Generate stepper-motor speed profiles in real time
A new algorithm for stepper-motor acceleration allows speed profiles to be parameterized and calculated in real time. This algorithm can run on a low-end microcontroller using only simple fixed-point arithmetic operations and no data tables. It develops an accurate approximation for the timing of a linear ramp with constant acceleration and deceleration.
It's commonly thought that the timing of a linear speed ramp for a stepper motor is too complex to be calculated in real time. The exact formula for the step delay is in Equation 8. The solution has been to store the ramp data in precompiled arrays, but this method is inflexible and wastes memory. The alternative has been to use a more powerful and expensive processor than otherwise needed or a high-level stepper-control IC. This article develops an accurate approximation that has been implemented in C using 24.8 fixed-point arithmetic on a mid-range PIC microcontroller.
Motor step signals can be generated by a 16-bit timer-comparator module as commonly integrated in microcontrollers. On the PIC, the CCP (capture/compare/pwm) performs this function. It allows steps to be timed to the resolution of one timer period. Each step advances the motor by a constant increment, typically 1.8 degrees on a hybrid stepper motor.
The timer frequency should be as high as possible while still allowing long delays as the motor is accelerated from stop. A timer frequency of 1MHz has been used. A maximum motor speed of 300rpm is then equivalent to a delay count of 1,000. It's necessary to have high timer resolution to give smooth acceleration at high speed.
Notation and basic formulas
Delay (sec) programmed by timer count c:
Equation 1
f = timer frequency (Hz).
Motor speed ω (rad/sec) at fixed timer count c:
Equation 2
α = motor step angle (radian).
1rad = 180/π = 57.3deg. 1rad/sec = 30/π = 9.55rpm.
Acceleration ω' (rad/sec^{2}) from adjacent timer counts c1 and c2:
Equation 3
Equation 3 assumes fixed-count speed (Equation 2) at the midpoint of each step interval (Equation 1), as on a linear ramp, Figure 1. Note that ω' resolution is inversely proportional to the cube of the speed.
Figure 1: Ramp geometry: move of m=12 steps
Linear speed ramp—exact
On a linear ramp, acceleration ω' is constant, and speed ω(t) = ω'.t. Integration gives the motor shaft angle θ(t):
Equation 4
n ≥ 0 step number (real). When the shaft is at θ = n.α, (integer n) it's time for the nth step pulse:
Equation 5
The exact timer count to program the delay between the nth and (n+1)th pulses (n ≥ 0) is:
Equation 6
The initial count c_{0} factorizes out to give Equations 7 and 8:
Equation 7
Equation 8
Note that c_{0} sets the acceleration, proportional to (1/c_{0})^{2} .
In real-time, Equation 8 would require calculation of a square-root for each step, with the added problem of loss of precision by subtraction.
Approximating linear ramp
Ratio of successive exact timer counts from Equation 8:
Equation 9
Taylor series:
Equation 10
Equation 11 is the second-order approximation to Equation 9 using Equation 10:
Equation 11
Equation 11 can be rearranged for faster calculation:
Equation 12
Finally, we can disconnect the physical step number, i, from the step number n on a ramp from zero, to give the general-purpose ramp algorithm shown in Equation 13. Here n determines the acceleration and increments with i for constant acceleration. To ramp up from stop, n_{i} = i, i=1,2, . . . :
Equation 13
Negative n-values give deceleration. In particular, Equation 14, with n_{i} = i - m, can be used to ramp any speed down to stop in the final steps of a move of m steps:
Equation 14
Table 1: Accuracy of the step-delay approximation
Step n | Exact (9) | Approx (11) | Relative error |
1 | 0.4142 | 0.6000 | 0.4485 |
2 | 0.7673 | 0.7778 | 0.0136 |
3 | 0.8430 | 0.8462 | 0.00370 |
4 | 0.8810 | 0.8824 | 0.00152 |
5 | 0.9041 | 0.9048 | 7.66E-4 |
6 | 0.9196 | 0.9200 | 4.41E-4 |
10 | 0.9511 | 0.9512 | 9.42E-5 |
100 | 0.9950 | 0.9950 | 9.38E-8 |
1,000 | 0.9995 | 0.9995 | 9.37E-11 |
Accuracy of approximation
Table 1 shows that the approximation is accurate even at low step number n and relative error decreases with n^{3}. However, n=1 has a significant inaccuracy. The inaccuracy at n=1 can be handled in two ways:
- Treat n=1 as a special case. Using c1 0.4056 c0 compensates for the inaccuracies at the start of the ramp and allows Equation 7 to be used to calculate c0.
- Ignore the inaccuracy. In place of Equation 7 use Equation 15:
Equation 15
The first alternative gives an almost perfect linear ramp. The second alternative starts with a fast step. This is to the good, as it helps keep the motor moving between step pulses 0 and 1-and establishes the angle error needed to generate torque. It also allows a wider range of accelerations to be generated with a 16-bit timer and has the advantage of simplicity. It's therefore recommended to ignore the inaccuracy at n=1.
Figure 2: Stepper-motor speed ramp
Figure 3: Start of ramp detail
Figure 4: End of ramp detail
Figures 2 through 4 compare the options for a target ramp from 0 to 120rpm in 1sec. For clarity, step changes in speed are shown, calculated from Equation 2. The true profile should be close to a straight line.
2.c/(4.n+1) in Equation 12 could be approximated by c/2.n. Some effects would be:
- The algorithm would still produce a linear ramp.
- c_{0} would be closer to the "exact" value shown in Equation 7: 88.6% instead of 67.6% for the same ramp acceleration.
- A single equation like Equation 13 could no longer be used for both acceleration and deceleration.
From Equations 4 and 5 we can obtain an expression for the step number n as a function of speed and acceleration:
Equation 16
Thus the number of steps needed to reach a given speed is inversely proportional to the acceleration:
Equation 17
This makes it possible to change the acceleration at a point on the ramp by changing the step number n in the ramp algorithm Equation 13. Moreover, using signed ω' values results in signed n-values that behave correctly in the algorithm. Only ω' = 0 needs special handling.
The n-value given by Equation 17 is correct for t_{n}. However c_{n} represents an average for the interval t_{n} .. t_{n}_{+1}. Equation 17 is usually adequate, but it's more accurate to add a half-step to n-values for use in the ramp algorithm:
Equation 18
The numerical example shown in Table 2 changes acceleration from 10 to 5 and to -20rad/sec^{2} from step 200. Complex speed profiles can be built up piecewise in this way.
Table 2: Acceleration changes
Step i | n_{i} | ci (13) | ω' (3) | notes |
198 | 198 | 2,813.067 | ||
199 |
199 398.5 -100.25 |
2,806.008 | 10 |
10.(199+.5) = 5.(398.5+.5) = -20.(-100.25+.5) |
200 | 399.5 | 2,803.498 | 5 | |
201 | 400.5 | 2,799.001 | 5 | |
200 | -99.25 | 2,820.180 | -20 | |
201 | -98.25 | 2,834.568 | -20 |
Deceleration ramp
For a short move of m steps, where the up-ramp at ω'_{1} meets the down-ramp at ω'_{2} before max speed is reached, the step number m at which to start decelerating is, from Equation 17:
Equation 19
ω'_{1} = acceleration, ω'_{2} = deceleration (positive). Round n to integer and calculate c_{n} .. c_{m-1} using Equation 14.
In other cases, Equations 17 or 18 can be used to calculate the number of steps n2 needed to stop at deceleration ω'_{2}, given that the present speed was reached at step n1 with acceleration ω'_{1}. Round n2 to integer and calculate c_{m-n2} .. c_{m-1} using Equation 14.
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