Nonlinear Regulation for Motorcycle Maneuvering

Transcription

1 Nonlinear Regulation for Motorcycle Maneuvering John Hauser Univ of Colorado in collaboration with Alessandro Saccon* & Ruggero Frezza, Univ Padova * asaccon@dei.unipd.it for dissertation

2 aggressive maneuvering we seek to understand dynamics and control issues of 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

3 motorcycles motorcycles possess unstable nonlinear dynamics coupling of inputs control vector field sign changes nonminimum phase response broad range of operation: mph, lateral g s rapidly changing trajectories: turn-in, chicane, accel, braking just plain fun! Note: we do not intend to replace rider

4 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.

5 aggressive Moto maneuvers are desired! Loris Capirossi

6 Circuit Catalunya

7 max acceleration and braking Loris Capirossi Valentino Rossi

8 complex Moto behaviors are possible! Isle of Man 1999

9 Hierarchy of models: motorcycle specifics - 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!

10 planning 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

11 test track

12 velocity profile

13 velocity and accel trajectory

14 maneuvers and maneuver regulation Given ẋ = f(x, u) and a trajectory (x(t),u(t)), t R, with ẋ(t) and ẍ(t) bdd and ẋ(t) bdd away from zero, the corresponding maneuver is the curve swept out by (x( ),u( )) together with local temporal separation. The maneuver has unique projection within a tube prop In practice, a maneuver is specified using a parametrized curve ( x(θ), ū(θ)), θ R The param θ could be time-like or arc-length s.

15 transverse dynamics Around a maneuver, choose transverse coordinates θ = 1+g 1 (ρ,u ū(θ)) ρ = A(θ)ρ + B(θ)(u ū(θ)) + g 2 (ρ,u ū(θ)) locally, we may eliminate time d dθ ρ = A(θ)ρ + B(θ)(u ū(θ)) + f 2(ρ,u ū(θ)) key: study stability, control, robustness of time-varying nonlinear control systems discuss

16 nonholonomic motorcycle model. nonholonomic car model ẋ = v cos ψ ẏ = v sinψ v = u 1 ψ = vσ σ = u 2 coupled roll dynamics R =1/σ h ϕ = g sinϕ ((1 hσ sinϕ)σv 2 + b ψ)cosϕ ψ ϕ (x, y) b h p δ

17 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

18 lifting to an executable Moto trajectory given the desired flatland traj, find a roll trajectory consistent with, roughly, h ϕ = g sinϕ a lat (t)cosϕ + u hog after dynamic embedding, we optimize away the hand of God for now, we do the whole trajectory

19 quasi-static 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!

20 achievable motorcycle trajectories problem: given a smooth velocity-curvature profile, find, if possible, an upright roll trajectory satisfying with h ϕ = g sinϕ a lat (t)cosϕ a lat (t) =[σv 2 + b( vσ + v σ)](t) 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

21 wanted: an upright soln of the geometric story ~Thm: if ϕ( ) is an upright soln, the phase traj lies in 6 phase plane pi/2 -pi/4 0 pi/4 pi/2

22 Thm: existence of an upright roll traj with a bdd that is const before some t 0 possesses an upright soln 6 phase plane pi/2 -pi/4 0 pi/4 pi/2

23 dynamics w.r.t.. quasi-static static roll 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.

24 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.

25 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]

26 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(.)

27 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

28 maneuver 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

29 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)

30 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...

31 Q = I, R = I not too interesting σ root locus too fast desired region

32 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

33 Q, R obtained by inverse opt heuristic σ root locus

34 Q, R obtained by inverse opt heuristic v root locus v root locus

35 example performance eval

36 remarks robustness: we have applied maneuver regulation (based on simple moto model) to regulation of high fidelity motorcycle model (multi-body)---with great success! Ale Saccon for details (in his dissertation).

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