DH parameters and inverse kinematics, with the UR5e as a worked example

Forward and inverse kinematics

Forward kinematics answers a well-defined question: given the six joint angles of a UR5e, where is the tool flange and how is it oriented? The answer is unique and comes from chaining one coordinate transform per joint.

Inverse kinematics asks the opposite: which joint angles put the tool at a required pose? A six-axis arm usually has several valid solutions, for example elbow up or elbow down, and none when the pose is out of reach. Controllers and motion planners choose between solutions using joint limits, the current configuration and obstacles.

The Denavit–Hartenberg convention

The Denavit–Hartenberg convention describes each link with four parameters: a link length (a), a link offset (d), a twist (α) and the joint angle (θ). For a rotating joint, θ is the variable and the other three are fixed by the design.

Universal Robots notes that its a parameter corresponds to the r parameter in some references, and that the standard convention is singular when describing parallel joints.

UR5e DH parameters

Universal Robots publishes the same table for UR5e and UR7e. Lengths are in metres. The table also gives each link’s centre of mass, which is reproduced on the UR5e machine page.

Using the parameters responsibly

These are nominal design values. A specific arm’s calibrated kinematics can differ slightly, which matters when positions are calculated rather than taught. Universal Robots lists ROS and ROS 2 among the UR5e controller’s supported interfaces, so many projects use existing kinematics packages rather than writing a solver from scratch.

Treat any inverse-kinematics result as a candidate: check it against joint limits, the safety configuration and collisions before it reaches a real arm.

• UR5e kinematics and full specifications
• Degrees of freedom explained
• Repeatability vs accuracy