Juan González-Gómez and Eduardo Boemo

School of Engineering

Universidad Autónoma de Madrid

8th International Conference on Climbing and Walking Robots. Clawar 2005

September 13-15. London. 2005.

Motion of Minimal Configurations of a Modular
Robot: Sinusoidal, Lateral Rolling and Lateral Shift

Introduction

Configuration I

Configuration II

Configuration III

Videos

Conclusion and future work

Index

Introduction (I):
Previous work

In Clawar 2004 we presented the Cube Revolutions worm-like robot

It is constructed by 8 similar modules, connected in the same orientation

It only could move along a straight line, using a 1D sinusoidal gait

Introduction (II):
Modules Y1

We designed a one-degree-of-freedom module, very cheap and easy to construct

Two modules can be attached one to each other in two ways:

Connection in the same orientation

One module move about the pitch axis and the other about the yaw axis

Introduction (III):
Goals

Complex robots can be constructed by attaching these modules

But, what we wonder is:

What is the minimum number of modules needed to achieve locomotion in 1D and 2D?

How do these modules have to be coordinated to achieve the locomotion?

In order to answer these questions, we have constructed three prototypes

Configuration I (Pitch-Pitch)
Description

Pitch axis

We call it Pitch-Pitch configuration (PP)

Two modules connected in the same orientation

They both move about the pitch axis

1D sinusoidal gait

Configuration I (Pitch-Pitch)
Coordination

Two sinusoidal waves are applied to each articulation

These waves only differ on the phase ( )

determines the coordination of the movement

1

2

j

2

Asin(

2

p

T

t+

Df

)

=

j

1

Asin(

2

p

T

=

t)

Df

Df

Configuration I (Pitch-Pitch)
Results

0

25

50

75

100

125

150

175

200

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

5,5

ΔS (cm/cycle)

When is in the range [100, 130] degrees, the speed is maximum and the coordination is the best.

Df

Df

Configuration I (Pitch-Pitch)
Results

A10

A20

A30

A40

A50

A60

A70

Amplitude

0

25

50

75

100

125

150

175

200

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

5,5

6

6,5

7

ΔX (S/cycle)

Speed can be controlled by modifying the amplitude.

Df

Configuration I (Pitch-Pitch)
Brief summary

Minimum number of modules needed: 2

Best coordination is achieved when:

The sign of determines the sense of the movement

The speed can be controlled modifying the amplitude

<130

1D Sinusoidal gait

Df

Df

100<

Configuration II (Pitch-Yaw-Pitch)
Description

Pitch axis

Pitch axis

Yaw Axis

Three modules: two rotating in the pitch axis and one in the yaw

We call it Pitch-Yaw-Pitch configuration (PYP)

1D and 2D sinusoidal gait

Lateral shift gait

Lateral rolling gait

1

3

2

Configuration II (Pitch-Yaw-Pitch)
1D sinusoidal gait

j

1

Asin(

2

p

T

=

t)

1

2

3

j

3

Asin(

2

p

T

t +

Df

)

=

j

2

0

=

The angle of articulation 2 fixed to 0 degrees

Articulations 1 and 3 coordinated in the same way that in the PP configuration

Sames results as in configuration PP

Configuration II (Pitch-Yaw-Pitch)
2D sinusoidal gait

j

1

Asin(

2

p

T

=

t)

1

2

3

j

3

Asin(

2

p

T

t +

Df

)

=

j

2

0

=

The same as in 1D sinusoidal gait, but the angle of articulation 2 different from 0 degrees

The trajectory of the robot is an arc

Configuration II (Pitch-Yaw-Pitch)
Lateral shift gait

1

2

3

j

2

Asin(

2

p

T

t +

)

=

A<=40

Module 1 and 3 are in phase

Module 2 is 90 degrees out of phase

The robot moves parallel to its body axis

j

3

=

j

1

p

2

j

1

Asin(

2

p

T

=

t)

Configuration II (Pitch-Yaw-Pitch)
Lateral rolling gait

The same coordination as in the lateral shift gait, but using and amplitude A>60 degrees.

The sense of rolling can also be controlled by changing the sign of the difference of phase

The robot rolls about its body axis

Configuration II (Pitch-Yaw-Pitch)
Brief summary

1D sin.

2D sin

Lateral shift

Lateral rolling

j

2

j

3

j

1

A

[100,130]

0

>0

Df

Asin(

2

p

T

t+

Asin(

2

p

T

)

Asin(

2

p

T

t)

0

=

[100,130]

>0

Asin(

2

p

T

)

Asin(

2

p

T

t)

Asin(

2

p

T

t)

p

2

Asin(

2

p

T

t+

)

Asin(

2

p

T

)

p

2

<40

j

1

p

2

Asin(

2

p

T

t+

)

Asin(

2

p

T

)

p

2

>60

j

1

Df

Asin(

2

p

T

t+

Asin(

2

p

T

)

Df

Configuration III: three-modules star
Description

1

2

3

Three modules in the same plane, moving about its pitch axis

The angle between the modules is 120 degrees (connected in a three-points-star form)

1D sinusoidal gait along six different directions

Rotation about the robot's yaw axis

Configuration III: three-modules star
1D sinusoidal gait

The robot can move along six different directions

Three sinusoidal waves are applied

Example: In order to move along the red direction:

1

2

3

j

2

j

3

=

=

Asin(

2

p

T

)

j

1

Df

Asin(

2

p

T

+

)

Df

Asin(

2

p

T

+

)

=

<130

100<

Df

Configuration III: three-modules star
Rotation about its yaw axis

Rotation about the robot yaw axis

Three sinusoidal waves are applied

1

2

3

j

1

Asin(

2

p

T

Asin(

2

p

T

)

=

j

2

Asin(

2

p

T

Asin(

2

p

)

=

2

p

3

2

p

+

j

3

Asin(

2

p

T

Asin(

2

p

)

=

4

p

3

p

+

Let's see some videos....

Conclusions and future work

Only two modules are needed to perform 1D sinusoidal gait.

Adding only one more module, three new gaits appear: 2D sinusoidal gait, lateral shift and lateral rolling.

In general, although the gaits are different, the coordination of the modules is very similar.

Further work

The construction of a new snake robot capable of moving on 3D

Generation of gaits using genetic algorithm

THANK you for your attention

I apologize for my bad English :-(

Juan González-Gómez and Eduardo Boemo

School of Engineering

Universidad Autónoma de Madrid

8th International Conference on Climbing and Walking Robots. Clawar 2005

September 13-15. London. 2005.

Motion of Minimal Configurations of a Modular
Robot: Sinusoidal, Lateral Rolling and Lateral Shift

2003-04-25T18:22:37