# Gait Analysis of Quadruped Robot Using the Equivalent Mechanism Concept Based on Metamorphosis

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## Abstract

The previous research regarding the gait planning of quadruped robot focuses on the sequence for lifting off and placing the feet, but neglects the influence of body height. However, body height affects gait performance significantly, such as in terms of the stride length and stability margin. We herein study the performance of a quadruped robot using the equivalent mechanism concept based on metamorphosis. Assuming the constraints between standing feet and the ground with hinges, the ground, standing legs and robot body are considered as a parallel mechanism, and each swing leg is regarded as a typical serial manipulator. The equivalent mechanism varies while the robot moves on the ground. One gait cycle is divided into several periods, including step forward stages and switching stages. There exists a specific equivalent mechanism corresponding to each gait period. The robot’s locomotion can be regarded as the motion of these series of equivalent mechanisms. The kinematics model and simplified model of the equivalent mechanism is established. A new definition of the multilegged robot stability margin, based on friction coefficient, is presented to evaluate the robot stability. The stable workspaces of the equivalent mechanism in the step forward stage of trotting gait under different friction coefficients are analyzed. The stride length of the robots is presented by analyzing the relationship between the stable workspaces of the equivalent mechanisms of two adjacent step forward stages in one gait cycle. The simulation results show that the stride length is larger with increasing friction coefficient. We herein propose a new method based on metamorphosis, and an equivalent mechanism to analyze the stability margin and stable workspace of the multilegged robot.

## Keywords

Quadruped robot Gait transference Metamorphosis Metamorphic mechanism Stability Stride length## 1 Introduction

Multilegged robots exhibit high adaptability to the environment because they do not require continuous support on the ground [1]. Therefore, they have become popular as a research topic in the recent decades. Multilegged robots include the biped robot, quadruped robot, hexapod robot, eight-legged robot, and other robots with more legs. A multilegged robot must rely on the support surface to walk. When the leg of the robot is in the stance phase, a particular constraint exists between the standing foot and support surface. While the standing leg transfers to the swing leg, the constraint is released. If we consider the constraint as a hinge, subsequently the system consists of a support surface, and the robot can be regarded as a specific mechanism. The equivalent mechanism of a multilegged robot exhibits a few similar characteristics with the metamorphic mechanism. The metamorphic mechanism is a novel mechanism proposed by DAI and JONES at the 25th ASME Biennial Mechanisms and Robotics Conference in 1998 [2]. This type of mechanism can change their topological configurations, effective links number, shapes, and degrees of freedom (DOF) [2, 3]. Unlike the traditional mechanism that only contains an unchanged topological configuration, the metamorphic mechanism has various topological configurations. Metamorphic mechanisms contain multiple configurations, and they can change their configuration to provide suitable and effective links and DOFs to adapt to different tasks. The metamorphic mechanism theory can be used to analyze the locomotion of multilegged robots [4, 5].

Quadruped robots not only exhibit better stability and greater load capacity than biped robots, but also has a simpler structure and easier control algorithms than hexapod robots and eight-legged robots. It can walk using statically stable gaits on complex terrains and walk quickly using dynamic stable gaits on even terrains. Research on quadruped robot focuses on structure design, kinematics analysis, dynamic analysis, gait planning, and walking control. The quadruped robot is in fact the “KUMO-I” robot [6], developed by Shigeo Hirose of the Tokyo Institute of Technology in Japan in 1976. This robot can walk by a statically stable gait. Boston Dynamics Engineering Company, in 2004, released a quadruped robot named “LittleDog” [7], that was used for machine learning, intuitive control, situational awareness, and studying topics such as irregular terrain. This company, in January 2006 and March 2008, developed two generations of quadruped robots named “BigDog” [8] for battlefield transports and, in February 2015, posted a video showing a man kicking the quadruped robot “Spot” to demonstrate the robot’s capacity to regain its balance [9]. The “HyQ” robot [10, 11, 12] that was created by the Italian Institute of Technology in Italy, is combined with an electronic motor and hydraulic drive system. It can achieve jumping gait with high dynamics, high speed, and other characteristics. In 2013, MIT released the quadruped robot named “MIT cheetah”, which can run at 22 km/h (6 m/s) with high efficiency [13, 14, 15, 16, 17, 18]. The Tokyo Institute of Technology developed a sprawling-type quadruped robot named TITAN-XIII in 2013 that is capable of high-speed and energy-efficient walking, and a wire-driven mechanism was used to move its joint [19, 20, 21]. In 2012, the Swiss Federal Institute of Technology Zurich (ETHZ) devised the StarlETH quadruped robot [21, 22] that can trot quickly using compliant joints. Additionally, in 2016, ETHZ published a quadruped robot named ANYmal to inspect oil and gas sites. It exhibits precise torque control, and is highly robust against impulsive loads during running or jumping with various sensors [23, 24, 25, 26]. The quadruped robot “Scalf” with a hydraulic drive system was built in Shandong university in 2012 [27, 28, 29, 30]. This robot can run with trotting gait using the established kinematics, inverse kinematics model, and gait planning algorithms. “The Baby Elephant” designed by Shanghai Jiao Tong University used a hybrid leg mechanism [31, 32, 33, 34, 35, 36, 37]. Chen et al. [38] researched the theory to minimize the energy expenditure by optimizing the frequency and length of the stride, and were implemented in “The Baby Elephant”.

A walking quadruped robot is a varying hybrid serial-parallel system [39]. In different periods, specific equivalent mechanisms are required. In the motion analysis of the quadruped robot, it is not suitable to regard the system composed of the robot and ground as a single specific mechanism [4, 40]. The locomotion of the quadruped robot can be considered as a series of moving hybrid serial-parallel mechanisms. One gait cycle of its gaits can be divided into several stages. In one stage, the system exhibits a specific mechanism. To investigate the gait performance of the quadruped robot, all the equivalent mechanisms in different stages must be studied. In the past, gait planning focused on the sequence for lifting off and placing the feet, but neglected the influence of body height. In fact, body height affects gait performance significantly, such as the stride length and stability margin.

In the context of the metamorphic theory, a new method is proposed to analyze the movement performance of multilegged robots herein. Assuming the constraints between standing feet and ground with hinges, one gait cycle is divided into several stages. In a particular stage, the ground, stance legs, and robot body form a parallel mechanism, and each swinging leg is regarded as a series manipulator. The whole system exhibits a hybrid serial-parallel mechanism. The walking movements of the multilegged robot are the motion and transition processes of these equivalent mechanisms. A new concept of stable margin is proposed to estimate the stability of gaits. A stable workspace of the equivalent mechanism in the step forward stage is analyzed. A new method to calculate the stride length of multilegged robots is presented by analyzing the relationship between the stable workspace of two adjacent mechanisms of the step forward stage in one gait cycle. The influence of friction coefficient on the stride size is analyzed, and a comparison of the stride length in different friction coefficients between the stance feet and ground are presented. The results of this study can be used to direct the motion planning of quadruped robots.

## 2 Structure and Gaits of Quadruped Robot

### 2.1 Structure of Quadruped Robot

### 2.2 Typical Gaits of Quadruped Robot

Other gaits exhibit similar characteristics with these two gaits; their walking can be regarded as a movement of a series of equivalent mechanisms. We do not discuss them herein.

### 2.3 Gait Transference

## 3 Kinematics of Quadruped Robot

This section describes the kinematics of the quadruped robot. Because of the different equivalent mechanisms of the robot in different periods, we must establish the general kinematics for the quadruped robot. For each equivalent mechanism, there exists the stance legs, swing legs, or these two types combined. Therefore, we establish the kinematics for the swing leg and stance leg, separately. The forward kinematics equations are established through the product of the exponential formula, and the inverse kinematics are derived by the geometric method.

### 3.1 Forward Kinematics of Swing Leg

*O*}, and the body reference frame {

*B*} that is fixed on the geometrical center of the robot body and moves with the body (Figure 9); the single leg reference frame {

*W*

_{i}}, which is established at the waist joint and fixed to the chassis of the robot, and the foot reference frame {

*A*

_{i}} which is located at the end of the swinging leg

*i*(foot of swinging leg

*i*). The configuration when all legs are fully extended is chosen as the home configuration. In this position, all the joint variables are assumed to be zero.

*B*} and frame {

*W*

_{i}} are both fixed on the robot body, the transformation matrix from frame {

*B*} to frame {

*W*

_{i}} is constant and can be given as follows:

The transformation matrix from frame {*W*_{i}} to frame {*A*_{i}} at the home configuration can be given as follows:

*j*th joint of the

*i*th leg. The twist for a rotation joint can be represented by

*i*is given by the product of the exponentials:

*O*} to frame {

*B*},

*O*}, and \(^{o} \varvec{p}_{B}\) is a 3 × 1 position vector, \(^{o} \varvec{p}_{B} \, = \,\left( {^{o} x_{b} ,\,^{o} y_{b} ,\,^{o} z_{b} } \right)^{\text{T}},\) that denotes the position of point B relative to frame {

*O*}.

### 3.2 Forward Kinematics of Standing Leg

*O*} is built as the global reference frame. The body reference frame {

*B*} is fixed on the robot body and moves with the body. The leg reference frame {

*W*

_{i}} is established at the waist joint and fixed to the body of robot, and the foot reference frame {

*A*

_{i}} is located at the foot of the supporting leg

*i*. The configuration when the leg is fully extended is regarded as the home configuration. In this position, all the joint variables are set to zero.

*l*

_{4}and

*l*

_{5}are parameters of the virtual sphere joint, they are equal to zero. The transformation matrix from frame {

*A*

_{i}} to frame {

*B*} at the home configuration is represented by

*i*is given in the following:

*O*} to frame {

*A*

_{i}},

*A*

_{i}} relative to frame {

*O*}, and \(^{o} \varvec{p}_{{A_{i} }}\) is a 3 × 1 position vector, \(^{o} \varvec{p}_{{A_{i} }} \, = \,\left( {^{o} x_{{A_{i} }} ,\,^{o} y_{{A_{i} }} ,\,^{o} z_{{A_{i} }} } \right)^{\text{T}},\) that denotes the position of point

*A*

_{i}relative to frame {

*O*}.

### 3.3 Inverse Kinematics of Robot

We used the geometric approach to obtain the inverse kinematics solution. In the geometric approach, the spatial geometry problem can be decomposed into several plane-geometry problems. Because of the simple structure of the robot leg, this can be performed easily. The variables of each leg joint (joint angles) can be solved using the tools of the plane geometry.

*i*with 3 DOFs as shown in Figure 11, the vectors \(^{o} \varvec{p}_{{W_{i} }}\) and \(^{o} \varvec{p}_{{A_{i} }}\) that are the position of points W

_{i}and A

_{i}, respectively, relative to the coordinate system {

*O*} are known, and the orientation of frame {

*W*

_{i}} relative to the frame {

*O*}, \({}_{{w{}_{i}}}^{o} \varvec{R}\) is given. The position of point

*A*

_{i}relative to {

*O*} can be obtained:

*O*}. Using the rotation transformation, Eq. (4) can be written as follows:

*A*

_{i}relative to frame {W

_{i}}. Therefore, we can obtain \({}^{{w_{i} }}\varvec{p}_{{A_{i} }}\) as follows:

*L*

_{i}is the projection length of leg

*i*in the

*xy*plane of frame {

*W*

_{i}}, and

*h*

_{i}is the projection length from foot

*A*

_{i}to the

*xz*plane of frame {

*W*

_{i}}.

*i*onto the

*xz*plane of {

*W*

_{i}}. Therefore, we can obtain the following equations:

*W*

_{i}}.

If the position of point *A*_{i} relative to frame {*O*}, and the position and orientation of the robot body relative to frame {*O*} are known, the variables of leg *i*, \(\alpha_{i} ,\;\beta_{i} ,\;\gamma_{i}\) can be solved by Eqs. (6)–(8).

## 4 Stability Analysis of Typical Gaits

In previous studies, the evaluation of statically stable gaits depends on the relationship between the vertical projection of the center of mass and the support polygon. If the vertical projection of the center of mass is located in the support polygon, the robot is stable; otherwise, the robot is not in the stable state. This approach ignores the impact of inertial force generated by the acceleration and deceleration of the robot. The “stability margin” is defined as the shortest distance from the vertical projection of the center of gravity to the boundaries of the support pattern in the horizontal plane [43]. However, in some cases, despite the vertical projection of the robot’s center of mass being in the support polygon, the reaction force of the ground required owing to the acceleration of the robot exceeds the force that the ground can provide. If the standing feet slips on the ground, the robot will not be in a stable state. Quadruped robots often use the dynamic stable gait to obtain high walking efficiency. The zero-moment point (ZMP) if often used to estimate the dynamic stability of robots [44]. When the ZMP is located in the standing foot, the robot is stable; otherwise, it is not. The “stability margin” of the dynamic stable gaits are defined as the shortest distance from the ZMP to the boundaries of the support pattern [45].

In fact, multilegged robots depend on the reactive force from the supporting face to maintain balance. If the supporting face can provide adequate reactive force, the robot can be stable. In this case, the ZMP of the robot is located in the support polygon. Otherwise, the robot is unstable.

*i*is shown in Figure 13. The reactive force is composed of one force vector \(\varvec{F}_{{A_{i} }} \, = \,\left( {f_{{A_{i} }}^{x}\: \,f_{{A_{i} }}^{y}\: \,f_{{A_{i} }}^{z} } \right)\), and one moment vector \(\varvec{\tau}_{{A_{i} }} \, = \,\left( {\tau_{{A_{i} }}^{x} \,\tau_{{A_{i} }}^{y} \,\tau_{{A_{i} }}^{z} } \right)\). The constraints between the standing foot and ground must satisfy the equation as follows:

*μ*is the static friction coefficient. Here, we choose the coefficient of sliding friction instead of the static friction coefficient.

*n*standing legs can be defined as

When \(S_{d} \, > \,0\), the robot is in the stable state. When \(S_{d} \, < \,0\), the robot is in the unstable state. When \(S_{d} \, = \,0\), the robot is in the critical stable state.

From Eq. (10) and Eq. (11), we can find that the stability of the multilegged robot depends primarily on the friction coefficient between the standing foot and ground.

On the CoM of the robot, a similar vector exists: \(\varvec{W}_{c} \, = \,\left( {f_{{x_{c} }} ,\,f_{{y_{c} }} ,\,f_{{z_{c} }} ,\,\tau_{{x_{c} }} ,\,\tau_{{y_{c} }} ,\,\tau_{{z_{c} }} } \right)^{\text{T}}\). If we ignore the influence of external forces, then \(\tau_{{x_{c} }} \, = \,\tau_{{y_{c} }} \, = \,\tau_{{z_{c} }} \, = \,0\), \(f_{{x_{c} }} \, = \,m_{c} a_{c}^{x}\), \(f_{{y_{c} }} \, = \,m_{c} a_{c}^{y}\), and \(f_{{z_{c} }} \, = \,m_{c} a_{c}^{z} \, + \,m_{c} g\).

*n*stance legs,

When the position and pose of the robot are known, and the acceleration of the CoM is measured, we can obtain the minimum friction forces to evaluate the stability of the robot, by calculating the least-squares solution of Eq. (13).

*C*is connected to axis

*A*

_{2}

*A*

_{4}by two rods. These two rods are connected together through one prismatic joint, and the rod farther away from mass point

*C*is connected to axis

*A*

_{2}

*A*

_{4}through one cylindrical pair in the center of pressure of the two footholds. The premise of this simplified equivalent mechanism is that the robot is in the stable state; in other words, the supporting surface can provide sufficient reaction force to the standing feet. After the switching stage, the robot moves into the next step forward stage. When the robot walks with leg 1 and leg 3 in the stance phase, its simplified equivalent mechanism is as shown as Figure 14(b).

*C*is obtained from following equation:

In this case, the ZMP of the robot is point *D*, which is the center point of the cylindrical joint on axis *A*_{2}*A*_{4}.

## 5 Stride Length of Trotting Gait

The relationship between the stable workspaces of the equivalent mechanisms of two adjacent step forward stages is determined by the robot movement. If there are no any intersections between these workspaces, the robot must walk unstably, or be running or jumping four feet off the ground. Instead, if some intersections exist between the two workspaces, a switching stage will occur in one gait cycle. In the trotting gait of the quadruped robot, switching stages exist. The stable workspaces of the equivalent mechanisms of two adjacent step forward stages must intersect together. Based on this principle, we can obtain the feasible stride length of the trotting gait under certain constraints.

## 6 Conclusions

A new method was proposed to analyze the performance of the multilegged robot. With regard to the metamorphic mechanism theory, one gait cycle of the robot was divided into several stages. Each stage of the system consisted of the supporting surface, and the robot exhibited a specific equivalent mechanism. By analyzing this series of equivalent mechanisms, the performance of the multilegged robots was obtained.

A new definition of multilegged robot stability margin was presented to evaluate the robot stability. This definition depended on the friction coefficient between the standing feet and supporting surface. It could be used to estimate whether the robot was in a stable state, regardless of statically stable gait or dynamic stable gait.

The stable workspace of the equivalent mechanism of the robot was defined and simulated based on the kinematics and simplified model of the robot in the step forward stage. Considering the change in body height, the maximum stride length was obtained by analyzing the stable workspaces of the equivalent mechanisms in two adjacent step forward stages of the trotting gait. The simulation results indicated that the stride length increased with the friction coefficient.

## Notes

### Authors’ Contributions

KX was in charge of the whole analyses and wrote the initial manuscript; PZ assisted with simulation analyses; XD revised the final manuscript. All authors read and approved the final manuscript.

### Authors’ Information

Kun Xu, born in 1981, is currently a lecturer at *Robotics Institute, Beihang University, China*. He received his PhD degree from *Beihang University, China*, in 2012. His research interests include legged robot and Space deployable structure.

Peijin Zi, born in 1993, is currently a PhD candidate at *Robotics Institute, Beihang University, China*.

Xilun Ding, born in 1967, is currently a professor and a PhD candidate supervisor at *Robotics Institute, Beihang University, China*. He received his PhD degree from *Harbin Institute of Technology, China*, in 1997. His research interests include the dynamics of compliant mechanical systems and robots, nonholonomic control of space robots, dynamics and control of aerial robots, and biomimetic robots.

### Competing Interests

The authors declare that they have no competing interests.

### Funding

Supported by National Natural Science Foundation of China (Grant Nos. 51775011, 91748201).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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