Appunti di Automation and control in electric and hybrid vehicles

Pneumatic suspensions

Also called constant-volume leveling, Fig. 3a.

Recall ∂ = Mg/Mk and ∂ = c/M. For typical values ∂ = 1 + ωξ p/pn atm, so:

Mg/Mk ≈ ωξ p/pn atm

Mk/M ≈ ωξ n Vc = ξ 2Mωn

∂ is invariant with respect to ω and n, so controllable damping is needed.

Hydro-Pneumatic suspensions

Also called constant-mass leveling, Fig. 3b.

Assume an isothermal transformation. pVn^n = pV^n. Recalling the usual formulas for ω and ξ:

ω = √(2γMg/n) and ξ = √(2ωMn/pVn)

This can be managed with controllable damping, however pneumatic suspensions are still better because dynamics doesn't change with

1. The low frequency relationship between flow and height is ∆z = therefore a simple1−φ z z φ,t sproportional controller is often enough to control the system.
2. The stiffness changes at different loads.
3. We could have regressive and progressive behavior depending on the stroke position.
4. Manual pre-loading adjusters or hydraulic height adjusters are possible as well.

1.8 Control - Full-active suspensions

Model - Slow-Active

Architecture 1 (Fig. 19a):“Mercedes”

 = )−k(z ∆ )−c( − − − −M z̈ ż ż z M g t t s= +c( ) + ∆ ) (z ∆ )− − − − − − −mz̈ ż ż k(z z k z mgt t p t s t t r t = )+F

 −c( − −M z̈ ż ż M gt ∆ )−F = 0

−k(z − −z p t s= +c( ) + ∆ ) (z ∆ )− − − − − − −mz̈ ż ż k(z z k z mgt t p t s t t r

t̴̴̴̴ = +−βF̴ Ḟ βF̴ inWith 5 2π. Notice the second equation is not dynamic and is only used to determine the plunger≈ ·βposition . The inputs are now = [z ].z u , Fp r inArchitecture 2 (Fig. 19b):“Audi” = ) ∆ )−c( − − − − −M z̈ ż ż k(z z M g t t s= +c( ) + ∆ ) (z ∆ )− − − − − − −mz̈ ż ż k(z z k z mgt t t s t t r t = +F ) ∆s) − − − − − −M z̈ c(ż ż k(z z M gt t = + ) + ∆ ) (z ∆ )−F − − − − − − −mz̈ c(ż ż k(z z k z mgt t t s t t r t = + −βFḞ βF inModel - Full-ActiveWe use the force instead of the damper. Also if we don’t use springs (Fig. 19c):F  = ) ∆ )−c( − − − − −M z̈ ż ż k(z z M g t t s= +c( ) + ∆ ) (z ∆ )− −

− − − − −mz̈ ż ż k(z z k z mgt t t s t t r t = −M z̈ F Mg = (z ∆ )−F − − − −mz̈ k z mgt t t r t = + −βFḞ βF inWith 20 2π.≈ ·β a Slow-Active b Slow-Active c Full-Active“Mercedes” “Audi”Figure 19is easier than semi-active suspensions: the system is linear so classical control design tools can beControlused (LQR, H , etc). For example we could minimize = lim [z̈ + (z ) + (z ) ]dt.T1 2 2 2− −RJ ρ z ρ z∞ →∞T t t r1 20TWith full-active electro-mechanical suspensions we could also recuperate energy, in this case two differentactuators can be used:Rotary PM motors: high power density but needs gearbox to transform rotational to linear motion.12Tubular PM actuators: lower power density but to linear movement.direct driveNew types of suspensions include mixed semi-active/active actuators.

piston has no orifices and oilflows through a motor-driven pump. In semi-active mode the motor is used as generator (the dampingforce can be controlled), in active mode the motor delivers energy to the system. Energy recuperation isin the order of 30W per corner.

1.9 Sensors configurations

The most used sensor configurations when height adjusting is needed uses 4 elongation sensors ∆z, 4accelerometers as well as a 6dof central IMU.z̈Another classical configuration when height adjusting is not needed uses 4 accelerometers for insteadz̈tof the elongation sensors.A lot of reduced sensors configurations can be used with minor loss of performance (Fig. 20).

Figure 20: Main sensors configurations

1.10 Appendix - Towards the autonomous car

As far as suspensions are concerned, in autonomous cars we need to emphasize on:

• Comfort performance rather than road-holding performance.
• Motion sickness reduction, usually around 0.2Hz.
• Minimize longitudinal and lateral accelerations.

technologies such as in-wheel motors and preview capabilities (both from sensors and from knowledge of the car trajectory).

132 Longitudinal dynamics control

General goal: control longitudinal wheel slip during braking or traction.

Related movements: forward movement.

2.1 Introduction

Define:

 if braking

v−ωr

=Longitudinal slip = v−ωr

vλ: λ if traction

max{v,ωr} v−ωr

 ωr

Side slip angle angle between the tire longitudinal axis and the wheel velocity vector.

α:Camber angle angle between the tire vertical axis and the direction orthogonal to the road. Inγ:motorcycles this is called lean angle.

2.1.1 Tire-road contact forces

Both and depend on a large number of features of the road, tyre, suspensions, etc but are often

F Fx ydescribed as: (F (λ,∼ ·F , λ, α, γ) F μ α)x z z x(F (λ,∼ ·F , λ, α, γ) F μ α)y z z y

directionality.λIn summary, given a value increasing transfers force from lateral to longitudinal force. This can beα, λbetter seen in the plane (Fig. 22).−μ μx y Figure 22: relation−μ μx y14To address accelerations, we study the lateral-longitudinal acceleration plot, also called g-g plot. Undersimplifying assumptions = = = so if aerodynamics is not important the−→ −→F F μ M a M gμ a gμ,znormalized g-g plot must be identical to the plot.−μ μx yRemarks:A perfect driver always drives at the limits of the g-g plot (Fig. 23a), while an everyday driver• stays on the two axes (Fig. 23b).• Aerodynamic forces allow us to greatly enlarge the g-g plot limits (Fig. 23c).a Professional driver b Everyday driver c g-g plot without aero(green) and with aero(red)Figure 232.1.2 Aerodynamic forcesWe know: 1= 2ρCSvFxdrag 21= 2F ρCSvzlift 2And define the efficiency as = .

Normally 0.3 and (0.2, while 5 in F1 cars.≈ ∈ −3), ≈e C /C C C ez x x zIn braking we must consider aerodynamic forces: = and = = (M )µ,−F −F −FM a F F µ gx zdrag brake brake lifttherefore if 0 we increase down force without adding weight!C <z2.1.3 Braking and acceleration forcesFigure 24: Load transfer L   += h=0 = + f M g M aFP F M g F F xzr  z zr zf L L+L +L−→ −→ r rf f= 0 + = 0 = hL −−F −P M g M aM L F L M a h F rzf f zr r x zf xyellow   L L+L +Lr rf fAnd the static load distribution is: L = fF Mgzr L +Lrf= LF Mgrzf L +LrfIn general load transfer is a negative effect: L• We can find the rollover limit condition: setting = 0 and = we get = However− fF F M g a g.zr zf x hLthe actual rollover also depends on friction: if rollover occurs before over-slip (like infµ >x,max hLmotorcycles), if over-slip occurs before rollover (like in