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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1

Safe Nonlinear Trajectory Generation for Parallel

Autonomy With a Dynamic Vehicle Model

Wilko Schwarting , Javier Alonso-Mora , Member, IEEE, Liam Paull, Member, IEEE,

Sertac Karaman, Member, IEEE, and Daniela Rus, Fellow, IEEE

Abstract—High-end vehicles are already equipped with safety

systems, such as assistive braking and automatic lane follow-

ing, enhancing vehicle safety. Yet, these current solutions can

only help in low-complexity driving situations. In this paper,

we introduce a parallel autonomy, or shared control, framework

that computes safe trajectories for an automated vehicle, based on

human inputs. We minimize the deviation from the human inputs

while ensuring safety via a set of collision avoidance constraints.

Our method achieves safe motion even in complex driving

scenarios, such as those commonly encountered in an urban

setting. We introduce a receding horizon planner formulated as

nonlinear model predictive control (NMPC), which includes the

analytic descriptions of road boundaries and the conﬁguration

and future uncertainties of other road participants. The NMPC

operates over both steering and acceleration simultaneously.

We introduce a nonslip model suitable for handling complex

environments with dynamic obstacles, and a nonlinear combined

slip vehicle model including normal load transfer capable of

handling static environments. We validate the proposed approach

in two complex driving scenarios. First, in an urban environment

that includes a left-turn across trafﬁc and passing on a busy

street. And second, under snow conditions on a race track with

sharp turns and under complex dynamic constraints. We evaluate

the performance of the method with various human driving styles.

Manuscript received November 27, 2016; revised June 20, 2017 and

October 9, 2017; accepted October 19, 2017. Toyota Research Institute (TRI)

provided funds to assist the authors with their research but this article solely

reﬂects the opinions and conclusions of its authors and not TRI or any other

Toyota entity. This article is a revised and extended version of the paper

entitled “Parallel Autonomy in Automated Vehicles: Safe Motion Generation

with Minimal Intervention” IEEE ICRA 2017, Singapore, June 2017 [1]. The

main extensions are a combined-slip dynamical vehicle model including load

transfer, the motivation of computationally tractable probabilistic collision

constraints for general planning in dynamic and uncertain environments, the

generalization of the minimal intervention principle to any human control

inputs, a thorough technical discussion including caveats, and in-depth expla-

nations of the NMPC, minimal intervention and the combination thereof, and

derivations and additional results including a general measure of intervention.

The Associate Editor for this paper was F. Lian. (Corresponding author:

Wilko Schwarting.)

W. Schwarting and D. Rus are with the Computer Science and Artiﬁcial

Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge,

MA 02139 USA (e-mail: [email protected]; rus@csail.mit.edu).

J. Alonso-Mora is with the Computer Science and Artiﬁcial Intel-

ligence Laboratory, Massachusetts Institute of Technology, Cambridge,

MA 02139 USA, and also with Cognitive Robotics, Delft University of Tech-

nology, 2628 CD Delft, The Netherlands (e-mail: j.alonsom[email protected]).

L. Paull is with the Computer Science and Artiﬁcial Intelligence Labora-

tory, Massachusetts Institute of Technology, Cambridge, MA 02139 USA,

and also with the Département d’Informatique et de Recherche Opéra-

tionnelle, Université de Montréal, Montréal, QC H3T 1J4, Canada (e-mail:

[email protected].ca).

S. Karaman is with the Laboratory of Information and Decision Systems,

Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:

[email protected]).

Color versions of one or more of the ﬁgures in this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TITS.2017.2771351

We consequently observe that the method successfully avoids

collisions and generates motions with minimal intervention for

parallel autonomy. We note that the method can also be applied

to generate safe motion for fully autonomous vehicles.

Index Terms— Advanced driver assistance systems (ADAS),

parallel autonomy, motion planning, collision avoidance, trajec-

tory generation, shared control, intelligent vehicles.

I. INTRODUCTION

LOBALLY, over 3,000 human lives are lost every

day [2] in vehicle-related accidents and over one hundred

thousand are injured or disabled on average. Worse still is that

this number is continuing to increase [3]. In the United States,

11% of accidents are caused by driver distraction (such as cell

phone use), 31% involve an impaired driver due to alcohol

consumption, 28% involved speeding, and an additional 2.6%

were due to fatigue [4]. This troubling trend has resulted in

the continued development of advanced safety systems by

commercial car manufacturers.

For example, systems exist to automatically brake in the

case of unexpected obstacles [5], maintain a car in a lane at a

given speed, and alert users of pedestrians, signage, and other

vehicles on the roadway [6]. However, the scenarios that these

systems are able to deal with are relatively simple compared to

the diverse and complicated situations that we ﬁnd ourselves

in as human drivers routinely. Human drivers are capable of

handling nearly all driving tasks well enough most of the time,

but are overwhelmed in key moments when quick and precise

actions are needed in fast and complex trafﬁc scenarios. A deer

crossing the road, a preceding crash on the highway, a missed

blind spot, or driver tiredness are only some of the ubiquitous

challenges human drivers face in everyday trafﬁc. To handle

these situations, an advanced autonomy system must leave the

driver in full control while ensuring safety when required.

In this work we propose a framework for advanced safety

in complex scenarios that we refer to as Parallel Autonomy.

In this framework we minimize the deviation from the human

input while ensuring safety. The design of the system has

two main objectives: (a) minimal intervention - we only apply

autonomous control when necessary, and (b) guaranteed safety

- the collision free state of the vehicle is explicitly enforced

through constraints in the optimization.

There are three types of collaborative autonomy: (1) series

autonomy, in which the human driver orders the vehicle to

execute a function, similar to most self-driving approaches to

date; (2) interleaved autonomy, in which the human driver and

the autonomous system intermittently take turns in operation

of the vehicle; and (3) Parallel Autonomy, in which the

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

2 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS

Fig. 1. Parallel Autonomy in complex driving scenarios: Human driver

(red) tries to accelerate into an intersection, as shown by the red bar in the

lower left inset. However, given the future uncertainty of the other vehicles’

positions the Parallel Autonomy system prevents the vehicle from continuing

and potentially inhibits a collision.

Fig. 2. A receding horizon planner computes control inputs based on human

inputs and an environment consisting of the road, and static and dynamic

obstacles.

autonomous system functions as a guardian angel in the back-

ground to ensure safety while the human driver is operating

the vehicle. Whether drivers are distracted by smartphones,

searching in their glove box, operating the navigation system,

or are simply overwhelmed by the difﬁculty of driving in

challenging scenarios, the Parallel Autonomy principles offer

additional safety due to redundancy.

We provide a formulation and algorithmic solution to Paral-

lel Autonomy based on a Nonlinear Model Predictive Control

(Nonlinear Model Predictive Control (NMPC)) policy, under

the assumption of known current conﬁguration of the ego

vehicle and the road boundaries, cf. Fig. 2. Uncertain current

and future conﬁgurations of other vehicles are described in

the form of a posterior distribution and we parametrize them

by their mean, the expected conﬁguration, and covariances.

We assume this information to be available from an inference

framework. Speciﬁcally,

• we incorporate the time-varying uncertainty related to the

dynamic obstacle predictions explicitly in the optimiza-

tion,

• the vehicle can follow the road by contour tracking with

additional constraints for the road boundaries,

• and we simultaneously optimize over steering and accel-

eration while maintaining the ability to plan online over

long time horizons (∼ 9s),

The basic operation of the controller is shown in Fig. 1,

where the driver attempts to cut in front of oncoming trafﬁc

to make a left turn, however the Parallel Autonomy system

prevents this action to avoid a collision with the incoming

vehicles.

This paper contributes the following:

• A formulation of Parallel Autonomy as a shared control

approach between humans and intelligent vehicles, which

adheres to the minimal intervention principle and is able

to handle complex driving scenarios

• The development of a real-time NMPC suitable for tra-

jectory generation in intelligent and autonomous vehicles,

which relies on a state of the art solver

•

Simulation of complex trafﬁc scenarios with real human

inputs of different driving styles, such as a left-turn across

trafﬁc and driving on a snowy race track

We will employ two motion models of different complexity:

• A dynamical nonlinear combined slip vehicle model

including load transfer for static environments

• A kinematic model for dynamic and complex environ-

ments

In Sec. II we summarize the related work in the ﬁeld

whereas in Sec. III we present the Parallel Autonomy control

approach. In Sec. IV we provide a concrete instantiation of

the framework and present the NMPC approach to solve it.

Finally, we show detailed simulation results and conclusions

in Sec. V and Sec. VI respectively.

II. R

ELATED WORKS

In this section we provide an overview of the related work

in the areas of general safety, shared control for autonomous

vehicles, and Model Predictive Control (MPC).

A. Safety of Autonomous Vehicles

In theory, safety can be guaranteed for deterministic systems

by computing the set of the states for which the vehicle

will inevitably have a collision and then ensuring that the

vehicle never enters that set. The set is referred to by different

terms in the literature, such as the capture set [7]–[10],

the inevitable collision states (ICS) [11]–[13], the region of

inevitable collision (RIC) [14], and the target set [15]. How-

ever, without some assumptions or limiting the applicability to

relatively simplistic scenarios, this set is difﬁcult to compute

analytically. Reference [16] apply a control barrier function to

guarantee never entering the infeasible set upon moving into

an avoidable set constructed from a polar algorithm in slow

speeds to avoid pedestrians. These ICS-inspired methods tend

to only intervene when the system is at the boundary of the

capture set, which can cause undesirable behavior, and toggle

between either the autonomous system input or the human

input. We follow the idea of [7], [12] and [13] and deﬁne a

set of probabilistic constraints for collision avoidance, which

produce a safe behavior. Yet, our method produces smooth

inputs for the vehicle, since it is always active, as a Parallel

Autonomy safety system.

B. Shared Control of Intelligent Vehicles

The most intuitive way of merging the human input with

the output of a safety system is by linear combination of the

two, as shown by [17] and [18], who proposed threat measures

based on the dynamic limitations of the vehicle. The human

input was overwritten by the trajectory based on the severity

of the threat. While they reasoned about the human drivers

intended homotopy class [17], their framework did not take

the human input directly into account. In a similar line [19],

computed safety margins from sampled trajectories clustered

We employ FORCES Pro by Embotech to autogenerate a fast NMPC solver

for our problem.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

SCHWARTING et al.: SAFE NONLINEAR TRAJECTORY GENERATION FOR PARALLEL AUTONOMY 3

into homotopy classes, but do not share the control with the

human driver.

In contrast, in this work we directly incorporate the human

inputs into an optimization framework in a minimally inva-

sive manner and also add a soft nudging behavior to guide

the driver. Our approach minimizes the deviation of the

autonomous system’s plan from the driver’s intent - current

steering and acceleration inputs - and provides small feedback

to the driver shortly before situations become critical. This is

important to prevent startling the driver and to decrease the

likelihood of unnecessary high intervention later, thanks to

slight intervention early on.

Similar to our approach, several constrained optimization

approaches for shared control have been formulated. Refer-

ence [20] directly minimized the difference from the human

predicted control input necessary to achieve safe trajectories,

and [21] minimized the difference in steering wheel angle.

Reference [22] minimize the deviation from desired front

wheel lateral force with an additional discount factor with

increasing time. We apply a discount factor in a similar

manner. Reference [23] minimized the deviation from human

inputs, in this case orientation and speed, via a convex con-

strained optimization to generate safe motion of a wheelchair.

In contrast, we present a more general approach where we

conduct the minimization jointly in both steering and acceler-

ation inputs, can blend in additional trajectory-speciﬁc costs

to provide a nudging behavior, model the dynamical model of

the vehicle, and strictly enforce safety constraints.

C. Receding Horizon Control for Shared Control

We formulate the constrained optimization as a receding

horizon control problem, typically referred to as Model Pre-

dictive Control (MPC). Related to our work [21], employed a

hierarchical MPC approach for avoidance of static obstacles

with motion primitives and path tracking, which switches con-

trol to and from the driver as a function of driver attentiveness.

Instead of planning paths ﬁrst and computing velocity proﬁles

separately, we directly plan full trajectories that also avoid

dynamic obstacles. Similar approaches include: a constrained

pathless MPC that blends human and controller steering com-

mands only, proposed by [17], a shared control MPC for

static unstructured robot environments avoiding circular obsta-

cles [24], and robust NMPC [21] to avoid static obstacles while

tracking the roads center line over a very short horizon of

less than 1.5s. Alternatively [22], deﬁned vehicle-stability and

environmental envelopes to supply safe steering commands at

constant speed in a discretized environment.

In contrast, our approach does not require linearization - we

solve a Nonlinear MPC (NMPC) problem directly - and can

handle complex road scenarios with dynamic maneuvers and

dynamic obstacles, with steering and acceleration control over

long horizons (∼ 9s).

The methods developed by [22] and [25] construct corridors

consisting of multiple convex regions to describe the area

that the vehicle will drive through. Based on the corridors

and constant velocity, they apply constraints on the lateral

position of the vehicle to avoid colliding with obstacles

yielding a convex optimization problem with global optimality.

Operating over both steering and acceleration in a non-convex

environment and solving our NMPC formulation inherits

the general limitations of non-convex optimization, such as

uncertain convergence and runtime, and lack of guarantee

of optimality. Nonetheless, braking is clearly an essential

function in vehicle safety.

We provide all costs and constraints to the solver in closed

form without pre-linearization, and we beneﬁt from recent

advances in efﬁcient Interior-Point solvers [26] to directly

solve the NMPC. To guide the planner along the road, we build

upon Model Predictive Contouring Control (MPCC) [27]–[29],

which approximates path progress inside a corridor, the road

in our application. By tracking the center of the lane and

remaining within the limits of the road our planner can be

employed for both Parallel Autonomy, where we minimize

the deviation from human input, and for fully autonomous

vehicles.

III. P

ROBLEM FORMULATION

The Parallel Autonomy problem is based on two overarch-

ing principles.

• Minimal intervention with respect to the human driver.

That is, the control inputs to the vehicle should be as

close as possible to those of the human driver.

• Safety. The probability of collision with respect to the

environment and other trafﬁc participants is below a given

threshold.

A. Deﬁnitions

We use the discrete time shorthand k  t

,wheret

= t



i=1

t

, with t

the current time and t

the i-th timestep

of the planner. Vectors are bold.

1) Ego Vehicle: We refer to the car driven by the human

as ego-vehicle. At time k, we denote the conﬁguration of

the ego-vehicle, typically position p

=[x

, y

], heading φ

longitudinal and lateral velocity v

x,k

, v

y,k

,yawrate

and

steering angle δ

,bythestatez

=[p

,φ

,δ

x,k

y,k

]∈Z.

Its control input, typically steering velocity

and longitudinal

acceleration ˙v

x,k

, is labeled u

, ˙v

x,k

]∈U.

The evolution of the state of a vehicle is then represented

by a general discrete dynamical system

k+1

= f (z

, u

), (1)

described in Sec. III-C.2. The arguably largest source of

uncertainty in the system operating over long time horizons

lies in the prediction of motions of other vehicles, which

dominates all other sources of uncertainty. The comparably

low uncertainty in the dynamic motions of the ego-vehicle

thus motivates the choice for a deterministic motion model.

The area occupied by the ego-vehicle at state z

is denoted by

B(z

) ⊂ R

. In particular, we model it as a union of circles

as shown in Sec. IV-E, Fig. 5.

2) Other Trafﬁc Participants: Other trafﬁc participants,

such as vehicles, pedestrians and bikes, are indexed by i =

{1,...,n}. Their conﬁguration and control input are described

as z

∈ Z

and u

∈ U

. To incorporate uncertainty, we

consider that an approximate posterior distribution describing

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