Abstract: The dynamic analysis of a novel 3-DOF robot shoulder joint based on orthogonal spherical 3-RRR parallel mechanism was presented. The kinematics constraint equation of this mechanism was established based on the geometric structure and the kinematics model of moving platform was derived. The dynamics model was established based on Lagrange method and the effective inertia, coupling inertia and driving torque of the shoulder joint were analyzed. The variation of the dynamics parameters with the change of the mechanism was discussed and the dynamic coupling relationship between branches was analyzed. The analysis results show that the posture change of the mechanism has a great influence on the dynamics parameters in the process of movement. The results will be useful to improve the control scheme of this mechanism and the selection of servo motor.
INTRODUCTION
Spherical 3-DOF (degrees of freedom) parallel mechanism is a limited-DOF parallel mechanism which can be widely used for different kinds of practical applications (Zhen et al., 2006). It both maintains the inherent advantages of parallel mechanisms and possesses several other advantages in terms of simple structure, low cost in manufacturing and operations. Therefore it has a potentially wide range of applications, such as motion simulator, bionic joint and machine tools. It is attracting the attention of the various researchers and many prototypes based on spherical 3-DOF parallel manipulator have been developed, such as agile eye (Gosselin and Hamel, 1994; Gosselin and St-Pierre, 1997), NC rotary table (Zeng et al., 2001), wrist joint (Sun et al., 2003), shoulder joint (Jin and Rong, 2007), waist joint (Li and Jin, 2007), bionic eye (Sellaouti et al., 2002) etc. The dynamics is an important research content of robot which plays an important role in the, trajectory planning, driver selection and control scheme of the robot (Craig, 2011) and the establishment of the dynamic model for mechanism is the premise and basis for the dynamics analysis of parallel mechanism (Feng et al., 2006; Ji et al., 2012; Chen et al., 2013).
On the basis of the optimal scale parameter values, a novel 3-DOF robot shoulder joint based on orthogonal spherical 3-RRR parallel mechanism is designed but the dynamics of the novel 3-DOF robot shoulder joint is not discussed (Zhang et al., 2015), so the dynamic analysis of the robot shoulder joint is presented in this study. Through the analysis and calculation of kinetic energy and potential energy of each component, the dynamics model of the relationship between driving torque and the motion of the system parameters is established based on Lagrange method. On the basis of the dynamics model, the effective inertia, coupling inertia and driving torque of the parallel mechanism system are analyzed.
MATERIALS AND METHODS
Description of the shoulder: The computer aided design model and schematic of the shoulder joint is shown in Fig. 1 and 2, respectively. The prototype of the shoulder joint in the study is a novel parallel 3RRR orthogonal spherical parallel mechanism. It consists of a fixed platform, a moving platform and three limbs which consist of a framed link and a connecting rod with identical structure. Each limb that consists of three revolute joints in series connects the fixed platform to the moving platform. Thus, the moving platform is attached to the fixed platform by three identical RRR linkages.
| Fig. 1: | Prototype of shoulder joint, 1: Connecting rod, 2: Frame link, 3: Frame, 4: Trunk connector, 5: Actuator, 6: Moving platform, 7: Big arm |
| Fig. 2: | Schematic diagram of shoulder joint, 1: Fixed base, 2: Frame connecting rod, 3: Connecting rod, 4: Mobile platform |
Nine revolute axes of the mechanism intersect at one point O which is the center of rotation of the spherical body. The axes of three revolute joints connected with the fixed platform are vertical each other and the axes of three revolute joints connected with the moving platform are vertical to each other and the revolute joint axes of framed links and connecting rods are vertical to each other.
Kinematics analysis of the shoulder: Taking the center point O as the origin of the system coordinates, as shown in Fig. 2, a fixed Cartesian frame O{X, Y, Z} connected with the fixed platform and a moving Cartesian frame O{x, y, z} connected with the moving platform are established, with the X, Y, Z axes in coincidence with U1, U2, U3 axes and the x, y, z axes in coincidence with V1, V2, V3 axes, respectively. In the initial posture, the fixed Cartesian frame is coincident with the moving Cartesian frame.
The moving platform only can rotate round the fixed platform, referring to Zhen et al. (2006) the transformation matrix between the motion coordinate and the fixed coordinate is given as follows:
| (1) |
where, α, β and γ are the posture angles of the mobile platform.
When the moving platform pose changes, the direction cosine U1, wi and vi of revolute joint of the i-th branch (i = 1, 2, 3) can be derived by:
| (2) |
where, Ui0, Wi and Vi0 are the direction cosines of revolute joint of the i-th branch (i = 1, 2, 3) when the mechanism is on the initial state.
Referring to Zhen et al. (2006) study the constraint equation for the mechanism can be written as follows:
| (3) |
Differentiating the constraint Eq. 3 with respect to time, yields:
| (4) |
where,
| Fig. 3: | Sketch of the i-th limb |
Dynamic analysis: The schematic diagram of the i-th limb is shown in Fig. 3. The length of aibi and diei is, l1, respectively, the length of bici is l2, the length of cidi is l3, their linear density is ρ. Then the mass of aici is mac = ρ(l1+l2), the mass of ciei is mce = ρ(l1+l3).
Kinetic energy analysis: The ri1, ri2, ri4 and diei are arbitrary dots in aibi, diei, diei and bici, respectively, as shown in Fig. 2 and their position vectors can be described as follows:
| (5) |
The kinetic energies of aibi, bici, cidi and diei are:
| (6) |
The kinetic energy of the i-th limb is:
| (7) |
Where:
The total kinetic energy of the three limbs is:
| (8) |
The moving platform only can rotate round the fixed platform, so its kinetic energy is:
| (9) |
where, I is a inertia tensor of the moving platform. Based on theoretical mechanics, the following equation can be derived:
| (10) |
By substituting Eq. 10 into Eq. 9, the following equation is obtained:
| (11) |
Where:
In view of Eq. 8 and 11, the total kinetic energy of the mechanism is:
| (12) |
Potential energy analysis: Let, position vectors of aici and ciei be pi1 and pi2, respectively, in view of Fig. 2, the position vectors are:
| (13) |
where, xac yac is the mass center coordinate of aici in the WiOVi plane, xce yce is the mass center coordinate of ciei in the WiOUi plane.
The potential energy of the i-th limb is:
| (14) |
Where:
| pi1z | = | xac Wiz+yacViz |
| pi2z | = | xceWiz+yceUiz |
The total potential energy of the three limbs is:
| (15) |
Where:
| k5 | = | macgxac+mecgxce |
Because the center of mass of the moving platform locates at the origin of the fixed coordinate system, the value of potential energy is 0. Then the total potential energy of the mechanism is:
| (16) |
Dynamic modeling: Based on Lagrange method, the dynamic model of shoulder joint can be derived that:
| (17) |
where, L = T-E, in which T is kinetic energy, E is potential energy, ψ = (α β γ)T is generalized coordinates, f = (f1 f2 f3)T is generalized force.
By substituting Eq. 5-16 into Eq. 17, the following equation is obtained:
| (18) |
The kinetic equation for the generalized coordinate α can be expressed as:
| (19) |
The kinetic equation for the generalized coordinate β can be expressed as:
| (20) |
The kinetic equation for the generalized coordinate γ can be expressed as:
| (21) |
Equation 18-20 can be rewritten into the general structure dynamic equation as:
| (22) |
where,
Let input generalized force vector for parallel mechanism be τ = (τ1 τ2 τ3)T. The following equation is obtained via virtual work principle:
| (23) |
where, G is force Jacobian matrix of the mechanism. According to dual relationship between motion and force transmissions in mechanism (Craig, 2011) yields:
| (24) |
Substituting Eq. 24 and 23 into Eq. 22, the following equation is obtained, which is the inverse dynamics equation of the mechanism:
| (25) |
Equivalent inertia analysis: Equivalent inertia has effect on the stability, precision and dynamic response of control system which plays a leading role when the mechanism is accelerated and decelerated. The diagonal elements of the equivalent inertia matrix are also called effective inertia which can be used to estimate the equivalent inertia load of the servo motor. The matrix off-diagonal elements are also called coupling inertia, directly affect the stability of the control system. So, the equivalent inertia determines the dynamic characteristics of mechanism. It is necessary to analyze the equivalent inertia of the mechanism.
The equivalent inertia matrix of parallel shoulder can be derived from Eq. 25, as follows:
| (26) |
Where:
According to the discussion above, the equivalent inertia matrix which closely correlates with the sizes of mechanism and density of component is a time-varying function of mechanism configuration and changes with the body posture. This determines the dynamic characteristics of the mechanism.
RESULTS AND DISCUSSION
The structural parameters and inertial parameters of the shoulder are listed in Table 1.
Let motion laws of moving platform of the shoulder joint be as follows:
| (27) |
MATLAB (Matrix laboratory) is an interactive software system for numerical computations and graphics. It's a very powerful tool for doing numerical computations with matrices and vectors. Also, it has a variety of capabilities to display information graphically and can be extended through programs written in its own programming language. MATLAB provides a range of numerical computation methods for analyzing data, developing algorithms and creating models. So, the dynamic characteristics indices of the shoulder in task space can be obtained by a developed MATLAB program.
Solving inverse kinematics models, the curves for angular velocities of driving bars are obtained, as shown in Fig. 4. The angular velocities of driving bars regularly change with the posture parameters.
| Table 1: | Structural parameters and inertial parameters of the shoulder |
| Fig. 4: | Curves for driving angular velocities |
The angular velocity maximum of the drive rod of branch 1 is 5.371 rad sec-1, the angular velocity maximum of the drive rod of branch 2 is 3.628 rad sec-1 and the angular velocity maximum of the drive rod of branch 3 is 4.689 rad sec-1.
Solving the inverse dynamics Eq. 25, the curves for driving torques of driving bars are obtained, as shown in Fig. 5. In the process of movement, the driving torques of the drive rods regularly vary with the changing of posture parameters. The driving torque maximum of the drive rod of branch 1 is 0.4723 Nm, the driving torque maximum of the driving rod of branch 2 is 0.2281 Nm and the driving torque maximum of the drive rod of branch 3 is 0.3558 Nm.
Curves for inertia terms of the mechanism are shown in Fig. 6. It can be observed from Fig. 5 that the equivalent inertia parameters are functions of the parameters of the orientation and change with the mechanism posture. The trends of the equivalent inertia parameters and driving torques are consistent. The variation of the diagonal elements D11 and D33 of the inertia terms are same, their maximum and minimum values are same, respectively.
| Fig. 5: | Curves for driving torques |
| Fig. 6(a-b): | Curves for inertia terms, (a) Diagonal element and (b) Off diagonal element |
The maximum value is 0.0043 kgm2, the minimum value is 0.0036 kgm2. The maximum value and minimum value of D22 is 0.0066 kgm2 and 0.0055 kgm2, respectively. The effective inertia of branch 2 is larger. The variation of the off-diagonal elements D12 and D23 of the inertia terms are same, their maximum and minimum values are same respectively, their maximum value is 0.0012 kgm2, the maximum value of D13 is 0.0029 kgm2. The coupling effect between the branch 1 and branch 3 is larger.
CONCLUSION
Through the analysis and calculation of kinetic energy and potential energy of each component of the novel 3-DOF robot shoulder joint, the dynamics model of the relationship between driving torque and the motion of the system parameters is established based on Lagrange method:
| • | On the basis of the dynamics model, the effective inertia, coupling inertia and driving torque of the parallel mechanism system are analyzed and the variation of the dynamics parameters of the mechanism is given, the dynamic coupling relationship between branches is analyzed with a developed MATLAB program |
| • | The results show that, the angular velocity and driving torque maximum of the drive rod of branch 1 are the largest, the effective inertia of branch 2 is larger. The coupling effect between the branch 1 and branch 3 is larger and the posture change of the mechanism has a great influence on the dynamics parameters. The results can provide theoretical basis for the mechanism of dynamics optimization design, trajectory planning and control scheme |