IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 3, JUNE 2015
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A Force Bounding Approach for Multi-Degree-of-Freedom Haptic Interaction Jong-Phil Kim, Sang-Yun Baek, Student Member, IEEE, and Jeha Ryu, Member, IEEE
Abstract—Stability and transparency are two major conflicting requirements in haptic interaction systems: some level of transparency is required for providing a realistic feeling while avoiding unstable behaviors that may completely destroy the contact realism of virtual environments or injure the human operator. In this paper, we propose a new multi-degree-of-freedom force bounding approach for a robustly stable and directionally transparent haptic interaction with any virtual environments. The proposed approach is based on two (less and more) conservative sufficient conditions for the passivity condition of sampled-data haptic systems. A less conservative sufficient condition contains, however, memory effect causing contact oscillations due to the accumulation of past remaining dissipation capability during free motion. In order to avoid contact oscillations due to the memory effect, a more conservative sufficient condition may be used for systematically resetting the past accumulated energy. We present experimental results to verify that the proposed approaches make the haptic interaction passive and increase haptic realism significantly. Index Terms—Haptics, haptic interaction control, passivity, stability.
I. INTRODUCTION TABILITY is one of the most critical major requirements in haptic interaction systems since the haptic interaction contains bidirectional energy flows. On the other hand, some degree of transparency is required also for realistic experiences of virtual environments (VEs). In haptic interaction, unwanted energy generated by the active motion of a haptic interface can be passed on to a human operator and may result in the deterioration of contact realism or, more importantly, human injury. Transparent and harmless delivery of haptic information to the human operator necessitates, therefore, careful handling of the energy generation and building stable haptic interaction systems. Among the many efforts to construct stable haptic interaction systems, the passivity-based approach remains attractive because it guarantees robust stability for many VEs. Colgate et al. [1] proposed a virtual coupling (VC) concept based on a passivity theorem to ensure haptic interaction stability. The VC restricted the impedance of a passive VE to within
S
Manuscript received January 20, 2014; accepted June 12, 2014. Date of publication June 11, 2014; date of current version May 18, 2015. Recommended by Technical Editor E. Richer. This work was supported by the National Research Foundation of Korea grant funded by the Korea government (2013-067321). (J. P. Kim and S. Y. Baek contributed equally to this work). (Corresponding author: Jeha Ryu.) J.-P. Kim is with the Imaging Media Research Center, Korea Institute of Science and Technology, Seoul 136-791, Korea (e-mail:
[email protected]). S.-Y. Baek and J. Ryu are with the Human-Robotics Lab, School of Mechatronics, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2014.2333537
the passivity-guaranteed impedance range. Hannaford and Ryu [2] and Ryu et al. [3] proposed a time-domain passivity algorithm (TDPA) composed of a passivity observer (PO) to monitor energy flow and a passivity controller (PC) to dissipate excessive energy. Kim and Ryu [4], [5] proposed the energy bounding approach (EBA) to restrict the energy generation of the sample and the hold operator within the physical energy dissipation capability in the haptic system. It also made the VE passive to guarantee robustly stable haptic interactions for any VEs. Lee and Lee [6] proposed an adjusting output-limiter (AOL) algorithm for stable haptic interaction with unknown, time varying, and/or nonlinear deformable objects. The AOL adjusted the maximum reflective force to guarantee the passivity of the haptic system. Lee and Huang [7] proposed the positionbased haptic control algorithm, the so called Passive-SetPosition-Modulation (PSPM). It sets the modulated position signal as close to the original position signal as possible, yet only to the extent permissible by the passivity constraint. In the related bilateral teleoperation framework, Franken et al. [8] compared characteristic features of TDPA, EBA, and PSPM with their proposed two-layer approach (TLA). These previous works concentrated mostly on the stability aspects while showing varying degrees of magnitude transparency in terms of the stably displayable stiffness, especially in steady states. In cases of multi-degree-of-freedom (multi-DOF) haptic interaction with 3-D objects, direction transparency is more critical than magnitude transparency. For example, when we trace a curved surface (e.g., circle or sphere), we may not feel the surface curvature correctly if the direction of the surface normal reaction force is not in line with the real geometric normal direction. Unlike the single-DOF haptic interaction, not many investigations have been made for the multi-DOF haptic interaction. Preusche et al. [9] extended the TDPA to multiDOF haptic interaction. They assigned adaptive damping along the direction of the desired force and torque vectors. Hertkorn et al. [10] generalized TDPA for multi-DOF haptic systems for the haptic interaction system with a small time delay. Kim et al. [11], [12] proposed a multi-DOF EBA for enhancing directional transparency while guaranteeing stability with multi-DOF haptic interactions. EBA or AOL is applied in each coordinate to guarantee passivity, and a simple projection method is used to correct the distortion of the direction of the rendered force and torque vectors. Recently, Ryu and Yoon [13] proposed a memory-based passivation approach (MBPA) that can also be applied to multi-DOF haptic interactions. In this paper, we propose a new multi-DOF force bounding approach (FBA) for robustly stable and directionally transparent haptic interaction with any 3-D (linear, nonlinear, delayed, etc.)
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VEs. Based on the passivity condition of a one-port network representation of sampled-data haptic systems, we propose two (less and more) sufficient conditions for the force that can be passively displayed. A less conservative sufficient condition, the basic idea of which had been presented in [14] for a single-DOF case, can display higher stiffness of virtual objects but shows a memory effect causing contact oscillations due to the accumulation of dissipation capability during the free motion. This memory effect is a common problem of all the time-domain passivity-based approaches. In order to avoid contact oscillations by the memory effect, we propose a more conservative sufficient condition for systematically removing the past accumulated energy. We present comprehensive experimental results to verify that the proposed approaches make the haptic interaction passive and increase haptic realism significantly. The proposed approach has a good common feature as in the single-DOF PO/PC, EBA, AOL, PSPM, TLA, MBPA: The design of the control force is not dependent on the parameters of VEs; therefore, it can be applied to any unknown VEs that may be linear, nonlinear, delayed. This point is critical because it is very likely that virtual objects are developed by noncontrol or nonhaptic engineers (i.e., content developers) who may specify unusually high stiffness values. In the meantime, the VC ensures stability only for passive VEs. Basic demerits of each approach can be summarized as follows: In the TDPA, the sampling rate should be sufficiently greater than system modes in order to ensure the assumption of a constant velocity from the current time step to the next future time step. The EBA divided the passivity condition of haptic system into two conditions; thus, it may be too conservative. In the AOL, memory effects due to the accumulation of past remaining dissipation capability could induce oscillations upon contact with a very stiff environment. The PSPM also assumed a fast servo rate of haptic devices for ensuring passivity. Otherwise, the magnitude transparency may be deteriorated significantly. TLA, PO/PC, and MBPA typically show oscillatory behaviors upon contact because of nonperfect monitoring of the active energy. For the multi-DOF case, all algorithms seem to guarantee directional transparency at least in theory. Basic assumptions or limitations can be summarized as follows: The multi-DOF TDPA [9], [10] may become singular for very low speeds that are common in contacting to stiff walls. The generalized multiDOF TDPA [10] requires also very accurate estimation of velocity and mass matrix. The multi-DOF EBA [11] and AOL [6] may show severe deterioration of magnitude transparency because the passivity condition is applied in each DOF. The multi-DOF MBPA [13] requires very precise bookkeeping of the force/position histories despite the merit of displaying very high stiffness. Even though the generalized multi-DOF TDPA [10], multi-DOF PSPM [7], and multi-DOF MBPA [13] were developed for multi-DOF cases, no experimental results were presented for the curved surface tracing, from which the directional transparency could not be judged at this time. The proposed FBA has fundamental and structural differences from the existing approaches. One major theoretical difference from existing approaches (PO/PC, PSPM, TLA, etc.) is that the proposed approach is derived from the passivity of a one-
IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 3, JUNE 2015
port network, while the other approaches were derived from the passivity of the two-port network. Therefore, the proposed FBA does not assume passivity of VEs, which means that the proposed FBA can be applied to active VE such as delayed VE. Other fundamental differences are: 1) The proposed FBA does not estimate future positions as in PO/PC, does not assume fast serve rates as in PSPM, and does not require a minimum artificial mass that is necessary for the discrete time passivity of the VE simulation as in PSPM and VC. This makes the proposed approach fundamentally robust against any magnitudes and variations of time delays in any time delayed systems including sampled-data and teleoperated systems. Therefore, nonreal-time controllers such as ordinary thread controls based on the multimedia time clock can be used. 2) The proposed FBA utilizes only the physical damping in the haptic systems. Dependence only on the physical damping, however, may reduce magnitude transparency. Large artificial or physical damping, however, may cause a sluggish motion in free space, which can be felt by force peaks before a contact occurs so that the user may not notice real contact time. 3) The proposed less conservative sufficient condition for a single-DOF case, in fact, is the same as the condition in [6], but it is derived in a very simple way, which should also be considered to be a theoretical contribution. Moreover, severe oscillations upon contact that was observed in the AOL for stiffer virtual walls can be removed in a systematic way by the proposed more conservative sufficient condition. The remainder of this paper is organized as follows. Section II summarizes the passivity condition of a singleDOF haptic system that is modeled as a sampled-data system. Section III presents the proposed multi-DOF FBA with two sufficient conditions (a less and a more conservative sufficient conditions) that satisfy the passivity condition for haptic interaction systems. Comprehensive experimental results are presented in Section IV to show the effectiveness of the proposed approaches. We conclude in Section V with listing some future works. II. PASSIVITY CONDITION FOR A SAMPLED-DATA SYSTEM In this section, a passivity condition of a single-DOF sampleddata haptic interaction system is summarized from [5] to help the reader better understand the remaining parts of this paper. In general, a haptic system is composed of a haptic device, a controller, and a VE. The haptic device contains mechanical linkages, actuators, and sensors. The controller contains numerical computations of forward kinematics, inverse kinematics, and Jacobian, as well as gravity and friction compensation algorithms. The VE contains a haptic rendering algorithm, which allows users to touch and feel the virtual objects in the VE. Fig. 1 shows the overall configuration of a haptic system with a single-DOF device. In the haptic system, a sample and hold operator, usually a zero-order holder (ZOH), is used to connect the discrete system to the continuous system. The loss of information in the sampling as well as the half sample delay in the ZOH can make the haptic system active, which may result in unstable haptic interaction. A common goal of the passivity-based haptic control is to make the haptic system passive, which enables users
KIM et al.: FORCE BOUNDING APPROACH FOR MULTI-DEGREE-OF-FREEDOM HAPTIC INTERACTION
Fig. 1.
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Overall configuration of a haptic system: single-degree-of-freedom case.
to interact with virtual objects passively as they do with passive objects in a real environment. For a passive haptic system with assumption of passive human operators, the energy flow-in (E(n)) to the HSCV system that is consisted of haptic device, sample & hold, controller, and VE in (1) should not be negative [1] such that t E(n) = 0 Fh (τ )vh (τ )dτ ≥ 0 (1) for t > 0 and all admissible Fh (t) where Fh (t) and vh (t) represent the applied force and velocity by a human operator, respectively. Note here that all the initial conditions are assumed to be zero for simple mathematical derivations without loss of generalities. For a single-DOF haptic device dynamics with mass (m) and damping (b) elements, (1) can be rewritten for 0 ≤ t < nT as nT mv˙ d (t)vd (t)dt E(n) = 0
nT
bvd2 (t)dt
+
nT
Fdh (t)vd (t)dt
+
0
≥0
(2)
ples and this term is much more dominant than the first term as time t increases (see [16]) even though the first term might also be dominant initially when the device moves with high acceleration. The third term in (2) indicates energy flow-in to the oneport network system (SCV subsystem) that is composed of the sample and hold, controller, and VE. When ZOH is used as a hold operator, it can be rewritten as nT n −1 (k +1)T Fdh (t)vd (t)dt = Fdh (t)vd (t)dt 0
n −1 = Fd (k)
=
n −1
The second term of (2) indicates energy dissipation by the viscous damper and can be rewritten using the Cauchy–Schwarz inequality (see [15]) as (k +1)T nT n −1 2 bvd (t)dt = b(k) vd2 (t)dt 0
kT
k =0
≥
n −1 b(k) k =0
=
n −1
T
2
(k +1)T
vd (t)dt kT
B(k)Δx2d (k + 1)
(k +1)T
vd (t)dt kT
k =0
Fd (k)Δxd (k + 1).
(5)
k =0
Neglecting the first term in (2), the passivity condition of the haptic system during 0 ≤ t < nT can be restated as
0
where Fdh (k) and vd (k) represent the hold actuator force by the digital-to-analog converter and velocity of the haptic device, respectively. The first term of (2) indicates energy storage via inertia. Assuming zero initial velocity, it has a finite nonnegative value for finite velocity motion that nT 1 mv˙ d (t)vd (t) = mvd2 (n) ≥ 0. (3) 2 0
kT
k =0
E(n) ≥ E1 (n) ≡
n −1
B(k)Δx2d (k + 1)
k =0
+
n −1
Fd (k)Δxd (k + 1).
(6)
k =0
Note that the condition (6) (E1 (n) ≥ 0) is a sufficient condition for the original passivity condition in (2) (E(n) ≥ 0) due to the use of the Cauchy–Schwarz inequality in (4) and due to neglect of the kinetic energy in (3). In addition to the loss of information as well as the half sample delay in the sample and hold, there are several other energy generation factors in the SCV subsystem: computation delay, explicit integration in simulating the VE, nonzero phase lag in position or velocity filters, communication delays when a haptic device is connected to the VE by a network, and the gravity or friction compensation algorithms. They can make the SCV subsystem active; thus, the second term (ΣFd (k)Δxd (k + 1)) in (6) can be negative. Therefore, the actuator force Fd (k) should be carefully bounded to guarantee the passivity condition in (6).
(4)
k =0
where B(k) = b(k)/T is the configuration and is time dependent. It can be assumed to be piecewise constant over kT ≤ t ≤ (k + 1)T . In (4), Δxd (k + 1) = [xd (k + 1) − xd (k)]. Note that B(k) implies energy dissipation capability between sam-
III. FORCE BOUNDING APPROACH This section presents a multi-DOF FBA. Two sufficient conditions (a less and a more conservative sufficient condition) are proposed for satisfying passivity condition in (6). Since the condition in (6) is a sufficient condition for the passivity condition
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in (1), the necessary and sufficient condition satisfying (6) is called the less conservative sufficient condition for the original passivity condition in (1). This less conservative sufficient condition, which is derived in Section III-A, however, usually induces severe contact oscillations due to the energy accumulation before wall contact. Therefore, Section III-B derives a more conservative sufficient condition that can remove those contact oscillations systematically.
then (10) is true. Putting the relation of E1 (n) in (12) into (8) generates the condition in (9). Proposition 2: Existence of Possible Region of Fd (n). Let Fd (i) satisfy (1/4)Fd (i)T B(i)−1 Fd (i) ≤ E1 (i) for all i ࢠ [1, n−1]. Then, Fd (i) always has a possible region satisfying (9) for i = n (i.e., E1 (n) ≥ 0). Proof: When i = 1, assuming Fd (0) = 0 because there is no force before contact, and there always exists a possible region of Fd (1) by (9)
A. Less Conservative FBA The single-DOF passivity condition in (6) can easily be extended to the multi-DOF condition as E1 (n) ≡
n −1
≥ T
Δxd (k + 1) B(k)Δxd (k + 1)
n −1
Fd (k)T Δxd (k + 1) ≥ 0
(7)
k =0
where B(n) is a positive-definite damping matrix that shows energy dissipation capability of the haptic interface in task (endeffector) space (see [17]), and where Δxd (n) is the generalized displacement in task space and Fd (n) is the associated generalized force in task space. Further, (7) can be rewritten as T
E1 (n + 1) = Δxd (n + 1) B(n)Δxd (n + 1) + Fd (n)T Δxd (n + 1) + E1 (n) ≥ 0.
(8)
The actuator force Fd (n) decided at t = nT should guarantee the passivity condition in (8) until the next time step (i.e., t < (n + 1)T ) since the force is maintained and cannot be changed. Note that Δxd (n + 1) is an unknown variable when t = nT . Proposition 1: Less Conservative Sufficient Condition of Fd (n). For all possible Δxd (n + 1), the passivity condition of the multi-DOF haptic system in (8) is satisfied if and only if 1 Fd (n)T B(n)−1 Fd (n) ≤ E1 (n). 4
(9)
Proof: For the positive-definite damping matrix B(n) (i.e., x(n)T B(n)x(n) > 0 for all nonzero vector x(n)), the passivity condition in (9) can be rewritten as 1 y(n)T y(n) + E1 (n) 4 1 − Fd (n)B(n)−1 Fd (n) ≥ 0 4
(10)
where y(n) = 2C(n)T Δxd (n + 1) + C(n)−1 Fd (n)
1 Fd (1)T B(1)−1 Fd (1). 4
(13)
Assume that E1 (i) ≥ (1/4)Fd (i)T B(i)−1 Fd (i) is true for i = n − 1 as
k =0
+
E1 (1) = Δxd (1)T B(0)Δxd (1)
(11)
with C(n) being a lower triangular matrix, which is obtained by the Cholesky decomposition of B(n) (i.e., B(n) = C(n)C(n)T ). Since the first term y(n)T y(n) in (10) is nonnegative for all possible Δxd (n + 1), if the following condition is true 1 (12) E1 (n) − Fd (n)T B(n)−1 Fd (n) ≥ 0 4
E1 (n − 1) ≥
1 Fd (n − 1)T B(n − 1)−1 Fd (n − 1). 4
(14)
Then, the combination of (8) and (14) gives E1 (n) ≥ Δxd (n)T B(n − 1)Δxd (n) + Fd (n − 1)T Δxd (n) 1 + Fd (n − 1)T B(n − 1)−1 Fd (n − 1). (15) 4 Through the Cholesky decomposition of B(n) (i.e., B(n) = C(n)C(n)T ), (15) can be rewritten as 1 y(n − 1)T y(n − 1) ≤ E1 (n) 4
(16)
since yT (n − 1)y(n − 1) ≥ 0, E1 (n) ≥ 0. Therefore, there al ways exists a possible region of Fd (n) for all n. In order to make a haptic system passive, the actuator force should satisfy the condition in (9) for all n. In other words, the actuator force in the current step n may be upper-bounded by the total energy that was accumulated until the previous stop n − 1. If the desired actuator force from the VE (Fe (n)) does not satisfy the condition in (9), to enforce the passivity condition in (9), the magnitude of the actuator force along the generalized force direction needs to be bounded as ud (n)T B(n)−1 ud (n) fd (n)2 ≤ E1 (n) 4
(17)
where fd (n) and ud (n) are the magnitude and the unit vector of the nonzero actuator force (i.e., Fd (n) = fd (n)ud (n)), respectively. Note that the condition in (17) can ensure the passivity without distortion of the desired force direction. This approach, therefore, does not use the projection method that was proposed in the previous approach, where the EBA (see [12]) was applied in each coordinate direction and then the resultant force vector was projected into the desired force direction. Instead, the proposed multi-DOF approach applies the FBA directly in the direction of the force and torque vectors so that direction transparency is always preserved and the magnitude transparency may also be increased.
KIM et al.: FORCE BOUNDING APPROACH FOR MULTI-DEGREE-OF-FREEDOM HAPTIC INTERACTION
The following control and bounding laws must then be used from (17) as Control Law: Fd (n) = fd (n)ud (n).
(18)
Bounding Laws: if (fe (n) > fd m ax (n)),
fd (n) = fd m ax (n),
else if (fe (n) < fd m in (n)),
fd (n) = fd m in (n),
n −1
β(k)Δx2d (k + 1) +
k =0
(19)
where fe (n) is the desired force magnitude in the direction of the desired generalized force direction and E1 (n) = Δxd (n)T B(n − 1)Δxd (n) + Fd (n − 1)T Δxd (n) + E1 (n − 1) fd m ax (n) = 4E1 (n)/α (n) fd m in (n) = − 4E1 (n)/α(n).
essary to avoid severe oscillatory behaviors during specific time periods. A simple way to derive the memoryless type of force condition for passivity is to systematically ignore the previous remaining dissipation. To do so, we may begin with the following singleDOF passivity condition instead of (6) as E2 (n) ≡
fd (n) = fe (n),
else
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(20)
n −1
Fd (k)Δxd (k + 1) ≥ 0
k =0
(22) where 0 < β(k) ≤ B(k) for all k ∈ [0, n − 1]. Note that condition (22) is sufficient for condition (6) since E1 (n) ≥ E2 (n). Note also that conditions (22) and (6) become exactly the same as each other when we choose β(k) = B(k) for all k ∈ [0, n − 1]. For multi-DOF haptic interaction, the passivity condition in (22) is rewritten as E2 (n + 1) = Δxd (n + 1)T β(n)Δxd (n + 1) + Fd (n)T Δxd (n + 1) + E2 (n) ≥ 0 (23)
(21)
In (21), a scalar value α(n) = ud (n)T B(n)−1 ud (n), which is configuration and time dependent, denotes a directional dissipation capability in the direction of the ud (n) vector. Note that E1 (n) can be computed recursively by (8) and represents a total sum of energy that is a combination of the dissipative energy from (Δxd (n + 1)T B(n)Δxd (n + 1)) and the potentially active energy from (Fd (n)T Δxd (n + 1) during all the interaction time, up to time n. B. More Conservative FBA Even though the less conservative sufficient condition in (9) guarantees the passivity condition, accumulation of past remaining dissipation that is represented as the first term (ΣΔxd (k + 1)T B(k)Δxd (k + 1)) in E1 (n) of (7) may induce contact oscillations. This accumulated energy induces a memory effect that may deteriorate contact realism. If, for example, a user moves in free space for a long time before contacting a virtual object, the accumulated energy may have a larger value before contact. Upon contact, the accumulated dissipation capability allows active behavior until it is counterbalanced by the generated energy. Note that this memory effect is a common problem of all the time-domain passivity-based approaches. This memory effect is different from that in PO/PC (see [3]), where the energy is accumulated only in the discrete virtual world. This memory effect may induce severe oscillations (that will be shown in the experimental results in the later section) and thus may not be acceptable for some applications requiring realistic contact transparency such as in medical simulators. A simple resetting of the accumulated energy before the contact (PO/PC [3]) may be complicated for the multipoint contact cases because of difficulties of checking exact contacts. The strategy having upper limits for the accumulated energy and/or exchanging the extra energy (PSPM [7]) requires careful tuning of related parameters that may not be an easy job for multi-DOF cases. A systematic resetting of the past accumulated energy, therefore, may be nec-
where β(n) = ξ(n)B(n) and 0 < ξ(n) ≤ 1 for all n. Proposition 3: More Conservative Sufficient Condition of Fd (n). For all possible Δxd (n + 1), the passivity condition of the multi-DOF haptic system in (23) is satisfied if Fd (n)T B−1 (n)Fd (n) ≤
ξ(n) Φ(n)T Φ(n) ξ(n − 1)
(24)
where Φ(n) = 2ξ(n − 1)C(n − 1)T Δxd (n) + C(n − 1)−1 Fd (n − 1)
(25)
for all interaction time n. Proof: Similar to Proposition 1 and its proof, it can easily be shown that for all possible Δxd (n + 1), the condition in (23) is satisfied if and only if n −1
ξ(k)Δxd (k + 1)T B(k)Δxd (k + 1)
k =0
+
n −1
Fd (k)T Δxd (k + 1)
k =0
≥
1 Fd (n)T B(n)−1 Fd (n). 4ξ(n)
(26)
The right side of (26) can be rewritten as n −1
ξ(k)Δxd (k + 1)T B(k)Δxd (k + 1)
k =0
+
n −1
Fd (k)T Δxd (k + 1)
k =0
≥
1 Fd (n)T B(n)−1 Fd (n). 4ξ(n)
(27)
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Then, the condition in (26) can be rewritten as n −1
Φ(k + 1)T Φ(k + 1) 4ξ(k) k =0 Fd (k + 1)T B(k + 1)−1 Fd (k + 1) − ≥ 0. 4ξ(k + 1)
(28)
If the actuator force Fd (i) satisfies the following condition for each i ࢠ [1, n]: 1 1 Φ(i)T Φ(i) ≥ Fd (i)T B(i)−1 Fd (i) ξ(i − 1) ξ(i)
(29)
then the bracket term of (28) has a nonnegative value for each k, and thus, the condition in (26) is satisfied. Note that there always exists a possible region of Fd (n) for satisfying the more conservative sufficient condition (24) since the right side of (24) always has a nonnegative value because ξ(i) > 0 and Φ(i)T Φ(i) > 0 for all i ࢠ [1, n]. The sufficient condition of Fd (n) in (24) provides a more conservative force region guaranteeing the passivity condition in (8) because of two facts: 1) The smaller dissipation β(n) ࢠ (0, B(n)] may be used to satisfy passivity condition in (23) instead of using the full dissipation B(n), and 2) condition (24) makes each term of (28) nonnegative to make the total sum nonnegative. The following control and bounding laws are, therefore, necessary in order for the actuator force Fd (n) decided at t = nT to guarantee the passivity condition until the next time step: Control Law: Fd (n) = fd (n)ud (n).
(30)
Bounding Laws: if (fe (n) > fd m ax (n)),
fd (n) = fd m ax (n),
else if (fe (n) < fd m in (n)),
fd (n) = fd m in (n),
(31)
fd (n) = fe (n),
else where ξ(n) =
ξ(n − 1)α(n)fe2 (n) for fe (n) = 0 Φ(n)T Φ(n) if (ξ(n) > 1)
ξ(n) = 1
(32)
(33)
ξ(n) Φ(n)T Φ(n) α(n)ξ(n − 1) ξ(n) Φ(n)T Φ(n). fd m in (n) = − α(n)ξ(n − 1)
fd m ax (n) =
(34)
In the above bounding rule, the required dissipation capability ξ(n) in (32) is computed first to display the desired force magnitude fe (n) transparently. If the required dissipation β(n)(= ξ(n)B(n)) exceeds the available dissipation capability B(n), then β(n) is limited to B(n) as in (33). The initial value of ξ(0) can be set to 1.
Fig. 2.
Custom-built 1-DOF haptic device.
IV. EXPERIMENTS This section presents experimental results to show that the proposed algorithm can enhance the haptic interaction transparency while guaranteeing robust stability. The experiments are performed first with a single-DOF haptic device for showing magnitude transparency and then with a multi-DOF haptic device for showing direction transparency. A. Virtual Wall 1) Experimental Setup: Experiments are performed with a custom-built single-DOF haptic device in Fig. 2. It has a maxon RE40 DC motor and an encoder with a resolution of 8000 pulses/rev, resulting in position resolution of 15 μm. The torque constant of the motor is 30.2 mN·m/A and the torque is amplified 12 times by a capstan mechanism which use cable transmission instead of gear for back-drivability. The proposed approach is implemented using the Microsoft Visual Studio 2008 operating in Windows 7. A sampling rate of about 1 kHz is used by the multimedia time clock, in which the sampling time is not exact. The physical damping (b) is identified as 0.03 N·s/mm; therefore, B = b/T = 30 N/mm. Three different VEs are considered: virtual walls with small stiffness, very large stiffness, and a large time-delay. The following virtual wall model is used: K{xd (n) − xwall }, for xd (n) ≥ xwall (35) Fe (n) = 0, for xd (n) < xwall where xwall is the position of the virtual wall that is located at xwall = 0 mm. 2) Small Stiffness Case: Fig. 3 shows the experimental results for various values of the virtual wall stiffness when no control law is applied (i.e., Fd (n) = Fe (n)). Up to the virtual wall stiffness of about 60 N/mm, the interaction is stable as seen in Fig. 3(a), (c), and (e). For higher virtual wall stiffness of 80 N/mm, the interaction is definitely unstable as seen in Fig. 3 (b), (d), and (f). Figs. 4 and 5 show the experimental results for the soft virtual wall stiffness (K = 50 ≤ 60 N/mm) with less [see (18)–(21)] and more [see (30)–(34)] conservative sufficient conditions, respectively. The displayed stiffness Fd (n)/{xd (n)–xwall } in
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Fig. 3. Experimental results for various stiffness values. (a) Position-60 N/mm. (b) Position-80 N/mm. (c) Force-60 N/mm. (d) Force-80 N/mm. (e) Energy-60 N/mm. (f) Energy-80 N/mm. Fig. 5. Experimental results for small stiffness (K = 50 N/mm) with a more conservative sufficient condition. (a) Position. (b) Force. (c) Energy. (d) Displayed stiffness. (e) βn.
Fig. 4. Experimental results for small stiffness (K = 50 N/mm) with a less conservative sufficient condition. (a) Position. (b) Force. (c) Energy. (d) Displayed stiffness.
Figs. 4(d) and 5(d) shows that both conditions display transparently the full stiffness of K = 50 N/mm because no force bounding occurs. For a more conservative condition, the initial value of β(n) is selected as B(n) as shown in Fig. 5(e), since B(n) is the maximum value satisfying the passivity condition of the haptic interaction system. After contact with the soft virtual wall, this value is updated in (20), which is smaller than B(n) as shown in Fig. 5(e). Note that β(n) is mostly constant before contact and at the steady state except during the short initial contact periods. Therefore, there is no confusion on the wall stiffness perception. 3) Large Stiffness Case: Fig. 6 shows the experimental results for the large virtual wall stiffness of K = 150 N/mm when the less conservative sufficient condition [see (18)–(21)] is used. During the free-space movement (0.393 ≤ t ≤ 0.648 s), the energy E1 (n) is accumulated and increases monotonically as shown in Fig. 6(c). For some duration (0.648 ≤ t ≤ 0.854 s) after the initial contact, severe oscillations that can be clearly felt are observed as shown in Fig. 6(e). In this period, E1 (n) further increases until Fe (n) reaches the Fm ax (n) in (21) and then starts to decrease to a steady state value. This phenomenon may be explained as follows: upon contact, the computed force Fm ax (n) in (21) that satisfies the less sufficient condition may be large due to the accumulated energy during the free motion.
Fig. 6. Experimental results for large stiffness (K = 150 N/mm) with a less conservative sufficient condition. (a) Position. (b) Force. (c) Energy. (d) Displayed stiffness. (e) Oscillation data.
The large computed force, therefore, allows the desired environmental force Fe (n) to pass as seen in the bounding law in (19). During this period, the user motion may be sprung back to the free space by the large reaction force from the wall. Then, the user tries to contact the wall implicitly. After contact, the energy flow-in (ΣFd (k)Δxd (k + 1)) in (5) becomes negative, meaning that energy generated by the sampling process. The positive dissipative energy E1 (n) that had been accumulated during the free motion then decreases not monotonously but sinusoidally as shown in the magnified view in Fig. 6(e) until the accumulated dissipative energy is counterbalanced by the generated negative energy. Because the large accumulated value of E1 (n) allows larger contact forces that still satisfy the passivity condition, the displayed stiffness may be high up to the virtual wall stiffness K = 150 N/mm during the oscillatory motion as shown in
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Fig. 10. Experimental results for stiffness (K = 10 N/mm) with a one-way time delay (50 ms) and more conservative sufficient condition. (a) Position. (b) Force. (c) Energy. (d) Displayed stiffness.
Fig. 7. Experimental results for large stiffness (K = 150 N/mm) with a more conservative sufficient condition. (a) Position. (b) Force. (c) Energy. (d) Displayed stiffness. (e) Energy.
Fig. 11.
Fig. 8. Experimental results for stiffness (K = 10 N/mm) with a one-way time delay (50 ms). (a) Position. (b) Force. (c) Energy.
Fig. 9. Experimental results for stiffness (K = 10 N/mm) with a one-way time delay (50 ms) and less conservative sufficient condition. (a) Position. (b) Force. (c) Energy. (d) Displayed stiffness.
Fig. 6(d). After the remaining dissipation is fully compensated by the generated energy, the displayed stiffness is reduced to a certain constant value at the steady state. It is interesting to note that the less conservative sufficient condition can provide a much larger stable contact stiffness than the value (60 N/mm) that is predicted from the Colgate’s passivity condition (K ≤ 2b/T ) [1]; thus, a much stiffer wall [about 93 N/mm at the steady state as shown in Fig. 6(d)] can be displayed passively than the
Experimental setup. (a) PHANToM Omni. (b) Virtual circle.
experimental results (60 N/mm) in Fig. 3. Note, however, that ensuring passivity condition does not mean an oscillation-free contact as evidenced by the experiments, which requires a more conservative passivity condition for eliminating the energy that had been accumulated during the free motion. This nonoscillating contact is definitely required for transparent contact realism. Fig. 7 shows the experimental results for the large virtual wall stiffness K = 150 N/mm when the more conservative sufficient condition [see (30)–(34)] is satisfied. The more conservative sufficient condition does not contain the memory effect, and thus, active behavior is rarely observed as shown in Fig. 7(a) and (b). The displayed stiffness is about K = 60 N/mm in the steady state as shown in Fig. 7(d), which clearly shows a more conservative stiffness value than that from the less conservative sufficient condition in Fig. 6(d). The dissipation capability β(n) is upper-bounded to B(n) for all n because of a very stiff wall contact. The energy E3 (n) in Fig. 7(c) shows that it becomes a constant value, meaning no further accumulation after contact. Note that the nonzero energy, until contact, is the accumulated energy by ΣB(k)Δx2d (k + 1) in free space motion. The proposed more conservative condition may deteriorate the magnitude transparency in the steady state if the stiffness of the virtual wall is very high but increases significantly the initial contact realism (i.e., oscillation-free contact). If the virtual wall stiffness is not high enough so that no bounding occurs, then 100% magnitude transparency is guaranteed in the proposed approach. The magnitude transparency degradation may also not be important for some virtual wall simulation since human may not feel very high stiffness beyond the perception capability. This magnitude transparency may be significantly improved by
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Fig. 14. Experimental results for large stiffness (K = 10 N/mm) with a less conservative sufficient condition. (a) Position. (b) Force. (c) Energy. (d) Displayed stiffness.
Fig. 12. Experimental results for virtual circle (K = 10 N/mm) with a less conservative sufficient condition. (a) Position. (b) Force. (c) Energy (x) (d) Energy (y). (e) Total energy (f) Displayed stiffness.
B. Multi-DOF 1) Experimental Setup: Experimental results are presented with a commercial 3-DOF haptic device, the PHANToM Omni in Fig. 11(a), whose basic specifications are a maximum force output of 3.3 N and an encoder resolution of 0.055 mm at the nominal position. In the multi-DOF haptic interaction scenario, a person should trace either a circular trajectory in the inner side of the circle in the vertical plane or a spherical trajectory of the outer surface of a sphere while keeping contact as firmly as possible. For contact simulation in the VE, virtual objects are modeled as a simple virtual spring as Fe (n) =
Fig. 13. Experimental results for virtual circle (K = 10 N/mm) with a more conservative sufficient condition. (a) Position. (b) Force. (c) Energy. (d) Displayed stiffness.
additional physical damping as suggested by Mehling et al. [18], Gosline and Hayward [19], and Lim et al. [20]. 4) Time Delay Case: One important application of a haptic interaction system is to simulate with time-delayed VEs that may occur in the networked haptic VEs or in computationintensive medical simulations. Fig. 8 shows the experimental results with a virtual spring (K = 10 N/mm) and a constant one-way time delay of 50 ms. It clearly shows that the time-delayed VE is active as seen in Fig. 8(c), which results in unstable behavior without any stability control algorithms. On the other hand, Figs. 9 and 10 show the stabilized experimental results for the time-delayed VEs using the less and more conservative sufficient conditions, respectively. These results show basically the same behaviors as those with large stiffness.
Kd(n),
for rd (n) ≥ rwall
0,
for rd (n) < rwall
(36)
where the penetration depth d(n) = rd (n) –rwall is along the normal direction on the contact surface, and rwall is the radius of the virtual objects: 40 mm for a circle and 20 mm for a sphere. In general, B(n) is configuration-dependent (see [17]) because the sources of physical damping are mainly from the backelectromagnetic force (EMF) of motor torques in the joint space, viscous damping in the mechanical joints, etc. Each equivalent physical damping value of a diagonal element can easily be measured in the Cartesian task space by the method (see [16]). Note that the constant equivalent physical damping values are different for different axes in the workspace of a haptic device. For uniform surface contact feeling along each Cartesian axis, the smallest values are used in the experiments. In the case of a circle, the equivalent physical damping values (bx , by ) in both Cartesian axes shown in Fig. 11(a) are selected as 1.0 N·s/m. In the case of a sphere, however, the equivalent physical damping values are all selected as 0.5 N·s/m. This value is smaller than that in the case of the circle because, in the case of a sphere, the third elbow motor that contribute z-axis damping mainly comes into play and it has the lesser dissipation capability than the first torso and the second shoulder motors.
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TABLE I PERFORMANCE COMPARISON FOR VIRTUAL CIRCLE
var(F d ) mean(K d i s p )
FBA with Less conservative
FBA with more conservative
1.63 6.61
0.13 5.80
TABLE II PERFORMANCE COMPARISON FOR VIRTUAL SPHERE
var(F d ) mean(K d i s p )
FBA with Less conservative
FBA with more conservative
1.75 3.62
0.23 3.47
Fig. 15. Experimental results for large stiffness (K = 10 N/mm) with a more conservative sufficient condition. (a) Position. (b) Force. (c) Energy. (d) Displayed stiffness.
V. CONCLUSION AND FUTURE WORKS 2) Virtual Circle With the Large Stiffness Case: Fig. 12 shows the experimental results with the proposed less conservative multi-DOF FBA for the virtual circle stiffness of K = 10 N/mm that is larger than the stably displable stiffness (K = 2 N/mm) without no stabilizing controllers. The total energy in Fig. 12(e) shows that the proposed multi-DOF FBA really makes the haptic interaction system passive by maintaining the total energy positive even though the energy in some axes (e.g., the x-axis) may not be positive as shown in Fig. 12(c). The direction transparency is perfect in that the associated force directions all point toward the center of the circle as shown in Fig. 12(b). Some oscillations, however, are felt as seen in Fig. 12(a) because of the memory effect that was explained in Section III-B. Fig. 13 shows the corresponding experimental results for the same virtual circle with the proposed more conservative multiDOF FBA. The position and displayed stiffness show no oscillations as shown in Fig. 13(a) and (d). There is also no distortion of direction transparency as shown in Fig. 13(b). The mean displayed stiffness in this case is 5.80 N/mm from Fig. 13(d), which is smaller than that from the less conservative multi-DOF FBA. 3) Virtual Sphere With the Large Stiffness Case: Fig. 14 shows the experimental results for the virtual sphere stiffness of K = 10 N/mm with the proposed less conservative multiDOF FBA. The direction transparency is also perfect as shown in Fig. 14(b). Some oscillations, however, are felt as seen in Fig. 14(a) even though the total energy in Fig. 14(c) shows that the proposed less conservative multi-DOF FBA makes the haptic interaction system passive. Fig. 15 shows the experimental results for the same virtual sphere with the more conservative sufficient condition. In this case, direction transparency shows also no distortion as shown in Fig. 15(b). Moreover, there are no oscillations as shown in Fig. 15(a) and (d). Tables I and II show performance comparison for circles and spheres. The mean displayed stiffness is slightly smaller, but the force variance is much smaller in the more conservative case. The more conservative multi-DOF FBA is, therefore, recommended for interaction with multi-DOF virtual objects.
This paper proposed FBAs for the stable haptic interaction with general 3-D virtual objects. Based on the passivity condition for sampled-data haptic systems, less and more conservative sufficient conditions for the force that can be passively displayed are derived. In order to systematically remove the memory effect due to the accumulation of past remaining dissipation in the less conservative approach, the more conservative multi-DOF FBA was necessary. The proposed approaches made perfect directional transparency while guaranteeing robust stability. A few points need to be investigated in future. 1) The current examples used only a constant damping matrix B. Since this matrix is, in general, configuration dependent and time varying and since a constant penetration depth cannot be maintained in the penalty-based haptic rendering methods, the surface of a 3-D virtual object may not be felt as a uniform surface that has the same stiffness because the force magnitude fd (n) in (31) is changing over space and time. This problem is a common problem to most existing methods. Research efforts that can make uniform feeling for uniform surfaces will, therefore, be investigated. 2) The proposed approaches will be applied to a teleoperation system since a teleoperation system can be viewed as a combination of two haptic interaction systems as in [21] due to analogy between the haptic and teleoperation systems. REFERENCES [1] J. E. Colgate, M. C. Stanley, and J. M. Brown, “Issues in the haptic display of tool use,” in Proc. IEEE/RSJ Conf. Int. Conf. Intell. Robot. Syst., 1995, pp. 140–145. [2] B. Hannaford and J. H. Ryu, “Time domain passivity control of haptic interfaces,” IEEE Trans. Robot. Autom., vol. 18, no. 1, pp. 1–10, Feb. 2002. [3] J. H. Ryu, Y. S. Kim, and B. Hannaford, “Sampled- and continuoustime passivity and stability of virtual environments,” IEEE Trans. Robot., vol. 20, no. 4, pp. 772–776, Aug. 2004. [4] J. P. Kim and J. Ryu, “Energy bounding algorithm based on passivity theorem for stable haptic interaction control,” in Proc. IEEE Symp. Hapt. Interf. Virt. Env. Teleop. Syst., 2004, pp. 351–357. [5] J. P. Kim and J. Ryu, “Robustly stable haptic interaction control using an energy-bounding algorithm,” Int. J. Robot. Res., vol. 29, no. 6, pp. 666–679, Apr. 2010. [6] K. Lee and D. Y. Lee, “Adjusting output-limiter for stable haptic rendering in virtual environments,” IEEE Trans. Control Syst. Technol., vol. 17, no. 4, pp. 768–779, Jul. 2009.
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[7] D. Lee and K. Huang, “Passive-set-position-modulation framework for interactive robotic systems,” IEEE Trans. Robot., vol. 26, no. 2, pp. 354– 369, Apr. 2010. [8] M. Franken, S. Stramigioli, S. Misra, C. Secchi, and A. Maccahelli, “Bilateral telemanipulation with time delays: A two-layer approach combining passivity and transparency,” IEEE Trans. Robot., vol. 27, no. 4, pp. 741– 756, Aug. 2011. [9] C. Preusche, G. Hirzinger, J. H. Ryu, and B. Hannaford, “Time domain passivity control for six degrees of freedom haptic displays,” in Proc. IEEE/RSJ Int. Conf. Intell. Robot. Syst., 2003, pp. 2944–2949. [10] K. Hertkorn, T. Hulin, P. Kremer, C. Preusche, and G. Hirzinger, “Time domain passivity control for multi-degree of freedom haptic devices with time delay,” in Proc. IEEE/RSJ Int. Conf. Robot. Autom., 2010, pp. 1313– 1319. [11] J. Kim, C. Seo, and J. Ryu, “Six degree-of-freedom energy bounding algorithm for stable and directionally transparent haptic interaction,” in Proc. Int. Conf. Cont. Autom. Syst., 2008, pp. 260–265. [12] J. Kim, J. P. Kim, C. Seo, and J. Ryu, “A directionally transparent energy bounding approach for multiple degree-of-freedom haptic interaction,” Int. J. Cont. Autom. Syst., vol. 8, no. 2, pp. 352–360, Apr. 2010. [13] J. H. Ryu and M.-Y. Yoon, “Memory-based passivation approach for stable haptic interaction,” IEEE/ASME Trans. Mechatronics, vol. 19, no. 4, pp. 1424–1435, Aug. 2014. [14] J. P. Kim, S. Y. Baek, and J. Ryu, “A force bounding approach for stable haptic interaction,” in Proc. IEEE World. Hapt. Conf., 2011, pp. 397–402. [15] J. J. Abbott and A. M. Okamura, “Effects of position quantization and sampling rate on virtual-wall passivity,” IEEE Trans. Robot. Autom., vol. 21, no. 5, pp. 952–964, Oct. 2005. [16] J. P. Kim, C. Seo, and J. Ryu, “Equivalent damping estimation for stable haptic interaction,” (in Korean), Kor. Robot. J., vol. 1, no. 2, pp. 135–139, Dec. 2006. [17] B. E. Miller, J. E. Colgate, and R. A. Freeman, “On the role of dissipation in haptic systems,” IEEE Trans. Robot., vol. 20, no. 4, pp. 768–771, Aug. 2004. [18] J. S. Mehling, J. E. Colgate, and M. A. Peshkin, “Increasing the impedance range of a haptic display by adding electrical damping,” in Proc. IEEE World. Hapt. Conf., 2005, pp. 257–262. [19] A. H. C. Gosline and V. Hayward, “Eddy current brakes for haptic interfaces: Design, identification, and control,” IEEE/ASME Trans. Mechatronics, vol. 13, no. 6, pp. 669–677, Dec. 2008. [20] Y. A. Lim, H-S. Ahn, and J. Ryu, “An analog input shaper for haptic interfaces,” IET Cont. Theory. Appl., vol. 3, no. 12, pp. 1553–1564, Dec. 2009. [21] C. Seo, J. P. Kim, J. Kim, H. S. Ahn, and J. Ryu, “Robustly stable bilateral teleoperation under time-varying delays and data losses: energy-bounding approach,” J. Mech. Sci. Tech., vol. 28, no. 8, pp. 2089–2100, Aug. 2011.
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Jong-Phil Kim received the B.S. degree in mechanical engineering from the Korea Advanced Institute of Science and Technology, Daejeon, Korea, in 1998, and the M.S. and Ph.D. degrees from Department of Mechatronics, Gwangju Institute of Science and Technology, Gwangju, Korea, in 2000 and 2007, respectively. He is currently a Postdoctoral Research Fellow in the Image Media Research Center, the Korea Institute of Science and Technology, Seoul, Korea. His research interests include haptics, robotics, and teleoperation.
Sang-Yun Baek (S’14) received the B.S. degree from the Department of Mechanical Systems Engineering, Chonnam National University, Gwangju, Korea, in 2010, and the M.S. degree from the School of Mechatronics, Gwangju Institute of Science and Technology (GIST), Gwangju. In 2012, he joined the Human Robotics Lab at GIST, Gwangju, Korea, as a Ph.D. student. His research interests include mechanical design, haptics, robotics, and teleoperation.
Jeha Ryu (M’10) received the B.S. degree from Seoul National University, Seoul, Korea, the M.S. degree from the Advanced Institute of Science and Technology, Seoul, and the Ph.D. degree from the University of Iowa, Iowa City, IA, USA, in 1982, 1984, and 1991, respectively, all in mechanical engineering. He is currently a Professor in the School of Mechatronics, Gwangju Institute of Science and Technology, Gwangju, Korea. He has published and presented more than 150 research articles and reports. His research interests include haptic interaction control, haptic modeling and rendering, haptic application for various multimedia systems, and teleoperation. Prof. Ryu is a Member of the American Society of Mechanical Engineers, Korean Society of Mechanical Engineers, and the Korean Society of Automotive Engineers.
