Notes The Incremental Motion Model:

Transcription

1 The Icremetal Motio Model: The Jacobia Matrix I the forward kiematics model, we saw that it was possible to relate joit agles θ, to the cofiguratio of the robot ed effector T I this sectio, we will see that it is possible to model the relatioship betwee the joit rates, θ, ad the velocity of the ed effector, ẋ, with a matrix as follows: 6 ẋ J( θ)θ or ẋ ẏ ż ω x J( θ) θ 1 θ 2... ω y θ N ω z ω i where is the ith compoet of agular velocity ad J ( θ) is a matrix of size 6 x N, ad N is the umber of joits o the robot. We will derive this matrix by calculatig ẋ as a fuctio of θ ad factorig out J( θ). But first, as further illustratio, let us cosider a simple plaar example.: F t,v t θ y τ ω θ x We ca see from this example that we ca resolve the velocity of the -68-

2 tip ito x ad y compoets as follows: v x rωsiθ v y rωcosθ which ca be expressed i a trivial matrix equatio as: or v x rsiθ v y rcosθ ω ẋ J( θ)θ Alteratively, we ca look at the tip force, ad the torque aroud the joit: τ rf x siθ+ rf y cosθ which agai gives a trivial matrix equatio: τ [ rsiθ, rcosθ] F x or F y τ J T ( θ)f Jumpig ahead from this simple example, we will see that the Jacobia Matrix ca also be used to relate tip forces to joit torques. -69-

3 Properties of Liear Trasformatios Source: Itroductio to Matrices ad Determiats, F.Max Stei, Facts: A 1 1. exists if ad oly if det[a] ad if det [ A], we say A is a sigular matrix. 2. The rak of A is the size of the largest square sub-matrix, S, for which det [ S] 3. If two rows or colums of A are equal or related by a costat, the det [ A]. 4. Ay sigular matrix has at least oe eigevalue equal to zero. 5. If A is o-sigular, ad λ is a eigevalue of A correspodig to the eigevector x, the x λ 1 A 1 x 6. If A is square, the A ad have the same eigevalues: i.e. Ax i A T λ i x i ad A T y i λ i y i 7. If the x matrix A is of full rak (i.e. r ) the the oly solutio to Ax is the trivial oe, x. 8. If r <, there are r liearly idepedet (i.e. orthogoal) solutios, x j ( 1 j r) for which Ax j (see also 4.) -7-

4 Properties of the Jacobia Matrix We have see ẋ J( θ)θ (1) ad θ J 1 ( θ)ẋ (2). Velocity Mappig Cosider the square 6x6 case. If rak ( J ( θ) ) < 6, (2) has o solutio (facts 1 & 2). Also, by fact 8, if the rak, r 5, there is oe otrivial solutio to ẋ J( θ)θ so there is a directio i joit velocity space for which joit motio produces o EE motio. We call ay joit cofiguratio, θ Q, for which rak ( J ( Q) ) < 6 a sigular cofiguratio. Furthermore, for certai directios of EE motio, ẋ i, 1 i 6, where ẋ i J( θ)θ λ i ( θ)ω i ( θ) λ i ω i ( θ) are eigevalues of J ( θ) ( θ)are eigevectors ofj ( θ) if J( θ) is full rak, we have:, ad ẋi ω i J 1 1 ( θ)ẋ i λ i (see fact 5). As we approach the sigular cofiguratio θ Q, there is at least oe eigevalue (j) for which λ j. Thus, ẋ j ω j --- I other words, motio i the particular directio ẋ j causes joit velocities to approach ifiity. -71-

5 Force Mappig Cosider τ J T ( θ)f Clearly, rak J T rak ( J), sice det [ A] det A T. Thus, at θ Q, there exists F such that J T ( Q)F I other words, a fiite force i a certai directio produces o torque at the joits (if rak [ J( Q) ] 5, there is just oe such force directio, see facts 4,6,7). -72-

6 The Jacobia by Differetiatio I additio to the method of velocity propogatio, the Jacobia ca be obtaied by differetiatio of the forward kiematics equatios. Y α y l 3 θ 3 l 2 θ 2 l 1 θ 1 x X Cosiderig the above plaar maipulator, defie the cofiguratio vector x [ x, y, α] T. The forward kiematic equatios of this arm are: x l 1 + l y l 1 s 1 + l 2 s 12 α θ 1 + θ 2 + θ 3 Differetiatig the first of these expressios, gives ẋ or l 1 s 1 θ 1 l 2 s 12 ( θ 1 + θ 2 ) ( θ 1 + θ 2 + θ 3 ) ẋ ( l 1 s 1 + l 2 s 12 )θ 1 ( l 2 s 12 )θ 2 ( l 3 )θ 3-73-

7 Similarly, ẏ ( l 1 + l )θ 1 + ( l )θ 2 23 ad α θ 1 + θ 2 + θ 3 Thus, ( )θ 123 J ( θ) l 1 s 1 l 2 s 12 l 2 s 12 l 1 + l l l Remember that we started with the expressio for the EE cofiguratio i frame so the resultig Jacobia is still i frame. I cotrast, the velocity propogatio method leaves us with a Jacobia i frame N. Either result ca be rotated later as ecessary. Example: Sigular Cofiguratio To look at a sigular cofiguratio of this maipulator, take for example, Q π -- 4 π Note that for θ Q, s 12 s 1, s 1, 2, 23. This gives J ( Q) ( l 1 l 2 )s 1 ( l 2 )s 1 l 3 s 1 ( l 1 + l 2 ) ( l 2 ) π If we deote the rows of J by {r1,r2, r3}, ad recall that for θ 1 --, 4 s 1.77 the we ote that r1 ( 1)r2 so that det [ J ( Q) ] (fact 3). -74-

8 This meas that Q is a sigular cofiguratio. Note that o closer examiatio, we see that J ( Q ) is sigular for ay value of θ 1! (This should be obvious from the geometry). This is because, for ay θ1, we still have θ 1 ad ta ( ) is a costat. Thus, π -- 4 π r1 r2 ( ta ( θ 1 )) is a sigular cofiguratio for ay. Force at a sigularity Q Sice τ J T F, at a sigular poit Q, we ca expect o trivial (i.e. o zero) forces such that F j J T I words, there will be some force vector or vectors which ca be applied to the tip which geerate o torques i the joits. So, i a sigular cofiguratio, the mechaism ca lock up with respect to tip forces or torques i certai directios. For example, suppose Q is defied as above, ad Note that this correspods to a force applied to the tip i the directio opposite to the outstretched arm, ad that o exteral torque is ap- θ 1 π θ 1 ( Q)F j F θ 1 cos F 1 Fsiθ 1-75-

9 plied to the tip. Now, writig ( Q) i a simplified form, J T where ( )F 1 τ J T Q bs 1 b 1 as 1 a 1 cs 1 c 1 F Fs 1 a l 1 l 2 b l 2 c l 3 ad τ afs 1 + afs 1 + bfs 1 + bfs 1 + This situatio is a old ad famous oe i mechaical egieerig. For example, i the steam locomotive, top dead ceter refers to the followig coditio Pisto Drive Wheel F rail The pisto force, F, caot geerate ay torque aroud the drive wheel axis because the likage is sigular i the positio show. -76-

10 Velocity at a sigularity Note that although θ J 1 ( Q)ẋ has o solutio i geeral, if we assume that J loses rak by oly oe (i.e. r -1), the there are -1 eigevectors i the task velocity space, ẋ j, for which solutios do exist. However, there will be multiple solutios. For example, if Vs 1 ẋ 1 V The force vector looks like this Y ẋ 1 V l 3 l 2 θ 2 l 1 θ 1 X We ca see (at least geometrically) that the valid solutios for the resultig joit velocities will be -77-

11 θ A V ( l 1 + l 2 ) θ B V l 2 θ C V l 3 or, liear combiatios of these such as θ a 1 θ A + a 2 θ B + a 3 θ C where a 1 + a 2 + a 3 1 This ca be verified by multiplyig ay of the solutios by J( θ) to obtai. ẋ 1 However, if there is ay compoet of tip velocity i the directio correspodig to the zero eigevalue of J, the at least oe joit rate will go to ifiity. -78-

12 Advaced Topic: Coordiate Free Jacobia Matrix Source: Lecture otes by Dr. Amir Fijay, USC/JPL, 199 We earlier described the elemets of J as the effective radius betwee task axis i ad the ed effector. Cosider the propagatio of agular velocity for a all revolute maipulator which we used to compute the bottom three rows of the Jacobia matrix ω 1 1 1R ω 1 Z θ + 1 This is also true idepedet of ay particular frame. That is ω ω 1 + Z 1 θ as log as all of the vector quatities i this equatio, ω, ω 1 ad Z 1 are ultimately evaluated i the same frame. This is a coordiate free represetatio of the propagatio of velocity. Sice the coordiate free form is so simple, let s expad the recursio for a six axis maipulator: ω 1 Z 1 θ 1 + ω 2 Z 1 θ 1 + Z 2 θ 2 etc. Which gives fially, which is ω Z θ θ 1 ω 6 Z 1 Z 2... Z 6 θ 2 θ 3... Note that the matrix formed by assemblig the vectors ito colums is simply the bottom three rows of the Jacobia matrix (also i coordiate free form). By a similar but slightly more complicated θ 6 Z -79-

13 derivatio: θ 1 ẋ 6 Z 1 P 16, Z 2 P 26,... Z 6 P 66, θ 2 θ 3... P, 6 P 66, Where is the vector coectig the origi of frame 6 with the origi of frame. Note that is zero ad so the last colum above should also be zero. This is correct because θ 6 should make o cotributio to the liear velocity of frame 6. The overall Jacobia i coordiate free form is obtaied by combiig the two velocity results: θ 6 where ẋ J 6 θ Z 1 P 16, Z 2 P 26,... Z 6 P 66, J 6 Z 1 Z 2... Z 6 The projectio of this Jacobia oto ay desired frame is simply obtaied by represetig each vector i the desired frame, e.g. J6 Z 1 P16, Z2 P... Z 26, 6 P 66, Z 1 Z2... Z6-8-

14 ad, J ca be trasformed to ay other frame by 6 m J6 m R J6 m R m where R is the rotatio matrix describig the relative orietatio betwee frames m ad ad is the 3x3 matrix of zero elemets. The two R s simply rotate the vectorial elemets of the two rows of the Jacobia matrix. Similarly, we ca chage the poit at which velocity is computed from the origi of frame 6 to ay poit, r, which is rigidly coected to by F 6 J r I Pˆ 6r, I J 6 where I is the 3x3 idetity matrix, ad Pˆ6r represets the skewsymmetric matrix which ecodes the operator projected ito some frame,,, P6r,. Whe Pˆ 6r, Pz Py P P z Py P x x -81-