Lecture Notes in Computer Science
Motion Control and Tracking Algorithm
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- A Generalised Entropy Based Associative Model
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4 Motion Control and Tracking Algorithm When connectivity r is su fficiently large, one random initial pattern converges into one of four limit cycle attractors as time evolves. By the coding transformation of motion functions, the corresponding motion of the tracker in 2-dimensional space becomes monotonic(see Fig.2). On the other hand, when connectivity r is quite small, chaotic dynamics occurs in the state space, correspondingly, the tracker moves chaotically (see Fig.3). If the updating of state pattern in chaotic regime is replaced by random 400-bit- pattern generator, the tracker shows random walk(see Fig.4). Obviously, chaotic motion is di
fferent from random walk, and has a certain dynamical structure. -1 0 1 2 3 4 5 6 -1 0 1 2 3 4 5 6 START r=399 Fig. 2. Monotonic motion -10
-8 -6 -4 -2 0 2 4 -10
-8 -6 -4 -2 0 2 4 START
r=30 (500 steps) Fig. 3. Chaotic walk -10
-8 -6 -4 -2 0 2 4 -10
-8 -6 -4 -2 0 2 4 START
Random walk (500 steps) Fig. 4. Random walk Therefore, when the network evolves, monotonic motion and chaotic motion can be switched by switching the connectivity r. Based on this idea, we proposed a simple algorithm to track a moving target, shown in Fig.5. First, an tracker is assumed to be tracking a target that is moving along a certain trajectory in two-dimensional space, and the tracker can obtain the rough directional information D 1 (t) of the moving target, which is called as global target direction. At a certain time t, the present position of the tracker is assumed at the point (p x (t) , p
(t)). This point is taken as the origin point and two-dimensional space can be divided into four quadrants. If the target is moving in the nth quadrant, D 1 (t) = n (n = 1, 2, 3, 4). Next, we also suppose that the tracker can know another directional information D 2 (t) = m (m = 1, 2, 3, 4), which is called global motion direction. It means that the tracker has moved toward the mth quadrant from time t − 1 to t, that is, in the previous step. Global target direction D 1 (t) and global motion direction D 2 (t) are time-dependent variables. If these information are taken as feedback to the network in real time, the connectivity r also becomes a time-dependent variable r(t) and is determined by D 1 (t) and D 2 (t). In Fig.5, R
is a su
fficiently large connectivity and R S is a quite small connectivity that can lead to chaotic dynamics in the neural network. Adaptive switching of connec- tivity is the core idea of the algorithm. When the synaptic connectivity r(t) is determined by comparing two directions,D 1 (t −1) and D 2 (t −1), the motion increments of the tracker are calculated from the state pattern of the network updated with r(t). The new motion 184 Y. Li and S. Nara Fig. 5. Control algorithm of tracking a moving target: By judging whether global target direc- tion D 1 (t) coincides with global motion direction D 2 (t) or not, adaptive switching of connectivity r between R S and R L results in chaotic dynamics or attractor’s dynamics in state space. Corre- spondingly, the tracker is adaptively tracking a moving target in two-dimensional space. causes the next D 1 (t) and D 2 (t), and produces the next connectivity r(t + 1). By repeat- ing this process, the synaptic connectivity r(t) is adaptively switching between R L and
R S , the tracker is alternatively implementing monotonic motion and chaotic motion in two-dimensional space.
In order to confirm that this control algorithm is useful to tracking a moving target, the moving target should be set. Firstly, we have taken nine kinds of trajectories which the target moves along, which are shown in Fig.6 and include one circular trajectory and eight linear trajectories. Suppose that the initial position of the tracker is the origin(0,0) of two-dimensional space. The distance L between initial position of the tracker and that of the target is a constant value. Therefore, at the beginning of tracking, the tracker is at the circular center of the circular trajectory and the other eight linear trajectories are tangential to the circular trajectory along a certain angle α, where the angle is defined by the x axis. The tangential angle α = nπ/4 (n = 1, 2, . . . , 8), so we number the eight linear trajectories as LT
, and the circular trajectory as LT 0 .
Arrow represents the moving direction of the target. Solid point means the po- sition at time t =0.
-20 -15
-10 -5 0 5 10
15 20
-20 -15 -10 -5 0 5 10
15 20
Object Target
Capture Fig. 7. An example of tracking a target that is moving along a circular trajectory with the simple algorithm. the tracker captured the moving target at the intersection point. Tracking a Moving Target Using Chaotic Dynamics 185
Next, let us consider the velocity of the target. In computer simulation, the tracker moves one step per discrete time step, at the same time, the target also moves one step with a certain step length S L that represents the velocity of the target. The motion increments of the tracker ranges from -1 to 1, so the step length S L is taken with an interval 0.01 from 0.01 to 1 up to 100 di fferent velocities. Because velocity is a relative quantity, so S L = 0.01 is a slower target velocity and S L = 1 is a faster target velocity relative to the tracker. Now, let us look at a simulation of tracking a moving target using the algorithm proposed above, shown in Fig.7. When an target is moving along a circular trajectory at a certain velocity, the tracker captured the target at a certain point of the circular trajectory, which is a successful capture to a circular trajectory.
To show the performance of tracking a moving target, we have evaluated the success rate of tracking a moving target that moves along one of nine trajectories over 100 initial state patterns. In tracking process, the tracker su fficiently approaching the target within a certain tolerance during 600 steps is regarded as a successful trial. The rate of successful trials is called as the success rate. However, even though tracking a same target trajectory, the performance of tracking depends not only on synaptic connectivity r, but also on target velocity or target step length S L. Therefore, when we evaluate the success rate of tracking, a pair of parameters, that is, one of connectivity r(1 ≤ r ≤ 60) and one of target velocity S L(0 .01 ≤ T ≤ 1.0), is taken. Because we take 100 different target velocity with a same interval 0.01, we have C 100 60
evaluated the success rate of tracking a moving target along di fferent trajectories. Two examples are shown as Fig.8(a) and (b). By comparing Fig.8(a) and (b), we are sure that tracking a moving target of circular trajectory has better performance than that of linear trajectory. However, to some linear trajectories, quite excellent performance was observed. On the other hand, the success rate highly depends on connectivity r and the target velocity S L even if the same target trajectory is set. In order to observe the performance clearly, we have taken the data 0 10 20 30 40 50 60 0 20
40 60
80 100
0 0.2
0.4 0.6
0.8 1 Success Rate Connectivity Target Velocity x10 -2
10 20 30 40 50 60 0
20 40
60 80
100 0 0.2 0.4 0.6
0.8 1 Success Rate Connectivity Target Velocity x10 -2
(b) linear target trajectory Fig. 8. Success rate of tracking a moving target along (a)a circle trajectory;(b)a linear trajectory: The positive orientation obeys the right-hand rule. The vertical axis represents success rate, and two axes in the horizontal plane represents connectivity r and target velocity S L, respectively.
186 Y. Li and S. Nara 0 0.2
0.4 0.6
0.8 1 0 20 40
60 80
100 Success rate Target Velocity ( x10 -2 ) 0 0.2
0.4 0.6
0.8 1 0 20 40
60 80
100 Success rate Target Velocity ( x10 -2 ) (a) r = 16: downward tendency (b) r = 51: upward tendency Fig. 9. Success rates drawn from Fig.8(a): We take the data of a certain connectivity and show them in two dimension diagram. The horizontal axis represents target velocity from 0.01 to 1.0, and the vertical axis represents success rate. of certain connectivities from Fig.8(a), and plot them in two-dimensional coordinates, shown as Fig.9. Comparing these figures, we can see a novel performance, when the target velocity becomes faster, the success rate has a upward tendency, such as r = 51. In other words, when the chaotic dynamics is not too strong, it seems useful to tracking a faster target.
In order to show the relations between the above cases and chaotic dynamics, from dy- namical viewpoint, we have investigated dynamical structure of chaotic dynamics. To a quite small connectivity from 1 to 60, the network performs chaotic wandering for long time from a random initial state pattern. During this hysteresis, we have taken a statis- tics of continuously staying time in a certain basin [8] and evaluated the distribution p(l , μ) which is defined by p(l , μ) = {the number of l | S(t) ∈ β μ in
μ (4)
and S( τ + l + 1) β μ , μ| μ ∈ [1, L]} β μ = K λ=1
B λ μ (5) T =
lp(l , μ)
(6) where l is the length of continuously staying time steps in each attractor basin, and p(l , μ) represents a distribution of continuously staying l steps in attractor basin L = μ within T steps. In our actual simulation, T = 10
5 . To di
fferent connectivity r=15 and r =50, the distribution p(l, μ) are shown in Fig.10(a) and Fig.10(b). In these figures, di fferent basins are marked with different symbols. From the results, we can know that continuously staying time l becomes longer and longer with increase of the connectivity r. Referring to those novel performances talked in previous section, let us try to consider the reason.
Tracking a Moving Target Using Chaotic Dynamics 187
1 10
100 1000
10000 100000
2 4 6 8 10
12 14
16 Frequency distribution of staying Continuously staying time steps Basin 1
Basin 2 Basin 3
Basin 4 1 10 100 1000
10000 100000
10 20
30 40
50 60
Frequency distribution of staying Continuously staying time steps Basin 1 Basin 2
Basin 3 Basin 4
(a) r = 15: shorter (b) r = 50: longer Fig. 10. The log plot of the frequency distribution of continuously staying time l: The horizontal axis represents continuously staying time steps l in a certain basin μ during long time chaotic wandering, and the vertical axis represents the accumulative number p(l , μ) of the same staying time steps l in a certain basin μ. continuously staying time steps l becomes long with the increase of connectivity r. First, in the case of slower target velocity, a decreasing success rate with the increase of connectivity r is observed from both circular target trajectory and linear ones. This point shows that chaotic dynamics localized in a certain basin for too much time is not good to track a slower target. Second, in the case of faster target velocity, it seems useful to track a faster target when chaotic dynamics is not too strong. Computer simulations shows that, when the target moves quickly, the action of the tracker is always chaotic so as to track the target. In past experiments, we know that motion increments of chaotic motion is very short. Therefore, shorter motion increments and faster target velocity result in bad tracking performance. However, when continuously staying time l in a certain basin becomes longer, the tracker can move toward a certain direction for l steps. This would be useful for the tracker to track the faster target. Therefore, when connectivity becomes a little large (r =50 or so), success rate arises following the increase of target velocity, such as the case shown in Fig.9. As an issue for future study, a functional aspect of chaotic dynamics still has context dependence. 8 Summary We proposed a simple method to tracking a moving target using chaotic dynamics in a recurrent neural network model. Although chaotic dynamics could not always solve all complex problems with better performance, better results often were often observed on using chaotic dynamics to solve certain ill-posed problems, such as tracking a moving target and solving mazes [8]. From results of the computer simulation, we can state the following several points. • A simple method to tracking a moving target was proposed • Chaotic dynamics is quite efficient to track a target that is moving along a circular trajectory. 188 Y. Li and S. Nara • Performance of tracking a moving target of a linear trajectory is not better than that of a circular trajectory, however, to some linear trajectories, excellent performance was observed. • The length of continuously staying time steps becomes long with the increase of synaptic connectivity r that can lead chaotic dynamics in the network. • Continuously longer staying time in a certain basin seems useful to track a faster target.
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Abstract. In this paper, a generalised entropy based associative memory model will be proposed and applied to memory retrievals with analogue embedded vectors instead of the binary ones in order to compare with the conventional autoassociative model with a quadratic Lyapunov functionals. In the present approach, the updating dynamics will be constructed on the basis of the entropy minimization strategy which may be reduced asymptotically to the autocorrelation dynamics as a special case. From numerical results, it will be found that the presently proposed novel approach realizes the larger memory capacity even for the analogue memory retrievals in comparison with the autocorrelation model based on dynamics such as associatron according to the higher-order correlation involved in the proposed dynamics.
During the past quarter century, the numerous autoassociative models have been extensively investigated on the basis of the autocorrelation dynamics. Since the proposals of the retrieval models by Anderson, [1] Kohonen, [2] and Nakano, [3] some works related to such an autoassociation model of the inter-connected neurons through an autocorrelation matrix were theoretically analyzed by Amari, [4] Amit et [5] and Gardner . [6] So far it has been well appreciated that the storage capacity of the autocorrelation model , or the number of stored pattern vectors, L , to be completely associated vs the number of neurons N, which is called the relative storage capacity or loading rate and denoted as
= L / N , is estimated as
~0.14
at most for the autocorrelation learning model with the activation function as the signum one ( sgn (x ) for the abbreviation) . [7,8] In contrast to the above- mentioned models with monotonous activation functions, the neuro-dynamics with a nonmonotonous mapping was recently proposed by Morita, [9] Yanai and Amari, [10] Shiino and Fukai . [11] They reported that the nonmonotonous mapping in a neuro- dynamics possesses a remarkable advantage in the storage capacity, c ~0.27,
superior than the conventional association models with monotonous mappings, e.g. the signum or sigmoidal function. In the present paper, we shall propose a novel approach based on the entropy defined in terms of the overlaps, which are defined by the innerproducts between the A Generalised Entropy Based Associative Model Masahiro Nakagawa Nagaoka University of Technology, Kamitomioka 1603-1, Nagaoka, Niigata 940-2188, Japan masanaka@vos.nagaokaut.ac.jp M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 189–198, 2008. © Springer-Verlag Berlin Heidelberg 2008 al .
state vector and the analogue embedded vectors instead of the previously investigated binary ones [1-16,25]. 2 Theory Let us consider an associative model with the embedded analogue vector e i (r ) (1 i
of embedded vectors. The states of the neural network are characterized in terms of the output vector s i (1 i N ) and the internal states i (1 i N ) which are related each other in terms of
s i =f i (1 i N),
(1)
where f • ( ) is the activation function of the neuron. Then we introduce the following entropy I which is to be related to the overlaps as
I=- 1 2 r=1 L m (r) 2 log m
(r) 2 ,
(2) where the overlaps m (r) (r=1,2,...,L) are defined by
m (r)
= i=1
N e †(r) i s i ;
(3) here the covariant vector e †(r ) i is defined in terms of the following orthogonal relation,
i=1 N e †(r) i e (s) i = rs (1 r,s L) ,
(4)
e †(r)
i = r'=1 L a rr' e (r')
i
(4a)
a rr' =(
-1 ) rr' ,
(4b)
= i=1
N e (r) i e (r') i .
(4c)
The entropy defined by eq.(2) can be minimized by the following condition
(5)
190 M. Nakagawa and
, (r)
= rs m (1 r,s L), r=1 L m (r) 2 = 1
(6)
That is, regarding m (r) 2 (1 r L) as the probability distribution in eq.(2), a target pattern may be retrieved by minimizing the entropy I with respect to m (r) or the state vector
to achieve the retrieval of a target pattern in which the eqs.(5a) and (5b) are to be satisfied. Therefore the entropy function may be considered to be a functional to be minimized during the retrieval process of the auto-association model instead of the conventional quadratic energy functional, E, i.e. E=- 1
i=1 N j=1 N w ij s † i s j ,
(6a) where
s †
is the covariant vector defined by s † i = r=1 L j=1
N e †(r) i e †(r) j s j ,
(6b) and the connection matrix
is defined in terms of w ij
r=1 L e (r) i e †(r) j .
(6c) According to the steepest descent approach in the discrete time model, the updating rule of the internal states
(1 i N ) may be defined by
i (t+1) =-
I s † i (1 i N) ,
(7)
where ( > 0) is a coefficient. Substituting eqs.(2) and (3) into eq.(7) and noting the following relation with aid of eq.(6b),
m (r) = i=1 N e †(r) i s i = i=1
N e (r) i s † i ,
(8)
one may readily derive the following relation. i (t+1)=-
I s † i =+ 1
2 s † i r=1 L m (r) 2 log m
(r) 2 = 1
2 s † i r=1 L j=1 N e (r) j s † j t 2 log j=1
N e (r) j s † j t 2 =
r=1 L e (r) i j=1
N e (r) j s † j t 1+log j=1 N e (r) j s † j t 2 =
r=1 L e (r) i m (r) 1+log m
(r) 2 .
(9) A Generalised Entropy Based Associative Model 191
Generalizing somewhat the above dynamics in order to combine the quadratic approach ( 0) and the present entropy one ( 1) , we propose the following dynamic rule, in a somewhat ad-hoc manner, for the internal states
i
r=1
L e (r) i j=1
N e †(r) j s j t 1+log 1-
+ j=1
N e †(r) j s j t 2 = r=1
L e (r) i m (r) t 1+log 1- + m (r)
t 2 . (10)
In practice, in the limit of 0 , the above dynamics will be reduced to the autocorrelation dynamics.
i (t+1)= lim 0 r=1
L e (r) i m (r) t 1+log 1- + m (r)
t 2
= r=1
L e (r) i m (r) t =- r=1
L e (r) i j=1
N e † (r) j s j (t) = j=1
N w ij s j (t) . (11)
On the other hand, eq.(10) results in eq.(9) in the case of 1 . Therefore one may control the dynamics between the autocorrelation ( 0 ) and the entropy based approach ( 1 ). Download 12.42 Mb. Do'stlaringiz bilan baham: 
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