Kinematics of Planar Robots with Revolut Joints

This post is part of my Robotics Modeling, Control, and Design Tutorial series.

Introduction

Kinematics is the branch of mechanics that describes the motion of systems without considering the forces that cause that motion. In robotics, kinematics focuses on the position, velocity, and acceleration of robot links and joints.

Forward Kinematics (FK): Given the joint variables (angles for revolute joints), compute the position and orientation of the robot’s end-effector. FK is usually analytical and straightforward for planar robots.

Inverse Kinematics (IK): Given the desired position (and possibly orientation) of the end-effector, compute the joint variables that achieve it. IK can be analytical for simple robots, numerical or iterative for more complex ones. Also, it may have multiple solutions, no solution, or infinite solutions

Let us examine planar robots with 2 and 3 revolute joints.

Planar Robot with 2 Revolute Joints

Forward Kinematics of 2R Plannar Robots

Consider a planar robot with two revolute joints, as shown in the figure below. Let:

$\chi_1,\, \chi_2 \in [-\pi, \pi)$ be the joint angles
$l_1,\, l_2 \in \mathbb{R}$ be the link lengths

**Figure:** Planar robot with 2 revolut joints.

Then the position of the end-effector is given by: $$ \begin{align} x_E &= l_1 \cos(\chi_1) + l_2 \cos(\chi_1 + \chi_2) \\ y_E &= l_1 \sin(\chi_1) + l_2 \sin(\chi_1 + \chi_2) \end{align} $$

This model represents a two-link planar manipulator in an elbow configuration.

Inverse Kinematics of 2R Plannar Robot

Let the target end-effector position be $\left(x_E, y_E\right)$. The goal is to solve for joint angles $\chi_1, \chi_2$ given the link lengths $l_1,\, l_2$.

Step 1: Compute $\chi_2$ using the law of cosines

Applying the law of cosines to the triangle $O_1O_2E$:

\[\begin{align} \cos \widehat{O_1O_2E} &= \dfrac{O_2O_1^2 + O_2E^2 - O_1E^2}{2O_1O_2},\\ \Rightarrow \cos(\pi - \chi_2) &= \dfrac{l_1^2 + l_2^2 - \left(x_E^2 + y_E^2\right)}{2l_1 l_2}. \end{align}\]

Noting that $\cos(\pi - \chi_2) = -\cos\chi_2$, we compute $\chi_2 \in [-\pi, \pi)$ as:

\[\begin{align} \chi_2 = \pm \arccos \left(\dfrac{x_E^2 + y_E^2 - l_1^2 - l_2^2}{2l_1 l_2}\right). \end{align}\]

There are two possible solutions for $\chi_2$ corresponding to the elbow-up and elbow-down configurations.

Step 2: Compute $\chi_1$ using the law of cosines

Let $\psi = \widehat{O_2O_1E}$. Then the sum $\chi_1 + \psi$ is the angle from the base to the target, and is given by:

\[\begin{align} \chi_1 + \psi = \text{arctan2 }(y_E, x_E). \end{align}\]

Again applying the law of cosines in triangle $O_2O_1E$:

\[\begin{align} \cos \widehat{O_2O_1E} &= \dfrac{O_1O_2^2 + O_1E^2 - O_2E^2}{2O_1O_2},\\ \Rightarrow \cos \psi &= \dfrac{x_E^2 + y_E^2 +l_1^2 - l_2^2}{2l_1 \sqrt{x_E^2 + y_E^2}},\\ \Rightarrow \psi &= \pm \arccos \left(\dfrac{x_E^2 + y_E^2 +l_1^2 - l_2^2}{2l_1 \sqrt{x_E^2 + y_E^2}}\right). \end{align}\]

Thus, $\chi_1$ is computed as:

\[\begin{align} \chi_1 = \text{arctan2 }(y_E, x_E) \pm \arccos \left(\dfrac{x_E^2 + y_E^2 +l_1^2 - l_2^2}{2l_1 \sqrt{x_E^2 + y_E^2}}\right). \end{align}\]

Final Solution

From the equations above, we see that there are four mathematical solutions for $(\chi_1, \chi_2)$. However, only two of them are physically feasible configurations for the robot. These are:

Elbow-down configuration $$ \begin{align} \chi_1 &= \text{arctan2 }(y_E, x_E) - \arccos \left(\dfrac{x_E^2 + y_E^2 +l_1^2 - l_2^2}{2l_1 \sqrt{x_E^2 + y_E^2}}\right),\\ \chi_2 &= \arccos \left(\dfrac{x_E^2 + y_E^2 - l_1^2 - l_2^2}{2l_1 l_2}\right). \end{align} $$ Elbow-up configuration $$ \begin{align} \chi_1 &= \text{arctan2 }(y_E, x_E) + \arccos \left(\dfrac{x_E^2 + y_E^2 +l_1^2 - l_2^2}{2l_1 \sqrt{x_E^2 + y_E^2}}\right),\\ \chi_2 &= -\arccos \left(\dfrac{x_E^2 + y_E^2 - l_1^2 - l_2^2}{2l_1 l_2}\right). \end{align} $$

Planar Robot with 3 Revolute Joints

Forward Kinematics of 3R Plannar Robot

Let:

$\chi_1,\, \chi_2,\, \chi_3$ be the joint angles
$l_1,\, l_2,\, l_3$ be the link lengths

The forward kinematics gives the position of the end-effector:

\[\begin{aligned} x_E &= l_1 \cos(\chi_1) + l_2 \cos(\chi_1 + \chi_2) + l_3 \cos(\chi_1 + \chi_2 + \chi_3) \\ y_E &= l_1 \sin(\chi_1) + l_2 \sin(\chi_1 + \chi_2) + l_3 \sin(\chi_1 + \chi_2 + \chi_3) \end{aligned}\]

This configuration introduces kinematic redundancy, which allows more flexibility in reaching a target point, and enables optimization for additional objectives such as obstacle avoidance or joint effort minimization.

Inverse Kinematics of 3R Plannar Robot

**Figure:** Planar robot with 3 revolut joints.

Because the robot has three degrees of freedom in a 2D space, the system is redundant — meaning that there are infinitely many combinations of $(\chi_1, \chi_2, \chi_3)$ that achieve the same end-effector pose. In other word, we can position not only the position $(x_E, y_E)$ but also the orientation $\theta_E$ of the end effector.

To find a solution, we can adopt the following approach:

Choose a desired orientation with respect to the base frame $\theta_E$ for the end-effector (either given or arbitrarily selected).
Compute: $\chi_3 = \theta_E - (\chi_1 + \chi_2)$
Define the wrist position $(x_W, y_W) = (x_{O_3}, y_{O_3})$ as:

\[\begin{align} x_W &= x_E - l_3 \cos(\theta_E) \\ y_W &= y_E - l_3 \sin(\theta_E) \end{align}\]

This reduces the problem to solving inversed kinematics for a 2R manipulator to reach the wrist position with links $l_1, l_2$. As derived in the previous section, we have

Elbow-down configuration $$ \begin{align} \chi_1 &= \text{arctan2 }(y_W, x_W) - \arccos \left(\dfrac{x_W^2 + y_W^2 +l_1^2 - l_2^2}{2l_1 \sqrt{x_W^2 + y_W^2}}\right),\\ \chi_2 &= \arccos \left(\dfrac{x_W^2 + y_W^2 - l_1^2 - l_2^2}{2l_1 l_2}\right),\\ \chi_3 &= \theta_E - (\chi_1 + \chi_2). \end{align} $$ Elbow-up configuration $$ \begin{align} \chi_1 &= \text{arctan2 }(y_W, x_W) + \arccos \left(\dfrac{x_W^2 + y_W^2 +l_1^2 - l_2^2}{2l_1 \sqrt{x_W^2 + y_W^2}}\right),\\ \chi_2 &= -\arccos \left(\dfrac{x_W^2 + y_W^2 - l_1^2 - l_2^2}{2l_1 l_2}\right),\\ \chi_3 &= \theta_E - (\chi_1 + \chi_2). \end{align} $$

Forward Kinematics of an $N$-R Planar Robot

The forward kinematics of a planar robot with $N$ revolute joints can be expressed in a compact and generalized form. Each link contributes to the end-effector position based on the cumulative joint angles up to that link:

\[\begin{aligned} x_E &= \sum_{i=1}^{N} l_i \cos\left( \sum_{j=1}^{i} \chi_j \right), \\ y_E &= \sum_{i=1}^{N} l_i \sin\left( \sum_{j=1}^{i} \chi_j \right). \end{aligned}\]

This formulation accounts for the sequential rotation of each joint and the cumulative nature of the angles in planar serial chains.

Summary

In this post, we explored the kinematics of planar robots with 2 and 3 revolute joints.

Kinematics describes how a robot moves without reference to forces. We examined both Forward Kinematics (FK) and Inverse Kinematics (IK).
For the 2R planar robot, we derived analytical expressions for both FK and IK. The IK problem yields two feasible configurations: elbow-up and elbow-down.
For the 3R planar robot, we introduced redundancy. The robot has three degrees of freedom in a 2D space, allowing it to reach the same point with infinite combinations of joint angles. By choosing a desired orientation of the end-effector, we reduce the problem to a 2R IK problem for the wrist position.
This approach is commonly used in practice and can be extended to optimize for secondary objectives such as minimizing joint displacement or avoiding obstacles.

This study provides a solid foundation for understanding kinematic modeling of planar manipulators. In future sections, we will generalize these methods using techniques such as Denavit–Hartenberg parameters (DH) and the Product of Exponentials (PoE) formulation for more complex spatial mechanisms.