Gaits-Transferable CPG Controller for a Snake-Like Robot

Science in China Series F: Information Sciences www.scichina.com

Gaits-transferable CPG controller for a snake-like robot

LU ZhenLi1,3,4†, MA ShuGen1,2, LI Bin1 & WANG YueChao1

1 Robotics Laboratory, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China; 2 Organization for Promotion of the COE Program, Ritsumeikan University, Shiga-Kan 525-8577, Japan; 3 Graduate School of the Chinese Academy of Sciences, Beijing 100039, China; 4 Shenyang Ligong University, Shenyang 110168, China With slim and legless body, particular ball articulation, and rhythmic locomotion, a nature snake adapted itself to many terrains under the control of a neuron system. Based on analyzing the locomotion mechanism, the main functional features of the motor system in snakes are specified in detail. Furthermore, a bidirectional cyclic inhibitory (BCI) CPG model is applied for the first time to imitate the pattern generation for the locomotion control of the snake-like robot, and its characteristics are discussed, particularly for the generation of three kinds of rhythmic locomotion. Moreover, we introduce the neuron network organized by the BCI-CPGs connected in line with unilateral excitation to switch automatically locomotion pattern of a snake-like robot under different commands from the higher level control neuron and present a necessary condition for the CPG neuron network to sustain a rhythmic output. The validity for the generation of different kinds of rhythmic locomotion modes by the CPG network are verified by the dynamic simulations and experiments. This research provided a new method to model the generation mechanism of the rhythmic pattern of the snake. snake-like robot, bidirectional cyclic inhibition (BCI), central pattern generator (CPG), stability analysis, locomotion control

1 Introduction

Snakes can have between 130―500 vertebrae in their cylindrical, legless bodies, as shown in Figure 1. The particular ball articulations can perform many kinds of rhythmic movements, such as serpentine locomotion, rectilinear locomotion, and side winding locomotion, and adapt itself to environments[1]. Its functional subdivision of motor system is familiar with other vertebrate, as

Received April 18, 2007; accepted July 19, 2007 doi: 10.1007/s11432-008-0026-0 †Corresponding author (email: [email protected]) Supported in part by the National Natural Science Foundation of China (Grant No. 60375029), the National Hi-tech Research and Development Plan (Grant No. 2001AA422360), and the Japan Society for the Promotion of Science Grants-in-Aid (Grant No. 15360129)

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shown in Figure 2. The key component of the motor system is the Central Pattern Generator (CPG), a neural circuit that can produce a rhythmic motor pattern with no need for sensory feedback. It is mainly located in the spinal cord of vertebrates or in relevant ganglia in invertebrates[2]. To explain the mechanism of such autonomic oscillatory activities, various models of CPG have been suggested[3]. Reiss[4] showed that a pair of reciprocally inhibiting neurons with fatigue can produce alternate bursts of firing. Kling[5] investigated rhythmic activities of circular networks with cyclic inhibitions. Matsuoka[6] gave mathematical condition for mutual inhibition networks represented by a continues-variable neuron model to generate oscillations, which was utilized for a bipedal walking model by Taga[7]. Kimura[8] uses the CPG model to control a quadruped walking robot. The mutual inhibitory CPG is also adopted to control the walking robots in Tsinghua University and Shanghai Jiaotong University[9,10].

Figure 1 Snake and its bone structure. (a) Snake; (b) bone.

Figure 2 Main functional features of motor system.

Snake-like robot that behaves to imitate the shape and locomotion of the snake is studied all over the world[11]. Conradt[12] realized the traveling wave on a real snake-like robot by connecting CPGs in series. However, the dynamics of neural model is not dealt with. Ma’s[13] group adopted a mutual inhibitory CPG to construct a control system for creeping locomotion of a snake-like robot. However, it is only suitable for 2D (2-dimensional) locomotion control. Ma’s group also proposed a cyclic inhibitory CPG for 3D (3-dimensional) locomotion control of a snake-like robot[14], which can also realize creeping locomotion by canceling the pitch joint controlling signal[15,16]. However, a rhythmic pattern cannot be switched to another pattern by temporarily changing the stimulus pattern in all above CPG control system for the locomotion of the snake-like robot. It

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is not resembled in the mechanism of nature snakes. In order to solve this problem, we proposed a bidirectional cyclic inhibitory (BCI) CPG for the locomotion control and pattern switch of a snake-like robot to imitate the mechanism of nature snakes. The mechanism of the rhythm generation in this BCI-CPG model is completely different from that in refs. [14―16]. It is a kind of relaxation oscillator, requiring the adaptation for rhythm generation. Most important of all, it can generate the multi-rhythmic patterns under different activations, which can be adopted to exhibit the automatic transform mechanism among the different typical gaits for the snake-like robot.

2 Construction of BCI-CPG for snake-like robot

Using a reciprocal inhibition among three neurons in ref. [6] and giving all synaptic weights equal values, we constructed a BCI-CPG model which is a completely symmetric structure in terms of graph theory. The BCI-CPG is composed of ny (yaw neuron), np (pitch neuron), and nm (modulate neuron). Here, we adopted a combined joint whose rotational axis connected perpen- dicularly to imitate the locomotion of the hemispherical condyle in snakes. The structure of the CPG and its control relation with the combined joint is shown in Figure 3.

Figure 3 BCI-CPG and its control relation with combined joint.

We utilize a transient-type analog neuron model to specify the CPG, as shown in Figure 4. The dynamics of the CPG is described in the following:

Tur,{y,p,m},ii {y,p,m}, =− u {y,p,m}, i − wywy {y,p,m}, ii {p,m,y}, − {y,p,m}, ii {m,y,p}, n −++βvs{y,p,m},ii 0,{y,p,m},Feed {y,p,m}, iijj +∑ wy {y,p,m}, , (1) j=1

Tva,{y,p,m},ii {y,p,m}, =,- v {y,p,m}, i+ y {y,p,m}, i (2)

yguguu{y,p,m},iii == (), {y,p,m}, ()max(0, {y,p,m}, {y,p,m}, i) , (3)

yyyyaw,ii = y,- m, i, (4)

yyypitch,ii = p,- m, i, (5) where u{y,p,m},i describes respectively membrance potentials of ny, np or nm of ith CPG; v{y,p,m},i describes respectively adaptation of ny, np or nm of ith CPG; Tr,{y,p,m},i shows respectively time constants of membrance potential of ny, np or nm of ith CPG; Ta,{y,p,m},i shows respectively time constants of adaptation of ny, np or nm of ith CPG; w{y,p,m},i shows respectively connection weight among ny, np or nm of ith CPG, here we set it as a constant wypm; β gives respectively connection

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weight inner ny, np or nm of ith CPG; wij is connection weight between ith and jth CPG; S0,{y,p,m},i are constant, positive input, each of which is the summation of all inputs to ny, np or nm by the weight of synaptic conjunction, excepting the output of the neuron in the ith CPG; y{y,p,m},i are outputs of ny, np or nm of ith CPG; y{yaw,pitch},i are CPG control signal of yaw or pitch joint of ith combined joint; Feed{y,p,m},i are sensory feedback of ny, np or nm of ith CPG.

Figure 4 BCI-CPG model.

3 Analysis of BCI-CPG mechanism for rhythm generation

In this section, we specify the theorems on the existence, uniqueness and boundedness of the solution of dynamics of the BCI-CPG (eqs. (1) and (2)). Theorem 1. A solution of eqs. (1) and (2) exists uniquely for any initial state and is bounded for t≥0. Proof. The existence and the uniqueness of the solution of eqs. (1), (2) can be proven by checking Lipschiz’s condition. Expressing eqs. (1), (2) in the following form: x =f(t,x). Here, we do not consider the stimulus from other CPG, wij=0; and no sensory feedback, Feed{y,p,m},i=0. 1 ftu(,{y,p,m},iii )=−− ( u {y,p,m}, w ypm u {p,m,y}, Tr,{y,p,m},i (6) −+−wuypm {m,y,p},ii s 0， {y,p,m},β v {y,p,m}, i), 1 ftv(,{y,p,m},iii )=+ (- v {y,p,m}, u {y,p,m}, ). (7) Tai,{y,p,m},

We assume the following: |u{y,p,m},i −u′{y,p,m},i|≤|u{y,p,m},i −u′{p,m,y},i|≤|u{y,p,m},i −u′{m,y,p},i|≤ρ1,

|v{y,p,m},i −v′{y,p,m},i |≤ρ2, |s0,{y,p,m},i − s′0,{y,p,m},i |≤ρ3, for every |t−t′|≤ρ. Here, ρ, ρ 1, ρ 2 and ρ3 are positive values. We obtain

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21wypm + |(,ftu{y,p,m},i )- ftu (,{y,p,m},′ i )||≤ T ri,{y,p,m},

βρ23- ρ + ||u{y,p,m},i -u{y,p,m},′ i |, (8) T ri,{y,p,m}, ρ1

ρ12- ρ |(,ftvv{y,p,m},ii )-- ftv (,{y,p,m},′′ii )||≤ || {y,p,m}, v {y,p,m}, |. (9) T ai,{y,p,m}, ρ2 Thus, |(,)f tX-- ftX (,)|′ ≤ L | X X′ |. (10)

⎛⎞21wypm + βρ- ρ ρρ- Here, L ≤max⎜⎟+ 23 , 12 . Thus, eqs. (1), (2) satisfied the ⎜⎟ ⎝⎠TTririai,{y,p,m}, ,{y,p,m},ρρ12 T ,{y,p,m}, Lipschitz’s condition.

To prove the boundedness, solve eq. (2) with respect to v{y,p,m},i(t)

-t T ai,{ y,p,m}, vv{y,p,m},ii()t = {y,p,m}, (0)e t (11) 1 -t T ai,{y,p,m},−x T ai ,{y,p,m}, + e(())ed.g u{y,p,m},i xx ∫0 T ai,{y,p,m},

Since g(())u{y,p,m},i x is nonnegative,

vv{,,ypm }, i(tt )≥≥ | {,, ypm }, i (0) |, 0. (12)

Solving eq. (1) with respect to u{y,p,m},i()t , we obtain

t / ⎛⎞11- T ri,{y,p,m}, utu( )=+ (0)e−t /T ri,{ y,p,m}, s (0) − e {y,p,m},ii {y,p,m}, 0,{y,p,m}, i⎜⎟ ⎝⎠TTriri,{y,p,m}, ,{y,p,m}, w t − ypm gu(())ed xx/T ri,{ y,p,m}, x ∫0 {p,m,y},i T ri,{y,p,m}, x / w t T ri,{y,p,m}, − ypm gu(())e x dx ∫0 {m,y,p},i T ri,{y,p,m},

β t x/T ri,{ y,p,m}, − vx()e d. x (13) ∫0 {y,p,m},i T ri,{y,p,m}, Applying eq. (12) to eq. (13), we obtain 1 β utu{y,p,m},ii( )≤ | {y,p,m}, (0) |++ s 0,{y,p,m}, i (0) v {y,p,m}, i (0) . (14) TTr,{y,p,m},iri ,{y,p,m}, Applying eq. (14) to eq. (11) gives similarly 1 ⎛ vtv()≤ | (0)|+ | u (0)|+ {y,p,m},ii {y,p,m},⎜ {y,p,m}, i T a,{y,p,m},i ⎝

1 β ⎞ sv (0)+ (0)⎟ . (15) 0,{y,p,m},ii {y,p,m}, ⎟ TTri,{y,p,m}, ri ,{y,p,m}, ⎠

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Applying eq. (14) and eq. (15) once again to eq. (13), we obtain 1 utu{y,p,m},ii()≥ −− | {y,p,m}, (0)| | s 0， {y,p,m}, i (0)| T ri,{y,p,m}, w ⎛⎞1 β ypm |us (0) | (0) v (0) −++⎜⎟{p,m,y},iii 0,{p,m,y}, {p,m,y}, T ri,{y,p,m}, ⎝⎠TTri,{y,p,m}, ri ,{y,p,m}, w ⎛⎞1 β −+ypm |(0)|u sv(0)+ (0) ⎜⎟{m,y,p},i 0， {m,y,p},ii {m,y,p}, T ri,{y,p,m}, ⎝⎠TTr,{y,p,m},,{y,p,m},iri β ⎪⎧ 1 ⎛ −+|(0)||(0)|vu ⎨ {y,p,m},ii⎜ {y,p,m}, TTri,{y,p,m},⎩⎪ a,{y,p,m}, i⎝ 1 β ⎞⎪⎫ ++sv(0) (0)⎟ . (16) 0,{y,p,m},ii {y,p,m}, ⎟⎬ TTri,{y,p,m}, ri ,{y,p,m}, ⎠⎭⎪ From eqs. (12), (14), (15) and (16), we can conclude that any solution of eqs. (1), (2) is bounded for t≥0. QED Thus, the solution of eqs. (1), (2) exists uniquely for any initial state and is bounded for t≥0. 00 0T Set yy{y,p,m}=[ {y,p,m},1," , y {y,p,m},n ] is a stable output of the CPG, and it must satisfy the following: 00 0 0T yRyy{y,p,m}= [ {y,p,m} ]=[rr1 ( {y,p,m} ), ..., n ( y {y,p,m} )] , (17)

000 0 where r i()=(ygwywys{y,p,m}−− ypm {p,m,y},iiii ypm {m,y,p}, + 0,{y,p,m}, −β y {y,p,m}, ), in=1,," .

0 In contrast, if y{y,p,m},i satisfied eq. (17), then

000 000 uwywysyvy{y,p,m},iiiiiii=+−− ypm {p,m,y}, ypm {m,y,p}, 0,{y,p,m}, −β {y,p,m}, ,= {y,p,m}, {y,p,m}, is a stable solution of eqs. (1) and (2). Theorem 2. Eqs. (1), (2) have at least one stationary solution. Proof. Define a bounded, convex region D in the 3n-dimensional Euclidean space R3n by

D=(0≤y{y,p,m},i≤s0,{y,p,m},i; i=1,...,n), since

00 0 0(≤≤gw−−ypm y {p,m,y},iiiii w ypm y {m,y,p}, + s 0,{y,p,m}, −β y {y,p,m}, ) s 0,{y,p,m}, (18)

0 for arbitrary y{y,p,m},i (i=1,...,n). F is a continuous, contractive mapping from D into D. According to Brower’s fixed point theorem, there must be a fixed point of F, and it is a stationary solution of eqs. (1) and (2). QED According to Theorems 1 and 2, we can conclude that any solution of eqs. (1), (2) is rhythmic output for t≥0. The rhythmic output of the neurons [on1, on2, on2] inner an isolated CPG under yaw neuron first fired is shown in Figure 5(a) (on1: output of first fired neuron; on2: output of two unfired neurons). The corresponding rhythmic output of [yyaw, ypitch] is shown in Figure 5(b) (oc1: CPG output by first fired and unfired neurons; oc2: CPG output by two unfired neurons). An interesting feature of this CPG is that it can generate the following rhythm patterns under

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different initial conditions:

Case 1. ny(on1)→(np(on2) ≡ nm(on2))→ny(on1) →(np(on2) ≡ nm(on2)) ... , (only ny is fired first), and the output of the CPG [yyaw, ypitch] is [oc1, oc2]. It is the rhythm pattern for the combined joint rotating in horizontal plan.

Case 2. np(on1)→(nm(on2) ≡ ny(on2))→np(on1) →(nm(on2) ≡ ny(on2)) ... , (only np is fired first), and the output of the CPG [yyaw, ypitch] is [oc2, oc1]. It is the rhythm pattern for the combined joint rotating in vertical plan.

Case 3. nm(on1)→(ny(on2) ≡ np(on2))→nm(on1) →(ny(on2) ≡ np(on2)) ... , (only nm is fired first), and the output of the CPG [yyaw, ypitch] is [oc2, oc2]. It is the rhythm pattern for the combined joint rotating in 3D plan. “→” indicates the firing sequence of neurons; “≡” indicates the same rhythm of neurons.

Figure 5 Step response output (Tr,{ y,p,m }=1, Ta,{ y,p,m }=5, s0,{ y,p,m }=1, β =2.5, w{y,p,m}=1.5, Feed{y,p,m}=0). (a) Neuron output; (b) CPG output.

4 Construction of BCI-CPG network control system for snake-like robot

Snake epaxial muscles are the complex interconnections and large cross-sectional of muscle- tendon networks which appear to enhance strength and reduce energy use[17]. The connections with long range are not considered in the research. The BCI-CPGs mentioned above are connected in series by the single excitatory synapse from head to tail, to construct the control system of the snake-like robot, as shown in Figure 6.

Figure 6 BCI-CPG control system of the snake-like robot.

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The dynamics of the CPG control system is described by eqs. (1)―(5). Here, we only consider the connection of front neighbor CPG, wij=0 (j≠i+1); there is no sensory feedback, Feed{y,p,m},i=0. The mechanism of the CPG network is that the neurons in the first CPG receive the control signal from the higher level control neuron to generate the corresponding rhythmic pattern. The neurons in the following CPGs are fired by the synapse connection from the corresponding neuron of its front CPG. The pattern of the rhythm is thus propagating from the head to the tail with a specific phasic difference. As a result, the corresponding signal for the rhythmic locomotion control of the snake-like robot is achieved.

We use the parameter ma{y,p},i to modulate the signal of the ith combined joint’s amplitude, and mz{y,p},i to modulate the phase of the signal to the ith combined joint, i=1,...,n. According to the theorem in ref. [18], we proposed a necessary condition for the CPG network to sustain a stable oscillatory output

wypm⎛⎞ ssssss 0,{y,p,m},iiiiii 0,{y,p,m}, 0,{p,m,y}, 0,{p,m,y}, 0,{m,y,p}, 0,{m,y,p}, ≤ min⎜⎟ , , , , , , (19) ⎜⎟ 1+ β ⎝⎠ssssss0,{p,m,y},iiiiii 0,{m,y,p}, 0,{m,y,p}, 0,{y,p,m}, 0,{y,p,m}, 0,{p,m,y},

⎛⎞TTTririri,{y,p,m}, ,{y,p,m}, ,{y,p,m}, w >+1max⎜⎟ , , . (20) ypm ⎜⎟ ⎝⎠TTTaiaiami,{y,p,m}, ,{p,m,y}, ,{ ,y,p},

5 Simulations and experiments for typical gaits transformation

According to the parameters of “Perambulator” shown in Table 1, we adopted ADAMS software to construct a 3D dynamic model of the snake-like robot.

Table 1 Parameters of “Perambulator” Joint number 8(Pitch4, Yaw4) Model parameters (U/m3) 0.07 × 0.033 × 0.055

Quantity (mu/kg) 0.2 Power (P/W) 1.21 -1 Maximal velocity (vmax/deg·s ) 273 Torque (T/Nm) 0.84 Joint work space (W/rad) ± π/2

Based on the theory in ref. [19] that the higher level control neuron just sends rhythm selection command, the CPG network can generate the corresponding rhythm pattern under it automatically. We use the CPG network to generate different rhythm pattern for serpentine locomotion

(only ny,1 first fired), concertina locomotion (only np,1 first fired), and side winding locomotion (only nm,1 first fired). 5.1 Parameter setting According to conditions (19) and (20), we specify the parameters of the CPG network, as shown in Table 2.

Under the condition that only ny,1 is fired firstly, the output of the neurons [ny,i, np,i, nm,i](i=1,..., 4) in CPG network is {[osn1, osn2, osn2], [osn3, osn4, osn4], [osn5, osn6, osn6], [osn7, osn8, osn8]}, as shown in Figure 7.

Here, [osn1, osn2]: the neuron output of CPG1, [osn3, osn4]: the neuron output of CPG2, [osn5, osn6]: the neuron output of CPG3, [osn7, osn8]: the neuron output of CPG4.

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Table 2 Parameters of the CPG network

Positive input s0,{y,p,m},i (i=1,…,4) 5

Time constant Tr,{y,p,m},i /s (i=1,…,4) 1

Time constant Ta,{y,p,m},i (i=1,…,4) 12 Connection weight inner a neuron β 2.5

Connection weight inner a CPG w{y,p,m},i (i=1,…,4) 1.5

Connection weight among CPGs wij (j=i+1; i=1,…,3) 0.1

Figure 7 Neurons’ output.

The output [yyaw,i, ypitch,i](i=1,...,4) of the CPG is {[osc1, osc5], [osc2, osc5], [osc3, osc5], [osc4, osc5]}, shown in Figure 8.

Here, osc1: the output of CPG1, osc2: the output of CPG2, osc3: the output of CPG3, osc4: the output of CPG4, osc5: the output of CPGi (i=1,..., 4). The run time in the simulation is 100 s, and the period of the rhythmic output is 20 s. From

Figures 7 and 8 we know that under the condition that only one neuron of CPG1 is first fired, the output of CPG1 from the first fired neuron and the other neuron is a rhythm pattern, and the output of CPG1 from the two first unfired neurons is zero. The outputs of the following CPGs have the similar rhythm pattern besides a specific phasic difference.

Figure 8 CPGs’ output.

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5.2 Simulations In the dynamic simulator, the contact between the robot and the environment is defined as a cou- lomb friction. The coefficient of the static friction is 0.3, and the coefficient of the dynamic friction is 0.1.

The higher level control neuron sends the command [ay,1, ap,1, am,1] to fire the neurons [ny,1, np,1, nm,1] in the first CPG, and other CPGs are unfired at the beginning. Output of the CPG network under different high level commands is shown as follows:

Case[ay,1, ap,1, am,1] is [100], the output of [ny,i, np,i, nm,i], (i=1,..., 4) is {[osn1, osn2, osn2], [osn3, osn4, osn4], [osn5, osn6, osn6], [osn7, osn8, osn8]}; the output of [yyaw,i, ypitch,i] (i=1,...,4) is {[osc1, osc5], [osc2, osc5], [osc3, osc5], [osc4, osc5]}. It is the rhythm pattern for serpentine locomotion.

Case [ay,1, ap,1, am,1] is [010], the output of [ny,i, np,i, nm,i], (i=1,..., 4) is {[osn2, osn1, osn2],

[osn4, osn3, osn4], [osn6, osn5, osn6], [osn8, osn7, osn8}；the output of [yyaw,i, ypitch,i] (i=1,...,4) is {[osc5, osc1], [osc5, osc2], [osc5, osc3], [osc5, osc4]}. It is the rhythm pattern for concertina locomotion.

Case [ay,1, ap,1, am,1] is [001], the output of [ny,i, np,i, nm,i], (i=1,..., 4) is {[osn2, osn2, osn1],

[osn4, osn4, osn3], [osn6, osn6, osn5], [osn8, osn8, osn7]}；the output of [yyaw,i, ypitch,i] (i=1,..., 4) is {[osc1, osc1], [osc2, osc2], [osc3, osc3], [osc4, osc4]}. It is the rhythm pattern for side winding locomotion. Here, “1” indicates to fire the neuron, and “0” indicates not to fire the neuron.

In order to verify the above result, we set ma{y,p}, i = 0.3, mz{y,p}, i = 1 (i=1,..., 4). Under different command [ay,1, ap,1, am,1] from the higher level control neuron, the data shown in Figure 8 are adopted as the joint angle input to the dynamic simulation model of the snake-like robot. The simulation results are shown in Figure 9. The run time for every gait is 60 s. The corresponding locomotion of the robot model is

Figure 9 Simulation results of gaits transformation. (a) Serpentine locomotion pattern; (b) concertina locomotion pattern; (c) side winding locomotion pattern.

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achieved. Case [ay,1, ap,1, am,1] is [100]: serpentine locomotion, and the corresponding locomotion speed is 0.002 m/s; Case [ay,1, ap,1, am,1] is [010]: concertina locomotion, and the corresponding locomotion speed is 0.0025 m/s; Case [ay,1, ap,1, am,1] is [001]: side winding locomotion, and the corresponding locomotion speed is 0.0028 m/s. 5.3 Experiments Some experiments were carried out on a snake-like robot named “Perambulator” to testify the validity of typical gaits transformation controlled by the BCI-CPG control system. The central- ized and wire control is adopted in “Perambulator.” The main computer in the system is applied to program the locomotion of the snake-like robot. The calculated control signal for every motor is sent out through the serial port to the single chip in the motor driver which can modulate the driving signal for the motors (for details, see ref. [20]). The robot moved on the smooth floor in the experiment. The coefficient of the static friction is 0.3, and the coefficient of the dynamic friction is 0.1. The parameters of the CPG network are shown in Table 2. Using the command [ay,1, ap,1, am,1]from high level neuron to fire [ny,1, np,1, nm,1] in the first CPG, the experimental results are shown in Figure 10.

Figure 10 Experiment results of gaits transformation. (a) Serpentine locomotion; (b) concertina locomotion; (c) side winding locomotion.

The run time is 60 s for every gait. “Perambulator” successfully realized the following gaits:

Serpentine locomotion (case [ay,1, ap,1, am,1] is [100]), and the corresponding locomotion speed is 0.0015 m/s;

Concertina locomotion (case [ay,1, ap,1, am,1] is [010]), and the corresponding locomotion speed is 0.0022 m/s;

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Side winding locomotion (case [ay,1, ap,1, am,1] is [001]), and the corresponding locomotion speed is 0.0025 m/s. The results of simulation and experiment are not so well in accordance with each other, due to the parameters in the simulation model and the fact that the real snake-like robot cannot be the same, where the main influence is the control board in the robot head and the wire among the joints. However, both the simulations and the experiments have shown the validity of the strategy of the BCI-CPG controller for the snake-like robot to realize typical gait transformation.

6 Conclusions

Based on the analyzing of the mechanism for generating typical gaits of snakes, the BCI-CPG was adopted for the first time to design the controller of the snake-like robot. Then, we investigated the stability of the BCI-CPG. Moreover, a numerical condition for the CPG controller to sustain a rhythm was presented. The dynamics simulations and experiments were carried out to verify the validity for performing serpentine locomotion, concertina locomotion, and side winding locomotion controller by the CPG controller. Both the simulation and experiment results have shown that the CPG control system can well realize the transformation among the typical gaits of the snake-like robot. Above all, this research provided a brand new method for the locomotion control of snake-like robots and modeling the generation mechanism of the rhythmic pattern in snakes. In future study, we will specify the function of the feedback in the BCI-CPG control system to achieve a more efficient locomotion of the snake-like robot in different terrains.

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