CPSC 8050 Advanced Computer Graphics - Google Search

CPSC 8050 Motion Planning

04 – Velocity Obstacles

January 25, 2017

Today

• Local Collision Avoidance (Part III)

• Continue from where we left off

• Resources (papers, code, etc) will be online

• Assign first homework (maybe?)

Velocity-Based Approaches

• Can be more intuitive than force-based approaches

• Formulate collision avoidance as an optimization problem

• Sampling-based

• Linear programming

A Sampling-Based Approach

• Consider an agent A having a goal velocity vg that has to navigate amidst other agents without collisions

• At each time step, let AV denote the set of admissible velocities for the agent. For example, if the agent is subject to a maximum speedυmax and a maximum acceleration αmax

• We can uniformly sample Ν candidate velocities from AV and select an optimal one according to a specific cost function


• Typically, we would like to choose a new velocity vn that minimizes the deviation from the goal velocity and the risk of collision with any of the other agents

• Here, tc denotes the minimum time that will take for the agent to collide with any of the other agents assuming that it selects a velocity vi

cand (perform time-to-collision computations for each neighbor). The constant γ controls the relative importance of the two cost terms


• Runtime complexity assuming n agents

• Naïve implementation scheme: Each agents checks interactions with the remaining n-1 agents O(n2)

• Using a sensing radius: Assuming that agents do not collide, there is an upper bound (k) on the # neighbors that each agent interacts with O(kn)

• Assuming k is constant, the runtime per time step is (asymptotically) equivalent to O(n)


• Performance is significantly affected by the computation of nearest neighbors

• Iterating over all other agents results in quadratic runtime

• Linear-time implementation for proximity queries can be employed based on spatial hashing schemes or grid maps. See, e.g., the work in http://www.red3d.com/cwr/papers/2000/pip.pdf


• Disadvantages

• Can be slow depending on the number Ν of candidate velocities that we evaluate

• Cannot guarantee collision-free motion due to sampling

Velocity Obstacles

• Introduced by [Fiorini and Shiller ‘98]

• Definition

Given two agents, A and B, the velocity obstacle VOA|B

(read: the velocity obstacle of A induced by B) is the set of all relative velocities that will lead to a collision between the two agents at some moment in time τ>=0

Velocity Obstacles

• From Cartesian to velocity plane

xB

xA

rB

rA

Velocity Obstacles


xA

rB

rA

xB-xA

xB

Velocity Obstacles


xB-xA

xA

rB

rA

xB

xB-xA

Velocity Obstacles

• What is the VO?

xB-xAxB-xA

Velocity Obstacles• Geometric Definition

A cone with its apex at the origin and its legs tangent to the disc of radius rA + rB centered at xB - xA

xB-xA

Velocity Obstacles

xB-xA

Velocity Obstacles

• Revising the VO Definition

Given two agents, A and B, the velocity obstacle VOA|B is the set of all relative velocities that will lead to a collision between the two agents at some moment in time τ>=0. Assuming A and B are approximated as discs of radius rA

and rB, respectively, VOA|B

where D(x,r) denote a disc centered at x having radius r

Velocity Obstacles

• This is basically a ray-disc intersection problem

𝜏𝒗 − (𝒙𝐵 − 𝒙𝐴) < 𝑟

Velocity Obstacles

• This is basically a ray-disc intersection problem

• which is ….

𝜏𝒗 − (𝒙𝐵 − 𝒙𝐴) < 𝑟

Last lecture: Time-to-Collision (τ)

• Formally, a collision exists if

𝒙 + 𝒗𝜏 = 𝑟

• Squaring and expanding, leads to the following quadratic equation for τ

𝒗 ⋅ 𝒗 𝜏2 + 2 𝒙 ⋅ 𝒗 𝜏 + 𝒙 ⋅ 𝒙 − 𝑟2 = 0

Time Horizon & Velocity Obstacles

Given two agents, A and B, the velocity obstacle VOA|B

(read: the velocity obstacle of A induced by B for time horizon τΗ) is the set of all relative velocities that will lead to a collision between the two agents at some moment within the next τΗ sec. Assuming A and B are approximated as discs of radius rA and rB, respectively

τΗ

τΗ

τΗ

Time Horizon & Velocity Obstacles• Geometric interpretation

A truncated cone with its apex at the origin and its legs tangent to the disc of radius rA + rB centered at xB – xA. The cone is truncated by an arc of a disc of radius (rA + rB)/τH centered at (xB – xA)/τH.

xB-xA

Time Horizon & Velocity Obstacles

Collision-Free Velocities

• Let CFVA|B denote the set of collision-free velocities for agent A induced by agent B.

xB

xA

rB

rA


xB

xA

rB

rA

• Let CFVA|B denote the set of collision-free velocities for agent A induced by agent B

• What is CFVA|B if B stands still?


xB

xA

rB

rA

• What is CFVA|B if B is moving at velocity vB

vB


xB

xA

rB

rA


vB

vB


xB

xA

rB

rA


vB


The set of collision-free velocities CFVA|B for agent A given an agent B is the set of all velocities that lie outside the VOA|B translated by B’s velocity vB. Formally

Where denotes the Minkowski sum. Given two sets X and Y:


• What if agent A has to avoid multiple agents?


• Agent A may also need to account for its set of admissible velocities AV if it is subject to kinodynamics constraints. Then


• Let’s assume that A is subject to maximum speed υmax

υmax


• Let’s assume that A is subject to maximum speed υmax

Selecting a Collision-Free Velocity

• Recall: For a given agent A, we need to compute a collision-free velocity that is as close as possible to its goal velocity

• Formulate this as an optimization problem

Sampling-Based Approach

• Sample admissible velocities and solve

• Already discussed issues that arise with this approach

Optimal Reciprocal Collision Avoidance

• Popular framework to reason about velocity-based collision avoidance

• Proposed by [van den Berg et al., 2011]

• Builds on the concept of Velocity Obstacles

• Widely known as ORCA

ORCA

• Anytime the relative velocity between agent A and agent B falls into VOA|B, we know that A heads into a collision with B.

ORCA

• Anytime the relative velocity between agent A and agent B falls into VOA|B, we know that A heads into a collision with B.

vA - vB

ORCA

• Let u denote the smallest change in relative velocity required to resolve the collision

• We can easily find u by simply projecting the relative velocity to the closest point on the boundary of the VO

vA - vBvA - vB

ORCA

• If A were to take vA + u for the next time step and nothing else changes, the collision will be resolved. But, by symmetry, B faces the exact same condition and will try to take vB - u• Waste a lot of “effort”

• Oscillations

vA - vBvA - vB

Reciprocity in ORCA

• Agents will split the effort to avert the collision. So, agent A can move by at least ½ ||u|| in the direction of u, that is velocities that get at least halfway out of VOA|B

vA - vB

ORCA Constraint

• The set ORCAA|B of permitted velocities for A for optimal reciprocal collision avoidance with B is given by

vA - vB

ORCA Constraint


vA - vB

vA - vB

ORCA Constraint


vA - vB

vA - vBvA - vB

ORCA Constraint

• In other words, ORCAA|B is a half-plane delimited by the line perpendicular to u that passes through the point vA + ½ u

• That is, agent B imposes a linear constraint on the velocities that A can select

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