Gait coordination control of bipedal modular reconfigurable robot based on adaptive neural network

Abstract

This paper proposed an adaptive neural network (NN) control method to track the desired trajectory of a bipedal modular reconfigurable robot (MRR), which can solve the gait coordination problem of bipedal MRR. The leg dynamic model and the body dynamic model of bipedal MRR are established based on the Newton–Euler iterative method, and the global dynamic model is subsequently established. Aiming at the gait coordination problem between double support phase (DSP) and single support phase (SSP), the desired trajectory is generated based on the zero-moment point (ZMP) method. The adaptive NN controller is designed to track the generated desired trajectory, which also compensates interconnected dynamic coupling (IDC) effects of the bipedal MRR. The stability of bipedal MRR system is proved by Lyapunov theory. In the end, the effectiveness of the control method is verified by comparative simulation. The simulation results show that the proposed adaptive NN method reduces the position tracking error by $~ 10 %$ and the control torque by $~ 15 %$ compared with the existing control methods.

Keywords

Bipedal modular reconfigurable robot neural network control zero-moment point gait coordination

Introduction

In the 21st century, with the development of science and technology, the walking robot industry has become a hot issue in social research. In addition, the walking robot has long entered the field of human work and life, replacing humans to complete many special tasks, and playing an important role in the manufacturing and industry (Biswal and Mohanty, 2021; Kar, 2003). Walking robot, also known as walking machine, integrates mechanical design, biological science, sensing technology, and information processing technology. At present, walking robot mainly includes bipedal robot and multi-legged robot (quadruped, hexapod and eight-legged robot, etc.) (Ding and Chen, 2016; Lyashenko, 2022). Among walking robot, bipedal robot has the similar leg structure to human beings; however, the traditional bipedal robot cannot adapt to the new environment and work task because of its poor flexibility. Consequently, it is expected that traditional bipedal robots will ultimately become obsolete.

Compared to traditional robots, modular reconfigurable robots (MRRs) offer superior structural flexibility and adaptability. When performing special tasks, MRR is more adaptable to the environment. The MRR system can be reassembled and disconnected according to the requirements of the mission environment (Pan et al., 2013). The recombination is not only a simple mechanical recombination but also a system recombination. So far, an increasing number of MRRs have been designed (Kotay et al., 1998; Murata et al., 1994). An et al. (2023) proposed a human motion intention estimation method based on a harmonic-driven compliance model, which forms the basis of MRR dynamic model. MRR has been extensively employed in disaster response, space exploration, medical emergency management, and rehabilitation training. Therefore, considering the high flexibility of the MRR system, the bipedal MRR system is proposed, which has high adaptability to the environment.

A growing numberf research teams around the world are working on bipedal robots, and their main research directions are gait coordination, foot force control, foot trajectory planning, and motion control. In order to apply bipedal robot walk stably, many researchers have been exploring and proposing different motion control strategies. Aiming at the gait trajectory of bipedal robot, Vukobratović and Borovac (2004) proposed the zero-moment point (ZMP) stability criterion, and they applied this concept to the bipedal walking robot with dynamic balance, tracking the ideal ZMP curve to achieve steady walking. After reviewing the traditional pattern generation method based on ZMP, Kajita et al. (2003) proposed a bipedal walking path generation method based on ZMP and designed a tracking servo controller. A few years later, in 2007, Kajita et al. (2007) proposed a running model generation and controller based on ZMP. Alcaraz-Jiménez et al. (2013) combined the ZMP stability criterion with step-timing control and applied it to humanoid robots, such as a control system allows robust walking and anti-interference. Ugurlu and Kawamura (2009) proposed a generating online jumping patterns method for single-legged or humanoid robots. Caron et al. (2017) designed a global controller to generate feasible contact motion of bipedal robot. At present, research groups around the world are exploring trajectory generation methods of bipedal robots based on ZMP theory, which has been promoted in the literature (Esfahani and Elahinia, 2020; Ferreira et al., 2012; Su and Zheng, 2007; Wong et al., 2020). For bipedal MRR system, the proposed ZMP method greatly reduces the computation. Even under the contact conditions of complex environment, the stability criterion can ensure the stable walking of the robot. In addition, how to design the controller to make the walking robot move smoothly is also important.

The dynamic model of bipedal MRR system is inherently intricate and characterized by a significant level of uncertainty, making it exceedingly challenging to obtain precise model information. Neural network (NN) control method has strong learning ability, and it is an effective tool for controlling complex nonlinear dynamic systems. The basic idea is using NN approximators to compensate unknown nonlinear dynamic terms (Gomi and Kawato, 1993). In addition, the method based on NN can deal with the control problem of nonlinear system, and the system does not need to be linearized (Zhang et al., 2016). In the 20th century, NN has been used in robot system modeling and control (Ge et al., 1998; Lewis et al., 1998). Nguyen et al. (2020) addressed an important aspect of deep reinforcement learning (DRL) related to situations that require multiple agents to communicate and cooperate to solve complex tasks. In recent years, NN control technology has developed rapidly. Chu et al. (2019) proposed an adaptive global synovial controller based on NN. Zhu et al. (2023) transformed the trajectory tracking control problem into zero-sum game-based optimal control problem based on adaptive dynamic programming (ADP) and NN algorithm. Yu and Feu (2014) solved the issue of robust switching tracking neural control of robotic arm with uncertainties and disturbances. Jiang et al. (2021) proposed a control method for a two-handed robotic system. A paper Nguyen et al. (2022) presented a survey of algorithms used to create deepfakes and, more importantly, methods proposed to detect deepfakes in the literature to date. Zhang et al. (2015b) proposed a scheme for generating circular motion of both arms based on neurodynamics method. Li et al. (2015) proposed a foot force optimization method based on recursive NN for quadruped walking robot. Li and Li (2020) proposed a mobile robot algorithm model based on backpropagation NN and reinforcement learning, and NN controller is used in many robot systems (Cui et al., 2017; He et al., 2016). Nguyen and Reddi (2023) incorporated deep learning into traditional RL, and DRL is highly capable of solving complex, dynamic, and especially high-dimensional cyber defense problems. However, above methods only consider the traditional robot system model, but for MRR system, interconnection dynamic coupling (IDC) effect will affect the system performance. Since traditional NN control method cannot handle IDCs, this paper seeks a novel NN approximation method to solve these problems, which can improve the overall motion performance and control accuracy of the bipedal robot.

The main work of scholars in recent years is listed in Table 1.

Table 1.

Main work in recent years.

Time	Author	Major contribution
2021	Biswak et al.	Wrote a review of quadruped robot and summarized the different control methods.
2021	Jiang et al.	Proposed a control method for a two-handed robotic system.
2022	Lyashenko	Wrote an overview of the new walking robot.
2022	Yao et al.	Proposed a gait planning method for underactuated bipedal robot on uneven and compliant terrain.
2023	An et al.	Proposed a human motion intention estimation method based on harmonic-driven compliance model.
2023	Zhu et al.	Transformed the trajectory tracking control problem into zero-sum game-based optimal control problem based on
		adaptive dynamic programming (ADP) and NN method.

The main contributions of this paper are as follows:

Unlike the traditional biped robot dynamic model (Yao et al., 2022), the flexibility of the robot system is not considered. In this paper, a bipedal MRR system is proposed for the first time, which solves the limited motion range problem of traditional robot, and it has broad application prospects because of the flexibility. Considering the IDC effects of bipedal MRR, the NN identifier-based compensation control policy is used to solve the IDC effects.

Different from the traditional ZMP stability criterion (Caron et al., 2017), the ZMP method generates the expected gait path without considering the effect of acceleration, which reduces the computational burden.

A bipedal MRR steady-state walking coordination control method based on adaptive NN is proposed. And for bipedal MRR system with high nonlinear and model uncertainty, an adaptive NN controller is used to approximate the model uncertainty, which ensures the steady-state walking ability of the bipedal MRR system.

Kinematic analysis and gait planning of bipedal MRR

Kinematic analysis of bipedal MRR

The stable walking of bipedal MRR mainly depends on the lower limb walking mechanism. Moreover, the bipedal MRR system has characteristics of complex model structure and more degrees of freedom; therefore, reasonable kinematic analysis is a good basis for bipedal MRR to achieve stable walking.

In this section, a series of analyses and hypotheses are made from the perspective of bionics combined with the way of human leg movement. When people walk, they mainly rely on the coordination of the ankle, knee, and hip joints. And the hip joint can perform three directions of internal and external rotation, forward and forward pitching and abductive convergence. The knee joint can perform forward and forward pitching, and the ankle joint can perform abductive convergence and forward and forward pitching.

This paper only analyzed the forward motion of the bipedal MRR and did not consider the lateral or backward motion, and the bipedal MRR can satisfy the walking motion in the plane area. In conclusion, kinematic analysis is of great significance for gait planning and flexibility control strategy design of bipedal MRR.

Gait design of bipedal MRR

Since the movement mode of bipedal MRR is similar to humans, the two legs move alternately and drive the body to walk forward in the process of movement and carry out periodic movement. When researching the kinematic gait of the robot, only one cycle of MRR is studied. There are two main phases in the walking process of the bipedal MRR: double support phase (DSP) and single support phase (SSP). Considering the symmetry of the machine leg, gait period is defined as the time interval from the support stage of one support leg to the standing stage of the other support leg.

The SSP can also be called the swing phase (SP), where one leg is in a supporting state while the other mechanical leg is in a free-space swinging state and all the force/torque is applied to the supporting leg. The DSP includes weight acceptance phase (WAP) and weight support phase (WSP), when one mechanical leg of the bipedal MRR is in WSP and the other mechanical leg is in WAP or SP.

ZMP stability criterion of bipedal MRR

ZMP is a point on the ground, and the torque of the ground reaction force on the supporting foot of the robot is zero along the two vertical directions in the horizontal plane. In this paper, ZMP method is used to generate the footprint of bipedal MRR, which can plan the contact position between the ground and the plantar. We assume that two soles of the bipedal MRR are composed of two support rectangles (SR), which refers to the minimum support rectangle of the plantar and contact ground travel of the bipedal MRR. In SSP, ZMP should always remain within the rectangular box. In DSP, ZMP may be outside the rectangular box. ZMP is an important basis to judge whether bipedal MRR can contact the ground and maintain stability during walking.

In calculation, assuming that the ground is in complete contact with the sole of bipedal MRR, then the normal force against the ground and foot can be expressed as $P$ , and the dynamic variable relationship between ZMP and bipedal MRR can be easily obtained. We know that

f_{j}^{ext} = [\begin{matrix} \sum_{j} {(f_{j}^{ext})}_{x} \\ \sum_{j} {(f_{j}^{ext})}_{y} \\ \sum_{j} {(f_{j}^{ext})}_{z} \end{matrix}] = \sum_{i} m_{i} [\begin{matrix} {\overset{\cdot\cdot}{x}}_{i} \\ {\overset{\cdot\cdot}{y}}_{i} \\ ({\overset{\cdot\cdot}{z}}_{i} + g) \end{matrix}] - [\begin{matrix} 0 \\ 0 \\ P \end{matrix}]

(1)

where $(x_{j}, y_{j}, z_{j})$ is the center of mass coordinates of the link, $m_{i}$ is the mass of each connecting rod of MRR, and $f_{j}^{ext}$ is the external force applied at $(x_{j}, y_{j}, z_{j})$ .

We can also get

\begin{matrix} \sum_{j} [- {(f_{j}^{ext})}_{y} z_{j} + {(f_{j}^{ext})}_{z} y_{j} + {(M_{j}^{ext})}_{x}] + P y_{zmp} \\ - \sum_{i} m_{i} g y_{i} = ({\overset{\cdot}{H}}_{0})_{x} \end{matrix}

(2)

\begin{matrix} \sum_{j} [- {(f_{j}^{ext})}_{x} z_{j} + {(f_{j}^{ext})}_{z} x_{j} + {(M_{j}^{ext})}_{y}] + P x_{zmp} \\ - \sum_{i} m_{i} g x_{i} = ({\overset{\cdot}{H}}_{0})_{y} \end{matrix}

(3)

where $M_{j}^{ext}$ is the moment applied at $(x_{j}, y_{j}, z_{j})$ , suppose $p$ is a fixed point on the ground, and $H_{0}$ is the angular momentum of bipedal MRR with respect to point $p$ , we can get the following equations

(H_{0})_{x} = \sum_{i} (- m_{i} {\overset{\cdot}{y}}_{i} z_{i} + m_{i} {\overset{\cdot}{z}}_{i} y_{i} + J_{ix} {\overset{\cdot}{θ}}_{ix})

(4)

(H_{0})_{y} = \sum_{i} (- m_{i} {\overset{\cdot}{x}}_{i} z_{i} - m_{i} {\overset{\cdot}{z}}_{i} x_{i} + J_{iy} {\overset{\cdot}{θ}}_{iy})

(5)

where ${\overset{\cdot}{θ}}_{ix}$ and ${\overset{\cdot}{θ}}_{iy}$ are angular velocity of link $i$ in the $x$ and $y$ directions, $J_{ix}$ and $J_{iy}$ are moments of inertia of link $i$ about the $x$ and $y$ axes.

From equations (1), (3), and (5), we can obtain

\begin{matrix} x_{zmp} = {\begin{matrix} \sum_{i} [m_{i} (- {\overset{\cdot\cdot}{x}}_{i} z_{i} + ({\overset{\cdot\cdot}{z}}_{i} + g) x_{i}) - J_{iy} {\overset{\cdot\cdot}{θ}}_{iy}] \\ + \sum_{j} [{(f_{j}^{ext})}_{y} z_{j} - {(f_{j}^{ext})}_{z} y_{j} - {(M_{j}^{ext})}_{x}] \end{matrix}} \\ \times \frac{1}{\sum_{i} m_{i} ({\overset{\cdot\cdot}{z}}_{i} + g) - \sum_{j} {(f_{j}^{ext})}_{x}} \end{matrix}

(6)

From equations (1), (2), and (4), we can obtain

\begin{matrix} y_{zmp} = {\begin{matrix} \sum_{i} [m_{i} (- {\overset{\cdot\cdot}{y}}_{i} z_{i} + ({\overset{\cdot\cdot}{z}}_{i} + g) y_{i}) - J_{ix} {\overset{\cdot\cdot}{θ}}_{ix}] \\ + \sum_{j} [{(f_{j}^{ext})}_{y} z_{j} - ({f_{j}^{ext}}_{z} y_{j} - {(M_{j}^{ext})}_{x}] \end{matrix}} \\ \times \frac{1}{\sum_{i} m_{i} ({\overset{\cdot\cdot}{z}}_{i} + g) - \sum_{j} {(f_{j}^{ext})}_{x}} \end{matrix}

(7)

Assuming that there is no influence of external force on the sole of the bipedal MRR, equations (6) and (7) can be redescribed as

x_{zmp} = \frac{\sum_{i} m_{i} (- {\overset{\cdot\cdot}{x}}_{i} z_{i} + (g + {\overset{\cdot\cdot}{z}}_{i}) x_{i}) - J_{iy} {\overset{\cdot\cdot}{θ}}_{iy}}{\sum_{i} m_{i} (g + {\overset{\cdot\cdot}{z}}_{i})}

(8)

y_{zmp} = \frac{\sum_{i} m_{i} (- {\overset{\cdot\cdot}{y}}_{i} z_{i} + (g + {\overset{\cdot\cdot}{z}}_{i}) y_{i}) - J_{ix} {\overset{\cdot\cdot}{θ}}_{ix}}{\sum_{i} m_{i} (g + {\overset{\cdot\cdot}{z}}_{i})}

(9)

If the walking speed of the MRR is slow, the acceleration term in the above equations can be ignored (Shin and Kim, 2015),and the ZMP of the robot during static walking can be written as

x_{zmp} = \frac{\sum_{i} m_{i} g x_{i}}{\sum_{i} m_{i} g}

(10)

y_{zmp} = \frac{\sum_{i} m_{i} g y_{i}}{\sum_{i} m_{i} g}

(11)

As can be seen from the above equations, if acceleration is ignored, the formula for solving ZMP becomes the projection point of the center of mass (COM) on the ground. Therefore, when bipedal MRR walks statically, ZMP coincides with the center of gravity (COG).

Dynamic model of bipedal MRR

Leg dynamic model

Considering that bipedal MRR exists IDC effects $I_{i} (q, \overset{\cdot}{q}, \overset{\cdot\cdot}{q})$ and each leg of the bipedal MRR has $n$ degrees of freedom (DOF), the dynamic model of each leg can be defined as

\begin{matrix} M_{i} (q_{i}) {\overset{\cdot\cdot}{q}}_{i} + C_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) {\overset{\cdot}{q}}_{i} + G_{i} (q_{i}) + I_{i} (q, \overset{\cdot}{q}, \overset{\cdot\cdot}{q}) \\ = τ_{i} - J_{gi}^{T} F_{gi} + J_{ei}^{T} F_{ei} \end{matrix}

(12)

\begin{matrix} I_{i} (q, \overset{\cdot}{q}, \overset{\cdot\cdot}{q}) = {\sum_{j = 1, j \neq i}^{n} [M_{ii} (q) - M_{i} (q_{i})] \overset{\cdot\cdot}{q} + M_{ij} (q) {\overset{\cdot\cdot}{q}}_{j} + M_{ij} (q) {\overset{\cdot\cdot}{q}}_{j}} \\ + {\sum_{j = 1, j \neq i}^{n} [- C_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) + C_{ii} (q, \overset{\cdot}{q})] {\overset{\cdot}{q}}_{i} + C_{ij} (q, \overset{\cdot}{q}) {\overset{\cdot}{q}}_{j}} \\ + [- G_{i} (q_{i}) + {\bar{G}}_{i} (q)] \end{matrix}

(13)

where $M_{i} (q_{i}) \in R^{n \times n}$ is mass matrix of the leg, $q_{i} \in R^{n}$ is the joint position vector, $C_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) \in R^{n \times n}$ is centripetal Coriolis matrix, $G_{i} (q_{i}) \in R^{n}$ is gravity vector, and $τ_{i} \in R^{n}$ is joint torque vector. $J_{gi}$ is the Jacobian matrix of the leg reaction to the ground, $J_{ei}$ is the Jacobian matrix of the mechanical leg-to-joint space, $F_{gi} = {[\begin{matrix} f_{xi} & f_{yi} & f_{zi} \end{matrix}]}^{T}$ is the vector of the ground reaction force of the leg, and $F_{ei}$ is the force between the leg and the body of the bipedal MRR.

The overall dynamic of the leg of the bipedal MRR can be formulated as

M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + I (q, \overset{\cdot}{q}, \overset{\cdot\cdot}{q}) = τ - J_{g}^{T} F_{g} + J_{e}^{T} F_{e}

(14)

Body dynamic model

The body dynamic model of bipedal MRR can be described as

M_{0} (x_{0}) {\overset{\cdot\cdot}{x}}_{0} + C_{0} (x_{0}, {\overset{\cdot}{x}}_{0}) {\overset{\cdot}{x}}_{0} + G_{0} (x_{0}) = F_{0}

(15)

where $x_{0} \in R^{n_{0}}$ is position/orientation vector of body, $M_{0} (x_{0}) \in R^{n_{0} \times n_{0}}$ is symmetric positive definite inertia matrix of the body, $C_{0} (q, \overset{\cdot}{q}) \in R^{n_{0} \times n_{0}}$ is centripetal Coriolis matrix of the body, $G_{0} (x_{0}) \in R^{n_{0}}$ is the gravity vector, and $F_{0} \in R^{n_{0}}$ is resultant force applied to COM of body.

Defining $J_{0} (x_{0})$ is Jacobian matrix from body coordinates to leg coordinates, $F_{0}$ can be redescribed as

F_{0} = - J_{0}^{T} (x_{0}) F_{e}

(16)

Through the above equation, we can obtain

F_{e} = - (J_{0}^{T} (x_{0}))^{+} F_{0}

(17)

where $(J_{0}^{T} (x_{0}))^{+}$ is the pseudoinverse matrix of $J_{0}^{T} (x_{0})$ .

Substituting equation (15) into equation (17), we get

F_{e} = - (J_{0}^{T} (x_{0}))^{+} (M_{0} (x_{0}) {\overset{\cdot\cdot}{x}}_{0} + C_{0} (x_{0}, {\overset{\cdot}{x}}_{0}) {\overset{\cdot}{x}}_{0} + G_{0} (x_{0}))

(18)

Defining $x_{ie}$ is the position vector of body joining points. Therefore, we can get

{\overset{\cdot}{x}}_{ie} = J_{ie} (q_{i}) {\overset{\cdot}{q}}_{i}

(19)

{\overset{\cdot}{x}}_{ie} = J_{io} (x_{0}) {\overset{\cdot}{x}}_{0}

(20)

Combined with equations (19) and (20), the following relationship can be obtained

J_{ie} (q_{i}) {\overset{\cdot}{q}}_{i} = J_{io} (x_{0}) {\overset{\cdot}{x}}_{0}

(21)

Through the above formula, ${\overset{\cdot}{x}}_{0}$ can be redescribed as

{\overset{\cdot}{x}}_{0} = J_{0}^{+} (x_{0}) J_{e} (q) \overset{\cdot}{q}

(22)

Derivation of the equation (22) with respect to time $t$ yields

{\overset{\cdot\cdot}{x}}_{0} = J_{0}^{+} (x_{0}) J_{e} (q) \overset{\cdot\cdot}{q} + \frac{d}{dt} (J_{0}^{+} (x_{0}) J_{e} (q)) \overset{\cdot}{q}

(23)

Substituting equation (17) into equation (14), we have

\begin{matrix} M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + I (q, \overset{\cdot}{q}, \overset{\cdot\cdot}{q}) + J_{e} (J_{0}^{+} (x_{0}))^{+} F_{0} \\ = τ - J_{g} F_{g} \end{matrix}

(24)

Substituting equations (22) and (23) into equation (15), we have

\begin{matrix} M_{0} (x_{0}) J_{0}^{+} (x_{0}) J_{e} (q) \overset{\cdot\cdot}{q} + (M_{0} (x_{0}) \frac{d}{dt} (J_{0}^{+} (x_{0}) J_{e} (q)) \\ + C_{0} (x_{0}, {\overset{\cdot}{x}}_{0}) J_{0}^{+} (x_{0}) J_{e} (q)) \overset{\cdot}{q} + G_{0} (x_{0}) = F_{0} \end{matrix}

(25)

Defining $L = J_{0}^{+} (x_{0}) J_{e} (q)$ , combining equation (15) with equation (14) and multiplying both sides of equation (15) by $L^{T}$ , the global dynamic model is

M_{G} (q) \overset{\cdot\cdot}{q} + C_{G} (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G_{G} (q) + I (q, \overset{\cdot}{q}, \overset{\cdot\cdot}{q}) + J_{g} F_{g} = τ

(26)

where $M_{G} (q) = M (q) + L^{T} M_{0} (x_{0}) L$ , $C_{G} (q, \overset{\cdot}{q}) = C (q, \overset{\cdot}{q}) + L^{T}$ $(m_{0} \overset{\cdot}{L} + C_{0} (x_{0}, {\overset{\cdot}{x}}_{0}) L)$ , and $G_{G} (q) = G (g) + L^{T} G_{0} (x_{0})$ .

Adaptive NN controller design of bipedal MRR

Problem statement

Because of the special properties of bipedal MRR, $M_{G} (q)$ , $C_{G} (q, \overset{\cdot}{q})$ , and $G_{G} (q)$ are all unknown. Let $x_{1} = [q_{1}, q_{2}, . . ., q_{m}]^{T}$ and $x_{2} = [{\overset{\cdot}{q}}_{1}, {\overset{\cdot}{q}}_{2}, . . ., {\overset{\cdot}{q}}_{m}]^{T}$ are state vectors of the system, the state space equation can be obtained

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = B^{- 1} (x) (F (x) - J_{g} F_{g} + τ) + ψ (x) \\ y = x_{1} \end{matrix}

(27)

where $x$ and $y$ are state vector and output of the bipedal MRR system, $B (x) = M_{G} (q)$ , $F (x) = - C_{G} (q, \overset{\cdot}{q}) \overset{\cdot}{q} - G_{G} (q)$ , and $ψ (x) = - M_{G} (q) I (q, \overset{\cdot}{q}, \overset{\cdot\cdot}{q})$ is IDC term.

Assumption 1. (An et al., 2023):The desired position, desired velocity, and desired acceleration ( $q_{d}$ , ${\overset{\cdot}{q}}_{d}$ , and ${\overset{\cdot\cdot}{q}}_{d}$ ) of the joint are all bounded.

The expression of an IDC term is defined as

ψ (x) = ψ (x, x_{d}) + Δ ψ (x, x_{d})

(28)

where $ψ (x, x_{d})$ represents a certain part of IDC, $Δ ψ (x, x_{d})$ represents replacement error, and $x_{d}$ represents the desired state of the robot coupling system, and equation (27) can be reexpressed as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = B^{- 1} (x) (F (x) - J_{g} F_{g} + ψ (x, x_{d}) + τ) + Δ ψ (x, x_{d}) \\ y = x_{1} \end{matrix}

(29)

Ignoring the impact of IDC $Δ ψ (x, x_{d})$ for now, defining $B_{d} (x) \in R^{m \times m}$ as a diagonal matrix, then define the following equation

B (x) = B_{d} (x) + Δ_{B}

(30)

where $Δ_{B}$ is unknown, in summary, with the above formula, we can also define

F (x) = F_{d} (x) + Δ_{F}

(31)

where $F_{d} (x)$ is the known term of $C_{G} (q, \overset{\cdot}{q}) \overset{\cdot}{q}$ and $G_{G} (q)$ , and $Δ_{F}$ is unknown term of $C_{G} (q, \overset{\cdot}{q}) \overset{\cdot}{q}$ and $G_{G} (q)$ , then we can get the following equation

B (x) {\overset{\cdot}{x}}_{2} = F_{d} (x) + Δ_{F} - J_{g} F_{g} + τ + ψ (x, x_{d})

(32)

From equation (30), we can know

(B_{d} (x) + Δ_{B}) {\overset{\cdot}{x}}_{2} = F_{d} (x) + Δ_{F} - J_{g} F_{g} + τ + ψ (x, x_{d})

(33)

The above equation can be reexpressed as

\begin{matrix} B_{d} (x) {\overset{\cdot}{x}}_{2} = (I - Δ_{B} B^{- 1} (x)) (F_{d} (x) + Δ_{F}) + τ - Δ_{B} B^{- 1} (x) τ \\ - (I - Δ_{B} B^{- 1} (x)) J_{g} F_{g} + (I - Δ_{B} B^{- 1} (x)) ψ (x, x_{d}) \\ = F_{F} + τ + g (τ) + η + Ψ \end{matrix}

(34)

where $F_{F} = (I - Δ_{B} B^{- 1} (x)) (F_{d} (x) + Δ_{F}) \in R^{m}$ , $Ψ = (I - Δ_{B}$ $B^{- 1} (x)) ψ (x, x_{d})$ , and $η = - (I - Δ_{B} B^{- 1} (x)) J_{g} F_{g}$ .

We can define an error function as follows

s_{i} = {\overset{\cdot}{e}}_{i} + λ_{i} e_{i}

(35)

e_{i} = y_{i} - y_{di} (i = 1, . . ., m)

(36)

where $λ_{i}$ is positive parameter. Equation (35) can be rewritten as

\overset{\cdot}{s} = B_{d}^{- 1} [F_{F} + τ + g (τ) + η + Ψ]

(37)

where $s = {[s_{1}, . . ., s_{m}]}^{T}$ .

Identification and approximation of IDC term

For $Δ ψ (x, x_{d})$ , this section mainly identifies and approximates this term, and $Δ ψ$ can be expressed by the following equation

Δ ψ = w_{h}^{T} σ (x_{h}, x_{D}) + ε_{h} (x_{h})

(38)

where $x_{h}$ represents the known NN state, $w_{h}$ represents unknown ideal weight of the NN, $σ (x_{h}, x_{D})$ represents activation function of the NN, and $x_{D} = {[x_{1 d}, x_{2 d}, . . ., x_{md}]}^{T}$ .

Considering a bipedal MRR system, the bounded input $u_{h}$ can be expressed as

{\overset{\cdot}{x}}_{h} = Δ ψ + u_{h} = w_{h}^{T} σ (x_{h}, x_{D}) + ε_{h} (x_{h}) + u_{h}

(39)

From equation (39), we can easily get the following expression

{\overset{\cdot}{\hat{x}}}_{h} = u_{h} + Δ \hat{ψ} = {\hat{w}}_{h}^{T} σ_{h} + ε_{h} + u_{h} + r_{h}

(40)

where ${\hat{x}}_{h}$ represents the identification state of $x_{h}$ , $Δ \hat{ψ}$ represents approximation of $Δ ψ$ , ${\hat{w}}_{h}$ is the weight estimation, and $r_{h}$ is the integral of the error feedback term sign

r_{h} = k_{h} e_{h} + v_{h}

(41)

where $e_{h} = x_{h} - {\hat{x}}_{h}$ is the identification error. $ζ_{h}$ represents generalized solution to

{\overset{\cdot}{ζ}}_{h} = (k_{h} α_{h} + γ_{h}) e_{h} + δ_{1} sgn (e_{h})

(42)

where $k_{h}$ , $α_{h}$ , $γ_{h}$ , and $δ_{h}$ represent the positive constants that have been determined, and $sgn (\cdot)$ means the signal function.

Considering equations (39) and (40), the identification error dynamic equation can be obtained

{\overset{\cdot}{e}}_{h} = Δ \tilde{ψ} = w_{h}^{T} σ_{h} - {\hat{w}}_{h} σ_{h} + ε_{h} - r_{h}

(43)

where $Δ \tilde{ψ} = Δ ψ - Δ \hat{ψ}$ . Therefore, the identification error function can be represented by the following equation

ρ_{h} = {\overset{\cdot}{e}}_{h} + α_{h} e_{h}

(44)

By equations (43) and (44), we can obtain

\begin{matrix} {\overset{\cdot}{ρ}}_{h} = w_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{x}}_{h} - {\overset{\cdot}{\hat{w}}}_{h}^{T} σ_{h} - {\hat{w}}_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{\hat{x}}}_{h} + {\overset{\cdot}{ε}}_{h} \\ - k_{h} s_{h} - γ_{h} e_{h} - δ_{1} sgn (e_{h}) + α_{h} {\overset{\cdot}{e}}_{h} \end{matrix}

(45)

where the weight update law of equation (46) is given as follows

{\overset{\cdot}{\hat{w}}}_{h} = proj (Γ_{h} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{\hat{x}}}_{h} {\tilde{x}}_{h}^{T})

(46)

where $Γ_{h} > 0$ is a constant matrix, $proj (\cdot)$ represents the projection of a vector equation (45) can be reformulated as follows

\begin{matrix} {\overset{\cdot}{ρ}}_{h} = {\tilde{P}}_{h 1} (w_{h}, {\hat{w}}_{h}, e_{h}) + P_{h 2} (w_{h}, {\hat{w}}_{h}, x_{h}) \\ + P_{h 3} ({\hat{x}}_{h}, {\tilde{w}}_{h}) - γ_{h} e_{h} - δ_{1} sgn (e_{h}) - k_{h} s_{h} \end{matrix}

(47)

And we have

\begin{matrix} {\tilde{P}}_{h 1} = α_{h} {\overset{\cdot}{e}}_{h} - {\overset{\cdot}{\hat{w}}}_{h}^{T} σ_{h} + \frac{1}{2} w_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{e}}_{h} + \frac{1}{2} {\hat{w}}_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{e}}_{h} \\ P_{h 2} = \frac{1}{2} w_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{x}}_{h} - \frac{1}{2} {\hat{w}}_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{x}}_{h} + {\overset{\cdot}{ε}}_{h} \\ P_{h 3} = \frac{1}{2} {\tilde{w}}_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{\hat{x}}}_{h} \end{matrix}

(48)

where ${\tilde{w}}_{h} = w_{h} - {\hat{w}}_{h}$ , ${\tilde{P}}_{h 3} = {\hat{P}}_{h 3} - P_{h 3}$ , and $P_{h} = P_{h 2} + P_{h 3}$ .

Combined with equations (44), (46), and (48), we can get the following boundary conditions

\begin{matrix} ‖ {\tilde{P}}_{h 1} ‖ \leq μ_{1} (‖ b ‖) ‖ b ‖, ‖ P_{h 2} ‖ \leq ξ_{1}, ‖ P_{h 3} ‖ \leq ξ_{2} \\ ‖ P_{h} ‖ \leq ξ_{3} + ξ_{4} μ_{2} (‖ b ‖) ‖ b ‖ \\ ‖ {\overset{\cdot}{e}}_{h}^{T} P_{h 3} ‖ \leq ξ_{5} {‖ e_{h} ‖}^{2} + ξ_{6} {‖ s_{h} ‖}^{2} \end{matrix}

(49)

where $b = {[e_{h}^{T} ρ_{h}^{T}]}^{T}$ , $μ_{1} (\cdot)$ , $μ_{2} (\cdot)$ are positive, nondecreasing, invertible functions. We define

Q_{h} = {[e_{h}^{T} ρ_{h}^{T} \sqrt{M} \sqrt{N}]}^{T}

(50)

where $M$ is the generalized solution to the differential equation

\overset{\cdot}{M} = - ρ_{h}^{T} (P_{h 2} - δ_{1} sgn (e_{h})) + {\overset{\cdot}{e}}_{h}^{T} P_{h 3} - δ_{2} μ_{2} (‖ b ‖) ‖ b ‖ ‖ e_{h} ‖

(51)

where $δ_{1}$ , $δ_{2}$ are chosen to satisfy that $M \geq 0$ .

And we have that

δ_{1} > max (ξ_{2} + ξ_{1}, ξ_{1} + \frac{ξ_{3}}{α_{h}}), δ_{2} > ξ_{4}

(52)

Through the above equations, auxiliary functions $N$ can be defined as follows

N = \frac{1}{4} α_{h} [tr ({\tilde{w}}_{h} Γ_{h}^{- 1} {\tilde{w}}_{h})]

(53)

where $tr (\cdot)$ represents the trace of a matrix and $N$ represents the generalized solution to the differential equation.

Through equations (40)–(42), $τ_{1}$ can be designed as

\begin{matrix} τ_{1} = - B^{- 1} (\int_{0}^{t} ((k_{h} α_{h} + γ_{h}) e_{h} + δ_{1} sgn (e_{h})) dt \\ + k_{h} e_{h} + {\hat{w}}_{h}^{T} σ_{h}) \end{matrix}

(54)

where weight ${\hat{w}}_{h}$ is updated by equation (46).

Theorem 1. The asymptotic identification of IDCs can be guaranteed using the NN identification shown in equation (39) and the weight update law shown in equation (46), then we can get

lim_{t \to \infty} ‖ Δ \tilde{ψ} ‖ = 0

(55)

provided $k_{h}$ , $γ_{h}$ are satisfied by the following conditions

k_{h} > ξ_{6}, γ > \frac{ξ_{5}}{ξ_{h}}

(56)

where $k_{h}, γ_{h}, α_{h}, ξ_{5}, ξ_{6}$ are in equations (42) and (49).

Proof. Define the Lyapunov function candidate $V_{h} (Q_{h}) : D_{h} \to R_{h}$

V_{h} (Q_{h}) = \frac{1}{2} ρ_{h}^{T} ρ + \frac{1}{2} γ_{h} e_{h}^{T} e_{h} + M + N

(57)

Moreover, it satisfies the following relation

U_{1} (Q_{h}) \leq V_{h} (Q_{h}) \leq U_{2} (Q_{h})

(58)

where $U_{1} (Q_{h})$ and $U_{2} (Q_{h})$ are continuous positive definite functions

\begin{matrix} U_{1} (Q_{h}) = \frac{1}{2} min (γ_{h}, 1) {‖ Q_{h} ‖}^{2} \\ U_{2} (Q_{h}) = max (γ_{h}, 1) {‖ Q_{h} ‖}^{2} \end{matrix}

(59)

□

The derivative of equation (57) with respect to time is

\begin{matrix} {\overset{\cdot}{V}}_{h} = \nabla V_{h}^{T} {[\begin{matrix} {\overset{\cdot}{ρ}}_{h}^{T} & {\overset{\cdot}{e}}_{h}^{T} & \frac{1}{2} M^{- \frac{1}{2}} \overset{\cdot}{M} & \frac{1}{2} N^{- \frac{1}{2}} N \end{matrix}]}^{T} \\ = [\begin{matrix} ρ_{h}^{T} & γ_{h} e_{h}^{T} & 2 M^{\frac{1}{2}} & 2 N^{\frac{1}{2}} \end{matrix}] [\begin{matrix} {\overset{\cdot}{ρ}}_{h}^{T} & γ_{h} e_{h}^{T} & 2 M^{\frac{1}{2}} & 2 N^{\frac{1}{2}} \end{matrix}] \\ < ({\tilde{P}}_{h 1} + P_{h 2} + {\hat{P}}_{h 3} - k_{h} ρ_{h} - δ_{1} sgn (e_{h}) - γ_{h} e_{h}) ρ_{h}^{T} \\ + γ_{h} e_{h}^{T} (ρ_{h} - α_{h} e_{h}) - ρ_{h}^{T} (P_{h 2} - δ_{1} sgn (e_{h})) \\ - {\overset{\cdot}{e}}_{h}^{T} P_{h 3} + δ_{2} μ_{2} (‖ b ‖) ‖ b ‖ ‖ e_{h} ‖ - \frac{1}{2} α_{h} [tr ({\tilde{W}}_{h}^{T} Γ_{h}^{- 1} {\overset{\cdot}{\hat{w}}}_{h}] \\ = - α_{h} γ_{h} e_{h}^{T} e_{h} - k_{h} ρ_{h}^{T} ρ_{h} + \frac{1}{2} α_{h} e_{h}^{T} {\tilde{w}}_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{\hat{x}}}_{h} \\ + δ_{2} μ_{2} (‖ b ‖) ‖ b ‖ ‖ e_{h} ‖ - \frac{1}{2} α_{h} [tr ({\tilde{w}}_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{\hat{x}}}_{h} e_{h}^{T})] \\ + e_{h}^{T} ({\hat{P}}_{h 3} - P_{h 3}) + \frac{1}{2} α_{h} e_{h}^{T} {\hat{w}}_{h}^{T} {\overset{\cdot}{σ}}_{h} {\overset{\cdot}{\hat{x}}}_{h} + ρ_{h}^{T} {\tilde{P}}_{h 1} \end{matrix}

(60)

By eliminating the common terms $k_{h} = k_{h 1} + k_{h 2}, γ_{h} = γ_{h 1} + γ_{h 2}$ in equation (60), using equation (49) and completing the square, the upper bound of equation (60) can be obtained as follows

\begin{matrix} {\overset{\cdot}{V}}_{h} \leq - (α_{h} γ_{h} - ξ_{5}) ‖ ρ_{h} ‖^{2} - (k_{1} - ξ_{6}) ‖ ρ_{h} ‖^{2} \\ + \frac{μ_{1} {(‖ b ‖)}^{2}}{4 λ_{2}} ‖ b ‖^{2} + \frac{δ_{2}^{2} μ_{2} {(‖ b ‖)}^{2}}{4 α_{h} γ_{h 2}} ‖ b ‖^{2} \end{matrix}

(61)

If equation (56) is satisfied, then equation (61) can be written as

\begin{matrix} {\overset{\cdot}{V}}_{h} \leq - λ_{1} {‖ b ‖}^{2} + \frac{μ {‖ b ‖}^{2}}{4 λ_{2}} {‖ b ‖}^{2} \leq - U (Q_{h}) \\ \forall Q \in D_{h} \end{matrix}

(62)

where $U (Q_{h}) = α_{c} ‖ b ‖^{2}$ , $λ_{1} = min {k_{1} - ξ_{6}, α_{h} γ_{h 1} - ξ_{5}}$ , and $λ_{2} = min (k_{2}, (α_{h} γ_{h 2} / δ_{2}^{2})}$ , in which a constant $α_{c} > 0$ is the positive semidefinite function defined as $D_{h} = {Q_{h} ‖ Q_{h} ‖ \leq μ^{- 1} (2 \sqrt{λ_{1} λ_{2}})}$ .

Let $R_{h} \in D_{h}$ and denote the set defined as

R_{h} = {Q_{h} \in D_{h} | U_{2} (Q_{h}) < \frac{1}{2} {(μ^{- 1} (2 \sqrt{λ_{1} λ_{2}}))}^{2}}

(63)

a_{c} ‖ b ‖^{2} \to 0

(64)

while $\forall Q_{h} (0) \in R_{h}, t \to \infty$ .

Based on the definition of $b$ , $ρ_{h}$ , and $Δ ψ$ , we can easily get that $‖ e_{h} ‖, ‖ ρ_{h} ‖, ‖ {\overset{\cdot}{e}}_{h} ‖ \to 0$ , while $‖ e_{h} ‖, ‖ ρ_{h} ‖, ‖ {\overset{\cdot}{e}}_{h} ‖ \to 0$ . According to equations (43) and (55), using the above conclusions, we can get $lim_{t \to \infty} ‖ Δ \tilde{ψ} ‖ = 0$ . The proof is completed.

Adaptive NN controller

The model-based controller does not have a good control effect for the unmodeled robot system, and this paper uses radial basis function (RBF) NN to approximate the unknown function, which could overcome the above problems.

RBFNN is used to approximate the unknown function in bipedal MRR dynamic, and we define a set $Ω_{z}$ , and RBFNN can approximate any continuous function $H (Z)$ on $Ω_{z}$ with the following precision

H (Z) = W^{* T} S_{i} (Z) + μ

(65)

where $S_{i} (Z) (i = 1, 2, . . ., l)$ is the Gaussian function, $W^{*}$ is the ideal weight, and $μ$ is the approximation error. And $S_{i} (Z)$ is defined as follows

S_{i} (Z) = \exp [\frac{- {(Z - c_{i})}^{T} (Z - c_{i})}{b_{i}^{2}}]

(66)

where $c_{i} = [c_{i 1}, c_{i 2}, . . ., c_{iq}]$ is the central vector of $i$ hidden layer neuron and $b_{i} > 0$ is the base width parameter of node $i$ . According to the above definition, there must exist a positive constant $δ$ , and $‖ S (Z) ‖ \leq δ$ with $δ > 0$ . $W^{*}$ is $S_{i} (Z)$ the optimal constant weight, and $μ$ is the minimum approximation error of RBFNN. According to the RBFNN approximation capacity, obviously, the approximation error is limited by a constant $μ^{*}$ , that is, $| μ | \leq μ^{*}$ , with a positive constant $μ^{*} > 0$ .

Remark 1. For any continuous function $H (x)$ and the compact set $Ω_{z}$ , there exists a neural network satisfying above formula, and $| μ | \leq \bar{μ}, \forall z \in Ω_{z} = {z | ‖ z ‖ \leq \bar{z}}$ , and the approximate value of $\bar{z}$ is 10. It is assumed that the system state keeps in a compact set $Ω_{z}$ . By the property of Gaussian basis function, we have that $| S_{i} (\cdot) | \leq 1$ and $‖ S (Z) ‖ \leq \sqrt{l}$ .

RBFNN can be used to approximate the function $F_{F} (x)$ as follows

F_{F} (x) = - W^{* T} S (Z) - ε

(67)

where $W^{*}$ is the optimal NN weight, $S (Z)$ is the radial basis function, and $ε$ is the approximation error.

Considering equations (34) and (36), it can be obtained that

B_{d} (x) {\overset{\cdot}{x}}_{2} = - W^{* T} S (Z) - ε + τ + g (τ) + η + Ψ

(68)

In order to effectively solve the issue of approximate error $ε$ , we could view $ε$ as part of the external disturbance of the system. We define $D = g (τ) + η - ε + Ψ$ , and equation (68) can be expressed as

B_{d} (x) {\overset{\cdot}{x}}_{2} = τ + D - W^{* T} S (Z)

(69)

Lemma 1. There exists a differentiable continuous function $ϑ (t)$ , $\forall t \in [t, t_{0}]$ satisfying $υ_{1} \leq ‖ ϑ (t) ‖ \leq υ_{2}$ , where $υ_{1}$ and $υ_{2}$ are positive constants and $\overset{\cdot}{ϑ} (t)$ is bounded.

Proof. It can be obtained that

\exists ξ \in (0, t) : \frac{v (t) - v (0)}{t - 0} = \overset{\cdot}{v} (ξ)

According to the mean value theorem, it is easy to know that the derivative $\overset{\cdot}{ϑ} (t)$ is also bounded because $υ_{1} \leq ‖ ϑ (t) ‖ \leq υ_{2}$ , and $υ_{1} - υ_{2} \leq ϑ (t) - ϑ (0) \leq υ_{2} - υ_{1}$ is bounded.

Assumption 2. (Zhang et al., 2015a):The external disturbance $‖ d (t) ‖ : R^{+} \to R^{m}$ of the system is bounded. In other words, the inequality $‖ d (t) ‖ \leq d_{M}$ can be satisfied if there exists an unknown positive constant.□

Assumption 3. (Zhang et al., 2015a):There is an indeterminacy positive constant $ω_{1}$ with $| \overset{\cdot}{η} (θ) | \leq ω_{1}$ for nonlinear and strongly coupled bipedal MRR system (33).

Remark 2. For the proposed method, the involved computation is dominated by the training algorithm for $W$ . Now, ${x (t), y (t); t = 1, . . ., N}$ is a set of data samples. The structure of the adaptive neural network is $T - K - 1$ ( $T$ is input variables, $K$ is hidden neurons, and a single-output neuron). It has been verified that the complexity of most traditional RBF neural networks is $O (N^{2})$ , but our proposed control method reduces it to $O (N)$ . In the simulation process, the proposed adaptive neural network control algorithm can reduce the computation time.

Considering Lemma 1, Assumption 2, and Assumption 3, the following equation can be obtained

‖ \overset{\cdot}{D} ‖ \leq ρ

(70)

For the design of the controller, the following auxiliary variables are defined

z = D - K x_{2}

(71)

where $K = K^{T} > 0$ is a matrix to be designed.

By combining equations (69) and (71) and taking the derivative of $z$ with respect to time, the following equation can be obtained

\begin{matrix} \overset{\cdot}{z} = \overset{\cdot}{D} - K {\overset{\cdot}{x}}_{2} \\ = \overset{\cdot}{D} - {KB}_{d}^{- 1} (x) [τ + D - W^{* T} S (Z)] \end{matrix}

(72)

In order to achieve the estimation of $D$ , the estimation of intermediate variable $z$ should first be obtained. Therefore, based on equations (70) and (72), the following equation is proposed

\overset{\cdot}{\hat{z}} = - {KB}_{d}^{- 1} (x) [τ + \hat{D} - {\hat{W}}^{* T} S (Z)]

(73)

where $\hat{W}$ is the estimate of $W^{*}$ and $\hat{D}$ is the estimate of $D$ .

The estimated value of system disturbance $D$ can be obtained from the equation (71) as

\hat{D} = \hat{z} + K x_{2}

(74)

Considering equations (71) and (74), the following equation can be obtained

\tilde{z} = z - \hat{z} = D - \hat{D} = \tilde{D}

(75)

Considering equations (72) and (73), and taking the derivative of equation (75) with respect to time $t$ , the following equation is obtained

\begin{array}{l} \dot{\tilde{D}} = \dot{\tilde{z}} = \dot{z} - \dot{\hat{z}} \\ = \dot{D} - K B_{d}^{- 1} (x) [\tilde{D} + {\tilde{W}}^{T} S (Z)] \end{array}

(76)

where $\tilde{W} = \hat{W} - W^{*}$ .

Through the above assumptions and derivations, the following adaptive NN controller is designed based on RBFNN

τ_{2} = {\hat{W}}^{T} S (Z) - K_{1} s - \hat{D}

(77)

where $K_{1} = K_{1}^{T} > 0$ is a constant matrix.

The updating law of adaptive NN can be designed in the following form

{\overset{\cdot}{\hat{W}}}_{i} = - Γ [S_{i} (Z) s_{i} + ς {\hat{W}}_{i}]

(78)

where $ς$ is a positive constant, and $Γ_{i} \in R^{m} (i = 1, 2, . . ., m)$ is the symmetric positive constant matrix.

Remark 3. For bipedal MRR system with high nonlinear and uncertainty, this adaptive neural network control method based on disturbance observer is proposed to compensate the uncertainty of the model, and the disturbance observer is integrated to compensate the nonlinear dynamic, neural network approximation error and external disturbance. Compared with the traditional neural network control method, the trajectory tracking performance of bipedal MRR is improved.

Theorem 2. Considering bipedal MRR system (29) with model uncertainty equation (67) and external disturbance $D$ , if Assumption 2 holds, $\tilde{D}$ , $\tilde{W}$ , and $s$ are semi-globally ultimately bounded with the RBFNN controller designed in equation (77) and the updating law of adaptive NN equation (78).

Proof. Define Lyapunov function candidate

V = \frac{1}{2} s^{T} B_{d} s + \frac{1}{2} {\tilde{D}}^{T} \tilde{D} + \frac{1}{2} \sum_{i = 1}^{m} {\tilde{W}}_{i}^{T} Γ_{i}^{- 1} {\tilde{W}}_{i}

(79)

Taking the derivative of $V$ with respect to time $t$ , the following equation can be obtained

\begin{matrix} \overset{\cdot}{V} = s^{T} B_{d} \overset{\cdot}{s} + {\tilde{D}}^{T} \overset{\cdot}{\tilde{D}} + \sum_{i = 1}^{m} {\tilde{W}}_{i}^{T} Γ_{i}^{- 1} {\overset{\cdot}{\tilde{W}}}_{i} \\ = s^{T} [F_{F} + τ + g (τ) + η + Ψ] + {\tilde{D}}^{T} \overset{\cdot}{\tilde{D}} + \sum_{i = 1}^{m} {\tilde{W}}_{i}^{T} Γ_{i}^{- 1} \overset{\cdot}{\tilde{W}} \end{matrix}

(80)

By combining equations (69) and (77), the following equation can be established

{\overset{\cdot}{V}}_{2} = s^{T} [\tilde{W} S (Z) + \tilde{D} - K_{1} s] + {\tilde{D}}^{T} \overset{\cdot}{\tilde{D}} + \sum_{i = 1}^{m} {\tilde{W}}_{i}^{T} Γ_{i}^{- 1} \overset{\cdot}{\tilde{W}}

(81)

Considering equations (70), (76), and (78), and the following conditions

s^{T} \tilde{D} \leq \frac{1}{2} s^{T} s + \frac{1}{2} {\tilde{D}}^{T} \tilde{D}

(82)

{\tilde{D}}^{T} \overset{\cdot}{D} \leq \frac{1}{2} {\tilde{D}}^{T} \tilde{D} + \frac{1}{2} ‖ \overset{\cdot}{D} ‖^{2}

(83)

\sum_{t = 1}^{m} {\tilde{W}}_{i}^{T} S_{i} (Z) s_{i} = s^{T} {\tilde{W}}^{T} S (Z)

(84)

Then, it can be obtained that

\begin{matrix} \overset{\cdot}{V} \leq - s^{T} (K_{1} - 0.5 I_{m * m}) s + \frac{1}{2} ρ^{2} - {\tilde{D}}^{T} ({KB}_{d}^{- 1} - 2 I_{m * m}) \tilde{D} \\ + \frac{1}{2} ς {‖ W^{*} ‖}^{2} - \frac{ς - {KB}_{d}^{- 1} δ}{2} \sum_{j = 1}^{m} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} \end{matrix}

(85)

where

{\tilde{D}}^{T} {KB}_{d}^{- 1} \tilde{W} S (Z) \leq \frac{1}{2} ‖ D ‖ + \frac{K B^{- 1} S (Z) {‖ \tilde{W} ‖}^{2}}{2}

(86)

- ς {\tilde{W}}_{i}^{T} {\hat{W}}_{i} = - ς ‖ {\tilde{W}}_{i} ‖^{2} - ς {\tilde{W}}_{i}^{T}

(87)

W_{i}^{*} \leq - \frac{1}{2} ς ‖ {\tilde{W}}_{i} ‖^{2} + \frac{1}{2} ς ‖ W_{i}^{*} ‖^{2}

(88)

We need to choose positive definite matrix $K_{1}$ , $K$ , and positive constant $ς$ to make ${KB}_{d}^{- 1} - 2 I_{m * m} > 0$ and $ς - {KB}_{d}^{- 1} δ > 0$ . Then, we can establish the following inequality

{\overset{\cdot}{V}}_{2} \leq - κ V_{2} + C

(89)

κ = min (λ_{min} ({KB}_{d}^{- 1} - 2 I_{m * m}) \frac{ς - {KB}_{d}^{- 1} δ}{λ_{max} (\sum_{i = 1}^{m} Γ_{i}^{- 1})}, 1)

(90)

C = \frac{ς {‖ W^{*} ‖}^{2}}{2} + \frac{ρ^{2}}{2}

(91)

Multiplying by $e^{κ t}$ and integrating over time $t$ , we get that

V_{2} \leq (V_{2} (0) - \frac{C}{κ}) e^{- κ t} + \frac{C}{κ} \leq V_{2} (0) + \frac{C}{κ}

(92)

From the above inequality, we can see that $V_{2}$ is ultimately bounded. Therefore, $\tilde{D}$ , $\tilde{W}$ , and $s$ are also bounded. The proof is completed. □

Simulation results

The simulation is performed under MATLAB software, and the dynamic model of bipedal MRR can be defined as follows

\begin{matrix} M_{G} (q) = [\begin{matrix} - 0.1166 \cos^{2} (q_{2}) + 0.17 & - 0.06 \cos (q_{2}) \\ - 0.06 \cos (q_{2}) & 0.01233 \end{matrix}] \\ C_{G} (q, \overset{\cdot}{q}) = [\begin{matrix} - 0.1166 \sin (2 q_{2}) {\overset{\cdot}{q}}_{2} & - 0.06 \sin (q_{2}) {\overset{\cdot}{q}}_{2} \\ 0.06 \sin (q_{2}) {\overset{\cdot}{q}}_{2} & - 0.06 \sin (q_{2}) {\overset{\cdot}{q}}_{1} \\ - 0.0583 \sin (2 q_{2}) {\overset{\cdot}{q}}_{1} \end{matrix}] \\ G_{G} (q) = [\begin{matrix} 0 \\ - 5.88 \sin (q_{1} + q_{2}) \end{matrix}] \end{matrix}

(93)

Suppose that input vector is $X = [x_{1}, x_{2}, . . ., x_{n}]^{T}$ , the radial basis vector is $S = [S_{1}, S_{2}, . . ., S_{m}]^{T}$ and the weight vector is $W = [w_{1}, w_{2}, . . ., w_{m}]^{T}$ . From equation (66), we can get

y_{m} (t) = w_{1} h_{1} + w_{2} h_{2} + \dots + w_{m} h_{m}

(94)

The system error of the controller is defined as

e_{m} (k) = y (k) - y_{m} (k)

(95)

According to the gradient descent method, the iterative algorithm of weight, node center, and node radial parameters are as follows

Δ w_{i} (k + 1) = χ e_{m} (k) h_{i} + ι Δ w_{i} (k)

(96)

Δ b_{i} (k + 1) = χ e_{m} (k) w_{i} h_{i} \frac{- {(X - c_{i})}^{T} (X - c_{i})}{b_{i}^{3}} + ι Δ b_{i} (k)

(97)

Δ c_{ij} (k + 1) = χ e_{m} (k) w_{i} h_{i} \frac{x_{j} - c_{ij} (k)}{b_{i}^{2} (k)} + ι Δ c_{ij} (k)

(98)

where $χ$ is the learning rate and $ι$ is the momentum gene. Through the above method, we can determine the parameters of adaptive NN. We choose three hidden layer nodes NN to approximate the unknown models in the robotic system. Let $W_{i} (0) = [0, 0 . . ., 0]$ be the initial values. The bipedal MRR system is in a static state, and we chose the initial position as [0.05 - 0.05], we chose the initial velocity as 0, and the design parameters are set to $K = diag [5, 3]$ , $K_{1} = diag [2, 2]$ , $λ_{1} = 6$ , $λ_{2} = 5$ .

By gradient descent method, we choose the hidden layer that has three hidden layer nodes, and c is random. Apply the proposed adaptive NN control method, and we select the following parameters: $b = 10$ . Apply the traditional trajectory tracking control method based on NN, and we select the following parameters: $b = 8$ . Apply the trajectory tracking control method based on PID, and we select the following parameters: $b = 5$ .

Assuming that the bipedal MRR moves in a plane, the whole simulation process is carried out in Cartesian space. When bipedal MRR is walking, the two legs move alternately, and the desired trajectory is generated based on ZMP method. As shown in Figure 1, we know that the desired trajectory is COM trajectory. If the bipedal MRR can track the desired trajectory, then it can perform stable walking.

Through the simulation comparison of two control methods, the effectiveness of the proposed method is verified: (1) the traditional trajectory tracking control method based on NN, for example, Chu et al. (2019) and Yu and Feu (2014), and (2) the proposed adaptive NN control method. By comparing the performance of the two control strategies for desired trajectory tracking, the difference between the two methods can be clearly seen.

Position and velocity tracking performance

Since the nonlinear and uncertainty of the bipedal MRR dynamic model affect the tracking performance, we use adaptive NN to approximate the modeling errors and uncertainties. This control method realizes the tracking control of bipedal MRR, and the high-precision trajectory tracking control of robot manipulators is realized. Through the following theoretical verification and simulation comparison experiments, we can intuitively see the superiority of our proposed method, and the tracking error is uniformly bounded.

After generating desired trajectory, we introduce an adaptive NN control algorithm to track the desired trajectory. Figure 1 is position tracking of desired trajectory via the proposed adaptive NN control method. When t=2.5 s, actual trajectory gradually tracks the desired trajectory, and the tracking effect is well. In the comparison experiment of tracking error, we add a PID control method. Figures 2 –4 show position tracking of desired trajectory in joint space via the proposed adaptive NN control method, traditional trajectory tracking control method based on NN, and tracking control method based on PID. Figures 5 –7 show velocity error via the proposed adaptive NN control method, traditional trajectory tracking control method based on NN, and trajectory tracking control method based on PID. Through simulation verification, we can clearly see the effectiveness of our proposed method, and the position tracking error and velocity error remain stable at last.

NN weight curve

The weight of the NN is obtained by training the network, and Figure 10 shows the weight update rate under the proposed adaptive NN control method, which can be obtained using the converged weight.

Control torque

Figures 8 and 9 show the control torque using the proposed adaptive NN control method and the traditional trajectory tracking control method based on NN. According to the results, under the traditional trajectory tracking control method based on NN, the control torque curve shows a strong buffeting effect, which may reduce the accuracy of trajectory tracking, and the control torque is large. In this paper, the instantaneous increase in the control torque is limited to a safe range. As we can see the control torque curve is smooth and the overall control performance is improved.

Conclusion

Aiming at bipedal MRR system with strong coupling, a trajectory tracking method based on adaptive NN is proposed to solve the gait coordination control issue of bipedal MRR, and the dynamic model of bipedal MRR system is established by Newton–Euler iteration method. In different gait phases, the desired trajectory of the bipedal MRR is generated based on the ZMP. On this basis, an adaptive NN control method is designed to track the desired trajectory, and the IDC effect of bipedal MRR is compensated. The algorithm improves the tracking performance of the robot system. In addition, the stability of the bipedal MRR system is verified using the Lyapunov theory. Finally, simulation results show that the proposed method is effective.

Limitations and future work

At present, a series of research works are only in the initial stage, and more research contents and directions need to be improved and explored, which are mainly reflected in the following aspects:

The desired trajectory of bipedal MRR is generated by ZMP method, which ensures the stability of the robot in the walking process and requires less computation. However, this method is greatly affected by the inertia force of the robot. In the future, ZMP method can be further improved by considering the inertia force of the robot.

Due to the limited conditions, we only consider the contact with the plane in the application scenario. In future applications, we should also consider discussion and research in different complex task contacts to make the whole study more universal.

In future works, more advanced deep learning control methods should also be considered for gait coordination of bipedal MRR.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant no. 62173047), the Scientific Technological Development Plan Project in Jilin Province of China (Grant no. 20230508159RC), and the Science and Technology project of Jilin Provincial Education Department of China during the 13th Five-Year Plan Period (Grant no. JJKH20220689KJ).

ORCID iD

Bing Ma

Availability of data and materials

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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