The purpose of this study lies in the design of landing posture control for the twin-body systems which feature two bodies connected by an active joint. Three types of controllers have been designed for achieving desired landing postures and acquiring low torque. Posture control has many applications such as the control of the attitude of artificial satellites, active impact defense of falling portable products, actions of prostheses or medical rehabilitation mechanisms, and locomotion of bionic mechanisms. They are composed of: (1) standard control applied to a twin-body system with a 3-DOF active joint; (2) track planning performed on a twin-body system with a 2-DOF active joint; and (3) optimal control employed in a twin-body system with a 2-DOF active joint.
First, a twin-body system with a 3-DOF active joint is discussed, and the transformation between outputs and state variables is established for the application of a standard controller. The joint adopted has three rotational degrees-of-freedom (DOFs). Either three actuators or a spherical motor is installed inside the joint to provide orthogonal input torques. A standard modeling procedure for robotic dynamics is employed to capture the twin-body system dynamics in terms of 312 Euler angles, on the basis of the Newton-Euler formulation. Here, the posture components, i.e., the included angle, tilt angle, and the body that lands first, are employed to represent the landing posture that is specially designed to prevent the fragile parts of the system from being damaged. To apply the standard controller, the three posture components should be transformed into three control outputs, which are linear functions of state variables. The input-output linearization and computed torque methods are utilized to determine the input torques. Finally, the feasibility of the proposed landing posture control is confirmed using simulations. This design successfully implemented a standard control method to realize landing posture control by establishing transformations from real landing posture components to simulated control outputs, and the controllable regions were completely defined.
The dynamics for the posture control of a system with a 3-DOF active joint is more complicated, and it is difficult to realize a mechanism that can provide three orthogonal torques. Hence, the active joint installed in the system is simplified to a 2-DOF one. This active joint can provide two orthogonal torques, one for changing the inclination angle between two bodies and the other for rotating about the self-axis. The dynamics of the twin-body system is described using Euler angles with the coordinate transformation of a homogeneous transformation matrix (HTM). The dynamic equations of the twin-body system are established based on the Lagrange-Euler formulation. The dynamic characteristics are investigated, and stability for the pre-defined control objectives is ensured. Proportional derivative (PD) and sliding controllers are applied to achieve the desired final-landing postures while trajectory planning is applied to reduce the input torque/energy. The input torques/energy and the tracking errors with three different trajectory functions are discussed. Trajectory optimization is adopted, and the relationship between input energy, falling height, and tracking error is shown.
Finally, an optimal control method is employed in the system to achieve the landing posture control. This is applied to achieve landing at the desired posture or the impact point/region, and the objectives of optimal control is to minimize the input torque. We adopt only two input torques in the control system with three outputs, which makes it an under-actuated control problem. In addition, optimal control is adopted to minimize the input energy. The necessary equations for optimal control programming are then established in accordance with the control objectives and constraints. In the numerical analysis, the backward-sweep algorithm is adopted to obtain the optimal control gain, and the system dynamics are expressed using simulations. The simulations reveal that optimal control is applicable to the landing with the desired posture of an under-actuated system.

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