# How can Denavit-Hartenberg representation with only 4 variables describe rototranslations with 6 DOF?

I am wondering how the Denavit-Hartenberg representation is able to describe rototranslations with 6 DOF with only 4 parameters?

Some context: Denavit-Hartenberg representations can be used to describe the rototranslation from frame 1 to frame 2. Frame 2 can be rotated as well as translated. Usually this involves 6 degrees of freedom (DOF).

I am pretty sure that's impossible and that I am missing something else.

EDIT (my current solution): Systems that can be described using DHs only use joints with one joint variable. For example, a prismatic joint which can only move along one axis or a rotational joint which can only rotation around one axis. Then you lose 2 degrees of freedom => 6-2 = 4, thus 4 free variables are sufficient to describe the system.

• Not able to write a proper answer right now, but the D-H representation uses 4 parameters, not 4 coordinates. Those parameters describe the geometric relationships between 2 coordinate frames. This paper gives a pretty clear explanation (particularly section 3.2). – sempaiscuba Mar 30 '19 at 12:24
• Thanks for pointing that out. It was more like a typo as I actually meant parameters. – siva Mar 30 '19 at 12:27

In general you need 6 parameters to describe the position and orientation of any joint with respect to a link coordinate frame. The DH parameterisation includes 2 constraints so only 4 parameters are required. The constraints are:

• axis $$x_j$$ intersects axis $$z_{j-1}$$
• axis $$x_j$$ is perpendicular to axis $$z_{j-1}$$

(see Robotics, Vision & Control, second edition, p.197)

Personally I don't think it is useful to save 2 parameters given the confusion it creates for almost everybody, especially beginners, but I still find them confusing. Saving 2 parameters is pointless given today's computing capability, but back in 1955 when this notation was invented it was perhaps more pertinent.

• I wouldn't underestimate the importance of sparing unnecessary parameters with a compact representation when for example it comes down to the goal of optimizing the structure with a computational approach. Other than that, DH also suggests a good practice in the choice of reference frames. Conventions and good practices are somewhat relevant to engineering. – Ugo Pattacini Mar 31 '19 at 13:38
• @Ugo in principle I agree, but when the parameterisation forces you to have link coordinate frames that are actually not in or on the link itself, I think it has gone from helpful to unhelpful. – Peter Corke Apr 1 '19 at 23:11
• Upvoted for the comment on the uselessnes of DH! – 50k4 Apr 2 '19 at 7:16

In a nutshell, 4 DOFs do clearly represent only a manifold on all the possible 6 DOFs spatial roto-translations. This is actually the manifold corresponding to the movements of the links of a mechanical structure that are bound together via prismatic and revolute joints.

The DH convention assumes to deal with such structures. Thereby, under these premises, the frame attached to the endpoint of a link sliding on a track (prismatic) or rotating around a given axis (revolute) can be readily specified in terms of only 4 parameters with respect to the frame of the link attached before/after on the same structure.

To sum up, to describe a roto-translation without assuming anything you need of course 6 parameters. Instead, if you impose constraints - as those present for example in a serial kinematic chain - you need only 4 parameters.

Here's a related Q&A on Robotics SE: https://robotics.stackexchange.com/a/4422/6941.