Robot Control on Riemannian Manifolds With Tangent-Space iLQC
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Solution Overview
Problem
Existing methods for controlling robots with nonlinear dynamics on Riemannian manifolds face challenges due to the lack of a global vector space, making it difficult to recursively solve dynamic programs efficiently.
Innovation Solution
A method involving Gauss-Newton Multiple Shooting and Riemannian iterative linear quadratic control (iLQC) is employed, using tangent space approximations and parallel transport to optimize control sequences on Riemannian manifolds, allowing for efficient control by approximating state and cost dependencies in tangent spaces.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If dynamic programming is applied to control robots on Riemannian manifolds, then control optimality is improved, but computational complexity increases due to lack of global vector space
Solution Approach 1:
The patent segments the Riemannian manifold control problem into multiple local Euclidean patches. Each patch is equipped with its own coordinate system and dynamics model, allowing dynamic programming to be applied locally where the lack of global vector space does not prevent recursive solution. The global optimal control is constructed by combining these local solutions through transition functions between patches.
Solution Approach 2:
The patent introduces Euclidean embeddings as intermediary spaces that map the Riemannian manifold states into higher-dimensional Euclidean spaces. This allows the application of standard dynamic programming algorithms in the Euclidean embedding space, which then map back to the original Riemannian manifold control problem, effectively mediating between the non-Euclidean problem structure and Euclidean solution methods.
2Productivity
If tangent space approximations are used for state and cost dependencies, then computational efficiency is improved, but approximation accuracy may be reduced
Solution Approach 1:
The patent employs adaptive tangent space approximations that dynamically adjust the approximation order and local coordinate transformations based on the current state and control trajectory. This allows the system to use simpler linear approximations when sufficient while switching to higher-order approximations when precision is critical, optimizing the trade-off between computational efficiency and accuracy throughout the control process.
Data Source
AI summary
A method for controlling a technical system. The method includes determining an initial control sequence comprising control information for each control time of a sequence of control times, determining value approximation parameters for approximating the value of a state of the technical system and determining, for each control time, an updated control sequence comprising updated control information for each control time by determining the updated values such that the values of the states of the state sequence which the technical system follows when being controlled according to the updated control sequence are maximized according to the value approximation parameters and controlling the technical system according to the updated control sequence.