Adaptive Fuzzy Dynamic Surface Sliding Mode Position Control for a Robot Manipulator with Friction and Deadzone

Abstract

Precise tracking positioning performance in the presence of both the deadzone and friction of a robot manipulator actuator is difficult to achieve by traditional control methodology without proper nonlinear compensation schemes. In this paper, we present a dynamic surface sliding mode control scheme combined with an adaptive fuzzy system, state observer, and parameter estimator to estimate the uncertainty, friction, and deadzone nonlinearities of a robot manipulator system. We design a dynamic surface sliding mode basic controller by systematic recursive design steps that yields several adaptive laws for the compensation of nonlinear friction, deadzone, and other unknown nonlinear dynamics. The boundedness and convergence of this closed-loop system are guaranteed by the Lyapunov stability theorem. Experiments on the Scorbot robot manipulator demonstrate the validity and effectiveness of the proposed control scheme.

1. Introduction

In recent decades, several advanced control approaches have been developed to solve complex control problems as industrial machines and devices have rapidly progressed, requiring higher performance control. Among these, a breakthrough nonlinear control method, adaptive backstepping, [1, 2] achieved stabilizing controllers for nonlinear system and guaranteed global or regional regulation and tracking properties. The cancellation of useful nonlinearities that occur with the feedback linearization techniques can be also avoided by using a step-by-step recursive algorithm. However, the application of the backstepping design method requires that nonlinear dynamic models be known either exactly or linearly parameterized with respect to known nonlinear functions. In real situations, this requirement is frequently difficult to accomplish since most uncertainty in a nonlinear system is unknown. To solve this problem, adaptive backstepping methods combined with fuzzy methods [3, 4] and neural networks (NNs) [5, 6] have been developed to approximate these unknown uncertainties. Thus, recently, this approximator-based backstepping method has become a very popular control scheme for dealing with a large class of nonlinear systems. However, repeated differentiation of the virtual control functions [7] gives rise to an explosion of complexity in the controller terms of the complex nonlinear system. Dynamic surface control (DSC) [7, 8] was developed to help a nonlinear systems overcome this “explosion of terms” by using a first-order filter of the synthetic input at each step of the backstepping design procedure. Thus, several adaptive DSCs combined with fuzzy methods [9] and NNs have been developed [10–12] because these controllers are relatively much simpler than backstepping-based ones. Another option is a model-free approach such as fuzzy methods, which is synthesized.

In a dynamic system consisting of the actuator of a robot manipulator, friction and deadzones are frequently encountered and are the main obstacles to high-performance positioning and tracking control. Friction between a moving part and a guide surface gives rise to problems such as stick slip, limit cycle, and steady-state error. Deadzone nonlinearity also causes inaccuracy in a control system [13]. A controller designed to compensate for friction or deadzone independently may perform poorly in the friction/deadzone overlap. Thus, compensation for both nonlinearities should be taken into consideration together. However, with the exception of [12], most DSC applications have focused largely on the compensation of linear or smooth nonlinear system.

The effect of nonlinear friction appears most strongly in a low-velocity regime, especially during velocity reversals. The LuGre [14] and Elastoplastic [15] models can construct a friction estimator relatively easily by virtue of their more systematic structure and lower complexity compared to other available modes. Lin and Chen [16], Yau and Yan [17], and Han et al. [18, 19] developed a sliding mode control and fuzzy logic scheme with the LuGre and Elastoplastic friction model to compensate for the nonlinear friction of a ball-screw and robot systems. For deadzone, several control schemes [20–23] have been developed. However, compensations for both deadzones and friction together have not often been considered until now.

Fuzzy technology [24, 25] has replaced many complex nonlinear control applications. One major feature of fuzzy logic is its ability to express an amount of ambiguity, similar to judgments based on human experiences or expert opinion. Thus, fuzzy logic is an alternative way to deal with the unknown mathematical model of a complex system due to its universal approximation property [24]. A fuzzy controller depends on the experience of experts to create a fuzzy rule base and parameters that are adjusted by adaptive laws for a specified control performance. Hence, adaptive fuzzy controls have been applied successfully in many nonlinear control systems and guarantee improved system performance and stability in the Lyapunov sense [26–28]. However, a specific performance decision table, complicated learning mechanisms, and/or a large amount of fuzzy rules require design by trial-and-error and make practical application difficult.

It is well known that the sliding mode control (SMC) technique is robust to system uncertainty due to its use of a sliding surface [29, 30]. To reduce the fuzzy rules in fuzzy control and significantly increase control performance, SMC is combined with fuzzy logic [31] and other control methods such as intelligent methods [32] and backstepping control [33]. The adaptive sliding mode backstepping control for a semistrict feedback system with unmatched uncertainty was proposed in [34, 35]. However, the backstepping control technique has an explosion of terms problem due to the repeated differentiation of the virtual control functions. This problem leads to a severe computational burden for real hardware implementations such as complex robotic systems. Thus, although the backstepping method is theoretically tractable, in real applications, its increasing complexity is an insurmountable obstacle that prevents its application to multiple state control systems.

We propose an adaptive fuzzy strict feedback positioning control for a robot manipulator based on a DSC design. SMC is applied to a DSC and FLC frame to enhance robustness for the compensation of uncertainty and an adaptive fuzzy system approximates the unknown nonlinear function. The main contributions of this paper are as follows. (1) The DSC scheme is introduced to overcome the drawback of backstepping control. (2) We show that both the deadzone and friction nonsmooth and nonlinear effects of a robot manipulator can be compensated for simultaneously. (3) We then detail and show how SMC is combined with an adaptive DSC and FLC system to enhance the performance robustness for lumped uncertainty and required fuzzy rules and can then be reduced to an approximation to reduce the controller complexity. (4) The proposed control approach is successfully applied to the problem of both reducing nonsmooth nonlinear effect and uncertainty of the robot manipulator in the presence of the friction and deadzone by experiment.

2. Problem Formulations

2.1. Description of the Nonlinear Plant

We consider a robot manipulator system in the presence of deadzone and friction including actuator dynamics whose dynamic equations [36, 37] are described by 𝐌(𝐪)𝐪̈+𝐂(𝐪,𝐪̇)𝐪̇+𝐆(𝐪)+𝐓_𝑓(𝐪,𝐪̇)+𝐓_𝐿=𝝉,𝝉=𝐧𝐤_𝑡𝐋𝐢,_𝑚𝑑𝐢𝑑𝑡+𝐑_𝑚𝐢+𝐤_𝑏𝐪̇=𝐕, (1)(2)(3) where 𝐪,𝐪̇,𝐪̈∈𝑅^𝑛×1 denote the joint position, velocity, and acceleration vectors, respectively; the moment of inertia matrix 𝐌(𝐪)∈𝑅^𝑛×𝑛 is a positive definite symmetric matrix; 𝐂(𝑞,𝑞)̇∈𝑅^𝑛×𝑛 is the centripetal Coriolis matrix; 𝐌̇(𝐪)−2𝐂(𝑞,𝑞)̇ is a skew-symmetric matrix; 𝐆(𝐪)∈𝑅^𝑛×1 is the gravity vector; 𝐓_𝑓(𝐪,𝐪̇)∈𝑅^𝑛×1 is the nonlinear friction torque vector; 𝐓_𝐿∈𝑅^𝑛×1 is an external disturbance; 𝝉∈𝑅^𝑛×1 is the deadzone control torque vector of the joint actuators; 𝐢 is the motor current vector; 𝐕 is the voltage vector applied to the motor drive; 𝐋_𝑚 and 𝐑_𝑚 are the inductance and resistance of the motor, respectively; and 𝐤_𝑏 is the back electromotive-force (emf) constant of the motor.

Considering the modeling uncertainties and external disturbances, the robot system in (1) can be reformulated as 𝐪̈=𝐟_𝑛(𝐪,𝐪̇)−𝐠_𝑛(𝐪)𝐓_𝑓+𝐠_𝑛(𝐪)𝐓_𝑢+𝐠_𝑛(𝐪)𝝉, (4) where the subscript 𝑛 represents the system parameters in the nominal condition, 𝐟_𝑛(𝐪,𝐪̇)=−𝐠_𝑛(𝐪)[𝐂_𝑛(𝐪,𝐪̇)+𝐆_𝑛(𝐪)], where 𝐠_𝑛(𝐪)=𝐌_𝑛⁻¹(𝐪); 𝐓_𝑢 is a lumped uncertainty defined as 𝐓_𝑢=−Δ𝐌(𝐪)𝐪̈−Δ𝐂(𝐪,𝐪̇)−Δ𝐆(𝐪)−𝐓_𝐿; Δ𝐌(𝐪), Δ𝐂(𝐪,𝐪̇), Δ𝐆(𝐪), and Δ𝐓_𝑓 represent the unknown uncertainties of 𝐌(𝐪), 𝐂(𝐪,𝐪̇), 𝐆(𝐪), and 𝐓_𝑓, respectively; and 𝐓_𝐿∈𝑅^𝑛×1 is the disturbance vector. The uncertainties of Δ𝐌(𝐪), Δ𝐂(𝐪,𝐪̇), Δ𝐆(𝐪), and Δ𝐓_𝑓 are bounded by some positive constants 𝜌_𝑖(𝑖=𝑚,𝑐,𝑔,𝑓) such that ‖Δ𝐌‖≤𝜌_𝑚, ‖Δ𝐂‖≤𝜌_𝑐, ‖Δ𝐆‖≤𝜌_𝑔, and ‖Δ𝐓_𝑓‖≤𝜌_𝑓. For the disturbance, it is assumed that 𝐓_𝐿∈𝐿₂[0,𝑇], for all 𝑇∈[0,∞), and 𝐓_𝐿 is bounded by some positive constant 𝜌_𝑑:‖𝐓_𝐿‖≤𝜌_𝑑. Thus, the lumped uncertainty is assumed to be bounded by a finite value. To guarantee more improved control performance, an elaborate nonlinear friction model should be considered. The state equations for (1)–(4) are represented as 𝐱̇₁₀=𝐱₂₀,𝐱̇₂₀=𝐟_𝑛𝐱₁₀,𝐱₂₀−𝐠_𝑛𝐱₁₀𝐓_𝑓+𝐠_𝑛𝐱₁₀𝐓_𝑢+𝐠_𝑛𝐱₁₀𝐱̇𝝉,₃₀=−𝐋_𝑚⁻¹𝐑_𝑚𝐱₃₀−𝐋_𝑚⁻¹𝐤_𝑏𝐱₂₀+𝐋_𝑚⁻¹𝐕, (5) where 𝐱₁₀=𝐪, 𝐱₂₀=𝐪̇, and 𝐱₃₀=𝐢.

The deadzone nonlinearities 𝝉 are shown in Figure 1(a) and their mathematical models are described by 𝝉(𝑡)(𝐮)=𝐦=𝐷^𝑟𝐮(𝑡)−𝐁_𝑟for𝐮(𝑡)≥𝐁_𝑟0for𝐁_𝑙<𝐮(𝑡)<𝐁_𝑟𝐦_𝑙𝐮(𝑡)−𝐁_𝑙for𝐮(𝑡)≤𝐁_𝑙, (6) where 𝐦_𝑟 and 𝐦_𝑙 denote the slope of the deadzone and 𝐁_𝑟 and 𝐁_𝑙 stand for the deadzone width parameters. In the control problem, the practical assumptions of the deadzone are as follows.

(a) Deadzone

(b) Inverse deadzone

Figure 1 

Description of deadzone.

Assumption 1. The deadzone outputs 𝝉(𝑡) are not available for measurement. Furthermore, the deadzone parameters 𝐦_𝑟, 𝐦_𝑙, 𝐁_𝑟, and 𝐁_𝑙 are unknown but their signs are known, 𝐦_𝑟>0, 𝐦_𝑙>0, 𝐁_𝑟≥0, and 𝐁_𝑙≤0.

Assumption 2. The deadzone slopes are bounded by known constants 𝐦_𝑟min, 𝐦_𝑟max, 𝐦_𝑙min, and 𝐦_𝑟max such that 0<𝐦_𝑟min≤𝐦_𝑟≤𝐦_𝑟max and 0<𝐦_𝑙min≤𝐦_𝑙≤𝐦_𝑙max.

The deadzone inverse technique is a useful method to compensate for the deadzone effect [13]. Letting 𝐮_𝑑(𝑡) be the signal from the controller that does not take into account the deadzone, the following control signal 𝐮(𝑡) is generated according to the certainty equivalence deadzone inverse described in Figure 1(b): 𝐮(𝑡)=𝐷⁻¹𝐮_𝑑=𝐦_𝑟⁻¹𝐮_𝑑(𝑡)+𝐁_𝑚𝑟𝜹+𝐦_𝑙⁻¹𝐮_𝑑(𝑡)+𝐁_𝑚𝑙(𝐈−𝜹), (7) where 𝐦_𝑟, 𝐦_𝑙, 𝐁_𝑚𝑟, and 𝐁_𝑚𝑙 are the estimates of 𝐦_𝑟, 𝐦_𝑙, 𝐦_𝑟𝐁_𝑟, and 𝐦_𝑙𝐁_𝑙, respectively, and 𝜹=𝐈if𝐮_𝑑(𝑡)≥𝟎𝟎if𝐮_𝑑(𝑡)<𝟎. (8) The resulting errors between 𝐮 and 𝐮_𝑑 are given by 𝝉(𝑡)−𝐮_𝑑(𝑡)=𝐁_𝑚𝑟−𝐦_𝑟⁻¹𝐮_𝑑(𝑡)+𝐁_𝑚𝑟𝐦_𝑟𝜹+𝐁_𝑚𝑙−𝐦_𝑙⁻¹𝐮_𝑑(𝑡)+𝐁_𝑚𝑙𝐦_𝑙(𝐈−𝜹)+𝜺_𝑑, (9) where 𝜺_𝑑(𝑡) is known as the bounded function for all 𝐮(𝑡) [13].

The nonlinear friction forces are assumed to be modeled as 𝐓_𝑓=𝝈₀𝐳+𝝈₁𝐳̇+𝝈₂𝐯+𝜺_𝑓, (10) where 𝝈₀>0 is the stiffness of the elastic bristle, 𝝈₁>0 is the damping coefficients in the presliding range, 𝝈₂>0 is the viscous damping coefficients, and 𝜺_𝑓 contains the bounded friction modeling errors. The presliding states 𝐳 are represented by the following Elastoplastic model [15]: 𝐳̇=𝐯−𝜽𝝈₀𝐡(𝐯)𝐳, (11) where ℎ_𝑖𝑣_𝑖=𝑣_𝑖𝐹_ci+𝐹_si−𝐹_ciexp−𝑣_𝑖/𝑣_si²,𝑖=1,…,𝑛, (12)𝐹_ci is the Coulomb friction, 𝐹_si is the stiction level, 𝑣_𝑖 is the relative velocity between two contact surfaces, 𝑣_si is the Stribeck velocity, and 𝜽 is the unknown coefficient related to the presliding friction behavior. The function ℎ_𝑖(𝑣_𝑖) is positive and depends on many factors such as material properties, lubrication, and temperature. As the state variables 𝐳 cannot be measured directly, we use the friction state observers to estimate 𝐳 as follows [19]: 𝐳̇̂=𝐯−𝜽𝝈₀𝐡(𝐯)𝐳̂, (13) where 𝐳̂, 𝜽, and 𝝈₀ are the estimations of 𝐳, 𝜽, and 𝝈₀, respectively. The estimations of the friction 𝐓_𝑓 can be expressed as 𝐓_𝑓=𝝈₀𝐳̂+𝝈₁𝐳̇̂+𝝈₂𝐯, (14) where 𝝈₁ and 𝝈₂ are the estimations of 𝝈₁ and 𝝈₂, respectively. From (10), (11), (13), and (14), we have ̇̃𝐳=−𝜽𝝈₀𝐡𝐳̃−𝜽𝐡𝐳̂𝝈₀−𝝈₀𝐡𝐳̂𝐓𝜽,_𝑓=𝐓_𝑓−𝐓_𝑓=𝝈₀−𝜽𝝈₁𝐡𝐳̃+𝐳̂−𝝈₁𝜽𝐳̂𝐡𝝈₀+𝐯−𝜽𝐳̂𝐡𝝈₁+𝐯𝝈₂−𝝈₁𝝈₁𝐳̂𝐡𝜽+𝜺_𝑓, (15) where 𝐳̃=𝐳−𝐳̂, 𝝈_(⋅)=𝝈_(⋅)−𝝈_(⋅), and 𝜽=𝜽−𝜽.

We can transform the above state model into the following form: 𝐱̇₁=𝐱₂,𝐱̇₂=𝐱₃+𝐟₂𝐱₂−𝐠_𝑛𝐓_𝑓+𝐠_𝑛𝐁_𝑚𝑟−𝐦_𝑟⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑟𝐦_𝑟𝜹+𝐠_𝑛𝐁_𝑚𝑙−𝐦_𝑙⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑙𝐦_𝑙(𝐈−𝜹)+𝐓_𝑑,𝐱̇₃=𝐟₃𝐱₃+𝐛₃𝐮_𝑣, (16)(17)(18) where 𝐱₃=[𝐱₁𝐱₂𝐱₃]^𝑇=[𝐱₁₀𝐱₂₀𝐠_𝑛𝐧𝐤_𝑡𝐱₃₀]^𝑇, 𝐟₂(𝐱₂)=𝐟_𝑛(𝐱₁₀,𝐱₂₀),𝐓_𝑑=𝐠_𝑛(𝐱₁)𝐓_𝑢+𝐠_𝑛𝜺_𝑑, 𝐛₃=𝐠_𝑛𝐧𝐤_𝑡𝐋_𝑚⁻¹, and 𝐟₃(𝐱₃)=−𝐠_𝑛𝐧𝐤_𝑡𝐋_𝑚⁻¹(𝐑_𝑚𝐱₃₀+𝐤_𝑏𝐱₂₀), 𝐱₂=[𝐱₁𝐱₂]^𝑇, and 𝐮_𝑣=𝐕.

2.2. Function Approximation Using a Fuzzy Logic System

The basic configuration of a fuzzy system consists of the fuzzifier, fuzzy rule base, fuzzy inference engine, and defuzzifier. The fuzzy inference engine performs a mapping from an input linguistic vector 𝑥=[𝑥₁,…,𝑥_𝑛]^𝑇∈𝑅^𝑛 to an output linguistic scalar variable 𝑦∈𝑅. The fuzzy rule base consists of a collection of fuzzy IF-THEN rules. The 𝑙th IF-THEN rules are described by 𝑅^(𝑙)∶IF𝑥₁is𝐹^𝑙₁and⋯and𝑥_𝑛is𝐹^𝑙_𝑛,then𝑦is𝐺^𝑙,𝑙=1,2,…,𝑀, (19) where 𝐹^𝑙_𝑖, 𝑖=1,…,𝑛, and 𝐺^𝑙 are fuzzy sets characterized by the fuzzy membership functions 𝜇_𝐹𝑙𝑖(𝑥_𝑖) and 𝜇_𝐺𝑙(𝑦), respectively, and 𝑀 is the number of rules in the fuzzy rule base. The output of a fuzzy system with a center-average defuzzifier, product inference, and singleton fuzzifier is expressed as 𝑦(𝑥)=∑^𝑀_𝑙=1𝑦^𝑙∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖∑^𝑀_𝑙=1∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖, (20) where 𝑦^𝑙 is the point at which 𝜇_𝐺𝑙(𝑦^𝑙)=1 (its maximum value). This equation can be rewritten as 𝑦(𝑥)=𝑊^𝑇_𝑜𝑋(𝑥), (21) where 𝑊^𝑇_𝑜=[𝑦¹,…,𝑦^𝑀]^𝑇 is a vector that groups all the consequence parameters, and 𝑋(𝑥)=[𝑋₁,…,𝑋_𝑛]^𝑇 is a set of fuzzy basis functions defined as 𝑋^𝑙(𝑥)=∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖∑^𝑀_𝑙=1∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖. (22) It has been proven that a fuzzy logic system can approximate any nonlinear continuous function to an arbitrary degree accuracy if enough rules are provided [24]. Thus, a fuzzy logic system performs a universal approximation in the sense that, given any real continuous function 𝑓(⋅)∶𝑅^𝑛→𝑅 on a sufficiently large compact set Ω⊂𝑅 and an arbitrary 𝜀_𝑚>0, there exists a fuzzy logic system 𝑦(𝑥) in the form of (21) such that sup_𝑥∈Ω𝑓(𝑥)−𝑦(𝑥)≤𝜀_𝑚. (23) Then the function 𝑓(𝑥) can be expressed as 𝑓(𝑥)=𝑊_𝑜^∗𝑇𝑋(𝑥)+𝜀,∀𝑥∈Ω⊂𝑅^𝑛, (24) where |𝜀|≤𝜀_𝑚, 𝜀^∗ is the error of the fuzzy approximation and 𝑊^∗ is chosen to be the value of 𝑊 that minimizes the fuzzy approximation error 𝜀; that is, 𝑊^∗_𝑜=argmin_{𝑊𝑜∈𝑅𝑀}sup_𝑥∈Ω𝑓(𝑥)−𝑊^𝑇_𝑜𝑋(𝑥). (25) Since 𝑊^∗_𝑜 is unknown, it is replaced by 𝑊_𝑜, an estimation of 𝑊^∗_𝑜. Adaptation laws are required to update the parameter 𝑊_𝑜 and other related fuzzy parameters online to asymptotically minimize the reference tracking error. The optimal fuzzy output function can be rewritten as 𝑊_𝑜^∗𝑇𝜉(𝑥)=𝑊^𝑇_𝑜𝜉(𝑥)+𝑊^𝑇_𝑜𝜉(𝑥), (26) where 𝑊_𝑜=𝑊^∗_𝑜−𝑊_𝑜.

3. Design of Controller and Nonsmooth Nonlinear Compensator

In this section, the adaptive laws and controller are derived via recursive DSC design procedures. The control objective for a robot manipulator system is to determine a state feedback control system such that the system output 𝐱₁ can track a desired trajectory 𝐲_𝑑. We add a final assumption to the system.

Assumption 3. The desired trajectory vectors are continuous and available, and [𝐲_𝑑,𝐲̇_𝑑,𝐲̈_𝑑]^𝑇∈Ω_𝑑 with the known compact set Ω_𝑑={[𝐲_𝑑,𝐲̇_𝑑,𝐲̈_𝑑]^𝑇∶‖𝐲_𝑑‖²+‖𝐲̇_𝑑‖²+‖𝐲̈_𝑑‖²≤𝛿_𝑦}, where 𝛿_𝑦>0 is a constant. The state feedback control system is designed step-by-step using a DSC technique as follows.

Step 1. We define the tracking error to be the first error 𝐒₁=𝐱₁−𝐲_𝑑, (27) where time derivative of (27) is 𝐒̇₁=𝐱₂−𝐲̇_𝑑. (28) We define the following Lyapunov function: 𝑉₁=12𝐒^𝑇₁𝐒₁, (29) and its time derivative is given as 𝑉̇₁=𝐒^𝑇₁𝐒̇₁=𝐒^𝑇₁𝐱₂−𝐲̇_𝑑. (30) We choose a virtual control law to be 𝜶₁=−𝐜₁𝐒₁+𝐲̇_𝑑, (31) where 𝐜₁>𝐈 is a design constant. We introduce the filtering virtual control 𝝃₂ and let 𝜶₁ pass through a first-order filter with a time constant 𝜍₂ as 𝜍₂𝝃̇₂+𝝃₂=𝜶₁,𝝃₂(0)=𝜶₁(0). (32) Setting 𝝀₂=𝝃₂−𝜶₁, from (25), it follows that 𝝃̇₂=−𝝀₂𝜍₂. (33) By using the definition of 𝐱₂=𝐒₂+𝝃₂, (30) becomes 𝑉̇₁≤−𝐜₁𝐒^𝑇₁𝐒₁+𝐒^𝑇₁𝐒₂+𝐒^𝑇₁𝝀₂. (34) From (24) and (26), it follows that 𝝀̇₂=𝝃̇₂−𝜶̇₁𝝀=−²𝜍₂+𝐜₁𝐒̇₁−𝐲̈_𝑑𝝀≤−²𝜍₂+𝝍₂𝐒₁,𝐒₂,𝝀₂,𝐲_𝑑,𝐲̇_𝑑,𝐲̈_𝑑,𝝀̇₂+𝝀₂𝜍₂≤𝝍₂𝐒₁,𝐒₂,𝝀₂,𝐲_𝑑,𝐲̇_𝑑,𝐲̈_𝑑, (35)(36) where 𝝍₂(𝐒₁,𝐒₂,𝝀₂,𝐲_𝑑,𝐲̇_𝑑,𝐲̈_𝑑) is a continuous function. From (35) and (36), we have 𝝀^𝑇₂𝝀̇₂𝝀≤−²²𝜍₂+𝝀₂𝝍₂𝝀≤−²²𝜍₂+𝝀₂²+14𝝍₂². (37)

Step 2. We consider the following expression: 𝐱̇₂=𝐱₃+𝐖^∗𝑇_𝑜2𝐗₂𝐱₂+𝜺^∗₂−𝐠_𝑛𝐓_𝑓+𝐠_𝑛𝐁_𝑚𝑟−𝐦_𝑟⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑟𝐦_𝑟𝜹+𝐠_𝑛𝐁_𝑚𝑙−𝐦_𝑙⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑙𝐦_𝑙(𝐈−𝜹)+𝐓_𝑑. (38) By defining 𝐒₂=𝐱₂−𝝃₂, the time derivative of 𝐒₂ is given by 𝐒̇₂=𝐱₃+𝐖^𝑇_𝑜2𝐗₂𝐱₂+𝐖^𝑇_𝑜2𝐗₂𝐱₂+𝜺^∗₂−𝐠_𝑛𝐓_𝑓−𝐠_𝑛𝐓_𝑓+𝐠_𝑛𝐁_𝑚𝑟−𝐦_𝑟⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑟𝐦_𝑟𝜹+𝐠_𝑛𝐁_𝑚𝑙−𝐦_𝑙⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑙𝐦_𝑙(𝐈−𝜹)+𝐓_𝑑−𝝃̇₂. (39) Next, we define the Lyapunov function candidate: 𝑉₂=12𝐒^𝑇₂𝐒₂. (40) By differentiating (40) with respect to time, we obtain the following equation: 𝑉̇₂=𝐒^𝑇₂𝐒̇₂≤𝐒^𝑇₂𝐱₃+𝐖^𝑇_𝑜2𝐗₂𝐱₂−𝐠_𝑛𝐓_𝑓−𝝃̇₂+𝝆₂tanh𝜿₂⁻¹𝐒₂𝝆₂+𝐒^𝑇₂𝐖^𝑇_𝑜2𝐗₂𝐱₂−𝐒^𝑇₂𝐠_𝑛𝐓_𝑓−𝐒₂𝝆₂+𝐒^𝑇₂𝐠_𝑛𝐁_𝑚𝑟−𝐦_𝑟⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑟𝐦_𝑟𝜹+𝐒^𝑇₂𝐠_𝑛𝐁_𝑚𝑙−𝐦_𝑙⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑙𝐦_𝑙(𝐈−𝜹)+𝜅^′₂, (41) where ‖𝜺_𝑖2+𝐓_𝑑‖≤‖𝝆₂‖,0≤|𝑆_2𝑖𝜌_2𝑖|−𝑆_2𝑖𝜌_2𝑖tanh(𝑆_2𝑖𝜌_2𝑖/𝜅_2𝑖)≤0.2785𝜅_2𝑖=𝜅^′_2𝑖, 𝑖=1,…,𝑛, 𝜅_2𝑖 is a design constant, and where 𝝆₂=𝝆₂−𝝆₂. We specify a virtual control 𝜶₂ to be as follows: 𝜶₂=−𝐜₂𝐒₂−𝐒₁−𝐖^𝑇_𝑜2𝐗₂𝐱₂+𝝃̇₂+𝐠_𝑛𝐓_𝑓−𝐒₂𝝆₂tanh𝜿₂⁻¹𝐒₂𝝆₂, (42) where 𝐜₂>𝐈 is a design constant and 𝝆₂ is the estimation of 𝝆₂. We introduce a new filtering virtual control 𝝃₃ and let 𝜶₂ pass through a first-order filter with a time constant 𝜍₃ as 𝜍₃𝝃̇₃+𝝃₃=𝜶₂,𝝃₃(0)=𝜶₂(0). (43) By setting 𝝀₃=𝝃₃−𝜶₂, from (36), it follows that 𝝃̇₃=−𝝀₃𝜍₃. (44) We define 𝐒₃=𝐱₃−𝝃₃. It then follows that 𝐱₃=𝐒₃+𝝀₃−𝐜₂𝐒₂−𝐒₁−𝐖^𝑇_𝑜2𝐗₂𝐱₂+𝝃̇₂+𝐠_𝑛𝐓_𝑓−𝝆₂tanh𝜿₂⁻¹𝐒₂𝝆₂. (45) Substituting (45) into (41), we obtain the following expression: 𝑉̇₂≤−𝐜₂𝐒^𝑇₂𝐒₂−𝐒^𝑇₂𝐒₁+𝐒^𝑇₂𝐒₃+𝐒^𝑇₂𝝀₃+𝐒^𝑇₂𝐖^𝑇_𝑜2𝐗₂𝐱₂−𝐒^𝑇₂𝐠_𝑛𝐓_𝑓−𝐒₂𝝆₂+𝐒^𝑇₂𝐠_𝑛𝐁_𝑚𝑟−𝐦_𝑟⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑟𝐦_𝑟𝜹+𝐒^𝑇₂𝐠_𝑛𝐁_𝑚𝑙−𝐦_𝑙⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑙𝐦_𝑙(𝐈−𝜹)+𝜿^′₂. (46) From (36), it follows that 𝝀̇₃=𝝃̇₃−𝜶̇₂𝝀=−³𝜍₃+𝐜₂𝐒̇₂+𝐖̇^𝑇_𝑜2𝐗₂𝐱₂+𝐖^𝑇_𝑜2𝜕𝐗₂𝐱₂𝜕𝐱₂𝐱̇₂−𝐓̇_𝑓−𝝃̈₂+𝑑𝝆₂tanh𝜿₂⁻¹𝐒₂𝝆₂𝑑𝐒₂𝐒̇₂𝝀≤−³𝜍₃+𝝍₃𝐒₁,𝐒₂,𝐒₃,𝝀₂,𝝀₃,𝐖_𝑜2,𝐓_𝑓,𝐲_𝑑,𝐲̇_𝑑,𝐲̈_𝑑,𝝀̇₃+𝝀_𝑖3𝜍₃≤𝝍₃𝐒₁,𝐒₂,𝐒₃,𝝀₂,𝝀₃,𝐖_𝑜2,𝐓_𝑓,𝐲_𝑑,𝐲̇_𝑑,𝐲̈_𝑑, (47) where 𝝍₃(⋅) is a continuous function. From (47), we obtain the following inequality: 𝝀^𝑇₃𝝀̇₃𝝀≤−³²𝜍₃+𝝀₃𝝍₃𝝀≤−³²𝜍₃+𝝀₃²+14𝝍₃². (48)

Step 3. The final control law is derived in this step. Consider 𝐱̇₃=𝐛₃𝐮_𝑣+𝐖^∗𝑇_𝑜3𝐗₃𝐱₃+𝜺^∗₃. (49) From the third error surface 𝐒₃=𝐱₃−𝝃₃, it follows that 𝐒̇₃=𝐱̇₃−𝝃̇₃=𝐛₃𝐮_𝑣+𝐖^∗𝑇_𝑜3𝐗₃𝐱₃+𝜺^∗₃−𝝃̇₃≤𝐛₃𝐮_𝑣+𝐖^𝑇_𝑜3𝐗₃𝐱₃−𝝃̇₃+𝐖^𝑇_𝑜3𝐗₃𝐱₃+𝝆₃−𝝆₃, (50) where ‖𝜺^∗₃‖≤‖𝝆₃‖, 𝝆₃ is a positive constant, and 𝝆₃ is an estimation of 𝝆₃.

The adaptive strict feedback dynamic surface control is modified to enforce robustness by adding a sliding mode control. The modification starts by defining the following sliding surface in terms of the error coordinates: 𝝌=𝜸₁𝐒₁+𝜸₂𝐒₂+𝐒₃. (51) We define the following Lyapunov function candidate: 𝑉=²_𝑘=1𝑉_𝑘+12𝝌^𝑇1𝝌+2²_𝑘=1𝝀^𝑇_𝑘+1𝝀_𝑘+1+³_𝑘=212𝜂_𝑤𝑘𝐖^𝑇_𝑜𝑘𝐖_𝑜𝑘+³_𝑘=212𝜂_𝜌𝑘𝝆^𝑇_𝑘𝝆_𝑘+12𝐠_𝑛𝐳̃^𝑇̃1𝐳+2𝜂₀𝐠_𝑛𝝈^𝑇₀𝝈₀+12𝜂₁𝐠_𝑛𝝈^𝑇₁𝝈₁+12𝜂₂𝐠_𝑛𝝈^𝑇₂𝝈₂+12𝜂_𝜃𝐠_𝑛𝝈₁𝜽^𝑇𝜽+12𝜂_𝑚𝑟𝐠_𝑛𝐦^𝑇_𝑟𝐦_𝑟+12𝜂_𝑚𝑙𝐠_𝑛𝐦^𝑇_𝑙𝐦_𝑙+12𝜂_𝑏𝑟𝐠_𝑛𝐁^𝑇_𝑚𝑟𝐁_𝑚𝑟+12𝜂_𝑏𝑙𝐠_𝑛𝐁^𝑇_𝑚𝑙𝐁_𝑚𝑙, (52) where 𝜂_𝑖𝑖 are positive constants. The time derivative of 𝑉 is calculated as 𝑉̇=²_𝑘=1𝑉̇_𝑘+𝝌^𝑇𝝌̇+²_𝑘=1𝝀^𝑇_𝑘+1𝝀̇_𝑘+1+³_𝑘=21𝜂_𝑤𝑘𝐖^𝑇_𝑜𝑘̇𝐖_𝑜𝑘+³_𝑘=21𝜂_𝜌𝑘𝝆^𝑇_𝑘̇𝝆_𝑘+𝐠_𝑛𝐳̃^𝑇̇̃1𝐳+𝜂₀𝐠_𝑛𝝈^𝑇₀̇𝝈₀+1𝜂₁𝐠_𝑛𝝈^𝑇₁̇𝝈₁+1𝜂₂𝐠_𝑛𝝈^𝑇₂̇𝝈₂+1𝜂_𝜃𝐠_𝑛𝝈₁𝜽^𝑇̇𝜽+1𝜂_𝑚𝑟𝐠_𝑛𝐦^𝑇_𝑟̇𝐦_𝑟+1𝜂_𝑚𝑙𝐠_𝑛𝐦^𝑇_𝑙̇𝐦_𝑙+1𝜂_𝑏𝑟𝐠_𝑛𝐁^𝑇_𝑚𝑟̇𝐁_𝑚𝑟+1𝜂_𝑏𝑙𝐠_𝑛𝐁^𝑇_𝑚𝑙̇𝐁_𝑚𝑙≤−²_𝑘=1𝐜_𝑘𝐒^𝑇_𝑘𝐒_𝑘+𝐒^𝑇₂𝐒₃+²_𝑘=1𝐒^𝑇_𝑘𝝀_𝑘+1+𝐒^𝑇₂𝐖^𝑇_𝑜2𝐗₂𝐱₂−𝐒₂𝝆₂−𝐒^𝑇₂𝐠_𝑛𝐓_𝑓+𝐒^𝑇₂𝐠_𝑛𝐁_𝑚𝑟−𝐦_𝑟⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑟𝐦_𝑟𝜹+𝐒^𝑇₂𝐠_𝑛𝐁_𝑚𝑙−𝐦_𝑙⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑙𝐦_𝑙(𝐈−𝜹)+𝜿^′₂+𝝌^𝑇𝝌̇+²_𝑘=1𝝀^𝑇_𝑘+1𝝀̇_𝑘+1+³_𝑘=21𝜂_𝑤𝑘𝐖^𝑇_𝑜𝑘̇𝐖_𝑜𝑘+³_𝑘=21𝜂_𝜌𝑘𝝆^𝑇_𝑘̇𝝆_𝑘+𝐠_𝑛𝐳̃^𝑇̇̃1𝐳+𝜂₀𝐠_𝑛𝝈^𝑇₀̇𝝈₀+1𝜂₁𝐠_𝑛𝝈^𝑇₁̇𝝈₁+1𝜂₂𝐠_𝑛𝝈^𝑇₂̇𝝈₂+1𝜂_𝜃𝐠_𝑛𝝈₁𝜽^𝑇̇𝜽+1𝜂_𝑚𝑟𝐠_𝑛𝐦^𝑇_𝑟̇𝐦_𝑟+1𝜂_𝑚𝑙𝐠_𝑛𝐦^𝑇_𝑙̇𝐦_𝑙+1𝜂_𝑏𝑟𝐠_𝑛𝐁^𝑇_𝑚𝑟̇𝐁_𝑚𝑟+1𝜂_𝑏𝑙𝐠_𝑛𝐁^𝑇_𝑚𝑙̇𝐁_𝑚𝑙. (53) From (51), we obtain the following relation: 𝐒₃=−𝛾₁𝐒₁−𝛾₂𝐒₂+𝝌. (54) By considering the previous results and (54), we obtain the following result: 𝑉̇≤−²_𝑘=1𝐜_𝑘𝐒^𝑇_𝑘𝐒_𝑘+𝐒^𝑇₂−𝜸₁𝐒₁−𝜸₂𝐒₂+²_𝑘=1𝐒_𝑘𝝀_𝑘+1+𝐒^𝑇₂𝐖^𝑇_𝑜2𝐗₂𝐱₂−𝐒₂𝝆₂−𝐒^𝑇₂𝐠_𝑛𝐓_𝑓+𝐒^𝑇₂𝐠_𝑛𝐁_𝑚𝑟−𝐦_𝑟⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑟𝐦_𝑟𝜹+𝐒^𝑇₂𝐠_𝑛𝐁_𝑚𝑙−𝐦_𝑙⁻¹𝐠_𝑛⁻¹𝐱₃+𝐁_𝑚𝑙𝐦_𝑙(𝐈−𝜹)+𝜿^′₂+𝝌^𝑇𝐒^𝑇₂+𝜸₁𝐒₂+𝝃₂−𝑦̇_𝑑+𝜸₂𝐒₃+𝝀₃−𝐜₂𝐒₂−𝐒₁+𝐛₃𝐮_𝑣+𝐖^𝑇_𝑜3𝐗₃𝐱₃−𝝃̇₃+𝝆₃+𝝌^𝑇𝜸₂𝐖^𝑇_𝑜2𝐗₂𝐱₂−𝐠_𝑛𝜸₂𝐓_𝑓+𝐠_𝑛𝜸₂𝐁_𝑚𝑟𝜹−𝐠_𝑛𝜸₂𝐦_𝑟⁻¹𝐁_𝑚𝑟𝐦_𝑟𝜹+𝐠_𝑛𝜸₂𝐁_𝑚𝑙(𝐈−𝜹)−𝐠_𝑛𝜸₂𝐦_𝑙⁻¹𝐁_𝑚𝑙𝐦_𝑙(𝐈−𝜹)+𝐖^𝑇_𝑜3𝐗₃𝐱₃−𝝆₃+𝜿^′₂−²_𝑘=1𝝀_𝑘+1²𝜍_𝑘+1+²_𝑘=1𝝀_𝑘+1²+14²_𝑘=1𝝍_𝑘+1²+³_𝑘=21𝜂_𝑤𝑘𝐖^𝑇_𝑜𝑘̇𝐖_𝑜𝑘+³_𝑘=21𝜂_𝜌𝑘𝝆^𝑇_𝑘̇𝝆_𝑘+1𝜂_𝑏𝑟𝐠_𝑛𝐁^𝑇_𝑚𝑟̇𝐁_𝑚𝑟+𝐠_𝑛𝐳̃^𝑇̇̃1𝐳+𝜂₀𝐠_𝑛𝝈^𝑇₀̇𝝈₀+1𝜂₁𝐠_𝑛𝝈^𝑇₁̇𝝈₁+1𝜂₂𝐠_𝑛𝝈^𝑇₂̇𝝈₂+1𝜂_𝜃𝐠_𝑛𝝈₁𝜽^𝑇̇1𝜽+𝜂_𝑚𝑟𝐠_𝑛𝐦^𝑇_𝑟̇𝐦_𝑟+1𝜂_𝑚𝑙𝐠_𝑛𝐦^𝑇_𝑙̇𝐦_𝑙+1𝜂_𝑏𝑟𝐠_𝑛𝐁^𝑇_𝑚𝑟̇𝐁_𝑚𝑟+1𝜂_𝑏𝑙𝐠_𝑛𝐁^𝑇_𝑚𝑙̇𝐁_𝑚𝑙. (55) We choose the control input to be 𝐮_𝑣=𝐛₃⁻¹−𝐒^𝑇₂−𝜸₁𝐒₂+𝝃₂−𝑦̇_𝑑−𝜸₂𝐒₃+𝝀₃−𝐜₂𝐒₂−𝐒₁−𝐖^𝑇_𝑜3𝐗₃𝐱₃+𝝆₃−𝝃̇₃−𝐊₁𝝌−𝐊₂sgn(𝝌), (56) where 𝐊₁>0 and 𝐊₂>0 are design constants. By using Young’s inequality expressed as ‖𝐒_𝑘‖‖𝝀_𝑘+1‖≤‖𝐒_𝑘‖²+(1/4)‖𝝀_𝑘+1‖², (56) becomes 𝑉̇≤−²_𝑘=1𝐜_𝑘−1𝐒^𝑇_𝑘𝐒_𝑘+𝐒^𝑇₂−𝛾₁𝐒₁−𝛾₂𝐒₂−𝐊₁‖𝝌‖²−𝐊₂‖𝝌‖+𝐖^𝑇_𝑜3𝝌^𝑇𝐗₃𝐱₃−1𝜂_𝑤3𝐖̇_𝑜3+𝝆^𝑇₃−𝝌^𝑇+1𝜂_𝜌3𝝆̇₃+𝐠_𝑛𝐦^𝑇_𝑟−𝐒^𝑇₂+𝜸₂𝝌^𝑇𝐦_𝑟⁻¹𝐁_𝑚𝑟1𝜹−𝜂_𝑚𝑟𝐦̇_𝑟+𝐠_𝑛𝐦^𝑇_𝑙−𝐒^𝑇₂+𝜸₂𝝌^𝑇𝐦_𝑙⁻¹𝐁_𝑚𝑙()−1𝐈−𝜹𝜂_𝑚𝑙𝐦̇_𝑙+𝐠_𝑛𝐁^𝑇_𝑚𝑟𝐒^𝑇₂+𝜸₂𝝌^𝑇1𝜹−𝜂_𝑏𝑟𝐁̇_𝑚𝑟+𝐠_𝑛𝐁^𝑇_𝑚𝑙𝐒^𝑇₂+𝜸₂𝝌^𝑇()−1𝐈−𝜹𝜂_𝑏𝑙𝐁̇_𝑚𝑙+𝐖^𝑇_𝑜2𝐒^𝑇₂+𝜸₂𝝌^𝑇𝐗₂𝐱₂−1𝜂_𝑤2𝐖̇_𝑜2+𝝆^𝑇₂−𝐒₂+1𝜂_𝜌2𝝆̇₂+𝐠_𝑛𝝈^𝑇₀−𝐒^𝑇₂+𝜸₂𝝌^𝑇̂1𝐳−𝜂₀𝝈̇₀+𝐠_𝑛𝝈^𝑇₁−𝐒^𝑇₂+𝜸₂𝝌^𝑇̇̂1𝐳−𝜂₁𝝈̇₁+𝐠_𝑛𝝈^𝑇₂−𝐒^𝑇₂+𝜸₂𝝌^𝑇1𝐯−𝜂₂𝝈̇₂+𝐠_𝑛𝝈₁𝜽^𝑇−𝐒^𝑇₂+𝜸₂𝝌^𝑇𝝈₁̂1𝐳𝐡−𝜂_𝜃𝜽̇+54²_𝑘=1𝝀_𝑘+1²+14²_𝑘=1𝝍_𝑘+1²+𝐠_𝑛𝜿^′₂−𝐠_𝑛𝐒^𝑇₂+𝜸₂𝝌^𝑇𝝈₀−𝜽𝝈₁𝐡𝐳̃+𝐠_𝑛𝐒^𝑇₂+𝜸₂𝝌^𝑇𝝈₁𝜽𝐳̂𝐡𝝈₀−𝐠_𝑛𝐒^𝑇₂+𝜸₂𝝌^𝑇𝜺_𝑓+𝐠_𝑛−𝜽𝝈₀𝐡𝐳̃^𝑇−𝜽𝐡𝐳̂𝝈₀−𝝈₀𝐡𝐳̂𝜽𝐳̃. (57) We specify the adaptive laws as follows: 𝐖̇_𝑜2=Proj_𝑤𝑜2𝜂_𝑤2𝐒^𝑇₂+𝜸₂𝝌^𝑇𝐗₂𝐱₂,𝐖̇_𝑜3=Proj_𝑤𝑜3𝜂_𝑤3𝝌^𝑇𝐗₃𝐱₃,𝝆̇₂=Proj𝜌₂⁺𝜂_𝜌2𝐒₂,𝝆̇₃=Proj𝜌₃⁺𝜂_𝜌3𝝌^𝑇,𝝈̇₀=Proj𝜎₀₊−𝜂₀𝐒^𝑇₂+𝜸₂𝝌^𝑇𝐳̂,𝝈̇₁=Proj𝜎₁₊−𝜂₁𝐒^𝑇₂+𝜸₂𝝌^𝑇𝐳̇̂,𝝈̇₂=Proj𝜎₂₊−𝜂₂𝐒^𝑇₂+𝜸₂𝝌^𝑇𝐯,̇𝜃𝜽=Proj₊−𝜂_𝜃𝐒^𝑇₂+𝜸₂𝝌^𝑇𝝈₁̂,𝐦̇𝐳𝐡_𝑟=Proj𝑚_𝑟+−𝜂_𝑚𝑟𝐒^𝑇₂+𝜸₂𝝌^𝑇𝐦_𝑟⁻¹𝐁_𝑚𝑟𝜹,𝐦̇_𝑙=Proj𝑚_𝑙⁺−𝜂_𝑚𝑙𝐒^𝑇₂+𝜸₂𝝌^𝑇𝐦_𝑙⁻¹𝐁_𝑚𝑙(,𝐁̇𝐈−𝜹)_𝑚𝑟𝐵=Proj_𝑚𝑟+𝜂_𝑏𝑟𝐒^𝑇₂+𝜸₂𝝌^𝑇𝜹,𝐁̇_𝑚𝑙𝐵=Proj_𝑚𝑙−𝜂_𝑏𝑙𝐒^𝑇₂+𝜸₂𝝌^𝑇(𝐈−𝜹), (58) where the projection mapping is defined as [35] ΩProj_𝑘+(Ξ)=0ifΩ_𝑘=Ω_𝑘maxΩandΞ>0Ξotherwise,Proj_𝑘±(Ξ)=0ifΩ_𝑘=Ω_𝑘maxΩandΞ>0or_𝑘=Ω_𝑘minΩandΞ<0Ξotherwise,Proj₋(Ξ)=0ifΩ=Ω_minandΞ<0Ξotherwise. (59) Substituting (58) into (57), we can obtain the following relation: 𝑉̇≤−𝐒₁𝐒₂Θ𝐒₁𝐒₂^𝑇−𝐊₁‖𝝌‖²−𝐊₂‖𝝌‖−²_𝑘=1𝝀_𝑘+1²𝜍_𝑘+1+54²_𝑘=1𝝀_𝑘+1²+14²_𝑘=1𝝍_𝑘+1²+𝐠_𝑛𝜿^′₂−𝐠_𝑛𝐒^𝑇₂+𝜸₂𝝌^𝑇𝝈₀−𝜽𝝈₁𝐡𝐳̃+𝐠_𝑛𝐒^𝑇₂+𝜸₂𝝌^𝑇𝝈₁𝜽𝐳̂𝐡𝝈₀−𝐠_𝑛𝐒^𝑇₂+𝜸₂𝝌^𝑇𝜺_𝑓+𝐠_𝑛−𝜽𝝈₀𝐡𝐳̃^𝑇−𝜽𝐡𝐳̂𝝈₀−𝝈₀𝐡𝐳̂𝜽𝐳̃≤−𝐊₁‖𝝌‖²+𝝌Γ+Δ_𝑓, (60) where Γ=−𝐠_𝑛𝜸₂𝝈₀−𝜽𝝈₁𝐡𝐳̃+𝐠_𝑛𝜸₂𝝈₁𝜽𝐳̂𝐡𝝈₀−𝐠_𝑛𝜸₂𝜺_𝑓,Δ_𝑓=𝐠_𝑛𝐒^𝑇₂𝝈₁−𝜽𝝈₁𝐡𝐳̃+𝐠_𝑛𝐒^𝑇₂𝝈₁𝜽𝐡‖𝐳̂‖𝝈₀+𝐠_𝑛−𝜽𝝈₀𝐡𝐳̃^𝑇−𝜽𝐡‖𝐳̂‖𝝈₀−𝝈₀𝐡‖𝐳̂‖𝜽𝐳̃+54²_𝑘=1𝝀_𝑘+1²+14²_𝑘=1𝝍_𝑘+1²+𝐠_𝑛𝐒₂𝜺_𝑓+𝐠_𝑛𝜿^′₂=−𝐠_𝑛𝐒₂−𝝈₁−𝜽𝝈₁𝐡2‖𝐳̃‖²−𝐠_𝑛𝝈₁−𝜽𝝈₁𝐡4‖𝐳̃‖²+𝐠_𝑛𝐒₂²−𝝈₁𝜽𝐡𝐳̂²2𝝈₀²²−𝐠_𝑛𝐒₂²−𝐠_𝑛𝝈₁𝜽𝐡𝐳̂²4𝝈₀²−𝐠_𝑛𝜽𝝈₀𝐡‖𝐳̃‖+12𝝈₀𝝈₀²+𝐠_𝑛𝜽4𝝈₀𝝈₀²−𝐠_𝑛𝜽²+𝝈₀𝐡𝐳̂²2‖𝐳̃‖²+𝝈₀𝐡𝐳̂²4‖𝐳̃‖²+54²_𝑘=1𝝀_𝑘+1²+14²_𝑘=1𝝍_𝑘+1²+𝐠_𝑛𝐒₂𝜺_𝑓+𝐠_𝑛𝜿^′₂, (61)(62) and the positive definite matrix Θ_𝑖 is described as 𝐜Θ=¹𝟎𝜸−1₁𝐜₂−1+𝜸₂. (63) Then (60) can be written as 𝑉̇=−12𝝌^𝑇2𝐊₁−𝐈𝜇²1𝝌−2𝐈𝜇𝝌−𝜇Γ^𝑇𝐈𝜇+1𝝌−𝜇Γ2𝜇²‖Γ‖²+Δ_𝑓1≤−2𝝌^𝑇1𝐐𝝌+2𝜇²‖Γ‖²+Δ_𝑓, (64) where 𝐐=(2𝐊₁−𝐈/𝜇²), 𝐊₁>𝐈/2𝜇², and 𝜇 is a positive constant. By integrating both sides of (64) from 𝑡=0 to 𝑡=∞, we obtain the following inequality: 𝑉(𝑇)(0)−1≤𝑉2𝜆_min(𝐐)^𝑇₀‖𝝌‖²+1𝑑𝑡2𝜇^2𝑇₀‖Γ‖²𝑑𝑡+𝐠_𝑛^𝑇₀𝐒₂−𝝈₁𝜽𝐡𝐳̂²2𝝈₀²+1𝑑𝑡4^𝑇_𝑜𝐠_𝑛𝜽𝝈₀𝝈₀²1𝑑𝑡+4^𝑇₀𝝈₀𝐡𝐳̂²‖𝐳̃‖²+𝑑𝑡_𝑇054²_𝑘=1𝝀_𝑘+1²+14²_𝑘=1𝝍_𝑘+1²𝑑𝑡+Δ, (65) for all 0≤𝑇<∞ with Δ=𝐠_𝑛‖𝐒₂‖‖𝜺_𝑓‖+𝐠_𝑛𝜿^′₂. This implies all the states and signals are bounded. Finally, we can conclude that Γ∈𝐿₂[0,∞)∩𝐿_∞[0,∞), ‖𝐒₂‖−(‖𝝈₁𝜽𝐡𝐳̂‖²/2)‖𝝈₀‖∈𝐿₂[0,∞)∩𝐿_∞[0,∞), (𝐠_𝑛𝜽/‖𝝈₀‖)‖𝝈₀‖²∈𝐿₂[0,∞)∩𝐿_∞[0,∞), ‖𝝈₀𝐡𝐳̂‖²‖𝐳̃‖²∈𝐿₂[0,∞)∩𝐿_∞[0,∞), ∑²_𝑘=1‖𝝀_𝑘+1‖²∈𝐿₂[0,∞)∩𝐿_∞[0,∞), ∑²_𝑘=1‖𝝍_𝑘+1‖²∈𝐿₂[0,∞)∩𝐿_∞[0,∞), and Δ∈𝐿₂[0,∞)∩𝐿_∞[0,∞). Then, 𝐒_𝑖→𝟎,𝝌→𝟎, 𝐖_𝑜𝑖→𝟎,𝝆_𝑖→𝟎,𝝈₀→𝟎,𝝈₁→𝟎, 𝝈₂→𝟎, 𝜽→𝟎, 𝐦_𝑟→𝟎, 𝐦_𝑙→𝟎,𝐁_𝑚𝑟→𝟎, and 𝐁_𝑚𝑙→𝟎 as 𝑡→∞ by Barbalat’s Lemma [30].

4. Experimental Example

The experiments to evaluate the proposed control scheme using the Scorbot robot system are described in this section. A photograph of the Scorbot robot is given in Figure 2, where the deadzone occurs in the timing belt. We select only two links (upper arm = link1 and forearm = link2) among the four links of the Scorbot robot manipulator to simplify the verification process of our position control. From (1) to (3), the dynamic equations for the two DOF (degree of freedom) links of the Scorbot robot manipulator are described as 𝐌(𝐪)𝐪̈+𝐂(𝐪,𝐪̇)𝐪̇+𝐆(𝐪)+𝐓_𝑓(𝐪,𝐪̇)+𝐓_𝐿=𝝉,𝝉=𝐧𝐤_𝑡𝐋𝐢,_𝑚𝑑𝐢𝑑𝑡+𝐑_𝑚𝐢+𝐤_𝑏𝐪̇=𝐕, (66) where 𝐌(𝐪)=𝑀₁₁𝑀₁₂𝑀₂₁𝑀₂₂,𝑀₁₁=𝑚₁+𝑚₂𝐿²₁+𝑚₂𝐿²₂+2𝑚₂𝐿₁𝐿₂cos𝑞₂,𝑀₁₂=𝑚₂𝐿²₂+𝐿₁𝐿₂𝑚₂cos𝑞₂,𝑀₂₁=𝑚₂𝐿²₂+𝐿₁𝐿₂𝑚₂cos𝑞₂,𝑀₂₂=𝑚₂𝐿²₂,𝐂(𝐪,𝐪̇)=−𝑚₂𝐿₁𝐿₂sin𝑞₂𝑞̇²₂−2𝑚₂𝐿₁𝐿₂sin𝑞₂𝑞̇₁𝑞̇₂𝑚₂𝐿₁𝐿₂sin𝑞₂𝑞̇²₁,𝐆(𝐪)=𝑚₂𝐿₂𝑔cos𝑞₁+𝑞₂+𝑚₁+𝑚₂𝐿₁𝑔cos𝑞₁𝑚₂𝐿₂𝑔cos𝑞₁+𝑞₂,𝐅_𝑓(𝐪̇)=𝜎𝐪,⁰¹𝑧₁+𝜎₁₁𝑧̇₁+𝜎₂₁𝑞̇₁𝜎₀₂𝑧₂+𝜎₁₂𝑧̇₂+𝜎₂₂𝑞̇₂. (67)

(a)

(b)

Figure 2 

Photograph of the Scorbot robot system: (a) manipulator; (b) timing belt in the actuator.

The parameter values chosen for each link and actuator are represented in Table 1. The sine wave joint and circle motions of the end effector are chosen to be the desired trajectory commands. The sine wave is chosen to be 𝐪_𝑑(𝑡)=0.005sin(1.2566𝑡)(rad). The direct kinematics for a circle trajectory in a task space is given by Φ(𝐪)=𝐿₁sin𝑞₁+𝐿₂sin𝑞₁+𝑞₂𝐿₁cos𝑞₁+𝐿₂cos𝑞₁+𝑞₂. (68) Then, the desired end-effector trajectory of the manipulator becomes 𝐘_𝑑(𝑡)=𝑥_𝑑𝑦_𝑑=𝑥_𝑐()𝑦+𝑅cos𝜔×𝑡_𝑐+𝑅sin(𝜔×𝑡), (69) where 𝑥_𝑐=𝑦_𝑐=−0.1m, 𝑅=2.5mm, and 𝜔=0.45rad/sec. This trajectory makes the manipulator tip trace a circle in the 𝑥₀−𝑦₀ plane with a radius of 𝑅=2.5mm. The desired trajectory 𝐘_𝑑 was translated to the corresponding joint space desired position trajectory 𝐪_𝑑 via the inverse kinematics of the simulated two DOF links manipulators: 𝐪_𝑑=Φ⁻¹𝐘_𝑑=𝑞_𝑑1𝑞_𝑑2=tan⁻¹𝑦_𝑑/𝑥_𝑑−tan⁻¹𝐿₂sin𝑞_𝑑2𝐿₁+𝐿₂cos𝑞_𝑑2tan⁻¹1−𝑥²_𝑑+𝑦²_𝑑−𝐿²₁−𝐿²₂²/2𝐿₁𝐿₂²𝑥²_𝑑+𝑦²_𝑑−𝐿²₁−𝐿²₂/2𝐿₁𝐿₂. (70) The design parameters of the controller are given in Table 2. The fuzzy membership functions for the link1 are chosen to be 𝜇_𝐹11=11+exp𝑆₁₁−4.5×10⁻³²,𝜇_𝐹21=exp−𝑆₁₁−3×10⁻³²,𝜇_𝐹31=exp−𝑆₁₁−1.5×10⁻³²,𝜇_𝐹41=exp−𝑆₁₁+0×10⁻³²,𝜇_𝐹51=exp−𝑆₁₁+1.5×10⁻³²,𝜇_𝐹61=exp−𝑆₁₁+3×10⁻³²,𝜇_𝐹71=11+exp𝑆₁₁+4.5×10⁻³². (71) The second inputs of the fuzzy system in link1 are chosen to be 𝑆̇₁₁ and the membership functions are the same as the above expressions. The fuzzy membership functions for link2 are chosen to be 𝜇_𝐹12=11+exp𝑆₂₁−9×10⁻⁴²,𝜇_𝐹22=exp−𝑆₂₁−6×10⁻⁴²,𝜇_𝐹32=exp−𝑆₂₁−3×10⁻⁴²,𝜇_𝐹42=exp−𝑆₂₁+0×10⁻⁴²,𝜇_𝐹52=exp−𝑆₂₁+3×10⁻⁴²,𝜇_𝐹62=exp−𝑆₂₁+6×10⁻⁴²,𝜇_𝐹72=11+exp𝑆₂₁+9×10⁻⁴². (72) The second inputs of the fuzzy system in link2 are chosen to be 𝑆̇₂₁ and the membership functions are the same as the above expressions. The designed controllers generated in the computer are implemented in the MATLAB RTI system using an MF624 board (Humusoft) [38]. The control signals were transferred into the DC servomotor of the Scorbot robot through the servodrive. The sample frequency was set to1kHz.

Table 1 

Manipulator parameters.

Table 2 

Controller parameters.

In (18), the term 𝐟₃(𝑥₃) is taken to be uncertain and is not approximated by the fuzzy system since this uncertainty can be compensated for by adding a SMC in (56). Firstly, we designed two controllers to evaluate the performance of the proposed system: the proposed adaptive fuzzy DSC system with SMC (AF_DSMC) and the conventional fuzzy DSC system without SMC (AF_DSC). The experimental results are shown in Figure 3. The tracking errors of AF_DSMC system in Figures 3(a) and 3(b) are significantly lower than those of the AF_DSC due to the addition of a SMC into the controller in spite of the similar control inputs as shown in Figures 3(c) and 3(d). These results validate the general robustness property of a SMC. These experimental results show that the proposed dynamic surface control combined with a SMC has a superior performance than that of the conventional DSC system.

(a)

(b)

(c)

(d)

Figure 3 

Experimental results of AF_DSC and AF_DSMC systems for the sine-wave command: (a) tracking errors in link1; (b) tracking errors in link2; (c) 𝑢𝑑1; (d) 𝑢𝑑2.

Next, we designed two additional control systems: an adaptive fuzzy dynamic surface sliding mode controller with deadzone compensation alone (AF_DSMC_D) and an adaptive fuzzy dynamic surface sliding mode controller with both deadzone and friction compensation (AF_DSMC_DF). Experimental results for sine-wave command inputs are shown in Figure 4, where it can be seen that the proposed AF_DSMC_DF has much better tracking performance than those of deadzone compensation alone (AD_DSMC_D) and no compensation (AF_DSMC). In Figure 5, the estimated results for the fuzzy output weights, uncertainty, friction, and deadzone parameters are given for a sine-wave command input.

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(b)

Figure 4 

Experimental results of AF_DSMC, AF_DSMC_D, and AF_DSMC_DF systems for the sine command: (a) tracking errors in link1; (b) tracking errors in link2.

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(b)

(c)

Figure 5 

Experimental results of AF_DSMC, AF_DSMC_D, and AF_DSMC_DF systems for the circle command: (a) tracking results; (b) tracking errors in link1; (c) tracking errors in link2.

To quantitatively evaluate the tracking performance of each control system, the RMS tracking errors for a sine-wave command are summarized in Table 3. The size of the RMS tracking error is decreased by as much as 36% in the proposed control scheme compared to that of the AF_ DSC system.

Table 3 

RMS tracking error for the sine command.

The circle trajectory tracking results are represented in Figure 6, where it can be seen that, due to compensation of friction and deadzone, the circle tracking errors in the AF_DSMC_DF system are much lower than the errors of the AF_DSMC and AF_DSMC_D systems. To evaluate robustness to external disturbance, an additional mass of 1.8 kg was attached to link2 as shown in Figure 7 and pulse disturbances of 2 and 0.2 V for 0.5 sec of pulse widths were applied to each motor drive at 5, 10, 15, and 20 sec, respectively. The two control schemes with the deadzone and friction compensators with SMC (AFE_DSMC_DF) and without SMC (AFE_DSC_DF) were evaluated. As can be seen in Figure 7, the proposed AFE _DSMC_DF system is more robust to disturbance because of the SMC.

Figure 6 

Photograph of the link2 with attached mass.

(a)

(b)

(c)

Figure 7 

Experimental results of AF_DSC_DF and AF_DSMC_DF systems for the circle command against disturbances: (a) tracking results; (b) tracking errors in link1; (c) tracking errors in link2.

The RMS tracking errors for a circle command are summarized in Table 4, where the magnitude of the RMS tracking error is decreased as much as 16.5% in the proposed control scheme compared to that of the AF_ DMSC system.

Table 4 

RMS tracking error for the circle command.

5. Conclusion

In this paper, a nonmodel based dynamic surface sliding mode control scheme has been developed to provide significantly enhanced position tracking performance of a MIMO robot manipulator system in the presence of both deadzone and friction on the part of the actuator. An adaptive fuzzy system is considered to approximate the unknown uncertainty of the complex manipulator dynamics. To enforce the robustness of the DSC scheme, a SMC is also considered as well to introduce estimators for the unknown parameters of the Elastoplastic friction model and deadzone. The recursive steps of the DSC design procedure provide adaptive laws for the controller, friction, and deadzone estimators. As an example application, a Scorbot robot manipulator in the presence of joint friction and deadzone was tested. The favorable positioning performance of the proposed control scheme has been experimentally validated to effectively compensate for deadzone, friction, and uncertainty.

Acknowledgment

This research was supported by The Ministry of Trade, Industry and Energy (MOTIE), Republic of Korea, under the Human Resources Development Program for Specialized Environment Navigation/Localization Technology Research Center support program supervised by the National IT Industry Promotion Agency (NIPA) (H1502-13-1001).