thesis.md

Combining heading and course

What is actually used as psi_prime

This controller is controlling $\psi'$, a linear combination of heading and course defined as:

$$ \psi' = \psi + w_\chi \cdot {\mathrm{wrap}}_\pi(\chi - \psi) \tag{1} $$

where $\psi$ is the heading, $\chi$ the course, and ${\mathrm{wrap}}\pi(\cdot)$ maps an angle onto $[-\pi, \pi)$. The wrapping is not cosmetic: a plain convex combination $(1-w\chi)\psi + w_\chi\chi$ would sweep the long way round whenever the two angles straddle the $\pm\pi$ branch cut. Written this way, $\psi'$ is a pure heading signal at $w_\chi = 0$, a pure course signal at $w_\chi = 1$, and the shortest-arc interpolation between them in between.

The blending weight

Unlike the reference formulation below, the weight is not scheduled on the angular velocity in the small-earth projection, but directly on the kite speed $v_k = \lVert {\mathbf{v}}_\mathrm{kite} \rVert$:

$$ w_\chi = \mathrm{clamp}\left( \frac{v_k - v_{k,\psi}}{v_{k,\chi} - v_{k,\psi}},\ 0,\ 1 \right) \tag{2} $$

with $v_{k,\psi} = 5,\mathrm{m/s}$ (v_kite_heading) and $v_{k,\chi} = 10,\mathrm{m/s}$ (v_kite_course). At or below $5,\mathrm{m/s}$ the loop feeds back heading alone, at or above $10,\mathrm{m/s}$ course alone, and the two are blended linearly in the band between. The motivation is the same as in the reference: the course estimate is derived from the kite's motion in the small earth projection and is meaningless when the kite is nearly stationary, while the heading is always observable.

Scheduling on translational speed rather than on angular velocity was chosen because the quantity that actually degrades the course estimate is how far the kite moves per sample, not how fast it turns. A kite flying straight and fast has a perfectly good course estimate but $\omega \approx 0$, and the $\omega$-schedule would wrongly fall back to heading there.

Course sign convention

The course taken from SysState is corrected by $\pi$ before it enters (1), in calc_steering:

course_shifted = wrap2pi(course + ccs.course_offset)   # course_offset defaults to π
psi_prime = heading + w_course * wrap2pi(course_shifted - heading)

The raw tangent-frame course has its zero pointing away from zenith, whereas the bearing convention used everywhere else in the guidance is 0 = towards zenith, positive towards larger azimuth. The $+\pi$ puts course and heading on the same zero and the same sign, which is a precondition for (1) being meaningful at all — blending two angles measured in different frames would produce a signal that is neither.

Optional bypass on the pattern

Path following is a course problem, so from phase 3 (transition and figure eight) onwards the speed schedule can be bypassed (fig8_pure_course) and the course fed back at any speed:

$$ w_\chi = 1 \quad \text{if bypass enabled} \wedge \text{phase} \ge 3 \tag{3} $$

The reason is that on the pattern the schedule only dips into the blending band during the slow part of a turn, i.e. it swaps the feedback signal mid-manoeuvre, which is exactly when a consistent signal matters most. The entry phases (dive and hold) always keep the schedule, since the kite genuinely is slow there. With the shipped settings (data/fc_settings.yaml, data/fc_settings_reelout.yaml) fig8_pure_course is false, so the speed schedule governs in every phase.

Use in the loop

$\psi'$ is the regulated variable, not the setpoint: the heading PID is driven by the wrapped error against the commanded course $\chi_\mathrm{cmd}$,

$$ e = {\mathrm{wrap}}_\pi(\psi' - \chi_\mathrm{cmd}) \tag{4} $$

and regulated against a zero reference (DiscretePID does not wrap, so the error is formed outside it). The proportional gain is scheduled on apparent wind speed, $K \sim 1/v_a$, because the plant turn rate is $\dot\psi = c_1 v_a u_s$.

In the logs, $w_\chi$ is recorded as var_08 (0 = heading, 1 = course) and $e$ as var_06, so var_06 equals heading - bearing at low speed and course - bearing at high speed.

Relation to the reference formulation

The form given in (1) and the $\mathrm{atan2}$ fusion of (6.17) below are two different interpolations between the same two angles: (6.17) normalises the convex combination of the two unit vectors (an nlerp), whereas (1) interpolates the angle itself (a slerp). They agree exactly at $w_\chi \in {0, 0.5, 1}$ and differ only slightly in between — below $0.15°$ for a $30°$ gap between heading and course, and about $4°$ at a $90°$ gap. For the V3's typical course-minus-heading drift angle of ${\sim}13°$ the difference is under $0.02°$, i.e. far below the noise on either input. Form (1) was preferred because it makes the endpoints and the clamp behaviour obvious by inspection.

The two remaining differences from the reference are deliberate:

	Reference (6.16/6.17)	This package
Scheduling variable	angular velocity $\omega$ in the small earth projection	kite speed $\lVert{\mathbf{v}}_\mathrm{kite}\rVert$
Weight limit	$k_\chi \le 0.85$	$w_\chi \le 1$
Interpolation	nlerp via $\mathrm{atan2}$	slerp via wrapped difference

The $0.85$ cap exists in the reference to stop the kite getting too slow when flying figures of eight at very high elevation. It is not carried over here: the elevation of the pattern is set by the guidance (the lemniscate and, during entry, the descent limiter entry_chi_max) rather than by leaving a residual heading component in the feedback signal, so the effect the cap guarded against is addressed upstream of $\psi'$.

Just as reference, not used in this form for this package

The accuracy of the controller can be improved further, if not only the heading angle is controlled in a feedback loop, but also the course angle: A difference between the course and the heading angles is induced (i) by the gravity forces and (ii) by wind turbulences. On the other hand the course of the kite can only be controlled when it is moving. For controlling a kite, that is moving only very slowly in the small earth projection the heading angle the must be used. Therefore a linear parameter varying (LPV) controller is used, that uses mainly the course angle at high angular velocities ($\omega > 2.275\omega_{up}$) and only the heading angle, for $\omega < \omega_{up}$ . Both angles are fused according to the following algorithm: $$ k_\chi = \begin{cases} \min\left(\dfrac{\omega - 0.8}{1.2},\ 0.85\right) & \text{if } \omega > \omega_{up} \[4pt] 0.0 & \text{else} \end{cases} \tag{6.16} $$

$$ \begin{aligned} x &= \sin\psi,(1 - k_\chi) + \sin\chi, k_\chi \\ y &= \cos\psi,(1 - k_\chi) + \cos\chi, k_\chi \\ \psi' &= \mathrm{atan2}(x, y) \end{aligned} \tag{6.17} $$

The value of $k_\chi$ is limited to 0.85. This avoids the kite to get too slow when flying figures of eight at a very high elevation angle. The block diagram of this controller is shown in Fig. 6.9.