_{k}be the frame associated with link k.

### 1. Define Z Axis from All Joints

From the axis of joint k+1, define the z^{k}axis for link k. If the joint is revolute, the z axis is in the direction of rotation as followed by the right hand rule for the rotations. If the joint is prismatic, the z-axis for the joint is along the direction of the linear movement.

### 2. Define Origins

The intersection of the z^{k}and z

^{k-1}axes is selected as the origin of L

^{k}. If they do not intersect, use the intersection of z

^{k}with a common normal between z

^{k}and z

^{k-1}. There is always one line mutually perpendicular to any two skew lines, called common normal, which is the shortest distance between them.

### 3. Define X-axis

Assign x^{k}in the direction of the common normal between z

^{k}and z

^{k-1}. If z-axes are intersecting, select x

^{k}to be orthogonal to both z

^{k}and z

^{k-1}(the direction of the cross-product of the two z-axes). If z

^{k}and z

^{k-1}are parallel, point x

^{k}away from z

^{k-1}colinear with the common normal of the previous joint.

### 4. Define Y-axis

Select y^{k}to form a right-handed orthonormal coordinate frame L

_{k}.

### 5. Define the Four Kinematic Parameters

*θ_{k}is defined as the angle between x

^{k-1}and x

^{k}axes about the z

^{k-1}axis. *d

_{k}is the distance between x

^{k-1}and x

^{k}axes along z

^{k-1}axis. *a

_{k}is the distance between z

^{k-1}and z

^{k}axes along x

^{k}axis. *α

_{k}is the angle between z

^{k-1}and z

^{k}axes about x

^{k}axis.

### 6. Transferring Frame k-1 to Frame k

*Rotation of L^{k-1}about the z

^{k-1}axis by θ

_{k}will make x

^{k-1}and x

^{k}parallel to each other. This is true because the common normals a

^{k-1}and a

^{k}are both perpendicular to z

^{k-1}axis. *Translation of L

^{k-1}along z

^{k-1}axis a distance of d

_{k}will make x

^{k-1}and x

^{k}colinear. *Translation of L

^{k-1}along x

^{k}axis a distance of a

_{k}will bring the origins of L

^{k-1}and L

^{k}together. *Rotation of L

^{k-1}about x

^{k}axis by α

_{k}will make z

^{k-1}and z

^{k}axes parallel. At this point, frames L

^{k-1}and L

^{k}will be exactly the same.

For simulation, a nice free software called Robotassist can be downloaded at www.kinematics.com

Ref:

Introduction to Robotics -Analysis, Control, Applications; Second Edition, Saeed Benjamin Niku

Fundamental of Robotics -Analysis & Control, Robert J. Schilling

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