Matthew J. BrusstarUS Environmental Protection Agency
Ann Arbor, Michigan 48105
ABSTRACTConstrained optimization control techniques with preview
are designed in this paper to derive optimal velocity trajectoriesin longitudinal vehicle following mode, while ensuring that thegap from the lead vehicle is both safe and short enough to pre-vent cut-ins from other lanes. The lead vehicle associated withthe Federal Test Procedures (FTP) [1] is used as an example ofthe achieved benefits with such controlled velocity trajectories ofthe following vehicle. Fuel Consumption (FC) is indirectly min-imized by minimizing the accelerations and decelerations as theautonomous vehicle follows the hypothetical lead. Implement-ing the cost function in offline Dynamic Programming (DP) withfull drive cycle preview showed up to a 17% increase in FuelEconomy (FE). Real time implementation with Model PredictiveControl (MPC) showed improvements in FE, proportional to theprediction horizon. Specifically, 20s preview MPC was able tomatch the DP results. A minimum of 1.5s preview of the leadvehicle velocity with velocity tracking of the lead was required toobtain an increase in FE.
The optimal velocity trajectory found from these algorithmsexceeded the presently allowable error from standard drive cy-cles for FC testing. However, the trajectory was still safe andacceptable from the perspective of traffic flow. Based on our re-sults, regulators need to consider relaxing the constant velocityerror margins around the standard velocity trajectories dictatedby the FTP to encourage FE increase in autonomous driving.
∗Address all correspondence to this author.
1 INTRODUCTIONVehicle autonomy is steadily increasing and in the coming
years several manufacturers would offer vehicles with the capa-bility for highly autonomous driving. Presently, adaptive cruisecontrol systems lack the ability to navigate through all trafficconditions and are recommended only at highway speeds forsafe operations [2]. However, autonomous vehicles at the veryleast would have longitudinal traffic navigation capabilities at allspeeds and traffic conditions. The navigation algorithms for au-tonomous vehicles can be designed to improve their fuel econ-omy (FE), as compared to the FE obtained by a human drivingthrough the same traffic conditions.
In [3] a reduction of 0.5%− 10% in CO2 emissions isachieved using an adaptive cruise control algorithm, acting inthe speed range 18 MPH-100 MPH. CO2 emissions can be fur-ther reduced by eco-driving or trace smoothing strategies, whichentails reducing the total accelerations and decelerations. In-deed, [4] conservatively estimated a reduction of 33 million met-ric tons of CO2 annually by adopting eco-driving strategies. Asan example of a trace-smoothing following algorithm, [5] em-ployed a time-headway based linear strategy that improved theFE by increasing the time headway. In [6] a set of linear equa-tions were used to reduce the accelerations and decelerations.Both showed significant improvements in FE. However, theseresults are obtained without imposing conditions on how far thefollower vehicle could fall behind the lead, resulting in possiblecut-ins that would change the velocity. Hence maintaining theappropriate gap between the vehicles is an important considera-
Proceedings of the ASME 2016 Dynamic Systems and Control Conference DSCC2016
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tion that limits the scope for trace smoothing.The authors in [7] compared a linear quadratic and a Model
Predictive Control (MPC) following algorithm, showing a fuelconsumption decrease of 8.8% if a 5s look-ahead capability isguaranteed. However, the authors employed time-invariant con-straints on position. These could result in an awkward trafficpattern since real-world driving varies the following gap basedon the speed. Moreover, they did not use standard drive cyclesand hence were unable to show how the controller would per-form in known and regulated conditions. Also their baseline wasnot a human drive cycle but a LQR derived trajectory.
In this paper, the Federal Test Protocols (FTP) and associ-ated drive cycles are considered using the hypothetical lead vehi-cle. The concept of a hypothetical lead vehicle for any standarddrive cycle was introduced in [1]. The idea was to find the leadvehicle velocity trajectory followed by the driver of a standarddrive cycle. Autonomous vehicles could follow the same leadwith various algorithms. This method would allow for simula-tion of actual traffic conditions associated with the Federal drivecycles and also provide a consistent comparison between howhumans follow traffic and how optimal controllers would followthe same traffic conditions. Obviously any vehicle with auto-matic longitudinal control can employ these optimal controls.
This paper will develop optimal control algorithms that usethe preview of the hypothetical lead vehicle to chart a velocitytrajectory for the autonomous vehicle. The constraints on theautonomous vehicle are imposed by the position and speed ofthe lead. The objective is to minimize fuel consumption of theautonomous vehicle as it navigates through different traffic con-ditions, represented by different drive cycles. Fuel consumptionis indirectly minimized by reducing energy consumed during ac-celerations and energy loss in decelerations. The objective thustranslates into a constrained optimization problem.
The remainder of this paper is organized as follows. Sec-tion 2 presents the model of the vehicle and the problem state-ment. Section 3 shows the application and results of DP for thefollowing problem. Section 4 presents three different formula-tions for MPC implementation and compares them. Section 5
-10
0
10
20
30
Velocity
[m/s]
UDDS Drive Cycle
0 200 400 600 800 1000 1200 1400
Time [s]
0
20
40
60
80
Distance
from
Lead[m
]
Upper Bound on Follower
Lower Bound on Follower
FIGURE 1. Position constraints to be applied on the following au-tonomous vehicle based on the position and velocity of the hypotheticallead
concludes the paper with a discussion on the results obtained.
2 Model DescriptionIn this paper, the dynamics of the autonomous vehicle are
described by a simple point mass Linear Time-Invariant (LTI)model, with position (p) and velocity (v) as states and accelera-tion (a) as the only input, namely
xk+1 =
[1 Ts0 1
]︸ ︷︷ ︸
A
xk +
[0.5T 2
sTs
]︸ ︷︷ ︸
B
uk (1)
with Ts = 0.1s the sampling time, x = [x1 x2] = [p v]′ ∈R2 com-pletely measurable and u = a ∈ R. The simple dynamics allowfor fast online controller implementation. The velocity state vcan be used offline in the ALPHA model to estimate the FuelConsumption and hence the Fuel Economy of the vehicle. De-fine U and Xk as polyhedral sets of constraints on inputs andstates respectively, such that
U= {u ∈ R |umin ≤ u≤ umax} (2a)
Xk = {x ∈ R2 | [xmin1,k xmin
2 ]′ ≤ xk ≤ [xmax1,k xmax
2 ]′}. (2b)
The acceleration and deceleration constraints are time-invariant. For this paper they have been derived from the stan-dard drive cycles to be umin ≡−6 m/s2, and umax ≡ 6 m/s2. The
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same is true for the state x2 or velocity v where vmin ≡ 0 m/s, andvmax ≡ 40 m/s. Constraints on the state x1 or position p however,are time varying as the position of the autonomous vehicle is de-termined by the position and velocity of the hypothetical lead andhence a speed dependent gap.
The gap between the hypothetical lead and the autonomousvehicle is constrained by an upper and a lower bound. The up-per bound is a safety limit and is the closest that the followervehicle can follow the lead vehicle. This is derived from be-ing 1 car length behind the lead vehicle for every 10 MPH. Thelower bound is derived from assuming a distance that would pre-vent safe cut-ins from adjacent lanes. This is kept at 4 ft/MPH or2.7 m/m/s. The constraints are further relaxed at low speeds ofless than 20 MPH to 10 ft/MPH or 2.7 m/m/s. Indeed, at such lowspeeds cut-ins are not expected and a longer gap reduces frequentstarts and stops, thus delivering better FE. Since the position con-straints are dependent on the lead vehicle’s states at that instant,these constraints are time varying. Fig. 1 shows the upper andlower bounds on position at different velocities of the lead vehi-cle. The constraints on position and speed are selected accordingto
xmin1,k = xL + vl L/10 (3a)
xmax1,k = xL +
{vl dmax if vl < 20MPHvl dmin otherwise
(3b)
xmin2 = 0 (3c)
xmax2 = 40 (3d)
where xL is the position of the lead vehicle, vl is the velocity ofthe lead vehicle, L is 1 car length, dmax is 10 ft and dmin is 4 ft.The time-invariant limits on the speed are reasonable on almostall U.S.A. roads [11].
3 Dynamic ProgrammingThe objective of this paper is to indirectly minimize the
FE of the autonomous vehicle by minimizing its accelerationsand decelerations. The optimal velocity trajectory has to becomputed while keeping the vehicle within the position con-straints governed by the lead vehicle. Assuming that all futurereferences, constraints, and disturbances are perfectly known,
0 100 200 300 400 500 600
0
10
20
30
40
US 06 Velocity [m/s]
Standard CylceDP Optimized CycleUpper and Lower Bounds
0 100 200 300 400 500 600
0
50
100
Distance From Lead [m]
0 100 200 300 400 500 600
-4
-2
0
2
4
Acceleration [m/s2]
0 100 200 300 400 500 600
Time [s]
0
1
2
3
Fueling Rate [mgal/s]
FIGURE 2. Comparison of standard US06 drive cycle with the DPoptimized velocity profile. The fueling rates are found from ALPHAmodel simulations [9].
dynamic programming (DP) can be used to find a non-causal,global-optimal input (acceleration) sequence that minimizes adefined cost function [12]. In this work the input or the accel-eration itself has to be minimized. The DP methodology wouldprovide the minimum input, acceleration and deceleration neces-sary to ensure that the vehicle adheres to its position constraints.
Despite its limitation as an offline technique, DP resultsserve as an upper bound on performance for the design of real-
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0 2 4 6 8 10 12 14 16 18 20
Prediction Horizon [s]
25
26
27
28
29
30
31
32
33
Fuel
Econom
y[M
PG]
UDDS Drive Cycle
MPCa MPG
MPCp MPG
MPCv MPG
DP
UDDS
FIGURE 3. Fuel economy using various horizons and cost functionsin the MPC formulation
time control strategies. The formulation is as follows
min.u
N f
∑k=1‖uk‖2
2 (4a)
s.t. xk+1 = Axk +Buk (4b)uk ∈ U, xk ∈ Xk (4c)
where N f is the final time-step, A, B are defined as in (1) and U,Xk are defined as in (2), constraints on Xk are defined as in (3).
4 Model predictive controlAlthough DP provides the optimal solution, several reasons
prevent its use for online control such as the high computational
520 540 560 580 600 620 640 660 680 700 7200
50
100
Distance From Lead (m)
Standard UDDS
DP Optmized
MPCa Np=4s
MPCa Np=8s
Upper and Lower Bound
520 540 560 580 600 620 640 660 680 700 720-5
0
5
Acceleration [m/s2]
520 540 560 580 600 620 640 660 680 700 720
Time [s]
0
10
20
Velocity [m/s]
FIGURE 4. Comparison of standard UDDS drive cycle with DP andMPCa velocity traces. Clearly for MPCa with Np = 4s, at Time = 530,605, 655 and 715 s the position is too close to the lower bound andthe MPCa controller applies significant acceleration to stay within thebounds. These accelerations are less pronounced for the Np = 8s caseand absent in the DP case.
burden and the complete knowledge about the future behaviorof the system. The MPC methodology is becoming the stan-dard technology to handle fast-sampled multivariable processes,especially in automotive, aerospace and power electronics con-trol [8, 15–17]. This methodology solves a constrained, finite-horizon, optimal control problem, and by following the reced-ing horizon policy it applies only the current inputs to the sys-tem. At each sampling time the procedure is repeated and anew open-loop optimal control problem is solved with a one-step shifted horizon. Even if MPC drastically reduces the com-putational complexity with respect to DP, the time required tosolve the optimization problem is still considerable, especially inhigh frequency sampled systems [18]. However, recent advancesin convex and embedded optimization are enabling fast onlineMPC implementations [19–21]. When dealing with LTI models,quadratic cost function and affine constraints, MPC simplifies tosolving the following optimization problem at each sampling in-stant
minu
Np
∑i=1‖Wx(xk+i|k− rk+i|k)‖2
2 +‖Wuuk+i|k‖22 (5a)
s.t. xk+i+1|k = Axk+i|k +Buk+i|k (5b)
xk|k = xk (5c)
uk+i|k ∈ U, xk+i|k ∈ Xk+i|k (5d)
where Np is the prediction horizon expressed in sampling in-stants, Wx and Wu are square, diagonal, weight matrices, xk+i|kdenotes the prediction of the variable x at time k+ i based on theinformation available at time k, and rk+i|k denotes the prediction
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of the reference r to be tracked at time k+ i, xk is the current state.The influence of the prediction horizon, Np = Np.Ts expressed inseconds, to the optimal velocity trajectory will be investigatedalong with three different MPC formulations. All three formula-tions share the same linear prediction model and the same con-straints, as defined in Eqs. (1) and (2), respectively. They differfrom each other for the cost function, and we will refer to themas
1. MPCa: penalty on acceleration;2. MPCp: penalty on acceleration and position tracking;3. MPCv: penalty on acceleration and velocity tracking.
MPCa can be considered as a reduced horizon version of theoptimal control problem (4), and its cost function is defined bythe following weights:
MPCa ,
{W a
x = diag [0 0]W a
u = wau = 1
(6)
The length of the prediction horizon plays an important role inthis control problem as not only the dynamics of the model arepredicted but also the constraints on the states and the eventualreference trajectory. This means that high prediction horizons arenot only computationally demanding, but can even be infeasibledepending on the preview, look-ahead, and overall connectivityresponsible for accurately predicting the future behavior of thesystem. Indeed, in [22], it was shown that while velocity tra-jectories for short prediction horizons can be predicted well forNp = 1.5s, a good accuracy for longer horizon is not very prob-able.
For autonomous driving it can be assumed that these pre-dictions come from different methods such as vehicle to vehiclecommunication or from a traffic monitoring system. Given dif-ferent prediction horizons, this work evaluates the potential in-crease in fuel economy. It must be stressed that for this work,perfect prediction of constraints along the prediction horizon isassumed for all cases.
Fig. 3 shows that MPCa gives satisfactory results only forNp ≥ 6s. This is a significantly long prediction horizon andwould not be feasible. For shorter prediction horizons Ns < 5s,MPCa gives an FE that is even worse with respect to the standardcycle. The reason is clarified in Fig. 4, which shows that with-out enough prediction of the position constrains, MPCa keeps thevehicle too close to the lower bound. This leads to accelerationspikes to keep the vehicle within the bounds, decreasing the FE.Fig. 3 also shows that, as expected, the performance of MPCa
equals the one obtained with DP after a certain prediction hori-zon, i.e. 20s, as the two formulations are equivalent but for theprediction horizon. The RMS error between the velocity trajec-tory in DP and MPCa vary as 2.24m/s for Np = 4s, 1.26m/s for
650 700 750 800 850 900 9500
50
100
Distance from lead [m]Standard UDDS
MPCp Np=4s
MPCp Np=8s
Upper and Lower Bound
650 700 750 800 850 900 950-2
0
2
Acceleration [m/s2]
650 700 750 800 850 900 950
Time [s]
0
0.5
1
Fueling Rate [mgal/s]
FIGURE 5. Comparison of standard UDDS drive cycle with twoMPC filtered velocity traces where the MPC has position tracking inthe cost function. Clearly with increased prediction horizon, the posi-tion tracking improves. However, the reduction in acceleration betweenthe two MPC cases is small leading to very slight improvement in FE
Np = 8s and 0.58m/s for Np = 20s thus showing how the MPCa
results approach the DP ones.
4.1 Model predictive control with tracking penaltyFrom the results of MPSa, the vehicle had to be prevented
from falling too close to the lower bound on position. Since thestate vector is completely measurable, the deviation from the po-sition or velocity reference can be penalized to ensure that thevehicle is within acceptable bounds. The MPC implementations,MPCp and MPCv introduce two different tracking penalties inthe cost function. It is straightforward to verify that penaliz-ing the velocity or position tracking error worsens the FE forlonger prediction horizons, where MPCa works well. However,as shown in Fig. 3, the performance improvement for shorter pre-diction horizons is significant.
In MPCp, the position state is forced to track the upperbound of the position constraints. This ensures that the positionof the vehicle is far from the lower bound. However, positiontracking limits the ability of the controller to utilize the entiregap between the bounds and minimize acceleration. An alterna-
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tive is MPCv, where the controller is forced to track the velocityof the lead vehicle. Tracking the lead velocity would ensure thatfor short prediction horizons, the FE at least matches that of thestandard cycle. Additionally, acceleration optimization wouldincrease FE to a value beyond that of the standard cycle. Thetradeoff between acceleration optimization and velocity trackingby employing appropriate weights for different prediction hori-zons are discussed below.
In MPCp the weights on the cost function are assumed to be
MPCp ,
{W p
x = diag[wp
x 0]
W pu = wp
u(7)
whereas the weights for MPCv are such that
MPCv ,
{W v
x = diag [0 wvx]
W vu = wv
u(8)
Since the main objective is minimization of acceleration, theweights were chosen such that wp
u > wpx and wv
u > wvx. This rule
ensures that the penalty on tracking is less than the penalty onhigh acceleration. In the proposed results, the weights are tunedfor each prediction horizon. For longer prediction horizons thetracking accuracy can be reduced, thus giving freedom for higheracceleration minimization. As an example, for the case Np = 1.5sthe weights are wp
u = wau = 1, wp
x = 0.8 and wvx = 0.2.
Fig. 5 shows MPCp results for two prediction horizons, i.e.4s and 8s. The upper bound is closely tracked by both cases andthe tracking performance increases slightly as prediction horizonincreases. The FE improves over the standard cycle even forshort prediction horizons. The close position tracking, however,reduces the scope for optimization and hence the FE for bothcases is nearly the same. Fig. 3 shows that for Np = 8s, theMPCa has a much better FE than MPCp.
It was found that MPCv showed an even better FE thanMPCp for short prediction horizons. In Fig. 6 it can be seen thatwhile staying within the position constraints, MPCv had loweracceleration and engine torque demands than MPCa and MPCp.Clearly the DP case with the entire drive cycle preview is the bestfor FE. But, for Np = 1.5s MPCv shows the best results comparedto other possible real-time solutions. MPCv is therefore selectedas our real-time following control implementation.
5 DiscussionThe objective of this paper was to find a controller that gen-
erates an optimal velocity trajectory such that the accelerationis minimized. Table 1 shows the improvements in FE for boththe DP and MPCv cases. Significant improvements were shown
0
20
40
Distance from Lead [m]Standard UDDS
DP Optimized
MPCa Np=4s
MPCp Np=4s
MPCv Np=4s
Upper & Lower Bounds
-5
0
5
Difference from Lead Velocity [m/s]
-2
0
2
Acceleration [m/s2]
0
100
200
Engine Torque [Nm]
180 185 190 195 200 205 210
Time [s]
5
10
15
20
25
Velocity [m/s]
FIGURE 6. Comparison of different drive cycles derived from follow-ing the hypothetical lead using DP and MPC algorithms with differentcost functions at Np=4s. The acceleration profile of DP is the best fol-lowed by MPC with velocity tracking, MPC with position tracking, thestandard cycle and finally MPC without reference. This is reflected inthe the engine torque and correspondingly the fueling rate.
in FE for the Dynamic Programming case between 13.1% and16.7% indicating a potential for high gain in FE for autonomousvehicles that enable real time optimization. Real-time imple-mentation with MPCv controllers shows improvements between5.3% and 11.8%, for the shortest prediction horizon of 1.5s.While obviously not performing as well as DP, MPCv is able toimprove upon the baseline. For very long prediction horizons ofNp ≥ 20s, MPCa is able to match the DP results for all the givencycles simulated in ALPHA for the selected vehicle.
The model used in the optimal controller was deliberatelykept as a simple point mass LTI acceleration model to generatethe optimal velocity trajectory without considering the vehiclepower train or engine dynamics. This was achieved by reduc-ing the total accelerations and decelerations while maintainingan acceptable distance from the lead vehicle based on acceptabletraffic patterns. This work shows that in meeting the simple ac-
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TABLE 1. MPG improvements with DP and MPC
CycleName
StandardCycle [MPG]
DP[MPG | Imp.]
MPCv (Np=1.5s)[MPG | Imp.]
UDDS 28.3 32.0 13.1% 29.8 5.3%
US06 24.5 28.6 16.7% 27.4 11.8%
LA92 26.0 30.7 15.3% 28.2 8.5%
SC03 27.7 31.8 14.8% 29.2 5.4%
celeration objectives, FE can also be increased while following ahypothetical lead vehicle with a velocity profile associated withthe federal test procedures [1]. An autonomous vehicle couldhave this FE objective along with other safety and traffic flowobjectives. Tailoring the FE optimization by minimizing fuelconsumption instead of the generic acceleration could result ineven more benefits, given that not all accelerations are equallyfuel consuming. This optimization will require engine and trans-mission data or models.
6 ConclusionThis paper shows the possibilities for improvement on
present vehicle technologies through autonomous driving. Of-fline DP results achieve up to 17% improvement in FE. Simula-tion of various MPC formulations with a reasonable horizon wasable to achieve 12% improvement in FE. The velocity trajectoriesthat achieved these improvements in autonomous vehicles weresignificantly different from the velocity profile of the standardcycles. The velocity difference was more than what is currentlyallowed by regulations. In light of this work, regulators need toreconsider the standard FE testing procedure that imposes a tightband around the velocity trace to encourage use of algorithmsthat increase FE, acceptable from both traffic and safety perspec-tives.
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