Adaptive Jacobian Based Trajectory Tracking for Redundant Manipulators with Model Uncertainties in Repetitive Tasks

Tracking control of manipulators, which is also called kinematic control, has always been a fundamental problem in robot control, especially for redundant robots with higher degrees of freedom. This problem would become more difficult for systems with model uncertainties. This chapter presents an adaptive tracking controller that considers uncertain physical parameters. Based on the realtime feedback of task-space coordinates, by updating the motion parameters online, a Jacobian adaptive control strategy that does not require cartesian velocity and joint acceleration is established, which makes the controller much simpler. Then the Jacobian pseudo-inverse method is used to obtain the optimal repetitive solution as a secondary task. Lyapunov theory is used to prove that the tracking error of the end effector could asymptotically converge to zero. Numerical simulations verify the effectiveness of the proposed method.


Introduction
Robots have been widely used in industrial, agricultural, aerospace and other fields. Therefore, research on robotics, especially robot control technology, has been a hot issue in recent decades [1][2][3][4][5]. In order to improve the operation accuracy of the robot, tracking control has always been a fundamental problem in robot control, which has attracted wide attention from researchers.
The tracking control of manipulators can be divided into two categories: joint space tracking and task space tracking. The target of joint space tracking is to design a controller to drive each joint of the robot to track a predetermined trajectory (see, for example [6,7] and references therein). Another direction of tracking control is task space tracking, which is to establish the desired trajectory in cartesian space. Since the control commands do not match the target (control commands are sent to the actuators of each joint, and then the end effector is controlled to execute in Cartesian space, the mapping between the two spaces is highly nonlinear), taskspace tracking is more difficult than joint space. Therefore, we should first solve the kinematic inverse problem, that is, obtain the required joint space position or speed, and realize task space tracking. This can be done off-line or online. The desired path 1.1 Introduction discussed. In Sect. 1.4, we provide examples and numerical simulations to verify the effectiveness of the proposed tracking method. Finally, Sect. 1.5 concludes the chapter. Before concluding this section, we emphasize the main contributions of this chapter as follows: • This chapter mainly studies the adaptive kinematic control in the presence of model uncertainties, which is of great significance in practical engineering. • In the controller design process, there is no need to measure the task space velocity and joint acceleration. Therefore, the controller can be easily applied in actual systems. • The contribution also lies in that the proposed repeatability optimization scheme is guaranteed by introducing a variable coefficient, which could ensure the continuity of the joint speed.

Problem Formulation
Without loss of generality, the robot manipulator studied in this chapter is selected as a serial robot, which is most commonly used in industrial applications. The kinematic model of a serial robot manipulator is where θ(t) ∈ R n is the joint angles, and x(t) ∈ R m is the vector describing the position and orientation of the end-effector in cartesian space. f (•) : R n → R m is the forward kinematic mapping of the robot. f (•) is a nonlinear function. Differentiating x(t) with respect to time t, the cartesian velocityẋ(t) is formulated aṡ where J (qθ(t), a k ) = ∂ f (θ (t), a k )/∂θ (t) ∈ R m×n is the Jacobian matrix. As to a redundant manipulator, n > m. a k ∈ R l denotes the vector of kinematic parameters, also called physical parameters, while in this chapter, mainly refers to length of each joint. Therefore, a k is considered as a constant vector. The movement of the end-effector J (θ (t), a k )θ(t) consists of two parts: physical parameter dependent term and joint angle-speed dependent term, and can be described in the linearization-in-parameter form [21]: where Y k (θ (t),θ(t)) ∈ R m×l is called kinematic regressor matrix.
In this chapter, to avoid measuring the task-space velocity, a low-pass filter is used as followsẏ where λ 1 is a positive constant and y is the filtered output of the task-space velocity with initial value y(0) = 0. Rewriting (1.4) leads to y = λ 1ẋ /( p + λ 1 ), (1.5) where p is the Laplace variable. Combining (1.3) and (1.5), we have where W k (0) = 0. For simplicity, we write J (θ ), Y k (θ,θ) as J and Y k , respectively.

Remark 1.1
In real applications, a k has two different forms, which correspond to different meanings. The first value is the actual value of a k , the other one is the nominal value of a n k . a n k is usually a non-calibrated measurement result of a k , which is usually provided by the manufacturer or manual measurement. The real value of a k is difficult to obtain in real applications, and a k is generally not the same as its nominal value of a n k . Due to assembly errors and long-term operations (such as friction, wear, etc.,) besides, robots may operate different tools to perform tasks, which also leads to motion uncertainty. In this case, the direct use of a n k control method can lead to control errors, which is unacceptable in high precision tracking control.

Main Results
In this section, we will show the detailed process of the controller design. Firstly, the ideal case where all parameters are known is considered, then the basic idea of parameter-updating-in-realtime is designed in the case where parameters are unknown, and is then expanded to repeated optimization in null space. Finally, the stability of the closed-loop system is discussed.

Adaptive Tracking Method
Define the tracking error in Cartesian space as (1) Known parameter case When the kinematic parameters a k is perfectly known, the accurate Jacobian matrix J can be obtained, therefore, the reference trajectory can be designed as where k 1 and k 2 are positive control gains. According to Eq. (1.2), the Eq.
Eq. (1.9) can be rewritten asë it is obvious that is e(t) will eventually converge to zero, if k 1 and k 2 are Hurwitz. Combining Eqs. (1.8) and (1.2), and letting initial joint velocityθ(0) be 0, one can easily derive the corresponding control signals of joint speed as beloẇ (2) Unknown parameter case In this situation, since the exact value of a k is unknown, J is unknown. Therefore, in this case, we useĴ instead of J by replacing a k with its estimationâ k , and let a k (0) = a n k , then the estimatedẋ(t) isx(t) =Ĵθ . By replacing a k byâ k , according to (1.3), the estimated cartesian speedx satisfieŝ The modified reference trajectory is thus designed aṡ Since the accurate feedback of cartesian velocityẋ is unavailable, the derivative of tracking errorė =ẋ −ẋ d is also unknown, therefore, we define the alternative value ofė by using the estimated Cartesian speedẋ: then the updating law of kinematic parameters is designed aṡ where Γ 1 is a positive definite diagonal matrix, k 1 , k 2 and k 3 are positive control gains.

Remark 1.3
Without loss of generality, the initial value of the estimated kinematic parameters can be selected according to the nominal value, which can be obtained from handbook or manual measurement. In fact, the adjustment ofâ k (0) does affect the tracking process, which can be verified in the next section. The greater the error betweenâ k (0) and a k , the greater the initial simulation error. However, according to (1.15), no matter what the exact value ofâ k (0), the estimated value ofâ k will eventually converge to a k , which can be verified by stability analysis and numerical experiments.
Now, we are ready to offer a theorem about the task-space tracking problem for robots with uncertain physical parameters using the proposed adaptive controller as below. (1.16) Taking the time derivative of Δx and combining Eqs. (1.14) and (1.16) derives whereã k = a k −â k represents the difference between the real value of physical parameters a k and the estimated oneâ k . Eq. (1.17) can be written as Select a Lyapunov function candidate as follows (1.20) Then we can obtain that Δx, e andã k are all bounded. Based on Eqs. (1.14) and (1.3),Ĵθ,â k and Y kãk are also bounded. Notably that W k (t)ã k is the output of a stable system with bounded input Y k (t)ã k , we have W k (t)ã k is also bounded. Then according to Eq. (1.15),ȧ k is bounded. Differentiating W k (t)ã k with respect to time,

Remark 1.4
We have proved the convergence of the tracking error under the condition of kinematic uncertainties. In fact, when a k is perfectly known, Eq. (1.13) will be degenerated aṡ (1.22) which has the similar form compared with Eq. (1.11). Therefore, known parameter case described in Eq. (1.11) can be considered as a special form of Eq. (1.13).

Remark 1.5
The control velocityq in Eq. (1.13) is not the final result of this chapter. The velocity component in null space is ignored, although it has no effect on the movement of end-effector as well as the stability proof, this part can not be neglected, because the redundancy mechanism is of great engineering significance to the manipulator.
Calculate e, y, and W k (t) by Equation (1.7)

(3) Repeatability optimization
In this subsection, in order to make full use the of redundant design of a redundant manipulator, a repeatability optimization scheme is developed in the null space of the Jacobian matrix, which is helpful to improve the stability and reliability of robots in periodic tasks.
Define a following function to describe a robot's repeatability as where K is a positive parameter scaling the weight of repeatability optimization, θ ini is the initial value of the joint angles. By using gradient descent method, a velocity component in null space can be calculated as where θ ini (i) and θ(i) represent the ith element of θ int and θ , respectively, i = 1, · · · , n. Then the complete form of the proposed adaptive controller iṡ Remarkable that at the beginning stage of the tracking cycle, the repeatability is less important, and then it rises as the task continues. To this end, we set K as a variable: . . are natural numbers, T is the period of cyclic motion. If t < N T + T /2, the robot has just left the initial state to perform a task, thus we let K = 0, this will cause α = 0, the joint control velocity is the same as (1.13). When t > N T + T /2, K increases from 0 to maximum value K max with time, forcing the robot to repeat the initial state. The change curve of K with time is shown in Fig. 1.1.

Remark 1.6
The main reason for this selection of K is to ensure the continuity of joint speed signals during a motion cycle. Notable that the discontinuities of K still appear at the moment T = N T . If the robot can repeat the initial joint state, θ − θ ini would converge to 0, so α can be also regarded as continuous. Therefore, the definition of K in (1.27) is acceptable.

Numeral Simulations
In this section, several groups of numerical experiments are carried out to show the effectiveness of the designed controller. Firstly, a comparative simulation is given to show that the adaptive tracking law could achieve a satisfing performance with the existence of kinematic uncertainties. Secondly, we will check the performance in periodic tasks. Finally, more general trajectories are discussed to show the adaptiveness and robustness of the control algorithm.

Simulation Settings
The vector of initial joint angles is selected as θ ini = [π/2, −π/2, 0, 0] T rad, and the corresponding cartesian position is x ini = [0.6, 0.3] T . Since the exact value of kinematic parameters (see d i in Fig. 1.2), we assume the nominal values to be a n k = [0.25, 0.25, 0.12, 0.18] T m, and letâ k (0) = a n k . The control gains k 1 , k 2 and k 3 are set to be k 1 = 50, k 2 = 50, k 3 = 50, Γ = 10. As to the repeatable tasks, the parameter scaling the velocity component in the null space is selected as K max = 10. The time constant of low-pass filter is λ = 40. It is notable that matrixĴ is essential in the proposed tracking controller, which is used to estimate the actual Jacobian matrix J (q, a k ). To further show the detail of the proposed controller, analytical expression ofĴ is given in Appendix.

Verification of Parameter Estimation
Comparative simulations is firstly carried out to show the effectiveness of the proposed updating law (1.15). The desired path to be tracked is defined as x d (t) = 0.4 + 0.2cos(2t), y d (t) = 0.3 + 0.2sin(2t). In the first simulation, the nominal values are used directly in the tracking control according to Eq. (1.13). By contrast,â k is updated using (1.15) in the comparable simulation, and α is set to be zero (i.e., we didn't use repeatability in this part). Simulation results are shown in Fig. 1.3. Both controllers ensure the boundedness of the tracking error. When a k is known, benefiting from the closed-loop control mechanism, the tracking errors along two axes are much less than 5 mm. The tracking errors with parameter estimation are less

Verification of Repeatability Optimization
Then we check the effectiveness of repeatability optimization. Based on the simulation of the previous part, we introduce the proposed repeatability optimization scheme (i.e., the controller is the same as the adaptive tracking controller in the previous part except α = 0.) Simulation results are shown in Fig. 1.4. The curve of tracking error e is the same as the one when α = 0, showing the property that the velocity component in null space have no influence on the cartesian movement ( Fig. 1.4a). The estimated kinematic parametersâ k are shown in Fig. 1.4b, which slowly converges to a k with time. The error norm ||Y kâk −ẋ|| 2 of the estimated cartesian speed reduced to zero quickly, as shown in Fig. 1.4c. The curve of the repeatability function is shown in Fig. 1.4d, we can observe that when t = T, 2T, · · · , ||q − q ini || 2 equals to zero. This is to say, when repeatability optimization is used, ||q − q ini || 2 changes periodically. Figure 1.4e shows the motion trajectory tracked by its end-effector of the robot manipulator, illustrating the precise tracking of the circle desired trajectory.

Cardioid Tracking
To further verify effectiveness of the proposed control scheme, the manipulator is required to track a cardioid trajectory in 2-D workspace. The desired path is defined as x d (t) = 0.1(2sin(2t) − sin(4t)) + 0.6 m, y d (t) = 0.1(2cos(2t) − cos(4t)) + 0.2 m. Simulation results are shown in Fig. 1.5. The motion trajectory achieved by its end-effector of the manipulator is shown in Fig. 1 Fig. 1.5b, showing that the robot successfully tracks the given trajectory. ||q − q ini || 2 is guaranteed to 0 when t = T, 2T, 3T ( Fig. 1.5e), and the estimated kinematic parameters are shown in Fig. 1.5c. All in all, the proposed controller ensures stable tracking under the condition of model uncertainties, and the repeatability is also achieved.

Summary
In this chapter, an adaptive tracking controller is designed for redundant manipulators. Model uncertainties and repeatability are considered. The control scheme requires neither joint accelerations nor cartesian velocity, which is more suitable in practical engineering. By using the pseudo-inverse method, repeatability is optimized in the null space of Jacobian, the continuous of joint speed is also guaranteed. Future studies will concentrate on the experimental validation of the proposed controller. The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.