Inevitable Collision States
A Step Towards Safer Robots?
Thierry Fraichard & Hajime Asama
Advanced Robotics, 18(10):1001-1024, 2004
Abstract
An inevitable collision state for a rob otic system can be defined as a state for which, no matter
what the future trajectory followed by the system is, a collision with an obstacle eventually occurs. An
inevitable collision state takes into account the dynamics of both the system and the obstacles, fixed
or moving. The main contribution of this paper is to lay down and explore this novel concept (and
the companion concept of inevitable collision obstacle). Formal definitions of the inevitable collision
states and obstacles are given. Properties fundamental for their characterisation are established. This
concept is very general and can be useful both for navigation and motion planning purposes (for its
own safety, a robotic system should never find itself in an inevitable collision state). To illustrate
the interest of this concept, it is applied to a problem of safe motion planning for a robotic system
subject to sensing constr aints in a partially known environment (ie that may contain unexpected
obstacles). In safe motion planning, the issue is to compute motions for which it is guaranteed that,
no matter what happens at execution time, the robotic system never finds itself in a situation where
there is no way for it to avoid collision with an unexpected obstacle.
Keywords — Safety, navigation, motion planning, sensing constraints, collision avoidance.
1 Introduction
The configuration
1
space of a robotic system is the appropriate framework to address path planning
problems where the focus is on the geometric asp ects of motion planning (no collision between the
system and the fixed obstacles of the workspace) [1, 2]. The state
2
space, on the other hand, is more
appropriate when it comes to address trajectory planning problems where the dynamics of the system
is taken into account [4, 5]. Similarly, the time-state space is appropriate to address trajectory planning
problems involving moving obstacles [6, 7, 8].
In the configuration space, the notion of forbidden or collision configurations, ie configurations yield-
ing a collision, is well-known and so is the notion of configuration obstacles, ie the set of configurations
yielding a collision between the system and a particular obstacle [1]. Transposing these notions in the
state space, it is straightforward to define collision states and state obstacles (idem in the time-state
space).
However, be it in state space or time-state space, it takes a simple example such as the one depicted
in Fig. 1 to illustrate the interest of extending these notions so as to take into account the dynamics of
the system by introducing the concept of inevitable collision states.
Consider Fig. 1, let P be a point mass that can only move to the right with a variable speed. A state
of P is characterised by its position (x, y) and its speed v. If the workspace W features a wall, the states
whose position corresponds to the wall are obviously collision states. On the other hand, assuming that
it takes P a certain distance d(v) to slow down and stop, the states corresponding to the wall and the
states located at a distance less than d(v) left of the wall are such that, when P is in such a state, no
matter what it does in the future, a collision will occur. These states are inevitable collision states for
1
The configuration of a robotic system is a set of independent variables that uniquely determines the position and
orientation of every point of the system [1].
2
The state of a robotic system is a set of variables such that the knowledge of these variables at time t
0
together with
the knowledge of the controls applied to the system for t ≥ t
0
completely determines the behaviour of the system for any
time t ≥ t
0
[3]
1
000
000
000
000
000
000
000
000
111
111
111
111
111
111
111
111
P
Wall
v
d(v)
Collision states
x
W
y
W
W
Inevitable collision states
Figure 1: Collision states vs inevitable collision states.
00000
00000
00000
00000
00000
11111
11111
11111
11111
11111
x
S
y
S
v
S
S
d(v)
v
Collision states
Inevitable collision states
Figure 2: Full representation in the xyv state space S of P of the inevitable collision states corresponding
to the situation depicted in Fig. 1.
P . Clearly, for P ’s own safety, when it is moving at speed v, it should never be in one of these inevitable
collision states. The size of the inevitable collision states region, ie the grey region located to the left
of the wall, depends on the distance d(v) which in turns depends on the current speed of P . Assuming
that d(v) varies linearly with v, the complete set of inevitable collision states is a prism embedded in the
state space S of P (Fig. 2).
In general, an inevitable collision state for a given robotic system can be defined as a state for which,
no matter what the future trajectory followed by the system is, a collision eventually occurs with an
obstacle of the environment. Similarly, it is possible to define an inevitable collision obstacle as the set
of inevitable collision states yielding a collision with a particular obstacle.
Except for a brief mention of it in [9], this concept does not seem to have been considered before
by the Robotics community. A close idea can be found in [10]: it introduces braking prisms, ie subsets
of the configuration space associated with a given state of a robotic system and known to contain the
braking trajectory corresponding to a given braking policy (picked up from a restricted set of braking
policies). A configuration without any braking prism included in the free configuration space must be
avoided because it would yield a collision no matter which braking policy is used. In a similar vein, [11]
describes a trajectory planning scheme for a robotic system with a limited field of view. It characterises
robust states as states for which the robotic system can safely stop simply by braking even when placed
in a environment with unknown moving obstacles (basically, the field of view is shrunken according to the
2
moving obstacles’ maximum possible speed). Also, inevitable collision states are to some extent related
to the danger zone concept that can be found in the Air Traffic Control literature [12, 13]. Outside such
danger zones, evasive manoeuvres are provably safe.
All these approaches share the same principle: one or several evasive manoeuvres are defined, and
every state for which no such evasive manoeuvres (usually braking manoeuvres) are collision-free is
labelled as being dangerous and is avoided. In a sense, the inevitable collision state concept encompasses
all these approaches. It is general, it takes into account the dynamics of both the robotic system and
the obstacles (fixed or moving, known or unknown), and it considers manoeuvres other than braking
manoeuvres (as a matter of fact, it considers all possible manoeuvres). We therefore believe that it is a
concept worth exploring and that it can be very useful be it for motion planning or navigation purposes.
Consider navigation first (by navigation, we basically mean the problem of determining the elementary
motion that the robotic system should perform during the next time-step). The primary concern of
navigation is to ensure the safety of the robotic system. In a environment featuring moving obstacles,
this safety concern is critical and it is important to take into account both the dynamics of the robotic
system and the future behaviour of the moving obstacles. A number of research papers have addressed
these issues recently [10, 14, 15, 16, 17, 18, 19]. In this framework, the interest of the inevitable collision
state concept is obvious. By design, inevitable collision states integrate the dynamics of both the robotic
system and the obstacles, fixed or moving.
When it comes to motion planning, the inevitable collision state concept is also useful. Consider the
problem of planning motions for a robotic system moving in a partially known environment. The system
is s ubject to sensing constraints (a limited field of view), and it moves in an environment containing
obstacles, some of them are known beforehand while others are not (imagine a surveillance robot, it has
a map of the building it must patrol but it does not know a priori the position of the small furniture or
if people are moving around). Based on the a priori information available, a nominal trajectory for the
robotic system can be computed. However, what if, at execution time, the robotic system finds itself
in a situation where an unknown obstacle is detected so late that avoiding it is impossible. The issue
here is to compute safe motions, ie motions for which it is guaranteed that, no matter what happens
at execution time, the robotic system never finds itself in a situation where there is no way for it to
avoid collision with an unexpected obstacle. This issue is related to the dependency that exists between
motion planning and navigation, dependency which is usually ignored by motion planning systems (with
the exception of [11]). We show on an example how this issue can be addressed using the inevitable
collision state concept and how safe motions (in the sense given above) can be planned.
The main contribution of this paper is to lay down and explore the concept of inevitable collision
states. To begin with, a formal definition of what inevitable collision states and inevitable collision
obstacles are is given. Properties that are fundamental for their characterisation are established (§3). To
illustrate the use of these properties, a basic example is studied (§4). Finally, an example of application
of the inevitable collision state concept to safe motion planning is given (§5).
2 Notations and Preliminary Definitions
Before defining the inevitable collision states and obstacles, useful definitions and notations are intro-
duced. Let A denote a robotic system. It is assumed that its dynamics can be described by a differential
equation such as: ˙s = f(s, u) 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 φ ∈ Φ denote a control input,
ie a time-sequence of controls. φ represents a trajectory for A. Starting from an initial state s
0
(at time
0) and under the action of a control input φ, the state of A at time t is denoted by φ(s
0
, t).
Given a control input φ and a state s
0
(at time 0), a state s is reachable from s
0
by φ iff ∃t, φ(s
0
, t) = s.
Let R(s
0
, φ) denote the set of states reachable from s
0
by φ. Likewise, R(s
0
) denotes the set of states s
reachable from s
0
, ie such that ∃φ, s ∈ R(s
0
, φ):
R(s
0
, φ) = {s ∈ S|∃t, φ(s
0
, t) = s}
R(s
0
) = {s ∈ S|∃φ, s ∈ R(s
0
, φ)}
Introducing φ
−1
(s
0
, t) to denote the state s such that φ(s, t) = s
0
, it is possible to define R
−1
(s
0
)
(resp. R
−1
(s
0
, φ)), as the set of states from which it is possible to reach s
0
(resp. to reach s
0
by φ):
R
−1
(s
0
, φ) = {s ∈ S|∃t, φ(s, t) = s
0
⇔ φ
−1
(s
0
, t) = s}
3
R
−1
(s
0
) = {s ∈ S|∃φ, s ∈ R
−1
(s
0
, φ)}
Let W denote the workspace of A (W = IR
2
or IR
3
), it contains a set of obstacles that are de-
fined as closed subsets of W. Let WB denote such an obstacle. When WB is moving, WB(t) rep-
resents the subset of W occupied by WB at time t. W hen WB is fixed, the time index is omitted:
∀t, WB(t) = WB(0) = WB. In the configuration space, every obstacle has an image called configu-
ration obstacle which is the set of configurations yielding a collision between the robotic system and
the obstacle considered [1]. Likewise, every obstacle has an image in the state space: the set of states
yielding a collision between the robotic system and an obstacle WB(t) determines the state obstacle of
WB(t) which is denoted B(t). B(t) = {s ∈ S|A(s) ∩ WB(t) 6= ∅}, where A(s) denotes the closed subset
of W occupied by A in state s. Once again, when WB is fixed, the time index is omitted. A state s is
a collision state at time t iff ∃B, s ∈ B(t). In this case, s is a collision state at time t with B.
The rest of the article places itself in the s tate space framework. For the sake of simplicity, state
obstacles are called obstacles only and the time index is indicated only when necessary.
3 Inevitable Collision States and Obstacles
Based on the definitions and notations introduced in the previous section, the inevitable collision states
and the inevitable collision obstacles are formally defined.
Def. 1 (Inevitable Collision State) Given a control input φ, a state s is an inevitable collision
state for φ iff ∃t such that φ(s, t) is a collision state at time t. Now, a state s is an inevitable
collision state iff ∀φ, ∃t such that φ(s, t) is a collision state at time t. Likewise, s is an inevitable
collision state with B for φ iff ∃t such that φ(s, t) is a collision state at time t with B. Finally, s is
an inevitable collision state with B iff ∀φ, ∃t such that φ(s, t) is a collision state at time t with B.
Def. 2 (Inevitable Collision Obstacle) Given an obstacle B and a control input φ, ICO(B, φ), the
inevitable collision obstacle of B for φ is defined as:
ICO(B, φ) = {s ∈ S|s is an inevitable collision state with B for φ}
= {s ∈ S|∃t, φ(s, t) is a collision state at time t with B}
= {s ∈ S|∃t, φ(s, t) ∈ B(t)}
Now, ICO(B), the inevitable collision obstacle of B, is defined as:
ICO(B) = {s ∈ S|s is an inevitable collision state with B}
= {s ∈ S|∀φ, ∃t, φ(s, t) is a collision state at time t with B}
= {s ∈ S|∀φ, ∃t, φ(s, t) ∈ B(t)}
Based upon the two definitions above, the following property can be established. It shows that
ICO(B) can be derived from the ICO(B, φ) for every possible control input φ.
Property 1 (Control Inputs Intersection)
ICO(B) =
\
Φ
ICO(B, φ)
Proof:
s ∈ ICO(B) ⇔ ∀φ, ∃t, φ(s, t) is a collision state at time t with B
⇔ ∀φ, s ∈ ICO(B, φ)
⇔ s ∈
\
Φ
ICO(B, φ)
4
Assuming now that B is the union of a set of obstacles, B =
S
i
B
i
, the following property can be
established. It shows that ICO(B, φ) can be der ived from the ICO(B
i
, φ) for every subset B
i
.
Property 2 (Obstacles Union)
ICO(
[
i
B
i
, φ) =
[
i
ICO(B
i
, φ)
Proof:
s ∈ ICO(
S
i
B
i
, φ) ⇔ ∃t, φ(s, t) is a collision state at time t with
S
i
B
i
⇔ ∃B
i
, ∃t, φ(s, t) is a collision state at time t with B
i
⇔ ∃B
i
, s ∈ ICO(B
i
, φ)
⇔ s ∈
S
i
ICO(B
i
, φ)
Combining the two properties above, the following property is derived. It is the property that permits
the formal characterisation of the inevitable collision obstacles for a given robotic system.
Property 3 (ICO Characterisation) Let B =
S
i
B
i
,
ICO(B) =
\
Φ
[
i
ICO(B
i
, φ)
Proof:
ICO(B)
1
=
\
Φ
ICO(B, φ)
2
=
\
Φ
[
i
ICO(B
i
, φ)
Consider property 1 (and property 3), it establishes that ICO(B) can be derived from the ICO(B, φ)
for every possible control input φ. In general, there is an infinite number of control inputs which leaves
little hope of being actually able to compute ICO(B). Fortunately, it is possible to establish a property
which is of a vital practical value since it shows how to compute a conservative approximation of ICO(B)
by using a subset only of the whole set of possible control inputs.
Property 4 (ICO Approximation) Let I denote a subset of the set of possible control inputs Φ,
ICO(B) ⊂
\
I
ICO(B, φ)
Proof:
ICO(B)
1
=
\
I∪
I
ICO(B, φ)
=
\
I
ICO(B, φ) ∩
\
I
ICO(B, φ)
⊆
\
I
ICO(B, φ)
The interes t of these properties to characterise inevitable collision obstacles appears in the next
sections.
5
