Optimization and Experimental Evaluation of a Legged Robot Mechanism Based on Task Space Partitioning

Abstract

This study analyzed the spatial distribution characteristics of the foot-end trajectory of a robotic leg mechanism during different gait phases. Based on this analysis, a task space partition-based dimensional parameter optimization method was proposed. To further evaluate the spatial distribution of the high-performance transmission regions after optimization, a box-counting dimension and lacunarity were introduced as supplementary characterization indices. First, according to the functional requirements of different gait phases, the task space of the mechanism is partitioned into stance, mid-swing, and swing-transition regions. A unified kinematic model and singularity criterion are then established for the planar five-bar mechanism, and mechanism performance indices for different task regions are constructed based on the Jacobian matrix to characterize the force and velocity transmission capabilities of the mechanism, as well as its singularity margin. A genetic algorithm is used to perform dimensional synthesis optimization of the mechanism parameters. Furthermore, a task space transmission performance field is introduced, and the area ratio, box-counting dimension, and lacunarity of regions with high performance are used to characterize the spatial structure of high-performance transmission regions before and after optimization. Finally, a series of theoretical calculations and physical experiments are conducted to verify that the differential characteristics of the mechanism have a significant influence on both its static and dynamic performance. The experimental results show that the optimized mechanism achieves lower normalized objective values in all task regions and outperforms the reference mechanism in load capacity, static power consumption, positioning accuracy, and trajectory consistency. The maximum static load capacity reaches 1.29 times that of the reference mechanism, while the static power consumption is reduced to approximately one half of that of the reference mechanism.

1. Introduction

1.1. Research Background and Motivation

Legged robots show strong adaptability in locomotion over complex terrain, load transportation, and dynamic operation. Their performance is commonly reflected in load capacity, power consumption, positioning accuracy, and trajectory-tracking capability. These aspects are jointly affected by actuation, control strategy, structural compliance, dynamic effects, and mechanism design [1]. Among these factors, the configuration and geometric parameters of the leg mechanism determine its velocity and force transmission characteristics in the task space. They therefore directly affect load capacity, energy consumption, and trajectory-tracking performance. Although these performance measures cannot be fully explained by kinematics alone, the force state, velocity distribution, and posture evolution of the robot during static loading and dynamic motion are always constrained by the kinematic configuration space of the mechanism. It is therefore important to use physically interpretable kinematic indices to guide the parameter design of robotic leg mechanisms.

The planar five-bar mechanism has a closed-chain structure, proximal actuation, and relatively high structural stiffness. It has therefore been widely used in robotic leg design [2,3,4]. Compared with serial leg mechanisms, a five-bar leg mechanism allows the main actuators to be placed near the base, which reduces swing inertia and improves motion response. However, due to its closed-chain topology, its velocity transmission, force transmission, and singularity margin vary markedly across the task space. For a robotic leg mechanism, the foot does not perform a single uniform task during one gait cycle. The stance phase requires strong vertical load transmission and static support capability. The mid-swing phase places greater emphasis on horizontal foot velocity output. The swing-transition phase requires sufficient manipulability and a margin from singular configurations. Parameter optimization based on a single global index or a regional mean index may not fully reflect these different local transmission requirements in different gait phases.

In addition, changes in mechanism parameters affect not only regional mean performance, but also the spatial distribution of high-performance transmission points in the task space. For a continuous foot trajectory, the adaptability of the mechanism to practical motion tasks depends on whether the high-performance regions cover the main task path, remain spatially continuous, and avoid excessive fragmentation. Existing mechanism optimization studies mainly focus on global or regional scalar indices. Less attention has been paid to the spatial organization of high-performance transmission regions after optimization. It is therefore useful to combine physically interpretable regional Jacobian performance indices with spatial characterization of high-performance regions, so as to examine how dimensional parameter optimization affects local transmission performance and its spatial organization.

1.2. Literature Review

In leg mechanism design, increasing attention has been paid to the coupling between actuator capability and mechanism parameters. Many legged robots no longer place the actuator directly at the knee joint [5]. Instead, they use proximal actuation to reduce swing inertia, improve transmission efficiency, and enhance dynamic performance [2,6,7,8]. In such designs, knee motion is usually transmitted through belts [9] or parallel four-bar mechanisms [7]. Nonlinear linkages, parallel mechanisms, and elastic actuation mechanisms have also been proposed to improve the mechanical performance of legged robots [10,11,12,13,14]. These studies show that leg performance is strongly affected by the combined effects of mechanism topology, geometric parameters, and actuation characteristics. Compliant linkages and redundant actuation have also been used to improve the energy efficiency of leg motion [15,16]. At the same time, studies on quasi-direct-drive robots by Wensing et al. and Kau et al. showed that leg structure, transmission layout, and actuator parameters have a significant influence on joint torque demand, distal inertia, energy consumption, and dynamic response [7,17]. Nizami et al. proposed a proximally actuated elastic leg mechanism and pointed out clear trade-offs among mechanical advantage, output displacement amplification, and actuator load [18]. Liu et al. proposed a task-oriented system design method for a heavy-duty electrically actuated quadruped robot; motor torque density, battery energy density, load capacity, and speed performance were included in the design process [19]. Liu et al. further used multi-objective optimization to select spring parameters in a dual-slide parallel elastic actuator, aiming to reduce peak torque, motor power, and motion energy consumption [20]. Together, these studies indicate that leg mechanism parameter optimization should consider force and velocity transmission under actuator-limited conditions.

For parameter optimization of robotic leg mechanisms, the Jacobian matrix provides an important tool for analyzing how mechanism configuration and geometric parameters affect velocity and force transmission. The velocity ellipse and force ellipse give intuitive descriptions of the mapping among joint velocity, foot velocity, actuator torque, and endpoint force [21,22]. These concepts have been widely used in mechanism analysis and synthesis to evaluate workspace properties, manipulability, isotropy, and proximity to singular configurations [23]. Recent studies of five-bar mechanisms have shown that the direction, scale, and axis ratio of the velocity ellipse field can strongly affect directional velocity transmission. These characteristics further influence actuation demand, force transmission capability, energy consumption, and motion accuracy [24,25]. At the same time, parameter optimization of robotic mechanisms has gradually shifted from single geometric index optimization to multi-objective optimization. Conventional mechanism synthesis often uses Jacobian-related indices, such as manipulability, condition number, workspace coverage, and singularity margin, to evaluate kinematic performance [26,27,28]. As robotic tasks become more complex, energy consumption, actuator torque, cable tension, structural size, stiffness, and task reachability have also been included in optimization objectives. Wu et al. used a niched Pareto genetic algorithm for dimensional synthesis of multi-linkage robots, with dexterity and energy consumption considered simultaneously [29]. Ben Hamida et al. proposed a multi-objective design method for a cable-driven parallel robot for rehabilitation tasks, considering cable tension, mechanism footprint, and workspace performance [30]. Molaei and Ghatrehsamani used global manipulability and isotropy indices for multi-objective kinematic optimization in the design of a grapevine-pruning robot [31]. These studies show that multi-objective optimization has become an important approach for robot mechanism parameter design. However, most existing methods still formulate optimization objectives from the viewpoint of the global workspace or overall task performance. For legged robots, the stance, swing, and transition phases correspond to different local functional requirements. Relying only on global manipulability, condition number, or regional mean indices may not fully capture the different transmission requirements of different functional segments of the foot trajectory.

Gait characteristics, simplified dynamic models, and foot trajectory planning can also be used to define task requirements for leg mechanisms. Simplified models, such as the linear inverted pendulum (LIP) and the spring-loaded inverted pendulum (SLIP), have been widely used for gait analysis, stance-phase approximation, and trajectory generation in legged robots [32,33,34]. However, their use in mechanism synthesis remains limited, because they are mainly used for gait interpretation and control design. Biological gait data and simplified dynamic models show that, in the hip joint coordinate frame, the stance-phase foot trajectory is not a simple straight line. Instead, it forms an arc-shaped distribution with clear curvature. In addition, foot trajectories in different gait phases show stable clustering features in the task space. This indicates that the foot does not perform a single task during one gait cycle; rather, it undertakes support, swing, and phase-transition functions in different spatial regions. These functional and spatial features motivate the use of task space partitioning in mechanism parameter design.

Existing Jacobian-based mechanism synthesis methods usually evaluate regional or global performance using scalar statistics, such as regional mean values, extrema, manipulability indices, or constraint satisfaction measures [35,36,37]. For legged robots, the foot trajectory evolves continuously through different gait phases. A small number of high-performance discrete points may not sufficiently represent the adaptability of the mechanism to a continuous motion task. In contrast, spatially continuous high-performance regions that cover the main task path can better reflect practical motion adaptability. Therefore, mechanism performance analysis should consider not only the magnitude of local kinematic indices, but also the spatial distribution of high-performance transmission regions in the task space after dimensional optimization. Fractal characterization provides a way to quantify such spatial structures from a multi-scale perspective. The box-counting dimension describes the spatial filling capacity of a point set at different scales and is commonly used to quantify complex spatial distributions [38,39]. Lacunarity complements the fractal dimension by describing gaps, clustering, and heterogeneity in spatial distributions. It can also distinguish point sets with similar fractal dimensions but different spatial organizations [40,41]. In leg mechanism synthesis, these indices can be used to evaluate whether high-performance transmission regions show larger coverage, better continuity, and lower fragmentation.

Table 1. Related studies and their research characteristics.

Research Category	Main Focus	Advantages	Limitations Relative to This Study
Actuator-aware leg design [2,6,7,8,9,10,11,12,13,14,17,18,19,20]	Actuator placement, torque demand, inertia, energy efficiency	Closely related to actuator limits and dynamic performance	Mainly focuses on system-level actuator or leg design; local task-space transmission distribution is not explicitly characterized
Jacobian-based mechanism optimization [21,22,23,24,25,26,27,28,29,30,31]	Manipulability, condition number, workspace, multi-objective optimization	Provides mature physical indicators for kinematic synthesis	Mostly uses global or averaged indices; gait-function differences among task regions are weakly represented
Gait-model-based task generation [32,33,34]	LIP/SLIP models and biological gait data	Provides physically meaningful task space trajectories	Mainly used for gait analysis and control rather than dimensional synthesis
Spatial/fractal characterization [35,36,37,38,39,40,41]	Box-counting dimension, lacunarity, spatial distribution	Quantifies multi-scale spatial filling and heterogeneity	Rarely linked with Jacobian-based mechanism optimization of robotic legs

presents a qualitative comparison of leg mechanism design, Jacobian performance optimization, and spatial distribution characterization methods related to this study.

Existing studies have developed mature methods for actuator layout, leg mechanism design, global kinematic optimization, and spatial structure characterization. However, in the parameter optimization of robotic leg mechanisms, most studies still focus on global kinematic indices, actuator capability, or overall dynamic performance. Less attention has been paid to the differentiated transmission requirements of the foot trajectory in different functional regions and to the spatial distribution of high-performance operating points after optimization. Based on these considerations, this study proposes a dimensional parameter optimization method for a planar five-bar robotic leg mechanism based on task space partitioning. First, the task space is partitioned according to the functional characteristics of the foot trajectory. Then, Jacobian-based performance indices are constructed for the stance, mid-swing, and swing-transition regions, respectively, and multi-objective dimensional parameter optimization is performed. Finally, based on the high-performance operating points in the discrete task space, the area ratio, box-counting dimension, and lacunarity are introduced as supplementary evaluation indices for the spatial distribution of high-performance transmission regions after optimization. These indices are used to describe the coverage, spatial filling degree, and distribution non-uniformity of the high-performance point set in the finite task space.

The remainder of this paper is organized as follows. Section 2 describes the generation of task space requirements and the partitioning of the foot workspace based on gait characteristics and simplified models. Section 3 establishes the kinematic model of the planar five-bar mechanism and analyzes its singularity characteristics. Section 4 presents the construction of partitioned kinematic indices, the parameter optimization model, and the characterization of high-performance transmission regions in the task space. Section 5 provides experimental validation through comparative prototype tests. Finally, the main work of this study is summarized in Section 6.

2. Task Space Formulation and Partitioning

Using the spatiotemporal characteristics of typical canine trot gaits at different locomotion speeds reported by Maes et al. [42] as reference inputs, and combining them with the SLIP/LIP dynamic models, the full cycle foot-end trajectories in the hip joint coordinate frame $\left\{H\right\}$ are generated, as shown in Figure 1a. The foot exhibits clear spatial clustering in the hip joint coordinate frame: the stance phase trajectory is mainly distributed in the lower arc-shaped region, the mid-swing trajectory is mainly located in the upper region, and the transition trajectories between the stance and swing phases are concentrated on the left and right sides. This distribution indicates that the foot does not perform a single uniform task over one gait cycle; instead, it serves different functions in different spatial regions, including load-bearing support, rapid swing motion, and phase transition. Based on these trajectory distribution characteristics, the foot task space is partitioned into four functional regions: the stance region ${\mathsf{\Omega }}_{\mathrm{s}\mathrm{t}}$, the left swing-transition region ${\mathsf{\Omega }}_{\mathrm{s}\mathrm{w}\mathrm{L}}$, the mid-swing region ${\mathsf{\Omega }}_{\mathrm{s}\mathrm{w}\mathrm{M}}$, and the right swing-transition region ${\mathsf{\Omega }}_{\mathrm{s}\mathrm{w}\mathrm{R}}$:

$$\left\{\begin{array}{c}{\mathsf{\Omega }}_{\mathrm{st}}=\{(x,y)\in \mathrm{X} | y\le y_0\}\\ {\mathsf{\Omega }}_{\mathrm{swL}}=\{(x,y)\in \mathrm{X} | y_0<y<y_1,x<-x_d\}\\ {\mathsf{\Omega }}_{\mathrm{swM}}=\{(x,y)\in \mathrm{X} | y_0<y<y_1,x_L\le x\le x_d\}\\ {\mathsf{\Omega }}_{\mathrm{swR}}=\{(x,y)\in \mathrm{X} | y_0<y<y_1,x>x_d\}\end{array}\right.$$

(1)

Here, $y_0$ and $y_1$ define the vertical boundaries between the stance and swing regions and the internal vertical subdivision of the swing region, respectively, whereas $x_d$ defines the effective horizontal width of the mid-swing region. Based on the spatial distribution of the foot trajectory in the hip joint coordinate frame, these parameters are set to $y_0=-0.35\hspace{0.17em}\mathrm{m}$, $y_1=-0.10\hspace{0.17em}\mathrm{m}$, and $x_d=0.09\hspace{0.17em}\mathrm{m}$. It should be noted that these partition boundaries are not unique; rather, they represent a set of engineering partitioning parameters determined from the foot trajectory envelope, the functional differences among gait phases, and the workspace distribution characteristics.

For each velocity sample, the stance phase and swing phase trajectories are distinguished according to the vertical contact force, and the distribution density of the foot trajectory in the hip joint coordinate frame is then evaluated, as shown in Figure 1b,c. The resulting partitioning serves as the basis for the subsequent construction of partitioned kinematic indices and the parameter optimization model.

3. Kinematics and Jacobian Mechanism

The end-effector motion of a planar, closed-chain five-bar mechanism is determined by the active joint variables, geometric parameters, and loop closure constraints. As a result of its closed chain topology, the local velocity transmission, force transmission, and singularity characteristics vary significantly across the task space. To quantify these properties, a kinematic model of the mechanism is established, followed by analyses of the Jacobian matrix, velocity ellipse, and manipulability. These analyses provide the foundation for the construction of partitioned kinematic indices and the subsequent dimensional synthesis.

3.1. Kinematic Modeling of a Planar Five-Bar Mechanism

The leg mechanism studied in this paper is a planar, closed-chain five-bar configuration, as shown in Figure 2a.

The mechanism under study consists of the base link $AC$, the left branch $A-B-E$, the right branch $C-D-E$, and the ternary link $BEP$, where $A$ and $C$ are fixed revolute joints and $P$ is the foot point. Two actuators are mounted on links $AB$ and $CD$, respectively. The active joint angles are denoted by ${\theta }_1$ and ${\theta }_3$, whereas the passive joint angles are denoted by ${\theta }_2$ and ${\theta }_4$. The mechanism parameter vector $\mathbf{p}=[l_1,l_2,l_3,l_4,l_5,l_6,{\theta }_6]$ is defined as shown in Figure 2a, and the counterclockwise direction is taken as positive for all joint angles. Both the symmetric five-bar mechanism and the coaxially symmetric five-bar mechanism can be regarded as special cases of this unified model under specific parameter constraints and can therefore be treated within the same kinematic framework.

In the base coordinate frame $\left\{O\right\}$, point $A$ is taken as the origin, and the base link $AC$ is aligned along the positive $x$ axis. Then

$${\mathbf{r}}_A=\left[\begin{array}{c}0\\ 0\end{array}\right],{\mathbf{r}}_C=\left[\begin{array}{c}{l}_5\\ 0\end{array}\right]$$

(2)

For convenience of notation, $\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_i\right)$ and $\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_i\right)$ are denoted by ${\mathrm{C}}_i$ and ${\mathrm{S}}_i$, respectively, and $\mathrm{c}\mathrm{o}\mathrm{s}({\theta }_i+{\theta }_j)$ and $\mathrm{s}\mathrm{i}\mathrm{n}({\theta }_i+{\theta }_j)$ are denoted by ${\mathrm{C}}_{ij}$ and ${\mathrm{S}}_{ij}$, respectively. the position of the closed chain point $E$ can be expressed in terms of the left and right branches as

$$r_E^{(L)}=\left[\begin{array}{l}{l}_1{\mathrm{C}}_1+l_2{\mathrm{C}}_{12}\hfill \\ l_1{\mathrm{S}}_1+l_2{\mathrm{S}}_{12}\hfill \end{array}\right],\hspace{1em}{r}_E^{(R)}=\left[\begin{array}{c}{l}_5+l_3{\mathrm{C}}_3+l_4{\mathrm{C}}_{34}\\ l_3{\mathrm{S}}_3+l_4{\mathrm{S}}_{34}\end{array}\right]$$

(3)

The closed-loop constraint requires the two branches to coincide at the connecting point, i.e., ${\mathbf{r}}_E^{\left(L\right)}={\mathbf{r}}_E^{\left(R\right)}$. The corresponding closed-loop constraint equations of the planar five-bar mechanism are written as

$$\left\{\begin{array}{c}{l}_1{\mathrm{C}}_1+l_2{\mathrm{C}}_{12}-l_3{\mathrm{C}}_3-l_4{\mathrm{C}}_{34}-l_5=0\\ l_1{\mathrm{S}}_1+l_2{\mathrm{S}}_{12}-l_3{\mathrm{S}}_3-l_4{\mathrm{S}}_{34}=0\end{array}\right.$$

(4)

The position of the foot point $P$ can be expressed as

$$r_P=r_E+l_6\left[\begin{array}{l}\cos \left({\theta }_1+{\theta }_2+{\theta }_6\right)\hfill \\ \sin \left({\theta }_1+{\theta }_2+{\theta }_6\right)\hfill \end{array}\right]$$

(5)

For given mechanism parameters $\mathbf{p}$ and active joint variables $\mathit{q}=[{\theta }_1,{\theta }_3{]}^T$, the passive joint angles ${\theta }_2$ and ${\theta }_4$ can be obtained from Equation (4), and the foot position can then be determined accordingly. Conversely, for a given foot position, the corresponding mechanism configuration can be obtained through inverse kinematics.

3.2. Velocity Ellipses and Manipulability of the Mechanism

The motion and force transmission capabilities of the mechanism near a given configuration are determined by its first-order kinematic properties, which can be described by the Jacobian matrix. The Jacobian matrix maps joint velocities to foot-end velocities. It also relates actuator torques to foot-end forces through the principle of virtual work. To analyze the transmission characteristics in different directions more intuitively, a velocity ellipse is used in this study. Although it is referred to as a velocity ellipse, it contains the corresponding information on force transmission as well.

The mechanism is a planar two-degree-of-freedom mechanism. It consists of two input variables, $\mathit{q}=[{\theta }_1,{\theta }_3{]}^T$, and two Cartesian output variables, $\mathit{x}=[x,\mathrm{y}{]}^T$. These variables are constrained by $\mathit{f}(\mathit{q},\mathit{x})=\mathbf{0}\in {ℝ}^2$. The total differential of $\mathit{f}$ can be written as

$$\left[{\displaystyle \frac{\partial \mathit{f}}{\partial \mathit{q}}}\right]d\mathit{q}+\left[{\displaystyle \frac{\partial \mathit{f}}{\partial \mathit{x}}}\right]d\mathit{x}=\mathbf{0}$$

(6)

The transformation from input velocity to output velocity is given by

$$\dot{\mathit{x}}=-{\left[{\displaystyle \frac{\partial \mathit{f}}{\partial \mathit{q}}}\right]}^{-1}\left[{\displaystyle \frac{\partial \mathit{f}}{\partial \mathit{x}}}\right]\dot{\mathit{q}}=\mathit{J}(\mathit{q},p)\dot{\mathit{q}}$$

(7)

where $\mathit{J}(\mathit{q},p)$ is the Jacobian matrix, abbreviated as $\mathit{J}$ in the following text for simplicity. The input torque is defined as $\mathit{\tau }=[{\tau }_1,{\tau }_3{]}^T$, whose elements correspond to those of $\mathit{q}$. The output force is defined as $\mathit{F}=[F_x,F_y{]}^T$, whose elements correspond to those of $\mathit{x}$. According to the principle of virtual work, the virtual work balance can be written as $\mathit{\tau }\cdot \delta \mathit{q}=\mathit{F}\cdot \mathit{J}\delta \mathit{q}$. Rearranging this expression gives $(\mathit{\tau }-{\mathit{J}}^T\mathit{F})\cdot \delta \mathit{q}=0$. Setting the term in parentheses to zero, the force transmission relationship of the mechanism can be expressed as

$$\mathit{\tau }={\mathit{J}}^T\mathit{F}$$

(8)

Equation (8) can be rewritten as $\mathit{F}={\mathit{J}}^{-T}\mathit{\tau }$. Therefore, the Jacobian matrix not only determines the local mapping from joint velocities to end point velocities, but also characterizes the transmission relationship between the endpoint force and the driving joint torques. As illustrated in Figure 3, the matrix $\mathit{J}$ maps the input velocity (unit: rad/s) to the output velocity (unit: m/s). The matrix ${\mathit{J}}^T$ maps the input force torque (unit: N·m) to the output force (unit: N). With a unit input velocity circle or a unit input torque circle, the transformed set through $\mathit{J}$ or ${\mathit{J}}^T$ forms the velocity ellipse or the force ellipse, respectively. When the major or minor semi-axis length of the velocity ellipse tends to infinity, the mechanism is in a global or local singular state in the output space. In this case, the output velocity of the mechanism becomes extremely sensitive along the corresponding axis direction, which indicates a singularity of $\partial \mathit{f}/\partial \mathit{x}$. When the minor semi-axis length of the velocity ellipse tends to zero, the mechanism is in a global or local boundary state in the output space. In this case, the mechanism cannot generate output velocity along the corresponding short-axis direction, which indicates a singularity of $\partial \mathit{f}/\partial \mathit{q}$. The size and orientation of the velocity ellipse and the force ellipse can be obtained from the singular value decomposition (SVD) of $\mathit{J}$ or ${\mathit{J}}^T$. Let $\mathit{J}=\mathit{U}\Sigma {\mathit{V}}^{\mathrm{T}}$, where $\mathit{U}$ and $\mathit{V}$ are orthogonal matrices and $\Sigma$ is a diagonal matrix. It follows that ${\mathit{J}}^{\mathrm{T}}=\mathit{U}{\Sigma }^{-1}{\mathit{V}}^{\mathrm{T}}$. Therefore, the force ellipse and the velocity ellipse have the same principal directions, but their corresponding semi-axis lengths are reciprocal to each other.

The velocity ellipse provides an intuitive description of the velocity amplification capability of the mechanism in different directions in the task space, and it is dual to the corresponding force transmission capability. It should be noted that this duality refers to the reciprocal relationship between the semi-axis lengths under the same norm definitions and unit convention. The velocity and force quantities still have different physical units, and their numerical values should be calculated using the Jacobian relationships in Equations (7) and (8). For leg mechanisms, the swing phase is more closely associated with directional velocity output, whereas the stance phase places greater emphasis on vertical load support and force transmission. The velocity ellipse provides a unified geometric basis for the subsequent construction of indices for different task regions.

For the two-dimensional, planar five-bar mechanism considered in this paper, the Jacobian matrix $\mathit{J}$ is a $2\times 2$ square matrix. To quantify how far the mechanism is from singular configurations, the manipulability index $\omega$ is defined as

$$\omega (\mathit{q},\mathbf{p})=\sqrt{\det (J(\mathit{q},\mathbf{p})J^T(\mathit{q},\mathbf{p}))}$$

(9)

This index reflects the local velocity-mapping capacity of the mechanism. A larger value of $\omega (\mathit{q},\mathbf{p})$ indicates a larger local velocity mapping area and a larger margin from singular configurations. When $\omega (\mathit{q},\mathbf{p})=0$, the Jacobian matrix becomes singular, and the mechanism reaches a singular state. For a closed-chain five-bar mechanism, singularity not only implies degeneration of the local velocity/force mapping, but also means that the corresponding operating point is excluded from subsequent performance evaluation.

4. Kinematic Indices and Parameter Optimization of the Mechanism

The local velocity and force transmission capabilities of the mechanism in the task space are determined by its first-order kinematic properties. In this section, performance indices with clear physical meanings are constructed for different functional regions. Specifically, a vertical velocity mapping index is used for the stance region to reflect force transmission capability in the vertical direction. A horizontal velocity mapping index is used for the mid-swing region to evaluate the horizontal velocity output capability of the foot. A manipulability index is used for the swing-transition regions to measure the margin from singular configurations. Based on these indices, dimensional optimization of the geometric parameters of the planar five-bar mechanism is performed.

4.1. Task Space Sampling and Screening of Valid Operating Points

Using the envelopes of the stance, left swing-transition, mid-swing, and right swing-transition regions as regional boundaries, the task space is uniformly discretized with a fixed step size to construct the corresponding regional task point sets:

$${\mathcal{P}}_{\alpha }=\left\{x_j\mid x_j\in {\mathsf{\Omega }}_{\alpha },j=1,2,\dots ,N_{\alpha }\right\},\hspace{1em}\alpha \in \{\mathrm{st},\mathrm{swL},\mathrm{swM},\mathrm{swR}\}$$

(10)

where $N_{\alpha }$ denotes the total number of sampled points in region ${\mathsf{\Omega }}_{\alpha }$. In this study, the sampling step sizes are set to $\mathsf{\Delta }x=\mathsf{\Delta }y=0.03\hspace{0.17em}\mathrm{m}$. Since the left and right transition regions play the same functional role, they are merged in the subsequent analysis, and the total number of valid operating points in the transition regions is defined as $N_{\mathrm{s}\mathrm{l}\mathrm{r}}=N_{\mathrm{s}\mathrm{l}}+N_{\mathrm{s}\mathrm{r}}$.

To avoid bias in regional performance evaluation caused by insufficient valid samples during dimensional parameter optimization, the number of valid operating points in each functional region is introduced as an explicit coverage constraint in the optimization model.

$${\displaystyle \frac{N_{\mathrm{st}}}{N_{\mathrm{st}}^r}}\ge {\eta }_{\min },\hspace{1em}{\displaystyle \frac{N_{\mathrm{sm}}}{N_{\mathrm{sm}}^r}}\ge {\eta }_{\min },\hspace{1em}{\displaystyle \frac{N_{\mathrm{slr}}}{N_{\mathrm{slr}}^r}}\ge {\eta }_{\min }$$

(11)

where $N_{\mathrm{s}\mathrm{t}}^r$, $N_{\mathrm{s}\mathrm{m}}^r$, and $N_{\mathrm{s}\mathrm{l}\mathrm{r}}^r$ denote the numbers of valid operating points of the reference mechanism in the stance region, the mid-swing region, and swing transition regions, respectively, and ${\eta }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is the lower bound of the regional coverage ratio. The lower bound ${\eta }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is used as a conservative feasibility constraint to prevent the optimization from improving the regional transmission indices by excessively reducing the feasible task space. It is not the desired final coverage ratio, but the minimum acceptable coverage in each task region. The actual coverage ratios of the final optimized mechanism are further checked after optimization.

4.2. Construction of Partitioned Kinematic Indices

Let the velocity Jacobian matrix of the planar five-bar mechanism at the current operating point be denoted by

$$J(\mathit{q},p)=\left[\begin{array}{ll}{J}_{11}\hfill & J_{12}\hfill \\ J_{21}\hfill & J_{22}\hfill \end{array}\right]$$

(12)

The vertical velocity component of the end point is given by

$$\dot{y}=\left[\begin{array}{ll}{J}_{21}\hfill & J_{22}\hfill \end{array}\right]\dot{\mathit{q}}$$

(13)

Under the unit joint velocity constraint, the vertical velocity mapping index for the stance region is defined as

$$j_{\mathrm{st}}(\mathit{x},p)=\sqrt{J_{21}^2+J_{22}^2}$$

(14)

This index represents the mapping strength from joint-space velocity to the vertical direction of the task space at task point $\mathit{x}$. According to the relationship between velocity and force transmission, a smaller $j_{\mathrm{s}\mathrm{t}}$ indicates a larger mechanical advantage in the vertical direction. It is therefore more favorable for the load-bearing requirement during the stance phase.

For the mid-swing region ${\mathsf{\Omega }}_{\mathrm{s}\mathrm{w}\mathrm{M}}$, the foot mainly performs a rapid forward swing motion and requires strong horizontal velocity output capability. The first row of the Jacobian matrix is used to define the horizontal velocity mapping index, which characterizes the local horizontal motion capability in this region:

$$j_{\mathrm{sm}}(\mathit{x},p)=\sqrt{J_{11}^2+J_{12}^2}$$

(15)

A larger value of $j_{\mathrm{s}\mathrm{m}}$ indicates stronger horizontal velocity output capability, which is favorable for the rapid forward swing of the foot in the mid-swing region.

For the left and right swing-transition regions, ${\mathsf{\Omega }}_{\mathrm{s}\mathrm{w}\mathrm{L}}$ and ${\mathsf{\Omega }}_{\mathrm{s}\mathrm{w}\mathrm{R}}$, the foot trajectory lies in the transition stage between the swing and stance phases. Compared with the stance and mid-swing regions, these regions place greater emphasis on local reachability and sufficient margin from singular configurations, so as to ensure trajectory continuity and motion stability during phase transition. The manipulability of the mechanism is expressed as

$$j_{\mathrm{slr}}(\mathit{x},p)=\sqrt{\det (J(\mathit{q},\mathbf{p})J^T(\mathit{q},\mathbf{p}))}$$

(16)

Equation (16) reflects the stability of the local velocity mapping at the current operating point. A larger value of $j_{slr}$ indicates that the mechanism is farther from singular configurations and possesses more balanced local motion capability, which is more favorable for the trajectory transition between the swing and stance phases.

4.3. Index Normalization and Parameter Optimization Model

To remove the scale differences among the regional indices, the regional performance of the reference mechanism is taken as the baseline, and each regional index is normalized accordingly. For the mechanism parameter $\mathbf{p}$, inverse kinematics and Jacobian calculation are first performed for each sampled point in the task space. If a point is unreachable, the Jacobian elements are not finite, or the singularity-related criteria are not satisfied, the point is regarded as invalid and is excluded from the calculation of regional mean indices. Let ${\mathcal{P}}_{\alpha }^v$ denote the set of valid points in the $\alpha -\mathrm{th}$ task region, and let $N_{\alpha }^v$ be the number of valid points in this region. The left and right swing-transition regions play the same functional role, corresponding to the transition process between the stance and swing phases. They are therefore merged into one transition region in the subsequent calculation. The regional mean index of the transition region is calculated over all valid points in the two transition regions as a whole, rather than by averaging the two regional means with equal weights. The corresponding regional mean indices of the candidate mechanism are then defined as

$$\left\{\begin{array}{c}{\overline{j}}_{\mathrm{st}}(p)={\displaystyle \frac{1}{N_{\mathrm{st}}^v}}{\displaystyle \sum _{{\mathit{x}}_j\in {\mathcal{P}}_{\mathrm{st}}^v(p)}{j}_{\mathrm{st}}}\left({\mathit{x}}_j,p\right)\\ {\overline{j}}_{\mathrm{sm}}(p)={\displaystyle \frac{1}{N_{\mathrm{sm}}^v}}{\displaystyle \sum _{{\mathit{x}}_j\in {\mathcal{P}}_{\mathrm{swM}}^v(p)}{j}_{\mathrm{sm}}}\left({\mathit{x}}_j,p\right)\\ {\overline{j}}_{\mathrm{slr}}(p)={\displaystyle \frac{1}{N_{\mathrm{slr}}^v}}\left({\displaystyle \sum _{x_j\in {\mathcal{P}}_{\mathrm{swL}}^v(p)}{j}_{\mathrm{slr}}}\left({\mathit{x}}_j,p\right)+{\displaystyle \sum _{x_j\in {\mathcal{P}}_{\mathrm{swR}}^v(p)}{j}_{\mathrm{slr}}}\left({\mathit{x}}_j,p\right)\right)\end{array}\right.$$

(17)

Let the regional mean indices of the reference mechanism in the stance region, mid-swing region, and transition regions be denoted by ${\overline{j}}_{\mathrm{st}}^r$, ${\overline{j}}_{\mathrm{sm}}^r$, and ${\overline{j}}_{\mathrm{slr}}^r$, respectively. The normalized objectives are defined as

$$f_{\mathrm{st}}(\mathbf{p})={\displaystyle \frac{{\overline{j}}_{\mathrm{st}}(\mathbf{p})}{{\overline{j}}_{\mathrm{st}}^r}}$$

(18)

$$f_{\mathrm{sm}}(p)={\displaystyle \frac{{\overline{j}}_{\mathrm{sm}}^r}{{\overline{j}}_{\mathrm{sm}}(p)}}$$

(19)

$$f_{\mathrm{slr}}(p)={\displaystyle \frac{{\overline{j}}_{\mathrm{slr}}^r}{{\overline{j}}_{\mathrm{slr}}(p)}}$$

(20)

Equations (18)–(20) define the normalized regional objectives in minimization form. The parameter optimization problem of the planar five-bar mechanism can be formulated as

$$\begin{array}{l}\underset{p}{\min }F(p)={\omega }_{\mathrm{st}}{f}_{\mathrm{st}}(p)+{\omega }_{\mathrm{sm}}{f}_{\mathrm{sm}}(p)+{\omega }_{\mathrm{slr}}{f}_{\mathrm{slr}}(p)\\ \mathrm{s}.\mathrm{t}.{\displaystyle \frac{N_{\mathrm{st}}}{N_{\mathrm{st}}^r}}\ge {\eta }_{\min },\hspace{1em}{\displaystyle \frac{N_{\mathrm{sm}}}{N_{\mathrm{sm}}^r}}\ge {\eta }_{\min },\hspace{1em}{\displaystyle \frac{N_{\mathrm{slr}}}{N_{\mathrm{slr}}^r}}\ge {\eta }_{\min },p_{\min }\le p\le p_{\max }\end{array}$$

(21)

where ${\omega }_{\mathrm{s}\mathrm{t}}$, ${\omega }_{\mathrm{s}\mathrm{m}}$, and ${\omega }_{\mathrm{s}\mathrm{l}\mathrm{r}}$ are the weighting coefficients for the objectives of the stance region, the mid-swing region, and the transition regions, respectively. Moreover, ${\eta }_{\mathrm{m}\mathrm{i}\mathrm{n}}=0.7, \mathrm{a}\mathrm{n}\mathrm{d} {\mathbf{p}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathbf{p}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ denote the lower and upper bounds of the mechanism design variables, respectively. The design variables and their ranges are listed in

Table 2. Design parameters of the mechanism and their ranges.

Parameter	Range (${\mathbf{p}}_{\mathbf{r}}$)	Range ($\mathbf{p}$)	Parameter	Range (${\mathbf{p}}_{\mathbf{r}}$)	Range ($\mathbf{p}$)
$l_1$ (m)	0.25	$[0.7l_{r1},1.3l_{r1}]$	$l_5$(m)	0.15	$[0.7l_{r5},1.3l_{r5}]$
$l_2$ (m)	0.25	$[0.7l_{r2},1.3l_{r2}]$	$l_6$(m)	\	$[l_5,1.3l_5]$
$l_3$ (m)	0.25	$[0.7l_{r3},1.3l_{r3}]$	${\theta }_6$(°)	\	$[0,10]$
$l_4$ (m)	0.25	$[0.7l_{r4},1.3l_{r4}]$

Note: “\” indicates that the corresponding mechanism does not have this parameter.

.

4.4. Transmission Performance Field and Spatial Distribution Evaluation of High-Performance Regions

Regional mean indices characterize the overall performance of the mechanism in different task regions, but they do not describe the spatial organization of high-performance regions in the task space. Since the foot trajectory evolves continuously in the task space, the continuity of high-performance regions and their coverage of the main task path are important for practical motion performance.

For the task region ${\mathsf{\Omega }}_{\alpha }$, the corresponding transmission performance field is denoted as

$${\mathsf{\Phi }}_{\alpha }(x,y)=\left\{\begin{array}{c}{j}_{\mathrm{st}}^2(x,y),(x,y)\in {\mathsf{\Omega }}_{\mathrm{st}}\\ {\displaystyle \frac{1}{j_{\mathrm{sm}}^2(x,y)+\epsilon }},(x,y)\in {\mathsf{\Omega }}_{\mathrm{swM}}\\ {\displaystyle \frac{1}{j_{\mathrm{slr}}^2(x,y)+\epsilon }},(x,y)\in {\mathsf{\Omega }}_{\mathrm{swR}/\mathrm{L}}\end{array}\right.$$

(22)

It should be emphasized that Equation (22) does not represent a direct extension of the same physical quantity across different regions. Instead, it defines a region-dependent scalar performance field according to the functional requirements of different task regions. Its purpose is to express favorable local transmission performance in each region as a smaller value, so that high-performance regions can be defined and their spatial distributions can be compared. The regularization parameter is set to $\epsilon ={10}^{-6}$. The high-performance transmission region $S_{\alpha }$ is defined as

$$S_{\alpha }=\left\{\left(x,y\right)\in {\mathsf{\Omega }}_{\alpha }|{\mathsf{\Phi }}_{\alpha }(x,y)\le T_{\alpha }\right\}$$

(23)

where $T_{\alpha }$ denotes the high-performance threshold for the corresponding region. For the stance region, Equations (8) and (22) give the following relationship between the unit vertical foot-end force and the norm of the joint torque:

$${\displaystyle \frac{‖\mathit{\tau }‖}{F_y}}=\sqrt{J_{21}^2+J_{22}^2}=j_{\mathrm{st}}(x,y)$$

(24)

Let the peak torque of the drive motor be ${\tau }_{\max }$, and let the required vertical force in the stance phase be obtained from the LIP model. The stance-region points that satisfy the stable walking requirement under the LIP model are defined as high-performance points. The high-performance threshold for the stance region is then given by

$$T_{st}={\left({\displaystyle \frac{\sqrt{2}{\tau }_{\max }}{k_v{F}_y}}\right)}^2$$

(25)

where the operating factor $k_v$ is used to describe the amplification of the actual dynamic support requirement relative to the nominal support requirement obtained from the simplified model. Previous biological gait studies have shown that the peak vertical ground reaction force during human walking can increase to about 1.5 times body weight as walking speed increases and can reach about 2.0–2.9 times body weight during running [43,44]. Rumph et al. measured the ground reaction forces of dogs during normal trot gait and found that the combined peak vertical force of a diagonal supporting foot pair was about 1.72 times body weight [45]. Bledt et al. showed that, for MIT Cheetah 3, the pure vertical ground reaction force capability of a single leg under typical conditions was about 1.6–2.2 times the robot body weight [8]. Since the gait information used in this study is mainly derived from quadruped dog locomotion studies, $k_v=1.72$ is adopted.

Similarly, for the swing region, let the allowable peak joint velocity of the drive motor be ${\dot{q}}_{\max }$, and the required peak horizontal foot velocity $v_x$ in the swing region is determined jointly by the LIP parameters and the quintic polynomial foot trajectory planning described above. The high-performance threshold for the mid-swing region can then be expressed as

$$T_{\mathrm{sm}}={\displaystyle \frac{1}{{\left(\raisebox{1ex}{$v_x$}\!\left/ \!\raisebox{-1ex}{$\left(\sqrt{2}{\dot{q}}_{\max }\right)$}\right.\right)}^2+\epsilon }}$$

(26)

For the swing-transition region, $T_{slr}$ is defined as the 30th percentile of the performance values of the reference mechanism ${\mathbf{p}}_{\mathbf{r}}$ in the corresponding task region. The area ratio of the high-performance region is defined as

$$A={\displaystyle \frac{N_{hp}}{N}}$$

(27)

where $N_{hp}$ and $N$ denote the numbers of high-performance points and valid operating points in ${\mathsf{\Omega }}_r$, respectively.

The spatial distribution characteristics of $S_r$ are quantified using the box-counting dimension $D_{box}$ and lacunarity $\mathsf{\Lambda }$. Let $N(\epsilon )$ denote the number of grid boxes containing high-performance points at the scale $\epsilon$. The box-counting relation is written as

$$\log N(\epsilon )=D_{box}\log ({\displaystyle \frac{1}{\epsilon }})+C$$

(28)

where $D_{box}$ characterizes the spatial filling capacity of the high-performance region. $m_i$ denotes the number of high-performance points in the $i_{th}$ grid box at the scale $\epsilon$. The lacunarity is defined as

$$\mathsf{\Lambda }(\epsilon )={\displaystyle \frac{M_2(\epsilon )}{M_1^2(\epsilon )}},(M_1(\epsilon )={\displaystyle \frac{1}{n}}{\displaystyle {\displaystyle \sum _{i=1}^n}{m}_i},M_2(\epsilon )={\displaystyle \frac{1}{n}}{\displaystyle {\displaystyle \sum _{i=1}^n}{m}_i^2})$$

(29)

The reported lacunarity is the average value over the selected box sizes:

$$\overline{\mathsf{\Lambda }}={\displaystyle \frac{1}{K}}{\displaystyle \sum _{k=1}^K\mathsf{\Lambda }({\epsilon }_k)}$$

(30)

Table 3. Drive motor and LIP model parameters.

${\mathit{\tau }}_{\mathbf{max}}$ (N·m)	${\dot{\mathit{q}}}_{\mathbf{max}}$ (rpm)	Total Mass (kg)	CoM Height (m)	Target Speed (m/s)	Gait Period (s)	Duty Factor	${\mathit{F}}_{\mathit{y}}$ (N)	${\mathit{v}}_{\mathit{x}}$ (m/s)
11	240	5.5	0.4	2.0	0.5	0.5	55	7.5

lists the parameters of the drive motor (M6010, StackForce, Shenzhen, Guangdong, China) and the LIP model. Substituting these parameters into Equations (25) and (26) gives $T_{\mathrm{st}}=0.0311$,$T_{\mathrm{sm}}=22.46$, and $T_{\mathrm{slr}}=209.89$. The thresholds used to identify high-performance regions are based on simplified gait assumptions, motor specifications, and literature-based estimates. They should therefore be regarded as application-oriented reference values rather than absolute physical limits. Different assumptions may change the absolute values of the area ratio, box-counting dimension, and lacunarity. However, since the same threshold definitions are applied to both the reference and optimized mechanisms, the comparison still reflects the relative change in the spatial distribution of favorable transmission regions with the selected task requirements.

The proposed idea may also be extended to spatial or multi-degree-of-freedom leg mechanisms. In that case, the task space should be partitioned according to three-dimensional gait functions, such as vertical support, forward swing, lateral stabilization, and transition motion. The regional performance indices can be constructed from the spatial Jacobian matrix to describe direction-dependent velocity and force transmission in 3D space. However, this extension requires new task space sampling strategies, higher-dimensional high-performance region definitions, and adapted box-counting or lacunarity calculations. Therefore, applying the present method to spatial leg mechanisms and complete multi-degree-of-freedom legged robots will be a direction for future work.

4.5. Parameter Sensitivity and Final Optimization Results

The optimization problem in Equation (21) is solved using a genetic algorithm. The population size is set to 200, the maximum number of generations is set to 20,000, the elite count is set to 6, and the crossover fraction is set to 0.70. Tournament selection is used as the selection operator, with a tournament size of 10. An adaptive feasible mutation operator is used as the mutation operator. The function tolerance is set to ${10}^{-20}$.

4.5.1. Sensitivity to Objective Function Weights

The optimization objective in Equation (21) is constructed as a weighted sum of three normalized objectives corresponding to the stance, mid-swing, and swing-transition regions. Since different gait phases have different functional requirements, the choice of weighting coefficients may affect the optimization results. To examine this effect, several weight combinations are used, and the mechanism parameters are re-optimized for each combination. These combinations include an equal-weight case, a stance-enhanced case, a mid-swing-enhanced case, a swing-transition-enhanced case, and combined cases in which two regions are enhanced simultaneously.

Table 4. Optimization results under different weight combinations.

Case	${\mathit{\omega }}_{\mathbf{st}}$	${\mathit{\omega }}_{\mathbf{sm}}$	${\mathit{\omega }}_{\mathbf{slr}}$	${\mathit{f}}_{\mathbf{st}}$	${\mathit{f}}_{\mathbf{sm}}$	${\mathit{f}}_{\mathbf{slr}}$	${\mathit{\eta }}_{\mathbf{st}}$	${\mathit{\eta }}_{\mathbf{sm}}$	${\mathit{\eta }}_{\mathbf{sl}}$	${\mathit{\eta }}_{\mathbf{sr}}$
1	0.33	0.33	0.33	0.993	0.566	0.518	0.903	1.000	1.143	0.836
2	0.50	0.25	0.25	0.824	0.731	0.647	0.790	1.000	1.114	0.808
3	0.25	0.50	0.25	0.991	0.527	0.573	0.903	1.000	1.143	0.836
4	0.25	0.25	0.50	0.989	0.594	0.495	0.903	1.000	1.143	0.836
5	0.40	0.40	0.20	0.960	0.476	0.671	1.323	1.000	1.143	0.973
6	0.40	0.20	0.40	0.891	0.674	0.539	0.903	1.000	1.129	0.836
7	0.20	0.40	0.40	0.990	0.558	0.552	0.903	1.000	1.129	0.836

gives the optimization results for different weight combinations under the task space boundary parameters of Case 2 in

Table 5. Optimization results under different task space partition boundaries.

Case	${\mathit{y}}_0$	${\mathit{y}}_1$	${\mathit{x}}_{\mathit{d}}$	${\mathit{f}}_{\mathbf{st}}$	${\mathit{f}}_{\mathbf{sm}}$	${\mathit{f}}_{\mathbf{slr}}$	${\mathit{\eta }}_{\mathbf{st}}$	${\mathit{\eta }}_{\mathbf{sm}}$	${\mathit{\eta }}_{\mathbf{sl}}$	${\mathit{\eta }}_{\mathbf{sr}}$
1	−0.32	−0.10	0.09	0.881	0.433	0.685	1.500	1.000	1.111	1.015
2	−0.35	−0.10	0.09	0.824	0.731	0.647	0.790	1.000	1.114	0.808
3	−0.38	−0.10	0.09	0.743	0.653	0.734	0.705	1.000	1.105	0.797
4	−0.35	−0.12	0.09	0.848	0.560	0.548	0.935	0.977	1.172	0.824
5	−0.35	−0.08	0.09	0.951	0.536	0.725	1.694	0.944	1.100	1.036
6	−0.35	−0.10	0.07	0.786	0.510	0.690	0.806	1.250	1.013	0.827
7	−0.35	−0.10	0.11	0.812	0.699	0.612	0.806	0.889	1.127	0.908

. The optimized mechanism parameters vary with the weight allocation, indicating that the objective weights affect the specific distribution of the mechanism dimensions. Across all tested cases, the optimization results show clear differences under different weighting coefficients. Nevertheless, the normalized objective values of the optimized mechanisms are generally lower than those of the reference mechanism in the main task regions, and the valid-point coverage ratios satisfy the prescribed constraints. The final weighting combination was selected according to the practical functional requirements of the leg mechanism. The stance region is directly related to vertical load support and is critical for stable operation. Therefore, a relatively higher improvement in this region is desirable in practical design. However, the support capability should not be improved at the expense of a substantial loss of valid task points or a severe degradation of the other task regions. Otherwise, the optimized design may become biased toward a single function. Based on the weight sensitivity results, ${\omega }_{\mathrm{st}}:{\omega }_{\mathrm{sm}}:{\omega }_{\mathrm{slr}}=0.5:0.25:0.25$ provides a relatively balanced improvement among the three regional objectives while maintaining reasonable valid-point coverage. Therefore, it is used as the final weighting scheme in this study.

4.5.2. Sensitivity Analysis of Task Space Partition Boundaries

To examine the influence of partition boundaries on the optimization results, this subsection uses the spatial distributions of foot trajectories at different locomotion speeds in the hip joint coordinate frame as references. Three boundary parameters in the task space partitioning, $y_0$, $y_1$, and $x_d$, are investigated, and parameter optimization is repeated under each boundary setting. Table 5 gives the normalized objective values and valid task-point coverage ratios with different boundary settings. The results show that changes in the boundary parameters alter the number of valid operating points in each task region. They also affect the optimized mechanism parameters and the regional objective values. Within the tested boundary ranges, the performance indices of the optimized mechanisms in different functional regions show certain variations. $f_{\mathrm{st}}$, $f_{\mathrm{sm}}$, and $f_{\mathrm{slr}}$ are all smaller than 1, indicating that the optimized mechanisms still maintain performance improvement relative to the reference mechanism. In addition, all cases satisfy the valid task-point coverage constraints, showing that the optimization results are not obtained by a clear loss of effective workspace. Considering the normalized objective values and the valid task-point coverage ratios, the boundary parameters corresponding to Case 2 are selected as the basis for subsequent optimization and experimental validation. It should be noted that task space partitioning is not intended to describe a unique and exact gait boundary. Rather, it is used to map different functional requirements to different mechanism performance indices. Therefore, as long as the partition boundaries preserve the spatial clustering relationships of the main stance trajectory, swing trajectory, and phase-transition trajectory, the overall optimization trend remains stable.

4.5.3. Sensitivity Analysis of the Regularization Parameter

In the definition of the transmission performance field, inverse forms are used to describe favorable transmission performance in the mid-swing and swing-transition regions. To avoid numerical divergence when the denominator approaches zero, the regularization parameter $\epsilon$ is introduced in Equation (22). This parameter is used only to improve numerical stability and is not involved in the construction of the mechanism parameter optimization objective.

Figure 4 shows the area ratio, box-counting dimension, and lacunarity of the high-performance regions under different values of $\epsilon$. The results show that the stance-region results are not affected by $\epsilon$, because the stance-region performance field does not contain an inverse regularization term. For the mid-swing and swing-transition regions, the threshold values and local performance field values vary slightly with $\epsilon$. However, the overall trends of the optimized mechanism relative to the reference mechanism remain consistent. Thus, the conclusions on the spatial distribution of high-performance regions do not depend on a specific value of $\epsilon$.

4.5.4. Analysis of Optimization Results

Considering the stochastic nature of the genetic algorithm, 10 independent optimizations are performed using different random seeds to examine the stability of the optimization results. All independent runs converge normally and satisfy the workspace coverage constraints in all task regions. The mean value of the weighted normalized objective function over the 10 runs is 0.7471, with a standard deviation of 0.0198. The minimum and maximum values are 0.7359 and 0.7992, respectively. These results indicate that the optimization results are not sensitive to the random initial population, and that the performance improvement trend is stable. The optimized parameters are listed in

Table 6. Optimized mechanism parameters.

Parameter	${\mathit{l}}_1$ (m)	${\mathit{l}}_2$ (m)	${\mathit{l}}_3$ (m)	${\mathit{l}}_4$ (m)	${\mathit{l}}_5$ (m)	${\mathit{l}}_6$ (m)	${\mathit{\theta }}_6$ (°)
Result	0.200	0.239	0.292	0.210	0.132	0.257	3.97

.

Compared with the symmetric reference mechanism, the optimized mechanism is asymmetric. After optimization, the mean objective value is reduced by 17.63% in the stance region, 26.94% in the mid-swing region, and 35.34% in the swing-transition regions.

To compare the influence of different optimization objective formulations on the optimized mechanism parameters, three reference optimization schemes are considered: global manipulability index optimization [26], global Jacobian condition number optimization [28], and global multi-objective optimization without task space partitioning. The corresponding objective functions are provided in the Supplementary Materials. To ensure a consistent comparison, all reference optimization models use the same design variables, parameter bounds, validity criteria, and workspace coverage constraints as the proposed method. The only difference lies in the formulation of the objective function.

Scheme S1: global manipulability optimization, in which the manipulability performance over the entire valid task space is used as the optimization objective. Scheme S2: global condition number optimization, in which the Jacobian condition number over the entire valid task space is used as the optimization objective. Scheme S3: global multi-objective optimization without task space partitioning, in which the vertical velocity mapping, horizontal velocity mapping, and manipulability-related quantities are considered over the entire valid task space and normalized to construct a global multi-objective function.

The optimization results with different optimization objectives are shown in

Table 7. Optimization results under different optimization objectives.

Scheme	${\mathit{F}}_{\mathbf{man}}$	${\mathit{F}}_{\mathbf{cond}}$	${\mathit{\eta }}_{\mathbf{all}}$	${\mathit{f}}_{\mathbf{st}}$	${\mathit{f}}_{\mathbf{sm}}$	${\mathit{f}}_{\mathbf{slr}}$	${\mathit{\eta }}_{\mathbf{st}}$	${\mathit{\eta }}_{\mathbf{sm}}$	${\mathit{\eta }}_{\mathbf{sl}}$	${\mathit{\eta }}_{\mathbf{sr}}$
This study	1.631	1.291	0.925	0.824	0.731	0.647	0.790	1.000	1.114	0.808
1	0.142	3.234	1.477	4.872	0.333	0.091	2.919	1.000	1.143	1.096
2	4.312	0.333	1.240	0.804	3.823	4.636	2.371	0.750	1.028	1.082
3	0.426	0.665	1.236	1.296	0.868	0.427	1.822	1.000	1.142	1.068

. Different global optimization objectives can significantly improve their corresponding global performance indices. However, the improvement is not necessarily distributed in the key task regions of the robotic leg mechanism. Global manipulability optimization reduces the global manipulability cost to 0.1416 of that of the reference mechanism. However, the normalized objective value in the stance region increases to 4.8721. This indicates that this scheme improves overall manipulability but weakens the vertical load transmission performance in the stance region. Global condition number optimization reduces the global condition number to 0.3334 of that of the reference mechanism. However, the normalized objective values in the mid-swing and swing-transition regions increase to 3.8237 and 4.6365, respectively. This shows that improvement in overall isotropy does not necessarily lead to better velocity transmission in the swing region or better manipulability in the transition regions. Global multi-objective optimization without task space partitioning improves the global combined indices to some extent. However, the normalized objective value in the stance region remains greater than 1, with a value of 1.2961. This suggests that a global averaged objective without task region distinction may weaken the constraint on local functional requirements. In contrast, the task space partition-based optimization method proposed in this study achieves normalized objective values smaller than 1 in the stance, mid-swing, and swing-transition regions. These results indicate that the performance improvement of the proposed method is not simply caused by relaxing the symmetry constraint, nor can it be achieved by optimizing a single global kinematic index within the enlarged design space. The role of task space partitioning is to guide the mechanism parameter optimization toward the transmission requirements of different functional regions, thereby achieving coordinated improvements in the stance, mid-swing, and swing-transition regions while maintaining effective workspace coverage.

Figure 5 compares the task space coverage and regional transmission performance fields of the reference mechanism and the optimized mechanism. As shown in Figure 5a, the numbers of valid operating points of the optimized mechanism in all functional regions satisfy the prescribed coverage constraints. This confirms that the optimized design maintains sufficient task space feasibility for the subsequent regional performance evaluation. Figure 5b,c show the task space distributions of the regional transmission performance fields. Compared with the reference mechanism, the optimized mechanism exhibits lower performance field values in the main task regions, indicating improved local transmission characteristics while preserving the required workspace coverage.

Figure 6 shows the distributions of high-performance transmission points for the reference mechanism and the optimized mechanism. The red points denote the discrete operating points satisfying the threshold criterion. In this study, the area ratio $A$ is approximated by the ratio of the number of high-performance points to the number of valid operating points in the corresponding task region. It should be noted that the box-counting dimension $D_{\mathrm{box}}$ and lacunarity $\overline{\mathsf{\Lambda }}$ are used only to describe the multi-scale spatial distribution of high-performance points in the finite discrete task space. In the stance region, the coverage of the high-performance region is markedly increased after optimization, with $A$ increasing from 25.81% to 48.98%. Meanwhile, $D_{\mathrm{box}}$ increases from 0.842 to 1.10, and $\overline{\mathsf{\Lambda }}$ decreases from 11.085 to 8.0282. This indicates that the high-performance load-bearing points after optimization have larger coverage, stronger spatial filling, and lower distribution non-uniformity in the finite task space. In the mid-swing region, where $A$ increases from 41.67% to 68.75%, $D_{\mathrm{box}}$ increases from 0.9533 to 1.2726, and $\overline{\mathsf{\Lambda }}$ decreases from 9.8306 to 8.3927. This shows that the high-performance points that are favorable for horizontal velocity transmission become more widely distributed and more spatially uniform after optimization. By contrast, the swing-transition regions exhibit limited spatial structure variation. The area ratios of the high-performance regions remain nearly unchanged, and the variations in $D_{\mathrm{box}}$ and $\overline{\mathsf{\Lambda }}$ are relatively small. Combined with the regional mean indices, this result indicates that the optimization effect in the swing-transition regions is mainly reflected in the reduction of local performance values, rather than a clear expansion of the high-performance point coverage. Overall, these results show that mechanism parameter optimization based on task space partitioning can change the multi-scale spatial distribution of high-performance operating points in the finite task space. Although $D_{\mathrm{box}}$ and $\overline{\mathsf{\Lambda }}$ are not included in the optimization objective function, they can serve as supplementary indices for evaluating the coverage, spatial filling degree, and distribution non-uniformity of high-performance point sets before and after optimization.

5. Experimental Validation

This section experimentally verifies whether the mechanism obtained using the proposed method achieves the expected improvements in load-carrying capability, motion transmission, and manipulability. The experiments include velocity ellipse characterization, static power consumption tests, limit load-carrying capacity tests, steady-state position error tests, and dynamic trajectory-tracking tests.

5.1. Experimental Setup and Test Methods

The experiments are conducted on a symmetric five-bar reference mechanism and an optimized asymmetric five-bar mechanism, whose physical prototypes are shown in Figure 7. The experimental setup consists mainly of the mechanism body, the drive motors, a loading module, and a data acquisition system. The link length parameters of the two mechanisms correspond to those of the reference mechanism and the optimized mechanism obtained above, respectively. All links are fabricated from carbon fiber material, and rolling bearings are installed at each revolute joint to reduce friction. To minimize the influence of variation among actuators on the experimental results, the same set of drive units is used for both mechanisms.

The drive units are integrated servo motors (M6010, StackForce, Shenzhen, Guangdong, China) supplied at 24 V, which provide real-time state information including position, velocity, torque, coil temperature, and MOS temperature. Power consumption is measured using an SUI-201 power/current meter. In the experiments, the end load is applied using weights, and the thermal state of the actuators is monitored simultaneously by an infrared thermal camera and motor temperature feedback. To ensure comparability among different experiments, all tests are conducted with identical control parameters and consistent initial thermal conditions, as far as possible. When the coil temperature or MOS temperature of any motor approaches the operating limit, the current test is suspended, and the system is allowed to cool naturally before the next test.

5.2. Validation of Velocity Ellipse Characteristics

The velocity ellipse experiment is conducted to examine whether the optimized mechanism exhibits, in the physical prototype, directional local transmission characteristics consistent with the theoretical predictions. For a given task point, the corresponding active joint angles are first obtained through inverse kinematics as ${\mathit{q}}_i=[{\theta }_{1,i},{\theta }_{3,i}{]}^T$, and a small circular perturbation trajectory of radius $r$ is then imposed in the joint space:

$$\mathit{q}(t)={\mathit{q}}_i+r\left[\begin{array}{l}\cos (\omega t)\hfill \\ \sin (\omega t)\hfill \end{array}\right]$$

(31)

where $\omega$ is the commanded angular frequency.

To ensure the validity of the local linear approximation, the perturbation radius is set to $r=0.02\hspace{0.17em}\mathrm{r}\mathrm{a}\mathrm{d}$, and the excitation frequency to $1\hspace{0.17em}\mathrm{H}\mathrm{z}$. The two drive motors are position controlled using identical PD gains, and the joint position responses are recorded during the experiment. The theoretical velocity ellipse is obtained from the Jacobian matrix at the selected task point under the unit joint velocity constraint $\parallel \dot{\mathit{q}}\parallel =1$, according to $\dot{\mathbf{x}}=\mathit{J}({\mathit{q}}_i,\mathbf{p})\dot{\mathit{q}}$. The trajectory predicted by the local linear approximation is obtained from $\mathsf{\Delta }\mathbf{x}\approx \mathit{J}({\mathit{q}}_i,\mathbf{p})\mathsf{\Delta }\mathit{q}$, whereas the experimental trajectory is reconstructed from the measured joint responses through forward kinematics. The comparisons between the theoretical and experimental velocity ellipses of the mechanisms before and after optimization at different task points are shown in Figure 8.

Figure 8 shows that the experimentally measured velocity ellipses are generally consistent with the theoretical results, and the local directional transmission characteristics predicted by the Jacobian analysis are reflected in the physical prototype. For the optimal load-bearing task point in the stance region, the minor axis of the velocity ellipse of the reference mechanism, corresponding to the second ellipse from the right in Figure 8a, is 4.600 mm, whereas that of the optimized mechanism, corresponding to the second ellipse from the left in Figure 8b, is 3.183 mm. According to the relationship between the velocity and force ellipses, a smaller velocity ellipse radius in a given direction corresponds to a larger force transmission capability in the same direction. This observation is consistent with the subsequent static load carrying and power consumption tests. Combined with the foregoing theoretical analysis, the optimized mechanism is expected to provide a vertical load-carrying capacity approximately 1.45 times that of the reference mechanism.

At static equilibrium, neither the actuators nor the mechanism produce mechanical output power, and the consumed electrical power is mainly dissipated as heat [46]. The thermal loss power of the motor drives is proportional to the square of the output torques:

$$P_e={\displaystyle \frac{R_m}{K_m^2}}({\tau }_1^2+{\tau }_2^2)$$

(32)

With identical drive units, the static power consumption of the optimized mechanism is predicted to be lower by approximately a factor of 2.1 than that of the reference mechanism.

5.3. Static Power Consumption Experiment

In this section, static power consumption experiments are conducted of the reference mechanism and the optimized mechanism. The steady-state electrical power at each valid task point is compared under different loading conditions. In the experiments, the two mechanisms use the same drive units and control parameters. The difference in static power consumption is mainly attributed to the different driving torques required by the two mechanisms to sustain the same external load. Accordingly, Equation (32) can be approximated as

$$P_e\propto ({\tau }_1^2+{\tau }_2^2)$$

(33)

For a given task point and load, the driving torques can be calculated from the Jacobian matrix at that point and the external load applied at the endpoint through the static equilibrium relationship. Therefore, the theoretical power distribution can be indirectly predicted from the local first-order kinematic characteristics of the mechanism. Because the two mechanisms use the same set of motors, the expression above is used here mainly to compare relative power consumption differences between mechanisms, rather than to provide an accurate estimate of the absolute electrical power.

In the experiments, both mechanisms are mounted in an inverted configuration, and weights are symmetrically applied on both sides of the endpoint to generate a stable vertical static load. The controller sends position commands sequentially to guide the endpoint through all valid task points, where it remains stationary for 15 s at each point while the steady state power is recorded using a power/current meter. To examine the influence of load variation on the experimental conclusions, the tests are repeated under four loading conditions: 0.5 kg, 1.0 kg, 1.5 kg, and 2.0 kg. During the experiments, the thermal state of the actuators is monitored simultaneously. When the coil temperature or MOS temperature of any motor approaches the operating limit, the experiment is suspended and resumed only after the system has cooled naturally. This procedure reduces the influence of thermal accumulation on the measurements.

Figure 9 presents the power distribution results of the two mechanisms with a vertical load of 0.5 kg, including both the theoretical estimates and the experimental measurements. Over the entire evaluated task space, the optimized asymmetric five-bar mechanism consistently exhibits lower static power consumption. Near the power peak, the static power consumption of the symmetric five-bar mechanism is approximately twice that of the optimized mechanism, which is consistent with the previous prediction from the velocity ellipse analysis that the static power consumption would be reduced by about a factor of 2.1.

To examine the robustness of this conclusion under different loading conditions, Figure 10 summarizes the static power results of the two mechanisms along the horizontal direction of the workspace under different loading conditions. The scattered data points and mean values show that the optimized asymmetric five-bar mechanism has lower average power consumption than the symmetric reference mechanism at all four load levels. This reduction is observed over most valid operating points, rather than only at a few extreme positions.

5.4. Limit Load Capacity

In this study, the limit static load capacity is defined as the maximum vertical load that the mechanism can sustain at a given task point before any actuator reaches its thermal safety limit. The stance task points with the highest theoretical load capacity are selected for the symmetric five-bar reference mechanism and the optimized asymmetric five-bar mechanism, respectively. During the experiments, the mechanisms are mounted in an inverted configuration, and weights are symmetrically applied on both sides of the endpoint to generate a stable vertical load. Because the limit load capacity is closely related to actuator thermal stability, after each change in load, the system is allowed to cool naturally to approximately room temperature to ensure comparable initial thermal conditions.

The internal temperature feedback of the motors is used as the criterion for evaluating the limit load capacity. According to the operating temperature limit provided by the motor manufacturer and the experimental observations, when either the coil temperature or the MOS temperature of any drive unit reaches approximately 70 °C, the motor first shows communication interruption. It then maintains the load output only for a short time, after which the load-holding capability decreases and the motor stops normal operation. Therefore, 70 °C is used as the thermal safety termination threshold for the present experimental protocol. In the experiments, both the reference mechanism and the optimized mechanism are tested from initial thermal states close to room temperature. They are compared under the same control parameter, loading method, and natural cooling conditions. For the reference mechanism, stable operation can be maintained for a relatively long time under a 3.0 kg load. When the load is increased to 3.5 kg, the temperature of one drive unit reaches 70 °C after approximately 75 s, and thermal failure is triggered, as shown in Figure 11a. Therefore, under the discrete loading conditions used in this study, 3.5 kg is taken as the thermally limited static load of the reference mechanism. For the optimized mechanism, as shown in Figure 11b, the load is further increased. With a 4.5 kg load, the mechanism can still maintain static equilibrium during the initial stage and reaches the thermal termination threshold of 70 °C after approximately 100 s. Therefore, 4.5 kg is taken as the thermally limited static load of the optimized mechanism using the present experimental protocol. It should be noted that the load is applied using discrete weight increments, so the exact critical load of the optimized mechanism corresponding to the same failure time as that of the reference mechanism cannot be obtained. Therefore, 4.5 kg is better interpreted as a conservative comparative value under the current experimental conditions, rather than the absolute mechanical load limit of the optimized mechanism.

The maximum static load capacity of the optimized asymmetric mechanism is approximately 1.29 times that of the symmetric reference mechanism, which is consistent with the trend predicted by the velocity ellipse analysis and the corresponding mechanical advantage. The measured improvement is lower than the theoretical factor of 1.45. This difference is mainly attributed to unmodeled practical effects, including distributed link mass, joint friction, assembly errors, controller limitations, and motor parameter drift at elevated temperatures. In addition, the load was applied using discrete weights, so the exact critical load could not be determined continuously. Therefore, the reported 4.5 kg load should be regarded as a conservative comparative value using the present experimental protocol.

5.5. Steady-State Position Error Experiment

With static loading, the steady-state position error at the endpoint is influenced by both the controller and the mechanical advantage of the mechanism in the main load-bearing direction. For two mechanisms using the same drive units and control parameters, a larger mechanical advantage in the vertical direction generally requires smaller joint deviations to balance the same external load, leading to a smaller steady-state endpoint error. This section compares the positioning performance of the symmetric five-bar reference mechanism and the optimized asymmetric five-bar mechanism with static loading.

The experimental procedure is the same as that used in the static power consumption test. Under PD position control, the endpoint is sequentially moved to each valid task point and held stationary for 15 s. During the test, the joint encoder feedback is recorded, and the actual endpoint position is reconstructed through forward kinematics. The Euclidean distance between the measured steady-state endpoint position and the reference task point is defined as the steady-state position error.

$$e_p=‖{\mathit{P}}_i-{\mathit{P}}_i^r‖$$

(34)

The actual sampling frequency of the system is approximately 80 Hz. The steady-state errors of different mechanisms are experimentally measured with loads of 0.5 kg, 1.0 kg, 1.5 kg, and 2.0 kg. Figure 12 shows the distributions of the steady-state position errors of the two mechanisms at different task points with a load of 1.0 kg. Although the theoretical and experimental results differ in absolute value, their spatial distribution patterns are generally consistent. Both results show that the optimized asymmetric five-bar mechanism has smaller steady-state position errors than the symmetric reference mechanism at most task points. For both mechanisms, the error increases as the task point moves upward toward the upper boundary of the workspace.

To examine the overall trend under different loading conditions, Figure 13 summarizes the steady-state error results of the two mechanisms over the entire workspace. The mean values, together with the scattered data points, show that, under the same load condition, the optimized asymmetric five-bar mechanism not only has a lower mean error, but also exhibits a markedly smaller error distribution at most task points than the symmetric reference mechanism. This advantage becomes more pronounced as the load increases. Taking the 2.0 kg load as an example, the maximum steady-state position error of the symmetric five-bar mechanism reaches 71.2 mm, with a minimum mean value of 28.3 mm; by contrast, the maximum steady-state position error of the optimized asymmetric five-bar mechanism is 45.0 mm, with a minimum mean value of 13.9 mm. These results indicate that, under the high load test conditions considered in this study, the optimized mechanism exhibits better static position-holding capability overall.

It should be noted that the discrepancies between the theoretical and experimental results mainly arise from model simplifications and nonideal factors in the actual system, including the approximate treatment of link mass distribution, joint friction, assembly errors, and sensor measurement noise. These factors, however, do not alter the relative relationship between the two mechanisms in terms of the steady-state error level.

5.6. Dynamic Trajectory-Tracking Experiment

In addition to static load-carrying capacity, the dynamic tracking performance of a robot leg mechanism during a periodic foot trajectory is also an important criterion for evaluating the mechanism. Although the dynamic response is influenced by such factors as control parameters, actuator bandwidth, link inertia, and friction, the first-order kinematic characteristics of the mechanism can still affect the dynamic performance indirectly through local velocity-mapping capability and mechanical advantage. On this basis, this section compares the tracking capabilities of the symmetric five-bar reference mechanism and the optimized asymmetric five-bar mechanism during periodic foot motion through dynamic trajectory-tracking experiments.

The target trajectory used in the experiment and its velocity distribution are shown in Figure 14. Figure 14a shows a periodic foot reference trajectory with a trot-like rhythm, for which the duty factor is set to 0.5 and the leg lifting height to 0.04 m. According to the task space partitioning results presented earlier, the entire trajectory is divided into the stance phase, the swing-transition phase, and the mid-swing phase. During the experiment, the mechanism is mounted upside down, and a load of 0.5 kg is applied at the endpoint. One complete traversal of the endpoint along the reference trajectory is defined as one cycle, and the motion speed is varied by adjusting the repetition frequency of the reference trajectory sent by the host computer. When the trajectory frequency is 1 Hz, the velocity components of the foot in the $x$ and $y$ directions are shown in Figure 14b.

During the experiment, the host computer computes the desired driving joint angles in real time according to the reference trajectory positions, the coordinate transformation relationships, and the inverse kinematics of the mechanism and sends position commands at a control frequency of 80 Hz, while simultaneously receiving the motor feedback states. The actual endpoint position of the mechanism is then obtained from the feedback joint positions through forward kinematics and compared with the reference trajectory at the same instant. In this study, the instantaneous trajectory tracking error is defined as

$$e_t\left(t_i\right)=‖{\mathit{P}}^c\left(t_i\right)-{\mathit{P}}^r\left(t_i\right)‖$$

(35)

where ${\mathit{P}}^c\left(t_i\right)$ is the endpoint position obtained from the actual joint feedback through kinematic reconstruction, and ${\mathit{P}}^r\left(t_i\right)$ is the reference trajectory position. By evaluating the tracking error over one complete cycle at different frequencies, the dynamic tracking performance of the mechanism under different operating conditions can be obtained.

As shown in Figure 14c, under low-frequency conditions, both mechanisms are able to follow the target trajectory satisfactorily, and the difference in performance is not significant below approximately 0.8 Hz. As the motion frequency increases, however, the tracking error of the symmetric five-bar mechanism grows more rapidly, and once the frequency exceeds 1.3 Hz, pronounced oscillation and overshoot appear, making stable tracking difficult to maintain. By contrast, the optimized asymmetric five-bar mechanism still preserves good trajectory consistency at higher frequencies and remains capable of stable tracking up to 1.5 Hz. This behavior is consistent with its stronger horizontal velocity-mapping capability in the mid-swing region and its larger singularity margin in the transition regions.

Figure 15 presents the actual trajectories of the two mechanisms at different frequencies. As the frequency increases, the trajectory of the symmetric five-bar mechanism gradually deviates from the reference trajectory, especially near the swing-transition region and phase-switching points. In contrast, the optimized asymmetric five-bar mechanism better preserves the overall shape of the reference trajectory under the same frequency conditions. The dynamic trajectory-tracking experiment was conducted without ground contact. The experiment mainly compared the local trajectory-tracking performance of the two mechanisms during a prescribed foot-end trajectory, rather than providing a complete validation of locomotion performance during legged robot motion. Ground impact, slippage, terrain interaction, and whole-body dynamics were not considered in this test. Nevertheless, using the same control parameters, load condition, and reference trajectory, the optimized mechanism showed better trajectory consistency at higher frequencies, which further supports the improvement of its local transmission characteristics from the perspective of dynamic performance.

6. Conclusions

This study proposed a task space partition-based dimensional parameter optimization method for a planar five-bar robotic leg mechanism. The method is based on the observation that the foot end does not perform a single task during one gait cycle; instead, it undertakes support, swing, and phase-transition functions in different spatial regions. Biological gait data and simplified dynamic models show that, in the hip joint coordinate system, the foot trajectory in the stance phase is not a simple straight line, but an arc-shaped curve with clear curvature. The foot trajectories of different gait phases also show stable spatial clustering characteristics in the task space. Based on these features, a kinematic model of the planar five-bar mechanism was established, and Jacobian-based performance indices were defined for the stance, mid-swing, and swing-transition regions according to the first-order kinematic characteristics of the mechanism.

During the optimization process, the effects of task space partition boundaries, objective function weights, and different objective function formulations were analyzed. The results show that the proposed method can improve the local transmission performance of the mechanism in different task regions while maintaining the required workspace coverage. Compared with the symmetric reference configuration, the optimized asymmetric mechanism reduces the normalized objective values in the stance, mid-swing, and swing-transition regions by 17.63%, 26.94%, and 35.34%, respectively. In addition, the area ratio, box-counting dimension, and lacunarity were used as supplementary measures to evaluate the spatial distribution of the high-performance transmission regions after optimization. These indices were not included in the optimization objective function. They were used to describe the coverage, spatial filling capacity, and distribution non-uniformity of high-performance point sets in the finite discrete task space. The results show that the optimized mechanism forms larger high-performance regions in the stance and mid-swing regions, with higher box-counting dimensions and lower lacunarity. This indicates broader coverage, stronger spatial filling, and a more uniform distribution of high-performance transmission points in the task space.

Finally, prototype experiments verified that the proposed geometric parameter optimization method improves both the static and the dynamic performance of the planar five-bar robotic leg mechanism. The experiments included static power consumption, limit load capacity, steady-state position error, and dynamic trajectory-tracking tests. Compared with the reference mechanism, the optimized asymmetric mechanism increased the maximum static load capacity approximately 1.29 times and reduced the static power consumption by about a factor of 2.0. It also showed smaller steady-state position errors and better trajectory consistency in high-frequency trajectory-tracking tests.

The present work focuses on a single planar five-bar leg mechanism. Future work will extend the proposed task space partitioning and high-performance region evaluation method to complete legged robot systems, where the performance improvement will be evaluated with terrain interaction, impact, complex dynamic conditions, and long-term operation.