Adaptive Task-Space Manipulator Control with Parametric Uncertainties in Kinematics and Dynamics

: This paper aims to deal with the problem of robot tracking control in the presence of parametric uncertainties in kinematics and dynamics. We propose a simple and e ﬀ ective adaptive control scheme that includes adaptation laws for unknown constant kinematic and dynamic parameters. In addition, instead of convolution-type ﬁltered di ﬀ erentiation, we designed a new observer to estimate velocity in the task space, and the proposed adaptive control requires no acceleration measurement in the joint space. Using the Lyapunov stability and Barbalat’s lemma, we show that by appropriately choosing design parameters, the tracking errors and estimation errors in task space can asymptotically converge to zero. Through numerical simulation on a two-link robot with a ﬁxed camera, we illustrate the design procedures and demonstrate the feasibility of the proposed adaptive control scheme for the trajectory tracking of robot manipulators.


Introduction
Through visual feedback, human beings can perform various tasks in an unknown environment without a priori knowledge. Human can adapt to unforeseen changes by fast visual responses and dexterous hand manipulations. For example, a person can easily move his hand to grip a tool at different positions and in varying orientations. It seems that human beings do not require thorough understanding of the kinematics and dynamics of the human eye-to-hand system. This flexibility to adapt to unknown variations might arise from the consequences of low-speed hand motions and high-gain sensorimotor control. Nevertheless, a well-designed and flexible robot control system should be able to adapt to parametric variations in kinematics and dynamics to meet the demands of industrial-robot applications that require high-speed manipulations for better productivity and low control energy for better efficiency [1].
Adaptive tracking control schemes in joint space have been proposed for more than two decades to deal with parametric uncertainties in dynamics [2][3][4][5]. For instance, Middelton and Goodwin [3] investigated the adaptive control of rigid link manipulator systems using linear estimation techniques together with a computed torque control. The proposed adaptive control algorithms were shown to be globally convergent and do not require acceleration measurements. Spong and Ortega [4] proposed an adaptive inverse dynamics control for rigid robots to relax the assumption that the inverse of the estimated inertia matrix must remain bounded. Furthermore, using linear parameterization and skew-symmetric properties from the inertia matrix, Slotine and Li [5] defined a sliding vector and proposed an adaptive control algorithm that does not require the measurement of joint acceleration.
In general, task-space control algorithms can use vision sensors to provide visual feedback and thus compute the joint control torque using the Jacobian matrix, defined as a transformation from the joint space to the image space. Assuming kinematics is known, adaptive control schemes in the joint space

Dynamics
The dynamics of n-link rigid robot manipulators with revolute joints can be described by the following Euler-Lagrange equations of motion: where q ∈ R n is the joint space vector, M(q) ∈ R n×n is the symmetric and positive definite inertia matrix, C q, q ∈ R n is the vector of centrifugal and Coriolis forces, G(q) ∈ R n is the vector of gravitational forces, and τ ∈ R n is the vector of joint torques. Let us use λ M (A), λ m (A) for the largest and smallest eigenvalues, respectively, of a matrix, A. We denote the Euclidean norm for an n × 1 vector x by x = √ x T x.

Properties
For the revolute robots, we state several fundamental properties of the dynamic equation of motion [19].

Property 1.
The inertia matrix is symmetric, positive definite, and is bounded by q is skew symmetric using the Christoffel symbols to define C q, . q . q.
where the dynamic regressor Y d q, q is an n × p d matrix and θ d is a p d × 1 vector of constant dynamic parameters.

Property 4. Kinematic regressor
where x ∈ R n is the task space vector, J(q, θ k ) ∈ R n×n is the Jacobian matrix from the joint space to the task space, the kinematic regressor Y k q, . q is an n × p k matrix, and θ k is a p k × 1 vector of constant kinematic parameters.
, and assume f : R n × R + → R n is locally Lipschitz in x and uniform in t. Let V : R n × R + → R + be a continuously differentiable function such that ∀t ≥ 0, ∀x ∈ R n , where γ 1 and γ 2 are class K ∞ functions and W is a positive semi-definite continuous function. Then, the solution of If W is positive definite, the equilibrium x = 0 is uniformly asymptotically stable.

Adaptive Tracking Control in the Task Space
In this section, we introduce the adaptive tracking control for robot manipulators. First, we define the estimation errors and the filtered variables for tracking and adaptation. Second, we proposed a controller for tracking control of robot manipulators and an adaptation law for dynamic and kinematic parameters.
Before deriving the tracking control, we will first define the estimation error and wherex is the estimated position, x d is the desired position, .x is the estimated velocity from the observer defined later, and .
x d is the desired velocity. DenoteĴ q,θ k as the estimate of Jacobian J(q, θ k ).
AssumingĴ −1 q,θ k is non-singular, we can define the reference signal . q r and .. ..
where α is a constant related to the convergence of the tracking error. Define the sliding vector as s ≡ . q − . q r (9) Notice that the dynamic regressor Y r can be expressed as We propose the following adaptive tracking control: where k v and k p are positive constants andθ d is the estimate of θ d . The update laws for dynamic and kinematic parameters are and .θ where Γ d and Γ k are diagonal and positive matrices.
The observer for the velocity in task space is where λ is a constant related to the convergence of the estimation error. The proposed adaptive control scheme is shown in Figure 1.

Stability Analysis
In this section, we will analyze the stability of the proposed control, the observer, and the system parameter adaptation scheme described in Section 3.
The estimation errors of dynamic and kinematic parameters are defined as follows and the vector Now, we can state the following theorem for the adaptive tracking control, the velocity observer, and parameter adaptation in the task space. Theorem 1. For dynamic system (1), using adaptive tracking control (11), parameter update law (12) and (13), and observer (14), the design parameters , ; feedback gains , ; and adaptation matrices , can be chosen such that the vector globally asymptotically converges to zero.
Proof. In order to perform stability analysis, we consider the Lyapunov function candidate Taking the time derivative of gives Using the definition of , we have Using the control in (11) and the definition in (15), we have Substituting ( ) from (20) with Property 2, the adaptive update law (12) and (13), the observer (14) into (19), and using the definition of in (16) with Property 4, we have From the definition (9) and (5)-(7), we can compute as follows:

Stability Analysis
In this section, we will analyze the stability of the proposed control, the observer, and the system parameter adaptation scheme described in Section 3.
The estimation errors of dynamic and kinematic parameters are defined as follows and the vector Now, we can state the following theorem for the adaptive tracking control, the velocity observer, and parameter adaptation in the task space. Theorem 1. For dynamic system (1), using adaptive tracking control (11), parameter update law (12) and (13), and observer (14), the design parameters α, λ; feedback gains k v , k p ; and adaptation matrices Γ d , Γ k can be chosen such that the vector y globally asymptotically converges to zero.
Proof. In order to perform stability analysis, we consider the Lyapunov function candidate Taking the time derivative of V gives Using the definition of s, we have Using the control in (11) and the definition in (15), we have s from (20) with Property 2, the adaptive update law (12) and (13), the observer (14) into (19), and using the definition of θ k in (16) with Property 4, we have From the definition (9) and (5)-(7), we can compute s as follows: Substituting s from (23) in (22), we have We can obtain the following sufficient condition for stability: Then, it follows from (24) that Using Lemma 1, we can conclude that y = e T x e T v x T T globally asymptotically converges to zero.
This completes the proof of Theorem 1.

Corollary 1. The estimation error in velocity
.
x asymptotically converges to zero.
Proof. Taking the derivative of x in (4) and using the observer (14), we have .
It follows that ..
It is straightforward to show that Y k q,  x in (29) is bounded because θ k and x are bounded from the stability analysis in Theorem 1. Consequently, from Barbalat's lemma, we can conclude that .
x asymptotically converges to zero.

Dynamic Model
To illustrate the design of the proposed adaptive control scheme in the task space, we give an example of a two-link rigid robot. The dynamics of the robot in joint space can be formulated by the following matrices in (1): . q 2 −m 2 l 1 l c2 sin(q 2 ) .
The image Jacobian matrix J I for a fixed camera can be given as: where v 1 and v 2 represent the camera ratio of scaling parameters over the depth with respect to the image frame. In this simulation, v 1 = 400 and v 2 = 500 are used. The Jacobian matrix J e can be derived for a two-link robot: The Jacobian matrix defined in Property 4 can be expressed as The velocity in the task space is .
The vector of kinematic parameters θ k can be expressed as follows: where Then, the kinematic regressor matrix Y k ∈ R 2×4 can be expressed as

Design
To test the performance of the proposed scheme, we used the initial conditions for the system as follows: The exact dynamic parameter vector could be obtained as The following initial parameter vector was used in the simulation: Subsequently, the design parameters were chosen as where I n represents an n × n identity matrix. It is easy to verify that the design parameters satisfy the stability condition in (26). The desired trajectories can be expressed using pixels in an image frame: x d1 = 420 + 120 cos(4t) (37) For the problem of set-point control, the desired position in pixels was x dp = [540 200] T Figures 2-9 show the simulation results of the proposed scheme on the two-link robot for trajectory tracking. The position trajectories in the task space are shown in Figure 2. The tracking error in terms of position and velocity is shown in Figures 3 and 4, respectively. Clearly, from the simulation results, without the velocity measurement in the task space, the proposed control scheme could successfully track the desired trajectory in the case of uncertain parametric dynamics and kinematics parameters. The convergence rate could be modified by the choice of design parameters. We can see from the parameter estimation shown in Figure 5 that the dynamic parameters were bounded and convergent. According to the stability analysis, it can be understood that there was no guarantee of convergence to the true value of dynamic parameters. In contrast to the results of dynamic parameters, the kinematic parameters shown in Figure 6 converged to the true values for trajectory tracking. The estimation tracking errors for position and velocity are shown in Figures 7 and 8, respectively. The observer successfully estimated the velocity in the task space without measurement. Figure 9 presents the control torques for the trajectory tracking in the task space.

Comparison
A comparative simulation of set-point control was conducted by using the proposed adaptive control schemes for the same model and the parameters described in Section 5.1. The simulation results are shown in Figures 10-17 for the set-point control. As shown in Figure 10 Figures 11 and 12, respectively. Figure 13 presents the estimation of dynamic parameters. In particular, from the simulation results shown in Figure 14, we can see that the kinematic parameters were not necessarily converging to the true values due to the rank deficiency of the kinematic regressor. Figures 15 and 16 show the estimation errors in terms of positions and velocities. Due to the set-point control, the control torques reached the steady state after 0.5 s, as shown in Figure 17.  Figures 2-9 show the simulation results of the proposed scheme on the two-link robot for trajectory tracking. The position trajectories in the task space are shown in Figure 2. The tracking error in terms of position and velocity is shown in Figures 3 and 4, respectively. Clearly, from the simulation results, without the velocity measurement in the task space, the proposed control scheme could successfully track the desired trajectory in the case of uncertain parametric dynamics and kinematics parameters. The convergence rate could be modified by the choice of design parameters. We can see from the parameter estimation shown in Figure 5 that the dynamic parameters were bounded and convergent. According to the stability analysis, it can be understood that there was no guarantee of convergence to the true value of dynamic parameters. In contrast to the results of dynamic parameters, the kinematic parameters shown in Figure 6 converged to the true values for trajectory tracking. The estimation tracking errors for position and velocity are shown in Figures 7  and 8, respectively. The observer successfully estimated the velocity in the task space without measurement. Figure 9 presents the control torques for the trajectory tracking in the task space.

Comparison
A comparative simulation of set-point control was conducted by using the proposed adaptive control schemes for the same model and the parameters described in Section 5.1. The simulation results are shown in Figures 10-17 for the set-point control. As shown in Figure 10 Figures 11 and 12, respectively. Figure 13 presents the estimation of dynamic parameters. In particular, from the simulation results shown in Figure 14, we can see that the kinematic parameters were not necessarily converging to the true values due to the rank deficiency of the kinematic regressor. Figures 15 and 16 show the estimation errors in terms of positions and velocities. Due to the set-point control, the control torques reached the steady state after 0.5 s, as shown in Figure 17.

Conclusions
In this paper, we present a simple solution to the problem of task-space robot tracking control under the parametric uncertainties in both kinematics and dynamics. We proposed an adaptive control scheme with adaptation laws and designed a new observer to estimate velocities in the task space. Using the Lyapunov stability and Barbalat's lemma, we show that the tracking errors and estimation errors in task space can asymptotically converge to zero. Furthermore, we used a two-link robot with a fixed camera to verify the proposed adaptive control and the task-space velocity observer. From the simulation results on the trajectory tracking and the set-point control, the proposed control scheme can effectively accomplish the control goal in the presence of kinematic and dynamic uncertainties. Note that this paper only considers non-singular Jacobian cases. How to identify and deal with the singular Jacobian situation to make the control operate appropriately requires further research. In the future, we will conduct experimental evaluations of the proposed adaptive control and extend the control scheme to networked robot systems.