modelling, simulation and control of an electric...
TRANSCRIPT
Modelling, simulation and control of an electric unicycle
D. Caldecott, A. Edwards, M. Haynes, M. Jerbic, A. Kadis, R. Madigan, Z. Prime and B. CazzolatoUniversity of Adelaide, Australia
AbstractIn this paper the outcomes from an honours projectto design, build and control a self-balancing electricunicycle are presented. The design of the mechan-ical and electrical components is discussed, fol-lowed by a derivation of the dynamics of a genericunicycle. A linear control strategy to stabilise theunicycle is implemented on the physcial system anda simulated model is derived from the system dy-namics. Finally, a comparison of the results fromsimulations conducted on a model of the unicycleas well as experimental results are presented.
1 IntroductionThe School of Mechanical Engineering at the University of
Adelaide has a long history of encouraging honours studentsto undertake complex systems engineering projects involvingthe design, build and control of nonlinear, inherently unsta-ble, under-actuated plants. Some of these are shown in Figure1 and include a double Furuta pendulum [Driver and Thorpe,2004], self-balancing inline scooters [Clark et al., 2005;Baker et al., 2006], a 6DOF VTOL platform [Arbon et al.,2006], a ballbot [Fong and Uppill, 2009] and an electric di-wheel [Dyer et al., 2009].
In March 2010, honours students in the School of Mechan-ical Engineering at the University of Adelaide commencedthe design, build and control of an electric self-balancing uni-cycle called MICYCLE. The dynamics of single wheeled’inverted pendulum’ vehicles, such as the two-wheel self-balancing robot [Yorihisa Yamamoto, 2009], ballbot [Fongand Uppill, 2009] and Segway clones [Clark et al., 2005;Baker et al., 2006] are well known. The manner in whichhumans regulate pedal powered unicycles has been identi-fied experimentally [Sheng and Yamafuji, 1997; Shimamuraet al., 2009]. Numerous electric unicycles have been builtthat demonstrate self-balancing capability, one of the first be-ing that of Schoonwinkel [1987]. Most early attempts to sta-bilise the system in the pitch direction used linear controllers:either classical PD controllers [Hofer, 2005; 2006; Black-well, ] or optimal state controllers [Schoonwinkel, 1987;
(a) Double Furuta pen-dulum.
(b) EDGAR scooter. (c) SON of EDGARscooter.
(d) 6DOF VTOL. (e) Ballbot. (f) Electric Diwheel.
Figure 1: Previous examples of unstable nonlinear under-actuated systems built by honours students at the Universityof Adelaide.
Huang, 2010]. Later control systems employ non-linearstrategies for both pitch and roll stabilisation in path track-ing [Naveh et al., 1999].
In this paper the dynamics of a generic unicycle using a La-grangian formulation are presented. A linear control law forthe stabilisation of the plant is presented. Details of the MI-CYCLE unicycle construction and hardware are provided. Thedynamic and control parameters of the MICYCLE are used ina numerical simulation which is compared with experimentalresults.
Figure 2: Rendered image of the MICYCLE.
2 Mechanical and electrical systemA rendered solid model of the MICYCLE is shown in Figure
2 and Figure 3 shows the as-built platform.The mechanical elements of the MICYCLE consist of a cen-
tral plate and a fork assembly that provides steering and holdsthe hub motor. The main plate houses the batteries, microcon-troller, power distribution board, inertial measurement unit(IMU) and motor controller. Additionally, the seat and poleon which the rider sits is mounted at the top of the plate. Thefork assembly connects the main plate to the hub motor andprovides a steering mechanism.
The steering mechanism is a major feature of the design. Itenables the rider to steer by placing pressure on the footpegs,rather than by twisting the frame in the unsteady motion re-quired by a regular unicycle. The mechanism incorporates atorsion spring to centre the steering arm and a rotary damperto eliminate overshoot and wobble. Additionally, a mechani-cal limit of ±15◦ prevents over-steering.
The system is actuated by a MagicPie MP-16F brushlessDC hub motor capable of delivering a maximum torque of30 Nm at a maximum speed of 250 rpm. The motor is con-trolled with a MaxonMotor EPOS2 70/10 motor controlleroperating in a current control mode, with a current loop band-width of 10 kHz. Electrical power is provided by a 36 V,10 Ah capacity, LiFePO4 battery pack.
The control logic is processed by a Wytec MiniDRAGON-Plus2 Development Board, with control state feedback pro-vided by two main sensors. A Microstrain 3DM-GX2 IMUoperating at 300 Hz provides the pitch angle position andpitch angular rate. Hall sensors built into the hub motor re-turn the angular velocity of the wheel, from which the transla-
Figure 3: The as-built MICYCLE.
tional velocity of the platform is estimated by a linear velocityapproximation.
3 Dynamics of the 2DOF systemThe dynamics of the Micycle are developed from the
inverted pendulum model used extensively in Driver andThorpe [2004] and further developed in Fong and Uphill[2009], Huang [2010] and Lauwers and Hollis [2006] to in-clude the translational motion of the pendulum. However,these derivations are inconsistent with regards to coordinateframes and non-conservative forces. Therefore an extensiveverification process was undertaken to derive the dynamics ofthe MICYCLE. This process was based on the dynamics de-rived in Nakajime et al. [1997] and through coordinate trans-forms, verified with the papers discussed above.
The following assumptions have been made in the deriva-tion of the dynamics, with reference to the coordinate systemand directions shown in Figure 4.
• Motion is restricted to the xy-plane.
• A rigid cylinder is used to model the chassis and a verti-cally orientated thin solid disk used to model the wheel.
• Coulomb friction arising from the bearings and tyre-ground contact is neglected, and hence only viscous fric-tion is considered.
• The motor is controlled via an intelligent controller in“current mode” such that the input into the plant is atorque command.
• 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 is locatedat the centre of the wheel, as shown in Figure 4. The posi-tive x-direction is to the left and positive y is upward. Anti-clockwise rotations about the z-axis are considered positive.The zero datum for the measurement of the frame angle φ iscoincident with the positive y-axis and the wheel angle θ ismeasured relative to φ .
Figure 4: The coordinate system used in the derivation of thesystem dynamics.
3.1 NomenclatureWith reference to Figure 4, Tables 1 and 2 define the con-
tants and variables used in the derivation of the plant dynam-ics that follows in Section 3.2.
Table 1: Constants used in modelling the MICYCLE.
Symbol Value Descriptionrw 0.203 m Radius of the wheel
r f 0.3 mDistance to the centre of
mass of the frame from theorigin
mw 7 kg Mass of the wheelm f 15 kg Mass of the frame
I f 0.45 kg.m2Moment of inertia of theframe w.r.t. its centre of
mass
Iw 0.1445 kg.m2Moment of inertia of the
wheel w.r.t. its own centreof mass
µφ 3.5Coefficient of rotationalviscous friction (bearingfriction and motor losses)
µθ 0.12Coefficient of translational
viscous friction (rollingresistance)
kτ 1.64 Nm/A Motor torque constantg 9.81 m/s2 Gravitational acceleration
Table 2: Variables used in modelling the MICYCLE.
Symbol Description
θAngular position of the wheel with respect
to the frame (anti-clockwise positive)θ Angular velocity of the wheelθ Angular acceleration of the wheel
φAngular position of the frame with respect
to the positive y-axis
φAngular velocity of the frame
(anti-clockwise positive)φ Angular acceleration of the frame
τTorque applied by the motor, exluding
frictioni Motor supply current
3.2 Non-linear dynamicsThe Euler-Lagrange equations describe the dynamic model
in terms of energy and are given by:
ddt
(∂L
∂ qi
)− ∂L
∂qi= Fi
where the Lagrangian L is the difference in kinetic and po-tential energies of the system, qi are the generalised coordi-nates (in this case θ and φ ) and Fi are the generalised forces.The kinetic and potential energies of the wheel and frame aredenoted Kw, Vw, K f and Vf respectively.
Kw =Iwθ 2
2+
mw(rwθ)2
2,
Vw = 0,
K f =m f
2(r2
wθ2 + r f φ cosφ
)2+
I f
2θ
2,
Vf = m f gr f cosφ .
If the generalised coordinates are q = [ θ , φ ]T , then
ddt
(∂L
∂ q
)− ∂L
∂q=
[0τ
]− D(q)
whereD(q) =
[µθ θ , µφ φ
]Tis the vector describing the viscous friction terms. Therefore,the Euler-Lagrange equations of motion can be expressed as
M(q)q + C(q, q)+G(q)+D(q) =[
0τ
].
Where the mass matrix, M(q), is
M(q) =[
Iw + r2w(m f +mw
)m f rwr f cosφ
m f rwr f cosφ I f + r2f cos2 φm f
],
the vector of centrifugal effects, C(q,q), is
C(q,q) =[
−m f r f rwφ 2 sinφ
−m f r2f φ 2 sinφ cosφ
],
and the vector of gravitational forces, G(q), is
G(q) =[
0−m f r f gsinφ
].
These are described in the standard non-linear state spaceform by defining the state vector, x =
[qT qT ]
, and theinput as u = τ . This, together with the above equations gives
x=
q
M(q)−1([
0τ
]−C(q, q)−G(q)−D(q)
) = f (x,u)
4 Electromechanical system dynamicsThe derivation of the mechanical dynamics assumes a mo-
tor torque input into the system. In practice, the torque iscontrolled by the motor controller operating in current mode.In this mode, the motor controller provides a PI-controlledcurrent output in response to an analogue voltage input.
The control system feedback signal which ranges from -100 V to +100 V is scaled to 0 V to 5 V through the con-troller’s DAC (digital to analogue converter). This analogue
voltage command is sent to the motor controller, where 0 Vrepresents maximum negative current, 5 V is maximum pos-itive current and 2.5 V is zero current. Values outside of thisrange are clipped to these values, representing saturation inthe motor.
There is a linear relationship between the motor supply cur-rent and the resulting torque:
τ = kτ i (1)
where τ is the motor torque, kτ = 1.64 Nm/A is the motortorque constant and i is the motor supply current. The torqueconstant was calculated experimentally.
Therefore, using the output range of the DAC on the con-troller, the current response to a given setpoint voltage on themotor controller and kτ and the range of the output DAC onthe controller, the relationship between a given error signaland the applied torque was established.
5 Control strategy5.1 PD controller
The control strategy employed here uses a standardproportional-derivative (PD) controller. A PID controller wasinitially trialed, however, it was found that that integral con-trol was not necessary due to the presence of a human in thecontrol loop. In this case, the human naturally acts to reducethe steady state error and the addition of integral control candegrade the performance of the control response [Clark et al.,2005].
A low pass filter was used on the derivative control to makethe controller proper and to filter out noise in the sensors inthe physical system. A SIMULINK model was built and thesystem was tuned using a tethered configuration of a DSPACEDS1104 rapid prototyping system. The parameters of thetuned control system are presented in Table 3. The transferfunction for the designed PD controller is:
GC = Kp(1+sTd
1+ s TdN
) (2)
Table 3: Parameters used in the PD Controller.
Parameter Value DescriptionKp 800 Proportional gainTd 0.1 s Derivative time constantTdN 0.01 s Low pass filter time
constant
6 Numerical simulationsNumerical simulations of the system dynamics were con-
ducted on the differential equations derived in Section 3within SIMULINK. These dynamics were then integrated with
a controller as seen in Figure 6. For validatation of the differ-ential equations and their implementation within SIMULINK,a parallel model was constructed using SIMMECHANICS asshown in Figure 5. Additionally, a virtual reality (VRML)model developed from the SIMULINK implemenation wasalso built to aid in the visualisation of the plant performance.
Figure 5: Wireframe view of SIMMECHANICS model show-ing linkages and centres of mass.
Figure 6: Simulated SIMULINK dynamics integrated with PDcontroller.
The control strategy detailed in Section 5 was also inte-grated into the SIMULINK model. The resulting simulation ispresented here.
6.1 Simulated open loop responseFigure 7 shows the open loop response of the frame’s pitch
angle φ and linear position x. The frame is initially tilted30◦ from the vertical and released. The frame comes to arest at 180◦ in an inverted position. The linear translationof the frame can also be observed as it moves whilst comingto a rest. The effects of both the linear and the rotationalfriction are observable in the damping of φ and x respectively.Note that the system is highly damped and this reflects thebehaviour observed in the actual MICYCLE.
0 1 2 3 4 5
0
1
2
3
4
Time
Am
pli
tud
e
Simulated Open Loop Response
Pitch Angle (Radians)
X Position (m)
Figure 7: Open loop response from a 30◦ pitch angle positionand released.
0 1 2 3 4 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time
Am
pli
tud
e
Simulated Closed Loop Response
Pitch Angle (Radians)
X Position (m)
Control Signal (V)
Figure 8: Closed loop response from a 30◦ pitch angle posi-tion and released.
6.2 Simulated closed loop simulated PD controlFigure 8 shows the simulated closed loop response from
the PD controller developed in Section 5. The frame initiallystarts at 30◦, from the vertical, in a slightly tilted positionand is released. The controller brings the frame up to 0◦,corresponding to the vertical position. Note that Figure 8 onlyshows part of the y axis. The control signal actually starts at-100 V, where it is saturated, and then increases to the valuesshown in the graph. Despite the saturation, the response isfast and there is minimal overshoot.
7 Experimental dataData was collected on the control systems of the fully
functional MICYCLE and compared to the results obtainedfrom simulations.
7.1 Testing methodologyAs the MICYCLE does not balance in the roll direction with-
out a rider, it was necessary to secure it so that it remainedlaterally stable throughout the experiments. The wheel of theMICYCLE was constrained so that it would not move. Thisphysical system is shown in Figure 9.
The experimental procedure involved offsetting the framefrom the vertical by a known angle and releasing it, whilemaintaining the x position at zero. The initial position of theframe acts as a disturbance and the performance of the controlsystem is assessed through the ability to return the system to0◦.
Although the embedded, untethered MICYCLE softwarewas fully functional, the testing was performed with theMATLAB SIMULINK controller implemented in a DSPACEDS1104 breakout box as a large amount of infromation canbe collected and plotted with this configuration.
The SIMULINK model of the system dynamics was modi-fied to reflect this constraint, the position of the wheel θ wasset to a fixed value of 0◦ and φ was set to the appropriateinitial position. An open loop simulation of the constrainedSIMULINK model was then run to verify the results and en-sure that the modelled system still behaved as expected. Itwas found that it did and this simulation can be seen in Fig-ure 10.
7.2 Experimental closed loop PD controllerThe above methodology was then applied to the physical
and simulated MICYCLE systems. The results can be seen inFigures 11, 12, 13 and 14.
There are several interesting discussion points from theseresults. Firstly, it should be noted that for both the physicaland the simulated systems, the response appears to be suffi-ciently fast and smooth. This is to be expected as the physicalsystem has been successfully ridden with these gains and con-troller configuration. The system appears to work and does soover a range of initial conditions.
Figure 9: The MICYCLE with the wheel constrained as usedin testing. Note the clamps at either end of the timber fix thetranslation position of the system while allowing the frame torotate about the wheel.
0 1 2 3 4 5
0
1
2
3
4
Time
Am
pli
tud
e
Simulated Open Loop Response − Wheel Constrained
Pitch Angle (Radians)
X Position (m)
Figure 10: Open loop response from a 30◦pitch angle posi-tion and released with the wheel constrained in the simulatedsystem. Note that there is little difference in the pitch an-gle response between here and the unconstrained response inFigure 7.
0 1 2 3 4 5
−100
−50
0
50
100
Time
Am
pli
tud
e
Closed Loop Response of Tested System
Pitch Angle (Degrees)
Control Signal ( V)
Figure 11: Closed loop response of the constrained physicalMICYCLE system when rotated to −15◦ and released.
0 1 2 3 4 5
−100
−50
0
50
100
Time
Am
pli
tud
e
Closed Loop Response of Tested System
Pitch Angle (Degrees)
Control Signal (V)
Figure 12: Closed loop response of the constrained physicalMICYCLE system when rotated to −30◦ and released.
0 1 2 3 4 5
−100
−50
0
50
100
Time
Am
pli
tud
e
Closed Loop Response of Simulated System
Pitch Angle (Degrees)
Control Signal (V)
Figure 13: Closed loop response of the constrained simulatedMICYCLE system when rotated to −15◦ and released.
0 1 2 3 4 5
−100
−50
0
50
100
Time
Am
pli
tud
e
Closed Loop Response of Simulated System
Pitch Angle (Degrees)
Control Signal (V)
Figure 14: Closed loop response of the constrained simulatedMICYCLE system when rotated to −30◦ and released.
Also noticeable is the fact that there is a slight differencein the response of the two systems. The physical system hasa slightly faster rise time but significantly more overshoot asevidenced by the oscillations in the control signal. The re-sponse of the simulated system does seem to be slightly bet-ter. In particular, the fact that there is a human riding thevehicle means that there should be as little overshoot as pos-sible so that the rider does not react to the overshoot. How-ever, the MICYCLE is not ridden in this experiment. With arider present and thus increases mass, m f , the effect of thefeedback is diminished, which is equivalent to smaller gains,and thus stability is improved. This issue needs to be furtherinvestigated.
A further point that should be raised is why the responsediffers between the two systems. One factor is that there isa significant amount of Coulomb friction in the motor thatis not being modelled. The motor has 28 pole pairs and acogging effect is created when the motor is turned by hand asthe windings are attracted to the next pole pair. This effectis quite pronounced; when the picture in Figure 9 was taken,the MICYCLE was unpowered, yet the Coulomb Friction wassufficient to prevent the frame from toppling over in the pitchdirection when undisturbed. It is possible that the overshootseen in the physical system is the motor having just passeda pole pair and thus overshooting until the next pole pair hasbeen reached and can actuate the system in reverse. This alsoneeds to be further investigated, ideally it should be modelledin the simulated system.
There is a further reason for why there exists a discrepancybetween the control response for the simulated and physicalsystems. This is that the physical system has a time delay inthe control system which is currently unmodelled in the simu-lated system. This would add phase lag in the control systemleading to more overshoot in the response. The most likelysource of this time delay is the IMU. The IMU operates at300 Hz, whilst all other digital components operate at morethan 10 kHz or are analogue. A solution would be to inves-tigate the period of all time delays and to model these in thesimulated system. Nevertheless, this is an area that needs tobe further investigated before any conclusions can be drawn.
Finally, it is worth highlighting the fact that the motor sat-urates in all cases in Figures 11, 12, 13 and 14. However,this does not appear to be particulary detrimental to the con-trol of the system. The saturated torque is still sufficient toovercome gravity and lift the frame.
8 Conclusion and future work
In this paper the dynamics of the unicycle were derivedand presented. Numerical simulations were used to verify thisderivation. This information was then used to simulate a PDcontroller for the system and the control performance of thesimulated system was then compared with the performanceof the physical MICYCLE system.
Future work will involve extending the model of the sys-tem dynamics to include Coulomb friction, to examine andmodel time delays in the control system and an investiga-tion of overshoot in the current PD implementation on thephysical system. Beyond this, a model based non-linear con-troller and a backstepping controller will be developed usingthe modelled system dynamics. These control strategies willbe compared and benchmarked, with the optimal strategy be-ing implemented into the final MICYCLE design.
AcknowledgmentsThe authors would like to acknowledge the support of the
Steve Kloeden, Phil Schmidt, Michael Riese, Norio Itsumiand Silvio De Ieso for their efforts in constructing the physi-cal MICYCLE used in this paper.
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