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Abstract

A new type of parallel robot ROBO_003 is presented. Its mechanisms, kinematics, and virtual prototype technology are introduced. The research of degrees of freedom (DOF) is based on screw theory, a set of screw is separated as a branch, which named as constrain screw. The type of three DOF gained by counting constrain screw, the moving platform’s frame, and base platform’s frame is set, respectively, a complete kinematic research including closed-form solutions for direct kinematic problem. The 3-D model of ROBO_003 is established using SOLIDWORKS; position and orientation of motion platform can be gained using ADMAS, which is a type of virtual prototype technology. The resultant shows that the structure of ROBO_003 is reasonable, three DOF of motion platform can be operated in a reasonable range, the solutions to the direct kinematics are right, and robot ROBO_003 can be used in many industrial fields. The research of this article provides a basis for the practical application of parallel robotics ROBO_003.

Keywords

ADAMS ROBO_003 screw theory virtual prototype technology

Introduction

Since the advent of robots, its development prospects have become wider and wider, especially in many industrial fields, which greatly improve labor productivity and product quality, reduce production costs, reduce labor intensity, and improve labor conditions.

Most of the traditional robotic manipulators are made up of joints in series, which are base, waist, upper arm, lower arm, wrist, and end-effector, so they are called serial robots. This kind of robot has the advantages of wide working range and flexible movement. However, if degree of freedom (DOF) is more and the arm is longer, the stiffness is poor and the bearing capacity is low, so there is a phenomenon of error accumulation and amplification. Therefore, people need to study a new type of robot with high position accuracy, good stiffness, strong bearing capacity, and simple structure.

Parallel robot was first proposed by Stewart in 1965, it is called Stewart mechanism.¹ It consists of a moving platform, a fixed platform, and six parallel branches with two platforms. Each branch can be retracted. The two ends of the branch are connected with two platforms by spherical or Hooke hinges. When the length of each branch changes, the moving platform can obtain three DOFs (Figure 1).

Figure 1.

Stewart platform.

In 1998, Professor Joachim in Germany designed a six-DOF parallel robot named triplanar (Figure 2). Triplanar is driven by three 2-DOF planar motors, ROBO_003 is the first model derived from triplanar since its birth (Figure 3). Its design conception comes from triplanar parallel robot and has its own obvious characteristics.²

Figure 2.

ROBO_003.

Figure 3.

Triplanar.

Three 2-DOF planar motors of triplanar parallel robot are replaced by three rotating discs around fixed axes in ROBO_003, which is equivalent to restricting two-DOF free motion of three planar motors to one-DOF along fixed circumference.

ROBO_003 is a new type of parallel robot with three DOF, so it can also be called pure rotation parallel robot. With its unique structural advantages, ROBO_003 parallel mechanism has great application prospects in many fields, such as satellite antenna, radar, photovoltaic tracker, and so on. However, there is almost no systematic research on the mechanism of this kind of parallel robot, so its mathematical modeling and simulation is a very innovative and challenging work, as well as laying a solid theoretical foundation for its practical application, which is of great significance.

In this article, the screw method is used to analyze the number and form of the DOF of the mechanism, the virtual prototype technology of SOLIDWORKS and ADMAS is used to carry out kinematics modeling and simulation. The theoretical analysis and software simulation of ROBO_003 parallel robot are also an innovative work, which has important practical significance.

Analysis of DoF form

Screw theory

Among many mathematical methods for analyzing space mechanisms, screw is a very effective tool.³ Firstly, it integrates six scalars, or two 3-D vectors, so that a screw can represent the direction and position of vectors at the same time, the velocity and angular velocity in rigid body motion, and the force and torque in rigid body mechanics. Therefore, screw with six scales is very useful for kinematics and dynamics analysis of spatial mechanisms. A vector in space is constrained on a line whose position is determined, the vector constrained by the line is called a line vector, which is expressed by a dual vector $(s; s_{0})$ , where s is the primitive part of the dual vector and s ₀ is the dual part of the dual vector. s is a space vector, $s_{0} = r \times s$ is the line distance of the space vector, having a unit of length, r is the radius from the origin to any point on the line in space; when $s \cdot s = 1$ , it is a unit line vector, s ₀ represents the position of the line vector in space, as $s \cdot s_{0} = 0$ , it is the characteristic of a line vector⁴ according to the definition of s ₀. If s is the angular velocity that an object rotates around an axis, s ₀ is defined as line velocity. The primitive and dual parts of dual vectors are not orthogonal in most cases. The nonorthogonal dual vectors are called screw, it is defined as $(s; s_{0})$ , $s \cdot s_{0} \neq 0$ if $s \cdot s = 1$ , it is a unit screw, $s \cdot s_{0} / s \cdot s = h$ is the pitch of screw. It has a unit of length if $h = 0$ , it is called line vector; if $h = \infty$ , it is called dual vector, which is defined as $(0; s)$ ; a screw can be transformed as follows: $(s; s_{0}) = (s; s_{0} - h s + h s) = (s; s + h s)$ ; a line position can be gained as $r \times s = s_{0} - h s$ , in which axis position of the Screw can be written as $(s; s_{0} - h s)$ ; the two vectors can be integrated as six scalars, and the six scalars is called Plucker coordinate. According to the above definition, any form of motion can be regarded as screw motion. If a screw motion is pure rotation, we call it velocity screw, whose pitch is zero, and the axis coincides with the rotating shaft of the rotating pair. A pure moving motion screw is a velocity screw with infinite pitch and straight line direction along the moving pair. Compound motion is a linear combination of the above two forms. Motion screw, also known as unit velocity screw, written as $s$ , its Plucker coordinate, has the form as following

s = [L M N; P Q R]

where L, M, and N can be expressed as unit angular velocity; P, Q, and R can be expressed as unit line velocity. Corresponding to r, $s$ is defined as constrain Screw as following

r s = [r L r M r N; r P r Q r R]

The first three components represent the principal vector of the force screw, and the last three components represent the principal moment of the force screw. For two screws, if the following formula is satisfied, they are called reciprocal screws

s \circ s^{r} = 0

According to the definition of reciprocal screw⁵, motion screw and force screw are a pair of reciprocal screw, and the sum of virtual power of external force system to stable rigid body is constant to zero at any time.

For an object connected to a fixed base by a pure series kinematic chain with n (n > 0) DOFs, its motion screw is the linear combination of the motion screw of all the motion pairs in the kinematic chain, which is expressed as follows

s = \sum_{i = 1}^{n} k_{i} s_{i}

where $s_{i}$ is the number i motion screw of the kinematic chain. Plucker coordinate can be expressed as follows

\begin{array}{l} s_{i} = (L_{i}, M_{i}, N_{i}; P_{i}, Q_{i}, R_{i}); \\ s = K J; \\ K = [k_{1}, k_{2}, \dots, k_{n}]; \\ J = [\begin{matrix} L_{1} & M_{1} & N_{1} & P_{1} & Q_{1} & R_{1} \\ L_{} & M_{2} & N_{2} & P_{2} & Q_{2} & R_{2} \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ L_{n} & M_{n} & N_{n} & P_{n} & Q_{n} & R_{n} \end{matrix}] \end{array}

where J is motion screw system of the kinematic chain, According to reciprocal screw theory, the anti-screw $s^{r}$ is the constraint anti-Screw $s$ of the kinematic chain on the object connected to its terminal, which is called constraint terminal.^6
–8 As $s \circ s^{r} = 0$ , so: $K J \circ s^{r} = 0$

K can take any value. Therefore, the necessary and sufficient condition for the establishment of the above formula is

J \circ s^{r} = 0

In formula (6)

s^{r} = [P^{r}; Q^{r}; R^{r}; L^{r}; M^{r}; N^{r}]

Therefore, for any given pure series kinematic chain, the terminal constraint (anti-Screw) can be obtained by this formula, and the terminal constraint is known. The unconstrained motion is the DOF of the terminal.

DOF analysis of ROBO_003

Structural diagram of ROBO_003, as shown in Figure 4, one of branched chain $P_{1} G_{1} Q_{1}$ of ROBO_003 was chosen, in this branched chain, P ₁ is a rotating pair, G ₁ is a spherical pair, Q ₁ is a rotating pair, coordinate system $P_{1} - x_{1} y_{1} z_{1}$ was established, as shown in Figure 5, the rotating joint in P ₁, its screw:

s_{_{1}} = (0 0 1; 0 0 0)

Figure 4.

ROBO_003 structural diagram.

Figure 5.

(a) ROBO_003 coordinate system and (b) the diagram of joint screw.

Spherical pairs in G ₁ can be decomposed into three orthogonal rotating pairs.^9,10 And they intersect the center of rotation of the spherical pair, the coordinate of the center is $(a b 0)$ , the screw of the three rotating pairs, receptively, separately is

s_{2} = (1 0 0; 0 0 - b)

s_{3} = (0 1 0; 0 0 a)

s_{4} = (0 0 1; b - a 0)

The direction and vector diameter of the axis of the rotating pair in Q ₁ cannot be determined; so, its Screw can be set as follows:

s_{5} = (d e f; g h i)

The motion Screw of $P_{1} G_{1} Q_{1}$ in the parallel robot ROBO_003 can be set as follows:

s = (s_{1} s_{2} s_{3} s_{4} s_{5})

It is expressed as the matrix:

s = [\begin{matrix} 0 & 1 & 0 & 0 & d \\ 0 & 0 & 1 & 0 & e \\ 1 & 0 & 0 & 1 & f \\ 0 & 0 & 0 & b & g \\ 0 & 0 & 0 & - a & h \\ 0 & - b & a & 0 & i \end{matrix}]

The rank of the matrix $s$ is 5, the five motion screw of the first branched chain is linearly independent. According to the reciprocal screw theory, when the rank of the screw system composed of all the kinematic pairs in the kinematic chain is less than 6, there exists an anti-screw, which is reciprocal to each screw in the screw system.^11,12 If the rank of the screw is W, the number of anti-screw is 6-W, it is an anti-screw system that has 6-W screws, there is only one basic screw in the constrained screw system of the parallel robot ROBO_003, it can be set as follows

s^{r} = [x 1 x 2 x 3; x 4 x 5 x 6]

According to the relation between the screw system, it is an anti-screw system, five equations are obtained as follows

s_{1} \circ s^{r} = 0

s_{2} \circ s^{r} = 0

s_{3} \circ s^{r} = 0

s_{4} \circ s^{r} = 0

s_{5} \circ s^{r} = 0

By solving the above equations, we finally get a group of equation solutions

\begin{array}{l} x 1 = 1 \\ x 2 = \frac{b}{a} \\ x 3 = \frac{g + \frac{b}{a} h}{a e - b d - i} \\ x 4 = \frac{b g + \frac{b^{2}}{a} h}{a e - b d - i} \\ x 5 = - \frac{a g + b h}{a e - b d - i} \\ x 6 = 0 \end{array}

After calculation, we get the equation:

x 1 \cdot x 4 + x 2 \cdot x 5 + x 3 \cdot x 6 = 0

We can conclude from the resultant that the anti-screw of the motion screw is a line vector, the anti-screw has a meaning of force, and its direction vector is $(x 1, x 2, x 3)$ . Similarly, the constraints of the other two branched chains of ROBO_003 on the moving platform are force vectors, it is proved that the three force vectors are linearly independent. Therefore, the three branches restrict the three DOF of the moving platform. For the spatial mechanism, only three DOF of rotation are unconstrained. Therefore, ROBO_003 is a pure rotating mechanism with three DOF, that is, rotating along the x, y, and z axes, respectively.

Kinematics

Kinematics research only considers motion condition, not force condition. Position analysis is the most basic task of ROBO_003 kinematics analysis. Its essence is to solve the position relationship between input and output components of mechanism. It can be divided into two subproblems: the forward and inverse position solutions.^6,7 The former involves the known joint variables to solve the operation variables, and the latter involves the known operation variables to solve the joint variables. For parallel mechanisms, the process of calculating all possible motion parameters of the driving joint by giving the position and attitude parameters of the moving platform relative to the static platform is the inverse position solution; and the known driving joint is the inverse position solution. The forward position solution is the process of solving the position and posture of all possible moving platforms relative to the basic platforms.

Establishment of coordinate system

As shown in Figure 5(a) and (b), two equilateral triangles represent the base and the moving platform, respectively. $O_{i} (i = 1, 2, 3)$ represent three vertices of the base triangle, $Q_{i} (i = 1, 2, 3)$ represent the midpoint of the three sides of the moving platform triangle, the axis of the rotating pair between the moving platform, and the connecting rod coincides with the three sides of the triangle of the moving platform. $P_{i} (i = 1, 2, 3)$ is on the drive turntable shaft, $G_{i} (i = 1, 2, 3)$ represent the center of spherical pair, $P_{1}, P_{2}, P_{3}, G_{1}, G_{2}, G_{3}$ locate in the same plane, inertial frame $O - x y z$ is established in the center point O of base platform, it is also called generalized frame, x-axis points O to O ₁, the z-axis is perpendicular to the base platform. The Y-axis conforms to the right hand, the z-axis is perpendicular to the base platform. The Y-axis conforms to the right-hand rule. Inertial frame $P_{i} - x_{i} y_{i} z_{i} (i = 1, 2, 3)$ is fixed on the input joint, the x_i -axis is in the same direction as the $\bar{O O_{i}}$ -axis, z_i -axis is rotating along revolving shaft, the y_i -axis conforms to the right-hand rule to describe the orientation of moving platform, moving frame $C - u v w (i = 1, 2, 3)$ is established, moving frame $C - u v w$ is fixed on moving platform, the origin point C is in the center, u-axis points from C to Q ₁,w-axis is perpendicular to the moving platform,and v-axis conforms to the right-hand rule. Moving frame $Q_{i} - u_{i} v_{i} w_{i} (i = 1, 2, 3)$ is fixed on the output joint, u_i -axis is in the same direction as $\bar{C Q_{i}}$ , v_i -axis is rotating along revolving shaft, w_i -axis conforms to the right-hand rule.

The input joint refers to the rotating pair between the base and the turntable, and the output joint refers to the rotating pair between the rod and the moving platform. The input and output refer to the input and output of motion. From the point of view of motion, the input joint is the driving joint and the output joint is the guiding joint, while the joint between the two, for example, the spherical hinges or pairs between the turntable and the rods are all transmission joints.

Quantitative analysis of motion requires setting motion parameters.⁸ Each moving joint transmits the corresponding motion parameters, in parallel mechanism ROBO_003, $β_{i} (i = 1, 2, 3)$ represents the input angle of no. i drive turntable, that is, quantitative number of input joint, $α_{i} (i = 1, 2, 3)$ , is counterclockwise angle from $\bar{Q G_{i}}$ to $\bar{C Q_{i}}$ , that is, output angle, that is, quantitative parameters representing output joint motion.

Position and orientation describing of moving platform

Position of moving platform

The position of the moving platform can be represented by the position vector of the center point C of the moving platform, the coordinate of point C, which relative to the base frame $O - x y z$ is $(X_{C}, Y_{C}, Z_{C})$ , the vector of C is expressed as follows

C = \bar{O C} = {[X_{C}, Y_{C}, Z_{C}]}^{T}

Orientation of moving platform

For orientation of moving platform, we describe it by rotational transformation of the moving frame $C - u v w$ relative to the inertial frame $O - x y z$ ^13
–15. Rotation transformation of moving frame relative to inertial frame: firstly, the direction of moving frame is in the coincidence of nertial frame, the moving frame rotates around x-axis, the rotating angle is ψ, then rotates around y-axis, the rotating angle is θ,then rotates around z-axis, the rotating angle is γ, we get the final frame $C - u v w$ . Let us multiply the basic rotation matrix $R_{X} (ψ)$ , $R_{Y} (θ)$ , $R_{Z} (γ)$ by left, in turn, the rotation transformation matrix $_{C}^{O} R$ can be obtained by moving frame $C - u v w$ rotating relative to inertial frame $O - x y z$ :

\begin{array}{l} _{C}^{O} R = R_{Z} (γ) * R_{Y} (θ) * R_{X} (ψ) \\ = [\begin{matrix} cos γ & - sin γ & 0 \\ sin γ & cos γ & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} cos θ & 0 & sin θ \\ 0 & 1 & 0 \\ - sin θ & 0 & cos θ \end{matrix}] \\ × [\begin{matrix} 1 & 0 & 0 \\ 0 & cos ψ & - sin ψ \\ 0 & sin ψ & cos ψ \end{matrix}] \\ = [\begin{matrix} c_{γ} c_{θ} & c_{γ} s_{θ} s_{ψ} - s_{γ} c_{ψ} & c_{γ} s_{θ} c_{ψ} + s_{γ} s_{ψ} \\ s_{γ} c_{θ} & s_{γ} s_{θ} s_{ψ} + c_{γ} c_{ψ} & s_{γ} s_{θ} c_{ψ} - c_{γ} s_{ψ} \\ - s_{θ} & c_{θ} s_{ψ} & c_{θ} c_{ψ} \end{matrix}] \end{array}

The three angles ψ,θ,γ are called RPY angle, they are called roll, pitch, yaw angle, respectively,⁹ which is the rotating angle of moving frame rotating around x-axis, y-axis, and z-axis.

Solutions for direct kinematic problem

Analytic method for close-form solutions

The mechanism parameters and input parameters of ROBO_003 parallel robot can be used to get the solution equation of moving frame, the closed solution is obtained, the mechanism parameters of ROBO_003 are provided in Table 1.

Table 1.

Parallel mechanism parameters.

Parameter	$\|O O_{i}\|$	$\|O_{i} P_{i}\|$	$\|P_{i} G_{i}\|$	$\|G_{i} Q_{i}\|$	$\|C Q_{i}\|$
Length	R	D	l	L	r

The kinematics of ROBO_003 studies the relationship between the input angle and the position and orientation of the moving platform. The input angle is the input parameter $β_{1, i} (i = 1, 2, 3)$ , and the position and orientation of the moving platform are the output parameter ( ${[X_{C}, Y_{C}, Z_{C}]}^{T}$ , $_{C}^{O} R$ ), as provided in Table 2.

Table 2.

Input parameters and output parameters.

Input parameters	Output parameters
$β_{1, i} (i = 1, 2, 3)$	${[X_{C}, Y_{C}, Z_{C}]}^{T}$	$_{C}^{O} R$

The selection of reference frame will not affect the distance between two points. Obviously, the distance between the center of the spherical hinge G_i and $G_{i + 1}$ is equal in the base frame and the moving frame, so there are the following constraint equations:^16,17

|G_{i + 1} - G_{i}| = |^{C} G_{i + 1} -^{C} G_{i}| (i = 1, 2, 3)

The position vector of G_i , that is, in frame $C - u v w$ is $^{C} G_{i}$ ; $λ_{i}$ is the angle between $\bar{O O_{i}}$ and $\bar{O x}$ , $λ_{i}$ reflects the position of the frame $P_{i} - x_{i} y_{i} z_{i}$ relative to the base frame $O - x y z$ , it can be obtained from the structure:

λ_{i} = (i - 1) \frac{2 π}{3}

Setting the initial state of ROBO_003 parallel robot motion, point G_i is on the x_i -axis of inertial frame $P_{i} - x_{i} y_{i} z_{i}$ , $_{P_{i}}^{O} T$ is the transformation matrix from frame $P_{i} - x_{i} y_{i} z_{i}$ to frame:

\begin{array}{l} _{P_{i}}^{O} T = [\begin{matrix} cos λ_{i} & - sin λ_{i} & 0 & R cos λ_{i} \\ sin λ_{i} & cos λ_{i} & 0 & R sin λ_{i} \\ 0 & 0 & 1 & d \\ 0 & 0 & 0 & 1 \end{matrix}], (i = 1, 2, 3) \end{array}

Setting $ϕ_{_{i}}$ is the angle between $\bar{P_{i} G_{i}}$ and $\bar{P_{i} x_{i}}$ . That is, the input angle of the drive turntable, $α_{i}$ , is the counterclockwise from $\bar{Q_{i} G_{i}}$ to $\bar{C Q_{i}}$ , that is, the output angle of platform position. The position vector of G_i point in the frame^18,19 $P_{i} - x_{i} y_{i} z_{i}$ is $^{P_{i}} G_{i}$ :

^{P_{i}} G_{i} = {[l cos ϕ_{i} l sin ϕ_{i} 0 1]}^{T}

The position vector of $G_{i} {[X_{G_{i}} Y_{G_{i}} Z_{G_{i}}]}^{T}$ , that is, in base frame can be obtained as in the following equation

G_{i} =_{P_{i}}^{O} T^{P_{i}} G_{i} = {[l cos (ϕ_{i} + λ_{i}) + R cos λ_{i} l sin (ϕ_{i} + λ_{i}) + R sin λ_{i} d]}^{T}

$β_{i} = φ_{i} + λ_{i}$ , the upper formula can be written as follows

G_{i} =_{P_{i}}^{O} T^{P_{i}} G_{i} = {[l cos β_{i} + R cos λ_{i} l sin β_{i} + R sin λ_{i} d]}^{T}

where $φ_{i}$ is the intersection angle of $\bar{C Q_{i}}$ and $\bar{C u}$ , $φ_{i}$ reflects the position of the frame $Q_{i} - u_{i} v_{i} w_{i}$ relative to the moving frame $C - u v w$ , we get

φ_{i} = (i - 1) \frac{2 π}{3}

$_{Q_{i}}^{C} T$ is the transformation matrix from frame $Q_{i} - u_{i} v_{i} w_{i}$ to frame $C - u v w$ , we get

_{Q_{i}}^{C} T = [\begin{matrix} cos φ_{i} & - sin φ_{i} & 0 & r cos φ_{i} \\ sin φ_{i} & cos φ_{i} & 0 & r sin φ_{i} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

The position vector of G_i in the frame $Q_{i} - u_{i} v_{i} w_{i}$ is $^{Q_{i}} G_{i}$ ,we get

^{Q_{i}} G_{i} = {[L cos α_{i} 0 - L sin α_{i} 1]}^{T}

The position vector of theG_i point in the frame $C - u v w$ can be obtained by transformation of the frame as follows^20
–22

^{C} G_{i} =_{Q_{i}}^{C} T^{Q_{i}} G_{i} = {[(r + L cos α_{i}) cos φ_{i} (r + L cos α_{i}) sin φ_{i} - L sin α_{i}]}^{T}

Setting m_i is the distance from G_i to $G_{i + 1}$ , that is, $m_{i} = |G_{i} G_{i + 1}|$ , so $m_{i}^{2} = {(X_{G_{i}} - X_{G_{i + 1}})}^{2} + {(Y_{G_{i}} - Y_{G_{i + 1}})}^{2} + {(Z_{G_{i}} - Z_{G_{i + 1}})}^{2}$ , when $ϕ_{i} (i = 1, 2, 3)$ is known, the coordinates of point G_i and $G_{i + 1}$ can be obtained from the formulas, so $m_{i}^{2}$ can be obtained; the formula $|G_{i + 1} - G_{i}| = |^{C} G_{i + 1} -^{C} G_{i}| (i = 1, 2, 3)$ is square, the left of the formula is $m_{i}^{2}$ . By substituting the coordinate expression of $^{C} G_{i}$ into the above formula, the following results can be obtained:

\begin{array}{l} A_{i} s α_{i} s α_{i + 1} + B_{i} c α_{i} c α_{i + 1} + C_{i} c α_{i} + C_{i} c α_{i + 1} + D_{i} = 0 \\ A_{i} = - 2 L^{2}; B_{i} = L^{2}; C_{i} = 3 L r; \\ D_{i} = 2 L^{2} + 3 r^{2} - m_{i}^{2} \end{array}

Therefore, the forward position analysis of ROBO_003 parallel mechanism can be reduced to the closed solution of the upper form.

Let $t_{i} = tan \frac{α_{i}}{2}$ , $cos α_{i} = \frac{1 - t_{i}^{2}}{1 + t_{i}^{2}}$ , and $sin α_{i} = \frac{2 t_{i}}{1 + t_{i}^{2}}$ .

Substitute the upper form and get it:

E_{i} t_{i}^{2} t_{i + 1}^{2} + F_{i} t_{i}^{2} + F_{i} t_{i + 1}^{2} + I_{i} t_{i} t_{i + 1} + J_{i} = 0 in the formula:

\begin{array}{l} E_{i} = 3 L^{2} + 3 r^{2} - 6 L r - m_{i}^{2} \\ F_{i} = L^{2} + 3 r^{2} - m_{i}^{2} \\ I_{i} = - 8 L^{2} \\ J_{i} = 3 L^{2} + 3 r^{2} + 6 L r - m_{i}^{2} \end{array}

When i = 1, 2, 3, the above formula is equivalent to the following equations:

\begin{array}{l} (E_{1} t_{1}^{2} + F_{1}) t_{2}^{2} + (I_{1} t_{1}) t_{2} + (F_{1} t_{1}^{2} + J_{1}) = 0 \\ (E_{2} t_{3}^{2} + F_{2}) t_{2}^{2} + (I_{2} t_{3}) t_{2} + (F_{2} t_{3}^{2} + J_{2}) = 0 \\ (E_{3} t_{1}^{2} + F_{3}) t_{3}^{2} + (I_{3} t_{1}) t_{3} + (F_{3} t_{1}^{2} + J_{3}) = 0 \end{array}

Let

\begin{array}{l} K_{1} = E_{1} t_{1}^{2} + F_{1}; M_{1} = I_{1} t_{1}; N_{1} = F_{1} t_{1}^{2} + J_{1} \\ K_{2} = E_{2} t_{3}^{2} + F_{2}; M_{2} = I_{2} t_{3}; N_{2} = F_{2} t_{3}^{2} + J_{2} \\ K_{3} = E_{3} t_{1}^{2} + F_{3}; M_{3} = I_{3} t_{1}; N_{3} = F_{3} t_{1}^{2} + J_{3} \end{array}

After simplification

\begin{array}{l} K_{1} t_{2}^{2} + M_{1} t_{2} + N_{1} = 0 \\ K_{2} t_{2}^{2} + M_{2} t_{2} + N_{2} = 0 \\ K_{3} t_{3}^{2} + M_{3} t_{3} + N_{3} = 0 \end{array}

To eliminate variables t ₂, the first two equations are multiplied by t ₂, two new equations are obtained. Consider the following two equations

\begin{array}{l} K_{1} t_{2}^{2} + M_{1} t_{2} + N_{1} = 0 \\ K_{1} t_{2}^{3} + M_{1} t_{2}^{2} + N_{1} t_{2} = 0 \\ K_{2} t_{2}^{2} + M_{2} t_{2} + N_{2} = 0 \\ K_{2} t_{2}^{3} + M_{2} t_{2}^{2} + N_{2} t_{2} = 0 \end{array}

The above equations are written in the form of matrices:

[\begin{matrix} 0 & K_{1} & M_{1} & N_{1} \\ K_{1} & M_{1} & N_{1} & 0 \\ 0 & K_{2} & M_{2} & N_{2} \\ K_{2} & M_{2} & N_{2} & 0 \end{matrix}] [\begin{matrix} t_{2}^{3} \\ t_{2}^{2} \\ t_{2} \\ 1 \end{matrix}] = 0

Only if the determinant of the fourth-order coefficient matrix is equal to 0, as following:

|\begin{matrix} 0 & K_{1} & M_{1} & N_{1} \\ K_{1} & M_{1} & N_{1} & 0 \\ 0 & K_{2} & M_{2} & N_{2} \\ K_{2} & M_{2} & N_{2} & 0 \end{matrix}| = 0

Expanded

\begin{array}{l} - K_{2} M_{1} M_{2} N_{1} + K_{1} M_{2}^{2} N_{1} + K_{2}^{2} N_{1}^{2} + K_{2} M_{1}^{2} N_{2} \\ - K_{1} M_{1} M_{2} N_{2} - 2 K_{1} K_{2} N_{1} N_{2} + K_{1}^{2} N_{2}^{2} = 0 \end{array}

Substitute $K_{1}, M_{1}, N_{1}, K_{2}, M_{2}, N_{2}$ into the upper form and simplify it

U_{1} t_{3}^{4} + U_{2} t_{3}^{3} + U_{3} t_{3}^{2} + U_{4} t_{3} + U_{5} = 0

To eliminate the variables t ₃,combine the above two formulas. Let the following six-order coefficient row number equals 0

|\begin{matrix} 0 & U_{1} & U_{2} & U_{3} & U_{4} & U_{5} \\ U_{1} & U_{2} & U_{3} & U_{4} & U_{5} & 0 \\ 0 & 0 & 0 & K_{3} & M_{3} & N_{3} \\ 0 & 0 & K_{3} & M_{3} & N_{3} & 0 \\ 0 & K_{3} & M_{3} & N_{3} & 0 & 0 \\ K_{3} & M_{3} & N_{3} & 0 & 0 & 0 \end{matrix}| = 0

The 16-degree polynomial equation of t ₁ obtained by expanding and merging the same terms

\begin{array}{l} T_{1} t_{1}^{16} + T_{2} t_{1}^{14} + T_{3} t_{1}^{12} + T_{4} t_{1}^{10} + T_{5} t_{1}^{8} \\ + T_{6} t_{1}^{6} + T_{7} t_{1}^{4} + T_{8} t_{1}^{2} + T_{9} = 0 \end{array}

where $T_{1} \sim T_{9}$ are functions of $ϕ_{i}, λ_{i}$ and structural parameters $R, l, r, L$ , formula (42) is eighth-order equation of $t_{1}^{2}$ . There are up to 16 real solutions of t ₁, all of which appear in pairs t ₁.

Numerical examples

The structural parameters of parallel mechanism ROBO_003 are provided in Table 3 (unit: meter).Set input angle

Table 3.

The value of structure parameter.

R	R	L	L	d	$λ_{1}$	$λ_{2}$	$λ_{3}$
0.7	0.3109	0.3	0.9290	0.36	0	$2 π / 3$	$4 π / 3$

φ_{1} = \frac{π}{4}; φ_{2} = \frac{π}{6}; φ_{3} = - \frac{π}{2}

Calculating forward kinematics solution based on MATLAB, obtaining the three points’ coordinate of G ₁, G ₂, and G ₃ respectively

\begin{array}{l} G_{1} = {[0.9121320343559 0.2121320343559 0.36]}^{T} \\ G_{2} = {[- 0.60980762113 0.7562177826491 0.36]}^{T} \\ G_{3} = {[- 0.6098076211353 - 0.4562177826491 0.36]}^{T} \end{array}

The 16th order polynomial equation is as follows:

\begin{array}{l} - 3.941311223 \times 10^{4} t_{1}^{16} + 1.395344728 \times 10^{5} t_{1}^{14} \\ + 6.725185981 \times 10^{5} t_{1}^{12} - 1.359223050 \times 10^{6} t_{1}^{10} \\ - 3.824110543 \times 10^{6} t_{1}^{8} + 2.010143645 \times 10^{6} t_{1}^{6} \\ + 6.462224314 \times 10^{6} t_{1}^{4} + 1.870048120 \times 10^{6} t_{1}^{2} \\ - 1.564682648 \times 10^{5} = 0 \end{array}

The eight real roots of t ₁ are obtained by solving the above equation. The corresponding value of $α_{1}$ is

\begin{array}{l} \pm 2.294124417; \pm 2.075031481; \\ \pm 1.827233337; \pm 0.5087138979 \end{array}

Two sets of solutions of ${X_{C}, Y_{C}, Z_{C}}$ can be obtained from each group of ${α_{1}, α_{2}, α_{3}}$ , eight sets of solutions of ${X_{C}, Y_{C}, Z_{C}}$ can be obtained from four group of ${α_{1}, α_{2}, α_{3}}$ , the eight sets of solutions of ${X_{C}, Y_{C}, Z_{C}}$ are corresponding to eight platform positions, and eight sets of real number solutions of ${X_{C}, Y_{C}, Z_{C}}$ are provided in Table 4.

Table 4.

Real number solutions of moving platform.

	1	2	3	4
X_C	0.2365084	0.23650842	0.18229494	0.18229494
Y_C	0.02301114	0.02301114	−0.17201958	−0.17201958
Z_C	0.06823364	0.65176635	0.34051953	0.37948046
	5	6	7	8
X_C	0.09371023	0.09371023	−0.05540199	−0.05540199
Y_C	0.54190801	0.54190801	0.16787604	0.16787604
Z_C	0.17393507	0.54606492	−0.36534417	1.08534417

The corresponding four rotation matrices are obtained as follows

^{O} R_{C} (1) = [\begin{matrix} - 0.7633 & - 0.3473 & - 0.9765 \\ - 0.2814 & 0.9455 & - 0.1818 \\ - 1.2072 & - 0.2544 & 0.0101 \end{matrix}]

^{O} R_{C} (2) = [\begin{matrix} - 7.8369 & - 5.2066 & 0.2076 \\ 6.9776 & 5.3107 & - 1.4441 \\ - 15.6863 & - 9.5149 & 2.2175 \end{matrix}]

^{O} R_{C} (3) = [\begin{matrix} 0.5549 & 0.0869 & - 0.9685 \\ 2.8340 & - 1.2046 & 0.0013 \\ 1.4639 & - 1.2341 & - 0.2779 \end{matrix}]

^{O} R_{C} (4) = [\begin{matrix} 0.9463 & - 0.0184 & 0.3314 \\ 0.0882 & 0.9915 & - 0.2500 \\ 0.1275 & - 0.0163 & - 0.4793 \end{matrix}]

The above results are the forward kinematics solutions of parallel mechanisms of ROBO_003.

ADMAS simulation

Introduction of ADMAS

Automatic dynamic analysis of mechanical systems (ADAMAS) is a virtual prototype analysis software developed by American MDI Company.²³ The start-up interface of ADMAS is shown in Figure 7. The simulation steps of ADAMS are provided in Table 5.

Figure 6.

The 3-D assembly model of ROBO_003.

Figure 7.

Effect drawing of adding constraints and motions.

Table 5.

Simulation step of ADMAS.

	Simulation step
1	Selecting the location of center of mass, selecting materials
2	Building maker point
3	Building joint constrain
4	Adding motion at each joint
5	Simulation of kinematics
6	Simulation of dynamics
7	Adding force or torque at each joint
8	Follow-up processing
8.1	Building measuring point
8.2	Loading simulation curve
8.3	Measurement of displacement, angle, velocity, and angular velocity

Kinematics simulation of ROBO_003 based on ADMAS

Before simulation, CAD software Solidworks is used to build 3-D assembly drawing of ROBO_003,²⁴ as shown in Figure 6.

The assembly drawings are established in Solidworks, the parts of the 3-D drawings are saved in the form of suffix x_t. Import ADMAS and reassemble in ADMAS.²⁵ The assembled model establishes the simulation model in ADMAS interface according to the simulation steps in Table 5. The simulation time and steps are set up. The results are shown in Figure 7.

Simulation results

Centroid position change curve of moving platform

Results and discussion

In this article, a new type of parallel mechanism ROBO_003 was presented, it is DOF form, kinematics, and virtual prototype technology have been studied.

The DOF of parallel mechanism is based on screw theory by choosing the joint motion screw of three branches of parallel mechanism, the screw system of each joint is obtained, and the constraint screw of each branch chain is obtained by matrix calculation. The constraint screw of the screw system consisting of the constraint screw of each branch chain is the motion screw of the moving platform. From the calculation results of formulas (1) to (10), it can be concluded that the motion form of ROBO_003 parallel mechanism moving platform is three DOF rotation, that is, rotation around X, Y, and Z axes, respectively (Figures 8 and 9).

Figure 8.

Centroid displacement changing curve (x-axis).

Figure 9.

Centroid displacement changing curve (y-axis).

For the kinematics research, first, we established the frame of the moving platform and the base platform, then the frame of each joint was established, choosing the rotation angle of the moving platform frame relative to the base platform frame respectively, they are roll, pitch, yaw angle, the rotation matrix is obtained using homogeneous matrix transform to obtain position and orientation of moving platform relative to base platform, choosing key points on moving platform and base platform, respectively, and setting constraint equation. After eliminating, the 16-degree equation is simplified, the closed solution is obtained by analytic method, the 16-degree equation is solved by programming, and 8 pairs of positive position solutions are obtained. After choosing or rejecting, the final position and orientation of the moving platform relative to the base platform are obtained.

ADMAS virtual prototyping technology is a simulation technology for kinematics and dynamics of robots. It can correctly analyze the kinematics of ROBO_003 parallel mechanism. According to theoretical calculation and ADMAS virtual prototyping technology, Figures 10 to 12 are displacement curves of the center of mass of moving platform in three directions of X, Y, and Z axes. Figures 13 to 15 are velocity curves of moving platform centroid in three directions of X, Y, and Z axes. Figures 14 and 15 are angular velocity and acceleration curves of moving platform relative to base platform orientation. The results show that the simulation results coincide with the actual theoretical calculation.

Figure 10.

Centroid displacement changing curve (z-axis).

Figure 11.

Centroid velocity changing curve (x-axis).

Figure 12.

Centroid velocity changing curve (y-axis).

Figure 13.

Centroid velocity changing curve (z-axis).

Figure 14.

Angular velocity changing curve of moving platform.

Figure 15.

Angular acceleration changing curve of moving platform.

Conclusions

In this article, the screw theory is used to study the DOF form of ROBO_003 parallel mechanism, the closed solution of forward kinematics is studied by analytical method, and the forward kinematics solution is obtained by simulation with ADMAS software. The results accord with the theoretical calculation value. The research results provide a theoretical basis for the practical application of ROBO_003 parallel mechanism.

For complex parallel mechanisms, the screw analysis method proposed in this article can easily solve the analysis of the DOF form of parallel mechanisms. It is concluded that the DOF form of ROBO_003 parallel mechanisms is three DOF rotation. The ADMAS virtual prototype can easily obtain the forward kinematics solution and the process of pose change.

The research of parallel mechanism has broad prospects. In the future, there will be more mathematical theories applied in mechanics, kinematics, and dynamics. ROBO_003 parallel mechanism can be applied to various industrial aerospace and aerospace fields, and its application prospect and values are very great.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publi-cation of this article.

Funding

The author(s) disclosed following financial support for the research, authorship, and/or publication of this article: This work was supported by the College Natural Science Foundation of Jiangsu Province (no. 17KJB460003), National Key Research Planning Project(No.2018YFC0309100), National Natural Science Foundation of China (no. 51375230), and Key Construction Subject Project of Jiangsu Province (no. (2016)9)

ORCID iD

Shi Baoyu

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Kinematics analysis of a new parallel robotics

Abstract

Keywords

Introduction

Analysis of DoF form

Screw theory

DOF analysis of ROBO_003

Kinematics

Establishment of coordinate system

Position and orientation describing of moving platform

Position of moving platform

Orientation of moving platform

Solutions for direct kinematic problem

Analytic method for close-form solutions

Numerical examples

ADMAS simulation

Introduction of ADMAS

Kinematics simulation of ROBO_003 based on ADMAS

Simulation results

Results and discussion

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References