Local vs Global Coordinate System - Kinematics

You start to get the hang of dynamics questions, and you know when to apply each equation. You draw your free-body diagram, apply linear and angular momentum laws, remember to use the parallel axis shift, and do everything you believe is correct. And the answer is wrong. You check the solution and find that the entire solution is done in a local coordinate system, and your global-system solution now seems overcomplicated. Furthermore, you can't even tell where you made a mistake. How do you pick when to use a local vs a global coordinate system?

In real engineering applications, the errors become much more extreme. If forces and torques on a drone are represented in the wrong reference frame, your drone will nosedive and spiral out of control during control testing, almost guaranteeing property damage as the forces aren't close to what reality expects.

Choosing coordinate systems trips up many students as it's not a fixed rule, but instead requires nuance. Both choices work, but one is usually easier. I'll provide you with the rules I used in my engineering life as a guideline.

What each one actually is

Global (inertial) coordinate system

A global (or inertial) coordinate system is one fixed frame for the whole problem. A stationary origin is selected and is simply fixed, all velocities and forces are then measured from that fixed reference frame. This frame is the one Newton's second law was initially written for. It is most commonly used for forces on a body that act in translation, and when an unambiguous real frame of reference of an observer is more important. Imagine a drone that's going through a trajectory: if I want the drone to go forward from its starting position, I don't want the current drone orientation to dictate which direction straight is. In cases like this, a global frame is feasible.

Local (body-fixed) coordinate system

A local (or body-fixed) coordinate system is attached to something that moves, this mainly involves translating or rotating bodies. A local coordinate system depends on body orientation and position, which change over time. This coordinate system is incredibly useful to simplify geometry. Again, imagine a drone starting to take flight. The moment generated by upwards thrust on the rotor is the cross product between the thrust and the lever arm to the drone body centre. In a global coordinate system, the lever arm is a time-dependent variable varying with drone position, speed, acceleration, attitude, angular velocity and angular acceleration. In the local coordinate system, it's a fixed vector since it moves with the body. For inertia calculations using the parallel axis theorem, having a local coordinate system moving with the body fixes the Steiner component, rather than making it dependent on multiple directional axes.

In actuality, the choice of the coordinate system is arbitrary. The same conclusion can be arrived at using either one. The process to get to that conclusion, though, takes significantly longer when a nonsensical coordinate system is used.

Common mistakes when using each coordinate system

Global coordinate system

• The vector position, velocity and acceleration will be provided in the global coordinate system with respect to a stationary point. Is this data really useful to you? Do you really care about the exact fixed current velocity vector dependent on an azimuth and altitude angle?
• Does the question actually become simpler in this frame, or are you just avoiding learning how to apply a rotating frame?
• If there is a required vector chain due to multiple kinematic components, can you actually describe what's going on at the end of the chain? Should you instead do a chain of moving coordinate systems to describe the kinematics?
• Is there ambiguity in your position or orientation? Are there potentially computational problems with 'dividing by zero' due to one of the angles being 0 or π/2

Moving coordinate system

• Can you actually apply certain laws to this? Does Newton's Second Law actually hold here?
• If the coordinate system is rotating, did you remember to do the Euler derivative?
• Does the location of the coordinate system only simplify the current calculation step you're doing, but figuring out the dynamics of this coordinate system itself becomes more difficult than another coordinate system choice? Did you overengineer the problem?
• Did you apply the relative linear and angular acceleration laws rather than absolute ones?

Everything that I wrote out here might make it seem like this is a very difficult skill to master. However, it just takes a bit of practice. Next time you solve a dynamics task, try understanding why the solution chose this specific coordinate system, and then try also solving it with the other one. Maybe you'll appreciate the simplified terms, maybe you'll come up with a better solution than your university. Either way, you will learn something, and that's the most important thing in engineering: figuring out your mistakes and improving from them.