Rigid-Body Dynamics Modeling and Core Functional Component Selection for Heavy-Duty Industrial Robots

Abstract

The selection and design of core functional components are a primary task in the engineering design of heavy-duty industrial robots, in which joint constraint forces and moments act as essential indicators for component selection. This paper proposes a general inverse rigid-body dynamics model for serial kinematic chains that explicitly incorporates joint constraint forces and moments. On this basis, a rigid-body dynamics model for heavy-duty industrial robots is established, which fully considers inertia, gravity, and balancing forces of the balance system, and is verified through dynamic simulations. Corresponding selection and design criteria are then formulated for joint motors, RV reducers, and balance systems. Simulation analyses and prototype full-load tests jointly confirm that the robot meets the 1000 kg load capacity requirement and validate the effectiveness of the proposed selection and design criteria. This study provides a reliable theoretical and engineering reference for the design and development of heavy-duty industrial robots.

1. Introduction

Heavy-duty industrial robots have gained widespread application in fields such as assembly and processing, logistics handling, and smelting and casting due to their advantages of large working space, strong load capacity, and excellent static/dynamic performance [1,2,3]. The selection and design of core functional components, such as joint motors, reducers, and balancing systems, are the primary tasks in the engineering design of such robots. Under full-load working conditions, it is essential to consider not only the substantial driving torque required in the joint drive directions but also the constraint forces and moments in the non-driven directions, which significantly impact the selection and design of core functional components for heavy-duty industrial robots. Therefore, at the outset of heavy-duty industrial robot engineering design, establishing a rigid-body dynamics model that accurately reflects joint constraint forces and moments is crucial. Additionally, the influences of balancing system inertia, gravitational forces, and balancing forces on the corresponding joint driving torques need to be incorporated into the dynamics modeling. Furthermore, formulating targeted criteria for the selection and design of core functional components, as well as realizing parameter optimization, are important research challenges to be addressed.

The purpose of rigid-body dynamics modeling is to establish a mapping relationship between the end-effector motion (displacement, velocity, acceleration) and the joint torques required to achieve that motion. This serves as a theoretical foundation for system dynamics analysis and the selection and design of core functional components. The mechanical principles available for rigid-body dynamics modeling mainly include the Newton–Euler Method [4,5], Lagrange’s Equations [6], and the Principle of Virtual Work [7]. Different mathematical tools, such as Lie group/Lie algebra and Screw Theory, can also be utilized to describe the same mechanical principles [8,9].

The Newton–Euler Method is a typical approach used for system dynamics analysis within the realm of vector mechanics. It aims to simultaneously solve for the hinge internal forces in kinematic pairs and the driving forces of the active joints. The primary calculation process can be summarized as follows: first, the Newton–Euler Equations for each free body subject to forces within the system are formulated. Next, the dynamic recursions among the various components of the system are investigated. Finally, the forces acting on each joint within the system are computed [10,11,12]. This method is advantageous for establishing the dynamic model of complex mechanical systems due to its strong intuitiveness and clear mechanical significance. However, it requires analyzing each free component one by one, resulting in a relatively cumbersome process. If the focus is solely on the joint driving forces or torques necessary to achieve a given motion, the dynamic equations of the system can be directly established using Lagrange’s Equations [13,14] or the Principle of Virtual Work [15,16,17]. Standard Lagrange equations provide no constraint forces. The Lagrange multiplier method can yield them but at the cost of solving a differential-algebraic equation system, which increases mathematical complexity. Consequently, the application of the Principle of Virtual Work in rigid-body dynamic modeling of robotic systems has become increasingly widespread. However, there is a paucity of literature that addresses the modeling of forces and torques in the non-driven directions of joints. Additionally, there is a lack of systematic studies that focus on the effects of the inertia, gravitational forces of the balancing system, as well as the balancing forces. These two issues remain important areas requiring further investigation.

The selection and design of core functional components is a critical step in achieving motion control and functional execution in robotic systems, significantly impacting the kinematic and dynamic performance indicators of robots. For the selection and design of core functional components, numerous studies are available for reference. In terms of selection criteria, existing research can be classified into two categories: experience-driven strategies and analysis-based optimization strategies. The former involves establishing specific constraints, such as the rated output speed constraint of servo motors, the rated output torque constraint of RV reducers, and geometric parameter constraints derived from installation space interference conditions, to achieve the selection and design of core functional components [18,19,20,21]. This method is convenient and efficient when dealing with simple mechanical systems or repetitive design tasks; however, it lacks the necessary adaptability when addressing complex mechanical systems or higher performance design requirements. On the basis of satisfying the above constraints, the latter approach introduces an explicit objective function, and systematically guides decision-making for the problem by means of optimization algorithms [22,23,24]. Compared to the experience-driven strategies, this method allows for the autonomous formulation of optimization objectives based on actual needs, such as minimizing mass or costs, resulting in greater flexibility in the selection and design process. However, the establishment of objective functions often requires support from additional mathematical models or computational resources, which increases the complexity of the selection and design problem. In addition, classified according to whether trajectory support is required, the selection and design methods for core functional components of robots can also be divided into trajectory-based methods and “worst-case”-based methods. The former employs one or more typical planned trajectories, uses kinematics and rigid-body dynamics to solve for the joint forces and torques of the robot, and takes these results as the reference for the selection and design of core functional components [25,26,27]. This method has a clear objective, often achieving optimal selection and design when the operational conditions are fixed. However, when faced with general-purpose operational tasks, it cannot guarantee that the selected core functional components will meet all requirements. The latter considers the “worst case” during robot operation, such as the possible maximum joint forces and moments, and uses this maximum value as a reference to guide the selection and design of core functional components [28,29,30]. This method is simple to operate and has high computational efficiency, but it often has the problem of design redundancy and cannot involve indicators related to the time-domain information of joint forces and moments (such as the root mean square value of torque).

However, aiming at the problem of large constraint forces and moments in the non-driven directions of robot joints, few studies have incorporated it into the selection and design indicators of core functional components. In addition, considering that the end load of the heavy-duty industrial robot changes over a wide range with working conditions, how to balance the influence of different loads on joint torques in the parameter design of the balancing system remains an issue that needs to be studied.

To address the aforementioned issues, this study primarily investigates the rigid-body dynamics modeling of heavy-duty industrial robots and the selection and design of their core functional components. Section 2 proposes a general inverse rigid-body dynamics modeling method for an n-DOF serial kinematic chain that incorporates joint constraint forces and moments. Section 3 integrates the effects of the inertia, gravity, and balancing forces of the balance system on the corresponding joint torques, thereby constructing the inverse rigid-body dynamics model for heavy-duty industrial robots. Section 4 presents selection and design criteria for core components, with particular attention to joint constraint forces and moments in reducer selection, and emphasizes the impact of end-effector load variations on balance system design. Section 5 conducts a validation analysis of the rigid-body dynamics model as well as the selection and design criteria for core functional components. Finally, Section 6 provides a summary of the entire study.

2. Inverse Rigid-Body Dynamics Modeling of n-DOF Serial Kinematic Chains

2.1. System Description

Without loss of generality, an n-DOF serial manipulator composed of a fixed base and n free rigid bodies is considered, where the fixed base and each free rigid body are connected by single-DOF ideal joints. For convenience, the fixed base and n free rigid bodies are sequentially labeled as link $i\left(i=0,1,\dots ,n\right)$, and the n ideal joints are sequentially labeled as joint $j\left(j=1,\dots ,n\right)$. The topological model of the n-DOF serial manipulator is illustrated in Figure 1.

As shown in Figure 1, a follower frame ${\mathcal{K}}_E$ is established at point $E$ on the end-effector link, with the coordinate axes of ${\mathcal{K}}_E$ required to coincide instantaneously with those of the global reference frame ${\mathcal{K}}_0$. A point $A_i\left(i=1,\dots ,n\right)$ is defined on the axis of joint $i$, and with $A_i$ as the origin, a body-fixed frame ${\mathcal{K}}_i$ is established for each link such that its ${\mathit{z}}_i$-axis is instantaneously coincident with the axis of joint $i$.

2.2. Inverse Rigid-Body Dynamics Modeling Incorporating Joint Constraint Forces and Torques

Given that traditional rigid-body dynamic models only need to characterize the one-dimensional torque in the joint driving direction, when employing the Principle of Virtual Work to establish the traditional rigid-body dynamic model of an n-DOF serial kinematic chain, it is sufficient to consider the one-dimensional virtual displacement corresponding to the joint driving direction for the i-th link in the kinematic chain, namely

$$\delta {\mathit{\xi }}_{t,i}={\displaystyle \sum _{j=1}^i\delta q_j{\widehat{\mathit{\zeta }}}_{t,j}}={\mathit{J}}_i\delta \mathit{q}$$

(1)

where

$${\mathit{J}}_i=\left[{\widehat{\mathit{\zeta }}}_{t,1} \cdots {\widehat{\mathit{\zeta }}}_{t,i} {\mathbf{0}}_{6\times \left(n-i\right)}\right]\in {ℝ}^{6\times n}, \delta \mathit{q}={\left(\begin{array}{cccc}\delta q_1& \delta q_2& \cdots & \delta q_n\end{array}\right)}^{\mathrm{T}}\in {ℝ}^{n\times 1}$$

herein, $\delta {\mathit{\xi }}_{t,i}$ denotes the variational motion twist of the i-th link measured in the frame ${\mathcal{K}}_E$; ${\widehat{\mathit{\zeta }}}_{t,i}$ denotes the unit infinitesimal displacement twist of the i-th link; $\delta \mathit{q}$ denotes the combination of variational motions of all joints in the driving direction; $\delta q_i$ denotes the variational motion of the i-th joint; and ${\mathit{J}}_i$ is the Jacobian matrix that maps $\delta \mathit{q}$ to $\delta {\mathit{\xi }}_{t,i}$.

Considering the actual need to accurately characterize the constraint forces and moments, a six-dimensional virtual displacement is introduced at the i-th joint, extending the modeling methodology of traditional rigid-body dynamics. This includes not only the variational motion corresponding to the joint driving direction, but also the remaining five-dimensional virtual displacement, as illustrated in Figure 2a.

After considering the six-dimensional virtual displacement, based on Screws Theory, the variational motion twist $\delta {\mathit{\xi }}_{t,i}$ of the i-th link in the serial kinematic chain can be expressed as

$$\delta {\mathit{\xi }}_{t,i}={\displaystyle \sum _{j=1}^i{\mathit{X}}_j\delta {\mathsf{\rho }}_{t,j}}={\mathit{W}}_i\delta {\mathsf{\rho }}_t$$

(2)

where

$${\mathit{W}}_i=\left[{\mathit{X}}_1 \cdots {\mathit{X}}_i {\mathbf{0}}_{6\times \left(6\times \left(n-i\right)\right)}\right]\in {ℝ}^{6\times \left(6\times n\right)}, \delta {\mathsf{\rho }}_t={\left(\begin{array}{ccc}\delta {\mathsf{\rho }}_{t,1}^{\mathrm{T}}& \dots & \delta {\mathsf{\rho }}_{t,n}^{\mathrm{T}}\end{array}\right)}^{\mathrm{T}}\in {ℝ}^{\left(6\times n\right)\times 1}$$

herein, ${\mathit{X}}_i\in {ℝ}^{6\times 6}$ denotes the adjoint transformation matrix between ${\mathcal{K}}_i$ and ${\mathcal{K}}_E$; the physical significance of the first to sixth column elements in ${\mathit{X}}_i$ is that they correspond to the unit infinitesimal displacement twists of the i-th link in the kinematic chain measured in the frame ${\mathcal{K}}_E$, considering the one-dimensional translational and rotational motions along and about the ${\mathit{x}}_i$-axes, ${\mathit{y}}_i$-axes, and ${\mathit{z}}_i$-axes of its body-fixed frame, respectively. ${\mathit{W}}_i$ denotes the generalized Jacobian matrix that maps $\delta {\mathsf{\rho }}_t$ to $\delta {\mathit{\xi }}_{t,i}$; $\delta {\mathsf{\rho }}_t$ denotes the combination of the six-dimensional variational motion twists of all joints in the kinematic chain; and $\delta {\mathsf{\rho }}_{t,i}$ denotes the six-dimensional variational motion twist of the i-th joint, as illustrated in Figure 2a.

Neglecting temporarily the external and conservative forces acting on the system, the Newton–Euler Equations for the i-th link about the reference point $E$ can be expressed as

$${\mathit{\xi }}_{w,i}={\mathit{M}}_i{\mathit{\xi }}_{a.i}-{\left[{\mathit{\xi }}_{t,i}\times \right]}^{\mathrm{T}}{\mathit{M}}_i{\mathit{\xi }}_{t,i}$$

(3)

where

$${}^i\mathit{M}_i=\left[\begin{array}{cc} m_i{\mathbf{1}}_3\hfill & m_i{\left[{}^i\mathit{p}_{A_i{C}_i}\times \right]}^{\mathrm{T}}\hfill \\ m_i\left[{}^i\mathit{p}_{A_i{C}_i}\times \right]\hfill & {}^{C_i}\mathit{I}_{C_i}+m_i\left[{}^i\mathit{p}_{A_i{C}_i}\times \right]{\left[{}^i\mathit{p}_{A_i{C}_i}\times \right]}^{\mathrm{T}}\hfill \end{array}\right]\in {ℝ}^{6\times 6}, {\mathit{M}}_i={\mathit{X}}_i^{-\mathrm{T}}{}^i\mathit{M}_i{\mathit{X}}_i^{-1}\in {ℝ}^{6\times 6}$$

herein, ${\mathit{\xi }}_{t,i}$ and ${\mathit{\xi }}_{a.i}$ denote the velocity twist and acceleration twist of the i-th link measured in the frame ${\mathcal{K}}_E$, respectively; ${}^i\mathit{M}_i$ and ${\mathit{M}}_i$ denote the spatial inertia matrices of the i-th link measured in the frame ${\mathcal{K}}_i$ and the frame ${\mathcal{K}}_E$, respectively; $m_i$ and ${}^{C_i}\mathit{I}$ denote the mass and the constant inertia matrix at its center of mass, respectively; and ${}^i\mathit{p}_{A_i{C}_i}$ denotes the position vector from the origin $A_i$ of the frame ${\mathcal{K}}_i$ to the center of mass of the i-th link, measured in the frame ${\mathcal{K}}_i$.

Furthermore, based on the Newton–Euler Method and the Principle of Virtual Work, the rigid-body dynamic model of the n-DOF serial kinematic chain incorporating the joint constraint forces and moments can be derived as

$${\tau }_w^n={\mathit{D}}^n\left(\mathit{q}\right)\ddot{\mathit{q}}+{\mathit{H}}^n\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+{\tau }_{wg}^n\left(\mathit{q}\right)+{\tau }_{wF}^n$$

(4)

where

$${\mathit{D}}^n\left(\mathit{q}\right)={\displaystyle \sum _{i=1}^n{\mathit{W}}_i^{\mathrm{T}}{\mathit{M}}_i{\mathit{J}}_i}\in {ℝ}^{\left(6\times n\right)\times n}, {\mathit{H}}^n\left(\mathit{q},\dot{\mathit{q}}\right)={\displaystyle \sum _{i=1}^n\left({\mathit{W}}_i^{\mathrm{T}}{\mathit{M}}_i{\dot{\mathit{J}}}_i-{\mathit{W}}_i^{\mathrm{T}}{\left[{\mathit{\xi }}_{t,i}\times \right]}^{\mathrm{T}}{\mathit{M}}_i{\mathit{J}}_i\right)}\in {ℝ}^{\left(6\times n\right)\times n}$$

$${\tau }_{wg}^n\left(\mathit{q}\right)=-{\displaystyle \sum _{i=1}^n{\mathit{W}}_i^{\mathrm{T}}{\mathit{\xi }}_{wg,i}}\in {ℝ}^{\left(6\times n\right)\times 1}, {\tau }_{wF}^n=-{\mathit{W}}_n^{\mathrm{T}}{\mathit{\xi }}_{wF}\in {ℝ}^{\left(6\times n\right)\times 1}$$

herein, $\mathit{q}\in {ℝ}^{n\times 1}$, $\dot{\mathit{q}}\in {ℝ}^{n\times 1}$, and $\ddot{\mathit{q}}\in {ℝ}^{n\times 1}$ denote the position, velocity, and acceleration vectors of all joints in the kinematic chain along their driving directions, respectively. ${\tau }_w^n\in {ℝ}^{\left(6\times n\right)\times 1}$ denotes the combination of joint forces and moments of all joints, where the six-dimensional force and moment ${\tau }_{t,i}\in {ℝ}^{6\times 1}$ of the i-th joint is illustrated in Figure 2b. ${\mathit{D}}^n\left(\mathit{q}\right)$ denotes the term related to the inertial forces; ${\mathit{H}}^n\left(\mathit{q},\dot{\mathit{q}}\right)$ denotes the term related to the centrifugal and Coriolis forces; ${\tau }_{wg}^n\left(\mathit{q}\right)$ denotes the term associated with the gravity forces; and ${\tau }_{wF}^n$ denotes the term related to external operational forces.

3. Inverse Rigid-Body Dynamic Modeling of Heavy-Duty Industrial Robots

3.1. System Description

Taking the number of DOF of the serial kinematic chain in Figure 1 above as 6, Figure 3 illustrates the 3D model of the heavy-duty industrial robot and its topological model.

As shown, the definitions of the links, joints, and frames in the serial kinematic chain of this robot model are consistent with those in Figure 1. The difference lies in that balance systems are mounted between link 1 and link 2, and between link 2 and link 3, respectively, which are referred to as balance system $u\left(u=1,2\right)$.

Figure 4 illustrates the 3D model of the balance system. The cylinder and piston rod in balance system $u$ are denoted as balance component $u,v\left(v=u,u+1\right)$, respectively. The revolute joint between cylinder $u$ and link $u$ is designated as joint $b_{u,u}$, and the cylindrical joint between cylinder $u$ and piston rod $u$ as joint $b_{u,u+1}$. Furthermore, $B_{u,v}$ is defined as the hinged center point between balance component $u,v$ and the adjacent link, with a body-fixed frame ${\mathcal{K}}_{B,u,v}$ established by taking this point as the origin, whose ${\mathit{z}}_{B,u,v}$-axis coincides instantaneously with the axis of joint $b_{u,v}$. In addition, the motion variables of each joint in the balance system are denoted as $q_{b,u,v}$.

3.2. Inverse Rigid-Body Dynamic Modeling Incorporating the Effects of the Balance System

3.2.1. Velocity Modeling

Denoting the velocity twist of balance component $u,v\left(u=1,2;v=u,u+1\right)$ about reference point E, measured in the frame ${\mathcal{K}}_E$, as ${\mathit{\xi }}_{tb,u,v}$, it follows that

$${\mathit{\xi }}_{tb,u,v}={\displaystyle {\displaystyle \sum _{j=1}^u}{\dot{q}}_j{\hat{\mathit{\zeta }}}_{t,j}}+{\displaystyle {\displaystyle \sum _{i=u}^v}{\dot{q}}_{b,u,i}{\hat{\mathit{\zeta }}}_{tb,u,i}}$$

(5)

where ${\widehat{\mathit{\zeta }}}_{t,j}$ denotes the infinitesimal unit displacement twist of link $j$; ${\hat{\mathit{\zeta }}}_{tb,u,v}$ denotes the infinitesimal unit displacement twist of balance component $u,v$; ${\dot{q}}_{b,u,v}$ denotes the time derivative of motion variable $q_{b,u,v}$, which is the unknown to be determined.

Using point $B_{u,u+1}$ as the reference point to construct the velocity coordination condition, and by solving for ${\dot{q}}_{b,u,v}$, Equation (5) can be rearranged into matrix form, which gives

$${\mathit{\xi }}_{tb,u,v}={\displaystyle {\displaystyle \sum _{j=1}^u}{\dot{q}}_j{\hat{\mathit{\zeta }}}_{t,j}}+{\displaystyle {\displaystyle \sum _{i=u}^v}{\dot{q}}_{b,u,i}{\hat{\mathit{\zeta }}}_{tb,u,i}}$$

(6)

where ${\mathit{J}}_{tb,u,v}$ denotes the Jacobian matrix that maps $\dot{\mathit{q}}$ to ${\mathit{\xi }}_{tb,u,v}$.

3.2.2. Acceleration Modeling

Let the velocity twist of balancing component $u,v\left(u=1,2;v=u,u+1\right)$ with respect to point $E$, measured in frame ${\mathcal{K}}_E$, be denoted as ${\mathit{\xi }}_{ab,u,v}$, it follows that

$${\mathit{\xi }}_{ab,u,v}={\displaystyle {\displaystyle \sum _{j=1}^u}{\ddot{q}}_j{\hat{\mathit{\zeta }}}_{t,j}}+{\displaystyle {\displaystyle \sum _{i=u}^v}{\ddot{q}}_{b,u,i}{\hat{\mathit{\zeta }}}_{tb,u,i}}+{\displaystyle {\displaystyle \sum _{j=1}^u}{\dot{q}}_j{\dot{\hat{\mathit{\zeta }}}}_{t,j}}+{\displaystyle {\displaystyle \sum _{i=u}^v}{\dot{q}}_{b,u,i}{\dot{\hat{\mathit{\zeta }}}}_{tb,u,i}}$$

(7)

where ${\ddot{q}}_{b,u,v}$ denotes the time derivative of velocity variable ${\dot{q}}_{b,u,v}$, which is the unknown to be determined.

Using point $B_{u,u+1}$ as the reference point to construct the acceleration coordination condition, and by solving for ${\ddot{q}}_{b,u,v}$, Equation (7) can be rearranged into matrix form, which gives

$${\mathit{\xi }}_{ab,u,v}={\mathit{J}}_{tb,u,v}\ddot{\mathit{q}}+{\mathit{J}}_{ab,u,v}\dot{\mathit{q}}$$

(8)

where ${\mathit{J}}_{ab,u,v}$ denotes the matrix that maps $\dot{\mathit{q}}$ to ${\mathit{\xi }}_{ab,u,v}$.

3.2.3. Inverse Rigid-Body Dynamic Modeling

Following Equation (3), the Newton–Euler Equation for balance component $u,v\left(u=1,2;v=u,u+1\right)$ is established, which gives

$${\mathit{\xi }}_{wb,u,v}={\mathit{M}}_{b,u,v}{\mathit{\xi }}_{ab,u,v}-{\left[{\mathit{\xi }}_{tb,u,v}\times \right]}^{\mathrm{T}}{\mathit{M}}_{b,u,v}{\mathit{\xi }}_{tb,u,v}$$

(9)

where all physical quantities are defined similarly to those above.

Considering the virtual displacement in the non-driven direction at joint $i$, the variational motion twist $\delta {\mathit{\xi }}_{tb,u,v}$ of the balance component $u,v\left(u=1,2; v=u, u+1\right)$ can be rewritten as

$$\delta {\mathit{\xi }}_{tb,u,v}={\mathit{J}}_{tb,u,v}\delta \mathit{q}={\mathit{J}}_{tb,u,v}{\mathit{\Phi }}_m\delta {\mathit{\rho }}_t$$

(10)

where

$${\mathit{\Phi }}_m=\mathrm{diag}\left[\begin{array}{cccccc}\mathit{P}& \mathit{P}& \mathit{P}& \mathit{P}& \mathit{P}& \mathit{P}\end{array}\right]\in {ℝ}^{6\times 36}, \mathit{P}=\left(\begin{array}{cc}{\mathbf{0}}_{1\times 5}& 1\end{array}\right)\in {ℝ}^{1\times 6}$$

herein, ${\mathit{\Phi }}_m$ denotes the matrix that maps $\delta {\mathit{\rho }}_t$ to $\delta \mathit{q}$.

Furthermore, by applying the principle of virtual work, the following equation holds:

$$\delta W_I+\delta W_g+\delta W_b+\delta W_F+\delta W_{\tau }=0$$

(11)

where $\delta W_I$, $\delta W_g$, $\delta W_b$, $\delta W_F$ and $\delta W_{\tau }$ denote the virtual work done by the inertial force, gravity, balancing force, end-effector operating force, and the forces and torques in all joint directions on the corresponding virtual displacements, respectively.

The mathematical model of inverse rigid-body dynamics for heavy-duty industrial robots that includes joint constraint forces and moments can be derived as

$${\tau }_w^b={\mathit{D}}^b\left(\mathit{q}\right)\ddot{\mathit{q}}+{\mathit{H}}^b\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+{\tau }_{wg}^b\left(\mathit{q}\right)+{\tau }_{wb}\left(\mathit{q}\right)+{\tau }_{wF}^6$$

(12)

where

$${\mathit{D}}^b\left(\mathit{q}\right)=\left({\mathit{D}}^6\left(\mathit{q}\right)+{\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=i}^{i+1}\left({\mathit{\Phi }}_m^{\mathrm{T}}{\mathit{J}}_{tb,i,j}^{\mathrm{T}}{\mathit{M}}_{b,i,j}{\mathit{J}}_{tb,i,j}\right)}}\right)\in {ℝ}^{36\times 6}$$

$${\mathit{H}}^b\left(\mathit{q},\dot{\mathit{q}}\right)=\left({\mathit{H}}^6\left(\mathit{q},\dot{\mathit{q}}\right)+{\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=i}^{i+1}\left({\mathit{\Phi }}_m^{\mathrm{T}}{\mathit{J}}_{tb,i,j}^{\mathrm{T}}{\mathit{M}}_{b,i,j}{\mathit{J}}_{ab,i,j}-{\mathit{\Phi }}_m^{\mathrm{T}}{\mathit{J}}_{tb,i,j}^{\mathrm{T}}{\left[{\mathit{\xi }}_{tb,i,j}\times \right]}^{\mathrm{T}}{\mathit{M}}_{b,i,j}{\mathit{J}}_{tb,i,j}\right)}}\right)\in {ℝ}^{36\times 6}$$

$${\tau }_{wg}^b\left(\mathit{q}\right)=\left({\tau }_{wg}^6\left(q\right)-{\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=i}^{i+1}{\mathit{\Phi }}_m^{\mathrm{T}}{\mathit{J}}_{tb,u,v}^{\mathrm{T}}{\xi }_{wbg,i,j}}}\right)\in {ℝ}^{36\times 1}, {\tau }_{wb}\left(\mathit{q}\right)=-{\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=i}^{i+1}{\mathit{\Phi }}_m^{\mathrm{T}}{\mathit{J}}_{tb,u,v}^{\mathrm{T}}{\mathit{\xi }}_{wbF,i,j}}}\in {ℝ}^{36\times 1}$$

herein, ${\tau }_w^b\in {ℝ}^{36\times 1}$ denotes the combination of all joint forces and torques of the heavy-duty industrial robot after considering the effects of the balancing system; ${\mathit{\xi }}_{wbg,i,j}$ and ${\mathit{\xi }}_{wbF,i,j}$ denote the gravity twist and balance force twist of the balance component $i,j$ measured in frame ${\mathcal{K}}_E$, respectively.

Compared with Equation (4), Equation (12) additionally accounts for the contributions of the balance systems.

Note that in the velocity and acceleration modeling of balancing components, the influence of virtual displacements in the joint constraint directions is not considered; that is, the joint constraint forces and moments shown in Equation (12) do not account for the inertia, gravity, and balancing forces of the balance components. Therefore, Equation (12) requires correction, namely by superposing a correction term in ${\tau }_{wb}$:

$${\tau }_{wb,\epsilon }=\left(\begin{array}{c}{0}_{6\times 1}\\ {\mathit{P}}_{\epsilon }{\mathit{X}}_2^{\mathrm{T}}{\mathit{\xi }}_{wbF,1,2}\\ {\mathit{P}}_{\epsilon }{\mathit{X}}_3^{\mathrm{T}}{\mathit{\xi }}_{wbF,2,3}\\ 0_{18\times 1}\end{array}\right)$$

(13)

where

$${\mathit{P}}_{\epsilon }=\left[\begin{array}{cc}{\mathbf{1}}_5& {\mathbf{0}}_{5\times 1}\\ {\mathbf{0}}_{1\times 5}& 0\end{array}\right]$$

For each joint, the generalized force vector obtained from the above inverse dynamics model (Equation (12)) consists of six components: one driving torque along the actuated direction, two bending moments, and three force components (e.g., thrust and shear forces) along the non-driven directions. These constraint forces and moments are crucial for the selection of core functional components, especially for reducers. Specifically, they directly determine the allowable thrust force and allowable bending moment constraints of the reducer, which will be used to establish the reducer selection criteria in Section 4.

4. Formulation of Selection Criteria for Core Functional Components

4.1. Selection Criteria for Servo Motors and Reducers

The selection criteria for servo motors mainly include rated torque constraints, maximum torque constraints, rated speed constraints, and inertia matching constraints. Similarly, the selection criteria for joint reducers mainly include rated output torque constraints, maximum allowable torque constraints, and rated output speed constraints. Relevant studies can be found in References [14,19,25].

Considering that heavy-duty industrial robots withstand significant forces and moments in the non-driven directions of joints, allowable thrust and bending moment constraints are the core indicators that must be emphasized in the selection of joint reducers for heavy-duty industrial robots. The allowable thrust and bending moment constraints define the upper limits of thrust and bending moment that RV reducers can withstand in the non-driven directions of joints, and their specific requirements are as follows

$$\left\{\begin{array}{l}{F}_{i,2,\mathrm{perm}}\ge F_{i,\max }^{\left(\mathcal{T}\right)}\\ m_{i,2,\mathrm{perm}}\ge m_{i,\max }^{\left(\mathcal{T}\right)}\end{array}\right.$$

(14)

where $F_{i,2,\mathrm{perm}}\left(m_{i,2,\mathrm{perm}}\right)$ denotes the allowable thrust and bending moment of reducer $i$; $F_{i,\max }^{\left(\mathcal{T}\right)}\left(m_{i,\max }^{\left(\mathcal{T}\right)}\right)$ denotes the maximum thrust and bending moment.

4.2. Parameter Design Criteria for the Balance System

In the parameter design of the balance system, the following two basic principles shall be satisfied:

(1) Principle of Minimizing the Maximum Absolute Value of Unbalanced Torque: Unbalanced torque is defined as the difference between the joint torque before balancing and the balancing torque, which is exactly the joint torque after balancing. This principle requires that the balancing torque should match the joint torque before balancing as closely as possible to reduce the joint torque after balancing.

(2) Principle of Minimizing the Maximum Fluctuation Value of Unbalanced Torque: The maximum fluctuation value of unbalanced torque is defined as the maximum difference between the peak and valley values of the unbalanced torque curve, which reflects the fluctuation characteristics of the joint torque after balancing. This principle requires that the joint torque after balancing should fluctuate within a narrow range to ensure the operational stability of the system.

In addition, considering that the end-load of heavy-duty industrial robots varies flexibly according to working conditions, if the parameter optimization of the balancing system is carried out only for the maximum load condition, the balancing system may not achieve optimal performance under other load conditions. Therefore, multiple groups of typical loads are uniformly selected within the load range designed for this robot. The joint output torques before and after adopting the balancing system under different load values are calculated separately, and their average values are taken as references to compute the absolute value and maximum fluctuation value of the unbalanced torque.

After considering various load conditions, the maximum absolute value and the maximum fluctuation value of the unbalanced torque can be obtained by the following formula:

$$\left\{\begin{array}{l}{\left|{\overline{\tau }}_{i,ub}^{\left(\mathcal{T}\right)}\right|}_{\max }={\left|{\overline{\tau }}_{i,bb}^{\left(\mathcal{T}\right)}-{\tau }_{b,i-1}^{\left(\mathcal{T}\right)}\right|}_{\max }\hfill \\ {\left({\tilde{\tau }}_{i,ub}^{\left(\mathcal{T}\right)}\right)}_{\max }={\left({\overline{\tau }}_{i,ub}^{\left(\mathcal{T}\right)}\right)}_{\max }-{\left({\overline{\tau }}_{i,ub}^{\left(\mathcal{T}\right)}\right)}_{\min }\hfill \end{array}\right.$$

(15)

where ${\tau }_{i,ub}$ denotes the unbalanced torque of joint $i$; ${\overline{\tau }}_{i,ub}^{\left(\mathcal{T}\right)}$ is the average value of ${\tau }_{i,ub}$ after applying multiple groups of typical loads; ${\tau }_{i,bb}$ denotes the joint torque before balancing of joint $i$; ${\overline{\tau }}_{i,bb}^{\left(\mathcal{T}\right)}$ is the average value of ${\tau }_{i,bb}$ after applying multiple groups of typical loads; ${\tau }_{b,i-1}^{\left(\mathcal{T}\right)}$ denotes the balancing torque output by the balance system $i-1$; ${\tilde{\tau }}_{i,ub}^{\left(\mathcal{T}\right)}$ denotes the fluctuation value of the unbalanced torque for joint $i$. Furthermore, it should be noted that the variables here are all derived from the complete inverse dynamics model, i.e., Equation (12).

The gas column length $l_{b,i,0}$, pressure $P_{b,i,0}$ of the gas inside the cylinder, and cylinder barrel diameter $d_{b,i}$ under the initial configuration are selected as the design parameters of the balance system. Converting the specified design criteria into optimization objectives, in combination with the selected design parameters, the design optimization model of the balance system can be constructed as

$$\left\{\begin{array}{l}\mathrm{find} {\mathit{X}}_{b,i}={\left(\begin{array}{ccc}{l}_{b,i,0}& P_{b,i,0}& d_{b,i}\end{array}\right)}^{\mathrm{T}}\\ \min f_{opt,b,i}^{\left(\mathcal{T}\right)}\left({\mathit{X}}_{b,i}\right)={\kappa }_{b,1,i}{\left|{\overline{\tau }}_{i+1,ub}^{\left(\mathcal{T}\right)}\right|}_{\max }+{\kappa }_{b,2,i}{\left({\tilde{\tau }}_{i+1,ub}^{\left(\mathcal{T}\right)}\right)}_{\max }\\ \mathrm{s}.\mathrm{t}. d_{b,i}\in \left[d_{b,i,\min },d_{b,i,\max }\right], P_{b,i,0}\in \left[P_{b,i,0,\min },P_{b,i,0,\max }\right], \\ l_{b,i,0}\in \left[l_{b,i,0,\min },l_{b,i,0,\max }\right]\end{array}\right.$$

(16)

where $f_{opt,b,i}^{\left(\mathcal{T}\right)}\left({\mathit{X}}_{b,i}\right)$ denotes the objective function for the optimal design of the balance system, $\min f_{opt,b,i}^{\left(\mathcal{T}\right)}\left({\mathit{X}}_{b,i}\right)$ is the mathematical expression of the two basic principles for balance system parameter design; ${\mathit{X}}_{b,i}$ denotes the set of parameters to be optimized; ${\kappa }_{b,1,i}$ and ${\kappa }_{b,2,i}$ denote the weights of the optimization objectives; $l_{b,i,0,\min }\left(l_{b,i,0,\max }\right)$, $P_{b,i,0,\min }\left(P_{b,i,0,\max }\right)$ and $d_{b,i,\min }\left(d_{b,i,\max }\right)$ denote the lower (upper) limits of the value range for design parameters $l_{b,i,0}$, $P_{b,i,0}$ and $d_{b,i}$, respectively.

5. Verification and Analysis

5.1. Rigid-Body Dynamic Simulation and Verification

To verify the rigid-body dynamic model established in this study, a typical trajectory $\mathcal{T}$ is planned, as shown in Figure 5a.

Trajectory $\mathcal{T}$ consists of three trajectory segments. Segment ① is an acceleration trajectory starting from point $P_0$ and passing through points $P_0\to P_1\to P_2\to P_3\to P_4$ in sequence. Segment ② is a uniform velocity trajectory starting from point $P_4$ and passing through Points $P_4\to P_5\to P_0\to P_6\to P_4$ in sequence. Segment ③ is a deceleration trajectory starting from point $P_6$ and passing through points $P_4\to P_7\to P_2\to P_8\to P_0$ in sequence. In addition, the trajectory $\mathcal{T}$ is required to coincide with the oblique diagonal plane of the largest inscribed cube within the robot workspace. Point $P_2$ coincides with the center of the oblique diagonal plane, and segment 2 of the trajectory is tangent to the edges of the oblique diagonal plane.

In the commercial software SolidWorks 2020, a kinematic simulation model of the robot is established. By inputting the trajectory $\mathcal{T}$, the operational trajectory of the simulation model can be obtained, as shown in Figure 5b.

At the same time, the dynamic parameters and joint motion laws in the mathematical model shown in Equation (12) are required to be consistent with those of the simulation model, and the joint force and torque information for both models under the typical trajectory is synchronously plotted in Figure 6.

For the sake of simplicity, only the joint force and torque information of joint 2 is presented. It can be seen from Figure 6 that the joint force and torque information obtained from the simulation model is basically consistent with that derived from the rigid body dynamic model, which verifies the correctness of the model established in this study.

5.2. Optimal Selection Results of Functional Components

Based on the above criteria, the model parameters of the core functional components obtained via optimal selection are presented in

Table 1. Selection Parameters of Motors and Reducers.

Parameter		Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
Motors	Model Number	Low-Inertia Motor ZLS			Medium-Inertia Motor ZMS
		3003	3004	2653	2152	1654	1652
	Power (kW)	21.2	23.3	15.6	6.0	7.5	5.4
	Mass (kg)	90	95	75	36	28	20
Reducers	Model Number	700C	700N	500N	320C	380N	200C
	Mass (kg)	140	102	57.2	79.5	44	55.6
	Allowable Thrust Force (N)	37,000	44,000	32,000	29,400	25,000	19,600
	Allowable Bending Moment (Nm)	29,400	15,000	11,000	20,580	7050	8820

and

Table 2. Design Parameters of the Balance System.

Parameter	$d_{b,1}$ (m)	$d_{b,2}$ (m)	$P_{b,1,0}$ (×106 Pa)	$P_{b,2,0}$ (×106 Pa)	$l_{b,1,0}$ (m)	$l_{b,2,0}$ (m)
Value	0.196	0.060	4.094	12.000	1.000	0.300

.

Based on the selection parameters given in Table 1 and Table 2, Figure 7 presents the evolution laws of the balancing torques over the entire workspace.

It can be seen that the maximum absolute values of the balancing torques output by balancing system 1 and balancing system 2 are $\mathrm{90,105.2} \mathrm{Nm}$ and $8882.9 \mathrm{Nm}$ respectively, with their variation ranges being approximately $\left[-9.0, 5.7\right]\times {10}^4 \mathrm{Nm}$ and $\left[-8.9, 6.9\right]\times {10}^3 \mathrm{Nm}$ respectively, and a smooth transition exists between the maximum and minimum value regions. Since the balancing torques are only related to the corresponding joint rotation angles, they thus exhibit distinct fan-shaped evolution laws. Among these, when the angle of joint 2 is fixed and the angle of joint 3 varies arbitrarily within its feasible range, the balancing torque output by balancing system 1 remains constant, whereas that of balancing system 2 behaves oppositely.

Figure 8 presents the torque comparison of joint 2 before and after balancing under a load of 1000 kg.

It can be seen from the figure that under this working condition, the maximum torque of joint 2 before balancing can reach $3.75\times {10}^4 \mathrm{Nm}$, with the maximum torque fluctuation up to $1.19\times {10}^4 \mathrm{Nm}$. After the installation of the balance system, the maximum torque is reduced to $1.47\times {10}^4 \mathrm{Nm}$, and its peak value has decreased by 60.76% compared with the torque before balancing, while the torque fluctuation value remains basically unchanged, which is conducive to the smooth selection of motors and reducers.

Furthermore, the energy consumption level at the joints before and after the application of the balancing torque is investigated, and an index $W_{bi}^{\left(\mathcal{T}\right)}$ is defined to evaluate the positive effect of the balance system on reducing joint energy consumption, which is given by:

$$W_{bi}^{\left(\mathcal{T}\right)}=1-{\displaystyle \frac{{\displaystyle {\int }_{t_0}^{t_3}\left|{\tau }_{i,ub}\left(t\right){\dot{q}}_i\left(t\right)\right|\mathrm{d}t}}{{\displaystyle {\int }_{t_0}^{t_3}\left|{\tau }_{i,bb}\left(t\right){\dot{q}}_i\left(t\right)\right|\mathrm{d}t}}}$$

(17)

where $t_0$ and $t_3$ denote the start time and end time of the trajectory, respectively.

Under a load of 1000 kg, the energy consumption of joint 2 was calculated to have decreased by 67.05% following the introduction of the balance system, which is of considerable practical significance for reducing production costs in industrial applications.

Figure 9 presents the variation laws of the thrust force, bending moment and driving-direction torque of joint 4 under a load of 1000 kg.

It can be seen that the maximum thrust force of joint 4 can reach approximately $1.65\times {10}^4 \mathrm{N}$, and the maximum bending moment can reach approximately $1.15\times {10}^4 \mathrm{N}$, while the torque in the driving direction of joint 4 is only approximately $571 \mathrm{Nm}$. The bending moment value is much higher than the torque value in the driving direction. Obviously, if only indicators such as the rated output torque and maximum output torque are referred to, the selected joint reducer will be insufficient to meet the requirements of this working condition. Therefore, it is necessary to incorporate the force and moment values in the non-driving direction of the joint into the selection indicators of the joint reducer, so as to ensure the correct selection of the core functional components.

Figure 10 presents the joint 3 torque under two balancing system design schemes: “multiple groups of typical loads” and “maximum load”.

In Figure 10, ${\left|{\tau }_{3,ub}^{\left(\mathcal{T}\right)}\right|}_{\max }$ denotes the maximum absolute value of the unbalanced torque of joint 3, and ${\left({\tilde{\tau }}_{3,ub}^{\left(\mathcal{T}\right)}\right)}_{\max }$ denotes the maximum fluctuation value of the unbalanced torque of joint 3. According to Equation (16), the second scheme achieves a better balancing effect under 1000 kg load, mainly because it is specially optimized for the 1000 kg load. However, when the load is 0 kg, 500 kg or within a wider load range, it can be seen from the figure that both the output torque and torque fluctuation value of the first scheme are lower than those of the second scheme, resulting in a better performance of the balance system. Therefore, for heavy-duty industrial robots with a large end-effector load range, conducting parameter optimization with consideration of multiple groups of loads during the design of their balance systems can yield a more excellent balancing effect.

5.3. Experimental Testing

Based on the above analysis, an engineering prototype of the heavy-duty industrial robot with a load capacity of 1000 kg has been developed, as shown in Figure 11a.

To verify the load capacity of this robot, 8 metal plates were selected as the end-effector load, which were fixedly mounted on the flange in sequence via bolted connections. Each metal plate has a mass of 130 kg, with a total load mass of 1040 kg.

A series of pose points were defined in the workspace, the robot was actuated to traverse each pose point in sequence at the designed rated speed, and the test was repeated three times. Some experimental poses are shown in Figure 11b.

During the robot test, the execution of control commands was smooth, the trajectory operation was stable, the pose point positioning was accurate, and no alarm or fault phenomena occurred. The test results show that the developed heavy-duty industrial robot meets the design requirements for a load of 1000 kg based on the aforementioned selection and design criteria.

6. Conclusions

This paper studies the rigid-body dynamics modeling method incorporating joint constraint forces and moments, formulates a series of criteria suitable for the selection and design of core functional components of heavy-duty industrial robots, and carries out simulation verification and experimental verification. The main conclusions are as follows:

• A novel inverse rigid-body dynamics modeling method for serial kinematic chains incorporating joint constraint forces and moments is proposed. By introducing multi-dimensional virtual displacements along the joint constraint directions and based on the principle of virtual work, the virtual work in the joint driving directions is extended to the non-driving directions, and a general rigid-body dynamics model capable of accurately characterizing all forces and moments at any joint in an n-DOF serial kinematic chain is constructed.
• The rigid-body dynamics model of the heavy-duty industrial robot is established. In view of the structural characteristics of heavy-duty industrial robots, the influences of the inertia, gravity and balancing forces of the balance system on the corresponding joint forces and moments are analyzed in focus, and the analysis results can provide a necessary theoretical basis for the selection/design of core functional components of such industrial robots.
• A series of selection/design criteria for core functional components suitable for heavy-duty industrial robots is formulated. Among them, indicators related to joint constraint forces and moments are taken into account in the selection of joint reducers, and the influence of variations in the robot’s end-effector load on the design parameters of the balance system is considered in the design of the balance system. Furthermore, based on the above criteria, an engineering prototype of a heavy-duty industrial robot with a 1000 kg load capacity is developed, and load capacity experimental tests are carried out on it. The test results show that the developed robot meets the design requirements for a 1000 kg load.