Kinematics

The function of a robot is to manipulate objects in its workspace. To manipulate objects means to cause them to move in a desired way (as determined by a particular application)

Typical Examples Picking up a box from point A and moving it to point B: Object: box Z XY. AB box.

Typical Examples Welding a seam on a curved surface: Object: torch Z XY torch seam

Typical Examples A mobile robot navigating a hallway: Object: mobile robot itself Y X mobile robot hallway...

In each case, the object being manipulated may be modeled as a rigid body, or non-deformable mass of material. xyzxyz yaw Z XY roll pitch Rigid body Position vector:

An unconstrained rigid body has six degrees of freedom –3 position variables: x y z –3 orientation variables: Roll Pitch Yaw

Suppose we attach a coordinate frame to the object being manipulated: Z XY Frame z (object) Frame 1 (workspace) Y Z X

The specification of the desired motion of the manipulated object relative to the robot’s workspace amounts to describing the position and orientation (and their rate of change including linear and angular velocities and accelerations) of frame z with respect to frame 1. Such a description of motion is called Kinematics. Kinematics concerns the geometry of motion only, without considering the forces and torques needed to actually cause the motion.

Putting the robot into the picture, the following is the standard kinematics diagram for robotics. Z X Y Z X Y Z X Y Links and joints (4) between Wrist and Base of robot; there are sensors and actuators at each joint Base frame User frame World frame (stationary) Z X Y Wrist frame Z X Y Tool frame

World frame: stationary; serves as common frame, e.g., if there are multiple robots Base frame: may be moving, e.g., a mobile robot; defined with respect to World frame Wrist frame: defined with respect to Base frame; position and orientation determined by link lengths and joint angles/offsets Tool frame: defined with respect to Wrist frame; position and orientation are fixed User frame: may be moving, e.g., a conveyor; defined with respect to World frame Knowledge of the positions and orientations of each of the different frames fully determines the state of the robot relative to its environment (workspace) at any given point in time.

To relate Wrist frame to Base frame: –Forward Kinematics: given link length/twist values and joint angle/offset values, determine the corresponding position and orientation of the Wrist frame –Inverse Kinematics: given a desired Tool frame position and orientation, determine the necessary joint angles/offsets (assuming known link lengths/twists) The solution is not always unique or feasible

Forward Kinematics The general description of a link and joint is given below. a: link length  : length twist  : joint angle d: joint offset n+1 links: 0-n n joints: 1-n Dennvit-Hartenberg rotation

Revolute joint: constant offset d, variable angle  Prismatic joint: variable offset d, constant angle  Convention for first and last links: –a 0 = 0 and  0 = 0 –a n = undefined and  n = undefined Convention for first and last joints: –If joint 1 is revolute: d 1 = 0,  1 has arbitrary zero position –If joint 1 is prismatic:  1 = 0, d 1 has arbitrary zero position –Similarly for joint n

Individual frames (right-handed) are attached to each link according to the following convention. Z i axis along joint i axis Origin of frame i: intersection of a i and joint i axis X i axis along a i from joint i to joint i+1  i is positive about + X i axis Y i specified so as to obtain a right- handed system

Frame 0 = Base frame –Convention: choose Z 0 along Z 1, and Frame 0 coincides with Frame 1 when  1 = 0 or d 1 = 0 Frame n = Wrist frame –Joint n revolute: X n lines up with X n-1 when  n = 0; origin chosen so that d n = 0 –Joint n prismatic: direction of X n chosen so that  n = 0; origin chosen at intersection of X n-1 and joint n axis when d n = 0

Then : –Link length, a i = distance from Z i to Z i+1 measured along X i –Link twist,  i = angle between Z i and Z i+1 measured about X i –Joint offset, d i = distance from X i-1 to X i measured along Z i –Joint angle,  i = angle between X i-1 and X i measured about Z i Therefore : –Determining the position and orientation of the Wrist frame with respect to the Base frame amounts to determining the position and orientation of Frame n with respect to Frame 0.

The position and orientation of one frame with respect to another frame can be represented by a 4 x 4 matrix (transform matrix) T = whereR: 3 x 3 rotation matrix P: 3 x 1 position vector last row: “filler” R P Can write n with respect to 0: (eq. 1) where = cos  i -sin  i 0a i-1 sin  i cos  i-1 cos  i cos  i-1 - sin  i-1 - sin  i-1 d i sin  i sin  i-1 cos  i sin  i-1 cos  i-1 cos  i-1 d i 0001 (eq. 2)

Rotation Matrices roll, , about X: pitch, , about Y: yaw, , about Z: cos  -sin  0 sin  cos  cos  sin  sin  cos  0 cos  -sin  0 sin  cos  R X (  ) R Y (  ) R Z (  )

R X (  ) · R Y (  ) · R Z (  ) = c·cc·c c  ·s  ·s  - s  ·c  c·s·c·s·sc·s·c·s·s s·cs·c s  ·s  ·s  + c  ·c  s·s·c·c·ss·s·c·c·s -s  c·sc·s c·cc·c Composite rotation, in order: 1.roll 2.pitch 3.yaw

Represent as r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 pitch yaw roll, then can determine roll, pitch, and yaw as: If  =90°  =0°  =arctan z(r 12, r 22 ) e.g., arctan z(-1,-1) = 135° arctan z(1, 1) = 45°

Example 1 Consider the following planar manipulator: i  i-1 a i-1 didi ii 1000 11 20L1L1 0 22 30L2L2 0 33

Suppose L 1 = 1m, L 2 = 1m,  1 = 30°,  2 = 60°, and  3 = -90°. Then: Alternate  1 = 90°  2 = -60°  3 = -30° Wrist x = 0.866m y = 1.5m z = 0  = 0 β = 0  = 0 = =

And therefore, = pitch, yaw, roll,