Vibration is a serious problem in mechanical systems that are required to perform precise motion in the presence of structural flexibility. Examples of such systems range from the positioning of a disk drive¡¯s head to large space structures, flexible manipulators and container cranes. In most cases, the residual vibration at the end of a move is the most detrimental and the extent of the residual vibration limits the performance of the system. The effective use of such systems can only be achieved when such vibration can be properly handled. As a result, there is active research interest in finding methods that will eliminate vibration for a variety of mechanical and structural systems.

Traditional closed-loop feedback can be used to reduce end-point vibration. The closed-loop system will then benefit from the inherent advantages of feedback, such as insensitivity to parameter variations, noise attenuation and disturbance rejection. However, such a feedback system can be difficult to implement in practice, as it requires reliable sensor information for feedback. Such sensor information may not be so easily available. For example, in the container crane system problem, it is not a trivial task (nor practical due to reliability of sensors and its environment) to devise a sensor to measure the position at the end-point. Another approach is input shaping technique, in which the input is pre-shaped such that the resulting residual vibration is reduced or eliminated. These methods are popular in industry because they are relatively simple to implement the pre-shaped input together with closed-loop feedback strategies to enjoy the benefits of both systems.
Input shaping is a feed-forward control technique for improving the settling time and the positioning accuracy, while minimizing residual vibrations, of computer-controlled machines. Input shaping is a strategy for the generation of time-optimal shaped commands using only a simple model, which consists of the estimates of natural frequencies and damping ratios . Input shaping is implemented by convolving a desired system command signal with a sequence of impulses, so-called input shaper, to produce a shaped input, as shown in below figure.

The amplitudes and time locations of the impulses are determined by numerically solving a set of constraint equations that are set to the dynamic response of the system, or in some cases, by using a closed-form solution. The input shaper is chosen in such a way that in the absence of control input, it itself would not cause residual vibration. The result of the convolution is then used to drive the system.

1. Input shaper to minimize the expected residual vibration
2. Application to a tracking control problem
3. Combine feedback control structure (or other technique) and input shaping
4. Hybrid input shaper
5. SI (specified insensitivity) shaper
6. TVIS (time-varying impulse shaping)
7. Application to a nonlinear and a time-varying system
8. Implementation to a real system using input shaping
9. Application to a MIMO system

1. Residual Vibration Constraint
2. Robustness Constraint
3. Requirement of Time-Optimality
4. Amplitude Constraints

• ZV shaper ZV shaper
  ZV shaper Basic constraints (1)=0, (3), and (4) are used to solve ZV shaper.
• ZV shaper ZVD shaper
  ZV shaper Basic constraints (1)=0, (2), (3), and (4) are used to solve ZV shaper.
• ZV shaper EI shaper
  ZV shaper If the sensitivity curve approach is introduced, EI shaper can be easily solved. The basic constraints (1)=tolerable vibration, (2), (3), and (4) are used.

1. Impulse Response
2. Sensitivity Curves
3. Pole-Zero Analysis
4. Vector Diagrams

The efficiency of cargo handling work at a port depends largely on the operation of container cranes. When a ship is unloaded, containers are first transferred from the ship to a waiting truck by a container crane. The truck then carries the container to an open storage area, where another crane stacks the container to a pre-assigned place. The bottleneck of this cycle lies in the transfer of the containers from the ship to the truck. Therefore, minimizing this transfer time will bring about a large cost saving. When a ship is loaded, the same problem is encountered.
Since a large swing of the container load during the transfer is dangerous, the problem is to transfer a container to the desired place as quickly as possible while minimizing the swing of the container during transfer as well as the swing at the end of transfer.
If the oscillation of the container load is ignored, time-optimal rigid-body (TORB) commands can be easily calculated. Unfortunately, TORB commands will usually result in large amplitude oscillations. Experienced crane operators attempt to eliminate vibration by causing a deceleration oscillation which cancels the oscillation induced during acceleration, or they may brush the payload against obstacles to damp out the vibration.
When the swing is considered, the time-optimal flexible-body (TOFB) commands that result in zero residual vibration can be generated. Hoisting of the load during the motion increases the difficulty of generating the control because the system is nonlinear. If the system model is linearized, then the associated frequency is time-varying. Optimal controls based on a nonlinear model can be difficult to generate. One method for developing optimal controls divides the motion into fundamental parts. The control for each part is then derived and pieced together. Even when optimal commands can be generated, implementation is usually impractical because the boundary conditions at the end of the maneuver (move distance) must be known at the start of the move. When feedback is available, both robust controllers and combination of open- and closed-loop controls are possible.

Fig. 2.1 shows a schematic diagram of container crane systems with hoisting of the load.

1. Nonlinear System
2. Linear Time-Varying System
3. Linear Time-Invariant System

• Fig. 2 Path planning: Four stages of operation cycle
• Fig. 2 Transportation sequence of the container

Generally, the anti-sway system (with feedback loop) shall bring the spreader to a stop within 30 mm at rope length of 25.0 meters.
It shall be capable of bringing of the spreader to a stop to within 2 swings or three seconds after the trolley is brought to a halt from full speed.

• Time-Optimal Control
• PID Control
• Gain Scheduling
• Fuzzy Control
• Nonlinear Control
• Model Reference Adaptive Control
• Input Shaping Control
• Operator-in-the-loop Control
  Assumptions :
  --The operator makes no attempt to eliminate the residual vibration.
  --If a maximum velocity is reached, the acceleration goes to zero automatically.
• Error Feedback Control
• Two-Stage Control: Fast Traveling Control + Residual Sway Control

• Fig. 4.1 Experimental setup
• Fig. 4.1 PNU 1:10 scale down pilot-crane
• Fig. 4.3 Crane operating room: CAOS and Multiverters
• Fig. 4.4 Control box
• Fig. 4.5 Beacon fixed at spreader
• Fig. 4.6 Main power