Road Condition Based Adaptive Model Predictive Control for Autonomous Vehicles

Xin Wang School of Construction Machinery

Chang’an University Xi’an, Shaanxi 710064, China

Department of Automotive Engineering

Clemson University Greenville, SC 29607, USA

wangxin@chd.edu.cn

Longxiang Guo Department of Automotive

Engineering Clemson University

Greenville, SC 29607, USA longxig@clemson.edu

Yunyi Jia* Department of Automotive

Engineering Clemson University

Greenville, SC 29607, USA yunyij@clemson.edu

ABSTRACT

Road conditions are of critical importance for motion

control problems of the autonomous vehicle. In the existing

studies of Model Predictive Control (MPC), road condition is

generally modeled with the system dynamics, sometimes

simplified as common disturbances, or even ignored based on

some assumptions. For most of such MPC formulations, the

cost function is usually designed as fixed function and has no

relations with the time-varying road conditions. In order to

comprehensively deal with the uncertain road conditions and

improve the overall control performance, a new model

predictive control strategy based on a mechanism of adaptive

cost function is proposed in this paper. The relation between the

cost function and road conditions is established based on a set

of priority policies which reflect the different cost requirements

under different road grades and friction coefficients. The

adaptive MPC strategy is applied to solve the longitudinal

control problem of autonomous vehicles. Simulation studies are

conducted on the MPC method with both the fixed cost

function and the adaptive cost function. The results show that

the proposed adaptive MPC approach can achieve a better

overall control performance under different road conditions.

Keywords

Model Predictive Control, Longitudinal control, Adaptive cost

function, Road condition.

1 INTRODUCTION

With the rapid development of autonomous driving

technology, many basic self-driving functions have been

implemented and applied to autonomous vehicles successfully.

*Corresponding Author, yunyij@clemson.edu

The latest autonomous vehicles have the functions of adaptive

cruise control, lane keeping, autonomous parking and crash

avoidance, etc., due to the applications of various control

methods and other information techniques. Many people

believe that autonomous vehicles will replace traditional

vehicles gradually and become a major transportation tool

someday in the future [1]. Nowadays, people are no longer

satisfied with merely the safety and the basic functions of the

autonomous vehicle. Higher demands are placed on the

driving/riding comfort and control efficiency. Hence, more

sophisticated control strategies and methods are expected to

improve the control performance with lower control costs.

In recent years, Model Predictive Control (MPC) method

has gained much attention in the field of autonomous vehicles,

due to its unique advantages in solving control problems that

are hard to model accurately and/or have constraints [2]. In

most of the autonomous driving problems, system constraints

are inevitable. These constraints include mechanical input

limitations, the acceleration ability and the safe distance

between the vehicle and other objects, etc. Naturally, MPC

becomes an ideal choice to solve such problems.

In motion control problems of the autonomous vehicle,

road condition is an ineligible factor which influences the

control performance significantly. For moving vehicles, road

conditions mainly refer to the road grade and the friction

coefficient. The former is primarily decided by the

physiographic condition and the design of the road, while the

latter is subject to many factors, including the road pavement

material, the age of the road, and even the weather [3]. Since

road conditions are time-varying and uncertain, comprehensive

modeling is difficult to realize. Therefore, it is meaningful to

deal with the road condition by other ways.

In the existing MPC studies of autonomous vehicles, road

conditions are considered more or less. In most cases, the road

Proceedings of the ASME 2018Dynamic Systems and Control Conference

DSCC2018September 30-October 3, 2018, Atlanta, Georgia, USA

DSCC2018-9095

1 Copyright © 2018 ASME

Attendee R

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opy

condition, especially the friction coefficient, is modeled with

the system dynamics, as parameters or disturbances. Borrelli et

al. in [4] applied nonlinear MPC to the autonomous steering

system. In their study, the bicycle model and the road-tire

model of the vehicle are combined to establish the system

dynamics. The road friction coefficient is considered as a

parameter of the system dynamics. Falcone et al. in [5] and [6]

also studied the active steering problem based on the same

system dynamics as [4], but using a successive online

linearization approach. In [7], Turri et al. studied the lane

keeping and obstacle avoidance problems based on linear MPC.

The extended bicycle mode is utilized and the friction

coefficient is modeled with the longitudinal tire force. Carvalho

et al. in [8] presented their design of a nonlinear MPC

controller, which can track the lane centerline while avoiding

collisions with obstacles. In their study, the friction coefficient

is also incorporated in the longitudinal tire force model. In [9],

Kamal et al. introduced an energy-saving MPC strategy and

applied it to the adaptive cruise control problem. A concept of

equivalent acceleration is proposed considering the effect of the

road grade. In some other MPC studies, road conditions are

treated as the system constraints. Yi et al. in [10] studied the

MPC-based trajectory planning for critical driving maneuvers.

They introduced a quadratic MPC control method considering

friction limits in evasion maneuvers. In [11], Jalalmaab et al.

proposed an MPC-based collision avoidance scheme for

autonomous vehicles. The maximum road friction coefficient is

considered in the acceleration constraints, and a particular

optimization method is developed to estimate the max road

friction coefficient. In [12], Beal et al. presented a vehicle

stabilization approach by utilizing a model predictive envelope

controller to bound the vehicle motion within the stable region

of the state space. The estimated road condition is used to

define the state boundaries. Among the above studies, the factor

of road grade is not mentioned in [4-8,10-12], while in [9], the

road friction coefficient is not taken into account.

Due to the difficulty of being detected, the information of

road conditions is always obtained through model-based

estimation techniques in many applications. Chen et al. in [13]

designed a recursive least square estimator to achieve the road

friction coefficient online. Kidambi, et al. in [14], developed a

mass and grade estimation method using longitudinal

acceleration. Similar studies can also be found in [15-19].

Whereas, in some of the MPC studies, due to the complexity of

the system, road conditions are not taken into account in the

control process, as in [20-21].

In spite of these existing studies, research and applications

regarding adaptive cost function strategies have rarely been

addressed. In most of the existing MPC studies, road conditions

are usually simplified based on some assumptions, and not all

the factors are modeled with system dynamics. Meanwhile, in

these MPC studies, traditional cost functions with fixed penalty

weights are used, and there is no special design for the cost

function. Therefore, the main contribution of this paper is the

development of a new MPC strategy with the adaptive cost

function in the receding optimization process based on the

time-varying road conditions. This approach keeps all the

advantages of MPC method and at the same time considers the

different requirements of the optimization objectives under

different road conditions. By using the proposed approach, the

control performance can be accomplished with a much lower

control cost.

The remainder of this paper is organized as follows:

Section II describes the control problem to be studied, and

formulates the system dynamics. The MPC controller is

designed and solved in Section III. In Section IV, the road

condition adaptive strategy is devised. Simulation study is

presented and discussed in Section V. Finally, conclusions and

future work of this paper are addressed in Section VI.

2 PROBLEM FORMULATION

The purpose of this paper is to investigate a new road

condition adaptive MPC strategy. To achieve this goal, the

longitudinal control problem is chosen as a preliminary study,

because it is less complicated than other motion control

problems. For the same reason, a mass point model is utilized

to describe the system dynamics.

In the longitudinal control problem, the actual velocity 𝑣

of the vehicle must well track the given reference velocity 𝑣𝑟 .

An MPC controller will be designed to implement this control

with consideration of the time-varying road conditions.

Figure 1. Force diagram of a moving vehicle on the upgrade road

Both the road grade and the friction coefficient are

considered in the control process. We use 𝜃 to represent the

road grade, as shown in Figure 1, which can be detected via the

attitude sensor equipped in the vehicle. The friction coefficient,

which has strong stochastic characteristics, is hard to detect by

existing sensors. Therefore, the slip ratio is employed to replace

the friction coefficient approximately in this study, which is

defined by

𝑠 = 1 − 𝑣𝑅𝜔 {𝑣 ≠ 0 for braking𝜔 ≠ 0 for driving (1)

where s is the slip ratio; 𝜔 is the angular speed of the vehicle

wheel; 𝑅 is the tire radius.

As shown in Figure 1, the autonomous vehicle is regarded

as a mass point in the longitudinal control problem. We can

establish its dynamic model by using Newton’s Second Law

directly [22, 23], and thus have

𝑚�̇� = 𝑚𝑢 − 12 𝐶𝑑𝜌𝑎𝐴𝑓𝑣2 − 𝐶𝑟𝛽(𝑠)𝑚𝑔cos(𝜃) − 𝑚𝑔sin(𝜃) (2)

where 𝑣 is the velocity of the vehicle; 𝑢 is the acceleration

control input per unit mass to accelerate or decelerate the car

which is derived from the control strategy; 𝑚 is the vehicle

mass; 𝜌𝑎 is the air density; 𝐶𝑑 is the drag coefficient; 𝐴𝑓 is

2 Copyright © 2018 ASME

the frontal area of the vehicle; 𝐶𝑟 is the rolling resistance

coefficient; 𝛽 is an influence factor which is related to the road

friction coefficient/slip ratio 𝑠.

In general, the system dynamics of a vehicle is always

non-linear. For the reason that a non-linear MPC problem is

harder to solve than a linear MPC problem, we can simplify the

non-linear model to a linear one. By linearization and

discretization, a linear CARIMA model can be obtained.

CARIMA model is the most common used transfer function

model in MPC [24]. It is a discrete and one-step-ahead

prediction model, and can be expressed by

𝑎(𝑧)𝑦𝑘 = 𝑏(𝑧)𝑢𝑘 + 𝑤𝑘𝛥 (3)

where 𝑢𝑘 , 𝑦𝑘 and 𝑤𝑘 are the sampled values of the input 𝑢(𝑡), the output 𝑦(𝑡) and the system disturbance 𝑤(𝑡) at

time instant 𝑘 , respectively; 𝑎(𝑧) and 𝑏(𝑧) are the

denominator and the numerator polynomial of the impulse

response transfer function of the system, which can be derived

from model (2) by using z Transform; 𝛥 is the sampling time

interval. Then (3) can be written as

𝐴(𝑧)𝑦𝑘 = 𝑏(𝑧)𝛥𝑢𝑘 + 𝑤𝑘 (4)

where 𝐴(𝑧) = 𝑎(𝑧)𝛥 = 1 + 𝐴1𝑧−1 + ⋯+ 𝐴𝑛𝑧−𝑛 , and 𝑏(z) = 𝑏1𝑧−1 + ⋯ + 𝑏𝑚𝑧−𝑚 , where 𝑛 and 𝑚 are the order

and the input number of the system, respectively. Consequently,

the discrete model of the system can be represented by

𝑦𝑘+1 + 𝐴1𝑦𝑘 + ⋯+ 𝐴𝑛𝑦𝑘−𝑛+1 = 𝑏1Δ𝑢𝑘 + 𝑏2Δ𝑢𝑘−1 + ⋯+𝑏𝑚Δ𝑢𝑘−𝑚+1 (5)

where Δ𝑢𝑘 is the input increment. The output 𝑦𝑘 is often

measurable in practice. Equation (5) is the basis of the

one-step-ahead prediction, which can be used recursively to get

an n-step-ahead prediction.

3 MPC CONTROLLER DESIGN

To get the solution of the control, the linear MPC problem

can be converted to a standard QP (Quadratic Programming)

problem, as shown by (6).

{min𝑥 𝐽 = 𝑥𝑇𝑆𝑥 + 𝑥𝑇𝑓 + 𝑐𝑠. 𝑡. 𝑀𝑥 ≤ 𝑑 (6)

where 𝑆 is the weight matrix; 𝑓 and 𝑐 are the coefficient

matrix of the first-order term, and the constant matrix,

respectively; 𝑀 and 𝑑 are the coefficient matrix and the

border matrix of the inequality constraint, respectively.

By minimizing the cost function 𝐽 the optimal solution 𝑥𝑜𝑝𝑡 can be achieved under the system constraints.

3.1 Output Prediction

Referring to [24], we can express the multi-step output

prediction as

�⃗�𝑘+1 = 𝐻∆�⃗⃗�𝑘 + 𝑃∆�⃖⃗�𝑘−1 − 𝑄�⃖�𝑘 (7)

where the notation “” represents “future”, and “” represents “past” in the prediction; 𝐻 = 𝐶𝐴−1𝐶𝑏 , 𝑃 = 𝐶𝐴−1𝐻𝑏

and 𝑄 = 𝐶𝐴−1𝐻𝐴, with the parameter matrices 𝐶𝐴, 𝐶𝑏, 𝐻𝐴 and 𝐻𝑏 defined as

𝐶𝐴 = [ 1 0 0 ⋯ 0𝐴1 1 0 ⋯ 0𝐴2 𝐴1 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ 1𝐴𝑛𝑦 ⋯ 𝐴2 𝐴1 1]

, 𝐶𝑏 = [𝑏1 0𝑏2 𝑏1 0 00 0⋮ ⋮𝑏𝑛𝑢 ⋯ 𝑏1 ⋮𝑏2 𝑏1

],

𝐻𝐴 = [ 𝐴1 ⋯ 𝐴𝑛−𝑛𝑦𝐴2 ⋯ 𝐴𝑛−𝑛𝑦+1 𝐴𝑛−𝑛𝑦+1 ⋯ 𝐴𝑛𝐴𝑛−𝑛𝑦+2 ⋯ 0⋮ ⋮ ⋮ 𝐴𝑛𝑦 ⋯ 𝐴𝑛−1 ⋮ ⋮ ⋮ 𝐴𝑛 ⋯ 0 ]

, and

𝐻𝑏 = [ 𝑏2 ⋯ 𝑏𝑚−𝑛𝑢𝑏3 ⋯ 𝑏𝑚−𝑛𝑢+1 𝑏𝑚−𝑛𝑢+1 ⋯ 𝑏𝑚𝑏𝑚−𝑛𝑢+2 ⋯ 0⋮ ⋮ ⋮ 𝑏𝑛𝑢+1 ⋯ 𝑏𝑚−1 ⋮ ⋮ ⋮ 𝑏𝑚 ⋯ 0 ]

and {�⃗�𝑘+1 = [𝑦𝑘+1 𝑦𝑘+2 … 𝑦𝑘+𝑛𝑦]𝑇 �⃖�𝑘= [𝑦𝑘 𝑦𝑘−1 … 𝑦𝑘−𝑛𝑦+1]𝑇 ∆�⃗⃗�𝑘 = [∆𝑢𝑘 ∆𝑢𝑘+1 … ∆𝑢𝑘+𝑛𝑢−1]𝑇

where 𝑛𝑦 and 𝑛𝑢 are the prediction horizon and the input

horizon, respectively.

3.2 Standardization of the Cost Function

Typically, the cost function acts as the performance index

of an optimization control process. In our problem, the cost

function consists of four penalty terms, including the terms of

vehicle velocity 𝑣, velocity rate ∆𝑣, acceleration control input 𝑢 and its rate ∆𝑢. Among them, ∆𝑣 is relative to the velocity

smoothness, 𝑢 is the control input and ∆𝑢 represents “vehicle

jerking”. We design our cost function 𝐽 based on the four

factors, and it is described as

𝐽 = 𝑊1(𝑣𝑟⃗⃗ ⃗⃗ 𝑘+1 − �⃗�𝑘+1)𝑇(𝑣𝑟⃗⃗ ⃗⃗ 𝑘+1 − �⃗�𝑘+1) +𝑊2�⃗�𝑘+1𝑇 �⃗�𝑘+1 + 𝑊3�⃗⃗�𝑘𝑇 �⃗⃗�𝑘 + 𝑊4∆�⃗⃗�𝑘𝑇∆�⃗⃗�𝑘 (8)

where 𝑊𝑖 (i = 1,2,3,4) are the weights of the four penalty

terms, respectively. Through mathematical deductions, the

optimization objective can be expressed by

min⏟∆�⃗⃗⃗�𝑘 𝐽 =min⏟∆�⃗⃗⃗�𝑘 ∆�⃗⃗�𝑘𝑇 [(𝑊1 + 𝑊2)𝐻𝑇𝐻 + (𝑊3 + 𝑊4)𝐼]∆�⃗⃗�𝑘 +2∆�⃗⃗�𝑘𝑇[(𝑊1 + 𝑊2)𝐻𝑇(𝑃∆�⃖⃗�𝑘−1 + 𝑄�⃖�𝑘) − 𝑊1𝐻𝑇𝑟𝑘+1 +𝑊3∆�⃖⃗�𝑘−1] (9)

Then, we can convert our problem to a standard QP problem as

defined by (6), with 𝑥 = ∆�⃗⃗�𝑘 and

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{𝑆 = (𝑊1 + 𝑊2)𝐻𝑇𝐻 + (𝑊3 + 𝑊4)𝐼 𝑓 = 2[(𝑊1 + 𝑊2)𝐻𝑇(𝑃∆�⃗⃖�𝑘−1 + 𝑄�⃖�𝑘) − 𝑊1𝐻𝑇𝑟𝑘+1 + 𝑊3∆�⃗⃖�𝑘−1] 𝑐 = 0

For the space limitation of this paper, the proof of (9) is

omitted here.

3.3 Standardization of the Constraints

In our problem, both the control input 𝑢𝑘 and its rate ∆𝑢𝑘

are physically limited. They are constrained by {𝑢 ≤ 𝑢𝑘 ≤ 𝑢, ∀𝑘 ∆𝑢 ≤ ∆𝑢𝑘 ≤ ∆𝑢, ∀𝑘 (10)

where 𝑢 and 𝑢 are the upper and lower limits of 𝑢𝑘 ,

respectively; ∆𝑢 and ∆𝑢 are the upper and lower limits of ∆𝑢𝑘, respectively. Referring to [25], we can combine the two

constraints by a compact matrix expression as

[𝐶∆𝑢𝐶𝑢𝐸] ∆�⃗⃗�𝑘 ≤ [𝑑∆𝑢𝑑𝑢 ] + [ 0−𝐶𝑢𝐿] 𝑢𝑘−1 (11)

where

𝐶𝑢, 𝐶∆𝑢 =[ 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 1−1 0 ⋯ 0 0 −1 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ]

, 𝐸 = [1 01 1 ⋯ 0⋯ 0⋮ ⋮1 1 ⋮ ⋮⋯ 1],

𝐿 = [11⋮1], 𝑑𝑢 =[ 𝑢⋮𝑢−𝑢⋮−𝑢]

, and 𝑑∆𝑢 =[ ∆𝑢⋮∆𝑢−∆𝑢⋮−∆𝑢]

. 4 STRATEGY OF ADAPTIVE COST FUNCTION

To make the cost function adaptive, a straightforward

method is to adjust the weight value of each penalty term with

the changing of road conditions. Before designing the adaptive

weights, weight priority policies are necessary, which should

reflect the relationship between the four penalty terms and the

road conditions. On the basis of human driving experience, we

propose a set of priority policies as the following.

4.1 Weight Priority Policies with respect of the Road Grade

Four basic policies for road grade are listed below.

(i) For flat roads, the velocity accuracy and the

driving/riding comfort (smoothness) are of high

priority. It is because the flat road is favorable and

also the most common situation.

(ii) For upgrade and downgrade scenarios, the priority of

the velocity accuracy should be decreased, which is

similar in human driving behaviors.

(iii) The fuel economy (acceleration) is less important

except in downgrade scenario.

(iv) Vehicle jerking should be avoided in most of the

scenarios.

4.2 Weight Priority Policies with respect of the Road Friction Coefficient

There are also four basic policies for road friction

coefficient/slip ratio:

(v) For roads with low slip ratio, the velocity accuracy

and the smoothness are both important due to the

advantageous road condition.

(vi) For roads with high slip ratio, e.g. the snowy or icy

road, the velocity accuracy is less important, while

the velocity rate is more important to guarantee the

safety.

(vii) For roads with high slip ratio, the fuel economy is less

important.

(viii)Vehicle jerking should be avoided in most of the

scenarios.

4.3 Determination Method of the Penalty Weights

Define 𝑊1𝑖 and 𝑊2𝑖 (𝑖 = 1,2,3,4) as the sub-weights in

terms of the road grade and the friction coefficient/slip ratio,

respectively. The weight priority can be described by Table I

and Table II.

TABLE I WEIGHT PRIORITY WITH RESPECT OF THE ROAD GRADE

Road grade range Typical road Weight priority Σ𝑖=14 𝑊1𝑖 𝑊11 𝑊12 𝑊13 𝑊14 𝜃 < −10° Downgrade LS LS VR VR 1.0 −10° ≤ 𝜃 ≤ 10° Flat VR LS LS VR 1.0 𝜃 > 10° Upgrade LS LS LS VR 1.0

TABLE II WEIGHT PRIORITY WITH RESPECT OF THE SLIP RATIO

Slip ratio range Typical road Weight priority Σ𝑖=14 𝑊2𝑖 𝑊21 𝑊22 𝑊23 𝑊24 𝑠 < 0 Braking LS IM LS VR 1.0 0.0 ≤ 𝑠 < 0.2 Newly Constructed IM VR LS VR 1.0 0.2 ≤ 𝑠 < 0.5 Dry asphalt LS LS LS VR 1.0 0.5 ≤ 𝑠 < 0.8 Snowy/Wet LS LS LS IM 1.0 0.8 ≤ 𝑠 ≤ 1.0 Icy LS VR VR VR 1.0

where the priority notations are defined as

LS: Less important

IM: Important

VR: Very important

The final value of each weight 𝑊𝑖 can be calculated by 𝑊𝑖(𝜃, 𝑠) = 𝐾𝑠𝑊1𝑖(𝜃) + (1 − 𝐾𝑠)𝑊2𝑖(𝑠), (0 < 𝐾𝑠 < 1, 𝑖 = 1,2,3,4) (12)

where 𝐾𝑠 is the allocation coefficient, which represents the

priorities of the two road condition factors in one problem.

It should be noted that the notation LS, IM and VR are not

exact values but just priorities. In a practical application, the

penalty weights should can be given as exact values (𝑊1𝑖(𝜃)

and 𝑊2𝑖(𝑠)) and must be consistent with their priorities. In our

design, the total sum of the four penalty weights, i.e. 𝛴𝑖=14 𝑊𝑖, must be 1.0, which implies a normalization process. In addition,

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The control input rates of the two strategies are shown in

Figure 2(e). It can be seen that the adaptive-weight strategy

behaves better than the fixed-weight strategy, still with a

smaller peak value.

The most important comparison of the two strategies are

presented by control costs. In the preceding optimization

process of MPC, the two curves of the cost function value

present quite different features in the amplitude. As we can

observe clearly in Figure 2(f), the total control cost of the

adaptive-weight strategy is about 49% lower than that of the

fixed-weight strategy. This percent is calculated by

𝐷𝐽 = ∑ {𝐽Fixed−weight(𝑘)−𝐽Adaptive−weight(𝑘)}𝑁𝑘=1 ∑ 𝐽Fixed−weight(𝑘)𝑁𝑘=1 × 100% (13)

where 𝑁 represents the total number of the time instants; 𝑘

represents the current time instant.

Therefore, the overall performance of the adaptive-weight

strategy is better than that of the fixed-weight strategy because,

for the same control problem the former has a better tracking

performance and at the same time a much lower control cost

than the latter.

7 CONCLUSIONS

In this paper, a road condition adaptive strategy of MPC

control has been devised based on a mechanism of adaptive

cost function. The proposed approach has been applied to the

longitudinal control problem of autonomous vehicles. The MPC

controller is solved after being converted to a QP problem. To

verify the effectiveness of the proposed approach, simulation

has been carried out on a simulated road with time-varying road

grades and friction coefficients. The results show that the

proposed approach is effective in handling different road

conditions. Using the new strategy, lower control cost can be

achieved compared with the strategy of fixed cost function.

This paper presents the application of the road condition

adaptive MPC method on the longitudinal control of

autonomous vehicles. As for future work, we will study to

apply the proposed approach to more other control problems in

autonomous vehicles such as platooning control and lateral

control, to further leverage the research outcomes.

ACKNOWLEDGEMENT

This work was supported by the National Science

Foundation Grant CRII-1755771.

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