# Inverse Jacobian Adaptive Tracking Control of Robot Manipulators with Kinematic, Dynamic, and Actuator Uncertainties

Authors: Bing Zhou; Liang Yang; Chengdong Wang; Yong Chen; Kairui Chen
Journal: Complexity
 (2020)

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## Abstract

In this paper, we mainly solve the adaptive control problem of robot manipulators with uncertain kinematics, dynamics, and actuators parameters, which has been a long-standing, yet unsolved problem in the robotics field, because of the technical difficulties in handling highly coupled effect between control torque and the mentioned uncertainties. To overcome the difficulties, we propose a new Lyapunov-based adaptive control methodology, which effectively fuses the inverse Jacobian technique and the actuator adaptation law, with which the chattering in tracking errors caused by actuator parameter perturbation is well suppressed. It is demonstrated that the asymptotic convergence of all closed-loop signals is guaranteed. Moreover, the effectiveness of our control scheme is illustrated through simulation studies.


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## Body

## 1. Introduction

Control of robot manipulators has attracted a great deal of attentions in the past few decades, due to its wide application in industrial manufacturing [1], military [2], medical [3], and other fields [4]. Some promising results studying robot manipulators have been reported in [5–14]. In [5], an adaptive strategy was developed for visual tracking problem of robot manipulators based on the image-based look-and-move structure, without using image velocity measurements. In [6], by using nonlinear model to depict friction and load change, a switched adaptive controller was designed to achieve asymptotic tracking control for robot manipulators with friction and changing loads. Moreover, in [15], an adaptive sliding mode control scheme based on delay estimation was proposed to deal with uncertainties and external disturbances, and an excellent tracking performance with small chattering effect was guaranteed.It is worth noting that the aforementioned control schemes are all based on joint space (using dynamics), which may be inconvenient in practical application compared with task-space control schemes (using dynamics and kinematics). To cope with this restriction, some interesting results have been reported in [16–18]. In [16], a task-space controller was developed to guarantee asymptotic tracking of end-effector position and orientation by utilizing a model-based observer. In [18], a new task-space controller based on regional feedback was proposed for various control problems in task-space such as singularity problem and limited sensing zone. However, in these aforementioned approaches, the kinematic parameters are assumed to be accurately known. As pointed out earlier in [19, 20], such assumption is hard to be satisfied in practice. For robot manipulator, neglecting the effects of uncertain kinematics and dynamics will lead to the degradation of control performance, especially in high precision control scenario.To address this challenging problem, much results have been achieved in [21–29]. These methods for designing controllers in task-space are all based on the Jacobian matrix. In [22], Galicki proposed a class of absolutely continuous Jacobian transpose robust controllers to solve the problems of uncertain dynamics and external disturbances. In [23], Cheah et al. designed an adaptive Jacobian control method to handle the control problem of robot manipulators with both kinematic and dynamic uncertainties. In [22, 23, 26] and [27], the design of the feedback controllers was based on the transposed Jacobian matrix, and all showed excellent stability characteristics. However, when the manipulator moves in a wide range, using the transposed Jacobian feedback method does not make the manipulator tracking maintain good performance. In [30], Craig proposed another control scheme based on inverse Jacobian matrix, and in fact, the stability of the control system based on inverse Jacobian feedback was reformulated and solved in [21], which theoretically explained that the mechanism of the inverse Jacobian control system can be stabilized. The results in [29, 31] demonstrate the effectiveness of using inverse Jacobian feedback control methods to resolve kinematic uncertainties.It is well known that uncertainty is an important and complex research topic in engineering, and much progress has been made in complex system (e.g., [32, 33]). However, how to deal with the uncertainties in robot system is still an open question. The first challenge stems from the coupling of uncertain kinematics, dynamics, and actuator. It is worth mentioning that, for linear time-invariant systems, we can design closed-loop poles properly to ensure the stability. However, it is not applicable for a nonlinear system (see, e.g., [34–37]), especially under the case of kinematic and dynamic parameters being unknown. Furthermore, there is no in-depth discussion on the decoupling of kinematics, dynamics, and actuator uncertainties. The second challenge arises from the perturbation in actuator parameters. As pointed out in [23], the actuator parameters may also change due to overheating of motor, which may degrade the control performance. In this case, even the dynamic and kinematic parameters can be calibrated, and the overall tracking error of the system cannot be guaranteed to achieve good convergence effect. It has been illustrated through simulation and experiment examples that the chattering phenomenon exists due to the perturbation in actuator parameters.Inspired by the above observation, this paper investigates the adaptive tracking control problem of robot manipulators, in which both kinematics and dynamics are uncertain. Actually, one of the most challenging difficulties in controller design is to search an effective adaptive approach to cope with the uncertainty of actuator model, whose parameters may change after long time running. To deal with this challenge, by designing new adaptation laws, an efficient inverse Jacobian adaptive control scheme is constructed. Our approaches in this paper can be summarized as follows:(1)
Unlike previous available results in the literature on tracking control of robot manipulators, e.g., [28, 29, 38], we take the perturbation of actuator parameters into account in our design, with particular interest in the compensation of unknown actuator uncertainty whose model may not be accurate due to the imprecise measurement. In our scheme, a new adaptive method is developed to cancel the overlarge actuator compensation error and the unknown disturbances, and the development of such a compensation mechanism can well handle the unknown-in-time perturbation of actuator parameters. Hence, our proposed adaptive controller is feasible in dealing with the highly coupled effect of uncertain dynamics, kinematics, and actuator model.(2)
Traditionally, the tracking control algorithms of robot manipulators are based on the transpose Jacobian matrix (see [23, 27, 28] for example), which may not be convenient because what we can only design is the joint velocity in this mode (see, e.g., [29]). By using the inverse Jacobian, a new joint reference velocity is defined to replace the joint velocity command for the control loop, and in addition, combined with new control law, the separation of the kinematics and dynamics is achieved. Moreover, with the fusion of inverse Jacobian and actuator adaptation law, the chattering phenomenon of tracking errors caused by actuator parameter perturbation is successfully suppressed.The remainder of this paper is organized as follows. In Section2, the model of robot manipulators and the control problem are illustrated. Section 3 is devoted to designing an adaptive controller and analyzing the stability of the system. In Section 4, the effectiveness of the proposed adaptive control method is illustrated by simulation experiments. Finally, the conclusions are given in Section 5.


## 2. Mathematical Model of Robot Manipulators and Problem Statement

### 2.1. Dynamics and Kinematics Model of Robot Manipulators

If each actuator of the robot is a direct current (DC) motor, the robot manipulators system can be modeled as [30](1)Mθθ¨+ℂθ,θ˙θ˙+Gθ=KTu,where Mθ∈Rn×n represents the inertia matrix, ℂθ,θ˙∈Rn×n denotes the Coriolis and Centrifugal matrix, and Gθ∈Rn is the vector of gravitational force. θ∈Rn is the joint-space position, and θ˙∈Rn and θ¨∈Rn denote the joint-space velocity and acceleration, respectively. u∈Rn is current input to DC motor, and KT∈Rn×n is a positive definite constant diagonal matrix that converts actuator inputs u into control torque. In real applications, the value of KT is usually unknown and varies due to external disturbance. In other words, actuator parameter perturbation phenomenon always exists.Letx∈Rn denote the position of the end-effector in task-space, and the mapping function between x and θ can be given as [30, 39](2)x=fθ,where fθ∈Rn⟶Rn is a nonlinear differentiable transformation describing the forward kinematics of the manipulator.By differentiating (2), we have(3)x˙=Jθθ˙,where Jθ=∂f/∂θ∈Rn×n is the differentiable manipulator Jacobian matrix. In most cases, Jθ is unknown (because kinematic parameters always vary while performing different tasks, e.g., [23]). Therefore, the position/velocity in task-space can not be directly obtained from the kinematics formula given above. Instead, the information on position/velocity in task space can be measured by utilizing cameras.To facilitate the design and stability analysis, four properties related to robot dynamics (1) and kinematics (3) are listed below [23, 39, 40].Property 1.
The inertia matrixMθ in the dynamic model (1) is symmetric and uniformly positive definite for all θ.Property 2.
The matricesℂθ,θ˙ and M˙θ satisfy XTM˙θ−2ℂθ,θ˙X=0, ∀X∈Rn, and M˙θ−2ℂθ,θ˙ is a skew-symmetric matrix.Property 3.
The left-hand side of dynamic (1) is linear in a set of physical parameters ϕd=ϕd1,ϕd2,…,ϕdlT, and thus(4)Mθξ˙+ℂθ,θ˙ξ+Gθ=Ydθ,θ˙,ξ,ξ˙ϕd,where ξ∈Rn is a differentiable vector, ξ˙ is the time derivative of ξ, and Ydθ,θ˙,ξ,ξ˙∈Rn×l is the dynamic regressor matrix.Property 4.
The right-hand side of kinematic (3) is linear in a set of kinematic parameters ϕk=ϕk1,ϕk2,…,ϕkmT, which leads to(5)Jθζ=Ykθ,ζϕk,where ζ∈Rn is a vector and Ykθ,ζ∈Rn×m is the kinematic regressor matrix.


### 2.2. Problem Statement

In real applications, when a robot manipulator grabs tools, the kinematic and dynamic parameters of robot will inevitably change. Meanwhile, the actuator model may be uncertain due to overheating of motor or changes in ambient. Moreover, the kinematics, dynamics, and uncertain actuator model are highly coupled, which makes the design of the controller more difficult. In the following content, our goal is to design an adaptive controller with separation characteristics to solve the tracking error or unstable response caused by uncertain parameters.Also, what needs to be explained is the measurable state parameters in practical application, joint-space positionθ, joint-space velocity θ˙ and task-space position x, but Jθ cannot be accurately obtained by measurement, which means that x˙ cannot be got directly through (3). In this paper, the control purpose is to make limt⟶∞xdt−xt=0, where xd denotes the desired trajectory.


## 2.1. Dynamics and Kinematics Model of Robot Manipulators

If each actuator of the robot is a direct current (DC) motor, the robot manipulators system can be modeled as [30](1)Mθθ¨+ℂθ,θ˙θ˙+Gθ=KTu,where Mθ∈Rn×n represents the inertia matrix, ℂθ,θ˙∈Rn×n denotes the Coriolis and Centrifugal matrix, and Gθ∈Rn is the vector of gravitational force. θ∈Rn is the joint-space position, and θ˙∈Rn and θ¨∈Rn denote the joint-space velocity and acceleration, respectively. u∈Rn is current input to DC motor, and KT∈Rn×n is a positive definite constant diagonal matrix that converts actuator inputs u into control torque. In real applications, the value of KT is usually unknown and varies due to external disturbance. In other words, actuator parameter perturbation phenomenon always exists.Letx∈Rn denote the position of the end-effector in task-space, and the mapping function between x and θ can be given as [30, 39](2)x=fθ,where fθ∈Rn⟶Rn is a nonlinear differentiable transformation describing the forward kinematics of the manipulator.By differentiating (2), we have(3)x˙=Jθθ˙,where Jθ=∂f/∂θ∈Rn×n is the differentiable manipulator Jacobian matrix. In most cases, Jθ is unknown (because kinematic parameters always vary while performing different tasks, e.g., [23]). Therefore, the position/velocity in task-space can not be directly obtained from the kinematics formula given above. Instead, the information on position/velocity in task space can be measured by utilizing cameras.To facilitate the design and stability analysis, four properties related to robot dynamics (1) and kinematics (3) are listed below [23, 39, 40].Property 1.
The inertia matrixMθ in the dynamic model (1) is symmetric and uniformly positive definite for all θ.Property 2.
The matricesℂθ,θ˙ and M˙θ satisfy XTM˙θ−2ℂθ,θ˙X=0, ∀X∈Rn, and M˙θ−2ℂθ,θ˙ is a skew-symmetric matrix.Property 3.
The left-hand side of dynamic (1) is linear in a set of physical parameters ϕd=ϕd1,ϕd2,…,ϕdlT, and thus(4)Mθξ˙+ℂθ,θ˙ξ+Gθ=Ydθ,θ˙,ξ,ξ˙ϕd,where ξ∈Rn is a differentiable vector, ξ˙ is the time derivative of ξ, and Ydθ,θ˙,ξ,ξ˙∈Rn×l is the dynamic regressor matrix.Property 4.
The right-hand side of kinematic (3) is linear in a set of kinematic parameters ϕk=ϕk1,ϕk2,…,ϕkmT, which leads to(5)Jθζ=Ykθ,ζϕk,where ζ∈Rn is a vector and Ykθ,ζ∈Rn×m is the kinematic regressor matrix.


## 2.2. Problem Statement

In real applications, when a robot manipulator grabs tools, the kinematic and dynamic parameters of robot will inevitably change. Meanwhile, the actuator model may be uncertain due to overheating of motor or changes in ambient. Moreover, the kinematics, dynamics, and uncertain actuator model are highly coupled, which makes the design of the controller more difficult. In the following content, our goal is to design an adaptive controller with separation characteristics to solve the tracking error or unstable response caused by uncertain parameters.Also, what needs to be explained is the measurable state parameters in practical application, joint-space positionθ, joint-space velocity θ˙ and task-space position x, but Jθ cannot be accurately obtained by measurement, which means that x˙ cannot be got directly through (3). In this paper, the control purpose is to make limt⟶∞xdt−xt=0, where xd denotes the desired trajectory.


## 3. Adaptive Tracking Control

We first discuss the controller design of robot manipulators with known model, then design adaptive laws and control law for the robot with uncertain kinematics, dynamics, and actuator model to realize trajectory tracking control. In addition, we assume thatxd, x˙d, and x¨d are all bounded.


### 3.1. Control Design of Robot Manipulators with Known Model

Following [23], we define the position tracking error as(6)Δx=x−xd,and task-space reference velocity as(7)x˙r=x˙d−αΔx,where xr is task-space reference position and α is a constant satisfying α>0.As kinematic parametersϕk and Jθ are known clearly, the joint-space reference velocity can be defined as(8)θ˙r=J−1θx˙r,where θr is the joint-space reference position and J−1θ is the inverse of Jθ. In addition, a sliding vector is designed as(9)s=θ˙−θ˙r.By differentiating (9) with respect to time, we further get(10)s˙=θ¨−θ¨r.Similarly, when the actuator parameters are known, the dynamics model of DC motor can be described asKTu=τ, where τ∈Rn is the joint control torque. Therefore, the task-space tracking control law can be given as(11)τ=−Kos+Mθθ¨r+ℂθ,θ˙θ˙r+Gθ,where Ko∈Rn×n is a positive definite symmetric matrix.By substituting (9)–(11) into (1), we have(12)Mθs˙+ℂθ,θ˙s=−Kos.Then, the Lyapunov function is chosen as(13)Vo=12sTMθs.By differentiating (13) with respect to time and using (12) and Property 1, we can obtain(14)V˙o=12sTM˙θs+sTMθs˙=sT12M˙θ−ℂθ,θ˙s−sTKos=−sTKos≤0.It implies thats⟶0 as t⟶∞. Multiplying Jθ to (9), and using (3) and (6), we can further obtain Δx˙+αΔx=Jθs⟶0 as t⟶∞. Therefore, we have Δx⟶0 and Δx˙⟶0 as t⟶∞, since α>0.


### 3.2. Control Design of Robot Manipulators with Unknown Model

In practice, we cannot getx˙ or θ˙ directly through (3) and (8) since kinematic parameters are uncertain. In this part, we design the controller by using task-space reference velocity, joint-space reference velocity, and estimated Jacobian matrix.From Property4, equation (2) can be further reformulated as(15)x˙r=J^θθ˙r=Ykθ,θ˙rϕ^k,where ϕ^k is the estimation of ϕk, and J^θ is the estimation of Jθ, which is computed via ϕ^k. Furthermore, we can redefine the joint-space reference velocity as [29].(16)θ˙r=J^−1θx˙r,(17)θ¨r=J^−1θx¨r−J^˙θθ˙r.Substituting (6), (15), and (16) into (9), we have(18)s=J−1θx˙−Jθθ˙r=J−1θΔx˙+αΔx+Ykθ,θ˙rΔϕk.From [41, 42], left-multiplying (18) by Jθ, we can further get(19)Δx˙=Jθs−αΔx−Ykθ,θ˙rΔϕk,where Δϕk=ϕ^k−ϕk is the estimation error of kinematic parameter.Substituting (9) and (10) into (1), the equations of dynamics can be expressed as(20)Mθs˙+ℂθs+Mθθ¨r+ℂθ,θ˙θ˙r+Gθ=KTu.From Property3, replace the vector ξ in (4) with θr; hence, (20) can be rewritten as(21)Mθs˙+ℂθ,θ˙s+Ydθ,θ˙,θ˙r,θ¨rϕd=KTu.Now, we define the control law as(22)u=−K−1Krs+K−1Ydθ,θ˙,θ˙r,θ¨rϕ^d+K−1Yaδoϕ^a,where K∈Rn×n is a positive definite constant approximate matrix of KT, Kr∈Rn×n is a positive definite symmetric matrix, and ϕ^d is the estimated vector of ϕd. Furthermore, Yaδoϕ^a is an adaptive term that is designed to deal with the uncertainty in actuator model, where Yaδo=diagδo1,δo2,…,δon, and δoi is the ith element of the vector δo, which is defined as(23)δo=Krs−Ydθ,θ˙,θ˙r,θ¨rϕ^d.The adaptation laws are designed as follows:(24)ϕ^˙k=ΓkYkTθ,θ˙rφΔx˙+Δx,(25)ϕ^˙d=−ΓdYdTθ,θ˙,θ˙r,θ¨rs,(26)ϕ^˙a=−ΓaYaTδos,where Γd∈Rl×l and Γk∈Rm×m are positive definite symmetric matrices, Γa∈Rn×n is a positive definite diagonal matrix, and φ is a design constant that satisfies φ≥0 and αφ≤1.Remark 1.
An assumption is firstly made that the robot manipulators will not reach the singular configuration, and its kinematic can be parameterized linearly as (15). Thus, the estimated Jacobian matrix J^θ and its inverse J^−1θ remain nonsingular while being updated by θ and ϕ^k.Remark 2.
In the control law (22), the first term is a feedback law contains task-space position and velocity errors and kinematic parameters estimation error, which can be further rewritten as −K−1KrJ−1θΔx˙+αΔx+Ykθ,θ˙rΔϕk. Hence, (22) can be interpreted as a controller using inverse Jacobian matrix feedback, rather than a transposed approximate Jacobian matrix feedback controller as [23]. The last two terms are estimated dynamic and actuator model compensation terms. The control law (22) expands the adaptive scheme of task space and kinematic parameter estimation error in [29] based on inverse Jacobian feedback and further increases the actuator parameter estimation error feedback, which provides it with the ability to handle the uncertainty of the actuator model.
By substituting (22) into (21), we get(27)Mθs˙+ℂθ,θ˙s=−Krs+Ydθ,θ˙,θ˙r,θ¨rΔϕd+KTK−1Yaδoϕ^a−KTK−1−Iδo,where Δϕd=ϕ^d−ϕd.
Following [23], since KT and K are all defined as diagonal matrices, we have(28)KTK−1−Iδo=diagδo1,δo2,…,δonϕ¯a=Yaδoϕ¯a,where ϕ¯a=kt1/k1−1,kt2/k2−1,…,ktn/kn−1, and kti and ki denote the ith diagonal elements of the KT and K, respectively. Therefore, the last two terms of (27) can be expressed as(29)KTK−1Yaδoϕ^a−KTK−1−Iδo=YaδoKTK−1ϕ^a−ϕ¯a.
By substituting (29) into (27), we have(30)Mθs˙+ℂθ,θ˙s=−Krs+Ydθ,θ˙,θ˙r,θ¨rΔϕd+YaδoΔϕ¯a,where Δϕ¯a=KTK−1ϕ^a−ϕ¯a. Thus, the closed-loop system can be described as(31)Δx˙=Jθs−αΔx−Ykθ,θ˙rΔϕk,Mθs˙+ℂθ,θ˙s=−Krs+Ydθ,θ˙,θ˙r,θ¨rΔϕd+YaδoΔϕ¯a.


### 3.3. Stability Analysis

Through the above efforts, the relationships among the stability of closed-loop system and design parameters have been successfully established, as shown in the following theorem.Theorem 1.
Consider the closed-loop system consisting of robot manipulator (1) with uncertain kinematics, dynamics, and actuator model. Under control of the adaptive controller (22), (24)–(26), the position and velocity tracking errors of the task-space converge to zero.Proof:.
Consider the following Lyapunov-like function in terms of dynamics:(32)V1=12sTMθs+12ΔϕdTΓd−1Δϕd+12Δϕ¯aTΓa−1KKT−1Δϕ¯a.
DifferentiatingV1 with respect to time and using Δϕ˙d=ϕ^˙d, Δϕ¯˙a=KTK−1ϕ^˙a, we have(33)V˙1=sTMθs˙+12sTM˙θs+ΔϕdTΓd−1ϕ^˙d+Δϕ¯aTΓa−1ϕ^˙a.
By substitutingMθs˙ from (31), dynamic adaptation law ϕ^˙d from (25), and actuator adaptation law ϕ^˙a from (26) into (33), we have(34)V˙1=−sTKrs≤0.
From Property1 and (34), we can obtain that V1 is bounded. This implies that s, Δϕd, and Δϕ¯a are bounded vectors and, hence, implies that ϕ^d and ϕ^a are bounded.
Since all the joints of the manipulator are rotatable, it can be concluded thatJθ is bounded, and hence we have Jθs∈ℒ2. Thus, according to ℒ2 norm properties [[43], p. 17], there exists a positive constant ψ such that ∫0tJθsTJθsdσ≤ψ. Therefore, following the result in [[44], p. 118] and [29], the quasi-Lyapunov function candidate can be considered as(35)V2=1−αφ2ΔxTΔx+12ΔϕkTΓk−1Δϕk+12αψ−∫0tJθsTJθsdσ.
By differentiatingV2 with respect to time, we have(36)V˙2=1−αφΔxTΔx˙+ΔϕkTΓk−1ϕ^˙k−12αsTJTθJθs=1−αφΔxT−αΔx−Ykθ,θ˙rΔϕk+Jθs+ΔϕkTΓk−1ϕ^˙k−12αsTJTθJθs=−α1−αφΔxTΔx−1−αφΔϕkTYkTθ,θ˙rΔx−1−αφ2αsTJTθJθs−φ2sTJTθJθs+ΔϕkTΓk−1ϕ^˙k+1−αφΔxTJθs.
UsingJθs=Δx˙+αΔx+Ykθ,θ˙rΔϕk from (19) and using kinematic adaptation law ϕ^˙k, we can write (36) as(37)V˙2=−α1−αφΔxTΔx+ΔϕkTYkTθ,θ˙rφΔx˙+αφΔx+1−αφΔxTJθs−1−αφ2αsTJTθJθs−φ2sTJTθJθs=−α1−αφΔxTΔx−φ2Δx˙+αΔxTΔx˙+αΔx+1−αφΔxTJθs−1−αφ2αsTJTθJθs−φ2ΔϕkTYkTθ,θ˙rYkθ,θ˙rΔϕk.
From Young’s inequality, we can derive the inequalityΔxTJθs≤α/2ΔxTΔx+1/2αsTJTθJθs. Therefore, (37), can be simplified as(38)V˙2≤−φ2Δx˙+αΔxTΔx˙+αΔx−α1−αφ2ΔxTΔx≤0.
Referring to the input-output properties of the exponentially stable and strictly proper closed-loop systems given in [[43], p. 59], the result of (38) implies that for αφ≤1, V2 is a nonincreasing function, and thus Δϕk and Δx are bounded.
Next, let us discuss the boundedness of other variables. From (7), we can obtain that x˙r is bounded if x˙d and Δx are bounded. Thus, θ˙r in (8) is bounded if J^θ is of full rank. Since s and θ˙r are bounded, θ˙ from (9) is also bounded, and x˙ from (3) is bounded. Thus, Δx is uniformly continuous and x¨r=x¨d−αΔx˙ is bounded. This implies that ϕ^˙k, J^˙θ and θ¨r are bounded. Hence, from the closed-loop robot manipulators system (31), we can obtain that s˙ is bounded. The boundedness of s˙ implies that θ¨ is bounded from (10), and x¨=Jθθ¨+J˙θθ˙ is also bounded. Thus, Δx¨ is bounded, and Δx˙ is uniformly continuous.
For simplicity, defineWt=Ykθ,θ˙rΔϕk, and we can obtain that Wt and W˙t are bounded since θ˙, θ˙r, θ¨r, Δϕk and Δϕ˙k are bounded. Differentiating (37) with respect to time, we further get(39)V¨2=−1αΔx˙Tα2φΔx˙+Δx¨−αΔxTΔx˙+φΔx¨−1−αφαΔx˙TW˙t−1−αφαWTtΔx¨−1αWTtW˙t,which implies that V¨2 is bounded since θ˙, θ˙r, θ¨r, Δϕk, Δϕ˙k, Δx, Δx˙ and Δx¨ are all bounded, and thus V˙2 is uniformly continuous. Finally, using the conclusions obtained above and Barbalat’s lemma, we obtain that Δx⟶0 and Δx˙⟶0 as t⟶∞.
In order to select design parameters more efficiently, we summarize some guidelines as follows:(1)
In the controller (22), the designed matrix K is required to be positive definite and diagonal, and the designed matrix Kr is required to be positive definite and symmetric.(2)
The designed parametersφ and α are positive constants, which are also required to satisfy the relations α>0,φ≥0 and αφ≤1.(3)
The designed matricesΓk and Γd in (24) and (25) are required to be positive definite symmetric. The designed matrix Γa is required to be positive definite diagonal. Since the diagonal matrix is symmetric, for convenience, Γk and Γd can be set as positive definite diagonal matrices.(4)
Supposing that the parameterφ and the matrices Γa and Γd are set to be smaller, but α, Kr and the matrix Γk are chosen larger, then the tracking error could be made smaller and transient performance could be improved.


## 3.1. Control Design of Robot Manipulators with Known Model

Following [23], we define the position tracking error as(6)Δx=x−xd,and task-space reference velocity as(7)x˙r=x˙d−αΔx,where xr is task-space reference position and α is a constant satisfying α>0.As kinematic parametersϕk and Jθ are known clearly, the joint-space reference velocity can be defined as(8)θ˙r=J−1θx˙r,where θr is the joint-space reference position and J−1θ is the inverse of Jθ. In addition, a sliding vector is designed as(9)s=θ˙−θ˙r.By differentiating (9) with respect to time, we further get(10)s˙=θ¨−θ¨r.Similarly, when the actuator parameters are known, the dynamics model of DC motor can be described asKTu=τ, where τ∈Rn is the joint control torque. Therefore, the task-space tracking control law can be given as(11)τ=−Kos+Mθθ¨r+ℂθ,θ˙θ˙r+Gθ,where Ko∈Rn×n is a positive definite symmetric matrix.By substituting (9)–(11) into (1), we have(12)Mθs˙+ℂθ,θ˙s=−Kos.Then, the Lyapunov function is chosen as(13)Vo=12sTMθs.By differentiating (13) with respect to time and using (12) and Property 1, we can obtain(14)V˙o=12sTM˙θs+sTMθs˙=sT12M˙θ−ℂθ,θ˙s−sTKos=−sTKos≤0.It implies thats⟶0 as t⟶∞. Multiplying Jθ to (9), and using (3) and (6), we can further obtain Δx˙+αΔx=Jθs⟶0 as t⟶∞. Therefore, we have Δx⟶0 and Δx˙⟶0 as t⟶∞, since α>0.


## 3.2. Control Design of Robot Manipulators with Unknown Model

In practice, we cannot getx˙ or θ˙ directly through (3) and (8) since kinematic parameters are uncertain. In this part, we design the controller by using task-space reference velocity, joint-space reference velocity, and estimated Jacobian matrix.From Property4, equation (2) can be further reformulated as(15)x˙r=J^θθ˙r=Ykθ,θ˙rϕ^k,where ϕ^k is the estimation of ϕk, and J^θ is the estimation of Jθ, which is computed via ϕ^k. Furthermore, we can redefine the joint-space reference velocity as [29].(16)θ˙r=J^−1θx˙r,(17)θ¨r=J^−1θx¨r−J^˙θθ˙r.Substituting (6), (15), and (16) into (9), we have(18)s=J−1θx˙−Jθθ˙r=J−1θΔx˙+αΔx+Ykθ,θ˙rΔϕk.From [41, 42], left-multiplying (18) by Jθ, we can further get(19)Δx˙=Jθs−αΔx−Ykθ,θ˙rΔϕk,where Δϕk=ϕ^k−ϕk is the estimation error of kinematic parameter.Substituting (9) and (10) into (1), the equations of dynamics can be expressed as(20)Mθs˙+ℂθs+Mθθ¨r+ℂθ,θ˙θ˙r+Gθ=KTu.From Property3, replace the vector ξ in (4) with θr; hence, (20) can be rewritten as(21)Mθs˙+ℂθ,θ˙s+Ydθ,θ˙,θ˙r,θ¨rϕd=KTu.Now, we define the control law as(22)u=−K−1Krs+K−1Ydθ,θ˙,θ˙r,θ¨rϕ^d+K−1Yaδoϕ^a,where K∈Rn×n is a positive definite constant approximate matrix of KT, Kr∈Rn×n is a positive definite symmetric matrix, and ϕ^d is the estimated vector of ϕd. Furthermore, Yaδoϕ^a is an adaptive term that is designed to deal with the uncertainty in actuator model, where Yaδo=diagδo1,δo2,…,δon, and δoi is the ith element of the vector δo, which is defined as(23)δo=Krs−Ydθ,θ˙,θ˙r,θ¨rϕ^d.The adaptation laws are designed as follows:(24)ϕ^˙k=ΓkYkTθ,θ˙rφΔx˙+Δx,(25)ϕ^˙d=−ΓdYdTθ,θ˙,θ˙r,θ¨rs,(26)ϕ^˙a=−ΓaYaTδos,where Γd∈Rl×l and Γk∈Rm×m are positive definite symmetric matrices, Γa∈Rn×n is a positive definite diagonal matrix, and φ is a design constant that satisfies φ≥0 and αφ≤1.Remark 1.
An assumption is firstly made that the robot manipulators will not reach the singular configuration, and its kinematic can be parameterized linearly as (15). Thus, the estimated Jacobian matrix J^θ and its inverse J^−1θ remain nonsingular while being updated by θ and ϕ^k.Remark 2.
In the control law (22), the first term is a feedback law contains task-space position and velocity errors and kinematic parameters estimation error, which can be further rewritten as −K−1KrJ−1θΔx˙+αΔx+Ykθ,θ˙rΔϕk. Hence, (22) can be interpreted as a controller using inverse Jacobian matrix feedback, rather than a transposed approximate Jacobian matrix feedback controller as [23]. The last two terms are estimated dynamic and actuator model compensation terms. The control law (22) expands the adaptive scheme of task space and kinematic parameter estimation error in [29] based on inverse Jacobian feedback and further increases the actuator parameter estimation error feedback, which provides it with the ability to handle the uncertainty of the actuator model.
By substituting (22) into (21), we get(27)Mθs˙+ℂθ,θ˙s=−Krs+Ydθ,θ˙,θ˙r,θ¨rΔϕd+KTK−1Yaδoϕ^a−KTK−1−Iδo,where Δϕd=ϕ^d−ϕd.
Following [23], since KT and K are all defined as diagonal matrices, we have(28)KTK−1−Iδo=diagδo1,δo2,…,δonϕ¯a=Yaδoϕ¯a,where ϕ¯a=kt1/k1−1,kt2/k2−1,…,ktn/kn−1, and kti and ki denote the ith diagonal elements of the KT and K, respectively. Therefore, the last two terms of (27) can be expressed as(29)KTK−1Yaδoϕ^a−KTK−1−Iδo=YaδoKTK−1ϕ^a−ϕ¯a.
By substituting (29) into (27), we have(30)Mθs˙+ℂθ,θ˙s=−Krs+Ydθ,θ˙,θ˙r,θ¨rΔϕd+YaδoΔϕ¯a,where Δϕ¯a=KTK−1ϕ^a−ϕ¯a. Thus, the closed-loop system can be described as(31)Δx˙=Jθs−αΔx−Ykθ,θ˙rΔϕk,Mθs˙+ℂθ,θ˙s=−Krs+Ydθ,θ˙,θ˙r,θ¨rΔϕd+YaδoΔϕ¯a.


## 3.3. Stability Analysis

Through the above efforts, the relationships among the stability of closed-loop system and design parameters have been successfully established, as shown in the following theorem.Theorem 1.
Consider the closed-loop system consisting of robot manipulator (1) with uncertain kinematics, dynamics, and actuator model. Under control of the adaptive controller (22), (24)–(26), the position and velocity tracking errors of the task-space converge to zero.Proof:.
Consider the following Lyapunov-like function in terms of dynamics:(32)V1=12sTMθs+12ΔϕdTΓd−1Δϕd+12Δϕ¯aTΓa−1KKT−1Δϕ¯a.
DifferentiatingV1 with respect to time and using Δϕ˙d=ϕ^˙d, Δϕ¯˙a=KTK−1ϕ^˙a, we have(33)V˙1=sTMθs˙+12sTM˙θs+ΔϕdTΓd−1ϕ^˙d+Δϕ¯aTΓa−1ϕ^˙a.
By substitutingMθs˙ from (31), dynamic adaptation law ϕ^˙d from (25), and actuator adaptation law ϕ^˙a from (26) into (33), we have(34)V˙1=−sTKrs≤0.
From Property1 and (34), we can obtain that V1 is bounded. This implies that s, Δϕd, and Δϕ¯a are bounded vectors and, hence, implies that ϕ^d and ϕ^a are bounded.
Since all the joints of the manipulator are rotatable, it can be concluded thatJθ is bounded, and hence we have Jθs∈ℒ2. Thus, according to ℒ2 norm properties [[43], p. 17], there exists a positive constant ψ such that ∫0tJθsTJθsdσ≤ψ. Therefore, following the result in [[44], p. 118] and [29], the quasi-Lyapunov function candidate can be considered as(35)V2=1−αφ2ΔxTΔx+12ΔϕkTΓk−1Δϕk+12αψ−∫0tJθsTJθsdσ.
By differentiatingV2 with respect to time, we have(36)V˙2=1−αφΔxTΔx˙+ΔϕkTΓk−1ϕ^˙k−12αsTJTθJθs=1−αφΔxT−αΔx−Ykθ,θ˙rΔϕk+Jθs+ΔϕkTΓk−1ϕ^˙k−12αsTJTθJθs=−α1−αφΔxTΔx−1−αφΔϕkTYkTθ,θ˙rΔx−1−αφ2αsTJTθJθs−φ2sTJTθJθs+ΔϕkTΓk−1ϕ^˙k+1−αφΔxTJθs.
UsingJθs=Δx˙+αΔx+Ykθ,θ˙rΔϕk from (19) and using kinematic adaptation law ϕ^˙k, we can write (36) as(37)V˙2=−α1−αφΔxTΔx+ΔϕkTYkTθ,θ˙rφΔx˙+αφΔx+1−αφΔxTJθs−1−αφ2αsTJTθJθs−φ2sTJTθJθs=−α1−αφΔxTΔx−φ2Δx˙+αΔxTΔx˙+αΔx+1−αφΔxTJθs−1−αφ2αsTJTθJθs−φ2ΔϕkTYkTθ,θ˙rYkθ,θ˙rΔϕk.
From Young’s inequality, we can derive the inequalityΔxTJθs≤α/2ΔxTΔx+1/2αsTJTθJθs. Therefore, (37), can be simplified as(38)V˙2≤−φ2Δx˙+αΔxTΔx˙+αΔx−α1−αφ2ΔxTΔx≤0.
Referring to the input-output properties of the exponentially stable and strictly proper closed-loop systems given in [[43], p. 59], the result of (38) implies that for αφ≤1, V2 is a nonincreasing function, and thus Δϕk and Δx are bounded.
Next, let us discuss the boundedness of other variables. From (7), we can obtain that x˙r is bounded if x˙d and Δx are bounded. Thus, θ˙r in (8) is bounded if J^θ is of full rank. Since s and θ˙r are bounded, θ˙ from (9) is also bounded, and x˙ from (3) is bounded. Thus, Δx is uniformly continuous and x¨r=x¨d−αΔx˙ is bounded. This implies that ϕ^˙k, J^˙θ and θ¨r are bounded. Hence, from the closed-loop robot manipulators system (31), we can obtain that s˙ is bounded. The boundedness of s˙ implies that θ¨ is bounded from (10), and x¨=Jθθ¨+J˙θθ˙ is also bounded. Thus, Δx¨ is bounded, and Δx˙ is uniformly continuous.
For simplicity, defineWt=Ykθ,θ˙rΔϕk, and we can obtain that Wt and W˙t are bounded since θ˙, θ˙r, θ¨r, Δϕk and Δϕ˙k are bounded. Differentiating (37) with respect to time, we further get(39)V¨2=−1αΔx˙Tα2φΔx˙+Δx¨−αΔxTΔx˙+φΔx¨−1−αφαΔx˙TW˙t−1−αφαWTtΔx¨−1αWTtW˙t,which implies that V¨2 is bounded since θ˙, θ˙r, θ¨r, Δϕk, Δϕ˙k, Δx, Δx˙ and Δx¨ are all bounded, and thus V˙2 is uniformly continuous. Finally, using the conclusions obtained above and Barbalat’s lemma, we obtain that Δx⟶0 and Δx˙⟶0 as t⟶∞.
In order to select design parameters more efficiently, we summarize some guidelines as follows:(1)
In the controller (22), the designed matrix K is required to be positive definite and diagonal, and the designed matrix Kr is required to be positive definite and symmetric.(2)
The designed parametersφ and α are positive constants, which are also required to satisfy the relations α>0,φ≥0 and αφ≤1.(3)
The designed matricesΓk and Γd in (24) and (25) are required to be positive definite symmetric. The designed matrix Γa is required to be positive definite diagonal. Since the diagonal matrix is symmetric, for convenience, Γk and Γd can be set as positive definite diagonal matrices.(4)
Supposing that the parameterφ and the matrices Γa and Γd are set to be smaller, but α, Kr and the matrix Γk are chosen larger, then the tracking error could be made smaller and transient performance could be improved.


## 4. Simulation and Analysis

### 4.1. Parameter Design

A planar manipulator with 2 degrees-of-freedom (DOFs) is considered, in which the lengths of the first and second links are roughly set to 0.31 m and 0.35 m, respectively. The mass of the first link together with actuator is approximately equal to 1 kg. The second link, with actuator and the payload attached, can be regarded as an augmented link with a mass of approximately 3 kg. Let the equivalent-length of the object held by the robot be 0.10 meters and the grab angle be45°. The desired tracking trajectory is given as xd=0.33+0.1sin0.54+3t,0.41+0.1cos0.54+3tT.According to [23], we have(40)x˙r=J^θθ˙r=Ykθ,θ˙rϕ^k=−s1θ˙r1−s12θ˙r1+θ˙r2−c12θ˙r1+θ˙r2c1θ˙r1c12θ˙r1+θ˙r2−s12θ˙r1+θ˙r2×l^1l^2+l^ocol^oso,where s1=sinθ1, c1=cosθ1, s12=sinθ1+θ2, c12=cosθ1+θ2, co=cosθ^o, and so=sinθ^o. l^1 and l^2 are the estimated link lengths, and l^o and θ^o are the estimated equivalent-length and grasping angle of the object, respectively.


### 4.2. Result and Analysis

At first, the robot is required to follow a typical kind of the reference trajectory in task space. In what follows, simulation studies are carried out to validate our control scheme.


#### 4.2.1. Trajectory Tracking without Actuator Parameters Perturbation

In this case, the link lengths were estimated asl^1=0.31m and l^2=0.32m. The actuator model can be equivalently set as a fixed matrix as KT=diag1.2,1. In the controller, the actuator model is estimated as K=diag0.8,0.9. Following the guidelines summarized in Section 4, the controller parameters α, φ, Γa, Γd, Γk and Kr are chosen as α=3.2, φ=0.11, Γa=0.001I2, Γd=0.001I4, Γk=2.8I3 and Kr=3.5I2, respectively. The initial values of the estimated parameters are selected as ϕ^d0=0,0,0,0T, ϕ^k0=0.26,0.08,0.07T and ϕ^a0=−19,−16T; in fact, their actual values are ϕd=0.6278,0.2875,0.1162,0.2013T, ϕk=0.30,0.31,0.087T and ϕa=−19.02,−16.01T.Simulation results are presented in Figure1. Figure 1(a) shows the position tracking performances in a horizontal plane. The tracking errors of end-effector in the task-space (Δx1 and Δx2) are presented in Figure 1(b). As seen from Figures 1(a) and 1(b), our proposed control scheme guarantees the convergence of the tracking error. The joint control torques of the tracking process are shown in Figure 1(c). As seen from the figures, the tracking controls are satisfactory in terms of the proposed method. It is noted that the added unknown actuator models do not degrade the tracking performances in the case of tracking control without actuator parameters perturbation.Figure 1
End-effector position tracking performance without actuator parameters perturbation. (a) Tracking performance. (b) Tracking errors. (c) Control torques.
(a)(b)(c)To validate the proposed method, the comparison with other controllers is conducted on robot manipulators. The following control laws from [23, 29] are employed to complete the tracking objective:(41)u′=−K−1J^Tθ,ϕ^kKvΔx˙+KpΔx+K−1Ydθ,θ˙,θ˙r,θ¨rϕ^d+K−1Yaδoϕ^a,(42)u″=−K−1Krs+K−1Ydθ,θ.,θ.r,θ..rϕ^d,where Kv and Kp are symmetric positive definite gain matrices. Specifically, the controller (41) is based on transpose Jacobian feedback (method in [23]), and the controller (42) does not contain the adaptive term for uncertain actuator model (method in [29]).The comparative results are presented in Figure2. As seen from Figures 2(a) and 2(b), the controllers (41) and (42) are not exactly tracking the desired trajectory. Moreover, with these two controllers, the tracking errors fail to converge to zero in a short time under the same control conditions. It can be further obtained by the qualitative analysis that the proposed controller (22) has a more stable tracking performance. Therefore, it can be concluded that the adaptive term Yaδoϕ^a designed in the controller (22) can effectively handle the uncertainty of the actuator model. In addition, the inverse-Jacobian-based controller (22) has better tracking performance than the transposed-Jacobian-based controller.Figure 2
Comparisons of end-effector position tracking performance. (a) Comparisons of tracking performance. (b) Comparisons of tracking errors. (c) Comparisons of control torques.
(a)(b)(c)


#### 4.2.2. Trajectory Control with Actuator Parameters Perturbation

In this case, actuator parameters may change according to the operating environment temperature. The control objective in this case is to design appropriate control torque to force the actual trajectory to converge to the desired one in the presence of actuator parameters perturbation. To simulate this dynamic procedure, the diagonal transmission matrixK is set to the next state values: KT=diag1.2,1 in 0,5s , KT=diag2,1.2 in 5,13s and KT=diag4,3 in 13,20s, and the designed parameter K is provided as K=diag0.8,0.9. The controller parameters α and Γk are chosen as α=3.5 and Γk=2.5I3, and the other parameters are the same as those of the first simulation.The performance results in the dynamic trajectory tracking are presented in Figure3. From the results shown in Figures 3(a) and 3(b), it is apparent that the proposed controller can guarantee that the tracking errors asymptotically converge to zero. Furthermore, the simulation results in Figures 3(a)–3(c) reveal that, with our control scheme, the effect on stability caused by actuator parameters perturbation is successfully suppressed.Figure 3
End-effector position tracking performance with actuator parameters perturbation. (a) Tracking performance. (b) Tracking errors. (c) Control torques.
(a)(b)(c)Similar to case 1, the comparisons of our controller (22) with controllers (41) and (42) have been carried out, and the results are presented in Figure 4. From Figures 4(a)–4(c), it is apparent that, with our method, a better tracking performance is guaranteed in the case that the actuator parameters perturbation exists.Figure 4
Comparisons of end-effector position tracking performance. (a) Comparisons of tracking performance. (b) Comparisons of tracking errors. (c) Comparisons of control torques.
(a)(b)(c)Moreover, in order to evaluate the quality of different control approaches, the tracking results ofx are also shown in Table 1 in terms of the relative Root Mean Squared Error (RMSE), which is defined:(43)RMSE=∑i=1Nxi−xdi2N,where xi represents the ith real trajectory of end-effector and xdi represents the ith desired trajectory of end-effector, respectively, and N is the number of samples.Table 1
RMSE values of proposed methods and other methods.
RMSEProposed case 1Proposed case 2Controller (41)Controller (42)RMSE ofx10.03450.03460.07650.0517RMSE ofx20.01260.01270.02580.0256As seen from Table1, in case 1 and case 2, similar RMSE values (only with a difference of 0.0001) are achieved, which indicates that the proposed controller can well solve the perturbation problem of actuator parameters. In particular, with proposed case 2, the RMSE values of x1 are reduced nearly 54.7% and 33.1% in comparison with controller (41) and controller (42), respectively. Analogously, the RMSE values of x2 are reduced nearly 50.7% and 50.4%, respectively. Therefore, it further reveals that our approach is more appropriate for the tracking tasks under the perturbation of actuator parameters.


## 4.1. Parameter Design

A planar manipulator with 2 degrees-of-freedom (DOFs) is considered, in which the lengths of the first and second links are roughly set to 0.31 m and 0.35 m, respectively. The mass of the first link together with actuator is approximately equal to 1 kg. The second link, with actuator and the payload attached, can be regarded as an augmented link with a mass of approximately 3 kg. Let the equivalent-length of the object held by the robot be 0.10 meters and the grab angle be45°. The desired tracking trajectory is given as xd=0.33+0.1sin0.54+3t,0.41+0.1cos0.54+3tT.According to [23], we have(40)x˙r=J^θθ˙r=Ykθ,θ˙rϕ^k=−s1θ˙r1−s12θ˙r1+θ˙r2−c12θ˙r1+θ˙r2c1θ˙r1c12θ˙r1+θ˙r2−s12θ˙r1+θ˙r2×l^1l^2+l^ocol^oso,where s1=sinθ1, c1=cosθ1, s12=sinθ1+θ2, c12=cosθ1+θ2, co=cosθ^o, and so=sinθ^o. l^1 and l^2 are the estimated link lengths, and l^o and θ^o are the estimated equivalent-length and grasping angle of the object, respectively.


## 4.2. Result and Analysis

At first, the robot is required to follow a typical kind of the reference trajectory in task space. In what follows, simulation studies are carried out to validate our control scheme.


### 4.2.1. Trajectory Tracking without Actuator Parameters Perturbation

In this case, the link lengths were estimated asl^1=0.31m and l^2=0.32m. The actuator model can be equivalently set as a fixed matrix as KT=diag1.2,1. In the controller, the actuator model is estimated as K=diag0.8,0.9. Following the guidelines summarized in Section 4, the controller parameters α, φ, Γa, Γd, Γk and Kr are chosen as α=3.2, φ=0.11, Γa=0.001I2, Γd=0.001I4, Γk=2.8I3 and Kr=3.5I2, respectively. The initial values of the estimated parameters are selected as ϕ^d0=0,0,0,0T, ϕ^k0=0.26,0.08,0.07T and ϕ^a0=−19,−16T; in fact, their actual values are ϕd=0.6278,0.2875,0.1162,0.2013T, ϕk=0.30,0.31,0.087T and ϕa=−19.02,−16.01T.Simulation results are presented in Figure1. Figure 1(a) shows the position tracking performances in a horizontal plane. The tracking errors of end-effector in the task-space (Δx1 and Δx2) are presented in Figure 1(b). As seen from Figures 1(a) and 1(b), our proposed control scheme guarantees the convergence of the tracking error. The joint control torques of the tracking process are shown in Figure 1(c). As seen from the figures, the tracking controls are satisfactory in terms of the proposed method. It is noted that the added unknown actuator models do not degrade the tracking performances in the case of tracking control without actuator parameters perturbation.Figure 1
End-effector position tracking performance without actuator parameters perturbation. (a) Tracking performance. (b) Tracking errors. (c) Control torques.
(a)(b)(c)To validate the proposed method, the comparison with other controllers is conducted on robot manipulators. The following control laws from [23, 29] are employed to complete the tracking objective:(41)u′=−K−1J^Tθ,ϕ^kKvΔx˙+KpΔx+K−1Ydθ,θ˙,θ˙r,θ¨rϕ^d+K−1Yaδoϕ^a,(42)u″=−K−1Krs+K−1Ydθ,θ.,θ.r,θ..rϕ^d,where Kv and Kp are symmetric positive definite gain matrices. Specifically, the controller (41) is based on transpose Jacobian feedback (method in [23]), and the controller (42) does not contain the adaptive term for uncertain actuator model (method in [29]).The comparative results are presented in Figure2. As seen from Figures 2(a) and 2(b), the controllers (41) and (42) are not exactly tracking the desired trajectory. Moreover, with these two controllers, the tracking errors fail to converge to zero in a short time under the same control conditions. It can be further obtained by the qualitative analysis that the proposed controller (22) has a more stable tracking performance. Therefore, it can be concluded that the adaptive term Yaδoϕ^a designed in the controller (22) can effectively handle the uncertainty of the actuator model. In addition, the inverse-Jacobian-based controller (22) has better tracking performance than the transposed-Jacobian-based controller.Figure 2
Comparisons of end-effector position tracking performance. (a) Comparisons of tracking performance. (b) Comparisons of tracking errors. (c) Comparisons of control torques.
(a)(b)(c)


### 4.2.2. Trajectory Control with Actuator Parameters Perturbation

In this case, actuator parameters may change according to the operating environment temperature. The control objective in this case is to design appropriate control torque to force the actual trajectory to converge to the desired one in the presence of actuator parameters perturbation. To simulate this dynamic procedure, the diagonal transmission matrixK is set to the next state values: KT=diag1.2,1 in 0,5s , KT=diag2,1.2 in 5,13s and KT=diag4,3 in 13,20s, and the designed parameter K is provided as K=diag0.8,0.9. The controller parameters α and Γk are chosen as α=3.5 and Γk=2.5I3, and the other parameters are the same as those of the first simulation.The performance results in the dynamic trajectory tracking are presented in Figure3. From the results shown in Figures 3(a) and 3(b), it is apparent that the proposed controller can guarantee that the tracking errors asymptotically converge to zero. Furthermore, the simulation results in Figures 3(a)–3(c) reveal that, with our control scheme, the effect on stability caused by actuator parameters perturbation is successfully suppressed.Figure 3
End-effector position tracking performance with actuator parameters perturbation. (a) Tracking performance. (b) Tracking errors. (c) Control torques.
(a)(b)(c)Similar to case 1, the comparisons of our controller (22) with controllers (41) and (42) have been carried out, and the results are presented in Figure 4. From Figures 4(a)–4(c), it is apparent that, with our method, a better tracking performance is guaranteed in the case that the actuator parameters perturbation exists.Figure 4
Comparisons of end-effector position tracking performance. (a) Comparisons of tracking performance. (b) Comparisons of tracking errors. (c) Comparisons of control torques.
(a)(b)(c)Moreover, in order to evaluate the quality of different control approaches, the tracking results ofx are also shown in Table 1 in terms of the relative Root Mean Squared Error (RMSE), which is defined:(43)RMSE=∑i=1Nxi−xdi2N,where xi represents the ith real trajectory of end-effector and xdi represents the ith desired trajectory of end-effector, respectively, and N is the number of samples.Table 1
RMSE values of proposed methods and other methods.
RMSEProposed case 1Proposed case 2Controller (41)Controller (42)RMSE ofx10.03450.03460.07650.0517RMSE ofx20.01260.01270.02580.0256As seen from Table1, in case 1 and case 2, similar RMSE values (only with a difference of 0.0001) are achieved, which indicates that the proposed controller can well solve the perturbation problem of actuator parameters. In particular, with proposed case 2, the RMSE values of x1 are reduced nearly 54.7% and 33.1% in comparison with controller (41) and controller (42), respectively. Analogously, the RMSE values of x2 are reduced nearly 50.7% and 50.4%, respectively. Therefore, it further reveals that our approach is more appropriate for the tracking tasks under the perturbation of actuator parameters.


## 4.2.1. Trajectory Tracking without Actuator Parameters Perturbation

In this case, the link lengths were estimated asl^1=0.31m and l^2=0.32m. The actuator model can be equivalently set as a fixed matrix as KT=diag1.2,1. In the controller, the actuator model is estimated as K=diag0.8,0.9. Following the guidelines summarized in Section 4, the controller parameters α, φ, Γa, Γd, Γk and Kr are chosen as α=3.2, φ=0.11, Γa=0.001I2, Γd=0.001I4, Γk=2.8I3 and Kr=3.5I2, respectively. The initial values of the estimated parameters are selected as ϕ^d0=0,0,0,0T, ϕ^k0=0.26,0.08,0.07T and ϕ^a0=−19,−16T; in fact, their actual values are ϕd=0.6278,0.2875,0.1162,0.2013T, ϕk=0.30,0.31,0.087T and ϕa=−19.02,−16.01T.Simulation results are presented in Figure1. Figure 1(a) shows the position tracking performances in a horizontal plane. The tracking errors of end-effector in the task-space (Δx1 and Δx2) are presented in Figure 1(b). As seen from Figures 1(a) and 1(b), our proposed control scheme guarantees the convergence of the tracking error. The joint control torques of the tracking process are shown in Figure 1(c). As seen from the figures, the tracking controls are satisfactory in terms of the proposed method. It is noted that the added unknown actuator models do not degrade the tracking performances in the case of tracking control without actuator parameters perturbation.Figure 1
End-effector position tracking performance without actuator parameters perturbation. (a) Tracking performance. (b) Tracking errors. (c) Control torques.
(a)(b)(c)To validate the proposed method, the comparison with other controllers is conducted on robot manipulators. The following control laws from [23, 29] are employed to complete the tracking objective:(41)u′=−K−1J^Tθ,ϕ^kKvΔx˙+KpΔx+K−1Ydθ,θ˙,θ˙r,θ¨rϕ^d+K−1Yaδoϕ^a,(42)u″=−K−1Krs+K−1Ydθ,θ.,θ.r,θ..rϕ^d,where Kv and Kp are symmetric positive definite gain matrices. Specifically, the controller (41) is based on transpose Jacobian feedback (method in [23]), and the controller (42) does not contain the adaptive term for uncertain actuator model (method in [29]).The comparative results are presented in Figure2. As seen from Figures 2(a) and 2(b), the controllers (41) and (42) are not exactly tracking the desired trajectory. Moreover, with these two controllers, the tracking errors fail to converge to zero in a short time under the same control conditions. It can be further obtained by the qualitative analysis that the proposed controller (22) has a more stable tracking performance. Therefore, it can be concluded that the adaptive term Yaδoϕ^a designed in the controller (22) can effectively handle the uncertainty of the actuator model. In addition, the inverse-Jacobian-based controller (22) has better tracking performance than the transposed-Jacobian-based controller.Figure 2
Comparisons of end-effector position tracking performance. (a) Comparisons of tracking performance. (b) Comparisons of tracking errors. (c) Comparisons of control torques.
(a)(b)(c)


## 4.2.2. Trajectory Control with Actuator Parameters Perturbation

In this case, actuator parameters may change according to the operating environment temperature. The control objective in this case is to design appropriate control torque to force the actual trajectory to converge to the desired one in the presence of actuator parameters perturbation. To simulate this dynamic procedure, the diagonal transmission matrixK is set to the next state values: KT=diag1.2,1 in 0,5s , KT=diag2,1.2 in 5,13s and KT=diag4,3 in 13,20s, and the designed parameter K is provided as K=diag0.8,0.9. The controller parameters α and Γk are chosen as α=3.5 and Γk=2.5I3, and the other parameters are the same as those of the first simulation.The performance results in the dynamic trajectory tracking are presented in Figure3. From the results shown in Figures 3(a) and 3(b), it is apparent that the proposed controller can guarantee that the tracking errors asymptotically converge to zero. Furthermore, the simulation results in Figures 3(a)–3(c) reveal that, with our control scheme, the effect on stability caused by actuator parameters perturbation is successfully suppressed.Figure 3
End-effector position tracking performance with actuator parameters perturbation. (a) Tracking performance. (b) Tracking errors. (c) Control torques.
(a)(b)(c)Similar to case 1, the comparisons of our controller (22) with controllers (41) and (42) have been carried out, and the results are presented in Figure 4. From Figures 4(a)–4(c), it is apparent that, with our method, a better tracking performance is guaranteed in the case that the actuator parameters perturbation exists.Figure 4
Comparisons of end-effector position tracking performance. (a) Comparisons of tracking performance. (b) Comparisons of tracking errors. (c) Comparisons of control torques.
(a)(b)(c)Moreover, in order to evaluate the quality of different control approaches, the tracking results ofx are also shown in Table 1 in terms of the relative Root Mean Squared Error (RMSE), which is defined:(43)RMSE=∑i=1Nxi−xdi2N,where xi represents the ith real trajectory of end-effector and xdi represents the ith desired trajectory of end-effector, respectively, and N is the number of samples.Table 1
RMSE values of proposed methods and other methods.
RMSEProposed case 1Proposed case 2Controller (41)Controller (42)RMSE ofx10.03450.03460.07650.0517RMSE ofx20.01260.01270.02580.0256As seen from Table1, in case 1 and case 2, similar RMSE values (only with a difference of 0.0001) are achieved, which indicates that the proposed controller can well solve the perturbation problem of actuator parameters. In particular, with proposed case 2, the RMSE values of x1 are reduced nearly 54.7% and 33.1% in comparison with controller (41) and controller (42), respectively. Analogously, the RMSE values of x2 are reduced nearly 50.7% and 50.4%, respectively. Therefore, it further reveals that our approach is more appropriate for the tracking tasks under the perturbation of actuator parameters.


## 5. Conclusion

In this paper, we cope with the adaptive control problem of robot manipulators by utilizing inverse-Jacobian-based technique. To remove the effect on tracking performance caused by the perturbation in actuator parameters, this paper proposes a new inverse Jacobian tracking control approach, which differs from those traditionally presented in that an actuator parameter transform matrix is additionally incorporated in the controller and corresponding adaptation laws are designed to deal with actuator uncertainty. Moreover, the asymptotic convergence of tracking error is proved by the strict Lyapunov stability analysis. Finally, the simulation and comparison results are illustrated to validate our control scheme.However, how to achieve finite time convergence and the optimal control performance is still a challenging problem in trajectory tracking control. Some interesting results have been reported in [45–48]. Specifically, the solution proposed in [47] gave a novel clue to address this challenge, and the finite-time stabilization technique may be the topic of our future research.



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*Source: 5070354-2020-08-24.xml*