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Hindawi Journal of Robotics Volume 2020, Article ID 8861847, 12 pages https://doi.org/10.1155/2020/8861847 Research Article Cascaded Control of Flexible-Joint Robots Based on Sliding-Mode Estimator Approach 1 2 1 Genliang Xiong , Jingxin Shi, and Haichu Chen School of Mechanical Engineering, Nanchang University, Nanchang XF999, China TTTech Germany GmbH, Shanghai 200070, China Correspondence should be addressed to Genliang Xiong; xionggenliang@ncu.edu.cn Received 7 July 2020; Revised 22 September 2020; Accepted 30 September 2020; Published 27 October 2020 Academic Editor: Arturo Buscarino Copyright © 2020 Genliang Xiong et al. 'is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 'e inherent highly nonlinear coupling and system uncertainties make the controller design for a ﬂexible-joint robot extremely diﬃcult. 'e goal of the control of any robotic system is to achieve high bandwidth, high accuracy of trajectory tracking, and high robustness, whereby the high bandwidth for ﬂexible-joint robot is the most challenging issue. 'is paper is dedicated to design such a link position controller with high bandwidth based on sliding-mode technique. 'en, two control approaches ((1) ex- tended-regular-form approach and (2) the cascaded control structure based on the sliding-mode estimator approach) are presentedforthelinkpositiontrackingcontrolofﬂexible-jointrobot,consideringthedynamicsofAC-motorsinrobotjoints,and compared with the singular perturbation approach. 'ese two-link position controllers are tested and veriﬁed by the simulation studies with diﬀerent reference trajectories and under diﬀerent joint stiﬀness. model (i.e., the joint ﬂexibility is generated by linear spring 1. Introduction and the kinetic energy of a joint is only generated by the 'e development of robotics in the past few years has been rotation of this joint), the resulting control algorithm is extended from the earlier standard applications of industrial rather complicated, due to the state transformation and the robots to new ﬁelds such as space, service robotics, medical, inverse calculation of the control input. 'e control algo- and force-feedback systems. 'is demand makes the re- rithm depends on the robot parameters, which are generally search directions on manipulators is to lighten the total unknown. 'e robust analysis about the feedback lineari- weightwhilekeepingthecontrolandoperationperformance zation approach can be found in [3]. unchanged. Especially, desire for higher performance from In general case of the ﬂexible-joint robot model, the the structure and mechanical speciﬁcations of chain-like static feedback linearization may not be realizable [4]. De Luca and Lucibello involve the previous system information mechanical manipulators has spurred designers to come up with ﬂexible-joint robots. 'e topic of control of ﬂexible- to form the so-called dynamic feedback linearization [5]. He joint robots has troubled control experts of the world several uses not only the actual states of the robot, but also the past decades. states; no global state transformation is required. 'e Most of the researchers start the control design for resulting control structure is of 2n(n − 1) order (with n ﬂexible-joint robots with the Spong model [1]. Since then, a being the number of robot joints). He pointed out the large amount of theoretical and experimental results are suﬃcient condition of dynamic feedback linearization: there developed. is no zero dynamics in the system. 'e authors in [5] won a Some descriptions about the state-space approach based best paper award during conference IRCA98 due to the on the feedback linearization have been given before. As theoretical contribution. For a simpliﬁed ﬂexible-joint robot proposed by Spong in [2], even using a simpliﬁed robot model, both static and dynamic feedback linearization can 2 Journal of Robotics It is clear that the advance control approaches for be applied. However, both linearization methods need the state variables which may not be measured in a practical ﬂexible-joint robots need the support of advance control theories. However, it seems that the model-based control robot system. An observer design for the unmeasured state variables has been proposed [6, 7]. However, these observer theories in the last 20 years have not got signiﬁcant progress design approaches increased the complexity of the control in the sense of solving real-world control problems, such as system and may make the feedback linearization the control of ﬂexible-joint robots as high-order, nonlinear, meaningless. uncertain MIMO systems. On the other hand, non-model- Singular perturbation approach is one of the promising based control approaches such as fuzzy-logic control and approaches to control the real-world lightweight robots, neutral network-based control have been tested everywhere and it is hoped that these control approaches are universal which solves the control problem in two time scalars: a fast joint torque control (often in form of a damping) term for andapplicabletoanydynamicsystem.However,thevalueof non-model-based control approaches is often over esti- the fast mode of the joint torque dynamics, and a slow joint torque feedforward term for the outer position control loop mated. As mentioned before, the non-model-based control approaches may not be applicable to the control of high- (related to the rigid-body dynamics of the robot arm) [8]. More about the research works of singular perturbation order systems. 'e above observations have motivated re- approach for ﬂexible-joint robots can be found in [9–12]. searchers to ﬁnd a middle way between model-based and Integral back-stepping approach is actually one of the non-model-based control designs. It is recognized, mean- pure cascaded control approaches and has the advantages while, that to design a good control system, the controller suchasnotsensitivetothejointstiﬀness;statevariables used designer has to possess a deep understanding about the for the control implementation are available. It provides a physic plant to be controlled, independent of which control approach is applied. As a result, for control engineers who systematic way, i.e., a step-by-step way to design a Lyapunov function for the overall control system. However, the have no “good feeling” about the controlled plant, a rough model which contains the basic bone structure of the dy- resulting controllers based on the basic version of integral back-stepping approach need system parameters; thus, the namic system is highly desirable, though there are some unmodeled dynamics, external disturbances, and parameter approach is sensitive to these parameters. To overcome this drawback, the Lyapunov function is often extended to in- uncertainties associated with this rough model. As a can- volve a parameter adaptation process and the system ro- didate of the control theories which are able to handle the bustness with respect to the parameter variations is basicbone-structuremodelwithahighdegreeofrobustness, theoretically ensured. However, the parameter adaptation variable structure control as well as sliding-mode control process makes the overall control system more complicated [18] (in this thesis, no diﬀerence is made between these two and may not be able to react on the fast changing of the closely related control approaches) has been selected for the control problems of the uncertain nonlinear systems system parameters. 'erefore, integral back-stepping ap- proach provides more theoretical contribution than it [19–22]. It well known that sliding-mode control theory can be applied to high-order, nonlinear, uncertain MIMO sys- practically does. Some works about the integral back-step- ping approach used for the control of ﬂexible-joints robots tems and the resulting controllers are generally simple can be found in [13, 14]. enough for the real-time implementation. Another advan- Passivity-based control approach uses the concept of tage of sliding-mode control theory is easy to understand for storage energy as well as storage energy changing (in time), “normal” control engineers. 'e major disadvantage asso- providing a suﬃcient condition for a dynamic system to be ciated with sliding-mode control is the chattering phe- stable. 'is control approach possesses some nice features: nomena due to the high-frequency switching of the physically interpretable, systematical Lyapunov stability discontinuous control input. However, if the chattering proof (using just the energy functions of the system), certain problem can be solved or the inherent discontinuous property of the ﬁnal control inputs (for the case of ﬂexible- degree of robustness with respect to system uncertainties, simple-form controller, applicable to the case when con- jointrobots,theﬁnalcontrolinputsaretheterminalvoltages on the stator windings of the electric motor used in robot tacting with environmental objects, etc. 'e author believes that passivity-based control approach is another promising joints) can be positive utilized, sliding-mode control theory control approach besides the singular perturbation ap- will be a good control design tool for the systems such as proach. Ott studied the passivity-based control approach for ﬂexible-joint robots. ﬂexible-joint robots systematically and showed the potential Besides using the control approaches discussed above, of this approach for diﬀerent control tasks [15]. From the adaptive control techniques [23], fuzzy logic and neural work of Ott, it seems that the only weak point of this control network approaches [24], and simple PD (or PID) con- trol[25]werealsousedtothecontrolofﬂexible-jointrobots. approach lies in the tracking control performance; this might be the price one has to pay for the nice features. Other If selecting the link position and the joint torque as state variables, the Spong model can be transformed into the research works about the passivity-based control approach used for the control of ﬂexible-joint robots can be found in block form of state-space description as follows: [16, 17]. In the literature about the passivity-based control approach for ﬂexible-joint robots, the dynamics of the M(q)€ q + C(q, q _)q _ + G(q) + F(q _) � τ, electric motor used in the robot joint were generally and (1) τ + A (t)τ + D (t) � B τ , τ τ τ m unfortunately not taken into account. Journal of Robotics 3 n×n n where M(q) ∈ R is the mass matrix, C(q, q _)q _ ∈ R is the A: second-order link position system vector including centrifugal and Coriolis forces, n n G(q) ∈ R is the gravity force vector, F(q) ∈ R is the fric- n n tion force vector, q ∈ R is the link position vector, τ ∈ R is − 1 − 1 τ τ the joint torque vector A (t) � K(J + M (q) ) D (t) � τ τ B: second-order joint torque system − 1 − 1 K(J τ + M(q) N(q, q _)) B � KJΓ N(q, q _) � C(q, q _)q _+ ds , τ , n n G(q) + F(q _), τ ∈ R is the motor torque vector, τ ∈ R is m ds n×n Figure 1: Two-block system of ﬂexible-joint robots. thedisturbance torquevector, J � [J ] ∈ R is thediagonal n×n joint inertia matrix, K � [k ] ∈ R is the diagonal joint n×n stiﬀness matrix, and Γ � [c ] ∈ R is the diagonal gear- ratio matrix. A: second-order link position system Equation (1) is actually a two-block system, block A and blockB,asshowninFigure1.'emotortorqueτ generates τ = K I m t q the joint torque τ, while the joint torque τ generates the B: second-order joint torque system motion of the link position q. In the joint torque dynamics, i.e., the second equation of (1), the inﬂuence of the link position and its time derivative exists. Normally, this in- U , U d q C: first-order motor current system ﬂuence is treated as system uncertainties when designing the joint torque controller, because the model parameters are generally unknown. Figure 2: 'ree-block system of ﬂexible-joint robots. It is recognized that the dynamics of block B should be faster than the ones of block A, otherwise it makes no sense. 'is assumption is true for all designed robot manipulators tracking control, a cascaded control structure with an inner so far, regardless of how large compliance the robot joints current control loop and an outer joint torque control loop have. For this two-block system, many control approaches would not work properly. 'erefore, we need a general have been developed for the problem of link position solution to control the joint torque, independent of which control, either trajectory tracking control or point-to-point block is faster among blocks B and C. Such a joint torque regulation. control approach has been presented in [26]. 'e basic problems of the existing control approaches For the link position control issues discussed in this can be summarized as follows: paper, two control approaches will be presented. In Section 2, we review the singular perturbation approach to work (1) 'e dynamics of the electric motor are generally not with the robust link position controller for rigid-body considered. In the literature, some researchers manipulators and with the direct sliding-mode current considered the motor dynamics (most of them used control. In Section 3, we integrate the robust link position only a DC-motor model instead of an AC one), but controller with the direct sliding-mode joint torque con- the physical properties of the electric motor were not troller. In Section 4, the joint torque controller based on the positively utilized to increase the system perfor- sliding-mode estimator is cascaded with the robust link mance. Instead, the motor dynamics was always as position controller. All these control approaches take joint negative eﬀect taken into account. torqueasinterfacevariable(orbettertosayasstatevariable); (2) 'e existing control approaches generally lack joint thus, the extension to the end-eﬀector force and impedance torque tracking control capability. Some of the control will be straightforward. However, the singular control approaches do not take joint torque as a state perturbation approach does not possess joint torque variable (or called interface variable) to be con- tracking controlcapability;instead, therequiredjointtorque trolled;thus,theextensiontotheend-eﬀectforceand for the slow dynamics (i.e., the dynamics about the link impedance control is not straightforward. position) is implemented in a way of open-loop control, or (3) Link position tracking control, joint torque tracking called feedforward control. Because the singular perturba- control, dynamics of AC-motors, and robustness tion approach is a simple and eﬀective control approach, with respect to the system uncertainties were not thus it is considered here as an alternative to the proposed considered simultaneously. controllers. In this paper, we will consider all aspects in point (3) simultaneously. As ﬁrst, we add the motor dynamics to form 2. Singular Perturbation Approach the so-called three-block formulation of ﬂexible-joint robots 'e composite control structure of singular perturbation as shown in Figure 2. approach for the slow and fast dynamics will be summarized As mentioned before, the dynamics of block B are faster as follows. than the ones of block A. However, the dynamics of block B 'e robust link position controller for rigid-body ma- may be faster or slower than the ones of block C (i.e., the nipulators is now taken as the controller for the slow dy- dynamics about the motor currents). If the dynamics of namics of the ﬂexible-joint robots: block B are faster than that of block C, for the joint torque 4 Journal of Robotics joint stiﬀness is changed, these control gain matrices need to τ � M (q) € q − K q _ − K q + N (q, q _), ⎧ ⎪ 0 0 d D e P e 0 be retuned accordingly). For the motor current control, the equations of the current controller are written as follows (note that this s � q _ + K q + K q (ξ)dξ − q _ (0) − K q (0) ⎪ e D e P e e D e th current controller is only for the control of i joint motor, and the subscript i is not used for simplicity): ⎪ t ⎪ − 1 s � i − i , + M (q) τ − τ dξ ⎧ ⎪ ⎪ d d 1 1av d 0 ⎪ ⎪ s � i − i , ⎪ q q q ⎪ s ⎪ Ω � s cosθ − s sinθ , ⎪τ � − Γ , 1 d a q a ⎪ 1 0 ‖s‖ ⎪ ⎨ Ω � s cosθ − s sinθ , ⎪ 2 d b q b (5) τ � lowpass τ , ⎪ 1av 1 ⎪ Ω � s cosθ − s sinθ , 3 d c q c ⎪ ⎪ ⎪ u � − u sign Ω , 1 0 1 τ � τ + τ , d 0 1av u � − u sign Ω , 2 0 2 (2) u � − u sign Ω , 3 0 3 _ _ where M (q) � M(q) − ΔM and N (q, q) � N(q, q) − ΔN, 0 0 with ΔM and ΔN being the unknown part of matrix M(q) with i and i are the stator currents in the (d, q) coordinate d q and vector N(q, q _), respectively; q frame; i is one of the components of compose controller � q − q is the link e d q n×n n×n ∗ position error vector; K ∈ R and K ∈ R are positive (3); and reference current component i � 0 for constant D P d deﬁnite diagonal gain matrices determining the closed-loop torque operation and i ≠0 for ﬁeld-weakening operation θ � θ ,θ � θ − (2π /3),θ � θ + (2π/3), and θ being the performance; τ represents the computed torque part of the 0 a e b e c e e controller; Γ is a positive constant (control gain may also rotor electrical angle of the PMSM used. 'is current controller does not need the motor parameters as well as the take other forms), and ‖s‖ denotes the norm 2 of s, i.e., ������������� � decoupling process; thus, it is a robust current controller. 2 2 2 ‖s‖ � s + s + . . . + s ;andcontrolterm τ serveshereas 1av 1 2 n From equations (2)∼(5), it can be recognized that this a perturbation compensator. 'e output of this controller link position control system needs very few information τ is quasicontinuous, which is the reference input for the from the controlled plant. 'is is the main advantage of this joint torque implementation. Normally, when using sin- control approach. 'e disadvantage is that the joint torque gular perturbation approach, the joint inertia matrix J has dynamics are not really controlled, but taken as disturbance to be considered in the link position controller by adding and rejected by a damping term. From theory point of view matrix J to the mass matrix of the robot arm M(q). andveriﬁedbythesimulationstudiesgivenlater,thiscontrol However, since our link position controller is a robust system is not robust with respect to the large change in the controller, implying that no exact parameters are required, joint stiﬀness. the information about the joint inertia is normally not necessary (the system robustness depends on the available 3. Extended-Regular-Form Approach control resource). 'e reference current vector for the most inner current In this section, we will combine the state-space joint torque control loop can be calculated from the torque commands controller using direct sliding-mode control with the robust n n for the slow and fast dynamics, i.e., τ ∈ R and τ ∈ R : slow fast link position controller for rigid-body manipulators to form ∗ − 1 − 1 − 1 I � K τ � K Γ τ + τ , (3) a robust link position controller for ﬂexible-joint robots. To m slow fast q t t achieve this design goal, we need some theoretical supports. where K is the diagonal torque constant matrix of the Two methods are often used in the control of nonlinear electric motors and Γ is the diagonal gear-ratio matrix; high-order uncertain systems: ∗ ∗ n I � [i ] ∈ R , i �1∼n, is the reference current vector in- q qi (1) 'e order reduction method, e.g., using singular cluding the reference q-axis currents for all joints; τ rep- perturbation theory resents the motor torque vector. 'e slow and fast joint torque commands can be given as (taken from reference (2) 'e pure cascaded control method [15]) 'e ﬁrst method may possess a relative higher band- width than the pure cascaded control method, but the τ � τ , ⎧ ⎪ slow d neglected high-frequency dynamics in the real controlled τ � − D τ, (4) fast SPB plant may be excited if high control gains are used (high orτ � − K τ − τ − D τ _ , control gains are often required by some robust control fast SPB d SPB approaches); thus, the bandwidth will be limited in turns. n×n n×n 'ecascadedcontrolmethodhastheadvantages:thecontrol with D ∈ R and K ∈ R being constant diagonal SPB SPB gain matrices to be determined by the control designer (if system is easier being set into operation and the state Journal of Robotics 5 variables used by the controller are measurable by some state vector x of the ﬁrst block according to some per- sensors. However, the bandwidth of the control system is formance criteria: limited by the cascaded control structure. 'eoretically, the (8) x � − s x . 0 1 state-space control structure based on the full-state feedback has a higher control performance (i.e., higher bandwidth). 'en,forthecontrolof x inthesecondblocktobeequal However, the state-space method may need the high-order to the one given above, we design the control u using the time derivatives of the sensor signal, which are diﬃcult to sliding-mode control theory to achieve the required ro- obtain, because bustness with respect to the system uncertainties including theinﬂuenceof x tothesecondblock,sonowwedealwitha (1) the sensor signal has always some noise reduced order problem of an uncertain system. At this (2) a low-pass ﬁlter would introduce some time delay second stage, discontinuous control u is to be designed to (3) an observer would need a dynamic model and as- enforce sliding mode in the manifold: sociated parameters s � Cx − x � 0, (9) As aresult,onehas todosome tradeoﬀs.InSection 1,we have presented the three-block formulation of the ﬂexible- m×l where C ∈ R is a designed constant matrix determining joint robots. Actually, we had combined blocks B and C to the system behavior in sliding mode (note that s may also control the joint torque in a way of state-space control. take other forms). In sliding mode, the system motion is Moreover, because we used the direct sliding-mode control governed by approach to implement the joint torque tracking control by x _ � f x , x , applying the discontinuous terminal voltages of the motor ⎧ ⎨ 1 1 1 2 (10) windings directly, the joint torque controller is free from the ⎩ d C x − x � 0. 2 2 chattering problem and is of a high robustness with respect to the system uncertainties. 'is control performance can 'e second system in (6)–(10) is of l − m order, the hardly be achieved by a normal cascaded control structure. convergence of x to x depends only on the designed For the link position control problem for ﬂexible-joint parameter matrix C, and theoretically, the poles of the robots dealt with in this section, we will use the state-space second system can be placed arbitrarily, implying that the jointtorquecontrollerastheinnercontrolloop.Astheouter system response can be designed as fast as required. position control loop, it is nature to use the robust link 'erefore, a fast convergence of x to x can be achieved by position controller for rigid-body manipulators. As a result, properly selecting matrix C (under the condition that the there are totally two control loops instead of three. Because sliding mode already occurs). If the control gains used in the dynamics of blocks A and B+C in Figure 2 are inter- controller (8) are not inﬁnitely high, the motion in (10) can connected, ifwe cascade the link position controllerwith the be classiﬁed into slow and fast dynamics (corresponding to joint torque controller, we need associated theoretical the ﬁrst and second equation of (10), respectively). support. For this purpose, we extend the so-called regular- Depending on singular perturbation theory, for the slow form approach in the context of sliding-mode control dynamics, i.e., the ﬁrst equation of (10) x � x can be theory. assumed. As a result, the slow dynamics will be stabilized by For a general nonlinear aﬃne system, the feedback control given in (8) and the following ﬁnal system will be stable as expected: x _ � f(x) + B(x)u, (6) n n×m m x _ � f x , − s x . (11) 1 1 1 1 where x, f(x) ∈ R , B(x) ∈ R , and u(x) ∈ R , and we propose now the concept of extended-regular-form. System For the control of ﬂexible-joint robots, x in (7) stands (6) can be rearranged or transferred to the following two- for the state vector of the link position system (which is a block system (see [23] for such kind of transformation, but second-order system) and x presents the state vector of the in case of ﬂexible-joint robots, the system equations are joint torque system (which is a third-order system including already in this form): the dynamics of motor current). x _ � f x , x , As mentioned before, we intend to use the link po- 1 1 1 2 (7) sition controller to control the link position and to use the x _ � f x , x + B x , x u, 2 2 1 2 2 1 2 joint torque controller to control the joint torque. Both n− l l controllers are robust controller based on sliding-mode where x ∈ R , x ∈ R , and B is an l × m matrix with 1 2 2 technique. 'e joint torque controller utilizes the l ≥ m. 'e ﬁrst block does not depend on control. switching property of the power converter (i.e., the in- Note that the classical regular-form approach requires verter), implying that we are not suﬀered from the thatthedimensionof x shouldbe equaltothedimensionof chattering problem. the control input, i.e., l � m, see [27]. Now, we extend this design concept to the case of l ≥ m. 'e control design is performed in two stages. At ﬁrst, 3.1.LinkPositionControlleroftheRobotArm. 'e controller the l-dimensional state vector x is handled as the control algorithm for this section can be summarized as follows: input for the ﬁrst block and designed as a function of the 6 Journal of Robotics (2) 'ejointtorquecontrolperformanceforlargeorfast τ � M (q) € q − K q _ − K q + N (q, q _), ⎧ ⎪ 0 0 d D e P e 0 changing reference torques is not as good as the one of direct sliding-mode control approach s � q _ + K q + K q (ξ)dξ − q _ (0) − K q (0) (3) It needs an inner current control loop for the control ⎪ e D e P e e D e of i ; thus, there are totally three control loops in- stead of two. ⎪ t ⎪ − 1 + M (q) τ − τ dξ, 1 1av 4.1.LinkPositionControlleroftheRobotArm. 'e controller ⎪ s algorithm for this section can be summarized as follows: ⎪τ � − Γ , ⎪ 1 0 ‖s‖ ⎪ _ _ τ � M (q) q − K q − K q + N (q, q), ⎪ ⎧ ⎪ 0 0 d D e P e 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ � lowpass τ , ⎪ ⎪ 1av 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s � q _ + K q + K q (ξ)dξ − q _ (0) − K q (0) ⎪ ⎪ e D e P e e D e ⎪ ⎪ ⎩ ⎪ τ � τ + τ . d 0 1av ⎪ t (12) ⎪ − 1 + M (q) τ − τ dξ, 0 1 1av See Section 2 for the deﬁnitions of variables and pa- rameters as equation (2). ⎪ s τ � − Γ , ⎪ 1 0 ‖s‖ th ⎪ 3.2.JointTorqueControllerof i RobotJoint(SubscriptiIsNot Used for Simplicity). τ � lowpass τ , 1av 1 e � τ − τ , ⎩ ⎧ ⎪ τ d τ � τ + τ . d 0 1av s � e + c e_ + c e , τ τ 1 τ 0 τ (14) ⎪ − 1 ∗ s � A L i − i , d d d See Section 2 for the deﬁnitions of variables and pa- ⎪ Ω � s cosθ − s sinθ , 1 d a τ a ⎪ rameters as equation (2). Ω � s cosθ − s sinθ , (13) 2 d b τ b ⎪ th Ω � s cosθ − s sinθ , 4.2.JointTorqueControllerof i RobotJoint(SubscriptiIsNot ⎪ 3 d c τ c Used for Simplicity). ⎪ u � − u sign Ω , ⎪ 1 0 1 ⎪ e � τ − τ , ⎧ ⎪ τ d ⎪ ⎪ u � − u sign Ω , ⎪ ⎪ 2 0 2 ⎪ ⎪ ⎪ − 1 u � − u sign Ω , ⎪ τ � b a τ + τ − c e_ − c e , 3 0 3 ⎪ f d 1 τ 0 τ ⎪ _ z(0) � τ(0), where e � τ − τ is the joint torque control error. 'e τ d − 1 parameter A L � kck /J is introduced in s to simplify the u � lowpass M sign(z − τ _ ), t d eq s (15) control design; the parameters c and c have to be provided 0 1 − 1 ⎪ by the control designer depending on the required closed- ⎪τ � b u , r eq loopperformanceofthejointtorquecontrol.'edeﬁnitions _ of other variables and parameters in the control system are _ z � τ − c e_ − c e + bτ − M sign(z − τ), d 1 τ 0 τ r s given by (12) and (13), see Section 2. τ � τ + τ , m f r 4. Cascaded Control Structure Based on the where z is the artiﬁcially introduced auxiliary variable, Sliding-Mode Estimator Approach which is actually an estimate of τ _. Parameters a , b, c , c , 0 1 and M , and the time constant of the low-pass ﬁlter have to In this section, we use the joint torque controller for the be provided by the control designer. 'e deﬁnitions of inner torque control loop and the robust link position others variables and parameters are given in Section 2. 'e controller for the outer link position control loop. 'is stability proof of controller (15) is similar to the literature combination has an advantage comparing to the approach [28]. given in the previous section, namely, signal τ is not re- A current controller for the control of i is required as quired. However, it has also some disadvantages: the most internal control loop (the same happens with the (1) 'e control algorithm needs the nominal value of control system presented in Section 2). 'e current con- some parameters troller used here could be the same as the one given by Journal of Robotics 7 equation system (5) (sure, a classical current controller with y conventional PWM may be employed too). 5. Simulation Studies M q 1 2 5.1. Plant Model Used for the Simulation. To verify the proposed control approaches, we use a two-link ﬂexible- joint robot as the plant model shown in Figure 3, which x x consistsofthetwo-linkrigid-bodyrobotmodelwhichcanbe given as Figure 3: Two-link manipulator with link lengths L and L and 1 2 concentrated link masses M and M (the manipulator is shown in m m € q d + g + f τ 1 2 11 12 1 1 1 1 1 + � , joint conﬁguration (q , q ), which leads to end-eﬀector position 1 2 m m q d + g + f τ 21 22 2 2 2 2 2 (x , y ) in world coordinates). w w m m 11 12 ie. M(q) � , Table 1: Arm parameters. m m 21 22 M M L L 1 2 1 2 d + g + f 1 1 1 N(q, q _) � , 4kg 2kg 0.5m 0.5m d + g + f 2 2 2 (16) Table 2: Parameters for motor1 and motor2. with L(H) R(Ohm) λ P k (Nm/A) I (A) U (V) 0 t q max 0 m � L M , 44.5 × 10 0.68 0.24 4 (3/2)Pλ 60 120 22 2 2 0 m � m � m + L L M cos q , 12 21 22 1 2 2 2 Table 3: Parameters of joint 1 and joint 2. m � L M + M + 2m − m , 11 1 1 2 12 22 J (kgm ) k (Nm/Rad) c k (Nm/(Rad/s)) d � − L L M 2q _ q _ − q _ sin q , 1 1 2 2 1 2 2 2 1.5 10000 40 1 (17) d � L L M q _ sin q , 2 1 2 2 1 2 for joint 1 and joint 2 are calculated according to equation g � L M gcos q + q , 2 2 2 1 2 (19). 'is reference trajectory will generate large and fast g � L M + M gcos q + g , 1 1 1 2 1 2 changing joint torques to be followed: f � k q _ + k sign q _ , 2 2 2 2 1 v1 1 c1 1 ⎧ ⎪ x + y − L − L w w 1 2 q � atan2(D, C), with C � , f � k q _ + k sign q _ . ⎪ 2L L 2 v2 2 c2 2 ⎪ 1 2 ����� � ⎪ 2 'e parameters of the two-link ﬂexible-joint robot used D � ± 1 − C , for the simulation are listed in Tables 1∼3. q � atan2 y , x − 2atan2 L sin q , L + L cos q . 1 w w 2 2 1 2 2 (19) 5.2. Reference Input for Testing the Link Position Controllers. For the link position tracking control, we demand the manipulator to move along a circular trajectory in its workspace; see the following equation: 5.3. Controller Parameters. 'e parameters for the outer position control loop of Sections 2∼4 are selected to be the x (t) � x + r cosψ , ⎧ ⎪ d d0 d d same, and they are 2.0 0 M (q) � , y (t) � y − r sinψ , ⎪ d d0 d d ⎪ 0 1 (18) ⎪ _ N (q, q) � , ⎪ 2π 2π 0 ψ (t) � t − sin t, ⎪ d ⎪ t t f f 120 0 K � , (20) ⎪ 0 120 0 ≤ t ≤ t . 50 0 K � , 0 50 'e parameters of the circle are given as r �1.0m, 200 0 x �0.5m, and y �0.5m. 'e simulation time is now d0 d0 Γ � . selected as t �4s in order to zoom-in the transition period. 0 200 'rough the inverse kinematics, the reference link positions 8 Journal of Robotics 'e joint torques of both joints are limited to 200Nm. c � 20000, ⎧ 01 'e time constant of the two low-pass ﬁlters to extract the equivalent control of τ ∈ R is 0.01s. To improve the ⎪ c � 200, control performance, this time constant is linearly increased from zero to 0.01s in the ﬁrst half second and remains ⎪ M � 2 × 10 , s1 constant thereafter, similar to the following equation: − 3 ⎪μ � 1 × 10 , 0.025 1 ⎧ ⎪ t, 0 ≤ t ≤0.5, ⎪ (24) 0.5 u(t) � (21) c � 20000, ⎪ 02 0.025, t >0.5. c � 200, For the singular perturbation approach described in ⎪ 6 M � 2 × 10 , Section 2, the simple form fast τ � − D τ _ is used for the s2 fast SPB ⎪ fast dynamics, where matrix D is selected as ⎪ SPB ⎩ − 3 μ � 1 × 10 . 0.002 0 D � . (22) SPB 0 0.002 5.4. Simulation Results and Discussion. Figures 4∼8 show For the extended-regular-form approach described in the simulation results of the link position tracking Section 3, the inner loop joint torque controller parameters control of the two-joint robot arm considering the joint are selected to be the same as those given by ﬂexibility and the AC-motor dynamics. Figures 4 and 5 are for the case of normal joint stiﬀness, i.e., c � 20000, ⎧ ⎪ 01 k �k �10000Nm/Rad; while Figures 6 and 7 are for the 1 2 ⎪ c � 200, ⎪ 11 ⎪ case of large joint compliance, i.e., very small joint ⎪ − 1 5 A L � 3.84 × 10 , ⎨ stiﬀness k �k �1000Nm/Rad (without changing the 1 2 (23) ⎪ controller parameters). c � 20000, ⎪ As one can see from Figures 4 and 5, for the normal ⎪ c � 200, 12 joint stiﬀness (i.e., k �k �10000Nm/Rad), the singular 1 2 ⎩ − 1 5 perturbation approach and two new presented control A L � 3.84 × 10 . approaches given in Sections 2∼4 have similar tracking control performance. However, for the case of large joint For the cascaded control structure based on the sliding- compliance (k �k �1000Nm/Rad), see Figures 6 and 7, mode estimator approach described in Section 4, the inner loop 1 2 the singular perturbation approach shows a poorer jointtorquecontrollerparametersarethesameasthosegivenby control performance, due to the lack of adaptation mechanisms to the changing of the joint stiﬀness. 'e 1.3J ⎧ ⎪ ⎪ J � , ⎪ m1 n 2 extended-regular-form approach based on the direct ⎪ c ⎪ 1 sliding-mode joint torque control and the cascaded 2 2 control structure based on the sliding-mode estimator J � 1.5m � 1.5L M + L + L M , l1 n 11|q �0 1 1 1 2 2 (for joint torque control) show similar control perfor- mance for the normal and the small joint stiﬀness at ﬁrst k � 0.5k , 1 n 1 glance. However, if we take a close look in the joint ⎪ 1.3J ⎪ torque tracking performance in the inner control loop J � , ⎪ m2 n ⎪ c (see Figure 8 for the joint torque tracking of joint 1 in ⎪ 2 zoomed time range of 1s), it can be found immediately ⎪ 2 J � 1.5m � 1.5L M , ⎪ that the direct sliding-mode joint torque controller l2 n 2 2 2 under the extended-regular-form approach has a much k � 0.5k , 2 n 2 better tracking performance as the one of the sliding- k k ⎧ ⎪ 1 n 1 n mode estimator approach under the cascaded control ⎪ a � + , ⎪ 2 ⎪ J structure. 'is result conﬁrms the theoretical c J ⎪ l1 n 1 m1 n expectation. ⎪ 1 n Actually, a simple high-gain controller is not ade- b � , ⎪ quately being used within a cascaded multiple-loop control c J ⎨ m1 n system,exceptforthemostinternalcontrolloop.'isisthe ⎪ k k ⎪ 2 n 2 n reason why we tried to reduce the number of control loops a � + , ⎪ 2 J and use the sliding-mode (i.e., high gain) controller in the ⎪ c J l2 n m2 n ⎪ 2 ⎪ most internal control loop, as done with the extended- 2 n regular-form approach. ⎪ ⎪ b � , ⎩ 2 2 c J 2 m2 n Journal of Robotics 9 0.8 0.6 0.4 0.2 –0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Reference signal Extended-regular-form control SPB-control Cascaded control based on SME (a) 0.8 0.6 0.4 0.2 –0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Error of SPB-control Error of extended-regular-form control Error of cascaded control based on SME (b) Figure 4: Position control of joint 1 (normal joint stiﬀness). (a) Angular position tracking of joint 1. (b) Position tracking error of joint 1. –0.5 –1 –1.5 –2 –2.5 –3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Reference signal Extended-regular-form control SPB-control Cascaded control based on SME (a) 0.5 –0.5 –1 –1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Error of SPB-control Error of extended-regular-form control Error of cascaded control based on SME (b) Figure 5: Position control of joint 2 (normal joint stiﬀness). (a) Angular position tracking of joint 2. (b) Position tracking error of joint 2. Position tracking Angular position (Rad) Position tracking Angular position (Rad) error (Rad) error (Rad) 10 Journal of Robotics 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Reference signal Extended-regular-form control SPB-control Cascaded control based on SME (a) 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Error of SPB-control Error of extended-regular-form control Error of cascaded control based on SME (b) Figure 6: Position control of joint 1 (large joint compliance). (a) Angular position tracking of joint 1. (b) Position tracking error of joint 1. –0.5 –1 –1.5 –2 –2.5 –3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Reference signal Extended-regular-form control SPB-control Cascaded control based on SME (a) 0.5 –0.5 –1 –1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) Error of SPB-control Error of extended-regular-form control Error of cascaded control based on SME (b) Figure 7: Position control of joint 2 (large joint compliance). (a) Angular position tracking of joint 2. (b) Position tracking error of joint 2. Position tracking error Angular position Position tracking error Angular position (Rad) (Rad) (Rad) (Rad) Journal of Robotics 11 –50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) Reference joint torque signal Real joint torque signal (a) –100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) Reference joint torque signal Real joint torque signal (b) Figure8:Jointtorquetrackingcontrolofjoint1(largejointcompliance).(a)Innerloopjointtorquecontrolbydirectsliding-modecontrol. (b) Inner loop joint torque control by sliding-mode estimator (grey solid line �reference joint torque signal; black solid line �real joint torque signal). torque signal with minimal time delay and, at the same time, 6. Conclusions reducing the sensitivity of the control algorithms with re- In this paper, the three-block formulation of the dynamics specttothenoisyjointtorquesignal.'eotherisseekingthe for ﬂexible-joint robots was introduced at ﬁrst. 'en, hardware solution for the direct sliding-mode joint torque singular perturbation approach and two new control ap- control (without using the build-in PWM in micro- proaches are presented for the link position tracking controllers or DSP) to achieve the advanced control per- control of this kind of robot. Among them, the singular formances provided by this control approach. perturbation approach (famous approach) is the simplest one for the real-time implementation, but it is sensitive to Data Availability the changing of joint stiﬀness, from a theory point of view and veriﬁed by the simulation studies. 'e extended-reg- 'e RAR data used to support the ﬁndings of this study are ular-form approach with the direct sliding-mode joint included within the supplementary information ﬁle. torquecontrolhasthehighestcontrolperformance,andthe implementation is also quite simple, except for the re- Additional Points quirementon the second-timederivative of the jointtorque signal. 'e cascaded control structure based on the sliding- 'e concept of extended-regular-form for the block-control mode estimator approach tries to avoid the second-time of high-order uncertain systems is proposed. 'e proposed derivative of the joint torque signal but possesses a more method breaks through the limitation (under certain given involved control structure and needs more controller pa- condition) that the dimension of the inner block must be rameters than the other two. equal to the dimension of the control input associated with 'ese comparative studies conﬁrm again that there is no the conventional regular-form approach. 'e proposed free lunch in the control of high-order uncertain systems, method serves as the theoretical support for cascading an unless to give up the intention of achieving high bandwidth. outer position control loop with the inner direct (state- 'e proposed extended-regular-form concept can also be space) sliding-mode joint torque control loop for the tra- applied to some other high-order, nonlinear, uncertain jectory tracking control of ﬂexible-joint robots. systems. Moreover, there are still some topics left which need to Conflicts of Interest be investigated in the future research works. One is achieving a usable second-time derivative of noisy joint 'e authors declare that they have no conﬂicts of interest. Torque (Nm) Torque (Nm) 12 Journal of Robotics [13] L. Zouari, H. Abid, and M. Abid, “Backstepping controller for Acknowledgments electrically driven ﬂexible joint manipulator under uncer- tainties,” International Journal of Applied Engineering Re- 'is project was supported by the National Natural Science search, vol. 10, no. 8, pp. 19885–19896, 2015. Foundation of China (NSFC, nos. 61763030 and 61263045), [14] Z. H. Jiang and K. Shinohara, “Workspace trajectory tracking the Jiangxi Province Science and Technology Support control of ﬂexible joint robots based on backstepping Project (20112BB550017), and the Jiangxi Province Natural method,” in Proceedings of the IEEE Region 10 Conference Science Fund Project (20132BAB201040). 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Journal of Robotics – Hindawi Publishing Corporation
Published: Oct 27, 2020
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