Explicit non-linear model predictive control for autonomous helicopters

Transcription

1 Loughborough University Institutional Repository Explicit non-linear model predictive control or autonomous helicopters This item was submitted to Loughborough University's Institutional Repository by the/an author. Citation: LIU, C., CHEN, W-H. and ANDREWS, J.D., Explicit nonlinear model predictive control or autonomous helicopters. Proceedings o the Institution o Mechanical Engineers, Part G: Journal o Aerospace Engineering, 226 (9), pp Additional Inormation: This article was published in Proceedings o the Institution o Mechanical Engineers, Part G: Journal o Aerospace Engineering [ c Sage and IMechE]. The deinitive version is available rom: Metadata Record: Version: Accepted or publication Publisher: Sage Publications c Institution o Mechanical Engineers (IMechE). Please cite the published version.

2 This item was submitted to Loughborough s Institutional Repository ( by the author and is made available under the ollowing Creative Commons Licence conditions. For the ull text o this licence, please go to:

3 Explicit Nonlinear Model Predictive Control or Autonomous Helicopters Cunjia Liu, Wen-Hua Chen, John Andrews Abstract Trajectory tracking is a basic unction required or autonomous helicopters, but it also poses chanlleges to controller design due to the complexity o helicopter dynamics. This paper introduces a closed-orm model predictive control (MPC) to solve this problem, which inherents the advantages o the nonlinear MPC but eliminates the time-consuming online optimisation. The explicit solution to the nonlinear MPC problem is derived by using Taylor expansion and exploiting the helicopter model. With the explicit MPC solution, the control signals can be calculated instantaneously to respond to the ast dynamics o helicopters and suppress the disturbances immediately. On the other hand, the online optimisation process can be removed rom the MPC ramework, which can accelerate the sotware development and simpliy the onboard hardware. Due to these advantages o the proposed method, the overall control ramework has a low complexity and high reliability, and it is easy to deploy on the small-scale helicopters. The explicit nonlinear MPC has been successully validated in simulations and inactual light tests using the Trex-250 helicopter. 1 Introduction Autonomous helicopters are increasingly attracting attentions or their potential applications, mainly due to their ability to hover, ly in very low altitudes, and take o and land almost everywhere. These properties make them suitable or a board range o tasks like surveillance, boundary patrol, search and rescue. However, due to their inherent instabilities and nonlinearities and a high dimensional model structure, the controller design or helicopter autonomous lights is a challenge. To this end, a number o control techniques have been applied to address this problem including the classic cascaded PID control [KS03], eedback linearisation [KS98], multivariable adaptive Control [KAM02], neural network adaptive control [JK05], state-dependent riccati equation (SDRE) control [BW07] and composite nonlinear eedback control [PCC + 09]. Recently, model predictive control (MPC) has been recognised a promising method in the unmanned aerial vehicle (UAV) community [OM04]. MPC is an optimal control strategy that uses a model to predict the uture behaviour o the plant over a prediction horizon. Based on these predictions, the perormance index deined to penalise the tracking errors or state errors is minimised with respect to the sequence o 1

4 uture inputs. Only the irst action in the optimised control sequence is applied into the plant, and this procedure is repeatedly executed in a receding horizon ashion to continuously generate control signals. The eatures o MPC make it as a suitable control technique or UAV applications. Firstly, it naturally takes into account the uture value o the reerence to improve path tracking perormance; secondly, it simpliies the control design by directly using the vehicle model in the control loop; thirdly, it considers both the kinematics and dynamics o UAVs as an entire system in an integrated guidance and control ashion so enhances the light agility. The essential procedure in the implementation o MPC algorithm is to solve the ormulated optimisation problem (OP). I the system model is linear and there are no constraints acting on the system, the explicit minimisation solution can be ound. Otherwise, MPC technique usually requires solving an optimisation problem numerically at every sampling instant, which poses obstacles on the real-time implementation due to the heavy computational burden. Although the development o the avionics and microprocessor technology makes the online optimisation possible, the implementation o computationally demanding MPC on small UAVs is very challenging. Most o the existing applications tend to use linear MPC so that the ormulated OP can be solved by eicient Quadratic Programming [KB06]. To it the nonlinear tracking problem into a linear setting, associated techniques like linearisation and eedback linearisation techniques are usually required. For nonlinear MPC, although it is more powerul the resulting optimisation problem is non-convex, which means the solving time is much longer and even not deterministic. Thereore, the nonlinear MPC is more likely to be seen in the guidance layer to enhance the autonomy o the UAVs rather than in the time-critical light control level [HK09]. The associated low bandwidth and computational delay o nonlinear MPC make it very diicult to meet the control requirement or systems with ast dynamics like helicopters. Only ew applications on helicopter light control have been reported in [KSS02, SKS03], where the authors adopt a high-level MPC to solve the tracking problem and rely on a local linear eedback controller to compensate the high-level MPC. Moreover, the ormulated nonlinear optimisation problem has to be solved by a secondary light computer. The extra payload and power consumption are quite luxury or a smallscale helicopter. To avoid using online optimisation and inherent the advantages o nonlinear MPC, this paper introduces an explicit nonlinear MPC (ENMPC) or trajectory tracking o autonomous helicopters. By approximating the tracking error and control eorts in the receding horizon using their Taylor expansion to a speciied order, an analytic solution to nonlinear MPC can be ound and consequently the closed orm controller can be ormulated without online optimisation [CBG03]. The beneits o using this MPC algorithm are not only the elimination o online optimisation and associated resource, but also a higher control bandwidth it can achieve which is very important or helicopters in aggressive light scenarios. The similar technique has been applied to a glider and a paraoil aircrat, and has shown promising results in the simulations [SKC06]. However, in both the applications, ENMPC was just used to control the vehicle attitude with only inner-loop dynamics under consideration. To 2

5 develop a ENMPC or trajectory tracking o autonomous helicopters, the entire helicopter model must be taken into account. A considerable eort is required to develop ENMPC tailored or autonomous trajectory tracking or unmanned helicopters. To veriy the perormance o the proposed ENMPC or trajectory tracking o autonomous helicopter, numerical simulations are carried out. The result is also compared with the conventional MPC algorithm when an online optimisation problem is solved at each time instant. Due to the eature o ENMPC, the overall control ramework has a low complexity and high reliability, and it is easy to be deployed on the small-scale helicopters. To demonstrate this, light tests are perormed on our indoor testbed using a Trex-250 helicopter. The remaining part o this paper is organised as ollows: Section 2 presents the mathematical model o small-scale helicopters and the simpliication or control design; in Section 3 the algorithm o ENMPC and its implementation on autonomous helicopters are discussed in detail; Section 4 provides some simulation and light experiment results, ollowed by a conclusion in Section 5. 2 Helicopter modelling A helicopter is a highly nonlinear system with multiple inputs and multiple outputs and complicated internal couplings. The complete model taking into account the lexibility o the rotors and uselage usually results in a model o high degrees-o-reedom. The complexity o such a model will make the ollowing system identiication much more diicult. However, the general dynamics o a small-scale helicopter can be captured by a six-degrees-o-reedom rigid-body model augmented with a simpliied rotor dynamic model [MTK02, GMF01], as shown in Fig.1. Hence, the kinematic relationship o the helicopter, i.e. the position and the orientation represented by Z-Y-X Euler angles, can be expressed as: [ ẋ ẏ ż ] T = R i b (φ, θ, ψ)[ u v w ]T (1) φ 1 sin φ tan θ cos φ tan φ p θ = 0 cos φ sin φ q (2) ψ 0 sin φ sec θ cos φ sec θ r where (x, y, z) describe the helicopter inertial position, (u, v, w) are the local velocities along three body axis, (p, q, r) are angular rates and (φ, θ, ψ) are the attitude angles and R i b is a transormation matrix rom body to inertial coordinates given in (3) with short notation c or cosine and s or sine. cθcψ sφsθcψ cφsψ cφsθcψ + sφsψ R i b (φ, θ, ψ) = cθsψ sφsθsψ + cφcψ cφsθsψ sφcψ (3) sθ sφcθ cφcθ In terms o the dynamics model, the helicopter is driven by the external orces and moments which are primarily generated by main and tail rotor thrusts, in and uselage drags. This means that they are dependent on both the rotor and the rigidbody states. The our control inputs, comprising longitudinal and lateral cyclic pitch, 3

6 Figure 1: Helicopter rame main rotor collective pitch and tail collective pitch, alter the states o the main rotor and the trail rotor and consequently exert their inluences on the helicopter uselage. In the control design, the external orces and moments can be approximated by the linear combination o states and control inputs using stability and control derivatives, but the other interactions remain a nonlinear relationship. The model structure is represented in (4). u = vr wq g sin θ + X u u + X a a v = wp ur + g cos θ sin φ + Y v v + Y b b ẇ = uq vp + g cos θ cos φ + Z w w + Z col δ col g ṗ = L a a + L b b q = M a a + M b b ṙ = N r r + N col δ col + N ped δ ped ȧ = q a τ + A lat τ ḃ = p b τ + B lat τ δ lat + A lon δ lon τ δ lat + B lon δ lon τ (4) where the dynamics o the main rotor is described by the lapping angles [ a b ] T with the eective time constant τ; u = [ δ lat δ lon δ ped δ col ] T is the control inputs including lateral and longitudinal cyclic pitch, tail and main rotor collective pitch respectively; the other parameters in the model structure are the stability and control derivatives, whose values are obtained by system identiication. In this model, the rotor lapping states a and b cannot be directly measured, which usually rely on a state observer. In order to reduce the complexity and ocus on the control design, we use steady state approximation as a measurement o the 4

7 lapping angles [BW07]: a = τq + A lat δ lat + A lon δ lon b = τp + B lat δ lat + B lon δ lon (5) whose values are inserted to the model to replace the state values o a and b, such that where ṗ = L pq + L lat δ lat + L lon δ lon q = M pq + M lat δ lat + M lon δ lon (6) L pq = τ(l a q + L b p), M pq = τ(m a q + M b p), L lat = L a A lat + L b B lat, M lat = M a A lat + M b B lat, L lon = L a A lon + L b B lon, M lon = M a A lon + M b B lon, On the other hand, the helicopter suers slightly unstable zero dynamics introduced by the couplings between the rotor and uselage [KS98], which are relected on derivatives X a and Y b in (4). Due to the small magnitudes o the lapping angles, these terms can be saely neglected such that the dominate orce is the main rotor thrust only. This simpliication is quite common in controller design o vertical take-o and landing (VTOL) vehicles [MN07]. The simpliied helicopter model by combining (1)-(6) can be expressed in the ollowing compact orm: ẋ = (x) + g(x)u y = h(x) where x = [ x y z u v w p q r φ θ ψ ] is the helicopter state and y is the output o the helicopter. In the trajectory tracking control o an autonomous helicopter, the interested outputs are the position and heading angle. Thus, y = [ x y z ψ ] T. 3 Closed-orm MPC Trajectory tracking is the basic unction required when an autonomous helicopters perorms a task. To this end, we need design a controller such that the output y(t) o the helicopter (8) tracks the prescribed reerence w(t). In the MPC strategy, tracking control can be achieved by minimising a receding horizon perormance index J = 1 2 T 0 (ŷ(t + τ) w(t + τ)) T Q(ŷ(t + τ) w(t + τ))dτ (9) where weighting matrix Q = diag{q 1, q 2, q 3, q 4 }, q i > 0, i = 1, 2, 3, 4. Note that the hatted variables belong to the prediction time rame. Conventional MPC algorithm requires solving o an optimisation problem at every sampling instant to obtain the control signals. To avoid the computationally intensive online optimisation, we adopt an explicit solution or the nonlinear MPC problem based on the approximation o the tracking error in the receding prediction horizon. 5 (7) (8)

8 3.1 Output approximation For a nonlinear MIMO system like the helicopter, it is well known that ater dierentiating the outputs or a speciic number o times, the control inputs appear. The number o times o dierentiation is deined as relative degree. For the helicopter with output y = [ x y z ψ ] and the corresponding input u = [ δ lon δ lat δ col δ ped ], the relative degree is a vector, ρ = [ ρ 1 ρ 2 ρ 3 ρ 4 ]. I continuously dierentiating the output ater the control input appears, the derivatives o control input appear, where the number o the input derivatives r is deined as the control order. Since the helicopter model has dierent relative degrees, the control order r is irst speciied in the controller design. The ith output o the helicopter in the receding horizon can be approximated by its Taylor series expansion up to order ρ i + r: τ r+ρ i ŷ i (t + τ) y i (t) + τẏ i (t) + + (r + ρ i )! y[r+ρ i] i (t) y i (t) [ ] τ = 1 τ r+ρ i ẏ i (t) (r+ρ i )! y [r+ρ i] i (t) (10) where i = 1, 2, 3, 4. In this way, the approximation o the overall output o the helicopter can be cast in a matrix orm: ˆx(t + τ) ŷ 1 (t + τ) ŷ(t + τ) = ŷ(t + τ) ẑ(t + τ) = ŷ 2 (t + τ) ŷ 3 (t + τ) ˆψ(t + τ) ŷ 4 (t + τ) y 1 (t) ẏ 1 (t) 1, τ,, τ r+ρ 1 (r+ρ 1 )! 0 1 (r+ρ4 +1) y [r+ρ 1] 1 (t) = 0 1 (r+ρ1 +1) 1, τ,, τ r+ρ 4 y (r+ρ 4 )! 4 (t) ẏ 4 (t) y [r+ρ 4] 4 (t) For each channel in the output matrix, the control orders r are the same and can be decided during the control design, whereas the relative degrees ρ i are dierent but determined by the helicopter model structure. Manipulating the output matrix (11) gives the ollowing partition: τ ρ4 ŷ(t + τ) = τ 1 τ r ρ1 τ 4 (12) [Ȳ1 (t) T Ȳ 4 (t) T Ỹ1(t) T Ỹ r (t) ] T T (11) 6

9 where and [ ] T Ȳ i = y i (t) ẏ i (t) y [ρ i 1] i, i = 1, 2, 3, 4 (13) [ ] T Ỹ i = y [ρ 1+i 1] 1 y [ρ 2+i 1] 2 y [ρ 4+i 1] 4, i = 1,..., r + 1 (14) [ ] τ τ i = 1 τ ρ i 1 (ρ i 1)!, i = 1, 2, 3, 4 (15) { τ = diag τ ρ 1 +i 1 τ (ρ 1 +i 1)! ρ 4 +i 1 (ρ 4 +i 1)! It can be observered rom Eq(12) that the prediction o the helicopter output ŷ(t + τ), 0 τ T, in the receding horizon needs the derivatives o each output o the helicopter up to r + ρ i order at time instant t. Except or the output y(t) itsel that can be directly measured, the other derivatives have to be derived according to the helicopter model (8). During this process the control input will appear in the ρ i th derivatives, where i = 1, 2, 3, 4. The irst derivatives can be obtained rom the helicopter s kinematics model: ẏ 1 ẋ u ẏ 2 = ẏ = R i b v (17) ẏ 3 ż w } (16) ẏ 4 = ψ = q sin φ sec θ + r cos φ sec θ (18) Dierentiating (17) and (18) with substitution o helicopter dynamics (4) yields the second derivatives: ÿ 1 ẍ 0 0 ÿ 2 = ÿ = R i b 0 + 0, (19) ÿ 3 z T g where T = Z w w + Z col δ col g is the nomarlised main rotor thrust (intermedia steps given in apendix A), and ÿ 4 = ψ = q cos φ cos θ φ sin φ sin θ + q cos 2 θ sin φ cos φ L pq cos θ + N r cos θ + [ sin φ L lat cos θ θ r sin φ cos θ φ cos φ sin θ + r θ cos 2 θ L lon sin φ cos θ N col cos φ cos θ N ped cos φ cos θ ] u (20) where u is the control vector. Note that although control input δ col appears in (19), the other control inputs do not, so we have to continue dierentiating the irst three outputs. To acilitate the derivation, we adopt the relationship Ṙi b = Ri bˆω by using skew-symmetric matrix ˆω R 3 3 : 0 r q ˆω = r 0 p. (21) q p 0 Thus, the third and ourth derivatives o the position output can be written in: y [3] 1 x [3] 0 0 y [3] = y [3] = R i bˆω 0 + R i z [3] b 0, (22) T Z w ẇ + Z col δcol 2 y [3] 3 7

10 and y [4] 1 y [4] 2 y [4] 3 = x [4] y [4] z [4] =R i bˆωˆω R i b 0 0 T M pq T L pq T Z w ẅ + 2R i bˆω + R i b 0 0 Z w ẇ + Z col δcol + M lat T M lon T 0 L lat T L lon T Z col δ lat δ lon δ col (23) At this stage, the control inputs explicitly appear in (23). Thereore, the vector relative degree or the helicopter is ρ = [ ]. Note that in the ormulation o (23) δ col is the new control input, whereas δ col and δ col are treated as the states which can be obtained by adding integrators. By invoking (17) -(22), we now can construct matrix Ȳi, i = 1, 2, 3, 4. However, in order to ind the elements in Ỹi, i = 1, 2,..., r + 1, urther manipulation is required. By combining (20) and (23) and utilizing the Lie notation [Isi95], we have: Ỹ 1 = y [ρ 1] 1 y [ρ 2] 2 y [ρ 3] 3 y [ρ 4] 4 = x [4] y [4] z [4] ψ [2] = h 1(x) h 2(x) h + A(x)ũ (24) 3(x) h 4(x) where ũ = [ δ lat δ lon δcol δ ped ]; nonlinear terms L ρ i h i(x), i = 1, 2, 3, 4, can be ound in the previous derivation, and where A(x) = L g1 L ρ 1 1 L g1 L ρ 1 1 L g1 L ρ 1 1 A 11 = R i b L ρ 1 L ρ 2 L ρ 3 L ρ 4 h 1 (x) L g4 L ρ 1 1 h 1 (x) h 2 (x) L g4 L ρ [ 1 1 h 2 (x) A11 A = 12 A 21 A 22 h 4 (x) L g4 L ρ 1 1 h 4 (x) A 21 = [ L lat sin φ cos θ M lat T M lon T 0 L lat T L lon T Z col L lon sin φ cos θ, A 12 = 0 3 1, 0 ], A 22 = N ped cos φ cos θ. ]. (25) Dierentiating (24) with respect to time together with substituation o the system s dynamics gives Ỹ 2 = y [ρ 1+1] 1 y [ρ 2+1] 2 y [ρ 3+1] 3 y [ρ 4+1] 4 = L ρ 1+1 L ρ 2+1 L ρ 3+1 L ρ 4+1 (26) h 1 (x) h 2 (x) h 3 (x) + A(x)ũ[1] + p 1 (x, ũ) (27) h 4 (x) 8

11 where p 1 (x, ũ) is a nonlinear vector unction o x and ũ. By repeating this procedure, the higher derivatives o the output and Ỹi, i = 1, 2,..., r, can be calculated and inally we have Ỹ r+1 = y [ρ 1+r] 1 y [ρ 2+r] 2 y [ρ 3+r] 3 y [ρ 4+r] 4 = L ρ 1+r L ρ 2+r L ρ 3+r L ρ 4+r h 1 (x) h 2 (x) h 3 (x) + A(x)ũ[r] + p r (x, ũ, ũ [1],..., ũ [r] ) (28) h 4 (x) So ar by exploiting the helicopter model the elements to construct Ȳ and Ỹ are available. Thereore, the output o the helicopter in the uture horizon y(t + τ) can be expressed by its Taylor expansion in a generalized linear orm with respect to the prediction time τ and current states as shown in Eq.(12). In the same ashion as in Eq.(12), the reerence in the receding horizon w(t + τ) can also be approximated by: w 1 (t + τ) w(t+τ) = w 2 (t + τ) w 3 (t + τ) = [ T ] [ T s W1 (t) T W4 (t) T W 1 (t) T Wr+1 (t) ] T T w 4 (t + τ) (29) where τ ρ4 T =..... (30) 0 1 ρ1 τ 4 and T s = [ τ 1 τ r+1 ] and the construction o W i (t), i = 1, 2, 3, 4, and W i, i = 1,..., r + 1, can reer to the structure o Ȳi(t) and Ỹi, respectively. 3.2 Explicit nonlinear MPC solution The conventional MPC needs to solve a ormulated optimisation problem to generate the control signal, where the control perormance index is minimised with respect to the uture control input over the prediction horizon. In this paper, ater the output is approximated by its Taylor expansion, the control proile can be deined as ũ(t + τ) = ũ(t) + τũ [1] (t) + + τ r (31) r! ũ[r] (t), 0 τ T (32) Thereby, the helicopter outputs depend on the control variables ū = { ũ, ũ [1],..., ũ [r] }. Recalling the perormance index (9) and the output and reerence approximation (12) and (29), we have: J = 1 [ ] 2 (Ȳ (t) W (t)) T T1 T 2 T2 T (Ȳ T (t) W (t)) (33) 3 9

12 where Ȳ (t) = [ Ȳ 1 (t) T Ȳ 4 (t) T Ỹ1(t) T Ỹ r (t) T ] T, (34) and W (t) = [ W1 (t) T W4 (t) T W 1 (t) T Wr+1 (t) T ] T, (35) T 1 = T 2 = T 3 = T 0 T 0 T 0 T T QT dτ, (36) T T QT sdτ, (37) T T s QT s dτ. (38) Thereore, instead o minimising the perormance index (9) with respect to control proile u(t + τ), 0 < τ < T directly, we can minimise the approximated index (33) with respect to ū, where the necessary condition or the optimality is given by J ū = 0 (39) Ater solving the nonlinear equation (39), we can obtain the optimal control variables ū to construct the optimal control proile deined by Eq.(32). As in MPC only the current control in the control proile is implemented, the explicit solution is ũ = ũ(t + τ), or τ = 0. The resulting controller is given by ũ = A(x) 1 (KM ρ + M 1 ) (40) where K R 4 (ρ 1+ +ρ 4 ) is the irst 4 row o the matrix T3 1 T2 T R4(r+1) (ρ 1+ +ρ 4 ) where the ijth block o T 2 is o ρ i 4 matrix, and all its elements are zeros except the ith column is given by [ T T q ρ i +j T i (ρ i +j 1)!(ρ i +j) q 2ρ i +j 1 i (ρ i +j 1)!(ρ i 1)!(2ρ i +j 1)] (41) or i = 1, 2, 3, 4 and j = 1, 2,..., r + 1, and ijth block o T 3 is given by { T diag q 2ρ 1 +i+j 1 1 (ρ 1 +i 1)!(ρ 1 +j 1)!(2ρ 1 +i+j 1),, q T 2ρ 4 +i+j 1 4 (ρ 4 +i 1)!(ρ 4 +j 1)!(2ρ 4 +i+j 1) or i, j = 1, 2,..., r + 1; the matrix M ρ R ρ 1+ +ρ 4 and matrix M i R 4 are deined as: Ȳ 1 (t) T W 1 (t) T M ρ =.. (43) Ȳ 4 (t) T W 4 (t) T and M i = L ρ 1+i 1 L ρ 2+i 1 L ρ 4+i 1 } (42) h 1 (t) h 2 (t). W i (t) T, i = 1, 2,..., r + 1. (44) h 4 (t) The detailed derivation is provided in the appendix. The overall controller structure is shown in Fig.2. 10

13 Figure 2: ENMPC structure 3.3 Implementation issue When the ENMPC is applied or trajectory tracking o autonomous helicopters, not only the reerence trajectory is required, the higher derivatives o the reerence trajectory with respect to time are also needed in the prediction. Although this can be achieved by using various modern path planning algorithms, there are still some applications where the dedicated path generator is not available. In these cases the reerence is more likely to be designed comprising only the demanded helicopter position and the associated heading angle. To address this problem, we adopt a simple but eective method o low-pass preilter (45), as shown in Fig. 3. G(s) = ω 2 n s 2 + 2ζω n s + ω 2 n (45) Given the appropriate parameters ζ and ω, the command preilter can provide irst and second derivatives o the original reerence, which is adequate or a smooth trajectory tracking. Figure 3: Command preilter 11

14 4 Simulation and experiment 4.1 Simulation The proposed ENMPC has been validated in simulations and light experiments. The simulation and experiment are based on the Trex-250 miniature helicopter that is a radio controlled helicopter with a main rotor diameter o 460mm and a trail rotor diameter o 108mm. Trex-250 has a collective pitch rotor and well designed Bell-Hiller stabilizer mechanism which is consistent with most o the small-scale helicopters mentioned in the literature. Moreover, the miniaturized size and aerobatic ability make it well-suited or indoor light test. The model parameters o Trex-250 have been obtained by comprehensive system identiication in the previous work. Numerical simulations were carried out irst to investigate the attained perormance o ENMPC and compare it with the conventional MPC supported by the online optimisation [LCA10]. In the simulation, the ull dynamic model with 20% parameter uncertainties was used as the plant, whereas the simpliied model was just or control synthesis purposes. The ENMPC is designed with the prediction horizon T = 4s, control order r = 4 and Q = diag { }. The command preilter parameters are chosen as: ζ = 0.7 and ω b = 10rad/s. In the conventional MPC, the prediction horizon is set to T = 1s and the weighting matrix is chosen as the same as in ENMPC, but to penalise the control eort, the control input weighting term u T Ru is added into the perormance index, where R = diag { }. The corresponding optimisation problem is solved by using Matlab unction mincon. In the simulation, the helicopter was required to track a mulit-section reerence connected by abrupt turns. The helicopter tracking perormance is given in Fig.4. It can be seen that the helicopter under both conventional MPC and ENMPC is able to track the reerence with very satisactory perormance. However, in the conventional MPC the average time or solving the ormulated OP is about 0.2s, which restricts the control bandwidth to 5Hz. The ENMPC tackles this problem by directly using the explicit solution and can easily reach the required control bandwidth. On the other hand, during the abrupt reerence changes, the conventional MPC can replan the local trajectory to adapt to the helicopter dynamics, whereas the ENMPC relying on the command preilter can also deliver a smooth tracking perormance. 4.2 Flight experiment The proposed ENMPC was urther validated through light experiments which took place on our indoor testbed. Consisting o Trex-250 helicopters, Vicon motion capture system and ground station computers, the testbed combines commercial-o-the-shel equipments eectively and integrates them into the Matlab environment (see Fig.5). By developing dedicated interaces, researcher can implement algorithms in Simulink to operate the real helicopter as normal as in the simulation. Thus, the testbed provides a seamless way rom the control analysis and design to the experimental validation [LCCAed]. The irst light test is to track a square trajectory with the heading angle ixed 12

15 Figure 4: Trajectory tracking results Figure 5: Structure o indoor testbed 13

16 at zero. During this process the helicopter exhibits its manoeuvrability along our directions respectively. The tracking result is shown in Fig.6 in a 3-dimensional view with the attitude indicated along the trajectory. With the predictive eature, the helicopter under the control o ENMPC has the smooth and stable tracking capability even i it comes across an abrupt turn. The roll and pitch angle history provided in Fig.7 shows how the lateral and longitudinal channels are coordinated by the controller. During the turning points, the roll angle and pitch angle change together to increase the translational speed at one direction and decrease at another. The corresponding control signals are also given in Fig.8. Figure 6: Square trajectory tracking Figure 7: Attitude angles or square tracking Another light test was carried out to track a eight-shape trajectory. This light test is used to show the coordinated heading and position tracking capability that is 14

17 Figure 8: Control signals or square tracking not demonstrated in the irst light test. Along the reerence trajectory the required heading angle w 4 is deined by w 4 (t) = arctan 2(ẇ 1 (t), ẇ 2 (t)) (46) where arctan 2 is the our-quadrant inverse tangent unction, w 1 and ẇ 2 are the reerence velocity in x and y directions, respectively. The helicopter tracking result is given in Fig.9 in a 2-dimensional view with heading angle indicated. The control signals are given in Fig Summary Designing an MPC based controller with oresee eature to support the trajectory tracking o autonomous helicopters is a promising but challenging work, as the helicopter is unstable, highly nonlinear and particularly exhibits ast dynamics. To inherent the advantages o the MPC technique and prevent time consuming online optimisation, we introduce a closed-orm MPC or the helicopter tracking problem. The explicit solution to the MPC problem is derived by using Taylor expansion and exploiting the helicopter model. With the explicit MPC solution, on one hand the control signals can be calculated instantaneously to respond to the ast dynamics o helicopters and suppress the disturbances immediately. On the other hand, the online optimisation process can be removed rom the MPC ramework, which can accelerate the sotware development and simpliy the onboard hardware. Due to these advantages o ENMPC, the overall control ramework has a low complexity and high reliability, and it is easy to deploy on the small-scale helicopters. The proposed ENMPC has been successully validated in simulations and actual light tests using the Trex-250 helicopter. ENMPC shows a competitive perormance 15

18 Figure 9: Eight-shape trajectory tracking Figure 10: Control signals or eight-shape tracking 16

19 against the conventional MPC in the simulation and it demonstrates the excellent capability in the practical light tests. A Derivation o second derivative o position By using the relationship in Eq.(21) the irst three equations in the helicopter dynamics (4) can be simpliied as ollows u u 0 0 v = ˆω w + R b i (47) ẇ w g T where R b i is a transormation matrix rom inertial to body coordinates with the act that R b i Ri b = I 3 3. On the other hand, dierentiating (17) gives ẍ u u ÿ = R i bˆω v + R i b v (48) z w ẇ By substituting Eq.(47) into Eq.(48) and cancelling the equivalent terms, one has the second derivative o position (19). B Derivation o explicit MPC solution By using the short notations o (43) and (44),we have the ollowing relationship: Ȳ W = M ρ M 1 M 2. M r (ρ1 + +ρ 4 ) 1 A(x)ũ A(x)ũ [1] + p 1 (x, ũ). A(x)ũ [r] + p r (x, ũ,, ũ [r 1] ) = [ Mρ M r ] + [ ] 0 H (49) Hence, the perormance index (33) can be urther manipulated as J = 1 2 = 1 2 ([ Mρ M r ] + [ 0 H [ ] T [ Mρ T1 T 2 M r T2 T T 3 1 [ 2 HT T2 T ] [ M T ρ 3 M r ]) T [ ] ([ ] [ ]) T1 T 2 Mρ 0 T2 T + T 3 M r H ] [ ] T Mρ + 1 [ ] T [ ] Mρ T2 H+ M r 2 M r T 3 ] HT T 3 H (50) It can be seen that the necessary optimal condition can be written as ( ) H T [ ] [ ] T T M 2 T ρ 3 + ū M r 17 ( ) H T T 3 H = 0 (51) ū

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