# Motor Control¶

This section describes the principles of motor control at a low level.

## Electrical Commutation¶

Electrical commutation is the process of generating rotating magnetic fields in an electromagnetic system. In electric motors this involves the generation of AC and DC signals. Some simple motors may do this mechanically, however for precision applications such as robotics and manufacturing it is done with digital control systems.

Many types of motors may be controlled with electrical commutation:

• Brushless DC

• Induction

• Stepper

• Switched Reluctance

There are of course many innovative topologies being developed every year, and many of these use the same operating principles as the above. RAPcores aims to provide a toolkit for controlling a wide range of motor types.

The two basic elements of a motor controller are:

• Power systems

• Control systems

The power systems element is the element that handle the (relatively) high power that drives the motor. Typically MOSFETs or IGBTs are used in full or H bridge inverter configurations to generate AC voltages. It may include additional sensors such as an encoder, voltage sensor, and current sensor. All three are preferred for closed-loop control.

The control systems element is the software or digital logic system managing inputs and outputs of the power circuitry, sensors, and input trajectories. In RAPcores the low levels of this are handled in FPGA or ASIC, with a microcontroller or CPU handling high level trajectory planning. The job of the control system is to ensure the motor is following the defined trajectory or set behavior.

## Bridge Control¶

Below is a simple motor control hierarchy:

Bridge control involves the timing of output signals such that the motor power circuitry creates the desires voltage and current in each coil of the motor. In addition, it needs to be fault aware and control timing to create smooth operation. Typically high-resolution PWM signals are used at supra-audible frequencies (>35kHz).

Due to the use of PWM and commutation, the voltages are non-contant. There are some timing considerations when switching the motor:

• Off time

• On time

• Blank time

Dead time is the time required for a transistor in the bridge to turn on or off. When switching polarities, it is important the controller wait this “dead time”, or delay, before changing the direction of the inverter otherwise breakage might occur.

Off time and On time are the duration the PWM signal is on or off. The relation of these two is relevant to current and voltage measurements of the motor. Measuring voltage and current over time allows for measurement of resistance and inductance in the motor.

Blank time is related to the propogation delay of measuring the outputs to the motor. Since a digital design can react on a cycle-to-cycle basis of the PWM, delays must be accounted in elements such as ADCs and comparators before measuring then reacting to a change in the power output.

## Motor Fault Types¶

The following faults may occur in a motor:

• Disconnection

• Overcurrent

• Undervoltage

• Thermal Shutdown

• Overvoltage

A good bridge design should be able to handle the majority of these fault conditions and report back to higher level controllers for rectification by the user or path planner. In RAPcores there are fault diagnostic registers for each motor channel.

## Space Vector Modulation¶

The key to smooth and efficient motor control is a high precision electrical commutator. To achieve this, precise currents and voltages must be generated. Moreover, they must rotate the magnetic field at a wide variety of speeds and accelerations. As mentioned, PWM signals are used to control the power output and to rotate the magnetic field with PWM, we use a Space Vector Modulator.

Two elements are required for Space Vector Modulation or SVM:

• Microstepping (subdividing the commutation table)

• Current regulation (limiting power output for efficiency)

To understand the concepts and mathematics of the rotating magnetic field, we will use vectors and phasors.

### Vector Concepts¶

Here we will present the mathematical ideas of how to model a rotating magnetic field in a bipolar stepper motor as a vector. This is known as Space Vector Modulation. To start we must understand some basic concepts from vector algebra and trigonometry.

Recall that sine and cosine are derived from the components of a vector rotating around the origin:

And recall the identity where:

$cos(\theta)^2+sin(\theta)^2=1$

Or alternatively:

$A \cdot cos(\theta)^2+A \cdot sin(\theta)^2=A$

The above equation is essential in our understanding of constructing the vector. Recall the phase table for the bipolar stepper:

A

B

+

-

-

-

-

+

+

+

If we visualize each phase as a 2D plot we can see these commutation steps form the corners of a square as shown below:

Microstepping is possible in this space (a square), however it yields undesirable effects since the total current in the motor varies due to a change of vector length as shown in the red arrow above. This can yield resonance and torque ripples. The objective is to create smooth motion as we traverse between the phases. Therefore we need to move our vector along a circle, such as the arrows in blue above. The trade-off is that we do not achieve the peak torque attainable in the corners of the square, but instead the motion is smooth and controllable.

### PWM Concepts¶

Below is a simple PWM module in verilog:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22  /* Simple PWM module */ module pwm #( parameter bits = 8 ) ( input clk, input resetn, input [bits-1:0] val, output pwm ); reg [bits-1:0] accum; assign pwm = (accum < val); always @(posedge clk) if(!resetn) accum <= 0; else if(resetn) accum <= accum + 1'b1; endmodule 

We can see that the PWM output frequency is a function of the base clock frequency (clk) and the number of bits used for the accumulator. E.g.:

$F_{PMW} = \frac{F_{clk}}{2^{bits}}$

For quiet operation and fast updates we want the PWM frequency to be superaudible, so a value greater than 30khz. Assume we use a PLL to achieve a higher operational frequency for PWM module at 150mhz. The bit resolution of the PWM can be calculated thus:

$bits = \log_2({F_{PMW}/F_{clk}})$

In our example of a 30khz output with a 150mhz accumulator clock we get ~12.3 bit resolution. For simplicity we will use 12 bits going forward.

Now the challenge is how to compute the value to the PWM such that we bring both the current and the microstep/phase angle into a single value. In the next section we will see this is a relatively straight forward process that falls out of the vector model.

### Applied Space Vector Modulation¶

Recall that a vector ($$\vec{A}$$) may be element-wise scaled by a given factor such that the length ($$\left\lVert\vec{A}\right\rVert$$) is scaled by the same factor. Our vector is formed from a given angle (or microstep position) $$\theta$$ as: $$(cos(\theta), sin(\theta))$$. Then scaling the current is simply multiplication of this vector by a factor C: $$(C \cdot cos(\theta), C \cdot sin(\theta))$$. Then using the above identify we known that the length of this vector is:

$\left\lVert(C \cdot cos(\theta), C \cdot sin(\theta))\right\rVert = C$

Then the matter of partitioning the 12 bit space of the PWM become quite simple. For example we may use 8 bits for the trigonometric functions (implemented as lookup tables in practice), and 4 bits for current. Which gives sufficient precision for 64 microsteps and 16 discrete current values.

So now we can do space vector modulation. But where do we put it? The answer is as a voltage reference or gate PWM input. For example we may use this output to create a reference for a 1-bit ADC by adding a RC filter to the output in a chopper drive. Or for a dead-reckoned approach this PWM can be used to quickly turn the gate drivers on and off. An example of this can be found in the RAPcores Dual H Bridge module.

### SVM in Three Phase¶

For the mathematically inclined, you may notice that the bipolar stepper is nice as the phases form an orthonormal basis in 2D space. In three phase or more motors, this is not the case, as the vector is embedded in three or more dimensional space.

The technique used in three phase control is to project the three dimensional vector down into a two dimensional space such that we can use simpler math for the control. In addition, the inverse may be applied, bringing us from the two dimensional frame of reference to the three dimensional frame of reference for the three phase motor.

The forward transform:

$\begin{split}i_{\alpha\beta}(t) = \frac23 \begin{bmatrix} 1 & -\frac12 & -\frac12\\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}\begin{bmatrix}i_a(t)\\i_b(t)\\i_c(t)\end{bmatrix}\end{split}$

The inverse transform:

$\begin{split}i_{abc}(t) = \frac32\begin{bmatrix} \frac23 & 0 \\ -\frac{1}{3} & \frac{\sqrt{3}}{3} \\ -\frac{1}{3} & -\frac{\sqrt{3}}{3} \end{bmatrix} \begin{bmatrix}i_\alpha(t)\\i_\beta(t)\end{bmatrix}\end{split}$

## Frequency Considerations¶

Below is a table showing some relevant stepping frequencies for a stock Prusa MK3S:

Prusa MK3S

X

Y

Z

E

Full Steps/rev

200

200

200

200

Microsteps

16

16

16

32

Homing Feedrate (mm/sec)

50

50

13.3

Max travel Rate (mm/sec)

200

200

12

120

(Steps*microsteps)/mm

100

100

400

280

Microsteps/sec (travel)

20000

20000

4800

33600

Full Steps/sec (travel)

1250

1250

300

1050

Microsteps/sec (homing)

5000

5000

5333.3

Full Steps/sec (homing)

312.5

312.5

333.3

A

B

+

-

-

-

-

+

+

+

A

B

C

+

-

+

-

+

-

-

+

-

+

-

+

A

B

C

D

E

+

-

-

+

+

-

+

-

-

+

+

-

-

+

-

-

+

+

-

-

+

-

+

+

-

-

+

+

-

+

## Analytical Models¶

### Stepper Motor¶

A model of a hybrid stepper motor is given by Bodson, et. al.: [1]

$\frac{d \theta}{dt} = \omega$
$\frac{d \omega}{dt} = ( -K_m i_a sin(N_r \theta) + K_m i_b cos(N_r \theta) - B \omega - \tau)/J$
$\frac{d i_a}{dt} = ( v_a - R i_a + K_m sin(N_r \theta))/L$
$\frac{i_b}{dt} = ( v_b - R i_b - K_m cos(N_r \theta))/L$

Where:

• $$\omega$$ - Angular Velocity

• $$\theta$$ - Angular Position

• $$J$$ - Rotor and Load Inertia

• $$K_m$$ - Motor Torque constant

• $$L$$ - Coil Inductance

• $$R$$ - Coil Resistance

• $$N_r$$ - Number of rotor teeth

• $$i_a, i_b$$ - Coil Current

• $$v_a, v_b$$ - Coil Current

From this a simpler model for the current can be developed:

$i_{qr} = \frac{J}{K_m} \alpha_r + \frac{B}{K_m} \omega_r$