Non-linear Model Predictive Control for Autonomous Vehicles

Muhammad Awais Abbas

Supervisor: Prof. Mikael Eklund

Co-Supervisor: Prof. Ruth Milman

Presentation Outline Autonomous Vehicles: Introduction Autonomous Vehicle Framework Vehicle Model Model Validation Gradient Descent Algorithm Cost Function Formulation Simulation Results

Obstacle Avoidance Tests Real-Time Analysis Reference Trajectory Criterion

Conclusion

Autonomous Vehicles: Introduction Capable of navigating on its own.

Human not required for operation of the vehicle.

Types of autonomous vehicles: Aerial Vehicles (UAVs), Ground Vehicles (UGVs) Surface Vehicles (ASVs), Underwater Vehicles (AUV)

Technology of future

Require advanced control systems and sensing

In the U.S. State of Nevada autonomous vehicle can be legally operated on roads.

Autonomous Vehicle Framework Layered Architecture.

Trajectory may need to be replanned. Unknown obstacles, vehicles.

Difference in type of information available to Low Level and Replanning Layer.

Difference in sampling frequencies,Controller intelligence.

Offline Trajectory Generation

Layer 1

Online Trajectory Replanning

Layer 2

Low Level Control Layer 3

Vehicle and Environment

U

y

Model Predictive Control: Introduction Optimal Control

Linear Quadratic Regulator (LQR) Model Predictive Control (MPC)

Ability to look into the future

Advantages Constraints Nonlinear systems Online control solution

Limitations Computation time

PLAN

PLAN

PLAN

DO

DO

DO

t

t

t

Step 1

Step 3

Step 2

t k

2t k

3t k

t k Nk

2t k Nk

3t k Nk

N

MPC Strategy

Cost Function

where , , and are weighing matrices.

1

0

( ) , ,N

N k k kk

J L u

0

Terminal Cost

( ) TNN NQ

Running Cost

, , T T Tk k k k k k k k kQL Su Ru u

k satu u 1

j

ki

P

Vehicle Modeling

[ , , , , ]

( ) [ ]f

t X Y

u t

State vector :

Input vector :

dsfsaasf

f ,

0 0 0 1 0. where

0 0 0 0 1

[ ]

t u t

C C

X Y

ξ  t

η ξ  t

η

1

.f ,

k k

s k k

t t

T t u t

ξ ξ2. Euler’s Discretization

1. Dynamic Model

Model Validation

20 /xv km h 40 /xv km h

Also known as steepest descent

Well known and simplest method

Finds a local minimum

Plant input is decision variable of algorithm

Gradient Descent Algorithm

Downhill direction

Initial guess

Step Size

.( )J

x

)(xf

( )f m

m

1 ,k k k ku u J x u

Model Predictive Control Setup

MPC Controller

Plant* ( )u t

( )t

( )tOptimizer

Cost Function+

Constraints

Mathematical Model

Simulation Environment

Simulation ResultsObstacle Avoidance 20km/h

Parameter

Value

Sampling Time

Steering Constraint

0.05sT

0Q 0.1,0;0,0.1

0.05

0 5 5

Q

RS

0.1,0;0,0.1

xv 20 /km h

20

Simulation ResultsObstacle Avoidance 60km/h

Parameter

Value

Sampling Time

Steering Constraint

0.05sT

0Q 0.1,0;0,0.1

0.05

0 5 5

Q

RS

0.1,0;0,0.1

xv 60 /km h

5

Simulation ResultsTurning Move With N=20,40,60,80,100

60 left turn at 50

90 right turn at 90

X m

X m

77% decrease in simulation cost

1797% increase in computation cost

Real-Time Analysis Cold start method Warm start

method

, 0,0,0,0...0init ku *, 1init k ku u

Parameter Cold Start

Warm Start

Comparison

Number of iterations for a 20s run 6539 4410 32.5% decrease

Avg. computation time per controller step

0.0393s

0.0275s

30% decrease

Number of 0.05s limit violations 70 38 45% decrease

Reference Trajectory Criteria

`

Start Point

Obstacle

1,k

Goal Point

1,,kd

3,,kd

2,,kd

3,k2,k

`

Start Point 1,k

Goal Point

Ykd ,1,,

Xkd ,1,,

2,k

Xkd ,2,,

Ykd ,2,,

Method 1 Method 2

No goal point information

Goal point information included

,Trajectory error variable k d k k

Reference Trajectory Criteria

Method 1 Method 2

Vehicle reaches goal point

successfully

Vehicle reaches goal point

successfully

Reference Trajectory Criteria

Method 1 Method 2

Vehicle retreat from a large obstacle

Vehicle reaches goal point

successfully

Conclusion Successfully controlled the vehicle dynamics for trajectory generation.

Used fully nonlinear CarSim vehicle model for simulations.

Various types of obstacles simulated.

NMPC controller was able to steer vehicle in an unknown environment with obstacles.

Tuning of weighing matrices time consuming process.

Value of horizon length N should be selected based on the available computation power and required tracking performance.

The controller is real-time implementable at shorter horizon lengths.

Steering constraints need to be tightened with an increase in the speed.

Method 2 for trajectory tracking is found to be superior to Method 1.

Overall, the controller works well in realistic simulations and can be used for practical implementation.

THANK YOU

QUESTIONS?