Differential Kinematics

Size: px

Start display at page:

Download "Differential Kinematics"

Elwin Stevenson
5 years ago
Views:

1 Lctu Diffntia Kinmatic Acknowgmnt : Pof. Ouama Khatib, Robotic Laboato, tanfo Univit, UA Pof. Ha Aaa, AI Laboato, MIT, UA Guiing Qution In obotic appication, not on th poition an ointation, but th vocit of th n-ffct i ao to b monito an conto. How can th vocit of th n-ffcto b cacuat? In o to mov th n-ffct in a pcifi iction with a pcifi p, it i nca to cooinat th p of th iniviua joint Funamnta mtho a to b vop fo achiving uch cooinat joint motion in mutip-joint obotic tm. W iv th iffntia ationhip btwn th joint ipacmnt an th n-ffct ocation, an thn ov fo th iniviua joint motion.

2 Diffntia Rationhip Gnaiz o-oinat, q

3 acobian: Dict Diffntiation En-ffcto poition En-ffcto ifction acobian: Dict Diffntiation Du to a ma movmnt of iniviua joint at th cunt poition q, th utant motion of th n-ffct i. acobian mati mati of patia ivativ at q to Tho ma movmnt a ivi b t to iv th ationhip btwn joint an atian vociti q q t t q,,, q

4 Eamp Fowa kinmatic of th pann manipuato. co co co co ψ Th iffntia ationhip i ψ q ψ 7 Eamp co co co, co, 8

5 acobian acobian povi th ationhip btwn th joint vociti an th utant n-ffct vocit acobian can b ov a foow [ [ In gna, ach coumn vcto of th acobian pnt th nffct vocit an angua vocit gnat b th iniviua joint vocit whi a oth joint a immobiiz,, z, φ, φ, φ p q L n q n Aignmnt : Vocit Pofi z T 9 Fo th pann two-ink manipuato hown, tmin th n-ffcto vocit pofi fo th foowing motion, T,9 T 9, T Aum L cm, L 8cm L L Wit Maab m-fi, to aw nffcto p in an iction

6 ingua Am onfiguation A ong a an a not aign, vociti of th two joint can b t accoing to mak th n-ffcto mov in an iction. Diction of an a configuation-pnant, an whn th a aign, n-ffcto i on movab aong that iction. uch am configuation a known a gua am configuation ingua Am onfiguation inguaiti occu in pann two-ink am whn,o 8, co co co At guait configuation 8 o o co co co co oumn vcto in up ach oth

7 Dtminant of an inguait, co co co co [ [ co co [ co co At gua am configuation. o 8 Inv Kinmatic of Diffntia Motion Rov n-ffcto vocit into vociti of iniviua joint. Whnv acobian i not gua, inv kinmatic can b ov a foow q o q Th oution i uniqu, unik th inv kinmatic of nffcto poition, wh mutip oution it. Thi mapping can b u fo obot manipuato conto a popo Rov Motion Rat onto. Dani Whitn 99

8 Motion Na inguaiti oni th two-ink pann aticuat obot am. W want to mov th npoint at a contant p aong a path taing at point A,, go co to th oigin though Bε, an,ε, an ach th fina point D, Aum ach am ink i of unit ngth an obtain th pofi of th iniviua joint vociti. ommnt on joint vociti in B gmnt of th motion Vocit Inv Kinmatic B invting vocit kinmatic co co co co co co [ co co [ co co co

9 oint Vociti Na inguaiti V high joint vociti a ut at point A an D, which a th am gua configuation o to th oigin 8, th vocit of th fit joint bcom v ag in o to quick tun th am fom B to, v co v v [co co v inguait Anai [ Whn th am i fu tn, A an D poition Fo poition A i i, an co j co j both joint gnat npoint vocit aong th -ai, thu, n-point can mov on aong -ai o of capabiit. Whn th am i f -8 B, an poition, an co co Fit joint o not pouc an contibution to npoint motion 7 8

10 acobian of a -Link Am Lock joint an, mov joint with unit ang at an fin npoint vocit [ Lock joint an, mov joint with unit ang at an fin npoint vocit [ Lock joint an, mov joint with unit ang at an fin npoint vocit [ Dtmin acobian a [ Fin guaiti b 9 inguait an Runanc omtim, uch gua configuation it in th mi of th wokpac iou gaing mobiit an manipuatabiit of th obot To ovcom thi ifficut, npoint tajctoi can b pann awa fom gua configuation. Atnativ, aitiona g of fom hou b incu o that vn whn om g of fom a ot at ctain configuation, th obot can ti maintain an aquat numb of g of fom Runant Manipuato. To ocat th npoint at an poition with an ointation, a pann manipuato n vaiab,,φ, wha a pacia manipuato n vaiab,,z,φ,φ,φ z. am numb of g of fom a qui fo nonunnt pann an pcia am.

11 acobian of th Am A Poition acobian c c c c c c c B Patia Diffntiation Poition acobian

12 A Ointation acobian [ [ z [ [ z z tanfo chinman Am B Patia Diffntiation

13 Rpntation 9 Tota acobian acobian pn on th pntation an am configuation

Gantry-Tau A New Three Degrees of Freedom Parallel Kinematic Robot

Gantry-Tau A New Three Degrees of Freedom Parallel Kinematic Robot Gant-au A w h Dg of Fom Paa Kinmatic Robot La Johannon, Vikto Bbuk, ogn Bogåh. Dpatmnt of Machin an Vhic Stm Cham Univit of chnoog, 4 9 Götbog, Swn -mai: a.johannon@m.cham.. Dpatmnt of Machin an Vhic Stm