Torque Bandwidth Map
While the torque-speed diagram considers static actuator operation, many robotic applications demand dynamic torque generation. A way to characterize the dynamic torque generation capability of an actuator consists of looking at the 3-dB cut-off frequency of the torque transfer function magnitude. Due to nonlinear friction, actuator compliance, current and voltage saturation as well as back-emf generation and the influence of the actual load, the achievable torque bandwidth of an actuator varies with the delivered torque amplitude and cannot be well characterized by just a single value.
A common technique to assess the achievable torque bandwidth of an actuator is to lock its output and control the actuator to track a sweep reference, i.e. a sinusoidal input signal with growing frequency. By determining the cut-off frequency of the resulting response, this technique yields the locked torque bandwidth of the actuator controlled by the implemented controller in the specific parameter tuning for the chosen sweep amplitude.
The illustration in the figure above emerges from an extended procedure detailed in [1]. For any electrically generated torque between zero and peak torque, the graphic shows the 3-dB cut-off frequency that can be ideally attained with the actuator within its physical limits, irrespective of any eventual controller.
The dotted light grey line represents the case where the motor generates a sinusoidal torque with the amplitude matching the actuator rated torque. Analogously, the black dotted line corresponds to sinusoidal torque generation with the amplitude matching the actuator peak torque. The two lines indicate, the same generated torque amplitude allows to deliver different torque amplitudes at different cut-off frequencies.
The grey solid line illustrates the operation feasibility boundary. It computes from the minimum of the peak torque generation curve (black dotted line) and the back-EMF generation limit (black dashed line). The grey dashed line is the natural actuator frequency. The colour shade encodes the thermally admissible operation time at each frequency and torque amplitude considering the operation starts from motor windings at ambient temperature. The thermal restrictions apply.
In a similar way, the figure below illustrates the controller independent achievable torque bandwidth across different load inertias. This addresses fact that the actuator in practice is supposed to drive a certain link mechanism as a load. The graphic exclusively considers peak operation. The line colours from green to blue encode the ratio of the load inertia to the actuator inertia. The red solid line corresponds to the locked output case illustrated in \fig{fig:torFreqLock}, where the ratio becomes infinite. The black solid curve represents a link inertia equal to the actuator inertia, where the ratio equals one. The dashed black line is again the back-EMF generation limit for a damping-free actuator compliance. The grey dotted lines illustrate the same limit for non-zero internal damping. The set of theoretically reachable operating conditions is indicated by the red shaded area.
