PaperlayerView exact revision on arXiv

Source-linked AI summary

Control Barrier Functions for Mechanical Systems: Theory and Application to Robotic Grasping

Wenceslao Shaw Cortez, Denny Oetomo, Chris Manzie, Peter Choong

arXiv:1903.09816v1cs.RO

TL;DR

The paper addresses the limited treatment of relative-degree-two, sampled-data mechanical systems with model uncertainty in existing control barrier function methods. It develops robust barrier functions and a grasp controller that works with nominal manipulation controllers, with simulation and hardware demonstrations enforcing key grasp constraints.

  • Problem

    Existing control barrier function results are mostly limited to continuous-time relative-degree-one systems, whereas mechanical robots require sampled-data, force/torque-level control for relative-degree-two position and velocity constraints.

  • Method

    The paper develops robust zeroing control barrier functions for sampled-data mechanical systems and a grasp controller that minimally modifies nominal control to enforce grasp-admissible states.

  • Results

    Simulation and hardware demonstrations validate the proposed control for robotic grasping while enforcing no slip, no over-extension, and no excessive rolling.

  • Takeaways & Limitations

    The method provides a systematic way to bound velocity near constraint boundaries and apply existing manipulation controllers while prioritizing grasp constraint satisfaction.

Abstract

from arXiv · show

Control barrier functions have been demonstrated to be a useful method of ensuring constraint satisfaction for a wide class of controllers, however existing results are mostly restricted to continuous time systems of relative degree one. Mechanical systems, including robots, are typically second-order systems in which the control occurs at the force/torque level. These systems have velocity and position constraints (i.e. relative degree two) that are vital for safety and/or task execution. Additionally, mechanical systems are typically controlled digitally as sampled-data systems. The contribution of this work is two-fold. First, is the development of novel, robust control barrier functions that ensure constraint satisfaction for relative degree two, sampled-data systems in the presence of model uncertainty. Second, is the application of the proposed method to the challenging problem of robotic grasping in which a robotic hand must ensure an object remains inside the grasp while manipulating it to a desired reference trajectory. A grasp constraint satisfying controller is proposed that can admit existing nominal manipulation controllers from the literature, while simultaneously ensuring no slip, no over-extension (e.g. singular configurations), and no rolling off of the fingertips. Simulation and experimental results validate the proposed control for the robotic hand application.

I. INTRODUCTION

The paper develops robust zeroing control barrier functions for relative-degree-two mechanical systems under sampling, perturbations, and model uncertainty, then applies them to robotic grasping. The approach bounds velocity near constraint boundaries and preserves grasp constraints while minimally modifying nominal manipulation control.

  • Motivation: Mechanical systems are second-order, force/torque-controlled systems whose position and velocity constraints are important for safe operation.The paper motivates barrier-function methods for robots and other physical systems with workspace constraints.
  • Limitations of Existing Methods: Existing zeroing control barrier functions mainly address continuous-time, relative-degree-one systems and impose restrictive conditions for locally Lipschitz quadratic-program controllers.These limitations motivate a formulation tailored to mechanical systems and digital implementation.
  • Limitations of Existing Methods: Near a constraint boundary, ignoring the position–velocity relationship can permit large velocities that require excessive control effort because of system inertia.The proposed formulation explicitly addresses this mechanical relationship.
  • Proposed Method: The proposed zeroing control barrier function formally handles relative-degree-two systems while remaining robust to sampling effects, perturbations, and model uncertainty.It also allows designers to tune velocity bounds near constraint boundaries and define a controller that stays minimally close to nominal control.
  • Robotic Grasping: The grasping application enforces no slip, no joint over-extension, and no excessive fingertip rolling while manipulating an object along a desired trajectory.These conditions define the grasp constraints required for successful in-hand manipulation.
  • Robotic Grasping: The grasp controller can operate alongside existing manipulation controllers, retaining nominal control when constraints are safe and intervening when violation is imminent.Simulation and hardware demonstrations support the proposed approach, while prior methods often rely on quasi-static assumptions, exact models, or substantial computation.
  • Proposed Method: The barrier-function construction can be extended to higher relative degrees by repeated application of its recursive design.The paper gives relative degree three as an example.

B. Control Barrier Functions for Sampled-Data Systems

The paper extends zeroing control barrier functions to relative degree two mechanical systems implemented as sampled-data systems, accounting for disturbances and inter-sampling effects. A nominal controller can be minimally modified to prioritize constraint satisfaction, although quadratic-program regularity conditions may be restrictive.

  • Implementation conditions: Linear-independence and strict-complementary-slackness properties used for quadratic-program regularity can be difficult to ensure when active constraints outnumber control variables.The paper identifies robotic grasping as a setting where slip and workspace constraints may exceed the available control inputs.
  • Sampled-data formulation: The sampled-data formulation uses a zero-order-held, piecewise-constant control updated from measurements every sampling period.The system samples x_k at t=kT and holds u_k constant over each interval.
  • Robust sampled-data barriers: The method introduces a margin ν(T) in the barrier condition to counteract inter-sampling effects.ν(T) is an extended class-K function used to preserve the required barrier inequality between samples.
  • Robust sampled-data barriers: Theorem 2 guarantees forward invariance of the constraint set under bounded disturbances and suitable robustness, sampling, and barrier margins.The result applies over t ∈[0,NT) for admissible piecewise-constant controls and states initialized in the admissible set.
  • Constraint-enforcing control: The proposed controller combines a sampled nominal control with barrier constraints, prioritizing constraint satisfaction over nominal stability or performance guarantees.The quadratic-program formulation is designed to remain minimally close to the nominal control while enforcing constraints.
  • Constraint-enforcing control: Theorem 3 guarantees constraint satisfaction for the proposed nominal-control modification under bounded disturbances over the sampled-data horizon.For a given sampling time, suitable nonnegative margins and tuning parameters ensure the state remains in the admissible set.
  • Comparison with alternatives: The sampled-data approach avoids the additional smoothness conditions required by emulation-based alternatives and therefore applies to more general systems.The paper contrasts this with emulation techniques that require sufficient smoothness of all controller components.

III. GRASP CONSTRAINT SATISFACTION

The proposed barrier method is applied to robotic grasping, where a hand must manipulate an object along a desired reference pose trajectory while maintaining grasp constraints. The application develops the analysis needed to ensure those constraints during manipulation.

  • Application scope: The grasping application requires a robotic hand to manipulate an object to a desired reference pose trajectory while satisfying grasp constraints.The paper treats this application as an additional contribution beyond the mechanical-system barrier method.

A. Hand-Object System

The hand-object model represents a multi-fingered hand, contact forces, object dynamics, and grasp geometry. Its constraints address slip, force-closure assumptions, and feasible hand configurations.

  • System model: The modeled system is a fully actuated multi-fingered hand grasping a rigid convex object at multiple contact points.Each finger has revolute joints and a smooth, high-stiffness convex fingertip.
  • System model: The hand dynamics use joint torques as control inputs and include inertia, Coriolis or centrifugal effects, contact forces, and disturbance torques.The hand Jacobian and concatenated vectors map individual finger quantities into the full system representation.
  • System model: The object dynamics include inertia, Coriolis and centrifugal terms, external wrench disturbances, and a grasp map that converts contact forces into net object wrench.The grasp map G maps the concatenated contact force to the wrench acting on the object.
  • Grasp constraints: Slip prevention requires each contact force to remain inside its friction cone, with the full admissible cone formed as their Cartesian product.The contact force is decomposed into two tangential components and one normal component, with friction coefficient μ.
  • Grasp assumptions: The grasp assumes at least 3n joints, a full-rank force-closure grasp, and smooth system dynamics and local contact surfaces.Force closure means every object wrench has a contact-force realization inside the friction cone.

B. Hand-Contact Kinematics

The hand-contact kinematics describe rolling contact through local surface coordinates and differential-geometric quantities. Metric, curvature, and torsion tensors parameterize the fingertip and object surfaces and support the contact-motion equations.

  • Contact parameterization: Rolling-contact kinematics are formulated using local coordinates on both the object surface and fingertip surface.The contact surfaces are parameterized separately for each contact.
  • Surface geometry: Metric, curvature, and torsion tensors provide the geometric parameters used to define rolling-contact kinematics.The same tensor construction is applied to fingertip and object surfaces by substituting their respective local coordinates.
  • Contact frames: The contact frame is defined using the Gauss frame, with the fingertip-to-contact rotation mapping the contact geometry into the fingertip frame.The geometry is represented through the rotation R_fci and the local surface quantities.
  • Contact dynamics: The local surface coordinates and contact angle dynamics are governed by the equations of motion for fingertip and object contact coordinates.The parameterization assumptions ensure the contact-motion equations are well-defined.

C. Problem Formulation

This section introduces the formal definition of the grasp constraint satisfaction problem.

  • The grasp constraint satisfaction problem is formally defined in this section.
  • The formal problem definition establishes the basis for specifying constraint-admissible grasp states.
  • The section frames grasp constraint satisfaction as a formal problem for subsequent control design.

Contact Force Set:

The grasp constraint-admissible state set combines no-slip contact forces, feasible joint angles, and fingertip contact locations. These constraints define the states in which the hand remains nonsingular and contacts remain inside the grasp.

  • Contact-force constraints: No slip is enforced by requiring contact forces to remain inside a pyramid approximation of the friction cone.The pyramid is defined by linear constraints on the contact forces.
  • Joint constraints: Joint over-extension is prevented by box constraints that keep every joint angle within feasible limits excluding singular hand configurations.The admissible joint-angle set consists of states satisfying the upper and lower angle constraints.
  • Contact-location constraints: Excessive rolling is addressed by constraining each contact location to the modeled workspace of its fingertip.The fingertip workspace is represented with geometric parameterizations and box constraints.
  • Constraint-admissible states: The full admissible state set combines feasible contact forces, joint configurations, and contact locations as H = ¯Cf × ¯Cq × ¯Cr.Within H, the hand configuration is nonsingular, contact points do not slip, and the grasp relation holds.

D. Proposed Grasp Constraint Satisfying Control

The proposed controller combines robust no-slip constraints with zeroing control barrier functions for joint and sampled-data constraints. Its margins account for model uncertainty, friction uncertainty, bounded errors, and sampling effects while preserving admissible manipulation behavior.

  • Controller construction: The controller combines no-slip, no-over-extension, and no-rolling constraints to ensure forward invariance of the grasp-admissible set.The constraints are combined into a controller addressing the formal grasp constraint satisfaction problem.
  • Robustness: Robustness is designed for limited sensing, model uncertainty, unknown disturbances, conservative friction estimates, and bounded approximation errors.The approach explicitly analyzes uncertain contact forces and uses assumptions on friction and error bounds.
  • No-slip constraint: A conservative friction coefficient and tuning margin produce a robust no-slip condition whose satisfaction makes the admissible force set forward invariant.Under Assumptions 1–6, an ε greater than ε* guarantees forward invariance of ¯Cf.
  • Joint constraints: Zeroing control barrier functions with robustness and sampling margins constrain joint dynamics while accounting for the control dependence of contact forces.The barrier functions are applied where the torque input enters the second-order joint dynamics.

Contact Location Constraint:

The contact-location constraint keeps every fingertip contact inside a conservative workspace. Robust zeroing control barrier functions and sampled-data torque constraints enforce forward invariance despite the derived contact-location dynamics.

  • Conservative contact workspace: Robustness margins shrink the fingertip workspace to define a conservative admissible contact-location set.Each boundary function is reduced by a nonnegative margin before the barrier constraints are formed.
  • Barrier functions: Robust zeroing control barrier functions constrain each fingertip boundary to prevent excessive rolling.The functions combine the derivative of each robust boundary function with an extended class-K term and a robustness margin.
  • Contact-point dynamics: The contact-location barrier functions are applied to the second-order dynamics of each contact point, where the torque input appears through differentiated kinematics and contact-force substitutions.The derivation proceeds through substitutions involving fingertip velocity, hand dynamics, and contact force.
  • Torque constraint: A linear torque constraint with a sampling-time margin is imposed to ensure forward invariance of the conservative contact-location set.The admissible torque set uses ˆAruk ≥ ˆbr + ˆνr1.

Actuator Constraints:

The paper defines bounded actuator torques and a controller that stays close to a nominal manipulation input while enforcing grasp constraints. Theorem 4 guarantees forward invariance of the constraint set under the stated assumptions, and simulations demonstrate constraint satisfaction despite model uncertainty.

  • Actuator constraints: Actuator torques are constrained between umin and umax to reflect finite real-world actuation limits.The bounded torque set contains controls satisfying umin ≤ uk ≤ umax.
  • Constraint-admissible control: The proposed controller minimizes deviation from the nominal torque while enforcing constraints preventing slip, joint over-extension, excessive rolling, and actuator-limit violations.The controller admits an existing nominal manipulation controller and prioritizes grasp-constraint satisfaction.
  • Theoretical guarantee: Under Assumptions 1–7, the control law guarantees that the hand-object state remains in the constraint set over the sampled-data horizon.The guarantee relies on feasible admissible torques, local Lipschitz conditions, robustness margins, and the stated class-K functions.
  • Robotic grasping: The grasping application addresses slip, joint-limit violation, and fingertip workspace violation using tactile-based blind grasping under uncertain model parameters.The controller uses contact locations and joint positions and velocities without requiring complete hand-object knowledge.
  • Simulation results: 0.928 s: the nominal controller without constraint satisfaction fails when contact location bcf2 exceeds the fingertip surface during Z-axis object manipulation.The failed grasp also exhibits slip, joint over-extension, and excessive rolling before the simulation stops.
  • Simulation results: The constraint-satisfying controller produces a successful grasp while tracking the reference and avoiding the grasp failures shown for the nominal controller.The simulation uses a nine-degree-of-freedom hand and a cube object, with the initial configuration shown in Figure 3.

B. Hardware Results

Hardware demonstrations show that the proposed controller preserves nominal manipulation when the reference is feasible and prioritizes grasp constraints when the reference is compromising. The controller prevents constraint-set violations under model uncertainty and sampling effects, though constraint enforcement can produce a steady-state reference offset.

  • Experimental setup: The Allegro Hand hardware implementation evaluates the proposed controller through three demonstrations using feasible and compromising references.The setup uses a 16-degree-of-freedom fully actuated hand with joint encoders and emulated tactile measurements.
  • Demonstration 2: For the compromising reference rψ = 0.7 ± 0.06 rad, nominal control alone becomes unstable, exceeds actuator capabilities, and causes grasp failure.The resulting hand configuration loses contact with the object.
  • Demonstration 3: With the same compromising reference, the proposed controller prevents ξcf from exceeding the constraint set despite model uncertainty and sampling-time effects.The controller intervenes through torque deviations from the nominal law and prioritizes constraint satisfaction over exact reference tracking.
  • Overall result: The paper concludes that the proposed controller supports no slip, no over-extension, and no excessive rolling while admitting an existing nominal controller.Simulation and hardware implementation are used to demonstrate the approach for robotic grasping.

VI. APPENDIX

The appendix lists constraint-related quantities for joint limits and contact locations, followed by a derivative expression used in the formulation. The supplied appendix passages do not provide enough context to interpret the displayed symbols further.

  • Joint constraints: The appendix identifies Aq and bq as the joint constraint-related terms.The supplied passage does not specify their dimensions or complete definitions.
  • Joint constraints: The appendix displays entries labeled bqmin1 through bqmaxm for the joint constraint terms.The displayed sequence indicates minimum and maximum bounds, but the passage does not define the indexing fully.
  • Contact constraints: The appendix identifies Ar and br as the contact location constraint-related terms.The supplied passage does not provide their complete construction.
Loading 1903.09816v1…