Closed-Form Minkowski Sums of Convex Bodies with Smooth Positively Curved Boundaries
CClosed-Form Minkowski Sums of Convex Bodies withSmooth Positively Curved Boundaries
Sipu Ruan a , Gregory S. Chirikjian a,b a Department of Mechanical Engineering, National University of Singapore,Singapore and the Laboratory for Computational Sensing and Robotics, JohnsHopkins University, Baltimore, USA b Address all correspondence to this author.
Abstract
This paper proposes a closed-form parametric formula of the Minkowski sumboundary for broad classes of convex bodies in d -dimensional Euclidean space.With positive sectional curvatures at every point, the boundary that encloseseach body can be characterized by the surface gradient. The first theorem di-rectly parameterizes the Minkowski sums using the unit normal vector at eachbody surface. Although simple to express mathematically, such a parameteri-zation is not always practical to obtain computationally. Therefore, the secondtheorem derives a more useful parametric closed-form expression using the gra-dient that is not normalized. In the special case of two ellipsoids, the proposedexpressions are identical to those derived previously using geometric interpre-tations. In order to further examine the results, numerical verifications andcomparisons of the Minkowski sums between two superquadric bodies are con-ducted. The application for the generation of configuration space obstacles inmotion planning problems is introduced and demonstrated. Keywords:
Minkowski sums, computer-aided design, computational geometry
1. Introduction
Minkowski sums between two solid bodies have been studied for decades, andhave wide applications in computer-aided design and manufacturing [1], com-putational geometry [2], robot motion planning [3], etc. This paper computesthe boundary of a d -dimensional Minkowski sum, which is closely related to the( d − contact space in the context of robotics[4]. As a particular example in robot motion planning, the configuration space(C-space) of a robot is constructed by computing the Minkowski sums between Email addresses: [email protected] (Sipu Ruan), [email protected] [email protected] (Gregory S. Chirikjian)
Preprint submitted to Elsevier January 1, 2021 a r X i v : . [ m a t h . M G ] D ec he robot parts at all possible orientations and the obstacles in the environ-ment [5]. In this case, the robot is shrunk into a point and the boundaries ofobstacles are inflated, resulting in configuration-space obstacles (C-obstacles) .Then a motion plan can be made for this point to traverse through the C-space.In addition, another class of applications includes the detection of contacts be-tween two rigid bodies [6], as well as their separation distance or penetrationdepth [7]. The contact status between two bodies can be determined by query-ing the relative location between the Minkowski sums boundary and the pointthat the other body is shrunk into [8]. This method gives equivalent resultswith the one that directly and simultaneously finding a pair of points on theboundary surfaces with anti-parallel and opposite normal vectors [9]. Using anexact Minkowski sum expression is advantageous due to the reduced numbersof variables since only the point on the Minkowski sums boundary is searched.In general, the computations of exact Minkowski sums can be very expensive[10]. A large amount of the investigations focus on bodies that are encapsulatedby polytopes, with discrete and faceted surfaces [2, 11]. They are simple to berepresented and stored in a modern computer, and can characterize a large rangeof convex or non-convex bodies through finite element methods. However, thistype of surface is generally not smooth and requires a large set of parameterslike vertices and faces information. In this paper, alternatively, we focus on aclass of convex bodies whose boundary surfaces have implicit and parametricexpressions. This class of surfaces is smooth and captures a wide variety ofshapes with just a small number of parameters. And it is shown that for thiswide class of convex bodies, it is possible to parameterize the boundary oftheir Minkowski sums in closed form. Note that the theoretical derivationsthroughout this paper are valid for any dimension, but for real applicationscenarios, we only discuss the 2D and 3D cases.Previous work has shown that such a closed-form Minkowski sum boundarycan be obtained between two ellipsoids [12], as well as when one of the two bod-ies is an ellipsoid and the other is enclosed by a general convex and differentiablesurface [13, 8]. The methods used there were based on geometric interpretationsby transforming the ellipsoid into a sphere and calculating an offset surface. Thisprocess limits at least one body to be an ellipsoid. In a broader class of bod-ies, this paper derives the closed-form boundary of Minkowski sums betweentwo convex bodies with smooth boundaries having positive Gaussian curvature.The surfaces are parameterized by outward normal vectors [14, 9], which canbe either normalized or un-normalized. Based on the two different parameteri-zations of the surfaces, two expressions for the closed-form Minkowski sums arederived, which are applicable to the bodies under general linear transformationssuch as rotation and shear. In the special case of ellipsoids, the proposed expres-sions of Minkowski sums boundary are identical with those from the previousgeometric-based derivations.The rest of this paper is organized as follows. Section 2 reviews related liter-ature on the computational techniques for Minkowski sums. Section 3 reviewssome useful properties of surfaces that are related to the derivations in thispaper. Section 4 introduces the main results to compute the exact closed-form2inkowski sums between two general bodies. Section 5 and 6 demonstrate theproposed expressions to the cases of ellipsoids and superquadrics. Section 7numerically verifies the proposed method and compares the performance withthe original definitions of Minkowski sums. Section 8 introduces an applicationof the proposed method in efficient generations of configuration-space obstaclesin motion planning problems. Finally, section 9 concludes the paper.
2. Related Work
This section reviews related work on the computations of Minkowski sums.In general, if two bodies in R are non-convex, the complexity of computing theirMinkowski sums can be as high as O ( m n ), where m and n are the features (i.e.the number of facets) of the two polytopes. And the exact complexity boundsof Minkowski sums are rigorously analyzed in [10]. To make the calculationstractable, a large amount of efficient algorithms have been proposed, whichcan be generally grouped as decomposition-based, point-based and convolution-based methods.The decomposition-based methods (either exact or approximated) segmentsthe general polytopes into convex components [2, 15], since the complexity canbe reduced to O ( mn ) in the case of convex polytopes. The core idea behindis that the union of Minkowski sums is the Minkowski sums of the union ofbodies. For this type of methods, the efficiency of convex decomposition affectsthe overall performance of the algorithms. In addition, point-based methodsavoid using the expensive convex decomposition and computing the union ofMinkowski sums [16, 17]. The major advantages are the ease of generating pointsthan meshes, and the possibility of parallelisms [18]. But the local propertiescannot be expressed by individual points themselves. Another type of methodsis based on convolutions of two bodies, with the fact that Minkowski sum of twosolid bodies is the support of the convolution of their indicator functions [19, 20,11, 21]. Convolution-based methods were also widely applied into Minkowskisum computations between surfaces or curves with algebraic expressions [22,23, 24, 25, 26]. Particularly, these methods constructs an offset surface or curvebased on the shapes of the two bodies, and trims the portions that fall into theinterior of the outer boundary.For the case of bounding surfaces of two convex bodies, such a trimming canbe omitted since their Minkowski sum is also convex. By applying this property,closed-form expressions of Minkowski sums have been proposed, with the caseswhen at least one body is ellipsoid [12, 13, 8]. These work calculated an offsetsurface in the affine space when one ellipsoid is transformed into a sphere. Theexpressions are exact and in closed-form. However, they require at least onebody to be an ellipsoid, which limits the ability to extend for more complexshapes.The goal of this paper is also to develop exact and closed-form expressionsof Minkowski sums. But the two bodies are allowed to be within a more generalclass of convex bodies with positive Gaussian curvatures. The expressions are3arameterized in a totally different way as compared to the previous geometric-based method. The results are the same in the special case of ellipsoids.
3. Minkowski Sums and Geometric Properties of Surfaces
This section reviews several definitions and properties of Minkowski sumsand surfaces that are necessary for the derivations throughout this paper.
Given two solid bodies B and B in d -dimensional Euclidean space, theirMinkowski sum is defined as B ⊕ B . = { x + y | x ∈ B , y ∈ B } . (1)In this paper, we specifically compute the boundary of Minkowski sums between B and − B , i.e. ∂ [ B ⊕ ( − B )], where B ⊕ ( − B ) . = { x − y | x ∈ B , y ∈ B } . (2)Geometrically, when placing B at this boundary surface, B and B touch eachother (without penetrating) outside. In the context of robotics, this boundary isalso called the contact space , which divide the space of collision and separation.Note that − B is the reflection of the original body B with respect to itscenter. And if B is central symmetric to itself (like an ellipsoid), Eqs. (1) and(2) are equivalent, i.e. B ⊕ B = B ⊕ ( − B ). Once this boundary is known, B ⊕ ( − B ) can also be parameterized trivially.If a set of points is sampled on the boundary of each surface, the Minkowskisums between these two bodies can be approximated as a discrete surface. Sinceboth boundary surfaces are assumed to be convex, a direct way to compute theapproximated Minkowski sums is to add all the points on different surfaces andtake the convex hull of the result [27]. This refers to a direct method using the definition (Eq. (2)) of the Minkowski sums, which is applied to be comparedwith our closed-form method to be proposed.Figure 1 shows the process of computing Minkowski sums using the originaldefinition as in Eq. (1) as well as our closed-form solution. The demonstrationsin the 2D case are shown in the figure, and the boundaries of both bodies aswell as their Minkowski sums are 1D curves. For both methods, the points onthe bounding surfaces are sampled based on a set of angles θ k ∈ [ − π, π ] thatparameterize an 1D unit circle. Suppose that the boundary of convex closed body B i is defined by the para-metric equation x = f ( u ) ∈ R d (3)where the unit vector u ∈ S d − is in turn parameterized by d − B i itself can be4
21 1 (cid:1769) ( (cid:3398) ) [ (cid:1769) ( (cid:3398) )] Convex hull (a) Computational process via the original Minkowski sums definition. (cid:3398)(cid:1828) (cid:2870) (cid:1828) (cid:2869) (cid:2034) [ (cid:1828) (cid:2869) (cid:1769) ( (cid:3398)(cid:1828) (cid:2870) )] (cid:2206) (cid:2869) ( (cid:2016) (cid:3038) ) (cid:2206) (cid:2869)(cid:2878)(cid:2870) ( (cid:2016) (cid:3038) ) (b) Computational process of the closed-form solution.Figure 1: Demonstration of the process to compute Minkowski sums using the original defini-tion and our closed-form solutions. In (a), the surfaces enclosing both bodies are sampled intopoint sets; then all the pairs of points in different surfaces are added to obtain the Minkowskisums, i.e. B ⊕ ( − B ); the boundary of Minkowski sums, i.e. ∂ [ B ⊕ ( − B )], is then computedvia convex hull operation. In (b), only one of the body (i.e. B ) surfaces is sampled, andthe parameters (i.e. θ k ) of the points on Minkowski sums boundary is the same with thoseon the sampled body; in other words, x ( θ k ) is computed based on x ( θ k ) in closed-form.Here, θ k ∈ [ − π, π ] is the sampled angle that parameterizes the boundary of a unit circle, i.e.[cos θ k , sin θ k ] T . x = r f ( u ) where r ∈ [0 , ∈ B i . Here f : S d − → R d can be thought of as an embedding of the spherewhich is deformed while ensuring the convexity of B i . Consequently, the linesegment connecting any two points in ∂B i is fully contained in B i .Assume that the corresponding implicit equation of B i exists and can bewritten as Ψ( x ) = Ψ( f ( u )) = 1 . (4)Thinking of u = u ( φ , ..., φ d − ), the tangent vectors can be computed fromEq. (3) as t i = ∂ x ∂φ i , which, in general, is not orthonormal. Moreover, the outward pointing unitnormal to the surface at the same point x can be calculated as n ( u ) . = ( ∇ x Ψ)( x ) (cid:107) ( ∇ x Ψ)( x ) (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) x = f ( u ) . (5)The Gauss map assigns to each point f ( u ) ∈ ∂B its normal n ( u ) ∈ S d − . Andthe function n : S d − → S d − is closely related. In this paper we will makeextensive use of Eq. (5), as well as the un-normalized gradient m ( u ) . = ( ∇ x Ψ)( x ) | x = f ( u ) . (6)The Gaussian curvature is the Jacobian determinant of the Gauss map.Consequently, if ∂B has positive Gaussian curvature everywhere, then the rela-tionship n = n ( u ) will be invertible as u = n − ( u ) . = u ( n ). Then it becomespossible, at least in principle, to re-parameterize positions on the surface usingthe outward normal as x = ˜ f ( n ) , (7)where ˜ f ( n ) = f ( u ( n )). As with u , it is possible to use angles to parameterize n . When such a parameterization is used,( ∇ x Ψ)( x ) (cid:107) ( ∇ x Ψ)( x ) (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) x =˜ f ( n ) = n . (8)
4. Main Results
In this section, the main results of the paper are presented. We first proposetwo possible expressions for closed-form Minkowski sums, along with the proofs.Both of these theorems assume two bodies are in their canonical forms. There-fore, in the following result, we show the expression when linear transformationsare applied to the two bodies. The expressions are demonstrated in the 2D casein Figs. 2 and 3. 6 .1. Closed-form Minkowski sums parameterized by outward unit normal
Theorem 1 : If the boundaries ∂B and ∂B of convex bodies B and B eachhave closed form parametric expressions of the form in Eq. (3) and their Gaussmaps can be obtained in closed form resulting in parametric equations x i =˜ f i ( n i ), then their Minkowski sum boundary ∂ [ B ⊕ ( − B )] can be parameterizedas x ( n ) = ˜ f ( n ) − ˜ f ( − n ) , (9)where n ∈ S d − can be parameterized using spherical coordinates which inturn parameterize x . Proof : Hold B fixed, and allow B to move such that it kisses B at anypoint on its surface. Record the surface that the origin of the reference frameattached to B traces out as B undergoes all translational motions for which ∂B ∩ ∂B consists of a single point. When this occurs, the normals of bothsurfaces must pass through the point of contact, lie along a common line, andhave opposite sense. The common point of contact between the two bodieswill appear as ˜ f ( n ) in the coordinate system of B (i.e., the world frame),and as ˜ f ( − n ) is the moving coordinate system attached to B . The latter isdescribed in the coordinate system in the world frame as the relative positionvector − ˜ f ( − n ). Adding both gives the result. (cid:3) Figure 2 demonstrates the relationships of the outward normal vectors at eachbody boundary (i.e. n ) as well as different terms in the closed-form Minkowskisums expression as in Eq. (9). Though this can be done in principle, in practiceit is often difficult to obtain in closed form expression for ˜ f i ( n i ).In such cases, it can be somewhat easier to parameterize x in terms of itsun-normalized gradient as x = ˜˜ f ( m ) , (10)where m is obtained from Eq. (6). Then it is possible to equate f ( u ) = ˜ f ( n ) = ˜˜ f ( m )because Eqs. (3), (7) and (10) are different parameterizations of the sameboundary, i.e. ∂B . Once ˜˜ f ( m ) is obtained analytically, an explicit surfaceparameterization can be constructed by setting m ( φ ) = ˜˜ f − ( f ( u ( φ ))) . (11)In the case of Eq. (7), there is greater choice: Either the analogous computationcan be done as n ( φ ) = ˜ f − ( f ( u ( φ ))) , (12)or we can simply choose to parameterize n in the same way as u ( φ ). That is,even though n is not u , in the final step u is no longer part of the descriptionand we can choose to parameterize the surface by letting n = u ( φ ). Hence these7 igure 2: Demonstration of Theorem 1. The two bodies touch each other at only a point,which is parameterized by the unit outward normal vectors, i.e. ˜ f ( n ) and ˜ f ( − n ). Thenormal vectors at the contact point are anti-parallel to each other. The green and blue patchedbodies are B and B respectively. c is the center of B , and B is placed at one of theMinkowski sums boundary point x . The red curve is a section of the computed closed-formMinkowski sums boundary. m ( φ ) because it is not a unit vector, and Eq. (11)) is used. Notethat by definition ( ∇ x Ψ)( x ) | x =˜˜ f ( m ) = m . (13)And this leads to our second main result. Theorem 2 : If the boundaries ∂B and ∂B of convex bodies B and B each have closed form parametric expressions of the form in Eq. (10) and ifit is possible to write (cid:107) m (cid:107) = Φ( m ), where Φ( · ) has closed form, then theMinkowski sum boundary ∂ [ B ⊕ ( − B )] can be parameterized in closed formas x ( m ) = ˜˜ f ( m ) − ˜˜ f (cid:18) − Φ( m ) (cid:107) m (cid:107) m (cid:19) , (14)where m ∈ R d can be parameterized using the spherical coordinates of theoriginal parametric expression for B . Proof : Using the same reasoning as in Theorem 1, for the bodies to kiss ata point their gradients must lie along the common normal line passing throughthe kissing point, and have opposite sense. Since m i = (cid:107) m i (cid:107) n i ( i = 1 , x ( m ) in the frame at-tached to B to kiss B at the point x ( m ) in the frame attached to B , canbe screened for by computing (cid:90) S d − δ ( n , n )˜˜ f ( −(cid:107) m (cid:107) n ) d n = ˜˜ f ( −(cid:107) m (cid:107) n ) . This can then be written as˜˜ f (cid:18) − (cid:107) m (cid:107)(cid:107) m (cid:107) m (cid:19) = ˜˜ f (cid:18) − Φ( m ) (cid:107) m (cid:107) m (cid:19) . Then, subtracting gives Eq. (14). (cid:3)
As a consequence of these theorems, it is easy to observe that when B = cB , x = (1 + c ) x . Figure 3 shows the relationships of each term in the expression derived in Eq.(14). Φ( m )In general, (cid:107) m (cid:107) = Φ( m ) can be obtained when a sufficient condition issatisfied. The relationships between m and m in closed-form can be computedas follows. 9 igure 3: Demonstration of Theorem 2. The body boundaries are parameterized by theunnormalized gradients ( m and m ), where m = − Φ( m ) (cid:107) m (cid:107) m . They are anti-parallel buthave different magnitudes. The green and blue patched bodies are B and B respectively. c is the center of B , and B is placed at one of the Minkowski sums boundary point x .The red curve is a section of the computed closed-form Minkowski sums boundary.
10t the contact point, the outward normal vectors of the two surfaces areanti-parallel, with the relationship being m (cid:107) m (cid:107) = − m (cid:107) m (cid:107) . (15)If m can be expressed in closed-form as a function of the unit vector u ,i.e. m = g ( u ), then Eq. (15) becomes g ( u ) (cid:107) m (cid:107) = − m (cid:107) m (cid:107) . (16)This means that g ( u ) is proportional to − m . If g ( k u ) = β ( k ) g ( u ), then u is proportion to g − ( − m ). Here, k is a constant scalar, and β ( k ) is anyscalar function of k , which is also a constant value. This condition is sufficientto derive the closed-form expression of (cid:107) m (cid:107) = Φ( m ).Since u is a unit vector, we get u = g − ( m ) (cid:107) g − ( m ) (cid:107) . (17)Then, (cid:107) m (cid:107) = (cid:107) g ( u ) (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13) g (cid:18) g − ( m ) (cid:107) g − ( m ) (cid:107) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . = Φ( m ) . (18) Under invertible linear transformations from R d defined by x (cid:48) = M x , tan-gents transform linearly as t i → M t i . Alternatively, gradients transform as( ∇ x (cid:48) Ψ)( M − x (cid:48) ) (cid:12)(cid:12) x (cid:48) = M x = M − T ( ∇ x Ψ)( x ) . (19)This can be written as m (cid:48) = M − T m , (20)where M − T . = ( M T ) − = ( M − ) T . This can be used together with ˜˜ f i ( m i ) to getthe Minkowski sum of linearly transformed parametric surfaces, by observingthat the result of transforming by M is˜˜ f (cid:48) i ( m (cid:48) i ) = M i ˜˜ f i ( m i ) = M i ˜˜ f i ( M Ti m (cid:48) i ) . Then substituting either of these into Eq. (14) and expanding gives x (cid:48) ( m (cid:48) ) = M ˜˜ f ( M T m (cid:48) ) − M ˜˜ f (cid:18) − Φ( M T m (cid:48) ) (cid:107) M T m (cid:48) (cid:107) M T m (cid:48) (cid:19) , (21)and x (cid:48) ( m ) = M ˜˜ f ( m ) − M ˜˜ f (cid:32) − Φ( M T M − T m ) (cid:107) M T M − T m (cid:107) M T M − T m (cid:33) . (22)11 . Demonstration with Ellipsoids This section demonstrates the proposed closed-form Minkowski sums be-tween two ellipsoids. The results are compared with the ones coming fromthe previous geometric interpretations, which are the same. But using thederivations from this paper, the expressions are symmetrical and reflects thecommutative property of Minkowski sums.The surface of an ellipsoid can be parameterized as f ( u ) = A u ∈ R d (23)where A = A T ∈ R d × d is a positive definite matrix, and u ∈ S d − , which canbe parameterized by spherical angles φ = ( φ , ..., φ d − ). The correspondingimplicit equation is Ψ( x ) = 1 where Ψ( x ) . = x T A − x . (24) Computing the normal as the normalized gradient, i.e. n = A − u (cid:107) A − u (cid:107) . This can be inverted to give u = A n (cid:107) A n (cid:107) . Then ˜ f ( n ) = A A n (cid:107) A n (cid:107) . Then, from Theorem 1, x ( n ) = A n (cid:107) A n (cid:107) + A n (cid:107) A n (cid:107) . (25)This is a variant of the formular given in [12] and the same as the formuladerived in [28]. The unit vector n ∈ S d − can be parameterized with sphericalangles φ = ( φ , ..., φ d − ) just like u ( φ ), though it should not be confused withit. An alternative would be to parameterize ellipsoids according to Theorem 2.For two ellipsoids, i.e. i = 1 ,
2, we have f i ( u i ) = A i u i (26)and ( ∇ Ψ i )( x ) = A − i x . x = ˜˜ f i ( m i ) gives A − i ˜˜ f i ( m i ) = m i or ˜˜ f i ( m i ) = A i m i . (27) B and B will kiss at a point when m (cid:107) m (cid:107) = − m (cid:107) m (cid:107) . There is not enough information here to solve for m in terms of m . But fromEqs. (26) and (27), m i = ˜˜ f − i ( f ( u i )) = A − i u i , (28)which means that the kissing condition can be written as A − u (cid:107) A − u (cid:107) = − A − u (cid:107) A − u (cid:107) . This equation says that u must be proportional to − A A − u , and the con-straint that u is a unit vector then specifies that u = − A A − u (cid:107) A A − u (cid:107) . Then using Eq. (28), we find that m = − m (cid:107) A m (cid:107) and so Φ( m ) . = (cid:107) m (cid:107) = (cid:107) m (cid:107)(cid:107) A m (cid:107) . Therefore, Φ( m ) (cid:107) m (cid:107) = 1 (cid:107) A m (cid:107) . Then Theorem 2 states that x = A m + A m (cid:107) A m (cid:107) . (29)Then substituting m = A − u with u parameterized as u ( φ ) gives x ( φ ) = A u ( φ ) + A A − u ( φ ) (cid:107) A A − u ( φ ) (cid:107) . (30)This is the same as the closed-form formula derived in a completely differentway in [12]. 13 .3. Demonstration of transformations Ellipsoids can be viewed as linearly transformed versions of spheres. For asphere, f ( u ) = u and since n = m = u we have ˜ f ( n ) = n and ˜˜ f ( m ) = m . Viewing the Minkowski sum of two ellipsoids as that of two transformed spheres,where M i = A i , then m (cid:48) i = A − i u and m i = u and Eqs. (21) and (22) bothreduce to (29).
6. Demonstration with Superquadrics
We now examine more general types of bodies, with superquadrics beingtypical examples. The Minkowski sums can also be parameterized by their un-normalized gradients. The two theorems as well as the transformed version inboth 2D and 3D cases are demonstrated with derivations of the intermediateexpressions necessary to use the general formulas in Eq. (21) and (22).
A planar superquadric can be parameterized as x ( u ( θ )) = (cid:32) a cos (cid:15) θb sin (cid:15) θ (cid:33) (31)where (cid:15) ∈ (0 ,
2) to ensure the convexity of the body. With u ( θ ) = (cid:32) cos θ sin θ (cid:33) , this means that f ( u ) = (cid:32) a u (cid:15) b u (cid:15) (cid:33) , where the un-bolded u i is the i -th entry of the vector u . The correspondingnormal vector is m ( θ ) = 2 (cid:15) (cid:32) a − cos − (cid:15) θb − sin − (cid:15) θ (cid:33) , and the square of its magnitude is (cid:107) m ( θ ) (cid:107) = a − cos − (cid:15) θ + b − sin − (cid:15) θ . x ) = (cid:16) x a (cid:17) /(cid:15) + (cid:16) x b (cid:17) /(cid:15) = 1 . (32)The gradient is then( ∇ Ψ)( x ) = (cid:15)a (cid:0) x a (cid:1) /(cid:15) − (cid:15)b (cid:0) x b (cid:1) /(cid:15) − . = m = (cid:32) m m (cid:33) . Raising each component of this to the power (cid:15)/ (2 − (cid:15) ), and solving for x , it iseasy to reparameterize Eq. (31) in terms of m as˜˜ f ( m ) = a (cid:0) (cid:15)a (cid:1) − (cid:15) − (cid:15) m (cid:15) − (cid:15) b (cid:0) (cid:15)b (cid:1) − (cid:15) − (cid:15) m (cid:15) − (cid:15) . Theorem 2 is used here to compute the closed-form expression of Minkowskisums by first obtaining the relationships between m and u as m = g ( u ) = (cid:32) a (cid:15) u − (cid:15) b (cid:15) u − (cid:15) . (cid:33) , (33)where u j = u · e j and e j is the j -th unit Cartesian coordinate basis, and theun-bolded a , b and (cid:15) denote the shape parameters of B . Observing that g ( k u ) = k − (cid:15) g ( u ), the sufficient condition for this mapping is satisfied. Sowe are able to express (cid:107) m (cid:107) as a function of m using Eq. (18). The reversefunction of g as a function of m is then explicitly computed as g − ( m ) = (cid:0) a (cid:15) m (cid:1) / (2 − (cid:15) ) (cid:0) b (cid:15) m (cid:1) / (2 − (cid:15) ) , (34)where m j = m · e j denotes the j -th entry of the gradient vector of B . Finally,substituting Eqs. (33) and (34) into Eq. (18) gives the closed-form expressionof (cid:107) m (cid:107) .Furthermore, using Eq. (22), we demonstrate the closed-form Minkowskisums under different linear transformations for the two superellipses, including: • pure rotations, i.e. M (1) i = rot ( α i ) = (cid:18) cos α i − sin α i sin α i cos α i (cid:19) , where α isthe rotational angle ; • pure shear, i.e. M (2) i = shear ( s i ) = (cid:18) s i (cid:19) , where s is the magnitudeof the shear transformation ; • a combination of them, i.e. M (3) i = rot ( α i ) shear ( s i ) .15 a) Canonical forms. (b) Pure rotations.(c) Pure shear transformations. (d) Combining rotation and shear.Figure 4: Demonstration of the closed-form Minkowski sums for two 2D superquadrics incanonical forms and under linear transformations. B is plotted in grey color at the center.The red curve represents the computed closed-form Minkowski sums boundary. B is plottedin blue color and placed at 10 different locations on the Minkowski sums boundary. Note that the right parenthesized superscript denotes the different types oftransformation, and the right subscript denotes the i -th body.Figure 4 demonstrates the expressions using different types of convex superel-lipses with (cid:15) ∈ (0 , B is drawn in the center as a black curve, B is drawn inblue and translates along the closed-form Minkowski sums boundary (in red) soas to kiss B . The demonstrations include superellipses in their canonical formsand under different types of linear transformations such as rotations and sheartransformations. The implicit expression for a 3D superquadric surface isΨ( x ) = (cid:18)(cid:16) x a (cid:17) /(cid:15) + (cid:16) x b (cid:17) /(cid:15) (cid:19) (cid:15) /(cid:15) + (cid:16) x c (cid:17) /(cid:15) = 1 , (35)16nd the gradient is( ∇ Ψ)( x ) = a(cid:15) [ ψ ( x , x )] (cid:15) /(cid:15) − (cid:0) x a (cid:1) /(cid:15) − b(cid:15) [ ψ ( x , x )] (cid:15) /(cid:15) − (cid:0) x b (cid:1) /(cid:15) − c(cid:15) (cid:0) x c (cid:1) /(cid:15) − . = m = m m m , (36)where ψ ( x , x ) = (cid:0) x a (cid:1) /(cid:15) + (cid:0) x b (cid:1) /(cid:15) denotes the part in the implicit expressionthat includes coordinates x and x , and (cid:15) , (cid:15) ∈ (0 ,
2) to ensure the convexityof the body enclosed by the bounding surface.Directly solving Eq. (36) for x , x , x gives˜˜ f ( m ) = a (cid:0) a(cid:15) m (cid:1) (cid:15) / (2 − (cid:15) ) [ γ ( m )] ( (cid:15) − (cid:15) ) / (2 − (cid:15) ) b (cid:0) b(cid:15) m (cid:1) (cid:15) / (2 − (cid:15) ) [ γ ( m )] ( (cid:15) − (cid:15) ) / (2 − (cid:15) ) c (cid:0) c(cid:15) m (cid:1) (cid:15) / (2 − (cid:15) ) , where γ ( m ) = 1 − (cid:16) c(cid:15) m (cid:17) / (2 − (cid:15) ) . On the other hand, Eq. (36) can be written with respect to the unit vector u ( η, ω ) = cos η cos ω cos η sin ω sin η . = u u u as ( ∇ Ψ)( η, ω ) = a(cid:15) cos − (cid:15) η cos − (cid:15) ω b(cid:15) cos − (cid:15) η sin − (cid:15) ω c(cid:15) sin − (cid:15) η . Substituting η, ω in terms of u , u , u , we get the explicit expressions thatare necessary to compute Φ( m ) as m = g ( u ) = a (cid:15) u − (cid:15) (cid:0) u + u (cid:1) ( (cid:15) − (cid:15) ) / b (cid:15) u − (cid:15) (cid:0) u + u (cid:1) ( (cid:15) − (cid:15) ) / c (cid:15) u − (cid:15) , (37)where a , b , c , (cid:15) and (cid:15) are the shape parameters of B , and u j = u · e j isthe j -th entry of the vector u of B . It can also be observed that, g ( k u ) = k − (cid:15) g ( u ), which satisfies the sufficient condition. Then the reverse function g − ( m ) can be computed as g − ( m ) = (cid:0) a (cid:15) m (cid:1) − (cid:15) [ ρ ( m , m )] (cid:15) − (cid:15) − (cid:15) (cid:0) b (cid:15) m (cid:1) − (cid:15) [ ρ ( m , m )] (cid:15) − (cid:15) − (cid:15) (cid:0) c (cid:15) m (cid:1) − (cid:15) , (38)17 a) Canonical forms. (b) Pure rotation.Figure 5: Demonstration of the closed-form Minkowski sums for two 3D superquadrics incanonical forms and with different orientations. B is shown in green color at the center. Theyellow surface represents the computed closed-form Minkowski sums boundary. B is plottedin blue color and placed at one location on the Minkowski sums boundary. where ρ ( m , m ) = (cid:16) a (cid:15) m (cid:17) − (cid:15) + (cid:18) b (cid:15) m (cid:19) − (cid:15) , and m ij = m i · e j is the j -th entry of the gradient vector m i of B i .The closed-form Minkowski sums for B and B in different orientations arevisualized here, as shown in Fig. 5. B is drawn in the center in green, and B isdrawn in blue and translates along the closed-form Minkowski sums boundary,which only kisses B at every point.
7. Numerical Validations and Benchmarks in Simulation for the Closed-form Minkowski Sums
To validate the correctness and evaluate the performance of the proposedclosed-form Minkowski sums, we conduct numerical simulations and comparewith the discrete Minkowski sums computations from definition as shown inFig. 1a.At first, a set of points on the Minkowski sums boundary is generated usingthe proposed exact closed-form expression. And we examine, when placing B at these points, whether there exists a kissing point that locates on bothbody boundary surfaces and the corresponding normal vectors are anti-parallel.To evaluate the efficiency of the proposed closed-form expression, we conductbenchmarks in the second simulation. The numbers of points on the boundariesof S and S are varied, and the running time to generate the point set on theMinkowski sums boundary are compared. All the simulations throughout thispaper are implemented in Matlab R2018b and performed in an Intel Core i7CPU at 2.70 GHz. 18 .1. Numerical validations for the closed-form Minkowski sums computations The general idea is to compute the kissing point of the two bodies whenplacing B at each point on Minkowski sum boundary. We are using the factthat the kissing point x kiss between two bodies must satisfies: • The kissing point lies on both surfaces. This can be achieved when theimplicit expression at the kissing point equals to one, i.e.Ψ ( x kiss ) = Ψ ( x kiss ) = 1 , (39)where Ψ i ( · ) is the implicit expression of the i -th body and i x kiss is thekissing point as viewed in the body frame of i -th body. • The normal vectors at the kissing point with respect to the two surfaces areanti-parallel. Specifically, we can compute the un-normalized gradients atthe kissing point on each surface, and verify that the angle between themis π .In practice, we first sample a set of points x ( φ k ) on ∂B based on the set ofangular parameters { φ k } . Then, this parameter set can also be used to generatethe points on Minkowski sum x ( φ k ) using our closed-form expression. Then,we place B at x ( φ k ), and verify that each x ( φ k ) is the kissing point.To validate the first condition that the kissing point locates on both boundingsurfaces, x ( φ k ) is transformed into the local frame of each body, then Eq.(39) can be examined. Note that, since x is computed from the parametricexpression of B , we only need to verify that the value of e implicit . = | Ψ ( x ) − | = 0 . (40)Then, since the gradient can be computed from the point on surface, the un-normalized gradients at the candidate kissing point on both bodies can be com-puted as m ( x ) and m ( x ) respectively. They are anti-parallel when e gradient . = (cid:12)(cid:12)(cid:12)(cid:12) m ( x ) · m ( x ) (cid:107) m ( x ) (cid:107)(cid:107) m ( x ) (cid:107) − cos( π ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (41)Once these two conditions are verified, we can say that x is the kissing pointwhen B is place at the Minkowski sum boundary.We conduct simulations to verify in both 2D and 3D cases. For 2D case, atotal of 1000 points are sampled on ∂B based on the angular parameter set { φ Dk } = { θ k } ; and for 3D case, 100 samples are generated for each angularparameter pair (i.e. { φ Dk } = { ( η k , ω k ) } ), so a total of 10 points are sampledon ∂B . And for both cases, 100 trials are simulated and the mean values ofEq. (40) and (41) for among all the trials are computed.For the 2D case, among all the trials, the mean values of e implicit and e gradient are 5 . × − and 6 . × − , respectively. And for the 3D case, the meanvalues of e implicit and e gradient are 1 . × − and 4 . × − , respectively.The results show that when B is centered at the Minkowski sum boundary19oints x ( φ ), which are computed by both our proposed closed-form expres-sion, there exists a kissing point x ( φ ). Ideally, the expressions in Eq. (40)and (41) should equal to zero since our proposed closed-form Minkowski sumsexpression is the exact solution. Due to the numerical precision issues, thereare discrepancies between the computational results and the ideal expectations.However, these discrepancies are all very small, i.e. less than the level of 10 − ,which is acceptable for the numerical verification purposes. Therefore, the pointon ∂B parameterized by φ k is the kissing point when placing B at x ( φ k ),with which the proposed closed-form Minkowski sum expression is numericalverified. To evaluate the performance of computing a discrete point set on the Minkowskisums boundary, the running time is measured for our proposed method and themethod using the original definition. Note that for the Minkowski sums com-puted from definition, we represent the discrete boundaries of both surfaces aspoint sets. Since the two bodies are convex, the corresponding Minkowski sumsboundary is the convex hull of the summations of all possible pairs from the twopoint sets. Although it is not an exact result, we could still get a sense of thequality of our closed-form computation compared to the definition of Minkowskisums.To give an impression of the computational speed of our proposed method,the running time for both 2D and 3D cases are compared. In principle, therunning time for ours only depend on the number of points sampled on ∂B ,while the brute-force method depends on the sampled points on both bodysurfaces. Therefore, we expect that ours can achieve a linear time complexitywith respect to the number of sampled vertices, while the brute-force methodwould be much slower. Figure 6 compares the running time for the two methodsof computing Minkowski sums at different discretization levels over the 100experiment trials.From the comparison results, our proposed closed-form expressions for Minkowskisums achieve a linear time complexity with respect to the number of verticeson one body surface. This outperforms the method that computes the dis-crete Minkowski sums by definition, which is increases much faster when morevertices are sampled. Furthermore, since we find a relationship based on theoutward normal at the touching point, the number of the points on Minkowskisum boundary only depends on one body where the normal is computed. Thisexplains the linear complexity with respect to the number of vertices on onebody surface. Through this nice property, the Minkowski sums boundary canbe computed in a very efficient way, which provides quite an advantage espe-cially in the 3D case. 20
00 200 300 400 500
Number of vertices R unn i ng t i m e / s Closed-formDefinition
100 105 110 115 12000.511.522.5 -3 × (a) 2D case.
150 152 154 156 158 160 162-0.0100.010.020.030.040.050.06 (b) 3D case.Figure 6: Running time comparisons for the closed-form and brute-force calculations ofMinkowski sums. For both sub-figures, curves represents the mean value of the running time,and the shaded regions show the range of the standard deviation. The blue curve denotes ourproposed method, while the orange curve denotes the method using definition. A zoom-inview of a small region when the number of vertices is small is shown below each sub-figure. a) Obstacles in workspace. (b) C-obstacles in different slices.Figure 7: Demonstration of the C-obstacles generation in SE(2). The figures show the trialwhen 500 obstacles are located arbitrarily in the environment and the corresponding C-obstacles for 10 sampled orientations of the robot. Obstacles in the workspace are representedby black superellipses, and the calculated Minkowski sums in each C-slice are represented inred.
8. Application on Point-based Configuration Space Obstacles Gener-ations
One of the important applications of Minkowski sums is to generate config-uration space obstacles [3] in robot motion planning algorithms. The idea is toshrink the robot into a single point and inflate all obstacles by the Minkowskisums boundaries. Then a motion planner can be developed for the point robotthat avoids the inflated obstacles, which are called configuration-space obstacles(C-obstacles) . If the robot is bounded by a circle or sphere, then, the currentspace is the configuration space of the robot. But if the robot is enclosed byother kinds of geometric shapes, i.e. the superquadrics, then the orientationmatters. In this case, multiple slices [29] of the configuration space are requiredto be generated, each of which requires the computations of the C-obstaclesusing Minkowski sums. Such a class of planners are effective when the robot isrigid, i.e. mobile robot, drone, underwater vehicle, etc, where the Minkowskisums computation take a very essential role.Here we show that using our proposed closed-form Minkowski sums, C-obstacles for rigid-body robots in multiple orientations can be calculated ef-ficiently in both 2D and 3D cases. Both the robot and obstacles are gener-ated as superquadric bodies with arbitrary shape parameters. We measure thecomputational time for generating the C-obstacle boundary points for a set oforientations of the robot and multiple numbers of obstacles. The Minkowskisums for all obstacles are computed in pairwise with the robot in a loop. Figure7 demonstrates the simulation for C-obstacles generation in SE(2) configura-tion space. This demonstration includes 500 obstacles in the environment witharbitrary shapes and poses. And 10 slices of the configuration space with thecomputed C-obstacles are shown. 22 able 1: Parameters and running time for C-obstacle generations in both 2D and 3Dworkspaces.
Dim. Num. Num. Num. Total Avg. timeobstacles points orientations time (s) per point ( µs )2D 50 50 50 0.41 3.2802D 1000 50 50 7.07 2.8282D 10000 50 50 71.36 2.8543D 50 100 30 1.07 7.1333D 1000 100 30 20.63 6.8773D 10000 100 30 199.13 6.638Table 1 summarizes the parameters and the running time results of generat-ing C-obstacles in both 2D and 3D cases. The parameters include the numbersof obstacles, computed Minkowski sum boundary points and the orientations ofthe robot. And for the running time, the total computational time for all thepoints generated on all obstacles and the averaged time for each point on theMinkowski sum boundary are measured.The simulation shows a potential application of our closed-form Minkowskisums computations in generating C-obstacles, which is essential in developingefficient motion planning algorithms for rigid-body robots. The performanceare based on a Matlab implementation without GPU or parallel computingaccelerations, which achieves a level of seconds for hundreds of obstacles andminutes for thousands of obstacles. On average, the closed-form computationfor each boundary point is in the level of microseconds. With these points onthe boundary of C-obstacles, the collision-free configuration space can then becharacterized, which leads to various efficient motion planners [29, 13].
9. Conclusion
This paper introduces a novel exact and closed-form parameterization ofMinkowski sums between two positively curved bodies in d -dimensional Eu-clidean space. The boundary surface of each body is parameterized as a func-tion of the gradient. It is shown that the Minkowski sums can be derived inclosed-form based on the surface gradient that can be either normalized (i.e.becomes the outward normal vector) or un-normalized. The results proposedin this paper, both in canonical form and with linear transformations, are iden-tical with those using geometrical interpretation in the case of two ellipsoids.And in general, these expressions can be applied to pairs of bodies that areenclosed by general smooth surfaces with positive Gaussian curvature at everypoint. Numerical simulations are conducted in the case of two superquadrics,showing the correctness and efficiency of the proposed closed-form expression.An application in generating configuration-space obstacles for motion planningalgorithms is introduced and demonstrated.23 cknowledgement This work was performed under U.S National Science Foundation grants IIS-1619050 and CCF-1640970 and U.S. Office of Naval Research Award N00014-17-1-2142. The ideas expressed in this paper are solely those of the authors.