Modified Denavit-Hartenberg Parameters

A variant of DH parameters that places each link's frame on the joint axis nearer the base, with a different transform order.

Modified DH parameters are the convention John J. Craig adopted in his textbook Introduction to Robotics: Mechanics and Control. Like the standard DH convention introduced by Denavit and Hartenberg in 1955, it describes the relationship between two adjacent frames using 4 parameters per joint: link length a, twist angle α, offset d, and joint angle θ. The difference is in where frame i is placed and in what order the transforms are composed: frame i sits on joint i's own axis (the proximal end of the link), whereas standard DH places it on joint i+1's axis (the distal end); the transform order becomes Rot_x(α_{i−1})·Trans_x(a_{i−1})·Rot_z(θ_i)·Trans_z(d_i) — rotate and translate about the previous x-axis first, then rotate and translate about this joint's own z-axis — which is why a and α in the parameter table carry an index one less than θ and d. The two parameter conventions cannot be mixed, or forward kinematics will come out wrong. Appendix C of Modern Robotics also uses this convention.

ExampleFranka Research 3's official documentation notes that its DH table 'follows the Craig convention' — for example, joint 1 has a = 0, d = 0.333 m, α = 0, and joint 4 has a = 0.0825 m, d = 0, α = π/2. Plugging this table directly into the standard-DH formula would give the wrong end-effector pose.

Also called

MDH, Craig DH, Craig Convention

Related

Denavit-Hartenberg (DH) Parameters · Homogeneous Transformation Matrix · Forward Kinematics (FK) · Product of Exponentials Formula · Kinematic Calibration · Franka Emika Panda / Franka Research 3

Sources

Wikipedia: Denavit–Hartenberg parameters（Modified DH parameters 一节） (Chinese)
Franka Robotics 文档：Robot and interface specifications（DH 参数，遵循 Craig 约定） (Chinese)
Lynch & Park, Modern Robotics（2017）附录 C：Denavit–Hartenberg Parameters (Chinese)