Here’s some techniques to train from Half-Guard bottom when your opponent has head and arm control on you.
These are demonstrated on the Submission Master grappling dummy.
As far as your legs go, they’ll either be crossed to the outside or inside of his leg.
You may use lockdown position as well, which controls and stretches your oppenent’s leg back, taking away his ability to have good balance for striking.
And, your opponent will either have double underhooks, head and arm control (this video), or just one shoulder control.
One of your best options is to get your own double underhooks on him and avoid his double underhooks. This protects your arms from attack and allows you to escape to his back.
If you can get to your side with your arms tucked, it’s difficult for him to get double underhooks, and allow you to go for techniques like kimura shoulder lock.
To get your own double underhooks, you can push his head up and to the opposite side, bring one arm underneath to brace against his chest and make space, then bring the other arm under his arms.
You can also put lockdown on his leg, drive him back with it, grip his wrist that’s around your neck, bump him forward and remove his arm from around your neck.
Or, with my legs crossed to the outside of his leg, I can underhook his thigh with my leg and flip him over, going to side control.
Next, I can again use a lockdown, grip his ankle with my hand, lift him, overhook his leg with my leg, and “elevator” him over to take mount.
In the next technique, my legs are crossed on the outside of his leg, I block his knee with my elbow or hand, and shrimp both directions to put him into guard.
If he follows you, you can push on his shoulders to make space and get the guard position.
Then, you can also put him into butterfly guard using a lockdown on the inside of his leg, put one hand to his hip and overwrap his other arm. Bring your foot to the outside of his leg and underhook his thihg. Then lift as you push away, giving you space to put him into butterfly guard position.
You can also start with the earlier technique where you put lockdown on his leg, grip his wrist that’s behind your head, pull him forward and remove his arm from behind your neck. Then block his knee with your hand so he can’t post it out. Lift him over and take top position inside his guard.
And finally, you can start with the earlier technique that pushes his head up to get double underhooks. From there you can take his back.