Inverse Kinematics Control Transfer Matrix for Singularity Avoidance
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Solution Overview
Problem
The inverse kinematics problem in robotics and computer graphics is computationally intensive and challenging due to the need for real-time computation of joint variables for articulated systems with a large number of degrees of freedom, especially when the Jacobian matrix becomes rank deficient, leading to singular states and excessive values for joint velocities.
Innovation Solution
A real-time method using a feedback loop with a control transfer matrix (K) that generates solutions even when the Jacobian is rank deficient, avoiding singular states by steering the trajectory away from singular directions, and requiring only multiply and accumulate operations, which can be performed on single or parallel processors.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional numerical methods are used to solve inverse kinematics by inverting the Jacobian matrix, then the joint variables can be computed, but the computational load becomes excessive and the system becomes unstable when the Jacobian matrix is rank deficient
Solution Approach 1:
The patent applies feedback by using the error between desired and current trajectory (e = dxd - dx) to generate corrective joint velocity commands. The control transfer matrix K processes this error signal in a feedback loop, continuously adjusting the solution to maintain stability even when the Jacobian is rank deficient, eliminating the need for damping factor adaptation.
Solution Approach 2:
The patent changes the parameter representation by using a control transfer matrix K with dimensions m×n instead of inverting the n×m Jacobian matrix. This parameter transformation allows the system to compute joint velocities directly from the error signal without encountering singularity issues, improving both reliability and computational efficiency.
2Measurement precision
If the Jacobian matrix is inverted to compute joint velocities, then the inverse kinematics solution can be obtained, but excessive values occur when the system is in singular states
Solution Approach 1:
The feedback mechanism using error signal e = dxd - dx ensures that joint velocity commands are generated based on trajectory deviation rather than direct Jacobian inversion. This feedback approach maintains measurement precision while avoiding excessive values in singular states, as the control transfer matrix K processes the error signal without being affected by Jacobian rank deficiency.
Solution Approach 2:
By changing from Jacobian matrix inversion to using a control transfer matrix K that operates on the error signal, the patent transforms the computation parameter from direct inverse kinematics to error-based control. This parameter change eliminates the singularity problem while maintaining accurate joint velocity computation.
3Reliability
If damping factor adaptation is used to handle singular states, then the system stability improves, but the device complexity and computational overhead increase
Solution Approach 1:
The patent uses feedback with a fixed control transfer matrix K to handle singular states without requiring damping factor adaptation. The feedback loop continuously processes the error signal e = dxd - dx through matrix K, providing stable joint velocity commands near singular configurations while keeping the algorithm simple and computationally efficient.
Solution Approach 2:
The control transfer matrix K is designed to inherently handle singular states without requiring additional complexity or adaptive parameters. The system serves itself by using the error signal and fixed matrix multiplication to automatically generate appropriate joint velocity commands, eliminating the need for damping factor adaptation mechanisms.
4Manufacturing precision
If real-time computation is performed for systems with large numbers of degrees of freedom, then the trajectory tracking accuracy improves, but the computational load becomes prohibitive
Solution Approach 1:
The feedback approach using error signal e = dxd - dx and control transfer matrix K enables real-time computation for high-DOF systems. By computing only the necessary matrix multiplications (K×e and Jt×result) rather than full Jacobian inversion, the system achieves both high trajectory tracking accuracy and computational efficiency suitable for real-time operation.
Solution Approach 2:
The patent changes the computational parameter from inverting an n×m Jacobian matrix to multiplying an m×n control transfer matrix K with the error signal. This parameter transformation reduces the computational complexity from O(n³) to O(mn), enabling real-time trajectory tracking for systems with large numbers of degrees of freedom while maintaining accuracy.
Data Source
AI summary
A real-time method for controlling a system, the system including a plurality of controlling means each having at least one variable parameter (q) and a controlled element having a trajectory which is controlled by the controlling means, wherein the trajectory is related to the variable parameters by a variable matrix, the method comprising defining a control transfer matrix (K) relating the variable parameters dq to the trajectory dx, and using a feedback loop in which a feedback term is computed that is dependent on an error (e) which is the difference between the desired trajectory (dxd) which can have an arbitrary dimension specified as (m) and a current trajectory (dx).