Trajectory prediction for flying objects is critical in domains ranging from sports analytics to aerospace. However, traditional methods struggle with complex physical modeling, computational inefficiencies, and high hardware demands, often neglecting critical trajectory events like landing points. This paper introduces a novel, hardware-efficient trajectory prediction framework that integrates environmental priors with a Dual-Transformer-Cascaded (DTC) architecture. We demonstrate this approach by predicting the landing points of tennis balls in real-world outdoor courts. Using a single industrial camera and YOLO-based detection, we extract high-speed flight coordinates. These coordinates, fused with structural environmental priors (e.g., court boundaries), form a comprehensive dataset fed into our proposed DTC model. A first-level Transformer classifies the trajectory, while a second-level Transformer synthesizes these features to precisely predict the landing point. Extensive ablation and comparative experiments demonstrate that integrating environmental priors within the DTC architecture significantly outperforms existing trajectory prediction frameworks
Figures & tables
The type of trajectory
Prediction model
Flying ball trajectory
RNN [ 13 ] ; LSTM [ 11 , 12 ] , [ 32 ]
Vehicle trajectory
Markov [ 24 ] ; LSTM [ 28 ] ;
Transformer [ 35 ]
Ship trajectory
GRU [ 29 ] , [ 38 ] ; LSTM [ 30 ]
TABLE I: Summary of data-driven trajectory prediction
Fig. 1: (a) The schematic overview of the trajectory data acquisition system. (b) Schematic diagram of the experimental scene.
Fig. 2: Flowchart of dataset construction.
Fig. 3: Overview of the trajectory prediction model - PIDTC.
Fig. 4: (a) Structure of the trajectory classification module. (b) Structure of the landing point prediction module.
Parameter
Classification
Prediction
Epoch
500
1000
Batch-size
10
10
Learning rate
0.0001
0.0001
Embedding number
128
500
dmodel
64
512
Dropout rate
0.1
0.1
TABLE II: Parameters for training
Fig. 5: The loss during training and testing processes. (a) The BCE loss of classification module. (b) The MSE loss of prediction module.
Fig. 6: The loss of ablation models versus training epochs. (a) The BCE loss. (b) The MSE loss.
CMN
CMP
Accuracy
52.86%
85.71%
Precision
52.85%
81.40%
Recall
100%
94.59%
TABLE III: Classification performance comparison of ablation models
PMN
PMP
PMC
MSE
1183.39
690.16
372.39
RMSE
34.40
26.27
19.30
Bias (pixel)
23.06
18.02
13.35
PhyBias (cm)
29.58
23.16
17.07
TABLE IV: Prediction performance comparison of ablation models
Fig. 7: The training loss across different models versus training epochs: (a) MSE, (b) RMSE, (c) Bias.
MSE
RMSE
Bias (pixel)
PhyBias (cm)
RNN [ 13 ]
1064.99
32.63
26.71
34.16
GRU [ 38 ]
3417.77
58.46
49.24
63.98
LSTM [ 11 ]
866.72
29.44
23.96
30.55
Transformer [ 35 ]
1170.42
34.21
22.48
27.74
PIDTC
372.39
19.30
13.35
17.07
TABLE V: Prediction Performance Comparison ACROSS DIFFERENT MODELS
20% Nt
40% Nt
60% Nt
80% Nt
MSE
499.41
547.52
542.15
372.39
RMSE
22.35
23.40
23.28
19.30
Bias (pixel)
15.65
17.10
16.22
13.35
PhyBias (cm)
19.98
21.57
20.65
17.07
TABLE VI: Prediction performance comparison for different training set sizes