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![Page 1: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/1.jpg)
Efficient Roadway Modeling and Behavior Control for Real-time
Simulation
Hongling WangDepartment Of Computer Science
University of Iowa
Oct. 28, 2004
![Page 2: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/2.jpg)
Overview
• Research introduction
• Motivation
• Model of roadways
• Behavior control on roadways
• Contributions
• Future work
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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
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Motivation
• Virtual environments– Laboratories for psychology– Understanding driver/rider behavior– Test future car concepts
• More applications
![Page 5: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/5.jpg)
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
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Ribbon
• Ribbon coordinate system– Distance, Offset, and loft (D,O,L)
• Provides a frame of reference for local spatial relationships
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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)))
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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' )()()(
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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
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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
![Page 11: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/11.jpg)
Experimental Results
(1) m=5 (2) m=10 Experimental curve(blue) and the derived curve (red) with
their knot points
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Experimental Results (cont.)
(1) m=5 (2) m=10
Match error of the derived curve
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Experimental Results (cont.)
(1) m=5 (2) m=10 Arc-length parameterization error of the derived curve
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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 *))(),(),((),(
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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
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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
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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
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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
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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
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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.
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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
![Page 22: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/22.jpg)
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.
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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.
![Page 24: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/24.jpg)
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.
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Results of Three Methods
Example curve and some spatial points
Statistics of three methods
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Intersections—Where Roads Join
• Shared regions of way• Non-oriented• Corridors splice together incoming and
outgoing lanes– Seen as single lane ribbons
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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
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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.
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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
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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
![Page 31: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/31.jpg)
Cruising Behavior
• Determine desired speed of an vehicle
• Proportional controller
)(* 11 td
tt VVKpa
![Page 32: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/32.jpg)
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
![Page 33: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/33.jpg)
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
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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
![Page 35: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/35.jpg)
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
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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)
![Page 37: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/37.jpg)
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
![Page 38: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/38.jpg)
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
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Behavior Combination
• Combine acceleration contributions from– Cruising behavior– Following behavior– Intersection behavior
• Combine steering angle contributions from– Tracking behavior– Lane changing behavior
![Page 40: Efficient Roadway Modeling and Behavior Control for Real-time Simulation Hongling Wang Department Of Computer Science University of Iowa Oct. 28, 2004.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bffb1a28abf838cc0ec7/html5/thumbnails/40.jpg)
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
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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
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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
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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