Physical properties of a singlelink flexible manipulator.
1. Introduction
Flexible manipulators are finding their way in industrial and space robotics applications due to their lighter weight and faster response time compared to rigid manipulators. Control of flexible manipulators has been studied extensively for more than a decade by several researchers (Book 1993, Cannon and Schmitz 1984, De Luca and Siciliano 1989, Siciliano and Book 1988, Vidyasagar and Anderson 1989, and Wang and etc. 1989). Despite their applications, control of flexible manipulators has proven to be rather complicated.
It is well known that stabilization of a flexible manipulator can be greatly simplified by collocating the sensors and the actuator, in which the inputoutput mapping is passive (Wang and Vidyasagar 1990), and a stable controller can be easily devised independent of the structure details. However, the performance of this collocated feedback turns out to be not satisfactory due to a week control of the vibrations of the link (Chudavarapu and Spong 1996). This initiated finding other noncollocated output measurements like the position of the endpoint of the link to increase the control performance (Cannon and Schmitz 1984). However, if the endpoint is chosen as the output and the joint torque is chosen as the input, the system becomes a nonminimum phase one, hence possibly behave actively. As a result, the small increment of the output feedback controller gains can easily make the closedloop system unstable. This had led many researchers to seek other outputs for which the passivity property is enjoyed.
Wang and Vidyasagar (1990) proposed the socalled reflected tip position as such an output. This corresponds to the rigid body deflection minus the deflection at the tip of the flexible manipulator. Pota and Vidyasagar (1991) used the same output to show that in the limit, for a non uniform link, the transfer function from the input torque to the derivative of the reflected tip position is passive whenever the ratio of the link inertia to the hub inertia is sufficiently small. Chodavarapu and Spong (1996) considered the virtual angle of rotation, which consists of the hub angle of rotation augmented with a weighted value of the slope of the link at its tip. They showed that the transfer function with this output is minimum phase and that the zero dynamics are stable.
Despite the fact that these previous efforts have succeeded in numerous kinds of applications, the critical drawback was that these are modelbased approaches requiring the system parameters or the dynamic structure information at the least. However, interesting systems are uncertain and it is usually hard to obtain the exact dynamic parameters and structure information.
In this paper, we introduce a different way of treating noncollocated control systems without any model information. Recently developed stability guaranteed control method based on timedomain passivity control (Hannaford and Ryu 2002, Ryu, Kwon, and Hannaford 2002) is applied.
2. Review of stability guaranteed control with time domain passivity approach
2.1. Network model
In our previous paper (Ryu, Kwon, and Hannaford 2002), the traditional control system view could be analyzed in terms of energy flow by representing it in a network point of view. Energy here was defined as the integral of the inner product between the conjugate input and output, which may or may not correspond to a physical energy. We partition the traditional control system into three elements, the trajectory generator (consisting of the trajectory generator), the control element (consisting of the controller, actuator and sensors) and the plant (consisting of the plant). The connection between the controller element and the plant is a physical interface at which, suitable conjugate variables define the physical energy flow between controller and plant. The connection between trajectory generator and controller, which traditionally consists of a oneway command information flow, is modified by the addition of a virtual feedback of the conjugate variable. For a motion control system, the trajectory generator output would be a desired velocity
To show that this consideration is generally possible for motion control systems, we physically interpret these energy flows. We consider a general tracking control system with a position PID and feed forward controller for moving a mass
2.2. Stability concept
From the circuit representation (Fig. 4), we find that the virtual input energy from the trajectory generator depends on the impedance of the connected network. If the connected network (controller and plant) with the trajectory generator is passive, the control system can remain passive (Desoer and Vidyasagar 1975) since the trajectory generator creates just the amount of energy necessary to make up for the energy losses of the connected passive network. This is just like a normal electric circuit. Thus we have to make the connected network passive to guarantee the stability of the control system since passivity is a sufficient condition for stability.
In addition, the plant is uncertain and has a wide variation range of impedance or admittance (from zero to infinite). Thus, the controller 2port should be passive to guarantee stability with any passive plant.
2.3. Time domain passivity approach
A new, energybased method has been presented for making large classes of control systems passive by making the controller 2port passive based on the timedomain passivity concept. In this section, we briefly review timedomain passivity control.
First, we define the sign convention for all forces and velocities so that their product is positive when power enters the system port (Fig. 4). Also, the system is assumed to have initial stored energy at
for admissible forces
Equation (1) states that the energy supplied to a passive network must be greater than negative
The conjugate variables that define power flow in such a computer system are discretetime values, and the analysis is confined to systems having a sampling rate substantially faster than the dynamics of the system so that the change in force and velocity with each sample is small. Thus, we can easily “instrument” one or more blocks in the system with the following “Passivity Observer,” (PO) for a twoport network to check the passivity (1).
where
Consider a twoport system which may be active. Depending on operating conditions and the specifics of the twoport element’s dynamics, the PO may or may not be negative at a particular time. However, if it is negative at any time, we know that the twoport may then be contributing to instability. Moreover, we know the exact amount of energy generated and we can design a timevarying element to dissipate only the required amount of energy. We call this element a “Passivity Controller” (PC). The PC takes the form of a dissipative element in a series or parallel configuration for the input causality (Hannaford and Ryu 2002).
For a 2port network with impedence causality at each port, we can design two series PCs (Fig. 5) in real time as follows:
Two series PCs can be designed for several cases
where each case is as follows:
Case 1: energy does not flow out
Case 2: energy flows out from the left port
Case 3: energy flows out from the right port
Case 4: energy flows out from the both ports: as we mentioned above, in this paper, we divide it into two cases. The first case is when the produced energy from the right port is greater than the previously dissipated energy:
in this case, we only have to dissipate the net generation energy of the right port as the second line in Eq. (4). The second case is when the produced energy from the right port is less than the previously dissipated energy:
in this case we don’t need to activate the right port PC, and also reduce the conservatism of the left port PC as the fist line of Eq. (3).
Please see (Ryu, Kwon and Hannaford 2002a, 2002b) for more detail about twoport time domain passivity control approach.
3. Implementation issues
This section addresses how to implement the time domain passivity control approach to flexible manipulator with noncollocated feedback. Consider a single link flexible manipulator having a planar motion, as detailed in Fig. 6.
3.1. Network modeling
When we feedback endpoint position to control the motion of the flexible manipulator, a network model (including causality) of the overall control system is depicted as in Fig. 7.
To solve this problem, we make the above network model suitable to our framework. The important physical fact is that the conjugate inputoutput pair
3.2. Designing the PO/PC
First, for designing the PO, it is necessary to check the realtime availability of the conjugate signal pairs at each port of the controller. The conjugate pair at the port that is connected with the trajectory generator is usually available since the desired trajectory
After designing the PO, the causality of each port of the controller should be determined in order to choose the type of PC for implementation. In a noncollocated flexible manipulator control system, the output of the trajectory generator is the desired velocity
From the result in Ryu, Kwon and Hannaford 2002b), the initial energy of the controller is as follows:
where
4. Simulation examples
Many researchers have used a flexible manipulator for testing newly developed control methods due to its significant control challenges. In this section, the proposed stability guaranteed control scheme for noncollocated control systems is tested for feasibility with a simulated flexible link manipulator.
The experimentally verified single link flexible manipulator model (Kwon and Book 1994) is employed in this paper. A single link flexible manipulator having a planar motion is detailed in Fig. 6. The rotational inertia of the servo motor, the tachometer, and the clamping hub are modeled as a single hub inertia
In this section, a stable tip position feedback control is achieved for a flexible manipulator by using the PO/PC.
The following PD controller gain is used
In this noncollocated feedback systems, the hub angle can be considered as a conjugate pair with joint torque to calculate physical energy output flow into the flexible manipulator (see Section 3.A).
Without the PC turned on, tipposition tracking control is simulated (Fig. 10). The desired tipposition trajectory is as follows:
Link  EI : stiffness (Nm^{2})  11.85  H : thickness (m)  47.63E4 

0.2457  L : length  1.1938  
Tip mass  M_{e} : mass (kg)  0.5867  Je : rot. Inertia (kgm^{2})  0.2787 
Hub  I_{h} : rot. Inertia (kgm2)  0.016 
The tipposition can not follow the desired trajectory, tip position and control input have oscillation which increases with time (Fig. 10a,b), the PO (Fig. 10c) grow to more and more negative values.
Stable tipposition tracking is achieved with the PC turned on. Tipposition tracks the desired trajectory very well (Fig. 11a), and the PO is constrained to positive values (Fig. 11c). The PC at the both side is active only when these are required, and dissipate the just amount of energy generation (Fig. 11d)
5. Conclusions
In this paper, we propose a stability guaranteed control scheme of noncollocated feedback control systems without any model information. The main contribution of this research is proposing a method to implement the PO/PC for a possibility active plant due to the noncollocated feedback. We separate the active plant into passive one and a transfer function from the collocated output to the noncollocated output. Therefore, the control system can be fit to our PO/PC framework. As a result, we can achieve stable control even for the noncollocated control system.
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