A human-simulated fuzzy membrane approach for the joint controller of walking biped robots

Abstract

To guarantee their locomotion, biped robots need to walk stably. The latter is achieved by a high performance in joint control. This article addresses this issue by proposing a novel human-simulated fuzzy (HF) membrane control system of the joint angles. The proposed control system, human-simulated fuzzy membrane controller (HFMC), contains several key elements. The first is an HF algorithm based on human-simulated intelligent control (HSIC). This HF algorithm incorporates elements of both multi-mode proportional-derivative (PD) and fuzzy control, aiming at solving the chattering problem of multi-mode switching while improving control accuracy. The second is a membrane architecture that makes use of the natural parallelisation potential of membrane computing to improve the real-time performance of the controller. The proposed HFMC is utilised as the joint controller for a biped robot. Numerical tests in a simulation are carried out with the planar and slope walking of a five-link biped robot, and the effectiveness of the HFMC is verified by comparing and evaluating the results of the designed HFMC, HSIC and PD. Experimental results demonstrate that the proposed HFMC not only retains the advantages of traditional PD control but also improves control accuracy, real-time performance and stability.

Keywords

Human-simulated intelligent control multi-mode control fuzzy control membrane computing

1. Introduction

Biped robots have attracted much attention due to their good mobility and agility, and they have a wide range of application prospects in civil rescue, military surveillance, etc. As the basis of biped robot movement, walking control has always been the focus of research [1, 2, 3, 4]. Generally, a biped robot walks by driving its joints to track the target trajectories [5, 6, 7]. With the development of control technology [8, 9, 10, 11, 12], requirements for the robustness, accuracy, timeliness and adaptability of robot joint control are continuously increasing. To meet control demands and ensure that the robot’s speed, displacement, foot placement and other variables of the robot are consistent with the desired trajectory, it is necessary to further study the motion-tracking control of the joint. A variety of control methods have been developed to address the motion trajectory tracking problem, such as PD control [13, 14, 15], fuzzy control [16, 17, 18], neural network [19, 20, 21, 22, 23, 24], model predictive control (MPC) [25, 26], human-simulated intelligent control (HSIC) [27, 28] or a variety of combination methods [29, 30, 31, 32, 33].

This paragraph summarises the main results. Kolathaya et al. established the stability results of the PD tracking control law of biped robots [13]. Chand et al. trained a neural network to realise the dynamic walking of a biped robot with unknown predetermined trajectories [19]. Huang et al. improved the motion performance of biped robots by using MPC to track the reference zero-moment point trajectory [25]. Liu et al. decomposed the leg posture of human walking and utilised HSIC to design the scheduled planning control scheme for periodic walking [27].

Although nonlinear control methods have made great progresses [34, 35, 36, 37, 38], they also have some limitations. For practical applications, PD control is the most popular choice. PD control [39, 40] is model-independent and easy to implement, but it often cannot meet high-precision control requirements. The neural network [41, 42, 43, 44, 45, 46] has a strong approximation ability, but it requires a large amount of computational time to train and learn. MPC [47, 48] has advantages in dealing with uncertainty and constraint problems; however, it requires a highly accurate model, and it suffers from weak generalisation. HSIC is an intelligent control algorithm based on logical reasoning, which mainly studies how to imitate the behaviour of experts [49, 50]. It adopts multi-mode control based on error phase plane dynamic adjustment, which has strong robustness and can realise fast identification and high-precision control without an accurate mathematical model. However, due to frequent and sudden mode switching, conventional HSIC usually produces significant chattering, which should be avoided in practice.

Therefore, in recent years, many researchers have adopted a combination of multiple control algorithms to improve the accuracy and robustness of tracking control. Huynh et al. proposed an intelligent self-organising double function-link fuzzy brain emotional control system, which can effectively reduce tracking errors [29]. Teng et al. proposed a PD-based fuzzy sliding mode control to deal with unmodelled dynamics and external disturbances in human-exoskeleton systems [30]. Azimi et al. proposed a robust sliding mode adaptive controller with great joint tracking performance [31].

However, with the increasing complexity of control problems, mass calculation increases the difficulty of real-time control, thus affecting the control quality. The above studies rarely consider this issue. In recent studies, parallel computing models of membrane computing (also known as P systems) have demonstrated a high potential in solving real-time problems [51, 52, 53, 54, 55]. Among them, a variant of P systems called enzymatic numerical P systems (ENPS) has been proven to be successful in mobile robotics [56, 57].

Within this context, HSIC can be seen as a compound control that combines the advantages of both traditional and modern/intelligent control methods. In addition, among the existing algorithms for chattering reduction [30, 58], fuzzy theory has been successfully used in practical applications [30]. Therefore, with the aim of achieving an easy-to-implement method with high performance in terms of accuracy, robustness and real-time functioning, this paper proposes a new human-simulated fuzzy membrane controller (HFMC), which integrates PD control, HSIC, fuzzy control and membrane computing into the joint angle tracking control of a biped robot, achieving excellent performance in stable walking.

HSIC that uses a multi-mode PD control algorithm can improve the control accuracy of the traditional PD algorithm, but the chattering phenomenon will be caused by multi-mode PD switching, which should be avoided in application. According to our research, fuzzy control has been successfully applied to address chattering problems, and fuzzy algorithms have been successful in hybridisations with membrane computing [59, 60]. Therefore, the fuzzy algorithm can not only solve the chattering problem of multi-mode PD switching, but it can also be used in membrane computing. At present, other similar algorithms have not been found in the application of membrane computing.

The key contributions of this article are as follows:

1)
An HF algorithm was designed by combining a multi-mode PD algorithm, based on HSIC, and a fuzzy controller. The designed HF algorithm solves the chattering phenomenon caused by traditional HSIC multi-mode switching while improving the control accuracy.
2)
For the first time, HFMC was developed by integrating membrane computing with the HF algorithm. As the container of the HF algorithm, the membrane system runs control rules in parallel. This parallelisation facilitates the control in real time and further improves stability and accuracy.
3)
HFMC was applied to the joint angle tracking control of biped robots. Extensive experimental simulations of a five-link biped robot walking on flat and sloped surfaces were carried out. The evaluation results fully demonstrate the stable and accurate control performance of HFMC in comparison to HSIC and PD control.

Figure 1.
The control framework of a walking biped robot.

The remainder of the article is organised as follows: Section 2 describes the model of a biped robot and the proposed control framework; Section 3 presents the proposed HFMC; Section 4 displays and discusses the results of the simulation tests and experiments; Section 5 contains the concluding remarks of this work.
2. Robot model and control framework

2.1 Dynamic model of biped robot

The model of an $n$ -degree of freedom ( $n$ -DOF) walking biped robot can be represented by a hybrid dynamic model, including a continuous single-leg support phase (SSP) and discrete impact events. In SSP, the robot lacks one degree of freedom. In the case of considering gravity, its dynamic expression can be obtained by the Lagrangian method [13]:

$\displaystyle D(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=\tau$ (1)

where $q,\dot{q},\ddot{q}\in R^{n}$ are the vectors of joint position, velocity and acceleration, respectively; $D(q)\in R^{n\times n}$ is the inertia matrix; $C(q,\dot{q})\in R^{n\times n}$ denotes the Coriolis and centripetal forces; $G(q)$ represents the gravitational term; and $\tau\in R^{n}$ is the control input torque applied at the joints.

The evolution of Lagrangian dynamics is discontinuous the instant a leg touches down to the ground. Let us define a function $H$ to express the height of the end of the swing leg above the ground. $S=\{(q,\dot{q})\mid H(q)=0,\nabla H(q)\dot{q}<0\}$ represents the set of state $(q,\dot{q})$ when the swinging leg collides with the ground.

The impact model from [13] is written as follows:

$\displaystyle\left\{\begin{array}[]{l}q(t^{+})\!=\!q(t^{-})\\ \dot{q}(t^{+})\!=\!Z(q(t^{-}))\dot{q}(t^{-}),(q(t^{-}),\dot{q}(t^{-}))\in S\\ \end{array}\right.$ (2)

( $-$ , $+$ ) and $Z$ represents the state values and transition matrix before and after the impact.

A complete hybrid dynamic model of the biped robot consists of Eqs (1) and (2).

2.2 Control framework

Similar to human walking, the movements of biped robots are driven by joints, and the walking process is composed of alternating SSPs and double-leg support phases.

In the double-leg support phase, an important task is maintaining the balance of the robot’s body posture. The balance control object is primarily the height of the centroid and the angle of the body posture. The centroid height is maintained at a constant value, controlled by the knee joint of the supporting leg. The hip joint controls the attitude angle to stabilise the body in the vertical position. In the SSP, the robot moves the swing leg forwards. That is to say, while the supporting leg maintains motion stability, the swinging leg selects the appropriate foot landing position based on the current and desired speed. Then the swing leg touches the ground according to the step cycle and switches to the supporting leg.

The joint controller is the core part of the robot’s gait, which provides the torque of the actuating joint. In the above analysis, it can be seen that the controlled object is the knee and hip joint angles of both legs. The desired joint angle of the supporting leg is a constant value, and the expected joint angle of the swing leg needs to be calculated by the inverse kinematics of the foot placement. Finally, the joint controller enables the joint to reach the target angle to achieve bipedal walking.

Thus, the motion planning and joint controller design are components of the walking control framework, as shown in Fig. 1.

2.2.1 Motion planning

From the above description, it can be seen that planning the appropriate foot placement is the premise of robot walking. As shown in Fig. 1, in the motion planning part, the velocity controller controls the real-time centroid velocity, $v$ , by feedback PD and outputs $U_{v}$ to the foot placement $X_{f}$ . After that, the foot placement is planned with the centroid velocity and single-step period.

To stabilise the robot’s starting and walking, the target speed $v_{d}$ and minimum speed $v_{\textit{min}}$ of the robot are set in Eq. (3), and their difference is divided evenly into $n$ parts, where $\kappa$ is the set speed scale value.

$\displaystyle n=(v_{d}-v_{\textit{min}})/\kappa$ (3) $\displaystyle v_{\textit{ref}}=v_{\textit{min}}+(n-n/\left\lfloor t\right% \rfloor)\kappa$ (4)

The $v_{\textit{ref}}$ in Eq. (4) means that the robot starts at the minimum speed and rises to the target speed uniformly with the increase of time. The purpose of this is to ensure the stability of the robot’s speed:

$\displaystyle U_{v}=K_{p}^{v}(v_{\textit{ref}}-v)+K_{d}^{v}(\dot{v}_{\textit{% ref}}-\dot{v})$ (5) $\displaystyle X_{f}=v_{\textit{ref}}T_{s}/2+U_{v}$ (6)

where $X_{f}$ represents the foot placement of the robot’s swing leg, and $T_{s}$ is one step period. Subsequently, the joint trajectory, which is the target joint angle $q_{r}$ can be obtained from Eq. (6) by inverse kinematics [61].

2.2.2 Joint controller

As an actuator, the joint controller is the nucleus of robot walking. As shown in Fig. 1, for $n$ driving joints, $n$ independent HFMCs are designed.

Information processing is a necessary step of the controller input, which processes the real-time joint angle $q$ and the target joint angle $q_{r}$ to get $e_{q}=q_{r}-q$ . $\dot{e}_{q}=\dot{q}_{r}-\dot{q}$ , $e_{q}$ is the angle error of the joint, and $\dot{e}_{q}$ is its derivative. After that, the error $e_{q}=e_{i}$ $(i=1,2,\ldots n)$ and error derivative $\dot{e}_{q}=\dot{e}_{i}$ $(i=1,2,\ldots n)$ of each joint angle are input to the joint controllers’ HFMC. Finally, the HFMC inputs the control torque $\tau_{i}$ $(i=1,2,\ldots n)$ to the drive motors of the joints, and the joints are controlled to approach the target trajectory and make the robot walk.

3. HF membrane approach

This section introduces the relevant background information about HSIC and P systems and describes the design of HFMC in detail. The proposed HFMC is designed based on HSIC and P systems. First, the HF control rules are developed by integrating HSIC and fuzzy control, which can significantly coordinate the conflicting control quality demands. Second, a P system is embedded into the HF to establish the HFMC, improving the real-time performance of the controller.

3.1 HSIC

HSIC is a branch of intelligent control methods originally proposed in [49]. In the following years, HSIC has been successfully applied to many fields, such as automatic driving [62] and robotics [27, 28].

According to the variation of control error, HSIC selects different control strategies for different characteristic states of the controlled system. This is called multi-mode control. Based on prior knowledge and control experience, HSIC has human logic reasoning function and does not rely on a model’s accuracy.

The prototype algorithm of HSIC solves the contradiction between the stability and accuracy of the conventional proportional-integral-derivative (PID) algorithm, as shown in the following Eq. (7):

$\displaystyle u\!=\!\begin{cases}\!K_{p}e+kK_{p}{\textstyle\sum_{i=1}^{n-1}}e_% {m,i}&\!e\!\cdot\!\dot{e}\!>\!0\cup e\!=\!0\\ \!kK_{p}{\textstyle\sum_{i=1}^{n}}e_{m,i}&\!e\cdot\!\dot{e}\!<\!0\cup\dot{e}\!% =\!0\end{cases}$ (7)

where, $u$ is the control output, $K_{p}$ is the proportionality coefficient, $k$ is the inhibition coefficient, $e$ is the error of the controlled system, $\dot{e}$ is the rate of change of error, $e_{m,i}$ is the $i-th$ peak error and $kK_{p}{\textstyle\sum_{i=1}^{n}}e_{m,i}$ is an extreme value sampling and holding control strategy.

Figure 2.

Feature and control modes in the error phase plane.

The prototype algorithm Eq. (7) first proposed the dual-mode control based on the combination of the open and closed loop and used the characteristics of the error phase plane to select and determine the control strategy online. As shown in Fig. 2, when a system error is in quadrant one or three, the controller will work in the proportional control mode; when an error is in quadrant two or four, the controller will work in the holding control mode. Paper [50] provides the details of the algorithm Eq. (7) for static and dynamic analysis.

Subsequently, with the research and development of Li et al. [63, 64, 65, 66], the dual-mode prototype algorithm evolved into a multi-mode control method combining multiple control strategies. Since the multi-mode control method can coordinate incompatible control quality requirements such as rapidity and smoothness, accuracy and robustness, it has reached high popularity. However, this approach causes chattering during mode switching. This situation is especially undesirable in practical applications.

3.2 ENPS and fuzzy reasoning spiking neural P system

Membrane computing (P System) is a branch of natural computing [67, 51, 68, 54, 69]. It is a computing model abstracted from the information processing cooperation of cell groups, such as cell function and structure, organs and tissues. It is an information processing inclusive space that can contain knowledge, such as control rules, and has the advantage of being easily parallelisable. This feature helps to improve the calculation speed of the controller [54]. Among the various categories of P systems, ENPSs and fuzzy reasoning spiking neural P systems are used for controller design in this paper; see [56].

3.2.1 ENPS

The ENPS [56, 57, 70] is a cellular P system that deals with numerical variables and allows multiple rules in each membrane. It is suitable to be used as the carrier of control rules in the design of a biped robot’s membrane controller.

The basic structure of a standard ENPS is defined as:

$\displaystyle\Pi=(m,H,\mu,(\textit{Var}_{1},E_{1},\textit{Pr}_{1},\textit{Var}% _{1}(0)),\ldots,(\textit{Var}_{m},E_{m},Pr_{m},\textit{Var}_{m}(0)))$ (8)

where,

(1)

$m$ is the number of layers of a membrane in a P system, usually $m\geqslant$ 1;

(2)

$H$ is the alphabet containing all the symbols in the $m$ -th membrane;

(3)

$\mu$ is the membrane structure;

(4)

$\textit{Var}_{i}$ is a set of variables in the membrane structure sub-region $i$ , here $1\leqslant i\leqslant m$ ;

(5)

$\textit{Var}_{0}$ is the initial value of variable $\textit{Var}_{i}$ ;

(6)

${E}_{i}$ is a set of enzyme variable objects in sub-region $i$ , $E_{i}\subseteq\textit{Var}_{i}$ ;

(7)

$\textit{Pr}_{i}$ is a set of rules within sub-region $i$ , composed of value generation rules and value allocation rules. The rule of ENPS has two different forms:

an enzyme-free form similar to a numerical P system:

$\displaystyle\textit{Pr}_{l,i}=(F_{l,i}(x_{l,i},\ldots,x_{k_{i},i}),$ (9) $\displaystyle\quad c_{l,1}|v_{1}+c_{l,2}|v_{2}+\ldots+c_{l,n_{i}}|v_{n_{i}})$

a form with enzymes:

$\displaystyle{Pr}_{l,i}=(F_{l,i}(x_{l,i},\ldots,x_{k_{i},i}),e_{j,i},$ (10) $\displaystyle\quad c_{l,1}|v_{1}+c_{l,2}|v_{2}+\ldots+c_{l,n_{i}}|v_{n_{i}})$

Here, $e_{j,i}\in E_{i}$ . The calculation process of ENPS is divided into the following steps:

(a)

Select rules that can be executed in each layer of the membrane structure.

(b)

Combine the activated rules (freely).

(c)

Operate these rules in all layers of the membrane system.

(d)

Reassign the calculated value results to the defined variable object according to the value distribution rule, ending the value consumption process.

3.2.2 Fuzzy reasoning spiking neural P system

The fuzzy reasoning spiking neural P system (FRSNPS) [59, 60, 71] is a type of computational model generated by embedding fuzzy theory into P systems with the aim to solve the uncertainties faced by practical problems in the P system. It has the characteristics of direct graphic representation, parallelism, dynamics and uncertainty. Among them, ‘and’ regular neurons and ‘or’ regular neurons can be used to represent the fuzzy membership relation between input and output in fuzzy control.

The FRSNPS can be expressed as:

$\displaystyle\Pi=(O,\sigma_{1},\ldots,\sigma_{m},\textit{syn},\textit{in},% \textit{out})$ (11)

where,

(1)

$O=\{a\}$ is a spiking set, where the spiking is represented by the letter $a$ ;

(2)

$\sigma_{1}\sim\sigma_{m}$ represent that the system has $m$ neurons, $\sigma_{i}=(\alpha_{i},\tau_{i},r_{i})$ , $1\leqslant i\leqslant m$ , where,

$\alpha_{i}$ represents the spiking value of $\sigma_{i}$ and takes the real number on the interval [0, 1];

$\tau_{i}$ represents the true value of $\sigma_{i}$ ;

$r_{i}$ represents the neuronal firing rule, and it is in the form of $E/a^{\alpha}\to a^{\beta}$ , where $\alpha$ and $\beta$ take the real number on the interval [0, 1];

(3)

$\textit{syn}\subseteq\{1,2,\ldots,m\}\times\{1,2,\ldots,m\}$ represents the connection between neurons, for all $(i,j)\in\textit{syn}$ , $1\leqslant i,j\leqslant m$ , $i\neq j$ ;

(4)

in and out represent the set of input and output neurons, respectively.

Figure 3.

(a) ‘And’ rule neurons; (b) ‘Or’ rule neurons.

As shown in Fig. 3, (a) and (b) are conditional ‘and’ and ‘or’ rule neurons, respectively, which are used to represent fuzzy production rules with ‘and’ and ‘or’ types in fuzzy knowledge bases. The determining factor of the rule is defined as $\zeta$ .

3.3 HFMC design

3.3.1 Design of HF algorithm

As mentioned above, the joint angle of a biped robot is the controlled object, which is suitable to be controlled by HSIC with the advantages of low model dependence, high accuracy and strong robustness. Based on HSIC, the phase plane of joint angle error is designed first, as shown in Fig. 4.

Figure 4.

The joint angle error phase plane, divided into three feature modes.

In Fig. 4, $e_{q}=q_{r}-q$ , $\dot{e}_{q}=\dot{q}_{r}-\dot{q}$ , where $q_{r}$ is the target value calculated from the foot placement by inverse kinematics, $q$ is the actual joint angle, $e_{q}$ is the angle error of the joint and $\dot{e}_{q}$ is its derivative. By following basic control theory, the phase plane is divided into three regions. Considering that PD control is the most commonly used motion control, three PD control laws are designed that correspond to the three phase plane regions, as shown in Table 1.

Table 1

Control law based on HSIC

Feature state	Control law	Remark
$\psi_{1}:e_{q}\geqslant e_{0}$	$K_{p1}e_{q}+K_{d1}\dot{e}_{q}$	PD $+^{\ast}$
$\psi_{2}:-e_{0}\leqslant e_{q}<e_{0}$	$K_{p2}e_{q}+K_{d2}\dot{e}_{q}$	PD ${}^{\ast}$
$\psi_{3}:e_{q}\leqslant-e_{0}$	$K_{p3}e_{q}+K_{d3}\dot{e}_{q}$	PD $-^{\ast}$

${}^{*}$ PD $+$ means strengthening PD control, PD means standard PD control and PD $-$ means weakening PD control.

In Table 1, three groups of PD coefficients are needed to set, where

(1)

$\psi_{1}$ :The error is large, and $q<q_{r}$ ; a large set of PD parameters are adopted.

(2)

$\psi_{2}$ : The error is small; a proper set of PD parameters are preferred.

(3)

$\psi_{3}$ : The error is large, and $q>q_{r}$ ; a small set of PD parameters are adopted.

In short, the selection of PD coefficients should follow the rules of $K_{p1}>K_{p2}>K_{p3}$ and $K_{d1}>K_{d2}>K_{d3}$ .

It can be seen from Fig. 4 that the three regions of the error phase plane have obvious boundaries. In contrast, while the errors and error change rates on both sides of the boundary are very small, different control strategies are adopted according to the control rules. Obviously, this control strategy is not conducive to the system’s stability, and it will make the control output chatter. Aiming to address this problem, several kinds of research on fuzzy rules to suppress the chattering of switching control have been investigated. Therefore, to obtain better control performance, the feature elements with a clear boundary are improved to fuzzy set representation, which aims to eliminate the influence of threshold boundary jump.

As shown in Fig. 5, considering the error threshold of $\psi_{2}$ is small, it is replaced directly with fuzzy control, and PD control is still used in $\phi_{1}$ and $\phi_{2}$ . The PD parameters of $\phi_{1}$ are bigger than $\phi_{2}$ . Overall, this is a new fuzzy PD multi-mode control algorithm to improve the control stability and accuracy of the robot joint angle.

Figure 5.

The joint angle error phase plane was improved by fuzzy control.

3.3.2 HFMC design

In an HFMC joint controller, ENPS is the container of the HF control rules. In the HF control rules, fuzzy control is designed by FRSNPS. The HFMC is shown in Fig. 6, and this nested membrane system accommodates control rules for three switching modes. In the HFMC, the fuzzy membrane controller (FuzzyMC) works as follows. Step 1: Membrane error (ME) and membrane error change (MEC) are parallel membranes and they can compute at the same time. After the calculation is completed, the membranes of ME and MEC are dissolved, and the results are released into the membrane of fuzzification. Meanwhile, the membrane of fuzzy_inference dissolves after the calculation is completed, and the results are released into mu_index. Step 2: the membranes of fuzzification and mu_index are parallel, they can compute simultaneously and the results are released to UF. Step 3: the membrane of UF dissolves after its calculation is completed, and the results are released into $\textit{UR}_{i}$ . Step 4: $\textit{UR}_{i}$ calculates and dissolves, releasing the results into HFMC.

PDMC_1, PDMC_2 and FuzzyMC are parallel relationships, and all three can compute simultaneously. PDMC_1 and PDMC_2 are similar to FuzzyMC in the computation process, and then, the three membranes dissolve and release the values into the HFMC. Ultimately, the HFMC makes the selection of control rules. In the calculation process of the membrane, it can see that the parallel relationship of membranes allows the calculation of multiple membranes to be performed simultaneously, which will save computation time.

For the convenience of design, the error $e_{i}$ is first transformed to the range [ $-$ 1, 1], and the threshold value $e_{0}$ is 0.4. When the error $e_{i}$ is in $\phi_{1}$ or $\phi_{2}$ , the PD membrane controller (PDMC) works, and the value $\textit{UP}_{i}$ is output to the outermost membrane HFMC after dissolution. When $e_{i}$ is in the region of Fuzzy, the FuzzyMC works, and the value $\textit{UR}_{i}$ is output to the outermost membrane HFMC after dissolution.

Finally, the dissolved HFMC outputs $\tau_{i}$ to the joint drive motors. Its iterative process is as follows:

Pr ${}_{1}$ :
$s_{1}\leqslant 0(-1\leqslant e_{i}<-0.4)$ , activate this rule and output $\textit{UP}_{i}$ to $\tau_{i}$ .
Pr ${}_{2}$ :
$s_{2}\leqslant 0(-0.4\leqslant e_{i}<0.4)$ , activate this rule and output $\textit{UR}_{i}$ to $\tau_{i}$ .
Pr ${}_{3}$ :
$s_{3}\leqslant 0(0.4\leqslant e_{i}<1)$ , activate this rule and output $\textit{UP}_{i}$ to $\tau_{i}$ .

where PDMC_1 corresponds to the phase plane $\phi_{1}$ , and PDMC_2 corresponds to the phase plane $\phi_{2}$ . The proportional parameter $\textit{Kp}_{i}$ and differential parameter $\textit{Kd}_{i}$ of PDMC_1 are larger than PDMC_2. As seen from Fig. 7, Pr ${}_{1}$ is a PD control rule that runs directly and sends the results to $\textit{UP}_{i}$ without conditions. The PDMC then dissolves.

Figure 6.
HFMC.

Figure 7.
PD membrane structure.

Figure 8.
The membrane structure of fuzzification.

The fuzzy control rules are as follows [18]: first, the fuzzy sets of inputs and outputs are established, and the fuzzy membership are designed; second, the fuzzy control rules are established; finally, the fuzzy decisions and defuzzification are specified. Based on this control rule, in this paper, the same fuzzy sets and membership are used for both inputs and outputs (see Fig. 9).

Figure 9.
Membership and fuzzy sets of inputs and outputs.

The design process of FuzzyMC is: 1) fuzzy inputting (fuzzification); 2) establishment of fuzzy control rules (mu_index); 3) formation of a fuzzy relationship UF; 4) obtainment of the fuzzy output set by fuzzy logic reasoning, and then the obtainment of accurate control output UR by the central average defuzzification method.

The UF of Fig. 8 contains two parts: one part is fuzzification for input fuzzification to establish fuzzy membership, and the other part, mu_index, is for output fuzzification to establish fuzzy membership. In fuzzification, ME and MEC fuzzify the input error and error derivative respectively and establish the membership (corresponding to Fig. 9). Meanwhile, in mu_index, fuzzy_inference establishes the fuzzy control rule (corresponding to Fig. 10), and then the fuzzy membership of the output rule is calculated by mu_index (corresponding to Fig. 9). After that, in UF, the ‘OR’ relation of input’s and output’s membership is the first step of fuzzy decision, the ‘OR’ relation in Fig. 11a is the second step of fuzzy decision and Fig. 11b is defuzzification.

1)
In fuzzification (see Fig. 8), the input $e$ , de and output u_range are transformed to the range [ $-$ 1, 1]. Then they are fuzzified, and the fuzzy set is $\{\rm NB(1),NS(2),ZO(3),PS(4),PB(5)\}$ ; they represent negative big, negative small, zero, positive small and positive big, respectively. The corresponding fuzzy subset domain is $\{-1,-2/3,\linebreak-1/3,0,1/3,2/3,1\}$ . Finally, the membership is obtained from the fuzzy set and the fuzzy subset domain (see Fig. 9).

Figure 10.
Fuzzy control rule list.

Figure 11.
The membrane structure of ur (a) and UR (b).

In Fig. 8, ME and MEC are the solving process of membership degree of input $e$ and de.
2)
The fuzzy_inference in Fig. 8 is the established fuzzy control rules of input and output, corresponding to 25 rules in Fig. 10. After this membrane is dissolved, the results are input to the mu_index to calculate the membership of the output.
3)
In Fig. 11a, $\textit{uf}_{i}$ ( $i=$ 1, 2, 3, 4) and ur are the fuzzy relations between input and output.
4)
Figure 11b is the defuzzification of

$\displaystyle\textit{UR}=\textit{sum}(\textit{u\_range}\times\textit{ur}^{% \prime})/\textit{sum}(\textit{ur})$ (12)

4. Experimental results

To verify the control effect of the HFMC, this paper designs the flat ground walking and slope walking simulation experiments and further compares them with PD and HSIC control methods. A five-link biped robot model (see Fig. 12) is built on Matlab R2019a based on [72]; the parameters are shown in Table 2.

Table 2
Parameters of the biped robot

Biped	Mass (kg)	Length (m)	Inertia (kg $\cdot$ m ${}^{2}$ ) Location of centre of mass (m)
Torso	14.79	0.486	0.033	0.282
Thigh	5.28	0.302	0.033	0.236
Shank	2.23	0.332	0.033	0.189

Figure 12.

Five-link biped robot model.

Figure 13.

Planar and slope walking diagrams of the biped robot. (a) Planar. (b) Uphill. (c) Downhill.

Figure 14.

Centroid velocity and displacement of slope walking. Angle tracking performance of joint $q_{1}$ (a), $q_{2}$ (b) and $q_{3}$ (c) under PD, HSIC and HFMC.

4.1 Experimental setup

The experimental model has four active joints and the corresponding control joint angles are $q_{1}$ , $q_{2}$ , $q_{3}$ and $q_{4}$ , as shown in Fig. 12. The input of the controller is the error $e_{i}$ ( $i=$ 1, 2, 3, 4) and the error derivative $\textit{de}_{i}$ ( $i=$ 1, 2, 3, 4) of the joint angle. The output is the joint torque $\tau_{i}$ . In experiments, the single-step period $T_{s}$ is set to 0.4 s, the desired speed $v_{d}$ is 0.4 m/s and the minimum speed $v_{\textit{min}}$ is 0.1 m/s, $n=$ 2. The target hip joint angle $q_{d3}$ and knee joint angle $q_{d4}$ of the swing leg are inversely calculated by $X_{f}$ . The target values of the knee and hip joints of the supporting legs are set to a fixed value: $q_{d1}=-0.5$ rad and $q_{d2}=0$ rad.

Afterwards, based on the current joint angle of the robot, the joint angle error $e_{i}$ and error derivative $de_{i}$ are calculated. In HSIC and HFMC, $e_{0}$ is 0.4. In HFMC, $e_{i}$ and $de_{i}$ are input to the PDMC_1, PDMC_2, ME and MEC of the corresponding joint controllers. When $e_{i}$ is greater than 0.4, PDMC_1 is adopted; when $e_{i}$ is less than $-$ 0.4, PDMC_2 is adopted, and FuzzyMC is adopted when $e_{i}$ is between $-$ 0.4 and 0.4. In PDMC, the corresponding torques are obtained through the calculation of PD control. In FuzzyMC, the joint torques are obtained through the summation and disjunction of $e_{i}$ and $de_{i}$ . Finally, the torques are input to achieve dynamic walking of the robot.

4.2 Experiment results

In this paper, three experiments are carried out: planar walking, uphill and downhill, as shown in Fig. 13.

4.2.1 Planar walking

The experiment lasted for 4 seconds, and the stability, accuracy and timeliness of HFMC control were proved by comparison with PD and HSIC. The experimental effect is illustrated by following the joint angle, joint angle error, centroid velocity and displacement, as shown in Figs 14–16.

Table 3
Mean and standard deviation of joint angle error and centroid velocity (denoted as CV) error (mean $\pm$ standard Deviation)

	PD [13, 14]	$W_{1}$ ${}^{\ast}$	HSIC [27, 28]	$W_{2}$ ${}^{\ast}$	HFMC
$q_{1}$	0.0918 $\pm$ 0.1158	$+$	0.1029 $\pm$ 0.1194	$+$	0.0969 $\pm$ 0.1027
$q_{2}$	0.0044 $\pm$ 0.0079	$+$	0.0041 $\pm$ 0.0115	$+$	0.0043 $\pm$ 0.0134
$q_{3}$	0.1495 $\pm$ 0.1662	$+$	0.1244 $\pm$ 0.2622	$+$	$-$ 0.0323 $\pm$ 0.1377
$q_{4}$	0.0996 $\pm$ 0.2153	$+$	$-$ 0.0718 $\pm$ 0.1724	$+$	$-$ 0.0388 $\pm$ 0.1911
CV	0.5491 $\pm$ 0.1643	$+$	0.8478 $\pm$ 0.3582	$+$	0.4180 $\pm$ 0.1536

${}^{*}$ This is the Wilcoxon test. $W_{1}$ is the result of the comparison between HFMC and PD, ‘ $+$ ’ indicates that HFMC is better than PD; $W_{2}$ is the result of the comparison between HFMC and HSIC and ‘ $+$ ’ ndicates that HFMC is better than HSIC.

Figure 15.

Angle tracking performance of joint $q_{4}$ under (a) PD, (b) HSIC and (c) HFMC.

Figures 14–15 are the joint angles following the results of $q_{1}$ , $q_{2}$ , $q_{3}$ and $q_{4}$ controlled by PD, HSIC and HFMC, respectively, where $q_{1}$ and $q_{2}$ are supporting leg joints, $q_{3}$ and $q_{4}$ are swing leg joints. The solid blue line is the desired value, and the dotted red line is the actual value.

In Fig. 14, it can be seen that the robot starts wobbling under PD control, while HSIC and HFMC start stably. In Figs 14c and 15, there is a control lag of PD, and the trajectory tracking effect is poor. The effect of HSIC is better than that of PD, but there is still the problem of control lag. It can be seen in Fig. 15 that the chattering is caused by the switching of the control mode of HSIC, which affects walking efficiency, while HFMC effectively inhibits the chattering. In general, the tracking accuracy and real-time performances of HFMC are optimal.

Figure 16.

(a) Centroid velocity and (b) displacement for planar walking.

Figure 16 shows the centroid velocity and displacement. By comparison, the robot in Fig. 16a is difficult to start under PD control, and the velocity fluctuates greatly. The robot can start stably under HSIC and HFMC. However, the speed of HSIC increases after 1 s, which is far greater than the desired speed of 0.4 m/s. HFMC effectively controls the velocity within the desired speed range. As depicted in Fig. 16b, the robot smoothly reaches the desired displacement under HFMC, while PD and HSIC exceed the desired value of 1.6 m due to inaccurate speed control.

Table 3 shows the static analysis of joint angle and centroid velocity tracking errors. A comparison of the mean and standard deviation error data show that the joint angle error of HSIC is smaller than PD, while HFMC is smaller than PD and HSIC. The velocity of HSIC is the fastest, PD is the second and HFMC is the closest to the target value. The Wilcoxon test shows that the performance of HFMC is better than that of HSIC and PD.

According to synthetical evaluation, it can be concluded that the robot has a superior accomplishment under HFMC. It has good timeliness and a small joint angle following error. The robot starts stably, and the speed and displacement are controlled accurately.

4.2.2 Slope walking

This section includes both uphill and downhill experiments. The uphill experiments are carried out on a slope of 0 ${}^{\circ}$ $\sim$ 20 ${}^{\circ}$ , the downhill experiments are carried out on a slope of $-$ 15 ${}^{\circ}$ $\sim$ 0 ${}^{\circ}$ , and the parameters were consistent with the planar walking. Beyond this angle range, the robot would fall down.

1) Uphill

In the uphill experiments, 90 independent runs of experiments were performed for each slope value, starting from 0 ${}^{\circ}$ and increasing by 1 ${}^{\circ}$ to 20 ${}^{\circ}$ , respectively. Experimental results for 5 ${}^{\circ}$ and 15 ${}^{\circ}$ have been selected displayed in Fig. 17 for illustration purposes.

Figure 17.

Performance of velocity, displacement and foot placement of uphill. (a–b) Centroid velocity and displacement at a slope angle of $\alpha=$ 5 ${}^{\circ}$ and $\alpha=$ 15 ${}^{\circ}$ ;. (c–d) Foot placement at a slope angle of $\alpha=$ 5 ${}^{\circ}$ and $\alpha=$ 15 ${}^{\circ}$ .

Figures 17a and 17b show the velocity and displacement of the centre of mass. In Fig. 17a, the robot’s starting speed increases rapidly under PD control, resulting in an unstable first step; HSIC is still not stable enough as the speed increases to 0.7m/s after the start and then decreases. On the other hand, the robot has a stable start and walks under HFMC. In Fig. 17b, the robot starts with a large swing under PD, which will cause the robot to fall easily. The robot starts relatively stable under HSIC, but the speed fluctuates significantly, rising to 0.8 m/s and then falling, while HFMC remains stable at the beginning and after walking.

In a comprehensive view of the displacement diagram, at 5 ${}^{\circ}$ , the velocity of PD and HSIC increase and then decrease, and the final displacements are below the target value of 1.6 m, but the fluctuation of PD is greater. Nevertheless, the displacement under HFMC reaches 1.6 m evenly and smoothly. Similarly, at 15 ${}^{\circ}$ , the displacement of PD shows the phenomenon of the robot back-swing, and the final displacement is significantly lower than the target value of 1.6 m. The final displacement of HSIC is higher than the target value because the speed is higher than the target speed of 0.4 m/s, while HFMC performs well. In general, PD control results in an unstable first step, and the speed and displacement exceed the target with HSIC. By comparison, HFMC is the most optimal, with the advantages of stability and high-precision control.

Figure 18.

Performance of velocity, displacement and foot placement of downhill. (a–b) Centroid velocity and displacement at a slope angle of $\alpha=$ 5 ${}^{\circ}$ and $\alpha=$ 15 ${}^{\circ}$ ; (c–d) Foot placement at a slope angle of $\alpha=$ 5 ${}^{\circ}$ and $\alpha=$ 15 ${}^{\circ}$ .

In Figs 17c and 17d, the black dotted line is the inclined plane, and the other three lines are the foot placement for the three controlled robots to walk along the inclined plane. At 5 ${}^{\circ}$ , PD’s velocity fluctuates the most, HSIC fluctuates less than PD and HFMC is relatively smooth. Although the final landing point is smaller than the target landing point [1.6, 0.14], HFMC is closer. At 15 ${}^{\circ}$ , the robot under PD has a serious back-swing, which leads the final foot placement to be much smaller than the target value. Due to the high speed of the robot under HSIC, the foot placement is larger than the target value. However, the foot placement of HFMC on two different slopes are more accurate than PD and HSIC.

2) Downhill

Similar to the uphill experiments, in the downhill experiments, $-$ 5 ${}^{\circ}$ and $-$ 15 ${}^{\circ}$ have been chosen to present the experiments carried out.

Figures 18a and 18b show the velocity and displacement. At $-$ 5 ${}^{\circ}$ , the velocity of PD is lower than the target value of 0.4 m/s before 2.7 s and starts to rise with sudden fluctuations at 2.7 s before falling, and the final displacement is lower than the target value of 1.6 m. The high speed of HSIC causes the displacement to be much higher than the target value. HFMC, on the other hand, has the smallest velocity fluctuation, and the displacement is closest to 1.6 m. At $-$ 15 ${}^{\circ}$ , the velocity tends to increase with all three methods due to the decreasing slope and the effect of gravity. PD is the fastest, HSIC is the second and HFMC is the smallest, which eventually leads to the same results for displacement. Comparatively, HFMC performs better than PD and HFMC.

Table 4

The mean centroid velocity (denoted as CV) (m/s) and displacement (denoted as D) (m) of the robot walking on the slope

Slope angle		PD [13, 14]	HSIC [27, 28]	HFMC
20 ${}^{\circ}$	CV	0.20	0.47	0.37
	D	0.81	2.01	1.62
15 ${}^{\circ}$	CV	0.20	0.45	0.37
	D	0.78	1.77	1.60
10 ${}^{\circ}$	CV	0.20	0.39	0.41
	D	0.77	1.34	1.57
5 ${}^{\circ}$	CV	0.34	0.34	0.34
	D	1.41	1.36	1.62
$-$ 5 ${}^{\circ}$	CV	0.35	0.77	0.38
	D	1.18	2.47	1.58
$-$ 10 ${}^{\circ}$	CV	0.32	0.50	0.42
	D	1.09	3.2	1.6
$-$ 15 ${}^{\circ}$	CV	1.24	0.83	0.81
	D	5.03	3.94	2.26

In Fig. 18c, due to the fluctuation of velocity, the foot placement of PD appears uneven and smaller than the target value of [1.6, $-$ 0.14]. HSIC is much larger than the target foot placement because the fast velocity, and HFMC is quite close to the target value. In Fig. 18d, all three methods lead to a final foot placement that is larger than the target value, but HFMC is the closest to [1.6, $-$ 0.43].

Table 5

Time (s) for PD, HSIC and HFMC on the various slope scenarios

Slope angle	PD [13, 14]	HSIC [27, 28]	HFMC
20 ${}^{\circ}$	0.5165	0.5607	0.2855
15 ${}^{\circ}$	0.5359	0.5645	0.2159
10 ${}^{\circ}$	0.5379	0.5603	0.1868
5 ${}^{\circ}$	0.5835	0.6206	0.2069
0 ${}^{\circ}$	0.6467	0.5811	0.2147
$-$ 5 ${}^{\circ}$	0.6553	0.6033	0.2158
$-$ 10 ${}^{\circ}$	0.5503	0.6115	0.2128
$-$ 15 ${}^{\circ}$	0.5392	0.6154	0.2069

In addition, in order to enrich the experimental results from multiple angles, the equal segment is divided with a difference of 5 ${}^{\circ}$ , and the results are shown in the Table 4. In the slope walking experiments with multiple angles, the target value of speed is 0.4 m/s, and the target value of displacement is 1.6 m. As can be seen from the table, no matter which angle, the robot’s speed and displacement with HFMC are the closest to the target value.

To sum up, it can be ascertained that under the control of HFMC, the robot can not only walk on flat ground but also walk stably and accurately on slopes with different inclination angles under the same control parameters, which is sufficient to demonstrate the superiority of HFMC control performance.

In order to emphasise the benefits in terms of computational time of the proposed HFMC, time results are reported in Table 5.

Table 5 is the running time of a single joint controller, and it shows that HFMC outperforms, in terms of time, both PD and HSIC. These results indicate the potential of HFMC for real-time control.

The following link contains videos of the experiments and displays the effect of the various controllers under consideration on the robot’s gait: https:// github.com/xy950/HFMC-for-robot-walking.

5. Conclusion

This paper proposes a novel joint controller for biped robots, indicated with HFMC. The proposed approach combines the advantages of PD control, HSIC, fuzzy control and membrane computing. The main conclusions of this study are the following:

Chattering reduction: the multi-mode PD based on HSIC can reduce control error, but the Chattering phenomenon caused by mode switching needs to be avoided for better control. The integration of fuzzy control rules effectively solves this problem.

Real-time control improvement: membrane computing has a parallel distributed structure and powerful computing power. The membrane controller HFMC combined with the control algorithm HF is a good choice to improve real-time control.

HFMC retains the advantages of traditional PD, enhances the accuracy, stability and real-time of biped robot joint control and ensures excellent walking performance.

Future work will address the following issues: first, how to extend the proposed method to complex bipedal robots; second, conducting further research on HSIC and membrane computing to solve the adaptability and stability of biped robot movement in sloped, uneven and other complex environments.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61972324) and the Beijing Advanced Innovation Center for Intelligent Robots and Systems (2019IRS14).

References

Collins

Ruina

Tedrake

Wisse

. Efficient bipedal robots based on passive-dynamic walkers. Science. 2005; 307(5712): 1082-5.

Alexander

. Walking made simple. Science. 2005; 308(5718): 58-9.

Chen

Huang

Zhang

Meng

Zhang

, et al. Gait Planning of Omnidirectional Walk on Inclined Ground for Biped Robots. IEEE Transactions on Systems, Man, and Cybernetics: Systems. 2016; 46(7): 888-97.

. Recognizing object surface materials to adapt robotic disinfection in infrastructure facilities. Computer-Aided Civil and Infrastructure Engineering. 2022.

Huang

Yokoi

Kajita

Kaneko

Arai

Koyachi

, et al. Planning walking patterns for a biped robot. IEEE Transactions on Robotics and Automation. 2001; 17(3): 280-9.

Kim

Spieler

Lupu

Ramezani

Chung

. A bipedal walking robot that can fly, slackline, and skateboard. Science Robotics. 2021; 6(59): 8136-50.

Abu-Zidan,

Nguyen

Mendis

Priyan

Setunge

Adeli

. Design of a smart prefabricated sanitising chamber for COVID-19 using computational fluid dynamics. Journal of Civil Engineering and Management. 2021; 27(2): 139-48.

Pegah

Touraj

. Seismic adaptive control of building structures with simultaneous sensor and damper faults based on dynamic neural network. Computer-Aided Civil and Infrastructure Engineering. 2022; 37(11): 1402-16.

Adeli

Saleh

. Optimal control of adaptive/smart bridge structures. Journal of Structural Engineering. 1997; 123(2): 218-26.

10.

Gutierrez Soto

Adeli

, Many-objective control optimization of high-rise building structures using replicator dynamics and neural dynamics model. Structural and Multidisciplinary Optimization. 2017; 56(6): 1521-37.

11.

Gutierrez Soto

Adeli

, Recent advances in control algorithms for smart structures and machines. Expert Systems. 2017; 34(2): 12205-19.

12.

Adeli

. Control methodologies for vibration control of smart civil and mechanical structures. Expert systems. 2018; 35(6): 12354-74.

13.

Kolathaya

. Local stability of PD controlled bipedal walking robots. Automatica. 2020; 114: 108841-57.

14.

Shishir

. PD tracking for a class of underactuated robotic systems with kinetic symmetry. IEEE Control Systems Letters. 2020; 5(3): 809-14.

15.

Ren

Luo

Chen

Wang

. Gait trajectory-based interactive controller for lower limb exoskeletons for construction workers. Computer-Aided Civil and Infrastructure Engineering. 2022; 37(5): 558-72.

16.

Yin

Chen

. Stackelberg-theoretic approach for performance improvement in fuzzy systems. IEEE Transactions on Cybernetics. 2018; 50(5): 2223-36.

17.

Kuo

Chen

Hung

Luan

Hsu

, et al. Fuzzy double deep Q-network-based gait pattern controller for humanoid robots. IEEE Transactions on Fuzzy Systems. 2020; 30(1): 147-61.

18.

Dong

Chen

Huang

. Adaptability Control Towards Complex Ground Based on Fuzzy Logic for Humanoid Robots. IEEE Transactions on Fuzzy Systems. 2022; 30(6): 1574-84.

19.

Chand

Veer

Poulakakis

. Interactive Dynamic Walking: Learning Gait Switching Policies With Generalization Guarantees. IEEE Robotics and Automation Letters. 2022; 7(2): 4149-56.

20.

Lele

Fang

Ting

Raychowdhury

. Learning to walk: bio-mimetic hexapod locomotion via reinforcement-based spiking central pattern generation. IEEE Journal on Emerging and Selected Topics in Circuits and Systems. 2020; 10(4): 536-45.

21.

Benamara

Val-Calvo

Sánchez

JRÁ

Díaz-Morcillo

de Vicente

JMF

Fernández-Jover

, et al. Real-time facial expression recognition using smoothed deep neural network ensemble. Integrated Computer-Aided Engineering. 2021; 28(1): 97-111.

22.

Macias-Garcia

Pérez

Medrano-Hermosillo

Bayro-Corrochano

. Multi-stage deep learning perception system for mobile robots. Integrated Computer-Aided Engineering. 2021; 28(2): 191-205.

23.

Wang

Adeli

. Self-constructing wavelet neural network algorithm for nonlinear control of large structures. Engineering Applications of Artificial Intelligence. 2015; 41: 249-58.

24.

Chen

Dong

Labi

. Graph neural network and reinforcement learning for multi-agent cooperative control of connected autonomous vehicles. Computer-Aided Civil and Infrastructure Engineering. 2021; 36(7): 838-57.

25.

Huang

Dong

Chen

, et al. Resistant Compliance Control for Biped Robot Inspired by Humanlike Behavior. IEEE/ASME Transactions on Mechatronics. 2022. doi: 10.1109/TMECH.2021.3139332.

26.

Zhang

Meng

Chen

Liu

Huang

. Stride Length and Stepping Duration Adjustments Based on Center of Mass Stabilization Control. IEEE/ASME Transactions on Mechatronics. 2022. doi: 10.1109/TMECH.2022.3159321.

27.

Liu

Xue

Deng

. Biped robot dynamic walking humanoid intelligent control. Journal of Chongqing University. 2013; 36(2): 40-50.

28.

Jing

Zhu

. Application of humanoid predictive control in biped robot walking control. Sensor and Microsystem. 2014; 33(3): 157-60.

29.

Huynh

Lin

Cho

Pham

Chao

. A new self-organizing fuzzy cerebellar model articulation controller for uncertain nonlinear systems using overlapped Gaussian membership functions. IEEE Transactions on Industrial Electronics. 2019; 67(11): 9671-82.

30.

Teng

Gull

Bai

. PD-based fuzzy sliding mode control of a wheelchair exoskeleton robot. IEEE/ASME Transactions on Mechatronics. 2020; 25(5): 2546-55.

31.

Azimi

Shu

Zhao

Gehlhar

Simon

Ames

. Model-based adaptive control of transfemoral prostheses: Theory, simulation, and experiments. IEEE Transactions on Systems, Man, and Cybernetics: Systems. 2019; 51(2): 1174-91.

32.

Siddique

Adeli

. Nature Inspired Computing: An Overview and Some Future Directions. Cogn Comput. 2015; 7(6): 706-14.

33.

Rafiei

Adeli

. A New Neural Dynamic Classification Algorithm. IEEE Trans Neural Networks Learn Syst. 2017; 28(12): 3074-83.

34.

Kim

Adeli

. Hybrid feedback-least mean square algorithm for structural control. Journal of Structural Engineering. 2004; 130(1): 120-7.

35.

Gutierrez Soto

Adeli

. Multi-agent replicator controller for sustainable vibration control of smart structures. Journal of Vibroengineering. 2017; 19(6): 4300-22.

36.

Gutierrez Soto

Adeli

. Vibration control of smart base-isolated irregular buildings using neural dynamic optimization model and replicator dynamics. Engineering Structures. 2018; 156: 322-36.

37.

Gutierrez Soto

Adeli

. Semi-active vibration control of smart isolated highway bridge structures using replicator dynamics. Engineering Structures. 2019; 186: 536-52.

38.

Hormozabad

Gutierrez Soto

Adeli

. Integrating structural control, health monitoring, and energy harvesting for smart cities. Expert Systems. 2021; 38(8): 12845-69.

39.

Zhang

Jing

. Model-Free Saturated PD-SMC Method for 4-DOF Tower Crane Systems. IEEE Transactions on Industrial Electronics. 2022; 69(10): 10270-80.

40.

Chen

Luo

. A Two-Degree-of-Freedom Controller Design Satisfying Separation Principle With Fractional-Order PD and Generalized ESO. IEEE/ASME Transactions on Mechatronics. 2021; 27(1): 137-48.

41.

Zahra

Navarro-Alarcon

Tolu

. A neurorobotic embodiment for exploring the dynamical interactions of a spiking cerebellar model and a robot arm during vision-based manipulation tasks. International Journal of Neural Systems. 2022; 32(8): 2150028.

42.

Ieracitano

Morabito

Hussain

Mammone

. A hybrid-domain deep learning-based BCI for discriminating hand motion planning from EEG sources. International journal of neural systems. 2021; 31(9): 2150038.

43.

Duan

Zhu

. A Robust 3D-Convolutional Neural Network-Based Electroencephalogram Decoding Model for the Intra-Individual Difference. International Journal of Neural Systems. 2022; 2250034-4.

44.

Macias-Garcia

Galeana-Perez

Medrano-Hermosillo

Bayro-Corrochano

. Multi-stage deep learning perception system for mobile robots. Integrated Computer-Aided Engineering. 2021; 28(2): 191-205.

45.

Benamara

Val-Calvo

Alvarez-Sanchez

Diaz-Morcillo

Ferrandez-Vicente

Fernandez-Jover

, et al. Real-time facial expression recognition using smoothed deep neural network ensemble. Integrated Computer-Aided Engineering. 2021; 28(1): 97-111.

46.

Wang

Hou

Wang

. Reinforcement learning-based bird-view automated vehicle control to avoid crossing traffic. Computer-Aided Civil and Infrastructure Engineering. 2021; 36(7): 890-901.

47.

Salem

Abido

Blaabjerg

. Common-Mode Voltage Mitigation of Dual T-Type Inverter Drives Using Fast MPC Approach. IEEE Transactions on Industrial Electronics. 2021; 69(8): 7663-74.

48.

Chen

Zhang

. Path Planning for Intelligent Vehicle Collision Avoidance of Dynamic Pedestrian Using Att-LSTM, MSFM, and MPC at Unsignalized Crosswalk. IEEE Transactions on Industrial Electronics. 2021; 69(4): 4285-95.

49.

Zhou

. A Self Tuning Algorithm Capable of Simulating Human Control Strategy. Journal of Chongqing University (Natural Science Edition). 1985; 1: 135-45.

50.

. Humanoid intelligent control. Beijing: National Defense Industry Press; 2003.

51.

Zhang

Pérez-Jiménez

Riscos-Núñez

Verlan

Konur

Hinze

, et al. Membrane computing models: implementations. Springer; 2021. vol. 10.

52.

Gatti

Leporati

Zandron

. On Spiking Neural Membrane Systems with Neuron and Synapse Creation. International Journal of Neural Systems. 2022; 32(8): 2250036-6.

53.

Zhang

Rong

Paul

Zhu

Neri

, et al. A layered spiking neural system for classification problems. International Journal of Neural Systems. 2022; 2250023.

54.

Zhang

Shang

Verlan

Martínez-Del-Amor

MÁ

Yuan

Valencia-Cabrera

, et al. An overview of hardware implementation of membrane computing models. ACM Computing Surveys (CSUR). 2020; 53(4): 1-38.

55.

Zhang

Pérez-Jiménez

Gheorghe

. Real-life applications with membrane computing. Springer; 2017. vol. 25.

56.

Wang

Zhang

Gou

Paul

Neri

Rong

, et al. Multi-behaviors coordination controller design with enzymatic numerical P systems for robots. Integrated Computer-Aided Engineering. 2021; 28(2): 119-40.

57.

Pérez-Hurtado

Martínez-del-Amor

MÁ

Zhang

Neri

Pérez-Jiménez

. A membrane parallel rapidly-exploring random tree algorithm for robotic motion planning. Integrated Computer-Aided Engineering. 2020; 27(2): 121-38.

58.

Wang

Adeli

. Algorithms for chattering reduction in system control. Journal of the Franklin Institute. 2012; 349(8): 2687-703.

59.

Peng

Wang

Pérez-Jiménez

Wang

Shao

Wang

. Fuzzy reasoning spiking neural P system for fault diagnosis. Information Sciences. 2013; 235: 106-16.

60.

Wang

Zhang

Zhao

Wang

Pérez-Jiménez

. Fault diagnosis of electric power systems based on fuzzy reasoning spiking neural P systems. IEEE Transactions on Power Systems. 2014; 30(3): 1182-94.

61.

Hernandez-Barragan

Lopez-Franco

Arana-Daniel

Alanis

Lopez-Franco

. A modified firefly algorithm for the inverse kinematics solutions of robotic manipulators. Integrated Computer-Aided Engineering. 2021; 28(3): 257-75.

62.

Chen

Sun

Zhao

Liu

. A New Lane Keeping Method Based on Human-Simulated Intelligent Control. IEEE Transactions on Intelligent Transportation Systems. 2021; 1-12.

63.

Zhang

Wen

Wang

. Human simulated intelligent control based on sensory-motor intelligent schemas. In: Fifth World Congress on Intelligent Control and Automation (IEEE Cat. No. 04EX788). IEEE; 2004. vol. 3. pp. 2423-7.

64.

Chen

Tang

. Parameter optimisation of human-simulated intelligent controller for a cart-double pendulum system. International Journal of Modelling, Identification and Control. 2010; 10(3-4): 194-201.

65.

Chen

Sun

Zhao

Liu

Jin

. Human-machine cooperative scheme for car-following control of the connected and automated vehicles. Physica A: Statistical Mechanics and its Applications. 2021; 573: 125949-67.

66.

Xiong

Zhu

Pan

Wang

. Research on Intelligent Trajectory Control Method of Water Quality Testing Unmanned Surface Vessel. Journal of Marine Science and Engineering. 2022; 10(9): 1252-78.

67.

Paun

. Computing with Membranes. Journal of Computer and System Sciences. 2000; 61(1): 108-43.

68.

Paun

. Membrane Computing: An Introduction. Natural Computing Series. Springer; 2002.

69.

Zhang

Rong

Paul

Neri

Pérez-Jiménez

. A complete arithmetic calculator constructed from spiking neural P systems and its application to information fusion. International Journal of Neural Systems. 2021; 31(1): 2050055-20071.

70.

Zhang

Xiao

Dong

Zhang

Neri

. Enzymatic Numerical Spiking Neural Membrane Systems and their Application in Designing Membrane Controllers. International Journal of Neural Systems. 2022; 2250055-5.

71.

Wang

Liu

Cabrera

Wang

Wei

Zang

. A novel fault diagnosis method of smart grids based on memory spiking neural P systems considering measurement tampering attacks. Information Sciences. 2022; 596: 520-36.

72.

Tzafestas

Raibert

Tzafestas

. Robust sliding-mode control applied to a 5-link biped robot. Journal of Intelligent and Robotic Systems. 1996; 15(1): 67-133.

A human-simulated fuzzy membrane approach for the joint controller of walking biped robots

Abstract

Keywords

1. Introduction

2.1 Dynamic model of biped robot

2.2.1 Motion planning

3. HF membrane approach

3.1 HSIC

3.2.1 ENPS

3.3.1 Design of HF algorithm

Table 2 Parameters of the biped robot

4.2 Experiment results

4.2.1 Planar walking

Table 3 Mean and standard deviation of joint angle error and centroid velocity (denoted as CV) error (mean ± standard Deviation)

Footnotes

Acknowledgments

References

Table 2
Parameters of the biped robot

Table 3
Mean and standard deviation of joint angle error and centroid velocity (denoted as CV) error (mean $\pm$ standard Deviation)