Denavit-Hartenberg Practice Laboratory

During this laboratory each engineering team will analysis motion in and then develop the full kinematic model for the 4-Axis cylindrical robot and the 5-Axis indirect-drive articulating robot in our laboratory. To achieve this goal, each person will be required to develop the ‘frame skeletons’, link-parameter tables, Ai matrices and forward kinematic solutions (in the most efficient form) for both of these machines.

As an ‘add on,’ try to develop the inverse kinematic solution for the 4-Axis cylindrical robot.

Some Thoughts on D-H Modeling:

During your laboratory analysis, consider the appropriateness of your models by performing physical verifications of each Ai matrix as well as the completed forward kinematic solution (Tno). When these proposed solutions are examined, do they appear to adequately represent the robot motion?

Could they be implemented in the controllers to drive the robots? How and where would they be used? What modifications would be needed?

As you review your inclass work, analyze and explain each step that is being performed – consider the acceptability of the procedure and the accuracy of the methods used.


The D-H Modeling Rules:

1)  Locate & Label the Joint Axes: Z0 to Zn-1

2)  Establish the Base Frame. Set Base Origin anywhere on the Z0 axis. Choose X0 and Y0 conveniently and to form a right hand frame.
Repeat Steps 3 to 5 for i= 1 through n-1:

3)  Locate the origin Oi where the common normal to Zi-1 and Zi intersects Zi. If Zi intersects Zi-1 locate Oi at this intersection. If Zi-1 and Zi are parallel, locate Oi at Joint i+1

4)  Establish Xi along the common normal between Zi-1 and Zi through Oi, or in the direction Normal to the plane Zi-1 – Zi if these axes intersect

5)  Establish Yi to form a right hand system
if i < n-1 repeat 3 to 5
______

6)  Establish the End-Effector (n) frame: OnXnYnZn. Assuming the n-th joint is revolute, set kn = a along the direction Zn-1. Establish the origin On conveniently along Zn, at center of gripper or tool tip. Set jn = o in the direction of gripper closure (opening) and set in = n such that n=oxa. Note if tool is not a simple gripper, set Xn and Yn conveniently to form a right hand frame.

7)  Create a table of “Link” parameters:

·  qi as angle about Zi-1 between X’s

·  di as distance along Zi-1

·  ai as angle about Xi between Z’s

·  ai as distance along Xi

8)  Form HTM matrices A1, A2, … An by substituting q, d, a and a into the general model

9)  Form T0n = A1*A2*…*An


Denavit – Hartenberg Modeling Issues

1.  Find and identify all Joint Axes, both revolute and prismatic.

2.  Assign ZI-1 to each of these axes starting with Joint 1 as Z0.

3.  Examine Model and REASON out a best Kinematic Home.

Best Kinematic Home is – for Revolute Joints – where XI-1 and XI are Parallel -- (not an issue for Prismatic Joints!)

4.  If it is possible, Rotate about ZI-1’s to align XI’s.

5.  Follow the Algorthimic steps to complete all model frames.

6.  Note: in a perfect model, the Number of Frames is number of Joints + 1 (0 to n). However, more frames may be needed to account for fixed angles and/or lengths.

7.  Once complete “Frame Skeleton” is constructed, develop a Link Parameter (LP) Table. Each row represents the transformations between consecutive Frames in the model: 001, 1-2, etc.

8.  Build AI Matrices from the information contained in each row of the LP Table.

9.  Build Forward Kinematic Solution: A1*A2* … *An. Note, consider the angular issues due to parallel joint axes during construction.


Doing the D-H Model of Robotic Linkages

  1. Locate the Joint Axes à These become the Z0, Z1, … Zn-1 directions
  2. Determine ‘Kinematic Home’ à this is critically important for Revolute Joint, not so important for Prismatic Joints
  3. Work through with Oi’s and Xi’s à Remembering that Xi is perpendicular to and intersecting Zi-1
    -- this allows for measurable qI’s and aI’s (so we may need ‘Dummy Frames’)
  4. Add Yi’s to form right hand systems à this is critically important for prismatic joints
  5. Build Link Parameter Table ‘as you go’ à check that they make sense
  6. Build Ai’s ‘as you go’ à do a physical verification to see if they make sense,
    -- remember that the Ai’s are the Transformations from Frame 0 to 1, 1 to 2, ··· and ending with n-1 to n
  7. Build the Forward Kinematic Solution (FKS) = A1 * ··· * An
    à write as a ‘Table’ and perform ‘Physical Verification’