MULTI DEGREE-OF-FREEDOM HYDRAULIC HUMAN POWER AMPLIFIER WITH RENDERING OF ASSISTIVE DYNAMICS

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1 Proceedings of Dynamic Systems and Control Conference DSCC 216 October 12-14, 216, Minneapolis, MN, USA DSCC MULTI DEGREE-OF-FREEDOM HYDRAULIC HUMAN POWER AMPLIFIER WITH RENDERING OF ASSISTIVE DYNAMICS Sangyoon Lee ERC for Compact and Efficient Fluid Power Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota Fredrik Eskilsson SAAB Aerospace Förmansgatan 11 Linköping, Sweden Perry Y. Li ERC for Compact and Efficient Fluid Power Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota ABSTRACT The hydraulic human power amplifier (HPA) is a tool similar to exoskeleton that uses hydraulic actuation to amplify the applied human force. The control objective is to make the system behave like a passive mechanical tool that interacts with the human and the environment passively with a specified power scaling factor. In our previous work, a virtual velocity coordination approach recasts the single degree-of-freedom human power amplifier control problem into a velocity coordination with a fictitious reference mechanical system. Force amplification becomes a natural consequence of the velocity coordination. In this paper, this control approach is extended for fully coupled multi- DoF systems. A passivity based control approach that uses the natural energy storage of the hydraulic actuator to take full account of the nonlinear pressure dynamics is used to define the flow requirement. Additional passive assistance dynamics are designed and implemented to enable the user to perform specific tasks more easily. Guidance is achieved using a passive velocity field controller (PVFC), and obstacle avoidance is achieved using a potential field. Experimental results demonstrate good performance on a 2-DoF Human Power Amplifier. 1 Introduction The 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 of Minnesota. machine is an extension of his body while amplifying the applied human effort. The control objective is identical to that of a wearable exoskeleton. The only difference is that the human can let go of the HPA, but is an exoskeleton is always attached. In both cases, since the operator participates physically and directly in controlling the machine, it is more intuitive than using a remote joystick. Because of the direct, physical interaction, energetic passivity, which limits the amount of energy that can be transferred to the human and the environment, is a useful property for HPAs to ensure coupling stability and safety. In [1] it was shown that a direct approach to control actuator force as an amplified human force leads to a positive velocity feedback which is not robust in the presence of uncertainty, slow sampling or feedback noise. In [2] and [3], an alternate controller was proposed which models the actuator as a combination of an ideal velocity source and a nonlinear spring, the latter captures the compressibility effects of the fluid medium. Instead of controlling the actuator to track the desired force directly, the controller coordinates the velocities of the system and of a fictitious virtual mass whose dynamics are influenced by the hydraulic actuator and the human force. The control law for achieving coordination is accomplished via a passive decomposition [4 6] into a shape system and a locked system. This control is more robust as the passivity property is enforced by the control structure itself. A variety of control laws can be designed to stabilize the shape system to coordinate the velocities of the actual and virtual systems. In particular, a control law was derived in [7] using the natural compressibility energy of the hydraulic actuator [8] 1 Copyright c 216 by ASME

2 A Force Handle B Pm1 rm Pm2 Hydraulic Motor, Reach xp Dm xθ Qθ Pθ, A1 PT, A2 Hydraulic Actuator, Pitch θp FIGURE 1. Picture of the Human Power Amplifier by: FIGURE 2. Schematic of the Human Power Amplifier instead of approximating the hydraulic actuator by a nonlinear mechanical spring. In these previous works, the control was developed for each individual degree-of-freedom assumed to be decoupled. In this paper, we expand the results from [7] to a fully coupled multi-dof HPA. In addition, we also develop additional human-machine shared control strategies that render useful passive 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 a specified velocity field. Obstacle avoidance is achieved by incorporating potential fields [1] to prohibit the machine from entering prohibited zones. With these task oriented passive dynamics, the operator can execute tasks accurately with less attention while remaining in direct control since the he/she must supply a portion of the physical power. The rest of paper is organized as follow. System dynamics and control objectives are stated in Section 2. In Section 3, the reformulation of the problem as a velocity coordination is reviewed, followed by the presentation for the proposed flow requirement. Guidance dynamics in the form of PVFC and obstacle avoidance are presented in sections 4 and 5. Experimental results and concluding remarks are given in sections 6 and 7. 2 System Description and Model We consider a 2-DoF human power amplifier (HPA) shown in Fig. 1-2 with generalized coordinate q = [θ p,x p ] T where θ p describes the angular position of the pitch movement and x p describes the linear position of the reach movement. The pitch angular motion is actuated by a linear hydraulic actuator whereas the reach linear motion is actuated by a hydraulic motor via a pulley and belt mechanism. The dynamics of the HPA are given M p (q) q +C p ( q,q) q + G p (q) = F human + F env + F a (1) where M p (q) R 2X2 is the symmetric and positive definite inertia matrix, C p ( q,q) R 2X2 is the Coriolis matrix such that Ṁ p (q) 2C p ( q,q) is skew-symmetric, and G p (q) R 2X1 is the gravity vector. F human is the generalized torque/force applied by the human on a handle instrumented with force sensors; F env is the force exerted by the environment which is not measured. F a is the generalized actuator force/torque: F a = [ Tθ F x ] = [ JA (θ p ) 1 rm ][ Fθ where T θ and F x are the torque and force applied to the pitch and reach directions. As seen in Figure. 2, the pitch torque T θ is generated by a hydraulic cylinder with force F θ given by T x ] (2) F θ = P θ A 1 P T A 2 (3) where A 1 and A 2 are the cap side and piston side areas, P θ and P T are the supply and tank pressures on the cap and rod sides of the actuator. The Jacobian J A (θ p ) is used to translate linear force generated into a generalized torque. The reach force F x is generated by a fixed displacement hydraulic motor with torque T x given by: T x = P x D m where D m is the motor displacement, P x = P m1 P m2 is the pressure across the motor. The motor is connected to a belt via a pulley with a radius r m. The cap side of the hydraulic cylinder for the pitch motion is connected to the output of a hydraulic transformer [7] with pressure dynamics given by: (4) Ṗ θ = β(p θ ) V 1 (x θ ) (Q θ A 1 ẋ θ ) (5) 2 Copyright c 216 by ASME

3 where Q θ is the flow input to the cap side chamber, V 1 (x θ ) = V 1 + A 1 x θ (6) is the fluid volume in the cap-side chamber and the hose dependent on the linear displacement of the cylinder x θ, β(p θ ) is the pressure dependent fluid bulk modulus. The rod side is connected to the lower common pressure rail so that P T is assumed to be constant. It is assumed that the gravity load (in F env ) is sufficiently large such that over-running load and cavitation will not occur. The pressure dynamics for the two sides of the hydraulic motor are: Ṗ m1 = β(p ( m1) V m Ṗ m2 = β(p m2) V m ) ẋ p r m ẋ p r m Q m1 D m ( Q m2 + D m where V m are the fixed fluid volumes in the motor and the hose which, for simplicity, are the same on both sides of the motor, β( ) is the pressure dependent bulk modulus. Flow Q m1 = Q m2 are the equal input and return flows in the servo valve. Let P x = P m1 P m2 and taking the difference between these dynamics, the motor pressure dynamics are: Ṗ x = Ṗ m1 Ṗ m2 = β e(p x,p m1 ) V m ) ( Q x D m ẋ p r m where β e (P x,p m1 ) = β(p m1 )+β(p m2 ) is the effective pressure dependent bulk modulus rising from the pressure difference across the motor, and Q x := Q m1 = Q m2 is the flow input to the motor. In the experiment, Q θ and Q x are respectively controlled by a custom built hydraulic transformer [11] and a servo valve. Readers are referred to [7, 11, 12] and [13] for details of how these flows are achieved with these devices. Control Objective The control objective is to control flow inputs to the hydraulic actuators Q θ and Q x in (5) and (9) such that the applied human force is amplified by a factor of (ρ + 1), which results in the target dynamics of a passive mechanical tool M L (q) q+c L ( q,q) q = (ρ +1)F human +F env G(q)+F guide (1) where M L (q) and C L ( q,q) are the apparent inertia and associated Coriolis matrix of the tool to be designed. The human would feel that he/she is interacting with an inertia and an environment force that are attenuated as M L /(ρ +1), F env /(ρ +1) and G(q)/(ρ +1) respectively. For M L (q) M p (q), this can be achieved if the generalized actuator force satisfies: F a ρf human ) (7) (8) (9) Fd=ρFhuman MV qv Fhuman+Fenv FIGURE 3. Model of hydraulic actuator with the ideal velocity command provided by the velocity of a virtual inertia, and the virtual inertia affected by the actuator force. In a human power amplifier F d = ρf human F guide is the additional task specific guidance dynamics to provide assistance to the user to operate the HPA. Fa 3 Virtual Coordination Control Approach to Force Tracking Instead of directly controlling the actuator force F a to track the desired force ρf human, the virtual coordination approach in [2] converts the problem into one of coordinating velocities of two coupled mechanical systems - the plant and a virtual inertia. Besides avoiding the need for positive velocity feedback, this approach can also be interpreted physically as an interconnection of passive components so that it is more robust and safer to operate. With the actuator compressibility represented by a springlike object that interacts with the inertia of the machine M p, the approach is to control F a such that the other end of the springlike object is interacting with a small virtual inertia M v R acted on by a set of desired forces (Fig. 3). Let the dynamics of a virtual inertia M v (implemented as part of the controller) be given by: qi MP M v q v = F d F a + w + F guide (11) where q v = [θ v,x v ] T is the generalized coordinate for the virtual inertia, F d = ρf human is the desired force, F a is the generalized actuator force (Eq. (2)) and w and F guide are the additional controls to be designed for the locked system or for task guidance. If exact coordination between the virtual inertia M v and M p (q), such that q v (t) q(t) (i.e. they become a single rigid inertia), then comparing (1) and (11), and the fact that w will be defined such that w when coordinated, the resulting dynamics becomes: (M v + M p (q)) q +C p ( q,q) q = (ρ + 1)F human + F env G p (q) + F guide which is the desired target dynamics in (1) with the apparent inertia being M L (q) = M v + M p (q), and C L (q, q) = C p (q, q). 1 For simplicity, M v is a constant inertia represented by a positive definite matrix. 3 Copyright c 216 by ASME

4 As will be seen, the guidance force F guide will be designed to satisfy a passivity property: t [ q T F guide ]dτ c 2 g (12) Then, after coordination q v (t) q(t), the closed loop system is energetically passive with respect to the scaled power input by the human and environment such that there exists c 2 > so that for all F human ( ), F env ( ), t q T [(ρ + 1)F human + F env ]dτ c 2 (13) We have used the fact that gravity is conservative such that G p (q) = V G(q) q (14) where V G (q) R is the gravitational potential field and q lies in a compact work space. In the following, we extend the virtual coordination controller in [7] that uses the natural energy storage function for the hydraulic actuators to fully coupled multi-dof systems. 3.1 Passive Decomposition into Locked and Shape Systems The coupled system of Eq. (1) and (11) is given by M p (q) q +C p ( q,q) q + G p (q) = F human + F env + F a M v q v = F d F a + w + F guide (15) where the generalized coordinates for the physical system are q = [θ p,x p ] T and for the virtual system are q v = [θ v,x v ] T. As we are interested in coordination between q and q v, i.e, V E := q q v it is desirable to study the problem in relative coordinates. However, we also do not wish to disturb the energetics of the desired target dynamics in Eq. (1). Therefore, we apply the passive decomposition [4 6] to transform the velocities into locked and shape coordinates: VL I φ φ q = (16) V E I I q v }{{} S(q) where I φ = [M p (q) + M v ] 1 M p (q) φ = [M p (q) + M v ] 1 M v and M L (q) = M p (q) + M v is the inertia corresponding to the locked system. V L (locked system velocity) is the velocity of the center of mass of the combined virtual and actual system, whereas V E (shape system velocity) is the velocity coordination error. The dynamics in the transformed coordinates are given by: ML (q) V L + M E (q) V E [ CL ( q,q) C LE ( q,q) C EL ( q,q) C E ( q,q) ][ VL V E ] = ψ (17) where the inertia matrix and Coriolis matrix are transformed according to the definition of the passive decomposition [4], ML (q) M E (q) = S T Mp (q) M v = CL ( q,q) C LE ( q,q) C EL ( q,q) C E ( q,q) + d dt S T [ Mp (q) M v S 1 [ Mp (q) + M v (I φ) T M v (I φ) ] (18) = S T Mp (q) d M v dt (S 1 ) ] S 1 + S T C( q,q) S 1 (19) Forces acting on the virtual and actual inertia are: ψ = S T Fhuman + F env + F a G p (q) F d F a + w + F guide The transformed system is represented as M L (q) V L +C L ( q,q)v L +C LE ( q,q)v E = (2) F d + F env + F human G(q) + w + F guide (21) M E (q) V E +C E ( q,q)v E +C EL ( q,q)v L = F a + φ(f env ) }{{} F E1 +φ(f human G(q)) (I φ)(f d + w + F guide ) (22) }{{} F E2 3.2 Shape System Control From (22), the shape system dynamics are: M E (q) V E +C E ( q,q)v E +C EL ( q,q)v L = F a + F E1 + F E2 (23) where F E2 contains measurable or known terms and F E1 is potentially unknown. We define the desired actuator force to achieve shape system control to be: Fa,d1 F a,d = = C EL ( q,q)v L λv E ˆF E1 F E2 (24) F a,d2 4 Copyright c 216 by ASME

5 where ˆF E1 is an estimate of F E1 and λ >. The nonlinear decoupling term C EL ( q,q)v L is needed to decouple the locked system dynamics from the shape system dynamics, which is not necessary for single DoF control. The following input flows for each degree of freedom in (5) and (9) are proposed: Q d θ = A 1J A (θ p ) θ v + V 1(x θ ) β(p d,θ )Ṗd,θ λ pθ P θ (25) Q d x = D m ẋ v r m + V x β e (P d,x )Ṗd,x λ px P x (26) where P d,θ and P d,x are the desired actuator pressures in the pitch and reach directions given by: P d,θ = 1 [ P T A ] F a,d1 (27) A 1 J A P d,x = r m D m F a,d2 (28) The estimate for the external force Ḟ E1 is obtained from the adaptation algorithm, ˆF E1 = σv E + Ḟ E1 (29) where Ḟ E1 is the best estimate of the time derivative of F E1 ; λ, λ pθ, λ px, and σ are all positive constants. To see that this control is appropriate, consider the following Lyapunov function, W = 1 2 V T E M E V E + 1 σ F T E1 F E1 +V 1 (x θ )W V ( P θ,p d,θ ) +V m W V ( P x,p d,x ) (3) where F E1 is the error in estimating the unknown external force; W V ( P θ,p d,θ ) is the volumetric pressure error energy density associated with compressing the fluid from pressure P d,θ to P d,θ + P θ as defined in [8]. Likewise, W V ( P x,p d,x ) is defined in the same manner. Differentiating the above Lyapunov function (See [8] and [7] for details and proof), and with λ p,θ and λ p,x sufficiently large we get, Ẇ V T E λv E m θ (λ p,θ ) P 2 θ m x(λ p,x ) P 2 x such that m θ (λ p,θ ) > and m x (λ p,x ) >. This in turn shows that V E, P θ, P x. 3.3 Locked System Control From (22), the Locked system dynamics are: M L (q) V L +C L ( q,q)v L +C LE ( q,q)v E = F d + F env + F human G(q) + w + F guide (31) We can design w to cancel out the coupling dynamics w = C LE ( q,q)v E (32) Note that w is also used in shape system control because of w in F E2 (see (22)). If desired, G(q) could be included in w to cancel out the effect of gravity. With F d = ρf human, and after coordination (i.e. V L = q = q v ): M L (q) q+c L ( q,q) q = (ρ +1)F human +F env G(q)+F guide (33) which is the target dynamics we wanted in Eq. (1) 4 Passive Velocity Field Controller In this and next section, we design F guide n Eq. (11) to provide 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 strategy will be discussed in section 5. An earlier attempt to implement useful dynamics for HPA can be found in [15]. In PVFC, passive dynamics are incorporated into the machine to guide the operator to follow a scaled copy of a desired velocity field - i.e. a desired velocity at each position, while allowing the machine to remain passive. This can guide the human operator to follow a desired path which is the flow of the velocity field. An example velocity field is shown in Fig. 4 which guides the HPA to converge to and follow a circle. The speed at which the field is followed is determined by the kinematic energy available in this system. In this way, it is possible to provide path guidance without violating passivity. In particular, the energy input into the system must be provided either by the human operator or the environment. As the operator is physically connected to the machine in HPA operation, PVFC works as a feedback to the operator informing whether he is on the right track. For safety and comfort, it is still important that the machine remains passive. An outline of how PVFC incorporates into HPA control is given below. Readers are referred to [9] [14] for detailed proofs. In this paper, we incorporate this guidance dynamics to the locked system dynamics in (31) and use the virtual coordination scheme rather than directly to the physical system. 4.1 Desired Velocity Field and Augmented System Let the desired velocity field be V (q) R 2 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 the user to perform a circular motion. In the absence of any human or environmental input, the PVFC controller will cause q(t) β(t)v (q(t)) where β 2 (t) is proportional to the kinetic energy in the system. 5 Copyright c 216 by ASME

6 Y movement [m] Velocity Field Desired Path X movement [m] FIGURE 4. Velocity field for tracing a circle such that the kinetic energy of the augmented system is constant when the augmented field is tracked. This can be accomplished by ensuring for all q R 2, Ē = 1 2 V T (q) M(q) V (q) where Ē is a sufficiently large constant. In other words, the desired flywheel velocity field is given by: V F (q) = 2 M F ( Ē 1 ) 2 V (q)t M L (q)v (q) (4) 4.2 PVF Controller With the augmented system and augmented velocity field, the coupling control τ can be designed as To define PVFC, we first augment the system dynamics with a 1 DoF fictitious flywheel dynamics: M F q F = τ F (34) where M F is the apparent inertia of this virtual flywheel, q F is the position of the flywheel, and τ F is the coupling control input to the flywheel. Combined with the locked system in (31), the augmented system becomes: M(q) q + C(q, q) q = τ + τ e (35) where q = [ V T L q F] T are the augmented velocity, τ = [ F T guide τ T F ] T being the augmented control input, τe = [ τ T e ] T being the augmented external force where τ e = F env + (ρ + 1)F human G(q) (36) ML (q) M(q) =, C(q, q) = M F CL (q, q) (37) are the augmented inertia matrix and the augmented Coriolis matrix. The kinetic energy of the augmented system is k( q, q) = 1 2 q T M( q) q = 1 2 V L T M L (q)v L + 1 }{{} 2 M F q 2 F }{{} Locked System flywheel (38) In order to control and utilize the virtual flywheel, the desired velocity field V (q) needs to be augmented as: V (q) = [ V (q) T V F (q) ] T (39) τ = Ω(q, q) q (41) where Ω(q, q) R (n+1) (n+1) is skew symmetric. This ensures that q T τ = so that the PVFC control is passive. The coupling force re-distributes energy between the locked system and the fictitious flywheel conservatively. To find suitable Ω(q, q), the followings 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 actual momentum, and w(q, q) is the covariant derivative of the desired momentum field. With these, the coupling control law is given by where or τ( q, q) = τ c ( q, q) + τ f ( q, q) (45) τ c = 1 2Ē ( w P T P w T ) q (46) τ f = γ( P p T p P T ) q (47) Ω(q, q) = 1 2Ē ( w P T P w T ) + γ( P p T p P T ) (48) τ c corresponds to a feedfoward control, giving information about the desired system dynamics, and τ f is a feedback control which vanishes when q = α(t) V (q(t)) for some scalar α(t), γ is a control gain that determines the convergence rate and the sense in which the desired velocity will be followed. 6 Copyright c 216 by ASME

7 The PVFC component of F guide is then the first two elements of the coupling control in (45) such that.12 Potential Field 1 9 [ FPV FC τ F ] = τ( q, q) (49) Y Properties of PVFC With the control in (41)-(48), the closed loop dynamics for the coupled augmented system can be written as M( q) q +Ȳ ( q, q) q = τ e (5) X where τ e is the augmented external force and Ȳ R 3X3 is: Ȳ (q, q) = C(q, q) 1 2Ē ( w P T P w T ) γ ( P p T p P T ) }{{}}{{} skew symmetric skew symmetric (51) Notice that since both control terms in (46) and (47) are skew symmetric, M 2Ȳ is also skew symmetric, just as Ṁ 2C is skew symmetric for the robot manipulator. Utilizing this skew symmetric structure and (5), to differentiate kinetic energy in (38), we obtain d dt k( q, q) = V T L (t)τ e (t) (52) where VL T τ e = q T τ e because τ e = [ τe T ] T. Integrating (52) w.r.t time gives t V T L (t)τ e (t)dτ c 2 (53) FIGURE 5. An example potential field for a point obstacle in Cartesian (workspace) coordinates. 5 Obstacle Avoidance The aim of Obstacle Avoidance Control is to prevent the machine from entering prohibited area in the workspace to protect itself or other objects. Here we utilize an artificial potential field approach [1] [16] to provide the operator a tactile feedback to repel the machine from the obstacle. The potential field is designed to be non-negative continuous and differentiable function that tends to infinity as the machine approaches the obstacle. It is also designed such that the influence of the potential field is limited to certain to avoid having undesirable perturbing forces beyond the obstacle s vicinity. For a point obstacle, an example potential field U oa (q) expressed in the Cartesian (workspace) coordinates (X(q),Y (q)) of the tip of the HPA is shown in (Fig. 5). This field is exponentially decaying (with distance) and is radially symmetric in the workspace coordinates. The force arising from this potential function is the negative gradient of the above function such that: This means that the closed loop dynamics of the augmented system given by (5) is passive with respect to the supply rate V T L τ e (the power produced by the external forces), and its kinetic energy in (38) is its storage function. With the PVFC controller, it can be shown that, in the absence of τ e, F OA = U oa(q) q The combined guidance control is: F guide = F PV FC + F OA (55) where q β(t) V (q(t)) (54) β(t) = sign(γ) k(q, q) Ē 6 Results and Discussion 6.1 Energetic Passivity Property With the total energy function: W total = 1 2 V T E M E V E + 1 σ F T E1 F E1 +V 1 (x θ )W V ( P θ,p d,θ ) Thus, as the kinetic energy of the system increases (such as with input by the human operator or the environment, the speed at which the desired velocity is tracked will also increase. +V m W V ( P x,p d,x ) V T L M L V L M F q 2 F +U oa (q) +V G (q) (56) 7 Copyright c 216 by ASME

8 1 Pitch 1 Reach [Nm] 5 Actual Desired [N] Actual Desired [rad/s] Virtual Actual Time [s] [m/s] Virtual Actual Time [s] Pitch motion θ p ; Top: torque tracking; Bottom: coordi- FIGURE 6. nation. Reach motion x p ; Top: force tracking; Bottom: coordi- FIGURE 7. nation. which includes the kinetic and potential energies of the physical system, the kinetic energies of the virtual inertia and the fictitious flyhweel of PVFC, and the obstacle avoidance potential field, it can be shown (by differentiating W total and integrating over time) that: t V T L [(ρ + 1)F human + F env ]dτ c 2. (57) This shows that after the coordination, i.e. V L q (ensured by the shape system control), and the closed loop system achieves the target energetic passivity in Eq. (13) with the supply rate being the scaled power input from the human and the environment DoF Virtual Coordination The controller in Section 3 has been experimentally implemented on a 2-DoF Human Power Amplifier (HPA) in Fig. 2. The pitch motion (Fig. 6) is actuated by a hydraulic transformer and the reach motion (Fig. 7) is actuated by a servo valve. Velocities of the virtual inertia and the actual system are coordinated for each DoF, showing RMS error of.86 rad/s and.11 m/s for the pitch and reach directions, respectively. With F d = ρf human, where ρ = 7, the pitch direction shows 6.9 Nm of RMS torque error and the reach direction shows 1.75 N of RMS force error. Y movement [m] Velocity Field.7 Circle Actual Path Range of Motion X movement [m] FIGURE 8. Guidance velocity field and resulting motion in Cartesian coordinates (of the tip). 6.4 Obstacle Avoidance Figure 1 shows the results for the Obstacle Avoidance control. A symmetric potential field was defined (in the Cartesian coordinates) and as a result, the machine is not allowed to enter into the circular prohibited region. 6.3 PVFC Figure 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 in Fig. 9, which shows RMS error of.15 rad/s for pitch movement,.69 m/s for reach movement, and.135 rad/s for the virtual flywheel. 7 Conclusions In this paper, the results of virtual coordination framework from [7] is extended for fully coupled multi-dof system. A passivity based control approach that uses natural energy storage of the hydraulic actuator is used to define the flow requirement. Additional passive dynamics that helps the user to per- 8 Copyright c 216 by ASME

9 .5 Pitch form specific tasks, previously implemented on direct force control framework, are implemented with the virtual coordination framework. Guidance is achieved using a passive velocity field controller (PVFC), while the obstacle avoidance is achieved using a potential field. Experimental results demonstrate good performance on a 2-DoF Human Power Amplifier Reach Flywheel V k( q, q) Ē Time [s] FIGURE 9. Y movement [m] Actual velocity vs scaled desired velocity HPA Movement.8 Obstacle Range of Motion X movement [m] FIGURE 1. Obstacle Avoidance q ACKNOWLEDGMENT This work is performed within the Center for Compact and Efficient Fluid Power (CCEFP) supported by the National Science Foundation under grant EEC Donation of components from Takako Industries is gratefully acknowledged. REFERENCES [1] Li, P. Y., 24. Design and control of a hydraulic human power amplifier. In ASME 24 International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, pp [2] Li, P. Y., 26. A new passive controller for a hydraulic human power amplifier. In ASME 26 International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, pp [3] Li, P. Y., and Durbha, V., 28. Passive control of fluid powered human power amplifiers. In Proceedings of the JFPS International Symposium on Fluid Power, no. 7-1,, pp [4] Lee, D. J., and Li, P. Y., 25. Passive bilateral control and tool dynamics rendering for nonlinear mechanical teleoperators. Robotics, IEEE Transactions on, 21(5), pp [5] Lee, D. J., and Li, P. Y., 27. Passive decomposition approach to formation and maneuver control of multiple rigidbodies. ASME Journal of Dynamic Systems, Measurement and Control, 129, September, pp [6] Lee, D. J., and Li, P. Y., 213. Passive decomposition of multiple mechanical systems under coordination requirements. IEEE Transactions on Automatic Control, 58, January, pp [7] Lee, S., and Li, P. Y., 215. Passive control of a hydraulic human power amplifier using a hydraulic transformer. In ASME 215 Dynamic Systems and Control Conference, American Society of Mechanical Engineers, pp. V2T27A4 V2T27A4. [8] Li, P. Y., and Wang, M., 214. Natural storage function for passivity-based trajectory control of hydraulic actuators. IEEE/ASME Transactions on Mechatronics, 19(3), July, pp [9] Li, P. Y., and Horowitz, R., Passive velocity field control of mechanical manipulators. Robotics and Automation, IEEE Transactions on, 15(4), pp [1] Khatib, O., Real-time obstacle avoidance for manipulators and mobile robots. The international journal of robotics research, 5(1), pp [11] Lee, S., and Li, P. Y., 215. Passivity based backstepping control for trajectory tracking using a hydraulic 9 Copyright c 216 by ASME

10 transformer. In ASME/BATH 215 Symposium on Fluid Power and Motion Control, American Society of Mechanical Engineers, pp. V1T1A64 V1T1A64. [12] Lee, S., and Li, P. Y., 214. Trajectory tracking control using a hydraulic transformer. 214 International Symposium on Flexible Automation, Awaji Island, Japan. [13] Li, P. Y., 26. A new passive controller for a hydraulic human power amplifier. In ASME 26 International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, pp [14] Lee, D., 24. Passive decomposition and control of interactive mechanical systems under motion coordination requirements. PhD thesis, University of Minnesota. [15] Eskilsson, F., 211. Passive control for a human power amplifier, providing force amplification, guidance and obstacle avoidance. Master s thesis, Linkping University. (Research performed as part of an international exchange at the University of Minnesota.). [16] Rimon, E., and Koditschek, D. E., Exact robot navigation using artificial potential functions. Robotics and Automation, IEEE Transactions on, 8(5), pp Copyright c 216 by ASME