Autonomous Systems Lab
Prof. Roland Siegwart
Studies on Mechatronics
Spherical Rolling Robots
Autumn Term 2013
Supervised by:
Author:
Péter Fankhauser
Jan Carius
Marco Hutter
Declaration of Originality
I hereby declare that the written work I have submitted entitled
Spherical Rolling Robots
is original work which I alone have authored and which is written in my own words.
1
Author(s)
Jan
Carius
Student supervisor(s)
Péter
Fankhauser
Marco
Hutter
Supervising lecturer
Roland
Siegwart
With the signature I declare that I have been informed regarding normal academic
citation rules and that I have read and understood the information on 'Citation eti-
quette' (https://www.ethz.ch/content/dam/ethz/associates/students/studium/
exams/files-en/plagiarism-citationetiquette.pdf).
The citation conventions
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The above written work may be tested electronically for plagiarism.
Place and date
Signature
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originality of the entire piece of written work in its nal form.
Contents
Abstract
v
Symbols
vii
1 Introduction
1
2 Mechanical Principles
3
2.1
Mechanics of the Sphere . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Spinning Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.3
Gyroscopes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3.1
Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3.2
Advantages and Disadvantages
2.4
2.5
Reaction Wheels
. . . . . . . . . . . . . . . . .
8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.4.1
Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2
Advantages and Disadvantages
. . . . . . . . . . . . . . . . .
3.2
Centre of Mass Shift . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.5.2
Advantages and Disadvantages
17
. . . . . . . . . . . . . . . . .
Overview
19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.1.1
Robot by University of Science and Technology Kraków
. . .
19
3.1.2
Sphero 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.1.3
RoBall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.1.4
Spherical Rolling Robot by University of Delaware . . . . . .
22
3.1.5
Spherical Mobile Robot by Indian Institute of Technolgy . . .
22
3.1.6
Robot by Massachusetts Institute of Technology
. . . . . . .
23
3.1.7
Gyrover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.1.8
Gyrobot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.9
Reactobot . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.1.10 Robot by Kobe University . . . . . . . . . . . . . . . . . . . .
28
Conclusions from Existing Designs
28
. . . . . . . . . . . . . . . . . . .
4 New Design Ideas
4.1
8
11
2.5.1
3 Existing Designs
3.1
3
31
Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.1.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.1.2
Fullling the Desired Requirements . . . . . . . . . . . . . . .
32
4.2
Omnidirectional Car-based
. . . . . . . . . . . . . . . . . . . . . . .
4.3
Complicated Pendulum with Flywheel
. . . . . . . . . . . . . . . . .
34
4.4
Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5 Conclusion
5.1
Outlook
33
39
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
39
A Gyroscope Dynamics
43
B SimMechanics Model
45
Bibliography
48
Abstract
This report summarizes our work on the analysis and design of round rolling robots.
This includes spherical as well as single wheel vehicles which are fully contained in
their round shell.
A mechanical analysis and modelling of dierent locomotion
techniques show that there are three main principles to achieve motion with these
type of robots: 1. Using the large angular momentum of gyroscopes for stabilization
and torque amplication, 2. manipulating the internal angular momentum by means
of reaction wheels and 3. taking advantage of a gravitational induced torque when
the centre of gravity lies outside the contact point. We show that a gyroscope is
capable of producing the largest torques, however a centre of mass shift is superior
when it comes to supplying a constant torque over a longer period of time. Reaction
wheels are suitable for storing and releasing momentum. Since all of these abilities
are usually desirable, a combination of two or more of the above mechanisms seems
promising for a highly agile vehicle. We summarize and compare ten robots that
have already been developed. Most of them use more than one propulsion principle,
the most popular way to generate torque is by shifting of an internal mass and this
is almost always realized by a hanging pendulum mass being pitched in the desired
direction. We develop a new idea where a pendulum mass is used dynamically as
an excentric reaction wheel. Such a design is proposed in the last section and could
be the basis for further research on this topic.
v
Symbols
Symbols
Ω
Absolute angular velocity of a rigid body
L
Absolute angular momentum of a rigid body
T
Torque
I, K
eIx
Ca
Inertial- resp. body-xed coordinate system
Unit base vector of the
I
system in the
Coordinate representation of vector
a
in
ABI
Coordinate transformation matrix from
ωIC
Angular velocity of system
g
C
x-direction
C
I
coordinates
to
B
system
relative to system
I
Acceleration due to gravity
Vectors and Matrices are denoted in bold (x) and scalars in light (h).
Acronyms and Abbreviations
ETH
Eidgenössische Technische Hochschule
RW
Reaction Wheel
CG
Centre of Gravity
DoF
Degree of Freedom
vii
Chapter 1
Introduction
Land robots may have dierent concepts of locomotion, for example walking on two
or more legs, driving by means of wheels or as a track vehicle, hopping or rolling
on parts of the robot or its entire surface. Although nature often resorts to legged
systems because of the diculty of rotating joints, most technical applications use
rolling as the primary way of locomotion. Rolling has the advantage of being very
ecient on a variety of dierent terrains. Furthermore dicult step length calculations and stability issues [1] can be avoided.
Of special interest in this report is the branch of rotund robots. In this context,
we dene a spherical robot to be a machine, whose outer surface has the shape
of a sphere, i.e., there are no extremities and the whole interior is contained in a
spherical hull.
Such systems are called encapsulated.
This design appears inter-
esting, since there is a priori no preferred orientation which means the robot can
move omni-directionally and the outer hull can be entirely sealed to provide protection to the interior parts. It is self-evident, that rolling will be the main mode
of locomotion, however this does not exclude any hopping manoeuvres or the like.
Another advantage of a spherical shape is that all positions can be statically stable,
allowing the robot to rest on any point of its surface.
Further investigation and
design considerations will show if this feature can be maintained when the internal
mechanisms come into play. In the past, propulsion by means of a controllable pendulum or hamster-wheel designs have enjoyed great popularity [2]. For exploration
purposes in large and at environments, wind-driven concepts have also been mentioned, however in this work we require much more active steering and acceleration
capabilities.
The motivation of this study is to nd out what manoeuvres round robots are
potentially capable of. The goal of this report is to present a number propulsion
mechanisms available to round robots and to identify concepts for high performance
manoeuvres. We want to explore what devices and mechanism can be employed to
provide extra capabilities such as fast acceleration and jumping over steep obstacles.
In chapter 2 the underlying mechanical properties will be analysed and modelled
and advantages and drawbacks shall be considered.
Subsequently chapter 3 will
give an overview over several previously existing round robots that are documented
in literature. They are ordered according to their locomotion method. In chapter 4,
requirements for a new design will be presented and conclusions from the previous
chapters are drawn to propose new ideas and designs for a round robot.
ideas will be analysed and modelled.
Several
Chapter 5 concludes with an outlook for
further work on this topic and how development may be continued.
1
Chapter 1.
Introduction
2
Chapter 2
Mechanical Principles
This chapter provides the basic mechanical analysis for dierent propulsion strategies.
This is to explore a variety of options how locomotion of a spherical robot
can be achieved and what advantages and disadvantages the concepts have to each
other.
As for the modelling, all linear and angular velocities as well as linear and angular
momenta are given with respect to an inertial (world) reference frame.
Unless
otherwise stated, all bodies in this text shall be considered rigid bodies and friction
is neglected.
The latter especially applies to all joints and couplings which are
considered to be ideal.
Before the specic systems are considered, we will at rst take a look at spherical
and spinning bodies in general.
2.1
Mechanics of the Sphere
A sphere in space (as any unconstrained body) has six degrees of freedom (DoF);
three translational and three rotational. When there is ground contact extra couplings need to be considered. Assuming no slipping and no taking o, the DoFs are
reduced to three: Two for motion (rolling) along the plane and one for the yaw axis
(turning in place). If the contact point is perfect, yaw rotation is only needed for
reorientation in case the device is not omnidirectional.
Usually, we are only interested in planar motion and hence in theory two independent actuators should be sucient to control both relevant DoFs independently.
Das and Mukherjee showed that by using a discontinuous control strategy one can
achieve global stability and exponential convergence of an arbitrary conguration
for a sphere [3].
The control inputs must be chosen in a rotating frame of refer-
ence which result in independent sweep and tuck manoeuvres. Sweep manoeuvres
make the sphere go along a circular arc (xed distance to a point on the plane)
whereas tuck manoeuvres make the sphere move in a radial direction towards or
away from the same point.
Therefore two actuators, if positioned correctly, are
in theory enough to make the rolling sphere fully controllable. In fact, there are
certain designs that only use two motors.
2.2
Spinning Devices
Spinning devices can be grouped into dierent categories. For the remainder of this
text, the following denitions are used:
3
Chapter 2.
Mechanical Principles
4
Figure 2.1: Simple Disc
ˆ
Flywheel
A ywheel is a rotating device used for energy storage. It is
often used to ensure a continuous energy supply for discontinuous motors by
providing a uniform rotational speed through its high inertia. In this text, a
ywheel is simply a spinning disc.
ˆ
Gyroscope
A gyroscope is an attitude control device. The spinning disc
is mounted in gimbals, which allow rotational motion. The number of DoFs
available depends on the specic setup and usually at least one DoF is actuated
and can be tilted. Through a large angular momentum, small input torques
get amplied and lead to large output torques in a perpendicular direction.
Usually the rotor speed is held constant and large torques can be produced
without excessive power consumption.
ˆ
Reaction Wheel
A reaction wheel (RW) is a device that is spun around
a xed axis. By accelerating it, a motor reacts against it thereby producing
a torque on its stator that is mounted to an external structure. By changing
the rotational speed it can therefore change the orientation of the robot and
may release a lot of energy very quickly that has been accumulated over a
longer period of time.
D,
R, thickness b and constant density ρ. Its centre of
introduce a coordinate system B as drawn in gure
In order to simplify further modelling, we will limit ourselves to a spinning disc
i.e., a short cylinder with radius
gravity is denoted with
S.
We
2.1. The mass of this disk is
m = ρV = ρbπR2 .
(2.1)
In the following, we will often use the inertia tensor, which will be derived below.
All integrations are done with respect to the
B -system.
By denition the tensor of
inertias is
 R
B ΘS
=
(y 2 + z 2 )dm
RD −(xy)dm
−(xz)dm
D
D
R
R
−(xy)dm
2
(x
+ z 2 )dm
D
R
−(yz)dm
D
R
D
R

RD −(xz)dm

R D −(yz)dm
2
2
(x + y )dm
D
(2.2)
5
with
2.3.
dm = ρdV .
Gyroscopes
By using polar coordinates to compute these volumetric integrals,
one nally arrives at

B ΘS
which shows that our
br
R2
4

= m 0
0
B -system
0
+
0
b2
12
2
R
4
0
0
+

2
b
12


(2.3)
consists of three principal axes. For very thin disks
this can be further simplied to
B ΘS
2.3
R2
2
=m
R2
· diag(2, 1, 1) .
4
(2.4)
Gyroscopes
2.3.1 Modelling
Gyroscopes are interesting to study in the aspect of dynamics and control because
they inherently exhibit some unintuitive behaviour.
They are highly agile and
powerful, however also dicult to control and handle. In order to understand their
behaviour, we take a general approach:
All bodies are reluctant to changes in their rotational momentum. If a body is at rest
and a torque acts on it, the body will begin to rotate in the direction of the torque.
In this simple case, the relation between torque
T
and rotational acceleration
ϕ̈
is
given by
ϕ̈ · Θ = T
where the constant of proportionality
Θ
(2.5)
is called moment of inertia (along the axis
of the torque or rotation). The moment of inertia plays a similar role as mass plays
in translational motion.
The peculiar properties of a gyroscope arise when the object subjected to a torque
is already spinning. In this case, the principle of angular momentum for rigid bodies
has to be considered to capture all eects. It links the time derivative of the angular
momentum
MO
L̇O
of a body with respect to a stationary point
O to the external torque
O:
acting on the same body also expressed with respect to the same point
L̇O = MO .
(2.6)
In basic terms, this equation states that the angular momentum vector will gradually
move (or increase) in the direction of the applied external torque. This results in
some very unintuitive behaviours if the angular momentum is not parallel to the
applied torque.
The full derivation of the equations of motion of a gyroscope can be found in the
appendix under A (page 43).
The following subsections only present the most
important ndings.
Precession
Precession is an inherent property of gyroscopes when subjected to external torques.
Mathematically it is described by
T =Θ·ω×Ψ.
T is
ω the
where
the torque on the gyroscope,
axis,
spin angular speed and
Ψ
Θ
(2.7)
the moment of inertia about the spin
the wheel precession rate.
Chapter 2.
Mechanical Principles
6
Figure 2.2: Single gimbaled gyroscope
Using a simulation of a spinning disc as developed in Appendix A (see (A.9)) this
behaviour can be clearly seen. For a ywheel spinning at a high angular velocity
around its axis of revolution (x-axis in this case), an input torque in the
rection makes the wheel turn around the
z -axis.
y
di-
The precession rate is inversely
proportional to the spin speed.
This eectively means that the spinning gyroscope does not `retreat' in the direction
one would expect a stationary body to do, but it actually turns around an axis
perpendicular to both the torque and the spin axis.
This could be exploited for
example by using an input torque engendered by gravitational forces to produce an
output torque without reaction on the sphere. An alternative approach to produce
torques without reaction on the hull is to use two counter-rotating ywheels that
can react against each other for tilting. If the gyroscope encounters resistance (for
instance damping forces or rigid joints) in the direction of precession, these resistive
torques make it eventually turn in the direction that one would intuitively think
of. In this case, large torques will be experienced by the suspension. An analysis of
this phenomena is provided in the following section.
Torque Amplication
One reason for using gyroscopes in the propulsion of robots (and also aerial or space
vehicles) is their torque amplication ability. With a relatively small motor input
torque, large torques can be applied to the system. This property will be derived
in the following. Consider gure 2.2 where
ϕ
describes the spinning motion and
α
accounts for the turning motion of the gimbal. There are three coordinate systems
to realize this motion. The
system
G
has its origin at
around the common
S
I
system is inertially xed in space; the intermediate
(the centre of gravity (CG) of the disk) and is rotated
x-Axis of I
and
axis of symmetry of the disk; the
the common
z -Axis
of
G
and
B.
G
Evaluating all equations in the
B
G by the angle α such that eG
z
is parallel to the
system is xed to the disk and rotated around
system is easiest in this case. The total angular
velocity can easily be deducted from the schematic and is given by

GΩ
The
G
= G ωIG + G ωGB

α̇
= 0 .
ϕ̇
(2.8)
system is a principle coordinate system and the moment of inertial tensor
with respect to
S
is constant in time when expressed in
G.
Due to symmetry the
7
2.3.
Gyroscopes
rst two entries are the same and hence

ΘA
 0
G ΘS =
0
0
ΘA
0

0
0  .
ΘB
(2.9)
It is straightforward to calculate the angular momentum of the disk as a function
of
α
and
ϕ
and their derivatives.
Using Euler-Dierentiation one can nd the
derivative of the angular momentum to be


ΘA α̇
 0 ,
G LS = G Θ̄S · G Ω =
ΘB ϕ̇


ΘA α̈
 −ΘB α̇ϕ̇  .
G (L̇S ) =
ΘB ϕ̈
Now the principle of angular momentum can be applied in the
G
(2.10)
system with
M
being the external torque on the disk which reads
 

MxG
ΘA α̈
 −ΘB α̇ϕ̇  =  MyG  .
ΘB ϕ̈
MzG

G (L̇S )
= GM
⇔
It is apparent that a torque in the
speed of the disk.
const , ϕ̈ = 0.
(2.11)
z -direction is only required to change the spinning
ϕ̇ =
In a pure gyroscope this us usually held constant, hence
This simplication leads to

 
MxG
ΘA α̈
 −ΘB α̇ϕ̇  =  MyG  .
0
MzG
(2.12)
z -component
shows that no torque is

We can conclude several things: First, the
required to keep the disk spinning (ideal frictionless case) because the other two
ϕ̈. Second, we look at how the input torque τin = MxG =
I
Mx relates to the output torque τout = MyG (which will be the torque on the whole
structure). Using the initial conditions that the gimbal is at rest and the gyroscope
torques cannot inuence
spins at a constant speed
ϕ̇ 6= 0
α = α̇ = 0
(2.13)
it follows from (2.12) that
τout (t) =
MyG (t)
−ΘB ϕ̇
= −ΘB ϕ̇α̇ =
ΘA
Zt
τin (t̂) dt̂ .
(2.14)
0
(2.14) matches the result by Brown and Peck [4] calculated for control moment
gyroscopes:
Tcmg = −φ̇ · g × hr
where
Tcmg
is the output torque,
gimbal frame),
g
φ̇
(2.15)
the gimbal rate (i.e., the rate of rotation of the
the direction of the gimbal axis and
hr
the rotor momentum.
Eectively a gyroscope `accumulates' and amplies the input torque and changes
its direction. This means an input torque lets the gimbal rotate even if it was only
applied for a limited amount of time. As long as the gimbal rotates (i.e., the integral
6= 0)
a torque acts on the structure. This torque rotates with respect to an inertial
frame of reference because it turns with the gimbal. Therefore the gimbal assembly
does not provide a torque in a constant direction which may potentially make it
dicult to control. The amplication is proportional to the spinning speed of the
ywheel.
The ratio of the moment of inertias is constant and for a thin disc is
ΘB /ΘA ≈ 2
(from (2.4)).
Chapter 2.
Mechanical Principles
8
The kinetic energy only changes insignicantly during torque amplication and the
power required for a manoeuvre is ideally only the useful output power [5]. In these
terms a gyroscope is more ecient than a reaction wheel but it largely depends
on how much power is lost just by maintaining a constant spinning speed of the
gyroscope disc when friction comes into play.
2.3.2 Advantages and Disadvantages
Below, reasons for and against using a gyroscope are listed:
Advantages
ˆ
Torques in all directions are possible when all three DoF of the gyroscope are
actuated. It can therefore turn in a spot.
ˆ
Can produce very high torques due to torque amplication properties.
ˆ
Naturally resistant against changes in attitude, hence the robot gets stabilized
and smooth motion can be expected.
ˆ
If mounted in a wheel instead of a sphere, gravity induced torques can be used
to generate precession.
Disadvantages
ˆ
Energy losses due to fast rotating disc even when not providing any torque.
ˆ
Dicult to control and to nd path planning algorithms; all DoFs become
coupled.
ˆ
Cannot stand on a slope for innite time.
ˆ
Lots of energy is stored in the ywheel which might be dangerous if control
fails.
ˆ
Torques are available only for a limited amount of time (until gyroscope becomes misaligned) and even during this time the torque will most likely not
be constant.
ˆ
Gyroscope is not eective at all times. Bringing it back into desired position
may induce undesired torques and takes time.
2.4
Reaction Wheels
2.4.1 Modelling
A reaction wheel is a relatively simple device and may be used in conjunction with
a motor to create a torque in one specic direction. It has the potential to amass
energy over a longer period of time and release it very quickly as well as providing
a constant torque for some time.
In order to demonstrate the most important properties of a reaction wheel, a simple
R, mass
M , moment of inertia IS ) with a central axis onto which a reaction wheel (radius
r, mass m, moment of inertia ID ) is mounted. A motor exerts a torque T on the
2D model (gure 2.3) shall be considered. It consists of a sphere (radius
reaction wheel and the equal but opposite torque is felt by the sphere hull. A no-slip
constraint with the ground is enforced. Both bodies have their centre of gravity at
their common geometric centre
S
whose horizontal position is captured by
angular orientation of the sphere is denoted with
ψ,
of the disk with
ϕ.
zS .
The
9
2.4.
Reaction Wheels
Figure 2.3: Sphere with central reaction wheel
For both rigid bodies, the principle of angular momentum with respect to
S
and
the principle of linear momentum are set up:
ID ϕ̈ = T ,
(2.16)
IS ψ̈ = −T − Fground · R ,
(2.17)
(m + M )z̈S = Fground .
(2.18)
Enforcing the no-slip condition
⇒
ϕ̇R = żS
ϕ̈R = z̈S
(2.19)
gives us uncoupled linear second order dierential equations as a function of the
driving torque
T:
ϕ̈ =
z̈ =
T
,
ID
IS
R
(2.20)
T
.
+ (m + M )R
(2.21)
There are several things to note from these equations: First, it is clear (and also
very intuitive) that the reaction wheel and the sphere accelerate in opposite angular directions. If both are initially at rest, they begin to spin in opposite directions
which may create unfortunate gyroscopic eects as seen in section 2.3.1 on precession.
It is desirable to keep
ϕ
as small as possible for reasons mentioned above.
From this viewpoint we prefer to have a high moment of inertia
ID
of the disk.
Another important thing to notice is that all second derivatives of the variables are
proportional to the applied torque.
This torque may be achieved by a motor or
alternatively by braking, i.e., trying to synchronize the ywheel rotational speed
with the one of the sphere.
What we are really interested in is the horizontal dynamics (2.21), since this describes how the sphere can move along the plane. For fast acceleration capabilities,
we would therefore like to have a small total mass
inertia of the sphere
IS .
Since for a spherical hull
(m + M ) and a small moment of
IS ∝ R2 , a small R is benecial
for large accelerations.
Overall, it would therefore be favourable to have a large but thin inner disk with
most of its mass at the outer edge and a very light spherical hull with small radius.
Chapter 2.
Mechanical Principles
10
Figure 2.4: Sphere on a slope
The components that are xed to the sphere should be placed as close to the centre
as possible.
Finally, let us consider the case where there acts a counter torque on the sphere, for
example through friction. In this case one has to replace the torque
an eective torque
Teff = Tmotor − Tlosses ,
T
in (2.21) by
i.e., the motor torque is simply reduced
by the counter torque. In the specic example of climbing a slope (angle of incline
γ,
see gure 2.4), there is a gravity induced torque:
Tgravity = (m + M )gR sin γ .
If we let
ξ
(2.22)
denote the position along the slope, we quickly nd the dynamics using
(2.21) to be
Teff − (m + M )gR sin γ
ξ¨ =
.
IS
R + (m + M )R
(2.23)
Instant Braking Mechanism
Inspired by Cubli [6], a cube which can jump on its edge and autonomously balance
on one edge or corner, ywheels have the potential to momentarily create very large
torques. The idea employed in Cubli is that by using brakes to quickly decelerate
fast moving ywheels, one can generate much higher torques than any motor of
acceptable size and mass would be capable of: A motor slowly accelerates a ywheel
up to given speed and keeps it at this angular velocity. When needed, brakes are
applied and almost instantly release the angular momentum stored in the ywheel,
i.e., transfer it to the whole system. This will not be very energy ecient because
of the energy lost in the brakes but may enable our robot to escape from holes
or similar places where large torques are needed.
Such a braking mechanism is
investigated in the following.
The sphere is modelled similarly to the section above and is initially resting on at
ground. The ywheel inside is spinning (ϕ̇init
> 0) and
P of
has an angular momentum
with respect to the xed ground contact point
Linit
= ID · ϕ̇init .
p
(2.24)
If now brakes are applied such that the relative angular speed between disk and
sphere is brought to zero (ϕ̇
= ψ̇ ) very quickly, a lot of angular momentum is trans-
ferred from the disk to the hull. By using the conservation of angular momentum
around
P
and the no-slip condition
ψ̇R = v
where
v
is the horizontal speed, we
11
2.4.
Reaction Wheels
nd:
no − slip
−−−−−−→
!
Linit
p
=
ID · ϕ̇init
=
⇔
=
v
Lfinal
= ψ̇(ID + IS ) + (m + M )Rv ,
p
v
(ID + IS ) + (m + M )Rv ,
R
ID ϕ̇init
.
ID +IS
+ (m + M )R
R
(2.25)
(2.26)
(2.27)
The speed given by (2.27) is the maximum possible speed after a braking manoeuvre.
However no-slip is very unlikely in such a situation and if the braking mechanism is
not fast enough, frictional losses also need to be considered. Again we see that low
total mass is desirable as well as a small moment of inertial of the spherical hull.
Again, a large moment of inertia of the reaction wheel is favourable.
2.4.2 Advantages and Disadvantages
There are several issues arising when using fast spinning ywheels which need to
be taken under consideration.
1. Energy consumption is usually unfavourable [5]. A change in rotational speed
of the reaction wheel brings about a change in its kinetic energy:
∆Ekin =
1
2
2
Θ(ωnew
− ωold
)
2
(2.28)
This energy could potentially be recuperated and thereby improving eciency
but this of course adds to the complexity of the design. It is likely that most
of the energy will be converted into heat when the wheel is decelerated.
2. Gyroscopic eects are unavoidable when the ywheels are spun quickly and
are turned around an axis other than their rotational axis. This makes control
more dicult.
T will cause the
T
. The power P of the actuator is
Θ
given by the product of torque and rotational speed P = ω · T . Since the total
3. Velocity saturation will become a problem: A motor torque
RW to accelerate as seen before:
ω̇ =
output power is limited, the motor cannot provide any more torque when the
ywheel is spinning too fast. Hence a maximum ywheel speed is reached.
Considering these problems, one must design a controller that aims to keep the rate
of rotation low whenever possible.
Below, reasons for and against using reaction wheels are tabulated:
Advantages
ˆ
Energy and momentum can be stored and released.
ˆ
When using braking mechanisms, extremely large torques may be generated
(quick transfer of momentum).
ˆ
Still relatively easy to control for 1 DoF as long as gyroscopic eects can be
ignored.
ˆ
Can potentially turn in place.
ˆ
Reaction times only limited by motor reaction; torque in desired direction is
achieved immediately (no nonminimumphase behaviour).
ˆ
Rather simple for 1 DoF and extendible to several DoFs.
Chapter 2.
Mechanical Principles
12
Disadvantages
ˆ
May give rise to undesired gyroscopic eects when spun quickly.
ˆ
Velocity saturation will likely be a problem, i.e., torque only available for
limited time.
ˆ
Only 1 DoF actuated per RW and hence the mass might become large if
several DoFs are to be controlled by RW.
ˆ
For smooth rolling, the CG must coincide with the geometric centre which
might be dicult to achieve in practice.
ˆ
With CG at the centre, the system is critically stable.
It cannot rest on a
slope for innite time.
ˆ
A lot of kinetic energy may become stored in the ywheels, i.e., not all power
is used for propulsion.
ˆ
2.5
Fast rotating ywheels dissipate energy and give rise to ineciencies.
Centre of Mass Shift
2.5.1 Modelling
Rolling
In a study on rolling motion, Armour and Vincent [7] have identied seven distinct
design principles that have been used so far in rolling robots. Their study shows
that a very popular way to create torques to actuate the outer hull is a shift of the
centre of gravity away from the point of contact.
A frequently used approach employs a pendulum with two DoFs, hanging from a
xed axle inside the spherical shell. It moves by rotating the pendulum around the
axis, creating a torque induced by the weight of the shifted mass which is no longer
acting through the contact point.
Steering is achieved by moving the pendulum
around the axis parallel to the direction of motion. Compared to other approaches
shown, steering and control of such a design appears simpler, however forward thrust
is clearly limited if we only rely on the gravity-induced torque. Another possibility
to shift mass is by using a moving cart inside the sphere (like a hamster wheel).
In the following a two dimensional dynamic model for a pendulum concept is derived. The basic setup and coordinates can be seen in gure 2.5. The spherical hull
(including all components attached to it) has mass
M,
moment of inertia
assumed to be axially symmetric with the CG at its geometric centre
motion is captured by the height over ground
lowest point).
horizontal.
The angle
ψ
h
S.
IS
and is
Its vertical
(i.e., distance between ground and
accounts for rotation and
ξ
is the position along the
In the middle of the sphere a massless pendulum arm of length
mounted (idealization).
The angle between the vertical and this arm is
motor can apply a torque
arm at point
P,
a mass
m
TS
α
r
is
and a
between arm and centre of sphere. At the end of this
with moment of inertia
Ip
(relative to its CG) is xed.
First we start with the kinematic relations:
ψ=
ξ
,
R
ψ̇ =
ξ˙
,
R
ψ̈ =
ξ¨
,
R
(No-Slip)
(2.29)
13
2.5.
Centre of Mass Shift
Figure 2.5: Mechanic model of a pendulum based system
I rOS =
I ṙOS
I r̈OS
−ξ
,
R+h
−ξ˙
=
,
ḣ
−ξ¨
=
,
ḧ
−ξ − r sin α
,
R + h − r cos α
−ξ˙ − rα̇ cos α
=
,
ḣ + rα̇ sin α
−ξ¨ − r(α̈ cos α − α̇2 sin α)
=
.
ḧ + r(α̈ sin α + α̇2 cos α)
I rOP =
I ṙOP
I r̈OP
Only if the sphere has ground contact (h
= 0),
(2.30)
the no-slip condition (2.29) applies
and normal and tangential forces from the ground adjust themselves such that the
kinematic relations are fullled. A rolling friction moment
TR
is also considered in
this case.
Now principles of linear and angular momentum can be applied to the sphere,
pendulum arm and pendulum mass.
Using the fact that the pendulum arm has
no mass, it must simply full static equilibrium. Therefore the remaining relevant
equations are
M ξ¨
M ḧ
IS ψ̈
mẍP
mÿP
Ip α̈
=
=
=
=
=
=
Px + FR
−M g − Py + N
TS − TR − FR R
.
Px
−mg + Py
TS + Px r cos α − Py r sin α
(2.31)
By eliminating the constraining forces one arrives at the equations of motion: For
N need to be eliminated, leaving two
equations of motion (2 DoF). Furthermore h = ḣ = ḧ = 0 in this case. Without
ground contact N = FR = TR = 0 and only Px and Py need to be eliminated,
the case of ground contact,
Px , FR , Py
and
leaving four equations of motion (4 DoF).
To allow for losing ground contact, the normal force
N
has to constantly be moni-
tored. Once it gets negative (i.e., the ground would have to hold the sphere back)
one has to switch into `ight' mode. The problem with landing on the ground again
is that elasticities have to be considered to capture the behaviour correctly. A solution to this problem is deferred for now and will be presented in section 4.4 on
page 35.
While remaining in ground contact, these second order dierential equations can be
decoupled and can be solved numerically.
Chapter 2.
Mechanical Principles
14
Figure 2.6: Nonminimumphase behaviour of the sphere's horizontal velocity
Nonminimumphase Behaviour
A further interesting behaviour catches the eye if the graph of the simulation is
considered immediately after a torque is applied. In gure 2.6 the horizontal speed
ξ˙
has been plotted over time.
With initial rest conditions and a step signal on
the torque, the sphere at rst moves in a dierent direction than intended.
system with such behaviour is called
nonminimumphase
A
which puts limits on its
performance (especially reaction times). In this case it is a result of the inertia of
the pendulum mass: At rst, this mass has to be accelerated towards the desired
movement direction. Due to conservation of momentum, the sphere initially moves
in the opposite direction before the gravity-induced torque dominates.
Jumping
A further very interesting ability of the robot is to jump by moving the centre of
mass around. This is relevant for our application since the robot can remain fully in
shape (i.e., the spherical hull does not need to deform) and no interfaces are needed.
Furthermore it will look rather enthralling to the outside observer if an encapsulated
system suddenly manages to loose ground contact and jump over small barriers.
From a mechanical perspective, the only external force acting on the sphere that
would allow it to jump is the normal reaction force from the ground. Therefore an
inner mechanism needs to make sure that this force is larger than the weight for some
time such that upward momentum can be accumulated internally. Alternatively one
could keep the centre of gravity more or less xed in space but the spherical hull
is pulled upwards. Both can be combined and as a rst idea and treatment of the
topic we will consider the following strategy:
1. Accelerate some mass upwards on the inside such that the centre of gravity
moves up. The force from the ground will exceed total weight during this time
and upward momentum is accumulated in the moving mass. Then this mass
is stopped relative to the sphere and transfers its momentum to the whole
system which will lift o the ground if stopped fast enough.
2. Once in the air, the spherical hull reacts against the inner moving mass and
pushes itself further up relative to the centre of mass. This does not change
the trajectories of the total CG but the distance between ground and sphere
will be increased. The theoretical maximum height that can be achieved with
this step only is twice the radius of the sphere, that is, if the hull had no mass
at all and was moved instantaneously.
15
2.5.
Centre of Mass Shift
Figure 2.7: Jumping dynamics
Stage 1
As drawn in gure 2.7, the spherical hull (radius
cluding the mobile mass
is
ys ,
m
the one of the mass is
R)
has mass
M,
ex-
inside of it. The vertical position of the sphere's centre
ym .
To begin with, the sphere rests on the ground and
the mobile mass is at the lowest position possible:
ys (0) = R
ym (0) = ym0
ẏs (0) = 0
ẏm (0) = 0
.
(Initial Conditions)
The inner mass is now accelerated upwards by a force
F1 > mg
(2.32)
that is provided by
an internal mechanism. Due to the initial rest condition the trajectory is
ym (t) =
F1
−g
m
t2
+ ym0 .
2
(2.33)
∆t(1) the mass will reach its highest position ym1 in the sphere. For
a := (ym1 − ym0 )/2 shall be the distance that the mobile mass can move a
At a time
clarity,
way from the sphere's centre in both directions. The time is given by
s
(1)
∆t
=
2(ym1 − ym0 )
=
F1
m −g
s
4a
.
−g
F1
m
(2.34)
At this point in time the mobile mass is stopped relative to the spherical hull and
its momentum is shared with the sphere. In an idealized model where the braking
happens instantaneously, the momentum remains conserved:
−
!
+
ẏm (∆t(1 ) ) · m = ẏs (∆t(1 ) ) · (M + m)
(2.35)
Using this initial condition for the trajectory of the sphere, one can nally nd the
maximum height
(1)
hmax
(distance between ground and lowest point of sphere) that
can be achieved by the rst stage of the mechanism.
h(1)
max
Stage 2
1
=
g
m
M +m
2 F1
− g 2a
m
(2.36)
Once the mass is up, it can be pushed down again so that the spherical
hull gains even more height. Again this is trough an internal mechanism and this
time
F2 > M g
(pointing in the opposite direction as
F
drawn in gure 2.7). Similar
Chapter 2.
Mechanical Principles
16
to the calculations above, we will consider the internal mass
by the force
F2
and its weight
mg .
m
being pushed down
The initial condition to this process is that the
internal mass has no velocity relative to the sphere. Integrating assuming that the
sphere remains in the air yields the time
∆t(2)
that it takes to bring the internal
mass to its lowest position again.
s
∆t
On the sphere, the same force
position at the time
∆t(2)
(2)
F2
=
F2
m
4a
+ FM2
(2.37)
acts upwards and its weight downwards.
Its
can be computed to
h(2)
max =
F2
−g
M
F2
m
2a
.
+ FM2
(2.38)
Using (2.36) and (2.38) together give the maximum height that can be achieved by
such a combined jumping mechanism:
2
2a
(2)
(1)
F1
F2
1
m
=
+
h
htot
=
h
max
max
max
g M +m
m − g 2a + M − g F2 + F2
m M
2
F
F
1
Mm
1
2
=
2a g1 Mm
.
+m
m − g + F2 M − g M +m
(2.39)
Jumping with a pendulum
The above mechanism assumes that a mass can be shifted along a straight line within
the sphere. In this section we will explore what changes if a mass on a pendulum
arm is used. The principle of exchanging momentum between an internal mass and
the outer hull remains the same. The dierence with a pendulum is simply that a
rotational motion can be steady (no large breaking or accelerating forces/torques
required) and still exchange a large amount of momentum because of the non-linear
path.
Since the dynamics become rather complicated, we will only consider a sphere resting on the ground and take a look at the normal force that acts between sphere
and ground. Using (2.31) with
h = ḣ = ḧ = 0
and solving for the normal force
N
(acting upwards on the sphere, see gure 2.5):
N = (M + m)g + mr(α̈ sin α + α̇2 cos α)
| {z } |
{z
}
static
(2.40)
dynamic
The static part is only due to the weight of the whole system while the dynamic
part has its origins in the swinging motion of the pendulum. If the dynamic part
is negative and outweighs the static part, the normal force becomes negative. This
is equivalent to saying the sphere would loose ground contact.
For simplicity we
assume a clockwise rotation of the pendulum (α̇
The dynamic part
> 0, gure 2.5).
can be further divided into two distinct eects:
mrα̇2 cos α
centripetal force
causes upward force when pendulum is
in the upper hemisphere due to the circular path.
mrα̈ sin α
inertial force
causes upward force when accelerating
in the right hemisphere and when decelerating in the left hemisphere
A possible jumping strategy could therefore be as follows:
17
2.5.
Centre of Mass Shift
1. While on the ground, the normal force should be as large as possible to gain
upward momentum. Therefore we would like the pendulum to be in the lower
hemisphere and decelerating on the right, accelerating on the left.
2. While in thie air, the `normal force' should be made as negative as possible.
Therefore we would like the pendulum to be in the upper hemisphere and
accelerate on the right, decelerate on the left.
Obviously the speed of rotation
α̇ has to be chosen in a way such that the pendulum
is in the correct position when in air and on the ground and hence allow for a
bouncing motion.
What will very likely cause diculty is the fact that without
ground contact and without no-slip condition, the pendulum reacts against the
sphere which begins to rotate, too. Determining the absolute angle of the pendulum
arm with the vertical will be challenging. Additionally, in this case it is actually
useful to have a spherical hull with a large moment of inertia.
This contradicts
earlier objectives and a design compromise needs to be established.
2.5.2 Advantages and Disadvantages
To sum this section up, reasons for and against using an actuated pendulum are
listed:
Advantages
ˆ
Easy control as long as the axis remains xed.
ˆ
Constant torque can be achieved until motor power output reached (no misalignment).
ˆ
A 2 DoF pendulum provides one mechanism for both accelerating and steering.
ˆ
Potentially little energy losses (none if at rest) because nearly all energy goes
into propulsion.
ˆ
Robot can recover from any position; robust against disturbances .
ˆ
System inherently statically stable since CG below centre of sphere. It can
rest in place without energy consumption if the ground is at.
ˆ
Can stay standing on a slope for innite time (although feedback loop required
for yaw stability).
ˆ
Extra eects (torques and forces) possible if actuated dynamically (swinging).
Disadvantages
ˆ
Usually not omnidirectional and cannot turn in place unless yaw dynamics
are used.
ˆ
Maximum torque (in steady-state operation) limited by the mass and the
distance over which it can be shifted, i.e., how far the CG can be moved away
from the contact point.
ˆ
Maximum speed limited by motor power.
ˆ
The closer the pendulum arm is to the horizontal, the more the roll stability
is lost.
ˆ
Relatively large space is required for 2 DoF pendulum that allows for narrow
turns.
Chapter 2.
ˆ
Mechanical Principles
Reaction times might be slow (e.g., from max forward to max backward
torque) because mass has to be moved.
ˆ
18
Sytem is not minimumphase (reacts in wrong direction at rst).
Chapter 3
Existing Designs
3.1
Overview
The following section aims to provide an overview about prior art of rolling robots.
The list is obviously not exhaustive but all relevant concepts are represented by at
least one robot.
In table 3.1 one can easily see that many dierent designs exist and dierent combinations of propulsion mechanisms have been tried before. The abbreviation N/A
means that this propulsion mechanism has not been used. The robots have been
ordered in terms of the mechanism used. Robots that use a combination are considered afterwards.
Alike to all robots is that they can be full enclosed by their outer shell which protects
the components on the inside and reduces the risk of getting stuck since there are
no extremities to the shape. Equipping the outer shell with special material that
allow the robot to roll over rough surfaces or even swim is possible in theory.
Performance overview
Table 3.2 provides an overview over the robots presented. Where available, data
from literature has been taken, however not all robots had quantitative performance
indicators available.
3.1.1 Single Wheel Robot by University of Science and Technology Kraków
The rst design is a single wheel robot that uses mass shifting for propulsion and
stability in a very interesting way. It has recently been developed by Buratowski
and Cie±lak [8].
Their aim was to build a robot that can remain upright for an
unlimited time both in motion and when standing.
Figure 3.1: Robot developed in Kraków
19
Chapter 3.
Existing Designs
20
Figure 3.2: Sphero 2.0
As it can be seen in gure 3.1, inside the big wheel, two independent mechanisms
ensure pitch actuation and roll stability: For forward and backward propulsion, a
cart can move inside the wheel and shift the centre of mass as required. The balancing mechanism rotates an arc-shaped mass around the roll axis and can therefore
be used for steering, too. What is interesting here is that the sideways mass shifting actually occurs at a point above the centre of gravity, similar to an inverted
pendulum. This might introduce further agility, however also poses a more dicult
problem in terms of roll stability. Unfortunately little further comments have been
made on this aspect.
In their paper, an important observation is made: The model of the single-wheel
robot in the lateral plane is equivalent to a double inverted pendulum whose rst
joint is underactuated
1 . Hence control algorithms from the inverted pendulum
problem might be useful in this case, too.
The authors note that controller that
stabilizes the system has been found and proven in practice.
3.1.2 Sphero 2.0
As the name suggests, Sphero is a robot of spherical shape of the size of a baseball
and can be wirelessly controlled using smartphone applications. It is a commercial
2 and hence openly available information and data is dicult to
Product by Orbotix
acquire. However from several reviews [9, 10, 11] on this product and online videos,
some properties can be inferred.
Although contained in a very durable polycarbonate spherical shell, it is not omnidirectional. When sending a drive command into a certain direction, it often has
to reorient rst before it can drive o. This reorientation is however relatively fast
3
and comes with nearly no lateral displacement . Sphero also appears to be able to
4
jump . Unfortunately no information could be found on how this is achieved.
Generally Sphero is very agile and it can achieve speeds up to 2 m/s which is already
a quick walking pace. The battery lasts about one hour and can be recharged using
an inductive charging unit which allows the hull to remain without any gateways.
Sphero is also capable of swimming on water, using its rotation to propel itself
1 Buratowski and Cie±lak in [8] p. 104
2 Orbotix, Sphero | Robotic Gaming
gosphero.com/
System for iOS and Android, 2013. See http://www.
3 This can for instance be seen in the video Show and Tell: Rollin' With Sphero 2.0 by Tested
on https://www.youtube.com/watch?v=078XSj1zKCs
4 Can be viewed in Sphero 2.0 Jump Test by Go Sphero on https://www.youtube.com/watch?
v=GQhRN5tTghA
21
3.1.
Overview
Figure 3.3: Interior of RoBall
in a given direction. A nubby cover around the sphere greatly enhances swimming
speed. According to the development team, the greatest diculties were fabricating
a perfectly round shell which is at the same time lightweight and resistant. They
seem to have overcome this issue very well, the shell is even strong enough to
withstand an adult person standing on it. Finding durable motors has also been
challenging as well as problems with electric discharge that inevitably happens for
example when Sphero moves over carpet oors.
Internally, Sphero uses the concept of a moving cart with a sprung central member.
This can be seen in gure
5 3.2. All relevant components are packed together and
contribute to the mass of the robot.
This includes for instance the two motors,
battery, computation and communication unit and sensors.
axis accelerometer and a gyroscope to sense movement.
Sphero has a three
Two small wheels with
rubber tires roll along the inside of the shell and can be controlled independently.
The normal contact force is provided by an arm that is extending in the opposite
direction and slides (slip bearing) against the inner shell. This way, the wheels never
loose contact even in positions where most mass is above then geometric centre of
the ball.
3.1.3 RoBall
With a diameter of 27 cm, Roball is a relatively large spherical robot. Kabaªa and
Wnuk [12] have employed a two DoF pendulum to shift the centre of mass of their
robot and hence use a gravity induced driving torque. The motor axle is xed inside
the sphere and in gure 3.3 one can see the interior components.
Roball is not omnidirectional as its forward or backward motion must be perpendicular to this xed axis in the hull. This means curves and reorientation around
the yaw axis can only be performed while in motion. Roball is able to move fully
autonomously with all sensing and control being done onboard.
Unfortunately little further information on the mechanical aspect has been found.
5 Image
by Christopher Harting, taken from [10]
Chapter 3.
Existing Designs
22
3.1.4 Spherical Rolling Robot by University of Delaware
At the University of Delaware, Bhattacharya and Agrawal [13] proposed a spherical
rolling robot that uses reaction wheels only. This construction has a spherical aluminium shell, assembled from two halves. Each contains a receiver, motor assembly,
rotor and battery. There are two (rigidly connected) rotors on an axis through the
geometric centre and one rotor perpendicular to this axis.
Special attention was
paid to ensure that each component is directly opposite to its pendant in the other
hemisphere, such that the centre of gravity is exactly above the contact point for
all orientations and the robot does not tip over. The position of the robot is determined by an overhead camera which means not all sensing capability is contained in
the robot. The computation is also done externally and signals are sent wirelessly
to the motor controls.
For modelling purposes, the system was conceived to be an assembly of three rigid
bodies, namely the shell and everything xed to it, the double-rotor and the single
rotor. Through imposing a no-slip-constraint and formulating angular momentum
conservation around the ground contact point, system equations were derived. The
inputs for control were the speeds of the rotors and an open-loop control design
was implemented, hence no feedback from the motor speeds. However there was,
of course, information available about the motion of the sphere by the overhead
camera. Minimum time and minimum energy trajectories have been calculated and
successfully implemented experimentally, both showing reasonable accuracy. The
authors noted an interesting eect with the minimum time specication: One motor
was used to provide forward thrust at full power, whilst the other only undertook
the steering.
For simple inputs simulation agreed well with the actual trajectory however for more
complex inputs experiment and numerical simulation results deviated signicantly.
The authors suggest the reasons for this to be an unequal mass distribution around
the centre of the sphere and the nature of the open-loop control.
Clearly, if the robot was to be employed in unknown terrain and outside the laboratory there must be an attitude and position estimation system incorporated since
no overhead camera will be available.
Although not explicitly stated, it is very
likely that the robot cannot follow a straight line forever due to velocity saturation
of the driving wheel.
3.1.5 Spherical Mobile Robot by Indian Institute of Technolgy
Another interesting concept was put forward by Joshi, Banavar and Hippalgaonkar
[14]. Their robot has a truly spherical shape is assembled from two halves. On the
inside, each hemispheres has a motor assembly attached to it (including a rotor) and
a battery pack with some dead weight. This robot is actuated using two independent
internal reaction wheels. The additional weights are necessary to ensure that the
centre of gravity is exactly at the geometric centre, such that the sphere is statically
stable in all orientations and cannot topple over.
The motors are
80 W
brushless DC, however no indication of performance of this
robot is given. The total mass of the assembly is
3.4 kg
which appears relatively
heavy when compared to the previously presented models. This might be due to
the dead weight, which should ideally be avoided at all since additional inertia is
never favourable for an agile robot.
The two rotors are placed in a way, that their axes of rotation are perpendicular to
each other. Locomotion of this robot is solely based on the principle of conservation
of angular momentum. A model of the robot was created by considering generalized
coordinates describing the contact point on the rolling plane and the orientation of
23
3.1.
Overview
Figure 3.4: Robot by the Indian Institute of Technology
the sphere. Enforcing a no slip constraint, there remain three DoFs.
Control inputs of this robot are the speeds of the two rotors. The authors showed
in their paper [14], that this system is fully controllable and can be taken from any
given position to any other desired point on the plane with a desired orientation.
They further noted that all existing path-planning algorithms cannot be applied
and latter must rst be developed.
An advantage of this design is its simplicity. It is remarkable, that using only two
perpendicular rotors, one obtains a fully controllable system. However it could be
the case that this type of locomotion is not very ecient, neither in time nor energy
for a given start and end point and it is questionable whether large torques can
be generated to make such a device overcome obstacles.
Furthermore if one was
to realize a similar design, a solution must be found where no additional mass is
required, since this will reduce the performance of the robot signicantly. At least,
one could replace that mass by additional batteries to lengthen the runtime of this
device.
3.1.6 Robot by Massachusetts Institute of Technology
In his Bachelor Thesis [15], Schroll aimed for designing a spherical vehicle that
overcomes the typical limitations of mass shifting approaches and came up with a
mechanism to manipulate angular velocity internally. Schroll identied the space
constraint of mass shifts and hence the limited gravity induced torque as a major
drawback of this propulsion method. Instead it was proposed to generate changes
in angular momentum internally and especially separate generation and usage of
angular momentum chronologically. After a few dierent approaches using a single
reaction wheel or a single gyroscope, Scholl proposed a design that uses a scissored of
gyroscopes together with a pendulum mass shift mechanism. This has the advantage
that when the gyroscopes are parallel and spun in opposite directions, they have
no net angular momentum and do not introduce undesired gyroscopic eects.
A
net torque can be generated by tilting the gyroscopes towards each other, so that
they both generate a precession torque in the same direction.
are schematically drawn in gure 3.5.
The gyroscopes
This should be relatively energy ecient
because the kinetic energy of the gyroscopes remains constant and only the angular
momentum is used.
There are however several problems associated with such a concept:
Chapter 3.
Existing Designs
24
Figure 3.5: Robot by MIT
Figure 3.6: Gyrover
Once the ywheels are not parallel anymore, their angular momenta do not cancel
out and inuence the dynamic behaviour. If they are misaligned, they cannot supply
any more torque and have to be brought back to their initial orientations. This may
cause undesired torques of the ywheels keep spinning or cause large energy losses
if ywheels are stopped.
Furthermore large stresses on the structure are to be
expected which needs to be built accordingly.
3.1.7 Gyrover
Gyrover is a single wheel rolling robot that uses a gyroscope to stabilize its motion
and orientation. It was developed by Brown and Xu [16, 17, 18] at Carnegie Mellon
University. This robot is not truly spherical but has an ellipsoid shape and Brown
and Xu opine that this shape is in fact more suitable since it exhibits natural
steering behaviour. Gyroscopic eects are used for steering and stability, but not
for propulsion. Figure 3.6 shows a picture and the internal components of Gyrover.
At the heart of Gyrover, there is a ywheel spinning with its axis parallel to the
roll axis in forward direction.
This leads to self-correcting behaviour due to the
eects of precession: Any deviation from the upright position (i.e., a lean to one
25
3.1.
Overview
side) will generate a torque on the gyroscope due to gravitational forces, making
the gyroscope precess around the yaw axis. This is a steering action, engendering
a curved path to be followed. Due to the centripetal force provided by the ground
through friction, there is a righting moment that will bring Gyrover back towards
its upright position. Since the angular momentum of the ywheel is not coupled
with the rotational speed of the outer surface, this eect is also present when the
robot is at rest. However due to the lack of a centripetal force (since there is no
motion), the robot will continue to precess around the yaw axis (i.e., turn in place)
and needs an inner mechanism such as shifting the CG if it was to achieve lateral
stability again without starting to move.
Active steering is achieved by a mechanism that allows the axis of the internal
ywheel to be tilted manually around the roll axis. If for example an actuator tries
to tilt the ywheel to the left, the outer hull leans entirely to the same side whilst
the ywheel remains roughly aligned, because it serves as an internal reference for
attitude due to its high mass and angular momentum. As the CG of Gyrover is now
left of the contact point, a torque by gravity trying to make it lean even more to
the left is experienced. This however is prevented by the gyroscope which starts to
precess leading to the correcting eect as described above. Moving the gyroscope
back into its nominal position nishes the manoeuvre.
Thrust is produced by means of a hanging pendulum suspended at a central axis,
xed to the outer hull.
A separate drive motor tries to shift the pendulum into
the desired direction (forward or backward), causing a gravity induced torque that
makes the device roll.
There are three working prototypes of Gyrover. The rst has a diameter of 29 cm
and a mass of 2.0 kg.
It shows good high-speed performance (10 km/h) even on
rough terrain, moved through a gravel pile and can also stand in place. The thrust
produced by the pendulum is approximately limited to 25 percent of the vehicle
weight which means it can climb slopes with a 25 % incline without dynamic eects.
Although the robot is not spherical and can therefore fall on its side, it has the ability
to right itself from the rest (side) position with the help of the tiltable ywheel.
Brown and Xu found out, that as forward speed increases, the tilting torque of the
gyroscope required for steering increases strongly, because the angular momentum
of both the wheel and the ywheel exhibit self-correcting properties. The authors
have identied a design trade-o between a high angular momentum suitable for
static stability and at low speeds and a lower angular momentum to allow steering at higher speeds. Therefore a ywheel with adjustable moment of inertia and
rotational speed would be advisable. The high energy consumption by the motor
driving the ywheel has been addressed in the second prototype by placing it in an
evacuated shell. This has cut down power consumption by 80%.
3.1.8 Gyrobot
Gyrobot [19, 20] is very similar to Gyrover (see 3.1.7) except Al Mamun made some
modications to the actuation mechanism.
Eectively the pendulum mass-shift
mechanism was combined with the gyroscope.
Gyrobot is a single wheel vehicle where a gimbaled gyroscope disc is hung from a
central axis.
This disc is made of carbon steel and rotates at 10 000 rpm.
a large angular momentum stabilizes the wheel in an upright position.
Such
Shifting
the disk forward moves the centre of gravity in front of the contact point and the
device starts to move.
When a tilting torque in the roll direction is applied to
the gyroscope, it begins to precess. Both the eect on the roll movement and the
precession are exploited for steering.
Figure 3.7 shows how the interior of Gyrobot looks like.
The ywheel is smaller
than the one from Gyrover, however the CG is also likely to be lower.
Chapter 3.
Existing Designs
26
Figure 3.7: Gyrobot
3.1.9 Reactobot
A concept that uses both a reaction wheel and mass shifting by means of a pendulum
is Reacobot, designed by Biswas, Bhartendu, Kadam and Seth [21, 22].
It is a
single wheel robot that has an actuated pendulum arm suspended from the axis of
revolution of the wheel. It is fully contained in a hollow disc. At the end of this
arm, there is an actuated reaction wheel mounted with its axis parallel to the roll
axis of the robot. Figure 3.8 shows all parts schematically.
For forward and backward motion, the pendulum arm is pitched forwards or backwards respectively, and the heavy reaction wheel at its end causes the centre of
gravity of the whole system to shift. Due to geometry and mass distribution, the
robot is inherently stable along the pitch axis but unstable in roll. Therefore the
reaction wheel is responsible for roll stability as well as steering.
Curves can be
accomplished by tilting the robot to one side while in motion.
Kadam and Seth [22] designed an LQR controller for the system that ensures roll
stability and brings the reaction wheel to rest at the same time. Using this controller, given trajectories can be followed. It is crucial that the reaction wheel is at
low speeds most of the time for three main reasons: 1. Avoiding gyroscopic eects,
2.
energy consumption and 3.
roll stability must be achieved before the power
output limit is reached. The latter becomes especially important because the robot
cannot recover once it has fallen onto its side. This means it has to be set up to
start rolling and any external disturbance that might cause the robot to tip over
will bring about a system failure.
One would expect that the faster the wheel is moving, the more stable the roll axis
is anyway, because the wheel exhibits gyroscopic properties, too. A problem with
this design is however, that while accelerating, the RW axis is not parallel to the
ground (pendulum lifted up) and therefore eects the yaw motion as well. Hence
several DoFs become coupled and pose a diculty in controlling.
On the positive side, this setup manages to follow given trajectories using only two
actuators.
27
3.1.
Figure 3.8: Reactobot
Figure 3.9: Robot by Kobe University
Overview
Chapter 3.
Existing Designs
28
3.1.10 Robot by Kobe University
A spherical rolling robot that uses one ywheel only for propulsion in a very elegant
way has been put forward by Urakubo, Osawa, Tamaki et al. from Kobe University
[23]. The centre of gravity of the entire system coincides with the geometric centre.
The system is composed of three main bodies: a gyroscope, a gyroscope case (inner
hull) and the outer shell. The most interesting part of this concept is that it uses
two concentric spherical hulls. Inside the inner hull, a gyroscope spinning at a large
angular velocity is rigidly mounted and hence this inner hull's attitude is stabilized
by the gyroscope. The connection between the hulls is realized through four rubber
rollers, two of which are actuated. These rollers can be driven by two independent
motors in directions perpendicular to each other and the gyroscope axis. This allows the outer hull to be rotated relative to the inner hull. With this mechanism,
the angular momentum from the gyroscope can eectively be transferred to the
outer hull and therefore provides a torque for movement of the entire system. An
important condition is that the rollers do not slip sideways which poses high requirements on the mechanical design. With another motor that directly drives the
gyroscope and reacts against the inner hull, there are three independent torques
that can be applied to the system. According to the considerations under 2.1 (page
3), all available DoFs should be controllable. The system equations are derived by
applying an overall conservation law of angular momentum over all three bodies
(spinning disc and both hulls). Through controlling all three input torques rolling
has been achieved.
One might expect nutation to be a problem, however this was not detected experimentally.
The authors suggested that this might be strongly damped and does
therefore not show. In experiment, the robot with a suitable controller was able to
track a given velocity prole with a small error.
A problem that was already identied by the developers was that rolling friction
will change the total angular momentum and should therefore not be neglected in
the model, which they did in their rst approach.
3.2
Conclusions from Existing Designs
From the above designs one can see that nearly all propulsion mechanisms can be
combined in dierent forms and there are still many more possibilities to try out.
It also became clear that two motors are in fact enough to adequately control a
spherical or single wheel robot. The most used concept was pendulum mass shift
and interestingly this goes well together with reaction wheels and gyroscopes. A
possible reason for the frequent use is the relative simple control design due to oneaxis actuation only and the ability that it can provide a constant torque without
issues of misalignment or the like. Very often is has been noted that these systems
are relatively dicult to control and previous algorithms for path planning usually
do not apply.
Therefore it is very likely when proposing a new design that one
cannot resort to existing controller software but must develop the latter specically
for this robot.
In almost all cases, the design goals are to achieve maximum torque for acceleration
whilst keeping mass of the vehicle within reasonable limits and permit good steering
capability [2].
Furthermore the robot should be able to overcome obstacles and
ideally also climb stairs.
Concerning size, one must consider that enough space
needs to be available for ywheels, other propulsion mechanisms including motors
and all electronics, for example micro-controllers and battery packs. Furthermore,
the larger the radius, the easier it will be to overcome obstacles and the lesser
chance to get stuck in a hole.
However a large hull will have a relatively high
29
3.2.
Conclusions from Existing Designs
moment of inertia and prohibits fast acceleration, whilst at the same time being
susceptible for external inuences, namely wind. Furthermore we might want the
robot to explore rather small cavities or rest on small platforms, therefore a good
compromise between the above arguments must be found.
Chapter 3.
Existing Designs
30
Table 3.1: Propulsion mechanisms of existing robots
Robot
Shifting Mass
Kraków
Mass moves like cart for
pitch
mass
torques,
can
shift
Reaction Wheel(s)
Gyroscope(s)
N/A
N/A
N/A
N/A
N/A
N/A
Two RW mounted perpen-
N/A
another
laterally
for roll stabilization and
steering (roll)
Sphero
Mass can move like a cart
inside
sphere
to
control
pitch and yaw, roll is inherently stable (passive).
Sprung
member
ensures
permanent contact
RoBall
Suspended mass (2 DoF
pendulum)
for
forward
and backward acceleration
and sideways steering
Delaware
N/A
dicularly control roll and
pitch
Indian Inst.
N/A
Two RW mounted perpen-
of Technology
MIT Robot
pitch
Suspended mass (1 DoF
pendulum)
for
N/A
Scissored
Suspended mass (1 DoF
pendulum)
for
pendulum)
for
Gyroscope used for steerity
N/A
Gyroscope
stability
forward
and backward acceleration
Reactobot
University
RW with axis in drive di-
pendulum)for forward and
rection used for roll stabi-
backward acceleration
lization and steering (roll)
of Kobe
used
and
for
N/A
A ywheel can be acceler-
Two
ated and decelerated
rotate the gyroscope rela-
independent
rollers
tive to the outer shell
Table 3.2: Performance of existing designs
Robot
Diameter [cm]
Mass [kg]
Speed [m/s]
Sphero
7.5
27
60
29
43
20.6
0.180
1.42
3.4
2.0
3.24
4.1
2.0
1.9
RoBall
Indian Inst. of Techn.
Gyrover
Gyrobot
University of Kobe
roll
steering
(yaw)
Suspended mass (1 DoF
N/A
gyro-
ing (yaw) and roll stabil-
forward
Suspended mass (1 DoF
of
ing torque (pitch)
N/A
and backward acceleration
Gyrobot
pair
scopes to supplement driv-
forward
and backward acceleration
Gyrover
N/A
dicularly control roll and
2.78
0.49
Chapter 4
New Design Ideas
Having seen a variety of dierent concepts that have already been designed and
build, we would like to improve on existing designs and propose some new ideas.
The rst step of this process would be to introduce the requirements that we want
our robot to full.
4.1
Requirements
1. Basic abilities
[a] Move along any given trajectory on at ground for an innite time
(only limited by battery power)
[b] Be able to move up slopes at constant speed for innite time
[c] There should be no dead spot on the spherical hull where the robot
cannot recover from
[d] Can stand and rest at a given position on a slope for innite time
without using excessive energy
2. Advanced abilities
[a] Make use of dynamic eects to climb steeper slopes and longer distances and recover from cavities
[b] Should be able to turn in place in order to move o in any direction
and make very sharp turns. (Requirement obsolete if the robot is inherently
omnidirectional)
[c] Capable of rapid acceleration, i.e., be very quick in responding to
direction and speed changes.
It should look very agile from the eyes of an
observer.
[d] Being able to overcome obstacles by jumping over them.
4.1.1 Conclusions
Very long straight-line movement can hardly be achieved by a RW or Gyroscope
only.
These will eventually get velocity saturated and/or misaligned so that no
further torque can be provided. Therefore some form of shifting mass mechanism
is necessary that allows for a constant gravity-induced torque.
Taking another
look at the review on rolling motion [7], car-driven (possibly in connection with
sprung central axis) and pendulum are the only feasible mechanisms to work inside
a constrained space. Let us have a closer look at those:
31
Chapter 4.
New Design Ideas
Car driven
32
If the car is suciently at, this appears to be a very space
ecient solution. The mobile cart must obviously be secured against falling over.
A cage or any other constraint that keeps its wheels on the inner spherical surface is
acceptable. The energy supply to the car appears to be critical. The mass should be
as far away from the centre of the sphere as possible to allow for maximum gravity
induced torque. Depending on the design of the car, it may be omnidirectional (car
must be able to go in both directions on the surface equally). The reaction time
might not be ideal because the cart has to move to the opposite side of the sphere
in the extreme case.
Pendulum
A pendulum would require much more space to move around,
especially if it is has two DoF. The mass distribution is not as good as with a car
driven system but in this case the mass itself could be used for another purpose, for
instance it could be a spinning wheel that exhibits gyroscopic properties.
4.1.2 Fullling the Desired Requirements
ˆ
Basic abilities
[a+b] The ability to climb slopes or roll on at ground with friction for
innite time is equivalent to the being able to provide a constant torque in
the same direction. This would be automatically fullled by both mechanisms
presented above. The slope incline is however strictly limited to a certain value
depending on how far the CG can be shifted (see section 4.4 for maximum
slope).
[c] Neither for car driven nor for a pendulum model there are any dead
spots. Both systems are furthermore inherently stable because their centre of
mass dose not coincide with the geometric centre.
[d] Similarly to the rst condition, standing on a slope for an innite time
requires a constant torque to react against the incline. A reaction wheel or
a gyroscope will eventually become velocity saturated respectively misaligned
which shows once again that a mass-shifting mechanism is necessary. It must
be noted that not only roll must be controlled but also yaw (which will be
unstable).
ˆ
Advanced abilities
[a] Any device can use its translational and rotational momentum to
climb up steeper slopes (i.e., gather momentum beforehand).
Furthermore,
both shifting mass mechanisms can perform a swinging-motion.
This leads
to non-constant torque which is likely to be stronger than the static torques
at its peaks, but whether this is the case on average must yet be determined.
A further reaction wheel could be accelerated in opposite direction of rolling
motion, thereby creating a time-limited torque to move the sphere further
up. If the reaction wheel was turning in direction of spin of the sphere, its
momentum could be utilized directly by synchronising the this wheel with the
sphere. Finally, gyroscopes can induce very large torques supplementing the
static shifting mass torque for some time. The reaction torque by the driving
motor will however cause over movements that need to be counteracted upon.
[b] Truly turning on spot is only possible if internally the angular momentum in the direction of the yaw axis is changed. This would be most easily
achieved with one reaction wheel aligned with this vertical axis or several
reaction wheels whose combined momentum is changed along this axis only.
However one must also consider that with an omnidirectional car, turning in
place is not necessary.
33
4.2.
Omnidirectional Car-based
Figure 4.1: Design Idea: Cart and Gyroscope
[c] This is again closely linked with the rst point and calls for a mechanism that provides extra torque in the drive or lateral direction when a control
input change happens (i.e., for quick responses).
[d] Jumping is a complex manoeuvre that requires either shifting the
internal centre of gravity in a vertial direction or being able to 'push away'
from the ground, or both (see section 2.5.1). A pendulum is more likely to
achieve this, but any design could be equipped with a suitable mechanism.
4.2
Omnidirectional Car-based
One possibility to achieve a highly agile yet powerful actuation for a sphere would
be to combine a car-based mass shifting mechanism with a spinning disc used as
reaction wheel and gyroscope. Due to the high mass of a ywheel, one would use
only a single one mounted in the centre of the spherical hull and actuated in all
three rotational axis (3 DoF). The car must be able to go in any direction and would
make the robot fully omnidirectional. For quick manoeuvres or extra propulsion the
ywheel would in turn provide extra torque for a limited time. A potential design
has been sketched in gure 4.1
To the inside of the spherical hull at two opposite points, rods are xed. On one
side, this rod has a rotational joint along the axis of the rod, on the other side there
is a motor. This allows a ring-track to be mounted such that it can turn around a
xed axis relative to the outer hull. On this ring track, an actuated cart is attached
that can move along the entire circumference of this round track. The ring track
must be attached to the rods in a way that allows the cart passing through these
points.
One possibility to do so has roughly been sketched.
Further inside the
sphere a ywheel is mounted using two gimbals, both of which are actuated. The
spinning speed of ywheel itself can be controlled by a further motor.
This setup would require ve actuators which is two more than DoFs the sphere
has (no-slip condition). One motor would need to be on the car itself to make it go
aground the ring track. The energy supply could be through an electrical potential
dierence between the two tracks of the ring which is picked up by the car. This
Chapter 4.
New Design Ideas
34
would however require sliding electrical contacts. Furthermore the energy supply
for the other motors is another dicult issue due to many moving parts and joints;
a separate battery for each actuator might be the only viable solution. Finally this
design has a high degree of mechanical complexity and would probably be dicult
to build and maintain.
In the following, advantages and disadvantages of such a design are listed:
Advantages
ˆ
A thin trolley on the ring track leaves most of the space inside the sphere for
other components, i.e., very space ecient.
ˆ
The centre of gravity is relatively high since the ywheel is mounted at the
centre. This allows the robot to overcome small steps relatively easily with
some inertia.
ˆ
Concept is omnidirectional. It can drive o in any direction without turning
beforehand.
ˆ
Gyroscopic eects can be completely 'turned o ' when not needed because the
rotational motion of the gyroscope is decoupled from the one of the sphere.
ˆ
Gyroscope may be used to actively stabilize the motion and reject disturbances, even when no extra torque for a specic manoeuvre is required.
ˆ
When the trolley and the gyroscope are used together, a gravity induced
torque can be amplied to go around curves without much eort.
Disadvantages
ˆ
A high CG is disadvantageous when climbing slopes.
ˆ
Five actuators are three more than really necessary and the energy supply of
the inner ones will likely pose diculties.
ˆ
The ywheel might be misaligned to give the desired torque at any time.
Furthermore a strong outside disturbance might lead to nutation and energy
and intelligent control is required to damp such undesired motion.
ˆ
The trolley has a dead spot if located on the axis of its ring track. If it wants
to go sideways at this point, the track has to react very quickly which creates
an undesired torque on the outer hull.
4.3
Complicated Pendulum with Flywheel
Another idea is to use a two DoF pendulum hung from a xed axis in the sphere
with the mass at the end being a ywheel that can be twisted and accelerated. This
idea is shown in gure 4.2. This approach would use four actuators and dierent
operation modes are possible. For instance, the ywheel could be positioned with its
axis in the direction of motion and therefore only used for roll stabilization whereas
the pendulum does all the rest. Further highly dynamic motions that exploit the
gyroscopic properties are imaginable.
listed below:
Advantages
Reasons for and against such a design are
35
4.4.
Simple Pendulum
Figure 4.2: Design Idea: Pendulum with Gyroscope
ˆ
Requires only four actuators
ˆ
CG is restively low so good slope climbing ability can be expected and larger
centre of gravity shifts are possible than with the trolley approach
Disadvantages
ˆ
Most torques by gyroscope act through the central actuator that connects the
upper pendulum arm with the axis in the sphere. This motor must therefore
be able to produce (withstand) extremely high torques
ˆ
This design is not fully omnidirectional, however it can easily employ the
gyroscope to align the pendulum for the desired forward motion direction.
ˆ
Nearly all space inside must be reserved for the ywheel to move in any
possible position
4.4
Simple Pendulum
Usually, a suspended mass used as pendulum is considered to have a maximum
torque when it is held horizontally, because then the shift of centre of mass is
greatest. For example the developers of GyroBot mention that acceleration torque
T
is limited by the suspended mass [19] :
T = mg · r · sin(α)
(4.1)
This is reasonable if we want a steady-state behaviour, i.e., keeping this torque for
a longer period of time. However we propose a new design in which this constraint
is being ignored. For the shell, the torque that is seen is exactly the motor torque
that acts on the pendulum arm and the central axis of the sphere. Nothing prevents
this motor to exceed the maximum torque by gravity. Obviously the pendulum arm
may lift further than
90◦
and some very interesting behaviour can be observed.
The basic equations that should capture these dynamics have been developed in 2.5
(page 12). However they do not accurately reect the jumping dynamics because the
Chapter 4.
New Design Ideas
36
switching between being airborne and ground contact is too strict and no elasticities
are considered.
Especially the latter are very much relevant when it comes to
bouncing.
As a dierent approach, a model of the sphere and a point mass at the end of
the pendulum has been implemented in a multi-body simulation program called
SimMechanics
x
ϕ gives the angular orientation.
by MATLAB/Simulink. The model is two-dimensional with the
direction pointing along the ground,
y
upwards and
By using the signum function


−1
sign(x) := 0


1
if
if
if
x < 0,
x = 0,
x > 0,
(4.2)
dierent aspects have been modelled as follows:
ˆ
Ground Normal Force:
The normal force exerted by the ground on the
sphere is modelled as a one-sided linear elastic spring:
N = k1 (|y| − y)
1
2
(4.3)
This formulation gives zero force when there is no ground contact (y
for contact (y
ˆ
≤ 0)
> 0) and
a force proportional to ground penetration.
Ground Damping:
To avoid perfectly (and unrealistic) elastic behaviour, a
damping force is added to the normal ground force from above:
Nd = −d1 ẏ sign(|y| − y)
ˆ
No-Slip:
(4.4)
The ground tangential force should prohibit any slipping while the
sphere has surface contact.
This is obviously not very realistic because in
reality, slipping is well possible and the friction force actually decreases once
traction is lost. In this case, the friction force is proportional to the relative
motion between ground and contact point of the sphere:
FR = −k2 sign(|y| − y)(ẋ + Rϕ̇)
ˆ
Friction:
(4.5)
All friction eects such as air resistance and other non-ideal be-
haviour is accounted for in a linear damper term that opposes any motion:
Ffriction = k3
−ẋ
−ẏ
(4.6)
Under section B, gure B.1 a SimMechanics model has been built using the above
force laws to simulate the ground.
Performance
This part is concerned with the performance of a simple pendulum approach. With
the conditions described in this section there are several indicators to consider:
The maximum driving torque that can be maintained without dynamic eects (i.e.,
only the gravity induced torque) is
gravity
Tmax
= mgr
(4.7)
37
4.4.
Simple Pendulum
assuming that the maximum motor torque exceeds this value. Considering a countertorque due to slopes with angle of elevation
γ
T slope = (m + M )gR sin γ
(4.8)
we can use (4.7) and (4.8) to calculate the maximum slope that can theoretically
(no-slip) be climbed using such a steady-state (α̇
!
gravity
Tmax
= T slope
⇒
γmax
= 0) arrangement:
mr
= arcsin
(M + m)R
(4.9)
If friction is still large enough, obviously steeper slopes can be climbed for a limited
amount of time using momentum and stronger motor torques.
The acceleration on at and frictionless ground that can be achieved by an arbitrary
motor torque
T drive
that acts on the sphere and the pendulum can be taken from
(2.23) to be
ẍ =
T drive
R(m + M ) +
ΘS
R
.
A good approximation for the moment of inertia of a spherical hull is
and hence
ẍ =
T drive
.
R(m + 35 M )
(4.10)
ΘS = 23 M R2
(4.11)
Chapter 4.
New Design Ideas
38
Chapter 5
Conclusion
This research project has shown that there are several options available to propel a
round robot. Among the three main categories gyroscopes, reaction wheels and
mass shifting the latter appears to be predominant for most applications, mainly
because it can provide constant torque which does not get exhausted. Previously
many dierent round robots have been built and most use more than one propulsion
mechanism. Especially combinations using mass shifting together with one out of
the other two categories appears to be popular because it adds extra stabilization
and torque capability to the vehicle. Some very elegant combinations are possible
and there are still numerous designs which are yet to be tried out. Several robots
presented are not spherical but rather ellipsoid like a single wheel and arguably
have better dynamic properties. However the inherent roll stability is lost and an
extra mechanisms is required to retrieve stability.
Encapsulated systems in general provide good protection to the (fragile) interior
parts and hence show promise to be more durable especially when operating in
a rough environment. The manufacturing of such a sealed hull as well as energy
supply remains a challenge and must be thoroughly considered when designing a
new robot . So far, only Sphero is capable of being charged up without opening the
hull, that is by induction charging.
The mechanical analysis as well as simulation models could be used in further
works to elaborate on a new concept for rolling robots. A rst idea is presented in
the following section. The ndings of previous works showed that it is crucial to
fully develop a concept into a prototype because one can hardly identify the full
dynamic behaviour without experiments. As controller design seems to be a major
hurdle, a good theoretical model and parameter identication using the prototype
are necessary to capture the system adequately.
5.1
Outlook
As starting point for further works, one might want to consider a two dimensional
model to explore the simple pendulum concept from 4.4. For instance a single wheel
robot with a bike wheel could be worth a try:
Typical bike wheel data
1 of a standard rim 700 wheel (29 inch diameter) with 32
spokes and including tire, tube and axle:
Polar moment of inertia
Mass
Ip
m
=
=
0.0885 kgm2
1.264 kg
(5.1)
1 Taken from: T. Compton. (1998). Analytic Cycling - Performance and Wheels Concepts [Online, Accessed 2013/12/30]. Available: http://www.analyticcycling.com/WheelsConcept_Disc.
html
39
Chapter 5.
Conclusion
40
Figure 5.1: Possible System using a bike wheel with test stand (side and front view)
The objectives for the prototype are
ˆ
Fast acceleration
ˆ
Being able to hop/jump dynamically
ˆ
Can jump with only little horizontal displacement (i.e., can stay in one place
while jumping)
Considering the dynamics of jumping from section 2.5.1 (p.
14) there obviously
needs to be a compromise between large wheel moment of inertia (prevents wheel
from rolling while jumping) and small mass (faster acceleration, higher jumps).
The basic setup of such a bike wheel setup could be as follows:
A lightweight but rigid pendulum arm is attached to the axle of a bicycle wheel
through a joint that permits free rotation around an axis collinear with the wheel
axis. A motor could be xed onto this arm with its rotor rigidly connected to the
wheel axle so that it can exert a torque between pendulum arm and bike wheel.
All batteries and computation units will be mounted on the arm, too.
A sensor
measuring the angular velocity between wheel and arm is required and one also
needs at least one inertial measurement unit on both wheel and pendulum. Since
a bike wheel is not stable in the roll direction, a test stand needs to be build which
removes this DoF. Figure 5.1 shows how this may look like.
What makes this concept dierent from previous ones is that is uses a pendulum
mass-shift mechanism to perform highly dynamic tasks by using it as an excentric ywheel.
This has never been tried out before.
Furthermore there are some
advantages:
ˆ
Relatively simple design compared to manufacturing a sphere - uses available
parts such as bike wheel.
ˆ
Should be able to jump which no robot but Sphero achieved so far.
41
5.1.
Outlook
ˆ
Can operate in both a steady-state and dynamic mode.
ˆ
An extension to the general 3D is case possible.
ˆ
Relatively lightweight since no ywheels or gyroscopes are necessary (and no
dead weight).
ˆ
No undesired gyroscopic eects and no energy losses due to keeping ywheels
spinning at high rotational speeds.
Chapter 5.
Conclusion
42
Appendix A
Gyroscope Dynamics
In this section the equations of motion of a gyroscope are derived.
As done in
previous parts of this work, only a spinning disc with constant density will be
considered.
The centre of gravity of this disk shall rest in space and due to symmetry, the
moment of inertia tensor can be simplied in the body-xed coordinate system
K ΘS
= diag(ΘA , ΘB , ΘB )
K:
(A.1)
Through three elementary rotations (XY
Z ) an arbitrary orientation of the bodyK -system can be achieved: The inertial system I is xed in space. By rotating
around a common x-Axis by an angle ψ we reach the B -system. The following C system is rotated around the y -axis by an angle α. Finally the K -system is obtained
by rotating around the z -axis by an angle β . These are all elementary rotations
xed
where the transformation matrices and rotational velocities can be found straight
away. For notation, please refer to the symbols section
(page vii). The coordinate
transforms are schematically drawn in gure A.1.

1
0
0
sin ψ 
=  0 cos ψ
0 − sin ψ cos ψ


cos α 0 − sin α

1
0
= 0
sin α 0 cos α


cos β
sin β 0
=  − sin β cos β 0 
0
0
1

ABI
ACB
AKC


ψ̇
 0 
I ω IB = B ω IB =
0


0
 α̇ 
B ω BC = C ω BC =
0


0
 0 
C ω CK = K ω CK =
β̇
Figure A.1: Gyroscope with coordinate system transform
43
(A.2)
(A.3)
(A.4)
Appendix A. Gyroscope Dynamics
44
We further dene the absolute angular velocity components of the disk in the bodyxed coordinate system to be


ωx
 ωy  .
K Ω :=
ωz
(A.5)
Now we are ready to compute the angular momentum of the disk with respect to
its resting CG. The derivative has been calculated using the Euler-derivative rule
for rotating frames of references:

K LS
K (L̇S )

ωx ΘA
= K ΘS K Ω =  ωy ΘB  ,
ωz ΘB

(A.6)

ω̇x ΘA
= K L̇S + K Ω × K LS =  ω̇y ΘB + (ΘA − ΘB )ωx ωz  .
ω̇z ΘB + (ΘB − ΘA )ωx ωy
(A.7)
This result (A.7) is exactly Eulers equation for a gyroscope [24].
The next step is to express the angular velocity vector K Ω in terms of the angles
For the sake of
(and their derivatives) used in our coordinate transformations.
clarity, some steps are omitted in the following calculation.


ωx
 ωy  = K Ω
ωz
:=
K ω IK
=
AKC ACB B ω IB + AKC C ω BC + K ω CK


ψ̇ cos α cos β + α̇ sin β
. . . =  −ψ̇ cos α sin β + α̇ cos β  .
ψ̇ sin β + β̇
=
= K ω IB + K ω BC + K ω CK
(A.8)
Now we can nally use the principle of angular momentum. This will be evaluated
in the body xed-coordinate system using (A.7) and (A.8). In order to shorten the
expression cosine and sine functions are abbreviated by



=

c
and
s
resprectively.
MS = K (L̇S )
(A.9)
K

ΘA ψ̈ cα cβ − ψ̇ α̇ sα cβ + β̇ cα sβ + α̈ sβ + α̇β̇ cβ


ΘB −ψ̈ cα sβ + ψ̇ α̇ sα sβ − β̇ cα cβ + α̈ cβ − α̇β̇ sβ


ΘB ψ̈ sβ + ψ̇ β̇ cβ + β̈ + (ΘB − ΘA ) ψ̇ cα cβ + α̇ sβ −ψ̇ cα sβ + α̇ cβ
This is the nal equation to simulate the behaviour of the gyroscope in the presence
of dierent external torques. If the external moments are given as a function of time,
this system can be solved numerically.
Singularity
One clear disadvantage of using Euler-rotations in describing the orientation of a
rigid body becomes evident here.
have the same inuence on the orientation while
the equations show a
and β anymore (they
α remains in this position). This is
singularity because one cannot tell the dierence
known as
α = 90◦
between ψ
For the case of
gimbal lock 1 . Using quaternion description might have been useful here.
1 Gimbal lock is a situation where two axes of rotation become aligned. This eectively results
in a loss of one DoF
Appendix B
SimMechanics Model
Figure B.1 shows the SimMechanics model used to visualize some of the properties
of the 2D shifting mass pendulum mechanism.
45
Appendix B. SimMechanics Model
Figure B.1: SimMechanics Model
46
Bibliography
[1] J. K. Hodgins and M. N. Raibert, Adjusting step length for rough terrain
Robotics and Automation, IEEE, vol. 7, no. 3, 1991.
T. Ylikorpi and J. Suomela, Ball-shaped robots, in Climbing & Walking
Robots, Towards New Applications, no. October, p. 546, 2007.
locomotion,
[2]
[3] T. Das and R. Mukherjee, Exponential stabilization of the rolling sphere,
Automatica, vol. 40, pp. 18771889, Nov. 2004.
AIAA Guidance, Navigation and Control Conference and Exhibit, pp. 114, Aug. 2008.
[4] D. Brown, Control Moment Gyros as Space-Robotics Actuators,
[5] D. Brown and M. Peck, Energetics of Control Moment Gyroscopes as Joint
Actuators,
Journal of Guidance, Control, and Dynamics, vol. 32, pp. 1871
1883, Nov. 2009.
[6] M. Gajamohan, M. Merz, I. Thommen, and R. D'Andrea, The Cubli: A cube
2012 IEEE/RSJ International Conference on
Intelligent Robots and Systems, pp. 37223727, Oct. 2012.
that can jump up and balance,
[7] R. H. Armour and J. F. V. Vincent, ScienceDirect Rolling in Nature and
Robotics :
A Review,
Journal of Bionic Engineering,
vol. 3, pp. 195208,
2006.
[8] T. Buratowski and P. Cie±lak, A self-stabilising multipurpose single-wheel
robot,
Journal of theoretical and applied mechanics, no. 390857, pp. 99118,
2012.
[9] E.
Ackerman,
Orbotix
Rolls
Out
Speedy
Next-Gen
http://spectrum.ieee.org/automaton/robotics/home-robots/
orbotix-rolls-out-speedy-nextgen-sphero, 2013.
Sphero.
http://www.technologyreview.com/
hack/426981/rolling-rolling-rolling/, 2012.
[10] S. Cass, Rolling, Rolling, Rolling ....
Lancet, Orbotrix Sphero 2.0 Review and Giveaway. http://www.
makeuseof.com/tag/orbotrix-sphero-2-0-review-and-giveaway/, 2013.
[11] Y.
[12] M. Kabaªa and M. Wnuk, Design and construction of RoBall, a spherical,
nonholonomic mobile robot, tech. rep., 2004.
[13] S. Bhattacharya and S. Agrawal, Spherical Rolling Robot:
Motion Planning Studies,
A Design and
IEEE Transactions on Robotics and Automation,
vol. 16, no. 6, pp. 835839, 2000.
[14] V. a. Joshi, R. N. Banavar, and R. Hippalgaonkar, Design and analysis of a
spherical mobile robot,
Mechanism and Machine Theory, vol. 45, pp. 130136,
Feb. 2010.
47
Bibliography
48
[15] G. C. Schroll, Design of a Spherical Vehicle with Flywheel Momentum Storage
for High Torque Capability, 2008.
[16] H. B. Brown and Y. Xu, A single wheel, gyroscopically stabilized robot,
Robotics and Automation, IEEE, 1997.
[17] Y. Xu, S. K. W. Au, and W. Yu, Control of a Gyroscopically Stabilized
Robot, 2000.
[18] Y. Xu and S. K. W. Au, Stabilization and path following of a single wheel
robot,
IEEE/ASME Transactions on Mechatronics, vol. 9, no. 2, pp. 407419,
2004.
[19] T. Saleh, A. Al Mamun, and V. Prahlad, Design of a gyroscopically stabi-
IEEE Conference on Robotics, Automation and
Mechatronics, 2004., vol. 2, pp. 904908, 2004.
lized single-wheeled robot,
[20] A. Al Mamun, Z. Zhen, P. Vadakkepat, and T. H. Lee, Tracking control
31st Annual
Conference of IEEE Industrial Electronics Society, 2005. IECON 2005., p. 6
of the gyrobot-a gyroscopically stabilized single-wheeled robot,
pp., 2005.
[21] J. Biswas and S. Bhartendu, Dynamic Stabilisation of a Reaction-Wheel Ac-
International Journal of Factory Automation, Robotics
and Soft Computing, no. 4, pp. 135140, 2008.
tuated Wheel-Robot,
[22] S. N. Kadam and B. Seth, LQR controller of one wheel robot stabilized by
reaction wheel principle, 2nd International Conference on Instrumentation
Control and Automation 2011, Badung, Indonesia, no. November, pp. 299
303, 2011.
[23] T. Urakubo, M. Osawa, H. Tamaki, Y. Tada, and S. Maekawa, Development
IEEE International
Conference on Mechatronics and Automation, pp. 16021607, Ieee, Aug. 2012.
of a Spherical Rolling Robot Equipped with a Gyro, in
[24] C. Glocker and A. Nievergelt, Mechanik III, tech. rep., 2011.