Study on the Dynamic Response Characteristics of Bionic Legs During Instantaneous Ground Contact

Study on the Dynamic Response Characteristics of Bionic Legs During Instantaneous Ground Contact | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Study on the Dynamic Response Characteristics of Bionic Legs During Instantaneous Ground Contact mengjun song, jinggong wei, Liping Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4894964/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Musculoskeletal system vibrations are initiated at paw-strike in animal’s high-speed running. The short ground contact moment suggests that there exists a transient dynamic response of the impact between the leg and the ground, which is a high nonlinear problem and not well understood. From the anatomical measurement data of a domestic cat, a musculoskeletal system model of the quadruped animal was constructed in this study. The changes of muscle forces and joint moments were computed based on a high-speed motion sequence. The elastic moduli were measured and calculated for different parts of the tibia by a nano-indentation technique. On the basis of the measured material parameters, the substructure technique for dynamics was employed to numerically solve the contact-impact behavior of bio-materials and bionic components. To record the contact-impact process, high-speed videos (more than 10,000 fps) were taken during the capture experiments. Results demonstrated that multiple impacts existed in the tibia and the PLA leg at the moment of contact-impact. The results from this paper further reveal that the multiple contact-impact phenomena are adapted to cats during running, which may provide a certain support for selecting bionic components and improving the performance of the bionic mechanism. Physical sciences/Engineering/Mechanical engineering Biological sciences/Zoology/Biomechanics Biomechanics Impact Bionic mechanism Tibia Transient dynamic response Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1. Introduction During the impact loading in activities such as walking and galloping, a contact-impact as a transient response process exists between the leg and the ground [ 1 , 2 ] . Typical time histories of the initial contact event during the impact are usually very short. The dynamic response wave produced by the impact force propagates through both skeletons and muscles [ 3 , 4 , 5 ] and must be attenuated by the musculoskeletal system during the impact landing [ 6 ] . In the musculoskeletal system, muscles are attached to tendons, which insert into the bones and cross different joints, transmitting muscle forces to the skeleton and protecting the bone from external invasions [ 7 , 8 ] . As a kind of soft tissues, a generic muscle (group) can be viewed as a set of fibers, which connects two tendons forming connection lines for mathematical models, to calculate the muscle forces according to the length change of the muscle [ 9 , 10 ] . Muscle forces can also be solved by other methods such as the experimental determination. The muscle forces can be transformed into joint moments multiplied by moment arm [ 11 ] . After being generated, the joint moments act on joints and other parts of the body, which then contact with the ground or objects [ 12 , 13 ] . During the initial ground contact, the joint contact force is usually larger than the body weight (BW) [ 14 ] , that produces high impacts on the joints and causes a dynamic response to the bones [ 14 ] . Experimental method can be used to measure the stress-strain properties to obtain the dynamic response of the bones [ 15 , 16 ] . Other methods such as constructing the three dimensional models and numerical models of the musculoskeletal system by using a special 3D capture system are also adopted [ 17 ] . The impact shock generated between the leg and the ground is attenuated primarily in the lower extremity [ 18 , 19 ] by regulating the leg(and the back) stiffness [ 20 , 21 ] when the bones and muscles of the musculoskeletal are not fatigued from the impact loading [ 22 , 23 ] . As a kind of soft tissues, muscles assist in the absorption of impact force, and muscle activity can be tuned to the impact force characteristics to control the soft-tissue vibrations [ 24 ] . Therefore, the impact loadings produced during galloping which have a higher influence on bone fatigue and stress fracture [ 25 ][ 26 ] , especially the initial impact with a brief duration and sizeable amplitude. In order for a mobile robot to survive through impact conditions, an impact reduction mechanism is also essential [ 27 , 28 ] . Therefore the bionic mechanism is usually designed with shock absorbing and cushioning properties [ 29 ] . In the above-mentioned studies, knowledge about the dynamic response of musculoskeletal system especially the limb bones could be useful particularly to improve the environment adaptability of the bionic mechanism [ 30 , 31 ] . However, there has been few work on numerical modeling and experimental analyzing the transient dynamic response of bones and soft tissues during impact loading, where the impact shock travelling through the musculoskeletal system. In this paper, the musculoskeletal system model of a quadruped was firstly constructed from anatomical measurement of a domestic cat. Then a nano-indentation technology was used to measure the elastic moduli of the tibia precisely. To obtain the transient response abilities of bones, soft tissues and bionic components, the substructure method was introduced in dynamic equations. Finally, the impact experiments of the tibia and the bionic leg were taken by a high-speed video system. 2 Materials and Methods As the primary body weight support mechanism, the musculoskeletal system is sensitive to the impact loading of the joints and the bones during walking and galloping, the analysis of the mechanical properties of the bones and the soft tissues can provide a reference to the design of bionic mechanisms. Based on the anatomical data, we construct a mathematical model for the musculoskeletal system of a domestic cat. 2.1 Musculoskeletal system modeling: In the anatomical data, the truck of the cat is 280 mm long, the shoulder is 60/70 mm wide, and the body weight is 1.4 kg. The other measured data is given in Table 1 . Table 1 Parameters of the limb bones and muscles skeleton and muscle length/mm diameter /mm scapula 44 -- humerus 69 6.8 radius 58 4 metacarpal 22 3.1 femur 80 6.9 tibia 82 6 metatarsal 35 3 Posterior tibia muscle group 90 -- Lateral femoral muscle group 85 -- Posterior femoral group muscle group 90 -- The distribution of the tendon attachment data is also obtained from the anatomical measurements, where the muscles are distributed along various joints and bones, and the muscle forces are correlated to the motions of different bones and joints to drive the bones and joints swinging, transforming the muscle contractions into the rotation around the joint of the bones. Under the parameters shown in Table 1 , the location of the attachment point of muscle tendon along the bone can be utilized to construct the kinematical model of the musculoskeletal system, while the bones and joints are simplified according to their functions of the body activity in this paper. Furthermore, the coordinate transformation method is employed for the kinematical modeling of the musculoskeletal system, as shown in the following equations: $${T_{hind\lim b}}={T_{hip}}{T_{knee}}{T_{ankle}}{T_{toe}}$$ 1 Eq. ( 1 ) is the kinematical equations of hind limbs, containing the relationship of the coordinate transformation from the toe end to the hip joint, where T hip , T knee , T ankle , T toe are the kinematical transform equations of the latter joint relative to the former joint. The kinematic equations of the other limbs are similar. Mradius : Torque acting on radius; Mhumerus : Torque acting on humerus; Mmetacarpal : Torque acting on metacarpal; Mscapula : Torque acting on scapula; Mtibia :Torque acting on tibia; Mfemur :Torque acting on femur: Mtarsal :Torque acting on tarsal Using the kinematical model of the musculoskeletal system (shown in Fig. 1 b) analyzed above and by measuring the running pattern [ 35 , 36 ] of the cat shown in Fig. 1 a, the change in each joint angle can be obtained. The change of the muscle length and contraction velocity can be solved analytically as well. The muscle forces can be calculated by solving Eq. ( 2 ): $${F_m}={F_a}+{F_p}={F_0}({f_1}{f_2}a(t)+{f_3})$$ 2 In Eq. ( 2 ), the distance change between the muscle tendon attachment points in Fig. 1 b varying with time is known, F m represents the size of the muscle force represented by the single connection lines shown in Fig. 1 b could be obtained., F a and F p represent the forces of the flexor muscle and extensor muscle, respectively. f 1 , f 2 , f 3 represent the exponential functions where are the inverse tangent function of the change of the muscle length and contraction velocity, respectively. Finally, the muscle force acting at each joint within the running time of 0.3 s is calculated. The muscle forces of the hind limbs are generally larger than those of the fore limbs, which shows that the hind limbs are the major contributors to body support during running. 2.2 Determination of biomaterial properties In Fig. 1 c, the torque acting on different joints is obtained by multiplying the muscle force (calculated in Eq. ( 2 )) and the moment arm, where the moment arm is the minimum distance between the connection line (shown in Fig. 1 b) and the rotation center of the joint [ 32 ] . As described above, hind limbs exert the major power for high speed movement [ 33 ] , and tibias bridge the preceding and the following for the transmission of movement. The mechanism of tibias can bear a certain bending moment and torque moment, can also withstand the radial impact pressure. Therefore, in this paper we select the tibia to study the dynamic response of bones of the domestic cat under impact loading. The tibia used in this study is 8.3 cm long, and its lateral ankle is 2.2 cm long and weighs 1.41 g, with a density of about 298.7 kg/m 3 . The truncated tibia is 4.1 cm long and weighs 1 g with a density of about 1940.9 kg/m 3 . In order to precisely calculate the dynamic response, a nano-indentation technique is employed to measure Young's moduli of the tibia. Figure 2 depicts the displacement-loading curves of the tibia material, where the abscissa represents the pressed depth, and the ordinate represents the press-in force. Figure 2 a shows the result for the outside surface of tibia, and Fig. 2 b for the inside of the tibia and the medial condyle. There are eight groups of reasonable measured values in Fig. 2 a, six in Fig. 2 b. We feed the measuring results into the O&P method [ 34 ] to obtain multi-group elastic moduli of tibia material, which are averaged to get the elastic moduli of the truncated tibia and the medial condyle as 7.45Gpa and 1.485Gpa respectively. 2.3 Numerical model of the dynamic response process To investigate the transient response of bones in this paper, the contact between the tibia and the tarsal is simplified: the tibia is simplified into an one-dimensional plane rod, and the tarsal into a rigid surface, as shown in Fig. 3 a, where under the action of the joint's moment M , the femur exerts a pressure on the tibia, and a brief contact-impact occurs between them to generate the dynamic response. To handle different elastic moduli between the cancellous bone and the compact bone, as shown in Fig. 3 b, the tibia is simplified into the collision rod of different materials, where the lower end of the rod represents the condyle. The cancellous bone is relatively abundant. The elastic moduli is E 1 . In the middle of tibia the elastic moduli is E 2 , and there is more compact bone. To solve the nonlinear problem of the dynamic response, the substructure technique for dynamics is employed in this paper. The method can reduce the modal order of the finite element method, and ensure the accuracy of the calculation. Based on the substructure method, we divide the collision rod into n substructure rods, each containing m two-node link elements, as shown in Fig. 3 c, where C represents a contact point, a , V is the acceleration and speed of the rod, respectively, u the physical displacement of the node. The motion equation of the substructure S (k) can be expressed as: $$\left[ {\begin{array}{*{20}{c}} {M_{{ii}}^{{\left( s \right)}}}&{M_{{ib}}^{{\left( s \right)}}} \\ {M_{{bi}}^{{\left( s \right)}}}&{M_{{bb}}^{{\left( s \right)}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {\ddot {u}_{i}^{{\left( s \right)}}} \\ {\ddot {u}_{b}^{{\left( s \right)}}} \end{array}} \right\}+\left[ {\begin{array}{*{20}{c}} {K_{{ii}}^{{\left( s \right)}}}&{K_{{ib}}^{{\left( s \right)}}} \\ {K_{{bi}}^{{\left( s \right)}}}&{K_{{bb}}^{{\left( s \right)}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {u_{i}^{{\left( s \right)}}} \\ {u_{b}^{{\left( s \right)}}} \end{array}} \right\}=\left\{ {\begin{array}{*{20}{c}} {F_{i}^{{\left( s \right)}}} \\ {F_{b}^{{\left( s \right)}}+R_{b}^{{\left( s \right)}}} \end{array}} \right\}$$ 3 Namely: (4) The mass matrix of Eq. (4) is a coordinated mass matrix, where F is the external load, and R the interfacial force. The physics displacement vector u (s) is divided into two parts: the internal node displacement u i (s) and the interface node displacement u b (s) . By calculating the transforming relationship between the physical displacement and the modal displacement, the modal matrix can be obtained, and Eq. (4) can be solved based on the known parameter matrices. Meanwhile, when the rod vertically impacts the ground, we assume no invasion between the rod and the ground, and no geometric dispersion along the rod. As shown in Fig. 3 c, during the solution process, the rod length L is divided into about 300 units, with every four nodes making a substructure unit. The convergence of the substructure method has been proved. 3. Results 3.1 Numerical solution of impact force On the basis of the above numerical model, the contact forces of the tibia under different conditions are solved. The amplitudes of the impact forces generated by the contact-impact between the tibias (the truncated, 41 mm long, and the not- truncated, 63 mm long) and the rigid surface under the action of five different loads are shown in Fig. 4 . It is obviously that with the increase of the load, the amplitudes of impact force also increase. The changes of the amplitude of the collision forces in Fig. 4 b and Fig. 4 c are similar, where there is an increase of both amplitudes. It is a result of the loading force acting on the other end of the rod. The amplitude of the collision stress wave is computed as: ρ * c * v , where ρ represents the density of the rod material, c (the square root of E / ρ ) represents the speed of the stress wave propagating along the rod. With this expression, even under different external forces the same material will always have the same end time of impact response, and the amplitude of the impact force depends on the magnitude of the original speed of the impact loading. In Fig. 4 c there is a significant increase in the amplitude, which is explained as follows. When conducting the numerical model calculation, the elastic moduli and the density as well as the other parameters between the two materials do not achieve a smooth transition, which results in a jitter in the result. There is a significant secondary impact for the two objects near the two cycles of the collision (around 8.43 x10 − 5 s and 1.235 x10 − 4 s). The impact force increased slightly again from zero, as shown in the windows inside Fig. 4 a and Fig. 4 b. The rods with different loads fall freely from the same height in Fig. 4 b and Fig. 4 d. The falling speeds are the same at the moment of the contact with ground. During the first half of the first contact-impact (about 2 x10 − 5 s and 3 x10 − 5 s), the truncated tibia (41 mm long) and long tibia (63mm long) have the same impact force. The amplitude of the impact force is less than that in the previous two cases. During the second half of the first contact-impact, due to the influence of the loading force, the impact forces rise slightly. However, after a certain time of separation, the contact-impact of the two materials under different loading is obviously doubled. For example, in the 500g loading conditions, the two materials rapidly impact and stably contact with the rigid surface again after achieving the first contact-impact. Except for tibia, other bionic components such as aluminum alloy, PLA (engineering material) also have the dynamic response ability. The two little windows in Fig. 4 e show the impact forces when the aluminum alloy and PLA rods touch the rigid surface once again at about 7.75x10 − 5 s and 2.45x10 − 4 s respectively. It further suggests that multiple impacts also exist between the common bionic mechanism materials and the ground, and that the generated vibrations last a short time. For example, the aluminum alloy with larger elastic moduli impacts and contacts the ground for a shorter time and responds more quickly than the PLA material. 3.2 Transient response wave of dynamics Because the impact force and the stress wave have the same propagation characteristic along the rod, the stress wave spreading from right to left will be in a rectangular wave as shown in Fig. 5 a and Fig. 5 b. Regardless of forward propagation or reverse propagation, the amplitude of the rectangular stress is relatively stable. The amplitude of the stress wave in Fig. 5 b changes greatly at 22 mm from the right end, which is resulted from the similar propagation velocities and quite different densities in the rods of the two materials. When the rod impacts the ground, the stress wave generated by load force ( σ l ) will also move from the left to the right in a rectangular wave, while the amplitude is relatively small. Compared with the stress wave, the velocity wave also has a rapidly changing wave surface, spreading toward the other end at a certain speed along the length, as shown in Fig. 5 c and Fig. 5 d. The amplitude of the velocity wave before 1x10 − 5 s is relatively larger after the velocity wave reflects back to the ankle bone segment, which is determined by the propagation characteristics of the velocity wave in different materials. The wave surface of the velocity changes very quickly, which is directly related to the change of the acceleration wave in the rod. The acceleration wave propagates along the length with two peak values, which are caused by the impact force and the load force respectively, propagating from inverse directions, as shown in Fig. 5 e and Fig. 5 f. The acceleration peak of each (mass) point in the rod changes very quickly, so the wave surface of the velocity can change rapidly to achieve a uniform shape. During the propagation, the peak value of the acceleration wave in Fig. 5 e and Fig. 5 f decreases as the wave width increases gradually. 3.3 Displacement wave propagating along the rod Different from the propagation characteristics of the velocity and acceleration wave, before reaching the loaded end, the front end of the displacement wave propagating along the rod is an almost linear curve as shown in Fig. 6 . In Fig. 6 b the propagation characteristics of the displacement wave shows that each point in the tibia moves forward during the contact-impact, while the tibia keeps contacting with the rigid surface simultaneously. At the time of 1.5 x10 − 4 s, the mass points in the tibia bounce with almost the same displacement after the tibia leaving the rigid surface. Aluminum alloy and PLA are commonly used in bionic mechanisms. Numerical analysis indicates that at the moment of contact-impact, the aluminum alloy rod and the PLA rod have similar dynamic response of the displacement wave, as shown in Fig. 6 c and Fig. 6 d. The one-dimensional rod of PLA responds more slowly compared to the other two materials, as shown in Fig. 6 c. The material properties of PLA are measured in this study, e.g., the Young's moduli of PLA leg: E = 3.093Gpa was experimentally determined. 4. Experiments In our experiments, we use a high-speed video system to record the dynamic response characteristics of the cat’s bones. The frequency of the high-speed video system is 10000 fps (frames per second), with 1:1 dedicated lens. With this frequency, in a millisecond (ms) numbered as u , there are 10 frames, which are labeled as u ms-1, u ms-2, …, u ms-10 in this paper. The capture results are shown in Fig. 7 . Under 100g loading, the freely falling tibia impacts and contacts with the rigid surface, impacting twice the time less than 2 ms in frames 725ms-4 to 725ms-10 for the first contact-impact, and in frames 726ms-1 to 726ms-7 for the second contact-impact. In the second contact-impact, the tibia keeps contacting and fits with the rigid surface. The time interval between the two contact-impact is about 5 x10 − 4 s, and which is similar to the numerical results shown in Fig. 4 b (impacting twice at about 5 x10 − 4 s).This suggests that the tibia impacts with the rigid face repeatedly at the moment of contact-impact. Figure 4 c and Fig. 4 d suggests that with different contact area, density, and elastic moduli, the materials will have different dynamic responses characteristics. Figure 5 and Fig. 6 further confirm this statement. In Fig. 8 , the tibia containing a small amount of soft tissues is used in the experiment, where the impacting and contacting take about 2 ms, a longer time than in Fig. 4 c and Fig. 6 b. The ankle side on the tibia is connected with the soft tissues. When the soft tissues first contact with the rigid surface, the contact-impact time of the ankle is extended, due to buffering functions of the soft tissues. As shown in Fig. 8 , the contact-impact process between the tibia and the rigid surface can be expressed by calibrating the distance between the tibia side and the rigid surface. In the figure, frames 213ms-1 to 215ms-2 show the contact and depressed process, frames 215ms-3 to 216ms-7 show the bounce process of tibia. The duration of the contact is about 10 times of the calculated value. In this study four mechanical legs of PLA are installed on the joint end of a quadruped robot, and a walking experiment is conducted. Figure 9 a depicts a quadruped robot and the mechanical legs of PLA. As shown in Fig. 9 d, in a short time period multiple contact-impacts occur between the leg and the ground. The end part of the leg impacts and contacts with the ground for the first time (between frames 17ms-1 and 26ms-6), then follow the second (frames 30ms-1 and 52 − 1) and third time (frames 54ms-10 and 86ms-1). During the whole process, the end part of the leg has bounce actions until the contact with the ground becomes stable. 5. Discussions The bionic mechanism is designed based on the biological properties, so the bionic performance of a robot is directly related to the capability of the bionic mechanism to imitate various biological properties. Therefore, the study of the musculoskeletal system, especially bones and soft tissues, plays an important role in the improvement of the performance of the bionic mechanism. As the primary motion part of the body for mammal, the musculoskeletal system is more sensitive to the stress response, it is necessary to study the dynamic response characteristics of the primary skeleton of a quadruped. The main purpose of this study is to model and analyze the transient dynamic response of bones and soft tissues during impact loading, where the impact shock travelling through the musculoskeletal system. In this study, a domestic cat was first dissected, and the anatomical data was obtained for constructing the kinematical model for the musculoskeletal system of the hind limbs. The tibia was selected to study the transient dynamic response by considering the important role that the tibia plays in hind limbs during high speed movement. To get precise results, the nano-indentation technology was employed to measure Young’s moduli of the tibia. The experimental measurements and calculations show that the outside surface of the tibia has a higher moduli (7.45Gpa) than inside of the tibia (1.2662Gpa), and that the middle part of the outside surface of the tibia has a higher moduli than the medial malleolus at the distal end of the tibia (2.6Gpa). It was also verified that the tibial surface kept moist by spraying physiological saline over the surface has a lower moduli than the relatively dry tibia (13.02Gpa) without being kept moist (for more than 24 hours after dissected), and that the tibia kept moist has a more uniform distribution of moduli than the relatively dry tibia. Since it is known that the stiffness can be calculated from the Young’s moduli and the structure parameters of the subject directly, the characteristics of the distribution for the moduli along the tibia may influence the movement performance of the hind limbs and the whole body. During galloping and hopping, the limbs behave as springs, and the whole body is often modeled with a ‘‘spring-mass model’’, thus the ability to modify the stiffness of these leg springs is essential to maintaining an efficient gait [ 32 , 36 ] . In this study, the Young’s moduli does not vary significantly between adjacent measurement regions (0.1 mm in width, 0.1 mm in length) of the outside surface of the tibia, especially of the tibia kept moist. Therefore, we conclude that the measurement region has relatively stable values of Young’s moduli during high speed movements. In this paper, to investigate the transient response of bones, the contact between the tibia and tarsal was simplified, and the substructure technique for dynamics was employed for solving the high nonlinear problem of dynamic response. From the results shown in Fig. 4 , it can be concluded that the body of the domestic cat experienced a brief but sizeable impact upon paw strike during galloping. Because the effects of soft tissues and cartilages were not considered, the frequency of the contact-impact is more higher than the frequency provided by other methods [ 23 , 26 ] , and the frequency content of the impact loading can be effectively obtained from the capture experiments by using the high-speed video system. Though the 1:1 dedicated lens were taken during the capture experiments, the change of the stress-strain property (as shown in Fig. 5 and Fig. 6 ) was hard to measure because of the high frequency during impact loading. Additionally, under the action of an external force and the existence of residual accelerations shown in Fig. 5 , secondary and multiple impacts generated between the rod and the rigid surface. This phenomenon indicates that the animal may use the energy produced by elastic deformation of the tibia, then the tibia has been hit to feed back. The process is similar to the release of elastic potential energy that is stored by the rod structure itself. During the impacts, the object being hit usually stores and releases a certain potential energy. In the process of moving contact between the end part of the robot leg and the soil, we can observe that there exists an energy feedback from the soil, which helps the rapid movement of cats and other quadrupeds. 6. Conclusions In this paper, the transient response for the dynamics of tibias and PLA legs during impact loading was numerically solved and experimentally verified. Our work confirmed that multiple impacts exist in tibias and PLA legs at the moment of the contact-impact, and that the transient dynamic response exists during the impact loading. The properties of the transient dynamic response of tibias and PLA legs can be employed to improve the future design for bionic mechanisms. Declarations Acknowledgement This study was supported by the Tianjin Education Commission Research Project (2022KJ124). Author Introduction Meng-jun Song received his doctor's degree from Hebei University of Technology, China, in 2013.He is currently a master's student of Vehicle Engineering in the College of Automobile and Transportation, Tianjin University of Technology and Education, China. His research interests include vehicle dynamics and control, vehicle path planning and vehicle active safety.He is an associate professor at the College of Automobile and Transportation of Tianjin Vocational and Technical Normal University and has published 20 articles. His main research approach is mobile robot kinematics. E-mail: [email protected] Jing-gogn Wei received his master's degree from Jilin University of Technology, China, in 2009.He is currently a Senior Engineer of Vehicle Engineering in the College of Automobile and Transportation, Tianjin University of Technology and Education, China. His research interests include Structural Lightweighting Optimization Design & Mechanical Lightweighting Optimization. E-mail:weijinggong@ tute.edu.cn Li-ping Zhang received his bachelor's degree from Shangrao Normal University, China, in 2010. She is currently a master's student in Nankai University, China. Her research interests include Sports Biomechanics, Robotics Kinematics. E-mail: [email protected] Author Contributions: First Author: Wrote and revised the manuscript, incorporating research findings and analysis. Second Author (Corresponding): Secured funding, supervised quality control, and managed submission. Third Author: Finalized formatting and style to meet journal requirements. Data Availability Statement : All data generated or analyzed during this study are included in this article. We welcome open communication and collaboration regarding our findings. No confidential or restricted data were utilized in this work, and therefore, all information presented is freely available for further research and discussion. Should any additional clarifications or details be required, readers are encouraged to contact the corresponding author. Declaration of Interest Statement The authors declare that they have no financial or personal relationships with other people or organizations that could inappropriately influence (bias) their work. Specifically, and have no financial interests, direct or indirect, in the subject matter or materials discussed in this manuscript. Furthermore, the authors have no relevant conflicts of interest to disclose with respect to the research, authorship, and/or publication of this article. References Chan, S. C., Yee, K. M., Chan, J. C. & Tan, Y. Q. 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Bionic Eng. 11 , 506–516 (2014). McNulty, A. L. & Guilak, F. Mechanobiology of the meniscus. J. Biomech. 48 (8), 1469–1478 (2015). Wang, X. et al. Influence of vibrations and shocks on the stability of biomimetic attachments. Colloids Surf., A , 682,(2024). Konow, N. & Roberts, T. J. Prepared for landing: A simple activation strategy scales muscle force to landing height. J. Biomech. , 165 , (2024). Malekipour, F., Hitchens, P. L., Whitton, R. C. & Lee, P. V. S. Effects of in vivo fatigue-induced subchondral bone microdamage on the mechanical response of cartilage-bone under a single impact compression. J. Biomech. , 100 , (2020). Edwards, W. B. et al. Effects of running speed on a probabilistic stress fracture model. Clin. Biomech. Elsevier Ltd . 25 , 372–377 (2010). Benjamin, D. et al. Shock Response Spectr. Anal. Fatigued Runners Sens. , 22 (6), (2022). Jarakorn Arnas, Kunlabud, A. et al. Impact reduction mobile robot and the design of the compliant legs. Robot. Auton. Syst. 62 , 38–45 (2014). Chen Jie, L. et al. Biomimetic Design and Optimal Swing of a Hexapod Robot Leg. J. Bionic Eng. 11 , 26–35 (2014). Mengjun Song1 Kun Wei1 Jinggong Wei1 Haijun Zhao. Dynamic substructure method for impacting response of bionic component. Mach. Des. Res. 39 (05), 72–75 (2023). Zong, H. et al. Bionic lightweight design of limb leg units for hydraulic quadruped robots by additive manufacturing and topology optimization. Bio-Design Manuf. 7 (1), 1–13 (2024). Soo, I. H., Oren, G. & Sheehan, F. T. The Effective Quadriceps and Patellar Tendon Moment Arms Relative to the Tibiofemoral Finite Helical Axis. J. Biomech. 48 , 3737–3742 (2015). McKay, J. L., Burkholder, T. J. & Ting, L. H. Biomechanical capabilities influence postural control strategies in the cat hindlimb. J. Biomech. 40 , 2254–2260 (2007). Oliver, W. C. & Pharr, G. M. Improved technique for determining hardness and elastic moduli using load and displacement sensing indentation experiments. J. Mater. Res. 7 , 1564–1583 (1992). Kim, Y. K., Park, J., Yoon, B. & Kim Kyung-Soo, Kim, S. The role of relative spinal motion during feline galloping for speed performance. J. Bionic Eng. 11 , 517–528 (2014). Song, M. et al. Research on Kinematics and Biomechanics of the Bionic Mechanism[J].IOP Conference Series: Earth and Environmental Science, 170(4), (2018). Author & Introduction. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4894964","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":357917611,"identity":"de846f03-224d-4a2f-b82a-e4d41da5074e","order_by":0,"name":"mengjun song","email":"","orcid":"","institution":"Tianjin University of Technology and Education","correspondingAuthor":false,"prefix":"","firstName":"mengjun","middleName":"","lastName":"song","suffix":""},{"id":357917614,"identity":"fa19d34d-76ae-40c9-b548-534b08ccca9e","order_by":1,"name":"jinggong wei","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/ElEQVRIiWNgGAWjYFACxgaDBCjrAVTIgGgtzAYHIKoJaUEANgmitBgcb24oeLij1q5fuv1a9ce2P3IM7M3bJBhq7uDWcuZgg0HimePJM+ecKbtxsM3AmIHnWJkEw7FnOLWY3UgEamk7lmxwIycNpCWxQSLHTIKx4TBuLfcfIrQUALXUN8i/IaDlBiNIS42dwY30YwxALQkMEjz4tdifATvsQILkjBxmiTPnjA3beNKKLRKO4dYi2X78meHPtjp7fon0hx8qyuTk+dkPb7zxoQa3FiBgA0bD4cQGBh5IdLCBiAR8GoCR/oCBoc6egYH9AX51o2AUjIJRMGIBAK7zXFA8XJ5eAAAAAElFTkSuQmCC","orcid":"","institution":"Tianjin University of Technology and Education","correspondingAuthor":true,"prefix":"","firstName":"jinggong","middleName":"","lastName":"wei","suffix":""},{"id":357917616,"identity":"69799df9-8040-4eb1-8af3-6677305d43bd","order_by":2,"name":"Liping Zhang","email":"","orcid":"","institution":"Nankai University","correspondingAuthor":false,"prefix":"","firstName":"Liping","middleName":"","lastName":"Zhang","suffix":""}],"badges":[],"createdAt":"2024-08-11 11:33:36","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4894964/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4894964/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":65187814,"identity":"da98eea5-b024-400e-af83-877ac4f276a8","added_by":"auto","created_at":"2024-09-24 14:07:47","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":136852,"visible":true,"origin":"","legend":"\u003cp\u003e(a) The calibration of motion sequence of a cat; (b) the motion curve of the hind limb and (c)the variation curve of the moment acting on the joints of Anterior and hind limb when the cat is running at a high speed\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/b743b88ad768b28b3f8fb8eb.png"},{"id":65188873,"identity":"cd8699c4-0c28-4e21-a1c9-cd45f43248cb","added_by":"auto","created_at":"2024-09-24 14:15:47","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":29470,"visible":true,"origin":"","legend":"\u003cp\u003eThe displacement-loading curves of truncated tibia and condyle\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/ff3c33620100152bfc2e206a.png"},{"id":65187820,"identity":"d8d30be5-29f3-4e13-9cec-299ee7585ca3","added_by":"auto","created_at":"2024-09-24 14:07:47","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":84247,"visible":true,"origin":"","legend":"\u003cp\u003e(a) The leg of a cat during running and (b) a simplified schematic figure of a tibial impact and (c) the substructure mehod used in the contact-impact model\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/81654243b1a37f48ccab9c0d.png"},{"id":65188871,"identity":"4a4cf9dd-3a6f-4f28-94db-a0bfade17803","added_by":"auto","created_at":"2024-09-24 14:15:47","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":44366,"visible":true,"origin":"","legend":"\u003cp\u003eThe calculation of the impact forces under diffenet materials and different loading conditons\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/c5a390f9b9b7a4fd7f5c4118.png"},{"id":65187815,"identity":"521fcee5-aeff-4deb-a77d-f2c9c4939410","added_by":"auto","created_at":"2024-09-24 14:07:47","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":36394,"visible":true,"origin":"","legend":"\u003cp\u003eThe propagation of velocity wave, stress wave and acceleration wave in tibia, red for forward wave, black for back wave\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/607ac438723f854ec48514ad.png"},{"id":65187817,"identity":"0151c334-39fb-4435-bcd1-8489047a02fa","added_by":"auto","created_at":"2024-09-24 14:07:47","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":31387,"visible":true,"origin":"","legend":"\u003cp\u003e(a) The change of the displacement wave in truncated tibia and (b) the change of the displacement wave in the tibia with considering the medial condyle, (c)the change of displacement wave in the mechanical legs of aluminum alloy and (d) PLA material\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/b40799c8ca3985f6e340045e.png"},{"id":65187818,"identity":"c8b8115c-1bd5-41e8-8215-e01e0961b53e","added_by":"auto","created_at":"2024-09-24 14:07:47","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":187870,"visible":true,"origin":"","legend":"\u003cp\u003eThe truncated tibia fell freely under 100g loading to impact the rigid surface, the number behind the short horizontal line is the first few frames in one millisecond and later figures following the same name rules\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/22d677ba29b8315be020cbbb.png"},{"id":65189354,"identity":"7e03b3c8-2a15-493e-8955-4c4e03f942e0","added_by":"auto","created_at":"2024-09-24 14:23:47","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":274082,"visible":true,"origin":"","legend":"\u003cp\u003eThe tibia with soft tissue fell freely under 100g loading to impact aganist the rigid surface\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/b678dc33cea5a9183c66581e.png"},{"id":65187822,"identity":"56161f6a-4a0a-4328-9d4d-bb82433a1986","added_by":"auto","created_at":"2024-09-24 14:07:47","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":246686,"visible":true,"origin":"","legend":"\u003cp\u003e(a) A quadruped mobile robot; (b) a PLA leg of the robot, (c) a picture of the end part of the leg touching the ground, (d) a frame sequence of the end part touching the ground.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/ac445b11ecb8ad34f48e1622.png"},{"id":87167269,"identity":"5de6ddd0-29d8-4a05-9b54-4f03b9feabad","added_by":"auto","created_at":"2025-07-21 06:39:13","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1987708,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4894964/v1/feeca908-d108-466a-af5a-7afdc01b4d1a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Study on the Dynamic Response Characteristics of Bionic Legs During Instantaneous Ground Contact","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eDuring the impact loading in activities such as walking and galloping, a contact-impact as a transient response process exists between the leg and the ground\u003csup\u003e[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]\u003c/sup\u003e. Typical time histories of the initial contact event during the impact are usually very short. The dynamic response wave produced by the impact force propagates through both skeletons and muscles\u003csup\u003e[\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]\u003c/sup\u003e and must be attenuated by the musculoskeletal system during the impact landing\u003csup\u003e[\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]\u003c/sup\u003e. In the musculoskeletal system, muscles are attached to tendons, which insert into the bones and cross different joints, transmitting muscle forces to the skeleton and protecting the bone from external invasions\u003csup\u003e[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]\u003c/sup\u003e. As a kind of soft tissues, a generic muscle (group) can be viewed as a set of fibers, which connects two tendons forming connection lines for mathematical models, to calculate the muscle forces according to the length change of the muscle\u003csup\u003e[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]\u003c/sup\u003e. Muscle forces can also be solved by other methods such as the experimental determination. The muscle forces can be transformed into joint moments multiplied by moment arm\u003csup\u003e[\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]\u003c/sup\u003e. After being generated, the joint moments act on joints and other parts of the body, which then contact with the ground or objects \u003csup\u003e[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]\u003c/sup\u003e. During the initial ground contact, the joint contact force is usually larger than the body weight (BW)\u003csup\u003e[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/sup\u003e, that produces high impacts on the joints and causes a dynamic response to the bones\u003csup\u003e[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/sup\u003e. Experimental method can be used to measure the stress-strain properties to obtain the dynamic response of the bones\u003csup\u003e[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]\u003c/sup\u003e. Other methods such as constructing the three dimensional models and numerical models of the musculoskeletal system by using a special 3D capture system are also adopted\u003csup\u003e[\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe impact shock generated between the leg and the ground is attenuated primarily in the lower extremity\u003csup\u003e[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]\u003c/sup\u003e by regulating the leg(and the back) stiffness\u003csup\u003e[\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]\u003c/sup\u003e when the bones and muscles of the musculoskeletal are not fatigued from the impact loading\u003csup\u003e[\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]\u003c/sup\u003e. As a kind of soft tissues, muscles assist in the absorption of impact force, and muscle activity can be tuned to the impact force characteristics to control the soft-tissue vibrations\u003csup\u003e[\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]\u003c/sup\u003e. Therefore, the impact loadings produced during galloping which have a higher influence on bone fatigue and stress fracture\u003csup\u003e[\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e][\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]\u003c/sup\u003e, especially the initial impact with a brief duration and sizeable amplitude. In order for a mobile robot to survive through impact conditions, an impact reduction mechanism is also essential\u003csup\u003e[\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e]\u003c/sup\u003e. Therefore the bionic mechanism is usually designed with shock absorbing and cushioning properties\u003csup\u003e[\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eIn the above-mentioned studies, knowledge about the dynamic response of musculoskeletal system especially the limb bones could be useful particularly to improve the environment adaptability of the bionic mechanism\u003csup\u003e[\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]\u003c/sup\u003e. However, there has been few work on numerical modeling and experimental analyzing the transient dynamic response of bones and soft tissues during impact loading, where the impact shock travelling through the musculoskeletal system.\u003c/p\u003e \u003cp\u003eIn this paper, the musculoskeletal system model of a quadruped was firstly constructed from anatomical measurement of a domestic cat. Then a nano-indentation technology was used to measure the elastic moduli of the tibia precisely. To obtain the transient response abilities of bones, soft tissues and bionic components, the substructure method was introduced in dynamic equations. Finally, the impact experiments of the tibia and the bionic leg were taken by a high-speed video system.\u003c/p\u003e"},{"header":"2 Materials and Methods","content":"\u003cp\u003eAs the primary body weight support mechanism, the musculoskeletal system is sensitive to the impact loading of the joints and the bones during walking and galloping, the analysis of the mechanical properties of the bones and the soft tissues can provide a reference to the design of bionic mechanisms. Based on the anatomical data, we construct a mathematical model for the musculoskeletal system of a domestic cat.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Musculoskeletal system modeling:\u003c/h2\u003e \u003cp\u003eIn the anatomical data, the truck of the cat is 280 mm long, the shoulder is 60/70 mm wide, and the body weight is 1.4 kg. The other measured data is given in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of the limb bones and muscles\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eskeleton and muscle\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003elength/mm\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ediameter /mm\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003escapula\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e--\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ehumerus\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e69\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6.8\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eradius\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e58\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003emetacarpal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003efemur\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003etibia\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e82\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003emetatarsal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePosterior tibia muscle group\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e90\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e--\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLateral femoral muscle group\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e85\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e--\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePosterior femoral group muscle group\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e90\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e--\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe distribution of the tendon attachment data is also obtained from the anatomical measurements, where the muscles are distributed along various joints and bones, and the muscle forces are correlated to the motions of different bones and joints to drive the bones and joints swinging, transforming the muscle contractions into the rotation around the joint of the bones. Under the parameters shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the location of the attachment point of muscle tendon along the bone can be utilized to construct the kinematical model of the musculoskeletal system, while the bones and joints are simplified according to their functions of the body activity in this paper. Furthermore, the coordinate transformation method is employed for the kinematical modeling of the musculoskeletal system, as shown in the following equations:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${T_{hind\\lim b}}={T_{hip}}{T_{knee}}{T_{ankle}}{T_{toe}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eEq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) is the kinematical equations of hind limbs, containing the relationship of the coordinate transformation from the toe end to the hip joint, where \u003cb\u003eT\u003c/b\u003e\u003csub\u003e\u003cb\u003ehip\u003c/b\u003e\u003c/sub\u003e, \u003cb\u003eT\u003c/b\u003e\u003csub\u003e\u003cb\u003eknee\u003c/b\u003e\u003c/sub\u003e, \u003cb\u003eT\u003c/b\u003e\u003csub\u003e\u003cb\u003eankle\u003c/b\u003e\u003c/sub\u003e, \u003cb\u003eT\u003c/b\u003e\u003csub\u003e\u003cb\u003etoe\u003c/b\u003e\u003c/sub\u003e are the kinematical transform equations of the latter joint relative to the former joint. The kinematic equations of the other limbs are similar.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003eMradius\u003c/b\u003e: Torque acting on radius; \u003cb\u003eMhumerus\u003c/b\u003e: Torque acting on humerus; \u003cb\u003eMmetacarpal\u003c/b\u003e: Torque acting on metacarpal; \u003cb\u003eMscapula\u003c/b\u003e: Torque acting on scapula; \u003cb\u003eMtibia\u003c/b\u003e:Torque acting on tibia; \u003cb\u003eMfemur\u003c/b\u003e:Torque acting on femur: \u003cb\u003eMtarsal\u003c/b\u003e:Torque acting on tarsal\u003c/p\u003e \u003cp\u003eUsing the kinematical model of the musculoskeletal system (shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e1\u003c/span\u003eb) analyzed above and by measuring the running pattern\u003csup\u003e[\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e]\u003c/sup\u003e of the cat shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e1\u003c/span\u003ea, the change in each joint angle can be obtained. The change of the muscle length and contraction velocity can be solved analytically as well. The muscle forces can be calculated by solving Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e):\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${F_m}={F_a}+{F_p}={F_0}({f_1}{f_2}a(t)+{f_3})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), the distance change between the muscle tendon attachment points in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e1\u003c/span\u003eb varying with time is known, \u003cb\u003eF\u003c/b\u003e\u003csub\u003e\u003cb\u003em\u003c/b\u003e\u003c/sub\u003e represents the size of the muscle force represented by the single connection lines shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e1\u003c/span\u003eb could be obtained., \u003cb\u003eF\u003c/b\u003e\u003csub\u003e\u003cb\u003ea\u003c/b\u003e\u003c/sub\u003e and \u003cb\u003eF\u003c/b\u003e\u003csub\u003e\u003cb\u003ep\u003c/b\u003e\u003c/sub\u003e represent the forces of the flexor muscle and extensor muscle, respectively. \u003cb\u003ef\u003c/b\u003e\u003csub\u003e\u003cb\u003e1\u003c/b\u003e\u003c/sub\u003e, \u003cb\u003ef\u003c/b\u003e\u003csub\u003e\u003cb\u003e2\u003c/b\u003e\u003c/sub\u003e, \u003cb\u003ef\u003c/b\u003e\u003csub\u003e\u003cb\u003e3\u003c/b\u003e\u003c/sub\u003e represent the exponential functions where are the inverse tangent function of the change of the muscle length and contraction velocity, respectively.\u003c/p\u003e \u003cp\u003eFinally, the muscle force acting at each joint within the running time of 0.3 s is calculated. The muscle forces of the hind limbs are generally larger than those of the fore limbs, which shows that the hind limbs are the major contributors to body support during running.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Determination of biomaterial properties\u003c/h2\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e1\u003c/span\u003ec, the torque acting on different joints is obtained by multiplying the muscle force (calculated in Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e)) and the moment arm, where the moment arm is the minimum distance between the connection line (shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e1\u003c/span\u003eb) and the rotation center of the joint\u003csup\u003e[\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e]\u003c/sup\u003e. As described above, hind limbs exert the major power for high speed movement\u003csup\u003e[\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]\u003c/sup\u003e, and tibias bridge the preceding and the following for the transmission of movement. The mechanism of tibias can bear a certain bending moment and torque moment, can also withstand the radial impact pressure. Therefore, in this paper we select the tibia to study the dynamic response of bones of the domestic cat under impact loading.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe tibia used in this study is 8.3 cm long, and its lateral ankle is 2.2 cm long and weighs 1.41 g, with a density of about 298.7 kg/m\u003csup\u003e3\u003c/sup\u003e. The truncated tibia is 4.1 cm long and weighs 1 g with a density of about 1940.9 kg/m\u003csup\u003e3\u003c/sup\u003e. In order to precisely calculate the dynamic response, a nano-indentation technique is employed to measure Young's moduli of the tibia. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e depicts the displacement-loading curves of the tibia material, where the abscissa represents the pressed depth, and the ordinate represents the press-in force. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea shows the result for the outside surface of tibia, and Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb for the inside of the tibia and the medial condyle. There are eight groups of reasonable measured values in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea, six in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb. We feed the measuring results into the O\u0026amp;P method\u003csup\u003e[\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e]\u003c/sup\u003e to obtain multi-group elastic moduli of tibia material, which are averaged to get the elastic moduli of the truncated tibia and the medial condyle as 7.45Gpa and 1.485Gpa respectively. \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Numerical model of the dynamic response process\u003c/h2\u003e \u003cp\u003eTo investigate the transient response of bones in this paper, the contact between the tibia and the tarsal is simplified: the tibia is simplified into an one-dimensional plane rod, and the tarsal into a rigid surface, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e3\u003c/span\u003ea, where under the action of the joint's moment \u003cb\u003eM\u003c/b\u003e, the femur exerts a pressure on the tibia, and a brief contact-impact occurs between them to generate the dynamic response.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo handle different elastic moduli between the cancellous bone and the compact bone, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e3\u003c/span\u003eb, the tibia is simplified into the collision rod of different materials, where the lower end of the rod represents the condyle. The cancellous bone is relatively abundant. The elastic moduli is \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e. In the middle of tibia the elastic moduli is \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e, and there is more compact bone.\u003c/p\u003e \u003cp\u003eTo solve the nonlinear problem of the dynamic response, the substructure technique for dynamics is employed in this paper. The method can reduce the modal order of the finite element method, and ensure the accuracy of the calculation. Based on the substructure method, we divide the collision rod into \u003cem\u003en\u003c/em\u003e substructure rods, each containing \u003cem\u003em\u003c/em\u003e two-node link elements, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e3\u003c/span\u003ec, where \u003cem\u003eC\u003c/em\u003e represents a contact point, \u003cb\u003ea\u003c/b\u003e, \u003cb\u003eV\u003c/b\u003e is the acceleration and speed of the rod, respectively, \u003cem\u003eu\u003c/em\u003e the physical displacement of the node. The motion equation of the substructure \u003cb\u003eS\u003c/b\u003e\u003csup\u003e\u003cb\u003e(k)\u003c/b\u003e\u003c/sup\u003e can be expressed as:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\left[ {\\begin{array}{*{20}{c}} {M_{{ii}}^{{\\left( s \\right)}}}\u0026amp;{M_{{ib}}^{{\\left( s \\right)}}} \\\\ {M_{{bi}}^{{\\left( s \\right)}}}\u0026amp;{M_{{bb}}^{{\\left( s \\right)}}} \\end{array}} \\right]\\left\\{ {\\begin{array}{*{20}{c}} {\\ddot {u}_{i}^{{\\left( s \\right)}}} \\\\ {\\ddot {u}_{b}^{{\\left( s \\right)}}} \\end{array}} \\right\\}+\\left[ {\\begin{array}{*{20}{c}} {K_{{ii}}^{{\\left( s \\right)}}}\u0026amp;{K_{{ib}}^{{\\left( s \\right)}}} \\\\ {K_{{bi}}^{{\\left( s \\right)}}}\u0026amp;{K_{{bb}}^{{\\left( s \\right)}}} \\end{array}} \\right]\\left\\{ {\\begin{array}{*{20}{c}} {u_{i}^{{\\left( s \\right)}}} \\\\ {u_{b}^{{\\left( s \\right)}}} \\end{array}} \\right\\}=\\left\\{ {\\begin{array}{*{20}{c}} {F_{i}^{{\\left( s \\right)}}} \\\\ {F_{b}^{{\\left( s \\right)}}+R_{b}^{{\\left( s \\right)}}} \\end{array}} \\right\\}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eNamely: \u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003e (4)\u003c/p\u003e \u003cp\u003eThe mass matrix of Eq.\u0026nbsp;(4) is a coordinated mass matrix, where \u003cb\u003eF\u003c/b\u003e is the external load, and \u003cb\u003eR\u003c/b\u003e the interfacial force. The physics displacement vector \u003cb\u003eu\u003c/b\u003e\u003csup\u003e\u003cb\u003e(s)\u003c/b\u003e\u003c/sup\u003e is divided into two parts: the internal node displacement \u003cb\u003eu\u003c/b\u003e\u003csub\u003e\u003cb\u003ei\u003c/b\u003e\u003c/sub\u003e\u003csup\u003e\u003cb\u003e(s)\u003c/b\u003e\u003c/sup\u003e and the interface node displacement \u003cb\u003eu\u003c/b\u003e\u003csub\u003e\u003cb\u003eb\u003c/b\u003e\u003c/sub\u003e\u003csup\u003e\u003cb\u003e(s)\u003c/b\u003e\u003c/sup\u003e. By calculating the transforming relationship between the physical displacement and the modal displacement, the modal matrix can be obtained, and Eq.\u0026nbsp;(4) can be solved based on the known parameter matrices. Meanwhile, when the rod vertically impacts the ground, we assume no invasion between the rod and the ground, and no geometric dispersion along the rod. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e3\u003c/span\u003ec, during the solution process, the rod length \u003cem\u003eL\u003c/em\u003e is divided into about 300 units, with every four nodes making a substructure unit. The convergence of the substructure method has been proved.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Numerical solution of impact force\u003c/h2\u003e \u003cp\u003eOn the basis of the above numerical model, the contact forces of the tibia under different conditions are solved. The amplitudes of the impact forces generated by the contact-impact between the tibias (the truncated, 41 mm long, and the not- truncated, 63 mm long) and the rigid surface under the action of five different loads are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. It is obviously that with the increase of the load, the amplitudes of impact force also increase. The changes of the amplitude of the collision forces in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb and Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ec are similar, where there is an increase of both amplitudes. It is a result of the loading force acting on the other end of the rod. The amplitude of the collision stress wave is computed as: \u003cb\u003eρ\u003c/b\u003e*\u003cb\u003ec\u003c/b\u003e*\u003cb\u003ev\u003c/b\u003e, where \u003cb\u003eρ\u003c/b\u003e represents the density of the rod material, c (the square root of \u003cb\u003eE\u003c/b\u003e/\u003cb\u003eρ\u003c/b\u003e) represents the speed of the stress wave propagating along the rod. With this expression, even under different external forces the same material will always have the same end time of impact response, and the amplitude of the impact force depends on the magnitude of the original speed of the impact loading. In Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ec there is a significant increase in the amplitude, which is explained as follows. When conducting the numerical model calculation, the elastic moduli and the density as well as the other parameters between the two materials do not achieve a smooth transition, which results in a jitter in the result. There is a significant secondary impact for the two objects near the two cycles of the collision (around 8.43 x10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e s and 1.235 x10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e s). The impact force increased slightly again from zero, as shown in the windows inside Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea and Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe rods with different loads fall freely from the same height in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb and Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ed. The falling speeds are the same at the moment of the contact with ground. During the first half of the first contact-impact (about 2 x10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e s and 3 x10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e s), the truncated tibia (41 mm long) and long tibia (63mm long) have the same impact force. The amplitude of the impact force is less than that in the previous two cases. During the second half of the first contact-impact, due to the influence of the loading force, the impact forces rise slightly. However, after a certain time of separation, the contact-impact of the two materials under different loading is obviously doubled. For example, in the 500g loading conditions, the two materials rapidly impact and stably contact with the rigid surface again after achieving the first contact-impact.\u003c/p\u003e \u003cp\u003eExcept for tibia, other bionic components such as aluminum alloy, PLA (engineering material) also have the dynamic response ability. The two little windows in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ee show the impact forces when the aluminum alloy and PLA rods touch the rigid surface once again at about 7.75x10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003es and 2.45x10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003es respectively. It further suggests that multiple impacts also exist between the common bionic mechanism materials and the ground, and that the generated vibrations last a short time. For example, the aluminum alloy with larger elastic moduli impacts and contacts the ground for a shorter time and responds more quickly than the PLA material.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Transient response wave of dynamics\u003c/h2\u003e \u003cp\u003eBecause the impact force and the stress wave have the same propagation characteristic along the rod, the stress wave spreading from right to left will be in a rectangular wave as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea and Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb. Regardless of forward propagation or reverse propagation, the amplitude of the rectangular stress is relatively stable. The amplitude of the stress wave in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb changes greatly at 22 mm from the right end, which is resulted from the similar propagation velocities and quite different densities in the rods of the two materials. When the rod impacts the ground, the stress wave generated by load force (\u003cb\u003eσ\u003c/b\u003e\u003csub\u003e\u003cb\u003el\u003c/b\u003e\u003c/sub\u003e ) will also move from the left to the right in a rectangular wave, while the amplitude is relatively small.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eCompared with the stress wave, the velocity wave also has a rapidly changing wave surface, spreading toward the other end at a certain speed along the length, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ec and Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ed. The amplitude of the velocity wave before 1x10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e s is relatively larger after the velocity wave reflects back to the ankle bone segment, which is determined by the propagation characteristics of the velocity wave in different materials. The wave surface of the velocity changes very quickly, which is directly related to the change of the acceleration wave in the rod.\u003c/p\u003e \u003cp\u003eThe acceleration wave propagates along the length with two peak values, which are caused by the impact force and the load force respectively, propagating from inverse directions, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ee and Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ef. The acceleration peak of each (mass) point in the rod changes very quickly, so the wave surface of the velocity can change rapidly to achieve a uniform shape. During the propagation, the peak value of the acceleration wave in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ee and Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ef decreases as the wave width increases gradually.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Displacement wave propagating along the rod\u003c/h2\u003e \u003cp\u003eDifferent from the propagation characteristics of the velocity and acceleration wave, before reaching the loaded end, the front end of the displacement wave propagating along the rod is an almost linear curve as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. In Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eb the propagation characteristics of the displacement wave shows that each point in the tibia moves forward during the contact-impact, while the tibia keeps contacting with the rigid surface simultaneously. At the time of 1.5 x10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003es, the mass points in the tibia bounce with almost the same displacement after the tibia leaving the rigid surface.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAluminum alloy and PLA are commonly used in bionic mechanisms. Numerical analysis indicates that at the moment of contact-impact, the aluminum alloy rod and the PLA rod have similar dynamic response of the displacement wave, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ec and Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ed. The one-dimensional rod of PLA responds more slowly compared to the other two materials, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ec. The material properties of PLA are measured in this study, e.g., the Young's moduli of PLA leg: E\u0026thinsp;=\u0026thinsp;3.093Gpa was experimentally determined.\u003c/p\u003e"},{"header":"4. Experiments","content":"\u003cp\u003eIn our experiments, we use a high-speed video system to record the dynamic response characteristics of the cat\u0026rsquo;s bones. The frequency of the high-speed video system is 10000 fps (frames per second), with 1:1 dedicated lens. With this frequency, in a millisecond (ms) numbered as \u003cem\u003eu\u003c/em\u003e, there are 10 frames, which are labeled as \u003cem\u003eu\u003c/em\u003e ms-1, \u003cem\u003eu\u003c/em\u003e ms-2, \u0026hellip;, \u003cem\u003eu\u003c/em\u003e ms-10 in this paper.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe capture results are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. Under 100g loading, the freely falling tibia impacts and contacts with the rigid surface, impacting twice the time less than 2 ms in frames 725ms-4 to 725ms-10 for the first contact-impact, and in frames 726ms-1 to 726ms-7 for the second contact-impact. In the second contact-impact, the tibia keeps contacting and fits with the rigid surface. The time interval between the two contact-impact is about 5 x10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e s, and which is similar to the numerical results shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb (impacting twice at about 5 x10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e s).This suggests that the tibia impacts with the rigid face repeatedly at the moment of contact-impact.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ec and Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ed suggests that with different contact area, density, and elastic moduli, the materials will have different dynamic responses characteristics. Figure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e further confirm this statement. In Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e, the tibia containing a small amount of soft tissues is used in the experiment, where the impacting and contacting take about 2 ms, a longer time than in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ec and Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eb. The ankle side on the tibia is connected with the soft tissues. When the soft tissues first contact with the rigid surface, the contact-impact time of the ankle is extended, due to buffering functions of the soft tissues.\u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e, the contact-impact process between the tibia and the rigid surface can be expressed by calibrating the distance between the tibia side and the rigid surface. In the figure, frames 213ms-1 to 215ms-2 show the contact and depressed process, frames 215ms-3 to 216ms-7 show the bounce process of tibia. The duration of the contact is about 10 times of the calculated value.\u003c/p\u003e \u003cp\u003eIn this study four mechanical legs of PLA are installed on the joint end of a quadruped robot, and a walking experiment is conducted. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003ea depicts a quadruped robot and the mechanical legs of PLA. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003ed, in a short time period multiple contact-impacts occur between the leg and the ground. The end part of the leg impacts and contacts with the ground for the first time (between frames 17ms-1 and 26ms-6), then follow the second (frames 30ms-1 and 52\u0026thinsp;\u0026minus;\u0026thinsp;1) and third time (frames 54ms-10 and 86ms-1). During the whole process, the end part of the leg has bounce actions until the contact with the ground becomes stable.\u003c/p\u003e \u003c/div\u003e"},{"header":"5. Discussions","content":"\u003cp\u003eThe bionic mechanism is designed based on the biological properties, so the bionic performance of a robot is directly related to the capability of the bionic mechanism to imitate various biological properties. Therefore, the study of the musculoskeletal system, especially bones and soft tissues, plays an important role in the improvement of the performance of the bionic mechanism. As the primary motion part of the body for mammal, the musculoskeletal system is more sensitive to the stress response, it is necessary to study the dynamic response characteristics of the primary skeleton of a quadruped.\u003c/p\u003e \u003cp\u003eThe main purpose of this study is to model and analyze the transient dynamic response of bones and soft tissues during impact loading, where the impact shock travelling through the musculoskeletal system. In this study, a domestic cat was first dissected, and the anatomical data was obtained for constructing the kinematical model for the musculoskeletal system of the hind limbs. The tibia was selected to study the transient dynamic response by considering the important role that the tibia plays in hind limbs during high speed movement. To get precise results, the nano-indentation technology was employed to measure Young\u0026rsquo;s moduli of the tibia. The experimental measurements and calculations show that the outside surface of the tibia has a higher moduli (7.45Gpa) than inside of the tibia (1.2662Gpa), and that the middle part of the outside surface of the tibia has a higher moduli than the medial malleolus at the distal end of the tibia (2.6Gpa). It was also verified that the tibial surface kept moist by spraying physiological saline over the surface has a lower moduli than the relatively dry tibia (13.02Gpa) without being kept moist (for more than 24 hours after dissected), and that the tibia kept moist has a more uniform distribution of moduli than the relatively dry tibia. Since it is known that the stiffness can be calculated from the Young\u0026rsquo;s moduli and the structure parameters of the subject directly, the characteristics of the distribution for the moduli along the tibia may influence the movement performance of the hind limbs and the whole body. During galloping and hopping, the limbs behave as springs, and the whole body is often modeled with a \u0026lsquo;\u0026lsquo;spring-mass model\u0026rsquo;\u0026rsquo;, thus the ability to modify the stiffness of these leg springs is essential to maintaining an efficient gait\u003csup\u003e[\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e]\u003c/sup\u003e. In this study, the Young\u0026rsquo;s moduli does not vary significantly between adjacent measurement regions (0.1 mm in width, 0.1 mm in length) of the outside surface of the tibia, especially of the tibia kept moist. Therefore, we conclude that the measurement region has relatively stable values of Young\u0026rsquo;s moduli during high speed movements.\u003c/p\u003e \u003cp\u003eIn this paper, to investigate the transient response of bones, the contact between the tibia and tarsal was simplified, and the substructure technique for dynamics was employed for solving the high nonlinear problem of dynamic response. From the results shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, it can be concluded that the body of the domestic cat experienced a brief but sizeable impact upon paw strike during galloping. Because the effects of soft tissues and cartilages were not considered, the frequency of the contact-impact is more higher than the frequency provided by other methods\u003csup\u003e[\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]\u003c/sup\u003e, and the frequency content of the impact loading can be effectively obtained from the capture experiments by using the high-speed video system. Though the 1:1 dedicated lens were taken during the capture experiments, the change of the stress-strain property (as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e) was hard to measure because of the high frequency during impact loading.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAdditionally, under the action of an external force and the existence of residual accelerations shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, secondary and multiple impacts generated between the rod and the rigid surface. This phenomenon indicates that the animal may use the energy produced by elastic deformation of the tibia, then the tibia has been hit to feed back. The process is similar to the release of elastic potential energy that is stored by the rod structure itself. During the impacts, the object being hit usually stores and releases a certain potential energy. In the process of moving contact between the end part of the robot leg and the soil, we can observe that there exists an energy feedback from the soil, which helps the rapid movement of cats and other quadrupeds.\u003c/p\u003e "},{"header":"6. Conclusions","content":"\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003cp\u003eIn this paper, the transient response for the dynamics of tibias and PLA legs during impact loading was numerically solved and experimentally verified. Our work confirmed that multiple impacts exist in tibias and PLA legs at the moment of the contact-impact, and that the transient dynamic response exists during the impact loading. The properties of the transient dynamic response of tibias and PLA legs can be employed to improve the future design for bionic mechanisms.\u003c/p\u003e \u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study was supported by the Tianjin Education Commission Research Project (2022KJ124).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Introduction\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eMeng-jun Song received his doctor\u0026apos;s degree from Hebei University of Technology, China, in 2013.He is currently a master\u0026apos;s student of Vehicle Engineering in the College of Automobile and Transportation, Tianjin University of Technology and Education, China. His research interests include vehicle dynamics and control, vehicle path planning and vehicle active safety.He is an associate professor at the College of Automobile and Transportation of Tianjin Vocational and Technical Normal University and has published 20 articles. His main research approach is mobile robot kinematics.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eE-mail:[email protected]\u003c/p\u003e\n\u003cp\u003eJing-gogn Wei received his master\u0026apos;s degree from Jilin University of Technology, China, in 2009.He is currently a Senior Engineer of Vehicle Engineering in the College of Automobile and Transportation, Tianjin University of Technology and Education, China. His research interests include Structural Lightweighting Optimization Design \u0026amp; Mechanical Lightweighting Optimization.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eE-mail:weijinggong@ tute.edu.cn\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eLi-ping Zhang received his bachelor\u0026apos;s degree from Shangrao Normal University, China, in 2010. She is currently a master\u0026apos;s student in Nankai University, China. Her research interests include Sports Biomechanics, Robotics Kinematics.\u003c/p\u003e\n\u003cp\u003eE-mail: [email protected]\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions:\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eFirst Author: Wrote and revised the manuscript, incorporating research findings and analysis.\u003c/p\u003e\n\u003cp\u003eSecond Author (Corresponding): Secured funding, supervised quality control, and managed submission.\u003c/p\u003e\n\u003cp\u003eThird Author: Finalized formatting and style to meet journal requirements.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability Statement\u003c/strong\u003e: All data generated or analyzed during this study are included in this article. We welcome open communication and collaboration regarding our findings. No confidential or restricted data were utilized in this work, and therefore, all information presented is freely available for further research and discussion. Should any additional clarifications or details be required, readers are encouraged to contact the corresponding author.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclaration of Interest Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no financial or personal relationships with other people or organizations that could inappropriately influence (bias) their work. Specifically, and have no financial interests, direct or indirect, in the subject matter or materials discussed in this manuscript. Furthermore, the authors have no relevant conflicts of interest to disclose with respect to the research, authorship, and/or publication of this article.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eChan, S. C., Yee, K. M., Chan, J. C. \u0026amp; Tan, Y. Q. 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Research on Kinematics and Biomechanics of the Bionic Mechanism[J].IOP Conference Series: Earth and Environmental Science, 170(4), (2018).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eAuthor \u0026amp; Introduction.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Biomechanics, Impact, Bionic mechanism, Tibia, Transient dynamic response","lastPublishedDoi":"10.21203/rs.3.rs-4894964/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4894964/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMusculoskeletal system vibrations are initiated at paw-strike in animal\u0026rsquo;s high-speed running. The short ground contact moment suggests that there exists a transient dynamic response of the impact between the leg and the ground, which is a high nonlinear problem and not well understood. From the anatomical measurement data of a domestic cat, a musculoskeletal system model of the quadruped animal was constructed in this study. The changes of muscle forces and joint moments were computed based on a high-speed motion sequence. The elastic moduli were measured and calculated for different parts of the tibia by a nano-indentation technique. On the basis of the measured material parameters, the substructure technique for dynamics was employed to numerically solve the contact-impact behavior of bio-materials and bionic components. To record the contact-impact process, high-speed videos (more than 10,000 fps) were taken during the capture experiments. Results demonstrated that multiple impacts existed in the tibia and the PLA leg at the moment of contact-impact. The results from this paper further reveal that the multiple contact-impact phenomena are adapted to cats during running, which may provide a certain support for selecting bionic components and improving the performance of the bionic mechanism.\u003c/p\u003e","manuscriptTitle":"Study on the Dynamic Response Characteristics of Bionic Legs During Instantaneous Ground Contact","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-09-24 14:07:42","doi":"10.21203/rs.3.rs-4894964/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"9c9bff10-8cb9-49ce-baa7-66f782c37f04","owner":[],"postedDate":"September 24th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":38075572,"name":"Physical sciences/Engineering/Mechanical engineering"},{"id":38075573,"name":"Biological sciences/Zoology/Biomechanics"}],"tags":[],"updatedAt":"2025-07-21T06:38:29+00:00","versionOfRecord":[],"versionCreatedAt":"2024-09-24 14:07:42","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4894964","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4894964","identity":"rs-4894964","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}