Technische Universität Wien

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1 Technische Universität Wien TU Geometric design of motions constrained by a contacting surface pair M. Hofer, H. Pottmann, B. Ravani Technical Report No. 97 December 2002 Institut für Geometrie A-1040 Wien, Wiedner Hauptstraße 8 10/113

2 Geometric design of motions constrained by a contacting surface pair Michael Hofer a,b,, Helmut Pottmann b, Bahram Ravani a a Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, USA b Institute of Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8-10/113, A-1040 Wien, Austria Abstract We discuss the following problem which arises in robot motion planning and NC machining: Given are a fixed surface Ψ and N positions Φ i of a moving surface Φ such that the Φ i are in contact with Ψ. Compute a smooth and fair gliding motion Φ(t) of the surface Φ on the surface Ψ which interpolates (or approximates) the given positions Φ i at time instances t i. We present a variational subdivision algorithm that uses a feature point representation of the moving surface and relies on known subdivision algorithms for curves, on instantaneous kinematics, and on ideas from line geometry. We use geometric methods for the numerical solution of the arising optimization problems. Key words: Motion design, Variational subdivision, Gliding motion, Kinematics, Line Geometry 1 Introduction Motion design is an important and widely studied problem, e.g., in robotics, manufacturing geometry, and computer animation. Contributions to the solution of the problem come from Computational Geometry (see e.g. Halperin et al. (1997); Sharir (1997)), Robotics and Kinematics (see e.g. Latombe (2001); Park and Ravani (1995)), Computer Graphics (see e.g. Shoemake (1985); Marchand and Courty (2002)) and Computer Aided Geometric Design (see e.g. Röschel (1998); Jüttler and Wagner (2002)). Corresponding author: phone , fax address: hofer@geometrie.tuwien.ac.at (Michael Hofer). URL: (Michael Hofer). Preprint submitted to Elsevier Science 17 December 2002

3 We are interested in the design of a smooth and fair motion of one surface (e.g. a toroidal cutter) gliding on another surface (e.g. a freeform surface) such that given positions already in contact are either interpolated or approximated. This has applications e.g. in the NC machining of complex shapes (cf. Marciniak (1991)) and in medical applications where the gliding motion of joints is studied (cf. Hoschek and Weber (1987)). Using variational subdivision, instantaneous kinematics, and the relation to line geometry, we show how to construct such motions. We confine ourselves to differentiable oneparameter motions in Euclidean 3-space. For applications it is important to be able to control the smoothness level of a motion. Forces and vibrations depend on second derivatives and jumps in the velocity should be avoided with NC machining. Smooth motions that make use of NURBS techniques have been investigated by e.g. Jüttler and Wagner (1996). To describe rigid body motions it is necessary to use rational rather than polynomial representations. However, rational representations are much less suitable for variational design and efficient optimization techniques than polynomial ones. Therefore, Hyun et al. (2001) recently investigated the construction of affine spline motions with minimal distortion. Motion design based on the quaternion representation of the spherical component and nonlinear extensions of spline constructions in affine spaces to the sphere (see the references in Röschel (1998)), are also difficult to deal with for optimization purposes. Algorithms for motion fairing using the quaternion representation have been proposed by Fang et al. (1998). Belta and Kumar (2002) recently presented a singular value decomposition projection method for interpolation on the group of Euclidean motions SE(3). Hofer et al. (2002a) have outlined two new approaches to motion design which deal with variational motion design and that allow to incorporate constraints into the motion planning process. The first approach has been studied in Hofer et al. (2002b), the second approach will be studied here and is extended to handle motions constrained by a contacting surface pair which is the main contribution of this paper. The singularities of motions constrained by a contacting surface pair have been studied by Pottmann and Ravani (2000). Here we present an algorithm for the design of a gliding motion for the entire motion cycle rather than around singularities. All other previous works have not considered gliding motions where a rigid moving body maintains contact with another body during its motion cycle. Wallner (2002b) has investigated gliding spline motions using an active motion approach. However, the resulting motions are in general only near-euclidean near-contact spline motions. Our work in terms of the formulation of the fairness is also different from previous approaches on un-constraint motion design in that our formulation of fairness is based on fair trajectories of chosen feature points of the moving surface. A rigid body motion is a curve in the Euclidean motion group and variational motion design could be based on expressing the fairness of that curve (see Park and Ravani (1997)). However, for the applications we have in mind, not the motion as such, but its action on a certain rigid body is 2

4 employed in fairness criteria. The paper is organized as follows. In Sect. 2 we review some aspects of Kobbelt s interpolatory variational subdivision algorithms for curves and extend them to approximating schemes and curves on surfaces. In Sect. 3 we summarize relations between spatial kinematics and line geometry that we use for our motion design algorithm. In Sect. 4 we present the subdivision algorithm for unconstraint motion design, which in Sect. 5 is extended to a subdivision algorithm for the design of a gliding motion of one surface on another surface such that given positions in contact are interpolated or approximated. We conclude the paper in Sect. 6 with an outlook towards future research. 2 Variational subdivision for curves Kobbelt (1996) has introduced a class of interpolatory variational subdivision schemes. Thereby, new points for the refinement of a given polygon are determined by solving an optimization problem that minimizes quadratic energy functionals. This is a familiar approach from non-discrete curve design where energy functionals are minimized for curve fairing. Kobbelt and Schröder (1998) have extended interpolatory variational subdivision from the uniform to the non-uniform parameter setting and discussed it in a multiresolution framework. These schemes are global, i.e., every new point depends on all points to be refined. Definition 1 An interpolatory subdivision scheme is formally described by (Kobbelt and Schröder, 1998) as an operator which maps a given sequence of points (a polygon) P (m) = [p (m) i p (m+1) 2i 1 ] to a refined sequence P (m+1) = [p (m+1) i ] with i of the polygon P (m) are contained = p (m) i. That is, the points p (m) in the refined polygon P (m+1) and thus the limit curve P ( ) interpolates the initially given points p (0) i. Definition 2 Variational approaches start with a quadratic fairness (energy) functional (Kobbelt, 1996) E(P (m+1) ) := i K ( p (m+1) i ) 2 with K ( p (m+1) i ) := j β j p (m+1) i+j. (1) K is an affinely invariant ( j β j = 0) measure of local strain energy and hence only finitely many β j do not vanish. The indices i and j are understood to range over all integers for which the associated variables p (m+1) are defined. Important examples of variational subdivision are found by minimizing differences. In the uniform (non-uniform) case we minimize forward (divided) differences, see Kobbelt and Schröder (1998). In our motion design algorithm we use the minimization of second differences, which minimize the change in velocity, on K polygons simultaneously. Therefore it is useful to explicitly state the curve case here such that its generalization later on will be straightforward. 3

5 In the uniform case we minimize second forward differences ( ) (m+1) K 2 p i := 2 p (m+1) i = p (m+1) i 2p (m+1) i+1 + p (m+1) i+2, (2) and in the non-uniform case, where every point p (m+1) i is associated with the parameter value t (m+1) i, we minimize second divided differences, K 2 ( p (m+1) i ) := p (m+1) t (m+1) i i ( (m+1) t i+1 + t (m+1) ) i + t (m+1) i+1 p (m+1) i+1 t (m+1) i+1 t (m+1) i p (m+1) i+2 ( (m+1) t i+1 + t (m+1) i ). (3) In our application the t i s will most likely be given, since they are related to the timing of the motion. If we have to estimate t (0) i, we may take it from a centripetal parameterization, i.e., t (0) i+1 t (0) i := p (0) i+1 p (0) i. (4) Remark 3 While the parameters t (m+1) 2i 1 for the points p (m+1) 2i 1 = p (m) i can be calculated e.g. by using Eq. (4) we need to choose the parameters t (m+1) 2i corresponding to the points p (m+1) 2i that are inserted in this refinement step. For simplicity we set t (m+1) 2i := 1 ( (m+1) t 2i t (m+1) ) 2i+1. (5) Further note that for uniform t (m+1) i, Eq. (3) becomes Eq. (2) and that the minimization of (K 2 ) 2, where K 2 is given by Eq. (2), is geometrically the same as minimization of the sum of squared distances shown in Fig. 1 (a). From now on we will drop the index (m+1) corresponding to the refinement step and simplify the notation by calling the given points f i := p (m+1) 2i 1 = p (m) i and the new points that are inserted in the current refinement step q i := p (m+1) 2i. With this new notation we derive explicit formulae in the uniform case for open and closed polygons, which we can later refer to when we describe the motion design algorithm. 2.1 Uniform interpolatory variational subdivision for curves In the uniform parameter setting for open polygons, the position of the new points q i is found by minimizing the objective function F 2 (q 1,..., q N 1 ) = N 1 f i 2q i + f i N 2 q i 2f i+1 + q i+1 2. (6) 4

6 F 2 is quadratic in the unknowns q i R d. The minimization of F 2 can be computed by setting the partial derivatives of F 2 with respect to the unknowns q i equal to zero, which leads to the linear system of N 1 equations q 1... = q N } {{ } =:A } {{ } =:B f 1.. (7) 4 4 f N 4 2 In the uniform parameter setting for closed polygons, the position of the new points q i is found by minimizing the objective function N N F 2 (q 1,..., q N ) = f i 2q i + f i q i 2f i+1 + q i+1 2, (8) where we identify f N+1 f 1 and q N+1 q 1. Minimizing F 2 leads to solving the following linear system of N equations: f q = (9) f N q N 4 4 f Similarly one can derive explicit formulae for the non-uniform variational subdivision for curves. Note that we use the same symbol F 2 for all quadratic objective functions we encounter in this paper. Remark 4 The refinement schemes are global, i.e., every new point depends on all points of the polygon to be refined. Interpolation is guaranteed since the old points belong to the newly calculated finer version. Kobbelt (1996) has shown that by minimizing second differences the limit curves P ( ) are at least C 2 curves. So we can use these schemes for variational motion design of a C 2 motion. 2.2 Uniform approximating variational subdivision for curves The previously discussed variational subdivision scheme is an interpolating one. The given points f i remain unchanged. However, we can pick some or all of these input points and allow a change as well. For those points which may 5

7 q i 1 q i 1 (f 2 i 1 + f i ) 1 (q 2 i 1 + q i ) 1 (f 2 i + f i+1 ) PSfrag replacements q i 1 q i f i f i+1 1 (f 2 i + f i+1 ) 1 (q 2 i 1 + q i ) 1 (f 2 i 1 + f i ) q i 1 f i f i+1 f n i 1 (f n 2 i 1 + fi n ) q i 1 (f 2 i n + fi+1) n 1 (q 2 i 1 + q i ) f n i+1 f i 1 f i 1 (a) f n i 1 (b) Fig. 1. (a) Uniform interpolatory variational subdivision. (b) Uniform approximating variational subdivision. be changed, we add a term λ i (f i f n i ) 2 to the functional (6) to be minimized. It may be seen as placing a spring between the given point f i and the new location f n i, see Fig. 1 (b). The influence of the spring is governed by the real number λ i > 0. The functional to be minimized is again quadratic, namely a discretization of the functional used for smoothing splines, F 2 (q 1,..., q N 1, f n 1,..., f n N) = + N 2 q i 2f n i+1 + q i N 1 N f n i 2q i + f n i+1 2 λ i f i f n i 2. (10) Thus, the minimization of F leads to a linear system of equations A a x = b where we collect the unknowns in a 2N 1 vector x = (q 1,..., q N 1, f n 1,..., f n N) T Note that this system has to be solved for every coordinate. The (2N 1) (2N 1) symmetric coefficient matrix A a has a band structure, is sparse and can be described as a block matrix, b is a 2N 1 vector: A a A = B T B, b = (0,..., 0, λ }{{} 1 f 1,..., λ N f N ) T. (11) L N 1 Non-bold f i denotes one coordinate of the vector f i. The block matrices A and 6

8 B are the (N 1) (N 1) and (N 1) N matrices of Eq. (6), and 1 + λ λ 2 1 L := R N N. (12) λ N λ N Note that, if a spring is weak, i.e., if we let λ i 0, then this point is neglected and only the remaining points are taken into consideration for the shape of the final curve. Thus, if we let λ i 0 for all i, then the limit curve will just be a straight line approximating the input points, see Fig. 2 (a). For λ i the corresponding input point f i is interpolated by the final curve. Thus, if we have strong springs at the first and last input point, i.e., λ 1, λ N, but all other springs are weak (λ 2,..., λ N 1 0), then the limit curve is a straight line connecting the first and last input point, see Fig. 2 (b). The most useful case is the one where the first and the last spring are very strong (λ 1, λ N ) and the intermediate ones pull the curve towards the input points (0 < λ i < for i = 2,..., N 1), see Fig. 2 (c). (a) (b) (c) (d) Fig. 2. Uniform approximating variational subdivision for curves: (a) λ i 0 for i = 1,..., N. (b) λ 1, λ N, λ i 0 for i = 2,..., N 1. (c) λ 1, λ N, 0 < λ i < for i = 2,..., N 1. (d) λ i for i = 1,..., N. If we let λ i for all i, then we get in the limit an interpolating curve, see Fig. 2 (d). In that case we will instead use interpolatory variational subdivision. Remark 5 It is straightforward to extend this approximating scheme to closed polygons and to the non-uniform case. 2.3 Variational subdivision for surface curves Given are now N points on a surface and we want to use variational subdivision to compute a surface curve that interpolates those points. First, in every subdivision step use uniform or non-uniform interpolatory variational 7

9 subdivision in Euclidean three-space E 3. And then, since the such inserted points will in general not be on the surface, project them orthogonally onto the surface. We call this the projection method. For a surface in parametric representation, a surface curve is found by mapping a curve in the parameter domain of the surface. Hence, one can use uniform or non-uniform interpolating or approximating variational subdivision in the parameter domain of the surface, and then map these points to get the surface curve. We call this approach the parameter domain method. Fig. 3 compares this approach with the projection method for a rectangular surface patch and an ellipsoid. (a) (b) Fig. 3. Variational subdivision surface curves interpolating given surface points: projection method (black curve), parameter domain method (gray curve). Although the parameter domain method is efficient to compute, it has several drawbacks if used to determine the coarse shape of a surface curve. Distortions under parametrization can result in undesirable shapes. Since the domain curve might overshoot the parameter domain, the surface curve might partially miss the surface patch (see Fig. 3 (a)). The surface curve produces unwanted results near singularities of the parametrization (see Fig. 3 (b) for an ellipsoid). Therefore we use a hybrid algorithm that employs the projection method until the coarse shape of the surface curve is determined (3 4 subdivision steps) and then, for further refinement, switches to the parameter domain method to save computation time. 3 The relation between spatial kinematics and line geometry 3.1 Instantaneous kinematics Consider a differentiable one-parameter rigid body motion in Euclidean 3- space. Introducing Cartesian coordinate frames Σ in the moving system and Σ 0 in the fixed system, the time dependent position x 0 (t) of a point x Σ in 8

10 the fixed system Σ 0 is given by x 0 (t) = R(t)x + r(t). (13) Here, the time dependent orthogonal matrix R(t) represents the spherical component of the motion, and r(t) describes the trajectory of the origin of the moving system. All arising functions shall be C 1. By differentiation we get the velocity vectors. It is well-known that the velocity vector field is linear at any time instant. More precisely, at any time instant there exist vectors c, c such that the velocity vector v(x) of any point x of the moving body can be computed as v(x) = c + c x. (14) Note that in this formula all arising vectors are represented in the same system; this may be the moving or the fixed system. The meaning of c, c is as follows: c represents the velocity vector of the origin, and c is the so-called Darboux vector. Only very special one-parameter motions have a constant, i.e., timeindependent velocity vector field. These motions are A translation with constant velocity (if c = 0), A uniform rotation about an axis (if c c = 0), A uniform helical motion (if c c 0). The most general case is that of a uniform helical motion, which is the superposition of a rotation with constant angular velocity about an axis A and a translation with constant velocity parallel to A. If the moving body rotates about an angle t, the translation distance is pt. The constant factor p is referred to as pitch of the helical motion. For more details on helical motions and its close relation to line geometry we refer to Pottmann and Wallner (2001). The Plücker coordinates (a, ā) of the axis A, the pitch p of the helical motion and the angular velocity ω are computed from (c, c) by a = c c, c pc ā = c, p = c c, ω = c. (15) c2 In our motion design algorithm we use the constant linear velocity vector field of an instantaneous helical motion to linearize a motion. Therefore we attach to each point x i the velocity vector v(x i ) = c + c x i of the still unknown constant velocity vector field of an instantaneous helical motion. Minimizing an objective function that is quadratic in c, c gives us those unknowns. Note however, that the transformation which maps x i to x i + v(x i ) is an affine map and not a rigid Euclidean transformation. Therefore we use the uniform helical motion that is uniquely determined by the velocity vector field represented by the pair (c, c). Applying a rotation about A through an angle of α = arctan( c ) and a translation parallel to A by the distance αp brings the points x i in a position x i close to the tips x i + v(x i ) of the velocity vectors used in the computation, see Fig. 4. 9

11 PSfrag replacements e i x i + v(x i ) A x i αpa p α x i Fig. 4. New position x i of a point x i by applying the helical motion with the velocity vector field described by (c, c). Note that α = 0, i.e., c = 0, corresponds to a pure translation. Then x i = x i + c and thus e i = 0. This expresses the fact that the velocity vector field of a pure translation coincides with the translation itself. If the pitch p = 0, we just apply a rotation by the angle α around the axis A. Remark 6 The presented estimation of an instantaneous motion has also been used by Bourdet and Clément (1976, 1988) for the solution of a registration problem in Computer Vision. However, they directly transform the body (in an affine way) with the velocity vectors, and do not use the suggested correction via an appropriate helical motion. Pottmann et al. (2002a) have used linearized motions in an industrial inspection algorithm that registers a point cloud of an object (e.g. obtained by using a 3D laser scanner) to a CAD model of that same object. Pottmann et al. (2002b) have also shown, how this method can be used for the simultaneous registration of multiple point clouds. 3.2 Line geometry In three-dimensional Euclidean space E 3 with a Cartesian coordinate system, a straight line L can be represented by a normalized direction vector l, l = 1, and its moment vector l := x l with respect to the origin. Thereby, x denotes the coordinate vector of an arbitrary point on L. Note that l is independent of the choice of x. The 6 coordinates of L are its so-called normalized Plücker coordinates. They satisfy the Plücker relation l l = 0. Any 6-tuple (l, l) R 6 with l = 1 and l l = 0 represents a line in E 3, where (l, l) and ( l, l) describe the same line. One of the fundamental relations between spatial one-parameter motions and line geometry is, that for every time instant t, the path normals (l, l), i.e., the lines through a point x normal to v(x), of such a motion lie in a linear line complex C. Such a linear complex is defined as the solution set of a linear homogeneous equation in Plücker coordinates, 0 = v l = ( c + c x) l = c l + c l. (16) The linear line complex is singular if either p = 0 or p =. We speak of the pitch p and the axis A of a linear complex which can be computed via Eq. (15), 10

12 since C is defined by (c, c). Conversely, any linear line complex C, defined as the solution set of a linear homogeneous equation in Plücker coordinates c l + c l = 0, (17) can be obtained as a path normal complex of a spatial one-parameter motion. We will use this in Sect. 5 for the gliding motion design algorithm. 4 Variational subdivision for motion design Hofer et al. (2002a) have outlined a variational subdivision algorithm for motion design which we study here. In Sect. 5 we will add linear contact constraints to that algorithm for the purpose of designing motions constraint by a gliding surface pair. Current motion design algorithms consider fairness criteria which possess full kinematic invariance. This means that the result does not change under the application of Euclidean displacements in the moving or fixed system, respectively. As a consequence of that, we cannot impose fairness criteria based only on the trajectories of points on the moved object. In contrast, we choose K feature points f 1,..., f K on the moving body Φ that represent it sufficiently well, and use them to employ fairness criteria in the motion design process. Fig. 5 shows a simplified robot gripper arm where the 24 vertices of the polyhedral representation are used as feature points. For the choice of feature points we refer the reader to Sect Fig. 5. Simplified robot gripper arm with 24 feature points. In Euclidean three-space we insert in an iterative manner rigid copies of a moving body between given copies of that body such that the feature point paths (polygons) simultaneously minimize one of the discrete energy functionals presented in Sect. 2. This is a nonlinear problem which has to be solved in an iterative manner. For the solution we use instantaneous kinematics as presented in Sect. 3. This allows us to linearize the motions, that are applied to the inserted rigid bodies, in the minimization. The advantage of working in Euclidean three-space is, that fairness criteria can be employed much easier than if we would work on the six-dimensional manifold of Euclidean motions. Furthermore, our approach can be used with the whole class of interpolatory variational subdivision algorithms introduced by Kobbelt (1996), and with the extension to approximating schemes as discussed 11

13 in Sect. 2. We present the algorithm in terms of the minimization of second forward differences, since in that case the formulae are shorter. The generalization to the non-uniform parameter setting is straightforward, although a bit more tedious. Variational subdivision for curves combines the insertion and position optimization (i.e., minimization of some quadratic energy functional) of new points in every subdivision step. In the motion design case we have to split this into two separate steps, insertion and optimization, since we do not just have a single point but a rigid body. These two steps are repeated until the required density of positions is reached: (1) Position insertion: Insert between each two adjacent given copies one Euclidean copy Φ (start) i i+1 of the moving body. For given N positions this gives N 1 initial intermediate positions. (2) Position optimization: Use Φ (start) i i+1 of the moving body as starting values for an iterative optimization algorithm that uses linearized motions and simultaneous variational subdivision on all K feature points. Terminate the iteration if the objective function reaches a minimum, or alternatively, if the sum of squared distances between the feature points in the current and in the following iteration step falls below a certain threshold. Then add the final positions to the sequence of given positions, update the number of given positions to 2N 1 and continue with step 1. Remark 7 By a position of a rigid body we mean the location in space determined by six parameters, three for translation and three for rotation. In the literature, a position sometimes is only defined by the three translation parameters, and the three rotation parameters are referred to as orientation of the rigid body. Now we discuss the individual steps of our motion design algorithm in more detail. 4.1 Position insertion Between each two adjacent given copies Φ i and Φ i+1 one new copy Φ (start) i i+1 of the moving body Φ is inserted (in the curve case one new point would be inserted between each two given points). We get Φ (start) i i+1 by applying the Euclidean rigid body transformation m to Φ i, Φ (start) i i+1 := m(φ i ) := R i (Φ i ) + t i, (18) where m is chosen such that Φ (start) i i+1 interpolates between Φ i and Φ i+1. The simplest choice of m is the identity transformation R i = I and t i = 0. But the algorithm needs less iteration steps in the optimization procedure if we apply reasonable t i and R i. We compute the translational and the rotational part separately. Consider the given positions s i of the barycenter s of the moving body Φ and apply one iteration step of interpolatory variational subdivision for curves to find 12

14 intermediate positions s i i+1. Then the obvious choice for the translational part of the Euclidean transformation m is, t i := s i i+1 s i. (19) Now we discuss the computation of R i. Translate Φ i+1 to a position Φ i+1 such that s i+1 = s i. Since Φ i and Φ i+1 are Euclidean copies of Φ there is a unique rotation with angle of rotation ρ i and direction vector d i of the rotation axis that rotates Φ i into Φ i+1. In our algorithm we rotate Φ i+1 through an angle of ρ i /2. Hence for the computation of R i we use the unit quaternion (a 0, a 1, a 2, a 3 ) = (cos ρ i 4, sin ρ i 4 d i). (20) The corresponding rotation matrix is a a 2 1 a 2 2 a 2 3 2(a 1 a 2 + a 0 a 3 ) 2(a 1 a 3 a 0 a 2 ) R i = 2(a 1 a 2 a 0 a 3 ) a 2 0 a a 2 2 a 2 3 2(a 2 a 3 + a 0 a 1 ). (21) 2(a 1 a 3 + a 0 a 2 ) 2(a 2 a 3 a 0 a 1 ) a 2 0 a 2 1 a a 2 3 Thus we find the starting positions Φ (start) i i+1 for the optimization step in the following way: For i = 1,..., N 1 first compute the translational part t i by Eq. (19), then the rotational part R i as described above, and finally apply the Euclidean transformation Eq. (18) to get the starting position Φ (start) i i+1. For easier reading of the following computations we denote the feature points in the positions Φ ( ) i i+1 by qk i for i = 1,..., N 1 and k = 1,..., K. 4.2 Position optimization This step in our motion design algorithm simultaneously optimizes the location of the inserted copies Φ ( ) i i+1. The iterative algorithm uses instantaneous kinematics and simultaneous interpolatory variational subdivision on the K sequences of feature points f1 k, q k 1, f2 k,..., fn 1, k q k N 1, fn k (k = 1,..., K). If we assume uniformly distributed time instances t i of our input positions, then we generalize uniform interpolatory variational subdivision for curves, see Sect To the feature points q k i (i = 1,..., N 1; k = 1,..., K) of each single copy Φ i i+1 we attach (see Fig. 6) vectors v k i := v(q k i ) = c i + c i q k i, (22) belonging to the linear velocity vector field of an instantaneous helical motion described by the pair (c i, c i ), see Sect The velocity vectors are used for first order estimates of the new positions. Simultaneous variational subdivision (with uniform parametrization) on the K sequences of feature points amounts to the minimization of the following objective function, which is quadratic in the unknowns c 1, c 1,..., c N 1, c N 1, 13

15 PSfrag replacements q k i 1 f k i q k i 1 + vi 1 k 1 (f k 2 i 1 + fi k ) q k i q k i + v k i 1 2 (f k i + f k i+1) f k i (qk i 1 + vi 1 k + q k i + vi k ) f k i 1 Fig. 6. points. Simultaneous variational subdivision on the K sequences of homologous F 2 (c 1, c 1,..., c N 1, c N 1 ) = + N 2 K k=1 N 1 f k i 2(q k i + v k i ) + f k i+1 2 (q k i + v k i ) 2f k i+1 + (q k i+1 + v k i+1) 2. (23) The minimization of (23) is found by setting the partial derivatives of F 2 with respect to the unknowns c i, c i equal to zero, F 2 c i = 0, F 2 c i = 0, i = 1,..., N 1. This leads to the linear system of equations A C = B (24) where 2(q k 1 f1 k ) + 4(q k 1 f2 k ) (q k 1 q k 2) 2f 1 k + 4f2 k 5q k 1 q k 2. K (q k i 1 q k i ) + 4(q k i fi k ) + 4(q k i fi+1) k (q k i q k i+1) B =, (25) k=1 4f k i + 4fi+1 k q k i 1 6q k i q k i+1. 2(q k N 1 fn) k + 4(q k N 1 fn 1) k (q k N 1 q k N 2) 2fN k + 4fN 1 k 5q k N 1 q k N 2 C = (c 1, c 1,, c i, c i,, c N 1, c N 1 ) T, (26) 14

16 5A k 11 5Q k 1 A k 13 Q k Q k 1 5I Q k 2 I 0 0 K A k A = 2i 1,2i 3 Q k i 6A k 2i 1,2i 1 6Q k i A k 2i 1,2i+1 Q k i k=1 Q k i 1 I 6Q k i 6I Q k i+1 I 0 0 A k 2N 3,2N 5 Q k N 1 5A k 2N 3,2N 3 5Q k N Q k N 2 I 5Q k N 1 5I.(27) The 3 3 blockmatrices A k ij of the symmetric matrix A are given by A k 11 = (q k 1) 2 I q k 1(q k 1) T A k 13 = (q k 1 q k 2)I q k 2(q k 1) T A k 2i 1,2i 3 = (q k i 1 q k i )I q k i 1(q k i ) T A k 2i 1,2i 1 = (q k i ) 2 I q k i (q k i ) T (28) A k 2i 1,2i+1 = (q k i q k i+1)i q k i+1(q k i ) T A k 2N 3,2N 5 = (q k N 2 q k N 1)I q k N 2(q k N 1) T A k 2N 3,2N 3 = (q k N 1) 2 I q k N 1(q k N 1) T and all other entries of the matrix A are zero. I is the 3 3 unit matrix and 0 is the 3 3 zero matrix. The skew symmetric 3 3 matrix Q k i corresponds to 0 (q k i ) z (q k i ) y q k i c = Q k i c = (q k i ) z 0 (q k i ) x c, (q k i ) y (q k i ) x 0 where (q k i ) ξ denotes the ξ-th coordinate of the point q k i with ξ {x, y, z}. The solution of (24) gives us (c i, c i ) for i = 1,..., N 1. In every iteration step we apply to each intermediate position Φ i i+1 the helical motion that is uniquely determined by the velocity vector field represented by the pair (c i, c i ). This means that we apply a rotation through an angle of α i = arctan( c i ) and a translation parallel to the axis (a i, ā i ) by the distance α i p i. For the computation of a i, ā i, p i from (c i, c i ) see Eq. (15). This transformation brings the points q 1 i,..., q K i to positions q 1 i,..., q K i that are close to the tips q 1 i + vi 1,..., q K i + vi K of the velocity vectors used in Eq. (23). 15

17 PSfrag replacements q k(final) i Φ k(final) 2 3 q k(start) i q k(final) i q k(start) i Φ 2 Φ (final) 1 2 Φ (start) 1 2 Φ (start) 2 3 Sfrag replacements Φ k(final) 2 3 Φ 3 Φ (final) 2 3 Φ 1 (a) (b) Fig. 7. One subdivision step in variational motion design: (a) Position insertion ( ), iterative position optimization ( ) and final position ( ) of the feature points. (b) The dashed line connects the barycenters ( ) in their final positions. Then we use the new positions q k i of the feature points as the starting positions for the next iteration step in the iterative optimization procedure. As a termination criterion for the iteration we use the sum of squared distances between the positions of the feature points before and after the repositioning, S := i,k q k i q k i 2. (29) If this value S falls below a certain threshold, we terminate the optimization and insert the last obtained positions into the sequence of the fixed positions of the moving body. This new sequence of fixed positions is the input for the next subdivision step in our motion design algorithm. Remark 8 Similarly to minimizing second forward differences in the motion design algorithm one generalizes non-uniform and approximating variational subdivision for motion design purposes. It is also possible to insert in each iteration step more than one intermediate position between known positions. 4.3 Higher-dimensional viewpoint and dependency on feature points An affine transformation α in Euclidean three-space E 3 is given by x a + Ax, a R 3, A R 3 3, det A 0. (30) We denote the 12-dimensional space of affine transformations by A 12, and the 6-dimensional manifold of Euclidean transformations by M 6. We define a Euclidean metric in A 12 via the sum of squared distances between feature points in homologous positions in E 3. 16

18 Φ 3 Φ 3 Φ 2 Φ 2 PSfrag replacements Φ 4 PSfrag replacements Φ 4 Φ 5 Φ 5 Φ 1 Φ 1 (a) (b) Fig. 8. Three variational subdivision steps for (a) open and (b) closed motion design to five input positions Φ i. Definition 9 Let α, β be two affine maps that are applied to a Euclidean body Φ, which is represented by feature points f 1,..., f K. We define the distance d(α, β) between α and β by K d 2 (α, β) := α(f i ) β(f i ) 2. (31) Remark 10 If a moving body is represented by a finite number of feature points x 1,..., x n of unit point masses, then the coordinate matrix J of the inertia tensor is given by n J = x i x T i. (32) Proposition 11 The Euclidean metric in A 12, defined via the distance d(α, β), only depends on the barycenter and on the inertia tensor of the rigid body Φ. PROOF. We rewrite d 2 (α, β) from (31) using the definition (30) of an affine map, K d 2 (α, β) = α(f i ) β(f i ) 2 K = (a + Af i b Bf i ) T (a + Af i b Bf i ) K := (d + Df i ) T (d + Df i ) K = (d T d + 2d T Df i + f it D T Df i ) 17

19 K K = Kd T d + 2d T D f i + f it Qf i (33) where d := a b, D := A B and Q := D T D. The second term in (33) contains the barycenter of the point cloud f 1,..., f K and the last term contains the components i f i jf i k of the inertia tensor described by the matrix J. Note that (33) can be rewritten as d 2 (α, β) = (â ˆb) T M(â ˆb) (34) with a positive definite matrix M. The vectors â and ˆb collect the twelve parameters (a, A) and (b, B) of the affine transformations α and β. Thus, the Euclidean metric we have defined is the same for all choices of feature points with the same barycenter s := 1/N N f i and the same inertia tensor. By a well-known result from mechanics we can replace the feature points f 1,..., f K by the six special points ± f i λi := s ± 2 e i, i = 1, 2, 3, (35) where λ 1, λ 2, λ 3 and e 1, e 2, e 3 are the eigenvalues and eigenvectors of J. PSfrag replacements F i M 6 Q i 1 Q i 1 Q i Q i Q i 1 Q i A 12 F i 1 F i+1 Fig. 9. The motion design algorithm sketched in A 12. We compare the motion design algorithm presented in this paper to a previously discussed algorithm: The algorithm for the design of a rigid body motion presented by Hofer et al. (2002b) first computes an affine motion α(t) (i.e., via a variational approach a curve in A 12 ) that interpolates M 6 in given points F i. Then α(t) is projected orthogonally onto M 6. This orthogonality is understood in the Euclidean metric defined in (34). 18

20 The approach we present in this paper designs via variational subdivision a curve on M 6 : Between given points F i M 6 we insert initial points Q i M 6 to which we attach unknown vectors V i in the 6-dimensional tangent space of M 6 at Q i (parameterized with help of (c i, c i )). Then we minimize a quadratic energy functional for the points..., F i, Q i := Q i + V i, F i+1,.... Since the such computed new positions Q i are in A 12 but not on M 6 we replace them by positions Q i on M 6 close to them. This is done by using the helical motion with the velocity vector field (c i, c i ). We iterate this two step procedure until the quadratic energy functional is minimized. Then we continue with the next subdivision step. Remark 12 Note that our algorithm for motion design is a special case of variational curve design (via subdivision) on arbitrary k-dimensional surfaces in R d. Another example follows in the next section, where a 5-dimensional surface contained in M 6 arises from a contact condition on the designed motion. 4.4 Smoothness analysis The smoothness analysis is so far only done numerically for the paths of the feature points. The variational interpolatory subdivision scheme we use in our approach produces at least C 2 -curves, as proved by Kobbelt (1996). However, we have slightly modified this scheme and do apply it to several input polygons simultaneously. Furthermore, we do not reposition the feature points by the velocity vectors used in the computation which would lead to affinely distorted copies of our moving body but with the underlying helical motion. Therefore it is more difficult to theoretically analyze the smoothness of the resulting motion. However, numerical tests suggest that the resulting paths of the feature points are C 2 (discontinuities first occur in the third numerical derivative), and hence we claim, that the computed motions are C 2 as well. 5 Gliding motion design In this section we present an algorithm that solves the following problem: Given N positions Φ i of a moving surface Φ (e.g. a toroidal cutter) in contact with a fixed surface Ψ, i.e., the surface normals of both surfaces coincide in the contact points, compute a motion such that Φ glides on Ψ and interpolates (or approximates) the given positions Φ i. The gliding motion of a surface Φ on a surface Ψ is a five-parameter motion. At every instance, the gliding motion of two surfaces has an instantaneous path normal complex which contains the common normal of the gliding surface pair. Up to first order this is the only constraint on a gliding surface pair, see e.g. Pottmann and Ravani (2000). Thus we extend the subdivision algorithm for motion design presented in Sect. 4 to the design of gliding motions by adding this gliding constraint. 19

21 Φ 2 Φ 1 PSfrag replacements Φ 3 Ψ Φ 4 Fig. 10. The fixed surface Ψ and the input positions Φ i of the moving surface Φ in contact with Ψ. The instantaneous motions used in the optimization procedure (23) are described by pairs (c i, c i ). The common normal n i of the two surfaces Ψ and Φ i in the instantaneous contact point has normalized Plücker coordinates (n i, n i ). The constraint, that n i is contained in the instantaneous path normal complex, follows from Eq. (17) and thus reads as c i n i + c i n i = 0, i = 1,..., N 1. (36) These linear constraints are added to the quadratic objective function (23) and the such obtained energy functional is again minimized. The optimization is again based on a feature point representation of the moving surface Φ, see Sect Before we discuss the individual steps of the algorithm in more detail we give an outline of the subdivision algorithm for gliding motion design: (1) Position insertion: Insert initial intermediate positions of the moving surface in contact with the fixed surface between each two given contact positions. (a) Insert with the hybrid algorithm for variational design of a surface curve new intermediate contact points on both, the fixed and the moving surface. Let new points with the same index correspond to each other. (b) Insert Euclidean copies Φ i i+1 such that the surface normals of fixed and moving surface coincide in the new contact points. (c) Apply an iterative optimization to all Φ i i+1 simultaneously by using velocity vector fields of pure rotations around the common surface normals in the new contact points. This gives us good starting positions for the following optimization with gliding constraints. (2) Position optimization with gliding constraint: (a) Minimize the quadratic energy functional (23) with the linear constraint (36) and apply the resulting helical motions to the inserted 20

22 copies Φ i i+1 to get Φ i i+1. (b) Note that after step (a) the moving surface might not be in contact with the fixed surface. Therefore we compute the common normals of Ψ and Φ i i+1 and translate each Φ i i+1 such that it is again in contact with Ψ. (c) Repeat steps (a) and (b) until the constraint objective function reaches a minimum. Then insert the final positions into the sequence of fixed positions of Φ and use these new 2N 1 copies as input to the next subdivision step. Remark 13 Note that the optimization is computed using a feature point representation of the moving surface. But once we have found the motions that perform the relocations of the feature points, we can also apply this motion to the moving surface itself (and vice versa). Now we discuss the individual steps of the algorithm in more detail. 5.1 Position insertion The input positions Φ i (i = 1,..., N) of the moving surface Φ are in contact with the fixed surface Ψ, i.e., the surface normals of Ψ and Φ i coincide. Between each two adjacent input positions Φ i and Φ i+1 we insert an initial intermediate position Φ i i+1 of the moving surface Φ in such a way that Φ i i+1 is in contact with Ψ and both surfaces have the same surface normal in the contact point. The Φ i i+1 are then the start positions for an optimization with gliding constraints. For the insertion we apply one subdivision step of the hybrid algorithm for surface curves presented in Sect We do this for both surfaces (fixed and moving) separately. In these corresponding surface points we compute the surface normal for each surface. Then we insert copies of the moving surface such that the inserted contact points of the moving surface are mapped to the inserted contact points of the fixed surface and that the surface normals of Ψ and Φ in those points coincide. The such obtained positions Φ i i+1 are not unique. For every position there is still one degree of freedom, the rotation around the common surface normal. We use this degree of freedom for every Φ i i+1 for a simultaneous optimization with a variant of the variational subdivision algorithm for motion design presented in Sect. 4. Each velocity vector field is restricted to coincide up to first order with a pure rotation around the common normal in the contact point of fixed and moving surface. This reduces the six free parameters (c i, c i ) for every moving position in Eq. (23) to only one, the rotation angle γ i := c i. Since for a pure rotation the pitch p is zero, we conclude from Eq. (15) that (c i, c i ) are of the form (c i, c i ) = c i (n i, n i ), i = 1,..., N 1. (37) The objective function (23) with the 6(N 1) unknowns c 1, c 1,..., c N 1, c N 1 21

23 now reduces to the following objective function, which is quadratic in the N 1 unknowns γ 1,..., γ N 1, where F 2 (γ 1,..., γ N 1 ) = + N 2 K k=1 N 1 f k i 2(q k i + v k i ) + f k i+1 2 (q k i + v k i ) 2f k i+1 + (q k i+1 + v k i+1) 2, (38) v k i := v(q k i ) = γ i ( n i + n i q k i ). (39) The minimization of (38) is found by solving the linear system of equations A r Γ = B r. We collect the unknowns in Γ, Γ = (γ 1,..., γ N 1 ), (40) and derive for i = 1,..., N 1 the i-th entry of the vector B r as B r (i) = B(6i 5,..., 6i 3) n i + B(6i 2,..., 6i) n i, (41) where B is given by (25). The entries on the main diagonal of the tridiagonal (N 1) (N 1) symmetric matrix A r are given by a jj = 5K n 2 j + 10( k q k j ) ( n j n j ) + 5( k q k jj ) n jj 10( k q k jj ) n jj, a ii = 6K n 2 i + 12( k q k i ) ( n i n i ) + 6( k q k ii ) n ii 12( k q k ii ) n ii, for j = 1, N 1 and i = 2,..., N 2. The elements on the upper diagonal are the real numbers a i,i+1 = K( n i n i+1 ) + ( k q k i+1) ( n i n i+1 ) + ( k q k i ) ( n i+1 n i ) + ( k q k i,i+1) n i,i+1 ( k q k i,i+1) n i+1,i ( k q k i+1,i) n i,i+1, (42) for i = 1,..., N 2. Thereby we denote the entries of a vector by q k i = (q1i, k q2i, k q3i) k T, n i = (n 1i, n 2i, n 3i ) T. (43) The superscripts,, denote special vectors, here defined in terms of n, n 1i n 1j n 1i n 3j n 2i n 2j + n 3i n 3j n ij := n 2i n 2j, n ij := n 2i n 1j, n ij := n 1i n 1j + n 3i n 3j. (44) n 3i n 3j n 3i n 2j n 1i n 1j + n 2i n 2j 22

24 Then we apply the underlying rotations through the computed angles γ i and iterate the procedure. This gives the initial intermediate positions Φ (start) i i+1 (i = 1,..., N 1) that are used as starting values for the constraint optimization described in Sect Position optimization with gliding constraint In this step of the algorithm we use the above computed positions of the moving surface as starting values for an iterative optimization algorithm. As in Sect. 4 we denote the feature points of Φ i i+1 by q k i and attach vectors v k i := v(q k i ) = c i + c i q k i, (45) belonging to the linear velocity vector field of an instantaneous helical motion described by the pair (c i, c i ), see Sect The velocity vectors are used for first order estimates of the new positions. The gliding motion of two surfaces has an instantaneous path normal complex which contains the common normal of the gliding surface pairs Ψ and Φ i i+1. Up to first order this is the only gliding constraint c i n i + c i n i = 0, i = 1,..., N 1. (46) Thereby, (c i, c i ) are the yet unknown representations of the velocity vector fields and (n i, n i ) are the Plücker coordinates of the contact normals in the initial intermediate positions. This gives N 1 linear constraints, one for each inserted position of the moving body. Using (46) and n = 1 one can eliminate 3(N 1) unknowns, since c i = (c i n i )n i, i = 1,..., N 1. (47) Substitution of (47) in the objective function (23) results in solving a quadratic objective function for the remaining 3(N 1) unknowns c i. However, we introduce Lagrange multipliers λ 1,..., λ N 1, since this allows us to describe the linear system of equations, that has to be solved, in terms of that of Sect The Lagrange function L consists of the quadratic objective function F 2 from (23) and λ i times the linear constraints (46), L(c 1, c 1,..., c N 1, c N 1, λ 1,..., λ N 1 ) = F 2 + N 1 λ i (c i n i + c i n i ).(48) Computing the partial derivatives of L with respect to C i := (c i, c i ) and λ i, L C i = 0, L λ i = 0, i = 1,..., N 1, (49) leads to a system of 7(N 1) linear equations A c C c = B c. (50) 23

25 We find B c, C c, A c by extending B, C, A from Eq. (25), (26), (27) respectively in the following way. The vector B c is equal to the vector B extended by (N 1) zeros, and the vector of unknowns C c is derived from C by adding the N 1 Lagrange multipliers λ i at the end, B c = (B T, 0,..., 0), }{{} C c = ( ) C T T, λ 1,..., λ N 1. (51) N 1 The symmetric matrix A c is a block matrix with A from Eq. (27) as an 6(N 1) 6(N 1) block in the upper left corner, the 6(N 1) (N 1) block matrix N in the upper right corner, its transpose N T and the (N 1) (N 1) zero matrix 0 in the lower two blocks, n T 1 n T 1 A c A N = with N T = (52) N T 0 n T N 1 n T N 1 Once we have solved (50) we use the helical motions corresponding to (c i, c i ), see Sect. 3, to separately reposition each intermediate position Φ i i+1 of the moving surface and of the feature points q k i. Note that after we have applied the helical motions, the resulting positions Φ i i+1 might not be in contact with Ψ anymore. Therefore we have to perform the following position correction, which we just briefly outline. 5.3 Position correction Compute the common normal (see e.g. Sohn et at. (2002) and the references therein) of Ψ and Φ i i+1 which intersects the two surfaces in points p i i+1,ψ and p i i+1,φ. Then translate Φ i i+1 by the vector t i i+1 := p i i+1,ψ p i i+1,φ. (53) Now the fixed and the copies of the moving surface are again in contact and we continue with the next subdivision step of the gliding motion design algorithm. Fig. 11 shows the inserted copies of the moving surface Φ after the first to fourth subdivision step. Fig. 12 shows the contact curves on the fixed surface Ψ and on the moving surface Φ. Remark 14 We have presented the subdivision algorithms for smooth and gliding motion design such that the computed motion interpolates the given positions. With the same approach, using a generalization of the approximating variational subdivision presented in Sect. 2.2, we can design smooth and gliding motions that approximate given input positions. 24

26 Φ2 Φ2 Φ1 Φ1 Φ3 PSfrag replacements PSfrag replacements Ψ Φ3 Ψ Φ4 Φ4 Φ2 Φ2 Φ1 Φ1 Φ3 PSfrag replacements PSfrag replacements Ψ Φ3 Ψ Φ4 Φ4 Fig. 11. The positions of the moving surface after the first to fourth subdivision step of the gliding motion design algorithm. PSfrag replacements PSfrag replacements Ψ Φ Ψ Φ Fig. 12. The contact curve on fixed and moving surface. 6 Conclusions The main contribution of this paper is a subdivision algorithm for geometric design of motions constraint by a contacting surface pair. The presented algorithm uses instantaneous kinematics, line geometry, and variational subdivision algorithms. The results can be extended to the design of smooth motions in the presence of obstacles. There are many algorithms to design smooth motions based on techniques from Computer Aided Geometric Design, and a variety of solutions for the avoidance of obstacles based on Computational Geometry. Whereas the latter approaches do not deal with spline-type smooth motions, the former are lacking an efficient treatment of obstacles. Using the 25