Recap: rigid motions

Transcription

1 Forward and Invere Knemat Chapter 3 Had Morad (orgnal lde by Steve from Harvard) Reap: rgd moton Rgd moton a ombnaton of rotaton and tranlaton Defned by a rotaton matrx (R) and a dplaement vetor (d) the group of all rgd moton (dr) known a the Speal Euldean group SE(3)

2 2 Reap: homogeneou tranform Ba tranform: Th t l t th t t Three pure tranlaton three pure rotaton b a b y a x Tran Tran β β β β β y x Rot Rot Tran z γ γ γ γ γ β β Rot z Example Euler angle: we have only dued ZYZ Euler angle. What the g y g et of all poble Euler angle that an be ued to repreent any rotaton matrx? XYZ YZX ZXY XYX YZY ZXZ XZY YXZ ZYX XZX YXY ZYZ

3 Example Compute the homogeneou tranformaton repreentng a tranlaton of 3 unt along the x-ax followed by a rotaton of π/2 about the urrent z- ax followed by a tranlaton of unt along the fxed y-ax Forward knemat ntroduton Challenge: gven all the jont parameter of a manpulator determne g g j p p the poton and orentaton of the tool frame 3

4 Conventon A n-dof manpulator wll have n jont (ether revolute or prmat) and n+ lnk (ne eah jont onnet two lnk) We aume that eah jont only ha one DOF. The o frame the nertal frame o n the tool frame Jont onnet lnk - and The o onneted to lnk Jont varable q q d f jont revolute f jont prmat Conventon We ad that a homogeneou tranformaton allowed u to expre the poton and orentaton of o j wth repet to o 4

5 5 Conventon Fnally the poton and orentaton of the tool frame wth repet to the nertal frame gven by one homogeneou tranformaton matrx: nertal frame gven by one homogeneou tranformaton matrx: For a n-dof manpulator ( ) ( ) ( ) n n n n n q A q A q A T o R H 2 2 The Denavt-Hartenberg (DH) Conventon Repreentng eah ndvdual homogeneou tranformaton a the produt of four ba tranformaton: produt of four ba tranformaton: Rot Tran Tran Rot x a x d z z a a a d A d a

6 Extene and unquene When an we repreent a homogeneou tranformaton ung the 4 DH parameter? Extene and unquene Proof:. We aume that R ha the form: R R R z 2. Ue DH to verfy the form of R x 6

7 7 Extene and unquene Proof: Ue DH2 to determne the form of o. Ue DH2 to determne the form of o Phyal ba for DH parameter a : lnk length dtane between the z and z (along x ) : lnk twt angle between z and z (meaured around x ) d : lnk offet dtane between o and ntereton of z and x (along z ) : jont angle angle between x and x (meaured around z ) potve onventon: x a x d z z A Rot Tran Tran Rot a d d a a a d

8 Agnng oordnate frame For any n-lnk manpulator we an alway hooe oordnate frame uh that DH and DH2 are atfed The hoe not unque but the end reult wll alway be the ame. Chooe z a ax of rotaton for jont + 2. Agn bae frame Agnng oordnate frame 3. Choe x y to follow the rght-handed onventon 4. Now tart an teratve proe to defne frame wth repet to frame - 8

9 . z - and z are non-oplanar Agnng oordnate frame Agnng oordnate frame. z - and z nteret 9

10 Agnng oordnate frame. z - and z are parallel Agnng tool frame The prevou agnment are vald up to frame n- The tool frame agnment mot often defned by the axe n a:

11 Example : two-lnk planar manpulator 2DOF: need to agn three oordnate frame Example : two-lnk planar manpulator Now defne DH parameter Frt defne the ontant parameter a Seond defne the varable parameter d lnk a d 2

12 Example 2: three-lnk ylndral robot 3DOF: need to agn four oordnate frame Example 2: three-lnk ylndral robot lnk a d Now defne DH parameter Frt defne the ontant parameter a Seond defne the varable parameter d 2 3 2

13 Example 3: pheral wrt 3DOF: need to agn four oordnate frame Example 3: pheral wrt Now defne DH parameter Frt defne the ontant parameter a Seond defne the varable parameter d lnk a d

14 Next la More example for ommon onfguraton 4