Mimicking the Behaviour of Idiotypic AIS Robot Controllers Using 
Probabilistic Systems 
 
 
Amanda M. Whitbrook, Uwe Aickelin and Jonathan M. Garibaldi 
Intelligent Modelling & Analysis Research Group,  
School of Computer Science, 
University of Nottingham, Nottingham, UK 
 
 
 
ABSTRACT 
 
Previous work has shown that robot navigation systems that 
employ an architecture based upon the idiotypic network theory 
of the immune system have an advantage over control 
techniques that rely on reinforcement learning only. This is 
thought to be a result of intelligent behaviour selection on the 
part of the idiotypic robot. In this paper an attempt is made to 
imitate idiotypic dynamics by creating controllers that use 
reinforcement with a number of different probabilistic schemes 
to select robot behaviour. The aims are to show that the idiotypic 
system is not merely performing some kind of periodic random 
behaviour selection, and to try to gain further insight into the 
processes that govern the idiotypic mechanism. Trials are carried 
out using simulated Pioneer robots that undertake navigation 
exercises. Results show that a scheme that boosts the probability 
of selecting highly-ranked alternative behaviours to 50% during 
stall conditions comes closest to achieving the properties of the 
idiotypic system, but remains unable to match it in terms of all 
round performance. 
 
Keywords: Artificial immune system (AIS), behaviour 
arbitration, idiotypic network theory, reinforcement learning 
(RL). 
 
1. 
INTRODUCTION 
 
An artificial immune system (AIS) is a computational algorithm 
that attempts to mimic properties of the vertebrate immune 
system in order to solve complex problems. There are a number 
of different types including clonal selection-based algorithms 
[4], negative selection-based algorithms [5] and idiotypic 
networks [6]. The latter group is inspired by Jerne’s network 
theory of the immune system [1], which asserts that suppression 
and stimulation between antibodies plays an important role in 
the immune response.   
Within the domain of mobile robotics, the idiotypic network has 
remained a popular choice of AIS, with most researchers opting 
to implement Farmer et al.’s computational model [2] of Jerne’s 
theory. This is largely because the network of stimulation and 
suppression between antibodies (analogous to behaviours in 
these systems) is thought to provide a means of achieving a 
decentralized behaviour-selection mechanism for the robot. 
Initial results, for example [7-13] are certainly very encouraging, 
but lack any sort of comparison with other systems to assert the 
idiotypic advantage. Furthermore, the complex system dynamics 
are poorly understood.  
However, in [3] the performance of a reinforcement learning 
(RL)-based idiotypic system is compared with a system that uses 
RL only, and provides statistical evidence that the idiotypic 
system is able to complete a task faster and with fewer stalls 
than the RL scheme. The paper also attempts to analyze the 
performance of both control systems in order to explain the 
idiotypic advantage. It suggests that the RL-only system 
provides a strategy that is too greedy, always selecting the 
behaviour 
(antibody) 
that 
best 
matches 
the 
current 
environmental situation (antigen). In contrast, the idiotypic 
system is much more flexible, allowing behaviours that are not 
necessarily the best to flourish. In particular, the paper proposes 
that when the robot is stalled the idiotypic mechanism is able to 
increase the rate of antibody change autonomously so that 
alternative behaviours are used instead of those already tried.  
Given these suggestions, it is possible to create robot controllers 
that attempt to mimic these properties using probabilistic 
behaviour selectors coupled with RL. Hence, the main aim of 
this paper is the construction and testing of such systems in 
order to facilitate further scrutiny of idiotypic dynamics. In a 
number of experiments, nine different probabilistic schemes 
compete with the idiotypic system in order to establish that the 
idiotypic mechanism is not merely performing the equivalent of 
random behaviour arbitration, but acting in a more intelligent 
way. Furthermore, the gradual use of more complex systems that 
apply probabilistic selection more intelligently should help to 
provide additional insight into the processes that govern 
idiotypic dynamics. 
This rest of this paper is arranged as follows. Section 2 provides 
some background information about Jerne’s idiotypic network 
theory, Farmer’s computation model of it, and the variation of 
the model used in [3]. Section 3 describes the architectures of 
the idiotypic and probabilistic systems that are used in this 
research, and illustrates the environments and problems used for 
testing them. The experimental procedures adopted are reported 
in Section 4, and Section 5 discusses the results obtained. 
Section 6 concludes the paper.  
 
 
2. BACKGROUND 
 
In the vertebrate immune system antibodies plays a central role 
in eliminating antigens (for example bacteria and viruses) from 
the body. In order to achieve this, an antibody’s combining site 
(paratope) must be able to bind to a region of the antigen called 
the epitope. According to the clonal selection theory of Burnet 
[14], antibodies with paratopes that possess a good degree of 
match to a given antigen epitope pattern proliferate within the 
system, i.e. are cloned (increase in concentration) and are kept in 
circulation. However, Jerne’s idiotypic network theory [1] 
proposes that antibodies also possess a set of epitopes called 
idiotopes, and that these are the mechanism by which antibodies 
recognize each other. He suggests that antibody concentration 
levels are also influenced by inter-antibody activity, i.e. 
antibodies that are recognized by others are suppressed and 
reduce in concentration and those that do the recognizing are 
stimulated and increase in concentration. 
Farmer et al. [2] go on to suggest that the dynamics of an 
idiotypic AIS system with L antigens [y1, y2,…, yL] and N 
antibodies [x1, x2,…, xL]  can be modelled with the following 
equation: 
 
),
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
1
1
1
1
i
L
=
j
N
=
p
p
i
ip
N
=
m
m
i
im
j
i
ij
i
x
C
k
x
C
x
C
W
+
x
C
x
C
V
k
y
C
x
C
U
b
=
x
C
−










−





 
(1) 
 
where C represents concentration and b, k1 and k2 are constants. 
In Eq. (1) the first sum in the square bracket models antibody 
stimulation due to antigens, the second sum models inter-
antibody suppression and the last sum models inter-antibody 
stimulation. The match specificities for these three kinds of 
interaction are given by some functions U, V and W respectively. 
N. B. The term outside the brackets embodies the natural 
antibody death rate. Here, the equation models both background 
antibody communication (i.e. that between all antibodies) and 
also active antibody communication. The latter is the stimulation 
and suppression that takes place between the antigenic antibody 

 (that with the best match to the presenting antigen) and any 
other antibody that matches the presenting antigen (i.e. any 
competing antibody). 
In [3] background communication is ignored for simplification 
and antigen concentrations are not needed since each 
environmental scenario (antigen) is ranked in order of 
importance and weighted accordingly, i.e. multiple antigens are 
allowed to present themselves but one is deemed dominant and 
given greater weighting. Furthermore, N × L matrices P and I, 
which represent the antibody paratope and idiotope respectively, 
are used. The elements of P are the current RL scores, which 
reflect the degree of match between each antibody and antigen, 
and the elements of I are fixed disallowed antibody-antigen 
combinations. Antibody communication is hence simulated by 
comparing the paratope of 
 with the idiotope of the competing 
antibodies (i.e. those that have nonzero match to the set of 
presenting antigens) and vice versa. The model thus reduces to 
Eq. (2) below: 
  
),
(
)
(
)
(
)
(
)
(
)
(
2
1
1
1
1
i
L
=
j
L
=
p
i
ip
L
=
m
i
im
ij
i
x
C
k
x
C
x
C
W
+
x
C
x
C
V
k
U
b
=
x
C
−










′
′
−



α
α


      (2) 
 
with matching functions U, V’ and W’ given by: 
 
,
)
(
]
[
j
i
j
i
ij
x
G
y
,
x
P
=
U
 
 
 
 
  (3) 
 
,
]
[
]
[
i
m
i
m
im
H
y
,
x
I
y
,
x
P
=
V
α
′
  
 
 
  (4) 
 
(
)
.
]
[
]
[
1
i
p
p
i
ip
H
y
,
x
I
y
,
x
P
=
W
α
−
′
 
 
 
  (5) 
 
In Eq. (3) G(xi) is an antigen array with value zero for non-
presenting antigens, value two for the dominant antigen yd if 
P[xi, yd] > 0, value zero for the dominant antigen if P[xi, yd] = 0, 
and value 0.25 for all other presenting antigens. In Eq. (4) and 
Eq. (5) H is an antibody array with value one for competing 
antibodies and value zero for noncompeting antibodies. 
However, Eq. (2) cannot be evaluated as a whole since 
 must 
be found first, so it is broken down into constituent parts: 
 
,
)
(
]
[
)
(
max
1
1










=

L
=
j
j
i
j
i
i
i
x
G
y
,
x
P
=
x
S
α
 
 
 
(6)                     
,)
(
)
(
]
[
]
[
)
(
1
2

L
=
m
i
i
m
i
m
i
x
C
x
C
H
y
,
x
I
y
,
x
P
=
x
S
α
α
          (7)                        
(
)
.)
(
)
(
]
[
]
[
1
)
(
1
3

−
L
=
p
i
i
p
p
i
i
x
C
x
C
H
y
,
x
I
y
,
x
P
=
x
S
α
α
                  (8) 
 
 
First, antibody 
 is computed from Eq. (6), then the suppression 
and stimulation factors are calculated from Eqs. (7) and (8) 
respectively. Finally, the global match strength Sg is determined 
from: 
 
                                  
 
(
)
.)
(
)
(
)
(
)
(
3
2
1
1
i
i
i
i
g
x
S
+
x
S
k
x
S
=
x
S
−
                                   (9) 
 
The concentration of each antibody is hence given by 
substitution of Eq. (9) into Eq. (2) giving: 
 
 
(
)
,)
(
)
(
)
(
2
i
i
g
i
x
C
k
x
S
b
=
x
C
−


                                                 (10) 
 
 
which is transformed to: 
 
(
)
,
)
(
)
(
)
(
)
(
2
1
t
i
t
i
g
t
i
t+
i
x
C
k
x
S
b
+
x
C
x
C
−
=
                             (11) 
 
  
upon discretization. The antibody  selected for execution is that 
with the highest normalized concentration, given by: 
 
.
)
(
max
1
+
t
i
i
x
C
=
β
 
 
 
            (12) 
 
In [3] experiments are performed that vary the values of the 
suppression-stimulation balancing constant k1 and the rate 
constant b using a simulated robot navigation exercise as a test 
bed.  Results show that when k2 is fixed at 0.05, the robot tends 
to perform best with b set approximately between 40 and 160 
and k1 set between 0.575 and 0.650.  In this region 
   (there is 
an idiotypic difference) approximately 20% of the time. The 
parameter k2 governs how quickly the antibodies reach zero 
concentration. In other systems this might lead to their removal 
and replacement with alternatives, but this particular architecture 
uses a fixed number of antibodies that are never replaced, so k2 
is deliberately kept low. 
It is worth noting that in [17] a slightly different idiotypic design 
is used that allows for only one presenting antigen per iteration, 
and more importantly employs a variable idiotope matrix with 
probabilistic components. This means that the idiotypic 
difference rate is much harder to predict for given values in the 
{b, k1, k2} space, and suggests that the findings in [3] may be 
altogether dependent on the choice of the fixed disallowed 
antibody-antigen combinations in the idiotope matrix. For this 
reason the same combinations used in the idiotope matrix in [3] 
are used here, see Section 3. 
 
 
 
 
3. TEST ENVIRONMENT AND  
SYSTEM ARCHITECTURE 
 
Throughout this research simulated Pioneer 3 robots are used 
with Player’s Stage 2.0.3 simulator [15]. The virtual robots 
possess eight sonar sensors at the rear and a laser sensor at the 
front that spans 180º. For convenience this 180º sector is 
subdivided into six 30º subsectors 1 to 6, with 1 and 2 
representing the left, 3 and 4 corresponding to the centre, and 5 
and 6 corresponding to the right of the robot. A frontal camera 
that can detect different coloured objects is also placed centrally, 
so that the robot is able to recognize cyan squares placed in the 
doorways. Its task is to use these as markers in order to navigate 
through the rooms in two different maze environments, M1 and 
M2 which are shown in Figures 1 and 2 respectively. Note that 
when a robot has passed a cyan marker its path back to the 
previous room is blocked off manually. 
As stated earlier, environmental information is modelled with 
antigens and robot behaviours are modelled as antibodies that 
possess a fixed action component, and an idiotope and paratope 
element value for each antigen.  For this purpose, eight antigens 
and sixteen antibodies are created as detailed in Tables 1 and 2 
respectively and as in [3] where justification for choosing them 
is also given. Table 1 shows the priority ranking of the antigens, 
with 0 the lowest (least urgent) and 5 the highest (most urgent). 
Detection of the various antigens is governed by several sensor 
reading metrics which include the minimum and average laser 
readings Zmin and Zav, the average rear sonar reading Eav, and the 
position of the minimum laser reading Rmin. The maximum laser 
reading Zmax is also used by antibody 11, see Table 2.  
 
 
 
Figure 1.  Simulated Maze World M1 
 
 
Figure 2.  Simulated Maze World M2 
 
Ten control systems are created of which nine are probabilistic. 
The other system ID uses the idiotypic architecture described in 
Section 2, with b set at 80, k1 set at 0.65, and k2 set at 0.05, as 
these parameter values fall within the region of {b, k1, k2} space 
where performance is optimal in [3].   
The ID controller begins by reading in the paratope matrix P and 
the idiotope matrix I. Matrix P is initially random, developing 
via RL as the code executes and the robot learns. Three different 
random paratope matrices are used D1, D2 and D3, and each has 
initial element values between 0.50 and 0.75 to try to avoid any 
bias. Matrix I is fixed and has value 0.50 for the antibody-
antigen pairs 0-0, 0-2, 10-0, 10-2, 14-0, 14-2, 15-0 and 15-2, and 
value 1.00 for the pairs 1-2, 2-0, 3-1, 4-0, 5-2, 6-2, 7-0, 8-0, 9-2, 
11-5, 12-5, 13-5, see Table 3. These pairs represent “disallowed” 
combinations, which effectively control the nature of the 
antibody connections within the idiotypic network.  
Next the sensors are read and the dominant antigen and antigen 
array (G) element values are determined so that S1 (the degree of 
match to antigen) can be calculated for each antibody and 
 can 
be determined using Eq. (6). Following this, Eqs. (7) and (8) are 
used to calculate suppression and stimulation respectively and 
thus deduce the global strength of match Sg using Eq. (9). The 
concentration of every antibody in the system is then calculated 
using (11) and normalized using: 
 
,
)
(
)
(
)
(
1
1
1
1

=
+
+
+ =
N
j
t
j
t
i
t
i
x
C
x
C
x
C
 
 
 
(13) 
 
so that the total number of antibody clones is kept constant to 
mimic the biology more closely and help prevent scaling 
problems. Note that the term concentration is used to mean the 
proportional number of clones of an antibody type in circulation 
if the total number of antibodies is held constant at N, with all 
concentration values beginning at 1.00. The final antibody 
selected is  given by (12), i.e.  is the antibody with the highest 
normalized concentration.  
 
TABLE 1 
THE ANTIGENS AND THEIR CONDITIONS 
Antigen (Priority) 
Conditions 
0 - Object left (2) 
Zmin < 0.55 m and Rmin = 1 or 2 (-90º to -30º) 
1 - Object centre (2) 
Zmin < 0.55 m and Rmin = 3 or 4 (-29º to 29º) 
2 - Object right (2) 
Zmin < 0.55 m and Rmin = 5 or 6 (30º to 90º) 
3 - Zav > threshold (0) 
Zav >= 0.45 m 
4 - Zav < threshold (3) 
Zav < 0.45 m 
5 - Robot stalled (4) 
Distance travelled = 0 
6 - Robot blocked behind (5) 
Distance travelled = 0 and Eav < 0.35 m 
7 - Door marker seen (1) 
A cyan marker has been detected by the camera 
 
 
TABLE 2 
THE ANTIBODIES AND THEIR ACTIONS 
 
Antibody 
Angular 
speed 
(°/s) 
Linear 
speed 
(m/s) 
Details 
0 
Reverse spin 1 
-90 
-0.15 
Fixed angular and linear movement 
1 
Slow right 15 
-15 
0.06 
Fixed angular and linear movement 
2 
Slow left 15 
15 
0.06 
Fixed angular and linear movement 
3 
Fast centre 
0 
M/2 
Fixed linear movement 
4 
Fast left 15 
15 
M/2 
Fixed angular and linear movement 
5 
Fast right 15 
-15 
M/2 
Fixed angular and linear movement 
6 
Slow right 35 
-35 
0.06 
Fixed angular and linear movement 
7 
Slow left 35 
35 
0.06 
Fixed angular and linear movement 
8 
Fast left 35 
35 
M/2 
Fixed angular and linear movement 
9 
Fast right 35 
-35 
M/2 
Fixed angular and linear movement 
10 
Reverse spin 2 
90 
-0.15 
Fixed angular and linear movement 
11 
Wander max 
Variable 
M 
Wander towards Zmax reading 
12 
Wander min 
Variable 
M/2 
Wander away from Zmin reading 
13 
Track blobs 
Variable 
M 
Align towards centre of cyan marker 
14 
Reverse 1 
-25 
-0.15 
Fixed angular and linear movement 
15 
Reverse 2 
25 
-0.15 
Fixed angular and linear movement 
M = Maximum speed permitted (2.0 m/s) 
Pioneer robot 
Cyan door marker 
Obstacles 
Finish Point 
Pioneer Robot 
Finish Point 
Obstacles 
Cyan door 
marker 
When an antibody is selected for execution it carries out its 
designated action and the result of that action (half a second 
later) is scored either positively or negatively using RL. This 
means that paratope element value Pd is adjusted upon every 
iteration using: 
 
(
)
(
)
],
,0
max[
5.0
1
+
+
+
=
t
t
d
t
d
P
P
τ
β
β
 
 
 
(14) 
 
where  is the positive or negative RL score awarded and d is the 
index of the dominant antigen. Further details on the particular 
RL scheme used here are provided in [3], and a general 
explanation of RL can be found in [16]. 
The nine probabilistic systems, R1 – R9 are summarized in Table 
4. They use the same essential architecture as the idiotypic 
system (described above), except that they compute antibody 
 
only, not , i.e. they omit the suppression and stimulation 
calculations in Eqs. (7) and (8). Having calculated 
 from the 
RL-scores of paratope matrix P using Eq. (6), systems R1 – R9 
either use it or simply select an alternative antibody . The rate 
of  selection and which alternative antibody is used both 
depend on pre-determined probability values. The idiotypic 
system is therefore mimicked by using a number of systems with 
probability values that simulate an approximate overall  rate of 
20%. Note that it is P
d that is scored using RL for the 
probabilistic systems, or Pd when 
 is rejected in favour of 
alternative antibody . Also, for systems R1 - R9 concentrations 
play no role in selecting the antibody that will execute its action. 
 
 
TABLE 3 
THE FIXED IDIOTOPE MATRIX 
 
0 
1 
2 
3 
4 
5 
6 
7 
0 
0.50 
0.00 
0.50 
0.00 
0.00 
0.00 
0.00 
0.00 
1 
0.00 
0.00 
1.00 
0.00 
0.00 
0.00 
0.00 
0.00 
2 
1.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
3 
0.00 
1.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
4 
1.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
5 
0.00 
0.00 
1.00 
0.00 
0.00 
0.00 
0.00 
0.00 
6 
0.00 
0.00 
1.00 
0.00 
0.00 
0.00 
0.00 
0.00 
7 
1.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
8 
1.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
9 
0.00 
0.00 
1.00 
0.00 
0.00 
0.00 
0.00 
0.00 
10 
0.50 
0.00 
0.50 
0.00 
0.00 
0.00 
0.00 
0.00 
11 
0.00 
0.00 
0.00 
0.00 
0.00 
1.00 
0.00 
0.00 
12 
0.00 
0.00 
0.00 
0.00 
0.00 
1.00 
0.00 
0.00 
13 
0.00 
0.00 
0.00 
0.00 
0.00 
1.00 
0.00 
0.00 
14 
0.50 
0.00 
0.50 
0.00 
0.00 
0.00 
0.00 
0.00 
15 
0.50 
0.00 
0.50 
0.00 
0.00 
0.00 
0.00 
0.00 
 
 
TABLE 4 
THE PROBABILISTIC SYSTEMS 
Probability of antibody selection (%) 
 
Description 

 
 
2nd  
best 
3rd 
best 
4th 
best 
R1 
20% any other antibody but  
- Equal probability 
20 
80 
- 
- 
- 
R2 
20% any other antibody but  
- Probability based on RL score 
20 
80 
- 
- 
- 
R3 
20% 2nd best antibody 
20 
80 
20 
- 
- 
R4 
20% 2nd or 3rd best 
20 
80 
10 
10 
- 
R5 
20% 2nd, 3rd or 4th best 
20 
80 
10 
5 
5 
If use of previous antibody 
deemed unsuccessful by RL 
28 
72 
14 
7 
7 
R6 
If use of previous antibody 
deemed successful by RL 
14 
86 
7 
3.5 
3.5 
If current or previous antigen 
was either 5 or 6 
33 
67 
16.5 
8.25 
8.25 
R7 
Otherwise 
15 
85 
7.5 
3.75 
3.75 
If current or previous antigen 
was either 5 or 6 
50 
50 
25 
12.5 
12.5 
R8 
Otherwise 
13 
87 
6.5 
3.25 
3.25 
If current or previous antigen 
was either 5 or 6 
75 
25 
37.5 
18.75 
18.75 
R9 
Otherwise 
2 
98 
1 
0.5 
0.5 
In the case of R1 there is a 20% chance of choosing any other 
antibody apart from 
, and these are selected with equal 
probability. System R2 is similar with a 20% chance of not 
selecting 
, but the alternative antibody is chosen based on 
probabilities derived from the paratope matrix, i.e. the RL-scores 
representing the match between each antibody and the dominant 
antigen are used. The probability of selection  of antibody xi is 
given by: 
 
,
)
(
1
=
=
N
j
jd
id
i
P
P
x
ν
  
 
 
 
(13) 
 
where N is the number of antibodies and d is the index of the 
dominant antigen. If 
 is chosen again the process repeats until  
is different to 
. 
Systems R3, R4 and R5 also have a 20%  rate, but when 
 is 
rejected R3 always uses the antibody that is second-best and R4 
uses either the second or third best-matched antibody with equal 
probability. System R5 uses either the second, third, or fourth 
best-matched antibody, but is twice as likely to use the second-
best. System R6 considers whether the previously-used antibody 
was deemed successful by the RL. If it was regarded as 
successful then there is only a 14% chance of selecting the 
second, third or fourth best-matched antibody. If it was marked 
as unsuccessful by the RL, then there is probably a greater need 
for a different antibody, so the probability of not selecting 
 
increases to 28%. In either case, bias is toward choosing the 
second best-matched antibody, rather than the third or fourth. 
Systems R7, R8 and R9 are similar but take into account whether 
the robot is currently stalled or was stalled on the previous 
iteration. This methodology is adopted as previous analysis of 
antibody selection in system ID has shown that the idiotypic 
difference rate tends to increase to around 30% during stall 
conditions. With R7, if there are no stall conditions then there is 
a 15% chance of not choosing 
. Again, bias is toward the 
second best-matched antibody, with the third and fourth best-
matched being only half as likely to be selected. However, if the 
robot is currently stalled or was stalled on the previous iteration 
there is probably a much stronger requirement for an alternative 
antibody, so the chance of not selecting 
 increases to 33%, 
(bias is still toward the second-best antibody). Systems R8 and 
R9 work in the same way as R7 but use 50% and 75%  rates 
respectively when the robot is stalled and 13% and 2%  rates 
otherwise. In systems R6 to R9 the probabilities are selected 
based on pre-trials to generate an approximate overall 20% 
observed  rate.  
 
 
4. EXPERIMENTAL PROCEDURES 
 
Each of the ten control systems is run twelve times in Maze 
World M1, six times starting with paratope D1 and the other six 
times starting with paratope D2. For each run, the time taken to 
complete the course T is recorded along with the number of 
robot stalls . A stall represents a collision with an obstacle or 
the walls and is determined either by detecting that the robot has 
come to a complete stand-still for more than one time-loop 
interval (antigen 5) or by recording stand-still coupled with a 
rear-sonar reading of less than 0.35 m (antigen 6). 
A fast robot that continually crashes or a careful robot that takes 
too long to complete the task is undesirable, so a fitness measure 
F, which combines T and  is computed for each run. This is 
given by: 
 
),
(
2
1
1σ
ϕ
+
=
T
F
  
 
 
 
(14) 
 
where 1 is the ratio of the mean task time to mean number of 
stalls over all the 120 experiments in M1: 
 
.
1
σ
ϕ
T
=
 
 
 
 
 
(15) 
 
Maze World M2 represents a more difficult task for the robot as 
there are more rooms with more obstacles and there is generally 
less space for the robot to move around in. The idiotypic system 
and the best-performing probabilistic controller from the 
experiments with M1 (i.e. that with the best fitness) are both 
used for robot navigation in M2, six times starting with D1 and 
six times using D3. Again, T and  are recorded for each run and 
F is calculated, this time using 2, the ratio of the mean task time 
to mean number of stalls over all the 24 experiments in M2. 
In both worlds, mean T,  and F values are computed for each 
control system and are compared using a 1-tailed t-test, with 
differences accepted as significant at the 99% level only. As 
another measure of task performance, runs with an above 
average fitness for each world are counted as good and those 
with fitness in the bottom 10% of all runs in each world are 
counted as bad. In addition, the  rate is noted for each run and 
the mean is calculated for each control system. 
 
 
5. RESULTS 
 
Table 5 shows the mean T, , F and  values for each control 
system in each world, and also the percentage of good and bad 
runs. Table 6 displays the significant difference levels when 
each of the systems is compared to the idiotypic controller. 
 
 
TABLE 5 
MEAN TASK TIME, COLLISIONS, FITNESS AND µ  RATE  
WITH PERCENTAGE OF GOOD AND BAD RUNS IN EACH SYSTEM 
Means 
Run % 
System 
Maze 
World 
T 

 
F 
 
(%) 
Good  
Bad  
ID 
218 
21 
180 
21 
92 
0 
R1 
414 
62 
419 
20 
25 
25 
R2 
317 
39 
293 
19 
50 
0 
R3 
295 
55 
335 
19 
42 
17 
R4 
290 
45 
298 
18 
58 
17 
R5 
296 
43 
296 
19 
58 
8 
R6 
313 
54 
342 
21 
42 
17 
R7 
302 
42 
296 
19 
58 
8 
R8 
259 
39 
263 
20 
67 
8 
R9 
M1 
 
293 
48 
312 
16 
50 
8 
ID 
339 
18 
258 
17 
83 
0 
R8 
M2 
 
398 
46 
479 
21 
25 
33 
 
 
The results show that none of the probabilistic controllers 
performs as well as ID in Maze World M1. The idiotypic system 
has the fastest completion time, the least number of stalls and 
the best fitness. All of these performances are significantly 
better than the probabilistic systems, except in the case of R8 and 
when comparing  values for system R7 and T values for R9. 
Furthermore, ID has the highest percentage of good runs (92%) 
and has no bad runs. Probabilistic system R2 also has no bad 
runs, but only 50% of its runs are considered good.  
 
 
TABLE 6 
SIGNIFICANT DIFFERENCE LEVELS BETWEEN THE PROBABILISTIC 
SYSTEMS AND THE IDIOTYPIC SYSTEM 
Significant Difference Level 
System 
Maze 
World 
T 

 
F 
R1 
100.0 
100.0 
100.0 
R2 
99.9 
99.5 
99.8 
R3 
99.6 
99.9 
99.9 
R4 
99.1 
99.2 
99.3 
R5 
99.1 
99.3 
99.4 
R6 
99.6 
99.9 
99.9 
R7 
99.2 
98.9 
99.2 
R8 
94.0 
98.5 
98.1 
R9 
M1 
 
96.8 
99.8 
99.6 
R8 
M2 
97.6 
100.0 
100.0 
 
 
Since system R8 is second best in terms of fitness, it is used in 
Maze World M2 for comparison with ID. However, in this world 
both its  and F values have significantly higher means than 
those observed with ID, and T is almost significant. In addition, 
33% of runs are deemed bad and only 25% are deemed good for 
system R8. In contrast, 83% of the idiotypic system’s runs are 
judged as good and none are judged as bad. 
All of the probabilistic systems show an overall mean  rate of 
approximately 20%, which validates the probability choices. 
 
 
Discussion 
 
The results achieved provide strong empirical evidence that the 
idiotypic system possesses a highly intelligent form of behaviour 
selection that cannot easily be mimicked using simple 
probabilistic systems. In fact, ID performs better even when a 
probabilistic system uses some form of inherent intelligence, for 
example basing the likelihood of antibody selection upon the 
current RL scores (as in R2) or boosting the probability of 
selecting an alternative to 
 under certain conditions (R6 – R9).  
The probabilistic controller that performs best in world M1 (and 
therefore comes closest to mimicking idiotypic dynamics) is 
system R8, which increases the theoretical  rate (probability of 
selecting either the second, third, or fourth best-matched 
antibody) to 50% under stall conditions. However, the mean 
number of stalls is still significantly higher than for the idiotypic 
system in world M2, which suggests that R8 is less able to deal 
with more complex environments.  
System R7 has a theoretical  rate of 33% during stall 
conditions, which is very close to the mean idiotypic difference 
rate recorded for ID under these circumstances (31%). However, 
its performance is inferior to ID and also to R8, which increases 
the  rate to 50% under stall conditions. This suggests that the 
idiotypic dynamics are doing more than merely raising the rate 
of antibody change when the robot is in difficulty. Indeed, [3] 
proposes that it is the increased RL success rate of the antibodies 
chosen during stall conditions that contributes to an idiotypic 
robot’s superior performance, and that the idiotypic process 
works by selecting antibodies of similar type to 
. In other 
words, as well as raising the  rate during stall conditions, the 
probabilistic systems also need a better mechanism for 
determining which alternative antibody should be selected. 
Presently, only the current second, third, and fourth best-
matched antibodies are considered, with the second-best being 
twice as likely to be selected as the third or fourth. This is a 
fundamental weakness in the probabilistic schemes, as an 
alternative antibody with a highly-ranked RL score for a 
particular antigen does not necessarily represent an antibody 
with similar properties to 
. Further research is therefore needed, 
in particular, a detailed examination of the alternative antibodies 
that are chosen under stall conditions in the idiotypic system, 
and how they rank in terms of matching to the presenting 
antigens. If a general pattern of selection could be identified and 
formalized into a probabilistic algorithm that approximates it, it 
might be possible to mimic the idiotypic dynamics much more 
closely. 
However, it is still questionable whether such a system would be 
able to equal or better the performance of ID. This is because the 
idiotypic mechanism is a dynamic process of continuous change, 
where the behaviour selected at a given time affects future 
selections, i.e. it represents a self-regulating system with 
feedback. In contrast, the probabilistic systems are only flexible 
in that they permit other antibodies to be chosen; in all other 
aspects they are inherently rigid. Furthermore, feedback in the 
idiotypic system is driven by the use of concentrations in the 
choice of alternative antibody as well as global strength of 
match to antigen, which means that it provides a kind of 
memory feature for past selection as well as considering current 
environmental information. In fact, it may be the balance 
between these two aspects that gives idiotypic robots their 
advantage. In Eq. (10) parameter b governs the weighting given 
to the global strength of match Sg when calculating new 
concentration values, and experiments that vary this parameter 
have shown that idiotypic robots show significantly better 
performance when b is within a certain region [3].  
A probabilistic scheme that aims to imitate the dynamics of an 
idiotypic system accurately would therefore need to: 
 
1. 
Incorporate some form of memory feature analogous 
to antibody concentrations that enables the system to 
record past antibody use.  
2. 
Utilize a mechanism that gives weighted consideration 
to both the memory and the strength-of-match to 
antigen when selecting alternative antibodies. This 
would introduce feedback into the system and provide 
a more dynamic selection process. 
3. 
Mimic the ideal idiotypic difference rates, both during 
stall conditions and when the robot is free. 
4. 
Imitate the patterns of alternative antibody selection 
inherent in the idiotypic dynamics during stall 
conditions, ideally by using a method that favours 
antibodies with similar properties to 
. 
 
This research has currently addressed item 3) only, which might 
explain why R8 came closest to reproducing the performance of 
ID. System R8’s theoretical  rate under stall conditions is greater 
(50%) than the corresponding idiotypic difference rate of ID 
(30%). The greater chance of switching to an alternative 
antibody may have provided some form of compensation for 
lack of the other features, and may account for R8’s superior 
performance to R7. However, it should be noted that system R9, 
which boosted the  rate to 75% under stall conditions, was 
inferior in performance to both R7 and R8. This suggests that 
there may be an optimal  rate under stall conditions for 
probabilistic systems that lack design specifications 1), 2) and 
4). Future research will investigate this further by determining 
the optimal value, incorporating the missing design features, and 
examining any changes in the optimal value once these are in 
place.  
6. CONCLUSIONS 
 
This research has compared the performances of an idiotypic 
AIS robot control system with nine other control systems that 
select robot behaviour using probability functions. It has 
provided substantial empirical evidence that the idiotypic 
selection mechanism is superior to any of these systems, which 
suggests that the idiotypic dynamics are facilitating more 
intelligent behaviour selection. The probabilistic system that 
comes closest to approximating these dynamics is one that 
boosts the likelihood of non-
 selection (i.e. increases the  rate) 
during stall conditions, although its performance is still inferior 
to the idiotypic system. This supports the notion that idiotypic 
behaviour arbitration incorporates an innate ability to recognize 
and respond effectively to situations in which the robot is 
trapped. 
Further research will aim to study the patterns of alternative 
antibody selection within the idiotypic system during stall 
conditions, in particular the strength-of-match rankings of 
antibodies chosen instead of 
. Study of these patterns might 
show how idiotypic systems are able to nominate more 
successful antibodies, and how the selection-mechanism is able 
to determine which ones have similar properties to 
. This might 
enable a more accurate probabilistic model of the idiotypic 
system to be created. Furthermore, a means of recording past 
antibody use is absent in the probabilistic systems constructed 
here, and may contribute to their inferior performance. Thus, an 
important aspect of future research will be the construction of a 
probabilistic algorithm that imitates this additional feedback 
feature. A detailed examination of the relationship between 
antibody concentrations, past use and time in the idiotypic 
system would greatly assist in this process.   
 
 
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