15.2 Project Analysis
Kinematic Analysis
The Basis for this project is the Klann Linkage, which is an extremely well understood and optimized Mechanism. Due to this there is not a lot of range available for us to improve its operation, however a detailed analysis of its function was still created.
This mechanism most closely resembles a insect/crab movement and requires only one primary crank input, making it perfect for our application. Due to its prevalence in existing research, we first looked at a general Klann linkage mechanism within MotionGen for dimensions (based on existing values) and kinematic analysis.
Animated Linkage
Where:
L1 = ground (triangle with dimensions 2.84” x 1.83” x 2.62”
L2 = 2.62” + 2.13” (10° offset)
L3 = 1.04”
L4 = 1.23”
L5 = 2.5” + 5” (10° offset)
L6 = 1.71”
To analyze mobility, we use Gruebler’s equation:
M = 3(N – 1) – 2J – H
Where:
N = 6 (including ground)
J = 7 (revolute joints)
H = 0
Thus,
M = 3(6 – 1) – 2(7) – 0
M = 1 DOF
Kinematic Profiles:
Ideally, the “foot” trajectory should follow a near-flat bottom profile (as seen in the last linear upwards trend in the displacement graph) with an airborne reset phase. From this linkage, we also see that linear acceleration is constant at the bottom which is ideal.
Matlab Analysis
We then moved on to Matlab analysis, where we plotted the linkages to optimize and visualize the different gait and flatness profiles.
As seen below, this is the resulting output motion of our Klann Linkage with the optimized lengths.
Footpath Analysis
In order to optimize the gait length and duration, we optimized different linkage lengths and angles to see which resulted in our desired output. In order to calculate the flatness, we chose two points 180 offset from each other (shown by two circled points in the following images) and calculated the standard deviation of the y values in this range in order to produce a flatness score. This score was compared across all the different link lengths in order to produce the motion path shown below.
Optimized Foot Path:
Optimization of Links
We first started with the optimization of the individual links, in order to gain and understanding of the working ranges of each link and the effect each link has on the overall output.
Link 3 | Link 4 | Link 5 | Link6 |
|---|---|---|---|
Optimization of Angles
We then moved on to two angles we deemed as impactful angles in order to visualize the changes of each angle on the output profiles.
Angle 4 | Angle of Ground Foot |
|---|---|
Optimization of Extensions
Finally, we compared the different extension lengths of Link 4 and the Extension of the foot so as to fine tune the final gait profile to the height and length of gait that we wanted.
L4 Extension | Foot Extension |
|---|---|
Oscillating Gear Box
In order to produce an oscillating motion with a desired gear reduction from a non-reversing motor, we developed an oscillating gear box.
To achieve this, we used a trimmed 35 tooth gear as seen below to drive the left and right reductor gear assemblies in turn.
Oscillation Driver Gear | Reductor Gear Assembly |
|---|---|
| |
The setup, as seen below, allows for the intermittent turning of either the right side or the left side. (Note: all the gears in the assembly will be rotating at any given time)
This intermittent engagement of the gears allows for the output shaft (connected to the central 19 tooth gear) to be driven in turn be either left or right gear assembly, with one gear assembly connected to an idler (gear with hex hole) so as to reverse direction.
The output shaft is not directly meshed to both gear assemblies as it may seem in the image below, rather one gear assembly is offset with an extra spacer so at to only mesh with an idler that in turn meshes with the output shaft.
In order to achieve the desired gear reduction, the gear assemblies essentially act as a compound gear train to reduce the resulting speed as seen by calculations below:
Speed of motor = 50 rpm
Using the compound gear train formula of
omegaout = omegain (Ndriver/Ndriven)
We get:
For the right side:
Input Speed | Motor to Oscillator | Oscillator to Gear Assembly | Gear Assembly to output | Klann Reduction | Output Steps: |
|---|---|---|---|---|---|
50/60 rps | (19/35) | (16/19) | (35/19) | (26/46) | 0.876 steps/linkage to the right per revolution at 0.4523 revolutions per second Resulting Speed: 0.396 Steps Per Second Per Linkage |
Since we an 180 degree phase offset for our legs (4 legs move together out of the 8 total), we essentially complete two gait cycles per rotation, resulting in close to 1.752 steps in one direction before the crab switches its direction.
Similarly for the left side:
Input Speed | Motor to Oscillator | Oscillator to Gear Assembly | Gear Assembly to idler | Idler to Output | Klann Reduction | Output Steps: |
|---|---|---|---|---|---|---|
50/60 rps | (19/35) | (16/19) | (35/19) | (19/19) | (26/46) | 0.876 steps/link to the left per revolution at 0.4523 revolutions per second Resulting Speed: 0.396 Steps Per Second Per Linkage |