See Avoiding Takedowns is Simple by Shamrock which is on YouTube.com: part 1, part 2, and part 3.
Some defenses he mentions include the sprawl, getting underhooks, jamming/jabbing the head or shoulder using your palm or fist, or using jamming their shoulders using your elbows and basically ending up with them in your clinch.
Relax your knees to lower your level.
At the same time, clasp your hands together and chicken-wing your arms up to form a frame at head height. Keep your arms up in this frame until explicitly told to move them in the steps below.
With your left foot, step back and to the right so that it is behind your opponent's right foot.
Bring your left arm under your opponent's right arm and around their back .
Duck your head under their right arm and press it against their upper back behind their shoulder.
Bring your right arm around the front of their body.
You should now have your opponent in a kind of standing side/back control.
Place your right hand on their left wrist (grip so that your hand is over their wrist, with your thumb in).
Slightly twist to the left and lower your level. This will create some space between their left arm and your left side.
With your left hand, reach under their left arm and grab onto your right wrist (figure-four style grip, palm down on top of your wrist, thumb underneath).
Now spin to your left and at the same time raise your hands (keeping the figure-four grip locked). This will twist your opponent's arm.
If your opponent ends up facing you to lessen the strain on his/her wrist, you can block their left leg with your own left leg (close the gap) and then fall back, kicking their left leg towards your left side so that they roll over you onto their back, and then you can come up into a side control position.
This assumes your opponent is standing in front of you, facing you, and you are both in a normal fighting stance with your left foot forward.
Drop your level (bending at the knees).
Drop forward onto your front (left) knee. Your knee should be pointing directly in between your opponents legs and quite close to them.
With your right foot, step forward. The foot should end up almost level and to the right of your opponent (i.e. on their left).
Place your head against their left hip (to your right).
Grapevine your hands around their legs at the knees.
“Turn the corner” by swinging your left foot around 90 degrees to the right and at the same time pointing your body to the left, into your opponent.
Drive forwards and upwards, at the same time swinging your hands to the right and maintaining good posture.
A common mistake is to forgot the first step (dropping your level). This won't be anywhere near as effective and puts a lot of strain on your knees!! Turning the corner is also important.
The double-leg takedown above can only be done if there is enough space between you and your opponent. If they are too close, you can do a front push-kick to their left hip or upper leg. This will create some space between you and off-balance them at the same time, setting you up for the double-leg.
