Nonlinear control for ground-air trajectory tracking by a hybrid vehicle: theory and experiments⁎

10th IFAC Symposium on Intelligent Autonomous Vehicles 10th Symposium on Autonomous 10th IFAC IFACPoland, Symposium on Intelligent Intelligent Autonomous Vehicles Vehicles Gdansk, July 3-5, 2019 10th IFAC Symposium on Intelligent Autonomous Vehicles Gdansk, Poland, Poland, July July 3-5, 3-5, 2019 2019 Available online at www.sciencedirect.com Gdansk, Gdansk, July 3-5, 2019 10th IFACPoland, Symposium on Intelligent Autonomous Vehicles Gdansk, Poland, July 3-5, 2019

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IFAC PapersOnLine 52-8 (2019) 19–24

Nonlinear control for ground-air trajectory Nonlinear Nonlinear control control for for ground-air ground-air trajectory trajectory tracking byfor a hybrid vehicle: Nonlinear control ground-air trajectory tracking by a hybrid vehicle: tracking by a hybrid vehicle:  theory anda experiments tracking by hybrid vehicle:  theory theory and and experiments experiments  theoryazquez and∗ experiments J. Colmenares-V´ P. Castillo ∗∗ N. Marchand ∗

∗∗ ∗∗ Colmenares-V´ a zquez ∗∗∗ P. Castillo Marchand ∗∗∗ ∗∗ N. Colmenares-V´ a P. N. Colmenares-V´ azquez zquez P. Castillo Castillo N. Marchand Marchand D. Huerta-Garc´ ıa ∗∗∗ ∗∗∗ ∗∗∗ ∗ ∗∗ D. Huerta-Garc´ ıa ∗ ∗∗ D. Huerta-Garc´ ıa ∗∗∗ N. Marchand ∗∗ Colmenares-V´ azquez P. Castillo D. Huerta-Garc´ ıa ∗∗∗ ∗∗∗ ∗ D.CNRS, Huerta-Garc´ ıa INP, Univ. Grenoble Alpes, Grenoble GIPSA-Lab, 38000 ∗ ∗ Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-Lab, 38000 ∗ Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-Lab, 38000 38000 Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-Lab, Grenoble, France (e-mail: [email protected], ∗ Grenoble, France (e-mail: [email protected], ∗ Univ. Grenoble Grenoble, France (e-mail: [email protected], Alpes, CNRS, Grenoble INP, GIPSA-Lab, 38000 Grenoble, France (e-mail: [email protected], [email protected]). [email protected]). [email protected]). ∗∗ Grenoble, France (e-mail:e de [email protected], [email protected]). Sorbonne Universit´ ees, Universit´ Technologie de Compi`eegne, ∗∗ ∗∗ Sorbonne Universit´ s, Universit´ eee de Technologie de ∗∗ Sorbonne Universit´ e s, Universit´ de Technologie de Compi` Compi`eegne, gne, [email protected]). Sorbonne Universit´ e s, Universit´ de Technologie de Compi` gne, Heudiasyc Lab., UMR 7253, France (e-mail: [email protected]) ∗∗ Heudiasyc Lab., UMR 7253, France (e-mail: [email protected]) ∗∗ Heudiasyc Lab., UMR 7253, France (e-mail: [email protected]) ∗∗∗ Sorbonne Universit´ e s, Universit´ e de Technologie de Compi` e gne, Heudiasyc Lab., UMR 7253, France (e-mail: [email protected]) Instituto Tecnol´ o gico de Tlalnepantla, Electrical Department, State ∗∗∗ ∗∗∗ Instituto Tecnol´ o gico de Tlalnepantla, Electrical Department, State ∗∗∗Heudiasyc Instituto Tecnol´ o gico de Tlalnepantla, Electrical Department, State Lab., UMR 7253, France (e-mail: [email protected]) Instituto Tecnol´ o gico de Tlalnepantla, Electrical Department, State of Mexico (e-mail: [email protected]) ∗∗∗ of Mexico (e-mail: [email protected]) ∗∗∗ of Mexico (e-mail: [email protected]) Instituto Tecnol´ o gico de Tlalnepantla, Electrical Department, State of Mexico (e-mail: [email protected]) Abstract: A nonlinear scheme for controlling a hybrid vehicle with capacities for terrestrial and of Mexico (e-mail: [email protected]) Abstract: A scheme a hybrid vehicle capacities for and Abstract: A nonlinear nonlinear scheme for forincontrolling controlling hybrid vehicle with with capacities for terrestrial terrestrial and Abstract: A nonlinear scheme for controlling hybrid vehicle with capacities for terrestrial and aerial displacements is proposed this paper.aa The nonlinear model of the vehicle is obtained aerial displacements is proposed in this paper. The nonlinear model of the vehicle is obtained aerial displacements is proposed in this paper. The nonlinear model of the vehicle is obtained Abstract: A nonlinear scheme forin controlling a The hybrid vehicle with capacities for terrestrial and aerial displacements is formalism proposed this into paper. nonlinear model of the vehicle is air). obtained from the Newton-Euler taking account two operation modes (ground and The from the the Newton-Euler formalism taking into account two operation operation modes (ground and air). The from Newton-Euler formalism taking into two modes (ground and The aerial displacements proposed in this paper. The model of the vehicle is air). obtained from the Newton-Euler formalism taking into account two full operation modes (ground and air). The attitude controller hasisbeen developed step by account step for nonlinear the complete system and stabilizes both attitude has been developed step by step for the full complete system and stabilizes both attitude controller has developed step by step for full complete system and both from the controller Newton-Euler formalism into two operation modes (ground and The attitude controller has been been developed step by step for the the full complete system andItstabilizes stabilizes both operation modes with the goal totaking produce a account seamless transition between them. has air). adaptive operation modes with the goal to produce a seamless transition between them. It has adaptive operation modes with the goal to produce a seamless transition between them. It has adaptive attitude controller has been developed step by step for the full complete system and stabilizes both operation modes with the goal to produce a seamless transition between them. It has adaptive properties that helps to counteract the effects produced by the nonlinear uncertainties in the properties modes that helps helps to counteract the effects effects produced by the the nonlinear nonlinearthem. uncertainties in the the properties to the produced by uncertainties in operation with position thecounteract goalcontrollers to produce a conceived seamless transition It has properties that helps to counteract the effects produced by the between nonlinear uncertainties in the model. Thethat nonlinear are considering the gravity force for adaptive tracking model. The Thethat nonlinear position controllers are conceived conceived considering the gravity gravity force for for tracking tracking model. nonlinear position controllers are considering the force properties helps to counteract the effects produced by the nonlinear uncertainties in the model. The position controllers conceivedand considering gravity force tracking an aerial or nonlinear ground trajectory. Numerical are simulations real-timethe experiments are for carried out an aerial or ground trajectory. Numerical simulations and real-time experiments are carried out an aerial or ground trajectory. Numerical simulations and real-time experiments are carried out model. The nonlinear position controllers are conceived considering the gravity force for tracking an or ground trajectory. simulations and real-time experiments are carried out for aerial validating and verifying the Numerical good performance of the proposed algorithms. for validating and the good of proposed algorithms. for aerial validating and verifying verifying the Numerical good performance performance of the the proposed algorithms. an or ground trajectory. simulations and real-time experiments are carried out for validating and verifying the good performance of the proposed algorithms. Keywords: non-linear control, vehicle, tracking, adaptive © 2019, IFAC (International Federation Automatic Control) Hosting byalgorithms. Elsevier Ltd.algorithm All rights reserved. for validating and verifying thehybrid good ofperformance of thestabilization, proposed Keywords: non-linear control, hybrid vehicle, tracking, stabilization, Keywords: non-linear non-linear control, control, hybrid hybrid vehicle, vehicle, tracking, tracking, stabilization, stabilization, adaptive adaptive algorithm algorithm Keywords: adaptive algorithm Keywords: non-linear control, hybrid vehicle, tracking, stabilization, adaptive algorithm terrestrial mode it consumes significantly less energy than 1. INTRODUCTION terrestrial mode it consumes significantly less energy than 1. INTRODUCTION terrestrial mode it significantly less than 1. INTRODUCTION mode it consumes consumes significantly less energy energy than 1. INTRODUCTION in the aerial mode. In Dudley et al. (2015), the authors As a result of the 1. technological progress, the autonomous terrestrial in the aerial mode. In Dudley et al. (2015), the authors authors in the aerial mode. In Dudley et al. (2015), the As a result of the technological progress, the autonomous terrestrial mode it consumes significantly less energy thana INTRODUCTION in the aerial mode. In Dudley et al. (2015), the authors present a hybrid vehicle based on a quadrotor inside As a a result of become the technological technological progress, the thenow autonomous As result of the progress, autonomous vehicles have more sophisticated, they are present a hybrid vehicle based on a quadrotor inside a present aa hybrid vehicle based quadrotor inside the aerial mode. In Dudley eton al.aa (2015), the authors vehicles have become more sophisticated, sophisticated, now they are are in present hybrid vehicle based on quadrotor inside a lightweight spherical exoskeleton. This is a small robot anda vehicles have become more now they As a result of the technological progress, the autonomous vehicles have more sophisticated, theyspots are lightweight capable to dobecome more complex tasks and tonow reach lightweight spherical exoskeleton. This is a small robot and spherical exoskeleton. This is a small robot and capable to do more tasks and to reach spots a hybrid vehicle based on inside lightweight spherical exoskeleton. This isquadrotor a smallinspections, robot anda its portability made it interesting fora pipeline capable to dobecome more complex complex tasksthat andwere tonow reach spots vehicles have more sophisticated, they are present capable do more complex tasks and to reach spots with poorto accessibility. Applications not possible its portability made it interesting for pipeline inspections, its portability made interesting for spherical exoskeleton. This isprototypes a smallinspections, robot with poor poortoaccessibility. accessibility. Applications that were not possible its portability made it it interesting for pipeline pipeline inspections, mapping and surveillance tasks. Similar are and dewith Applications not possible capable do more tasksthat andwere to had reach spots lightweight with Applications that were not possible beforepoor are accessibility. feasible nowcomplex because these vehicles to adopt mapping and and surveillance surveillance tasks. tasks. Similar Similar prototypes prototypes are are dedemapping before are feasible now because these vehicles had to adopt its portability made it interesting for pipeline inspections, mapping and surveillance tasks. Similar prototypes scribed in Kalantari and Spenko (2013); Bachmann et deal. beforepoor areoperation feasible now now because these vehicles had to adopt scribed in Kalantari and Spenko (2013); Bachmannare with accessibility. Applications were not to possible before are feasible because these vehicles had adopt different modes in order to that face efficiently diverse et al. scribed in Kalantari and Spenko (2013); Bachmann et al. mapping and surveillance tasks. Similar prototypes are dedifferent operation modes in order order to face face efficiently diverse scribed in Kalantari and Spenko (2013); Bachmann et al. (2009); Peterson et al. (2011) where the authors implement different operation modes in to efficiently diverse before are feasible now because these vehicles had to adopt different operation modes incan order to face situations. These vehicles behave, forefficiently example,diverse as an (2009); (2009); Peterson et al. (2011) where the authors implement Peterson et al. (2011) where the authors implement situations. These vehicles can behave, for example, as an scribed in Kalantari and Spenko (2013); Bachmann et (2009); Peterson et al. (2011) where the authors implement a cage to move over ground or add other mechanisms in situations. These vehicles can behave, forefficiently example, as an an a cage to move over ground or add other mechanisms al. different operation modes incan order face diverse situations. These behave, for example, as aerial vehicle or asvehicles a terrestrial onetoduring the achievement in aaorder cage to move over or add other mechanisms in (2009); Peterson et operation al.ground (2011) where the authors implement aerial vehicle vehicle or as asvehicles a terrestrial terrestrial one during during the achievement cage to move over ground or add other mechanisms in to change its mode. aerial or a one the achievement situations. These can behave, for example, as an aerial as a terrestrial one during of its vehicle assignedor work. Hence, when a wall the or achievement obstacle are order order to change its operation mode. to change its operation mode. of its assigned work. Hence, when a wall or obstacle are a cage to move over ground or add other mechanisms in order to change its operation mode. of its its vehicle assigned work. Hence, when a wall wall or achievement obstacle are aerial or work. as a could terrestrial one during of assigned Hence, a or obstacle found, the vehicle fly when instead of the wheeling, or are in order These torobots seem to converge to a hybrid vehicle using change its operation mode. found, the vehicle could fly instead of wheeling, or in These robots seem to converge to a hybrid vehicle using found, the vehicle could fly instead of wheeling, or in These robots seem to converge to a hybrid vehicle using of its assigned work. Hence, when a wall or obstacle are found, thewhen vehicle could fly instead of wheeling, or in These situations the spot is reduced, the vehicle could switch a quadrotor additionalto passive used robots with seem an to converge a hybridstructure vehicle using situations when the spot is the vehicle could switch with an additional used situations when the spot is reduced, reduced, the vehicle could switch aaforquadrotor quadrotor with an additional passive structure used seem mode. to converge to passive a bases, hybridstructure vehicle using found, themode. vehicle could fly ininstead of wheeling, or in aThese situations when the spot is reduced, vehicle could switch to the car Nevertheless, thethe design of such vehicles, quadrotor with an additional passive structure used therobots terrestrial On these our proposed to the car mode. Nevertheless, in the design of such vehicles, for the terrestrial mode. On these bases, our proposed to the car mode. Nevertheless, in the design of such vehicles, for the terrestrial mode. On these bases, our proposed situations when the spot is reduced, the vehicle could switch to carcharacteristics mode. Nevertheless, the design vehicles, thethe best of eachinmode must of besuch retained and afor quadrotor with an additional passive structure used the terrestrial mode. On these bases, our proposed prototype uses a quadrotor without additional servos and the best best characteristics characteristics of of each each mode mode must must be be retained retained and and prototype uses a quadrotor without additional servos and the prototype uses aa quadrotor without additional and mode. On bases, ourservos proposed to carcharacteristics mode. Nevertheless, inmode the design vehicles, thethe best of each must of beassuch retained and resulting device must work efficiently it was two for prototype uses passive quadrotor without additional servos and withthe onlyterrestrial two wheels tothese move over ground. These the resulting device must work efficiently as it was two with only two passive wheels to move over ground. These resulting device must work efficiently as it was two with only two passive wheels to move over ground. These the best characteristics of each mode must be retained and resulting device must work efficiently as it was two prototype separated vehicles. the uses a quadrotor without additional servos and with only two passive wheels to move over ground. These wheels serve as a protection for collisions and represent separated vehicles. vehicles. wheels serve as a protection for collisions and represent separated wheels serve as aa protection for collisions and represent the resulting device must work efficiently as it was two with separated vehicles. only two passive wheels to move over ground. These wheels serve as protection for collisions and represent a smaller structure compared to the cage or wings used An example of these hybrid vehicles is the DALER (Deploy- aa smaller smaller structure structure compared compared to to the the cage cage or or wings wings used used separated vehicles. An example example of these these hybrid hybrid vehicles vehicles is is the the DALER DALER (Deploy(Deploy- wheels serve as a protection collisions andwings represent smaller structure compared for toAmong the cage or used roll in precedent prototypes. its advantages, we An of ableexample Air Land al. (2013). This ato to roll in precedent prototypes. Among its advantages, we An of Exploration these hybrid Robot) vehiclesDaler is theet DALER (Deployto roll in precedent prototypes. Among its advantages, we able Air Land Exploration Robot) Daler et al. (2013). This a smaller structure compared to the cage or wings used to roll in precedent prototypes. Among its advantages, we can mention an increase of the energy efficiency because able Air Land Exploration Robot) Daler et al. (2013). This An example of these hybrid vehicles is the DALER (Deployable Airacting Land Exploration et al.Take-Off (2013). This vehicle at first as aRobot) VTOLDaler (Vertical and can can mention an increase of the energy efficiency because mention an increase of the energy efficiency because vehicle acting at first as a VTOL (Vertical Take-Off and to roll in precedent prototypes. Among its advantages, we can mention an increase of the energy efficiency because the wheels reduce the contact points with the ground and vehicle acting at first as a VTOL (Vertical Take-Off and able Airacting Land Exploration et short al.Take-Off (2013). This vehicle at first as wings aRobot) VTOL (Vertical and the Landing) can also use its for Daler walking distances. reduce the points with the and the wheels wheels reduce the contact contact points with the ground ground and Landing) can also use its wings for walking short distances. can mention an increase of the energy efficiency because the wheels reduce the contact points with the ground and consequently this reduces the resistance to roll. Also, using Landing) can also use its wings for walking short distances. vehicle acting at first as a VTOL (Vertical Take-Off and Landing) canstructure also use has its wings for walking Reusing its the advantage toshort keep distances. the same consequently this reduces reduces the the resistance resistance to to roll. roll. Also, Also, using using this Reusing its structure has the advantage advantage toshort keep distances. the same same consequently wheels reduce the contact points with the feet ground and consequently this reduces the resistance to roll. Also, using wheels allows faster displacements than using and the Reusing structure the keep the Landing) can also its wings for Reusing its structure has theand advantage tothe keep the same structuralits mass of use the has robot towalking reduceto drawback of the wheels allows faster displacements than using feet and the wheels allows faster displacements than using feet and the structural mass of the robot and to reduce the drawback of consequently this reduces the resistance to roll. Also, using wheels allows faster displacements than using feet and the air flow passing through the helices is less perturbed than structural mass of the robot and to reduce the drawback of Reusing its structure has the advantage to keep the same structural mass of thebyrobot and toparts. reduceFurthermore, the drawbackthe of air performance caused additional air flow passing through the helices is less perturbed than flow passing through the helices is less perturbed than performance caused by additional parts. Furthermore, the wheels allows faster displacements than using feet and the air flow passing through the helices is less perturbed than the flows in the prototypes using a cage for the terrestrial performance caused by additional parts. Furthermore, the structural mass of thebyrobot and toparts. reduce thethe drawback of the performance caused additional the robot can easily overcome obstacles foundFurthermore, in terrestrial flows in prototypes using aa cage for the the flow flowspassing in the the through prototypes using cage forperturbed the terrestrial terrestrial robot can easily overcome obstacles found in the terrestrial air the helices is less than the flows in the prototypes using a cage for the terrestrial mode. robot can easily overcome obstacles found in the terrestrial performance caused by additional parts. Furthermore, the robot easily overcome foundAnother in the terrestrial mode can by switching to the obstacles VTOL mode. platform mode. mode by switching switching to the the obstacles VTOL mode. mode. Another platform mode. flows in the prototypes using a cage for the terrestrial mode. mode by to VTOL Another platform robot can easily overcome found(2014) in themoves terrestrial mode by switching toand thed’Andrea VTOL mode. Another platform developed by Thorel Novel over the Different models for aerial and ground vehicles can be found developed by Thorel and d’Andrea Novel (2014) moves over mode. developed by Thorel and d’Andrea Novel (2014) moves over Different models for for aerial aerial and and ground ground vehicles vehicles can can be be found found mode by switching to the VTOL mode. Another platform Different models developed by four Thorel and d’Andrea Novel (2014) over Different ground using little supports. This vehicle is amoves quadrotor models for aerial anddescribing ground vehicles cancharacterbe found in the literature, some ones specific ground using four little supports. This vehicle is a a quadrotor quadrotor ground four little supports. This vehicle is in the the literature, literature, some ones describing specific characterdeveloped Thorel and d’Andrea Novel over Different ground using four little supports. This vehicle is amoves quadrotor suitableusing forbyindoor exploration because of(2014) its maneuverabilin some ones describing specific charactermodels for aerial and ground vehicles can be found in theand literature, someneglecting ones describing specific characteristics other ones nonlinear parameters to suitable for indoor exploration because of its maneuverabilsuitable for exploration because of ground four little supports. This vehicle isconceived a quadrotor and other ones neglecting nonlinear parameters to suitable for indoor exploration because of its itsismaneuverabilmaneuverability and using its indoor hover possibility. This vehicle to istics istics and other ones neglecting nonlinear parameters to in the literature, some ones describing specific characteristics and other ones neglecting nonlinear parameters to ity and its hover possibility. This vehicle is conceived to design the control strategies, more details about them can ity and its hover possibility. This vehicle is conceived to suitable indoor exploration because ofnecessary, itsismaneuverability and for its hover possibility. This vehicle conceived to design displace on ground and to fly only when and this the control strategies, more details about them can design the control strategies, more details about them can istics and other ones neglecting nonlinear parameters to displace on ground and to fly only when necessary, and this design the control strategies, more details about them can be seen in Castillo et al. (2005); Azzam and Wang (2010); displace on ground and to fly only when necessary, and this ity and its hover possibility. This vehicle is conceived to displace and to of flythe onlyoperation when necessary, and this leads to on an ground improvement time because in design be seen seenthe in Castillo Castillo et al. al. (2005); (2005); Azzam and Wang (2010); be in et Azzam and Wang (2010); control (2014); strategies, more details about them can leads to an improvement of the operation time because in be seen in Castillo et al. (2005); Azzam and Wang (2010); leads to an improvement of the operation time because in Page and Pounds Manecy et al. (2015); Marchand displace and to of flythe onlyoperation when necessary, and this leads to on an ground improvement time because in be Page and Pounds (2014); ManecyAzzam et al. al. and (2015); Marchand  This work was supported by CONACYT of the Mexican govPage and Manecy Marchand in Pounds Castillo et al. (2005); Wang (2010); and Pounds (2014); Manecy etexists al. (2015); (2015); Marchand andseen Alamir (2003).(2014); Similarly, thereet a large literature  leads to an improvement of the operation time because in Page  This work was supported by CONACYT of the Mexican govand Alamir (2003). Similarly, there exists aa large literature This work work wasPERSYVAL-Lab supported by by CONACYT CONACYT of the the Mexican Mexican gov and Alamir (2003). Similarly, there exists large literature ernment, LabEx (ANR-11-LABX-0025), Equipex This was supported of govPage and Pounds (2014); Manecy et al. (2015); Marchand and (2003). Similarly, there exists a large literature that Alamir covers methods to counteract nonlinear uncertainties ernment, LabEx PERSYVAL-Lab (ANR-11-LABX-0025), Equipex ernment, LabEx PERSYVAL-Lab (ANR-11-LABX-0025), Equipex  that covers to nonlinear uncertainties ROBOTEX (ANR-10-EQPX-44-01) and by GIPSA-lab in France. This work wasPERSYVAL-Lab supported by CONACYT of the Mexican government, LabEx (ANR-11-LABX-0025), Equipex that Alamir covers methods methods to counteract counteract nonlinear uncertainties and (2003). Similarly, there exists a large literature that covers methods to counteract nonlinear uncertainties ROBOTEX (ANR-10-EQPX-44-01) and by GIPSA-lab in France. ROBOTEX (ANR-10-EQPX-44-01) and by GIPSA-lab in France. ROBOTEX (ANR-10-EQPX-44-01) and by GIPSA-lab in France. ernment, LabEx PERSYVAL-Lab (ANR-11-LABX-0025), Equipex that covers methods to counteract nonlinear uncertainties ROBOTEX (ANR-10-EQPX-44-01) and by GIPSA-lab in France.

J. J. J. J.

ROBOTEX and by GIPSA-lab in France. 2405-8963 © ©(ANR-10-EQPX-44-01) 2019, IFAC IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 Copyright © under 2019 IFAC IFAC Peer review responsibility of International Federation of Automatic Control. Copyright © 2019 Copyright © 2019 IFAC 10.1016/j.ifacol.2019.08.042 Copyright © 2019 IFAC

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3. CONTROL SCHEME

dynamics, techniques to improve the performance of a vehicle and also algorithms to regulate the position of a wheeled vehicle, some of them can be analyzed in Zachary Thompson Dydek (2010); Brezoescu et al. (2013); Low and Ng (2011); Gao et al. (2014); Khebbache and Tadjine (2013); Shojaei et al. (2010); Canigur and Ozkan (2012); Tian et al. (2009).

The control algorithm for the hybrid vehicle is obtained in two steps: firstly the orientation dynamics are stabilized and secondly the translational dynamics are controlled for air-ground path tracking. The position algorithm will generate desired angles that will be used as references for the attitude scheme.

In our vehicle, the developed model considers a body with perfect symmetry where the mass center and the rotation center are at the same location, see Section 2. We neglect the propellers flexibility and the discharge of battery during the operation time. Here, the control design is focused in developing a xy -position control law with the same structure for both operation modes that allows to have a seamless transition when passing from ground to air and vice versa. Instead of using two control laws separately, the objective is to pass from a parameter set to the other during the transition. The height of the vehicle is governed by a PID controller in the aerial mode. Besides, we implement an adaptive technique in the attitude regulation in order to reduce the negative effects of the neglected dynamics, see Section 3. The convergence of the algorithm is discussed and validated by simulations and by real experiments, see Sections 4 and 5. For the experimental validation a prototype is built, the main characteristics of the platform are given also in Section 5. At the end of this paper (Section 6), some conclusions about this work are given. 2. MATHEMATICAL MODEL From Figure 1 and using the Newton-Euler approach, the mathematical equations for the hybrid vehicle can be written as

3.1 Attitude algorithm From Eqn. (1), the orientation model of the vehicle includes the term δτ that could contain nonlinear uncertainties and asymmetries of the vehicle. This fact motivate us for designing a robust adaptive algorithm capable of estimating and compensating these unknown dynamics. It is supposed, for control design, that δτ varies so slowly that could be considered constant. Define the attitude error as e˙ η = η˙ − η˙ r eη = η − ηr =⇒ = B(η)ω − η˙ r where ηr denotes the attitude reference. The matrix B(η) has the following form   0 sin φ/ cos θ cos φ/ cos θ cos φ − sin φ B= 0 1 sin φ · tan θ cos φ · tan θ ψ , θ and φ are the yaw, pitch and roll angles. The matrix B(η) is not singular if and only if cos(θ) = 0. It is proposed the following positive definite function 1 VLη = eTη eη 2 and by choosing conveniently a desired angular velocity ω as ωd = B−1 (η˙ r − Kη eη ) with Kη as a positive diagonal constant matrix, it yields (2) V˙ Lη |ω=ω = −eTη Kη eη ≤ 0 ∀ t ≥ 0 d

Now, let us define the angular velocity error as eω = ω − ωd =⇒ e˙ ω = ω˙ − ω˙ d and keep in mind that, ω = ωd + e ω , ω˙ = J−1 (τ − [ω]× Jω + δτ ) (3) Then, consider the following candidate Lyapunov function for the attitude system 1 VLω = VLη + eTω eω 2 hence, V˙ Lω = V˙ Lη + eTω e˙ ω and using (2) and (3), it results V˙ Lω = −eTη Kη eη + eTη Beω + eTω e˙ ω

Fig. 1. Unmanned Ground and Aerial Vehicle - UGAV prototype. m¨ r = RF + F g + δu η˙ = B(η)ω (1) Jω˙ = τ − [ω]× Jω + δτ The bold letters represent vectors. F means the thrusts generated by the helices, Fg is the gravity force and δu is a force term that changes depending on the operation mode of the vehicle. η stands for the vector of Euler angles, ω means the angular velocity in the body frame. m indicates the mass of the drone and r defines the position of the mass center in the inertial system, R describes the rotation matrix generated in the order yaw-pitch-roll. B(η) represents the matrix that relates the angular velocity and the derivative of the Euler angles. J is the inertia matrix of the drone. [ω]× means the skew symmetric matrix of angular velocity, τ defines the torques applied to the vehicle and δτ represents a torque term related to the vehicle operation mode.

Therefore, by choosing   τ = −δˆτ + [ω]× Jω + J ω˙ d − BT eη − Kω eω

with δˆτ as the estimate of the unknown dynamics in the attitude model, it yields V˙ Lω = −eTη Kη eη − eTω Kω eω + eTω J−1 δ˜τ δ˜τ is defined as δ˜τ = δτ − δˆτ , Thus where δτ is supposed to ˙be constant. ˙ δ˜τ = −δˆ τ Considering the augmented candidate Lyapunov function 1 VLω2 = VLω + δ˜τT Γ−1 δ˜τ 2 with Γ as a positive diagonal matrix, it follows 20

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21

the vehicle must follow to be aligned with the target and β as the desired inclination which produces a well-defined movement to the target, as follows α = arg (ex + ˆı · ey ) (7) ˙ β = asin (kpg d + kdg d)

˙ V˙ Lω2 = −eTη Kη eη − eTω Kω eω + eTω J−1 δ˜τ − δ˜τT Γ−1 δˆτ

and if the desired dynamics for δˆτ is taken as ˙ δˆτ = ΓJ−1 eω then, V˙ Lω2 = −eTη Kη eη − eTω Kω eω ≤ 0 ∀ t ≥ 0

where ˆı is a unitary vector and kpg and kdg must be chosen in order to satisfy |kpg d + kdg d|˙ < π/2

The matrices Kη , Kω and Γ are chosen in order to tune the attitude controller. This control algorithm will let us to counteract some slow dynamics when the vehicle is in aerial mode or the frictions torques due to the ground in terrestrial mode. 3.2 Translational control Two modes are considered, first one is when the vehicle is moving on the ground and the other will be when the vehicle is operating as an aerial vehicle.

Therefore for the ground mode, the Euler angles references and thrust can be chosen as follows ψgr = {α, α + π : cos(ψgr − ψ) > 0} θgr = sign(cos(ψgr − ψ)) β (8) φ gr = 0 f < m·g f is chosen constant and in such a way as to avoid takeoff, ψgr will generate the minimal effort to align the vehicle with the target, θgr will produce an inclination always to the desired position and φgr = 0 because the vehicle is moving on a flat surface.

Ground operation mode The strategy will be composed following the natural movements of a ground vehicle, firstly its heading will be oriented directly to the target position, see Figure 2, and finally the translational dynamics will be controlled for trajectory tracking. The longitudinal movement will be produced by changing the pitch angle of the vehicle that will depend on the distance from the mass center to the target position and on the approaching speed to the target. Therefore, for far distance to the target, the pitch angle will be increased implying high translational speed. Hence a well-tune of the gain controllers is necessary for having a good performance of the vehicle when tracking the desired path.

Stability and Convergence Transforming (4) to polar coordinates and using Figure 3, it follows that 1 d¨ − ϑ˙ 2 d = (−f sin β cos γg − δuxy sin γg ) m (9) 1 ˙ uxy cos γg ) dϑ¨ + 2d˙ϑ˙ = (f sin β sin γg − sign(ϑ)δ m where d and ϑ represents the polar coordinates, δuxy = (δux , δuy ) and from (8), cos γg > 0. final orientation

final rientation

v

v

target

initi orienta

v target

target

Fig. 3. The origin of polar coordinates is the target position. a) Orientation towards target

δuxy is considered as a lateral force that prevents the lateral displacement. γg represents the difference between the desired orientation α and the instant orientation ψ of the vehicle.

b) Moving towards target

Fig. 2. Ground operation mode of the hybrid vehicle. First the vehicle turns around itself in order to get aligned to the target. Then, it changes its pitch angle in order to move to the target.

Observe that θgr has been chosen in such a way the vehicle moves toward the target and this leads to a negative d˙. Then, the second equation of (9) can be rewritten as 1 1 ˙ uxy cos γg + 2d| ˙ ϑ|) ˙ dϑ¨ = (f sin β sin γg ) − sign(ϑ)(δ m m (10) ˙ > 0, the If δuxy is big enough such that δuxy cos γg + 2d|˙ ϑ| dynamics described by (9) is stable. Hence, the angular velocity ϑ˙ will have the form (11) ϑ˙ = kϑ sin β sin γg + o() where kϑ is a gain that depends on the value of ˙ ϑ|) ˙ and the term o() represents the tran(δuxy cos γg + 2d| sitory dynamics. Also, γg will be zero when ψ → α or when β → 0 which implies ϑ˙ → o() and this, in turn, implies ϑ will be constant. From this result, it is concluded that the vehicle will move straightforward to the target. Notice that this result is based on the assumption that δuxy is big enough, thus it is important to choose a convenient material for the wheels in order to have enough friction with the floor. Furthermore, consider to choose a thrust not too big so as to not diminish the grip of the wheels.

From (1) notice that the term δu in ground operation mode is not zero. This term should compensate the difference between the thrust and the vehicle weight. Consider now that the vehicle is moving over a perfect flat and it is always in contact with the floor, this implies that φ = 0 and the altitude is constant, thus the translational dynamics can to   bereduced    x ¨ cos ψ δux m = · f sin θ + (4) y¨ sin ψ δuy where x and y stand for the xy position in an inertial frame, δux and δuy are the xy components of δu and f indicates the total thrust of motors. These terms represent the forces which prevent the lateral displacement in terrestrial mode. Now,let us  define  the error  position as: ex xr − x exy = = (5) yr − y ey and

d = exy  (6) where xr and yr are the desired path in the horizontal plane x − y and d defines the distance of the vehicle with respect to the target. Define now, α as the direction that 21

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Introducing the expression for β in (7) into the first equation of (9), it yields 1 (12) d¨ + a1 d˙ + a0 d = − δuxy sin γg m 1 with a0 = f cos γg kpg − ϑ˙ 2 m 1 a1 = f cos γg kdg m From the previous analysis, ϑ˙ → 0 and γg → 0. Note that (12) is stable iff a0 and a1 are positive. Then, it can be concluded that d → 0. Also, observe that if the target is only a desired constant position, the vehicle will turn around itself when it enters in a very small neighborhood centered at the target. This phenomenon is caused by the definition of α, this problem can be solved making constant the yaw angle when the vehicle enters into this neighborhood. 3.3 Aerial operation mode The goal here is that the vehicle continues to track the aerial desired trajectory. The previous algorithm will be extended for the aerial mode, in this mode no friction on the wheels is considered, in addition, a new term to compensate lateral movements to the line connecting the mass center with the target position need to be defined. This term will be proportional to the velocity of the vehicle. In Figure 4, the challenge for this mode is presented graphically.

zb =



sin ψ sin φ + cos ψ cos φ sin θ cos φ sin ψ sin θ − cos ψ sin φ cos θ cos φ



=



sin β cos α sin β sin α cos β



(16) Therefore, from previous equality, the references the vehicle have to follow during the trajectory tracking in aerial mode are θar = atan(cos(ψ − α) tan β) φar = asin(sin(ψ − α) sin β) (17) ψar ∈ [−π, π] m · g + uz f = cos φ cos θ The value for ψar can be assigned arbitrarily and the expression for f is derived from (13). The expression for α and β are indicated in (15). Stability and Convergence Transforming equation (13) into cylindrical coordinates, it follows that 1 d¨ − ϑ˙ 2 d = (−f sin β cos γa ) m 1 (18) dϑ¨ + 2d˙ϑ˙ = (f sin β sin γa ) m m¨ z = f cos φ cos θ − m · g

Taking the expression for f from (17) and substituting it in the third equation of (18), it yields:  t

ez dτ = 0

m¨ ez + kdz e˙ z + kpz ez + kiz

(19)

0

which is stable iff kpz > m·kiz /kdz and hence ez → 0. Using ˙ , the first and second (15) for β and γa = −|γa |sign(ϑ) equation of (18) can be rewriting as     f f (20) cos γa kda d˙ + cos γa kpa − ϑ˙ 2 d m m    f ˙ d˙ + f sin|γa |kp d ˙ sin|γa |kda + 2|ϑ| dϑ¨ = −sign(ϑ) a m m (21) 0 = d¨ +

target

Therefore, the two conditions for assuring the convergence of d → 0 and ϑ˙ → 0 are f cos γa kpa − ϑ˙ 2 > 0 (22) m f sin|γa |kpa d (23) d˙ > − ˙ f sin|γa |kd + 2m|ϑ|

Fig. 4. Graphical representation for the challenge in the aerial mode. From figure, α is used to compensate the lateral movement, β represents the desired inclination for reaching the target. γa is the difference between the direction α and the direction of the line connecting the vehicle and the target.

a

where |γa |  0. Observing the Figure 4, if |γa | = 0 this means, ϑ˙ = 0, and hence the vehicle moves straightforward to the target, and by consequence d → 0. In the case, |γa | > ˙ of the approaching 0, it is necessary the magnitude (|d|) speed to the target does not exceed the value given by (23). If (23) holds, then ϑ˙ → 0. Therefore, it is necessary to tune the parameters kda and kpa to impose a speed performance satisfying (23) and consequently this leads to d → 0.

In aerial mode the vehicle becomes a UAV. From (1), its position model can be written as m¨ r = RF + F g (13) The error position in the horizontal plane xy is taken as in (5). The z position error is defined as ez = zr − z , where z is the altitude of the vehicle and zr represents the desired height. In order to stabilize the altitudea PID controller is  t proposed uz = kpz ez + kdz e˙ z + kiz ez dτ (14)

4. NUMERICAL SIMULATION The goal for this part is that the hybrid vehicle tracks an airground trajectory autonomously. This trajectory is divided in four parts; the first part begins at (x, y, z) = (0.5, 0, 0) in meters, and describes a part of a sinusoidal curve, connecting the initial point with (0.5, 0.30, 1) m. The second path is a half circle finishing at (−0.5, 0.3, 1) m. The third trajectory is similar to the first one, beginning at the last point of the second trajectory and finishing at (−0.5, 0, 0) m. The fourth path is also a half circle connecting the third trajectory and finishing at the initial point (0.5, 0, 0) m. In the first trajectory, the desired orientation is at 90◦ or perpendicular to the x axis, the same is for the third path.

0

with kpz , kdz and kiz as positive constants. Also, it is proposed the following expressions for α, β as α = arg ( (ex + kv e˙ x ) + ˆı · (ey + kv e˙ y ) ) (15) ˙ β = asin (kpa d + kda d) where d is defined in (6), kv represents a positive constant which counteracts the lateral movement and kpa and kda are positive constants satisfying |kpa d + kda d|˙ < π/2

From Figure 4, the coordinates of zb expressed in terms of the zyx Euler angles and also computed using the angles α and β , yield 22

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R ground station equipped with MATLAB . This algorithm generates the references (ψr , θr , φr , f ) according to the operation mode. Two sets of parameter values were obtained during the tuning of each operation mode. The switching between these two sets is made according to the height of the desired trajectory. The initial position is [1, 0, 0] m, the height of the mass center of the vehicle is 0.15 m and the maximum height is 0.7 m, the semicircle radius is 1 m. The orientation algorithm is programmed in the flight controller and it is the same for the two modes. The parameters used for the position control law were: kv = 0.75, kpa = 0.1, kda = 0.1, kpz = 0.7, kdz = 0.4, kiz = 0.01, kpg = 0.1, kdg = 5, fg = 2.8 Nm, m = 0.350 kg. The results of this experiment are shown in Figure 6.

In the second and four path, the vehicle direction will follow the sense of displacement. Figure 5 introduces the behavior of the vehicle when it tracks the trajectory. When the vehicle moves from ground to air and vice versa, the expressions for the references ψr (ψgr , ψar ), θr (θgr , θar ), φr (φgr , φar ) and f change according to the operation mode.

a)

a)

b)

b)

c)

c)

d)

Fig. 5. a) System performance, r, when following the hybrid trajectory rr . The initial position is (0.5,0,0) in meters. b) Roll and pitch angles behavior. φ = 0 when the vehicle is in ground operation. c) Heading performance of the vehicle. The yaw angle is constant during the taking-off and landing. d) Thrust behavior. In terrestrial mode has a constant value, f = 3.26 N , . d)

5. PLATFORM AND EXPERIMENTAL RESULTS The prototype used in this test is shown in Figure 7. The Table 1 shows its specifications. The challenge is to repeat the same behavior obtained in the numerical simulations. The vehicle must track the aerial-ground trajectory defined in section 4. The position control algorithm is computed in a

Fig. 6. a) Real-time performance of the hybrid vehicle when following the air-ground trajectory. The initial position is (1,0,0) in meters. b) Roll and pitch angles performances. φ = 0 when the vehicle is in ground operation. c) Heading behavior of the vehicle during the path tracking. d) Thrust evolution. In terrestrial mode, the thrust is constant, f = 2.8 N .

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Castillo, P., Lozano, R., and Dzul, A.E. (2005). Modelling and Control of Mini-Flying Machines, chapter 2,3. Springer-Verlag, London. Daler, L., Lecoeur, J., Hanlen, P.B., and Floreano, D. (2013). A flying robot with adaptive morphology for multi-modal locomotion. In 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, 1361–1366. doi:10.1109/IROS.2013.6696526. Dudley, C.J., Woods, A.C., and Leang, K.K. (2015). A micro spherical rolling and flying robot. In Intelligent Robots and Systems (IROS), 2015 IEEE/RSJ International Conference on, 5863–5869. doi:10.1109/IROS.2015.7354210. Gao, Q., Yue, F., and Hu, D. (2014). Research of stability augmentation hybrid controller for quadrotor uav. In Control and Decision Conference (2014 CCDC), The 26th Chinese, 5224–5229. Kalantari, A. and Spenko, M. (2013). Design and experimental validation of hytaq, a hybrid terrestrial and aerial quadrotor. In Robotics and Automation (ICRA), 2013 IEEE International Conference on, 4445–4450. doi: 10.1109/ICRA.2013.6631208. Khebbache, H. and Tadjine, M. (2013). Robust fuzzy backstepping sliding mode controller for a quadrotor unmanned aerial vehicle. Journal of Control Engineering and Applied Informatics, CEAI, 15(2), 3–11. Low, C.B. and Ng, Q.S. (2011). A flexible virtual structure formation keeping control for fixed-wing UAVs. In 9th IEEE International Conference on Control and Automation (ICCA), 621–626. Manecy, A., Marchand, N., Ruffier, F., and Viollet, S. (2015). X4-mag: a low-cost open-source micro-quadrotor and its linux-based controller. International Journal of Micro Air Vehicles, 7(2), 89–109. doi:10.1260/17568293.7.2.89. Hal-01099975. Marchand, N. and Alamir, M. (2003). Discontinuous exponential stabilization of chained form systems. Automatica, 39(2), 343–348. doi:10.1016/S0005-1098(02)00229-7. Page, J.R. and Pounds, P.E.I. (2014). The quadroller: Modeling of a uav/ugv hybrid quadrotor. In 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, 4834–4841. doi:10.1109/IROS.2014.6943249. Peterson, K., Birkmeyer, P., Dudley, R., and Fearing, R.S. (2011). A wing-assisted running robot and implications for avian flight evolution. Bioinspiration and Biomimetics, 6(4), 046008. Shojaei, K., Shahri, A., and Tabibian, B. (2010). Adaptiverobust feedback linearizing control of a nonholonomic wheeled mobile robot. In Advanced Intelligent Mechatronics (AIM), 2010 IEEE/ASME International Conference on, 497–502. Thorel, S. and d’Andrea Novel, B. (2014). Hybrid terrestrial and aerial quadrotor control. In IFAC Proceedings Volumes, volume 47, 9834–9839. doi:10.3182/201408246-ZA-1003.00378. Tian, Y., N. Sidek, N., and Sarkar, N. (2009). Modeling and control of a nonholonomic wheeled mobile robot with wheel slip dynamics. In IEEE Symposium on Computational Intelligence in Control and Automation, 7–14. doi:10.1109/CICA.2009.4982776. Zachary Thompson Dydek (2010). Adaptive Control of Unmanned Aerial Systems. Ph.D. thesis, Massachusetts Institute of Technology. Ch. 3.

Fig. 7. Protoype Table 1. Hybrid Vehicle Parameters

Parameter mass payload length height helix diam battery motor

Value 0.350 kg 0.070 kg 0.32 m 0.30 m 0.125 m 1200 mAh 2S 7.4V 30C Li-Po Brushless 28000 kv 13 gr 6. CONCLUSIONS

In this paper, a nonlinear controller is proposed for tracking an air-ground path using a hybrid vehicle. Analyzing the performance of the vehicle, a nonlinear dynamic model was defined and used for conceiving the control scheme. The controller was obtained in two parts, firstly an attitude controller was developed using the Backstepping methodology and later the translational dynamics were used to define references for the tracking of the hybrid trajectory. For the ground mode, it was observed the translational dynamics depends on the natural friction between the wheels and the ground. In the aerial mode, the vehicle can be seen as a classical quadcopter vehicle. Several simulations were carried out showing a good performance of the controller in closed-loop system. In addition, experimental results have confirmed the well behavior of the controller in both modes (aerial and ground). REFERENCES Azzam, A. and Wang, X. (2010). Quad rotor arial robot dynamic modeling and configuration stabilization. In 2nd International Asia Conference on Informatics in Control, Automation and Robotics (CAR), volume 1, 438–444. Bachmann, R.J., Vaidyanathan, R., and Quinn, R.D. (2009). Drive train design enabling locomotion transition of a small hybrid air-land vehicle. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 5647–5652. doi:10.1109/IROS.2009.5354102. Brezoescu, A., Lozano, R., and Castillo, P. (2013). Bank to turn approach for airplane translational motion in unknown wind. In International Conference on Unmanned Aircraft Systems (ICUAS), 1022–1029. Canigur, E. and Ozkan, M. (2012). Model reference adaptive control of a nonholonomic wheeled mobile robot for trajectory tracking. In International Symposium on Innovations in Intelligent Systems and Applications, 1– 5. doi:10.1109/INISTA.2012.6247005. 24