As electro-mechanical systems become commonplace on modern machines, the engineers who design those systems increasingly have to answer some tough questions about the positioning accuracy, speed, potential for noise and robustness. All these electro-mechanical performance characteristics revolve around the dynamic properties of the machine, but too many engineers consider dynamics an exclusive property of the motion controller.
There is something to that view. As motion is controlled by properly converting the electrical energy into mechanical energy at the motor side, poor dynamic behavior is often associated with de-tuned controllers, bad drive systems and sluggish or under-sized motors. Of course, dynamics is a property of mechanical systems too. Their physical dimensions, material properties and interconnections will eventually be the key factors in determining the overall system dynamics.
Motion control systems can shape the mechanical energy delivered by the motor to reach almost any dynamic profile attainable with the existing system's size — as determined by mass inertia ratios and available power. But at the other end of the power train, failure to pay attention to the load side of the system can have negative effects — including overshoot, oscillation, instability and noise.
There is a simple reason why motion controlled at the motor is never identical to motion obtained at the load — compliance, which is essentially a loss of stiffness. A mechanical system which couples the motor with the final actor is elastic in nature. This elasticity between the motion actuator and the final controlled element creates at least a two-mass oscillator system and changes the overall dynamic picture.
There are at least two ways to analyze electro-mechanical systems such as these (Figure 1). Note: All figures are in an image gallery at the end of this article. The first involves a simple, lumped-mass representation of the entire powertrain. Each component of the powertrain is separately considered as a dimensionless mass, connected to the adjacent component via a spring-damper element. A more complex approach uses finite elements to model the mechanical structure. Axial and torsional modes of the connections between the system elements are completed with structural modes of the components themselves.
Both methods have their advantages. We'll focus here on the first of these mechatronic analysis techniques, which is fast and avoids complex analysis software. Remember, though, that its results and accuracy depend very much on the experience of the modeler. The FEA-based technique, although more elaborate, gives the highest degree of precision and reveals phenomena that might escape a simple, lumped-mass analysis.
Lumped Mass Analysis
Using the first approach and simplifying the system, we get the classic two-mass oscillator representation (Figure 2) in which the load and the motor are coupled with a spring-damper element. The motor itself is coupled to the ground via a spring representing the stiffness of the electrical-to-mechanical energy converter. This stiffness can be controlled and is usually much higher than the weakest spring between the motor and the load. Thus the set position in our stiff controller may be considered equal to the motor position at all times. We need then to know how the load position (XLoad) will vary when the motor position or the setpoint is changed.
To characterize the dependency between the position at the load and the motor position, we use the frequency domain representation. The motor position is following a sinusoid of unity amplitude and various frequencies. At each frequency the load position is recorded, and two values are plotted on a Bode-plot: the ratio between the amplitudes of the signals and their phase difference.
At low frequencies (Figure 3), the load is accurately following the motor. There is almost no phase shift, while the amplitudes remain equal (XLoad/ XMotor = 1.1=0.1 dB). If the motor is running at 10 Hz (Figure 4), the load enters resonance and moves with much higher amplitude. The phase is already shifted 90 degrees. The resonant frequency is a characteristic of the mechanical system and can be shifted upward or downward only by structural modifications: changing elements' mass-inertia or the stiffness of their interconnections. At high frequencies beyond resonance, the motor movement is hardly "noticed" by the load and the motion is dramatically attenuated. The load, now as much as 180 degrees out of phase, becomes quasi-decoupled from the motor, meaning it is no longer controllable.
The purpose of identifying the resonance of the system is clear: the bandwidth of the position-controlled powertrain is going to be limited by this lowest natural frequency. Such a frequency response characteristic is like the system's DNA sequence. By simply looking at its magnitude and phase characteristics we can predict, based on the desired motion profile, whether the machine is going to perform well within given specifications.
||Analysis of encoder placement in a torque motor drive — a) Finite Element Model of the cylinder and torque motor; b) Plant frequency response; c) Closed loop speed controller frequency response; d) Compliance frequency response.
For example, at a step-position setpoint the system in Figure 5 will oscillate into position with its natural frequency of 30 Hz and initial amplitude (overshoot) depending on the damping in the elastic element. This oscillation will always take place (with higher or lower amplitude) as long as the input signal contains the 30 Hz harmonic. Since a step signal is, according to Fourier, the sum of all spectral harmonics, evidently the lowest natural frequency will be excited and oscillations into position will happen. The positioning window will be reached sooner or later, depending on the amount of mechanical damping in the system.
Small values of the natural frequency and damping lead to slow positioning loops and consequently low overall dynamics. Can we influence or change this fact by means of control?
Sometimes we can use the capabilities of the speed and position controllers to obtain a better shape of the dynamic response. There are several control parameters to be modified to obtain the required accuracy and eliminate oscillation into position.
First, the position controller might be set too hard. The Kv gain is a function of the equivalent time of the speed control loop at a given positioning sampling time. This equivalent time of the speed loop is itself depending on the lowest eigenfrequency of the system. We may have to lower the Kv gain to accommodate the resonance.
A second option is to use a damping optimal setting in the speed controller. It requires a lower proportional gain in the speed loop and higher integrator time. Drives adjusted this way may be used when the load requires a little damping — in applications like robotics, manipulators or other processes without load retroaction.
The third possibility would be to apply a speed setpoint filter in the position loop controller. Such a filter would prevent the speed command value to increase too fast and, thus, attenuate the oscillations in position at the target frequency.
One of the most commonly used ways to improve the precision and behavior of a motion profile is by modifying the setpoint signal parameters. Limiting the jerk, modifying the acceleration profile or limiting the maximum speed all are measures to smooth-out the undesirable oscillations in the output. All these measures have a similar effect: to filter out the unwanted frequency from the excitation signal such that the mechanical eigenfrequencies are not excited and evident in the output.
To decrease the overshoot on axes with a low natural frequency, a low-jerk limitation should be used. But this leads to a considerable increase in process time and consequently the idle time for positioning. With the limitation in jerk the maximum acceleration may not be reached.
Probably the best approach to reduce overshoot and the oscillations into position is to consider altering the acceleration profile itself. The command acceleration translates to torque and thus the excitation of our mechanical system. If we could "extract" from the frequency content of the excitation signal the mechanical lowest natural frequency, the positioning would no longer be affected by unwanted oscillations. In the sample motion profile shown in Figure 6, a square acceleration profile was used. The acceleration was limited to 20,000 deg/sec². When transformed into a Fourier series, our square-shaped profile contains the harmonic at 15 Hz.
What if, contrary to any intuitive guess, we double the acceleration limits, leaving everything unchanged: controller parameters, length of motion, maximum speed, a high-jerk limitation. With a higher acceleration limit of 40,000 deg/sec², the duration of the positioning process is shorter. The harmonic content of the acceleration or command torque profile now lacks the harmonic at 15 Hz. This acts like a filtering effect on this natural frequency. The time response shown in Figure 7 is now not only faster but smoother and almost perfect. The 15 Hz frequency has been subtracted, or filtered-out, from the excitation frequency content. Thus the natural mode is no longer present in the system response.
If the requirements in the motion profile and controller dynamics cannot be changed as they were specified, the only option is to redesign the mechanical system. Sometimes there is just no way to obtain the required dynamics by simply tuning the controllers. As explained before, the mechanical dynamics is limited by the lowest natural frequency and the damping of the mechanics. The higher it is, the better the response dynamics. There are two ways to increase this frequency: lowering the load inertia or increasing the stiffness of the weakest link.
Our weakest link is the belt-pulley system. Its stiffness depends on the belt material, width and length. If the best solution has been already selected, and there is no way these parameters are changed, then the belt-pulley is likely not the best mechanical choice for controlling the large-inertia cylinder with a high dynamic profile.
One of the possible radical design changes is getting rid of all compliant transmission elements — such as the coupling and belt-pulley — and connecting a torque motor directly to the cylinder shaft. Often, in such direct-drive conversions, a question arises: Where is the best place to connect the feedback encoder? At the torque-motor shaft or at the other end of the cylinder? A mechatronic analysis can easily provide the answer to this question.
Here is a short overview of an analysis done for a customer in the printing industry (Figure 8). The speed controller was first closed at the motor side (blue frequency response) and then at the load side (red frequency response). In the second scenario, the gain of the speed controller could be increased 20 times higher than for the system with the encoder mounted at the motor side. A higher gain in the speed controller and a lower integrator time implies the system is stiffer, and the controller dynamic is much better than in the first case. The compliance (inverse of stiffness) characteristic is plotted in Figure 8D. When the encoder is mounted at the load the stiffness increases dramatically in the range of 0-20 Hz.
The analysis proves the direct drive solution is feasible and is going to give better results when the encoder is mounted at the load side.