Gait Transition in Artiﬁcial Locomotion Systems using Adaptive Control

Jonas Kr¨aml and Carsten Behn

Department of Technical Mechanics, Technical University of Ilmenau, Max-Planck-Ring 12, 98693 Ilmenau, Germany

Keywords:

Bio-inspired Locomotion System, Gait Transition, Adaptive Control, Uncertain System, Gait Generation.

Abstract:

This paper deals with the modeling, analysis and controlled gait transitions of terrestrial artiﬁcial locomotion

systems. These systems are inspired by the motion of earthworms and are ﬁrstly moving unidirectionally. In

contrast to the analyzed systems in literature, the mechanical model in this paper consists of a chain of 10

discrete mass points. The theory is not restricted to a speciﬁed number of mass point, just to a ﬁxed, but

arbitrary number. Recent results from literature present investigations of short worms (n < 4). The movement

of the whole system is achieved by shortening and lengthening of the distances between consecutive mass

points, while they can only move in forward direction. To inhibit the backward movement, a spiky contact

to the ground using ideal spikes – preventing velocities from being negative – are attached to every mass

point realizing the ground contact. The changes of the distances combined with the ground contact results in

a global movement of the system, called undulatory locomotion. But, to change the distances, viscoelastic

force actuators link neighboring mass points and shall control desired distances in using adaptive control

strategies. Speciﬁc gaits are required to guarantee a controlled movement that differ especially in the number of

resting mass points and the load of actuators and spikes. To determine the most advantageous gaits, numerical

investigations are performed and a weighting function offers a decision of best possible gaits. Using these

gaits, a gait transition algorithm, which autonomously changes velocity and number of resting mass points

depending on the spike and actuator force load, is presented and tested in numerical simulations.

1 INTRODUCTION

Worm-like locomotion systems play an increasing

role in current mechanics literature, see for example

(Miller, 1988), (Hirose, 1993), (Ostrowski and Bur-

dick, 1996), (Vaidyanathan et al., 2000), (Liu et al.,

2006), and ﬁnd their place in teaching and educa-

tion of students, see textbooks (Zimmermann et al.,

2009) and (Steigenberger and Behn, 2012). These

systems have the advantage of little space require-

ments due to their unidirectional motion. Hence, they

are used in narrow places. Possible applications are,

e.g., minimally invasive surgery (Dario et al., 1996),

service and maintenance robots (Fatikow and Rem-

bold, 1997) or drilling robots (Kubota et al., 2007).

Previous publications deal with worm-like loco-

motion systems with 3 or 4 mass points (Schwebke

and Behn, 2013). This paper supplements the knowl-

edge of the dynamic behavior of worm-like locomo-

tion systems with 10 mass points with the goal to ex-

pand the gained results to snake-like locomotion sys-

tems in further works.

Firstly, the mechanical model of a worm-like lo-

comotion system and the adaptive control scheme

are presented. Afterwards, the generation of suit-

able gaits considering current literature is introduced.

These gaits are used by a gait transition algorithm that

changes velocity and number of resting mass points

depending on the load of spikes and actuators, like the

biological paradigm does (Merz and Edwards, 1998).

Finally, simulations are carried out to demonstrate the

functionality of the scheme.

2 MODELING & CONTROL

Modeling: The model is identical to (Steigenberger

and Behn, 2012). The kinematic model comprises

a chain of discrete mass points m

as shown in Fig-

ure 1, where x

(t)(i = 0, n) are the coordinates of the

mass points (single degree of freedom). The distance

of neighboring mass elements is l

(t) := x

i−1

(t) −

(t). To inhibit backward movement, ideal spikes are

mounted at each segment.

A segment here is a mass point, but it can

also appear as balloon-like or bellows-like elements

(possibly ﬂuid ﬁlled), see (Slatkin et al., 1995),

Kräml, J. and Behn, C.

Gait Transition in Artiﬁcial Locomotion Systems using Adaptive Control.

DOI: 10.5220/0006003001190129

In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 2, pages 119-129

ISBN: 978-989-758-198-4

119

Figure 1: Chain of mass points with spikes, due to (Schwe-

bke and Behn, 2013).

(Vaidyanathan et al., 2000), (Meier et al., 2004) and

(Liu et al., 2006).

To allow a movement of the worm, the distances

between the mass points have to be shortened and

lengthened. This can be done in adjusting their longi-

tudinal or radial dimensions (Nakamura et al., 2006)

or, as here, the distances between adjacent elements

(Behn and Zimmermann, 2011a) and (Behn and Zim-

mermann, 2011b). For this purpose, viscoelastic ac-

tuators are assumed between the segments in the dy-

namic model. The applied forces on a mass point are:

• Spring Forces F

c,i

= c

i−1

− x

− l

0,i

) and

−F

c,i+1

= −c

i+1

− x

i+1

− l

0,i+1

), where l

0,i

and

0,i+1

are the detensioned lengths of the springs;

• Damping Forces F

d,i

= d

( ˙x

i−1

− ˙x

) and

−F

d,i+1

= −d

i+1

( ˙x

− ˙x

i+1

);

• Actuator Forces u

and −u

i+1

;

• Spike Forces F

Z,i

;

• Weight F

Gx,i

in x-direction;

• Friction Force F

R,i

= 0.

At this stage of investigations, the assumed zero fric-

tion force is only a special case. The interaction to

the ground is modeled via ideal spikes. In future

work, we have to replace these spikes by anisotropic

Coulomb friction, see also (Steigenberger and Behn,

2012).

According to (Steigenberger and Behn, 2012), the

ideal spikes have to fulﬁll these conditions:

˙x

≥ 0, F

Z,i

≥ 0, ˙x

· F

Z,i

= 0 (1)

These conditions can be fulﬁlled by the following

equation, where F

is the sum of all remaining applied

forces:

Z,i

= −

(1− sign( ˙x

)) · (1− sign(F

)) · F

(2)

Now, the coupled differential equations for movement

of the segments can be formulated:

¨x

= +c

i−1

− x

− l

0,i

)

− c

i+1

− x

i+1

− l

0,i+1

)

+ d

( ˙x

i−1

− ˙x

) − d

i+1

( ˙x

− ˙x

i+1

)

+ u

− u

i+1

+ F

Z,i

+ F

Gx,i

+ F

R,i

(3)

with c

= c

n+1

= d

n+1

= u

n+1

= 0. The

DoF of the system is N.

To inﬂuence the system, acuators have to apply

forces on the mass points. They serve as inputs to the

crawling system to control the distances between the

segments.

Control: To follow a given motion pattern, to react to

changes of the environment and to deal with unknown

or uncertain system parameters, an adaptive controller

is used that generates necessary actuator forces on its

own. The forces depend on the error e

(t):

• l

(t) := x

j−1

(t) − x

(t), the distance between

neighboring mass points, which are the system

outputs;

• l

ref, j

(t), the predeﬁned time-variant reference dis-

tance functions;

• e

(t) := l

(t) − l

ref, j

(t), error of the output.

The used controller is described in (Behn, 2013). It

contains regular PD-control, which adapts the gain of

P and D elements depending on the 2-norm of the er-

ror ke(t)k. The controller’s goal is to track a reference

function of the outputs and to keep the error within a

certain tolerance λ-tube. This kind of λ-tracking in

combination with an adaptive controller is described

in (Behn and Loepelmann, 2012):

e(t) := l(t) −l

ref

(t)

u(t) = k(t) e(t) +k(t)κ

e(t) = k(t) · (e(t) + κ

e(t))

k(t) =











γ· (ke(t)k−λ)

, ke(t)k ≥ λ +1

γ· (ke(t)k−λ)

0.5

, λ+1 > ke(t)k ≥ λ

0, (ke(t)k < λ)

∧(t − t

< t

)

−σk(t), (ke(t)k < λ)

∧(t − t

≥ t

)

k(t

) = k

(4)

with γ > 1, κ > 0, σ > 0, t

≥ 0, λ ≥ 0, k

> 0,

determined in pre-simulations and set in Table 2.

It is obvious that the proposed controller is based

on the availability of the error velocity. This is some-

times quite hard to arrange, therefore, see (Behn,

2011) and (Ye, 1999) for controllers without deriva-

tive measurement of the output.

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120

Controller (4) works as follows: if the error norm

is higher than λ, k(·) increases either quadratic or with

a square-root function depending on the amount of ex-

ceeding. The variable t

is the point in time, when the

error norm latest entered the λ-tube. If the error norm

is smaller than λ and t

is smaller than the parameter

, k(·) is staying constant. Ift

is bigger than t

, there

is an exponential decrease of k(·).

3 GENERATION OF GAITS

Firstly, it is demonstrated for a system of N = 3 mass

points, how the controller works and that suitable

gaits are necessary. Suitable means some kind of op-

timality with respect to resting phases of mass points

and speed of the whole system. Therefore, arbitrar-

ily chosen reference distance functions l

ref, j

(t) and

their time derivatives

ref, j

(t) are deﬁned according to

(Schwebke, 2012):

ref,1

(t) = l

sin(ωt)

ref,2

(t) = l

sin



ωt +



ref,1

(t) =

ωcos(ωt)

ref,2

(t) =

ωcos



ωt +



(5)

Figure 2 shows, how the gain k(t) behaves de-

pending on the error norm ke(t)k . Regarding the

worm movement x

(t), the system is able to move for-

ward successfully. However, the velocities of the seg-

ments ˙x

(t) are non-uniform,which results in different

resting times of the mass points. It is not possible to

create a deﬁned gait with a certain sequence of active

spikes (i.e., resting mass points) by using this kind of

reference distance functions (5), which is important

for gait transition. Moreover, due to the arbitrarily

chosen gait, the speed of the worm system is rather

low (keep this in mind for the upcoming simulations,

where the speed of the system is increased in using

appropriate developed gaits).

Hence, gaits haveto be designed systematically, as

described in (Steigenberger and Behn, 2011). For fur-

ther algorithms relying on binary segment states (con-

tracted, extended), see (Slatkin et al., 1995), (Chen

et al., 1999a), (Chen et al., 1999b) and (Chen et al.,

2001).

First of all, gaits differ in the number of active

spikes a ∈ {1, . . . ,n}. Furthermore, there is a peri-

odic sequence of active spikes A(t), e.g., for a sys-

tem with N = 3 mass points a possible sequence is

A(t) = {0} → {1} → {2}. With this information,

100

200

300

time [s]

gain k [N/m]

0.00

0.05

0.10

0.15

error norm kek [m]

gain

error norm

λ-tube

(a) Gain and error norm.

−2

time [s]

worm movement x

[m]

mass point m

(b) Worm movement.

0.2

0.4

0.6

0.8

time [s]

velocity ˙x

[m]

Figure 2: Worm-like locomotion of a system with 3 mass

points: arbitrary distance functions (5).

it can be deduced whether a distance l

(t) has to be

shortened or lengthened at a certain point of time.

Following the recommendation from (Steigenberger

and Behn, 2011), the sequence of active spike should

move to left or to the right (only by one segment)

like the worm does. A possible sequence is A(t) =

{0, 1} → {1, 2} → {2, 3} → {3, 0} for a system with

N = 4 segments, while A(t) = {0, 1} → {2, 3} →

{1, 2} → {3, 0} is not recommended. Hence, allowed

gaits can be described explicitly by the beginning se-

quence A

of the resting mass points of a time pe-

riod and the direction dir of the wave of active spikes,

which can be ”l” for left or ”r” for right.

The reference distance functions are built as de-

scribed in (Steigenberger and Behn, 2011). The time

intervals are deﬁned as:

Gait Transition in Artiﬁcial Locomotion Systems using Adaptive Control

121

t ∈



, (p+ 1)



, p ∈ N

To ensure a smooth movement of the system, i.e.,

there are no jerks to the mass points, we use sin

(·)-

functions to describe the link lengths of the mass

points, while τ = t − p

(τ) = εl

2N f sin

(πfNτ)

(τ) = l

0∗

+ εl

N fτ−

2π

εl

sin(2π fNτ) ,

(6)

• |ε| ∈ (0;1) is the relative factor of the maximum

distance change,

• f is the frequency of the A(t)-sequence with its

periodic time T =

, chosen in simulation to

avoid a rigid-body-movement of the whole worm

system,

• l

> 0 is the initial distance (detensioned spring),

• l

0∗

is the distance at the beginning of the time

interval (τ = 0), depending on the previous

interval either l

, l

(1+ε) or l

(1−ε) (Schwebke

and Behn, 2013).

The allowed gaits also differ in their load of the spikes

and actuators. To ﬁnd the most advantageous (i.e.,

lowest load of actuators and spikes) gaits for transi-

tion, numerical simulations are executed. Gaits with

equal number of active spikes a (i.e., equally quick)

are rated with a weighting function:

S,g

:=w

max,S

+ w

∑

i=0

Z,i,max,S

+ w

∑

j=1

j,max,S

T,g

:=w

max,T

+ w

∑

i=0

Z,i,max,T

+ w

∑

j=1

j,max,T

S,g,sc

S,g

S,min

; W

T,g,sc

T,g

T,min

g,tot

T,g,sc

S,g,sc

T,g,sc

S,g,sc

T,g,sc

(7)

Because of transient effects at the beginning of the

simulation, this function considers ﬁrstly the maxi-

mum load of actuators u

(·) and spikes F

Z,i

(·), and the

maximum gain parameter k(·) for a transient interval

T,g

as well as a stationary intervalW

S,g

. The weight-

ing factors are chosen as w

= 1.0m/N, w

= 4.0N

−1

= 4.0N

−1

to have a bigger inﬂuence of the load of

actuators and spikes. Afterwards, the values W

S,g

and

T,g

are scaled to the minimum value W

S,min

respec-

tively W

T,min

within the gaits with equal number of

active spikes a. Finally, transient and stationary parts

are weighted against each other. The minimum value

g,tot

within gaits with equal number of active spikes

a identiﬁes the most advantageous gait.

This leads to the result shown in Table 1 for a sys-

tem with N = 10 mass points:

Table 1: Most advantageous gaits.

gait

1 A

= {1}, dir = r

= {2, 3}, dir = r

3 A

= {0, 1, 2}, dir = r

= {6, 7, 8, 9}, dir = l

5 A

= {2, 3, 4, 5, 6}, dir = l

= {5, 6, 7, 8, 9, 0}, dir = l

7 A

= {2, 3, 4, 5, 6, 7, 8}, dir = l

= {1, 2, 3, 4, 5, 6, 7, 8}, dir = l

9 A

= {1, 2, 3, 4, 5, 6, 7, 8, 9}, dir = l

These gaits are used for gait transition.

4 GAIT TRANSITION

Changes of the environment, e.g., change of the slope,

malfunction of an actuator or failing of spikes, lead

to different loads of (the remaining) actuators and

spikes. To react to such circumstances, the system has

to change the gait and its frequency autonomously,

i.e., on its own. This is addressed to the following

investigations. Analogous example: driving a car –

increasing the frequency can be compared to acceler-

ating while gait changing is similar to gear shifting.

The frequency shall only be changed after con-

cluding a single period, i.e., when a part of the se-

quence A(t) is ﬁnished. Changing the frequency has

a great inﬂuence on the loads of actuators and spikes.

To affect their amount, a P-controller is used for the

frequency adjustment. Additionally, it is possible to

weight the load of actuators and spikes against each

other with the factors w

and w

(1+ k

p,Fz

z,set

− F

z,act

))

(1+ k

p,u

set

− u

act

))

, (8)

with k

p,Fz

and k

p,u

as the gain parameters for spikes

and actuators, f

as the previous frequency and f

the newly adjusted frequency. The setpoints F

z,set

and

set

are predeﬁned, while the actual values are within

a single period:

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0 10 20 30 40

0.00

0.05

0.10

0.15

0.20

time [s]

frequency f [Hz]

number of active spikes a [-]

frequency active spikes

(a) Frequency and number of active spikes.

0 10 20 30 40

300

600

900

time [s]

gain k [N/m]

0.00

0.10

0.20

0.30

error norm kek [m]

gain

error norm

λ-tube

(b) Gain and error norm.

0 10 20 30 40

−10

time [s]

worm movement x

[m]

Figure 3: Worm-like locomotion of a system with 10 mass

points: without transition (part 1).

z,act

= max{F

z,0

, F

z,1

, ..., F

z,9

}

¯u

t−T

(τ)dτ

act

= max{ ¯u

, ¯u

, ..., ¯u

}

(9)

The value for the frequency has to be limited to f

max

According to (Steigenberger and Behn, 2012), there

would occur rigid-body-movement, if the frequency

exceeded f

max

. The system would be uncontrollable.

The maximum frequency, from a kinematical theory

according to (Steigenberger and Behn, 2012), is given

by:

0 10 20 30 40

0.00

0.25

0.50

0.75

1.00

1.25

time [s]

mean velocity x

[m/s]

(a) Mean velocity.

0 10 20 30 40

−100

−50

100

time [s]

actuator forces u

[N]

(b) Actuator forces.

0 10 20 30 40

100

time [s]

spike forces F

Z,i

[N]

Figure 4: Worm-like locomotion of a system with 10 mass

points: without transition (part 2).

max

(a) =

gsin(α)

2πεl

N(N − a)

(10)

The system changes the number of active spikes after

ﬁnishing a total period T, i.e., when the sequence of

active spikes would start again. The model upshifts

(decrease the number of active spikes), if the maxi-

mum frequency f

max

of a gait is reached. It down-

shifts (increases the number of active spikes), if the

current reference velocity (11), according to (Steigen-

berger and Behn, 2012), is also reachable with the

next slower gait without exceeding the maximum fre-

quency of the slower gait:

¯v

ref

(a, f ) = (N − a)εl

f (11)

Gait Transition in Artiﬁcial Locomotion Systems using Adaptive Control

123

0 10 20 30 40

0.00

0.10

0.20

0.30

0.40

0.50

time [s]

frequency f [Hz]

number of active spikes a [-]

frequency active spikes

(a) Frequency and number of active spikes.

0 10 20 30 40

200

400

600

800

time [s]

gain k [N/m]

0.00

0.05

0.10

0.15

0.20

error norm kek [m]

gain

error norm

λ-tube

(b) Gain and error norm..

0 10 20 30 40

−10

time [s]

worm movement x

[m]

Figure 5: Worm-like locomotion of a system with 10 mass

points: with transition (part 1).

This downshift frequency f

min

is:

¯v

min,a

= ¯v

max,a+1

(N − a)εl

min

= [N − (a+ 1)]εl

max,a+1

min

N − (a+ 1)

N − a

max,a+1

(12)

To guarantee the same velocity before and after a gait

transition, the frequency has to be adapted. The anal-

ogy to car driving is the adaption of the engine speed.

The frequency after the transition is:

¯v

new

= ¯v

old

(N − a

new

)εl

new

= (N − a

old

)εl

old

new

N − a

old

N − a

new

old

(13)

0 10 20 30 40

0.00

0.25

0.50

0.75

1.00

time [s]

mean velocity x

[m/s]

(a) Mean velocity.

0 10 20 30 40

−100

−50

100

time [s]

actuator forces u

[N]

(b) Actuator forces.

0 10 20 30 40

100

time [s]

spike forces F

Z,i

[N]

Figure 6: Worm-like locomotion of a system with 10 mass

points: with transition (part 2).

5 SIMULATIONS

Example 1: Worm System with Constant Slope

without Gait Transition: First of all, a simulation

without transition and frequency controlling is pre-

sented to get familiar with the basic functionality of

the system. The worm crawls up a ramp with a slope

of 30

◦

with the maximum frequency f

max

= 0.147Hz

according to (10) of the fastest gait with a = 1. The

used parameters for each simulations are shown in Ta-

ble 2.

One can clearly see in Figure 3(c) a typical worm

movement with the reference functions according

to (6), the ﬁrst mass point travels 18m in 40s. The

adaptive controller works reliably; the gain parameter

reaches its stationary state after 7s and oscillates

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100

150

200

250

300

0.00

0.12

0.24

0.36

0.48

0.60

time [s]

frequency f [Hz]

number of active spikes a [-]

frequency active spikes

(a) Frequency and number of active spikes.

100

150

200

250

300

400

800

1,200

time [s]

gain k [N/m]

0.00

0.05

0.10

0.15

error norm kek [m]

gain

error norm

λ-tube

(b) Gain and error norm.

100

150

200

250

300

−20

time [s]

worm movement x

[m]

Figure 7: Worm-like locomotion of a system with 10 mass

points: with transition and changing slope (part 1).

around a value of 700N/m, see Figure 3(b). The max-

imum spike force is 98.3N, see Figure 4(c), while the

maximum actuator force is 90.3N, see Figure 4(b).

Thus, the maximum values F

Z,max

respectively u

max

are set as 100N for the spike force and 90N for the

actuator force for the upcoming simulations with gait

transitions.

Example 2: Worm System with Constant Slope

and with Gait Transition: Now, the system will

change the gait and the frequency, while the weighing

factors in (8) are w

= w

= 1. The worm crawls up

a ramp with a slope of 30

◦

again.

As expected, the gait is changed depending on the

100

150

200

250

300

0.00

0.25

0.50

0.75

1.00

time [s]

mean velocity x

[m/s]

(a) Mean velocity.

100

150

200

250

300

−100

−50

100

time [s]

actuator forces u

[N]

(b) Actuator forces.

100

150

200

250

300

120

time [s]

spike forces F

Z,i

[N]

Figure 8: Worm-like locomotion of a system with 10 mass

points: with transition and changing slope (part 2).

Table 2: Parameters for simulations.

end

= 40s m

= 1.0kg

= 10.0N/m d

= 5.0kg/s

= 1.0m ε = 0.4

λ = 0.05m κ = 1s

= 2.0s γ = 500

σ = 0.2s

−1

= 10N/m

p,Fz

= 0.02N

−1

p,u

= 0.02N

−1

g = 9.806m/s

α = 30

◦

loads, see Figure 5(a). Due to changing only one gait,

it takes a long time until the system ﬁnds a suitable

gait. E.g., the system requires 40s to change from

a = 9 to a = 1. To solve this problem, we restrict the

usable number of gaits from 9 to 3 afterwards.

Gait Transition in Artiﬁcial Locomotion Systems using Adaptive Control

125

100

150

200

250

0.00

0.10

0.20

0.30

0.40

0.50

time [s]

frequency f [Hz]

number of active spikes a [-]

frequency active spikes

(a) Frequency and number of active spikes.

100

150

200

250

400

800

1,200

time [s]

gain k [N/m]

0.00

0.05

0.10

0.15

error norm kek [m]

gain

error norm

λ-tube

(b) Gain and error norm.

100

150

200

250

−20

time [s]

worm movement x

[m]

Figure 9: Worm-like locomotion of a system with 10 mass

points: limitation of actuator forces (part 1).

Example 3: Worm System with Changing slope

and with Gait Transition: Here, the worm also

crawls up a ramp with a slope of 30

◦

, but when the

worm covered the mean distance of 25m, there is

a change of it to 60

◦

for each segment (in the plots

marked with a red vertical line). After a distance

of 50m, it is changed to 30

◦

again (also marked

with a red vertical line). To get the setpoints F

Z,set

respectively u

set

for actuator and spike force, the

maximum values F

Z,max

and u

max

are multiplied from

now on with a safety factor s = 0.8.

The system adapts the frequency and gait after the

change of the slope to reduce/increase the loads of

actuators and spikes, see Figure 7(a). Similiar to

100

150

200

250

0.00

0.25

0.50

0.75

1.00

time [s]

mean velocity x

[m/s]

(a) Mean velocity.

100

150

200

250

−100

−50

100

time [s]

actuator forces u

[N]

(b) Actuator forces.

100

150

200

250

120

time [s]

spike forces F

Z,i

[N]

Figure 10: Worm-like locomotion of a system with 10 mass

points: limitation of actuator forces (part 2).

Example 2, it takes too much time to adjust the gait.

E.g., the system requires 53s to change from a = 1

to a = 9. Furthermore, there occur values above the

maximum permissible values F

Z,max

and u

max

due to

slope of 60

◦

, see Figure 8(b) and 8(c). This problem

is faced below.

Example 4: Worm System with Changing Slope

and with Gait Transition for Limitation of Actu-

ator Forces: To limit the actuator forces, a limita-

tion factor l = 0.99 is used, that is multiplied with

the maximum actuator force u

max

. Actuators are now

not able to exceed this value. In practice, this could

be realized with a current limit function. In contrast,

the spike forces cannot be limited by any function and

hence, the spike load has to be estimated.

The actuator forces do not exceed their maximum

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126

100

150

200

250

0.00

0.08

0.16

0.24

0.32

0.40

time [s]

frequency f [Hz]

number of active spikes a [-]

frequency active spikes

(a) Frequency and number of active spikes.

100

150

200

250

600

1,200

1,800

time [s]

gain k [N/m]

0.00

0.10

0.20

0.30

error norm kek [m]

gain

error norm

λ-tube

(b) Gain and error norm.

100

150

200

250

−20

time [s]

worm movement x

[m]

Figure 11: Worm-like locomotion of a system with 10 mass

points: transition with three gaits (part 1).

value, see Figure 10(b). However, the spike forces

are still exceeded, see Figure 10(c). Spikes have

to be designed adequately solid and should not be

overstrained in practice.

Example 5: Worm System with Changing Slope

and with Gait Transition using Only Three

Gaits: As mentioned above, the number of gait

transitions has to be reduced. For this purpose, the

system can change only three gaits at a time. Pos-

sible gaits are now those with a = 2, 5, 8 from Table 2.

The number of transitions can be reduced signiﬁ-

cantly with this solution, see Figure 11(a). The sys-

tem is able to ﬁnd a suitable gait more quickly. The

100

150

200

250

0.00

0.25

0.50

0.75

1.00

1.25

time [s]

mean velocity x

[m/s]

(a) Mean velocity.

100

150

200

250

−100

−50

100

time [s]

actuator forces u

[N]

(b) Actuator forces.

100

150

200

250

120

time [s]

spike forces F

Z,i

[N]

Figure 12: Worm-like locomotion of a system with 10 mass

points: transition with three gaits (part 2).

loads of spikes and actuators are not inﬂuenced by this

method.

6 CONCLUSION & OUTLOOK

The main focus of the presented work was laid on the

design of an artiﬁcial worm-like locomotion system

with 10 mass points using adaptive control for gait

transition and performing simulations with it. There-

fore, gaits had to be generated, which differ in the load

of actuators and spikes. In order to ﬁnd those with the

smallest load for gait transition, a weighting function

was used that considers stationary and transient parts.

After determining these most advantageous gaits, a

scheme for frequency-control and gait transition was

presented. The frequency-controlleruses actuator and

Gait Transition in Artiﬁcial Locomotion Systems using Adaptive Control

127

Figure 13: Prototype of a cascaded locomotion system consisting of 4 mass points on a conveyor belt with changing slope,

(Otterbach, 2016).

spike forces as input, which can be weighted against

each other. A foregoing simulation without transi-

tion proved the suitability of the adaptive controller

for this system and also served to determine the max-

imum forces for actuators and spikes for the follow-

ing simulations. With this, a gait transition simulation

was performed. The system changed the frequency

and gait depending on the actuators’ and spikes’ load.

However, the maximum values for actuator and spike

force were exceeded. So, a limitation of the actua-

tor forces was implemented, which prevents the ac-

tuator forces of exceeding, but cannot keep the spike

forces below their maximum value. Also, the system

requires too much time to reach the suitable gait. To

deal with this problem, the number of gaits was re-

duced to three. This solution reduces the number of

gait transitions successfully.

Future work shall be directed to ﬁnd a solution to

limit the spike forces; consideration of sliding fric-

tion; experimental veriﬁcation of these theoretical in-

vestigations; expand the system to a 2D-snake-like-

movement based on (Behn et al., 2015), which deals

only with the adaptive movement without gait transi-

tion.

Furtheron, we have to focus on experimental re-

sults, because the working principle of the gait ad-

justing algorithm is just validated by numerical sim-

ulations. For this, we developed a prototype of (until

now) 4 mass points moving on a conveyor belt whose

slope can be controlled/pre-adjusted, see Figure 13.

Experimental results will be generated in near future.

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