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 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

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 Moto Moto maneuvers are desired! maneuvers are desired! aggressive Loris Capirossi

Circuit Catalunya Catalunya Circuit

max acceleration and braking max acceleration and braking Loris Capirossi Valentino Rossi

complex Moto Moto behaviors are possible! behaviors are possible! complex 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 – – maneuvering objectives maneuvering objectives planning - 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 accel accel trajectory trajectory velocity and

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

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

nonholonomic motorcycle model motorcycle model . nonholonomic . ϕ h nonholonomic car model ˙ = v cos ψ x δ ˙ = v sin ψ y ˙ = u 1 v ˙ b = v σ ψ p ( x, y ) ˙ = u 2 σ ψ R = 1 / σ coupled roll dynamics ϕ = g sin ϕ − ((1 − h σ sin ϕ ) σ v 2 + b ¨ h ¨ ψ ) cos ϕ

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

quasi- -static roll trajectory static roll trajectory quasi 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 h ¨ ϕ = g sin ϕ − a lat ( t ) cos ϕ with a lat ( t ) = [ σ v 2 + b ( ˙ v σ + 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

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

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

dynamics w.r.t w.r.t. quasi . quasi- -static roll static roll traj traj dynamics 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. ³ h ( t ) ≈ α / 2 e − α | t | ´ . (note: above optimization can also be used)

maneuver regulationg regulationg maneuver 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 not too interesting Q = I, R = I not too interesting σ root locus 25 too fast 20 15 10 5 0 -5 desired region -10 -15 -20 -25 -50 -40 -30 -20 -10 0 10

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 obtained by inverse opt heuristic Q, R obtained by inverse opt heuristic σ root locus 6 4 2 0 -2 -4 -6 -12 -10 -8 -6 -4 -2 0 2

Q, R obtained by inverse opt heuristic Q, R obtained by inverse opt heuristic v root locus 6 4 2 v root locus 0 6 -2 4 -4 2 -6 0 -12 -10 -8 -6 -4 -2 0 2 -2 -4 -6 -14 -12 -10 -8 -6 -4 -2 0 2 4

example performance eval eval … … example performance