Kinematics 제어시스템 이론 및 실습 2014.05.20 조현우 ※참고: John J. Graig, “Introduction to robotics: Mechanics and control,” 3rd Ed. Persion.

A position in 3-dimensional Space Orthonormal basis: 1 2 Cf. Orthogonal basis: Only Satisfies 1 Vector space: Set of all linear combinations of the basis The basis span the vector space

A position in 3-dimensional Cartesian coordinate Basis of Cartesian coordinate: A position in frame A: an element of the vector space

Relationship under coordinate rotation Coordinate frame A Coordinate frame B Rotational relationship Their origins coincide with each other

Relationship under coordinate rotation Coordinate frame B

Relationship under coordinate rotation Coordinate frame A ※ The Position is the same!!! Only the reference is changed to frame A Position w.r.t frame A Position w.r.t frame B

Relationship under coordinate rotation The relationship between the coordinates of and Linear Transformation Rotation matrix 3 x 3

Relationship under translation Coordinate frame A Coordinate frame B translation

Relationship under translation

Combination Step 1: Express w.r.t frame A By Step 2: Translate

Coordinate Transformation Matrix The same position

Coordinate Transformation Matrix

Coordinate Transformation Matrix Coordinate w.r.t. B  Coordinate w.r.t. A Coordinate of w.r.t frame A Coordinate of w.r.t frame A Coordinate of w.r.t frame A

Coordinate Transformation Matrix <Example> (0,1) Basis 2 Basis 1 Frame B (1,0) Frame A

Coordinate Transformation Matrix <Example> (0,1) Basis 2 Basis 1 Frame B (1,0) Frame A

Coordinate Transformation from frame B to frame A Augmented Matrix Form

Coordinate Transformation from frame B to frame A describe frame B relative to frame A maps  Multiplication Find position C in frame C w.r.t frame A: Frame C  B  A Inversion We know Find (a) (b)

Euler Angle Roll Rotation about Rotation about Pitch Rotation about Yaw Rotation about

Manipulator Kinematics -Kinematics: The science of motion that threats the subject without regard to the force that cause the motion -Dynamics: The relationships between the motions and the forces (or torques) -Forward Kinematics: The static geometrical problem of computing the position and orientation of the end-effector (tool frame) relative to the base frame, given a set of joint variables (joint angles or link offset)

Denavit-Hartenberg notation -Degree of freedom: The number of independent parameters requires to specify the position & orientation  The number of joints of manipulator -Link: a rigid body that defines the relationship between neighboring joint axes of manipulator -Joint axis i: a line in space about which link i rotates relative to link i-1 -Link Length: the distance from to measured along -Link Twist: the angle from to measured about -Link Offset: the distance from to measured along -Joint Angle: the angle from to measured about Link i-1 Link i

Denavit-Hartenberg notation (link length) is positive value (link twist, link offset, joint angle) are signed quantities is variable if joint i is prismatic is variable if joint i is revolute Prismatic joint Revolute joint

Kinematic Example All revolute joint manipulator i 1 L2 L1 2 3 L2 L1

Kinematic Example L1 L2 i 1 L1 2 90 3 L2

Manipulator Kinematics 1. Construct the transform defining frame {i} relative to the frame {i-1} Cf) {i-1} Rotate around {R} Translate along {Q} Intermediate frames: {R}, {Q}, {P} Rotate around {P} Translate along {i} 2. Concatenating link transformations

Manipulator Kinematics

Manipulator Kinematics Base frame {B} Station frame {S} Wrist frame {W} Tool frame {T} Goal frame {G} Other names Frame {0} Frame {N} Description  Nonmoving part of the robot  Base of the manipulator  Task frame  World frame  Universe frame  Last link of manipulator  End of any tool origin between the finger tips  {T} should coincide with {G} at the end of motion