arXiv:1704.00866v1  [cs.RO]  4 Apr 2017
Indirect Shared Control of Highly Automated Vehicles for
Cooperative Driving between Driver and Automation
Renjie Li, Yanan Li, Shengbo Eben Li, Etienne Burdet and Bo Cheng
Abstract—It is widely acknowledged that drivers should re-
main in the control loop of automated vehicles before they
completely meet real-world operational conditions. This paper
introduces an ‘indirect shared control’ scheme for steer-by-
wire vehicles, which allows the vehicle control authority to be
continuously shared between the driver and automation through
unphysical cooperation. This paper first balances the control
objectives of the driver and automation in a weighted summation,
and then models the driver’s adaptive control behavior using
a predictive control approach. The driver adaptation modeling
enables off-line evaluations of indirect shared control systems
and thus facilitates the design of the assistant controller. Unlike
any conventional driver model for manual driving, this model
assumes that the driver can learn and incorporate the controller
strategy into his internal model for more accurate path following.
To satisfy the driving demands in different scenarios, a sliding-
window detector is designed to continuously monitor the driver
intention and automatically switch the authority weights between
the driver and automation. The simulation results illustrate
the advantages of considering the driver adaptation in path-
following and obstacle-avoidance tasks, and show the effectiveness
of indirect shared control for cooperative driving.
I. INTRODUCTION
In recent years, several ambitious projects of automated
vehicles (or self-driving cars, driverless cars), e.g., Google’s
self-driving car and Tesla’s Autopilot, have realized rapid pro-
gresses towards commercialization. While driverless cars are
considered as an effective approach to relieve drivers through
advanced sensing and navigation technologies, various factors
need to be addressed before their ultimate deployment. These
factors include technical requirements, safety issues, ethical
dilemma [1], and subsequent harsh government regulations
[2]. In addition, previous studies have shown that imperfect
automation may lead to severe human factor problems such
as loss of situation awareness, overreliance, distrust, etc [3],
[4]. These problems are usually considered to be caused by
drivers being kept out of the operating loop during autonomous
driving [5]. τ In light of the problems brought by imperfect
self-driving technology, there is an increasing interest to keep
the human driver in the control loop, who may have superior
capabilities for handling complicated situations. For instance,
This research is supported by NSF China with U1664263 and 51575293,
and National Key R&D Program in China with 2016YFB0100906. All
correspondence should be sent to S. Li (e-mail: lisb04@gmail.com).
R. Li is with the State Key Lab of Automotive Safety and Energy,
Tsinghua University, Beijing, China 100084. He is currently visiting the
Department of Bioengineering, Imperial College London, UK SW7 2AZ (e-
mail: lirenaxe@gmail.com).
S. Li and B. Cheng are with the State Key Lab of Automotive
Safety and Energy, Tsinghua University, Beijing, China 100084 (e-mail:
lisb04@gmail.com; chengbo@tsinghua.edu.cn).
Y. Li and E. Burdet are with the Department of Bioengineering, Im-
perial College London, UK SW7 2AZ (e-mail: yanan.li@imperial.ac.uk;
e.burdet@imperial.ac.uk).
in some copilot systems [6]–[8], the human driver assumes
control most of the time, while the assistant controller only
intervenes if it reckons that the vehicle is at risk. Therefore,
the support provided by the automation is limited to a short
proportion of the time. By contrast, in shared control [9],
or cooperative control depending on the context [10], the
automation remains in the control loop and provides contin-
uous support to the driver, thus significantly reducing their
workload.
The shared control concept is commonly implemented on
conventional vehicles equipped with mechanical steering sys-
tems as the haptic shared control, in which the driver and
assistant controller simultaneously apply a control torque on
the steering wheel [9]. Under such a framework, the driver and
automation control the vehicle steering wheel cooperatively
through physical interaction. The performance enhancement of
drivers in haptic shared control has been broadly reported in
literature (see the survey of Petermeijer et al. [11]) including
experimental evaluations [12] and controller design aspects
[13], [14]. In haptic shared control, drivers can deny the
assistant control torque by stiffening the hands using muscles
contraction [15]–[17] thus keeping the final control authority
(if the assistant torque is designed not to exceed the human
resistance limit). The mechanism of haptic shared control has
been studied in other fields such as human-robot interaction
[18]–[20] where it is referred to as “motor interaction” or
“joint motor action” [21], [22].
In contrast to mechanical steering systems, steer-by-wire
technology allows the mechanical decoupling of the steering
wheel and road wheels. With steer-by-wire it is possible for
the assistant controller to modulate the driver’s commanded
steering input by the lower-level steering actuator. Therefore,
steer-by-wire vehicles are an ideal platform to implement
various forms of shared control, wherein the controller can
complement the driver’s steering input behind the scenes
without introducing physical interaction. In this scheme, the
driver can only control the vehicle indirectly through the
controller, and his control authority entirely depends on how
the automation assimilates his control input. Therefore, such
a scheme is called “indirect shared control”. Compared with
haptic shared control, indirect shared control can further
minimize the driver effort since the driver is not obligated to
provide full steering. Moreover, the removal of direct physical
interaction may reduce driver discomfort which has been
observed in haptic shared control [23].
Important questions that need to be addressed in order to
establish indirect shared control include: 1) how to design
the controller algorithm such that it can respect the driver
input while exploiting its own path-tracking ability; 2) how
to model the driver’s adaptive behavior in presence of a given
controller such that the system can be evaluated through off-
line simulation. Concerning the first problem, Manabu et al.
[24] proposed a weighted summation method to combine
the driver input with the automation’s desired input, which
was implemented on a steer-by-wire vehicle and validated
through field tests. In this work the authority weights are
static during driving, which does not allow the driver to gain
more control authority in a critical situation that cannot be
properly handled by the automation. Concerning the second
problem, to our knowledge no previous work has modeled
the driver adaptation. Some studies on haptic shared control
[13], [25] included a driver model in controller design and
simulation, but they did not consider the driver adaptation,
which has however been observed [26], [27]. Na and Cole
[28] employed game theory to describe the driver-automation
co-adaptation, whereas it required the automation to exactly
predict the driver’s reference path. In fact, numerous studies
have documented the adaptation inherent to sensorimotor
control, which has been modeled through the formation of
an internal inverse or forward model [29]. This leads to an
intuitive conjecture that the driver adaptation in indirect shared
control can be interpreted as the inclusion of the assistant
controller in their internal model.
Contributions: This paper proposes a novel indirect shared
control scheme to maintain the driver-in-the-loop function in
highly automated steer-by-wire vehicles, which allows an un-
physical cooperation between the driver and automation. The
assistant controller follows the weighted summation method as
in [24] to balance the interests of the driver and automation,
whereas the control authorities can be automatically switched
with respect to the driver intention. To better assess the
performance of indirect shared control, this paper proposes
a method to model the driver adaptation in presence of the
assistant controller. This method relies on the assumption that
the human driver can identify the controller strategy through
sensorimotor learning. Cole et al. [30] first suggested that
driver steering behavior can be properly modeled by model
predictive control (MPC), in which the internal model of vehi-
cle dynamics serves as the predictor. Here we develop an MPC
model of the driver’s adaptive behavior which incorporates the
identified controller strategy into the driver’s internal model for
predictive path tracking. The modeling of the driver adaptation
enables off-line evaluations of indirect shared control systems
and thus facilitates the design of the assistant controller.
This paper is organized as follows. Section II describes the
framework of indirect shared control including the vehicle
model to be used. Section III formulates the strategies of the
controller and the driver based on two fundamental assump-
tions, and Section IV derives the analytical solution of the
driver strategy formulated in Section III. Section V introduces
an automatic weight switching method which can overcome
the trade-off brought by static weighting. Section VI validates
the proposed driver model and the automatic weight switching
by simulation, and Section VII concludes the paper.
II. SYSTEM DESCRIPTION
A. General Framework
The frameworks of haptic shared control and indirect shared
control are compared in Fig. 1, where rD and rA are the
reference paths of the driver and automation, respectively,
x is the vehicle state governed by the vehicle dynamics
˙x = f(x, u), which is to be specified in Section II-B. In
haptic shared control, the driver and controller simultaneously
apply a control torque (denoted by TD and TA) on the steering
wheel and the final steering input u (steering wheel angle)
is determined by the resultant torque. The two parties jointly
control the vehicle through physical interaction on the steering
wheel. In indirect shared control, the steering wheel angle
uD is solely commanded by the driver, whereas it can be
transformed by the controller into the final steering input u
before being delivered to the vehicle. uD and u are connected
by the controller’s transformation function u = g(x, uD, rA).
In other words, the driver input uD can be viewed as the
intermediate input of the vehicle. The automation blends its
own control objective with the driver input through the trans-
formation u = g(x, uD, rA). If the transformation function is
u ≡uD, indirect shared control degrades to manual driving;
if u is independent of uD, the vehicle becomes autonomous.
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Fig. 1. Block diagrams of two shared control frameworks: a) haptic shared
control; b) indirect shared control.
The intermediate input transformation u = g(x, uD, rA) is
the crucial part of indirect shared control, which exploits the
automation’s path-tracking ability and respects the driver input
at the same time. To put it another way, the driver can indi-
rectly influence the vehicle control through u = g(x, uD, rA).
The driver’s control authority and responsibility depend on
how largely u relies on uD. In this sense, the driver and the
automation share the control authority and guide the vehicle
in a cooperative fashion. This scheme guarantees that the
driver is actively involved in the loop because he is obligated
to convey control input throughout driving. Meanwhile, the
driver’s control effort can be partly relieved by virtue of the
assistant controller.
B. Vehicle Dynamics
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Fig. 2. Linearized dynamic bicycle model.
The vehicle model used in this paper is the single-track
dynamic bicycle model [30] shown in Fig. 2. This model has
high fidelity if the vehicle sideslip angle is small, i.e., the
tires work in a linear region. If the vehicle is running at a
constant longitudinal velocity U and the vehicle yaw angle
from X-axis is small, its dynamics can be described by the
linear time-invariant equation
˙x(t) = Acx(t) + Bcu(t)
z(t) = Ccx(t)
(1)
where the state x(t) =

v(t)
ω(t)
y(t)
ψ(t)
T (v: lateral
velocity, ω: yaw rate, y: lateral displacement, ψ: yaw angle),
u(t) = δ(t) is the steering wheel angle, z(t) =
y(t)
ψ(t)T
is the output used to evaluate the path-following performance
and provide feedback. The subscript “c” denotes “continuous-
time”. Ac and Bc are constant matrices related to the vehicle’s
intrinsic properties and the constant velocity U:
Ac =


−(Cf + Cr)
mU
−(aCf −bCr)
mU
−U
0
0
−(aCf −bCr)
IzU
−(a2Cf + b2Cr)
IzU
0
0
1
0
0
U
0
1
0
0


Bc =
 Cf
ism
aCf
isIz
0
0
T
, Cc =

0
0
1
0
0
0
0
1


(2)
where Cf is front cornering stiffness, Cr rear cornering
stiffness, m the vehicle mass, a the distance from center of
mass to front axle, b the distance from center of mass to rear
axle, Iz the polar moment of inertia, and is the steering ratio.
The continuous-time vehicle dynamics (1) can be converted to
the discrete-time equation
x(k + 1) = A x(k) + Bu(k)
z(k) = Cx(k)
(3)
where k denotes discrete time index.
III. INDIRECT SHARED CONTROL
This section formulates the strategies of the controller and
driver in indirect shared control. The controller’s transfor-
mation strategy u = g(x, uD, rA) is first introduced, which
should ensure that the interests of both the driver and automa-
tion are respected. Second, given this transformation strategy,
the driver’s adaptive control strategy is modeled.
A. Controller Strategy
We assume that the controller can observe the driver input
uD at every time. The controller needs to follow a reference
path rA while considering the driver input. In practice, the
reference path may be generated by a path planner, although
this is beyond the scope of this paper. We follow the weighted
summation method [24] to blend the driver input with the
automation’s desired input:
u(k) = λD uD(k) + λA uA(k),
λD, λA ≥0 ,
(4)
where u is the final input delivered to the vehicle, uD the
instant driver input, uA the automation’s desired input, and
λD, λA the authority weights assigned to the driver input and
the automation’s desired input, respectively. λD + λA ≡1 is
often assumed to avoid conflict and force escalation, as well
as to simplify the control authority management using only
one parameter.
The second part of the controller strategy is to specify
how to generate the desired input uA at each time. In this
study, uA is solely based on the automation’s reference path
and control objective, which means that uA is independent of
uD. We adopt MPC to calculate uA, which uses the vehicle
dynamics (3) as the predictor and repeatedly programs the
optimal input sequence over a predictive horizon associated
with a cost function:
uA(k) = u∗
A,(1)(k)
(5a)
u∗
A(k) =
arg min
{uA(k),uA(k+1),··· ,uA(k+Nc−1)}
JA(k)
(5b)
x(k + 1) = Ax(k) + BuA(k)
(5c)
z(k) = Cx(k)
(5d)
where uA is the automation’s desired input following MPC,
Nc the control horizon, u∗
A the optimal input sequence over
the control horizon, and JA the automation’s cost function.
u∗
A,(1) denotes the first element of u∗
A used as the command
at each time. In this paper, we consider the following quadratic
cost to minimize the tracking error and effort:
JA(k) ≡
Np
X
i=1
∥z(k + i) −rA(k + i)∥2
QA
+
Nu−1
X
i=0
∥uA(k + i)∥2
RA
(6)
where rA(k) is the automation’s reference path at each time
step, Np the predictive horizon, and QA, RA constant positive
definite weighting matrices of appropriate dimensions.
B. Driver Strategy
The driver strategy is modeled considering two main as-
sumptions:
1) The driver has identified the controller’s transformation
strategy through motor learning.
2) The driver incorporates learned transformation strategy
into their internal model for predictive control.
The above assumptions state that the driver employs MPC-type
strategy to perform path-following task [30] but will adapt the
controller’s transformation strategy (4) into the predictor after
motor learning. Accordingly, the driver’s control strategy is
modeled as
uD(k) = u∗
D,(1)(k)
(7a)
u∗
D(k) =
arg min
{uD(k),uD(k+1),··· ,uD(k+Nc−1)}
JD(k)
(7b)
s.t. u(k) = λDuD(k) + λAuA(k)
(7c)
x(k + 1) = Ax(k) + Bu(k)
(7d)
z(k) = Cx(k)
(7e)
where uD is the driver input following MPC strategy, Nc the
control horizon which is assumed identical to the controller,
u∗
D is the optimal input sequence up to the control horizon,
and JD is the driver’s cost function. The key difference
between (7) and the MPC strategy for manual driving in
[30] is that the driver hereby incorporates the controller’s
transformation strategy (4) into the prediction.
Cost function JD is critical for modeling the driver behavior
in indirect shared control. According to [29], [30], the cost
function for manual path following takes a quadratic form
similar to (6). In this paper, we assume that the driver’s cost
function preserves the form in indirect shared control, i.e.,
JD(k) ≡
Np
X
i=1
∥z(k + i) −rD(k + i)∥2
QD
+
Nu−1
X
i=0
∥uD(k + i)∥2
RD
(8)
where rD(k) is the driver’s reference path at each step, Np the
predictive horizon which is assumed identical to the controller,
and QD and RD constant positive definite weighting matrices
of appropriate dimensions. Similar to the automation, the
driver also seeks to minimize the path-tracking error and
control effort.
The authority weights λD and λA determine the control
authorities of the driver and automation. If one of them is set
to zero, indirect shared control becomes either manual driving
or fully autonomous driving:
1) λD = 0, λA = 1 means that the controller does not take
the driver input into account, i.e., the driver has no control
authority, and the vehicle becomes autonomous.
2) λD = 1, λA = 0 means that the controller completely
complies with the driver, i.e., the driver retains direct
control authority, and the vehicle becomes manually
driven.
IV. ANALYTICAL SOLUTION
In this section, we derive the analytic expressions of the
strategies, i.e., uA(k), uD(k), and u(k), formulated in Section
III. By doing this, we can better observe how indirect shared
control shapes the driver’s control strategy in comparison
with manual driving cases. More specifically, we can obtain
the influence of authority weights λD and λA on the driver
behavior, which can be used to improve the design of indirect
shared control. The calculation of the automation’s desired
input uA(k) is actually an unconstrained MPC-tracking prob-
lem. We directly give its analytic expression based on existing
theoretical solution. The focus of this section lies in the
derivation of the driver strategy uD(k) from which uA(k)
and thus u(k) can be computed through (4). For real-time
implementation, there are some efficient MPC computation
methods such as in [31], [32].
For simplicity, we set Np ≡Nc ≡N, and introduce several
notations to make the derivation more concise:
x(k) =


x(k + 1)
x(k + 2)
...
x(k + N)

,
uA(k) =


uA(k)
uA(k + 1)
...
uA(k + N −1)


uD(k) =


uD(k)
uD(k + 1)
...
uD(k + N −1)

,
z(k) =


z(k + 1)
z(k + 2)
...
z(k + N)


rA(k) =


rA(k + 1)
rA(k + 2)
...
rA(k + N)

,
rD(k) =


rD(k + 1)
rD(k + 2)
...
rD(k + N)


A =


A
A2
...
AN

,
B =


B
0
· · ·
0
AB
B
· · ·
0
...
...
...
...
AN−1B
AN−2B
· · ·
B


C =


C
C
...
C









N,
Φ = CA =


CA
CA2
...
CAN−1


Θ = CB =


CB
0
· · ·
0
CAB
CB
· · ·
0
...
...
...
...
CAN−1B
CAN−2B
· · ·
CB


QA (QD) =


QA (QD)
QA (QD)
...
QA (QD)









N
RA (RD) =


RA (RD)
RA (RD)
...
RA (RD)









N
We first present the analytical solution of uA(k) to the MPC
problem (5,6). This is an unconstrained MPC-tracking problem
with solution
uA(k) = eT
1 KAεA(k) .
(9)
where eT
1 ≡
N
z
}|
{
1
0
· · ·
0
, KA ≡
√QAΘ
√RA
† √QA
0

(‘†’
denotes pseudo-inverse) is a constant matrix and εA(k) ≡
rA(k)−Φx(k) is associated with the reference path up to the
predictive horizon rA(k) and the current vehicle state x(k).
The focus of this section is to derive uD(k), given the
problem (7, 8) and the solution of uA(k) in (9). Substituting
(9) into (7c), and then into (7d), we can rewrite the driver’s
prediction equation as
x(k + 1) = Ax(k) + λDBuD(k)
+ λABeT
1 KA[rA(k) −Φx(k)].
(10)
Rearranging the terms, we have
x(k + 1) = (A −λABeT
1 KAΦ)x(k) + λDBuD(k)
+ λABeT
1 KArA(k).
(11)
Denoting
˜A ≡A −λABeT
1 KAΦ
(12)
and
wA(k) ≡eT
1 KArA(k) ,
(13)
we can rewrite (11) as
x(k + 1) = ˜Ax(k) + λDBuD(k) + λABwA(k).
(14)
Iterating (14) for N −1 times and stacking the results, yield


x(k + 1)
x(k + 2)
...
x(k + N)

=


˜A
˜A2
...
˜AN

x(k)
+ λD


B
0
· · ·
0
˜AB
B
· · ·
0
...
...
...
...
˜AN−1B
˜AN−2B
· · ·
B




uD(k)
uD(k + 1)
...
uD(k + N −1)


+ λA


B
0
· · ·
0
˜AB
B
· · ·
0
...
...
...
...
˜AN−1B
˜AN−2B
· · ·
B




wA(k)
wA(k + 1)
...
wA(k + N −1)


(15)
Using the notations
wA(k) =


wA(k + 1)
wA(k + 2)
...
wA(k + N)

,
˜
A =


˜A
˜A2
...
˜AN


˜B =


B
0
· · ·
0
˜AB
B
· · ·
0
...
...
...
...
˜AN−1B
˜AN−2B
· · ·
B

, ˜Φ = C ˜
A =


C ˜A
C ˜A2
...
C ˜AN


˜Θ = C ˜B =


CB
0
· · ·
0
C ˜AB
CB
· · ·
0
...
...
...
...
C ˜AN−1B
C ˜AN−2B
· · ·
CB

.
Eq. (15) can be rewritten as
x(k) = ˜
Ax(k) + λD ˜BuD(k) + λA ˜BwA(k)
(16)
and the output is
z(k) = Cx(k) = C ˜
Ax(k) + λDC ˜BuD(k) + λAC ˜BwA(k)
= ˜Φx(k) + λD ˜ΘuD(k) + λA ˜ΘwA(k).
(17)
The driver’s cost function (8) can be expressed as
JD(k) = ∥z(k) −rD(k)∥2
QD + ∥uD(k)∥2
RD .
(18)
According to (17), the path-tracking error can be stated as
z(k)−rD(k) = ˜Φx(k)+λD ˜ΘuD(k)+λA ˜ΘwA(k)−rD(k).
(19)
Let us set
εD(k) ≡rD(k) −˜Φx(k) −λA ˜ΘwA(k) ,
(20)
then we have
z(k) −rD(k) = λD ˜ΘuD(k) −εD(k) .
(21)
The driver’s cost function can be then expressed as
JD(k) =



λD ˜ΘuD(k) −εD(k)



2
QD + ∥uD(k)∥2
RD
=




√QD[λD ˜ΘuD(k) −εD(k)]
√RDuD(k)




2
.
(22)
The receding optimization problem (7) is converted to
u∗
D(k) = arg min
uD(k)




√QD[λD ˜ΘuD(k) −εD(k)]
√RDuD(k)




2
.
(23)
The solution of (23) is identical to the least-square solution of
the problem

λD
√QD ˜Θ
√RD

uD(k) =
√QD
0

εD(k),
(24)
which can be stated as
u∗
D(k) =

λD
√QD ˜Θ
√RD
† √QD
0

εD(k) = KDεD(k), (25)
where KD ≡

λD
√QD ˜Θ
√RD
† √QD
0

is a constant matrix,
εD(k) is a vector associated with the driver’s reference path
up to the predictive horizon rD(k), the current vehicle state
x(k), and wA(k). Note that
wA(k) =


wA(k)
wA(k + 1)
...
wA(k + N −1)

=


eT
1 KArA(k)
eT
1 KArA(k + 1)
...
eT
1 KArA(k + N −1)

,
(26)
which actually contains the automation’s reference path up to
horizon 2N −1, because rA(k + N −1) encompasses rA(k +
N −1), · · · , rA(k + 2N −1).
Finally, the driver’s control strategy is to use the first
element of u∗
D(k) as motor command, which is
uD(k) = eT
1 KDεD(k) .
(27)
We can then compute u(k) using equations (4, 9, 27):
u(k) = λDeT
1 KDεD(k) + λAeT
1 KAεA(k).
(28)
We can briefly validate (27) by examining two special cases,
i.e., manual driving (λD = 1, λA = 0) and fully automated
driving (λD = 0, λA = 1):
1) Let λD
= 1 and λA = 0, which transforms indi-
rect shared control into manual driving. In this case,
˜A = A, then
˜
A = A, ˜B = B, ˜Φ = Φ, ˜Θ = Θ.
KD =
√QDΘ
√RD
† √QD
0

, εD(k) = rD(k) −Φx(k).
The strategy (27) becomes the manual driving strategy in
[30]. This is consistent with our intuition.
2) Let λD
= 0 and λA = 1, which transforms indi-
rect shared control into automated driving. In this case,
according to (22), the driver’s cost function becomes
JD(k) = ∥−εD(k)∥2
QD + ∥uD(k)∥2
RD. Because εD(k)
is not subject to uD(k), the driver can only minimize
∥uD(k)∥2
RD, which yields the trivial solution u∗
D(k) = 0
and uD(k) = 0. This means that if the driver realizes that
he has no control authority, he will not take any action
but stays relaxed, which is consistent with our intuition.
V. AUTOMATIC WEIGHT SWITCHING
Intuitively, the authority weights λD and λA should vary
with the specific situation. When the automation is consistent
with the driver intention, i.e., they are following an identical
path, λA could be smaller (λD could be larger) to relieve
the driver. However, if the driver intends to divert the vehicle
from the original path, λD should increase (λA should be
decreased) in order to follow the driver intention. A typical
situation is when there is an obstacle ahead of the vehicle that
is undetected by the automation, the driver should be assigned
a larger λD (i.e., a smaller λA) in order to avoid the obstacle
easily.
A simple approach to achieve this objective would be to
allow the driver to set the authority weights manually, for
example by pressing a button to inform the automation that
more control authority is required. However there may not
be sufficient time for the driver to do the notification, and
this manual setting approach places a heavy burden to the
driver if the request is frequent. Therefore, it is desirable that
the authority weights λD and λA can change automatically
according to the driver intention.
We hereby use a model-based method to detect whether the
automation corresponds to the driver intention. We assume that
the driver’s reference path is identical to the automation, i.e.,
rD ≡rA, if their intentions are matched, and the automation
knows the driver’s QD through either prior knowledge or real-
time estimation. Therefore, the automation can estimate the
expected driver input at every time step based on the driver
model (27):
ˆuD(k) = eT
1 KDεD(k).
(29)
In reality, when the driver intention is consistent with the
automation, this should result in some oscillatory error be-
tween the actual driver input and expected one, known as
the model error. However, if the driver intention becomes
inconsistent with the automation, this error steadily increases
in one direction. To detect such inconsistency between the
driver and automation we thus monitor the mean cumulative
error within a constant-length sliding window:
δ(k) = 1
H

k
X
j=k−H+1

˜uD(j) −ˆuD(j)


,
(30)
where H is the window length, and ˜uD the actual driver input.
A threshold-based rule is adopted to detect the driver intention
and switch the weights accordingly:
(
λD(k + 1) = λ+
D , λA(k + 1) = λ−
A
if δ(k) ≥δ∗
λD(k + 1) = λ−
D , λA(k + 1) = λ+
A
if δ(k) < δ∗, (31)
where λ+
D (λ+
A) and λ−
D (λ−
A) denote the constant larger and
smaller weights of the driver (automation), respectively. δ∗is
a predefined threshold to judge if the driver intention is still
consistent with the automation. This rule indicates that if the
mean cumulative error between ˜uD(k) and ˆuD(k) within a
constant-length window exceeds a certain threshold, the driver
is deemed to hold a different intention with the automation,
and the authority weights will be switched accordingly.
The selection of δ∗should rely on the prior knowledge of
the model error and inevitably raises some trade-off. If δ∗is
too large, the detection becomes increasingly insensitive such
that there is a latency of weight switching in response to the
driver’s intention change; otherwise if δ∗is too small, the
detection is unnecessarily sensitive which leads to frequent
false switching. In practice, δ∗should be tuned until satisfac-
tory performance is achieved; it will typically depend on each
individual.
VI. SIMULATION
To validate the proposed approach, we carried out simula-
tions of indirect shared control in two scenarios: path following
and obstacle avoidance. In the path-following scenario, the
controller assists the driver to follow the reference path (usu-
ally designated as the lane centerline). It is expected that under
such a circumstance indirect shared control can reduce the
driver’s control effort and enhance the vehicle path-tracking
performance. In the obstacle-avoidance scenario, the driver
attempts to control the vehicle in order to avoid an obstacle
TABLE I
SIMULATION PARAMETERS
Front wheel cornering
stiffness
Cf
12000 [N/rad]
Rear wheel cornering stiffness
Cr
8000 [N/rad]
Distance from mass center to
front axle
a
0.92 [m]
Distance from mass center to
rear axle
b
1.38 [m]
Mass
m
1200 [kg]
Polar moment of inertia
Iz
1500 [kg·m2]
Steering ratio
is
16
Longitudinal velocity
U
20 [m/s]
Sampling time
T
0.02 [s]
Predictive horizon
N
50
Automation weighting matrix
QA

1.5
0.6

Driver weighting matrix in
path following
QD (PF)

0.036
0.02

Driver weighting matrix in
obstacle avoidance
QD (OA)

36
20

which is undetected by the automation. Indirect shared control
enables the driver to avoid the obstacle without turning off the
automation. In each scenario, the model proposed in this paper
is compared with a conventional driver model without adap-
tation. Finally, this section compares how automatic weight
switching facilitates the obstacle-avoidance task relative to
static weighting.
The parameters for simulation are given in Table I, which
include the parameters of vehicle dynamics and of the driver’s
(automation’s) predictive control. Note that QA is larger than
QD (PF) because the automation generally has superior path
tracking capability to a human driver. Moreover, a small QD
(PF) also indicates that the driver tends to relax the control by
compromising tracking performance in path-following tasks.
Nevertheless, QD (OA) is much larger than QD (PF) because
in emergency situations the driver prefers to guide the vehicle
along his desired path, even at the cost of a large control effort.
A. Path Following
This part examines the benefits of indirect shared control in
a path-following scenario. Fig. 3(a) illustrates the trajectory of
lateral displacement y in different degrees of shared control
(manual driving is a special case of shared control with
λD = 1 and λA = 0). Corresponding driver control effort
is depicted in Fig. 3(b). The reference path is designed as
a continuous curve. According to our assumption, the driver
tracks the same reference path as the automation. Additionally,
the driver model without adaptation (denoted as “conventional
model” in the figures) is also investigated. This is done by
setting λD = 1 and λA = 0 when calculating the driver input
according to (27). It can be seen in Fig. 3(a) that the vehicle
path-tracking performance is enhanced as the automation’s
control authority λA increases. The model difference is not
evident in terms of the path-tracking error. Fig. 3(b) shows
that the driver’s control effort is significantly reduced with the
increase of the automation’s control authority λA. However,
the conventional driver model requires more control effort than
the proposed model as it does not consider the controller
assistance when steering. This result validates the proposed
driver model. Intuitively, in path following tasks, drivers would
be less engaged in vehicle control if they are aware of the
controller assistance.













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Fig. 3.
Path-following scenario: a) vehicle lateral displacement; b) driver’s
control effort.
B. Obstacle Avoidance
This part investigates how shared control reacts when the
driver intention is inconsistent with the automation. In partic-
ular, we simulate the case of an obstacle undetected by the
automation. Figs. 4(a) and 4(b) illustrate the vehicle lateral
displacement y and the driver control effort uD, respectively.
In this scenario, the driver wants to avoid the obstacle, whereas
the automation is unaware of it and still sticks to the original
reference path. We see in Fig. 4(a) that a larger λA impairs
the obstacle-avoidance performance, as the driver’s reference
path is more difficult to track. This is because the controller
attempts to tug the vehicle back to its reference path when
the vehicle deviates. Fig. 4(b) shows that the driver needs
more effort to compensate for the controller’s tug with a
larger λA. Therefore, a larger λA is undesired when the driver
intentionally departs from the original reference path. Fig.
4(a) also shows that if the driver does not take the controller
behavior into account as in the conventional model, they
will not require more control effort to compensate for the
controller’s tug. Consequently, the obstacle-avoidance perfor-
mance further deteriorates. This however contradicts to our
intuition as drivers are expected to override the automation’s
“malicious” effort if they are well aware of it. Again, the
proposed driver model is validated through this scenario.
















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Fig. 4.
Obstacle-avoidance scenario: a) vehicle lateral displacement; b)
driver’s control effort.
C. Automatic Weight Switching
The above simulation results show that the control authority
weights should vary with respect to the specific situation. If the
driver intention is consistent with the automation, a larger λA
(smaller λD) will help relieve the driver’s control effort and
enhance path-tracking performance; otherwise a smaller λA
(larger λD) is desirable to allocate more authority to the driver.
A possible strategy to implement this is given by the automatic
weight switching method proposed in Section V. This part
verifies the effectiveness of the automatic weight switching
algorithm (31) in a complex scenario. The complex scenario is
exactly a temporal combination of the path-following scenario
in Section VI-A and the obstacle-avoidance scenario in Section
VI-B. The parameters for automatic weight switching are
chosen as: window length H = 50, threshold δ∗= 0.1 rad,
λ+
D = λ+
A = 0.7, λ−
D = λ−
A = 0.3. To take the driver model
error into account, the estimated driver weighting matrix for
path following is QD = diag(0.028, 0.015) compared with
the actual one QD = diag(0.036, 0.02).
Fig. 5 illustrates the benefits of automatic weight switching
relative to static weighting. Static weighting can hardly avoid
the trade-off between the performances in path following and
obstacle avoidance, whereas weight switching is demonstrated
to be a promising solution to it. To clearly show how the
automatic weight switching works, we zoom in the uD around
the switching point, as shown in Fig. 5(b). It is observed that
the driver’s intention change is detected 1s after it occurs,
which exactly equals to the length of the sliding window. In
fact, the detection time is highly correlated with the selected
δ∗. We can anticipate that a larger δ∗will lead to a longer
detection time. Besides, we can see a drop in uD immediately
after the weights are switched, which is inherently caused by
the discontinuity of λD and λA. This is a drawback of the
weight switching method, which calls for future work.

















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Fig. 5. Complex scenario: a) vehicle lateral displacement; b) driver’s control
effort around the switching point.
VII. CONCLUSION
This paper investigated an indirect driver-automation shared
control method for steer-by-wire vehicles. A weighted summa-
tion approach was used to balance the interests of the driver
and automation, where the control authorities are defined as
the weights assigned to their inputs. An MPC-based method
was used to model the driver steering behavior in indirect
shared control. The resulting algorithm considers the driver
adaptation, assuming that they have identified the transfor-
mation strategy of the controller through motor learning, and
incorporates it into the internal model for predictive control.
To avoid the performance trade-off between different situations
(whether the driver intention is consistent with the automation
or not) stemming from static control authority allocation, this
paper proposed a sliding-window-based approach to monitor
the driver intention in real time, and switches the authority
weights automatically. The simulation result showed that:
1) For path-following tasks, the driver control effort is re-
duced and the vehicle tracking performance is improved,
with higher automation authority and lower driver au-
thority. Compared with conventional driver model, the
modeled driver becomes less engaged in control with
control authority partly undertaken by the automation.
2) For obstacle-avoidance tasks, the driver has to use a
larger control effort to avoid forward collision when
the automation authority is higher. This shows that if
the driver intention is inconsistent with the automation,
indirect shared control will hinder his completion of
the task. Compared with conventional driver model, the
proposed model suggests that the driver manages to avoid
the obstacle more smoothly by using a larger control
effort as he has included the controller impedance into
his internal model.
3) The proposed automatic weight switching method can
effectively address the performance trade-off caused by
static weighting. However, it is observed that the driver
input experiences a sudden drop after the authority
weights are switched.
Limitations of the proposed driver modeling and weight
switching method include the assumption that drivers can
accurately identify the transformation strategy of the controller
through learning. While motor control studies have shown
that humans are generally able to identify complex dynamics
[29], it remains to test how well the vehicle dynamics and
the controller strategy can be identified in practice. Second,
the weight switching approach requires improvement in terms
of robustness for different drivers and to deal with the driver
input discontinuity. This entails the identification of a subject
specific threshold and the development of a smooth weight
shifting process.
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