Robust Tracking Control of Robot Manipulators Using Only Joint Position Measurements

Abstract

This paper concerns the tracking control of a robot manipulator with unknown uncertainties and disturbances. It presents a new control method that uses only joint position measurements to design a tracking controller. The controller has two parts. One is based on a feedback linearization technique; it makes the nominal model of a manipulator asymptotically track a desired trajectory. The other is based on the idea of equivalent input disturbance (EID); it compensates for uncertainties and disturbances. Together they enable a robot manipulator to precisely track the desired trajectory. The new control algorithm is applied to a two-link robot manipulator, and simulation results demonstrate the validity of this method.

1. Introduction

Robot manipulators are widely used in many fields. They are especially useful in areas where it is impractical or undesirable for a human to go, for example, undersea exploration, radioactive environments, and defusing explosive devices. Interest in the control of robot manipulators has been increasing over the past few years [1–3], and it is now a central issue in robotics.

If an exact dynamic model of a robot manipulator is known, the motion control problem is easy to solve by the computed-torque-control (CTC) method [4]. It uses nonlinear state feedback to cancel the nonlinear terms and a simple PD controller for motion control. Although this method is simple and effective, the requirement of an exact model limits its practicality because it is usually impossible to obtain an exact, or even reasonably accurate, dynamic model in practical applications. For example, an actual plant inevitably contains structured and unstructured uncertainties, and a robot manipulator may be influenced by unpredictable external disturbances when the operating environment changes. Since these uncertainties and disturbances may greatly affect control performance, it is necessary to consider their effects in the study of the motion control of robot manipulators.

A number of strategies have been developed to solve the problem of controlling the motion of a robot manipulator with uncertainties and disturbances. They include a Lyapunov-based method [5], a neural-network-based method [6, 7], an adaptive neural network 𝐻_∞ strategy [8], an adaptive switching learning PD (ASL-PD) method [9], a parameter-dependent nonlinear observer approach [10], and a variable-structure PID control method [11]. However, all of them require measurement of both the displacement and velocity of joints.

Generally speaking, joint displacement can be accurately measured with an encoder. However, velocity is typically measured with a tachometer, and the results usually contain noise, which can affect the control precision and performance of a closed-loop system. So, both practically and theoretically, it is meaningful to devise a motion control method for robot manipulators which relies only on the measurement of joint position. Various strategies have been developed to solve this challenging problem. One is a controller-observer combination strategy. It has a two-step design procedure: (1) construct an observer for a robot manipulator based on joint information and (2) use joint displacement and the state of the observer to design a state feedback controller. Many control methods based on this strategy have been reported [12–14]. The problem with it is that the stability of both the observer and the controller is of a local nature. Another strategy [15–17] involves using a linear or nonlinear compensator to obtain substitutes for the velocity variables. It enables the global tracking control of a robot manipulator, but the addition of external state variables complicates the design of the control system. A third is an adaptive tracking control strategy that includes an output feedback scheme (OFS) [18] and an iterative learning scheme (ILS) [19], but the OFS-based controller only locally ensures the asymptotic stability of the joint position error, and the ILS-based controller makes the system only track the same task iteratively.

This paper presents a new tracking control approach for a robot manipulator with unknown uncertainties and disturbances. Its advantage is that the design of the tracking controller relies only on the measurements of joint position, not velocity, and the tracking control is global. It is based on the concept of an equivalent input disturbance (EID), which was first presented in [20] to deal with disturbance rejection in a linear servo system. The EID approach has been validated through application to several mechatronic systems [20–22]. In this study, it was used to design a global robust tracking controller for a robot manipulator. The controller has two parts: one makes the nominal model of a manipulator asymptotically track a desired trajectory and the other compensates for uncertainties and disturbances. Together they enable a robot manipulator with unknown uncertainties and disturbances to precisely track the desired trajectory.

The rest of this paper is organized as follows. Section 2 describes the model and formulates the problem. Section 3 explains the design of an EID-based tracking controller. Section 4 discusses a numerical example for a two-link robot manipulator. Finally, Section 5 presents some concluding remarks.

2. Model Description and Problem Formulation

For a robot manipulator with 𝑛 serial links, we take 𝑞,̈𝑞,̇𝑞∈ℝ^𝑛 to be the acceleration, velocity, and position vectors, respectively, of the joints. Choose the Lagrangian of the robot system to be 𝐿𝑞̇=1𝑞,2𝑞̇^𝑇𝑀𝑞𝑞̇−𝑃𝑞, (1) where 𝑀(𝑞)∈ℝ^𝑛×𝑛 is a positive-definite symmetric inertia matrix and 𝑃(𝑞) is the potential energy of the system. The equation of motion of the manipulator is obtained from the Euler-Lagrange equation: 𝑑𝑞̇𝑑𝑡𝜕𝐿𝑞,𝜕𝑞̇−𝜕𝐿𝑞,𝑞̇𝜕𝑞=𝜏−𝜏_𝑓+𝜏_ext, (2) where 𝜏∈ℝ^𝑛×1 is the vector of control torques, 𝜏_𝑓∈ℝ^𝑛×1 is the vector of friction torques, and 𝜏_ext∈ℝ^𝑛×1 is the vector of the external disturbances imposed on the joints. We rewrite (2) in the general form 𝑀𝑞𝑞̈+𝐻𝑞,𝑞̇+𝑁𝑞=𝜏−𝜏_𝑓+𝜏_ext, (3) where 𝐻(𝑞,𝑞)̇∈ℝ^𝑛×1 is the vector of Coriolis and centrifugal forces and 𝑁(𝑞)∈ℝ^𝑛×1 is the gravity vector.

Due to the unmodeled dynamics, measurement error, and changes in environment, it is difficult to obtain precise values for the masses and lengths of the links, the moments of inertia of the links, and other physical parameters. The measured values of these parameters are usually not very accurate. Thus, the values of the matrices 𝑀(𝑞),  𝐻(𝑞,𝑞)̇, and 𝑁(𝑞) in (3) are 𝑀𝑞=𝑀₀𝑞𝑞,𝐻𝑞̇+Δ𝑀𝑞,=𝐻₀𝑞̇𝑞̇,𝑁𝑞𝑞,+Δ𝐻𝑞,=𝑁₀𝑞+Δ𝑁𝑞, (4) where 𝑀₀(𝑞), 𝐻₀(𝑞,𝑞)̇, and 𝑁₀(𝑞) are the nominal values of 𝑀(𝑞), 𝐻(𝑞,𝑞)̇, and 𝑁(𝑞), respectively, and Δ𝑀(𝑞), Δ𝐻(𝑞,𝑞)̇, and Δ𝑁(𝑞) are the corresponding additive uncertain terms. Consequently, (3) becomes 𝑀₀𝑞𝑞̈+𝐻₀𝑞,𝑞̇+𝑁₀𝑞=𝜏+𝑑, (5) where 𝑑=𝜏_ext−Δ𝑀𝑞𝑞̈−Δ𝐻𝑞,𝑞̇−Δ𝑁𝑞−𝜏_𝑓. (6)

We assume that there is no prior information about Δ𝑀(𝑞), Δ𝐻(𝑞,𝑞)̇, Δ𝑁(𝑞), 𝜏_𝑓, or 𝜏_ext. Thus, 𝑑 is an unknown disturbance of the nonlinear system (5).

Let 𝑞_𝑟∈ℝ^𝑛×1 be the desired trajectory of the manipulator, and let 𝑒=𝑞−𝑞_𝑟 be the tracking error of the trajectory. If we take the control law, 𝜏, in (5) to be 𝜏₀=𝑀₀𝑞𝑞̈_𝑟−𝐾_𝐷𝑞̇−𝑞̇_𝑟−𝐾_𝑃𝑞−𝑞_𝑟+𝐻₀𝑞,𝑞̇+𝑁₀𝑞, (7) where 𝐾_𝑃 and 𝐾_𝐷∈ℝ^𝑛×𝑛 are two given positive-definite diagonal matrices, then (5) and (7) give 𝑒̈+𝐾_𝐷𝑒̇+𝐾_𝑃𝑒−𝑀₀⁻¹𝑞𝑑=0. (8)

Assume that Δ𝑀(𝑞), Δ𝐻(𝑞,𝑞)̇, Δ𝑁(𝑞), 𝜏_𝑓, and 𝜏_ext are all zero, which means that there are no parameter perturbations and external disturbances acting on the manipulator. Then, combining (6) and (8) yields 𝑒̈+𝐾_𝐷𝑒̇+𝐾_𝑃𝑒=0. (9)

It is easy to obtain lim_𝑡→∞𝑒=0 and lim_𝑡→∞𝑒̇=0 for (9). So, the state variables of the system, [𝑞^𝑇,𝑞̇^𝑇], asymptotically approach the desired trajectory, [𝑞^𝑇_𝑟,𝑞̇^𝑇_𝑟], for this case.

However, in practice it is difficult to acquire exact knowledge of a robot manipulator; uncertainties and disturbances can greatly reduce the tracking precision. So, we need to consider the tracking control problem for the perturbed robot system (5). This paper presents an EID-based tracking controller for (5) that relies only on measured joint position, 𝑞.

3. Design of EID-Based Tracking Controller

For the nonlinear system (5), we take 𝜏=𝜏₀+𝑀₀𝑞𝐵𝑢, (10) where 𝜏₀ is defined in (7); 𝐵=[𝑏₁,𝑏₂,…,𝑏_𝑛]^𝑇∈ℝ^𝑛×1, 𝑏_𝑖 (𝑖=1,2,…,𝑛) are constants; and 𝑢∈ℝ is a new input that is used to compensate for the disturbance, 𝑑. Combining (5) and (10) yields 𝑒̈=−𝐾_𝐷𝑒̇−𝐾_𝑃𝑒+𝐵𝑢+𝑀₀⁻¹𝑞𝑑. (11)

Let 𝑥=[𝑒^𝑇,𝑒^𝑇̇]^𝑇∈ℝ^2𝑛×1; the state space form of (11) is 𝑥̇=𝐴𝑥+𝐵𝑢+𝜀, (12) where 0𝐴=^𝑛×𝑛𝐼_𝑛−𝐾_𝑃−𝐾_𝐷0,𝐵=^𝑛×1𝐵,0𝜀=^𝑛×1𝑀₀⁻¹𝑞𝑑, (13) where 𝐼_𝑛 is the 𝑛×𝑛 identity matrix and 0_𝑖×𝑗 is an 𝑖×𝑗 zero matrix for positive integers 𝑖 and 𝑗.

Since only the measured 𝑞 is available, we take the output of (12) to be 𝑦=𝐶𝑥=𝑒,𝐶=𝐼_𝑛,0_𝑛×𝑛. (14)

It is easy to verify that (𝐶,𝐴) is observable. If we denote Φ(𝑠)=𝐾_𝑃+𝑠𝐼_𝑛+𝐾_𝐷𝑠𝐼_𝑛⁻¹, (15) then 𝐶(𝑠𝐼−𝐴)⁻¹𝐵=𝐼_𝑛,0_𝑛×𝑛×−Φ(𝑠)𝑠𝐼_𝑛+𝐾_𝐷Φ(𝑠)−𝐼_𝑛−𝑠𝐼_𝑛Φ(𝑠)𝑠𝐼_𝑛+𝐾_𝐷𝑠𝐼_𝑛Φ(𝑠)𝐵=−Φ(𝑠)𝑠𝐼_𝑛+𝐾_𝐷(𝑠𝐵=0,Φ)_𝑛×1,−Φ(𝑠)𝐵. (16) Equation (16) tells us that (𝐴,𝐵,𝐶) has no zeros on the imaginary axis because Φ(𝑠)𝑏𝐵=¹𝜙₁,𝑏(𝑠)𝜙²₂𝑏(𝑠),…,^𝑛𝜙_𝑛(𝑠)^𝑇, (17) where 𝜙_𝑖(𝑠)=𝑠²+𝐾_𝑑𝑖𝑠+𝐾_𝑝𝑖 (𝑖=1,2,…,𝑛), and 𝐾_𝑝𝑖 and 𝐾_𝑑𝑖 are the diagonal elements of the matrices 𝐾_𝑃 and 𝐾_𝐷, respectively.

In addition, we choose 𝑏_𝑖 in 𝐵 such that (𝐴,𝐵) is controllable. Then, as shown in [20], there always exists an EID, 𝑑_𝑒, on the control input channel. It produces the same effect on the output, 𝑦, as the disturbance, 𝜀, does. The perturbed plant (12) and (14) can be considered to be an EID-based plant: 𝑥̇=𝐴𝑥+𝐵𝑢+𝑑_𝑒,𝑦=𝐶𝑥. (18)

In the configuration of the EID-based control system (see Figure 1), 𝐾_𝑓 is the feedback gain; 𝐿 is the observer gain; 𝑑_𝑒 is an estimate of the EID, 𝑑_𝑒; and 𝐹(𝑠) is a low-pass filter that limits the angular frequency band of the disturbance estimate. Their design is discussed below.

Figure 1 

Configuration of an EID-based control system.

3.1. Estimation of Equivalent Input Disturbance

First, we construct a full-order Luenberger observer for (18): 𝑥̇=𝐴𝑥+𝐵𝑢_𝑓+𝐿𝑦−𝐶𝑥, (19) where 𝑢_𝑓=𝑢+𝑑_𝑒. Letting Δ𝑥=𝑥−𝑥 (20) and substituting that into (18) yield 𝑥̇=𝐴𝑥+𝐵𝑢+𝑑_𝑒, (21) where 𝑑_𝑒=𝑑_𝑒̇+Δ𝑑,𝐵Δ𝑑=Δ𝑥−𝐴Δ𝑥. (22)

Note that (21) has the same form as (18). We take 𝑑_𝑒 to be an estimate of the actual EID, 𝑑_𝑒. From (18), (21), and (22), it is clear that the difference between the state of the plant and that of the observer is equivalent to the difference between 𝑑_𝑒 and 𝑑_𝑒. Combining (19) and (21) yields 𝐵𝑢+𝑑_𝑒−𝑢_𝑓=𝐿𝑦−𝐶𝑥. (23)

We solve (23) for 𝑑_𝑒 and obtain a least-squares solution: 𝑑_𝑒=𝐵⁺𝐿𝑦−𝐶𝑥+𝑢_𝑓−𝑢, (24) where 𝐵⁺=𝐵^𝑇𝐵⁻¹𝐵^𝑇. (25)

To ensure the estimation accuracy, the low-pass filter 𝐹(𝑠)=1𝑇𝑠+1 (26) is used to select the angular frequency band for disturbance estimation, where 𝑇 is the time constant. The filtered disturbance estimate, 𝑑_𝑒, is given by 𝐷_𝑒(𝑠)=𝐹(𝑠)𝐷_𝑒(𝑠), (27) where 𝐷_𝑒(𝑠) and 𝐷_𝑒(𝑠) are the Laplace transforms of 𝑑_𝑒 and 𝑑_𝑒, respectively.

We take control law 𝑢 to be 𝑢=𝑢_𝑓−𝑑_𝑒,𝑢_𝑓=𝐾_𝑓𝑥, (28)(29) where 𝐾_𝑓 is a feedback gain that makes 𝐴+𝐵𝐾_𝑓 stable. Assume that the observer (19) is stable. Then, control law (29) makes the output of plant (18) asymptotically converge to zero. According to the definition of EID, control law (28) asymptotically stabilizes the output of (12), which is 𝑒, at the origin. Thus, the tracking control objective of system (12) is achieved.

Remark 1. From (7) and (10), the velocity, 𝑞̇, is needed to calculate the control input, 𝜏. Since we assume that the measured value of 𝑞̇ is unavailable, we use information of the observer (19) to obtain a substitute for 𝑞̇ in (7): 𝑞̇=𝑥₂+𝑞̇_𝑟, (30) where 𝑥₂∈ℝ^𝑛, 𝑥=[𝑥^𝑇₁,𝑥^𝑇₂]^𝑇 is the state vector of the observer (19), and 𝑥₁∈ℝ^𝑛.

3.2. Design of State Observer

Since the separation theorem holds for an EID-based control system [20], we can separately design the feedback gain 𝐾_𝑓 and the observer gain 𝐿. Since (𝐴,𝐵) is controllable, it is easy to design a 𝐾_𝑓 by any appropriate method (pole placement, optimal control, etc.). So, here we focus on the design of 𝐿.

The design of 𝐿 should first ensure the stability of the state observer (19). We take 𝐿=𝐾^𝑇_𝜌,𝐾_𝜌=𝑅_𝐿⁻¹𝐶𝑆,𝐴𝑆+𝑆𝐴^𝑇−𝑆𝐶^𝑇𝑅_𝐿⁻¹𝐶𝑆+𝜌𝑄_𝐿=0, (31) where 𝑄_𝐿 and 𝑅_𝐿 are two given positive-definite matrices and 𝜌 is a positive scalar. Since (𝐶,𝐴) is observable, (𝐶^𝑇,𝐴^𝑇) is controllable. Thus, the observer gain, 𝐿, designed in (31) makes 𝐴−𝐿𝐶 stable, which means that the state observer (19) is stable.

On the other hand, we tackle the stability issue by first letting 𝑑=0. From (19), (21), and (22), we have Δ𝑥̇=(𝐴−𝐿𝐶)Δ𝑥+𝐵𝑑_𝑒. (32)

Combining (24) and (10) yields 𝑑_𝑒=−𝐵⁺𝐿𝐶Δ𝑥+𝑑_𝑒. (33)

From (32) and (24), we obtain the transfer function from 𝑑_𝑒 to 𝑑_𝑒: 𝐺(𝑠)=1−𝐵⁺𝐿𝐶[𝑠𝐼−(𝐴−𝐿𝐶)]⁻¹𝐵=𝐵⁺(𝑠𝐼−𝐴)[𝑠𝐼−(𝐴−𝐿𝐶)]⁻¹𝐵. (34)

Note that the transfer function from 𝑑_𝑒 to 𝑑_𝑒 is 𝐹(𝑠) (Figure 2). The small-gain theorem [23] tells us that the condition ‖𝐺𝐹‖_∞<1 (35) must be satisfied to guarantee the stability of the control system, where ‖𝐺𝐹‖_∞∶=sup_{0≤𝜔≤∞}𝜎_max[𝐺(𝑗𝜔)𝐹(𝑗𝜔)] and 𝜎_max(⋅) is the maximum singular-value function.

Figure 2 

Stability condition (35).

Since the number of inputs of the plant (𝐴^𝑇,𝐶^𝑇,𝐵^𝑇) is not less than the number of outputs, and since 𝐴^𝑇−𝐶^𝑇𝐾_𝜌 is stable, according to [24, Theorems 1 and  3], we have lim_𝜌→∞𝑠𝐼−𝐴−𝐾^𝑇_𝜌𝐶⁻¹𝐵^𝑇=0. (36)

Note that [𝑠𝐼−(𝐴−𝐾^𝑇_𝜌𝐶)]⁻¹𝐵 is part of 𝐺(𝑠). So, for a given 𝐹(𝑠) in (26), the observer gain, 𝐿, designed in (31) makes the condition (35) true provided that 𝜌 is large enough.

4. Numerical Example

We applied the EID-based tracking control strategy to a two-link rigid robot manipulator (Figure 3) to demonstrate its validity.

Figure 3 

Two-link robot manipulator.

The simulations employed the parameters (Table 1) of a PUMA 560 manipulator [7]. The matrices for the dynamics of the nominal model are 𝑀₀𝑞=𝑎₁+𝑎₂+2𝑎₃cos𝑞₂𝑎₂+𝑎₃cos𝑞₂𝑎₂+𝑎₃cos𝑞₂𝑎₂,𝐻₀𝑞,𝑞̇=−𝑎₃2𝑞̇₁+𝑞̇₂𝑞̇₂sin𝑞₂𝑎₃𝑞̇²₁sin𝑞₂,𝑁₀𝑞=−𝑏₁sin𝑞₁−𝑏₂sin𝑞₁+𝑞₂−𝑏₂sin𝑞₁+𝑞₂, (37) where 𝑎₁∶=𝑚₁𝐿²_𝑐1+𝑚₂𝐿²₁+𝐽₁,𝑎₂∶=𝑚₂𝐿²_𝑐2+𝐽₂,𝑎₃∶=𝑚₂𝐿₁𝐿_𝑐2,𝑏₁∶=𝑚₁𝐿_𝑐1+𝑚₂𝐿₁𝑏𝑔,₂∶=𝑚₂𝐿_𝑐2𝑔, (38)𝑞₁ is the angle of the first link relative to the vertical, 𝑞₂ is the angle of the second link relative to the first link, 𝑚_𝑖 is the mass of the 𝑖th link (𝑖=1,2), 𝐿_𝑖 is the length of the 𝑖th link (𝑖=1,2), 𝐿_𝑐𝑖 is the distance from the 𝑖th joint to the center of mass (COM) of the 𝑖th link (𝑖=1,2), 𝐽_𝑖 is the moment of inertia around the COM of the 𝑖th link (𝑖=1,2), 𝜏_𝑖 is the torque applied to the 𝑖th joint (𝑖=1,2), and 𝑔 is the gravitational constant (9.80665 m/s²).

Table 1 

Parameters of the two-link manipulator for simulations.

Let the desired trajectory 𝑞_𝑟=[𝑞_1𝑟,𝑞_2𝑟]^𝑇 be 𝑞_1𝑟𝑞=0.5cos𝑡+0.2sin3𝑡,_2𝑟=−0.2sin2𝑡−0.5cos𝑡. (39)

First, when 𝑑=0, we choose 𝐾_𝑃=1001,𝐾_𝐷=2001 (40) for (7) and (9). Figure 4 shows the tracking control results for the initial condition 𝑞₁,𝑞₂,𝑞̇₁,𝑞̇₂^𝑇=[𝜋,0,0,0]^𝑇. (41) Notice that the actual trajectories converge to the desired trajectories in less than 10 seconds.

(a)

(b)

(c)

(d)

Figure 4 

Tracking control results without uncertainties or disturbances.

Next, we consider the uncertainties in the physical parameters by letting 𝑚₁ and 𝐿_𝑐2 be 5% larger than their nominal values, letting 𝑚₂ and 𝐿_𝑐1 be 5% smaller than their nominal values, and letting 𝐽₁ and 𝐽₂ be 10% larger than their nominal values. We also added two types of torques to the joints of the manipulator: (a) a viscous friction torque, 𝜏_𝑓=[𝜏_1𝑓,𝜏_2𝑓]^𝑇: 𝜏_1𝑓=0.5𝑞̇₁,𝜏_2𝑓=0.5𝑞̇₂, (42) and (b) an external disturbance torque, 𝜏_ext=[𝜏_1ext,𝜏_2ext]^𝑇 (Figure 5): 𝜏_1ext(),𝜏=0.5sin4𝜋𝑡+cos2𝜋𝑡_2ext=0.5(sin𝜋𝑡+cos0.5𝜋𝑡). (43)

(a)

(b)

Figure 5 

Disturbance to torque of joints added to the two-link manipulator.

The simulation results (Figure 6) for (5), (7), (40), and (41) show that the tracking performance was much worse, and a large steady-state tracking error appeared. That means that control law (7) by itself does not force the robot manipulator to track the desired trajectory when uncertainties and disturbances are present.

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(b)

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(d)

Figure 6 

Simulation results for control law (7) when uncertainties and disturbances are present.

Finally, we applied the EID-based control strategy. We chose 𝐵=[10,10]. It is easy to verify that (𝐴,𝐵) is controllable. Letting all of the poles of 𝐴+𝐵𝐾_𝑓 be −2.5 yields the feedback gain 𝐾_𝑓=[−2.3625,−1.4437,−1.8563,1.1563]. (44) The design parameters for (26) and (31) were chosen to be 𝑇=0.1,𝑄_𝐿=10𝐼₄,𝑅_𝐿=50𝐼₂,𝜌=10⁶. (45)

That gave 𝐿=447.44630104.107800447.62450183.8319^𝑇,‖𝐺𝐹‖_∞=0.9186<1. (46)

The simulation results (Figure 7) for (5), (10), (28), (40), (41), and (45) show that the tracking performance was much better than that in Figure 6. The EID-based controller (28) almost completely rejects the effect of the uncertainties and disturbances. So, the perturbed manipulator system (3) precisely tracks the desired trajectory. Since measurement noise is very common in actual control engineering applications, we also added white noise (peak value: ±0.2) to the measured 𝑞. The simulation results (Figure 8) show that the robot manipulator precisely tracks the desired trajectory even in this case.

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(b)

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Figure 7 

Simulation results for EID-based control when uncertainties and disturbances are present.

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Figure 8 

Simulation results for EID-based control when uncertainties, disturbances, and measurement white noise are present.

5. Conclusion

This paper presents an EID-based control strategy that solves the tracking control problem for a robot manipulator with unknown uncertainties and disturbances. It uses only joint position measurements in the design of the tracking controller. The controller has two parts: one part uses an exact linearization technique to guarantee the asymptotical stability of the nominal model and the other is based on the idea of EID, which compensates for the effects of parameter uncertainties and exogenous disturbances. The combination makes a robot manipulator precisely track the desired trajectory. Simulation results show the validity of this control strategy.

Acknowledgments

This work was supported in part by the National Science Foundation of China under Grants nos. 61304023, 61273012 and 61074112, by the Project of Shandong Province Higher Educational Science and Technology Program under Grant nos. J12LI58 and J13LI11, and by the Applied Mathematics Enhancement Program of Linyi University. The authors declare that there is no conflict of interests regarding the publication of this paper.