Control on Ben Bokser's Blog

Intro to Robot Simulation, Part 4: Rigid Bodies

Sat, 08 Aug 2026 00:00:00 +0000

In the last part of this series, we learned how to simulate a point mass making frictional contact with the ground plane.

This time, we’ll do the same thing, but with a three-dimensional cube.

Modeling a Floating Rigid Body

Let’s start by making our cube float in a zero-G environment.

Dynamics

To simulate a three-dimensional rigid body, our new state-space system must encapsulate not just the object’s position and linear velocity, but also its orientation/attitude and angular velocity. That leads us to the following state vector:

Intro to Robot Simulation, Part 3: Friction

Mon, 20 Jan 2025 00:00:00 +0000

In this post, we will add 1-dimensional coulomb friction to our simulation.

First off, we’re going to need to allow movement in the horizontal direction. Our new state-space system will be a 2-dimensional system subject to both horizontal and vertical forces.

$$ \begin{equation} \underbrace{\begin{bmatrix} \dot{x} \\ \dot{z} \\ \ddot{x} \\ \ddot{z} \end{bmatrix}}_{\dot{X}} = \underbrace{\begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}}_A \underbrace{\begin{bmatrix} x \\ z \\ \dot{x} \\ \dot{z} \end{bmatrix}}_X + \underbrace{\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \frac{1}{m} & 0 \\ 0 & \frac{1}{m} \end{bmatrix}}_B \underbrace{\begin{bmatrix} f_x \\ f_z \end{bmatrix}}_F + \underbrace{\begin{bmatrix} 0 \\ 0 \\ 0 \\ -9.81 \end{bmatrix}}_G \end{equation} $$

Modeling Friction

For 1-dimensional Coulomb friction, our friction force, $f_x$, is related to our vertical force $f_z$ by the following equation.

Intro to Robot Simulation, Part 2: Contact

Tue, 31 Dec 2024 00:00:00 +0000

Okay, so we can simulate a floating point mass subject to arbitrary forces. Let’s now think about what happens when our point mass hits the ground.

Assuming 100% stiff, inelastic impact, we would get the following position, velocity, and acceleration trajectories:

Notably, the acceleration at impact approaches infinity, which would be very difficult to simulate–it would require infinitesimal timesteps. Of course, in reality there is no such thing as a perfectly stiff and inelastic collision, but in the field of robotics we are often concerned with relatively stiff collisions which cause decelerations on the order of microseconds.

Intro to Robot Simulation, Part 1: Point Masses

Sun, 18 Feb 2024 00:00:00 +0000

So, you want to write your own dynamic robot simulation from scratch.

Let’s start with one of the simplest possible dynamical systems: a vertically constrained point mass. Think of it as a rocket locked to a linear rail.

The rocket cannot rotate or translate laterally; it can only move up or down, and can it only control this by either exerting an upward force or letting gravity take its course.

Deriving the Manipulator Equations from the Euler-Lagrange Equation

Mon, 06 Mar 2023 00:00:00 +0000

The equations of motion for a robot manipulator and their relationship to the Euler-Lagrange equation are well known in the field of robotics, and yet I was unable to find a single source that provides the derivation. I suppose it’s one of those “the solution is trivial and is left as an exercise to the reader” situations. As such, I’m providing one here for dummies like me who need it spelled out step-by-step.

Stable Bipedal Walking in Simulation

Tue, 20 Jul 2021 00:00:00 +0000

In late 2019, I designed a custom QDD gearbox. Then I designed a bipedal robot with said gearing. By early 2020 I was spending about an hour per day after work coding a controller for the bipedal robot I had designed. In that time I learned a great deal of Python, and my controls proficiency skyrocketed. Then I started grad school and had to put this project on hold for about six months. But now, over a year and a half later, I’ve finally achieved stable bipedal walking in simulation.

Model Predictive Control for a Legged Robot

Mon, 12 Oct 2020 00:00:00 +0000

I have recently implemented a model predictive controller (MPC) to calculate the necessary reaction forces for a legged robot. The work presented here is based on this paper by Kim et al. If you don’t know what model predictive control is, I recommend this excellent explanation by Steve Brunton. I also found this guide to model predictive control with CasADI to be extremely helpful. CasADi is an open source nonlinear optimization tool which I’m using for MPC.

Solving for the Jacobians of a Robot Leg

Mon, 04 May 2020 00:00:00 +0000

I’m currently working on the Python code to control a simulated version of my latest bipedal robot design in PyBullet. My focus over the past few weeks was getting the operational space control to work (many thanks to Travis DeWolf’s incredibly helpful blog). After finally getting it to work properly, I have decided to share my math in the hopes of providing a useful example for anyone else having trouble with this. There really aren’t enough resources on the internet that explain this in a succinct manner.