Optimization of Global Chassis Control Variables

1
Optimization of Global Chassis Control Variables

• Josip Kasac, Joško Deur, Branko Novakovic,
• Matthew Hancock, Francis Assadian
• University of Zagreb, Faculty of Mech. Eng.
  Naval Arch., Zagreb, Croatia (e-mail
  josip.kasac_at_fsb.hr, josko.deur_at_fsb.hr,
  branko.novakovic_at_fsb.hr).
• Jaguar Cars Ltd, Whitley Engineering Centre,
  Coventry, UK
• (e-mail fassadia_at_ford.com, mhancoc1_at_jaguar.com).

2
Introduction

• Introduction of new actuators - active rear
  steering (ARS), active torque vectoring
  differential (TVD), active limited-slip
  differential (ALSD), offers new possibilities of
  improving active vehicle stability and
  performance
• However, the control system becomes more complex
  (Global Chassis Control GCC), which calls for
  application of advanced controller optimization
  methods
• Benefits of using the nonlinear open-loop
  optimization
• assessment on the degree of GCC improvement
  achieved by introducing different actuators
• gaining an insight on how the state controller
  can be extended by feedforward and/or gain
  scheduling actions to improve the performance.
• In this paper a gradient-based algorithm for
  optimal control of nonlinear multivariable
  systems with control and state vectors
  constraints is proposed
• GCC application - double lane change maneuver
  executed by using control actions of active rear
  steering and active rear differential actuators.

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Optimal control problem formulation

• Find the control vector u(t) that minimizes the
  cost function

Time-dicretization

• subject to the nonlinear MIMO dynamics process
  equations

Euler time-dicretization

• with initial and final conditions of the state
  vector

• subject to control state vector inequality and
  equality constraints

4
Extending the cost function with
constraints-related terms
Basic cost function defined above
Penalty for final state condition
Weighting factors
Penalty for inequality constraints
Penalty for equality constraints
Final problem formulation
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Comparison with nonlinear programming based
algorithms

• Penalty functions

• Plant equation constraints

• Nonlinear programming approach

• Advantage vs. Nonlinear Programming based
  algorithms Process equations constraints (ODE)
  are not included in the total cost function as
  equality constraints

• The control and state vectors are treated as
  dependent variables, thus leading to backward in
  time structure of algorithm ? similar to BPTT
  algorithm from NN

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Exact gradient calculation

• Implicit but exact calculation of cost function
  gradient

- chain rule for ordered derivatives

• BPTT algorithm time generalisation of BP
  algorithm

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Backward-in-time structure of the algorithm
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Modified gradient algorithm - convergence
speed-up

• The gradient algorithm with the constant
  convergence coefficient ? and a linear gradient
  is characterized by a slow convergence.
• Small value of the gradient near the optimal
  solution is the main reason for the slow
  convergence.

• a sliding-mode-based modification of the
  gradient algorithm
• provides a stronger influence of the gradient
  near the optimal solution, and better convergence

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Definition of vehicle dynamics quantities
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1. State-Space Subsystem

• 1.1 Longitudinal, lateral, and yaw DOF

Fxi, Fyi, - longitudinal and lateral forces M -
vehicle mass, Izz - vehicle moment of inertia,
b - distance from the front axle to the CoG, c
- distance from the rear axle to the CoG, t -
track
U, V - longitudinal and lateral velocity, r
- yaw rate, X, Y - vehicle position in the
inertial system ? - yaw angle
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• 1.2 The wheel rotational dynamics

?j - rotational speed of the i-th wheel, Fxti
- longitudinal force of the i-th tire, Ti -
torque at the i-th wheel, Iwi - wheel moment
of inertia, R - effective tire radius.

• 1.3 Delayed total lateral force (needed to
  calculate the lateral tire load shift)

• 1.4 The actuator dynamics

- rear wheel steering angle,
- rear differential torque shift,
- actuator time constants.
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2. Longitudinal Slip Subsystem
3. Lateral Slip Subsystem
4. Tire Load Subsystem
l - wheelbase hg - CoG height
5. Tire Subsystem
µ - tire friction coefficient B, C, D - tire
model parameters
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6. Rear Active Differential Subsystem
?Tr - differential torque shift control
variable, Ti - input torque (driveline
torque) and Tb - braking torque

• Active limited-slip differential (ALSD)

• Torque vectoring differential (TVD)

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GCC optimization problem formulation

• Nonlinear vehicle dynamics (process) description

• Control variables (to be optimized) ?r (ARS)
  and ?Tr (TVD/ALSD)
• Other inputs (drivers inputs) ?f
• State variables U, V, r, ?i (i 1,...,4), ?,
  X, Y

• Cost functions definitions

Reference trajectory

• Path following (in external coordinates)
• Control effort penalty
• Different constraints implemented
• control variable limit
• vehicle side slip angle limit
• boundary condition on Y and dY / dt

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Example Double line change maneuver (22 m/s, ?1)
Reference trajectory for next optimizations

• Front wheel steering optimization results for
  asphalt road (? 1)

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• Optimization results for ARSTVD control and ?
  0.6

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• Optimization results for ARS control and ?
  0.6

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• Optimization results for TVD control and ?
  0.6

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• Optimization results for ALSD control and ?
  0.6

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• Optimization results for different actuators
  (?0.6)

ARSTVD
ARS
TVD
ALSD

• ARS and TVD gives comparable results no
  advantage of combined ARS/TVD (except for lower
  control effort) ALSD less effective due to lack
  of oversteer generation

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• Optimization results for different actuators
  (?0.3)

ARSTVD
ARS
TVD
ALSD

• At low-? surface the lateral optimizer limits
  lateral acceleration to stabilize vehicle as a
  result trajectory tracking is worsen

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Conclusions

• A back-propagation-through-time (BPTT) exact
  gradient method for optimal control has been
  applied for control variable optimization in
  Global Chassis Control (GCC) systems.
• The BPTT optimization approach is proven to be
  numerically robust, precise (control variables
  are optimized in 5000 time points), and rather
  computationally efficient
• Recent algorithm improvement
• numerical Jacobians calculation
• implementation of higher-order Adams methods
• The future work will be directed towards
• use of more accurate tire model
• introduction of a driver model for closed-loop
  maneuvers
• model extension with roll, pitch, and heave
  dynamics
• implementation of different gradient methods for
  convergence speed-up