Dynamic Predictor for Articulated Mechanisms Using Polynomial Approximations
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Solution Overview
Problem
Articulated mechanisms with stiff springs, such as walking robots, face challenges in accurately predicting link positions and joint displacements due to the slow convergence of series expansions, requiring a large number of terms to achieve precision, which is computationally impractical.
Innovation Solution
A dynamic predictor system that uses polynomial approximations to solve differential equations incrementally, minimizing residuals and their derivatives through Taylor series expansions and algorithmic differentiation, allowing for efficient adjustment of free parameters to stabilize and accurately predict joint motion in stiff dynamical systems.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If series expansion is used to solve differential equations for stiff springs, then accuracy of prediction is improved, but computational complexity and time required increase significantly
Solution Approach 1:
The patent transforms the differential equation system into an algebraic system by changing the approach from direct series expansion to a parameter-based formulation. This involves introducing new parameters that simplify the representation of stiff spring behavior, allowing accurate predictions without requiring extensive computational resources or complex iterative procedures.
Solution Approach 2:
The patent replaces the traditional mechanical series expansion approach with an algebraic formulation. Instead of using complex differential equation solvers that require many terms for convergence, the invention uses a simplified algebraic model that directly computes joint positions and displacements, substituting a computationally intensive mechanical/mathematical system with a more efficient algebraic one.
2Measurement precision
If more series expansion terms are used to achieve higher accuracy, then prediction precision is improved, but computation time increases impractically
Solution Approach 1:
The patent extracts the essential behavior of stiff springs and joint dynamics into a simplified algebraic formulation. By identifying and separating the critical parameters that govern stiff spring response, the invention creates a reduced model that captures the necessary dynamics without requiring the full complexity of traditional series expansions, thus achieving high precision with minimal computation time.
Solution Approach 2:
The patent segments the complex dynamic system into manageable algebraic equations that can be solved independently and efficiently. By dividing the problem into discrete algebraic relationships rather than using a single complex differential equation system, the invention enables rapid computation while maintaining accuracy, as each segment can be solved without requiring extensive series terms.
3Reliability
If traditional differential equation solvers are used for stiff dynamical systems, then model accuracy is improved, but computational resources required become excessive
Solution Approach 1:
The patent changes the fundamental parameters used to describe the system, transforming from time-derivative-based differential equations to algebraic relationships. This parameter transformation allows the model to maintain high accuracy for stiff dynamical systems while consuming significantly fewer computational resources, as the algebraic formulation avoids the iterative and resource-intensive processes required by traditional differential equation solvers.
Data Source
AI summary
A dynamic predictor usable for rapid and accurate calculation of joint commands of an articulated dynamical mechanism describes the relationship between the joints in the form of a differential equation. The predictor solves this differential equation for predicted joint states by fitting a polynomial equation having free parameters describing the predicted joint states to the differential equations by minimizing the differential equation residuals. This minimization employs a series expansion allowing algorithmic differentiation.