This chapter discusses the design and modelling of a spherical flying robot. The main objective is to control its hovering and omnidirectional mobility by controlling the air mass differential pressure between two asynchronous coaxial rotors that are aligned collinearly. The spherical robot design has embedded a gyroscopic mechanism of three rings that allow the rotors’ under-actuated mobility with 3DOF. The main objective of this study is to maintain the thrust force with nearly vertical direction. The change in pressure between rotors allows to vary the rotors’ tilt and pitch. The system uses special design propellers to improve the laminar air mass flux. A nonlinear fitting model automatically calibrates the rotors’ angular speed as a function of digital values. This model is the functional form that represents the reference input to control the rotors’ speed, validated by three types controllers: P, PI, and PID. The robot’s thrust and induced forces and flight mechanics are proposed and analysed. The simulation results show the feasibility of the approach.
- flight mechanics
- flying control
- robot modelling
- thrust force
Nowadays, unmanned aerial vehicle (UAV) robots are being deployed at an increased rate for numerous applications falling into a variety of engineering fields. There exist numerous kinds of rotary-wing-based robotic technologies in particular with active devices. Robots with rotating wings are capable to self-control over lift, propulsion, landing, hovering And take-off tasks [1–4]. Overcoming vertical flight (with minimum energy cost) is fundamental to accomplish autonomous precise tasks. One fundamental aspect in controlling and designing rotary-wing-based intelligent machines is to consider under-actuated issues to reduce the number of actuators. Under-actuated flying robots perform motion tasks more naturally, taking advantage of the inertial and gravitational forces, consequently, reducing the use of electrical energy. Biological flying birds are instances of extremely efficient under-actuated bodies. Therefore, in order to design flying machines with a reduced number of actuators, it is essential to model and understand the mechanical nature of the robot mechanics, the fluids and their physics-based relationship.
The present work has foundations on the prototype of a home-made spherical aerial robot. Some experiments can be viewed at https://www.youtube.com/watch?v=rrGH1Oh_beM. Nevertheless, the purpose of this chapter is not on showing and discussing experimental results, but on mathematically sustaining the hypothesis of the robot’s flight mechanics and control. Unlike, known spherical design approaches [5–7], rather than deploying aileron-like propellers, we proposed yaw, pitch and roll changes through under-actuation exerting an inner gyroscopic mechanism. In the present chapter, the authors are particularly interested in disclosing the physical model of a dual rotary-wing spherical robot with an under-actuated gyroscopic mechanism. The model has been divided into four major areas: the robot’s flight mechanics with direct and inverse solution, the thrusting or induced force model, the rotors control model and a proportional integral derivative (PID) based control with non-stationary reference values.
This chapter is organised as follows. In Section 2, the design and mechanical aspects of the aerial robot are presented. Section 3 presents the kinematic direct and inverse solutions of the flight mechanics. In Section 4, the acceleration components and forces involved in the robot’s aerodynamics are discussed. Section 5 presents the robot’s thrusting force model that involves two collinear induced forces. Section 6 presents the rotors’ actuator speed models that are proposed from empirical measurements, and subsequently, the analytical solution is obtained. In Section 7, the actuators’ feedback linear control is described. Finally, in Section 8, conclusions are drawn.
2. Spherical gyroscopic robot
3. Flight mechanics model
Flight mechanics refers to the study of geometry of flight of a heavier-than-air aircraft, considering aerodynamic aspects. Expressions (1)–(3) model the three-dimension robot’s Cartesian kinematic components that describe its motion. The components, namely, x, y and z, are the space positions w.r.t. the location of the robot’s starting flight. The proposed kinematic model is constrained with initial posture as the inertial frame origin, where d is the distance between the robot’s instantaneous 3D position and its Cartesian origin. Azimuth angle ϕ0 is w.r.t. the plane XY, and the elevation angle ϕ1 is w.r.t. the Y-axis:
The y component (vertical) is expressed as
And, the z component is expressed as
The vertical component is expressed as
And, the z component is expressed as
Figure 2 centre depicts the robot’s flying space, which is spherical with the Cartesian origin at robot’s starting flying task.
Expressing in the matrix form, the first-order derivative dp/dt w.r.t. time is
By simplifying, the linear equation of direct kinematic is denoted by the Jacobian matrix J and the first-order vector of independent variables,
In order to obtain a recursive functional form equivalent to previous state variables, the derivatives are expressed in the following manner:
Time differentials are eliminated, and the integration operators complete the remaining differentials dp and dϕ:
Thus, to solve for the robot’s Cartesian position a recursive form is obtained by algebraically reordering. Next, robot’s position pt+1 is obtained by successive approximations of ϕt until ϕf:
In addition, the kinematic inverse solution requires the inverse-squared Jacobian matrix, assuming that it is an invertible and non-singular matrix, with non-zero determinant:
Therefore, the first-order inverse kinematic is obtained by an algebraic approach:
As described earlier, we complete the differentials dp and dϕ:
By integrating both sides of the equality, respectively, a recursive inverse solution in terms of the rotor’s angular speed is obtained,
where pt is the actual robot 3D position and pf represents the final desired position in space. To achieve such location, the robot recursively approximates the next desired rotor’s controlled velocity. The next figure depicts a simulation result where the aerial robot successively approached the final desired position, staring from the Cartesian origin.
4. Aerodynamic robot’s model
The aerodynamic robot’s model refers to the application of the Newton’s second law of motion in three dimensions to infer the thrusting force T and other involved forces that produce the Cartesian accelerations:
In the matrix form, the following equation represents the direct kinematic solution for the Cartesian accelerations:
From the previous expression, the vector acceleration is substituted into the Newton’s second law of motion,
where the Cartesian force components defined in robot’s local inertial frame are expressed as follows:
The force component along the robot’s local y component is expressed as
And, the force component along the robot’s local Z component is expressed as
By simplifying the Cartesian force components in the matrix form
And, by substituting the functional form of the vector force f into the Newton’s second law,
Thus, by reordering the previous equation, we substitute the vector constraints wT =(C1S0, S1, C1C0). The acceleration vector d2p/dt2 is a function of the next position pt+1 and the rotors variables:
By dropping off the induced robot’s force T,
So far, in this expression, the total thrusting induced force T represents the robot’s global flying force. Thus, T is an arithmetic result produced by the sum of the top rotor’s induced force T1 and the below rotor’s induced force T2 according to the following governing constraints:
For T1 = T2, the inflow air mass is same throughout both rotors, hence T = T1 + T2.
For T1 > T2, speed and air mass below rotor 1 are greater than rotor 2 inflow, T = T1 + α2 T2.
For T1<T2, opposed to constraint (b), then T2 = α2T1 and T = T1(1+α2).
Here, the numerical factors α1,2 are gains denoting rotors’ speed-rate differences. For either constraint (b) or (c), the gyroscopic mechanism angles’ tilt and pitch are affected, consequently changing the robot’s azimuth and elevation angles.
5. Induced force model
According to the depictions of Figure 3, the rotors are continuously pushing the air down. As per Newton’s third law, an equal and opposite reaction force, denoted as rotor thrust, is acting on the rotor due to air. The induced force model refers to the thrusting force exerted to accelerate the robot. And at a constant velocity the quasi-static hovering is achieved [1–4].
The momentum conservation is obtained by relating the induced force T2 to the rate of momentum change. It is the mass rate and the far-field wake-induced velocity vw below rotor 2, where dm/dt=ρAv2 and the rotor disk area A = πR2. Thus, the moment conservation
The energy conservation per unit time
To obtain a relationship between v and vw, let us substitute T and dm/dt,
Hence, substituting vw into T,
Dropping off v, the following expression is obtained;
The propulsive power Pw is the thrusting force T capable to move the robot at a given velocity (distance over time):
The induced power per unit thrust for a hovering rotor can be written as
The above expression indicates that, for a low inflow velocity, the efficiency is higher. This is possible if the rotor has a low disk loading (T/A). Note that the parameter determining the induced power is essentially T/(ϱA). Therefore, the effective disk loading increases with an increase in altitude and temperature.
From previous analysis, let us now precisely define the thrusting force for rotors 1 and 2, according to Figure 3 (left side). For rotor 1, the air mass flow is
Hence, considering only rotor 1, the induced force is
The energy conservation principle for rotor 1 is expressed as
And finding a relationship between v1 and v2 in accordance with Figure 3 (left side),
The induced air velocity induced by rotor 1 is modelled by
Similarly, modelling both the induced force T2 and velocity v2 for rotor 2, the following analysis is developed. The airflow rate,
And the induced force T2 considers the inflow air mass and the far-field wake-induced velocity vw,
The energy conservation for the second rotor is expressed as
The relationship between far-field wake-induced air velocity vw and v3 is given by vw=2v3, and
Therefore, the second rotor’s air induced velocity is
From the three previous postulates of Figure 3, let us deduce the conditions when both rotors, although asynchronous, simultaneously induce the airflow equally, when v1=v3 (Figure 3a). For this case, the total robot’s thrusting force T=T1+T2,
For case 1, let us assume A1=A2 and v2=v3 through the second rotor’s disc area. And,
Rewriting total T as a function of v1, thus we have
The air mass variation is denoted as the mass derivative w.r.t. time,
Now, for the case v1>v3,
Let us substitute our T1 and T2 models previously analysed,
In addition, A1=A2, v2=2v1 and vw=2v3, but v2>v3 hence v3=α2v2. Thus, vw=2(α2 2(2v1)), and we obtain the relationship between the far-field wake-induced velocity and v1:
By substituting v1 into expression T, developing and factorising algebraically,
Similarly, for the case when v1<v3, T=T1(1+α2),
for this case, A1=A2, v2=2v1 and vw=2v3. However, just above and below rotor 2, v2<v3, and therefore v2=α2v3, then v3=2v2/α2.
Factorizing and algebraically arranging,
Solving the parameter α2 for each of the three cases,
Therefore, we synthesise the total thrusting force T for all cases by
Therefore, from Eq. (60), now we have a functional form for the thrusting force T. Thus, to reach a controlled rotors’ velocity, we must establish a relationship between the induced velocity v1 and the rotor angular velocity dϕ/dt using the tip speed of the rotor blade as reference. The rotor inflow is represented in non-dimensional form as
where CT is the thrust coefficient modelled by
Therefore, the following equality allow us to deduce CT with more
Thus, substituting (Texto) into v1,
And subsequently, v12 is substituted into T,
Our objective is to find an analytical solution for the rotor speed required to reach the induced velocity v1 and the induced force T, which is governed by the flight mechanics law. The induced air mass velocity v1 can be expressed in terms of the rotor speed that is controlled to obtain the desired angular velocity,
Thus, it follows a set of empirical temperature measurements where some experimental hovering experiments were carried out. The plots in Figure 4 depict how the air pressure is affected as the temperature varies over time.
6. Actuators’ speed model
The actuators’ self-calibration speed model is discussed in this section. The real rotary velocity in a range from minimal to maximal values approached a logarithmic model. It considered the empirical set of angular speed measurements w.r.t. digital controls. The inherent physical variations, such as temperature, air pressure, density and air dust particles, affected the actuators’ performance. Since the angular speed value capable to hover the spherical robot’s body is disturbed, a self-calibration is required to maintain position control as accurate as possible. From experiments, the empirical models that obtained (Figure 5) φ vs. d are fitted according to the next model. The parameters A and β are unknown and must fit the speed measurements φ, w.r.t to digital word d,
We temporally substitute d’=ln(d), and thus the rotary speed
To estimate the unknown parameter β, a linear mean-squared method is applied,
Subsequently, the previous parameter solution is used in the next expression A,
By substituting the parameters numerical values, the rotary speed model is obtained as follows:
In addition, in order to obtain the inverse solution, we algebraically drop off the variable d from Eq. (72),
In order to compare how our theoretical model fits the empirical model, both inverse and direct operation control modes are depicted in Figure 5.
7. Rotor’s speed control
From the previous section, the actuators’ speed model is now used to formulate a feedback linear control. Let us assume that a rotor control variable (i.e., angle, velocity and acceleration) should ideally be equal to the real sensed control variable, as expressed by the next equality (75). Nevertheless, in a realistic scenario real and ideal control variables are different due to a number of factors, such as frictions, inertial and gravity forces, motor electromagnetic performance and so on. Thus, both variables are approached by a multiplicative gain or factor alpha, which approximates both numerical values according to the relation
Assuming an arbitrary actual derivative order, the equation is equivalently expressed as
The time differentials are eliminated and the remaining differentials are obtained by solving the following definite integrals,
By solving the definite integrals,
In this case, is the expected or the reference value to be reached ideally, . Thus, by adjusting the times sub-interval labels, and algebraically reordering, the next recursive numerical successive approximation equation is expressed as follows:
From the previous expression, the error ket is spanned into the past (angular displacement), the actual (rotary speed) and the future (angular accelerations) errors in order to cover the whole error history. And the general constant gain k is proportional to kp, kI and kd. Thus,
Therefore, the next feedback proportional error ep (rad/s) with proportional gain kp (dimensionless) is obtained with the observation ϕt measured online. And the reference model (Texto) is established in terms of the instantaneous control word δt:
For illustrative purpose, an accelerative rotor’s task to exert robot’s propulsion was performed. Figure 6 (left) depicts both the reference model ϕref and the observation ϕ(t).
In addition, Figure 6 (right) shows the proportional error behaviour ϕref-ϕ(t) without kp. Furthermore, the feedback integral error eI (rad/s) with integral gain kI (dimensionless) is expressed by the time integration of the difference of (d2ϕ/dt2 – d2ϕ(t)/dt2).
The observation model d2ϕ/dt2 was obtained online by the numerical derivatives of the optical encoder according to the following relationship:
In addition, since the observation model inherently poses perturbations, an analytical reference model d2ϕref/dt2 was obtained using a nonlinear regressive fitting process for parameters identification,
where the previous expression is similarly expressed as
And by solving vector x,
Hence, the reference model is a theoretical nonlinear function of time,
Therefore, for the sake of the integral control uI (rad/s), the angular acceleration reference model (Texto) is substituted next in its general form:
Finally, in order to keep data homogeneity (numerical data subtraction), time integration is obtained by the trapezoid rule for numerical integration,
In addition, the derivative control ud=kded (rad/s) with feedback derivative error ed, and with derivative gain kd (dimensionless), improves the closed-loop stability as follows:
In order to obtain the time derivative observation model, the rotor’s angle evolution ϕ(t) (rad) is observed online using an optical encoder during the time slot where velocity and acceleration are also measured. As the measurements are read with noise, the analytical reference model is fitted as a nonlinear polynomial of the following form:
Figure 8 (left) depicts both reference ϕref(t) and observation ϕ(t) models together. Although both curves are apparently fitted, the vertical scale is provided in thousands of radians. Figure 8 (right) shows the derivative error scale.
Generally, the PID controller is expressed by the next expression
Therefore, the controlled rotor’s velocity that is recursively calculated by dφt+1/dt= dφt/dt + ut is applied, and we established the following controller choices: proportional (P), proportional integral (PI) and proportional-integral-derivative. Figure 9 (left) depicts the rotors’ angular speed without control and with three types of controllers. The constant parameters were adjusted accordingly to obtain such results. We can see that after 25 s the responses P and PI gradually converge w.r.t. the raw rotor’s speed (Figure 9, left).
In addition, with the controlled rotor’s speed output, the induced force is iteratively calculated by
Thus, Figure 9 (right) depicts the induced component forces that are produced using three types of controllers.
This work briefly introduced the design of an aerial spherical robot with under-actuated gyroscopic mechanism. Although the purpose of this study was not to describe the robot flying and physical capabilities, the main objective was to demonstrate the induced force model deploying dual collinear rotary wings with no steering actuators. This study describes the following four major areas: the robot flight mechanics, the model of the induced thrusting forces, the self-calibration actuators’ Speed model validated with three types of controllers (P, PI, PID) to drive the rotors’ motor speed. The proposed aerodynamic mechanism poses neither ailerons nor propellers for steering control. Although the platform is a home-made laboratory prototype with special arrangements, the main focus of this chapter is to model the hypothesis of controlling the robot’s directions by varying rotors’ asynchronous speed. To achieve this, the under-actuated gyroscopic mechanism provides the ability to control its inner yaw, pitch and roll angles. Until this stage, the robot development is under an early control capability. However, this study presents mathematical solutions and simulation results to demonstrate the proposed aerodynamic hypothesis.