Adaptive robust sliding mode trajectory tracking control for 6 degree-of-freedom industrial assembly robot with disturbances

Li Pan (Key Laboratory of E&M, Zhejiang University of Technology, Ministry of Education & Zhejiang Province, Zhejiang Hangzhou, China)
Guanjun Bao (Key Laboratory of E&M, Zhejiang University of Technology, Ministry of Education & Zhejiang Province, Zhejiang Hangzhou, China)
Fang Xu (Key Laboratory of E&M, Zhejiang University of Technology, Ministry of Education & Zhejiang Province, Zhejiang Hangzhou, China)
Libin Zhang (Key Laboratory of E&M, Zhejiang University of Technology, Ministry of Education & Zhejiang Province, Zhejiang Hangzhou, China)

Assembly Automation

ISSN: 0144-5154

Publication date: 6 August 2018

Abstract

Purpose

This paper aims to present an adaptive robust sliding mode tracking controller for a 6 degree-of-freedom industrial assembly robot with parametric uncertainties and external disturbances. The controller is used to achieve both stringent trajectory tracking, accurate parameter estimations and robustness against external disturbances.

Design/methodology/approach

The controller is designed based on the combination of sliding mode control, adaptive and robust controls and hence has good adaptation and robustness abilities to parametric variations and uncertainties. The unknown parameter estimates are updated online based on a discontinuous projection adaptation law. The robotic dynamics is first formulated in both joint spaces and workspace of the robot’s end-effector. Then, the design procedure of the adaptive robust sliding mode tracking controller and the parameter update law is detailed.

Findings

Comparative tests are also conducted to verify the effectiveness of the proposed controller, which show that the proposed controller achieves significantly better dynamic trajectory tracking performances as compared with conventional proportional derivative controller and sliding mode controller under the same conditions.

Originality/value

This is a new innovation for industrial assembly robot to improve assembly automation.

Keywords

Citation

Pan, L., Bao, G., Xu, F. and Zhang, L. (2018), "Adaptive robust sliding mode trajectory tracking control for 6 degree-of-freedom industrial assembly robot with disturbances", Assembly Automation, Vol. 38 No. 3, pp. 259-267. https://doi.org/10.1108/AA-02-2017-026

Download as .RIS

Publisher

:

Emerald Publishing Limited

Copyright © 2018, Emerald Publishing Limited


1. Introduction

At present, industrial robots are reshaping the manufacturing industry and are becoming a common fixture for situations that require high output and no errors in factories (Nof et al., 2012). Industrial robots can help manufacturers become more competitive and efficient while reducing work related injuries caused by repetition on the assembly line. Among various types of industrial robots, an industrial serial robot is a type of widely used manipulator that is designed and used to move materials, assembly components and perform a variety of programmed tasks in manufacturing and production areas that are usually dangerous or unsuitable for human workers (Nof et al., 2012). For example, the industrial assembly robot is a type of industrial serial robot used for small part transfer and assembly. Therefore, accurate motion tracking control in the face of uncertainties and variations is generally of great importance to effective assembly operations, particularly for industrial assembly robot, in various working environments.

Many methodologies have been developed or applied on robotic systems in the recent literature to improve the motion tracking performances for industrial robots. In Hsu and Fu (2005), an adaptive decentralized controller for trajectory tracking of robot manipulators was proposed. However, such a decentralized controller cannot be designed precisely for each robot joint because control inputs and states of and all the robot joints are strongly interconnected. In Yang et al. (2012), a disturbance observer was added to the adaptive decentralized controller to compensate for the coupled uncertainties for each joint. However, the boundedness of the robot control system was only ensured by using some special nonlinear damping terms, and only simulation results were used to confirm the theoretical claims, which is not convincing. In Yang et al. (2016), an optimal adaptive radial basis function neural network (RBFNN) controller was designed for nonlinear discrete time robot manipulators with uncertain dynamics. The controller was developed based on the feedback principles without non-causal problems and its closed-loop uniformly ultimate boundedness was justified by using rigorous Lyapunov analysis and simulation studies. However, the practical implementation of the proposed controller requires rather large amount of computation, accurate dynamic parameters and the burden of difficulty in specifying effective parameters for the RBFNN. In Fateh et al. (2014) and Zhang et al. (2014), a robot dynamics model and a discrete time adaptive controller were developed based on RBFNN to address the problems from dynamics uncertainties. However, the closed-loop stability conditions were only established in continuous time, whereas the overall convergence proof when using digital control was not analyzed. An adaptive fuzzy controller with on-line the fuzzy rules was designed in Mendes and Neto (2015) for contact force control between the end-effector and working environment in an industrial robot in the presence of robot uncertainties. Even though the developed controller has relatively high stability and robustness, the controller requires high computational burden for practical implementation and also cannot be used for trajectory tracking control. A fuzzy logic controller with its membership functions optimized by a particle swarm optimization (PSO) method was presented in Bingül and Karahan (2011) to improve motion tracking control for a 2 degree-of-freedom (DOF) planar robot. However, only simulation results were presented to demonstrate its effectiveness without parametric uncertainties. In Nikdel et al. (2017), an adaptive backstepping controller was developed for a 2-DOF serial robot manipulator to increase motion tracking performance and response characteristics in the existence of parameter uncertainties and nonlinearities. The stability of the robot control system was guaranteed based on the Lyapunov theory with parameter update laws. However, the use of backstepping would cause serious explosion of complexity due to the repeated time differentials of virtual control inputs. Adaptive controllers were also designed for serial robot manipulators in He et al. (2016c) by using neural networks, in Homayounzade and Keshmiri (2014) based on complete state measurements, in He et al. (2016a) to compensate for output constraint and dead zone and in He et al. (2016b) for impedance control with input saturation. An artificial neuro fuzzy adaptive controller was proposed in Vijay and Jena (2016) to achieve robust control for a 2-DOF rigid robot manipulator. The parameters of the controller were optimized by minimizing quadratic performance indices based on PSO. However, the controller is generally computation consuming and cannot be extendable to general n-DOF serial industrial robot.

A kinematic control algorithm was proposed in Leylavi Shoushtari et al. (2016) to control a redundant robotic manipulator. The performances of the proposed algorithm were evaluated based on a set of sample trajectories. However, only simulations were conducted to confirm the continuity and accuracy of the algorithm for given end-effector trajectories, whereas experimental results were not presented. A model reference impedance control was proposed in Jamil et al. (2016) to permit an accurate and a flexible grasping for soft objects. However, the control approach was only verified by using simulation results, whereas practical implementation of the control method was not mentioned. The control of the movement of industrial robots with virtual and real-time variable time delay was performed and applied in the simulation and real-time in Soyguder and Abut (2016). Although position tracking performance and stability of the robotic system were analyzed, only visual interface was designed to provide visual feedback, which needs further verifications. A motion trajectory tracking method based on infrared array module was proposed in Ye et al. (2016) for assembling omni-directional mobile robot. The different patterns of the trajectory width were optimized for getting more trajectory tracking accuracy. However, the trajectory tracking control algorithm was not verified based on comparative experiments, and its superiorities were not explicitly clarified. A force/position controller was proposed in Chaudhary et al. (2016) to control a 6-DOF PUMA robot manipulator. Even though performance comparison were made between the proposed controller and other control approaches, the validity of the proposed controller was not promising for real time implementation due to its complexity and large computational burden. A model reference adaptive control was presented in Alqaudi et al. (2016) for a combined human–robot system. The controller used no prescribed trajectory and accounted for the human operator dynamics. However, the controller was of a nonstandard form and was verified based on only simulations in a repetitive point-to-point motion task. An optimal control law was designed in Mian et al. (2016) to control and stabilize motions to track the desired trajectory for a two link robot manipulator. A reference sine wave was given as a reference to the robot manipulator and the desired response was plotted showing that the robot manipulator provided the desired trajectory tracking. However, only simulations results were provided, whereas experimental results were not provided to show actual trajectory from the desired reference. A Gaussian process regression-based deformation prediction and compensation method was presented in Wan et al. (2017) to improve the robot motion accuracy for assembly robots in assembling large-scale heavy-weight components. Although the proposed method provided a path toward hard-measuring easy deformation assembly task, its effectiveness was not verified in practical implementations due to the stochastic nature of the Gaussian process. A disturbance observer based control technique was proposed in Ajwad et al. (2015) to improve the trajectory tracking performance and eliminate the effect of uncertainties and disturbances for a 6-DOF robotic manipulator. However, the effects from the highly coupled dynamics, internal and external perturbation forces, joint friction and parameter variations were not taken into account in the controller design, which significantly degrade the performance of the controller in motion tracking control. An algorithm was proposed in Zhao et al. (2015) for the accurate path tracking of industrial robots. However, the designed algorithm was based on the identification of robot kinematic parameters, which were not precise in general. Also, the identified robot kinematic parameters were only constrained in a local working zone, which was not convincing.

By using the above-mentioned control methods, it is generally difficult to achieve accurate motion tracking control in practice due to the technological limitations of current controllers, coupled with the demanding requirements that are needed in cases where robots are operating in unstructured environments. As a result, when stringent motion tracking performance is of major concern, the aforementioned methods are often inadequate as actual robots are always subjected to certain uncertainties (e.g. the change of payloads), whereas these control techniques did not explicitly address the effect of parametric uncertainties and uncertain nonlinearities.

Therefore, this paper presents an adaptive robust sliding mode tracking controller for a 6-DOF industrial assembly robot under both parametric uncertainties and external disturbances. Unlike the existing trajectory tracking controllers for industrial robots, the proposed controller not only can maintain accurate trajectory tracking control but also can adequately address parametric uncertainties and external disturbances at once. Specifically, a discontinuous-projection-based parameter update law is also presented. The proposed approach explicitly takes into account the specific characteristics of the industrial robot in the controller designs and uses the work space coordinate of the end-effector for motion tracking control. Comparative results are also obtained based on a dual-robot assembly platform. The results verify the significantly better motion tracking performance of the proposed controller in an actual implementation in spite of various parametric uncertainties and uncertain disturbances.

2. Robotic dynamics and transformation

The dynamics of the 6-DOF industrial assembly robot can be naturally formulated in joint space coordinates as following (Bruno et al., 2010):

(1) M(θ)θ̈+C(θ,θ̇)θ̇+G(θ)=τ+d
where M(θ) is the 6 × 6 inertia matrix, θ is the 6 × 1 vector of joint angles, C(θ,θ̇) is the 6 × 6 centrifugal and Coriolis matrices and C(θ,θ̇)θ̇ gives the Coriolis and centrifugal force terms in the equations of motion. G(θ) includes gravity terms and other forces which act at the joints, d is a 6 × 1 vector of external disturbances and τ is the 6 × 1 vector of actuator torques for each joint.

The above dynamics can be equivalently transformed into the end-effector’s workspace coordinate to describe the dynamics of the end-effector by using smooth and invertible Jacobian matrix J(θ). Thus:

(2) Me(x)ẍ+Ce(x,ẋ)ẋ+Ge(x)=F+de
where x is the actual position of the end-effector and its orientation with respect to the workspace coordinate for specifying an assembly task. Me(x), Ce(x,ẋ) and Ge(x) represent the corresponding matrices in terms of the workspace coordinates, F is the equivalent end-effector force, de is the equivalent disturbances in the workspace of the end-effector and de ≤ δde, where δde is the upper bound of the disturbances.

Thus:

(3) Me(x)=JT(θ)M(θ)J1(θ)
(4) Ce(x,ẋ)=JT(θ){C(θ,θ̇)J1(θ)+M(θ)ddtJ1(θ)}
(5) Ge(x)=JT(θ)G(θ)
(6) {ẋ=J(θ)θ̇θ̇=J1(θ)ẋ
(7) {ẍ=J(θ)θ̈+J̇(θ)θ̇θ̈=J1(θ)ẍ+ddtJ1(θ)ẋ
(8) {τ=JT(θ)FF=JT(θ)τde=JT(θ)d

The above equations explicitly describe the relationships between joint and end-effector variables and hence formulate the forward and inverse dynamics for the robot. The matrix Me(x) is a symmetric positive definite matrix with Me(x) ≤ μI, where μ is a positive scalar. The matrices Me(x) and Ce(x,ẋ), which summarize the inertial properties of the manipulator, and satisfy the following structural properties:

P1: Me(x) is symmetric and positive definite; thus, MeT(x)=Me(x).

P2: Ṁe(x) − 2Ce(x,ẋ) is a skew-symmetric matrix; thus, xT[Ṁe(x) − 2Ce(x,ẋ)]x = 0.

3. Adaptive robust tracking control

The primary control objective of the 6-DOF industrial assembly robot should be accurate dynamic tracking of the desired trajectory with sufficient robustness and adaptability with respect to parametric uncertainties and external disturbances. As such, an adaptive robust sliding mode controller is designed for the 6-DOF robot to maintain accurate trajectory tracking, sufficient robustness and adaptability under parametric uncertainties and external disturbances (Edwards and Sarah, 1998). The controller is designed based on the combination of sliding mode control, adaptive and robust controls and hence has good adaptation and robustness abilities to parametric variations and uncertainties. The adaptive control term is used to synthesize a projection type parameter adaptation law in this controller, which can be constructed to reliably and accurately estimate unknown robotic parameters on-line to achieve a guaranteed transient and final tracking accuracy even in the presence of uncertain nonlinearities. By involving the robust term in this controller, the parametric model uncertainties and external disturbances can be sufficiently attenuated to guarantee the enough robustness of the 6-DOF robot under different assembly conditions. The controller design also emphasizes the effective combination of sliding mode control, adaptive and robust controls to preserve excellent performances of the three design approaches while meeting various conflicting performance requirements such as fast transient response and improved final tracking accuracy–asymptotic tracking or zero final tracking error.

3.1 Controller design

The dynamic trajectory tracking error of the robot’s end-effector can be defined as:

(9) e=xxd

A sliding mode manifold s is defined as:

(10) s=ė+Λe=ẋẋeq
where Λ is a constant positive definite diagonal matrix, xeq is the equivalent position of the end-effector and its orientation with respect to the workspace coordinate for specifying an assembly task and xd is the input reference or desired motion trajectory:
(11) ẋeqẋdΛe

Define a positive semidefinite Lyapunov function as:

(12) V(t)=12sTMe(x)s

Differentiating V(t) with respect to time and noting equations (2), (9) and (10), one obtains:

(13) V̇(t)=12sTṀes+sTMeṡ=sTMeṡ+sTCes=sT(MeẍMeẍeq+Ces)=sT[FMeẍeqCeẋeqGe+de]
where P2 is used to eliminate the term 12sTṀes.

Define the following equation as:

(14) Φ(x,ẋ,ẋeq,ẍeq)γ=MeẍeqCeẋeqGe
where Φ(x, ẋ, ẋeq, ẍeq) is a regressor vector and γ is the robot parameter vector to be updated online:
(15) Φ(x,ẋ,ẋeq,ẍeq)=[ẍeq,ẋeq,1]
(16) γ=[Me,Ce,Ge]T

Substituting the above equations (14)-(16) into equation (13) gives:

(17) V̇(t)=sT[F+Φ(x,ẋ,ẋeq,ẍeq)γ+de]

Considering the structure of equation (17), the controller can be reasonably designed as:

(18) F=Fa+Fs
where Fa and Fs are model compensation term and proportional robust control term, respectively:
(19) Fa=Φ(x,ẋ,ẋeq,ẍeq)γ^
(20) Fs=Fs1+Fs2
(21) Fs1=Ks
where γ̂ denotes the estimate of γ, K is a symmetric positive definite matrix, Fs1 is a proportional feedback term with K being a symmetric positive definite matrix in this case, Fs2 is a robust feedback term used to attenuate the effect of parametric uncertainties and external disturbances and can be synthesized later.

Substituting equations (19), (20) and (21) into equation (17) yields:

(22) V̇(t)=sT[FsΦ(x,ẋ,ẋeq,ẍeq)γ˜+de]=sTKs+sT[Fs2Φ(x,ẋ,ẋeq,ẍeq)γ˜+de]

The robust term Fs2 can be designed such that the following two conditions are satisfied:

(23) {sT[Fs2Φ(x,ẋ,ẋeq,ẍeq)γ˜+de]ξsTFs20
where γ͂ denotes the estimate error vector of γ,γ͂ = γ̂ − γ,ξ is a positive scalar.

3.2 Parameter update law

The parameter estimate γ̂ can be updated online based on a discontinuous projection adaptation law. Thus:

(24) γ^̇=Projγ^[ΓΦT(x,ẋ,ẋeq,ẍeq)s]
where Γ is a positive diagonal matrix that can be properly selected to guarantee the asymptotic stability of the closed-loop robot control system and improve dynamic response.

The discontinuous projection mapping Proj (•) can be represented as (Xie et al., 2014):

(25) Projγ^[ΓΦT(x,ẋ,ẋeq,ẍeq)s]={0     if  γ^=γmax and ΓΦT(x,ẋ,ẋeq,ẍeq)s>00     if  γ^=γmin  and  ΓΦT(x,ẋ,ẋeq,ẍeq)s<0ΓΦT(x,ẋ,ẋeq,ẍeq)s  Otherwise

For this parameter update law, the known upper and lower bounds of the parameter vector is used to confine the on-line parameter estimates within their known ranges. When the initial parameter estimate vector is set within the known bounds, i.e. γminγ̂(0) ≤ γmax, the discontinuous projection mapping equation (20) has the following two properties:

P3:

(26) γminγ^γmax

P4:

(27) γ˜T{Γ1Projγ^[ΓΦT(x,ẋ,ẋeq,ẍeq)s]ΦT(x,ẋ,ẋeq,ẍeq)s}0

Theorem 1: The proposed controller with the control terms in equation (18) can achieve the following results for the whole robot control system.

  1. Guaranteed transient performance and stability: In general, all signals are bounded; furthermore, the positive definite function V(t) defined by equation (12) is bounded by:

    (28) V(t)exp(λt)V(0)+ξλ[1exp(λt)]

    where λ=2σmin(K)μ, σmin(K) is the minimum eigenvalue of the matrix K.

  2. Zero final trajectory tracking error: Supposing there exist parametric uncertainties only after a finite time, i.e. de = 0. Then, zero final contouring error is also achieved, i.e. e → 0, s → 0 when time t.

Proof: Noting equations (22) and (23), the derivative of V(t) is given by:

(29) V̇(t)σmin(K)s2+ξ=λV(t)+ξ
which leads to equation (28) and thus proves the results in (1) of Theorem 1.

To verify the overall convergence and stability of the robot control system, the Lyapunov function in equation (12) can be augmented as:

(30) Va(t)=12sTMes+12γ˜TΓ1γ˜

The time derivative of the augmented Lyapunov function in equation (30) can be written as follows when considering equation (23):

(31) V̇a(t)=sTKssTΦ(x,ẋ,ẋeq,ẍeq)γ˜+γ˜Γ-1γ^̇

By considering property P4, equation (31) can be re-formulated as:

(32) V̇a(t)=-sTKs+γ˜T[Γ-1γ^̇ΦT(x,ẋ,ẋeq,ẍeq)s]-sTKs0.

Based on Barbalat’s lemma (Hou et al., 2010), equation (32) indicates that the both s and ṡ are bounded and uniformly continuous, and s → 0 when time t → . Hence, zero final trajectory tracking error will also be achieved, e → 0 as time t → despite parameter variations and uncertainties.

Remark: One smooth example of Fs2 satisfying equation (23) can be chosen as the following:

(33) Fs2=Δ2s4ξ
where Δ ≥ ‖γM‖‖Φ(x, ẋ, ẋeq,ẍeq )‖ + δde and γM = γmax – γmin.

4. Comparative verifications and discussions

Comparative results have been obtained based on an industrial robot experimental platform to evaluate the capability and effectiveness of the proposed adaptive robust tracking control in dynamic trajectory tracking control accuracy.

4.1 Experimental setup

As shown in Figure 1, the experimental platform mainly consists of two 6-DOF MITSUBISHI RV-4F industrial assembly robots, which are typically used for assembling low-voltage electrical appliances, such as air switches. Each robot has the mass of 39 kg, the maximum payload of 4 kg, the maximum operating speed of 9 m/s, the maximum actuation range of 515 mm, the positional repeatability precision of 0.02 mm and the allowable inertia of 0.2 kgm2. Each joint of the robots is powered by 400 W AC servo motor with position encoders capable of providing the joint angles in real time for the tracking control. The external disturbances for each robot mainly stem from the assembling interferences between the two robots, which may deteriorate the motion tracking performances of each robot.

The main parameters of the tracking control are initially set as: Λ = diag{6.2, 6, 6, 6.7, 6, 6}, K = diag{12.8, 12.8, 12.6, 12.2, 12.1, 12}, Γ = diag{26.2, 26.1, 24.3, 22.8, 21.8, 21.9}, and the initial positions of the joints and end-effector are set as ẋ = θ̇ = [0, 0,0,0,0]T,x = θ = [0, 0,0,0,0]T.

The proposed dynamic control algorithms have been first designed and simulated in MATLAB Toolboxes. The proposed control was also compared with a commonly used proportional derivative (PD) controller with fixed control parameters (Vijay and Jena, 2016) and a conventional sliding mode controller without adaptive and robust terms. The results for free-trajectory tracking under disturbances were repeated for several times, and the test data have been averaged over a long time to get averaged stable tracking error results.

The conventional sliding mode controller for this robot can be reasonably designed as (Utkin et al., 2009):

(34) F=Φ(x,ẋ,ẋeq,ẍeq)γδdeKssgn(s)
where sgn(s) is a sign function and can be defined as:
(35) sgn(s)={1ifs>00ifs=01ifs<0

The performance index, the averaged relative error (ARE), has been used to measure the quality of each control algorithm.

ARE: The ratio value of the trajectory tracking error with respect to the reference trajectory value.

The ARE for the trajectory tracking of the end effector can be defined as:

(36) ARE=16Ni=16j=1Nxijxdijxdiji=1,2,..6,j=1,2,,N

The ARE for the trajectory tracking of each joint angle can be defined as:

(37) ARE=1Nj=1Nθjθdjθdjj=1,2,,N
where N denotes the number of sampling data of the trajectory tracking, xij and xdij denote the j th sampling data of the i th element of the end-effector’s trajectory vector x and reference trajectory vector xdij, respectively, θj and θdj denote the j th sampling data of the joint angle trajectory θ and reference trajectory vector θd, respectively.

The system uncertainties and external disturbances mainly stem from modelling uncertainties, variable payloads and collisions with environment. These uncertainties can be reasonably calculated based on actual measurements and modelling data and thus can be described asδ = −Φ(x, ẋ, ẋeq,ẍeq)γ͂ + dewith the maximum value as Δ.

The averaged relative uncertainties can also be formulated as follows to describe the system uncertainties. Thus:

(38) ARU=16Ni=16j=1NδijΔiji=1,2,..6,j=1,2,,N
where δij and Δij denote the j th sampling data of the i th element of the system uncertainties vector δ and the maximum uncertainties vector Δ, respectively.

4.2 Comparative results without external disturbances

Figure 2 (a)-(f) illustrates the AREs of the six joint angles for the three different controllers. The ARE varies significantly between 9 and −9 per cent by using the traditional PD controller and the ARE for the conventional sliding mode controller also fluctuates significantly between −7 and 7 per cent. On the contrary, the ARE can be well maintained around −4 and 4 per cent when the proposed adaptive robust sliding mode controller is applied. The simulations results indicate that the proposed controller can be more effective in reducing tracking errors of joint angles as compared with PD controller and sliding mode controller, which generally do not have parametric adaption capabilities.

As shown in Figure 3, the ARE of the end-effector’s trajectory fluctuates considerably by using conventional PD controller and sliding mode controller. The AREs for PD control and sliding mode control are, respectively, −9.5-10 per cent and −6-6 per cent, which are significantly higher than the that for the proposed controller with ARE of −3.5-4 per cent. The end-effector’s trajectory also converges fast to the desired values when using the proposed controller, which demonstrates that the proposed controller can be used to significantly improve the trajectory tracking efficiency and effectiveness as compared with PD control and conventional sliding mode control.

4.3 Comparative results with external disturbances

As described in Figure 4, the ARE fluctuates dramatically between −10 and +10 per cent by using the PD control, between −6 and 6 per cent by using conventional sliding mode control, whereas the averaged relative tracking errors of each joint can be readily regulated within −4 and +4 per cent by using the proposed adaptive robust tracking control and the tracking errors are generally stable. Therefore, the free trajectory tracking error can be better maintained by using the adaptive robust tracking control over a wide range of time period as compared with the widely used PD control and sliding mode control.

As illustrated in Figure 5, the averaged position tracking error of the end effector oscillates frequently between −10 and +10 per cent with PD control and between −5.8-6 per cent with sliding mode control, whereas the averaged tracking error will be significantly decreased into −3 and +3 per cent and exhibits significant convergence by using the proposed adaptive robust tracking control with high robustness and adaptability. Thus, the proposed adaptive robust tracking control is capable of better maintaining the desired dynamic tracking error of the end effector as compared to the PD control.

Figure 6 shows the averaged system uncertainties and external disturbances, which oscillate significantly between the maximum value and 15 per cent of the maximum value. These averaged system uncertainties generally have higher dynamics and frequencies than the robotic dynamics and can be considered as additional noises to the system control. Generally, these uncertainties and external disturbances can be well compensated by using the proposed model compensation term Fa and proportional robust control term Fs in the designed controller, whereas these uncertainties cannot be well compensated by using the conventional PD control or sliding mode control because these controllers do not have parametric adaption and robust terms.

5. Conclusion

This paper has presented an adaptive robust sliding mode tracking control strategy for a 6-DOF industrial assembly robot. The adaptive robust tracking controller has been designed for both free trajectory tracking and parameter compensation under external disturbances. A discontinuous-projection-based parameter update law has also been incorporated into the sliding mode controller to improve the motion tracking control accuracy under various parametric uncertainties. The proposed controller has been demonstrated achieving significant performance improvement as compared with PD controller. The comparative results verify that the proposed dynamic control is capable for maintaining higher precision motion tracking control, higher robustness and parameter adaptability than PD control and conventional sliding mode control, even in the presence of parametric uncertainties and unknown external disturbances.

Figures

Experimental platform of 6-DOF assembly industrial robots

Figure 1

Experimental platform of 6-DOF assembly industrial robots

Relative tracking errors for joint angle

Figure 2

Relative tracking errors for joint angle

Relative tracking errors for the end-effector

Figure 3

Relative tracking errors for the end-effector

Relative tracking errors for joint angle

Figure 4

Relative tracking errors for joint angle

Relative trajectory tracking error of the end effector

Figure 5

Relative trajectory tracking error of the end effector

The averaged system uncertainties and external disturbances

Figure 6

The averaged system uncertainties and external disturbances

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant No. U1509212 and the National High Tech Support Fund of China under Grant No. 2015BAF01B02.

Corresponding author

Libin Zhang can be contacted at: lbz@zjut.edu.cn