Screw Dynamics of a Multibody System by a Schoenflies Manipulator

Abstract

This paper presents a screw dynamics method for multi-rigid-body systems. It establishes the relationship between velocity screw (twist) and force screw (wrench) in the theorem of momentum screw. Then, the structure of a 2UPS + 2UPU parallel manipulator is introduced as an example of application. By analyzing the constraint wrench of each limb and the twist of the moving platform in the theorem of momentum screw, the dynamics equation for solving the driving force makes the method very convenient for computer programming. The dynamics equation can be reorganized into a non-homogeneous linear equation and establishes the relationship between the constraint wrench of each limb and the twist of the moving platform in screw coordinate. The outstanding advantage of the algorithm proposed in this paper is that the average calculation time is only 83.21% of that of the Newton–Euler method for the 2UPS + 2UPU parallel manipulator on the same computer. This methodology provides a convenient canonical form for the dynamics analysis of multi-rigid-body systems.

1. Introduction

At present, parallel manipulators are widely used in machine tools because of their high stiffness and loading capacity, fewer driving branches, and easy control. With the application of parallel robots in machine tools, the dynamics analysis of a parallel manipulator becomes increasingly important. The mobility characteristics of the Schoenflies motion include three independent translations and one rotation around a fixed axis. The parallel mechanism that performs the Schoenflies motion can be used in special manipulation. For example, it can be used for product grabbing, assembly, and the processing production line. However, a Schoenflies parallel manipulator has a structure of multiple closed loops, which leads to a very complicated dynamics model. Inverse dynamics is utilized to calculate the driving force as an input to control the motion required by the end effector. Compared with their series counterparts with perfect dynamics modeling methods, the dynamics of parallel mechanisms is still a problem [1]. Therefore, this paper aims to propose an effective dynamics method for such multi-rigid-body systems.

In the field of robot dynamics, research on the dynamics modeling of parallel robots has made significant progress. The main method includes the Newton–Euler equation, the Euler–Lagrange equation, the Kane method, and the virtual work principle [2]. The Newton–Euler equation is widely used to establish the dynamics model of a robot. Considering the inertial force and moment, the Newton equations and Euler equations of all rigid bodies can be obtained to solve the closed-loop dynamics of systems such as the Stewart platform [3], a self-reconfiguring chain [4], and multilink cable-driven hyper-redundant robots [5]. The Newton–Euler equations take all joint forces, including the torques and forces, and input and output into account, which is one major advantage of this method. However, a large number of algebraic and differential equations increase the calculation cost for the Newton–Euler method. From the perspective of analytical mechanics, some scholars utilized the Lagrange method to investigate the dynamics of manipulators [6,7,8]. The Lagrange equation of the first type takes the Lagrange multiplier and treats the constraint as an additional equation [9]. The Lagrange equation of the second type allows one to establish the ordinary differential equations of a mechanical system and obtain the solution of the equation by using a numerical algorithm [10]. For example, the inverse dynamics of the Stewart platform were derived by using the Lagrange method [11,12]. Kumar et al. [13] proposed a modular analysis method for the dynamics of hybrid robots and demonstrated its application under a software framework. The Kane method takes the generalized rate as an independent variable and can eliminate constrained forces/moments [14]. Enferadi et al. [15] derived a closed-form dynamics formula for a spherical parallel manipulator by using the Kane method. The virtual work principle provides a universal method for solving the dynamics problems of complex mechanical systems [16]. In the dynamic simulation and control of robots, these methods have higher computational efficiency [17,18]. However, when conducting inverse dynamics and structure design, constraints need to be easily calculated. When analyzing the inverse dynamics of robots, these methods all have their individual limitations. In addition, the natural orthogonal complement method [19,20], the particle dynamics and a recursive approach [21], a parallel Hamiltonian formulation [22], and a symbolic–numerical method [23] are adopted to establish the dynamics models of closed-loop systems. The existing methods for the dynamics of complicated multi-closed-loop systems often face severe difficulties [2,24,25]. In the aforementioned research, seldom were the dynamics discussed in the coordinates of a momentum screw.

This paper aims at the dynamics of a multi-rigid-body system with the example of the Schoenflies parallel manipulator from the aspect of a momentum screw. It derives the dynamics equation of the momentum screw in one coordinate frame to solve the multi-rigid-body system. The theorem of the momentum screw proposed in this paper provides a vector dynamics algorithm and reveals the relationship between the velocity screw and the force screw. It brings a new idea for the research of wrench and twist. The real-time calculation of the Schoenflies parallel manipulator is performed by programming the algorithm on a computer. When solving the input force(s), we eliminate the redundant internal force and joint constraint force/torque, which effectively reduces the calculation load. This paper proposes an inverse dynamics algorithm based on the theorem of the momentum screw. Compared with the traditional methods, our algorithm has higher computation efficiency and lower computation cost.

2. Momentum Screw

Screw theory is a useful mathematical tool used to represent motion and force of rigid bodies. In this paper, the concept of a momentum screw to connect the force screw (wrench) and velocity screw (twist) provides a new theoretical basis for the dynamics analysis of a multi-rigid-body system.

Suppose that the velocity of a differential mass $dm$ on the rigid body is $\mathit{v}$, then its momentum is

$$d\mathit{p}=\mathit{v}dm$$

(1)

where $d\mathit{p}$ represents the differential of momentum.

In the coordinate frame shown in Figure 1, the differential moment of the momentum of the differential mass is indicated by $d\left({}_o\mathit{L}\right)$ where ${}_o\mathit{L}$ represents the moment of momentum of the particle about the origin of the coordinate frame:

$$d\left({}_o\mathit{L}\right)=\mathit{r}\times \mathit{v}dm$$

(2)

where $\mathit{r}$ represents the position vector of the differential mass $dm$.

Define the differential momentum screw as

$$d\mathit{\Gamma }=\left[\begin{array}{c}d\mathit{p}\\ d\left({}_o\mathit{L}\right)\end{array}\right]=\left[\begin{array}{c}\mathit{v}dm\\ \mathit{r}\times \mathit{v}dm\end{array}\right]$$

(3)

where $\mathit{\Gamma }$ represents the momentum screw.

Therefore, the momentum screw of a rigid body is

$$\mathit{\Gamma }=\left[\begin{array}{c}{\displaystyle \underset{\Omega }{\int }\mathit{v}dm}\\ {\displaystyle \underset{\Omega }{\int }\mathit{r}\times \mathit{v}dm}\end{array}\right]$$

(4)

where $\Omega$ represents the whole space of the rigid body. Because

$$\mathit{p}={\displaystyle \underset{\Omega }{\int }\mathit{v}dm}=m{\mathit{v}}_C$$

(5)

where $C$ represents the center of mass, ${\mathit{v}}_C$ is the linear velocity of the mass center. At the same time,

$$\begin{array}{cc}{}_o\mathit{L}& =\rho {\displaystyle \underset{\Omega }{\int }\left({\mathit{r}}_C+{\mathit{r}}_{CA}\right)\times \left({\mathit{v}}_C+\mathit{\omega }\times {\mathit{r}}_{CA}\right)d\Omega }\hfill \\ & =\rho {\mathit{r}}_C\times {\mathit{v}}_C{\displaystyle \underset{\Omega }{\int }d\Omega }+\rho {\displaystyle \underset{\Omega }{\int }{\mathit{r}}_{CA}\times \left(\mathit{\omega }\times {\mathit{r}}_{CA}\right)d\Omega }\hfill \\ & ={\mathit{r}}_C\times m{\mathit{v}}_C+\left\{{\displaystyle \underset{\Omega }{\int }\left[\left({\mathit{r}}_{CA}^T{\mathit{r}}_{CA}\right)\mathit{I}-\left({\mathit{r}}_{CA}{\mathit{r}}_{CA}^T\right)\right]\rho d\Omega }\right\}\mathit{\omega }\hfill \end{array}$$

(6)

where $\mathit{\omega }$ is the angular velocity of the rigid body, ${\mathit{r}}_{CA}$ is the vector from point $C$ to point $A$, and ${\mathit{J}}_C={\displaystyle \underset{\Omega }{\int }\left[\left({\mathit{r}}_{CA}^T{\mathit{r}}_{CA}\right)\mathit{I}-{\mathit{r}}_{CA}{\mathit{r}}_{CA}^T\right]\rho d\Omega }$. So, we obtain that

$${}_o\mathit{L}={\mathit{r}}_C\times \mathit{p}+{\mathit{J}}_C\mathit{\omega }$$

(7)

As a result, the momentum screw can be defined as

$$\mathit{\Gamma }=\left[\begin{array}{c}\mathit{p}\\ {}_o\mathit{L}\end{array}\right]$$

(8)

where $\mathit{p}=m\mathit{v}$, which is the momentum of a rigid body; ${}_o\mathit{L}={\mathit{r}}_C\times \mathit{p}+{\mathit{J}}_C\mathit{\omega }$, which represents the moment of momentum of the rigid body; and the left subscript $o$ represents the quantity in the coordinate frame $oxyz$.

Let the resultant force screw acting on the rigid body be

$$\mathsf{\$}=\left[\begin{array}{c}\mathit{F}\\ {\mathit{r}}_A\times \mathit{F}\end{array}\right]$$

(9)

where ${\mathit{r}}_A$ represents the position vector of an arbitrary point on the action line of the force in the coordinate frame. According to Newton’s second law, there is

$$\frac{d\mathit{\Gamma }}{dt}=\mathsf{\$}$$

(10)

Therefore, separating the variables of differential Equation (10) and integrating both sides of the equation yields

$$\mathit{\Gamma }\left(t_2\right)-\mathit{\Gamma }\left(t_1\right)={\displaystyle {\int }_{t_1}^{t_2}\mathsf{\$}dt}$$

(11)

When $\mathsf{\$}\equiv 0$, it can be obtained from Equation (11) that

$$\mathit{\Gamma }\left(t_2\right)-\mathit{\Gamma }\left(t_1\right)=0$$

(12)

Equation (12) shows the conservation of momentum and the conservation of the moment of momentum of a rigid body when the resultant wrench of the external forces acting on the rigid body is zero.

For a system composed of $n$ rigid bodies, the theorem of momentum screw is expressed as

$${\displaystyle \sum _{i=1}^n\left[{\mathit{\Gamma }}_i\left(t_2\right)-{\mathit{\Gamma }}_i\left(t_1\right)\right]}={\displaystyle \sum _{j=1}^l{\displaystyle {\int }_{t_1}^{t_2}{\mathsf{\$}}_{j}dt}}$$

(13)

where $l$ represents the total number of the external forces.

When ${\displaystyle \sum _{j=1}^l{\displaystyle {\int }_{t_1}^{t_2}{\mathsf{\$}}_{j}dt}}=0$, the rigid-body system is in equilibrium. Otherwise,

$$\frac{d}{dt}\left[{\displaystyle \sum _{i=1}^n\left[{\mathit{\Gamma }}_i\left(t\right)\right]}\right]={\displaystyle \sum _{j=1}^l{\mathsf{\$}}_j}$$

(14)

where $i$ represents that the momentum screw is about the $i\mathrm{th}$ rigid body, $n$ represents the total number of rigid bodies, and $j$ represents the resultant external force screw on the $j\mathrm{th}$ rigid body.

Equation (14) jointly states the theorem of momentum and the theorem of moment of momentum of a multi-rigid-body system in screw coordinate. In this paper, we call Equation (14) the theorem of momentum screw.

3. Screw Kinematics and Dynamics of a Limb of a Schoenflies Manipulator

3.1. Description of the Schoenflies Manipulator

A spatial Schoenflies parallel manipulator is made up of two UPU kinematic chains and two UPS kinematic chains as shown in Figure 2. Here U represents a universal joint, P represents a prismatic joint, and S represents a spherical joint.

In the absolute coordinate frame $oxyz$, the z-axis is perpendicular to the plane $A_1{A}_2{A}_3{A}_4$, the origin is located at the geometric center of $A_1{A}_2{A}_3{A}_4$, the x-axis is perpendicular to $A_1{A}_2$, and the y-axis is determined by the right-hand rule. The local coordinate frame $o_M{x}_M{y}_M{z}_M$ is established on the end effector $C_1{C}_2{C}_3{C}_4$. The origin of the coordinate frame, $O_M$, is seated at the geometric center of the plane $C_1{C}_2{C}_3{C}_4$. The $z_M$-axis is perpendicular to the plane $C_1{C}_2{C}_3{C}_4$, the $x_M$-axis is perpendicular to line $C_1{C}_2$, and the $y_M$-axis is determined in accordance to the right-hand rule.

3.2. Kinematic Analysis of a Limb

According to the theory of degree of freedom analysis [26,27], in a nonsingular space, the end effector of the 2UPS + 2UPU mechanism has four DoFs, which allows it to perform Schoenflies motion. In this section, the kinematics of each limb will be analyzed. The displacement analysis of the UPS limb of the parallel mechanism is shown in Figure 3.

The connection point $C_i\left(i=1,2,3,4\right)$ in the platform coordinate frame can be transformed to the base frame:

$${\mathit{r}}_{C_i}=\mathit{R}{\mathit{r}}_{O_M{C}_i}^{\prime }+{\mathit{r}}_{O_M}$$

(15)

where ${\mathit{r}}_{C_i}$ and ${\mathit{r}}_{O_M{C}_i}^{\prime }$ are the position vectors in the absolute coordinate frame and platform coordinate frame. We obtain that

$${\mathit{r}}_{A_{\mathrm{i}}{C}_{\mathrm{i}}}={\mathit{r}}_{O_M}+{\mathit{r}}_{O_M{C}_{\mathrm{i}}}-{\mathit{r}}_{A_{\mathrm{i}}}$$

(16)

or

$${\mathit{r}}_{C_{\mathrm{i}}}={\mathit{r}}_{O_M}+{\mathit{r}}_{O_M{C}_i}={\mathit{r}}_{A_{\mathrm{i}}}+{\mathit{r}}_{A_i{C}_i}$$

(17)

which is the inverse displacement equation for the Schoenflies parallel manipulator. Supposing that the length of limb $A_i{C}_i$ is $l_i$, and its unit direction vector is ${\mathit{e}}_i$, we have

$$l_i=‖{\mathit{r}}_{A_i{C}_i}‖$$

(18)

and

$${\mathit{r}}_{A_i{C}_i}=l_i{\mathit{e}}_i$$

(19)

The time derivative of the limb vector gives the velocity of the connection point of the limb and platform:

$${\dot{\mathit{r}}}_{A_i{C}_i}=\mathit{\omega }\times {\mathit{r}}_{O_M{C}_i}+{\mathit{v}}_{O_M}$$

(20)

where $\mathit{\omega }$ is the angular velocity of the moving platform around the z-axis, and ${\mathit{v}}_{O_M}$ is the linear velocity of the moving platform.

The linear velocity $v_{C_{iT}}$ between two connecting parts of the limb is specified by the component of this velocity along the limb:

$$v_{C_{iT}}={\mathit{e}}_i\cdot {\dot{\mathit{r}}}_{A_i{C}_i}$$

(21)

The linear velocity perpendicular to the limb is related to the angular velocity of the limb:

$${\mathit{v}}_{C_{iR}}={\dot{\mathit{r}}}_{A_i{C}_i}-\left({\mathit{e}}_i\cdot {\dot{\mathit{r}}}_{A_i{C}_i}\right){\mathit{e}}_i={\mathit{\omega }}_i\times {\mathit{r}}_{A_i{C}_i}$$

(22)

where ${\mathit{\omega }}_i$ is the angular velocity of the first UPS limb perpendicular to the second limb, and

$${\mathit{\omega }}_i=\frac{{\mathit{e}}_i\times {\dot{\mathit{r}}}_{A_i{C}_i}}{l_i}$$

(23)

which is shown in Figure 4.

Hence, the twist of each limb ${\mathit{\xi }}_i\left(i=1,2,3,4\right)$ may be uniformly expressed by:

$${\mathit{\xi }}_i=\left[\begin{array}{c}{\mathit{\omega }}_i\\ {\dot{l}}_i{\mathit{e}}_i+{\mathit{r}}_{A_i{C}_i}\times {\mathit{\omega }}_i\end{array}\right]$$

(24)

3.3. Screw Dynamics Analysis of a Limb

The momentum screw of the lower limb is

$${\mathit{\Gamma }}_l=\left[\begin{array}{c}\mathit{p}\\ {\mathit{r}}_C\times \mathit{p}+{\mathit{J}}_C\mathit{\omega }\end{array}\right]=\left[\begin{array}{c}{m}_{\mathrm{I}}\left({\mathit{\omega }}_i\times {\mathit{r}}_{\mathrm{I}}\right)\\ {\mathit{J}}_{\mathrm{I}}{\mathit{\omega }}_i\end{array}\right]$$

(25)

where ${\mathit{r}}_{\mathrm{I}}$ is the position vector of the center of mass $C_{\mathrm{I}}$ of the lower limb in the absolute coordinate frame. ${\mathit{J}}_{\mathrm{I}}$ is the mass moment of inertia matrix of the lower limb relative to $A_i\left(i=1,2,3,4\right)$ in the absolute coordinate frame. It is worth noting that the given ${\mathit{J}}_{\mathrm{I}}^{\prime }$ is the mass moment of inertia matrix of the lower limb relative to its center, $C_{\mathrm{I}}$, in the local coordinate frame, and they meet that ${\mathit{J}}_{\mathrm{I}}={{\mathit{R}}^{\prime }}^T{\mathit{J}}_{\mathrm{I}}^{\prime }{\mathit{R}}^{\prime }+m_{\mathrm{I}}\left({\mathit{r}}_{\mathrm{I}}^2\mathit{I}-{\mathit{r}}_{\mathrm{I}}{\mathit{r}}_{\mathrm{I}}^T\right)$. For a kinematic chain, ${\mathit{R}}^{\prime }$ is a transformation matrix from its local coordinate frame to the absolute coordinate frame.

The momentum screw of the upper limb is

$${\mathit{\Gamma }}_u=\left[\begin{array}{c}\mathit{p}\\ {\mathit{r}}_C\times \mathit{p}+{\mathit{J}}_C\mathit{\omega }\end{array}\right]=\left[\begin{array}{c}{m}_{\mathrm{I}\mathrm{I}}{\mathit{v}}_{C_1}+m_{\mathrm{I}\mathrm{I}}\left({\mathit{\omega }}_i\times {\mathit{r}}_{\mathrm{I}\mathrm{I}}\right)\\ {\mathit{J}}_{\mathrm{I}\mathrm{I}}{\mathit{\omega }}_i\end{array}\right]$$

(26)

where ${\mathit{r}}_{\mathrm{I}\mathrm{I}}$ is the position vector of the center of mass $C_{\mathrm{I}\mathrm{I}}$ of the upper limb in the absolute coordinate frame. ${\mathit{J}}_{\mathrm{I}\mathrm{I}}$ is the mass moment of inertia matrix of the upper limb relative to $A_i\left(i=1,2,3,4\right)$ in the absolute coordinate frame. Similarly, ${\mathit{J}}_{\mathrm{I}\mathrm{I}}^{\prime }$ meets ${\mathit{J}}_{\mathrm{II}}={{\mathit{R}}^{\prime }}^T{\mathit{J}}_{\mathrm{II}}^{\prime }{\mathit{R}}^{\prime }+m_{\mathrm{II}}\left(r_{\mathrm{II}}^2\mathit{I}-{\mathit{r}}_{\mathrm{I}\mathrm{I}}{\mathit{r}}_{\mathrm{I}\mathrm{I}}^T\right)$.

The force screw acting on the single limb is

$${\mathsf{\$}}_i=\left[\begin{array}{c}{\mathit{F}}_{A_i}+{\mathit{F}}_{C_i}\\ {\mathit{r}}_{A_i{C}_i}\times {\mathit{F}}_{C_i}+\left(T_{A_i}+T_{C_i}\right){\mathit{e}}_{p_i}\end{array}\right]$$

(27)

Substituting Equations (25)–(27) into Equation (14) yields

$$\frac{d}{dt}\left({\mathit{\Gamma }}_l+{\mathit{\Gamma }}_u\right)=\mathsf{\$}$$

(28)

or

$$\frac{d}{dt}\left[\begin{array}{c}{m}_{\mathrm{I}}\left({\mathit{\omega }}_i\times {\mathit{r}}_{\mathrm{I}}\right)+m_{\mathrm{I}\mathrm{I}}{\mathit{v}}_{C_1}+m_{\mathrm{I}\mathrm{I}}\left({\mathit{\omega }}_i\times {\mathit{r}}_{\mathrm{I}\mathrm{I}}\right)\\ {\mathit{J}}_{\mathrm{I}}{\mathit{\omega }}_i+{\mathit{J}}_{\mathrm{I}\mathrm{I}}{\mathit{\omega }}_i\end{array}\right]={\mathsf{\$}}_i+{\mathsf{\$}}_{m\mathit{g}}$$

(29)

where ${\mathsf{\$}}_i$ is the sum of the external force screw, including the constraints of joints $A_i$ and $B_i$. ${\mathsf{\$}}_{m\mathit{g}}$ is the force screw of gravity.

Here, the viscous friction between the prismatic joint belongs to the internal force and therefore does not appear on the right side of the momentum screw Equation (14).

Suppose $D=\left({\mathit{J}}_{\mathrm{I}}+{\mathit{J}}_{\mathrm{II}}\right){\dot{\mathit{\omega }}}_1-{\mathit{r}}_{\mathrm{I}}\times m_1\mathit{g}-{\mathit{r}}_{\mathrm{II}}\times m_{\mathrm{II}}\mathit{g}$, we obtain that

$$D={\mathit{r}}_{A_i{C}_i}\times {\mathit{F}}_{C_i}+\left(T_{A_i}+T_{C_i}\right){\mathit{e}}_{p_i}$$

(30)

Next, the dynamics equations of the UPU and UPS kinematic chains will be discussed separately.

• UPU kinematic chains

According to the physical meaning of a reciprocal screw, ${\mathsf{\$}}_{A_i{B}_i{C}_i}^r\left(i=1,4\right)$ stands for the constraint torque around ${\mathit{e}}_{p_i}$ as shown in Figure 5 for $i=1$. The terminal constraint of the UPU limbs is a torque parallel to the vector ${\mathit{e}}_{p_i}\left(i=1,4\right)$. Therefore, the constraint torques of the UPU limbs can be expressed by $\left(T_{A_i}+T_{C_i}\right){\mathit{e}}_{p_i}\left(i=1,4\right)$.

The constraint torques $\left(T_{A_i}+T_{C_i}\right){\mathit{e}}_{p_i}\left(i=1,4\right)$ can be eliminated from Equation (30) by taking the cross products on both sides of the equation with ${\mathit{e}}_{p_i}$ as

$${\mathit{e}}_{p_i}\times D={\mathit{e}}_{p_i}\times \left({\mathit{r}}_{A_i{C}_i}\times {\mathit{F}}_{C_i}\right)$$

(31)

Therefore, the constraint force ${\mathit{F}}_{C_i}$ is obtained from Equation (31):

$${\mathit{F}}_{C_i}=\frac{{\mathit{r}}_{A_i{C}_i}}{{\mathit{e}}_{p_i}\cdot {\mathit{r}}_{A_i{C}_i}}{x}_i-\frac{{\mathit{e}}_{p_i}\times D}{{\mathit{e}}_{p_i}\cdot {\mathit{r}}_{A_i{C}_i}}$$

(32)

where $x_i\left(i=1,4\right)$ is the component of the force ${\mathit{F}}_{C_i}$ at the universal joint along the limb.

2.

UPS kinematic chain

For the UPS kinematic chain, the constraint torques $T_{C_2}$ and $T_{C_3}$ are equal to zero at the spherical joint. $D$ can be obtained as

$$D={\mathit{r}}_{A_i{C}_i}\times {\mathit{F}}_{C_i}+T_{A_i}{\mathit{e}}_{p_i}$$

(33)

Similarly, the force ${\mathit{F}}_{C_i}\left(i=2,3\right)$ can be obtained by the cross products ${\mathit{e}}_{p_i}$ on both sides of Equation (33). Therefore, the force ${\mathit{F}}_{C_i}$ can be expressed as follows:

$${\mathit{F}}_{C_i}={\mathit{K}}_i{x}_i-{\mathit{H}}_i$$

(34)

where $x_i\left(i=1,2,3,4\right)$ is the component of the force ${\mathit{F}}_{C_i}$ at the spherical joint or universal joint along the limb ${\mathit{K}}_i=\frac{{\mathit{r}}_{A_i{C}_i}}{{\mathit{e}}_{p_i}\cdot {\mathit{r}}_{A_i{C}_i}}$, and ${\mathit{H}}_i=\frac{{\mathit{e}}_{p_i}\times D}{{\mathit{e}}_{p_i}\cdot {\mathit{r}}_{A_i{C}_i}}$.

The spherical joints at $C_2$ and $C_3$ only contain constraint forces. The constraint forces are ${\mathit{F}}_{C_2}={\mathit{K}}_2{x}_2-{\mathit{H}}_2$ and ${\mathit{F}}_{C_3}={\mathit{K}}_3{x}_3-{\mathit{H}}_3$. For the universal joints at $C_1$ and $C_4$, there are both constraint forces and torques. The constraint forces are ${\mathit{F}}_{C_1}={\mathit{K}}_1{x}_1-{\mathit{H}}_1$ and ${\mathit{F}}_{C_4}={\mathit{K}}_4{x}_4-{\mathit{H}}_4$. The constraint torques ${\mathit{T}}_1$ and ${\mathit{T}}_4$ can be expressed by $x_5{\mathit{e}}_{p_1}$ and $x_6{\mathit{e}}_{p_4}$, where $x_5$ and $x_6$ are the magnitudes of two constraint torques. It is worth noting that $x_i\left(i=1,2,3,4\right)$ is not the magnitude of force ${\mathit{F}}_{C_i}\left(i=1,2,3,4\right)$, while $x_5$ and $x_6$ are the magnitudes of moments ${\mathit{T}}_1$ and ${\mathit{T}}_4$. $x_i$ is the component of constraint force ${\mathit{F}}_{C_i}$ along the limb direction.

3.4. Screw Dynamics Analysis of the Moving Platform

The constraint force screws from the four joints to the end actuator are demonstrated in Figure 6.

It can be seen from Equation (14) that the momentum screw of the moving platform is

$$\mathit{\Gamma }=\left[\begin{array}{c}m{\mathit{v}}_{O_M}\\ {\mathit{r}}_{O_M}\times \left(m{\mathit{v}}_{O_M}\right)+{\mathit{J}}_M\mathit{\omega }\end{array}\right]$$

(35)

The sum of the external force screws acting on the moving platform is

$$\mathsf{\$}={\displaystyle \sum _{i=1}^4{\mathsf{\$}}_{x_i}+{\mathsf{\$}}_{M\mathit{g}}}$$

(36)

Substituting Equations (38) and (39) into Equation (14) presents

$$\frac{d}{dt}\left[\begin{array}{c}m{\mathit{v}}_{O_M}\\ {\mathit{r}}_{O_M}\times \left(m{\mathit{v}}_{O_M}\right)+{\mathit{J}}_M\mathit{\omega }\end{array}\right]=\left[\begin{array}{c}-{\mathit{F}}_{C_1}-{\mathit{F}}_{C_2}-{\mathit{F}}_{C_3}-{\mathit{F}}_{C_4}+M\mathit{g}\\ \begin{array}{l}{\mathit{r}}_{C_1}\times \left(-{\mathit{F}}_{C_1}\right)+x_5{\mathit{e}}_{p_1}+{\mathit{r}}_{C_2}\times \left(-{\mathit{F}}_{C_2}\right)+\\ {\mathit{r}}_{C_3}\times \left(-{\mathit{F}}_{C_3}\right)+{\mathit{r}}_{C_4}\times \left(-{\mathit{F}}_{C_4}\right)+x_6{\mathit{e}}_{p_4}+{\mathit{r}}_{O_M}\times M\mathit{g}\end{array}\end{array}\right]$$

(37)

The constraint force ${\mathit{F}}_{C_i}\left(i=1,2,3,4\right)$ can be transformed by Equations (32) and (34). Therefore, Equation (37) contains six scalar equations in the unknowns denoted by a vector $\mathit{x}={\left[\begin{array}{llllll}{x}_1\hfill & x_2\hfill & x_3\hfill & x_4\hfill & x_5\hfill & x_6\hfill \end{array}\right]}^T$. From Equations (32), (34), and (37), we obtain that

$$\frac{d}{dt}\left[\begin{array}{c}m{\mathit{v}}_{O_M}\\ {\mathit{r}}_{O_M}\times \left(m{\mathit{v}}_{O_M}\right)+{\mathit{J}}_M\mathit{\omega }\end{array}\right]=-\left[\begin{array}{c}{\displaystyle \sum _{i=1}^4{\mathit{K}}_i{x}_i}+{\displaystyle \sum _{i=1}^4{\mathit{H}}_i}+M\mathit{g}\\ {\displaystyle \sum _{i=1}^4\left({\mathit{r}}_{C_i}\times {\mathit{K}}_i\right)x_i}+{\displaystyle \sum _{i=1}^4\left(\begin{array}{l}{\mathit{r}}_{C_i}\times {\mathit{H}}_i-x_5{\mathit{e}}_{p_1}\\ -x_6{\mathit{e}}_{p_4}-{\mathit{r}}_{O_M}\times M\mathit{g}\end{array}\right)}\end{array}\right]$$

(38)

Rearranging Equation (38) yields

$$\left[\begin{array}{c}{\displaystyle \sum _{i=1}^4{\mathit{K}}_i{x}_i}\\ \begin{array}{l}{\displaystyle \sum _{i=1}^4\left({\mathit{r}}_{C_i}\times {\mathit{K}}_i\right)x_i}\\ -x_5{\mathit{e}}_{p_1}-x_6{\mathit{e}}_{p_4}\end{array}\end{array}\right]=-\frac{d}{dt}\left[\begin{array}{c}m{\mathit{v}}_{O_M}\\ {\mathit{r}}_{O_M}\times \left(m{\mathit{v}}_{O_M}\right)+{\mathit{J}}_M\mathit{\omega }\end{array}\right]-\left[\begin{array}{c}{\displaystyle \sum _{i=1}^4{\mathit{H}}_i}+M\mathit{g}\\ {\displaystyle \sum _{i=1}^4{\mathit{r}}_{C_i}\times {\mathit{H}}_i}-{\mathit{r}}_{O_M}\times M\mathit{g}\end{array}\right]$$

(39)

Thus, Equation (39) can be expressed in matrix multiplication form,

$$\mathit{A}\mathit{x}=\mathit{b}$$

(40)

where $\mathit{A}$ is the coefficient matrix, and $\mathit{b}$ is a column vector: $\mathit{A}=\left[\begin{array}{c}{\mathit{A}}_1\\ {\mathit{A}}_2\end{array}\right]=\left[\begin{array}{c}\begin{array}{llllll}{\mathit{K}}_1\hfill & {\mathit{K}}_2\hfill & {\mathit{K}}_3\hfill & {\mathit{K}}_4\hfill & 0\hfill & 0\hfill \end{array}\\ \begin{array}{llllll}{\mathit{r}}_{C_1}\times {\mathit{K}}_1\hfill & {\mathit{r}}_{C_2}\times {\mathit{K}}_2\hfill & {\mathit{r}}_{C_3}\times {\mathit{K}}_3\hfill & {\mathit{r}}_{C_4}\times {\mathit{K}}_4\hfill & -{\mathit{e}}_{p_1}\hfill & -{\mathit{e}}_{p_4}\hfill \end{array}\end{array}\right]$ and $\mathit{b}=-\frac{d}{dt}\left[\begin{array}{c}m{\mathit{v}}_{O_M}\\ {\mathit{r}}_{O_M}\times \left(m{\mathit{v}}_{O_M}\right)+{\mathit{J}}_M\mathit{\omega }\end{array}\right]-\left[\begin{array}{c}{\displaystyle \sum _{i=1}^4{\mathit{H}}_i}+M\mathit{g}\\ {\displaystyle \sum _{i=1}^4{\mathit{r}}_{C_1}\times {\mathit{H}}_i-\left({\mathit{r}}_{O_M}\times M\mathit{g}\right)}\end{array}\right]$.

From Figure 5, we know that ${\mathit{e}}_{p_1}$ and ${\mathit{e}}_{p_4}$ are located in the $xoy$ plane. Therefore, only $x_i\left(i=1,2,3,4\right)$ need to be calculated because they are required in the calculation of driving forces.

3.5. Solution of the Driving Forces

For manipulator control, the driving forces in the limbs need to be fully determined. After obtaining $\mathit{x}$, the driving force, $f_i$, of each limb can be found by

$$f_i+{\mathit{F}}_{C_i}\cdot {\mathit{e}}_i+m_{\mathrm{II}}\mathit{g}\cdot {\mathit{e}}_i={\dot{\mathit{p}}}_{\mathrm{II}}\cdot {\mathit{e}}_i$$

(41)

where ${\mathit{p}}_{\mathrm{II}}=m_{\mathrm{II}}{\mathit{v}}_{C_1}+m_{\mathrm{II}}\left({\mathit{\omega }}_i\times {\mathit{r}}_{\mathrm{II}}\right)$ is the momentum of the upper limb. We can resort to Equation (41) to solve the driving force of each limb.

4. Simulation and Discussion

The Schoenflies parallel mechanism can be used in machine tools to fabricate complex workpieces. To verify the dynamics equations of the 2UPS + 2UPU parallel manipulator, the desired trajectory function that changes over time is given as follows:

$$\left\{\begin{array}{l}{x}_{O_M}=0.5sin\left(\pi t\right)\hfill \\ y_{O_M}=0.3cos\left(2\pi t\right)\hfill \\ z_{O_M}=0.2sin\left(3\pi t\right)+1\hfill \\ {\theta }_{O_M}=0.2sin\left(\pi t\right)\hfill \end{array}\right.$$

(42)

The trajectory changes over time, and the reason for choosing this trajectory is that it represents reciprocating motion. It should be noted that the trajectory at the left and right sides of the intersection point are identical, and the diagram displays their periodic motion.

The geometry parameters of the structure and dynamics parameters of the Schoenflies parallel manipulator are listed in

Table 1. The structure parameters of the Schoenflies parallel manipulator.

Component of the Manipulator	Meaning of the Components	Parameter	Value
Moving platform	Mass	$m\left(\mathrm{kg}\right)$	4.196
	Inertia matrix	$J\left(\mathrm{kg}{\mathrm{m}}^2\right)$	$\mathrm{diag}\left(0.0713,0.0357,0.0357\right)$
	Circumradius	$r\left(\mathrm{m}\right)$	0.212
Lower limb I	Mass	$m_{\mathrm{I}}\left(\mathrm{kg}\right)$	0.367
	Inertia matrix	$J_{\mathrm{I}}^{\prime }\left(\mathrm{kg}{\mathrm{m}}^2\right)$	$\mathrm{diag}\left(0.011,0.011,0.000097\right)$
	length	$l_{\mathrm{I}}\left(\mathrm{m}\right)$	0.6
Upper limb II	Mass	$m_{\mathrm{I}\mathrm{I}}\left(\mathrm{kg}\right)$	0.367
	Inertia matrix	$J_{\mathrm{I}\mathrm{I}}^{\prime }\left(\mathrm{kg}{\mathrm{m}}^2\right)$	$\mathrm{diag}\left(0.015,0.015,0.00007\right)$
	length	$l_{\mathrm{I}\mathrm{I}}\left(\mathrm{m}\right)$	0.7
Base platform	Circumradius	$R\left(\mathrm{m}\right)$	0.707

.

The first step in solving the inverse dynamics is to solve the joint motion based on the motion equation of the end effector. According to the inverse kinematics algorithm, we obtained the velocity of the actuators and plotted them in Figure 7.

To evaluate the computation efficiency of this method, the computation cost is evaluated by programming the algorithm on a computer. The computation cost of the rigid-body dynamics algorithm means the number of operations of the algorithm [28]. The computation cost of an algorithm can be quantified by the calculation time. The momentum screw method only eliminates unnecessary constraints, which is similar to but different from the Newton–Euler equation.

To compare the momentum screw method with the existing methods, we also used the Newton–Euler method to obtain the dynamics equations of the same Schoenflies manipulator. Figure 8 illustrates the simulation results obtained by the momentum screw method and the Newton–Euler method. From Figure 8, it can be seen that the final dynamics results obtained using these two methods are the same. Next, we will discuss three aspects: Computational results discussion, the real-time computing power, and a comparison of the computational advantages.

(1) 

Computational results discussion

From Figure 8, it can be seen that the amplitude of the force calculated using our method show good consistency with the results carried out through Newton-Euler method. The results validate our method.

Next, we will discuss the difference in calculation costs when using the Newton–Euler method and the momentum screw method proposed in this paper for calculation. A comparison of the Newton–Euler method and the proposed momentum screw method in terms of the elapsed time in computation is listed in

Table 2. Comparison of elapsed time by utilizing the momentum screw method versus the Newton–Euler method.

Parameter	Momentum Screw Method	Newton–Euler Method
Time interval (ms)	0.001	0.001
Number of period node	6000	6000
Elapsed times (s)	0.768	0.923

.

(2) 

The real-time capabilities

To demonstrate the real-time capabilities of the codes, the runtime of the codes is calculated using the tic/toc method in MATLAB. The dynamics modeling algorithms for these two methods were implemented via programming on MATLAB. These programs are executed on Windows 11 computers equipped with Intel Core i7-12700H CPUs. The calculation time for solving dynamics equations utilizing the momentum screw method is 0.767529 s, while the calculation time for the Newton–Euler method is 0.923187 s. The codes run within 6 s. Thus, the real-time advantage of the momentum screw method for the Schoenflies parallel manipulator is validated.

(3) 

Comparison of computational advantages

Using the Newton–Euler method, there are three constraint forces and one constraint torque at a universal joint. Therefore, when using the Newton–Euler equation, each UPS limb will provide 13 unknowns: 3 constraint forces and 3 constraint torques at the prismatic joint, 3 constraint forces and 1 constraint torque at the universal joint, and 3 pure constraint forces at the spherical joint. Similarly, each UPU limb will provide 14 unknowns. The difference between the UPU limb and the UPS limb is that the universal joint connecting the leg and the end effector will provide an additional constraint torque. A large number of second-order differential equations and calculations for the Newton–Euler method are inevitable. Appendix A provides the calculation process of the Newton–Euler method.

Unlike the traditional Newton–Euler method, the screw formulation avoids second-order differential equations, which leads to lower computation cost. It is worth noting that there are two unknowns in the equations of the UPU limb and one unknown in the equation of the UPS limb for the true effective constraint of the end effector. Therefore, there are six unknowns provided by the four limbs to the moving platform with a definite direction. The six unknowns can be solved by adopting the momentum screw equation for the moving platform. Compared with the Newton–Euler method, we only need to analyze the constraint forces of the end effector and convert the joint forces to be four unknowns (Equation (40)), which greatly reduces the computation cost. Therefore, the algorithm proposed in this paper is effective, has lower computation cost, and is therefore more convenient in on-time control.

5. Conclusions

This paper presents a momentum screw method to study the dynamics of multi-rigid-body systems through an example of a Schoenflies unsymmetrical parallel manipulator. The momentum screw theorem provides a new perspective between the velocity screw and force screw. Therefore, a set of momentum screw equations of the multi-rigid-body system are derived in the screw coordinate. The kinematics model of each limb is established by vectors, and the velocity screw of each limb is obtained. By linearizing the constraint force of each joint, the momentum screw equation of a rigid body is established. The dynamics equation can be reorganized into a non-homogeneous linear equation. By analyzing the relationship between the constraint wrench of each leg and its twist in the theorem of the momentum screw, the dynamics for solving the driving force makes the method very suitable for computer programming. The dynamics equation in the theorem of the moment of momentum of the screw establishes the relationship between the wrench of each limb and the twist of the moving platform in the screw coordinate. Using a Schoenflies parallel manipulator as a case for dynamics modeling, we validate the merit of efficiency of the screw dynamics of multibody systems. Compared with the traditional methods, this algorithm has a lower cost of computation.