Denavit-Hartenberg Parameters

The Denavit-Hartenberg convention describes any serial arm with a table of four numbers per joint. Each row moves from one joint frame to the next with a fixed recipe: rotate about z, slide along z, slide along x, rotate about x.

θ₁ (base turn) θ₂ (shoulder) θ₃ (elbow) Rows applied

Drag the 3D view to look from another angle.

One row, four steps

Row $i$ of the table holds $(\theta_i, d_i, a_i, \alpha_i)$. It turns frame $i-1$ into frame $i$ with

$$T_i = R_z(\theta_i)\;T_z(d_i)\;T_x(a_i)\;R_x(\alpha_i)$$

and the whole arm is the product of the rows:

$$T_{0,3} = T_1\,T_2\,T_3$$

For a revolute joint, $\theta_i$ is the joint angle and the other three numbers are fixed by the arm's shape. $d$ is an offset along the joint axis, $a$ is the link length between axes, and $\alpha$ is the twist between axes.

What to notice

Set "Rows applied" to 0, then step it up. Row 1 lifts the frame 0.4 m and twists it 90° so the next z axis points sideways, ready for the shoulder. Rows 2 and 3 each slide along x by a link length. In every frame, the z axis (amber) is the axis the next joint turns about. That rule is the heart of DH.

DH is compact, but frame placement has some freedom and two common versions exist, standard and modified. The product of exponentials is a popular alternative.