2 Mechanical Knuckle Boom Crane Model

Energy Saving Potential in Knuckle Boom Cranes using a Novel Pump Controlled Cylinder Drive

S. Ketelsen L. Schmidt V. H. Donkov T. O. Andersen

Department of Energy Technology, Aalborg University, 9220 Aalborg, Denmark. E-mail: [email protected]

Abstract

This paper is considering the application of a novel pump controlled cylinder drive, the so-called Speed- variable Switched Differential Pump (SvSDP), for knuckle boom crane actuation. Especially the control system for the SvSDP drive is considered, and aiming on improving energy efficiency a refinement of the existing control structure is proposed. An energy efficient sizing algorithm for the SvSDP drive is developed, and fundamental differences between the achievable operating range for the SvSDP drive compared to a conventional valve-cylinder drive are discussed. A case study is conducted with knuckle boom crane actuation, and compared to a conventional valve actuation. Simulation results show that the motion tracking performance is on a similar level compared to the valve actuation approach, while the energy consumption is drastically decreased. For the given test trajectory the valve actuation system consumes 0.79 kWh of electrical energy, while the SvSDP drive consume 0.06 kWh, if ideal energy recovery and storage is assumed.

Keywords: Energy efficient hydraulic actuation, pump controlled cylinder, cylinder direct drive, offshore cranes, multivariable control

1 Introduction

The usage of hydraulics for low-speed high-force linear actuation is a well established standard in many in- dustries. Hydraulic actuation are traditionally selected due to the high power and force density they can of- fer. Conventionally, hydraulic cylinders are controlled using proportional valves, which achieve the desired motion control performance by throttling the flows in and out of the cylinder chambers, which in turn is a major source of losses in hydraulic systems. To re- duce the throttling losses, a load sensing pump may be installed. This is often the case in knuckle boom crane actuation systems, but a drawback of this strat- egy is that the supply pressure is determined by the demand of the consumer requiring the largest pressure.

This may lead to situations where a fast moving un- loaded cylinder, requires a large flow, while another small flow-high pressure consumer determines the sup-

ply pressure, leading to large throttling losses for the fast moving cylinder. With increasing demands for en- ergy efficieny in most industries, research communities as well as industry are trying to identify other meth- ods for improving the overall efficiency of linear actua- tion in machinery, such as replacing hydraulic cylinders with self-contained electromechanical cylinders Hagen et al. (2017). Other researchers are looking into so- called digital hydraulics. Examples of this approach include multi-chamber and/or multi-pressure cylinders Hedegaard Hansen et al.(2018),Linjama et al.(2009) and Huova et al. (2017), hydraulic buck converters Kogler and Scheidl (2016), hydraulic power manage- ment conceptsVukovic et al.(2016),Linjama and Huh- tala (2010) etc. Most of these technologies are still in the development phase, with the predominant chal- lenges being low reliability under high load operating conditions for self-contained electromechanical cylin- ders Hagen et al. (2017), or the demand for faster

xP

M A C B

M_eq

Geq

Cx_P²

ω_m,ref ω_m

QA QC

QB

QAC

QvA QvB

xvB

xvA

uvA uvB

PA PB

A_A A_B

Figure 1: Asymmetric cylinder controlled by SvSDP- system.

on/off valves for hydraulic buck convertersKogler and Scheidl (2016). Another approach for reducing throt- tling losses is direct hydraulic cylinder drives/pump controlled cylinders. The main idea is here to control the flow to the cylinder chamber directly by the pump without any valves in the main transmission lines. A fundamental challenge for this concept is that an asym- metric hydraulic cylinder requires different chamber volume flows. One way of compensating the unequal flow rates is by throttling only the differential volume flow using pilot operated check valvesRahmfeld(2002) or via an inverse shuttle valve C¸ aliskan et al. (2016) and Michel and Weber (2012). Another approach is to use an asymmetric pump unit Quan et al. (2014), which effectively also can be obtained by using two fixed displacement pumps such as investigated inPed- ersen et al.(2014) andJ¨arf et al.(2016). Here the two pumps are connected to a common shaft, but rotate in opposite directions. Furthermore, they are sized to match the area ratio of the cylinder.

The drive concept presented inPedersen et al.(2014) demonstrated good hydraulic efficiency, but it was found that the effective pump displacement- and the cylinder area ratios, cannot be matched in the entire operating range due to pump leakage, resulting in ir- regular performance. These issues led to the develop- ment of the Speed-variable Switched Differential Pump (SvSDP) drive, which was introduced and investigated in Schmidt et al. (2015). The concept adds a third pump, only delivering flow in certain situations. In Schmidt et al.(2017) the control structure of this con- cept was further developed with the aim to decouple

Figure 2: Knuckle Boom Crane example provided by National Oilwell Varco. c

the motion control and chamber pressure control.

The SvSDP drive topology is depicted in Fig. 1. At positive shaft speeds, pumps A and C provide flow to chamber A of the cylinder, while pump B withdraws fluid from cylinder chamber B. At negative shaft speeds pump C idles, effectively providing no flow to the cylin- der A chamber. Hence a surplus flow into the cylinder is present in both directions of operation, and a pres- sure increase will appear for shaft speeds where pump flow exceeds internal leakageSchmidt et al.(2015). In order to maintain chamber pressures at reasonable lev- els this flow mismatch may be bled off via the 2/2 pro- portional valves. Hence the system has three inputs ω_m,ref, u_vA and u_vB. As it is not possible to control the motion of the cylinder and both chamber pressures independently, only two sensible outputs may be de- fined, leaving the systemover-actuated. In the current investigation, the SvSDP concept fromSchmidt et al.

(2017) will be investigated for actuation of a knuckle boom crane similar to the one seen in Fig. 2. A model based approach to the sizing of the SvSDP drives is presented, and a control strategy targeting at a high energy efficiency is established. The resulting perfor- mance is compared to that of a conventional valve op- eration approach and simulation results demonstrate similar motion performance, while the SvSDP drive ap- proach shows a highly improved energy efficiency com- pared to the valve actuation system.

2 Mechanical Knuckle Boom Crane Model

A knuckle boom crane may be illustrated as depicted in Fig. 3. Due to the often large cylinders, the masses of cylinder tubes and rods may be significant compared to the crane booms and payload, and may therefore be taken into account. Hence the mechanical system may be depicted as having seven centers of mass (CMs), all moving relative to each other. As illustrated, the

cylinder tube and piston CMs are equivalated into a combined CM for each cylinder, resulting in five CMs to be included in the model. Defining the joint angles ϕ1,ϕ2andϕ3(generalised coordinates), the CMs may be described by:

PCM1=A1S1, PCM2=A1S2+A2S3 (1) PCM3=A1S2+A2S4+A3S4, (2) P_CMcyl1=A₄S₆, P_CMcyl2=A₁S₂+A₅S₇ (3)

Here, matrices and vectors are given by:

A₁=

cos(ϕ1) −sin(ϕ1) sin(ϕ1) cos(ϕ1)

A2=

cos(ϕ1+ϕ2) −sin(ϕ1+ϕ2) sin(ϕ1+ϕ2) cos(ϕ1+ϕ2)

A3=

cos(ϕ1+ϕ2+ϕ3) −sin(ϕ1+ϕ2+ϕ3) sin(ϕ1+ϕ2+ϕ3) cos(ϕ1+ϕ2+ϕ3)

A4=

cos(ϕ1−α8−α1) −sin(ϕ1−α8−α1) sin(ϕ1−α8−α1) cos(ϕ1−α8−α1)

A5=

cos(ϕ₁+ϕ₂−α₁₀+α₂) sin(ϕ₁+ϕ₂−α₁₀+α₂)

−sin(ϕ₁+ϕ₂−α₁₀+α₂) cos(ϕ₁+ϕ₂−α₁₀+α₂)

S₁=

LCCM1x

LCCM1y

, S₂=

LCF

0

, S₃=

LFCM2x

LFCM2y

S4= LFJ

0

, S5=

LJCM3

0

S₆=

L_Ccyl1 0

, S₇=

L_Fcyl2 0

The total kinetic energyKmay be expressed as:

K= m1

2

P˙^T_CM1P˙CM1+m2

2

P˙^T_CM2P˙CM2 (4) +m₃

2

P˙^T_CM3P˙CM3+m_cyl1 2

P˙^T_CMcyl1P˙CMcyl1

+mcyl2

2

P˙^T_CMcyl2P˙CMcyl2+J1

2 ϕ˙²₁+J2

2 ( ˙ϕ1+ ˙ϕ2)² +J3

2 ( ˙ϕ1+ ˙ϕ2+ ˙ϕ2)²+Jcyl1

2

˙

ϕ1−∂α8

∂ϕ₁ϕ˙1

+Jcyl2

2

˙

ϕ1+ ˙ϕ2−∂α10

∂ϕ2

˙ ϕ2

The total potential energyP may be expressed as:

P =g^T(PCM1+PCM2+PCM3+PCMcyl1 (5) +P_CMcyl2), g= [0 g]^T

Hence, the LagrangianL=K−Pmay by formed, from which the joint torques may be established as:

τ1= d dt

∂L

∂ϕ˙1

− ∂L

∂ϕ1

(6) τ₂= d

dt

∂L

∂ϕ˙2

− ∂L

∂ϕ2

(7) τ3= d

dt

∂L

∂ϕ˙3

− ∂L

∂ϕ3

(8)

Lcyl

xmax = 2xmin

xmin/2 xmin

C

D

CM2

CM3 G

F

H

α1 J

B φ1

-φ2

α2

A α4

α5

α6

α7

α9

α8

α10

LFcyl2

-φ3

CM1

CMcyl1

CMcyl2

Payload xP

xmin/2 Equivalent center of mass for cylinders

CMtube CMpiston CMcyl

α3

Figure 3: Illustration of a knuckle boom crane with two hydraulic cylinders.

Definingq= [ϕ1 ϕ2 ϕ3]^T where ϕ1=α1+α3(xP1) + α4−π/2,ϕ2=α5(xP2) +α6−α2−π, the joint torque τ= [τ1 τ2 τ3]^T may be expressed as:

τ=D(q)¨q+C(q,q) +˙ G(q) (9) Noting ˙q=Jx, ˙˙ x = [ ˙xP1 x˙P2 0]^T the dynamic model may be described in cylinder space as:

F=J^TD(x)J¨x+J^T(D(x) ˙Jx˙ +C(x,x))˙ (10) +J^TG(x)

Here,Fis the linear mechanical output force given by:

F=Fhyd−Bx,˙ B=

B_v1 0 0 B_v2

(11)

3 SvSDP Cylinder Drive

The two structurally identical SvSDP drives needed for knuckle boom crane actuation are subscripted with i= 1,2 (SvSDP₁and SvSDP₂).

3.1 Nonlinear Model

Considering Fig. 1, the SvSDP drive is described by Eq. (12)-(20), assuming ideal check valves, no cylin- der cross port leakage, nonlinear friction phenomena absent, and defining V_Ai = V_A0i +A_Aix_Pi, V_Bi = V_B0i−A_Bix_Pi,α_i=A_Bi/A_Ai. i= 1,2

¨

xP_i= AA_i(PA_i−αiPB_i)−x˙P_iBv_i−f(x,x)˙ M_eq

i

(12) P˙A_i= βA_i

VA_i

(QAC_i−QvA_i−x˙P_iAA_i) (13) P˙_Bi= β_Bi

VBi

( ˙x_Piα_iA_Ai−Q_Bi−Q_vBi) (14) QAC_i=

Q¯A_i(ωm_i)ηvA_i+ ¯QC_i(ωm_i)ηvC_i , ωm_i ≥0 Q¯A_i(ωm_i)/ηvA_i , ωm_i <0 (15) Q_Bi=

Q¯B_i(ωm_i)/ηvB_i , ωm_i ≥0

Q¯B_i(ωm_i)ηvB , ωm_i <0 (16) QvA_i=KvA_ixvA_i, QvB_i=KvB_ixvB_i (17)

¨

x_vAi=ω²_vA

iu_vAi−2ζ_vAiω_vAix˙_vAi−ω²_vAx_vAi (18)

¨

x_vBi=ω²_vB

iu_vBi−2ζ_vBiω_vBix˙_vBi−ω_vB² x_vBi (19)

¨

ωm_i=ω²_viωm,ref_i−2ζv_iωv_iω˙m_i−ω_v²ωm_i (20) f(x,x) contains the gravitational load and the Coriolis˙ force. ¯QA_i, ¯QB_i and ¯QC_i are leakage free pump flows, ηvA_i, ηvB_i and ηvC_i are pressure dependent volumet- ric pump efficiencies, PA_i andPB_i chamber pressures, xvA_i, xvB_i the valve spool positions, ωm_i the motor shaft speed andxP_i the cylinder piston position. The

valve inputs uvA_i, uvB_i and the motor shaft reference speed ωm,ref_i are the three system inputs. Addition- ally,Bv,iis a viscous friction coefficient,ζvA_i,ζvB_i,ζv_i

damping ratios,ωvA_i, ωvB_i, ωv_i bandwidths,βA_i, βB_i

the effective bulk moduli and KvA, KvB are the flow gains of the 2/2 pressure compensated proportional valves.

Furthermore, the hydraulic forces are given by:

F_hyd=

AA₁(PA₁−α1PB₁) AA₂(PA₂−α2PB₂)

(21)

3.2 Linear Model

The inverse flow characteristics of the 2/2 proportional valves are used to compensate the input (uvA_i and uvB_i), meaning that ideallyQvA_i =QvA,ref_i, QvB_i = QvB,ref_i, if neglecting valve dynamics. Considering the gravitational load as a disturbance and assuming the velocity dependent Coriolis forces negligible and equal effective bulk moduli (i.e. β_Ai =β_Bi =β_i), the linear model is given by Eq. (22), when defining relations ρ_i =V_Bi/V_Ai, ρ_0i =V_0Bi/V_0Ai where V_0Ai =V_Ai|_x0

i, V_0Bi = V_Bi|_x0

i and x_0i is the state vector at the lin- earisation point.

x˙c =Ac_ixc_i+Bc_iuref_i, y_p

i =Cc_ixc_i

xc_i = xp_i

xui

, Ac_i=

Ap_i Bp_iCu_i

0 Aui

Bci = 0

B_ui

, Cci = C_p

i 0

(22) x_p

i = [x_Pi x˙_Pi p_Ai p_Bi]^T

x_ui = [ω_mi ω˙_mi q_vAi q˙_vAi q_vBi q˙_vBi]^T u_refi = [ω_m,refi q_vA,refi q_vB,refi]^T

y_p

i = [x_Pi p_Ai p_Bi]^T, C_p

i =





1 0 0 0 0 0 1 0 0 0 0 1





A_p

i =













0 1 0 0

0 −_M^Bvi

eqi

AAi

M_eqi −^α_M^iAAi

eqi

0 −^β_V^iAAi

0Ai

−^βi_V^KAqpi

0Ai

0 0 ^β_ρ^iαiAAi

0iV_0Ai 0 ^β_ρ^iKBqpi

0iV_0Ai













B_p

i =











0 0 0

0 0 0

β_iK_Aqi V0Ai

−_V^βi

0Ai

0

−_ρ^βiKBqi

0iV0Ai

0 −_ρ ^βi

0iV0Ai











KAqp_i = ∂QAC_i

∂P_{Ai x}

0

KBqp_i= ∂QB_i

∂P_{Bi x}

0

K_Aqi = ∂QAC_i

∂ω_{mi x}

0

K_Bqi = ∂QB_i

∂ω_{mi x}

0

Au_i =





Au1_i 0 0 0 Au2_i 0 0 0 Au3_i





Au1_i =

0 1

−ω_v²_i −2ζviω_vi

Au2_i =

0 1

−ω_vA²

i −2ζ_vAiω_vAi

Cui =





1 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 1 0





B_ui =

















0 0 0

ω²_v

i 0 0

0 0 0

0 ω²_vA

i 0

0 0 0

0 0 ω_vB²

i

















The combined linear model Eq. (22) may be ex- pressed by the transfer function matrix Eq. (24)

y_p(s) =G_c(s)u_ref(s) (23) Gc(s) =Cc(sI−Ac)⁻¹Bc (24) The transfer functions for the plant and actuator dy- namics may respectively be obtained as:

y_p

i(s) =Gp_i(s)up_i(s)

Gp_i(s) =Cp_i(sI−Ap_i)⁻¹Bp_i (25) y_u

i(s) =G_ui(s)u_refi(s)

Gu_i(s) =Cu_i(sI−Au_i)⁻¹Bu_i (26) up_i= [ωm_i qvA_i qvB_i]^T

uref_i = [ωm,ref_i qvA,ref_i qvB,ref_i]^T

In Schmidt et al.(2017) the significance of the dy- namic couplings was studied using a relative gain array (RGA)-analysis. For the considered input-output com- binations it was found that severe dynamic couplings are present, especially close to the system eigenfre- quency. Due to these couplings a decentralised control strategy may not be utilised directly on the system.

3.3 Control Strategy

In Schmidt et al. (2017) a drive control strategy has been developed to handle the dynamic couplings, with the overall structure depicted in Fig. 4. The funda- mental idea is to transform the input- and output vari- ables using ˜y = W2y_p, ˜u = W⁻¹₁ uref. By choosing the transformation matrices W1 and W2 properly it

is shown possible to decouple the transformed system states, as shown in Eq. (27) and Eq. (28).

u_p=G_uu_ref, u_ref=W₁u˜ ⇒ u_p=G_uW₁u˜ (27)

˜

y=W2y_p, y_p=Gpup ⇒ y˜ =W2Gpup (28) Substituting Eq. (27) into Eq. (28), gives the trans- formed system as:

˜

y=W2GpGuW1u˜ = ˜Gcu,˜ (29) Note that the indexiis omitted in this section, as the developed control strategy is identical for both cylinder drives.

3.3.1 Output Transformation

As mentioned, it is only sensible to control two of the non-transformed outputs. As three inputs are avail- able the system is said to beover-actuated. InSchmidt et al. (2017) it is found desirable to formulate an out- put transformation (W₂) such that more appropriate states than the actual chamber pressures may be con- sidered. Theseappropriatestates are selected to be the piston position, the virtual load pressure P_L, and the level pressureP_δ. The level pressure can be considered a weighted sum of the chamber pressures:

PL=PA−αPB, Pδ =PA+δPB, δ >0 (30) UsingP_L,P_δ thenP_A,P_B may be written as:

PA= αPδ

α+δ+ δPL

α+δ, PB = Pδ

α+δ− PL

α+δ (31) The nonlinear dynamics of the load and level pressure are described in Eq. (32) and Eq. (33).

P˙_L= ˙P_A−αP˙_B (32)

= β

ρVA

(ρ(Q_AC−Q_vA) +α(Q_B+Q_vB)

−A_A(α²+ρ) ˙x_P)

P˙δ = ˙PA+δP˙B+ ˙δPB (33)

= β

ρVA

(ρ(Q_AC−Q_vA)−δ(Q_B+Q_vB)

−AA(ρ−αδ) ˙xP) + δ˙

δ+α(Pδ−PL)

In Eq. (33) δ is chosen as ρ/α in order to decouple volume flow from the level pressure dynamics. Doing so, Eq. (32) and Eq. (33) become:

P˙L= β ρVA

(ρ(QAC−QvA) +α(QB+QvB) (34)

−AA(α²+ρ) ˙xP

P˙δ= β

ρVA

(ρ(QAC−QvA)−δ(QB+QvB) (35)

−x˙_PA_Aρ β

δ+ 1

δ+α(P_δ−P_L)

Level Pressure Ref. Generator

Inverse Flow Compensator Valve B flow ref.

Motor speed reference

Pδ

δ-update

xP

pB

pA Motion

Reference

xP

xP

Pδ

PL

Gain Gain

+- ++

++ -+

xref

.

Drive Controller

xP,ref

Pressure controller Position

controller

Pδ,ref

Output trans.

(W2) Input trans.

(W1) PL

Valve A flow ref. Valve signal A Valve signal B

Transformed / virtual states Physical states

M A C B

xP

pB

pA

SvSDP Drive

QL

Qδ

Figure 4: Schematic of the complete drive control system. Schmidt et al.(2017).

The linear pressure dynamics may be obtained as:

˙

pδ₀= β V0A(α+δ0)

(α+δ0)

K∆ωωm−qvA−qvB

α

−(K_δpδ+K_δp)p_δ−(K_δpL−K_δp)p_L) (36)

−Kδxdx˙P−KδxpxP

˙

pL= β(α+δ0) V0Aδ0

δ0

α+δ0

KΛωωm−qvA+qvB

δ0

−AAx˙_P− δ₀K_Lpδ

(α+δ0)²p_δ− δ₀K_LpL (α+δ0)²p_L

(37) K∆ω=KAq−KBq

α , Kδpδ =αKAqp−KBqp

α KδpL=δ0KAqp+K_Bqp

α , KΛω=KAq+K_Bq δ0

K_Lpδ=αK_Aqp+K_Bqp δ0

, K_LpL=δ₀K_Aqp−K_Bqp δ0

Kδp= x˙P0AA(δ0+ 1)

β , Kδxd= ∂P˙δ

∂x˙_{P x}

0

Kδxp= ∂P˙δ

∂x_P x0

From the above, the output transformation may be es- tablished as:

˜y=W2y, y˜=



 xP

pL

p_δ



, W2=





1 0 0

0 1 −α 0 1 δ₀



 (38) Virtual inputs in terms oflevel flow qδandload flow qL are defined based on the level and load pressure dynamics in Eq. (36) and Eq. (37), as defined in Eq.

(39) and Eq. (40):

qδ = (α+δ0)

K∆ωωm−qvA−qvB

α

(39) qL = δ0

α+δ0

KΛωωm−qvA+qvB

δ0

(40)

˙

pδ0 = β

V0A(α+δ0)(qδ−(Kδpδ+Kδp)pδ−

(KδpL−Kδp)pL)−Kδxdx˙P−KδxpxP (41)

˙

pL =β(α+δ0) V_0Aδ₀

qL−AAx˙P− δ0KLpδ

(α+δ₀)²pδ (42)

− δ0KLpL

(α+δ₀)²pL

The choice of q_δ and q_L is such that the non- transformed inputs in terms of shaft velocity and pro- portional valve flows does not directly influence the pressure level and load pressure gradients, but are con- tained inqδ andqL. Control structures using these new inputs can thereby be designed independently of the value and sign of the shaft velocity and of the valve signal allocation.

The input transformation matrix W1 is used to transformqδ andqL to the original input signals.

u=W1u˜ (43) From Eq. (39) and Eq. (40) the inverse input trans- formation matrix can be obtained as Eq. (44):

˜

u=W⁻¹₁ u

u˜= [qL qδ q0]^T, u= [ωm qvA qvB]^T W⁻¹₁ =





δ₀K_Λω

α+δ0 −_α+δ^δ0

0

1 α+δ0

(α+δ₀)K_∆ω −(α+δ₀) −^α+δ_α⁰ v₃₁ v₃₂ v₃₃



 (44)

The entriesv31,v32,v33may be chosen arbitrarily. The flowq0is aflow constraintwhich is chosen based on the desired distribution of the valve signals. The simplest flow constraint isq0= 0, which is chosen here.

InSchmidt et al.(2017) main focus is on motion per- formance, i.e. de-emphasising energy efficiency. The input transformation matrix was constructed such that the valve flows did not influence the load flow in Eq.

(40) i.e. ideally the piston motion is only driven byω_m. This can be obtained by choosing parametersv₃₁= 0, v₃₂= 1,v₃₃=−1/δ₀, andq₀ according to:

q₀=q_vA−q_vB δ0

= 0 (45)

The resulting inputuis then given by:

u^xP=W^x₁^Pu˜ =







α+δ₀ δ₀K_Λωq_L

K_∆ωα

δ₀K_ΛωqL−_(α+δ^α

0)²qδ K_∆ωα

K_Λω q_L−_(α+δ^αδ0

0)²q_δ





 (46) The superscript x_P is added to emphasise that the in- put transformation is derived to improve motion per- formance, and is referred to as theoriginalinput trans- formation in the remainder.

In Schmidt et al. (2017) an RGA-analysis of the transformed system, ˜Gcin Eq. (29) using the transfor- mation matricesW^x₁^P andW2showed that an almost perfect decoupling in the frequency range below the actuator bandwidths was achieved. A decentralised control strategy of the transformed system is there- fore reasonable, where the transformed inputsqL and qδ are used to control the transformed inputs xP and pδ respectively. These decentralised controllers are de- signed based on a generic analytical linear controller design approach, presented in Schmidt et al. (2017), capable of calculating appropriate controller parame- ters regardless of SvSDP drive size. This is done by including physical parameters such as cylinder areas and pump displacements combined with desired rel- ative stability margins in the controller design algo- rithm. The level pressure ref. generator seen in Fig. 4, is used to generate aPδ reference. The reference is gen- erated based on which chamber pressure to keep at a reasonable valuepset= 20 bar and the piston position.

InSchmidt et al.(2017) experimental results prove that the SvSDP drive and presented control strategy are ca- pable of maintaining a minimum chamber pressure at

≈psetwhile achieving a motion performance at least on the same level as a conventional servo-valve controlled system.

3.3.2 Energy Efficient Valve Utilisation

For the presented input transformation matrix, oil is si- multaneously throttled through both 2/2 valves, which

obviously is not the optimal valve utilisation in terms of energy efficiency. For an energy efficient valve util- isation only oil from the low pressure side should be throttled, which may be achieved by changing the in- put transformation. It is notable that only the input transformation needs to be changed while the output transformation, controller parameters etc. remain un- changed, as these are used for controlling transformed variables. As such the input transformation is only used to allocate physical inputs from transformed in- puts. The input transformation, which only allows oil through the B-side valve can be obtained by defining v31= 0,v32= 1, v33= 0 in Eq. (44) andq0as:

q0=qvA= 0 (47) The resulting inputu^qvB is then given by:

u^qvB =W^q₁^vBu˜=







α+δ0

αK_∆ω+δ₀K_ΛωqL+_(α+δ ^α

0)(αK_∆ω+δ₀K_Λω)qδ

0

(α+δ0)K∆ωα

α K_∆ω+δ₀K_ΛωqL−_(α+δ ^δ0KΛωα

0)(α K_∆ω+δ₀K_Λω)qδ





 (48) When only allowing oil through the A-side valve the input transformation can be obtained by definingv₃₁= 0,v₃₂= 0,v₃₃= 1 andq₀ as:

q0=qvB= 0 (49) resulting in the following input transformation:

u^qvA=W^q₁^vAu˜ =







−_(K ^α+δ0

∆ω−KΛω)δ₀qL+_(α+δ ¹

0)(K_∆ω−KΛω)qδ

−_(K^{(α+δ0)K∆ω}

∆ω−KΛω)δ₀qL+_(α+δ ^KΛω

0)(K_∆ω−KΛω)qδ

0





 (50) Which input transformation Eq. (48) or Eq. (50) to be used depends on which chamber pressure to be con- trolled to the minimum chamber pressure, p_set. This switching condition is defined from the measured load pressurePL using Eq. (51) settingPA=PB =Pset

pL_sw =Pset−αPset= (1−α)Pset (51) As illustrated in Fig. 5(a), ideally P_A=P_set forp_L<

p_Lsw using the input transformation in Eq. (50) and PB=PsetforpL> pL_sw using the input transformation in Eq. (48). To avoid abrupt jumps in the utilised input transformation method a switching variableZ is defined as:

Z =











0 , pL <(pLsw−Zband)

P_L−p_Lsw+Z_band

2Z_band , (pL_sw +Zband)> pL >

(pL_sw−Zband) 1 , pL >(pL_sw+Zband)

(52)

Z¯ = 1−Z; (53)

In Fig. 5(b) the switching variables are seen as a func- tion of the load pressure. The non-transformed inputs u are then defined as a weighted sum between u^qvA andu^qvB according to:

u=Zu^qvB+ ¯Zu^qvA (54) Eq. (54) is referred to as the energy efficient input transformation in the remainder.

-50 0 50

(a) Load Pressure [Bar]

0 50 100 150

Chamber Pressures [Bar]

-50 0 50

(b) Load Pressure [Bar]

0 0.5 1

Switching Condition [-]

Figure 5: (a) Ideally controlled chamber pressures as a function of load pressure for pset = 20 bar and α = 0.5. (b) Input transformation switching variables as a function of load pres- sure forZband= 5 bar

3.4 Energy Efficient Pump Sizing

Assuming that the considered knuckle boom crane should be retrofitted with SvSDP-actuated cylinder drives, it is likely that the hydraulic cylinders remain unchanged thus only replacing the conventional HPU including valves with SvSDP drives. This necessitates that the SvSDP sizing should aim at delivering ap- proximately the same flow amount in the same pres- sure range as achievable with the conventional Valve Cylinder Drives (VCDs). InSchmidt et al.(2017), ex- ternal gear pumps have been used as flow suppliers in the SvSDP drive. As these generally operate in a lim- ited pressure range, internal gear pumps are suggested for larger power applications as considered here. The pump sizes selected for the system are of crucial im- portance in terms of energy efficiency, as these heavily affect both electrical and hydraulic losses. Ten different pump sizes ranging from 16 cm³/rev to 125.2 cm³/rev (Rexroth,2010) and (Rexroth,2013) have been consid- ered for each pump, yielding a total of 1000 different pump size combinations. In this section a selection algorithm aiming on selecting an energy efficient and

feasible pump combination is proposed. For doing so, the dominant system losses must be described.

3.4.1 Dominating Losses

In Fig. 6the main losses during operation are shown.

E

E

E

E

Electrical losses

Eβ

Mechanical

losses Volumetric

losses Throttling losses Figure 6: Main losses during operating of the SvSDP-

actuated cylinder. E˙β is power due to the compressibility of the oil.

The electrical losses in the frequency converter and iron losses in the electrical motor are assumed ne- glectable and thus the only electrical loss included are the Ohmic losses in the electrical motor described by (Willkomm et al.,2014):

E˙l,Ω= 3RcuI²= 3Rcu

τshaft

τnom

Inom

2

(55) R_cu is the winding resistance, and I_nom is the nomi- nal current at the nominal motor torque, τ_nom. These parameters are available from datasheets. τ_shaft is cal- culated by:

τshaft=

(KAQ+KCQ)PA−KBQPB , ωm≥0 K_AQP_A−K_BQP_B , ω_m<0

(56) where KAQ, KBQ and KCQ are the theoretical pump displacements [m³/rad].

The mechanical losses due to pump friction have been neglected as no information for the considered in- ternal gear pumps are available. The volumetric losses in the pump are described by:

E˙l,v = (Kl,v1∆P+Kl,v2∆P²)∆P (57) where ∆P is the pressure difference across the pump, and Kl,v1 and Kl,v2 are leakage parameters. From pump datasheet (Rexroth, 2013), a flow curve for a 16 cm³/rev pump at 1450 RPM is available. This has been used to fit the leakage related parameters in Eq.

(57), assuming leakage-free flow at ∆P=0. The co- herence between the flow curve and the model is seen

in Fig. 7(a). In Fig. 7(b) corresponding volumetric efficiencies for different speed levels are shown. Identi- cal volumetric efficiencies have been used for all pump sizes considered. This may be a conservative estima- tion as volumetric efficiencies are assessed to improve for larger pump sizes.

0 50 100 150 200 250 300

(a) P [Bar]

22 23 24

Q [L/min] Datasheet_@1450RPM

Model_@1450RPM

0 50 100 150 200 250 300

(b) P [Bar]

0.6 0.8 1

vol [-] ^{200 RPM}

500 RPM 1000 RPM 2500 RPM

Figure 7: (a) Pump flow model compared to datasheet.

(b) Pump volumetric efficiency.

Throttling losses in pipes and hoses as well as over the ideally modelled check valves are neglected. There- fore the throttling losses only involve the oil through the 2/2 proportional valves, described by:

E˙l,th=QvA·PA+QvB·PB (58) 3.4.2 Pump Selection Algorithm

The Valve Cylinder Drives (VCD) used as a benchmark (see Section4) for the proposed SvSDP, produces max- imum flows of 160 L/min to each cylinder. Due to load sensing it is assumed that this can be done indepen- dently of the cylinder load pressure. This translates to piston velocities ranging from -88 mm/s to 43 mm/s for cylinder 1 and from -113mm/s to 54 mm/s for cylinder 2.

A fundamental difference between a (symmetrical)- valve controlled asymmetric cylinder system and the proposed SvSDP system, is that for the SvSDP the pump flows are matched to the cylinder area ratio caus- ing the achievable velocities to be somewhat symmet- rically distributed around 0 mm/s, if utilising the 2/2 valve on the A-side only.

However for energy-efficient valve utilisation it is de- sired to throttle from the low pressure chamber, which may be any chamber, affecting the achievable piston velocities. E.g. if always throttling from the B-side, the steady state forward velocity is determined by the

combined flow of pump C and A, whereas the retract- ing velocity is determined by the flow of pump A only as the C-pump is idling. This causes the maximum velocities to be asymmetrical around 0 mm/s. For the given example the forward velocity is larger than the retracting velocity, exactly opposite of a VCD.

These considerations show that a SvSDP drive sized to achieve VCD comparable retracting velocities, may be heavily oversized in the forward direction, due to idling of the C pump. The flow requirements (FR) are therefore relaxed in the retracting direction and formulated as:

FR^ωm≥0=







1 , (Q_A+Q_C)≥160_min^L and QB/α≥160_min^L 0 , otherwise

(59)

FR^ωm<0=







1 , −QA≥0.9·160_min^L and −QB/α≥160_min^L 0 , otherwise

(60) whereQ_A,Q_B,Q_Care pump flows evaluated at maxi- mum allowed pressure and maximum positive/negative pump speed for FR^ωm≥0and FR^ωm<0respectively. For a combination of pumps to be feasible FR^ωm≥0 = 1 and FR^ωm<0 = 1 is required along with a match- ratio (χ) larger than 1 evaluated at ±500 RPM and 225 bar. Forχ >1 a surplus of flow into the cylinder is present. χis defined as:

χ= ( _Q

A+Q_C

Q_B α , ωm≥0

Q_B

Q_Aα , ωm<0 (61) Evaluating Eq. (59), (60) and Eq. (61) gives a num- ber of feasible designs. Assuming oil always to be throttled from the low pressure side and by neglect- ing pump leakage the static mismatch flow for each feasible combination can be calculated and throttling losses evaluated using Eq. (58), as a function of shaft speed andp_set. Assuming a constant minimum cham- ber pressure, the shaft torque may be calculated and the Ohmic losses may be evaluated using Eq. (55) as a function of load pressure. By sweeping over the en- tire pump velocity and load pressure range, the pump combination with the smallest average loss is chosen.

The pump selection algorithm is summarised in Fig. 8.

Construct Design Vector

1000 designs

Evaluate χ

@ 500 RPM / 225 Bar

Evaluate flow requirements

@ 3000 RPM / 315 Bar Reject design

>1

< 1

( 177 / 170 ) Evaluate mean

losses

Satisfied ( 45 / 41 )

Unsatisfied

Select design with smallest mean loss

Figure 8: Pump selection algorithm. In parenthesis are shown the number of feasible designs for SvSDP₁and SvSDP₂respectively.

As the two cylinders have almost equivalent area ra- tios and the same flow requirement is imposed the se- lected pump combination is the same for SvSDP1 and SvSDP2. Pump A is selected as having a displacement of 50.7 cm³/rev and Pump B and C are both having a displacement of 32.7 cm³/rev. The resulting match ratioχfor SvSDP1 is seen in Fig9(a).

-3000 -2000 -1000 0 1000 2000 3000 (a)

m1

[RPM]

0.8 1 1.2

1 [-] _{10 Bar}

100 Bar 300 Bar

-100 -50 0 50 100

(b) dx

P1

[mm/s]

0 200 pL1 [bar]

VCD Operating Range

-100 -50 0 50 100

(c) dx

P2

[mm/s]

-200 0 200

pL2 [bar]

SvSDP w/ Low Pressure Throttling SvSDP w/ High Pressure Throttling

Figure 9: (a) Match ratio for the selected pump com- bination. (b) Operating range for SvSDP1

compared to VCD. (c) Operating range for SvSDP2 compared to VCD.

The obtained operating ranges, assuming that a suit- able electrical motor which does not saturate at corner power requirements, is seen in Fig. 9(b) and (c). The operating ranges are evaluated by assuming that the control structure can maintain a minimum chamber pressure of 20 bar during motion, and using a max- imum operating pressure of 315 bar for the internal gear pumps.

Noting that due to the crane structure chamber A will always be load carrying for SvSDP1 while the SvSDP2 should carry the load in both directions, this translates to load forces ranging from 10 to 305 bar (62 kN to 1.88 MN) for SvSDP₁and -132 to 305 bar (-0.65 MN to 1.5 MN) for SvSDP₂ respectively.

Note, that in Fig. 9 operating ranges obtainable if throttling from the high pressure chamber are also de- picted. Doing so, alters the achievable operating range, at the cost of larger throttling losses.

4 Benchmark System

As mentioned, the benchmarks for the two SvSDP drives are Valve Cylinder Drives (VCDs), convention- ally used for actuation of a knuckle boom crane. The hydraulic system structure is shown in Fig. 10, and has also been used in Donkov et al. (2018). The motion of the cylinders is controlled by the directional pro- portional valves. The two directional valves are pres- sure compensated and produce maximum flows of 160 L/min. Furthermore, the supply pump has load sens- ing capabilities. The larger pressure in an inlet cham- ber selects the outlet pressure setting for the pump (Ps = Pmax + 35 bar). The counterbalance valves (CBV) are used to prevent the load from overrunning.

For details on the modeling of the benchmark sys- tem, component sizes etc., seeDonkov et al.(2018).

M

Meq xP

PA1 1

Meq Cylinder 1

Load Sensing Pump CBV

P_B1 AA1 AB1

xP

PA₂

2 Cylinder 2

P_B2 AA₂ AB₂

Figure 10: Knuckle Boom Crane hydraulic circuit from Donkov et al.(2018).

5 Simulation Results

To compare the performance of the SvSDP concept with the conventional system a simulation study has been conducted. A test case trajectory has been se- lected where a 5000 kg payload begins and ends in the same place. In tool center space this can be seen in Fig. 11taken fromDonkov et al. (2018). The trajec- tory consists of starting the load at point 1, moving it to point 2 and returning it to point 1.

In actuator space the trajectory is shown in Fig. 12.

Key parameters, such as component sizes, crane di- mensions and masses used for the simulation study are found in theKey Parameter List on page88.