Basic Methods

This section explains some essential theorems and approaches that have had a significant impact in the field of computational geometry and collision detection, and which have a direct application in collision computation; Figure 2.6 illustrates them. Note that all these and similar basic methods are commonly broken down to algebraic operations between basic elements in Euclidean geometry, such as points, lines, planes, or surfaces expressed in closed form. These operations are considered prior knowledge and are not reviewed in this chapter; in this respect, the interested reader might consult [AM01],

[Moe97], and [Ebe99].

Separating Axis Theorem (SAT)

The Separating Axis Theorem (SAT) states that two convex objects overlap with each other if and only if there exits no axis on which their projections are disjoint; or, in other words: two convex objects do not overlap in case there exists an axis on which their projections are disjoint (see Figure 2.6(a)). An axis with disjoint projections accounts for the existence of a separating plane that fences off the two convex objects in two separate half-spaces. The application of the theorem consists in finding a separating axis among relevant candidates; in practice, for 3D convex polyhedra it suffices to evaluate the projections on the directions of face normals and their cross products.

tween simple BVs, such as OBBs [GLM96] or AABBs [vdB97]. The amount of candidate axes to be checked increases quadratically with the number of features, thus, it has been proposed to reduce the pool of candidate axes to those which lead to highest probabil- ities of correct test results [vdB97]. The notion of the separating plane inherent to the SAT has also been applied to more complex convex geometries using machine learning techniques [Zac00]. The idea consists in incrementally improving a plane which would separate the vertex set of one convex object from the other’s with a perceptron; the method stops if the plane is found or after a maximum number of iterations. Although not as efficient as the previous ones, linear programming or optimization approaches can be used to obtain the coefficients of the separating plane [LM04].

Minkowski Sums, Minkowski Subtractions

The Minkowski subtraction of two convex objects A and B is the Minkowski sum of one object (A) and the mirrored of the other (−B), defined as

A ⊕ −B = {a − b : a ∈ A, b ∈ B}, (2.5) where a, b are primitive features (e. g., vertices) of objects A, B, respectively (see Fig- ure2.6(b)). This structure is called Configuration Space Obstacle (CSO) in the literature and has two important properties:

(i) convex objects A and B overlap if and only if the origin point O of the space in which they are defined is inside their Minkowski subtraction or CSO,

(ii) the distance from the origin O to the surface of the CSO is either the distance or the penetration between A and B, depending on whether O is outside or inside the CSO.

Hence, the problem of collision and proximity computation between two convex objects (A and B) is reduced to checking a point against the convex hull that encloses the CSO. The popular Gilbert-Johnson-Keehrti or GJK algorithm [GJK88] exploits these properties to determine iteratively the distance between convex polyhedra. The approach is similar to the well-known Dantzig’s simplex algorithm [Dan63] employed in linear programming. First, a small subset of witness or support points located on the surface of the convex set is defined. The key idea is to iteratively update this subset walking on the surface until the solution point is found. The walking direction is given by the point in the subset which minimizes the distance to the origin.

Several enhancements have been applied to the original GJK algorithm, such as, using hill-climbing by starting with the vertex that yielded the solution in the previous

cycle [Cam97], support of several geometric shapes in addition to polyhedra (i. e., boxes, cones, spheres, cylinders) [vdB99], and computing penetration depth [vdB01].

Feature Tracking Based on Exterior Voronoi Marching

Given a convex hull A, the exterior Voronoi region of any of its features fi (i. e., vertex,

edge, face) located on its surface ∂A is the set of exterior points that are closest exclusively to that feature; thus, the exterior Voronoi region of fiin 3D space can be formally defined

as

V (fi) = {x ∈ R3, x /∈ A : d(x, fi) < d(x, fj); ∀i 6= j, ∀fi, fj ∈ ∂A}, (2.6)

being d(·, ·) the selected distance function (usually the two-norm for the Euclidean dis- tance); Figure 2.6(c) illustrates the case for a triangle.

The pair of closest features of convex polyhedra has an interesting property: each closest feature must be fully or partially contained by the exterior Voronoi region of its closest pair. The popular Lin-Canny (LC) algorithm [LC91] exploits that property for detecting and tracking closest features. In a preprocessing step, a mesh of features is constructed so that each feature is aware of its direct neighbor features. Additionally, exterior Voronoi cells are computed for each feature by setting orthogonal planes to the face features. In runtime, closest pairs of features are iteratively determined by marching on the exterior Voronoi regions. If the candidate pair determined in the previous iteration satisfies the aforementioned property for closest features, the candidate features are the solution and the seed or initial guess for the next cycle. If not, a new candidate pair is defined by selecting neighbor features; this selection happens following the direction of the Voronoi cell boundaries (planes) that are violated by any of the candidate features. Since object configurations change minimally from frame to frame, a greedy walk in the neighborhood of the current approximation leads to a local solution, which turns out to be global due to the convexity of the polyhedra. Hence, the method is able to achieve fast and near to constant computation times after initialization.

The approach works with non-convex geometries after applying convex decomposi- tion (see, for instance [GME+_{00]), but with the cost of quadratic complexity on the} number of convex parts. Similarly, multi-resolution hierarchies of convex hulls have been successfully used with the method [EL00]. A significant improvement of the method was presented with the V-Clip (Voronoi Clip) algorithm [Mir98]. The key idea on which the method builds up consists in clipping parametrized candidate closest edges with planes of the Voronoi regions. Depending on the resulting intersection points, further decisions on marching directions can be taken, converging ultimately to the solution. This im- provement avoids iteratively computing distances between features or performing unsta-

ble divisions, which boosts computational speed and robustness. Moreover, penetration computation is also supported. Altogether, the LC algorithm and its derivates have been and are still widely employed, as recorded in Table2.1 and Table2.2.

Other Basic Methods

A large body of works uses implicit representations for collision and proximity queries, and to lesser extent, parametric geometry descriptions. As introduced in Section 2.3.1, implicit fields map space points x to scalar values through a global or local field function f (x) = 0– the most common scalar distance values are considered in this section. Given f (x), it is straightforward evaluating the collision state and distance/penetration of a point. Instead, the most complex tasks consist in (i) selecting the relevant features that are checked in the field function, and (ii) building or locally computing the field function itself.

For the first task, the aforementioned methods can be used, often in combination with Bounding Volume Hierarchies (BVH) explained in Section2.3.2.5. For the second, several methods have been presented:

• If f(x) is a global closed-form function, no additional computations must be carried out to use the field function [ST97].

• When regular grids (e. g., voxelmaps) are used to discretely store distance fields, linear [SSeS14a] or trilinear interpolation [BJ08] can be used to locally approxi- mate the distance field function f(x). That requires fast and robust access to the neighbor cell values. Many works based on the principles of the widely extended Voxelmap-Pointshell (VPS) algorithm [MPT99], [RPP+_{01], [MPT06], follow these} methods, such as the approach presented in Chapter3. In these approaches, points sampling the surface of an object are tested against the discrete implicit distance field of the other object.

• Depth maps computed from different perspectives (e. g., three orthogonal direc- tions) also implicitly encode a discrete field that can be locally processed to render iso-surfaces [KLR15]; similarly, support planes or height maps set on the faces or vertices of the convex hull of the object have also been proposed [Mou15]: each discretely sampled point of a support plane maps to a distance value related to the proximity of object’s surface, and, in runtime, point penetrations are obtained after evaluating them in support plane mappings.

• In the case of point clouds, it is possible to fit planes or surfaces in regions close to the interaction tool, and to build distance functions to them [KZ04]; similarly, a

weighted average of these points can be computed to obtain the local approximation to the scalar field function and its gradient [LCS12]. In order to narrow down the subset of points close to the tool, several techniques have been proposed, such as sphere trees [KZ04], k-d trees [LCS12], or, in the case of streamed point clouds, simply accessing the points inside the projection of the tool’s bounding box on the depth map [RC13b].

Note that an important property of the field function is that its gradient ∇f(x) accounts for the normals of its iso-surfaces (i. e., geometric loci of same field value); thus, the closest path to collision should be aligned with ∇f(x).

As far as parametric representations are considered, it is common exploiting the con- trol mesh as a first step for collision queries, either by projecting interaction points on

it [TJC97], or by using BVs that enclose mesh parts [JC98]. When closest or collid-

ing mesh points are found, these can be easily mapped onto underlying smooth sur- faces [TJC97], and, afterwards, tracked in function of the object motion [NJC99]. To that purpose, first, the movement of the endeffector point in the Euclidean space can be projected onto the tangent plane of the parametric surface; this projection can be then used to compute a first order approximation of the parametric velocity of the previous closest point, which leads to the current closest point on the surface.