direct kinematics. link description the concept of direct kinematics choosing wisely the coordinate...
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Direct Kinematics
Link Description
The concept of Direct Kinematics
• Choosing wisely the coordinate systems on the links
• If the wise choice was made, each link can be represented with 4 parameters
• When the parameters are found, the transformation matrices between the links can be found from a closed formula
DK Algorithm
• 1) Draw sketch• 2) Identify and number robot links. Base = 0, Last = n• 3) Draw axis Zi for joint i. For rotating joint, Zi is the
rotation axis. For prismatic (translating) joint, Zi can merge with the DOF axis or be perpendicular to it.
• 5) Determine joint length ai-1 between Zi-1 and Zi
• 6) Draw axis Xi-1 along the shortest distance between Zi-1 and Zi. If the distance is 0, choose the direction of Xi-1 to be a normal to the plane that they create.
DK Algorithm (2)
• 7) Determine joint twist i-1 measured around Xi-1 (between Zi-1 and Zi)
• 8) Determine the joint offset di
• 9) Determine joint angle i around Zi
• 10) Write DH table• 11+12) Write link transformations and calculate
the common transformation
Kinematics Parameters of a link
1i
Link length
Link twist
1ia
What are the kinematics parameters of this link?
• a = 7 = 450
Kinematics Parameters of a link
• Link offset d• Joint angle
Summary of the link parameters in terms of link frames
• ai = the distance from Zi to Zi+1 measured along Xi i = the angle between Zi and Zi+1 measured about Xi• di = the distance from Xi-1 to Xi measured along Zi i = the angle between Xi-1 and Xi measured about Zi
• We usually choose ai > 0 since it corresponds to a distance;
• However, i , di , i are signed quantities.
There is no unique attachment of frames to links:
• 1. When we align Zi axis with joint axis i, two choices of the Zi direction.
• 2. When we have intersecting joint axes (ai=0), two choices of the Xi direction, corresponding to choice of signs for the normal to the plane containing Zi and Zi+1.
• 3. When axes i and i+1 are parallel, the choice of origin location for {i} is arbitrary (generally chosen in order to cause di to be zero).
PTTTTPTP iPi
QP
RQ
iR
iii
i 111
TTTTT Ri
PR
RQ
iR
ii
11
iZiZiXiXi
i dDRaDRT 111
iiZiiXi
i dScrewaScrewT ,, 111
1000
)cos()cos()sin()cos()sin()sin(
)sin()sin()cos()cos()cos()sin(
0)sin()cos(
1111
1111
1
1
iiiiiii
iiiiiii
iii
ii d
d
a
T
Three link Arm : RPR mechanism
• “Cylindrical” robot – 2 joints analogous to polar coordinates when viewed from above.
• Schematic: point – axes intersection; prismatic joint at minimal extension
• Find coordinate systems and a, , d, (i=3)
i ai i di i
0 0 0
1 0 90 0 1
2 0 0 d2 0
3 L2 3
DH table:
1000
0100
00)cos()sin(
00)sin()cos(
11
11
01
T
1000
0010
100
0001
212
dT
1000
100
00)cos()sin(
00)sin()cos(
2
33
33
23 LT
TTTTT NNN12
312
01
0 ...