Simulated Zero-Force Trajectory (sZFT)

• Simulated Zero-Force Trajectory (sZFT) is a reconstructed equilibrium estimate derived from observed task-space motion and external wrench, enabling precise impedance adaptation.
• It employs a Transformer-based conditional diffusion model to accurately reconstruct both translational and rotational equilibrium states under contact conditions.
• The approach dynamically adjusts stiffness along task-relevant and non-task axes, ensuring robust performance in applications like obstacle traversal and peg insertion.

Searching arXiv for papers mentioning “simulated zero-force trajectory” and closely related formulations. arxiv_search(query="\"simulated zero-force trajectory\" OR sZFT OR \"Zero-Force Trajectory\" impedance manipulation", max_results=10, sort_by="relevance") Refining the search to capture the manipulation and control contexts where “Zero-Force Trajectory” is used explicitly. arxiv_search(query="\"Zero-Force Trajectory\" manipulation OR impedance OR trajectory", max_results=10, sort_by="relevance") Simulated Zero-Force Trajectory (sZFT) denotes, in its explicit contemporary use, a reconstructed equilibrium trajectory for contact-rich manipulation. In "Diffusion-Based Impedance Learning for Contact-Rich Manipulation Tasks," the sZFT, denoted $(\hat{\mathbf p}_0,\hat{\mathbf q}_0)$, is inferred online from observed task-space motion and external wrench, and functions as a contact-consistent estimate of the task-space equilibrium rather than either the actual contact trajectory or the nominal free-space path (Geiger et al., 24 Sep 2025). The term extends the underlying Zero-Force Trajectory (ZFT) notion from impedance control, where the equilibrium is the unique end-effector pose at which the interaction wrench vanishes. In this formulation, the simulated variant is introduced because contact perturbs the observed motion away from the programmed equilibrium, making direct impedance adaptation from the nominal reference inadequate.

1. Definition and operational role

The paper that explicitly introduces sZFT distinguishes two trajectories: the nominal ZFT and the simulated ZFT. The nominal ZFT is the programmed equilibrium motion in the unconstrained case, while the sZFT is the reconstructed equilibrium trajectory inferred from real interaction. The distinction is operationally strict. The controller continues to track the nominal ZFT as the motion source, whereas the reconstructed sZFT is used only for impedance adaptation (Geiger et al., 24 Sep 2025).

This usage makes sZFT neither a direct motion command nor a post hoc smoothing of the measured path. The observed end-effector pose $(\mathbf p,\mathbf q)$ is displaced from equilibrium by environmental interaction through the virtual spring-damper relation of impedance control, and the sZFT estimates that hidden equilibrium. The paper therefore characterizes sZFT as a denoised, contact-consistent equilibrium estimate and, in classical control terms, as a learned moving equilibrium estimate.

The motivation follows directly from contact-rich execution. If stiffness adaptation is computed from deviation relative to the nominal free-space ZFT alone, then the deviation can reflect environmental blockage rather than the task-consistent equilibrium under contact. The sZFT is introduced to resolve that ambiguity by reconstructing an equilibrium trajectory that is conditioned on the current physical interaction.

2. Equilibrium formulation in task space

The mathematical setting is task-space impedance control. The task-space displacement is defined as

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$

with translational and rotational equilibrium pose $(\mathbf p_0,\mathbf q_0)$ (Geiger et al., 24 Sep 2025).

The virtual spring potential is

$$\mathcal U(\Delta \mathbf x) = \tfrac12\, \mathbf e_t^\top \mathbf K_t \mathbf e_t + \tfrac12\, \mathbf e_r^\top \mathbf K_r \mathbf e_r,$$

and the wrench follows from

$$\mathbf F = \frac{\partial \mathcal U}{\partial \Delta \mathbf x}.$$

Within this formulation, the equilibrium pose is the point at which the virtual wrench vanishes. The sZFT is introduced because the equilibrium needed for adaptation during contact is not, in general, identical to the originally programmed free-space reference.

The reconstruction target is therefore the hidden equilibrium itself. For translation, the full noise is

$$\mathbf p_{\text{noise}} = \mathbf p - \mathbf p_0,$$

and at inference the model predicts $\hat{\mathbf p}_{\text{noise}}$ and reconstructs

$$\hat{\mathbf p}_0 = \mathbf p - \hat{\mathbf p}_{\text{noise}}.$$

For rotation, the full noise is the relative quaternion

$$\mathbf q_{\text{noise}} = \mathbf q\,\mathbf q_0^{-1},$$

and the reconstructed equilibrium quaternion is

$(\mathbf p,\mathbf q)$0

These equations formalize the sZFT as an inferred equilibrium rather than a simulated zero-load trajectory in the sense of unconstrained dynamics.

3. Diffusion reconstruction and rotational geometry

The reconstruction module is a Transformer-based conditional diffusion model with cross-attention to external wrench. Its inputs are observed translations and unit quaternions over a sequence, a timestep embedding $(\mathbf p,\mathbf q)$1, and external wrench context $(\mathbf p,\mathbf q)$2 projected into context tokens. Pose tokens are embedded and augmented with learned positional and timestep embeddings, interaction signals are projected into context tokens, and a multi-head cross-attention layer uses trajectory tokens as queries and wrench tokens as keys and values. The output head predicts $(\mathbf p,\mathbf q)$3D noise per token: $(\mathbf p,\mathbf q)$4 translational and $(\mathbf p,\mathbf q)$5 quaternion components (Geiger et al., 24 Sep 2025).

The diffusion model is presented in the standard forward and reverse forms,

$(\mathbf p,\mathbf q)$6

$(\mathbf p,\mathbf q)$7

with

$(\mathbf p,\mathbf q)$8

and the simple denoising loss

$(\mathbf p,\mathbf q)$9

In the conditional version, the context is the external wrench, and the denoised clean sample is the equilibrium trajectory sequence.

Rotational reconstruction is a central technical point. The paper states that additive Euclidean Gaussian noise on quaternion coordinates would violate the unit-norm constraint and distort rotational geometry. It therefore introduces a SLERP-based quaternion noise scheduler. A unit quaternion is represented in axis-angle form as

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$0

and the rotational displacement for control is

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$1

The scheduler uses spherical linear interpolation

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$2

where

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$3

This preserves unit norm, follows valid rotational geodesics, and allows the rotational noise magnitude to be scheduled across diffusion steps.

4. Energy-based impedance estimation from sZFT

Once the sZFT has been reconstructed, the controller computes displacement relative to that equilibrium: $$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$4 These quantities, together with measured external wrench, drive an energy-based estimator for stiffness reduction (Geiger et al., 24 Sep 2025).

For translation, the spring energy and external work are

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$5

which yields

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$6

To reduce oscillation, the paper defines a damped error

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$7

Per axis, the update is

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$8

followed by

$$\Delta \mathbf x = \begin{pmatrix} \mathbf e_t \ \mathbf e_r \end{pmatrix} \in \mathfrak{se}(3), \qquad \mathbf e_t := \mathbf p - \mathbf p_0 \in \mathbb R^3, \qquad \mathbf e_r := \log\!\big(\mathbf q_0 \mathbf q^{-1}\big) \in \mathbb R^3 \simeq \mathfrak{so}(3),$$9

For rotation, the analogous quantities are

$(\mathbf p_0,\mathbf q_0)$0

leading to

$(\mathbf p_0,\mathbf q_0)$1

with damped rotational error

$(\mathbf p_0,\mathbf q_0)$2

The controller uses diagonal positive semidefinite stiffness matrices. In the reported experiments,

$(\mathbf p_0,\mathbf q_0)$3

The directional rule is defined directly from the sZFT-based displacement. For translation,

$(\mathbf p_0,\mathbf q_0)$4

and for rotation,

$(\mathbf p_0,\mathbf q_0)$5

Final adapted stiffnesses are

$(\mathbf p_0,\mathbf q_0)$6

Axes that contribute strongly to the intended displacement receive little reduction; axes that contribute weakly receive stronger reduction. This is the formal mechanism behind the claim that impedance is reduced along non-task axes while rigidity is preserved along task directions.

At execution level, the robot runs task-space impedance control on a KUKA LBR iiwa via Fast Robot Interface in torque mode at $(\mathbf p_0,\mathbf q_0)$7 ms sample time, with gravity and Coriolis compensation active. Translational force and rotational moment are

$(\mathbf p_0,\mathbf q_0)$8

with damping proportional to stiffness,

$(\mathbf p_0,\mathbf q_0)$9

The important control-theoretic point is that the nominal ZFT remains the programmed reference motion, while the sZFT informs the gain adaptation law.

5. Empirical validation

The reported training data were collected by teleoperation using Apple Vision Pro. For the parkour dataset, the paper records 17,457 samples; for therapy, 55,838 samples. Sampling was at $$\mathcal U(\Delta \mathbf x) = \tfrac12\, \mathbf e_t^\top \mathbf K_t \mathbf e_t + \tfrac12\, \mathbf e_r^\top \mathbf K_r \mathbf e_r,$$0, i.e. $$\mathcal U(\Delta \mathbf x) = \tfrac12\, \mathbf e_t^\top \mathbf K_t \mathbf e_t + \tfrac12\, \mathbf e_r^\top \mathbf K_r \mathbf e_r,$$1 Hz. With the selected Transformer-diffusion model of hidden dimension 512, 4 attention heads, and 6 Transformer layers, the reconstruction accuracy on the parkour dataset was 0.994 mm positional loss, 0.249° angular magnitude error, and 0.003° axis deviation. On the combined parkour + therapy dataset, accuracy improved to 0.883 mm positional loss, 0.233° angular magnitude error, and 0.002° axis deviation (Geiger et al., 24 Sep 2025).

The parkour experiment demonstrates the control role of sZFT. With fixed stiffness, the end-effector failed immediately at the first obstacle, energy accumulated in the spring, and release caused acceleration into the safety stop; forces up to 15 N were observed. With diffusion-based impedance learning, the robot traversed all obstacles smoothly without violating velocity or force limits. The paper attributes this to stiffness adaptation based on the reconstructed sZFT, which prevented jamming in directions inconsistent with the task while preserving rigidity where motion was intended.

The peg-insertion results are the paper’s most direct task-generalization evidence. The training data contained no peg-in-hole demonstrations, yet the controller achieved 30/30 success for cylindrical, square, and star peg insertions. Under constant stiffness, the results were 30/30 for cylindrical, 4/30 for square, and 0/30 for star. The reported behavior is consistent with the directional rule: large force peaks along task-relevant axes can lead to only marginal stiffness reduction, while smaller forces on less task-relevant axes can induce stronger reduction.

The ablation study establishes the necessity of sZFT within the method. Uniform stiffness adaptation failed in both parkour and peg insertion. Directional adaptation computed from the nominal ZFT also failed in contact-rich settings: parkour hit safety bounds and stopped, and peg insertion succeeded only for cylindrical pegs while failing for square and star. The authors therefore conclude that directional factors must be derived from the reconstructed, contact-consistent sZFT rather than from the nominal ZFT.

6. Related formulations and non-equivalent uses

The term sZFT is explicit in contact-rich impedance learning, but adjacent literatures use related zero-force or zero-interaction concepts in non-equivalent ways.

Context	What is present	Relation to sZFT
Impedance learning (Geiger et al., 24 Sep 2025)	Reconstructed equilibrium trajectory from pose and wrench	Exact use of sZFT
Social-force trajectory prediction (Zhang et al., 2023)	No explicit sZFT; force features condition a Transformer/CVAE	Analogue only
Simulated tempering in force (Hartmann et al., 2019)	Zero-force ensemble reweighting within a force range	Equilibrium analogue
Epsilon-trajectory tracking (Ferrin et al., 2020)	Pre-compensated reference for zero steady-state error	Geometric analogue
Swept-volume planning (Wang et al., 2024)	Continuous collision-free trajectory generation	Non-contact analogue
Zero-propellant maneuver guidance (Zhang et al., 2017)	Off-line nominal reference adjusted online	Operational analogue

In pedestrian trajectory prediction, ForceFormer integrates social forces into a Transformer-based stochastic generative model, but the forces are auxiliary conditioning signals rather than the basis of a direct force rollout. The paper does not define “sZFT,” “zero-force trajectory,” or a future trajectory obtained by setting all social forces to zero and simulating motion. Its closest analogues are the no-force-input baseline AgentFormer, the goal-conditioned no-force variant ForceFormer-Goal, and ForceFormer-Dr, which uses driving force only without repulsive force input (Zhang et al., 2023).

In molecular simulation, Infinite Switch Simulated Tempering in Force supports recovery of the zero-force equilibrium ensemble, zero-force free-energy surfaces, and zero-force observables by reweighting, provided $$\mathcal U(\Delta \mathbf x) = \tfrac12\, \mathbf e_t^\top \mathbf K_t \mathbf e_t + \tfrac12\, \mathbf e_r^\top \mathbf K_r \mathbf e_r,$$2 lies in the sampled force interval. However, the propagated dynamics follow an effective force and an effective potential rather than the unbiased zero-force equations of motion, so the raw trajectory is not a true zero-force dynamical trajectory (Hartmann et al., 2019).

In autonomous-vehicle tracking, the so-called $$\mathcal U(\Delta \mathbf x) = \tfrac12\, \mathbf e_t^\top \mathbf K_t \mathbf e_t + \tfrac12\, \mathbf e_r^\top \mathbf K_r \mathbf e_r,$$3-trajectory is a transformed reference for a point located a fixed distance $$\mathcal U(\Delta \mathbf x) = \tfrac12\, \mathbf e_t^\top \mathbf K_t \mathbf e_t + \tfrac12\, \mathbf e_r^\top \mathbf K_r \mathbf e_r,$$4 in front of the vehicle. It is designed so that tracking this transformed trajectory yields asymptotic convergence of the body point to the original desired path with zero steady-state error. This is a geometric and kinematic pre-compensation method, not a force-based construction (Ferrin et al., 2020).

In continuous collision avoidance, the swept-volume signed distance field gives a geometric route to trajectories that remain continuously non-intersecting with obstacles. This suggests a zero-contact interpretation of “zero force,” since no collision event occurs if the swept volume avoids obstacles, but the method does not model contact forces, compliance, friction, or impact (Wang et al., 2024).

In zero-propellant maneuver guidance, the off-line planned nominal attitude trajectory plays a role functionally analogous to an sZFT-like baseline reference. The contribution there is a robust online correction layer that adjusts the nominal trajectory using total-angular-momentum feedback before tracking it, rather than an explicit simulated zero-force reconstruction (Zhang et al., 2017).

A plausible implication is that “Simulated Zero-Force Trajectory” currently has one exact meaning in the literature—interaction-aware equilibrium reconstruction for impedance adaptation—and several adjacent meanings that concern zero-force ensembles, zero-contact motion, pre-compensated references, or robustly adjusted nominal trajectories. These formulations are technically related, but they are not interchangeable. The explicit sZFT of contact-rich manipulation is distinguished by three properties: it is conditioned on measured external wrench, it reconstructs a hidden equilibrium rather than a free trajectory, and it is used to adapt impedance while leaving the nominal motion source intact.