Denavit-Hartenberg Transformation Matrix Calculator

Robotics kinematics & multibody mechanics: Compute the $4 imes 4$ homogeneous transformation matrix ($^{i-1}T_i$) from standard Denavit-Hartenberg (D-H) parameters ($a_i, alpha_i, d_i, heta_i$).

Denavit-Hartenberg (D-H) Parameters

Joint Angle θ_i (°): Rotation about Z_{i-1} axis

Link Offset d_i (mm): Translation along Z_{i-1} axis

Link Length a_i (mm): Common normal length along X_i

Link Twist α_i (°): Rotation about X_i axis

Convention Form: Transformation sequence

Local Point P_i (X, Y, Z mm): Local point to transform

4×4 Homogeneous Transformation Matrix

          
            ----
----
----
0001

Frame Origin Position (X, Y, Z)

In parent reference frame (mm)

Transformed Point P_{i-1}

T · [P_i; 1]

Euler Orientation (Roll, Pitch, Yaw)

ZYX intrinsic convention

Matrix Determinant det(R)

1.0000

Orthonormal SO(3) check

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Frequently Asked Questions

What is the difference between Standard D-H and Modified (Craig) D-H?

In Standard D-H, the coordinate frame i is fixed to the link i at the outer joint. In Modified D-H (John Craig convention), frame i is attached at the inner joint (axis i), changing the transformation order to Rot(x) -> Trans(x) -> Rot(z) -> Trans(z).

How are D-H matrices chained for a 6-DOF robot?

The complete end-effector pose relative to the robot base is computed by simple sequential matrix multiplication: ⁰T₆ = ⁰T₁ · ¹T₂ · ²T₃ · ³T₄ · ⁴T₅ · ⁵T₆.

Why can D-H representation fail for parallel joint axes?

When two consecutive joint axes are parallel, the common normal line is not unique. Slight geometric errors in manufacturing can cause sudden discontinuous jumps in the D-H parameters. Alternative methods like Hayati-Roberts or Screw Theory (Product of Exponentials) avoid this singularity.