ROBOTICS Forward Kinematics TEMPUS IV Project 158644 JPCR

ROBOTICS Forward Kinematics TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS

Forward Kinematics • Modeling assumptions • Review: – Spatial Coordinates • Pose = Position + Orientation 1 st part – Rotation Matrices – Homogeneous Coordinates • Frame Assignment – Denavit Hartenberg Parameters • Robot Kinematics 2 nd part – End-effector Position, – Velocity, & – Acceleration TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 2

Industrial Robot sequence of rigid bodies (links) connected by means of articulations (joints) TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 3

Robot Basics: Modeling • Kinematics: – Relationship between the joint angles, velocities & accelerations and the end-effector position, velocity, & acceleration TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 4

Modeling Robot Manipulators • Open kinematic chain (in this course) • One sequence of links connecting the two ends of the chain (Closed kinematic chains form a loop) • Prismatic or revolute joints, each with a single degree of mobility • Prismatic: translational motion between links • Revolute: rotational motion between links • Degrees of mobility (joints) vs. degrees of freedom (task) • Positioning and orienting requires 6 DOF • Redundant: degrees of mobility > degrees of freedom • Workspace • Portion of environment where the end-effector can access TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 5

Modeling Robot Manipulators • Open kinematic chain – sequence of links with one end constrained to the base, the other to the end-effector End-effector Base 6

Modeling Robot Manipulators • Motion is a composition of elementary motions Joint 2 Joint 1 End-effector Joint 3 Base 7

Kinematic Modeling of Manipulators • Composition of elementary motion of each link • Use linear algebra + systematic approach • Obtain an expression for the pose of the end-effector as a function of joint variables qi (angles/displacements) and link geometry (link lengths and relative orientations) Pe = f(q 1, q 2, ¼, qn ; l 1¼, ln, 1¼, n) TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 8

Pose of a Rigid Body • Pose = Position + Orientation • Physical space, E 3, has no natural coordinates. • In mathematical terms, a coordinate map is a homeomorphism (1 -1, onto differentiable mapping with a differentiable inverse) of a subset of space to an open subset of R 3. – A point, P, is assigned a 3 -vector: AP = (x, y, z) where A denotes the frame of reference TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 9

P Z BP AP X = (x, y, z) Z = (x, y, z) A Y B Y X TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 10

Pose of a Rigid Body • Pose = Position + Orientation How do we do this? TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 11

Pose of a Rigid Body • Pose = Position + Orientation • Orientation of the rigid body – Attach a orthonormal FRAME to the body – Express the unit vectors of this frame with respect to the reference frame XA YA ZA TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 12

Pose of a Rigid Body • Pose = Position + Orientation • Orientation of the rigid body – Attach a orthonormal FRAME to the body – Express the unit vectors of this frame with respect to the reference frame XA YA ZA TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 13

Rotation Matrices • OXYZ & OUVW have coincident origins at O – OUVW is fixed to the object – OXYZ has unit vectors in the directions of the three axes ix, jy, and kz – OUVW has unit vectors in the directions of the three axes iu, jv, and kw • Point P can be expressed in either frame: TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 14

P Z AP = (x, y, z) BP = (u, v, w) W V X O Y U TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 15

P Z AP = (x, y, z) BP = (u, v, w) W V X O Y U TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 16

P Z AP = (x, y, z) BP = (u, v, w) W V X O Y U TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 17

P Z AP = (x, y, z) BP = (u, v, w) W V X O Y U TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 18

Rotation Matrices TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 19

Rotation Matrices 1 X axis expressed wrt Ouvw 20

Rotation Matrices 1 Y axis expressed wrt Ouvw 21

Rotation Matrices 1 Z axis expressed wrt Ouvw 22

Rotation Matrices TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 23

Rotation Matrices X axis expressed wrt Ouvw Z axis expressed wrt Ouvw TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS Y axis expressed wrt Ouvw ROBOTICS 24

Rotation Matrices 1 U axis expressed wrt Oxyz 25

Rotation Matrices U axis expressed wrt Oxyz V axis expressed wrt Oxyz W axis expressed wrt Oxyz 26

Properties of Rotation Matrices • Column vectors are the unit vectors of the orthonormal frame – They are mutually orthogonal – They have unit length • The inverse relationship is: – Row vectors are also orthogonal unit vectors TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 27

Properties of Rotation Matrices • Rotation matrices are orthogonal • The transpose is the inverse: • For right-handed systems – Determinant = -1(Left handed) • Eigenvectors of the matrix form the axis of rotation TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 28

Elementary Rotations: X axis Z X Y TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 29

Elementary Rotations: X axis Z X Y TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 30

Elementary Rotations: Y axis Z X Y TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 31

Elementary Rotations: Z-axis Z X Y 32

Composition of Rotation Matrices • Express P in 3 coincident rotated frames TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 33

Composition of Rotation Matrices • Recall for matrices AB ¹ BA (matrix multiplication is not commutative) Rot[Z, 90] Rot[Y, -90] TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 34

Composition of Rotation Matrices • Recall for matrices AB ¹ BA (matrix multiplication is not commutative) Rot[Z, 90] Rot[Y, -90] TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 35

Rot[Z, 90] Rot[Y, -90] Rot[Z, 90] TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 36

Rot[z, 90]Rot[y, -90] ¹ Rot[y, -90] Rot[z, 90] TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 37

Decomposition of Rotation Matrices • Rotation Matrices contain 9 elements • Rotation matrices are orthogonal – (6 non-linear constraints) 3 parameters describe rotation • Decomposition is not unique TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 38

Decomposition of Rotation Matrices • Euler Angles • Roll, Pitch, and Yaw TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 39

Decomposition of Rotation Matrices • Angle Axis TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 40

Decomposition of Rotation Matrices • Angle Axis • Elementary Rotations 41

Pose of a Rigid Body • Pose = Position + Orientation Ok. Now we know what to do about orientation…let’s get back to pose TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 42

Spatial Description of Body • position of the origin with an orientation Z B X A Y TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 43

Homogeneous Coordinates • Notational convenience TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 44

Composition of Homogeneous Transformations • Before: • After TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 45

Homogeneous Coordinates • Inverse Transformation TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 46

Homogeneous Coordinates • Inverse Transformation Orthogonal: no matrix inversion! 47

Literature: Richard M. Murray, Zexiang Li, S. Shankar Sastry: A mathematical Introduction to Robotic Manipulation, University of California, Berkeley, 1994, CRC Press, pp. 93 -95. An electronic edition of the book is available from: http: //www. cds. caltech. edu/~murray/mlswiki TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 48