Robot Kinematics Logics of presentation Kinematics what Coordinate

Robot Kinematics Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems 2020 -10 -07 handout 3 1

Robot Kinematics: what? Kinematics: study of the relationship between any two Variable 1 Variable 2 displacement variables in a dynamic system. Robot kinematics: a robot is a dynamic system, and Motor variable kinematics is to study the motor variable and the endeffector variable End-effector variable 2

Robot Kinematics: what? A robot consists of a set of servomotors which drive the end-effector: (a) the motion of the end-effector, and (b) the motion of the servomotors. These two are related, and further Given (b) to find (a): forward kinematics (process 1) Given (a) to find (b): inverse kinematics (process 2) (b) Motor 2020 -10 -07 1 (a) z y 2 Endeffector x handout 3 3

Robot Kinematics Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems 2020 -10 -07 handout 3 4

Robot Kinematics: coordinate system The general idea to describe motion: Coordinate systems, as it provides a reference upon which the motion of an object can be quantitatively described. 2020 -10 -07 handout 3 5

Robot Kinematics: coordinate system There are two coordinate systems to measure the types of motions (joint level and end-effector level), respectively: q motor or joint coordinate system for joint level motions (see Fig. 2 -11). q world coordinate system for end effector level motions (see Fig. 2 -12). Fig. 2 -11 Fig. 2 -12 2020 -10 -07 handout 3 6

Robot Kinematics: coordinate system 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). XB YB Yw P M i O 2020 -10 -07 Fig. 2 -13 Xw handout 3 7

Robot Kinematics: coordinate system 2020 -10 -07 handout 3 8

Robot Kinematics: coordinate system 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 orientation and location of the object with respect to the reference coordinate system reduce to the relation between the local coordinate system and the reference coordinate system. 2020 -10 -07 hand out 3 9

Robot Kinematics: coordinate system q Denote the relation between A and B as Rel (A-B) q Suppose {A} is the LCS of Object A, and {B} is a reference coordinate system. q Note that by defining {A} on an object, we imply that details of the object are known with respect to {A} in this case. q Orientation and location of the object (i. e. , A) in the reference coordinate system thus reduces to the relation of {A} with respect to {B}. 2020 -10 -07 hand out 3 10

Robot Kinematics Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems 2020 -10 -07 handout 3 11

Robot Kinematics: relation between two coordinate systems {A} and {B} q Case 1: two origins are coincident q Case 2: two coordinate systems are in parallel 2020 -10 -07 hand out 3 12

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. 2020 -10 -07 hand out 3 13

Two origins of the frames are coincident 2020 -10 -07 hand out 3 14

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) = 2020 -10 -07 handout 3 15

Equation (2 -1) can be further written as To be given in the classroom (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 2020 -10 -07 handout 3 (2 -3) 16

How A with respect to B ? To be given in the classroom B with respect to A 2020 -10 -07 handout 3 17

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) 2020 -10 -07 Orthogonal matrix 18

When frame A and frame B are not at the same location (see Fig. 2 -14), the difference between A and B is captured by The origin of {B} in Frame {A} P 2020 -10 -07 handout 3 19

When frame A and frame B are not at the same location nor in parallel, 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 location but {B} rotates with respect to {A}. The relation between {A} and {B} in this case is: 2020 -10 -07 hand out 3 20

Fig. 2 -14 2020 -10 -07 hand out 3 21

So the total relationship between {A} and {B} is captured by 2020 -10 -07 hand out 3 22

Fig. 2 -15 2020 -10 -07 hand out 3 23

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) = 2020 -10 -07 hand out 3 24

Point P at different frames Fig. 2 -16 shows that the same point, P, is expressed in two different frames, A and B. P Fig. 2 -16 2020 -10 -07 hand out 3 25

Case 1: Frame A and Frame B are in parallel but at different locations (see Fig. 2 -17) P Fig. 2 -17 In this case, we have the following relation 2020 -10 -07 hand out 3 26

(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) 2020 -10 -07 hand out 3 27

Fig. 2 -18 2020 -10 -07 hand out 3 28

See Fig. 2 -16, A and B are at different locations as well as with different orientations We have: (2 -9) We can further write equation (2 -9) into a frame-like form, namely a kind of mapping (2 -10) The matrix T has the following form: 2020 -10 -07 hand out 3 29

q 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). q For example, for Fig. 2 -15, {A}, {B}, {C}, {U}, we have for P in the space: 2020 -10 -07 hand out 3 30

Notation helps to verify the correctness of the expression 2020 -10 -07 hand out 3 31

Example 1: Fig. 2 -19 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} 2020 -10 -07 hand out 3 32

Fig. 2 -19 10 30 o 2020 -10 -07 hand out 3 33

Solution: To be given in the classroom 2020 -10 -07 hand out 3 34

To be given in the classroom (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 2020 -10 -07 hand out 3 35

Fig. 2 -20 YB YA P (3, 7, 0) XB 30 o 5 XA 10 2020 -10 -07 hand out 3 36

Solution: To be given in the classroom 2020 -10 -07 hand out 3 37

Summary q Forward kinematics versus inverse kinematics. q Motion is measured with respect to coordinate system or frame. q Frame is attached with an object. q Every detail of the object is with respect to that frame, local frame. q Relation between two frames are represented by a 4 by 4 matrix, T, in general. 2020 -10 -07 hand out 3 38

Summary (continued) q When two frames are in the same location, T is expressed by q When two frames are in parallel but different locations, T is expressed by 2020 -10 -07 hand out 3 39