Project Prototype (11)

Project Prototype

Kinematic Analysis

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Fig. 1: The bevel gearset used to control rotation and assist in elevation.

Bevel Gearset

The gearset under the crank consists of 3 bevel gears held together by a central block as depicted in figure 1. The central bottom bevel is grounded, allowing the other two side bevels and central block to rotate. Rotating the side bevels in the same direction with respect to their own frame of reference (looking at the bevel from behind its wide end) will rotate the central block. Attempting to rotate them in the opposite directions wrt (with respect to) their frames of reference will cause the entire gearset to lock up since mv=-N1N2-N2N3=N1N2N2N3. The positive velocity ratio indicates that the gears 1 and 3, which are the side gears, will want to rotate in the same direction relative to their bases. Rotation in opposite directions is impossible and this fact allows us to provide a zero velocity input to the sun gears in the planetary arrangement described below to elevate the arm.

All three bevel gears are identical and thus no torque or velocity ratio is achieved in this particular mechanism.

Planetary Gearset

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Fig. 2: A planetary gearset has now been added over the bevel gearset. The crank forms the arm of the gearset.

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Fig. 3: The planetary gearset; all other components have been hidden for clarity.

A crank arm is added, which shares the same joint as the side bevel gears. However, it does not share a shaft with the side bevel gears and the bevel’s and crank’s rotations are not directly related to each other. 

On the other hand, the sun gear in the planetary arrangement does share a shaft with the side bevel gear and its rotation is locked with that of the bevel gear’s. This means that, as mentioned above, when the motors attempt to rotate in the opposite directions wrt the bevels’ frame of reference, the bevel gears will provide a velocity input of zero to the planetary gears. With the other input velocity being provided by the motor to the planet gear, the output velocity of the arm, which controls up and down motion, can be found. 

Each full rotation of the planet gear moves the arm’s angle by 90 degrees, since the pitch circle of the planet is one-fourth that of the sun’s; thus the angular velocity of the arm is one-fourth that of the planet’s. Due to the conservation of power T_in x w_in = T_out x w_out, if w_in = 4w_out, then T_in x 4w_out=T_out x w_out which simplifies to T_out / T_in=4. 

Linkage

The initial linkage was intended as a non-Grashoff crank rocker 4 bar. However, the Kutzbach modification of the Gruebler Equation M=3(L-1)-2J1-J2 shows that the classic 4 bar only has one DoF, which was found to be insufficient for achieving the desired 180 degree hemispherical (or close to) range of motion. As such, one of the pin joints was changed to a slider joint, yielding the degree of freedom required. This can be seen by solving the Gruebler equation for a classic 4 bar, where M=3(4-1)-24-0=9-8=1. If one of the pin joints is replaced by a slider, we lose a full joint and gain a half joint. The Gruebler equation now yields M=3(4-1)-23-1=2. With 2 DoF, we can modify the design to get closer to the desired range of motion.

The initial design was to drive the crank, which would be located at the front, with an unpowered rocker in the rear. The hypothetical end effector would be attached to the connecting rod which would extend far out in front of the crank. The arrangement is depicted below.

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Fig. 4: Initial robot arm idea involving a classic four bar linkage. Analysis revealed that the arm would not have the desired range of motion with this linkage.

The analysis that identified the lack of range of motion is depicted below and was performed using a slightly modified version of the 4 bar plotting program given in take home quiz 3. Two positions were taken, one with the crank at 0 degrees, and the other with the crank at 180 degrees, representing the extreme ends of the arm’s range. A parallel four bar linkage was examined as a baseline to which the other two variations (longer/shorter crank) were compared. The heights that the end effector can reach in this configuration are limited by the length of the crank relative to the rocker. If the crank is longer than the rocker, then the arm can reach up well but has limited range of motion below the crank’s grounded joint and vice versa.

Open Fig. 5: Matlab plot of the parallel 4 bar. The tip of the red extension represents the range of motion of the arm.	Open Fig. 6: Matlab plot of a 4 bar where the crank (red) is longer than the rocker(green). Note the poor downward range of motion.
Open Fig. 7: Matlab plot of the same 4 bar in figure 6. Note the improved upwards range of motion.	Open Fig. 8: Matlab plot of a 4 bar where the crank is shorter than the rocker. Note the significantly improved downward range of motion compared to the 4 bar depicted in figure 6.

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Fig. 9: Matlab plot of the same 4 bar in figure 8. Note that the linkage’s ability to reach up has been reduced to only 45 degrees.

Figures 6 and 7 show the same 4 bar, where the crank is longer than the rocker. This results in a poor ability to reach downwards compared to upwards. Note that while this 4 bar can technically reach the ground (y=0), it can only do so by reaching out very far in front (fig 6), making it difficult to manipulate objects close to the arm’s base. While it can reach very high (fig 7), it quickly begins reaching behind itself, which is not very useful.

Figures 8 and 9 show a different 4 bar, one where the crank is shorter than the rocker. This results in a much improved ability to reach down but the ability to reach up is significantly reduced. Again, while it can reach down quite well, it can only reach far out in front.

In order to improve the range of motion and allow the arm to reach up high while also reaching far out in front of itself as well as reach low while reaching close to itself, the pin joint between the rocker and connecting rod will be changed to a slider joint. This effectively allows L3 (length of connecting rod) to vary increasing the number of potential positions that the arm’s end effector can take.

While kinematic analysis was performed for the position, velocity and acceleration are of little interest since the arm will not move with any significant velocity or acceleration. Velocity control is done by varying the inputs from the motors rather than linkage design and the determination of acceleration for identifying interlinkage forces is of lesser importance than the position analysis.

The gearset that will be used to transfer power up the rocker arm, through the slider joint, and to the wrist is still undetermined. Its kinematic analysis will be added at a later date, although it is expected to be just as simple as the ones for the existing gearsets. Additionally, the gearset used to power the rocker link is simply an inversion of the existing gearset used to power the crank.

Slider 4-Bar Analysis

The joint between the rocker and crank (blue and green) was replaced with a slider. In matlab, this was modeled as a link 3 of variable length. There are now two inputs, theta2 and theta3 which are the angles of the crank and rocker, respectively. The figures below show the increased range of motion. As with the standard pin jointed 4 bar plot, the tip of the connecting rod (green) shows the range of motion of the arm.

Open Fig. 10: The rocker (blue, not shown) is fixed at some angle while the crank’s (red) position is theta2 = 0. Note the almost vertical orientation of the connecting rod.	Open Fig. 11: The same linkage in figure 10 is shown, but with the crank at theta2 = 180. Note the arm’s increased ability to reach down in front of itself.
Open Fig. 12: Now, the crank (red) is fixed at some angle while the rocker (blue) is free to rotate. This image depicts the rocker at theta4 = 0.	Open Fig. 13: The same linkage as figure 12 is depicted. The rocker is at theta4 = 180. Note the significantly increased reach of the connecting rod.

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Video demonstrating the increased range of motion of the 4 bar once a pin joint is converted to a slider.

Force Analysis

Using the parallel 4 bar as a starting point, we assume a 1kg load at the end of the arm at a starting position where the crank is at 180 degrees. Since there is no rotation of the connecting rod in a parallel 4 bar, the only consideration is the downwards force exerted by the load and its resulting torque about the crank’s grounded pin joint. A 1 kg load exerts 9.8N of force. With a crank length of 50mm, the required torque is 0.49Nm. Assuming we are using the same motors as those in the Robot Car project, which have 0.15 Nm of torque, we will need a 1:4 ratio. 

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Fig. 14: Notebook analysis of the required torque with a sketch of the initial conditions.

The torque calculation for the planetary gearset done in the kinematics section is repeated here. “Each full rotation of the planet gear moves the arm’s angle by 90 degrees, since the pitch circle of the planet is one-fourth that of the sun’s; thus the angular velocity of the arm is one-fourth that of the planet’s. Due to the conservation of power T_in x w_in = T_out x w_out, if w_in = 4w_out, then T_in x 4w_out=T_out x w_out which simplifies to T_out / T_in=4. ” If the premise of the calculation is correct, this torque ratio of 4 should apply to raising the arm as well as rotating it.

Force analysis is yet to be done on the gear teeth. The spur gears in the planetary arrangement were obtained via an online gear generator to quickly mock up a gearset for the prototype, and key parameters of the teeth are missing, such as tooth thickness. The final spur gears will be derived from scratch to better account for the needs of this specific application and this will ensure we have the full set of parameters required to do a full force analysis on the gear teeth.

The bevel gears follow an octoid profile rather than an involute one. Since I am unsure as to what the octoid profile entails, I am unsure what the properties of the bevel teeth are. I will need to contact the professor for more clarification on octoid teeth.

A friction and sliding force analysis will need to be done on the slider joint between the rocker and connecting rod once the revisions to the design are complete. Bending stresses will also need to be examined before the final arm is delivered.

Iterations

Prototype 1

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Fig. 15: Prototype 1.

Prototype 1 is a looks-like prototype designed to capture the approximate dimensions and arrangement of the bevel gear arrangement. The bevels are modeled by smooth cones made of TPU. It was hoped that friction between the cones would cause them to rotate each other but unfortunately they were too far apart. This was not a very big deal since the prototype was never supposed to simulate function.

Prototype 2

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Fig. 16: Prototype 2

Prototype 2 had identical form to prototype 1, with the smooth cones replaced by toothed bevel gears. This upgrades it to a works-like prototype in addition to it being a looks-like prototype. It serves the same purpose as prototype 1 but its functionality makes it more useful in understanding the kinematics of the gearset.

This prototype was made to confirm the functionality of the gear teeth, ensuring that they mesh well without interference and roll across each other smoothly. Informal evaluation for backlash was also performed.

Prototype 3

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Fig. 17: Prototype 3

Prototype 3 is simply prototype 2 with the rest of the arm’s mechanism added around it. The grounded bevel has been extended to form a solid base while the 4 bar and a planetary gearset has been placed around and over the bevel gearset. This prototype is a looks-like and works-like prototype for the core mechanism of the arm. While the final product will place the bevel and sun gear on a shared shaft allowing for torque transmission, the prototype uses a screw joint for simplicity which does not provide any link between sun and bevel gear. 

This prototype allows for the evaluation of the planetary gearset to evaluate the quality of mesh and rolling of the gears. The whole assembly is also useful to ensure that the correct tolerances have been set for the chosen manufacturing method (FDM). 

This prototype also serves as a prototype for the gearset that will be used to actuate the rocker of the 4 bar, as that gearset is simply an inversion of the current one, where the center cube is grounded and the middle bevel is allowed to spin freely. 

BOM