A NUMERICAL COMPARISON OF LANGRANGE AND KANE S METHODS OF AN ARM SEGMENT

Transcription

1 Internatonal Conference Mathematcal and Computatonal ology 0 Internatonal Journal of Modern Physcs: Conference Seres Vol World Scentfc Publshng Company DOI: 0.4/S A NUMERICAL COMPARISON OF LANGRANGE AND KANE S METHODS OF AN ARM SEGMENT AZMIN SHAM RAMELY School of Mathematcal Scences, Faculty of Scence & Technology, Unverst Kebangsaan Malaysa, ang Selangor, Malaysa asr@ukm.my NORHAFIZA A. HALIM School of Mathematcal Scences, Faculty of Scence & Technology, Unverst Kebangsaan Malaysa, ang Selangor, Malaysa feeza@ukm.my ROKIAH ROZITA AHMAD School of Mathematcal Scences, Faculty of Scence & Technology, Unverst Kebangsaan Malaysa, ang Selangor, Malaysa rozy@ukm.my A -D model of a two-lnk knematc chan s developed usng two dynamcs equatons of moton, namely Kane s and Lagrange Methods. The dynamcs equatons are reduced to frst order dfferental equaton and solved usng modfed Euler and fourth order Runge Kutta to approxmate the shoulder and elbow jont angles durng a smash performance n badmnton. Results showed that Runge-Kutta produced a better and exact approxmaton than that of modfed Euler and both dynamc equatons produced better absolute errors. Keywords: Arm segment; Kane s method; numercal approach.. Introducton Modelng of an upper lmb of human has attracted many researches n the feld of human movement. Several studes on modelng of the arm have been done by Ref. -5. In the modelng process, knematc data of real human movement s used and a smulaton of the movement can be performed to study the causal factors that lead to the observed movements. However, durng the smulaton process errors can be accumulated profoundly. In ths study, a two-dmensonal knematc chan model for two segments of an arm s developed. The arm model represents segments of the upper arm and forearm n the sagttal plane. The equaton of motons for ths model s wrtten usng two methods, Kane s and Lagrange. The dynamc equaton s reduced to frst order dfferental 68

2 A Numercal Comparson of Langrange and Kane s Methods of an Arm Segment 69 equatons and s solved usng the modfed Euler method and classcal Runge-Kutta of order four to approxmate angles of shoulder and elbow durng a performance of badmnton smash. Thus the objectve of the paper s to nvestgate the accuracy of two numercal ntegraton methods, the modfed Euler and classcal Runge-Kutta of order four.. Kane s Method Kane s equaton s a dynamc equaton for mult-body systems developed by Ref. 6. Kane s method of dynamc equatons uses vector cross and dot products of vectors. Ths method ntroduces a concept of generalzed speed, partal velocty, partal angular velocty, generalzed actve forces and generalzed nertal forces. To generate knematc equatons, expresson of the angular velocty and the angular acceleraton of rgd body should be derved. Angular velocty and angular acceleraton can be obtaned through the cosne table 7. Generalzed actve forces s a scalar quantty whch ncludes the contrbuton of actve forces to the dynamc equatons of moton, whle the generalzed nerta force s a scalar quantty whch ncludes the contrbuton quanttes of nerta nerta force and torque for the dynamc equatons of moton... Kane s dynamc equatons of moton Dynamc equaton of moton nvolves two forces, namely the generalzed actve forces and generalzed nertal forces. These equatons are solved smultaneously and reduced to frst order dfferental equatons as a functon of angle and angular velocty. The dynamc equatons of moton for the frst chan s, F F = [ qɺɺ I m ρ ml m ρl cos q m ρ I * * A A A A * A qɺɺ m ρl cos q m ρ I m ρl qɺɺ q sn q A * A m ρl qɺ sn q ] m gρ cos q m g l cos q A A A A ρ cos qɺ qɺ fl sn q fl sn q q f Whle that of the second chan s gven by, A l Acos q f cos q q T. l F F = [ qɺɺ m ρl cos q m ρ I qɺɺ m ρ I * A * * mρl Aqɺ sn q ] m gρcos q q fl sn q q fl cos q q T... Lagrange s dynamc equatons of moton Lagrangan formulaton uses the Lagrangan functon, L, for a system whch s defned to be the dfference between knetc and potental energes expressed as a functon of postons and veloctes. The dynamc equaton of moton for the frst lnk s,

3 70 A. S. Rambely, N. A. Halm and R. R. Ahmad Al A A l A ρ l Al l Aρ ρ ɺɺ l Al sn T = m I m m m cosq I qɺɺ m cosq m I q m q qɺɺ q l ml ρ sn q qɺ m gρ m g cosq m gρ cos q q A A A A 3 and that of the second lnk s gven by, 3. Solvng the Dfferental Equaton System for the Smplfed Model of the Arm Segment The resultng dynamc equatons of moton are reduced to frst order dfferental equatons to be solved usng numercal methods, namely Modfed Euler method and fourth order Runge-Kutta method. Ths can be done by ntroducng the method of state space wth four new varables x, x, x 3, and x 4 as x = q, x =ɺ q, x3 = q, and x4 =ɺ q. The dynamc equaton of moton wll form four frst order dfferental equatons as follows: where * * A xɺ = x C3cos x3 C A C A xɺ = C3cos x3 C C C3cos x3 C xɺ = x 3 4 C3cos x3 C 3 3 A C C cos x A xɺ 4 = C3cos x3 C C C C3cos x3 A = T fl sn q fl sn q q fl cos q fl cos q q A A C3x4 sn x3 C3 x x4 sn x3 C4cos x C5cos x x3 C x sn x A C cos x x f q q f cos q q T C = I I ml m ρ m ρ 3 3 = 5 3 l sn l C = I m ρ C 3 * = ml l A C = m gρ m gl C l Aρ ρ ɺɺ ρ T = m cosq m I q m I qɺɺ ml ρ sn q qɺ m gρ cos q q A 4 A A A 5 = m gρ. A A A A 4 5 6

4 3.. Modfed Euler method A Numercal Comparson of Langrange and Kane s Methods of an Arm Segment 7 The Modfed Euler method s gven by x = x k k where k = hf t, x k = hf t, x k The Fourth Order Runge-Kutta Method The Runge-Kutta equaton s gven by x = x k k k3 k4 6 where k = hf t, x h k k = hf t x h k k3 = hf t x k = hf t h, x k.,, 4 3 The expanson of the Modfed Euler and Runge-Kutta methods produces new values for x, x, x3, and x 4 and requres two functons g t, x, x, x3, x 4, and j t, x, x, x3, x4 wth ntal values g j q 0 x, x, x, and x where s a functon of the frst knematc lnk s a functon of the second knematc lnk = x s shoulder angle q = x s shoulder angular speed q 0 = x s elbow angle, and 3 0 q = x s elbow angular speed Results and Dscusson The ntegraton of the dfferental equatons system that represents the reduced model of the arm segment has been made n MatLab 7.0, usng the above notatons and the two numercal methods. Graphc representatons of the actual values from expermental procedures and predcted values of shoulder and elbow angles from Modfed Euler and Runge-Kutta methods have been obtaned for both Kane s Fg. and Lagrange s method Fg.. 8

5 7 A. S. Rambely, N. A. Halm and R. R. Ahmad a b c Fg.. a Actual values for shoulder and elbow angles b Predcted values from Modfed Euler method c Predcted values from fourth order Runge-Kutta Method va Kane s Method. a b c Fg.. a Actual values for shoulder and elbow angles b Predcted values from Modfed Euler method c Predcted values from fourth order Runge-Kutta Method va Langrange s Method. From the graphs obtaned n Fg. and, results show that the modfed Euler method s not sutable for the ntegraton of dynamc equatons for both methods. The method gves a less accurate soluton than that of the angles obtaned by fourth order Runge- Kutta method. To calculate the accuracy of the numercal methods used, errors are calculated for the two numercal methods. Ths error s the absolute error, whch s calculated based on the formula, Absolute error = true value - approxmate value. Ths error s the dfference between the actual and the approxmated angles, obtaned from the ntegraton usng the modfed Euler and Runge-Kutta methods for both Kane s and Lagrage s Methods, as shown n Fg. 3 and 4, respectvely.

6 A Numercal Comparson of Langrange and Kane s Methods of an Arm Segment 73 Fg. 3. Calculated errors for both shoulder and elbow angles through modfed Euler and fourth order Runge Kutta of Kane s Method. Fg. 4. Calculated errors for both shoulder and elbow angles through modfed Euler and fourth order Runge Kutta methods of Lagrange s Method. ased on the graphs above, the Runge-Kutta method produces a smaller error than that of the modfed Euler method. It s concluded that the fourth order Runge-Kutta method gves a more accurate approxmaton of the jont values compared to that of the modfed Euler method. Snce both equatons of moton are found to be smlar n predcted values through an ntegraton process usng the Runge-Kutta method, a comparson of the errors produced between both equatons of motons are observed. Fgure 5 shows the comparson of the errors of each jont angle.

7 74 A. S. Rambely, N. A. Halm and R. R. Ahmad Fg. 5. Comparson of errors for both equatons of moton va Runge-Kutta Method. ased on the graph, Kane s method s found to produce a smaller error than that usng the Lagrange method. Kane s method takes nto account the external forces n producng the dynamc equaton of moton whch gves more accurate results than that of Lagrange. Lagrange method nvolves calculatng the knetc and potental energy regardless of external forces actng on the body. However, the errors obtaned for both methods of Kane and Lagrange are not much dfferent. Thus t s concluded that both methods are sutable n producng the dynamc equatons. 5. Concluson The paper dscusses a comparson of predcted values of jont angles durng a performance of badmnton smash, solved by an ntegraton of frst order dfferental equaton usng modfed Euler and fourth order Runge-Kutta numercal methods. The moton was modeled by a planar two-lnk knematc chan va Kane s and Lagrange methods. The accuracy of the numercal methods used s compared n terms of the errors obtaned. Results showed that the modfed Euler method s not sutable for the ntegraton of dynamc equatons of moton for ths method gves a less accurate approxmaton than the graph of the angle obtaned by that of Runge-Kutta method. The calculated errors showed that Kane s method produce a smaller error than that of Lagrange equaton of moton. In concluson, Kane s method produced a smaller error n the smulaton of an arm segment, specfcally n the smash performance n badmnton games. Acknowledgments The research s funded by the FRGS grant UKM-ST-07-FRGS from the Mnstry of Hgher Educaton, Malaysa.

8 A Numercal Comparson of Langrange and Kane s Methods of an Arm Segment 75 References. E. Pennestr, R. Stefanell, P. P. Valentn and L. Vta, Vrtual musculo-skeletal model for the bomechancal analyss of the upper lmb, J. omech. 40, T. Flash and N. Hogan, The coordnaton of arm movements: An Expermentally Confrmed Mathematcal Model, The J. of Neuroscence, 57, T. J. McCue, C. E. Guse and R. L. Dempsey, Upper Extremty Pan Seen Wth Fly-Castng Technque: A Survey of Fly-Castng Instructors, Wlderness and Env. Med., 5, F. H. M. Arff and A. S. Rambely, Modelng of an arm va Kane s method: An nverse dynamc approach, Euro. J. of Sc. Research, 33, W. R. Wan Dn, A. S. Rambely and A. A. Jeman, omechancs of a Rfle-Frng Model: Effects of Rfle Dynamcs on Target Accuracy, Int. J. Appl. Maths. & Stats. 3, D T. R. Kane and D. A. Levnson. Dynamcs: Theory and Applcatons, McGraw-Hll, New York, G. T. Yamaguch. Dynamc Modellng of Musculoskeletal Moton: A Vectorzed Approach for omechancal Analyss n Three Dmensons, Sprnger, New York, 006.