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There is another option overlooked, of detecting the cat inside the litter box. If the box is tall enough, that is the place you could potentially put an ultrasonic sensor, by having it inside looking down, but due to ringing issues these have a minimum detection distance which you may or may not be able to achieve. Such a method would definitely point to software interpretation - you need software range-gating to ignore the floor of mounded litter, and see the cat as a closer reflection; then you probably also need to entirely ignore the situation for a bit after triggering the cleaning mechanism, so you don't re-trigger off that.

There is another option overlooked, of detecting the cat inside the litter box. If the box is tall enough, that is the place you could potentially put an ultrasonic sensor, by having it inside looking down, but due to ringing issues these have a minimum detection distance which you may or may not be able to achieve. Such a method would definitely point to software interpretation - you need software range-gating to ignore the floor of mounded litter, and see the cat as a closer reflection; then you probably also need to entirely ignore the situation for a bit after triggering the cleaning mechanism, so you don't re-trigger off that. As mentioned in comments, the response of the cat to an ultrasonic system should be evaluated before committing to such a path. At the cost of higher power, identifying a cat inside the litter box (or at least a "blob") should also be well within the capability of computer vision algorithms watching a webcam given suitable IR illumination.

If you had an ultrasonic system with direct software signal interpretation of the received signal, then you could fire it out the door and look for a reflection from an approaching cat in addition to the spurious one from the door aperture. But the cheap "Arduino project" modules just give you the output of the first strong reflection, and you may have a lot of trouble making the door aperture not be that. Doppler ultrasound could work, too, but again points to having a far more custom circuit where you can actually see the analog response, and not just get an pre-(mis)interpreted output as from a little HC-SR04 module.

If you had an ultrasonic system with direct software signal interpretation of the received signal, then you could fire it out the door and look for a reflection from an approaching cat in addition to the spurious one from the door aperture. But the cheap "Arduino project" modules just give you the output of the first strong reflection, and you may have a lot of trouble making the door aperture not be that. Doppler ultrasound could work, too, but again points to having a far more custom circuit where you can actually see the analog response, and not just get an pre-(mis)interpreted output as from a little HC-SR04 module. One advantage of detecting the cat inside the box is that you have longer to do it, so you could have a battery powered system which polled at a much lower rate to preserve battery life, but designing systems which are truly low power without unintended power drain when they are supposed to be asleep is practically far more tricky than it first looks.

In terms of actual components, you'll want to use an IR LED toggled on and off at 38 KHz, and a matching receiver which looks only for that, and ignores ambient light. Typical receiver modules also include an AGC as they are looking for bursts of pulses (eg from a TV remote) so you'll either need to get a receiver without AGC, or else have your transmitter output a series of 38 KHz pulses, give a gap, then another series so that it looks not unlike a TV remote control signal. Such a pattern is easily implemented in a small MCU, which can also help with the detection rules. So basically you have something along the lines of an "Arduino project" (/ATtiny, Cortex M0, MSP430, whatever) to pulse the LED, monitor the receiver, and then probably activate a small relay to trigger the mechanism (unless you can reverse engineer its circuitry enough to drive directly).

In terms of actual components, you'll want to use an IR LED toggled on and off at 38 KHz, and a matching receiver which looks only for that, and ignores ambient light. Typical receiver modules also include an AGC as they are looking for bursts of pulses (eg from a TV remote) so you'll either need to get a receiver without AGC, or else have your transmitter output a series of 38 KHz pulses, give a gap, then another series so that it looks not unlike a TV remote control signal - try something like 5-10 pulses, then a gap of the same amount of time. Such a pattern is easily implemented in a small MCU, which can also help with the detection rules. So basically you have something along the lines of an "Arduino project" (/ATtiny, Cortex M0, MSP430, whatever) to pulse the LED, monitor the receiver, and then probably activate a small relay to trigger the mechanism (unless you can reverse engineer its circuitry enough to drive directly).