Optimal Control of an Under-Actuated System for Landing With Desired Postures

Control over landing posture can effectively prevent structural damage when a portable device is accidentally dropped. Given size and cost constraints, the number of actuators should be limited. This paper presents an optimal posture control method that allows an under-actuated system to land with the desired posture. A simplified model of a portable computer/telephone is considered, comprising two rigid bodies and an active joint. The objective is to minimize the input torque produced by the actuator to achieve the desired posture. A two-point boundary value problem is formulated; i.e., the initial and final angular positions and velocities are predetermined, and the inequality constraints are established on the basis of the capacity of the actuator and acceptable level of state variables. In the numerical analysis, the backward-sweep algorithm is applied to determine the appropriate Lagrange multiplier, and the falling dynamics are explored using MATLAB. The optimal controller design is presented, together with simulation results confirming that the system is capable of performing landing posture control with minimum input torque.