1. Introduction
Humanoid manipulators have the characteristics of light weight, small size, strong nonlinearity, coupling, friction, and clearance [
1,
2,
3]. Among them, friction and clearance are disturbances that significantly influence the control performances of humanoid manipulators. As sliding mode control has the excellent characteristic of efficiently disturbances settlement [
4,
5,
6,
7], it is widely used for humanoid manipulators’ position-tracking control [
8,
9,
10]. A low-cost humanoid manipulator further constrains its hardware configuration conditions. Complex and high-precision sensors or motors cannot be used when designing a humanoid manipulator because they are generally large in volume. The sensors and motors with small volumes and high precision are generally quite expensive. Teleoperation systems are also not included for cost-saving purposes [
11]. All the above features aggravate the nonlinearity, coupling, and complexity of the low-cost humanoid manipulator’s dynamics. Benefiting from their variable structure characteristics, sliding mode controllers possess excellent robustness and anti-disturbance ability [
12,
13]. However, the alternation between the reaching phase and the sliding phase of a sliding mode controller always accompanies by a phenomenon called chattering [
14]. The production of the chattering is related to the sliding mode boundary layer thickness [
15]. The system states trajectory always has a certain speed when reaching the manifold, and the inertia makes the system states move across the manifold. After that, the controller input generates a reverse signal to pull the system states back to the sliding mode manifold. However, it must take time. This time difference makes the system states trajectory deviate from the sliding mode manifold for a certain distance and intensifies the thickness of the boundary layer. In this way, the system states trajectory repeatedly passes through the sliding mode manifold, which is reflected in the macroscopic view as a chattering phenomenon [
16,
17]. Chattering may cause energy consumption and mechanical system damage problems. Therefore, the chattering problem must be first solved when designing sliding mode controllers, especially for a low-cost humanoid manipulator.
The sign function term in a sliding mode reaching law is not only the key to the rapid convergence of sliding functions but also one of the main causes of chattering [
18]. Replacement of the sign function with the saturation function is one way to suppress chattering [
19]. Hiding the sign function term in a higher-order term is another effective way to eliminate chattering [
20,
21,
22,
23,
24]. As the system states enter the quasi-sliding mode, the super-twisting sliding mode (STSM) gains the characteristics of the sign function term and ensures that the sliding function and its derivative converge to zero within a limited time [
25,
26,
27]. The STSM control which belongs to the second-order sliding mode control only requires information on the sliding function [
28]. It contains a power rate term that can reduce the time of the system states arriving in quasi-sliding mode and suppress the chattering more effectively compared with the proportional term [
29]. Moreno and Osorio used the Lyapunov stability theory to analyze the stability of the super-twisting algorithm (STA) [
30]. The proof process has been used in many STSM controllers. Kali et al. proposed an STSM control for manipulators with uncertainties. [
31]. Tayebi-Haghighi et al. designed a high-order STSM controller to address multiple control challenges of robots with verification. [
20].
Apart from introducing super-twisting algorithms, adding a fractional-order operator to the sliding mode controller is also an effective way to suppress chattering and improve motion control performance [
32]. Applying a fractional-order (FO) operator to a sliding mode controller can enhance the flexibility of the controller, and improve its control performances [
32,
33,
34,
35,
36]. Among all the fractional-order controllers, fractional-order nonsingular terminal sliding mode (FONTSM) controllers are widely used in manipulators to obtain precise position-tracking performance [
37,
38,
39,
40,
41,
42,
43]. Tran et al. designed an adaptive fuzzy FONTSM control strategy for a two-degree-of-freedom manipulator and used simulation results to illustrate its control performances. [
40]. Nojavanzadeh et al. proposed an adaptive FONTSM controller to control robots with uncertainty and external interference and used simulation to verify its effectiveness [
41]. Su et al. proposed an adaptive FONTSM controller for a cable-driven manipulator with external interferences, which was verified by simulations. [
42]. Wang et al. formulated an adaptive sliding mode controller that integrates the advantages of PID and FOTNSM manifold [
43]. However, most of the FONTSM controllers were verified by simulations. Only a few papers showed the experimental verification of the FONTSM control for manipulators [
44,
45,
46]. In the previous work of the authors, a dynamic FONTSM controller was proposed for a class of second-order nonlinear systems with simulation verification [
47].
To summarize, super-twisting sliding mode control can effectively accelerate the speed of system states reaching the quasi-sliding mode and suppress the chattering. FONTSM control has superiority in motion control accuracy. The fractional-order nonsingular super-twisting sliding mode (FONTSM-STA) control combines the advantages of the FONTSM manifold and the STA, which not only meets the demand for high tracking accuracy but also suppresses the chattering [
45]. However, simply combining these two methods has little effect on the joint error converge speed which is a vital factor for humanoid manipulators to mimic arm movements.
Aiming at the above-mentioned problems, a dynamic fractional-order nonsingular terminal super-twisting sliding mode (DFONTSM-STA) control scheme is proposed for a low-cost humanoid manipulator in this study. The main contributions of this paper are summarized as follows:
- (1)
A DFONTSM-STA controller is formulated for the control of the low-cost humanoid manipulator by combining the dynamic fractional-order nonsingular terminal sliding mode (DFONTSM) manifold with the super-twisting reaching law, which can effectively improve the control accuracy, quickly force the tracking error of each joint to converge and significantly suppress the chattering. The stability and convergence of the low-cost humanoid manipulator control system are proven based on the Lyapunov stability theory.
- (2)
Experiments illustrate the superiority and feasibility of the proposed DFONTSM-STA control for the low-cost humanoid manipulator. Compared with FONTSM and FONTSM-STA control, the DFONTSM-STA controller has superior control performance. Its terminal position tracking accuracy is increased by 53.3% and 23.7% respectively. Its chattering of joints one to four is decreased by 54.1%, 51.1%, 46.2%, and 55.1% compared with FONTSM control. Its error convergence speed is accelerated significantly. Joints one and two are converged at the beginning, and joints three and four are accelerated by 43.5% and 33.6%, 72.7%, and 54.6% respectively compared with the FONTSM and FONTSM-STA control.
2. Model Description and Problem Analysis
The low-cost humanoid manipulator is limited by cost, size, and weight, which uses hall sensors that generate three pulses in one turn to read the angle position information. The position is detected by sensors whose resolution converted to the joint is 0.3°.
Figure 1 shows the three dimensional (3D) model of the low-cost humanoid manipulator and the Denavit-Hartenberg (D-H) coordinate of each joint.
Table 1 shows its D-H parameters. Coordinate
represents the base coordinate;
and
represent the coordinate of shoulder joints one and two;
and
represent the coordinate of elbow joints three and four;
represents the terminal coordinate. The dynamic equation of the four degrees-of-freedoms (DOFs) low-cost humanoid manipulator with friction model is described as follows:
where
, and
ϵ R
4 respectively represent the position, velocity, and acceleration vectors of every joint;
is the non-singular inertia matrix;
is the centrifugal and Coriolis matrix;
represents the gravitational vector;
is the vector of the viscous friction torque at joints;
denotes the torque input vectors;
denotes every joint coulomb friction torque;
denotes every joint viscous friction coefficient.
The low-cost humanoid manipulator contains four joints to mimic shoulder and elbow movements. To suppress chattering and improve both the error convergence speed and the tracking accuracy, it is necessary to design a new sliding mode control method. As the STSM control hides the sign function in higher order, it possesses a better effect on chattering suppression. Apart from the chattering suppression requirement, the low-cost humanoid manipulator also needs precise trajectory tracking accuracy for mimicking human arm movements. The DFONTSM manifold can enhance the entire control performance by dynamically changing the position of the sliding mode manifold [
47]. The steady-state accuracy can be improved because the fractional-order operator has the characteristics of heredity and memory, which can describe detailed information and increase the flexibility of the control law. Therefore, to solve the trajectory tracking performances of the low-cost humanoid manipulator, this paper proposes a DFONTSM-STA control method that combines a DFONTSM manifold with a super-twisting reaching law.
3. DFONTSM-STA Control for the Low-Cost Humanoid Manipulator
This section will briefly present some definitions and lemmas that are needed in the stability proof of the proposed DFONTSM-STA controller.
Definition 1. The general representation of the FO derivative-integral operator is expressed by Equation (3), [48]where is the FO and t0 is the initial time. The operator is the symbol for the FO, integral and constant operator. Definition 2. The th-order Riemann–Liouville (RL) fractional and integral are presented as Equations (4) and (5) [48].wherem −
1 <
<
m,
m∈
N.
Property 1. [48] if α∈ C, β∈ C,ℜe(α) > 0,ℜe(β) > 0 and f(ɑ,b)∈ Lp(a,b) (1 ≤ p ≤ ∞), then , ; , .
Lemma 1. [44]. Supposing parameters ; are positive, then the following inequality about and can be obtained: Lemma 2. [38,44]. For a Lyapunov function V(x), assuming that V0 is its initial value and α∈ C, β∈ C, thenThe corresponding settling time can be calculated as The FONTSM manifold parameter transforms into the function
that calculates the error by exponential function is designed as Equation (9). Then the DFONTSM manifold (8) is formulated for the system described in the model (1) [
47].
where 0 <
λi < 1,
i = 1∼4,
Dλ is the FO operator;
= diag(
) is the diagonal matrix and
ϵ (0,1);
= diag(
) and
= diag(
) are tuning matrices and
,
ϵ(0, + ∞);
denotes the tracking error vector between the target joint rotation angle and the actual joint rotation angle.
is a positive diagonal matrix.
can be chosen from one to four.
where
is the exponential parameter, which satisfies the relation:
. A super-twisting reaching law from [
31] is adopted.
where
= diag(
),
= diag(
) are constant matrices and 0 <
b =
b1 = ··· =
b4 < 1,
i = 1∼4; With the above reaching law (10), the error vector
can be forced to zero. Derivating the DFONTSM manifold (8), we have:
where:
Combining (11) with (10), the control law (14) can be obtained by Equation (13).
4. Stability Proof of DFONTSM-STA Control
The states
can be obtained from the dynamic model (1):
where
. The constant diagonal matrix of
represents the friction interference upper boundary. Substituting (14) and (15) into (13), yields:
By simplifying the above equation, error dynamics (16) can be determined:
Rewrite error dynamics (16) as the form as follows:
where
ε is formulated from (16) and (17) as
, satisfying the inequality equation
, where
. For each joint, the error dynamics can be written as:
where
, and
satisfying
, where
. The following Lyapunov function is chosen:
Then the Lyapunov function
of each joint of the low-cost humanoid manipulator can be determined as:
where
i represents joint
i, and
satisfies
.
Supposing
and
, then (20) can be rewritten as:
satisfies the following inequality equation:
where
denotes minimum eigenvalues of
and
denotes the maximum eigenvalue.
is the Euclidean norm. By rearranging Equation (22), we obtain:
Derivating the Lyapunov function,
can be expressed as:
Substituting error dynamics (18) into the above equation, yields:
Simplify the above equation into matrix multiple forms:
Supposing:
,
,
, then Equation (24) can be rewritten as:
If
,
, then
satisfies:
By substituting inequality (26) into rearranged
(25), we obtain the following inequality (27):
where:
If
and
satisfy the following conditions:
then the error dynamics for each joint (18) meet the demands of the Lyapunov stability and the matrix
is symmetrical and positive. Obviously
, substituting Equation (23) into Equation (27),
satisfies:
where term
denotes the minimum eigenvalues of the matrix
,
denotes the maximum eigenvalues. Thus, the converge time
for each joint of the low-cost humanoid manipulator satisfies the following inequality which can be obtained through lemmas 1 and 2:
Therefore, the control law (14) can ensure the error dynamics of the low-cost humanoid manipulator to be stable.
7. Conclusions
A DFONTSM-STA control was proposed for a low-cost humanoid manipulator by combining the dynamic fractional-order nonsingular terminal sliding mode manifold with the super-twisting reaching law. Experiments were conducted to control the low-cost humanoid manipulator and showed that the proposed control method can effectively improve error convergence speed, tracking accuracy, and chattering suppression ability. In error convergence speed, the errors of joints one and two were converged at the beginning while joint three was speeded up by 43.5% and 33.6%, and joint four was speeded up by 72.7% and 54.6% compared with FONTSM and FONTSM-STA controllers. In chattering suppressing, suppression performances of joints, one to four were enhanced by 54.1%, 51.1%, 46.2%, and 55.1%, respectively, compared with FONTSM control. In trajectory tracking performance, compared with FONTSM control and FONTSM-STA control, the tracking accuracy of DFONTSM-STA control was promoted by 53.3% and 23.7% respectively. Simulation and experimental results illustrate the superiority of the proposed DFONTSM-STA control for the low-cost humanoid manipulator.
The proposed DFONTSM-STA controller is applicable to humanoid manipulators that require fast error convergence and accurate trajectory tracking movements. The dynamic model of the low-cost humanoid manipulator is complex, and there exists the problem of big calculations in solving control inputs. Therefore, the adaptive fuzzy method will be studied to estimate model uncertainties and disturbances in future research work.