SCARA – Forward Kinematics

Use the DH Algorithm to assign the frames and kinematic parameters

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=0

Compute 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

Joint d a Home q

1          

2          

3          

4