nonlinear regulation for motorcycle...

Nonlinear RegulationNonlinear Regulationforfor

Motorcycle ManeuveringMotorcycle Maneuvering

John HauserUniv of Colorado

in collaboration withAlessandro Saccon* & Ruggero Frezza, Univ Padova* email asaccon@dei.unipd.it 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 asaccon@dei.unipd.itfor details (in his dissertation)

.