NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

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1 NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

2 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!! We wll now consder mass, nerta, and gravty The Forward Dynamc Prolem Gven the jont torques (τ s), or tool forces (F), what root moton (.e. q, q, q or pose, pose, pose) would ths gve rse to? Moton Smulaton Forward Dynamcs qqq,, The Inverse Dynamc Prolem Gven a desred root moton (.e. q, q, q or pose, pose, pose) what jont torques (τ s), or tool forces (F), would e requred to cause such moton? Control qd, qd, qd Inverse Dynamcs ME 482/582: Rootcs Engneerng Slde 2 / 17

3 7. Root Dynamcs 7.2 Intro to Root Dynamcs Two Common Approached to Constructng the Dynamc Model of a Root 1. The Newton-Euler Approach Velocty propagaton from ase to tool, and Force propagaton from tool to ase 2. The Euler-Lagrange Approach Derved va applcaton of the Euler-Lagrange equaton Based on a result from the Calculus of Varatons Mnmze the energy of the system ME 482/582: Rootcs Engneerng Slde 3 / 17

4 7. Root Dynamcs 7.3 An Intro. To Calculus of Varatons Gven, Fnd a soluton y such that I has a statonary value. Let y t = y(t) e the requred soluton. Therefore, any other admssle curve can e descred y: Where. ε s an artrarly small quantty. y( a) y( a) ( a) ( ) 0 y( ) y( ). η(t) s ndependent of ε Therefore, I f ( t, y( t), y( t)) dt a y( t) y( t) ( t) I( ) f ( t, y, y ) dt a (7.1) ME 482/582: Rootcs Engneerng Slde 4 / 17

5 7. Root Dynamcs 7.3 An Intro. To Calculus of Varatons From (5.1), we know that the requred soluton occurs at ε = 0. At a statonary pont di dε = 0. di( ) f ( ) dy f ( ) dy dt 0 d y d y d 0 a 0 f( ) f( ) dt 0 (7.2) y y a 0 Now, ntegratng equaton (7.2) y parts a f ( ) f ( ) d f ( ) dt dt y y dt y a a d f() dt dt y a ME 482/582: Rootcs Engneerng Slde 5 / 17

6 7. Root Dynamcs 7.3 An Intro. To Calculus of Varatons Returnng to equaton (5.2), we have Ths can only e true f whch s to say that di( ) f ( ) d f ( ) dt 0, admssle ( t) d 0 y dt y a f ( ) d f ( ) y dt y 0 f ( ) d f ( ) y dt y y y y y 0 0 Ths result suggests that the optmal path wll oey ths equaton! 0 ME 482/582: Rootcs Engneerng Slde 6 / 17

7 7. Root Dynamcs 7.3 An Intro. To Calculus of Varatons Ths result can e extended to the case of n-ndependent varales (q = q 1, q 2,, q n T, say), n the presence of an external force τ (say), gvng rse to f ( q, q) d f ( q, q), 1,..., n (7.3) q dt q Ths s the so-called Euler-Lagrange equaton, n n-varales. ME 482/582: Rootcs Engneerng Slde 7 / 17

8 7. Root Dynamcs 7.3 An Intro. To Calculus of Varatons A Smple Example: Let s mnmze the energy of ths smple system, and therey determne the equatons that govern the moton of the mass m. mn ( x, x) dt where L denotes the total energy of the system. The knetc energy s 1 2 KE mx 2 The (change n) potental energy s PE 0 The energy stored n the sprng s 1 2 Sprng E kx mx 2 kx (energy of the system) g ME 482/582: Rootcs Engneerng Slde 8 / 17

9 7. Root Dynamcs 7.3 An Intro. To Calculus of Varatons Applyng the Euler-Lagrange, we get d d kx ( mx) 0 x dt x dt kx mx 0 (a force alance equaton) If an external force F s appled n the +ve x-drecton we get mx kx F At ths pont we can solve the aove dfferental equaton to determne the moton resultng from a partcular F(t). ME 482/582: Rootcs Engneerng Slde 9 / 17

10 7. Root Dynamcs 7.4 Lagrangan Dynamcs We can develop a generalzed verson of the dynamc model for (rgd) roots y usng the results from Calculus of Varatons and mnmzng the ntegral of total energy of the rootc system. where L К υ s known as the Lagrangan. mn ( К υ) dq mn Ths s referred to as the Euler-Lagrange approach. dq ME 482/582: Rootcs Engneerng Slde 10 / 17

11 7. Root Dynamcs Knetc and Potental Energy Consder a sngle (rgd) lnk of a n-lnk manpulator root arm. The knematc energy of ths lnk conssts of oth lnear (translatonal moton) and rotary (orentatonal moton) components, and can e stated ntutvely as: KE mv I 2 2 or n vector notaton as 1 T 1 T KE mvc vc 2 2 (7.4) wheren the term v( v c ) refers to the translatonal velocty of the center of mass of the lnk and the (3 3) matrx s the nerta matrx of the lnk and ω the cumulatve (or total) angular velocty. The frst term 1 2 v ct vc m s ndependent of the orentaton of the coordnate frame n whch t s expressed. ME 482/582: Rootcs Engneerng Slde 11 / 17

12 7. Root Dynamcs Knetc and Potental Energy To prove ths, consder Independent of orentaton, ascally the length (squared) of the vector v c. In the case of the second term 1 2 ωt ω, t s a good dea to compute the nerta matrx w.r.t. a coordnate frame attached to the ody (.e., lnk) Thus, wll e ndependent of the moton of the ody (constant w.r.t. tme). vc Rv 1 c 1 T 1 T T vc v 1 c m v 1 c R Rvcm T vc vcm 2 It s, however, necessary to compute the angular velocty w.r.t. that same reference (n whch was computed)!! ME 482/582: Rootcs Engneerng Slde 12 / 17

13 7. Root Dynamcs Knetc and Potental Energy Let's defne the angular velocty of the th lnk, descred n frame { } y R R (. e., R R) (7.5) where frame { } refers to the coordnate system (on lnk, usually the center of mass) n whch the nerta matrx was computed n. Let's try to smplfy equaton (7.5). Recallng the secton on moton knematcs we know that 0 0 ( ) z1 1 z 2 2 z Note that s a constant rotaton matrx!! R Also, f we compute the nerta matrx n the same orentaton as lnk frame {} ths results n I. Ths makes lfe smple!! R ME 482/582: Rootcs Engneerng Slde 13 / 17

14 7. Root Dynamcs Knetc and Potental Energy Illustraton of the lnk & center of mass & Inerta See handout for Inerta matrces of some common shapes MomentsOfInertaCommonShapes.pdf h yˆ l P c xˆ zˆ ˆ y c ˆ x c R c ˆ z c P P R P c c v 0 0 c d dt P c c 1 mh ml 0 1 mh 12 l 2 2 ME 482/582: Rootcs Engneerng Slde 14 / 17

15 7. Root Dynamcs Knetc and Potental Energy Fnally, equaton (7.4) can e appled to all of the n lnks of the root. Therefore, the total knematc energy of the n lnks s smply (7.6) Through knowledge of the moton knematcs the velocty v c n equaton (7.6) can e related to the jont rates q and v c n К К 1 1 n T T [ v v m ] J q v R c 1 c c 2 0 R J q ME 482/582: Rootcs Engneerng Slde 15 / 17

16 7. Root Dynamcs Knetc and Potental Energy Thus, (5.11) can e rewrtten as 1 T n T T T К q ( Jv (7.7) 1 c Jv ) c m J R R J q 2 1 T q D( q) q (7.8) 2 Equaton (7.8) descres the knematc energy of the root n terms of the jont varales (q) and rate of change of jont varales (q). T T T Note that, the matrx D( q) ( Jv Jv m J R R J ) s the sum of n symmetrc matrces, and s therefore a symmetrc matrx!! 1 It s also only a functon of q and NOT q!! n c c The potental energy of a ody s ntutvely m g h, or n vector form: where 0 g υ g 0 T 0 P m c s the gravtatonal vector (expressed n frame {0} coordnates). ME 482/582: Rootcs Engneerng Slde 16 / 17

17 7. Root Dynamcs Knetc and Potental Energy Hence, the potental energy of the n-lnks of the root s n υ υ 1 Assumng that g s expressed w.r.t. frame {0}. (7.9) Each 0 Pc can e expressed usng the forward knematcs as a functon of (q 1, q 2,, q ) n 1 g P m υ( q) 0 T 0 c The center of mass of lnk (.e., 0 Pc ) s smply a fxed pont n {}, and therefore does not depend on (q +1,, q n ), or the q 's. ME 482/582: Rootcs Engneerng Slde 17 / 17

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