Team Members
- David Moiseev [dmoiseev]
- Preeti J. Pillai [preetijp]
1 Project Description & Objective
Our project focuses on the kinematic behavior of a three slider linkage mechanism, wherein the endpoints of each slider is connected to a freely moving revolute joint. The mechanism has three moving links, one fixed and one fixed link and four revolute joints of which two are prismatic as well . This mechanism it transfers rotary motion into translation
The movements of all the three links are in perfect co-ordination with each other, so that the motion of one link causes the adjacent connected link to make a rotation as well as a translation movement. Our primary objective was to conclude the total number of variables and develop a set of constraint equations that would enable us to find the Jacobian of the manipulator. The most challenging problem of this project was the unpredictable nature of the contour traced by the follower sliders when either one or two sliders are powered up while the remaining one acts as a follower. For example, if slider Link3 is displaced by a distance d3, then the corresponding movement might be either a change in the rotation angles of β & θ1 or a translation along Link2 by some distance d2.
In this planar linkage mechanism, of all the linkages, two are fixed, Hence, the total degrees of freedom for this structure is:
DOF = 3(4-2) = 6
In this planar linkage mechanism, of all the linkages, two are fixed, Hence, the total degrees of freedom for this structure is:
DOF = 3(4-2) = 6
The mobility of a linkage is the number of input parameters which must be controlled independently in order to bring the planar mechanism to a set position. This factor is given by,
m = 3 (n-1) - 2 j 1 - j 2
= 3(4-1)-2*1-2
= 3(3)-2-2=5
m=5
Where, j 1 - number of joints with one degree of freedom
j 2 - number of joints with two degrees of freedom
Figure 1 Original Design
Figure 2 Final Modified Design
2 Implementation
The linkage structure was set up as per the figure. We set Link4 to greater than that of Link2. Link3 was fixed to the base in the same way as Link 4. The lengths of Link1 and Link2 are the same, and no changes were made to modify these lengths.
The output voltage from the HBridges were given to the sensor connections on the sliders. In order to manually adjust the link lengths, the sensor connections were given a separate power supply.
3 Circuit Diagram
4 Project Model
Figure 4 Complete Manipulator
5 Angular Position Analysis
Figure 5 The arrows indicate the direction of motion
In the above linkage structure, we now have three displacement variables and three joint rotations.
Consider a reference point at the movable end of Link3 , Let this point be (X,Y). We will derive the angular position constraint equations from this point's reference.
6 Equations
6.1 Forward Kinematics
6.2 Inverse Kinematics
6.3 Analytical Jacobian Analysis
7 Results
The initial attempts to make the linkage end effector work, were not completely successful due to the following reasons:-
- The most significant reason was the slow interface between the arduino and matlab.
- The second cause was that, only proportional control was used.
The nature of the plots indicate why perfect position control was unachievable, and mainly due to the cycle speed , since each blue circle indicates the update of new measurement and path correction.
Figure 7 The Blue circles indicate each data point and the red point is the desired position. Implemented with P control. The Gain Ki is the integral gain.
After observing these results, and as per Prof. Gillespie's suggestion, we thought of improving the the position control by integrating it with integral control. Though the results displayed an improvement over the previous outputs, the linkage still continued to finally settle down in the same place as before. We believe that this occurs due to the slow response time and the nature of the linkage itself.
By comparing the output plots, we can conclude that though the integral control does not help much with the final position, there's certainly an improvement in the position control which is indicated when the linkage end effector exactly passes through the desired point. This is due to the reduction in error by the with integral controlled feedback.
8 Final Exam Question & Solution
