Articulated Robot Kinematics

Hello,

Your description seems a little too vague for me. Do you have a DH table?

Thanks!

James Moliere

Dear Mr. James Moliere.

I would like to know the meaning of DH table.

Thanks a lot.

Raul Ernesto Conti

Raul,

Here's a lesson on the DH (Denavit-Hartenberg) table...

http://www.cs.cmu.edu/afs/cs/academic/class/15384/web/lectures/lecture09_18sep06.pdf

You'll need a little background in Linear Algebra and understand that you are constructing a 4x4 matrix where the first 3x3 components of the matrix represent rotation and the 1x3 (last column) represents the position of the arm (or joint) in 3D space.

James Moliere

James:

Thanks a lot for your answer.

I would like to know if there is a close or analytical solution to inverse kinematics for articulated manipulaors of general geometric configuration. The Denavit-Hartenberg Chart is similar to the following:

LINK...... di........ai............ αi

1 ...........0....... a1......... -80

2 ..........d2....... a2....... +123

3 ..........d3....... a3........ +14

4 ..........d4....... a4........ - 5

5 ..........d5....... a5........ +95

6 ..........d6....... a6........ +12

where all di and ai are diferent to zero, and alpha-i are diferent to 0, or +/-90 or +/- 180°.

Sincerely yours.

Raul Conti

Raul,

I need more specific parameters for di and ai or there will be an infinite number of solutions.

The DH table normally looks like

LINK...... di........ai............ αi...........ti

1 ...........0....... a1......... -80...........t1

2 ..........d2....... a2....... +123..........t2

3 ..........d3....... a3........ +14...........t3

4 ..........d4....... a4........ - 5............t4

5 ..........d5....... a5........ +95...........t5

6 ..........d6....... a6........ +12...........t6

where t represents the actual joint angle. If you look at the number of variables above, there are 18 (for d, a , & t). there are simply too many variables and too many variables means that there is a possiblity for an infinite number of solutions -- which is not good.

James Moliere

Just curious, but are you talking about some sort of "snake-like" device?

Not exactly. I am talking about robot arms used in industrial processes, like automotive industry.

An articulated arm is comformed by rigid links connected by rotative joints.

However an articulated robot is similar a "snake-like" device usually limited to six joints.

The robot you're talkin' about is a six axis jointed spherical manipulator. They are used in the automotive industry for assembly. They have closed loop feed back for each axis motor, usually using syncro or resolver (analog) or absolute optical (digital) encoders. Some fine ones are ABB (Asea Brown Baveri) GMF (General Motors Fanuc) Motoman. They're used in welding, waterjet cutting, adhesive laying, painting, and material handling. I've been playin' with em for 20 years. I love em.

Have a good one.

Rick

Automation Eng.

Rick

I would like to know what kind of mathematical formulations or software is used in calibration of robot arms.

Thanks a lot,

Raul Conti

Raul,

Most robotic arms are calibrated or synchronized, mechanically. They are jogged to set positions and their feedback units are electrically synchronized with the push of a button on their teach pendants. Some have sync offset and motor commutator angle offset numbers to enter into their teach pendants. This balances or aligns the servo motors with the feedback units. Hey Raul, Thanks for asking.

Rick

Raul,

It appears that you are describing a PUMA robotic arm. If so, YES, there is a general Inverse Kinemtic solution (to your question) for a robotic arm with 6 joints. Beware, the math can get a bit intense.

To be honest, I usually have to do the math over a few times to ensure my calculations are correct. Once you have the solution, it's pretty awesome. The best I've seen is 16 possible solutions per arm -- at one time I was convinced it was 32 solutions. If you can find a solution better than 16, let me know!!!!

If you want to contact me to discuss this more, please do

James Moliere
jmoliere@ucsd.edu

James.

I am working to find a general solution for manipulators with no restrictions on their geometrical parameters. Solutions to PUMA, STANDFORD and other robotic arms are based on simplified geometry (most distances between links are equal to zero, some lenghts of links are equal to zero, and torsional link angles take values of zero, or +/- 90 or +/- 180 °).

When robotis arm is litle diferent to these values is required aditional software to correct position or orientation errors.

Raul Conti

Raul,

Remember, 6 joints means 6 variables to ultimately equal 1 known position in space -- remember in math for 6 unknowns you need 6 equations and if you only have 5 equations your solutions can be infinite?. When alpha, d, and a parameters are set (constants), there are tricks you can do to find multiple solutions that will guarantee at least 1 solution will work as long as the end point is within range of the arm. The trick is, what number of multiple solutions can you live with? With the PUMA, there are 16 solutions. With arms that have no restrictions on their parameters (parameters become variables such as alpha, d, a, and theta), there will be an infinite number of solutions for a given position in space.

This is one of those questions where the devil is in the details. Assuming that you are looking for a closed form solution -- no infinite solutions, the DH table that you posted will require a constant real number for the parameters di, ai, and alphai in the calculations.

I looked at the DH table in your prior post. I haven't solved the equation for a robotic arm where the 5th joint length is NOT 0 (zero). there is a little trick in robotics that allow you to determine the joint angles of the last 3 joints using Z-Y-Z Euler angles. I must admit, I haven't tried solving the IK for a robotic arm like yours. I can tell you one thing, the math will be intense if the last 3 joints aren't rotated in 0, +90,-90, or 180 degrees to each other. The common trick is to extract the angles out of the 3 X 3 matrix that is formed when multiplying the last 12 DH parameters -- which ultimately (mathematically) implements the ZYZ Euler angles.

I hope my explanation made sense.

James Moliere

James,

Thanks a lot. Your explanation is very clear.

In fact, the parameters alpha, d, and a are constants. Then I am looking for a close form solution of IK for a robotic arm with a DH table like the following, where d5=0 and the last three joints are rotated +/- 90 degrees to each other.

LINK......... di...........ai............... αi...............ti

1 ...........598.0......53.50.........+263.0.......... 0

2 ...........232.0......93.40..........+91.5........... 0

3 ...........678.1......82.20......... -76.4............ 0

4 ...........453.2.....112.00......... +90.0........... 0

5 ..............0.0.......61.60......... -90.0........... 0

6 .............72.0......12.50......... +90.0.......... 0

Faithfully yours,

Raul Conti

Yes, both forward and inverse (although a little bit messier and more complicated than the forward one) kinematics are known. I suggest reading Spong and Vidyasagar's famous and precious "Robot Dynamics and Control" (1989) book for the formulations and stuff. That is for me, the "must read" book of robotics.

Thanks for your answer. I have studied the Spong and Vidyasagar's "Robot Dynamics and Control" book and I also think is a precious book.

But I am looking for a close form solution to inverse kinematics of a six degree of freedom robot arm when all distances between links, di, and all lenghts of links, ai, are diferent to zero, and when all torsional angles, alpha-i, are diferent to zero or +/- 90° or +/- 180°.

Raul Conti.

Raul,

Did you find the solution that you were looking for? I certainly hope you have!!!

James Moliere

Merry Christmas and Happy New Year!