Forward and Inverse Kinematics

Kinematics relates a robot's joint angles to the position and orientation of its tool. The UR5E has 6 joints, so its configuration is a vector of 6 angles, and its tool pose is a position together with an orientation, written $[\text{position},\text{orientation}]$.

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The Six Joints

Named from the base outward:

• base and shoulder place the arm in the workspace
• elbow sets the reach
• wrist 1, wrist 2, and wrist 3 orient the tool

A degree of freedom is an independent joint that can move to produce a unique point in space, so 6 joints give the UR5E 6 DOF, the minimum needed to reach any position and orientation in 3D.

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Forward Kinematics

Forward kinematics takes the joint values and computes where the tool ends up:

• input is the joint vector $q$, the 6 joint angles
• output is the TCP pose $p$, the tool position and orientation
• the direction is $q\to p$
• in URScript this is get_forward_kin

There are two different ways to perform Forward Kinematics:

1. Denavit-Hartenberg Approach (which is the Frame per link used in my autonomous tank project)
2. Product of Exponentials (PoE)

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Inverse Kinematics (IK)

Inverse kinematics goes the other way, from a desired tool pose back to joint angles:

• input is the desired TCP pose $p$
• output is the joint vector $q$ that reaches it
• the direction is $p\to q$
• in URScript this is get_inverse_kin

Forward Kinematics and Inverse Kinematics are the math the UR5E uses to move between joint space and Cartesian tool space.