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Human-powered vehicles come in many flavours, not just ordinary bikes. The following article provides a brief overview of the multitrack kind.
As a recumbent, this is by far the most numerous design: rear wheel driven, front wheels steered (with some exceptions, of course: this one is all wheel drive, this is FWD/RWS). Brakes are either on front wheels only, each of them operated by its own lever (e.g. Azub and others), or one lever controls coupled front brakes and the other controls the rear (e.g. Štercl and others). Overall proportions are usually chosen so that each wheel carries about a third of the weight and the centre of gravity is as low as possible. Tadpoles are good for fast road riding because they are sufficiently stable during braking in corners, and because they fit nicely into a teardrop-shaped aerodynamic shell. Their weak point is rear wheel's grip: it can slip in rough terrain or cause a flip at high speed (tadpole oversteers, so if the rear wheel loses grip, rear end goes sideways and you go rubber side up). Turning radius is limited by steering lock of front wheels. Good news for homebuilders is simple one-wheel drivetrain, bad news is rather complex steering geometry (inner wheel must turn sharper than outer one). Cargo can be hung around the rear wheel like on a classic bicycle.
Upright tadpoles are used almost exclusively as cargo haulers: a big box sits between the front wheels while the rear end looks like a standard bike. They can't ride too fast because their centre of gravity is too high for high speed cornering. Some designs steer with the front wheels (Nihola), some turn the whole front axle together with the loadbed (Christiania), some have the front wheels driven and steer with the rear (Jernhest).
Most common front wheel geometry looks like this:
Rearward tilt of the kingpins creates trail (t), which provides dynamic centering moment. Inward tilt provides static centering moment and minimizes or eliminates the value of r and therefore bump steer. In a corner, it's good to have axes of all three wheels intersecting in one common point, otherwise tyres scrub and bounce unpredictably. It is called Ackermann geometry:
Someone found it works good enough if lines drawn through the steering arms intersect somewhere at the rear axle level:
Getting closer to perfection requires more complex designs with various pushrods and rockers and it is not trivial to get the dimensions right.
Most common design in the recumbent world is long wheelbase (front wheel ahead of pedals), driven rear wheels or at least one and steered front wheel (examples: Kettwiesel, Anura and others). Brakes are often installed at the rear only. Rider sits slightly ahead of the rear axle and rear wheels carry most of the weight, so the trike is sufficiently stable when braking in corners, and understeers if skidded. Good news is good offroad grip and excellent steering lock (the little front wheel can be turned close to right angle if needed), worse news is not so good aerodynamics (the triangular shape is not easy to envelope in a teardrop, although it is not impossible). An advantage for homebuilders is simple steering, disadvantage is the more complex drivetrain (if you want to drive two wheels, you need either a differential or a double freewheel). Space behind the seat is easy to adapt for bulky cargo. Deltas are often designed to be able to stand on their rear end to take less parking space.
Short wheelbase recumbent variant (pedals ahead of front wheel) makes sense if the front wheels is driven. Depending on how far back you put the rear axle, you partially sacrifice either cornering stability or front wheel grip. Main advantage is maximum simplicity: no differential, no Ackermann, no long chain. And an unparalleled cargo space behind the seat. An example of commercial design is SUV by Organic Engines (the whole front part with rider's seat tilts when steering), homebuild examples are Hipparion and Python. FWD deltas are well suited for hand propulsion - these are called handbikes.
Upright deltas with rear wheel drive are used mostly as utility vehicles - "pickup trucks" or rickshaws (transport in Asia, Cycles Maximus etc.). Cornering stability depends on how much load you put on the rear axle, high speeds are usually not expected. Another application is racing: if you replace the single rear wheel of a standard road bike with two, it still conforms to UCI rules, so people with balance issues can join the fun (cornering instability is solved by leaning upper body into turns). Sometimes you can see a small utility trike with shopping basket between the rear wheels (instability is solved by not riding fast).
If you don't insist on symmetry, you have many other options. A bike can be converted to a trike by attaching a sidecar to it. It may be difficult to turn left without flipping over if the sidecar is empty, but you can load it with very long and bulky items that stick out both fore and aft.
Or you can make one left wheel driven, two right wheels steered and sit somewhere in the middle (Coventry). Simple drivetrain and steering geometry, good stability when turning left, slightly worse when turning right, slightly better grip than tadpoles (half the weight on driven wheel), excellent steering lock. The only drawback is very little cargo space.
And of course, nobody says the three wheels cannot be inline. But this means a singletrack vehicle with all the usual pros and cons.
Cornering tipover stability is an inherent problem of trikes. It can be improved by replacing the single wheel at the front or back with two. This combines all advantages of tadpoles and deltas: good stability and grip under all conditions and lots of luggage space. However, disadvantages combine as well: complex drivetrain and steering. Commercial product examples: Brox, Quattrovelo. Homebuilders can easily apply a conversion kit to their trike.
Dicycle is more of a curiosity than a practical vehicle adaptable for human power. It steers by changing the speed ratio of its two wheels. Longitudinal stability is solved either by having huge wheels with centre of gravity under their axles (you can still roll over if you brake too hard), or by active electronic balancing (Segway).
Multitrack vehicles beat singletrack vehicles in not falling over in low speeds or when stationary, but they lose in tipping over if cornering at high speed. Could we combine the best of the two worlds? Yes, and many trails have already been blazed. Deltas with front wheel drive are used most frequently because the leaning mechanism doesn't have to mix with drive or steering. Often a standard lowracer is used as a base and its rear wheel is replaced by some contraption:
Tilting axle (1) is simple and can accomodate a lot of luggage, if you don't mind the lateral stress on the wheels. Parallelogram (2) is the best solution geometrically (track width relative to road doesn't change when leaning, so there is no tyre scrub), but is fairly complex (lots of joints) and bulky. Swingarms connected with a rocker arm (3) add a little bit of tyre scrub, but they easily combine with suspension and provide lots of free space between the wheels. A special case of swingarms is the iLean system (4): one swingarm points forward, the other rearward and they rotate on a common axle. It is the simplest design of all, but doesn't mix well with brakes (they roll the axle forward, causing lean or possibly rollover).
If a trike leans freely, it behaves exactly like a bicycle (including falling over when stopped) and doesn't give you any advantages except more comfort (only a half of each bump is transmitted to the frame). There are several solutions. The most obvious one is some lock that would keep the lean centered (usually implemented by peg and hole or a modified disk brake). But the problem is it effectively reverses your steering when toggled: singletrack vehicles initiate a turn by a short countersteer in the opposite direction while multitracks turn directly where you want to go. In addition, a lock is just a mechanical part that can fail. The most reliable solution is to design the leaning geometry so that the centre of gravity of the whole vehicle rises when leaning: inner wheel must sink more than the outer wheel rises (Jetrike). This provides a static centering moment which keeps you upright when stopped or riding straight ahead, and is overcome by stronger steering moment in turns. More or less the same effect can be achieved by a spring or a rubber block which returns the lean back to central position. The only difference is a spring can fail, geometry can't. Another approach is to couple the lean with steering: if you hold the handlebars firmly, you sit upright; if you steer, you lean as well (Organic Engines). It is simple to build, but the fixed lean to steer ratio is only correct for one particular speed and the rest must be compensated by leaning out of the seat. Another way is to control the lean independently of steering: either by two separate sticks (e.g. Tripendo or this), or by bracing on the handlebars (Vacuum). It is probably the most complex solution and hardest to get used to, but there are no other disadvantages to it.
A comprehensive overview of perhaps all trikes ever made can be found at Murray's trike links page.