What are Denavit-Hartenberg (D-H) parameters and how are frames assigned to robot links?

In one line

D-H parameters are four numbers per link (\( \theta, d, a, \alpha \)) that describe how each joint frame is placed relative to the previous one, using a standard rule for attaching frames.

Explained simply

Giving directions between two buildings needs only a few facts: turn by this angle, go up this many floors, walk this far, and tilt by this much. The D-H convention reduces the relation between any two robot links to exactly four such facts.

Technically, frame i is attached so that \( z_{i-1} \) is the axis of joint i, and \( x_i \) lies along the common normal between \( z_{i-1} \) and \( z_i \). The link transform is \( A_i = \mathrm{Rot}_z(\theta_i)\,\mathrm{Trans}_z(d_i)\,\mathrm{Trans}_x(a_i)\,\mathrm{Rot}_x(\alpha_i) \).

How it works

1. Assign \( z_{i-1} \) along the axis of joint i (rotation axis for R, sliding direction for P).
2. Place \( x_i \) along the common normal from \( z_{i-1} \) to \( z_i \); y completes a right-handed frame.
3. \( \theta_i \) (joint angle): rotation about \( z_{i-1} \) from \( x_{i-1} \) to \( x_i \); variable for a revolute joint.
4. \( d_i \) (link offset): distance along \( z_{i-1} \); variable for a prismatic joint.
5. \( a_i \) (link length): distance along \( x_i \) between the z-axes.
6. \( \alpha_i \) (link twist): rotation about \( x_i \) from \( z_{i-1} \) to \( z_i \).
7. Fill a D-H table (one row per link), then multiply the \( A_i \) matrices.

Key formula

\( A_i = \begin{bmatrix} \cos\theta_i & -\sin\theta_i\cos\alpha_i & \sin\theta_i\sin\alpha_i & a_i\cos\theta_i \\ \sin\theta_i & \cos\theta_i\cos\alpha_i & -\cos\theta_i\sin\alpha_i & a_i\sin\theta_i \\ 0 & \sin\alpha_i & \cos\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix} \); \( a, d \) in m, \( \theta, \alpha \) in degrees or radians.

Quick example

Planar 2R arm (both joint axes parallel to z): row 1: \( \theta_1, d = 0, a = L_1, \alpha = 0 \); row 2: \( \theta_2, d = 0, a = L_2, \alpha = 0 \). Only \( \theta_1, \theta_2 \) vary, matching its 2 DOF.

Common mistakes

• Mixing up \( a \) (along x) and \( d \) (along z).
• Writing a non-zero \( \alpha \) for parallel joint axes (it is 0).
• Marking the fixed parameter as the variable (for revolute joints \( \theta \) varies, for prismatic d varies).

How MU asks it

Theory 4 to 5 marks: 'Explain the D-H parameters with a neat sketch'. Numerical: 'Write the D-H table for the 2-DOF robot shown and find the end-effector position'. Draw frames clearly, define all four parameters, and identify the joint variables in the table.