ICCAS 2001
International Conference on Control, Automation and Systems
On the Design Method of a Haptic Interface Controller with Virtual Coupling Keehoon Kim W. K. Chung Y. Youm Dept. of Mechanical Engineering, Pohang University of Science and Technology, Pohang, Korea Tel : +82-54-279-2844, Fax : +82-54-279-5899, E-mail : {khk,wkchung,yyoum}@postech.ac.kr )
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Abstract: A haptic interface can be a passive system with virtual coupling as a filter. Virtual coupling has been designed for satisfying passivity. However, it affects transparency of haptic interface as well as stability. This paper suggests new design criterion of a haptic interface controller by considering transparency. As a result, sampling time and the range of impedance or admittance should be considered as well as virtual coupling for desired performance of haptic display. And experiments show that the suggested design criterion can be applied successfully for desired performance. Keywords: haptic device, haptic interface, virtual coupling *
-vde
vh
+
+
human haptic * operator Fh device Fde
Real World (Continuous Time)
virtual environment
vh
-
Fig. 1. Interaction between an operator and virtual reality in the ideal case
*
*
vdh
vdh
+ human haptic operator Fh device Fdh
Fdh
*
CO
-
Virtual World (Discrete Time)
ZOH
-
-vde
virtual + virtual * environcoupling Fde ment -
Fig. 2. Structure of haptic interface and information interaction 2. Haptic Interface for Stability
1. Introduction
T
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O
N
O
In recent days, a haptic interface becomes a new application of tele-operation. It can be assumed to have no time delay. However, since the virtual reality is generated in discrete world and a human operator feels the virtual environment in continuous world, the transmission of information between them is distorted. In this condition, the stability of haptic interface can not be guaranteed. Generally, the stability is guaranteed by using passivity theory. In passivated haptic interface, there is always critical tradeoff between sampling rate, virtual wall stiffness, and viscosity of haptic device[1]. The Z-width which is the dynamic range of achievable impedance was introduced[2] and it represents the range of impedance when the passivity is guaranteed. The virtual coupling was also used as a bridge that connects a haptic display and virtual environment[3]. However, virtual coupling affects the transparency while it should be used for the passivity of a haptic interface. In other word, the design methods for transparency has not been considered. This paper presents the relation among design factors for transparency. Transparency can be expressed as the rising time to achieve impedance of virtual environment and the range of expressible impedance within fixed time. When a haptic interface is developed by using virtual coupling, the response time is determined by the virtual coupling, the generated environment impedance, and sampling rate. This can be a criterion for desired dynamic response when virtual coupling is used. Especially, it is useful when the impedance of environment has wide range.
October 17-21, 2001, Cheju National Univ. Jeju, Korea
Fh ∗ −vde
=
Zh =
Fh vh
0 −1 =
1 0
∗ Fde ∗ vde
vh ∗ Fde
(1)
= Ze
A haptic system consists of an operator, virtual reality, and a haptic interface. In Figure 1, impedance control at an operator side and admittance control at virtual reality are applied. An operator commands vh and virtual ∗ reality accepts it as desired velocity, vde . Virtual real∗ ity returns reaction force, Fde , and an operator feels Fh . Eq.(1) represents an ideal haptic system as shown in Figure 1. In Eq.(1), the commanded velocity and reaction force are transmitted perfectly and the impedance of an operator, Zh , is coincident with the impedance of virtual reality, Ze . However, an ideal haptic system can not be realized since information is always distorted between continuous time and discrete time domain. In other words, the information distortion is inevitable because of digitizer and zero order hold which connect the two sides. Virtual coupling is needed in order to overcome the problem. Figure 2 shows impedance haptic display. Fh and vh mean reaction force from haptic device and velocity commanded ∗ ∗ from an operator. vde and Fde are desired velocity transmitted to virtual reality and desired reaction force to an ∗ operator. Asterisk means discrete value. vdh is discrete value of vdh through digitizer and Fdh is continuous value ∗ through zero order hold, ZOH. Virtual coupling of Fdh ∗ makes desired velocity to virtual reality, vde , and desired ∗ ∗ ∗ force to display, Fdh with vdh and Fdh . 2.1. Virtual Coupling for Admittance Display Admittance haptic display means that an operator commands force to virtual reality and velocity of virtual reality returns through a haptic device. Though Conference Web Site: http://www.iccas.org
*
*
Fdh Fh
+ -
haptic device
Fd KPI
vh
Fde
*
vdh
+ +
virtual coupling +
ZOH
-
vde
*
continuous time
CA1
CA4
*
* vdh
CA2
Fdh
discrete time
Fig. 3. Admittance display
* -vde
CA3
+ +
*
Fde
Fig. 4. Virtual coupling for admittance display
this method usually needs force sensor, it is more realistic because a human being controls an object by using admittance display. In Figure 3, force information is transmitted to virtual reality and velocity information comes to an operator through a haptic device. Therefore, in order to control a haptic device, a controller is needed. KP I means PD controller. By using the controller, a haptic device can be in desired position.
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Following condition should be satisfied to guarantee the passivity of Eq.(10). Re(CA1 (z)) ≥ 0 (11) 1 Re ZOH(z)G(z)CA4 (z) + Zd (z)+KP I (z) ≥ 0 (12) 1 2Re(CA1 (z))Re ZOH(z)G(z)CA4 (z) + Zd (z)+K ≥ P I (z) 2 CA2 (z)+ZOH(z)G(Z)CA3 (z) ∀ω ≥ 0 (13) 2
Kp + Kd |s=(z−1)/T z (2) Stability of the suggested admittance haptic display is s guaranteed by how to design virtual coupling factors, Kp and Kd mean p-gain and d-gain. CA1 (z), CA2 (z), CA3 (z), and CA4 (z). Virtual coupling In Figure 3, if there is no virtual coupling, ∗ has been designed in order to satisfy only passivity in ∗ Fdh 0 1 −vdh = . many papers. However, this equation is only for stability. vh −ZOH(z)G(z) 1/(Zd (z) + KP I ) Fh Transparency as well as stability should be considered for (3) haptic display. Zd (z) means impedance of haptic device.
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KP I (z) =
Zd (z) = (ms + b)|s= 2
.
(4)
3. New Virtual Coupling Design of Admittance Display for transparency In this section, design method of virtual coupling is considered to guarantee transparency as well as stability. From Eq.(10), following shows the relation between admittances of an operator and virtual reality. ZOH(z)KP I (z)CA4 (z) + 1 vh Yh = = Fh Zd (z) + KP I (z) CA2 CA3 ZOH(z)G(z)Ye (14) − 1 + CA1 Ye
T
z−1 T z+1
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A haptic device is modelled by ms + b and m and b mean mass and damping coefficient. ms + b can be converted to discrete time domain model through Tustin’s method. T means sampling time. ZOH is zero order hold which has 90◦ phase lag and unit gain. 1z+1 . 2 z
(5)
N
ZOH(z) =
G(z) is a transfer function which includes a PD controller and zero order hold. KP I (z) , Zd (z) + KP I (z)
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G(z) =
Yh and Ye mean admittance of an operator and virtual reality. In Eq.(14), the first part of right side is determined if haptic device, PD controller, and virtual coupling factors are given. The second part of right side is dependent on admittance of virtual reality, Ye . Therefore, the second part decides transparency with virtual coupling factor.
(6)
To guarantee passivity of Eq.(3),
D
Re(1/(Zd (z) + KP I ) ≥ 0 2 0 ≥ 1−ZOH(z)G(z) 2
(7) ∀ω ≥ 0.
(8)
Eq.(7) is always satisfied since Zd (z) and KP I are always positive. However, Eq.(8)can not be satisfied for all frequency, ω. Therefore, a haptic interface should include virtual coupling for stability. To design generalized virtual coupling, the form of Figure 4 is suggested. ∗ ∗ Fde CA1 (z) CA2 (z) −vde = . (9) ∗ ∗ vdh CA3 (z) CA4 (z) Fdh
D(z)
=
October 17-21, 2001, Cheju National Univ. Jeju, Korea
(15)
D(z) is assumed to converge Ye . The convergence rate depends on the location of poles which mean transparency. bAi (z) (i = 1, 2, 3, 4) aAi (z) aAi (z) and bAi (z) are polynomials. Let CAi =
From Eq.(3) and Eq.(9), the relation between input and output of haptic interface becomes ∗ ∗ Fde CA1 (z) CA2 (z) −vde = (10) −vh ZOH(z)G(z)CA3 (z) X(z) Fh X(z) = ZOH(z)G(z)CA4 (z) + 1/(Zd (z) + KP I )
ZOH(z)KP I (z)CA4 (z) + 1 Zd (z) + KP I (z) CA2 CA3 ZOH(z)G(z)Ye − 1 + CA1 Ye Yh −
,
.
D(z)
=
−
(16)
(z + 1)aA1 (z)bA2 (z)bA3 (z)G(z)Ye (17) 2z(aA1 (z) + bA1 (z)Ye )aA2 (z)aA3 (z) Conference Web Site: http://www.iccas.org
* vdh
* vde k1
c2
*
Fdh
Assumption 1: A haptic device is so stiff that it can not be controlled by an operator. Assumption 2: Control frequency is high enough and PD controller is optimized. Now, Eq.(20) is always satisfied. Stability and transparency can be determined by followings with Assumption 1 and Assumption 2 by which G(z) → 1 and 1 → 0. Zd (z)+KP I (z)
k2 m2
m1 c1 * Fde
Fig. 5. Passive system
2Re(ZAvc (z))Re ZOH(z)
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aA1 (z) + bA1 (z)Ye = 0,
aA2 (z) = 0,
aA3 (z) = 0, and G(z) = 0.
Therefore, D(z) has the best performance when aA2 (z) and aA3 (z) are real constants. CA2 (z) and CA3 (z) should be real constants for causality. Besides, CA2 (z) and CA3 (z) are 1 and −1 in order for coincidence of signs ∗ ∗ ∗ ∗ among Fde , vdh , Fdh , and vde .
(28) (29) (30)
0 In Eq.(28),ZAvc (z) and ZAvc (z) can be designed to satisfy the following for stability. 1 − ZOH(z) 2 jw ≤ 0.25. 2 z=e
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D(z) =
ZAvc (z)0 2 1 − ZOH(z)G(Z) ≥ ∀ω ≥ 0 2 (z + 1)Ye D(z) = 2z(1 + ZAvc(z) Ye )
From Eq.(17), poles of D(z) are on points which satisfy that z = 0,
1
(z + 1)G(z)Ye 2z(1 + CA1 (z)Ye )
(18)
(z + 1)Ye (1/c2 + T /m2 )z − 1/c1 2z (1/c2 + T /m2 + Ye )z − (1/c1 + Ye ) (31) From Eq.(31), DC-gain of D(z) is Ye and convergence rate is determined by (1/c2 + Ye )/(1/c2 + T /m2 + Ye ). It means that a haptic interface is affected by admittance of virtual reality, Ye , and sampling time, T , as well as virtual coupling factors. In other words, there are limits of expressible admittance and increasing sampling time does not mean better performance without exception. The relation among sampling time, the range of expressible admittance, and virtual coupling factors will be explained. Though admittance of virtual reality changes from 0 to infinity randomly, admittance is assumed to be fixed to clarify the relation. Eq.(31) can be derived as the following. D(z) =
Eq.(18) should satisfy the followings for stable haptic interface.
O
T
Re(CA1 (z)) ≥ 0 (19) 1 Re ZOH(z)G(z)CA4 (z) + Zd (z)+K ≥0 (20) P I (z) 1 2Re(CA1 (z))Re ZOH(z)G(z)CA4 (z) + Zd (z)+K ≥ P I (z) 2 1−ZOH(z)G(Z) ∀ω ≥ 0 (21) 2
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A passive system as shown in Figure 5 is considered in order to satisfy Eq.(19). ∗ ∗ ∗ ∗ (Fdh − Fde − m1 svde ) k21/s + c12 + m12 s = vde (23) ∗ ∗ ∗ Fdh = (vdh − vde ) ks1 + c1 (24)
∗ Fde
=
(25)
∗ vdh
=
(26)
D
Let m1 = 0 since CA2 (z) = 1 and CA3 (z) = −1. Besides, ∗ ∗ if k2 = 0 in which Fde is not fluctuated when Fdh = 0, ∗ ∗ Fdh − ZAvc (z)vde 1 ∗ ∗ vde + 0 Fdh ZAvc
(22)
1/c2 + Ye D[n − 1] 1/c2 + Ye + T /m2 Ye T /m2 = 1/c2 + Ye + T /m2 n 1/c2 + Ye D[n] = Ye 1 − (32) 1/c2 + Ye + T /m2 D[n] −
T /m2 1/c2 + Ye N T means the time in which D(z) reaches Let QA =
where
ZAvc (z)
=
YAvc (z)
=
ZAvc (z)0
=
1 YAvc (z) 1 1 + |s=(z−1)/T z c2 m2 s k1 + c1 |s=(z−1)/T z s
(1 − X )Ye from 0 when admittance of virtual reality is Ye . (27)
For example, if D(z) reaches 95% of Ye in 0.1(sec) and sampling time is 0.01, N = 10 and X = 0.05.
From Eq.(25) and Eq.(26), CA1
=
CA4
=
ZAvc (z) 1 ZAvc (z)0
October 17-21, 2001, Cheju National Univ. Jeju, Korea
1 1 + QA
N ≤X − 1
QA ≥ X N − 1
(33)
Conference Web Site: http://www.iccas.org
Fig. 6. c2 = 100, m2 = 0.01, and T = 0.005
Fig. 7. c2 = 100, m2 = 0.1, and T = 0.005
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QA has lower limit when N and X are determined in Eq.(33). ln X (34) N ≥ ln(1/(1 + QA ))
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Eq.(34) shows that more than N is needed to converge (1 − X )Ye when virtual coupling factors are determined. − 1 T /m2 ≥ (1/c2 + Ye ) X N − 1 (35) Virtual coupling factor, m2 , and sampling time, T should be determined as shown in Eq.(35) to converge (1 − X )Ye in N step for given c2 and Ye . − 1 − 1 Ye ≤ T /m2 − X N − 1 /c2 / X N − 1 (36)
[1]
[2]
References M. Minsky, M. Ouh-Young, O. Steele, F.P. Brooks, and M. Behensky,“Feeling and seeing issues in force display” Comput. Graph, Vol.24, No.2, pp.235-243, 1990. J. E. Colgate and J. M. Brown, “Factors Affecting the Z-Width of a Haptic Display,” Proc. of IEEE Int. Conf. on Robotics and Automation, Los Alamitos, CA, pp.3205-3210, 1994. R. J. Adams and B. Hannaford,“Stable Haptic Interaction with Virtual Environments,” IEEE Trans. on Robotics and Automation, Vol.15, No.3, pp.465-474, June, 1999.
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T
If virtual coupling factors and sampling time are given, Ye should satisfy Eq.(36) in order to converge (1−X )Ye in N step. Eq.(36) represents that there is limit of expressible admittance when other factors are given.
Fig. 8. c2 = 100, m2 = 1, and T = 0.005
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4. Experiment for Admittance Display In this section, the results stated above are confirmed through experiments, qualitatively. Only admittance display is considered because of limit of page. In the simulations, it is assumed that admittance is constant in order to analyze easily and to prove relation among factors which consist of haptic interface. Though admittance changes from 0 to infinity, fluctuation of it is so small in short sampling time that relation mentioned above can be shown in real haptic display. For example, it is obvious that a haptic interface has upper limit of admittance to display for the selected sampling time and virtual coupling factors. In other words, sampling time and virtual coupling factors should be changed according to inequalities mentioned above to express desired admittance.
[3]
Figure 6-10 show admittance of virtual reality and an operator when m2 is changed from 0.01 to 100, c2 = 100, and T = 0.005. Circles and crosses represent admittance of an operator and virtual reality, each other. From Eq.(35)-Eq.(36), it can be expected that increasing m2 affects transparency negatively. Figure 6 shows that admittance of an operator is coincident with that of virtual reality. However, the coincidence is almost broken especially in high admittance as shown in Figure 7 and 8. Finally, Figure 9 and Figure 10 show that transparency is not guaranteed anymore. October 17-21, 2001, Cheju National Univ. Jeju, Korea
Fig. 9. c2 = 100, m2 = 10, and T = 0.005
Fig. 10. c2 = 100, m2 = 100, and T = 0.005
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