Published on 12, July, 2020
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.
can you give me a good example of where 0 displacment would be an issue?
Benefits of the Modified Formula:
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?
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.
My point about zero displacement issues is more about practical considerations in real-world scenarios:
Even if the system is designed to return to =x=a, practical sensors might have limitations in precision or be affected by noise, potentially causing them to miss or inaccurately register very small displacements.
Moreover, In computational implementations, rounding errors or limitations in numerical precision can sometimes result in tiny displacements being treated as zero, which might impact the control system’s performance.
Practical systems might face challenges such as dead zones or limitations in control resolution that could affect the handling of very small deviations.
While the control system you described is robust, addressing these practical aspects helps ensure that even the smallest movements are detected and managed accurately, which is crucial in high-precision applications.
I hope this clarifies why zero displacement issues can be relevant in specific contexts.
ok I'm still not seeing the issue. Supose I have a 1 axis robotic system. I want the arm to move to a given point a and stay there. There is a displasment sensor on the arm so I program an equation for the force like so
f=-b(x-a) (or -c sign(x-a) if abs(b(x-a))>c)
and of course f = m d^2x/dt^2 so even if an external force comes along and displaces the arm or if the sensor has some noise it will keep returning to aproxomatly point a because what we have is mathermaticly equivalent to a spring.
The computer controle system will constantly monitor the displacement sensor and apply the apropreat force. If x=a it will just apply 0 force and the moment it drifts the computer will apply force again ... I don't see the issue here.
Surgical Robots
Micro-assembly
Satellite Positioning
Spacecraft Maneuvering
CNC Machines
Laser Cutting and Engraving
Microscopy
Experimental Physics
Character Animation
Virtual Reality (VR).. Etc
My world too..
Thank you for your question! You’re right that introducing a small ϵ won’t solve issues like numerical instability or fundamentally change how the Euler method operates, especially in stiff systems or complex ODE/PDE solvers. However, the purpose of my approach isn’t to fix broader issues like stability, but rather to handle specific cases where zero displacement becomes problematic.In applications such as robotics, animation, or position tracking, zero displacement can lead to issues with motion continuity and path tracking. By introducing a small ϵ, I ensure that even minimal movements are registered, preventing the system from stalling due to rounding errors or small velocities.
For instance, in a robotic system, where small movements are significant for tasks like assembly or navigation, adding ϵ helps maintain continuous motion. This is especially important in real-world scenarios where small but detectable displacements matter. The modified formula prevents the system from treating very small movements as zero, which could affect performance
I completely agree that ϵ doesn’t address the broader limitations of methods like explicit Euler, particularly in cases where numerical stability is a concern (e.g., stiff systems). More advanced methods like Runge-Kutta or implicit solvers would be more appropriate for those cases.
In short, my proposal focuses on solving a specific problem of non-zero displacement in certain applications, rather than addressing the fundamental limitations of numerical methods.