Differentiable collision detection for polytopes, capsules, cylinders, cones, spheres, and polygons.
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July 2022

A library for differential collision detection, as implemented from Differentiable Collision Detection for a Set of Convex Primitives. The core algorithmn, DCOL, computes collision information between the following convex primitives:

  • polytopes
  • capsules
  • cylinders
  • cones
  • spheres
  • ellipsoids
  • padded polygons

A limited version of DCOL is available in Python/JAX as dpax.


DCOL works by creating a struct for each shape, and calling a function to query a proximity value between them.


Each primitive is implemented as a struct in DCOL. The defining dimensions for each primitive is described in the paper, and the primitives can be constructed as the following:

import DifferentiableCollisions as dc

polytope = dc.Polytope(A, b)   # polytope is described by Ax <= b
capsule  = dc.Capsule(R, L)    # radius R, length L
cylinder = dc.Cylinder(R, L)   # radius R, length L
cone     = dc.Cone(H, β)       # height H, half angle β
sphere   = dc.Sphere(R)        # radius R
ellips   = dc.Ellipsoid(P)     # x'*P*x ≦ 1
polygon  = dc.Polygon(A, b, R) # polygon is described by Ay <= b, cushion radius R

where all of these structs are ::AbstractPrimitive, and use a quaternion for attitude. The position and attitude of a primitive P1::AbstractPrimitive are updated in the following way:

using StaticArrays
P1 =  dc.Polytope(A,  b)::AbstractPrimitive
P1.r = SA[1, 2, 3.0]     # position in world frame W
P1.q = SA[1.0, 0, 0, 0]  # quaternion ᵂqᴮ

MRP Support

In cases where a three-parameter attitude parameterization is more convenient, a Modified Rodrigues Parameter (MRP) can be used in the following way:

P1 = dc.PolytopeMRP(A, b)::AbstractPrimitiveMRP
P1.r = SA[1, 2, 3.0]    # position in world frame W
P1.p = SA[0.0, 0, 0]    # MRP ᵂpᴮ

Proximity Functions

DCOL exposes a function proximity for collision detection, as well as proximity_jacobian for collision detection and derivatives. Two optional arguments are included that pertain to the optimization solver under the hood, verbose turns on logging for this solver, and pdip_tol is the termination criteria.

# return min scaling α and intersection x
α, x = dc.proximity(P1, P2; verbose = false, pdip_tol = 1e-6)

# return min scaling α and gradient of α wrt configurations 
α, dα_dstate = dc.proximity_gradient(P1, P2; verbose = false, pdip_tol = 1e-6)

# return min scaling α, intersection x, and jacobian J (*)
α, x, J = dc.proximity_jacobian(P1, P2; verbose = false, pdip_tol = 1e-6)

These functions output $\alpha$ as the minimum scaling, with the following significance:

  • $\alpha \leq 1$ means there is a collision between the two primitives
  • $\alpha &gt;1$ means there is not a collision between the two primitives

Also, returned is x which is the intersection point between the scaled shapes (see algorithm for significance), and a Jacobian J which is the following:

$$ \begin{align*} J &= \frac{\partial (x,\alpha) }{\partial (r_1,q_1,r_2,q_2)} \end{align*} $$

In the case where AbstractPrimitiveMRP's are used, proximity_jacobian will automatically return the following Jacobian:

$$ \begin{align*} J &= \frac{\partial (x,\alpha) }{\partial (r_1,p_1,r_2,p_2)} \end{align*} $$


All of the primitives (both quaternion and MRP) can be visualized in MeshCat. Below is an example of visualization for a cone:

import DCOL as dc
import MeshCat as mc

vis = mc.Visualizer()

cone = dc.Cone(3.0, deg2rad(22))
cone.r = @SVector randn(3)
cone.q = normalize((@SVector randn(4)))

# build primitive scaled by α = 1.0
dc.build_primitive!(vis, cone, :cone; α = 1.0, color = mc.RGBA(1,0,0,1.0))

# update position and attitude
dc.update_pose!(vis[:cone], cone)


DCOL calculates the collision information between two primitives by solving for the minimum scaling applied to both primitives that result in an intersection. This is done by forming an optimization problem with the following primal variables:

  • $\alpha \in \mathbb{R}$, the scaling applied to each primitive
  • $x \in \mathbb{R}^3$, an intersection point in the world frame

The following optimization problem solves for the minimum scaling α such that a point x exists in the scaled versions of two primitives P1 and P2.

$$ \begin{align*} \underset{x,\alpha}{\text{minimize}} & \quad \alpha \\ \text{subject to} & \quad \alpha \geq 0, \\ & \quad x \in P_1(\alpha),\\ & \quad x \in P_2(\alpha) \end{align*} $$

This problem is a convex optimization problem with conic constraints, and is solved with a custom primal-dual interior-point method inspired by cvxopt. If the minimum scaling α > 1, then there is no collision because each primitive had to be scaled up in order to find an intersection. Alternatively, this means that if α ≤ 1, the two primitives are in contact. The solution to this optimization problem can be differentiated with respect to the position and orientation of each primitive using the implicit function theorem. By using a primal-dual interior-point method and returning a solution at a pdip_tol of [1e-4,1e-8], the log barrier will effectively smooth out the corners of the primitives to return useful and smooth derivatives.

Paper Examples

Examples from our paper are available here:

  • examples/trajectory_optimization/piano_mover.jl
  • examples/trajectory_optimization/cluttered_hallway_quadrotor.jl
  • examples/trajectory_optimization/cone_through_wall.jl
  • examples/contact_physics/ncp_contact_simulation.jl

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