'Zigzag / zagzig' and 'zagzig / zagzig' robots (Walking Machines V)

I have not discussed the theme of walking machines for a long time: the last post on that subject was published back in 2010, but dealt with 'radial robots', meant as toys. As you will know the word robot was derived from 'worker', so it is fitting to go back to posts on robots meant to do proper work (here).

The lack of interest was not because no progress is being made; quite the contrary. There is so much improvement that robots are slowly taking on aspects of animals, and the reason for that is en ever-increasing subtlety of control of the movement. The word 'cybernetics' has its origin in a Greek word for 'steersman', and 'steering' all aspects of movement is a key concept in walking, regardless of whether you are talking about an animal or a machine. The other major theme is mechanics, of course. For some background on leg design, see here and here.

You are probably familiar with 'Big Dog', a walking robot made by Boston Dynamics. That company has developed a range of robots meant to aid the military. The video above shows the level of control their robots have these days: clearly, this thing, the 'legged squad support system' (LS3) can hold its won on difficult terrain and follows its human master on its own. There are more YouTube videos (and with better quality) that are found on YouTube after a search for 'Boston Dynamics'.

Here is another one: a 'cheetah' running very fast on a treadmill. It is tethered and the power source is external, but is still an amazing sight. It is interesting to see how the engineers handled the problem of elongating stride length. Running mammals generally have legs with  three major segments; the foot can be seen as a fourth, minor segment. Cheetahs obtain an additional lengthening of their strides by flexing and extending their bodies as well during the stride. Now compare that to the cheetah robot: it does have a flexing body, but the legs have only two segments and the foot appears to be something like a rubber ball only.

Their latest attempt is called 'WildCat', which is apparently based on the cheetah design. This time both the front legs and the hind legs seem to be linked to the main body by a flexible joint, instead of the legs being fixed to the body directly as was the case for LS3. As a result, the setup is beginning to resemble the setup of shoulder and pelvic girdles common to vertebrates.

The legs still consist of two segments only. Seen from a level of control, having only two segments makes it much easier to find out where the foot should be, as only two angles have to be controlled, and each foot position can be reached with only one combination of angles. If you add another segment, each foot position can be reached in an infinite combination of joint angles. I wonder whether the lack of a foot is due to similar considerations: adding another segment, even a short one, probably adds a considerable computing overhead. 

Another interesting aspects is how the engineers chose the directions where the knees and elbows point to. In an earlier post I discussed whether legs should start with a segment pointing forwards (a 'zig') or backwards (a 'zag'). The next segment than points the other way. The upper arm (humerus) of mammal front legs points backwards and the forearm forwards, so the mammal front leg is a 'zagzig'. Hind legs, in the same jargon, are 'zigzags'. The formula for the entire mammal is a 'zagzig / zigzag'. Are you still there? (Mind you, this is just a simplification paving the way to look at robots; if you include the scapula, -a zig!- and label all three segments, mammals are 'zigzagzig /zigzagzig' animals.)

Now have a look at the LS3 again. Its mid leg joints point away from the body, just the opposite of the mammalian leg bone pattern. The LS3 is a 'zigzag / zagzig' walker. WildCat, in contrast, is a 'zagzig / zagzig' walker. I have no idea why the engineers  chose the designs they did, but the results strengthens my feeling that there is no basic overwhelming advantage inherent in the current mammal pattern. During evolution sideways-pointing legs were turned to have the plane of the leg parallel to that of the body, and in this turn front legs turned backwards and hind legs forwards. Evolution might well have resulted in a different pattern, that of LS3. At least it prevents knocking elbows into knees! Those who wish to add more 'alienosity' to their animals might consider departing from the Earth vertebrate pattern. Have a look at LS3, WildCat, or, of course, at an older post in this blog to see what might be done.

Finally, a word on gaits. Walking consists of a repeated cycle of leg movements, and a gait is nothing more than the phase differences between the various legs. The basic gait of LS3 is a trot, in which front left and right hind legs move together as one pair, and the other two legs from the other pair, moving exactly half a cycle out of phase. If this is confusing go the Furaha 'walking with...' page, where the major gaits are explained. The engineers of Boston Dynamics have managed to proceed beyond the trot, so the thing can bound and gallop as well. I am very surprised though that it seems to use a trot when it is walking very slowly. You would expect a 'walk', which in this context also is a defined gait. The various gaits used by animals have important energy consequences, and a trot is more expensive than a gait. I wonder how much further 'evolution' will take these robots. More segments? More efficient gaits? More legs, even?  

Layers of leaves (Alien plants V)

The last post on alien plants went into some fairly technical details about photosynthesis, and took a look at where the process could be adapted to make it more alien. Today's post has a closer look at just one aspect: leaves.

Photosynthesis obviously depends on catching light and is therefore a process that takes place on the surface of a plant. How much of a surface is needed will depend on many things, such as how much energy is needed. As related before, C3 photosynthesis can only make use of up to 25% of the light falling on them (well, at noon in the tropics, that is). Photosynthesis becomes saturated, doing nothing with that extra light.

For now, let's assume the presence of leaves on a planet of choice as very similar to Earth's broad leaves; flat structures of, say, 10 cm across. They need light, and so face the sun. But not every leaf of every plant will receive full sunlight: the sun moves across the sky (well, as far as the plant is concerned it does), other plants may be in the way, and even its own leaves, placed higher up, will take light away from lower leaves. A typical Earth leaf transmits only 5% of the light striking it. The next leaf down in turn absorbs 95% of the -little- light striking it, leaving only 0.05 times 0.05 of the sun light, or 0.25% of the light striking it.* On Earth the critical level for photosynthesis to be of any use is at about 1% of full sunlight.

It is therefore reasonable to assume that plants would have only one or perhaps two layers of leaves, right? Additional leaves would not contribute anything, and yet trees typically have many more layers of trees. The answer to this riddle is found in the efficacy of photosynthesis and a fact that you might not have considered interesting in this respect: the size of the sun.

Click to enlarge; source: http://en.wikipedia.org/wiki/Umbra

The image above shows the umbra and penumbra as commonly illustrated in astronomy books. The sun is not a point source of light but a sphere much larger than the Earth. Rays of light depart in all directions from any point on its surface, to the effect that there is a conical volume of space behind the Earth where the rays cannot reach, or, in other words, from where an observer can see no part of the apparent disk of the sun. That conical volume of space is the 'umbra' , simply meaning shadow in Latin. Around it there is an area from which an observer can see part of the sun's disk, so that area receives some direct sunlight, but not full sunlight: the 'penumbra' (nearly shadow). Everywhere else receives full sunlight. The length of the umbra cone depends on the diameters of the Sun and the Earth and the distance between them, as a few minutes experimenting with some sketches will show you.

The same applies for objects closer by. All you have to do is to look at the shadow of your hand as you raise it from the ground on a sunny day. Leaves also cast an umbra, an area without direct sunlight, where it would be best not to place another leaf. The length of the umbra can be calculated as explained above, and for Earth the calculations that the umbra is about 108 times the width of a leaf. For a 10 cm leaf that would boil down to 1080 cm, or 10 meters. As we will see the distance is shorter in practice. Leaves may receive enough light in the penumbra to work well. Remember that on Earth photosynthesis is already saturated at 25% of full sunlight, so photosynthesis can work at full capacity even with a fair amount of shade.

I wrote a Matlab program to have a look at how the umbra and penumbra could look for some artificial leaves. The distance between the Earth and the sun is 149,597,870,700 meters and the diameter of the sun is 1,392,684,000 m., both according to Wikipedia. A leaf takes up half the area of a 10 by 1-0 cm square area. In the program, this meant that I could paint half the pixels in a square area black denoting the leaf. All the program does is to cast ray from a raster of points on the sun's disk to all points on the leaf area, and to see which rays are intercepted and which are not. I did that for three distances behind the leaf: 0.5, 1 and 5 meters.

Click to enlarge; copyright Gert van Dijk

And here is the result of a simple roughly circular leaf. Half a meter away our leaf casts a recognizable shadow. I calculated how large the area is that receives less than 25% of full sunlight. That value of 25% is randomly chosen but helps to indicate deep shadow. Depending on the efficacy of photosynthesis, the value could indicate the lower limit of light for photosynthesis to work if it is particularly inefficient, or perhaps the point at which its efficacy becomes impaired. At half a meter an area of 85% of the original leaf receives less than 25% of full light, while at one meter the area decreases to 69%; at 5 meters it is 0%.

Click to enlarge; copyright Gert van Dijk

Let's try with a differently shaped leaf. After all, the umbra depends on the width of the leaf, so a leaf with a thinner shape should do better. This cross-shaped leaf was somewhat disappointing, as its values for 25% full light were only slightly better than for the circular leaf. For 0.5 meter the value was 81% of leaf area, for 1 meter it was 65% and at 5 meters it is 0%. Clearly, some more shape experimentation is needed.

Click to enlarge; copyright Gert van Dijk

This 'clover' has more space between its petals. Does it work better? Yes it does: 0.5 m results in 62%, 1 m in 31%, and 5 meters as usual results in 0%.

Click to enlarge; copyright Gert van Dijk

Finally, here is the ultimate feathery leaf, designed to have thin strands, while its area is still the same as that of the others. Here are the values: for 0,5 meter, only 11% of the leaf area receives less than 25% of full light, and at 1 and 5 meters the value is 0%.

Click to enlarge; source here

So, what does all this mean for the design of trees on other worlds? Firstly, like on Earth, you can have multiple layers of leaves and still have enough light trickling down for lower leaves to be useful. The shape of leaves is also important. Apparently some Earth trees use this effect: the outer or upper leaves of olive trees are thinner than the leaves lower down, which makes sense in view of the experiments above.

An interesting consequence is that the distance between leaf layers would depend on the apparent diameter of the sun's diameter as seen from a planetary surface. Doubling the diameter would half the length of the umbra, so leaves could be closer together and still receive an adequate amount of light. 

Should your alien trees have a few layers of leaves or multiple ones? Theoretical considerations on Earth suggest that fewer layers work better when the amount of light is low to start with: the absolute level decays very quickly with the number of layers. For alien worlds, 'low light' can probably be rephrased as a low capacity to make use of available light. That could be low light with good photosynthesis or good light with poor photosynthesis. The effect of changing the saturation point is more difficult to predict. On Earth, where photosynthesis saturates at only 20-25% of full light, shadows may still leave enough light. But if photosynthesis could use up to 75% of all light, the top layers might generate all the energy needed, so more layers would be superfluous. Then again, the plant might well have evolved to use all that energy, so perhaps lower layers would still be useful.

No doubt, additional demands, such as transport of metabolites and structural stiffness will complicate the picture. Nevertheless, on Furaha various plants carry their leaves in umbrella-like shapes, in which two or three layers of leaves form a nearly completely closed canopy, closing off the sky to potential competitors that might grow up beneath them.                               

* This example is taken from the excellent book 'The life of a leaf' by Steven Vogel