This paper presents the design and development of a single actuator tendon driven two-finger linkage gripper that can perform both strong pinch and adaptable cylindrical grasp. The gripper mechanism consists of an anthropomorphic linkage finger with an additional revolute joint driven by a single actuator and a fixed thumb. The gripper can achieve a maximum pinch force of 11.7 N and an adaptable grasping ranging from 30 mm to 145 mm diameter, making it suitable for various applications, such as pick-and-place tasks in robotics and automation. Moreover, complaint design makes it suitable for the safe physical human robot interaction. In addition, proposed linkage finger’s characteristics were evaluated through kinematic analysis, simulation and experimental tests of prototype. The proposed gripper design is simple, low-cost, and easy to implement, making it an attractive alternative to more complex and expensive gripper designs.

In-hand pivoting is one of the important manipulation skills that leverage robot grippers' extrinsic dexterity to perform repositioning tasks to compensate for environmental uncertainties and imprecise motion execution. Although many researchers have been trying to solve pivoting problems using mathematical modeling or learning-based approaches, the problems remain as open challenges. On the other hand, humans perform in-hand manipulation with remarkable precision and speed. Hence, the solution could be provided by making full use of this intrinsic human skill through dexterous teleoperation. For dexterous teleoperation to be successful, interfaces that enhance and complement haptic feedback are of great necessity. In this paper, we propose a cutaneous feedback interface that complements the somatosensory information humans rely on when performing dexterous skills. The interface is designed based on five-bar link mechanisms and provides two contact points in the index finger and thumb for cutaneous feedback. By integrating the interface with a commercially available haptic device, the system can display information such as grasping force, shear force, friction, and grasped object's pose. Passive pivoting tasks inside a numerical simulator Isaac Sim is conducted to evaluate the effect of the proposed cutaneous feedback interface.

Microinjection technology is applied widely in biomedical research for the purposes of gene manipulation and microinsemination. Generally, microinjection is performed under an optical microscope environment through image presentation of the targets. To perform the microinjection process, it is necessary to place multiple cells in the same droplet and perform multiple injections. This process requires observation at different magnifications for the injection and embryo transfer processes, with the operator required to change the magnification and light intensity each time. The complexity of the process can lead to variations in the accuracy, reproducibility, and productivity of the course of multiple microinjections. To simplify the microinjection process and reduce the workload on the operator, we propose a micromanipulation system that enables both wide-range and high-resolution video shooting with free viewpoint selection. The effectiveness of the proposed system is verified through microinjection experiments using porcine embryos.

In recent years, vision-based analysis systems of micro-objects in a microchannel have been actively developed. However, it is difficult to focus on high-speed micro-objects in a microchannel because the general height of a microchannel is approximately 10-100 μm, whereas the depth of focus of the objective lens is approximately 1-4 μm. Therefore, we propose a high-speed well-focused image-capturing microscope, which is a system with an objective lens attached to a vibration machine that moves the focus position rapidly by oscillating it up and down to capture well-focused images using a histogram-based algorithm. The proposed microscope system is verified experimentally to capture well-focused images of moving micro- objects.

Recent progress in LOC technology have made it possible to analyze cells in microfluidic devices in a microscopic environment. Vision-based cell analysis systems have a limitation in their analysis range owing to the small view area of the microscopic objective lens; thus, an expansion of the view area of microscopic observation is required. In this paper, we propose a view expansion system for photomicroscopy using the observing point movement of a Galvano mirror. We develop a prototype of the proposed view expansion system that consists of a microscope, Galvano mirror, and vision system. Then, the visual field of the developed microscopic view expansion system is derived experimentally. Finally, our system is demonstrated through panoramic image developing experiments. The effectiveness of the proposed system is verified by comparing it with a normal microscopic video shooting area.

In this paper, we propose a design method for the mechanism of a robotic arm suitable for a multicopter. The motion of an arm attached to a multicopter can possibly disturb the multicopter attitude. Our aim is to suppress this attitude disturbance by mechanical methods. In the proposed design method, the arm has the following features for suppressing the disturbance of the multicopter attitude. 1. The robotic arm can adjust its center of gravity (COG). 2. In the simplified model, the sum of the angular momentum of the rotating parts constituting the robot arm is equal to zero. These features can compensate for both the displacement of the COG of the arm and the counter torque generated from rotating parts, which cause the disturbance to the multicopter attitude. Since the disturbance of the multicopter attitude can be suppressed mechanically, the robotic arm does not require a special flight control law. In other words, this robotic arm has the advantage of being attached to a multicopter using a common, off-the-shelf, multipurpose flight controller. Furthermore, we discuss the mass distribution, power transmission method, and the moment of inertia of rotating parts, such as the speed reducer, motor, and arm. Additionally, we fabricate a prototype robotic arm with four joints and three degrees of freedom (DOFs), based on the proposed design method. Finally, the experiments showed that the robotic arm suppressed the disturbance of the multicopter attitude caused by the arm motion.

The research field of nongrasping manipulation is a maturing area in robotic motion control. However, the common principles of motion planning for nongrasping manipulation systems have not yet been established. This paper proposes the concept of virtual connecting manipulation as a generalized motion planning framework for nongrasping manipulation systems. As a preliminary result, we had previously realized a flower-stick juggling task called “propeller motion” using an actual experimental system. In this paper, we apply the virtual connecting manipulation concept to a flower-stick juggling task and analyze the generated motion from the view point of analytical methodology. We conduct a stability analysis of the generated cyclic motion of the flower stick by using a Poincaré map, and the analytical results show that the generated cyclic motion is asymptotically stable.

Conventional humanoids have achieved walking and running by independent controllers, even though a transition between these independent motions should be added to connect them while ensuring stability. In contrast, human selects walking/running at low/high speed in terms of energy efficiency, and transits between them naturally. This fact implies that human gait shares the inherent controller among walking and running despite their quite different appearances. Hence, we propose a “unified bipedal gait,” which includes walking, running, and the transition. The unified bipedal gait has the inherent controller: passive dynamic autonomous control (PDAC) with a damping and spring loaded inverted pendulum (D-SLIP) model. The PDAC constrains the humanoid natural dynamics by a virtual holonomic constraint (VHC) that degenerates the natural manifold of the states for stabilization. Compliance in the D-SLIP is capable to yield the required characteristics of walking/running: low/high compliant legs for walking/running. Thus, a novel VHC is designed to extract the required characteristics of walking/running from the D-SLIP dynamics and form the proper manifold. As a result, we achieved the unified bipedal gait that bifurcates to walking and running via the natural transition. The high energy efficiency was confirmed in this unified bipedal gait at any gait speed.

This paper proposes a quasi-passive dynamic autonomous control (Q-PDAC) for a three-dimensional (3-D) bipedal gait of humanoid robots from start points to goal points. The major approach for 3-D traveling is currently footstep planning by a constantly stable gait with an emphasis on its accurate and secure traveling. However, energy would potentially be wasted when the robot accurately travels according to the planned footsteps. In contrast, a limit-cycle-based gait possesses the good efficiency by a gait speed control, although the accurate and secure traveling is difficult for it. Its gait speed control is unfortunately not enough to freely travel on 3-D spaces: shortages of a turning speed control, trackability of horizontal speed, and stability of the bipedal gait. Hence, the Q-PDAC supplies three proper angular momenta by hip and ankle joints to achieve the turning motion and enhance the trackability of horizontal motion. Three angular momenta are simply designed consistent in the PDAC dynamics, and achieved the sufficient gait speed control for 3-D traveling. As a result, the robot can efficiently travel from the start point to the goal point while following a leader point not to collide with walls.

We propose a novel scheme for manipulating a passive object using an active plate. The objective of this study is to control an object's orientation with respect to the gravitational force direction by using an active plate for realizing hitherto unrealized object motion. In this context, motions of the object and active plate are designed to be cyclic. A state vector composed of the object's angle and angular velocity is defined, and the cyclic motion is expressed as a nonlinear discrete system. Fixed points of the state vector are searched for in the designed cyclic motion. A stability analysis around the fixed points is conducted using a Poincaré map. As a result, the fixed points are shown to be asymptotically stable. Finally, experimental results are used to verify that the object's angle can be manipulated with the designed cyclic motion using the plate.

Walking capability composed of stability and efficiency is one of the most important issues in the field of humanoid robots. An effective swing of the arms is expected to enhance the walking capability under the constraints from the limited body. We propose an arm-swing method to enhance the stability and efficiency by selecting optimal arm-swing strategy depending on the walking conditions. In this research, we select the optimal strategy between the support of the center of gravity (COG) tracking for stability and the walk without arm-swing for efficiency. To support the COG tracking, we employ a predictive control. States are defined as an inverted pendulum model and inputs are given as an inertial force of arm-swing. Input and output weights in the predictive control are adjustable by a support weight introduced in this paper. Selection of the optimal support weight by a selection algorithm for locomotion (Su-SAL) switches the two strategies by adjusting the ratio of input and output (I/O) weights. Su-SAL maximizes the efficiency while keeping the stability in comparison with the case of the constant support weight.

Flower stick juggling is a dexterous task done by skillful jugglers. We aim to realize dexterous tasks done by humans using robotic systems. This work focuses on flower stick juggling and proposes a feedback control strategy for a flower stick juggling task called “propeller” as one of the robotic dexterous manipulations. The propeller motion is modeled by considering combined flower stick and a robotic manipulator. We developed a control strategy that allows stable cyclic rotation of the flower stick in the air. The control parametars in the control strategy are searched through numerical simulation. Finally, the flower stick propeller motion is realized using an actual robotic system.

This paper proposes a novel concept for simultaneous cell shape and motion analysis in fast microchannel flows by implementing a multiobject feature extraction algorithm on a frame-straddling high-speed vision platform. The system can synchronize two camera inputs with the same view with only a tiny time delay on the sub-microsecond timescale. Real-time video processing is performed in hardware logic by extracting the moment features of multiple cells in 512 × 256 images at 4000 fps for the two camera inputs and their frame-straddling time can be adjusted from 0 to 0.25 ms in 9.9 ns steps. By setting the frame-straddling time in a certain range to avoid large image displacements between the two camera inputs, our frame-straddling high-speed vision platform can perform simultaneous shape and motion analysis of cells in fast microchannel flows of 1 m/s or greater. The results of real-time experiments conducted to analyze the deformabilities and velocities of sea urchin egg cells fast-flowing in microchannels verify the efficacy of our vision-based cell analysis system.

We present a high-speed vision-based morphological analysis system for fast-flowing cells in a microchannel. Real-time video processing is performed in hardware logic by extracting the moment features and bounding boxes of multiple cells in 512 × 256-pixel images at 2000 fps. The extracted cell regions are pushed into a first-in-first-out (FIFO) buffer for real-time image-based morphological analysis after being shrunk proportionally to a certain size. By extracting the bounding boxes of the cell regions using hardware logic and shrinking the cell region to a certain size to reduce processing time, our high-speed vision system can perform fast morphological analysis of cells at 500 cells/s in fast microchannel flows. Moreover, snapshots of all passed cells under the microscope are stored for offline verification. The results of real-time experiments conducted to analyze the size, eccentricity, and transparency of sea urchin embryos flowing fast in microchannels verify the efficacy of our vision-based cell analysis system.

An environmentally specific type of locomotion (e.g., bipedal or quadrupedal walking) is effective only under the specified environments. However, other conditions could cause physical body constraints and decrease mobility. Despite these constraints, legged robots are desired with high overall mobility such that they can walk under various conditions. Thus, a combination of types of locomotion is needed to maximize overall mobility. We have developed a gorilla-type robot, which can switch between bipedal and quadrupedal walking. A selection technique to optimize locomotion choice would be beneficial to the robot, which will experience challenging situations when walking through complex terrains, receiving disturbances, or malfunctioning. We present a selection algorithm for locomotion (SAL) that improves overall mobility by autonomously selecting the optimal locomotion. The falling risk of each locomotion mode is evaluated with a Bayesian network to represent the robot's situation. The evaluation function for the SAL determines the optimal locomotion choice based on falling risk and moving speed. In this paper, the SAL is used for two state variables of locomotion: gait (Ga-SAL) and speed (Sp-SAL). Both the simulations and experiments validated that the robot traveled efficiently in complex environments.

In this paper, we report on the development of a projection mapping system that can project RGB light patterns that are enhanced for three-dimensional (3-D) scenes using a GPU-based high-frame-rate (HFR) vision system synchronized with HFR projectors. Our system can acquire 512×512 depth images in real time at 500 fps. The depth image processing is accelerated by installing a GPU board for parallel processing of a gray-code structured light method using infrared (IR) light patterns projected from an IR projector. Using the computed depth images, suitable RGB light patterns to be projected are generated in real time for enhanced application tasks. They are projected from an RGB projector as augmented information onto a 3-D scene with pixel-wise correspondence even when the 3-D scene is time-varied. Experimental results obtained from enhanced application tasks for time-varying 3-D scenes such as (1) depth-based color mapping and (2) augmented reality (AR) spirit level, confirm the efficacy of our system.

This paper proposes a novel method for simultaneous cell shape and motion analysis in rapid microchannel flows based on a multi-object feature extraction algorithm with a frame-straddling high-speed vision platform. This system can synchronize two camera inputs that share the same view with only a very small sub-microsecond time delay. Real-time video processing is performed using the hardware logic by extracting the moment features of multiple cells at 2000 fps or more, which are obtained from the two camera inputs, and their frame-straddling time can be adjusted from 0 to 0.5 ms in 9.9 ns steps. After setting the frame-straddling time within a certain range to avoid large image displacements between the two camera inputs, the frame-straddling high-speed vision platform can perform simultaneous shape and motion analysis of cells in fast microchannel flows of 1 m/s or greater. The results of real-time experiments conducted to analyze the deformabilities, velocities, and shapes of fast-flowing sea urchin egg cells in straight and L-type microchannels verified the efficacy of our vision-based cell analysis system.

A manipulator which enables noncontact control of the position and attitude of an object on a plane by controlling a fluid field is proposed. The manipulator comprises 2 boards with grid pattern arrangements of holes and a parallel link robot which is capable of translational motion with 2 degrees of freedom and rotational motion with 1 degree of freedom in a plane. Air jets are discharged from the grid of holes. The direction of the air jets from the holes can be controlled by changing the positional relationship of the 2 boards by means of the parallel link robot. This makes it possible to form an air flow with a unidirectional fluid field or a vortex-like fluid field. When an object is placed in the fluid field, translational motion and rotational motion of the object are possible. If the position and attitude of the object are photographed and calculated using high speed camera and feedback control is performed, the position and attitude of the object can be controlled. The structure of the manipulator and the principle of control of the fluid field are described, and the possibility of controlling the translational motion and rotational motion of an object is demonstrated experimentally.

In this paper, a high-speed vision-based morphological analysis system for fast-flowing cells in a microchannel implementing a multi-object feature extraction algorithm on a high-speed vision platform is proposed. Real-time video processing is performed in hardware logic by extracting the moment features and bounding boxes of multiple cells in 512×256-pixel images at 2000 fps. The extracted cell regions are pushed into a first-in-first-out (FIFO) buffer for real-time image-based morphological analysis after being shrunk proportionally to a certain size. By extracting the bounding boxes of the cell regions using hardware logic and shrinking the cell region to a certain size to reduce processing time, our high-speed vision system can perform fast morphological analysis of cells at 2 ms/cell in fast microchannel flows. The results of real-time experiments conducted to analyze the size, eccentricity, and transparency of fertilized sea urchin eggs fast flowing in microchannels verify the efficacy of our vision-based cell analysis system.

Improved robot mobility is necessary in order to realize the expected uses of robots in the human life environment, which includes obstacles such as stairs and narrow passages. Because objects which humans require are often located on desks or tables, which are at some height above ground level, a long vertical dimension is also necessary in the robot design. To meet these requirements, this study focused on a type of mobile robot using an inverted pendulum, which enables a robot with a long vertical dimension to negotiate narrow passages. Since it is difficult for inverted pendulum robots to climb steps, this study proposes a planetary wheel mechanism which makes it possible for an inverted pendulum robot to ascend differences in level. This mechanism has an extremely simple structure, comprising a belt, pulley, and arm, and has the advantage of not requiring addition of a new actuator. This paper describes the composition of the proposed mechanism and its principle of operation, and also describes the control method. A prototype of the robot was produced, and it was shown experimentally that the robot was capable of climbing stairs with a height of 120 mm-130 mm at a rate of approximately 2.4 s per one step.

We propose a novel dot-pattern-projection three-dimensional (3-D) shape measurement method that can measure 3-D displacements of blink dots projected onto a measured object accurately even when it moves rapidly or is observed from a camera as moving rapidly. In our method, blinking dot patterns, in which each dot changes its size at different timings corresponding to its identification (ID) number, are projected from a projector at a high frame rate. 3-D shapes can be obtained without any miscorrespondence of the projected dots between frames by simultaneous tracking and identification of multiple dots projected onto a measured 3-D object in a camera view. Our method is implemented on a field-programmable gate array (FPGA)-based high-frame-rate (HFR) vision platform that can track and recognize as much as 15×15 blink-dot pattern in a 512×512 image in real time at 1000 fps, synchronized with an HFR projector. We demonstrate the performance of our system by showing real-time 3-D measurement results when our system is mounted on a parallel link manipulator as a sensing head.

This paper shows improvement of stability and efficiency for mobility using locomotion selection strategy. First strategy is the selection of a gait relying on locomotion rewards. The locomotion reward has been proposed as an indicator for selection algorithm based on Falling Risk and the moving speed. This strategy has achieved a capability of large changes of uncertainties, such as a steep slope. Second strategy is adjustment of moving speed by the extended locomotion reward that explicitly shows the relationship between the moving speed and Falling Risk. The robot aims at the maximum moving speed without a falling, and removes small changes of uncertainties as a result. We performed an experiment in order to confirm effects of two strategies in an environment that includes a rough terrain as a small uncertainty and two steps as a large uncertainty. The robot improved the moving speed about 37.5% from the case of only using the gait selection strategy.

We conducted high-frame-rate (HFR) video mosaicing for real-time synthesis of a panoramic image by implementing an improved feature-based video mosaicing algorithm on a field-programmable gate array (FPGA)-based high-speed vision platform. In the implementation of the mosaicing algorithm, feature point extraction was accelerated by implementing a parallel processing circuit module for Harris corner detection in the FPGA on the high-speed vision platform. Feature point correspondence matching can be executed for hundreds of selected feature points in the current frame by searching those in the previous frame in their neighbor ranges, assuming that frame-to-frame image displacement becomes considerably smaller in HFR vision. The system we developed can mosaic 512×512 images at 500 fps as a single synthesized image in real time by stitching the images based on their estimated frame-to-frame changes in displacement and orientation. The results of an experiment conducted, in which an outdoor scene was captured using a hand-held camera-head that was quickly moved by hand, verify the performance of our system.

We have developed a fast target tracking system that utilizes four cameras with lenses of different focal lengths to track an object without blurring images, even when the object moves in the depth direction away from the cameras in three-dimensional (3-D) space. This system can maintain a well-focused camera view by switching among the four input images, instead of using lens motor control. The multi-camera system was mounted on a two-axis mechanical active vision platform. The active vision control and camera-view switching are executed by processing color 512 × 512 images from the four camera inputs at 500 fps in real time on a high-speed vision platform. The performance of our system was verified by its tracking results for objects moving rapidly in 3-D space.

This paper deals with a method of locomotion selection based on Falling Risk and moving efficiency. The robot estimates information from sensors by solving state equation. The robot evaluates the Falling Risk as an indicator of uncertainty. Falling Risk is derived from measured information by using Bayesian Network. Locomotion selection during walking is modeled as a Semi-Markov Decision Process and the most appropriate locomotion is selected by using the greedy algorithm. As a result, the robot can move in the environment that is difficult to travel by single locomotion mode, maintaining the maximum moving efficiency.

For multi-contact robot motion, a closed chain is formed by robot links and the environment. This paper proposes a new methodology named “key joint method” for reducing the energy cost by adjusting an internal stress inside a closed chain. Firstly, we analyze the internal stress theoretically taking the degrees of freedom (DOF) and the number of position actuated joints into consideration, then a practical key joint method is proposed by changing a suitable redundant position controlled joint to be force control. After that, a parametric family is introduced for representing various of possible motions subjected to the robot dynamics and other constraints. Finally, a general optimization method is proposed for planning an energetically efficient multi-contact robot motion taking the motion trajectories and internal stress into consideration. As an example, the pace gait ladder decent motion is taken to explain the principle and realization of the proposed method. As experimental evaluation shows, the key joint method is effective for reducing the energy cost in the multi-contact motion.

We consider a 24-degrees-of-freedom monkey robot that is supposed to perform brachiation locomotion, i.e. swinging from one row of a horizontal ladder to the next one using the arms. The robot hand is constructed as a planar hook so that the contact point about which the robot swings is a passive hinge. We identify the 10 most relevant degrees of freedom for this underactuated mechanical system and formulate a tractable search: (a) introduce a family of coordination patterns to be enforced on the dynamics with respect to a path coordinate; (b) formulate geometric equality constraints that are necessary for periodic locomotion; (c) generate trajectories from integrable reduced dynamics associated with the passive hinge; (d) evaluate the energetic cost of transport. Moreover, we observe that a linear approximation of the reduced dynamics can be used for trajectory generation which allows us to incorporate the gradient of the cost function into the search algorithm.

This paper describes a 3-D biped walking over rough terrain. The robot is modeled as the special 3-D inverted pendulum that can change the length. The dynamics of the 3-D inverted pendulum is modeled as 2-D autonomous system by applying the Passive Dynamic Autonomous Control (PDAC) that is based on the assumption of point-contact of the robot foot and the virtual holonomic constraint as to robot joints. We analyze the stability of the 2-D autonomous system by use of Poincarée map, and derive the stable range over rough terrain. By applying the virtual compliance control to an actual robot “Gorilla Robot III”, the angle of the pendulum is modified. Finally, the 3-D biped walking over rough terrain is realized by use of the Gorilla Robot III.

This paper describes walk-to-brachiate transfer of a multi-locomotion robot (MLR). The MLR has multiple types of locomotion such as biped walking, quadruped walking and brachiation. This transfer is carried out through vertical ladder climbing as the robot must raise its body to start brachiating. As a result we have designed two stable transfer motions from walk to climb and from climb to brachiate, while contact situations and constraints of the robot are changing during the transfers. In addition, we have proposed a control algorithm by considering the reaction force from environment, and the setting of parameter is based on a kinetic model of the robot in order to tolerate relative position errors between the robot and its environments such as rungs of the ladder. The robustness of the designed motions with error corrections is experimentally verified.

The applicative field of activities of robots which have only one locomotion strategy is limited. As a mean of enhancing the mobile range, it is necessary to have various locomotion modes. Therefore, we focus on dynamic transitions between several kinds of locomotion modes adapting to environmental changes. In this paper, we aim to realize a stable locomotion along some unknown test courses with transition between biped and quadruped walks. To achive this transition, we propose a method to get environmental information and internal conditions. Robot plans locomotion based on recognition of test courses and estimate stability of walking using Bayesian Network. The effectiveness of proposed method is verified by experiments.

This paper proposes a three-dimensional biped dynamic walking algorithm based on passive dynamic autonomous control (PDAC) which is previously proposed. The robot dynamics is modeled as an autonomous system of a three-dimensional inverted pendulum by applying the PDAC concept that is based on the assumption of point contact of the robot foot and the virtual constraint as to robot joints. Due to autonomy, there are two conservative quantities named ldquoPDAC constantrdquo, that determine the velocity and direction of the biped walking. We also propose the convergence algorithm to make PDAC constant converge to arbitrary value, so that walking velocity and direction is controllable. Numerical simulation results validate proposed algorithm.

This paper addresses a three-dimensional biped dynamic walking control based on Passive Dynamic Autonomous Control (PDAC). In our previous work, the robot dynamics is modeled as a two-dimensional autonomous system of a three-dimensional inverted pendulum by applying the PDAC concept. In addition, the convergence algorithm based on conservative quantities named ¿PDAC constant¿ was proposed, so that walking velocity and direction is controllable. In this paper, we apply our control framework to an experimental robot ¿Multi-locomotion Robot¿; then the performance and the efficiency of the proposed control algorithm are verified by experiments.

An animal has a characteristic ratio of forefoot and rear legs so that its morphology can adapt to the living environment. Likewise, the structure of robot should be better fitted the locomotion in the working environment. This paper derives an optimal structure of the quadruped robot, which minimizes the sum of joint torques of the robot. Minimization of the joint torque allows to reduce the joint acceleration in walking motion, and hence to reduce energy consumption. Numerical simulation analyzed joint torques in each limb length and slope angle under walking on a slope. The optimal rate of rear leg length (RRL) is derived by the simulation as the physical structure. Our analysis suggests that the joint torque will increase as the slope angle becomes steeper in the case that the rear legs are shorter than forelegs. On the other hand, the joint torque will decrease as the slope angle is declined in the case that the forelegs are shorter than the rear legs. Experimental results validated the simulation analysis.