the largest consistent error in longitudinal velocity occurs when the reference velocity increases, between about 47 and 53 seconds. This build up of error in longitudinal velocity likely contributed to the error seen in the other states in Fig. 13. Fig. 15 shows the stabilizing LQR inputs. The deviations between the equilibrium and commanded input values result from the controller stabilizing the drift even when the measured state does not match the desired state.
Despite the limitations of depending on brakes to control longitudinal velocity and parameters accurate enough to avoid significant steady state error, this controller performs remarkably well overall. The success of this approach for a trajectory with a plethora of operating points demonstrates the linearity and smoothness of the dynamics in the drifting region of the state space.

D. Tuning LQR Weights
A final test of the appropriateness of linearization is to see whether or not the linearized system responds to gain changes in an intuitive manner. One way to fine tune an LQR controller is to choose covariances corresponding to the maximum acceptable deviations from equilibrium values, as shown in (14) and 15. For the inputs, the maximum acceptable deviations from equilibrium values are essentially the maximum acceptable feedback.  Since changing any covariance in the Q or R matrices adjusts all of the gains, a simple test is to vary the maximal acceptable feedback on one input and see if the closed-loop system compensates with the other input. Figs. 16 and 17 show the results of an experiment stabilizing an equilibrium while the maximum acceptable feedback of F xr changes along the trajectory. The Since all weights in LQR design are relative, decreasing the maximum acceptable feedback of one input will require the controller to rely more heavily on the other. The experimental results demonstrate exactly this trade off between using F xr and δ to stabilize the vehicle. As shown in Fig. 16, as the maximum acceptable feedback value of F xr increases, the variability of the F xr command increases, and the variability of the δ command decreases. By adjusting only one element of the R matrix, the controller responds in an intuitive way. Fig. 17 shows the four LQR states from the R matrix tuning experiment. Of the four different maximum acceptable feedback values of F xr , the first maximum acceptable feedback value seems too low. The error in each of the four states is greater in this section since the controller requires more F xr to decrease the state errors. The following three maximum acceptable feedback values of F xr yield satisfactory state tracking results. The decision of maximum acceptable feedback of F xr is then merely a question of desired variability in the two input commands and a value can be chosen anywhere in this range. The experiment begins using the nonlinear controller described in [5] and then switches over to the linear controller described in this paper at the time indicated by the magenta vertical line.

VI. DISCUSSION
Compared to other drift controllers that track a desired quasiequilibrium path [6], [5], this controller achieves more accurate path tracking results-specifically much lower RMS lateral path tracking error. Goh et al. achieved a RMS tracking error of 18 cm [5] and Goel et al. achieved a RMS tracking error of 42 cm [6] while this controller achieved RMS tracking error of only 3.7 cm. The strength of the approach we present lies in the wide array of control techniques available for linear models. While Goh and Goel rely on nonlinear models and specifically chosen error dynamics, the controller developed here is an LQR controller based on a linear model plus an outer proportional feedback plus feedforward loop. The LQR technique does not require strict adherence to arbitrary error dynamics and hence produces better tracking results. Additionally, this straightforward controller makes it easy to intuitively choose and tune input weights. Fig. 18 shows a more direct comparison between the nonlinear controller described in [5] and the linear controller developed here. The experiment begins with the nonlinear controller then switches to the linear controller at approximately 23.4 seconds. While the nonlinear controller employed at the beginning of this dataset succeeds in achieving and maintaining the desired states, the path tracking improves significantly when the linear controller takes over. With the linear design it is straightforward to emphasize path tracking performance. In contrast, the link between gain choice and path tracking is much less clear in the nonlinear controller. This clarity is what enables us to find a set of linear gains for a simple LQR controller that outperforms the baseline nonlinear controller.

VII. CONCLUSION AND FUTURE WORK
The dynamics in a significant region around drift equilibria can be reasonably approximated by linearized dynamics. This allows for classical linear control techniques such as LQR to be used to precisely control the states and trajectory of a drifting vehicle.
One avenue for future work is to incorporate actuator dynamics into the vehicle model. Not all systems on the vehicle-engine, brakes, and steering-respond at the same rate. Incorporating the actuator dynamics would prevent actuators from working against each other and could potentially lead to better results. The vehicle used to demonstrate this controller was electric and could provide near instantaneous torque; however, an internal combustion vehicle would have non-negligible delays in providing the necessary torque. This could require additional modeling, particularly to complete more transient maneuvers. In order to capture the dynamics of an internal combustion engine, we would likely need to augment the state vector to include wheelspeed or engine speed, and then test this new controller on an internal combustion vehicle.
The linearized model presented here could also be implemented in a model predictive control (MPC) framework. Since we have shown that we can control this nonlinear system after linearizing it, we can now use many analysis tools that were not compatible with a nonlinear system. Here the linearized model would allow us to study uncertainty; tube MPC is one possible avenue for quantifying and making claims about uncertainty. Additionally, a MPC framework could be used to gain more control over longitudinal velocity if we add longitudinal velocity as a state and brake force as an input.
The accuracy of the state and path tracking achieved by this controller show that drifting is not a region of the state space that needs to be avoided. By implementing a controller based on a linearized model, we have achieved better path tracking results than controllers developed with more complicated nonlinear models. There is still much to be understood about controlling drifting vehicles, and the linearized models presented here are an important step forward in understanding and, ultimately, harnessing this unstable region of the state space.

APPENDIX FORCE CALCULATION
Since direct measurements of the tire forces are not available, we estimate the tire forces as follows.
We use the angle between the longitudinal axis of the tire and the total force vector defined by Goh et al. in [5] as the thrust angle, γ, to decompose the total force on the rear axle into the longitudinal and lateral forces. The thrust angle is a function of the current states, including the rear wheel speed, ω r , and the vehicle parameters including the rear wheel radius, R r .
To relate the tire forces via the thrust angle we assume that the rear tires are sliding and that F r = μF zr . We then calculate the rear lateral and longitudinal forces. Rearranging the equations of motion, (1)-(3), we have a linear system of three equations and two unknowns.
I zṙ + bF yr = aF yf cos δ + aF xf sin δ (25) Using a Savitzky-Golay filter in MATLAB, we calculate the derivatives of U x , U y , and r. Then we solve the new equations of motion, (23)-(25), for F yf and F xf using least squares for each time step in the data set.