Efficient Roadway Modeling and Behavior Control for Real-time

Simulation

Hongling WangDepartment Of Computer Science

University of Iowa

Oct. 28, 2004

Overview

• Research introduction

• Motivation

• Model of roadways

• Behavior control on roadways

• Contributions

• Future work

Research Introduction

• Dynamic Virtual Environment– Vehicles, pedestrians, etc…– Lots of them!

• Roadway Modeling– Put some activities on roadways

• Behaviors– Control the activities happing on

roadways

Motivation

• Virtual environments– Laboratories for psychology– Understanding driver/rider behavior– Test future car concepts

• More applications

Roadway Modeling• Ribbon network

– Modeling roads, streets, sidewalks, and other navigable ways as ribbons

• Ribbon defines geometry and orientation of navigable surface– Centerline curve

– Ribbon twisting around centerline

– Boundaries on two sides

– Orientation

Ribbon

• Ribbon coordinate system– Distance, Offset, and loft (D,O,L)

• Provides a frame of reference for local spatial relationships

Ribbon Centerline

• Modeled by cubic spline Q(t)=(x(t),y(t),z(t))• Arc-length parameterization

– Compute arc length s as a function of parameter t

– Compute the inverse function

t=A-1(s)– Replace parameter t with

A-1(s)

P(s)=(x(A-1(s)),y(A-1(s)),z(A-1(s)))

Arc-length Parameterization

• Generally integral for A(t) does not integrate

s=A(t)=• Function t=A-1(s) is not elementary function• Numeric methods impractical for real-time

applications

Solution: Approximately arc-length

parameterized cubic spline curve

t

tdttztytx

0

2'2'2' )()()(

Approximately Arc-length Parameterized Cubic Spline Curve

Compute length of input curve Find m+1 equally spaced points on

input curve Interpolate the equally space points to

arc length s to derive a new cubic spline curve

Errors Analysis

• Match error – Misfit of the derived curve from an input curve– Measured by difference between the two

curves at corresponding points, |Q(t)-P(s)|

• Arc-length parameterization error– Deviation of the derived curve from arc-length

parameterization– Measured by formula

0.1)()()( 222 ds

dz

ds

dy

ds

dx

Experimental Results

(1) m=5 (2) m=10 Experimental curve(blue) and the derived curve (red) with

their knot points

Experimental Results (cont.)

(1) m=5 (2) m=10

Match error of the derived curve

Experimental Results (cont.)

(1) m=5 (2) m=10 Arc-length parameterization error of the derived curve

A parametric model for ribbons

• Through any point on a ribbon passes a line that lies on it and is perpendicular to the central axis– Intersection between the line and

the central axis (x(s),y(s),z(s))– Unit normal vector v on the line

pointing to left side– A parametric surface model

vwszsysxwsp *))(),(),((),(

Mapping between Ribbon and Cartesian coordinates

• Some computations are most naturally expressed in Cartesian coordinates (D,O,L)– Kinematics code computing object motion

• Other computations require object locations expressed in ribbon coordinates (X,Y,Z)– Behavior code tracking roads

• Efficient and robust code to map between ribbon and Cartesian coordinates

Mapping DOL to XYZ

• Compute p1 with distance coordinate Dp

• Compute p2 with p1 and offset coordinate Op

• Compute p with p2 and loft coordinate Lp

Conclusion: this mapping is very efficient

Mapping XYZ to DOL

• Locate the closest point p1 and get Dp

• Compute p2, the projection of p

• Offset Op is |p1-p2|• Loft Lp is |p-p2|

Problem: computation of the closest point

Closest Point Computation

• Modeled as an optimization problem of computing the minimum distance between a spatial point and a parametric spatial curve

– Quadratic minimization– Newton’s method– Combining quadratic minimization and Newton’s

method

20

20

20 ))(())(())(()( zszysyxsxsD

Method 1: Quadratic Minimization

Let s1, s2, and s3 be estimates of s*

1) Compute a quadratic polynomial p(s) that interpolates D(s) at s1, s2, and s3

2) Solve s4 that minimizes p(s)3) if

then s* s4

else { si s4 with i such that p(si) = ( p(sj) )

repeat }

max3,2,1j

|| ji ssmax4,3,2,1, ji

Observation of Quadratic Minimization

Rates of slow convergence and divergence make this method unacceptable by itself.

• Fails on seemingly simple cases.• In these cases the method usually makes

progress in the initial iterations and then stalls.

Method 2: Newton’s Method

Solve the rootfinding problem

Let s0 be initial estimate of s*

repeat

until

0)( sD

|| 1 ii ss

)(

)(1

i

iii sD

sDss

Observation of Newton’s Method

Infrequent divergence causes unacceptable failure rate.

• Unpredictably diverges for some points• With a good initial estimate converges in 1 or 2

iterations.

Method 3: Combining Quadratic Minimization and Newton’s Method

Exploits the complementary strengths of the

two optimization techniques

• Run the quadratic method for a small number of steps (typically about 4).

• Run Newton’s method initialized with the result from the quadratic method.

Observation of Composite Method

• Reliable and rapid convergence– Quadratic method provides a good estimate to

initialize Newton’s method– Newton’s method robustly converges (usually in 1

or 2 iterations.)

• The method has undergone rigorous testing in the Hank Simulator– We have had no failures.

Results of Three Methods

Example curve and some spatial points

Statistics of three methods

Intersections—Where Roads Join

• Shared regions of way• Non-oriented• Corridors splice together incoming and

outgoing lanes– Seen as single lane ribbons

Limitations of ribbons

• Transition between ribbons is hard– Different ribbons represent different local

coordinate systems– Hard to understand the spatial relationship of

positions on different ribbons

• Solution: a uniform ribbon called a path to unite connected, aligned ribbons– Lanes on roads and corridors on intersections

are seen ribbons

Path

• Single-lane ribbon overlaid on the road network– Easy transition between a

road and an intersection

• An interface between behaviors and the environment– The path relates behaviors to

environment

• Augmented dynamically– The vehicle is never behind

or ahead of its path.

A Path as a Basis for Building Behaviors

• A path is a frame of reference for tracking– Aim for a succession of pursuit points on the

path

• A frame of reference for local spatial relationships

Tracking Behavior

)0,,( OD• Ribbon coordinates

• Pursuit point

• Project pursuit point onto the vehicle’s local XY plane

• Compute a circular track

• Move the vehicle to a new position on the circular track

• Project the new position onto ribbon surface

)0,,( oOdD

Cruising Behavior

• Determine desired speed of an vehicle

• Proportional controller

)(* 11 td

tt VVKpa

Path Based Following Behavior

• Query the leader on path• Compute relative distance

and relative speed• Proportional-derivative

controller

• Discarded if positive otherwise applied

)()(1

rt

VKd*dt

Drt

DKp*t

a

Intersection Behavior

Gates access to a shared region of roads– An intersection is a resource

Decision of action selection– Going forward/stopping– Stop a vehicle on a desired position

Right-of-way rules and social conventions embedded in environment database

Regulate the motion of a vehicle before it enters an intersection

Intersection Behavior (Cont’)

Solve deadlock problem– Two vehicles yield right of

way to other two vehicles to block them at the same time

Solve starvation problem– A vehicle yielding right of

way gets stuck if vehicles having right of way come in a continuous stream

)2/(2 sva

Limitations of a Path

• An action-oriented geometric steering guide– A path between the

current and goal positions does not always exist

• Solution: a goal-oriented topological directional steering guide called a route

Route

• A succession of roads and intersections

• A global, strategic goal of an agent– The route is determined ahead of the path– The path is updated according to the

requirements of the route

• Support lane changing behaviors– Discretional lane change (DLC)– Mandatory lane change (MLC)

Route Based Lane Changing Decision Making

• The route forms constraints for choice of lane on a road

• Lane change decisions subject to the constraints– A DLC must consider route constraints– An MLC must enforce route constraints

Path Based Lane Changing Action

• A lane changing gap determined by the spatial relationship between the vehicle and nearby vehicles

• The path forms a frame of reference to deviate the pursuit point from the current lane to the target lane

Behavior Combination

• Combine acceleration contributions from– Cruising behavior– Following behavior– Intersection behavior

• Combine steering angle contributions from– Tracking behavior– Lane changing behavior

Solve Disturbances between Component Behaviors

• The switch in leaders when a vehicle leaves one lane and enters another– Abrupt acceleration change– Start two copies of following behavior

• Following behavior stops lane changing progress– Relaxing following distance

Solve Disturbances between Component Behaviors (Cont.)

• Following behavior unnecessarily slows down lane changing process– Disable following

behavior in the original lane when it has a clear trajectory to the target lane

– Visibility computation in DO plane

Contributions

• An accurate, efficient, robust roadway model– Ribbon network– Arc length parameterization– Efficient mapping between ribbon and Cartesian

coordinates

• A framework for modeling behaviors– Ribbon based tracking– Path based behaviors– Route as a strategic goal

Future Work

• Accuracy, efficiency, and robustness of geometric computations for off-road objects

• Efficient model for non-oriented navigable surfaces, i.e., intersections

• Good pursuit point control

• Behavior diversity

• Non autonomous behaviors