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Differential Flatness of Quasi-Static Slider-Pusher Models with Applications in Control

Sander De Witte, Tom Lefebvre, Thomas Neve, Andras Retzler, Guillaume Crevecoeur

TL;DR

This work develops a differential, quasi-static, frictionless model for planar slider-pusher systems, valid for generic slider shapes and circular pushers. By analyzing differential flatness, the authors show that polygon sliders exhibit flatness with the center of mass as a flat output, enabling two tracking strategies: a cascaded quasi-static feedback controller and a dynamic feedback linearization approach. They validate the modeling and control design through both simulations with perturbations and real experiments using a finger-like pusher and vision-based state estimation, observing good cross-domain transfer of gains. The results establish differential flatness as a practical tool for planning and tracking in pushing-based manipulation, with clear pathways to more automatic tuning and broader geometry applicability.

Abstract

This paper investigates the dynamic properties of planar slider-pusher systems as a motion primitive in manipulation tasks. To that end, we construct a differential kinematic model deriving from the limit surface approach under the quasi-static assumption and with negligible contact friction. The quasi-static model applies to generic slider shapes and circular pusher geometries, enabling a differential kinematic representation of the system. From this model, we analyze differential flatness - a property advantageous for control synthesis and planning - and find that slider-pusher systems with polygon sliders and circular pushers exhibit flatness with the centre of mass as a flat output. Leveraging this property, we propose two control strategies for trajectory tracking: a cascaded quasi-static feedback strategy and a dynamic feedback linearization approach. We validate these strategies through closed-loop simulations incorporating perturbed models and input noise, as well as experimental results using a physical setup with a finger-like pusher and vision-based state detection. The real-world experiments confirm the applicability of the simulation gains, highlighting the potential of the proposed methods for

Differential Flatness of Quasi-Static Slider-Pusher Models with Applications in Control

TL;DR

This work develops a differential, quasi-static, frictionless model for planar slider-pusher systems, valid for generic slider shapes and circular pushers. By analyzing differential flatness, the authors show that polygon sliders exhibit flatness with the center of mass as a flat output, enabling two tracking strategies: a cascaded quasi-static feedback controller and a dynamic feedback linearization approach. They validate the modeling and control design through both simulations with perturbations and real experiments using a finger-like pusher and vision-based state estimation, observing good cross-domain transfer of gains. The results establish differential flatness as a practical tool for planning and tracking in pushing-based manipulation, with clear pathways to more automatic tuning and broader geometry applicability.

Abstract

This paper investigates the dynamic properties of planar slider-pusher systems as a motion primitive in manipulation tasks. To that end, we construct a differential kinematic model deriving from the limit surface approach under the quasi-static assumption and with negligible contact friction. The quasi-static model applies to generic slider shapes and circular pusher geometries, enabling a differential kinematic representation of the system. From this model, we analyze differential flatness - a property advantageous for control synthesis and planning - and find that slider-pusher systems with polygon sliders and circular pushers exhibit flatness with the centre of mass as a flat output. Leveraging this property, we propose two control strategies for trajectory tracking: a cascaded quasi-static feedback strategy and a dynamic feedback linearization approach. We validate these strategies through closed-loop simulations incorporating perturbed models and input noise, as well as experimental results using a physical setup with a finger-like pusher and vision-based state detection. The real-world experiments confirm the applicability of the simulation gains, highlighting the potential of the proposed methods for

Paper Structure

This paper contains 30 sections, 1 theorem, 75 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Modeling
  3. Kinematics
  4. Quasi-static model
  5. Force-motion model
  6. Motion-motion model
  7. Smooth dynamics
  8. Rectangular slider
  9. Circular slider
  10. Transition and point contact dynamics
  11. Full dynamics
  12. Differential flatness
  13. Centre of mass
  14. Other points on the sliders
  15. Flat slider-pusher systems
  16. ...and 15 more sections

Key Result

Theorem 1

System (eq:ss) is dynamic feedback linearizable if and only if it is differentially flat.

Figures (12)

  • Figure 1: Snapshots of one of the cameras, showing dynamic feedback linearization on the set-up. The reference trajectory is shown in blue, while the executed path is yellow.
  • Figure 2: Geometry of the slider-pusher system.
  • Figure 3: Geometry of the transition and point contact dynamics.
  • Figure 4: Example of finite state machine describing the dynamics of a $2$-faced slider. The conditions for which a transition between either smooth or point dynamics is triggered by the events denoted above the edges. The nodes refer to the equations of motion corresponding the motion regime that is triggered by the events.
  • Figure 5: Illustration of flat slider geometries.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Definition 1
  • Theorem 1
  • Definition 2: Dynamic feedback linearization lee2022linearization