E11.1 The ability to balance actively is a key ingredient in the mobility of a device that hops and runs on one springy leg, as shown in Figure E11.1 [8]. The control of the attitude of the device uses a gyroscope and a feedback such that u t( ) = Kx( )t , where

Determine a value for k so that the response of each hop is critically damped. 

-k 0 K 0 -3k and x(t) = Ax(t) + Bu(t), where A= and B = I. -20