Theory corner

The control concepts behind the driver's modes, for readers who want to know why a setting does what it does. Nothing here is needed to use the driver; the chapters say what to type.

Field-oriented control of a brushless motor

A brushless motor has three stator windings and a rotor of permanent magnets with \(p\) pole pairs: one mechanical revolution is \(p\) electrical cycles, 11 for a typical 5208 gimbal motor. The torque the rotor receives depends on the angle between the stator field and the rotor field, the load angle; it is largest when the stator field stands a quarter of an electrical cycle ahead of the rotor and zero when the two are aligned. A hobby ESC does not measure the rotor angle: it steps the field around in six fixed states and infers the rotor's position from the back-EMF of the winding it is not driving, which works at speed and fails at standstill. That is why an ESC cannot hold a position or produce a controlled torque at low speed.

Field-oriented control (FOC) measures the rotor angle instead. The encoder on the motor shaft gives the mechanical angle \(\theta\); with the pole pairs \(p\) and the angle offset \(\theta_0\) found by calibration, the electrical angle is \(\theta_e = p\,\theta + \theta_0\). The three phase currents, which sum to zero, are reduced to two (the Clarke transform) and then expressed in the rotor's frame of reference (the Park transform):

\[ I_\alpha = I_U,\qquad I_\beta = \frac{I_U + 2\,I_V}{\sqrt{3}} \]

\[ I_d = I_\alpha \cos\theta_e + I_\beta \sin\theta_e,\qquad I_q = -I_\alpha \sin\theta_e + I_\beta \cos\theta_e \]

\(I_q\), the quadrature current, is the part of the current that stands at right angles to the rotor field and produces torque: \(\tau = K_t\, I_q\). \(I_d\), the direct current, is aligned with the field and produces none. Two PI controllers regulate \(I_q\) to the torque demand and \(I_d\) to zero by choosing the voltages \(U_q\) and \(U_d\); the inverse transforms turn those into three phase voltages, applied by PWM at 25 kHz with space-vector modulation, which gets about 15 % more phase voltage out of the supply than a plain sine would. The loop runs at 5 kHz.

The consequences are what makes a FOC drive behave like a servo: the current is only what the torque needs, so the motor runs cool and quiet; the field is always placed where it produces the most torque per ampere; the motor produces its full torque at standstill and holds a position; and the torque is known from the current, which is what makes impedance control and force-limited motion possible.

Calibration supplies the constants this needs. The resistance \(R\) and inductance \(L\) of a winding are measured. The current loop gains follow from them and the chosen bandwidth \(\omega_c = 2\pi f_c\) (#Cbw): \(K_p = L\,\omega_c\) and \(K_i = R\,\omega_c\) cancel the winding's electrical time constant and give a first-order response at \(f_c\). The pole pairs come from an open-loop spin: the field is turned a known number of electrical cycles and the encoder says how far the rotor went. The angle offset is found by driving a current along the \(d\) axis and reading where the rotor settles. The velocity constant \(K_V\) comes from the back-EMF the spinning motor generates, and the torque constant follows from it: in SI units \(K_t = 1/K_V\) (with \(K_V\) in rpm/V, \(K_t \approx 9.55/K_V\) Nm/A).

The cascade of loops

Position, velocity and current are controlled by three loops nested inside one another:

\[ \text{position loop: } \omega_d = K_{pp}\,(\theta_d - \theta) \qquad \text{velocity loop: } I_{q,d} = K_{pv}\,(\omega_d - \omega) + K_{iv}\!\int(\omega_d - \omega)\,dt \qquad \text{current loop: } U_q = \text{PI}(I_{q,d} - I_q) \]

The position loop is proportional: its output is a speed demand, limited by the velocity limit. The velocity loop is a PI controller: its output is a current demand, limited by the current limit; the integral term removes the steady-state error a load would otherwise leave. The current loop is the innermost and fastest. Each loop must be slower than the one inside it, by a factor of five to ten, for the inner loop to look like an ideal actuator to the outer one; that is why the loops are tuned from the inside out, and why a current loop bandwidth of 500 Hz leaves room for a velocity loop of a few tens of hertz and a position loop below that.

A command picks where the cascade is entered: a current setpoint uses the current loop alone, a speed setpoint the two inner loops, a position setpoint all three. The feedforward terms the CAN motion frames carry are added to the demands rather than computed from an error: a speed feedforward is added to the position loop's output, a current feedforward to the velocity loop's output. A host that knows the trajectory can command it directly and leave the loops to correct what is left, which gives a faster, smoother response than letting the loops produce the whole demand from the error.

An integrator accumulates error over time, and keeps accumulating while the output is at its limit; the overshoot this causes when the limit is released is windup. The velocity loop's integrator is clamped to the current limit, and #Rstint resets the integrators on every new setpoint, which suits a host that sends one setpoint after another in large steps.

Impedance control

Impedance control does not prescribe a motion or a force. It prescribes the relation between the two: how much force the motor produces when it is displaced from its setpoint and how fast. The driver's PD mode implements a spring and a damper around the setpoint:

\[ \tau = K_P\,(\theta_d - \theta) + K_D\,(\dot\theta_d - \dot\theta) + K_t\, I_{ff} \]

with the stiffness \(K_P\) in Nm/rad, the damping \(K_D\) in Nm·s/rad, and a current feedforward \(I_{ff}\) for a known load such as gravity. The torque is produced by the current loop (\(I_q = \tau / K_t\)), which is what makes FOC a prerequisite: a drive without the rotor angle cannot produce a commanded torque at low speed.

Pushed off its setpoint by an external force \(F\), the motor yields by \(\Delta\theta = \tau_{ext}/K_P\) and pushes back with that force; let go, it returns, with the damper settling the motion. A small \(K_P\) makes a soft joint, a large one a stiff joint; \(K_D\) without \(K_P\) is a pure damper. The setpoint can move, in which case the virtual spring drags the rotor along a trajectory with a bounded force — the property legged robots, hand-guided arms and anything that touches people or an uncertain environment need. The impedance loop is also what a position loop is not: the position loop's integral action removes any steady-state deflection and so makes the joint as stiff as the current limit allows, which is the right thing for a machine tool and the wrong thing for a leg.

The full impedance model includes an inertia term, \(M\ddot\theta\); the driver, like most joint controllers, leaves it out, so the apparent inertia is the motor's own.

Admittance control

Admittance is the inverse relation: the controller measures the force and commands a motion,

\[ \dot\theta_d = \frac{\tau_{ext} - K\,(\theta - \theta_0)}{B} \]

on top of a stiff position controller. It is the natural form for a machine that has a force sensor and a position loop that cannot be made soft, such as an industrial arm; impedance control is the natural form for a machine whose actuator is a torque source, such as a FOC motor. The driver has no force sensor and implements impedance; a host that wants admittance behaviour can estimate the external torque from \(K_t I_q\) minus the known load and command positions accordingly, with the limits of that estimate (friction, cogging, the current loop's accuracy).

Compliance and stiffness

Stiffness \(K\) is the force per unit of displacement; compliance is its inverse, \(C = 1/K\), the displacement per unit of force. Passive compliance is in the mechanism — a spring, a flexible finger, a belt; active compliance is in the controller — the impedance loop above. The driver offers three points on that scale: the position mode, stiff up to the current limit; the PD mode, as stiff as \(K_P\); and the current mode, which is no stiffness at all, a torque source the host shapes as it likes.

The current limit #Ilim is a force limit in every mode: the largest torque the motor will produce before it yields, whatever the loops ask. Set it for the motor's rating, and for the mechanism's.

Which mode

Mode	Behaviour under an external force	Use
Position	holds the position up to the current limit, then yields	positioning, machines, anything that must not deflect
Velocity	holds the speed up to the current limit	wheels, conveyors, spindles
PD (impedance)	deflects in proportion to the force, returns when released	legs, arms that touch things or people, hand guiding, teleoperation
Current (torque)	produces the commanded torque, moves as the load dictates	a host that closes its own loop, gravity compensation, force-limited pressing

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