Feedback-based Digital Higher-order Terminal Sliding Mode for 6-DOF Industrial Manipulators
Zhian Kuang, Xiang Zhang, Liting Sun, Huijun Gao, Masayoshi Tomizuka
FFeedback-based Digital Higher-order Terminal Sliding Modefor 6-DOF Industrial Manipulators
Zhian Kuang , , Xiang Zhang , Liting Sun , Huijun Gao and Masayoshi Tomizuka Abstract — The precise motion control of a multi-degree offreedom (DOF) robot manipulator is always challenging dueto its nonlinear dynamics, disturbances, and uncertainties.Because most manipulators are controlled by digital signals,a novel higher-order sliding mode controller in the discrete-time form with time delay estimation is proposed in this paper.The dynamic model of the manipulator used in the designallows proper handling of nonlinearities, uncertainties anddisturbances involved in the problem. Specifically, parametricuncertainties and disturbances are handled by the time delayestimation and the nonlinearity of the manipulator is addressedby the feedback structure of the controller. The combinationof terminal sliding mode surface and higher-order controlscheme in the controller guarantees a fast response with a smallchattering amplitude. Moreover, the controller is designed witha modified sliding mode surface and variable-gain structure,so that the performance of the controller is further enhanced.We also analyse the condition to guarantee the stability of theclosed-loop system in this paper. Finally, the simulation andexperimental results prove that the proposed control schemehas a precise performance in a robot manipulator system.
I. INTRODUCTIONManipulators have been extensively used in the industryto manufacture products. However, with nonlinearities, highcouplings, and significant parametric uncertainties existingin the dynamics, the manipulator’s precise control is alwayschallenging [1]. Moreover, with more and more industrialscenarios involving manipulators, different kinds of distur-bances and high accelerations are applied, making it harderto design a precise and robust controller for a manipulator.For the past several years, researchers have proposedseveral kinds of manipulator controllers, such as feedforwardand PID control [2], sliding mode control [3], adaptive robustcontrol [4]. Among these methods, sliding mode controlis proven to be straightforward to be applied in complexelectromechanical system [5]. More importantly, becauseof the sliding mode surface, this control method showsoutstanding robustness to disturbances and uncertainties [6].Researchers have proposed several kinds of modified SMCto make up the original drawbacks like the slow convergenceon the sliding surface [7], [8]. For instance, Yong proposed anon-singular terminal sliding mode control (TSMC) method, Zhian Kuang and Huijun Gao are with the Research Insti-tute of Intelligent Control and Systems, Harbin Institute of Technol-ogy, 150001, Harbin, P.R. China. [email protected],[email protected] Xiang Zhang, Liting Sun, and Masayoshi Tomizuka are withthe Department of Mechanical Engineering, University of California,Berkeley, CA 94720, USA. [email protected],[email protected], [email protected] Zhian Kuang is also with the Department of Mechanical Engineering,University of California, Berkeley, CA 94720, USA. which makes the states converge in finite time on the slidingsurface [9].Despite that TSMC has precise and robust performancein theory, it cannot be applied directly to the manipulatorsfor two reasons. One reason is that the digital processoris always used to control the manipulator nowadays, di-rectly applying the continuous-time controller to the discrete-time environment results in large chattering amplitude oreven instability [10]. To address this problem, researchershave put forward the discrete-time terminal sliding modecontrol (DTSMC) [11] and apply it to motion controlsystems [12], [13]. Compared with the traditional linearsliding mode surface, the terminal sliding surface has a fasterresponse benefiting from the finite-time convergence [14],[15]. Nevertheless, this kind of dynamics also enlarges thechattering amplitude, which makes the precision of themanipulator decrease. A proper way to solve this problemis to raise the order of the sliding mode [16], [17]. Forexample, Xu proposed a second-order terminal sliding modestrategy [18]. However, the implementation is on a linearpositioning model, which cannot be applied instantly tothe manipulator. Moreover, the order of the sliding modecontroller is fixed, which cannot be changed based on theperformance of the practical application. Raising the orderis beneficial to reduce the chattering, but may also reducethe controller’s robustness. Thus we need to build a new andflexible structure that can balance the chattering amplitudeand the robustness at the same time.The other obstacles to improving manipulators’ perfor-mance are the nonlinearity, the significant parametric uncer-tainties, and disturbances. They need to be handled appropri-ately; otherwise, the tracking precision will deteriorate, espe-cially when the manipulator is in operation with large accel-eration [19]. A popular way is to introduce the feedforwardterm to compensate for the nonlinear dynamics and use thecontroller to suppress the uncertainties and disturbances [2].As the feedforward structure is open-loop, this method is notsufficient enough when there are parametric uncertainties onthe manipulator’s dynamic model. Another method to handlethe nonlinearity is the time-delay estimation (TDE) [20],[21]. By using TDE, the nonlinear dynamics, the presentlumped disturbances, and uncertainties are estimated by theprevious states [1]. When the sampling frequency of thesystem is high, the delay time is appropriately chosen,and the disturbances change relatively slowly, this methodhas excellent performance in [22] and [23]. However, asTDE highly rely on the state at a particular time point,extra disturbances are easy to be introduced. Moreover, a r X i v : . [ c s . R O ] F e b hen the manipulator operates with fast speed and massiveacceleration, the estimation error will increase due to the fastchange of dynamics. Thus we need a new structure to makethe best use of TDE to achieve better precision.This paper proposes a novel feedback-based digital higher-order terminal sliding mode control (DHTSMC) scheme.This controller is designed based on the manipulator’s nom-inal dynamics, including the linear and nonlinear parts. Themanipulator’s nonlinear dynamics are directly handled by theequivalent control part of the controller in a feedback way. Asan assistant, TDE is used to compensate for the uncertaintiesand disturbances in the system. For the controller, a newterminal sliding mode surface is designed to improve thestates’ dynamics on the sliding mode surface. We also giveout a universal and flexible form of a higher-order switchingcontroller. A specific order can be selected to balance thechattering phenomenon and the robustness of the controller.To address the performance reduction when there are largeaccelerations, we mainly use the controller’s variable gain.More importantly, we implement the proposed controller to a6-DOF manipulator in the environments of both simulationsand experiments, whose results show that our proposedmethod has an advantage over previous ones.The remainder of this paper is as follows. Section II intro-duces the model of manipulators and the proposed controller.In Section III, the stability of the closed-loop control systemis analyzed. Section IV introduces the practical applicationof the proposed controller to manipulators and the results ofsimulations and experiments.II. CONTROLLER DESIGNThe continuous-time Lagrange model of a manipulatorwith n-DOFs is written as M ( q ) ¨ q + C ( q , ˙ q ) ˙ q + G ( ˙ q ) + F f ( q , ˙ q ) + d = τ (1)where M ( q ) ∈ R n × n is the inertia matrix, C ( q , ˙ q ) ∈ R n × n denotes the Coriolis matrix, G ( ˙ q ) ∈ R n and F f ( q , ˙ q ) ∈ R n are the gravity vector and the friction vector respectively, ¨ q , ˙ q and q are respectively the angular acceleration, velocity andposition of each joint, d ∈ R n stands for the disturbances,and τ ∈ R n is the torque vector.Because the parametric uncertainties extensively exist inpractical systems, we use ¯ • to denote the matrices or vectorscalculated from the nominal values, then (1) is rewritten as τ = ¯ M ( q ) ¨ q + ¯ C ( q , ˙ q ) ˙ q + ¯ G ( ˙ q ) + ¯ F f ( q , ˙ q )+ H ( q , ˙ q , ¨ q ) (2)with H ( q , ˙ q , ¨ q ) = ˜ M ( q ) ¨ q + ˜ C ( q , ˙ q ) ˙ q + ˜ G ( ˙ q )+ ˜ F f ( q , ˙ q ) + d (3)where ˜ • = • − ¯ • stands for the uncertain part of each matrixor vector. If the manipulator is sampled with a sampling interval T ,then the discrete-time model at time t is described as τ k = ¯ M ( q k ) ¨ q k + ¯ C ( q k , ˙ q k ) ˙ q k + ¯ G ( ˙ q k ) + ¯ F f ( q k , ˙ q k )+ H ( q k , ˙ q k , ¨ q k ) (4)with H ( q k , ˙ q k , ¨ q k ) = ˜ M ( q k ) ¨ q k + ˜ C ( q k , ˙ q k ) ˙ q k + ˜ G ( ˙ q k )+ ˜ F f ( q k , ˙ q k ) + d k (5)where t = kT , ˙ q k = q k +1 − q k T , and ¨ q k = ˙ q k +1 − ˙ q k T .We use TDE to estimate H ( q k , ˙ q k , ¨ q k ) , i.e., we calculateout the uncertain part in the previous sampling point and usethis value to approximate the uncertain part in the present.It is formulated as ˆ H ( q k , ˙ q k , ¨ q k ) = H ( q k − , ˙ q k − , ¨ q k − )= τ k − − ¯ M ( q k − ) ¨ q k − − ¯ C ( q k − , ˙ q k − ) ˙ q k − − ¯ G ( ˙ q k − ) − ¯ F f ( q k − , ˙ q k − ) . (6)The estimation error between H ( q k , ˙ q k , ¨ q k ) and ˆ H ( q k , ˙ q k , ¨ q k ) is presented as ˇ H ( q k , ˙ q k , ¨ q k ) = H ( q k , ˙ q k , ¨ q k ) − H ( q k − , ˙ q k − , ¨ q k − ) .Define the tracking error of the joints as ˇ q k = q k − r k (7)where r k is the reference vector of the joints.To obtain precise, fast and robust performance, we designa novel discrete-time terminal sliding surface as s k = a ˇ q k + a sig β k ˇ q k + ˙ˇ q k (8)with sig β k ˇ q k = [ sig β k, ˇ q k, sig β k, ˇ q k, · · · sig β k,n ˇ q k,n ] T , β k,i = | ˇ q k,i | +0 . | ˇ q k,i | +1 , i = 1 , , · · · , n , where sig • ∗ = |∗| • sgn ( ∗ ) , a = diag [ a ,i ] , i = 1 , , · · · , n , a = diag [ a ,i ] , i =1 , , · · · , n , a ,i and a ,i are positive constants.We design the reaching law of r-order variable-gain slidingmode controller as s k +1 ,i = − b ,i (¨ q k,i ) T s ηk,i − b ,i (¨ q k,i ) T s ηk − ,i − . . . − b r,i (¨ q k,i ) T s ηk − r,i (9)where < η < is a positive constant, b j,i (¨ q k,i ) , j =1 , , · · · , r is a function of ¨ q k,i .Based on the reaching law (9),the discrete-time r-ordervariable-gain terminal sliding mode controller with TDE isobtained as τ k = 1 T ¯ M ( ˙ r k +1 − a ( ˙ q k T + q k − r k +1 ) − a sig β k ( ˙ q k T + q k − r k +1 ) − ˙ q k ) + ¯ C ( q k , ˙ q k ) ˙ q k + ¯ G ( ˙ q k )+ ¯ F f ( q k , ˙ q k ) + ˆ H ( q k , ˙ q k , ¨ q k )+ ¯ M ((1 − b ( ¨ q k ) T ) s k − b ( ¨ q k ) T s k − − b ( ¨ q k ) T s k − − · · · − b r ( ¨ q k ) T s k − r ) . (10)II. S TABILITY A NALYSIS
The following theorem states the sufficient condition toguarantee the control system stable:
Theorem 1:
For the system (4) under the controller (10),if there are a series positive constants α , α , · · · , α r satisfying > α > α > · · · > α r > and b n,i (¨ q k,i ) ≤ T (cid:114) α n − α n +1 r + 2 (11)then s k − m,i will converge to the region Γ i = (cid:8) s k − m,i (cid:12)(cid:12) | s k − m,i | < γ i (cid:9) (12)where γ i = ( r + 2) E k,i + (cid:80) rj =0 ( b j,i (¨ q k,i ) T ) α m − α m +1 − ( r + 2)( b m,i (¨ q k,i ) T ) (13) | s k − m,i | = max {| s k − r,i | , | s k − r +1 ,i | , · · · , | s k,i |} (14)and E k,i is the elements of E k = ¯ M − ˇ H ( q k , ˙ q k , ¨ q k ) . Proof:
Substituting (10) into (4), we obtain thedynamics of s k +1 ,i as s k +1 ,i = − b ,i (¨ q k,i ) T s ηk,i − b ,i (¨ q k,i ) T s ηk − ,i − · · · − b r,i (¨ q k,i ) T s ηk − r,i + E k,i . Construct a Lyapunov candidate function U k,i as U k,i = s k,i + α s k − ,i + α s k − ,i + · · · + α r s k − r,i . (15)Then the difference between U k +1 ,i and U k,i is ∆ U k = U k +1 ,i − U k,i =( α − s k,i + ( α − α ) s k − ,i + . . . + ( α r − α r − ) s k − r +1 ,i + s k +1 ,i − α r s k − r,i . (16)When s i > , we have s ηi < s i . When s i ≤ , we have s ηi ≤ . Thus, it holds that s ηi ≤ s i + 1 . (17)Based on the fact that ab ≤ a + b , a, b ∈ R , we have s k +1 ,i = b ,i (¨ q k,i ) T s ηk,i + 2 b ,i (¨ q k,i ) b ,i (¨ q k,i ) T s ηk,i s ηk − ,i + ( b ,i (¨ q k,i ) T ) s ηk − ,i + 2 b ,i (¨ q k,i ) b ,i (¨ q k,i ) T s ηk − ,i s ηk − ,i + ( b ,i T ) s ηk − ,i + · · · + ( b r,i (¨ q k,i ) T ) s ηk − r,i + ( b r,i (¨ q k,i ) T ) s ηk − r,i E i + E i ≤ ( r + 2)[( b ,i (¨ q k,i ) T ) s ηk,i + ( b ,i (¨ q k,i ) T ) s ηk − ,i + · · · + ( b r,i (¨ q k,i ) T ) s ηk − r,i + E i ] . (18)According to (17) further, we have s k +1 ,i < ( r + 2) (cid:0) ( b ,i (¨ q k,i ) T ) s k,i + ( b ,i (¨ q k,i ) T ) s k − ,i + · · · + ( b r,i (¨ q k,i ) T ) s k − r,i (cid:1) + ( r + 2) E i + r (cid:88) j =0 ( b j,i (¨ q k,i ) T ) . (19) If we define α = 1 and α r +1 = 0 , then there is ∆ U k,i = (cid:0) α − α + ( r + 2)( b ,i (¨ q k,i ) T ) (cid:1) s k,i + (cid:0) α − α + ( r + 2)( b ,i (¨ q k,i ) T ) (cid:1) s k − ,i + . . . + (cid:0) α r − α r − + ( r + 2)( b r − ,i (¨ q k,i ) T ) (cid:1) s k − r +1 ,i + ( α r +1 − α r + ( r + 2)( b r,i (¨ q k,i ) T ) ) s k − r,i + ( r + 2) E . (20)From (11), we obtain that α n +1 − α n + ( r + 2)( b n,i (¨ q k,i ) T ) s k,i ≤ . (21)Assume that s k − m,i = max { s k,i , s k − ,i , . . . , s k − r,i } , thenwe have ∆ U k,i ≤ (cid:0) α m +1 ,i − α m,i + ( r + 2)( b m,i (¨ q k,i ) T ) (cid:1) s k − m,i + ( r + 2) E + r (cid:88) j =0 ( b j,i (¨ q k,i ) T ) . (22)When | s k − m | is not in the region Γ i , i.e., there is s k − m,i > ( r + 2) E + (cid:80) rj =0 ( b j,i (¨ q k,i ) T ) α m − α m +1 − ( r + 2)( b m,i (¨ q k,i ) T ) (23)it is obtained that ∆ U k,i < . (24)This completes the proof of Theorem 1. Remark 1:
From (11), we can infer that when the sam-pling interval T is smaller, b n,i (¨ q k,i ) has a larger range ofvalues. In other words, the stability of the system determinesa maximum value of the sampling interval. Remark 2:
Equation (13) gives some clues on the sourceof errors. Firstly, a larger T results in a larger region Γ i .Secondly, a larger E k,i also causes a larger convergenceregion of s k − m,i , and worse tracking precision further. As E k,i is related to the estimation error of TDE, we can getthat a precise TDE is the precondition of the precise tracking.Traditional methods such as [20] and [24], only rely onTDE to compensate for the nonlinearity. They have a largerestimation error than our method under the situations that thenonlinearity is large. Thus our method has better precisiontheoretically.IV. SIMULATIONS AND EXPERIMENTS (a) (b) Fig. 1. The simulation setup of the FANUC LR Mate 200iD industrialrobot, including (a) the frames and (b) the reference trajectory.ig. 2. Structure of the closed-loop system where DHTSMC is applied torobots ( R r stands for the reduction ratios of gears).TABLE ID-H P ARAMETERS OF
ATE I D R
OBOT i α i − (rad) a i − (mm) d i (mm) θ i (rad)1 - π
50 0 q π
440 0 q − π π
35 0 q π q π q π q A. Simulation
The simulation environment is developed based on MAT-LAB R2019b and Simulink. With the help of the tool ofMechanics Explorer, a complete model of FANUC LR Mate200iD industrial robot (shown in Fig. 1) is established.Coordinates of the manipulator are set as shown in Fig. 1 (a),and corresponding D-H parameters are displayed in Table I.Other related physical parameters of the simulation modelare given in Table II.The control structure of the proposed method is shownin Fig. 2. First of all, the reference positions and rota-tions in Cartesian space are generated. The initial posi-tion is p (0 . , , . , and the initial rotation angle is ψ (0 , , π ) in the order of Z Y X . Then it moves to p (0 . , . , . with rotation angle ψ ( π/ , π/ , π/ . Atpoint p (0 . , . , . with rotation angle ψ ( π/ , π/ , π/ ,after that, the manipulator returns to the initial po-sition and rotation via p (0 . , . , . with rotation ψ ( π/ , π/ , π/ , as shown in Fig. 1 (b).Then the motion planning rules are conducted to smooththe trajectories and realize some additional functions onrobots. The simple S-curve motion planning rule is con-sidered in the simulation to make the acceleration anddeceleration of the robots are performed gently and steadily.After that, we input the planning results to the inversekinematics, which is derived based on the frames and De-navit–Hartenberg (D-H) parameters of the robot manipulator,to obtain the reference q . Then the proposed controller isapplied in the joint space.We set the maximum acceleration as m/s from p to p and as m/s from p to p . Before every motion,the manipulator will wait for . s , and the sampling timeis set as the same as the controller, ms . After that, Time (s) -150-100-50050100 A n g l e ( d e g r ee ) r r r r r r Fig. 3. The reference angle of each joint in the simulation.
Time (s) -0.1-0.0500.05 J o i n t ( d e g r ee ) FF-TSMCProposed Method
Time (s) -2024 J o i n t ( d e g r ee ) -3 Time (s) J o i n t ( d e g r ee ) -3 Time (s) -0.0100.01 J o i n t ( d e g r ee ) Time (s) -505 J o i n t ( d e g r ee ) -3 Time (s) -0.0500.05 J o i n t ( d e g r ee ) Fig. 4. The tracking error in the joint space in the simulation. we calculate the inverse kinematics based on the framesshown in Fig. 1 (a) and D-H parameters shown in Table. I.After the planning and calculation of inverse kinematics, thetrajectory of each joint is shown in Fig. 3. We introducethe band-limited white noise as an additional disturbance,whose sampling time is set as . s and other parametersremain default. For the controller, parameters are tuned as a = diag [10 , , , , , , a = 0 . I , b =4 . × + 0 . q i , b = 2 . × , r = 1 and I isthe unit matrix.Apart from the proposed DHTSMC method, we alsoimplement the traditional feedforward control scheme withTSMC (noted as FF-TSMC) with TDE proposed in [1] as acomparison. The results of the simulations are illustrated inFig. 4. For each joint, there are four peaks or valleys in thetrajectory of tracking error, corresponding to the processesmoving from one point to the other. When the joint is waitingin a position, tracking errors of both method can convergeto a relatively small value. This phenomenon means thatwhen the joint is accelerating or decelerating, the joint tendsto have a more massive tracking error. However, with theproposed method, the tracking error is significantly reduced ABLE IIP
HYSICAL P ARAMETERS OF
ATE I D R
OBOT
NO. Gear Ratio Link Mass Motor Inertia Link Coulomb Link Viscous Motor Coulomb Motor Viscous(kg) (kg m ) Friction (N · m) Coefficient (N · s/m) Friction (N · m) Coefficient (N · s/m)1 114.6 2.4 8.9 × − × − × − × − × − × − × − × − × − × − × − × − when the joints change their positions, which is about two-thirds of the traditional FF-TSMC. This shows that comparedwith the feedforward structure, the proposed method canmake better use of the dynamics, to get better performance. Fig. 5. The experimental setup.
Time (s) -0.1-0.050 J o i n t ( d e g r ee ) FF-TSMCProposed Method
Time (s) -0.200.2 J o i n t ( d e g r ee ) Time (s) -0.2-0.100.1 J o i n t ( d e g r ee ) Time (s) -0.100.10.2 J o i n t ( d e g r ee ) Time (s) -0.200.2 J o i n t ( d e g r ee ) Time (s) -0.200.2 J o i n t ( d e g r ee ) Fig. 6. The tracking error in the joint space in the experiments.
B. Experiments
Practical experiments are also conducted to verify theeffectiveness of the proposed method further. As shown inFig. 5, the experimental setup consists of the host computer,
Time (s) -101 X ( mm ) FF-TSMCProposed Method
Time (s) -2-10 Y ( mm ) Time (s) -101 Z ( mm ) Time (s) -0.2-0.100.1 R o ll ( d e g r ee ) Time (s) -0.200.20.4 P i t c h ( d e g r ee ) Time (s) -0.2-0.100.1 Y a w ( d e g r ee ) Fig. 7. The tracking error in the Cartesian space in the experiments. the target computer, the digital servo adapter (DSA), andthe FANUC LR Mate 200iD robots. The control algorithmis programmed on the host computer and applied to theDSA and robot via Simulink Real-Time running on the targetcomputer. The parameters of the robot are the same as themodel in the previous simulation. The sampling interval ofthe robot is set as 0.25 ms, and that of the controller is setas 1 ms.We also compare the FF-TSMC and the proposed controlmethod in the experiments. The reference trajectory of therobots is directly given in joint space. All the joints are setto rotate from ◦ to ◦ and then return to ◦ . Parameters ofthe controller are set as a = diag [1 , , , , , , a =0 . I , b = 1 × + 0 . q i , b = 0 . × and r = 1 .Fig. 6 shows the tracking error of each joint under thetwo controllers. We can see that the tracking error of theproposed method is overall smaller in each joint comparedwith the previous FF-TSMC. We have also noticed that forboth methods, there is an offset after the manipulator reachesthe target position. We think this is caused by the unmodelleddynamics of the robot. It is easy to find that for most joints,the proposed method has a smaller offset than the FF-TSMC.Further, the performance of the robots in the Cartesianspace is shown in Fig. 7. The position and pose of the end-ffector are calculated by using the joint position and theforward kinematics. In particular, the pose is described byEuler angles, which rotates in the order of Z Y X . FromFig. 7, we can see that the proposed method has a smallererror in both positions and poses. The position error of theproposed method is around 1 mm, and the Euler angularerror is below 0.2 ◦ , which are half of the previous method.V. CONCLUSIONSA control method of DHTSMC with TDE was pro-posed and implemented on the manipulators in this paper.The controller was derived with the nonlinear dynamicsof robots taken into the feedback loop directly, so thatthe nonlinearities, the uncertainties, and disturbances wereadequately handled. Specifically, we improved the structureof the sliding surface to achieve a faster response, and wegave a universal control method of different order slidingmode controllers. By using the variable gains, the conflictbetween the fast response and small chattering amplitudewas further settled. We also analyzed the stability of theclosed-loop system. Results of simulations and experimentsshowed that the proposed controller was much more effectivethan conventional DTSMC methods in both joint space andCartesian space. ACKNOWLEDGMENTThe authors would like to thank Dr. Yu Zhao for his con-tribution to the MATLAB simulator of the robot manipulatorused in this work. R EFERENCES[1] Y. Wang, K. Zhu, B. Chen, and M. Jin, “Model-free continuous nonsin-gular fast terminal sliding mode control for cable-driven manipulators,”
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