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Feedback Control of the Pusher-Slider System: A Story of Hybrid and Underactuated Contact Dynamics
Francois Robert Hogan, Alberto Rodriguez
TL;DR
Robotic manipulation through contact lacks general feedback-control methods that use sensed information to adapt motion, despite hybrid contact modes and friction-limited forces. The paper combines model predictive control with integer programming and introduces Family of Modes for real-time optimization. Simulations and robotic experiments validate stabilization through single-point contact, while extending the approach to multiple-contact manipulation remains future work.
Problem
Robotic manipulation lacks general feedback-control methodology for using sensed information to adapt motion through contact, where hybrid modes and friction-constrained forces complicate control.
Source: §1 · p. 1Method
The controller combines model predictive control with integer programming to represent contact constraints and uses Family of Modes to speed optimization for real-time control.
Source: §1 · p. 2, §5.1 · p. 10, §7 · p. 14Results
Numerical simulations and experiments validate the Family of Modes controller for stabilizing a sliding object through single-point contact.
Source: §1 · p. 2, §7 · p. 15Takeaways & Limitations
The validation is for single-point pushing; applying Family of Modes to multiple-contact manipulation is identified as future work.
Source: §7 · p. 15The quasi-static assumption applies at low velocities, where frictional contact forces dominate and inertia is not decisive.
Source: §4.2 · p. 5
Abstract
from arXiv · showhide
This paper investigates real-time control strategies for dynamical systems that involve frictional contact interactions. Hybridness and underactuation are key characteristics of these systems that complicate the design of feedback controllers. In this research, we examine and test a novel feedback controller design on a planar pushing system, where the purpose is to control the motion of a sliding object on a flat surface using a point robotic pusher. The pusher-slider is a simple dynamical system that retains many of the challenges that are typical of robotic manipulation tasks. Our results show that a model predictive control approach used in tandem with integer programming offers a powerful solution to capture the dynamic constraints associated with the friction cone as well as the hybrid nature of the contact. In order to achieve real-time control, simplifications are proposed to speed up the integer program. The concept of Family of Modes (FOM) is introduced to solve an online convex optimization problem by selecting a set of contact mode schedules that spans a large set of dynamic behaviors that can occur during the prediction horizon. The controller design is applied to stabilize the motion of a sliding object about a nominal trajectory, and to re-plan its trajectory in real-time to follow a moving target. We validate the controller design through numerical simulations and experimental results on an industrial ABB IRB 120 robotic arm.
1 INTRODUCTION
The paper addresses the lack of feedback control methods for manipulation through contact, where hybrid contact modes and friction-limited forces complicate control. It develops and tests a real-time pusher-slider controller using model predictive control and integer programming.
1 INTRODUCTION
- Robotic manipulation remains far from human dexterity, and robots often cannot use sensor information for real-time correction of open-loop motion.The paper identifies a lack of general feedback-controller methodology for manipulation in which objects move relative to the end effector.Source: §1 · pp. 1–2
- Hybrid mode transitions create discontinuous dynamics, while friction-cone-limited contact forces prevent arbitrary slider acceleration and require finite-horizon planning.The pusher-slider system is used to study these challenges in controlling a sliding object with a point pusher.Source: §1 · p. 2
- Numerical simulations and robotic experiments show that the Family of Modes controller can run in real time and stabilize a slider through single-point contact.The controller is designed to handle contact's hybrid nature and friction-constrained dynamics.Source: §1 · p. 2
- Model predictive control combined with integer programming represents friction-cone constraints and hybrid contact; Family of Modes simplifies optimization for real-time control.FOM simulates the system using a selected family of contact-mode schedules.Source: §1 · p. 2
2 RELATED WORK
Prior manipulation research includes grasping, in-hand control, regrasping, and contact-aware trajectory optimization. The paper distinguishes its approach by explicitly modeling hybrid contact in feedback control.
2 RELATED WORK
- In-hand dexterous-control methods typically rely on complex hands and sticking contacts, while regrasp strategies can exploit sliding but require offline planning and accurate contact models.Conventional grasping approaches establish geometric closure and then treat the object as a rigid extension of the arm.Source: §2 · pp. 2–3
- Trajectory optimization has included contact forces and motions as decision variables for planning contact-rich paths in high-degree-of-freedom systems.The paper shares the motivation of including contact forces in an optimization program.Source: §2 · p. 3
- Prior feedback-control work on discontinuous contact used smoothed contact models, whereas this paper considers contact hybridness explicitly.The application of feedback control to discontinuous contact systems is described as relatively unexplored.Source: §2 · p. 3
- Pusher-slider research developed planar-pushing mechanics, stable pushing without sensor feedback, and tactile-feedback control for a point pusher.Source: §2 · p. 3
3 CHALLENGES OF CONTROL THROUGH CONTACT
Control through contact is challenging because contact interactions can switch modes and transmit only physically feasible forces. These properties make smooth-model feedback and instantaneous actuation insufficient for many manipulation tasks.
3 CHALLENGES OF CONTROL THROUGH CONTACT
- Contact control combines two difficulties: hybridness, from changing interaction modes, and underactuation, from limited contact forces and torques.Source: §3 · p. 3
3.1 Hybridness
- Manipulation mode schedules are often unknown in advance, and execution errors can require changes; unlike periodic gait, a fixed sequence may be impractical.Mode transitions can make the dynamics non-smooth, complicating feedback design that relies on smooth models.Source: §3.1 · pp. 3–4, Figure 1 · p. 4
3.2 Underactuation
- Contact forces are physically limited—for example, contact can push but not pull—so control must plan over a finite horizon when goal-directed forces are not currently feasible.Source: §3.2 · p. 5
4 PUSHER-SLIDER SYSTEM
The paper studies a sliding object controlled through one frictional point contact. Its quasi-static formulation simplifies the relation between slider motion and pusher motion for experimental implementation.
4 PUSHER-SLIDER SYSTEM
- The pusher-slider is a nonprehensile manipulation system for controlling a sliding object through a single frictional contact point.It serves as a test case for controller design where actuation arises from friction.Source: §4 · p. 5
4.1 Kinematics
- The system description specifies the slider pose by its center-of-mass coordinates and orientation, alongside the pusher's position relative to point b.Figure 3 depicts the kinematics for a single point of contact.Source: §4.1 · p. 5, Figure 3 · p. 5
4.2 Quasi-Static Approximation
- At low velocities, the quasi-static assumption treats frictional contact forces as dominant over inertia, yielding a direct mapping from slider motion to pusher motion.The authors say this mapping is desirable for implementing the controller with a position-controlled manipulator; an inertial term could also be integrated into the formulation.Source: §4.2 · pp. 5–6
- The motion equations follow a quasi-static formulation with uniform pressure distribution and use concepts including the limit surface and motion cone.Source: §4.2 · p. 6
4.3 Limit Surface
The limit surface relates frictional forces to object motion under a quasi-static assumption, while the motion cone uses frictional contact behavior to distinguish sticking from sliding.
4.3 Limit Surface
- Under the quasi-static assumption, the limit surface maps an applied frictional force on the object to its resulting velocity.It is a convex surface bounding the frictional forces and moments that the interface can sustain.Source: §4.3 · p. 6
- The paper uses an ellipsoidal limit-surface approximation with semi-principal axes fmax, fmax, and mmax.The passage identifies µg, m, and g as the ground friction coefficient, object mass, and gravitational acceleration; it also defines r_b as the position of dm relative to the origin of Fb.Source: §4.3 · p. 6
4.4 Motion Cone
- The motion cone boundaries are constructed from the slider’s contact-point velocity under frictional forces on the friction-cone boundary; pusher velocity within them indicates sticking, otherwise sliding.The limit-surface model used for flat-faced objects approximates the boundaries.Source: §4.4 · p. 6, Figure 4 · p. 6
Sticking
The motion cone constraints distinguish sticking from sliding up or down according to the pusher’s velocity relative to the cone boundaries.
Sticking
- For sticking, the pusher velocity must lie within the motion-cone boundaries, and relative tangential velocity between pusher and object is zero.The constraint is denoted u ∈ MC.Source: Sticking · p. 7
Sliding Up
- For sliding up, the pusher velocity must lie above the motion cone’s upper boundary.This is the stated condition for sliding in the tangential direction relative to the object.Source: Sliding Up · p. 7
Sliding Down
- For sliding down, the pusher velocity must lie below the motion cone’s lower boundary.This is the stated condition for sliding in the tangential direction relative to the object.Source: Sliding Down · p. 7
4.5 Motion Equations
The motion equations represent contact as hybrid dynamics, with the governing mode determined by pusher velocity relative to the motion cone.
4.5 Motion Equations
- The motion equations use f1, f2, and f3 for pusher velocities within, above, and below the motion cone, respectively.These correspond to sticking, sliding up, and sliding down; separation is not considered.Source: §4.5 · pp. 7–8
- The pusher velocity u = [vn vt]T is resolved in the contact frame Fb.The passage also identifies the mode-dependent dynamics as functions of state x and input u.Source: §4.5 · p. 7
- The model excludes separation, treating it as least relevant under the assumption of small forces with low impact.The remaining modes are sticking, sliding up, and sliding down.Source: §4.5 · p. 8
4.6 Linearization
The motion equations and contact constraints are linearized about a feasible nominal trajectory, yielding a formulation intended for real-time optimization.
4.6 Linearization
- The linearization yields linear equations that are computationally tractable for real-time execution of the controller.It is applied to the motion equations and motion-cone constraints about a feasible nominal trajectory.Source: §4.6 · p. 8
- Linearization about nominal state x⋆(t) and input u⋆(t) yields the perturbation dynamics ˙¯x = Aj(t)¯x + Bj(t)¯u.The perturbations are defined as ¯x = x − x⋆ and ¯u = u − u⋆.Source: §4.6 · p. 8
- The sticking and sliding contact constraints are perturbed about the nominal trajectory and rearranged as linear matrix inequalities.The sliding-up and sliding-down inequalities are E2(t)¯x + D2(t)¯u ≤ g2(t) and E3(t)¯x + D3(t)¯u ≤ g3(t), with higher-order perturbations neglected.Source: §4.6 · pp. 8–9
5 MODEL PREDICTIVE CONTROL
The MPC formulation optimizes perturbed states and control inputs over a finite horizon, subject to motion and contact constraints. Integer mode decisions and Family of Modes scheduling address the resulting hybrid search.
5 MODEL PREDICTIVE CONTROL
- MPC minimizes a finite-horizon cost over perturbed states and controls, subject to system dynamics and contact-mode-dependent motion-cone constraints.The cost weights error state, terminal error, and control input.Source: §5 · p. 9
5.1 Mixed-Integer Quadratic Program
- The MIQP uses integer variables for sticking, sliding up, and sliding down, with big-M constraints encoding mode-dependent conditions.A binary mode indicator is assigned to each of the three contact interactions.Source: §5.1 · p. 10
5.2 Family of Modes (FOM) Scheduling
- FOM replaces exhaustive mode combinations with preset schedules, each solved as a convex program, while feedback reselects the schedule at every step.For the pusher-slider, the example family includes slide-up then stick, slide-down then stick, or sticking throughout; the method does not scale well to systems with more degrees of freedom or contact points.Source: §5.2 · pp. 11–12
6 RESULTS
The controller is evaluated in straight-line tracking with an external perturbation and moving-target tracking, using simulations and experiments. Both tests assess its ability to guide the slider through contact.
6 RESULTS
- The evaluation covers two tracking scenarios in simulation and experiments with an ABB IRB 120, while Vicon tracks the slider pose.A metallic rod attached to the arm pushes an aluminum slider on a flat plywood surface.Source: §6 · p. 12, Figure 7 · p. 12
6.1 Straight Line Tracking
- Simulated and experimental responses drive perturbations from the nominal straight-line trajectory to zero.The experiment applies an impulsive force with a handheld poker; the pusher also corrects for the slider’s imperfect initial placement.Source: §6.1 · p. 13, Figure 8 · p. 13
6.2 Trajectory Tracking
- Both simulated and experimental trajectories reach all three successive targets within the specified tolerance.When the slider comes within 0.01 m of a target, tracking advances to the next; the controller favors sliding to rotate the object, then sticking to push it toward the target.Source: §6.2 · p. 14
7 CONCLUSION
The paper presents a contact-based feedback controller that combines model predictive control with integer programming, including Mixed-Integer Quadratic Programming and Family of Modes methods. Simulations and experiments validate Family of Modes on a single-point pusher-slider system, while extension to multiple-contact tasks remains future work.
7 CONCLUSION
- Model predictive control with integer programming incorporates contact dynamics and hybrid interactions; Family of Modes is proposed to speed up the integer program for real-time control.The paper describes both Mixed-Integer Quadratic Programming and Family of Modes as ways to solve the optimization program.Source: §7 · p. 14
- Simulations and experiments validate stabilization of a sliding object through single-point contact using Family of Modes.The figure caption reports that both simulation and experiment reach all three targets within the specified tolerance.Source: Figure 11 · p. 15, §7 · p. 15
- Extending Family of Modes to multiple-contact manipulation, including in-hand manipulation with extrinsic dexterity, is identified as future work.The paper notes that key mode sequences may be identified from physical insight or offline planning, with symmetry useful for some tasks.Source: §7 · p. 15