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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Real-Time B´zier Trajectory Deformation for e Potential Fields Planning Methods L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ e o September 25-30, 2011 San Francisco International Conference on Intelligent Robots and SystemsL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 1/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e 1 Introduction. 2 B´zier Trajectory Deformation (BTD) in Mobile Robots and e Obstacles. 3 Simulations Results. 4 Conclusions and Future Works.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 2/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e B´zier Trajectory Deformation e The objective of this work: A New Technique for obtaining a Flexible Trajectory Free of Collisions based on the Deformation of a B´zier curve through a e Field of Vectors. BTD This technique is called B´zier Trajectory Deformation (BTD) . eL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 3/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Trajectory Generation Problem 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. It is a useful property for the Trajectory Generation Problem in Mobile Robots. 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. Our algorithm BTD 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´ — BTD for Potential Fields Planning Methods e o 4/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e 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´ — BTD for Potential Fields Planning Methods e o 5/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e 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´ — BTD for Potential Fields Planning Methods e o 6/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Trajectory Generation+Collision Avoidance 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. This technique 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. We called: B´zier Trajectory Deformation, BTD. e 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´ — BTD for Potential Fields Planning Methods e o 7/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Deﬁnitions Deﬁnition A B´zier Curve is deﬁned as, e n α(u) = Pi · Bi,n (u) (1) i=0 n is the Order of the B´zier curve. e n Bi,n (u) = i u i (1 − u)n−i Bernstein Basis u ∈ [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´ — BTD for Potential Fields Planning Methods e o 8/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Deﬁnitions The intrinsic parameter, u, is non-dimensional. In order to use a B´zier curve as a trajectory, this parameter must be e redeﬁned as a time variable, associating each curve position (robot position) with a time instant t ∈ [t0 , tf ], where t0 and tf are the initial and ﬁnal trajectory instants. The deﬁnition of the B´zier curve has to change: e Deﬁnition n α(t) = Pi · Bi,n (t); t ∈ [t0 , tf ] (2) i=0 In this case, the deﬁnition of the Bernstein Basis is: −t Bi,n (t) = n ( tt−t00 )i ( ttff−t0 )n−i such that i = 0, 1, · · · , n i f −tL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 9/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Deﬁnitions Deﬁnition A Modiﬁed B´zier curve is deﬁned as, e n Sε (α(t)) := (Pi + εi ) · Bi,n (t); t ∈ [t0 , tf ] (3) i=0 To deform a given B´zier curve describing a Trajectory, the control e 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´ — BTD for Potential Fields Planning Methods e o 10/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e 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, tf 2 α(t) − Sε (α(t)) 2 dt (4) t0L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 11/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Optimization Function Disadvantage A B´zier curve is numerically unstable if the B´zier curve has a large e e number of control points. In Mobile Robots, it is necessary to concatenate some B´zier e curves to obtain the complete trajectory. So the optimization function is redeﬁned. Deﬁnition The optimization function using k-B´zier curves is deﬁned as, e k (l) tf 2 g := αl (t) − Sε (αl (t)) 2 dt (5) (l) l=1 t0L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 12/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e The set of Constraints First Constraint The robot is guided to non-collision positions. For that reason, The Modiﬁed B´zier, Sε (αi (t)), passes through the Target Point, Ti . The e vectors joining the Start point and the Target Point are the ﬁeld of Forces computed through the PFP. 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´ — BTD for Potential Fields Planning Methods e o 13/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e The set of Constraints Second Constraint The Trajectory of the robot must be a smooth Trajectory. So, 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´ — BTD for Potential Fields Planning Methods e o 14/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e The set of Constraints Third Constraint The continuity between the Present position and the predicted Future position is ensured. 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´ — BTD for Potential Fields Planning Methods e o 15/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e 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´ — BTD for Potential Fields Planning Methods e o 16/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e 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 Advantages The Computational Cost is reduced if the order of the B´zier curve is maintained e invariable, because the matrix can be computed in advanced. It is possible to include more 8 B´zier curves are used.The e constraints to improve the modiﬁed trajectory is computed in algorithm. 0.23 ms in a Pentium IV 2.4 GhzL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 17/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e PFP+BTD BTD requires an algorithm that generates an Initial Trajectory and a Field of Vectors to modify it in case obstacles are detected, ensuring that the deformed trajectory is collision-free. The BTD technique is evaluated through a predictive PF path planning method, (see M.C.Mora et al., 2007-2008): the Potential Field Projection (PFP) method. Potential Field Projection This method is based on the combination of: The Classical PF (see O. Khatib, 1986) The Multi-rate Kalman Filter estimation (see J.Tornero et al., 1999 and R. Piz´, 2003)) aL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 18/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Potential Field Projection Potential Field Projection This method takes into account: Uncertainties on locations. The future trajectory of the robot and the obstacles. Multi-rate information supplied by sensors.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 19/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Potential Field Projection Predicted future positions and uncertainties are obtained from the prediction equations of the Multi-rate Kalman ﬁlter for every object in the environment. The predicted positions and their uncertainties are used in the generation of a PF that guides the robot to the goal avoiding the obstacles in the environment. These PF generate forces in every prediction instant. The set of repulsive forces are transformed into displacements. These displacements aﬀect the shape of the initial parametric trajectory. The displacements are used in BTD algorithm and the robot is guided to the goal without colliding with the obstacles.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 20/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Simulations ResultsL. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 21/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Conclusions The Robot’s Trajectory is computed with a parametric curve.(in this case B´zier curves). e The BTD algorithm has been devoloped to compute the deformation of the Trajectory through a ﬁeld of vectors. This algorithm needs a set of vectors. In this case, the ﬁeld of repulsive forces necessary to modify the Trajectory (the B´zier e curve) are obtained by PFP. The Modiﬁed Trajectory avoids the obstacles. It is the FIRST technique joining PFP with Parametric Curves.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 22/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Future Works The Improvement of the algorithm using Tensorial Notation. This structure improves the Computational Cost. To include more constraints in the algorithm, for example, the curvature. To develop this algorithm in three dimensions. Design trajectories free of collisions in 3D, for example in a UAV (Unmanned Aerial Vehicle) or in a Robot Arm.L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 23/24
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Introduction. B´zier Trajectory Deformation (BTD) in Mobile Robots and Obstacles. Simulations Results. Conclusions and Future Works. e Thank you for your attention! Questions?L. Hilario, N. Mont´s, M.C.Mora, A. Falc´ — BTD for Potential Fields Planning Methods e o 24/24
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