Since the filesystems for P1-P4 are stored on the Controller it's even easier to setup SSH keys to log into the Pi Zeros without using a password from the Controller.

ssh-keygen -t rsa -N "" -f ~/.ssh/id_rsa
mkdir -p /var/lib/clusterctrl/nfs/p{1,2,3,4}/home/pi/.ssh/
cat ~/.ssh/id_rsa.pub >> /var/lib/clusterctrl/nfs/p1/home/pi/.ssh/authorized_keys
cat ~/.ssh/id_rsa.pub >> /var/lib/clusterctrl/nfs/p2/home/pi/.ssh/authorized_keys
cat ~/.ssh/id_rsa.pub >> /var/lib/clusterctrl/nfs/p3/home/pi/.ssh/authorized_keys
cat ~/.ssh/id_rsa.pub >> /var/lib/clusterctrl/nfs/p4/home/pi/.ssh/authorized_keys

You should now be able to log into the Pi Zeros.

ssh pi@p1.local

To run a command on all Pi Zeros sequentially you can reuse a portion of the above command.

for I in 1 2 3 4; do echo -n "p$I:";ssh pi@p$I.local uptime;done

You should see something similar to the output below which shows how long each Pi Zero has been booted, number of users, etc.

p1: 21:43:32 up 24 min,  3 users,  load average: 0.01, 0.02, 0.00
p2: 21:43:33 up 6 min,  2 users,  load average: 0.07, 0.08, 0.05
p3: 21:43:33 up 5 min,  2 users,  load average: 0.00, 0.07, 0.04
p4: 21:43:34 up 4 min,  2 users,  load average: 0.02, 0.11, 0.06