17.2 - Kinematic Analysis

Overview

Our objective is to design and analyze a mechanism prototype that can locate the tab of a can and produce enough force to open the tab. In this section, the kinematics analysis of three distinct mechanisms are shown to decide which motion pattern, velocity, and force output will be best to lift the tab of a canned beverage. Our analysis will be focused on 2 variations of the four bar mechanism, where one is in an open configuration and the other is a closed configuration.  Our goal is to find the best combination of position, velocity, and mechanical advantage that is able to position the last link under the minimal space under a tab and produce enough force to lift the tab up. We hypothesize that the desired path of the end effector will look similar to the path shown in the image below. Through research and empirical testing, we have determined that our mechanism will need to output approximately 30 Newtons of force to reliably pop open the tab on a can. To achieve our force requirement, we need to minimize the velocity of the end-effector on link 3 and maximize the mechanical advantage during its interaction with the tab. 

                             Figure 1:  Open 4-bar configuration (Mechanism 1).                                                   Figure 2:  Closed 4-bar configuration (Mechanism 2). 

Kinematic Analysis

In this section, each mechanism will be thoroughly analyzed to find the specific kinematic profiles that fits our needs for this project. Our intention is to follow the path shown in Figure 3 and constrain the position to fit the area on the top of the can. The end-effector needs to first begin at the initial contact within the 8 mm area between the lip of the can and the tab. Then, the end-effector is positioned within that area, a sliding motion will transfer the end-effector to be under the 25 mm tab. After that sliding motion, the mechanism will need to pull the tab open with enough force to open the can. The force to input angular velocity relationship needs to be defined with a high mechanical advantage at that specific time frame from the initial contact to when the mechanism pulls up. Once we define the link lengths and configuration of the best mechanism, we can continue with creating the prototype.

Figure 3: Position diagram of how and when the motion of the end-effector on link 3 in the 4-bar open configuration will open the tab of the can.

Mechanism 1: Open 4-Bar Configuration

Position Analysis

Pictured below are the varying position profiles of the end-effector link for mechanism 1. We iterated through different lengths for each link, besides link 2, the length of link 2 had to remain constant as it determined the length that the end effector was in contact with the can. We also wanted to only use lengths that would satisfy the Grashof condition and produce a path that was closer to the desired position (drawn above) to be able to get under the tab of the can. Due to the size of the can, this path is very small and must be very precise. The mechanism achieved the desired path about halfway through each of the iterations, with the link lengths established in the table below.

Figure 4: The position analysis found by varying the lengths of links 1, 3, and 4.

Table 1: Lengths of the links in mechanism 1 that produced the best path profile.

Velocity Analysis

We further analyzed the velocity of the mechanism with these link lengths to better understand the end effector's motion.

                                                                                       Figure 5: The velocity analysis of the optimal mechanism 1.

The original value of the angle of link one was 0 degrees, which produced the first velocity plot shown above for the mechanism's end link. From our analysis in Motion Gen we know that the end-effector (attached to link 3 as shown in the photo above) interacts with the tab of the can when the crank angle is around 180 degrees. In the first plot we can see that with a theta1 = 0 degrees the velocity is decreasing as it pulls the tab up, but it is not the slowest that it could be. When we change our mechanism such that theta1 = 120 degrees, the lowest velocity is achieved where the end-effector needs to impart the most force.

Mechanical Advantage

                                                                       Figure 6:  Analysis of the mechanical advantage after changing the angle of link 1.

 Originally, with an initial angle of link 1 at 0 degrees, the greatest mechanical advantage was achieved when the crank angle was less than 80 degrees. As we adjusted the angle of link 1, the location of the peak mechanical advantage changed. With the angle of link 1 at 45 degrees, the mechanism will have the greatest mechanical advantage at a crank angle around 180 degrees and 330 degrees. This is closer to where the end effector will interact with the tab of the can allowing us to impart more force with less motor power.

Mobility and Grashof Analysis:

To verify our design we need to check the Grashof and mobility equations to ensure that the mechanism achieves the motion and position profile of our end effector link that will lift the tab of our can. The open 4 bar mechanism has 1 DOF and meets the Grashof condition. 

[The Grashof and mobility equations were used to calculate if the preferred mechanism is able to have full continuous motion about the input joint. The mobility of the crossed 4-bar mechanism is 1, which is described by the rotational movement that is created. Furthermore, the mechanism is able to produce a continuous motion about the input link as it is Grashof. ]

Figure 7: Mobility and Grashof Calculations

Final Design

After the kinematic analysis of our open 4-bar mechanism, pictured below is the resulting final design modeled in MotionGen.

Figure 8: Final open 4-bar mechanism after kinematic analysis

Mechanism 2: Closed 4-Bar Configuration

For the Crossed 4-Bar configuration, the team has found a patent that modeled the segments of the human finger with a crossed 4-bar linkage. The idea is that the crossed 4-bar is able to induce a rotation of the the coupler linkage and the extension on the coupler linkage is able to curve similar to a human finger. For our model, we designed the crossed 4-bar linkage similar to the picture shown in Figure 8 and attached another link at position 104 to describe the tip of the finger. There were link lengths given within the patent that the team used as a starting point to the mechanism design. These link length values values are given in Table 1.

                        Figure 9: 4-bar linkage model for human finger (Patent)

Table 2: Link lengths given from patent

Position Analysis

Our team modified the link lengths individually to find the best position profile that can curl the end-effector under the tab and lift the tab. In figure x, each link is varied to find a position profile that best matches our desired profile. 

            a)                                                                                                                                            b)

          c)                                                                                                                                           d)

                                                      Figure 10: Position profiles of the Mechanism with varying lengths for a) Link 1, b) Link 2, c) Link 3, and d) Link 4

We found that by altering link 2 to 2.5 cm, the position profile creates a hooking motion that is able to curl under the tab and provide a lifting motion when under the tab. The position profile as further altered by changing the theta 1 position so that the lifting motion was direct more vertically. After multiple theta 1 values, we settled for an angle of 45 degrees to best model the hooking motion. 

                                                                                     Figure 11: Position Analysis with Theta 1 of 0 Degrees and 45 Degrees

Velocity Analysis

For the velocity analysis, the team calculated the velocity profiles of the corresponding link length modifications and Figure X depicts the velocity profile for the different link lengths. 

      a)                                                                                                                                    b)

      c)                                                                                                                                    d)

                                                         Figure 12: Velocity profiles of the Mechanism with varying lengths for a) Link 1, b) Link 2, c) Link 3, and d) Link 4

The velocity profile of the configuration with the best link lengths was also plotted to describe the compare the motion between a theta 1 of 0 degrees and 45 degrees. In Figure x, the plots illustrate that the minimum velocity after rotating link 1 by 45 degrees is approximately zero at 280 degrees. 

                                                                              Figure 13: Velocity Analysis with Theta 1 of 0 Degrees and 45 Degrees

Mechanical Advantage

For the mechanical advantage, we calculated the ratio between the angular velocity of the input link to the output velocity calculated in the above section and Figure X depicts the mechanical advantage as the link lengths are altered.

       a)                                                                                                                               b)       

      c)                                                                                                                                    d)

                                              Figure 14: Mechanical Advantage of the mechanism with varying lengths for a) Link 1, b) Link 2, c) Link 3, and d) Link 4

The mechanical advantage of the configuration with the best link lengths was also plotted to describe the compare the motion between a theta 1 of 0 degrees and 45 degrees. In Figure x, the plots illustrate that the peak mechanical advantage after rotating link 1 by 45 degrees is above 300 at 280 degrees. 

                                                                                     Figure 15: Mechanical Advantage with Theta 1 of 0 Degrees and 45 Degrees

Mobility and Grashof Analysis 

Similar to the previous mechanism, the Grashof and mobility equations were used to calculate if the preferred mechanism is able to have full continuous motion about the input joint. The mobility of the crossed 4-bar mechanism is 1, which is described by the rotational movement that is created. however, the preferred mechanism design is not Grashof, highlighted with the non-continuous motion. However, because we are looking for a hooking mechanism, this non-Grashoff design we created is still able to complete the goals we need.

Figure 16: Mobility and Grashof Calculations

Final Design

After the kinematic analysis of our crossed 4-bar mechanism, pictured below is an approximated final design modeled in MotionGen.

Figure 17: Final crossed 4-bar mechanism after kinematic analysis

Selected Design

Our team has chosen to continue with the open 4-bar design as the position profile is closely matching the continuous motion we described at the beginning. Not only is the path very similar, the distance that the end effector needs to travel when in contact with the can is more easily controlled with the open 4-bar. Furthermore, the velocity and resulting mechanical advantage from the best open 4-bar configuration provides a desirable location to apply a high force to the tab. Following this kinematic analysis is documentation of the prototype iterations we have created. 

Bill of Materials

Our bill of materials are given within this spreadsheet . We expect to collect all the parts by 4/15.