Transition Flight Control of the Quad-Tilting Rotor ... - IEEE Xplore

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high-speed cruise capabilities of a conventional airplane with the hovering capabilities of a helicopter by tilting their four rotors. Changing the flight condition ...
2013 International Conference on Unmanned Aircraft Systems (ICUAS) May 28-31, 2013, Grand Hyatt Atlanta, Atlanta, GA

Transition Flight Control of the Quad-Tilting Rotor Convertible MAV Gerardo Flores† and R. Lozano⋆

Abstract— This paper presents a model of a particular class of a convertible MAV with fixed wings. This vehicle can operate as a helicopter as well as a conventional airplane, i.e. the aircraft is able to switch their flight configuration from hover to level flight and vice versa by means of a transition maneuver. The paper focuses on finding a controller capable of performing such transition via the tilting of their four rotors. The altitude should remain on a predefined value throughout the transition stage. For this purpose a nonlinear control strategy based on saturations and Lyapunov design is given. The use of this control law enables to make the transition maneuver while maintaining the aircraft in flight. Numerical results are presented, showing the effectiveness of the proposed methodology to deal with the transition stage.

Fig. 1.

Quad-plane experimental platform schema.

I. I NTRODUCTION In the last decade some convertible MAV experimental configurations have been investigated. In [1] and [2] the authors described the development (modeling, control architecture and experimental prototype) of a two-rotor tailsitter. The control architecture features a switching logic of classical linear controllers to deal with the vertical, transition and forward flight. [3] presents a classical airplane configuration to perform both operational modes. The hover flight is autonomously controlled by an onboard control flight system while the transition and cruise flight is manually controlled. A standard PD controller is employed during hover flight to command the rudder and elevator. In [4] some preliminary results are presented for the vertical flight of a two-rotor MAV as well as a low-cost embedded flight control system. In [5] an optimal transition maneuver for the tail-sitter V/STOL aircraft is presented, showing numerical results. In [6] the authors describe the development of robust, multi-variable H ∞ control systems for the cruise and hover operating points of an experimental tilt-wing aircraft. There exist some tilt-rotor vehicles with a quad-rotor similar structure as the Boeing’s V44 [7], [8] and the QTW UAV [9]. In [10] the authors present the progress of their ongoing project, an aircraft with four tilting wings. Most of the available work concerning the convertible MAV [3], [4], [1], [2], [11], has not addressed the control problem of the transition maneuver. On the aforementioned works an analysis is presented considering the airplane and the helicopter dynamics in a separate way. Thus, the controllers are developed in the same manner, assuming the change of dynamics using a switching condition but without any analysis between those flying regimes. The main contribution of the paper is to develop a control technique in order to perform the transition form helicopter mode to airplane mode by tilting the four rotors forward while maintaining the desired altitude. Unlike the previous

Helicopters and fixed-wing aircrafts have their advantages and shortcomings respectively. Helicopters can take off and land vertically, but they cannot fly forward at high speed carrying big payloads. On the other hand, fixed-wing aircrafts can fly forward at high speed carrying large payloads. However, they cannot take-off and land vertically. Take-off and landing capabilities of helicopters and forward flight efficiency of fixed-wing aircrafts, can be combined in a single aircraft, the so-called convertible aerial vehicle. In this paper, a novel convertible MAV called Quad-plane is presented (Fig 1), having a combination of both the quadrotor and the fixed-wing aircraft. This vehicle combines the high-speed cruise capabilities of a conventional airplane with the hovering capabilities of a helicopter by tilting their four rotors. Changing the flight condition between hover and cruise flight mode is called transition maneuver. When the aircraft hovers, takes off or lands, it can be controlled as a classical quad-rotor. On the other hand during high speed flight, the Quad-plane’s configuration is similar to the one of an airplane, in which the aerodynamic surfaces generate the lift force necessary to compensate the gravity force. While the Quad-plane is a very promising concept, it also comes with significant challenges. Indeed it is necessary to design controllers which will work over the complete flight envelope of the vehicle: from low-speed vertical flight through highspeed forward flight. The main challenge, besides understanding the detailed aerodynamics, is to deal with the large variation in the vehicle dynamics between these two different flight regimes. This work was partially supported by the Institute for Science & Technology of Mexico City (ICyTDF). † is with the Heudiasyc UMR 6599 Laboratory, University of Technology of Compi`egne, France. (email: [email protected]) ⋆ is with the Heudiasyc UMR 6599 Laboratory, UTC CNRS France and LAFMIA UMI 3175, Cinvestav, M´exico. (email: [email protected])

978-1-4799-0817-2/13/$31.00 ©2013 IEEE

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A. Kinematics transformations •







Fig. 2. Coordinate systems: Inertial frame (F i ) and Body-fixed frame (F b )

F i denotes the inertial earth-fixed frame with origin, Oi , at the earth surface. This frame is associated to the vector basis {ii , ji , ki }. F b denotes the body-fixed frame, with origin, Ob , at the center of gravity CG. This frame is associated to the vector basis {ib , jb , kb }. F a denotes the aerodynamic frame, with origin, Oa , at the center of gravity CG. This frame is associated to the vector basis {ia , ja , ka }. The orthonormal transformation matrices Rbi and Rab , respectively used to transform a vector from F b → F i and F a → F b within the longitudinal plane (pitch axis), are given by: 

cos θ Rbi =  0 − sin θ

  cos α 0 sin θ 1 0  , Rab =  0 − sin α 0 cos θ

 0 sin α 1 0  0 cos α

B. Aerodynamic forces

Fig. 3. MAV.

It is important to consider the aerodynamic forces properly because they are fundamentally affected by the vehicle’s motion and thus they alter the basic dynamics involved. The analysis used in the present paper will be based on a combination of a low-order aerodynamic model coupled with a simple actuator disc model of the flow induced by the propellers. In order to proceed with the aerodynamic analysis, it is worth to mention the following assumptions: • A1. The vehicle is a rigid body, i.e. the flexibility of the aircraft wings or fuselage will be neglected. • A2. Non-varying mass is considered (m(t) ˙ = 0). • A3. The aerodynamic center (AC) and the center of gravity (CG) coincide. The forces consist of a lift force L, perpendicular to the total air flow vector Vt , a drag force D parallel to Vt , and the airfoil’s pitching moment M , about the positive cartesian y-axis. The above discussion can be summarized by:

Free-body scheme showing the forces acting on the Quad-tilting

work [12], in which a switching between change of dynamics has considered, in this paper a control strategy suitable for handling the transition maneuver in a continuous manner is proposed. The controller is stable in the sense of Lyapunov for all the envelope, going from hovering flight to high speed flight. The paper is organized as follows. Section II presents the mathematical model of the Quad-plane aircraft using the Newton formulation. Section III introduces the flight envelope description and the problem statement. Section IV details the nonlinear control strategy based on saturations and Lyapunov design simplified linear. Section V shows the numerical results obtained when applying the proposed controller. Conclusions and future work are finally presented in section VI.

Cl Cd Cm

= = =

C lα α C dp + C di C mα α

where the above equations are standard aerodynamic nondimensional lift, drag and moment coefficients respectively. To obtain the lift and drag forces and the pitching moment on the aircraft it is only necessary to obtain the total wind velocity vector Vt , the angle of attack α and the aerodynamic parameters Clα , Clδ , Cd , Cmδ , Cmα which depend on the geometry of the vehicle. The equations representing the lift, drag and pitching moment are given as

II. DYNAMIC M ODEL This section presents the longitudinal equations of motion as well as the aerodynamics of the vehicle. We distinguish three operation modes on the flight profile of the Quad-plane. (1) Hover Flight (HF): the aircraft behaves as a rotary-wing platform (γ = 0), (2) Transition Flight (TF): (0 < γ < π2 ) and finally (3) Fast-Forward Flight (FFF): where the aerial robot behaves as a pure airborne vehicle (γ = π2 ).

L D M

= = =

1 2 2 Cl ρVt S 1 2 2 Cd ρVt S 1 2 c 2 Cm ρVt S¯

(1)

where S and c¯ are the area and the wing chord respectively. The angle of attack α and the magnitude of Vt are obtained 790

thrust is written as T = T3,4 + T1,2 , and the difference of these thrusts is Td = T1,2 − T3,4 . The mathematical representation (8) also can be seen in [5].

from the following equations α = arctan(vw /vu ) p 2 + v2 Vt = v w u

(2)

III. F LIGHT ENVELOPE DESCRIPTION AND P ROBLEM S TATEMENT

where vw and vu are the components of the vehicle velocity expressed in the body-fixed frame.

A. Flight envelope description

C. Forces exerted on the Quad-plane

The flight envelope of the vehicle encompasses three different flight conditions achieved by means of the collective angular displacement of the rotors. Indeed, tilting the four rotors forward will produce the transition from helicopter mode to airplane mode. These flight conditions are explained below: 1) During the HF mode the 3D vehicle’s motion relies only on the rotors. Within this phase the vehicle features VTOL flight profile. The controller for this regime disregards the aerodynamic terms due to the negligible translational speed. 2) It is possible to distinguish an intermediate operation mode, the transition maneuver TF which links the two flight conditions, HF and FFF. This is probably the most complex dynamics and in this work we will focus on this regimen. 3) FFF regime mode (Aft position), at this flight mode the aircraft has gained enough speed to generate aerodynamic forces to lift and control the vehicle motion. In this mode the vehicle behaves like a common airplane.

Applying the second Newton’s law to the Quad-plane (Fig. 3 ) we obtain mξ¨ = Rbi T b + Rbi Rab Aa + W i

(3)

T

where, ξ = (x, y, z) is the CG’s position vector in F i , T b = (0, 0, T )T is the collective thrust in F b , Aa = (−D, 0, L)T is the vector of aerodynamic forces in F a and finally W i = (0, 0, mg)T denotes the weight of the vehicle in F i , where m is the mass of the vehicle and g is the gravitational acceleration. Note that the vector of aerodynamic forces Aa is not only involved in translational motion, but also in the rotational motion of the vehicle, as is shown in Section II-D. The four propellers produce the collective thrust T , which can be modeled as T = Kl

i=4 X

ωi2

(4)

i=1

where ωi is the angular velocity of ith -rotor, Kl is a lift factor depending on the aerodynamic parameters of the propeller.

B. Problem statement Considering the aerodynamic equations (8) we take the following assumptions: 1) θ ≈ 0, i.e. the pitch angle is zero. In HF mode this assumption is achieved using the control input Td . In FFF mode the pitch control is obtained via the elevator deflection. 2) α ≈ 0, i.e. the angle of attack is zero. 3) From (1) an approximation for the lift and drag equations can be taken as L = lx˙ 2 and D = dx˙ 2 . 4) The total thrust T will be set to a constant value. 5) The term ue x˙ 2 representing the elevator deflection is introduced in the aircraft dynamics model which plays a role just after the speed reaches the required value to compensate the weight, i.e. when

D. Moments acting on the Quad-plane The forces shifted away from the CG induce moments causing rotational motion. The moment exerted about the CG can be written as b τ b = τTb + τM

(5)

b where τM is the airfoil’s pitching moment, τTb is the induced moment due the difference of thrust between T3,4 and T1,2 . τTb is obtained as

τTb = l1 (−T3,4 cos γ + T1,2 cos γ)jb

(6)

where, l1 is the distance from the CG to the rotors shown in b is obtained from Fig. (3). The airfoil’s pitching moment τM the airfoil’s Cm slope. b τM = M jb

lx˙ 2 = mg

(7)

ue is a control input for the FFF mode. Considering all the aforementioned assumptions, (8) reduces to x ¨ = T sin (γ) − dx˙ 2 (10) z¨ = T cos (γ) + lx˙ 2 + ue x˙ 2 − mg

Since the present paper concentrates on the longitudinal flight of the vehicle, we will neglect the drag torque due to the propeller drag force and the gyroscopic moment. The corresponding equations modeling the forces and moments applied to the vehicle are written as:

The vehicle will change during the transition from HF mode with γ = 0 to FFF mode with γ = π2 . The control objective is to perform a transition from HF mode to FFF mode such that the translational speed x˙ varies from 0 to some value x˙ max while the altitude z remains constant and equal to a desired value zd .

m¨ x =T sin (θ + γ) + L sin (θ − α) − D cos (θ − α) m¨ z =T cos (θ + γ) + L cos (θ − α) + D sin (θ − α) − mg

(9)

(8)

where γ represents the rotors angle with respect to the fuselage and will be considered as control input. The total 791

When the condition lx˙ 2 = mg is satisfied, the transition maneuver has finished and the vehicle changes to FFF mode. Once the velocity is high enough, the lift reaches a value which compensates the weight of the vehicle. Furthermore the elevator deflection has an impact on the lift presented on the airplane. Thus, at this time the effect of the input control ue affects the aircraft dynamics.

and the only solution is p 2mgl + (2mgl)2 + 4(d2 + l2 )(T 2 − m2 g 2 ) 2 x˙ s = (16) 2(d2 + l2 ) From (10) it follows that the lift force lx˙ 2 will compensate the weight mg as long as the speed is large enough, i.e. when x˙ 2 > mg/l which means that the aircraft is operating in the plane mode. From (16) it follows that

IV. C ONTROL S TRATEGY In this section we will present a control strategy for the transition phase of the convertible plane. The control algorithm objective is to make the transition from hover mode to forward flight by tilting the rotors in such a way that the aircraft altitude remains constant. The total thrust should certainly be larger that the aircraft weight and the exces thrust is used for altitude control as well as for increasing the aircraft speed. Interestingly enough the strategy proposed in this section is a continuous-time control law. In order to stabilize the altitude z with a bounded control input, we will use the nested saturation control approach z¨ = uz = −ǫσ1 (z˙ + σ2 (z + z˙ − zd ))

x˙ 2s >

2l2 >1 + l2 )

or l>d which means that the lift force should be larger than the drag force, which is normally the case. Notice also that the RHS of (12) varies from a value close to 1 at the begining of the transition to a value close to 0 at the end of the transition, which means that the tilting angle control input γ varies from 0 to π2 . When lx˙ 2 = mg, there exists sufficient velocity to generate the adequate lift in order to maintain the vehicle flying. Until that moment the input control ue begins to affect the system dynamics. Such control input is given by

(11)

−(T cos γ + lx22 − mg) − k1e (z − zd ) − k2e z˙ (19) x˙ 2 where k1e and k2e are positive real numbers. The stability of (14) can be studied using the following positive definite function ue =

mg − lx˙ 2 + uz (12) T In order to complete the transition stage, and from the above discussion the thrust should satisfy

V =

(13)

for some small constant ǫ1 > 0. Let us now compute the velocity corresponding to the control input (12). For simplicity we will assume that ǫ1 is very small and we will neglect it. The convergence of z to the desired value zd will occur at a slow rate, but the altitude dynamics will remain asymptotically stable. Introducing (12) into (10) we get =

2

(18)

(d2

cos (γ) =

x ¨

(17)

Therefore the final steady state p velocity x˙ s will be larger than the plane stall velocity mg/l if

where ǫ is the maximum amplitude of the control input uz and σi (·) is a saturation function such that |σi (·)| ≤ Mi for i = 1, 2 and ζ1 is a function of the states (z, z) ˙ properly chosen to ensure global stability. The stability analysis of (11) and the value of ζ1 , can be seen in the Appendix. The above control input is such that z converges to zd , and it will be used before the Quad-plane achieves the FFF mode. Before the Quad-plane completes the transition maneuver ue = 0 and thus, from (10) and (11) it follows that

T > mg + ǫ1

4mgl 2l2 mg = 2 2 2 2 2(d + l ) (d + l ) l

1 (x˙ − x˙ s )2 2

whose derivative with respect to (14) is given by V˙ = (x˙ − x˙ s )υ

(20)

where υ is the RHS of (14) 1

υ = (T 2 − (mg − lx˙ 2 )2 ) 2 − dx˙ 2

1

x˙ 2 2 ) ) − dx˙ 2 T (1 − ( mg−l T

Let us study the derivative of υ

or x ¨

=

1

(T 2 − (mg − lx˙ 2 )2 ) 2 − dx˙ 2

(14) υ˙ =

2

Notice that in view of (13), T > (mg − lx˙ ). Notice also that the velocity subsystem above is stable. The steady state velocity is reached when the right hand side is zero, i.e. when T 2 − (mg − lx˙ 2 )2 = (dx˙ 2 )2

4(mg − lx˙ 2 )lx˙ 1 − 2dx˙ 2 (T 2 − (mg − lx˙ 2 )2 ) 12

Evaluating the above at x˙ = x˙ s , see (15) and (16), we have υ˙ |x= ˙ x˙ s =

(15)

1 4(mg − lx˙ 2 )lx˙ − 2dx˙ 2 dx˙ 2

or

or

υ˙ |x= ˙ x˙ s =

(d2 + l2 )x˙ 4 − 2mglx˙ 2 − (T 2 − m2 g 2 ) = 0 792

2(mg − lx˙ 2s )l − 2dx˙ s dx˙ s

TABLE I T HE PARAMETERS OF THE CONVERTIBLE MAV CONSIDERED FOR SIMULATION ANALYSIS .

Parameter

Value

m g l d T zd

1.1 kg 9.8 m/s2 1 0.1 10 N 15 m

when lx˙ 2 = mg, this condition occurs in t = 1 and is showed in Figure (7).

Fig. 4. A prototype of convertible MAV developed in HEUDIASYC Laboratory.

From (17) and (18) it follows that (mg−lx˙ 2s ) < 0 and thus υ˙ |x= ˙ x˙ s < 0. Therefore we conclude that in the neigborhood of x˙ = x˙ s we have

16.5

z [m]

16

V˙ = (x˙ − x˙ s )υ ≤ 0

15.5 15 14.5 14 0

1

2

3

4

5

3

4

5

3

4

5

3

4

5

time [s]

which proves the (local) stability of the horizontal velocity subsystem (14). z˙ [m/s]

1

V. N UMERICAL R ESULTS In this section some simulations showing the performance of the algorithm proposed in section IV are presented. The following numerical results have been obtained by using the parameters in table I under the initial conditions x = x˙ = 0, z = 15 and z˙ = 1. The parameters used for the dynamic model are based on preliminary measurements for a Quad-plane scale model MAV developed in the HEUDIASYC Laboratory (Fig. 4). The simulations results demonstrate the stability of the closed loop system with the transition maneuver where the autopilot changes the tilting angle γ from zero to 90 degrees. We will suppose that the vehicle begins the transition maneuver at an altitude equal to 15 meters which in fact will be the desired altitude during all the transition stage. The evolution of the dynamic states of the nonlinear system is shown in Figure (5). The transition begins in t = 0 supposing a stabilized position carried out by the helicopter mode. The input control uz is the responsible for begin the transition maneuver, his behavior is shown in Figure (6). When the transition begins, there exists an increment in the velocity x˙ due to the tilting of the rotors, such increase continues growing until the vehicle achieves the desired altitude and the transitions maneuver has finished. In t = 1 the angle γ achieves the value of 90 degrees, as we can see in Figure (7), indicating the end of the transition maneuver. However, at this time the velocity x˙ continues to increase until the altitude is stabilized. In order to stabilize the z dynamics just after γ = 90, the controller ue is introduced, as one can see in Figure (6). As we have mention in assumption (5) of section (III-B), the vehicle can operate as an airplane

0 −1 0

1

2 time [s]

x [m]

40 20 0 0

1

2 time [s]

x˙ [m/s]

10

5

0 0

1

2 time [s]

Fig. 5. The state z is affected by tilting of the rotors. z converges to the desired value zd when the velocity x˙ achieves the maximum value corresponding to T = 10 N.

VI. C ONCLUSION This work has focused on the problem of designing a transition maneuver for a class of convertible MAV. An approximate longitudinal dynamic model of the aircraft is presented, including the aerodynamics and dynamics of the tilting mechanism. A control strategy is proposed to handle 793

Note that if |z| ˙ > M2 then V˙ < 0, i.e. ∃T1 such that |z| ˙ ≤ M2 for t > T1 . We define

0.02

uz

0.01

ν = z + z˙

(23)

ν˙ = z¨ + z˙ = z˙ − σ1 z˙ + σ2 (ζ1 )

(24)

M1 ≥ 2M2

(25)

0

Differentiating (23)

−0.01 −0.02 0

1

2

3

4

5

time [s]

Let us choose

0

From the definition of σ(·) we can see that |σi (·)| ≤ Mi . This implies that in a finite time, ∃T1 such that |z| ˙ ≤ M2 for t ≥ T1 . Therefore, for t ≥ T1 , |z˙ + σ2 ζ1 | ≤ 2M2 . Thus, it follows that, ∀t ≥ T1

ue

−0.5 −1 −1.5 0

1

2

3

4

5

σ1 (z˙ + σ2 (ζ1 )) = z˙ + σ2 (ζ1 )

time [s]

(26)

Using (24) and (26) finally we obtain

Fig. 6. The control uz is the responsible to begin the transition stage, then, when condition (9) is accomplished. Just after such condition, the elevator deflection ue , can manipulate the altitude of the airplane.

ν˙ 1 = −σ2 (ζ1 )

(27)

We propose ζ1 in the following form ζ1 = ν 1

(28)

50

ν˙ 1 = −σ2 (ν1 )

(29)

0

this implies that ν1 → 0, and from (28) ζ1 → 0. We can see from (22) that z˙ → 0 and from (23) that z → 0. Finally using (23) and (28) we can rewrite (11) as

γ [deg]

100

then

lx˙ 2, mg

0

0.2

0.4

0.6 time [s]

0.8

1.0

1.2

30

uz = −ǫσ1 (z˙ + σ2 (z + z˙ − zd ))

20

R EFERENCES

10

[1] H. Stone, “Control architecture for a tail-sitter unmanned air vehicle,” in 5th Asian Control Conference, 2004. [2] R. H. Stone, “Aerodynamic modelling of a wing-in-slipstream tailsitter uav,” in AIAA Biennial International Powered Lift Conference and Exhibit, Williamsburg, Nov. 2002. [3] W. Green and P. Oh, “Autonomous hovering of a fixed-wing micro air vehicle,” in International Conference on Robotics and Automation, 2006. [4] J. Escareno, S. Salazar, and R. Lozano, “Modeling and control of a convertible vtol aircraft,” in 45th IEEE Conference on Decision and Control, 2006. [5] R. Naldi and L. Marconi, “Optimal transition maneuvers for a class of v/stol aircraft,” Automatica, vol. 47, no. 5, pp. 870–879, 2011. [6] D. R. Mix, J. S. Koenig, K. M. Linda, O. Cifdaloz, V. L. Wells, and A. A. Rodriguez, “Towards gain-scheduled h∞ control design for a tilt-wing aircraft,” in Proc. IEEE Conference on Decision and Control (CDC’2004), Atlantis, Paradise Island, Bahamas, Dec. 2004, pp. 1222– 1227. [7] D. Snyder, “The quad tiltrotor: Its beginning and evolution,” in Proceedings of the 56th Annual Forum, American Helicopter Society, 2000. [8] D. A. Ta, I. Fantoni, and R. Lozano, “Modeling and control of a tilt tri-rotor airplane,” in Proc. IEEE American Control Conference (ACC’2012), Montral, Canada, Jun. 2012, pp. 131–136. [9] K. Nonami, “Prospect and recent research and development for civil use autonomous unmanned aircraft as uav and mav,” in Journal of System Design and Dynamics, vol. 1, 2007. [10] K. Oner, E. Cetinsoy, M. Unel, M. Aksit, I. Kandemir, and K. Gulez, “Dynamic model and control of a new quadrotor unmanned aerial vehicle with tilt-wing mechanism,” in International Journal of Engineering and Applied Sciences, 2009, vol. 5:2. [11] J. Escareno, H. Stone, A. Sanchez, and R. Lozano, “Modeling and control strategyfor the transition of a convertible uav,” in European Control Conference (ECC07), 2007. [12] G. Flores, J. Escareno, R. Lozano, and S. Salazar, “Quad-tilting rotor convertible mav: Modeling and real-time hover flight control,” Journal of Intelligent and Robotic Systems, vol. 65, no. 1-4, pp. 457–471, 2012.

0 0

0.2

0.4

0.6 time [s]

0.8

1.0

1.2

Fig. 7. The transition maneuver is achieved in approximately one second when lx22 = mg = 10.

both operational regimes which feature two nonlinear control algorithm used to stabilize the dynamic system around the origin in both modes. The proposed control strategy was evaluated in numerical simulations using the nonlinear dynamic model obtaining satisfactory results. The proposed control law is suitable for embedded applications and it does not require a high computational cost. A PPENDIX The stability proof of (11) is presented in this appendix. For this purpose, we propose the following Lyapunov function 1 (21) V = z˙ 2 2 Differentiating (21) with respect to time and evaluating on the trajectories of (11) one obtains V˙ = −zǫσ ˙ 1 (z˙ + σ2 (ζ1 ))

(22) 794

(30)