Efficient Roadway Modeling and Behavior Control for Real-time Simulation

1
Efficient Roadway Modeling and Behavior Control
for Real-time Simulation

• Hongling Wang
• Department Of Computer Science
• University of Iowa
• Oct. 28, 2004

2
Overview

• Research introduction
• Motivation
• Model of roadways
• Behavior control on roadways
• Contributions
• Future work

3
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

4
Motivation

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

5
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

6
Ribbon

• Ribbon coordinate system
• Distance, Offset, and loft (D,O,L)
• Provides a frame of reference for local spatial
  relationships

7
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
• tA-1(s)
• Replace parameter t with A-1(s)
• P(s)(x(A-1(s)),y(A-1(s)),z(A-1(s)))

8
Arc-length Parameterization

• Generally integral for A(t) does not integrate
• sA(t)
• Function tA-1(s) is not elementary function
• Numeric methods impractical for real-time
  applications
• Solution Approximately arc-length
• parameterized cubic spline curve

9
Approximately Arc-length Parameterized Cubic
Spline Curve

• Compute length of input curve
• Find m1 equally spaced points on input curve
• Interpolate the equally space points to arc
  length s to derive a new cubic spline curve

10
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

11
Experimental Results

• (1) m5
  (2) m10
• Experimental curve(blue) and the derived curve
  (red) with their knot points

12
Experimental Results (cont.)

• (1) m5
  (2) m10
• Match error of the derived curve

13
Experimental Results (cont.)

• (1) m5
  (2) m10
• Arc-length parameterization error of the
  derived curve

14
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

15
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

16
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

17
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

18
Closest Point Computation

• Modeled as an optimization problem of computing
  the minimum distance between a spatial point and
  a parametric spatial curve
• Quadratic minimization
• Newtons method
• Combining quadratic minimization and Newtons
  method

19
Method 1 Quadratic Minimization

• Let s1, s2, and s3 be estimates of s
• Compute a quadratic polynomial p(s) that
  interpolates D(s) at s1, s2, and s3
• Solve s4 that minimizes p(s)
• if
• then s s4
• else si s4 with i such that p(si)
  ( p(sj) )
• repeat

20
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.

21
Method 2 Newtons Method

• Solve the rootfinding problem
• Let s0 be initial estimate of s
• repeat
• until

22
Observation of Newtons Method

• Infrequent divergence causes unacceptable
  failure rate.
• Unpredictably diverges for some points
• With a good initial estimate converges in 1 or 2
  iterations.

23
Method 3 Combining Quadratic Minimization and
Newtons Method

• Exploits the complementary strengths of the
• two optimization techniques
• Run the quadratic method for a small number of
  steps (typically about 4).
• Run Newtons method initialized with the result
  from the quadratic method.

24
Observation of Composite Method

• Reliable and rapid convergence
• Quadratic method provides a good estimate to
  initialize Newtons method
• Newtons method robustly converges (usually in 1
  or 2 iterations.)
• The method has undergone rigorous testing in the
  Hank Simulator
• We have had no failures.

25
Results of Three Methods
Statistics of three methods

• Example curve and some spatial points

26
IntersectionsWhere Roads Join

• Shared regions of way
• Non-oriented
• Corridors splice together incoming and outgoing
  lanes
• Seen as single lane ribbons

27
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

28
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.

29
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

30
Tracking Behavior

• Ribbon coordinates
• Pursuit point
• Project pursuit point onto the vehicles 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

31
Cruising Behavior

• Determine desired speed of an vehicle
• Proportional controller

32
Path Based Following Behavior

• Query the leader on path
• Compute relative distance and relative speed
• Proportional-derivative controller
• Discarded if positive otherwise applied

33
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

34
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

35
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

36
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)

37
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

38
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

39
Behavior Combination

• Combine acceleration contributions from
• Cruising behavior
• Following behavior
• Intersection behavior
• Combine steering angle contributions from
• Tracking behavior
• Lane changing behavior

40
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

41
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

42
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

43
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