1.2 Kinematic Analysis

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.

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.

A very desirable wall climbing motion can be seen in Figure 5. Point P, represented by a yellow dot, has a near linear path when it's further away from the origin (Pressing up against the wall). The specific foot angle and link lengths needed to obtain this particular motion resulted from multiple stages of analysis. First, we had to confirm the linkage's rotational range and degrees of freedom. Next, we had to observe the end effector's position, velocity, and acceleration profiles, tweaking lengths and angles in order to manipulate different characteristics of the path. Finally, we estimated the pressing force of the legs against the walls, making sure the resulting friction force would overcome the gravitational force on the climber. Each of these analysis stages are presented in the following subsections.

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

Using MATLAB, our group was able to illustrate the motion cycle of the end effector P, along with its accompanying velocity and acceleration values. We used this visualization to correct the path's flatness, length, and thickness.

NOTE: For each of the graphs below with theta 2 as their x component, the angle is that of L2 with respect to the global X axis shown in Figure 4, not the ground link.

  Figure 6: X vs Y position of Point P                                     Figure 7: Theta 2 vs Velocity of Point P                                                                           Figure 8: Theta 2 vs Acceleration of Point P

In Figure 6, the Y axis represents the contacted wall. According to the plot, the end effector makes contact with the wall for approximately 30mm (3cm). This length is much less than half the overall path length of the end effector, meaning that two sets of legs aren't sufficient enough to cover contact for a full rotation. However, the contact length looks larger than at least a third of the overall length, so three sets of legs might be enough to maintain constant contact. If not, compliance in the foot could be added so that the contact length grows (Allowing the path to effectively overlap the y-axis and increase the length from foot contact to release). 

Leg Force Analysis

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 weight of the robot and friction at the feet 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)

gravitational acceleration = 9.81 m/s^2

µ = 0.75 (Estimation based on the static friction coefficient between wood and rubber, assuming we make the walls out of wood and the feet out of rubber)

k = 820 N/m (Estimation based on the springs we plan to order)

x = 0.015 m (Estimation based on how far the parallel walls are spaced apart. Can be adjusted within reason)

Fg = mass * gravitational acceleration = 1.2kg * 9.81 m/s^2 = 11.772 N

N = k*x = 820 N/m * 0.015m = 12.3 N

f (Single Leg) = µ*N = 0.75 * 12.3N = 9.225 N

There is one set of legs in contact at all times. One set has four legs moving in sync, which means that there are four contact points where friction is generated.

f (Total) = 4*f = 4 * 9.225N = 36.9 N

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, 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.