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Science, A Modified Displacement Formula for Non-Zero Displacements

Noon..
12 days ago

Helllo there, 

I always dreamed of becoming a scientist, but I couldn't get into university. Still, I managed to modify the displacement law, which had been troubling me at zero. I hope you enjoy the modification

Introduction

The standard displacement formula, Δx=xf−xi​, can result in zero displacement when xf=xi This limitation is particularly problematic in applications such as simulations, numerical analyses, and contexts requiring continuous motion. For instance, in robotics and animation, zero displacement can cause issues with position tracking and visual continuity. To address this challenge, we propose a modified displacement formula designed to ensure non-zero displacement, which is crucial for accurate modeling and simulation.

Methodology

Redefining Initial Position:

To avoid zero displacement, we introduce a small positive constant ϵ\epsilonϵ. The modified initial position xi′ ​ is defined as:

xi′=xf+ϵ 

where ϵ is a small value, selected to be contextually appropriate. The displacement formula then becomes:

Δx′=xf−(xf+ϵ)=−ϵ

This modification ensures that the displacement Δx′ is always non-zero, thus preventing zero displacement in scenarios where continuous motion is required.

Parents
  • 11 days ago

    can you give me a good example of where 0 displacment would be an issue?

  • 8 days ago in reply to Peter

    Benefits of the Modified Formula:

    • Non-Zero Displacement: Ensures that even minimal movements are represented, which is essential for applications like robotics where continuous tracking of position is critical. For example, in a robotic arm simulation, ensuring non-zero displacement can help in accurate path planning and obstacle avoidance.
    • Numerical Stability: Helps prevent computational artifacts in simulations where zero displacement could lead to errors or instability. For instance, in numerical fluid dynamics simulations, ensuring non-zero displacement helps maintain stability in iterative calculations.
  • 8 days ago in reply to Noon..

    Well minimal movements you can detect are going to be corected for. Frankly I still don't see the issue. How exactly do you think a robotic arm or numerical ODE/PDE solver will behave difrently using these infimatesimals you are adding on to 0s? How is the numerical algorithem suposed to treat them difrently? for example, Lets consider the explicit eular apromation (en.wikipedia.org/.../Euler_method) for solving ODEs for simplicities sake. How do you supose your proposal would modify it's function?

  • 5 days ago in reply to Noon..

    not really. Adding an infinitesamal to 0 won't fix dead zone or sensor noise issues. For starters how do you know you are corecting in the right direction. Depending on the sign or your error maybe you should be adding or subtracting. Maybe your 'very small number' is actually bigger than any error once in a while. All real world controle systems are aproxomate. There is no mathermatical trick for getting around the inherent precision limits of sensors and motors.

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  • 5 days ago in reply to Noon..

    not really. Adding an infinitesamal to 0 won't fix dead zone or sensor noise issues. For starters how do you know you are corecting in the right direction. Depending on the sign or your error maybe you should be adding or subtracting. Maybe your 'very small number' is actually bigger than any error once in a while. All real world controle systems are aproxomate. There is no mathermatical trick for getting around the inherent precision limits of sensors and motors.

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