DYNAMIC MODELING OF A MOMENT EXCHANGE UNICYCLE

ROBOT

S. Langius

Department of Mechanical Engineering, University of Twente, Enschede, 7500 AE, The Netherlands

R. A. de Callafon

Department of Mechanical and Aerospace Engineering, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0411, U.S.A.

Keywords:

Unicycle, Robotics, Modeling.

Abstract:

In this paper a three dimensional non-linear model has been derived to describe the dynamics of an unstable

moment exchange unicycle robot. The robot uses a driving wheel to provide stabilization in longitudinal direc-

tion, while a second moment exchange wheel with a large inertia is used for stabilization in lateral direction.

For validation purposes, the resulting equations of motion are compared independently against the simulation

results of a ﬁnite element package called SPACAR. The model includes the coupling between the lateral and

longitudinal motion, which makes it possible to control the yaw angle and the model can be used to design a

stabilizing feedback controller.

1 INTRODUCTION

Using basic principles of kinematics and dynamics,

dynamic models of robotic systems can be derived in-

dependently by hand or by an automated computer

program to ensure cross model validity. This pa-

per illustrates this approach on an inherently unstable

Moment Exchange Unicycle Robot (MEUR) depicted

in Figure 1 that requires stabilization in both a lat-

eral and longitudinal direction. Using Newtonian me-

chanics and a ﬁnite element package called SPACAR,

three dimensional non-linear models that incorporates

the coupling between the lateral and longitudinal mo-

tion of the MEUR are derived. The resulting dynamic

model captures the dynamics of a wide variety of mo-

ment exchange unicycle robots and can be used to de-

sign stabilizing feedback control algorithms.

Earlier work on dynamic modeling of robotic uni-

cycles can be distinguished by their mechanism to

achieve lateral stabilization (van Pommeren, 2007).

An inertia wheel moving in the horizontal plane

for stabilization is used in many earlier applications

(Schoonwinkel, 1987; Vos and Flotow, 1990; Naveh

et al., 1999). This approach is comparable to the

twisting torso motion of a real unicyclist (Ohsaki

et al., 2008; Sheng and Yamafuji, 1997). The gyro-

scopic effect of two inertias spinning in opposite di-

rection in the horizontal plane is used in (Zenkov

et al., 1999). A pendulum movingin the vertical plane

perpendicular to that of the driving wheel is used in

(Dao and Liu, 2005) and an inverted pendulum in the

same plane is used in (Nakajima et al., 1997) where

lateral stabilization is further improved by a barrel

shaped wheel. The work of (Au and Xu, 1999) makes

use of the gyroscopic effect for stabilization.

inertia wheel

body

driving wheel

Figure 1: Schematic drawing of a Moment Exchange Uni-

cycle Robot (MEUR).

The three dimensional model of the MEUR de-

rived in this paper includes the coupling between the

lateral and longitudinal motion, which makes it pos-

sible to control the yaw angle similar as in (Majima

et al., 2006). For validation purposes of the dy-

namic model, the simulation results obtained from

the equations of motion are compared independently

216

Langius S. and A. de Callafon R. (2010).

DYNAMIC MODELING OF A MOMENT EXCHANGE UNICYCLE ROBOT.

In Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, pages 216-221

DOI: 10.5220/0002938402160221

 SciTePress

against the simulation results of a ﬁnite element pack-

age called SPACAR (Aarts et al., 2008).

2 EQUATIONS OF MOTION

2.1 Model Structure and Orientation

The conﬁguration and position of the the MEUR in

Figure 1 can be described by six independent coor-

dinates displayed in Table 1. The six coordinates do

not include the pitch angle of the inertia wheel, as we

assume that the inertia wheel is simply replaced by

a moment acting on the body. This is a simpliﬁca-

tion of the problem, discarding the quadratic velocity

terms caused by the (small) pitch velocity of the in-

ertia wheel. Figure 2 shows all the transformations

needed to apply Newtonian mechanics, where Table

2 describes all the coordinate systems involved.

Table 1: Deﬁnition of independent coordinates.

description applies to axis

ψ Yaw angle wheel & body a

γ Roll angle wheel & body b

ϕ Body pitch angle wheel & body c

θ Wheel pitch angle wheel d

X x-position point Q wheel & body a

Y y-position point Q wheel & body a

ϕ θ

A to B B to C C to D D to E

Figure 2: All coordinate system transformations.

Table 2: Deﬁnition of the coordinate systems (CS).

CS description unit vectors

A Inertial CS a

, a

B Contact force CS b

, b

C No-slip constraint CS c

, c

D Body CS d

, d

E Wheel CS e

, e

The inertial coordinate system will be used to ap-

ply Newton’s second law of motion, since this law is

only valid when observing translational accelerations

from this coordinate system. The contact force coor-

dinate system is used to deﬁne the forces acting in the

contact point. The no-slip constraint coordinate sys-

tem is used for the no-slip constraint since the con-

tact point is a stationary point only in this coordinate

system. The body and wheel coordinate systems are

used to apply Euler’s equations, since their rotational

inertias remain constant in these coordinate systems

(Hughes, 1986).

The transformations B = Aθ

, C = Bθ

, D =

Cθ

and E = Cθ

between the different coordinate

systems is captured by the rotation matrices





−S

0 0 1





, θ





1 0 0

0 C

−S

0 S









0 S

0 1 0

−S

0 C





, θ





θ+ϕ

0 S

θ+ϕ

0 1 0

−S

θ+ϕ

0 C

θ+ϕ





where the cos(x) and sin(x) terms in the rotation ma-

trices are shortened to respectively C

and S

. For

dynamic analysis, the derivatives of the rotation ma-

trices are given by

= θ

and

= θ

where





0 −

ψ 0

ψ 0 0

0 0 0









0 0 0

0 0 −

γ 0









0 0

0 0 0

−

ϕ 0 0









0 0

0 0 0

−

α 0 0





where

α =

θ+

ϕ.

2.2 Wheel Equations

The angular velocity vector of the wheel ω can be ex-

pressed as the sum of all individual angular velocities

each deﬁned in their own coordinate system. When

all these angular velocities are transformed to coordi-

nate system C, leads to an expression

= θ

























+ θ







θ+







(1)

and correspondingly,

is a matrix composed of the

elements of vector ω

given by





0 −ω

−ω





(2)

Vector r

is shown in Figure 3 and is the vector

going from point P (the contact point of the wheel) to

point Q (the center of the wheel), pointing in the c

direction and its length is equal to radius of the wheel

QP,C

= {0 0 R}

. Since no slip is assumed, the veloc-

ity at the contact point satisﬁes v

P,C

= 0. The velocity

of the center of the wheel expressed in coordinate sys-

tem C is equal to the velocity of the contact point plus

the relative velocity due to rotation

Q,C

= v

P,C

+ ω

× r

QP,C

DYNAMIC MODELING OF A MOMENT EXCHANGE UNICYCLE ROBOT

217

following (Kolve, 1993) to describe the derivative of

the rotation matrices and replacing the cross product

by a matrix multiplication. The same velocity can be

transformed to coordinate system A, which is the ve-

locity v

Q,A

with respect to the ﬁxed world given by

Q,A

= θ



P,C

QP,C



(3)

Writing out the right hand side of (3) leads to the dif-

ferential equations for the components of v

Q,A

˙x = R



θ+

ϕ+

ψS



γC



(4)

˙y = R



θ+

ϕ+

ψS



−

γC



(5)

˙z = −R

γS

(6)

Solving (4) and (5) for x and y results in the solution

for the ﬁrst 2 independent coordinates. This can only

be done numerically since these equations are non-

integrable. The solution for z = RC

can be obtained

from the analytical solution of the integral of (6).

To ﬁnd an expression for the acceleration of point

Q, ﬁrst (1) will be differentiated to obtain

= −













+ θ

























. . .

···+ θ







θ+







+ θ







θ+







and again,

is a matrix composed of the elements

of vector

given by





0 −

−





Differentiating (3) now leads to an expression for the

acceleration of point Q given by

˙v

Q,A

= θ



+ θ



QP,C

, M

, τ

, M

Figure 3: Forces and moments acting on the wheel.

The gravity force vector F

G,B

does not move with

any of the angles and since its direction is unaffected

by rotation matrix θ

it can be deﬁned in the coordi-

nate system B via F

G,B

= {0 0 − m

because of

its simplicity later on. The contact force vector F

C,B

is chosen to be deﬁned in coordinate system B, so that

it always rotates with the yaw angle ψ but stays in the

same plane as the ground on which the wheel moves.

In this way f

always points in driving direction and

points in perpendicular direction. The third ele-

ment working on the same point is the normal force

creating F

C,B

= { f

}

. Finally, the reac-

tion force vector F

R,D

= { f

}

is chosen to be

deﬁned in coordinate system D, so that it moves with

the body.

Newton’s second law of motion yields

∑

= m

˙v

Q,A

where the left hand side is the sum of previously men-

tioned forces, transformed to the inertial coordinate

system A, given by

∑

= θ



C,B

+ F

G,B

+ θ

R,D



and leads to an expression for the unknown contact

force vector

C,B

= θ

˙v

Q,A

− F

G,B

− θ

R,D

(7)

The reaction moment vector M

R,D

moves with the

body, just like the reaction force vector, and is there-

fore deﬁned in coordinate system D. The second el-

ement of M

R,D

is equal to the torque applied by the

motor between the driving wheel and body leading

to M

R,D

= {M

}

. Since the wheel is rotation

symmetric about the c

-axis, the pitch angles have no

inﬂuence on the inertias and Euler’s equation

∑

M =

h (8)

applies in coordinate system C. The angular momen-

tum is expressed in coordinate system C by

h = J · ω =





· (Cω

) = CJ

where





0 0

0 J

0 0 J





The derivative of h can then be written as

h =

+CJ

= C(

+ J

) (9)

and substituting (9) into (8) with

∑

M = C

∑

yields

∑

+ J

(10)

Finally, the left hand side of (10) can be written as

∑

= θ

R,D

− ˜r

QP,C

C,B

(11)

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

218

where

˜r

QP,C





0 −r

QP,C3

QP,C2

QP,C3

0 −r

QP,C1

−r

QP,C2

QP,C1





As a ﬁnal note it can be observed from Figure 3

that the only force vector creating a moment on point

Q is the contact force vector and the arm for this mo-

ment is in opposite direction of r

QP,C

. For the ﬁnal

moment balance, the reacting force and moment vec-

tor have to be transformed to coordinate system C

ﬁrst.

2.3 Body Equations

Figure 4 shows the forces and moments acting on the

body, where r

RQ,D

= {0 0 d

}

is the vector going

from point Q to R and the components of F

and M

are deﬁned in positive direction on the wheel and thus

work in opposite direction on the body. The gravity

force vector F

GB,A

= {0 0 − m

acts on point R,

the center of gravity for the body and inertia wheel

together. The inertia wheel is simply replaced by a

moment, M

I,D

= {τ

0 0}

, where the assumption is

made that low inertial wheel velocity allows moments

caused by quadratic velocity terms to be discarded.

, τ

, M

Figure 4: Forces and moments acting on the body.

The angular velocity of the body is equal to that

of the wheel

b,C

= θ

























+ θ













however it does not rotate with angle θ. The velocity

of point R is equal to the velocity of point Q plus the

relative velocity due to rotation

R,A

= v

Q,A

b,A

RQ,A

(12)

where

b,A

and r

RQ,A

in (12) can be expressed by the

known equations for

b,C

and r

RQ,D

given by

b,A

= θ

b,C

(13)

RQ,A

= θ

RQ,D

(14)

and substitution of (13) and (14) into (12) yields

R,A

= v

Q,A

+ θ

b,C

RQ,D

(15)

Finally, differentiating the right hand side of (15)

yields

˙v

R,A

= ˙v

Q,A

+ θ

(

b,C

+ θ

b,C

. . .

. . . +θ

b,C

+ θ

b,C



RQ,D

(16)

that can be used in Newton’s second law

∑

= m

˙v

R,A

(17)

for the translational motion of the body. The left hand

side of (17) can be written as

∑

= −θ

R,D

+ F

GB,A

(18)

and substitution of (18) into (17) and rearranging

leads to an expression for the reaction force vector

R,D

= θ



GB,A

− m

˙v

R,A



(19)

Application of Euler’s equation on the rotational

motion of the body will result in

∑

b,D

+ J

b,D

where the left hand side can be written as the sum of

all moments

∑

= −r

RQ,D

× −F

R,D

− M

R,D

+ M

I,D

acting on point R. Rewriting this last expression leads

the reaction moment vector

R,D

=˜r

RQ,D

R,D

+ M

I,D

−

b,D

. . .

. . . − J

b,D

(20)

where

and ω

b,D

in (20) can be written as

= θ





(21)

b,D

= θ

b,C

(22)

3 SPACAR

To validate the model derived in this paper, a sec-

ond independent model is created using SPACAR

(Aarts et al., 2008). SPACAR is based on the non-

linear ﬁnite element theory for multi-degree of free-

dom mechanisms and runs in a Matlab environment

and capable of analyzing the dynamics of planar and

spatial mechanisms and manipulators with ﬂexible

links. The code listed below constructs the model of

the MEUR in this paper. The SPACAR model con-

sists of a spatial rigid beam element, a spatial wheel

element and four spatial hinge elements deﬁned by

the commands

RBEAM

WHEEL

and

HINGE

. The ﬁrst

number after the command is the element number.

The next two or three numbers are the coordinates,

where the numbers 1 to 5 are rotational coordinates

and 6 to 8 are translational coordinates. The last three

numbers represent the initial orientation.

DYNAMIC MODELING OF A MOMENT EXCHANGE UNICYCLE ROBOT

219

HINGE 1 1 2 0.0 0.0 1.0

HINGE 2 2 3 1.0 0.0 0.0

HINGE 3 3 4 0.0 1.0 0.0

WHEEL 4 6 5 7 0.0 1.0 0.0

HINGE 5 4 5 0.0 1.0 0.0

RBEAM 6 6 4 8 0.0 0.0 1.0

X 6 0.0 0.0 0.0495

X 7 0.0 0.0 0.0

X 8 0.00001 0.0 1.0495

FIX 1

DYNE 1 1

DYNE 2 1

DYNE 3 1

DYNE 5 1

KINX 6 1 2

END

HALT

XM 6 0.1

XM 5 0.000021213 0 0 0.00003 0 0.000021213

XM 8 0.2

XM 4 0.8000 0 0 0.002 0 0.4000

GRAVITY 0 0 -9.81

STARTDE 1 1 0 0

STARTDE 2 1 0.6 0

STARTDE 3 1 3.741592653589793 0

STARTDE 5 1 0 0

TIMESTEP 4 400

END

The initial values are deﬁned in the second block

by the command

and assigns the robot dimensions.

In the third block the ﬁrst rotational coordinate is

ﬁxed to the world with

FIX

DYNE

deﬁnes the degrees

of freedom, being the ﬁrst deformation of the ele-

ments 1, 2, 3 and 5, equal to respectively ψ, γ, θ and

φ.

KINX

deﬁnes two coordinates where the no-slip

condition holds.

deﬁnes the point masses in coor-

dinate 6 and 8 and the inertia’s along the x, y and z

axis.

GRAVITY

takes care of the external forces act-

ing on the masses. The third and fourth number of

STARTDE

deﬁnes the initial conditions for ψ, γ, φ and

θ and the initial conditions for their time derivatives.

TIMESTEP

deﬁnes the simulation time followed by the

amount of time steps.

4 SIMULATIONS

To cross validate the model derived in Section 2 and

the model provided by SPACAR in Section 3, time

domain simulations are carried out using the numeri-

cal values listed in Table 3. During the simulation, the

trajectory of the center of the driving wheel is chosen

as a measure for the cross validation.

Table 3: Numerical values of MEUR parameters.

Wheel Body

0.1 kg m

0.2 kg

R 0.0495 m d

1 m

0.00002 kgm

b,xx

0.8 kgm

0.00003 kgm

b,yy

0.002 kgm

0.00002 kgm

b,zz

0.4 kgm

g 9.81 ms

−2

Different non-zero initial conditions and constant

motor torques are used and listed in Table 4. Time

simulations are computed using a non-stiff differen-

tial equation solver (Cooper, 2004) and implemented

via

ode45

in Matlab. The initial values under #1 in

Table 4 and the deﬁnition of the angles in Table 1 are

chosen such that the model starts as a stable mechani-

cal system during the simulation. In addition, a small

initial roll angle γ = 0.6 is chosen to demonstrate the

coupling effects between the lateral and longitudinal

motion of the MEUR.

Table 4: Non-zero initial conditions for the simulations.

set #1 set #2

γ(0) 0.6 0.6 rad

ϕ(0) π+ 0.6 0 rad

0 0.1 Nm

0 0 Nm

It can be observed from the coinciding simulation

results depicted in Figure 5 that the time trajectory

(x(t), y(t)) of the center of the driving wheel starting

in point (0, 0) undergoes a periodic oscillation in the

x direction due to the initial non-zero body pitch an-

gle ϕ(0) = π+ 0.6. Interesting to see is also the small

motion in the y direction of the center of the driving

wheel due to the a initial roll angle γ(0) = 0.6, causing

a small change in yaw angle ψ(t). With simulation re-

sults of the body pitch angle ϕ(0) = 0 and the roll an-

gle γ(0) = 0.6 depicted in Figure 6, the MEUR model

now starts in upward and unstable direction and dur-

ing the simulation we assume the MEUR can fall and

oscillate through the base plane. In addition, a con-

stant torque τ

= 0.1 is applied between the driving

wheel and body. In Figure 6, both models follow the

same trajectories very closely for some time but di-

verge eventually. This is due to the fact that the model

of the mechanical system is unstable at the initial con-

dition, and small numerical errors in either initial con-

ditions or numerical integration leads to exponentially

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

220

increasing differences in the simulation.

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−0.15

−0.1

−0.05

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Newtonian model 3−D

SPACAR model 3−D

Figure 5: Simulation results of the trajectory of the center

of the driving wheel starting in point (0, 0) using the param-

eters of set #1 in Table 4.

−8 −6 −4 −2 0 2

Newtonian model 3−D

SPACAR model 3−D

Figure 6: Simulation results of the trajectory of the center

of the driving wheel starting in point (0, 0) using the param-

eters of set #2 in Table 4.

5 CONCLUSIONS

This paper presents a three dimensional nonlinear dy-

namic model for a Moment Exchange Unicycle Robot

(MEUR). The model is derived using both Newto-

nian mechanics and a non-linear ﬁnite element pack-

age for multi-degree of freedom mechanisms called

SPACAR. The simulation results presented in this pa-

per cross validate the Newtonian and the SPACAR

model, as simulations of the center of the driving

wheel coincide. Differences is simulations attributed

to small errors in the in either initial conditions or nu-

merical integration can only be observed in case of an

unstable initial condition. In addition, the simulation

results demonstrate the coupling between lateral and

longitudinal motion the center of the driving wheel.

Coupling effects are small only in the case of limited

(stabilized) motions of the MEUR.

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