University of Southern Denmark

Virtual Motoneuron Activation for Goal-directed Locomotion of a Hexapod Robot

Zhu, Yaguang; Zhang, Liang; Manoonpong, Poramate

Published in:2020 5th IEEE International Conference on Advanced Robotics and Mechatronics (ICARM)

DOI:10.1109/ICARM49381.2020.9195387

Publication date:2020

Document version:Accepted manuscript

Citation for pulished version (APA):Zhu, Y., Zhang, L., & Manoonpong, P. (2020). Virtual Motoneuron Activation for Goal-directed Locomotion of aHexapod Robot. In 2020 5th IEEE International Conference on Advanced Robotics and Mechatronics (ICARM)(pp. 380-386). IEEE. https://doi.org/10.1109/ICARM49381.2020.9195387

Go to publication entry in University of Southern Denmark's Research Portal

Terms of useThis work is brought to you by the University of Southern Denmark.Unless otherwise specified it has been shared according to the terms for self-archiving.If no other license is stated, these terms apply:

• You may download this work for personal use only. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying this open access versionIf you believe that this document breaches copyright please contact us providing details and we will investigate your claim.Please direct all enquiries to puresupport@bib.sdu.dk

Download date: 28. Nov. 2021

Γ

Abstract— Walking animals have shown energy efficiency and

versatile locomotion ability when adapting to their environments.

Different methods, including machine learning, model control

and biologically inspired control, have been applied in artificial

legged locomotion systems to imitate these natural

characteristics. In this paper, a virtual motoneuron activation

module (VMAM) is built to convert goal-directed behavioral

instructions into nerve signal strength, which leads to animal-like

signal processing mechanisms and behavioral responses. The

system is composed of a reticulospinal neuron module, a central

pattern generator (CPG) and a virtual motoneuron network

(VMN). The VMN receives the rhythm signals from the CPG

and generates different motions that are controlled by global and

local factors of muscle activation and goal-directed points; then,

through an inverse kinematics module, the signals for driving the

joint motors of the robot are obtained. Finally, four groups of

experiments with different global and local factors of muscle

activation are tested on a bionic hexapod robot to validate the

effectiveness. The results prove that the method can generate

active adaption and emergent behavior according to signal

feedback and improve the locomotion agility, coordination, and

diversity of the robot.

I. INTRODUCTION

Movements of walking animals are elegant and versatile. These capabilities are the result of coupling biomechanics and neural control. The noumenon structure of robots and the corresponding neural control architecture involved in the research of neuro-inspired robots have received increasing attention. This kind of robot mainly responds to external

This work was supported in part by National Natural Science Foundation

of China under Grant 51605039, in part by Thirteenth Five-Year Plan

Equipment Pre-research Field Fund under Grant 61403120407, in part by the

China Postdoctoral Science Foundation under Grant 2018T111005, in part by the Shaanxi International Science and Technology Cooperation Project

under Grant 2020KW-064, in part by the Fundamental Research Funds for

the Central Universities under Grant 300102259308 and 300102259401, and in part by the China Scholarship Council. (Corresponding author: Yaguang

Zhu)

Y. Zhu is with the Key Laboratory of Road Construction Technology and Equipment of MOE, Chang’an University, Xi’an, 710064, China, and with

Embodied AI and Neurorobotics Laboratory, Centre for BioRobotics, Mærsk

Mc-Kinney Møller Institute, University of Southern Denmark, Odense 5230, Denmark. (e-mail: zhuyaguang@chd.edu.cn).

L. Zhang is with the Key Laboratory of Road Construction Technology

and Equipment of MOE, Chang’an University, Xi’an, 710064, China (e-mail: liangzhang@chd.edu.cn).

P. Manoonpong is with the Embodied Artificial Intelligence and

Neurorobotics Laboratory, SDU Biorobotics, the Mærsk Mc-Kinney Møller Institute, The University of Southern Denmark, Odense 5230, Denmark, and

with the College of Mechanical and Electrical Engineering, Nanjing

University of Aeronautics and Astronautics, Nanjing 210016, China, and with the School of Information Science & Technology, Vidyasirimedhi

Institute of Science & Technology, Rayong 21210, Thailand (e-mail:

poma@mmmi.sdu.dk).

sensing information and its own sensing information through neural control architecture and realizes various behaviors. Based on early central pattern generator (CPG) architecture, the behavior of this kind of robot is more complex, the sensing system is more abundant, and the capability of the upper layer is superior. However, in practical applications, the existing neuro-inspired robots already have locomotion and motor control systems [1], learning and memory systems [2, 3], action selection and value systems [4, 5], sensory perception [6, 7], etc. because the constructed neural control system can achieve animal-like signal processing mechanisms and behavioral responses.

Different locomotor gaits in animals, such as walking or galloping, are produced by coordinated activity in neuronal circuits in the neuronal systems. The basic pattern of neural activity controlling locomotion in animals is generated by neural circuits [8, 9]. These circuits instigate numerous muscle fibers to complete intralimb flexor–extensor alternation by generating rhythm and signal coordination and finally achieve all aspects of locomotor activity. The mechanisms and sites of the locomotion neural system are numerous, and Fig. 1 shows some of the possibilities. Sensory inputs from the leg reach the spinal cord and project to motoneurons directly or through interneurons and the CPG. These mechanisms are only shown for motoneurons in the spinal cord and for one idealized cell in supraspinal structures. In Fig. 1, special senses (visual, vestibular, and auditory) interact dynamically with pattern-generating networks through descending pathways [10]. As important inputs, cutaneous and proprioceptive afferents on the limbs course through the spinal cord, reaching supraspinal structures (telencephalon, brain stem) through ascending spinal pathways. There is thus a dynamic transformation of a sensory signal into a motor command, which relies on intrinsic membrane properties of brain stem neurons [11]. Three key mechanisms are essential for locomotion control systems. (1) Central mechanisms. CPGs are activated through tonic signals from supraspinal regions. Basically, networks of CPGs located at the spinal level are capable of autonomously producing motor patterns without sensory feedback or any rhythmic inputs [12]. These CPG networks control much of the timing, pattern, amplitude and rhythm of muscle activation when activated by simple commands. Many neuron-based models [13, 14] and oscillator-based models [15, 16] have been used as CPGs of neural control robots. (2) Interneurons. At the spinal cord level, sets of interneurons allow or block transmission during a given task or control transmission in a phase-dependent manner. This is achieved either by inputs processed by those interneurons implicated directly in the pattern generation or through interneurons whose excitability is modulated cyclically by the CPGs. Interneurons in the spinal cord are considered to be cyclically influenced by the CPG but are not part of the rhythm generation process. These mechanisms are

Virtual Motoneuron Activation for Goal-directed Locomotion of a

Hexapod Robot

Yaguang Zhu, Liang Zhang and Poramate Manoonpong

typically used as internal forward models for sensory prediction, walking-state estimations and coordination. Several instances of this mechanism have been successfully applied in biological legged systems. (3) Motoneurons and Muscles. Muscle fibers have the ability to decrease their length, which in turn produces muscle contraction, and act as animal actuators. There are numerous types of muscle tissue in animals that work together to accomplish a variety of different behaviors, and neuro-inspired robots must use limited driving joints to complete various behaviors. Scientists complete the actuation of robots by decoding causal motor neuron behavior [17]. However, this method is still an abstraction of a neuro-muscular function in animals, and the interaction between skeletal muscle contraction and sympathetic nerve activation in animals remains unknown at this time. At present, most neural-inspired robots combine CPG signals with coordination modules and motion modules to directly drive joint actuators, leading to a coupling relationship between the driving strength and the rhythm signals. It is difficult to consider both the coordination of a robot and independent limb control. Thus, if a neural signal strength processing mechanism can be built separately, the control of limbs and overall coordination can be achieved at the same time.

IN IN

F E

F

E

Telencephalon

and

Brain stem

CPG

Interneurons

Motoneurons

Spinal Cord

visual, vestibular,

and auditory

Muscles

Figure 1. Mechanism of locomotion neural system

Neural control systems and coordination have been applied to salamander-like robots [18], quadruped robots [19], hexapod robots [20] and bat robots [21]. These robots have shown excellent performance in terms of behavioral diversity and limb coordination. We proposed a CPG module for a hexapod [22] and a synchronization network for a quadruped robot parallel torso [23] in previous works, which function as a partial base of research in this paper. Taking all these mechanisms into account for the design of our adaptive neural locomotion control system for a SmartHex [24], we focus on the interaction between actuator and nerve activation in this study. We construct a neural control system with a goal-directed mechanism for a hexapod robot. In the system, we propose a virtual motoneuron network (VMN) to convert the goal-directed behavioral command signals into neural signal strength. The behavior commands are generated by a reticulospinal neuron (RS) module, the generation of rhythm

signals and overall coordination are completed by the CPG module, and the dynamic amplitude change of the actuator is achieved by the neural signal strength module. Under this framework, the instructions issued by the RS module can address the overall behavior with a coordination relationship, a disconnected limb coordination relationship, or the goal-directed behavior of a single limb. In addition, the proposed virtual motoneuron network can not only provide stable geometric characteristics but also locally adjust the behavior trajectory according to the shape parameters. Finally, we conduct several groups of experiments to validate the method.

II. NEURAL CONTROL SYSTEM

The neural control framework proposed in this paper includes an RS module, a CPG coordination module, and a virtual motoneuron activation module (VMAM), as shown in Fig. 1(a). The initiation and regulation of the activity of these networks is determined from the brainstem locomotor command systems, the mesencephalic locomotor region (MLR) and the diencephalic locomotor region (DLR). This regulation occurs via RS, which regulates the activity level of the CPGs [25]. This activity is not addressed in this paper; only the command is used for the parameters of other modules. The CPG coordination module generates rhythm signals with phase differences after receiving the commands of the RS. These signals are transmitted to the VMAM to execute coordinated motions of different limbs or joints. The motoneuron activation module receives behavior commands from the RS, and the rhythm signals from the CPG coordination module are processed to obtain muscle nerve strength signals for controlling the robot joint motor. The signal transmission of the neural control architecture is shown in Fig. 2. After the CPG signals pass through the phase processing neural module, they are sent to the VMN to generate behavior signals, which are converted into joint control signals to be transmitted to the joint motors through the inverse mapping module. The active adaption and emergent behavior in the VMAM is shown in Fig. 2(c).

Different types of oscillators can be used as CPGs for coordination without affecting the overall control architecture. In this paper, the σ-Hopf oscillator is used, since it has a symmetrical circular limit cycle, and the waveform is regular and stable. Different from other oscillators, the frequency, amplitude, and duty ratio of the σ-Hopf oscillator can be adjusted independently, which contributes to the control of different moving speeds, motion ranges, forward-back durations and fast synchronization capabilities; see details in [24]. These advantages can enhance the behavioral diversity and agility of bionic robots, which is of great significance for bionic control. The form of synchronous coupling of σ-Hopf is:

1

1

2 2

1

( ) ( )( ) ( )

( ) ( )

( ( 1) ) / ((1- ) ( 1) )

(( ) ( ) )

y y

x a y bxt t

y b x ay

e e

x a y b

υ g u

(1)

Reticulospinal neurons

τL+ττ

2τ

τL

RR1

HR1

KR1

R1

IMM

BN

BN

PN

RR2

HR2

KR2

R2

IMM

BN

BN

PN

RR3

HR3

KR3

R3

IMM

BN

BN

PN

RL1

HL1

KL1

L1

IMM

BN

BN

PN

RL2

HL2

KL2

L2

IMM

BN

BN

PN

RL3

HL3

KL3

L3

IMM

BN

BN

PN

1 1x y

μ2 μ1

CPG coordination module

Inverse Mapping Module

Joint angles

Robot motors

Envir

onm

ent

Sen

sors

Active

adaption

Emerge

behavior

Virtual motoneuron network

Vir

tual

mo

ton

euro

n

acti

vat

ion

mo

du

le

(b) (c)(a)

Active

adaption

wBOi

wP

wBO1

wP

...

wBOn

wP

n

Pw

wP

...

...

... ...

...

0.5

0.5

1 1x y

μ1

0

1

i

n

Cpost Cpost

CpostCpost

CpostCpost

τL+2τ

Cpost

Opost

λn

Pn

λn-1

Pn-1

λ0

P0

λn-i

Pn-i

Excitatory synapse

Inhibitory synapse

Cpost

μ2

Figure 2. Neural control architecture. (a) Neural system diagram. (b) Signal transmission diagram. (c) Motoneuron activation module.

where x and y are the state variables. a, b is the center of the

limit cycle. μ is the amplitude of the oscillations. The

bifurcation parameter α can switch from -1 to 1 such that this

would change the stable limit cycle dynamics to the dynamics

with a globally stable equilibrium point. In σ(ρ, λ, t), λ is the

strength, and φ is the period factor. 0<ρ<1 denotes the duty

factor and determines the transformation speed between the

ascending and descending phases. In (1), the parameters φ

and ρ are uncorrelated. Thus, the movement period will not be

influenced by a change in the duty factor. g(t) is the coupling

input for a single oscillator g(t)=0. u(t)=−sign(y), and u is the

external input. Different phase shifts between different legs

lead to different gait patterns. The oscillation signal is

transmitted to each leg with a specific phase difference. The

definitions of τ and τL in Fig. 2(b) are given by

L(1 ) and k (2)

where the parameter k controls the gait pattern of the left and

right groups of legs. Hexapod robots have the most gait

patterns with k=3. ζ is the phase shift coefficient and

determines the phase of the output signal of the oscillator,

which is used to directly control the gait pattern. According to

the principle of bionics, by substituting (1) in the output

signal of the oscillator, the control signal of each leg is given

by

( (1 ) )( )

( (4 )

( (1 ) (1 ) ) ( 1,2,3)

( ( 4 ) (1 ) ) ( 4,5,6)

CPG

CPG

j CPG

L

CPG

CPG

x t jx t

x t j

x t j j

x t k j j

, (3)

where j=1, …, 6 represent the LF, LM, LH, RF, RM, and RH,

respectively. Since the Hopf oscillator has the ability to

rapidly converge to a limit cycle, when the parameters change,

they do not dramatically influence the stability of the robot.

According to our previous research [26], the switching

process can basically be performed in a gait cycle. Because

the parameters of the Hopf oscillator have been decoupled

and each value corresponds to a movement state, we can

transform the movement state by changing the parameters.

III. MOTONEURON ACTIVATION FOR GOAL-DIRECTED

LOCOMOTION

As mentioned earlier, different behavioral patterns of organisms produce rhythms and intensities of different neural signals. In this section, the function of motoneuron activation is refined, and it is combined with the CPG and the network to form basic neural signals to control the rhythm and coordination of each limb of the robot. The VMN is used to generate basic behavioral actions.

A. Virtual Motoneuron Network

The quasi-Bezier curve with shape parameters shows that a desired curve or surface can be flexibly chosen from a set of curves or surfaces that differ either locally or globally by suitably modifying the values of the shape parameters when the control polygon is maintained [27]. On this premise, we propose a neural network with virtual motoneuron activation for goal-directed locomotion of legged robots. The quasi-Bezier curve can be written as:

21 1

0 11

2

( ) ( ) (1 ) (1 ) ,

n

n nn n i i n i i

i i i i i i

ni ii

r PB a a

(4)

where ( ) (1 )n n n i i

i iB , 0,1, ,i n , [0,1]T= ,

i 1= ,n n

i i

, 1,2,..., , 2

ni

i 1 ,n n

i i

,

+1,...,2

ni n

, Pi (i=0, 1,…, n) ∈ℝ2 or ℝ3. Then, we

define

( ) , ( ) 11

iii i

B Pw w

, (5)

( 1) ( 1)

, 0,1, ,!

i

n

n n n iC i n

i

. (6)

Thus, ( )n

iB can be written as

( ) ( ) ( ), 0,1, ,n n i i

i P n BB w C w i n (7)

and (4) can be written as

0

1, 2( ) ( ) ( ) ( ) ,

1, 2

nn i i

P B i n i i P

i

i nr w w PC a w

i n

(8)

For limb control, the excitation signal rhythm is provided by the CPG oscillator. The output of the CPG oscillator is

transformed as the excitation signal, so ( )CPG

jx t is used to

replace in the quasi-Bezier curve. Thus, we can obtain

( )

( ) 0.5 ( ) 0.5, 1,2,...,6

Cpost

j

Cpost CPG

j j

x t

x t x t j

T. (9)

The above equations are combined and rewritten as a neural network. The input of the entire network is

0( ( )) ( ( ))Cpost Cpost n

j n n P j n nx t a w x t P C . (10)

The output of the first layer can be written as

1 0 1( ( )) ( ( )) ( ( )) ( ( ))Cpost Cpost Cpost Cpost

j B j j jO x t w x t x t x t . (11)

The ith layer output is

1( ( )) ( ( )) ( ( )) ( ( ))

( ( )) ( ( ))

Cpost Cpost Cpost Cpost

i j B j i j i j

Cpost Cpost n i

i j n i n i P j n i n

O x t w x t O x t x t

x t a w x t P C

. (12)

The output of the ith layer is postprocessed as

0

( ( )) ( ( )) ( ( )) ( ( ))n

Cpost n Cpost i Cpost Cpost

post j P j B j n i j

i

O x t w x t w x t x t

, (13)

where 1, 2

1, 2

i n

i n

. The final network architecture with

signal propagation of the VMN is shown in Fig. 3. i is the

output of the input neuron. Oi is the interneuron output of the

ith layer and represents the activation of the neuron. Opost is

the output of the motoneuron and represents the activation of

motoneuron. TPi and SPi denote the goal-directed points of

stance and swing phase. Tλi and Sλi denote local factors of

muscle activation to adjust the activation of the motoneurons

and behavior trajectory. The CPG signals are generated by (3).

After the CPG postprocessing module performs the scale

transformation, the postprocessing CPG signals are used as

the excitation signals of the VMN, and then the calculation of

(9)-(13) is used to derive the behavioral motion (e.g., the

points Pi given by the RS module are used as parameters of

the network). There are several input neurons i (i=0, 1, 2,…,

n), interneurons Oi (i=1, 2,…, n), and one motoneuron Opost.

The local factor of muscle activation λi corresponding to

different Pi is designed for adjusting the behavior. Obviously,

the general behavioral movement is determined by the three

directions in Cartesian space and leads to three groups of

networks to control the directions.

wBOi

wP

wBO1

wP

...

wBOn

wP

n

Pw

wP

...

...

... ...

...

0.5

0.5

1 1x y

μ1

0

1

i

n

Opost

λn

Pn

λn-1

Pn-1

λ0

P0

λn-i

Pn-i

Cpost

μ2

wP

Pn-i

wBOi

λn-i

i

wP

Pn-i

λn-i

i

2i n

2i n

i

Oi-1

Excitatory synapse

Inhibitory synapse

Figure 3. Virtual motoneuron network

The properties of the network output are also investigated for motion control. For the positions, velocities and accelerations of the initial and final moments, the outputs can be written as

0

1 1 0 1

2 1 0 1 1 0 2 2 1

1 2 1 1 1 2

(0) , (1)

(0) ( )( ), (1) ( )( ),

(0) ( 1)( 2 ) 2 ( ) 2 ( ),

(1) ( 1)( 2 ) 2 ( ) 2 ( )

post post n

post post n n n

post

post n n n n n n n n n

O P O P

O n P P O n P P

O n n P P P n P P P P

O n n P P P n P P P P

. (14)

For the terminal properties P0, P1,…, Pn and Pn, Pn-1,…, P0, define the same trajectory in a different parametrization. Thus, the output can be

1 1

1 01 1

2 2 2

( ); ;

1 ( ); ,..., , ( 1) , ,..., ;

n nCpost

post j i ii i

nCpost n

post j n n in n n i

O x t P

O x t P

.(15)

With the above theorem, different trajectories of the swing phase and stance phase can be obtained by adjusting the

parameters. In this paper, the rising process of ( )CPG

jx t from

-1 to 1 represents the swing phase, and the descending process

of ( )CPG

jx t represents the stance phase. For geometric

invariance, the shape of a VMN is independent of the choice of coordinate, i.e., (8) satisfies the following equations:

0 1 0 11 1

0 1 0 11 1

( ; ; , ,..., ) ( ; ; , ,..., )

( ; ; , ,..., ) ( ; ; , ,..., )

n n

j n j nj j

n n

j n j nj j

r t P q P q P q r t P P P q

r t P P P r t P P P

, (16)

where q denotes the shift factor, and κ denotes the proportional factor. This property can be used to quickly modify limb motion with different amplitudes. The details of using these factors to modify the trajectory are illustrated in the next part.

B. Behavior control based on VMN

Excitatory synapse

Inhibitory synapse

OpostZn

Pw

ZO8

wP

1 wB8

Z

ZP0

Zλ0ZO7

wP

1 wB7

Z

ZP1

Zλ1ZO6

wP

1 wB6

Z

ZP2

Zλ2ZO5

wP

1 wB5

Z

ZP3

Zλ3ZO4

wP

1 wB4

Z

ZP4

Zλ4ZO3

wP

1 wB3

Z

ZP5

Zλ5

Cpost0.5

0.5

ZO2

wP

1 wB2

Z

ZP6

Zλ6ZO1

wP

1 wB1

Z

ZP7

Zλ7

wP

10

Z

ZP8

Zλ8

OpostYn

Pw

YO8

wP

1 wB8

Y

YO7

wP

1 wB7

Y

YO6

wP

1 wB6

Y

YO5

wP

1 wB5

Y

YO4

wP

1 wB4

Y

YO3

wP

1 wB3

Y

Cpost0.5

0.5

YO2

wP

1 wB2

Y

YO1

wP

1 wB1

Y

wP

10

Y

OpostXn

Pw

XO8

wP

1 wB8

X

XO7

wP

1 wB7

X

XO6

wP

1 wB6

X

ZO5

wP

1 wB5

X

XO4

wP

1 wB4

X

XO3

wP

1 wB3

X

Cpost0.5

0.5

XO2

wP

1 wB2

X

XO1

wP

1 wB1

X

wP

10

X

1 1x y

μ1

Excitatory synapse

Inhibitory synapse

μ2

IMM

R H K

YP0

Yλ0

YP1

Yλ1

YP2

Yλ2

YP3

Yλ3

YP4

Yλ4

YP5

Yλ5

YP6

Yλ6

YP7

Yλ7

YP8

Yλ8

XP0

Xλ0

XP1

Xλ1

XP2

Xλ2

XP3

Xλ3

XP4

Xλ4

XP5

Xλ5

XP6

Xλ6

XP7

Xλ7

XP8

Xλ8

Figure 4. VMN for locomotion control of a single leg

According to the above properties and architecture, the output signals of the VMN are transmitted to the inverse mapping module (IMM) and then sent to the joint motors. The number of joint motors is related to the physical structure of the robot, and different structures lead to different expressions of the IMM module. In this paper, a VMN with eight layers (n=8) is constructed in three directions—the X direction (lateral direction), the Y direction (forward direction) and the Z direction (height direction)—and is used for the behavior control of an 18-joint hexapod walking robot. The network for a single leg is shown in Fig. 4. It is also possible to use fewer or more layers to meet the requirements of applications. The VMN is composed of goal-directed point vectors TPi (

XTPi, YTPi,

ZTPi), SPi (

XSPi, YSPi,

ZSPi) and local factors of muscle activation Tλi (

XTλi, YTλi,

ZTλi), Sλi (

XSλi, YSλi,

ZSλi). Once the relationship between goal-directed points is determined, it cannot be easily modified. Instead, the behavioral trajectory can be locally adjusted by Sλi and Tλi. The shape parameters only affect the trajectory near the adjacent position point, and the entire behavior trajectory can be adjusted in segments without changing the properties of the network, as shown in Fig. 5(a-c).

-20 0 20 40 60-40 -20 0 20 40 60-40-20 0 20 40 60-40

(c)(a)Y(mm) Y(mm)

-20 0 20 40 60-40-60

-200

-220

-260

Z(m

m)

-240

-200

-220

-260

Z(m

m)

-240

-20 0 20 40 60-40-60 80 0 50 100100 50

-20 0 20 40 60-40 0 50-50 -20 0 20 40 60-40

(d)Y(mm)

(e)Y(mm)

(f)Y(mm)

(g)Y(mm)

(h)Y(mm)

(j)Y(mm)

(b)Y(mm)

(-37.5, -250)

(-25, -230) (6.25, -220)

(-25, -210)

(0, -215)

(12.5, -210)

(37.5, -250)

(37.5, -230)

(50, -240)

-200

-220

-260

Z(m

m)

-240

1 2 3

Figure 5.Properties of VMN. (a-c) Trajectories with different local factors. (d) Distributions of SPi. (e) Translation. q1=0, q2=-12.5 and q3=12.5. (f) Stride length. κ1=1, κ2=1.5 and κ3=2. (g) Height. H1=40, H2=50, H3=30. (h) Start

position. YSP01=-37.5mm, YSP02=-50mm and YSP03=-25mm. (j) final

position. ZSPn1=250mm, ZSPn2=-260mm and ZSPn3=-240mm.

We then introduce the design of goal-directed points TPi and SPi. The goal-directed points are usually given by the RS module, but some learning methods can be helpful in determining the optimal points. Taking SPi in the swing phase as an example, SP0 and SP8 are the start point and the final point in the swing phase. SP6 and SP7 are set to smooth the landing process. SP1 is set for a quick lifting of the leg in the swing phase. SP2 and SP5 are set for obstacle crossing. SP3 and SP4 are the transition points. Generally, SP0,

SP1 and SP2 are responsible for the lifting segment of the trajectory. SP2,

SP3, SP4,

SP5 and SP6 are responsible for the middle segment. SP6, SP7 and SP8 are responsible for the trajectory of the touchdown segment. The distribution of the goal-directed points is shown in Fig. 5(d), and TPi is uniformly distributed between TP0 = (-37.5, 0, -250) and TP8 = (37.5, 0, -250). Fig. 5(e-j) illustrates the geometric invariant properties of the module indicated by (16). The step length, trajectory height, and foot tip position of the legged robot can be changed through q, κ, TPi,

SPi, and H (H represents the trajectory height and H= ZSP5 -

ZSP0, since

ZSP5 is the highest point and ZSP0 is the start point). In fact, the shape parameters have basically maintained the adjustment ability of each part of the trajectory, which lays the foundation for adaptive locomotion.

IV. RESULTS

Figure 7. Bioinspired hexapod robot, SmartHex.

The proposed locomotion control architecture in this paper is tested through walking experiments to verify the influence of motoneuron activation in the VMN on robot behaviors. All

the experiments are conducted on a bioinspired hexapod robot SmartHex, as shown in Fig. 7. The six legs are symmetrically distributed on both sides of the body of the robot. Each joint is driven by a servo motor, and the servo motors are controlled by an ARM controller (see [24] for more details).

The experiments include one group of global activation adjustments and three groups of local activation adjustments of virtual muscle. Group I is the global activation experiment, and the global factor H in the VMN is adjusted to accomplish the change of the lifting height and the overall foot-tip trajectory. The factor H equals 40 mm in 1-3 s, 55 mm in 3.5-6 s and 70 mm in 6.5-9 s. Groups II to IV are the local activation experiments, and the local muscle activation factor Sλi is changed to achieve local changes in the locomotion trajectory. Since obstacle avoidance is mainly related to the lifting height of the leg in the Z direction, we only take local factors ZSλi of the swing phase into consideration. In Group II, the local factors ZSλ2=0 (ZSλ5=0 in Group III and ZSλ7=0 in Group IV) in 1-3 s, ZSλ2=125 (ZSλ5=125 in Group III and ZSλ7=125 in Group IV) in 3.5-6 s and ZSλ2=250 (ZSλ5=0 in Group III and ZSλ7=250 in Group IV) in 6.5-9 s. 3-3.5 s and 6-6.5 s are the transition periods of different parameter values. Tλi=0, the rest of ZSλi is equal to 0, and the values of TPi and SPi are the same as in Fig. 5 (d). Snapshots of the walking process in Group II are shown in Fig. 8.

ZSλ2=0

ZSλ2=125

ZSλ2=250

Figure 8. Walking process of robot with different ZSλ2

0 731 2 4 5 6 8 9Time (s)

-180

-200

-220

-50 0 50 -50 0 50 0 50 -50 0 50

(a)

H=40

ZSλ2=125ZSλ2=250

ZSλ5=125ZSλ5=250

ZSλ7=125ZSλ7=250

H=55

H=70

-240

-50

Original

Trajectory

ZSλ2ZSλ5

ZSλ7H-180

-200

-220

-240

Y(mm)

Z(m

m)

Y(mm) Y(mm) Y(mm)

Opost

(b)

Figure 9. (a) Snapshots and curves of trajectories. (b) Output of Opost. For original trajectory, H=40 (blue line). H = 100 mm (red line) , ZSλ2 (orange

line), ZSλ5 (purple line), and ZSλ7 (green line) change from 0 to 125 at 3s and

change from 125 to 250 at 6s.

The adjustment of global and local factors of the VMN has different effects on the motion trajectory of the robot. As shown in Fig. 9(a), different H, ZSλ2, ZSλ5 and ZSλ7 lead to different limb trajectories, which are also related to the motoneuron activation Opost. In Fig. 9(b), when the global factor is adjusted, the motoneuron activation Opost changes globally (see the red line). Similarly, local factors can partially change the output of motoneuron activation Opost. To some extent, the motoneuron activation Opost as the output of interneurons directly demonstrates the motion trajectory.

4 80

4 80 4 80 4 80

Time (s)

0

×104

-1.4

-1.6

×104

-1.2

-1.4

×104

Time (s) Time (s) Time (s)

4 80Time (s)

4 80Time (s)

4 80Time (s)

-1.8

-1.9

-2.0

-249.5

-250.5

×103 ×103

3

4 80

-6.5

-7.0

Time (s)

×103

1

4 5

6 7 8

Original

Trajectory

XSλ2XSλ5

XSλ7H

4 80Time (s)

-6.02

-1.75

-250

-249.5

-250.5

1.0×103

0

-1.0

-2.0

-1.0

-1.2

-1.4

-1.0

-2.0

-4.0

-6.0

0

-7.0

×103

3.2 4

-1.9

-1.8

-2.05.2 6 7.2 8

-250

Figure 10. Output of input neurons with different factors.

Actually, input neuron i plays a very important role in

the system. Both global and local factors of muscle activation are processed in the input neuron according to (12), as shown in Fig. 10. The change in H influences the value of goal-directed point SP1-SP7, which in turn causes the change in

the intensity of the input neurons 1 7- in swing phase. This

affects the overall activation of motoneuron Opost. In contrast, the value of goal-directed point SP1-SP7 remains constant even with different ZSλ2, ZSλ5 and ZSλ7. Only the activation intensities

of 6,

3 and

1 in the swing phase are affected by the local

factors. When the limb of the robot is in a support state, there

is no activation intensity change of i.

V. DISCUSSION AND CONCLUSION

In recent years, robots that are based on neural control architecture have received increasing attention, enabling robots to implement excellent signal processing mechanisms and fast behavioral responses. Trajectory planning is a precondition for a robot to maintain a good motion state (including joint space planning and Cartesian space planning) [29], but obstacles and emergencies will result in an unsuitable trajectory in the actual environment and can easily disrupt the original established balance of the motion state. A

goal-directed neural control system for a hexapod robot based on the biological neuromuscular theory is proposed to adjust the behavioral trajectory by changing the global and local factors of muscle activation in a VMAM. With this control system, the robot can quickly adjust its motion trajectory to safely cross the obstacles and instantly recover to the original motion state. The proposed VMN is based on the quasi-Bezier curve, which has many mathematical properties (i.e., geometric invariance and terminal properties). Thus, the VMAM not only has the ability to change behavioral motion according to the complex environment but also has the potential learning ability of neural networks. Through these learning methods, the global and local factors of muscle activation and goal-directed points can be modified according to the external environment. Such modification will be shown in future studies. The activation of the proposed VMAM can be adjusted locally to achieve a specific motion trajectory of the robot and enable the robot to respond to emergencies in a timely manner. As the complexity of the VMN increases, the computation will not have much impact on the motion adjustment. In addition, since the generated trajectories based on the muscle activities are achieved by the joint actuators with position control, other control methods can be used to achieve better performance to regulate the locomotion [30].

In summary, the factor of muscle activation offers flexible adjustment and can be used to determine the output of the network depending on the trajectory required by the actual environment. The experimental results demonstrate the effectiveness of the proposed neural control system, which can be widely applied to motion networks for bioinspired robots in terms of control accuracy and local adjustment of motion trajectory. In the future, adaptive behavior and network learning strategies will be applied to a VMAM to optimize robot control. Sensory feedback will be used to adaptively adjust muscle activation and motion trajectories [31].

REFERENCES

[1] A. J. Ijspeert, “Central pattern generators for locomotion control in

animals and robots: A review,” Neural Networks, vol. 21, no. 4, pp. 642-653, Mar. 2008.

[2] M. Rucci, D. Bullock, & F. Santini, “Integrating robotics and

neuroscience: brains for robots, bodies for brains,” Advanced Robotics, vol. 21, no. 10, pp. 1115-1129, 2007.

[3] B. R Cox, & J. L. Krichmar, “Neuromodulation as a robot controller a brain-inspired strategy for controlling autonomous robots,” IEEE

Robot Autom Mag, vol. 16, no. 3, pp. 72-80, 2009.

[4] J. L. Krichmar, and G. M. Edelman, “Machine psychology: autonomous behavior, perceptual categorization, and conditioning in a

brain-based device,” Cerebral Cortex, vol. 12, no. 8, pp. 818-830,

2002.

[5] K. Doya, and E. Uchibe, “The cyber rodent project: exploration of adaptive mechanisms for self-preservation and self-reproduction,” Adp.

Behav., vol. 13, no. 2, pp. 149-160, Jun. 2005.

[6] F. Santini, and M. Rucci, “Active estimation of distance in a robotic system that replicates human eye movements,” J Robot Auton. Syst.,

vol. 55, no. 2, pp. 107-121, Feb. 2007.

[7] J. U. Duncombe, “Infrared navigation—Part I: An assessment of feasibility (Periodical style),” IEEE Trans. Electron Devices, vol.

ED-11, pp. 34–39, Jan. 1959.

[8] S. Grillner, “biological pattern generation: the cellular and

computational logic of networks in motion,” Neuron, vol. 52, no. 5, pp. 751-766, 2006.

[9] O. Kiehn, “Development and functional organization of spinal

locomotor circuits,” Curr. Opin. Neurobiol., vol. 21, no. 1, pp.100-109, Feb. 2011.

[10] S. Rossignol, R. Dubuc, J.-P. Gossard, “Dynamic sensorimotor interactions in locomotion,” Physiolog. Rev., vol. 86, no. 1, pp. 89-154,

2006.

[11] A. J. McComas, “Hypothesis: Hughlings Jackson and presynaptic inhibition: is there a big picture?” J. Neurophysiol, vol. 116, no. 1, pp.

41-50, Jul. 2016.

[12] O. Kiehn, “Locomotor circuits in the mammalian spinal cord,” Annu. Rev. Neurosci, vol. 29, no. 1, pp. 279-306, 2006.

[13] S. H. Strogatz, "From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators," Physica D, vol.

143, no. 1-4, pp. 1-20, Sep. 2000.

[14] S. J. Chung, J. J. Slotine, “On synchronization of coupled

Hopf-Kuramoto oscillators with phase delays.” in 49th IEEE Conference on Decision and Control (CDC), in IEEE, Atlanta, GA,

USA, 2010, pp. 3181-3187.

[15] K. Matsuoka, “Analysis of a neural oscillator,” Biol. Cybern., vol. 4-5, no. 104, pp. 297-304, 2011.

[16] H. R. Wilson, J. D. Cowan, “A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue,” Kybernetik, vol. 13,

no. 2, pp. 55-80, 1973.

[17] M. Sartori, U. Ş. Yavuz, D. Farina, “In vivo neuromechanics: decoding causal motor neuron behavior with resulting musculoskeletal function,”

Sci. Rep., vol. 7, no. 1, pp. 1-14, Oct. 2017. [18] ljspeert.A. Crespi, D. Ryczko, and J.-M. Cabelguen, “From swimming

to walking with a salamander robot driven by a spinal cord model,”

Science, vol. 315, no. 5817, pp. 1416-1420, 2007.

[19] S. Aoi, D. Katayama, S. Fujiki, et al., “A stability-based mechanism for

hysteresis in the walk-trot transition in quadruped locomotion,” J. R. Soc. Interface, vol. 10, no. 81, 2013.

[20] S. Steingrube, M. Timme, P. Manoonpong, et al. “Self-organized adaptation of a simple neural circuit enables complex robot behavior,”

Nat. Phys., vol. 6, no. 3, pp. 224-230, 2010.

[21] A. Ramezani, S.-J. Chung, S. Hutchinson, “A biomimetic robotic platform to study flight specializations of bats,” Sci. Robot., vol. 2, no.3,

pp. 1-13, 2017.

[22] Y. Zhu, Y. Wu, Q. Liu, et al., “A backward control based on σ-Hopf

oscillator with decoupled parameters for smooth locomotion of

bio-inspired legged robot,” Robot Auton Syst, vol. 106, pp. 165-178, 2018.

[23] Y. Zhu, S. Zhou, D. Gao, et al., “Synchronization of non-linear oscillators for neurobiologically inspired control on a bionic parallel

waist of legged robot,” Frontiers in Neurorobotics, vol. 13, pp. 59.

2019.

[24] Y. Zhu, B. Jin, Y. Wang, et al. “Trajectory correction and locomotion

analysis of a hexapod walking robot with semi-round rigid feet,” Sensors, vol. 16, no. 9, pp. 1392, 2016.

[25] M. Shimamura, S. Grillner, V. R. Edgerton, and L. M. Jordan,

“Brainstem and spinal cord mechanisms for the initiation of locomotion,” In Neurobiological Basis of Human Locomotion, Tokyo,

Japan Scientific Societies, 1991.

[26] Y. Zhu, T. Guo, Q. Liu, “A study of arbitrary gait pattern generation for

turning of a bio-inspired hexapod robot,” Robot Auton Syst, vol. 97, pp.

125-135, 2017.

[27] X. Han, Y. Ma, X. Huang, “A novel generalization of Bézier curve and

surface,” J. Comput. Appl. Math., vol. 217, no. 1, pp. 180-193, 2008.

[28] Y. Zhu, B. Jin, W. Li, et al, “Optimal design of a hexapod walking robot

based on energy consumption and workspace,” T. Can. Soc. Mech. Eng., vol. 38, no. 3, pp. 305-317, 2014.

[29] A. W. Winkler, C. D. Bellicoso, M. Hutter, et al., “Gait and trajectory

optimization for legged systems through phase-based end-effector parameterization,” IEEE Robot. Autom. Lett, vol. 3, no.3, pp.

1560-1567, 2018.

[30] D. Kim, J. Di Carlo, B. Katz, et al., “Highly dynamic quadruped locomotion via whole-body impulse control and model predictive

control,” Robotics, Sep. 2019.

[31] P. Manoonpong, F. Pasemann, F. Wörgötter, “Sensor-driven neural

control for omnidirectional locomotion and versatile reactive behaviors of walking machines,” Robot. Auton. Syst., vol. 56, no. 3, pp. 265-288,

2008.