Part 2: The Skeleton of Robotics (Kinematics)

Welcome to Part 2 of our "Robotics Zero to Hero" series. In our last post, we discussed the mathematics of rigid body transformations. Now, we string those transformations together to form the "skeleton" of a robot. This is the study of Kinematics.

The Math: Forward Kinematics and D-H Parameters

Forward Kinematics (FK) is the process of calculating the position and orientation of the robot's end-effector (the "hand" or "tip") given the current angles or displacements of its joints.

If a robot has $n$ joints, we can describe the transformation from the base frame to the end-effector frame as a chain of matrix multiplications:

Where $q_i$ is the variable for joint $i$ (an angle for a revolute joint, or a distance for a prismatic joint).

Denavit-Hartenberg (D-H) Parameters

To standardize how we define these transformations, roboticists use Denavit-Hartenberg (D-H) parameters. For any pair of adjacent links, we can define the transformation using exactly four parameters:

• $\theta_i$: Joint angle (rotation about $z_{i-1}$)
• $d_i$: Link offset (translation along $z_{i-1}$)
• $a_i$: Link length (translation along $x_i$)
• $\alpha_i$: Link twist (rotation about $x_i$)

The standard D-H transformation matrix from frame $i-1$ to frame $i$ is:

Python Implementation: Building a D-H Chain

Let's build a simple Python class to evaluate a kinematic chain based on D-H parameters.

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Focus on the Octopus: Finding the Tip

Traditional industrial robots have rigid links and discrete joints, making D-H parameters highly effective. Our metallic continuum octopus robot, however, bends continuously.

To approximate the forward kinematics of an octopus tentacle, we treat the continuum arm as a series of many tiny rigid segments, each capable of bending slightly. We can define the "joint angles" $q$ as the curvature parameters of each segment. By chaining together dozens of these small transformation matrices (similar to our D-H loop above), we can accurately calculate the 3D coordinate of the tentacle tip based on the pneumatic or cable-driven actuation lengths.

In Part 3, we flip the problem on its head: if we know where we want the tentacle tip to go, how do we calculate the necessary joint angles? Enter Inverse Kinematics.