Cartesian Control. Analytical inverse kinematics can be difficult to derive Inverse kinematics are not as well suited for small differential motions

Transcription

1 Catesian Contol Analtical invese kinematics can e diicult to deive Invese kinematics ae not as well suited o small dieential motions Let s take a look at how ou use the acoian to contol Catesian position

2 Catesian contol Let s contol the position (not oientation) o the thee link am end eecto: z q l s c lc lc c lc lc l c l c s l c l c l c l c l c l c l c s s z q z l q l z We can use the same stateg that we used eoe: q q q q

3 Catesian contol joint ctl z q l joint position senso q l z z l z q q q Howeve, this onl woks i the acoian is squae and ull ank All ows/columns ae lineal independent, o Columns span Catesian space, o Deteminant is not zeo

4 What i ou want to contol the twodimensional position o a thee-link manipulato? Catesian contol q z q q l l l c l c l c l c l c l c l s l s l s l s l s l s l q q q q q wo equations o thee vaiales each his is an unde-constained sstem o equations. multiple solutions thee ae multiple joint angle velocities that ealize the same EFF velocit.

5 I the acoian is not a squae mati (o is not ull ank), then the invese doesn t eist what net? Genealized invese l q l q We have: q We ae looking o a mati # such that: z q l q # q

6 Mooe-Penose Pseudoinvese Undeconstained manipulato: that minimizes suject to Oveconstained manipulato: that minimizes Reminde: -nom o

7 Contolling Catesian Position joint ctl joint position senso Old method

8 Contolling Catesian Position joint ctl joint position senso New method

9 Contolling Catesian Position joint ctl joint position senso Pocedue o contolling position:. Calculate position eo:. Multipl the velocit acoian pseudoinvese:

10 Contolling Catesian Position joint ctl joint position senso DEMO!

11 Calculating the pseudoinvese he pseudoinvese can e calculated using two dieent equations depending upon the nume o ows and columns: # # # Undeconstained case (i thee ae moe columns than ows (m<n)) Oveconstained case (i thee ae moe ows than columns (n<m)) I thee ae an equal nume o ows and columns (n=m) hese equations can onl e used i the acoian is ull ank; othewise, use singula value decomposition (SVD):

12 Calculating the pseudoinvese using SVD Singula value decomposition decomposes a mati as ollows: V U U V n # Fo an unde-constained mati, is a diagonal mati o singula values: m m n n m n U V # n V U

13 Calculating the pseudoinvese using SVD Image: wikimedia

14 Popeties o the pseudoinvese Mooe-Penose conditions: # # # # # # # # Genealized invese: satisies condition Releive genealized invese: satisies conditions and Pseudoinvese: satisies all ou conditions Othe useul popeties o the pseudoinvese: # # # #

15 Contolling Catesian Oientation How does this stateg wok o oientation contol? Suppose ou want to each an oientation o R d You cuent oientation is You ve calculated a dieence: How do ou tun this dieence into a desied # angula velocit to use in? R c q R R cd c R d R d # joint ctl R c FK(q) joint position senso

16 Contolling Catesian Oientation How does this stateg wok o oientation contol? Suppose ou want to each an oientation o R d You cuent oientation is You ve calculated a dieence: How do ou tun this dieence into a desied # angula velocit to use in? R c q R R cd c R d Answe: convet into ais angle epesentation HOW?

17 Ais-angle epesentation R SO R [, ) heoem: (Eule). An oientation,, is equivalent to a otation aout a ied ais,, though an angle (also called eponential coodinates) Ais: k k k k z Angle: Conveting to a otation mati: R k e S k I S k sin S k cos

18 Ais-angle epesentation heoem: (Eule). An oientation,, is equivalent to a otation aout a ied ais, R, though an angle [, ) Rodigues omula (also called eponential coodinates) Ais: k k k Soln to dieential equation: Conveting to a otation mati: k z R SO Den o angula velocit: p S p Angle: R k e S k I S k sin S k cos

19 Ais-angle epesentation Conveting to ais angle: Magnitude o otation: Ais o otation: Whee: kˆ R k sin cos tace( R) and: tace( R)

20 Contolling Catesian Oientation R d Conv aisangle # joint ctl R c FK(q) joint position senso

21 acoian anspose Contol he sto o Catesian contol so a:. q # q.

22 acoian anspose Contol Hee s anothe appoach: e e q e e e e q q q q q e q v e q q e Stat with a squaed position eo unction (assume the poses ae epesented as ow vectos) Position eo: Gadient descent: take steps popotional to in the diection o the negative gadient. e e

23 acoian anspose Contol he same appoach can e used to contol oientation: q cu k e oientation eo: ais angle oientation o eeence pose in the cuent end eecto eeence ame: cu ke

24 acoian anspose Contol So, evidentl, this is the gadient o that q e e e e acoian tanspose contol descends a squaed eo unction. Gadient descent alwas ollows the steepest gadient

25 acoian anspose v Pseudoinvese What gives? Which is moe diect? acoian pseudoinvese o tanspose? q o q # he do dieent things: anspose: move towad a eeence pose as quickl as possile One dimensional goal (squaed distance meteic) Pseudoinvese: move along a least squaes eeence twist tajecto Si dimensional goal (o whateve the dimension o the elevant twist is)

26 acoian anspose v Pseudoinvese he pseudoinvese moves the end eecto in a staight line path towad the goal pose using the least squaed joint velocities. he goal is speciied in tems o the eeence twist Manipulato ollows a staight line path in Catesian space d he tanspose moves the end eecto towad the goal position In geneal, not a staight line path in Catesian space Instead, the tanspose ollows the gadient in joint space d

27 Using the acoian o Statics Up until now, we ve used the acoian in the twist equation, q Inteestingl, ou can also use the acoian in a statics equation: w oint toques Wench: w m oce moment (toque)

28 Supplementa

29 Genealized invese wo cases: Undeconstained manipulato (edundant) Oveconstained Genealized invese: o the undeconstained manipulato: given, ind an vecto that minimizes s.t. q q o the oveconstained manipulato: given, ind an vecto s.t. Is minimized q q

30 acoian Pseudoinvese: Redundant manipulato Psuedoinvese deinition: (undeconstained) Given a desied twist,, ind a vecto o joint velocities, q, that satisies d while minimizing d ( q ) q q q q l l q Minimize joint velocities (z) g( z) Minimize suject to : Use lagange multiplie method: ( z) g( z) z z z q l his condition must e met when suject to g( z) (z) is at a minimum

31 acoian Pseudoinvese: Redundant manipulato z ( z) g( z) z ( q) q q g( q ) q Minimize Suject to ( q ) q g( q ) q q q q

32 acoian Pseudoinvese: Redundant manipulato q q q q q # q # I won t sa wh, ut i is invetile is ull ank, then So, the pseudoinvese calculates the vecto o joint velocities that satisies d q while minimizing the squaed magnitude o joint velocit ( ). heeoe, the pseudoinvese calculates the least-squaes solution. q q

33 Contolling Catesian Oientation You can t do this: Convet the dieence to ZYZ Eule angles: Multipl the Eule angles a scaling acto and petend that the ae an angula velocit: q # Rememe that in geneal: q R d # joint ctl R c FK(q) joint position senso

34 he Analtical acoian I ou eall want to multipl the angula acoian the deivative o an Eule angle, ou have to convet to the analtical acoian: q l l q q A A A q s c s s c c s z q Fo ZYZ Eule angles l Gimal lock: using an analtical acoian instead o the angula velocit acoian, ou intoduce the gimal lock polems we talked aout ealie into the acoian this essentiall adds singulaities (we ll talk moe aout that in a it )

35 Contolling Catesian Oientation he easiest wa to handle this Catesian oientation polem is to epesent the eo in ais-angle omat k q q l l q Ais angle delta otation Pocedue o contolling otation: z q l. Repesent the otation eo in ais angle omat: e. Multipl a scaling acto: e e. Multipl the angula velocit acoian # pseudoinvese: q e

36 Using the acoian o Statics It tuns out that oth wenches and twists can e undestood in tems o a epesentation o displacement known as a scew. heeoe, ou can calculate wok integating the dot poduct: W v m v m Wok in Catesian space W q Wok in joint space Consevation o eneg: q v m

37 Using the acoian o Statics Incemental wok (vitual wok) v m q q m q m m w w q vs Wench-twist dualit:

38 Note that twist can e epesented in dieent eeence ames: v k k k v Conside two eeence ames attached to the same igid od: v v wist: conveting etween eeence ames

39 wist: conveting etween eeence ames v v v I S I v v R R I S I R R v v R R S R v wist in ame wist in ame

40 Wench: conveting etween eeence ames Wench can also e epesented in dieent eeence ames: w m k w k k m

41 Use the vitual wok agument to deive the elationship: Wench: conveting etween eeence ames v m v m v m v R R S R m m R R S R m m R R S R m

42 Conveting wenches: Eample Use a 6-ais load cell isecting the second link to calculate wenches at the end eecto (the tip o the last link) e e R senso senso c s l s c l c l s e q l e l senso 6 ais load cell senso

43 Conveting wenches: Eample senso senso senso senso senso e senso e senso e e senso e e e e e m R R S R m, 6 ais load cell l l q e e senso senso senso senso senso senso e e e e m s l c l c l s l c s c l l s c s l c s s c m