System Design, Modelling, and Control of a Four-Wheel-Steering Mobile Robot
|
|
- Bertina Ilene Dorsey
- 5 years ago
- Views:
Transcription
1 System Desgn, Modellng, and Control of a Four-Wheel-Steerng Moble Robot Maxm Makatchev Dept. of Manufacturng Engneerng and Engneerng Management Cty Unversty of Hong Kong Hong Kong John J. McPhee Dept. of Systems Desgn Engneerng Unversty of Waterloo Ontaro Canada NL 3G1 S. K. Tso Sherman Y. T. Lang Centre for Intellgent Desgn, Automaton and Manufacturng Cty Unversty of Hong Kong Hong Kong Integrated Manufacturng Technologes Insttute Natonal Research Councl of Canada London, Ontaro, Canada N6G 4X8 Abstract Ths paper descrbes a dynamc model and crosscouplng control of a moble robot wth four ndependently steered and drven wheels. The dynamc model ncorporates the components of the wheel mechansms, backlash effects of the wheel actuators and nonlnear forces produced by the tre-ground nteracton. The problem of computatonally effcent control of the vehcle s solved usng a low-level control loop that cross-couples steerng and drvng motors and s utlzed n conjuncton wth a knematcs-based trajectorytrackng controller. The results of smulaton show sgnfcant reducton of the vehcle slp wth moderate trajectory-trackng errors. 1 Introducton Research on modellng and control of wheeled vehcles can be dvded nto two major categores: one that s orented towards automobles and terran vehcles, the other s orented towards ndoor wheeled moble robots (WMRs). Ths dvson s based on the dfferences n the desgn of the vehcles, operatonal and envronmental condtons. To menton a few: wheels of WMRs are often ndependently actuated, as opposed to mechancally coupled pars of steerng wheels of automobles. a wheel plane of a typcal WMR s normally perpendcular to the ground plane, whle a wheel plane of an automoble may have a nonzero nclnaton angle (camber) [, 1]. The work s supported by a CERG grant 944 and a Cty Unversty of Hong Kong grant 769 n Hong Kong. ndoor WMRs usually operate wthn a lower speed range compared to automobles. Mechancal couplng of the wheels n automobles often allows the use of a smplfed, sngle-track model of two-wheel-steerng and four-wheel-steerng cars for the desgn of moton controllers [1, 4]. In modellng and control of moble robots wth ndependently steered and/or drven wheels t s requred to explctly account for each of the wheels of the vehcle. Such complex models of four-wheel vehcles has been developed n works [3, 7, 8]. However the controllers n these works are stll desgned under the assumpton that the wheels are steered n pars. The models also utlze smplfed wheel actuator dynamcs that may not be approprate for modellng of some types of WMRs, e. g. heavy vehcles wth DC motor actuated wheels, for whch the effects of backlash and wheel mechansm dynamcs may be sgnfcant. Such WMRs are often utlzed as autonomous guded vehcles (AGVs) n tght manufacturng envronments havng hgh requrements on maneuverablty. These requrements mply the necessty of actve utlzaton of the capablty of such WMRs for ndependent wheel steerng. Knematcs-based control of WMRs wth multple steerng wheels has receved consderable attenton over the last decade. Effcent trajectory-trackng controllers are developed usng dynamc feedback [9]. To the authors best knowledge, controllers that utlze complex dynamc models of the WMRs wth multple steerng wheels are not used n practce due to ther computatonal nfeasblty and dffcultes related to analytcal nvestgaton of ther propertes.
2 One of the possble ways to deal wth the tradeoff between computatonal effcency of control and a granulaton of a correspondng model s to augment a knematcs-based controller wth a controller that accounts for a partcular dynamc effect. In ths work we descrbe such a controller that ams specfcally at mnmzaton of slppage due to wheel msalgnment. The controller extends the approach presented n [6] by () cross-couplng all four wheels of the vehcle, nstead of couplng them parwse, () ncorporatng not only steerng but also drvng actuator dynamcs, () generalzng the control objectves so that the wheels are able to track a lnear approxmaton of the vehcle knematc constrants durng transent states of control. Ths couplng scheme allows the four wheels to be steered at arbtrary angles, not necessarly the same angles as n [6]. The performance of the cross-couplng controller s evaluated wth the dynamc model developed for the AGV prototype of the project. The dynamc model of the vehcle dffers from those descrbed n [3, 7, 8] n that, n addton to nonlnear forces at the contact between the tre and the ground, t ncorporates the dynamcs of the components of the wheel mechansms and a more realstc model of the wheel actuators ncludng backlash effects of DC motors. The paper s organzed as follows. Secton descrbes the knematcs of the WMR. The dynamc model s presented n secton 3. The cross-couplng controller and smulaton results are addressed n secton 4. Concludng remarks are gven n secton. Knematc model A dagram of the vehcle s presented n fgure 1. Assumng that the vehcle frame s a rgd body, the veloctes v of the wheel base ponts W are related to the velocty v c of the reference pont C of the frame as v = v c + θk CW, = 1,..., 4. (1) Pure rollng condton mples that the wheel s rotaton rate φ and steerng angle ψ wth respect to the frame coordnate system are such that φ = v r, tan ψ = v y v x, = 1,..., 4. () The knematc equaton relatng acceleraton a of the wheel base pont W and acceleraton a c of the vehcle reference pont C s a = a c + θk CW θ CW. (3) Usng the fact that a c = ( v x θv y ) + ( v y + θv x )j we have a = a x + a y j, where a x = v x θv y CW x θ CW y θ, (4) a y = v y + θv x CW y θ + CW x θ. () Y O W 3 X y A y4 W 4 T ψ4 y 1 C W 1 A x4 v 1 W Slp angle 1 Fgure 1: Notaton of knematcs of the WMR and dynamcs of the frame. 3 Dynamc model The dynamc model of the vehcle s derved by consderng dynamcs of the vehcle frame, wheel struts, wheels, and wheel motors. 3.1 Frame dynamcs The frame lateral, longtudnal and yaw dynamc equatons are: ma x = m( v x θv y ) = ma y = m( v y + θv x ) = I θ = ψ 1 θ x 1 x v c A x, (6) A y, (7) P (8) respectvely, where A x and A y are the x-component and the y-component of reacton at the steer jont W, P = T ψ +CW x A y CW y A x s the vertcal component of the total torque appled to the frame va the steer jont W. We neglect horzontal components P x, P y of the torque appled to the vehcle frame va the steer jont, assumng that all wheels of the vehcle are always n contact wth the ground and that the change of normal reacton at the wheel-ground contact pont s neglgble. 3. Wheel strut dynamcs We assume that for each wheel gravty centers of the strut and the wheel, the pont of the wheel contact wth the ground, geometrc center of the wheel, and steer jont W are all on a sngle vertcal lne. As wth the frame, we neglect horzontal components of the total moment of the wheel strut.
3 3.3 Wheel dynamcs m s a x = S x A x, (9) m s a y = S y A y, (1) w s = S z A z, (11) I s ( θ + ψ ) = T ψ D z. (1) The forces appled to the wheel are related as follows: m w a x = F x S x, (13) m w a y = F y S y, (14) w w = N S z. (1) The total moment produced by couples and forces appled to wheel s M = T φ + D z E + D x rk (F x + F y j ) = (D x + rf y ) + (T φ rf x )j + (D z E )k. (16) We assume the wheel to be cylndrcal so that the axes x, y, z are ts prncpal axes of nerta. The angular moments of the wheel are: M x = I x x ω x + (I z z I y y )ω y ω z, (17) M y = I y y ω y + (I x x I z z )ω x ω z, (18) M z = I z z ω z + (I y y I x x )ω x ω y, (19) where I x x = I z z = I t s the transverse moment of nerta and I y y = I a s the axal moment of nerta of the wheel. Assumng that the x -component of the angular velocty of the wheel s zero (no camber), the angular velocty of the wheel s ω = ( θ + ψ )k + φ j. () Substtuton of () nto (17) (19) and usng (16) and (1) gves us the expressons for the steerng and drvng torques respectvely: T ψ = (I s + I t )( θ + ψ ) + E, (1) T φ = I a φ + rf x. () The lateral and longtudnal tre frcton forces F y, F x, and the algnng torque due to frcton E are modelled as commonly done n the automotve ndustry [], [1]. 3.4 Dynamcs of wheel motors Assumng that the armature-wndng nductance of the motors s neglgble, the equatons governng the dynamcs of the () steerng and () drvng motors of the wheel are as follows: () steerng motors k 1 e ψ = k 3 ψ + R ψ, (3) k ψ = (I s + I t )( θ + ψ ) + b ψ + E, (4) where (I s +I t )( θ+ ψ)+e = T ψ s the steerng torque appled to the wheel, and b ψ s the energy loss due to the frcton n the motor and n the gear tran; () drvng motors k 1e φ = k 3 + R φ, () k φ = I a φ + b φ + rf x, (6) where I a φ +rf x = T φ s the drvng torque appled to the wheel, and b φ s the energy loss due to the frcton n the motor and n the gear tran. 3. Vehcle Dynamcs Combnng (6) (8), (9) (1), (13), (14), (4), () wth (3) (6), (1), (), the state-space model of the vehcle can be presented as: v x v ÿ θ ψ 1 4 φ 1 4 = M 1 Q(v x, v y, θ, ψ 1 4, φ 1 4, ψ 1 4, φ 1 4, e ψ1 4, e φ1 4 ), (7) where matrces M and Q are specfed n the Appendx. 4 Control 4.1 Cross-couplng controller We consder the problem of control of the moble robot equpped wth multple steerng and drvng wheels. The effcent knematcs-based trajectorytrackng control va lnearzng dynamc feedback s proposed n [9]. To preserve the computatonal feasblty of knematcs-based control and to account for tre slp due to msalgnment of the wheels, we propose to use a cross-couplng controller n conjuncton wth a knematcs-based trajectory-trackng controller. The proposed cross-couplng controller s based on the dea of a pecewse lnear approxmaton of the pure rollng condton () wth the node ponts n the space of steerng angles and drvng rates generated by a hgher level trajectory trackng controller [, 6]. For example, the lnear approxmaton of the curve n the space of steerng angles s specfed by the relatonshp ψ ψ d δ = ψ j ψ jd δ j, δ, δ j, (8)
4 Lateral Slp Angle Tme (s) Lateral Slp Angle Tme (s) 1 - Lateral Slp Angle Tme (s) Lateral Slp Angle Tme (s) Fgure : Lateral slp angles correspondng to the four wheels wth the dfferent control schemes: dash-dotted lne uncoupled control; dashed lne partally coupled control (uncoupled controller for the steerng motors, wth the drvng motors coupled wth the respectve steerng motors); sold lne cross-couplng control of the steerng and drvng motors. Concluson In ths paper we develop a dynamc model of a moble robot wth four ndependently steered and drven wheels. Detaled modellng of wheel plants allows utlzaton of the model for smulaton nvolvng such dynamc effects as slppage due to wheel msalgnment and drvng rate errors. To mnmze slppage due to msalgnment of the wheels, a cross-couplng controller s proposed to follow a pecewse lnear approxmaton of the trajectory correspondng to the pure rollng condton n the space of steerng and drvng actuators. The smulaton performed wth the dynamc model shows sgnfcant reducton of lateral slp wth moderate trajectorytrackng errors. Acknowledgments The authors would lke to thank N. Lee for hs assstance wth the AGV prototype and K. F. Man for hs advce on smulaton. Appendx A Nomenclature where ψ s and ψ d are the ntal and desred steerng angles respectvely, δ = ψ s ψ d are the ntal errors. We refer to (8) as the condton of proportonal errors between the steerng actuators and j. A smlar relatonshp can couple the drvng and steerng actuators. The LQR-based controller that couples the steerng and drvng actuators accordng to the condton of proportonal errors s descrbed n detal n []. 4. Smulaton results The performance of the cross-couplng controller s evaluated by smulaton wth the dynamc model mplemented n Matlab. The reference trajectory conssts of a straght-lne nterval and a crcular arc of the radus R = m wth the center closer to wheels 1 and 4. The vehcle undergoes the turnng maneuver so that durng the lne-nterval followng all reference steerng angles are equal to zero, and durng the crcular-arc followng the reference steerng angles are related as ψ 1d = ψ 3d = 38.8, ψ d = ψ 4d =.1. Lateral slp angles obtaned for dfferent couplng schemes are presented n fgure. The smulated and the deal trajectory n the space of steerng angles are shown n fgure 3. The trajectory trackng errors under the crosscouplng control are larger than under the uncoupled control, but consdered moderate (more detals n []). reacton force and torque at steer jont A, T ψ mass and moment of nerta of the frame m, I reacton force and torque at wheel-strut jont S, D mass and weght of the strut m s, w s moment of nerta of the strut I s lateral tre frcton force F y longtudnal tre frcton (tractve) force F x normal force at the tre-ground contact N algnng torque due to frcton E steerng torque T ψ drvng torque T φ mass and weght of the wheel m w, w w radus of the wheel r nput voltage of the motors e ψ, e φ amplfer constants k 1, k 1 motor torque constants k, k back emf constants of the motors k 3, k 3 armature current of the motors ψ, φ resstance of the motors R, R vscous-frcton coeffcents b, b B Generalzed mass and force matrces The elements of matrx of generalzed masses M are as follows: M 1,1 = M, = m + (m s + m w), M 1,3 = M 3,1 = (m s + m w)cw y, M,3 = M 3, = (m s + m w)cw x,
5 3 3.. Tme (s) 1. 1 Tme (s) Steerng angle (deg) Steerng angle 1 (deg) 1 1 Steerng angle (deg) Steerng angle 1 (deg) (a) Uncoupled control. (b) Coupled control. Fgure 3: Steerng angles of wheels 1 and under dfferent control schemes. For each subfgure the sold lne corresponds to the actual steerng angles of the wheels, the x-marked lne s the projecton of the 3D trajectory represented by the sold lne on the D space of steerng angles (wthout tme), and the o-marked lne s the trajectory n the space of steerng angles that corresponds to the no-slp steerng. M 3,3 = I + (m s + m w)cw, M 4,3 = M 4,4 = I s + I t, M, = I a, and the rest of elements are zeros. The elements of the vector of generalzed forces Q 1 are as follows: Q 1,1 = mv y θ + ( + Fx + (m s + m ( w) v y θ )) + CWx θ, Q,1 = mv x θ + ( + Fy (m s + m ( w) v x θ )) + CWy θ, Q 3,1 = (T ψ + (CW x F y CW y F x ) θ (m s + m w) (CW x v x + CW y v y) ), Q 4,1 = T ψ E, Q,1 = T φ rf x, where steerng and drvng torques are derved from (3) (6) wth the armature voltage e ψ, e φ of the steerng and drvng motors gven. References [1] J. Ackermann, Robust Decouplng, Ideal Steerng Dynamcs and Yaw Stablzaton of 4WS Cars, Automatca, 1994, vol. 3, no. 11, pp [] T. D. Gllespe, Fundamentals of vehcle dynamcs. Warrendale: Socety of Automotve Engneers, 199. [3] W. Langson, A. Alleyne, Multvarable Blnear Vehcle Control Usng Steerng and Indvdual Wheel Torques, Journal of Dynamc Systems, Measurement and Control, December 1999, vol. 11, pp [4] A. Y. Lee, A Prevew Steerng Autoplot Control Algorthm for Four-Wheel-Steerng Passenger Vehcles, Journal of Dynamc Systems, Measurement, and Control, vol. 114, September 199, pp [] M. Makatchev, S. Y. T. Lang, S. K. Tso, J. J. McPhee, Cross-Couplng Control for Slppage Mnmzaton of a Four-Wheel-Steerng Moble Robot, Proc. of the 31st Int. Symposum on Robotcs (ISR ), Montreal, Canada, May 14 17,, pp [6] N. Matsumoto, H. Kuraoka, M. Ohba, An expermental study on vehcle lateral and yaw moton control, Proc. of Int. Conf. on Industral Electroncs, Control and Instrumentaton (IECON), 1991, vol. 1, pp [7] N. Matsumoto, M. Tomzuka, Vehcle Lateral Velocty and Yaw Rate Control Wth Two Independent Control Inputs, Trans. of the ASME, Vol. 114, December 199, pp [8] S.-H. Yu, J. J. Moskwa, A Global Approach to Vehcle Control: Coordnaton of Four Wheel Steerng and Wheel Torques, Journal of Dynamc Systems, Measurement and Control, vol. 116, December 1994, pp [9] B. Thulot, B. d Andréa-Novel, A. Mcael, Modelng and Feedback Control of Moble Robots Equpped wth Several Steerng Wheels, IEEE Trans. on Robotcs and Automaton, vol. 1, no. 3, June 1996, pp [1] J. Y. Wong, Theory of Ground Vehcles. New York: Wley, 1993.
Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationCross-Coupling Control for Slippage Minimization of a Four-Wheel-Steering Mobile Robot
Cross-Coupling Control for Slippage Minimization of a Four-Wheel-Steering Mobile Robot Maxim Makatchev Dept. of Manufacturing Engineering and Engineering Management City University of Hong Kong Hong Kong
More informationIterative General Dynamic Model for Serial-Link Manipulators
EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for Seral-Lnk Manpulators In ths set of notes, we are gong to develop a method for computng a general
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationStudy on Active Micro-vibration Isolation System with Linear Motor Actuator. Gong-yu PAN, Wen-yan GU and Dong LI
2017 2nd Internatonal Conference on Electrcal and Electroncs: echnques and Applcatons (EEA 2017) ISBN: 978-1-60595-416-5 Study on Actve Mcro-vbraton Isolaton System wth Lnear Motor Actuator Gong-yu PAN,
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationResearch Report. Eiichi Ono, Yoshikazu Hattori, Yuji Muragishi. Abstract. Special Issue Estimation and Control of Vehicle Dynamics for Active Safety
Specal Issue Estmaton and Control of Vehcle Dynamcs for Actve Safety 7 Research Report Estmaton of Tre Frcton Crcle and Vehcle Dynamcs Integrated Control for Four-wheel Dstrbuted Steerng and Four-wheel
More informationModelling of the precise movement of a ship at slow speed to minimize the trajectory deviation risk
Computatonal Methods and Expermental Measurements XIV 29 Modellng of the precse movement of a shp at slow speed to mnmze the trajectory devaton rsk J. Maleck Polsh Naval Academy, Poland Faculty of Mechancs
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationNote 10. Modeling and Simulation of Dynamic Systems
Lecture Notes of ME 475: Introducton to Mechatroncs Note 0 Modelng and Smulaton of Dynamc Systems Department of Mechancal Engneerng, Unversty Of Saskatchewan, 57 Campus Drve, Saskatoon, SK S7N 5A9, Canada
More informationDesign and Analysis of Landing Gear Mechanic Structure for the Mine Rescue Carrier Robot
Sensors & Transducers 214 by IFSA Publshng, S. L. http://www.sensorsportal.com Desgn and Analyss of Landng Gear Mechanc Structure for the Mne Rescue Carrer Robot We Juan, Wu Ja-Long X an Unversty of Scence
More informationResearch on the Fuzzy Control for Vehicle Semi-active Suspension. Xiaoming Hu 1, a, Wanli Li 1,b
Advanced Materals Research Onlne: 0-0- ISSN: -9, Vol., pp -9 do:0.0/www.scentfc.net/amr.. 0 Trans Tech Publcatons, Swterland Research on the Fuy Control for Vehcle Sem-actve Suspenson Xaomng Hu, a, Wanl
More informationA Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph
A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationAmplification and Relaxation of Electron Spin Polarization in Semiconductor Devices
Amplfcaton and Relaxaton of Electron Spn Polarzaton n Semconductor Devces Yury V. Pershn and Vladmr Prvman Center for Quantum Devce Technology, Clarkson Unversty, Potsdam, New York 13699-570, USA Spn Relaxaton
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More informationTHE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS OF A TELESCOPIC HYDRAULIC CYLINDER SUBJECTED TO EULER S LOAD
Journal of Appled Mathematcs and Computatonal Mechancs 7, 6(3), 7- www.amcm.pcz.pl p-issn 99-9965 DOI:.75/jamcm.7.3. e-issn 353-588 THE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS
More informationA Multi-Axis Force Measurement System for a Space Docking Mechanism
3rd Internatonal Conference on Materal, Mechancal and Manufacturng Engneerng (IC3ME 215) A Mult-Axs orce Measurement System for a Space Dockng Mechansm Gangfeng Lu a*, Changle L b and Zenghu Xe c Buldng
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationDynamics of Rotational Motion
Dynamcs of Rotatonal Moton Torque: the rotatonal analogue of force Torque = force x moment arm = Fl moment arm = perpendcular dstance through whch the force acts a.k.a. leer arm l F l F l F l F = Fl =
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationSWITCHING PROCESS IN LIMITED SLIP DIFFERENTIAL
6th Internatonal DAAAM Baltc Conference INDUSTRIAL ENGINEERING 24-26 Aprl 2008, Tallnn, Estona SWITCHING PROCESS IN LIMITED SLIP DIFFERENTIAL Resev, J.; Roosmölder, L.; Stas, M. Abstract: The energy flow
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15
NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More informationIrregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More informationParameter Estimation for Dynamic System using Unscented Kalman filter
Parameter Estmaton for Dynamc System usng Unscented Kalman flter Jhoon Seung 1,a, Amr Atya F. 2,b, Alexander G.Parlos 3,c, and Klto Chong 1,4,d* 1 Dvson of Electroncs Engneerng, Chonbuk Natonal Unversty,
More informationEmbedded estimation of the tire/road forces and validation in a laboratory vehicle
- 090 0080490 Embedded estmaton of the tre/road forces and valdaton n a laboratory vehcle Moustapha DOUMIATI*(Correspondng author, Gullaume BAET*, Danel LECHNER**, Alessandro VICTORINO*, Al CHARARA*, *
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationResource Allocation with a Budget Constraint for Computing Independent Tasks in the Cloud
Resource Allocaton wth a Budget Constrant for Computng Independent Tasks n the Cloud Wemng Sh and Bo Hong School of Electrcal and Computer Engneerng Georga Insttute of Technology, USA 2nd IEEE Internatonal
More informationAP Physics 1 & 2 Summer Assignment
AP Physcs 1 & 2 Summer Assgnment AP Physcs 1 requres an exceptonal profcency n algebra, trgonometry, and geometry. It was desgned by a select group of college professors and hgh school scence teachers
More informationDO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.
EE 539 Homeworks Sprng 08 Updated: Tuesday, Aprl 7, 08 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. For full credt, show all work. Some problems requre hand calculatons.
More informationAssessment of Site Amplification Effect from Input Energy Spectra of Strong Ground Motion
Assessment of Ste Amplfcaton Effect from Input Energy Spectra of Strong Ground Moton M.S. Gong & L.L Xe Key Laboratory of Earthquake Engneerng and Engneerng Vbraton,Insttute of Engneerng Mechancs, CEA,
More informationPath Planning and Force Control of a 4WD4WS Vehicle
Proceedngs of Australasan Conference on Robotcs and Automaton, -4 Dec 14, The Unversty of Melbourne, Melbourne, Australa Path Plannng and Force Control of a 4WD4WS Vehcle Pengle Da and Jay Katuptya Unversty
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton
More informationOFF-AXIS MECHANICAL PROPERTIES OF FRP COMPOSITES
ICAMS 204 5 th Internatonal Conference on Advanced Materals and Systems OFF-AXIS MECHANICAL PROPERTIES OF FRP COMPOSITES VLAD LUPĂŞTEANU, NICOLAE ŢĂRANU, RALUCA HOHAN, PAUL CIOBANU Gh. Asach Techncal Unversty
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationEPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski
EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on
More informationAppendix B: Resampling Algorithms
407 Appendx B: Resamplng Algorthms A common problem of all partcle flters s the degeneracy of weghts, whch conssts of the unbounded ncrease of the varance of the mportance weghts ω [ ] of the partcles
More informationSolving Nonlinear Differential Equations by a Neural Network Method
Solvng Nonlnear Dfferental Equatons by a Neural Network Method Luce P. Aarts and Peter Van der Veer Delft Unversty of Technology, Faculty of Cvlengneerng and Geoscences, Secton of Cvlengneerng Informatcs,
More informationThe classical spin-rotation coupling
LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spn-rotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd
More informationAn Algorithm to Solve the Inverse Kinematics Problem of a Robotic Manipulator Based on Rotation Vectors
An Algorthm to Solve the Inverse Knematcs Problem of a Robotc Manpulator Based on Rotaton Vectors Mohamad Z. Al-az*, Mazn Z. Othman**, and Baker B. Al-Bahr* *AL-Nahran Unversty, Computer Eng. Dep., Baghdad,
More informationOn the Multicriteria Integer Network Flow Problem
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of
More informationEN40: Dynamics and Vibrations. Homework 4: Work, Energy and Linear Momentum Due Friday March 1 st
EN40: Dynamcs and bratons Homework 4: Work, Energy and Lnear Momentum Due Frday March 1 st School of Engneerng Brown Unversty 1. The fgure (from ths publcaton) shows the energy per unt area requred to
More informationPart C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis
Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More information] (1) Fuzzy Trajectory Control Design for Underwater Robot. Abstract. 2 Equations of Motion. 1 Introduction
EUSFLAT - LFA 5 Fuzzy Trajectory Control Desgn for Underwater Robot Jerzy Garus Department of Electroncs and Electrcal Engneerng Naval Unversty 8-3 Gdyna, ul. Śmdowcza 69 Poland j.garus@amw.gdyna.pl Zygmunt
More informationLesson 5: Kinematics and Dynamics of Particles
Lesson 5: Knematcs and Dynamcs of Partcles hs set of notes descrbes the basc methodology for formulatng the knematc and knetc equatons for multbody dynamcs. In order to concentrate on the methodology and
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationSpring 2002 Lecture #13
44-50 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallel-as Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the md-term
More informationModeling and Simulation of a Hexapod Machine Tool for the Dynamic Stability Analysis of Milling Processes. C. Henninger, P.
Smpack User Meetng 27 Modelng and Smulaton of a Heapod Machne Tool for the Dynamc Stablty Analyss of Mllng Processes C. Hennnger, P. Eberhard Insttute of Engneerng project funded by the DFG wthn the framework
More informationSCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS SEMESTER 2 EXAMINATIONS 2011/2012 DYNAMICS ME247 DR. N.D.D. MICHÉ
s SCHOOL OF COMPUTING, ENGINEERING ND MTHEMTICS SEMESTER EXMINTIONS 011/01 DYNMICS ME47 DR. N.D.D. MICHÉ Tme allowed: THREE hours nswer: ny FOUR from SIX questons Each queston carres 5 marks Ths s a CLOSED-BOOK
More informationThe Finite Element Method
The Fnte Element Method GENERAL INTRODUCTION Read: Chapters 1 and 2 CONTENTS Engneerng and analyss Smulaton of a physcal process Examples mathematcal model development Approxmate solutons and methods of
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationA NUMERICAL COMPARISON OF LANGRANGE AND KANE S METHODS OF AN ARM SEGMENT
Internatonal Conference Mathematcal and Computatonal ology 0 Internatonal Journal of Modern Physcs: Conference Seres Vol. 9 0 68 75 World Scentfc Publshng Company DOI: 0.4/S009450059 A NUMERICAL COMPARISON
More informationEN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics
N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?.
More informationPower law and dimension of the maximum value for belief distribution with the max Deng entropy
Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy Bngy Kang a, a College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Abstract Deng
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationStudy Guide For Exam Two
Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 01-06 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationParabola Model with the Application to the Single-Point Mooring System
Journal of Multdscplnary Engneerng Scence and Technology (JMEST) ISSN: 58-93 Vol. Issue 3, March - 7 Parabola Model wth the Applcaton to the Sngle-Pont Moorng System Chunle Sun, Yuheng Rong, Xn Wang, Mengjao
More informationAnalysis of Dynamic Cross Response between Spindles in a Dual Spindle Type Multi-Functional Turning Machine
Journal of Power and Energy Engneerng, 2013, 1, 20-24 http://dx.do.org/10.4236/jpee.2013.17004 Publshed Onlne December 2013 (http://www.scrp.org/journal/jpee) Analyss of Dynamc Cross Response between Spndles
More informationThe optimal delay of the second test is therefore approximately 210 hours earlier than =2.
THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More informationFirst Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.
Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act
More informationConservation of Angular Momentum = "Spin"
Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
More informationA large scale tsunami run-up simulation and numerical evaluation of fluid force during tsunami by using a particle method
A large scale tsunam run-up smulaton and numercal evaluaton of flud force durng tsunam by usng a partcle method *Mtsuteru Asa 1), Shoch Tanabe 2) and Masaharu Isshk 3) 1), 2) Department of Cvl Engneerng,
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationHomework 2: Kinematics and Dynamics of Particles Due Friday Feb 7, 2014 Max Score 45 Points + 8 Extra Credit
EN40: Dynamcs and Vbratons School of Engneerng Brown Unversty Homework : Knematcs and Dynamcs of Partcles Due Frday Feb 7, 014 Max Score 45 Ponts + 8 Extra Credt 1. An expermental mcro-robot (see a descrpton
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationPhysics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More informationAngular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004
Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a
More informationLecture 23: Newton-Euler Formulation. Vaibhav Srivastava
Lecture 23: Newton-Euler Formulaton Based on Chapter 7, Spong, Hutchnson, and Vdyasagar Vabhav Srvastava Department of Electrcal & Computer Engneerng Mchgan State Unversty Aprl 10, 2017 ECE 818: Robotcs
More informationLATERAL AUTOPILOT DESIGN FOR A UAV USING COEFFICIENT DIAGRAM METHOD
th INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES LATERAL AUTOPILOT DESIGN FOR A UAV USING COEFFICIENT DIAGRAM METHOD Ru Hrokawa, Koch Sato MtsubshElectrcKamakuraWorks Keywords: Flght Control Systems,Unmanned
More informationImportant Dates: Post Test: Dec during recitations. If you have taken the post test, don t come to recitation!
Important Dates: Post Test: Dec. 8 0 durng rectatons. If you have taken the post test, don t come to rectaton! Post Test Make-Up Sessons n ARC 03: Sat Dec. 6, 0 AM noon, and Sun Dec. 7, 8 PM 0 PM. Post
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationInvestigation on the Wheel/Rail Contact and Longitudinal Train/Track Interaction Forces
Investgaton on the Wheel/Ral Contact and Longtudnal Tran/Track Interacton Forces BY ALI AFSHARI B.S., Iran Unversty of Scence and Technology, 2004 M.S., Sharf Unversty of Technology, 2006 THESIS Submtted
More informationMODEL OF HYDROPNEUMATIC THREE POINT HITCH
ENINEERIN FR RUR DEVEPMENT Jelgava, 3.-4.05.03. MDE F YDRPNEUMTI TREE PINT IT Jans acekls-bertmans, Erks Kronbergs atva Unversty of grculture jans.lacekls@llu.lv, erks.kronbergs@llu.lv bstract. Ths paper
More informationRESEARCH REGARDING FRICTION INFLUENCE OF WIRES TO JOINTS INTERIOR ON PRECISION POSITIONING OF A ROBOTIC ARM
Internatonal Journal of Modern Manufacturng Technologes ISSN 2067 3604, Vol. VIII, No. 1 / 2016 RESEARCH REGARDING FRICTION INFLUENCE OF WIRES TO JOINTS INTERIOR ON PRECISION POSITIONING OF A ROBOTIC ARM
More informationChapter 3. r r. Position, Velocity, and Acceleration Revisited
Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector
More informationAn efficient algorithm for multivariate Maclaurin Newton transformation
Annales UMCS Informatca AI VIII, 2 2008) 5 14 DOI: 10.2478/v10065-008-0020-6 An effcent algorthm for multvarate Maclaurn Newton transformaton Joanna Kapusta Insttute of Mathematcs and Computer Scence,
More information