1. Introduction
The tracking control of robot manipulators is a mature field but still has many research possibilities, and a straightforward control scheme is known as computed-torque control [
1]. Computed-torque control generally performs well in practice when the robot arm parameters are accurately known [
1]. When uncertainties and unknown disturbances occur, conventional robust stability analysis shows that if the nominal system is exponentially stable, the system can tolerate “small” system uncertainties and external disturbances [
2], therefore restricting the application of linear control to robot manipulators, which is a highly nonlinear system. In this case, adaptive control [
3,
4,
5], sliding-mode control [
6,
7,
8,
9,
10,
11], and neural network control [
12,
13,
14] were proposed to solve the problem.
Sliding mode control is known for its robustness against large system uncertainties and external disturbances [
15,
16]. However, the sliding mode control has a disadvantage of control chattering, which is due either to switching time delay [
17] or unmodelled dynamics [
18,
19]. Boundary layer design has been proposed as a solution, in which the switching function is replaced with a continuous interpolation function [
6,
20]. In boundary layer design, control accuracy and control smoothness are ensured by a small and large boundary layer width, respectively, and thus trade off each other.
Other approaches have also been proposed to reduced control chattering in sliding mode control, such as higher-order sliding-mode (HOSM) control [
21]. The application of HOSM control to robot manipulator can be found in [
22,
23,
24,
25]. However, the modified sliding mode controls are complicated. Moreover, the boundary layer control [
26] and HOSM control [
27] were proved to be sensitive to measurement noise; the control signal will inevitably have undesired chattering when the state or the estimation is corrupted by a stochastic noise, and only uniformly ultimate boundedness is guaranteed for both designs in a noisy environment.
The purpose of this paper is to demonstrate that simple linear control can deal with system uncertainties and external disturbances as effectively as sliding mode control can, especially in the robot control task. Furthermore, linear control has no side effect of control chattering, and its control law is simple. Disagreeing with the conventional belief that linear control can only cope with small linear or nonlinear system uncertainties [
28,
29], this paper formulates a framework where linear control can cope with large linear or nonlinear system uncertainties and suppress large external disturbances. With the new stability analysis, the linear control design is proposed as a modified computed-torque control that does not require the information of system parameters. Moreover, a noise-free control scheme is presented to efficiently eliminate the noise-induced chattering that often occurs in sliding-mode [
27] of boundary layer controls [
26].
The remainder of this paper proceeds as follows. The problem is formulated in
Section 2, preliminary lemmas are presented in
Section 3, the stability of the proposed linear control is analyzed in
Section 4, the noise-free control scheme design is presented in
Section 5, an application to a two degree-of-freedom (DOF) robot manipulator is demonstrated in
Section 6, and conclusions are presented in
Section 7.
2. Problem Description
Consider the dynamic equation of an
n-DOF link robot
where
is joint position,
,
are the joint velocity and acceleration vectors;
,
,
,
F are the inertia matrix, Coriolis matrix, gravity matrix, and frictional matrix with proper dimensions,
is the input torque vector. Defining the desired joint position
, velocity
, and acceleration
, the position errors for each joint are given as
for all
, and the error vector
is composed. When the system matrices
,
,
,
F in (
1) are accurately known, the computed-torque control [
1] gives
where
is the gain matrix to be determined with
for all
, and
is a
zero matrix. Substituting the torque commend (
2) into (
1) yields
which is the basic formulation of impedance control of a robot manipulator [
30] with constant diagonal matrices
and
.
When the system matrices
,
,
,
F in (
1) are unknown, the error dynamic of the state vector
x is described as
on the basis of (
1), where
and the system matrices are
with
In this paper, one assumes
is a uniformly bounded and differentiable unknown disturbance that satisfies the upper bound
and the unknown nonlinearity
satisfies the Lipschitz condition:
with Lipschitz constant
and the nonlinearity is assumed to be differentiable. This paper does not force the nonlinearity to be small; hence,
h can be a large number. Conventionally, when given the uncertain system (
5), one would most likely use switching sliding mode control [
15,
16] or boundary layer control [
6,
20]
to stabilize the system and to suppress the disturbance, where
is an upper bound of the uncertainties,
s is the sliding variable, and
is the boundary layer width. However, the aforementioned sliding mode control and the boundary layer control have undesirable side effects, as discussed in the Introduction section. This paper therefore considers the possibility of dispensing with the nonlinear switching function
or boundary layer interpolation function
. The proposed control law is simply a linear state-feedback control
where
K places the eigenvalues of
in the left-half-plane. With the linear feedback control, system (
5) becomes the following:
The goal of this paper is to show that (1) given any large Lipschitz nonlinearity
, one can always stabilize the closed-loop system with the proposed simple linear control (
11) and (2) the simple linear control is sufficient to suppress the effects of large disturbance
d on the system. No complex, nonlinear sliding mode control is required to deal with the system uncertainties and external disturbances in (
5).
Remark 1. It is worthy noting the system Equation (5) describes the error dynamic of (1) as in [9], and the system matrices (7) in (5) and (12) are therefore exact according to the definition of the error dynamic [9]. However, if there is a structured uncertainty in the closed-loop system (12), the proposed design naturally requests the uncertainty to fulfill a matching condition [31]. 5. Noise-Free Control Design
In the previous section, we showed that the linear state-feedback control can suppress the disturbance as sliding-mode control. However, the control law based on a large parameter
results in high-gain control that is sensitive to the measurement noise. To address this disadvantage, a design structure inspired by [
34] is proposed.
The concept of noise-free control design is illustrated by a single input system. As depicted in
Figure 2, an integrator is placed in front of the controlled system in the new design, and the control parameter now satisfies
. Intuitively, the control parameter
u, which is treated as an output signal of the integrator, carries no noise-induced chattering. In the new design, one can construct an augmented system:
where
v is still a simple linear state feedback design
and
is the state feedback gain stabilizing the augmented system (
32) when
.
Theorem 3. The control design v in (33) drives the original system state x to an arbitrarily small region around the origin with any large disturbance upper bound D and large Lipschitz constant h in (9) and (10). Proof. The pair
is described in the controllable canonical form
and the control parameter
u can be represented as
where
is a set of bounded parameters that satisfies
Note that the set of first-order differential equations (
32) can be described as an
th-order differential equation
if the state feedback gain in (
33) is designed to place the closed-loop system poles to
with the design parameter
. Correspondingly, one obtains
Define a new time index
. The preceding equation becomes -4.6cm0cm
The differential equation can be written in the controllable canonical form
where
Note that the subscript of
stands for the transformed state of the augmented system. Because
is a stable matrix, one has
according to Lemma 1,and the augmented system state
satisfies the relation
From the preceding equation, one obtains the inequalities
where (
9), (
10), (
35), and (
43) are used to derive (
44). Using the procedure for evaluating Theorem 2, one obtains
where
and
From (
45), the transformed state is uniformly bounded for all time points
. Because
,
,
are constants, one can always choose the design parameter
that satisfies
With this design parameter
, all exponents in (
45) decay to zero as time approaches infinity. Thus, one has
and, by combining (
43) and (
49), one obtains
As per (
48), the denominator of (
50) is always positive. As evident in the preceding equation, when a sufficiently large
is specified, the right-hand side of the inequality becomes an arbitrary small quantity; thus, the system state
converges to a small region around the origin. Substituting (
49) into (
44), one verifies the following bound for the control signal as time approaches infinity:
where the fraction term vanishes with a sufficiently large
and where the bound is dominated by the upper bound of the disturbance
d. □
Remark 5. When the control system has multiple control inputs , the controller canonical form of system is combined by m realizations [33], and the state-space realization of the noise-free design can be written as m differential equations with the structure of (37). Therefore the result for single input system can be generalized to the multiple input case; the augmented system (32) becomesfor the multiple inputs system, and the convergency of system state (50) and boundedness of control signal (51) hold for (52) as well. 6. Application to a Two DOF Robot Manipulator
A two-link robot studied by [
9] is used here to illustrate the efficiency of the proposed control design. As shown in
Figure 3, the manipulator is in the vertical position, and the parameters are shown in
Table 1. System matrices in (
1) are defined as in [
9]:
and the desired trajectories are
In this case, the system matrices (
53) are assumed to be unknown and considered as system uncertainties and disturbances in (
5), and the system matrices in the closed-loop system (
5) are
Conventionally, the sliding-mode control design [
17] for the uncertain system (
5) is constructed as
where
is the state feedback gain that places eigenvalues of
in the open left-half plane. As the nominal closed-loop system matrix
is stable, a positive definite matrix
P exists, satisfying the Lyapunov equation [
35]
In the control law (
56),
s is the sliding variable
where
P is obtained from the Lyapunov equation (
57) and the constant
is an upper bound of unknown disturbances.
Figure 4 shows that the system outputs track desired references when the time exceeds 4 s with the sliding-mode control design, and
Figure 5 shows the time history of the control signals. It can be seen that the control inputs suffer from the chattering phenomenon because the discontinuous switching function
is used in the control design (
56). By contrast, the robust linear control design (
11) is employed to deal with the uncertain system (
5), where
K is the state feedback gain that places the eigenvalue of
in (
12) to
, and the design parameter
observes the scheduling law
where
,
and
to provide a similar convergent speed as the sliding-mode control design (
56). In
Figure 6, the tracking performance with the robust linear control (
11) is depicted, and
Figure 7 presents the control signals. It is seen the robust linear control (
11) performs the robustness as the sliding-mode control design (
56) with a much more straightforward design algorithm; moreover, the chattering phenomenon in
Figure 5 is absent, and the undesirable peaking phenomenon [
17] is eliminated.
When a uniform distributed measurement noise in the interval
is added to the state measurement of
x, the control performance of proposed robust linear control is shown in
Figure 8 and
Figure 9.
Figure 8 shows that the control mission is achieved even if the measurement is corrupted by a stochastic noise. However, as depicted in
Figure 9, when the design parameter
in (
59) is increasing, the interference of measurement noise is increased in the control signals; as discussed in Remark 4, the control signal is coupled with an undesirable, high-frequency oscillation when the state feedback gain
K is correspondingly large.
Because the measurement process always couples with measurement noise, the undesirable oscillation occurs when the control design (
11) is implemented. Therefore, developing the robust noise-free control design is necessary for eliminating the noise-induced chattering. As an intuitive augmentation of (
11), the state feedback gain
K of the robust noise-free control in (
33) is designed to place the poles of system (
52) to
, and the design parameter
are scheduled as in (
59). The controlled system response is depicted in
Figure 10. Following the same concept of pole placement, the exponent
in (
41) coincides with
in (
28); the convergent speed of robust linear control (
30) and robust noise-free linear control (
45) are dominated by the same singular value when
is large (
Figure 11). On the other hand, because of the high-frequency oscillation is filtered out by the integrator, the control inputs of the noise-free design are smooth even if the state measurement is corrupted by a stochastic noise and the control gain
K is scheduled to a high level; moreover, the peaking phenomenon [
17] which often occurs in high-gain control is absent due to the design control structure and the scheduling law (
59).