1. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. A Tensor Calculus Approach for Bzier Shape e Deformation L. Hilario, N. Monts, M.C.Mora, A. Falceo June 18-22, 2012 Valencia SIAM Conference on Applied Linear AlgebraL. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e1/40

2. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. 1 Introduction. 2 Bzier 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. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e2/40 3. 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-Bzier 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. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e3/40 4. 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 Bzier curve through a eld ofe vectors. Applications There are two applications of the BSD: Mobile Robots and Liquid Composite Moulding.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e4/40 5. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Parametric curves The parametric curves (Bzier, 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 Bzier curves. They are a polynomial curves and e they possess a number of mathematical properties which facilitate their manipulation and analysis.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e5/40 6. 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. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e6/40 7. 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. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e7/40 8. 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. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e8/40 9. 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 Articial 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 lling the robots workspacewith an articial potential eld in which therobot is attracted by the goal and repelledby the obstacles.APF produces a eld of vectors that guidesthe robot to non-collision positions.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e9/40 10. 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 modies the parametric curve through a eld of vectors.The shape of the Bzier curve is modied.eThe 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. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e10/40 11. 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 dened as FLOW FRONT. The ow 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 lling to take decision on-line.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e11/40 12. 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 ow fronts representation. The result is a discrete set of points (nodes). However, the resins ow front is a continuous smooth curve. A continuous ow front is proposed using parametric curves, in this case Bzier curve. eL. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e12/40 13. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Liquid Composite Moulding The ow front is updated with the BSD algorithm because the ow front is modied through a eld of vectors, in this application, velocity vectors. These velocity vectors are obtained throughout the Darcys Law applying Finite Element Methods Simulation. Darcys Lawk v=P(1)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e13/40 14. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Denitions Denition A Bzier Curve is dened as,en(t) =Pi Bi,n (t)(2)i=0 n is the Order of the Bzier curve.e n Bi,n (t) =i t i (1 t)ni Bernstein Basis t [0, 1] is the Intrinsic Parameter. (n + 1) Control Points, Pi such that i = 0, 1, , n.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e14/40 15. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Denitions Denition A Modied Bzier curve is dened as,en S ((t)) := (Pi + i ) Bi,n (t); t [0, 1](3)i=0 To deform a given Bzier curve describing a Trajectory or a Flowe Front, the control points must be changed and the perturbation, i , of every control point must be computed.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e15/40 16. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Constrained optimization problem. This problem is solved dening a constrained optimization problem. It is solved with the Lagrange Multipliers Theorem. The optimization function minimizes the distance between the orginal Bzier curve, (t), and the modied Bzier curve, S ((t)).ee Thus, this function minimizes the changes of the shape.(Wu et al.2005) Denition The optimization function is dened as, 1 2(t) S ((t)) 2 dt (4) 0L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e16/40 17. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Optimization Function Disadvantage A Bzier curve is numerically unstable if the Bzier curve has a largee e number of control points. It is necessary to concatenate some Bzier curves to obtain the e complete trajectory or the complete ow front. So the optimization function is redened. Denition The optimization function using k-Bzier curves is dened as,e k1 2 g := l (t) S (l (t)) 2dt (5) l=10L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e17/40 18. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. The set of Constraints First Constraint The Modied Bzier, S (i (t)), passes through the Target Point, Ti .e Mathematical Formulation krl(l) (l)r1 = , Tj S (l (tj ))(6) l=1 j=1L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e18/40 19. 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 eld of forces computed through the PFP. LCM In LCM, this constraint means that the ow front is modied during lling the mould.The ow front evolution is updated by the BSD. In this case, the led of vectors are the velocity vectors obtained with the Darcys Law.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e19/40 20. 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 Formulationk1(l) (l+1) r2 =, S (l (tf )) S (l+1 (t0)) (7)l=1k1(l) (l+1) r3 =, S (l (tf )) S (l+1 (t0)) (8)l=1L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e20/40 21. 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 resins ow front is a continuous smooth curve, so it is necessary this restriction.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e21/40 22. 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 = , 1 (t0 ) S (1 (t0 )) + , k (tf ) S (k (tf )) (9)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e22/40 23. 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. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e23/40 24. 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 dened in 5 including the set of constraints dened below. Lagrange FunctionL((1) , , (k) , ) = 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)(l) L=0 (12) L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 24/40 25. 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 = (, ) computes the perturbation of every control point. ExampleExample Mobile RobotsLCM !L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e25/40 26. 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. Denition The Kronecker Product of A Rn1 n1 and B Rn2 n2 , written A B, is the tensor algebraic operation dened as a11 B a12 B a1n1 B a21 B a22 B a2n B Rn1 n2 n1 n2 .1AB = .. ..... . . .. . an1 1 B an1 2 B an1 n1 BL. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e26/40 27. 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. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e27/40 28. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Mathematical Formulation Denition The denition of the Bzier curve, (1), it is written in equivalent matrix e form,n (u) = Pn (t) Bn (u); u [0, 1] t (13) where Pn (t) = P0 (t) nPn (t) n R2(n+1)(14) T Bn (u) =B0,n (u) Bn,n (u) R(n+1)1 .(15) Denition Its standard euclidean norm is dened asn (u) t 2 2 = (Pn (t) Bn (u))T Pn (t) Bn (u)(16)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 28/40 29. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Mathematical Formulation Denition For each xed t, the energy of the u-parametrized curve n int L2 ([0, 1], R2 ) is given by, 11/2 11/2 nt 2 = n (u) 2 dut 2 = (Pn (t)Bn (u))T Pn (t)Bn (u)du . 00 (17) We consider a nite set of Target Points T0 , . . . , Tr D, a rr connected and compact set in R2 . We move from an initial Bzier curve, denoted by n andet characterized by the set of its control points Pn (t), to a curve, denoted by n t+t by means a set, of perturbations for each control point, namelyXn =X0 n Xnn R2(n+1) . (18)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e29/40 30. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. Mathematical Formulation Denition The resultant Bzier curve net+t is given by,n 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 n totn .t+tMoreover, this transformed curve passes through the targetpoints for a given set 0 = u1 < u2 < < urr 1 < urr = 1, ofrrparameter values. Optimization Problem solved with Lagrange Multipliers Theorem min n nt+t t22 (20) s. t. nrjt+t (uj ) = Tr for 1 j r and r n 1.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 30/40 31. 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 =T1r Tr r R2r . (21) andr Bn = r Bn (u1 ) rBn (ur ) R(n+1)r(22) Finally, we consider the matrix function 1n (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) n (Xn ) (24)r s. t. (Pn (t) + Xn )Bn = TrL. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 31/40 32. 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 operatormin(vec Xn )R2(n+1)1 n (vec Xn ) (25) rs. 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 n is dened over a convex set. Thus, by proving the convexity of n , each critical point of (25) will give us an absolute minimum.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 32/40 33. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions. A Concatenate T-BSD Now, we consider that the curve t in now described by a nite set of concatenate Bzier curves n1 , . . . , nk of degrees n1 , . . . , nk , ett respectively. Thus, it is necessary to include the constraints explained before, in the slide 18. Optimization Problem Concatenating k Bzier Curves e We would to compute Xni R2(ni +1) for 1 i k satisfyingk min(Xn1 ,...,Xnk ) (Xn1 , . . . , Xnk ) = i=1 ni (Xni ) rs. t.(Pni (t) + Xni )Bnii = Tri 1 i k ni X ni = X0i+1 , 1 i k 1nni (Xnii Xnii 1 )nn= ni+1 (X1i+1 X0i+1 ), 1 i k 1n nn1 (X11 X01 ) n n=0 nk (Xnk Xnk 1 ) nk n k=0 (26)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 33/40 34. 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 deneRni = 0 0 I2 R22(ni +1) , (27) Rni =0 0 I2I2 R22(ni +1) ,(28)Lni =I2 0 0 R22(ni +1)(29) andLi = n I2I2 0 0 R22(ni +1) .(30)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 34/40 35. 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 ni (vec Xni )k ri T i=1 (i )((Bnii )T I2 )vec Xni vec Tri + vec (Pni (t)Bnii )r rk1 i=1T [Rni vec Xni Lni+1 vec Xni+1 ]i T n1 L1 vec Xn1 k n T nk Rnk vec Xnk k+1k1 i=1Ti+1+k ni Rni vecXni ni+1 Lni+1 vec Xni+1 (31)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 35/40 36. 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 thei Lagrangian Functions is obtained a linear system equation dened as follow, Linear System using Tensorial Structure The linear system, Az = f(32) The A matrix is dened as follows, A Rpp(33)k k k p=2(ni + 1) + 2 ri + 2(k 1) + 4 + 2(k 1) = 2(ni + ri ) + 6k.i=1 i=1i=1 (34)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 36/40 37. 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 r 1 1 . z=. Rp1 .(35) . rk k 1 .. . 2k The solution computes the perturbation, vecXni , of every control point.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e 37/40 38. 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 Bzier curves is increased, BSD grows exponentially, e whereas T-BSD grows linearly.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e38/40 39. 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 eld of vectors. This algorithm needs a set of vectors.In Mobile Robots, the eld of forces necessary to modify the Bzier ecurve are obtained by PFP. It is the FIRST technique joining PFPwith the Parametric Curves.In LCM, the eld of velocity vectors are obtained by Darcys Law. Itis the FIRST time that the ow front is represented with acontinuous curve and it is updated with the velocity vectors.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e39/40 40. Introduction. BSD in engineering applications. Tensorial Representation. Comparison BSD and T-BSD. Conclusions.Thank you for your attention! Questions?L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e40/40