Inverse Function Control for Machine Axis Movement
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Solution Overview
Problem
Higher-order polynomials used to control movement sequences in machine elements often result in overshooting, leading to undesirable negative values or backward movements, particularly when dealing with unfavorable normalized speed boundary values, which is unacceptable in industries like printing where travel limits must be adhered to.
Innovation Solution
The method involves determining a polynomial of nth order that describes the functional relationship between a master and slave axis, modifying and inverting boundary values, especially normalized speeds, to form an inverse function that prevents overshooting by ensuring the inverse function's boundary values match the original speed values, thereby avoiding excessive overshoot.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If higher-order polynomials are used to control movement sequences, then the functional relationship between master and slave axes can be precisely described taking into account boundary conditions, but overshooting occurs leading to negative values or backward movements
Solution Approach 1:
The patent applies inversion by determining the polynomial function from the inverse perspective: instead of directly determining the slave axis position from master axis position, the method first determines the inverse function that maps slave axis position back to master axis position. This inverted approach fundamentally changes the calculation direction and prevents overshooting while maintaining precision.
Solution Approach 2:
The patent changes the parameter representation by inverting the boundary values during polynomial determination. Specifically, the boundary values for the polynomial coefficients are inverted (e.g., using 1/v instead of v for velocity boundaries), which transforms the problem into an equivalent form that eliminates overshooting behavior while preserving the essential movement constraints.
2Ease of operation
If polynomial functions are used with unfavorable normalized speed boundary values, then the movement sequence can be controlled, but excessive overshoot occurs which is unacceptable in applications like printing
Solution Approach 1:
The patent applies preliminary action by pre-inverting the boundary values before determining the polynomial function. This preparatory step ensures that the polynomial is constructed with inverted boundary conditions that inherently prevent overshooting, rather than attempting to correct overshooting after it occurs. The inversion is performed in advance as part of the polynomial determination process.
Solution Approach 2:
The patent transforms the problematic normalized speed boundary values into inverted parameters (e.g., using reciprocal values). This parameter transformation changes the mathematical characteristics of the boundary conditions, converting an unstable problem prone to overshooting into a stable problem that naturally adheres to travel limits while maintaining ease of control.
3Reliability
If the inverse function approach is used to prevent overshooting, then travel limits are adhered to, but the calculation complexity increases requiring numerical methods like Newton method
Solution Approach 1:
The patent implements feedback by using the inverse function relationship to continuously verify and adjust the movement sequence. The inverse function acts as a feedback mechanism that ensures the calculated slave axis position, when mapped back through the inverse relationship, consistently satisfies the master axis position and boundary conditions, thereby guaranteeing travel limit adherence.
Solution Approach 2:
The patent replaces direct mechanical constraint enforcement with a mathematical substitution approach. Instead of using complex mechanical mechanisms or iterative numerical optimization to prevent overshooting, the method substitutes the problem into an equivalent inverse mathematical form that naturally prevents overshooting through its structure, reducing the need for complex computational iterations.
Data Source
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AI summary
A method for controlling the motion sequence of a machine element, wherein the control of the motion sequence of the machine element is based on a functional relationship between a master axis (2) and a follower axis (4), and the functional relationship (3) is determined taking into account several conditions of this motion sequence. According to the invention, the functional relationship has at least a first section formed by a polynomial of order n and at least a second section, which is at least partially separated from the first section and formed by a polynomial of order a, where a is less than n.