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

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.


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][6][7][8][9][10][11][12][13][14]. In [5], an adaptive strategy was developed for visual tracking problem of robot manipulators based on the image-based look-andmove 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][17][18]. In [16], a task-space controller was developed to guarantee asymptotic tracking of end-effector position and orientation by utilizing a modelbased 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.
ese 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. e 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. e 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][35][36][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. e 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. e remainder of this paper is organized as follows. In Section 2, 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.

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] where M(θ) ∈ R n×n represents the inertia matrix, C(θ, _ θ) ∈ R n×n denotes the Coriolis and Centrifugal matrix, and G(θ) ∈ R n is the vector of gravitational force. θ ∈ R n is the joint-space position, and _ θ ∈ R n and € θ ∈ R n denote the joint-space velocity and acceleration, respectively. u ∈ R n is current input to DC motor, and K T ∈ R n×n is a positive definite constant diagonal matrix that converts actuator inputs u into control torque. In real applications, the value of K T is usually unknown and varies due to external disturbance. In other words, actuator parameter perturbation phenomenon always exists.
Let x ∈ R n 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 Complexity where f(θ) ∈ R n ⟶ R n is a nonlinear differentiable transformation describing the forward kinematics of the manipulator. By differentiating (2), we have where J(θ) � (zf/zθ) ∈ R n×n is the differentiable manipulator Jacobian matrix. In most cases, J(θ) is unknown (because kinematic parameters always vary while performing different tasks, e.g., [23]). erefore, 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.
e inertia matrix M(θ) in the dynamic model (1) is symmetric and uniformly positive definite for all θ.
where ξ ∈ R n is a differentiable vector, _ ξ is the time derivative of ξ, and Y d (θ, _ θ, ξ, _ ξ) ∈ R n×l is the dynamic regressor matrix.

Property 4.
e right-hand side of kinematic (3) is linear in a set of kinematic parameters ϕ k � (ϕ k1 , ϕ k2 , . . . , ϕ km ) T , which leads to where ζ ∈ R n is a vector and Y k (θ, ζ) ∈ R n×m is the kinematic regressor matrix.

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 position θ, joint-space velocity _ θ and task-space position x, but J(θ) 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

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 that x d , _ x d , and € x d are all bounded.

Control Design of Robot Manipulators with Known Model.
Following [23], we define the position tracking error as and task-space reference velocity as where x r is task-space reference position and α is a constant satisfying α > 0. As kinematic parameters ϕ k and J(θ) are known clearly, the joint-space reference velocity can be defined as where θ r is the joint-space reference position and J − 1 (θ) is the inverse of J(θ). In addition, a sliding vector is designed as By differentiating (9) with respect to time, we further get Similarly, when the actuator parameters are known, the dynamics model of DC motor can be described as K T u � τ, where τ ∈ R n is the joint control torque. erefore, the taskspace tracking control law can be given as where K o ∈ R n×n is a positive definite symmetric matrix. By substituting (9)-(11) into (1), we have en, the Lyapunov function is chosen as By differentiating (13) with respect to time and using (12) and Property 1, we can obtain Complexity It implies that s ⟶ 0 as t ⟶ ∞. Multiplying J(θ) to (9), and using (3) and (6), we can further obtain Δ _ x + αΔx � J(θ)s ⟶ 0 as t ⟶ ∞. erefore, we have Δx ⟶ 0 and Δ _ x ⟶ 0 as t ⟶ ∞, since α > 0.

Control Design of Robot Manipulators with Unknown
Model. In practice, we cannot get _ x 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 Property 4, equation (2) can be further reformulated as where ϕ k is the estimation of ϕ k , and J(θ) is the estimation of J(θ), which is computed via ϕ k . Furthermore, we can redefine the joint-space reference velocity as [29].
Substituting (9) and (10) into (1), the equations of dynamics can be expressed as From Property 3, replace the vector ξ in (4) with θ r ; hence, (20) can be rewritten as Now, we define the control law as where K ∈ R n×n is a positive definite constant approximate matrix of K T , K r ∈ R n×n is a positive definite symmetric matrix, and ϕ d is the estimated vector of ϕ d . Furthermore, Y a (δ o )ϕ a is an adaptive term that is designed to deal with the uncertainty in actuator model, where Y a (δ o ) � diag δ o1 , δ o2 , . . . , δ on , and δ oi is the ith element of the vector δ o , which is defined as e adaptation laws are designed as follows: where Γ d ∈ R l×l and Γ k ∈ R m×m are positive definite symmetric matrices, Γ a ∈ R n×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). us, the estimated Jacobian matrix J(θ) and its inverse J − 1 (θ) 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 (22) can be interpreted as a controller using inverse Jacobian matrix feedback, rather than a transposed approximate Jacobian matrix feedback controller as [23]. e last two terms are estimated dynamic and actuator model compensation terms. e 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 where Following [23], since K T and K are all defined as diagonal matrices, we have where ϕ a � [(k t1 /k 1 ) − 1, (k t2 /k 2 ) − 1, . . . , (k tn /k n ) − 1], and k ti and k i denote the ith diagonal elements of the K T and K, respectively. erefore, the last two terms of (27) can be expressed as By substituting (29) into (27), we have 4 Complexity where Δϕ a � K T K − 1 ϕ a − ϕ a . us, the closed-loop system can be described as

Stability Analysis.
rough 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: Differentiating V 1 with respect to time and using By substituting M(θ) _ s from (31), dynamic adaptation law _ ϕ d from (25), and actuator adaptation law _ ϕ a from (26) into (33), we have  [29], the quasi-Lyapunov function candidate can be considered as By differentiating V 2 with respect to time, we have Using (19) and using kinematic adaptation law _ ϕ k , we can write (36) as From Young's inequality, we can derive the inequality Δx T J(θ)s ≤ (α/2)Δx T Δx + [1/(2α)]s T J T (θ)J(θ)s.
erefore, (37), can be simplified as 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, V 2 is a nonincreasing function, and thus Δϕ k and Δx are bounded.
Next, let us discuss the boundedness of other variables. From (7) x and Δ € x 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 Δ _ x ⟶ 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 K r is required to be positive definite and symmetric. (2) e designed parameters φ and α are positive constants, which are also required to satisfy the relations α > 0, φ ≥ 0 and αφ ≤ 1. (3) e designed matrices Γ k and Γ d in (24) and (25) are required to be positive definite symmetric. e designed matrix Γ a is required to be positive definite diagonal. Since the diagonal matrix is symmetric, for convenience, Γ k and Γ d can be set as positive definite diagonal matrices.
(4) Supposing that the parameter φ and the matrices Γ a and Γ d are set to be smaller, but α, K r and the matrix Γ k are chosen larger, then the tracking error could be made smaller and transient performance could be improved. According to [23], we have where s 1 � sin θ 1 , c 1 � cos θ 1 , s 12 � sin(θ 1 + θ 2 ), c 12 � cos(θ 1 + θ 2 ), c o � cos θ o , and s o � sin θ 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.

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.

Trajectory Tracking without Actuator Parameters
Perturbation. In this case, the link lengths were estimated as l 1 � 0.31 m and l 2 � 0.32 m. e actuator model can be equivalently set as a fixed matrix as K T � diag 1.2, 1 { }. In the controller, the actuator model is estimated as K � diag 0.8, 0.9 { }. Following the guidelines summarized in Section 4, the controller parameters α, φ, Γ a , Γ d , Γ k and K r are chosen as α � 3.2, φ � 0.11, Γ a � 0.001I 2 , Γ d � 0.001I 4 , Γ k � 2.8I 3 and K r � 3. Simulation results are presented in Figure 1. Figure 1(a) shows the position tracking performances in a horizontal plane. e tracking errors of end-effector in the task-space (Δx 1 and Δx 2 ) 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. e 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.
To validate the proposed method, the comparison with other controllers is conducted on robot manipulators. e following control laws from [23,29] are employed to complete the tracking objective: 6 Complexity where K v and K p 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]). e comparative results are presented in Figure 2. 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. erefore, it can be concluded that the adaptive term Y a (δ o )ϕ 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-Jacobianbased controller.

Trajectory Control with Actuator Parameters
Perturbation. In this case, actuator parameters may change according to the operating environment temperature. e  and Γ k are chosen as α � 3.5 and Γ k � 2.5I 3 , and the other parameters are the same as those of the first simulation. e performance results in the dynamic trajectory tracking are presented in Figure 3. 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    (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.
Moreover, in order to evaluate the quality of different control approaches, the tracking results of x are also shown in Table 1 in terms of the relative Root Mean Squared Error (RMSE), which is defined: where x(i) represents the ith real trajectory of end-effector and x d (i) represents the ith desired trajectory of end-effector, respectively, and N is the number of samples.
As seen from Table 1, 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 x 1 are reduced nearly 54.7% and 33.1% in comparison with controller (41) and controller (42), respectively. Analogously, the RMSE values of x 2 are reduced nearly 50.7% and 50.4%, respectively. erefore, it further reveals that our approach is more appropriate for the tracking tasks under the perturbation of actuator parameters. Controller (22) Controller (41) Controller (42) Desired Controller (22) Controller (41) Controller (

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][46][47][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.

Data Availability
Some or all data, models, or codes generated or used during the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.