Proceedings of Dynamic Systems and Control ConferenceDSCC 2016

October 12-14, 2016, Minneapolis, MN, USA

DSCC2016-9781

MULTI DEGREE-OF-FREEDOM HYDRAULIC HUMAN POWER AMPLIFIER WITHRENDERING OF ASSISTIVE DYNAMICS

Sangyoon LeeERC for Compact and Efficient Fluid Power

Department of Mechanical EngineeringUniversity of Minnesota

Minneapolis, Minnesota 55455Email: sylee@umn.edu

Fredrik Eskilsson ∗SAAB AerospaceFormansgatan 11

Linkoping, Sweden 58737Email: f.eskilsson@gmail.com

Perry Y. LiERC for Compact and Efficient Fluid Power

Department of Mechanical EngineeringUniversity of Minnesota

Minneapolis, Minnesota 55455Email: lixxx099@umn.edu

ABSTRACTThe hydraulic human power amplifier (HPA) is a tool sim-

ilar to exoskeleton that uses hydraulic actuation to amplify theapplied human force. The control objective is to make the systembehave like a passive mechanical tool that interacts with the hu-man and the environment passively with a specified power scal-ing factor. In our previous work, a virtual velocity coordinationapproach recasts the single degree-of-freedom human power am-plifier control problem into a velocity coordination with a ficti-tious reference mechanical system. Force amplification becomesa natural consequence of the velocity coordination. In this pa-per, this control approach is extended for fully coupled multi-DoF systems. A passivity based control approach that uses thenatural energy storage of the hydraulic actuator to take full ac-count of the nonlinear pressure dynamics is used to define theflow requirement. Additional passive assistance dynamics aredesigned and implemented to enable the user to perform specifictasks more easily. Guidance is achieved using a passive veloc-ity field controller (PVFC), and obstacle avoidance is achievedusing a potential field. Experimental results demonstrate goodperformance on a 2-DoF Human Power Amplifier.

1 IntroductionThe goal of the human power amplifier (HPA) is to enable

a human operator to directly interact with the machine as if the

∗Fredrik Eskilsson was an international exchange student at the University ofMinnesota.

machine is an extension of his body while amplifying the appliedhuman effort. The control objective is identical to that of a wear-able exoskeleton. The only difference is that the human can letgo of the HPA, but is an exoskeleton is always attached. In bothcases, since the operator participates physically and directly incontrolling the machine, it is more intuitive than using a remotejoystick. Because of the direct, physical interaction, energeticpassivity, which limits the amount of energy that can be trans-ferred to the human and the environment, is a useful property forHPAs to ensure coupling stability and safety.

In [1] it was shown that a direct approach to control actua-tor force as an amplified human force leads to a positive velocityfeedback which is not robust in the presence of uncertainty, slowsampling or feedback noise. In [2] and [3], an alternate controllerwas proposed which models the actuator as a combination of anideal velocity source and a nonlinear spring, the latter capturesthe compressibility effects of the fluid medium. Instead of con-trolling the actuator to track the desired force directly, the con-troller coordinates the velocities of the system and of a fictitiousvirtual mass whose dynamics are influenced by the hydraulic ac-tuator and the human force. The control law for achieving coor-dination is accomplished via a passive decomposition [4–6] intoa shape system and a locked system. This control is more robustas the passivity property is enforced by the control structure it-self. A variety of control laws can be designed to stabilize theshape system to coordinate the velocities of the actual and vir-tual systems. In particular, a control law was derived in [7] usingthe natural compressibility energy of the hydraulic actuator [8]

1 Copyright c© 2016 by ASME

FIGURE 1. Picture of the Human Power Amplifier

instead of approximating the hydraulic actuator by a nonlinearmechanical spring.

In these previous works, the control was developed foreach individual degree-of-freedom assumed to be decoupled.In this paper, we expand the results from [7] to a fully cou-pled multi-DoF HPA. In addition, we also develop additionalhuman-machine shared control strategies that render useful pas-sive dynamics to assist the human to execute the task more easily.Guidance is achieved using the Passive Velocity Field Controller(PVFC) [9] to guide the HPA to move along the direction of aspecified velocity field. Obstacle avoidance is achieved by incor-porating potential fields [10] to prohibit the machine from enter-ing prohibited zones. With these task oriented passive dynam-ics, the operator can execute tasks accurately with less attentionwhile remaining in direct control since the he/she must supply aportion of the physical power.

The rest of paper is organized as follow. System dynam-ics and control objectives are stated in Section 2. In Section3, the reformulation of the problem as a velocity coordinationis reviewed, followed by the presentation for the proposed flowrequirement. Guidance dynamics in the form of PVFC and ob-stacle avoidance are presented in sections 4 and 5. Experimentalresults and concluding remarks are given in sections 6 and 7.

2 System Description and ModelWe consider a 2-DoF human power amplifier (HPA) shown

in Fig. 1-2 with generalized coordinate q = [θp,xp]T where θp

describes the angular position of the pitch movement and xp de-scribes the linear position of the reach movement. The pitch an-gular motion is actuated by a linear hydraulic actuator whereasthe reach linear motion is actuated by a hydraulic motor via apulley and belt mechanism. The dynamics of the HPA are given

AB

Force Handle

rmxp

xθ

θp

PT, A2

Pθ, A1

Qθ

Pm1 Pm2

Dm

Hydraulic Motor,Reach

Hydraulic Actuator,Pitch

FIGURE 2. Schematic of the Human Power Amplifier

by:

Mp(q)q+Cp(q,q)q+Gp(q) = Fhuman +Fenv +Fa (1)

where Mp(q) ∈ ℜ2X2 is the symmetric and positive definite in-ertia matrix, Cp(q,q) ∈ ℜ2X2 is the Coriolis matrix such thatMp(q)− 2Cp(q,q) is skew-symmetric, and Gp(q) ∈ ℜ2X1 is thegravity vector. Fhuman is the generalized torque/force applied bythe human on a handle instrumented with force sensors; Fenv isthe force exerted by the environment which is not measured. Fais the generalized actuator force/torque:

Fa =

[Tθ

Fx

]=

[JA(θp) 0

0 1rm

][Fθ

Tx

](2)

where Tθ and Fx are the torque and force applied to the pitchand reach directions. As seen in Figure. 2, the pitch torque Tθ isgenerated by a hydraulic cylinder with force Fθ given by

Fθ = Pθ A1−PT A2 (3)

where A1 and A2 are the cap side and piston side areas, Pθ andPT are the supply and tank pressures on the cap and rod sides ofthe actuator. The Jacobian JA(θp) is used to translate linear forcegenerated into a generalized torque.

The reach force Fx is generated by a fixed displacement hy-draulic motor with torque Tx given by:

Tx = PxDm

2π(4)

where Dm is the motor displacement, Px = Pm1−Pm2 is the pres-sure across the motor. The motor is connected to a belt via apulley with a radius rm.

The cap side of the hydraulic cylinder for the pitch motionis connected to the output of a hydraulic transformer [7] withpressure dynamics given by:

Pθ =β (Pθ )

V1(xθ )(Qθ −A1xθ ) (5)

2 Copyright c© 2016 by ASME

where Qθ is the flow input to the cap side chamber,

V1(xθ ) =V10 +A1xθ (6)

is the fluid volume in the cap-side chamber and the hose de-pendent on the linear displacement of the cylinder xθ , β (Pθ ) isthe pressure dependent fluid bulk modulus. The rod side is con-nected to the lower common pressure rail so that PT is assumedto be constant. It is assumed that the gravity load (in Fenv) is suf-ficiently large such that over-running load and cavitation will notoccur.

The pressure dynamics for the two sides of the hydraulicmotor are:

Pm1 =β (Pm1)

Vm

(Qm1−

Dm

2π

xp

rm

)(7)

Pm2 =β (Pm2)

Vm

(−Qm2 +

Dm

2π

xp

rm

)(8)

where Vm are the fixed fluid volumes in the motor and the hosewhich, for simplicity, are the same on both sides of the motor,β (·) is the pressure dependent bulk modulus. Flow Qm1 = Qm2are the equal input and return flows in the servo valve. Let Px =Pm1−Pm2 and taking the difference between these dynamics, themotor pressure dynamics are:

Px = Pm1− Pm2 =βe(Px,Pm1)

Vm

(Qx−

Dm

2π

xp

rm

)(9)

where βe(Px,Pm1)= β (Pm1)+β (Pm2) is the effective pressure de-pendent bulk modulus rising from the pressure difference acrossthe motor, and Qx := Qm1 = Qm2 is the flow input to the motor.

In the experiment, Qθ and Qx are respectively controlledby a custom built hydraulic transformer [11] and a servo valve.Readers are referred to [7, 11, 12] and [13] for details of howthese flows are achieved with these devices.

Control ObjectiveThe control objective is to control flow inputs to the hy-

draulic actuators Qθ and Qx in (5) and (9) such that the appliedhuman force is amplified by a factor of (ρ +1), which results inthe target dynamics of a passive mechanical tool

ML(q)q+CL(q,q)q= (ρ+1)Fhuman+Fenv−G(q)+Fguide (10)

where ML(q) and CL(q,q) are the apparent inertia and associatedCoriolis matrix of the tool to be designed. The human would feelthat he/she is interacting with an inertia and an environment forcethat are attenuated as ML/(ρ+1), Fenv/(ρ+1) and G(q)/(ρ+1)respectively. For ML(q) ≈ Mp(q), this can be achieved if thegeneralized actuator force satisfies:

Fa ≈ ρFhuman

qI qv

MPMV

Fa

Fd=ρFhuman

Fhuman+Fenv

FIGURE 3. Model of hydraulic actuator with the ideal velocity com-mand provided by the velocity of a virtual inertia, and the virtual inertiaaffected by the actuator force. In a human power amplifier Fd = ρFhuman

Fguide is the additional task specific guidance dynamics to pro-vide assistance to the user to operate the HPA.

3 Virtual Coordination Control Approach to ForceTrackingInstead of directly controlling the actuator force Fa to track

the desired force ρFhuman, the virtual coordination approach in[2] converts the problem into one of coordinating velocities oftwo coupled mechanical systems - the plant and a virtual inertia.Besides avoiding the need for positive velocity feedback, this ap-proach can also be interpreted physically as an interconnection ofpassive components so that it is more robust and safer to operate.

With the actuator compressibility represented by a spring-like object that interacts with the inertia of the machine Mp, theapproach is to control Fa such that the other end of the spring-like object is interacting with a small virtual inertia Mv ∈ ℜ2×2

1 acted on by a set of desired forces (Fig. 3). Let the dynamicsof a virtual inertia Mv (implemented as part of the controller) begiven by:

Mvqv = Fd−Fa +w+Fguide (11)

where qv = [θv,xv]T is the generalized coordinate for the virtual

inertia, Fd = ρFhuman is the desired force, Fa is the generalizedactuator force (Eq. (2)) and w and Fguide are the additional con-trols to be designed for the locked system or for task guidance.

If exact coordination between the virtual inertia Mv andMp(q), such that qv(t) ≡ q(t) (i.e. they become a single rigidinertia), then comparing (1) and (11), and the fact that w will bedefined such that w→ 0 when coordinated, the resulting dynam-ics becomes:

(Mv +Mp(q))q+Cp(q,q)q =

(ρ +1)Fhuman +Fenv−Gp(q)+Fguide

which is the desired target dynamics in (10) with the apparentinertia being ML(q) = Mv +Mp(q), and CL(q, q) =Cp(q, q).

1For simplicity, Mv is a constant inertia represented by a positive definitematrix.

3 Copyright c© 2016 by ASME

As will be seen, the guidance force Fguide will be designedto satisfy a passivity property:∫ t

0[qT Fguide]dτ ≤ c2

g (12)

Then, after coordination qv(t) ≡ q(t), the closed loop system isenergetically passive with respect to the scaled power input bythe human and environment such that there exists c2 > 0 so thatfor all Fhuman(·), Fenv(·),∫ t

0qT [(ρ +1)Fhuman +Fenv]dτ ≥−c2 (13)

We have used the fact that gravity is conservative such that

Gp(q) =∂VG(q)

∂q(14)

where VG(q) ∈ℜ is the gravitational potential field and q lies ina compact work space.

In the following, we extend the virtual coordination con-troller in [7] that uses the natural energy storage function for thehydraulic actuators to fully coupled multi-DoF systems.

3.1 Passive Decomposition into Locked and ShapeSystems

The coupled system of Eq. (1) and (11) is given by

Mp(q)q+Cp(q,q)q+Gp(q) = Fhuman +Fenv +Fa

Mvqv = Fd−Fa +w+Fguide (15)

where the generalized coordinates for the physical system areq = [θp,xp]

T and for the virtual system are qv = [θv,xv]T . As we

are interested in coordination between q and qv, i.e,

VE := q− qv→ 0

it is desirable to study the problem in relative coordinates. How-ever, we also do not wish to disturb the energetics of the desiredtarget dynamics in Eq. (10). Therefore, we apply the passivedecomposition [4–6] to transform the velocities into locked andshape coordinates: [

VLVE

]=

[I−φ φ

I −I

]︸ ︷︷ ︸

S(q)

[qqv

](16)

where

I−φ = [Mp(q)+Mv]−1Mp(q)

φ = [Mp(q)+Mv]−1Mv

and ML(q) = Mp(q) + Mv is the inertia corresponding to thelocked system. VL (locked system velocity) is the velocity ofthe center of mass of the combined virtual and actual system,whereas VE (shape system velocity) is the velocity coordinationerror.

The dynamics in the transformed coordinates are given by:[ML(q) 0

0 ME(q)

][VLVE

]+

[CL(q,q) CLE(q,q)

CEL(q,q) CE(q,q)

][VLVE

]= ψ

′

(17)

where the inertia matrix and Coriolis matrix are transformed ac-cording to the definition of the passive decomposition [4],[

ML(q) 00 ME(q)

]= S−T

[Mp(q) 0

0 Mv

]S−1

=

[Mp(q)+Mv 0

0 (I−φ)T Mv(I−φ)

](18)

[CL(q,q) CLE(q,q)

CEL(q,q) CE(q,q)

]= S−T

[Mp(q) 0

0 Mv

]ddt(S−1)

+ddt

S−T[

Mp(q) 00 Mv

]S−1 +S−T

[C(q,q) 0

0 0

]S−1 (19)

Forces acting on the virtual and actual inertia are:

ψ′ = S−T

[Fhuman +Fenv +Fa−Gp(q)

Fd−Fa +w+Fguide

](20)

The transformed system is represented as

ML(q)VL +CL(q,q)VL +CLE(q,q)VE =

Fd +Fenv +Fhuman−G(q)+w+Fguide (21)ME(q)VE +CE(q,q)VE +CEL(q,q)VL = Fa +φ(Fenv)︸ ︷︷ ︸

FE1

+φ(Fhuman−G(q))− (I−φ)(Fd +w+Fguide)︸ ︷︷ ︸FE2

(22)

3.2 Shape System ControlFrom (22), the shape system dynamics are:

ME(q)VE +CE(q,q)VE +CEL(q,q)VL = Fa +FE1 +FE2 (23)

where FE2 contains measurable or known terms and FE1 is poten-tially unknown. We define the desired actuator force to achieveshape system control to be:

Fa,d =

[Fa,d1Fa,d2

]=CEL(q,q)VL−λVE − FE1−FE2 (24)

4 Copyright c© 2016 by ASME

where FE1 is an estimate of FE1 and λ > 0.The nonlinear decoupling term CEL(q,q)VL is needed to de-

couple the locked system dynamics from the shape system dy-namics, which is not necessary for single DoF control.

The following input flows for each degree of freedom in (5)and (9) are proposed:

Qdθ = A1JA(θp)θv +

V1(xθ )

β (Pd,θ )Pd,θ −λpθ Pθ (25)

Qdx =

Dm

2π

xv

rm+

Vx

βe(Pd,x)Pd,x−λpxPx (26)

where Pd,θ and Pd,x are the desired actuator pressures in the pitchand reach directions given by:

Pd,θ =1

A1

[PT A2 +

1JA

Fa,d1

](27)

Pd,x = rm2π

DmFa,d2 (28)

The estimate for the external force FE1 is obtained from the adap-tation algorithm,

˙FE1 = σVE + FE1 (29)

where FE1 is the best estimate of the time derivative of FE1; λ ,λpθ , λpx, and σ are all positive constants.

To see that this control is appropriate, consider the followingLyapunov function,

W =12

V TE MEVE +

1σ

FTE1FE1 +V1(xθ )WV (Pθ ,Pd,θ )

+Vm ·WV (Px,Pd,x) (30)

where FE1 is the error in estimating the unknown external force;WV (Pθ ,Pd,θ ) is the volumetric pressure error energy density asso-ciated with compressing the fluid from pressure Pd,θ to Pd,θ + Pθ

as defined in [8]. Likewise, WV (Px,Pd,x) is defined in the samemanner. Differentiating the above Lyapunov function (See [8]and [7] for details and proof), and with λp,θ and λp,x sufficientlylarge we get,

W ≤−V TE λVE −mθ (λp,θ )P2

θ −mx(λp,x)P2x

such that mθ (λp,θ )> 0 and mx(λp,x)> 0. This in turn shows thatVE , Pθ , Px→ 0.

3.3 Locked System ControlFrom (22), the Locked system dynamics are:

ML(q)VL +CL(q,q)VL +CLE(q,q)VE =

Fd +Fenv +Fhuman−G(q)+w+Fguide (31)

We can design w to cancel out the coupling dynamics

w =CLE(q,q)VE (32)

Note that w is also used in shape system control because of win FE2 (see (22)). If desired, G(q) could be included in w tocancel out the effect of gravity. With Fd = ρFhuman, and aftercoordination (i.e. VL = q = qv):

ML(q)q+CL(q,q)q= (ρ+1)Fhuman+Fenv−G(q)+Fguide (33)

which is the target dynamics we wanted in Eq. (10)

4 Passive Velocity Field ControllerIn this and next section, we design Fguide n Eq. (11) to pro-

vide useful dynamics to assist the human operator in his/her task.This section discusses the use of Passive Velocity Field Control(PVFC) to impart guidance; [9] [14]. Obstacle avoidance strat-egy will be discussed in section 5. An earlier attempt to imple-ment useful dynamics for HPA can be found in [15].

In PVFC, passive dynamics are incorporated into the ma-chine to guide the operator to follow a scaled copy of a desiredvelocity field - i.e. a desired velocity at each position, while al-lowing the machine to remain passive. This can guide the humanoperator to follow a desired path which is the flow of the veloc-ity field. An example velocity field is shown in Fig. 4 whichguides the HPA to converge to and follow a circle. The speedat which the field is followed is determined by the kinematic en-ergy available in this system. In this way, it is possible to providepath guidance without violating passivity. In particular, the en-ergy input into the system must be provided either by the humanoperator or the environment.

As the operator is physically connected to the machine inHPA operation, PVFC works as a feedback to the operator in-forming whether he is on the right track. For safety and comfort,it is still important that the machine remains passive. An out-line of how PVFC incorporates into HPA control is given below.Readers are referred to [9] [14] for detailed proofs. In this pa-per, we incorporate this guidance dynamics to the locked systemdynamics in (31) and use the virtual coordination scheme ratherthan directly to the physical system.

4.1 Desired Velocity Field and Augmented SystemLet the desired velocity field be V (q) ∈ ℜ2 which defines

at each configuration q a desired velocity V (q). In this paper,the example velocity field depicted in Fig. 4 is used to assist theuser to perform a circular motion. In the absence of any humanor environmental input, the PVFC controller will cause q(t)→β (t)V (q(t)) where β 2(t) is proportional to the kinetic energy inthe system.

5 Copyright c© 2016 by ASME

0.4 0.5 0.6 0.7 0.8 0.9 1

−0.55

−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

X movement [m]

Y m

ovem

ent [

m]

Velocity FieldDesired Path

FIGURE 4. Velocity field for tracing a circle

To define PVFC, we first augment the system dynamics witha 1 DoF fictitious flywheel dynamics:

MF qF = τF (34)

where MF is the apparent inertia of this virtual flywheel, qF isthe position of the flywheel, and τF is the coupling control inputto the flywheel.

Combined with the locked system in (31), the augmentedsystem becomes:

M(q) ¨q+C(q, ˙q) ˙q = τ + τe (35)

where ˙q =[V T

L qF]T are the augmented velocity, τ =[

FTguide τT

F]T

being the augmented control input, τe =[τT

e 0]T

being the augmented external force where

τe = Fenv +(ρ +1)Fhuman−G(q) (36)

M(q) =[

ML(q) 00 MF

], C(q, ˙q) =

[CL(q, q) 0

0 0

](37)

are the augmented inertia matrix and the augmented Coriolis ma-trix.

The kinetic energy of the augmented system is

k(q, ˙q) =12

˙qT M(q) ˙q =12

V TL ML(q)VL︸ ︷︷ ︸

Locked System

+12

MF q2F︸ ︷︷ ︸

flywheel

(38)

In order to control and utilize the virtual flywheel, the de-sired velocity field V (q) needs to be augmented as:

V (q) =[V (q)T VF(q)

]T (39)

such that the kinetic energy of the augmented system is constantwhen the augmented field is tracked. This can be accomplishedby ensuring for all q ∈ℜ2,

E =12

V T (q)M(q)V (q)

where E is a sufficiently large constant. In other words, the de-sired flywheel velocity field is given by:

VF(q) =

√2

MF

(E− 1

2V (q)T ML(q)V (q)

)(40)

4.2 PVF ControllerWith the augmented system and augmented velocity field,

the coupling control τ can be designed as

τ = Ω(q, ˙q) ˙q (41)

where Ω(q, q) ∈ R(n+1)×(n+1) is skew symmetric. This ensuresthat ˙qT τ = 0 so that the PVFC control is passive. The couplingforce re-distributes energy between the locked system and thefictitious flywheel conservatively. To find suitable Ω(q, q), thefollowings are defined:

P(q) = M(q)V (q) (42)p(q, ˙q) = M(q) ˙q (43)

w(q, ˙q) = M(q) ˙V (q)+C(q, ˙q)V (q) (44)

where P(q) is the desired momentum field, p(q, ˙q) is the actualmomentum, and w(q, ˙q) is the covariant derivative of the desiredmomentum field.

With these, the coupling control law is given by

τ(q, ˙q) = τc(q, ˙q)+ τ f (q, ˙q) (45)

where

τc =1

2E(wPT − PwT ) ˙q (46)

τ f = γ(PpT − pPT ) ˙q (47)

or

Ω(q, ˙q) =1

2E(wPT − PwT )+ γ(PpT − pPT ) (48)

τc corresponds to a feedfoward control, giving information aboutthe desired system dynamics, and τ f is a feedback control whichvanishes when ˙q = α(t)V (q(t)) for some scalar α(t), γ is a con-trol gain that determines the convergence rate and the sense inwhich the desired velocity will be followed.

6 Copyright c© 2016 by ASME

The PVFC component of Fguide is then the first two elementsof the coupling control in (45) such that[

FPV FCτF

]= τ(q, ˙q) (49)

4.3 Properties of PVFCWith the control in (41)-(48), the closed loop dynamics for

the coupled augmented system can be written as

M(q) ¨q+ Y (q, ˙q) ˙q = τe (50)

where τe is the augmented external force and Y ∈ℜ3X3 is:

Y (q, ˙q) = C(q, ˙q)− 12E

(wPT − PwT )︸ ︷︷ ︸skew symmetric

−γ (PpT − pPT )︸ ︷︷ ︸skew symmetric

(51)

Notice that since both control terms in (46) and (47) are skewsymmetric, ˙M− 2Y is also skew symmetric, just as M− 2C isskew symmetric for the robot manipulator. Utilizing this skewsymmetric structure and (50), to differentiate kinetic energy in(38), we obtain

ddt

k(q, ˙q) =V TL (t)τe(t) (52)

where V TL τe = ˙qT τe because τe =

[τT

e 0]T . Integrating (52) w.r.t

time gives

∫ t

0V T

L (t)τe(t)dτ ≥−c2 (53)

This means that the closed loop dynamics of the augmented sys-tem given by (50) is passive with respect to the supply rate V T

L τe(the power produced by the external forces), and its kinetic en-ergy in (38) is its storage function. With the PVFC controller, itcan be shown that, in the absence of τe,

˙q→ β (t)V (q(t)) (54)

where

β (t) = sign(γ)

√k(q, ˙q)

E

Thus, as the kinetic energy of the system increases (such as withinput by the human operator or the environment, the speed atwhich the desired velocity is tracked will also increase.

X

Y

Potential Field

0.7 0.75 0.8 0.85−0.3

−0.28

−0.26

−0.24

−0.22

−0.2

−0.18

−0.16

−0.14

−0.12

1

2

3

4

5

6

7

8

9

10

FIGURE 5. An example potential field for a point obstacle in Carte-sian (workspace) coordinates.

5 Obstacle AvoidanceThe aim of Obstacle Avoidance Control is to prevent the ma-

chine from entering prohibited area in the workspace to protectitself or other objects. Here we utilize an artificial potential fieldapproach [10] [16] to provide the operator a tactile feedback torepel the machine from the obstacle.

The potential field is designed to be non-negative continuousand differentiable function that tends to infinity as the machineapproaches the obstacle. It is also designed such that the influ-ence of the potential field is limited to certain to avoid havingundesirable perturbing forces beyond the obstacle’s vicinity.

For a point obstacle, an example potential field Uoa(q) ex-pressed in the Cartesian (workspace) coordinates (X(q),Y (q)) ofthe tip of the HPA is shown in (Fig. 5). This field is exponen-tially decaying (with distance) and is radially symmetric in theworkspace coordinates.

The force arising from this potential function is the negativegradient of the above function such that:

FOA =−∂Uoa(q)∂q

(55)

The combined guidance control is:

Fguide = FPV FC +FOA

6 Results and Discussion6.1 Energetic Passivity Property

With the total energy function:

Wtotal =12

V TE MEVE +

1σ

FTE1FE1 +V1(xθ )WV (Pθ ,Pd,θ )

+VmWV (Px,Pd,x)+12

V TL MLVL +

12

MF q2F +Uoa(q)+VG(q)

(56)

7 Copyright c© 2016 by ASME

70 75 80 85 90 95 100 105 110−50

0

50

100Pitch

[Nm

]

70 75 80 85 90 95 100 105 110

−0.2

−0.1

0

0.1

0.2

0.3

[rad

/s]

Time [s]

ActualDesired

VirtualActual

FIGURE 6. Pitch motion θp; Top: torque tracking; Bottom: coordi-nation.

which includes the kinetic and potential energies of the physicalsystem, the kinetic energies of the virtual inertia and the fictitiousflyhweel of PVFC, and the obstacle avoidance potential field, itcan be shown (by differentiating Wtotal and integrating over time)that: ∫ t

0V T

L [(ρ +1)Fhuman +Fenv]dτ ≥−c2. (57)

This shows that after the coordination, i.e. VL → q (ensured bythe shape system control), and the closed loop system achievesthe target energetic passivity in Eq. (13) with the supply rate be-ing the scaled power input from the human and the environment.

6.2 2-DoF Virtual CoordinationThe controller in Section 3 has been experimentally imple-

mented on a 2-DoF Human Power Amplifier (HPA) in Fig. 2.The pitch motion (Fig. 6) is actuated by a hydraulic transformerand the reach motion (Fig. 7) is actuated by a servo valve. Veloci-ties of the virtual inertia and the actual system are coordinated foreach DoF, showing RMS error of 0.0860 rad/s and 0.110 m/s forthe pitch and reach directions, respectively. With Fd = ρFhuman,where ρ = 7, the pitch direction shows 6.9 Nm of RMS torqueerror and the reach direction shows 1.75 N of RMS force error.

6.3 PVFCFigure 8 shows PVFC converging to a desired path through

desired direction and tracing a circle continuously afterwards.This is achieved through following a desired velocity shown inFig. 9, which shows RMS error of 0.15 rad/s for pitch movement,0.069 m/s for reach movement, and 0.0135 rad/s for the virtualflywheel.

70 75 80 85 90 95 100 105 110−150

−100

−50

0

50

100

[N]

Reach

70 75 80 85 90 95 100 105 110

−0.2

−0.1

0

0.1

0.2

0.3

[m/s

]

Time [s]

ActualDesired

VirtualActual

FIGURE 7. Reach motion xp; Top: force tracking; Bottom: coordi-nation.

0 0.2 0.4 0.6 0.8 1−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

X movement [m]

Y m

ovem

ent [

m]

Velocity FieldCircleActual PathRange of Motion

FIGURE 8. Guidance velocity field and resulting motion in Cartesiancoordinates (of the tip).

6.4 Obstacle AvoidanceFigure 10 shows the results for the Obstacle Avoidance con-

trol. A symmetric potential field was defined (in the Cartesiancoordinates) and as a result, the machine is not allowed to enterinto the circular prohibited region.

7 ConclusionsIn this paper, the results of virtual coordination framework

from [7] is extended for fully coupled multi-DoF system. Apassivity based control approach that uses natural energy stor-age of the hydraulic actuator is used to define the flow require-ment. Additional passive dynamics that helps the user to per-

8 Copyright c© 2016 by ASME

170 175 180 185 190 195 200−0.5

0

0.5Pitch

170 175 180 185 190 195 200−0.4

−0.2

0

0.2

0.4Reach

170 175 180 185 190 195 2002.5

3

3.5Flywheel

Time [s]

˙q

V

√

k(q, ˙q)

E

FIGURE 9. Actual velocity vs scaled desired velocity

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

X movement [m]

Y m

ovem

ent [

m]

HPA MovementObstacleRange of Motion

FIGURE 10. Obstacle Avoidance

form specific tasks, previously implemented on direct force con-trol framework, are implemented with the virtual coordinationframework. Guidance is achieved using a passive velocity fieldcontroller (PVFC), while the obstacle avoidance is achieved us-ing a potential field. Experimental results demonstrate good per-

formance on a 2-DoF Human Power Amplifier.

ACKNOWLEDGMENTThis work is performed within the Center for Compact and

Efficient Fluid Power (CCEFP) supported by the National Sci-ence Foundation under grant EEC-05040834. Donation of com-ponents from Takako Industries is gratefully acknowledged.

REFERENCES[1] Li, P. Y., 2004. “Design and control of a hydraulic human

power amplifier”. In ASME 2004 International MechanicalEngineering Congress and Exposition, American Societyof Mechanical Engineers, pp. 385–393.

[2] Li, P. Y., 2006. “A new passive controller for a hydraulichuman power amplifier”. In ASME 2006 International Me-chanical Engineering Congress and Exposition, AmericanSociety of Mechanical Engineers, pp. 1375–1384.

[3] Li, P. Y., and Durbha, V., 2008. “Passive control of fluidpowered human power amplifiers”. In Proceedings of theJFPS International Symposium on Fluid Power, no. 7-1, ,pp. 207–212.

[4] Lee, D. J., and Li, P. Y., 2005. “Passive bilateral control andtool dynamics rendering for nonlinear mechanical teleoper-ators”. Robotics, IEEE Transactions on, 21(5), pp. 936–951.

[5] Lee, D. J., and Li, P. Y., 2007. “Passive decomposition ap-proach to formation and maneuver control of multiple rigid-bodies”. ASME Journal of Dynamic Systems, Measurementand Control, 129, September, pp. 662–677.

[6] Lee, D. J., and Li, P. Y., 2013. “Passive decompositionof multiple mechanical systems under coordination require-ments”. IEEE Transactions on Automatic Control, 58, Jan-uary, pp. 230–235.

[7] Lee, S., and Li, P. Y., 2015. “Passive control of a hy-draulic human power amplifier using a hydraulic trans-former”. In ASME 2015 Dynamic Systems and ControlConference, American Society of Mechanical Engineers,pp. V002T27A004–V002T27A004.

[8] Li, P. Y., and Wang, M., 2014. “Natural storage functionfor passivity-based trajectory control of hydraulic actua-tors”. IEEE/ASME Transactions on Mechatronics, 19(3),July, pp. 1057–1068.

[9] Li, P. Y., and Horowitz, R., 1999. “Passive velocity fieldcontrol of mechanical manipulators”. Robotics and Au-tomation, IEEE Transactions on, 15(4), pp. 751–763.

[10] Khatib, O., 1986. “Real-time obstacle avoidance for ma-nipulators and mobile robots”. The international journal ofrobotics research, 5(1), pp. 90–98.

[11] Lee, S., and Li, P. Y., 2015. “Passivity based back-stepping control for trajectory tracking using a hydraulic

9 Copyright c© 2016 by ASME

transformer”. In ASME/BATH 2015 Symposium on FluidPower and Motion Control, American Society of Mechani-cal Engineers, pp. V001T01A064–V001T01A064.

[12] Lee, S., and Li, P. Y., 2014. “Trajectory tracking controlusing a hydraulic transformer”. 2014 International Sympo-sium on Flexible Automation, Awaji Island, Japan.

[13] Li, P. Y., 2006. “A new passive controller for a hydraulichuman power amplifier”. In ASME 2006 International Me-chanical Engineering Congress and Exposition, AmericanSociety of Mechanical Engineers, pp. 1375–1384.

[14] Lee, D., 2004. “Passive decomposition and control of in-teractive mechanical systems under motion coordination re-quirements”. PhD thesis, University of Minnesota.

[15] Eskilsson, F., 2011. “Passive control for a human poweramplifier, providing force amplification, guidance and ob-stacle avoidance”. Master’s thesis, Linkping University.(Research performed as part of an international exchangeat the University of Minnesota.).

[16] Rimon, E., and Koditschek, D. E., 1992. “Exact robot nav-igation using artificial potential functions”. Robotics andAutomation, IEEE Transactions on, 8(5), pp. 501–518.

10 Copyright c© 2016 by ASME