An empirical evidence of Braitenberg vehicle 2b behaving as a ...

An empirical evidence of Braitenberg vehicle 2b
behaving as a billiard ball.
I. Rañó
Computer Sciences and Systems Engineering Dept.
University of Zaragoza,
Spain.
Abstract. Braitenberg vehicles have been used for decades on an empirical
basis to implement different tasks. Grounded on a mathematical
model of vehicle 2b, this paper presents empirical evidence that some
theoretical results from billiard ball dynamics can be used to analyse the
behaviour of the vehicle. The possibility to use such vehicles to cover a
region is presented as a consequence of the its behaviour.
1 Introduction
Braitenberg vehicles have been used for decades in robotics on an empirical
and intuitive basis. Each vehicle displays a different behaviour according to a
thought experiment presented in [3]. Complexity on their behaviour emerges
from the interaction of the vehicle with an external stimulus, not from the internal
mechanisms that generate the behaviour. They were used as a justification
of the simplicity of the behaviour based approach to robotics [4]. Braitenberg
vehicles were also successfully used in earlier works of artificial life [14] [17], but
they can be used to model animal behaviour as well.
Different kinds of Braitenberg vehicles have been used to provide robots with
several abilities like obstacle avoidance [2], local navigation [18], chemical source
seeking [6] [9], and even complex tasks like football playing [10], yet always
based on empirical evidence. However, since the original work is both qualitative
and very intuitive there has been no effort in trying to build a quantitative
framework for a further development. In fact, besides purely empirical implementations,
Braitenberg vehicles are used as test-bed for evolutive behaviour
generation [15] [7], where usually neural networks act as controllers of the vehicles
[8]. Neural networks are selected as controllers so they can be evolutively
adjusted to perform well, but also because of their properties to deal with noise
and approximate any function with no need of design knowledge.
As we will see later, one of the applications of Braitenberg 2b vehicles is
coverage, a well studied problem on robotics with many applications. In [5] a
classification of coverage techniques as heuristic or complete is performed. Complete
techniques treat coverage as a multiple sequential planning problem, the
execution of a sequence of planned trajectories, which turns it into a time expensive
and complexity growing problem. On the other hand, heuristic techniques

are based on sequencing behaviours or random motions, making hard to treat
them formally or to measure its performance. The use of Braitenberg vehicle 2b
for coverage could be included in the heuristic category, though it can be treated
theoretically under some circumstances. This paper will show that a single mechanism,
the Braitenberg vehicle 2b, can be used to generate covering trajectories
in simple scenarios. Although the simplicity of the mechanism, the resulting behaviour
is difficult to analyse, in the most general case, since it is modelled as a
nonlinear differential equation with no equilibrium point. However, we will see
experimentally, that mathematical theory of billiard can be used to explain its
behaviour under some circumstances. This allows the application of theoretical
results to the behaviour of Braitenberg vehicles.
The rest of the paper is organised as follows: Section 2 reviews Braitenberg
vehicle 2b and presents some results of its behaviour under the influence of a
bounded stimulus with a single point maximum. When the stimulus maxima can
be approximated by a region a new behaviour appears as presented in Section 3.
Conclusions and further work lines are drawn in Section 4. Each section includes
its corresponding simulated results.
2 Braitenberg vehicle 2b in bounded stimuli with
point-like maximum
The Braitenberg vehicle 2b consists on a dual-drive wheeled vehicle with two
point-like frontal sensors arranged symmetrically. The connections between the
sensors and the motors are shown in Figure 1. Each sensor is linked to the
wheel on the opposite side in an increasing way. The sensors capture some scalar
stimulus S(x) from the environment D. The stronger the stimulus is perceived
on the sensor, the faster the opposite wheel spins, that is what the ‘+’ sign on
the figure means. The overall effect is making the robot turn in the direction
of the stimulus. Moreover, as the stimulus intensity grows, the linear velocity of
the vehicle increases, since both wheels turn faster. Therefore, the vehicle heads
the stimulus while increases its velocity. If the stimulus is generated by a point
source, the vehicle will eventually reach the source with a maximal speed, this is
the reason why the behaviour was originally named aggression. Before drawing
any conclusion of the behaviour, we will present the formal model of the vehicle
and derive some theoretical results.
Let D ⊂ R 2 denote an open set such that D ∪ ∂D is connected and compact,
this will represent the workspace of the vehicle and its boundary. A stimulus
function can be modelled as a C 2 function S : D∪∂D → R + ∪{0} (a function with
second order continuous derivatives), a single absolute maximum and such that
S(∂D) = 0 but S(D) ≠ 0. This represents a scalar stimulus from a point source
with a bounded effect, vanishing as we move away from the source. The stimulus
is time independent, i.e., even there is spatial variation, it remains constant in
time. A Braitenberg 2b connection function is a function F (s), F : R + ∪ {0} →
R + dF (s)
∪{0}, with second order continuous derivatives such that
ds
> 0∀s ∈ R + ∪
{0} and F (0) = 0. This function models the increasing connection between the

E L
E R
δ
VL
+ +
VR
d
Fig. 1. Internal structure of the Braitenberg vehicle 2b.
sensors and the motors. The fact that it cannot take negative values implies the
vehicle cannot move backward. This has a biological justification since animals
usually do not move backward. The condition F (0) = 0 can be replaced by
F (S min ) = 0, which makes the vehicle move until it reaches a stimulus threshold.
Under these conditions ∂D will be replaced by a closed curve Γ defined by the
level set on S(Γ ) = S min . These two functions are the basis for modelling the
differential equation governing the behaviour of the Braitenberg vehicle 2b. The
actual state space of the vehicle is R 2 ×S 1 since its heading has to be considered.
We will denote the Cartesian coordinates x = (x, y) and θ will be the vehicle
heading. Therefore, the whole state will be (x, θ) = (x, y, θ).
Assuming the wheel radius is a multiplying factor included in F (s), the
velocities of the right and left wheels of the vehicle are v r = F (S(x l )) and
v l = F (S(x r )), where x r and x l are the positions of the right and left sensors
respectively. Since both, functions S(x) and F (s) are C 2 , the velocities can be
approximated as a first order Taylor series around the midpoint between the
sensors x as:
v r ≈ F (S(x)) + δ 2 ∇F (S(x)) · ê p (1)
v l ≈ F (S(x)) − δ 2 ∇F (S(x)) · ê p (2)
where δ is the distance between the sensors, ∇F (S(x)) is the gradient of the
compound connection-stimulus function and ê p = [− sin θ cos θ] T is a unitary
vector orthogonal to the vehicle’s head direction pointing to its left. Assuming
a point like differential-drive vehicle, its state evolves according to the following
system of differential equations:
ẋ = F (S(x)) cos θ (3)

ẏ = F (S(x)) sin θ (4)
˙θ = δ d ∇F (S(x)) · ê p (5)
where d is the wheelbase of the vehicle (see [13] for more details). It is worth
noting that the dynamical system describing the behaviour of the vehicle has no
stability point in D since equations (3) and (4) never vanish simultaneously.
Intuitively the vehicle will never stop but it will slow down as it approaches
∂D since F (s) ∈ C 2 and F (S(∂D)) = 0. It can be proved that the vehicle will
not move outside D, i.e. that the solution x(t), θ(t) of the Cauchy problem (3),
(4), (5), x 0 ∈ D and θ 0 ∈ (−π, π] stays in D for all t. In fact, the solution of the
Cauchy problem will stay on the configuration space D×S 1 . For a solution going
outside the configuration space, the flow defined by the dynamical system (3),
(4) and (5) should have a component pointing outside the surface ∂D×S 1 . Since
the first two components of the flow are zero at the boundary (F (S(∂D)) = 0),
and the normal to the boundary has no angular component, the dot product of
the flow and the normal vector to ∂D × S 1 is zero, and therefore either the flow
vanishes in ∂D×S 1 or is tangent to ∂D×S 1 . From this, we can deduce that there
can be stability points of the dynamical system on the boundary of D where the
gradient ∇F (S(x)) is orthogonal to ê p . This means that the vehicle can, under
some circumstances, move towards some point of ∂D, though it can be shown
that these points are unstable equilibria of the dynamical system. Formally, the
points (x, θ) ∈ ∂D × S 1 where ∇F (x) · ê p = 0 are unstable equilibria of the
dynamical system.
A special case with analytic solution to the Braitenberg vehicle 2b model
appears when the stimulus function has circular symmetry. This is common for
some real stimulus and is treated in [12]. It can be shown that if a positive real
value r 0 such that F (S(r)) + r δ ∂F (S(r))
d ∂r
= 0 exists, two circular trajectories
of radius r 0 are solutions of the differential equations with the proper initial
conditions. If equations (3), (4) and (5) are converted to polar coordinates, they
can be linearised around the trajectory r 0 (t) = r 0 , ψ 0 (t) = ω 0 t + ψ 0 and θ 0 (t) =
ω 0 t + ψ 0 − π/2, where (r, ψ) are the polar coordinates. This leads to the system
of linear differential equations
⎡ ⎤ ⎡
⎤ ⎡ ⎤
˙˜r
⎢ ⎥ (1
0
)
−v 0 v 0 ˜r
⎣
˙˜ψ ⎦ = ⎣ +
d ω0
δ r 0
0 0 ⎦ ⎣ ˜ψ ⎦ (6)
d ˙˜θ
2 F (r 0)
dr
0 0 ˜θ
2
where the matrix has constant coefficients, v 0 = F (S(r 0 )), ω 0 = F (S(r0))
r 0
and
˜r(t) = r(t) − r 0 , ˜ψ(t) = ψ(t) − ψ 0 (t) and ˜θ(t) = θ(t) − θ 0 (t) are the incremental
variables. The eigenvalues λ i =
{
0, ∓
√v 0
(
δ
d
d 2 F (r 0)
dr 2
− (1 + d δ ) ω0
r 0
) } of this matrix
establish the behaviour of trajectories close to the equilibrium one. When
the factor inside the square root is negative, the linearised solution oscillates

around the circular trajectory. Moreover, if the quotient of ω 0 and the eigenvalue
is an irrational number, the resulting trajectory will be dense in D. This
means that for any point in D there is always a trajectory passing arbitrarily
close to it, so the vehicle will cover that part of the workspace.
Algorithm 1 Fixed Point of Poincaré Map P (x)
Require: P (x), x 0
Ensure: x m = P (x m)
repeat
j ← 1
x 0 ← x m
S = {x 0}
repeat
x j ← P (x j−1)
S ← S ∪ {x j}
j ← j + 1
until points in S form a closed trajectory
x m ← average(S)
until error > ɛ
return x m
In the general case of non symmetric stimulus, finding an analytic solution to
the differential equation is a hard problem. However, some numerical methods
exist to find periodic solutions of nonlinear differential equations [11]. Unfortunately,
either they need knowledge about the period of the solution or they only
work for attracting limit cycles. This is not the case of the system at hand, since
the oscillation has constant amplitude around the circle (eigenvalues are pure
imaginary), and therefore the equilibrium is not an attractor. We therefore used
an approximated way of computing the periodic solutions.
The trajectories being the superposition of two periodic trajectories means
that close to the equilibrium, the trajectory is diffeomorphic to a 2D torus in
the state space (x, θ), and the periodic solution is contained inside the torus.
For such a trajectory, the Poincaré map (the intersection of the 2D torus and a
plane on the state space) is diffeomorphic to a circle. Moreover, the closer the
initial condition is to the equilibrium, the more the intersection will look like a
circle. Taking advantage of this, we implemented Algorithm 1, which averages
several points of the Poincaré map. This algorithm works even for unstable
periodic solutions provided that the initial point is close to the real solution and
the divergent component of the map is small enough. The real map will not
be exactly a circle since there will be a divergent component and therefore the
trajectory will look more like the sampling of a logarithmic increasing spiral.
However, when averaging the mean point will get close to the periodic solution,
a fixed point in the corresponding Poincaré map.

2.1 Simulations for point like stimulus
To test the results of the previous section we simulated the model on a stimulus
such that the function F (S(x)) was:
[ ]
F (S(x)) = g 0 − ax T Σx = g 0 − ax T 1 0
x (7)
0 α
where g 0 and a are positive constants and α is a parameter of the dynamical
system. For this function, the domain D is defined by the equation g 0 > ax T Σx,
and ∂D = {x|g 0 = ax T Σx}. When α = 1 the stimulus function has circular
symmetry and therefore a periodic solution exists for r 0 =
√
g0
a(1+δ/d) . For α ≠
1 the periodic solution was computed numerically using algorithm 1. In the
simulations we used g 0 = 2.5, a = 0.1 and δ/d = 0.85.
As an initial guess for the position of the periodic solution the closest known
periodic solution is used. The algorithm was iterated for different (increasing
and decreasing) values of α starting from α close to 1 with initial condition
(3.04, 0, π/2), and the plane to compute the Poincaré map was y = 0. On each
iteration the value of the parameter α was slightly changed and the previous
solution of the fixed point of the Poincaré map was used as initial guess on the
algorithm. The Poincaré map was computed using the time halving algorithm
presented in [11].
4
4.0
3
3.5
2
3.0
1
2.5
0
−4 −3 −2 −1 0 1 2 3 4
y
2.0
−1
−2
−3
1.5
1.0
−4
(a) Periodic trajectories.
0.5
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
(b) x coordinate of the periodic trajectories.
Fig. 2. Periodic trajectories and initial conditions for different α values.
Figure 2(b) shows the resulting x coordinate for the periodic solution. This
coordinate is the fixed point of the corresponding Poincaré map, the other coordinates
were θ = π/2 and y = 0. Figure 2(a) shows the simulation of periodic
trajectories for different values of the parameter α. The trajectories were computed
by integration of the system of differential equations with initial conditions
(x 0 , 0, π/2) using a fixed step Runge-Kutta4 algorithm. As the value of α moves

away from 1 the trajectories differ from the circle. We saw that for circular symmetric
stimulus the trajectories close to the circle oscillate around it, however
nothing can be said for a parabolic shaped stimulus. To obtain information about
the trajectories close to the periodic solutions we computed the characteristic
multipliers of the solutions. If we integrate the Jacobi field obtained from the
derivative of the vector flow along a periodic trajectory the characteristic multipliers
are the eigenvalues of the Jacobian matrix. The eigenvalues of this matrix
are related to the stability of the corresponding Poincaré map and, therefore,
also to the stability of the periodic solution [1]. In our case all the trajectories
had two complex conjugate eigenvalues and one real. Figure 3(a) shows the absolute
value of the three characteristic multipliers as a function of the parameter
α, the complex ones actually generate the same plot. In the case of circular symmetric
stimulus all the eigenvalues fall into the unit circle of the complex plane,
this matches the theoretical result of oscillating trajectories around the circular
solution. For α ≠ 1 the absolute value of at least one of the eigenvectors has
value larger than one, making the Poincaré map and the corresponding differential
equation unstable. In fact, this is the reason why we had to implement
Algorithm 1. However, since at least one of the eigenvalues has absolute value
smaller than one, the fixed point of the Poincaré map is actually a saddle point.
1.0025
2.0e−04
1.0020
1.5e−04
1.0015
1.0e−04
1.0010
5.0e−05
1.0005
1.0000
0.0e+00
0.9995
−5.0e−05
0.9990
−1.0e−04
0.9985
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
(a) Characteristic Multipliers.
−1.5e−04
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
(b) Lyapunov Exponents.
Fig. 3. Characteristic multipliers and Lyapunov exponents of the periodic solution for
different values of α.
Figure 3(b) shows the Lyapunov exponents for the periodic trajectories as
a function of the parameter α. If the sum of all the exponents is positive but
some of them are negative, the corresponding dynamical system will be chaotic.
The relation between the Lyapunov exponents and the characteristic multipliers
is given by the expression λ = 1 T
ln ν, where λ is the Lyapunov exponent
corresponding to the characteristic multiplier ν, and T is the period of the solution
trajectory (see [1]). The period can be easily obtained while computing
the characteristic multipliers, since it is the simulation time needed to get the
Poincaré map. The dotted line represents the sum of all Lyapunov exponents.

Since its sum is close to zero we can deduce that the behaviour of the vehicle is
not chaotic for the given functions and parameters.
3 Wandering on a flat stimulus
So far we saw some results for point like stimulus sources. If the stimulus has no
isolated or single maximum the behaviour of the vehicle is different and other
analysis techniques need to be used. A special case is a constant stimulus, for
which ∇F (S(x)) = 0 in some subset of D. Clearly, since the rotational speed
of the vehicle is proportional to this gradient it will not turn, but it will still
move forward since F (S(x)) > 0. However, some of the previous results are
still valid; the vehicle will move in a bounded area provided F (S(x)) = 0 for
x ∈ ∂D. In fact, the area does not need to be simply connected, since the
contour of the obstacles can belong to ∂D, and therefore ∂D will be a set of
closed curves inside a bigger one. Since F (S(x)) ∈ C 2 and vanishes at ∂D, if
there is a region where ∇F (S(x)) = 0, there must also be a region close to
the boundary where ∇F (S(x)) ≠ 0. Therefore, there will be a region where
the vehicle will potentially turn. One way of building a stimulus on a bounded
area with obstacles is to generate a function of the distance to the obstacles
with continuous derivative. Such function should take a zero value close to the
obstacles and the boundary, and a constant value far from them.
To approximate a flat stimulus keeping the function in C 2 we will compute
the hyperbolic tangent of the previously chosen function F (S(x)). Obviously
this function has also a maximum at the same point as before, however when
the vehicle is far enough from the contour, the gradient is so small that it can
be considered zero and the trajectory can be approximated by a straight line.
Having a close look at the trajectories, we could see they are not straight lines
since the derivative of tanh(F (S(x))) does not vanishes. In sum, the trajectory
will almost follow a straight line and will only blend close to ∂D. This behaviour
looks like the one of balls on a billiard table with elastic collisions. In fact, the
trajectories described by billiard balls are not C 2 functions, and they follow a
geometric reflection law on the border of the table.
It is well known that billiard balls on a circular table can follow periodic
trajectories after several bounces when the angle between the tangent to the
table and the trajectory (φ) is a rational multiple of π, otherwise the trajectory
is dense [16]. We simulated Braitenberg vehicle 2b on a circular stimulus with an
almost flat region in the centre and found that it also follows a periodic trajectory
when the proper initial conditions are selected. Figure 4(a) shows the trajectory
described by the vehicle starting with initial conditions (1, 0, π/2.715), which
is clearly periodic. Another result for unit radius circular billiard table is that
the trajectories are tangent to a circle with radius r int = cos φ [16]. Figure 4(b)
shows that, even the trajectory of the Braitenberg vehicle 2b is not the same
as the billiard ball, it is tangent to some circle. Given the initial condition of
the vehicle (x, θ) = (1, 0, π/4) and the size of the circle r = 2 we can compute
the angle of a bouncing billiard ball φ = 65 ◦ and the radius of the inner circle

−2.0
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
−0.5
−0.5
−1.0
−1.0
−1.5
−1.5
5
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0
(a) Periodic trajectory.
−2.0
3
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0
(b) Internal circle.
4
3
2
2
1
1
0
0
−1
−2
−1
−3
−2
−4
−5
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0
(c) Parabolic table.
−3
−4 −3 −2 −1 0 1 2 3 4
(d) Parabolic table and
parabolic obstacle.
Fig. 4. Braitenberg vehicle as a billiard ball.
r cos φ = 0.85, which roughly matches the simulation results for the Braitenberg
vehicle. In fact, the actual radius of the internal circle of the figure is r = 0.705
slightly smaller. Neither the apparently straight trajectories far from the circle
boundary are straight lines nor the bounces are real bounces, though there is
a clear experimental match between the expected results from the ball and the
simulation of the Braitenberg vehicle pointing to a new way of looking at them
and analysing their behaviour.
Other results from billiard theory also apply to Braitenberg vehicle 2b, like
the ones related to trajectories on a elliptic billiard table. However, these results
apply only to a certain limit. Figure 4(c) shows the trajectory of a simulated
vehicle on an elliptic stimulus with quite different lengths on the principal axis.
Theoretically the resulting trajectory should be tangent to an hyperbola or an
ellipse, but it is clearly not true in this case. It seems that we moved away from
the parametric region where the billiard ball approximation is valid. Figure 4(d)
shows the simulation of a Braitenberg vehicle inside an elliptic region with an
elliptic obstacle inside. The trajectory is complex to analyse but the simulation
points that it is dense on the free space making the control mechanism look
suitable for coverage tasks.
4 Conclusions and Further Work
This paper shows that there is empirical evidence to apply the mathematical
results of billiard balls to Braitenberg vehicle 2b. For simple shapes of the en-

vironment the control technique of the vehicle can be used to generate dense
trajectories, the ones suited on covering tasks. Furthermore, this technique is
clearly much simpler than using algorithms for sequential planning of the trajectories.
Obviously these results need a formal support and the conditions under
which the billiard theory can be applied need to be identified.
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