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Number the joints 1 to n starting with the base and ending with the tool yaw, pitch and roll in that order.
Note: There is no tool pitch or yawno tool pitch or yaw in this case
1
2 3
4-Tool Roll
Assign a right-handed orthonormal frame L0 to the robot base, making sure that z0 aligns with the axis of joint. Set k=1
z0
x0
y0
k=0
z0
x0
y0
z1
2
Align zk with the axis of joint k+1.Locate the origin of Lk at the intersection of the zk and zk-1axesIf they do not intersect use the the intersection of zk with a common normal between zk and zk-1.(can point up or down in this case)
Common Normal
k=1
z0
x0
y0
z1
Select xk to be orthogonal to both zk and zk-1. If zk and zk-1are parallel, point xk away from zk-1.
Select yk to form a right handed orthonormal co-ordinate frame Lk
x1y1
k=1
z0 y0
z1
x1y1
Align zk with the axis of joint k+1.
Vertical Extension
Again zk and zk-1 are parallel the so we use the intersection of zk
with a common normal.
Common Normal
z2
x0
k=2
z0 y0
z1
x1y1
z2
Select xk to be orthogonal to both zk and zk-1. Once again zk and zk-1are parallel, point xk away from zk-1.
x2
y2
Select yk to complete the right handed orthonormal co-ordinate frame
x0
k=2
z0 y0
z1
x1y1z2
x2
y2
Align zk with the axis of joint k+1.
4
Locate the origin of Lk at the intersection of the zk and zk-1axes
z3
x0
k=3
z0 y0
z1
x1y1z2
x2
y2
z3
Select xk to be orthogonal to both zk and zk-1. Again xk can point in either direction. It is chosen to point in the
same direction as xk-1
x3
Select yk to complete the right handed orthonormal co-ordinate frame
y3
x0
k=3
z0 y0
z1
x1y1z2
x2
y2
z3
x3
Set the origin of Ln at the tool tip. Align zn with the approach vector of the tool.
z4
Align yn with the sliding vector of the tool.
y3
y4
Align zn with the normal vector of the tool.
x4
x0
k=4
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
With the frames assigned the kinematic parameters can be determined.
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
Locate point bk at the intersection of the xk and zk-1 axes. If they do not intersect, use the intersection of xk with a common normal
between xk and zk-1
b4
k=4
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
Compute k as the angle of rotation from xk-1 to xk measured about zk-1
It can be seen here that the angle of rotation from xk-1 to xk about zk-1 is 90 degrees (clockwise +ve) i.e. 4 = 90º
But this is only for the soft home position, 4 is the joint variable.
4
k=4
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1
b4
d4
Compute ak as the distance from point bk to the origin of frame Lk along xk
In this case these are the same point therefore a4=0
4
k=1
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b4
d4
Compute k as the angle of rotation from zk-1 to zk measured about xk
It can be seen here that the angle of rotation from z3 to z4 about x4 is zero i.e. 4 = 0º
4
k=4
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b3
d4
Locate point bk at the intersection of the xk and zk-1 axes. If they do not intersect, use the intersection of xk with a common normal
between xk and zk-1
4
k=3
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b3
d4
Compute k as the angle of rotation from xk-1 to xk measured about zk-1
It can be seen here that the angle of rotation from xk-1 to xk about zk-1 is zero i.e. 3 = 0º
4
k=3
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b3
d4
Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1
Compute ak as the distance from point bk to the origin of frame Lk along xk
In this case these are the same point, therefore ak=0
d3
Since joint 3 is prismatic d3 is the joint variable
4
k=3
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b3
d4
d3
Compute k as the angle of rotation from zk-1 to zk measured about xk
It can be seen here that the angle of rotation from z2 to z3 about x3 is zero i.e. 3 = 0º
4
k=3
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b2
d4
d3
Once again locate point bk at the intersection of the xk and zk-1 axes If they did not intersect we would use the intersection of xk with a
common normal between xk and zk-1
4
k=2
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b2
d4
d3
Compute k as the angle of rotation from xk-1 to xk measured about zk-1
It can be seen here that the angle of rotation from x1 to x2 about z1 is zero i.e. 2 = 0º
But this is only for the soft home position, 4 is the joint variable.
4
2
k=2
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b2
d4
d3
Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1
In this case these are the same point therefore d2=0Compute ak as the distance from point bk to the origin of frame Lk along xk
a2
4
2
k=2
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b2
d4
d3
a2
Compute k as the angle of rotation from zk-1 to zk measured about xk
It can be seen here that the angle of rotation from z1 to z2 about x2 is zero i.e. 2 = 0º
4
2
k=2
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b1
d4
d3
a2
For the final time locate point bk at the intersection of the xk and zk-1
axes
4
2
k=1
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
bk
d4
d3
a2
Compute k as the angle of rotation from xk-1 to xk measured about zk-1
It can be seen here that the angle of rotation from x0 to x1 about z0 is zero i.e. 1 = 0º
But this is only for the soft home position, 1is the joint variable.
1
4
2
k=1
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b1
d4
d3
a2
Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1
Compute ak as the distance from point bk to the origin of frame Lk along xk
d1
a1
1
4
2
k=1
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
b1
d4
d3
a2
d1
a1
Compute k as the angle of rotation from zk-1 to zk measured about xk-1
It can be seen here that the angle of rotation from z0 to z1 about x1 is 180 degrees
1
4
2
k=1
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
d4
d3
a2
d1
a1
1
4
2
From this drawing of D-H parameters can be compiled
z0 y0
z1
x1y1z2
x2
y2
z3
x3
z4
y3
y4
x4
x0
d4
d3
a2
d1
a1
1
4
2
Joint d a Home q
1 1 d1 a1 180º 0º
2 2 0 a2
0º 0º
3 0º
d3 0 0º dmax
4 4 d4 0
0º 90º