New Feasible Optimal Harmonic Multi-rotor Flight Trajectories

For a realistic, feasible control input of multi-rotor UAVs, we must not only consider equation (54), but also (55) and (58), so the designed UAV path has to be such that the displacement pop (𝝃⁽⁶⁾) must be continuous and the body snap transient has to be feasible by a BLDC: 𝝃(𝑡)⁽⁴⁾~𝝎(𝑡).

As described in [s6], [s7], [s13], [s14] to have realistic, feasible torques along a trajectory, which are efficiently controllable without chattering, we need smooth torque changes. The term (n times) smooth is used as in being equivalent to having continuous (n^th) time derivative.

THESIS IV.b - DEFINITION:

For a realistic, feasible multi-rotor trajectory one must design realistic, feasible control system inputs, such that the planned body displacement sixth time derivative, the pop (𝜉⁽⁶⁾) must be continuous and proportional to the third derivative of the motor shaft rotational displacement as in equation (62).

For any trajectory we have a target position and generally a limit to the maximum velocity ξ⁽¹⁾ < V_max for safety. Usually there is also a limit | ξ⁽²⁾| < A_max to the acceleration and deceleration, too – either for power source capacity, constructional integrity or passenger safety reasons. The jerk is to be bounded| ξ⁽³⁾| < J_max as it has already been concluded by many researchers either to reduce structural vibrations or just for passenger comfort.

The important message of the proposal of this paper is that we also must not overlook a simple physical constraint of the actuator motor either: the rotation speed of the rotor is bounded – this limit in case of a multi-rotor vehicle is equivalent to limiting the trajectory displacement crackle | ξ⁽⁵⁾| < C_max.

As already highlighted all currently named “optimal” trajectory planning methods, be it minimum time, acceleration bang-bang, minimum energy or fuel, minimum jerk, minimum torque or minimum acceleration trajectories, they all suffer from the same physical infeasibility issue: existence of supposed discontinuity in the second or third derivative of the displacement, which translates to a discontinuity target in the actuator rotor position time derivatives of second or third order. In case of a multi-rotor vehicle not even a minimum snap trajectory qualifies as physically feasible sound trajectory – in terms of targeting continuous actuator actions up to the third time derivative of the actuator rotor displacement.

It is easy to overlook these issues as we have the power of the feedback controller at our disposal. A well-tuned strong, fast feedback loop efficiently copes with system identification deficiencies and unmodelled, random perturbations, so we readily use them - and by the way we mask our physically infeasible trajectories. This is so natural an approach that the most basic tool for measuring the performance of a controller is the step response function. The goal is usually to have the fastest response with just a small overshoot, which induces only limited system vibrations lasting for a planned settling time. But we have to pay the cost of extra energy used for control – a luxury that we cannot allow for battery powered flying bodies.

The important message of the proposal of this paper based on [s6] is that we must not overlook the physical capabilities, constraint of neither the system nor the actuator itself. For multi-rotors their body torques and matching rotation speed of rotors and their transient behaviour is limited – these constraints are proportional to properties of the trajectory displacement snap 𝒔(𝑡) = 𝝃⁽⁴⁾(𝑡). The snap is required to be 2 times smooth, equivalent to pop 𝒑(𝑡) = 𝛏⁽⁶⁾(𝑡) being continuous. Also by equation (61) the

transient behaviour of rotor 𝝎_𝒕(𝑡) rotation speed has to be proportional to 1 + 𝑡𝑎𝑛ℎ (^𝜋_𝑃(𝑡 −^𝑃₄)).

The proposal of this paper is to use for the snap transient a base function in the form of:

(5) 2𝜋

where 𝑃 is the design parameter responsible for the trajectory duration and also the acceleration and deceleration, too – either for power source capacity, constructional integrity or passenger well-being reasons. For advanced projects also the jerk is to be limited| ξ⁽³⁾| < 𝐽_𝑚𝑎𝑥 as it has already been concluded by many researchers either to reduce structural vibrations or just for passenger comfort.

THESIS IV.c - DEFINITION:

A realistic, feasible multi-rotor trajectory parametrisation of continuous body displacement sixth time derivative pop (𝜉⁽⁶⁾), such that the snap (𝜉⁽⁴⁾) is proportional to the motor shaft rotational velocity as in equation (62) can be designed by selecting:

𝒑_𝒕(𝑡) = 𝝃_𝒕⁽⁶⁾(𝑡) = 𝐺 ∙^2𝜋_𝑃 sin (^2𝜋_𝑃 𝑡). (65)

𝒄_𝒕(𝑡) = 𝝃_𝒕⁽⁵⁾(𝑡) = 𝐺 ∙ (1 − 𝑐𝑜𝑠 (^2𝜋_𝑃 𝑡)), (64)

where P is either measured, as in equation (62), or calculated based on equation (63) 𝝎_𝒕(𝑡) =^{𝜔𝑠𝑡𝑎𝑡}₂ (1 + 𝑡𝑎𝑛ℎ (^𝜋_𝑃(𝑡 −^𝑃₄))), (62)

𝑃 =^2𝜋_𝐴 = 2𝜋√(^𝐶_𝐵²₂+^𝐵𝐷_𝐵2𝐸)⁻¹= ^{2𝜋𝐿𝑒(𝐽𝑀+𝐽𝑅)}

√(𝑅_𝑒(𝐽𝑀+𝐽_𝑅)+𝐿𝑒𝛾_𝑀)²+𝐸𝐿_𝑒(𝐽𝑀+𝐽_𝑅)𝐿𝑒𝐾_{𝑑 (63)} The proposal of this paper is to use for multi-rotors a parameterised single sinus wave 𝒑(𝑡) = 𝐺^2𝜋_𝑃 sin (^2𝜋_𝑃 𝑡) as the base function for the displacement pop to reach the desired smooth crackle as 𝒄(𝑡) = ∫ 𝒑(𝑡)𝑑𝑡 = 𝐺 (1 − 𝑐𝑜𝑠 (^2𝜋_𝑃 𝑡)), which is of transient characteristics physically feasible to match by a BLDC motor. P is the period of 𝒑(𝑡) and by this it must match the dynamics of the actuated system. G can be an arbitrary positive real value, which controls the amplitude of the pop base function and thus trajectory displacement length. The integral of a full period 𝒄(𝑡) for t=1..P is to be used for the ascending part of the jerk function 𝒋⁺(𝑡) = ∫ 𝒄(𝑡)𝑑𝑡, for simplicity we take 0 for the integral constant value. For 𝒋⁻(𝑡) descending part of jerk the integral of –c(t) is taken. In case that the acceleration 𝒂(𝒕) = ∫(𝒋⁺(𝑡) + 𝒋⁻(𝑡 + 𝑃))𝑑𝑡 does not reach the desired level, a constant 𝒋_𝒎𝒂𝒙 interval is to be inserted between 𝒋⁺ and 𝒋⁻ intervals. The velocity is planned in an analogous manner, by integrating the rising acceleration and the falling deceleration interval, with optional inclusion of a constant acceleration interval to reach the desired maximum velocity, all this without overshooting the reached acceleration limit. By keeping the velocity constant in the middle of the

trajectory we ensure feasible time optimally reaching the desired displacement without exceeding the speed limit.

5.2.3 Implementation of the New Feasible Optimal Harmonic Multi-rotor