![]() If we have an end-effector, we can add that in as well however for now, this should be sufficient to get a rough dynamical model. Finally, we offset the third frame by the length of the second link about the x-axis (red) of the second frame and rotate about the y-axis by angle θ₃ to get to the third frame. We then rotate about the y-axis (green) by angle θ₂ to get to the second frame. To get from the robot’s base frame (fixed) to the first frame, we’ll rotate about the z-axis (blue) by angle θ₁. ![]() This way, the translations between these joints are zero vectors. We’ll place the origins of the robot’s base frame, first frame and the second frame at the same point - at the intersection of the robot’s first and second joint axes. Generally, there are conventions such as the Denavit-Hartenberg convention for where to place coordinate frames but we’re going to keep it simple to minimize the number of variables. In order to do this, we need to place coordinate frames at each joint. We start with defining the generalized coordinates of the system. More papers can be found on the Research page of our website that describe dynamical modeling for control systems design. I’ll be referencing equations from the paper in this article. I describe the important equations in deriving the dynamical model of a robotic system in this paper on designing an omnidirectional controller for a spherical robot. You can install it using the terminal command: pip3 install sympy You need the sympy library to process symbolic math in Python. In this example, we’ll consider an Anthropomorphic Arm without a wrist to keep it simple. Once we derive the equations of motion, we’re going to develop a simple simulator for it which will allow us to develop and test controllers. I can also write another article in the future that shows an example of modeling another kind of robot. If you’re interested in modeling a mobile robot or a drone, this is still very important in order to understand the process and the core equations behind the dynamics. In this article, we’ll use Python to derive the closed form dynamic equations of motion of a simple three-link robotic manipulator. However, MATLAB is not free so we’ll use Python 3 - a great, free programming language with a ton of libraries and toolkits. Robotics programs at most Universities use MATLAB to derive the equations of motion of a robot. If you’re reading this, you probably either want to design a robot or you have a robot and want to develop a simulator or controller for it. It is also very helpful to understand Kinematics of robotic systems - specifically the forward and inverse kinematics, before you dig into dynamics. You need to understand linear algebra, coordinate frames, and basic Calculus. ![]() I’ve also provided links to good courses or materials on these topics if you’re missing something. Prerequisites: Yes, there is definitely prior knowledge you must have before we dive into this! This article is ideal for those with a Bachelor’s degree in an engineering, physics or mathematics field. This article specifically goes over how you can derive the equations of motion of a robot. They are a set of equations that allow you to calculate the next state of the robot in a dynamic setting, given the current state and some input. The dynamic equations of motion of a system address this. In order to do either of those, it’s critical to understand the dynamics of the robot - that is, how the robot responds to certain inputs that cause it to be in motion. What I’m attempting to do is teach some basic robotics math - sufficient to develop your own robots or controller. Generally if users would like to develop their own controller, they would need to be a robotics engineer or have extensive knowledge of robot forward and inverse kinematics, dynamics, control systems design and parameter identification. Many of these come with open-source software development kits (SDKs) and libraries to program the controller for custom applications. Lots of desktop robotic manipulators are also popping up such as the DOBOT Magician and the 7Bot. This includes robotic manipulators used in assembly lines and the ISS, mobile robots and drones. This is overly generalized, so for the purposes of this series of articles, we’re going to stick to any actuated set of joints designed to do specific tasks. The word “robot” stems from the Crezh word “robotnik” and Slavonic word “rabota”, both which mean “slave”. By robots, I don’t just mean humanoids or robotic arms. Robotics is a very fast growing industry and with the availability of low-cost motors, actuators, computers and open-source software libraries, it’s easier than ever for hobbyists to design and build their own robots.
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