Jacobian Range Space

Transcription

1 Kinematic Redundanc A manipuator ma have more DOFs than are necessar to contro a desired variabe What do ou do w/ the etra DOFs? However, even if the manipuator has enough DOFs, it ma sti be unabe to contro some variabes in some configurations

2 acobian Range Space Before we think about redundanc, et s ook at the range space of the acobian transform: he veocit acobian maps joint veocities onto end effector veocities: v v v : Q V Space of joint veocities his is the domain of : D v Space of end effector veocities his is the range space of : R v

3 acobian Range Space v : Q V In some configurations, the range space of the acobian ma not span the entire space of the variabe to be controed: v V, v R v R spans V if v V, v R v Eampe: a and b span this two dimensiona space: v a b

4 acobian Range Space his is the case in the manipuator to the right: In this configuration, the acobian does not span the direction or the direction) V, R v

5 Let s cacuate the veocit acobian: c c c c c c s s s s s s v oint configuration of manipuator: v v here is no joint veocit,, that wi produce a veocit, herefore, ou re in a singuarit. acobian Range Space

6 acobian Singuarities In singuar configurations: does not span the space of Cartesian veocities oses rank v ) v ) est for kinematic singuarit: If is ero, then manipuator is in a singuar configuration ) ) det something det det ) ) det Eampe:

7 acobian Singuarities: Eampe he four singuarities of the three-ink panar arm:

8 acobian Singuarities and Cartesian Contro Cartesian contro invoves cacuating the inverse or pseudoinverse: # However, in singuar configurations, the pseudoinverse or inverse) does not eist because is undefined. As ou approach a singuar configuration, joint veocities in the singuar direction cacuated b the pseudoinverse get ver arge: # s big In acobian transpose contro, joint veocities in the singuar direction i.e. the gradient) go to ero: s Where is a singuar direction. s s

9 acobian Singuarities and Cartesian Contro So, singuarities are most a probem for acobian pseudoinverse contro where the pseudoinverse bows up. Not much of a probem for transpose contro he worst that can happen is that the manipuator gets stuck in a singuar configuration because the direction of the goa is in a singuar direction. his stuck configuration is unstabe an motion awa from the singuar configuration wi aow the manipuator to continue on its wa.

10 acobian Singuarities and Cartesian Contro One wa to get the best of both words is to use the damped east suares inverse aka the singuarit robust SR) inverse: k I * Because of the additiona term inside the inversion, the SR inverse does not bow up. In regions near a singuarit, the SR inverse trades off eact trajector foowing for minima joint veocities. BW, another wa to hande singuarities is simp to avoid them this method is preferred b man More on this in a bit

11 Kinematic redundanc A genera-purpose robot arm freuent has more DOFs than are strict necessar to perform a given function in order to independent contro the position of a panar manipuator end effector, on two DOFs are strict necessar If the manipuator has three DOFs, then it is redundant w.r.t. the task of controing two dimensiona position. In order to independent contro end effector position in -space, ou need at east DOFs In order to independent contro end effector position and orientation, at east 6 DOFs are needed the have to be configured right, too )

12 Kinematic redundanc he oca redundanc of an arm can be understood in terms of the oca acobian he manipuator contros a number of Cartesian DOFs eua to the number of independent rows in the acobian j j j j You use three joints to contro two Cartesian DOFs n=) j j Since there are two independent rows, ou can contro two Cartesian DOFs independent m=) Since the number of independent Cartesian directions is ess than the number of joints, m<n), this manipuator is redundant w.r.t. the task of controing those Cartesian directions.

13 Kinematic redundanc What does this redundant space ook ike? At first gance, ou might think that it s inear because the acobian is inear But, the acobian is on oca inear he dimension of the redundant space is the number of joints the number of independent Cartesian DOFs: n-m. For the three ink panar arm, the redundant space is a set of one dimensiona curves traced through the three dimensiona joint space. Each curve corresponds to the set of joint configurations that pace the end effector in the same position. Redundant manifods in joint space

14 Kinematic redundanc oint veocities in redundant directions causes no motion at the end effector hese are interna motions of the manipuator. Redundant joint veocities satisf this euation: ) the nu space of ) N ) Q : ) Compare to the range space of ) : R ) X : Q, ) Redundant manifods in joint space

15 Nu space and Range space oint space Q SO n ) Cartesian space m X R N ) R ) N ) Q : ) Nu space Motions in the nu space are interna motions You can t generate these motions R ) X : Q, ) Range space

16 Doing hings in the Redundant oint Space Motions in the redundant space do not affect the position of the end effector. Since the don t change end effector position, is there something we woud ike to do in this space? Optimie kinematic manipuabiit? Sta awa from obstaces? Something ese?

17 Doing hings in the Redundant oint Space # # I Nu space projection matri: I # his matri projects an arbitrar vector into the nu space of : Zero end-effector veocities his makes it eas to do things in the redundant space just cacuate what ou woud ike to do and project it into the nu space.

18 Doing hings in the Redundant oint Space Assume that ou are given a joint veocit,, ou woud ike to achieve whie aso achieving a desired end effector twist, Reuired objective: Desired objective: d d f ) g ) f ) g ) Minimie subject to : Use agrange mutipier method: f ) g )

19 Doing hings in the Redundant oint Space f g f ) g ) # # I

20 hings You Might do in the Nu Space Avoid kinematic singuarities:. Cacuate the gradient of the manipuabiit measure:. Project into nu space: # # I det Avoid joint imits:. Cacuate a gradient of the suared distance from a joint imit: #. Project into nu space: where is the joint configuration at the center of the joints m and is the current joint position m I #

21 hings You Might do in the Nu Space d obstace Avoid kinematic obstaces:. Consider a set of contro points nodes) on the manipuator:,,. Move a nodes awa from the object: i i obstace. Project desired motion into joint space: inodes i i 4. Project into nu space: # I #

22 Manipuabiit Eipsoid Can we characterie how cose we are to a singuarit? Yes imagine the possibe instantaneous motions are described b an eipsoid in Cartesian space. Can t move much this wa Can move a ot this wa

23 Manipuabiit Eipsoid he manipuabiit eipsoid is an eipse in Cartesian space corresponding to the twists that unit joint veocities can generate: A unit sphere in joint veocit space # # Project the sphere into Cartesian space he space of feasibe Cartesian veocities

24 Manipuabiit Eipsoid You can cacuate the directions and magnitudes of the principe aes of the eipsoid b taking the eigenvaues and eigenvectors of he engths of the aes are the suare roots of the eigenvaues v v Yoshikawa s manipuabiit measure: You tr to maimie this measure Maimied in isotropic configurations det his measures the voume of the eipsoid

25 Manipuabiit Eipsoid Another characteriation of the manipuabiit eipsoid: the ratio of the argest eigenvaue to the smaest eigenvaue: v v Let be the argest eigenvaue and et be the smaest. n hen the condition number of the eipsoid is: k n he coser to one the condition number, the more isotropic the eispoid is.

26 Manipuabiit Eipsoid Isotropic manipuabiit eipsoid NO isotropic manipuabiit eipsoid

27 Force Manipuabiit Eipsoid You can aso cacuate a manipuabiit eipsoid for force: F A unit sphere in the space of joint torues F F F F he space of feasibe Cartesian wrenches

28 Manipuabiit Eipsoid Principe aes of the force manipuabiit eipsoid: the eigenvaues and eigenvectors of: has the same eigenvectors as : v v i v f i But, the eigenvaues of the force and veocit eipsoids are reciprocas: f i v i herefore, the shortest principe aes of the veocit eipsoid are the ongest principe aes of the force eipsoid and vice versa

29 Veocit and force manipuabiit are orthogona! Force eipsoid Veocit eipsoid his is known as force/veocit duait You can app the argest forces in the same directions that our ma veocit is smaest Your ma veocit is greatest in the directions where ou can on app the smaest forces

30 Manipuabiit Eipsoid: Eampe Sove for the principe aes of the manipuabiit eipsoid for the panar two ink manipuator with unit ength inks at c c c s s s v Principe aes: v v v

31 Suppementar

32 Nu space and Range space Degree of manipuabiit: dim dim ) Degree of redundanc: dim N ) R N ) dim R ) tota DOF of manipuator ) N ) R )

33 Nu space and Range space As the manipuator moves to new configurations, the degree of manipuabiit ma temporari decrease these are the singuar configurations. here is a corresponding increase in degree of redundanc. ) N ) R )

34 ) N ) R ) Nu space and Range space N R ) ) R N ) ) F ) Remember the acobian s appication to statics:

35 ) N ) R ) Nu space and Range space in the Force Domain R ) N ) F ) ) R ) N N ) R )

36 Nu space and Range space in the Force Domain R ) N ) F ) ) R ) N N R ) ) R N ) ) A Cartesian force cannot generate joint torues in the joint veocit nu space.