STOCHASTIC MODELING AND TRACKING CONTROL FOR A TWO-LINK PLANAR RIGID ROBOT MANIPULATOR. Received February 2012; revised June 2012

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1 International Journal of Innovative Computing, Information and Control ICIC International c 03 ISSN Volume 9, Number 4, April 03 pp STOCHASTIC MODELING AND TRACKING CONTROL FOR A TWO-LINK PLANAR RIGID ROBOT MANIPULATOR Ming-Yue Cui, Zhao-Jing Wu and Xue-Jun Xie, Institute of Automation Qufu Normal University No. 57, West Jingxuan Rd., Qufu 7365, P. R. China mingyuecmy@6.com Corresponding author: xuejunxie@6.com School of Mathematics and Informational Science Yantai University No. 3, Qingquan Rd., Laishan District, Yantai 64005, P. R. China wuzhaojing00@88.com Received February 0; revised June 0 Abstract. In this paper, the problem of stochastic modeling and tracking control of a two-link planar rigid robot manipulator is considered. A stochastic Lagrangian model is constructed to describe the motion of the manipulator in random vibration environment. Based on the constructed model, a state feedback controller is designed such that the error system is 4-th moment exponentially practically stable. The simulation result demonstrates the efficiency of the proposed scheme. Keywords: Two-link planar rigid manipulator, Stochastic Lagrangian model, Tracking control. Introduction. In recent several decades, the robot control and its application are very popular research topics in control field. It is well known that robot manipulators are often subjected to random vibration, which affects the performance of manipulators and even damages the structure. In this paper, we consider a well-used robotic system, the two-link planar rigid manipulator, in the random vibration environment as shown in Figure, which is connected to O on the floor by a freely moveable joint. For i =,, m i denotes the mass of i-th link, l i denotes the length of i-th link, l ci denotes the distance from the previous joint to the center of mass of i-th link, q i denotes the i-th joint angle and u i denotes the torque with respect to i-th joint, whose units are kg, m, rad and N m, respectively. As in Section 8..4 of [], let ξ, ξ denote the acceleration of the point O in horizontal and vertical directions, which can be seen as independent white noises. The reasonable dynamic modeling is a necessary premise to design efficient control strategies. For the deterministic case, i.e., ξ i = 0, various modeling methods can be found in many monographes such as [-4]. A common dynamic model is derived from the general Lagrangian equation method to describe the relationship between force and motion, i.e., Mq) q + Cq, q) q + hq) = Gq)u, ) where q = q, q ) T, u = u, u ) T, the symmetric and positive definite matrix Mq) R is the inertia matrix generalized mass), Cq, q) q is the vector of centrifugal and Coriolis force, hq) = P with P is the potential energy of the system, and Gq)u is q the generalized force caused by the control u. However, for the stochastic case, the deterministic model ) cannot well describe the motion of the manipulator. One objective 769

2 770 M.-Y. CUI, Z.-J. WU AND X.-J. XIE y O l lc q u m l A lc q u B m g x ξ O ξ Figure. Two-link planar rigid manipulator subjected to random vibration of this paper is to construct a reasonable model to describe the motion of the manipulator in the random vibration environment. A basic problem in controlling robots [5-8] is to make the manipulator follow a desired trajectory q r t) C R ), i.e., design a controller u such that the tracking error e t) = qt) q r t) ) tends to zero. For the deterministic case, based on Lagrangian model ), great efforts have been made in developing desirable schemes to achieve the tracking control objectives. In many references such as [9-], some effective control strategies such as PID control, computed torque control, robust control, adaptive control were considered. However, for the stochastic case, to the best of the authors knowledge, there is no result. Hence, another objective of this paper is to design a tracking controller for the manipulator in the random vibration environment based on the constructed model. The main work consists of the following two aspects: i) Since the manipulator is subjected to random vibration, the main difficulty for stochastic dynamic modeling is how to transform the random vibration in environment to the mass points along the links between them. By using the Lagrangian mechanics and the equivalence principle of mechanics, a stochastic model is established to describe the motion of the two-link planar rigid robot manipulator in random vibration environment. ii) Based on the constructed model, a state feedback controller is designed such that the tracking error system is 4-th moment exponentially practically stable, the mean square of tracking error tends to an arbitrarily small neighborhood of zero by tuning design parameters, and so does its derivative. The simulation result demonstrates the efficiency of the proposed scheme. This paper is organized as follows: The mathematical preliminaries are given in Section. Section 3 constructs a stochastic model. In Section 4, controller design and stability analysis are addressed, following a simulation result in Section 5. The paper is concluded in Section 6. Notations: The following notations are used throughout the paper. For a vector x, x T denotes its transpose and x denotes its usual Euclidean norm. For a matrix X, X denotes its inverse and X F denotes its Frobenius norm which is defined by X F = Tr{XX T }) /, where Tr ) denotes the square matrix trace; R + denotes the set of all nonnegative real numbers; R n denotes the real n-dimensional space; R n r denotes the real n r matrix space; C i R n ) denotes the set of all functions with continuous i-th partial derivative on R n, C, R n R +, R + ) denotes the family of all nonnegative functions V x, t) on R n R + which are C in x and C in t and K denotes the set of all functions:

3 STOCHASTIC MODELING AND TRACKING CONTROL 77 R + R +, which are continuous, strictly increasing and vanish at zero. For simplicity, sometimes the arguments of functions are omitted.. Mathematical Preliminaries. Consider the following stochastic nonlinear system dxt) = fxt), t)dt + gxt), t)dw t), 3) where xt) R n is the state of system, W t) is an r-dimensional independent standard Wiener process or Brownian motion), and the underlying complete probability space is taken to be the quartet Ω, F, F t, P ) with a filtration F t satisfying the usual conditions i.e., it is increasing and right continuous while F 0 contains all P -null sets). Both functions f : R n R + R n and g : R n R + R n r are locally Lipschitz in x R n and piecewise continuous in t, namely, for any R > 0, there exists a constant C R 0 such that fx, t) fx, t) + gx, t) gx, t) F C R x x, t R +, 4) for any x, x U R = {ξ : ξ R}. Moreover, f0, t) and g0, t) are bounded a.s.. For V x, t) C, R n R +, R + ), the infinitesimal generator is defined by LV x, t) = V t x, t) + V x x, t)fx, t) + Tr { g T x, t)v xx x, t)gx, t) }, 5) ) where V t = V, V V t x = x, V x,..., V x n and V xx = V x i x j )n n. For stability analysis, in view of the concept of p-th moment exponential stability in [,3], p-th moment exponential practical stability is reasonably introduced in the following. Definition.. The system 3) is said to be p-th moment exponentially practically stable if there exist positive constants λ, d and function κ K such that E xt) p κ x 0 )e λt t 0) + d, t t 0, x 0 R n. 6) When p =, we also say that it is exponentially practically stable in mean square. The criterion for p-th moment exponential practical stability is given as follows. Lemma.. For system 3), assume that there exist a function V x, t) C, R n R +, R + ), positive constants k i, k i, p i, p i, c and d c such that n i= k i x i p i V x, t) n i= k i x i p i, 7) LV x, t) cv x, t) + d c. 8) Then, there exists a unique strong solution xt) = xt; x 0, t 0 ) of system 3) for each xt 0 ) = x 0 R n and system 3) is p-th moment exponentially practically stable where p = min{p,, p n }. Proof: For any k > 0, define the first exit time η k = inf{t : t t 0, xt) > k} with the special case inf Ø =, and set t k = η k t for any t t 0. According to Theorem 4. of [4], there exists a unique strong solution xt) = xt; x 0, t 0 ) of system 3) for each xt 0 ) = x 0 R n, that is, η k a.s. as k. In view of 7), the infinitesimal generator of the function e ct V x, t) is Le ct V x, t)) = e ct LV x, t) + cv x, t)) e ct d c. 9) Hence, by Lemma 3.3. in [], for t k, we have Ee ct k V xtk ), t k )) e ct 0 V x 0, t 0 ) + E t k t 0 e cs d c ds. 0) Since t k t, according to the dominated convergence theorem, letting k, one gets EV xt), t) V x 0, t 0 )e ct t 0) + d c c, t t 0. )

4 77 M.-Y. CUI, Z.-J. WU AND X.-J. XIE From 7), for i =,,, n, this estimate yields E x i t) p i n k k j x 0 p j e ct t 0 ) + d c i k i c. ) j= According to Jensen s inequality, from p i p, one has E x i t) p E x i t) p ) p i p p ) p i, which, together with ) and c r inequality, implies that then E x i t) p E xt) p n p n p k i n j= k j x 0 p j n E x i t) p i= n i= k i j= ) p p i e cp p i t t 0 ) + ) p p n i k j x 0 p j e cp t t p 0 ) i + n p d c k i c ) p p i, 3) n i= ) p dc p i k i c κ x 0 )e λt t0) + d, where κs) = n p n ) p n i= k i j= k js p p i j cp, λ = and d = n p n max i {p i } i= system 3) is p-th moment exponentially practically stable. d c k i c 4) ) p p i, that is, 3. Stochastic Dynamic Modeling for the Manipulator. In this section, the stochastic model of the manipulator is established by deriving the kinetic and potential energy of the manipulator and then using Lagrangian equation method [3] and the equivalence principle of mechanics [5]. 3.. Kinetic and potential energy. Consider the system of particles consisting of two links which are regarded as the mass points and select q, q ) as the generalized coordinate. The base coordinate frame Oxy is shown in Figure, then the natural coordinates of particles are { x = l c cos q y = l c sin q, { x = l cos q + l c cosq + q ) y = l sin q + l c sinq + q ). From 5), the total kinetic energy of the system is K = m ẋ + ẏ) + m ẋ + ẏ) = qt Mq) q, where the inertia matrix generalized mass) Mq) R is [ ] m lc Mq) = + m l + m lc + m l l c cos q m lc + m l l c cos q m lc + m l l c cos q m lc. 6) Since the k, j)-th element of the matrix Cq, q) is defined as c kj = ) i= c ijkq) q i with the Christoffel symbols c ijk = mkj q i + m ki q j m ij q k see P.07 of []), then [ ] m l Cq, q) = l c q sin q m l l c q sin q m l l c q sin q. 7) m l l c q sin q 0 The total potential energy of the system equals to P = m gy + m gy = m l c + m l )g sin q + m gl c sinq + q ), then [ ] hq) = P = m l c + m l )g cos q + m l c g cosq + q ). 8) q m l c g cosq + q ) If ϕ is a convex function and x is a random variable on Ω, then ϕex) Eϕx). a + b r c r a r + b r ), r > 0, where c r = {, r r, r. 5)

5 STOCHASTIC MODELING AND TRACKING CONTROL 773 y F B F q O q A x Figure. The decomposition of forces a O O ξ a ξ O O ) A A A aa aa a O A ) B B Figure 3. The stochastic force 3.. The generalized forces. In order to obtain the control force τ c, it is supposed that the control u i, i =,, is the torque F i acting on the barycenter of i-th link, i.e., u i = F i l ci, as shown in Figure. Based on the generalized forces formula 3 see P.4 in [6]), the generalized forces corresponding to the control u and u are τ c x y = F sin q + F cos q F sinq + q ) x + F cosq + q ) y q q q q = F l c + F l cos q + F l c = u + u + l ) cos q, l c τ c x y = F sin q + F cos q F sinq + q ) x + F cosq + q ) y q q q q = F l c = u, which means that the vector of generalized control forces τ c = Gq)u with [ + l ] Gq) = l c cos q. 0) 0 In order to obtain the random excitation force τ e, O is regarded as the reference point. As shown in Figure 3), decomposing ξ and ξ at the point O, we have a O = ξ sin q + ξ cos q, a O = ξ cos q + ξ sin q. 3 The generalized force along the j-th generalized coordinates is defined by Q j = N i= F ix x i y q j +F i iy q j + z F i iz q j ), j =,,, n, where F ix, F iy, F iz ) is the force acted on the i-th point. The transformation from the natural coordinates r i = x i, y i, z i ) to the generalized coordinates {q, q,, q n } is represented as r i = r i q, q,, q n, t). 9) )

6 774 M.-Y. CUI, Z.-J. WU AND X.-J. XIE Next, decomposing a O at the point A see Figure 3)), one has a A = ξ cos q + ξ sin q ) sin q, a A = ξ cos q + ξ sin q ) cos q. According to the principle of relative motion [7], the stochastic force ˇF i in the direction of F i is introduced such that ˇF i /m i describes the stochastic motion of i-th particle, i.e., ˇF = m a O = m sin q ξ m cos q ξ, ˇF = m a A = m cos q sin q ξ + m sin q sin q ξ. Based on the generalized forces formula, following the same line as 9), the generalized stochastic forces are τ e = Λ q)ξ + Λ q)ξ, τ e 4) = Λ q)ξ + Λ q)ξ, with Λ q) = m l c sin q +m l c cos q sin q + m l cos q sin q, Λ q) = m l c cos q + m l c sin q sin q + m l sin q sin q, Λ q) = m l c cos q sin q and Λ q) = m l c sin q sin q. This means that the vector of random excitation force τ e = τ e, τ e ) T = Λq)ξ with Λq) = Λ ij q)) and ξ = ξ, ξ ) T. The generalized force τ consists of the control force τ c = Gq)u caused by the control u acting on the system and the random excitation force τ e = Λq)ξ caused by the independent white noise ξ Stochastic model of the manipulator. On the basis of the above two subsections, the stochastic dynamic equation of the manipulator is obtained ) 3) Mq) q + Cq, q) q + hq) = Gq)u + Λq)ξ. 5) By replacing ξ with db and viewing dt qt, q T ) T as state, the Stratonovich stochastic differential equation SDE) of 5) can be obtained dq = qdt, d q = M q)cq, q) q + hq)) + M q)gq)u ) dt + M q)λq) db, where B is an r-dimensional independent Wiener process and M q) = m m lcl c + m ll c sin q [ ] m lc m lc + m l l c cos q ) m lc + m l l c cos q ) m lc + m l + m lc. + m l l c cos q Since the diffusion function of q subsystem does not depend on q, the Wong-Zakai correction term see 6..3) in [8]) equals to zero, i.e., ) + ) q q 0 + =, 7) q 0 q where = 0 and = M q)λq). The corresponding Itô SDE is dq = qdt, d q = M q)cq, q) q + hq)) + M q)gq)u ) dt + M q)λq)db, i.e., the final stochastic dynamic model of the manipulator is constructed. 6) 8)

7 STOCHASTIC MODELING AND TRACKING CONTROL 775 Remark 3.. By reasonably introducing the random noise ξ to describe the random vibration, and using the Lagrangian mechanics and the equivalence principle of mechanics, the stochastic model 8) is established to describe the motion of the two-link planar manipulator in random vibration environment. 4. Tracking Control via State Feedback. The configuration q and its velocity q can be measured by adding some measuring equipments to the manipulator. It is easy to verify that the functions Mq), Cq, q), hq), Gq) and Λq) are smooth, and Gq) is invertible. 4.. Tracking error system. Before the tracking controller is designed, the tracking error system will be developed. Similar to [9], we define the filtered tracking error as e t) = ė t) + c e t) = qt) q r t) + c e t), 9) which is measurable due to the measurability of q and q, where c > 0 is a design parameter. In view of 8), one has de = c e + e )dt, de = M q)cq, q) q + hq)) q r c e + c e + M q)gq)u ) dt + M q)λq)db. Suppose that the power spectral density PSD) of white noise ξ equals to Σ, which is π equivalent to the fact db = ΣdW, where Σ = r ij ) is a positive matrix and W is a -dimensional independent standard Wiener process. Then 30) can be rewritten as de = c e + e )dt, de = M q)cq, q) q + hq)) q r c e + c e + M q)gq)u ) dt + M q)λq)σdw. Here, the underlying complete probability space is taken to be the quartet Ω, F, F t, P ) with a filtration F t satisfying the usual conditions. 4.. Tracking controller design. For the Lyapunov function 30) 3) V = 4 et e ) + 4 et e ), 3) the infinitesimal generator of V along 3) satisfies LV = c e T e e T e + e T e e T e + e T e e T ψq, q, qr, q r ) + M q)gq)u ) + Tr { Σ T Λ T q)m q)e e T + e T e I)M q)λq)σ }, 33) where ψq, q, q r, q r ) = M q)cq, q) q + hq)) q r c e + c e. Applying Young s inequality 4 to the second term in 33), one has e T e e T e c 4 et e ) + 7 e T 4c 3 e ). 34) 4 For any vectors x, y R n and any scalars ɛ > 0, p >, there holds x T y ɛp p x p + qɛ y q, where q q = p p.

8 776 M.-Y. CUI, Z.-J. WU AND X.-J. XIE From the definition of Λq), it is learned that Λ q) Λ q r )) = m l c sin q sin q r ) + m l c cos q cos q r ) sin q + m l c cos q r sin q sin q r ) + m l cos q cos q r ) sin q ) + m l cos q r sin q sin q r ) 3m lc + 6m lcsin q + cos q r ) + 3 ) m lsin q + 4 cos q r ) q q r. Similarly, one has Λ q) Λ q r )) 3m lc + 6m lccos q + sin q r ) + 3 ) m lcos q + 4 sin q r ) q q r, Λ q) Λ q r )) m lcsin q + cos q r ) q q r, Λ q) Λ q r )) m lccos q + sin q r ) q q r. 35) 36) By the definition of the Frobenius norm, it follows from 35) and 36) that Λq) Λq r ) F = Λ ij q) Λ ij q r )) δq, q r ) q q r, 37) i= j= where δq, q r ) = 6m l c + 6m l c + 5 m l. Then, the Frobenius norm of Λq) satisfies Λq) F Λq) Λq r ) F + Λq r ) F δq, q r )e + Λq r ) F, 38) with Λq r ) F = m l c+m l c sin q r + 4 m l sin q r +m l l c sin q r sin q r. According to the definition of Frobenius norm, the norm compatibility and Young s inequality, by 38), the last term of 33) yields Tr { Σ T Λ T q)m q)e e T + e T e I)M q)λq)σ } 3 e δq, q r ) e + Λq r ) F ) M q) F Σ F c 4 et e ) + 9 c δ q, q r ) + 9 ɛ Λq r) 4 F ) M q) 4 F Σ 4 F e T e ) + ɛ 4, 39) where M q) F = m m lc l c +m l l c sin q ) m lc 4 +m lc +m l l c cos q )) +m lc + m l + m lc + m l l c cos q ) ), Σ F = r + r + r + r and ɛ > 0 is a design parameter. Substituting 34) and 39) into 33), one leads to LV c 7 et e ) + e T e e T e 4c 3 + ψq, q, q r, q r ) + M q)gq)u 9 + δ q, q r ) + 9 ) ) c ɛ Λq r) 4 F M q) 4F Σ 4F e + ɛ4 40).

9 STOCHASTIC MODELING AND TRACKING CONTROL 777 Since Gq) is a nonsingular square matrix, the control u is designed as u = G q)mq) c e 7 e 4c 3 ψq, q, q r, q r ) 9 δ q, q r ) + 9 ) ) 4) c ɛ Λq r) 4 F M q) 4F Σ 4F e where c > 0 is a design parameter and [ G l ] q) = l c cos q. 4) 0 Since q and q are measurable signals and q r is the reference signal, the control u can be implemented. Therefore, the infinitesimal generator of V satisfies LV c et e ) c et e ) + ɛ 4 cv + ɛ 4, 43) where c = min{c, c }. Just for simplicity, c and c are selected as scalars. In fact, they can be chosen as any positive definite constant matrix. Up to now, from 3) and 4), the following error system is got: de = c e + e )dt, c de = δ q, q 4c 3 r ) + 9 ) ) c ɛ Λq r) 4 F M q) 4 F Σ 4 F e dt + M q)λq)σdw, based on which, stability analysis will be given Stability analysis. Theorem 4.. For the stochastic model 5) and the reference signal q r, under the statefeedback controller 4), i) Error system 44) has a unique strong solution on [t 0, ) and 4-th moment exponentially practically stable for initial values e t 0 ), e t 0 ) R ; ii) The tracking error e t) and ė t) satisfy lim E e t) ɛ t c ), lim E ė t) + c ) ɛ t c where the right-hand side can be made small enough by choosing appropriate design parameters. Proof: Denoting et) = e T t), e T t)) T, by 3), it is obvious that V e) C R 4 ) and 8 et) 4 V e) 4 et) 4. 46) Since the functions of the error system 44) satisfy the local Lipschitz condition, by Lemma., 43) and 46), there exists a unique strong solution of system 44) for each et 0 ) R 4. System 44) is 4-th moment exponentially practically stable, and ), 44) 45) E et) 4 e ct t 0) et 0 ) 4 + ɛ c. 47) In view of ė t) = e t) + c e t) ) e t) + c e t), 45) holds from 47). Considering c = min{c, c }, it is clear that the right-hand side of 45) can be made small enough by choosing c, c large enough and ɛ small enough.

10 778 M.-Y. CUI, Z.-J. WU AND X.-J. XIE Remark 4.. By Chebyshev s inequality and 45), for any ε > 0 and ε 0 > 0, there exists T > 0 such that when t > T P { e t) > ε} ε ε0 + ɛ c ) ) ε, P { ė t) > ε} ε ε c ) ɛ c ) ) ε, where ε can be made small enough by tuning design parameters. At the expense of large control effort, the asymptotic tracking in probability in some sense can be achieved. Therefore, the effect between the tracking error and allowable control effort must be carefully compromised in the tracking controller design. Remark 4.. Due to the diffusion term and the nonvanishing signal q r, it is difficult to achieve the asymptotic tracking. The parameter ɛ is introduced to deal with the term caused by nonvanishing signal q r in Hessian term. When q r = 0, ɛ can be avoided, then the stabilization of q and q can be achieved. 5. Simulation. Choose the reference signal q r t) = sin t, 0.5 cos t) T, whose unit is rad. The state feedback tracking control law is given by 4). The simulation is performed under the following conditions: the PSD of the white noise Σ = r ij ) with r ii = 0. and r ij = 0 i j); the system parameters m = 0.5kg, m = 0.5kg, l c = 0.8m, l c = m, l = m, l =.5m and g = 9.8m/s ; the initial values q 0) = 0.4rad, q 0) = 0.85rad, q 0) = 0rad/s and q 0) = 0rad/s; the design parameters c = 8, c = 5 and ɛ = 0.. The simulation result demonstrates the effectiveness of the control scheme; see Figure 4. 48) 00 u u Controller N m) TimeSec) Tracking error rad) e =q q r e =q q r TimeSec) The time derivative of tracking error rad/s) e =q q r e =q q r TimeSec) Figure 4. The responses of closed-loop system

11 STOCHASTIC MODELING AND TRACKING CONTROL Conclusions. For the two-link planar rigid manipulator in the random vibration environment, by analyzing the effect of the random vibration, a stochastic model is reasonably constructed. Based on the model, a state feedback controller is designed such that the mean square of tracking error tends to an arbitrarily small neighborhood of zero. There are some remaining problems to be investigated: ) Find other control methods such as [0-3] to solve this problem and compare their effectiveness. ) Consider the output feedback control, adaptive tracking control and robust control by extending the deterministic control methods [4,5], etc.) to the stochastic case. Acknowledgment. This work is partially supported by the National Natural Science Foundation of China Nos. 6735, 6738), the Specialized Research Fund for the Doctoral Program of Higher Education No ), the Shandong Provincial Natural Science Foundation of China No. ZR0FM08), the Project of Taishan Scholar of Shandong Province. REFERENCES [] W. Q. Zhu, Nonlinear Stochastic Dynamics and Control: A Framework of Hamiltonian Theory, Science Press, Beijing, 003 in Chinese). [] M. W. Spong, S. Hutchinson and M. Vidyasagar, Robot Dynamics and Control, John Wiley & Sons Inc, New York, 004. [3] F. L. Lewis, D. M. Dawson and C. T. Abdallah, Robot Manipulator Control-Theory and Practice, Marcel Dekke, New York, 004. [4] L. Sciavicco and B. Siciliano, Modelling and Control of Robot Manipulator, Springer-Verlag, London, 000. [5] R. Kelly, V. Santibez and A. Lora, Control of Robot Manipulators in Joint Space, Springer-Verlag, London, 005. [6] F. L. Lewis, S. Jagannathan and A. Yeildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems, Taylor & Francis, Inc, Bristol, 998. [7] S. S. Ge, T. H. Lee and C. J. Harris, Adaptive Neural Network Control of Robotic Manipulators, World Scientific Publishing Co. Pte. Ltd., Singapore, 998. [8] A. C. Huang and M. C. Chien, Adaptive Control of Robot Manipulators: A Unified Regressor-Free Approach, World Scientific Publishing Co. Pte. Ltd., Singapore, 00. [9] H. D. Patino, R. Carelli and B. R. Kuchen, Neural networks for advanced control of robot manipulators, IEEE Trans. on Neural Networks, vol.3, no., pp , 00. [0] J. Kasac, B. Novakovic, D. Majetic and D. Brezak, Passive finite-dimensional repetitive control of robot manipulators, IEEE Trans. on Control Systems Technology, vol.6, no.3, pp , 008. [] K. Najim, E. Ikonen and E. Gomez-Ramírez, Trajectory tracking control based on a genealogical decision tree controller for robot manipulators, International Journal of Innovative Computing, Information and Control, vol.4, no., pp.53-6, 008. [] R. Z. Khas minskii, Stochastic Stability of Differential Equations, S & N International Publisher, Maryland, 980. [3] X. Mao, Stochastic Differential Equations and Applications, Horwood, New York, 997. [4] H. Deng, M. Krstić and R. J. Williams, Stabilization of stochastic nonlinear systems driven by noise of unknown covariance, IEEE Trans. on Automatic Control, vol.46, no.8, pp.37-53, 00. [5] J. M. Knudsen and P. G. Hjorth, Elements of Newtonian Mechanics: Including Nonlinear Dynamics, Springer-Verlag, New York, 00. [6] M. G. Calkin, Lagrangian and Hamiltonian Mechanics, World Scientific Publishing Co. Pte. Ltd., Singapore, 996. [7] D. L. Goodstein, R. P. Olenick and T. M. Apostol, The Mechanical Universe: Introduction to Mechanics and Heat, Cambridge University Press, Cambridge, 008. [8] B. Øksendal, Stochastic Differential Equations An Introduction with Applications, 6th Edition, Springer-Verlag, New York, 003. [9] M. M. Bridges, D. M. Dawson, Z. Qu and S. C. Martindale, Robust control of rigid-link flexible-joint robots with redundant joint actuators, IEEE Trans. on Systems, Man, and Cybernetics, vol.4, no.7, pp , 994.

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