Kinematics - PowerPoint PPT Presentation

About This Presentation
Title:

Kinematics

Description:

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 ... – PowerPoint PPT presentation

Number of Views:174
Avg rating:3.0/5.0
Slides: 25
Provided by: ComputerS
Category:

less

Transcript and Presenter's Notes

Title: Kinematics


1
Kinematics
2
  • 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)

3
Typical Examples
  • Picking up a box from point A and moving it to
    point B

Object box
4
Typical Examples
  • Welding a seam on a curved surface

5
Typical Examples
  • A mobile robot navigating a hallway

hallway
Y
mobile robot
Object mobile robot itself
X
6
  • In each case, the object being manipulated may be
    modeled as a rigid body, or non-deformable mass
    of material.

7
  • An unconstrained rigid body has six degrees of
    freedom
  • 3 position variables
  • x
  • y
  • z
  • 3 orientation variables
  • Roll
  • Pitch
  • Yaw

8
  • Suppose we attach a coordinate frame to the
    object being manipulated

Frame z (object)
X
Y
Z
Frame 1 (workspace)
9
  • The specification of the desired motion of the
    manipulated object relative to the robots
    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.

10
  • Putting the robot into the picture, the following
    is the standard kinematics diagram for robotics.

11
  • 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.
12
  • 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

13
Forward Kinematics
  • The general description of a link and joint is
    given below.
  • a link length
  • ? length twist
  • ? joint angle
  • d joint offset
  • n1 links 0-n
  • n joints 1-n

Dennvit-Hartenberg rotation
14
  • Revolute joint constant offset d, variable angle
    ?
  • Prismatic joint variable offset d, constant
    angle ?
  • Convention for first and last links
  • a0 0 and ?0 0
  • an undefined and ?n undefined
  • Convention for first and last joints
  • If joint 1 is revolute d1 0, ?1 has arbitrary
    zero position
  • If joint 1 is prismatic ?1 0, d1 has arbitrary
    zero position
  • Similarly for joint n

15
  • Individual frames (right-handed) are attached to
    each link according to the following convention.
  • Zi axis along joint i axis
  • Origin of frame i intersection of ai and joint i
    axis
  • Xi axis along ai from joint i to joint i1
  • ?i is positive about Xi axis
  • Yi specified so as to obtain a right-handed system

16
  • Frame 0 Base frame
  • Convention choose Z0 along Z1, and Frame 0
    coincides with Frame 1 when ?1 0 or d1 0
  • Frame n Wrist frame
  • Joint n revolute Xn lines up with Xn-1 when ?n
    0 origin chosen so that dn 0
  • Joint n prismatic direction of Xn chosen so that
    ?n 0 origin chosen at intersection of Xn-1 and
    joint n axis when dn 0

17
  • Then
  • Link length, ai distance from Zi to Zi1
    measured along Xi
  • Link twist, ?i angle between Zi and Zi1
    measured about Xi
  • Joint offset, di distance from Xi-1 to Xi
    measured along Zi
  • Joint angle, ?i angle between Xi-1 and Xi
    measured about Zi
  • 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.

18
  • The position and orientation of one frame with
    respect to another frame can be represented by a
    4 x 4 matrix (transform matrix)

R
P
where R 3 x 3 rotation matrix P 3 x 1 position
vector last row filler
T
0 0 0 1
Can write n with respect to 0 (eq. 1)
where
cos ?i -sin ?i 0 ai-1
sin ?i cos ?i-1 cos ?i cos ?i-1 - sin ?i-1 - sin ?i-1 di
sin ?i sin ?i-1 cos ?i sin ?i-1 cos ?i-1 cos ?i-1 di
0 0 0 1
(eq. 2)
19
Rotation Matrices
1 0 0
0 cos ? -sin ?
0 sin ? cos ?
roll, ? , about X
RX (?)
cos ? sin ? 0
0 1 0
-sin ? cos ? 0
pitch, ?, about Y
RY (?)
cos ? -sin ? 0
sin ? cos ? 0
0 0 1
yaw, ?, about Z
RZ (?)
20
  • Composite rotation, in order
  • roll
  • pitch
  • yaw

c?c? c?s?s? - s?c? c?s?c?s?s?
s?c? s?s?s? c?c? s?s?c?c?s?
-s? c?s? c?c?
RX (?) RY (?) RZ (?)
21
r11 r12 r13
r21 r22 r23
r31 r32 r33
Represent as
, then can determine roll, pitch, and yaw as
e.g., arctan z(-1,-1) 135 arctan z(1, 1)
45
pitch
yaw
roll
If ?90 ?0 ?arctan z(r12, r22)
22
Example 1
  • Consider the following planar manipulator

i ?i-1 ai-1 di ?i
1 0 0 0 ?1
2 0 L1 0 ?2
3 0 L2 0 ?3
23
Suppose L1 1m, L2 1m, ?1 30, ?2 60, and
?3 -90. Then
0.866 -0.5 0 0
0.5 0.866 0 0
0 0 1 0
0 0 0 1
Wrist x 0.866m y 1.5m z 0 ? 0 ß 0 ? 0
0.5 -0.866 0 1
0.866 0. 5 0 0
0 0 1 0
0 0 0 1
Alternate ?1 90 ?2 -60 ?3 -30
24
0 1 0 1
-1 0 0 0
0 0 1 0
0 0 0 1
And therefore,
0.5 -0.866 0 1
0.866 0. 5 0 0
0 0 1 0
0 0 0 1
pitch,
yaw,
roll,
Write a Comment
User Comments (0)
About PowerShow.com