In order to maintain balance and traction, the motion paths and cyclical timings of a wall climber's legs must be fine tuned. The leg's end effector, or foot, must maintain a steady and prolonged contact with the wall. Push in too much and the linkage will stall, push in too little and the foot will slip, so a near linear motion profile is desired. A compliant foot, such as one made of rubber or springs, could allow for more flexibility in the system, but excessive compliance might diminish the foot's traction against the wall. With these challenges in mind, our group has chosen a leg geometry that we believe strikes the balance between contact force and compliance. A sketch of this linkage is shown below.
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Figure 4: Sketch of Leg Linkage
Linkage Parameters
L1 = 30mm
L2 = 10mm
L3 = 30mm
L4 = 12mm
LP = 65mm
Offset angle theta = 56°
Foot angle delta = 80°
L1 is the ground link that will be connected statically to the rest of the climber body. L2 is the driving link that will be rotated around the origin point at a constant angular velocity. The ground link is offset by theta = 56° from the X axis so that the outer path of position P is as parallel to the Y axis as possible. To help visualize this motion, an animation of the linkage is shown below.
Figure 5: Animation of the leg linkage and the end effector point P. Dotted arrows represent the velocities at each point.
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Linkage Mobility Calculations
For each leg linkage, we need just one degree of freedom for the system and a full rotation range for L2. The two equations below were used to analyze both conditions.
Gruebler’s Equation
M = 3*(L-1) - 2*J1 - J2
M = 3*(4-1) - 2*(4) - 0
M = 9 - 8
M = 1
Our 4-bar linkage system has 1 degree of freedom.
Grashof Condition
S + L ≤ P + Q
10 + 30 ≤ 12 + 30
40 ≤ 42
Since this mathematical statement is true, at least 1 link in our linkage system can make a full rotation.
Kinematic Analysis of End Effector
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36.9 N > 11.772 N, which means that the total static friction force is greater than the gravitational force and slipping will be prevented. This is an ideal scenario with approximated variables, so we understand that this result may not ensure our results may not guarantee our mechanism's functionality. Regardless, our calculated friction force exceeded gravitational force by a large margin, and if the real friction somehow falls short, these calculations give us the framework to know what variables to modify.