Control Of Aerial Robots.pdf

Lorenzo Marconi and Roberto Naldi

Control of Aerial Robots Hybrid force and position feedback for a ducted fan

U

nmanned aerial vehicles (UAVs) attract the interest of many fields of engineering, including control, aerospace and aeronautics, electronics, and materials. The research interest in the area of control engineering is mainly focused on the control of the vehicle, in a fully autonomous way or with a partial human supervision, to fly through prespecified paths [1], to synchronize with other vehicles to form coordinated fleets [2], to perform acrobatic maneuvers [3], to reconstruct unknown environments [4], and to perform other operations. Indeed, research in the field is driven by domains of application in which UAVs are typically employed, such as surveillance and data acquisition in areas that are dangerous for human operators and inaccessible to ground vehicles. Many civil [5], [6] and military [7], [8] applications show their use in these contexts. The ability to fly within possibly unstructured environments explains why UAVs are also referred to as flying robots (see [9, Chap. 44]), a terminology inspired by ground robots, the latter identifying vehicles moving autonomously on the ground (see [9, Chap. 17]). However, we are witnessing research approaches that shift the focus to application-oriented domains where UAVs are not merely used as vehicles able to fly autonomously, but rather as vehicles that can physically interDigital Object Identifier 10.1109/MCS.2012.2194841 Date of publication: 12 July 2012

1066-033X/12/$31.00©2012ieee

act, in a constructive way, with the surrounding environment. Their aim is to accomplish in midair real robotic tasks, such as manipulating objects, acquiring data by contact, picking samples, and repairing and assembling objects. Examples of this research can be found in [10]–[14]. The European project AIRobots (innovative aerial service robots for remote inspections by contact, [15]) supported by the European Community, fits into this research scenario. The goal of AIRobots is to develop a new generation of service robots capable to support human beings in all those activities that require the ability to interact actively and safely with environments not constrained on ground but, indeed, airborne. The aerial platform is conceived to be remotely supervised by an operator with the use of haptic devices, with force and visual feedback strategies that are developed to transform the aerial platform in a “flying hand,” suitable for aerial manipulation. This article presents control solutions for UAVs physically interacting with the environment. The aerial platform of interest is a ducted fan UAV (see Figure S3 and “Ducted Fan Tail-Sitter Miniature Aerial Vehicles”), which lends itself to be used in contexts in which physical interaction is required. The addressed problem concerns the development of a hybrid force and position feedback, by which the UAV is required to slide along a vertical surface by tracking a vertical reference signal and by applying a desired contact force to the surface. More details can be found in “Force AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  43

Feedback in Robotics,” an overview of force feedback ­strategies available in the robotic literature. The analysis developed in this work places emphasis on the intrinsic nonminimum-phase behavior characterizing the controlled system and presents state feedback strategies needed to handle the hybrid force and position control problem.

The principles of the ducted fan and possible operative modes Ducted fan aerial vehicles are a particular class of tail-sitter aircraft. More details about this class of aircraft can be found in “Ducted Fan Tail-Sitter Miniature Aerial Vehicles.” As illustrated in Figure S3, the ducted fan mechanical configuration is composed of two main parts, a rotor and a

Ducted Fan Tail-Sitter Miniature Aerial Vehicles

T

ail-sitter aircraft are aerial vehicles that are able to take off and land by sitting on their tail. Historically, the first prototypes were produced approximately in the early 1950s [S1]. In particular, it is worth mentioning the NAVY/Convair XFY-1 “Pogo,” the NAVY/Lockheed XFV-1 and the USAF/Ryan X-13 “VertiJet” (see Figure S1), some of the first aerial vehicles able to change their flight configuration from hover to level flight and vice-versa. Since then, the goal of the tail-sitter design has been to combine both the maneuverability of a helicopter, and in particular the capability of achieving stationary flight, with the flight efficiency of an airplane, achieved by sustaining the level flight by means of the lift force produced by the main wings. With tail-sitter aircraft, the change of configuration is not obtained by tilting the rotors or the wings by using additional actuators, as it happens for tilt-rotors and tiltwings aircraft, but rather by changing the attitude of the entire airframe and then using the same actuators adopted to stabilize the standard flight mode. In this respect, when compared to other configurations capable of similar flight qualities, the tail-sitter configuration is characterized by low mechanical complexity and can then be scaled down to design miniature aerial vehicles [S2]. Ducted fan miniature aerial vehicles (see Figure S2) are a particular class of tail-sitter aircraft that owes its name to the presence of an annular fuselage, defined as “duct,” around the propeller. The duct offers relevant advantages. On the one hand, it helps in protecting the propeller from undesired contacts with surrounding objects. This feature allows the employment of this kind of vehicles in potentially cluttered environments or even in proximity of a person. On the other hand, the aerodynamic characteristics of the duct can be optimized to increase the efficiency of the thrust generation at hover [S3], and, at the same time, it is tuned to sustain the high speed flight producing a considerable amount of lift force [17], [S4]. In this latter case it is worth noting how the duct itself plays the role that additional fixed wings have been playing in earlier tail-sitter design. Besides the duct, the mechanical layout of this class of vehicles is solely based on two different subsystems. The first subsystem is responsible for generating the main thrust required to sustain the hover flight of the vehicle and can be produced by a propeller or a rotor driven by an electric motor or an endothermic engine. In some configurations counter rotating propellers are adopted to balance the momentum around

44  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

(a)

(b)

(c)

Figure S1  (a) The Navy/Convair XFY-1 Turboprop, (b) the USAF/Ryan X-13 VertiJet Turbojet, and (c) the Navy/Lockheed XFV-1 Turboprop.

(a)

(b)

(c)

Figure S2  Unmanned ducted fan miniature aircraft (from left to right): (a) Allied Aerospace iSTAR ducted fan, (b) Bertin Technologies ducted fan, and (c) Honeywell ducted fan. the propeller spin axis. The second subsystem is composed of a set of control vanes capable of producing the aerodynamic forces required to govern vehicle’s dynamics (see “Force and Torque Generation Mechanism in the Ducted Fan”). References [S1] B. W. McCormick, Aerodynamics of V/STOL Flight. San Diego: Dover, 1998. [S2] P. Castillo, R. Lozano, and A. E. Dzul, Modeling and Control of Mini Flying Machines. London: Springer-Verlag, 2003. [S3] A. Ko, O. J. Ohanian, and P. Gelhausen, “Ducted fan UAV modeling and simulation in preliminary design,” in Proc. AIAA Modeling and Simulation Technologies Conf. and Exhibit, SC, 2007, no. AIAA-20076375. [S4] E. N. Johonson and M. A. Turbe, “Modeling, control and flight testing of a small ducted fan aircraft,” J. Guid. Control Dyn., vol. 29, no. 4, pp. 769–779, 2006.

This article presents control solutions for UAVs physically interacting with the environment. set of actuated surfaces, referred to as control vanes, which are positioned below the rotor. Both parts are placed within a rigid structure, called the duct, which improves the overall aerodynamical efficiency of the system and protects the moving parts of the vehicle from the surrounding environment. The angular velocity of the rotor is one of the control inputs of the vehicle. In the chosen configuration the pitch of the blades remains constant, allowing the rotor to provide a vertical thrust, denoted by T in Figure S3, which is always directed upward, along the rotor axis. Besides generating the vertical thrust T essential to dominate the force of gravity, the rotor generates an air flow, directed downward within the duct, which, by impacting on the control vanes, generates forces and, con-

sequently, torques. Specifically, the airflow impacting on a particular vane generates a lift force, directed perpendicularly to the vane’s rotational axis, whose amplitude depends on the speed airflow and on the tilt angle of the vane, as shown in Figure S4. As a side effect, a vertical drag force is also generated. The amplitude of the drag force, however, can be minimized by keeping the vane’s tilt angle small and by designing aerodynamically efficient vane profiles. The overall actuation mechanism of the control vanes is designed in a way that the combination of the lift forces of all the vanes produces three main effects. The first is a torque around the vertical axis, denoted by N z in Figure S4, by which it is possible to control the yaw attitude of the vehicle and, in turn, to counteract the aerodynamic torque

Force Feedback in Robotics

I

nteraction between robots and the surrounding environment requires the capability to handle physical contacts. During contacts, the environment applies forces on the end-effector or even constraints on the geometric path that can be followed. Accordingly, feedback strategies need to be taken into account in relation to the force that the robot exchanges with the environment during its motion to bound the actuators effort and thus avoid situations that are potentially dangerous for both the robot and the surrounding objects. In this respect, force control strategies have been widely studied in robotics, in the context of telemanipulation [S7], human-robot interaction [S8], and other robotic applications. The available approaches can be divided into two main groups, referred to as the direct and the indirect force control [S9]. The main difference between the two is that the former actually implements a force feedback loop, while the latter addresses the problem by solely considering a motion control loop. In the case of direct force control, a measure of the force applied to the environment is calculated by using force sensors. The force signal can be obtained by measuring the deformation of a certain compliant material, through optical measurements, or by employing strain measurements. In the case of indirect force control, a force measure is not strictly required. The main idea, instead, is to design a control algorithm by imposing a certain relation between the position error and the force applied by the robot, to govern the force implicitly by means of the motion control loop. For instance, in the impedance control, the error between the desired reference and the actual motion of the robot is related to the contact force through a virtual mechanical impedance [S10].

In many applications, both the position of the end-effector and the forces applied to the environment need to be governed simultaneously. This kind of problem can be addressed by adopting the hybrid force and position control [S11]. The main idea of this control paradigm is to make use of force control schemes only along the directions in which the robot is constrained by the presence of the contacts, while using motion control techniques for the remaining degree-of-freedom. The performances of any force control paradigm are strongly influenced by the model of the interaction scenario that is used in the design of the controller. This model, in turn, depends on the characteristics of the environment in which the robot operates, such as the physical properties of the materials, the geometry of the contact surfaces, and the mechanical properties of the robot. As a consequence, many approaches have been developed (see [S9] and references therein) to address the diverse situations arising in the different application ­scenarios. References [S7] K. B. Shimoga, “A survey of perceptual feedback issues in dexterous telemanipulation—Part II: Finger force feedback,” in Proc. 1993 IEEE Virtual Reality Annual Int. Symp., pp. 263–270. [S8] A. De Santis, B. Siciliano, and A. Bicchi, “An atlas of physical human-robot interaction,” Mech. Machine Theory, vol. 43, no. 3, pp. 253–270, 2008. [S9] L. Villani and J. De Shutter, “Force control,” in Springer Handbook of Robotics, B. Siciliano and O. Khatib, Eds. Berlin, Germany: SpringerVerlag, 2008. [S10] N. Hogan, “Impedance control: An approach to manipulation—Parts I–III,” ASME J. Dyn. Syst. Meas. Control, vol. 107, no. 1, pp. 1–24, 1985. [S11] O. Khatib, “A unified approach for motion and force control of robot manipulators: The operational space formulation,” IEEE J. Robot. Automat., vol. 3, no. 1, pp. 45–53, 1987.

AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  45

The ducted fan configuration presents features that make the vehicle versatile and suitable to operate in many contexts, some of them unusual when dealing with flying vehicles.

generated by the rotor. The other two effects are two resulting forces, denoted by F xL and F Ly in Figure S4, directed along the lateral and longitudinal directions. The latter, acting at a certain distance from the center of mass of the vehicle, generate two torques, denoted by N x and N y , around the longitudinal and lateral axis through which the roll and pitch attitudes of the vehicle are controlled. The

(a)

(b) Figure  1  The ducted fan as an aerial robot interacting with the environment. In this scenario the ducted fan prototype, equipped with nondestructive testing devices, inspects by contact a structure to detect possible cracks or small defects. The area in need of inspection can be rapidly reached by taking advantage of the maneuverability of the vehicle in free-flight mode. 46  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

torque N z and the forces F xL and F Ly , and thus the resulting torques N x and N y , are controllable by means of three ­control inputs governing the overall vanes’ tilt angles. The mechanical features of the vehicle and the physical principles governing the force and torque ­generation mechanism are described in “Force and Torque Generation Mechanism in the Ducted Fan.” The ducted fan configuration presents features that make the vehicle versatile and suitable to operate in many contexts, some of them unusual when dealing with flying vehicles. We discuss below, by presenting possible operative contexts, the main features characterizing the vehicle. Besides the typical operative mode of the vehicle in freeflight, the fact that all the moving and actuated parts are protected by the duct makes the vehicle suitable to physically interacting with the environment. In particular, the vehicle lends itself to dock a vertical surface and to perform operations, such as tracking reference profiles in the vertical and longitudinal direction, while keeping contact with the inspected surface. A typical operative context where such a capability might be employed, is where large structures, such as power plants or wind turbines, not easily accessible by human operators or ground robots, must be inspected by means of nondestructive testing devices (NTDs) [16] to detect possible cracks or small defects in the structure. In this context, the UAV, equipped with the required NTD sensors, could fly close to the surface in need of inspection, dock, and start the inspection by contact. Figure 1 presents a graphical sketch of this operative mode. As a further operative mode, it can be noted how the vehicle is potentially able to behave as a fully actuated ground robot, with full control authority in the lateral and longitudinal direction and around the vertical axis without necessarily taking off. Specifically, by bearing in mind the force and torque generation mechanism explained above and detailed in “Force and Torque Generation Mechanism in the Ducted Fan,” the UAV, equipped with swivel wheels, can move laterally and longitudinally under the action of the forces F xL and F Ly and rotate under the action of the torque N z . In this scenario, the angular velocity of the rotor is controlled in a way that T remains lower than the force of gravity although the generated airflow is sufficient for the described vanes’ force and torque generating mechanism to work properly. This feature makes it possible the accomplishment of missions in which the UAV combines tasks

Force and Torque Generation Mechanism in the Ducted Fan

T

wo main subsystems characterize the ducted fan and its force-torque generation mechanism: a fixed-pitch rotor driven by an electric motor (labelled as (b) in Figure S3) and a set of control vanes positioned below the main propeller (labelled as (c1)–(c2) in Figure S3). Both subsystems are positioned within a duct specifically designed to improve the overall aerodynamical efficiency of the system and to protect the moving parts of the vehicle from the surrounding environment. The goal of the rotor is to generate the main thrust required to counteract gravity force. The thrust generated by the propeller is directed along the vertical body axis and, according to [S5], it is T = k T w 2P , with w P being the angular speed of the rotor and k T a constant parameter collecting all the aerodynamic coefficients. The rotation of the blades also generates an aerodynamic resistance torque equal to N = k N T , with k N being a constant coefficient collecting aerodynamical parameters. From Fraud’s momentum theory [S6], the induced air velocity generated by the propeller inside the duct is

Vi =

T tS disk

whereby S disk is the area of the propeller’s disk and t the air density. Velocity Vi can be considered perpendicular to the propeller disk, namely, aligned with the body Z-axis (see Figure S3). The rotational component of this velocity, caused by the propeller’s angular speed, is in fact compensated by the vanes positioned just below the propeller. The control vanes positioned below the main propeller are governed to achieve full controllability of the vehicle’s attitude, replacing the role that the tail rotor and the cyclic pitches have in a conventional helicopter. X Specifically, each control vane, considered as a wing immersed in a relative wind Fi Vi [S5], generates lift and drag forces, L

N C = 1 tc L S L1 V 2i c d T , 2 2



with d T /2 being the lever arm of the lift force, which is applied to the center of pressure of the vane, and S L1 being the overall surface. The role of the first level is to control the yaw attitude dynamics and, in turn, to counteract the aerodynamic torque N. The second level, depicted at the bottom of Figure S4, is composed of two independent control vanes, actuated by two independent motors, in which the angles of attack are named, respectively, a and b. The resultant lift forces, denoted by F Lx and F Ly , are directed, respectively, along the body X- and Yaxes of the vehicle and are given by F Lx = 1 tc L S L2 aV i2 , 2



in which S L2 is the surface of each independent control vane on the second level. The application points of the two forces F Lx and F Ly form, with respect to the center of mass of the ­vehicle, a lever arm of length d, see Figure S3. In this way, the

T

where c L, c D, rameters that file (which, in all the vanes)

C D = c D a 2 + c D0 , 

(a)

(S1)

Z

(b) c.m.

where S is the vane’s surface and C L, C D are, respectively, the lift and drag coefficients (see Figure S4). In case of small angles of attack, C L = c L a,

F Ly = 1 tc L S L2 bV i2 2

Y

and D, which can be calculated as L = 1 tSC L V 2i , D = 1 tSC D V 2i ,  2 2

of ­attack. A possible mechanical configuration is formed by the cascade of two different levels of control vanes, denoted, respectively, as (c1) and (c2) in Figure S3. In the first level, depicted at the top of Figure S4, the vanes are disposed radially around the propeller spin axis and, actuated by a single servo, are constrained to have the same angle of attack c with respect to the air flow. In this way, by (S1) and (S2), the aerodynamic lift forces generate a torque contribution equal to

(S2)

and c D0 are constant padepend on the airfoil prothis case, is the same for and a is the vane’s angle

d (c1) c.p.

dT

(c2)

Figure S3  The considered ducted fan aerial vehicle. The system is composed of two main subsystems: a fixed-pitch propeller (b) and a set of control vanes (c1), (c2). Full attitude control authority is the effect of the lift forces resulting from the air flow impacting on the vanes.

AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  47

and (S2). Indeed, by considering forces F Lx and F Ly negligible, compared to the contribution of main thrust T, the previous relations can be approximated as

Fb Vi

X Fb

Z

L

Nz

cp

L

D

Y c

Fb

dT X

Fb

L

Vi

Z

FyL Y

cp FxL

L

D a(b)

Figure S4  Control vanes subsystem. The notation cp denotes the center of pressure of the aerodynamic forces on the control vanes. The actuation mechanism of the control vanes is designed to produce three torques around the vertical, lateral, and longitudinal body axes. ­ enerated two torques control the roll and pitch attitude dyg namics. Overall, by considering the contributions of the control vanes and the propeller, the controlled forces and torques generated by the described mechanism are

b f control =f

F Lx F Ly p, - T + FD

- F Ly d b x control = f F Lx d p N + Nc

in which FD denotes the sum of all drag forces of the control vanes on both levels, which can be calculated by using (S1)

executed by moving on the ground, with advantages in terms of consumed energy, as well as tasks executed midair according to the required capabilities. Figure 2 sketches this operative mode. A further advantage of the considered aerial configuration comes from the possibility of combining the main flight characteristics of rotary wing aircraft, in terms of the ability of taking off/landing vertically and hovering over target points, and the flight features of fixed-wing aircraft, in terms of flight efficiency in covering long distances. In particular, two flight envelopes are feasible for this kind of vehicles, a helicopter-like and an airplane-like flight envelope. The helicopter-like flight envelope, depicted on the left of Figure 3, is characterized by high angles of attack, with the thrust T acting as the main lift force. On the other hand, in the airplane-like flight envelope, depicted on the 48  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

where v = ^a b chT and A (T), B (T) are defined as

X(Y )

0 b f bcontrol . f 0 p, x control . A (T) v + B (T) -T

0 - a1 0 A (T) = T f a 1 0 0 p, 0 0 - a2

0 B (T) = f 0 p N

with a 1 = F Lx d/aT and a 2 = N c /cT . Overall, the control authority of the ducted fan in the vertical direction is achieved by properly controlling the main thrust T through the angular speed of the main rotor. On the other hand, the control authority of the vehicle in the lateral and longitudinal direction is guaranteed by controlling, through the torques N x and N y, the roll and pitch angles of the UAV, namely, by projecting the main thrust T in the desired direction. Finally, the control authority of the yaw dynamics is enforced by properly controlling the torque N z . In brief, full control authority of the UAV in the vertical, lateral, and longitudinal direction and yaw attitude can be obtained by means of the four control inputs, the first corresponding to the angular speed of the main rotor, and the other three governing the vanes’ tilt angles. The combination of the airflow generated by the main rotor and the position of the vanes provides the ability to control the attitude of the vehicle and, as a consequence, to obtain full controllability. References [S5] R. F. Stengel, Flight Dynamics. Princeton, NJ: Princeton Univ. Press, 2004. [S6] H. Schlichting and K. Gersten, Boundary Layer Theory. London: Springer-Verlag, 1979.

right of Figure 3, the UAV moves at high lateral speed with small angles of attack. In this configuration the UAV is equipped with an actuated front canard to improve the overall controllability [17] and with the duct properly shaped to improve its aerodynamical properties. The main lift force comes from the aerodynamic lift generated by the duct, which behaves as a fixed wing, with the thrust T mainly contributing to generate longitudinal acceleration and, in turn, longitudinal speed. This capability makes it possible to figure out operative scenarios in which the ducted fan takes off vertically, performs a flight transition to reach the airplane-like configuration that allows the UAV to cover long distances efficiently and quickly, and then eventually performs a transition back to the helicopter-like configuration to land vertically at the desired destination. It is worth noting how the mentioned transitions

Figure 2  The ducted fan as a ground robot. The ducted fan, equipped with swivel wheels, is capable of combining ground and aerial characteristics. On the ground, in particular, the angular velocity of the main rotor is controlled in a way that the control vanes’ force/torque generating mechanism is active without necessarily taking off. Accordingly, the vehicle moves laterally and longitudinally under the action of the forces F Lx and F Ly, rotating under the action of the torque N z .

from high to small angles of attack hide challenging tasks in terms of modeling and controlling the vehicle in the different flight envelopes. Research attempts in this direction can be found in [18]– [22].

Modelling the aerial vehicle in contact with the environment The variety of operative modes described in the previous section make the modeling phase dependent on the specific mode in which the UAV operates. This fact explains why, in [13], hybrid automata [23], [24] are considered as a natural modeling tool to describe the different dynamics characterizing the vehicle in all possible operative modes. This article focuses on the operative mode in which the system interacts with a vertical surface, by sliding along it while applying a certain force. To make the presentation more tractable, yaw, roll, and longitudinal dynamics are neglected. In other words, we consider the planar dynamics depicted in Figure 4, in which the vehicle is allowed to slide vertically by rotating around the contact point p V while remaining in physical contact with the surface. In deriving the model of the system, particular attention is paid to keep trace of the position of the system center of gravity. To a certain extent, this position can be considered a degree-of-freedom in the mechanical design of the vehicle. The aim is to show how the performance of the controlled system can be influenced by the mechanical design of the vehicle and, specifically, by the position of the center of mass. By bearing in mind the content of “Force and Torque Generation Mechanism in the Ducted Fan,” the system under study is subject to the effect of two main controlled forces, the thrust T, directed along the vertical body axis,

Figure  3  The ducted fan during the transition maneuver. The ducted fan is potentially capable of combining both the flight advantages of a helicopter, in terms of high maneuverability, with those of an airplane, in terms of efficiency during level flight, by performing a transition maneuver. To improve the stability and the efficiency during level flight, the vehicle is equipped with additional aerodynamic surfaces.

and the control vanes’ lift force F, directed along the lateral body axis. The thrust T is proportional to the square of the propeller angular speed while the force F can be approximately considered as proportional to C L V 2i , where C L denotes the lift coefficient associated to the control vanes and Vi is the relative velocity of the wind impacting on the vanes. Given the fact that C L can be taken to be approximately proportional to the vane’s angle of attack a, and Vi

zi zb

xi

Fi

Fb cg

xb g

γT

T T

γg γF

pV

F

θ F

Figure 4  Schematic of the mechanical layout of a ducted fan in contact with a vertical rigid surface. The dynamic model of the system depends on the relative position of the contact point pV with respect to the points of application of the relevant force contributions, namely, the force of gravity, the thrust generated by the propeller (T), and the aerodynamic force produced by the control vanes (F). AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  49

Ducted fan aerial vehicles are a particular class of tail-sitter aircraft.

proportional to T , it follows that F = k F a T with the ­constant k F defined in “Force and Torque Generation Mechanism in the Ducted Fan.” In the following paragraphs the thrust T and the force F are regarded as the control inputs in place of the physical inputs given by the propeller’s angular speed and the vane’s angle of attack, the latter analytically retrievable from (T, F) . Physical constraints on the real inputs limit the admissible values for T and F. In particular, the propeller thrust, which can be applied in one direction and is limited by the maximum power characterizing the electric motor, ranges in the set T ! [ T, Tr ], with 0 1 T 1 Tr and Tr greater than the gravity force. On the other hand, because of mechanical and aerodynamical limitation, the limits on the maximum vane’s angle of attack result in upper bounds on the amplitude of the force F, which ranges in the set F ! [- Fr , Fr ] , with Fr 2 0. More details, relating to numerical values used for T, Tr , and Fr , can be found in the “Simulation Results” section. With reference to Figure 4, let , T and , F be, respectively, the length of the segments connecting the contact point p V and the points where the forces T and F are applied, with c T and c F set as the angles that those segments form with the UAV horizontal axis. The values of these parameters are fixed according to mechanical design specifications. Furthermore, let (x, z) be the position of the center of mass of the vehicle, denoted as cg in Figure 4, expressed in the inertial reference frame Fi . The position of the center of mass is parameterized by the angle cg that the segment connecting cg with p V , whose length is denoted by , g , forms with the UAV horizontal axis. Note that , g cos cg = , T cos c T = , F cos c F is a constant. In modeling the system we follow a Lagrangian approach [25] by identifying two Lagrangian generalized coordinates. Natural generalized coordinates are the angle i by which the UAV is tilted with respect to the vertical direction and the position b of the contact point p V along the z i axis. The latter is

b = z - , g sin (i + c g). 

(1)

axis and within the body of the vehicle, is placed within the vertical surface. By considering a constant viscous friction force along the vertical direction, the generalized forces Fi (T, F) and Fb (z, zo , i, io , T, F) acting on the vehicle with respect to the generalized coordinates i and b are c



T Fi m = G (i) c m - K bo  F Fb

(3)

in which

G (i) = c

, T cos c T - , F sin c F 0 m, K = c m  cos i - sin i mV

(4)

where m V is the viscous friction of the vertical surface. Note that, as , T cos c T = , F cos c F , the matrix G (i) is singular for i = c F . Physically, in this configuration the generalized forces generated by T and F are linearly dependent and, as a consequence, it is not possible to arbitrary assign vertical and angular accelerations (bp , ip ) by choosing the control inputs. In this configuration the system loses full control authority, as only a linear combination of the vertical and angular acceleration can be assigned arbitrarily. With an eye on Figure 4, let us assume that the mass M of the vehicle is concentrated in the point cg. By considering kinetic and potential energies, the Lagrangian function of the system can be computed as

o o + , 2g io 2 h L (b, bo , i, io ) = 1 M ^ bo 2 + 2, g cos (i + c g) bi 2 - Mg ^ b + , g sin (i + c g)h  (5)

and it is governed by the Lagrangian equations d 2L - 2L = F , i dt 2io 2i

d 2L - 2L = F . b dt 2bo 2b

By using (3)–(5), the Lagrangian equations can be explicitly written as g, g cos (i + c g) T ip o  (6) L (i) e p o = G (i) c m - K bo - M e F g - , g sin (i + c g) io 2 b in which

Regarding the angle i, note that, with the help of Figure 4, angular configurations that are physically plausible for the mechanical system are characterized by

i 1 r - cg . 2

(2)

As a matter of fact, i $ r/2 - cg corresponds to a configuration in which the point cg, which is located on the vertical 50  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

L (i) := M e

, 2g , g cos (i + c g) o 1 , g cos (i + c g)

and G (i) and K are defined as in (4). It is a fourth-order system with state (b, bo , i, io ) (or, alternatively, with state (z, zo , i, io )) , and inputs T and F. As the matrix L (i) is singular for i = - c g , the dynamics (6) fall into the class of the descriptor systems [26]. Indeed, for the purposes of this article, we do

not approach the study of the previous dynamics in terms of descriptor systems but rather we consider i = - c g as a singular configuration to be avoided by the closed loop solution. This fact allows us to consider the matrix L (i) invertible for all the values of i in the range of interest and, in turn, suggests a preliminary choice of the control inputs (T, F) as T F1 c m = Q (i) c m  F F2



(7)

where (F1, F2) are residual control inputs and Q (i) := G (i) -1 L (i). For all the values of i for which G (i) and L (i) are not singular, Q (i) is not singular and thus (F1, F2) can be alternatively used as control inputs instead of (T, F) . The choice (7) is meant to decouple i and b dynamics in terms of control inputs. Specifically, using (7) in (6) and the fact that L (i) is not singular for the i of interest, it appears that (6) transforms as ip = F1 - , i (i, io , bo ),



bp = F2 - , b (i, io , bo ) 

(8)

where

cos (i + c g) o 2 cos (i + c g) o , i (i, io , bo ) := i - mV b, sin (i + c g) M, g sin 2 (i + c g)

(9)

,g mV , b (i, io , bo ) := g io 2 + bo .  (10) sin (i + c g) M sin 2 (i + c g)  The underlying assumption behind the previous developments is that the UAV remains in contact with the vertical surface, namely, that the lateral movement of the vehicle is indeed constrained. In this respect, we now identify sets of the input and state space guaranteeing that the underlying contact condition holds. Note that the condition under which the contact is preserved is that the overall lateral force generated by the rotor and control vanes’ mechanism points toward the vertical surface. Keeping Figure 4 in mind, the mentioned lateral force is FEnv = T sin i + F cos i , namely, by bearing in mind (7),

FEnv (i, F1, F2) = ^sin i cos i h Q (i) c

F1 m.  F2

(11)

Hence, contact is preserved, and thus all the previous analysis are validated, if i and (F1, F2) are such that FEnv (i, F1, F2) 2 0. Quantitative considerations can be made to simplify the previous condition. Specifically, under the assumption that T 22 F, it follows that the contact condition is guaranteed if i $ ir where ir := arctan (Fr / T) . In summary the model of the system is given by (8)–(10) under the condition that

i ! ` ir , r - c g j , i ! c F , i ! - c g .  2

(12)

This model is used in the following section to design hybrid  force and position feedbacks. Note that, since

, g = , F cos c F/cos c g , the model is fully parameterized in terms of c g , namely, in terms of the position of the center of mass of the vehicle.

Hybrid force and position control while interacting with the environment The goal is to design a hybrid force and position feedback (see “Force Feedback in Robotics” and the references therein) by controlling the force applied perpendicularly to the surface and the position of the contact point to track desired reference signals, which, in the following, ref are denoted respectively by y ref force (t) and y pos (t) . A possible scenario behind this control problem is where the UAV, equipped with NTD sensors, is required to scan a surface to detect possible cracks or small defects. In this case, the desired force applied to the environment is typically required to be constant and equal to the ideal contact pressure given by the specifications of the NTD r ref sensor. Motivated by this, consider y ref force (t) / y force , in ref which yr force 2 0 is the constant desired force setpoint. The reference signal of the contact point, instead, might be imposed in real time by a human who teleoperates the aerial vehicle by probing the surface around an operative point. In other scenarios, the position reference signal might be generated according to predetermined scanning policies. In this respect it is convenient to express the desired position of the contact point as ref ref r ref y ref pos (t) = y pos + b (t) , in which b (t) is the time-varying displacement of the desired position from a fixed operative point denoted by yr ref pos . We consider system (8) and two outputs y 1 and y 2 , representing the force applied to the environment and the position of the contact point, defined as y 1 = R (i) c

F1 m, y2 = b F2

with R (i) := ^sin i cos i h Q (i). To avoid algebraic loops for the controlled system due to the presence of the control inputs in the expression of the first output, add an integrator to the two input channels, namely,

go 1 = u 1 ,

F1 = g 1 , 

(13)



go 2 = u 2 ,

F2 = g 2 , 

(14)

where u 1 and u 2 are the new control inputs. The overall system is a nonlinear system with state x = (i, io , b, bo , g 1, g 2) ! R 6 , control inputs u = (u 1, u 2) ! R 2, and output y = (y 1, y 2) ! R 2 described in compact form as

.

x = f (x) + Gu ,

y = h (x) , 

(15)

AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  51

with J N x2 K x 5 - , (x 1, x 2, x 4) O i K O x5 04 # 2 x4 K O R (x 1) c mp c m f , , = = ( )   f (x) = K G h x x6 . x 6 - , b (x 1, x 2, x 4)O I2 x3 K O 0 KK OO 0 L P  (16) The state variable x 1 ranges in the set of values specified in (12), guaranteeing that the contact between the UAV and the surface is preserved and that other singularities are

avoided. Furthermore, associate to the system a setpoint error variable describing the displacement of the outputs from the reference setpoints and defined as

γT T

γF

γg

pV

(17)

By using the expressions in (9)–(10), a simple calculation shows that this system has equilibrium points at x 1 = x 1* , x 3 = x 3* , x 2 = x 4 = x 5 = 0 , and x 6 = g , with u 1 = u 2 = 0 , where x *1 and x *3 are arbitrary values, with x *1 being in the allowed set of angular configurations. At these equilibrium points the values of the two outputs are

T

yr force yu = hu (x), hu (x) := h (x) - yr ref , yr ref := e ref o.  yr pos ref



y *1 = } c g (x *1) , y *2 = x *3 

(18)

with the function } c g ($) defined as } c g (x 1* ) = T * (, F cos (x *1 - c F) - , g cos (x *1 + cg ))

F

F

zi

Fi

where T * is the value of the control input T at the equilibrium resulting from

g

c

zb

θ

Fb cg

xb

xi

Figure 5  Schematic of the mechanical layout of a ducted fan in contact with a vertical rigid surface. With respect to the layout shown in Figure 4, this configuration has the center of gravity placed below the point where F is applied c g 1 -c F . In this mechanical configuration, the avionics and the battery pack, which contribute substantially to the weight of the vehicle, are placed in the lower section of the vehicle. As shown in the article, this mechanical configuration is unable to fly in contact with the vertical surface.

γg = π/3 γg = π/4(=γT) γg = π/8

y1ref (N)

20 18 16 14 12 10 8 6 4 2 0 0

γg = 0 γg = −π/8

0.1

0.2

0.3

0.4

0.5

0.6

ref

x1 (rad)

Figure 6  Relation between the force applied to the environment, ref y ref 1 , and the pitch angle of the vehicle, x 1 , resulting from cF = cT = π ∕4, ℓT = ℓF = 0.5, M = 1.5 and varying the value of cg, as specified in the different plots. The figure reveals how the force applied to the environment at a given attitude configuration of the pitch is strongly influenced by the mechanical layout of the vehicle, and, in particular, by the position of the center of gravity parameterized by cg. 52  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

0 T* m = Q (x *1) e o. g F*

Note that, from the physics of the system, T * 2 0 . The first of (18) reveals the relation between the force y *1 applied to the environment at the equilibrium and the coefficient c g parametrizing the position of the center of gravity. Specifically, if c g = - c F , and thus , g = , F , namely, if the center of gravity is located where the control vane’s force F is applied, necessarily y *1 = 0 for all x *1 . On the other hand, for c g 2 -c F, the function y *1 is positive and strictly increasing with x *1 . Similarly, for c g 1 -c F , the function y *1 is negative and strictly decreasing with x *1 ; see Figure 5. The previous considerations show the structural inability of the mechanical system, when the center of mass is located below the point where F is applied, to maintain an equilibrium configuration with the applied force pointing towards the vertical surface, namely, with y *1 2 0. In other words, any mechanical configuration characterized by c g 1 - c F has an equilibrium configuration that tends to detach the vehicle from the vertical surface. Hence, in the following sections, the values of c g are chosen to be 2 - c F . In these cases Figure 6 shows the plot } c g (x *1) for different values of c g 2 - c F. As shown by the plots, the force applied to the environment at a constant value of the angle increases with the value of c g . Furthermore, for any fixed value of c g, the function } c g (x *1) is invertible with the inverse yielding the required pitch angle of the UAV to apply a specific constant force to the environment. Among the possible equilibrium points of the system, select the one associated to the reference setpoints. In parref ref r ref r ref ticular, let x ref pos . 1 be such that y force = } c g (x 1 ) and x 3 = y ref ref ref Then the point x = col (x 1 , 0, x 3 , 0, 0 , g) is the equilibrium point of system (15) at which yu = 0 .

The mechanical design of the vehicle, and in particular the position of the center of gravity, influences the capability of the system to fly in contact with the vertical surface. Relative Degree, Normal Form, and Zero Dynamics This section discusses the properties of the system, regarded as a system with input u and output yu , in terms of the existence of the vector relative degree and zero dynamics, and computes its normal form (see “Vector Relative Degree and Normal Form for Multi-Input, Multi-Output Nonlinear Systems” and “Zero Dynamics”). This analysis is instrumental to the control design described in the next section. Let hu (x) and G in (16) and (17) be partitioned as hu (x) = (hu 1 (x), hu 2 (x))T and G = (g 1, g 2) T with hu i (x) being real-valued functions, and g i ! R 3 # 1 . By bearing in mind the theory in “Vector Relative Degree and Normal Form for Multi-Input, Multi-Output Nonlinear Systems” and by using the expressions in (16) and (17), observe that L g 1 hu 1 (x) = R 1 (x) and L g 2 hu 1 (x) = R 2 (x) , having denoted by R 1 (x) and R 2 (x) the two components of R (x) . Furthermore, k = 0, 1 , while L g i L kf hu 2 (x) / 0 for 2 u L g 1 L f h 2 (x) = 0 and L g 2 L 2f hu 2 (x) = 1 . Hence the system has vector relative degree {1, 3} if the matrix A (x) = e



p 11 + yr ref force - R 2 (h 1) (p 23 + , b (h 1, h 2, p 22)) , R 1 (h 1) x 6 = p 23 + , b (h 1, h 2, p 22). x5 =

We observe that the diffeomorphism is well-defined for all x 1 such that R 1 (x 1) is not zero, namely, where the vector relative degree is well-defined. Applying the change of variable results in the normal form

R 1 (x 1) R 2 (x 1) L g 1 hu 1 (x) L g 2 hu 1 (x) m o=c 0 1 L g 1 L 2f hu 2 (x) L g 2 L 2f hu 2 (x)

is not singular, namely, if the first element of the vector R (x 1) is not zero, for all x 1 in the range of interest. Since R (x 1) depends on c g through L (x 1) , it is possible to spot the locus of points in which the system has not vector relative degree (namely, where R 1 (x 1) = 0 ) in the plane c g - x 1 . The locus is marked in red in Figure 7 for values of physical parameters specified in Table 1. Apart from this red locus, the system has a well-defined vector relative degree, namely, the first time-derivative of yu 1 and the third time-derivative of yu 2 can be arbitrarily assigned through an appropriate choice of the control inputs. Now continue the analysis under the assumption that R 1 (x ref 1 ) ! 0 , namely the system has a well-defined vector relative degree at x ref . This implies that the system in question is diffeomorphic, locally with respect to x ref , to a system in special normal form (see “Vector Relative Degree and Normal Form for Multi-Input, Multi-Output Nonlinear Systems”), which is computed in the next part of the section. The normal form is the starting point of the control design stage addressed in the forthcoming section. Define

and let z 1 (x) : = p 11, z i+1 (x) : = p 2i, i = 1, 2, 3 . The change of variables is completed with two additional functions z 5 (x) and z 6 (x) such that L g j z i (x) = 0, j = 1, 2, i = 5, 6, and U (x) : = (z 1 (x), z 2 (x), f, z 6 (x)) is a diffeomorphism in the neighborhood of the equilibrium point x ref . These properties can be fulfilled by choosing z 5 (x) = x 1 and z 6 (x 1) = x 2 . The change of variables is thus (p, h) = U (x) , p = (p 11, p 21, p 22, p 23), h = (h 1, h 2) , with the inverse x = U -1 (p, h) defined as x 1 = h 1, x 2 = h 2, x 3 = p 21 + yr ref pos, x 4 = p 22 , and

x5 ref , p 11 : = hu 1 (x) = R (x 1) c m - yr force x6 ref p 21 : = hu 2 (x) = x 3 - yr pos , p 22 : = L f hu 2 (x) = x 4 , p 23 : = L 2f hu 2 (x) = x 6 - , b (x 1, x 2, x 4),

o h1 e o = q 0 (h 1, h 2) + q 1 (h 1) (p 11 + yr ref force) + q 2 (h 1) p 22 + q 3 (h 1) p 23 , ho 2 

(19) .

.

f . 21 p = f . 22 p , p

p

p 22

p 23

.

f . 11 p = b (p 11, p 22, p 23, h 1, h 2) + A (h 1) u p

p 23



(20)

where q 0 (h 1, h 2) = f

h2 , i (h 1, h 2, 0) p , 1 m ^R 1 (h 1) R 2 (h 1) h c , b (h 1, h 2, 0) R 1 (h 1)





A (h 1) = c

R 1 (h 1) R 2 (h 1) m, 0 1

q 1 (h 1) = f

0 1 p, R 1 (h 1)

N J 0 K J m cos (h 1 + c g) NO K OO K V 2 q 2 (h 1) = K R (h 1) K , g M sin (h 1 + c g) OO , K R 1 (h 1) K m V OO 1 KK OO K2 M sin (h 1 + c g) O PP L L 0 q 3 (h 1) = f - R 2 (h 1) p R 1 (h 1) AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  53

Vector Relative Degree and Normal Form for Multi-Input, Multi-Output Nonlinear Systems

T

he change of coordinates in the state space is an important tool very often used to transform nonlinear systems in special forms that make easier the interpretation of structural properties of the system and more straightforward certain design procedures. Among the possible changes of coordinates, an important role is played by the one originally proposed in [S14] transforming the system into a “special normal form.” The latter has been shown [S18] to be a privileged starting point for the solution of many control problems such as feedback linearization, noninteracting control, state and output feedback stabilization, output regulation, and tracking by system inversion. The main notions and steps that are needed to obtain the special normal form for multi-input, multi-output nonlinear systems are presented below. Consider the nonlinear system



xo = f (x) + y j = h j (x),

m

/ g i (x) u i ,

i=1

x ! R n, u i ! R y j ! R, j = 1, f, m 

(S3)

with the same number m $ 1 of inputs u = (u 1, f, u m) T and outputs y = (y 1,f, y m) T . The state x = (x 1, f, x n) T is assumed to belong to an open set U 3 R n and the functions f : U " R n, g i : U " R n, i = 1, f, m , and h j : U " R, j = 1, f, m , are assumed to be smooth. With 1 # j # m and k $ 1, denote by L kf h j (x) the kth derivative of h j (x) along f (x) , which is recursively defined as L f h j (x) =



dh j (x) f (x) dx

and



L of h j (x) =

dL of - 1 h j (x) f (x) for all o = 2,f, k . dx

Similarly, with 1 # i # m, 1 # j # m and k $ 1, define L g i L kf h j (x) as

L g i L kf h j (x) =

dL kf h j (x) g i (x). dx

Instrumental to computing the normal form of (S3) is the notion of vector relative degree of the system. System (S3) is said

and b (p 11, p 22, p 23, h 1, h 2) is a function having the property that b (0, 0, 0, x ref 1 , 0) = 0 . Note that this system, with (u 1, u 2) = (0, 0) , has equilibrium points at (p ref, h ref) with p ref = (0, 0, 0, 0) and h ref = (x ref 1 , 0) . This special normal form clearly shows also the zero dynamics of the system (see “Zero Dynamics”), namely, the dynamics governing the internal behavior of the system when the initial state and the inputs are chosen in such a 54  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

to have vector relative degree {r1, f, rm} at a point x 0 ! R n if the following two properties hold: i) L g i L kf h j (x) = 0 for all 1 # i # m , for all 1 # k 1 r j , for all 1 # j # m and for all x in a neighborhood of x 0 ii) the m # m matrix



J L g 1 L rf 1 - 1 h 1 (x) K r2 - 1 K L g 1 L f h 2 ( x) A (x) = K g K rm - 1 L L g 1 L f h m ( x)

g L g m L rf 1 - 1 h 1 (x) N O g L g m L rf 2 - 1 h 2 (x) O O g g O g L g m L rf m - 1 h m (x)P

is nonsingular at x = x 0 . Properties i) and ii) can be interpreted as follows. Let x (t) and u i (t), i = 1, f, m , be the value of the state and of the control inputs at a time t. Property i) is equivalent to say that the jth output y j (t) and its first r j - 1 time derivatives y (jk) (t), k = 1, f, r j - 1, at time t are not influenced by any of the control inputs u i (t) as long as x (t) is in the neighborhood of x 0 . Moreover, by property ii) and by the continuity of the functions in the matrix A (x) , note that A (x (t)) is nonsingular for all x (t) in the neighborhood of x 0 . This fact implies that property ii) is equivalent to the condition that the r j th time derivatives of the outputs y j (t), j = 1, f, m , can be arbitrarily assigned through an appropriate choice of the control input u (t) = col ^u 1 (t) f u m (t ) h .

It is worth emphasizing that the definition of vector relative degree is local about a point x 0 and that there could be points in the state space where the vector relative degree is not defined. Furthermore, if a system of the form (S3) has vector relative degree {r1, f, rm} at a point x 0 , then necessarily r1 + g + rm # n (see [S18, Prop. 5.1.2]). If a system of the form (S3) has vector relative degree {r1, f, rm} at a point x 0 , it can be transformed in a special normal form by means of a change of variables defined in the neighborhood of x 0 . In particular, with r = r1 + g + rm , let U p (x) : U " R r be the smooth function defined as

U p (x) = col ^h 1 (x) g L rf 1 h 1 (x) f h m (x) g L rf m h m (x) h.

It turns out (see [S18, Prop. 5.1.2]) that there always exists a smooth map U h (x) : U " R n - r , which degenerates into the empty map if r = n , such that U (x) = col (U p (x), U h (x)) is a local

way as to constrain the output yu to be identically zero. In fact they are ho 1 = h 2    R 1 (h 1) ho 2 = - ^R 1 (h 1) R 2 (h 1)h c , i (h 1, h 2, 0) m + yr ref force . (21) , b (h 1, h 2, 0) Note that this system has an equilibrium point at h ref = (x ref 1 , 0) . Furthermore, observe that the previous

diffeomorphism about x 0 and thus qualifies as a ­local change of variables. Specifically, by changing the coordinates as η = q (ξ, η)

U p (x) p m = U (x), x 7e o=c U h (x) h



ho po j1

= = g =

yj

= p j1 ,



ym ξm1

ξ12

ξm 2

....

m

/ a ij (p, h) u i ,

i=1

j = 1, f m



(S4)

where q (p, h), p (p, h) and b j (p, h), j = 1, f, m , are properly defined smooth functions and the functions a ij (h, p) are such that J a 11 (p, h) K K a 21 (p, h) K g K La m1 (p, h)

ξm

ξ11

q ( p, h) + p ( p, h) u , p j2 , , p jr j ,

po jr j - 1 po jr j = b j ( p , h) +

ξ1

y1

and by partitioning p as p = col (p 1, f, p m) with p j = (p j1, f, p jr j) ! R r j, j = 1, f, m , system (S3) locally transforms into the special normal form

f a 1m (p, h) N O f a 2m (p, h) O -1 O = A (U (p, h)). g g O f a mm (p, h)P 

ξ1r1

ξmrm m

u

ξ jr = bj (ξ, η) + j

(S5)

The h -dynamics, which are absent if r = n , can be further refined if the function U h (x) is chosen in appropriate way. Specifically, denoting by U j (x), j = r + 1, f, n , the components of U h (x) , it turns out that if U h (x) is chosen so that L g i U j (x) = 0 for all i = 1, f, m , for all j = r + 1, f, n , and for all x in a neighborhood of x 0 , then p (p, h) = 0 for all (p, h) in a neighborhood of U (x 0) . Sufficient conditions under which the U h (x) can be chosen in this way can be found in [S18, Prop. 5.1.2]. From a graphical viewpoint the normal form can be interpreted as in Figure S5. Generalization of the presented local tool to normal forms that are globally defined can be found in [S12], for singleinput single-output systems, and in [S15], for multi-input, multioutput systems. References [S12] W. P. Dayawansa, W. M. Boothby, and D. Elliott, “Global state and feedback equivalence of nonlinear systems,” Syst. Contr. Lett., vol. 6, no. 4, pp. 229–234, 1985.

dynamics depend on c g entering in the definition of , b ($) and , i ($) . The stability properties of the zero dynamics play a role in the design of the control input, by impacting on the achievable performance of the controlled system. In this respect, of particular interest is the ­influence that the position of the center of mass of the ­vehicle, parameterized in the model at hand by c g , has on the stability properties of the zero dynamics. The aim is to identify

∑ a (ξ, η) u ij

i

η

i=1

j = 1,...,m

Figure S5  A graphical sketch of the multi-input, multi-output normal form. The normal form can be interpreted as m chains of integrators, each one of length rj, which are driven by the state variables ( p 1r1, g, p mrm ) and whose state p drives the dynamics ho = q (p, h) . The m outputs of the system coincide with the state variables on the top of the m chains, namely y j = p j1, j = 1,…, m. The state variables p jr j (t) at the bottom of the chains, evaluated at a time t, coincide with the rjth time derivative of the outputs y j (t),  j = 1,…, m. Due to (S5) and to the fact that A (U -1 (p, h)) is nonsingular, the m state variables p 1r1 (t), f, p mrm (t)) at a generic time t can be arbitrarily assigned by an appropriate choice of the input u(t).

[S13] A. Isidori, Nonlinear Control Systems. London: Springer-Verlag, 1995. [S14] A. Isidori, A. J. Krener, C. Gori Giorgi, and S. Monaco, “Nonlinear decoupling via feedback: A differential geometric approach,” IEEE Trans. Automat. Contr., vol. AC-26, no. 2, pp. 331–345, 1981. [S15] B. Schwartz, A. Isidori, and T. J. Tarn, “Global normal forms for MIMO nonlinear systems, with applications to stabilization and disturbance attenuation,” Math. Control Signals Syst. (MCSS), vol. 12, no. 2, pp. 121–142, 1999.

possible mechanical solutions that make the control design easier and improve the possible performance of the controlled vehicle. To study the stability properties of the zero dynamics, start by analyzing the linear approximation of (21) at the equilibrium point h ref . The Jacobian of the system is J=c

0 1 m d1 d2

AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  55

Zero Dynamics

T

he concept of zero dynamics of a nonlinear system was introduced in [S16] as generalization of the concept of transmission zeros of a transfer function. According to the definition given in [S16], zero dynamics are “the dynamical system that characterizes the internal behavior of a system once initial conditions and inputs are chosen in such a way as to constrain the output to be identically zero.” One of the most important consequences of the introduction of differential geometric tools in the analysis and design of nonlinear feedback systems [S17] was the possibility to extend, to nonlinear systems, the notion of transmission zero. The extension was possible since the notion of zero for linear systems can be naturally characterized in terms of geometric techniques, with the zeros being the eigenvalues associated to a subsystem that can be rendered unobservable by means of a feedback. This characterization has been extended to nonlinear systems in [S17] and independently adopted in [S16] and [S21] to show that nonlinear systems having the subsystem being asymptotically stable can be stabilized by means of high-gain feedback, by extending in this way a property that is well known to hold for linear minimum-phase systems. The relevance in problems of asymptotic stabilization, in fact, is the reason why nonlinear systems are typically classified as minimum- or nonminimum-phase according to the as-

ref and recalling that yr ref force = R 2 (x 1 ) g , the two coefficients d 1 and d 2 are

where, by noting that , i (x ref 1 , 0, 0) = 0 , , b (0, 0, 0) = g , 2, b 2, = i = 2x 2 h ref 2x 2

h ref

2, i 2x 1

h

ref

=

2, b 2x 1

h ref

x1 (rad)

0.2

e

0.3 0.4

Stable

Unstable

0.5 0.6 γF 0.7 –1

–γF

–0.5

0 γg (rad)

0.5

ref

x1

,

d 2 = 0.

1

Figure 7  The red locus identifies values of the pair (cg, x1) for which the vector relative degree of the system is undefined. According to the notion of vector relative degree, these points correspond to configurations in which the first and the third derivatives of the two outputs y1 and y2 cannot be arbitrarily assigned by using control inputs. The locus divides the plane in regions, whereby the linear approximation of the zero dynamics of the system is, respectively, stable and unstable. 56  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

g dR 2 R 1 (x ref 1 ) dx 1

The fact that d 2 = 0 implies that the linear approximation is never asymptotically stable. In particular, it is stable (two complex eigenvalues on the imaginary axis) if d 1 1 0 and unstable (with one positive and one negative eigenvalue) otherwise. The fact that d 1 depends on x ref 1 and on c g leads us to study how the stability properties of the Jacobian vary in the plane x 1 - c g . Figure 7 shows how the plane is divided in regions whereby J is stable, namely, d 1 2 0, and J unstable, namely, d 1 # 0 . Changes in the stability properties of the Jacobian occur at the points, marked in red in Figure 7, in which the vector relative degree of the system is not defined.

bl

ta

s Un

d1 = -

= 0,

0 0.1

ymptotic stability properties of their zero dynamics. The concept of zero introduced in this way, then, was shown to have many other characterizations associated to the concept of transmission zero and to be relevant in many other analysis and design control contexts, such as in feedback linearization, noninteracting control, output regulation, and in characterizing the limit of performances for nonlinear systems. An historical perspective and actual research directions in this field can be found in [S20]. In general terms, zero dynamics can be computed by means of an algorithm, usually referred to as the zero dynamics algorithm [S18, p. 294], that computes the locally maximal output zeroing manifold associated with a nonlinear system, which is the nonlinear counterpart of the largest controlled invariant subspace contained in the kernel of the output map of a linear system. If the system possesses a vector relative degree, and thus it can be transformed in the special normal form (see “Relative Degree and Normal Form for Multi-Input, Multi-Output Nonlinear Systems”), the computation of zero dynamics is sensibly simpler and can be described as follows. Consider systems of the form (S4). Without loss of generality, let the point x 0 about which the special normal form is computed be such that x 0 = 0 , and assume that f (0) = 0 and h j (0) = 0, j = 1, f, m , namely, the origin of (S4) is an

Table 1 Numerical values of the UAV’s physical parameters used in simulation. The values refer to a ducted fan prototype available at the University of Bologna and described in [38]. ℓF = 0.5 m

ℓT = 0.5 m

M = 1.5 kg

Fr = 3 N

c F = r rad 4

c T = r rad 4

T = 12 N

Tr = 20 N

­ quilibrium point for the unforced system at which the outputs e are vanishing. In this case the local diffeomorphism putting the system in normal form can be constructed so that U (0) = 0. By the normal form (S4), the constraint that the outputs y j (t), j = 1, f, m , are identically zero is equivalent to the constraint p ji (t) / 0, i = 1, f, r j, j = 1, f, m , namely p (t) / 0 , and po jr (t) / 0, j = 1, f, m . This implies that the initial condition of j

the system that is compatible with outputs identically zero is of the form x 0 = U -1 (0, h 0) for any h 0 sufficiently close to the origin. Furthermore, the control input that is compatible with the previous constraints is

-1

-1

u (t) = - A ( U (0, h (t))) b (0, h (t)) 

(S6)

in which b (0, h) = col (b 1 (0, h), f, b m (0, h)) and h (t) is the solution of   ho (t) = q (0, h (t)) - p (0, h (t)) A -1 ( U -1 (0, h (t))) b (0, h (t))  (S7) with initial condition h 0. Note that h = 0 is an equilibrium point of (S7) and that the previous dynamics are well defined as long as h is sufficiently close to the origin, namely, as long as the normal form is well defined. Dynamics (S7) precisely charac-

Figure 7 can be interpreted as a bifurcation diagram of the zero dynamics, where both the equilibrium angle x ref 1 , related to the desired force yr ref force , and the design parameter c g can be considered as bifurcation parameters. Specifically, by scanning Figure 7 in the horizontal direction at a certain value of the angle x ref 1 , namely, of the force applied to the environment, c g can be interpreted as a bifurcation parameter that can be adjusted, at the mechanical design stage, to infer certain stability properties to the zero dynamics. Similarly, by scanning the figure in the vertical direction with a fixed c g , the diagram shows how the stability properties of the zero dynamics depend on the possible equilibrium point, namely, on the entity of the force applied to the environment. In Figure 8 the phase portraits of the zero dynamics associated to different points with the same x ref 1 = 0.4 and different c g are sketched. In the regions where the Jacobian is unstable, the equilibrium point (x ref 1 , 0) of the nonlinear zero dynamics is, of course, unstable. On the other hand, in the other parts of the plane, the zero dynamics exhibit a set of limit cycles centered at (x ref 1 , 0) that tend to shrink to the equilibrium point as (x ref 1 , c g) approaches the boundary regions. As clarified in the next part presenting the control design, it is beneficial to focus on operating points where the Jacobian is stable so that the zero dynamics can be stabilized by means of minimal control efforts. This fact, in turn, makes it possible to achieve the hybrid force and position tracking

terize the internal behavior of the system once initial conditions and inputs are chosen in such a way as to constrain the output to be identically zero, namely (S7) are the zero dynamics of the system. In this case the system is minimum phase if the equilibrium point h = 0 of (S7) is locally asymptotically stable and nonminimum-phase otherwise. A global and coordinate-free version of the computation in question can be found in [S18, Par. 9.1] for single-input, singleoutput systems and in [S19, Par. 11.5] for a rather general class of multivariable systems. References [S16] C. I. Byrnes and A. Isidori, “A frequency domain philosophy for nonlinear systems,” in Proc. IEEE Conf. Decision and Control, Las Vegas, 1984, pp. 1569–1573. [S17] A. Isidori, A. J. Krener, C. Gori Giorgi, and S. Monaco, “Nonlinear decoupling via feedback: A differential geometric approach,” IEEE Trans. Automat. Contr., vol. AC-26, no. 2, pp. 331–345, 1981. [S18] A. Isidori, Nonlinear Control Systems. London: Springer-Verlag, 1995. [S19] A. Isidori, Nonlinear Control Systems II. London: Springer-Verlag, 1999. [S20] A. Isidori, “The zero dynamics of a nonlinear system: From the origin to the latest progresses of a long successful story,” in Proc. 30th Chinese Control Conf. (CCC’2011), Yantai, China, pp. 18–25. [S21] R. Marino, “High-gain feedback in nonlinear control systems,” Int. J. Contr., vol. 42, no. 6, pp. 1369–1385, 1985.

objectives with a low degradation of performance, the latter due to the nonminimum phase behavior of the system.

Control The goal is to have the force y 1 , applied by the vehicle to the environment, and the vertical position y 2 of the contact r ref point, tracking respectively reference signals y ref force (t) / y force ref ref ref and y pos (t) = yr pos + b (t) . We start by designing a preliminary state feedback of the form u = A (h 1) -1 ^v - b (p 11, p 22, p 23, h 1, h 2)h 



(22)

with v = (v 1, v 2) residual control inputs, aiming to partially feedback linearize system (19)–(20) and to decouple the two outputs (y 1, y 2) from the two inputs (v 1, v 2) . Indeed, system (19)–(20) controlled by (22) assumes the form

e

ho 1 o = q 0 (h 1, h 2) + q 1 (h 1) (p 11 + yr ref force) + q 2 (h 1) p 22 + q 3 (h 1) p 23 , ho 2

 $

p 11 = v 1,

J$ N J$ N Kp 21O Kp 22O K$ O K$ O Kp 22O = Kp 23O  Kp$ O L v 2 P L 23P

(23) (24)

namely, it is given by two chains of integrators driven by the two inputs, with the two controlled outputs on the top, AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  57

0.5 0.45

x1 0.4 –1.5

–1

0

–0.5

x2

x1

–3

–2

–1

0.4 0.405 0.41 0.415 x1 0.39 0.395

0.4

–0.15

–0.1

–0.05

0

0.05

0.1

0.15

–0.2

–0.1

0

0.1

0.2

0.3

0.36 0.38

x2

0.4

0

1

2

3

–γF

–0.5 –1 0.7

γF

0.6

0.5

0.42 0.44 0.46 0.48 x1 γg = –0.6

0.4

0.3

0.2

0.1

0

x2

γg = –0.68

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0 γg

0.5

γg = –0.17

1

0.5

1.5

–0.2 –0.3

0

–0.1

0.1

0.3

0.2

1

0.26 g=

0.5 0.4

0.35

γg = 0.26

39 92 94 96 98 .4 02 04 06 08 41 0. 0.3 0.3 0.3 0.3 0 0.4 0.4 0.4 0.4 0. x1 γg = 0.24

x2

x2

Figure 8  Phase portraits of the zero dynamics associated to points in the plane x1 – cg with the same x ref 1 = 0.4 and different cg (specified in the plots). As expected, in the regions where the Jacobian is unstable (see Figure 7), the phase portraits show how the trajectories diverge from the equilibrium point ( x ref 1 , 0). In other sections of the plane, the zero dynamics exhibit a set of limit cycles centered at ref ( x ref 1 , 0) that tend to shrink to the equilibrium point as ( x 1 , cg) approaches the boundary regions. 58  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

The stability properties of the zero dynamics play a role in the design of the control input, by impacting on the achievable performances of the controlled system. feeding the zero dynamics of the system. A graphical sketch of system (23)–(24) is shown in Figure 9. The design of (v 1, v 2) must be accomplished according to two main control objectives. On the one hand, the control inputs must be designed to guarantee the tracking of the reference signals. On the other hand, the control design is required to stabilize the zero dynamics that, if left uncontrolled, might produce unstable angular motions of the vehicle leading it to move away from the vertical surface. For the system (23)–(24), two stabilizing channels for the zero dynamics can be identified: a force channel, in which the state variable p 11 = y 1 - yr ref force is used as virtual control input to stabilize the (h 1, h 2) dynamics through the gain q 1 (h 1) , and a position channel, in which the state variables (p 22, p 23) = (yo 2, yp 2) are used to this purpose through the gains q 2 (h 1) and q 3 (h 1) . This fact suggests that two different control strategies are, in principle, possible depending on the channel that is used to stabilize the zero dynamics. The first one in which the control input v 1 is designed to enforce the desired setpoint yr ref force in the output y 1 and, simultaneously, to stabilize the zero dynamics, while the control input v 2 is employed to enforce the desired position reference y ref pos (t) . The second one, in which the position channel is used to stabilize the zero dynamics, with v 2 designed to enforce the desired position reference y ref pos (t) while stabilizing the (h 1, h 2) dynamics, and v 1 used entirely to enforce the force ­setpoint yr ref force . In the following analysis, let S be the set where h 1 and h 2 are required to range to fulfill physical requirements. In particular, according to the results obtained while modeling the system, (h 1, h 2) ! S requires that h 1 , which is equal to i , fulfills the constraints (12), while h 2 is limited according to physical bounds underlying the angular speed of the vehicle.

regarded as a system with state (h 1, h 2) , control input v ! R , and exogenous bounded disturbances d 1, d 2 , and d 3 . For this system, formulate the design requirement as follows.

Stabilizability Requirement Design a smooth state feedback of the form v = Y (h 1, h 2) 



(26)

r ref with Y (x ref 1 , 0) = y force , so that, for some compact set X ! R # R and positive numbers D i, i = 1, 2, 3 , all the ­possible trajectories of the closed-loop system (25)–(26) originating from initial conditions (h 1 (0), h 2 (0)) ! X under the action of bounded d i fulfilling d i 3 # D i, i = 1, 2, 3 , satisfy (h 1 (t), h 2 (t)) ! S for all t $ 0 and lim sup h (t) - h ref # c lim sup (d 1 (t), d 2 (t), d 3 (t)) t"3

t"3

for some positive c. The stated requirement is that the zero dynamics (25) can be rendered input-to-state stable (ISS) at the equilibrium point h ref (see [30] and [31]) with respect to bounded

(η1,η2) y2 = ξ21

y1 = ξ11

v1

ξ22

ξ23

Using the Force Channel to Stabilize the Zero Dynamics In designing the controller, we follow a backstepping strategy [28], [29] that starts from the design of a stabilizer for the (h 1, h 2) subsystem by considering p 11 as virtual control input. Then, the virtual control input is back-stepped to design the real control inputs. For this purpose, refer to the reduced-order system

e

ho 1 o = q 0 (h 1, h 2) + q 1 (h 1) (v + d 1) + q 2 (h 1) d 2 + q 3 (h 1) d 3  (25) ho 2

v2 Figure 9  System (23)–(24) obtained by partially feedback linearizing system (19)–(20) by means of (22). The system is transformed into two chains of integrators driven by the two residual inputs (v1, v2), with the two controlled outputs at the top, driving the zero dynamics. The control inputs (v1, v2) must be designed to track chosen reference profiles by the two outputs, while stabilizing the zero dynamics of the system to prevent the vehicle detachment from the surface. AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  59

disturbances matched with the three input channels, with nonzero restrictions D i on the amplitude of the disturbances and with a certain linear asymptotic gain c [32], [33]. The virtual state feedback Y (h 1, h 2) is then back stepped to the true input v 1 by means of the following arguments. Define the error variable

lim sup y 1 (t) - yr ref force

pu 11 = y 1 - Y (h 1, h 2) = p 11 + yr ref force - Y (h 1, h 2) ,

t"3

= lim sup pu 11 (t) + Y (h 1 (t), h 2 (t)) - yr ref force t"3

and choose the control input v 1 as v 1 = 2Y h 2 + 2Y ho 2 - lpu 11  2h 1 2h 2



the force output is used to stabilize the zero dynamics leads to a degradation of the asymptotic tracking ­performance of the setpoint yr ref force by the output y 1 . The degradation index depends on the Lipschitz constant of the control law Y (h 1, h 2) in the set S and on an upper bound of the asymptotic norm of (bo ref, bp ref) . More ­specifically, by using the fact that pu 11 is asymptotically zero and (29),

= lim sup pu 11 (t) + Y (h 1 (t), h 2 (t)) - Y (x ref 1 , 0) t"3

(27)

where l is a positive design parameter. The design of v 2 can be carried out by controlling the chain of three integrators governing the output y 2 . By bearing in mind the definition of y ref pos and the definition of p 21 , let ref pu 21 = y 2 - y pos (t) = p 21 - b ref (t) , ref u o p 22 = p 22 - b (t) , pu 23 = p 23 - bp ref (t) ,

= lim sup Y (h 1 (t), h 2 (t)) - Y (x ref 1 , 0) t"3

# { lim sup h (t) - h ref t"3

# { c lim sup (bo ref (t), bp ref (t)) t"3

where { is the Lipschitz constant of Y ($) on S , namely, the constant such that Y (hl1 , hl2 ) - Y (hm1 , hm2 ) # { (hl1 , hl2 ) - (hm1 , hm2 ) for all (hl1 , hl2 ), (hm1 , hm2 ) ! S .

and choose v 2 = - l 1 pu 21 - l 2 pu 22 - l 3 pu 23 



(28)

where l i, i = 1, 2, 3 are design parameters such that m 3 + l 1 m 2 + l 2 m + l 3 is a Hurwitz polynomial. System (23)–(24) in closed loop with (27), (28) reads as

e

ho 1 o = q 0 (h 1, h 2) + q 1 (h 1) (Y (h 1, h 2) + pu 11) ho 2 + q 2 (h 1) (pu 22 + bo ref) + q 3 (h 1) (pu 23 + bp ref) ,

ou p 11 = - lpu 11 ,

J uo N N Kp 21O J pu 22 , O K ou O K u O. p 23 , Kp 22O = K Kpou O K- l 1 pu 21 - l 2 pu 22 - l 3 pu 23O P L 23P L

With l and l i set as above, it follows that pu 11 (t), pu 2i (t), i = 1, 2, 3 tend to zero asymptotically for any possible initial condition pu 11 (0) and pu 2i (0), i = 1, 2, 3 . Furthermore, the properties behind the design of Y (h 1, h 2) guarantee that if pu 11 (0) , pu 2i (0) , i = 1, 2, 3, bo ref 3 and bp ref 3 are sufficiently small (according to the values of the D i ’s) and the initial condition of h-subsystem satisfies (h 1 (0), h 2 (0)) ! X, the trajectories (h 1 (t), h 2 (t)) remain in the set S for all t and

lim sup h (t) - h ref # c lim sup (bo ref (t), bp ref (t)) .  (29) t"3

t"3

While the fact that pu 21 (t) tends to zero asymptotically guarantees that the output y 2 tracks asymptotically the reference y ref pos (t) (with transient performance that can be assigned by an appropriate design of the l i ’s), the fact that 60  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

Asymptotic tracking of the force setpoint yr ref force is thus guaranteed only in case of constant reference signal in the vertical position. In all other cases, a steady-state error in the force output must be accepted. Its amplitude, though, can be rendered small by keeping the vertical reference speed and acceleration small and by designing Y so that the product { c is minimized. Regarding the latter, the computation of the minimum value of { c that is attainable by a proper design of Y is an interesting control issue that goes beyond the scope of this article. We just observe how control designs aiming to decrease the asymptotic gain c of the system, namely, the asymptotic effect of bounded disturbances, necessarily lead to “high-gain” control laws, characterized by high Lipschitz constants {. On the other hand, design strategies leading to low-gain control laws necessarily result in closed-loop systems with high sensitivity to disturbances, namely, high asymptotic gain c. The product { c is thus bounded from below by a number that necessarily depends on the stability properties of the open-loop system, with highly unstable open-loop systems characterized by more restrictive static limit of performance, namely, higher attainable values of { c . In this respect, mechanical configurations of the UAV resulting in critically stable zero dynamics, namely, critically stable dynamics of (25) with u = d 1 = d 2 = d 3 = 0 , must be preferred if less restrictive static performance limitations are sought. This section is concluded by observing how improved tracking performance for the output y 1, in presence of variable vertical reference position b ref , might be achieved by possibly restricting the allowed b ref (t) and

by properly modifying the stabilizability requirement formulated above. Specifically, suppose that bo ref (t) and ref bp ref (t) are such that there exist bounded h ref 1 (t) and h 2 (t) solution of

e

ref ref r ref q 0 (h ref ho ref 1 (t), h 2 (t)) + q 1 (h 1 (t)) y force 1 (t) o = ref ref o ref p ref  ho 2 (t) + q 2 (h ref 1 (t)) b (t) + q 3 (h 1 (t)) b (t)

(30)

and satisfying the constraint ref (h ref 1 (t), h 2 (t)) ! S 



(31)

for all t $ 0 . Consider now the system  

e

ho 1 o = q 0 (h 1, h 2) + q 1 (h 1) (v + d 1) + q 2 (h 1) (bo ref + d 2) ho 2  + q 3 (h 1) (bp ref + d 3) (32)

with control input v and exogenous disturbances d i, i = 1, 2, 3 . Furthermore, consider the following modification of the stabilizability requirement.

Stabilizability Requirement (Modified) Design a smooth state feedback of the form v = Y (h 1, h 2) 



(33)

ref r ref with Y (h ref 1 (t), h 2 (t)) / y force , in a way that, for some compact set X ! R # R and positive numbers D i, i = 1, 2, 3 , all the possible trajectories of the closed-loop system (32)–(33) originating from initial conditions (h 1 (0), h 2 (0)) ! X under the action of bounded d i ’s fulfilling d i 3 # D i and lim d i (t) = 0, i = 1, 2, 3 , satisfy (h 1 (t), h 2 (t)) ! S for all t"3 t $ 0 and



lim h 1 (t) - h ref lim h 2 (t) - h ref 1 (t) = 0 , 2 (t) = 0.  t"3

t"3

(34)

It appears that, by defining pu 11 := p 11 + yr ref force - Y (h 1, h 2) and pu 2i, i = 1, 2, 3 , as done previously, the control law presented above guarantees that if pu 11 (0) and pu 2i (0) are sufficiently small (according to the value of the D i ’s) and if (h 1 (0), h 2 (0)) ! X, then the closed-loop trajectories (h 1 (t), h 2 (t)) remain in the set S for all t $ 0 , (34) is true, and lim y 1 (t) - yr ref lim y 2 (t) - y ref pos (t) = 0 . force = 0 , t"3 t"3 Hence, simultaneous perfect asymptotic tracking for the two outputs can be achieved at the expenses of restricting the class of vertical reference signal b ref (t) to those sigref nals that guarantee a bounded solution (h ref 1 (t), h 2 (t)) of (30) satisfying (31). In this respect, it is worth noting that ref the computation of (h ref solution of (30) 1 (t), h 2 (t)) amounts to computing a bounded inverse of the system

dynamics. Although the topic of dynamic inversion is not  further developed here, we do observe that the ­computation of bounded dynamic inverses is a crucial issue for systems with critically stable zero dynamics [34], [35]. Furthermore, we observe that the constraint (31) makes the problem at hand even more critical and, in turn, imposes severe restrictions on the set of admissible ­vertical references. In short, in all the cases in which a bounded inverse does not exist, or one is not willing to embark in the computation of a bounded solution of (30), the practical tracking result presented in the first part of the paragraph can be pursued. In this respect, the developed analysis emphasizes an intrinsic limit of static performance, imposed by the nonminimum-phase behavior of the system, which characterizes the state-feedback tracking problem whenever the reference signals are not compatible with a bounded inverse.

Using the Position Channel to Stabilize the Zero Dynamics In all contexts requiring high precision in terms of tracking a force setpoint, and tolerating a degradation in the vertical position tracking performance, an alternative design strategy can be adopted, by using the position channel to stabilize the zero dynamics. As above, backstepping design strategies can be followed to obtain the state feedback control law. In this case, the first design stage consists of designing a state feedback for the zero dynamics by using the state variables (p 22, p 23) as virtual control inputs. Toward this end, focus on the auxiliary system po 21 = p 22, 

e

(35)

ho 1 o = q 0 (h 1, h 2 h + q 1 (h 1) (y ref + d 1) + q 2 (h 1) (p 22 + d 2) force ho 2  (36) + q 3 (h 1) (v + d 3), .

p 22 = v + d 3 

(37)

regarded as a system with state (p 21, p 22, h 1, h 2) ! R 4 , control input v ! R , and exogenous bounded disturbances d 1, d 2 , and d 3 . Note that this system has an equilibrium point at (p 21, p 22, h 1, h 2) = (0, 0, h ref 1 , 0) with v = 0, d i = 0, i = 1, 2, 3 . For this system, formulate the following design requirement.

Stabilizability Requirement Design a smooth state feedback

v = Y (p 21, p 22, h 1, h 2) 

(38)

with Y (0, 0, h ref 1 , 0) = 0 , so that, for some compact set N ! 0 # 0 and X ! 0 # 0 , and positive numbers D i, i = 1, 2, 3 , (35)–(38) originating from ((p 21 (0), p 22 (0)), (h 1 (0), AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  61

h 2 (0))) ! N # X under the action of  bounded d i fulfilling d i 3 # D i, i = 1, 2, 3, satisfy (h 1 (t), h 2 (t)) ! S for all t $ 0 and

assigned by an appropriate design of l 1 ), a degradation in the tracking of the reference b ref by the output y 2 is observed. Specifically, ref lim sup y 1 (t) - yr ref pos - b (t) t"3

lim sup (p 21 (t), p 22 (t), h (t)) - (0, 0, h ref) t"3 # c lim sup (d 1 (t), d 2 (t), d 3 (t)) t"3

for some positive c. This stabilizability requirement can be interpreted in the ISS perspective as shown in the previous paragraph. In this case the system (35)–(37) exhibits a block feedforward structure that allows the application of the design methodology of forwarding [36], [37] to meet the desired requirements. As previously done, to design the real control input, backstep the virtual control law Y (p 21, p 22, h 1, h 2) through an integrator. Specifically, as above, define pu 21 = p 21 - b ref, pu 22 = p 22 - bo ref , pu 23 = p 23 - Y (pu 21, pu 22, h 1, h 2) - bp ref, and design the control input v 2 as v 2 = 2uY p 22 + 2uY po 22 + 2Y h 2 + 2Y ho 2 + b * (3) - l 3 pu 23 2h 1 2h 2 2p 21 2p 22 with l 3 a positive constant. Furthermore, design v 1 so that the output y 1 tracks the setpoint yr ref force . With this aim, choose v 1 as v 1 = - l 1 p 11 with l 1 a positive constant. The overall system (23)–(24) in closed loop with this v 1 and v 2 is o pu 21 = pu 22 , ho 1 e o = q 0 (h 1, h 2) ho 2 u o ref + q 1 (h 1) (yr ref force + p 11) + q 2 (h 1) (p 22 + b ) ref u u u p + q 3 (h 1) (Y (p 21, p 22, h 1, h 2) + p 23 + b ) ou p 22 = Y (pu 21, pu 22, h 1, h 2) + pu 23 , po 11 = - l 1 p 11 , o pu 23 = - l 3 pu 23 . Since l 1 and l 3 are positive, pu 11 (t) and pu 23 (t) tend to zero asymptotically for any possible initial condition pu 11 (0) and pu 23 (0) . Furthermore, the properties behind the design of Y (pu 21, pu 22, h 1, h 2) guarantee that if p 11 (0) , pu 23 (0) , bo ref 3 , and bp ref 3 are sufficiently small (depending on the values of the D i ’s) and the initial condition satisfies (h 1 (0), h 2 (0)) ! X and (pu 21 (0), pu 22 (0)) ! N , the trajectories (h 1 (t), h 2 (t)) remain in the set S for all t and lim sup (pu 21 (t), pu 22 (t), h 1 (t), h 2 (t)) - ( 0, 0, h ref 1 , 0) t"3

# c lim sup (bo ref (t), bp ref (t)) . t"3

While the fact that p 11 (t) tends to zero guarantees that the output y 1 tracks asymptotically the reference setpoint yr ref force (with transient performance that can be 62  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

= lim sup pu 21 (t) # c lim sup (bo ref (t), bp ref (t)) . t"3

t"3

Similarly to what has been discussed above, this relation highlights a limit of static performance in the position tracking that can be achieved whenever the position channel is used to stabilize the zero dynamics. The limit can be quantified by means of the minimum value attainable by the asymptotic gain c, with a proper design of the control law Y . As above, the result is that mechanical configurations of the UAV resulting in critically stable zero dynamics must be preferred if less restrictive static performance limitations are sought. Finally, observe that simultaneous asymptotic tracking for the  two outputs might be achieved by computing a bounded inverse of the systems associated to the reference b ref (t) and by properly modifying the stabilizability ­requirement, similarly to what has been presented in the ­previous paragraph.

Simulation results The values of the physical parameters concern a ducted fan prototype available at the University of Bologna and described in [38]. Table 1 reports the values relating to the considered UAV that are relevant to the results presented in this article. The center of gravity of the UAV is with c g = 0 rad and , g = , F cosc F . 0.35 m. Considering Figure 8, this value of c g makes the vector relative degree of the system welldefined for all value of i in the range of interest. Furthermore, the zero dynamics are locally stable. The aim is to control the UAV in a way that the force applied to the environment is constant and equal to yr ref force = 8 N. The inversion of the first relation in (18), with y *1 = yr ref force , allows the computation of the value of the angle at the desired equilibrium point that, for the considered value of c g , shows to be h ref 1 . 0.34 rad, see Figure 6. Now compute the region where h 1 is required to range, according to (12), to fulfill the physical constraints characterizing the system. By using the values of Fr and T in Table 1, the h 1 angle is constrained to fulfill h 1 $ arctan (Fr / T) . 0.24  rad, in order for the force generated by the vehicle to always point towards the vertical surface. In relation to the singularity of the matrix L, it is required that h 1 ! - c g . The further constraints on the angle h 1 are h 1 1 r/2 - c g and h 1 ! c F . With the numerical values of c g and c F , and the desired equilibrium point h ref 1 , it can be observed that all the constraints are fulfilled if h 1 1 c F . Hence, the set S is of the form

S = ^arctan (Fr / T), c F h # R . (0.24 rad, 0.78 rad) # R .

, i (h 1, h 2, 0) m , b (h 1, h 2, 0) - R 1 (h) (k 1 (h 1 - h ref 1 ) + k 2 h 2)

Y (h 1, h 2) = ^R 1 (h 1) R 2 (h 1)hc



(39)

where k 1 and k 2 are positive numbers. Note that T u = (h 1 - h ref r ref Y (h ref 1 , h 2) , 1 , 0) = y force as required. By letting h the previous choice transforms system (25) into hou = Hhu + d (h) 



(40)

where H=c

0 1 m , d (h) = q 1 (h 1) d 1 + q 2 (h 1) d 2 + q 3 (h 1) d 3 . - k1 - k2

ISS arguments can be used to show that the stabilizability requirement presented in the previous section is fulfilled, and to predict the set X and the restrictions D i . Specifically, let P = P T 2 0 be the solution of the Lyapunov equation PH + H T P = -I and let V (hu ) = hu T Phu be a candidate ISS Lyapunov function. The derivative of V along the solutions of (40) can be estimated as V = -hu T hu + 2hu T Pd # - hu ^ hu - 2 P d (h) h. .

It follows that hu $ 2 P d (h) implies that Vo # 0 . Now let r 2 0 be such that the set {hu : V (hu ) # r} is the largest level set of V contained in S . Furthermore, set D so that | d (h) | # D for all h ! S and set all d i so that d i # D i, i = 1, 2, 3 . By using the fact that V (hu ) # mr P hu 2 with mr P the largest eigenvalue of P, it follows that if D # r/ (4mr P P 2) then necessarily Vo (hu ) # 0 for all hu such that V (hu ) = r . The previous results suggest that the set X can take the form of ref T X = {(h 1, h 2) ! S : ( h 1 - h ref 1 , h 2 ) P ( h 1 - h 1 , h 2 ) # r}

while the restrictions D i on the disturbances d i, i = 1, 2, 3 can be fixed so that

max d (h) #

| d i | # D i, h ! X

r . 4mr P P 2

(41)

According to the previous formulas, both X and the D i ’s can be estimated numerically to identify restrictions on the initial state and the vertical reference profile that characterize the control solution. Figure 10 provides a graphical sketch behind the mentioned restrictions. The overall control law, resulting from (7), (13), (14), (22), and (27), (28), (39), is tuned according the values reported in

η2 (rad/s)

Now let us proceed to design the control law according to the previous theory using the force channel to stabilize the zero dynamics. Toward this end, start by designing a control law for the auxiliary system (25). By bearing in mind the expressions of F (h 1, h 2) and of L 1 (h 1), a possible choice for Y (h 1, h 2) is the feedback linearizing control law

0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1

S Ω (η1ref,η2ref)

0.2

0.3

0.4

0.5 η1 (rad)

0.6

0.7

0.8

Figure 10  The invariant set Ω for the choice h ref 1 = 0.4 and the control gains specified in Table 2. The set Ω, which identifies the restrictions on the initial state of (h 1, h 2) , represents the largest level set of the Lyapunov function V (hu ) = hu T Phu that is contained in S and that is invariant for system (40) provided that d (h) fulfills (41).

Table 2. Control parameters used in the simulations. l 1 = 12

l 2 = 20

l3 = 9

l = 10

k 1 = 20

k2 = 9

Table 2, with the values of T and F saturated according to the values of Tr , T , and Fr . The position reference signal b ref (t) is chosen as a ramp with constant speed bo ref (t) / 0.2 m/s ( bp ref / 0 ) in the time interval [0, 7] s and a constant signal ( bo ref / 0 ) in the time interval [7, 15] s. The system is initialized at i (0) = 0.3 rad, io (0) = 0 rad/s, b (0) = 0.2 m, bo (0) = 0 m/s. Figure 11(a), (b), (c), and (d) presents, respectively, the resulting force FEnv (t) applied to the environment, the control inputs (T (t), F (t)) , the angle h 1 (t) and angular speed h 2 (t) , and the vertical position of the UAV b (t) . As shown by the plot in Figure 11(a) and (d), while the vertical reference speed is tracked by b (t) , a steady-state error appears in the force applied to the environment, whenever bo ref is not zero.

Conclusions This article has dealt with the modeling and control of a class of aerial robots capable of interacting with the environment to accomplish robotic operations midair rather than being constrained to the ground. Discussed in detail are the design of hybrid force and position control laws for a ducted fan aerial vehicle. Particular attention has been placed on keeping track of how the stability properties of the system’s zero dynamics are affected by the position of the vehicle’s center of gravity, which can be mechanically designed to infer desired stability properties. The described control laws are state feedback and rely upon partial feedback linearizing AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  63

18

8

16

7.5

14

7

12 N

N

20

y1

8.5

6.5

8

6

6

5.5

4

5

2 2

4

6 8 10 Time (s) (a)

0.5

12

0

14

4

6 8 10 Time (s) (b)

1

0.2

0.8

0.1

0.6

0

0.4

−0.1

0.2 5

10

14

1.2

0.3

−0.2 0

12

1.4

η1 η2

0.4

2

β β*

m

Rad (Blue), rad/s (Green)

T F

10

15

Time (s) (c)

0 0

5

10

15

Time (s) (d)

Figure 11  Simulation results obtained with the control strategy stabilizing the zero dynamics by using the force channel. The results refer to the mechanical configuration with the physical parameters specified in Table 1 and are calculated by using the control parameters given in Table 2. The different figures show, respectively, (a) the force applied to the environment, (b) the control inputs, (c) the behavior of internal dynamics states i (t) and io (t) , and (d) the vertical position of the contact point of the UAV, overlapped to the reference b ref (t) . As shown in (a), and according to the theory developed in the article, a steady-state tracking error between the force applied ref to the environment and the desired reference yr force = 8 N is observed in the case bo ref ! 0 . The tracking error asymptotically vanishes as soon as the position reference signal becomes constant.

techniques. In this respect, future extensions concern the development of output feedback control strategies, whereby the nonminimum-phase behavior of the system imposes fundamental limitations [39] to the achievable tracking performance, regardless of the kind of control strategy adopted. The present article fits in a broader research context in which aerial vehicles are considered to be a support to human beings in all those activities that require the ability to interact safely with airborne environments. In this context, future research attempts are directed to develop teleoperation algorithms, according to which a human operator can remotely supervise the motion of the UAV by means of haptic devices. Experimental activities in the above interaction scenarios are also planned in the near future. 64  IEEE CONTROL SYSTEMS MAGAZINE  »  AUGUST 2012

Acknowledgments The authors would like to thank Prof. Alberto Isidori for the many suggestions given during the preparation of the manuscript.

Author Information Lorenzo Marconi ([email protected]) graduated in 1995 in electrical engineering from the University of Bologna. Since 1995 he has been with the Department of Electronics, Computer Science, and Systems at the University of Bologna, where he obtained his Ph.D. degree in 1998. Since 1999 he has been a faculty member in the same department in which he has been a professor since 2008. He has coauthored more than 150 publications on linear and nonlinear feedback control design. In 2005, one of his papers in Automatica received the Outstanding Application Paper

Award. He has served on the IEEE Control System Society Conference Editorial Board and the IEEE Control System Society Technical Committee on Nonlinear Systems and Control and the IFAC Technical Committee on Safety and Supervision in Technical Processes. He is chair of the IFAC Technical Committee on Nonlinear Control Systems. His research interests include nonlinear control, output regulation, control of autonomous vehicles, fault detection and isolation, and fault-tolerant control. He can be contacted at D.E.I.S., Università di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. Roberto Naldi graduated in computer science engineering from the University of Bologna in 2004 and received his Ph.D. degree from the Department of Electronics, Computer Science and Systems at the same university in 2008, in which he is currently an assistant professor. In 2007 he was a visiting scholar in the Laboratory for Information and Decision Systems (LIDS) at the Massachusetts Institute of Technology. He is a member of the Center for Research on Complex Automated Systems (CASY) at the University of Bologna. He has coauthored more than 40 journal and conference proceedings papers. His research interests are in nonlinear and hybrid control with applications to unmanned aerial vehicles.

References [1] I. Kaminer, A. Pascoal, E. Hallberg, and C. Silvestre, “Trajectory tracking for autonomous vehicles: An integrated approach to guidance and control,” J. Guid. Control Dyn., vol. 21, no. 1, pp. 29–38, 1998. [2] A. Richards, J. Bellingham, M. Tillerson, and J. How, “Coordination and control of multiple UAVs,” in Proc. AIAA Guidance, Navigation, and Control Conf. and Exhibit, Monterey, CA, 2002, no. AIAA 2002-4588. [3] V. Gavrilets, E. Frazzoli, B. Mettler, M. Piedimonte, and E. Feron, “Aggressive maneuvering of small autonomous helicopters: A human-centered approach,” Int. J. Robot., vol. 20, no. 10, pp. 795–807, 2001. [4] A. Bachrach, R. He, and N. Roy, “Autonomous flight in unknown indoor environments,” Int. J. Micro Air Vehicles, vol. 1, no. 4, pp. 217–228, 2009. [5] R. W. Beard, T. W. McLain, D. B. Nelson, D. Kingston, and D. Johanson, “Decentralized cooperative aerial surveillance using fixed-wing miniature UAVs,” Proc. IEEE, vol. 94, no. 7, pp. 1306–1324, 2006. [6] L. Merino, F. Caballero, J. R. Martnez de Dios, and A. Ollero, “Cooperative fire detection using unmanned aerial vehicles,” in Proc. IEEE/RSJ Int. Conf. Intelligent Robots and Systems, Barcelona, Spain, 2005, pp. 1884–1889. [7] R. W. Beard, T. W. McLain, M. Goodrich, and E. P. Anderson, “Coordinated target assignment and intercept for unmanned air vehicles,” IEEE Trans. Robot. Automat., vol. 18, no. 6, pp. 911–922, 2002. [8] B. B. Yenne, Attack of the Drones: A History of Unmanned Aerial Combat. St. Paul, MN: Zenith Press, 2004. [9] B. Siciliano and O. Khatib, Eds., Springer Handbook of Robotics. Berlin, Germany: Springer-Verlag, 2008. [10] A. Albers, S. Trautmann, T. Howard, T. A. Nguyen, M. Frietsch, and C. Sauter, “Semi-autonomous flying robot for physical interaction with environment,” in Proc. IEEE Conf. Robotics Automation and Mechatronics (RAM), Singapore, 2010, pp. 441–446. [11] A. L. Desbiens and M. Cutkosky, “Landing and perching on vertical surfaces with microspines for small unmanned air vehicle,” J. Intell. Robot. Syst., vol. 1, no. 4, pp. 313–327, 2009. [12] N. Michael, J. Fink, and V. Kumar, “Cooperative manipulation and transportation with aerial robots,” Auton. Robots, pp. 329–341, 2010. [13] R. Naldi and L. Marconi, “Modeling and control of the interaction between flying robots and the environment,” in Proc. 8th IFAC Symp. Nonlinear Control Systems, Bologna, Italy, 2010.

[14] R. Naldi, L. Marconi, and L. Gentili, “Robust takeoff and landing for a class of aerial robots,” in Proc. 48th IEEE Conf. Decision and Control, Shanghai, China, 2009, pp. 3436–3441. [15] Airobots project website. (2010, Feb.). [Online]. Available: http://www. airobots.eu [16] G. Hayward, M. Friedrich, and W. Galbraith, “Autonomous mobile robots for ultrasonic NDE,” in Proc. IEEE Ultrasonics Symp., Vancouver, BC, Canada, 2006, pp. 902–906. [17] I. Guerrero, K. Londenberg, P. Gelhausen, and A. Myklebust, “A powered lift aerodynamic analysis for the design of ducted fan UAVs,” in Proc. 2nd AIAA “Unmanned Unlimited” Systems, Technologies, and Operations, San Diego, CA, 2003, no. AIAA-2003-6567. [18] D. Chu, J. Sprinkle, R. Randall, and S. Shkarayev, “Simulator development for transition flight dynamics of a VTOL MAV,” Int. J. Micro Air Vehicles, vol. 2, no. 2, pp. 69–89, 2010. [19] A. Frank, J. S. McGrew, M. Valenti, D. Levine, and J. P. How, “Hover, transition, and level flight control design for a single-propeller indoor airplane,” in Proc. AIAA Guidance, Navigation and Control Conf., Hilton Head, 2007, pp. 3470–3475. [20] W. E. Green and P. Y. Oh, “A MAV that flies like an airplane and hovers like a helicopter,” in Proc. 2005 IEEE/ASME Int. Conf. Advanced Intelligent Mechatronics, Monterey, pp. 693–698. [21] R. Naldi and L. Marconi, “Optimal transition maneuvers for a class of V/STOL aircraft,” Automatica, vol. 47, no. 5, pp. 870–879, 2011. [22] R. H. Stone, P. Anderson, C. Hutchison, T. Allen, P. Gibbens, and K. C. Wang, “Maneuver-based motion planning for nonlinear systems with symmetries,” J. Aircr., vol. 45, no. 2, pp. 673–685, 2008. [23] R. Goebel, R. G. Sanfelice, and A. R. Teel, “Hybrid dynamical systems,” IEEE Control Syst. Mag., vol. 29, no. 2, pp. 28–93, 2009. [24] J. Lygeros, K. H. Johansson, S. N. Simic, J. Zhang, and S. Sastry, “Dynamical properties of hybrid automata,” IEEE Trans. Automat. Contr., vol. 48, no. 1, pp. 2–17, 2003. [25] H. Goldstein, Classical Mechanics. Boston: Addison-Wesley, 1980. [26] E. Yip and R. Sincovec, “Solvability, controllability, and observability of continuous descriptor systems,” IEEE Trans. Automat. Contr., vol. 26, no. 3, pp. 702–707, 2002. [27] A. Isidori, Nonlinear Control Systems. London: Springer-Verlag, 1995. [28] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ: Prentice-Hall, 1996. [29] M. Krstic, P. Kokotovic, and I. Kanellakopoulos, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [30] E. D. Sontag, “Input to state stability: Basic concepts and results,” in Nonlinear and Optimal Control Theory, P. Nistri and G. Stefani, Eds. Berlin, Germany: Springer-Verlag, 2007, pp. 163–220. [31] A. Isidori, Nonlinear Control Systems II. London: Springer-Verlag, 1998. [32] A. Teel, “A nonlinear small gain theorem for the analysis of control systems with saturations,” IEEE Trans. Automat. Contr., vol. 41, no. 9, pp. 1256–1270, 1996. [33] A. Isidori, L. Marconi, and A. Serrani, Robust Autonomous Guidance: An Internal Model Approach. London: Springer-Verlag, 2003. [34] S. Devasia, D. Chen, and B. Paden, “Nonlinear inversion-based output tracking,” IEEE Trans. Automat. Contr., vol. 41, no. 7, pp. 930–942, 1996. [35] A. Isidori and C. I. Byrnes, “Steady-state behaviors in nonlinear systems with an application to robust disturbance rejection,” Annu. Rev. Control, vol. 32, no. 1, pp. 1–16, 2008. [36] F. Mazenc and L. Praly, “Adding integrations, saturated controls and global asymptotic stabilization for feedforward systems,” IEEE Trans. Automat. Contr., vol. 41, no. 11, pp. 1559–1578, 1996. [37] R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. London: Springer-Verlag, 1997. [38] R. Naldi, L. Gentili, L. Marconi, and A. Sala, “Design and experimental validation of a nonlinear control law for a ducted-fan miniature aerial vehicle,” Control Eng. Pract., vol. 18, no. 7, pp. 747–760, 2010. [39] M. M. Seron, G. C. Goodwin, and J. Braslavsky, Fundamental Limitations in Filtering and Control. London: Springer-Verlag, 1997. AUGUST 2012  «  IEEE CONTROL SYSTEMS MAGAZINE  65