Clearly, the proportional valve control method can tame a wild machine axis. However, the system as presented thus far is not perfect. The most serious deficiency is the uncertainty as to where the cylinder and load will eventually stop. There is no means to back up should there be overtravel. Therefore, almost all the random factors that affect the stopping point of the discrete directional system will be visited upon this, the proportional system.

A system that is proportional for its shock control, but also monitors and controls position on a full-time basis, is called the electrohydraulic positional servomechanism. It possesses all the desirable characteristics for effective motion control, but because it is a bona-fide feedback control system, it introduces its own set of challenges. However, when properly designed and tuned, this system provides the ultimate solution to the most demanding motion control application challenges.

The positional servomechanism is normally considered a positioning system. However, when the concept of profiling is superimposed, the result is motion control - total control of the machine. To be successful, motion control must have three things designed into it:

• a command motion profile
• sufficient power to manage the load at the required speeds, and
• sufficient closed-loop bandwidth to meet accuracy and stability requirements.

The motion-control profile is crucial to the motion control system, and represents the most significant contribution of the digital computer to the motion-control process. The profile is the means by which acceleration, velocity, and position are controlled simultaneously. Profiles are very easily generated by a digital computer (motion controller card or circuit board), and are formulated at design time to determine the peak speeds that must be attained by the actuator based on a stated productivity rate.

In determining these peak speeds, we must first collect data on the productivity needs of the application. For example, suppose the application requires stamping out 30 packets of paper plates per minute. Immediately, we know that the cycle rate is 12 cycle per second. Further, if we are given that cylinder stroke must be, say, 4 in., and that the cylinder must remain extended for 0.5 sec, then we know that the actuator must fully extend and retract within 1.5 sec. We can add some reasonable acceleration times, plus take into account that the cylinder does not naturally want to retract as fast as it extends. (Yes, this statement is correct for the valve-controlled cylinder.) To compensate for this, we can rob a few milliseconds from the extension time and give it to the retraction time. This allows us to reach conclusions about the maximum speed needed in order to achieve the required productivity rate.

This example leads us to the conclusion that the peak extend speed will have to be about 10 in./sec. Once we are given the maximum thrust force, we can design for a given supply pressure, which makes it possible to completely size the critical components in the system. Our example introduces the idea of a motion profile, which is a means of defining the speed, acceleration, and position of the cylinder and load at any instant within each cycle and throughout each and every cycle for the entire life of the machine.

The relationship between acceleration, velocity, and position - which are drawn against a common time axis - is illustrated in Figure 5. Newton revealed through the calculus that:

• velocity is the integral of acceleration
• position is the integral of velocity
• velocity is the derivative of position, and
• acceleration is the derivative of velocity.

Even if you never studied calculus (see box below), these are mathematical facts of our physical world. What they mean to the motion control system designer is that specifying any one of these three parameters for all time automatically specifies the other two as well. This is because of their undeniable relationship to each other — they are all integrals or derivatives of each other.

This relationship makes it possible to achieve the simultaneous control of acceleration, velocity, and position, even though the final system closes the loop with position feedback only. It works like this: The position profile, the plot at the bottom of Figure 5, is the actual command to the motion-control servomechanism. The slope of the position profile represents the velocity of the output at each and every instant in time. The slope of the velocity profile is the acceleration profile, which is actually carried in the curvature of the position profile. So the designer controls acceleration by controlling the curvature of the position command profile.

The final result is this: For merely having fed the position command profile to the motion control servo axis, all three dynamic variables are implicit in that profile. Furthermore, designing a servo axis that faithfully follows the command profile, controls all three parameters. This is the sum and substance on modern motion control philosophy.

At design time, the design methodology recommends that the entire motion control profile, acceleration, velocity, and position, be synthesized based on productivity and other motion needs. In contrast, at commissioning time the programmer of a dedicated motion controller card may only specify that the motion must go from, say, position x to position y with maximum speed of v and maximum acceleration of a. The digital motion controller interprets these as instructions, then generates the actual profile commands on the fly. Consequently, the programmer may never see the entire profile itself.

### Calculus simplified

Simple algebra defines the slope of any line as the difference between two points on the y axis divided by the difference between two points on the x axis. Because the y axis represents position, the difference between two points is distance. Likewise, the x axis represents time, so the difference between two points is the time it took to travel the respective distance.

So the slope of a line plotted as position versus time is distance travelled divided by time travelled, which, by definition, is velocity. Likewise, the slope of velocity is acceleration. Calculus simply extends this concept further by calculating slope of a line or curve at any instant by breaking it into infinitely small steps instead of at one place.

Jack L. Johnson, PE, is an electrohydraulic specialist, consultant, former director of the Fluid Power Institute at the Milwaukee School of Engineering, and contributing editor for Hydraulics & Pneumatics. He can be reached by phone at (414) 236-5350 or via email at jack@idaseng.com.