automotive research center robotics and mechatronics a nonlinear tracking controller for a haptic...

Automotive Research CenterRobotics and MechatronicsRobotics and Mechatronics

A Nonlinear Tracking Controller for a A Nonlinear Tracking Controller for a Haptic Interface Steer-by-Wire SystemsHaptic 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

CLEMSONU N I V E R S I T Y


Presentation OutlinePresentation 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


System DescriptionSystem DescriptionSteer-by-wire system with haptic interfaceConventional system

Primary Subsystem

T1

Feedback Motor

Secondary Subsystem

2

T2

Drive Motor a2

a

Tire/Road interface forces

Driver input torque

I aÄµa + Na

³µa; _µa

´= ®1¿1 + ®2¿2


Problem MotivationProblem 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


Haptic Interface - GoalsHaptic 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


Past ResearchPast 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)


System ModelSystem Model

I 1Äµ1 + N1

³µ1; _µ1

´= ®1¿1 + T1

I 2Äµ2 + N2

³µ2; _µ2

´= ®2¿2 +T2

Secondary Subsystem

Primary Subsystem

I1 , I2 - Lumped inertia of Primary

and Secondary subsystems

N1

³µ1; _µ1

´= YN 1

³µ1; _µ1

´ÁN1 Damping and

Friction effects N2

³µ2; _µ2

´= YN 2

³µ2; _µ2

´ÁN 2

®1 ®2, - Scaling factors (gear ratios)

T1

Feedback Motor

2

T2

Drive Motor


Reference Model - ConceptReference Model - Concept

User feels no difference between these two cases

“Impedance Control Technique”


• 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).

Reference ModelReference Model

I T Äµd1 +NT

³µd1; _µd1

´= ®T 1¿1 +®T 2¿2

Target Conventional system

d2

d

I 1Äµ1 + N1

³µ1; _µ1

´= ®1¿1 + T1

T1

Primary Subsystem

µ1(t) µd1(t)I T NT

³µd1; _µd1

´


• 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

r1 = _e1 +¹ 1e1

r2 = _e2 +¹ 2e2

e1 = µd1 ¡ µ1

e2 = µ1 ¡ µ2:

Filtered Tracking Errors

I 1 _r1 = Y1Á1 ¡ T1

I 2 _r2 = Y2Á2 ¡ T2

Adaptive ControlAdaptive Control

Driver Experience Tracking error

Locked Tracking error

T1 = k1r1 + Y1Á̂1

T2 = k2r2 + Y2Á̂2

¢

Á̂1= ¡ 1Y T1 r1

¢

Á̂2= ¡ 2Y T2 r2

Parameter Update Laws


• 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

limt! 1

e1 (t) ;e2 (t) = 0:

Adaptive ControlAdaptive Control

I 1 _r1 = ¡ k1r1 + Y1~Á1

I 2 _r2 = ¡ k2r2 + Y2~Á2

V (t)

V =12

I 1r21 +

12I 2r2

2 +12

~ÁT1 ¡ ¡ 1

1~Á1 +

12

~ÁT2 ¡ ¡ 1

2~Á2

~Á1 = Á1 ¡ Á̂

~Á2 = Á2 ¡ Á̂2Parameter estimation errors

_V · ¡ k1r21 ¡ k2r2

2


• 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

Elimination of Torque MeasurementsElimination of Torque Measurements

e1 = µd1 ¡ µ1

e2 = µ1 ¡ µ2:

Äe1 =µ

1IT

¶(¡ NT (¢) + ®T 1¿̂1 + ®T 2¿̂2) ¡

µ1I1

¶(¡ N1 (¢) + ®1¿1 + T1)

Äe2 =µ

1I 1

¶(¡ N1 (¢) + ®1¿1 + T1) ¡

µ1I 2

¶(¡ N2 (¢) + ®2¿2 + T2) :

T1 = N1 (¢) +µ

I 1

I T

¶(¡ NT (¢) + ®T1¿̂1 + ®T 2¿̂2) ¡ ®1¿̂1

T2 = N2 (¢) +µ

I 2

I 1

¶(¡ N1 (¢) +T1 +®1¿̂1) ¡ ®2¿̂2:

Torque Observers(to be designed)


Elimination of Torque MeasurementsElimination 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 1 = 1 and 2 = 2 (Identification of tire road forces).

• The filtered tracking errors are redefined for this problem as

Äe1 =¡µ

®1

I 1

¶(¿1 ¡ ¿̂1)

Äe2 =µ

®1

I1

¶(¿1 ¡ ¿̂1) ¡

µ®2

I 2

¶(¿2 ¡ ¿̂2)

^ ^

s1 = Äe1 + (̄ 1 + 1) _e1 +¯1e1

s2 = Äe2 + (̄ 1 + 1) _e2 +¯1e2s1 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

.. ..



• 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

_s1 = ¡ K ss1 ¡®1

I 1

µ_¿1¡

¢¿̂1 ¡ (¯1 + K s + 1)(¿1 ¡ ¿̂1)

¶+ (¯1 + K s (¯1 +1)) _e2 + K s¯ 1e2

¢¿̂1= ¡ (¯1 + K s + 1) ¿̂1 ¡

I 1

®1[(¯1 + K s (¯1 + 1)) _e1 +K s¯1e1 + ½1sgn(p1)]

_s1 = ¡ K ss1 ¡ ´1 ¡ ½1sgn (p1)

Standard Signum function (sign function in matlab)

Feedback term

´1 =µ

®1

I 1

¶(_¿1 +(¯1 + K s + 1) ¿1)

Unmeasurable Disturbance

Robust control like term

p1 = _e1 + ¯1e1

Add and subtract (s1(t) is NOT measurable)



• 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 Babalat’s Lemma,

Va1 =12

s21

_Va1 = ¡ K ss21 +( _p1 + p1)(¡ ´1 ¡ ½1sgn (p1))

Va1 (t) · Va1(t0) ¡ K s

tZ

t0

s21 (¾) d¾+ ³01

s1 2 L 1 \ L 2 s2 2 L 1 \ L 2

limt! 1

e1 (t) ;e2 (t) = 0: limt! 1

¿̂1 =¿1 and limt! 1

¿̂2 = ¿2


Simulation ResultsSimulation Results

• Simulated system was assumed to have the following parameters

I 1Äµ1 + N1

³µ1; _µ1

´= ®1¿1 + T1

I 2Äµ2 + N2

³µ2; _µ2

´= ®2¿2 +T2

I1 = 6.8 X 10-2 Kg-m2

B1 = 1 X 10-5 Kg-m2/sK1 = 1 X 10-7 N-m = 11 = 5t exp(-0.005t)

®1

T1

2

T2

®2

I2 = 54.2 Kg-m2

B2 = 1 X 10-2 Kg-m2/sK2 = 1 X 10-4 N-m = 12 = -200 tanh(2)

Nx(.) = Bxqx + Kxqx .


• The target dynamics were generated using

• Further to evaluate performance, a conventional system was simulated

Simulation ResultsSimulation Results

IT = 2 Kg-m2

BT = 1 Kg-m2/sKT = 1 N-mT1 = 1T2 = 0.1

I T Äµd1 +NT

³µd1; _µd1

´= ®T 1¿1 +®T 2¿2

d2

d

I aÄµa + Na

³µa; _µa

´= ®1¿1 + ®2¿2

Ia = I1 + I2 = 54.268 Kg-m2

Ba = B1 + B2 = 1.001 X 10-2 Kg-m2/sKa = K1 + K2 = 1.001 X 10-4 N-m1 = 12 = 1


Simulation Results - Adaptive ControlSimulation Results - Adaptive Control

0 50 100 150 200

0

20

40

time (s)

1 (

N-m

)

-0.05

0

0.1

0.2

0.3

0.4

Ang

ular

Dis

plac

emen

t (ra

d)

d1

a


-14-12

-8

-4

0

46

x 10-3

Tra

ckin

g E

rror

(ra

d)

e1

e2

-10

0

20

40

60

70

Con

trol

Tor

ques

(N

-m)

T2

T1

0 50 100 150 200

time (s)

Simulation Results - Adaptive ControlSimulation Results - Adaptive Control


-10

0

20

40

60

70

Con

trol

Tor

ques

(N

-m)

0 50 100 150 200

time (s)

T1

T2

-0.06

-0.04

0

0.04

0.08

Tor

que

Obs

erva

tion

Err

ors

(N-m

)

2

1

Simulation Results - EMK ExtensionSimulation Results - EMK Extension


Experimental Results - EMK ExtensionExperimental Results - EMK Extension

Steering Wheel

Hydraulic Damper LVDT

Drive Motor

Feedback Motor Rack

Torque Sensors

Preamplifiers

Current Sensors



• 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/sK1 = 0 N-m

I2 = 2.5 X 10-3 Kg-m2

B2 = 2 X 10-3 Kg-m2/sK2 = 0 N-m

IT = 2 Kg-m2

BT = 0.3 Kg-m2/sKT = 0 N-mT1 = 10T2 = 1

1 = 500 Ks = 700 1 = 1 2 = 10



-0.4

-0.2

0

0.2

0.4

e 1, e

2 (r

ad)

0 10 20 30 40 50time (s)

0 10 20 30 40 50-2

-1

0

1

2

time (s)

d , 1

, 2

(ra

d)

-3

-2

-1

0

1

2

3

T1 ,

T2

0 10 20 30 40 50time (s)


-4

-3

-2

-1

0

1

2

3

4

1 ,

1 (N

-m)

^

0 10 20 30 40 50time (s)

-3

-2

-1

0

1

2

3

4

5

2 ,

2 (N

-m)

^




• Torque sensor measurements

– Noisy

– Drift

– Low resolution

• Target system dynamics involves twice integrating the torque signals for Adaptive control

• Gearing factor 1 and 2

• Torque capacity of the Feedback motor

• Repeatability of driver input - Choice of – larger value control torques have to change quickly (motors are

inductive systems)


Concluding RemarksConcluding 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

automotive research center robotics and mechatronics a nonlinear tracking controller for a haptic...

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