15/09/2015handout 31 Robot Kinematics Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems

15/09/2015handout 32 Robot Kinematics: what Kinematics: study of the relationship between any two displacement variables in a dynamic system. Robot kinematics: a robot is a dynamic system.

15/09/2015handout 33 Robot Kinematics: what A robot consists of a set of servomotors which drive the end- effector. Therefore we have: (a) the motion of the end- effector, and (b) the motion of the servomotors. These two are related. Given (b) to find (a): forward kinematics (process 1) Given (a) to find (b): inverse kinematics (process 2) 1 z x y 2 (b) (a)

15/09/2015handout 34 Robot Kinematics Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems

15/09/2015handout 35 Robot Kinematics: coordinate system The general principle to describe motion: Coordinate systems, as it provides a reference upon which motion of an object can be quantitatively described.

15/09/2015hand ut 36 There are two coordinate systems to measure two types of motions (joint level and end-effector level), respectively:  motor or joint coordinate system for joint level motions (see Fig. 2-11).  world coordinate system for end effector level motions (see Fig. 2-12). Robot Kinematics: coordinate system Fig Fig. 2-12

15/09/2015handout 37 The relationship of the attached coordinate system with respect to the world coordinate system completely describes the position and orientation of that body in the world coordinate system (Fig. 2-13). Xw Fig M i P Yw Robot Kinematics: coordinate system

15/09/2015hand out 38 Attached coordinate system (also called local coordinate system, LCS) replaces the object and represents it with respect to the world coordinate system (WCS) or reference coordinate system (RCS) The motion of the object with respect to the reference coordinate system reduces to the relation between the local coordinate system with respect to the reference coordinate system Robot Kinematics: coordinate system

15/09/2015hand out 39  Denote the relation between A and B as Rel (A-B)  Suppose {A} is the LCS of Object A, and {B} is a reference coordinate system,  Note that by defining {A} on an object, we imply that details of the object are defined, with respect to {A} in this case,  Motion of the object (i.e., A) in the reference coordinate system is thus the relation of {A} with respect to {B}.  The above thinking process shows how the motion of an object becomes the relation between two coordinate systems Robot Kinematics: coordinate system

15/09/2015hand out 310  As well, how a point P on Object A should be represented with respect to {B}

15/09/2015handout 311 Robot Kinematics Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems

15/09/2015hand out 312 Robot Kinematics: relation between two coordinate systems {A} and {B}  Case 1: two origins are coincident  Case 2: two coordinate systems are in parallel

15/09/2015hand out 313 Remark 1: Motion is also related to velocity and acceleration. The general idea is that they should be obtained by the differentiation of the transformation matrix. Remark 2: coordinate system is also called frame.

15/09/2015hand out 314 Two origins of the frames are coincident

15/09/2015handout 315 Unit vectors giving the principal directions of {B} as When these vectors are written in terms of {A}, we denote Stack these three together, and call rotation matrix = (2-1)

15/09/2015handout 316 Equation (2-1) can be further written as (2-2) The components in equation (2-2) are simply the projections of that vector onto the axes of its reference frame. Hence, each component of equation (2-2) can be within as the dot product of a pair of unit vectors as To be given in the classroom (2-3) To be given in the classroom

15/09/2015handout 317 B with respect to A How A with respect to B ? To be given in the classroom

15/09/2015hand out 318 The inspection of equation (2-3) shows that the rows of the matrix are the column of the matrix ; as such we have (2-4) To be given in the classroom It can be further verified that the transpose of R matrix is its inverse matrix. As such, we have (2-5)

15/09/2015hand out 319 When frame A and frame B are not at the same location (see Fig. 2-14), we will consider two steps to get the relationship between {A} and {B}: Step 1: Consider that {A} and {B} are in parallel first. Then, {B} translates to the location which is denoted as : The origin of {B} in Frame {A} Step 2: Imagine that {A} and {B} are at the same origin but {B} rotates with respect to {A}. The relation between {A} and {B} in this case is:

15/09/2015hand out 320 So the total relation between {A} and {B} is:

15/09/2015hand out 321 Fig. 2-14

15/09/2015hand out 322 Further, if we have three frames, A, B, C, (Fig. 2-15) then we have a chain rule such that (see the figure in the next slide) =

15/09/2015hand out 323 Fig. 2-15

15/09/2015hand out 324 Point P at different frames Fig shows that the same point, P, is expressed in two different frames, A and B. P Fig. 2-16

15/09/2015hand out 325 Case 1: Frame A and Frame B are in parallel but at different locations (see Fig. 2-17) P Fig In this case, we have the following relation

15/09/2015hand out 326 (2-7) in {A} in {B} Case 2: A and B are at the same location but with different orientations (see Fig. 2-18). In this case, we have (2-8)

15/09/2015hand out 327 Fig. 2-18

15/09/2015hand out 328 We have: We can further write equation (2-9) into a frame-like form, namely a kind of mapping (2-10) (2-9) The matrix T has the following form: See Fig. 2-16, A and B are both at different locations and with different orientations

15/09/2015hand out 329  T matrix is a 4 x 4 matrix, and it make the representation of P in different frames {A} and {B} a bit convenient, i.e., equation (2-10).  For example, for Fig. 2-15, {A}, {B}, {C}, {U}, we have for P in the space:

15/09/2015hand out 330 Notation helps to verify the correctness of the expression

15/09/2015hand out 331 Example 1: Fig shows a frame {B} which is rotated relative to frame {A} about is an axis perpendicular to the sheet plane Please find: (1) Representation of Frame {B} with respect to Frame {A} (2) (3) Representation of P with respect to Frame {A}

15/09/2015hand out 332 Fig o

15/09/2015hand out 333 Solution: To be given in the classroom

15/09/2015hand out 334 (2-11) Example 2: Fig.2-20 shows a frame {B} which is rotated relative to frame {A} about Z by 30 degrees, and translated 10 units in XA and 5 units in YA. Find where To be given in the classroom

15/09/2015hand out 335 XB XA YB YA o P (3, 7, 0) Fig. 2-20

15/09/2015hand out 336 To be given in the classroom Solution:

15/09/2015hand out 337 Summary  Forward kinematics versus inverse kinematics.  Motion is measured with respect to coordinate system or frame.  Frame is attached with an object.  Every details of the object is with respect to that frame, local frame.  Relation between two frames are represented by a 4 by 4 matrix, T, in general.

15/09/2015hand out 338 Summary (continued)  When two frames are in the same location, T is expressed by  When two frames are in parallel but different locations, T is expressed by