Forward kinematic analysis of a 3-RRRS mechanism with subordinate driving variables using a computer-aided geometric method

Abstract

The 3-RRRS mechanism (RRRS refers to kinematic pairs of a branched chain consisting of three R pairs (Rotational pairs) and one S pair (Spherical pair), successively) is used between a moving platform and a static platform, through which six-dimensional motion of the moving platform relative to the static platform can be achieved. Selecting six independently drivable joints from nine active joints makes it problematic to deal analytically with the kinematics of the 3-RRRS mechanism. In this article, a novel computer-aided geometric method for kinematic analysis is developed. This method can automatically detect the independently drivable joints for arbitrary kinematic chains. This method can be easily implemented compared to the analytical method of the forward kinematics. Based on the constraint relationship of the 3-RRRS mechanism, a general 3-RRRS mechanism digital model is built in the SolidWorks Application Program Interface embedded Visual Basic environment, in which the platform sizes and active driving angles are driven by the parameterized model, to make the moving platform move to the corresponding pose. After the pose of the moving platform is confirmed, the coordinate system is built in a preliminary sketch. The parameters are measured by the SolidWorks measuring functions, and the pose of the moving platform is obtained by combining homogeneous matrices. Using the computer-aided geometric method, the detailed kinematics formula is not required. The accuracy and efficiency of the computer-aided geometric method were assessed with some examples of kinematic analysis for the 3-RRRS mechanism. The results showed that the proposed method obtained competitive precision and calculation time to the analytical method and is beneficial as a convenient solving process. By using Visual Basic programming, a reachable poses analysis of the mechanism can be merged into the kinematics analysis system of the computer-aided geometric method. The computer-aided geometric method could be widely applied to kinematics analysis of mechanisms.

Introduction

To achieve variations of a pose between the platforms above them, several studies have been carried out on the movement of branched chains connecting the moving and static platform. Regarding branched chains, for example, Hunt ¹ proposed a 3-RPS mechanism (RPS refers to kinematic pairs of a branched chain being a R (Rotational pair) pair, P pair (Prismatic pair), and S pair (Spherical pair), successively) to realize two rotations and a translation. Fang et al. ² studied the differential kinematics problem of a 3-RPS parallel mechanism and analyzed the instantaneous motion characteristics of the parallel mechanisms, such as 3-RPS and 3-RRS. Based on the symmetric structure of 3-RSR parallel robot, Fang et al. ³ studied a position display analytical solution of a three degrees of freedom (DOF) mechanism. Luo et al. ⁴ studied a 3-SRR/SRU (SRU refers to kinematic pairs of a branched chain being an S pair, R pair, and U pair (Hooke pair), successively) 3-DOF coupling parallel mechanism. Research on the Stewart parallel mechanism is a very typical and in-depth method to form a mechanism with branched chains. ⁵ In our research, to form the 6-DOF motion of the moving platform, a new 3-RRRS branched chain connecting a moving platform and a static platform is proposed, which has the characteristics of a redundant drive.

When the mobility of the mechanism is increased up to the required DOFs of the moving platform, the manipulator is a kinematically redundant parallel manipulator. ^6,7 On the other hand, a parallel manipulator is a redundantly actuated manipulator when the number of actuators is greater than the mobility of the mechanism. ^8,9 In general, it is believed that the redundancy can usually improve the ability and performance of the parallel manipulator, for example, by avoiding kinematic singularities, increasing workspace, improving dexterity, and increasing load capability. ^{6 –9} Lee et al. ¹⁰ presented a novel energy-saving method of a robotic system using a parallel mechanism by redundant actuation that can reduce the peak torque of the actuating joints. Shin et al. ¹¹ investigated how to remove parallel singularities through redundant actuation. Moreover, it has been reported that a redundant parallel manipulator has a better rigid dynamic performance than that of its nonredundant counterpart. ¹²

Research on the 3-RRRS mechanism has proven that it is very difficult to select six independently drivable joints from nine active joints in the forward kinematics analysis as the other parallel robot mechanism because forward kinematics analysis on the 3-RRRS mechanism may lead to a univariate 16th order polynomial equation that is difficult to solve.

Forward kinematics analysis of the parallel robot mechanism is very typical on a Stewart platform. Various studies ^{13 –15} have suggested that forward kinematics analysis on a Stewart platform eventually causes a univariate 16th order polynomial equation to solve. To solve the direct kinematics of a 3-RS parallel mechanism, Kim et al. ¹⁶ used Sylvester’s dialytic elimination method; the direct kinematic problem is simplified to the solution of a 16th order polynomial equation with a single variable.

Deducing the univariate 16th order polynomial equation is rather complex, requiring both the variable substitution and the Bezout method, and thus efficiently solving such a high order polynomial equation is even more challenging. Ku ¹⁷ proposed a numerical iterative algorithm based on the Newton–Raphson method for the position positive solution of the 6-3 type Stewart platform mechanism. Song et al. ¹⁸ analyzed three kinds of common methods for position positive solutions on the Stewart platform based on the polynomial method, the numerical iterative method, and the additional sensors method. ¹⁹

Since the solving procedure involves a complex computer program, the requirements of parallel mechanism real-time control are difficult to meet, and finding the solution is highly time-consuming. Therefore, finding better forward solutions for the parallel mechanism remains an ongoing challenge up to now.

For the kinematics analysis of the 3-RRRS mechanism, the most commonly used method to solve robot kinematics is based on the Denavit–Hartenberg notation. Yu et al. ²⁰ proposed an effective approach to solve the inverse displacement analysis for a class of modular three-legged parallel robots by Paden–Kahan subproblems. Liu et al. ²¹ discussed the forward and inverse displacement analyses of the 3/3-RRRS parallel manipulator, a novel geometrical method referred as equivalent mechanism was proposed for the forward displacement analysis of the manipulator. Derived from the basis of screw theory, Goutham et al. ²² studied a new improved inverse kinematic approach for a 6-DOF 3-RRRS parallel manipulator. Along with Paden–Kahan subproblems, a new and improved subproblem is used.

We proposed a novel dimensionality-reduction method to improve forward kinematic analysis of a reptile-like four-legged walking robot with 4-RRR motion chain. ²³ For the reptile-like four-legged walking robot, the basis of forward kinematics analysis and solving workspace of the robot was acquisition of the subordinate driving angles. For solving these angles, a set of 16 nonlinear equations was obtained by Professor Chen Xuedong ²⁴ after complex deductions in the analysis of forward kinematics of the robot. Abel theorem shows that quintic equations of higher order have no general algebraic method of obtaining solutions and that they can be solved only by numerical method. As the robot uses redundant drives, forward kinematics analysis using analytical method is difficult. ²⁴ We proposed a computer-aided geometric (CAG) method for forward kinematics analysis of the robot to solving the stable workspace of quadruped bionic robot with hand–foot-integrated function. ^25,26

In our further research, we recognized that the forward kinematics analysis of the 3-RRRS mechanism needs the completed derivation of the kinematics equations. Solving the equations of motion is very complicated; however, using computer software to model the mechanism is a very effective way to simplify the task.

For a configuration of the reptile-like four-legged walking robot, a motion branch chain of each leg is a RRR open chain, which is not quite the same as the 3-RRRS mechanism. When using a CAG method for forward kinematics analysis of the reptile-like four-legged walking robot, we have not considered a motion range of each branched chain, especially, the motion range and position change of S pair. The other problems, such as the forward kinematic analysis of the velocity and acceleration, singular (or special) configuration of the mechanism, have not been studied, also. We need to further improve and systematize the previous work to developing a CAG method to analyze the motion of the 3-RRRS mechanism. ^25,26

Developing a CAG method to analyze the motion of the mechanism for reducing the difficulty of the analysis process and reducing the solving time is our research goal.

Many scholars have researched robot movement analysis using geometric methods. For example, Lin et al. ²⁷ adopted a geometric method mixed with an analytical method algorithm to solve the inverse kinematics of a modular manipulator robot with shoulder offset. Yang et al. ²⁸ presented a geometric method for solving the inverse and forward kinematics of a Delta robot. The forward kinematics are simplified as solving the intersection point of two circles and then transforming the coordinates system to obtain the final solution.

The geometric method is also used in the study of redundant drive mechanisms. Shim et al. ²⁹ proposed a novel optimal torque distribution method for a redundantly actuated 3-RRR parallel robot using a geometric approach, and a geometric analysis based on screw theory was carried out to calculate the stiffness matrix of the 3-RRR parallel robot.

In most instances, the geometric method has been used as an auxiliary method in the analysis of the mechanism motion. Using the CAG method, which establishes a three-dimensional digital driven mechanism model and combines with a variety of software technologies, the mechanism motion pose is not directly involved.

In this article, a CAG method for analyzing the 3-RRRS mechanism is developed. Based on the constraint relationship of the mechanism, a SolidWorks digital model of the mechanism is established, and a general digital model of the 3-RRRS mechanism is formed by using the SolidWorks program from the Visual Basic (VB) environment. The main innovative points include:

By using the digital model, the platform sizes and independently driving angles are driven by a parameterized model. The parameterized model can automatically detect other drivable joint angles for arbitrary kinematic chains to make the moving platform move according to the corresponding pose.

The parameters are automatically measured with the measure functions in SolidWorks. The pose of the moving platform is obtained by combining homogeneous matrices, and the forward kinematics of the mechanism can be solved. The CAG method makes it so that the detailed kinematics formula is not needed, which provides a heuristic thought for motion analysis of the mechanism.

By using VB programming, a velocity analysis, an acceleration analysis, a singularity analysis, and a reachable poses analysis of the mechanism can be merged into the kinematics analysis system of the CAG method. These can greatly facilitate the characteristics analysis of the mechanism.

This article is organized as follows: In the second section, the configuration of the 3-RRRS mechanism is described. In the third section, the difficulty of the analytical method of forward kinematics for the 3-RRRS mechanism is reviewed, and the CAG method is proposed in the fourth section. In the fifth section, some examples of the analysis are presented. Some issues about the velocity and acceleration analysis, the singularity of the mechanism, and the application of the CAG method in a real system are discussed in the sixth section. Conclusions are drawn in the seventh section.

The configuration of a 3-RRRS mechanism and its application

A configuration schematic diagram of a 3-RRRS mechanism connecting the moving platform and the static platform is shown in Figure 1. The motion axis of the R pair, which connects to the moving platform, is vertical to the moving platform. The motion axes of the two R pairs in the middle are parallel to the moving platform. An S pair of the RRRS branched chain is connected to the static platform. Each R pair of the three branched chains has a drive motor. The moving platform can form six-dimensional movement. Due to the 6-DOFs of the moving platform, nine driving pairs in the three branched chains exist, and only six of these pairs are independently driving pairs.

Figure 1.

A configuration schematic diagram of a 3-RRRS mechanism.

Applications of the 3-RRRS mechanism are very extensive. For example, if remove the elements contacting the static platform from the S pairs, the S balls of the S pairs will contact the static platform directly. Another RRRS branched chain is added to connect the moving platform, and the moving platform is transferred as a body installing controllers, and thus a typical configuration of the reptile-like quadruped robot can be formed.

Yu proposed that there are two kinds of configurations for three-legged parallel robots. One is 3-RRRS type whose active joints are all revolute joints, another one is 3-RPRS type whose active joints include both revolute and prismatic joints. ²⁰ Yang et al. ³⁰ researched a self-calibrated three-legged (6-DOF, RRRS) parallel robot. Yu et al. ³¹ designed a bionic robot, by 3/3-RRRS parallel mechanism driving teeth up and down, which achieved full chewing of food to study a masticatory movement of mankind for food processing.

The forward kinematics of the mechanism need complex derivation of the kinematic equations and six independently drivable joints are selected from nine active joints. Solving the kinematic equations is very complicated. A deeper understanding of kinematics analysis is of universal significance.

The difficulty of the analytical method of forward kinematics for a 3-RRRS mechanism

A kinematic sketch of a 3-RRRS mechanism is shown in Figure 2. The connection points between the S pair and the static platform are $A_{J_0}$ , $A_{K_0}$ , and $A_{L_0}$ ; ${\Sigma }_o$ is a global reference frame set on the static platform; $B_J$ , $B_K$ , and $B_L$ are points connecting the moving platform to the RRRS branched chains. To simplify the problem, $\Delta B_K{B}_L{B}_J$ are set to a right triangle. ${\Sigma }_c$ is a coordinate frame fixed to the moving platform, and its base point coincides with the midpoint of line segment $B_K{B}_J$ . The x, y directions of the coordinate frame ${\Sigma }_c$ are parallel to $B_K{B}_L$ (x-direction) and $B_K{B}_J$ (y-direction), respectively. The size of the triangle is $2m\times 2n$ (see Figure 2; 2m and 2n are the side-length of the moving platform $\Delta B_K{B}_L{B}_J$ ).

Figure 2.

A kinematic sketch of the 3-RRRS mechanism.

The distance between the sphere center of the S pairs and the static platform is h_L , and it is the same and perpendicular to the static platform in the three branched chains. The link lengths of each branched chain (from top to bottom) are l₁ , l₂ , and l₃ , respectively. ${\Sigma }_{B_i}$ (B_i -xyz) presents a relative coordinate frame fixed to B_i (i= J, K, L) while the z-axis is co-axial with regard to the spin axis of the pair.

Because the points $A_{J_0}$ , $A_{K_0}$ , and $A_{L_0}$ are invariant, the following equations can be obtained

$d_{i_0{j}_0}^2={({}^{\text{o}}x{ }_{A_{i_{{ }_0}}}-{}^{\text{o}}x{ }_{A_{j_{{ }_0}}})}^2+{({}^{o}y{ }_{A_{i_{{ }_0}}}-{}^{o}y{ }_{A_{j_{{ }_0}}})}^2+{({}^{o}z{ }_{A_{i_{{ }_0}}}-{}^{\text{o}}z{ }_{A_{j_{{ }_0}}})}^2\hspace{0.17em}\text{\hspace{1em}(}{i}_0=J_0,K_0,L_0)\hspace{0.17em}(j_0=J_0,K_0,L_0,j_0\ne i_0)$

1

$d_{ij}^2={({}^{o}x{ }_{A_i}-{}^{o}x{ }_{A_j})}^2+{({}^{o}y{ }_{A_i}-{}^{o}y{ }_{A_j})}^2+{({}^{o}z{ }_{A_i}-{}^{o}z{ }_{A_j})}^2\hspace{1em}(i=J,K,L)\hspace{0.17em}(j=J,K,L,j\ne i)$

2

where $d_{i_o{j}_o}$ is the distance from points $A_{i_0}$ to $A_{j_0}$ and d_ij is the distance from points $A_i$ to $A_j$ . ${}^{o}x{ }_{A_{i_0}}$ , ${}^{o}y{ }_{A_{i_0}}$ , ${}^{o}z{ }_{A_{i_0}}$ and ${}^{o}x{ }_{A_i}$ , ${}^{o}y{ }_{A_i},$ ${}^{o}z{ }_{A_i}$ (i = J, K, L) are known coordinates of the points $A_{i_0}$ and $A_i$ in ${\sum }_o$ .

A schematic of part of the three branched chains (from A_i to B_i (i = J, K, L)) is shown in Figure 3. The variables ${\varphi }_i$ , ${\phi }_i$ , and ${\chi }_i$ refer to variables of the driving joints. Both L_i and H_i are the stretch length of the branched chain i (from A_i to B_i (i =J, K, L)) in the x–y plane and the z-axis of ${\sum }_{B_i}$ , respectively. The following equations can be obtained

Figure 3.

A schematic of part of the three branched chains.

$\{\begin{array}{l}{L}_i=l_1+l_2\text{cos}{\phi }_i+l_3\text{cos}({\phi }_i+{\chi }_i)\\ H_i=l_2\text{sin}{\phi }_i+l_3\text{sin}({\phi }_i+{\chi }_i)\end{array}$

3

The projection of 3-RRRS mechanism in the x–y plane of ${\Sigma }_c$ is shown in Figure 4. In ${\Sigma }_c$ , the mechanism satisfies the following equations

$\left[\begin{array}{c}{}^{c}x{ }_{A_i}\\ {}^{c}y{ }_{A_i}\\ {}^{c}z{ }_{A_i}\end{array}\right]=\left[\begin{array}{c}{}^{c}x{ }_{B_i}-L_i\text{cos}{\varphi }_i\\ {}^{c}y{ }_{B_i}+L_i\text{sin}{\varphi }_i\\ {}^{c}z{ }_{B_i}-H_i\end{array}\right]$

4

Figure 4.

The projection of the mechanism in x–y plane of ${\sum }_c$ .

where i = J, K, L. ${}^{c}x{ }_{B_i}$ , ${}^{c}y{ }_{B_i}$ , and ${}^{c}z{ }_{B_i}$ are functions about m and n.

Because the distance between the centers of the sphere of the S pair are the same in both ${\Sigma }_o$ and ${\Sigma }_c$ , the following equations can be obtained

$d_{ij}^2={({}^{c}x{ }_{A_i}-{}^{c}x{ }_{A_j})}^2+{({}^{c}y{ }_{A_i}-{}^{c}y{ }_{A_j})}^2+{({}^{c}z{ }_{A_i}-{}^{c}z{ }_{A_j})}^2$

5

where (i = J, K, L), (j = J, K, L, and i ≠ j), ${}^{c}x{ }_{A_i}$ , ${}^{c}y{ }_{A_i}$ , and ${}^{c}z{ }_{A_i}$ (i = J, K, L) are coordinates of point A_i in ${\Sigma }_c$ .

The moving platform has six DOFs. In total, nine driving joints in three branched chains exist, and only six of the joints can be independent driving joints; the remaining are subordinate driving joints. Without any loss of generality, it can be assumed that the joint variables ${\phi }_i$ and ${\chi }_i$ of the three branched chains are the independently driving joint variables, and the joint variables ${\varphi }_i$ of the three branched chains are subordinate driving variables, and here i = J, K, L. ${\varphi }_i$ are unknown variables which need to be solved using constraint relation between the mechanism parameters and the independently driving joint variables.

By substituting equations (3) and (4) into equation (5), the following equations can be obtained

$\{\begin{array}{l}{({}^{c}x{ }_{B_K}-{}^{c}x{ }_{B_L}-L_K\text{cos}{\varphi }_K+L_L\text{cos}{\varphi }_L)}^2+{({}^{c}y{ }_{B_K}-{}^{c}y{ }_{B_L}+L_K\text{sin}{\varphi }_K-L_L\text{sin}{\varphi }_L)}^2+{(H_K-H_L)}^2=d_{KL}^2\\ {({}^{c}x{ }_{B_J}-{}^{c}x{ }_{B_L}-L_J\text{cos}{\varphi }_J+L_L\text{cos}{\varphi }_L)}^2+{({}^{c}y{ }_{B_J}-{}^{c}y{ }_{B_L}+L_J\text{sin}{\varphi }_J-L_L\text{sin}{\varphi }_L)}^2+{(H_J-H_L)}^2=d_{JL}^2\\ {({}^{c}x{ }_{B_K}-{}^{c}x{ }_{B_J}-L_K\text{cos}{\varphi }_K+L_J\text{cos}{\varphi }_J)}^2+{({}^{c}y{ }_{B_K}-{}^{c}y{ }_{B_J}+L_K\text{sin}{\varphi }_K-L_J\text{sin}{\varphi }_J)}^2+{(H_K-H_J)}^2=d_{KJ}^2\end{array}$

6

Using equation (6), we obtain

$\{\begin{array}{l}{a}_1\text{sin}{\varphi }_K\text{sin}{\varphi }_L+a_2\text{cos}{\varphi }_K\text{cos}{\varphi }_L+a_3\text{sin}{\varphi }_K+a_4\text{cos}{\varphi }_K+a_5\text{sin}{\varphi }_L+a_6\text{cos}{\varphi }_L+a_7=0\\ b_1\text{sin}{\varphi }_J\text{sin}{\varphi }_L+b_2\text{cos}{\varphi }_J\text{cos}{\varphi }_L+b_3\text{sin}{\varphi }_J+b_4\text{cos}{\varphi }_J+b_5\text{sin}{\varphi }_L+b_6\text{cos}{\varphi }_L+b_7=0\\ c_1\text{sin}{\varphi }_J\text{sin}{\varphi }_K+c_2\text{cos}{\varphi }_J\text{cos}{\varphi }_K+c_3\text{sin}{\varphi }_J+c_4\text{cos}{\varphi }_J+c_5\text{sin}{\varphi }_K+c_6\text{cos}{\varphi }_K+c_7=0\end{array}$

7

where a_i , b_i , and c_i are functions of known variables and they are expressed as follows

$$\begin{gathered} a_1=-2L_K{L}_L,\hspace{0.17em}{a}_2=-2L_K{L}_L,\hspace{0.17em}{a}_3=2({}^{c}y{ }_{B_K}-{}^{c}y{ }_{B_L})L_K,\hspace{0.17em}{a}_4=-2({}^{c}x{ }_{B_K}-{}^{c}x{ }_{B_L})L_K,\hspace{0.17em}{a}_5=-2({}^{c}y{ }_{B_K}-{}^{c}y{ }_{B_L})L_L,\hspace{0.17em} \\ {a}_6=2({}^{c}x{ }_{B_K}-{}^{c}x{ }_{B_L})L_L,\hspace{0.17em}{a}_7=-d^2{ }_{KL}+{(H_K-H_L)}^2+{({}^{c}x{ }_{B_K}-{}^{c}x{ }_{B_L})}^2+{({}^{c}y{ }_{B_K}-{}^{c}y{ }_{B_L})}^2+L^2{ }_K+L^2{ }_L \end{gathered}$$

$$\begin{gathered} b_1=-2L_J{L}_L,\hspace{0.17em}{b}_2=-2L_J{L}_L,\hspace{0.17em}{b}_3=2({}^{c}y{ }_{B_J}-{}^{c}y{ }_{B_L})L_J,\hspace{0.17em}{b}_4=-2({}^{c}x{ }_{B_J}-{}^{c}x{ }_{B_L})L_J,\hspace{0.17em}{b}_5=-2({}^{c}y{ }_{B_J}-{}^{c}y{ }_{B_L})L_L,\hspace{0.17em} \\ {b}_6=2({}^{c}x{ }_{B_J}-{}^{c}x{ }_{B_L})L_L,\hspace{0.17em}{b}_7=-d^2{ }_{JL}+{(H_J-H_L)}^2+{({}^{c}x{ }_{B_J}-{}^{c}x{ }_{B_L})}^2+{({}^{c}y{ }_{B_J}-{}^{c}y{ }_{B_L})}^2+L^2{ }_J+L^2{ }_L \end{gathered}$$

$$\begin{gathered} c_1=-2L_J{L}_K,\hspace{0.17em}{c}_2=-2L_J{L}_K,\hspace{0.17em}{c}_3=2({}^{c}y{ }_{B_J}-{}^{c}y{ }_{B_K})L_J,\hspace{0.17em}{c}_4=-2({}^{c}x{ }_{B_J}-{}^{c}x{ }_{B_K})L_J,\hspace{0.17em}{c}_5=-2({}^{c}y{ }_{B_J}-{}^{c}y{ }_{B_K})L_K,\hspace{0.17em} \\ {c}_6=2({}^{c}x{ }_{B_J}-{}^{c}x{ }_{B_K})L_K,\hspace{0.17em}{c}_7=-d^2{ }_{JK}+{(H_J-H_K)}^2+{({}^{c}x{ }_{B_J}-{}^{c}x{ }_{B_K})}^2+{({}^{c}y{ }_{B_J}-{}^{c}y{ }_{B_K})}^2+L^2{ }_J+L^2{ }_K \end{gathered}$$

For solving the variables of the subordinate driving joints in equation (7), let $t_1=\text{tan}({\phi }_J/2)$ , $t_2=\text{tan}({\phi }_K/2)$ , and $t_3=\text{tan}({\phi }_L/2)$ , and then $\text{sin}{\phi }_J=2t_1/(1+t_1^2)$ , $\text{cos}{\phi }_J\text{=(1}-t_1^2\text{)}/(1+t_1^2)$ , $\text{sin}{\phi }_K=2t_2/(1+t_2^2)$ , $\text{cos}{\phi }_K=\text{(1-}{t}_2^2\text{)}/(1+t_2^2)$ , $\text{sin}{\phi }_{\text{L}}=2t_3/(1+t_3^2)$ , and $\text{cos}{\phi }_{\text{L}}\text{=(1-}{t}_3^2\text{)}/(1+t_3^2)$ . After substituting all of them into equation (7), we obtain

$\{\begin{array}{c}(d_1{t}_2^2+d_2{t}_2+d_3)t_3^2+(d_4{t}_2^2+d_5{t}_2+d_6)t_3+(d_7{t}_2^2+d_8{t}_2+d_9)=0\\ (e_1{t}_1^2+e_2{t}_1+e_3)t_3^2+(e_4{t}_1^2+e_5{t}_1+e_6)t_3+(e_7{t}_1^2+e_8{t}_1+e_9)=0\\ (f_1{t}_1^2+f_2{t}_1+f_3)t_2^2+(f_4{t}_1^2+f_5{t}_1+f_6)t_2+(f_7{t}_1^2+f_8{t}_1+f_9)=0\end{array}$

8

where d_i , e_i , and f_i are functions of the known variables. Their detailed expressions are expressed as follows: $d_1=a_2-a_4-a_6+a_7$ , $d_2=2a_3$ , $d_3=-a_2+a_4-a_6+a_7$ , $d_4=2a_5$ , $d_5=4a_1$ , $d_6=2a_5$ , $d_7=-a_2-a_4+a_6+a_7$ , $d_8=2a_3$ , $d_9=a_2+a_4+a_6+a_7$ ; $e_1=b_2-b_4-b_6+b_7$ , $e_2=2b_3$ , $e_3=-b_2+b_4-b_6+b_7$ , $e_4=2b_5$ , $e_5=4b_1$ , $e_6=2b_5$ , $e_7=-b_2-b_4+b_6+b_7$ , $e_8=2b_3$ , $e_9=b_2+b_4+b_6+b_7$ ; $f_1=c_2-c_4-c_6+c_7$ , $f_2=2c_3$ , $f_3=-c_2+c_4-c_6+c_7$ , $f_4=2c_5$ , $f_5=4c_1$ , $f_6=2c_5$ , $f_7=-c_2-c_4+c_6+c_7$ , $f_8=2c_3$ , $f_9=c_2+c_4+c_6+c_7$ .

At this step, variables t ₂ and t ₃ can be eliminated from equation (8) using the Bezout method. The final equation only has the variable t ₁, which is expressed as follows

$\begin{array}{l}{v}_1{t}_1^{16}+v_2{t}_1^{15}+v_3{t}_1^{14}+v_4{t}_1^{13}+v_5{t}_1^{12}+v_6{t}_1^{11}+v_7{t}_1^{10}\\ +v_8{t}_1^9+v_9{t}_1^8+v_{10}{t}_1^7+v_{11}{t}_1^6+v_{12}{t}_1^5+v_{13}{t}_1^4+v_{14}{t}_1^3\\ +v_{15}{t}_1^2+v_{16}{t}_1+v_{17}=0\end{array}$

9

where v_i (i =1, 2,…, 17) are the coefficients which are functions of the known variables, and their specific expression are omitted here. It should be noted that the solving process and the coefficients are very complicated. Equation (9) is a 16th order equation, and according to Abel’s theorem, there exists no universal algebraic solution for this equation. Only numerical solutions for fifth order or higher polynomial equations exist. In other words, solving the equation of the 3-RRRS mechanism is challenging in practice.

As stated in the first section, many methods have shortcomings for solving this kind of equation, and the computational efficiency of many methods is very low. Sometimes, for a 16th order equation, only 16 imaginary roots can be obtained with a longer calculation time, and there is no real solution. A goal of this study is developing a novel method for solving this kind of problem. After using a relevant method to obtain the subordinate driving variables ${\varphi }_i$ $(i=J,K,L)$ of equation (7), the orientation matrix of the moving platform can be further calculated. The following equations can be obtained from the mechanism in Figures 2 and 3

$\{\begin{array}{l}{}^o\mathit{p}{ }_{A_i}-{}^o\mathit{p}{ }_{A_j}={\mathit{R}}_c\left({}^c\mathit{p}{ }_{A_i}-{}^c\mathit{p}{ }_{A_j}\right)\\ {}^o\mathit{p}{ }_{A_i}={}^o\mathit{p}{ }_{A_{i_{{ }_0}}}+h_L{}^o\mathit{z}\end{array}$

10

where (i = J, K, L), (j = J, K, L), and i ≠ j. R _c is an orientation matrix of the moving platform to the static platform. Let

$\{\begin{array}{l}{}^o\mathit{p}{ }_{A_i}-{}^o\mathit{p}{ }_{A_j}={[e_{ij}^o,f_{ij}^o,g_{ij}^o]}^T\\ {}^c\mathit{p}{ }_{A_i}-{}^c\mathit{p}{ }_{A_j}={[e_{ij}^c,f_{ij}^c,g_{ij}^c]}^T\end{array}$

11

where (i = J, K, L), (j= J, K, L), and i ≠ j. R _c is expressed as follows

${\mathit{R}}_c=\left[\begin{array}{l}{r}_{11}\hspace{0.17em}\hspace{0.17em}{r}_{12}\hspace{0.17em}\hspace{0.17em}{r}_{13}\\ r_{21}\hspace{0.17em}\hspace{0.17em}{r}_{22}\hspace{0.17em}\hspace{0.17em}{r}_{23}\\ r_{31}\hspace{0.17em}\hspace{0.17em}{r}_{32}\hspace{0.17em}\hspace{0.17em}{r}_{33}\end{array}\right]$

12

Considering the features of the standard orthogonal matrix R _c , by simultaneously solving the equations from (10) to (12), the elements r_ij (i=1, 2, 3; j=1, 2, 3) are solved.

Then, the moving platform’s trajectory (shown by the motion locus of origin C in ${\sum }_o$ ) can be obtained as follows

${}^o\mathit{p}{ }_c={}^o\mathit{p}{ }_{A_i}-{\mathit{R}}_c {}^c\mathit{p}{ }_{A_i}(i=J,K,L)$

13

By substituting equation (10) into equation (13), we obtain

${}^o\mathit{p}{ }_c={}^o\mathit{p}{ }_{A_{i_0}}+h_L{}^o\mathit{z}-{\mathit{R}}_c {}^c\mathit{p}{ }_{A_i}$

14

where ${}^o\mathit{p}{ }_{A_{i_{{ }_0}}}$ is a known quantity, h_L is a structural parameter, R _c is an orientation matrix, and ${}^c\mathit{p}{ }_{A_i}$ is obtained from equation (4).

Note that the solving of equation (9) using the analytical method is cumbersome. A CAG method for the mechanism kinematics is proposed below.

A CAG method of 3-RRRS mechanism kinematics

In a kinematic analysis system based on the CAG method, a VB program was used to join the SolidWorks system for constructing a 3D parameterized model of the mechanism. Different parameters of the mechanism were input using the VB program to drive the model to change its dimension, position, and posture in SolidWorks.

Using the CAG method for mechanism kinematic analysis, the parameterization technology is used .The parameterization technology means that a model size of the mechanism will automatically change when a user input different the mechanism parameter. That is, if a user changes a parameter, the system can update the mechanism’s geometry model automatically and configuration to maintain consistency of the model.

The flow chart for the kinematics analysis of the mechanism is shown in Figure 5. The independently driven angle was driven by this parameterized model to make the moving platform move to the corresponding pose. After the pose of the moving platform was confirmed, the coordinate system was built in a preliminary sketch. The parameters were automatically measured using SolidWorks functions, and the pose of the moving platform was obtained by combining homogeneous matrices; the reachable poses of the platform can be shown in SolidWorks if need be. The kinematics analysis of the mechanism was then achieved.

Figure 5.

The CAG method flow chart. CAG: computer-aided geometric.

Principle of the CAG method

Achieving the passive drive of the incompletely constrained components in SolidWorks is important for developing and applying a digital model of the mechanism. In the assembly of SolidWorks, except for the fixed part, each active part has six DOFs, and the number of DOFs of the parts is reduced as the constraints increase. When part A increases with the cooperative relationship to change its position or posture, part B, which has constraint relationship with part A, also passively changes its position and posture with the new increased cooperative relationship. However, if part B has been fully constrained, then no matter how it moves to a position or lays out in whatever kind of posture that cannot meet the new increased cooperative relationship, then the SolidWorks system shows a warning. In addition, the relationship between the original position and posture of the part remains unchanged in the whole system.

In our 3-RRRS mechanism’s analysis system, developed by the VB program driving SolidWorks software, a 3D model of each part is established. Based on the theoretical model of the 3-RRRS mechanism, the 3D model of the mechanism is assembled in the assembly file. At this time, the assembly model of the mechanism only has a basic cooperative relationship, that is, the motion of the kinematic pairs, including the R pair and S pair, of each module is a pure rotation. For the whole mechanism model, it has six DOFs, and the most basic constraint relationship is formed on the structure. After structural constraints establish the independently driven angles and the positions of the spherical pair S, all six DOFs of the mechanism model are subject to constraints.

For the 3-RRRS mechanism, the structural parameters of the mechanism model include the length 2m and the width 2n of the moving platform; the link lengths l₁ , l₂ , and l₃ of the branched chains; and the height h_L and positions $A_{K_0}$ , $A_{J_0}$ , and $A_{L_0}$ of the spherical pairs. These structural parameters are digitally driven in the developed mechanism’s analysis system. Specific values for each parameter are input from a program interface using the VB program. When different parameter values are input, the VB program calls the 3-D model in SolidWorks, and the corresponding change is made for the size and pose of the model.

The digital model of the overall mechanism structure is formed. Without a loss of generality, suppose the rotation angles ${\phi }_i$ and ${\chi }_i$ (see Figure 3) are independently driven angles, which are completely constrained when their values are input from the VB program interface. According to the passive driving principle of components, which is not completely constrained in SolidWorks, with the constraint of the structure and the independent drive, the mechanism assembly model is driven to reach the required posture and position. In this process, because the positions of the spherical pairs and branched chains rotation angles restrain the R pairs connecting the moving platform and the branched chains, passive constraint angles (subordinate driving angles ${\phi }_i$ in Figure 3) are confirmed. For confirming the angles of the passive drive, some parameters of the mechanism’s assembly are measured in the SolidWorks with the measure function. Meanwhile, the posture’s matrix of the moving platform and position coordinates of the points in the moving platform are obtained and kinematic analysis of the mechanism is realized with the acquired data and the homogeneous matrices.

The technology of the application software

In the developed kinematic analysis system, the data access between the VB environment and SolidWorks can be visualized as shown in Figure 6.

Figure 6.

The application software.

The API function feature of SolidWorks

The modeling process mainly adopts design features of a 3-D entity model based on the characteristic parameters and an assembly model based on constraints in SolidWorks. This software has hundreds of Application Program Interface (API) functions that are Object Linking and Embedding for SolidWorks or a Component Object Model (COM) Interface, which support a Windows application program to operate other Windows procedures by acquiring object control. In this design, the API functions are used to develop the program. In the API help manual, SolidWorks provides API objects at all levels as well as the parent–child relationships.

When using an API function, before controlling a sub_object, the control right of its father_object must first be taken; a control right of each sub_object belongs to its father_object. Each object has its specific functions or attributes. Most of the functions of normal operation in SolidWorks are encapsulated in the object at all levels. Programmers who gain the control right of the object can modify the attributes of the object or perform its function to realize the basic process.

Using the VB environment to call the API function

In the application of the mechanism modeling, the VB environment and Windows application program are used to build an interface for calling the SolidWorks API function and developing the required functionality.

Detailed technology of CAG method

Establishing a 3-D model of the 3-RRRS

According to the theoretical model of the 3-RRRS, the 3-D model is first established and it is mated in SolidWorks. Figure 7 depicts an assembly with the basic cooperating relationship. During assembly, the position of a spherical pair is set on the ground; the center of the ground coincides with the center of the assembly, and each reference plane coincides between the ground and the assembly.

Figure 7.

A 3-D assembly model of the 3-RRRS mechanism.

A fixed coordinate system is set as the acquiesced coordinate system of the assembly. At this step, positions of the spherical pairs in the fixed coordinate system and in the acquiesced coordinate system of assembly coincide. The position control for the spherical pair controls the distance between the point and each reference plane in the acquiesced coordinate system of the assembly, which belongs to a cooperative control of distance type.

Positions $A_{K_0}$ , $A_{J_0}$ , and $A_{L_0}$ are not fixed at this step, and each motion pair only has a rotation constraint relationship, that is, the angle constraint exists between two adjacent member bars of the motion pairs. Because the constraints are restrained for key operations, the model needs to be further controlled regarding the position of the spherical pair and the independently driven angles. In this case, the mechanism is driven to change its state.

The parameterization of the parts

Parts parameterization means automatically changing part models size following the different input parameters.

To establish the universal digital model of the3-RRRS mechanism, the length of the mechanism’s link; the sizes of the moving platform; and the positions $A_{K_0}$ , $A_{J_0}$ , and $A_{L_0}$ are parameterized. The VB program is adopted to drive the digital model of the 3-RRRS mechanism in SolidWorks. When a parameterization technology is used to drive the length of the link, the sizes of the moving platform, and the positions of $A_{K_0}$ , $A_{J_0}$ , $A_{L_0}$ in the model sketch, then particular attention should be paid to the size constraint relations within the structure. Details about the size constraint relations within the structure are discussed in Appendix 1.

If the sizes of the moving platform and positions $A_{K_0}$ , $A_{J_0}$ , and $A_{L_0}$ are random values, then the mechanism model may not keep the assembly relations. Under normal circumstances, the range of $A_{J_0}$ is restricted to the first quartile of ${\sum }_O$ , the range of $A_{K_0}$ is restricted to the third quartile of ${\sum }_O$ , and the range of $A_{L_0}$ is restricted to the fourth quartile of ${\sum }_O$ .

For the system developed in this article, first, a suitable range of the moving platform size and points $A_{K_0}$ , $A_{J_0}$ , and $A_{L_0}$ is chosen; secondly, parameterization technology is used to drive the link length (ensure it changes within a certain range); and then, after forming the assembly relationship of the specific mechanism model, parameter technology is used to drive the independently driven angles of the mechanism model and make mechanism model form a specific pose. The minimum size of the mechanism’s part in the model is set based on the design methods of mechanism model and constraint relationships of the structure.

Stretching the length is used to control each link’s size of the mechanism model when using parameterization technology to drive the length of a link. In achieving parameterization of parts, the SolidWorks API function is needed. First, it is necessary to obtain control rights for the superior object of the function. There are two methods for acquiring the control rights of the highest level object of API function in SolidWorks (i.e. the control rights of SolidWorks). For a more detailed method, see the SolidWorks user’s guide. ³² After gaining the control rights of the highest level application object, the controlled object can be opened and handled. In the design, it is the 3-D assembly of a theoretical model of the 3-RRRS mechanism.

In the system built using the CAG method, the draft dimension or characteristic dimension are driven to realize the parts’ parameterization. Details about realizing parts’ parameterization are given in Appendix 2.

A certain cooperative relationship of the structure should be controlled whether the mechanism model is formed and the sizes are confirmed or not. It is very important for the CAG method to successfully control the cooperative relationships of the mechanism’s independent driving angles and the cooperative relationship of the spherical pair’s position.

The key functions of adding a cooperative relationship in SolidWorks are used to control the cooperative relationship of the defined components. Details regarding adding a cooperative relationship are discussed in Appendix 3.

When the programs are used to control the position of the spherical pair and the independently driven angles, the sequence of the drive should be considered. In the forward kinematic analysis, the independently driven angles are coopted first, and then the spherical pair’s positions are coopted. On the contrary, assembly errors are prone to occur.

All coordinate relations between components in the mechanism model are the angle component and distance coordination.

Using the VB program to input parameters

The parameters of the variable are input using the VB program after the digital mechanism model is set by secondary development of Solid Works. Figure 8 shows the interface of the forward kinematics system of the mechanism using the VB program to input the parameters.

Figure 8.

The interface of the forward kinematics system of the mechanism for Example A.

Driving the 3-D assembly model of the 3-RRRS mechanism

Dimensions and positions of the components change along with the different input value by using the VB program to call the digital model of the mechanism. The adoption of the corresponding method confirms the mechanism model’s six independently driven angles and the three fixed positions of the spherical pair. The assembly of the mechanism model can form the position and the pose of the moving platform which matches the given conditions.

Measurement of the related parameters

When the mechanism model moves to the specified pose under the SolidWorks’ drive, the mechanism kinematics relationship is formed with a specified cooperation relationship. The related parameters are measured, and results are obtained as output.

In the proposed analysis system, the internal measurement system in SolidWorks is used. In Figure 7, the default coordinate system in SolidWorks is set as a fixed coordinate system. The reference coordinate systems are set in the center point C of the moving platform. Specific implementation processes are given below.

Using SolidWorks, a “static platform” part is inserted in the mechanism’s assembly. The default coordinate system of the “static platform” part and the mechanism’s assembly are set to coincide by the cooperative relationship of the assembly so that the “static platform” part is fixed. The center of the “static platform” in a default coordinate system is the geometric center. In measurements, the default datum plane in the “static platform” is used as a reference, and the vertical distances from points to each datum plane are measured. These are the coordinate values of the points in the fixed coordinate system.

The floating coordinate systems (i.e. coordinate system ${\sum }_c$ ) are fixed in the center point of the triangle’s hypotenuse of the moving platform of the mechanism model. The method of drawing sketches is used to set the coordinate system. As shown in Figure 7, the origin of the coordinate system ${\sum }_c$ is set at the center point of the triangle’s hypotenuse of the moving platform of the mechanism model, and sketches are drawn in the horizontal plane that include the origin. Two points are selected, each at 100 mm distance, measured along the 2m and 2n directions, respectively, parallel to the triangle’s right side of the moving platform with the center point of triangle’s hypotenuse as the origin. The x- and y-directions of the floating coordinate system ${\sum }_c$ are shown with the two points (marked as points x and y). Another point is selected 100 mm distance away from the origin along the vertical direction of the moving platform, and it is given as the z-direction (point z).

Assume that ${\mathit{p}}_c$ is the vector from the origin of system ${\sum }_O$ , denoting the center point of the floating coordinate system, and that lines from the origin of ${\sum }_O$ to the other three points (marked as points x, y, and z, which constitute the x-, y-, and z-direction for the floating coordinate system) are vectors ${\mathit{p}}_x, \hspace{0.17em}{\mathit{p}}_y, \hspace{0.17em}{\mathit{p}}_{\mathit{z}}$ in the fixed coordinate system. Obviously, the length of ${\mathit{p}}_c$ is the distance between the original points of the two coordinate systems. By constraining the relationship of the mechanism model, the measurement function regarding the positions of the points in the SolidWorks API functions is automatically called to measure the position of points.

Calculation of the mechanism’s pose

The vector expressions of the floating coordinate system in the fixed coordinate system are ${\mathit{p}}_x-{\mathit{p}}_c$ , ${\mathit{p}}_y-{\mathit{p}}_c$ , and ${\mathit{p}}_{\mathit{z}}-{\mathit{p}}_c$ . By normalizing and combining them with the coordinate of the moving platform’s point C according to the column, the homogeneous coordinate matrix of two coordinate systems can be obtained as equation (15)

${}^o\mathit{T}{ }_c=\left[\begin{array}{cccc}[p_x(x)-p_c(x)]/100& [p_y(x)-p_c(x)]/100& [p_{\mathit{z}}(x)-p_c(x)]/100& p_c(x)\\ [p_x(y)-p_c(y)]/100& [p_y(y)-p_c(y)]/100& [p_{\mathit{z}}(y)-p_c(y)]/100& p_c(y)\\ [p_x(\mathit{z})-p_c(\mathit{z})]/100& [p_y(\mathit{z})-p_c(\mathit{z})]/100& [p_{\mathit{z}}(\mathit{z})-p_c(\mathit{z})]/100& p_c(\mathit{z})\\ 0& 0& 0& 1\end{array}\right]$

15

In equation (15)

$$\begin{gathered} {\mathit{p}}_x={(p_x(x),\hspace{0.17em}{p}_x(y),\hspace{0.17em}{p}_x(\mathit{z}))}^T;{\mathit{p}}_y={(p_y(x),\hspace{0.17em}{p}_y(y),\hspace{0.17em}{p}_y(\mathit{z}))}^T; \\ {\mathit{p}}_{\mathit{z}}={(p_{\mathit{z}}(x),\hspace{0.17em}{p}_{\mathit{z}}(y),\hspace{0.17em}{p}_{\mathit{z}}(\mathit{z}))}^T;{\mathit{p}}_c={(p_c(x),\hspace{0.17em}{p}_c(y),\hspace{0.17em}{p}_c(\mathit{z}))}^T \end{gathered}$$

The physical meaning of equation (15) is shown in Figure 9.

Figure 9.

The physical meaning of equation (15).

Equation (15) can be changed as follows

${}^{O}T{ }_c=\left[\begin{array}{cc}{}_C{}^O\mathit{R}& {}^{O}P{}_C \\ 0& 1\end{array}\right]$

16

Vector P has a transformation relationship between the coordinate system ${\Sigma }_C$ and ${\Sigma }_O$

${}^{O}P=\left[\begin{array}{cc}{}_C{}^{O}R& {}^{O}P{ }_C\\ 0& 1\end{array}\right]•{}^{C}P$

17

From equation (17), we have

${}^{O}P={}_C{}^{O}R{}^{C}P+{}^{O}P{ }_C$

18

Let $P=P_{A_i}$ , then equation (18) transforms into equation (13).

Obviously, using the CAG method eliminates a complicated formula derivation, and obtains the same results as using the analytical method.

Using the SolidWorks API functions to measure position coordinates of the relevant points, each element of ${}^o\mathit{T}{ }_c$ of equation (15) is calculated, and positions and postures of the moving platform are obtained.

The development of the computing system

Based on the above ideas, after the mechanism’s digital model is established, the sizes of the links and the moving platform, and the spherical pair’s positions can be changed and controlled. The cooperative relationship in the mechanism model is controlled based on the mechanism’s independent driving angles and the position of the static platform, which makes the mechanism model form the posture and position under all restrictions. Because the VB program automatically calls the SolidWorks measure function to measure the position of points and calculates the homogeneous coordinate matrix of the two coordinate systems, the posture of the moving platform is obtained. The forward kinematics solution of the mechanism can be obtained by the output functions of the VB program. Figure 8 shows the interface of the VB program of the developed forward kinematics system.

Calculation examples and discussion

Example A

The structure and motion parameters are as follows (see Figure 2): $l_1=0.0650$ , $l_2=0.2000$ , $l_3=0.2660$ , ${}^{o}P{ }_{A_{J_{{ }_0}}}={[0.3827\hspace{0.17em}\hspace{0.17em}0.3278\hspace{0.17em}\hspace{0.17em}0]}^T$ , ${}^{o}P{ }_{A_{K_{{ }_0}}}={[-0.3827\hspace{0.17em}\hspace{0.17em}-0.3278\hspace{0.17em}\hspace{0.17em}0]}^T$ , ${}^{o}P{ }_{A_{L_{{ }_0}}}={[0.3827\hspace{0.17em}\hspace{0.17em}-0.3278\hspace{0.17em}\hspace{0.17em}0]}^T$ , $m=0.1970$ , $n=0.1400$ , ${\text{h}}_L=0.0500$ (all in meters). The following six independently driving variables are known at any given moment (see Figure 3): ${\phi }_J={30}^{\circ }$ and ${\chi }_J={54.2186}^{\circ }$ ; ${\phi }_K={30}^{\circ }$ , ${\chi }_K={54.2186}^{\circ }$ , and ${\phi }_L={30}^{\circ }$ , ${\chi }_L={54.2186}^{\circ }$ . The pose of the mechanism needs to be determined.

Calculation using the analytical method

From equation (10), we have: ${}^{o}P{ }_{A_J}={[0.3827\hspace{0.17em}\hspace{0.17em}0.3278\hspace{0.17em}\hspace{0.17em}0.0500]}^T,$ ${}^{o}P{ }_{A_K}={[-0.3827-0.3278\hspace{0.17em}\hspace{0.17em}0.0500]}^T,$ and ${}^{o}P{ }_{A_L}={[0.3827\hspace{0.17em}\hspace{0.17em}-0.3278\hspace{0.17em}\hspace{0.17em}0.0500]}^T$ .

As shown in Figure 2, we can obtain

$\{\begin{array}{c}{}^{c}x{ }_{B_J}=0.1970\\ {}^{c}y{ }_{B_J}=0.1400\\ {}^c\mathit{z}{ }_{B_J}=0\end{array}\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{B_K}=-0.1970\\ {}^{c}y{ }_{B_K}=-0.1400\\ {}^c\mathit{z}{ }_{B_K}=0\end{array}\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{B_L}=0.1970\\ {}^{c}y{ }_{B_L}=-0.1400\\ {}^c\mathit{z}{ }_{B_L}=0\end{array}$

According to equation (2), we have: $d_{JK}=1.0079$ , $d_{JL}=0.6556$ , and $d_{KL}=0.7654$ .

According to equation (3), we have

$\{\begin{array}{c}{L}_J=0.2650\\ H_J=0.3646\end{array}\hspace{1em}\{\begin{array}{c}{L}_K=0.2650\\ H_K=0.3646\end{array}\hspace{1em}\{\begin{array}{c}{L}_L=0.2650\\ H_L=0.3646\end{array}$

Using the above results and equation (4) yields

$\{\begin{array}{c}{}^{c}x{ }_{A_J}=0.1970-0.2650\text{cos}{\varphi }_J\\ {}^{c}y{ }_{A_J}=0.1400+0.2650\text{sin}{\varphi }_J \\ {}^c\mathit{z}{ }_{A_J}=-0.3646 \end{array}\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{A_K}=-0.1970-0.2650\text{cos}{\varphi }_K\\ {}^{c}y{ }_{A_K}=-0.1400+0.2650\text{sin}{\varphi }_K\\ {}^c\mathit{z}{ }_{A_K}=-0.3646 \end{array}\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{A_L}=0.1970-0.2650\text{cos}{\varphi }_L \\ {}^{c}y{ }_{A_L}=-0.1400+0.2650\text{sin}{\varphi }_L\\ {}^c\mathit{z}{ }_{A_L}=-0.3646 \end{array}$

Using the above results yields the equations the same as in equation (6).We obtain the equations the same as in equation (7). Considering the structure of the mechanism, the motion ranges are given for ${\varphi }_J$ , ${\varphi }_K$ , and ${\varphi }_L$ (see Table 1).

Table 1.

Example A: The subordinate driving variables.

Driving variables	Range of motion	Result
${\varphi }_J$	30°–160°	152.4469°
${\varphi }_K$	225°–345°	338.9160°
${\varphi }_L$	210°–330°	245.4437°

The function ‘fsolve’ in MATLAB can be used to solve the equations the same as in equation (7). ³³ Subordinate driving variables ${\varphi }_i$ ( $i=J,K,L$ ) can be calculated and the output is summarized in Table 1. According to the results, from using equations (10) to (12) and equation (14), we have

${\mathit{R}}_c=\hspace{0.17em}\left[\hspace{0.17em}\begin{array}{ccc}0.9817& -0.1904& 0\\ 0.1904& 0.9817& 0\\ 0& 0& 1.0000\end{array}\right]\hspace{1em}{}^o\mathit{p}{ }_c=\left[\begin{array}{c}0.0087\\ -0.0122\\ 0.4146\end{array}\right]$

Calculation using the CAG method

Using the VB program to input the parameters into the system, the interface is shown in Figure 8 and the output calculation results are shown in Figure 8

$\{\begin{array}{l}{\varphi }_J=152.4453°\\ {\varphi }_K=338.9164°\\ {\varphi }_L=245.4402°\end{array}\hspace{1em}{\mathit{R}}_c=\hspace{0.17em}\left[\hspace{0.17em}\begin{array}{ccc}0.9817& -0.1904& 0\\ 0.1904& 0.9817& 0\\ 0& 0& 1.0000\end{array}\right]\hspace{0.17em}\hspace{1em}{}^o\mathit{p}{ }_c=\left[\begin{array}{c}0.0086\\ -0.0122\\ 0.4146\end{array}\right]$

Example B

The structure and motion parameters are as follows (see Figure 2): $l_1=0.1360$ , $l_2=0.1660$ , $l_3=0.1900$ , ${}^{o}P{ }_{A_{J_0}}={[0.2852\hspace{0.17em}\hspace{0.17em}0.2519\hspace{0.17em}\hspace{0.17em}0]}^T\hspace{0.17em}$ , ${}^{o}P{ }_{A_{K_0}}={[-0.3219\hspace{0.17em}\hspace{0.17em}-0.3552\hspace{0.17em}\hspace{0.17em}0]}^T\hspace{0.17em}$ , ${}^{o}P{ }_{A_{L_0}}={[0.2838\hspace{0.17em}\hspace{0.17em}-0.3577\hspace{0.17em}\hspace{0.17em}0]}^T$ , $m=0.0900$ , $n=0.0900$ , and $h_L=0.0500$ (all in meters). The following six independently driving joints are known at a certain moment (see Figure 3): ${\phi }_J={35}^{\circ }$ and ${\chi }_J={45}^{\circ }$ ; ${\phi }_K={35}^{\circ }$ , ${\chi }_K={45}^{\circ }$ and ${\phi }_L={35}^{\circ }$ , ${\chi }_L={45}^{\circ }$ . The goal at this point is to determine the pose of the mechanism at this moment. The motion range for ${\varphi }_J$ , ${\varphi }_K$ , and ${\varphi }_L$ are the same as in Example A.

Calculation using the analytical method

The details of this example are omitted here since it is like Example A. The outputs of the subordinate driving variables are summarized in Table 2.

Table 2.

Example B: The subordinate driving variables.

Driving variables	Range of motion	Result
${\varphi }_J$	30°–160°	149.5534
${\varphi }_K$ ${\varphi }_L$	225°–345°	329.9263
	210°–330°	240.2469

The final calculation results are

${\mathit{R}}_c=\left[\begin{array}{ccc}0.9836& -0.1808& 0\\ 0.1807& 0.9835& 0\\ 0& 0& 1.0000\end{array}\right]\hspace{1em}{}^o\mathit{p}{ }_c=\left[\begin{array}{c}-0.0177\\ -0.0523\\ \hspace{0.17em}\hspace{0.17em}0.3323\end{array}\right]$

Calculation using the CAG method

Using the VB program to input the parameters into the system, the interface and the calculation results are shown in Figure 10.

Figure 10.

The interface of the forward kinematics system of the mechanism for Example B.

$\{\begin{array}{l}{\varphi }_J=149.5532°\\ {\varphi }_K=329.9267°\\ {\varphi }_L=240.2456°\end{array}\hspace{1em}{\mathit{R}}_c=\left[\begin{array}{ccc}0.9835& -0.1808& 0\\ 0.1808& 0.9835& 0\\ 0& 0& 1.0000\end{array}\right]\hspace{1em}{}^o\mathit{p}{ }_c=\left[\begin{array}{c}-0.0177\\ -0.0524\\ \hspace{0.17em}\hspace{0.17em}0.3323\end{array}\right]$

Example C

The structure and motion parameters are (see Figure 2): $l_1=0.1360$ , $l_2=0.1660$ , $l_3=0.1900$ , ${}^{o}P{ }_{A_{J_0}}={[0.2852\hspace{0.17em}\hspace{0.17em}0.2519\hspace{0.17em}\hspace{0.17em}0]}^T\hspace{0.17em}$ , ${}^{o}P{ }_{A_{K_0}}={[-0.3219\hspace{0.17em}\hspace{0.17em}-0.3552\hspace{0.17em}\hspace{0.17em}0]}^T\hspace{0.17em}$ , ${}^{o}P{ }_{A_{L_0}}={[0.2838\hspace{0.17em}\hspace{0.17em}-0.3577\hspace{0.17em}\hspace{0.17em}0]}^T$ , $m=0.0900$ , $n=0.0900$ , $h_L=0.0500$ (all in meters). The following six independently driving joints are known in a certain moment (see Figure 3): ${\phi }_J=\text{14.3640}°$ , ${\phi }_K\text{=85.6695}°$ , ${\phi }_L=\text{320.9403}°$ and ${\chi }_J=\text{71.4461}°$ , ${\chi }_K=\text{291.3749}°$ , ${\chi }_L=\text{96.1402}°$ . The goal now is to determine the pose of the mechanism at this moment. The motion range for ${\varphi }_J$ , ${\varphi }_K$ , and ${\varphi }_L$ are the same of Example A.

Calculation using the analytical method

Since this example is similar to Example A, the details are omitted here. The outputs of subordinate driving variables are summarized in Table 3.

Table 3.

Example C: The subordinate driving variables.

Driving variables	Range of motion	Result
${\varphi }_J$	30°–160°	$\text{137.7895}°$
${\varphi }_K$	225°–345°	$\text{282.3500}°$
${\varphi }_L$	210°–330°	$\text{210.8140}°$

The final calculation results are

${\mathit{R}}_C=\left[\begin{array}{l}\text{ 0.9352 0.225}0\text{ 0.2734}\\ -\text{0.1433 0.9466 }-\text{0.2890}\\ -\text{0.3238 0.2311 0.9175}\end{array}\right]\hspace{1em}{}^O\mathit{p}{ }_c=\left[\begin{array}{l}-\text{0.0184}\\ -\text{0.0516}\\ \text{ 0.2963}\end{array}\right]$

Calculation using the CAG method

Using VB program to input the parameters into the system, the interface and the calculation results are shown in Figure 11

Figure 11.

The interface of forward kinematics system of mechanism for Example C.

${\mathit{R}}_C=\left[\begin{array}{l}\text{ 0.9352 0.2250\hspace{1em} 0.2734}\\ -\text{0.1433 0.9466 }-\text{0.2890}\\ -\text{0.3238 0.2311 0.9175}\end{array}\right]\hspace{1em}{}^O\mathit{p}{ }_c=\left[\begin{array}{l}-\text{0.0184}\\ -\text{0.0517}\\ \text{ 0.2963}\end{array}\right]\hspace{1em}\{\begin{array}{l}{\varphi }_J=\text{137.7936}°\\ {\varphi }_K=\text{282.3532}°\\ {\varphi }_L=\text{210.8177}°\end{array}$

The reachable poses of the platform can be shown in SolidWorks if need be. Here, we output the reachable poses of the platform for Example C using the output button “Show of model” in the interface of forward kinematics system of mechanism, shown as Figure 12.

Figure 12.

The reachable poses of the platform for Example C.

Discussion

To compare the accuracy of both methods, equation (7) can be rewritten as follows

$\{\begin{array}{l}{f}_1=a_1\text{sin}{\varphi }_K\text{sin}{\varphi }_L+a_2\text{cos}{\varphi }_K\text{cos}{\varphi }_L+a_3\text{sin}{\varphi }_K+a_4\text{cos}{\varphi }_K+a_5\text{sin}{\varphi }_L+a_6\text{cos}{\varphi }_L+a_7\\ f_2=b_1\text{sin}{\varphi }_J\text{sin}{\varphi }_L+b_2\text{cos}{\varphi }_J\text{cos}{\varphi }_L+b_3\text{sin}{\varphi }_J+b_4\text{cos}{\varphi }_J+b_5\text{sin}{\varphi }_L+b_6\text{cos}{\varphi }_L+b_7\\ f_3=c_1\text{sin}{\varphi }_J\text{sin}{\varphi }_K+c_2\text{cos}{\varphi }_J\text{cos}{\varphi }_K+c_3\text{sin}{\varphi }_J+c_4\text{cos}{\varphi }_J+c_5\text{sin}{\varphi }_K+c_6\text{cos}{\varphi }_K+c_7\end{array}$

19

The calculated values of the variables ${\varphi }_J$ , ${\varphi }_K$ , and ${\varphi }_L$ are substituted into equation (19). Then, the values for $f_1$ , $f_2$ , and $f_3$ are calculated. Their absolute values are smaller when calculation accuracy is higher.

Table 4 compares the results of the two methods for Examples A, B, and C, through which the precision of the mechanism motion obtained by the CAG method is not lower than that of the analytical method. The reason for this is that the calculation using the analytical method requires multiple mathematical transformations and data transfers of the intermediate variable while calculation using the CAG method is concise and visual. Table 4 shows the calculation times are observably less when using CAG method than using analytical method, which is very meaningful for meeting the requirements of the mechanism real-time control.

Table 4.

Comparison between the calculation precisions and calculation times of the two methods.

	Example A		Example B		Example C
	Precision	Time/s	Precision	Time/s	Precision	Time/s
Analytical method	f 1 = −2.020 × 10−5	5.96	f 1 = −8.063 × 10−6	6.58	f 1 = −6.180 × 10−8	6.96
	f 2 = 5.900 × 10−6		f 2 = 5.428 × 10−6		f 2 = 2.296 × 10−7
	f 3 = −2.135 × 10−4		f 3 = 1.420 × 10−7		f 3 = 2.611 × 10−6
CAG method	f 1 = 1.400 × 10−6	0.38	f 1 = 4.270 × 10−7	0.46	f 1 = 2.650 × 10−8	0.97
	f 2 = 1.200 × 10−6		f 2 = 1.059 × 10−6		f 2 = 1.241 × 10−7
	f 3 = −2.135 × 10−4		f 3 = 3.400 × 10−8		f 3 = 3.430 × 10−8

Discussions on relevant issues

In this section, some issues about the analysis of the velocity and acceleration, the singularity of the mechanism, and the application of the CAG method in a real system are discussed.

Forward kinematic analysis of the velocity and acceleration of the 3-RRRS mechanism

Forward kinematic analysis of the velocity and acceleration of the 3-RRRS mechanism is to determine the motion velocity and acceleration of the moving platform, based on analyzing the position and attitude of the mechanisms and giving the velocity and acceleration of independent driving joint. The proposed forward kinematic analysis of a 3-RRRS mechanism using a CAG method is focused on solving the complex derivation of the kinematics equation and the solving process in the pose analysis. In the forward kinematic analysis of the velocity and acceleration of the 3-RRRS mechanisms, the relevant expressions of the velocity and acceleration analysis need to be deduced. The obtained joint variables using the CAG method are substituted into the velocity and acceleration analysis equations of the mechanism, and then the velocity and acceleration of the moving platform are solved. Here, the velocity analysis of the moving platform is used as an example, and the relevant expressions are given.

The velocity analysis of the moving platform is to determine $\stackrel{•}{{}^o\mathit{p}{ }_c}$ and $\omega$ of the moving platform. It is assumed that the joint variables ${\phi }_i$ and ${\chi }_i$ of the three branched chains are the independently driven joint variables; therefore, for the mechanism shown in Figures 2 and 3, ${\varphi }_i$ , ${\phi }_i$ , ${\chi }_i$ , ${\stackrel{•}{\phi }}_i$ , and ${\stackrel{•}{\chi }}_i$ (i = J, K, L) are known variables. According to equation (3), we have

$\left[\begin{array}{c}{\stackrel{•}{L}}_i\\ {\stackrel{•}{H}}_i\end{array}\right]=\left[\begin{array}{cc}-H_i& -l_3\text{sin}\left({\phi }_i+{\chi }_i\right)\\ L_i-l_1& l_3\text{cos}\left({\phi }_i+{\chi }_i\right)\end{array}\right]\left[\begin{array}{c}{\stackrel{•}{\phi }}_i\\ {\stackrel{•}{\chi }}_i\end{array}\right]$

20

The velocities of the subordinate driving joints can be obtained by differentiating with respect to time for equation (6), and then we have

$\mathit{B}\left[\begin{array}{c}{\stackrel{•}{\phi }}_J\\ {\stackrel{•}{\phi }}_K\\ {\stackrel{•}{\phi }}_L\end{array}\right]=\mathit{C}\left[\begin{array}{c}\begin{array}{c}{\stackrel{•}{L}}_J\\ {\stackrel{•}{H}}_J\\ {\stackrel{•}{L}}_K\end{array}\\ {\stackrel{•}{H}}_K\\ {\stackrel{•}{L}}_L\\ {\stackrel{•}{H}}_L\end{array}\right]$

21

where $\mathit{B}=\left[\begin{array}{ccc}{\mathit{B}}_1& {\mathit{B}}_2& {\mathit{B}}_3\end{array}\right]$ , $\mathit{C}=\left[\begin{array}{ccc}{\mathit{C}}_1& {\mathit{C}}_2& {\mathit{C}}_3\end{array}\right]$ .

${\mathit{B}}_1=\left[\begin{array}{c}0\\ \Delta x_{JL}{L}_J\text{sin}{\varphi }_J+\Delta y_{JL}{L}_J\text{cos}{\varphi }_J\\ \Delta x_{JK}{L}_J\text{sin}{\varphi }_J+\Delta y_{JK}{L}_J\text{cos}{\varphi }_J\end{array}\right]\hspace{1em}{\mathit{B}}_2=\left[\begin{array}{c}\Delta x_{KL}{L}_K\text{sin}{\varphi }_K+\Delta y_{KL}{L}_K\text{cos}{\varphi }_K\\ 0\\ -\Delta x_{JK}{L}_K\text{sin}{\varphi }_K-\Delta y_{JK}{L}_K\text{cos}{\varphi }_K\end{array}\right]\hspace{1em}{\mathit{B}}_3=\left[\begin{array}{c}-\Delta x_{KL}{L}_L\text{sin}{\varphi }_L-\Delta y_{KL}{L}_L\text{cos}{\varphi }_L\\ -\Delta x_{JL}{L}_L\text{sin}{\varphi }_L-\Delta y_{JL}{L}_L\text{cos}{\varphi }_L\\ 0\end{array}\right]$

${\mathit{C}}_1=\left[\begin{array}{cc}\begin{array}{c}0\\ -\Delta x_{JL}\text{cos}{\varphi }_J+\Delta y_{JL}\text{sin}{\varphi }_J\\ -\Delta x_{JK}\text{cos}{\varphi }_L+\Delta y_{JK}\text{sin}{\varphi }_J\end{array}& \begin{array}{c}0\\ H_J-H_L\\ H_J-H_K\end{array}\end{array}\right]{\mathit{C}}_2=\left[\begin{array}{cc}\begin{array}{c}\Delta x_{KL}\text{cos}{\varphi }_K-\Delta y_{KL}\text{sin}{\varphi }_K\\ 0\\ -\Delta x_{JK}\text{cos}{\varphi }_K+\Delta y_{JK}\text{sin}{\varphi }_K\end{array}& \begin{array}{c}-H_K+H_L\\ 0\\ -H_K+H_J\end{array}\end{array}\right]\hspace{1em}{\mathit{C}}_3=\left[\begin{array}{cc}\begin{array}{c}-\Delta x_{KL}\text{cos}{\varphi }_L+\Delta y_{KL}\text{sin}{\varphi }_L\\ -\Delta x_{JL}\text{cos}{\varphi }_L+\Delta y_{JL}\text{sin}{\varphi }_L\\ 0\end{array}& \begin{array}{c}{H}_K-H_L\\ H_J-H_L\\ 0\end{array}\end{array}\right]$

$\Delta x_{KL}={}^{c}x{ }_{B_K}-{}^{c}x{ }_{B_L}-L_K\text{cos}{\varphi }_K+L_L\text{cos}{\varphi }_L\hspace{1em}\Delta y_{KL}={}^{c}y{ }_{B_K}-{}^{c}y{ }_{B_L}+L_K\text{sin}{\varphi }_K-L_L\text{sin}{\varphi }_L$

$\Delta x_{JL}={}^{c}x{ }_{B_J}-{}^{c}x{ }_{B_L}-L_J\text{cos}{\varphi }_J+L_L\text{cos}{\varphi }_L\hspace{1em}\Delta y_{JL}={}^{c}y{ }_{B_J}-{}^{c}y{ }_{B_L}+L_J\text{sin}{\varphi }_J-L_L\text{sin}{\varphi }_L$

$\Delta x_{JK}={}^{c}x{ }_{B_J}-{}^{c}x{ }_{B_K}-L_J\text{cos}{\varphi }_J+L_K\text{cos}{\varphi }_K\hspace{1em}\Delta y_{JK}={}^{c}y{ }_{B_J}-{}^{c}y{ }_{B_K}+L_J\text{sin}{\varphi }_J-L_K\text{sin}{\varphi }_K$

According to the joint variables of the three branched chains, it is easy to solve ${{}^o\stackrel{•}{\mathit{p}}}_c$ and $\omega$ of the moving platform.

Firstly, by differentiating with respect to time for equation (10) and considering ${{}^o\stackrel{•}{\mathit{p}}}_{Ai}={{}^o\stackrel{•}{\mathit{p}}}_{A_i}=0$ (i = J, K, L), then the following can be deduced

${\stackrel{•}{\mathit{R}}}_c\left({}^c\mathit{p}{ }_{A_i}-{}^c\mathit{p}{ }_{A_j}\right)+{\mathit{R}}_c({{}^c\stackrel{•}{\mathit{p}}}_{A_i}-{{}^c\stackrel{•}{\mathit{p}}}_{A_j})=0$

22

Differentiating with respect to time for equations (3) and (4), then they can be combined to obtain

${{}^c\stackrel{•}{\mathit{p}}}_{A_i}={\mathit{A}}_i\left[\begin{array}{c}{\stackrel{•}{\varphi }}_i\\ {\stackrel{•}{\phi }}_i\\ {\stackrel{•}{\chi }}_i\end{array}\right]$

23

$\text{where}\hspace{1em}{\mathit{A}}_i=\left[\begin{array}{ccc}{L}_i\text{sin}{\varphi }_i& H_i\text{cos}{\varphi }_i& l_3\text{cos}{\varphi }_i\text{sin}\left({\phi }_i+{\chi }_i\right)\\ L_i\text{cos}{\varphi }_i& -H_i\text{sin}{\varphi }_i& -l_3\text{sin}{\phi }_i\text{sin}\left({\phi }_i+{\chi }_i\right)\\ 0& -L_i+l_1& -l_3\text{cos}\left({\phi }_i+{\chi }_i\right)\end{array}\right]$

24

Then according to the basic definition of velocity, we have

$\stackrel{•}{\mathit{R}}\left(t\right)=\mathit{S}\left(\mathit{\omega }\right)\mathit{R}=\mathit{\omega }\times \mathit{R}$

25

By using equations (22) to (25), let $\stackrel{•}{{\varphi }_{ i}}={\left[\begin{array}{ccc}{\stackrel{•}{\varphi }}_i& {\stackrel{•}{\phi }}_i& {\stackrel{•}{\chi }}_i\end{array}\right]}^T$ and $\stackrel{•}{{\varphi }_{ j}}={\left[\begin{array}{ccc}{\stackrel{•}{\varphi }}_j& {\stackrel{•}{\phi }}_j& {\stackrel{•}{\chi }}_j\end{array}\right]}^T$ , then the following equation can be deduced

$\mathit{\omega }\times {\mathit{R}}_c\left({}^c\mathit{p}{ }_{{\mathit{A}}_i}-{}^c\mathit{p}{ }_{{\mathit{A}}_j}\right)=\left[\begin{array}{cc}-{\mathit{R}}_c{\mathit{A}}_i& {\mathit{R}}_c{\mathit{A}}_j\end{array}\right]\left[\begin{array}{c}{\stackrel{•}{\varphi }}_i\\ {\stackrel{•}{\varphi }}_j\end{array}\right]$

26

$\text{Let\hspace{1em}\hspace{1em}\hspace{1em}}{\mathit{\varpi }}_{ij}={\mathit{R}}_c\left({}^c\mathit{p}{ }_{A_i}-{}^c\mathit{p}{ }_{A_j}\right)=\left[\begin{array}{c}{\varpi }_1\\ {\varpi }_2\\ {\varpi }_3\end{array}\right]$

27

The above equation can be determined according to the position and pose solutions obtained previously. Equation (26) can be rewritten and we have

$-{\mathit{\varpi }}_{ij}\times \mathit{\omega }=\left[\begin{array}{cc}-{\mathit{R}}_c{\mathit{A}}_i& {\mathit{R}}_c{\mathit{A}}_j\end{array}\right]\left[\begin{array}{c}{\stackrel{•}{\varphi }}_i\\ {\stackrel{•}{\varphi }}_j\end{array}\right]$

28

The following antisymmetric matrices are established:

$\mathit{S}\left({\mathit{\varpi }}_{ij}\right)=\left[\begin{array}{ccc}0& -{\varpi }_3& {\varpi }_2\\ {\varpi }_3& 0& -{\varpi }_1\\ -{\varpi }_2& {\varpi }_1& 0\end{array}\right]$

29

Therefore, equation (28) can be rewritten as follows

$-\mathit{S}\left({\mathit{\varpi }}_{ij}\right)\mathit{\omega }=\left[\begin{array}{cc}-{\mathit{R}}_c{\mathit{A}}_i& {\mathit{R}}_c{\mathit{A}}_j\end{array}\right]\left[\begin{array}{c}{\stackrel{•}{\varphi }}_i\\ {\stackrel{•}{\varphi }}_j\end{array}\right]$

30

If $det \mathit{S}\left({\mathit{\varpi }}_{ij}\right)\ne 0$ , then we have

$\mathit{\omega }={\mathit{S}}^{-1}\left({\mathit{\varpi }}_{ij}\right)\left[\begin{array}{cc}{\mathit{R}}_c{\mathit{A}}_i& -{\mathit{R}}_c{\mathit{A}}_j\end{array}\right]\left[\begin{array}{c}{\stackrel{•}{\varphi }}_i\\ {\stackrel{•}{\varphi }}_j\end{array}\right]$

31

It should be noted that if $\text{det }\mathit{S}\left({\mathit{\varpi }}_{ij}\right)=0$ , that is, $\mathit{S}\left({\mathit{\varpi }}_{ij}\right)$ is not a full rank matrix. Then, in order to construct a full rank matrix, some unique equations similar to equation (30) must be added.

Then, differentiating with respect to time for equation (13), we have

${{}^o\stackrel{•}{\mathit{p}}}_c+{\stackrel{•}{\mathit{R}}}_c{}^c\mathit{p}{ }_{A_i}+{\mathit{R}}_c{{}^c\stackrel{•}{\mathit{p}}}_{A_i}=0$

32

Similarly, the following equation is easy to obtain

${{}^o\stackrel{•}{\mathit{p}}}_c=-{\mathit{R}}_c {\mathit{A}}_i {\stackrel{•}{\varphi }}_i-\mathit{S}\left(\mathit{\omega }\right) {\mathit{R}}_c{}^c\mathit{p}{ }_{{\mathit{A}}_i}$

33

Thus far, using equations (31) and (33), $\omega$ and ${{}^o\stackrel{•}{\mathit{p}}}_c$ can be solved.

The derivation process of acceleration expression for the moving platform is like that of the velocity. The specific expressions are omitted here.

By using VB programming, these calculation equations and the solving process can be merged into the kinematics system of the mechanism, as shown in Figure 8. More specific research will be conducted in our future work.

Here, we use a computational example to illustrate the problem how to construct a full rank matrix to solve $\omega$ in equations (30) and (31) if $\text{det }\mathit{S}\left({\mathit{\varpi }}_{ij}\right)=0$ .

The structure and motion parameters are (see Figure 2): $l_1=0.2$ , $l_2=0.8$ , $l_3=1$ , ${}^{o}P{ }_{{\mathit{A}}_{J_0}}={[3\hspace{0.17em}\hspace{0.17em}1\hspace{0.17em}\hspace{0.17em}0]}^T\hspace{0.17em}$ , ${}^{o}P{ }_{{\mathit{A}}_{K_0}}={[0\hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}0]}^T\hspace{0.17em}$ , ${}^{o}P{ }_{{\mathit{A}}_{L_0}}={[2\hspace{0.17em}\hspace{0.17em}-1\hspace{0.17em}\hspace{0.17em}0]}^T$ , $m=0.5$ , $n=0.5$ , $h_L=0.05$ (all in meters). The following six independently driving joints are known in a certain moment (see Figure 3): ${\phi }_J=\text{0}°$ , ${\phi }_K\text{=0}°$ , ${\phi }_L=0°$ and ${\chi }_J=\text{90}°$ , ${\chi }_J=90°$ , ${\chi }_L=90°$ . The value of velocity are given for ${\stackrel{•}{\phi }}_J=0$ , ${\stackrel{•}{\chi }}_J=-0.05$ , ${\stackrel{•}{\phi }}_K=0$ , ${\stackrel{•}{\chi }}_K=-0.05$ , ${\stackrel{•}{\phi }}_L=0$ , ${\stackrel{•}{\chi }}_L=-0.05$ (unit: rad/s). The goal now is to determine the pose and motion velocity of the mechanism at this moment.

From equation (10), we have: ${}^{o}P{ }_{{\mathit{A}}_J}={[3\hspace{0.17em}\hspace{0.17em}1\hspace{0.17em}\hspace{0.17em}0.05]}^T,$ ${}^{o}P{ }_{{\mathit{A}}_K}={[0\text{ 0 0.05}]}^T,$ ${}^{o}P{ }_{{\mathit{A}}_L}={[2\hspace{0.17em}\hspace{0.17em}-1\hspace{0.17em}\hspace{0.17em}0.05]}^T.$

As shown in Figure 2, we can obtain

$\{\begin{array}{c}{}^{c}x{ }_{B_J}=0.5\\ {}^{c}y{ }_{B_J}=0.5\\ {}^c\mathit{z}{ }_{B_J}=0\end{array}\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{B_K}=-0.5\\ {}^{c}y{ }_{B_K}=-0.5\\ {}^c\mathit{z}{ }_{B_K}=0\end{array}\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{B_L}=0.5\\ {}^{c}y{ }_{B_L}=-0.5\\ {}^c\mathit{z}{ }_{B_L}=0\end{array}$

According to equation (2), we have: $d_{JK}=\sqrt{5}$ , $d_{JL}=\sqrt{5}$ , $d_{KL}=\sqrt{10}$ .

According to equation (3), we have

$\{\begin{array}{c}{L}_J=1\\ H_J=1\end{array}\hspace{1em}\{\begin{array}{c}{L}_K=1\\ H_K=1\end{array}\hspace{1em}\{\begin{array}{c}{L}_L=1\\ H_L=1\end{array}$

Using the above results and equation (4), yields

$\{\begin{array}{c}{}^{c}x{ }_{{\mathit{A}}_J}=0.5-\text{cos}{\varphi }_J\\ {}^{c}y{ }_{{\mathit{A}}_J}=0.5+\text{sin}{\varphi }_J\\ {}^c\mathit{z}{ }_{{\mathit{A}}_J}=-1 \end{array}\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{{\mathit{A}}_K}=-0.5-\text{cos}{\varphi }_K\\ {}^{c}y{ }_{{\mathit{A}}_K}=-0.5+\text{sin}{\varphi }_K\\ {}^c\mathit{z}{ }_{{\mathit{A}}_K}=-1\end{array}\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{{\mathit{A}}_L}=0.5-\text{cos}{\varphi }_L\\ {}^{c}y{ }_{{\mathit{A}}_L}=-0.5+\text{sin}{\varphi }_L\\ {}^c\mathit{z}{ }_{{\mathit{A}}_L}=-1 \end{array}$

Using the CAG method, yields

$\{\begin{array}{l}{\varphi }_J=180°\\ {\varphi }_K=0°\\ {\varphi }_L=270°\end{array}\hspace{1em}{}^o\mathit{p}{ }_c=\left[\begin{array}{l}1.5\\ 0.5\\ 1.0\end{array}\right]\hspace{1em}{\mathit{R}}_c=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{{\mathit{A}}_J}=1.5\\ {}^{c}y{ }_{{\mathit{A}}_J}=0.5\\ {}^c\mathit{z}{ }_{{\mathit{A}}_J}=-1 \end{array}\{\begin{array}{c}{}^{c}x{ }_{{\mathit{A}}_K}=-1.5\\ {}^{c}y{ }_{{\mathit{A}}_K}=-0.5\\ {}^c\mathit{z}{ }_{{\mathit{A}}_K}=-1\end{array}\hspace{1em}\{\begin{array}{c}{}^{c}x{ }_{{\mathit{A}}_L}=0.5\\ {}^{c}y{ }_{{\mathit{A}}_L}=-1.5\\ {}^c\mathit{z}{ }_{{\mathit{A}}_L}=-1 \end{array}$

According to equations (20) and (21), we can get

$\left[\begin{array}{l}{\stackrel{•}{L}}_J\\ {\stackrel{•}{H}}_J\end{array}\right]=\left[\begin{array}{l}{\stackrel{•}{L}}_K\\ {\stackrel{•}{H}}_K\end{array}\right]=\left[\begin{array}{l}{\stackrel{•}{L}}_L\\ {\stackrel{•}{H}}_L\end{array}\right]=\left[\begin{array}{c}0.05\\ 0\end{array}\right]\hspace{1em}B=\left[\begin{array}{ccc}\text{0}& \text{1}& -2\\ -2& 0& 1\\ -1& -1& 0\end{array}\right]\hspace{1em}C=\lfloor \begin{array}{cccccc}0& 0& -2& 0& -1& 0\\ -1& 0& 0& 0& -2& 0\\ -3& 0& -3& 0& 0& 0\end{array}\rfloor$

Then from $B{[{\stackrel{•}{\phi }}_J {\stackrel{•}{\phi }}_K {\stackrel{•}{\phi }}_L]}^T=C{[0.05\text{ 0 }0.05\text{ 0 }0.05\text{ 0}]}^T$ , we can solve for $\stackrel{•}{{\varphi }_J}=\stackrel{•}{{\varphi }_K}=\stackrel{•}{{\varphi }_L}=0.15$ (unit: rad/s). From equation (27), we have

${\mathit{\varpi }}_{JK}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left(\left[\begin{array}{c}1.5\\ 0.5\\ -1\end{array}\right]-\left[\begin{array}{c}-1.5\\ -0.5\\ -1\end{array}\right]\right)=\left[\begin{array}{c}3\\ 1\\ 0\end{array}\right]\hspace{1em}\mathit{S}({\mathit{\varpi }}_{JK})=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& -3\\ -1& 3& 0\end{array}\right]$

Since $\text{det}(\mathit{S}({\mathit{\varpi }}_{JK}))=0$ , we need to do the same thing to get $\mathit{S}({\mathit{\varpi }}_{JL})$

$\mathit{S}({\mathit{\varpi }}_{JL})=\left[\begin{array}{ccc}0& 0& 2\\ 0& 0& -1\\ -2& 1& 0\end{array}\right]$

According to equation (24), we can obtain

${\mathit{A}}_J=\left[\begin{array}{ccc}0& -1& -1\\ -1& 0& 0\\ 0& -0.8& 0\end{array}\right]\hspace{1em}{\mathit{A}}_K=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 0\\ 0& -0.8& 0\end{array}\right]\hspace{1em}{\mathit{A}}_L=\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 1\\ 0& -0.8& 0\end{array}\right]$

Therefore, according to equation (30), when i = J, i = K, and i = J, i = L, we can get

$-\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& -3\\ -1& 3& 0\end{array}\right]\left[\begin{array}{c}{w}_x\\ w_y\\ w_{\mathit{z}}\end{array}\right]=\left[\begin{array}{cccccc}0& 1& 1& 0& 1& 1\\ 1& 0& 0& 1& 0& 0\\ 0& 0.8& 0& 0& -0.8& 0\end{array}\right]{\left[\begin{array}{cccccc}0.15& 0& -0.05& 0.15& 0& -0.05\end{array}\right]}^T$

$-\left[\begin{array}{ccc}0& 0& 2\\ 0& 0& -1\\ -2& 1& 0\end{array}\right]\left[\begin{array}{c}{w}_x\\ w_y\\ w_{\mathit{z}}\end{array}\right]=\left[\begin{array}{cccccc}0& 1& 1& -1& 0& 0\\ 1& 0& 0& 0& 1& 1\\ 0& 0.8& 0& 0& -0.8& 0\end{array}\right]{\left[\begin{array}{cccccc}0.15& 0& -0.05& 0.15& 0& -0.05\end{array}\right]}^T$

Using the above results, yields: $\{\begin{array}{c}2w_x-w_y=0\\ w_x-3w_y=0\\ w_{\mathit{z}}=0.1\end{array}$ .

Namely, w_x = 0, w_y = 0, w_z = 0.1 (unit: rad/s), $\mathit{S}(\mathit{\varpi })=\left[\begin{array}{ccc}0& -0.1& 0\\ 0.1& 0& 0\\ 0& 0& 0\end{array}\right]$ .

Accordingly, when i = J, the following results can be obtained from equation (33). (unit: rad/s)

${{}^o\stackrel{•}{\mathit{p}}}_c=-\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& -1& -1\\ -1& 0& 0\\ 0& -0.8& 0\end{array}\right]\left[\begin{array}{c}0.15\\ 0\\ -0.05\end{array}\right]-\left[\begin{array}{ccc}0& -0.1& 0\\ 0.1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}1.5\\ 0.5\\ -1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Singularity analysis

Singular (or special) configuration is a complex problem in the study of a closed-loop mechanism, especially a parallel mechanism. For a long time, there have been many scholars that have given much attention to the study of singular configuration. ³⁴

Singularities can be divided into three forms: boundary singularity, local singularity, and structural singularity. Singular configuration is an inherent property of a mechanism, which has a serious impact on the working performance of the mechanism. When the mechanism is in a specific configuration, its velocity Jacobian matrix is a singular matrix, that is to say, the determinant value of the velocity Jacobian matrix is zero, infinite, or uncertain. At this time, the configuration of the mechanism is called the singular configuration.

Boundary singularity: When the determinant value of the velocity Jacobian matrix is equal to zero, then the mechanism is in the boundary singular configuration.

Local singularity: When the determinant of the velocity Jacobian matrix tends to be infinite, then the mechanism is in a local singular configuration. The local singularity means that there is an uncontrollable DOF at the end of the mechanism.

Structural singularity: When the determinant of the velocity Jacobian matrix tends to be zero, then the mechanism is in a structural singular configuration. The structural singularity is also a unique characteristic of the parallel mechanisms. Only when the special sizes of the mechanisms are satisfied can the structural singularity be generated.

For this study, a CAG method is used to analyze the motion of the mechanism. When the structure and motion parameters of the mechanism are input, then the forward kinematics system could automatically form and display the real-time structure of the mechanism. It is more convenient for researchers to judge the singular configuration of the mechanism and improve the structural and motion parameters of the mechanism in the design and control of the mechanism by calculating and analyzing the determinant value of the velocity Jacobian matrix and observing the real-time configuration of the mechanism.

Specifically, the determinant values for the coefficient matrix $\mathit{S}\left({\mathit{\varpi }}_{ij}\right)$ on the left side and $\left[\begin{array}{cc}-{\mathit{R}}_c{\mathit{A}}_i& {\mathit{R}}_c{\mathit{A}}_j\end{array}\right]$ on the right side can be calculated for equation (30).

If the determinant value of the left coefficient matrix $\mathit{S}\left({\mathit{\varpi }}_{ij}\right)$ is not zero and that of the right coefficient matrix $\left[\begin{array}{cc}-{\mathit{R}}_c{\mathit{A}}_i& {\mathit{R}}_c{\mathit{A}}_j\end{array}\right]$ is zero, then the mechanism may have the boundary singularity.

If the determinant value of the left coefficient matrix $\mathit{S}\left({\mathit{\varpi }}_{ij}\right)$ is zero and that of the right coefficient matrix $\left[\begin{array}{cc}-{\mathit{R}}_c{\mathit{A}}_i& {\mathit{R}}_c{\mathit{A}}_j\end{array}\right]$ is not zero, then the mechanism may have the local singularity.

If the determinant values of the left coefficient matrix $\mathit{S}\left({\mathit{\varpi }}_{ij}\right)$ and the right coefficient matrix $\left[\begin{array}{cc}-{\mathit{R}}_c{\mathit{A}}_i& {\mathit{R}}_c{\mathit{A}}_j\end{array}\right]$ are both zero, then the mechanism may have the structural singularity.

By using VB programming, these criteria can be merged into the kinematics system of the mechanism, as shown in Figure 8. The system can automatically prompt and display the real-time configuration of the mechanism when the mechanism configurations meet the above conditions. Researchers can observe and judge the singularity configurations of the mechanism more intuitively.

More systematic and specific research on this will be conducted in our follow-up work.

The implementation of the proposed method in a real system

When the structure of each component of the mechanism is determined and assembled for a real structure of the 3-RRRS mechanism, then the structural parameters of each component are determined. After the three branched chains are connected to the static platform, the coordinates of the connection points in a global reference ${\Sigma }_o$ are determined in accordance with the connecting position between the three branched chains and static platform. In a real structure of the 3-RRRS mechanism, the three branched chains are equipped with servo drive motors in the place of 9 R joints, and a rotary encoder is correspondingly installed on each motor. The rotation angles, speeds, and accelerations for the motors can be detected and calculated using rotary encoders and control system, which are the input variables of the driving joints. The motion poses and parameters of the moving platform for the 3-RRRS mechanism can be determined by inputting the structural parameters of each component and input variables of each driving joint into the motion analysis system as shown in Figure 8.

As a specific application for the 3-RRRS mechanism (the kinematic sketch is shown as Figure 2), let h_L = 0. That is, removing the elements contacting the static platform from the S pairs, the S balls of the S pairs will contact the static platform directly, another branched chain is added, and the moving platform is transferred as body installing controllers; the reptile-like, quadruped MiniQuad-I robot was developed by Huazhong University of Science and Technology. Figure 13 shows the structure.

Figure 13.

The structure of the MiniQuad-Irobot.

Conclusions

This article presented a 3-RRRS mechanism with subordinate drive characteristics. A CAG method was developed for the kinematics analysis of this mechanism. Based on the CAG method, the forward kinematics system of the mechanism was implemented by using a VB program to make a secondary development of SolidWorks. When using the proposed method for kinematics analysis of the mechanism, the detailed derivation of the kinematics formulas is not required. Only the constraint relationships of the mechanism should be explicit, and the active and passive constraints should be determined. When the mechanism’s digital model is established and the cooperated relations are considered, then the position and posture of the mechanism can be expressed clearly by the proposed forward kinematics calculation system for the 3-RRRS mechanism. The logic of this method could be widely applied to the motion analysis of mechanisms with subordinate drives, and when using the analytical method, it overcomes the difficulty in the forward kinematics.