6DOF serial robot (PUMA robot) dynamic model equations

Question

I'm trying to find the most simplified (though accurate) calculations for the dynamic model of a standard 6DOF serial robot (PUMA robot).

I found this great paper: https://www-cs.stanford.edu/group/manips/publications/pdfs/Armstrong_1986.pdf

The paper contains the full equations with 28 constants of the system, and it specifies the calculation for each constant.

However, I have a problem regarding the 6th link. It has the inertia symmetric values - Ixx6 Iyy6 Izz6, but Iyy6 never appears in any constant. The specific robot in the paper (and actually it should be for every robot) has a 6th link that Ixx6 = Iyy6, but the same is true for Ixx5 and Iyy5 and still they both appear in the constants.

Do you think this is a mistake? I need the exact equations has I want to expand them to the case of a payload - I want to replace m6 and I6 with updated values that include the payload mass and transferred inertia and then the equilibrium Ixx6 = Iyy6 will not necessarily exist.

Do you know of any other source of the full equations? Or maybe you have a solution to understand which of the Ixx6 relates to Ixx6 or Iyy6?

Answer 1

There are ten inertial parameters for a rigid link, namely, mass ($m$), product of mass and center of mass ($m a_x$, $m a_y$, $m a_z$), inertia tensor ($I_{xx}$, $I_{yy}$, $I_{zz}$, $I_{xy}$, $I_{yz}$, $I_{zx}$).

Case 1: Consider a rigid link, which can freely translate and rotate in space in 3D. For such a link each and every inertial parameter appears in the dynamic model and affects it simply because the body is allowed to move in all the 6 dimensions (3 translational + 3 rotational).

Case 2: Let us take the same rigid link and fix to an immovable base with a revolute joint, referred to as a 1-DoF manipulator. Such a fixture constraints the motion of the rigid link. Constraining the motion of the manipulator to only a few dimensions renders some of the inertial parameters to be purposeless to the dynamic model.

Dynamic model of the 1-DoF manipulator is:

$\tau = (I_{zx}+I_{yz}+I_{zz})\ddot{\theta_{1}} + mg(a_{x} cos(\theta_{1}) + a_{y}sin(\theta_{1}))$

In the above dynamic model, 5 out of the 10 inertia parameters have no effect. Similarly, when multiple links are connected to each other with 1-DoF joints it constraints the relative motion of the links, which renders some inertial parameters futile and they do not appear int he dynamic model.

To verify the equations of motions you obtained, you may use Recursive Dynamics Simulator (ReDySim) symbolic module, which generates the symbolic equations of motion.