Analyzing D-O's Movements

Project Overview

Over the past several weeks, I have been working on a functional D-O droid from Star Wars. It is comprised of a chassis attached to two parallel wheels, with a cone shaped head on top. The purpose of this project is to analyze the physics behind the droid as well as other rolling bodies, and convert the analysis into a Snap Simulation.

The Designs

The Chassis

D-O's chassis contains the drive gears, servos to manipulate the droid's arm, and the drive electronics. For this simulation, we will consider this a single mass

The Wheels

D-O has two wheels holding the chassis over the ground. The wheels were be regarded as circular disks in derivations and Snap code.

Initial Data Collection

D-O's Dimension Sheet Based off the CAD Designs

After referencing the dimension sheet of the Droid and collecting the mass of each component in the droid's chassis, I got the following values:

Droid Total Mass = 1.614 kg
- Droid Chassis Mass = 1.250 kg
- Droid Wheel Mass = 0.182 kg
Droid wheel Radius = 0.225 m

For this experiment, we will only look at the bottom part of the droid.

Free Body Diagram

Free Body Diagram Including Chassis and Wheel

This initial diagram shows two stages of motion in the droid. The top image shows the droid at rest, where the only forces acting on it is gravity and the normal force. The bottom image shows the droid when it is accelerating. When the droid accelerates, the chassis "rolls back" due to its inertia. This image displays the chassis at some angle theta with the linear force Flin acting on it, pushing the droid forward.

Why Does the Droid Move?

To understand how D-O drives forward, we need to observe the droids center of mass (COM). Imagine the droid being like a hamster ball, with the chassis acting as the hamster and the wheels acting as the ball.

When it is not accelerating, the chassis is exactly level with the ground, and the COM is in the middle of the droid (horizontally). The only forces acting on it is gravity (Fg) and the normal force (N). Since there is no horizontal forces, the droid does not accelerate in any direction. This does not meaning it is stopped! The droid could have a velocity, but since no horizontal forces are acting on it, the velocity's rate of change is zero.

Now if we look at an accelerating droid, you can see the COM has changed. This is because the chassis pushing forward against the wheel has moved the COM to a side. Think of the hamster running in the ball; it pushes against the sides to move forward/backward. When the COM moves out of the center (in terms of x-axis), a torque is enacted on the wheel. Since a torque is simply a rotational force, the wheel begins to accelerate and starts spinning.

But how is this rotational acceleration turned into linear movement? And what is that frictional force doing there?

The answer is that the frictional force IS the linear force! If there was no friction between the ground and the wheel, the wheel would simply spin in place and wouldn't go anywhere. This frictional force acts on the point where the wheel touches the ground, allowing the droid to move forward (to the left side of this page in this case. This is why the point of contact, which I will denote as P, has an instantaneous velocity of 0 relative to the ground. The friction force is preventing it from "skidding". However, if the frictional force is too little, or if the wheels acceleration is too high, skidding will occur and the rotating body will slide on the surface. This is what happens when a car is driving on ice (low frictional force), or when a street racer performs a "burnout" (high acceleration).

Applications of this Concept

The use of rotational motion and friction is used in nearly every machine. A common use of this is in automotive technology, where wheels and their connection to the ground are critical. This is why cars with 4 wheel drive are more powerful than cars with 2 wheel drive; more torque applied means more powerful drive trains. Manufacturers utilize rotational motion data in Traction Control Systems, where cars adjust the torque and acceleration of their wheels to avoid slipping on the road. Everywhere you go, you can find machines utilizing rotational bodies and torque

Deriving a = alpha * R

Observing the free body diagram of the wheel, we can see Vrel, V0, and Vp(the velocity at point P). Since Vp must be 0 assuming the wheel is not skidding, Vrel and V0 must be equal to each other, due to them being opposing vectors. This means that both V0 = Vrel = wR (velocity at center of mass = angular velocity * radius). If we diffrentiate the equation, we will find that a = alpha * R (acceleration at center of mass = tangential acceleration * radius)

Tracker Analysis

Tracker Analysis High Speed

When the droid was accelerated at a high speed, the effects of what was described in the "Why Does the Droid Move" becomes clear. When the droid starts moving, its chassis begins to roll up the sides of the wheels, similar to a hamster pushing up against the sides of a ball. At approximately 0.5 seconds the chassis reaches its maximum height and begins to come down. At around 0.6 seconds, the velocity of the droid begins to level off and the acceleration stops.

Y Position of Drive Gear Over Time

Velocity of Droid Over Time

Tracker Analysis Low Speed

Similar to the high speed test, the droid's chassis rose and dropped sharply, correlating with the velocity increasing and leveling off. While there was far more noise in this data (slower speed meant closer together points) the results appear to exhibit the same conclusion.

Y Position of Drive Gear Over Time

Velocity of Droid Over Time

Snap Simulation V1 (Updated)

This is the first complete version of my snap simulation. It contains a rolling ball, which can be controlled via changes in torque. It also contains 6 data graphs displaying both linear and angular acceleration, position and velocity.

To view controls, scroll the Notes Section.

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