STATIC OPTIMIZATION: BASICS

Transcription

1 STATIC OPTIMIZATION: BASICS 7A- Lecture Overvew What s optmzaton? What applcatons? How can optmzaton be mplemented? How can optmzaton problems be solved? Why should optmzaton apply n human movement? How can muscle forces be estmated? Example I What objectve functons are relevant? Evaluaton Crtera Example II Example III

2 WHAT IS OPTIMIZATION? 7A- Optmzaton s the process of mnmzng or maxmzng the costs/benefts of some acton. Example. If you have money to nvest you would try and optmze your return by maxmzng the nterest you get on your money. Example. You have money to nvest, but the hgher nterest accounts nvolve rsk, so have two competng functons to balance to obtan maxmum return (nterest and rsk). [Statc Optmzaton refers to the process of mnmzng or maxmzng the costs/benefts of some acton for one nstant n tme only.]

3 WHAT APPLICATIONS? 7A-3 Optmzaton has many applcatons:- Economcs Mechancal Desgns Geographcal Models Parameter Estmaton Analyss of Human Movement and many others

4 HOW CAN OPTIMIZATION BE IMPLEMENTED? 7A-4 There are well establshed mathematcal technques for solvng statc optmzaton problems. These problems are formulated n the followng way Objectve Functon Mnmze U ( X X,..., ) subject to, X n Equalty constrants Φ ( X X..., X ) 0 k,, n = k =, e Inequalty constrants Φ ( X X,..., X ) 0 I, n I =, f Where n - number of ndependent varables X - ndependent varables e - number of equalty constrants, f - number of nequalty constrants.

5 HOW CAN OPTIMIZATION BE IMPLEMENTED? 7A-5 Mnmzaton versus Maxmzaton If t s a functon U, then s equvalent to Maxmze U ( X X,..., ), X n Mnmze U ( X, X,..., ) X n Global Mnmum mnmum possble value of a functon. Local Mnmum mnmum possble value of a functon n a gven neghborhood.

6 HOW CAN OPTIMIZATION PROBLEMS BE SOLVED? 7A-6 There are many algorthms for solvng optmzaton problems. Lnear Objectve Functons If the objectve functon s lnear the commonest technque s the Smplex algorthm (Dantzg, 963). Lmtatons The nature of the soluton to these lnear problems requres that the number of non-zero varables s less than the total number of constrants (equalty and nequalty) plus one. Therefore wthout the mposton of equalty constrants a lnear objectve functons wll only maxmze/mnmze one varable.

7 HOW CAN OPTIMIZATION PROBLEMS BE SOLVED? 7A-7 There are a varety of numercal technques for solvng non-lnear optmzaton problems. Technques nclude:- The drect search technque of Hooke and Jeeves (96). Ths procedure does not account for constrants, therefore another routne has to be run n conjuncton wth t so that a penalty functon constrans the soluton (Facco and McCormck, 968). Another popular technque s the Downhll Smplex (Nelder and Mead, 965). The most powerful class of these technques requres evaluaton of the functon and ts dervatves, gradent search technques (e.g. Powell, 970). The determnaton of a global mnmum for non-lnear objectve functons cannot be guaranteed usng numercal technques (Sddall, 98).

8 HOW CAN OPTIMIZATION PROBLEMS BE SOLVED? 7A-8 It s possble usng Lagrangan multplers and formng the resultant Lagrangan Functon to fnd an analytcal soluton to the problem to be mnmzed for certan classes of objectve functons (Bertsekas, 976). Example The general problem can be defned as fndng the mnmum of U ( ) ( ) F M F M = n = P P > The mnmum must be found subject to the constrant g n r = ( F ) = M M. F M The Lagrangan Functon s:- L F, λ = U F + λ. g = 0 ( ) ( ) ( F ) M M Where λ s a Lagrangan multpler. M

9 HOW CAN OPTIMIZATION PROBLEMS BE SOLVED? 7A-9 Soluton of the Lagrangan Functon for the force n the j th muscle gves F J = M. r. n j = r r j P P Ths s analytcal soluton to how the muscle forces would be shared to satsfy a gven resultant jont moment when the task s to mnmze the sum of the all the muscle forces producng the moment.

10 WHY SHOULD OPTIMIZATION APPLY IN HUMAN MOVEMENT? 7A-0 Chao and An (978) "However, n examnng the voluntary functons of an anatomcal structure, there s a strkng ablty for t to produce consstent functonal actvtes. Such unque ablty must be controlled by certan physologcal laws based on whch proper solutons to the redundant problem may be obtaned. Consequently, t would be mportant for the bomechancans to quanttate these laws, not only for the purpose of seekng basc understandng of how musculoskeletal systems functon but also to establsh workable models and theores for the analyss of abnormal functons n the hope of provdng valuable clncal nformaton." (page 59).

11 WHY SHOULD OPTIMIZATION APPLY IN HUMAN MOVEMENT? 7A- Examples of Optmzed Movement Cotes and Meade (960) examnng horzontal treadmll walkng showed that the strde frequency selected by the walkers requred the least energy expendture, compared wth walkng at strde frequences less than or greater than the selected frequency. Cavanagh and Wllams (98) examnng runnng at a constant speed showed subjects selected a strde length whch mnmzed the energy cost for runnng at that speed. McMahon, Valant, and Frederck (987) presented evdence to suggest that the preferred style of runnng also requres the lowest energy cost. Brett (965) presented evdence to suggest that swmmng salmon also select the most effcent style. Consder natural selecton.

12 HOW CAN MUSCLE FORCES BE ESTIMATED? Consder the followng case 7A- F M F M r M r M T J Two muscles crossng jont wth moment arms rm = 00. m rm = 003. m TJ = 30 N. m (flexon) T J = 30 = 0.0 FM FM Soluton (?) FM = N FM = N

13 HOW CAN MUSCLE FORCES BE ESTIMATED? 7A-3 Objectve Functon Mnmze U ( X X,..., ) Mnmze F = F + F M M M, X n Equalty constrants Φ ( X X..., X ) 0 k,, n = 0.0 F M FM 30 = 0. ( ) 0 k =, e Inequalty constrants Φ ( X X,..., X ) 0 I, n. Muscle can only generate force therefore F M 0N F M 0N I =, f. Muscles has lmted maxmum muscle force, assume both muscle have same cross-sectonal area, therefore F M 800 N F M 800 N Soluton F M = 600 N F M = 800 N

14 7A-4 WHAT OBJECTIVE FUNCTIONS ARE RELEVANT? MacConall (967) "Prncple of Mnmal Total Muscular Force states that 'no more total muscular force s used than s both necessary and suffcent for the task to be performed, whether ths be one of supportng some weght or carryng out a movement, the resstance to whch may vary from zero upwards.'" (page 43). Penrod, Davy, and Sngh (974) proposed that muscles are recruted to satsfy a functon whch mnmzes the sum of the muscle forces, whch they clamed was equvalent to mnmzng the muscular effort. They clamed that ths functon "...s ntutvely appealng and may represent an accurate pcture for a normal, healthy system famlar wth the loads t sustans." (page 8) Crownnsheld and Brand (98) proposed that mnmzng the muscle stress effectvely ncreased endurance tme, so was relevant.

15 7A-5 WHAT OBJECTIVE FUNCTIONS ARE RELEVANT? Muscle Force U = F = U = 3 F = Muscle Stress U = = σ U = = = F CSA = = F CSA 3 = σ 3 Relatve Muscle force U = = F F MAX = = F REL U = = F F MAX 3 = = F 3 REL

16 EVALUATION CRITERIA 7A-6 Two Prncple Methods EMG - Use EMG sgnal to ndcate when the muscles compare ths wth the actvty of the muscles as predcted by the optmzaton procedure. Drect Measurement compare optmzaton predcted muscle forces wth drectly measured muscle forces.

17 EXAMPLE II 7A-7 Objectve Functon U = F = Based on Prncple of Mnmal Total Muscular Force (MacConall, 967). Muscle Data Muscle Moment Arm Maxmum Force Maxmum Moment Bceps m N.6 N.m Brachals 0.0 m N.0 N.m Brachoradals m 6. N 3.9 N.m Results Jont Moment Bceps Force Brachals Force Brachoradals Force N 0.0 N 9.6 N N 0.0 N 85. N N 0.0 N 77.8 N N 0.0 N N

18 EXAMPLE II 7A-8 Objectve Functon U = F = Queston: What s effect of mposng constrants? Bceps force < N Brachals force < N Brachoradals force< 6. N Note Basmajan and Latf (957) Integrated actons and functons of the chef flexors of the elbow: A detaled electromyographc analyss. Journal of Bone and Jont Surgery 39A, 06-8.

19 EXAMPLE III 7A-9 Source: Herzog, W., and Leonard, T.R. (99) Valdaton of optmzaton models that estmate the forces exerted by synergstc muscles. Journal of Bomechancs 4, Suppl., Functons Evaluated. U = End.Tme =. U = = 3. U = = 4. U = = F CSA F F F 3 MAX 3 F 5. U = = σ = CSA =

20 REVIEW QUESTIONS 7A-0 ) What s meant by optmzaton? ) Wrte an objectve functon, equalty constrant(s), and nequalty constrant(s) whch could be used to estmate muscle forces. 3) What optons are avalable for valdatng optmzaton based routnes for estmatng muscle forces? 4) Suggest three objectve functons whch could be used n estmatng muscle forces. What s the problem(s) wth constranng these objectve functons usng each muscles maxmum sometrc force? 5) Wth reference to work you have read descrbe two applcatons for statc optmzaton problems.