Robotic Systems Using SoftMax Function. Networks ... SoftMax functions or normalized Gaussian functions [2] ..... left and right camera angles respectively.
Adaptive Output Tracking of Partly Known Robotic Systems Using SoftMax Function Networks Sisil Kumarawadu, Keigo Watanabe, Kazuo Kiguchi, and Kiyotaka Izumi Department of Advanced Systems Control Engineering, Graduate School of Science and Engineering, Saga University, 1-Honjomachi, Saga 840-8502, Japan Abstract–In this paper, a neural-network-based adaptive control scheme is presented to solve the output-tracking problem of a robotic system with unknown nonlinearities. The control scheme ingeniously combines the conventional Resolved Velocity Control (RVC) technique and a neurallyinspired adaptive compensating paradigm constructed using SoftMax function networks and Neural Gas (NG) algorithm. Results of simulations on our active binocular head are reported. The neural network (NN) model is constructed to have two neural subnets to separately take care of robot head’s neck and eye control simplifying the design and making for faster weight tuning algorithms. I. Introduction Artificial neural networks (NNs) have appeared as powerful tools for learning static and highly nonlinear dynamic systems, and recently as an identification based, indirect control technique. It is believed that NNs perform better when they are asked to learn lesser [1]. In this article we describe a neurally-inspired adaptive compensating model for performance improvement of a partially known dynamical system and an application of binocular visual tracking using that model. SoftMax functions or normalized Gaussian functions [2] are used as the nonlinear activation function. We use our own version of a SoftMax function with variable altitude Gaussian functions and an on-line altitude-tuning algorithm. The SoftMax basis function network with variable altitude Gaussian functions guarantees a small network topology and an improved performance in terms of faster convergence. A suitable network size is of utter importance especially when real time weight tuning algorithms are engaged. We determined a minimum network topology by the use of classical pruning procedures and Neural Gas [3] which is an unsupervised network model is engaged to learn important topological relations in the input domain. Although NG inspired position adaptation of neural units does not contribute to tracking error minimization, it gives the network an important robustness
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when the network size is unsuitable to deal with different situations by making sure that the NN covers every part of its input space. The NN model makes a good use of the effectiveness of the NG in distributing the neural units within the input space, and the good approximation properties of SoftMax basis functions. When information processing is complex, it is often judicious to decompose it, for instant, into partitioned NNs or neural subnets, each carrying out a partial, elementary processing. Biological brains organize themselves vastly on this principle and NN applications in the field of robotics should be of no exception. Though the scheme of neural subnets described in this paper to separately take care of the neck and eye control is not a general solution to this line of research, it seems very well suited to similar looking control problems. When the nature of model uncertainties is different in different parts of the robot, the uncertainty on how to initialize the NN weights leads to the necessity for preliminary off-line tuning. The weights of the individual neural subnets can be differently initialized and separately tuned, making for a faster weight update procedure. II. Adaptive Tracking Using SoftMax Function Networks A. SoftMax function networks with variable altitude Gaussian functions SoftMax function with variable altitude Gaussian functions has the analytic expression: G (x − ϕi , ai ) Ui (x, ϕi , ai ) = j G x − ϕj , aj where
x2 G(x, a) = exp f ln a − 2 . 2σ
(1)
(2)
Here, f and ai are the fan-in and the altitude of the ith neural unit respectively. In (1), the Gaussian functions, G(·)s, are of equal variance but different altitudes, and · denotes Euclidian norm. ϕi is the feature’s prototypical or center state, where Ui (·), which is a type of
hyperradial basis function has a maximum value and x is the f-dimensional input vector. The normalizing factor in Ui is assimilable to a lateral inhibition mechanism because highly activated neural units tend to depress other units by largely contributing to their denominator. In the sequel we use the notations · and · F to denote Euclidean and Forbenius norms respectively. B. Robot head dynamics Rigid body dynamics of the head can be written as ˙ τ = M (q)¨ q + h(q, q).
(3)
Substituting the computed torque for τ in (3), yields
the stability of the closed-loop system and weight converˆ ·U ˆ into ˆ is bounded. Expanding v ˆ=W gence, for then v a Taylor series for a given x, yields
ˆ )·U ˆ +W ˆ ·U ˆ · (a − a ˆ )T + ζ ˆ = (W − W v−v
(7)
ˆ ≡ dU /da | ˆ , ζ (ζ < ζN ) represents newhere U a=a ˆ ,a ˆ the current values of the NN glected terms and W weights as provided by the tuning algorithm. With the ideal weights W (W F ≤ Wmax ), a (aF ≤ amax ) that we wish to be reached, define the weight estimation errors as ˜ = W −W ˆ W (8) ˜ = a−a ˆ. a Combining (6), (7), and (8), we have
ˆ (q)(¨ ˆ) ˙ =M M (q)¨ q + h(q, q) q d + K e˙ + v
(4)
ˆ (q) is an approximate inertia matrix, and v ˆ (= where M ˆ ˆ W · U (x, ϕ, ˆ a)) is the NN signal vector, which modifies the resolved acceleration vector aiming to compensate head uncertainties. Torque command is computed using RVC velocity command and an approximate inertia matrix. The RVC velocity command is given by u = q˙ d + Ke where e = q d − q is the tracking error in joint space, and K ∈ n×n is a diagonal constant gain matrix, i.e., K = diag (k1 , · · · , kn). Differentiating and then adding NN signal vector yields
˜U ˆ +W ˆU ˆ a ˜ T + ε + ζ. v˜ = W ˆ ∈ N×N , in which N is the number where U units, is given by dU1 /da1 dU1 /da2 · · · dU1 /daN dU2 /da1 dU2 /da2 · · · dU2 /daN ˆ = U .. .. .. .. . . . . dUN /da1
dUN /da2
(9) of neural .
· · · dUN /daN
ˆ can be calculated as The entries of U Y − Xi f Xi dUi /dai = Y2 ai −Xi f dUi /daj = i = j, X j ; Y2 aj
u˙ = q¨d + K e˙ + vˆ .
where Xi and Y represent the numerator and denominator terms of Ui in (1) respectively. Rewriting the terms in equation (5), yields
Thus, torque command vector is computed as ˆ (q)(¨ ˆ ). τ =M q d + K e˙ + v
˜ E˙ = A E + B v where
A rearrangement of (4) yields ¨ + K e˙ e
=
ˆ −1 (q)M (q) − I]¨ q [M −1 ˆ ˙ −v ˆ +M (q)h(q, q)
=
˜ v
E=
(6)
where v and ε are the desired NN signal vector and approximation error vector (ε < εN ) respectively. C. The neural networks model We give here the mathematical formalism and weighttuning algorithm of the NN. It is required to guarantee
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e e˙
, A=
0n×n 0n×n
In −K
, B=
0n×n In
.
(5)
˜ is the uncertainty compensation error, which can where v also be written as ˜ = (v + ε) − v ˆ v
(10)
Define the signal vector to the NN as r = e + e˙ = C E
(11)
where C = [In , In ] ∈ n×2n . Equations (10) and (11) jointly represent the state-space model for the joint error dynamics. Simple computations prove that whenever kj ≥ 1, for any joint j, Re Tj (s− ) of the transfer function of error dynamics Tj (s) is non-negative, for a real > 0, if the following condition holds (kj + 1)(σ − )2 + (kj − 1)ω 2 σ− >− . (12) (σ − )2 + ω2 + kj
This confirms that Tj (s − ) is positive real for some real > 0, and hence Tj (s) is strictly positive real. Therefore, strictly positive realness of the transfer function of error dynamics for the complete system can be guaranteed by properly selecting the entries of K.
Using the property B T A = tr (ABT ) = tr (B T A) for any A, B ∈ n×1 , we can write
D. Weight updates for guaranteed tracking performance
Now
Let the NN weight tuning be provided by ˆ˙ = F rU ˆ T − κF EW ˆ W ˆU ˆ H − κˆ ˆ˙ = rT W aEH a
˜U ˆ rT W T ˆ ˆ T ˜ r WU a
ˆ rT W ˜) = tr (U T ˆ ˆ T ˜ ). = tr (r W U a
T ˜) ˜˙ F −1 W −E T QE + 2(CE)T (ε + ζ) + 2tr (W T T −1 ˆr W ˜ ) + 2tr (a ˜˙ H a ˜ ) +2tr (U
L˙ = (13)
ˆ a ˆU ˜ T ). +2tr (rT W
where F = F T > 0, H = H T > 0 are any constant matrices and κ > 0 a design parameter. Then the position error and error rate E is bounded and the NN weight convergence is guaranteed with practical bounds. In order to prove the above results, define Lyapunov function candidate as
˜˙ = −W ˆ˙ and a ˜˙ = −a ˆ˙ , with tuning rule (13) Since W
˜ ,a ˜ T F −1 W ˜ ) + tr (˜ ˜ ) = E T P E + tr (W ˜T ) L(E, W aH −1 a (14) where tr(·) denotes the trace of matrix (·). P is a positive definite solution of the Lyapunov equation AT P + P A + Q = 0, for any positive definite matrix Q. Differentiating (14) yields
Since tr [AT (B −A)] ≤ AF BF −A2F for any A, B ∈ ˆ )T ] = tr [ˆ ˆ )], there results m×n and tr [ˆ a(a − a aT (a − a ˆ T (W − W ˆ )] ≤ W ˆ F W F − W ˆ 2 tr [W F (15) T 2 ˆ ) ] ≤ ˆ tr [ˆ a(a − a aF aF − ˆ aF .
L˙
T ˜˙ F −1 W ˜) = E˙ T P E + E T P E˙ + 2tr (W T −1 ˜ ). ˜˙ H a +2tr (a
L˙ =
T
ˆ ) ]. +2κEtr [ˆ a(a − a
And also T
−E T QE + 2(CE) (ε + ζ)
√ ≤ −Qmin E2 + 2 2(εN + ζ N )E (16)
Substitution for E˙ from (10) and rearranging yields L˙
=
E T (AT P + P A)E + 2E T (P B)˜ v T
˜ ) + 2tr (a ˜˙ F −1 W ˜˙ H −1 a ˜ T ). +2tr (W
where Qmin is the minimum singular value of Q. Because of the inequalities (15) and (16), there results L˙
From Kalman-Yakubovich-Popov (KYP) lemma [4], when the transfer function is made strictly positive real, for the system described by (10) and (11) there exist two positive-definite symmetric matrices P and Q, satisfying KYP system AT P + P A + Q = 0,
P B = CT .
Now L˙
=
L˙
T
=
T ˜ ˆ ˆU ˆ a ˜ + ε + ζ) −E T QE + 2(CE) (W U +W
=
T ˜ ) + 2tr (a ˜˙ F −1 W ˜˙ H −1 a ˜T ) +2tr (W ˜U ˆ +W ˆU ˆ a ˜T ) −E T QE + 2r T (W T ˜) ˜˙ F −1 W +2(CE)T (ε + ζ) + 2tr (W T −1 ˜ ). ˜˙ H a +2tr (a
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≤
=
ˆ F (Wmax − W ˆ F ) −Qmin E2 + 2κEW aF ) +2κEˆ aF (amax − ˆ √ +2 2(εN + ζ N )E ˆ F (W ˆ F − Wmax ) −E[QminE + 2κW √ +2κˆ aF (ˆ aF − amax ) − 2 2(εN + ζ N )],
which is negative if the term within braces (TWB) is positive. We can write TWB
˜ −E T QE + 2E T C T v T ˜ ) + 2tr (a ˜˙ F −1 W ˜˙ H −1 a ˜ T ). +2tr (W
˜ from (9) gives Substitution for v
T
−E T QE + 2(CE) (ε + ζ) ˆ T (W − W ˆ )] +2κEtr [W
=
2 ˆ F − Wmax /2)2 − κWmax /2 2κ(W 2 +2κ(ˆ aF − amax /2) √ −κa2max /2 + Qmin E − 2 2(εN + ζ N ),
which is guaranteed to be negative as long as either √ 2 κWmax /2 + κa2max /2 + 2 2(εN + ζ N ) E > (17) Qmin or ˆ F > Wmax /2 W √ 2 /4 + a2 + Wmax 2(εN + ζ N )/κ (18) max /4 +
∑ + +
r
I n 0 n ×n ] 1
1
0 n ×n I n ]
(Kinematic Controller)
Set-point Generator
1
qd
q d
∑ + − q
∑ + − q
e
K
∑ + +
2
u
rN
2
vˆ N
rC
2
s
vˆC
+ vˆ ∑ +
u
NN1 NN2
Mˆ
2
q
Head q
s
q
e
Fig. 1. Overview of the control approach, where NN1 is the neck subnet and NN2 is the camera subnet. n1 and n2 are neck’s degrees-offreedom and the number of cameras respectively
or ˆ aF > amax /2 √ 2 /4 + a2 2(εN + ζ N )/κ. + Wmax max /4 +
(19)
Therefore, convergence is guaranteed and the system of error dynamics is stable in the sense that the size of the state vector is bounded. E. Partitioned neural networks
= =
[θpd − θp , θtd − θt ]T [θld − θl , θrd − θr ]T .
(20)
The NN signal vectors from the two subnets are given by ˆN v ˆC v
= [ˆ vp , vˆt ]T = [ˆ vl , vˆr ]T .
(21)
The procedure involves separately tuning of individual subnets, making for a faster weight updates and independent weight initializations. Now the weight update law (13) is ˆ˙ N W ˆ˙ N a
T
ˆN ˆ − κN FN EN W = FN r N U N
ˆNU ˆ H N − κN a ˆ N EN HN = rN T W N
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=
ˆ − κC FC EC W ˆC FC r C U C
ˆ˙ C a
=
ˆ CU ˆ H C − κC a ˆ C EC HC , rC T W C
T
(22)
where EN = [eTN , e˙ TN ]T , EC = [eTC , e˙ TC ]T , rN = (eN + e˙ N ) and rC = (eC + e˙ C ). The partitioned procedure makes it possible for each subnet to adaptively take care of the performance of the part of the robot, which it is dedicated to. Fig. 1 shows the overview of overall control approach. III. Simulation Results
The NNs approach described in the previous sections allows design in terms of neural subnets. As dynamics of the binocular robotic head is highly coupled, each sub¨ T ]T , net is provided with the same input vector [q T , q˙ T , q T where q = [θp , θt , θl , θr ] ; θp , θt , θl and θr are pan, tilt, left and right camera angles respectively. The tracking error vectors for the neck and for the cameras are given by eN eC
ˆ˙ C W
The proposed compensating NN model is used in our active binocular head controller to enhance the tracking performance by compensating for its inherent dynamic uncertainties. The actual dynamics model used in the dynamic computer simulations is highly coupled and nonlinear. But, in our controller, we only used largely approximated (80%) diagonal elements of the inertia matrix alone. The binocular head kinematics allows both cameras move independently about a vertical axis (independent vergence) and a common tilt motion. Furthermore the neck is capable of a pan independent motion. The head tries to track and fixate a moving object, whose image position is recovered by the imaging and visual processing system. The desired head positions and velocities to be used by the servo controller are computed using a suitable kinematics model [5]. The goal of the learning algorithm is that of maintaining the head position error and error ˆ vector to be rate close to zero by generating a proper v added to modify the motor commands. Computer simulations are carried out for simulated repetitive elliptical target trajectories through (0, ±90, 200) and (±120, 0, 200), and for target motions along arbitrarily selected edges and diagonals of a cuboid
0.04
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2
2
MSE [rad ]
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Fig. 2. Evaluating the performance of individual subnets for elliptical motion, where dotted line: without NN; solid line: with NN
performing intermittent linear motions in the world space. Fig. 2 and Fig. 3 show an evaluation of contribution of the individual neural subnets in terms of mean squared error (MSE) and sum of squared error (SSE) of joint positions for elliptical and for intermittent linear motions respectively. As the tracking performance of the neck and of the cameras depends on each other, when computing the squared error of one subnet, the other subnet is always kept active. The parameters of the NNs are selected as FN = diag(4.0, 4.0), HN = diag(0.2, 0.2, 0.2), κN = 0.001, FC = diag(8.0, 8.0), HC = diag(0.3, 0.3, 0.3), κC = 0.001. Three neural units are used in each subnet NN1 and NN2. Connecting weights are initialized in the ranges [-0.5, 0.5] and [-1.0, 1.0] for NN1 and NN2 respectively, and altitudes in the range [0.8, 1.2] for both subnets. RVC gain matrix is selected as K = diag(30.0, 30.0, 50.0, 50.0). The parameters of the NG algorithm are set alike for both subnets according to Martinetz et al. [6]. Fig. 4 shows the overall performance of the total con-
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Fig. 3. Evaluating the performance of individual subnets for linear motion, where dotted line: without NN; solid line: with NN
troller and the effect of altitude adaptation on NN performance. The neural model keeps the tracking error, which is clearly unbounded in its absence close to zero. The contribution by altitude adaptation is clearly seen for the elliptical case. When the altitudes are constant at unity, the controller goes unstable in the tenth cycle. Altitude estimates for the neural units are given in Fig. 5. IV. Conclusions This paper has presented an adaptive neural networktracking controller. According to the results of computer simulations on our active binocular tracking scheme, the actual and desired head trajectories were accurate and close enough to each other to keep the target images well within the 6.54 × 4.89 [mm] imaging planes of our CCD cameras. The SoftMax basis function network with variable altitude Gaussian functions guaranteed a small network topology and fast convergence. On-line weight tuning algorithm guaranteed tracking with small errors and error rates as well as bounded NN weights. The actual dynam-
Altitudes
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Time t [s]
Fig. 4. Overall performance of the controller, where dotted line: without NN; dashed line: with constant altitude NN; solid line: altitude adapting NN
ics model used in the simulation was highly coupled and nonlinear. But, we have only used largely approximated inertia matrix consisting of diagonal elements alone in the controller and shown that it can still handle the tracking problem to a greater accuracy. NG-based topology learning gives the network an added robustness and makes it possible to efficiently engage a minimal network topology. In standard use in robotics, the signal vector to the NN, also known as filtered tracking error is computed as ˙ where Λ = ΛT > 0 with its entries being r = e + Λe, usually larger values depending on the application. The use of an identity matrix instead gives a better guarantee for a positive definite matrix and a simpler condition to be satisfied for its existence. The unfavorable effects on controller performance of using an identity matrix can be compensated by setting larger values in F and H matrices. References [1] A. Ishiguro, T. Furuhashi, and S. Okuma, “A neural network compensator for uncertainties of robotics manipulators,” IEEE
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0
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Fig. 5. Altitude estimates, where solid line: neck subnet NN1; dashed line: camera subnet NN2
Trans. Industrial Electronics, vol. 39, no. 6, pp. 565–569, 1992. [2] P. Morasso and V. Sanguineti, “Self-organizing body schema for motor planning,” Journal of Motor Behavior, vol. 27, pp. 52–66, 1995. [3] T. M. Martinetz and K. J. Schulten, “ A ‘neural-gas’ network learns topologies,” in Artificial Neural Networks, T. Kohonen, K. M¨ akisara, O. Simula, and J. Kangas, Eds. Amsterdam, The Netherlands: North-Holland, 1991, pp. 397–402. [4] M. Vidyasagar, Nonlinear Systems Analysis—2nd Edition. Englewood Cliffs, NJ: Prentice-Hall, 1993. [5] S. Kumarawadu, K. Watanabe, K. Kiguchi, and K. Izumi, “An application of active vision head control using model-based compensating neural networks controller,” in Proc. of Int. Conf. on Control, Automation and Systems (ICCAS 2001), Cheju, Korea, Oct. 2001, pp. 879–882. [6] T. M. Martinetz, S. G. Berkovich, and K. J. Schulten, “Neuralgas network for vector quantization and its application to timeseries prediction,” IEEE Trans. Neural Networks, vol. 4, no. 4, pp. 558–569, 1993. [7] F. L. Lewis, K. Liu, and A. Yesildirek, “Neural net robot controller with guaranteed tracking performance,” IEEE Trans. Neural Networks, vol. 6, no. 3, pp. 703–715, 1995. [8] F. L. Lewis, A. Yesildirek, and K. Liu, “Multilayer neural-net robot controller with guaranteed tracking performance,” IEEE Trans. Neural Networks, vol. 7, no. 2, pp. 388–399, 1996.
