Two Link Kinematics

Forward Kinematics

The Cartesian location of the elbow joint between the links:

$$\vec{\textrm{elbow}} = (L_1 \sin \theta_1, L_1 \cos \theta_1)$$

The Cartesian location of the end point:

$$\vec{p} = \vec{\textrm{elbow}} + (L_2 \sin (\theta_1+\theta_2), L_2 \cos (\theta_1+\theta_2))$$

Inverse Kinematics

Given an endpoint expressed in polar coordinates:

$$\textrm{end} = (R, \theta)$$

Internal angles can be labeled as follows:

$$\begin{align}\begin{aligned}\theta = \theta_1 + \alpha\\\beta = \pi - \theta_2\end{aligned}\end{align}$$

Applying the law of cosines and algebra:

$$\begin{align}\begin{aligned}R^2 = L_1^2 + L_2^2 - 2 L_1 L_2 \cos \beta\\\beta = \arccos {R^2 - L_1^2 - L_2^2 \over - 2 L_1 L_2}\end{aligned}\end{align}$$

Applying the law of sine and algebra:

$$\begin{align}\begin{aligned}{R \over \sin \beta} = {L_2 \over \sin \alpha}\\\alpha = \arcsin {L_2 \sin \beta \over R}\end{aligned}\end{align}$$

Calculating the joint angles. Note that zero, one, or two solutions may exist.

$$\begin{align}\begin{aligned}\theta_1 = \theta - \alpha\\\theta_2 = \pi - \beta\end{aligned}\end{align}$$

Supporting Material

• documentation of our Webots Two-Link Robot Model

• my Fall 2020 video lecture on Kinematics of the Two-Link Arm

• SymPy derivation of two-link inverse kinematics two-link-kinematics.py

• sample Python code for two-link forward and inverse kinematics: two-link-ik.py