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Force/Torque-Based Kinematic Adaptation for Robotic Manipulation Tasks

Carl Glen Henshaw

arXiv:2608.21592v1cs.RO

TL;DR

Contact-rich manipulation requires kinematic models that can change with tools and contact modes, but these relationships are often unavailable without exteroceptive sensing. The paper develops a force/torque-only adaptive scheme with stability guarantees for rigid and compliant cases, then validates it in simulated peg-in-hole insertion. Its results also delimit identification by motion excitation and expose practical stability and observability boundaries.

  • Problem

    Contact-rich manipulation needs accurate, changing joint-to-task kinematics, yet estimating these relationships without exteroceptive task-space sensing remains limited.

  • Method

    The paper estimates kinematics online from joint encoders and wrist force/torque feedback using a composite tracking-plus-prediction adaptation law, including a compliant inner loop and QP formulation.

  • Results

    The rigid and compliant cases operate successfully in three experiments, including simulated peg seating, with identification confined to excited directions and passive excitation supplied by compliance.

  • Takeaways & Limitations

    The scheme supports online adaptation to robot, hand, or tool kinematics while preserving the distinction between commanded excitation and excitation generated by compliance.

  • Takeaways & Limitations

    Identification cannot recover offset components that produce no contact-lever signal, including axial tool length during a pure rigid insertion push.

Abstract

from arXiv · show

Contact-rich robotic manipulation requires an accurate model of the kinematic relationship between a robot's joints and the task features it senses. This relationship is rarely known exactly: it changes with each tool the robot picks up and shifts, sometimes almost instantaneously, as contact modes change --- especially for multi-fingered hands that make and break contact at points that are not exactly prescribed, as in full-hand grasping. This paper develops an adaptive scheme that estimates that relationship online, using only joint-angle sensing and a wrist-mounted force/torque sensor, with no exteroceptive measurement of the tool tip. We derive a provably stable kinematic update law that identifies the kinematics of an unknown tool from force/torque feedback alone, and prove stability of both the rigid case and the case with a compliance controller as an inner loop. We show that identification is confined to the directions the motion excites --- so that, for example, a tool's length is unobservable under a rigid insertion push, while a compliant loop's passive yielding partially excites it; and that with a second-order admittance the compliant certificate holds unconditionally in continuous time. We also pose the combined control and estimation problem as a Quadratic Program (QP): the formulation yields the prediction term of the update law exactly but, instructively, cannot reproduce the tracking adaptation term. We validate the scheme in simulation on a peg-in-hole insertion. This work is the first step in a research program aimed at factoring manipulation learning into a task policy which can be learned in isolation of the robot, for instance by reinforcement learning, and an adaptive kinematic component that adapts online to the particular robot, hand, or tool in use.

1 Background

Online kinematic estimation becomes important as robots manipulate human-designed tools and encounter varied contacts, unlike traditional adaptive control that usually assumes known kinematics. The paper focuses on force/torque-based estimation and the Jacobian relationships linking joint motion, end-effector motion, and generalized forces.

  • Motivation: Traditional adaptive manipulator controllers typically estimate unknown dynamics while assuming the robot’s kinematics are known.Unknown terms include masses, inertias, gravity, Coriolis effects, friction, and sometimes motor and gear dynamics.
  • Motivation: Complex tools, mechanisms, and varied contacts make the kinematic relationship between joint motion and environmental changes increasingly important.Industrial robots historically had fixed, easily quantified kinematics, whereas modern manipulation often uses human-designed tools and affordances such as hinges and screws.
  • Force/Torque-Based Estimation: The proposed estimation uses wrist force–torque sensing and joint encoders instead of task-space position sensors, with identification limited to motion-excited directions.The contribution targets settings where exteroceptive, typically visual, task-space measurements are unavailable.
  • Jacobian Relationships: The Jacobian maps joint velocities to end-effector velocities and, through its transpose, maps end-effector generalized forces to joint torques.This dual relationship follows from equality of virtual work and supports Jacobian-transpose control.
  • Jacobian Relationships: Force-based task-space feedback can be converted into joint torques through Jacobian transpose control.The controller produces a virtual task-space force, which the Jacobian transpose converts into virtual joint torques.

2 Adaptive Jacobian Control

The section develops adaptive Jacobian-transpose control for unknown robot or tool kinematics, using Lyapunov analysis to expose a regulation failure and motivate a composite tracking–prediction update. The resulting law preserves boundedness and task regulation under a full-row-rank estimate, while parameter convergence remains excitation-dependent.

  • Adaptive Jacobian Control: The controller estimates an unknown task-to-joint Jacobian online when robot, tool, or grasp kinematics are uncertain.The estimate is parameterized through known configuration-dependent terms and unknown kinematic parameters.
  • Failure of Tracking-Only Adaptation: The tracking-only update can drive the commanded motion and adaptation to zero before regulation, steering the estimate toward a singular set.Its command decays exponentially at rate γ∥ψ∥2∥Ke∥2, independently of whether the task error has converged.
  • Composite Adaptation Law: The composite update adds prediction-error adaptation, which cancels Lyapunov cross-terms and opposes rank loss along excited motion directions.Its prediction contribution is sign-definite, −κ(t)∥ε∥2, for positive κ(t).
  • Stability and Identifiability: The composite law guarantees bounded error and parameter-estimation dynamics, while task-error convergence still requires the estimated Jacobian to retain full row rank.Persistent excitation is additionally required for full parameter convergence.
  • Gain Design: The prediction gain may be positive, time-varying, or state-dependent, and normalized LMS scaling makes adaptation less sensitive to feature and command magnitudes.A positive εnorm keeps the normalized law well posed as ∥˙q∥ approaches zero.
  • Parameterization: The analytic regressor parameterization can yield globally correct kinematics with a small learning problem, but requires the designer to derive potentially complex kinematic equations.The design choice therefore trades learning simplicity against derivation effort.

3 Force/Torque–Only Tool–Offset Identification: The Rigid Case

The rigid case identifies an unknown tool offset from force/torque feedback and joint encoders without measuring tool-tip position. Stability and convergence hold along excited directions, while bias and unexcited motion constrain identification.

  • Problem and factorization: The sensor Jacobian factors into known arm and sensor maps surrounding an unknown wrist-to-tool transform parameterized by the tool offset.The estimated offset enters through the adjoint, making the Jacobian error linear in the offset error.
  • Force/torque-only prediction error: A value-gradient controller and composite tracking-plus-prediction update law use force/torque residuals instead of differentiated task-space position.Force and moment channels are measured by the wrist sensor, while insertion depth is proprioceptive; no task-space position is measured.
  • Stability and identification: Theorem 1 guarantees bounded sensor state and offset error, with command magnitude O(√κ¯b) and residual offset size O(¯b/σmin) on persistently excited directions.Unexcited directions remain unchanged, and full identification requires excitation.
  • Observability limits: Static wrench balance cannot identify the offset reliably because contact couples contaminate the wrist moment; motion–wrench covariation supplies the usable signal.The prediction residual detects the offset through its motion-dependent signature rather than instantaneous moment balance.
  • Observability limits: A pure insertion push cannot identify the axial tool length because that offset component produces no changing contact lever under rotation-free motion.The limitation is structural and independent of excitation.

4 The Compliant Case

The compliant case places a second-order admittance between the command and motion, adding mechanical storage and dissipation to the force/torque-only adaptation analysis. In continuous time, the resulting certificate is unconditional, while sampled implementations face a virtual-mass stability bound.

  • Compliant control law: The compliant controller is a virtual mass–damper driven by the value force and sensed contact wrench in joint space.The virtual inertia is positive definite, damping is positive semidefinite, and the unknown tool length is absent from the measured wrist wrench and known arm Jacobian.
  • Energy structure: The virtual mass–damper and elastic contact form coupled non-negative storages linked through power continuity.The kinetic storage cancels the drive cross-term, while the contact storage makes the interconnection power-continuous.
  • Stability and identification: Theorem 2 certifies bounded state, offset error, and joint velocity unconditionally in continuous time under the second-order admittance.The joint velocity enters a ball of radius O(√κ¯b), and excited-direction offset error converges to a residual set of size O(¯b/σmin) under persistent excitation.
  • Practical boundary: A sampled admittance requires virtual inertia large enough relative to contact stiffness and control period to remain passive.If virtual mass is too small, the discretized mass–contact interconnection can inject energy at each step.
  • Stability and identification: The compliant proof retains the composite adaptation law and prediction dissipation while adding kinetic storage and damping dissipation.The prediction contribution remains −κ∥˜J ˙q∥2 with the same O(¯b/σmin) bias floor as in the rigid case.

5 Results

The experiments validate force/torque-only insertion, directional identification, and robust composite adaptation across a wide gain range. Compliance supplies otherwise absent excitation while preserving stable task behavior.

  • 5.1 Force/torque-only insertion: The peg reaches the 45 mm target while force remains below 50 N, wrist moment below 2 N·m, and λmin( ˆJ ˆJ⊤) stays at least 10−6.These criteria depend only on the sensor vector and provide the online rank gate for the regulation certificate.
  • 5.1 Force/torque-only insertion: The representative rollout seats the peg using sensed depth, lateral force, and wrist moment, with all success channels closing at approximately 5.1 s.Sensor depth closely overlays forward-kinematic depth, indicating a small bias floor.
  • 5.2 Directional identification: A rigid insertion push leaves the axial tool-length direction structurally unexcited, whereas compliant reaction motion raises axial excitation during contact.The lateral direction identifies quickly; axial identification remains roughly 50 times poorer and the out-of-plane direction is slower.
  • 5.3 Robustness across adaptation gain: The composite adaptation trace overlays prediction-only behavior at Γ = 100 and Γ = 8000 while descending to its energy floor.The tracking term neither destabilizes the loop nor changes the gain-flat seat transition across the tested range.
  • 5 Results: Together, the experiments confirm force/torque-only seating, excitation-limited identification with compliance-generated excitation, and successful adaptation across two decades of gain.The estimated Jacobian retains the rank required by the regulation conclusion.

6 Future Work

Future work extends touch-only kinematic adaptation toward fully tactile manipulation and general task-space policies. The proposed direction combines proprioceptive, force/torque, and fingertip tactile sensing with policies that accommodate changing tools and grasps.

  • Fully tactile manipulation: The authors hope to extend touch-only kinematic updates to fully tactile manipulation using proprioceptive, force/torque, and fingertip tactile sensors.The stated aim is to infer more about complex manipulation tasks without relying solely on exteroceptive sensing.
  • General task-space policies: They also hope to generalize F to a task-space policy that can adapt across changes in grip, tool mass, and tool length.The motivation is rapid adaptation when using different tools and changing grasps.

7 Appendix: Two Families of RMRC Laws Derived as Quadratic Programs

The appendix derives two resolved-motion rate-control formulations as QPs: exact task-space tracking with minimum joint-velocity effort, and regularized tracking with hard constraints. Their solutions expose the trade-off between exact tracking and singularity robustness.

  • Family 1: exact tracking: The first QP minimizes joint-velocity effort subject to exact task-space velocity tracking, yielding the Moore–Penrose pseudoinverse solution.A null-space term can project an auxiliary velocity without disturbing end-effector tracking.
  • Family 2: constrained regularization: The second QP minimizes tracking error with regularization while enforcing hard constraints such as joint limits and obstacle avoidance.Its unconstrained solution is the damped least-squares, or Levenberg–Marquardt, solution.
  • Family 2: constrained regularization: Damped least squares avoids singularity-induced velocity blow-up but permits nonzero task-space error.The hard constraints are enforced directly by the QP solver.

7.3 Augmented QP Including Jacobian Estimation

The augmented QP can reproduce the prediction component of Jacobian adaptation but cannot recover the stabilizing tracking term. This mismatch shows that instantaneous optimality does not guarantee the longer-horizon stability supplied by the composite law.

  • Augmented QP: The augmented per-sample QP either assigns no tracking adaptation or selects its destabilizing sign, so it cannot derive the tracking term correctly.The failure reflects the difference between horizon-local optimality and stability over longer horizons.
  • Augmented QP: The prediction term is obtained exactly from the QP, with parameter updates penalized using Γ−1, the same metric used in the Lyapunov parameter-error term.This shared metric makes the QP and Lyapunov comparisons meaningful without making their answers agree.

7.4 KKT Conditions

The KKT conditions reproduce the Jacobian-transpose control law but force the instantaneous parameter update to zero. Because the tracking constraint depends on the current estimate rather than its rate, adaptation increases the objective without helping the current sample.

  • The stationarity condition for joint velocity gives ˙q = ˆJ⊤λ, reproducing the Jacobian-transpose control law.
  • The parameter-rate stationarity condition gives ˙ˆθ = 0, so the QP does not adapt its parameters.
  • The tracking constraint contains ˆθ but not ˙ˆθ, leaving parameter motion penalized without any single-sample benefit.
  • For regulation, choosing λ = Ke identifies the control law with Jacobian-transpose control.

7.6 The Myopic Adaptation Law, and Why It Destabilizes

Making the tracking constraint depend on updated parameters produces a bilinear program whose stationary update has the opposite sign from the stabilizing adaptation law. That sign reversal destroys the Lyapunov cancellation and can actively destabilize the system.

  • Replacing the tracking constraint with updated parameters makes the program bilinear in ˙q and ∆ˆθ, so it is no longer a QP.
  • The resulting stationary update has the opposite sign from the correctly signed Cheah–Liu–Slotine law.
  • The correct sign cancels the tracking cross-term and yields ˙V = −∥ˆJ⊤Ke∥2 ≤ 0.
  • With the reversed sign, ˙V becomes sign-indefinite and the update is actively destabilizing.
  • An instantaneous one-sample optimum cannot guarantee stability over the longer horizon represented by the error dynamics.

7.7 The Prediction Term as an Exact Identification QP

The prediction update is exactly the solution of a strictly convex per-sample identification QP. Unlike tracking adaptation, its greedy objective aligns with the long-horizon goal of consistency along excited directions.

  • The prediction term is the exact solution of a per-sample QP, unlike the tracking adaptation term.
  • The QP minimizes parameter motion in the Γ−1 metric while explaining the measured prediction error through the regressor Φ.
  • Strict convexity gives the stationarity equation (Γ−1 + κΦ⊤Φ)∆ˆθ = κΦ⊤ε.
  • The closed-form minimizer reproduces the prediction term of the composite law exactly, with εnorm = 1/(κγ).
  • In the hard-constraint limit κ →∞, the update becomes the minimum-norm projection consistent with the latest measurement.
  • The augmented QP separates control and prediction subproblems, reproducing the control law and prediction update without shared decision variables.
  • For identification, per-sample fidelity to observed kinematics matches the long-horizon target ˆJ → J along excited directions.

7.8 Interpretation

The augmented QP precisely marks what instantaneous optimization can provide: control, constraints, and prediction adaptation, but not the stabilizing tracking adaptation term. Neural-network Jacobians retain the provable linear structure only when adaptation is restricted to the output layer.

  • 7.8 Interpretation: The augmented QP delivers the control law, the prediction term, and hard-constraint extensions, but not the tracking adaptation term.
  • 7.8 Interpretation: A one-sample QP is not a stability proof because stability depends on error dynamics across samples.
  • 7.8 Interpretation: Soft constraints yield a damped-least-squares control law while leaving the identification subproblem and tracking-adaptation conclusions unchanged.
  • 7.10 Extension to Neural Network Jacobians: A full deep network does not satisfy the required linearity-in-parameters condition for the composite law.
  • 7.10 Extension to Neural Network Jacobians: Adapting only output-layer weights restores linearity and supports a hybrid offline/online strategy with provable stability.
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