This is a tough problem and I empathize with your dilemma. Unfortunately, I am not aware of any easy way to pick up all the goat-heads. One way to remove them from the soil surface would be to bury them by working the soil (as with a rototiller). This would normally be done when installing a new landscape but for renters I suspect that this is not a viable option. You would need to water the tilled soil to settle and compact the surface and the water would stimulate many weed seeds to germinate. You would then need to spray or pull these weeds to stop new seed production. Every time it rained or the surface was watered new weeds would emerge and need to be controlled. (This is probably happening every time it rains now.) A tough bermudagrass turf could be planted but it would be a lot of work to get it established and to protect it from the dogs. It would also require a significant amount of water that you may not want to purchase through your utilities. The only other options would be to try and figure out some mechanical way to pick up the goat-heads mimicking the way your shoes pick them up. Maybe you could find a piece of carpet or other material that you could attach to some kind of roller that would pick up the goat-heads. Then roll the device across the surface and remove the goat-heads into a container using a comb. This will involve some experimentation and a lot of work. I am uncertain how well it could work but as you have figured out you have a difficult problem. I wish you the best of luck. By way of encouragement, I will say that many years ago I saw an elementary school playground in mid-town that had a grassy area that was not used by the children because it contained so many goat-heads. An eagle scout project was done where a large group of scouts and other volunteers formed a line and went across the turf on hands and knees and picked up all the goat-heads. This was followed by an effective weed management program by the school groundskeepers to control goat-head (also know as puncturevine or Tribulus terrestris L.) that emerged later. For more help with this issue you might consider contacting the Pima County Cooperative Extension Office at 4210 N. Campbell Avenue, Tucson, AZ 85719-1109. Phone:<personal data hidden>. You could speak to a county agent or to master gardeners.