1998 Summer Q2 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.

The location of frame origins is ascertained only when the staps of the algorithm are carried out.

1

2

34-Tool Pitch

5-Tool Pitch

Note: There is no tool yawno tool yaw in this case

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 y0

x0

Frame 0

z0 y0

x0

Align zk with the axis of joint k+1.

z1

Locate the origin of Lk at the intersection of the zk and zk-1axes

Frame 1

z0 y0

x0

z1

Select xk to be orthogonal to both zk and zk-1.

Select yk to form a right handed orthonormal co-ordinate frame Lk

y1 will be hidden after this for the purpose of clarity

x1

y1Frame 1

z0 y0

x0

z1x1

Align zk with the axis of joint k+1.

This is the line of action of the prismatic joint

z2

Frame 2

z0 y0

x0

z1x1

z2

Select xk to be orthogonal to both zk and zk-1.

Select yk to complete the right handed orthonormal co-ordinate frame

x2

y2Frame 2

z0 y0

x0

z1x1

z2y2

Align zk with the axis of joint k+1.

Locate the origin of Lk at the intersection of the zk and zk-1axes

z3

x2

Frame 3

z0 y0

x0

z1x1

z2y2

z3

Select xk to be orthogonal to both zk and zk-1.

x3

Select yk to complete the right handed orthonormal co-ordinate frame

y3

x2

Frame 3

z0 y0

x0

z1x1

z2y2

z3

x3

y3

Align zk with the axis of joint k+1, the tool roll joint

The origin is actually at the same point as that of the tool pitch joint

z4

x2

Frame 4

z0 y0

x0

z1x1

z2y2

z3

x3

y3

z4

Select xk to be orthogonal to both zk and zk-1.

x4

Select yk to complete the right handed orthonormal co-ordinate frame

y4

x2

Frame 4

z0 y0

x0

z1x1

z2

y2

z3

x3

y3

z4

x4

y4

Set the origin of Ln at the tool tip. Align zn with the approach vector of the tool.

Align yn with the sliding vector of the tool.

Align xn with the normal vector of the tool.

z5

y5x5

x2

Frame 5

With the frames assigned the kinematic parameters can be determined.

z0 y0

x0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

z0 y0

x0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

Locate point bk (b5) 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

b5k = 5

z0 y0

x0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2 bk

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 i.e. 5 = -90º

But this is only for the soft home position, 5 is the joint variable.

5

k = 5

z0 y0

x0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2 b5

5

Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1

d5

Remember the roll joint frame is just moved out of position for clarity

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 a5=0

k = 5

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2 b5

5

d5

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 z4 to z5 about x5 is zero i.e. 5 = 0º

k = 5

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

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

b4

5

d5

k = 4

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

b4

5

d5

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. 4 = 0º

k = 4

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

b4

5

d5

As the origin of both frames are at the same point the a and d values are zero in this case, ie a4=0 and d4=0

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 -90º i.e. 4 = -90º

k = 4

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2bk

5

d5

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

k = 3

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2bk

5

d5

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 x2 to x3 about z2 is 180º i.e. 3 = 180º

k = 3

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2b3

5

d5

Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1

This is the joint variable for joint 3 which is the prismatic joint

d3

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 a3=0

k = 3

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2b3

5

d5d3

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 90º i.e. 3 = 90º

k = 3

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

b2

5

d5d3

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

k = 2

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

bk

5

d5d3

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 90º i.e. 2 = 90º

k = 2

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

b2

5

d5d3

Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1

This is zero in this case, as bk is at the origin of frame Lk-1 therefore d2 =0

Compute ak as the distance from point bk to the origin of frame Lk along xk

a2

k = 2

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

b2

5

d5d3

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 90º i.e. 2 = 90º

k = 2

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

b1

5

d5d3

a2

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

x0

k = 1

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

b1

5

d5d3

a2

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 zero i.e. 1 = 0º

k = 1

Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

bk

5

d5d3

a2

x0

d1

Compute ak as the distance from point bk to the origin of frame Lk along xk

This is zero in this case, therefore, a1 = 0.

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2

b1

5

d5d3

a2

x0

d1

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 zk-1 to zk about xk-1 is 90º i.e. 1 =90º

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2 5

d5d3

a2

x0

d1

From this drawing a table of D-H parameters can be compiled

z0 y0

z1

x1

z2

y2

z3 x3

y3

z4

x4

y4

z5

y5x5

x2 5

d5d3

a2

x0

d1

Link θ a d α Home q 1 q1 0 d1 = 0.8m 90o 0o 2 q2 a2=0.15m 0 90o 90o 3 180o 0 q3=d3=0.6+l1 90o 0.6+l1 4 q4 0 0 -90o 0o 5 q5 0 d5=0.55m 0o -90o