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9th National Congress on Theoretical and Applied Mechanics, Brussels, 9-10-11 May 2012

Simulation framework of modular robots using EasyDyn

Ir. Christophe Chariot1, Prof. Enrico Filippi2, Prof. Olivier Verlinden3

1,2Department of Machine Design and Production Engineering, University of Mons, Faculty of Engineering

Rue du Joncquois, 53, 7000 MONS - Belgium3Department of Theoretical Mechanics and Dynamics and Vibrations, University of Mons, Faculty of Engineering

Boulevard Dolez 31, 7000 MONS - Belgium

email: [email protected], [email protected], [email protected]

Abstract This paper presents a multibody simulation

framework developed to study the locomotion control of

modular robots. The resulting model aims at featuring the

main parts that characterize actual modular robots includ-

ing kinematics, inertia properties, actuators and the associ-

ated controllers. As the framework is dedicated to modular

robots, we spend some time to describe its core element, the

module, and the extend of its related model. We then show

how we exploited this modular approach to quickly build

a modular robot multibody model. Finally, we illustrate

the application of the framework on two configurations : a

caterpillar and a 6-legged robot.

Keywords Multibody simulation, modular robots

I. INTRODUCTION

THE simulation framework presented in this papercomes within the scope of research about the loco-motion control of modular robots. More specifically, thestudy aims at defining generic control algorithms that lead

to stable gaits on several robot configurations.

The principle of modular robotics is to build a robot

from elementary blocks called modules whose assembly

defines the shape and functionalities of the structure. The

modules are able to reorganize themselves so that they

form an active structure that can adapt its shape in order

to achieve a specific mission.

A. Motivation

Direct tests on prototypes would be specially time con-suming or could cause hardware damages. We therefore

take benefit from multibody simulation to perform prelim-

inary analyses in favourable conditions.

In particular, we use the tool as virtual experiment en-

gine to avoid the use of hardware that brings its own prob-

lems or is simply unavailable. Simulations are also conve-

nient to check the feasibility of a given gait with the actual

robot specifications so that we can prevent possible hard-

ware damage.

Furthermore, working in simulated environments pro-

vides complete and ideal instrumentation. During experi-ments, one can directly monitor any simulated variable or

even integrate a sensor model to assess its performances.

Similarly, we keep control on the experimental conditions

so that the comparison between several runs is not sub-

jected to random changes.

Lets also notice that computational resources are par-

ticularly convenient to run several tests in parallel and to

automate parameters-dependent simulations as used in op-

timization procedures.

B. Approach

A modular robot is an assembly of modules, which

means that the global structure is a repetition of identical

parts. A great interest of such architectures is that a new

robot is simply a combination of existing building blocks

so that there is no need to re-design the robot all over again.

This principle can be extended to the related multibody

model, we therefore have favoured a modular approach to

build the model, based on a recurrent subsystem : the mod-

ule. In the following, we thus detail the model of a single

module (see section II) and describe the way we build the

complete model from there in section III.

This approach is convenient to quickly study the be-

havior of a given control model on several configurations.

This makes the establishement of a generic methodology

easier, which is particularly relevant in the case of modular

robots.

I I . BAS E M ODEL

In this paper, we present the framework applied to the

M-Tran project developed at AIST in Japan. This sec-tion first presents this module and then details the different

parts of the considered model.

A. Module Description

The mechanical design of the module defines its specific

motion and interconnection abilities. In the case of the

M-Tran (see schematic in figure 1), each module is built

from three bodies, it has two controlled degrees of free-

dom, six connection surfaces, it measures 65x65x130 mm

and weighs 400 grams.

As far as the kinematics is concerned, a central part, the

link, has two rotational joints with parallel axes that link

it to the blocks. Therefore, each of the latter, can execute

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Fig. 1. Description of the M-Tran module from [1] : each

module results from the assembly of two blocks on a

central link. It features two rotational degrees of free-

dom (dof) about two parallel axes with a full range of

radians each

rotations within a range of rad with respect to the central

link.

Several modules can connect to each other to create a

modular robot thanks to the connection surfaces located

on the blocks. Even though these blocks are geometrically

identical, they have different functions when it comes to

connections between modules. Indeed, one of the blocks

is active, which means that it embeds active latch systems,

and the other one is passive, that is to say it has specific

surfaces on which the latch is able to grip. Therefore, only

connections between active and passive blocks can be con-

sidered and each of them has three connection surfaces so

that a module features three active and three passive con-

nection surfaces. Lets also note that the connection mech-

anism only allows docking for relative orientations multi-

ple of/2 radians so that there are four different connec-tion positions between two surfaces.

The distance between the joint axes is half the module

length so that modules can form a regular 3D grid with

hinge values of 0,/2 and/2 radians.

B. Extended Multibody Model

The simulation framework mainly aims at studying lo-

comotion control of modular robots by use of dynamics

simulations. We have chosen to establish an extended

multibody model that takes the various parts of the whole

control loop into account, to be representative of an ac-

tual robot. We therefore consider kinematics and inertia

properties of the module as well as its actuators and their

related position control loop. Consequently, the model in-

puts are the joints desired positions, values that are pro-

vided by the pattern generator of the locomotion algorithm(that part is covered later in section II-C). In this section,

we first give an overview of the framework, we present the

main tool EasyDyn and then we individually detail each

part of the model.

As depicted in figure 2, the joint angle of each controlled

hinge (two per module) is compared to the value given

by the locomotion algorithm so that the relative error is

filtered by a PID controller. The latter outputs a voltage

command that drives the motor whose electro-mechanical

model computes the resulting torque to be applied on the

related bodies.

Fig. 2. Simulation model framework : each motor gener-

ates a torque that is computed from its electrical and

mechanical equations whose supply voltage is deter-

mined by a PID position control loop.

B.1 EasyDyn

The framework is based on the EasyDyn simulation

tool developed at the Department of Theoretical Mechan-

ics, Dynamics and Vibrations of the University of Mons.

EasyDyn is an open source C++ library which provides sev-

eral modules to achieve multibody simulations (and more

generally problems described by second order differential

equations), that is to say :

the tools to handle vector calculus ; routines for integration of second-order differential

equations ;

a front-end to build the equations of motion from kine-

matics and applied efforts based on generalized coordi-

nates ;

scene definition of moving bodies for visualization.

Besides theses C++ modules, EasyDyn also includes a

MuPAD script that establishes the kinematics from the dif-

ferentiation of position matrices.

The library features several useful functions providing

specific applied efforts like contacts, springs, dampers,tyres,...

A great advantage ofEasyDyn is that the user always keeps

control of his model that doesnt rely on black box routines

and can thus be easily adapted to a specific situation. In

our case, we take advantage of this by adding customized

equations to extend the multibody model.

B.2 Kinematics

The multibody model is directly built from the module

description and thus includes three bodies. As the mod-

ular robot is defined by the relative connections betweenmodules, we similarly use the composition of motion to

describe the relative motion of every body. So, the block

that connects to a parent module (see section III for more

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details) is considered as reference for the link motion and

subsequently, the second block refers to the link. Each

module has two degrees of freedom1, which means that

we attribute two configuration parameters to each module

that corresponds to the rotational joints.

B.3 Actuator Model

In multibody simulations, the contribution of actuators

is included through applied forces and torques. In our case

we compute the resulting torque of the electrical motors

from their discretized electro-mechanical model2 so that

the input signal is the motor voltageu :

i = u KSn

R(1)

T = KTin (2)

with :

ithe current driven through the motor coil ;

KSthe motor speed constant ;

nthe gearbox reduction ratio ;

Rthe terminal resistance of the motor coil ;

the rotational speed of the body linked to the output

shaft of the gearbox ;

the efficiency of the gearbox ;

KTthe motor torque constant ;

T the torque applied on the body linked to the output

shaft of the gearbox.One can see that the motor model is deeply related to the

multibody simulation since the back electromotive force

contributionKSn is computed using the motion state of

the related body.

The parameters of this model have been extracted from the

datasheet of a specific motor whose selection was based

on the module technical specifications available on the

M-Tran website [2]. Moreover, the motor limitations are

taken into account : the input voltage is saturated accord-

ing to the driver supply and we monitor the current to re-

spect the absolute electrical ratings. Those two constrainstherefore act as speed and torque limitations for the consid-

ered motor so that the actual performances of the module

are used during simulation.

B.4 Position Control Loop

To achieve locomotion, we want the joints to execute

specific motions. More specifically, the high-level loco-

motion algorithm gives the time history of position (i.e.

angular value) for each controlled degree of freedom of

1Apart from the reference module of the whole robot that features six

extras degrees of freedom to represent its position and orientation in the

simulation environment.2Lets note that we neglect the inductance effect Ldi/dt since the

inductance valueL of the motor coil is very small.

the robot. We therefore need to determine the voltage to

be applied on the motors so that the outputs (i.e. the block

rotations) follow the algorithm order. This is managed by a

PID filter (one for each motor) that takes the desired posi-

tion as input and send out the voltage to the motor accord-

ing to the current position of the geared motor shaft. To

fit the numerical environment, the custom PID controller

is described by the following discrete equations (the ex-

ponent k refers to the current sampling period, typically

15ms):

Pk = Kpk (3)

Ik = Ik1 + Kitk1 (4)

Dk = KdD

k1 + KdN(k k1)

(Kd +Nt) (5)

u

k

= P

k

+I

k

+D

k

(6)with :

Pthe proportional contribution ;

Dthe derivative contribution ;

Ithe integral contribution ;

uis the output of the controller ;

the position error ;

Kp is the proportional gain ;

Kdis the derivative gain ;

Kiis the integral gain ;

Nis a constant used to filter the high frequency noise of

the derivative term.

C. Pattern Generation

Considering the global control scheme illustrated in fig-

ure 2, the algorithm part that provides the angular setpoint

for each joint is managed at a higher level that depends on

the subject of the considered simulation.

The presented simulation framework comes within the

scope of a study of CPG-based locomotion control algo-

rithms. Without going into details, a CPG is a bio-inspired

network of coupled non-linear oscillators. It generates

periodic signals that can be used as locomotion pattern

with the ability to fit distributed architectures. The mod-

ule model is therefore completed by adding the consid-

ered CPG equations whose output feeds the position con-

trol loop. As an example, for a simple phase oscillator (a

CPG model), the output value i related to joint i is de-

scribed by the following equations :

i = +j

i j (7)

ri = r(Ri ri) (8)

xi = x(Xi xi) (9)i = xi + ri sin(i) (10)

with:

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a fixed body to interact with the robot as the ground.

Lets note that the built model still features open param-

eters for the pattern generation that are loaded from an ex-

ternal file at the beginning of the simulation. We therefore

dont need to rebuild a new model for each iteration when

doing a parametric study or in the scope of optimizations.

C. Contact Forces

As far as the contact forces are concerned, the robot

builder takes several arguments, including an option to

generate the code related to specific contact forces for the

foot bodies. The latter are identified in the building tree as

parts that belong to the last module of every branch, that is

to say, on modules that have no child connection.

We therefore dont generate fully generic code because

we would have to manage contacts between all the bod-

ies in the environment. This would considerably slow

down the simulations at runtime which is something we

try to avoid. Indeed, as previously brought up, the model

is likely to be run many times (typically several hundred

runs) with various parameters, for the sake of parametric

studies or to tune a gait by means of optimization algo-

rithm.

However, The normal and tangential contact forces FnandFtare then evaluated according to the contact between

specific points of the considered module and the ground

using the following model :

Fn = Knpk +Cdamp.

pdd

dt (12)

Ft=

f Fnvgvlimg

(if||vg||

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where the coupling weight wi j is defined by :

wi j =

1 if j = i 10 if j= i 1

(16)

and with a unique phase shift parameter :

i j = 1.2 i,j (17)

As this specific coupling law is only valid for two sub-

sequent modules, the limit cycle (i.e. after the transient

phase) for joints 0 andi takes the following form :

0(t) = 0.6sin(/2t+ cst) (18)

i(t) = 0.6sin(i1 + 1.2) (19)

The oscillator output valueirelated to joint i is the desired

angular value as shown in figure 6.

Fig. 6. Physical interpretation of the oscillator output iapplied to the caterpillar structure.

A.2 Contact Forces

In the scope of a caterpillar gait, every module is likely

to have contact with the ground. We therefore added spe-

cific contact forces for each block by using two different

contact models : point-plane and sphere-plane contacts.

Indeed, on the contact point of view, the shape of each

block is defined by four points located on the vertices that

define the straight edges of the square area and two spheres

centered to fit the round edges of the block (see figure 7).

Fig. 7. Representation of the contact force elements for

the caterpillar simulation : four point-plane relations

and two sphere-plane (the plane being the ground) re-

lations for both blocks of the module.

A.3 Results

The resulting gait is represented in figure 8 in which one

can observe the behaviour of the transport wave.

Fig. 8. Transport wave gait of the caterpillar configura-

tion. Top picture represents the state of the caterpillar

at time t so that the system has already reached the

limit cycle. The next pictures are the states, respec-

tively, one, two, three and four seconds later. The black

arrow points out the joint 7 detailed in figure 9

Typical information extracted from simulation is the

torques required to achieve the considered gait. For ex-ample, figure 9 shows the time evolution of the torque re-

lated to joint 7 so that the angular position respects the sine

pattern generated by the phase oscillator.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

6 8 10 12 14-2-1.5-1-0.500.511.52

jointangula

rvalue[rad]

jointtorque[Nm]

time [s]

Time history of the position and torque values

Fig. 9. Time history of torque and angular position values

for joint 7. The solid line (left scale) shows the joint

position whereas the dashed line (right scale) repre-

sents the related torque. One can see that the joint an-

gular value follows a 0.6 magnitude sine trajectory asdescribed by the oscillator limit cycle (equation (18)).

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B. 6-legged configuration

For the second example, we consider a simple 6-legged

structure based on eleven modules whose configuration is

presented in figure 10. The corresponding build file is :

11modules-6legged.dat

1 1 4 0 1

1 2 4 1 3

1 3 4 2 1

1 4 4 3 3

0 5 3 0 3

2 6 5 0 1

0 7 3 2 3

2 8 5 2 1

0 9 3 4 3

2 10 5 4 1

This configuration features a total of 28 degrees of free-

dom and requires 33 differential equations to compute the

equations of motion.

Fig. 10. Example of a 6-legged configuration generated

with the framework : five modules are used to form

the spine and six modules are added to model the

limbs.

B.1 Gait Definition

For this example, we only used the framework to gener-

ate the physical model (kinematics, inertia, actuators and

the position controllers) and generated the gait algorithm

manually. The latter is based on an alternate tripod gait as

described in figure 11.

B.2 Contact Forces

The considered gait only requires a few contacts that

always involves the same parts of the robot. Indeed, the

Fig. 11. Alternative tripod gait applied to a 11 modules

configuration. The red areas indicate the limbs thathave contact with the ground while the dashed arrows

show the direction of motion. From top left to bot-

tom right : 1) the robot stands in low position on the

front right (FR), the rear right (RR) and the middle

left (ML) limbs with the 3 other limbs in back-side

position ; 2) the robot is still standing on FR, RR and

ML and goes up while bringing the other tripod to the

front - the latter does not touch the ground thanks to

an additional ankle movement ; 3) the robot reaches

the low position again, still standing on FR, RR and

ML with the front left (FL), the rear left (RL) and themiddle right (MR) limbs in front-side position ; 4) the

robot now stands on the FL-RL-MR tripod and goes

up again while bringing the FR-RR-ML tripod to the

front to repeat the pattern.

interaction between the robot and the ground is always lo-

cated at the lower plane area of the feet (on the bottom of

every lower green blocks of the limb modules). This con-

tact is fully defined by four points located at the vertices ofthe square area (see figure 12) and for which the model of

contact between a point and a plane is applied.

B.3 Results

For this gait, it appears that the highest torque required

is reached in the spine joint related to the block to which

the central limbs modules are connected (see figure 13).

For this joint, the gait algorithm wants the position con-trol loop to hold a constant value during the hole cycle so

that the spine remains straight. The resulting torque and

angular position are shown in figure 14.

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Fig. 12. Contact definition of the foot area : four con-

tact points delimit the square surface of each limb that

is used for ground interaction for the alternate tripod

gait.

Fig. 13. The red arrow points to the central joint of the

spine that corresponds to the most stressed actuator.

V. CONCLUSION & OUTLOOK

Our simulation framework provides an extended multi-body model to study the locomotion of modular robots.

It takes the kinematics and inertia properties into account

but also features the electro-mechanical model of the actu-

ators and the corresponding position control loops so that

the resulting simulation features many aspects of an actual

experiment. A specific CPG module is also included so we

can deal with the particularities of modular robot locomo-

tion.

Even if the framework can directly manage some con-

tact force cases, it generally needs the user to implementit manually. A more general approach is therefore desir-

able but it also requires a higher computational cost at

runtime. Besides, it is generally difficult to fit a contact

-0.1

-0.05

0

0.05

0.1

4 4.5 5 5.5 6 6.5 7 7.5 80

0.2

0.4

0.60.8

1

1.2

jointangularva

lue[rad]

jointtorque

[Nm]

time [s]

Time history of the position and torque values

Fig. 14. Time history of torque and angular position val-

ues for the central spine joint. The solid line (left

scale) shows the joint position whereas the dashed line

(right scale) represents the related torque. The joint isdesired to hold a position of zero during the hole cy-

cle and one can see the effect of the position control

loop : the generated torque compensates for the ef-

forts applied on the joint during the gait.

model to experimental data but in our case we dont focus

on quantitative results about contacts. Our contact model

remains simplistic, but it is sufficient to evaluate the effect

of a given kinematics pattern to provide a stable gait.

Moreover, the CPG module is limited to predefinedCPG models hard-coded in the robot builder, the user

therefore needs to update the module manually so that it

suits to its own model. We plan to generalize the CPG

module so that the user could automatically incorporate a

specific CPG just by providing its equations in a configu-

ration file.

REFERENCES

[1] Satoshi Murata and Haruhisa Kurokawa, Self-reconfigurable

robots, IEEE Robotics & Automation Magazine, pp. 7178,

March 2007.

[2] http://unit.aist.go.jp/is/frrg/dsysd/mtran3/, .

[3] C. Chariot, E Filippi, and O. Verlinden, Multibody simulation

of modular robots : performances and feasibility of simplified 6-

legged and caterpillar gait, inProceedings of the Multibody Dy-

namics 2011. ECCOMAS Thematic Conference, June 2011.

[4] O. Verlinden, G. Kouroussis, and C. Conti, Easydyn: A frame-

work based on free symbolic and numerical tools for teaching

multibody systems, in Proceedings of the Multibody Dynamics

2005. ECCOMAS Thematic Conference, June 2005.

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