Adaptive robust control for biped walking under uncertain external forces

1. INTRODUCTION

Recently, intelligent robots reflect the deep integration of new-generation information technologies, such as intelligent control and high-performance computing, and stand for major direction for the development of the new-generation artificial intelligence strategy of the country. As the ultimate form of intelligent robots, humanoid robots require complex mechanical control systems, environmental awareness, and motion planning capabilities, which are widely used in service entertainment, disaster rescue, rehabilitation medicine, and so on^[1-3]. Specially, flexible and robust walking is the most basic guarantee for various tasks. However, uncertain disturbances are inevitable in biped walking, which can affect walking stability or even periodic motion. Hence, it is of great significance to explore advanced high-performance robot control systems, break through the application bottleneck for fieldwork robots, and promote the development of humanoid robots.

Stability analysis is a convincing demonstration of robust walking. By analyzing the conditions for the existence of stable equilibrium points, the stability analysis will transform into a mathematical problem for the existence of limit cycles. Various stability criteria have been proposed. Early scholars proposed the zero moment point (ZMP) stability criterion, which means that the robot is considered stable when the ZMP falls within the support area. Goswamiti designed a sole flipping indicator^[4]. Huang further discussed the determination of the stable region^[5], and then a large number of 3D bipedal solid robots were manufactured based on this criterion. However, a series of problems arose subsequently, such as stiff movement and poor anti-interference ability. Considering stable bipedal walking exhibits periodic motion, the restricted Poincare Return Map can be used to analyze the stability of the system, which transforms the target of stable biped walking into the problem of stabilization of periodic orbits. The main purpose of using the Poincare return map is to analyze the stability of periodic orbits in low dimensions. Besides, it is not constrained by motion speed and, thus, is suitable for various foot structures. Tedrake analyzed the walking stability of a partially passive robot with drive only at the ankle joint^[6]. Grizzle et al. developed new jumping and running postures for point-legged robots^[7].

Compared to the walking ability of humans, biped robots still have a long way to go. They can be regarded as multi-variable, variable structure, and strong coupling nonlinear systems, possessing the characteristics of strong environmental adaptability, complex structure, and difficult motion control^[8-11]. Recent years have witnessed the rapid development of robust control of dynamic biped walking. The traditional quasi-static walking control based on ZMP has practicability; however, it is required that ZMP always falls in the support polygon, which is inconsistent with human walking^[12-14]. For the dynamic walking of biped robots under external forces, Ames^[14] proposed a hybrid zero dynamics control method and gave the analytical conditions for stable dynamic walking. Since SMC is insensitive to parameter changes and disturbances and has a fast response, it has become a research focus for robot control. Active force control is achieved by adding sensors to detect the external forces on the robot and designing corresponding force control algorithms to achieve the robot's active compliance with external forces. The classical force control usually includes impedance control and hybrid position and force control. Yadukumar et al. achieved the robot AMBER walking by collecting and analyzing human gait data and combining it with hybrid zero dynamics^[15]. The classic force control strategy is applicable to static environments in which environmental information is determined. The foundation of force control is the perception of external forces. There are generally two ways to measure external forces: one is to directly obtain the interaction force with the environment through external sensors; Another approach is to use the dynamic model of the robot to obtain external forces. Dai et al. took the ground as an external disturbance, quantified the robustness of the robot to ground disturbances through gain L2, and realized the robot's walking based on robust control^[16].

In the references^[17-20], the control method of a second-order sliding mode is introduced systematically, including a twisting algorithm, sub-optimal algorithm, and terminal sliding mode algorithm. Then, a motion/force hybrid control method based on recurrent neural networks (RNNs) was proposed afterward. Spong et al. proposed a continuous controller design method for dynamic walking on the uneven road^[21]. However, it is difficult to adjust control parameter items. Ravichandran et al. proposed a neural network control method with the inverted pendulum model^[22]. In view of the strong approximation ability of neural networks, they are usually utilized to approximate complex nonlinear systems. Particularly, combining with the self-adaptive technology, network weight coefficient identification and updating learning factors can be realized. It has a good generalization ability and can approach any nonlinear function with the required accuracy, which is suitable for real-time and online control of signal processing and robot control^[23-26]. Although there are numerous advanced results on biped dynamic walking^[27,28], there are still some unresolved issues worth studying, such as the robustness of walking and mobility flexibility. In this paper, we concentrate on adaptive robust control for bipedal robots under uncertain external forces.

The main content of the article is arranged as follows. Section 2 describes the dynamic model of the biped robot. It is modeled as a nonlinear impulsive system. An adaptive sliding-mode controller is proposed in Section 3. In Section 4, a primary RNN with self-stabilizing ability is utilized to deal with the complicated optimization problem. The hybrid robust control is then proposed to approximate unknown dynamic functions, and the network weights are adaptive. The simulation results are shown in Section 5, and Section 6 further proposes future work.

2. DYNAMIC MODELS

The biped robot model discussed in this paper is depicted in Figure 1, which includes a torso and two legs with revolute knees. $$ q=\left[q_{1}, q_{2}, q_{3}, q_{4}, q_{5}, q_{6}\right]^{T} $$ represents the angle of each joint. According to geometrical constraints on the biped robot joint coordinates, the constraints in the double-support phase are holonomic.

Figure 1. The nine-link model of biped robots.

Assumption 2.1. The swinging foot and the ground are completely elastic collisions.

Assumption 2.2. The joint angle remains the same, while the angular velocity changes immediately since the impact occurs instantaneously.

Assumption 2.3. The swinging leg did not slip or rebound with the ground during the collision.

Remark 1: The Assumptions 2.1-2.3 are general. These assumptions make it necessary to establish a relationship between the walking process and the dynamic model. Additionally, these assumptions have also been used in^[29,30].

The motion equation in the double support phase is described as,

where $$ M(q)\in R^{n \times n} $$ is the positive-definite inertia matrix, $$ G(q)=\partial P / \partial q \in R^{n} $$ is the gravity matrix, and $$ J=\partial \Phi / \partial q \in R^{n} $$ is the Jacobian matrix. $$ \Phi $$ represents robot constraints, and $$ \Phi(q)=0 $$. The nonlinear dynamic equation of the biped robot can be described as a second-order differential equation. $$ C(q, \dot{q}) \in R^{n \times n} $$ is the centrifugal force and Coriolis force terms, and

where N is the length of the generalized configuration vector. $$ u \in R^{n} $$ is the input torques, and $$ u_{d} \in R^{n} $$ is the external disturbance. Considering the external force exerted on the robot foot during an impact, which is defined as $$ F_{\text {ext }}=\int_{t^{-}}^{t^{+}} \delta F_{\text {ext }}(\tau) d(\tau) $$, (1) can be described as

where $$ q_{\mathrm{e}}=\left[\begin{array}{llll}q_{1} & q_{2} & p_{x} & p_{y}\end{array}\right]^{T} $$, and $$ \left(p_{x}, p_{y}\right) $$ shows the hip position in Cartesian coordinates. $$ M_{e}\left(q_{\mathrm{e}}\right) $$, $$ C_{e}\left(q_{\mathrm{e}}\right) $$, and $$ G_{e}\left(q_{\mathrm{e}}\right) $$ are inertia matrix, Coriolis force matrix, and gravity matrix in the double support phase, respectively. The collision mapping can be written as

Describing (2) and (4) as the form of state space, as shown in (5), demonstrates that the walking system is hybrid.

where $$ x(t)=[q, \dot{q}]^{T} $$ is the defined state variable, and $$ x^{-}(t)=\left[q^{-}, \dot{q}^{-}\right]^{T} $$ and $$ x^{+}(t)=\left[q^{+}, \dot{q}^{+}\right]^{T} $$ represent state variables before and after the impact, respectively. $$ f(x(t)) $$ and $$ g(x(t)) $$ are bounded nominal nonlinear functions; $$ \Delta f(x(t)) $$ and $$ \Delta g(x(t)) $$ represent uncertainties. Through differential homeomorphism transformation, the continuous part of the hybrid system (5) can be expressed as a nonlinear system with uncertainties,

where $$ \delta(x(t)) $$ is the system uncertainty, which includes system model uncertainties and external disturbances. Suppose $$ \delta(x(t)) $$ is bounded; that is, $$ \|\delta(x(t))\| \leq \rho $$, where $$ \|\cdot\| $$ is the Euclid norm, and $$ \rho $$ is a positive constant.

3. ADAPTIVE SLIDING MODE CONTROL

An adaptive sliding mode controller for uncertain disturbances is proposed in this section. The control system block diagram is shown in Figure 2. The basic control idea is to design a sliding mode controller, which makes the state of the system converge to the sliding mode surface when the robot is subjected to uncertain disturbances. In general, the controller design can be divided into two steps. A sliding surface is established to make the controlled system reach its control target.

Figure 2. Basic principle diagram of adaptive sliding mode control (SMC).

Due to the adaptability of error amplitude, the controller does not need to accurately estimate the amplitude of external disturbance. Let $$ \Delta q=q-q_{d} $$ describe the joint control error, where $$ q_{d} $$ is the reference track of each joint. Thus, the control target can be expressed as $$ s=\Delta \dot{q}+\alpha \Delta q \rightarrow 0 $$. However, in the actual robot operation, it will lead to chattering. In order to guarantee smooth operation, the regularization method in the boundary layer is utilized to solve the chattering problem. A controller is designed to make the state trajectory of the robot converge to a thin boundary layer, which is about the constant $$ \rho=\left\{(q, \dot{q}) \mid\|s\|<\Phi I_{8 \times 1}\right\} $$; that is to say, for $$ \forall i>0 $$, the robot will meet the following sliding mode conditions when $$ |s|>\Phi $$, and

$$ \frac{1}{2} \frac{\mathrm{d}}{d t} s_{i}^{2}=\dot{s}_{i} s_{i}<0 $$

Considering the coefficients of the Hurwitz polynomial $$ c_{1}+c_{2} \lambda+\cdots+c_{n-1} \lambda^{n-2}+\lambda^{n-1} $$, the sliding mode function can be selected as

where the constants $$ c_{l}(l=1, 2, \cdots, n-1) $$ are to be selected to meet the requirement of sliding surface, and the derivative of $$ s $$ about time $$ t $$ is

where $$ \alpha=\left[\begin{array}{lll} \alpha_{n} & \alpha_{n-1} \cdots \alpha_{n-1} \end{array}\right]^{T} $$, $$ \beta=\left[\begin{array}{lll} 0 & 0 \cdots I \end{array}\right]^{T} $$, and $$ E(t)=\left[e^{T}(t) \dot{e}^{T}(t) \cdots e^{(n-1) T}(t)\right] $$ will converge to the origin when $$ s_{i}=0 $$.

For a nonlinear system with bounded uncertainties (6), the designed sliding mode controller is proposed to ensure the system is asymptotically stable. It includes two parts: the former is used to realize input/output linearization, and the latter is used for robust compensation. The control law is designed as (9),

where $$ \operatorname{sgn}(\cdot) $$ is the sign function. A simple proof of convergence of the controller is as follows. Firstly, taking the splitting operation of the control law (9),

Substitute the control law (11) into the nonlinear system (7); it will be denoted by

thus, it can get $$ \rho \operatorname{sgn}(s(t))+\delta(x(t))=-\alpha^{T} E(t)-\beta \dot{E}(t)=-\dot{s}(t) $$.

The Lyapunov method is utilized to verify the stability of the control system. We select the candidate Lyapunov function as follows:

By taking the derivative of $$ V_{s}(s(t)) $$ with respect to $$ t $$, it follows that

According to the assumption that $$ \delta(x(t)) $$ is bounded, where $$ \|\delta(x(t))\| \leq \rho $$, so that $$ \dot{V}_{S}(s(t)) \leq 0 $$. Therefore, the SMC law (9) can make the robot system (6) asymptotically stable.

4. HYBRID MOTION/ FORCE CONTROL BASED ON RNN

Considering the uncertainty of the upper-bound error in the controller, a primary RNN is utilized. By designing adaptive laws of network weights based on the Lyapunov stability theory, the parameters of learning factors in neural networks are adjusted. In addition, the boundary value estimation algorithm is utilized to compensate for the estimation error. In order to analyze the system stability, the Poincare return map is utilized, in which the manifold of a continuous subsystem can pass through the impact cross section.

4.2. Design of a hybrid force/motion controller

The control structure of a hybrid motion/force controller is shown in Figure 3. In order to derive optimized contact force and motion, hybrid motion/force control is proposed based on RNNs to approximate dynamic functions. Several assumptions are made in advance.

Figure 3. The control structure.

Assumption 4.1. According to the approximation principle of neural networks, suppose that there exists the desired weight $$ W^{*} $$, Gauss function $$ \Phi^{*} $$, desired center value $$ m^{*} $$, width $$ \sigma^{*} $$, and recurrent gain coefficient $$ r^{*} $$, which makes an RNN approach any smooth nonlinear function $$ \delta $$.

(17) $$ \begin{equation} \delta=\delta^{*}+\xi=W^{*} \Phi^{*}\left(m^{*}, \sigma^{*}, r^{*}\right)+\xi, \end{equation} $$

where $$ W^{*}, \Phi^{*}, m^{*}, \sigma^{*}, r^{*} $$ are the desired values of network parameters - $$ W, \Phi, m, \sigma, r $$. Suppose that $$ W^{*}, \Phi^{*}, m^{*}, \sigma^{*}, r^{*} $$ are all bounded, $$ \left\|W^{*}\right\| \leq \bar{W} $$, $$ \left\|\Phi^{*}\right\| \leq \bar{\Phi} $$, $$ \left\|m^{*}\right\| \leq \bar{m} $$, $$ \left\|\sigma^{*}\right\| \leq \bar{\sigma} $$, and $$ \left\|r^{*}\right\| \leq \bar{r} $$. $$ \bar{W} $$, $$ \bar{\Phi} $$, $$ \bar{m} $$, $$ {\bar{\sigma}} $$, $$ \bar{r} $$ are the corresponding upper bound of each parameter, and $$ \xi $$ is the approximation error, which satisfies $$ \left\|\xi^{*}\right\| \leq \bar{\xi} $$. However, the desired values $$ W^{*}, \Phi^{*}, m^{*}, \sigma^{*}, r^{*}, \delta^{*} $$ are not available.

Inspired by pure-motion tracking, some notations are defined as,

(18) $$ \begin{equation} \begin{array}{l} x^{(n)}=f(x)+g(x) u+\hat{\delta} \\ \hat{\delta}=\hat{W} \hat{\Phi}(\hat{m}, \hat{\sigma}, \hat{r}) \end{array} \end{equation} $$

where $$ \hat{W}, \hat{\Phi}, \hat{m}, \hat{\sigma}, \hat{r} $$ are the estimated values of $$ W^{*}, \Phi^{*}, m^{*}, \sigma^{*}, r^{*} $$. $$ \hat{\delta} $$ is the estimation of the system error.

By adjusting the adaptive parameters of RNNs, the hybrid motion/force controller will approximate unknown dynamic functions. From the part of error estimation $$ \hat{\delta}=\hat{W} \hat{\Phi}(\hat{m}, \hat{\sigma}, \hat{r}) $$, the error is defined by

(19) $$ \begin{equation} \begin{array}{l} \widetilde{\delta}=\delta-\hat{\delta}=\delta^{*}+\xi-\hat{\delta} \\ =W^{* T} \Phi^{*}-\hat{W}^{T} \hat{\Phi}+\xi \\ =W^{* T} \Phi^{*}-\left(W^{*}-\tilde{W}\right)^{T}\left(\Phi^{*}-\widetilde{\Phi}\right)+\xi \text {, } \\ =W^{* T} \widetilde{\Phi}-\widetilde{W}^{T} \Phi^{*}+\widetilde{W}^{T} \widetilde{\Phi}+\xi \\ =\widetilde{W}^{T} \hat{\Phi}+\left(\widetilde{W}^{T}+\hat{W}^{T}\right) \widetilde{\Phi}+\xi \\ =\tilde{W}^{T} \widetilde{\Phi}+\widetilde{W}^{T} \hat{\Phi}+\hat{W}^{T} \widetilde{\Phi}+\xi \end{array} \end{equation} $$

where $$ \widetilde{W}=W^{*}-\hat{W} $$, and $$ \widetilde{\Phi}=\Phi^{*}-\hat{\Phi} $$.

Utilizing with the Taylor expansion of nonlinear functions, we can get that

(20) $$ \begin{equation} \widetilde{\Phi}=\Phi_{m} \widetilde{m}+\Phi_{\sigma} \widetilde{\sigma}+\Phi_{r} \widetilde{r}+o(\cdot) \text {. } \end{equation} $$

Substituting (20) into (19), we get

(21) $$ \begin{equation} \begin{array}{l} \widetilde{\delta}=\widetilde{W}^{T} \widetilde{\Phi}+\widetilde{W}^{T} \hat{\Phi}+\hat{W}^{T} \widetilde{\Phi}+\xi \\ =\widetilde{W}^{T} \widetilde{\Phi}+\widetilde{W}^{T} \hat{\Phi}+\hat{W}^{T}\left(\Phi_{m} \widetilde{m}+\Phi_{\sigma} \widetilde{\sigma}+\Phi_{r} \widetilde{r}+o(\cdot)\right)+\xi \\ =\widetilde{W}^{T} \widetilde{\Phi}+\xi+\hat{W}^{T} o(\cdot)+\widetilde{W}^{T} \hat{\Phi}+\hat{W}^{T} \Phi_{m} \widetilde{m}+\hat{W}^{T} \Phi_{\sigma} \widetilde{\sigma}+\hat{W}^{T} \Phi_{r} \widetilde{r} \end{array} \end{equation} $$

Define $$ \Theta=\widetilde{W}^{T} \widetilde{\Phi}+\xi+\hat{W}^{T} o(\cdot) $$; since $$ \|\xi\| \leq \bar{\xi} $$, it is assumed that $$ \Theta $$ is bounded, and $$ \|\Theta\| \leq \mathfrak{I} $$,

where $$ \mathfrak{I} $$ is a constant. Therefore,

(22) $$ \begin{equation} \widetilde{\delta}=\widetilde{W}^{T} \hat{\Phi}+\hat{W}^{T} \Phi_{m} \widetilde{m}+\hat{W}^{T} \Phi_{\sigma} \widetilde{\sigma}+\hat{W}^{T} \Phi_{r} \widetilde{r}+\Theta . \end{equation} $$

Theorem 4.1. For the nonlinear system (6) with bounded uncertainties, if the control

law is designed in the form of

(23) $$ \begin{equation} \left.u(t)=g^{-1}(x(t)) \mid-f(x(t))+x_{d}^{(n)}(t)+\alpha^{T} E(t)-\hat{\delta}\right\rfloor+g^{-1}(x(t)) \hat{)} \operatorname{sgn}(s(t)), \end{equation} $$

and adaptive regulation law is adopted as

(24) $$ \begin{equation} \left\{\begin{array}{l} \dot{\hat{W}}=-K_{w} s_{l} \hat{W} \\ \dot{\hat{m}}=-K_{m} \Phi_{m}^{T} \hat{W} s \\ \dot{\hat{\sigma}}=-K_{\sigma} \Phi_{\sigma}^{T} \hat{W} s \\ \dot{\hat{r}}=-K_{\gamma} \Phi_{r}^{T} \hat{W} s \\ \dot{\hat{\mathfrak{J}}}=K_{\mathfrak{\Im}}\|s\| \end{array}\right. \end{equation} $$

where $$ l=1, 2 \cdots n_{R} $$, and $$ \hat{\mathfrak{I}} $$ is the estimated value of $$ \mathfrak{I} $$; the robot walking system will be asymptotically stable.

Proof. By substituting the control law (30) into (7) $$ x^{(n)}(t)=f(x(t))+g(x(t)) u(t)+\delta(x(t)) $$, one can get

(25) $$ \begin{equation} \begin{array}{l} x^{(n)}(t) =f(x(t))-f(x(t))+x_{d}^{(n)}(t)+\alpha^{T} E(t)-\hat{\delta}+\delta+\hat{\Im} \operatorname{sgn}(s(t)) \\ =\alpha^{T} E(t)+\hat{\mathfrak{I}} \operatorname{sgn}(s(t))+x_{d}^{(n)}(t)-\hat{\delta}+\delta \end{array} \end{equation} $$

that is,

$$ \begin{array}{l} \alpha^{T} E(t)+\beta \dot{E}(t)+\hat{\mathfrak{J}} \operatorname{sgn}(s(t))-\hat{\delta}+\delta=0 \\ \dot{s}(t)=-\hat{\mathfrak{I}} \operatorname{sgn}(s(t))-(-\hat{\delta}+\delta) \end{array} $$

The Lyapunov stability theory is used to analyze the stability of the system, and the candidate Lyapunov function is selected as

$$ \begin{array}{c} V_{R}(s, \tilde{W}, \tilde{m}, \tilde{\sigma}, \tilde{\gamma}, \widetilde{\mathfrak{I}})=\frac{1}{2} s^{T} s+\frac{1}{2 K_{w}} \operatorname{tr}\left(\tilde{W}^{T} \tilde{W}\right)+\frac{1}{2 K_{m}} \tilde{m}^{T} \tilde{m}+\frac{1}{2 K_{\sigma}} \tilde{\sigma}^{T} \tilde{\sigma}+\frac{1}{2 K_{\gamma}} \tilde{\gamma}^{T} \tilde{\gamma}+\frac{1}{2 K_{\mathfrak{I}}} \tilde{\mathfrak{J}}^{2} \\ \dot{V}_{R}(s, \tilde{W}, \tilde{m}, \tilde{\sigma}, \tilde{\gamma}, \widetilde{\mathfrak{I}})=s^{T} \dot{s}+\frac{1}{K_{w}} \operatorname{tr}\left(\widetilde{W}^{T} \dot{\tilde{W}}\right)+\frac{1}{K_{m}} \tilde{m}^{T} \dot{\tilde{m}}+\frac{1}{K_{\sigma}} \tilde{\sigma}^{T} \dot{\tilde{\sigma}}+\frac{1}{K_{\gamma}} \tilde{\gamma}^{T} \dot{\tilde{\gamma}}+\frac{1}{K_{\mathfrak{J}}} \tilde{\mathfrak{I}} \dot{\tilde{\mathfrak{J}}} \\ s^{T} \dot{s}=s^{T}(-\hat{\mathfrak{I}} \operatorname{sgn}(s)-(-\hat{\delta}+\delta)) \\ =s^{T}\left(-\hat{\mathfrak{J}} \operatorname{sgn}(s)-\left(\widetilde{W}^{T} \hat{\Phi}+\hat{W}^{T} \Phi_{m} \widetilde{m}+\hat{W}^{T} \Phi_{\sigma} \widetilde{\sigma}+\hat{W}^{T} \Phi_{r} \widetilde{r}+\Theta\right)\right) \\ =-\hat{\mathfrak{J}}\|s\|-s^{T} \Theta-s^{T} \tilde{W}^{T} \hat{\Phi}-s^{T} \hat{W}^{T} \Phi_{m} \tilde{m}-s^{T} \hat{W}^{T} \Phi_{\sigma} \tilde{\sigma}-s^{T} \hat{W}^{T} \Phi_{r} \tilde{r} \end{array} $$

Considering the structural characteristics of RNNs, $$ \operatorname{tr}\left(\widetilde{W}^{T} \dot{\tilde{W}}\right)=\sum_{l=1}^{n_{R}} \widetilde{w}_{l}^{T} \dot{\widetilde{w}}_{l} $$, and $$ s^{T} \tilde{W}^{T} \hat{\Phi}=\sum_{l=1}^{n_{R}} s_{l} \widetilde{w}_{l}^{T} \hat{\Phi} $$, we can obtain that

(26) $$ \begin{equation} \begin{array}{l} \dot{V}_{R}(s, \widetilde{W}, \widetilde{m}, \widetilde{\sigma}, \tilde{\gamma}, \widetilde{\Im})=-K_{\Im}\|s\|-s^{T} \Theta-s^{T} \widetilde{W}^{T} \hat{\Phi}-s^{T} \hat{W}^{T} \Phi_{m} \widetilde{m}-s^{T} \hat{W}^{T} \Phi_{\sigma} \widetilde{\sigma}-s^{T} \hat{W}^{T} \Phi_{r} \widetilde{r} \\ +\frac{1}{K_{w}} \operatorname{tr}\left(\tilde{W}^{T} \dot{\tilde{W}}\right)+\frac{1}{K_{m}} \tilde{m}^{T} \dot{\tilde{m}}+\frac{1}{K_{\sigma}} \tilde{\sigma}^{T} \dot{\tilde{\sigma}}+\frac{1}{K_{\gamma}} \tilde{\gamma}^{T} \dot{\tilde{\gamma}}+\frac{1}{K_{\mathfrak{I}}} \tilde{\widetilde{\Im}} \dot{\tilde{\Im}} \\ =-\hat{\Im}\|s\|-s^{T} \Theta-\sum_{l=1}^{n_{R}} s_{l} \widetilde{w}_{l}^{T} \hat{\Phi}-s^{T} \hat{W}^{T} \Phi_{m} \widetilde{m}-s^{T} \hat{W}^{T} \Phi_{\sigma} \widetilde{\sigma}-s^{T} \hat{W}^{T} \Phi_{r} \widetilde{r} \\ +\frac{1}{K_{w}} \sum_{l=1}^{n_{R}} \widetilde{w}_{l}^{T} \dot{\tilde{w}}_{l}+\frac{1}{K_{m}} \widetilde{m}^{T} \dot{\tilde{m}}+\frac{1}{K_{\sigma}} \widetilde{\sigma}^{T} \dot{\widetilde{\sigma}}+\frac{1}{K_{\gamma}} \tilde{\gamma}^{T} \dot{\tilde{\gamma}}+\frac{1}{K_{\mathfrak{J}}} \tilde{\widetilde{J}} \dot{\tilde{J}} \\ =-\sum_{l=1}^{n_{R}} \widetilde{w}_{l}^{T}\left(s_{l} \hat{\Phi}-\frac{1}{K_{w}} \dot{\widetilde{w}}_{l}\right)-K_{\mathfrak{\Im}}\|s\|+\frac{1}{K_{\mathfrak{I}}} \hat{\mathfrak{\Im}} \dot{\widetilde{\mathfrak{J}}}-s^{T} \Theta \\ +\widetilde{m}^{T}\left(\frac{1}{K_{m}} \dot{\widetilde{m}}-\Phi_{m}^{T} \hat{W}^{T} S\right)+\widetilde{\sigma}^{T}\left(\frac{1}{K_{\sigma}} \dot{\tilde{\sigma}}-\Phi_{\sigma}^{T} \hat{W}^{T} s\right)+\tilde{\gamma}^{T}\left(\frac{1}{K_{\gamma}} \dot{\widetilde{\gamma}}-\Phi_{\gamma}^{T} \hat{W}^{T} S\right) \end{array} \end{equation} $$

According to the basic principle of SMC, there is $$ s(t) \rightarrow 0 $$ when $$ t \rightarrow \infty $$, thus, $$ \dot{\tilde{\widetilde{w}}_{l}}=-\dot{\hat{\hat{W}}}, \dot{\tilde{\tilde{m}}}=-\dot{\hat{m}}, \dot{\tilde{\sigma}}=-\dot{\hat{\sigma}}, \dot{\tilde{\gamma}}=-\dot{\hat{\gamma}}, \dot{\widetilde{\mathfrak{\Im}}}=-\dot{\hat{\mathfrak{J}}} \text {. } $$

Let the upper bound error be defined as $$ \widetilde{\mathfrak{I}}=\mathfrak{I}-\widehat{\mathfrak{I}} $$; on the other hand, by substituting the adaptive regulation law (23) into (26), we can obtain that

(27) $$ \begin{equation} \begin{array}{l} \dot{V}_{R}(s, \widetilde{W}, \widetilde{m}, \widetilde{\sigma}, \widetilde{\gamma}, \widetilde{\Im}) \\ =-\sum_{l=1}^{n_{R}} \widetilde{w}_{l}^{T}\left(s_{l} \hat{\Phi}-\frac{1}{K_{w}} K_{w} s_{l} \hat{W}\right)-\hat{\mathfrak{\Im}}\|s\|+\frac{1}{K_{\mathfrak{S}}} \widetilde{\widetilde{\Im}} K_{\mathfrak{\Im}}\|s\|-s^{T} \Theta \\ +\widetilde{m}^{T}\left(\frac{1}{K_{m}} K_{m} \Phi_{m}^{T} \hat{W} s-\Phi_{m}^{T} \hat{W}^{T} s\right)+\widetilde{\sigma}^{T}\left(\frac{1}{K_{\sigma}} K_{\sigma} \Phi_{\sigma}^{T} \hat{W}_{s}-\Phi_{\sigma}^{T} \hat{W}^{T} s\right)+\widetilde{\gamma}^{T}\left(\frac{1}{K_{\gamma}} K_{\gamma} \Phi_{r}^{T} \hat{W} s-\Phi_{\gamma}^{T} \hat{W}^{T} s\right) \\ =(\widetilde{\mathfrak{\Im}}-\hat{\mathfrak{\Im}}) \mid s \|-s^{T} \Theta \leq 0 \end{array} \end{equation} $$

Hence, from (41), it can be seen that the Lyapunov function is not increasing but bounded in its domain of definition; that is, $$ V_{R}(0)=V(s(0), \widetilde{W}(0), \widetilde{m}(0), \widetilde{\sigma}(0), \widetilde{\gamma}(0), \widetilde{\Im}(0)) $$, and $$ V_{R}(t)=V(s(t), \widetilde{W}(t), \widetilde{m}(t), \widetilde{\sigma}(t), \widetilde{\gamma}(t), \widetilde{\Im}(t)) $$.

Let $$ R(t)=(\|\Theta\|-\mathfrak{J}) S(t) $$. Considering $$ s(t) $$ is bounded, there exists

(28) $$ \begin{equation} \begin{array}{l} R(t) \leq\|\Theta \Theta\|-\Im)\|(t)\| \leq-\dot{V}_{R}(s, \widetilde{W}, \widetilde{m}, \widetilde{\sigma}, \tilde{\gamma}, \widetilde{\Im}) . \\ \int_{0}^{t} R(t) d t \leq-\int_{0}^{t} \dot{V}_{R}(s, \widetilde{W}, \widetilde{m}, \widetilde{\sigma}, \widetilde{\gamma}, \widetilde{\exists}) d t=V_{R}(0)-V_{R}(t) \text {. } \end{array} \end{equation} $$

According to the Barbarat lemma, we obtain $$ \lim _{\bf{t} \rightarrow \infty} R(t)=0 $$; hence, the system can be asymptotically stable when $$ \lim _{t \rightarrow \infty} s(t)=0 $$. This completes the proof of Theorem 1.

5. EXPERIMENTAL RESULTS AND DISCUSSION

To verify the control methods given in Section 3 and Section 4, simulations are implemented. The configuration parameters of robots are shown in Table 1.

Figure 4 shows that the absolute joint angles qi vary with the time of the biped robot in the duration of walking. It shows that the phase of each joint is reset based on the foot contact information at the beginning of each step. The trajectories of each joint are smooth and periodic. $$ q1 $$ and $$ q2 $$ have a jump in every period, indicating the switch between the swing leg and support leg. Figure 4 illustrates a phase diagram of the joint angle $$ qi $$ and joint angular velocity $$ qi $$ during the walking process, which are all limited circles to prove that the walking process can achieve asymptotic stability. In Figure 5, the straight lines indicate the discrete instance of the walking gait. It stands for the fact that the swinging leg has an impulsive action on the ground, and the joint angular velocity $$ qi $$ has a sudden change at the same time.

Figure 4. The absolute joint angles $$ q_{i} $$ vary with time.

Figure 5. Phase diagram of joint angle $$ q_{i} $$ and joint angular velocity $$ \dot{q}_{i} $$.

Figure 6 shows the total energy of the system changes over time in the body and inertial frame, respectively. Figure 7 shows the stride length. Figure 8 shows the hip position in the body and inertial frame, and its velocity is shown in Figure 9.

Figure 6. Energy plot.

Figure 7. Stride length.

Figure 8. Hip position.

Figure 9. Hip velocity.

Let external disturbance $$ \tau_{d}=[\exp (-0.1 t)]_{6 \times 1} $$ be exerted on the link 2 when t = 2.5 s. This will lead to the changes of the inertia matrix, Coriolis force matrix, and gravity matrix of the robot system, which is equivalent to introducing the the model uncertainty. Set the error upper bound $$ \beta=1 $$, and the boundary layer thickness is set as 0.01. The simulation results are shown in Figure 10 and Figure 11.

Figure 10. The tracking effect of joint motion in the adaptive sliding mode control (SMC).

Figure 11. The comparisons of system input torque. (A) with adaptive sliding mode control (SMC); (B) without adaptive SMC.

Figure 10 depicts the tracking effect of each joint. The blue solid line and the red dotted line represent the actual position and desired position of each joint in the walking process, respectively. The simulation results show that satisfactory excessive control chattering exists due to the fault of mass change. It can be seen that the designed adaptive sliding mode controller can meet the tracking requirement of the desired trajectory. Figure 11 describes the comparisons of system input torque. The comparisons of joint errors with time are shown in Figure 12. It indicates that the joint error finally tends to 0, which shows that the system can converge to the sliding surface and the robot can realize asymptotically stable walking. These figures illustrate that the proposed adaptive SMC control system can achieve the purpose of tracking reference trajectories. Therefore, the tracking errors of the proposed RNN hybrid control system converge more quickly than without adaptive SMC.

Figure 12. The comparisons of joint errors with time. (A) with adaptive sliding mode control (SMC); (B) without adaptive SMC.

6. CONCLUSIONS

The problem of external disturbance uncertainties will affect the stability during dynamic walking, which has greatly limited the application and efficiency of robots. In this paper, the robust and efficient walking of biped robots is investigated. The robust walking model will be optimized, which provides a theoretical basis for flexible and stable humanoid walking. The focus is on the following aspects: (1) analyzing the mechanism of disturbances and studying robust control strategies from the perspective of theoretical analysis; (2) transforming the target of stable biped walking into the problem of stabilization of periodic orbits and through stability analysis; (3) constructing the autonomous evolution mechanism based on hybrid robust control to realize adaptive optimization of walking models. The verification of the proposed control method is conducted by simulations. In future work, a more human-like walking gait will be designed to achieve more efficient walking. The external disturbance has been considered as an unknown uncertainty, and an uncertainty observer will be designed for efficient learning and dynamic response.

DECLARATIONS