Configuration Synthesis and Performance Analysis of 1T2R Decoupled Wheel-Legged Reconfigurable Mechanism

Abstract

A method for configuration synthesis of a reconfigurable decoupled parallel mechanical leg is proposed. In addition, a configuration evaluation index is proposed to evaluate the synthesized configurations and select the optimal one. Kinematic analysis and performance optimization of the selected mechanism’s configuration are carried out, and the motion mode of the robot’s reconfigurable mechanical leg is selected according to the task requirements. Then, the robot’s gait in walking mode is planned. Firstly, based on bionic principles, the motion characteristics of a mechanical leg based on a mammalian model and an insect model were analyzed. The input and output characteristics of the mechanism were analyzed to obtain the reconfiguration principle of the mechanism. Using type synthesis theory for the decoupled parallel mechanism, the configuration synthesis of the chain was carried out, and the constraint mode of the mechanical leg was determined according to the constraint property of the chain and the motion characteristics of the moving platform. Secondly, an evaluation index for the complexity of the reconfigurable mechanical leg structure was developed, and the synthesized mechanism was further analyzed and evaluated to select the mechanical leg’s configuration. Thirdly, the inverse position equations were established for the mechanical leg in the two motion modes, and its Jacobian matrix was derived. The degrees of freedom of the mechanism are completely decoupled in the two motion modes. Then, the workspace and motion/force transmission performance of the mechanical leg in the two motion modes were analyzed. Based on the weighted standard deviation of the motion/force transmission performance, the global performance fluctuation index of the mechanical leg motion/force transmission is defined, and the structural size parameters of the mechanical leg are optimized with the performance index as the optimization objective function. Finally, with the reconfigurable mechanical leg in the insect mode, the robot’s gait in the walking operation mode is planned according to the static stability criterion.

1. Introduction

Wheel-legged robots have the advantages of both wheeled robots and legged robots, meaning they can not only move quickly on flat terrain but also have the ability to overcome obstacles. Compared with the serial mechanical leg, the parallel mechanical leg has the advantages of structural stability and a high load capacity, but it entails significant motion coupling and its control is complex compared with the series mechanism. The performance requirements of mobile robots differ depending on the terrain and working environment. For example, in the face of complex terrain, the robot needs to be flexible and changeable and, in the face of heavy-duty tasks, it must have high load-bearing performance. A reconfigurable decoupled parallel mechanism can switch between different motion modes according to the requirements of the task and its control difficulty is low, which can meet the needs of the mobile robot leg.

Robots with both wheels and legs can be divided into two types: those in which the wheels and legs are combined and those in which they are independent of one another [1]. In robots with a wheel–leg combination, the wheel is directly installed at the end of the leg, and switching between the wheel and leg motion modes is realized through the motion control of the leg pairs and the locking and unlocking of the wheels [2,3,4,5]. Switching between the wheel and leg modes is relatively simple, which reduces the complexity of the control of the mechanical leg on different terrains. Tedeschi et al. [6] designed a six-wheel-legged robot named Cassino Hexapod III, which adopts a wheel–leg combination. Each leg of the robot has three revolute pairs, which enable the robot to cross obstacles under the premise of limiting the wheels’ movement. In the wheel mode of a robot in which the wheels and legs work independently, the legs are folded while the wheels are in contact with the ground, relying on the rolling of the wheels to realize rapid movement. In the leg mode, the wheels are separated from the ground, and the movement is realized by the coordinated movement of the joints [7,8,9]. Xu et al. [10] proposed a six-wheel-legged mobile robot called NOROS. Each leg has three driving pairs and adopts a combination of independent wheels and legs, where the wheels are mounted on the mechanical legs.

The input and output of the decoupled parallel mechanism maintain a one-to-one correspondence, which can significantly reduce the complexity of the control of the parallel mechanism, so it can perform more complex actions. Zeng et al. [10] proposed a 2CRR + RRRR (C denotes a cylindrical pair; R denotes a revolute pair) decoupled parallel mechanism and derived its Jacobian matrix via the inverse position solution. The results show that the Jacobian matrix of the mechanism is a diagonal matrix, which verifies the decoupled characteristics of the mechanism. Cao et al. [11] proposed a method for the type synthesis of a 1T3R decoupled parallel mechanism based on the kinematic characteristics of the mechanism and screw theory. Xu et al. [12] conducted configuration synthesis of a 2R1T parallel mechanism and a 2R parallel mechanism with two completely decoupled rotational degrees of freedom based on the relationship between the constrained screw and the rotation axis of the mechanism. Qu et al. [13] carried out the configuration synthesis of a 2R1T parallel mechanism with redundant constraints based on the redundant constraints of the parallel mechanism. Zhang et al. [14] established a model for mapping the input and output vectors of a 1T2R fully decoupled parallel mechanism based on screw theory and synthesized a series of 1T2R completely decoupled parallel mechanisms. Li et al. [15] proposed a method for configuration synthesis of a high-stiffness 3T completely decoupled parallel mechanism based on screw theory, constructing closed-loop units in the chains to enhance the stiffness of the parallel mechanism. Wang et al. [16] proposed a method for configuration synthesis of a 3T2R decoupled hybrid mechanism based on screw theory and the atlas method and put forward the decoupled conditions for 3T and 2R parallel mechanisms according to their degrees of freedom and decoupled characteristics.

The traditional parallel mechanism has certain degrees of freedom. When facing the requirements of a task on diversified terrain, the number of degrees of freedom required by the mechanism will change, which may increase the complexity and control difficulty of the mechanism. A reconfigurable parallel mechanism can improve the adaptability to the working environment by changing its configuration. Wang et al. [17] proposed an 8R metamorphic mechanism based on origami folding, which can switch between 2 and 5 degrees of freedom, and designed a quadruped mobile robot with this metamorphic mechanism as a reconfigurable torso. Hu et al. [18] proposed a lockable parallel spherical pair S_dm based on screw theory and constructed a 2-SdmPU/SPS parallel mechanism by combining the S_dm pair, a prismatic pair, and a spherical pair, which can realize multi-mode motion of fixed-axis rotation and a one-dimensional variable axis. Palpacelli et al. [19] proposed a lockable spherical pair S_r. By locking the rotating shafts of S_r in different directions, the mechanism can switch between three kinematic pair modes: a spherical pair, universal pair, and revolute pair. Ye et al. [20] proposed the 3SvPS reconfigurable parallel mechanism by combining the vA (variable-Axis) pair, the prismatic pair, and the spherical pair. When the vA pair is in different phases, the mechanism can switch between four motion modes: 3T3R, 2T3R, 1T3R, and 1T2R. Yuan et al. [21] proposed the TS pair based on the traditional universal pair; they then proposed a 3(TS)P(TS) reconfigurable mechanism based on the TS pair that can switch between 1T, 1T1R, 2T1R, and other motion modes by changing the phase of the motion axis of the TS pair. Inspired by single-vertex origami, Kuang et al. [22] proposed a 5R spherical mechanism and combined a reconfigurable kinematic pair with prismatic and spherical pairs to construct a 3-(5R)PS reconfigurable parallel mechanism.

Based on the functional requirements of a reconfigurable four-wheel-legged robot, this study proposes a method for configuration synthesis of a reconfigurable decoupled wheel-legged mechanical leg. In the second section, the relationship between the input and output of the mechanical leg and the motion characteristics of the horse and ant limb structures is analyzed, and the reconfiguration principle of the mechanical leg is obtained. In the third section, the configuration synthesis of the chain is carried out based on a lockable universal pair. According to the constraint properties of the chain and the motion characteristics of the moving platform, the mechanism’s constraint mode is determined and the chain type is extended. In the fourth section, an evaluation index of the structural complexity of the reconfigurable mechanical leg is proposed, and the synthesized mechanism is further analyzed and evaluated to determine the configuration of the mechanical leg. In the fifth section, the inverse position equations of the mechanical leg in the two motion modes are established; in addition, the Jacobian matrix of the mechanical leg is derived, and its decoupled motion characteristics are analyzed. In the sixth section, the workspace and motion/force transmission performance of the mechanical leg in the two motion modes are analyzed and compared. The global performance fluctuation index of motion/force transmission of the mechanical leg is defined based on the weighted standard deviation of motion/force transmission performance. The particle swarm optimization algorithm is used to optimize the structural size parameters of the mechanical leg. As shown in the seventh section, the insect mode is selected as the motion mode of the mechanical leg when the robot is in the walking operation mode. Finally, based on the stability criterion of the robot, the robot’s gait is planned.

2. Configuration Synthesis of the Reconfigurable Decoupled Mechanical Leg

When a mobile robot needs to complete heavy tasks, its legs need a high load capacity; meanwhile, when it faces complex terrain, its legs need to be capable of flexible movement. The mammalian configuration has the advantages of a high load capacity and good stability, but it has the disadvantages of a small support area and a high center of gravity. The advantage of the insect-type configuration is that the support area is large and flexible, while its disadvantage is that its load capacity is low [23]. The reconfigurable mechanical leg with two motion modes has the above advantages and can switch between different motion modes according to the task requirements. In this study, the horse and ant are selected as models for designing the bionic structure of the reconfigurable mechanical leg.

2.1. Analysis of Biological Limbs’ Motion Characteristics

Compared with other four-legged mammals, horses have a unique physiological structure and motion patterns. It has strong adaptability to environments with complex terrain and a high load-bearing capacity, and it was the main means of cargo transportation for thousands of years. Thus, the horse is chosen as the bionic research object for the mechanical leg in the mammalian mode, and its skeleton and joint pairs are analyzed and studied, as shown in Figure 1.

It can be seen in Figure 1 that the front and hind legs of the horse have three joints: the femoral, knee, and foot joints. The coordinate system is set at the femoral pair, as shown in Figure 2. The twist system of the mammalian one-legged configuration is

$$\begin{array}{lllllllll}{\mathsf{\$}}_{q11}& =& (1\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \\ {\mathsf{\$}}_{q12}& =& (0\hfill & 1\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \\ {\mathsf{\$}}_{q13}& =& (0\hfill & 1\hfill & 0\hfill & ;\hfill & m_{13}\hfill & 0\hfill & n_{13})\hfill \end{array}$$

(1)

where m₁₃ and n₁₃ are parameters related to the position of the kinematic pair of the mechanical leg.

The reciprocal wrench of the mammalian one-legged configuration is

$$\begin{array}{lllllllll}{\mathsf{\$}}_{q11}^r& =& (-n_{13}\hfill & 0\hfill & m_{13}\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \\ {\mathsf{\$}}_{q12}^r& =& (0\hfill & 1\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \\ {\mathsf{\$}}_{q13}^r& =& (0\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 1)\hfill \end{array}$$

(2)

As can be seen from Equation (2), the direction of the translational degree of freedom changes with the movement of the mechanical leg. The translation along the Z-axis enables the robot to adjust its leg length according to the height of the ground to adapt to a change in terrain. Considering that an unstructured terrain in a dangerous and complex environment requires the ability to cross obstacles, and given the technological conditions of the actual mechanical structure, rotation about the X-axis, rotation about the Y-axis, and translation along the Z-axis are selected as the degrees of freedom of the mammalian leg.

In the mammalian mode, the twist system of the mechanical leg is

$$\begin{array}{l}{\mathsf{\$}}_{11}=(\begin{array}{lllllll}1\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 0\hfill \end{array})\\ {\mathsf{\$}}_{12}=(\begin{array}{lllllll}0\hfill & 1\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 0\hfill \end{array})\\ {\mathsf{\$}}_{13}=(\begin{array}{lllllll}0\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 1\hfill \end{array})\end{array}$$

(3)

Over a long period of evolution, insects have developed the ability to move flexibly over various complex terrains. As a common hexapod insect, ants can switch gaits according to the required walking speed and the terrain and have flexible motion performance. Therefore, ants are selected as the bionic research object for the mechanical leg in the insect mode. The ant’s body and leg structures are shown in Figure 3.

It can be seen in Figure 3 that the main joints of the ant leg are the coxa, coxa-femoral, and femoro-tibial joints. The configuration of the insect-type single leg is analyzed, as shown in Figure 4. The twist system of the insect-like leg structure is

$$\begin{array}{lllllllll}{\mathsf{\$}}_{q21}& =& (0\hfill & 0\hfill & 1\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \\ {\mathsf{\$}}_{q22}& =& (0\hfill & 1\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \\ {\mathsf{\$}}_{q23}& =& (0\hfill & 1\hfill & 0\hfill & ;\hfill & m_{23}\hfill & 0\hfill & n_{23})\hfill \end{array}$$

(4)

where m₂₃ and n₂₃ are parameters related to the position of the kinematic pair of the mechanical leg.

The reciprocal wrench of the insect one-legged configuration is

$$\begin{array}{lllllllll}{\mathsf{\$}}_{q21}^r& =& (-n_{23}\hfill & 0\hfill & m_{23}\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \\ {\mathsf{\$}}_{q22}^r& =& (0\hfill & 1\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \\ {\mathsf{\$}}_{q23}^r& =& (0\hfill & 0\hfill & 0\hfill & ;\hfill & 1\hfill & 0\hfill & 0)\hfill \end{array}$$

(5)

As can be seen from Equation (5), the direction of the translational degree of freedom changes with the movement of the mechanical leg. Considering that an unstructured terrain in a dangerous and complex environment requires the ability to cross obstacles, and given the technological conditions of the actual mechanical structure, rotation about the Z-axis, rotation about the Y-axis, and translation along the Z-axis are selected as the degrees of freedom of the insect legs.

In the insect mode, the twist system of the mechanical leg is

$$\begin{array}{l}{\mathsf{\$}}_{21}=(\begin{array}{lllllll}0\hfill & 0\hfill & 1\hfill & ;\hfill & 0\hfill & 0\hfill & 0\hfill \end{array})\\ {\mathsf{\$}}_{22}=(\begin{array}{lllllll}0\hfill & 1\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 0\hfill \end{array})\\ {\mathsf{\$}}_{23}=(\begin{array}{lllllll}0\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 1\hfill \end{array})\end{array}$$

(6)

Equations (3) and (6) indicate that when the mechanical leg is in the mammalian mode, the degrees of freedom are R_XR_YT_Z; when the mechanical leg is in the insect mode, the degrees of freedom are R_ZR_YT_Z.

2.2. Input–Output Analysis of 1T2R Decoupled Parallel Mechanism

The instantaneous motion of the moving platform of the parallel mechanism can be expressed by the twist [14] of the mechanism’s chain, namely,

$$V=\left[\begin{array}{c}\mathit{v}\\ \mathsf{\omega }\end{array}\right]={\displaystyle \sum _{j=1}^{F_i}{ \dot{\mathit{q}}}_{ij}}{{\displaystyle \mathsf{\$}}}_{ij}\hspace{1em}\hspace{1em}\hspace{1em}\left(i=1,2,\cdots ,n\right)$$

(7)

where V is the velocity vector of the moving platform; v denotes the linear velocity of the moving platform; ω represents the angular velocity of the moving platform; F_i is the connectivity of the i-th chain; n is the number of chains of the mechanism;${ \dot{\mathit{q}}}_{ij}$ represents the linear or angular velocity of the j-th kinematic pair in the i-th chain; and $_ij is the twist of the j-th single-degree-of-freedom kinematic pair in the i-th chain.

The chain constraint wrench limits the translational degree of freedom of the moving platform parallel to the X- and Y-axes, such that any component parallel to the X- and Y-axes in v is always 0.

The transmission wrench screw $_T [15] represents the generalized force transmitted by the driving pair to the moving platform, which has a zero reciprocity product with all other twists except for the actuated twist. Taking the reciprocal product of both sides of Equation (7) with $_Ti yields

$${\mathsf{\$}}_{\mathrm{T}i}\circ \mathit{V}={ \dot{\mathit{q}}}_{ij}{\mathsf{\$}}_{\mathrm{T}i}\circ {\mathsf{\$}}_{ij}$$

(8)

The matrix form of Equation (8) is

$$J_{\mathrm{dir}}\mathit{V}=J_{\mathrm{inv}}\dot{q}$$

(9)

where J_dir is the mechanism’s output Jacobian matrix and J_inv is the its input Jacobian matrix.

If J_inv is invertible, Equation (9) can be written as

$$\dot{q}=J_{\mathrm{inv}}^{-1}{J}_{\mathrm{dir}}\mathit{V}=J\mathit{V}$$

(10)

where J is the Jacobian matrix of the mechanism.

When J is a diagonal matrix, the mechanism is a completely decoupled parallel mechanism. It can be seen from Equation (10) that J_inv is a diagonal matrix. According to the matrix operation rules, $J_{\mathrm{inv}}^{-1}$ is still a diagonal matrix. When J_dir is a diagonal matrix, J is a diagonal matrix, and the mechanism is a completely decoupled parallel mechanism.

2.3. Reconfiguration Principle of Mechanical Leg

According to the analysis of the input and output of the mechanical leg in Section 2.2, the degrees of freedom of the mechanical leg in mammalian and insect modes are R_XR_YT_Z and R_ZR_YT_Z, respectively. According to the independent driving rule of the chain and the Jacobian matrix of the complete decoupled parallel mechanism, the output degrees of freedom of the moving platform controlled by the chain can be determined. In the mammalian model, the three chains control the moving platform’s rotation around the X-axis, rotation around the Y-axis, and translation along the Z-axis, respectively. In the insect mode, the three chains control the moving platform’s rotation around the Y-axis, rotation around the Z-axis, and translation along the Z-axis, respectively.

In this mechanism, the number of degrees of freedom and characteristics of chains I and II remain unchanged in the two motion modes. The chain with variable degrees of freedom is chain III. Chains I and II, respectively, control the translation of the moving platform along the Z-axis and the rotation of the moving platform around the Y-axis. Chain III controls the rotation of the moving platform around the X-axis and Z-axis in the two motion modes, respectively. According to the selection principle of the driving pair of the decoupled parallel mechanism, the axis of the driving pair of chain III is parallel to the X-axis and Z-axis, respectively, in the two motion modes. Therefore, the reconfigurable kinematic pair can be set as a lockable universal pair (Ur), which consists of two rotation axes, R₁ and R₂, as shown in Figure 5. When the mechanical leg is in the mammalian mode, R₂ is locked, U_r is in the R_X mode, and R₁ provides the rotational drive in the X-axis direction; when the mechanical leg is in the insect mode, R₁ is locked, U_r is in the R_Z mode, and R₂ provides the rotational drive in the Z-axis direction.

3. Chain Type Synthesis Process of Mechanism

3.1. Configuration Synthesis of Chain I

Chain I controls the moving platform’s translation along the Z-axis direction. The transmission wrench screw acting on the moving platform is a force line vector with one direction, always along the Z-axis direction.

When J_dir and J_inv satisfy the conditions of a diagonal matrix, J must be a diagonal matrix, and the mechanism is a completely decoupled mechanism. Therefore, [J_dir]₁₁ in the first row of J_dir must be the only non-zero element in that row, so $_T1 can only be

$${\mathsf{\$}}_{\mathrm{T}1}=(\begin{array}{lllllll}0\hfill & 0\hfill & 1\hfill & ;\hfill & 0\hfill & 0\hfill & 0\hfill \end{array})$$

(11)

According to the selection principle of the driving pair of the decoupled parallel mechanism [24], chain I has three types of driving pairs: a prismatic pair along the Z-axis, a revolute pair with the axis parallel to the X-axis, and a revolute pair with the axis parallel to the Y-axis. The twist expression of the driving pair is

$$\begin{array}{lllllllll}{\mathsf{\$}}_{111}& =& (0\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 1)\hfill \\ {\mathsf{\$}}_{112}& =& (1\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & Q_{112}\hfill & R_{112})\hfill \\ {\mathsf{\$}}_{113}& =& (0\hfill & 1\hfill & 0\hfill & ;\hfill & P_{113}\hfill & 0\hfill & R_{113})\hfill \end{array}$$

(12)

In the first case, substituting $_T1 and $₁₁₁ into [J_inv]₁₁₁ gives

$${[J_{\mathrm{inv}}]}_{111}={\mathsf{\$}}_{\mathrm{T}1}\circ {\mathsf{\$}}_{111}=1\ne 0$$

(13)

At this time, [J_inv]₁₁₁ is always non-zero and meets the condition.

In the second case, substituting $_T1 and $₁₁₂ into [J_inv]₁₁₂ gives

$${[J_{\mathrm{inv}}]}_{112}={\mathsf{\$}}_{\mathrm{T}1}\circ {\mathsf{\$}}_{112}=R_{112}$$

(14)

In the third case, substituting $_T1 and $₁₁₃ into [J_inv]₁₁₃ gives

$${[J_{\mathrm{inv}}]}_{113}={\mathsf{\$}}_{\mathrm{T}1}\circ {\mathsf{\$}}_{113}=R_{113}$$

(15)

It can be seen from Equations (14) and (15) that R₁₁₂ and R₁₁₃ are parameters related to the position of the kinematic pair. As long as the position of the kinematic pair is adjusted so that R₁₁₂ and R₁₁₃ are not zero, [J_inv]₁₁ is non-zero in all three cases.

3.1.1. The First Case

The twists $₁₁₁ and $_T1 of the chain I are

$$\begin{array}{l}{\mathsf{\$}}_{111}=(\begin{array}{lllllll}0\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 1\hfill \end{array})\\ {\mathsf{\$}}_{\mathrm{T}1}=(\begin{array}{lllllll}0\hfill & 0\hfill & 1\hfill & ;\hfill & 0\hfill & 0\hfill & 0\hfill \end{array})\end{array}$$

(16)

The chains of the parallel mechanism must have the motion characteristics of the moving platform. The mechanical leg needs to switch between the R_XR_YT_Z and R_ZR_YT_Z motion modes. The chain’s degree-of-freedom type should include 1T3R, so the basic structure type of the first type of chain I is 1T3R. The underlined kinematic pair represents the driving pair. The subscripts X, Y, and Z represent the axial direction of the kinematic pair at the initial position. If two or more kinematic pairs have the same right-hand subscript letter, then the relative kinematic pairs are parallel to one another. Otherwise, they are orthogonal. P represents the prismatic pair, R represents the revolute pair, and the detailed chain structure is shown in the first category in

Table 1. Structure of Basic Chain I.

Driving Category	Characteristics of Degrees of Freedom	Kinematic Pair Type	Chain Type
The first case	1T3R	1P3R	PZRXRYRZ
	2T3R	2P3R	PZRXRYRZPX; PZRXRYRZPY
		1P4R	PZRXRYRZ1RZ2
	3T3R	3P3R	PZPXPYRXRYRZ
		2P4R	PZPXRXRYRZ1RZ2; PZPYRXRYRZ1RZ2
		1P5R	PZRXRYRZ1RZ2RZ3;
The second case	2T3R	4R1P	RX1RX2RYRZPY
		5R	RX1RX2RX3RYRZ
	3T3R	4R2P	RX1RX2RYRZPXPY
		5R1P	RX1RX2RX3RYRZPX; RX1RX2RY1RY2RZPY RX1RX2RYRZ1RZ2PY; RX1RX2RYRZ1RZ2PX
		6R	RX1RX2RYRZ1RZ2RZ3; RX1RX2RX3RYRZ1RZ2 RX1RX2RY1RY2RZ1RZ2; RX1RX2RX3RY1RY2RZ
The third case	2T3R	4R1P	RY1RY2RXRZPX
		5R	RY1RY2RXRZ1RZ2
	3T3R	4R2P	RY1RY2RXRZPXPY
		5R1P	RY1RY2RXRZ1RZ2PY; RY1RY2RXRZ1RZ2PX
		6R	RY1RY2RXRZ1RZ2RZ3

. In order to simplify the structure and make the analysis intuitive, it is assumed that the axes of adjacent kinematic pairs are orthogonal or parallel.

3.1.2. The Second Case

The twists $₁₁₂ and $_T1 of the chain I are

$$\begin{array}{lllllllll}{\mathsf{\$}}_{112}& =& (1\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & Q_{112}\hfill & R_{112})\hfill \\ {\mathsf{\$}}_{\mathrm{T}1}& =& (0\hfill & 0\hfill & 1\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \end{array}$$

(17)

When the driving pair is a revolute pair with the axis parallel to the X-axis direction, it is equivalent to replacing the translation drive along the Z-axis with a 2R parallel revolute pair. Therefore, the basic degree-of-freedom type for this form of chain I is 2T3R, and the specific chain structure is shown in the second category in Table 1.

3.1.3. The Third Case

The twists $₁₁₃ and $_T1 of the chain I are

$$\begin{array}{lllllllll}{\mathsf{\$}}_{113}& =& (0\hfill & 1\hfill & 0\hfill & ;\hfill & P_{113}\hfill & 0\hfill & R_{113})\hfill \\ {\mathsf{\$}}_{\mathrm{T}1}& =& (0\hfill & 0\hfill & 1\hfill & ;\hfill & 0\hfill & 0\hfill & 0)\hfill \end{array}$$

(18)

When the driving pair is a revolute pair with the axis parallel to the Y-axis direction, it is equivalent to replacing the translation drive along the Z-axis with a 2R parallel revolute pair. Therefore, the basic degree-of-freedom type for this form of chain I is 2T3R, and the specific chain structure is shown in the third case in Table 1.

Table 1 shows the possible structural types of chain I. It should be noted that in order to simplify the structure of the mechanism, the existence of redundant kinematic pairs is not considered for the chain described in this paper. In addition, due to space limitations, only one arrangement case is given for each type of chain. Changing the order of kinematic pairs in the chain is relatively simple, and such cases are not listed one by one.

3.2. Configuration Synthesis of Chain II

Chain II controls the moving platform’s rotation around the Y-axis. The transmission wrench screw $_T2 acting on the moving platform is a couple parallel to the Y-axis direction.

$${\mathsf{\$}}_{\mathrm{T}2}=(\begin{array}{lllllll}0\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & 1\hfill & 0\hfill \end{array})$$

(19)

The twist of the driving pair with a rotation component around the X-axis and Z-axis generates output that will affect the decoupled characteristics of the moving platform’s rotation around the Y-axis. The twist of the driving pair is

$${\mathsf{\$}}_{21}=(\begin{array}{lllllll}0\hfill & 1\hfill & 0\hfill & ;\hfill & P_{21}\hfill & 0\hfill & R_{21}\hfill \end{array})$$

(20)

Thus,

$${[J_{\mathrm{inv}}]}_{22}={\mathsf{\$}}_{\mathrm{T}2}\circ {\mathsf{\$}}_{21}=1\ne 0$$

(21)

According to the characteristics of the chain’s degrees of freedom, chain II’s structure can be obtained, as shown in

Table 2. Structure of Basic Chain II.

Connectivity	Characteristics of Degrees of Freedom	Kinematic Pair Type	Chain Type
4	1T3R	3R1P	RYPZRXRZ
5	2T3R	3R2P	RYPYPZRXRZ; RYPXPZRXRZ
		4R1P	RYPYRX1RX2RZ; RYPZRX1RX2RZ
		5R	RYRX1RX2RX3RZ
6	3T3R	3R3P	RYPXPYPZRXRZ
		4R2P	RYPYPXRX1RX2RZ; RYPZPXRX1RX2RZ; RYPYPZRXRZ1RZ2; RYPXPZRXRZ1RZ2
		5R1P	RYPYRX1RX2RZ1RZ2; RYPXRX1RX2RX3RZ; RYPXRX1RX2RZ1RZ2; RYPZRX1RX2RZ1RZ2; RYPZRXRZ1RZ2RZ3
		6R	RY1RX1RX2RZ1RZ2RZ3; RYRX1RX2RX3RZ1RZ2

.

3.3. Configuration Synthesis of Chain III

Chain III can be equivalent to a lockable universal pair in series with other kinematic pairs, which is used to control the rotation of the moving platform around the X-axis and Z-axis in the mammalian and insect modes. For the convenience of analysis, the other kinematic pairs in the variable-degree-of-freedom chain, except for the lockable universal pair, are called constrained chains.

Chain III controls the rotation of the moving platform around the X-axis and Z-axis, respectively, in the two motion modes. When the mechanical leg is in the mammalian mode, the transmission wrench screw acting on the moving platform via the variable-degree-of-freedom chain $_T31 is a couple parallel to the X-axis. When the mechanical leg is in the insect mode, the transmission wrench screw acting on the moving platform via the chain with variable degrees of freedom $_T32 is a couple parallel to the Z-axis, which is expressed as

$$\begin{array}{l}{\mathsf{\$}}_{\mathrm{T}31}=(\begin{array}{lllllll}0\hfill & 0\hfill & 0\hfill & ;\hfill & 1\hfill & 0\hfill & 0\hfill \end{array})\\ {\mathsf{\$}}_{\mathrm{T}32}=(\begin{array}{lllllll}0\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & 0\hfill & 1\hfill \end{array})\end{array}$$

(22)

It can be seen that the twist of the lockable universal pair in the mammalian mode is $₃₁₁. The twist of the lockable universal pair in the insect mode is $₃₁₂, which is expressed as

$$\begin{array}{l}{\mathsf{\$}}_{311}=(\begin{array}{lllllll}1\hfill & 0\hfill & 0\hfill & ;\hfill & 0\hfill & Q_{311}\hfill & R_{311}\hfill \end{array})\\ {\mathsf{\$}}_{312}=(\begin{array}{lllllll}0\hfill & 0\hfill & 1\hfill & ;\hfill & P_{312}\hfill & Q_{312}\hfill & 0\hfill \end{array})\end{array}$$

(23)

When the mechanical leg is in the mammalian model, [J_inv]₃₃₁ of chain III is

$${[J_{\mathrm{inv}}]}_{331}={\mathsf{\$}}_{\mathrm{T}31}\circ {\mathsf{\$}}_{311}=1\ne 0$$

(24)

When the mechanical leg is in the insect mode, [J_inv]₃₃₂ of chain III is

$${[J_{\mathrm{inv}}]}_{332}={\mathsf{\$}}_{\mathrm{T}32}\circ {\mathsf{\$}}_{312}=1\ne 0$$

(25)

It can be seen from Equations (24) and (25) that [J_inv]₃₃₁ and [J_inv]₃₃₂ are always non-zero, which meets the requirements of the Jacobian matrix for the fully decoupled parallel mechanism.

Because the reciprocal product of the transmission wrench screw and the twist system of the constraint chain is 0 and the motion characteristics of the chain include the motion characteristics of the moving platform, in the configuration synthesis process of the constrained chains, attention should be paid to the following:

(1)

The motion characteristics of the constraint chain do not include rotation around the X-axis and Z-axis, but they must include rotation around the Y-axis.

(2)

The intersection between the motion characteristics of the constrained chain and those of chains I and II is the translation along the Z-axis.

It can be seen from the analysis that there are three cases—2, 3, and 4 degrees of freedom—for the constrained chain. According to the screw-theory-based method, the constrained chain structure can be obtained via configuration synthesis of the constrained chain, as shown in

Table 3. Constrained chain structure.

Connectivity	Characteristics of Degrees of Freedom	Kinematic Pair Type	Chain Type
2	1T1R	1P1R	PZRY
3	2T1R	2P1R	PXPZRY; PZPXRY; PZPYRY
		1P2R	PXRY1RY2; PZRY1RY2; RY1PYRY2
		3R	RY1RY2RY3
4	3T1R	3P1R	PXPYPZRY
		2P2R	PYPXRY1RY2; PYPZRY1RY2
		3R1P	PYRY1RY2RY3

.

3.4. Feasible Constraint Model

The moving platform is subjected to a constraint force ${\mathsf{\$}}_{\mathrm{F}1}^r$ along the X-axis direction and a constraint force ${\mathsf{\$}}_{\mathrm{F}2}^r$ along the Y-axis direction in both the mammalian and insect modes. In the mammalian mode, the moving platform is subjected to a constraint couple ${\mathsf{\$}}_{\mathrm{C}1}^r$ around the Z-axis. In the insect mode, the moving platform is subjected to a constraint couple ${\mathsf{\$}}_{\mathrm{C}2}^r$ around the X-axis.

When a chain provides a constraint, the chain’s constraint type [25] may be

$$A={\mathsf{\$}}_{{\mathrm{F}}_1}^r; B={\mathsf{\$}}_{{\mathrm{F}}_2}^r; C={\mathsf{\$}}_{{\mathrm{C}}_1}^r/{\mathsf{\$}}_{{\mathrm{C}}_2}^r$$

(26)

When a chain provides two constraints, the chain’s constraint type may be

$$\left\{\begin{array}{l}D=AB={\mathsf{\$}}_{{\mathrm{F}}_1}^r\cup {\mathsf{\$}}_{{\mathrm{F}}_2}^r\hfill \\ E=AC={\mathsf{\$}}_{{\mathrm{F}}_1}^r\cup {\mathsf{\$}}_{{\mathrm{C}}_1}^r/{\mathsf{\$}}_{{\mathrm{C}}_2}^r\hfill \\ F=BC={\mathsf{\$}}_{{\mathrm{F}}_2}^r\cup {\mathsf{\$}}_{{\mathrm{C}}_1}^r/{\mathsf{\$}}_{{\mathrm{C}}_2}^r\hfill \end{array}\right.$$

(27)

where $\cup$ represents the union operation.

When a chain provides three constraints, the chain’s constraint type may be

$$G=ABC={\mathsf{\$}}_{{\mathrm{F}}_1}^r\cup {\mathsf{\$}}_{{\mathrm{F}}_2}^r\cup {\mathsf{\$}}_{{\mathrm{C}}_1}^r/{\mathsf{\$}}_{{\mathrm{C}}_2}^r$$

(28)

When the chain is unconstrained, its constraint type is

$$H=O$$

(29)

In the configuration synthesis of the mechanical leg, a constraint force and a constraint couple exist in each chain at the same time. The geometric conditions of the space required are complex, so a feasible constraint mode with only three independent constraints is selected. The feasible constraint modes that meet these requirements are ABC, CDH, AEH, and GHH. The constraint characteristics of each chain meeting requirements in different feasible constraint modes are shown in

Table 4. Constraint characteristics of each chain in feasible constraint mode.

Feasible Constraint Pattern	Chain	Degree-of-Freedom Type	Constraint Type
ABC	1	2T3R	A/B
	2	2T3R	B/A
	3	3T2R	C
CDH	1	1T3R/3T3R	D/H
	2	3T3R/1T3R	H/D
	3	3T2R	C
AEH	1	2T3R/3T3R	A/H
	2	3T3R/2T3R	H/A
	3	2T2R	E
GHH	1	3T3R	H
	2	3T3R	H
	3	1T2R	G

.

3.5. Selection of Feasible Constraint Mode

The mechanical leg is a 1T2R-type parallel mechanism in the mammalian and insect modes. It may generate parasitic motion in the rotation process, which increases the difficulty and complexity of planning the mechanism’s trajectory. Based on the topology design theory of the parallel mechanism, this study analyzes the motion characteristics of the moving platform in the four feasible constraint modes determined in Section 3.4.

The POC set calculation equations [26] for series and parallel mechanisms are

$$M_b=\underset{i=1}{\overset{m}{\cup }}{M}_{J_i}$$

(30)

$$M_{Pa}=\underset{i=1}{\overset{v+1}{\cap }}{M}_{b_i}$$

(31)

where $M_{J_i}$ is the POC set of the i-th kinematic pair; $M_b$ is the POC set of the serial chain; M_Pa is the POC set of the moving platform of the mechanism; $M_{b_i}$ is the POC set of the end of the i-th chain; m is the number of kinematic pairs; and v is the number of independent loops.

Because the motion characteristics of the moving platform relative to the static platform are related to the position of the base point, the center point of the moving platform is generally selected as the reference point of a moving coordinate system; thus, the center point of the moving platform is selected as the base point. The center point O₁ of the moving platform is positioned on the axis of the kinematic pair at the tail end of the chain with the fewest degrees of freedom among the three chains, which will reduce the occurrence of non-independent motion characteristics. Therefore, the base point O₁ is located on the axis of the kinematic pair at the end of the chain with the fewest degrees of freedom.

3.5.1. Motion Characteristics of Moving Platform in ABC Constraint Mode

The degree-of-freedom type for chains I and II is 2T3R, and the base point O₁ can be located on the axis of the kinematic pair at the end of chain I or chain II.

The base point O₁ is located on the axis of the end pair of chain I. The POC set of the chain I is

$$M_1=\left[\begin{array}{c}{t}^2\left(\perp {\mathrm{R}}_{11}\right)\\ r^3\end{array}\right]$$

(32)

where t and r represent translational and rotational degrees of freedom, respectively, and the superscript is the number of translational or rotational degrees of freedom.

The base point O₁ is located on the axis of the end pair of chain II. The POC set of the chain II is

$$M_2=\left[\begin{array}{c}{t}^2\cup \left\{t^1\left(\perp \rho \right)\right\}\\ r^3\end{array}\right]$$

(33)

where {*} denotes the parasitic translations induced by rotations.

The degree-of-freedom type for chain III is 3T2R, and its POC set is

$$M_3=\left[\begin{array}{c}{t}^3\\ r^2\end{array}\right]$$

(34)

As can be seen from Equations (32)–(34), M_Pa is

$$M_{pa}=M_1\cap M_2\cap M_3=\left[\begin{array}{c}{t}^2\\ r^3\end{array}\right]\cap \left[\begin{array}{c}{t}^2\cup \left\{t^1\left(\perp \rho \right)\right\}\\ r^3\end{array}\right]\cap \left[\begin{array}{c}{t}^3\\ r^2\end{array}\right]=\left[\begin{array}{c}{t}^2\left(\perp {\mathrm{R}}_{11}\right)\\ r^2\end{array}\right]$$

(35)

where t²(⊥R₁₁) denotes that there are two finite translations in the plane perpendicular to the axis of R₁₁.

Given that the mechanism has three DOFs, the POC set has only three independent elements, and any three of the four elements of M_pa can be taken as independent elements. Thus, the mechanism has one dependent parasitic motion.

3.5.2. Motion Characteristics of Moving Platform in CDH Constraint Mode

It can be seen from Table 4 that the degree-of-freedom type for chain I or chain II is 1T3R. When the degree-of-freedom type for chain I is 1T3R, the base point O₁ is located on the axis of the end pair of chain I. The POC set of chain I is

$$M_1=\left[\begin{array}{c}{t}^1\left(\parallel Z\right)\\ r^3\end{array}\right]$$

(36)

When the degree-of-freedom type for chain II is 1T3R, the base point O₁ is located on the axis of the end pair of chain II. The POC set of chain II is

$$M_2=\left[\begin{array}{c}{t}^2\left(\perp {\mathrm{R}}_{21}\right)\\ r^3\end{array}\right]$$

(37)

where t²(⊥R₂₁) denotes that there are two finite translations in the plane perpendicular to the axis of R₂₁.

When the base point O₁ is located on the axis of the end pair of chain I, the number of independent elements of the POC set for the end member is equal to the number of degrees of freedom of the chain.

When the degree-of-freedom type for chain I is 1T3R, that for chain II is 3T3R, and its POC set is

$$M_2=\left[\begin{array}{c}{t}^3\\ r^3\end{array}\right]$$

(38)

The constraint type of chain III is C. Similarly, the POC set of chain III is

$$M_3=\left[\begin{array}{c}{t}^3\\ r^2\end{array}\right]$$

(39)

As can be seen from Equations (36), (38) and (39), M_Pa is

$$M_{pa}=M_1\cap M_2\cap M_3=\left[\begin{array}{c}{t}^1\left(\parallel Z\right)\\ r^3\end{array}\right]\cap \left[\begin{array}{c}{t}^3\\ r^3\end{array}\right]\cap \left[\begin{array}{c}{t}^3\\ r^2\end{array}\right]=\left[\begin{array}{c}{t}^1\left(\parallel Z\right)\\ r^2\end{array}\right]$$

(40)

Given that the mechanism has three DOFs, the POC set has three independent elements, and the moving platform has no dependent parasitic motion.

3.5.3. Motion Characteristics of Moving Platform in AEH Constraint Mode

The constraint type of chain III is E, the degree-of-freedom type is 2T2R, and the base point O₁ is located on the axis of the end pair of chain III. The POC set of chain III is

$$M_3=\left[\begin{array}{c}{t}^2\\ r^2\end{array}\right]$$

(41)

The constraint type of chain I is A, and the degree-of-freedom type is 2T3R. The POC set of chain I is

$$M_1=\left[\begin{array}{c}{t}^2\cup \left\{t^1\left(\perp \rho \right)\right\}\\ r^3\end{array}\right]$$

(42)

The constraint type of chain II is H, and the degree-of-freedom type is 3T3R. The POC set of chain II is

$$M_2=\left[\begin{array}{c}{t}^3\\ r^3\end{array}\right]$$

(43)

As can be seen from Equations (41)–(43), M_Pa is

$$M_{pa}=M_1\cap M_2\cap M_3=\left[\begin{array}{c}{t}^2\\ r^2\end{array}\right]\cap \left[\begin{array}{c}{t}^2\cup \left\{t^1\left(\perp \rho \right)\right\}\\ r^3\end{array}\right]\cap \left[\begin{array}{c}{t}^3\\ r^3\end{array}\right]=\left[\begin{array}{c}{t}^2\\ r^2\end{array}\right]$$

(44)

Given that the mechanism has three DOFs, the POC set has only three independent elements, and any three of the four elements of M_pa can be taken as independent elements. The mechanism has one dependent parasitic motion.

3.5.4. Motion Characteristics of Moving Platform in GHH Constraint Mode

The constraint type of chain III is G, the degree-of-freedom property is 1T2R, and the base point O₁ is located on the axis of the end pair of chain III. The base point O₁ is not located on the axis of the lockable universal pair [26], so the POC set of chain III is

$$M_3=\left[\begin{array}{c}{t}^1\left(\parallel {\mathrm{P}}_{31}\right)\cup \left\{t^1\left(\perp \rho \right)\right\}\\ r^2\end{array}\right]$$

(45)

The POC set of chain I is

$$M_1=\left[\begin{array}{c}{t}^3\\ r^3\end{array}\right]$$

(46)

The POC set of chain II is

$$M_2=\left[\begin{array}{c}{t}^3\\ r^3\end{array}\right]$$

(47)

As can be seen from Equations (45)–(47), M_Pa is

$$M_{pa}=M_1\cap M_2\cap M_3=\left[\begin{array}{c}{t}^1\left(\parallel {\mathrm{P}}_{31}\right)\cup \left\{t^1\left(\perp \rho \right)\right\}\\ r^2\end{array}\right]\cap \left[\begin{array}{c}{t}^3\\ r^3\end{array}\right]\cap \left[\begin{array}{c}{t}^3\\ r^3\end{array}\right]=\left[\begin{array}{c}{t}^1\left(\parallel {\mathrm{P}}_{31}\right)\cup \left\{t^1\left(\perp \rho \right)\right\}\\ r^2\end{array}\right]$$

(48)

Given that the mechanism has three DOFs, the POC set has only three independent elements, and any three of the four elements of M_pa can be taken as independent elements. The mechanism has one dependent parasitic motion.

According to this analysis, in the CDH constraint mode, the moving platform has three independent elements in the POC set in the two motion modes. M_pa indicates that there is no dependent parasitic motion on the moving platform. Therefore, CDH is chosen as the constraint mode of the mechanical leg.

3.6. Extension of Chain Structure Type

The chain can combine a single-degree-of-freedom kinematic pair into a multi-degree-of-freedom kinematic pair. For example, the revolute pair and the prismatic pair can be combined into a cylindric pair, the spherical pair can replace a plurality of revolute pairs, and the single-degree-of-freedom kinematic pair can be combined into a composite kinematic pair, such as a Pa pair. The structure of the extended chain is shown in

Table 5. Structure of chain I.

Chain	Basic Chain Structure	Chain Structure with Multi-Degree-of-Freedom Kinematic Pair	Chain Structure Containing Closed-Loop Structures
Chain I	PZRXRYRZ	PZS; CZUXY	—

,

Table 6. Structure of chain II.

Chain	Basic Chain Structure	Chain Structure with Multi-Degree-of-Freedom Kinematic Pair	Chain Structure Containing Closed-Loop Structures
Chain II	RYPXPYPZRXRZ	RYPXPYPZUXZ	RYPXPYPaYUXZ; RYPX/YPaX/YPaX/YUXZ; RYPaY/XPaX/YPaX/YUXZ
	RYPXPYPZRXRZ	CYPXPZUXZ; CYCZCX	CYPX/ZPaYUXZ; CYPaYPaYUXZ
	RYPYPXRX1RX2RZ	RYPXPYRXUXZ; RYPYCXUXZ; CYCXUXZ	RYPXPaZRXUXZ; RYPaZPaZRXUXZ; RYPaZCXUXZ
	RYPZPXRX1RX2RZ	RYPZPXRXUXZ; RYPZCXUXZ; RYRXCZCX	RYPZPaYRXUXZ; RYPaYPaYRXUXZ; RYPaYCXUXZ
	RYPYPZRXRZ1RZ2	RYPYPZRZUXZ; RYPYCZUXZ; CYCZUXZ	RYPYPaXRZUXZ; RYPaXPaXRZUXZ; RYPaXCZUXZ
	RYPXPZRXRZ1RZ2	RYPXPZRZUXZ; RYPXCZUXZ; RYRZCXCZ	RYPXPaYRZUXZ; RYPaYPaYRZUXZ; RYPaYCZUXZ
	RYPYRX1RX2RZ1RZ2	RYPYRXRZUXZ; RYPYU1XZU2XZ; CYPYU1XZU2XZ	RYPaX/ZRXRZUXZ; RYPaX/ZU1XZU2XZ;
	RYPXRX1RX2RX3RZ	RYPXRX1RX2UXZ; RYRXCXUXZ;	RYPaY/ZRX1RX2UXZ
	RYPXRX1RX2RZ1RZ2	RYPXRXRZUXZ; RYPXU1XZU2XZ; RYRZCXUXZ	RYPaY/ZRXRZUXZ; RYPaY/ZU1XZU2XZ;
	RYPZRX1RX2RZ1RZ2	RYPZRXRZUXZ; RYPZU1XZU2XZ; RYRXCZUXZ	RYPaY/XRXRZUXZ; RYPaY/XU1XZU2XZ
	RYPZRXRZ1RZ2RZ3	RYPZRZ1RZ2UXZ; RYCZRZUXZ	RYPaY/XRZ1RZ2UXZ
	RYRX1RX2RZ1RZ2RZ3	RYRXRZ1RZ2UXZ; RYRZU1XZU2XZ	—
	RYRX1RX2RX3RZ1RZ2	RYRZRX1RX2UXZ; RYRXU1XZU2XZ	—

and

Table 7. Structure of chain III.

Chain	Basic Chain Structure	Chain Structure with Multi-Degree-of-Freedom Kinematic Pair	Chain Structure Containing Closed-Loop Structures
Chain III	UrPXPYPZRY	UrPXPZCY	UrPXPaYCY; UrPaXPaYCY; UrPaXPaYPaZRY
	UrPYPXRY1RY2	UrPXRYCY	UrRYPaZCY; UrPX/YPaZRY1RY2; UrPaZPaZRY1RY2
	UrPYPZRY1RY2	UrPZRYCY	UrRYPaXCY; UrPYPaXRY1RY2; UrPaXPaXRY1RY2
	UrPYRY1RY2RY3	UrRY1RY2CY	UrPaXRY1RY2RY3

. The configurations of the parts of chains I, II, and III are shown in Figure 6, Figure 7 and Figure 8.

4. Evaluation of Mechanical Leg Mechanism Configuration

In order to ensure the performance and use effect of the mechanical leg, it is necessary to evaluate and optimize the configuration of each chain of the mechanical leg [27,28]. In the design process of a parallel mechanism, the aim is to ensure that it has a compact structure and can be easily protected while also reducing the difficulty of prototype design. The factors that affect the compactness of the structure, the difficulty of protection, and the difficulty of prototype fabrication mainly depend on the selection of kinematic pairs.

The fabrication difficulty SC₁ of the chain in the mechanism can be expressed as

$$S{C}_1={\displaystyle \frac{k_{\mathrm{p}}{n}_{\mathrm{P}}+k_{\mathrm{R}}{n}_{\mathrm{R}}+k_{\mathrm{U}}{n}_{\mathrm{U}}+k_{\mathrm{S}}{n}_{\mathrm{S}}+k_{\mathrm{Pa}}{n}_{\mathrm{Pa}}+k_{\mathrm{C}}{n}_{\mathrm{C}}}{n}}$$

(49)

where n_P, n_R, n_U, n_S, n_Pa, and n_C represent the number of prismatic pairs, revolute pairs, universal pairs, spherical pairs, Pa pairs, and cylindrical pairs in the chain; N is the total number of kinematic pairs, n = n_P + n_R + n_U + n_S + n_Pa + n_C; and k_P, k_R, k_U, k_S, k_Pa, and k_C are difficulty indices of kinematic pair fabrication.

The smaller the value of the fabrication difficulty index SC₁, the easier the fabrication of the kinematic pair. The fabrication of the revolute and prismatic pairs is relatively easy, so k_P and k_R are taken as 1. The spherical and universal pairs can be produced through the continuous combination of revolute pairs, so k_U and k_S are taken as 2. The Pa and cylindrical pairs have strict requirements for installation position and rod length in the manufacturing process, so k_Pa and k_C are taken as 3.

The compactness SC₂ of the chain in the mechanism can be expressed as

$$S{C}_2={\displaystyle \frac{j_{\mathrm{p}}{n}_{\mathrm{P}}+j_{\mathrm{R}}{n}_{\mathrm{R}}+j_{\mathrm{U}}{n}_{\mathrm{U}}+j_{\mathrm{S}}{n}_{\mathrm{S}}+j_{\mathrm{Pa}}{n}_{\mathrm{Pa}}+j_{\mathrm{C}}{n}_{\mathrm{C}}}{n}}$$

(50)

where j_P, j_R, j_U, j_S, j_Pa, and j_C are the compactness indices of kinematic pairs.

A small value of the compactness index SC₂ indicates that the chain is more compact. Spherical and universal pairs have more compact structures and more degrees of freedom, so the value of j_U and j_S is 1. If all the required degrees of freedom of the chain are realized by the revolute pair, the overall volume of the mechanism will be huge, so j_R and j_C are taken as 2. The Pa pair is a planar closed-loop kinematic pair composed of four coaxial revolute pairs, which can replace a basic prismatic pair. The mechanical structure of the moving pair is generally large and complex, so j_Pa and j_P are taken as 3.

During movement, the mechanical leg is often impacted and rubbed by other objects, so it is necessary to select kinematic pairs that are easy to protect in their design [27]. The degree of difficulty of protecting the chain of the mechanism, SC₃, is

$$S{C}_3={\displaystyle \frac{f_{\mathrm{p}}{n}_{\mathrm{P}}+f_{\mathrm{R}}{n}_{\mathrm{R}}+f_{\mathrm{U}}{n}_{\mathrm{U}}+f_{\mathrm{S}}{n}_{\mathrm{S}}+f_{\mathrm{Pa}}{n}_{\mathrm{Pa}}+f_{\mathrm{C}}{n}_{\mathrm{C}}}{n}}$$

(51)

where f_P, f_R, f_U, f_S, f_Pa, and f_C are indices of the degree of difficulty of protecting the kinematic pair.

The smaller the value of the protection difficulty index SC₃, the easier the chain’s protection. The structure of the revolute pair is compact and easy to protect [29]. The spherical and universal pairs have more degrees of freedom and compact structures, so f_R, f_U, and f_S are taken as 1. The prismatic pair usually consists of a sliding block and a guide rail, and the latter is usually intermittently exposed to the outside during the operation of the prismatic pair. The guard rail surface is very easily damaged when hit or scratched by other objects, and it is not easy to protect. The Pa pair can be equivalent to a prismatic pair. Therefore, f_Pa and f_P are taken as 2. The cylindrical pair can be equivalent to a prismatic pair and a revolute pair, and the axes of the two pairs coincide, so f_C is taken as 3.

As can be seen from Equations (49)–(51), the smaller the values of the three indices SC₁, SC₂, and SC₃, the lower the complexity of the representative mechanism in the manufacturing and protection processes. Considering the influence of the three factors, the evaluation index WSC_DM of the complexity of the structure is defined as

$$WS{C}_{DM}=w_{1}S{C}_1+w_{2}S{C}_2+w_{3}S{C}_3$$

(52)

where w₁, w₂, and w₃ are the weights of SC₁, SC₂, and SC₃, respectively, with w₁ = 0.3, w₂ = 0.4, and w₃ = 0.3.

From the analysis in Section 3, the complexity of the chains in different combinations can be obtained, as shown in

Table 8. Complexity of chain I.

Chain Type	Kinematic Pair Type	Complexity
Chain structure containing multi-degree-of-freedom kinematic pair	PZS	1.7
	CZUXY	1.95

,

Table 9. Complexity of chain II.

Chain Structure	Complexity	Chain Structure Containing Multi-Degree-of-Freedom Kinematic Pair	Complexity
RYPXPYPZRXRZ	1.75	RYPXPYPZUXZ; CYPXPZUXZ; CYCZCX	1.8 2.03 2.6
RYPYPXRX1RX2RZ	1.63	RYPXPYRXUXZ; RYPYCXUXZ; CYCXUXZ	1.66 1.85 2.167
RYPZPXRX1RX2RZ	1.63	RYPZPXRXUXZ; RYPZCXUXZ; RYRXCZCX	1.66 1.85 2
RYPYPZRXRZ1RZ2	1.63	RYPYPZRZUXZ; RYPYCZUXZ; CYCZUXZ	1.66 1.85 2.167
RYPXPZRXRZ1RZ2	1.63	RYPXPZRZUXZ; RYPXCZUXZ; RYRZCXCZ	1.66 1.85 2
RYPYRX1RX2RZ1RZ2	1.52	RYPYRXRZUXZ; RYPYU1XZU2XZ; CYU1XZU2XZ	1.52 1.525 1.733
RYPXRX1RX2RX3RZ	1.52	RYPXRX1RX2UXZ; RYRXCXUXZ	1.52 1.675
RYPXRX1RX2RZ1RZ2	1.52	RYPXRXRZUXZ; RYPXU1XZU2XZ; RYRZCXUXZ	1.52 1.525 1.675
RYPZRX1RX2RZ1RZ2	1.52	RYPZRXRZUXZ; RYPZU1XZU2XZ; RYRXCZUXZ	1.52 1.525 1.675
RYPZRXRZ1RZ2RZ3	1.52	RYPZRZ1RZ2UXZ; RYCZRZUXZ	1.52 1.675
RYRX1RX2RZ1RZ2RZ3	1.4	RYRXRZ1RZ2UXZ; RYRZU1XZU2XZ	1.38 1.35
RYRX1RX2RX3RZ1RZ2	1.4	RYRZRX1RX2UXZ; RYRXU1XZU2XZ	1.38 1.35

,

Table 10. Complexity of chain II with closed-loop structure.

Chain II Structure Containing Closed-Loop Structures	Complexity
RYPXPYPaYUXZ	1.92
RYPX/YPaX/YPaX/YUXZ	2.04
RYPaY/XPaX/YPaX/YUXZ	2.16
CYPX/ZPaYUXZ	2.175
CYPaYPaYUXZ	2.325
RYPXPaZRXUXZ; RYPZPaYRXUXZ; RYPYPaXRZUXZ; RYPXPaYRZUXZ	1.78
RYPaZPaZRXUXZ; RYPaYPaYRXUXZ; RYPaXPaXRZUXZ; RYPaYPaYRZUXZ	1.9
RYPaZCXUXZ; RYPaYCXUXZ; RYPaXCZUXZ; RYPaYCZUXZ	2
RYPaX/ZRXRZUXZ; RYPaY/ZRX1RX2UXZ; RYPaY/ZRXRZUXZ; RYPaY/XRZ1RZ2UXZ; RYPaY/XRXRZUXZ	1.64
RYPaX/ZU1XZU2XZ; RYPaY/ZU1XZU2XZ; RYPaY/XU1XZU2XZ	1.675

and

Table 11. Complexity of chain III.

Chain Structure	Complexity	Chain Structure Containing Multi-Degree-of-Freedom Kinematic Pair	Complexity	Chain Structure Containing a Closed-Loop Structure	Complexity
UrPXPYPZRY	1.82	UrPXPZCY	2.05	UrPXPaYCY; UrPaXPaYCY; UrPaXPaYPaZRY	2.2 2.35 2.18
UrPYPXRY1RY2	1.68	UrPXRYCY	1.875	UrRYPaZCY; UrPX/YPaZRY1RY2; UrPaZPaZRY1RY2	2.025 1.8 1.92
UrPYPZRY1RY2	1.68	UrPZRYCY	1.875	UrRYPaXCY; UrPYPaXRY1RY2; UrPaXPaXRY1RY2	2.025 1.8 1.92
UrPYRY1RY2RY3	1.54	UrRY1RY2CY	1.7	UrPaXRY1RY2RY3	1.66

.

When chain I is P_ZS, its complexity is 1.7, which is the smallest value for chain I that meets the requirements. When chain II is R_YU_1XZU_2XZR_Z or R_YU_1XZU_2XZR_X, its complexity is 1.35, which is the smallest value for chain II that meets the requirements. When chain Ⅲ is U_rR_Y1R_Y2R_Y3P_Y, its complexity is 1.54, which is the smallest value for chain Ⅲ that meets the requirements.

The reconfigurable mechanical leg has two configurations: in configuration I, chain II is R_YU_1XZU_2XZR_Z, as shown in Figure 9a, and in configuration II, chain II is R_YU_1XZU_2XZR_X, as shown in Figure 9b.

When the reconfigurable mechanical leg is in configuration I, the chain coordinate system o₂-x₂y₂z₂ is established, where the y₂ axis is parallel to the R_Y axis of chain II and the z₂ axis is oriented upward. If the R_Z axis coincides with the U_1XZ axis, chain II will have a passive degree of freedom. The twist of the chain II is

$$\begin{array}{lllllllll}{\mathsf{\$}}_{211}& =& (0& 1& 0& ;& d_1& 0& 0)\\ {\mathsf{\$}}_{221}& =& (1& 0& 0& ;& 0& e_2& f_2)\\ {\mathsf{\$}}_{231}& =& (0& 0& 1& ;& d_3& e_3& 0)\\ {\mathsf{\$}}_{241}& =& (1& 0& 0& ;& 0& e_4& f_4)\\ {\mathsf{\$}}_{251}& =& (0& 0& 1& ;& d_5& e_5& 0)\\ {\mathsf{\$}}_{261}& =& (0& 0& 1& ;& d_3& e_3& 0)\end{array}$$

(53)

Here, $₂₃₁ and $₂₆₁ are linearly dependent, and there is a reciprocal screw. Chain II is no longer an unconstrained chain and does not meet the requirements of the feasible constraint mode. Therefore, configuration II is selected as the configuration of the reconfigurable mechanical leg.

The sequence and layout of the kinematic pairs have an important influence on the performance and manufacturing complexity of the mechanical leg. From the analysis in Section 3, it can be seen that the only arrangement order of chain I is P_ZS. There is no need to adjust the kinematic pair sequence of chain I. The arrangement order of chain II is R_YR_XU_1XZU_2XZ, R_YU_1XZR_XU_2XZ, or R_YU_1XZU_2XZR_X. When chain II is R_YR_XU_1XZU_2XZ or R_YU_1XZR_XU_2XZ, it is difficult to ensure that the rotation axis of R_X is perpendicular to that of R_Y during movement. Therefore, the arrangement order of chain II is R_YU_1XZU_2XZR_X. The order of chain III is U_rP_YR_Y1R_Y2R_Y3, U_rR_Y1R_Y2R_Y3P_Y, or P_YU_rR_Y1R_Y2R_Y3. When chain III is U_rP_YR_Y1R_Y2R_Y3 or U_rR_Y1R_Y2R_Y3P_Y, the motion range of the kinematic pair is limited by the chain or the moving platform. When installed on the moving platform, the horizontal prismatic pair will increase the weight and inertia of the moving platform and increase the difficulty of its motion control. When chain III is P_YU_rR_Y1R_Y2R_Y3, the horizontal prismatic pair is installed on the static platform, so the high stiffness and stability of the static platform can be used to reduce the motion error caused by the deformation of the chain or the moving platform. A wide range of horizontal translation can be achieved, thereby expanding the mechanism’s working space. Therefore, P_YU_rR_Y1R_Y2R_Y3 is selected as the arrangement order of chain III. The parallel mechanical leg is a PS + RUUR + PU_rRRR reconfigurable parallel mechanism, and its model diagram is shown in Figure 10. The three-dimensional model of the reconfigurable four-wheel-legged robot is shown in Figure 11, where the legs are numbered.

5. Kinematic Analysis of the Reconfigurable Mechanical Leg

5.1. Inverse Position Solution

Obtaining the inverse position solution entails solving each driving parameter by determining the position vector of the end point of the foot in the fixed coordinate system. The closed-loop vector method and the Denavit–Hartenberg (D-H) parameter method are used to establish the inverse position equations of the mechanical leg in the mammalian and insect modes.

Firstly, the inverse position solution of chain I is analyzed by using the closed-loop vector method. The position vector of the center point O₁ of the moving platform in the static coordinate system O-XYZ is OO₁ = [0 0 z]^T, and the center point A₁₁ of the spherical pair is set to coincide with O₁, as shown in Figure 12. The position vector of point A₁₁ in the moving coordinate system O₁-X₁Y₁Z₁ is [0 0 0]^T.

The attitude transformation matrix T of the moving coordinate system relative to the static coordinate system is

$$\begin{array}{l}\mathit{T}=\mathrm{Rot}\left(Z,\gamma \right)\mathrm{Rot}\left(Y,\beta \right)\mathrm{Rot}\left(X,\alpha \right)\\ =\left[\begin{array}{ccc}\mathrm{c}\alpha \mathrm{c}\beta & \mathrm{c}\alpha \mathrm{s}\beta \mathrm{s}\gamma -\mathrm{s}\alpha \mathrm{c}\gamma & \mathrm{c}\alpha \mathrm{s}\beta \mathrm{c}\gamma +\mathrm{s}\alpha \mathrm{s}\gamma \\ \mathrm{s}\alpha \mathrm{c}\beta & \mathrm{s}\alpha \mathrm{s}\beta \mathrm{s}\gamma +\mathrm{c}\alpha \mathrm{c}\gamma & \mathrm{s}\alpha \mathrm{s}\beta \mathrm{c}\gamma -\mathrm{c}\alpha \mathrm{s}\gamma \\ -\mathrm{s}\beta & \mathrm{c}\beta \mathrm{s}\gamma & \mathrm{c}\beta \mathrm{c}\gamma \end{array}\right]\end{array}$$

(54)

where s represents sin, c represents cos, and γ, β, and α are the rotation angles of the moving platform around the Z-axis, Y-axis, and X-axis, respectively.

The position vector of ${\mathit{A}}_{11}^{O_1}$ is transformed into the static coordinate system O-XYZ through the attitude transformation matrix T as follows:

$$\mathit{O}{\mathit{A}}_{11}^O=\mathit{T}{\mathit{O}}_1{\mathit{A}}_{11}^{O_1}+\mathit{O}{\mathit{O}}_1={[\begin{array}{ccc}0& 0& z\end{array}]}^{\mathrm{T}}$$

(55)

The position vector l₁₁ of chain I can be expressed as

$${\mathrm{l}}_{11}=\mathit{O}{\mathit{A}}_{11}^O={[\begin{array}{ccc}0& 0& z\end{array}]}^{\mathrm{T}}$$

(56)

The D-H parameter method is used to obtain the inverse position solution of chain II. The D-H coordinate system of chain II is established, as shown in Figure 13, and the corresponding D-H parameters are shown in

Table 12. D-H parameters of chain II.

i	αi−1/(°)	ai−1/mm	θi/(°)	di/mm
1	0	0	θ21	0
2	−90	l21	θ22 + 90	0
3	−90	0	θ23 + 90	0
4	0	a24	θ24	l24
5	−90	0	θ25	0
6	0	l25	θ26	0

.

α_i-1 is the degree of rotation from z_i-1 to z_i about the x_i-1 axis; a_i-1 is the length of movement from z_i-1 to z_i in the direction of the x_i-1 axis; θ_i is the number of degrees of rotation from x_i-1 to x_i about the z_i axis; and di is the length of movement from x_i-1 to x_i in the direction of the z_i axis.

The data in Table 12 are substituted into the following matrix transformation equation:

$${}_i{}^{i-1}\mathit{T}=\left[\begin{array}{cccc}\mathrm{c}{\theta }_i& -\mathrm{s}{\theta }_i& 0& a_{i-1}\\ \mathrm{s}{\theta }_i\mathrm{c}{\alpha }_{i-1}& \mathrm{c}{\theta }_i\mathrm{c}{\alpha }_{i-1}& -\mathrm{s}{\alpha }_{i-1}& -d_i\mathrm{s}{\alpha }_{i-1}\\ \mathrm{s}{\theta }_i\mathrm{s}{\alpha }_{i-1}& \mathrm{c}{\theta }_i\mathrm{s}{\alpha }_{i-1}& \mathrm{c}{\alpha }_{i-1}& d_i\mathrm{c}{\alpha }_{i-1}\\ 0& 0& 0& 1\end{array}\right]$$

(57)

The positional relationship between the initial coordinate system o₀-x₀y₀z₀ of chain II and the static coordinate system O-XYZ is shown in Figure 14a. The positional relationship between the moving coordinate system O₁-X₁Y₁Z₁ and chain II’s end coordinate system o₆-x₆y₆z₆ is shown in Figure 14b.

The transformation matrix ${}_0{}^O\mathit{T}^2$ from the initial coordinate system o₀-x₀y₀z₀ of chain II to the static coordinate system O-XYZ is

$${}_0{}^O\mathit{T}^2=\left[\begin{array}{cccc}1& 0& 0& -R/2\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

(58)

The transformation matrix ${}_{O_1}{}^6\mathit{T}^2$ from the moving coordinate system O₁-X₁Y₁Z₁ to chain II’s end coordinate system o₆-x₆y₆z₆ is

$${}_{O_1}{}^6\mathit{T}^2=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& r/2\\ 0& 0& 0& 1\end{array}\right]$$

(59)

The transformation matrix ${}_{O_1}{}^O\mathit{T}^2$ from the moving platform to the static platform can be obtained as follows:

$${}_{O_1}{}^O\mathit{T}^2={}_0{}^O\mathit{T}^2{}_1{}^0\mathit{T}^2{}_2{}^1\mathit{T}^2{}_3{}^2\mathit{T}^2{}_4{}^3\mathit{T}^2{}_5{}^4\mathit{T}^2{}_6{}^5\mathit{T}^2{}_{O_1}{}^6\mathit{T}^2=\left[\begin{array}{cccc}{M}_{11}^2& M_{12}^2& M_{13}^2& M_{14}^2\\ M_{21}^2& M_{22}^2& M_{23}^2& M_{24}^2\\ M_{31}^2& M_{32}^2& M_{33}^2& M_{34}^2\\ M_{41}^2& M_{42}^2& M_{43}^2& M_{44}^2\end{array}\right]$$

(60)

In the mammalian mode, the transformation matrix ${\mathit{T}}_{YX}^2$ of the moving platform is

$${\mathit{T}}_{YX}^2=\left[\begin{array}{cccc}\mathrm{c}\beta & \mathrm{s}\alpha \mathrm{s}\beta & \mathrm{c}\alpha \mathrm{s}\beta & x\\ 0& \mathrm{c}\alpha & -\mathrm{s}\alpha & y\\ -\mathrm{s}\beta & \mathrm{c}\beta \mathrm{s}\alpha & \mathrm{c}\alpha \mathrm{c}\beta & z\\ 0& 0& 0& 1\end{array}\right]$$

(61)

In the insect mode, the transformation matrix ${\mathit{T}}_{YZ}^2$ of the moving platform is

$${\mathit{T}}_{YZ}^2=\left[\begin{array}{cccc}\mathrm{c}\beta \mathrm{c}\gamma & -\mathrm{c}\beta \mathrm{s}\gamma & \mathrm{s}\beta & x\\ \mathrm{s}\gamma & \mathrm{c}\gamma & 0& y\\ -\mathrm{c}\gamma \mathrm{s}\beta & \mathrm{s}\beta \mathrm{s}\gamma & \mathrm{c}\beta & z\\ 0& 0& 0& 1\end{array}\right]$$

(62)

The transformation matrices ${}_{O_1}{}^O\mathit{T}^2$ of chain II in the mammalian and insect modes are equal to the elements of the corresponding position of the transformation matrix of the moving platform.

In the mammalian mode,

$${}_{O_1}{}^O\mathit{T}^2={\mathit{T}}_{YX}^2$$

(63)

In the insect mode,

$${}_{O_1}{}^O\mathit{T}^2={\mathit{T}}_{YZ}^2$$

(64)

After simplified calculations, the solution of chain II’s driving θ₂₁ can be obtained as follows:

$${\theta }_{21}=\beta$$

(65)

The D-H parameter method is used to obtain the inverse position solution of chain III. The D-H coordinate system of chain III in the mammalian and insect modes is established, as shown in Figure 15.

The D-H parameters of chain III in the mammalian and insect models are shown in

Table 13. D-H parameters of chain III in mammalian model.

i	αi−1/(°)	ai−1/mm	θi/(°)	di/mm
1	0	0	0	d31
2	90	l31	θ32 − 90	0
3	−90	l32	θ33	0
4	0	l33	θ34	0
5	0	l34	θ35	0

and

Table 14. D-H parameters of chain III in insect model.

i	αi−1/(°)	ai−1/mm	θi/(°)	di/mm
1	0	0	0	d31
2	90	0	θ32	l31
3	−90	0	θ33	l32
4	0	l33	θ34 − 90	0
5	0	l34	θ35	0

, respectively.

The positional relationship between the initial coordinate system o₀-x₀y₀z₀ of chain III and the static coordinate system O-XYZ is shown in Figure 16a; the positional relationship between the moving coordinate system O₁-X₁Y₁Z₁ and chain III’s end coordinate system o₅-x₅y₅z₅ is shown in Figure 16b.

The transformation matrix from the initial coordinate system o₀-x₀y₀z₀ of chain III to the static coordinate system O-XYZ is

$${}_0{}^O\mathit{T}^3=\left[\begin{array}{cccc}1& 0& 0& R/2\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

(66)

The transformation matrix from the moving coordinate system O₁-X₁Y₁Z₁ to chain III’s end coordinate system o₅-x₅y₅z₅ is

$${}_{O_1}{}^5\mathit{T}^3=\left[\begin{array}{cccc}0& 0& 1& 0\\ 1& 0& 0& r/2\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

(67)

The transformation matrix from the moving platform to the static platform can be obtained as follows:

$${}_{O_1}{}^O\mathit{T}^3={}_0{}^O\mathit{T}^3{}_1{}^0\mathit{T}^3{}_2{}^1\mathit{T}^3{}_3{}^2\mathit{T}^3{}_4{}^3\mathit{T}^3{}_5{}^4\mathit{T}^3{}_{O_1}{}^5\mathit{T}^3=\left[\begin{array}{cccc}{M}_{11}^3& M_{12}^3& M_{13}^3& M_{14}^3\\ M_{21}^3& M_{22}^3& M_{23}^3& M_{24}^3\\ M_{31}^3& M_{32}^3& M_{33}^3& M_{34}^3\\ M_{41}^3& M_{42}^3& M_{43}^3& M_{44}^3\end{array}\right]$$

(68)

In the mammalian mode, the transformation matrix ${\mathit{T}}_{XY}^3$ of the moving platform is

$${\mathit{T}}_{XY}^3=\left[\begin{array}{cccc}\mathrm{c}\beta & 0& \mathrm{s}\beta & x\\ \mathrm{s}\alpha \mathrm{s}\beta & \mathrm{c}\alpha & -\mathrm{s}\alpha \mathrm{c}\beta & y\\ -\mathrm{c}\alpha \mathrm{s}\beta & \mathrm{s}\alpha & \mathrm{c}\alpha \mathrm{c}\beta & z\\ 0& 0& 0& 1\end{array}\right]$$

(69)

In the insect mode, the transformation matrix ${\mathit{T}}_{ZY}^3$ of the moving platform is

$${\mathit{T}}_{ZY}^3=\left[\begin{array}{cccc}\mathrm{c}\beta \mathrm{c}\gamma & -\mathrm{s}\gamma & \mathrm{s}\beta \mathrm{c}\gamma & x\\ \mathrm{c}\beta \mathrm{s}\gamma & \mathrm{c}\gamma & \mathrm{s}\beta \mathrm{s}\gamma & y\\ -\mathrm{s}\beta & 0& \mathrm{c}\beta & z\\ 0& 0& 0& 1\end{array}\right]$$

(70)

The transformation matrices ${}_{O_1}{}^O\mathit{T}^3$ of chain III in the mammalian and insect modes are equal to the elements of the corresponding position of the transformation matrix of the moving platform.

In the mammalian mode,

$${}_{O_1}{}^O\mathit{T}^3={\mathit{T}}_{XY}^3$$

(71)

In the insect mode,

$${}_{O_1}{}^O\mathit{T}^3={\mathit{T}}_{XY}^3$$

(72)

After simplified calculations, the driving solution of chain III in the mammalian mode can be obtained as follows:

$${\theta }_{32}=\alpha$$

(73)

In the mammalian mode, d₃₁ is

$$d_{31}=\left(Z-{\displaystyle \frac{r}{2}}\mathrm{c}\alpha \mathrm{s}\beta -l_{31}\right)\mathrm{t}\alpha$$

(74)

where t represents tan.

After simplified calculations, the driving solution of chain III in insect mode can be obtained as follows:

$${\theta }_{32}=\gamma$$

(75)

In the insect mode, d₃₁ is

$$d_{31}={\displaystyle \frac{R}{2}}\mathrm{t}\gamma$$

(76)

5.2. Velocity Analysis

In this section, the Jacobian matrix of the mechanism in the two modes is obtained by deriving the inverse solution of the position of the mechanism in the mammalian and insect modes.

When the mechanical leg is in the mammalian mode, the derivatives of Equations (56), (65), and (73) with respect to time t are obtained:

$$\left[\begin{array}{c}{\dot{l}}_{11}\\ {\dot{\theta }}_{32}\\ {\dot{\theta }}_{21}\end{array}\right]={\mathit{J}}^1\left[\begin{array}{c}\dot{z}\\ \dot{\alpha }\\ \dot{\beta }\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}\dot{z}\\ \dot{\alpha }\\ \dot{\beta }\end{array}\right]$$

(77)

where J¹ is the Jacobian matrix of the mechanism in the mammalian mode.

When the mechanical leg is in the insect mode, the derivatives of Equations (56), (65), and (75) with respect to time t are obtained:

$$\left[\begin{array}{c}{\dot{l}}_{11}\\ {\dot{\theta }}_{21}\\ {\dot{\theta }}_{32}\end{array}\right]={\mathit{J}}^2\left[\begin{array}{c}\dot{z}\\ \dot{\beta }\\ \dot{\gamma }\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}\dot{z}\\ \dot{\beta }\\ \dot{\gamma }\end{array}\right]$$

(78)

where J² is the Jacobian matrix of the mechanism in the insect mode.

It can be seen from Equations (77) and (78) that the Jacobian matrix of the mechanical leg in both the mammalian and insect modes is a unit diagonal matrix. It is shown that the degrees of freedom of the reconfigurable mechanical leg are completely decoupled in the two motion modes.

6. Performance Analysis and Optimization Design of the Reconfigurable Mechanical Leg

6.1. Workspace Analysis

The working space of the mechanical leg determines the step length and the obstacle-crossing ability of the mobile robot. Its size and shape represent the range of motion of the mechanical leg’s end point, which is an important index for measuring the kinematical performance of the mechanical leg. The translation distance of the prismatic pair is [200, 500] mm, and the rotation range of the revolute pair is [−50, 50]°. The maximum rotation angle of the spherical pair is 50°. In the mammalian mode, the search space of the mechanical leg is Z ∈ [150, 500] mm, α ∈ [−60, 60]°, β ∈ [−60, 60]°; in the insect mode, the search space of the mechanical leg is Z ∈ [150, 500] mm, β ∈ [−60, 60]°, γ ∈ [−60, 60]°. Under the condition that the constraint conditions are satisfied, the workspace of the mechanical leg in the mammalian and insect modes is obtained, as shown in Figure 17.

It can be seen from Figure 17 that the reconfigurable mechanical leg has a large reachable working space range in both the mammalian and insect modes. The workspace is continuous and has no holes. When the reconfigurable mechanical leg is in the mammalian mode, the ranges of α, β, and Z are [−43, 43]°, [−45, 45]°, and [200, 400] mm, respectively. The rotation angle of the moving platform decreases with the increase in Z. When the reconfigurable mechanical leg is in the insect mode, the ranges of γ, β, and Z are [−23, 23]°, [−45, 45]°, and [200, 400] mm, respectively, which indicates a larger workspace than that in the mammalian mode. The shape is a regular cuboid, which is more suitable for fast walking over complex terrain.

The performance index W_V of the reachable workspace can be defined as

$$W_{\mathrm{V}}={\displaystyle \frac{W_i}{Q}}$$

(79)

where Q is the number of points in the search space, and W_i is the number of points satisfying the constraint condition.

The value range of WV is [0, 1], and the larger the value is, the larger the workspace volume is.

6.2. Motion/Force Transmission Performance Analysis

The motion/force transmission performance index of the structure comprises input and output transmission performance indices. The input transmission performance index is the result of the interaction between the input twist and the transmission wrench screw, which indicates the efficiency with which the driver’s input energy is transferred to the chain. The output transmission performance index is the result of the interaction between the transmission wrench screw and the output twist, which indicates the efficiency with which the force spiral on the chain is transferred to the moving platform.

The equations for calculating the input/output transfer performance indices of a non-redundant parallel mechanism [30,31] are

$${\lambda }_i={\displaystyle \frac{\left|{\mathsf{\$}}_{\mathrm{T}i}\circ {\mathsf{\$}}_{\mathrm{I}i}\right|}{{\left|{\mathsf{\$}}_{\mathrm{T}i}\circ {\mathsf{\$}}_{\mathrm{I}i}\right|}_{\max }}}$$

(80)

$${\eta }_i={\displaystyle \frac{\left|{\mathsf{\$}}_{\mathrm{T}i}\circ {\mathsf{\$}}_{\mathrm{O}i}\right|}{{\left|{\mathsf{\$}}_{\mathrm{T}i}\circ {\mathsf{\$}}_{\mathrm{O}i}\right|}_{\max }}}$$

(81)

$$\mathrm{\Gamma }=\min \left\{{\lambda }_i,{\eta }_i\right\}$$

(82)

where λ_i is the input transfer performance index of chain i, η_i is the output transfer performance index of chain i, Γ is the local transfer performance index, $_Ii is the input twist of chain i, $_Ti is the transmission wrench screw of chain i, and $_Oi is the output twist of chain i.

The input and output transfer performance indices represent the transfer of energy from the drive input to the chain and from the chain to the output, respectively. According to the definition, the value ranges of λ_i, η_i, and Γ are between 0 and 1, and due to the dimensionless nature of the index, its size is independent of the selection of the coordinate system. The closer the value of Γ is to 1, the better the motion/force transmission performance of the mechanism is. On the contrary, it indicates that the mechanism is closer to singularity.

Taking the reconfigurable mechanical leg in the general configuration as the research object and taking chain I as an example, $_I1, $_T1, and $_O1 are calculated. The twist system of chain I in the general configuration is

$$\begin{array}{lllll}{\mathsf{\$}}_{11}& =& (0& ;& \mathit{Z})\\ {\mathsf{\$}}_{12}& =& (\mathit{X}& ;& {\mathit{r}}_{A_{11}}\times \mathit{X})\\ {\mathsf{\$}}_{13}& =& (\mathit{Y}& ;& {\mathit{r}}_{A_{11}}\times \mathit{Y})\\ {\mathsf{\$}}_{14}& =& (\mathit{Z}& ;& {\mathit{r}}_{A_{11}}\times \mathit{Z})\end{array}$$

(83)

The input twist $_I1 of chain I is the twist of the driving pair, specifically

$${\mathsf{\$}}_{\mathrm{I}1}={\mathsf{\$}}_{11}=\left(\begin{array}{ccc}0& ;& \mathit{Z}\end{array}\right)$$

(84)

where Z is the direction vector of the Z-axis.

Similarly, the input twists $_I2 and $_I3 of chains II and III are, respectively,

$$\begin{array}{lllllll}{\mathsf{\$}}_{\mathrm{I}2}& =& {\mathsf{\$}}_{21}& =& (\mathit{Y}& ;& {\mathit{r}}_{A_{21}}\times \mathit{Y})\\ {\mathsf{\$}}_{\mathrm{I}3}& =& {\mathsf{\$}}_{32}& =& ({\mathit{u}}_{32}& ;& {\mathit{r}}_{A_{31}}\times {\mathit{u}}_{32})\end{array}$$

(85)

where Y is the direction vector of the Y-axis, ${\mathit{r}}_{A_{21}}$ and ${\mathit{r}}_{A_{31}}$ are the position vectors of A₂₁ and A₃₁, and u₃₂ is the unit vector of the rotation axis of the lockable universal pair.

Assuming that the driving pair P_Z of chain I is locked, $₁₁ will be removed from the twist of chain I, and by taking the reciprocal product of the kinematic twist of the remaining kinematic pairs, the transmission wrench screw $_T1 of chain I is obtained as follows:

$${\mathsf{\$}}_{\mathrm{T}1}=\left(\begin{array}{ccc}\mathit{Z}& ;& 0\end{array}\right)$$

(86)

In the same way, the transmission wrench screw $_T2 of chain II can be obtained as follows:

$${\mathsf{\$}}_{\mathrm{T}2}=\left(\begin{array}{ccc}0& ;& \mathit{Y}\end{array}\right)$$

(87)

When the reconfigurable mechanical leg is in the mammalian mode, the transmission wrench screw $_T31 of chain III is

$${\mathsf{\$}}_{\mathrm{T}31}=\left(\begin{array}{ccc}0& ;& \mathit{X}\end{array}\right)$$

(88)

where X is the direction vector of the X-axis.

When the reconfigurable mechanical leg is in the insect mode, the transmission wrench screw $_T32 of chain III is

$${\mathsf{\$}}_{\mathrm{T}32}=\left(\begin{array}{ccc}0& ;& \mathit{Z}\end{array}\right)$$

(89)

When the reconfigurable mechanical leg is in the mammalian mode, the drive of chains II and III is locked, whereas the drive of chain I is retained. Each chain can generate a five-dimensional constraining force wrench composed of $\left[\begin{array}{lllll}{\mathsf{\$}}_{{\mathrm{F}}_1}^r,\hfill & {\mathsf{\$}}_{{\mathrm{F}}_2}^r,\hfill & {\mathsf{\$}}_{{\mathrm{C}}_1}^r,\hfill & {\mathsf{\$}}_{\mathrm{T}2},\hfill & {\mathsf{\$}}_{\mathrm{T}31}\hfill \end{array}\right]$ to act on the moving platform. In this case, the mechanism has a single degree of freedom, and the output twist $_O11 of the chain I is

$${\mathsf{\$}}_{\mathrm{O}11}=\left(\begin{array}{ccccccc}0& 0& 0& ;& 0& 0& 1\end{array}\right)$$

(90)

Similarly, the output twists $_O21 and $_O31 of chains II and III can be obtained as follows:

$$\begin{array}{lllllllll}{\mathsf{\$}}_{\mathrm{O}21}& =& (0& \mathrm{c}\alpha & \mathrm{s}\alpha & ;& -Z_{C1}\mathrm{c}\alpha & 0& 0)\\ {\mathsf{\$}}_{\mathrm{O}31}& =& (1& 0& 0& ;& 0& Z_{C1}& 0)\end{array}$$

(91)

where Z_A11 is the coordinate of A₁₁ on the Z-axis.

When the reconfigurable mechanical leg is in the insect mode, the output twist of each chain is

$$\begin{array}{lllllllll}{\mathsf{\$}}_{\mathrm{O}12}& =& (0& 0& 0& ;& 0& 0& 1)\\ {\mathsf{\$}}_{\mathrm{O}22}& =& (\mathrm{c}\gamma & \mathrm{s}\gamma & 0& ;& -Z_{C1}\mathrm{s}\gamma & Z_{C1}\mathrm{c}\gamma & 0)\\ {\mathsf{\$}}_{\mathrm{O}32}& =& (0& 0& 1& ;& 0& 0& 0)\end{array}$$

(92)

When the reconfigurable mechanical leg is in the mammalian mode, the input and output transmission indices of each chain are, respectively,

$$\begin{array}{l}{\lambda }_{11}={\lambda }_{21}={\lambda }_{31}=1\\ {\eta }_{11}={\eta }_{31}=1\\ {\eta }_{21}=\left|\mathrm{c}\alpha \right|\end{array}$$

(93)

When the reconfigurable mechanical leg is in the insect mode, the input transmission index and the output transmission index of each chain are, respectively,

$$\begin{array}{l}{\lambda }_{12}={\lambda }_{22}={\lambda }_{32}=1\\ {\eta }_{12}={\eta }_{32}=1\\ {\eta }_{22}=\left|\mathrm{s}\gamma \right|\end{array}$$

(94)

The local transmission index of the mechanical leg in the mammalian mode is

$${\mathrm{\Gamma }}_1=\min \left\{{\lambda }_{11},{\lambda }_{21},{\lambda }_{31},{\eta }_{11},{\eta }_{21},{\eta }_{31}\right\}=\left|\mathrm{c}\alpha \right|$$

(95)

The local transmission index of the mechanical leg in the insect mode is

$${\mathrm{\Gamma }}_2=\min \left\{{\lambda }_{12},{\lambda }_{22},{\lambda }_{32},{\eta }_{12},{\eta }_{22},{\eta }_{32}\right\}=\left|\mathrm{s}\gamma \right|$$

(96)

From Equations (95) and (96), the motion/force transmission performance index of the reconfigurable mechanical leg in the mammal and insect modes can be obtained, as shown in Figure 18.

In Figure 18, different colors represent the magnitude of the Γ value. When the reconfigurable mechanical leg is in the mammalian mode, Γ increases with the increase in Z. When the reconfigurable mechanical leg is in the insect mode, the fluctuation of Γ on the β axis is small, and the motion performance is stable.

The distribution of Γ in the plane cannot be directly observed in Figure 18. Therefore, the three-dimensional distribution of Γ in the section at Z = 250 mm is taken to present the variation in Γ within the workspace, as shown in Figure 19.

When the robot is subjected to external loads, the smaller the internal force required inside the robot, the better its motion/force performance, that is, the higher its carrying capacity. It can be seen from Figure 19 that at Z = 250 mm, the motion/force transmission performance of the mammalian model is significantly better than that of the insect model. In the high-load application scenario, the mammalian mode is selected to meet the task requirements.

As can be seen from Equation (82), Γ only reflects the motion/force transmission performance of the parallel mechanism at a certain point in the workspace. It cannot be used to evaluate the global motion/force transmission performance of the mechanism. Therefore, in the workspace of the parallel mechanism, the global motion/force transmission performance evaluation index $\overline{\zeta }$ is defined as

$$\overline{\zeta }={\displaystyle \frac{{\displaystyle \sum _{i=1}^{W_i}\mathrm{\Gamma }}}{W_i}}$$

(97)

From the definition of $\tilde{\zeta }$, the global motion/force transfer performance index is an average of the local motion/force transfer performance index Γ over the entire workspace, such that its size reflects the average transmission performance of the mechanism in the whole working space. However, it cannot reflect the fluctuation of the mechanism’s performance across the whole working space. During the movement of the reconfigurable mechanical leg, the mechanism is usually expected to have good average performance while maintaining stability. Based on the weighted standard deviation of motion/force transmission performance, a global motion performance fluctuation index of the parallel mechanism is proposed in this paper. The equation for evaluating the degree of fluctuation of the mechanism’s performance index $\tilde{\zeta }$ is

$$\tilde{\zeta }={\displaystyle \frac{{\displaystyle \sum _{i=1}^{W_i}\mathrm{\Gamma }\sqrt{{\left(\mathrm{\Gamma }-\overline{\zeta }\right)}^2}}}{W_i}}$$

(98)

According to the definition of standard deviation, the larger the standard deviation $\tilde{\zeta }$, the larger the fluctuation of the mechanism’s transmission performance in the working space. A smaller $\tilde{\zeta }$ indicates more stable motion/force performance of the mechanism in the workspace.

From Equations (95) and (96), it can be seen that when the reconfigurable mechanical leg is in the mammalian mode, the performance index of global motion/force transmission $\overline{\zeta }$ is 0.9752, and $\tilde{\zeta }$ is 0.0212. When the reconfigurable mechanical leg is in the insect mode, the global motion/force transmission performance index $\overline{\zeta }$ is 0.2021, and $\tilde{\zeta }$ is 0.0201, indicating that when subjected to external loads, the mammalian configuration requires less transmission force than the insect configuration.

Through a comparative analysis of the workspace and motion/force transmission performance of the reconfigurable mechanical leg in the mammalian and insect modes, it can be seen that the insect mode is superior to the mammalian mode in terms of the workspace; in terms of motion/force transmission performance, the mammalian model is stronger than the insect model. Therefore, for the reconfigurable four-wheel-legged robot, if a large and stable carrying capacity is required during the movement, the mammalian mode is selected, whereas the insect mode is selected if a large step size is required during movement.

6.3. Optimized Design

In this section, the workspace and motion/force transfer performance of the reconfigurable mechanical leg in the two motion modes are taken as the objective function, and the particle swarm optimization algorithm is used to optimize the geometric size parameters of the mechanism.

6.3.1. Design Variables

Based on the above analysis, the performance index of the reconfigurable mechanical leg is related to the side length R of the static platform and the length of the links l₃₁ and the side length r of the moving platform. Therefore, the parameters for dimension optimization are R, l₃₁, and r.

6.3.2. Objective Function

In the process of optimization design, the superiority of the mechanism’s performance and stability in the workspace should be taken into account. Therefore, in the kinematical optimization design of the mechanism, $\overline{\zeta }$ and $\tilde{\zeta }$ should be considered comprehensively, with the aim of making $\overline{\zeta }$ as large as possible and $\tilde{\zeta }$ as small as possible. Therefore, the comprehensive kinematic performance index ζ of the mechanism is proposed, that is,

$$\zeta =\left(1-\tilde{\zeta }\right)\overline{\zeta }$$

(99)

For the parallel mechanism, under the premise of meeting design requirements, it is often desirable to maximize the reachable workspace for optimal performance. From Equations (79) and (99), it can be seen that the workspace performance index and the comprehensive kinematic performance index range from 0 to 1. The larger the value of the two, the larger the workspace, and the better the transmission performance of the mechanical leg. When the reconfigurable mechanical leg is optimized, the reference value of the objective function of the workspace and the comprehensive kinematic performance index is set to 1 so that the objective function is minimized. The single objective optimization function of the workspace performance index and the comprehensive kinematic performance index of the reconfigurable mechanical leg in the mammalian and insect modes is

$$\left\{\begin{array}{l}{f}_1=1-W_{V1}\\ f_2=1-{\zeta }_1\\ f_3=1-W_{V2}\\ f_4=1-{\zeta }_2\end{array}\right.$$

(100)

where W_V1 and ζ₁ are the performance indices in the mammalian mode, and W_V2 and ζ₂ are those in the insect mode.

The reference target distance method can be used to transform the four objective functions into a single objective function. This method is similar to the multi-objective approach. The same direction processing is simple and widely used. Using this method, the objective function can be defined as

$$F\left(x\right)={\displaystyle \frac{1}{4}}\left({\displaystyle \sum _{i=1}^4\left|Z_i-f_i\left(X\right)\right|}\right)$$

(101)

where Z_i is the reference value of the objective function and is set as 1.

6.3.3. Constraints

According to the structural parameters of the mechanical leg, the constraint range of each design parameter is given, as shown in

Table 15. Constraint ranges of design parameters.

Design Variables	Constraint Range
R (mm)	[250, 450]
l31 (mm)	[50, 150]
r (mm)	[100, 220]

.

6.3.4. Optimization Examples

The parameters of the particle swarm optimization algorithm are shown in

Table 16. Parameters of particle swarm optimization algorithm.

Parameter	Population Size	Number of Iterations	Social Learning Factor	Inertia Weight
Numerical value	50	80	2.0	0.99

. The optimization results for the mechanism’s size parameters and performance indices are shown in

Table 17. Optimization results of structural parameters.

Structural Parameters	R/mm	l31/mm	r/mm
Before optimization	300	90	160
After optimization	441.4536	143.5765	122.2757

and

Table 18. Performance index optimization results.

Mode	Performance Indicators	WV	ζ
Mammalian mode	Before optimization	0.4144	0.9332
	After optimization	0.7310	0.8870
Insect mode	Before optimization	0.5165	0.1981
	After optimization	0.5385	0.2059

, respectively.

It can be seen from Table 17 that R and l₃₁ are increased, while r is reduced. Because the precision and manufacturing cost of the mechanism will be limited in the actual processing, the design parameters of the optimized mechanism in Table 17 are rounded, and the results are R = 440 mm, l₃₁ = 140 mm, and r = 120 mm. The rounded size parameters are substituted into the workspace and comprehensive motion performance index formulas for the two modes. The optimized mechanism performance indices can be obtained, as shown in Table 18. In the mammalian mode, the workspace index of the mechanism is increased by 76.4%, and the comprehensive motion performance index is reduced by 5.0%. In the insect mode, the workspace index of the mechanism is increased by 4.3%, and the comprehensive motion performance index is increased by 3.9%.

6.3.5. Comparison of Results

According to the optimized structural parameters, the kinematic performance index of the mechanism is recalculated, as shown in Figure 20.

Figure 20 shows that, compared with the size before optimization, the workspace range has increased and the comprehensive motion performance index has decreased. In the insect mode, the workspace performance and motion/force transmission performance are improved compared with before optimization. The overall kinematic performance index of the mechanical leg is improved.

7. Research on Stability of Reconfigurable Four-Wheel-Legged Robot

In the walking operation mode, the reconfigurable four-wheel-legged robot needs to have a large step space so that it can adapt to a complex and changeable unstructured terrain. As shown in Section 6, the volume of the workspace of the mechanical leg in the insect mode is larger than that in the mammalian mode. Therefore, when the robot is in the walking mode, it is better to use the insect mode as the configuration of the reconfigurable mechanical leg. It is necessary to analyze the robot’s stability while moving so that it remains stable during movement. When the quadruped robot enters the static walking mode, one leg is used as the swing leg, and the other three legs are used as support legs. The robot’s center of mass must always be located in the triangular support area composed of the three support legs, as shown in Figure 21.

The stability margin d_M [32] of the robot is

$$d_{\mathrm{M}}=\min \left\{\begin{array}{ccc}{d}_1,& d_2,& d_3\end{array}\right\}$$

(102)

where d_i is the distance from the center of gravity to these three sides.

When d_M > 0, the robot is in a stable state; when d_M = 0, the robot is in a critical state, and a slight external disturbance can cause the body to become imbalanced; and when d_M < 0, the robot is in an unstable state.

In the static gait, there are ${}_4{}^4\mathrm{A}=24$ kinds of robot steps, among which the 4-1-3-2 gait is the best step order for a quadruped robot in static gait [33]. As the robot moves, it is necessary to adjust the position of its center of gravity to prevent it from overturning. In Figure 22, ◌ represents the foot drop point at the previous moment, ● represents the foot drop point, ○ represents the point where the foot is about to fall, ◑ represents the center of gravity of the body, L is the distance between the center points of adjacent foot ends, s is the robot step size, and λ is the stride. The static gait of the robot is planned, as shown in Figure 22.

Before the robot walks, the four legs are in an upright state, as shown in Figure 22a. The center of gravity of the fuselage is located at the geometric center of the quadrilateral support area composed of legs 1, 2, 3, and 4.

In order to ensure that the robot remains static and stable when leg 4 swings, the body’s center of gravity needs to be adjusted before leg 4 moves. The body moves λ/2 forward along the x-axis and moves ∆L upward along the y-axis, as shown in Figure 22b.

The robot moves leg 4 and moves s along the x-axis. The body’s center of gravity is located in the triangular support area composed of legs 1, 2, and 3, as shown in Figure 22c.

The robot moves leg 1 and moves s along the x-axis. The body’s center of gravity is located in the triangular support area composed of legs 2, 3, and 4, as shown in Figure 22d.

In order to ensure that the robot remains static and stable when leg 3 swings, the body’s center of gravity needs to be adjusted before leg 3 moves. The body continues to move λ/2 forward along the x-axis and moves 2∆L downward along the y-axis, as shown in Figure 22e.

The robot moves leg 3 and moves s along the x-axis. The body’s center of gravity is located in the triangular support area composed of legs 1, 2, and 4, as shown in Figure 22f.

The robot moves leg 2 and moves s along the x-axis. The body’s center of gravity is located in the triangular support area composed of legs 1, 3, and 4, as shown in Figure 22g.

After leg 2 touches the ground, the body moves up ∆L along the y-axis, and the robot returns to the initial position, as shown in Figure 22h.

The body’s center of gravity moves forward along the x-axis with a stride λ. The center of gravity is located at the geometric center of the quadrilateral support area composed of legs 1, 2, 3, and 4 as support legs, as shown in Figure 22i.

8. Conclusions

(1)

Based on bionic principles and configuration synthesis theory for a decoupled parallel mechanism, a method for configuration synthesis of a reconfigurable decoupled mechanical leg was proposed. The mechanical leg switches between mammalian and insect modes through a change in the lockable universal pair rotating shaft.

(2)

Based on the difficulty of fabricating and protecting the chain, as well as its compactness, an evaluation index for the complexity of the chain of the reconfigurable mechanical leg was proposed, and a series of synthesized chain configurations were evaluated. Then, the configuration of each chain of the reconfigurable mechanical leg was determined.

(3)

Based on the weighted standard deviation of motion/force transmission performance, a global performance fluctuation index of the motion/force transmission of the mechanical leg was proposed. The proposed index can reflect the fluctuation of the mechanism’s motion performance in the workspace and evaluate its stability.

(4)

When the robot is in the walking operation mode, the insect mode is used as the configuration of the reconfigurable mechanical leg. The static stability criterion is used to plan the robot’s gait such that it can meet the needs of the task environment.