A new Control Strategy for Trajectory Tracking of Fire Rescue Turntable Ladders

Transcription

1 Proceedings o the 17th World Congress The International Federation o Automatic Control Seoul, Korea, July 6-11, 28 A new Control Strategy or Trajectory Tracking o Fire Rescue Turntable Ladders N. Zimmert, A. Kharitonov, O. Sawodny Institute or System Dynamics, University o Stuttgart, Paenwaldring 9, D-7569 Stuttgart, Germany ( {zimmert, kharitonov, sawodny}@isys.uni-stuttgart.de. Abstract: Modern ire-rescue turntable ladders are constructed in a lightweight mode to increase their maximum operation velocities, maximum length, and outreach respectively. Hence, the ladder has a limited stiness and will be more and more subject to oscillations o delection along with dominant overtones. This paper deals with the active oscillation damping o such ladders. A new eedorward and eedback control strategy is applied. The eedorward control is calculated through system inversion o a multi-body system utilizing its dierentially latness. The design o the eedback is based on partial dierential equations (PDE describing a Euler Bernoulli model o a beam with a concentrated point mass at the end. The modal representation o the system is constructed based on the analytical orm o the eigenunctions. For active oscillation damping by eedback without a dynamical observer the ladder was equipped with a gyroscope additionally to strain gauges. Due to computational eorts and measurement noise a reduced state vector is disposed or stabilization. The proposed control approach allows damping the undamental oscillation as well as the irst dominant overtone and asymptotically stabilizing the system around the reerence trajectory. Measurement results rom the IVECO DLK 55 CS ire-rescue turntable ladder validate the good perormance o the control. Keywords: dierentially latness; system inversion; modal transorm; distributed-parameter system; oscillation damping. 1. INTRODUCTION In this paper a new control strategy or active damping o a ire-rescue turntable ladder is presented. Lightweight construction is applied to modern turntable ladders such as the IVECO DLK 55 CS (Fig. 1 in order to increase its maximum operation speeds, maximum ladder s length, and its horizontal outreach. Hence the ladder has a very limited stiness and will be more and more subject to oscillations along with dominant overtones. The IVECO DLK 55 CS turntable ladder is characterized by a maximum extension o the ladder set extension o L = 53.2 m and a maximum outreach o 22 m (at min. ϕ A = 68 respectively. The erection angle covers a space o ϕ A = [ ]. In the vertical plane the ladder is driven by hydraulic cylinders, which provide a maximum angular velocity o the erection angle ϕ A = 3 /s. The cage has a maximal loading o 3 kg, which corresponds to three ire ighters with ull equipment. In the proposed paper we shell use dierent mathematical models or eedorward and eedback design. For the eedorward loop the dierential equations o motion are derived by applying the Lagrangian ormalism (Banerjee [25], dewit:96. The dynamics o the hydraulic actuators are approximated by a 1st order transer unction. The design o the eedorward loop is ormulated based on simpliications o this dynamical model. An Euler- Bernoulli model o a beam with a point mass at the end is considered by Kharitonov et al. [27]. Based on the analytical eigenunctions the modal description o the vehicle gyroscope strain gauges cylinders L(t cage ϕ A (t Fig. 1. turntable ladder: IVECO Magirus DLK 55 CS (approach angle ϕ A 68, ladder length L = 53.2 m plant is constructed, but considering only the irst and the second mode. On this basis a eedback law is derived. This Ansatz is enhanced to dierent ladder s lengths. The plant is equipped with gyroscopes in addition to the strain gauges. Hence an observer is unnecessary because all states are determined by solving a system o algebraic equations /8/$2. 28 IFAC / KR

2 17th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-11, 28 Magnitude [normed] 1.5 strain gauges gyroscope Frequency [Hz] Fig. 2. Discrete ast ourier transorm o a step response with a length o L = 53.2 m o the two measurement systems. The control concept has been applied to a IVECO DLK 55 CS ire rescue turntable ladder. The ladder is driven with a micro controller working with ix-point arithmetic. Though computational costs have to be taken into account during the development o the control concept. In recent years the task o active oscillation damping o turntable ladders with the length up to 3 m was considered in some works. In Lambeck and Sawodny [27], Lambeck et al. [26], Sawodny et al. [22] a trajectory tracking control based on a dynamic model o the ladder and decentralized control strategy has been developed. In these works the inormation about the system state is limited to the erecting angle and the strain gauges at a point close to the hub. By using these signals, a eedback was designed within ramework o a multibody system model. The designed controller is able to reduce eectively the swaying concerning the undamental oscillation, but it cannot damp undesirable high requency oscillations (overtones, which become large especially or large lengths o the ladder. In Zuyev and Sawodny [25] the turntable ladder was considered as a lexible manipulator model with passive joints based on the Euler Bernoulli beam concept. The eedback design is based on the Galerkin approximation. In Zuyev and Sawodny [26, 27] a similar approach is applied on a Timoshenko model o a beam. For the realization o the designed eedback law it is proposed to use the dynamical observer needing more computational power o a micro controller. In the paper an overview on the control structure is given in section 2. Section 2.1 and 2.2 deal with the details o the control design. In section 2.1 a mathematical model o the ladder is derived and the eedorward control is ormulated based on ordinary dierential equations (ODE. The calculation o the eedback by using partial dierential equations (PDE, which describe a Euler Bernoulli model o a beam, is presented in section 2.2. Measurement results o the IVECO DLK 55 CS ire rescue turntable ladder are obtained and analyzed in section 3. In Section 4 concluding remarks are given and aspects o uture work are discussed. 2. CONTROL STRATEGY The control consists o a eedorward and a eedback loop. The structure is presented in Fig. 3. The eedorward control is designed within the ramework o a multi-body system model utilizing a latness based approach or system inversion. The reerence input to the closed loop system is ẏ re (t, which is the signal coming rom the hand lever o ẏ re z re latness based eed-orward control trajectory generation u parameter - u b stabilization based on pole placement x 1 x 3 parameter eigenvalues Fig. 3. Scheme o the control structure algebraic reconstruction parameter u strain gauges m 1 m 2 gyroscope irerescue turntable ladder the operator. Thereore a trajectory generator is needed to provide easible reerence trajectories ( z re (t. The stabilizing eedback is dimensioned with pole placement on the modal state space description o the Euler Bernoulli model o a beam. The states (x 1 (t, x 3 (t o the system (amplitudes o the two modes are constructed by solving a system o algebraic equations by utilizing the two measurements (m 1 (t, m 2 (t. Due to the act that the ladder can be extended to dierent lengths the parameters o the ladder have to be updated in every time step (e. g. stiness, eigenrequencies and damping coeicients o the dominant modes, et cetera. Fortunately the ladder s length is changing very slowly, so it can be considered as a parameter. The time variance o the system can be neglected or the controller design, because all the parameters depend on the length o the ladder. But a gain scheduling depending on the ladder s length is necessary to obtain a good perormance within the whole range. Although the oscillation o delection will be considered in the vertical plane, the action o gravity on the concentrated mass is neglected. Since the mathematical model is linear or small angles and the steady state solution taking into account the action o gravity and the prestressing o the ladder can be always subtracted or the task o stabilization. 2.1 Feedorward Control Multi-Body Model The dynamic model o the turntable ladder can be derived by using the Lagrange ormalism on a multi-body system with elasticity and damper-elements (Fig. 4, where the ladder set together with the cage and the vehicle are approximated by two equivalent masses (m h... vehicle, m l... ladder and cage and the arm elasticity is approximated by spring-damper elements. Thus the ladder is considered as a massless lea spring with the length L, the stiness coeicient c v, the damping coeicient d v, and at the end o it the point mass m l. The delection at the end o the ladder is named v z. The turntable ladder has some more degrees o reedom such as the turning motion o the hub (ϕ R, but as mentioned beore the ocus o the paper is on the motion in the vertical plane. = m l v z (t + m l L ϕ A (t + d v v z (t + c v v z (t + + (L ϕ R (t + v y (t 2 ( sin 2ϕ A (t + 2 2L L v z (t }{{} p R (t Even though the Coriolis-orce does not bother during the controller design, the eect o the rotary motion on (1 87

3 17th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-11, 28 ϕ R (t z L v z (t m l d v (L c v (L ỹ = h (x = x 1 + x 3 /L with the relative degree o r = 4 is obtained. The dierence between the control output and the dierentially lat output are negligible. By using the Byrnes-Isidori normal orm S : ỹ = z 1, ż 1 = z 2,..., ż r 1 = z r [ ż r = L r h + L g L r 1 the system comes up in the orm ] hu φ 1 (z = a (z + b (z u m h ϕ A (t Fig. 4. Multi-body system model with elastic degree o reedom as model or eedorward control design the erecting motion is neglected (p R (t, because the rotary motion is quiet slowly ( ϕ R 1 and this axis is actively damped by a separate controller ( v z 1 as well. For the sake o less computational costs this simpliication is quiet convenient. The dynamics o the actuators is approximated by 1st order transer unction, because there is a subsidiary low control o the cylinders hydraulics. The nonlinear dc gain depending on the erecting angle k A (ϕ A is determined rom the construction. ϕ A = 1 ϕ A + k A (ϕ A u (2 τ A τ A By choosing the state vector as x (t = [ ϕ A (t ϕ A (t v z (t v z (t ] T the ollowing state space is the result ẋ = (x + g (x u x 2 1 x 2 = τ A x + 4L x 2 c v x 3 d v x 4 τ A m l m l y = h (x = x 1 + arctan ( x3 L k A (x 1 τ A Lk A (x 1 τ A u(3a (3b Model Inversion To invert the model mentioned in (3a a dierentially lat output with the relative degree r = n has to be ound. The relative degree r is deined by the ollowing conditions: L g L i h (x = i =,..., r 2 L g L r 1 h (x x R n (4 The dierential operator L represents the Lie derivative o the argument along the vector ield and L g along the vector ield g respectively. The real output mentioned in (3b has a relative degree o r = 2. Thus y is not a dierentially lat output, because the system is o 4th order. But the delection proportional to the ladder s length is very small. I we assume v z /L 1 a new output z 1 = x 1 + L 1 x 3 z 2 = x 2 + L 1 x 4 z 3 = c v m 1 L x 3 z 4 = c v m 1 L x 4 c v ż 4 = τ A m 2 1 L (m 1Lx 2 τ A c v x 3 m 1 Lk A u. Via a dieomorph state transorm z = φ (x, z i = φ i (x = L i 1 h (x i = 1,..., r (5a (5b (5c (5d (5e the model can be inverted with respect to the dierentially lat output. The new control input is deined as n-th derivative with respect to the time o the dierentially lat output ν = ż 4 = ỹ (IV and so the control signal u is determined by a (z + ν u = b (z 1 u = c v ka (z 1, z 3 (c vz 2 + c v τ A z 3 + m l z 4 + τ A m l ν. (6 By the model inversion we obtain a chain o our integrators (system s order with the input ν and the dierentially lat output ỹ = z Feedback Control Applying the eedorward control law (6 with easible trajectories or ν and z accordingly the load sway will be much less compared to an uncontrolled motion. But or the reason o model mismatches and assumptions, which have been made in the design, the load sway is not eliminated totally. Perturbances such as wind and people moving or working on the platorm o course cannot be compensated by eedorward control. So a eedback loop is needully to stabilize the system around the reerence trajectories to ensure a minimum o load sway. Euler Bernoulli model The mathematical model o the two-body system (ladder and cage corresponds to a hybrid system consisting o a distributed as well as a lumped parameter system due to the concentrated mass at the end o the ladder (Fig. 5. It leads to the boundary conditions which have to obey the dynamics o this concentrated mass. The ladder is driven by a control torque M(t at its end z =. L is the length o the ladder, ϕ A (t corresponds to the angle between the moving axis Oz and the horizontal direction, w(z, t is the delection rom the center line o the beam describing the ladder. M p and J p are the mass 871

4 17th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-11, 28 z = M(t J h, m h w(z, t ϕ A (t L z J p, m p S(L, I(L, E(L, ρ Fig. 5. Euler Bernoulli beam with a point mass at the end as model or eedback control design and moment o inertia o the cage with respect to its center o mass. Assuming that the angular velocity ϕ A (t is small and neglecting Coriolis orces, the ladder concerning the distributed part o the plant can be described as the Euler Bernoulli beam (Moallem et al. [2], Wit et al. [1996] by the ollowing partial dierential equation EI 4 w(z, t 4 + ρs 2 [ϕ A (tz + w(z, t] t 2 =, z (, L, t >, (7 where E is the Young s modulus, I is the moment o inertia o the cross section, ρ is deined as the density, and S is the cross section area o the ladder. These parameters were evaluated through experimental identiications. The major eects could be reproduced by simulations o the ladder as a single lexible link. All except ρ are varying with the length o the ladder. But the ladder s length is changing very slowly, thereore the eedback can be derived as i the ladder had a constant length. The calculations just need to be repeated or every change in the ladder s length. The boundary conditions at the end z = o the beam have the ollowing orm (ixed end w(, t =, t >, (8 w(, t =, t >. (9 The boundary conditions at the end z = L describing the connection between the distributed and lumped parts o the plant can be written as ollows (Aoustin et al. [1997], Moallem et al. [2] EI 2 w(l, t d 2 ( w(l, t 2 =J p dt 2 ϕ A (t +, t >,(1 EI 3 w(l, t d 2 ( 3 =M p dt 2 ϕ A (tl + w(l, t, t >.(11 The terms in the right part describe the moment and orce provided by the moving concentrated mass, respectively. Introducing the new depending value V (z, t = ϕ A (tz + w(z, t, the system equation (7 together with the boundary conditions (8 (11 take the orm EI 4 V (z, t 4 + ρs 2 V (z, t t 2 =, (12 V (, t=, (13 V (, t =ϕ A (t, (14 EI 2 V (L, t d 2 ( V (L, t 2 =J p dt 2, (15 EI 3 V (L, t 3 =M p d 2 V (L, t dt 2. (16 For the simplicity, in this paper the control input u(t is introduced as ollows u(t = ϕ A (t = V (, t. (17 The dynamics o the drive mechanism will not be taken into account here, although the actual hub torque at the end z = o the beam is to be calculated by M(t = J h d 2 ϕ A (t dt 2 EI 2 w(, t 2, (18 where J h is the hub inertia. Feedback Design Via a modal transorm o the system in (13 (16, which will be presented in detail by Kharitonov et al. [27], one obtains ( d 2 Vk (t dt 2 + ωkv 2 k (t = ωk 2 k (IV k u(t. (19 η where η = ρs/ei, k (IV and k are series expansions in the basis o eigenunctions (Z k, and ω i are the circular requencies corresponding to the eigenvalues λ i. All these parameters depend on the ladder s length. The task o stabilization will be considered. Further the equilibrium V (z, t z [, L] will be treated as the operating point without loss o generality. For the technical implementation o the eedback law one proposes to develop a eedback based only on the irst two modes. Indeed, the limiting requency provided by the hydraulic cylinders is 3 Hz but the requency corresponding to the third mode is higher especially or shorter ladder s lengths. I we consider the irst two dominant modes the state values will be introduced as ollows x 1 (t = V 1 (t, x 2 (t = ẋ 1 (t = V 1 (t, x 3 (t = V 2 (t, x 4 (t = ẋ 3 (t = V 2 (t. Even so internal damping is neglected in the Euler Bernoulli model o beam, it can be observed in reality. Through experimental measurements the damping coeicients o the two irst modes (D 1, D 2 have been identiied and (19 was augmented respectively. The system description in the state space takes the orm 872

5 17th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-11, 28 ẋ 1 ẋ2 ẋ3 ẋ4 = 1 ω 2 1 2D 1 ω 1 1 ω 2 2 2D 2 ω 1 }{{} =A x 1 x 2 x 3 x 4 + (IV ω η u(t. (2 (IV ω η }{{} =B In order to be able to stabilize the plant state values must be available or a eedback. For this the necessary state values can be obtained by a Luenberger type observer or measured by some sensors. The IVECO DLK 55 CS turntable ladder was equipped with strain gauges located at the point z = z 1 providing m 1 (t = 2 w(z 1, t 2 = 2 V (z 1, t 2 and a gyroscope located at z = z 2 sensing m 2 (t = 2 V (z 2, t. t I only two dominant modes are taken into account, the values to be measured can be represented as ollows m 1 (t = 2 V (z 1, t 2 = ˆx 1 (tz 1(z 1 + ˆx 3 (tz 2(z 1, m 2 (t = 2 V (z 2, t = ˆx 2 (tz t 1(z 2 + ˆx 4 (tz 2(z 2. In order to evaluate all state values one needs additional inormation. These can be obtained as ollows (the gauge signal m 1 (t is assumed to be suicient smooth t dm 1 (t dt = 3 V (z 1, t 2 t m 2 (τdτ = V (z 2, t = ˆx 2 (tz 1(z 1 + ˆx 4 (tz 2(z 1, = ˆx 1 (tz 1(z 2 + ˆx 3 (tz 2(z 2. The estimates o the state values can be obtained in this case as a solution o the ollowing system o linear algebraic equation Z 1(z 1 Z 2(z 1 ˆx 1 (t Z 1(z 2 Z 2(z 2 ˆx 2 (t Z 1(z 1 Z 2(z 1 ˆx 3 (t = Z 1(z 2 Z 2(z 2 ˆx 4 (t m 1 (t m 2 (t dm 1 (t/dt = t. (21 m 2 (τdτ From the numerical point o view it is evident, the points or measurements z 1 and z 2 must be chosen in such a way, that the condition number with respect to inversion o the system matrix (21 becomes small. Obviously in the system matrix (21 the states ˆx 1 and ˆx 3 are not interdependent with the ˆx 2 and ˆx 4. For computational eorts and measurement noise a reduced state vector is introduced x(t = [ ˆx 1 (t ˆx 3 (t ] T. So a system o linear algebraic equation o 2nd order ollows. The states can be calculated by [ ] [ ] m 1 (t ˆx1 (t Z = 1(z 1 Z 2(z 1 1 ˆx 3 (t Z 1(z 2 Z 2(z t 2 m 2 (τdτ. (22 Using only two estimates o the state values the state eedback can be realized as u b (t = K [ ˆx 1 (t ˆx 2 (t ˆx 3 (t ˆx 4 (t ] T, with K = [ k 1 (t k 3 (t ] where the controller gain matrix K can be constructed by standard means, e. g. by means o pole placement along the root locus o the matrix (A BK. The asymptotic stability o the equilibrium is assured, since the Kalman controllability criterion or the system (2 with its parameters is satisied through all lengths o the ladder, i. e. det [ B AB A 2 B A 3 B ] [ = b 2 2b 2 ω ω2 4 + ( 4D D2 2 2 ω1ω 2 2 ] ( D 1 D 2 ω 3 1 ω 2 + ω 1 ω2 3 L. 3. MEASUREMENT RESULTS In this section we present experimental results, which could be achieved with the proposed control strategy. In the experiment the ladder has been erected rom 66.5 to 71 and lowered again without any controller in action. Aterward (t > 119 s the same motion was repeated with the proposed active oscillation damping. The ladder has a length o 53.2 m. Figure 6 shows the angular position o the ladder s tip. The solid line is the reerence trajectory o the dierentially lat output (z 1,re = ỹ re y re and dash dotted line is the sensed position ( z 1 = ϕ A + v z L 1, v z (t w(l, t. So, the ladder is damped within hal the periodic time o the irst mode. The overshoot is about.3 m o delection at the end o the ladder (see Fig. 7. In general there is a reduction o approximately 4 % compared to the uncontrolled motion. The oscillations are damped during motion and very low residual load sway is achieved as well. The time response o the two dominant modes presented in Fig. 8 reveals that the amplitudes o the second mode can be slightly higher with the controller in action. This is because the eedorward control excites higher amplitudes, which is due to the act that multiple modes were not taken into account during the design. But obviously, both modes are damped actively by the stabilizing eedback. 4. CONCLUSION In this paper a control approach or erecting motion o a ire rescue turntable ladder was presented. A model 873

6 17th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-11, 28 Angular position o cage [ ] uncontrolled controlled z 1,re z Time [s] Fig. 6. The angular position o the cage at the end o the ladder in degrees Delection at the end o the ladder [m] uncontrolled controlled Time [s] Fig. 7. The delection at the end o the ladder in m Mode 2 [ ] Mode 1 [ ] uncontrolled controlled Time [s] Fig. 8. Estimated states (ˆx 1, ˆx 3 o the two dominant modes inversion based on a dierentially lat output or eedorward control is applied to a simple linear model o the ladder. The eedback loop is designed on Euler Bernoulli model o a beam with a concentrated mass at the end. Based on the analytical orm o the eigenunctions the modal description o the plant was derived. The control law is realized on a micro controller system with ix-point arithmetic and limited computational power. Thereore only two state values were constructed rom a algebraic equation taking the two measurements into account. The controller gain was calculated by the means o pole placement. The properties o the ladder depending on its length are taken into account as parameters and are updated in every time step o the micro controller system. In contrast to earlier works the proposed approach can be used or ladders with length over 3 m and it is adaptive to varying ladder lengths without additional requirements on a micro controller. At the present moment the proposed approach is veriied at the IVECO DLK 55 CS turntable ladder. For uture work the non-linear dynamics o the hydraulic cylinders will be taken into account more precisely and the eedorward control will be derived within the ramework o the Euler Bernoulli model o a beam. REFERENCES Y. Aoustin, M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Theory and practice in the motion planning and control o a lexible robot arm using mikusinski operators. In Proceedings o the 5th IFAC Symposium on Robot Control, pages , Nantes, France, S. Banerjee. Dynamics or Engineers. John Wiley & Sons, Ltd, 25. A. Kharitonov, N. Zimmert, and O. Sawodny. Active oscillation damping o the ire-rescue turnable ladder. In Proceedings o the IEEE Conerence on Control Applications 27, pages , Singapore City, Singapore, S. Lambeck and O. Sawodny. Trajectory generation and oscillation damping control or a lexible link robot. In Proceedings o the 33rd Annual Conerence o the IEEE Industrial Electronics Society, Taipei, Taiwan, S. Lambeck, O. Sawodny, and E. Arnold. Trajectory tracking control or a new generation o ire rescue turntable ladders. In Proceedings o the IEEE Conerence on Robotics, Automation and Mechatronics 26, pages 1 6, Bangkok, Thailand, doi: 1.119/RAMECH M. Moallem, P.V. Patel, and K. Khorasani. Flexible-link Robot Manipulators: Control Techniques and Structural Design. Springer-Verlag, London, 2. O. Sawodny, S. Lambeck, and A. Hildebrandt. Trajectory generation or the trajectory tracking control o a ire rescue turntable ladder. In Proceedings o the Third International Workshop on Robot Motion and Control. RoMoCo 2, pages , C.C. de Wit, B. Siciliano, and G. Bastin. Theory o Robot Control. Communications and control engineering series. Springer-Verlag, London, A. Zuyev and O. Sawodny. Stabilization o a lexible manipulator model with passive joints. In Proceedings o the 16th IFAC World Congress, Prague, Czech Republic, A. Zuyev and O. Sawodny. Observer design or a lexible manipulator model with a payload. In Proceedings o the 45th IEEE Conerence on Decision and Control, pages , San Diego, CA, USA, doi: 1.119/CDC A. Zuyev and O. Sawodny. Stabilization and observability o a rotating Timoshenko beam model. Mathematical Problems in Engineering, 27:Article ID 57238, 19 pages, 27. doi: /27/