Optimization for the Balancing Cylinder of a 3-DOF Planar Manipulator

Abstract: 3-DOF planar manipulator is used for loading and unloading materials and as a accessory for machine tool. A new approach is presented for design and optimization for the balancing cylinder. The working environment of the manipulator is introduced and according to the loading and unloading working processes, path planning is designed and expressed by piecewise function. In a period of motion of the manipulator, every component generates the torque caused by gravity. The static total torque is calculated by the theoretical analysis and kinematics simulation used ADAMS. Then the balancing cylinder is designed and the relevant parameters are optimized. The results show that the method of the optimization is right and effective.


Introduction
3-DOF planar manipulator is a special mechanism as a accessory for machine tool to load and unload materials.It is different from general industrial robot because of being installed in the machine, obviously more compact and cheaper.Especially, a numerical control machine mainly processes a class of part, this manipulator will be very suitable for this occasion.
The manipulator consists of big arm, small arm and gripper.The motor and reducer that drive the big arm rotate relative to the base must output big torque because of the gravity of load and every component.So balancing cylinder is very important to neutralize the total torque caused by the gravity of them.
About balancing cylinder, many experts and scholars have conducted a lot of research, and some results have been achieved.These results are very helpful for this research.Jakob Weström presented an automated approach in optimal design of the spring balancing cylinder of an industrial robot using multi-disciplinary and multi-objective design optimization, and the developed methodology was robust and efficient enough for use in the engineering practice [1].Simon Lessard designed a new medical parallel robot and analyzed its static balancing to enhance the safety of the proposed robot [2].In this article, five-bar assembly with torsion springs were presented and its main objective of static balancing is the safety enhancement of the proposed robot structure.Because the static balancing can reduce motor torque, it is very useful for increasing safety in operating mode and the mobility of the robot (by decreasing the motor weight).Professor W.J. Zhang researched on force balancing in robot mechanism design and a force balancing method called adjusting kinematic parameters (AKP) for robotic mechanisms or real time controllable (RTC) mechanisms is proposed [3].The key idea in the AKP method is that when a system is not in a force balancing status, change of the kinematic parameters can make the system balanced.V. Arakelian proposed a new approach for balancing of spatial parallel manipulators that involves connecting a secondary mechanical system to the initial robot, which generates a vertical force applied to the platform of the manipulator [4].The suggested balancing mechanism was designed on the base of the multiloop pantograph linkage introduced between the robot base and the platform.The minimization of the input torques was carried out by constant and variable forces for static and dynamic modes of operation.The study conducted by Alberto Martini dealt with the compensation of gravity loads in closed-loop mechanisms as a possible strategy for enhancing their working performance [5].Dan Zhang designed dynamic balanced legs that were combined to synthesize parallel mechanisms [6].These researchers are very enlightening and in this article a 3-DOF manipulator is designed and related parameters are optimized.So the motor and reducer can become much lighter and smaller and the safety can be enhanced.

3-DOF manipulator and its path planning
The numerical control machines VTC3240 have configured 3-DOFmanipulator according to the characteristics of the processed object, as shown in Figure 1.The manipulator designed is 3-DOF planar series mechanism.Panasonic servo motors and reducers are used in three joints.The pneumatic gripper is Schunk imported from Germany.The working process of the manipulator is shown in Figure 2, and can be described as: workpiece conveyor belt moves along vertical paper direction, left side is the placing area for processing parts and right side is the placing area for processed parts, the gripper firstly grab the workpiece to be processed, then moves to the triangle chuck of machine and put it on the right position, at last the manipulator returns to the initial position waiting for processing the workpiece.When the process is over, the manipulator moves to grab the part and put it on the right side of the conveyor belt, and then moves to grab another new workpiece again along the same track.The manipulator moves to grab the workpiece from original position, then put it on the triangle chuck and then returns back to the original position.This process is a working period.In this period, the trajectory of the gripper palm is "A→B→C→B→A→D→E→D→A", as shown in Figure 3.The running time of AB, AD, DE are all 3 s, and BC is 4 s, so the full working period is 26 s.According to the characteristics of movement and function of the manipulator, we can make the workpiece always kept upright posture during the movement process of loading and unloading.So if the trajectory coordinates of palm point M(xM,yM)are given, the pose of workpiece during the whole movement process can be obtained.The coordinate system is established according to Figure 3, the motion trajectory equation of point M can be expressed as the form of piecewise function: (unit:m) (1)

Static torque of shaft I
The Figure 3 shows that the manipulator is a three-bar mechanism, upper arm, lower arm and gripper from top to bottom, and the axes of rotation are I, II and III.During the process of loading and unloading, the motion is slow and steady, so although the acceleration at the start of the moment is a little high and should be considered seriously, the servo motor can bear a lot of instantaneous overload and static torque is more important if we want to design and optimize balancing cylinder.According to the related parameters, the instantaneous acceleration can be get by experiment, and the maximum instantaneous dynamic torque can be calculated.If this value is lower than the times of instantaneous overload of servo motor, it is reasonable not to consider this factor.We can calculate static torques of the 3 axes.Among them, the static torque of axis I is maximum, and it is due to the gravity of workpiece, gripper, lower arm, upper arm and all of the motors and reducers.
At any time, the total static torque is formed by every component's gravity acting on the axis I, as shown in Figure 4. Assuming that clockwise is positive and anticlockwise is negative, equation ( 2) is obtained: where M z is the total static torque on axis I, G b is the gravity of upper arm, x cb is the horizontal coordinate of the centre of gravity of upper arm C b , G sm is the gravity of motor and reducer of axis II, x A is the horizontal coordinate of A, G s is the gravity of lower arm, x cs is the horizontal coordinate of the centre of gravity of lower arm C s , G hm is the gravity of motor and reducer of axis III, xB is the horizontal coordinate of B. Measure the parameters of each part of the manipulator, as shown in Table 1.m i is the mass of every component, l i is the length of upper arm and lower arm, l ci is the distance to O and A from upper arm gravity center and lower arm gravity center.The horizontal coordinates of B, C s , A and C b are obtained by kinematics simulation according to the target track of the workpiece, in a period, as shown in Figure 5.

Gh
And then according to the formula (2), the total static torque around axis I, M z , can be obtained, in a period, as shown in Figure 6.

Parameter design and optimization
According to the installation space of actual device, a balanced cylinder system, as shown in Figure 7, is used.One end of the balancing cylinder is hinged at the S point, and the other end is hinged at the C b point.Theoretically S point can be change on the whole plane, but considering the actual structure and the symmetry about Y axis of movement, S point should be located on the Y axis, and its coordinate can be optimized.C b point can't be changed and is a constraint condition.The spring stiffness of the balance cylinder system is determined according to the empirical formula (3) given by the author which is proved to be effective: where k is the spring stiffness; l 1 and l 2 are respectively the distances between point S and C b when Mz reaches the maximum value Mz max and minimum value Mz min .Further, establish the objective function to optimize the coordinates of S point: where n represents the number of discrete points in Figure 5 and 6; (x cbi , y cbi ) are the coordinates of point C b at the i position in the trajectory, and ys is the vertical coordinate of the point S; l 0 is the original length of the balancing cylinder spring; Mz i is the total static torque at the i position in the trajectory.
The optimization goal is f = f min , and then calculate the parameters k and y s .According to the empirical value and actual position of assemble, there are another constraint conditions: y s ∈[0.1, 0.5] (unit: m).
The program is established and optimized with MATLAB software according to the formula (3) and ( 4).The result is that y s = 0.312 m.So we can further get the change curve of balancing cylinder output force F c , as shown in Figure 8.The torque M bc produced by the balancing cylinder on the I axis can be calculated according to the formula (5): Further, the synthetic moment M zz on the I axis after adding balancing cylinder can be calculated according to the formula (6): The change curve of M zz in a movement cycle is shown as in Figure 9. Before and after adding the balancing cylinder system the total static torques on the I axis are shown as in Figure 10.We use ΔM to express the level of reducing torques needed by balancing cylinder, and the curve of ΔM is shown as in Figure 11.

Conclusion
In conclusion, aiming at a 3-DOF planar manipulator used for loading and unloading for machine, a method of parameters optimization has been proposed.The simulation results show that the balancing cylinder can effectively reduce total static torque on axis I, so the manipulator can be much lighter and safer.The method of balancing cylinder design and parameters optimization for a 3-DOF manipulator is right and effective, the whole process of analyzing is complete and can play an important role in development of manipulator for numerical control machine.

Figure 2 .
Figure 2. Loading and unloading working process.

Figure 3 .
Figure 3. Path planning of the manipulator.

Figure 4 .
Figure 4. Composition of total static torque.

Figure 5 .
Figure 5. x-coordinate of every point during a period.

Figure 6 .
Figure 6.M z around I axis during a period.

Figure 8 .
Figure 8. Curve of output force of balancing cylinder.

Figure 11 .
Figure 11.Curve of the level of reducing torques.

Table 1 .
Parameters of manipulator model