LEGO MINDSTORMS - Building Robots Part 9 docx

280 Chapter 15 • Building Robots That Walk Introduction So far in this book, we have discussed in depth many mobility configurations, all of them based upon one of the most important inventions of humankind: the wheel. In this chapter, we will try and emulate what nature invented long before the wheel to provide humankind with a mode of transportation—legs! Legged robots are rather unpractical for all but some special applications, but there’s much to learn in designing and building a walking robot, and the matter is both challenging and fascinating. This chapter owes a lot to all the great designers who published their creations on the net, and patiently explained their choices through text and pictures, including Joe Nagata, Jin Sato, Kazuhiro Umeda, Miguel Agullo, and many others. The Theory behind Walking How can one define walking? It’s the process of lifting a leg from the ground while one or more other legs support the body.When the leg has been lifted, it gets advanced and lowered back to the ground. From there the process continues with another leg, and so on. The crucial point is this:What prevents a creature from falling down when a leg is lifted? To discover this, we need to introduce some basic concepts from a branch of physics called statics, which explains the laws of balance. The weight of an object is the resulting effect of the force of gravity against the mass of the object.To describe a force, you need to determine three variables: its intensity, its direction, and its point of application. For example, if you want to move a piece of furniture in your room, the intensity is the amount of strength you must apply to make it move, the direction is the bearing of the course you’re pushing it on, and the point of application is where you place your hands to apply the force. Returning to gravity, its intensity is proportional to the mass of the object. Its direction points vertically downward, but where is its application point? To answer this question, you should consider an object as being the sum of a very large number of very small particles, each one having its own mass.The gravity exerts a force upon every particle, and thus all of them can be considered a point of application. However, physics teaches that a combination of forces can be interpreted as a single force—called the resultant—which has its own intensity, direction, and point of application.The resultant of the force of gravity has an intensity which corresponds to the weight of the objects, a direction pointing downward, and a point of application called the center of gravity (COG) of the object (Figure 15.1). www.syngress.com 174_LEGO_15 10/29/01 4:15 PM Page 280 www.syngress.com The force of gravity acts on any object and tries to move its COG as close as possible to the ground; this is why objects fall and shift until they reach a stable position. But what makes a position stable? Statics teaches that a body becomes stable when the vertical passing for its center of gravity falls inside its supporting base.The supporting base is the surface whose perimeter results from connecting the supporting points with straight lines, where the supporting point is any point on the object which is in contact with the ground or with any other stable object (like the floor of your room or your desk). For example, a book placed on a table has the whole surface of its cover in touch with the table, and that defines its supporting base.A table has four legs, each one having a small surface in touch with the floor: its supporting base is the area delimited by the legs, which includes points untouched by the table (Figure 15.2). Every child learns this rule by experience when building towers of stacked blocks: while the COG remains within the supporting base, the tower is stable; as soon as it falls outside the base, the tower itself falls down (Figure 15.3). Okay, now you know the rule, but where’s the COG of an object? For objects that are symmetrical in shape and density, the COG coincides with their geometrical center, but in more complex objects the COG is not very easy to find, and it is not guaranteed to be inside the object.A table is again a good example:The COG of a typical table lies somewhere below its top, as demon- strated by the fact that it has more than just one stable position (Figure 15.4). Building Robots That Walk • Chapter 15 281 Figure 15.1 The Center of Gravity of an Object 174_LEGO_15 10/29/01 4:15 PM Page 281 282 Chapter 15 • Building Robots That Walk www.syngress.com Figure 15.2 The Supporting Base of a Table Figure 15.3 Stable and Unstable Piles of Bricks 174_LEGO_15 10/29/01 4:15 PM Page 282 Building Robots That Walk • Chapter 15 283 Fortunately, you don’t need to find the actual position of the COG of your robots.You are actually interested in the position of the vertical line that passes through the COG, in order to see if it falls inside the supporting base.This is easier to find. If your robot is mainly symmetrical, this line will pass very close to its geo- metrical center.Thus, what you really need is to look at your robot from the top, to figure out if the COG falls over the supporting base delimited by the legs. For example, in Figure 15.5 you see a scheme that represents a robot with four large legs (top view). One of the legs is lifted, and you see that the COG falls inside the surface delimited by the other three legs.Thus, the robot is stable. www.syngress.com Figure 15.4 The COG of an Object May Lie Outside It 174_LEGO_15 10/29/01 4:15 PM Page 283 284 Chapter 15 • Building Robots That Walk The same robot can stay balanced even with just two legs, because the COG still falls inside the supporting base (Figure 15.6). When the robot advances the two lifted legs, part of its mass moves forward, and the COG moves forward a bit, too. But the large contact surfaces of the legs delimit a zone wide enough to make the COG fall within the boundaries (see Figure 15.7). Using more than four legs, you don’t need to rely on their size anymore.A six-legged robot, for example, can walk with very thin feet provided it always has at least three of them touching the ground (Figure 15.8). On the contrary, when reducing the number of legs, things become more complicated.The making of two-legged (biped) robots requires a very careful design.A little trick is to build U-shaped legs that partly interlace, providing a large support for the robot (Figure 15.9). LEGO suggested a similar approach in one of its Idea Books (8891, back in 1991). www.syngress.com Figure 15.5 A Four-Legged Robot with One Leg Lifted Figure 15.6 A Four-Legged Robot with Two Legs Lifted 174_LEGO_15 10/29/01 4:15 PM Page 284 Building Robots That Walk • Chapter 15 285 www.syngress.com Figure 15.7 A Four-Legged Robot with Two Legs Lifted and Advanced Figure 15.8 A Six-Legged Robot with Three Legs Lifted Figure 15.9 A Two-Legged Robot with Interlaced Legs 174_LEGO_15 10/29/01 4:15 PM Page 285 286 Chapter 15 • Building Robots That Walk Though this works, it’s a bit like cheating! If you want to emulate the way we human beings walk, you must understand what happens in the human body. Let’s do a simple experiment. Stand still, being sure to distribute your weight evenly over your feet. Keep your arms at your side and keep all your muscles relaxed. Now slowly try and lift one leg: your body tends to fall to that side.While walking under normal conditions, you unwittingly move your COG over one foot before lifting the other.This gives you balance and stability and prevents you from falling. This is the behavior you have to replicate to build a true biped robot.You have to shift its COG over one foot before lifting and advancing the other (Figures 15.10 and 15.11). Actually, the way human beings and animals walk follows not only the rules of statics but also those of dynamics, the branch of physics which deals with matter in motion.When a man runs, for example, he is in dynamic balance, producing forces that oppose gravity and temporarily violate the rules of statics.To understand how this happens, you can study how you walk, and also look carefully at how animals phase their walking (bipeds, four-legged animals, insects, and arachnids). For example, elephants and other very large animals only lift one leg when walking slowly, to keep the static COG within the triangle bounded by their remaining legs. Once the pace picks up, the opposing gait takes over, which is similar to the sequence we described in Figures 15.6 and 15.7. Most four-legged animals use this www.syngress.com Figure 15.10 A Biped Robot Standing Figure 15.11 A Biped Robot Shifts Its COG over One Foot before Lifting the Other 174_LEGO_15 10/29/01 4:15 PM Page 286 Building Robots That Walk • Chapter 15 287 scheme when trotting.At further increases of speed, like in galloping, dynamic sta- bility is more important than static: only one leg needs to contact the ground, and this allows the animal to cover more ground with every cycle. Building a robot that walks or runs using dynamic balance is a very compli- cated task, and for this reason in this chapter we will stay inside the comforting walls of statics. Building Legs Whatever kind of walking robot you’re going to build, you must find a way to convert the rotary motion provided by the electric motors into the proper sequence of movements necessary for a leg to work.Animals and human beings use a very complex geometry operated by an impressive number of independent muscles.You must stick to the constraints imposed by the MINDSTORMS system, thus finding simpler solutions. Figure 15.12 illustrates an initial idea: a leg mounted on two gear wheels of the same size, which are then connected in phase through a third gear. It’s very important that the leg attaches to two corresponding holes of the gears, otherwise it won’t work because the holes will change their spacing as the gears turn www.syngress.com Figure 15.12 This Leg Always Remains Vertical and Follows a Circumference 174_LEGO_15 10/29/01 4:15 PM Page 287 288 Chapter 15 • Building Robots That Walk By driving any of the three gears, this simple leg will go up and down, for- ward and back, always in a circle.The leg always remains vertical. Figure 15.13 shows a slightly different approach, where only one point of the leg is attached to the wheel, and the leg itself slides freely into a rotating support (fulcrum). In this assembly, the terminal point of the leg describes an ellipse—a flattened circle—whose height is equal to the distance between the uppermost and lower- most positions of the point where the leg is attached to the wheel, and whose length is a function of the distance between the fulcrum and the wheel.The closer the fulcrum to the wheel, the longer the ellipse and, consequently, the stride of the leg.You can adjust this distance to make your robot take longer or shorter steps, affecting its speed.We invite you to experiment with this setup, changing the distance between the wheel and the fulcrum, to understand the effect on the stride. Later in the chapter, we’ll use this feature to provide a legged robot with turning ability. www.syngress.com Figure 15.13 This Leg Describes an Ellipse 174_LEGO_15 10/29/01 4:15 PM Page 288 Building Robots That Walk • Chapter 15 289 More complex leg geometries are also possible (see Figure 15.14). Designing legs is almost an art—it requires good intuition, and a lot of patience to test and improve your initial idea. Building a Four-Legged Robot Let’s start by building a robot with four legs in order to demonstrate the center of gravity principle explained in Figures 15.5 to 15.7.The architecture is very simple, and symmetrical: keep the COG as close as possible to the center (Figure 15.15). We built it solely from RIS parts. Removing the RCX, you’ll notice there’s a single motor which, through two worm gears, provides motion to the front and rear leg assemblies (Figures 15.16 and 15.17).The other thing to notice is the phase of the legs: they are diagonally paired.The front left goes together with the rear right, while the front right accompanies the rear left, which implements the walking scheme shown in Figures 15.6 and 15.7. www.syngress.com Figure 15.14 A Leg with a More Complex Geometry 174_LEGO_15 10/29/01 4:15 PM Page 289 [...]... and discuss some more challenging projects www.syngress.com 291 174 _LEGO_ 15 292 10/ 29/ 01 4:15 PM Page 292 Chapter 15 • Building Robots That Walk Figure 15.18 The Front Left Leg Figure 15. 19 Front View, the Robot Stands on Two Legs www.syngress.com 174 _LEGO_ 15 10/ 29/ 01 4:15 PM Page 293 Building Robots That Walk • Chapter 15 Building a Six-Legged Steering Robot By increasing the number of legs, you can...174 _LEGO_ 15 290 10/ 29/ 01 4:15 PM Page 290 Chapter 15 • Building Robots That Walk Figure 15.15 Our Four-Legged Robot Figure 15.16 Top View (RCX Removed) www.syngress.com 174 _LEGO_ 15 10/ 29/ 01 4:15 PM Page 291 Building Robots That Walk • Chapter 15 Figure 15.17 Bottom View The legs follow the scheme of Figure 15.12,... legs make contact As a result, its walking is rather irregular and jolting www.syngress.com 293 174 _LEGO_ 15 294 10/ 29/ 01 4:15 PM Page 294 Chapter 15 • Building Robots That Walk Figure 15.21 Top View (RCX Removed) Figure 15.22 The Left Leg Group www.syngress.com 174 _LEGO_ 15 10/ 29/ 01 4:15 PM Page 295 Building Robots That Walk • Chapter 15 What could you do to smooth the walking motion? Using two sensors... lowered, and longer at the other, thus making the robot turn www.syngress.com 295 174 _LEGO_ 15 296 10/ 29/ 01 4:15 PM Page 296 Chapter 15 • Building Robots That Walk Figure 15.24 Top View (RCX Removed) Figure 15.25 Three Legs Are Always in Contact with the Ground (Side View) www.syngress.com 174 _LEGO_ 15 10/ 29/ 01 4:15 PM Page 297 Building Robots That Walk • Chapter 15 Figure 15.26 Rear View The front and rear... a sort of sled pushed by the robot (see Figure 15. 29) www.syngress.com 174 _LEGO_ 15 10/ 29/ 01 4:16 PM Page 299 Building Robots That Walk • Chapter 15 Figure 15. 29 The Light Sensor Was Always at the Same Height from the Ground Designing Bipeds Biped robots are among the most challenging projects you can ever face In a biped, the position of any single part, any single gram of mass, is critical to a stable... Interlacing Legs Let’s start with a biped based on the technique shown in Figure 15 .9 using interlacing legs.The feet must be U-shaped and large enough to support the weight of the whole robot (Figure 15.30, MINDSTORMS parts only) www.syngress.com 299 174 _LEGO_ 15 300 10/ 29/ 01 4:16 PM Page 300 Chapter 15 • Building Robots That Walk Figure 15.30 A Biped with Interlacing Legs This robot uses a simple... drives or front-wheel drive cars (Figure 15.42) A second axle at the rear provides motion to the legs, through two crankshafts and two liftarms www.syngress.com 174 _LEGO_ 15 10/ 29/ 01 4:16 PM Page 307 Building Robots That Walk • Chapter 15 Figure 15.40 An Ankle-Bending Walker Figure 15.41 Detail of One Foot www.syngress.com 307 174 _LEGO_ 15 308 10/ 29/ 01 4:16 PM Page 308 Chapter 15 • Building Robots That... 15.34 Front View Don’t forget to add some decorative parts to your walker! We used some of the parts left over in the MINDSTORMS box to provide the robot with a dinosaur-like appearance In Figures 15.30 and 15.31, you can clearly distinguish its fierce-looking head and the short front legs COG Shifting At the beginning of 199 9, we built our first COG-shifting biped, S6, challenged by the widespread belief... black pegs close it when the sled is either at its left or right limit www.syngress.com 303 174 _LEGO_ 15 304 10/ 29/ 01 4:16 PM Page 304 Chapter 15 • Building Robots That Walk Figure 15.36 The Robot Standing on One Leg Figure 15.37 Left Side View www.syngress.com 174 _LEGO_ 15 10/ 29/ 01 4:16 PM Page 305 Building Robots That Walk • Chapter 15 In the walkers seen so far there have been no need for sensors, but... sides of the swinging chassis are operated through a long joined axle and two 1 x 2 bricks with an axle hole (Figure 15.27) Figure 15.27 Bottom View www.syngress.com 297 174 _LEGO_ 15 298 10/ 29/ 01 4:16 PM Page 298 Chapter 15 • Building Robots That Walk This robot needs no sensors to control its motion.When you want to make it turn, switch the motor on for a few seconds to change the geometry, then brake . Figure 15. 29) . www.syngress.com Figure 15.28 Dodi, Our 12-Legged Line Follower 174 _LEGO_ 15 10/ 29/ 01 4:16 PM Page 298 Building Robots That Walk • Chapter 15 299 Designing Bipeds Biped robots are. Geometry 174 _LEGO_ 15 10/ 29/ 01 4:15 PM Page 2 89 290 Chapter 15 • Building Robots That Walk www.syngress.com Figure 15.15 Our Four-Legged Robot Figure 15.16 Top View (RCX Removed) 174 _LEGO_ 15 10/ 29/ 01. (8 891 , back in 199 1). www.syngress.com Figure 15.5 A Four-Legged Robot with One Leg Lifted Figure 15.6 A Four-Legged Robot with Two Legs Lifted 174 _LEGO_ 15 10/ 29/ 01 4:15 PM Page 284 Building Robots