# Adaptive robust control for biped walking under uncertain external forces

*Intell Robot*2023;3(3):479-94.

## Abstract

Adaptability and robustness are important expressions of the intelligent walking ability of biped robots. This paper is concerned with the problem of dynamical biped walking and robust control of biped robots under external forces. Due to the characteristics of strong coupling and hybrid, the robotic system is modeled as a rigid kinematic chain with Lagrange equations. A novel adaptive feedback controller is proposed based on sliding mode control (SMC) and hybrid zero dynamics. The novelty of the proposed work lies in taking the uncertainty of upper-bound error into consideration. The hybrid robust control is mentioned to approximate unknown dynamic functions with the adaptive weight. The restricted Poincare return map is utilized to analyze the stability of a nonlinear impulsive system. It ensures that the flow of the continuous subsystem can pass through the impact cross section. Finally, the simulation results illustrate that the proposed adaptive SMC control system can favorably track the reference trajectories, even when a fault occurs, which verifies the effectiveness of the proposed method.

## Keywords

*,*impulse hybrid system

*,*adaptive sliding mode control

*,*stability analysis

## 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.

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

where N is the length of the generalized configuration vector.

where

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

where

where

## 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.

Due to the adaptability of error amplitude, the controller does not need to accurately estimate the amplitude of external disturbance. Let

Considering the coefficients of the Hurwitz polynomial

where the constants

where

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

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

thus, it can get

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

According to the assumption that

## 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.

### 4.1. Principle and structure of RNN

A primary RNN is adopted to deal with the complicated optimization problem. It will remember previous information and use it to affect the output of subsequent nodes. The mapping equation of an RNN is expressed as follows:

where

The

where

Define the weight memory space matrix

### 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.

Assumption 4.1. According to the approximation principle of neural networks, suppose that there exists the desired weight

where

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

where

By adjusting the adaptive parameters of RNNs, the hybrid motion/force controller will approximate unknown dynamic functions. From the part of error estimation

where

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

Substituting (20) into (19), we get

Define

where

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

law is designed in the form of

and adaptive regulation law is adopted as

where

Proof. By substituting the control law (30) into (7)

that is,

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

Considering the structural characteristics of RNNs,

According to the basic principle of SMC, there is

Let the upper bound error be defined as

Hence, from (41), it can be seen that the Lyapunov function is not increasing but bounded in its domain of definition; that is,

Let

According to the Barbarat lemma, we obtain

## 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.

Configuration parameters of robots

Parameter | Torso length | Thigh length | Calf length | Trunk mass | Thigh mass | Calf length | Thigh rotational inertia | Calf rotational inertia |

Value unit | 0.204 m | 0.412 m | 0.385 m | 5.9 kg | 13.2 kg | 7.7 kg | 0.56 | 0.28 |

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.

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.

Let external disturbance

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.

## 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

### Authors' contributions

Made substantial contributions to the conception and design of the study and performed data analysis and interpretation: Wang H

Performed data acquisition and provided administrative, technical, and material support: Wang H, Chen Q

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

This work is supported by the National Natural Science Foundation of China (No. 61733013; No. 61573260; No. 62073245; No. U1713211).

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2023.

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## Cite This Article

## How to Cite

Wang, H.; Chen Q. Adaptive robust control for biped walking under uncertain external forces. *Intell. Robot.* **2023**, *3*, 479-94. http://dx.doi.org/10.20517/ir.2023.26

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