1. Introduction
Multilegged robots possess superior mobility in challenging environments and uneven terrain where wheeled and tracked vehicles cannot reach [
1]. Compared with quadruped robots, hexapod robots are able to own high stability and capacity at the expense of certain dynamic performance owing to their tripod gaits [
2]. With the help of hydraulic actuation, hydraulic hexapod robots (HHR) can provide large force output, as well as high power density and strong robustness, which is suitable for heavy load occasions and disaster-rescue tasks. However, higher complexity and cost, as well as the low energy efficiency, are the critical issues restricting the widespread use of hydraulic legged robots [
3].
Since Boston Dynamics created a hydraulic quadruped BigDog in 2005 [
4], various research centers, universities, and industries have proposed their hydraulic quadruped robot, such as HyQ series [
5,
6,
7], JINPOONG [
8], SCalf series [
9,
10,
11], Baby-elephant [
12], MBBOT [
13], NUDT quadruped robot [
14], BIT quadruped robot [
15], etc. In terms of HHR, COMET series [
16,
17], HexaTerra [
18], LSHDSL-robot [
19], etc. Dynamic motion is the fundament for hydraulic walking robots performing different tasks; therefore, various control strategies have been proposed. Cunha et al. [
20] combined the gain scheduling algorithm with PID controller in the HyQ leg prototype. Focchi et al. [
21] compared the control performance of the conventional PID, the linear quadratic regulator (LQR) controller, and the feedback linearization (FL) controller in the HyQ leg prototype. Bin et al. [
22] applied self-tuning fuzzy-PID in the hydraulic quadruped robot leg prototype. Ke et al. [
23] designed a feed-forward paralleled Active Disturbance Rejection Controller (ADRC) for foot-end-force control of the support leg of NUDT quadruped robot. Barai et al. [
24] proposed a robust adaptive fuzzy controller with self-tuned adaptation gain in the hydraulic hexapod robot COMET-III. Irawan et al. [
17] designed a position-based impedance controller for the leg and a center of mass-based impedance controller for the hydraulic hexapod robot COMET-IV. Wei et al. [
25] designed a robust adaptive dynamic surface controller in the hydraulic quadruped robot hip joint. Wang et al. [
26] adopted a fuzzy sliding mode controller in the hydraulic quadruped robot. Quan et al. [
27] used a decoupling control strategy based on the diagonal matrix method in the hydraulic drive unit of a quadruped robot. Gao et al. [
28] designed a neural network (NN) model reference decoupling controller to reduce the influence of the coupling of the hydraulically driven quadruped robot. Liu et al. [
29] proposed a PID compound controller with velocity feedforward compensation (VFC) in the hydraulic wheeled-legged robot WLBOT. Apart from that, much research has also been done on force control and compliance control.
Though hydraulic legged robots have unparalleled advantages in high power density and robustness, their energy efficiency is still worse than animals with similar masses [
30]. Nowadays, methods of reducing the energy consumption of hydraulic legged robots are as follows: (1) Optimization of mechanical and hydraulic system structure; (2) Optimization of motion planning; (3) Energy-saving and energy-recovery control strategies of the hydraulic system. Zhai et al. [
31] proposed an archive-based micro genetic algorithm (AMGA) to optimize the mechanical structure and gait parameters, which shows a 40% energy-consumption decrease compared with the original structure. Barasuol et al. [
32] designed a hydraulic integrated smart actuator (ISA) V5 to realize a power saving of approximately 112 W per actuator. Hua et al. [
33] designed a hydraulic servo actuator with passive compliance (HPCA) in the hydraulic quadruped robot, which can help save more than 80 J energy in two gait cycles. Dong et al. [
34] proposed a centroid fluctuation gait that can save more than 10% energy. Yang et al. [
35] studied a foot trajectory based on the Fourier series to reduce about 7.55% joint energy consumption. Deng et al. [
36] proposed a low energy cost foot trajectory planning method to realize a constant velocity of the body of a hydraulic hexapod robot which could reduce about 39% peak power. Tani et al. [
37] proposed a method of taking the characteristics of the limited powered pump into consideration when designing the walking trajectory of a hydraulic legged robot, which could improve the energy efficiency and achieve power matching. Guglielmino et al. [
38] established a hydraulic equivalent of the DC-DC switching Buck converter for the HyQ leg prototype to save about 75% energy. Xue et al. [
39] designed a double-stage energy supply system using small accumulators to meet the instant high-pressure demands of hydraulic legged robots.
On the above issues of hydraulic legged robots, this paper concentrates on the design and control strategies of a hydraulic hexapod robot (HHR) with a two-stage supply pressure hydraulic system (TSS), especially on joint sliding mode repetitive control (SMRC) and energy-saving efficiency. In summary, the main contributions made are as follows:
A SMRC controller is designed to improve the joint trajectory tracking performance;
The high order sliding mode differentiator (HOSMD) is designed to help get the angular velocity and acceleration of HHR;
A two-stage supply pressure hydraulic system (TSS) is utilized in HHR to save the energy of legs in the swing phase.
The remainder of this paper is structured as follows:
Section 2 gives an overview of the HHR system, including its mechanical structure, hydraulic system, and control system.
Section 3 establishes the kinematics model and hydraulic model of HHR.
Section 4 introduces the configuration of different valves in TSS and SMRC joint controllers.
Section 5 describes the effectiveness of the control algorithm and energy-saving in ADAMS and MATLAB/Simulink co-simulation. A conclusion is offered in
Section 6.
4. Controller Design
In the HHR control system, the onboard industrial personal computer PXIe-8861 can obtain signals of joint angle, load force, hydraulic cylinder pressure, and foot force from different sensors, thus implementing the control algorithm to achieve the high-precision and steady locomotion of the robot. Combined with the SMISMO system as
Figure 5 shows, all the valves need to be appropriately configured, not only to get better control performance but to save energy as well.
4.1. Valve Configuration in TSS
Three valves are configured as
Table 2 to deal with different load forces in the stance phase and swing phase. Owing to the foot force sensors installed on the robot feet, the foot force will be gained to easily distinguish which leg is in the stance phase. When the contact detection identifies that the leg is in the stance phase, the control voltage of
will be set to zero, which will turn off
. In this case,
and
can independently control the flow rate of the chamber without the rod and with the rod, respectively, which will form a SMISMO control system automatically.
is connected to the high-pressure resource
, oil tank, and the chamber without the rod, while
is connected to the low-pressure resource
, oil tank, and the chamber with the rod. Through the control algorithm designed, the hydraulic cylinder can implement the exact trajectory tracking and periodic reciprocating motion. When the leg is in the swing phase, the control voltage of
will be set to zero, which will turn off
. Thus,
and
will control the movement of the hydraulic cylinder. Because
is connected to the low-pressure resource
instead of
, energy in the hydraulic cylinder extension can be saved.
4.2. Sliding Mode Repetitive Control
Firstly, a sliding mode control (SMC) is proposed to realize joint angle control of HHR. Assuming the leg is in the stance phase, only
and
are, thus, working. From Equations (5), (6) and (12), we can easily get
where
Define a set of new parameters as
where
The control voltage
of
can be calculated. Define state variables and establish state-space equations as
For simplicity, the following practical assumption is made.
Assumption 1. The extent of parametric uncertainties and uncertain nonlinearities are known, i.e.,whereis uncertain nonlinearity and other disturbance from environments, is known. Let
where
is the desired hydraulic cylinder displacement.
The sliding surface can be designed as
where
,
,
,
can be chosen such that
is Hurwitz.
A positive semi-definite Lyapunov function can be written as
The derivative
can be expressed as
The control input
of valve
can be designed as
where
is feedback linearization compensation component,
is a sliding mode switching component.
where
determines the exponential convergence speed of the error on the sliding surface;
;
is a high-slope saturation function which can replace the signum function to eliminate chattering,
is the thickness of the boundary layer.
The closed-loop system is stable according to Lasalle’s invariant principle.
Similarly, the control
of valve
can be designed as
where
Secondly, the repetitive control (RC) will be combined with SMC to form SMRC to improve tracking accuracy. RC, based on the internal model principle, is regarded as a simple learning control because the control input is calculated using the information of the error signal in the preceding periods. RC is often utilized to track periodic signals, which can effectively suppress periodic load interference. The schematic of SMRC is illustrated in
Figure 6.
where
is SMC output;
is RC output.
where
is the transfer function from position error
to RC output
;
is the compensation term to ensure system stability, which is always chosen as a constant near 1 or a low-pass filter;
is time delay element,
is the delay time;
is the transfer function of a proportional-integral-derivative (PID) controller;
is the stabilization compensation term for amplitude and phase correction of the controller.
As a comparison, a conventional proportional-integral-derivative (PID) controller is designed as follows:
where
is the tracking error;
,
,
are the proportional gain, integral gain, and derivative gain, respectively.
4.3. High-Order Sliding Mode Differentiator
Although the joint angle can be obtained easily through the angle encoders, it is difficult to get the joint angular velocity, and acceleration for the direct differential will bring in and amplify noise. Thus, a high-order sliding mode differentiator (HOSMD) is proposed to gain accurate joint angular velocity and acceleration.
The HOSMD can be expressed as
where
can be the joint angle
;
is the calculated joint angular velocity;
is the calculated joint angular acceleration;
,
,
are the designed parameters, respectively.
In order to prove the stability of HOSMD, the new parameters can be defined as
,
; thus, Equation (36) can be rewritten as
where
,
is a designed known Lipschitz constant;
,
,
are the designed parameters, respectively.
A positive semi-definite Lyapunov function can be written as
The derivative
can be expressed as
when the parameters are chosen as
HOSMD can be stable. Through HOSMD and mechanical structure Equation (1), the joint angle, angular velocity, and acceleration can be straightforwardly calculated and transferred into the hydraulic cylinder’s displacement, which can be utilized in the SMRC algorithm.
6. Conclusions
In this paper, the system design and control strategies of a HHR ZJUHEX01 with a two-stage supply pressure system (TSS) are introduced in detail. An overview of the mechanical system, hydraulic system, and control system is given, as well as the kinematics model and hydraulic system model. A SMRC controller for SMISMO hydraulic TSS system is proposed to help HHR get better control performance compared with the conventional PID, and the absolute maximum tracking error of hip and knee can be reduced by 59.58% and 63.53%, respectively. Also, the RMS tracking error of the hip and knee can be reduced by 65.64% and 74.74%, respectively, which brings about great improvement. Apart from that, a HOSMD is designed to get the joint angular velocity and acceleration for the use of the control algorithm. In the co-simulation model of ADAMS and MATLAB/Simulink, HHR can walk at a speed of 0.75 m/s in a tripod gait, where the effectiveness of the control strategy is also verified. Additionally, the energy consumption of TSS is compared with that of OSS to show a great energy-saving effect of 51.94%.
Although SRMC has a better performance in joint trajectory tracking than the conventional PID controller, it does not consider compliance when the feet are in contact with the ground. Thus, active compliance control, such as impedance control and virtual model control (VMC), is the research focus. Furthermore, based on the proposed SMISMO hydraulic TSS system, other energy-saving strategies can also be implemented. Future work will concentrate on the experiments of the active compliance control and energy-saving strategies in the HHR.