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Proc. R. Soc. A
doi:10.1098/rspa.2009.0559
Published online
On the stability and control of unicycles
BY ROBIN S. SHARP*
Department of Mechanical, Medical and Aerospace Engineering, Faculty
of Engineering and Physical Sciences, University of Surrey, Guildford, UK
A mathematical model of a unicycle and rider, with a uniquely realistic tyre force and
moment representation, is set up with the aid of multibody modelling software. The
rider’s upper body is joined to the lower body through a spherical joint, so that wheel,
yaw, pitch and roll torques are available for control. The rider’s bandwidth is restricted
by low-pass filters. The linear equations describing small perturbations from a straightrunning state are shown, which equations derive from a parallel derivation yielding the
same eigenvalues as obtained from the first method. A nonlinear simulation model and
the linear model for small perturbations from a general trim (or dynamic equilibrium)
state are constructed. The linear model is used to reveal the stability properties for
the uncontrolled machine and rider near to straight running, and for the derivation of
optimal controls. These controls minimize a cost function made up of tracking errors
and control efforts. Optimal controls for near-straight-running conditions, with left/right
symmetry, and more complex ones for cornering trims are included. Frequency responses
of some closed-loop systems, from the former class, demonstrate excellent path-tracking
qualities within bandwidth and amplitude limits. Controls are installed for path-following
trials. Lane-change and clothoid manoeuvres are simulated, demonstrating good-quality
tracking of longitudinal and lateral demands. Pitch torque control is little used by the
rider, while yaw and roll torques are complementary, with the former being more useful in
transients, while the latter has value also in steady states. Wheel torque is influential on
lateral control in turning. Adaptive control by gain switching is used to enable clothoid
tracking up to lateral accelerations greater than 1 m s−2 . General control of the motions of
a virtual or robotic unicycle will be possible through the addition of more comprehensive
adaptation to the control scheme described.
Keywords: unicycle; dynamics; stability; control; preview; tracking
1. Introduction
A unicycle has one wheel and direct drive to that wheel by the rider. Typical
contemporary machines can be seen at http://www.jugglingstore.com/. The
wheel is usually quite small, with the saddle conveniently located just above the
wheel. The rider provides most of the mass of the man–machine system and it is
evident that the uncontrolled system is unstable in both longitudinal and lateral
*[email protected]
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2009.0559 or via
http://rspa.royalsocietypublishing.org.
Received 21 October 2009
Accepted 16 December 2009
1
This journal is © 2010 The Royal Society
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R. S. Sharp
directions. The stability of the combination depends on the control skills of the
rider. A monocycle is dynamically similar to a unicycle but, in this case, the rider
sits inside a wheel of sufficiently large diameter and a motor is fitted (Cardini
2006). Carvallo (1899) showed, using a very simple analysis, that a monocycle
is longitudinally stable if the rider assembly mass centre lies below the wheel
centre. In both cases, lateral symmetry of the straight-running state implies that
longitudinal and lateral problems are decoupled at first-order level near to straight
running (Meijaard et al. 2007; Sharp 2008).
Skilled riding of unicycles can be observed at many Internet sites, for example
http://www.unicyclist.com/. Much of the riding typically displayed is trickriding, involving much jumping from object to object, balancing when stationary,
riding backwards and so on, but the present concern is with accurate control of
the path of the unicycle when the ground is flat and level. It can be seen that
expert riders can follow a narrow path, along the top of a wall, for example, even
tracking along a horizontal tree trunk of diameter, perhaps, 0.3 m. When they
do this, they typically extend both arms and use vigorous arm movements in
yaw and roll for control purposes. It is also apparent that the lean angle of the
unicycle frame is invariably no more than a few degrees in these trials, implying
that control is sufficiently challenging that the tyre is unlikely to be exercised
very far from its free rolling state.
Most of what is currently known about the stability and control of the unicycle
comes from a doctoral study by Vos (1992) and Vos & Von Flotow (1990), a
paper by Naveh et al. (1999) and a short paper by Zenkov et al. (1999). Each of
these studies was motivated mainly by interest in control systems design, with
the unicycle providing a challenging plant, making effective control far from easy.
The present study is more concerned with the means by which real riders stabilize
and guide their machines to follow desired paths. Any implications for robotic
unicycles resulting are somewhat coincidental.
Vos built an autonomous unicycle and was able to demonstrate its
rather tentative stable running along a straight path at very low speed
(see http://www.rockwellcollins.com/athena/demos/unicycle/). His unicycle was
fitted with two servo-motor actuators, one to drive the road wheel and the other
to rotate an inertia-wheel in yaw, and his model includes only such actuation.
The mathematical model used for control system design assumed pure rolling
longitudinally, which constitutes a non-holonomic constraint on the motion, and a
yawing moment linearly dependent on side-slip and on yaw rate up to a maximum,
friction-saturation level. In the author’s view, this modelling is seriously flawed;
there is neither experimental nor theoretical support for such a representation of
the tyre force and moment system, so that all the findings must be viewed with
some suspicion. In particular, it is well known that the rolling contact between an
elastic tyre and a rigid ground involves a finite contact length and, for low force
levels, features primarily non-sliding contact between tread rubber and ground,
with elastic deformation of the tyre carcass to accommodate slip (departures
from pure-rolling kinematics) (Clark 1981; Pacejka & Sharp 1991; Pacejka 2002;
Gent & Walter 2005). The consequences of such elastic deformation are that the
small-slip, rolling tyre generates a longitudinal force in proportion to longitudinal
slip ratio, a lateral force linearly dependent on lateral-slip ratio, turn-slip ratio
and camber angle, and a yawing moment proportional to lateral-slip, turn-slip
and camber angle. Also, to represent the migration of the contact patch around
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Unicycle dynamics
3
the tyre cross section as the camber angle changes, a rolling moment proportional
to the camber angle can be incorporated (Pacejka 2002; Sharp 2008).
Nevertheless, Vos was able to point out correctly that the lateral dynamics of
the unicycle are strongly influenced by travel speed and considered it necessary
to employ gain scheduling over a number of basically linear controllers with speed
as a scheduling parameter, to deal with such sensitivity. Naveh et al. inherited the
mechanical modelling and no-longitudinal-slip assumptions of Vos but employed
an even stranger lateral tyre force description. They argued that linear controllers
miss some essential ingredients of the real problem and that nonlinear control
terms are essential.
Zenkov et al. assumed point contact between tyre and ground and pure rolling
of the tyre over the surface but their ‘actuator’ acted in roll, like a lateral
pendulum. Zenkov’s output-feedback controller is linear but the stability of the
closed-loop, nonlinear system was checked by the Liapunov–Malkin theorem.
None of these previous studies included preview of the path in the controller
design but a large body of research, certainly stretching back to Bender (1968)
and Tomizuka & Whitney (1975), indicates the enormous advantages that can
be gained from using preview information for control, if the future desired path
of the plant is known in advance. This is very much the case in steering a freeranging vehicle along a path that the driver can see or knows by some other
means.
Over the last several years, linear optimal preview control theory has been
applied to car driving and to motorcycle and bicycle riding (Sharp & Valtetsiotis
2001; Sharp 2005, 2006, 2007a–e, 2008; Thommyppillai et al. 2009a,b) and
this article is about the application of the theory to the longitudinal and
lateral unicycle stabilization and path-following-control problem. The strategy
for dealing with varying speed and higher amplitude nonlinear motion is to
design linear preview controllers that are optimal for the locality, in a state-space
sense, of a trim state and to schedule, using suitable indicators of the appropriate
neighbouring trim states, over the gain sets and trim states, such that the current
control is always well matched to the present operating conditions. Also, the
path points demanded will be close to the trim values, such that tracking errors
remain small.
The work is motivated by an interest in how humans control difficult machines,
so that the modelling is more elaborate than has been attempted before. The
notional rider is able to yaw, pitch and roll his/her upper body, relative to the
unicycle frame. Low-pass filter properties are also built in to these ‘actuators’ to
represent the bandwidth limitations of human controllers. Tyre force and moment
descriptions are consistent with conventional wisdom from vehicle dynamics and
tyre mechanics research. Consequently, automated multibody mechanics software
is almost essential to the model-building task and the full nonlinear equations of
motion are too lengthy and complex to show. However, the linear equations for
small perturbations from a straight-running trim are manageable and provide
some insight into the system dynamics.
In the next section, the unicycle and rider model is set up. Following that,
there is a brief review of the relevant linear-optimal-preview-control theory and a
description of how it is applied to a vehicle-driving problem. Control design and
closed-loop system results are then shown and interpreted and some path-tracking
simulations are discussed. Conclusions are drawn at the end.
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R. S. Sharp
2. Mathematical model of unicycle and rider
The mathematical model of the unicycle and rider is built using the multibodymodelling software VEHICLESIM, formerly called AUTOSIM (Mousseau et al. 1992;
Sayers 1999; Sharp et al. 2005; also see http://www.carsim.com). The package
has been employed to develop the widely used commercial models TRUCKSIM,
CARSIM and BIKESIM. VEHICLESIM capabilities are described in Thommyppillai
et al. (2009b) and in the electronic supplementary material, appendix A. To
check the model building and to obtain the linear equations of motion for small
perturbations from straight running at a given speed, the equations of motion are
also derived using Lagrange’s energy method for quasi-coordinates (Pacejka 2002)
and symbolic Matlab eigenvalues from the two approaches prove to be identical.
The equations of motion generated by VEHICLESIM, the system parameter
values and desired outputs can be written automatically into a simulation code,
with the aid of a ‘C’ or ‘Fortran’ compiler, or they can be linearized for small
perturbations about a general trim state and written into a Matlab ‘M’-file.
Typically, for linear analysis, the nonlinear simulation program is used to find
trim states and the equilibrium values of states and inputs are passed to Matlab
to set up the numerical state-space form of the linear system equations. If the
uncontrolled system is unstable, some form of stabilizing controller must be
incorporated in the simulation model to find trim states. An alternative that
has not been explored here is to solve the cornering equilibrium equations using
a Newton–Raphson process, say.
Unicycle and rider are represented as follows. The main component of the
system is the rigid frame, which includes the lower body of the rider. The freedoms
of the frame are defined as follows: a massless body, the yawframe, has freedom
to translate along x- and y-axes, in the road plane and to yaw about the vertical
z-axis. A massless rollframe has only a roll degree of freedom relative to its parent
body, the yawframe, with a common point between the two frames at ground level,
that is, at the notional contact point. The axisymmetric wheel has spin freedom
relative to the rollframe and it makes point contact with the flat and level ground.
The unicycle frame pitches with respect to the rollframe, with a common point
at the wheel spindle. The tyre can slip longitudinally and laterally. The frame
rotations comprise yaw, jf ; roll, ff ; and pitch, qf , so that the tyre camber angle
relative to the ground is ff . The tyre generates longitudinal and lateral forces, an
aligning moment in response to slip and camber and an overturning moment in
response to camber, according to conventional wisdom for small slip ratios and
camber angles (Clark 1981; Pacejka & Sharp 1991; Pacejka 2002; Sharp 2007d,
2008). The rider’s upper body is joined to the frame by a spherical joint, so that
this body has yaw, jr ; pitch, qr ; and roll, fr , freedoms relative to the frame.
Fairly weak springs and parallel dampers act between upper and lower bodies
of the rider, representing the rider’s structure in a simple way. The mechanical
model is supported by figures 1 and 2.
For the description of the tyre longitudinal-, lateral- and turn-slips and,
through them, shear forces and steering moment, the rolling velocity is needed
(Pacejka & Sharp 1991; Pacejka 2002; Sharp 2008). This is the forward velocity of
the point on the unicycle frame that coincides with the contact centre, ur , say. The
longitudinal slip ratio is the forward velocity of the theoretical ground-contact
point P (figure 1; P being a material point on the tyre periphery), up , divided by
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Unicycle dynamics
side
view
rider upper
body
rear
view
spherical joint
x
x
frame and
rider lower
body
frame
pitch
wheel
hf
hj
z
z
hr
rw
roll
P
x
y
Figure 1. Diagrammatic unicycle model showing frame, wheel and rider upper body with
axis directions.
yawframe; translate
(x, y), rotate z
inertial body
rollframe; rotate x
wheel; rotate y
tw
unicycle frame;
rotate y
ty
z
y
tq
tf
rider upper body;
rotate (z, y, x)
x
Figure 2. Bodies, freedoms and controls of the unicycle model. The yawframe and the rollframe
are massless. Separating the freedoms as shown causes the forward and lateral velocities of the
unicycle reference point to be chosen as generalized speeds in preference to the absolute velocities.
ur , the lateral slip ratio is the lateral velocity of P, vp , divided by ur , while the
turn-slip ratio is the yaw rate of the frame, rf , divided by ur . Tyre shear forces
and moments are specified by
up
,
ur
vp
rf
Fy = −Cfy + Cmz − Cf ff ,
ur
ur
vp
rf
Mz = Cmz − Crz − Cmzg ff
ur
ur
Mx = −Cmx ff .
Fx = −Cfx
and
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R. S. Sharp
The tyre forces and moments are linear functions of the slips and camber,
naturally limiting the range in which they describe the real tyre accurately. It
turns out that losing control is likely to occur well before tyre effects show signs
of saturating, so that, within the context of this study, the linear tyre is considered
adequate. Tyre data specifically for unicycle tyres is thought not to exist, so that
numerical values have to be estimated based on knowledge of bicycle tyres (Sharp
2008), accounting for the load carried.
The rider can exert four control torques. The first is the wheel torque, tw , which
acts between the frame and wheel. The other three are yawing, tj ; pitching, tq ;
and rolling, tf , torques acting on the frame and reacted by the rider’s upper body.
Each of the four ‘actuators’ that applies a control torque has a second-order
Butterworth low-pass filter associated with it, to represent the response-time
limitations of real riders. Each filter equation is of the form
ẗ = un2 tdem −
√
2un ṫ − un2 t,
where t is the actuator torque, tdem is the torque demand and un is the filter
bandwidth. The system is holonomic, smooth and differentiable and has nine
velocity states, nine displacement states and eight auxiliary states associated with
the low-pass filters. The ignorable coordinate for the spin angle of the road wheel
does not appear in the equations of motion, so that the (25 × 1) state vector is
z = [x y jf ff qf jr qr fr ṫw tw ṫj tj ṫq tq ṫf tf u v rf pf qf rr qr pr qw ]T ,
in which the angular velocities rf , pf , qf , rr , qr and pr are related to the
corresponding displacements as shown below.
The linear equations of motion for small perturbations from straight running,
derived from the symbolic Matlab approach and omitting the rider low-pass
filters, using mt = mf + mw + mr , hfw = hf − rw , hrw = hr − rw , hrj = hr − hj , mth =
mf hf + mw rw + mr hr , Itx = Ifx + Iwx + Irx + mf hf2 + mw rw2 + mr hr2 , Ity = Ify + Iry +
2
2
+ mr hrw
, Irxe = Irx + mr hr hrj and Irye = Iry + mr hrj hrw , with u0 and qw0 the
mf hfw
trim speed and the trim wheel-spin speed respectively, are
Cfx
Cfx rw ˙
u + (mf hfw + mr hrw )q¨f + mr hrj q¨r −
qw = 0,
u0
u0
Cfy
Cmz
mt v̇ +
j̇f − mth f̈f + Cf ff − mr hrj f̈r = 0,
v + m t u0 −
u0
u0
Cmz
Crz
−
v + (Ifz + Iwz + Irz )j̈f +
j̇f + Iwy qw0 ḟf + Cmzg ff + Irz j̈r = 0,
u0
u0
mt u̇ +
−mth (v̇ + u0 j̇f + gff ) − Iwy qw0 j̇f + Itx f̈f + Cmx ff + Irxe f̈r − mr hrj gfr = 0,
(mf hfw + mr hrw )(u̇ − gqf ) + Ity q¨f + Irye q¨r − mr hrj gqr = −tw ,
Irz j̈f + Irz j̈r + cy j̇r + ky jr = −tj ,
mr hrj [u̇ − g(qf + qr )] + Irye q¨f + (Iry + mr hrj2 )q¨r + cp q˙r + kp qr = −tq ,
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Unicycle dynamics
Table 1. Unicycle and rider model parameter definitions and values.
parameter and symbol
value
wheel radius (rw )
frame mass centre height (hf )
frame to rider upper body joint height (hj )
rider mass centre height (hr )
0.25 m
0.6 m
0.8 m
1.4 m
frame mass (mf )
wheel mass (mw )
rider upper body mass (mr )
gravitational acceleration (g)
35 kg
3 kg
40 kg
9.80665 m s−2
wheel camber inertia (Iwx )
wheel spin inertia (Iwy )
frame inertias (Ifx , Ify , Ifz )
rider upper body inertias (Irx , Iry , Irz )
0.025 kg m−2
0.04 kg m−2
(4.2, 4.5, 1.4) kg m−2
(1.4, 1.1, 0.9) kg m−2
tyre
tyre
tyre
tyre
tyre
tyre
tyre
7000 N
5000 N
15 Nm rad−1
80 Nm
8 Nm rad−1
750 N rad−1
5 Nms rad−1
long-slip stiffness (Cfx )
side-slip stiffness (Cfy )
rolling moment coefficient (Cmx )
aligning moment/side-slip coefficient (Cmz )
aligning moment/camber coefficient (Cmzg )
camber stiffness (Cf )
aligning moment/turn-slip coefficient (Crz )
rider
rider
rider
rider
rider
rider
upper/lower
upper/lower
upper/lower
upper/lower
upper/lower
upper/lower
body
body
body
body
body
body
yaw stiffness (ky )
pitch stiffness (kp )
roll stiffness (kr )
yaw damping coefficient (cy )
pitch damping coefficient (cp )
roll damping coefficient (cr )
wheel actuator bandwidth (un )
rider yaw actuator bandwidth (un )
rider pitch actuator bandwidth (un )
rider roll actuator bandwidth (un )
100 Nm rad−1
100 Nm rad−1
100 Nm rad−1
10 Nms rad−1
10 Nms rad−1
10 Nms rad−1
12.6 rad s−1
12.6 rad s−1
12.6 rad s−1
12.6 rad s−1
−mr hrj [v̇ + u0 j̇f + g(ff + fr )] + Irxe f̈f + (Irx + mr hrj2 )f̈r + cr ḟr + kr fr = −tf
and
−
Cfx rw
Cfx rw2 ˙
u + Iwy q¨w +
qw = tw .
u0
u0
Kinematically,
x˙ = u cos jf − v sin jf ; y˙ = u sin jf + v cos jf ; j̇f = rf ; ḟf = pf ;
˙
˙
qf = qf ; qw = qw ; j̇r = rr ; q˙r = qr ; ḟr = pr .
The nonlinear simulation model outputs all the states, since one of its functions
is to find equilibrium running (trim) conditions, while the linear model used
for control system design only outputs those quantities that appear in the cost
function, namely x and y, the absolute position coordinates of the unicycle’s
reference point. Parameter definitions and values, representing a typical unicycle
and rider, are collected in table 1. Most of these parameters are non-critical to
the study, since it is in the nature of the optimal preview control theory to be
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R. S. Sharp
10
9
0.1 m/s
0.152 m/s
0.231 m/s
0.351 m/s
0.534 m/s
0.811 m/s
1.233 m/s
1.875 m/s
2.85 m/s
4.33 m/s
6.583 m/s
10 m/s
filter modes
eigenvalue imaginary part (rad s–1)
8
7
yaw/roll
6
5
4
3
2
1
0
longitudinal eigenvalues:
−8.20;
−4.719;
2.372;
5.789
roll
roll/yaw
−10
−8
−6
roll/yaw
roll
−4
−2
0
eigenvalue real part (s–1)
2
4
6
Figure 3. Segment of root-locus plot for the uncontrolled unicycle and rider with speed varying in
the range of 0.1–10 m s−1 .
applied that the inverse dynamics of the plant are represented in the controls.
If the plant is altered within reason, then the controls change to compensate.
However, it turns out that care has to be taken with the value of Crz , the
coefficient relating the tyre aligning moment to the turn-slip, as will be explained
subsequently.
If the trim state for the linearized model involves straight running, the system
has left/right symmetry and only the forward speed and the wheel-spin velocity
are non-zero. The wheel-spin velocity is the speed divided by the wheel radius,
since no slip is necessary in the absence of aerodynamic drag and tyre rolling
resistance. The normal modes consist of one set for longitudinal motions and
another set for lateral motions. A segment of a corresponding root-locus plot
for the uncontrolled system with speed varying in the range of 0.1–10 m s−1 is
shown in figure 3. Outside the plot space, there are two numerically large negative
eigenvalues. The lateral modes vary with speed, as noted by Vos (1992), while
the longitudinal modes represented by real eigenvalues at −8.20 s−1 , −4.719 s−1 ,
2.372 s−1 and 5.789 s−1 are hardly affected by speed variations. The longitudinal
divergence with eigenvalue 5.789 s−1 shows the difficulty associated with the
stabilization by a human rider. A uniform rod of length 0.439 m balancing on
its point has a similar time constant. From this viewpoint, stabilizing the lateral
motions is less onerous and it gets a little easier as the speed rises.
A parameter set describing a typical monocycle and its root-locus through a
somewhat extended speed range is given in the electronic supplementary material,
appendix B. The root-locus corresponding to that above for the unicycle but
under the assumption that the tyre constrains the motion by rolling perfectly
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Unicycle dynamics
9
without slip is shown in the electronic supplementary material, appendix C. The
longitudinal modes are almost exactly the same as above but the lateral modes
are significantly different.
3. Optimal linear preview control theory
(a) General observations
Riding a unicycle can be viewed as a problem in optimal control, with
optimization by reinforcement learning (Gurney 1997; Sutton & Barto 1998).
Restricting attention to mild manoeuvring, it is a problem in linear-optimal
control. More vigorous motions can be treated by gain-scheduling over several
linear control schemes, each designed for the neighbourhood of a trim state.
The problem involves preview of the path ahead as an essential feature and
it involves time delays. Solutions for the control of systems that include time
delays tend to be easier in discrete time, and that is the approach preferred
here. Appropriate theory for optimal-linear-preview control exists (Tomizuka &
Whitney 1975; Tomizuka 1976; Tomizuka & Rosenthal 1979; Louam et al. 1988,
1992; Prokop & Sharp 1995), which has been applied to steering and speed
control of various road vehicles (Sharp & Valtetsiotis 2001; Sharp 2005, 2006,
2007a,c,d,e, 2008; Thommyppillai et al. 2009a,b). The detailed theory required is
included in Thommyppillai et al. (2009b) and it is replicated here as the electronic
supplementary material, appendix D. It is now reviewed briefly and then applied
to longitudinal and lateral control of the unicycle.
(b) Optimal linear preview control theory background
First, the nonlinear unicycle model is linearized for small perturbations about
an equilibrium running, or trim, condition. The absolute longitudinal and lateral
displacements of a reference point, the tyre–ground contact point, are outputs
from the model. Then the relevant linear unicycle equations, with state-vector
z, input u and output vector y are expressed in state-space form. The problem
is converted to discrete-time form after selection of the sampling interval, Ts . A
parallel discrete dynamic system, describing the target motions in (x, y, t) form,
is joined to the unicycle description. Sample values of the longitudinal and lateral
displacement demands through time, at intervals Ts , are each moved through a
shift-register with the passage of a time interval. The oldest samples depart the
problem, all the register contents move one step closer to the unicycle and the
two samples, which were previously the input to the system, enter the registers.
New (x, y) samples, previously just outside the problem compass, provide the
new input to the problem.
With the conjoining of the dynamics of the unicycle to those of the
displacement demands, the full state vector contains the appropriate unicycle
states for the former and 2n preview values for the latter, where n is the number
of preview points to be used. The first pair of preview samples in the problem at
any time instant are the x- and y-displacements that the unicycle should have at
this particular time and the cost function to be minimized in the optimal control
calculations contains the sum of the squares of the differences between the demand
and the actual at this instant. The cost includes a summation over infinite future
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R. S. Sharp
shift register; n = 14
xdem
ydem
unicycle and rider
with low-pass
pedal, yaw, pitch
and roll torque
actuators
x
y
pedal torque demand
yaw torque demand
pitch torque demand
roll torque demand
K2
K1
unicycle states
Figure 4. Structure of the four-control, x- and y-input preview tracking system. xdem and ydem are
the previewable longitudinal and lateral displacement signals. K 1 represents the full unicyclestate feedback, while K 2 represents the preview control, in the form of feedback of the shift
register states.
time of this sum of squares. Control power also has to be included in the cost,
with weighting coefficients defining the relative importance of tracking errors and
control efforts. Implicit in the optimal control theory, the x- and y-demands are
regarded as consisting of sample values from independent white noise processes
but it is known that the optimal controls continue to be optimal if the white
noise is low-pass filtered and sufficient preview of the disturbance is available
(Tomizuka & Whitney 1975; Sharp 2005). The problem structure and optimal
controls are illustrated in figure 4.
The optimal controls are conveniently found through Hazell’s Matlab toolbox
(Hazell 2008). The toolbox requires only the setting up of the standard statespace (A, B, C, D) matrices, the setting of weights on tracking errors and control
efforts, the discrete-time step and the number of preview points, for the optimal
preview controls to be computed (see http://code.google.com/p/preview-controltoolbox/). The preview gains K 2 fall to zero as the preview distance increases,
corresponding to the fact that preview information can be too far ahead of the
current situation to be useful. Therefore, the number of preview points included
can be chosen, by trials, so that the full benefit available is effectively obtained.
This is referred to as ‘full’ preview. With less than full preview, the control is
suboptimal and is not of much interest, so that only full preview control will be
used here.
4. Optimal controls
Examples of optimal controls are shown in this section. Each set of controls to
be generated requires choices of: (i) the time step to be used in the problem
discretization and the number of preview points to be used, chosen here always to
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Unicycle dynamics
11
give full preview, (ii) the trim state from which small perturbations are considered
to occur, and (iii) x- and y-tracking error and control power weights relating to
wheel, yaw, pitch and roll torque demands.
When the trim state of the unicycle involves straight running, the only nonzero trim values are those for forward speed and wheel spin velocity, but, when
the trim involves turning, a control sufficient to stabilize the turning motion is
needed to enable simulation of the condition, to determine the trim. The simplest
solution is obtained by first designing controls for straight running and then using
them to simulate turning. If the turning becomes too vigorous, of course the
control will break, so the ambition must be curtailed. If the turning motion is
only quasi-steady, the first control derived will be only approximate. Then, a
steady turn defining a perfect trim state can be obtained either (i) by running
the simulation model under feedback-control only or (ii) by tracking a circular arc
of the required radius (with preview), with linearization for small perturbations
from such a state allowing new locally optimal controls to be found. Further
simulations in the same vein enable optimal controls to be determined for turns
of increasing vigour. Straight running will be illustrated here and more optimal
controls for both straight running and turning will be shown in the electronic
supplementary material, appendix E.
For straight running, take Ts = 0.01 s and R = diag[1, 10, 10, 10], the latter
to represent the idea that wheel torque control is relatively easy for the rider
to provide, while yaw, pitch and roll torques rank the same as each other but
are harder to provide than wheel torque. Then, we choose 10 s preview and use
trial and error to determine a set of weights on tracking errors to give ‘full’
preview, yielding q = diag[5e2, 1e4]. Feedback gains obtained in discrete time are
asymptotic to the corresponding continuous-time LQR-optimal gains as the time
step is reduced, so that continuous-time gains are shown in table 2 and preview
gains in figure 5. These results show that the speed of the unicycle has hardly
any influence on the optimal longitudinal controls for straight running. Tighter
controls designed for preview times of 5 s and 2 s with q = diag[2.5e3, 2e5] and
q = diag[1e5, 1e7], respectively, are shown in a similar fashion in the electronic
supplementary material, appendix E.
With a set of optimal controls installed, the unicycle becomes stable by virtue
of the state feedback and autonomous, being capable of tracking a desired path,
using the preview control. The frequency responses of the closed-loop system
(Sharp 2007c,d,e, 2008; Thommyppillai et al. 2009a,b) show the path-tracking
capability to be perfect within amplitude and bandwidth limits, which depend
on the tightness employed in the control design (figures 6 and 7). Figure 6 gives
the longitudinal response to an x-displacement demand, while figure 7 shows the
lateral response to a y-displacement demand. Results are represented in Bode
diagram form, showing gain and phase against circular frequency. The input in
these trials is at the furthest extent of the preview from the unicycle. That is,
it is at the preview horizon. The unicycle is required to track what the rider
can see at the horizon and it will take some time to arrive there. The system
contains a transport lag, indicated by the plot symbols. Perfect tracking requires
a gain of unity and a phase lag corresponding to the transport delay. This phase
lag amounts to 180nTs u/p degrees, where n is the number of preview points, Ts
the discrete time step and u the circular frequency of the perturbation. Results
are given only for perturbations from a straight-running trim state with speed of
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R. S. Sharp
Table 2. Continuous-time optimal state-feedback gains in SI units for straight-running trims at 1,
2, 4 and 8 m s−1 with R = diag[1, 10, 10, 10] and q = diag[5e2, 1e4]. Because of the symmetry of the
straight-running trims, in-plane variables feed back only to in-plane actuators (upper rows) and
conversely (lower rows).
u
1
2
4
8
qf
tw
tq
tw
tq
tw
tq
tw
tq
u
1
2
4
8
tj
tf
tj
tf
tj
tf
tj
tf
−769.9
−91.86
−770.2
−91.87
−770.8
−91.90
−771.9
−91.97
qr
−376.1
−131.5
−376.2
−131.5
−376.4
−131.5
−376.9
−131.5
ff
jr
4376
−1296
1601
−693.4
690.5
−305.3
315.4
−224.9
242.3
−73.25
200.5
−98.52
156.3
−93.04
100.8
−55.61
u
qf
qr
qw
−38.97
6.625
−38.97
6.626
−38.97
6.629
−38.96
6.633
−170.0
−23.44
−170.0
−23.44
−170.2
−23.44
−170.4
−23.45
−90.27
−25.42
−90.30
−25.43
−90.35
−25.43
−90.45
−25.43
0.7962
0.0922
0.7961
0.0922
0.7959
0.0922
0.7955
0.0922
fr
v
rf
pf
1705
−747.6
704.9
−619.0
299.9
−519.9
67.60
−441.8
−1282
361.1
−466.2
184.5
−196.6
67.98
−89.10
37.06
293.2
−126.0
217.3
−184.9
126.6
−212.5
14.98
−175.8
1665
−533.2
634.5
−336.4
271.7
−201.2
105.4
−148.9
rr
110.0
−47.94
81.34
−70.97
46.56
−82.89
2.888
−70.15
pr
622.5
−236.0
247.4
−176.5
105.4
−133.9
31.14
−110.2
2 m s−1 but four different levels of control tightness are illustrated. Corresponding
weights are q = diag[5e2, 1e4], diag[5e3, 1e5], diag[5e4, 1e6] and diag[5e5, 1e7],
with R = diag[1, 10, 10, 10] in each case.
With respect to each direction, the system phase lag matches the transport
lag closely up to a circular frequency just above 3 rad s−1 . The lack of phase
precision for higher frequencies implies worsening tracking performance whatever
the control tightness. With loose control, gain attenuation sets the limit for good
tracking at lower frequencies than this but, if the disturbance-input frequency is
low enough in relation to the control tightness, the tracking capability is excellent.
Two cornering trim states, involving frame roll angles of 3.19◦ and 6.58◦ and
the optimal controls corresponding to them, now cross-coupled, are discussed in
the electronic supplementary material, appendix E.
5. Path-tracking simulations
Each tracking simulation run starts with the definition of the path to be followed
in the form of (x, y, t) points, with t values conveniently separated by Ts , the
sampling interval. The trim state and the initial conditions define the course of
the unicycle if only the trim controls are utilized, so that differences between the
(x, y) positions implied by the trim and those demanded by the path are used,
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(a)
0.10
gain (N )
Unicycle dynamics
0.05
x-errors
speeds 1, 2, 4, 8 m s–1
0
−0.05
gain (N )
(b)
gain (N )
(c)
y-errors to yaw torque
0.2
0.1
0
−0.1
−0.2
y-errors to roll torque
0.10
0.05
0
−0.05
−0.10
0
1
2
3
4
5
6
preview time (s)
7
8
9
10
Figure 5. Preview gain sequences for the straight-running unicycle at 1, 2, 4 and 8 m s−1 , with q =
diag[5e2, 1e4], R = diag[1, 10, 10, 10] and 1000 preview points. The lack of cross-coupling between
longitudinal and lateral problems for this symmetric trim implies that the gains not shown are
all zero. (a) Solid line, wheel torque: dashed line, pitch torque. (b,c) Solid line, 1 m s−1 ; dashed
line, 2 m s−1 ; dash dotted line, 4 m s−1 ; dotted line, 8 m s−1 .
together with the optimal gains, to derive the control perturbations necessary to
track the path. The optimal controls are obtained in a fixed frame of reference,
owing to the simplicity of the road model in such a reference system, but general
path-tracking is feasible only if the controls are applied in a local, rider’s view,
reference frame (Sharp & Valtetsiotis 2001; Sharp 2005, 2006, 2007a,b,d). At
each time step, Ts , therefore, the position and orientation of the unicycle are
used to transform the road data belonging to the current time up to the preview
horizon, nTs ahead, into the local frame of the rider. The reference axes are
shifted to coincide with the unicycle, so that there is no problem in tracking paths
which turn through large angles. Figure 8 illustrates the situation. Relationships
between the feedback gains for longitudinal, lateral- and attitude-angle position
errors and summations over the preview gains ensure invariance of the controls
when the reference axes are moved (Sharp & Valtetsiotis 2001).
First, a lane-change manoeuvre is performed with trim state, straight running
at 2 m s−1 , and controls described in table 2 and figure 5. In the path description,
the x-points are separated in the x-direction by a constant 0.02 m, so that modest
speed variations are called for. Results are shown in figures 9–11, where it can
be seen that the tracking errors are generally less than 0.2 m, being similar in
longitudinal and lateral senses. Corner-cutting is in evidence and the main errors
derive from that. Tighter control could be used to reduce the errors but there
would be an increased risk of loss-of-control occurring in the absence of adaptation
(Thommyppillai et al. 2009a). Control torques used are very modest, with the
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R. S. Sharp
(a)
0
gain (db)
−10
−20
−30
−40
−50
(b)
0
phase (º)
−500
−1000
−1500
−2000
−2500
10−1
100
circular frequency (rad s–1)
101
Figure 6. (a,b) Frequency responses to longitudinal demands of closed-loop systems with q =
diag[5e2, 1e4] (solid line), diag[5e3, 1e5] (dashed line), diag[5e4, 1e6] (dash dotted line) and
diag[5e5, 1e7] (dotted line), with R = diag[1, 10, 10, 10] in each case, using 1000, 500, 250 and
200 preview points, respectively.
pitch torque small, and yaw and roll torques of similar sizes in this case. It should
be clear that the yaw torque control is more useful under transient than steadystate conditions, since it operates through the inertia of the rider’s upper body
but there is no such limitation with respect to the roll-torque control. If only
one of these controls were allowed (Vos & Von Flotow 1990; Vos 1992; Naveh
et al. 1999; Zenkov et al. 1999), it is likely that performance would be prejudiced
substantially.
Secondly, a clothoid manoeuvre is set up to demonstrate tracking capability
up to a lateral acceleration of about 1.17 m s−2 through the use of gain-switching
control. The clothoid has curvature increasing in proportion to the distance
travelled along the track (Bronshtein & Semendyayev 1971). The speed demand is
a constant 2 m s−1 . At the start, the same straight-line controls as used for the lane
change are installed. Then, after every 40 s interval, new controls designed for the
path curvature now current are installed, and the manoeuvre is continued. The
total distance covered is 720 m, with duration 360 s. Excellent tracking is evident
in figure 12. x- and y-tracking errors are given in figure 13, where it can be seen
that the errors grow as the manoeuvre becomes more demanding. The largest
errors are of the order of 0.2 m both longitudinally and laterally. The switching
points are clear in the control plots (figure 14). Interpolating between the controls
(Thommyppillai et al. 2009a,b) would avoid these discontinuous actions but the
results would be less interesting in such a case. Frame roll and pitch angles and the
rider upper body relative angles are shown in figure 15, where the consequences
of control switching can also be seen.
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Unicycle dynamics
(a)
0
gain (db)
−10
−20
−30
−40
−50
(b)
0
phase (º)
−500
−1000
−1500
−2000
−2500
10−1
100
circular frequency
101
(rad s–1)
Figure 7. Frequency responses to lateral demands of closed-loop systems with q = diag[5e2, 1e4],
diag[5e3, 1e5], diag[5e4, 1e6] and diag[5e5, 1e7], with R = diag[1, 10, 10, 10] in each case, using
1000, 500, 250 and 200 preview points, respectively. (a,b) Solid line, q = [5e2 0; 0 1e4]; dashed line,
q ∗ 10; dash dotted line, q ∗ 100; dotted line, q ∗ 1000.
y
unicycle
road
trim state
trim sideslip angle
path tange
nt
x
Figure 8. Frozen-time depiction of discrete-time tracking problem, with the reference-axis origin
placed at the unicycle. The picture depicts a cornering trim state. x- and y-components of error
vectors are used for preview control. Arrows show error vectors whose x- and y-components are
used for preview control. Circle, trim trajectory points evenly spaced along circular arc for intervals
Ts ; square, ideal path points for intervals Ts .
The clothoid path was chosen partly to enable the determination of trim
states through simulation. The rate of change of curvature is low with the
parameters selected. The unicycle and rider therefore pass through nearequilibrium states in the (virtual) tracking experiment. The tyre aligning moment
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(a)
4
y-displacement (m)
R. S. Sharp
3
2
1
10
0
20
30
x-displacement (m)
40
50
60
position error (m)
(b) 0.3
0.2
0.1
0
−0.1
−0.2
0
5
10
15
20
25
time (s)
Figure 9. Path-tracking of the unicycle in a lane-change manoeuvre with controls from table 2
and figure 5. (a) Shows the paths of the roadway and the unicycle, while (b) shows longitudinal
and lateral position errors. (a) Solid line, unicycle path; dashed line, ideal path. (b) Solid line,
longitudinal; dashed line, lateral.
(a)
torque (Nm)
5
0
−5
(b)
torque (Nm)
10
5
0
−5
−10
0
5
10
15
20
25
time (s)
Figure 10. Control torques for a lane-change manoeuvre with controls from table 2 and figure 5.
(a) Solid line, wheel; dashed line, pitch. (b) Solid line, yaw; dashed line, roll.
must be near zero and the tyre side-force must account for the lateral acceleration
at any time. The aligning moment has contributions from side-slip, turn-slip
and camber. The path curvature and speed define a turn-slip, governing this
contribution to the aligning moment and more or less determining the tyre
side-slip required, since the camber contribution is small. The side-slip, in turn,
determines the relevant contribution to the tyre side-force and the remaining force
required mainly derives from camber. This sets the wheel camber angle. Then, for
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Unicycle dynamics
4
rider roll
angle (º)
2
0
−2
frame roll
−4
−6
−8
0
5
10
15
20
25
Figure 11. Frame and rider upper body relative angles in a lane-change manoeuvre with controls
from table 2 and figure 5. Dashed line, frame pitch; dash dotted line, rider yaw; dotted line, rider
pitch.
60
y-displacement (m)
50
40
30
20
10
0
10
20
30
40
50
60
70
x-displacement (m)
Figure 12. Path-following performance of loosely controlled unicycle at 2 m s−1 following a clothoid
with control switching every 40 s. Solid line, unicycle path; dashed line, ideal path.
proper balance in roll, the rider upper body angle follows. In the development of
the results shown, the parameter Crz , the coefficient relating turn-slip to aligning
moment, has been varied widely and, through the mechanism described above, its
value has been found to have a strong influence on the relationship between the
rider upper body lean and the frame lean angles. The value chosen (table 1), gives
nice behaviour with respect to both steady turning and controllability. It would
be of interest to determine Crz experimentally for unicycle tyres, or alternatively
to study the lean angles adopted by unicycles and riders in steady turns of various
sorts. Nice behaviour is characterized by very small side-slips in steady turning.
Then, the camber angle is such that the tyre develops the necessary side-force for
turning and the rider’s body needs to lean with the frame, implying that fr will
be small in steady state.
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R. S. Sharp
position error (m)
0.3
0.2
0.1
0
−0.1
−0.2
0
50
100
150
200
time (s)
250
300
350
Figure 13. x- and y-tracking errors for clothoid manoeuvre at 2 m s−1 with control switching at 40 s
intervals. Solid line, longitudinal; dashed line, lateral.
torque (Nm)
(a) 4
2
0
−2
torque (Nm)
(b)
2
0
−2
−4
−6
0
50
100
150
200
time (s)
250
300
350
Figure 14. Control torques for clothoid manoeuvre at 2 m s−1 with control switching at 40 s
intervals. (a) Solid line, wheel; dashed line, pitch. (b) Solid line, yaw; dashed line, roll.
4
angle (º)
2
rider roll
0
−2
−4
frame roll
−6
0
50
100
150
200
250
300
350
Figure 15. Frame and rider upper body relative angles in a clothoid manoeuvre with control
switching at 40 s intervals. Dashed line, frame pitch; dash dotted line, rider yaw; dotted line,
rider pitch.
6. Conclusions
A uniquely representative mathematical model of a unicycle and rider has been
constructed and tested through simulation. The tyre force and moment model,
in particular, is descriptive of real tyres operating at modest slip ratios and
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Unicycle dynamics
19
camber angles. A full set of possibilities for actuation by the rider is included,
with bandwidth limitations representative of real riders. The symbolic equations
of motion, for small perturbations from straight running, have been presented.
Representative sets of parameter values have allowed descriptions of a unicycle
and a monocycle and root-locus results have been provided for each one. Care
has to be exercised in choosing a model and parameter values yielding feasible
steady turning behaviour.
Identical eigenvalues have been obtained through more or less independent
processes for generating the equations of motion, one involving automated
multibody software based on Kane’s equations, the other using Lagrange’s energy
method and symbolic Matlab. The assumption that the tyres roll without slip,
providing constraints on the motion as opposed to developing forces and moments,
has been examined in root-locus form.
Linear optimal preview control theory developed in the context of car driving
and motorcycle and bicycle riding has been shown to apply straightforwardly
to the unicycle, in the form of the above model. Longitudinally and laterally
decoupled controls for small perturbations from straight-running trim states and
cross-coupled controls appropriate to cornering trims have been demonstrated.
Rider-controlled tracking capabilities have been shown for varying levels of
control tightness by means of frequency-response calculations. Path-tracking
has been illustrated and simulation results shown for lane-change and clothoid
manoeuvres. The lane change demonstrates a transient capability while the
clothoid case involves very slow changes of path curvature but extends to a
lateral acceleration of nearly 1.2 m s−2 . Gain switching to give a rudimentary
adaptation of the controls to the running conditions has been illustrated
for the clothoid.
A new appreciation of the way in which unicycle riders control their machines
comes from the results obtained. The first step in learning to ride a unicycle
concerns coping with the rather rapid divergence in pitch of the uncontrolled
system, mainly by pedalling. The second step concerns the cross-coupling of the
longitudinal wheel-torque control to the lateral motions when cornering. Yaw
torque and roll torque controls are both useful for manoeuvring, with the former
having more of a transient value and the latter being more useful in steady-state
conditions. Pitch torque control is of lesser value.
The theory and results described provide a predictive capability for robotic
unicycles, which can be used to guide the design of such machines. It appears
essential to include wheel actuation and desirable to include both yaw and roll
actuators. Linear optimal preview control with gain scheduling to cover speed
and both longitudinal and lateral acceleration ranges provides an excellent basis
for control design.
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