Task Space Trajectory Tracking Control of Robot Manipulators with Uncertain Kinematics and Dynamics

To improve the tracking precision of robot manipulators’ end-effector with uncertain kinematics and dynamics in the task space, a new control method is proposed. The controller is based on time delay estimation and combines with the nonsingular terminal sliding mode (NTSM) and adaptive fuzzy logic control scheme. Kinematic parameters are not exactly required with the consideration of kinematic uncertainties in the controller. No dynamic models or numerous parameters of the robot manipulator system are requiredwith the use of TDE.Thus, the controller is simple structure and suitable for practical applications. Furthermore, errors caused by time delay estimation are compensated by the adaptive fuzzy nonsingular terminal sliding mode scheme. The simulation is performed on a 2-DOF robot manipulator with three cases in the task space. The results show that the proposed controller provides faster convergence rate and higher tracking precision than TDE based NTSM and improved TDE based NTSM controller.


Introduction
Robot manipulators play an important role in the industry automation field in recent years [1][2][3].One of the irreplaceable capabilities of robot manipulations is the high accuracy and high-speed performance of trajectory tracking.Therefore, engineers have sought to realize their automatic control by various control methods, such as sliding mode control (SMC) [4][5][6][7], adaptive control [8], neural network methods [9], and fuzzy logic control [10].The automatic control of robot manipulators presents a unique challenge from the control aspect, which is caused by inherent large nonlinearities and external uncertainties in system dynamics.Generally, to obtain a satisfactory control performance, the above control methods are mostly either model-free types introducing numerous parameters or model-based types requiring nominal model [11].However, robot manipulators, similar to many other mechanical systems, are complex and nonlinear.It is difficult to establish an accurate dynamic model of a robot manipulator system, which limits the practical applications of the control methods.
It is noteworthy that time delay estimation (TDE) provides a simple way to solve the above problems.The main idea of the TDE is to estimate unknown dynamics and disturbances by intentionally using time delayed information [12].The main advantage of the TDE is the mitigation of tedious modeling burden of complex system [13].Due to its advantages, TDE has been widely used in the design process of various controllers and provides satisfactory results [14][15][16].Lee et al. proposed an adaptive robust controller using TDE and adaptive integral sliding mode control, which was proved to be robust, chattering-free, and highly accurate [17].Roy et al. proposed a new adaptive robust control strategy with time delay control to remove prerequisite of system model and to alleviate the over-or underestimation problems of the switching gain [13].A systematic method was proposed using time delay estimation to simplify the tune process of fuzzy PID controller by Kim et al. [18].
Controllers designed based on TDE are typically composed of two elements.One element is the TDE element, which cancels nonlinear dynamics.The other element is an injecting element, which endows desired error dynamics.
Linear error dynamics are widely used as the desired error dynamics in TDE based controllers, such as time delay control (TDC) [19].Through extensive research, controllers based on TDC have been developed to improve the control performance.To suppress estimation error in TDC, fuzzy logic system was introduced in TDC as the third element by Bae et al., and satisfactory results were obtained [20].To realize the automatic tuning of TDC parameters, adaptive time delay control was proposed by Jin et al. [21] and Cho et al. [22], respectively.To improve the convergence rate and tracking precision, the nonlinear error dynamics such as nonsingular terminal sliding mode (NTSM) were used together with TDE by recent work [23,24].Jin et al. employed this control method in the trajectory tracking of robot manipulators by simulations and experiments.The tracking results showed that the controller is highly accurate, model-free, simple to implement, and robust.However, there are two aspects to be improved.(1) Kinematic uncertainties are not considered in the controller design process.When the desired trajectory is planned in the task space such as Cartesian space, the Jacobian matrix from joint space to Cartesian space and the kinematic parameters of the robot manipulator are assumed exactly to be known.Satisfactory performance can be obtained in the task space without the feedback of the end-effector position with the assumption.But in most practical applications, the kinematics parameters may not be exactly known, which is caused by the interaction between manipulator and different environments and the imprecise measurements of physical parameters.(2) TDE can not eliminate the nonlinearities such as Coulomb friction perfectly.Those nonlinearities may cause TDE error, which reduces the precision of trajectory tracking [25][26][27].
In order to solve the above problems, a novel control method is proposed for trajectory tracking of robot manipulator's end-effector in the task space.The controller is based on TDE and combines with the NTSM and adaptive fuzzy logic control scheme.The contributions of this controller are listed as follows.(1) The kinematic uncertainties of the manipulator are taken into consideration without the prior knowledge of system.(2) No dynamic models of the robot manipulator system are required.(3) The adaptive fuzzy logic scheme is implemented as the third element to compensate the TDE error and eliminate the undesired chattering in NTSM.(4) The controller is easy to implement with simple structure, and it is suitable for practical applications.

TDE Based NTSM Control with Kinematic and Dynamic Uncertainties
In this section, the conventional TDE based NTSM controller is improved with the consideration of kinematic uncertainties.The control objective of this controller is to make the end-effector of manipulator follow a desired trajectory in the task space with unknown dynamics and uncertain kinematics.
The dynamical equation of -link robot manipulator can be shown as (1) where , θ , θ ∈ R  are vector of position, velocity, and acceleration of the joints respectively;  ∈ R  denotes the actuator torque; and M() ∈ R × represents the inertia matrix; V(, θ ) ∈ R  stands for the Coriolis and centrifugal matrix; G() ∈ R  is the gravitational vector; F ∈ R  is the friction term; and D ∈ R  denotes the disturbance torques.
Equation ( 1) can be rewritten as follows after defining a constant diagonal matrix M: where The trajectory of manipulator end-effector in the task space can be expressed as where h() is the transformation relationship between the task space and the joint space; x is the trajectory of endeffector in the task space, which can be measured by laser sensor or vision sensors.ẋ is the velocity in task space and it is related to θ in the joint space, which can be expressed as where J() is the Jacobian matrix.The differential of (4) can be expressed as It should be noted that the inverse of Jacobin matrix will be an ill conditioned matrix when the manipulator is at the vicinity of a singular configuration, and the singularity problem exists in the control of robot manipulators.Some methods have been proposed to solve this problem [28].In this paper, it is assumed that the Jacobian matrix is nonsingular, and then (5) can be rewritten as Substituting ( 6) into (2), it can be obtained as Because the kinematic and dynamic parameters of robot manipulators are not exactly known in the practical applications, the parameters in (8) are substituted by the estimated value.
Replacing ẍ by the acceleration of the desired trajectory ẍ d , the equivalent control input can be expressed as To compensate the kinematic and dynamic uncertainties and guarantee the stability, the NTSM scheme  ntsm is implemented as The NTSM scheme is designed by two steps.The first step is the design of a sliding surface, and the second step is the design of a control law to guarantee the existence of sliding mode.
Then, sliding mode surface is chosen as where e is the tracking error between the desired position and actual position of the manipulator end-effector in the task space, and e = x d − x; s represents an  × 1 vector; K denotes an  ×  defined positive constant matrix;  and  are positive odd integers and 1 < / < 2.
To guarantee the existence of sliding mode, the NTSM scheme is designed as where K sw is a constant matrix to be designed.Then, the control law can be expressed as In this control law, Ĥ is the estimation of all the nonlinearities kinematic and dynamic uncertainties, which makes it difficult to establish model and identify parameters.TDE is used to estimate Ĥ in this controller, which can be expressed as where • − denotes the time delayed value of •, and if the time delay L is set as infinitesimally small, an estimation of H would be possible by TDE.
Then the output of the controller can be expressed as Substituting controller equation ( 15) into dynamics equation (2), it can be obtained as In (6), the Jacobin matrix is substituted by the estimated value.It can be expressed as Substituting ( 17) into (16), it can be obtained as where  = Ĵ()M −1 (H − − H) and it is defined as TDE error.
The TDE error is mainly caused by the finite time delay L under nonlinearities such as Coulomb friction.
Lyapunov function is chosen to prove the stability of the system, which is expressed as Then, the derivative of V is expressed as According to the study of Jin et al. [24],  is proved to be bounded.In (20), p and q are positive odd integers and 1 < p/q < 2; then ė   /  −1 > 0 for ė ̸ = 0. To guarantee the stability of the system, K sw should be chosen as Then, s  (p/q)K ė p/q−1 [ − K sw sgn(s)] < 0 and V < 0. Thus, the controller considering the kinematic and dynamic uncertainties is proposed as (15).However, it still has two drawbacks.One is the chattering problem caused by sign function.Although the replacement of the sign function by saturation function can eliminate the chattering, it reduces the tracking precision [29,30].The other problem is the TDE errors, which is shown in (18).TDE errors may cause the tracking error of the system.

TDE Based AFNTSM Control with Kinematic and Dynamic Uncertainties
In order to solve the above problems, adaptive fuzzy logic control scheme is used to eliminate the chattering and improve the tracking precision.In this section, the controller is presented and the stability is proved based on the Lyapunov method.The new controller structure is designed as Figure 1 demonstrates the block diagram of the proposed controller.It is composed of three terms, including the equivalent control term, the adaptive fuzzy nonsingular terminal sliding mode term, and time delay estimation term.The main difference of controller equation (22) and equation ( 15) is the replacement of sign function by an adaptive fuzzy logic control scheme k + s, where k is the adaptive fuzzy logic controller and s is its compensator. is designed as a diagonal positive definite matrix and  = diag[ 1 + 1 , . . .,   +   , . . .,   +   ], where   is a positive constant and   is a positive value.

Fuzzy Logic Control Scheme.
It is designed that k has the same sign as that of   .Substituting ( 22) into (20), it is obtained as Because (p/q)K ė p/q−1 > 0, the term is very small, which has little effect on the value of V.Then, small |  | is allowed to avoid chattering.When |  | is zero, s  [ − k − s] is zero and |  | can be zero.From these analyses, the rule base is defined as where   the input of fuzzy system and   is the output of the system.They are partitioned into five fuzzy subsets: positive big (PB), positive small (PS), zero (ZE), negative small (NS), and negative big (NB).They are Gaussian membership function defined as where the subscript  denotes the fuzzy sets such as PB, . . ., NB;   donates   and   ;  is the center of ; and  is the width of .
Choosing the product inference engine, singleton fuzzification, and center average defuzzification, then,   can be written as where .    is chosen as the parameter to be updated.   (  ) can be regard as the weight of the parameter vector.22) into (2), it can be expressed as

Adaptive Scheme. Substituting (
*   is defined; then   =  *    Ψ   (  ) is the optimal estimation for .The optimal estimation error   > 0 exists satisfying Define and then The upper boundary of compensator is defined as Define and then Choose the adaptive law as where     and    are positive constants.Choose the Lyapunov function as Then, the derivative of V can be obtained as As   = γ     (  ) +  *       (  ) and     =     +     , (36) can be expressed as Since the adaptive law is and Since Then (39) becomes Since   is a positive constant, therefore V < 0 when ė ̸ = 0 and  ̸ = 0.For ė = 0, it is obtained as (41) by substituting ( 22) into (2).
Therefore, ė = 0 is not an attractor in the reaching phase, and V = 0 only when  = 0. Thus, it is proved that the adaptive law in (34) drives the tracking error to converge to zero in finite time.Therefore, the actual trajectory of the manipulator end-effector x converges to the desired trajectory x d in the task space.

Simulation Setup.
In order to verify the effectiveness of the proposed controller by simulation, a 2-DOF robot manipulator is adopted, which is shown in Figure 2.
The Jacobian matrix of this manipulator is shown as where  1 and  2 denote the length of first link and second link, respectively;  1 and  2 are the joint position of the two links.
The dynamic model of the robot manipulator system is given for simulation.The dynamic model is given as (1).Details of the model are shown as The friction term affects the control performance of robot system seriously.Thus, the friction term is chosen as where 1 ,  2 = / 1 ;  1 denotes the mass of first link;  1 is the distance between the mass center of first link and the first joint;  1 is the moment of inertia of the first link;   is the mass of second link with payload;   is the distance between the mass center of second link and the second joint;   is the moment of inertia of the second link;   is the angle relative to the original second link.The physical parameters of the robot manipulator are shown in Table 1.
The desired trajectory of end-effector in the task space is designed as a circle, which is expressed as Three controllers are chosen to control the manipulator plant.Controller 1 is conventional TDE based NTSM control method.According to the study of Jin et al. [24], controller 1 is expressed as where and it is the replacement function of sign function to reduce the chattering; e  is the tracking error and e  =  d − .Controller 2 is the improved controller based on controller 1, which is expressed as (15).Similarly, to reduce the chattering, the sign function existing in controller is replaced by saturation function.Controller 3 is the proposed controller, which is expressed as (22).After the tuning process, the gains of controller 1 are M = 0.15I, K = 0.9I, p = 5I, q = 3I, K sw = 10I, and Φ = 0.04I.The gains of controller 2 are M = 0.15I, K = 0.9I, p = 5I, q = 3I, K sw = 10I, and Φ = 0.04I.The gains of proposed controller are M = 0.15I, K = 0.9I, p = 5I, q = 3I,     = 10, and    = 200;   = 200.Three controllers are designed based on TDE.They are easy to implement and no information about the dynamic model is required.Because kinematics uncertainties are considered in controllers 2 and 3, the values of  1 and  2 are required.In this simulation, it is assumed that the values of  1 and  2 are not exactly known, and they are estimated as The other parameters of the robot manipulator are not required.
The simulation is carried out by three cases.Case 1 is the simulation with the external kinematics uncertainties.Case 2 is the simulation with external dynamic uncertainties.Case 3 is the simulation with measurement noise.

Case 1 with External Kinematic
Uncertainties.In this case, the simulation is carried out on the plant model with kinematic uncertainties.It is assumed that the size of the two links varies as  1 = 1.2 + 0.1 sin(2),  2 = 1.3 + 0.1 sin(2).
Three controllers are applied to control the plant, respectively.The results are shown in Figures 3-7.Maximum values (MAX) and root mean square (RMS) of tracking errors are shown in Figures 8 and 9. MAX and RMS are calculated by the simulation data in the second period.From Figures 3-7, it can be observed that the desired trajectory can be tracked well under controller 2 and the proposed controller, while the trajectory can not be tracked well under controller 1.The reason is that controller 1 does not consider the kinematic uncertainties, and the kinematic parameters are estimated.When errors exist between real size and estimation size of the link, the large tracking errors occur in the trajectory of the manipulator end-effector in the task space.From Figures 4 and 5, it is observed that peaks of tracking error exist during the control procedure of  controller 2 and proposed controller.Those peaks are mainly caused by TDE error, due to Coulomb friction.It can also be observed that the peaks under the proposed controller are the smallest, which benefited from the compensation by adaptive fuzzy logic control scheme.The corresponding results are also shown in Figure 7.The fuzzy logic scheme output of proposed control provides relative larger control signal when the error peaks occur, which is essential for obtaining better control performance under large errors.It is obvious that the proposed controller has the highest tracking precision and fastest convergence rate among the three controllers, and the conventional TDE based NTSM controller has the relative worst performance.As shown in Figures 8 and 9, the tracking error of proposed controller has the smallest MAX under kinematic uncertainties.The MAX of proposed controller is 3.2% and 59.4% of those from the other two controllers.The RMS of proposed controller is 1.1% and 45.1% of those from the other two controllers.From the above results, best performance is got by the proposed controller under kinematic uncertainties in the task space.Furthermore, no chattering is observed using those three controllers.

Case 2 with External Dynamic
Uncertainties.In this case, the simulation is carried out on the plant model with dynamic uncertainties.The external dynamic uncertainties are shown in Figure 10.Three controllers are applied to control the plant, respectively.The results are shown in convergence rate among the three controllers.As shown in Figures 15 and 16, the MAX of proposed controller is 3.1% and 60.5% of those from other two controllers during the tracking error of end-effector.The RMS of proposed controller is 1.1% and 43.1% of those from other two controllers.

Case 3 with Measurement Noise.
In practical applications, the measurement noise is unavoidable.Thus, to prove the practicality of the proposed controller, the measurement noise is introduced into the position measurement of robotic manipulator based on the above simulation.A band-limited white noise module is used as the noise effect.The noise power is set as 4×10 −8 , and other parameters remain at default values.After being filtered by the discrete filter module, the simulation results are shown in Figures 17-20.
As shown in Figures 17, 18, and 20, controller 2 and the proposed controller can ensure small tracking error under measurement noise.Controller 1 has a relative large tracking error due to the estimation of  1 and  2 .The control input of the joint becomes noisy due to measurement noise.From the tracking result, the proposed controller still has the best control performance among the controllers.Moreover, it can be observed that the MAX and RMX of tracking errors of the proposed controller are smallest.
In summary, from those simulation results, it can be concluded that the proposed control ensures faster convergence rate and higher tracking precision under external kinematic and dynamic uncertainties.Furthermore, due to the application of TDE, the proposed controller is easy to implement and it does not require the dynamical model of robot system.
It is estimated that the good performance results from the controller structure closely.First, the controller considers the external kinematic uncertainties during the design procedure.Thus, the controller is not sensitive to the varying of kinematic uncertainties.The effectiveness of this consideration is showed by the comparison between controller 1 and the proposed controller.Second, TDE cancels most of the kinematic and dynamic uncertainties by the scheme  − − M θ − .No prior knowledge of the robot dynamic and offline identification is required with TDE, which ensures the controller to be simple and easy to implement.Third, the high tracking precision is benefited from the NTSM and adaptive fuzzy logic scheme.NTSM is selected as the desired error dynamics, which improves the tracking precision and convergence rate.The implanting of fuzzy logic control scheme can eliminate the chattering caused by NTSM.Some peaks can be observed in the tracking error, which is caused by the by Coulomb friction when the sign of velocity changes.This scheme can compensate the TDE error effectively.Furthermore, it can also compensate the external kinematic and dynamic uncertainties.The advantage of using this scheme is verified by the comparison between controller 2 and the proposed controller.
It should be pointed out that simplicity is the critical virtue of the TDE based controllers.The combination of adaptive fuzzy logic scheme may introduce more parameters.The consideration about this problem in our controller design is discussed as follows.In the design process, adaptive scheme is implemented in the controller, which has the following advantages.First, the parameters of the factors in fuzzy mechanism and the compensator output value are adapted online.Second, the rule number in the FLC is reduced using the adaptive fuzzy nonsingular terminal sliding mode.Third, kinematic parameters of the system are not required exactly.Thus, with the above advantages, only one more parameter is introduced compared with the number of parameters in TDE based NTSM.The parameter tuning step is similar to TDE based NTSM.First, the parameters in terminal sliding mode should be selected.Then, the parameter M can be tuned from small positive value to obtain a satisfactory control performance.After that, one can tune the three parameters in adaptive scheme to achieve greater performance.Since the tuning process of the controller is by trial and error method, further research will focus on the automatic tuning of controller parameters.

Conclusions
In this paper, a new control method is proposed for the trajectory tracking of robot manipulators in the task space under uncertain kinematics and dynamics.The controller is designed based on TDE.Kinematic parameters and dynamic models or numerous parameters of the robot manipulator systems are not exactly required in this controller.TDE scheme is used to cancel most of the uncertainties.The NTSM scheme endows desired error dynamics to provide finite time convergence.Adaptive fuzzy logic scheme is used to compensate the TDE error and eliminate the  chattering, which reduces the tracking error.The performance of proposed controller is verified through simulation on a 2-DOF robot manipulator.Three cases simulations are designed to track the desired trajectory in the task space using the controllers.The comparison results show that the proposed controller provides faster convergence rate and higher tracking precision than TDE based NTSM and improved TDE based NTSM controller.Furthermore,

Figure 1 :
Figure 1: Block diagram of the proposed controller.

Figure 5 :
Figure 5: Tracking error of  and  directions with kinematic uncertainties.

Figure 6 :
Figure 6: Control input of joint 1 and joint 2 with kinematic uncertainties.

Figure 7 :
Figure 7: Fuzzy output of k + s with kinematic uncertainties.

Figure 8 :
Figure 8: MAX of tracking errors with kinematic uncertainties.

Figure 9 :
Figure 9: RMS of tracking errors with kinematic uncertainties.

Figure 11 :
Figure 11: Tracking error of manipulator end-effector with dynamic uncertainties.

Figure 12 :
Figure 12: Tracking error of  and  directions with dynamic uncertainties.

Figure 17 :
Figure 17: Tracking error of manipulator end-effector with measurement noise.

Figure 18 :
Figure 18: Tracking error of  and  directions with measurement noise.

Figure 19 :
Figure 19: Control input of joint 1 and joint 2 with measurement noise.

Figure 20 :
Figure 20: MAX and RMS of tracking errors with measurement noise.

Table 1 :
The physical parameters of the robot manipulator.