Adaptive neural control of quadruped robots with input deadzone

doi:10.1016/j.neucom.2018.09.032

Neurocomputing

Volume 329, 15 February 2019, Pages 486-494

https://doi.org/10.1016/j.neucom.2018.09.032 Get rights and content

Abstract

In this paper, neural network control are presented for a quadruped robot with input deadzone. The study is aiming at the compensation for input deadzone and the control of trajectory using radial basis function neural networks under both full state feedback and output feedback. Neural networks are utlized to estimate the unknown parameters in the quadruped robot model and compensate for input deadzone. The closed-loop stability is proved by Lyapunov’s stability theorem. The extensive simulations are conducted to show the validity of the proposed control.

Introduction

Quadruped robots can keep balance more easily with multiple contacts to interact with the environment compared with biped robots. Unlike wheeled robots, the small interface with the ground makes them good at coping with various situations. Therefore, the control of quadruped robots has received a lot of attention recently [1], [2], [3]. The most basic skill of quadruped robots is to move from a place to another one, which may require the study of machine learning for fast quadruped locomotion [4] and the control of a quadruped robot foot trajectory [5], [6]. A distributed system of adaptive oscillators is proposed to realize the gait adaptation in a quadruped robot [7], and such a system can enhance itself by interacting with the environment. Exploiting body dynamics makes it more difficult to control the behavior of quadruped robot as it enhances the complexity of the robot design and affects the speed of the robot [9], [10], [11]. Some researches discuss the body dynamics in order to solve this issue [12], [13]. To meet acquirement of various requirements, some studies propose control algorithms to make the robot run stably [14], [15], [16]. The fuzzy learning algorithms and dynamic surface control approaches are combined in dynamic balance control and optimization of a quadruped robot with flexible joints [17], [18], [19]. When abstract biological principles are used in designing the control of robots, the overall performance of the robot can be improved, and a canine-inspired quadruped robot is designed by using the data of joint range of motion from canines [20], [21].

Adaptive control plays an important role in system control [22], [23]. A general method for adaptive control is presented in [24], [25], which proposes different switching and tuning schemes. The approach is introduced to deal with large and rapidly varying parameters that exist in control systems [26], [27]. A non-smooth feedback framework is introduced to cope with nonlinearly parameterized systems in [28], [29], [30], where a constructive design method is proposed based on a new parameter separation method and a way of adding a power integrator. Linear robust adaptive controller and neural network based adaptive controller are used in a nonlinear discrete time dynamical system [31], [32].

In the area of adaptive neural network control, the nonlinear constrains existing in control input will severely affect the performance of the system. The nonlinear effects complicate the system and lead to some undesired results [33], [34]. Fuzzy logic system is used in some researches to approximate the unknown functions existing in the nonlinear system [35], [36], and some MIMO nonlinear systems [37], [38]. The deadzone is one of the most complicated nonlinearities, and the method of eliminating the negative effects resulted from nonlinear deadzone has been proposed. Some papers propose output feedback adaptive control to compensate for the deadzone in nonlinear non-discrete time system [39], [40]. Robust adaptive control is a significant method to guarantee the performance in a DC servo system with deadzone [41], [42], which can be proved according to the small tracking error of the system. Similarly, fuzzy logic control is an alternative to eliminate the effects caused by deadzone in a DC motor system [43], [45], and the tuning algorithm makes the system adaptive and stable. Considering that the spacing of the deadzone is unknown, an adaptive update law is proposed to estimate the spacing [44], [46]. Adaptive neural networks are proved to be an effective method to approximate the deadzone function [47], [48], and generalized regression neural network [49], [50], [51], tracking control based on recurrent neural networks [52], [53] are used in some papers to design model control.

During the past decades, the techniques about neural network control have been rapidly developing. Numerous studies have shown that neural network is good at dealing with the problem of function approximation and uncertainties, and many significant advances about neural network have been made in dealing with the problem in continuous-time uncertain nonlinear systems [54], continuous-time nonlinear interconnected systems [55], [56], MIMO nonlinear systems [57], [58], nonlinear discrete-time systems [59], [60], low-triangular-structured nonlinear systems [61], [62], discrete-time affine nonlinear systems [60] and strict-feedback nonlinear systems [63]. The bio-basis function neural network technique is proved to be practical in the detection of proteins. Neural networks are practical to be used as parametric structure to estimate the costate function and the corresponding control law [64], [65] or facilitate the implementation of the proposed iterative algorithm [8], [66], [67].

In this paper, we discuss the trajectory tracking control of a quadruped robot. Considering the capability of neural networks in function approximation and uncertainties, we use neural networks to approximate the unknown parameters of the dynamic and compensate for the deadzone of the input in the robot system. The high-gain observer is designed to approximate the unknown state vectors. The output trajectory error is proved to converge to a small range via Lyapunov’s stability theorem with the proposed control. We highlight the contributions of this paper as follows:

(i)
The neural networks controller is designed to approximate the unknown parameters of the quadruped robot system and improve the stability of the quadruped robot control system.
(ii)
Compensation controller based on neural networks is designed to cope with the problem of the input deadzone, which can solve the negative effect caused by nonlinear sector in a control system, and convert the system into a generalized linear system.
(iii)
Lyapunov’s theorem is utilized to prove the stability of the system. The control system states eventually converge to a compact set by appropriately choosing the designed parameters.

The rest of the paper is organised as below: we discuss the dynamics of the quadruped robot and the model of the deadzone respectively in Section 2, and the design of the controller, when the circumstances that full state information is known or unknown in Sections 3 and 4. The validity of the proposed method is proved in Section 5, and a conclusion is drawn finally in Section 6.

Section snippets

Dynamics

Considering the two degrees of freedom each leg of a quadruped robot which is shown in Fig. 1, the dynamics of k legs can be expressed as [68] $\begin{matrix} M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G = τ_{t} - J_{g}^{T} F_{g} + J_{e}^{T} F_{e}, \end{matrix}$ where $q \in R^{3 k}$ is a joint position vector, $M (q) \in R^{3 k \times 3 k}$ is a 3k × 3k mass matrix of the legs, $C (q, \dot{q}) \in R^{3 k \times 3 k}$ denotes the centrifugal and Coriolis terms, $G \in R^{3 k}$ denotes a vector of gravitational force, and $τ_{t} \in R^{3 k}$ denotes a vector of joint torques, $J_{e} \in R^{3 k \times 3 k}$ denotes the Jacobian matrix from the connect point of the quadruped

Full state feedback neural network control design

As we can see in Fig. 3, “NN1” represents the neural networks that are utilized to estimate the unknown parameters. “NN2” represents the neural networks that are dsigned to cope with the problem of the input deadzone. Full state information q_h and ${\dot{q}}_{h}$ are assumed to be available, then we define a generalized tracking error variable as $z_{1} = q_{h} - q_{h}^{d},$ and thus we have ${\dot{z}}_{1} = {\dot{q}}_{h} - {\dot{q}}_{h}^{d}$ .

Introducing a virtual control variable α(t), then we can define a second error variable as $z_{2} = {\dot{q}}_{h} - α (t),$ where $α (t) = - K_{1} z_{1} + q$

Output feedback neural network control design

As we can see in Fig. 4, “NN1” denotes the neural networks that are utilized to estimate the unknown parameters. “NN2” denotes the neural networks that are dsigned to cope with the problem of the input deadzone. Considering that the state information is unknown for lack of velocity sensors, a high-gain observer is used here to approximate the state vector ${\ddot{q}}_{h}$ .

From Lemma 1, we can consider that $\frac{π_{k + 1}}{ɛ^{k}}$ converges to the $x_{1}^{(k)},$ and the kth order derivative of x₁ such as ξ_k asymptotically converges

Simulation

In this section, some simulation results are listed to prove the effectiveness of the proposed neural network control. The parameters of the simulation are presented as follows. The quadruped robot model is shown in Fig. 1. Each leg of the quadruped robot is composed of two links and thus it has two degrees of freedom. The parameters of the simulation are listed as: $g = 9.8,$ the initial vector $L = [1.0, 1.0],$ $θ_{1} = θ_{2} = 0.01,$ $Γ_{1} = 50,$ $Γ_{2} = 100,$ $K_{1} = diag [500, 100],$ $K_{2} = diag [2000, 2000]$ . As for the quadruped

Conclusion

In this paper, we have used neural networks to estimate the unknown parameters and design the controller in a quadruped robot system. Considering the existing of nonlinear deadzone in the system input, we use neural networks to compensate for the deadzone. Then we design the controller in the lack of some velocity sensors, and high-gain observer is proposed to estimate the unknown state vectors. Finally, the stability of the system is proved through Lyapunov’s stability theorem, and the

Shuang Zhang received the Ph.D. degree from the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, in 2012, and the M.Eng. degree from the School of Automation Science and Engineering, South China University of Technology, Guangzhou, China, in 2009. She is currently an Associate Professor with the School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, China. Her current research interests include

References (71)

ZhangS. et al.
Active vibration control for a flexible string system with input backlash
IET Control Theory Appl.
(2016)
HeW. et al.
Cooperative control of a nonuniform gantry crane with constrained tension
Automatica
(2016)
LiuY.-J. et al.
Adaptive fuzzy control for a class of nonlinear discrete-time systems with backlash
IEEE Trans. Fuzzy Syst.
(2014)
LiuY.-J. et al.
Adaptive fuzzy identification and control for a class of nonlinear pure-feedback Mimo systems with unknown dead zones
IEEE Trans. Fuzzy Syst.
(2015)
YuM. et al.
Robust adaptive control design for dc servo system with friction and dead-zone compensation
J. East Chin. Univ. Sci. Technol. (Nat. Sci. Ed.)
(2010)
ZhonghuaW. et al.
Robust adaptive deadzone compensation of dc servo system
IEE Proc. Control Theory Appl.
(2006)
YangC. et al.
Neural-learning-based telerobot control with guaranteed performance
IEEE Trans. Cybern.
(2017)
HeW. et al.
Adaptive boundary iterative learning control for an euler–bernoulli beam system with input constraint
IEEE Trans. Neural Netw. Learn. Syst.
(2018)
WangM. et al.
Learning from adaptive neural dynamic surface control of strict-feedback systems
IEEE Trans. Neural Netw. Learn. Syst.
(2015)
H. Kimura et al.
Adaptive dynamic walking of a quadruped robot on natural ground based on biological concepts
Int. J. Robot. Res.
(2007)

S. Hirose et al.

Development of quadruped walking robot Titan-Viii and motor driver Titech driver

J. Robot. Soc. Jpn.

(2005)

G.S. Hornby et al.

Autonomous evolution of dynamic gaits with two quadruped robots

IEEE Trans. Robot.

(2005)

N. Kohl et al.

Machine learning for fast quadrupedal locomotion

Proceedings of the AAAI

(2004)

YiX. et al.

Mechanical design and trajectory planning of leg structure of bionic quadruped robot

J. Mech. Trans.

(2013)

LuY. et al.

Research on improved cycloid foot trajectory of quadruped robot

Applied Mechanics and Materials

(2013)

M.A. Lewis et al.

Gait adaptation in a quadruped robot

Autonom. Robot.

(2002)

HeW. et al.

Iterative learning control for a flapping wing micro aerial vehicle under distributed disturbances

IEEE Trans. Cybern.

(2018)

GuoQ. et al.

Backstepping control of electro-hydraulic system based on extended-state-observer with plant dynamics largely unknown

IEEE Trans. Ind. Electron.

(2016)

HeW. et al.

PDE model-based boundary control design for a flexible robotic manipulator with input backlash

IEEE Trans. Control Syst. Technol.

(2018)

LiZ. et al.

Adaptive impedance control for an upper limb robotic exoskeleton using biological signals

IEEE Trans. Ind. Electron.

(2017)

F. Iida et al.

Exploiting body dynamics for controlling a running quadruped robot

Proceedings of the Twelfth International Conference on Advanced Robotics. ICAR’05

(2005)

HeW. et al.

Vibration control of a flexible robotic manipulator in the presence of input deadzone

IEEE Trans. Ind. Inf.

(2017)

D.P. Krasny et al.

Generating high-speed dynamic running gaits in a quadruped robot using an evolutionary search

IEEE Trans. Syst. Man Cybern. Part B (Cybern.)

(2004)

S. Talebi et al.

Quadruped robot running with a bounding gait

Experimental Robotics VII

(2001)

HeX. et al.

Boundary vibration control of variable length crane systems in two-dimensional space with output constraints

IEEE/ASME Trans. Mechatron.

(2017)

LiZ. et al.

Dynamic balance optimization and control of quadruped robot systems with flexible joints

IEEE Trans. Syst. Man Cybern. Syst.

(2016)

HeW. et al.

A survey of human-centered intelligent robots: issues and challenges

IEEE/CAA J. Autom. Sin.

(2017)

M. Hamdy et al.

A novel inverted fuzzy decoupling scheme for Mimo systems with disturbance: a case study of binary distillation column

J. Intell. Manuf.

(2016)

K.S. Aschenbeck et al.

Design of a quadruped robot driven by air muscles

Proceedings of the First IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics. BioRob 2006

(2006)

SunC. et al.

Fuzzy neural network control of a flexible robotic manipulator using assumed mode method

IEEE Trans. Neural Netw. Learn. Syst.

(2018)

HeW. et al.

Adaptive neural network control of a flapping wing micro aerial vehicle with disturbance observer

IEEE Trans. Cybern.

(2017)

YangC. et al.

Neural control of bimanual robots with guaranteed global stability and motion precision

IEEE Trans. Ind. Inf.

(2017)

R.E. Bellman

Adaptive Control Processes: A Guided tour

(2015)

YangC. et al.

Neural network-based motion control of an underactuated wheeled inverted pendulum model

IEEE Trans. Neural Netw. Learn. Syst.

(2014)

K.S. Narendra et al.

Adaptive control using multiple models, switching and tuning

Int. J. Adap. Control Signal Process.

(2003)

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Donghao Zhang received the B. E. degree in automation from the School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, China, in 2017, where he is currently pursuing the M.E. degree. His current research interests include neural networks, underactuated reflex hand.

Cheng Chang received the B. E. degree from the School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, China, in 2016. He is currently pursuing the M.degree with the School of Automation, Huazhong University of Science and Technology, Wuhan, China. His research interests includes neural network control and robotics.

Qiang Fu received the B.S. degree in thermal energy and power engineering from Beijing Jiaotong University, Beijing, China, in 2009, and the Ph.D. degree in control science and engineering from Beihang University (formerly Beijing University of Aeronautics and Astronautics), Beijing, China, in 2016. He is currently a lecturer in the School of Automation and Electrical Engineering, University of Science and Technology Beijing. His main research interests include vision-based navigation and 3-D vision.

Yu Wang received the B.E. degree in automation from the Beijing Institute of Technology, Beijing, China, in July 2011, and the Ph.D. degree in control theory and control engineering at the State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing, in July 2016. He is currently a Research Assistant in the State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences. His research interests include intelligent control, robotics, and biomimetic robots.

^☆: This work is partially supported by the National Natural Science Foundation of China (61873297, 61803025), the Beijing Science and Technology Project under Grant (Z181100003118006) and the Fundamental Research Funds for the China Central Universities of USTB under Grant (FRF-BD-17-002A).

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Adaptive neural control of quadruped robots with input deadzone☆

Abstract

Introduction

Section snippets

Dynamics

Full state feedback neural network control design

Output feedback neural network control design

Simulation

Conclusion

IET Control Theory Appl.

Automatica

IEEE Trans. Fuzzy Syst.

IEEE Trans. Fuzzy Syst.

J. East Chin. Univ. Sci. Technol. (Nat. Sci. Ed.)

IEE Proc. Control Theory Appl.

IEEE Trans. Cybern.

IEEE Trans. Neural Netw. Learn. Syst.

IEEE Trans. Neural Netw. Learn. Syst.

Adaptive dynamic walking of a quadruped robot on natural ground based on biological concepts

Int. J. Robot. Res.

Development of quadruped walking robot Titan-Viii and motor driver Titech driver

J. Robot. Soc. Jpn.

Autonomous evolution of dynamic gaits with two quadruped robots

IEEE Trans. Robot.

Machine learning for fast quadrupedal locomotion

Proceedings of the AAAI

Mechanical design and trajectory planning of leg structure of bionic quadruped robot

J. Mech. Trans.

Research on improved cycloid foot trajectory of quadruped robot

Applied Mechanics and Materials

Gait adaptation in a quadruped robot

Autonom. Robot.

Iterative learning control for a flapping wing micro aerial vehicle under distributed disturbances

IEEE Trans. Cybern.

Backstepping control of electro-hydraulic system based on extended-state-observer with plant dynamics largely unknown

IEEE Trans. Ind. Electron.

PDE model-based boundary control design for a flexible robotic manipulator with input backlash

IEEE Trans. Control Syst. Technol.

Adaptive impedance control for an upper limb robotic exoskeleton using biological signals

IEEE Trans. Ind. Electron.

Exploiting body dynamics for controlling a running quadruped robot

Proceedings of the Twelfth International Conference on Advanced Robotics. ICAR’05

Vibration control of a flexible robotic manipulator in the presence of input deadzone

IEEE Trans. Ind. Inf.

Generating high-speed dynamic running gaits in a quadruped robot using an evolutionary search

IEEE Trans. Syst. Man Cybern. Part B (Cybern.)

Quadruped robot running with a bounding gait

Experimental Robotics VII

Boundary vibration control of variable length crane systems in two-dimensional space with output constraints

IEEE/ASME Trans. Mechatron.

Dynamic balance optimization and control of quadruped robot systems with flexible joints

IEEE Trans. Syst. Man Cybern. Syst.

A survey of human-centered intelligent robots: issues and challenges

IEEE/CAA J. Autom. Sin.

A novel inverted fuzzy decoupling scheme for Mimo systems with disturbance: a case study of binary distillation column

J. Intell. Manuf.

Design of a quadruped robot driven by air muscles

Proceedings of the First IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics. BioRob 2006

Fuzzy neural network control of a flexible robotic manipulator using assumed mode method

IEEE Trans. Neural Netw. Learn. Syst.

Adaptive neural network control of a flapping wing micro aerial vehicle with disturbance observer

IEEE Trans. Cybern.

Neural control of bimanual robots with guaranteed global stability and motion precision

IEEE Trans. Ind. Inf.

Adaptive Control Processes: A Guided tour

Neural network-based motion control of an underactuated wheeled inverted pendulum model

IEEE Trans. Neural Netw. Learn. Syst.

Adaptive control using multiple models, switching and tuning

Int. J. Adap. Control Signal Process.