Applying kinematics to motion control
Non-Cartesian actuators are improving applications where they weren't even feasible before. Full definition of their dynamics and kinematics helps designers extract maximum performance from these systems.
In addition to these forces are the effects due to centrifugal and Coriolis forces. The centrifugal terms on one joint are proportional to the square of the velocity of the other joint, and are at a maximum when the two links are perpendicular. The Coriolis term is proportional to the product of the two joint velocities, and is also at a maximum when the two links are perpendicular.
In most controllers, the servo loop for each axis (really, each joint) is closed independently, without regard to the action of other axes. Effects caused by other axes are just treated by the servo loop as disturbances to be rejected. But of course, before they can be rejected, they must create real errors in the axis — which can compromise the quality of motion.
In more sophisticated controllers these predictable interaxis effects can be used as crosscoupled feedforward terms. The purpose of feedforward is to anticipate the required torque for a given trajectory, leaving the feedback loop to handle only unanticipated disturbances and errors in the prediction. Most widely used are single-axis feedforward terms based on one axis' own trajectory — particularly its velocity and acceleration feedforward components. The same principle can be extended to the dynamic effects on cross-coupled axes.
Employing cross-coupled feedforward terms using the instantaneous commanded velocity and acceleration of the other axis in addition to the standard feedforward terms dramatically reduces the disturbances induced by these effects. This can lead to more accurate actual trajectories or even increased dynamic capabilities while maintaining a given level of accuracy. Tables above show the cross-coupled gain terms required for this mechanism.
Acceleration feedforward gains Here, the components of acceleration feedforward gain are regrouped for clarity.
Married links
Here are the crosscoupled gain terms for our SCARA robot example, broken out of the Contributing terms chart. The terms show the interdependencies that can be exploited for control improved with feedforwarding.
In addition to the standard, constant moment-of-inertia terms that can be multiplied by the commanded acceleration to produce an accelerationfeedforward term (as in a Cartesian system), our example SCARA robot also has a variable component in its moment of inertia about the shoulder joint. As already mentioned, it changes with the angle of the elbow joint. This means that every elbow movement changes the optimal acceleration feed-forward gain. The table below shows the constant and variable components of the acceleration feed-forward gain for the shoulder and elbow joints.
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Live and learn In the academic world, much attention is focused on automatic adaptive control algorithms that attempt, at least, to continually identify changing system characteristics from servo performance, and to adjust the gains accordingly. Most users in the industrial world don't trust these types of algorithms, and for good reason; they often give unstable results by misidentifying the system parameters. In industrial controls, a more popular method of adaptive control is the one we just explored: gain scheduling. In this strategy, gain changes made are not based on closedloop performance, but rather in an open-loop fashion based on the system's state. (In the case we just considered this is the elbow position and its resultant effect on the inertial moment about the shoulder.) While this might seem to be an inferior solution because there is no loop closed around the adaptation algorithm,-it is more predictable because its possible-range of actions can be precisely determined ahead of time. Further, it cannot be fooled by unmodeled dynamics, as when friction is mistaken for higher inertia when movement is less than expected, for example. A controller that can directly compensate for these effects in non-Cartesian mechanisms can dramatically improve performance. This capability is especially useful on wafer-handling robot arms. Due to the fragile nature of silicon wafers, end effectors cannot tightly grip them. To prevent dropping, smooth motion is critically important. The effects of the uncompensated dynamics have for a long time been a key limiting factor in the performance of these robots. Now however, implementation of the cross-coupled feedforward terms and scheduled gain terms yields dramatic improvements in performance of these systems. |
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© 2012 Penton Media Inc.
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