Second-Order Sliding Mode Control with Adaptive ...

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Second-order Sliding Mode Control with Adaptive Control Authority for the Tracking Control of Robotic Manipulators Luca M. Capisani ∗ Antonella Ferrara ∗ Alessandro Pisano ∗∗ ∗

Dept. of Computer Engineering and Systems Science, Univ. of Pavia, Pavia, Italy; e-mail: {luca.capisani,antonella.ferrara}@unipv.it. ∗∗ Dept. of Electrical and Electronic Engineering, Univ. of Cagliari, Cagliari, Italy; e-mail: [email protected].

Abstract: In this work, the joint position tracking control problem of industrial robots is tackled. To cope with the model uncertainties and external disturbances affecting the robot, the Inverse Dynamic Controller (IDC) is combined with an approach based on higher order Sliding Mode Control (SMC) technique. We make use, in particular, of the so-called “Twisting” Second Order Sliding Mode Controller. Higher order SMC techniques transfer the inherent discontinuities to the time derivative of the input torque and this allows to obtain a continuous profile for the input torque, which is computed through integration of an appropriate discontinuous switching signal. Despite the chattering phenomenon is strongly attenuated, some residual problems (vibration and acustical noise) are still observed during the experimental implementation of such an approach in its standard formulation. To improve the system performance we suggest in this work an adaptation mechanism to adjust on-line the authority of the SMC. The logic is driven by a “sliding-mode indicator” that detects, on line, the occurrence of a sliding mode behaviour and uses this information for adaptation purposes. When large and fast control activity is demanded (e.g. to track fast reference trajectories) the adaptation unit reacts by automatically increasing the control authority of the SMC. On the other hand when small control authority is sufficient the control magnitude is lowered. Such a bidirectional adaptation logic significantly reduces the chattering. The proposed technique is theoretically analyzed and experimentally tested, and the results of comparative experiments are discussed in the paper. 1. INTRODUCTION Motion control of industrial plants is a well-known problem in robotics (Sciavicco and Siciliano [2000]). In fact, industrial robots often have many degrees of freedom and complex kinematics and dynamics, thus nonlinear and unmodelled effects may be significant. The accuracy of the trajectory tracking also depends on the particular trajectory to be followed, (Capisani et al. [2009]), and on the presence of unknown additional loads on the manipulator structure. Unmodelled friction effects are more evident when slow trajectories are imposed. Previous research and experimental activities have shown that different approaches may be adopted in order to solve these problems, such as, for instance, decentralized control, see Asada and Slotine [1986], Koivo [1989], Chiacchio et al. [1993], Sciavicco and Siciliano [2000]; feedback linearization, see Kreutz [1989], Kuo and Wang [1989], Abdallah et al. [1991], Spong et al. [1993], Ferrara and Magnani [2007], Calanca et al. [2007]; model predictive control, see Richalet et al. [1997], Juang and Eure [1998], Poignet and Gautier [2000]; sliding mode control, see Guldner et al. [1995], Shyu et al. [1996], Chen and Chang [1999], Utkin et al. [1999], Jafarov et al. [2000], Bartolini et al. [2000], Bartolini et al. [2003], Bartolini and Pisano [2003], Davila et al. [2005], and Capisani et al. [2009]; discrete-time sliding mode control, see Capisani et al. [2010]; adaptive Copyright by the International Federation of Automatic Control (IFAC)

control, see, for instance, Balestrino et al. [1983], Craig [1988], Ortega and Spong [1989], Liu [1999], Colbaugh et al. [2000], Perk et al. [2001], and Cheah et al. [2006]. In this paper, once the model of the system has been formulated and identified, see Calanca et al. [2010] and Capisani et al. [2007], feedback linearization is adopted to linearize and decouple the dynamics of the manipulator, see Sciavicco and Siciliano [2000] and Capisani et al. [2009], such as to define an auxiliary input. In the new inputoutput relation, a sliding mode controller is designed on the basis on the linear and decoupled plant. In practice, it is difficult to achieve a global linearization, since the nonlinear dynamics compensation is sensitive to uncertainties which arise from imprecise knowledge of the dynamics and other unknown effects on the actual system. This is the reason why robust control methodologies are often adopted, see Abdallah et al. [1991]. The higher order sliding mode control represents a suitable approach in robotics, see Bartolini et al. [2003], Bartolini and Pisano [2003], and Capisani et al. [2009]. By adopting this solution, it is possible to attenuate the vibrations generated by the so-called chattering effect while keeping the robustness properties typical of conventional sliding mode control. The practical implementation of these techniques often requires an experimental tuning of the parameters because of the presence of unknown unmodelled

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effects in the plant to be controlled. In this proposal, the Twisting second order sliding mode control approach (see Levant [1993]) is considered to design the auxiliary input to the decoupled linearized plant. An adaptation unit that adjusts on-line the discontinuous control effort is then proposed and tested. The logic is driven by a “slidingmode indicator” that detects, on line, the occurrence of a sliding mode behaviour and uses this information for adaptation purposes. When large and fast control activity is demanded (e.g. to track fast trajectories) the adaptation unit reacts by automatically increasing the control authority of the SMC. On the other hand, when small control authority is sufficient the control magnitude is lowered. This is achieved by decreasing or increasing the control gain stepwise, at the end of a the adjacent time intervals of equal and pre-specified length T , on the basis of the number of zero crossings of the sliding variable. Such a bidirectional adaptation logic significantly reduces the chattering. It is shown that due to this machinery the system trajectories remain confined to a small boundary layer of the sliding manifold of size O(T 2 ). Note that the proposed approach presents the advantage of reducing the chattering effect by suitably adjusting the control authority. This advantage is independent of the use of a feedforward component. Clearly, the presence of the feedforward term further improves the performance. 2. SYSTEM MODEL AND CONTROL OBJECTIVE The dynamical model of a n-joints robot manipulator in the joint space, convenient as a model to be used to design motion controllers can be written, by using the Lagrangian approach, as τ (t) = B(q(t))¨ q (t) + n(q, q) ˙ n(q, q) ˙ = C(q(t), q(t)) ˙ q(t) ˙ + g(q(t)) + Fv q(t) ˙

(1)

(see Sciavicco and Siciliano [2000]), where q ∈ Rn represents the system configuration vector, q T = [q1 , . . . , qn ], τ ∈ Rn is the vector of motor control torques, B(q) ∈ Rn×n is the inertia matrix, C(q, q) ˙ q˙ ∈ Rn represents centripetal and Coriolis torques, Fv ∈ Rn×n is the viscous friction diagonal matrix, and g(q) ∈ Rn is the vector of gravitational torques, see also Spong et al. [1993]. The control objective considered in the paper is to track a sufficiently smooth pre-specified joint trajectory qd (t). (3) Let q˙d (t), q¨d (t) and qd (t) exist and be bounded almost everywhere. More precisely, the aim is to steer the tracking error qd (t) − q(t) within a small boundary of the origin. The identification of the parameters of this model follows the guidelines of a recently devised procedure, see Capisani et al. [2007] and Calanca et al. [2010]. Let the term n ˆ (q, q) ˙ denote the identified centripetal, Coriolis, gravity and friction torques terms, while the inertia matrix B(q) is assumed to be known. 3. SECOND-ORDER SLIDING MODE CONTROL WITH ADAPTIVE CONTROL EFFORT We select the input torque as follows τ = B(q)y(t) + n ˆ (q, q) ˙

(2)

where y(t) is a fictitious input variable having the physical dimension of an acceleration.

Then, the compensated system becomes q¨ = y(t) + B(q)−1 n ˜ (q, q) ˙ = y(t) − η(t)

(3)

where η = −B −1 (q)˜ n(q, q) ˙ and n ˜ (q, q) ˙ =n ˆ (q, q) ˙ − n(q, q). ˙ The y(t) signal is obtained by combining a feedforward and a feedback component y(t) = u(t) + q¨d (t) − c(q˙ − q˙d ), c>0 (4) where u(t) has to be designed so as to enforce the robust tracking. Denote the sliding vector as σ(t) = q˙ − q˙d + c(q − qd ). (5) The control objective can be achieved by steering the vector σ(t) as close as possible to the origin. Trivial computations show that the second order dynamics of the sliding vector σ is a perturbed double integrator of the form σ ¨ (t) = u(t)− ˙ η(t). ˙ From now on let us denote through the subindex i the generic component of the n-dimensional vectors involved in the system dynamics. Let η˙i (t) and η¨i (t) of the uncertain perturbation vectors η(t) ˙ and η¨(t) be globally bounded by known constants Ni (i = 1, 2, ..., n) according to |η˙i (t)| ≤ Ni |¨ ηi (t)| ≤ Pi . (6) Let σi (t) and ui (t) denote the i − th entry of vectors σ(t) and u(t), respectively. We shall now define an adaptive discontinuous control law for the derivative u˙ i (t). The actual control torque will be then computed by (2). The standard (i.e. non-adaptive) dynamical twisting algorithm can be implemented as Levant [1993] u˙ i (t) = −αi UMi signσ + UMi signσ, ˙ i = 1, 2, ..., n (7) where αi > 3 is an arbitrary constant and the gain parameter UMi must fulfill the next inequality called “dominance condition”: UMi > Ni . (8) The initial condition ui (0) is irrelevant (it can be customarily set to zero). It is well known (see Levant [1993]) that the control algorithm (7)-(8) steers to zero in finite time the sliding vector σm σ, ˙ which are governed by the uncertain double integrator dynamics (3)-(6). The transient trajectory in the σ-σ˙ plane features contractive rotations (“twists”) similar to those reported in the Fig. 1.

Fig. 1. Twisting Algorithm Trajectories in the σ-σ˙ plane. The algorithm considered in this paper performs a real time adaptation of the UMi parameter, and it has been firstly introduced in Bartolini et al. [1999]. The basis of the adaptation mechanism is the so-called “sliding mode indicator ”. We consider adjacent time intervals of fixed length Ti , and we compute, at the end of

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every interval, the number of zero crossings of the sliding quantity during the interval. When this number is large, it reveals the occurrence of the sliding mode behaviour, which, in turns, implies that the control authority is large enough. If the sliding mode is correctly taking place then it is sensible to reduce the control gain. On the other hand, when the sliding mode is not occurring the control gain is increased. Upper and lower saturation thresholds are also taken into account. As compared to (Bartolini et al. [1999]), here we apply the same adaptation mechanism to a different controller configuration, and furthermore we test the algorithm experimentally. The adaptation procedure is formalized as follows. Starting from the initial time moment we consider the sequence of adjacent time intervals Tij of width Ti : Tij ≡ [(j − 1)Ti , jTi ) j = 1, 2, . . . (9) and we suggest a piece-wise constant form for the adaptive coefficient UMi which becomes adaptive according to j UMi (t) = UMi , t ∈ Tij . (10) j We modify the control amplitude UMi at the end of each time interval Tij according to 1 UMi = U(Mi j j max(UMi − Λ1i Ti , 0) if Nsw,i ≥ Ni∗ j+1 UMi = j j min(UMi + Λ2i Ti , UMi ) if Nsw,i < Ni∗

(11)

j in which Nsw,i is the number of sign commutations of σi in the interval Tij and Ni∗ , i = 1, 2, 3 are appropriate integer numbers. Roughly speaking, at the end of each time interval Tij we decrement the control magnitude stepwise of Λ1 Ti while the frequency of sign commutations of σ is sufficiently high, otherwise we increment it stepwise of Λ2 Ti . The adaptation logic also include lower and upper bounds for the control magnitude (0 and UMi , respectively). Λ1 > 0 is free to be chosen while Λ2 should be sufficiently large. Each parameter Ni∗ can be evaluated by taking into account the following Lemma. Lemma 1

Consider the second-order sliding variable dynamics σ ¨ (t) j and let Nsw,i be the number of zero crossings of σi during j the time interval Tij of length Ti . If condition Nsw,i ≥ 3 is satisfied, and, in the same interval Tij , |¨ σi | ≤ a1 for some a1 > 0, then inequalities (12) |σi | ≤ a1 Ti2 |σ˙ i | ≤ a1 Ti

with Ni∗ ≥ 3 and parameter UMi chosen in accordance with (8). Then the following inequalities are achieved after a finite-time transient process for some positive constants b1 , b2 and for all i = 1, 2, ...n |σi (t)| ≤ b1 Ti2 ,

|σ˙ i (t)| ≤ b2 Ti .

(15)

Proof of Theorem 1 Let the initial conditions σi (0), σ˙ i (0) be sufficiently large in magnitude. During the initial transient there are obviously no frequent sign commutations of σ(t), hence the magnitude of the discontinuous control effort keeps constant at the value UMi which is large enough to counteract the worst case uncertainties and, therefore, according to the convergence properties of the Twisting algorithm Levant [1993], enforces contractive rotations of the system trajectories in the σ − σ˙ plane as shown in the Fig. 1. As long as the actual system trajectory will be approaching the origin featuring subsequent crossings of the σ = 0 and σ˙ = 0 axes, the frequency of the sign commutations of σ will correspondingly and progressively j increase. Thereby condition Nsw,i ≥ 3 is kept at some finite j = M1 , and the stepwise reduction of UMj is then activated starting from the end of the time interval TiM1 . M1 ≥ Ni∗ , at the end of By relying on the fact that Nsw,i the time interval TiM1 (i.e., at the time instant t = M1 Ti ) M1 the actual value of UMi will be “dominating” the actual M1 upperbound of η˙ i (t) in accordance with UMi > |η(M ˙ 1 Ti )|. The dominance over the uncertainties (formalized by condition (8)) will be lost after a finite number of intervals, j and at some j = M2 > M1 the 2-sliding criterion Nsw,i ≥3 will be violated.

It implies that at the end of the preceding time interval Ti,M2 −1 this dominance inequality holds: M2 −1 UMi > |η((M ˙ 2 − 1)Ti )|.

(16)

By Lemma 1, at the end of such time interval Ti,M2 −1 the variables σ and σ˙ are bounded as in (12) with M2 −1 σi (t)| = Ni + αi UMi . a1 = supt∈Ti,M2 −1 |¨

(17)

At the end of the interval Ti,M2 +1 , i.e. one interval after j the violation of the 2-sliding criterion Nsw,i ≥ 3, the magnitude of the uncertainty η(t) ˙ will be such that |η((M ˙ ˙ 2 + 1)Ti )| ≤ |η((M 2 − 1)Ti )| + 2Pi Ti

(18)

hold in the whole time interval Tij . Proof of Lemma 1 . If function σi has more than two zero crossings within the interval Tij , then, by virtue of the Rolle theorem, its first derivative σ˙ i has more than one zero crossing within the same interval. Since |¨ σi | < a1 by assumption, simple time-integration yields (12). 2

which is trivially derived by taking into account (6). On the other hand, the adaptive magnitude will be increased at the end of the interval Ti,M2 − 1, and decreased at the end of the successive interval Ti,M2 +1 , which means that

Summarizing, the proposed control algorithm has the following form j j u˙ i (t) = −αi UMi signσ + UMi signσ, ˙ t ∈ Tij . (13)

Therefore, considering (16) and (19), provided that the Λ2 parameter is such that Λ2 > Λ1 + 2Pi . It follows that the dominance condition (8) will be already restored at the end of the interval Ti,M2 +1 , i.e. one interval after the violation j of the 2-sliding criterion Nsw,i ≥ 3. The maximal deviation undergone by |σ| and |σ| ˙ can be evaluated by studying the limit trajectories obtained starting from the initial condition (12). The resulting algebraic computations, skipped

The following Theorem can be demonstrated. Theorem 1. Consider system σ ¨ (t) = u(t) ˙ − η(t) ˙ satisfying Assumption (6), choose the control input as in (13), (11). Let Λ1i > 0, Λ2i > Λ1i + 2Pi (14)

M2 −1 M2 +1 + Ti (Λ2 − Λ1 ). > UMi UMi

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

The standard (i.e. non adaptive) Twisting controller will be denoted as TC, while its adaptive version will be denoted as A-TC. In the first test (TEST 1), sinusoidal reference trajectories for the controlled joints are considered. The TC has been implemented with the parameters αT = [316.6, 175.0, 1065.0] T UM

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(21)

= [UM1 , UM2 , UM3 ] = [0.0015, 0.0142, 0.0226].(22)

The A-TC controller has been also implemented with the parameters Λ1 = 0.00003, Λ2 = 0.00004, T1 = T2 = 100ms, T3 = 32ms, N ∗ = [N1∗ , N2∗ , N3∗ ] = [5, 5, 3]T . The plots in Fig. 3 show the joint reference and actual trajectories, and the corresponding tracking errors, when

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The plots in Fig. 5 shows the ui (t) command signals when the TC is applied (on the left) and when the A-TC is applied (on the right) to perform TEST 1. The comparison clearly shows that the chattering effect is suitably attenuated using the adaptation procedure. This is also confirmed by the reduction of the audible noise produced by the actuators. Table 2 reports a

Fig. 2. The COMAU SMART3-S2 robot. It consists of six-DOF actuated by six brushless electric motors. Six 12 bit resolvers supply accurate angular position measurements. In this paper, a three-DOF planar manipulator is considered moving on a vertical plane. That is, for our purposes, joints 1, 4 and 6 of the robot have been locked so that only joints 2, 3 and 5 are used. The three considered joints are numbered as {1, 2, 3}. The height of the robot is about 3m.

Joint 1 0.00041049 0.00026154 -36.2855 %

Table 1. RMS of the Tracking Errors in the TEST 1 for the TC and A-TC.

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4. EXPERIMENTAL RESULTS The COMAU SMART3-S2 industrial anthropomorphic rigid manipulator, located at the Department of Electrical Engineering of the University of Pavia, is shown in Fig. 2.

Exp. TC A-TC Diff. %

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Remark 1 Theorem 1 does not guarantee anymore the occurrence of the ideal sliding mode behaviour σi (t) = σ˙ i (t) = 0 but only the fact the proposed adaptation globally steers the systems towards an invariant boundary layer depending on, and vanishing with, the Ti parameter. The main benefit associated with the proposed adaptation logic is the reduction of the chattering phenomenon.

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j While UMi continues to grow, the contractive rotations of the system trajectories around the origin of the σ − σ˙ plane preserve the inequalities (15). The process of loose and successive restoring of the dominance will continue iteratively in the subsequent time intervals, so the Theorem is proved. 2

the TEST 1 is executed using the standard TC. The plots in Fig. 4 show the same signals when the TEST 1 is executed using the A-TC. From the comparative analysis of the two figures it follows that the order of magnitude of the tracking accuracy appears to be similar in the two cases. However, the waveforms obtained using the A-TC appears smoother than those obtained using the standard TC. To investigate this fact, the RMS value of the tracking errors has been computed and the corresponding values are reported in the Table 1. A reduction of near 35% can be observed when the A-TC is implemented. Note that these results are significantly improved w.r.t. those obtained on the same experimental setup by applying other chattering reduction scheme (see previous works in references of Capisani and Ferrara).

e [deg]

for the sake of brevity, yield conditions (15). A conservative estimate of b1 is given by   Ni + αi UMi . (20) b1 = (Ni + αi UMi ) 1 + 2 (αi UMi − Ni )

Joint 1 96.1365 84.7422 -11.8523 %

Joint 2 21.1743 15.9736 -24.5613 %

Joint 3 5.4308 4.3319 -20.2349 %

Table 2. Mean actuator power consumption in the TEST 1 for the TC and A-TC. tests has demonstrated that the adaptive control law attenuates chattering and reduces the power consumption as compared with the non-adaptive case.

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Fig. 5. TEST 1. Comparison between the ui (t) signals using TC (left) and A-TC (right). A second series of test (TEST 2) has been made using a more complicated profile for the joint reference trajectories. The proposed A-TC has been implemented with the same parameters as those used in the previous experiments. The task of TEST 2 is to give a better understanding of the mechanism that make the proposed adaptation mechanism work. The plots in Fig. 6 shows the actual and desired joint trajectories and the tracking error obtained during TEST 2. It can be seen that the tracking accuracy is much higher in the central phase of the experiment when the joint reference trajectory is constant, see Fig. 7, which shows the UMi (t) signals (on the left) and the u(t) ˙ signals (on the right). In the central phase smaller control authority is sufficient, and the control gain is indeed lowered by the adaptation unit. It is not surprising that the control authority can tend to zero when the set point is constant since the proposed overall controller (2), (4) embeds the compensation of gravity which is not shown in the figure 7. On the other hand, when the reference trajectories becomes faster the control gain is promptly increased. 5. CONCLUSION A novel chattering-free sliding mode control algorithm for the tracking control of robotic manipulators has been

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Fig. 7. TEST 2. Discontinuous input gain UMi (left) and the correspondent u(t) ˙ signals (right) with the A-TC controller. proposed and experimentally tested. The main novelty is the implementation of an adaptive mechanism that adjusts on-line the authority of the discontinuous control component. Interestingly, the proposed method gives rise to a bidirectional adaptation policy. A significant chattering reduction, and some other benefits, are observed during the experiments. The proposed methodology seems to be applicable to adjust the gain in more general discontinuous control algorithm as well (e.g the super-twisting or suboptimal 2-SMC algorithm), and next activities will be devoted to explore this opportunity, not necessarily in the area of robotics. REFERENCES C. T. Abdallah, D. M. Dawson, P. Dorato, and M. Jamshidi. Survey of robust control for rigids robots. IEEE Control Systems Magazine, 11(2):24–30, Feb. 1991. H. Asada and J. J. E. Slotine. Robot Analysis and Control. John Wiley & Sons, New York, USA, 1986. A. Balestrino, G. De Maria, and L. Sciavicco. An adaptive model following control for robotic manipulators. ASME Journal of Dynamic Systems, Measurement, and Control, 105:143–151, Sep. 1983.

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

G. Bartolini and A. Pisano. Global output-feedback tracking and load disturbance rejection for electrically-driven robotic manipulators with uncertain dynamics. International Journal of Control, 76(12):1201–1213, Aug. 2003. G. Bartolini, A. Levant, A. Pisano, and E. Usai. 2-sliding mode with adaptation. In Proc. IEEE Mediterranean Conference on Control and Systems, pages 2421–2429, Haifa, Israel, Jun. 1999. G. Bartolini, A. Ferrara, and E. Punta. Multi-input second-order sliding-mode hybrid control of constrained manipulators. Dynamics and Control, 10(3):277–296, Jul. 2000. G. Bartolini, A. Pisano, E. Punta, and E. Usai. A survey of applications of second-order sliding mode control to mechanical systems. International Journal of Control, 76(9):875–892, Jun. 2003. A. Calanca, L. M. Capisani, A. Ferrara, and L. Magnani. An inverse dynamics-based discrete-time sliding mode controller for robot manipulators. In K. Kozlowski, editor, Robot Motion and Control 2007, (LNCiS) Lecture Notes in Control and Information Sciences n. 360, ch. 12, pages 137–146. Springer-Verlag, London, UK, 2007. A. Calanca, L. M. Capisani, A. Ferrara, and L. Magnani. Mimo closed loop identification of an industrial robot. IEEE Transactions on Control Systems Technology, to appear 2010. L. M. Capisani, A. Ferrara, and L. Magnani. MIMO identification with optimal experiment design for rigid robot manipulators. In Proc. IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pages 1–6, Z¨ urich, Switzerland, Sep. 4-7 2007. L. M. Capisani, A. Ferrara, and L. Magnani. Design and experimental validation of a second-order sliding-mode motion controller for robot manipulators. International Journal of Control, 82(2):365–377, Feb. 2009. L. M. Capisani, T. Facchinetti, and A. Ferrara. Real-time networked control of an industrial robot manipulator via discrete-time second-order sliding modes. International Journal of Control, 83(8):1595–1611, Aug. 2010. C. C. Cheah, C. Liu, and J. J. E. Slotine. Adaptive tracking control for robots with unknown kinematic and dynamic properties. The International Journal of Robotics Research, 25(3):283–296, Mar. 2006. Y. Chen and J. L. Chang. Sliding-mode force control of manipulators. National Science Council ROC(A), 23 (2):281–289, 1999. P. Chiacchio, F. Pierrot, L. Sciavicco, and B. Siciliano. Robust design of independent joint controllers with experimentation on a high-speed parallel robot. IEEE Transactions on Industrial Electronics, 40(4):393–403, Aug. 1993. R. D. Colbaugh, E. Bassi, F. Benzi, and M. Trabatti. Enhancing the trajectory tracking performance capabilities of position-controlled manipulators. In Proc. IEEE Industry Applications Conference, volume 2, pages 1170– 1177, Rome, Italy, Oct. 2000. J. J. Craig. Adaptive Control of Mechanical Manipulators. Addison-Wesley, New York, USA, 1988. J. A. Davila, L. M. Fridman, and A. Levant. Second-order sliding-mode observer for mechanical systems. IEEE Transactions on Automatic Control, 50(11):1785–1789, Nov. 2005.

A. Ferrara and L. Magnani. Motion control of rigid robot manipulators via first and second order sliding modes. Journal of Intelligent and Robotic Systems, 48(1):23–36, Jan. 2007. J. Guldner, V. I. Utkin, H. Hashimoto, and F. Harashima. Obstacle avoidance in rn based on artificial harmonic potential fields. In Proc. IEEE Conference on robotics and automation, volume 3, pages 3051–3056, Nagoya, Aichi, Japan, May 1995. E. M. Jafarov, A. M. N. Parlak¸ci, and Y. Istefanopulos. A new variable structure pid-controller design for robot manipulators. IEEE Transactions on Control Systems Technology, 13(1):122–130, Jan. 2000. J. N. Juang and K. W. Eure. Predictive feedback and feedforward control for systems with unknown disturbance. NASA/Tm-1998-208744, pages 1–35, Dec. 1998. A. J. Koivo. Fundamentals for Control of Robotic Manipulators. Wiley & Sons, New York, USA, 1989. K. Kreutz. On manipulator control by exact linearization. IEEE Transactions on Automatic Control, 34(7):763– 767, Jul. 1989. C. Y. Kuo and S. P. T. Wang. Nonlinear robust industrial robot control. Transactions of the ASME Journal of Dynamic Systems, Measurement and Control, 111(1): 24–30, Mar. 1989. A. Levant. Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58(6): 1247–1263, Aug. 1993. M. Liu. Decentralized control of robot manipulators: Nonlinear and adaptive approaches. IEEE Transactions on Automatic Control, 44(2):357–363, Feb. 1999. R. Ortega and M. W. Spong. Adaptive motion control of rigid robots: a tutorial. Automatica, 25(6):877–888, Nov. 1989. J. S. Perk, G. S. Han, H. S. Ahn, and D. H. Kim. Adaptive approaches on the sliding mode control of robot manipulators. Transactions on Control, Automation and Systems Engineering, 3(2):15–20, Mar. 2001. P. Poignet and M. Gautier. Nonlinear model predictive control of a robot manipulator. In Proc. 6th International Workshop on Advanced Motion Control, pages 401–406, Nagoya, Japan, Mar. 2000. J. Richalet, E. Abu, S. Ata-Doss, C. Arber, H. B.Kuntze, A. Jacubash, and W. Schill. Predictive functional control. application to fast and accurate robots. In Proc. 10th IFAC World Congress, pages 251–258, Munich, Germany, Jul. 1997. L. Sciavicco and B. Siciliano. Modelling and Control of Robot Manipulators. Springer-Verlag, London, UK, second edition, 2000. K. K. Shyu, P. H. Chu, and L. J. Shang. Control of rigid robot manipulators via combination of adaptive sliding mode control and compensated inverse dynamics approach. In Proc. IEE Control Theory and Application, volume 143, pages 283–288, May 1996. M. W. Spong, F. L. Lewis, and C. T. Abdallah. Robot Control: Dynamics, Motion Planning, and Analysis. IEEE Press, Piscataway, New Jersey, USA, 1993. V. I. Utkin, J. Guldner, and J. Shi. Sliding Mode Control in Electromechanical Systems. Taylor & Francis, London, UK, 1999.

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