I spent 13 years designing electronics of this exact nature: three phase induction motor reduced voltage soft starters and variable frequency AC drives. I spent the last few of those years as a VFD applications engineer helping customers select and configure this equipment for various loads and industries around the country as well.
You will not be able to build something that is cheap and safe. The voltages and currents involved are well beyond the safety margin of a hobbyiest, especially someone who is openly avoiding buying commercial units in order to save money. Don't do it!
While the theory behind AC motor control is very straightforward, the detail level work (heat sink sizing, snubbering, gate drive requirements, de-sat protection, motor overload calculations, bus capacitor protection, etc.) can be quite tricky to get down, especially with heavy duty cycling and regenerative power modes which a carnival ride will CERTAINLY be generating. I strongly caution you against trying to build something of this nature unless you have significant experience not only in microcontrollers and embedded systems design but also significant experience in power electronics and three phase circuitry. People get hurt and killed building this stuff.
My first question for you is whether speed control is really required, or if you only require a soft start up and slow down. Do you vary the speed of the motor once it is started? If not, you may be able to get away with a MUCH cheaper reduced voltage soft starter. These units act like three phase light dimmers; they only adjust the applied voltage to the motor. You will not have a lot of torque at low speeds, but with the right design of motor (NEMA class D) you can achieve exactly what you're after with a fraction of the cost and maintenance.
If you really do need to vary the full-load speed of the motor then you are more or less stuck using a variable frequency drive. As you are aware these are expensive and if you buy cheap you are likely to replace them sooner due to your high surge current (they call this "constant torque") application. What I would definitely recommend doing if this is the case would be to contact various manufacturers (Allen-Bradley, Cutler-Hammer, SAF drives, Benshaw, Yaskawa, etc.) and ask for reconditioned units. Ask for a drive capable of delivering 150% rated current for 30s (this is usually known as heavy duty) or size the drive 30-50% larger than your nominal current rating. You will also likely be running off of generator power which is notorious for being undersized and prone to brownouts and surges as the load requirements change with the state of the equipment being run. Drives don't like that (voltage sags cause current spikes as the motor starts slipping and surges can cause you to overvoltage the bus capacitors) and have a tendency to either fault out or blow up.
I am all about the little guy building something and saving a buck, but this is not the type of project to do this on. If you really want to build a three phase AC drive, start with a little 10HP 480V motor with a hand brake on a test bench. You have all the potential for experiencing the pants-filling sensation of an H-bridge failure or a bus capacitor explosion two feet from your head but without the potential lawsuits and loss of life (except perhaps your own).
The positioning control (real industrial stuf) is done in cascade with multiple controllers: Position controller (P-regulator), Speed controller (PI-regulator), current controller (PI-regulator).
https://www.google.si/search?q=position+control+loop&biw=1391&bih=683&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwj0gMXLo-rJAhWDfxoKHaUmCTMQsAQIIA#imgrc=N1aiYN6Li7XBCM%3A
When you ommit certain stages, the control gets worse. For example if you don't have current transducer, you can control the motor directly from speed controller, that would be voltage control. If the load wants to move the motor at standstill, the PI regulator "will find" such voltage that, will feed such current into the motor, that will produce such torque at shaft, that will put the motor back to standstill.
If you have a current transducer, then you can make two PI-regultors. First is speed control which gives the setpoint to second one PI-reg. the current controller. Now the output of current controller has to control PWM and measures a current feedback with the current transducer. With extra current feedback the overall dynamics will be faster compared to previous method without transducer.
So the rough conclusion is: the torque control is actualy the current control of the motor.
Best Answer
Note that your question states that you want to reduce the starting torque, but you actually want to reduce the starting current.
There are a few ways to reduce the starting current of a three-phase induction motor. In rough order of cost:
Reduced Voltage starters
These methods apply a reduced voltage (say 50%) to the motor on first starting. This reduces the current draw to 50%. After the motor attains enough speed, the voltage is increased from 50% to 100%.
Note that the torque is reduced as the square of the voltage, so with 50% voltage the motor will only develop 25% torque. You need to ensure that this torque is enough to turn the motor's load from a standing start.
Soft starter - an electronic device that starts the motor at a low voltage, with corresponding low current, and continuously increases the voltage until full speed is reached. The difference between this and rotor resistance, star-delta, and autotransformer starting is that the voltage is increased steadily, as opposed to in one sudden step. This is more gentle to the motor and load.
Variable speed drive (also known as VSD, VFD, VVVF) - an electronic device that starts the motor using full rated voltage, but at a lower frequency. The most expensive of all methods, but also the best, as it can apply very high torque from a standing start.
As the name suggests, VSD's also allow variable speed control, which no other motor starting method does.
It doesn't sound like you have a wound-rotor induction motor, so I will neglect to mention rotor resistance starting.
More on this topic can be found by searching for "three-phase induction motor starting". You should be able to find application notes and guides written by equipment manufacturers, that will go into more detail about how the above methods work and what equipment you can use to implement them.