Abstract
In this chapter, methods for expanding the workspace of parallel robots are introduced. Firstly, methods for expanding the translational workspace of the parallel robot are discussed. The parallel robot has multiple solutions of the inverse and forward displacement analysis. By changing its configurations from one solution to another, the parallel robot can expand its translational workspace. However, conventional nonredundant parallel robot encounters singularity during the mode change. Singularity-free mode changes of the parallel robot by redundant actuation are introduced. Next, methods for expanding the rotational workspace of the parallel robot are shown. In order to achieve the large rotation, some mechanical gimmicks by gears, pulleys, and helical joints have been embedded in the moving part. A novel differential screw-nut mechanism for expanding the rotational workspace of the parallel robot is introduced.
Keywords
- parallel robot
- workspace
- mode change
- kinematics
- singularity
- differential
- helical joint
1. Introduction
The parallel robot has excellent characteristics such as high speed, high precision, and high rigidity [1]. However, mechanical collisions between limbs and complexly existing singular configurations restrict its workspace. In this chapter, firstly, methods for expanding the translational workspace of the parallel robot are discussed. The parallel robot has multiple solutions of the inverse and forward displacement analysis. By changing its configurations from one solution to another, the parallel robot can expand its translational workspace. However, conventional nonredundant parallel robot encounters singularity during the mode change. Singularity-free mode changes of the parallel robot by redundant actuation are introduced. Next, methods for expanding the rotational workspace of the parallel robot are shown. In order to achieve the large rotation, some mechanical gimmicks by gears, pulleys, and helical joints have been embedded in the moving part. A novel differential screw-nut mechanism for expanding the rotational workspace of the parallel robot is introduced.
2. Expanding the translational workspace
2.1. Mode change of the parallel robot
Schematic model of a planar two-dof (degree-of-freedom) serial robot and a planar two-dof parallel robot is illustrated in Figures 1 and 2, respectively.
2.2. Kinematics of the parallel robot
As mentioned before, the parallel robot can expand its workspace by the mode change. However, parallel robot encounters the singular configuration between the mode changes. In this section, kinematics and singular configuration of a 2
Equation (1) is the loop closure equation of the parallel robot. Next, the forward displacement analysis and inverse displacement analysis of the parallel robot are going to be derived.
Eq. (1) is deformed as
The unit direction vector
As shown in Figure 3,
Substituting Eq. (4) into Eq. (3), one obtains a quadratic equation about the displacement of the actuator as,
Solutions of the inverse displacement analysis are given by the solution of Eq. (5) as,
Eq. (6) represents that there are two solutions for the inverse displacement analysis. It means that the parallel robot has two working modes. Figure 2(a) (i) and (iii) represents the configurations of the parallel robot of the two working modes.
In the forward displacement analysis, positions
Each equation of (7) represents that a circle of the central position is (
When
2.3. Singularity analysis of the parallel robot
2.3.1. The Jacobian of the parallel robot
Differentiating the both sides of Eq. (1) with respect to time, one obtains
In Eq. (1),
Differentiating the both sides of Eq. (9) with respect to time, one obtains
Multiplying the both sides of Eq. (8) by
The displacements of all actuators
2.3.2. Second kind of singularity
When the Jacobian matrix
Then, the parallel robot is in the second kind of singularity [2] or in direct kinematics singularity [3]. The 2
This case corresponds to that two rods of the parallel robot lay in one line as shown in Figure 2(b) (ii). As shown in Figure 2(b), the parallel robot encounters the second kind of singularity during the assembly mode change.
2.3.3. First kind of singularity
When the Jacobian matrix
Then, the parallel robot is in the first kind of singularity [2] or in inverse kinematics singularity [3]. The 2
As shown if Figure 2(a), the parallel robot encounters the first kind of singularity during the working mode change.
2.3.4. The third kind of singularity
The third kind of singularity [2] occurs when both
2.4. Passing the singular configuration during the mode changes
2.4.1. Passing the second kind of singularity during the assembly mode change
2.4.1.1. Using the inertia
As shown in Figure 5(a), if the parallel robot has some speed just before the singular configuration, the robot keeps moving according to the law of inertia; then, the parallel robot passes the second kind of singularity. Note here if the parallel robot stops at the singular configuration, it is impossible to pass the singular configuration by the inertia.
2.4.1.2. Using the gravity
As shown in Figure 5(b), when the gravity force acts to the lower direction, the parallel robot can pass the singular configuration even if the parallel robot stops at the singular configuration. However, the robot cannot move to the upper direction against the gravity. When the robot moves on the horizontal plane where the gravity force does not act on the robot, the parallel robot cannot pass the singular configuration.
2.4.1.3. Using the redundancy
A robot has actuation redundancy when it is driven by number of actuators greater than the degree of freedom. The actuation redundancy may increase the cost of the robot and complexity of control. However, the actuation redundancy is one of the most effective methods for avoiding the singularity during the mode change.
Figure 6 represents 3
where
As shown in Eq. (18), one minor is singular and the other two minors are nonsingular; namely, the
Now, the parallel robot loses the redundancy but keeps the nonsingularity. The parallel robot can pass the singular configuration of Figure 6 (iii). In the same way, the parallel robot can pass the singular configuration of Figure 6 (ii). Thus, the parallel robot achieves the assembly mode change from Figure 6 (i) to (iv).
2.4.2. Passing the first kind of singularity during the working mode change
The parallel robot encounters the first kind of singularity during the working mode change. In the velocity control of a robot, the velocity of the actuator is controlled to trace the desired velocity of the actuator
However,
2.5. Researches on mode change of the parallel robot
Researches on expanding the workspace by the mode change of the parallel robot have been reported for nonredundant two-dof planar robot [4, 5], nonredundant three-dof spatial translational robot [6], redundantly driven three-dof translational and rotational planar robot [7], and redundantly driven three-dof spatial translational robot [8]. Redundant actuation is the actuation in one of the most effective methods for avoiding the singularity. However, the redundancy is not always the answer to avoiding the singularity. Additional ingenuities, for example, path planning for mode change, or asymmetrical design for the robot, are required.
3. Expanding the rotational workspace
3.1. Conventional methods
In this section, methods for expanding the rotational workspace of the parallel robot are introduced. Figure 7(a) represents the Stewart Platform [9], six-dof parallel robot with three-dof translations and three-dof rotations. The moving plate is driven by linear actuator embedded six limbs. Each limb is paired by a passive universal joint (
In order to cope with the problem, Clavel invented a novel parallel robot DELTA [11] as shown in Figure 8. The DELTA generates three-dof translation and one-dof rotation around the z-axis.
Three control arms are rotated by three motors located on the base plate. Each arm is connected to the moving part by a parallelogram rod with spherical joints. The gripper on the moving part is connected to the motor on the base plate with a telescopic arm via universal joints (Figure 9).
Researches on expanding the rotational workspace of the DELTA like parallel robot have been reported from research group at the Montpelier University. The basic idea is to convert the translational motion to the rotational motion by the mechanical gimmicks such as pulley, rack and pinion, and screw [12]. Crank embedded moving part has been proposed by McGill University for two-limb parallel robot [13] and by Fraunhofer IPA for cable-driven parallel robot [14].
3.2. Differential screw drive system
Recently, differential screw drive systems composed of two screw-nut pairs of different leads have been applied to robot systems [15, 16]. There are four driving methods of the differential drive systems;
Rotations of the two nuts are converted to the rotation and translation of the screw pair, which is coaxially arranged with one end interconnected with each other.
Translations of the two nuts are converted to the rotation and translation of the screw pair, which is coaxially arranged with one end interconnected each other.
Rotations of the two screws are converted to the rotation and translation of the nut pair, which is coaxially arranged with one end interconnected each other.
Translations of the two screws are converted to the rotation and translation of the nut pair, which is coaxially arranged with one end interconnected each other.
In this section, kinematics of the differential drive system (b) as shown in Figure 10 is discussed for enlargement of the rotational workspace of the parallel robot.
Let
Equation (20) is given in the matrix form as
Equation (21) gives the inverse kinematics of the differential drive system. The forward kinematics of the system is derived by inverting Eq. (21) as
If two leads of the screw-nut pair are identical as
then the matrix in Eq. (22) becomes singular. It is necessary that the leads of the two screw nuts are different from the differential drive system.
Figure 11 represents a 4-dof parallel robot that the differential drive system (a) embedded in the moving part [15, 16].
4. Conclusion and future works
After commercialization of the DALTA robot, a lot of parallel robots can be found in machine factory and other industries. Recently, one can get the kit of a 3D printer machine by linear DELTA mechanism under 500 USD. It may be said that the first-generation parallel robot such as DELTA and Stewart Platform is getting to reach the mature stage. In this chapter, expanding the workspace of the parallel robot was introduced. The translational workspace was expanded by singularity-free mode change using the actuation redundancy. The differential drive mechanism converts the translational motion to the rotational motion, which expands the rotational workspace of the parallel robot. These parallel robots are expected as the next-generation robot.
At the end of this chapter, an example of the next-generation parallel robot is introduced. Figure 12(a) represents a novel two-limb six-dof parallel robot [17]. As shown in Figure 12(b), a differential screw mechanism is embedded in the output link for enlarging the rotational workspace of the gripper. This parallel robot extends its degree of freedom from four (three translations and one rotation as shown in Figure 11) to six (three translations and three rotations) as shown in Figure 13. We are working on the kinematic analysis and detailed mechanical design of the parallel robot [18].
Acknowledgments
This work was supported by JSPS KAKENHI Grant Number 15K05918 and 24560314.
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