1. 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 can be extended to the adaptive tracking control in the task space. Therefore, kinematic uncertainties such as unforeseen changes of object positions and/or orientation and unknown camera parameters must be considered for accurate tracking control in the task space. The dynamic effects in visual servo control can be ignored if the robot’s inertia is small and the tracking velocity is low. However, for high-speed trajectory tracking, a task-space robot control scheme that can adapt to the kinematic and dynamic uncertainties is important for improving the performance of future robots.
Recently, researchers have considered dynamic and kinematic uncertainties in robot manipulators and have proposed many means of task-space adaptive tracking control [
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18]. In particular, Cheah et al. [
7] extended the classic Slotine–Li adaptive algorithms [
5] for manipulator control to task-space adaptive algorithms that can deal with dynamic uncertainties and unknown kinematic properties such as uncertain camera parameters. Nevertheless, the formulation of the dynamic regressor requires the exact Jacobian matrix that may not be available due to the parametric uncertainty in kinematics. Cheah [
11] presented a unified approach and discussed the duality property of the transpose Jacobian and inverse Jacobian PD (Proportional-Derivative) control for the task-space regulation of robot manipulators. For the transpose Jacobian approach, in general, it is assumed that the Jacobian matrix is nonsingular for the control schemes in the task space. Li and Cheah [
17] developed a global task-space adaptive robot control strategy to address the problems of singularity of the Jacobian matrix and limited sensing zone. The authors proved the global dynamic stability of the proposed task-space control strategy that can smoothly transit between regional feedback and avoid discontinuous switching. Considering the uncertainties in dynamics and kinematics, Wang and Xie [
12] developed an adaptive inverse dynamics robot control strategy. Their schemes require the measurement of acceleration in the dynamics regressor and use the filtered differentiation to obtain the end-point velocity in the task space. Recently, Wang [
18] proposed a passivity-based adaptive scheme to solve the problem of the synchronization of network robots with uncertain kinematics and dynamics. Tracking errors in the task space and synchronization errors of the networked robots were shown to be convergent to zero through Lyapunov stability analysis.
In this paper, the problem of task-space tracking control for robot manipulators is addressed under parametric uncertainties in kinematics and dynamics. First, we propose a simple and effective adaptive control scheme that includes adaptation laws for unknown constant kinematic and dynamic parameters. Moreover, we design a new observer to estimate velocity in the task space, and the proposed adaptive control requires no acceleration measurement in the joint space. We use the Lyapunov stability and Barbalat’s lemma to prove that the tracking errors and estimation errors in the task space can asymptotically converge to zero. Finally, we illustrate the design procedures on a two-link robot with a fixed camera and demonstrate the feasibility of the proposed adaptive control scheme for the trajectory tracking of robot manipulators.
The contribution of this paper is three-fold: First, it offers a simple and easy-to-implement adaptive algorithm to deal with parametric uncertainties in both kinematics and dynamics. Second, it presents the clear design of the task-space velocity observer for manipulator tracking control. Third, it provides the stability analysis to show the asymptotically convergent tracking errors and estimation errors.
This paper is organized as follows:
Section 2 discusses the robot dynamics and properties.
Section 3 introduces the proposed adaptive task-space manipulator control.
Section 4 presents the stability analysis. In
Section 5, numerical simulations are performed to apply the proposed adaptive tracking control scheme to a two-link robot manipulator with a fixed camera.
Section 6 presents some conclusions.
3. 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
where
is the estimated position,
is the desired position,
is the estimated velocity from the observer defined later, and
is the desired velocity.
Denote as the estimate of Jacobian .
Assuming
is non-singular, we can define the reference signal
and
:
where
is a constant related to the convergence of the tracking error.
Define the sliding vector as
Notice that the dynamic regressor
can be expressed as
We propose the following adaptive tracking control:
where
and
are positive constants and
is the estimate of
.
The update laws for dynamic and kinematic parameters are
and
where
and
are diagonal and positive matrices.
The observer for the velocity in task space is
where
is a constant related to the convergence of the estimation error.
The proposed adaptive control scheme is shown in
Figure 1.
4. 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 vectorglobally 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:
Substituting
from (23) in (22), we have
where
We can obtain the following sufficient condition for stability:
Then, it follows from (24) that
Using Lemma 1, we can conclude that globally asymptotically converges to zero.
This completes the proof of Theorem 1. □
Corollary 1. The estimation error in velocityasymptotically converges to zero.
Proof. Taking the derivative of
in (4) and using the observer (14), we have
It is straightforward to show that and are bounded: is bounded because is bounded, is bounded because and are bounded, and is bounded from (1) because is bounded. Then, the vector in (29) is bounded because and are bounded from the stability analysis in Theorem 1. Consequently, from Barbalat’s lemma, we can conclude that asymptotically converges to zero. □