A Nonlinear Tracking Controller for a Haptic Interface Steer-by-Wire Systems P. Setlur, D. Dawson, J. Chen, and J. Wagner Departments of Mechanical and Electrical/Computer Engineering Conference on Decision and Control, December 2002, Las Vegas

1
A Nonlinear Tracking Controller for a Haptic
Interface Steer-by-Wire Systems P. Setlur, D.
Dawson, J. Chen, and J. WagnerDepartments of
Mechanical and Electrical/Computer Engineering
Conference on Decision and Control, December
2002, Las Vegas
2
Presentation Outline

• Introduction
• System Description and Problem Statement
• Problem Motivation
• Past Research
• Model Development
• System model
• Reference model concepts
• Adaptive Control Design
• Error Definitions
• Control Design
• Stability Proof
• Extension to Eliminate Torque Measurements
• Numerical Simulation Results
• Experimental Results
• Setup
• Preliminary Results
• Conclusion

3
System Description
Steer-by-wire system with haptic interface
Conventional system
Primary Subsystem
Secondary Subsystem
4
Problem Motivation

• Advent of Hybrid Vehicles is due to scarcity in
  fossil fuel and environmental concerns
• engine may be cycled on/off Hydraulic steering
  systems not feasible
• power limitations mandate efficient
  technologies
• Steer-by-wire systems provide
• improved vehicle response ( electrical systems
  are faster)
• ability to use additional driver input devices (
  joystick)
• Varied preferences in amount of feedback and feel
• most important feedback to the driver, after
  vision
• Flexibility in vehicle design

5
Haptic Interface - Goals

• Accurate reproduction of driver commands at the
  wheel
• Provide force feedback to the driver
• Use feedback motor in steer-by-wire systems
• Ability to scale inputs
• Displacement of the driver input device should be
  governed by a set of target dynamics
• Tunable dynamics that permit various choices of
  road feel
• Adaptive techniques to compensate for unknown
  system parameters
• Elimination of force measurement
• Identification of tire/road interface forces

6
Past Research

• Liu et al. - worked on estimating the effect of
  force feedback in a driving simulator
• (1995)
• Gillespie et al. - proposed use of force
  reflecting joysticks to cancel feedthrough
• dynamics in
  aircrafts (1999)
• Qu et al. - showed how a dynamic robust-learning
  control scheme can compensate
• for disturbances that are
  bounded and sufficiently smooth (2002)
• Lewis et al. - detailed description of the
  impedance control technique (1993)
• Setlur et al. - controller to achieve trajectory
  tracking for steer-by-wire systems (2002)
• Mills et al. - developed detailed models for
  steer-by-wire systems (2001)

7
System Model
Primary Subsystem
I1 , I2 - Lumped inertia of Primary
and Secondary subsystems
Damping and Friction effects
Secondary Subsystem
8
Reference Model - Concept
User feels no difference between these two cases
Impedance Control Technique
9
Reference Model
Target Conventional system
Primary Subsystem

• If follows , then the driver
  feels as if he were driving a conventional
• vehicle with inertia , damping and
  friction function .
• Target system parameters are chosen so that the
  reference trajectories remain bounded
• at all times (reference system dynamics
  are BIBO stable).

10
Adaptive Control

• To quantify the control objective, the following
  error signals are defined
• After taking the time derivatives of the filtered
  tracking errors, the open-loop error system can
  be rewritten as
• To achieve the control objectives outlined, the
  control torques are designed as

Filtered Tracking Errors
Driver Experience Tracking error
Locked Tracking error
Parameter Update Laws
11
Adaptive Control

• After substituting the control in the open-loop
  error system, the closed-loop error system can be
  written as
• A non-negative function is defined as
• After differentiating the above function with
  respect to time, and substituting the
  closed-loop error systems, we obtain

Parameter estimation errors
12
Elimination of Torque Measurements

• For this extension, all system parameters are
  assumed to be known. The target dynamics are
  generated using estimated torques. The tracking
  error signals are defined as before
• After taking second derivative with respect to
  time and using the system and reference dynamics,
  we obtain the open-loop error system
• The control torques, T1 and T2 are designed as

Torque Observers (to be designed)
13
Elimination of Torque Measurements

• After substituting the control design in the
  open-loop error system, the closed-loop error
  system can be written as
• Clearly, if e1 e2 0 then t1 t1 and t2 t2
  (Identification of tire road forces).
• The filtered tracking errors are redefined for
  this problem as

.
s1 0 e1, e1, e1 0
Analysis will be presented only for the Primary
System. The analysis for the secondary system is
based on similar arguments
14
Elimination of Torque Measurements

• After taking the first time derivative and using
  the system and reference dynamics, we obtain the
  open-loop error system
• Based on the above structure, the torque observer
  is designed as
• After substituting the observer in the open-loop
  error system, the closed-loop error system can be
  written as

Add and subtract (s1(t) is NOT measurable)
Standard Signum function (sign function in
matlab)
Robust control like term
Feedback term
15
Elimination of Torque Measurements

• A non-negative function Va1(t) is defined as
• After differentiating the above function with
  respect to time, and substituting the
  closed-loop error system, we obtain
• After integrating both sides and performing some
  manipulations, we obtain
• So, . Similarly, we can
  show . From Babalats
  Lemma,

and
16
Simulation Results

• Simulated system was assumed to have the
  following parameters

17
Simulation Results

• The target dynamics were generated using
• Further to evaluate performance, a conventional
  system was simulated

IT 2 Kg-m2 BT 1 Kg-m2/s KT 1 N-m aT1
1 aT2 0.1
Ia I1 I2 54.268 Kg-m2 Ba B1 B2 1.001
X 10-2 Kg-m2/s Ka K1 K2 1.001 X 10-4 N-m a1
1 a2 1
18
Simulation Results - Adaptive Control
40
20
0
0.4
q
d1

0.3
0.2
Angular Displacement (rad)
0.1
q

a
0
-0.05
0
50
100
150
200
time (s)
19
Simulation Results - Adaptive Control
6
4
e

2
0
e
1

-4
Tracking Error (rad)
-8
-12
-14
70
60
T

2
40
Control Torques (N-m)
20
T
1
0
-10
0
50
100
150
200
time (s)
20
Simulation Results - EMK Extension
70
60
40
T
2
Control Torques (N-m)
20
T
1
0
-10
0.08

t
2
0.04

t
Torque Observation Errors (N-m)
1
0
-0.04
-0.06
0
50
100
150
200
time (s)
21
Experimental Results - EMK Extension
Drive Motor
Steering Wheel
Torque Sensors
Feedback Motor
Rack
LVDT
Hydraulic Damper
Current Sensors
Preamplifiers
22
Experimental Results - EMK Extension

• Tests were performed to identify the parameters
  of the system. The following results were
  obtained
• The target system was chosen to have the
  following parameters
• The control gains were chosen to be

I1 0.0725 Kg-m2 B1 0.3 Kg-m2/s K1 0 N-m
I2 2.5 X 10-3 Kg-m2 B2 2 X 10-3 Kg-m2/s K2
0 N-m
IT 2 Kg-m2 BT 0.3 Kg-m2/s KT 0 N-m aT1
10 aT2 1
b1 500 Ks 700 r1 1
r2 10
23
Experimental Results - EMK Extension
24
Experimental Results - EMK Extension
0
10
20
30
40
50
time (s)
25
Experimental Results - EMK Extension

• Torque sensor measurements
• Noisy
• Drift
• Low resolution
• Target system dynamics involves twice integrating
  the torque signals for Adaptive control
• Gearing factor a1 and a2
• Torque capacity of the Feedback motor
• Repeatability of driver input - Choice of r
• larger value control torques have to
  change quickly (motors are
• inductive systems)

26
Concluding Remarks

• Presented Vehicle Steering System Model for the
  Steer-by-wire configuration.
• Presented the Adaptive tracking control algorithm
  to ensure that
• vehicle follows driver commands
• driver is provided a haptic feedback
• Proposed an EMK extension that eliminates the
  need for torque sensor measurements
• identified tire/road interface forces
• Simulation Results verify the efficacy of the
  proposed control laws
• Preliminary Experimental Results were presented
  to discuss practical issues
• Future work would involve
• Control algorithm to compensation of parametric
  uncertainties without measurement of torque
• Incorporation of visual feedback for
  driver-in-loop tests