Discriminative Impact Point Predictor (DIPP)

• The paper introduces DIPP, which significantly reduces impact point error by 25–30% through early discriminative embedding of object trajectories.
• DIPP employs a novel LSTM-based Discriminative Feature Embedding module with an Impact-Point-Enhanced loss to cluster trajectories by aerodynamic dynamics.
• Experimental results on enriched motion-capture datasets and real-robot tests demonstrate near-perfect catching success rates under challenging aerodynamic conditions.

The Discriminative Impact Point Predictor (DIPP) is a machine learning framework for early and accurate impact point prediction of diverse in-flight objects, particularly for robotic catching applications. Developed to address the complexities of unsteady aerodynamics and the lack of representative real-world datasets, DIPP employs a discriminative neural architecture and a principled training regime, exhibiting significant improvements in impact prediction accuracy and real-robot catching success under challenging, aerodynamic-rich settings (Nguyen et al., 18 Sep 2025).

1. Problem Formulation and Dataset

DIPP addresses the task of predicting the landing (impact) point $$\boldsymbol{p}_\text{imp} = (x_\text{imp}, y_\text{imp}, z_\text{plane})$$ for an in-flight object, using only a brief early segment of its observed trajectory. The underlying dynamics are given by

$$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$

where $f$ represents complex aerodynamic forces (including lift, drag, Magnus effect, vortex shedding) and $\boldsymbol{w}(t)$ is an unmodeled disturbance. Only partial state information—position, velocity, and acceleration at discrete times $t_i$—is available for prediction.

To support this framework, a real-world motion-capture dataset was constructed:

Parameter	Value/Description
Number of object shapes	20 (e.g., boomerang, pinwheel, toy airplane, fan blade)
Raw trajectories	100 throws/object × 20 objects = 2,000
Augmented trajectories	400/object × 20 = 8,000 (random 3D translations/yaw)
Sampling rate	120 Hz (OptiTrack)
Measured features per frame	Position $\boldsymbol{x}$, velocity $\boldsymbol{v}$, acceleration $\boldsymbol{a}$
Spatial statistics, horizontal distance	2.9–4.2 m
Apex height	2.5–2.8 m
Diversity (Parabola Deviation Score)	DIPP: 0.12 m; prior NAE-rosDataset: 0.04 m

The dataset exhibits greater trajectory diversity and aerodynamic complexity than prior work, as reflected in its higher mean Parabola Deviation Score (PDS) (Nguyen et al., 18 Sep 2025).

2. Discriminative Feature Embedding (DFE) Module

The DFE module encodes observed trajectory segments into discriminative feature vectors in $\mathbb{R}^{128}$, enabling separation of trajectories by their underlying aerodynamic dynamics at an early stage. Formally, the input sequence $$\boldsymbol{s}_{t-T:t} = [\boldsymbol{s}_{t-T}, \dots, \boldsymbol{s}_t]$$ (each $$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$0) is processed by a single-layer LSTM encoder $$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$1 with input size 9 and hidden size 128.

$$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$2

The DFE does not employ an explicit contrastive loss; instead, discriminativity is induced by a joint Impact-Point-Enhanced (IPE) loss, specifically its trajectory-alignment component: $$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$3

Training the LSTM encoder under this loss yields embeddings that naturally cluster trajectories with similar aerodynamic signatures (e.g., objects with common lift/Magnus dynamics). Visualizations using t-SNE show that discriminative separation emerges after as few as five frames. This property enables generalization to unseen objects whose early dynamics resemble those of known classes (Nguyen et al., 18 Sep 2025).

3. Impact Point Predictor (IPP) Architectures

The IPP module takes high-level features from the DFE and estimates the impact point via two variants:

Neural Acceleration Estimator (NAE)–Based Method

• Architecture: Two-layer LSTM decoder (hidden size 128), generating future state predictions $$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$4 autoregressively.
• Impact point calculation: Identify first future $$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$5-coordinate intersection with $$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$6 and return $$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$7.

Direct Point Estimator (DPE)–Based Method

• Architecture: Single fully connected layer $$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$8 (no autoregression), returning impact coordinates $$\dot{\boldsymbol{x}}(t) = \boldsymbol{v}(t), \quad \dot{\boldsymbol{v}}(t) = \boldsymbol{a}(t), \quad \dot{\boldsymbol{a}}(t) = f(\boldsymbol{x}(t), \boldsymbol{v}(t), \boldsymbol{a}(t)) + \boldsymbol{w}(t)$$9 directly on the catching plane.

Loss functions are constructed as follows:

$f$0

$f$1

4. Training Protocol and Data Regimes

Models are trained in PyTorch using Adam optimization with a batch size of 512 and up to $f$2 epochs. Learning rates are $f$3 for the NAE baseline and $f$4 for DPE and DIPP variants. Data augmentation consists of random translations (±0.2 m) and yaw rotations (±30°), generating 8,000 unique trajectories.

For evaluation, 15 objects are allocated for training (80%/10%/10% split for train/val/test), and 5 distinct objects are held out for testing generalization (Nguyen et al., 18 Sep 2025).

5. Quantitative Evaluation and Results

Three primary metrics are used:

• Impact Point Error (IE): Euclidean distance between ground truth and predicted impact points.
• Catching Success Rate (SR): Fraction of trials where $f$5 (basket radius).

Key findings at 0.4 seconds before impact:

Method	IE (Seen)	IE (Unseen)
Newton	0.35	0.40
SVR	0.30	0.33
NAE	0.25	0.28
DPE	0.20	0.22
DIPP‐NAE	0.18	0.19
DIPP‐DPE	0.19	0.20

DIPP-based methods exhibited a 25–30% reduction in error at early prediction times (statistical significance $f$6).

Catching success rates for simulated trials (with basket radius $f$7 in meters):

Method	r=0.05	r=0.10	r=0.15	r=0.20
Newton	0.02 / 0.04	0.06 / 0.16	0.11/0.32	0.25/0.44
SVR	0.06 / 0.06	0.17 / 0.17	0.35/0.32	0.52/0.47
NAE	0.11 / 0.08	0.36 / 0.26	0.66/0.46	0.83/0.66
DPE	0.51 / 0.13	0.79 / 0.33	0.92/0.55	0.97/0.72
DIPP‐NAE	0.51 / 0.16	0.84 / 0.38	0.96/0.59	0.98/0.78
DIPP‐DPE	0.54 / 0.16	0.80 / 0.35	0.94/0.55	0.98/0.71

In real-robot catching experiments using a Boston Dynamics Spot quadruped (basket radius 0.15 m), DIPP‐NAE caught all throws for both seen and unseen boomerang and pinwheel objects, while NAE failed to catch either.

6. Discriminative Embedding and Early Prediction

Early discrimination of trajectory dynamics is achieved via the DFE, allowing substantial reduction in impact point error during the initial portion of flight (e.g., at 0.4 seconds to impact). This capability was shown to translate directly into improved catching performance; for example, simulation success rates increased from 0.66 to 0.96 (seen, $f$8) and from 0.46 to 0.59 (unseen). In physical robot tests, this deterministic early discrimination enabled robust interception of challenging objects for which baseline models failed.

A plausible implication is that such early-stage discriminative embeddings can facilitate real-time control responses in time-critical robotic catching or interception applications where prior methods may be limited by slow adaptation to aerodynamic heterogeneity.

7. Significance and Context within the Field

DIPP introduces a principled, dataset-rich, and architecturally discriminative solution for impact prediction in robotic catching under complex aerodynamics. Its empirical advances center on:

• The use of an extensive, high-diversity motion capture dataset incorporating objects with non-parabolic trajectories, offering broader generalization than prior datasets.
• The DFE module—a joint embedding mechanism governed by an alignment-promoting loss—which differentiates among diverse aerodynamic signatures rapidly.
• Dual prediction heads (NAE, DPE) for separate explicit trajectory extrapolation or direct point estimation, offering flexible performance profiles.

Previous approaches (Newtonian models, SVR, etc.) are outperformed, particularly in early prediction regimes and on unseen objects, marking a notable performance milestone in the field of robotic interception and real-world learning under complex, unsteady object dynamics (Nguyen et al., 18 Sep 2025).