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05 February 2001

A Study in Polynomial Motion

by Dennis L. Klipp

Using polynomial techniques takes effort but is worthwhile for challenging motion problems.

The advent of servo-driven motion control has unquestionably provided incredible and exciting possibilities for solving motion problems. Input drive problems for mechanical motion devices, such as cams and linkage mechanisms, have been eliminated and replaced with a few cables. Positioning the motion actuation where it needs to be has never been easier. When one considers the flexibility to tailor the motion for unforeseen issues, to accommodate specification changes, or to enhance performance, our motion control capabilities have enjoyed a paradigm shift.

It seems, however, that we've lost sight of the proper criteria for motion kinematics and have ignored the resultant dynamic issues. The lessons learned from years of cam motion design haven't been transferred to servo drive technology on a broad scale. One can speculate why this has occurred. A likely reason may be the limited capabilities of the early stepping motor and servo drive controllers. The only motion type available was a combination of "velocity ramping" and constant velocity, also known as "trapezoidal motion" (taking its name from the trapezoidal shape of the velocity curve). There are long-used cam motions called "trapezoid" and "modified trapezoid," named for the shape of the acceleration, rather than the velocity, curve. Velocity ramping is really the parabolic, or constant acceleration, motion that was widely used for cams starting in the 1920s, and later abandoned by experienced designers in the early 1960s. The application of velocity ramping in the early controllers produced acceptable results, however, and that has simply carried through to modern devices.

Motion Development Criteria

The design and evaluation of a motion form focuses first on the acceleration function. Two characteristics of the acceleration curve that must be considered are the magnitude of the peak acceleration and the shape of the curve. The acceleration magnitude directly impacts peak inertia force, or torque, by Newton's Second Law, F = ma. The acceleration curve's pattern relates to the motion's vibrational characteristics. Parabolic acceleration does a great job of reducing acceleration magnitude. In fact, it produces the lowest possible acceleration magnitude for given distance and time parameters. This low acceleration magnitude has always been the major justification for using parabolic motion. Minimum acceleration means minimum force. Figure 1 shows the velocity, acceleration, and power curves for a complete parabolic motion. Examining the acceleration pattern reveals discontinuities at the beginning, middle, and end of the motion period.

The "Fundamental Law for Cam Motion Design" requires an acceptable motion to be mathematically continuous through the first and second derivatives (velocity and acceleration) for the entire motion period. Stated another way, the pulse, or jerk, function (slope of the acceleration curve) must be finite over the entire motion period. The concern with the manner in which the acceleration and resultant force are applied to the output mechanism relates to the mechanism's dynamic response or vibration.

Figure 2 shows the velocity, acceleration, and power curves for a standard 3-4-5 polynomial motion. While the acceleration magnitude is necessarily greater than that of parabolic motion, note that it begins at zero, builds smoothly to its maximum value, transitions smoothly to its negative peak, and ends at zero. An acceleration comparison between the 3-4-5 polynomial and parabolic motions is displayed in Figure 3. Whether the motion for a mechanism is generated from a cam or a servo drive, this fundamental law applies. Parabolic motion (velocity ramps) fails this requirement totally.

The velocity of a motion for a servo drive is closely scrutinized because the peak velocity determines the gear ratio that may be used while staying within the maximum motor speed. The size selection of the motor and amplifier is also related to the peak velocity because we must stay within the motor's torque envelope capabilities. The common method for managing the maximum velocity is to introduce a period of constant velocity to the motion. This works very well in reducing the peak velocity magnitude, but it has a profound effect on the value of peak acceleration and the resulting torque required of the servo drive. Using an excessively high portion of constant velocity can adversely affect system performance at the beginning and end of the motion and subject the mechanical components of the mechanism to high stresses and subsequent service life problems. Another reason to consider velocity magnitude: The kinetic energy developed is a square function of velocity. Figure 4 graphically compares parabolic motion's peak velocity to that of the 3-4-5 polynomial. The 3-4-5 polynomial's lower peak velocity is an advantage.

A factor that has generally been ignored in evaluating a motion for cams was input torque, or power curve. Magnitude is always considered, but for a cam system, the pattern is much more important than the peak value. A cam drive's input system has, by practical necessity, backlash and a degree of flexibility. The transition period from peak positive power to peak negative power greatly influences the mechanism's performance. For the inertia load component, the power curve shape is the product of the acceleration and velocity functions. Figure 5 compares the parabolic and 3-4-5 polynomial power functions. Note that the parabolic power curve has zero transition time, going from peak positive to peak negative power instantaneously. No wonder cam designers abandoned this motion! While the mechanical aspects of the cam system's input drive train are replaced by electronics with a servo system, the pattern of current demand follows this power curve. My own training and experience is in mechanical engineering, not electronics. However, it's a logical assumption that a smooth and gradual application of current demand is easier on the electronic components of the amplifier/controller.

Defining a Polynomial

I've referenced the 3-4-5 polynomial motion and compared it with parabolic motion. One can refer to numerous cam design books and likely find its description. It's a very fine motion function with a balance of acceleration, velocity, and power for both magnitude and pattern characteristics. The real opportunity that polynomial functions can offer the motion designer is the ability to create near-optimum solutions for motion situations. I use polynomial techniques almost exclusively to design motions for both servo- and cam-driven systems.

The general form of polynomial motion equations follows:

y=C₀+C₁T+C₂T²+C₃T³+C₄T⁴+…+C_nTⁿ

where

The ratio of intermediate time to total time is used for two reasons. First, it controls the magnitude of T and the powers of T in the equations. Second, cam design texts and polynomial-generating computer programs use this arrangement so the coefficients computed for standard polynomials such as the 3-4-5 will match the values presented in these references.

The following relationships must be noted:

To create a motion using polynomial equations, the designer determines a number of boundary conditions or, as I prefer, kinematic events. One coefficient and one equation are required for each kinematic event defined. The order of the polynomial is equal to the number of coefficients. The degree of the polynomial is equal to the order of the equation minus one. The set of simultaneous equations is solved, typically with matrix methods, to determine the coefficients for the polynomial that will yield the defined kinematic events.

For example, let's set up the six kinematic events for a standard 3-4-5 polynomial motion with a 1-inch rise. The motion time is unimportant for calculating the coefficients.

The kinematic events are

Setting up the six equations,

Solving for the three nonzero coefficients, we obtain

The motion equations for the standard 3-4-5 polynomial with a 1-inch displacement are

The real power of polynomial motion techniques lies in the ability to create curves that meet special kinematic requirements. With this power comes the responsibility to ascertain that the results are acceptable. If unreasonable kinematic events are specified, the resultant polynomial could take you on a wild ride. Assuming that the mathematics are properly executed, a polynomial will always produce the kinematic events that were specified. It's the designer's task to examine the entire curve for unanticipated wanderings. For example, the specification of more than 10 kinematic events seldom produces an acceptable motion unless the events are well chosen. Intermediate events or conditions not at the start or end of the motion period cam have a profound influence on the overall results, especially if one or more displacement events are specified. Finally, note that a sixth-order polynomial is needed to adhere to our fundamental law.

The software I've developed to design polynomial motions for cams has been modified to produce the coordinate tables for downloading to a servo controller. This program is written in Visual Basic, and it uses polynomial motions exclusively. A question that should come to mind is how constant velocity is managed. Constant velocity must be used to match other machine elements for tracking applications and is also important for reducing maximum velocity if required. Constant velocity is a second-order polynomial defined by specifying only the displacement at the start and end of the motion period. In fact, a "velocity ramp," or parabolic acceleration, is a third-order polynomial. For parabolic motion, we must define two separate polynomials: one for the acceleration period and one for the deceleration period.

The Case Study

A pick-and-place mechanism is needed to transfer a group of cylindrical products from a collection area to containers. Because of the product's nature, we desire a very gentle motion to help maintain control of the parts with minimum necessary air pressure on the grippers. A horizontal displacement of 46 inches and a vertical displacement of 2.75 inches are the stroke specifications. The vertical stroke is the minimum safe displacement to extract the product from the collecting area pockets. The size of the product area, certain special material requirements dictated by the client, and the transfer distance resulted in a total carriage weight of more than 500 pounds. The weight of the vertically displacing components was more than 200 pounds. The real challenge comes with an allowable time of 4.5 seconds for the entire pick-and-place function. Of this, 2 seconds are allocated to the actual placing motion that lifts and transfers the product. This case study focuses on that placing motion. The servo controller has a "cam" facility for driving the two axes in a coordinated fashion from tabular data.

The motion solution for this problem used the standard 3-4-5 polynomial for the horizontal axis. The time duration for the horizontal move was selected in conjunction with the development of a single eighth-order polynomial that generates the lift-and-place vertical function. Figure 6 displays the kinematics of both motions, and Figure 7 shows their coordinated resultant path. The performance of these motions was outstanding, with absolutely no vibration throughout the cycle. The process of this motion development and the strategies employed will be explained in detail.

First and foremost, the acceleration curves for both motions strictly adhere to the fundamental law of motion design. This is accomplished by specifying the acceleration magnitude equal to zero at the beginning and end of each motion. The time duration for the placing motion was set to 2 seconds to favor this motion over the return motions. The return motion isn't addressed here, as the placing motion is much more interesting.

The next strategy for any motion design is aggressively maximizing the motion time period while maintaining the critical functions of the process. Acceleration magnitude is inversely proportional to the square of the time period. For a given displacement, twice the time period reduces acceleration by one quarter. Recall that the vertical stroke required to adequately clear the pocket containing the product is 2.75 inches. From Figure 6, the maximum displacement of the vertical motion is 6 inches at 1.1 seconds, and the vertical rise and fall is defined by a single eighth-order polynomial. Concerning clearance issues, a longer stroke will permit a longer motion period. Conventional wisdom tells us to minimize stroke, however, because acceleration increases in direct proportion to the increase in displacement. This is true for a fixed time period. By experimenting with the displacement magnitude and the time position of the maximum displacement, we can obtain a relatively rapid extraction of the product from the nest and a quick placement into the container cavity. In this case, it's necessary only to insert the lower part of the product into the container and let gravity do the rest—this is why the height at the deposit position is 0.75 inches higher than the picking position.

The vertical motion is an example of another important motion strategy: eliminating unnecessary dwells. Don't stop unless it's really needed for the process. Furthermore, notice that the acceleration isn't defined to be zero at the peak displacement position. Allowing the acceleration to follow this pattern minimizes vibration and also lowers the acceleration values for the entire motion.

The final version for the vertical motion is defined by the following kinematic events.

An iterating process established the time period and positioning of the horizontal motion relative to the vertical motion. The goal was to maximize the time duration for the horizontal motion and provide a resultant path for the product that adequately clears both the pockets and the container cavity. As with any iteration, the designer's judgment must be used to decide on the final arrangement.

Success in any endeavor is a function of making good decisions, which are themselves a function of experience. How does one get experience? Experience is obtained by making bad decisions. The use of polynomial techniques for developing motions isn't required for rudimentary motion applications. This process takes effort but is very worthwhile for more challenging problems. With the ability to change a servo-driven system's motions, there are likely many existing applications that offer opportunities for productivity improvements through the creative application of these techniques. MC

Figures and Graphics

• Figure 1. Parabolic (constant acceleration) motion profile.
• Figure 2. 3-4-5 polynomial motion profile.
• Figure 3. Comparative acceleration profiles for parabolic and 3-4-5 polynomial motions.
• Figure 4. Comparative velocity profiles for parabolic and 3-4-5 polynomial motions.
• Figure 5. Power comparison of the parabolic and 3-4-5 polynomial motion profiles.
• Figure 6. Motion profiles for our pick-and-place kinematics example.
• Figure 7. Resultant motion path for our pick-and-place kinematics example.

Author Information

Dennis L. Klipp is an engineering consultant specializing in mechanisms and motion development. Contact him at 26 Cherry Hill Terrace, Waterville, ME 04901; tel: (207) 873-5824; fax: (207) 873-5835.