Abstract
In this paper, an orthogonal decomposition-based state observer for systems with explicit constraints is proposed. State observers have been an integral part of robotic systems, reflecting the practicality and effectiveness of the dynamic state feedback control, but the same methods are lacking for the systems with explicit mechanical constraints, where observer designs have been proposed only for special cases of such systems, with relatively restrictive assumptions. This work aims to provide an observer design framework for a general case linear time-invariant system with explicit constraints, by finding lower-dimensional subspaces in the state space, where the system is observable while giving sufficient information for both feedback and feed-forward control. We show that the proposed formulation recovers minimal coordinate representation when it is sufficient for the control law generation and retains non-minimal coordinates when those are required for the feed-forward control law. The proposed observer is tested on a flywheel inverted pendulum and on a quadruped robot Unitree A1.
Highlights
State estimation is needed when values of some of the state variables required by the control law are not directly available from the measurements
While many mechanical systems and robots can be effectively described by systems of ordinary differential equations, making it possible to use the first term in Taylor expansion of their dynamics to apply aforementioned linear state estimation methods, some are better described as differential-algebraic equations (DAE); these include systems with explicit mechanical constraints [1]
There are a number of reasons why this description can be favored: it allows for maintaining the structure of the equations while contacts are being acquired and released, it explicitly includes contact forces, allowing to place them under additional constraints, such as a friction cone, and, in some cases, it is difficult to find a singularity-free minimal coordinate representation for multi-body dynamics
Summary
State estimation is needed when values of some of the state variables required by the control law are not directly available from the measurements. Control law requires full state information, while only some of the states are directly measured This problem has been addressed in great detail, leading to solutions in the form of, among others, optimal Dynamic Output Feedback, Luenberger State Observers, and Kalman Filters. While many mechanical systems and robots can be effectively described by systems of ordinary differential equations, making it possible to use the first term in Taylor expansion of their dynamics to apply aforementioned linear state estimation methods, some are better described as differential-algebraic equations (DAE); these include systems with explicit mechanical constraints [1]. Dynamics represented as a DAE poses a problem when classical control and state estimation methods are to be applied
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