Goal Force: Dynamics in Robotics & Planning

• Goal force is a mathematically formalized vector or tensor that directs agents toward target states and prescribed effects using force-based dynamics.
• It integrates physical constraints, real-world feedback, and hierarchical control strategies across domains like robotics, video modeling, and simulation.
• It enhances task precision and robustness, proving essential in applications such as humanoid manipulation, tactile grasping, and multi-agent trajectory forecasting.

A goal force is a mathematically and algorithmically formalized vector or tensor within a dynamical or planning system that acts to drive an agent, object, or process toward a prescribed state, location, or effect. The explicit representation and computation of goal force underpins a diverse range of fields including physics-conditioned video modeling, humanoid robot manipulation, pedestrian dynamics, vehicle trajectory forecasting, and tactile robotic grasping. Across these domains, goal force serves both as an attractive directive (pulling a system toward its goal) and as a rich interface for integrating physical constraints, real-world feedback, and policy conditioning.

1. Mathematical and Algorithmic Formulation

At its core, goal force generalizes the notion of a terminal or waypoint-driven action in a dynamical system by converting the result—a spatial, velocity, or effect-based goal—into a specification of force, impulse, or equivalent dynamical target.

Canonical Formulation:

• Direct Attraction: For a particle or agent at position $p(t)$ with goal $p^T$, the goal force may be written as

$$F_{\rm goal}(t) = \frac{1}{\tau}\bigl(v_{\rm des}^t - \dot p(t)\bigr),$$

where $v_{\rm des}^t$ aims directly at $p^T$ and $\tau$ adjusts reaction time (Gan et al., 2024).

• Mixed-Mode Planning: In robotic manipulation, the goal force may comprise both an end-effector kinematic goal (pose and orientation) and a direct force or torque to be applied at contact, e.g., $G_F = \{F_A, F_G\}$ for applied wrist force and grip force (Collins et al., 2023).
• Force-Optimal Grasping: For multi-contact manipulation, the problem becomes a constrained QP:

$$\min_{f} \sum_{i=1}^n \|f_i\|_2^2,\quad \text{s.t.}~\sum_{i=1}^n G_i f_i + w_{\text{ext}} = 0;\ f_i^n \geq 0;\ \|f_i^t\| \leq \mu f_i^n,$$

ensuring the minimal set of contact forces that maintain equilibrium (Lyu et al., 3 Nov 2025).

Hierarchical Integration:

Goal force is often computed or adapted within hierarchical controllers. In humanoid pushing, a high-level vision-conditioned policy sets desired Cartesian velocities and implicit force setpoints, while a low-level QP controller reconciles these with real-time actuation limits, dynamics, and observed resistance, automatically adjusting the end-effector force as required (Hu et al., 2 May 2026).

2. Goal Force in Video World Models

The Goal Force framework extends the specification of goals for video generation tasks by conditioning generative models not merely on text or target frames but on explicit force vectors encoding user-intended physical outcomes. Users provide a multi-channel tensor,

$$\tilde{\pi} \in \mathbb{R}^{F \times 3 \times H \times W},$$

where dedicated channels encode direct (applied) force, goal (desired effect) force, and mass, localized and oriented as 2D spatial Gaussians. Conditioning a diffusion video model on this tensor enables both simulation (cause → effect) and planning (effect → antecedent) (Gillman et al., 9 Jan 2026).

Unlike standard text/image conditionings, this design allows the model to:

• Plan antecedent physical actions that generate a desired force outcome, e.g., finding a sequence of interactions that topples a specific domino.
• Infer and propagate mass-dependent dynamics, generalizing to novel objects/events.
• Serve as an implicit, differentiable physics engine capable of generating physically plausible plans in complex, multi-object scenes without access to explicit external simulation.

Quantitative human preference and physical constraint selectivity studies confirm that this approach yields superior task adherence and planning diversity compared to text- or image-based controls.

3. Robotic Manipulation and Force-Adaptation

Goal force is central to closed-loop, physics-adaptive control for contact-rich manipulation. Recent systems integrate goal force at multiple control strata:

• Visual-Force Goal Prediction: State-of-the-art networks predict both kinematic and force goals ($G = (G_K, G_F)$), such that robot controllers can simultaneously track pose and apply the correct magnitude/direction of interaction force (e.g., when pulling a drawer or executing a precision grasp) (Collins et al., 2023). Excluding force objectives from control reduces success rates on contact-rich subtasks by a factor of 2–3.
• Force-Optimal Grasping: In tactile manipulation, a goal force is the minimal set of contact forces satisfying both grasp equilibrium and friction-cone constraints. Target tactile imprints are predicted via latent diffusion models, with feedback controllers adjusting gripper state until measured tactile data matches the force-optimal prediction (Lyu et al., 3 Nov 2025).

An explicit, adaptive goal force markedly improves robot robustness under uncertainty in object mass, friction, or unmodeled external disturbances (Hu et al., 2 May 2026).

4. Social Force Models in Pedestrian and Vehicle Trajectory Planning

Goal force originated in social force models for pedestrian dynamics but now underpins interpretable, long-term trajectory prediction in multi-agent scenarios.

Pedestrian Crowd Dynamics:

• In SG-SFM, the goal force is generalized from a fixed-destination "destination force" to a navigational force that dynamically selects locally feasible sub-goals based on the environment. The resultant force drives the pedestrian toward a sequence of sub-goals, balancing attraction to the destination with repulsion from obstacles and vehicles. This unified force field yields lower collision rates and trajectory errors in mixed-traffic simulations (Yang et al., 2021).

Vehicular Trajectory Generation:

• The Goal-based Neural Physics model (GNP) decomposes vehicle motion into (a) prediction of a terminal spatial goal and (b) integration of social-force ODEs. The goal force pulls the vehicle toward the forecast destination, while physics-parameterized repulsive forces ensure multi-agent feasibility and safety. Visualization of goal force vectors at each prediction step enhances interpretability (Gan et al., 2024).
• Quantitatively, explicit goal force architectures halve 5-s trajectory prediction RMSE compared to prior LSTM and attention-based baselines.

5. Training Objectives, Losses, and Policy Learning

Goal force extraction and propagation require joint consideration of global objectives and local physical dynamics. Training methodologies often include:

• Matching to Ground-Truth Force Trajectories: Networks regress goal force vectors or their parameterizations using task-conditioned loss terms, e.g., $p^T$0 for applied force and grip, or control policy rewards that explicitly penalize motion or force deviations (Collins et al., 2023, Lyu et al., 3 Nov 2025).
• Diffusion/Generative Modeling with Physics-Consistent Curriculum: Video models are fine-tuned on synthetic datasets with masked causal primitives, forcing the network to learn both forward and inverse dynamics (Gillman et al., 9 Jan 2026).
• Hierarchical Policy Distillation: Teacher policies with privileged (full state) force objectives are distilled into student policies operating on raw egocentric or visual input, preserving force-optimality in the learned action distributions (Hu et al., 2 May 2026).

6. Quantitative Evaluation and Empirical Impact

Empirical studies across domains demonstrate that explicit goal force formulation improves both task completion rates and physical plausibility:

Domain	Success Measure	Baseline	With Goal Force	Source
Humanoid pushing	Real-world object push rate	<50%	80–90%	(Hu et al., 2 May 2026)
Visual-force grasp	Contact-rich task success	45%	90%	(Collins et al., 2023)
Tactile grasp	FOSG (minimal-force stable)	40%	60%	(Lyu et al., 3 Nov 2025)
Pedestrian sim	Adj. Final Displacement Err.	1.111	0.967	(Yang et al., 2021)
Vehicular traj.	5 s RMSE (NGSIM, meters)	3.67–4.55	1.86	(Gan et al., 2024)

In all cases, goal force adds not only robustness but interpretability—enabling online adjustment, visualization, and diagnostics not possible with end-to-end policy learning alone.

7. Limitations and Future Directions

Despite the broad success of goal force schemes, several limitations persist:

• Generalization to domains with unmodeled physics (e.g., fluids, soft bodies) is limited by the representational assumptions built into physics-conditioned models (Gillman et al., 9 Jan 2026).
• Many implementations rely on per-domain normalization and do not yet incorporate calibration to absolute physical units.
• Handling of rare or visually ambiguous mass/force scenarios remains less robust, especially in implicit video models.
• Extensions to richer spatial representations (3D, multi-view, multisensory) and integration into downstream, policy-extracting control loops remain active areas for research.

A plausible implication is that as generative neural models expand their physics priors and as closed-loop control policies incorporate more explicit force representations, goal force will serve as a foundational organizing principle for interpretable, robust, and generalizable intelligent agents in physically grounded environments.