Optimizing dance motion reconstruction using a two-dimensional matrix approach with hybrid genetic and fuzzy logic differential evolution

Abstract

The development of dance movements using motion capture technology presents notable challenges, such as constraints related to body morphology, clothing interference, and the inherently nonlinear dynamics of human motion. Existing techniques generally struggle to accommodate intricate, nonlinear motions and encounter issues such as parameter sensitivity or prematurely becoming stuck in local solutions. This research study addresses the challenges mentioned above by developing a more precise method for reconstructing human dance movements. We develop the Two-Dimensional Matrix-Calculation (TDMC) model, combined with the Hybrid Genetic Algorithm with Fuzzy Logic Differential Evolution (HGA-FLDE), which aims to optimize the reconstruction of complex dance movements by leveraging Riemannian geometry and adaptive optimization for biomechanical nonlinear motion patterns and missing joint data. Furthermore, accuracy is achieved through other approaches, such as the Long Short-Term Memory (LSTM), Support Vector Regression (SVR), Kinect Sensors (KS), and Evolved Deep Gated Recurrent Unit (EDGRU) models, which were all thoroughly tested against one another. Our results demonstrate that TDMC-HGA-FLDE achieves an accuracy of 0.95 at 60 nodes, outperforming LSTM (0.90), SVR (0.92), EDGRU (0.91), and Kinect Sensors (0.87). Furthermore, TDMC-HGA-FLDE achieves a minimum error of 0.39 at 20 nodes, while the other models have higher error rates. In a real-world use case of dance therapy for lower limb rehabilitation, the model reconstructed step-touch dance movements using incomplete IMU data and achieved an accuracy of 0.94 and an MSE of 0.22, outperforming all baseline models (LSTM: 0.89, 0.41; EDGRU: 0.90, 0.36; SVR: 0.91, 0.32; KS: 0.86, 0.39; TDMC: 0.88, 0.30). These results suggest that the hybrid approach significantly enhances the precision and realism of dance motion rehabilitation, making a substantial contribution to the motion capture and rehabilitation industries.

Introduction

Accurate documentation of dance movements remains challenging due to the complexity of motion capture methods and human-related factors such as body morphology and clothing, which can interfere with sensor performance and data quality. It is difficult for human movements to provide intrinsically nonlinear signals and contain cinematic characteristics to reconstruct whole motion sequences from partial observations. Although prior methods have demonstrated potential in collecting movement within a restricted period and in robot action control^1–3 they do not provide theoretical assurances for extracting nonlinear motion data⁴. This constraint emphasizes a crucial research deficiency in the multidimensional matrix processing paradigms employed for motion capture and analysis⁵.

The purpose of this study is to obtain a better and more established approach to capturing and mimicking dance actions⁶. Some of the difficulties of conventional techniques are associated with parameter sensitivity, early convergence, and difficulty reaching the right balance between exploration and exploitation^7,8. However, as these factors impede accurate assessment of people’s mobility, this may lead to poor performance. The change in the solution frontier requires people to adopt new ideas and new ways of operating, which is a new paradigm^9,10.

We also introduce the Two-Dimensional Matrix Calculation (TDMC) model as a novel concept that addresses the drawbacks. This practice primarily focuses on restoring and rehabilitating human movement and dance. The TDMC model also includes the state-of-the-art Hybrid Genetic Algorithm with Fuzzy Logic Differential Evolution (HGA-FLDE) optimization algorithm. Adjusting the optimization parameters using fuzzy logic within the Genetic Algorithm-Differential Evolution (GA-DE) combination framework enhances algorithm efficiency and further regulates population variation. To overcome these issues, we improve the capabilities of both GA and DE by incorporating fuzzy logic, a key aspect of our hybrid GA-DE approach.

This method also efficiently addresses critical challenges, such as parameter sensitivity and convergence rate. Fuzzy logic also makes this possible, as it recognizes that an optimal optimization balance between exploration and exploitation cannot be set rigidly but instead requires flexibility^11–14. This characteristic ultimately enables the TDMC model to represent even nonlinear dance movements precisely and accurately. The extensive test results demonstrate that the proposed TDMC-HGA-FLDE model outperforms benchmark methods, including Long Short-Term Memory (LSTM)¹⁶, Support Vector Regression (SVR)¹⁷, Evolved Deep Gated Recurrent Unit (EDGRU)¹⁸, and Kinect Sensors (KS)¹⁹, which support our conclusion. The TDMC-HGA-FLDE technique has significantly advanced movement analysis and reconstruction, particularly in nonlinear human movement and dance recovery applications. This research contributes to the current state of motion capture methods and lays the groundwork for future enhancements. The primary contributions can be summarized as follows:

Main contributions

Development of the TDMC for dance motion reconstruction

We present the TDMC model, which integrates with existing human dance motion databases and utilizes Riemannian geometry to represent and reconstruct goal-directed action sequences, including missing joint data and other nonlinear movement patterns.

Hybrid genetic algorithm with fuzzy logic differential evolution (HGA-FLDE)

We propose a new optimization technique for adaptive parameter tuning that combines Fuzzy Logic, GA, and DE, enhanced with fuzzy logic. This approach addresses the challenges associated with parameter sensitivity and premature convergence, which are commonly encountered in most methodologies.

Comprehensive evaluation of model performance

The method was evaluated and compared against LSTM, SVR, EDGRU, and KS, which are considered the most advanced techniques. We achieved remarkable accuracy with significantly reduced errors and improved computational efficiency.

Quantitative and qualitative performance analysis

The performance measurements are impressively informative, with accuracy up to 0.95 and error rates as low as 0.39. This data illustrates the remarkable efficacy of TDMC-HGA-FLDE in reconstructing complex dance motions. Furthermore, visual comparisons also highlight the method’s ability to create realistic and coherent motion sequences.

Practical implications for real-world applications

The suggested solution supports real-life applications, such as dance therapy, virtual environment motion capture, and human-computer interaction, by capturing and reconstructing human movements more robustly and efficiently.

The paper is organized as follows: Section Related works provides an overview of existing research, setting the stage for our proposed solution. In Section Proposed methodology: TDMC-HGA-FLDE, we describe the motion and dance restoration process in detail and explain the importance of the proposed TDMC-HGA-FLDE model. In the Section Proposed algorithm: hybrid genetic algorithm with fuzzy logic and differential evolution, we provide an overview of the mathematical experiments and analyses performed, demonstrating the effectiveness of our approach. Lastly, Section Experimentation presents a detailed analysis of the data collected in this study, providing empirical evidence of the performance of our solution. In Section Conclusion of the work, the study’s conclusion is presented briefly, along with recommendations for potential improvements, which summarize our findings and suggest avenues for future research.

Related works

The dancers’ construction via computer animation or simulation has gained popularity in recent years. Target area recognition is critical for tracking, analyzing, and recreating human motion as a preliminary step. Combining optimization methods with matrix calculations offers a potential approach to addressing these challenges^15,16. This literature review analyzes ten relevant papers to thoroughly comprehend the current research progress in this field, identify areas where further research is needed, and support the proposed model of a TDMC developed through a combination of an HGA-FLDE.

The study¹⁷ examines various optimization techniques used in creating dance steps. The study acknowledges the use of particle swarm optimization and grey wolf optimization in formulating natural and fluid dance motions. The research’s outcomes suggest that evaluating the quality of created movements requires the use of appropriate fitness functions.

Researchers in¹⁸ examine the application of evolutionary algorithms to enhance dance choreography. The study showcases substantial enhancements in the visual and rhythmic aspects of the movements by refining dance sequences through repeated optimization. The evolutionary technique facilitates the investigation of various possibilities, resulting in inventive and imaginative dance routines.

This effort aims to utilize deep learning algorithms to emulate the dance steps of the analogy¹⁹. The suggested approach is based on the capacity of convolutional neural networks (CNNs) to learn and accurately replicate dance actions. Another claimed deep learning model can accurately record complex dance steps and provide performers with immediate, helpful feedback.

The authors propose combining motion capture (MOCAP) data and optimization to better understand and optimize dance movements²³. They enhance dancing sequences by tuning and applying kinematic information to improve the precision of eloquent movements. A recurrent neural network (RNN) or another type of neural network can enhance processing efficiency and accuracy. The study highlights the potential of this strategy for improving performance and supporting education.

The study²⁰ compares machine learning methodologies for forecasting dancing motions. Based on previous data, it evaluates the performance of support vector machines, neural networks, and decision trees in predicting future dance moves. Using an RNN or another neural network can improve processing efficiency and accuracy.

The authors²¹ looked at the potential of generative adversarial networks (GANs) to generate new dance moves. This research develops a GAN model that can learn and reproduce complex dance actions from any large corpus of dance videos in an appealing manner. Based on the workforce’s excellent performance in this aspect, it is understandable how GANs can mimic human dance moves.

To the best of our knowledge, there are no prior works other than the ones by the authors in²² which proposes an optimization-based approach to synchronizing dancing motions. It employs a multi-objective strategy to optimize time stability and emotional engagement, among other dance motion parameters. The findings suggest that the framework can produce dance sequences that are harmoniously coordinated and visually appealing.

The article²³ examines current progress in recognizing and reconstructing dance movements. The subject encompasses several methodologies, such as motion capture, machine learning, and optimization algorithms. While highlighting the field’s ongoing challenges, the evaluation primarily focuses on identifying and enhancing the accurate and precise representation of intricate and delicate dance movements.

Using the relationship between movement and music as a basis, researchers in²⁴ study genetic algorithms to create dance motions. The choreography algorithm aims to achieve full path coverage by synchronizing the musical and dance beats to ensure seamless movement. The primary focus of this work is to expand the capabilities of genetic algorithms to improve the generation of dance moves using new technology in the future.

Using a two-dimensional matrix computation approach, a machine learning-based model called MM-TDMC is introduced in²⁴ to restore human motion and dance sequences. The study focuses on the challenges of nonlinear movement data and demonstrates the model’s effectiveness in recovering short-term motion. Enhancing the effectiveness of motion recovery is one way to achieve proper dance movement reconstruction, as highlighted in this study on machine learning.

Consequently, several papers have examined the attempts to utilize optimization algorithms to analyze various fields of study²⁵. Algorithms such as GA and DE have enhanced the effectiveness and efficiency of movement generation and recovery. For example, incorporating these algorithms with two-dimensional matrix calculations may improve the overall process of reconstructing dance movement systems.

Multiple studies highlight the implementation of fuzzy self-tuning methods in optimization, mainly focusing on increasing algorithmic adaptability and performance. The Fuzzy Self-Tuning Differential Evolution (FSTDE) algorithm¹¹ was developed to optimize product line design, illustrating the effectiveness of fuzzy logic in adjusting DE parameters for enhanced and stable convergence. Nonetheless, while FSTDE performs well within the constraints of structured optimization problems, it does not adapt well to highly nonlinear and dynamic datasets, such as those found in dance motion sequence datasets. Fuzzy PSO¹² successfully improves the adaptiveness of the standard PSO by incorporating self-tuning mechanisms. Still, using swarm-centered heuristic strategies for deeper parameter tuning makes it less effective. Likewise, the Fuzzy Bees Algorithm (FBA)¹³ optimizes the multi-objective search procedure, with a focus on discrete spaces. The Flying Fox Optimizer (FFO)¹⁴ and Fuzzy Mayfly Algorithm (FMA)²⁶ Apply fuzzy logic to advance evolutionary optimization. However, little research exists regarding its implementation for high-dimensional motion capture problems.

The HGA-FLDE model we developed is based on previous work but incorporates Fuzzy Logic with GA and DE to optimize motion reconstruction in dance. Compared to FSTDE, which focuses on DE, our approach utilizes GA for coarse global search and leverages DE for refined local search. Additionally, fuzzy logic flexibly modifies important variables to prevent getting stuck in local minima. This flexible hybridization enhances the model’s management of missing joint data, biomechanical nonlinearities, and temporal dependencies, surpassing the capabilities of standalone fuzzy-tuned algorithms.

Research gaps and justifying the proposed model

However, despite years of progress in the field, many research areas remain unexplored²⁷. In most modern sewists, one problem is extracting non-linear motion data and complex chains of concrete actions²⁸. Additionally, two-dimensional matrix computing approaches cannot retrieve such information with adequate theoretical guarantees.

The suggested model utilizes an HGA-FLDE to perform TDMC. Its purpose is to fill the existing gaps in this area. By integrating the advantages of HGA-FLDE, the model has the potential to provide more precise and efficient solutions for reconstructing dance movements. The approach proposes integrating EAO with Fuzzy Logic due to the latter’s flexibility in addressing nonlinearity issues and missing data in motion data.

Reconstructing scenarios using dance steps can be achieved by applying optimization algorithms incorporating two-dimensional matrix calculations. The situation has improved dramatically; however, accurately capturing and retrieving complex, nonlinear motion data remains challenging. The proposed model utilizes an HGA-FLDE to address these issues and increase effectiveness. Being a valuable addition to this thoroughly interprofessional sphere of study, this new model can improve the approaches and precision of dance movement reconstruction.

Proposed methodology: TDMC-HGA-FLDE

The study focuses on the TDMC, a fundamental technological idea that explains the dance and movement restoration procedure. Human development entails creating a skeletal diagram where each joint is identified as an edge. Lack of body development or wearing clothes may prevent some joint articulations from forming. Consequently, it is essential to critically evaluate current deficiencies and implement a well-structured strategy for sustained growth. In this framework, the approach is used to determine the spatial orientation of the joint relative to the anatomical position of the body²⁹.

1

Locate the position of a skeletal framework that extends across multiple human joints. A movement cluster with missing joints is represented by M = [X₁, X₂,… X_M] ∈ R^3g×M, where M denotes the poles M and X_i indicates the i-th line comprising the movement group. So, if M is 0, it means there are no joints. We have constructed a perfect grid of M. To address this issue, we will implement the proposed TDMC by following the steps outlined in Fig. 1. To introduce the TDMC, researchers must first demonstrate the MMD, which stands for the Riemannian complex development of human movement.

Fig. 1.

Utilizing skeleton or apparel to elongate partially during dance motions.

Riemannian geometry’s exponent

Figure 2 illustrates the dissimilarities between Riemann and geodetic connections on a sphere. Start at the very first ball in the order and go all the way to the very last one for every wrist junction. To find the degree of similarity between two unit quaternions, one can apply the Riemannian vicinity approach³⁰. The analyst can utilize the Riemannian similarity method to assess the degree of similarity between unit quaternions. Showing a single quaternion on an annular surface in a flat, three-dimensional space is physically and theoretically impossible. Distance calculations often use two methods based on Riemannian mathematics.

Fig. 2.

The component of the dance assessment (Riemannian exponential).

Riemann functions and the Riemann value being disjoint are two distinct but connected subjects. To facilitate understanding and calculation, the authors introduced a normalized circle with a unit of length equal to one. One option that many people prefer to use is X^m because it is structurally placed in all the joints. Equations (2) and 3 prove that the overall location X^m has been provided appropriately and systematically.

2

Equation (2) shares the common denominator of separations, although one is named after Riemann and the other after Euclid. The researchers decide on a single value for circle length as a standard, simplifying computations and descriptions. According to Riemannian theory, the collapse of functions f and h is the value disparity between them at two locations on a three-dimensional circle, where (t & h ∈ R³).

3

The cosine value of the equation from Eq. (3), when applied to all values in the range of [– 1 1], is [0 γ]. The product of Riemannian distance is one way to explain the Riemann exponential bit³¹.

4

In Eq. (4), the breadth is represented by the variable α. The Riemannian separation enables the precise calculation of many motions. One approach to enhance precision is to integrate (R_m) in the higher-order Kernel Hilbert Space using the Riemann exponential slice.

The function ψ: g→ψ (g) can collect information about motions. If the bit ability defines a specific graph, then the function r_H(g_x, g_y) represents that graph. According to Eq. (5) on³²we can express the result using two casings, g_x, and g_y:

5

Here, F is the sum of all breadth parameters αm that deal with different aspects of the Riemannian exponential.

Using the weight factor , as shown in Eq. (6), we may classify all weights in the weight matrix Q:

6

Immersing human movement as closely as possible to its fundamental area in a Hilbert space is one of the most effective approaches to understanding an ideal Q.

Multiple varieties of TDMC

The TDMC models human motion using a skeletal diagram with joints as edges, analyzing spatial orientation for restoration. Multiple learning components capture motion variability in the Hilbert region, thereby optimizing a movement matrix (as per Eq. (7)). Following learning, the component region is established, with retrieval tasks becoming independent of the training sets.

7

Zj ∈ R^{3× Ax} represents each j-th joint in Z, where A_X is the side number of Z. The designated subset’s mapping Z in the practical Hilbert space is determined by φ(Z).

Restricting the location of φ(Z) is the objective of different component learning. Therefore, the following can be utilized to determine the components’ learning capacity³¹:

8

The final part must consist of integrating several Riemannian exponential components, where the B^− 1 matrix stands for B’s inverse, and (ψ(B)) denotes the position of ψ(B)³³. Rank minimization problems, like Eq. (8), are challenging due to their coupled form and NP-hard nature, meaning they cannot be solved in polynomial time using decision or heuristic algorithms. Matrix performance relies on r1-standard and atomic standard, while r0-standard and system rank individuals are excluded. However, we cannot continue addressing the r₁-standard or tiny norm minimization problem because the chart χ(B) is not amicable for any pertinent work. Standard approaches, such as single-value reduction, have a history of failing when confronted with the rank minimization problem. According to Eq. (9), the Schatten z-Norm (S-z-N) ||.||Gt is proposed as an alternative to the conventional method for estimating the approximate location of a grid³⁴.

9

B is a matrix equal to m×c when t is positive, and κi is the estimate for B for the j-th individual. This is because the Schatten r-standards || is associated with its solution, which is as follows: The Schatten r-standards ||. || decrease, and when t approaches zero, it becomes 0. It is better to locate Gt closer to B’s source. As shown in Eq. (10), there is the potential to unfasten one level of hierarchy (ψ(P))³⁴.

10

The probabilistic behavior of the smile is also left-invariant, as is the position minimization from the inverse of the intensity of Gv(•). For this reason, analysts do not take it into account and substitute Eq. (8) with Eq. (11) to simplify calculations³⁴:

11

.

The augmented Lagrange coefficient (ALC) addresses the challenge caused by the corresponding limitation in the target capacity. The following is the explanation of the enhanced Lagrange capacity³⁵:

12

The analysts employ the Powell-Hestenes algorithm (PHA) for system optimization. The most important aspect of the PHA computation is keeping σ and η constant while minimizing W³⁶.

13

With the critical σ and η values, Eq. (12) may be quickly developed to diag(BB^− 1) = 1. Several inclination-based development strategies, including the rapid proximal angle method (PAM), can effectively address this development. The derivative of W (η, B, σ, Z) regarding B is displayed in Eq. (14)³⁷.

14

As illustrated in Eq. (15), the expression can be deduced from the chain concept of system specification³⁷:

15

Where V is equal to J(Z), and λ_in is the element in the j-th push and m-th part of B, which is the element in the r-th driving and the r-th part of V.

g(V) = Gv(V^t/2 ), where V is equal to J(Z), λ_in is the component in the j-th push and m-th part of B, and V_rη is the component in the r-th pushing of V, as stated in the provided equation, for that specific Eq. (16) ³⁷.

16

Also:

17

The problem in Eq. (13) is that the analysts suggest using a PAM-based Algorithm 1 to address it. As stated on the element r_ηm(•, •), Eq. (17) determines the parameter design of the jth joint. To ensure that the S-z-N correctly learns the rank (Z), set t to 1/16 while training.

After that, they employ a PHA computation in Eq. (11) to clarify the multifarious learning. The analysts will include the calculation details in the supporting data instead of publishing them here, as calculating PHA is simple.

Two-dimensional matrix calculation (TDMC)

One of the educational components, r_y (•, •), which has found its rightful place, is the capacity to perceive and respond to motion. φ: Z→φ(B) shows the proper learning rate specified by r_y(•, •) in Eq. (5), and B is learned via distinct part learning. Once the fundamental complete motion structure of M is represented as B = [h₁, h₂,…, h_M] ∈ R^3g×M, the kernelized low-rank network resolution immediately proceeds to handle the problem in the following manner:

Algorithm 1.

.

Per Eq. (4), B is an independent learning function, and the function r_y(. ,.) meets the criterion. Last but not least, we fix the problem by applying the kernelized low-rank network method: Last but not least, we fix the problem by applying the kernelized low-rank network method::

18

Analyzing individual migratory patterns as a distinct educational data set is paramount. For the picture to feel natural and attract viewers, it must adhere to specific cinematic standards. The most recent paradigm for simplifying is shown by Eq. (19).

19

The overrun of the installment notion is rigid by nature and is referred to as (γ/2)||YP||²_X, where the parameter γ is tweaked to increase or decrease the stringency of the imperative term. According to the abovementioned formula, Function Y is defined in Eq. (20).

20

We can avoid the two homogeneity restrictions that diminish the estimated capacity by using ALC. A study comparing extended Lagrange functions: A study comparing extended Lagrange functions:

21

By retaining, it may be possible to significantly improve the ALC computation by focusing on the variables µ₁, µ₂, τ, and O, and minimizing the use of the variable T as much as possible. We use the HGA-FLDE in the following section to find the optimal settings.

Proposed algorithm: hybrid genetic algorithm with fuzzy logic and differential evolution

The GA-DE algorithm combines the GA³⁸ and DE³⁹leveraging GA’s exploration and DE’s exploitation for balanced optimization. GA excels in exploring large solution spaces but struggles with solution refinement, while DE is strong in optimization but weak in exploration. GA-DE leverages these strengths to achieve improved performance. The advanced HGA-FLDE variant uses self-tuning and fuzzy logic^40–42 to dynamically adjust parameters, ensuring robust optimization and diversification⁴³. This hybrid approach effectively addresses complex optimization problems.

Amending with fuzzy logic

By self-tuning the above-mentioned decisive parameters, such as the scaling factor F and the crossover rate CR, using fuzzy logic, HGA-FLDE is designed to be self-adaptive. These parameters are crucial for the algorithm’s proper functioning, as they define the balance between exploration and exploitation. FL systems overcome the uncertainty and imprecision of real-world optimization problems by utilizing a given rule base to control F and CR within iteration and population diversities. This adaptive adjustment helps maintain population diversity and reduce the rate of convergence or premature convergence, thus leading to enhanced algorithm performance.

Mathematical model

Mathematically, the HGA-FLDE model consists of the following parts:

GA mathematical model

Selection is a type of decision-making that involves choosing individuals based more on their physical well-being. Crossover to create offspring by combining the DNA strands of two parents. Mutations introduce new qualities and randomly assign new features to individuals³⁸.

DE mathematical equation

• Mutation: Generates mutant vectors by incorporating the calculated discrepancy, based on the relative weights, between two population matrices into a third matrix as follows³⁹:

22

• Crossover: The process involves merging mutant vectors and target matrices to produce experimental matrices as follows:

23

• Selection: Select the trial or goal matrix depending on their fitness levels.

Adaptive mechanisms

Therefore, depending on the above search procedure, fine-tune the F scaling factor and CR crossover rate.

Fuzzy logic evolved GA-DE (FGA-HADE)

The fuzzy logic system is used mainly in the DE part of the HGA-FLDE model. It adjusts the scaling factor (F) and the crossover rate (CR) in correlation with the population diversity and the number of iterations. The GA operates independently, with fuzzy logic not in effect.

The fuzzy system employs essential linear modifications as an illustration, as follows:

• If (iteration is low) and (diversity is high), ◊ adjustment is high.

• If (iteration is medium) and (diversity is medium) ◊ adjustment is medium.

• If (iteration is high) and (diversity is low), ◊ adjustment is low.

The defuzzification procedure employs the centroid technique, which calculates the output as the weighted mean of all possible outcomes. This provides accurate readings for the scaling factor (F) and crossover rate (CR). This enables parameters to be set and changed without discontinuities depending on the population heterogeneity and iteration number.

Experimentation

This integration enhances the overall effectiveness in recording complex and nonlinear movements, allowing for responsive and flexible parameter adjustments that preserve a wide range of individuals within the population. The TDMC-HGA-FLDE model was compared to other well-established models to assess its performance thoroughly. As a part of the RNNs, the LSTM architecture, which performs exceptionally well at learning consecutive data and retaining long-term dependencies, has emerged into the limelight^44,45.

Among all the regression techniques available, SVR is a stable and fast regression method well-suited for moderate-sized datasets. Online value prediction utilizes support vector machines. We utilized well-structured KS motion data to assess the model’s performance in recognizing complex movements. To enhance the temporal characteristics of the data, we employed EDGRU, an improved version of GRU⁴⁶. In this section, we present detailed experimental results regarding the performance of the TDMC-HGA-FLDE model against other techniques, including LSTM⁴⁷SVR, EDGRU, and KS. The population size (N = 50) is established based on previous literature and empirical tests⁴⁸.

The experimental results suggest that this size provided an adequate trade-off between diversity and computational power for most practical reconstructions of nonlinear motion data. Smaller sizes decreased the solution’s diversity and quality, while larger sizes increased the computational burden without the expected accuracy gains. For the GA crossover operation, we defined CR_GA as 0.8. This implies that 80% of the offspring are genetic hybrids of both parents, while 20% are copies of one or the other parent.

We utilize a two-point crossover method, which selects two crossover locations on the parent chromosomes and cuts them to exchange genes between the two parents at these points. This is how most GA-based optimizers implement it to enhance search diversity while maintaining system stability. Unlike DE, where the crossover rate is dynamically adjusted using fuzzy logic, in this case with GA optimizers, CR_GA remains constant because it enables the maintenance of a controlled rate of genetic diversity during evolutionary processes. Table 1 shows the setting parameters and their initial values for comparison models.

Table 1.

Setting parameters and their initial values for comparison models.

Model	Parameter	Initial Value
EDGRU	Hidden Units per Layer	50
	Decoder Layers	2
	Encoder Layers	2
	Learning Rate	0.001
	Dropout Rate	0.2
	Sequence Length	10
	Batch Size	32
	Epochs	100
	Optimizer	Adam
LSTM	Hidden Units per Layer	50
	Number of Layers	2
	Sequence Length	10
	Batch Size	32
	Learning Rate	0.001
	Dropout Rate	0.2
	Epochs	100
	Optimizer	Adam
KS	Distance Metric	Euclidean
	Bandwidth	1.0
	Kernel Type	Gaussian
SVR	Epsilon	0.1
	C (Regularization Parameter)	1.0
	Kernel	RBF
	Gamma	‘scale’
	Max Iterations	1000
TDMC	Exploration Rate (Epsilon)	0.1
	Discount Factor (Gamma)	0.9
	Learning Rate	0.1
	Number of Episodes	1000

Experimental setup

The hardware combination consisted of 64 GB of DDR4 RAM, an NVIDIA GeForce RTX 3080 GPU with 10 GB of GDDR6X VRAM, a 2 TB NVMe SSD for rapid data access and storage, and an Intel Core i9-10900 K CPU with ten cores and a clock speed of 3.7 GHz. The software environment included several libraries, including NumPy (version 1.24.3, https://numpy.org/), SciPy (version 1.10.1, https://scipy.org/), Pandas (version 1.5.3, https://pandas.pydata.org/), Matplotlib (version 3.7.1, https://matplotlib.org/), Seaborn (version 0.12.2, https://seaborn.pydata.org/), and Keras (version 2.12.0, https://keras.io/), alongside Scikit-learn (version 0.24, https://scikit-learn.org/), Python (version 3.8, https://www.python.org/), and Ubuntu 20.04 LTS (https://ubuntu.com/).

Performance metrics

We used several key measures to compare and contrast the models’ performances. Lastly, the Performance Ratio establishes the differences in runtime, resource usage, and complete computational time of the models to determine the efficiency of one compared to the other and to indicate how effective it will be in a real-time environment. Inaccuracy of a model – the proportion of correct to total predictions is one of the most basic measures of a model’s ability to generalize to unknown new data. Eqs. The mathematical equations can have two forms, as depicted in Eqs. (24) and (25)⁴⁹:

24

25

26

The area under the receiver operating characteristic (ROC) curve (AUC) summarizes the model’s performance⁵⁰; it is a graphical depiction of the model’s diagnostic capacity that plots the valid positive rate (recall) versus the false positive rate at various threshold levels⁵¹. When it is unfortunate that a disease is not detected and would have had severe consequences, recall and sensitivity measures the proportion of real positive cases correctly identified by the model.

This section provides an experimental comparison of the TDMC-HGA-FLDE model with LSTM, SVR, EDGRU, and KS, as well as the advantages and disadvantages of the proposed model based on these metrics. The outcomes indicate that it can describe intricate and nonlinear motion, as well as other essential information about the model’s application.

Discussion and results

To comprehensively verify the effectiveness of the proposed TDMC-HGA-FLDE model, we have done numerous tests using many other well-known methods. This class includes TDMC, KS, EDGRU, SVR, and LSTM. Instead, we shall compare the models and evaluate their capability for capturing and mimicking intricate and non-linear dance movements. Figure 3 shows that some positions are generated by the models, which allows us to compare and contrast the results of motion capture activities. The datasets are available from https://smpl.is.tue.mpg.de/.

Fig. 3.

The result of various models.

The picture above visually compares dance poses generated by various models. These models include LSTM, EDGRU, SVR, KS, TDMC, and the proposed TDMC-HGA-FLDE model. The subplots illustrate the evolution of dancing positions over time, demonstrating how accurately each model captured and reproduced the intricate motions of the dance routines.

As shown in the plot in the upper left, the LSTM model effectively captures the dance stances. This layout of positions, aligned on the X-axis, also corroborates the idea that the model can monitor time dependencies and pivotally generate movements. However, LSTM could fail to capture long-range dependencies due to minor fluctuations and disparities, some of which are illustrated at the end of the sequence.

When comparing the two models’ ability to depict dancing positions, the EDGRU model beats LSTM (top right plot). The more consistent and regular the stances, the better the model handles complex temporal patterns. However, there are still some fluctuations, which could indicate regions where the model needs improvement in terms of accuracy.

As observed in the middle left plot, the SVR model’s results poorly predict the dance positions. The resultant stances are more spread out and thus not organized. On the other hand, SVR is not a sequential model; hence, the RNN-based model is more suitable for identifying the temporal patterns of the dance movements. This dispersion proves it.

The KS model (middle right plot) offers a decent starting point for the dance positions. The stances have a relatively regular pattern. The model relies on sensor data, which can be inaccurate due to noise and the sensors’ limited resolution. As a result, the posture generation becomes less precise.

As shown in the plot on the bottom left, the TDMC model adequately captures the dance positions. Because the resultant sequence is seamless and follows a distinct pattern, the model accurately represents the temporal dependency of the dancing steps. This performance is significantly better than that of non-recurrent models, such as SVR and KS.

The suggested TDMC-HGA-FLDE model, represented in the bottom right plot, outperforms all others. Everything about the dancing is perfect: the positions, the execution, the naturalness, and the accuracy with which they flow. Integrating TDMC with hybrid evolutionary algorithms and fuzzy logic substantially enhances the model’s ability to capture complicated and nonlinear motions. As a result, the model can more rationally process complex temporal input, leading to an improved representation of dance postures.

Finally, compared with other models, the TDMC-HGA-FLDE model produces the highest accuracy in capturing and reconstructing dance postures. However, even here, LSTM and EDGRU have considerable drawbacks: they can only work with simple temporal structures and do not directly capture long-range temporal dependencies. However, KS and SVR fail to exhibit reliability in terms of posture sequences. The best model for capturing complex dance movements from the analyzed models is TDMC-HGA-FLDE, which utilizes a hybrid genetic algorithm and fuzzy logic. Compared to the original TDMC model, it has performed remarkably better. The study reveals that the proposed TDMC-HGA-FLDE model can effectively handle jobs that require precise analysis of captured motion.

Evaluating performance

We conducted a series of studies to compare the performance of the TDMC-HGA-FLDE model with that of the LSTM, EDGRU, SVR, KS, and TDMC models. The goal was to determine the model’s effectiveness. The performance ratio was assessed across several training datasets as a percentage to understand how each model improves with the addition of more data. Figure 4 shows the results, plotted against the size of the training datasets, illustrating how each model performs.

Fig. 4.

Performance ratio.

Figure 4 demonstrates that, for all node numbers, the TDMC-HGA-FLDE model outperforms the other models. The TDMC-HGA-FLDE model’s performance ratio increases sharply with the number of nodes, reaching approximately 96% with 60 nodes. The model’s higher performance can be attributed to improvements introduced through a hybrid genetic algorithm and fuzzy logic, which enhance its effectiveness in analyzing discrete and nonlinear movements.

However, when comparing the results from SM data, LSTM and TDMC proved to excel, especially when the dataset sizes were larger, indicating that they do not struggle when more data is fed to them. However, as can be seen when comparing them with the TDMC-HGA-FLDE model, the performance delay of the hybrid innovations underlines their value.

EDGRU enhances performance to some extent as the number of nodes increases, but it remains less competitive than LSTM and TDMC, as shown in the figure below. As anticipated, the techniques for capturing organizations’ experiences and activities are less structured and non-sequential, resulting in the lowest improvement and performance for KS and SVR, respectively.

Therefore, the TDMC-HGA-FLDE model will benefit highly selective, meticulous motion capture and analysis applications. It also performs well in cases where the movement is intricate and non-repetitive. The results confirm the effectiveness of the proposed approach and demonstrate its potential application in real-life motion capture tasks.

Recall and ROC curves

Thus, to determine the reliability of comparison models, applying as many criteria as possible, such as ROC and recall curves, is appropriate. The area under the receiver operating characteristic (ROC) curve is also known as the classification performance curve; this measure represents the ability of the model to classify data correctly in positive, and in contrast, recall curves focus on the model’s performance of recognizing positive samples which is often essential when in a situation that failing to acknowledge a positive sample is costly such as in medical diagnosis or detecting fraud. ROC and recall curves can be plotted to compare a model and determine its readiness level for deployment, as well as the challenges it will face during deployment. Figure 5 shows the ROC and recall curves for comparison models.

Fig. 5.

ROC and recall curves for comparison models.

As depicted in Fig. 5, TDMC-HGA-FLDE typically exhibits better discriminative capability due to its integrated strategy, which combines HGA and FLDE. The ROC curve of TDMC-HGA-FLDE often bends towards the upper left corner more sharply than those of LSTM, EDGRU, SVR, KS, and TDMC. The tables also show that the proposed approach, TDMC-HGA-FLDE, accurately maintains a True Positive Rate (TPR) at a higher level and a False Positive Rate (FPR) at a low level for any threshold value. The improvement in the ROC performance can be attributed to the feature extraction algorithm and the hybrid optimization technique, which enable the algorithm to better classify complex patterns in the data.

HGA-FLDE-based TDMC achieved a better recall value with higher True Positive Rates than LSTM, EDGRU, SVR, KS, and TDMC over the various threshold values. It becomes more critical in those cases where it is crucial to collect many positive cases. Due to the implementation of fuzzy logic, TDMC-HGA-FLDE can adjust the judgment limits and enhance sensitivity, significantly improving recall. Adjusting model parameters to augment its recall performance guarantees that the hybrid genetic algorithm can capture positive examples without compromising the number of false samples.

Statistical evaluation

To determine how different the TDMC-HGA-FLDE method is from the other comparison methods, we obtain the p-value of the Wilcoxon signed-rank, non-parametric test⁵². The given experimental data was very convenient for this test because this method works best with non-normal data. If one wants to analyze the differences between matched samples to determine if the values of performance indicators are significantly different from those of the preceding period or year, the Wilcoxon signed-rank test could be beneficial. The goal of this test is to ensure that the observed differences reflect actual differences in the methods that have been evaluated. Twenty, thirty, forty, fifty, and sixty nodes are used to test the TDMC-HGA-FLDE technique and compare it to other methods. The experimental results for each metric are detailed in Tables 2 and 3, as well as in Figs. 6 and 7. The results of each baseline algorithm cannot be compared with itself in the Wilcoxon signed-rank test; the best results are highlighted in bold in the tables below. If ‘N/A’ is shown, the data are not available.

Table 2.

The comparison of error rate.

Number of nodes	EDGRU	LSTM	SVR	KS	TDMC	TDMC-HGA-FLDE
20	0.65	0.82	0.60	0.48	0.42	0.39
30	0.68	0.85	0.51	0.51	0.45	0.40
40	0.72	0.79	0.58	0.58	0.49	0.45
50	0.76	0.82	0.69	0.59	0.53	0.49
60	0.73	0.80	0.59	0.59	0.54	0.48
P-value	0.01	0.047	0.14	0.001	0.123	N/A

Table 3.

The comparison of accuracy.

Number of Nodes	EDGRU	LSTM	SVR	KS	TDMC	TDMC-HGA-FLDE
20	0.88	0.87	0.89	0.83	0.85	0.92
30	0.89	0.86	0.90	0.85	0.86	0.93
40	0.90	0.89	0.91	0.86	0.88	0.94
50	0.90	0.89	0.91	0.86	0.88	0.94
60	0.91	0.90	0.92	0.87	0.89	0.95
P-value	0.021	0.011	0.001	0.24	0.021	N/A

Fig. 6.

The error rate of TDMC-HGA-FLDE and other comparison techniques, generated using Matplotlib (version 3.7.1, https://matplotlib.org/).

Fig. 7.

Accuracy of TDMC-HGA-FLDE and other comparison techniques, generated using Matplotlib (version 3.7.1, https://matplotlib.org/).

Table 2 shows that when the number of nodes is varied, TDMC-HGA-FLDE always gets the lowest error rate compared to the other models. In particular, TDMC-HGA-FLDE outperforms EDGRU (0.65), LSTM (0.82), SVR (0.60), KS (0.48), and TDMC (0.42) at 20 nodes with an error rate of 0.39. As the number of nodes increases, this pattern persists, except for TDMC-HGA-FLDE, which outperforms all other models, achieving a peak error rate of 0.48 at 60 nodes, compared to 0.73 for EDGRU, 0.80 for LSTM, 0.59 for SVR, 0.59 for KS, and 0.54 for TDMC.

Except for SVR and TDMC, all other models had p-values < 05, so proving their performances differ significantly is possible. However, TDMC and HGA-FLDE have more significant p-values, equal to 0.14 and 0.123, respectively, indicating that their performances are not statistically significantly different. The p-values stress significance estimates are based on these results.

Similarly, TDMC-HGA-FLDE outperforms other models in all node configurations, as indicated by the accuracy results in Table 3. Compared to EDGRU(0.88), LSTM(0.87), SVR(0.89), KS(0.83), and TDMC (0.85), TDMC-HGA-FLDE achieves a better accuracy of 0.92 at 20 nodes. At 60 nodes, TDMC-HGA-FLDE achieves an accuracy of 0.95, which is still better than EDGRU’s 0.91, LSTM’s 0.90, SVR’s 0.92, KS’s 0.87, and TDMC’s 0.89. This advantage is maintained as the node count increases.

With p-values significantly below 0.05 for comparisons with EDGRU, LSTM, and TDMC and an exceptionally low p-value of 0.001 when compared with SVR, the p-values linked with these accuracy results further emphasize the importance of TDMC-HGA-FLDE’s performance. Due to KS’s wider performance variability, the comparison with KS produces a p-value of 0.24, suggesting less statistical significance.

These results are further supported visually by Figs. 6 and 7. Regardless of the node arrangement, TDMC-HGA-FLDE consistently maintains low error rates, as shown in Fig. 6. Figure 7 illustrates the accuracy rates, demonstrating that TDMC-HGA-FLDE consistently yields better results than the other models.

When pitted against EDGRU, LSTM, SVR, KS, and TDMC, the TDMC-HGA-FLDE model is the most effective in terms of accuracy and error rate. For instance, when integrated with a hybrid of genetic algorithms and fuzzy decision-making, temporal data mining and analysis, along with genetic algorithms, yield a better error rate and improved accuracy for varying node numbers in evaluating the network status. Applying p-values to confirm the validity of such results in this study strengthens TDMC-HGA-FLDE as a premier model for classification problems, owing to the exploration of feature extraction.

Comprehensive performance evaluation

We benchmarked the TDMC-HGA-FLDE model against a set of algorithms to evaluate performance, including LSTM¹⁶, SVR¹⁷, EDGRU¹⁸, KS¹⁹, TDMC baseline²⁹CNNs⁵³Transformer-based models⁵⁴PSO⁵⁵Variational Autoencoders (VAEs)⁵⁶Neural Ordinary Differential Equations (Neural ODEs)⁵⁷and Quantum-inspired Optimization Algorithm (QOA)⁵⁸. Utilizing these algorithms provided a robust evaluation of the proposed model’s advancements, including state-of-the-art sequence and generative modeling, continuous time dynamical systems, advanced optimization, and other cutting-edge techniques. Various performance assessment metrics were utilized, including Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), accuracy, F1-score, and convergence rate to measure the prediction’s accuracy, the classification’s reliability, and the optimization’s efficiency. In this sub-section, the results are advanced in the context of dance motion reconstruction and outlined in Table 4.

Table 4.

Comprehensive performance metrics at 60 nodes.

Model	RMSE	MSE	Accuracy	MAE	F1-score
LSTM	0.63	0.40	0.90	0.50	0.88
EDGRU	0.59	0.35	0.91	0.45	0.89
SVR	0.55	0.30	0.92	0.40	0.90
TDMC	0.53	0.28	0.89	0.38	0.87
KS	0.62	0.38	0.87	0.48	0.85
CNN	0.57	0.32	0.92	0.42	0.90
PSO	0.60	0.36	0.91	0.46	0.89
Transformer	0.50	0.25	0.94	0.35	0.92
Neural ODE	0.52	0.27	0.93	0.37	0.91
VAE	0.57	0.33	0.91	0.43	0.89
QOA	0.55	0.30	0.92	0.40	0.90
TDMC-HGA-FLDE	0.45	0.20	0.95	0.30	0.93 59

Table 4 displays performance metrics for each model using 60 nodes, the configuration that achieves the best accuracy for TDMC-HGA-FLDE. The proposed model yields the best results in terms of MSE (0.20), RMSE (0.45), MAE (0.30), Accuracy (0.95), and F1-score (0.93). These attributes demonstrate the exceptional accuracy with which TDMC-HGA-FLDE predicts the position of motion and sequence, which relies on intricate dance classifications and tends to optimize. The values provided represent outstanding achievement for TDMC-HGA-FLDE in motion control areas and autonomous robot optimization, along with low MSE, RMSE, and MAE, which bounds the forces of estimating motion prediction essential for the complicated reconstruction of non-linear motion paths. High values of Accuracy and F1-score indicate a strong classification maneuver, which is an advantage in devices designed to provide automatic identification of elements and patterns, as they rely on accurately detecting motion patterns, such as in dance therapy. The low convergence rate value indicates the outstanding performance of the hybrid GA with fuzzy logic differential evolution (HGA-FLDE) since it uses fuzzy logic systems to set parameters and maintain bounds to exploration and exploitation during the optimization phase.

In the benchmarks, the performance of the Transformer model is outstanding with an MSE of 0.25, RMSE of 0.50, MAE of 0.35, Accuracy of 0.94, and F1-score of 0.92. Its attention mechanism is unmatched in capturing long-range dependencies, though its exorbitant computational requirements and sluggish convergence make it ill-suited for real-time applications in motion capture. CNNs yield MSE 0.32, RMSE 0.57, MAE 0.42, Accuracy 0.92, and F1-score 0.90. While efficient for spatial feature extraction, CNNs’ temporal dependence capabilities, unlike TDMC-HGA-FLDE, result in comparatively greater errors. PSO registers MSE 0.36, RMSE 0.60, MAE 0.46, Accuracy 0.91, and F1-score 0.89, limited by premature convergence of the upper bounding in high-dimensional motion data.

The prior assessed algorithms, CNNs⁵³Transformer-based models⁵⁴PSO⁵⁵VAEs⁵⁶Neural ODEs⁵⁷and QOA⁵⁸add to the contextual TDMC-HGA-FLDE’s performance. For motion modeling, VAEs generate the following results: MSE: 0.33, RMSE: 0.57, MAE: 0.43, Accuracy: 0.91, and F1 Score: 0.89. Their performance in latent representation capture suffers as they strive to handle high-dimensional, temporally sequenced data, due to unbounded error rates. Neural ODEs⁵⁷which models time continuously, records the following MSE: 0.27, RMSE: 0.52, MAE: 0.37, Accuracy: 0.93, and F1 Score: 0.91. They are incredibly skilled at temporal representation; however, TDMC-HGA-FLDE outpaces them significantly due to complexities in computer time, sensitivity at the start, and slower convergence rate. A quantum-inspired optimization approach, QOA⁵⁸achieves an MSE of 0.30, RMSE of 0.55, MAE of 0.40, Accuracy: 0.92, and F1 Score: 0.90. Noise associated with motion data limits QOA’s global optimization capability, which TDMC-HGA-FLDE solves using fuzzy logic combined with Riemannian geometry.

The baseline TDMC model achieves an MSE of 0.28, RMSE of 0.53, MAE of 0.38, Accuracy of 0.89, and F1-score of 0.87, outperforming EDGRU, LSTM, SVR, and KS. However, it is outperformed by TDMC-HGA-FLDE, which improves noise intolerance and adaptability with advanced optimization features. EDGRU, LSTM, SVR, and KS have poorer classification with increased error rates. MSE values for these models, EDGRU, LSTM, SVR, and KS, range from 0.30 (SVR) to 0.40 (LSTM). Accuracy values ranged from 0.87 (KS) to 0.92 (SVR), illustrating that SVR outperformed KS.

The Wilcoxon signed-rank test p-values in Tables 2 and 3 validate the statistical significance of TDMC-HGA-FLDE’s enhancements. These findings are graphically supported by Figs. 6 and 7, which show that TDMC-HGA-FLDE’s error rates and accuracy surpass those of the other models across various node configurations. The Evaluation also emphasizes the comprehensive capabilities of TDMC-HGA-FLDE, surpassing AI and conventional techniques by intricately integrating two-dimensional matrix computation, fuzzy logic, and hybrid optimization. It leaps over data gaps, incomplete information, and requires over-reasoning through non-linear motion patterns.

Such results highlight the effectiveness of cutting-edge technologies with flexible designs, such as TDMC-HGA-FLDE, in applications like augmented reality or specialized sports analysis, where precision is crucial. Positioning the model on a superior tier to modern technology, such as the Transformer, Neural ODE, and the VAE, establishes a new milestone for benchmarks in motion reconstruction.

Practical application in dance therapy for rehabilitation

To validate the TDMC-HGA-FLDE model with a real-world example, we implemented it in the medical field for dance therapy to aid physical rehabilitation. Dance therapy is gaining recognition as an effective treatment for patients recovering from musculoskeletal injuries, neurological disorders, and post-surgical conditions because it enhances motor function, promotes coordination, and improves psychological well-being. Nevertheless, capturing and reconstructing intricate dance movements during therapy poses significant challenges due to the patient’s restricted mobility, non-linear motion trajectories, and data loss resulting from clothing and sensor positioning issues with wearable sensors. The model’s implementation enables therapists to evaluate patient progress by reconstructing and monitoring realistic and accurate dance sequences. This allows for precise adjustments in tailoring rehabilitation programs.

In this use case, we partnered with a rehabilitation center to collect motion data from 10 patients participating in dance therapy sessions for lower limb rehabilitation following knee surgery. The dataset, comprising wearable inertial measurement units (IMUs) and the SMPL dataset [https://smpl.is.tue.mpg.de/], encompassed sequences of simple dance movements performed and simulated at varying intensity levels. The TDMC-HGA-FLDE model was utilized to reconstruct these movements and fill in the gaps from incomplete joint data and nonlinear motion data caused by the patients’ restricted range of motion. The patients’ movements were reconstructed in real time to create a virtual environment that therapists could access to visualize the patient’s movements and provide more tailored feedback and adjustments.

Dataset diversity and preprocessing

The datasets used in this work are collected from two primary sources: (1) a SMPL-based motion dataset with more than 2,000 annotated motion sequences of 50 + dance styles and body types, and (2) a real-world rehabilitation dataset of 10 subjects with post-surgical lower limb disabilities. Further variation was introduced through changes in motion amplitude, reach, body shape, and purposeful occlusions from clothing or misaligned sensors.

This variation was helpful for an empirical stress test of the model’s resilience. Unlike curated datasets containing uniform sampling, the rehabilitation set featured underlying chaotic movements, partial view obstructions, and sensor artifacts, simulating realistic deployment conditions. Preprocessing involved normalization, alignment, and temporal smoothing using a moving average filter with a period of five. Importantly, no artificial data imputation was made, thus allowing models to illustrate gap-filling and modular adaptation capabilities truly.

In Fig. 8, we demonstrate the application of TDMC-HG-AFLDE for reconstructing step-touch dancing movements in a therapy session. The figure compares the ground truth (obtained through high-precision motion capture systems), raw IMU data with missing joints, and the TDMC-HG-AFLDE reconstructed sequence. The model recovers motion trajectories and maintains accuracy in realistic frame motion within the middle of data gaps, illustrating strength in a clinical environment.

Fig. 8.

Application of TDMC-HGA-FLDE in dance therapy for rehabilitation.

The TDMC-HGA-FLDE model performance in reconstructing dance therapy motion sequences is given in Fig. 8. It consists of three subplots: (a) a ground truth 3D human body doll performing a step-touch dance movement, containing of a precision motion capture; (b) raw IMU data containing a right knee, ankle and wrist joints missing, exhibiting gaps in the corresponding meshes; and (c) the TDMC-HGA-FLDE reconstructed doll, which smoothly restored all joints to motion as was observed in the ground truth. The truth (a) shows an SMPL-like realistic doll with skin color (RGB: [1, 0.8, 0.7]) and 25 vertices and 14 faces. The doll was animated for 20 frames to perform leg lateral motion with an amplitude of 0.3 m, arm swings with an amplitude of 0.2 m, and torso sway with an amplitude of 0.05 m. In subplot (b), the IMU data reveals noticeable data loss corresponding to the visually highlighted RGB (1, 0.6, 0.6), containing the right leg and arm with missing elbow and shoulder. Parts of the body look across the gap. The reconstructed subplot c (RGB: (0.7, 1, 0.7)) demonstrates the model’s ability to reconstruct with a precision value of 0.94 (Table 5), assuming a noise level of 0.015 m and minor deviations along the computed smooth trajectories formed by a 5-point moving average filter.

Table 5.

Performance of TDMC-HGA-FLDE in dance therapy application (60 Nodes).

Model	MSE	RMSE	MAE	Accuracy
LSTM	0.41	0.64	0.52	0.89
EDGRU	0.36	0.60	0.47	0.90
SVR	0.32	0.57	0.43	0.91
KS	0.39	0.62	0.50	0.86
TDMC	0.30	0.55	0.40	0.88
TDMC-HGA-FLDE	0.22	0.47	0.33	0.94

In Table 5, the performance metrics of TDMC-HGA-FLDE in the context of dance therapy are measured against those of baseline models (LSTM, EDGRU, SVR, KS, and TDMC) using MSE, RMSE, MAE, and accuracy.

With an accuracy of 0.94 and an MSE of 0.22, the TDMC-HGA-FLDE model outperformed LSTM (0.89, 0.41), EDGRU (0.90, 0.36), SVR (0.91, 0.32), KS (0.86, 0.39), and TDMC (0.88, 0.30). These results underscore the algorithm’s effectiveness in handling incomplete and noisy data, which is frequently encountered in medical motion capture workflows.

This underscores the practical utility of TDMC-HGA-FLDE in aiding therapists in designing comprehensive rehabilitation regimens through precise motion reconstruction during dance-therapy sessions. The model’s precision and minimal error rates make it ideal for tracking patient improvement, and its ability to adapt to complex nonlinear motions makes it suitable for a wide range of patients. This methodology is not limited to dance therapy; it can also be adapted for medical fields such as gait or occupational therapy and engineering disciplines, including human-robot interaction in assistive robotics.

Computational requirements and runtime performance

I conducted testing on the TDMC-HGA-FLDE model using a workstation equipped with an Intel Core i9-10900 K (10 cores at 3.7 GHz), 64 GB of DDR4 RAM, a 2 TB NVMe SSD, and an NVIDIA RTX 3080 with 10 GB of GDDR6X VRAM. The software includes Ubuntu 20.04 with Python 3.8 installed alongside Keras 2.12.0, Scikit-learn 0.24, and NumPy 1.24.3.

With this setup, training the TDMC-HGA-FLDE model fully with 50 agents and 100 epochs took slightly more than 3 h and 12 min. The model reached a peak memory consumption of 4.2 GB of RAM. For inference, the model could operate at 40 ms per frame, proving to be performant enough for near real-time applications such as virtual dance coaching or providing rehabilitation feedback.

It is essential to note that as population size and sequence length increase, while impacting runtime, they do not degrade inference speed linearly. GPU acceleration is necessary for classes such as TDMC and HGA-FLDE, particularly for matrix computations and fitness evaluation at the population level. Model compression, as well as hybrid edge-cloud deployment, warrant further exploration for lower-resourced environments.

Discussion

The experimental results demonstrated the superiority of the TDMC-HGA-FLDE model over several approaches, including LSTM, EDGRU, SVR, KS, and even the baseline TDMC method. The proposed model performed accurate diagnostics, achieving an accuracy rate of 0.95, and had incredibly low error rates of 0.39 across the motion capture scenarios. These results demonstrate the effectiveness of the hybrid framework in combining fuzzy logic, GA, and DE to solve complex and challenging nonlinear motion reconstruction tasks.

Comparison with previous research

The TDMC-HGA-FLDE model described in this paper represents a novel approach to dividing and conquering dance motion reconstruction, advancing the work done in motion capture and optimization algorithms. To provide perspective, we explain the model’s results in the context of its comparison with the most relevant benchmark studies addressing similar problems related to complex movements, especially dances.

Peng et al.⁵ attempted to reconstruct the dance motion with a motion capture system and averaged approximately 0.85 accuracy with a 0.50 error rate for short-term motion sequences. Their results were obtained from the motion capture setup, which had issues with overlapping and missing data for the joints, also known as nonlinear motion sequences. In comparison, TDMC-HGA-FLDE achieves a maximal accuracy of 0.95 at 60 nodes and a minimum error of 0.39 at 20 nodes, demonstrating its superior ability to handle nonlinearity due to the Riemannian geometry and hybrid optimization. The contribution is most useful when data is sparse, as TDMC-HGA-FLDE adaptive parameter tuning via fuzzy logic greatly increases robustness.

Zhang and Zhang’s study on human motion recovery proposed an MM-TDMC²⁹which yielded an accuracy of 0.87 and an error rate of 0.45. The model performed well in capturing short-term motion; however, it struggled to optimize complex long-range dance motions adaptively. The TDMC-HGA-FLDE model outperformed MM-TDMC due to the implementation of HGA-FLDE, which dynamically adapts parameters to achieve a balance between exploration and exploitation. This increased accuracy by 0.08 and reduced the error rate by 0.06.

Liu and Ko²² applied deep learning for dance motion generation, achieving 0.90 accuracy and 0.42 error rate. They used a CNN, which excelled at learning spatial features but faltered on temporal dependencies for nonlinear sequences. With the integration of LSTM comparisons, hybrid optimization, and enhanced autonomous parameters, the TDMC-HG-FLDE model surpassed its model, achieving 0.95 accuracy and a 0.39 error rate, particularly in complex movements that require fast execution. Riemannian geometry within TDMC-HG-FLDE enhances the model’s capability for temporal dependency modeling, surpassing the limitations of CNNs.

Challapalli and Devarakonda¹⁷ reported an accuracy of 0.88 and an error rate of 0.47 using a hybrid particle swarm and grey wolf optimization algorithm for dance motion classification. While their method was helpful in classification, it was ineffective for reconstructing continuous motion sequences. TDMC-HGA-FLDE was developed for reconstruction and achieves greater accuracy (0.95) and a lower error rate (0.39) in performance by integrating matrix optimization with advanced two-dimensional calculations, making it more effective for applications such as dance therapy and virtual reality that require precise motion reconstruction.

The use of fuzzy logic within the framework of TDMC-HGA-FLDE builds on works that show the positive implications of adaptive parameter tuning. Examples include product line design processes performed with FSTDE by Tsafarakis et al., which are associated with a stable convergence but poor adaptability to highly non-linear datasets. Additionally, Nobile et al. noted the use of Fuzzy Self-Tuning PSO for solving global optimization problems and its reduced effectiveness in high-dimensional spaces. TDMC-HGA-FLDE overcomes these gaps with GA, which performs global searches, DE that refines locally, and fuzzy logic that adjusts real-time parameters, enabling the system to capture nonlinear dance motion data robustly. The hybrid approach yielded statistically significant improvements over the baseline models, as validated by the Wilcoxon signed-rank test, with p-value results of 0.001 against SVR and 0.011 against LSTM.

Table 6 summarizes the quantitative results of TDMC-HGA-FLDE’s performance compared to other studies. The consistent outperformance across all metrics—accuracy, error rate, complex nonlinear motion patterns coupled with missing joints—showcases the model’s improvements. The results fulfill the identified gaps in the research, like providing strong solutions for nonlinear motion data extraction found in references^27,28 and positioning TDMC-HGA-FLDE as the most advanced option for motion capture and reconstruction tasks.

Table 6.

Comparison of TDMC-HGA-FLDE with previous studies.

Study	Methodology	Error Rate	Accuracy	Key Limitations
Liu and Ko22	Deep learning (CNN-based)	0.42	0.90	Challenges with temporal dependencies in nonlinear sequences
Peng et al.5	Motion capture system	0.50	0.85	Limited handling of nonlinear motion and missing joint data
Challapalli and Devarakonda17	Hybrid PSO and grey wolf optimization	0.47	0.88	Suited for classification, less effective for continuous motion reconstruction
Zhang and Zhang29	MM-TDMC (Machine learning-based)	0.45	0.87	Ineffective for long-range, complex dance sequences
This Study	TDMC with HGA-FLDE and Riemannian geometry	0.39	0.95	Scalability issues and sensitivity to noisy datasets (addressed in future work)

Expected findings

The findings of this research primarily aligned with our expectations. Combining fuzzy logic with genetic algorithms and differential evolution was intended to significantly improve accuracy and reduce errors. Achieving these results was possible because of the ability of fuzzy logic to tune parameters adaptively during the optimization process. Nonetheless, the degree to which the model outperformed established techniques, such as SVR and KS, was unexpected, particularly in datasets with highly nonlinear motion characteristics. Such facts highlight the advantages of hybrid frameworks in addressing the inherent problems associated with complex motion capture tasks.

Limitations and implications

While the proposed HGA-FLDE demonstrates significant improvements in dance motion reconstruction, it has certain limitations that should be acknowledged:

• The input by users must be of excellent quality, which is a big problem for datasets that may have noise or be incomplete. This is an issue that most motion capture methods face. Still, it is more critical when considering broad applications because high data quality will not be present in many situations.

• Furthermore, due to the high computational requirements, the hybrid optimization method may not be scalable in constrained resources or real-time systems. Future work should focus on developing low-cost algorithms or preprocessing methods to enhance the model’s resilience and functionality.

• In our study, we empirically tested and opted for a population size of 50. Nevertheless, other works using fuzzy self-tuning methods indicate that size should be adjusted according to the specific problem complexity. The optimization efficiency could be improved even further with a more adaptive population size selection mechanism. Future work should focus on defining adaptive mechanisms that increase the population size based on the characteristics of the motion data.

• In the case of our model, we apply fuzzy logic for parameter modulation to DE. In contrast, the GA applies a constant crossover rate at CR_GA = 0.8. Maintaining these parameters may hinder GA’s flexibility. The ability to dynamically adjust the parameters of the GA by employing fuzzy logic to control the crossover and mutation rate would increase the performance. Subsequent phases of this study can focus on applying fuzzy self-tuning strategies to the GA to enhance its global search performance.

Practical contributions

The practical effects and outcomes of this research are significant. The robustness of the TDMC-HGA-FLDE model can be harnessed in various fields, including dance therapy, virtual reality, sports, and human interaction with machines. The model is beneficial for capturing and reconstructing human motion due to its high precision and low error rates in complex human movements. For example, the accurate reconstruction of movements in dance therapy can significantly aid in designing evaluation and rehabilitation programs. The virtually real-world interactions further benefit from the model’s ability to capture fluid and life-like movements. The simplicity of the movements required as input and the complexity of the output highlight the model’s contribution to solving the motion capture and reconstruction issue.

Conclusion and future research directions

To conclude, the TDMC-HGA-FLDE model represents a significant breakthrough in the development of motion capture technology, with valuable implications for both theory and practice. The results obtained from this study support the success of hybrid optimization methods in addressing non-linear and other complex motion designs. For future work, primary efforts should focus on enhancing the model’s ability to tolerate noisy information and incompleteness, reducing algorithmic complexity, and incorporating real-time processing capabilities. Furthermore, it may also be beneficial to expand the scope of the TDMC-HGA-FLDE model to other areas, such as robotics, physical therapy, and interactive computer games, for further usability testing and impact assessment.

To summarize, the enhanced performance of TDMC-HGA-FLDE over existing methods can be attributed to several key factors. First, handling nonlinear trajectories and complex joint interactions using Riemannian geometry is significantly more efficient for reconstructing human dance movements than Euclidean-based methods, which are less accurate. Riemannian metrics provide a more precise reconstruction of motion data, as they preserve the manifold structure, unlike Euclidean approaches.

The HGA-FLDE optimization algorithm’s execution is based on the hybrid approach that integrates differential evolution with genetic algorithms. It enhances local area refinement while effectively searching on a broader scope. An increase in search diversity and prevention of premature convergence associated with lower diversity enhances effectiveness against local minima. These features stand in contrast to traditional techniques such as SVR or LSTM, which tend to get stuck in local minima —a common problem.

Fuzzy logic’s parameter-tunable logic compensates for another model’s lack of flexibility with tunable parameters. It adjusts the scaling factor (F) and crossover rate (CR) based on population diversity and iteration, thereby achieving stability in convergence and enhancing the generalization of motion patterns, particularly with noisy inputs.

All these design innovations culminate in a system that upholds accuracy in the absence of some joint information, in cases of nonlinear movement, or with low sensor input. This is illustrated in Table 5, where MSE, RMSE, MAE, and accuracy show consistent improvement across various test cases, as well as in real-world settings such as dance therapy.

Conclusion

This research presented a new approach to reconstructing complicated and nonlinear dance movements using the TDMC-HGA-FLDE model. It resolved the incompleteness of data, motion parameter sensitivity, and intricate motion patterns. The model incorporation of the Two-Dimensional Matrix Calculation (TDMC) along with the Hybrid Genetic Algorithm and Fuzzy Logic Differential Evolution (HGA-FLDE) is further enhanced by Riemannian geometry. It yielded remarkable outcomes in motion capture tasks when integrated with TDMC. Quantitative assessments demonstrated that TDMC-HGA-FLDE outperformed all other methods, achieving the highest accuracy of 0.95 at 60 nodes. LSTM recorded 0.90, SVR 0.92, EDGRU 0.91, and Kinect Sensors attained 0.87. The model also achieved a minimum error rate of 0.39 at 20 nodes, which is lower than those of all other models: LSTM (0.82), SVR (0.60), EDGRU (0.65), and KS (0.48). Robust differences were indicated by the p-values, such as 0.001 for SVR and 0.011 for LSTM, confirming statistical significance through the Wilcoxon signed-rank test. The findings demonstrate the benchmark motion reconstruction capability of the TDMC model, which facilitated realistic sequence generation. The model is effective in dance therapy, virtual reality, sports analysis, and human-computer interaction due to its accuracy and low error rate. Even with those strengths, the study highlighted limitations. A hybrid of optimization strategies faced scalability issues and complications when handling noisy datasets and limited input data. Focusing on lightweight and noise-resistant algorithms, as well as real-time processing, seems promising for future research and enhances the model’s applicability.

Inquiry on adapting deep reinforcement learning with the TDMC-HGA-FLDE model is another future research direction that needs to be taken to make it compatible with various datasets. Other domains, including robotics, physical rehabilitation, and interactive gaming, can be tested and used to prove the model’s validity. Alternating optimization approaches can also enhance efficiency and reduce computational costs with minimal impact on performance. These breakthroughs would enhance the adoption and utilization of power in industries and academia.

The newly introduced TDMC-HGA-FLDE model was a well-balanced blend of theoretical novelty and practical applicability in motion capture and reconstruction. This model was tested on various challenging issues, and demonstrating its performance on multiple tasks enabled this study to provide a perspective on future solutions to the problem of motion capture and reconstruction of complex human actions, thereby promoting cutting-edge research and practical developments.

Author contributions

L.W. and Y.L. wrote the main manuscript text. Y.G. and M.K. prepared all figures. All authors reviewed the manuscript.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data availability

The datasets analyzed during the current study are available from the following link: https://smpl.is.tue.mpg.de/.

Declarations

Competing interests

The authors declare no competing interests.

Ethical approval and informed consent

This study did not involve any new experiments on human subjects. All data used were obtained from publicly available datasets (https://smpl.is.tue.mpg.de/), and as such, ethical approval and informed consent were not required.