Title: Kinematics
1Kinematics
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)
3Typical Examples
- Picking up a box from point A and moving it to
point B
Object box
4Typical Examples
- Welding a seam on a curved surface
5Typical 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
13Forward 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)
19Rotation 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 (?)
21r11 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)
22Example 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
23Suppose 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,