Teoria - Automation and Control in Vehicles

Automation and Control in Vehicles (ACV)

28/02/2013

Chapter 1: Suspension Control

• three main models

Introduction (the problem):

The chassis of the car → single RIGID BODY (3 directions)

Suspensions directly influence three main movements:

• Heave (displacement)
• Roll (rotation)
• Pitch (rotation)

Indirectly, sometimes influence residual movements:

• Yaw, Sway, Surge

M_susp: sprung mass (body)

C: damping coefficient of damper

K: spring coefficient

m_u: unsprung wheel mass

K_t: tire stiffness _pneumatic

We have two mainly important signals:

• Z_r: road profile INPUT
• Z: chassis height OUTPUT

(the car is split in four pieces, one for each wheel.)

The suspension is a low-pass filter:

Objectives:

1. COMFORT: small body acceleration

Z_.. must be minimized

Filter → I/O transfer function (road to body)

Abs. Amplitude frequency response of transfer function Z/Z_r (low-pass filter with 2 resonances)

Ideal comfort: perfectly flat (Z=0)

Real comfort: low-frequency component → pass, high-frequency component → stop

Body resonance, wheel resonance

• Is perfect disturbance cancellation possible? almost possible
• Main limitations?
  • We have to enlarge the bandwidth with active suspensions that allow us to deal with higher cancellation reducing the delay of control feedback
  • Actuators capability: we must have quick electric motors with large forces that bring us in a lot of power consumptions.
  • Available travel of suspensions comparable to the size of the disturbances.

2) HANDLING/PERFORMANCE/SAFETY: Small road variations

The force F is splitted in 3 directions

• Fx = Fz • μx
• Fy = Fz • μy

μ: friction coefficient depending on road conditions

In order to increase Friction, Fz must be maximize

Fz = M₀ g + Dynamic Load + Aerodynamic Load

Fz negative part very bad → F = ∅ means loose of contact (we don't have Fx, Fy anymore)

I want to stay near the nominal part → SMALL ROAD VARIATION (PERFECTLY FOLLOW THE OBSTACLE)

3) STROKE LIMITATION

In order to avoid destruction of performance 1-2

• Stroke limitation depends in every situation (S ≈ 50 cm)
• End-stop bushes: made of rubber to avoid contact between steel-steel

But where is the damper?

• Since the pipe line has a continuous flow we can see a dissipation effect such as the one caused by the orifices in the piston of the damper.
• We can control pipe line diameter electronically.

Comparison:

• Size/space
• Air hollow requires higher volume
• Very simple design managing

Static Friction ➔ Initial force to go over friction is higher than the dynamic one.

Model:

• z_ℒ: where suspension is connected to the wheel
• z: where upper part of suspension is connected to the body
• t: means tires, a rubber bubble (spring effect)

What are we neglecting in the model?

Quater-car mathematical model

M z̈_ℒ(t) = -c(ż_ℒ(t) - ż_b(t)) -k (z_ℒ(t) - z_b(t) - Δs) - Mg

m z̈_b(t) = +c(ż_ℒ(t) - ż_b(t)) + K (z_ℒ(t) - z_b(t) - Δs) -k_t (z_ℒ(t) - z_r(t) - Δt) - mg

(mass acceleration = ∑(forces) ➔ only 1 direction (Z))

x = [ ^z/₂ℒ ^z/₂ ^z/_ℒ ]

u = z_r

4^th order, linear, t-invariant differential equation

where ^z/_ℒ

are unloaded deltas

Limits of validity?

1. Available stroke
2. Tire part of the second equation cannot be negative

z_r is a disturbance not measurable

Full model: sensitivity w.r.t. C

we normally have 3 important C points

1. - W = sqrt(k/m)

the other two are difficult to find

Full model: sensitivity w.r.t. wheel mass

• Fac (comfort)
• Finacc (comfort force)

• on heavy wheel is much more comfortable
• lighter wheel worse comfort

green line useful for sport due to the advantage in contact at higher frequencies

remark on wheel mass

• In-wheel motor: motor built (integrated) in the wheel
• The disadvantage is about less contact force – good for city-car like camper.

Motorcycle:

• Older fork
• Newer fork

Disadvantages:

• Less contact forces means less friction on the road

Advantages:

• Simpler design management

Advantages:

• Damping mass attached to the big mass of the body
• Good for contact forces, so for motor sports
• Less wheel mass

Static equilibrium equation for a gas-spring:

Mg = (p - patm) A

Stiffness of a gas-spring

For fast movements (higher than 0.1 Hz) we can assume adiabatic compression

pV^γ = const

Total differential:

⇒ dp = -pδV / V

Volume V = (z - zℓ) A

⇒ δF = A · dp

Example:

• Same area/same pressure
• Same volume

Pneumatic (gas) spring - levelling

Nominal condition

Increased load not leveled

Increased load leveled

Hydro-pneumatic spring

Some gas-spring but with oil injection instead of air

Constant volume levelling

Typical values:

• patm = 300 kPa
• p = 400-100 kPa

Constant mass levelling

V · p = V_N · p_N + V_P = V_N p_N / p

This is not an adiabatic compression, no mass variation.

Stroke-Threshold (ST) control (short-stroke applications)

We have two-state switched damper.

C(Lt) = Cmax if |Z-dot-Ze| > Te C(Lt) = Cmin if |Z-dot-Ze| < Te

The priority is to avoid contact, useful when we have short stroke availability.

Cmin in the central part or Cmax in the higher/lower part It's mechanically feasible? Yes The higher the C the lower the stroke

Remark K 1. On semi-active algorithms

It is an high level point of view. The system and the controller are non-linear.

zr: disturbance c(t): control Z-hat: output comfort

We have two sensors in feedback { stroke { acceleration

PROBLEMS:

• Non linearity
• Non smooth non-linearity (saturation)
• Linearization C(t) not anymore controllable

We can try to minimize the error, we don't directly design (optimize) a just studied strategy. Direct design is extremely difficult (only ADD but with no real assumption)

Remark 2. About Skyhook algorithm sensors

Our physical sensors are

• Body accelerometer: Z-double-dot + da
• Stroke sensor (elong.): (z-Ze)+de

Thinking about Skyhook algorithm, we need velocity and stroke speed, so we need to deal with an integration we can filter through the Transfer Function.

PROBLEM: BODY SPEED ESTIMATION

We have also a bias problem disturbance (constant) due to an imperfect vertical position of the sensor.

Trick: Instead of pure integration we can filter 1 over s+epsilon where epsilon is one decade below the first resonance

V(s) = A(s) 1 over s+epsilon + D(s) 1 over s+epsilon where epsilon = 2Pi . 0.1 (or beyond 0.1 Hz)