Modelling, simulation and control of an electric unicycle

A. Kadis, D. Caldecott, A. Edwards, M. JerbicR. Madigan, M Haynes, B. Cazzolato and Z. Prime,

The University of Adelaide, Australiaandrewkadis@gmail.com

Abstract

In this paper the outcomes from a one yearhonours project to design, build and control aself-balancing electric unicycle, known as theMicycle, are presented. The design of the sys-tem along with the associated mechanical andelectrical components is presented. This is fol-lowed by a derivation of the system dynamics us-ing the Lagrangian method which are simulatedusing Simulink. A linear control strategy tostabilize in the pitch direction is then proposed.Finally, a comparison between the control sys-tems results from the Simulink simulations andexperimental results from the physical systemis presented.

1 Introduction

The University of Adelaide’s School of Mechanical Engi-neering has a history of encouraging honours students todevelop complex systems engineering projects involvingthe design, build and control of nonlinear, inherentlyunstable plants. The Micycle, shown in Figure 1, isthe last in a long series of such projects. These can beseen in Figure 2 and include a double Furuta pendulum[Driver and Thorpe, 2004], self-balancing inline scooters[Clark et al., 2005; Baker et al., 2006], a 6DOF VTOLplatform [Arbon et al., 2006], a ballbot [Fong and Uppill,2009] and an electric diwheel [Dyer et al., 2009].

In March 2010, honours students in the School of Me-chanical Engineering at the University of Adelaide com-menced the design, build and control of an electric selfbalancing unicycle called the Micycle. The dynam-ics of single wheeled ‘inverted pendulum vehicles, suchas the two wheel self-balancing robot [Yamamoto andChikamasa, 2009], ballbot [Fong and Uppill, 2009] andSegway clones [Clark et al., 2005; Baker et al., 2006]are well known. The manner in which humans regulatepedal powered unicycles has been identified experimen-tally [Sheng and Yamafuji, 1997]. Numerous electric

Figure 1: Photograph of the Micycle.

unicycles have been built that demonstrate self-balancingcapability, one of the first being that of Schoonwinkel[1987]. Most early attempts to stabilise the system in thepitch direction used linear controllers: either classical PDcontrollers [Hofer, 2005, 2006; Blackwell, 2007] or opti-mal state controllers [Schoonwinkel, 1987; Huang, 2010].Later control systems employ non-linear strategies forboth pitch and roll stabilisation in path tracking [Navehet al., 1999].

(a) Double Furutapendulum.

(b) EDGAR scooter.

(c) SON of EDGARscooter.

(d) 6DOF VTOL.

(e) Ballbot. (f) Electric Diwheel.

Figure 2: Previous examples of unstable nonlinear under-actuated systems built by honours students at The Uni-versity of Adelaide.

In this paper, the Micycle system is discussed indetail. This involves a discussion of the mechanical andelectrical components used in the system. The mechan-ical steering mechanism, used to steer to and to assistthe rider in balancing in the roll direction is discussed.Following this, the dynamics of a generic unicycle in thepitch direction are derived using a Lagrangian formu-lation and simulated in Simulink. A linear PD controllaw which stabilizes the plant in the pitch direction isthen proposed. This controller is implemented on thesimulated and physical systems and data from both isused to quantitatively assess the control system.

2 Mechanical and electrical system

The design purview of the Micycle was to provide afully rideable self-balancing electric unicycle which couldbe learnt to ride in under an hour. To accomplish this,a control system was used to balance the system in thepitch direction and a mechanical steering linkage wasdesigned to assist the rider balancing in the roll direction.The mechanical design, including the steering, and theelectrical components, required to implement an embed-ded control system, are discussed here. These can beseen in a rendered solid model of the Micycle, shownin Figure 3.

Figure 3: A rendered view showing the of the componentsof the Micycle.

The mechanical elements of the Micycle consist of acentral plate and a fork assembly to hold the motor. Themain plate houses the batteries, microcontroller, powerdistribution board, inertial measurement unit (IMU) andmotor controller. Additionally, the seat and pole onwhich the rider sits is mounted at the top of the plate.The fork assembly connects the main plate to the hubmotor and incorporates a steering mechanism.

The steering mechanism is a major feature of the design.The fork assembly acts as a mechanical linkage betweenthe footpegs and the steering mechanism. The mechanismincorporates a torsion spring to centre the steering armand a rotary damper to eliminate overshoot and wobble.Additionally, a mechanical limit of −15◦ prevents over-steering. The end result of this is that when pressure isplaced on either of the footpegs, the wheel will turn inthat direction independent of the frame. A discussion ofthe specific geometry of the steering mechanism is beyond

the scope of this paper, but the important dimensionscan be seen in Figure 4.

Figure 4: Diagram showing steering geometry designangles.

The steering provides two functions. Firstly, it assiststhe rider in balancing in the roll direction. When fallingto the left, one simply presses on the right footpeg tohelp stabilise one’s self. Although this sounds counter-intuitive, it is very effective in practice. The secondfunction of the steering is that it makes it easy to turn.Typically, a unicyclist must use their arms and hipsto twist for turning but the steering mechanism allowssmooth turning controlled by simply pressing on the footpegs.

The system is actuated by a Golden Motors MagicPieMP-16F brushless DC hub motor capable of deliveringa maximum torque of 30 Nm at a maximum speed of250 rpm. The motor is controlled with a maxon motorEPOS2 70/10 motor controller operating in a currentcontrol mode, with a current loop bandwidth of 10 kHz.Electrical power is provided by a 36 V, 10 Ah capacity,LiFePO4 battery pack.

The control logic is processed by a WytecMiniDRAGON-Plus2 Development Board microcon-troller, with control state feedback provided by two mainsensors. A MicroStrain 3DM-GX2 IMU operating at300 Hz provides the pitch angle position and pitch an-gular rate. Hall sensors built into the hub motor returnthe angular velocity of the wheel, from which the trans-lational velocity of the platform is estimated by a linearvelocity approximation. The microcontroller was alsoused to implement a number of safety features into the

system using various safety logic cutoffs. A dSPACEDS1104 prototyping board was also used to design andtune the control system and it is this system and not theembedded one which was used to collect data from thephysical system in this paper.

3 Dynamics of the 2DOF system

The dynamics of the Micycle are developed from theinverted pendulum model used extensively in Driver andThorpe [2004] and further developed in Fong and Uppill[2009]; Huang [2010]; Lauwers et al. [2006] to include thetranslational motion of the pendulum. However, thesederivations are inconsistent with regards to coordinateframes and non-conservative forces. Therefore an ex-tensive verification process was undertaken to derive thedynamics of the Micycle. This process was based on thedynamics derived in Nakajima et al. [1997]; Nagarajanet al. [2009] and through coordinate transforms, verifiedwith the papers discussed above.

The following assumptions have been made in thederivation of the dynamics, with reference to the coordi-nate system and directions shown in Figure 5.

Figure 5: The coordinate system used in the derivationof the system dynamics.

• Motion is restricted to the xy-plane

• A rigid cylinder is used to model the chassis and avertically orientated thin solid disk used to modelthe wheel

• Coulomb friction arising from the bearings and tyre-ground contact is neglected, and hence only viscousfriction is considered

• The motor is controlled via an intelligent controllerin ‘current mode’ such that the input into the plantis a torque command

Table 1: Constants used to model the Micycle.

Symbol Value Description

rw 0.203 m Radius of the wheelrf 0.300 m Distance to the centre of mass

of the frame from the originmw 7.00 kg Mass of the wheelmf 15.0 kg Mass of the frameIf 0.450 kgm2 Moment of inertia of the

frame w.r.t its centre of massIw 0.145 kgm2 Moment of inertia of the w.r.t

its own centre of massµφ 0.08

Nm/(rad/s)Coefficient of rotationalviscous friction (bearing

friction and motor losses)µθ 0.05

Nm/(rad/s)Coefficient of translational

viscous friction (rollingresistance)

kτ 1.64 Nm/A Motor torque coefficientg 9.81 m/s2 Gravitational acceleration

• There is no slip between the tyre and the ground

The model is defined in terms of coordinates φ and θ,where:

• φ — rotation of the frame about the z-axis

• θ — rotation of the wheel relative to the frame angle

The origin of the right-handed coordinate frame islocated at the centre of the wheel, as shown in FigureFigure 5. The positive x-direction is to the right andpositive y is upward. The two angular quantities, φ andθ, have been chosen such that anti-clockwise rotationsabout the z-axis are considered positive. The zero datumfor the measurement of the frame angle φ is coincidentwith the positive y-axis and the wheel angle θ is measuredrelative to φ.

3.1 Nomenclature

With reference to Figure 5, Tables 1 and 2 define thecontants and variables used in the derivation of the plantdynamics that follows in Section 3.2 .

3.2 Non-linear dynamics

The Euler-Lagrange equations describe the dynamicmodel in terms of energy and are given by:

d

dt

(∂L∂qi

)− ∂L∂qi

= Fi

where the Lagrangian L is the difference in kinetic andpotential energies of the system, qi are the generalised co-ordinates (in this case θ and φ) and Fi are the generalised

Table 2: Variables used in modelling the Micycle.

Symbol Description

θ Angular position of the wheelwith respect to the frame(anti-clockwise positive)

θ Angular velocity of the wheel

θ Angular acceleration of thewheel

φ Angular position of the framewith respect to the positive

y-axis

φ Angular velocity of the frame(anti-clockwise positive)

φ Angular acceleration of theframe

τ Torque applied by the motor,exluding friction

i Motor supply current

forces. The kinetic and potential energies of the wheeland frame are denoted Kw, Vw, Kf and Vf respectively.

Kw =Iwθ

2

2+mw(rwθ)

2

2,

Vw = 0,

Kf =mf

2

(r2wθ

2 + rf φ cosφ)2

+If2θ2,

Vf = mfgrf cosφ.

If the generalised coordinates are q = [ θ, φ ]T , then

d

dt

(∂L∂q

)− ∂L∂q

=

[0τ

]− D(q)

where

D(q) =[

µθ θ , µφφ]T

is the vector describing the viscous friction terms. There-fore, the Euler-Lagrange equations of motion can beexpressed as

M(q)q + C(q, q) +G(q) +D(q) =

[0τ

].

Where the mass matrix, M(q), is

M(q) =

[Iw + r2w (mf +mw) mfrwrf cosφ

mfrwrf cosφ If + r2f cos2 φmf

],

the vector of centrifugal effects, C(q,q), is

C(q,q) =

[−mfrfrwφ

2 sinφ

−mfr2f φ

2 sinφ cosφ

],

and the vector of gravitational forces, G(q), is

G(q) =

[0

−mfrfg sinφ

].

These are described in the standard non-linear statespace form by defining the state vector, x =

[qT qT

],

and the input as u = τ . This, together with the aboveequations gives

x =

q

M(q)−1

([0τ

]−C(q, q)−G(q)−D(q)

) = f(x, u)

4 Electromechanical system dynamics

The derivation of the mechanical dynamics assumes amotor torque input into the system. In practice, thetorque is controlled by the motor controller operating incurrent mode. There exists a linear relationship betweenthe motor supply current and the resultant motor torque:

t = kti

where τ is the motor torque, kt = 1.64 Nm/A is the motortorque constant and i is the motor supply current. Thetorque constant was deduced experimentally.

Figure 6: Wireframe view of the SimMechanics modelshowing linkages and centres of mass.

Table 3: Variables used in modelling the Micycle.

Parameter Value Description

Kp 800 A/rad Proportional gainTd 0.1 sec Derivative time

constantTd

N 0.01 sec Low pass filter timeconstant

5 State Estimation

There are two states which the control system is requiredto measure. These are φ and φ, the angular positionand angular rate of the frame respectively. Figure 7shows how these states are read from the IMU. Note thatthe φ value is read directly from the IMU rather thandifferentiating the φ value. This is because there is lesslatency in the φ filters than the φ filters. However, thefilter implemented with the IMU are proprietary. It isknown that the φ filters are slower than the φ filters,but no other specifics are known about their structure orfrequency response. Thus, the φ was read directly fromthe IMU rather than differentiating φ.

6 PD controller

The control strategy employed here uses a standardproportional-derivative (PD) controller. The implementa-tion of this controller can be seen in Figure 7. The reasonfor why a PID controller was not used is that a humannaturally acts to reduce the steady state error and theaddition of integral control can degrade the performanceof the controlled response [Clark et al., 2005].

A low pass filter was used on the derivative controlterm to make the controller proper and to filter out noisefrom the sensors in the physical system. The parametersof the tuned control system are presented in Table 3. Thetransfer function for the designed PD controller is pre-sented in Equation (1). Note that strictly speaking, thesystem actually consists of two distinct transfer functions,one for φ and one for φ, as different sensors are used foreach state. Nevertheless, this is a PD controller andEquation (1) represents the effective transfer functionwith the two feedback terms combined.

Gc = Kp(1 +sTd

1 + sTd

N

) (1)

7 Numerical simulations

The differential equations derived in Section 3 were mod-elled within Simulink. For validatation of this simulation,a parallel model was constructed using SimMechanicsas shown in Figure 6. Additionally, a virtual reality

Figure 7: Simulink block diagram representing the state estimation and the PD controller.

(VR) model developed from the Simulink implementationwas also built to aid in the visualisation of the plantperformance.

The control strategy detailed in Section 6 was thenintegrated into the Simulink model. This was used togenerate numerical simulations of the closed loop systemresponse, presented in Section 8.

8 Experimental data

Data was collected on the physical Micycle system andcompared to the results obtained from the correspondingnumerical simulations.

8.1 Testing methodolgy

Despite the fact that the Micycle is fully functional asan embedded system, it is not possible to collect datawhen it is being run off a microcontroller. Thus it wasnecessary to run the control system in Simulink through adSPACE DS1104 prototyping board for data acquisition.As the Micycle does not balance in the roll directionwithout a rider, it was necessary to constrain the wheelso that it would not move. This experimental setup isshown in Figure 8.

The experimental procedure involved offsetting theframe from the vertical axis by a known angle and re-leasing it. The initial position of the frame, φ, acts asa disturbance to the plant and the performance of thecontrol system is assessed through the ability to returnthe system to 0◦.

To allow the numerical simulations to correlate withthe physical data, the Simulink model of the systemdynamics was modified to reflect this constraint. Thewheel angle, θ, was set to a fixed value of 0◦ and φ wasset to the appropriate initial position.

It should be noted that although this does not quite rep-resent real world operation of the Micycle, this method-ology offered a means of being able to quantiatively assessthe performance of the physical control system comparedto the simulated system. In riding the Micycle it hasbeen found that the control system does indeed possess a

limited ability to reject disturbances such as bumps andslopes. Nevertheless, these are difficult to quantify andthus the chosen method was adopted.

Figure 8: Photograph of the Micycle with the wheelconstrained as used in testing. Note the clamps at eitherend of the timber fix the translational position of thesystem while allowing the frame to rotate about thewheel.

8.2 Experimental closed loop PDcontroller

The above methodology was then applied to the physicaland simulated Micycle systems. The results can beseen in Figures 9, 11, 10 and 12.

Control signal (A)

Pitch angle (Deg)

Amplitude

Time (sec)

0 0.5 1 1.5 2−30

−20

−10

0

10

20

30

Figure 9: Closed loop response of the constrained physicalMicycle system when rotated to 20◦ and released.

Control signal (A)

Pitch angle (Deg)

Amplitude

Time (sec)

0 0.5 1 1.5 2−30

−20

−10

0

10

20

30

Figure 10: Closed loop response of the constrained phys-ical Micycle system when rotated to 5◦ and released.

Control signal (A)

Pitch angle (Deg)

Amplitude

time (sec)

0 0.5 1 1.5 2−30

−20

−10

0

10

20

30

Figure 11: Closed loop response of the constrained simu-lated Micycle system when rotated to 20◦ and released.

Control signal (A)

Pitch angle (Deg)

Amplitude

time (sec)

0 0.5 1 1.5 2−30

−20

−10

0

10

20

30

Figure 12: Closed loop response of the constrained simu-lated Micycle system when rotated to 5◦ and released.

These results demostrate three main points. The firstis that the control system suffices for both the simulatedand physical systems. The second is that the response ofthe simulated system is superior to that of the physicalsystem. Finally, the controller has a strong tendency tosaturate.

Both the physical and simulated systems present a con-trol system which is able to maintain control authorityof an inherently unstable plant in response to a distur-bance. Rise times are less than a quarter of a secondand there is minimal overshoot, rendering this a suitablecontrol system to be used in a self-balancing unicycle tobe ridden by a human. This corellates with what hasbeen experienced in riding the Micycle, where it is fullyrideable and has the ability to resist small disturbances.

The response of the simulated system is superior to thatof the physical system in that it exhibits less overhsootand a faster response. It is thought that the physicalsystem has a time delay in the control system whichis currently unmodelled in the simulated system. Thiswould add phase lag in the control system leading to moreovershoot in the response. The simulated system wasrerun with time delays modelled in the φ and φ feedbackterms and it was possible to generate a similar responseto the physical system. However, it is suspected thatthe discrepancy is not merely the result of a pure timedelay, but also effected by the frequency response of thefilters used on the IMU sensor values. Unfortunately,these filters are proprietary and the specific details oftheir implementation is not known.

Finally, it is worth highlighting the fact that the motorsaturates in all cases. However, this does not appear to beparticulary detrimental to the control of the system. Thesaturated torque is still sufficient to overcome gravity andlift the frame. It is important to realise that in riding theMicycle, a more typical use case, there will not be sucha propensity for the motor to saturate. This is becausea rider will not typically see angular disturbances aslarge as 20◦. Smaller angular disturbances with a largermf , due to the additional mass of the rider, are morecommon and these will be less likely to saturate. The highdegree of saturation observed in the presented results isa consequence of the large proportional error inducedin this specific testing methodology. That said, it isrecognised that there is a strong tendency for the actuatorto saturate and a higher degree of control authority isdesirable. This is mainly due to the difficulty in procuringa higher capacity brushless DC motor controller and issomething that needs to be further investigated.

9 Conclusion and future work

In this paper the dynamics of the unicycle were derivedand presented. Numerical simulations were used to ver-ify this derivation. This information was then used to

simulate a PD controller whose performance was thencompared with that of the physical Micycle system.

Future work includes the development of a model basednon-linear controller and a backstepping controller. Thesecontrol strategies will be compared and benchmarked,with the optimal strategy being implemented into theMicycle design. A higher capacity motor controllershall also be integrated into the system to alleviate thehigh tendency to saturate. Another planned developmentis the addition of active stabilisation in the roll direction.This will use either a reaction wheel or a control momentgyroscope and this actuator will allow the Micycle tobe a completely self-balancing electric unicycle.

Acknowledgements

The authors would like to acknowledge the support of theSteve Kloeden, Phil Schmidt, Dr. Michael Riese, NorioItsumi and Silvio De Ieso for their efforts in constructingthe physical Micycle used in this paper.

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