Morphlight theory inspired by raptors: musculoskeletal modeling and muscle control in Falco peregrinus wing flapping

Falco peregrinus can achieve maneuverable flight by utilizing its morphing wing structure integrated with the musculoskeletal system. However, limited research has been conducted on computed muscle control of F. peregrinus wing musculoskeletal system. Consequently, the computed muscle control of a F. peregrinus was quantitatively analyzed to estimate the morphing wing mechanism.

The wing plays a critical role in maneuvering flight in birds; however, the anatomy structure of raptors is rarely reported. Fortunately, avian anatomy has been well investigated in species such as pigeons, which can serve as a good reference. Hieronymus (2016) used rock pigeon wings as samples for observation through standard dissection, micro-computed tomography, and tissue sectioning. They found that abundant smooth muscles play a role in efficiently maintaining folded wing posture, tuning wing shape, and regulating aeroelastic behavior during flight. Additionally, skeletal muscle tissue from the right wing of the rock pigeon was obtained. Meyers (1992, 1996) dissected the shoulder and forearm muscles of the American kestrel, analyzed muscle function in flight by studying the histomorphology, and mapped the location of each muscle in the shoulder and forearm. Tobalske and Biewener (2008) measured muscle length and activity in pigeons using sonomicrometry and electromyography techniques. Dial et al. (1991) used radiography and electromyography to study skeletal offsets and the activity of 11 shoulder muscles in purple-winged starlings. Bribiesca-Contreras and Sellers (2017) performed µCT scanning of sparrowhawk wings and proposed a three-dimensional (3D) model of the sparrowhawk wing musculoskeletal system. However, if the pulley and tendon sheath, composed of connective tissue, are severed during dissection, the pulling direction of the muscles will change, and the observed movement may become inaccurate (Vazquez, 1995). Moreover, conducting live flight experiments on raptors is challenging, and biomechanical data are difficult to measure experimentally, making further simulation studies necessary. Models and simulations can assess how changes in each input (e.g. anatomy, kinematics) independently influence function (Heers et al., 2018).

As many as 36 major pelvic limb muscle groups were incorporated into a 3D biomechanical computational model of the ostrich (Hutchinson et al., 2015). This model was used to investigate the function of the pelvic limb muscles in locomotion, verifying that the ostrich achieves movement through muscle-tendon optimization and static weight support. Subsequently, the location, length, excitation, force, etc. of each muscle are critical parameters for musculature modeling that must be determined in detail. Heers et al. (2018) modeled and simulated the bird musculoskeletal system using OPENSIM, controlled muscle parameters separately, and assessed their influence on movement by calculating muscle control, thereby predicting the functions of different muscles during movement.

In fundamental research on raptors, the mechanisms underlying wing deformation, and their flight capabilities have been observed, and these deformable wings are referred to as morphing wings (Tang et al., 2023, 2024). We integrate the shape-shifting ability and flight capabilities to propose an original scientific theory, Morphlight Theory, in which the F. peregrinus wing musculoskeletal system and muscle control are key components. F. peregrinus exhibits superior flight performance compared to other birds, maintaining high maneuverability even at elevated speeds. This maneuverability necessitates greater skeletal stability and a more advanced musculoskeletal structure; however, limited research and experiments have been conducted. Therefore, a detailed analysis of the wing musculoskeletal system and the corresponding computed muscle control is necessary to estimate the super-maneuvering mechanisms of raptors.

The axial, sagittal, and coronal sections of the bone were obtained through CT scanning (Fig. 2A-D). The mass of each bone in the wing skeleton was measured (Table S1). Smoothing algorithms were applied to reconstruct the bone with high precision, and cross-sections of the four main wing bones (humerus, ulna, radius, and metacarpus) were analyzed at the proximal, middle, and distal ends. As a result, a detailed 3D model of the F. peregrinus wing skeleton was obtained (Fig. 2E).

The six selected muscles were dissected to gather key information about their origin, insertion and attachment points, and muscle length (Table 1). These data are derived from our experimental measurements. Proximal refers to the side of the bone closest to the body, while distal refers to the side farthest from the body.

Based on the muscle location data from the CT scan and anatomy information, as well as ventral and dorsal views of the muscle tissues, bionic muscles were mounted on the 3D-printed skeletal model using Play-Doh colored clay (Fig. 2F,G). 1.75 mm white T-PLA resin was used as the 3D printing material. Blue clay substitutes for M. subscapularis, and brown clay substitutes for M. scapulotriceps, both originating from the scapula. M. subscapularis terminates at the proximal end of the humerus, and M. scapulotriceps terminates at the proximal end of the radius. Light green clay represents M. extensor digitorum communis, dark green clay represents M. flexor carpi ulnaris, and purple clay represents M. extensor metacarpi radialis, all originating from the distal end of the humerus to the proximal end of the metacarpal. M. extensor carpi ulnaris is represented by pink clay, extending from the radius to the metacarpal.

The residuals for each case were computed using the optimization algorithm (Table S2). The values of these residuals are derived from calculations. Where S represents a spherical hinge, U represents a universal joint, and R represents a revolute joint. The lower the residuals, the higher the computational accuracy achieved. The residuals of the spherical-spherical joint reached their minimum value due to its high degrees of freedom. As a balance between bionic accuracy and complexity, the revolute-universal joint was selected for the flapping study, and the kinematic vice was modeled as a two-degree-of-freedom joint with a residual of 0.5932 mm. The optimal computational results were selected as follows: the rotational center of the shoulder joint was optimized to −11.2, 0.1, 0.1, and the rotational center of the wrist joint was optimized to −17.3, 81.4, 39.2.

The joint rotation angles of F. peregrinus flapping wing motion were accurately measured and digitized (Fig. 3B,C). The joint rotation angles of F. peregrinus flapping wing motion are derived from experimental measurements. The flapping motion of the shoulder joint is denoted as xl_shoulder, while the wrist joint flapping motion is denoted as xl_wrist. Additionally, the extension and flexion movements of the wrist joint are denoted as zl_shoulder. In this study, the motion of the wing joints is represented using a Fourier series, and the flapping angle and its associated deviation band within a complete flapping cycle are obtained (Fig. 3B). Previous studies, such as Liu et al. (2006), have demonstrated the effectiveness of using Fourier series to approximate the motion of wing joints. This method is particularly useful for modeling periodic motions where precise, continuous data from live specimens may not always be available. Bird wing joints may not exhibit perfectly smooth, repetitive motion in all situations. However, the Fourier series provides a reasonable approximation given the available data. The flexibility of the Fourier series allows it to model the periodic nature of wing flapping while accounting for the variability in the motion pattern. This approach strikes a balance between model simplicity and the need for accuracy, making it a useful tool for simulating wing joint dynamics in the absence of more complex motion data.

During the 0-50% of the cycle, the shoulder joint exhibits a downward flapping motion. Conversely, the shoulder joint moves upward during the terminal 50-100% of the cycle. This flapping motion has an angular amplitude of 64°. On the other hand, the wrist joint flaps downward during 0-47% and 84-100% of the cycle, and flaps upward during 47-84% of the cycle. Meanwhile, a 75° rotation was also found for zl_wrist, indicating that the outer wing was not only flapped but also folded simultaneously during a flap cycle. The outer wing folds during the 34-84% of the cycle. This biomechanical observation can be further used to analyze the physiological characteristics of F. peregrinus.

The angle of flapping wing motion was converted into OPENSIM motion files, which were then used to drive the musculoskeletal model to flap according to the set angle data. This not only replicated the wing gestures of the F. peregrinus but also captured dynamic flapping motions of the wing (Fig. 3D). Cycle at 0%, 25%, 50%, and 75% were selected as critical frames to dynamically analyze the movement process of F. peregrinus wings during a flapping cycle. Both head and abdominal views were presented to illustrate the musculoskeletal gestures.

To validate the established musculoskeletal model, we compared the root mean square (RMS) of F. peregrinus’ muscle activities, calculated via the computed muscle control method, against experimental values from the American kestrel (Fig. 4). Meyers (1993) studied the root mean square of electromyographic activity in the shoulder muscles of the American kestrel (Falco sparverius), specifically M. supracoracoideus and M. triceps brachii, during wind tunnel flights. F. peregrinus and F. sparverius, both members of the genus Falco, share a close evolutionary relationship. Anatomical investigations have found significant similarities in the muscle physiology of both species. Therefore, the primary functions of M. supracoracoideus and M. triceps brachii during flapping motion can be considered analogous between the two species, providing a robust basis for comparing their electromyographic signals. For quantitative analysis of muscle functions, the root mean square of the two muscle activities computed in this study was compared with the root mean square of electromyographic activity of the two muscles (Meyers, 1993). Muscle activity data were obtained from OPENSIM 4.3 simulations, whereas electromyographic activity data were sourced from Meyers (1993). To quantitatively assess agreement, we normalized both the simulated root mean square values of F. peregrinus’ muscle activities (16 and 37) and the electromyographic root mean square values (0.1628 and 0.3478), subsequently calculating key statistical metrics. Mean absolute error (MAE), percentage difference (PD), and proportional error (PE) metrics were employed to evaluate model accuracy. The results were: MAE=0.02195, PD=9.66%, and PE=7.62%. These results indicate that both the computed myoelectric signals and the measured electromyographic activity accurately represent muscle functions. Additionally, the OPENSIM musculoskeletal model of the F. peregrinus wing was established, and its accuracy was verified.

To elucidate the intricacies of muscular function in flapping wing dynamics, a comprehensive analysis of muscle-tendon length variations was conducted across a range of flapping amplitudes. The experimental design ensured a constant temporal threshold to achieve peak amplitude at each angular increment during the wing flapping cycle. The flapping amplitudes 0.4, 0.6, 0.8, 1.0, and 1.2 were selected to investigate muscle functions. Notably, the 1.2-fold amplification approximated the upper limit of the angular range achievable by F. peregrinus during flapping. Consequently, this amplitude was selected as the upper boundary for our investigation. The resultant data, which encapsulate the temporal progression of two distinct muscles, were presented graphically (Fig. 5). This figure illustrates the intricate relationship between muscle-tendon length and wing flapping kinematics across the varied amplitudes. The muscle-tendon length values were derived from OPENSIM 4.3 simulation results.

To enable clearer observation of muscular alterations at different temporal intervals, the aforementioned muscles were analyzed through both static and dynamic dissection during the entire wing flapping cycle (Fig. 6). This approach allows for a thorough assessment of muscle behavior in distinct phases of wing motion, thereby improving our understanding of their functional dynamics.

The wrist joint plays a crucial role in wingbeat, however, limited observations of muscle functions around the wrist joint were found, as they were too small to measure directly in experimental. As an alternative, the computed muscle control results were used to evaluate their performance. Similarly, muscle-tendon length (Fig. 7), static anatomy, and dynamic anatomy (Fig. 8) were further analyzed for the shoulder muscle. The muscle-tendon length values were derived from OPENSIM 4.3 simulation results.

A more intuitive visualization of the muscles and their associated morphological changes during flapping motion has been developed.

To assess the robustness and reliability of our musculoskeletal model, we conducted a sensitivity analysis focusing on key biomechanical parameters, notably the optimal fiber length and tendon slack length. We varied these parameters within physiologically realistic ranges to observe their impact on model predictions. Each parameter was altered by ±20%, ±40%, and ±60% from its baseline value, and simulations were rerun to evaluate changes in muscle-tendon length. The muscle-tendon length remained consistent across parameter variations of ±20%, ±40%, and ±60%. When the optimal fiber length was reduced by 60%, the change in muscle-tendon length was less than 0.1‰, whereas a 60% reduction in tendon slack length resulted in a change of less than 0.1‰. This is because muscle-tendon length is influenced by muscle path, skeletal geometry, and joint motion (Delp et al., 1990; Dvinskikh et al., 2010). These results suggest that our model is stable within reasonable ranges of parameter uncertainty.

In addition to the sensitivity analysis, we further explored how muscle-tendon length varied within a single flapping cycle across different motion amplitudes of the same movement type. This analysis aimed to assess the sensitivity of our model to changes in motion amplitude, a critical factor in dynamic biomechanical simulations. The results showed that muscle-tendon length exhibited consistently similar trends across different motion amplitudes, and the muscle contraction period was highly coincident with each other (Table 2), confirming consistency in our model's predictions. Due to the unique geometry of the skeletal structure and the specific muscle paths of M. scapulotriceps and M. extensor carpi ulnaris, these two muscles exhibited less change in their activation periods when the motion amplitudes was altered. This consistency in muscle-tendon length response under varying conditions. Above all, the musculoskeletal model's reliability and parameter sensitivities were irrefutable verified.

The M. subscapularis is positioned cranially to the scapulohumeralis caudalis and spans from the scapula to the humerus, caudal to the shoulder joint (Fig. 6). This anatomical configuration is crucial for its function. Intriguingly, the muscle-tendon length variation of the M. subscapularis during wing flapping exhibited distinct patterns, possibly due to the M. subscapularis starting to bypass the wrapper object. Activation of the M. subscapularis occurs during the shoulder flapping phase, suggesting its significant role in shoulder flapping during wing motion. This function aligns with the description of the M. subscapularis in American kestrel by Meyers (1992), who utilized the retractor of the humerus to elucidate its role.

The M. scapulotriceps, a muscle extending from the scapula to the ulna, caudal to the shoulder joint (Fig. 6), exhibited activation during the shoulder overthrow phase of movement. This temporal correlation with the shoulder's kinematic profile suggests a mechanistic role for the M. scapulotriceps in shoulder overthrow, consistent with Biewener's (2011) analysis of M. scapulotriceps of pigeons.

The M. extensor digitorum communis extended from the distal humerus to the alula and major digit. Its apex attaches to a process on the lateral epicondyle of the distal humerus, between the origins of the M. extensor carpi radialis and the M. extensor carpi ulnaris. The tendon emerged from the sulcus at the distal end of the carpometacarpus, passed through a tendon loop, and made a 90-degree turn ventrally between the carpometacarpus and the proximal phalanx of the major digit. It then passed deep to the sesamoid in the tendon of the M. extensor longus digiti majoris. The M. extensor digitorum communis inserted onto the proximal cranio-ventral corner of the proximal phalanx of the major digit (Meyers, 1992) (Fig. 8). This muscle is activated during the wrist's downward and extension phases, suggesting that the M. extensor digitorum communis plays a significantly role in the wrist's downward and extension during the flapping cycle. This observation is consistent with the findings of Bribiesca-Contreras et al. (2021), who identified the role of this muscle in wrist flexion in aquatic birds.

The M. flexor carpi ulnaris originates from the medial epicondyle of the humerus and extends distally toward the wrist (Fig. 8). During the contraction phase of wrist joint movement, the M. flexor carpi ulnaris shows prominent activation. This suggests that the M. flexor carpi ulnaris plays a key role in mediating wrist flexion. The temporal alignment of the M. flexor carpi ulnaris activation with the wrist joint's contractile phase indicates a direct mechanistic contribution to the flexion movement.

The musculotendinous unit of the M. extensor metacarpi radialis, a prominent antebrachial muscle in the genus F. peregrinus, is located along the cranial aspect of the forearm (Fig. 8). Notably, a decrease in the length of this musculotendinous unit occurs during the initial 0-34% and terminal 84-100% phases of the wing flapping cycle, indicative its active involvement during these phases. This suggests a key role for the M. extensor metacarpi radialis in wrist joint extension during wing flapping. The analysis of the function of M. extensor metacarpi radialis and M. flexor carpi ulnaris in wing flapping is consistent with the findings of Vazquez (1994) in pigeons.

The M. extensor carpi ulnaris, a muscle with parallel fibers, is located on the caudal border of the dorsal antebrachium. It lies deep to the M. extensor digitorum communis and spans from the distal radius to the proximal metacarpus (Fig. 8). Given its anatomical position and activation pattern, it is hypothesized that the M. extensor carpi ulnaris plays a crucial role in wrist flapping and extension during the wing flapping.

F. peregrinus is classified as a second-class protected animal in China, where any sale, purchase, or use of this wild species is strictly prohibited. Future research, under proper authorization, could involve a larger sample size to enhance the robustness and generalizability of the findings. Additionally, we believe the data obtained from this specimen provide valuable insights into the musculoskeletal mechanics of F. peregrinus.

Due to the absence of direct EMG data for F. peregrinus, we compared our computed muscle activity with experimental data from the F. sparverius. We acknowledge differences between the two species in body size, wing morphology, flight dynamics, and muscle properties, all of which may influence muscle activation patterns. To account for these differences, we applied normalization techniques in our model. However, we acknowledge the inherent limitations of this cross-species comparison. Future research will aim to obtain species-specific EMG data for F. peregrinus to further validate and refine the model.

Given the complexity of the anatomical intricacies of raptors and the limitations of the modeling approach, it is important to acknowledge the simplifications made in the model. Specifically, during the modeling process, the scapula and coracoid were treated as rigid bodies, and the wings were simplified into a two-bar model, which has been shown to be reasonable in previous studies (Tang et al., 2023; Liu et al., 2006), although deviations remain when compared to real birds. Additionally, the joint mechanics used in this study are idealized and based on revolute joint models. In real birds, joint rotation may involve more complex interactions than those accounted for by a single-axis revolute joint. Especially raptors like the F. peregrinus, have highly specialized joint, muscle antagonism and bone structures that may not be fully captured by the foundational human joint models in OPENSIM. This study does not consider the nonlinearities in biological joints, such as cartilage deformation and ligament elasticity, which may influence muscle activity. These simplifications inevitably affect the accuracy of the simulations. Fortunately, Heers et al. (2018) did not incorporate the nonlinearities of joint mechanics in their study either. However, their simulation results were highly consistent with the muscle activity of real birds, leading us to hypothesize that these factors have minimal impact on muscle activity. Furthermore, this study reconstructed the flapping motion with the guidance of ornithologists to ensure accuracy. However, we did not perform control tests on alternative subjects, such as non-raptor species. One limitation of the flapping experiment is that the motion of the sample differs from its natural state; only the geometric pose data has reference value, and dynamic information such as acceleration and frequency cannot be reproduced through a static specimen. More valuable dynamic information needs to be collected in future studies involving live samples (e.g. pigeons, parrots) for a more comprehensive understanding. This will allow for the collection of reliable dynamic data such as acceleration and frequency. Therefore, the model in this study is limited to the flapping motion of F. peregrinus wing under idealized joint mechanics and does not fully represent the dynamic flight patterns exhibited by real birds, particularly under varying environmental conditions. Future work should incorporate more complex joint mechanics and anatomical differences to improve the accuracy of the simulations and better simulate the biomechanics of raptor flight.

F. peregrinus possesses exceptional maneuverability due to its unique musculoskeletal system. However, limited research has been conducted on its muscle function, leaving its flight capabilities largely unexplained. Therefore, we developed a numerical musculoskeletal model to investigate the underlying mechanisms of its flight. In this study, we present the following findings:

1. A wing musculoskeletal system was established, and a computed muscle control method was introduced to further analyze wing motion and muscle function during a flapping cycle.

2. The flapping angle of the F. peregrinus wing was measured to replicate the wing flapping motion, which is valuable for studies on bird flapping.

3. Muscle activity and muscle-tendon length were quantitatively assessed to analyze muscle function during a flapping cycle. The M. subscapularis and the M. scapulotriceps contribute to supra-shoulder flapping during wing flapping. The M. supracoracoideus plays a crucial role in the wing beating. The M. extensor digitorum communis contributes to wrist downward and extension during flapping, while the M. extensor metacarpi radialis facilitates wrist extension. M. extensor carpi ulnaris and M. flexor carpi ulnaris, functioning as antagonist muscles, play a significant role in wrist flexion.

We believe our observations provide both experimental and analytical methods for raptor flapping research and offer novel insights into the musculoskeletal system of the F. peregrinus. By understanding specific muscle functions during flapping, we can enhance our understanding of avian locomotion. This study has implications for the biomechanics of raptor wing flapping and muscle function.

The CT scanning experiment was conducted and discussed in our previous research (Tang et al., 2023). We performed CT scans of F. peregrinus using the GE Optima 540 16-slice CT scanner. Subsequently, the bone mass of each wing skeleton was measured, and the scanned DICOM data were processed using the 3D SLICER. Each bone, muscle, and feather were identified using density thresholds, then segmented, isolated, and reconstructed. The relatively small-sized bones of the F. peregrinus, with a wingspan of 0.8 m may contribute to this deviation. Subsequently, smoothing algorithms were applied to reconstruct the bones with high precision. Cross-sections of the four major wing bones (humerus, ulna, radius, and metacarpal) were examined at the proximal, middle, and distal ends. As a result, a refined 3D model of the F. peregrinus wing skeleton was obtained.

To study the wing flapping of a F. peregrinus, the wing skeleton model was simplified to a two-bar model with two joints. Therefore, the rotations of both the shoulder and the wrist joint were considered. Previous research (Liu et al., 2006) has shown that using these critical degrees of freedom to reconstruct a bird's flapping kinematics is reasonable. Thus, in the subsequent biomechanical study, only the muscles influencing the shoulder and wrist joints were considered. The muscles of the F. peregrinus wing were screened, and a total of six muscles around the shoulder and wrist joints were selected. The wing muscles of the F. peregrinus were dissected to investigate the functions of these muscles.

A muscle generates an active force up to a maximum value, depending on its fiber length and activity in the contractile element. In contrast, the generated passive force depends on the fiber length of the parallel element. Finally, the active force and passive force constitute the total muscle force. Therefore, the optimal fiber length, pennation angle, and tendon slack length should be well defined, as they are critical for computed muscle control simulation of F. peregrinus.

Optimal fiber length refers to the specific length at which a muscle fiber or a muscle-tendon unit operates with maximum mechanical efficiency (Heers et al., 2018). The pennation angle is the angle formed between the muscle fibers and the line of action of the muscle force. Muscle fibers attach to a tendon at a pennation angle, which scales the force they transmit to the tendon. To measure the optimal fiber length and pennation angle of the F. peregrinus wing, the muscles were photographed, and the pennation angle and length of all the muscle fibers were measured using ImageJ. For parallel fibers, the pennation angle was 0 degrees, and fiber length was measured at the midpoint of the muscle. For the pinnated fibers, measurements were taken at both ends of the tendon. The pennation angle was qualitatively consistent with the results of F. peregrinus forelimb muscles (Meyers, 1992, 1996).

Tendon slack length (L_ts) is defined as the length at which the tendons associated with a muscle begin to resist stretch and generate force. This parameter is crucial because it delineates the extent to which the force exerted by a muscle-tendon unit is actively produced by muscle contraction and passively by the tendon's series elastic component during stretch. The algorithm provided by Manal and Buchanan (2004) was used to calculate the tendon slack length for each muscle. The minimum and maximum lengths of both muscle and tendon (L_mt=muscle length L_m+tendon length L_t) were measured during joint motions, and the average pennation angle α of the muscle fibers along with the normalized fiber lengths were calculated.

The properties of the musculoskeletal system and its interactions with the environment are described by a set of ordinary differential equations, a forward dynamic simulation is performed using there equations. Feedforward and feedback control were introduced to guide the kinematic trajectory of dynamic model toward a set of measured experimental kinematics The computed muscle control algorithm consists of four calculation steps.

In the fourth stage, muscle excitation is introduced into the forward dynamic model, and the numerical integration approach is used to advance the state to the next time step. By looping through stage 1 to 4, muscle dynamic properties can be computed.

Path information, weight, and length were recorded to construct numerical muscles according to the CT experiment of F. peregrinus wing muscle from our previous research (Tang et al., 2023), as well as the anatomical data from Meyers (1992, 1996). Initially, data processing was performed to create the supporting objects required for musculoskeletal modeling, followed by the creation of OPENSIM objects and corresponding models (Fig. 1). Specifically, this process can be divided into the following nine parts:

1. Import bone surface model. CT scan data were used to generate 3D surface models of each bone for defining the model.

2. Create skeleton point cloud. Skeleton points at the joints define the reference coordinate system for the shoulder and wrist joints. In the global coordinate system, the shoulder joint is located at −11.2, 0.1, 0.1 and the wrist joint at −17.3, 81.4, 39.2.

3. Create joint reference coordinate system. Two reference coordinate systems are created for each joint, one for the parent and one for the child. These are aligned with the global coordinate system and do not undergo additional rotation.

4. Create muscle points. Muscle paths are defined by key points (origin, insertion, and intermediate attachment), forming the geometric shape of the muscle.

5. Create rigid bodies. The scapula, coracoid, humerus, ulna, radius, and metacarpus surfaces are modeled as rigid bodies. Input the corresponding bone mass, and the inertial characteristics are automatically calculated by the software and stored in the VME.

6. Define ground-relative joint. The shoulder joint is defined as fixed relative to the ground.

7. Create joints. The shoulder, elbow, and wrist joints are defined with appropriate parent-child body relationships and specified joint types (hinge, fixed, and universal).

8. Create wrap object. Wrap objects (e.g. spheres, ellipsoids, cylinders) are placed on the bone surfaces to prevent muscle crossing during movement.

9. Create muscles. Muscles are created based on their geometric paths, defined by the origin, insertion, and attachment points. OPENSIM muscle objects are generated from the data.

A motion optimization algorithm based on the skeletal point cloud was established (Hutchinson et al., 2015) and was subsequently used to analyze the degrees of freedom of the flapping wing skeleton. Subsequently, various bionic joints were used to represent the natural wing skeleton joints, replicating wing flapping motions. In the model, the coracoid and scapula were included to represent the body skeleton (fixed, global frame), and rotations of the humerus, ulna, radius, and metacarpal bones were calculated relative to the global frame. The wing postures, including extension, upward flap, and downward flap, were categorized as j=1∼3 in the quantitative analysis. The humerus, ulna, and metacarpal were denoted as i=1, 2 and 3, respectively. The radius was not considered in the current research. The shoulder joint, elbow joint, and wrist joint were denoted as k=1, 2, and 3. The proximal frame Ap(i) and distal local frame Ad(i) were also attached to each bone to describe its motions relative to adjacent bones.

Similarly, the motions (both axis and angular) of the ulna relative to the humerus and the rotations of the carpometacarpus relative to the ulna, were measured using a step-by-step point cloud fitting approach. Additionally, the motions of the humerus relative to the scapula were also measured using the step-by-step point cloud fitting approach to represent shoulder joint rotations. The average distance between different point clouds was calculated, and the optimization algorithm, utilizing the global optimization method Multi-Start in MATLAB software, was employed to measure the relative rotations during flapping.

The flapping motions of the wing skeleton can be computed using the optimization algorithm. Additionally, in our previous research (Tang et al., 2023), it was found that the scapula and coracoid can be treated as a rigid body to represent the entire F. peregrinus. Thus, the frame on the coracoid was treated as the global coordinate, and the flapping motions were represented by the relative rotations of each bone in the global coordinate system. Specifically, the shoulder joint rotation was calculated by the motion of the humerus relative to the scapula. The shoulder and wrist joints were modeled using three types of joints: spherical, universal, and revolute joint. The residuals for each case were then calculated using the optimization algorithm.

The Zhejiang Museum of Natural History provided a specimen of a F. peregrinus for this study. The F. peregrinus was 0.45 m in length, with a wingspan of 1.2 m and a weight of 0.96 kg. The body and wing structure of the F. peregrinus were complete, with no signs of trauma, and all joint movements were flexible. The flapping motions were artificially reconstructed under the professional guidance of ornithologists. Six angle sensors were attached to the wing surface of F. peregrinus specimens to document the details of their wing kinematics. Ornithologists approximated the flapping process by manually actuating the wing motions. A total of eight cycles of flapping motion were simulated. To ensure the continuity of the flapping motion, only the second to seventh cycles were selected for study. During the tests, three sensors were mounted on each F. peregrinus wing to capture the shoulder joint, elbow joint, and wrist joint rotations (Fig. 3A).

All data and figures in this study are original, except for those with special references.

Ethical review and approval were waived for this study due to the F. peregrinus in this research was the specimen, provided by Zhejiang Museum of Natural History.

The author D. T. took the lead in carrying out the research on super maneuvering flight of various large birds of prey and proposed the Morphlight theory inspired by raptors. The authors would like to thank the reviewers for their good and valuable suggestions, which greatly improved this paper, and we also thank the team of Zhongyong Fan for all their contributions on the F. peregrinus specimen. The authors gratefully acknowledge the technical support provided by the Medical Imaging Core Facility at XiXi Hospital, where CT imaging experiments were conducted using the GE Optima 540 16-slice CT scanner.