The Application of Spline Functions and Bézier Curves to AGV Path Planning
|
|
- Shanon McBride
- 6 years ago
- Views:
Transcription
1 IEEE ISIE 2005, June 20-23, 2005, Dubrovnik, Croatia The Application of Spline Functions and Bézier Curves to AGV Path Planning K. Petrinec, Z. Kova i University of Zagreb / Faculty of Electrical Engineering and Computing, Zagreb, Croatia kresimir.petrinec@fer.hr, zdenko.kovacic@fer.hr Abstract The paper describes a path planning method for an automated guided vehicle (AGV). The method consists of two steps, the calculation of a path by using Bézier curves and the calculation of a path trajectory by using a Ho and Cook spline-interpolation algorithm. The generation of parallel path curves for left-hand and right-hand AGV wheels is elaborated. The solution for the generation of sharp curves is derived. The proposed path planning method has been tested on tricycle-type AGVs with a front stabilization wheel and two rear powered wheels. I. INTRODUCTION Shop floor layouts usually contain a number of robotic manufacturing cells and a number of automated guided vehicles (AGVs). From the operational point of view they involve many repetitive operations. For example, an AGV travels from one node to another through determined paths and executes the sequence of actions like loading, unloading, parking or docking. A path following in such an environment can be accomplished by using a pure pursuit algorithm, whose main characteristics are marked simplicity and computational efficiency [1, 2]. A pure pursuit algorithm is an on-line method originally devised as a method for calculating the arc necessary to get a mobile robot back on a path. The role of a mission controller is to control the motion of AGVs along the paths and execute related operations determined during the shop floor layout design. The control involves a search for obstacle-free paths and in the same time, it cares about prevention of AGV collisions. In such circumstances, a Ho and Cook trajectory planning that uses cubic and fourth-degree spline-functions can be used [3]. The method was originally designed for off-line trajectory planning for industrial robots. In general, off-line methods have an advantage of allowing the knot points to be specified more closely in time, thus increasing the accuracy of a trajectory and allowing a shorter sampling period [4]. All the data for an application is calculated only once, before the robot runs the application for the first time. This data is stored and retrieved by a robot controller at run time. If a robot is executing some preplanned trajectory and an external stimulus is received, the controller may simply switch to the data for another preplanned trajectory. The method by Ho and Cook cannot be directly applied to AGVs path planning, as cubic and fourthdegree spline-functions cannot describe path segments with a required precision. In order to be able to apply the method, before the calculation of spline-functions a continuous curve through a desired path must be created. For this purpose a simple cubic equation developed by Pierre Bézier, usually called a Bézier curve, can be used [5]. In [6] an on-line piecewise cubic Bézier curves trajectory generation algorithm is used to create a smooth trajectory, but the method is based on the distance and not on the time. The paper is organized in the following way. First we describe the calculation of Bézier curves that define path knot points for a two wheel-driven AGV. Then we demonstrate a Ho and Cook trajectory generation method that uses these path knot points. An emphasis is given to the problem of planning of sharp turn paths. A method of reducing a number of inserted knot points related to the given path planning accuracy is explained and illustrated by simulation results. Comments and directives for the future work conclude the paper. II. PATH PLANNING The proposed path planning method for autonomous guided vehicles is a two-step method. The first step is generating the reference path points by using so-called Bézier curves and the second step is creating a trajectory through the reference points by applying a Ho and Cook trajectory planning method. A. The application of a Bézier curve As shown in Fig. 1, a Bézier curve is defined by four points: T 0 (x 0, y 0 ), T 1 (x 1, y 1 ), T 2 (x 2, y 2 ) and T 3 (x 3,y 3 ). Point T 0 is the origin endpoint, while point T 3 is the destination endpoint. Points T 1 and T 2 serve as control points. A cubic Bézier curve x, y can be created by using the following equations: 3 2 x axbxcx x0 3 2 y aybycy y where cx 3( x1 x0) bx 3( x2 x1) cx ax x3 x0 cx bx. cy 3( y1 y0) by 3( y2 y1) cy a y y c b y 3 0 y y 0, 0 1, (1) /05/$ IEEE 1453
2 Assuming that the mass center of the vehicle is moving along the Bézier curve [x( ), y( )], constructed curves parallel to, L [x L ( ), y L ( )]={T Li (x Li, y Li ), 0 i n} and R [x R ( ), y R ( )]={T Ri (x Ri, y Ri ), 0 i n}, describe the paths of the left-hand and right-hand wheels, respectively: vy xl x vc 2 vx yl y vc 2, (3) vy xr x vc 2 vx yr y v 2 c Figure 1. Bézier curve. Differentiation of (1) gives: dx 2 vx 3ax2bx cx d dy 2 vy 3ay2by c. (2) y d 2 2 v v v c x y In order to move a two wheel-driven vehicle along a Bézier curve, two parallel curves must be created, one for the left-hand wheel and another for the right-hand wheel. Except for the trivial cases of coincident parallel curves or straight lines, generally, it is impossible to derive a Bézier curve that is parallel to another Bézier curve. However, parallel Bézier curves can be constructed, as shown in Fig. 2. Points T L and T R lie on the normal to the tangent of the Bézier curve in point T. Points T L and T R are equidistant to point T. where w b is a wheel base length. Let us split -interval [0, 1] into n subintervals 1 i i 1, i 1, n, n 0 0, n 1, (4) Now, let us calculate n-1 points of the Bézier curve (1) and n+1 points of the left-hand and right-hand wheel paths (3). For n, calculated points create parallel curves H L, and H R. When n is small, curves H L, and H R can be approximated by piecewise linear segments which results in their quasi-parallelism (see Fig. 3). In other words, the piecewise linear approximation of path segments will cause an error in the calculation of the parallel curves. 6 5 T 3 4 y 3 T T 0 T L T R x T Figure 2. Construction of left-hand wheel and right-hand wheel paths parallel to the central Bézier curve. T 2 Figure 3. Bézier curve is quantized and parallel curves are constructed. The distances between two neighboring points of the left-hand and the right-hand parallel curves shown in Fig. 4 are calculated using the following equations: ( 1) ( 1) 2 2 d x x y y Lk Lk L k Lk L k ( 1) ( 1) 2 2 d x x y y Rk Rk R k Rk R k, 1 k n. (5) 1454
3 Equation (6) can be rewritten as: s s d Lk L( k 1) Lk s s d Rk R( k 1) Rk. (7) Figure 4. Distances between the neighboring points of the lefthand and the right-hand parallel curve. Besides the distance, it is necessary to find a motion direction of the wheels, i.e. the sign of d Lk and d Rk values. The motion direction can be detected by calculating the scalar product of vectors o 1, o 2 and o 3 defined by related neighboring points as shown in Fig. 5. Vector o 2 presents an orientation of a vehicle, while vectors o 1 and o 3 present oriented position changes of the left-hand and right-hand wheel. If o 1 o 2 < 0, d Lk must be taken with a negative sign. Similarly, if o 2 o 3 < 0, parameter d Rk has a negative sign. T L(k-1) T Lk o 1 o 2 B. The application of spline functions Now, the Ho and Cook trajectory planning method can be applied [3]. The method is based on the calculation of fourth-degree and third-degree polynomials that concatenated describe a continuous motion of an AGV between given knot points T Li (x Li, y Li ) and T Ri (x Ri, y Ri ), 0 i n. The developed path through n+1points contains n polynomials. The calculation of polynomials is an iterative process which involves intense operations with high-order matrices. Third-degree polynomials describe intermediate segments; while fourth-degree polynomials describe the first and last segments (we assume that velocity and acceleration at the start-points and end-points are equal to zero). In order to keep reference trajectories within physical limitations of AGV s wheel drives, the calculation of polynomial coefficients must account for actual AGV s position, velocity and acceleration constraints. Under assumption of the velocity constraint v m =1.2 m/s and the acceleration constraint a m =10 m/s 2, Fig. 7 shows simulation results of the AGV driving along the planned path. One can see that the planned path is continuous in position, velocity and acceleration, and planned velocities and accelerations are within physical limits. The traverse time from T 0 to T 3 is 6.4 s. The AGV is accurately positioned into the desired position, as shown in Fig. 8. T L(k-1) o 3 T lk Figure 5. Detection of the motion direction of the wheels. Figure 6. A reference path of the left-hand wheel and of the right-hand wheel. When the neighboring distances are calculated and the motion direction is detected, reference paths can be created (see Fig. 6). The reference path is a cumulative sum of the distances between points of the parallel curves: k s d, s d Lk Li Rk Ri i 1 i 1 k, 1 k n. (6) Figure 7. The planned AGV trajectory: v m =1.2m/s, a m =10 m/s
4 Figure 10. Distances between the neighboring points of the lefthand and the right-hand parallel curve (low density quantization is applied). Figure 8. The motion of the AGV driving along the planned trajectory. III. PLANNING OF PATHS WITH SHARP TURNS The Bézier curve-based planning of a path with sharp turns can result in a position error due to lower density of knot points in the turns (see Figs. 9 and 10). Lower density of points causes higher approximation errors of distances between two neighboring points lying on the parallel curves H L and H R. A problem can be solved by increasing the quantization of -interval and thus increasing a number of knot points (Figs. 11 and 12), but this can result in a longer traverse time and oscillations in the velocity and acceleration responses (see Fig. 13). As a compromise, the quantization of -interval could be higher at sharp turns, while it could remain lower on the straight parts of the path curves. Figure 11. Distances between the neighboring points of the lefthand and the right-hand parallel curve (high density quantization is applied). Figure 9. A position error due to low density quantization of the parallel curves. Figure 12. A position error due to high density quantization of the parallel curves. 1456
5 As shown in Figs. 15 and 16, the removing of excess points resulted in the significant reduction of a number of reference points. Comparing the numbers of reference points from Fig. 11 and Fig. 16, it can be seen that the described procedure reduced the number of points from k =100 to k =22. Besides on the density of the -interval quantization directed by the angle values of sharp turns, a final number of reference points primarily depends on the magnitude of a bounded deviation error parameter. In practice, the requirements on are not rigorous, which makes the choice of its acceptable value much easier. So obtained reference points for the paths with sharp turns become the knot points for the Ho and Cook trajectory planning procedure. As shown in Fig. 17, in spite of a reduced number of points, the positioning error remained the same, while the traverse time of the reference trajectory was reduced from 6.4 s to 4.9 s, which is more than 20% faster. Figure 13. The planned AGV trajectory: v m =1.2m/s, a m =10 m/s 2 (high density quantization is applied). Removing superfluous points from densely quantized curves H L and H R can be done by searching and removing those points whose distance from the line which connects neighboring points is smaller than a bounded deviation error. As can be seen in Fig. 14, the distance of point T Lk from the line T ( -1) T and the distance of point T Lk Lk ( +1) Rk from the line TRk ( -1) TRk ( +1) are less than. Therefore, points T Lk and T Rk can be removed from the list of the reference path points, but the lengths of the paths T ( -1) T and Lk Lk ( +1) T ( -1) T must be included in the Rk Rk ( +1) length of the reference path as d Lk +d L(k+1) and d Rk +d R(k+1). Figure 15. Reference paths obtained after dense quantization of parallel curves and after removing excess points. T L(k+ 1) T Lk d L(k 1) + T L(k-1) d Lk T Rk d R + (k 1) T R(k+ 1) d Rk T R(k-1) Figure 16. Distances between the adjacent points of the left-hand and the right-hand wheel paths after removing excess points. Figure 14. Removing superfluous points from densely quantized curves H L and H R. 1457
6 A problem arises when paths with sharp turns must be planned. Due to a lower density of knot points at the turns, higher approximation errors of distances between two neighboring points on the wheel paths occur. The problem was solved in two ways: first, by increasing the level of path quantization and thus getting a larger number of knot points at the turns; and second, by removing excess reference points using a bounded deviation error method. Reference path points obtained in this way become the knot points for the so-called Ho and Cook trajectory planning procedure, which calculates position, velocity and acceleration profiles for each AGV wheel using linked fourth-degree and third-degree polynomials (spline functions). The proposed path planning method has been tested by simulations on a tricycle-type AGV with a front stabilization wheel and two rear-powered wheels. The results of the planned motion indicate smooth changes of all wheel variables within given limits. REFERENCES Figure 17. The planned AGV trajectory: v m =1.2m/s, a m =10 m/s 2 (removing of excess points is applied). IV. CONCLUSIONS A two-step path planning method for automated guided vehicles (AGV) is described. The first step is the calculation of the geometric path through which the mass center of the vehicle should pass by using parameterdependent cubic Bézier curves. Since the case study deals with a tricycle type of an AGV equipped with a front stabilization wheel and two rear powered wheels, the generation of parallel path curves for left-hand and righthand AGV wheels is elaborated. [1] R.C. Coulter, Implementation of the Pure Pursuit Path Tracking Algorithm, The Robotics Institute Carnegie Mellon University Pittsburgh, Pennsylvania 15213, Jan [2] K.Petrinec, Z.Kovacic and A.Marozin, "Simulator of Multi AGV Robotic Industrial Environments", The CD-ROM Proceedings of the IEEE Internatioanl Conference on Industrial Technology ICIT'03, Maribor, Slovenia, December, 2003 [3] C.Y. Ho and C.C. Cook, The application of spline functions to trajectory generation for computer controlled manipulators, Digital Systems for Industrial Automation, 1 (4): , [4] P.G. Ranky and C.Y. Ho, Robot Modelling Control and Applications with Software, Springer-Verlag, IFS (Publications) Lts, UK, [5] J. Bloomenthal, R.E. Barnhill, B.A. Barsky, P. Bézier, R. Forrest,N. Max, D.M. Palyka, D.F. Rogers, H. Rushmeier, A.R. Smith, R. Stock, N.M. Thalmann and D. Thalmann, Graphics remembrances, Annals of the History of Computing, IEEE, vol. 20, issue 2, pp , April-June 1998 [6] J.-H. Hwang, R.C. Arkin and D.-S. Kwon, "Mobile robots at your fingertip: Bézier curve on-line trajectory generation for supervisory control", Proc. IROS 2003, vol. 2, pp , Las Vegas, NV, October
08 - Designing Approximating Curves
08 - Designing Approximating Curves Acknowledgement: Olga Sorkine-Hornung, Alexander Sorkine-Hornung, Ilya Baran Last time Interpolating curves Monomials Lagrange Hermite Different control types Polynomials
More informationComputer Graphics Curves and Surfaces. Matthias Teschner
Computer Graphics Curves and Surfaces Matthias Teschner Outline Introduction Polynomial curves Bézier curves Matrix notation Curve subdivision Differential curve properties Piecewise polynomial curves
More information3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2013 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels
More informationMulti-objective Evolutionary Fuzzy Modelling in Mobile Robotics
Multi-objective Evolutionary Fuzzy Modelling in Mobile Robotics J. M. Lucas Dept. Information and Communications Engineering University of Murcia Murcia, Spain jmlucas@um.es H. Martínez Dept. Information
More informationManipulator trajectory planning
Manipulator trajectory planning Václav Hlaváč Czech Technical University in Prague Faculty of Electrical Engineering Department of Cybernetics Czech Republic http://cmp.felk.cvut.cz/~hlavac Courtesy to
More informationChapter 3 Path Optimization
Chapter 3 Path Optimization Background information on optimization is discussed in this chapter, along with the inequality constraints that are used for the problem. Additionally, the MATLAB program for
More information3D Modeling Parametric Curves & Surfaces
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2012 3D Object Representations Raw data Point cloud Range image Polygon soup Solids Voxels BSP tree CSG Sweep Surfaces Mesh Subdivision
More informationVideo 11.1 Vijay Kumar. Property of University of Pennsylvania, Vijay Kumar
Video 11.1 Vijay Kumar 1 Smooth three dimensional trajectories START INT. POSITION INT. POSITION GOAL Applications Trajectory generation in robotics Planning trajectories for quad rotors 2 Motion Planning
More informationSpline Curves. Spline Curves. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1
Spline Curves Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1 Problem: In the previous chapter, we have seen that interpolating polynomials, especially those of high degree, tend to produce strong
More informationProf. Fanny Ficuciello Robotics for Bioengineering Trajectory planning
Trajectory planning to generate the reference inputs to the motion control system which ensures that the manipulator executes the planned trajectories path and trajectory joint space trajectories operational
More informationSUBDIVISION ALGORITHMS FOR MOTION DESIGN BASED ON HOMOLOGOUS POINTS
SUBDIVISION ALGORITHMS FOR MOTION DESIGN BASED ON HOMOLOGOUS POINTS M. Hofer and H. Pottmann Institute of Geometry Vienna University of Technology, Vienna, Austria hofer@geometrie.tuwien.ac.at, pottmann@geometrie.tuwien.ac.at
More informationCGT 581 G Geometric Modeling Curves
CGT 581 G Geometric Modeling Curves Bedrich Benes, Ph.D. Purdue University Department of Computer Graphics Technology Curves What is a curve? Mathematical definition 1) The continuous image of an interval
More information10/11/07 1. Motion Control (wheeled robots) Representing Robot Position ( ) ( ) [ ] T
3 3 Motion Control (wheeled robots) Introduction: Mobile Robot Kinematics Requirements for Motion Control Kinematic / dynamic model of the robot Model of the interaction between the wheel and the ground
More information1 Trajectories. Class Notes, Trajectory Planning, COMS4733. Figure 1: Robot control system.
Class Notes, Trajectory Planning, COMS4733 Figure 1: Robot control system. 1 Trajectories Trajectories are characterized by a path which is a space curve of the end effector. We can parameterize this curve
More informationTrajectory Planning for Automatic Machines and Robots
Luigi Biagiotti Claudio Melchiorri Trajectory Planning for Automatic Machines and Robots Springer 1 Trajectory Planning 1 1.1 A General Overview on Trajectory Planning 1 1.2 One-dimensional Trajectories
More informationCurve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur
Curve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur Email: jrkumar@iitk.ac.in Curve representation 1. Wireframe models There are three types
More informationJane Li. Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute
Jane Li Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute (3 pts) How to generate Delaunay Triangulation? (3 pts) Explain the difference
More informationA Path Tracking Method For Autonomous Mobile Robots Based On Grid Decomposition
A Path Tracking Method For Autonomous Mobile Robots Based On Grid Decomposition A. Pozo-Ruz*, C. Urdiales, A. Bandera, E. J. Pérez and F. Sandoval Dpto. Tecnología Electrónica. E.T.S.I. Telecomunicación,
More informationSpline Methods Draft. Tom Lyche and Knut Mørken
Spline Methods Draft Tom Lyche and Knut Mørken January 5, 2005 2 Contents 1 Splines and B-splines an Introduction 3 1.1 Convex combinations and convex hulls.................... 3 1.1.1 Stable computations...........................
More informationSpline Methods Draft. Tom Lyche and Knut Mørken
Spline Methods Draft Tom Lyche and Knut Mørken 24th May 2002 2 Contents 1 Splines and B-splines an introduction 3 1.1 Convex combinations and convex hulls..................... 3 1.1.1 Stable computations...........................
More informationFriday, 11 January 13. Interpolation
Interpolation Interpolation Interpolation is not a branch of mathematic but a collection of techniques useful for solving computer graphics problems Basically an interpolant is a way of changing one number
More informationDerivative. Bernstein polynomials: Jacobs University Visualization and Computer Graphics Lab : ESM4A - Numerical Methods 313
Derivative Bernstein polynomials: 120202: ESM4A - Numerical Methods 313 Derivative Bézier curve (over [0,1]): with differences. being the first forward 120202: ESM4A - Numerical Methods 314 Derivative
More informationRemark. Jacobs University Visualization and Computer Graphics Lab : ESM4A - Numerical Methods 331
Remark Reconsidering the motivating example, we observe that the derivatives are typically not given by the problem specification. However, they can be estimated in a pre-processing step. A good estimate
More informationAlmost Curvature Continuous Fitting of B-Spline Surfaces
Journal for Geometry and Graphics Volume 2 (1998), No. 1, 33 43 Almost Curvature Continuous Fitting of B-Spline Surfaces Márta Szilvási-Nagy Department of Geometry, Mathematical Institute, Technical University
More informationSpline-based Trajectory Optimization for Autonomous Vehicles with Ackerman drive
Spline-based Trajectory Optimization for Autonomous Vehicles with Ackerman drive Martin Gloderer Andreas Hertle Department of Computer Science, University of Freiburg Abstract Autonomous vehicles with
More informationSpline Methods Draft. Tom Lyche and Knut Mørken. Department of Informatics Centre of Mathematics for Applications University of Oslo
Spline Methods Draft Tom Lyche and Knut Mørken Department of Informatics Centre of Mathematics for Applications University of Oslo January 27, 2006 Contents 1 Splines and B-splines an Introduction 1 1.1
More informationNon-holonomic Planning
Non-holonomic Planning Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering http://users.wpi.edu/~zli11 Recap We have learned about RRTs. q new q init q near q rand But the standard
More informationSpline Guided Path of a Mobile Robot with Obstacle Avoidance Characteristics
Spline Guided Path of a Mobile Robot with Obstacle Avoidance Characteristics D. K. Biswas TIC, CMERI, Durgapur, India (Email: dbiswas@cmeri.res.in) Abstract Path planning of a mobile robot is a wide field
More informationFall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.
Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.hao-li.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve
More informationCurves and Surfaces 1
Curves and Surfaces 1 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric Bi-Cubic Surfaces Quadric Surfaces Specialized Modeling Techniques 2 The Teapot 3 Representing
More informationKeyword: Quadratic Bézier Curve, Bisection Algorithm, Biarc, Biarc Method, Hausdorff Distances, Tolerance Band.
Department of Computer Science Approximation Methods for Quadratic Bézier Curve, by Circular Arcs within a Tolerance Band Seminar aus Informatik Univ.-Prof. Dr. Wolfgang Pree Seyed Amir Hossein Siahposhha
More informationFurther Graphics. Bezier Curves and Surfaces. Alex Benton, University of Cambridge Supported in part by Google UK, Ltd
Further Graphics Bezier Curves and Surfaces Alex Benton, University of Cambridge alex@bentonian.com 1 Supported in part by Google UK, Ltd CAD, CAM, and a new motivation: shiny things Expensive products
More informationSpace Robot Path Planning for Collision Avoidance
Space Robot Path Planning for ollision voidance Yuya Yanoshita and Shinichi Tsuda bstract This paper deals with a path planning of space robot which includes a collision avoidance algorithm. For the future
More informationSplines. Parameterization of a Curve. Curve Representations. Roller coaster. What Do We Need From Curves in Computer Graphics? Modeling Complex Shapes
CSCI 420 Computer Graphics Lecture 8 Splines Jernej Barbic University of Southern California Hermite Splines Bezier Splines Catmull-Rom Splines Other Cubic Splines [Angel Ch 12.4-12.12] Roller coaster
More informationSplines. Connecting the Dots
Splines or: Connecting the Dots Jens Ogniewski Information Coding Group Linköping University Before we start... Some parts won t be part of the exam Basically all that is not described in the book. More
More information1498. End-effector vibrations reduction in trajectory tracking for mobile manipulator
1498. End-effector vibrations reduction in trajectory tracking for mobile manipulator G. Pajak University of Zielona Gora, Faculty of Mechanical Engineering, Zielona Góra, Poland E-mail: g.pajak@iizp.uz.zgora.pl
More informationCurves and Surfaces Computer Graphics I Lecture 9
15-462 Computer Graphics I Lecture 9 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] February 19, 2002 Frank Pfenning Carnegie
More informationMotion Control (wheeled robots)
Motion Control (wheeled robots) Requirements for Motion Control Kinematic / dynamic model of the robot Model of the interaction between the wheel and the ground Definition of required motion -> speed control,
More informationHorizontal Flight Dynamics Simulations using a Simplified Airplane Model and Considering Wind Perturbation
Horizontal Flight Dynamics Simulations using a Simplified Airplane Model and Considering Wind Perturbation Dan N. DUMITRIU*,1,2, Andrei CRAIFALEANU 2, Ion STROE 2 *Corresponding author *,1 SIMULTEC INGINERIE
More informationRobotics Project. Final Report. Computer Science University of Minnesota. December 17, 2007
Robotics Project Final Report Computer Science 5551 University of Minnesota December 17, 2007 Peter Bailey, Matt Beckler, Thomas Bishop, and John Saxton Abstract: A solution of the parallel-parking problem
More informationAnimation. Computer Graphics COMP 770 (236) Spring Instructor: Brandon Lloyd 4/23/07 1
Animation Computer Graphics COMP 770 (236) Spring 2007 Instructor: Brandon Lloyd 4/23/07 1 Today s Topics Interpolation Forward and inverse kinematics Rigid body simulation Fluids Particle systems Behavioral
More informationObjects 2: Curves & Splines Christian Miller CS Fall 2011
Objects 2: Curves & Splines Christian Miller CS 354 - Fall 2011 Parametric curves Curves that are defined by an equation and a parameter t Usually t [0, 1], and curve is finite Can be discretized at arbitrary
More informationNOISE AND OBJECT ELIMINATION FROM AUTOMATIC CORRELATION DATA USING A SPLINE FUNCTION IRINEU DA SILVA
NOISE AND OBJECT ELIMINATION FROM AUTOMATIC CORRELATION DATA USING A SPLINE FUNCTION IRINEU DA SILVA University of Sao Paulo, at Sao Carlos School of Engineering 13.560 - Sao Carlos, Sao Paulo, Brasil
More informationAppendices - Parametric Keyframe Interpolation Incorporating Kinetic Adjustment and Phrasing Control
University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science 7-1985 Appendices - Parametric Keyframe Interpolation Incorporating Kinetic Adjustment and
More informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationCMPUT 412 Motion Control Wheeled robots. Csaba Szepesvári University of Alberta
CMPUT 412 Motion Control Wheeled robots Csaba Szepesvári University of Alberta 1 Motion Control (wheeled robots) Requirements Kinematic/dynamic model of the robot Model of the interaction between the wheel
More informationLecture 8. Divided Differences,Least-Squares Approximations. Ceng375 Numerical Computations at December 9, 2010
Lecture 8, Ceng375 Numerical Computations at December 9, 2010 Computer Engineering Department Çankaya University 8.1 Contents 1 2 3 8.2 : These provide a more efficient way to construct an interpolating
More informationEfficient Path Interpolation and Speed Profile Computation for Nonholonomic Mobile Robots
Efficient Path Interpolation and Speed Profile Computation for Nonholonomic Mobile Robots Stéphane Lens 1 and Bernard Boigelot 1 Abstract This paper studies path synthesis for nonholonomic mobile robots
More informationMOTION TRAJECTORY PLANNING AND SIMULATION OF 6- DOF MANIPULATOR ARM ROBOT
MOTION TRAJECTORY PLANNING AND SIMULATION OF 6- DOF MANIPULATOR ARM ROBOT Hongjun ZHU ABSTRACT:In order to better study the trajectory of robot motion, a motion trajectory planning and simulation based
More informationMultiple-Choice Test Spline Method Interpolation COMPLETE SOLUTION SET
Multiple-Choice Test Spline Method Interpolation COMPLETE SOLUTION SET 1. The ollowing n data points, ( x ), ( x ),.. ( x, ) 1, y 1, y n y n quadratic spline interpolation the x-data needs to be (A) equally
More informationAUTONOMOUS PLANETARY ROVER CONTROL USING INVERSE SIMULATION
AUTONOMOUS PLANETARY ROVER CONTROL USING INVERSE SIMULATION Kevin Worrall (1), Douglas Thomson (1), Euan McGookin (1), Thaleia Flessa (1) (1)University of Glasgow, Glasgow, G12 8QQ, UK, Email: kevin.worrall@glasgow.ac.uk
More informationOptimal Trajectory Generation for Nonholonomic Robots in Dynamic Environments
28 IEEE International Conference on Robotics and Automation Pasadena, CA, USA, May 19-23, 28 Optimal Trajectory Generation for Nonholonomic Robots in Dynamic Environments Yi Guo and Tang Tang Abstract
More informationParametric Curves. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Parametric Curves University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Parametric Representations 3 basic representation strategies: Explicit: y = mx + b Implicit: ax + by + c
More informationFlank Millable Surface Design with Conical and Barrel Tools
461 Computer-Aided Design and Applications 2008 CAD Solutions, LLC http://www.cadanda.com Flank Millable Surface Design with Conical and Barrel Tools Chenggang Li 1, Sanjeev Bedi 2 and Stephen Mann 3 1
More informationA NON-TRIGONOMETRIC, PSEUDO AREA PRESERVING, POLYLINE SMOOTHING ALGORITHM
A NON-TRIGONOMETRIC, PSEUDO AREA PRESERVING, POLYLINE SMOOTHING ALGORITHM Wayne Brown and Leemon Baird Department of Computer Science The United States Air Force Academy 2354 Fairchild Dr., Suite 6G- USAF
More informationminutes/question 26 minutes
st Set Section I (Multiple Choice) Part A (No Graphing Calculator) 3 problems @.96 minutes/question 6 minutes. What is 3 3 cos cos lim? h hh (D) - The limit does not exist.. At which of the five points
More informationNatural Quartic Spline
Natural Quartic Spline Rafael E Banchs INTRODUCTION This report describes the natural quartic spline algorithm developed for the enhanced solution of the Time Harmonic Field Electric Logging problem As
More informationCOLLISION-FREE TRAJECTORY PLANNING FOR MANIPULATORS USING GENERALIZED PATTERN SEARCH
ISSN 1726-4529 Int j simul model 5 (26) 4, 145-154 Original scientific paper COLLISION-FREE TRAJECTORY PLANNING FOR MANIPULATORS USING GENERALIZED PATTERN SEARCH Ata, A. A. & Myo, T. R. Mechatronics Engineering
More informationDesign considerations
Curves Design considerations local control of shape design each segment independently smoothness and continuity ability to evaluate derivatives stability small change in input leads to small change in
More informationINSTITUTE OF AERONAUTICAL ENGINEERING
Name Code Class Branch Page 1 INSTITUTE OF AERONAUTICAL ENGINEERING : ROBOTICS (Autonomous) Dundigal, Hyderabad - 500 0 MECHANICAL ENGINEERING TUTORIAL QUESTION BANK : A7055 : IV B. Tech I Semester : MECHANICAL
More informationNumerical Methods in Physics Lecture 2 Interpolation
Numerical Methods in Physics Pat Scott Department of Physics, Imperial College November 8, 2016 Slides available from http://astro.ic.ac.uk/pscott/ course-webpage-numerical-methods-201617 Outline The problem
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More informationGeometric Modeling of Curves
Curves Locus of a point moving with one degree of freedom Locus of a one-dimensional parameter family of point Mathematically defined using: Explicit equations Implicit equations Parametric equations (Hermite,
More informationImproved Non-linear Spline Fitting for Teaching Trajectories to Mobile Robots
Improved Non-linear Spline Fitting for Teaching Trajectories to Mobile Robots Christoph Sprunk Boris Lau Wolfram Burgard Abstract In this paper, we present improved spline fitting techniques with the application
More informationParametric Curves. University of Texas at Austin CS384G - Computer Graphics
Parametric Curves University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Parametric Representations 3 basic representation strategies: Explicit: y = mx + b Implicit: ax + by + c
More informationA Curve Tutorial for Introductory Computer Graphics
A Curve Tutorial for Introductory Computer Graphics Michael Gleicher Department of Computer Sciences University of Wisconsin, Madison October 7, 2003 Note to 559 Students: These notes were put together
More informationBezier Curves, B-Splines, NURBS
Bezier Curves, B-Splines, NURBS Example Application: Font Design and Display Curved objects are everywhere There is always need for: mathematical fidelity high precision artistic freedom and flexibility
More informationNAVIGATION AND RETRO-TRAVERSE ON A REMOTELY OPERATED VEHICLE
NAVIGATION AND RETRO-TRAVERSE ON A REMOTELY OPERATED VEHICLE Karl Murphy Systems Integration Group Robot Systems Division National Institute of Standards and Technology Gaithersburg, MD 0899 Abstract During
More informationParameterization. Michael S. Floater. November 10, 2011
Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point
More informationThe goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a
The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a coordinate system and then the measuring of the point with
More informationDeficient Quartic Spline Interpolation
International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 2 (2011), pp. 227-236 International Research Publication House http://www.irphouse.com Deficient Quartic
More informationRobots are built to accomplish complex and difficult tasks that require highly non-linear motions.
Path and Trajectory specification Robots are built to accomplish complex and difficult tasks that require highly non-linear motions. Specifying the desired motion to achieve a specified goal is often a
More informationCOMP3421. Global Lighting Part 2: Radiosity
COMP3421 Global Lighting Part 2: Radiosity Recap: Global Lighting The lighting equation we looked at earlier only handled direct lighting from sources: We added an ambient fudge term to account for all
More informationCentral issues in modelling
Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction includes: manual modelling; fitting to
More informationParametric curves. Reading. Curves before computers. Mathematical curve representation. CSE 457 Winter Required:
Reading Required: Angel 10.1-10.3, 10.5.2, 10.6-10.7, 10.9 Parametric curves CSE 457 Winter 2014 Optional Bartels, Beatty, and Barsky. An Introduction to Splines for use in Computer Graphics and Geometric
More informationOMNI Interpolator for High-performance Motion Control System Part I: Algorithm of Interpolator
OMNI Interpolator for High-performance Motion Control System Part I: Algorithm of Interpolator Chengbin Ma, Wei Li, Yadong Liu, Kazuo Yamazaki Abstract This paper presents the OMNI-Interpolator design
More informationWaypoint Navigation with Position and Heading Control using Complex Vector Fields for an Ackermann Steering Autonomous Vehicle
Waypoint Navigation with Position and Heading Control using Complex Vector Fields for an Ackermann Steering Autonomous Vehicle Tommie J. Liddy and Tien-Fu Lu School of Mechanical Engineering; The University
More informationKnow it. Control points. B Spline surfaces. Implicit surfaces
Know it 15 B Spline Cur 14 13 12 11 Parametric curves Catmull clark subdivision Parametric surfaces Interpolating curves 10 9 8 7 6 5 4 3 2 Control points B Spline surfaces Implicit surfaces Bezier surfaces
More informationInterpolation by Spline Functions
Interpolation by Spline Functions Com S 477/577 Sep 0 007 High-degree polynomials tend to have large oscillations which are not the characteristics of the original data. To yield smooth interpolating curves
More informationSimple Keyframe Animation
Simple Keyframe Animation Mike Bailey mjb@cs.oregonstate.edu This work is licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives 4. International License keyframe.pptx Approaches to
More informationCS770/870 Spring 2017 Curve Generation
CS770/870 Spring 2017 Curve Generation Primary resources used in preparing these notes: 1. Foley, van Dam, Feiner, Hughes, Phillips, Introduction to Computer Graphics, Addison-Wesley, 1993. 2. Angel, Interactive
More informationCurve and Surface Basics
Curve and Surface Basics Implicit and parametric forms Power basis form Bezier curves Rational Bezier Curves Tensor Product Surfaces ME525x NURBS Curve and Surface Modeling Page 1 Implicit and Parametric
More informationCHAPTER 3 MATHEMATICAL MODEL
38 CHAPTER 3 MATHEMATICAL MODEL 3.1 KINEMATIC MODEL 3.1.1 Introduction The kinematic model of a mobile robot, represented by a set of equations, allows estimation of the robot s evolution on its trajectory,
More informationTime Optimal Trajectories for Bounded Velocity Differential Drive Robots
Time Optimal Trajectories for Bounded Velocity Differential Drive Robots Devin J. Balkcom Matthew T. Mason Robotics Institute and Computer Science Department Carnegie Mellon University Pittsburgh PA 53
More informationGraphics and Interaction Transformation geometry and homogeneous coordinates
433-324 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationCOMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates
COMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationCOMPUTER AIDED ENGINEERING DESIGN (BFF2612)
COMPUTER AIDED ENGINEERING DESIGN (BFF2612) BASIC MATHEMATICAL CONCEPTS IN CAED by Dr. Mohd Nizar Mhd Razali Faculty of Manufacturing Engineering mnizar@ump.edu.my COORDINATE SYSTEM y+ y+ z+ z+ x+ RIGHT
More informationMotion Planning in Urban Environments: Part I
Motion Planning in Urban Environments: Part I Dave Ferguson Intel Research Pittsburgh Pittsburgh, PA dave.ferguson@intel.com Thomas M. Howard Carnegie Mellon University Pittsburgh, PA thoward@ri.cmu.edu
More informationCO-OPERATIVE ACTION OF EXTRA-OCULAR MUSCLES*
Brit. J. Ophthal. (1962) 46, 397. CO-OPERATIVE ACTION OF EXTRA-OCULAR MUSCLES* BY PAUL BOEDER Iowa City, Iowa, U.S.A. IN every excursion of the eyes, two times six muscles act with extraordinary speed,
More informationBézier Splines. B-Splines. B-Splines. CS 475 / CS 675 Computer Graphics. Lecture 14 : Modelling Curves 3 B-Splines. n i t i 1 t n i. J n,i.
Bézier Splines CS 475 / CS 675 Computer Graphics Lecture 14 : Modelling Curves 3 n P t = B i J n,i t with 0 t 1 J n, i t = i=0 n i t i 1 t n i No local control. Degree restricted by the control polygon.
More informationInterpolation and Splines
Interpolation and Splines Anna Gryboś October 23, 27 1 Problem setting Many of physical phenomenona are described by the functions that we don t know exactly. Often we can calculate or measure the values
More informationConstrained Optimization Path Following of Wheeled Robots in Natural Terrain
Constrained Optimization Path Following of Wheeled Robots in Natural Terrain Thomas M. Howard, Ross A. Knepper, and Alonzo Kelly The Robotics Institute, Carnegie Mellon University, 5000 Forbes Avenue,
More informationMA 323 Geometric Modelling Course Notes: Day 10 Higher Order Polynomial Curves
MA 323 Geometric Modelling Course Notes: Day 10 Higher Order Polynomial Curves David L. Finn December 14th, 2004 Yesterday, we introduced quintic Hermite curves as a higher order variant of cubic Hermite
More informationArc-Length Parameterized Spline Curves for Real-Time Simulation
Arc-Length Parameterized Spline Curves for Real-Time Simulation Hongling Wang, Joseph Kearney, and Kendall Atkinson Abstract. Parametric curves are frequently used in computer animation and virtual environments
More informationAn Integrated Vision Sensor for the Computation of Optical Flow Singular Points
An Integrated Vision Sensor for the Computation of Optical Flow Singular Points Charles M. Higgins and Christof Koch Division of Biology, 39-74 California Institute of Technology Pasadena, CA 925 [chuck,koch]@klab.caltech.edu
More information2D Spline Curves. CS 4620 Lecture 13
2D Spline Curves CS 4620 Lecture 13 2008 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes [Boeing] that is, without discontinuities So far we can make things with corners
More informationLecture 1.1 Introduction to Fluid Dynamics
Lecture 1.1 Introduction to Fluid Dynamics 1 Introduction A thorough study of the laws of fluid mechanics is necessary to understand the fluid motion within the turbomachinery components. In this introductory
More informationRailway car dynamic response to track transition curve and single standard turnout
Computers in Railways X 849 Railway car dynamic response to track transition curve and single standard turnout J. Droździel & B. Sowiński Warsaw University of Technology, Poland Abstract In this paper
More informationParametric curves. Brian Curless CSE 457 Spring 2016
Parametric curves Brian Curless CSE 457 Spring 2016 1 Reading Required: Angel 10.1-10.3, 10.5.2, 10.6-10.7, 10.9 Optional Bartels, Beatty, and Barsky. An Introduction to Splines for use in Computer Graphics
More informationCS130 : Computer Graphics Curves (cont.) Tamar Shinar Computer Science & Engineering UC Riverside
CS130 : Computer Graphics Curves (cont.) Tamar Shinar Computer Science & Engineering UC Riverside Blending Functions Blending functions are more convenient basis than monomial basis canonical form (monomial
More information