a new adaptive neuro-fuzzy controller for trajectory tracking of robot ...
International Journal of Robotics and Automation, Vol. 26, No. 1, 2011
A NEW ADAPTIVE NEURO-FUZZY CONTROLLER FOR TRAJECTORY TRACKING OF ROBOT MANIPULATORS Dimitrios C. Theodoridis,∗,∗∗ Yiannis S. Boutalis∗,∗∗∗ and Manolis A. Christodoulou∗∗∗∗,∗∗∗∗∗
Abstract
1. Introduction
In this paper, an adaptive control method for trajectory tracking
Neuro-fuzzy controllers have been successfully applied in the past two decades by many researchers to nonlinear systems. The motivation for this utilization stems from two fundamental properties of neuro-fuzzy systems: (1) the ability of neural fuzzy networks (NFN’s) for low-level learning and computation power of neural networks and (2) the high-level human-like thinking and reasoning of fuzzy theory. Furthermore, it has been established that neural networks and fuzzy inference systems are universal approximators [1–3], i.e., they can approximate any nonlinear function to any prescribed accuracy provided that sufficient hidden neurons and training data or fuzzy rules are available. Trajectory tracking control is one of the fundamental tasks in robotic control applications [4–10]. Traditional control approaches, like computed-torque control, rely on an exact knowledge of the robot dynamics and may lead to poor results under parametric and dynamic uncertainties. Therefore, adaptive control of robotic manipulators has gained significant interest being an active area of research for many decades [5, 11–17]. A basic assumption in adaptive control is that the robot dynamics are expressed in linear with respect to the parameters (LIP) form [5], that is
of robot manipulators, based on new neuro-fuzzy modelling is presented. The proposed control scheme uses a three-layer neural fuzzy network (NFN) to estimate system uncertainties. The function of robot system dynamics is first modelled by a fuzzy system, which in the sequel is approximated by a combination of high order neural networks (HONNs).
The overall representation is
linear in respect to the unknown NN weights leading to weight adaptation laws that ensure stability and convergence to unique global minimum of the error functional. Due to the adaptive neurofuzzy modelling, the proposed controller is independent of robot dynamics, since the free parameters of the neuro-fuzzy controller are adaptively updated to cope with changes in the system and the environment. Adaptation laws for the network parameters are derived, which ensure network convergence and stable control. A weight hopping technique is also introduced to ensure that the estimated weights stay within pre-specified bounds. The simulation results show very good approximation performance of the proposed representation as compared with a simple NN approximator and very good tracking abilities under disturbance torque compared to conventional computed torque PD control.
Key Words Neuro-fuzzy systems, parameter estimation, parameter hopping,
f (x) = R(x)ξ
trajectory tracking, robot manipulators
(1)
where f (x) is a nonlinear robot function, R(x) is a regression matrix of known robot functions and ξ is a vector of unknown robot parameters (usually masses and friction coefficients). Linearity in respect to the unknown parameters leads to parameter adaptation laws that ensure stability and convergence to the unique global minimum of the representation’s error functional. In conventional LIP forms the regressor matrix R(x) depends on f (x) and must be recomputed for each different f (x). This means that for each different type of robot arm, one must recompute R(x). To overcome this restriction, Lewis and his associates introduced the use of neural networks to approximate the nonlinear robot function [13]. In the one-layer neural representation [13, 18, 19], which assumes a LIP
∗
Department of Electrical and Computer Engineering, Democritus University of Thrace, 67100 Xanthi, Greece; e-mail: {dtheodo, ybout}@ee.duth.gr ∗∗ Department of Industrial Informatics, Technological Educational Institute of Kavala, 65404 Kavala, Greece; e-mail: [email protected] ∗∗∗ Department of Electrical, Electronic and Communication Engineering, Chair of Automatic Control, University of Erlangen-Nuremberg, 91058 Erlangen, Germany; e-mail: [email protected] ∗∗∗∗ Department of Electronic and Computer Engineering, Technical University of Crete, 73100 Chania, Greece; e-mail: manolis@ ece.tuc.gr ∗∗∗∗∗ Dipartimento di Automatica et Informatica, Politecnico di Torino, 10129 Torino, Italy; e-mail: [email protected] Recommended by Prof. C. Abdallah (10.2316/Journal.206.2011.1.206-3401)
64
the filtered-error approximation-based control of robotic manipulators, while Section 4 introduces the proposed neuro-fuzzy representation in estimating the unknown robot function. Section 5 develops the neuro-fuzzy based control scheme, proposes its weight adaptation law and proves its stability using Lyapunovs direct method. Section 6 provides simulation results, while Section 7 gives the conclusions.
from the robot function is represented by f (x) = W T φ(x)
(2)
where W is a vector with neural weights and φ(x) is a vector with a basis set of neural functions, which is independent of the robot manipulator at hand. This approach has also been extended to the two-layer neural network controller [13, 20]. Moreover, some implementation considerations are given in [13], which include input signal preprocessing and partitioning of the NN structure into a number of probably simpler NNs, each one being dedicated in approximating a particular part of the robotic function. These considerations lead to improved results but assume some a priori knowledge on the various parts of the general type of the function to be approximated (here the robotic function f (x)). In this paper a new neuro-fuzzy representation is proposed for the filtered-error approximation-based control of robotic manipulators. It is assumed that the manipulator is composed of rigid-link arms moving in free space and the control objective is to force the end effector to follow a desired trajectory. It is also assumed that the system states are measurable. The new representation is based on the assumption that the unknown robotic function f (x) can be initially modelled by a fuzzy system. In the sequel, the fuzzy system is functionally represented by using the so-called “indicator functions” (IF), which in turn are approximated by appropriate high order neural network functions (HONNs). The same principle has been applied by the authors in [21–23] for the indirect and direct control of general affine in the control unknown nonlinear systems. Following this approach the approximator results in a neuro-fuzzy scheme, which is linear in respect to the tunable neural weights. However, unlike the NN partitioning made in [13] no a priori knowledge on the mathematical form of the approximated function is required. The advantage of this method compared with approaches that use only fuzzy system approximation is that, the a priori information related to the underlying fuzzy system turns to be minimal, since the only requirement is the knowledge of the centres of the output fuzzy variables of the underlying fuzzy system. All the other information related to the input fuzzy variables, the corresponding membership functions and the fuzzy rules is automatically captured by the HONNs. Another advantage against the use of simple RHONNs (or NNs in general) is that the resulting neurofuzzy representation can also be considered as a combination of HONN, each one being specialized in a particular part of the function to be approximated and associated with each centre of the fuzzy output partitions.This way, instead of having one large network trying to approximate “everything”, we have many, probably smaller, specialized networks. Conceptually, this strategy is expected to present better approximation results. The remaining paper is organized as follows. Section 2 introduces notation and preliminaries on the functional representation of fuzzy systems using indicator functions and the ability of HONN to act as fuzzy rule approximators. Section 3 presents the robot arm dynamics and
2. Fuzzy System Description using Rule Indicator Functions and HONN Consider a nonlinear function f (x) ∈ p , x ∈ X ⊂ n defined on a compact set Ψ ⊂ n and approximately described by a Mamdani-type fuzzy system (FS). In a dynamic system of the form x˙ = f (x), f (x) represents the dynamics of the system. The following mild assumptions are required for the approximation problem to make sense and to guarantee the existence and uniqueness of solution of x˙ = f (x) for any finite initial condition. Assumption 1. Ψ is a compact set. Notice that since Ψ ⊂ n then Ψ is closed and bounded set. Also, it is noted that even if Ψ is not compact we may assume that there is a time instant T such that x(t) remain in a compact subset of Ψ for all t < T ; i.e. ΨT := {x(t) ∈ Ψ, t < T }. The interval ΨT represents the time period over which the approximation is to be performed. Assumption 2. The vector field f is continuous with respect to its arguments and satisfy a local Lipchitz condition so that the solution x(t) is unique for any finite initial condition. l ,l ,...,l Let Ωj11 ,j22 ,...,jpn be defined as the subset of x ∈ X belonging to the (j1 , j2 , . . . , jn )th input fuzzy patch and pointing – through the vector field f (·) – to the subset which belong to the l1 , l2 , . . . , lp th output fuzzy patch. In l ,l ,...,l other words, Ωj11 ,j22 ,...,jpn contains input values x that are associated through a fuzzy rule with output values f (x). Furthermore, the FS receiving as input the n − tuple of x = (x1 , x2 , . . . , xn ) gives as output an approximate of f (x) using fuzzy rules and well-known fuzzy inference procedure. Definition 1. According to the above notation the rule firing indicator function (RFIF) or simply IF conl ,l ,...,l nected to Ωj11 ,j22 ,...,jpn is defined as follows: ⎧ ⎨ α(x(t)) if x(t) ∈ Ωl1 ,l2 ,...,lp l ,l ,...,l j1 ,j2 ,...,jn Ij11 ,j22 ,...,jpn (x(t)) = (3) ⎩0 otherwise where α(x(t)) denotes the firing strength of the rule. According to standard fuzzy system description, this strength depends on the membership value of each xi in the corresponding input membership functions μF ji and more specifically [3], α(x(t)) = min[μF j1 (x1(k)), . . . , μF jn (xn (k))]. Then, assuming a standard defuzzification procedure (e.g. weighted average), the functional representation of the fuzzy system can be written as l ,l2 ,...,lp l ,l2 ,...,lp f (x(t)) = (¯ xf )j11 ,...,j × (I )j11 ,...,j (x(t)) (4) n n 65
where the summation is carried over all the available fuzzy l ,l2 ,...,lp rules. Also, (xf )j11 ,...,j is any constant vector consisting n of the centers of fuzzy partitions of f determined by l ,l2 ,...,lp l1 , l2 , . . . , lp and (I )j11 ,...,j (x(t)) is a weighted IF (WIF), n being actually the IF defined in (3) divided by the sum of all IF participating in the summation of 4. According to [23], functions of HONNs are capable of approximating discontinuous functions such as WIF. HONNs. are defined as follows: Definition 2. A HONN is defined as: N (x(t); w, L) =
L
whot
hot=1
d (hot)
Φj j
techniques, such as fuzzy C-means clustering [24], mountain clustering [25] and subtractive clustering [26], based on gathered data. Any other information related to the input membership functions and the rules of the underlying fuzzy system is not necessary because it is replaced by the HONNs. 3. Filtered Error Approximation-based Robot Controller Consider an n–DOF robot manipulator, which can be freely moved in its working space and its end-effector is following a trajectory. The manipulator dynamics are described by the following nonlinear differential equation [13, 27]
(5)
j∈Ihot
where Ihot = {I1 , I2 , . . . , IL } is a collection of L not-ordered subsets of {1, 2, . . . , n}, dj (hot) are non-negative integers. Φj are the elements of the following vector, Φ = [Φ1 , . . . , Φn ]T = [s(x1 ), . . . , s(xn ) ]T
¨ + Vm (θ(t), θ(t)) ˙ ˙ + G(θ(t)) τ (t) = M (θ(t))θ(t) θ(t) ˙ + F (θ(t)) + τd (11)
(6)
˙ θ¨ are vectors of joint torques, positions, velocwhere τ, θ, θ, ities, and accelerations, respectively. Also, M, Vm , G, and F are the inertia matrix, the coriolis/centripental matrix, the gravity vector and the friction vector, respectively (see [7, 27] for the physical meaning of these matrices) and τd is a disturbance term. Furthermore, we can define the following skew symmetry property [27]. Definition 3. The Coriolis/Centripetal matrix can always be selected so that the matrix
where s(x) denotes the sigmoid function defined as: s(x) = a
1 −γ 1 + e−βx
(7)
with a, β, γ being positive real numbers and w := [w1 , . . . , wL ]T are the HONN weights. (5) can also be written as N (x(t); w, L) =
L
˙ ≡ M˙ (θ) − 2Vm (θ, θ) ˙ S(θ, θ) whot shot (x(t))
(8)
is skew symmetric. Therefore, xT Sx = 0 for all vectors x. This is a statement of the fact that the fictitious forces in the robot system do no work. Consider now the following tracking error e given by
hot=1
where shot (x(t)) are high order terms of sigmoid functions of x. l ,l2 ,...,lp Following the above notation (I )j11 ,...,j in (4) can be n
e(t) = θd − θ
l ,l ,...,l
2 p approximated by Nj11,...,j (x) = N (x(t); wj1 ,...,jn ;l1 ,l2 ,...,lp , n Lj1 ,...,jn ;l1 ,l2 ,...,lp ). So, (4) can be rewritten as
f (x(t)) =
l ,l ,...,l
l ,l ,...,l
2 p 2 p (¯ xf )j11 ,...,j × Nj11,...,j (x(t)) n n
(12)
(13)
where θd is the desired joint variables trajectories. The filtered version r of the tracking error, given by
(9)
r = e˙ + Λe
where the summation is carried over all the available fuzzy rules. To simplify the model structure, since some rules may result in the same output partition, we may replace the NNs aasociated with the rules having the same output with one NN, and therefore, the summation in (9) is carried out over the number of the corresponding output partitions, that is N pf f (x(t)) = x ¯fl × Nl (x(t)) (10)
(14)
where Λ is a positive definite design parameter matrix. The robot dynamics are expressed in terms of this error as [13] M r˙ = −Vm r + f (x) + τd − τ
(15)
where, taking into consideration (13), (14), the unknown nonlinear function f (x) is defined as
l=1
˙ θ˙d + Λe) + G(θ) + F (θ) ˙ ˙ + Vm (θ, θ)( f (x) = M (θ)(θ¨d + Λe) (16)
From the above definitions and (9) it is obvious that the accuracy of the approximation of f (x) depends on the approximation abilities of HONNs and on an initial estimate of the centres of the output membership functions. These centres can be obtained by experts or by off-line
˙ z2 = (θ˙d + Λe) and One may also define z1 = (θ¨d + Λe), x ≡ [z1T z2T sin(θT ) cos(θT ) θ˙T ]T 66
(17)
A general sort of approximation-based controller is derived by setting
where Mf is a sufficiently large positive constant. Let us also define the modelling error μ as follows:
τ = fˆ + Kv r − υ(t)
μ(x) = f (x) − Xf Wf∗ sf (x)
Δ
(18)
where fˆ is an estimate of f , Kv r = Kv e˙ + Kv Λe an outer PD tracking loop, and υ(t) an auxiliary signal to provide robustness in the face of disturbance and modelling errors. The filtered error approach relies on PD compensation as well as on proper estimation of f . Simple PD controllers have been shown to be able to control the motion of a robot under partial or zero knowledge of its dynamics [13, 28]. However, PD controllers rely on the proper selection of the PD gains, which may vary according to the robot manipulator used. Arbitrarily increasing the gains may result in good performance producing, however, large control signals, which for implementation reasons are sometimes undesirable. Moreover, simple PD schemes usually lack stability proofs. Therefore, proper estimation of f using a short of adaptive technique may result in good performance even in the presence of unknown dynamics and disturbances, providing at the same time mathematical proofs for stability and performance. Moreover, such advanced techniques may be directly extended to more complicated control objectives such as force control for grinding, polishing, etc., where straight PD methods are inadequate [13].
with the modelling error bounded by μ < μN . From the above two equations, we have that the robot dynamics satisfies f (x) = Xf Wf∗ sf (x) + μ
(20)
Note that the term μ can be made arbitrarily small and consequently the high order neuro-fuzzy network (NFN) (19) can approximate any smooth function arbitrarily close by appropriately selecting the number of high order collections LF and the number and values of the centers of fuzzy partitions. Then the control law (18) becomes τ = Xf Wf sf (x) + Kv r − υ(t)
(21)
It is now necessary to show how to tune the NFN weights Wf on line to guarantee stable tracking. To this end, define ˜ f = Wf − W ∗ . Then, the the weight estimation error as W f closed loop filtered error dynamics (15) becomes ˜ f sf (x) + (μ + τd ) + υ(t) (22) M r˙ = −(Kv + Vm )r − Xf W
4. Neuro-Fuzzy Representation for the Unknown Nonlinear Robot Dynamics
5. Neuro-Fuzzy Controller and Error Dynamics
In this section, we present the proposed NFN representation, for a general serial-link robot arm. ˙ θ) ¨ ∈ Ω, where Ω is a bounded simple Let (τ, θ, θ, connected set of n and f (·) : Ω → p . Also we assume that the desired trajectory is inside the workspace of the robot manipulator and moreover all of the points of the trajectory are reachable by the robot’s end-effector. Define C p (Ω) as the space of continuous functions f (·). Since f is a continuous smooth function it can be approximated by the neuro-fuzzy representation given in Section 2. Therefore, according to (10) we can approximate the unknown robot dynamics by several HONNs with LF high-order terms [23]
Suppose that a NFN is used to approximate the nonlinear robot function (16) according to (20) with Wf∗ the optimum approximating weights. The optimum weights are unknown and may even be nonunique. Let also assume, that the desired trajectory θd (t) is bounded by θ¯B a known scalar bound and the initial tracking error is bounded as well. We can now distinguish two possible cases. Case 1: Neuro-fuzzy controller in an ideal case. Suppose the NFN functional reconstruction error μ and unmodelled disturbances τd (t) are equal to zero. In Theorem 1, weight updating laws are given, which can serve the control objectives. It has to be mentioned that according to [21, 23] gradient-type updating laws of the form given in Theorem 1 have been found sufficient for the approximation of general nonlinear functions using the neuro-fuzzy representation of (19). Theorem 1. The control scheme for (11) be given by (21) (with υ(t) = 0), the filtered error system
fˆ(x) = Xf Wf sf (x)
(19)
where Xf is a n × n · m block diagonal matrix Xf = diag{Xf1 , . . . , Xfn } with the ith, i = 1, . . . , n diagonal block element being a row vector containing the m fuzzy output partition centers of fi . Here, for notational simplicity, it is assumed that each fi has the same number m of partitions. Wf is a n · m × LF matrix of synaptic weights, which can be considered as a block matrix Wf = [Wfil ], i = 1, . . . , n, l = 1, . . . , Lf , where each Wfil is a m-dimensional column vector. Finally, sf is a LF dimensional vector with each element sfl (x) being a high order combination of sigmoid functions of the state. Define now the optimal weight vector Wf∗ as follows: Δ Wf∗ = arg sup |f (x) − Xf Wf sf | min |Xf Wf |≤Mf
˜ f sf (x) M r˙ = −(Kv + Vm )r − Xf W
System
(23)
and the weight updating law ˙ f = (Xf )T df ri sl (x) W i i il guarantees the following properties. ˜ f ∈ L∞ , r ∈ L2 • r, W ˜˙ f (t) = 0 • limt→∞ r(t) = 0, limt→∞ W
x∈Ω
67
(24)
Proof: Consider the Lyapunov function candidate 1 1 ˜ T −1 ˜
V = rT M r + tr W (25) f Df W f 2 2
to zero. However, in actual systems there are disturbances. Moreover, the approximation error increases as the number of high order terms or fuzzy centers decreases. In non-ideal case it can be seen that if the approximation error and system disturbance are not zero but bounded, then our NF controller still works under an additional assumption of persistence of excitation (PE). In this case, the tracking error does not converge to zero, but is bounded by small enough values to guarantee good tracking performance. The high order sigmoid terms could be chosen to be persistently exciting by appropriately selecting parameters α, γ in (8). However, if the vector sf in (19) is not persistently exciting, it is well known from adaptive control theory [29] that the phenomenon of parameter drifting may occur. To correct this problem, one may use techniques from adaptive control, including dead-zone, σ or switching σ-modification and e-modification [29]. Alternatively, one could use a modified weight updating algorithm, that includes a weight hopping [22] when, during the adaptation, term Xf Wf in (19) exceeds a pre-specified bound. Theorem 2. The control scheme for (11) given by (21) (with υ(t) = 0, τd = 0, μ = 0), the filtered error system
where Df is a positive definite gain matrix which is selected by the designer. Taking the time derivatives of the Lyapunov function candidate (25) we obtain
1 ˜˙ T D−1 W˜f V˙ = rT M˙ r + rT M r˙ + tr W (26) f f 2 and after substituting (23) we have 1 ˜ f sf (x) V˙ = −rT Kv r + rT (M˙ − 2Vm )r − rT Xf W 2
˜˙ T D−1 W˜f + tr W (27) f
f
But according to Definition 3, (M˙ − 2Vm ) is skew symmetric and therefore rT (M˙ − 2Vm )r = 0, ∀r ∈ n . Let us consider that
˜˙ T D−1 W ˜ f = r T Xf W ˜ f sf (x) tr W f f then the above equation results in the following learning law ˙ f = Df X T rsT W f f
˜ f sf (x) + τd + μ M r˙ = −(Kv + Vm )r − Xf W
(28) and the modified learning law
Taking into account the block representation of Wf given in (19) the above law can also be written in respect to the updating of each Wfil according to (24), where dfi is a m × m gain matrix, which for simplicity could be taken to be diagonal and it’s elements equal. Then V˙ assumes the form V˙ = −rT Kv r ≤ −λmin (Kv )r2
(30)
˙ f = HP (Xf )T df ri sl (x); W i i il f
(Xfi )T dfi ri sl (x)
κdfi r(Xfi Wfil (Xfi )T ) − (31) tr{(Xfi )T Xfi }
Δ
(29)
where Pf = {Xfi ·Wfil : |Xfi · Wfil | < Mfi }, κ a positive constant selected from the designer and HPf represents the switching criterion between the two updating laws inside the square brackets guarantees the uniform ultimate boundedness of r with respect to the set
Since V > 0 and V˙ ≤ 0, we conclude that V ∈ L∞ , which ˜ f ∈ L∞ . Furthermore, Wf = W ˜ f + W ∗ are implies that r, W f also bounded. Since V is a non-increasing function of time and bounded from below, limt→∞ V = V∞ exists; therefore, by integrating V˙ from 0 to ∞ we have ∞ λmin (Kv ) r2 dt ≤ [V (0) − V∞ ] < ∞
=
0
(τB + μ ¯N ) r(t) : r(t) ≤ , λmin (Kv ) = 0 λmin (Kv )
Proof: Consider the Lyapunov function candidate (25). Taking its time derivatives and taking into account (22) with the disturbance τd and the NFN reconstruction error μ different than zero we obtain
which implies that r ∈ L2 . ˜ f , sf are bounded, r ∈ L∞ . Since Xf , W Since r ∈ L2 ∩ L∞ , using Barbalat’s Lemma we conclude that limt→∞ r(t) = 0. Hence, given that Λ in (14) is positive definite, we also have that
1 ˜ f sf (x) V˙ = −rT Kv r + rT (M˙ − 2Vm )r − rT Xf W 2
˜˙ T D−1 W˜f + rT (τd + μ) + tr W (32) f
lim e(t) = 0
f
t→∞
The skew symmetric property of (M˙ − 2Vm ) together with the learning law (24) (which is equivalent to (31) if no hopping occurs) results to the following form of V˙
Now, using the boundness of Xf , sf and the convergence of ˜˙ f also converges to zero [29]. r(t) to zero, we have that W Case 2: Neuro-fuzzy controller in non-ideal case. It has just been seen that under the ideal case of no modelling errors or unmodelled disturbances, our algorithm through time suffices to make the tracking error to go
V˙ ≤ −λmin (Kv )r2 + r(τB + μ ¯N ) ≤ −(λmin (Kv )r − (τB + μ ¯N ))r ≤ 0 68
(33)
Remark 1. The previous analysis reveals that in the case where we have a modelling error and disturbance different from zero at θ = 0, our adaptive algorithm can guarantee at least uniform ultimate boundedness of all signals in the closed loop. In particular, Theorem 2 shows that if τB + μ ¯N is sufficiently small, or if the design constant Kv is chosen appropriately, then r(t) can be arbitrarily close to zero and in the limit as Kv → ∞, actually becomes zero but implementation issues constraint the maximum allowable value of Kv . Remark 2. The switching condition HPf in (31) could be simply a threshold function determining whether Δ Xfi · Wfil ∈ Pf = {Xfi · Wfil : |Xfi · Wfil | < Mfi } or not. In a more elegant version one could also take into account the direction of the weight updating. In case the threshold function is activated but the weight updating is in the direction that moves Xfi · Wfil toward Pf , the switching is finally not activated. Details for this more elegant version can be found in [21, 22]. Remark 3. The weight hopping condition arrives from the desire to keep |Xf Wf sf | < Mf , where Mf is an k upper bound. Since Xfi Wfi sf = Xfi Wfil sfl (x) and l=1 k |Xfi Wfi sf (x)| ≤ l=1 |Xfi Wfil sfl (x)|, one is allowed to have a better insight in the algorithm if specialized bounds are put, each one for each component of Xf Wf sf (x). Each specialized bound may be expressed by |Xfi · Wfil | < Mfi , with Mfi being again a design parameter determining an external limit for Xfi · Wfil . In the sequel, we consider the forbidden hyperplanes being defined by the equation |Xfi · Wfil | = Mfi . Note that the direction of the weight updating (24) is perpendicular to the forbidden hyperplane. When the weight vector reaches one of the forbidden hyperplanes |Xfi · Wfil | = Mfi and the direction of updating is towards the forbidden hyperplane, a new hopping is introduced which moves the weights back to the desired hyperspace Pf , allowing thus the algorithm to search the entire space for a better weight solution. This procedure is depicted in Fig. 1, in a simplified 2dimensional representation. The magnitude of hopping is −κdfi r(Xfi Wfil (Xfi )T )/tr{(Xfi )T Xfi } being determined by following the vectorial proof in [22] and dfi is the ith element of the gain matrix Df .
provided that r >
(τB + μ ¯N ) λmin (Kv )
(34)
and λmin (Kv ) = 0 which is valid from definition. Inequality (34) demonstrates that the trajectory of r(t) is uniformly bounded with respect to the arbitrarily small (since Kv can be chosen sufficiently large), set shown below ¯N ) (τB + μ = r(t) : r(t) ≤ , λmin (Kv ) = 0 λmin (Kv ) In case the complete modified weight updating law (31) is applied, that is when the switching condition is occasionally met, then the second updating law inside the square brackets of (31) is applied. In this case and since the weight jump −κdfi r(Xfi Wfil (Xfi )T ) / tr{(Xfi )T Xfi } = −κdfi rhb can also be expressed in respect to the error weight vector w ˜ w ˜fil as −κdfi rhb w˜ffil [21, 22], V˙ satisfies the following il inequality ¯N ) − rΘ V˙ ≤ −λmin (Kv )r2 + r(τB + μ
(35)
with Θ being a positive constant expressed as ˜fi )T w˜fi )/w ˜fi , where the summation inΘ= κdfi hb ((w cludes all weight vectors which require hopping. Therefore, the negativity of V˙ is actually enhanced by the presence of Θ, with Θl ≤ Θ ≤ Θu where Θl stands for the case where only one hopping occurs and Θu where all possible hopping occur. After substituting with the worst case of Θ = Θl , V˙ becomes V˙ ≤ −λmin (Kv )r2 + r(τB + μ ¯N ) − Θl r ≤ −(λmin (Kv )r + Θl − (τB + μ ¯N ))r ≤ 0 (36) provided that r >
¯ N ) − Θl (τB + μ λmin (Kv )
(37)
Finally, inequality (37) demonstrates that the trajectory of r(t) is uniformly bounded with respect to the arbitrarily small (since Kv can be chosen sufficiently large, and the presence of Θl can make the ratio even smaller), set shown below (τB + μ ¯ N ) − Θl = r(t) : r(t) ≤ λmin (Kv )
6. Simulation Results The planar two-link revolute arm shown in Fig. 2 is used extensively in the literature for easy simulation of robotic controllers, with results that can be easily illustrated in 2-dimensional plots. Its dynamics are given as [7]
Thus, θ(t) is also bounded with respect to r(t). Furthermore, we have
x˙ 1 = x3 x˙ 2 = x4
˜ f sf (x) + τd + μ M r˙ = −(Kv + Vm )r − Xf W
[x˙ 3 x˙ 4 ]
˜ f is assured by the use of the Since the boundedness of Xf W modified weight hopping algorithm and τd + μ is bounded by definition, we conclude that r˙ ∈ L∞ .
T
= −M −1 (N + τ )
where x1 = θ1 is the angular position of joint 1, x2 = θ2 is the angular position of joint 2, x3 = θ˙1 is the angular 69
neural approximation (2). The weight updating laws for the simple NN approximation are given as follows [13]: ˙ = F φ(x)rT W
(38)
where F is a positive definite gain matrix. The weight updating law for the neuro-fuzzy approximation is given by (28) or equivalently by (24). In both approaches only first order sigmoid neural functions were used. The number of output membership partitions required by the neurofuzzy approach was arbitrarily selected to be m = 5 and the centres of these partitions were again arbitrarily selected to be: Xf1 = [−35 Xf2 = [10
Figure 2. Two-link planar elbow arm. velocity of joint 1 and x4 = θ˙2 is the angular velocity of joint 2. Also, the matrices M , N have the following form ⎡ ⎤ m2 a22 + m2 a1 a2 cos(x2 ) (m1 + m2 )a21 + m2 a22 ⎢ ⎥ ⎢ ⎥ M = ⎢ +2m2 a1 a2 cos(x2 ) ⎥ ⎣ ⎦ m2 a22 + m2 a1 a2 cos(x2 ) m2 a22 ⎢ ⎢ N =⎢ ⎣
−m2 a1 a2 (2x3 x4 + x24 ) sin(x2 )
30
75
20 110
75
100]
170]
That is, no actual expert’s knowledge has been used. No other fine tunning has been followed regarding the functions of the regression vector apart from the selection of sigmoid parameters and adaptation gains, which were selected by trial and error such that both representations give their best performance. Under these conditions both approaches are expected to have significant modelling errors, which in turn influence the performance of the controller and its tracking abilities. Nevertheless, to compare the approximation abilities of the two approaches, the robustifying term in (21) was selected to be 0. The controller parameters were taken as Kv = diag{20, 20}, Λ = diag{5, 5}, Df = diag{1.8, 1.5}, and F = diag{50, 50}. The sigmoid values for the NFN approach were α = 0.5, β = 0.4, and γ = 0.3 and for the NN approach were α = 0.6, β = 0.5, and γ = 0.4, respectively. Figure 3(a) shows the desired and the actual trajectories produced by the NFN and the NN approaches, respectively. It can be observed that the NFN approach outperforms the simple NN one. The MSE’s were 0.0843 for the NFN and 0.1526 for the NN representation. Similar performance is demonstrated in Figure 3(b), where the tracking of the second joint is illustrated. The MSE’s are 0.0274 and 0.0443, respectively.
Figure 1. Pictorial representation of the outer hopping.
⎡
−15
6.2 Scenario 2
⎤
Next, the performance of the proposed NFN representation was compared to conventional computed torque PD (CT-PD) control [7], where the PD gains were selected to be Kp = 100, Kv = 20. The conventional computed torque controller assumes the exact knowledge of the robot model and its parameters. The NFN model was the same as in the previous experiments but in the control law a simple robustifying term υ(t) = 300 e(t) was added to account for modelling error and/or disturbances. A constant disturbance term τd = 20 was added in both cases and the performance of both approaches were tested under the presence of this disturbance. Figure 4(a) and (b) shows the performance of both CT-PD and NFN approaches in following the desired trajectories for joints 1 and 2, respectively, when there is not any disturbance (τd = 0). As it was expected the CT-PD controller performs much
⎥ ⎥ +(m1 + m2 )ga1 cos(x1 ) + m2 ga2 cos(x1 + x2 ) ⎥ ⎦ m2 a1 a2 x23 sin(x2 ) + m2 ga2 cos(x1 + x2 )
We took the arm parameters as a1 = a2 = 1 m, m1 = m2 = 1 kg and the desired trajectory x1d = sin(t), x2d = cos(t). To test the applicability and the performance of the proposed technique a number of simulations were carried out according to the following scenarios. 6.1 Scenario 1 First the approximation abilities of the proposed neurofuzzy representation was tested against a simple one-layer 70
Figure 3. (a) Response of NFN and NN algorithms in trajectory tracking of joint angle 1 and (b) response of NFN and NN algorithms in trajectory tracking of joint angle 2, against the desired trajectory.
Figure 4. (a) Response of NFN and CT-PD algorithms in trajectory tracking of joint angle 1 and (b) response of NFN and CT-PD algorithms in trajectory tracking of joint angle 2, against the desired trajectory without any disturbance in both schemes.
in both cases (NFN method and CT-PD approach) but with the superiority of the NFN method to remain.
better, since it has complete knowledge of the robot model and its parameters. Figure 5(a) and (b) shows the performance of both approaches when there is the disturbance term τd = 20. Figure 6(a) and (b) shows the respective tracking errors. It can be seen that the performance of the CT-PD is seriously affected, while the performance of the NFN approach is only slightly affected.
6.3 Scenario 3 In this case, we assume that the robot is tracking a trajectory and it suddenly drops the load it is carrying. This means, for example, that the mass m2 is reduced from 1, to 0.8 kg after the first 5 s. Figure 8(a) and (b) shows the respective tracking errors. It can be seen that the performance of the CT-PD is seriously affected, especially in the
In the same scenario, we increased the proportional gain Kp to 400 and we took Figure 7(a) and (b), which shows a sufficient improvement to the trajectory tracking 71
Figure 5. (a) Response of NFN and CT-PD algorithms in trajectory tracking of joint angle 1 and (b) response of NFN and CT-PD algorithms in trajectory tracking of joint angle 2, against the desired trajectory with disturbance in both schemes.
Figure 6. (a) Tracking error of the first joint and (b) tracking error of the second joint, for the NFN and CT-PD approaches in both schemes.
7. Conclusion
second joint angle x2 , while the performance of the NFN approach is only slightly affected.
This paper proposed a new neuro-fuzzy approximation of unknown robotic functions f , which is used in a filterederror approximation-based robotic controller. The unknown nonlinear robotic function is first modelled as a fuzzy system, which is functionally expressed by using fuzzy rule indicating functions (IF). Each IF is in the sequel approximated by a HONN leading thus in a representation, which combines a number of HONN each one being specialized in approximating a part of f associated with a fuzzy output partition of the underlying fuzzy system. Using this representation the filtered-error approximation-based robotic controller for trajectory tracking was presented and NN weight updating laws were derived. A weight hopping technique was also introduced to ensure that the estimated weights stay within pre-specified bounds. It was shown
6.4 Scenario 4 In this case, we assume that the robot is tracking a trajectory and a sudden disturbance is applied. This happened between 10 and 20 s of simulation time and the disturbance had the following form τd = 10 + 5 · sin(18 · π · t)
(39)
Figure 9(a) and (b) shows the respective tracking errors. It can be seen that the performance of the CT-PD is seriously affected especially the joint angle x2 , while the performance of the NFN approach is only slightly affected. 72
Figure 7. (a) Tracking error of the first joint and (b) tracking error of the second joint, for the NFN and CT-PD approaches in both schemes when Kp = 400.
Figure 8. (a) Tracking error of the first joint and (b) tracking error of the second joint, for the NFN and CT-PD approaches when the mass m2 is reduced to 0.8 kg in both schemes.
Figure 9. (a) Tracking error of the first joint and (b) tracking error of the second joint, for the NFN and CT-PD approaches when a sudden disturbance is applied between 10 and 20 s in both schemes. 73
that in the ideal case, where the modelling error and the disturbances are zero, the tracking error goes asymptotically to zero. In case the modelling error and the unmodelled disturbances are not zero, it was shown that the tracking error remains bounded. Simulation results were presented on a 2-link revolute joint arm and comparisons were made with the performance of a simple NN approach as well as with the performance of a conventional computed torque PD control under the presence of unknown disturbance.
[22]
[23]
[24] [25]
References [26] [1] K. Hornik, M. Stinchcombe, & H. White, Multilayer feedforward networks are universal approximators, Neural Networks, 2, 1989, 359–366. [2] K. Passino & S. Yurkovich, Fuzzy control (Addison, 1998). [3] L. Wang, Adaptive fuzzy systems and control: Design and stability analysis (NJ: Prentice Hall, 1994). [4] D. Braganza, W.E. Dixon, D.M. Dawson, & B. Xian, Tracking control for robot manipulators with kinematic and dynamic uncertainty, International Journal of Robotics and Automation, 23 (8), 2008, 117–126. [5] J. Craig, Adaptive control of mechanical Manipulators (Reading, MA: Addison-Wesley, 1985). [6] M.I. EL-Hawwary & A.L. Elshafei, Robust adaptive fuzzy control of a two-link robot arm, International Journal of Robotics and Automation, 21 (4), 2006, 266–272. [7] F.L. Lewis, C.T. Abdallah, & D.M. Dawson, Control of Robot Manipulators (New York: Macmillan, 1993). [8] K. Najim, E. Ikonen, & E.G. Ramirez, Trajectory tracking control based on a genealogical decision tree controller for robot manipulators, International Journal of Innovative Computing, Information and Control, 4 (1), 2008, 53–62. [9] W. Yu & X. Li, Pd control of robot with velocity estimation and uncertainties compensation, International Journal of Robotics and Automation, 21 (1), 2006, 1–9. [10] P. Dorato C.T. Abdallah, D. Dawson, & M. Jamshidi, Survey of robust control for rigid robots, IEEE Control Systems Magazine, 11, 1991, 24–30. [11] M.M. Fateh & A. Azarfar, Improving fuzzy control of robot manipulators by increasing scaling factors, ICIC Express Letters, 3 (3(A)), 2009, 513–518. [12] C.W. Lai, P.K. Chen, Y.N. Chung, & C.H. Hsu, Applying adaptive estimator to maneuvering tracking system, ICIC Express Letters, 3 (3(A)), 2009, 427–432. [13] F. Lewis, S. Jaganathan, & A. Yesildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems (Taylor and Francis, 1999). [14] B. Li, X. Yang, J. Zhao, & P. Yan, Minimum time trajectory generation for a novel robotic manipulator, International Journal of Innovative Computing, Information and Control, 5 (2), 2009, 369–378. [15] D.M. Dawson, M.M. Bridges, & C.T. Abdallah, Control of rigidlink flexible joint robots: A survey of backstepping approaches, Journal of Robotic Systems, 12 (3), 1995, 199–216. [16] K.-C. Chiou & S.-J. Huang, An adaptive fuzzy controller for robot manipulators, Mechatronics, 15 (2), 2005, 151–177. [17] Y. Jin, Decentralized adaptive fuzzy control of robot manipulators, IEEE Transactions on SMCPART B, 28 (1), 1998, 47–57. [18] F.L. Lewis, K. Liu, & A. Yesildirek, Neural net robot controller with guaranteed tracking performance, IEEE Transactions on Neural Networks, 6 (3), 1995, 703–715. [19] F.L. Lewis, A. Yesildirek, & K. Liu, Neural net robot controller: Structure and stability proofs, Journal of Intelligent and Robotic Systems, 12, 1995, 277–299. [20] F.L. Lewis, A. Yesildirek, & K. Liu, Multilayer neural net robot controller: Structure and stability proofs. IEEE Transactions on Neural Networks, 7 (2), 1996, 1–12. [21] Y.S. Boutalis, D.C. Theodoridis, & M.A. Christodoulou, A new neuro fds definition for indirect adaptive control of unknown
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nonlinear systems using a method of parameter hopping, IEEE Transactions on Neural Networks, 20 (4), 2009, 609–625. D.C. Theodoridis, Y.S. Boutalis, & M.A. Christodoulou, Direct adaptive control of unknown nonlinear systems using a new neuro-fuzzy method together with a novel approach of parameter hopping, Kybernetica, 45 (3), 2009, 349–386. D.C. Theodoridis, Y.S. Boutalis, & M.A. Christodoulou, A new neuro-fuzzy dynamical system definition based on high order neural network function approximators, European Control Conference ECC-09, Budapest, Hungary, 2009. S.L. Chui, Fuzzy model identification based on cluster estimation, Journal of Intelligent and Fuzzy Systems, 2, 1994. J. Zhang & A.J. Morris, Recurrent neuro-fuzzy networks for nonlinear process modeling, IEEE Transactions on Neural Networks, 10 (2), 1999, 313–326. S. Mitra & Y. Hayashi, Neuro-fuzzy rule generation: Survey in soft computing framework, IEEE Transactions on Neural Networks, 11, 2000, 748–768. M.W. Spong & M. Vidyasagar, Robot dynamics and control (New York: Wiley, 1989). S. Arimoto & F. Miyazaki, Stability and robustness of pid feedback control for robotics manipulators of sensory capabilities, 1st Int. Symp. Robotics Research, 1984, 385–407. P. Ioannou & B. Fidan, Adaptive control tutorial (SIAM, 2006).
Biographies Dimitrios C. Theodoridis was born in Solingen (Germany) in 1974. He received a diploma in Physics and a M.Sc. degree in Electrical Physics (Radioelectrology) from the Aristotle University of Thessaloniki (AUTH) in 1997 and 1999, respectively. He is currently a Ph.D. candidate at the Democritus University of Thrace (DUTH). Since 2005 he served as a lecturer in the Department of Industrial Informatics in the Technological Educational Institute of Kavala. His main research interests lie in the field of neuro-fuzzy networks, neural networks, adaptive control, fuzzy logic control, intelligent control, nonlinear control systems, robot manipulators, and applications to electric drive systems. He is a member of the Hellenic Physical Society and member of the Hellenic Electronic PhysicalRadioelectrology Society. He has served as a reviewer for scientific journals in the area of neural networks, control, robotics, and automation as well as member of scientific conference committees (session chair). Yiannis S. Boutalis received his diploma of Electrical Engineer in 1983 from Democritus University of Thrace (DUTH), Greece and the Ph.D. degree in Electrical and Computer Engineering (topic Image Processing) in 1988 from the Computer Science Division of National Technical University of Athens, Greece. Since 1996, he serves as a faculty member at the Department of Electrical and Computer Engineering, DUTH, Greece, where he 74
International Journals. He is managing and cooperating on various research projects in Greece, in the European Union in collaboration with the United States. He has held many administrative positions such as the Vice Presidency of the Technical University of Crete, Chairman of the office of Sponsored research and a member of the board of governors of the University of Peloponnese. He is a member of the Technical Chamber of Greece. He has been active in the IEEE CS society as the founder and first Chairman of the IEEE Control Systems Society Greek Chapter, which received the 1997 Best Chapter of the Year Award and as the founder of the IEEE Mediterranean Conference on Control and Automation, which became an annual event. Dr. Christodoulou received the MCA Founders award in 2005. He is a member of the board of governors of the Mediterranean Control Association since 1993.
is currently an associate professor and director of the Automatic Control Systems lab. Currently, he is also a Visiting Professor for research cooperation at ErlangenNuremberg University of Germany, chair of Automatic Control. He served as an assistant visiting professor at University of Thessaly, Greece, and as a visiting professor in Air Defence Academy of General Staff of airforces of Greece. He also served as a researcher in the Institute of Language and Speech Processing (ILSP), Greece, and as a managing director of the R&D SME Ideatech S.A, Greece, specializing in pattern recognition and signal processing applications. His current research interests are focused in the development of Computational Intelligence techniques with applications in Control, Pattern Recognition, Signal and Image Processing Problems. Manolis A. Christodoulou was born in Kifissia, Greece, in 1955. He received diploma degree (EE’78) from the National Technical University of Athens, Greece, the M.S. degree (EE’79) from the University of Maryland, College Park the engineer degree (EE’82) from the University of Southern California, Los Angeles and the Ph.D. degree (EE’84) from the Democritus University, Thrace, Greece. He joined the Technical University of Crete, Greece in 1988, where he is currently a Professor of Control. He has been a Visiting Professor at Georgia Tech., Syracuse University, the University of Southern California, Tufts University, Victoria University and the Massachusetts Institute of Technology. He has authored and co-authored more than 200 journal articles, book chapters, books, and conference publications in the areas of control theory and applications, robotics, factory automation, computer integrated manufacturing in engineering, neural networks for dynamic system identification and control, in the use of robots for minimally invasive surgeries and recently in systems biology. Dr. Christodoulou is the organizer of various conferences and sessions of IEEE and IFAC and guest editor in various special issues of
75
A NEW ADAPTIVE NEURO-FUZZY CONTROLLER FOR TRAJECTORY TRACKING OF ROBOT MANIPULATORS Dimitrios C. Theodoridis,∗,∗∗ Yiannis S. Boutalis∗,∗∗∗ and Manolis A. Christodoulou∗∗∗∗,∗∗∗∗∗
Abstract
1. Introduction
In this paper, an adaptive control method for trajectory tracking
Neuro-fuzzy controllers have been successfully applied in the past two decades by many researchers to nonlinear systems. The motivation for this utilization stems from two fundamental properties of neuro-fuzzy systems: (1) the ability of neural fuzzy networks (NFN’s) for low-level learning and computation power of neural networks and (2) the high-level human-like thinking and reasoning of fuzzy theory. Furthermore, it has been established that neural networks and fuzzy inference systems are universal approximators [1–3], i.e., they can approximate any nonlinear function to any prescribed accuracy provided that sufficient hidden neurons and training data or fuzzy rules are available. Trajectory tracking control is one of the fundamental tasks in robotic control applications [4–10]. Traditional control approaches, like computed-torque control, rely on an exact knowledge of the robot dynamics and may lead to poor results under parametric and dynamic uncertainties. Therefore, adaptive control of robotic manipulators has gained significant interest being an active area of research for many decades [5, 11–17]. A basic assumption in adaptive control is that the robot dynamics are expressed in linear with respect to the parameters (LIP) form [5], that is
of robot manipulators, based on new neuro-fuzzy modelling is presented. The proposed control scheme uses a three-layer neural fuzzy network (NFN) to estimate system uncertainties. The function of robot system dynamics is first modelled by a fuzzy system, which in the sequel is approximated by a combination of high order neural networks (HONNs).
The overall representation is
linear in respect to the unknown NN weights leading to weight adaptation laws that ensure stability and convergence to unique global minimum of the error functional. Due to the adaptive neurofuzzy modelling, the proposed controller is independent of robot dynamics, since the free parameters of the neuro-fuzzy controller are adaptively updated to cope with changes in the system and the environment. Adaptation laws for the network parameters are derived, which ensure network convergence and stable control. A weight hopping technique is also introduced to ensure that the estimated weights stay within pre-specified bounds. The simulation results show very good approximation performance of the proposed representation as compared with a simple NN approximator and very good tracking abilities under disturbance torque compared to conventional computed torque PD control.
Key Words Neuro-fuzzy systems, parameter estimation, parameter hopping,
f (x) = R(x)ξ
trajectory tracking, robot manipulators
(1)
where f (x) is a nonlinear robot function, R(x) is a regression matrix of known robot functions and ξ is a vector of unknown robot parameters (usually masses and friction coefficients). Linearity in respect to the unknown parameters leads to parameter adaptation laws that ensure stability and convergence to the unique global minimum of the representation’s error functional. In conventional LIP forms the regressor matrix R(x) depends on f (x) and must be recomputed for each different f (x). This means that for each different type of robot arm, one must recompute R(x). To overcome this restriction, Lewis and his associates introduced the use of neural networks to approximate the nonlinear robot function [13]. In the one-layer neural representation [13, 18, 19], which assumes a LIP
∗
Department of Electrical and Computer Engineering, Democritus University of Thrace, 67100 Xanthi, Greece; e-mail: {dtheodo, ybout}@ee.duth.gr ∗∗ Department of Industrial Informatics, Technological Educational Institute of Kavala, 65404 Kavala, Greece; e-mail: [email protected] ∗∗∗ Department of Electrical, Electronic and Communication Engineering, Chair of Automatic Control, University of Erlangen-Nuremberg, 91058 Erlangen, Germany; e-mail: [email protected] ∗∗∗∗ Department of Electronic and Computer Engineering, Technical University of Crete, 73100 Chania, Greece; e-mail: manolis@ ece.tuc.gr ∗∗∗∗∗ Dipartimento di Automatica et Informatica, Politecnico di Torino, 10129 Torino, Italy; e-mail: [email protected] Recommended by Prof. C. Abdallah (10.2316/Journal.206.2011.1.206-3401)
64
the filtered-error approximation-based control of robotic manipulators, while Section 4 introduces the proposed neuro-fuzzy representation in estimating the unknown robot function. Section 5 develops the neuro-fuzzy based control scheme, proposes its weight adaptation law and proves its stability using Lyapunovs direct method. Section 6 provides simulation results, while Section 7 gives the conclusions.
from the robot function is represented by f (x) = W T φ(x)
(2)
where W is a vector with neural weights and φ(x) is a vector with a basis set of neural functions, which is independent of the robot manipulator at hand. This approach has also been extended to the two-layer neural network controller [13, 20]. Moreover, some implementation considerations are given in [13], which include input signal preprocessing and partitioning of the NN structure into a number of probably simpler NNs, each one being dedicated in approximating a particular part of the robotic function. These considerations lead to improved results but assume some a priori knowledge on the various parts of the general type of the function to be approximated (here the robotic function f (x)). In this paper a new neuro-fuzzy representation is proposed for the filtered-error approximation-based control of robotic manipulators. It is assumed that the manipulator is composed of rigid-link arms moving in free space and the control objective is to force the end effector to follow a desired trajectory. It is also assumed that the system states are measurable. The new representation is based on the assumption that the unknown robotic function f (x) can be initially modelled by a fuzzy system. In the sequel, the fuzzy system is functionally represented by using the so-called “indicator functions” (IF), which in turn are approximated by appropriate high order neural network functions (HONNs). The same principle has been applied by the authors in [21–23] for the indirect and direct control of general affine in the control unknown nonlinear systems. Following this approach the approximator results in a neuro-fuzzy scheme, which is linear in respect to the tunable neural weights. However, unlike the NN partitioning made in [13] no a priori knowledge on the mathematical form of the approximated function is required. The advantage of this method compared with approaches that use only fuzzy system approximation is that, the a priori information related to the underlying fuzzy system turns to be minimal, since the only requirement is the knowledge of the centres of the output fuzzy variables of the underlying fuzzy system. All the other information related to the input fuzzy variables, the corresponding membership functions and the fuzzy rules is automatically captured by the HONNs. Another advantage against the use of simple RHONNs (or NNs in general) is that the resulting neurofuzzy representation can also be considered as a combination of HONN, each one being specialized in a particular part of the function to be approximated and associated with each centre of the fuzzy output partitions.This way, instead of having one large network trying to approximate “everything”, we have many, probably smaller, specialized networks. Conceptually, this strategy is expected to present better approximation results. The remaining paper is organized as follows. Section 2 introduces notation and preliminaries on the functional representation of fuzzy systems using indicator functions and the ability of HONN to act as fuzzy rule approximators. Section 3 presents the robot arm dynamics and
2. Fuzzy System Description using Rule Indicator Functions and HONN Consider a nonlinear function f (x) ∈ p , x ∈ X ⊂ n defined on a compact set Ψ ⊂ n and approximately described by a Mamdani-type fuzzy system (FS). In a dynamic system of the form x˙ = f (x), f (x) represents the dynamics of the system. The following mild assumptions are required for the approximation problem to make sense and to guarantee the existence and uniqueness of solution of x˙ = f (x) for any finite initial condition. Assumption 1. Ψ is a compact set. Notice that since Ψ ⊂ n then Ψ is closed and bounded set. Also, it is noted that even if Ψ is not compact we may assume that there is a time instant T such that x(t) remain in a compact subset of Ψ for all t < T ; i.e. ΨT := {x(t) ∈ Ψ, t < T }. The interval ΨT represents the time period over which the approximation is to be performed. Assumption 2. The vector field f is continuous with respect to its arguments and satisfy a local Lipchitz condition so that the solution x(t) is unique for any finite initial condition. l ,l ,...,l Let Ωj11 ,j22 ,...,jpn be defined as the subset of x ∈ X belonging to the (j1 , j2 , . . . , jn )th input fuzzy patch and pointing – through the vector field f (·) – to the subset which belong to the l1 , l2 , . . . , lp th output fuzzy patch. In l ,l ,...,l other words, Ωj11 ,j22 ,...,jpn contains input values x that are associated through a fuzzy rule with output values f (x). Furthermore, the FS receiving as input the n − tuple of x = (x1 , x2 , . . . , xn ) gives as output an approximate of f (x) using fuzzy rules and well-known fuzzy inference procedure. Definition 1. According to the above notation the rule firing indicator function (RFIF) or simply IF conl ,l ,...,l nected to Ωj11 ,j22 ,...,jpn is defined as follows: ⎧ ⎨ α(x(t)) if x(t) ∈ Ωl1 ,l2 ,...,lp l ,l ,...,l j1 ,j2 ,...,jn Ij11 ,j22 ,...,jpn (x(t)) = (3) ⎩0 otherwise where α(x(t)) denotes the firing strength of the rule. According to standard fuzzy system description, this strength depends on the membership value of each xi in the corresponding input membership functions μF ji and more specifically [3], α(x(t)) = min[μF j1 (x1(k)), . . . , μF jn (xn (k))]. Then, assuming a standard defuzzification procedure (e.g. weighted average), the functional representation of the fuzzy system can be written as l ,l2 ,...,lp l ,l2 ,...,lp f (x(t)) = (¯ xf )j11 ,...,j × (I )j11 ,...,j (x(t)) (4) n n 65
where the summation is carried over all the available fuzzy l ,l2 ,...,lp rules. Also, (xf )j11 ,...,j is any constant vector consisting n of the centers of fuzzy partitions of f determined by l ,l2 ,...,lp l1 , l2 , . . . , lp and (I )j11 ,...,j (x(t)) is a weighted IF (WIF), n being actually the IF defined in (3) divided by the sum of all IF participating in the summation of 4. According to [23], functions of HONNs are capable of approximating discontinuous functions such as WIF. HONNs. are defined as follows: Definition 2. A HONN is defined as: N (x(t); w, L) =
L
whot
hot=1
d (hot)
Φj j
techniques, such as fuzzy C-means clustering [24], mountain clustering [25] and subtractive clustering [26], based on gathered data. Any other information related to the input membership functions and the rules of the underlying fuzzy system is not necessary because it is replaced by the HONNs. 3. Filtered Error Approximation-based Robot Controller Consider an n–DOF robot manipulator, which can be freely moved in its working space and its end-effector is following a trajectory. The manipulator dynamics are described by the following nonlinear differential equation [13, 27]
(5)
j∈Ihot
where Ihot = {I1 , I2 , . . . , IL } is a collection of L not-ordered subsets of {1, 2, . . . , n}, dj (hot) are non-negative integers. Φj are the elements of the following vector, Φ = [Φ1 , . . . , Φn ]T = [s(x1 ), . . . , s(xn ) ]T
¨ + Vm (θ(t), θ(t)) ˙ ˙ + G(θ(t)) τ (t) = M (θ(t))θ(t) θ(t) ˙ + F (θ(t)) + τd (11)
(6)
˙ θ¨ are vectors of joint torques, positions, velocwhere τ, θ, θ, ities, and accelerations, respectively. Also, M, Vm , G, and F are the inertia matrix, the coriolis/centripental matrix, the gravity vector and the friction vector, respectively (see [7, 27] for the physical meaning of these matrices) and τd is a disturbance term. Furthermore, we can define the following skew symmetry property [27]. Definition 3. The Coriolis/Centripetal matrix can always be selected so that the matrix
where s(x) denotes the sigmoid function defined as: s(x) = a
1 −γ 1 + e−βx
(7)
with a, β, γ being positive real numbers and w := [w1 , . . . , wL ]T are the HONN weights. (5) can also be written as N (x(t); w, L) =
L
˙ ≡ M˙ (θ) − 2Vm (θ, θ) ˙ S(θ, θ) whot shot (x(t))
(8)
is skew symmetric. Therefore, xT Sx = 0 for all vectors x. This is a statement of the fact that the fictitious forces in the robot system do no work. Consider now the following tracking error e given by
hot=1
where shot (x(t)) are high order terms of sigmoid functions of x. l ,l2 ,...,lp Following the above notation (I )j11 ,...,j in (4) can be n
e(t) = θd − θ
l ,l ,...,l
2 p approximated by Nj11,...,j (x) = N (x(t); wj1 ,...,jn ;l1 ,l2 ,...,lp , n Lj1 ,...,jn ;l1 ,l2 ,...,lp ). So, (4) can be rewritten as
f (x(t)) =
l ,l ,...,l
l ,l ,...,l
2 p 2 p (¯ xf )j11 ,...,j × Nj11,...,j (x(t)) n n
(12)
(13)
where θd is the desired joint variables trajectories. The filtered version r of the tracking error, given by
(9)
r = e˙ + Λe
where the summation is carried over all the available fuzzy rules. To simplify the model structure, since some rules may result in the same output partition, we may replace the NNs aasociated with the rules having the same output with one NN, and therefore, the summation in (9) is carried out over the number of the corresponding output partitions, that is N pf f (x(t)) = x ¯fl × Nl (x(t)) (10)
(14)
where Λ is a positive definite design parameter matrix. The robot dynamics are expressed in terms of this error as [13] M r˙ = −Vm r + f (x) + τd − τ
(15)
where, taking into consideration (13), (14), the unknown nonlinear function f (x) is defined as
l=1
˙ θ˙d + Λe) + G(θ) + F (θ) ˙ ˙ + Vm (θ, θ)( f (x) = M (θ)(θ¨d + Λe) (16)
From the above definitions and (9) it is obvious that the accuracy of the approximation of f (x) depends on the approximation abilities of HONNs and on an initial estimate of the centres of the output membership functions. These centres can be obtained by experts or by off-line
˙ z2 = (θ˙d + Λe) and One may also define z1 = (θ¨d + Λe), x ≡ [z1T z2T sin(θT ) cos(θT ) θ˙T ]T 66
(17)
A general sort of approximation-based controller is derived by setting
where Mf is a sufficiently large positive constant. Let us also define the modelling error μ as follows:
τ = fˆ + Kv r − υ(t)
μ(x) = f (x) − Xf Wf∗ sf (x)
Δ
(18)
where fˆ is an estimate of f , Kv r = Kv e˙ + Kv Λe an outer PD tracking loop, and υ(t) an auxiliary signal to provide robustness in the face of disturbance and modelling errors. The filtered error approach relies on PD compensation as well as on proper estimation of f . Simple PD controllers have been shown to be able to control the motion of a robot under partial or zero knowledge of its dynamics [13, 28]. However, PD controllers rely on the proper selection of the PD gains, which may vary according to the robot manipulator used. Arbitrarily increasing the gains may result in good performance producing, however, large control signals, which for implementation reasons are sometimes undesirable. Moreover, simple PD schemes usually lack stability proofs. Therefore, proper estimation of f using a short of adaptive technique may result in good performance even in the presence of unknown dynamics and disturbances, providing at the same time mathematical proofs for stability and performance. Moreover, such advanced techniques may be directly extended to more complicated control objectives such as force control for grinding, polishing, etc., where straight PD methods are inadequate [13].
with the modelling error bounded by μ < μN . From the above two equations, we have that the robot dynamics satisfies f (x) = Xf Wf∗ sf (x) + μ
(20)
Note that the term μ can be made arbitrarily small and consequently the high order neuro-fuzzy network (NFN) (19) can approximate any smooth function arbitrarily close by appropriately selecting the number of high order collections LF and the number and values of the centers of fuzzy partitions. Then the control law (18) becomes τ = Xf Wf sf (x) + Kv r − υ(t)
(21)
It is now necessary to show how to tune the NFN weights Wf on line to guarantee stable tracking. To this end, define ˜ f = Wf − W ∗ . Then, the the weight estimation error as W f closed loop filtered error dynamics (15) becomes ˜ f sf (x) + (μ + τd ) + υ(t) (22) M r˙ = −(Kv + Vm )r − Xf W
4. Neuro-Fuzzy Representation for the Unknown Nonlinear Robot Dynamics
5. Neuro-Fuzzy Controller and Error Dynamics
In this section, we present the proposed NFN representation, for a general serial-link robot arm. ˙ θ) ¨ ∈ Ω, where Ω is a bounded simple Let (τ, θ, θ, connected set of n and f (·) : Ω → p . Also we assume that the desired trajectory is inside the workspace of the robot manipulator and moreover all of the points of the trajectory are reachable by the robot’s end-effector. Define C p (Ω) as the space of continuous functions f (·). Since f is a continuous smooth function it can be approximated by the neuro-fuzzy representation given in Section 2. Therefore, according to (10) we can approximate the unknown robot dynamics by several HONNs with LF high-order terms [23]
Suppose that a NFN is used to approximate the nonlinear robot function (16) according to (20) with Wf∗ the optimum approximating weights. The optimum weights are unknown and may even be nonunique. Let also assume, that the desired trajectory θd (t) is bounded by θ¯B a known scalar bound and the initial tracking error is bounded as well. We can now distinguish two possible cases. Case 1: Neuro-fuzzy controller in an ideal case. Suppose the NFN functional reconstruction error μ and unmodelled disturbances τd (t) are equal to zero. In Theorem 1, weight updating laws are given, which can serve the control objectives. It has to be mentioned that according to [21, 23] gradient-type updating laws of the form given in Theorem 1 have been found sufficient for the approximation of general nonlinear functions using the neuro-fuzzy representation of (19). Theorem 1. The control scheme for (11) be given by (21) (with υ(t) = 0), the filtered error system
fˆ(x) = Xf Wf sf (x)
(19)
where Xf is a n × n · m block diagonal matrix Xf = diag{Xf1 , . . . , Xfn } with the ith, i = 1, . . . , n diagonal block element being a row vector containing the m fuzzy output partition centers of fi . Here, for notational simplicity, it is assumed that each fi has the same number m of partitions. Wf is a n · m × LF matrix of synaptic weights, which can be considered as a block matrix Wf = [Wfil ], i = 1, . . . , n, l = 1, . . . , Lf , where each Wfil is a m-dimensional column vector. Finally, sf is a LF dimensional vector with each element sfl (x) being a high order combination of sigmoid functions of the state. Define now the optimal weight vector Wf∗ as follows: Δ Wf∗ = arg sup |f (x) − Xf Wf sf | min |Xf Wf |≤Mf
˜ f sf (x) M r˙ = −(Kv + Vm )r − Xf W
System
(23)
and the weight updating law ˙ f = (Xf )T df ri sl (x) W i i il guarantees the following properties. ˜ f ∈ L∞ , r ∈ L2 • r, W ˜˙ f (t) = 0 • limt→∞ r(t) = 0, limt→∞ W
x∈Ω
67
(24)
Proof: Consider the Lyapunov function candidate 1 1 ˜ T −1 ˜
V = rT M r + tr W (25) f Df W f 2 2
to zero. However, in actual systems there are disturbances. Moreover, the approximation error increases as the number of high order terms or fuzzy centers decreases. In non-ideal case it can be seen that if the approximation error and system disturbance are not zero but bounded, then our NF controller still works under an additional assumption of persistence of excitation (PE). In this case, the tracking error does not converge to zero, but is bounded by small enough values to guarantee good tracking performance. The high order sigmoid terms could be chosen to be persistently exciting by appropriately selecting parameters α, γ in (8). However, if the vector sf in (19) is not persistently exciting, it is well known from adaptive control theory [29] that the phenomenon of parameter drifting may occur. To correct this problem, one may use techniques from adaptive control, including dead-zone, σ or switching σ-modification and e-modification [29]. Alternatively, one could use a modified weight updating algorithm, that includes a weight hopping [22] when, during the adaptation, term Xf Wf in (19) exceeds a pre-specified bound. Theorem 2. The control scheme for (11) given by (21) (with υ(t) = 0, τd = 0, μ = 0), the filtered error system
where Df is a positive definite gain matrix which is selected by the designer. Taking the time derivatives of the Lyapunov function candidate (25) we obtain
1 ˜˙ T D−1 W˜f V˙ = rT M˙ r + rT M r˙ + tr W (26) f f 2 and after substituting (23) we have 1 ˜ f sf (x) V˙ = −rT Kv r + rT (M˙ − 2Vm )r − rT Xf W 2
˜˙ T D−1 W˜f + tr W (27) f
f
But according to Definition 3, (M˙ − 2Vm ) is skew symmetric and therefore rT (M˙ − 2Vm )r = 0, ∀r ∈ n . Let us consider that
˜˙ T D−1 W ˜ f = r T Xf W ˜ f sf (x) tr W f f then the above equation results in the following learning law ˙ f = Df X T rsT W f f
˜ f sf (x) + τd + μ M r˙ = −(Kv + Vm )r − Xf W
(28) and the modified learning law
Taking into account the block representation of Wf given in (19) the above law can also be written in respect to the updating of each Wfil according to (24), where dfi is a m × m gain matrix, which for simplicity could be taken to be diagonal and it’s elements equal. Then V˙ assumes the form V˙ = −rT Kv r ≤ −λmin (Kv )r2
(30)
˙ f = HP (Xf )T df ri sl (x); W i i il f
(Xfi )T dfi ri sl (x)
κdfi r(Xfi Wfil (Xfi )T ) − (31) tr{(Xfi )T Xfi }
Δ
(29)
where Pf = {Xfi ·Wfil : |Xfi · Wfil | < Mfi }, κ a positive constant selected from the designer and HPf represents the switching criterion between the two updating laws inside the square brackets guarantees the uniform ultimate boundedness of r with respect to the set
Since V > 0 and V˙ ≤ 0, we conclude that V ∈ L∞ , which ˜ f ∈ L∞ . Furthermore, Wf = W ˜ f + W ∗ are implies that r, W f also bounded. Since V is a non-increasing function of time and bounded from below, limt→∞ V = V∞ exists; therefore, by integrating V˙ from 0 to ∞ we have ∞ λmin (Kv ) r2 dt ≤ [V (0) − V∞ ] < ∞
=
0
(τB + μ ¯N ) r(t) : r(t) ≤ , λmin (Kv ) = 0 λmin (Kv )
Proof: Consider the Lyapunov function candidate (25). Taking its time derivatives and taking into account (22) with the disturbance τd and the NFN reconstruction error μ different than zero we obtain
which implies that r ∈ L2 . ˜ f , sf are bounded, r ∈ L∞ . Since Xf , W Since r ∈ L2 ∩ L∞ , using Barbalat’s Lemma we conclude that limt→∞ r(t) = 0. Hence, given that Λ in (14) is positive definite, we also have that
1 ˜ f sf (x) V˙ = −rT Kv r + rT (M˙ − 2Vm )r − rT Xf W 2
˜˙ T D−1 W˜f + rT (τd + μ) + tr W (32) f
lim e(t) = 0
f
t→∞
The skew symmetric property of (M˙ − 2Vm ) together with the learning law (24) (which is equivalent to (31) if no hopping occurs) results to the following form of V˙
Now, using the boundness of Xf , sf and the convergence of ˜˙ f also converges to zero [29]. r(t) to zero, we have that W Case 2: Neuro-fuzzy controller in non-ideal case. It has just been seen that under the ideal case of no modelling errors or unmodelled disturbances, our algorithm through time suffices to make the tracking error to go
V˙ ≤ −λmin (Kv )r2 + r(τB + μ ¯N ) ≤ −(λmin (Kv )r − (τB + μ ¯N ))r ≤ 0 68
(33)
Remark 1. The previous analysis reveals that in the case where we have a modelling error and disturbance different from zero at θ = 0, our adaptive algorithm can guarantee at least uniform ultimate boundedness of all signals in the closed loop. In particular, Theorem 2 shows that if τB + μ ¯N is sufficiently small, or if the design constant Kv is chosen appropriately, then r(t) can be arbitrarily close to zero and in the limit as Kv → ∞, actually becomes zero but implementation issues constraint the maximum allowable value of Kv . Remark 2. The switching condition HPf in (31) could be simply a threshold function determining whether Δ Xfi · Wfil ∈ Pf = {Xfi · Wfil : |Xfi · Wfil | < Mfi } or not. In a more elegant version one could also take into account the direction of the weight updating. In case the threshold function is activated but the weight updating is in the direction that moves Xfi · Wfil toward Pf , the switching is finally not activated. Details for this more elegant version can be found in [21, 22]. Remark 3. The weight hopping condition arrives from the desire to keep |Xf Wf sf | < Mf , where Mf is an k upper bound. Since Xfi Wfi sf = Xfi Wfil sfl (x) and l=1 k |Xfi Wfi sf (x)| ≤ l=1 |Xfi Wfil sfl (x)|, one is allowed to have a better insight in the algorithm if specialized bounds are put, each one for each component of Xf Wf sf (x). Each specialized bound may be expressed by |Xfi · Wfil | < Mfi , with Mfi being again a design parameter determining an external limit for Xfi · Wfil . In the sequel, we consider the forbidden hyperplanes being defined by the equation |Xfi · Wfil | = Mfi . Note that the direction of the weight updating (24) is perpendicular to the forbidden hyperplane. When the weight vector reaches one of the forbidden hyperplanes |Xfi · Wfil | = Mfi and the direction of updating is towards the forbidden hyperplane, a new hopping is introduced which moves the weights back to the desired hyperspace Pf , allowing thus the algorithm to search the entire space for a better weight solution. This procedure is depicted in Fig. 1, in a simplified 2dimensional representation. The magnitude of hopping is −κdfi r(Xfi Wfil (Xfi )T )/tr{(Xfi )T Xfi } being determined by following the vectorial proof in [22] and dfi is the ith element of the gain matrix Df .
provided that r >
(τB + μ ¯N ) λmin (Kv )
(34)
and λmin (Kv ) = 0 which is valid from definition. Inequality (34) demonstrates that the trajectory of r(t) is uniformly bounded with respect to the arbitrarily small (since Kv can be chosen sufficiently large), set shown below ¯N ) (τB + μ = r(t) : r(t) ≤ , λmin (Kv ) = 0 λmin (Kv ) In case the complete modified weight updating law (31) is applied, that is when the switching condition is occasionally met, then the second updating law inside the square brackets of (31) is applied. In this case and since the weight jump −κdfi r(Xfi Wfil (Xfi )T ) / tr{(Xfi )T Xfi } = −κdfi rhb can also be expressed in respect to the error weight vector w ˜ w ˜fil as −κdfi rhb w˜ffil [21, 22], V˙ satisfies the following il inequality ¯N ) − rΘ V˙ ≤ −λmin (Kv )r2 + r(τB + μ
(35)
with Θ being a positive constant expressed as ˜fi )T w˜fi )/w ˜fi , where the summation inΘ= κdfi hb ((w cludes all weight vectors which require hopping. Therefore, the negativity of V˙ is actually enhanced by the presence of Θ, with Θl ≤ Θ ≤ Θu where Θl stands for the case where only one hopping occurs and Θu where all possible hopping occur. After substituting with the worst case of Θ = Θl , V˙ becomes V˙ ≤ −λmin (Kv )r2 + r(τB + μ ¯N ) − Θl r ≤ −(λmin (Kv )r + Θl − (τB + μ ¯N ))r ≤ 0 (36) provided that r >
¯ N ) − Θl (τB + μ λmin (Kv )
(37)
Finally, inequality (37) demonstrates that the trajectory of r(t) is uniformly bounded with respect to the arbitrarily small (since Kv can be chosen sufficiently large, and the presence of Θl can make the ratio even smaller), set shown below (τB + μ ¯ N ) − Θl = r(t) : r(t) ≤ λmin (Kv )
6. Simulation Results The planar two-link revolute arm shown in Fig. 2 is used extensively in the literature for easy simulation of robotic controllers, with results that can be easily illustrated in 2-dimensional plots. Its dynamics are given as [7]
Thus, θ(t) is also bounded with respect to r(t). Furthermore, we have
x˙ 1 = x3 x˙ 2 = x4
˜ f sf (x) + τd + μ M r˙ = −(Kv + Vm )r − Xf W
[x˙ 3 x˙ 4 ]
˜ f is assured by the use of the Since the boundedness of Xf W modified weight hopping algorithm and τd + μ is bounded by definition, we conclude that r˙ ∈ L∞ .
T
= −M −1 (N + τ )
where x1 = θ1 is the angular position of joint 1, x2 = θ2 is the angular position of joint 2, x3 = θ˙1 is the angular 69
neural approximation (2). The weight updating laws for the simple NN approximation are given as follows [13]: ˙ = F φ(x)rT W
(38)
where F is a positive definite gain matrix. The weight updating law for the neuro-fuzzy approximation is given by (28) or equivalently by (24). In both approaches only first order sigmoid neural functions were used. The number of output membership partitions required by the neurofuzzy approach was arbitrarily selected to be m = 5 and the centres of these partitions were again arbitrarily selected to be: Xf1 = [−35 Xf2 = [10
Figure 2. Two-link planar elbow arm. velocity of joint 1 and x4 = θ˙2 is the angular velocity of joint 2. Also, the matrices M , N have the following form ⎡ ⎤ m2 a22 + m2 a1 a2 cos(x2 ) (m1 + m2 )a21 + m2 a22 ⎢ ⎥ ⎢ ⎥ M = ⎢ +2m2 a1 a2 cos(x2 ) ⎥ ⎣ ⎦ m2 a22 + m2 a1 a2 cos(x2 ) m2 a22 ⎢ ⎢ N =⎢ ⎣
−m2 a1 a2 (2x3 x4 + x24 ) sin(x2 )
30
75
20 110
75
100]
170]
That is, no actual expert’s knowledge has been used. No other fine tunning has been followed regarding the functions of the regression vector apart from the selection of sigmoid parameters and adaptation gains, which were selected by trial and error such that both representations give their best performance. Under these conditions both approaches are expected to have significant modelling errors, which in turn influence the performance of the controller and its tracking abilities. Nevertheless, to compare the approximation abilities of the two approaches, the robustifying term in (21) was selected to be 0. The controller parameters were taken as Kv = diag{20, 20}, Λ = diag{5, 5}, Df = diag{1.8, 1.5}, and F = diag{50, 50}. The sigmoid values for the NFN approach were α = 0.5, β = 0.4, and γ = 0.3 and for the NN approach were α = 0.6, β = 0.5, and γ = 0.4, respectively. Figure 3(a) shows the desired and the actual trajectories produced by the NFN and the NN approaches, respectively. It can be observed that the NFN approach outperforms the simple NN one. The MSE’s were 0.0843 for the NFN and 0.1526 for the NN representation. Similar performance is demonstrated in Figure 3(b), where the tracking of the second joint is illustrated. The MSE’s are 0.0274 and 0.0443, respectively.
Figure 1. Pictorial representation of the outer hopping.
⎡
−15
6.2 Scenario 2
⎤
Next, the performance of the proposed NFN representation was compared to conventional computed torque PD (CT-PD) control [7], where the PD gains were selected to be Kp = 100, Kv = 20. The conventional computed torque controller assumes the exact knowledge of the robot model and its parameters. The NFN model was the same as in the previous experiments but in the control law a simple robustifying term υ(t) = 300 e(t) was added to account for modelling error and/or disturbances. A constant disturbance term τd = 20 was added in both cases and the performance of both approaches were tested under the presence of this disturbance. Figure 4(a) and (b) shows the performance of both CT-PD and NFN approaches in following the desired trajectories for joints 1 and 2, respectively, when there is not any disturbance (τd = 0). As it was expected the CT-PD controller performs much
⎥ ⎥ +(m1 + m2 )ga1 cos(x1 ) + m2 ga2 cos(x1 + x2 ) ⎥ ⎦ m2 a1 a2 x23 sin(x2 ) + m2 ga2 cos(x1 + x2 )
We took the arm parameters as a1 = a2 = 1 m, m1 = m2 = 1 kg and the desired trajectory x1d = sin(t), x2d = cos(t). To test the applicability and the performance of the proposed technique a number of simulations were carried out according to the following scenarios. 6.1 Scenario 1 First the approximation abilities of the proposed neurofuzzy representation was tested against a simple one-layer 70
Figure 3. (a) Response of NFN and NN algorithms in trajectory tracking of joint angle 1 and (b) response of NFN and NN algorithms in trajectory tracking of joint angle 2, against the desired trajectory.
Figure 4. (a) Response of NFN and CT-PD algorithms in trajectory tracking of joint angle 1 and (b) response of NFN and CT-PD algorithms in trajectory tracking of joint angle 2, against the desired trajectory without any disturbance in both schemes.
in both cases (NFN method and CT-PD approach) but with the superiority of the NFN method to remain.
better, since it has complete knowledge of the robot model and its parameters. Figure 5(a) and (b) shows the performance of both approaches when there is the disturbance term τd = 20. Figure 6(a) and (b) shows the respective tracking errors. It can be seen that the performance of the CT-PD is seriously affected, while the performance of the NFN approach is only slightly affected.
6.3 Scenario 3 In this case, we assume that the robot is tracking a trajectory and it suddenly drops the load it is carrying. This means, for example, that the mass m2 is reduced from 1, to 0.8 kg after the first 5 s. Figure 8(a) and (b) shows the respective tracking errors. It can be seen that the performance of the CT-PD is seriously affected, especially in the
In the same scenario, we increased the proportional gain Kp to 400 and we took Figure 7(a) and (b), which shows a sufficient improvement to the trajectory tracking 71
Figure 5. (a) Response of NFN and CT-PD algorithms in trajectory tracking of joint angle 1 and (b) response of NFN and CT-PD algorithms in trajectory tracking of joint angle 2, against the desired trajectory with disturbance in both schemes.
Figure 6. (a) Tracking error of the first joint and (b) tracking error of the second joint, for the NFN and CT-PD approaches in both schemes.
7. Conclusion
second joint angle x2 , while the performance of the NFN approach is only slightly affected.
This paper proposed a new neuro-fuzzy approximation of unknown robotic functions f , which is used in a filterederror approximation-based robotic controller. The unknown nonlinear robotic function is first modelled as a fuzzy system, which is functionally expressed by using fuzzy rule indicating functions (IF). Each IF is in the sequel approximated by a HONN leading thus in a representation, which combines a number of HONN each one being specialized in approximating a part of f associated with a fuzzy output partition of the underlying fuzzy system. Using this representation the filtered-error approximation-based robotic controller for trajectory tracking was presented and NN weight updating laws were derived. A weight hopping technique was also introduced to ensure that the estimated weights stay within pre-specified bounds. It was shown
6.4 Scenario 4 In this case, we assume that the robot is tracking a trajectory and a sudden disturbance is applied. This happened between 10 and 20 s of simulation time and the disturbance had the following form τd = 10 + 5 · sin(18 · π · t)
(39)
Figure 9(a) and (b) shows the respective tracking errors. It can be seen that the performance of the CT-PD is seriously affected especially the joint angle x2 , while the performance of the NFN approach is only slightly affected. 72
Figure 7. (a) Tracking error of the first joint and (b) tracking error of the second joint, for the NFN and CT-PD approaches in both schemes when Kp = 400.
Figure 8. (a) Tracking error of the first joint and (b) tracking error of the second joint, for the NFN and CT-PD approaches when the mass m2 is reduced to 0.8 kg in both schemes.
Figure 9. (a) Tracking error of the first joint and (b) tracking error of the second joint, for the NFN and CT-PD approaches when a sudden disturbance is applied between 10 and 20 s in both schemes. 73
that in the ideal case, where the modelling error and the disturbances are zero, the tracking error goes asymptotically to zero. In case the modelling error and the unmodelled disturbances are not zero, it was shown that the tracking error remains bounded. Simulation results were presented on a 2-link revolute joint arm and comparisons were made with the performance of a simple NN approach as well as with the performance of a conventional computed torque PD control under the presence of unknown disturbance.
[22]
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Biographies Dimitrios C. Theodoridis was born in Solingen (Germany) in 1974. He received a diploma in Physics and a M.Sc. degree in Electrical Physics (Radioelectrology) from the Aristotle University of Thessaloniki (AUTH) in 1997 and 1999, respectively. He is currently a Ph.D. candidate at the Democritus University of Thrace (DUTH). Since 2005 he served as a lecturer in the Department of Industrial Informatics in the Technological Educational Institute of Kavala. His main research interests lie in the field of neuro-fuzzy networks, neural networks, adaptive control, fuzzy logic control, intelligent control, nonlinear control systems, robot manipulators, and applications to electric drive systems. He is a member of the Hellenic Physical Society and member of the Hellenic Electronic PhysicalRadioelectrology Society. He has served as a reviewer for scientific journals in the area of neural networks, control, robotics, and automation as well as member of scientific conference committees (session chair). Yiannis S. Boutalis received his diploma of Electrical Engineer in 1983 from Democritus University of Thrace (DUTH), Greece and the Ph.D. degree in Electrical and Computer Engineering (topic Image Processing) in 1988 from the Computer Science Division of National Technical University of Athens, Greece. Since 1996, he serves as a faculty member at the Department of Electrical and Computer Engineering, DUTH, Greece, where he 74
International Journals. He is managing and cooperating on various research projects in Greece, in the European Union in collaboration with the United States. He has held many administrative positions such as the Vice Presidency of the Technical University of Crete, Chairman of the office of Sponsored research and a member of the board of governors of the University of Peloponnese. He is a member of the Technical Chamber of Greece. He has been active in the IEEE CS society as the founder and first Chairman of the IEEE Control Systems Society Greek Chapter, which received the 1997 Best Chapter of the Year Award and as the founder of the IEEE Mediterranean Conference on Control and Automation, which became an annual event. Dr. Christodoulou received the MCA Founders award in 2005. He is a member of the board of governors of the Mediterranean Control Association since 1993.
is currently an associate professor and director of the Automatic Control Systems lab. Currently, he is also a Visiting Professor for research cooperation at ErlangenNuremberg University of Germany, chair of Automatic Control. He served as an assistant visiting professor at University of Thessaly, Greece, and as a visiting professor in Air Defence Academy of General Staff of airforces of Greece. He also served as a researcher in the Institute of Language and Speech Processing (ILSP), Greece, and as a managing director of the R&D SME Ideatech S.A, Greece, specializing in pattern recognition and signal processing applications. His current research interests are focused in the development of Computational Intelligence techniques with applications in Control, Pattern Recognition, Signal and Image Processing Problems. Manolis A. Christodoulou was born in Kifissia, Greece, in 1955. He received diploma degree (EE’78) from the National Technical University of Athens, Greece, the M.S. degree (EE’79) from the University of Maryland, College Park the engineer degree (EE’82) from the University of Southern California, Los Angeles and the Ph.D. degree (EE’84) from the Democritus University, Thrace, Greece. He joined the Technical University of Crete, Greece in 1988, where he is currently a Professor of Control. He has been a Visiting Professor at Georgia Tech., Syracuse University, the University of Southern California, Tufts University, Victoria University and the Massachusetts Institute of Technology. He has authored and co-authored more than 200 journal articles, book chapters, books, and conference publications in the areas of control theory and applications, robotics, factory automation, computer integrated manufacturing in engineering, neural networks for dynamic system identification and control, in the use of robots for minimally invasive surgeries and recently in systems biology. Dr. Christodoulou is the organizer of various conferences and sessions of IEEE and IFAC and guest editor in various special issues of
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