Nonlinear Equations Nonlinear system of equations Robotic arms - PowerPoint PPT Presentation

Nonlinear Equations

Nonlinear system of equations

Robotic arms https://www.youtube.com/watch?v=NRgNDlVtmz0 (Robotic arm 1) https://www.youtube.com/watch?v=9DqRkLQ5Sv8 (Robotic arm 2) https://www.youtube.com/watch?v=DZ_ocmY8xEI (Blender)

Inverse Kinematics Given : n a i.ee#----*fflQl--oI Him .is f- , ( Oi , 02,8 ) = O . :* : : : iterative =

Nonlinear system of equations Goal: Solve P Q = R for P: ℛ ; → ℛ ; = = Elk ) e - - :c : : * ⇒ if ÷ : ink . x. x . . . . . . sew - ¥4 - fi -42-24 × 2 - o ? - 4 - It ? If - → = - 2X , -3 × 2+5--0 f- z

( ND ) Newton’s method Approximate the nonlinear function P Q by a linear function using Z Taylor expansion: 1- ( Its ) - I t.IE ) g - Ef 'T vector - vector - vector i :::÷÷÷÷÷i÷÷÷÷ vector :÷÷÷÷÷¥÷÷

f- Ets ) EEE ) t EE ) E ÷ O O ICI t EG ) E - on for → solve E |IEIE=-f system Linear of equation - , I - vector Io - initial - Intf IKH

IE ) - - e Newton’s method Algorithm: : initial guess Io . Is :÷÷:÷÷÷ for i = I , R , - - - an .÷÷÷÷÷¥¥÷÷ ri - evaluate Jkr ) - . :S of equation Convergence: • Typically has quadratic convergence - ④ • Drawback: Still only locally convergent O r Cost: • Main cost associated with computing the Jacobian matrix and solving - # the Newton step.

↳ E- Example Consider solving the nonlinear system of equations 2 = 2) + + µ 4 = + $ + 4) $ What is the result of applying one iteration of Newton’s method with the following initial -0¥ 't guess? . :D - % = 1 0 its :] :¥¥¥**i::F÷¥÷÷ E- rats . . =L 'oItfz .FI?oIs ] es .

Newton’s method ! 0 = #$#%#&' ()*++ ,-. / = 1,2, … Evaluate 4 = 5 ! 1 Evaluate 6 ! 1 Factorization of Jacobian (for example 78 = 5 ) we Solve using factorized J (for example 78 9 1 = −6 ! 1 - Update ! 123 = ! 1 + 9 1

Newton’s method - summary q Typically quadratic convergence (local convergence) - o q Computing the Jacobian matrix requires the equivalent of S ) function evaluations for a dense problem (where every function of P Q depends on every component of Q ). q Computation of the Jacobian may be cheaper if the matrix is sparse. E q The cost of calculating the step T is U S < for a dense Jacobian matrix (Factorization + Solve) q If the same Jacobian matrix V Q . is reused for several consecutive iterations, the convergence rate will suffer accordingly (trade-off between cost per iteration and number of iterations needed for convergence)

Inverse Kinematics X , y , p → 9 , Oz , 03 = Nifty ✓ ^ ¥i¥ : a a.bgivezga.a.no : Eat im r i . ← *# T fi-E-a-bt2abcos.bz#b2--atE-2ac = of tb c ' - Iab cos Oz → cos a , → ¥=b?aIt2accos(Oi-af=J - da ) = 360T 4/80-0 , -0 e) to z t p +904940 fffs=-9-0ztOstpTI - 69¥ If ? If , I