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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. A Tensor Calculus Approach for B´zier Shape e Deformation L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ e o June 18-22, 2012 Valencia SIAM Conference on Applied Linear AlgebraL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 1/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. 1 Introduction. 2 B´zier Shape Deformation (BSD) in engineering applications. e 3 Tensorial Representation of the BSD Algorithm (T-BSD). 4 Comparison BSD and T-BSD. 5 Conclusions.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 2/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Tensorial Structure of the BSD The objective of this work: The objective of this work is the reformulation of an algorithm using Tensorial Notation T-BSD This technique is called Tensor-B´zier Shape Deformation (T-BSD) . e Computational Cost One of the most important facts in engineering applications is the cost in computational time because some algorithms are applied in real-time.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 3/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Tensorial Structure of the BSD Computational cost There are an increased in numerical methods that make use of tensors. It is useful to reduce the numerical cost. (See Falc´ A. 2010, see o Hackbusch W., see Kolda T.G. et al 2009...) BSD The BSD computes the deformation of a B´zier curve through a ﬁeld of e vectors. Applications There are two applications of the BSD: Mobile Robots and Liquid Composite Moulding.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 4/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Parametric curves The parametric curves (B´zier, B-Splines, NURBS, RBC) are the e most widely used in computer graphics and geometric modelling since points on the curve are easily computed. The representation of this kind of parametric curves is a SMOOTH CURVE. Our algorithm BSD is developed with B´zier curves. They are a polynomial curves and e they possess a number of mathematical properties which facilitate their manipulation and analysis.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 5/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Problems in Mobile Robots Trajectory generation problem This problem consists in computing a feasible trajectory between a start and a goal state-time, for a given robotic system. The trajectories should be a continuous and a smooth curve. It is necessary to avoid slipping of the wheels. Collision Avoidance problem The smooth and continuous trajectory should be free of collisions. CPU time The algorithms are applied in real-time, for that reason the cost in computational time of the algorithms must be the lowest and the best. Realistic cluttered scenarios A realistic scenario is considered to be unknown, dynamic and sometimes cluttered with mobile obstacles. For that reason, the reduction of the execution time is necessary in limit situations.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 6/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Trajectory Generation Problem The smoothness of the parametric curve is a useful property for the Trajectory Generation Problem in Mobile Robots. The parametric curves represent in an appropiate manner the Trajectory of the Robot. A lot of researchers consider parametric curves in the construction of trajectories for wheeled robots, (see for example, Choi et. al, 2008-2009, Skrjanc and Klancar, 2007), etc.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 7/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Collision Avoidance Problem Collision avoidance is a fundamental problem in many areas such as robotics. An extreme situation of collision avoidance.......L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 8/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Collision Avoidance Problem The generation of the path can be properly done using reactive path planning methods adapting to environmental changes. One of the most popular reactive methods is Artiﬁcial Potential Fields(APF) (see Khatib, 1986), that is the basis of the Potential Field Projection method (PFP) (see Mora and Tornero, 2007) used in this work. APF consists in ﬁlling the robot’s workspace with an artiﬁcial potential ﬁeld in which the robot is attracted by the goal and repelled by the obstacles. APF produces a ﬁeld of vectors that guides the robot to non-collision positions.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 9/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Trajectory G.+ Collision A. Design and Modify a Parametric Curve is an important research issue, (Wu et al.2005, Xu et al. 2002) One of these techniques (Wu et al., 2005) has been adapted for its use in path planning for Holonomic Robots. BSD modiﬁes the parametric curve through a ﬁeld of vectors. The shape of the B´zier curve is modiﬁed. e The changes of the shape are minimized from the original one. These vectors are computed with PFP. The Repulsive Forces will modify the Original Trajectory to avoid every obstacle. The First Technique joining: Trajectory Generation using Parametric Curves Avoiding the Obstacles using Potential Field methodsL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 10/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Liquid Composite Moulding (LCM) The geometric line between the dry and the wet area of the preform is deﬁned as FLOW FRONT. The ﬂow front advance computation is used in Liquid Composite Moulding (LCM) simulation because is a common tool to compute the control actions in advanced composite manufacturing during ﬁlling to take decision on-line.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 11/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Liquid Composite Moulding Finite Element Method (FEM) techniques are used to compute the ﬂow front’s representation. The result is a discrete set of points (nodes). However, the resin’s ﬂow front is a continuous smooth curve. A continuous ﬂow front is proposed using parametric curves, in this case B´zier curve. eL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 12/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Liquid Composite Moulding The ﬂow front is updated with the BSD algorithm because the ﬂow front is modiﬁed through a ﬁeld of vectors, in this application, velocity vectors. These velocity vectors are obtained throughout the Darcy’s Law applying Finite Element Methods Simulation. Darcy’s Law k v=− P (1) µL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 13/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Deﬁnitions Deﬁnition A B´zier Curve is deﬁned as, e n α(t) = Pi · Bi,n (t) (2) i=0 n is the Order of the B´zier curve. e n Bi,n (t) = i t i (1 − t)n−i Bernstein Basis t ∈ [0, 1] is the Intrinsic Parameter. (n + 1) Control Points, Pi such that i = 0, 1, · · · , n.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 14/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Deﬁnitions Deﬁnition A Modiﬁed B´zier curve is deﬁned as, e n Sε (α(t)) := (Pi + εi ) · Bi,n (t); t ∈ [0, 1] (3) i=0 To deform a given B´zier curve describing a Trajectory or a Flow e Front, the control points must be changed and the perturbation, εi , of every control point must be computed.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 15/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Constrained optimization problem. This problem is solved deﬁning a constrained optimization problem. It is solved with the Lagrange Multipliers Theorem. The optimization function minimizes the distance between the orginal B´zier curve, α(t), and the modiﬁed B´zier curve, Sε (α(t)). e e Thus, this function minimizes the changes of the shape.(Wu et al.2005) Deﬁnition The optimization function is deﬁned as, 1 2 α(t) − Sε (α(t)) 2 dt (4) 0L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 16/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Optimization Function Disadvantage A B´zier curve is numerically unstable if the B´zier curve has a large e e number of control points. It is necessary to concatenate some B´zier curves to obtain the e complete trajectory or the complete ﬂow front. So the optimization function is redeﬁned. Deﬁnition The optimization function using k-B´zier curves is deﬁned as, e k 1 2 g := αl (t) − Sε (αl (t)) 2 dt (5) l=1 0L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 17/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. The set of Constraints First Constraint The Modiﬁed B´zier, Sε (αi (t)), passes through the Target Point, Ti . e Mathematical Formulation k rl (l) (l) r1 = λ, Tj − Sε (αl (tj )) (6) l=1 j=1L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 18/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. The set of Constraints First Constraint Mobile Robots In Mobile Robots, this constraint means that the robot is guided to non-collision positions. The vectors joining the Start Point and the Target Point are the ﬁeld of forces computed through the PFP. LCM In LCM, this constraint means that the ﬂow front is modiﬁed during ﬁlling the mould.The ﬂow front evolution is updated by the BSD. In this case, the ﬁled of vectors are the velocity vectors obtained with the Darcy’s Law.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 19/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. The set of Constraints Second Constraint Continuity and derivability is necessary to impose on the joined points of the concatenated curves. Mathematical Formulation k−1 (l) (l+1) r2 = λ, Sε (αl (tf )) − Sε (αl+1 (t0 )) (7) l=1 k−1 (l) (l+1) r3 = λ, Sε (αl (tf )) − Sε (αl+1 (t0 )) (8) l=1L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 20/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. The set of Constraints Second Constraint Mobile Robots In Mobile Robots, the Trajectory of the robot must be a smooth Trajectory, for that reason it is imposed this constraint. LCM In LCM, the actual resin’s ﬂow front is a continuous smooth curve, so it is necessary this restriction.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 21/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. The set of Constraints Third Constraint Therefore, derivative constraints on the start and end points of the resulting concatenated curves are imposed. Mathematical Formulation (1) (1) (k) (k) r4 = λ, α1 (t0 ) − Sε (α1 (t0 )) + λ, αk (tf ) − Sε (αk (tf )) (9)L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 22/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. The set of Constraints Third Constraint Mobile Robots In Mobile Robots, this constraint is necessary because the continuity between the Present position and the predicted Future position is ensured. LCM In LCM, this constraint is useful to maintain the derivative property of the curve.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 23/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. The Lagrange Multipliers Lagrange Multipliers The Lagrange Multipliers Theorem has been applied to solve the constrained optimization problem. The idea is to minimize the function deﬁned in 5 including the set of constraints deﬁned below. Lagrange Function L(ε(1) , · · · , ε(k) , λ) = g + r1 + r2 + r3 + r4 (10) The solution of the problem In order to obtain the Minimum of this convex function, we only to compute the stationary point of the Lagrangian derivative. ∂L = 0; (l) = 1, · · · , k (11) ∂ε(l) ∂L =0 (12) ∂λL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 24/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. The solution A square linear system of equations is obtained: A · X = b. It is solvable and the solution X = (ε, λ) computes the perturbation of every control point. Example Example Mobile Robots LCM !L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 25/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Tensorial Notation The BSD model has been reformulated using Tensorial Structure (T-BSD) reducing the critical point in engineering applications: The cost computational time to get a suitable Real-Time control. We introduce some of the notation used in this presentation. Deﬁnition The Kronecker Product of A ∈ Rn1 ×n1 and B ∈ Rn2 ×n2 , written A ⊗ B, is the tensor algebraic operation deﬁned as a11 B a12 B · · · a1n1 B a21 B a22 B · · · a2n B ∈ Rn1 n2 ×n1 n2 . 1 A⊗B = . . . . .. . . . . . . an1 1 B an1 2 B · · · an1 n1 BL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 26/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Tensorial Notation Let A = [A1 · · · An ] be an m × n matrix where Aj is its j-th column vector. Then vec A is the mn × 1 vector A1 vec A = . . . . An Thus the vec operator transform a matrix into a vector by stacking the columns of the matrix one underneath the other.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 27/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Mathematical Formulation Deﬁnition The deﬁnition of the B´zier curve, (1), it is written in equivalent matrix e form, αn (u) = Pn (t) Bn (u); u ∈ [0, 1] t (13) where Pn (t) = P0 (t) · · · n Pn (t) n ∈ R2×(n+1) (14) T Bn (u) = B0,n (u) · · · Bn,n (u) ∈ R(n+1)×1 . (15) Deﬁnition Its standard euclidean norm is deﬁned as αn (u) t 2 2 = (Pn (t) Bn (u))T Pn (t) Bn (u) (16)L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 28/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Mathematical Formulation Deﬁnition For each ﬁxed t, the energy of the u-parametrized curve αn in t L2 ([0, 1], R2 ) is given by, 1 1/2 1 1/2 αn t ∆2 = αn (u) 2 du t 2 = (Pn (t)Bn (u))T Pn (t)Bn (u)du . 0 0 (17) We consider a ﬁnite set of Target Points T0 , . . . , Tr ⊂ D, a r r connected and compact set in R2 . We move from an initial B´zier curve, denoted by αn and e t characterized by the set of its control points Pn (t), to a curve, denoted by αn t+∆t by means a set, of perturbations for each control point, namely Xn = X0 n ··· Xn n ∈ R2×(n+1) . (18)L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 29/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Mathematical Formulation Deﬁnition The resultant B´zier curve αn e t+∆t is given by, αn t+∆t (u) = (Pn (t) + Xn ) Bn (u); u ∈ [0, 1]. (19) To compute Xn : We minimize the energy used by the curve to move from αn to t αn . t+∆t Moreover, this transformed curve passes through the target points for a given set 0 = u1 < u2 < · · · < urr −1 < urr = 1, of r r parameter values. Optimization Problem solved with Lagrange Multipliers Theorem min αn n t+∆t − αt 2 ∆2 (20) s. t. αn r j t+∆t (uj ) = Tr for 1 ≤ j ≤ r and r ≤ n − 1.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 30/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Mathematical Formulation We write (20) in equivalent matrix form as follows. Let Tr = T1 r ··· Tr r ∈ R2×r . (21) and r Bn = r Bn (u1 ) · · · r Bn (ur ) ∈ R(n+1)×r (22) Finally, we consider the matrix function 1 Φn (Xn ) = Bn (u)T Xn Xn Bn (u) du, T (23) 0 then (20) can be written in matrix form as: Matrix Form of the Optimization Problem minXn ∈R2×(n+1) Φn (Xn ) (24) r s. t. (Pn (t) + Xn )Bn = TrL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 31/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Mathematical Formulation If we take the vec operator and we reformulate the constraint with the Kronecker Product in (24) we obtain the equivalent minimization program, Matrix Form of the Optimization Problem with the vec operator min(vec Xn )∈R2·(n+1)×1 Φn (vec Xn ) (25) r s. t. ((Bn )T ⊗ I2 ) vec Xn = vec Tr − vec (Pn (t)Bn ) r We note that the set of constrains of the problem (25) is linear, in consequence the map Φn is deﬁned over a convex set. Thus, by proving the convexity of Φn , each critical point of (25) will give us an absolute minimum.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 32/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. A Concatenate T-BSD Now, we consider that the curve αt in now described by a ﬁnite set of concatenate B´zier curves αn1 , . . . , αnk of degrees n1 , . . . , nk , e t t respectively. Thus, it is necessary to include the constraints explained before, in the slide 18. Optimization Problem Concatenating k B´zier Curves e We would to compute Xni ∈ R2×(ni +1) for 1 ≤ i ≤ k satisfying k min(Xn1 ,...,Xnk ) Φ(Xn1 , . . . , Xnk ) = i=1 Φni (Xni ) r s. t. (Pni (t) + Xni )Bnii = Tri 1 ≤ i ≤ k ni X ni = X0i+1 , 1 ≤ i ≤ k − 1 n ni (Xnii − Xnii −1 ) n n = ni+1 (X1i+1 − X0i+1 ), 1 ≤ i ≤ k − 1 n n n1 (X11 − X01 ) n n =0 nk (Xnk − Xnk −1 ) nk n k =0 (26)L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 33/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Now, we would to write (26) with a more compact notation. To this end we use the following four block matrices. For 1 ≤ i ≤ k we deﬁne Rni = 0 · · · 0 I2 ∈ R2×2(ni +1) , (27) ∗ Rni = 0 ··· 0 −I2 I2 ∈ R2×2(ni +1) , (28) Lni = I2 0 ··· 0 ∈ R2×2(ni +1) (29) and L∗i = n −I2 I2 0 ··· 0 ∈ R2×2(ni +1) . (30)L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 34/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Lagrangian Function The Lagrangian function associated to (26) can be written as follows, k L = i=1 Φni (vec Xni ) k ri T − i=1 (λi ) ((Bnii )T ⊗ I2 )vec Xni − vec Tri + vec (Pni (t)Bnii ) r r k−1 − i=1 µT [Rni vec Xni − Lni+1 vec Xni+1 ] i −µT n1 L∗1 vec Xn1 k n ∗ −µT nk Rnk vec Xnk k+1 k−1 ∗ ∗ − i=1 µT i+1+k ni Rni vecXni − ni+1 Lni+1 vec Xni+1 (31)L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 35/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. ∂L Making zero the partials, ∂ (vec Xn )T = 0; 1 ≤ i ≤ k, of the i Lagrangian Functions is obtained a linear system equation deﬁned as follow, Linear System using Tensorial Structure The linear system, Az = f (32) The A matrix is deﬁned as follows, A ∈ Rp×p (33) k k k p=2 (ni + 1) + 2 ri + 2(k − 1) + 4 + 2(k − 1) = 2 (ni + ri ) + 6k. i=1 i=1 i=1 (34)L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 36/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Solution of the T-BSD The solution of this system is the follow vector, vec Xn1 . . . vec Xnk λr 1 1 . z= . ∈ Rp×1 . (35) . rk λ k µ 1 . . . µ2k The solution computes the perturbation, vecXni , of every control point.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 37/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Comparison A picture is worth a thousand words If the number of the B´zier curves is increased, BSD grows exponentially, e whereas T-BSD grows linearly.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 38/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Conclusions With T-BSD the reduction of the computational cost of the BSD algorithm is achieved.Tensorial Algebra reduces drastically the cost computational time to apply the BSD model in Real-Time. (see Ammar et al., 2009, Falc´ A., 2010, Kolda et al., 2009) o The BSD algorithm has been devoloped to compute the deformation of a parametric curve through a ﬁeld of vectors. This algorithm needs a set of vectors. In Mobile Robots, the ﬁeld of forces necessary to modify the B´zier e curve are obtained by PFP. It is the FIRST technique joining PFP with the Parametric Curves. In LCM, the ﬁeld of velocity vectors are obtained by Darcy’s Law. It is the FIRST time that the ﬂow front is represented with a continuous curve and it is updated with the velocity vectors.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 39/40
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Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Thank you for your attention! Questions?L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — A Tensor Calculus Approach for B´zier Shape Deformation e o e 40/40
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