Minimization of Power Loss as a Design Criterion for the Optimal Synthesis of Loader Drive Mechanisms

Abstract

As energy efficiency becomes a significant performance indicator in mobile machines, power losses are recognized as an important criterion in the design and optimization of these systems. This paper analyses the loads and power loss due to friction in the revolute joints of the manipulator drive mechanisms during all phases of the loader manipulation task, based on dynamic simulations of the loader model with different variants of Z-kinematics manipulator drive mechanisms, using the MSC ADAMS 2020 software. The analysis is based on a general dynamic mathematical model of the loader, which enables the assessment of the influence of the parameters of the manipulator mechanisms on the functional, structural, and tribological characteristics of the revolute joints within the manipulator’s kinematic chain. Based on the analysis results, a minimum power loss criterion was defined as part of a multi-criteria optimal synthesis procedure for the manipulator drive mechanisms, with the objective of maximizing energy efficiency by minimizing power loss caused by friction in the revolute joints of the manipulator drive mechanisms.

1. Introduction

Loaders represent a group of mobile machines primarily designed for the cyclic transport of bulk materials. Their function typically involves repetitive loading, transport, and unloading operations. Due to the demanding nature of these tasks, modern development of loaders of all sizes relies on modular design and manufacturing processes built for high reliability and maintainability. Equally important are considerations of operator ergonomics and industrial design, as well as a commitment to sustainable development in new loader models [1,2]. For instance, electric loaders now offer low emissions and reduced noise to meet environmental regulations (aligning with sustainability goals), while also improving the operator’s working environment [1]; at the same time, refined control interfaces (e.g., optimized joystick and lever designs) help minimize driver fatigue during repetitive operations [3]. In recent years, this evolution has further accelerated with the introduction of electrified drivetrains and intelligent control systems, all aimed at meeting higher performance demands [4]. As mobile machinery continues to advance under these pressures, energy efficiency and durability have become paramount criteria in mechanism design. Traditional hydraulic drivetrains, for example, suffered significant power losses and poor fuel economy (e.g., in torque converters), prompting the adoption of innovative solutions like hydrostatic transmissions to improve overall efficiency [5].

Accurate load determination is a critical step in the optimization of mobile machines, not only for the correct sizing of the bearing, but also for understanding the frictional conditions and wear mechanisms that affect its service life. Excessive or poorly distributed loads can lead to increased contact pressures between the bearing surfaces, resulting in elevated friction levels, material degradation, and accelerated wear [6,7].

While earlier optimization of loader manipulators largely focused on kinematic performance (for instance, achieving desired bucket trajectories and maximum reach), modern approaches must increasingly account for tribological behavior and power dissipation. Frictional losses in revolute joints—such as the pin joints between the boom, linkage, and bucket—can lead not only to energy waste but also to severe wear at these high-load pivot points [8]. Recent studies have accordingly integrated tribological considerations into the design process, identifying the bucket and arm hinges as critical locations and proposing design improvements (e.g., better pin materials, lubrication, or dust seals) to reduce joint wear.

By addressing such frictional losses, designers aim to achieve more realistic, durable, and energy-efficient loader mechanisms. In one case, for example, incorporating a hydraulic energy-recovery system into an electric loader’s boom actuation allowed more than half of the boom’s gravitational potential energy to be recaptured, reducing overall energy consumption by nearly 40% [9]. These examples illustrate how a holistic optimization of the machine’s subsystems—including hydraulic, electrical, and mechanical elements—can yield substantial gains in efficiency and performance. Rather than optimizing solely for ideal kinematics, the current trend indicates a move toward multi-criteria design frameworks that select the most suitable solution for real-world use, balancing raw energy efficiency with other key factors such as controllability, operator needs, cost, and component size [10]. This integrated approach ensures that modern loaders meet stringent performance targets without compromising reliability or usability.

However, individual optimization criteria often come into conflict due to the complexity and mutual influence of transmission and transformation parameters of drive mechanisms [11,12]. Petrović et al. [12] propose a multi-criteria approach for synthesizing loader manipulator drive mechanisms using a robust group decision-making framework. Criterion weights play a crucial role in the MCDM process, as they determine the relative importance of each objective and directly affect the final ranking or selection of optimal mechanism design variants [13,14]. Petrović et al. integrate several performance measures into a unified optimization strategy, establishing a comprehensive design methodology. However, the study does not explicitly address the effects of joint friction on power losses, leaving room for further refinement by incorporating tribological effects. The study by Bai et al. [15] confirms the importance of the tribological characteristics of joints within the machine’s kinematic chain, as well as the need to minimize power losses due to friction, as it focuses on analyzing joint wear in mechanical systems using a numerical model based on Archard’s wear equation. The research shows that the presence of clearance in revolute joints leads to increased contact forces, vibrations, and localized wear, which significantly affects the service life and precision of mechanisms. The study by Backman et al. [16] investigates autonomous control of underground loaders using deep reinforcement learning, emphasizing the importance of optimizing energy efficiency and bucket filling performance through intelligent control strategies. Although their approach differs from structural optimization, their application of multi-agent systems that adaptively manage the loading process based on real-time sensor data highlights the growing relevance of energy-focused methods in the design of loaders. Lu et al. [17] develop a collaborative optimization approach to reduce energy consumption in excavators using multi-objective algorithms that integrate kinematic and energy-based criteria. Their methodology can be adapted to consider friction-based losses as part of a broader efficiency evaluation.

However, along with setting the criteria, the optimization process must also take into account numerous kinematic constraints that include specified working reaches, predetermined bucket angles during transport and unloading, and the requirement to ensure the declared digging force. One of the major constraints that must be considered is the elimination of “dead points”—kinematic positions in which controlled force transmission is lost in the manipulator drive mechanisms [18].

Research in the field of optimal synthesis of loader manipulator mechanisms has primarily focused on mechanisms with Z-kinematics, due to their widespread use in practice. This kinematic configuration is analyzed in paper [19], which presents an optimization procedure for the manipulator drive mechanism based on a single criterion: minimizing the change in the bucket angle to prevent uncontrolled material spillage during the transition from the transport to the unloading position. In addition to the objective function, the paper also defines the kinematic constraint functions in vector loop form, which makes it possible to apply the Newton–Raphson method in solving the defined optimization problem.

Within the optimal synthesis process, the objective functions are modeled as nonlinear equations, resulting from the complex interconnections between the geometric, kinematic, and dynamic parameters of the manipulator drive mechanisms, while the constraints are modeled in the form of inequalities. The authors of study [20] define six nonlinear objective sub-functions for optimizing manipulator drive mechanisms. Optimal solutions are obtained using a genetic algorithm, recognized as an effective method for exploring complex solution spaces

In [21], the authors present an algorithm for the multi-criteria optimization of Z-kinematics drive mechanisms, where the following criteria are defined as the objective functions: minimum change in the bucket angle, maximum mechanism ratio of the arm and bucket drive mechanisms, and minimum strokes of the boom and bucket hydraulic cylinders. The study makes a significant contribution to the field of multi-objective and adaptable optimization of Z-bar mechanisms, employing a fully parameterized kinematic model and an enhanced complex optimization algorithm. However, this work focuses solely on geometric and functional performance, without considering friction, wear, or energy losses in the joints. In contrast, our research introduces tribological parameters and a minimum power loss criterion, thereby overcoming a key limitation of previous studies and enabling a more energy-efficient and durable design.

In [22], the optimization procedure for the drive mechanisms of manipulators with parallelogram kinematics is presented using the MSC Adams optimization module, based on two objective functions: minimum change in the bucket angle and maximum mechanism ratio of the manipulator drive mechanisms. The study focuses on the optimization of an eight-bar loader mechanism based on simulation, with emphasis on improving the bucket angle and transmission ratio. Its main advantage lies in the use of dynamic simulation and the functional enhancement of the mechanism’s geometry. Although the optimization results are valuable for functional characteristics, the study does not include force analysis, load evaluation, or the influence of friction, which limits the broader applicability of the results. In contrast, our research integrates tribological parameters and a minimum power loss criterion, thereby contributing to the development of a more reliable and energy-efficient design.

As part of the research in [23], the authors conduct a sensitivity analysis to identify the parameters with the greatest impact on the objective functions. The following objective functions are selected for the multi-criteria optimization: minimum change in the bucket angle, maximum vertical reach, and minimum horizontal reach. The integration of fuzzy logic with weighted objective evaluation represents a methodological contribution in the field of complex mechanical system optimization. Although the optimization framework is thoroughly developed, the study lacks a connection to practical operational and maintenance requirements, such as wear, reliability, or durability. In contrast, our research explicitly incorporates tribological effects and energy dissipation as optimization criteria, thereby enhancing the practical applicability and functional reliability of the mechanism design. In previous research, the authors have normalized the multi-criteria optimization problem into a single criterion by using the Weighted Sum Method and the Penalty function.

The maximum mechanism ratio as an objective function of the optimal synthesis, presented in previous works, does not take into account the importance of the position of the manipulator in the working range. Research [24,25,26,27,28,29] has shown that the manipulator drive mechanisms are most heavily loaded during the material loading operation. This phase involves high resistance forces resulting from the interaction between the bucket and the excavation material, which significantly increases the forces transmitted through the kinematic chain and, consequently, the stress on the joints, hydraulic actuators, and structural elements. Due to these demanding working conditions, it is necessary to optimize the manipulator mechanism specifically for the loading operation phase.

In [30], the authors propose a directed digging force criterion that defines the normalized limit digging force, determined on the basis of the loader stability conditions, with the aim of achieving maximum digging forces in the zone of the manipulator’s working range, in which the material loading operation is most often performed.

Therefore, the existing literature in the field of working mechanism optimization for construction machinery primarily focuses on kinematic characteristics, geometric parameters, and multi-objective approaches that do not include a detailed analysis of actual energy losses. Most studies overlook the influence of joint friction and do not incorporate tribological parameters as part of the optimization function, leaving a clear gap in addressing practical energy efficiency and the long-term durability of mechanical systems.

This study addresses that gap by introducing a systematic approach that integrates joint friction as a quantitative criterion for energy optimization. In addition, dynamic simulation in the MSC Adams environment enables realistic modeling of the mechanism under load, while coupling with EDEM analysis allows for the assessment of actual material resistance in the bucket. In this way, the research contributes to the development of more energy-efficient and reliable designs, advancing previous studies. Considering the increasing need for higher energy efficiency and longer machine service life, power loss minimization is proposed as one of the primary criteria in this paper, achieving a comprehensive optimization approach.

A mathematical model of the loader is presented in the subsequent chapters, along with the tribological parameters for the revolute joints in the kinematic pairs. Using the developed model, the minimum power loss criterion is assessed by comparing manipulator drive mechanism variants with identical transformation parameters and differing transmission parameters and vice versa.

2. Mathematical Model of the Wheel Loader

A mathematical model of the loader has been developed to analyze the influence of manipulator drive mechanism parameters on the tribological characteristics of the joints within the kinematic chain. The model comprises the general configuration of the wheel loader’s kinematic chain, which consists of the following members: the rear structural-motion member L₁ (Figure 1), equipped with the wheels L₁₁ and L₁₂; the front structural-motion member L₂, with the wheels L₂₁ and L₂₂; and the manipulator with Z-kinematics that includes the arm L₃, bucket L₄, double-arm lever L₅, coupling rod L₆, hydraulic cylinders C₃ for the boom drive mechanism, and hydraulic cylinder C₄ for the bucket drive mechanism.

The mathematical model of the wheel loader is based on the following assumptions: (1) the manipulator’s kinematic chain is planar in configuration, meaning that the axes of the revolute joints are mutually parallel and the centers of the joints lie within the same plane, (2) the support surface and all members of the loader’s kinematic chain are modeled as rigid bodies, (3) the kinematic chain of the loader is characterized by an open configuration, where the final member, the bucket, is subjected to material loading forces Wx, Wy acting at its center of gravity, (4) the loader’s kinematic chain is subjected to gravitational and inertial forces arising from the members of the kinematic chain and members of the drive mechanisms, (5) the coefficient of friction between the revolute joint elements of the manipulator’s kinematic pairs is considered constant, while friction within the hydraulic cylinders is neglected.

The loader model space is defined by the absolute coordinate system OXYZ (Figure 1), with the unit vectors i, j, and k aligned with the coordinate axes OX, OY, and OZ, respectively. The support surface of the loader lies in the horizontal plane OXZ of the absolute coordinate system, while the vertical axis OY coincides with the axis of the L₁–L₂ kinematic pair.

Each member L_i of the loader’s kinematic chain is defined in its own local coordinate system O_ix_iy_iz_i, with corresponding unit vectors. Each link is characterized by a set of vectors and parameters: $\widehat{{\mathit{e}}_{\iota }}$ is the unit vector along the joint axis at O_i, $\widehat{{\mathit{s}}_{\iota }}$ is the kinematic length of the member L_i; $\widehat{{\mathit{t}}_{\iota }}$ is the position vector from the center of the revolute joint to the center of mass of the member L_i, which has the mass m_i, $\widehat{{\mathit{J}}_{\iota }}$ is the inertia tensor defined about the center of mass. Vectors expressed in the local coordinate systems of the model are denoted with a hat symbol [31,32].

Vectors r_i represent the positions of the joint centers of the kinematic chain members, while vectors r_ti denote the positions of the centers of mass of the corresponding members in the loader’s kinematic chain.

$${\mathit{r}}_1={\mathit{r}}_2,{\mathit{r}}_i={\sum }_{j=1}^{i-1}{A}_{jo}\widehat{{\mathit{s}}_J}\forall i=2,\dots ,4,{\mathit{r}}_5={\mathit{r}}_3+A_{3o}\widehat{{\mathit{s}}_5}$$

(1)

$${\mathit{r}}_{ti}={\mathit{r}}_i+A_{io}\widehat{{\mathit{t}}_{\iota }}\forall i=1,\dots ,5$$

(2)

where A_jo and A_io are the transformation matrices used to convert vectors from the local coordinate systems to the absolute (global) coordinate system of the loader’s mathematical model. These matrices are defined as functions of the internal (generalized) coordinates of the kinematic chain model θ = [θ₁, θ₂, θ₃, θ₄]^T.

2.1. Tribological Parameters of Revolute Joints

The members of the loader manipulator’s kinematic chain and drive mechanisms are linked through fifth-class revolute joints, forming kinematic pairs. The general structure of the revolute joint elements (Figure 2) consists of a clevis pin 1 and a sliding sleeve 2 which in the kinematic pair of members L_i–L_i+1 form a sliding contact joint with an appropriate clearance fit. The clevis pin is firmly attached to one member of the kinematic pair, and the sliding sleeve to the other, while the sliding sleeve can have relative movement in relation to the pin and vice versa.

Functional parameters of joints refer to the relative velocities and the loads acting on the joints during a manipulation task. Structural parameters of joints refer to the geometric dimensions of the joint elements, while tribological parameters relate to the moments of frictional resistance and the forces required to overcome this resistance within joints.

During manipulation tasks, tribological phenomena—friction and wear between joint elements—occur due to relative motion and loading of kinematic chain members and mechanisms. Friction in the joints increases resistance, resulting in power loss and reduced energy efficiency of the manipulator’s drive mechanisms.

Wear of the joint elements increases the clearances in the loose fit of the sliding joint. As a result, when the direction of relative movement between the members of the kinematic pair changes, dynamic shocks occur, leading to increased stress and reduced service life of the components in the kinematic chain of the manipulator mechanism.

It is characteristic that during certain operations of loader manipulation tasks, the members of the kinematic chain and mechanisms, including joint elements, have significant relative motion leading to increased wear and dynamic loading [33,34]. Previous analyses have shown that during the material loading and unloading operations, the total range of relative motion of the bucket in both directions is approximately θ_4o ≅ 100°, while the total range of relative motion of the arm during the transport of material from the loading to the unloading position and its return to a new loading position is approximately θ_4o ≅ 90°. The range of relative motion of the two-arm lever and coupling rod is smaller compared to that of the arm and bucket.

These significant ranges of relative motion, particularly during the material loading phase, are accompanied by the highest forces acting on the manipulator’s joints. During the loading operation, the manipulator must overcome the resistance of the material, which results in large reaction forces transmitted through the kinematic chain. The joints located near the end-effector, such as those of the bucket and the arm, are especially exposed to these loads.

2.2. Revolute Joint Loads

By performing a virtual cut of the kinematic chain at the joint O_i (i = 3,4) and applying the equilibrium conditions to the separated segment, the resultant loads at the center of the joint O_i are determined using the vector of the total reduced force and moment (Figure 2):

$${\mathit{F}}_i={\mathit{F}}_{ri}+{\mathit{F}}_{pi}\forall i=\mathrm{3,4}$$

(3)

$${\mathit{M}}_i={\mathit{M}}_{ri}+{\mathit{M}}_{pi}\forall i=\mathrm{3,4}$$

(4)

where F_ri and M_ri are the vectors of the total force and total moment of all forces acting on the separated part of the kinematic chain of the manipulator, while F_pi and M_pi are the vectors of the force and moment of the drive mechanism of the kinematic pair at the joint O_i.

The total force and total moment vectors of all forces acting on the separated part of the manipulator’s kinematic chain are determined by the following equations:

$${\mathit{F}}_{ri}={\mathit{F}}_{gi}+{\mathit{F}}_i+\mathit{W}\forall i=\mathrm{3,4}$$

(5)

$${\mathit{M}}_{ri}={\mathit{M}}_{gi}+{\mathit{M}}_i+{\mathit{M}}_W\forall i=\mathrm{3,4}$$

(6)

here, F_gi is the resultant of gravitational forces acting on the members of the separated part of the chain, F_i is the resultant of inertial forces caused by the motion of the members of the separated part of the chain, W is the resistance force of the material loaded by the bucket, M_gi represents the resultant moment of gravitational forces, M_i is the resultant moment of inertia due to the motion of the members of the separated part of the chain, and M_W is the resistance moment of the material loaded by the bucket.

The components of the total reduced force at the joint O_i in the local coordinate system O_ix_iy_iz_i of the driven member L_i of the kinematic pair are determined by the following equation:

$$F_{ix}={\mathit{F}}_i·{\mathit{i}}_i,{\mathit{F}}_{iy}={\mathit{F}}_i·{\mathit{j}}_i,{\mathit{F}}_{iz}={\mathit{F}}_i·{\mathit{k}}_i$$

(7)

These components load the joint elements with the components F_ix and F_iy, which are normal to the joint axis O_iz_i, generating friction between the joint elements.

The components of the total reduced moment at the joint O_i in the local coordinate system O_ix_iy_iz_i of the member L_i of the kinematic pair are determined by the following equations:

$$M_{ix}={\mathit{M}}_i·{\mathit{i}}_i,{\mathit{M}}_{iy}={\mathit{M}}_i·{\mathit{j}}_i,{\mathit{M}}_{iz}={\mathit{M}}_i·{\mathit{k}}_i=0$$

(8)

These components load the joint elements, except for the component M_iz, since the load moment component M_rz is counteracted by the required moment of the drive mechanism, such that M_pi = −M_rz.

In the further analysis of the influence of mechanism parameters on the tribological characteristics of the joints, the resultant force F_in, composed of the components F_ix and F_iy acting normal to the joint axis, is considered as resulting from all applied loads:

$$F_{in}={\left(F_{ix}^2+F_{iy}^2\right)}^{{\displaystyle \frac{1}{2}}}$$

(9)

The basic structural parameters of the revolute joint elements are the pin diameter d_is (Figure 2) and the sleeve length l_is. Due to the action of the resultant normal force F_in, the primary stresses in the elements of the kinematic pairs of the manipulator mechanism are shear, surface pressure, and bending. Shear stresses arise primarily from the transmission of tangential forces through the pin, while surface pressure is generated at the contact interfaces between the pin and the sleeve, influencing wear and fatigue life. Bending stresses occur due to eccentric loading or misalignment of the joint elements, which can induce additional moments on the pin.

Based on the primary loads and the allowable stresses of the joint material, the diameter d_is of the joint bolt is determined:

$$d_{is}=\left\{\begin{array}{c}{\left({\displaystyle \frac{2·F_{inmax}}{n_{is}·\pi ·{\tau }_{sd}}}\right)}^{{\displaystyle \frac{1}{2}}}\\ {\left({\displaystyle \frac{F_{inmax}}{n_{is}·e_{is}·p_{sd}}}\right)}^{{\displaystyle \frac{1}{2}}}\\ {\left({\displaystyle \frac{8·F_{inmax}\left(L_{is}-l_{is}\right)}{n_{is}·\pi ·{\sigma }_{sd}}}\right)}^{{\displaystyle \frac{1}{3}}}\end{array}\right.$$

(10)

where F_inmax is the maximum normal force acting on the joint, n_is represents the number of joints in the kinematic pair of the mechanism, e_is = l_is/d_is is the ratio of the sliding sleeve length to the clevis pin diameter, L_is is the span between the clevis pin supports; p_sd, τ_sd, σ_sd are the allowable surface pressure, shear stress, and bending stress of the joint material, respectively.

As a result of the action of the force F_in, which acts perpendicularly to the revolute joint axis O_i of the mechanism’s kinematic pair, key tribological parameters arise:

-

the friction-induced moment:

$$M_{it}=-sign\left(\dot{{\theta }_{\iota }}\right)·F_{in}·{\mu }_t{\displaystyle \frac{d_{is}}{2}}$$

(11)

-

and joint power loss caused by friction:

$$N_{it}=M_{it}·\dot{{\theta }_{\iota }}$$

(12)

where $\dot{{\theta }_{\iota }}$ is the angular velocity of the relative motion of the driven member in a kinematic pair of a mechanism, and μ_t is the coefficient of friction between the joint elements [35,36,37].

In the joint model, a friction coefficient of μ_t = 0.1 is employed, which reflects lubricated contact between steel and bronze components, typical for clevis pin/sliding sleeve interfaces in heavy machinery [38]. The joint model does not include effects of clearance or wear, and this simplification is justified by the focus on short-term power loss, where dry Coulomb friction is the dominant contributor. While long-term effects such as wear may alter joint behavior over time, they are beyond the scope of this study. Future work may include an extended model with clearance and degradation effects to evaluate their long-term influence.

The previously defined Equations (11) and (12) indicate that the structural and tribological parameters of the manipulator joints depend, among other factors, on the component F_in of the total joint load force.

The magnitude of this component is primarily influenced by the force generated by the hydraulic cylinders of the drive mechanisms. These forces must be sufficient to overcome the digging forces acting at the bucket during material loading. In the general position of the manipulator’s kinematic chain, the maximum forces that a hydraulic cylinder can generate are determined based on the following equation:

$$F={\displaystyle \frac{\left|M_{rzi}\right|}{i_i}}=\left\{\begin{array}{c}{n}_{ci}\left[{\displaystyle \frac{D_i^2·\pi }{4}}p-{\displaystyle \frac{D_i^2-d_i^2}{4}}\pi ·p_o\right]{\eta }_{cm}\\ n_{ci}\left[{\displaystyle \frac{D_i^2-d_i^2}{4}}\pi ·p-{\displaystyle \frac{D_i^2·\pi }{4}}·p_o\right]{\eta }_{cm}\end{array}\right.$$

(13)

where i_i is the mechanism ratio of the manipulator drive mechanism, p and p_o are the pressures in the main and return lines of the hydraulic cylinder, respectively, and D_i and d_i are the diameters of the piston and piston rod, respectively.

3. Analysis of Joint Loading Under Different Drive Mechanism Variants

A comprehensive analysis has been conducted to evaluate how variations in drive mechanism parameters affect the tribological behavior and friction-induced power loss in the revolute joints of a loader manipulator.

The generation of variant solutions and the application of design constraints represent the first stage in the procedure for optimal synthesis of loader manipulator mechanisms. This process is based on a previously developed mathematical model of the loader. However, it is not the primary focus of this paper and is discussed in more detail in [12].

The process of generating variant solutions begins with the definition of a set of design parameters D_p:

$$D_p=\left[V,X_i,Y_i,Y_d,Y_t,t_p,t_i,t_s,F_d,p_{max}\right]$$

(14)

where V is the bucket volume, X_i, Y_i, Y_d, Y_t are the working area of the manipulator, α_t, α_i are the angles of the bucket position during transport and unloading, t_p, t_i, t_s is the time of operation for the manipulation task, F_d is the declared bucket breaking force, and p_max is the maximum pressure of the manipulator hydrostatic drive system.

The design (optimization) space is defined by the permissible variations of the transmission parameters of the mechanisms within the range:

$$e_{3j}^{min}\le e_{3j}\le e_{3j}^{max}\forall j=1,\dots ,4;e_{4j}^{min}\le e_{4j}\le e_{4j}^{max}\forall j=1,\dots ,10$$

(15)

where, $e_{3j}^{min},e_{4j}^{min},e_{3j}^{max},e_{4j}^{max}$ represent the minimum and maximum values of the transmission parameters, defined on the basis of the available installation space and the requirement for undisturbed relative motion between the components of the manipulator drive mechanisms and the members of the loader’s kinematic chain.

In the synthesis of manipulator drive mechanisms, two types of variables are considered:

• Transmission parameters (continuous variables): include the coordinates of the joints and the lengths of the levers within the mechanism. These parameters define the geometric configuration and are optimized using a genetic algorithm. Possible variant solutions for the transmission part of the mechanism are selected on the basis of a set of geometric constraints that account for the relative positions of the mechanism’s members in specific manipulator configurations.
• Transformation parameters (discrete variables): refer to the diameters of the pistons and piston rods in the hydraulic cylinders that actuate the manipulator. As discrete design choices, they are selected on the basis of performance requirements, manufacturing constraints, and compatibility with the overall kinematic characteristics of the mechanism.

As an example, various drive mechanism solutions were proposed for a loader manipulator with a Z-kinematics linkage system, a mass of 15,000 kg and a bucket capacity of 2.7 m³. The selected parameters are based on the Komatsu WA320-6 loader, a typical mid-sized wheel loader which is widely used in construction, mining, and material handling applications. This configuration reflects standard industry practice and provides a realistic foundation for evaluating the performance of various drive mechanism solutions.

The optimization was conducted within a defined search space and included specific design constraints. This procedure generated 300 design variants for the transmission components of the manipulator’s drive system. After applying additional constraints related to the hydraulic cylinder characteristics, 26 feasible configurations were selected. These configurations exhibited notable differences in transmission and transformation parameters, as shown in

Table 1. Selected generated variant of the arm drive mechanism.

Variant	D3	d3	b3x	b3y	a3x	a3y	c3p	c3k
	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]
V.001	125	90	0	−524	1655	−129	1340	2058
V.108	150	100	−16	−362	1360	−86	1179	1618
V.135	125	90	137	−506	1764	−115	1340	2073

and

Table 2. Selected generated variant of the bucket drive mechanism.

Variant	D4	d4	b4x/b4y	s5x/s5y	a6x/a6y	a4x/a4y	b6x/b6y	c6	c4p	c4k
	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]
V.001	150	100	206/−209	1703/577	13/369	−182/676	0/−751	756	1340	1897
V.108	180	125	236/−83	1703/577	−13/299	−75/835	0/−840	637	1397	1727
V.135	150	100	206/−207	1703/577	12/354	−196/725	0/−751	756	1393	1862

Where: b_3x, b_3y are the coordinates of the joint O₂₃ where the arm hydraulic cylinder c₃ is connected to the front structural-motion member L₂ defined in the local coordinate system of the member L₂ (Figure 1), a_3x, a_3y are the coordinates of the joint O₃₃ defined in the local coordinate system of the arm L₃ where the arm hydraulic cylinder c₃ is connected; c_3p, c_3k are the initial and final lengths of the arm hydraulic cylinder c₃, respectively; b_4x/b_4y are the coordinates of the joint O₂₄ in the local coordinate system of the front structural-motion member L₂ where the bucket hydraulic cylinder c₄ is connected; s_5x/s_5y are the coordinates of the joint O₅ where the two-arm lever L₅ is connected to the arm L₃ defined in the local coordinate system of the arm L₃; a_6x/a_6y are the coordinates of the joint O₄₆ where the coupling rod L₆ is connected to the bucket L₄ defined in the local coordinate system of the bucket L₄; a_4x/a_4y are the coordinates of the joint O₅₄ in the local coordinate system of the double-arm lever L₅ where the bucket hydraulic cylinder c₄ is connected; b_6x/b_6y are the coordinates of the joint O₅₆ where the two-arm lever is connected to the coupling rod defined in the local coordinate system of the double-arm lever L₅; c₆ is the length of the lever L₆; c_4p, c_4k are the initial and final lengths of the bucket hydraulic cylinder, respectively.

.

Selected variants of the manipulator mechanisms have different transmission parameters. The variants V.001 and V.135 share the same transformation parameters (piston/piston rod diameters: D₃/d₃ = 125/90 mm and D₄/d₄ = 150/100 mm) compared to the variant V.108 (D₃/d₃ = 150/100 mm and D₄/d₄ = 180/125 mm).

Each variant of the drive mechanism was analyzed based on the forces generated during the material loading process. The components of the loading resistant force vector W representing the material loaded by the bucket during the loader’s manipulation task were determined using the Discrete Element Method (DEM) [39,40,41].

The forces of loading resistance were determined using a bucket with a nominal capacity of 2.7 m³ and a width of 2750 mm. From the EDEM 2021 software library, a material representing gravel was selected. The material properties used in the simulation were as follows: material density of 1934 kg/m³, coefficient of restitution of 0.75, coefficient of static friction of 0.32, and coefficient of rolling friction of 0.2. These values correspond to the typical behavior of coarse granular materials and were used as input parameters for the EDEM simulation environment. The bulk material consisted of 100,000 discrete particles whose sizes were generated according to a normal distribution, constrained within a range of 51.5 mm to 71.5 mm. These particles collectively formed a bulk material with an approximate volume of 30 m³, distributed on a horizontal surface.

Based on the simulation using the Discrete Element Method (DEM), the resistance force vectors during the loading operation were determined. Figure 3 shows the W_x and W_y components of the resistance force as a function of time during the simulated loading operations. The results are presented for two types of bucket penetration trajectories into the material: (a) stepwise (W_xstep, W_ystep) and (b) parabolic (W_xpar, W_ypar). The duration of the simulated loading operation cycle was 6 s for the stepwise trajectory and 7 s for the parabolic trajectory.

For assessment of the influence of manipulator mechanism parameters on the tribological characteristics of mechanism revolute joints, three selected variant solutions were analyzed using numerical dynamic simulation in MSC Adams. The models shared identical parameters for the L_i kinematic chain members but differed in the manipulator mechanism configurations.

For the purpose of dynamic simulation in MSC Adams, a 3D model of the loader was developed based on the Komatsu WA320-6, with its components connected via rotational and translational joints. The members of the loader’s kinematic chain are subjected to gravitational forces and digging resistance forces obtained through the DEM simulation. The resistance forces W_x and W_y act at the center of the cutting edge tip of the bucket. The actuator displacement and the resistance force were defined using step functions to ensure a more realistic representation of the actuator behavior and external loading, while also improving the numerical stability and convergence of the simulation.

Based on the simulation results, Figure 4 shows the variation of the normal forces F_in (i = 3, 4, 5) at the joints O₃, O₄, and O₅ of the kinematic pairs for each generated manipulator mechanism variant V.001, V.108, and V.135—in a function of the duration of the manipulation task.

The results indicate that the manipulator mechanism variant V.108, which has larger transformation parameters (larger size of hydraulic cylinders), exhibits significantly higher normal forces F_in, compared to the variant V.135, which has smaller transformation parameters.

During the loading operation (t = 0–6 s), the maximum intensities of normal forces occur across all joints. The highest normal force is F_3n, acting at the joint O₃ (Figure 4a), which belongs to the kinematic pair between the front structure-motion member and the arm (L₂–L₃).

The normal force F_5n (Figure 4c) acting at the joint O₅ in the kinematic pair connecting the arm and the double-arm lever (L₃–L₅) is noticeably lower. The lowest intensity is observed in the normal force F_4n (Figure 4b) acting at the joint O₄ of the kinematic pair between the arm and the bucket (L₃–L₄).

In addition to normal forces, the simulation determines the required power N_ut(V.001), N_ut(V.108), and N_ut(V.135) (Figure 5), which is necessary to overcome frictional resistance in the joints of the kinematic pairs for the selected manipulator mechanism variants.

These power losses were analyzed during manipulation tasks involving a loading method based on a stepped bucket trajectory (Figure 5a) and a loading method based on a parabolic bucket trajectory (Figure 5b). The results indicate that the manipulator mechanism variants exhibit differing levels of frictional resistance during the simulated tasks. The V.108 variant, characterized by larger transformation parameters, requires significantly greater power to overcome joint friction compared to the V.135 variant, which has smaller transformation parameters, under identical manipulation conditions.

The conducted analysis indicates that the transmission parameters of drive mechanisms significantly influence the tribological performance of kinematic pair joints and that a minimum power loss criterion enables the identification of drive mechanism variants that ensure both energy efficiency and reduced power loss in the revolute joints of the kinematic pairs.

4. Minimum Power Loss Criterion

Based on the previously established mathematical model and the conducted analysis, a custom software was developed to calculate the minimum power loss criterion based on joint angular velocities and maximum reaction forces obtained from the stability analysis of the loader that represents a worst-case scenario. Using the derived equations, the program computed the frictional power losses at each joint according to the defined mathematical model of the loader.

The optimization criterion was defined for the optimal synthesis of the drive mechanisms of the loader manipulator. The goal was to minimize the power N_ti required to overcome the resistance caused by friction in the joints of the kinematic pairs of the manipulator mechanisms during manipulation tasks.

The objective function, which represents the minimum power loss, is defined by the following equation:

$$f_t=min{\displaystyle \frac{1}{n_p}}\sum _i^{n_p}{N}_{ti}$$

(16)

where n_p represents the number of manipulator positions throughout the manipulation task.

A program was developed to determine tribological criterion indicators for evaluating the generated variant solutions of manipulator mechanisms. The criterion indicator was determined based on an analytical estimation of power losses due to friction under a constant worst-case load.

The program receives as input a file containing variant solutions of the loader, which includes parameters of the kinematic chain members and manipulator mechanisms, and the simulation conditions, i.e., the parameters of the manipulation task:

$$U_{st}=\left\{{\theta }_{31},{\theta }_{35},{\theta }_{41,}{\theta }_{43},{\alpha }_i,t_z,t_p,t_i\right\}$$

(17)

here, θ₃₁, θ₃₅ are the angles of the arm position during the material loading and unloading operations, respectively, θ₄₁, θ₄₃ are the angles of the bucket’s relative position at the beginning and end of the loading operation, respectively; α_i is the bucket angle during the unloading of material, while t_z, t_p, and t_i are the durations of the loading, material transfer and unloading phases, respectively.

When determining the tribological parameters in the manipulator joints, the possible loading resistance force, derived from the loader’s stability conditions, acts on the manipulator mechanism throughout the entire manipulation task.

By iteratively varying the duration t_c of the manipulation task, the program determines the positions of the kinematic chain and drive mechanisms. At each position, it calculates the tribological parameters of the revolute joints in the manipulator’s kinematic pairs: angular velocity ω_i, normal forces F_n, frictional resistance moments M_it, and the power N_it required to overcome frictional resistance between joint elements.

The results obtained (

Table 3. Objective functions and indicators of the tribological criterion of manipulator mechanisms.

Manipulator Variant Ev	Piston/Piston Rod Diameter				Objective Function	Criterion Indicator
	D3 [mm]	d3 [mm]	D4 [mm]	d4 [mm]	ft	pt = ft min/ft
V.001	125	90	150	100	425.137	0.875
V.014	125	80	160	100	541.580	0.687
V.020	125	80	140	90	446.604	0.833
V.026	110	80	170	115	473.667	0.785
V.028	110	80	180	115	513.465	0.725
V.033	140	90	160	110	634.874	0.586
V.036	125	90	150	100	424.215	0.877
V.051	150	100	170	110	708.934	0.552
V.053	125	90	150	100	498.660	0.746
V.064	140	90	150	100	598.755	0.621
V.095	125	90	140	100	448.396	0.830
V.108	150	100	180	125	674.644	0.532
V.111	110	80	170	115	470.975	0.790
V.117	140	90	125	090	476.138	0.781
V.135	125	90	150	100	425.254	0.875
V.147	140	90	160	100	641.743	0.580
V.150	125	90	160	100	515.947	0.721
V.178	150	100	150	100	602.219	0.618
V.223	125	80	150	100	509.715	0.730
V.236	110	80	170	110	476.795	0.780
V.245	150	100	140	90	615.591	0.604
V.267	125	80	170	115	559.207	0.665
V.271	110	80	140	100	ft min = 372.014	pt max = 1.000
V.278	110	80	160	100	444.064	0.838
V.290	125	90	160	100	524.807	0.709
V.295	125	90	150	100	516.179	0.721

) show that the generated variants of the manipulator mechanisms exhibit significantly different values of the objective functions fₜ, i.e., the indicators pₜ of the tribological criterion. It is notable that the variants of the manipulator mechanisms with smaller transformation parameters (smaller diameters of the piston/piston rod of the hydraulic cylinder) have lower values of the objective functions fₜ, and consequently higher indicators pₜ, than the variants with larger transformation parameters. For example, the variant V.271 (D₃/d₃ = 110/80 mm, D₄/d₄ = 140/100 mm) has the highest indicator (p = 1), while the variant V.108 (D₃/d₃ = 150/100 mm, D₄/d₄ = 180/125 mm) has the lowest (pₜ = 0.532).

5. Conclusions

The research results presented in this paper show that the loads and power losses due to friction in the joints of the loader manipulator mechanisms depend on the transformation and transmission parameters of the mechanisms. The transformation parameters refer to the diameters of the pistons and piston rods of the hydraulic cylinders, while the transmission parameters represent the coordinates of the connections between the hydraulic cylinders and the linkage levers for individual members of the manipulator’s kinematic chain.

All generated variant solutions satisfy the required functional constraints: declared digging force, geometry limits, and avoidance of dead points.

Using MSC Adams, the operation of the developed loader model was simulated with different variants of the Z-kinematic manipulator drive mechanisms. The simulation was performed under identical parameters for the manipulation task operations: loading, material transfer, and unloading. During the loading operation, the material resistances were modeled using the resistance force and resistance moment vectors and determined using the discrete element method (DEM).

The analysis results show that the variants of the loader manipulator mechanisms with smaller transformation parameters, compared to those with larger ones, have lower power losses caused by friction between the elements of the kinematic pairs. This is because, under the same manipulation task conditions, the forces in the joints of the mechanisms are smaller. Due to the reduced forces, the dimensions (diameters) of the joint bolts are smaller, and therefore the moments and friction forces between the joint elements are also smaller.

While larger hydraulic cylinders are typically used to achieve higher digging forces, smaller hydraulic cylinders offer benefits such as reduced size and potential cost savings. However, their use involves several important trade-offs that must be carefully considered. These include increased leakage risks due to higher operating pressures, greater heat generation, reduced durability of seals and components, and a potentially slower dynamic response. As such, selecting cylinder size involves balancing performance requirements with efficiency, reliability, and system-level impacts.

Based on the results of the conducted research, a minimum power loss criterion was defined within the procedure for multi-criteria optimal synthesis of manipulator drive mechanisms. The objective function aims for the mechanisms to achieve maximum energy efficiency, i.e., minimal power losses due to friction in the joints.

In future research, a multi-objective optimization approach is planned, which would include, in addition to the tribological criterion, other key factors such as the digging force criterion, time efficiency (cycle effectiveness), minimal mass of the mechanism components, and dynamic stability during operation. This would enable a comprehensive evaluation of the mutual influences and significance of each criterion in the context of the overall system efficiency.