Post by Wade Vagle
​As we learned from our TrotBot Scaleup attempt, not all robot's legs will walk by pushing or pulling the robot. When we pushed TrotBot Ver 0 the cranks would initially rotate, and then the linkage would freeze and the feet would skid on the pavement. So, we had to manually rotate TrotBot's cranks to make it walk.

The same thing happened when we tried to push our LEGO Klann walkers with the motors disengaged - the feet would skid and the cranks would not rotate. In both cases, this was due to the linkage being at a sort of reverse Dead-Point. As you can see in the image of Klann's linkage below, pushing Klann's feet at this point in the crank's rotation would not cause the crank to rotate. Instead, pushing the robot would only cause the legs to bend and the feet to skid.

However, pushing a Klann robot with the crank in the below position would cause the crank to rotate:

If another pair of legs were added to each side of Klann, as done in the simulation of Strider below, then maybe its legs could be driven by pushing the robot (although one of Klann's feet would still skid at the corner of the foot-path, so it may not work so well - maybe adding feet that could slide or rotate a little would help?)

Here's a test to see how easily this variation of Strider's legs can be driven by an external force - gravity in this case:

Notice how the robot wavers slightly to the left and right as it descends, due to the foot-speed varying a little.

Strider's legs can also be driven by pushing the robot when built in an 8-leg version, although not as efficiently as 12-leg versions:

Posted by Wade Vagle
Like how we appreciate Nike's when running on concrete, large-scale walkers benefit from shock-absorbing feet. By increasing the spring's travel, shock-absorbing feet may also be able to sufficiently increase the percentage of ground-contact of each leg to smooth the gaits of high-stepping walkers like TrotBot and Strider. And of course this would create new problems to be solved! 

First, here's the Mondo Spider's feet in action, which provide some shock absorption, and they also slide on the smooth concrete, which helps with turning and with smoothing Klann's speed:

It walks amazingly fluidly considering how Klann's foot-path comes to a stop at each end, and the springs probably smooth the transition between feet somewhat:

Watching the video raises some questions, like:

• how much power is consumed by friction when the feet slide? 
• is there a way for the feet to slide (or roll) when turning on rough terrain?

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Implementing Klann's linkage without shock-absorbing feet that slide results in a more halting gait, as can be seen in this version of the Walking Beast:

​​Next, here are some shock-absorbing feet ideas from Mechanical Walker pioneer, Professor Joseph Shigley:

​Feet with such springs extend  the feet toward the ground. So, in addition to absorbing shocks, the springs also increase the percentage of ground contact per crank rotation of each leg. 

Taken to an extreme, feet with very long springs could theoretically increase an 8-legged Strider's  ground-contact of the legs to the point that it always had one foot on the ground at each corner of robot, and do so without causing the robot's height to drop when the feet touching the ground switch. Since both of the feet will be near the ground at the foot transition, two springs will be pushing the robot up, reducing how far the robot falls at foot transitions. 

However, a few of the (probably numerous) issues of long-spring feet are:

• if the energy used to compress the foot's spring isn't returned to the mechanism when the foot lifts, then the power requirements to walk would skyrocket, similar to how post-holing when walking in deep snow is exhausting, and the motors would quickly stall.

• on the other hand, if the springs weren't damped in order to save energy, then the walker would be less stable, and vibration at a resonance frequency could be disastrous - can you imagine how much more violent this Klann's shaking could be if it had long, un-damped springs for feet?

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• if the horizontal foot-speed at the curled up ends of the foot-path are slower, the feet will skid, or the robot will lurch unless the speed variability is managed in other ways (like the Mondo Spider did with it's sliding feet)
• if the force required to compress the springs is too low or too high, the gait won't be as smooth
• the springs will be fully extended when foot is lifted and returned to the front of its foot-path, so long springs will lower step-heights

So, springs long enough to convert Strider to an 8-legged walker wouldn't be feasible, but we'll try to put together an experiment to check if shock-absorbing feet can smooth a 12-legged Strider's gait efficiently by testing it (or TrotBot) with a heavy load. For now, here are simulations of adding shock-absorbing feet to 12-leg versions of TrotBot and Strider (assuming perfectly elastic springs without damping that comply with Hook's Law, and ignoring inertia and spring oscillation):

Maybe the smoothest solution for 12-leg Striders would be to add shock-absorbing pads to the bottom of Strider's feet with toes, like the toes simulated below? Pads with a smooth, hard surface to facilitate sliding while turning like the Mondo Spider's feet? Note: the more refined dimensions of this non-LEGO version of Strider's linkage can be found here.

Strider's linkage variation 7 may also be smoothed by shock-absorbing foot pads, since the bump in it's gait occurs at the foot transition, and this should be easier to test since variation 7 can be built in LEGO.

Post by Wade Vagle
If you are looking for ideas on how to create highly functional mechanical walkers, Professor Joseph Shigley's 1960 feasibility study for the army is a great resource. It combines engineering rigor and sound reasoning to illuminate many of the challenges, and potential capabilities of mechanical walkers.  

To meet the requirements of walking tanks, Shigley sought a mechanism that could:

• carry heavy loads efficiently
• traverse rugged terrain
• walk at the high speeds that tanks require

all while being mechanically reliable, which is a pretty rigorous set of requirements for a mechanical walker!

To meet the rugged terrain and speed requirements of tanks, much of Shigley's study focused on the foot-paths of mechanisms. Below is Shigley's diagram of foot-path types, with type E representing his ideal type for rugged terrain and fast speeds.

When Creating New Mechanisms, Start By Investigating 4-Bar Linkages
In general, the fewer bars in a linkage, the better (lower costs and build-times, less mechanical complication and friction, etc.). So, searching for a mechanical solution based on a 4-bar linkage is usually a wise first step. Shigley advised searching thru Hrones-Nelson's atlas of hundreds of 4-bar linkage coupler curves as a "good first approach to such a problem", which must have been a vital resource for mechanical engineers in the days before personal computers. Below is about the best 4-bar linkage configuration Dr. Shigley found in Hrones-Nelson's atlas, but it doesn't step high enough for rugged terrain.

​Shigley described how the step-height could be increased​ by allowing a joint to slide along a cam groove, but we wanted a mechanism that could be prototyped in LEGO. So, instead we experimented with pairing two 4-bar linkages into a combined 10-bar linkage with front and back legs, such that the rear leg lifted the front foot and vice versa. This allowed us to increase the step-height significantly. 

​We found that the boat-shape of Strider's foot-path mitigates this problem, since the foot's horizontal speed during the lift and lower phases is similar to the foot-speed of the drive phase. This can be seen in the above GIF of Strider walking across the screen. Notice how the foot's horizontal position doesn't change much during the bottom halves of its lift and lower phases. So, when a foot steps on an obstacle halfway down its lower phase, it will be less likely to skid or cause the robot's forward motion to stop while stepping up onto the obstacle. 

The impact of the boat-shape can be exaggerated by reducing the number of legs from 12 to 8, causing the feet to contact the ground well above the bottom drive phase of a 12 leg Strider:

​You can see this in action in the below video of an 8-legged Strider walking on rocky terrain.  Strider's feet often contact the rocks well above the bottom drive phase, yet the robot's speed remains fairly consistent as it walks across the rocks, and it doesn't come to a stop with each step like our 8-legged Klann walkers do on such terrain.  

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​As you can see in the below simulation of one corner of a 12-legged Strider on an obstacle course, the robot's horizontal speed is fairly consistent as the terrain's elevation changes:

In contrast, the feet of robots with triangular foot-paths will be going the wrong way when stepping on/off larger obstacles, or when the terrain's slope changes abruptly. If the feet can't skid, then the change in the terrain's elevation can push the robot backward, as can be seen in the simulation of Klann's Mechanical Spider below. If the rear legs aren't stepping on a similar obstacle at the same moment, then they'll be pushing the robot forward as usual, fighting the front legs and potentially jamming the mechanism.

Foot-Path Symmetry
​The front-to-back symmetry of Strider's foot-path results in fairly consistent horizontal speed when the terrain's elevation changes on either side of its foot-path - e.g. when either ascending or descending the stilts in these simulations. In contrast, TrotBot's foot-path is somewhat teardrop-shaped. This causes TrotBot's horizontal speed to drop when encountering large elevation changes on the shorter, inner side of its foot-path. This can be seen in the below simulation when TrotBot steps down to a lower stilt:.

​Tank-Scale Functionality on Rugged Terrain?
Strider's high-stepping, boat-shaped foot-path allows it to meet Shigley's rugged terrain requirements fairly well at LEGO-scale. However, tank-scale functionality on rugged terrain is another matter, which would require the robot to be strengthened dramatically due to the lower strength-to-weight-ratios of larger scales. And, increasing strength typically involves adding structure and motor power, which further increases weight, which requires even more structure, etc.

The legs in particular would need to be strengthened, since walking along the side of a hill, or turning the walker tank-style while on rugged terrain would put a tremendous amount of sideways forces on the long legs - if it were even feasible at such large scales. However, adding too much structure and weight to fast-moving legs would create other problems. Instead, perhaps the lower half of the legs could be supported laterally by rails connected to the frame? Rails that created channels that the leg's path should follow? Positioned below the legs' knee and "hamstring" joints to reduce sideways forces on these weak points? Eh......for rugged terrain I think I'll stick to smaller-scale walkers, which also benefit from Shigley's insights.

High Speed Walking
As Shigley described, an ideal walking mechanism for a tank should have the constant horizontal speed of a wheel, while also being able to step over obstacles that block wheels. Strider has relatively constant foot-speed, resulting in an efficient gait, and is the only mechanism that we've tested which can walk with a 1:1 gear ratio without the LEGO motors stalling: 

High Speed Vibration
Shigley also described how the vertical and horizontal inertia forces of the mechanism should be balanced for a tank to walk fast without vibrating itself to death. Such vibration isn't much of a problem at LEGO-scales, but it gets worse as scale increases. If you've ever driven a vehicle where the steering wheel shakes due to its wheels being out of balance, and seen how it's corrected by attaching small metal weights to the wheels, then you can probably imagine how much worse the vibration problems could be for a vehicle with large mechanical legs running at high RPMs.  

Strider's mechanism would almost certainly need to be modified in order for a large-scale version to walk at high RPMs with limited vibration. If curious, Shigley's study below shows how to mathematically analyze inertia forces, and gives some suggestions for balancing them.
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Load Bearing and the Linkage's Bar Count
​​The more bars a linkage has, the more joints it needs. Each joint adds friction. Furthermore, the joints will always have some play, which can cause long legs with many joints to bend sideways under loads, especially when turning them tank-style. Like Klann's 6-bar linkage, Strider's linkage has relatively few bars per leg with 10 bars per leg pair, versus TrotBot's and Strandbeest's 8 bars per leg. If implemented well, this implies that Strider's linkage has the potential to be relatively energy efficient and should be able to handle loads relatively well.   
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Below tests linkage variation #6 with a 25 pound load. The plastic parts bend and shift somewhat under the load, increasing the difficulty of carrying the load, but the LEGO motors didn't stall. Also, we were planning on uploading a video with Strider self-destructing at the end by attempting to turn it while carrying 25 pounds. Some of our other walkers self-destructed while carrying much less weight, so we were a bit surprised that Strider survived with its legs intact. Strider's low bar count per leg does appear to help with carrying heavy loads.

Before performing this test the plastic LEGO axles were replaced with steel axles to handle the torque. Other than that, and the 2 steel support rods, all of the parts are plastic LEGO parts connected by LEGO pins (no glue). ​

Strider's linkage dimensions can be found here, and other configurations of Strider can be found here where you can also create your own customized Strider linkage with an embedded simulator.
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​Below is Prof Shigley's feasibility study, a 1960 Popular Science article discussing it, and the Hrones Nelson  Atlas of four bar linkage coupler curves.

As you can see in the following images, Klann's foot-speed slows down significantly at each corner of its foot-path:

​This causes robots using the Klann linkage to have a halting gait, which can be a problem on higher-friction terrain at higher speeds. One solution is to add feet that passively rotate as Klann's speed varies, like Strandbeest uses. 
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​As you can in the following video, rotating feet also reduce how much the feet drag during turns, since the inner and outer feet can rotate at different speeds and function somewhat like a differential. 

In case you were wondering, Klann's violent shaking on the carpeting did cause it to flip over a few times, but at least it didn't do this.

Rotating feet also improve performance on rugged terrain, since the feet are less likely to catch on obstacles. Prior to adding wheels to the feet, the Klann below couldn't take one step on those air filters without the toes catching and jamming the linkage, and if LEGO's XL motors had been used, jamming the linkage could have broken the gears.  

Strandbeest's foot-speed also slows at each corner of its foot-path, but less so than Klann's:

This may be part of the reason that Theo added rotating feet to Strandbeest? ( Jeez that thing is cool!)

Similar to Klann robots, adding rotating feet to LEGO Strandbeests smooths their speed:

As can be seen in the following simulation of one side of an 8-legged Strandbeest, the variation in Strandbeest's foot-speed is exaggerated when built with only 8 legs since the foot transition occurs when the feet are already lifted off the ground:

​​I rushed my Klann Ver 2 build and didn't build the outer frames in an optimal way:

The only thing keeping this frame's corners at right angles are the two 3x5 L-shaped LEGO parts.  If this frame were put under a lot of force, like would happen at a larger scale, the corners would be subjected to torque that could easily cause the rectangle to collapse. To avoid this diagonals need to be added, which will convert the rectangles into triangles and lock the corner's angles.
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The challenge with walkers in LEGO is we often need diagonals for rectangles that define the linkage's parameters.  In other words, these diagonals often need to be the hypotenuses of right triangles.   As you can see below, my Klann's upper support rods are 7 holes above the motor's axle, and the lower support rods are 2 holes below the motor's axle.  Neither of these lengths work with a 3-4-5 or 6-8-10 right triangle with LEGO parts for hypotenuses. What can we do using the integer lengths of LEGO's beams?  

NOTE:  When determining the length of LEGO beams the first hole is always counted as 0!  If you don't measure LEGO beam lengths in this way you won't be able to use the Pythagorean theorem to calculate which beams to use as hypotenuses.
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Fortunately, with LEGO we don't have to be at precise integer numbers, and we can use hypotenuses that are close enough. 
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1.  Hypotenuses for rectangles of height 7. 
Plugging in a 90 degree angle with a side length of 7, plus a few other whole number sides into the Pythagorean theorem yields this near integer number triangle:

​I used this triangle for Klann's inner frame where two 9 hole beams create hypotenuses of length 8:

I also used this triangle for TrotBot's frame:

​2. Hypotenuses for rectangles of height 9. 
We can also connect the top and bottom beams of Klann's frame with a hypotenuse:

Plugging a 90 degree angle with a side length of 9, plus a few other whole number sides into the Pythagorean theorem yields another near integer number triangle:

So, 13 hole LEGO beams can be used to lock Klann's outer frames into triangles, like this:

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Here are a few other useful triangles for LEGO frames:

​​Also, the below 5-3 bent beam can be used as a hypotenuse.

You can create more triangles by lengthening a side of the above parts by attaching a LEGO beam to it.

Klann's Mechanical Spider was the first walker I ever built.  I like it since it's robust and not too complicated (as mechanical walkers go).  

Here's a simulation using the ver 2 LEGO beams as the bar lengths.  The python script that made this sim can be downloaded here.

The Mechanical Spider can bear more stress than other walking mechanisms, why?