Milchiorri (2006) stated that the robot performances are mainly influenced by the mechanical design and by the actuation system. He also explained that there are two types of robot control schemes: Decentralized (or independent) control schemes and Centralized control schemes. The decentralized control scheme also known as independent joint control (IJC) model has Single Input Single Output (SISO) configuration while the centralized control scheme has a Multiple Input Multiple Output (MIMO) configuration. From the review, SISO configuration (which can equally be in
32
the form of multiple SISO) is more common and simpler than the MIMO in practice. According to Alassar (2010), the basis of IJC is that the robotic manipulator is treated as a set of independent actuators working independently. This means that each link of the robotic manipulator is considered as single input single output (SISO) system with an independent controller.
Independent joint control scheme is widely adopted in most industrial manipulator controller because of its simplicity (Voung and Ang, 2009).
According to Richter, to derive the independent joint control model, it is assumed that the DC motor is connected to a gear reduction of ratio r : 1 and moment of inertia Jg. The reduced-speed shaft drives a rotational inertia Jl, which represents the link driven by the motorized joint. The motion of the other links should influence the DC motor as well. For the independent joint model, however, the influences of the other links are treated as disturbances and then the controller is designed to be robust (tolerant) against them ( Richter). When applying the decentralized control scheme, Melchiorri (2006) stated that each joint of the robotic manipulator is considered independently, and the term d (= ๐๐๐๐
๏ฟฝ๐๐) is considered as an external disturbance. These considerations can be applied with proficiency when there is no direct coupling between the actuator and the joint (i.e., when there are gears).
Figure 2.10b: The lumped model of a single link with actuator/gear train (Melchiorri, 2006; Spong, 2006; Bozma, 2015)
33
Figure 2.10b illustrates the independent control scheme where the actuator and link dynamics are considered. Where Ja, Jg and Jl are respectively, the actuator, gear, and load inertias. Bm is the coefficient of motor friction and includes the friction in the brushes and gears. ๐๐๐๐ is the link torque, r is the gear ratio. Considering a DC motor actuator, the independent joint model of the robotic manipulator is derived in (Melchiorri, 2006; Spong, 2006;
Bozma, 2015). Figure 2.10c shows the block diagram of a robotic manipulator. Figure 2.10d represents a third order system from input voltage V(s) to output position ๐๐๐๐.
Figure 2.10c: Robotic manipulator block diagram (SISO)
Figure 2.10d: Block diagram of actuator model with link (Melchiorri, 2006;
Spong et al, 2006; Bozma, 2015) 2.7 Robotic Manipulator System Uncertainties
The robotic manipulator is a physical system which is modeled mathematically in many research works for control purpose and to enhance the performance of the system. Garulli et al stated that when modeling physical systems for control purposes, it is necessary to provide model
34
descriptions that capture the main features of the system behavior and are mathematically tractable at the same time. An extremely accurate model of a physical process may turn out to be unsuitable for application of the available analysis and design techniques. By contrast, an over-simplified model, which misses significant information on the system structure, may lead to unacceptable design performance. They opined that the discrepancy between the system and the adopted nominal model is usually represented as a perturbation on the nominal model. The resulting model, which is therefore composed of the nominal one and the perturbation, is usually referred to as the uncertain model or model set.
Uncertainties are unavoidable in physical systems and they are classified into two categories: disturbance signals and dynamic perturbations. The former includes input and output disturbance, sensor noise and actuator noise, etc. The latter represents the discrepancy between the mathematical model and the actual dynamics of the system in operation. A mathematical model of any real system is always just an approximation of the true, physical reality of the system dynamics. Typical sources of the discrepancy include unmodelled (usually high-frequency) dynamics, neglected nonlinearities in the modeling, effects of deliberate reduced-order models, and system-parameter variations due to environmental changes and torn-and-worn factors. These modeling errors may adversely affect the stability and performance of a control system. The consideration of the uncertainties in the design of a control system gave rise to the development of the robust control. Dynamic perturbations are usually described so that they can be well considered in system robustness analysis and design.
Scherer (2001) stated that all mathematical models of a physical system suffer from inaccuracies that result from non-exact measurements or from
35
the general inability to capture all phenomena that are involved in the dynamics of the considered system. Even if it is possible to accurately model a system, the resulting descriptions are often too complex to allow for a subsequent analysis, not to speak of the design of a controller. Hence one rather chooses for a simple model and takes a certain error between the simplified and the more complex model into account. Therefore, there is always a mismatch between the model and the system to be investigated. In control engineering this mismatch is referred to as uncertainty. The uncertainties in a physical system model can be classified into disturbance signals and dynamic perturbation. However, the dynamic perturbation is used to describe the mismatch between the model and the system.
According to Gu et al (2005), many dynamic perturbations that may occur in different parts of a system can, however, be lumped into one single perturbation block ฮ, for instance, some unmodelled, high-frequency dynamics which is referred to as โunstructuredโ uncertainty. The unstructured dynamics uncertainty in a control system can be described in different ways, such as shown in figures 2.11 and 2.12, where Gp(s) denotes the actual, perturbed system dynamics and Go(s) a nominal model description of the physical system.
Figure 2.11: Additive perturbation (Gu et al, 2005)
36
Figure 2.12: Input multiplicative perturbation configuration (Gu et al, 2005) The additive uncertainty represents the absolute error between the actual manipulator dynamics and the nominal model representation of the system, while the multiplicative representations show relative errors. According to Garulli et al, the unstructured uncertainty representation is used to describe unmodelled or difficult to model dynamics and it is usually given as a bound on some measure of the error signal between system and nominal model outputs for a chosen class of input signals. However, dynamic perturbations in many industrial control systems may also be caused by inaccurate description of component characteristics, torn-and-worn effects on plant components, or shifting of operating points, etc. Such perturbations may be represented by variations of certain system parameters over some possible value ranges. They affect the low-frequency range performance and are called parametric uncertainties (Gu et al, 2005). The structured uncertainty is a dynamic uncertainty represented by an element such as a finite dimensional vector or an operator, in some pre-specified uncertainty set of a suitable space. The uncertain elements in this case may be the coefficients of a transfer function, or the components of system matrices in a state-space realization.