Intelligent Hybrid Control Strategy for Trajectory Tracking of Robot Manipulators

Abstract

We address the problem of robust tracking control using a PD-plus-feedforward controller and an intelligent adaptive robust compensator for a rigid robotic manipulator with uncertain dynamics and external disturbances. A key feature of this scheme is that soft computer methods are used to learn the upper bound of system uncertainties and adjust the width of the boundary layer base. In this way, the prior knowledge of the upper bound of the system uncertainties does need not to be required. Moreover, chattering can be effectively eliminated, and asymptotic error convergence can be guaranteed. Numerical simulations and experiments of two-DOF rigid robots are presented to show effectiveness of the proposed scheme.

1. Introduction

Robot control has been an active research field during the past decade. A main concern of the control task is to find an effective controller to achieve accurate tracking of desired motions. However, robotic manipulators have to face many uncertainties in their dynamics, such as payload parameter, friction, and disturbance. Yet, the work using PD-plus-feedforward on robust control of robot [1–6] shows great attraction to researches. In [1], a PD-plus-feedforward controller is firstly proposed for robot manipulators which guarantees exponential convergence. The adaptive version of this controller is addressed later [2] considering the robustness of velocity measurement. However, the main drawback of them is that they require high controller gains in order to overcome the uncertainty in the initial parameter errors. Combining the ideas of [1, 3], Berghuis [4] put forward a modified plan ensuring global exponential convergence. Wen [5] treats the analysis of a robot controller based on energy-like Lyapunov function, which leads to a new class of global stability control design, but is still a high-gain method. Kelly [6] designs a class of PD-plus-feedforward controller on the assumption that parameters are precisely known, nevertheless obtaining local but not global stability results. Unfortunately, all above works do not consider the external disturbances and unmodeled dynamics.

Recently, considerable research efforts have been directed towards applying intelligent control methods to robot control [7–14]. In [9–11], intelligent controllers are proposed for the compensation of effects yielded by nonlinearities and uncertainties. Some theses [8, 12, 13] use soft computer methods to approximate the whole robot dynamics rather than its nonlinear components as done by static intelligent means. Man [7] and Qi [15] make use of intelligent systems to replace the constant switching control gain and the width of the boundary layer. However, [7] has the shortcoming that the inverse of inertia matrix needs to be calculated.

In this note, motivated by the PD-plus-feedforward control idea and Man's work [7], a new intelligent adaptive robust control scheme is proposed for robot manipulators with model uncertainties and external disturbance. The novel control scheme proposed in this paper contains a PD-plus-feedforward part and an intelligent robust compensational part. As compared with afore-mentioned results, not only does we suppose that the mathematical model of robot system is totally unknown, but we do not require calculating the inverse of inertia matrix. Moreover, the parameter of the proposed dynamic compensator is adaptively updated by using the output of intelligent system that is used to learn the unknown upper bound of system uncertainties, and a fuzzy logic is used to adjust the width of the boundary based on the state. In this way, the vibrancy of the controller will be effectively reduced. Application to a two-link robotic manipulator is presented. Numerical simulations and experiments are proposed to show the excellent tracking performance of our controller.

2. Preliminaries

In this section, we firstly give some descriptions of intelligent systems, and after that, present the robot manipulator model and some dynamic properties which are crucial to our work.

2.1. Description of Fuzzy-Logic Systems

The basic configuration of the fuzzy system is constructed from the fuzzy if-then rules using some specific inference, fuzzification, and defuzzification strategies [16]. Let 𝑥=(𝑥₁,…,𝑥_𝑁)^T∈𝑈⊂𝑅^N and 𝑦∈𝑉⊂𝑅 be the input and output of the fuzzy system. The fuzzifier maps a crisp point in 𝑈 into a fuzzy set in 𝑈. The fuzzy rule base consists of a collection of fuzzy if-then rules.

𝑅^(𝑙): If 𝑥₁ is 𝐹^𝑙₁ and ⋯ and 𝑥_𝑁 is 𝐹^𝑙_𝑁, then 𝑦 is 𝐺^𝑙, in which 𝐹^𝑙_𝑖 and 𝐺^𝑙 are fuzzy sets, and 𝑙=1,…,𝑀, where 𝑀 denotes the number of fuzzy if-then rules. The fuzzy inference engine performs a mapping from fuzzy sets in 𝑈 to fuzzy sets in 𝑉. The defuzzifier maps a fuzzy set in 𝑉 to a crisp point in 𝑉. A more detailed description of these systems can be found in [16].

The fuzzy systems with center-average defuzzifier, product inference, and singleton fuzzifier can be expressed as in the following form: ∑𝑦(𝑥)=^𝑀𝑙=1𝜃_𝑙(∏^𝑁_𝑖=1𝜇_𝐹𝑙𝑖(𝑥_𝑖))∑^𝑀_𝑙=1(∏^𝑁_𝑖=1𝜇_𝐹𝑙𝑖(𝑥_𝑖)), (1) where 𝜇_𝐹𝑙𝑖(⋅) is the numbership function of the fuzzy set 𝐹^𝑙_𝑖 and 𝜃_𝑙 is the point at which 𝜇_𝐺𝑙 achieves its maximum value (it is assumed here that 𝜇_𝐺𝑙(𝜃_𝑙)=1). By the universal approximation theorem [16], for any given real continuous function 𝑓(𝑥) on a compact subset 𝑈⊂𝑅^N and arbitrary 𝜖>0, there exists a fuzzy system 𝑦(𝑥) in the form of (1) such that max_𝑥∈𝑈‖𝑓(𝑥)−𝑦(𝑥)‖<𝜖.

2.2. Description of Neural-Network Systems

The basic configuration of the neural network system is implemented by using massive connections among processing units. Let 𝑥∈𝑈⊂𝑅^N be the input of the neural network system. A two-layer network has a network output given by [17, 18] 𝑦(𝑥)∶=^𝑀_𝑙=1𝜃_𝑙𝐻_𝑙(^𝑀_𝑗=1𝜔_𝑗𝑙𝑥_𝑗+𝑚_𝑙), (2) where 𝑀 is the number of hidden neurons, 𝐻_𝑙(⋅) denotes the activation function which must be a nonconstant, bounded, and monotonically increasing continuous function (e.g., hyperbolic tangent function), 𝜔_𝑗𝑖 is the first-to-second layer interconnection weights, 𝑚_𝑙 is the biases, and 𝜃_𝑖 is the second-to-output interconnection weights.

The neural network system is also a popular class of function approximators that can be used in a similar setting as the fuzzy logic system described above. That is, for any given real continuous function 𝑓(𝑥) and arbitrary 𝜖>0, there exists a neural network system 𝑦(𝑥) in the form of (2) such that max_𝑥∈𝑈‖𝑓(𝑥)−𝑦(𝑥)‖<𝜖.

2.3. Description of Robot Manipulator Model

The basics of robot dynamics and control are sufficiently well known by now that we will be brief in our derivation of the control algorithm. Thus, given the Euler-Lagrange dynamic equations for an 𝑛-link robot [19] ̈̇̇̇̇𝑀(𝑞)𝑞+𝑁(𝑞,𝑞)=𝜏,𝑁(𝑞,𝑞)=𝐶(𝑞,𝑞)𝑞+𝐺(𝑞)+𝐹_𝑑(𝑞)̇+𝐹_𝑠(𝑞)̇+𝜏_𝑑(𝑡,𝑞,𝑞)̇, (3) where 𝑞,𝑞,̇𝑞̈∈𝑅^𝑛 are the joint position, velocity, and acceleration vectors, respectively; 𝑀(𝑞) denotes the inertia matrix; 𝐶(𝑞,𝑞)̇ expresses the coriolis, centripetal matrix; 𝐺(𝑞) is the gravity vector; 𝐹_𝑑,𝐹_𝑠(𝑞)̇∈𝑅^𝑛×𝑛 represent the dynamic friction coefficient matrix and static friction vector, respectively; 𝜏_𝑑(𝑡,𝑞,𝑞)̇ is the vector of disturbances and unmodeled dynamics; 𝜏 is the control vector representing the torque exerting on joints. For our purposes, we assume (3) to have the following properties.

Property 1. 𝑀(𝑞) is a positive symmetric matrix defined by 𝑀_𝑚≤‖𝑀(𝑞)‖≤𝑀_𝑀 with 𝑀_𝑚,𝑀_𝑀>0 being known constants.

Property 2. 𝐶(𝑞,𝑞)̇ defined by using the Christoffel symbols satisfies that
(a)𝑀̇(𝑞)−2𝐶(𝑞,𝑞)̇ is Skew symmetry;(b)‖𝐶(𝑞,𝑞)̇‖≤𝐶_𝑀‖𝑞‖̇ and 𝐶(𝑞,𝑥)𝑦=𝐶(𝑞,𝑦)𝑥,𝑥,𝑦∈𝑅ⁿ,𝐶_𝑀>0.

Property 3. Linear parameterization of the local dynamics of each subsystem: 𝑀(𝑞)𝑞̈+𝐶(𝑞,𝑞)̇𝑞̇+𝑔(𝑞)=𝑌(𝑞,𝑞,̇𝑞)̈𝜃, (4) where 𝜃 is a constant 𝑝-dimensional vector of inertia parameters and 𝑌 is an 𝑛×𝑝 matrix called regressor, which contains known functions and is uniformly continuous in its arguments.

Moreover, without losing generality, the following assumptions on model (3) are made.

Assumption. The bounding function of the external disturbances and unmodeled dynamics 𝜏_𝑑(𝑡,𝑞,𝑞)̇ is assumed to be ‖𝜏_𝑑(𝑡,𝑞,𝑞)̇‖≤𝑐₀+𝑐₁‖𝑞‖+𝑐₂‖𝑞‖²+𝑐₃‖𝑞‖̇+𝑐₄‖𝑞‖̇², (5) where 𝑐_𝑖(𝑖=0,…,4) are positive constants.

Assumption. The desired trajectory 𝑞_𝑑,𝑞̇_𝑑, and 𝑞̈_𝑑 are uniformly bounded.

3. Ntelligent Adaptive Robust Controller Designs

We now come to the question of trajectory tracking of robot dynamic system (2). Setting ^0𝑥0𝑞^02𝑑𝑐=̇𝑞−𝑞,𝑞^{0𝑥002𝑑𝑐}=𝑞̇−𝑞,̇̇𝑠=𝑞^{0𝑥002𝑑𝑐}+Λ𝑞^{0𝑥002𝑑𝑐},𝑞̇_𝑟=𝑞̇_𝑑−Λ𝑞^{0𝑥002𝑑𝑐},𝑞̇₀=𝑞̇−Λ𝑞^{0𝑥002𝑑𝑐}, with Λ a positive definite matrix. Let us consider the following robust controller: 𝜏=𝑌(𝑓₁,…,𝑓₅)𝜃−𝐾_𝜈̇‌_{0𝑥002𝑑𝑐}𝑞−𝐾𝑞_𝑝^{0𝑥002𝑑𝑐}+𝑢(𝑡),𝑌(𝑓₁,…,𝑓₅)𝜃=𝑀(𝑓₁)𝑞̈+𝐶(𝑓₂,𝑓₃)𝑓₄+𝑓₅,̇𝜃=Proj[−Γ⁻¹𝑌^T(𝑓₁,…,𝑓₅)𝑠], (6) where (⋅)̂ denotes the estimate of (⋅); 𝑓₁,𝑓₂,𝑓₅∈{𝑞,𝑞_𝑑},𝑓₃,𝑓₄∈{𝑞,̇𝑞̇_𝑑,𝑞̇_𝑟,𝑞₀}; Γ is a positive definite matrix; Proj[•] expresses a discontinuous projection mapping [20], and particularly, in our paper, it is defined as (7); 𝑢(𝑡) represents the nonlinear control component used to stabilize the closed-loop system and enhance the system robustness against the external disturbances and unmodeled dynamics. The schematic diagram of overall control system is shown in Figure 1 (set 𝑓=₄^def𝑞̇_𝑑 for ease of illustration). In the following part, we consider how to design the robust compensator 𝑢(𝑡) so that the controller (6) can ensure the asymptotic convergence of tracking error: Proj[𝜔_𝑖𝜔]=^𝑖,𝜃_𝑖<𝜃_𝑖<𝜃_𝑖,𝜃_𝑖≥𝜃_𝑖,𝜔_𝑖𝜃≤0,_𝑖≤𝜃_𝑖,𝜔_𝑖𝜃≥0,0,^𝑖≥𝜃_𝑖,𝜔_𝑖𝜃>0,_𝑖≤𝜃_𝑖,𝜔_𝑖<0, (7) where 𝜔_𝑖=−Γ_𝑖⁻¹∑^𝑟_𝑗=1𝑌_𝑗𝑖𝑠_𝑗,𝜃_𝑖<𝜃_𝑖(0)<𝜃_𝑖,𝑖=1,2,…,𝑝.

Figure 1 

Schematic diagram of proposed PD plus feedforward robust control.

Substituting (6) into (3) yields the error dynamic system 𝐸̇=𝐴𝐸+𝐵(𝑢(𝑡)+𝑌(𝑓₁,…,𝑓₅)𝜃^{0𝑥002𝑑𝑐}+𝑁(𝑡,𝑓₁,…,𝑓₅)), (8) where 𝐸=[𝑞^{0𝑥002𝑑𝑐},_0𝑥002̇_𝑑𝑐𝑞]^T,𝐵=[0,𝑀⁻¹(𝑞)]^T,^0𝑥0𝜃^02𝑑𝑐=𝜃−𝜃, and 𝐴=0𝐼−𝑀⁻¹(𝑞)(𝐾_𝑝+𝜆𝐶(𝑞,𝑞)̇)−𝑀⁻¹(𝑞)(𝐾_𝑣̇,+𝐶(𝑞,𝑞)+𝜆𝑀(𝑞))𝑁(𝑡,𝑓₁,…,𝑓₅)=𝑌(𝑓₁,…,𝑓₅)𝜃−[𝑀(𝑞)(𝑞̈_𝑑̇−𝜆𝑞^{0𝑥002𝑑𝑐})+𝐶(𝑞,𝑞(̇𝑞̇_𝑑−𝜆𝑞^{0𝑥002𝑑𝑐}))+𝐺(𝑞)+𝐹_𝑠(𝑞)̇+𝜏_𝑑(𝑡,𝑞,𝑞)̇]. (9)

Due to the model uncertainty and external disturbance, 𝑁(𝑡,𝑓₁,…,𝑓₅) in (8) is an uncertain unknown nonlinear function. In order to obtain the upper bound of 𝑁(𝑡,𝑓₁,…,𝑓₅), inspired by the idea of [21], we give the following lemma.

Lemma 1. Using Properties 1–3 and Assumptions 1-2, one obtains the upper bound of 𝑁(𝑡,𝑓₁,…,𝑓₅) as follows: 𝑁(𝑡,𝑓₁,…,𝑓₅)≤𝑁(𝑡)=𝑑₁+𝑑₂‖𝐸‖+𝑑₃‖𝐸‖²+𝑑₄‖𝐸‖³ (10) for some 𝑑_𝑖>0,𝑖=1,2,3,4.

Proof. Using Properties 1, 2 and Assumption 1, we have 𝑁(𝑡,𝑓₁,…,𝑓₅)≤𝑁(𝑡)=𝜉₁+𝜉₂‖𝑓₁−𝑞‖+𝜉₃‖𝑞̇−𝑓₃‖⋅‖𝑓₄‖+𝜉₄‖𝑞−𝑓₂‖⋅‖𝑞‖̇⋅‖𝑓₄‖+𝜉₅‖𝑓₄‖⋅‖𝑞‖̇+𝜉₆‖𝑞‖̇⋅‖𝑓₄−𝑞̇_𝑑‖+𝜉₇‖𝑓₅−𝑞‖+𝜉₈‖_0𝑥002̇_𝑑𝑐𝑞‖+𝜉₉‖𝑞‖̇⋅‖𝑞^{0𝑥002𝑑𝑐}‖+‖𝜏_𝑑(𝑡,𝑞,𝑞)̇‖, (11) where 𝜉_𝑖(𝑖=1,…,9) are known constants.
Due to the definition of 𝑓_𝑖(𝑖=1,…,5) and Assumption 2, the following inequalities can be obtained immediately: ‖𝑓_𝑖−𝑞‖≤‖𝑞^{0𝑥002𝑑𝑐}‖≤‖𝐸‖,𝑖=1,2,5,‖𝑓_𝑖̇−𝑞‖≤‖𝑞^{0𝑥002𝑑𝑐}‖+‖Λ𝑞^{0𝑥002𝑑𝑐}‖≤(1+Λ)‖𝐸‖,Λ=𝜆_max(Λ),𝑖=3,4,‖𝑓₄‖≤𝑏₁+𝑏₂‖̇̇‖𝐸‖,𝑞‖≤‖𝑞^{0𝑥002𝑑𝑐}‖+‖𝑞̇_𝑑‖≤𝑏₃+‖𝐸‖,‖𝑞‖≤‖𝑞^{0𝑥002𝑑𝑐}‖+‖𝑞_𝑑‖≤𝑏₄+‖𝐸‖, (12) where 𝑏_𝑖(𝑖=1,2,3,4) are known constants, 𝜆_max denotes the maximum eigenvalue. Substituting (12) into (11) yields (10).

Remark 2. There are a lot of literature [22, 23], even when a small perturbation exists in system, pointing out that parameter drift occurs and devastates the total control scheme. Therefore, a nonlinear projection mapping Proj[•] is led into our controller for purpose of guaranteeing globally asymptotical converges and distinct transient performance.

Remark 3. According to the differential choice of 𝑓_𝑖,𝑖=1,…,5, as many as 2³×4²=128 kinds of global robust stability controllers could be obtained. It can be seen that our control scheme encompasses a wide range of circumstances.

Remark 4. From error (2), it is known that the key to synthesis controller (3) is how to design a suitable adaptive control compensator 𝑢(𝑡) so that the effect of uncertain function 𝑁(𝑡,𝑓₁,…,𝑓₅) can be effectively eliminated.

3.1. Routine Robust Compensator

When knowing the upper bound function 𝑁(𝑡) of concentrated uncertainty, the following continued dynamic compensator can be designed as follows: 𝑢(𝑡)=−𝑁(𝑡)𝑠‖𝑠‖𝑁(𝑡)+𝛼‖𝐸‖², (13) where constant 𝛼>0 satisfies 𝛽∶=𝜆_min(𝑄)−𝛼>0,𝜆_min(⋅) denotes the minimum eigenvalue of matrix, respectively, 𝑄 is a symmetrical positive matrix chosen as 𝑄=𝜆𝐾_𝑝00𝐾_𝜈. (14) Apparently, 𝜆_min(𝑄)=min(𝜆_min(𝜆𝐾_𝑝),𝜆_min(𝐾_𝜈)).

The following theorem gives the dynamic performance of closed loop system based on controller (6) with compensator (13).

Theorem 1. Consider the error dynamic model (8) of the robot system (3) with Assumptions 1-2. If the dynamic compensator of controller (6) is given by expression (13), then, the close-loop system is exponentially stable.

Proof. Choose the Lyapunov function as 1𝑉=2𝐸^T1𝑃𝐸+2^{0𝑥002𝑑𝑐}𝑞^T𝑃𝑞^{0𝑥002𝑑𝑐}+12^{0𝑥002𝑑𝑐}𝜃^TΓ𝜃^{0𝑥002𝑑𝑐}, (15) with 𝑃 the unique symmetrical positive matrix solution to Lyapunov function 𝐴^T𝑃+𝑃𝐴=−𝑄, (16) where 𝑄 is defined as (14).
Differentiating 𝑉 with respect to time along solution trajectory of (8), and using Property 3, we get 𝑉̇=𝐸^T𝑃𝐸̇+12𝐸^T𝑃̇𝐸+𝑞^{0𝑥002𝑑𝑐T}𝑃_0𝑥002̇_𝑑𝑐𝑞+𝜃^{0𝑥002𝑑𝑐T}Γ_0𝑥002̇_𝑑𝑐𝜃=𝐸^T𝑃(𝐴𝐸+𝐵(𝑢(𝑡)+𝑌(𝑓₁,…,𝑓₅)𝜃^{0𝑥002𝑑𝑐}+𝑁(𝑡,𝑓₁,…,𝑓₅+1)))2𝐸^T𝑃̇𝐸+𝑞^{0𝑥002𝑑𝑐T}𝑃_0𝑥002̇_𝑑𝑐𝑞+𝜃^{0𝑥002𝑑𝑐T}Γ_0𝑥002̇_𝑑𝑐𝜃=−𝐸^T𝑄𝐸+𝐸^T00−𝐾_𝑝0𝐸+𝑞^{0𝑥002𝑑𝑐T}𝑃_0𝑥002̇_𝑑𝑐𝑞+𝜃^{0𝑥002𝑑𝑐T}Γ_0𝑥002̇_𝑑𝑐𝜃+𝐸^T𝐼𝜆𝐼(𝑢(𝑡)+𝑌(𝑓₁,…,𝑓₅)𝜃^{0𝑥002𝑑𝑐}+𝑁(𝑡,𝑓₁,…,𝑓₅))=−𝐸^T𝑄𝐸+𝑠^T(𝑢(𝑡)+𝑁(𝑡,𝑓₁,…,𝑓₅))≤−𝜆_min(𝑄)‖𝐸‖²+𝑠^T𝑢(𝑡)+‖𝑠‖𝑁(𝑡)=−𝜆_min(𝑄)‖𝐸‖²−𝑁(𝑡)²‖𝑠‖²‖𝑠‖𝑁(𝑡)+𝛼‖𝐸‖²+‖𝑠‖𝑁(𝑡)≤−(𝜆_min(𝑄)−𝛼)‖𝐸‖², (17) that is, 𝑉̇≤−𝛽‖𝐸‖². (18) From expression (15), the following inequality can be obtained: (𝜆_min(𝑃)+𝜆_min(𝐾_𝑝))‖𝐸‖²≤2𝑉≤(𝜆_max(𝑃)+𝜆_max(𝐾_𝑝))‖𝐸‖². (19) Setting 𝜆₁=𝜆_min(𝑃)+𝜆_min(𝐾_𝑝),𝜆₂=𝜆_max(𝑃)+𝜆_max(𝐾_𝑝), (20) and then from expressions (18) and (19), we get 𝜆‖𝐸‖≤(²𝜆₁)^1/2‖𝐸(𝑡₀𝛽)‖exp(−𝜆₂𝑡). (21) Hence, the close loop system is globally exponentially stable.

Remark 5. In dynamic compensator (13), the prior knowledge on the upper bound 𝑁(𝑡) of system uncertainty is required. However, in general, precise bound on system uncertainty is difficult to compute; and moreover, the conventional estimating method may lead to a conservative design. Therefore, we introduce a new dynamic compensator based on intelligent systems due to the excellent nonlinear mapping.

3.2. Intelligent Adaptive Robust Compensator

Consider the upper bound 𝑁(𝑡) in (10) that cannot be directly available in the control design. An adaptive universal approximation system 𝑁(𝑥_𝐸,Θ)∈𝑅^𝑚 with input vector 𝑥_𝐸∈Ω for some compact set Ω∈𝑅^𝜅,𝜅>0, is proposed here to approximate the behavior of 𝑁(𝑡) where 𝑥_𝐸 is a function of 𝐸 and 𝑡, and Θ is a vector containing the tunable approximation parameters. Let 𝑁_𝑖(𝑥_𝐸,Θ_𝑖) denote the 𝑖 th component of 𝑁(𝑥_𝐸,Θ). As in many previous studies [16–18], the linearly parameterized fuzzy model and neural network model are adopted in the approximation procedure, and so 𝑁_𝑖(𝑥_𝐸,Θ_𝑖) can be expressed as 𝑁_𝑖(𝑥_𝐸,Θ_𝑖)=ℎ^T_𝑖(𝑥_𝐸)Θ_𝑖,𝑖=1,…,𝑚, (22) where Θ_𝑖=[Θ_𝑖1,…,Θ_𝑖𝑚𝑖]^T for some 𝑚_𝑖>0 is a parameter vector and ℎ_𝑖(𝑥_𝐸)=[ℎ_𝑖1(𝑥_𝐸),…,ℎ_𝑖𝑚𝑖(𝑥_𝐸)]^T is a regressive vector with the regressor defined for the adaptive fuzzy approximator as follows: ℎ_𝑖𝑙(𝑥_𝐸∏)=^𝜅𝑗=1𝜇_𝑁^𝑙_𝑗(𝑥_𝐸𝑗)∑^𝑀_𝑙=1(∏^𝑀_𝑗=1𝜇_𝑁^𝑙_𝑗(𝑥_𝐸𝑗)),𝑙=1,…,𝑚_𝑖, (23) and for the adaptive neural network approximator ℎ_𝑖𝑙(𝑥_𝐸𝑒)=^{𝜎𝑖𝑙(𝑥𝐸)}−𝑒^{−𝜎𝑖𝑙(𝑥𝐸)}𝑒^{𝜎𝑖𝑙(𝑥𝐸)}−𝑒^{−𝜎𝑖𝑙(𝑥𝐸)},𝑙=1,…,𝑚_𝑖. (24) The membership functions 𝜇_𝐻𝑙𝑗(𝑥_𝐸𝑗) in the fuzzy system and the activation functions 𝜎_𝑖𝑙(𝑥_𝐸) in the neural network system are specified beforehand.

Consequently, the totally adaptive approximation system 𝑁(𝑥_𝐸,Θ) can be expressed as 𝑁(𝑥_𝐸,Θ)=𝑊(𝑥_𝐸)Θ, (25) where 𝑊(𝑥_𝐸)=diag[ℎ^T₁(𝑥_𝐸),…,ℎ^T_𝑚(𝑥_𝐸)] denotes a basis matrix and Θ=[Θ^T₁,…,Θ^T_𝑚]. According to the universal approximation theorem [16–18], there is an optimal approximation parameter Θ such that 𝑁(𝑥_𝐸,Θ) can approximate 𝑁(𝑡) as best as possible. Therefore, the upper bound 𝑁(𝑡) can be expressed as 𝑁(𝑡)=𝑊(𝑥_𝐸)Θ+Δ𝑁(𝑡), (26) where Δ𝑁(𝑡) denotes the optimal approximation error.

Remark 6. Since both adaptive fuzzy system and adaptive neural network system possess the property of universal approximation, the structure of upper bound 𝑁(𝑡) can be allowed to be entirely unknown. On the other hand, suppose that the structure of 𝑁(𝑡) is known, then according to the design philosophy of classical adaptive control, 𝑁(𝑡) can be expressed as 𝑁(𝑡)=𝑊(𝐸,𝑡)Θ+Δ𝑁(𝑡), that is, as in the form of (10), in which 𝑁(𝑡) is a regressor matrix of known functions, Θ is a constant vector with components depending on the plant parameters, and Δ𝑁(𝑡) denotes the residual part that cannot be linearly parameterized.

For further analysis, the following assumptions are needed.

Assumption 3. Given an arbitrary small positive constant 𝜔, there exists an optimal weight vector Θ^∗ so that the approximation error Δ𝑁(𝑡) of intelligent systems satisfies |Δ𝑁(𝑡)|=|Θ^∗𝑇𝑊(𝑥_𝐸)−𝑁(𝑡)|<𝜔. (27)

Assumption 4. For 𝜔 in expression (27), the upper bound 𝑁(𝑡) of system uncertainty 𝑁(𝑡,𝑓₁,…,𝑓₅) satisfies 𝑁(𝑡)−‖𝑁(𝑡,𝑓₁,…,𝑓₅)‖>𝜔. (28)

Now, we use intelligent systems (25) to design the adaptive robust compensator 𝑢(𝑡), and get the following theorem.

Theorem 2. Consider the error equation (8) for the robot manipulator system (3) with Assumptions 1–4. If the compensator 𝑢(𝑡) in controller (6) is designed as 𝑢(𝑡)=−(𝑊(𝑥_𝐸)Θ)²𝑠‖𝑠‖(𝑊(𝑥_𝐸)Θ)+𝛼‖𝐸‖², (29) and the weight vector Θ is updated by the following adaptive mechanism: Θ̇=𝜂𝑊(𝑥_𝐸)‖𝑠‖, (30) with 𝜂>0 the adaptive rate, then controllers (6), (29) can ensure that the tracking error of robot system asymptotically converges to zero.

Proof. Define the following Lyapunov function 1𝑉=2𝐸^T1𝑃𝐸+2^{0𝑥002𝑑𝑐}𝑞^T𝑃𝑞^{0𝑥002𝑑𝑐}+12^{0𝑥002𝑑𝑐}𝜃^TΓ^{0𝑥002𝑑𝑐}1𝜃+2^{0𝑥002𝑑𝑐}Θ^T𝜂Θ^{−10𝑥002𝑑𝑐}, (31) where ^0𝑥0Θ^02𝑑𝑐=Θ^∗−̇Θ,Θ^{0𝑥002𝑑𝑐}=−Θ̇, matrix 𝑃 is defined by (15).
Differentiating 𝑉 with respect to time along trajectory of error equation (8), similar to the proof of Theorem 1, we get 𝑉̇=−𝐸^T𝑄𝐸+𝑠^T(𝑢(𝑡)+𝑁(𝑡,𝑓₁,…,𝑓₅))+Θ^{0𝑥002𝑑𝑐T}𝜂⁻¹̇‌_{0𝑥002𝑑𝑐}Θ≤−𝐸^T𝑄𝐸+𝑠^T𝑢(𝑡)+‖𝑠‖⋅‖𝑁(𝑡,𝑓₁,…,𝑓₅)‖+Θ^{0𝑥002𝑑𝑐T}𝜂⁻¹̇‌_{0𝑥002𝑑𝑐}Θ=−𝐸^T𝑄𝐸+𝑠^T𝑢(𝑡)+‖𝑠‖⋅(‖𝑁(𝑡,𝑓₁,…,𝑓₅)‖−𝑁(𝑡))+‖𝑠‖⋅𝑁(𝑡)+Θ^{0𝑥002𝑑𝑐T}𝜂⁻¹̇‌_{0𝑥002𝑑𝑐}Θ. (32) Using both expressions (27) and (30), and considering Assumption 4, we have 𝑉̇≤−𝐸^T𝑄𝐸+𝑠^T𝑢(𝑡)+‖𝑠‖⋅(‖𝑁(𝑡,𝑓₁,…,𝑓₅)‖−𝑁(𝑡))+‖𝑠‖⋅(𝑊(𝑥_𝐸)Θ^∗T+Δ𝑁(𝑡))+(Θ^∗−Θ)^T𝜂⁻¹̇‌_{0𝑥002𝑑𝑐}Θ≤−𝐸^T𝑄𝐸+𝑠^T𝑢(𝑡)+‖𝑠‖⋅(−𝜔+Δ𝑁(𝑡))+Θ^T𝑊(𝑥_𝐸)‖𝑠‖≤−𝜆_min(𝑄)‖𝐸‖²−(𝑊(𝑥_𝐸)Θ)²‖𝑠‖²‖𝑠‖(𝑊(𝑥_𝐸)Θ)+𝛼‖𝐸‖²+Θ^T𝑊(𝑥_𝐸)‖𝑠‖≤−(𝜆_min(𝑄)−𝛼)‖𝐸‖²=−𝛽‖𝐸‖². (33) Hence, it is easily concluded that 𝐸∈𝐿₂∩𝐿_∞,Θ^{0𝑥002𝑑𝑐}∈𝐿_∞, and then 𝑠∈𝐿_∞. Furthermore, from expression (8), we obtain that 𝐸̇∈𝐿_∞. Using Barbarlat lemma, it follows that lim_𝑡→∞̇‌_{0𝑥002𝑑𝑐}𝑞=0.

In the dynamic compensator (29), 𝛼‖𝐸‖² represents the width of the boundary layer which is chosen to be a constant. The constant-width boundary layer design can reduce chattering of the control signals, but it decreases the control accuracy. The compromise between the smoothness of control signals and the accuracy of the control results is dictated by the choice of the boundary layer width. Through simulation we know that, for the control with a small boundary layer width, chattering occurs only during the transient stage when the system state is far from the origin. When the state gets closer and closer to the origin, the chattering phenomenon gradually disappears even though the boundary layer width is enough small. Based on the observations, this note proposes a method, which makes use of fuzzy boundary layer control [24].

We selected ‖𝐸‖ and ‖𝑑𝐸/𝑑𝑡‖ as the control input, and the boundary layer width ℎ as the control output. The fuzzy control rules are set as follows.

If ‖𝐸‖ is PB AND ‖𝑑𝐸/𝑑𝑡‖ is PB, then ℎ is PM.

If ‖𝐸‖ is PB AND ‖𝑑𝐸/𝑑𝑡‖ is PS, then ℎ is PB.

If ‖𝐸‖ is PS AND ‖𝑑𝐸/𝑑𝑡‖ is PB, then ℎ is PM.

If ‖𝐸‖ is PS AND ‖𝑑𝐸/𝑑𝑡‖ is PS, then ℎ is PS.

Where “PB, PM, PS" stand for positive big, positive middle, and positive small. “AND” is defined by “𝜇_𝐴 AND 𝜇_𝐵=𝜇_𝐴×𝜇_𝐵” for any two membership values 𝜇_𝐴 and 𝜇_𝐵 over the fuzzy subsets 𝐴 and 𝐵, respectively. Product-sum-gravity method of the inference technique is introduced for fuzzy logic control system.

Now the control law is proposed to be 𝑢(𝑡)=−(𝑊(𝑥_𝐸)Θ)²𝑠‖𝑠‖(𝑊(𝑥_𝐸)Θ)+ℎ. (34)

Theorem 3. Consider the error equation (8) for the robot manipulator system (3) with Assumptions 1–4. If the compensator 𝑢(𝑡) in controller (6) is designed as 𝑢(𝑡)=−(𝑊(𝑥_𝐸)Θ)²𝑠‖𝑠‖(𝑊(𝑥_𝐸)Θ)+ℎ, (35) where constant ℎ denotes the fuzzy boundary layer width, and the weight vector Θ is updated by the following adaptive mechanism: Θ̇=𝜂𝑊(𝑥_𝐸)‖𝑠‖, (36) with 𝜂>0 the adaptive rate, then controllers (6), (34) can ensure that the tracking error of robot system asymptotically converges to zero.

Proof. The proof is similar to that of Theorem 1 above, thus we choose to omit the details.

Remark 7. In compensator (13), 𝛼 is a small positive constant. Moreover, it can be seen from the above proof that feedback gain matrix 𝐾_𝑝 and 𝐾_𝜈 are only required to satisfy 𝜆_min(𝑄)−𝛼>0. Hence, the proposed control scheme does not require the higher feedback gain matrix, whose robustness is guaranteed by intelligent adaptive dynamic compensator that effectively eliminated the effects of system uncertainties.

Remark 8. It has been shown that the proposed control scheme does need to calculate the inverse of inertia matrix as it was in [7].

4. Computer Simulation

Consider (3) of a two-DOF planar robot arm, in which 𝐺(𝑞)=0 (no gravity torque) for simplicity. The entries of 𝑀(𝑞),𝐶(𝑞,𝑞)̇,𝐹_𝑑,𝐹_𝑠(𝑞)̇, and 𝜏_𝑑(𝑡,𝑞,𝑞)̇ are 𝜃𝑀(𝑞)=¹+𝜃₂+2𝜃₃cos(𝑞₂)𝜃₂+𝜃₃cos(𝑞₂)𝜃₂+𝜃₃cos(𝑞₂)𝜃₂,𝐶(𝑞,𝑞)̇=−𝜃₃sin(𝑞₂)𝑞̇₂−𝜃₃sin(𝑞₂)(𝑞̇₁+𝑞̇₂)𝜃₃sin(𝑞₂)𝑞̇₁0, (37)𝐹_𝑑=diag[2,2],𝐹_𝑠(𝑞)̇=sgn(𝑞)̇,𝜏_𝑑(𝑡,𝑞,𝑞)̇=[𝑞̇₁sin𝑡,𝑞̇₂cos𝑡]^T, and one parameterization of this robot is given by 𝑌(𝑞,𝑞,̇𝑞)̈𝜃=𝜏,𝜃∈𝑅³,𝜃₁=𝑚₁𝑙²_𝑐1+𝑚₂𝑙²₁+𝐼₁,𝜃₂=𝑚₂𝑙²_𝑐2+𝐼₂,𝜃₃=𝑚₂𝑙₁𝑙_𝑐2,𝑦₁₁=𝑞̈₁,𝑦₂₁𝑦=0,₁₂=𝑞̈₁+𝑞̈₂,𝑦₂₂=𝑞̈₁+𝑞̈₂,𝑦₁₃=cos(𝑞₂)(2𝑞̈₁)−sin(𝑞₂)(𝑞̇²₂+2𝑞̇₁𝑞̇₂𝑦),₂₃=cos(𝑞₂)𝑞̈₁+sin(𝑞₂)𝑞̇²₁. (38)

In the equations above, physical parameters and their numerical values are shown in Table 1. The reference trajectories specified are 𝑞_𝑑1𝑞=sin𝑡+0.1sin3𝑡−0.2sin4𝑡,_𝑑2=−0.1sin2𝑡−0.2sin3𝑡+0.1sin4𝑡, (39) with selection of parameters 𝑇=1ms,𝐾_𝜈=diag[20,20],𝐾_𝑝=diag[100,100],Λ=diag[5,5],Γ=diag[50,10,10]. In simulation, the nominal robot parameter is 𝜃=[2.9,1.0,0.9]^T, with choice of its bound 0≤𝜃₁≤4,0≤𝜃₂≤2,0≤𝜃₃≤2. For ease of illustration, let 𝑓₁=𝑓₂=𝑓₅=𝑞_𝑑,𝑓₃=𝑓₄=𝑞̇_𝑑.

Table 1 

For the case of adopting conventional robust controller, namely, the upper bound on the parametric uncertainty is known, we have the following simulation results as shown in Figure 2(𝛼=10) and Figure 3(𝛼=100), by the choose of the upper bound function as 𝑁(𝑡)=20+7.5‖𝐸‖+5.0‖𝐸‖².

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Figure 2 

Response using proposed conventional robust control law (𝛼=10).

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Figure 3 

Response using proposed conventional robust control law (𝛼=100).

The results of the intelligent hybrid robust control algorithm, while the upper bound is assumed to be unknown a priori, are also presented here in Figure 4(𝛼=10) and Figure 5(𝛼=100) as a comparison, with selection of the adaptive rate 𝜂=0.5.

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Figure 4 

Response using proposed intelligent robust control law (𝛼=10,𝜂=0.5).

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Figure 5 

Response using proposed intelligent robust control law (𝛼=100,𝜂=0.5).

It can be observed from the above simulation results that both the conventional robust control method and the proposed intelligent hybrid robust control algorithm have satisfying robust results, and the latter achieves more excellent tracking performance than the former, for the reason that chattering of the control signals is obviously weakened by the fuzzy boundary layer, and the control precision is improved significantly. In addition, for the conventional robust control method, simulation results indicate that larger 𝛼 (e.g., 𝛼=100) may yield better performance in attenuating the chattering of control torque than the case of smaller 𝛼 (e.g., 𝛼=10); to the proposed intelligent hybrid robust control algorithm, yet the system behavior in the case of larger 𝛼 (e.g., 𝛼=100) is almost the same as the case of smaller 𝛼 (e.g., 𝛼=10). This implies that our proposed intelligent hybrid robust control algorithm has better robust performance in its insensitivity to parameter variation than the conventional robust control method.

5. Experiments

In this section, the experimental setup and system realization are described. Then experimental data is discussed and analyzed.

5.1. Hardware

A 2-DOF direct-drive ShinMaywa 𝑅3𝐶 robot arm is used as testbed, see Figure 6. It is a rigid-link, low-friction, planar robot with two joints.

Figure 6 

High-performance robot used as testbed.

The dimensions of the robot are given in Table 2, where subindexes 1 and 2 stand for the first and second links, respectively, and 𝐿_𝑐 stands for the center of mass, Par means parameters, 𝐼 stands for inertia, and Δ defines joint limits. Joint angles 𝑞 are measured with optical encoders at a high resolution of 120 000 pulse/rad, and joint-velocity signal 𝑞̇ is estimated from position signals using a first-order filter with a constant 𝜏=10.0.

Table 2 

5.2. Firmware

A Digital Signal Processor Loughborough Sound Images Ltd. 𝐷𝑃𝐶/𝐶40𝐵 board control system was integrated on a 16-bit expansion bus slot of a 𝐷𝑋—486 personal computer. 𝑇𝑀𝑆320 floating point 𝐷𝑆𝑃 C v.1.0.1 compiler provided the programming environment. The control input is transmitted to the servomotors through the Shinmaywa servosystem, which powers the pulse-wide modulators motor drives at each joint.

The performance of the proposed controller is shown in comparison with the neurofuzzy adaptive controller [12] and fuzzy supervisory sliding-mode controller [14]. Experiments are carried out at high velocities in order to show the performance of the system at inertial dominated dynamics.

Tuning the parameters is not obvious, and special attention is paid to avoid misleading conclusions. Since we present a comparison among similar but structurally different controllers, we set common gains to the same value. All feedback gains, including those of [12, 14], are given as ,,𝑟=120012,𝐾=1000080𝐼=1001,Λ=1005𝐶=(𝑟,𝐼),𝑟₀=.0.12000.12 (40)

Comparative experiments are difficult to obtain because any comparative result can be dangerously biased. Besides, it is difficult to qualitatively compare controllers that are structurally different. Therefore, we show experimental results obtained “at first run,” that is, when reasonable performance, as dictated by the theory, is obtained without perhaps obtaining the best plots.

5.3. Experimental Results

For better visualization of the plots, some figures are shown in two windows. Window 1 shows the time interval [0,1) s, while Window 2 shows the time interval [1,5] s.

Figure 7 shows a comparative plot of position tracking errors at high velocities of the neurofuzzy adaptive controller [12] and our intelligent hybrid control. Desired paths at each joint are 𝑞_𝑑1=0.5sin(6𝜋𝑡) and 𝑞_𝑑2=0.2sin(𝜋𝑡+𝜋/2). Initial conditions are set to −5.0^∘ with zero initial velocity. After a transient of 1 second, our controller yields tracking errors threefold and fourteenfold smaller at each joint, respectively. In this figure, we can observe that our controller yields better tracking accuracy with smooth control input, even when [3] is started with smaller position tracking errors, see controllers in Figure 8.

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Figure 7 

Comparative plots for tracking errors of both joints between the controller [12] (.–) and our proposed controller (–).

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Figure 8 

Comparative plots for control inputs of both joints between the controller [12] (.–) and our proposed controller (–).

Figure 9 shows a comparative plot of position tracking errors at high velocities of the the fuzzy supervisory sliding-mode controller [14] and our Intelligent hybrid control. Desired paths at each joint are 𝑞_𝑑1=1.25−(7/5)𝑒^−𝑡+(7/20)𝑒^−4𝑡 and 𝑞_𝑑2=1.25+𝑒^−𝑡−(1/4)𝑒^−4𝑡. Initial conditions are set to 1.0 with zero initial velocity. In this figure, we can observe that our controller yields better tracking accuracy with smooth control input, even when [14] is started with smaller position tracking errors, see controllers in Figure 10.

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Figure 9 

Comparative plots for tracking errors of both joints between the controller [14] (.–) and our proposed controller (–).

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Figure 10 

Comparative plots for control inputs of both joints between the controller [14] (.–) and our proposed controller (–).

6. Conclusion

A new adaptive intelligent hybrid robust controller based on sliding mode is proposed in this paper for the trajectory tracking of robot manipulators with unknown nonlinear dynamics. We use intelligent systems to approximate the upper bound on the parametric uncertainty, and propose the idea of fuzzy bound later used to weaken the chattering effect. The system global stability and the tracking error convergence are proved by Lyapunov techniques. Simulation results and experimental data show that the intelligent hybrid controller make a significant improvement in the tracking performance.

Acknowledgments

This work was supported by National Natural Science Foundation of China (10771055, 60775047), the Specialized Research Fund for the Doctoral Program of Higher Education (20050532023), Major Subject of Ministry of Education of China (706043), and National High Technology Research and Development Program of China (863 Program: 2007AA04Z244, 2008AA04Z214).