Polynomial ZMP Trajectory Generation for Mobile Robot Stability
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Solution Overview
Problem
Conventional methods for generating a desired ZMP trajectory for mobile robots face challenges in efficiently fulfilling multiple constraint conditions, such as maintaining stability and ensuring the ZMP exists within a permissible region, often restricting the degree of freedom in gait generation and impairing motion smoothness.
Innovation Solution
A device that determines a polynomial function-based ZMP trajectory using a quadratic programming method, incorporating constraint conditions expressed as linear inequalities and equalities, to generate a trajectory that meets stability, region, and smoothness requirements.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Stability of the object's composition
If a reference ZMP trajectory is corrected with a trapezoidal trajectory pattern to generate a desired gait that keeps the mobile robot continuously in a stable state, then the robot stability is improved, but the degree of freedom in setting a desired position of the ZMP is restricted and the ZMP existence permissible region constraint may not be fulfilled
Solution Approach 1:
The patent changes the mathematical representation of the ZMP trajectory from piecewise linear (polygonal) to polynomial functions. This parameter change in the trajectory formulation allows continuous adjustment of trajectory parameters while satisfying constraints, thereby maintaining robot stability through polynomial coefficients optimization while preserving degree of freedom in ZMP position setting within the supporting polygon.
2Stability of the object's composition
If a reference ZMP trajectory is corrected with a trapezoidal trajectory pattern to ensure continuous stability, then the robot stability is improved, but the desired ZMP trajectory may not fulfill the constraint condition regarding the ZMP existence permissible region, requiring regeneration
Solution Approach 1:
The patent formulates the ZMP trajectory generation as a polynomial function with predetermined coefficients that are calculated in advance to satisfy all constraint conditions simultaneously. This preliminary calculation approach ensures that the generated trajectory inherently fulfills both stability requirements and ZMP existence permissible region constraints, eliminating the need for iterative correction and regeneration.
3Stability of the object's composition
If a desired ZMP trajectory is generated as a polygonal line, then the trajectory can be generated to fulfill stability constraints, but the smoothness of mobile robot motion near break points is impaired
Solution Approach 1:
The patent replaces the polygonal line trajectory with polynomial function-based trajectories that exhibit continuous curvature. This curvature-based approach eliminates the sharp break points inherent in polygonal trajectories, providing smooth motion transitions while maintaining the stability characteristics required for robot operation.
4Reliability
If multiple constraint conditions are simultaneously imposed on the desired ZMP trajectory, then the trajectory can fulfill stability and region constraints, but the complexity of trajectory generation increases
Solution Approach 1:
The patent merges multiple constraint conditions into a unified polynomial optimization framework. By formulating all constraints (stability, ZMP existence permissible region, smoothness) as conditions on polynomial coefficients, the system simultaneously satisfies multiple requirements through a single mathematical formulation rather than separate processing steps.
Data Source
AI summary
A device for generating a desired ZMP trajectory for a mobile robot includes a polynomial function coefficient group determining section (53a) which determines, by regarding the desired ZMP trajectory as a trajectory expressed by a polynomial function, a desired coefficient group composed of desired values of coefficients in respective terms of the polynomial function. The polynomial function coefficient group determining section uses a quadratic evaluation function including square values of the coefficients included in the desired coefficient group as variables and a plurality of constraint conditions each configured by a linear equality or linear inequality about the coefficients, to determine the desired coefficient group, by a solution method for a quadratic programming problem, in such a way as to minimize a value of the evaluation function while fulfilling the constraint conditions.