This paper introduces an asymmetrical parallel robotic wrist, which can generate a decoupled unlimited-torsion motion and achieve high positioning accuracy. The kinematics, dexterity, and singularities of the manipulator are investigated to visualize the performance contours of the manipulator. Using the method of Lagrange multipliers and considering all the mobile components, the equations of motion of the manipulator are derived to investigate the dynamic characteristics efficiently. The developed dynamic model is numerically illustrated and compared with its simplified formulation to show its computation accuracy.

The parallel typed wrist mechanisms, also called spherical parallel manipulators (SPMs), have found their applications in camera-orientating [

The SPMs have been extensively studied on many aspects, such as workspace [

This paper introduces an asymmetrical parallel robotic wrist and presents the kinematic/dynamic analysis. The isocontours of the dexterity over workspace are visualized to show an image of the overall performance and singularity is analyzed. Using the method of Lagrange multipliers and considering all the mobile components, the equations of motion of the manipulator are derived to compute the power consumption efficiently. The developed dynamic model is numerically illustrated and compared with its simplified model.

The proposed asymmetrical wrist mechanism is displayed in Figure

The asymmetrical spherical parallel manipulator: (a) CAD model; (b) coordinate system.

The coordinate system

The orientation of the outer ring of the mobile platform is presented by Euler angles

Under the prescribed coordinate system, unit vector

Unit vector

Unit vector

Following the kinematic constraint equations below,

The input angle displacements can be solved as

According to the motion of U-joint [

Differentiating (

Since the angle rates can be transformed into the angular velocity, namely,

Equation (

This section presents the velocity equations for the mobile bodies of the manipulator and the acceleration equations can be derived by differentiating the velocity equations with respect to time.

Due to the decoupled motion of the outer ring of the mobile platform, its angle rates can be extracted from (

Referring to Figure

The motions of the legs: (a) active; (b) passive.

As the motion of the passive leg is confined in the vertical plane, the passive leg is considered as a planar four-bar linkage as shown in Figure

After knowing the

The workspace (WS) is one of the most important design issues as it defines the working volume of the robot/manipulator and determines the area that can be reached by a reference frame located on the moving platform or end-effector [

The spherical surface of a regular workspace.

Dexterity is another utmost important concern, which is usually evaluated by the condition number of the kinematic Jacobian matrix. The studies on the SPMs show that smaller angles

The dexterity distribution over the workspace with

To evaluate the kinematic performance over the regular workspace, a commonly used index is the global conditioning index (GCI) [

The GCI corresponding to the six designs in Figure

(a) | (b) | (c) | (d) | (e) | (f) | |
---|---|---|---|---|---|---|

GCI | 0.230 | 0.305 | 0.258 | 0.310 | 0.139 | 0.280 |

From Figure

For Type I singularity, it occurs when at least one of the diagonal elements in matrix

Examples of Type I singularity of the asymmetrical SPM.

From the forward Jacobian matrix, it is seen that the SPM is at Type II singularity when the normals to the planes of the distal links are linearly dependent, that is, the two distal links and the mobile platform being coplanar, which yields

Examples of Type II singularity of the asymmetrical SPM.

Based on the above analysis, although the set of parameters in Figure

In order to compute the power consumption effectively, all the mobile bodies will be taken into account for the dynamic modeling. The dynamics of the SPM under study can be solved by using the Lagrange equations [

Due to the decoupled rotation between the inner and outer rings of the mobile platform, the Lagrangian of the mobile platform includes two parts as follows:

Since all the mobile bodies of SPMs rotate about the center of rotation, from the motions of the active leg derived in Section

Let

The Lagrangian of the passive leg is derived as

Substituting the previous Lagrangian into (

In accordance with Section

The velocity and acceleration profiles for the simulations: (a) trajectory 1; (b) trajectory 2.

The corresponding simulation results, with the comparison of a simplified model where passive and distal links are not considered, are illustrated with Figures

The dynamic simulation results of trajectory 1: (a) actuator torque; (b) actuator power.

The dynamic simulation results of trajectory 2: (a) actuator torque; (b) actuator power.

Figure

The dynamic simulation results of trajectory 2 with external moment: (a) actuator torque; (b) actuator power.

This paper introduces an asymmetrical parallel robotic wrist which admits a large orientation workspace. Its unique structure allows that the manipulator can generate a decoupled unlimited-torsion rotation and ensures high positioning accuracy. This proposed manipulator can be used as a tool head in the complicated surface machining, such as milling, and also can work as an active spherical joint.

The kinematic performance of the proposed manipulator is studied and performance contours of the dexterity over the workspace are visualized for the optimum design in the further study. Moreover, the singularity is analyzed by solving the geometric constraint equations. Using the classical method of Lagrange multipliers, the equations of motion for the manipulator under study were derived. All the moving bodies are taken into account to describe this dynamic system effectively and clearly. Numerical simulations show that some simplified models that neglect the weight of intermediate bodies cannot characterize the dynamics of the system appropriately. The developed dynamic model can be used for the actuation selection or dynamics evaluation. In the future, the optimal design of the manipulator will be conducted for improvement of dexterity and reduction of energy consumption.

The Lagrangian of the robotic system

The angular velocities of the inner/outer rings and the

Vector of the actuator torque

Output angle of the four-bar linkage of the passive leg

The angle rates of the inner ring

Forward and backward kinematic Jacobians of the inner ring

Constraint Jacobian matrix

Kinematic Jacobian matrix

Forward and backward kinematic Jacobians of the outer ring

Rotation matrix of the outer and inner rings of the mobile platform

Transformation matrix of the

Unit vector of the input axis in the

Unit vector of the intermediate revolute joint in the

Unit vector of the top revolute joint in the

The

The author declares that there is no conflict of interests regarding the publication of this paper.