DH Parameters

Overview

The Denavit-Hartenberg (DH) convention provides a systematic method for assigning coordinate frames to robot links.

Standard DH Parameters

Four parameters for each joint:

Parameter	Description	Axis
\(\theta_i\)	Joint angle	Z (revolute)
\(d_i\)	Link offset	Z
\(a_i\)	Link length	X
\(\alpha_i\)	Link twist	X

Transformation Matrix

\[_{i-1}^{i}T = R_z(\theta_i) \cdot T_z(d_i) \cdot T_x(a_i) \cdot R_x(\alpha_i)\]

\[= \begin{bmatrix} \cos\theta_i & -\sin\theta_i\cos\alpha_i & \sin\theta_i\sin\alpha_i & a_i\cos\theta_i \\ \sin\theta_i & \cos\theta_i\cos\alpha_i & -\cos\theta_i\sin\alpha_i & a_i\sin\theta_i \\ 0 & \sin\alpha_i & \cos\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}\]

Convention Steps

1. Label joints 1 to n from base to end-effector
2. Align Z-axis with joint axes
3. Choose base frame origin at joint 1 origin
4. For each link:
5. \(a_i\): Distance from \(Z_i\) to \(Z_{i+1}\) along \(X_i\)
6. \(\alpha_i\): Angle from \(Z_i\) to \(Z_{i+1}\) about \(X_i\)
7. \(d_i\): Distance from \(X_{i-1}\) to \(X_i\) along \(Z_i\)
8. \(\theta_i\): Angle from \(X_{i-1}\) to \(X_i\) about \(Z_i\)

Modified DH Parameters

Alternative convention placing frame \(i\) at joint \(i\) (before transformation):

\[_{i}^{i-1}T = R_x(\alpha_{i-1}) \cdot T_x(a_{i-1}) \cdot R_z(\theta_i) \cdot T_z(d_i)\]

Best Practices

• Be consistent with convention choice
• Document which convention you're using
• Verify with known configurations

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