Relaxing the Inevitable Collision State Concept to Address
Provably Safe Mobile Robot Navigation with Limited Field-of-Views
in Unknown Dynamic Environments
Sara Bouraine
†
and Thierry Fraichard
‡
and Hassen Salhi
?
Abstract— This paper addresses the problem of provably
safe navigation for a mobile robot with a limited field-of-
view placed in a unknown dynamic environment. In such a
situation, absolute motion safety (in the sense that no collision
will ever take place whatever happens in the environment) is
impossible to guarantee in general. It is therefore settled for
a weaker level of motion safety dubbed passive motion safety:
it guarantees that, if a collision is inevitable, the robot will be
at rest. The primary contribution of this paper is a relaxation
of the Inevitable Collision State (ICS) concept called Braking
ICS. A Braking ICS is a state for which, no matter what the
future trajectory of the robot is, it is impossible to stop before
a collision takes place. Braking ICS are designed with a passive
motion safety perspective for robots with a limited field-of-view
in unknown dynamic environments. Braking ICS are formally
defined and a number of important properties are established.
These properties are then used to design a Braking ICS checker,
i.e. an algorithm that checks whether a given state is a Braking
ICS or not. In a companion paper, it is shown how the Braking
ICS checker can be integrated into a reactive navigation scheme
whose passive motion safety is provably guaranteed.
I. INTRODUCTION
Robotics technology is now mature and Autonomous
Ground Vehicles (AGVs) are becoming a reality: consider
the successes of the DARPA Challenges or the VisLab
Intercontinental Autonomous Challenge. They demonstrate
robotics systems traveling significant distances at high speed
in complex and realistic environments. However such sys-
tems remains prone to accidents (see [1]). While moving
(especially at high speed), AGVs (and other robotic systems
as well) can be potentially dangerous should a collision
occur; this is a critical issue if such systems are to transport
or share space with human beings.
Roboticists have long been aware of the motion safety
issue; there is a rich literature on collision avoidance and
collision-free navigation. Nonetheless, motion safety has for
a long time remained a taken-for-granted and ill-defined
notion (see [2]). Demonstrating that a robot avoids collision
on a limited set of experiments is not enough. If autonomous
robots are ever to be deployed among human beings on a
large scale, there is a need to design collision avoidance
and navigation schemes for which motion safety can be
characterized or even guaranteed. The literature review of §II
shows that the Robotics community is displaying a growing
interest in designing such provably safe collision avoidance
and navigation schemes. It also shows that motion safety in
†
CDTA (AL);
‡
INRIA (FR);
?
Blida Univ. (AL).
the real world remains an open problem as soon as the term
real world implies that:
1) The environment features both fixed and moving ob-
jects whose future behaviour is unknown.
2) The robot has only a partial knowledge of its surround-
ings because of its sensory limitations.
The purpose of this paper is precisely to address such a
problem, i.e. that of provably safe motion for a robot with
sensors having a limited field-of-view in an unknown en-
vironment featuring moving objects whose future behaviour
is unknown; it is merely assumed that their speed is upper-
bounded. It can be argued that in such a situation, absolute
motion safety (in the sense that no collision will ever take
place whatever happens in the environment) is impossible to
guarantee in general (unless questionable assumptions con-
cerning the robot and its environment are made). In theory,
absolute motion safety requires a complete knowledge of
the future, up to infinity in some singular situations (see the
motion safety criteria laid down in [2] and the discussion on
motion safety of [3]). To cope with that issue, a practical
stance is taken in this paper; it is settled for a weaker level
of motion safety; in other words, it is guaranteed that, if a
collision takes place, the robot at hand will be at rest. As
per [3], this motion safety level is dubbed passive motion
safety.
The primary contribution of this paper is a relaxation of the
Inevitable Collision State (ICS) concept developed in [4]. An
ICS is a state for which, no matter what the future trajectory
of the robot is, a collision eventually occurs. ICS were
originally defined with an absolute motion safety perspective
which is incompatible with the assumptions made in this
paper, i.e. limited field-of-view and limited knowledge of the
future. The relaxed ICS, henceforth called Braking ICS, are
defined with a passive motion safety perspective. A Braking
ICS is a state for which, no matter what the future trajectory
of the robot is, it is impossible to stop before a collision
takes place. Braking ICS are formally defined and a number
of important properties are established. These properties are
then used to derive a Braking ICS checker, i.e. the passively
safe version of the ICS-checking algorithm presented in [5].
In itself, the central idea behind passive motion safety, i.e.
using braking trajectories, is not new, it has been used before
in different contexts (see Section II). However, to the best
of the authors’knowledge, it is the first time it is given a
formal treatment in as general a context as possible whether
it concerns the robot’s dynamics, its field-of-view, or the
knowledge (or lack thereof) about the future behaviour of
the moving objects. As limited as it may appear, passive
motion safety is interesting for two reasons: (1) it allows to
provide at least one form of motion safety guarantee in such
challenging scenarios. , and (2) if every moving object in the
environment enforces it then no collision will take place at
all.
The paper is organized as follows: a review of the relevant
literature is done in §II. §III discusses motion safety issues
and defines passive motion safety. The adaptation of the
ICS concept to passive motion safety is done in §IV. The
passively safe version of the ICS-checking algorithm is
detailed in §V. Finally, experimental results obtained in
simulation are presented in §VI.
II. RELATED WORKS
As mentioned above, the Robotics literature is teeming
with works concerned with collision avoidance but most
of them do not offer an explicit formulation of the safety
guarantees they provide or the conditions under which they
must operate (see [2]).
The earliest relevant works addressed the so-called “As-
teroid Avoidance Problem” (wherein objects traveling at a
constant linear velocity must be avoided): in the 3D case,
[6] shows that collision avoidance is always possible if the
robot’s velocity is greater than the asteroids’ velocities and
if the robot is not initially in the “shadow” of an asteroid.
In the 2D case, [7] shows that collision avoidance is always
possible iff the asteroids appear beyond a “threat horizon”,
i.e. a distance which is a function of the number, size and
velocity of the asteroids. Likewise, [8] shows that, for a robot
operating in a planar environment with arbitrarily moving
objects, collision-free motion is guaranteed iff the maximum
velocity of the robot is a multiple of the maximum velocity of
the objects. Such results are very interesting. Unfortunately,
they rely on assumptions that rarely occur in the real world.
A related family of research works are those seeking to
coordinate the motion of a set of robots. Different distributed
coordination schemes have been proposed for which collision
avoidance is guaranteed, e.g. [9], [10]. However, this guaran-
tee is lost if the environment contains uncontrolled moving
objects.
General motion safety issues have been studied thanks
to the Inevitable Collision States (ICS) concept developed
in [4]. An ICS is a state for which, no matter what the future
trajectory of the robot is, a collision eventually occurs. ICS
provides insight into the complexity of guaranteeing motion
safety since it shows that it requires to reason about the
future evolution of the environment and to do so with an
appropriate lookahead
1
that can possibly be infinite. Such
conditions being next to impossible to obtain in the real
world plus the fact that ICS characterization is very complex
has led a number of authors to consider relaxations of ICS
such as:
1
I.e. how far into the future the reasoning is done.
• ICS approximation, e.g. [11], [12]: such approximations
being not conservative, the motion safety guarantee is
lost.
• τ-Safety, e.g. [13], [14]: the robot is guaranteed to
remain in states where it is safe for a given duration
(hopefully sufficient to compute an updated safe trajec-
tory. . . ).
• Evasive trajectories, e.g. [15], [16]: they guarantee that
the robot can only be in states where it is possible to
execute an evasive trajectory, e.g. a braking manoeuvre
for a car or a circling manoeuvre for a plane.
Recently, authors have proposed probabilistic versions of
the ICS concept, e.g. [17], [18], so as to better capture
the uncertainty that prevails in real world situations, in
particular the uncertainty concerning the future behaviour of
the moving objects. These approaches are interesting but they
offer no strict motion safety guarantees since probabilistic
models are used.
There are a few research works taking into account sensory
limitations. For instance, the occlusion problem, i.e. the
existence of regions that are hidden by other objects, is
addressed in a coarse manner in [19] and in a more principled
manner in [20]. The occlusion and the limited field-of-view
problems are addressed in [4] and [21]. [4] addresses the
case of a mobile robot moving in a static environment; its
approach is general and ICS-based. While [21] considers
dynamic environments, it does so primarily with a path-
velocity decomposition perspective. The contribution of this
paper is the extension of [4] to the dynamic environment
case.
III. SAFETY ISSUES
A. Outline of the Problem
(a) (b)
Fig. 1: (a) Robot with a limited field-of-view in an unknown
environment with fixed and moving objects.. (b) Its corre-
sponding field-of-view ∂ FOV. FOV is the grey area; ∂ FOV
and FOV
c
have two connected components..
As mentioned in §I, this paper addresses the problem of
provably safe motion for a mobile robot with sensors having
a limited field-of-view in an unknown environment featuring
fixed and moving objects with upper-bounded velocity and
unknown future behaviour. Let A denote the mobile robot at
hand. It operates in a 2D workspace W ; a state of A is de-
noted by s with s ∈ S , the state space of A . Assuming that
A is equipped with range sensors such as laser telemeters
or range cameras, it can only perceive a subset of W ; this
subset is A ’s field-of-view; its shape is arbitrary; it depends
on the current surroundings of A and the maximum range
of its sensors. It is henceforth denoted FOV. Accordingly,
W is partitioned in three subsets: (1) FOV, (2) FOV
c
, the
part which is unseen (FOV
c
= W \ cl(FOV)) and (3) ∂ FOV,
the boundary between the two. Both FOV and FOV
c
are
open sets. It seems reasonable to assume that A is “looking
around itself”; in other words that A (s) ⊂ FOV where A (s)
denotes the region of W occupied by A when it is in s. To
account for the existence of 3D range sensors, e.g. Velodyne
LIDAR or PrimeSensor range camera, FOV can contain
“holes” representing objects entirely perceived by the sensory
system of A . Accordingly, FOV,∂ FOV and FOV
c
are not
necessarily singly connected (see Fig. 1b). FOV represents
the region of W which is free of objects at the sensing time.
This generic field-of-view model can further be enriched
if the sensors of A can differentiate the fixed vs the moving
objects. In that case, ∂ FOV can be partitioned into three parts
respectively corresponding to fixed objects, moving objects
and so-called “unseen” objects, i.e. the sensing limits and
the occluding lines:
∂ FOV = ∂ FOV
f
∪ ∂ FOV
m
∪ ∂ FOV
u
(1)
When the sensors of A cannot differentiate between fixed
and moving objects, ∂ FOV = ∂ FOV
u
.
B. Modeling the Future
Fig. 2: Models of the future (from left to right): fixed object
(1); moving object with constant velocity (known future
motion) (2); conservative models for a moving point with
unknown future motion and upper-bounded velocity (3), and
upper-bounded acceleration (4).
The ICS concept brings to light two things: the first one
is that there is more to motion safety than the simple fact
that A ’s trajectory be collision-free; it must be ICS-free, i.e.
A must always be in a state for which an evasive trajectory
is available. The second one is that motion safety is always
defined wrt the model of the future that is used. When dealing
with objects whose future behaviour is unknown, what model
of the future should be used? The answer is to be conser-
vative: one must consider all possible future motions for
the object at hand. Consider the case of a point object with
upper-bounded velocity whose future behaviour is unknown.
Given the initial position of the object, the region of the
workspace that is possibly not collision-free is modeled by a
disc that grows over time with a growth rate corresponding
to the maximal velocity of the object [22]. In space×time, it
is represented as an inverted cone (see Fig. 2). Such a cone
is the reachable set [23, Chap. 14] of a point object whose
dynamics is characterized by infinite acceleration and upper-
bounded velocity capabilities. In general, reachable sets can
be used to represent all possible future motions for object
with arbitrary dynamics, e.g. an object with upper-bounded
velocity and acceleration (see Fig. 2).
Fig. 3: Conservative model of the future (partially repre-
sented for visualisation purposes) for the scenario of Fig. 1a.
Now, in a situation such as the one depicted in Fig. 1b,
how does one take into account the unseen parts of W that
belongs to ∂ FOV
u
or FOV
c
? Walking in the footsteps of [4]
or [21], the answer is once again to be conservative and to
treat every point of ∂ FOV
u
or FOV
c
as a potential moving
object with unknown future behaviour. In conclusion, the
space×time model of the future for A can be defined as
follows for the different components of A ’s field-of-view
(see Fig. 3):
• ∂ FOV
u
∪ FOV
C
(the unseen objects): every point in this
set is modeled as a disc that grows as time passes (i.e.
a cone in space×time).
• ∂ FOV
f
(the fixed objects): every point in this set
remains constant over time (i.e. a vertical line in
space×time).
• ∂ FOV
m
(the moving objects): if the information about
their future behaviour is available and reliable, every
point in this set is modelled accordingly (i.e. a curve in
space×time), otherwise it is treated as an unseen object
and modeled as a growing disc.
This of course is the case when the sensors of A can
differentiate between fixed and moving objects. If it is not
the case then every point in ∂ FOV is modeled as a disc that
grows as time passes (i.e. a cone in space×time). Within
such a model of the future, it is worth noting that the region
of W which is free of objects at the sensing time, i.e. FOV,
gradually shrinks as time passes and eventually vanishes (see
Fig. 3). Henceforth, FOV(t) denotes the region of W which
is free of objects at time t in the conservative model of the
future. Likewise, ∂ FOV(t) denotes its boundary.
C. Absolute vs. Passive Motion Safety
The ICS concept laid down in [4] guarantees absolute
motion safety in the sense that, for a state not to be an
ICS, there must exist a collision-free trajectory of infinite
duration. Now, an object with unknown future behaviour
is a challenge. If it is modeled conservatively as above
then, at some point in the future, the whole workspace is
entirely covered by the growing disc representing it. At that
moment, the whole state space of the robot is forbidden and
it becomes impossible to find a collision-free trajectory of
infinite duration. This is a situation where the ICS concept
becomes ineffective. In the authors’opinion, the only answer
to this challenge is to settle for a weaker level of motion
safety; the rationale being: better guarantee something than
guarantee nothing. The choice here is to guarantee that, if a
collision takes place, the robot will be at rest. This motion
safety level, dubbed passive motion safety in [3], seems a
reasonable choice given the harsh constraints imposed by a
limited field-of-view. It yields the following definition:
Def. 1: given a model of the future workspace evolution, a
passively safe or p-safe state for A is a state s such that there
exists one braking trajectory starting at s which is collision-
free until A has stopped.
IV. FROM ICS TO BRAKING ICS
Using braking trajectories in order to evaluate the safety of
a given state has been done before, e.g. [15], [16], [24]–[26].
The focus in this paper is to do it in the formal framework
of the ICS concept. The concept of Braking ICS (ICS
b
) is
first derived from the original ICS concept. It is then used
to design ICS
b
-CHECK, i.e. the corresponding variant of the
ICS checking algorithm proposed in [5]. ICS
b
-CHECK is
detailed in §V.
A. Notations
The dynamics of the robot A is generally described by
differential equations of the form:
˙s = f (s,u) subject to g(s, ˙s) ≤ 0 (2)
where s ∈ S is the state of A , ˙s its time derivative and u ∈
U a control. S and U respectively denote the state space
and the control space of A . Let A (s) denote the closed
subset of the workspace W occupied by A when it is in s.
Let ˜u : [0,t
f
] −→ U denote a control trajectory, i.e. a time-
sequence of controls, t
f
is the duration of ˜u. The set of all
possible control trajectories is denoted
˜
U . Starting from an
initial state s
0
at time 0, a state trajectory ˜s, i.e. a time-
sequence of states, is derived from a control trajectory ˜u by
integrating (2); ˜s(s
0
, ˜u,t) denotes the state reached at time t.
A control trajectory ˜u
b
∈
˜
U such that ˜s
b
(s
0
, ˜u
b
,t
b
) is a
state where A comes to a halt (and remains so) is a braking
trajectory for s
0
and t
b
is its braking time. The set of all
possible braking trajectories for s
0
is denoted
˜
U
s
0
b
.
In a situation such as the one depicted in Fig. 1b, the
open subset FOV is the free part of the workspace while
∂ FOV
f
,∂ FOV
m
,∂ FOV
u
and FOV
c
represent objects (seen
and unseen). Let B
i
denote the space×time model of the
future evolution of the corresponding object (according to the
modeling rules defined in §III-B). At time 0, i.e. the sensing
time, B
i
(0) corresponds to a subset of ∂ FOV
f
, ∂ FOV
m
,
∂ FOV
u
or FOV
c
. B
i
(t) denotes the subset of W occupied
by B
i
at a particular time t in the conservative model of
the future. It is assumed that each B
i
(t) is a closed subset
of W and that the total number of objects is n. Likewise
B
i
([t
1
,t
2
]) denote the space×time region occupied by the
object during the interval [t
1
,t
2
]. To ease notations, it is
assumed that B
i
≡ B
i
([0,∞)).
B. Braking ICS Definition
A Braking ICS (ICS
b
) is informally defined as a state for
which no matter what the future braking trajectory followed
by A is, a collision occurs before A is at rest. Hence the
following formal definition:
Def. 2 (Braking ICS): s is a ICS
b
iff ∀ ˜u
b
∈
˜
U
s
b
,∃t ∈
[0,t
b
[, ˜s(s, ˜u
b
,t) is a collision state at time t.
It is worth noting that when A is in a state s where A is
at rest,
˜
U
s
b
reduces to ˜u
◦
b
that denotes the braking trajectory
where a null control is applied to A . Accordingly, s is always
p-safe (even if A (s) is in collision).
It is then possible to define the set of ICS
b
yielding a
collision with a particular object B
i
:
ICS
b
(B
i
) = {s ∈ S |∀ ˜u
b
∈
˜
U
s
b
,∃t ∈ [0,t
b
[,
A ( ˜s(s, ˜u
b
,t)) ∩ B
i
(t) 6= /0} (3)
Likewise, the ICS
b
set yielding a collision with B
i
for a
given trajectory ˜u
b
(or a given set of trajectories E ⊆
˜
U
b
) is
defined as:
ICS
b
(B
i
, ˜u
b
) = {s ∈ S |∃t ∈ [0,t
b
[,
A ( ˜s(s, ˜u
b
,t)) ∩ B
i
(t) 6= /0} (4)
ICS
b
(B
i
,E ) =
\
˜u
b
∈E
ICS
b
(B
i
, ˜u
b
) (5)
C. Braking ICS Properties
The first two ICS
b
properties that can be shown are the
equivalent of two key ICS properties established in [4] and
seminal in the design of an ICS checking algorithm. Let
B =
S
n
1
B
i
. The first property shows that ICS
b
(B) can be
derived from ICS
b
(B
i
, ˜u
b
) for every object B
i
and every
possible braking trajectory ˜u
b
.
Property 1 (ICS
b
Characterization):
ICS
b
(B) =
\
˜u
b
∈
˜
U
b
n
[
i=1
ICS
b
(B
i
, ˜u
b
)
Proof: The two-stage proof of property 1 is done in a
straightforward manner as in [4]. It is first established that:
s ∈ ICS
b
(B) ⇔ s ∈
\
˜u
b
∈
˜
U
s
b
ICS
b
(B, ˜u
b
)
and then that:
s ∈ ICS
b
(
n
[
i=1
B
i
, ˜u
b
) ⇔ s ∈
n
[
i=1
ICS
b
(B
i
, ˜u
b
)
Combining the two properties above yields property 1.
The next property permits to compute a conservative
approximation of ICS
b
(B) by using a subset only of the
whole set of possible braking trajectories.
Property 2 (ICS
b
Approximation):
ICS
b
(B) ⊆ ICS
b
(B,E )
with E ⊆
˜
U
b
, a subset of the whole set of possible braking
trajectories.
Proof:
ICS
b
(B) =
\
˜u
b
∈E
ICS
b
(B, ˜u
b
) ∩
\
˜u
b
∈
˜
U
b
\E
ICS
b
(B, ˜u
b
)
⊆
\
˜u
b
∈E
ICS
b
(B, ˜u
b
)
One distinctive feature of the ICS concept is that trajecto-
ries of infinite duration are checked for collision, i.e. it has an
infinite lookahead (it is this infinite lookahead that guarantees
safety). While the ICS
b
concept also considers trajectories
˜u
b
of infinite duration, collision checking is limited to the
time interval [0,t
b
[ where t
b
is the braking time of ˜u
b
. For
an arbitrary subset E of the whole set of possible braking
trajectories, a finite lookahead T
h
exists:
T
h
= max
˜u
b
∈E
{t
b
} (6)
T
h
is a valid lookahead in the sense that, in order to
compute ICS
b
(B,E ), it suffices to consider the model of
the future up to time T
h
. This is established by the following
property:
Property 3 (ICS
b
Lookahead):
ICS
b
(B,E ) = ICS
b
(B([0,T
h
[),E )
Proof: Property 3 stems from the very definition of a
Braking ICS which, for a given braking manoeuvre ˜u
b
, is
only concerned with collisions taking place before t
b
< T
h
.
Finally, recall from §III-A that, for the case of a robot
with a limited field-of-view, B comprises ∂ FOV and FOV
c
,
i.e. the unseen part of W . From a motion safety perspective,
the next property is very important since it establishes that
FOV
c
can be ignored in the computation of ICS
b
(B). In
other words, considering ∂ FOV suffices to guarantee motion
safety.
Property 4 (Field-of-View Boundary):
ICS
b
(B) = ICS
b
(∂ FOV ∪ FOV
c
) = ICS
b
(∂ FOV)
Proof: The equality between ICS
b
(B) and
ICS
b
(∂ FOV) is done is two stages. Let s denote a
collision-free state whose corresponding position is located
inside FOV and such that s ∈ ICS
b
(∂ FOV). As per
Definition 2, it stems that:
∀B
i
,∀B
j
, ICS
b
(B
i
) ⊆ ICS
b
(B
i
∪ B
j
)
Accordingly:
s ∈ ICS
b
(∂ FOV) ⇒ s ∈ ICS
b
(∂ FOV ∪ FOV
c
).
It is assumed now that s ∈ ICS
b
(FOV
c
), it means that
∀ ˜u
b
∈
˜
U
s
b
,∃t ∈ [0,t
b
[ such that ˜s(s, ˜u
b
,t) is in collision with
a point of FOV
c
(t). Since s is located inside FOV, it takes
a simple topological argument to realize that ∃t
0
< t such
that ˜s(s, ˜u
b
,t
0
) is in collision with a point of ∂ FOV(t
0
).
Accordingly s ∈ ICS
b
(∂ FOV) and the following holds:
s ∈ ICS
b
(∂ FOV ∪ FOV
c
) ⇒ s ∈ ICS
b
(∂ FOV).
In other words, it suffices to consider ∂ FOV in order to
compute ICS
b
(B).
V. BRAKING ICS CHECKING
ICS
b
-CHECK is an algorithm that checks whether a given
state is a Braking ICS or not. It is the passively safe
version of the ICS checking algorithm (called ICS-CHECK)
presented in [5]. The passively safe version of ICS-CHECK
can be designed because Properties 1 and 2 are verified for
Braking ICS. The steps involved in checking whether a given
state s
c
is a ICS
b
are given in Algorithm 1. Besides the state
to be checked, the algorithm takes as input the model of
the environment and the conservative space× model of the
future (see §IV-A). Steps 2, 3 and 4 are the direct translation
of Property 1.
Algorithm 1: General ICS
b
Checking Algorithm.
Input: s
c
, the state to be checked; B
i
,i = 1...n.
Output: Boolean value.
Select E ⊂
˜
U
s
c
b
, a set of braking trajectories for s
c
;1
Compute ICS
b
(B
i
, ˜u
b
) for every B
i
and every ˜u
b
∈ E ;2
Compute ICS
b
(B, ˜u
b
) =
S
n
i=1
ICS
b
(B
i
, ˜u
b
) for every3
˜u
b
∈ E ;
Compute ICS
b
(B,E ) =
T
˜u
b
∈E
ICS
b
(B, ˜u
b
);
4
if s
c
∈ ICS
b
(B,E ) then5
return TRUE;6
else7
return FALSE;8
end9
As in [5], when A is planar, it becomes possible to design
ICS
b
-CHECK, i.e. an efficient version of Algorithm 1. In
that case, a state s of A can be rewritten s = (x,y,
ˆ
z) with
(x,y) the workspace coordinates of A ’s reference point,
and
ˆ
z the rest of the state parameters. The primary design
principle behind ICS
b
-CHECK is to compute the ICS
b
set
corresponding to a 2D slice of the state space S of A
(instead of attempting to perform computation in the fully-
dimensioned state space), and then to check if s
c
belongs to
this set. Assuming the state to be checked is s
c
= (x
c
,y
c
,
ˆ
z
c
),
the 2D slice considered is the
ˆ
z
c
-slice and it is possible to
define the ICS
b
set of the
ˆ
z
c
-slice considered that yields a
collision with B
i
at a particular time t ∈ [0,t
b
[ for the braking
trajectory ˜u
b
:
ICS
b
ˆ
z
c
(B
i
, ˜u
b
,t) = {s ∈
ˆ
z
c
-slice| (7)
A ( ˜s(s, ˜u
b
,t)) ∩ B
i
(t) 6= /0}
Likewise:
ICS
b
ˆ
z
c
(B
i
, ˜u
b
) =
[
t∈[0,t
b
[
ICS
b
ˆ
z
c
(B
i
, ˜u
b
,t) (8)
