Nonlinear RegulationNonlinear Regulationforfor
Motorcycle ManeuveringMotorcycle Maneuvering
John HauserUniv of Colorado
in collaboration withAlessandro Saccon* & Ruggero Frezza, Univ Padova* email [email protected] for dissertation
aggressive maneuveringaggressive maneuvering
we seek to understanddynamics and control issues
of aggressively maneuvering systems
an opinion: maneuvering is one of the mostcommon and interesting ways that nonlineareffects are seen in control systems
examples include aircraft, motorcycles, skiers
motorcyclesmotorcycles
motorcycles possess
– unstable nonlinear dynamics– coupling of inputs– control vector field sign changes– nonminimum phase response– broad range of operation:
40-220 mph, 1.2-1.5 lateral g’s– rapidly changing trajectories:
turn-in, chicane, accel, braking
just plain fun!
Note: we do not intend to replace rider …
motorcycles: engineering objectivesmotorcycles: engineering objectives
provide strategies to test-drivevarious virtual prototypes:– human rider is not able to evaluate virtual– needed: a virtual rider (a control system)
to enable complex maneuveringnear the limits of performance(max roll, max lateral accel)
and that can exploit input coupling
better understand performance tradeoffs:what setup(bike geometry, tires, suspension, …)
is best for different circuits.
aggressive aggressive MotoMoto maneuvers are desired!maneuvers are desired!
Loris Capirossi
Circuit Circuit CatalunyaCatalunya
max acceleration and brakingmax acceleration and braking
Loris Capirossi Valentino Rossi
complex complex MotoMoto behaviors are possible!behaviors are possible!
Isle of Man 1999
motorcycle specificsmotorcycle specificsHierarchy of models:
- nonholonomic motorcycle infinitely sticky tires, simplified geometry
- sliding plane motorcyclemore realistic contact forces,simplified geometry
...- articulated motorcycle
include suspension, chain, flexible frame, semi-empirical tire models, … art / magic!
planning planning –– maneuvering objectivesmaneuvering objectives
- track specificationinner and outer track boundariesgo fast … stay on track
- path or race line specificationarc length parametrized curvego fast … on this line
- ground trajectory specificationtime parametrized curve… leads to a desired maneuvering objective
test tracktest track
velocity profilevelocity profile
velocity and velocity and accelaccel trajectorytrajectory
maneuvers and maneuver regulationmaneuvers and maneuver regulation
Given and a trajectorywith and bdd and bdd away from zero,the corresponding maneuver is the curve swept outby together with local temporal separation.
The maneuver has unique projection within a tube prop
In practice, a maneuver is specified using a parametrizedcurve
The param could be time-like or arc-length .
x = f(x,u) (x(t), u(t)), t ∈ R,
(x(·), u(·))
x(t) x(t) x(t)
(x(θ), u(θ)), θ ∈ R
θ s
transverse dynamicstransverse dynamics
Around a maneuver, choose transverse coordinates
locally, we may eliminate time
key: study stability, control, robustness oftime-varying nonlinear control systems
… discuss
θ = 1+ g1(ρ, u− u(θ))ρ = A(θ)ρ+B(θ)(u− u(θ))+ g2(ρ, u− u(θ))
ddθρ = A(θ)ρ+B(θ)(u− u(θ))+ f2(ρ, u− u(θ))
nonholonomicnonholonomic motorcycle modelmotorcycle model ..
nonholonomic car model
coupled roll dynamics
x = v cosψy = v sinψv = u1ψ = vσσ = u2
hϕ = g sinϕ − ((1− hσ sinϕ)σv2+ bψ) cosϕ
R = 1/σψ
(x, y)
δ
ϕh
pb
to get a trajectory to get a trajectory ……
• path and velocity profile directly provide anonholonomic car trajectory
• the desired motorcycle maneuver is determined bylifting
the car trajectory to a moto traj, adding a roll traj
• in this fashion, theclass of motorcycle trajectories
is parametrized by thefamily of smooth curves
in the plane
lifting to an lifting to an executableexecutable MotoMoto trajectorytrajectory
given the desired flatland traj, find a roll trajectoryconsistent with, roughly,
after dynamic embedding, we optimize away thehand of God
for now, we do the whole trajectory …
hϕ = g sinϕ − alat(t) cosϕ + uhog
quasiquasi--static roll trajectorystatic roll trajectory
when the desired flatland traj is a constant speed, constant radius circle, there is a
static roll trajectorygiven by
for more dynamic flatland trajectories, we define thequasi-static roll trajectory
according to
we expect (hope) that the desired roll traj is close to this!
achievable motorcycle trajectoriesachievable motorcycle trajectories
problem: given a smooth velocity-curvature profile,find, if possible, an upright roll trajectory satisfying
with
in fact, such inverted pendulum dynamics is always a part of the dynamics of every motorcycle
also, the lateral acceleration will, in general, be much more complicated and may not be smooth
hϕ = g sinϕ − alat(t) cosϕ
alat(t) = [σv2+ b(vσ+ vσ)](t)
the geometric storythe geometric storywanted: an upright soln of
~Thm: if is an upright soln, the phase traj lies in
-pi/2 -pi/4 0 pi/4 pi/2-6
-4
-2
0
2
4
6phase plane
ϕ(·)
existence of an upright roll existence of an upright roll trajtrajThm: with a bdd
that is const before some t0 possesses an upright soln
-pi/2 -pi/4 0 pi/4 pi/2-6
-4
-2
0
2
4
6phase plane
dynamics dynamics w.r.tw.r.t. quasi. quasi--static roll static roll trajtrajdefining the quasi-static roll angle and total acceleration
the roll dynamics is given by
inverted pendulum dynamics with gravity that varies in strength and direction
we seek a bounded traj of the driven unstable system
.
bounded solutions: dichotomybounded solutions: dichotomy
when will a system like
have a bounded solution? [and with upright roll]
the unique bounded solution of the LTI system
is given by
.
bounded solutions: dichotomy bounded solutions: dichotomy ……
can we find a bounded solution for thetime-varying linear system
?
the LTI system is hyperbolic
for time-varying systems, we seek a dichotomy
[this will be used to show the TV nonlinear sys has a soln].
bounded solutions: dichotomy bounded solutions: dichotomy ……Thm: the unique soln of
is given by the noncausal bounded operator
where c(.) and d(.) are nonl filtered versions of α(.)
solution algorithmsolution algorithmFact: under some conditions, the unique soln of
can be computed by the algo
and, furthermore,
is small.. (note: above optimization can also be used)
³h(t) ≈ α/2 e−α|t|
´
maneuver maneuver regulationgregulationg
with an executable trajectory in hand (reparametrized by arclength), we may write the system dynamics
in transverse maneuver coordinates
so that the transverse dynamics are given by
maneuver regulation maneuver regulation ……
MP maneuver regulation may then be implemented using
possibly subject to some constraints (e.g., lateral accel)
a first order controller
may be obtain by solving a TV Riccati equation(where time is arclength)
cost function designcost function design
how should we choose Q and R?– the many heuristics suggested in the literature did not seem
effective to us …– performance requires a certain speed of response– physical motion requires a restricted speed of response– nonlinearities (seem to) require a certain uniformity of response
under aggressive maneuvering– … plus all the usual control performance expectations ...
Q = I, R = IQ = I, R = I not too interestingnot too interesting
-50 -40 -30 -20 -10 0 10
-25
-20
-15
-10
-5
0
5
10
15
20
25
σ root locus
too fast
desired region
another heuristic for Q & R designanother heuristic for Q & R design
• get a desired lateral response first for SS system(e.g., place poles for driving in a high g circle)
• solve, if able, an inverse optimal control problem(must satisfy return difference ineq…)
requiring Q, R > 0 (resulting 5x5 Q is far from diagonal)[can be done as a convex problem---we use SeDuMi]
• augment the lateral Q, R with a choice of Q, R for the (scalar) longitudinal subsystem
• evaluate over a range of velocity and lateral acceland iterate …
• reasonable results have been obtained fornonholonomic motorcycle
Q, RQ, R obtained by inverse opt heuristicobtained by inverse opt heuristic
-12 -10 -8 -6 -4 -2 0 2
-6
-4
-2
0
2
4
6
σ root locus
Q, RQ, R obtained by inverse opt heuristicobtained by inverse opt heuristic
-14 -12 -10 -8 -6 -4 -2 0 2 4
-6
-4
-2
0
2
4
6
v root locus
-12 -10 -8 -6 -4 -2 0 2
-6
-4
-2
0
2
4
6
v root locus
example performance example performance evaleval ……
remarksremarks
robustness: we have applied maneuver regulation (based on simple moto model) to regulation of high fidelity motorcycle model (multi-body)---with great success!
email Ale Saccon [email protected] details (in his dissertation)
.
