Exam - TTK 4190 Guidance & Control Eksamen - TTK 4190 Fartøysstyring

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1 Page 1 of 6 Norges teknisk- naturvitenskapelige universitet Institutt for teknisk kybernetikk Faglig kontakt / contact person: Navn: Morten Pedersen, Universitetslektor Tlf.: Exam - TTK 4190 Guidance & Control Eksamen - TTK 4190 Fartøysstyring 4. juni 2014, 09:00 13:00 Support material: A - All printed and hand-written support material is allowed. All calculators are allowed. Hjelpemidler: A - Alle trykte og håndskrevne hjelpemidler tillatt. Alle kalkulatorer tillatt. You may answer in Norwegian or English. Du kan svare på Norsk eller Engelsk.

2 Page 2 of 6 Exercise 1 (25 %) An instrument platform is mounted on the deck of a ship. platform in body-coordinates is: The position of the r b i = [10, 0, 15] T [m] The NED position and attitude of the ship, at a time t = t 0, are given by 1 : p n b/n (t 0) = [200, 400, 0] T [m], Θ nb (t 0 ) = [ 0.1, 0.2, 0.8] T [rad] a)[5%] What is the NED position of the instrument platform, r n i/n (t 0)? b)[10%] A body fixed accelerometer and angular rate sensor, mounted at the origin of the body frame, reads: a b imu(t 0 ) = [ 0.5, 0.2, 11.0] T [m/s 2 ], ω b imu(t 0 ) = [1, 0.3, 0] T [rad/s] at t = t 0. The gravitational acceleration is g n = [0, 0, 9.81] T [m/s 2 ]. Assuming perfect measurements, what is the spatial acceleration of the instrument platform in NED coordinates, r n i/n (t 0)? c)[5%] The ship s navigation system is implemented using quaternions. Why would the designers favor this approach over Euler-angles? d)[5%] Two navigation computers are onboard the ship, denoted computer A and B, both implemented using quaternions. One of them is known to be faulty. Given the readouts 2 : q A (t 0 ) = , q B(t 0 ) = with Euler-angles being those stated above, can you identify the faulty navigation computer? 1 The SNAME convention applies. 2 Rounded numbers

3 Page 3 of 6 Exercise 2 (25 %) Consider the platform illustrated in the figure below 3. The cylindrical pontoons towers are placed symmetrically fore-aft and side to side. The pontoons themselves have quadratic cross-sections. The body frame is located at the waterline in the center of the platform. L r l p,x y b l p,y CG z g b b z b CB z b T Top view Side view The dimensions are: l p,x = 11 [m], l p,y = 6 [m], r = 1 [m], L = 25 [m], b = 2 [m] The center of gravity and draft are respectively: r b g = [0, 0, 3] T [m], T = 4 [m] Water density is that of salt water: ρ = 1025 [kg/m 3 ]. a)[2%] What is the buoyancy B, weight W and mass m of the platform? b)[5%] Compute the body frame coordinates r b b of the center of buoyancy CB. c)[8%] The restoring force matrix for a linearized longitudinal plane model is: G = Z z Z θ 0 M z M θ Estimate the coefficients in the matrix and determine whether the platform is stable in pitch. d)[5%] as Iy CG Let the pitching moment of inertia around the center of gravity be given = mrg 2 where r g = 4 [m]. What is the longitudinal plane rigid-body mass 3 The drawing is not to scale.

4 Page 4 of 6 matrix M CG RB about the center of gravity? What is the mass matrix about the origin of the body frame M CO RB? e)[5%] Assume that the hydrodynamic added mass and damping are negligible. Use the longitudinal mass and restoring force matrices to predict the periods of the pitch and heave motions. Would a more realistic model including added-mass and damping tend to increase or decrease the predicted periods? Justify your answer. Exercise 3 (25 %) Consider the vessel illustrated in the figure below. l 3 l 4 F 1 = ku 1 l l F 2 = ku 2 F3 = k3u3 y b F4 = k4u4 A model describing motion in the plane is: η = R(ψ)ν, M ν + C(ν)ν + D(ν)ν = τ where it has been assumed that there are no external disturbances. The matrices in the model satisfy: M = M T > 0, C(ν) = C(ν) T, D(ν) 0 a)[5%] The vessel moves about using four independent actuators taking inputs u i. Identify the force coefficient matrix K and thrust coefficient matrix T so that: τ = TKu b)[2%] Define the input matrix B TK. What properties must B satisfy in order to guarantee full actuation of the vessel? c)[2%] Suppose that tunnel thruster in the bow is shut down. Is the vessel still fully actuated? If yes, do you foresee any practical problems with such a configuration? d)[10%] A DP-system is to be implemented on the vessel. A requirement for the control system is the ability to steer the generalized coordinates η(t) towards a constant reference value η d, such that: lim η(t) = 0, η(t 0), t η(t) η(t) η d

5 Page 5 of 6 when the system is initialized at t = t 0. Starting with the Lyapunov function candidate: V = 1 2 νt Mν ηt K p η, K p = K T p > 0 Propose a nonlinear PD-controller for τ that you can prove will fulfill the requirements on the control system. e)[6%] The generalized force vector τ (t) commanded by the system needs to be generated by inputs applied to the individual actuators u i. In many cases it is preferable to use expensive actuators less than their cheaper counterparts. Let the cost of each actuator be denoted w i. For the present system, it is possible to write: u(t) = Lτ (t), L R 4 3 Propose an, in some sense optimal, way of computing L. Take into account the cost indices w i. Justify your answer. Exercise 4 (25 %) An unmanned aerial vehicle is executing a steady coordinated turn, keeping constant altitude h(t) = h 0. Φ Vertical plane L y b C z b W U 0 The forces indicated in the figure are the weight W, Lift L and the Coriolis force C. Assume that the forward velocity of the aircraft U 0 = V T is constant, and let there be no sideslip β.

6 Page 6 of 6 a)[4%] Draw a free-body diagram and identify the forces in the vertical plane in terms of the mass m, gravitational constant g, velocity U 0, yaw rate ψ and roll angle Φ. b)[5%] Show that the path curvature in the horizontal plane κ[m 1 ] may be expressed as: κ = g U0 2 tan(φ) Demonstrate that a small-angle approximation for the Bank-to-turn equation is: ψ g U 0 Φ c)[8%] Let an approximate model for the roll dynamics be given as: T Φ + Φ = bδ A where δ A is the aileron deflection. Let T = 2 [s], U 0 = 30 [m/s] and b = 1. Design an autopilot capable of tracking a desired constant heading ψ d. Assume full state feedback. Use pole-placement to set the gains and explain your choice. d)[4%] Suppose now that your autopilot has to be implemented using only compass measurements. Is it, in principle, possible to estimate Φ? Give your reasons. e)[4%] The UAV is tasked with moving from pose A to pose B where: p A = N A E A ψ A = 0 [m] 0 [m] 0 [rad], p A = N B E B ψ B = 200 [m] 200 [m] 0 [rad] Let the forward speed of the aircraft be U 0 = 30 [m/s] and assume that the minimum turning radius is R min = 50 [m]. Sketch the path that will give the smallest travel time, given the constraint on turning radius. Compute also the time it takes to travel from pose A to pose B.