Rigid Dynamic Model and Analysis of 5-DOF Parallel Mechanism

Abstract

This paper introduces a novel 4-UPS-RPS spatial parallel mechanism, which can achieve three rotation degrees and two translation degrees of freedom. A rigid dynamic model is established and analysis of the parallel mechanism is carried out. The kinematics of the RPS and UPS driving limbs are analysed and the velocity-mapping relationships between the driving limbs and the other parts are built. The load conditions of the parts are analysed, and the rigid dynamic model of the 4-UPS-RPS parallel mechanism is derived by the virtual work principle approach. Using the example of the parallel mechanism movement, a driving forces analysis of the 4-UPS-RPS parallel mechanism is carried out, and the correctness of the rigid dynamic model is verified by numerical calculation and virtual simulation.

Keywords

parallel mechanism rigid dynamics model 5-DOF virtual work principle

1. Introduction

The spatial parallel mechanism, as a new-structure mechanism, has been applied in many fields, such as machine building, aerospace, and industrial robotics [1–3]. The rigid dynamic model and the analysis of the parallel mechanism, which form the theoretical basis of design, motor control, simulation and performance optimization [4–8], must be investigated.

Due to the closed-loop mechanism, the rigid dynamic modelling of the parallel mechanism is quite complicated. Nowadays, the main existing rigid modelling approaches for parallel mechanisms are the virtual work principle method, the Newton-Euler method and the Lagrange method [9–18]. The virtual work principle method, which has advantages of concision, less redundant information, and high efficiency, has been paid most attention by scholars.

In recent years, due to fewer driving elements, lower cost, and more compact structure, lower-mobility parallel mechanisms have shown great potential for application, and have has gradually attracted the attention of scholars both at home and abroad. The robot in this paper is a new 5-DOF (degrees of freedoms) parallel mechanism invented by our team. According to the screw theory, it can achieve three rotation degrees and two translation degrees of freedom. Since the 4-UPS-RPS (see Figure 1) 5-DOF spatial parallel mechanism has excellent kinematics performance and good application prospects, the parallel mechanism is taken as the object. The rigid dynamics equation of the 4-UPS-RPS parallel mechanism, which consists of four UPS (universal joints/prismatic pairs/spherical joints) driving limbs, one RPS (revolution joints/prismatic pairs/spherical joints) driving limb, a fixed platform and a moving platform, is derived by the virtual work principle in this paper. The driving forces of the 4-UPS-RPS parallel mechanism are analysed by numerical calculation and virtual simulation, respectively.

Figure 1.

Virtual prototype model of 4-UPS-RPS mechanism

2. Kinematic analysis of driving limbs for 4-UPS-RPS parallel mechanism

2.1 Linear velocities and angular velocities analysis of driving limbs

As shown in Figure 2, ${}^{A}ω_{B}$ represents the angular velocity of the moving platform in the fixed-coordinate system. ${}^{A}V_{B O}$ is the linear velocity of the moving platform in the fixed-coordinate system. ${}^{A}r_{s i}$ is the radius vector of kinematics pairs S₁ relative to O_B in the fixed-coordinate system, where ${}^{A}r_{s 1}$ is the radius vector of spherical joints S₁ relative to O_B, and joint S₁,S₂,S₃,S₄ and S₅ are made in turn anticlockwise on the moving platform, and ${}^{A}r_{s 2}$ , ${}^{A}r_{s 3}$ , ${}^{A}r_{s 4}$ as well as ${}^{A}r_{s 5}$ are surely made in turn anticlockwise. The speed of spherical joints s_i on the moving platform in the fixed-coordinate system can be expressed as

Figure 2.

The coordinate systems of the 4-UPS-RPS mechanism

{}^{A}V_{S i} = {}^{A}ω_{B} \times {}^{B}r_{S i} + {}^{A}V_{B O}

(1)

Because the projection of ${}^{A}V_{s 1}$ on ${}^{A}u_{1}$ equals zero $({}^{A}V_{u 1} = 0)$ , we can acquire

0 = {}^{A}V_{s 1} \cdot {}^{A}u_{1} = [\begin{matrix} {}^{A}u_{1}^{T} & {({}^{A}γ_{s 1} \times {}^{A}u_{1})}^{T} \end{matrix}] \cdot [\begin{matrix} {}^{A}V_{B O} \\ {}^{A}ω_{B O} \end{matrix}]

(2)

Equation (2) can be simplified as

0 = J_{u A} \cdot [\begin{matrix} {}^{A}V_{B O} \\ {}^{A}ω_{B} \end{matrix}]

(3)

where $J_{u A} = [\begin{matrix} {}^{A}u_{1}^{T} & {({}^{A}γ_{s 1} \times {}^{A}u_{1})}^{T} \end{matrix}]$ is the constraint mapping matrix.

The value of linear velocity and angular velocity of the RPS limb can be expressed as

{\dot{l}}_{1} = {}^{A}n_{1} \cdot ({}^{A}ω_{B} \times {}^{A}r_{S 1}) + {}^{A}n_{1} \cdot {}^{A}V_{B O} = [\begin{matrix} {}^{A}n_{1}^{T} & {({}^{A}r_{S 1} \times {}^{A}n_{1})}^{T} \end{matrix}] \cdot {[\begin{matrix} {}^{A}V_{B O} & {}^{A}ω_{B} \end{matrix}]}^{T}

(4)

{}^{A}ω_{1} = \frac{{}^{A}n_{1}}{l_{1}} \times ({}^{A}ω_{B} \times {}^{A}r_{S 1} + {}^{A}V_{B O}) = \frac{1}{l_{1}} [\begin{matrix} {}^{A}{\hat{n}}_{1}^{} & - {}^{A}{\hat{n}}_{1} {}^{A}{\hat{r}}_{S 1} \end{matrix}] \cdot {[\begin{matrix} {}^{A}V_{B O} & {}^{A}ω_{B} \end{matrix}]}^{T}

(5)

where

${}^{A}n_{1}$ is the unit vector, which points from U_i (Hooke joint U_i on the fixed platform) to S_i (spherical joint S_i on the moving platform).

${}^{A}n_{1}$ is a three-dimensional vector, and ${}^{A}n_{1} = {[\begin{matrix} {}^{A}n_{1 x} & {}^{A}n_{1 y} & {}^{A}n_{1 z} \end{matrix}]}^{T}$ .

${}^{A}{\hat{n}}_{1} = [\begin{matrix} 0 & - {}^{A}n_{1 z} & {}^{A}n_{1 y} \\ {}^{A}n_{1 z} & 0 & - {}^{A}n_{1 x} \\ - {}^{A}n_{1 y} & {}^{A}n_{1 x} & 0 \end{matrix}]$ is the skew-symmetric operator of vector ${}^{A}n_{1}$ .

${}^{A}r_{S 1}$ is a three-dimensional vector, and ${}^{A}r_{S 1} = {[\begin{matrix} {}^{A}r_{S 1 x} & {}^{A}r_{S 1 y} & {}^{A}r_{S 1 z} \end{matrix}]}^{T}$ .

${}^{A}{\hat{r}}_{S 1} = [\begin{matrix} 0 & - {}^{A}r_{S 1 z} & {}^{A}r_{S 1 y} \\ {}^{A}r_{S 1 z} & 0 & - {}^{A}r_{S 1 x} \\ - {}^{A}r_{S 1 y} & {}^{A}r_{S 1 x} & 0 \end{matrix}]$ is the skew-symmetric operator of vector ${}^{A}r_{S 1}$ .

The value of linear velocity and angular velocity of the UPS limb can be expressed as

{\dot{l}}_{i} = {}^{A}n_{i} \cdot ({}^{A}ω_{B} \times {}^{A}r_{S i}) + {}^{A}n_{i} \cdot {}^{A}V_{B O} = [\begin{matrix} {}^{A}n_{i}^{T} & {({}^{A}r_{S i} \times {}^{A}n_{i}^{})}^{T} \end{matrix}] \cdot {[\begin{matrix} {}^{A}V_{B O} & {}^{A}ω_{B} \end{matrix}]}^{T}

(6)

{}^{A}ω_{i} = \frac{{}^{A}n_{i}}{l_{i}} \times ({}^{A}ω_{B} \times {}^{A}r_{S i} + {}^{A}V_{B O}) = \frac{1}{l_{i}} [\begin{matrix} {}^{A}{\hat{n}}_{i} & - {}^{A}{\hat{n}}_{i} \cdot {}^{A}{\hat{r}}_{S i} \end{matrix}] \cdot {[\begin{matrix} {}^{A}V_{B O} & {}^{A}ω_{B} \end{matrix}]}^{T}

(7)

where

${}^{A}{\hat{r}}_{S i}$ is the skew-symmetric operator of vector ${}^{A}r_{S i}$ .

${}^{A}{\hat{n}}_{i}$ is the skew-symmetric operator of vector ${}^{A}n_{i}$ .

$J_{1 A} = [\begin{matrix} {}^{A}n_{i}^{T} & {({}^{A}r_{S i} \times {}^{A}n_{i})}^{T} \end{matrix}]$ is the velocity transfer matrix of the parallel mechanism.

For all the five driving limbs, combining equation (3) with equation (6), we can get

[\begin{matrix} {\dot{L}}_{i} \\ 0 \end{matrix}] = J \cdot [\begin{matrix} {}^{A}V_{B O} \\ {}^{A}ω_{B O} \end{matrix}]

(8)

where

${\dot{L}}_{i} = {[\begin{matrix} {\dot{l}}_{1} & {\dot{l}}_{2} & {\dot{l}}_{3} & {\dot{l}}_{4} & {\dot{l}}_{5} \end{matrix}]}^{T}$ is the linear velocity of five $U_{i} P_{i} S_{i}$ (when $i = 1$ , $R P S$ ) driving limbs.

$J = [\begin{matrix} J_{1 A} \\ J_{u A} \end{matrix}] = [\begin{matrix} {}^{A}n_{1}^{T} & {({}^{A}r_{S 1} \times {}^{A}n_{1})}^{T} \\ {}^{A}n_{2}^{T} & {({}^{A}r_{S 2} \times {}^{A}n_{2})}^{T} \\ {}^{A}n_{3}^{T} & {({}^{A}r_{S 3} \times {}^{A}n_{3})}^{T} \\ {}^{A}n_{4}^{T} & {({}^{A}r_{S 4} \times {}^{A}n_{4})}^{T} \\ {}^{A}n_{5}^{T} & {({}^{A}r_{S 5} \times {}^{A}n_{5})}^{T} \\ {}^{A}u_{1}^{T} & {({}^{A}r_{S 1} \times {}^{A}u_{1})}^{T} \end{matrix}]$ is the general mapping matrix which combines $J_{1 A}$ and $J_{uA}$ .

Converting $\dot{α}$ , $\dot{β}$ , $\dot{γ}$ to the fixed coordinate system {AS} the derivative of Euler angles, which can be used to express the rotational angular velocity of moving platform, is given by

\begin{array}{l} {}^{A}ω_{B} = R (Z, α) \cdot [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] \dot{α} + R (Z, α) R (Y, β) \cdot [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] \dot{β} + \\ + R (Z, α) R (Y, β) R (X, γ) \cdot [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] \dot{γ} = [\begin{matrix} 0 & - \sin α & \cos α \cos β \\ 0 & \cos α & \sin α \cos β \\ 1 & 0 & − \sin β \end{matrix}] \cdot [\begin{matrix} \dot{α} \\ \dot{β} \\ \dot{γ} \end{matrix}] \end{array}

(9)

where

R(X, γ) is the rotation matrix, which means the angle γ rotates around the X axis. After the rotation, a new coordinate system O₁ − X₁Y₁Z₁ is obtained. R(Y, β) is the rotation matrix, which means the angle β rotates around the Y₁ axis. Then, a new coordinate system O₂ − X₂Y₂Z₂ is obtained. R(Z, α) is the rotation matrix, which means the angle γ rotates around the Z₂ axis.

The spherical joint S₁ on the moving platform has no moving displacement along the Z direction, and the velocity of the moving platform along the Z axis is given by

{}^{A}V_{z B O} = - {}^{B}R_{s 1} \dot{β} \cos γ \sin β - {}^{B}R_{s 1} \dot{γ} \cos β \sin γ

(10)

where

${}^{B}R_{s 1}$ is the distance between S₁ (spherical joint S₁ on the moving platform) and O_B (the centre of the moving platform).

The six-dimensions velocity of the moving platform can be expressed as

[\begin{matrix} {}^{A}V_{B O} \\ {}^{A}ω_{B} \end{matrix}] = J_{2 A} [\begin{matrix} {}^{A}V_{x B O} \\ {}^{A}V_{y B O} \\ \dot{α} \\ \dot{β} \\ \dot{γ} \end{matrix}]

(11)

where

J_{2 A} = [\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - {}^{B}R_{s i} \cos γ \sin β & - {}^{B}R_{s i} \cos β \sin γ \\ 0 & 0 & 0 & - \sin α & \cos α \cos β \\ 0 & 0 & 0 & \cos α & \sin α \cos β \\ 0 & 0 & 1 & 0 & - \sin β \end{matrix}]

${}^{B}R_{s i}$ is the distance between S_i (centre of spherical hinge on the moving platform) and O_B (centre of the moving platform).

When i=1,…,5, combining equation (8) with equation (11), the linear velocities of five driving limbs can be written as

{\dot{L}}_{i} = [\begin{matrix} {\dot{l}}_{1} \\ {\dot{l}}_{2} \\ {\dot{l}}_{3} \\ {\dot{l}}_{4} \\ {\dot{l}}_{5} \end{matrix}] = {\bar{J}}_{A} [\begin{matrix} {}^{A}V_{x B O} \\ {}^{A}V_{y B O} \\ \dot{α} \\ \dot{β} \\ \dot{γ} \end{matrix}]

(12)

where

${\bar{J}}_{A 5 \times 5}^{} = J_{1 A} J {}_{2 A}\in R^{5 \times 5}$ is the speed transfer matrix in Euler angle form.

When i=1,…,5, combining equation (11) with equation (7), the linear velocities of five driving limbs can be written as

{}^{A}ω_{i} = \frac{1}{l_{i}} [\begin{matrix} {}^{A}{\hat{n}}_{i}^{} & - {}^{A}{\hat{n}}_{i} {}^{A}{\hat{r}}_{S i} \end{matrix}] J_{2 A} [\begin{matrix} {}^{A}V_{x B O} \\ {}^{A}V_{y B O} \\ \dot{α} \\ \dot{β} \\ \dot{γ} \end{matrix}]

(13)

2.2 Linear-accelerations and angular-accelerations analysis of driving limbs

Calculating the derivative of equation (12), the linear acceleration of each driving limb can be given by

a_{L i} = (\begin{matrix} {}^{A}n_{i}^{T} & {({}^{A}r_{S i} \times {}^{A}n_{i})}^{T} \end{matrix}) \cdot (\begin{matrix} a \\ ε \end{matrix}) + (\begin{matrix} {}^{A}V_{B O}^{T} & {}^{A}ω_{B}^{T} \end{matrix}) \cdot h_{i} \cdot (\begin{matrix} {}^{A}V_{B O} \\ {}^{A}ω_{B} \end{matrix})

(14)

where

a and ∊ are the linear acceleration and angular acceleration of the moving platform, respectively.

h_{i} = \frac{1}{l_{i}} {(\begin{matrix} - {}^{A}{\hat{n}}_{i}^{2} & {}^{A}{\hat{n}}_{i}^{2} \cdot {}^{A}{\hat{r}}_{S i} \\ - {}^{A}{\hat{r}}_{S i} \cdot {}^{A}{\hat{n}}_{i}^{2} & l_{i} \cdot {}^{A}{\hat{r}}_{S i} \cdot {}^{A}{\hat{n}}_{i} + {}^{A}{\hat{r}}_{S i} \cdot {}^{A}{\hat{n}}_{i}^{2} \cdot {}^{A}{\hat{r}}_{S i} \end{matrix})}_{6 \times 6}

Calculating the derivative of equation (13), the angular accelerations of driving limbs are given by

\begin{array}{l} {}^{A}ε_{i} = \frac{1}{l_{i}} {\begin{matrix}  \end{matrix} [\begin{matrix} {}^{A}n_{i}^{T} & - {}^{A}n_{i} {}^{A}{\hat{r}}_{S i} \end{matrix}] \cdot {[\begin{matrix} {}^{A}a_{B O} & {}^{A}ε_{B} \end{matrix}]}^{T} - {}^{A}{\hat{n}}_{i} {}^{A}{\hat{ω}}_{B}^{T} {}^{A}{\hat{r}}_{S i} {}^{A}ω_{B} \begin{matrix}  \end{matrix}} \\ - \frac{2}{l_{i}^{2}} {\begin{matrix}  \end{matrix} [\begin{matrix} {}^{A}n_{i}^{T} & - {}^{A}n_{i} {}^{A}{\hat{r}}_{S i} \end{matrix}] \cdot {[\begin{matrix} {}^{A}V_{B O} & {}^{A}ω_{B} \end{matrix}]}^{T} [\begin{matrix} {}^{A}n_{i}^{T} & {({}^{A}{\hat{r}}_{S i} {}^{A}n_{i})}^{T} \end{matrix}] \cdot {[\begin{matrix} {}^{A}V_{B O} & {}^{A}ω_{B} \end{matrix}]}^{T} \begin{matrix}  \end{matrix}} \end{array}

(15)

2.3 Velocity analysis of barycentre on components of driving limbs

Driving limbs can be divided into two parts. One part is called the oscillating rod, which is connected to the fixed platform, and its barycentre is denoted as $C_{u i}$ . $h_{u i}$ is the distance between C_ui and U_i (the universal joint or revolution joint on the fixed platform). Another part is called the expansion link, which is connected to the moving platform, and its barycentre is denoted as $C_{s i}$ . $h_{s i}$ is the distance between C_si and S_i (the spherical joints on the moving platform).

The linear velocities of the barycentre denoted as ${}^{A}V_{C u i}$ on oscillating rods can be expressed as

{}^{A}V_{C u i} = {}^{A}ω_{i} \times {}^{A}n_{i} h_{u i}

(16)

The linear accelerated velocities of the barycentre denoted as ${}^{A}a_{C u i}$ on oscillating rods can be expressed as

{}^{A}a_{C u i} = {}^{A}ε_{i} \times {}^{A}n_{i} \cdot h_{u i} + {}^{A}ω_{i} ({}^{A}ω_{i} \times {}^{A}n_{i}) \cdot h_{u i}

(17)

The linear velocities of the barycentre denoted as ${}^{A}V_{C s i}$ on expansion links can be expressed as

{}^{A}V_{C s i} = {}^{A}n_{i} {\dot{l}}_{i} + {}^{A}ω_{i} \times {}^{A}n_{i} (l_{i} - h_{s i})

(18)

The linear accelerated velocities of the barycentre denoted as ${}^{A}a_{C s i}$ on expansion links can be expressed as

{}^{A}a_{C s i} = {}^{A}n_{i} {\ddot{l}}_{i} + {}^{A}ε_{i} \times {}^{A}n_{i} (l_{i} - h_{s i}) + {}^{A}ω_{i} ({}^{A}ω_{i} \times {}^{A}n_{i}) \cdot (l_{i} - h_{s i}) + 2 ({}^{A}ω_{i} \times {}^{A}n_{i}) \cdot {\dot{l}}_{i}

(19)

The relationship between velocities of the barycentre on oscillating rods and velocity of the moving platform is given by

[\begin{matrix} {}^{A}V_{C u i} \\ {}^{A}ω_{i} \end{matrix}] = J_{U i} [\begin{matrix} {}^{A}V_{B O} \\ {}^{A}ω_{B} \end{matrix}]

(20)

The relationship between velocities of the barycentre on expansion links and velocity of the moving platform is given by

[\begin{matrix} {}^{A}V_{C s i} \\ {}^{A}ω_{i} \end{matrix}] = J_{S i} [\begin{matrix} {}^{A}V_{B O} \\ {}^{A}ω_{B} \end{matrix}]

(21)

where

J_{U i} = [\begin{matrix} - \frac{h_{u i} {}^{A}{\hat{n}}_{i}^{2}}{l_{i}} & \frac{h_{u i} {}^{A}{\hat{n}}_{i}^{2} {}^{A}{\hat{r}}_{S i}}{l_{i}} \\ \frac{{}^{A}{\hat{n}}_{i}^{}}{l_{i}} & - \frac{{}^{A}{\hat{n}}_{i} {}^{A}{\hat{r}}_{S i}}{l_{i}} \end{matrix}]

J_{S i} = [\begin{matrix} I + \frac{h_{s i} {}^{A}{\hat{n}}_{i}^{2}}{l_{i}} & - ({}^{A}{\hat{r}}_{S i} + \frac{h_{s i} {}^{A}{\hat{n}}_{i}^{2} {}^{A}{\hat{r}}_{S i}}{l_{i}}) \\ \frac{{}^{A}{\hat{n}}_{i}^{}}{l_{i}} & - \frac{{}^{A}{\hat{n}}_{i} {}^{A}{\hat{r}}_{S i}}{l_{i}} \end{matrix}]

3. Velocity-mapping matrix between each component and driving limb

According to equation (12), we can get

[\begin{matrix} {}^{A}V_{B O} \\ {}^{A}ω_{B} \end{matrix}] = \tilde{J} \cdot \dot{L}

(22)

where

$\tilde{J} = J_{2 A} \cdot {\bar{J}}_{A}^{- 1}$ is the velocity-mapping matrix between the moving platform and driving limbs

Putting equation (22) into equation (20), we can get

[\begin{matrix} {}^{A}V_{C u i} \\ {}^{A}ω_{i} \end{matrix}] = {\bar{J}}_{u i} \cdot \dot{L}

(23)

where

${\bar{J}}_{u i} = J_{u i} \cdot \tilde{J}$ is the velocity-mapping matrix between the oscillating rods and driving limbs

Putting equation (22) into equation (21), we can get

[\begin{matrix} {}^{A}V_{C s i} \\ {}^{A}ω_{i} \end{matrix}] = {\bar{J}}_{s i} \cdot \dot{L}

(24)

where

${\bar{J}}_{s i} = J_{s i} \cdot \tilde{J}$ is the velocity-mapping matrix between the expansion links and driving limbs.

4. Dynamic modelling of the 4-UPS-RPS parallel mechanism

4.1 Force analysis of the moving platform

The inertia force of the moving platform in the fixed coordinate system is given by

{}^{A}f_{B} = - m_{B} {}^{A}a_{B}

(25)

where

m_B is the mass of the moving platform.

The inertia moment of the moving platform in the fixed coordinate system is given by

{}^{A}n_{B} = - {}_{B}^{A}R \cdot {}^{B}I_{B} \cdot {}_{B}^{A}R^{T} {}^{A}ε_{B} - {}^{A}ω_{B} \times ({}_{B}^{A}R \cdot {}^{B}I_{B} \cdot {}_{B}^{A}R^{T} \cdot {}^{A}ω_{B})

(26)

where

${}^{B}I_{B}$ is the inertia of the moving platform in the moving-coordinate system;

${}_{B}^{A}R$ is the rotational transformation matrix of the moving-coordinate system relative to the fixed-coordinate system.

The gravity of the moving platform in the fixed-coordinate system {A} is given by

{}^{A}G_{B} = m_{B} {}^{A}g

(27)

The force and moment of the moving platform in the fixed-coordinate system {A} are written as

{}^{A}F_{B} = {}_{B}^{A}R {}^{B}F_{B}

(28)

{}^{A}M_{B} = {}_{B}^{A}R {}^{B}M_{B}

(29)

4.2 The force distribution of components on the driving limb RPS

As shown in Figure 2, the coordinate system {C_i} is set up on the driving limb RPS. Let the Z_Ci axis have the same direction with the unit vector of each driving limb; then, the direction of X_i axis is expressed as $X_{i} = {}^{A}n_{i} \times {}^{A}p_{i}$ , where ${}^{A}p_{i} = \frac{{}^{A}p_{u i}}{| {}^{A}p_{u i} |}$ is a unit vector pointed at the spherical joint on the moving platform in the fixed coordinate system. Then, the direction of Y_i axis can be expressed as $Y_{i} = Z_{i} \times X_{i}$ .

The coordinate transformation matrix ${}_{C i}^{A}R$ of coordinate {Ci} relative to the fixed coordinate system {A} can be expressed as

{}_{c i}^{A}R = {[\begin{matrix} {}^{A}n_{i} \times {}^{A}p_{i} & {}^{A}n_{i} \times ({}^{A}n_{i} \times {}^{A}p_{i}) & {}^{A}n_{i} \end{matrix}]}_{3 \times 3}

(30)

For driving limb RPS, where i=l, m_u1 is the mass of the oscillating rod. Therefore, the inertia force of oscillating rod in the fixed-coordinate system {A} can be expressed as

{}^{A}f_{u 1} = - m_{u 1} {}^{A}a_{u 1}

(31)

Defining ${}^{1}I_{u 1}$ as the inertia of the oscillating rod for the RPS driving limb, the inertia moment of the oscillating rod in the fixed coordinate system {A} can be expressed as

{}^{A}n_{u 1} = - {}_{D 1}^{A}R {}^{1}I_{u 1} {}_{D 1}^{A}R^{T} {}^{A}ε_{1} - {}^{A}ω_{1} \times ({}_{c 1}^{A}R {}^{1}I_{u 1} \cdot {}_{c 1}^{A}R^{T} \cdot {}^{A}ω_{B})

(32)

The gravity of the oscillating rod for the RPS driving limb in the fixed coordinate system {A} can be expressed as

{}^{A}G_{u 1} = m_{u 1} {}^{A}g

(33)

For the RPS driving limb, m_s1 is the mass of the expansion link. Therefore, the inertia force of the expansion link in the fixed coordinate system {A} can be expressed as

{}^{A}f_{s 1} = - m_{s 1} {}^{A}a_{s 1}

(34)

Let ${}^{1}I_{s 1}$ be the inertia of the expansion link of the RPS, and the inertia moment of expansion link in the fixed coordinate system {A} can be expressed as

{}^{A}n_{s 1} = - {}_{c 1}^{A}R {}^{1}I_{s 1} \cdot {}_{c 1}^{A}R^{T} \cdot {}^{A}ε_{1} - {}^{A}ω_{1} \times ({}_{c 1}^{A}R {}^{1}I_{s 1} \cdot {}_{c 1}^{A}R^{T} \cdot {}^{A}ω_{B})

(35)

The gravity of the expansion link of the RPS in the fixed coordinate system {A} can be expressed as

{}^{A}G_{s 1} = m_{s 1} {}^{A}g

(36)

4.3 The force distribution of components on the UPS driving limb

For each UPS driving limb, where i=2,…,5, m_ui is the mass of the oscillating rod. The inertia force of the oscillating rod in the fixed coordinate system {A} can therefore be expressed as

{}^{A}f_{u i} = - m_{u i} {}^{A}a_{u i}

(37)

Let ${}^{i}I_{u i}$ be the inertia of the oscillating rod of the UPS, where i=2,…,5, and the inertia moment of the oscillating rod in the fixed-coordinate system {A} can be expressed as

{}^{A}n_{u i} = - {}_{D i}^{A}R {}^{i}I_{u i} {}_{D i}^{A}R^{T} {}^{A}ε_{i} - {}^{A}ω_{i} \times ({}_{c i}^{A}R {}^{i}I_{u i} {}_{c i}^{A}R^{T} {}^{A}ω_{B})

(38)

The gravity of the oscillating rod of the UPS in the fixed-coordinate system {A} can be expressed as

{}^{A}G_{u i} = m_{u i} {}^{A}g

(39)

For each UPS driving limb, m_si is the mass of the expansion link, where i=2,…,5. Therefore, the inertia force of the expansion link in the fixed-coordinate system {A} can be expressed as

{}^{A}f_{s i} = - m_{s i} {}^{A}a_{s i}

(40)

Let ${}^{i}I_{s i}$ be the inertia of the expansion link of the UPS, where i=2,…,5, and the inertia moment of the expansion link in the fixed-coordinate system {A} can be expressed as

{}^{A}n_{s i} = - {}_{c i}^{A}R {}^{i}I_{s i} {}_{c i}^{A}R^{T} {}^{A}ε_{i} - {}^{A}ω_{i} \times ({}_{c i}^{A}R {}^{i}I_{s i} {}_{c i}^{A}R^{T} {}^{A}ω_{B})

(41)

The gravity of the expansion link of the UPS in the fixed-coordinate system {A} can be expressed as

{}^{A}G_{s i} = m_{s i} {}^{A}g

(42)

4.4 The establishment of dynamic model

In the fixed coordinate system {A}, firstly, the external forces on the main components of the spatial parallel mechanism should be simplified to the six-dimensional resultant force vector on the barycentre of each part. Then, the equivalent driving force, which is converted to five driving limbs, should be calculated.

The equivalent driving forces on five driving limbs are given by

F_{B} = {\tilde{J}}^{T} [\begin{matrix} {}^{A}f_{B} + {}^{A}G_{B} + {}^{A}F_{B} \\ {}^{A}n_{B} + {}^{A}M_{B} \end{matrix}]

(43)

The external forces on each driving limb U_iP_iS_i(i=1,…,5, when i=1, the driving limb is RPS) converted to five driving limbs can be expressed as

F_{i} = {\bar{J}}_{u i}^{T} [\begin{matrix} {}^{A}f_{u i} + {}^{A}G_{u i} \\ {}^{A}n_{u i} \end{matrix}] + {\bar{J}}_{s i}^{T} [\begin{matrix} {}^{A}f_{s i} + {}^{A}G_{s i} \\ {}^{A}n_{s i} \end{matrix}]

(44)

According to the virtual work principle, the driving forces of five driving limbs for the spatial parallel mechanism are equal to the equivalent driving force. We can then get

F + F_{B} + \sum_{i = 1}^{5} F_{i} = 0

(45)

Then, the dynamic model of the spatial parallel mechanism can be expressed as

F = - F_{B} - \sum_{i = 1}^{5} F_{i}

(46)

where i = 1,2,3,4,5

F = [f₁ f f₃ f₄ f₅]^T is the driving forces of five driving limbs.

5. Examples of dynamic analysis of the 4-UPS-RPS parallel mechanism

5.1 Structural parameters of the 4-UPS-RPS spatial parallel mechanism

The mass of the moving platform is 36.28 kg. The distance between the barycentre on the oscillating rod and the Hooke hinge (or the revolution joint) on the fixed platform is denoted as h_ui, and h_ui =0.378 m. The mass of the oscillating rod is denoted as m_ui, and m_ui =26.15 kg. The distance between the bary centre on the expansion link and the spherical joints on the moving platform is denoted as h_si, and h_si =0.452 m. The mass of the expansion link is denoted as m_si, and m_si =8.45 kg. The rotational inertia of the moving platform (unit: kg · m²) is denoted as ${}^{B}I_{B}$ , and ${}^{B}I_{B} = [\begin{matrix} 0.932 & 0 & 0 \\ 0 & 0.682 & 0 \\ 0 & 0 & 0.682 \end{matrix}]$ . The rotational inertia of the oscillating rod (unit: $k g \cdot m^{2}$ ) is denoted as ${}^{i}I_{u i}$ , and ${}^{i}I_{u i} = [\begin{matrix} 1.280 & 0 & 0 \\ 0 & 1.280 & 0 \\ 0 & 0 & 0.035 \end{matrix}]$ . The rotational inertia of the expansion link (unit: $k g \cdot m^{2}$ ) is denoted as ${}^{i}I_{s i}$ , and ${}^{i}I_{s i} = [\begin{matrix} 0.528 & 0 & 0 \\ 0 & 0.528 & 0 \\ 0 & 0 & 0.00167 \end{matrix}]$

5.2 Dynamic analysis results of the spatial parallel mechanism

The motion of the 4-UPS-RPS spatial parallel mechanism is described as follows:

{\begin{array}{l} x = 0.920 + 0.03 \cdot \cos 2 t \\ y = - 0.15 + 0.03 \cdot \sin 2 t \\ z = 0.202 \cdot \cos β \cdot \sin γ \\ α = \frac{π}{72} \cdot t \\ β = \frac{π}{72} \cdot t \\ γ = \frac{π}{2} + \frac{π}{72} \cdot t \end{array} (0 \leq t \leq 4)

(47)

According to the rigid dynamic equation (46), the driving force of the 4-UPS-RPS spatial parallel mechanism under no-load condition, as acquired by numerical calculation, is shown in Figure 3(a), and the driving force of the 4-UPS-RPS spatial parallel mechanism under no-load condition, as acquired by the simulation of ADAMS2010, is shown in Figure 3(b). When the moving platform is under load with force (unit: N) ${}^{B}F_{B} = {[\begin{matrix} 104 & - 124 & - 279 \end{matrix}]}^{T}$ and with moment (unit: Nm) ${}^{B}M_{B} = {[\begin{matrix} - 1.24 & - 124 & - 60 \end{matrix}]}^{T}$ , the driving force of the 4-UPS-RPS spatial parallel mechanism under loading condition, as acquired by numerical calculation, is shown in Figure 4(a), and the driving force of the 4-UPS-RPS spatial parallel mechanism under loading condition, as acquired by the simulation of ADAMS2010, is shown in Figure 4(b).

Figure 3.

Five driving forces of spatial parallel mechanism under no-load condition

Figure 4.

Five driving forces of spatial parallel mechanism under loading condition

According to Figures 3(a), 3(b), 4(a) and 4(b), the theoretical calculation results obtained by Matlab7.0 are basically consistent with the simulation results obtained by ADAMS2010; therefore, the correctness of the theoretical calculation and the virtual simulation is verified.

6. Conclusion

The kinematics of the RPS and UPS driving limbs of the 4-UPS-RPS spatial parallel mechanism were analysed and the relationships between the velocity of the parts and the velocity of the driving limbs were established.

The rigid dynamic equation of the 4-UPS-RPS spatial parallel mechanism was established.

The dynamic equation was analysed by numerical calculation and virtual simulation, and its correctness verified.

Footnotes

7.

This research is supported by the National Natural Science Foundation of China (grant no. 51005138, 51375282 11272167), the Shandong Young Scientists Award Fund (grant no. BS2012ZZ008), the Programme for Changjiang Scholars and Innovative Research Teams in Universities (IRT1266), Taishan Scholarship Project of Shandong Province (No. tshw20130956), the Science Foundation of SUST (grant no. 2011KYJQ102), and the project of Jiangsu Key Laboratory of Digital Manufacturing Technology (grant no. HGDML-1104).

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