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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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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Many of our design assumptions rely on the implementation of a compliant foot with springs and a rubber material. This will determine how much force each leg applies to the wall, and whether or not it will be able to resist slipping during its period of contact. Ideally, at the moment Point P makes contact with the wall, it becomes a static pin joint that the climber can use to pull itself up. This static characteristic is determined by the gravitational force acting on the climber (Fg = mass*gravitational acceleration) being overcome by the static frictional forces generated by the legs. This friction force f is defined by the equation f = µ*N, where N is normal force and µ is the static friction coefficient between the foot material and the wall material. Since we're utilizing springs, normal force can be found using the spring force equation. Thus, N = k*x, where k is the spring constant of our particular spring and x is the distance it's compressed. Using these equations, we were able to quantify the frictional forces created by the leg contacts and ensure that their summation was greater than the force of gravity. Calculations are provided below.
Note: Our leg linkages are ran by two gear trains. A motor is connected to these gear trains by a 3:1 gear ratio. We do not consider this part of the mechanism in our leg force analysis because it does not directly determine the mechanism's ability to climb. More so, it determines how long the legs can operate as the linkages work against internal friction. The friction at the feet and weight of the robot have a much greater influence over the mechanism's climbing ability. Our gear system is relatively simple and is unrelated to the kinematic motion, so we've left out its analysis.
Parameters:
mass = 1.2 kg (Estimation based on prototypes)
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36.9 N > 11.772 N, which means that the total static friction force is greater than the gravitational force. Slipping will be prevented, and the climber will be able to maintain its grip on the walls as it ascends. This is an ideal scenario with approximated variables, so we understand that 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 can give us the framework to know what variables to modify. .
Final Design Considerations
It should be noted that for our final design, we changed our configuration from vertical to horizontal because the assembly was growing heavier than expected. The mechanism would no longer be working against the force of gravity, and so we decided to simplify our assembly and swap the rubber-ended spring feet for foam feet with less friction. Although the leg force calculations above no longer apply to our final design, they should still be considered if our project were to ever be adapted back to a vertical scenario.