Nonlinear Regulation Nonlinear Regulation for for Motorcycle - - PowerPoint PPT Presentation

Nonlinear Regulation Nonlinear Regulation for for Motorcycle Maneuvering Motorcycle Maneuvering

John Hauser

Univ of Colorado

in collaboration with

Alessandro Saccon* & Ruggero Frezza, Univ Padova

* email asaccon@dei.unipd.it for dissertation

aggressive maneuvering aggressive maneuvering

we seek to understand dynamics and control issues

• f aggressively maneuvering systems

an opinion: maneuvering is one of the most common and interesting ways that nonlinear effects are seen in control systems

examples include aircraft, motorcycles, skiers

motorcycles motorcycles

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 objectives motorcycles: engineering objectives

provide strategies to test-drive various virtual prototypes: – human rider is not able to evaluate virtual – needed: a virtual rider (a control system) to enable complex maneuvering near 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 Moto Moto maneuvers are desired! maneuvers are desired!

Loris Capirossi

Circuit Circuit Catalunya Catalunya

max acceleration and braking max acceleration and braking

Loris Capirossi Valentino Rossi

complex complex Moto Moto behaviors are possible! behaviors are possible!

Isle of Man 1999

motorcycle specifics motorcycle specifics

Hierarchy of models:

• nonholonomic motorcycle

infinitely sticky tires, simplified geometry

• sliding plane motorcycle

more realistic contact forces, simplified geometry ...

• articulated motorcycle

include suspension, chain, flexible frame, semi-empirical tire models, …

art / magic!

planning planning – – maneuvering objectives maneuvering objectives

• track specification

inner and outer track boundaries go fast … stay on track

• path or race line specification

arc length parametrized curve go fast … on this line

• ground trajectory specification

time parametrized curve … leads to a desired maneuvering objective

test track test track

velocity profile velocity profile

velocity and velocity and accel accel trajectory trajectory

maneuvers and maneuver regulation maneuvers and maneuver regulation

Given and a trajectory with and bdd and bdd away from zero, the corresponding maneuver is the curve swept out by together with local temporal separation. The maneuver has unique projection within a tube prop In practice, a maneuver is specified using a parametrized curve 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 dynamics transverse dynamics

Around a maneuver, choose transverse coordinates locally, we may eliminate time key: study stability, control, robustness of time-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))

d dθρ = A(θ)ρ + B(θ)(u − ¯

u(θ)) + f2(ρ, u − ¯ u(θ))

nonholonomic nonholonomic motorcycle model motorcycle model . .

nonholonomic car model coupled roll dynamics

˙ x = v cos ψ ˙ y = v sinψ ˙ v = u1 ˙ ψ = vσ ˙ σ = u2 h¨ ϕ = g sinϕ − ((1 − hσ sinϕ)σv2 + b ¨ ψ) cosϕ

R = 1/σ

ψ

(x, y)

δ

ϕ

h p b

to get a trajectory to get a trajectory … …

• path and velocity profile directly provide a

nonholonomic car trajectory

• the desired motorcycle maneuver is determined by

lifting the car trajectory to a moto traj, adding a roll traj

• in this fashion, the

class of motorcycle trajectories is parametrized by the family of smooth curves in the plane

lifting to an lifting to an executable executable Moto Moto trajectory trajectory

given the desired flatland traj, find a roll trajectory consistent with, roughly, after dynamic embedding, we optimize away the hand of God for now, we do the whole trajectory …

h¨ ϕ = g sinϕ − alat(t) cosϕ + uhog

quasi quasi-

• static roll trajectory

static roll trajectory

when the desired flatland traj is a constant speed, constant radius circle, there is a static roll trajectory given by for more dynamic flatland trajectories, we define the quasi-static roll trajectory according to we expect (hope) that the desired roll traj is close to this!

achievable motorcycle trajectories achievable 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

h¨ ϕ = g sinϕ − alat(t) cosϕ alat(t) = [σv2 + b( ˙ vσ + v ˙ σ)](t)

the geometric story the geometric story

wanted: an upright soln of ~Thm: if is an upright soln, the phase traj lies in

• pi/2
• pi/4

pi/4 pi/2

• 6
• 4
• 2

2 4 6 phase plane

ϕ(·)

existence of an upright roll existence of an upright roll traj traj

Thm: with a bdd that is const before some t0 possesses an upright soln

• pi/2
• pi/4

pi/4 pi/2

• 6
• 4
• 2

2 4 6 phase plane

dynamics dynamics w.r.t w.r.t. quasi . quasi-

• static roll

static roll traj traj

defining 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: dichotomy bounded 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 the time-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 algorithm solution algorithm

Fact: under some conditions, the unique soln of can be computed by the algo and, furthermore, is small. . (note: above optimization can also be used)

³

h(t) ≈ α/2 e−α|t| ´

maneuver maneuver regulationg regulationg

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 design cost 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 = I Q = I, R = I not too interesting not too interesting

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10

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5 10 15 20 25

σ root locus

too fast desired region

another heuristic for Q & R design another 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 accel

and iterate …

• reasonable results have been obtained for

nonholonomic motorcycle

Q, R Q, R

• btained by inverse opt heuristic
• btained by inverse opt heuristic
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2

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2 4 6

σ root locus

Q, R Q, R

• btained by inverse opt heuristic
• btained by inverse opt heuristic
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2 4

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2 4 6 v root locus

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2

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2 4 6 v root locus

example performance example performance eval eval … …

remarks remarks

robustness: we have applied maneuver regulation (based

• n simple moto model) to regulation of high fidelity

motorcycle model (multi-body)---with great success! email Ale Saccon asaccon@dei.unipd.it for details (in his dissertation) .