bicycles - a mechatronics view · bicycles - a mechatronics view k. j. Åström lund university 1....
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Bicycles - A Mechatronics View
K. J. ÅströmLund University
1. Introduction
2. Modeling
3. Stabilization
4. Rear wheel steering
5. Maneuvering and stabilization
6. Conclusions
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 1
But don’t forget Control!c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 2
The US Army Motor Bike - A Design Disaster
F. R. Whitt and D. G. Wilson (1974) Bicycling Science - Er-gonomics and Mechanics. MIT Press Cambridge, MA.
“Many people have seen theoretical advantages in the fact thatfront-drive, rear-steered recumbent bicycles would have simplertransmissions than rear-driven recumbents and could have thecenter of mass nearer the front wheel than the rear. The U.S.Department of Transportation commissioned the constructionof a safe motorcycle with this configuration. It turned out to besafe in an unexpected way: No one could ride it.”
The Santa Barbara Connection
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 3
Why Bicycles are Interesting?
• Very common efficient means of transportation andrecreation
• Interesting non-trivial behavior
• Suitable to illustrate many issues in mechatronics
– Modeling– Dynamics and control– Integrated systems and control design– The role of sensors and actuators
• Useful for education
– Simple and illustrative experiments– High student attraction
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 4
Some Interesting Questions
• How do you stabilize a bicycle?
– By steering or by leaning?
• Do you normally stabilize a bicycle when you ride it?
• Why is it possible to ride without touching the handle bar?
• How is stabilization influenced by the design of the bike?
• Why does the front fork look the way it does?
• The main message:
– A bicycle is a feedback system, the action of the frontfork is the key!
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 5
Bicycles - A Mechatronics View
K. J. Åström
1. Introduction
2. Modeling
3. Stabilization
4. Rear wheel steering
5. Maneuvering and stabilization
6. Conclusions
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 6
Bicycle Modeling
Sommerfeld (1952) Mechanics. Lectures on TheoreticalPhysics, Vol 1. Academic Press 1952.
F. Klein and A. Sommerfeld Theorie d’s Kreisels, Vol IV, pp880.
“A bicycle is a doubly non-holonomic system; it has fivedegrees of freedom in finite motion, but only three suchdegrees i infinitesimal motion (rotation of the rear wheel in itsinstantaneous plane, to which the rotation of the front wheelis coupled by the condition of pure rolling; rotation aboutthe handle bar axis; and common rotation of front and rearwheel about the line connecting their points of contact with theground), as long as we do not consider the degrees of freedomof the cyclist himself.”
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 7
Arnold Sommerfeld on Gyroscopic Effects
That the gyroscopic effects of thewheels are very small comparedwith these (centripetal forces) canbe seen from the constructionof the wheel: if one wanted tostrengthen the gyroscopic effects,one should provide the wheelswith heavy rims and tires insteadof making them as light as possi-ble. It can nevertheless be shownthat these weak effects contributetheir share to the stability of thesystem.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 8
Bicycle Modeling
Whitt and Wilson Bicycling Science MIT Press:
“The scientific literature (Timoshenko, Young, DenHartog et al)shows often complete disagreement even about fundamentals.One advocates that a high center of mass improves stability,another concludes that a low center of mass is desirable.”
F. J. W Whipple, The stability of motion of a bicycle.Quart. JMath vol 30, 1899
Jones, The Stability of the bicycle. Physics Today. April 1970
More references later
Control theory gives very nice insights!
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 9
Coordinates and Variables
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 10
Major Simplifications
• Neglect elasticities.
• Five subsystems: frame, wheels, rider and front fork
• Configuration space has dimension 8: position of rearwheel x and y orientation ψ , roll of frame and front wheelϕ an ϕ f , steering angle δ , wheel angles γ r and γ f .
• Four non-holonomic constraints reduces degrees offreedom to 3.
• Assume constant forward velocity (-1 dof)
• Assume that rider is fixed to the bicycle (-1dof)
• Let steering angle δ be the control variable (-1dof)
• Resulting model has one dof, roll angle ϕ• Assume small tilt angles ; linear model.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 11
Tilt Dynamics
Linearized momentum balance around ξ -axis
Jd2ϕdt2 − D
dωdt= mnhϕ + hF, ω = V
bδ , F = mV 2
bδ
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 12
Linearized Tilt Dynamics
Jd2ϕdt2 − D
dωdt= mnhϕ + hF, ω = V
bδ , F = mV 2
bδ
Substituting and rearrangign gives
Jd2ϕdt2 −mnhϕ = mhV 2
bδ + D
bdδdt
Transfer function: P(s) = DVbJ
s+mhV/Ds2 −mnh/J
• Poles: p = ±√
mnh/J � ±√n/h
• Zero: z = −mhV/D � V/a
• Gain: k = amhVbJ
� aVbh
Approximations J � mh2, D � mahc& K. J. Åström, Mekatronikmöte, Göteborg, 030828 13
Summary
• Steering angle is the control variable
• Two states tilt and tilt rate
• Transfer function with one zero and two poles one in theRHP
• What is the model good for?
• The model cannot explain that it is possible to ride withouttouching the handle bar
• What assumptions should be changed?
• Teaching kids to bike (Don’t use stiff arms!)
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 14
Bicycles - A Mechatronics View
K. J. Åström
1. Introduction
2. Modeling
3. Stabilization
4. Rear wheel steering
5. Maneuvering and stabilization
6. Conclusions
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 15
A Slightly More Complicated Model
• It is not a good idea to assume that the steering angle isthe control variable!
• The driver influences the bicycle primarily by applyingtorques to the handle bar.
• The front fork creates a feedback because leaning createsa torque on the front fork.
• A simple experimental verification.
• Important to understand the basic mechanisms!
• Improved model
– Two subsystems: frame and front fork– One control variable: Torque on front fork– Dynamic model for frame static model for front fork
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 16
Block Diagram of a Bicycle
Control variable: Handlebar torque T
Process variables: Steering angle δ , tilt angle ϕ
PSfrag replacements
T
ϕδ
?
T
Front fork
Frame?
−1
Σ
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 17
Overview
• The model of the frame should describe how the thetilt of the frame ϕ is influenced by the steering angle δinfluences (already done)
– Dynamics is essential– What is the time scale? (ω =
√mnh/J)
– Storage of angular momentum (ϕ and dϕ/dt)
• The model of the front fork should tell how the steeringangle ϕ is influenced by the torque on the handle bar Tand the tilt angle ϕ– A static model will be used since the front fork dynam-
ics faster than tilt dynamics– Tire-road interaction and gyroscopic effects will be
neglected
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 18
The Front Fork
The front fork has many interesting features that were devel-oped over a long time. Its behavior is complicated by geometry,the trail, tire-road interaction and gyroscopic effects. We willdescribe it by a very simple static linear model.
With a positive trail the front wheellines up with the velocity, castercamber. The trail also creates atorque that turns the front fork intothe lean. This torque is opposedby the caster-camber action andby torques generated by tire roadinteraction. A simple model is
δ = k1T − k2ϕ
d
PSfrag replacements
t
Experimental verifications?c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 19
Block Diagram of a Bicycle
PSfrag replacements
T
ϕδ
k1
Handlebar torque T
Front fork
k(s+V/a)s2−mnh/Jk2
−1
Σ
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 20
The Closed Loop System
Combining the equations for the frame and the front fork gives
Jd2ϕdt2 −mnhϕ = DV
bdδdt+ mhV 2
bδ
δ = −k2ϕ + k1T
we find that the closed loop system is described by
Jd2ϕdt2 +
k2 DVb
dϕdt+mnh
(k2V 2
bn −1)
ϕ = k1 DVb
(dTdt+mhV
DT)
This equation is stable if
V > Vc =√
bn/k2
where Vc is the critical velocity. Physical interpretation. Thinkabout this next time you bike!
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 21
Gyroscopic Effects
Gyroscopic effects has a little influence on the front fork
Jd2ϕdt2 −mnhϕ + DV
bdδdt+ mhV 2
bδ
δ = −k2ϕ−kndϕdt+ k1T
we find that the closed loop system is described by(
J + knDVb
)d2ϕdt2 +
(k2 DVb
+ mhknV 2
bn)dϕ
dt+mnh
(k2V 2
bn − 1)
ϕ
= k1 DVb
(dTdt+ mhV
DT)
Damping can be improved drastically, but the stability conditionis the same as before
V > Vc =√
bn/k2
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 22
Summary
Modeling and feedback analysis gives good insight into sta-bilization of bicycles. A model of second order captures theessence.
• Simple models
– Stiff rider– Static front fork model
• Interesting conclusion
– Bicycle is self stabilizing if velocity is larger than thecritical velocity. Design of the front fork is the key!
• Model can be augmented in several ways
– Better front fork model centrifugal forces ...– Tire road interaction
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 23
Major Drawback of Simple Model
Parameters k1 and k2 in front fork model δ = k1T − k2ϕ arevelocity dependent. A static model givesXS
k1 =b
acmn cos λ(V 2 cos λ
bn − sin λ) , k2 =
1V 2 cos λ
bn − sin λ
Closed loop system(V 2 cos λ
bn − sin λ)
Jd2ϕdt2 +
DV cos λb
dϕdt+(
mnh sin λ − acmn cos λb
)ϕ
= DV hacmn
dTdt+ V 2h
acn T
Bifurcations when velocity changes are very different.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 24
Root LociEmpirical front fork model Physical front fork model
Effects of additional dynamics
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 25
More Complicated Models
• Front fork dynamics gives a fourth order model
• Linearized models have been obtained by many tech-niques
– Complicated error proned calculations– Computational tools highly desirable
• Other factors
– Rider lean– Tire road interaction
• Nonlinear models and parameter dependencies
• Tools like Modelica are indispensable
• Expressions for critical velocity complicated
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 26
Bicycles - A Mechatronics View
K. J. Åström
1. Introduction
2. Modeling
3. Stabilization
4. Rear wheel steering
5. Maneuvering and stabilization
6. Conclusions
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 27
Rear Wheel Steering
F. R. Whitt and D. G. Wilson (1974) Bicycling Science - Er-gonomics and Mechanics. MIT Press Cambridge, MA.
“Many people have seen theoretical advantages in the fact thatfront-drive, rear-steered recumbent bicycles would have simplertransmissions than rear-driven recumbents and could have thecenter of mass nearer the front wheel than the rear. The U.S.Department of Transportation commissioned the constructionof a safe motorcycle with this configuration. It turned out to besafe in an unexpected way: No one could ride it.”
The Santa Barbara Connection
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 28
The Tilt Equation for Rear Wheel Steering
PS
fragreplacem
entsORCA
Jd2ϕdt2 −mnhϕ = mhV 2
bδ−D
bdδdt
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 29
Compare with Front Wheel Steering
Front wheel steering:
Jd2ϕdt2 −mnhϕ = mhV 2
bδ+D
bdδdt
Rear wheel steering:
Jd2ϕdt2 −mnhϕ = mhV 2
bδ−D
bdδdt
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 30
The Linearized Tilt Equation
The linearized equation becomes
Jd2ϕdt2 −mnhϕ = mhV 2
bδ−D
bdδdt
The transfer function of the system is
P(s) = DVbJ
−s+ mhVD
s2 − mnhJ
One pole and one zero in the right half plane.
Physical interpretation.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 31
Rear Wheel Steering
The transfer function from steering angle to tilt is
P(s) = DVbJ
−s+ mhVD
s2 − mnhJ
One RHP pole at p =√
mnhJ
� 3 rad/s
One RHP zero at z = mhVD
Pole position independent of velocity but zero proportional tovelocity. When velocity increases from zero to high velocity youpass a region where z = p and the system is uncontrollable.The system is impossible to control when z = p.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 32
Does Feedback from Rear Fork Help?
Combining the equations for the frame and the rear fork gives
Jd2ϕdt2 −mnhϕ − amhV
bdδdt+ mhV 2
bδ
δ = −k2ϕ + k1T
we find that the closed loop system is described by
Jd2ϕdt2 −
k2
bdϕdt+mnh
(k2V 2
bn − 1)
ϕ = k1Vb
(dTdt+ mhV
DT)
where Vc =√
bn/k2. This equation is unstable for all k2.There are several ways to turn the rear fork but it makes littledifference.
Can the system be stabilized robustly with a more complexcontroller?
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 33
Limitations due to Non-minimum PhaseDynamics
Factor process transfer function as P(s) = Pmp(s)Pnmp(s) suchthat hPnmp(iω )h = 1 and Pnmp has negative phase. Requiring aphase margin ϕ m we get
arg L(iω nc) = arg Pnmp(iωnc) + arg Pmp(iωnc) + arg C(iω nc)≥ −π +ϕm
With Pmp(s)C(s) = (s/ω nc)n (Bode’s ideal loop transferfunction) we get arg PmpC = nπ/2 and
arg Pnmp(iω nc) ≥ −π +ϕ m − nπ2
This equation holds approximatively for other designs if n is theslope at the crossover frequency.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 34
Bode Plots Should Look Like This
10−1
100
101
10−2
10−1
100
101
102
10−1
100
101
−300
−200
−100
0PSfrag replacements
Gain curve
ω
hP(iω)h
Phase curve
ω
arg
P(iω)
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 35
Summary of Limitations - Part 1
• A RHP zero z gives an upper bound to bandwidth
ω nc
z≤{
0.5 for Ms, Mt < 20.2 for Ms, Mt < 1.4.
Slow RHP zeros difficult! Why e−sT � 1−sT/21+sT/2 , s = 1/2T
• A RHP pole p gives a lower bound to bandwidth
ω nc
p≥{
2 for Ms, Mt < 25 for Ms, Mt < 1.4.
Fast RHP poles difficult!
• RHP poles and zeros must be sufficiently separated
zp≥{
6.5 for Ms, Mt < 214.4 for Ms, Mt < 1.4.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 36
System with RHP Pole and Zero Pair
Pnmp(s) =(z− s)(s+ p)(z+ s)(s− p)
For z > p the cross over frequency inequality becomes
ωnc
z+ p
ωnc≤ (1− p
z) tan
(π2− ϕm
2+ nnc
π4
)
ϕm < π + nncπ2− 2 arctan
√p/z
1− p/z
With nnc = −0.5 we get
z/p 2 2.24 3.86 5 5.83 8.68 10 20
ϕm -6.0 0 30 38.6 45 60 64.8 84.6
Systems with RHP poles and zeros can be controlled robustlyonly if poles and zeros are well separated
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 37
Return to Rear Wheel Steering
The zero-pole ratio is
zp= V
D
√mhJn � V
a
√hn
The system is difficult to control robustly if this ratio is greaterthan 6.
To make the ratio large you can
• Make a small by leaning forward
• Make V large by biking fast (takes guts)
• Make h large by standing upright
• Move back and sit down when the velocity is sufficientlylarge
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 38
Richard Klein and His Bikes
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 39
Bicycles in Education
• Not High Tech but all students are familiar with them
• Good illustration of many interesting issues in control
• Special projects in Control Lund 1968, 1972, Richard KleinIllinois ACC Atlanta 1988 and IFAC CE Boston 1991
• Summer course for European student in Lund 1996
• Regular use in introductory courses at Lund and UCSB,Hannover 2000 (Lunze), Michigan (Stefanopoulou)
• How bicycles are used?
– Motivation– Concrete illustration of ideas and concepts– Experiments
• Very high student attractionc& K. J. Åström, Mekatronikmöte, Göteborg, 030828 40
The Santa Barbara experience
• Why should a mechanical engineer learn control
• Argue the design of a recliner
• Demo
• Motivate limitations and why it is worth while to knowabout poles and zeros
• Student feedback
– Control theory really gives insight– I did not know that a bike could be self stabilizing– Now I know why RHP poles and zeros are important– Even if I will not specialize in control I have a feel for
the questions I should ask if I became involved with acontrol system
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 41
The UCSB Bike
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 42
The Hamburg-Bochum Bike
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 43
Bicycles - A Mechatronics View
K. J. Åström
1. Introduction
2. Modeling
3. Stabilization
4. Rear wheel steering
5. Maneuvering and stabilization
6. Conclusions
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 44
Maneuvering and Stabilization - A ClassicProblem
Lecture by Wilbur Wright 1901:
Men know how to construct airplanes.Men also know how to build engines.
Inability to balance and steer still confrontsstudents of the flying problem.
When this one feature has been worked out,the age of flying will have arrived, for
all other difficulties are of minor importance.
The Wright Brothers figured it out and flew the Kitty Hawk onDecember 17 1903!
Minorsky 1922:
It is an old adage that a stable ship is difficult to steer.c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 45
Maneuvering
Having understood stabilization of bicycles we will now investi-gate steering for the bicycle with a rigid rider.
• Key question: How is the path of the bicycle influenced bythe handle bar torque?
• Steps in analysis, find the relations
– How handle bar torque influences steering angle– How steering angle influences velocity– How velocity influences the path
We will find that the instability of the bicycle frame causessome difficulties in steering (dynamics with right half planezeros). This has caused severe accidents for motor bikes.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 46
Block Diagram Analysis Useful
PSfrag replacements
T ϕδ
k1
Handlebar torque T
Front fork
k(s+V/a)s2−mnh/J
k2
−1
Σ
Transfer function from T to δ is
k1
1+ k2P(s) =k1
1+ k2k(s+V/a)
s2−mnh/J
= k1s2 −mnh/J
s2 + amhk2VbJ s+ mnh
J
(V2
V2c− 1)
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 47
Steering Angle to Path
dydt= Vψ
dψdt
= Vb
δ
Lδ = Gδ TLT
Gδ T =k2
1+ k1Gϕδ (s)
Transfer function from steering torque to path deviation
GyT =k1V 2
b(1+ k1Gϕδ (s))=
k1V 2(
s2 − mnhJ
)
bs2
(s2 + amhk1V0
bJ s+ mnhJ
(V2
0V2
c− 1))
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 48
Simulation
Assume that the bicycle is driven along a straight line and thata constant torque is applied to the handle bar. What path willthe bicycle take?
0 0.5 1 1.5 2 2.5 3−4
−3
−2
−1
0
1
2
PSfrag replacements
x
y
Physical interpretation!
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 49
Summary
• The simple second order model with a rigid rider canexplain stabilization. The model indicates that steeringis difficult due to the right half plane zero in the transferfunction from handle bar torque to steering angle.
• The right half plane zero has some unexpected conse-quences which gives the bicycle a counterintuitive behav-ior. This has caused many motorcycle accidents.
• How can we reconcile the difficulties with our practicalexperience that a bicycle is easy to steer?
• The phenomena depends strongly on the assumption thatthe rider does not lean.
• The difficulties can be avoided by introducing an extracontrol variable (leaning).
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 50
Coordinated Steering
An experienced rider uses both lean and the torque on thehandle bar for steering. Intuitively it is done as follows:
• The bicycle is driven so fast so that it is automaticallystabilized.
• The turn is initiated by a torque on the handle bar, therider then leans gently into the turn to counteract thecentripetal force which will tend to lean the bike in thewrong direction. This is particularly important for motorbikes which are much heavier than the rider.
A proper analysis of a bicycle where the rider leans requirea more complex model because we have to account for twobodies instead of one. There are also two inputs to deal with.Accurate modeling of a bicycle also has to consider tire roadinteraction and a more detailed account of the mechanics.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 51
Sensors, Actuators Poles and Zeros
• Watch our for time delays, and RHP poles and zeros
• Zeros can be eliminated by adding sensors and actuators!The zeros of the standard linear system (A,B,C) are thevalues of s where the matrix
sI − A B
C 0
looses rank
• Big difference between using observers and sensors as faras robustness are concerned!
• A naive observation on the rear steered bike
Jd2ϕdt2 −
k2 DVb
dϕdt+mnh
(k2V2
bn − 1)
ϕ = k1 DVb
(dTdt+ V
aT)
Adding a rate gyro solves the problem!c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 52
Bicycles - A Mechatronics View
K. J. Åström
1. Introduction
2. Modeling
3. Stabilization
4. Rear wheel steering
5. Maneuvering and stabilization
6. Conclusions
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 53
Conclusions
• Bicycles illustrate many issues in mechatronics
– Modeling– Integrated system and control design– The role of sensors and actuators
• Bicycles are cheap and very useful in education
– Stabilization and steering– Nice way to illustrate zeros
• Lesson 1: Dynamics is important! A system may look OKstatically but un-tractable because of dynamics.
• Lesson 2: A system that is difficult to control because ofzeros in the right half plane can be improved significantlyby introducing more sensors and actuators.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 54
References
Bode, H. W. (1945): Network Analysis and Feedback AmplifierDesign. Van Nostrand, New York. F. R. Whitt and D. G. Wilson(1974) Bicycling Science - Ergonomics and Mechanics. MIT PressCambridge, MA.
R.E. Klein (1988) Novel Systems and Dynamics Teaching Tech-niques Using Bicycles. Proc ACC, Atlanta, GA.
R.E. Klein (1989) Using bicycles to teach system dynamics. IEEEControl Systems Magazine, CSM Vol 9 No 3 pp, 4–9.
Seron, M. M., J. H. Braslavsky, and G. C. Goodwin (1997): Funda-mental Limitations in Filtering and Control. Springer, London.
K. J. Åström (2000) Limitations on control system performance.European Journal of Control Vol 6 No 1 pp 2–20.
c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 55