Types of Static and Dynamic Analyses
Statics analysis in robotics as commonly taught is as simple as relating the external static wrench applied to the robot’s end-effector to the joint torque through the Jacobian of the robot. By using the robot’s Jacobian, there is duality relation between statics and velocities. In other words, the robot’s Jacobian relates not only the joint velocity to the end-effector velocity but also relates the joint torque to the end-effector’s external static wrench.
Static analysis in mechanics, on the other hand, has a bit different meaning. In mechanics, there are three static analyses commonly performed. First, the static analysis aiming at evaluating the reaction forces and moments given some load applied to the structure (with certain boundary conditions). Second, the static analysis aiming at evaluating the elastic deformation of the structure given some load to the structure (with certain boundary conditions). Third, extending the second static analysis to the evaluation of stresses in the structure. The first static analysis does not take the material’s property and cross section into account. On the other hand, the second and third static analyses take the material’s property, i.e. modulus of elasticity, and cross section into account. Hence, the second and third static analyses are related to the strength of the structure’s material in a static setting.
Dynamic analysis in robotics as commonly taught is rigid body dynamics which consists of two types of dynamic analysis: forward dynamic analysis and inverse dynamic analysis. The forward dynamic analysis is defined as “given the joint torque, find the corresponding wrench delivered at the end-effector”. On the other hand, the inverse dynamic analysis is defined as “given the end-effector wrench, find the required joint torque”. Looks like the static analysis discussed earlier? No, this is different. This one is in a dynamic setting. But notice that the static analysis in robotics can be seen as a special case of the dynamic analysis. When the acceleration is zero, the dynamic analysis turns to static analysis.
Dynamic analysis in mechanics is usually flexible body dynamics which can be typically classified into: modal analysis, frequency response analysis, and time response analysis. What are the differences between these three analyses?
Modal analysis aims at evaluating the natural frequencies and the corresponding mode shapes. Natural frequencies are frequencies at which a structure will have resonance if an excitation at the same frequency is given to the structure. Resonance is typically to be avoided since it can lead to failure and damage to the structure due to uncontrolled, large vibration. But, modal analysis does not provide any information about how much the magnitude of the response (displacement) will be. Although we know that there will be large displacements at the natural frequencies, we do not know how large the displacements are. In fact, the displacements at different natural frequencies do not have the same magnitude. Furthermore, modal analysis does not depend on the applied force. In fact, modal analysis is performed without any applied force; it is an unforced system. It is also not much affected by damping. Hence, only the inertia and stiffness of the structure is typically taken into account in the modal analysis.
To get the magnitude of the response of a dynamic system, either frequency response analysis or time-response analysis can be performed. Between these two, the frequency response analysis is simpler and easier. The frequency response analysis is typically represented by a two dimensional plot in which the X-axis is the frequency whereas the Y-axis is the displacement magnitude. In this plot, it will be shown that large displacements occur at the natural frequencies of the system. The magnitudes of the displacements are shown in this plot.
How is the frequency response analysis performed? It is performed by applying an external load to the system at a certain frequency and the displacement is evaluated at this frequency. This represents the steady-state response of the system at this frequency. This is done for a range of frequency values. As a result, the frequency-vs-displacement plot is obtained. The frequency response analysis is performed without solving the system’s differential equations.
On the other hand, the time response analysis is represented by a two-dimensional plot in which the X-axis is the time whereas the Y-axis is the displacement magnitude. Similar to frequency response analysis, the time response analysis also requires the applied force to be included in the system. To obtain the time-domain response, the system’s differential equations should be solved by either analytical or numerical methods. This analysis shows the displacement of the system over a certain range of time. This solution includes both the transient and steady-state displacements of the system over time. The response starts with a transient behavior followed by its steady-state behavior.