Incorporating Geometric Algorithms in Impedance- and Admittance-Type Haptic Rendering

Size: px

Start display at page:

Download "Incorporating Geometric Algorithms in Impedance- and Admittance-Type Haptic Rendering"

Pearl Beasley
5 years ago
Views:

1 Incororating Geometric Algorithms in Imedance- and Admittance-Tye Hatic Rendering Ryo Kikuuwe and Hideo Fujimoto Nagoya Institute of Technology, Nagoya, , Jaan Abstract Geometric (roxy-based) hatic rendering algorithms are widely used in imedance-tye hatic rendering. Such methods are useful for develoing comlex virtual environment because they reduce mechanistic roblems of determining reaction forces into geometric roblems of udating roxy ositions. This aer aims to aly geometric algorithms to wider variations of hatic rendering systems. First, we resent a method to include viscous daming in imedance-tye hatic rendering. Second, we roose a method to utilize geometric algorithms in admittance-tye hatic rendering. The results of imlementation exeriments are resented. 1. Introduction The control schemes of hatic devices can be broadly classified into two tyes: the imedance tye and admittance tye. In imedance-tye hatic rendering, the actuator force is determined according to the osition of the tool (the grasing ortion of a hatic device) measured with osition sensors such as otical encoders. This scheme is suited for sufficiently back-drivable and lightweight hatic devices. On the contrary, in admittance-tye hatic rendering, the tool osition is controlled to follow a desired osition that is determined according to the force measured with force sensors. This control scheme is used with devices having a large inertia or high-friction joints (e.g., [1, 2, 3]) because such dynamics can be suressed by stiff osition controllers. In both of imedance- and admittance-tye hatic rendering, it is generally difficult to construct consistent algorithms to treat geometrically comlex virtual objects. In the imedance tye, it is convenient to use a secial virtual object (a roxy) connected to the tool through a virtual sring-like element (a virtual couling). The roxy moves together with the tool in the virtual environment, but it remains on the surface of a virtual object when the tool comes inside the virtual object. Moreover, friction and textures of virtual surfaces can be dislayed by aroriately limiting the moving seed of the roxy. In this scheme, the mechanistic 1 roblem of determining the reaction force can be relaced by a geometric roblem of udating the roxy osition. It is convenient for develoers (rogrammers) of virtual environments because the algorithms can be develoed without considering hysical forces. It however includes at least two roblems. One is that it is unclear how to include daming in the virtual couling in imedance-tye hatic rendering. The other is that an admittance-tye hatic rendering algorithm accets a force inut although a geometric algorithm reuires a osition inut. This aer rovides solutions for the above two roblems. First, we roose an imedance-tye hatic rendering scheme in which a geometric algorithm is combined with viscous daming. Second, we roose an admittancetye hatic rendering scheme incororating a geometric algorithm. The use of the roosed methods allows the use of common geometric algorithms both in imedance- and admittance-tye hatic rendering. In what follows, section 2 reviews the geometric aroach that have been used in imedance-tye hatic rendering. Its roblems and limitations are also discussed. Section 3 resents a mechanistic interretation on geometric algorithms and derives new techniues. Section 4 describes exerimental results and Section 5 rovides concluding remarks. virtual object roxy virtual couling K O tool Figure 1. Imedance-tye hatic rendering using roxy. 1 We use the adjective mechanistic to mean of mechanics. Mechanics includes dynamics and statics, which concern hysical forces.

2 2. Geometric Hatic Rendering Algorithms n new = G( K,, ) 2.1. Proxy-based Imedance-tye Hatic Rendering new = G( K,, ) The concet of roxy was first roosed by Zilles and Salisbury [4]. They used a oint-like virtual object to reresent the hatic device (tool) in the virtual environment, and termed it as a god object. Rusini et al. [5] extended this concet to dislay surface deformations, textures, and friction. They termed the virtual object as a roxy. The following discussion assumes that the roxy is a oint-like element and the virtual environment is an n- dimensional Cartesian sace. Most of recent hatic rendering algorithms using roxies can be described in the following general form: (k) =G (K, (k), (k 1)) f(k) =K((k) (k)) (1a) (1b) where k is a discrete-time index, R n and R n are the ositions of the tool and the roxy, resectively, and K>isthestiffness of the virtual couling. The function G : R R n R n R n is a geometric algorithm that accets the current tool osition (k) and the roxy osition (k 1) of the one timeste ago as inuts, and rovides a new roxy osition (k) as the outut. The function G takes the stiffness K as an argument so that its comutation may deend on K. The actuator force f(k) R n of the hatic device is determined by the ositional relation between the tool and the roxy as indicated in (1b). As a simle examle, let us consider a unilateral constraint imosed by a flat object surface including the origin. Then, the function G can be written as follows: { if n G(K,, ) = T > nn T if n T (2) where n R n ( n =1) is the outward normal direction of the surface. Fig 2 illustrates the relation among,,and new = G(K,, ) under this definition. The new roxy osition new coincides with the tool osition as long as the tool is outside the object (i.e., n T > ). When is inside the object, new is the orthogonal rojection of onto the surface, and thus the roxy remains on the surface. The following definition of the function G is another examle, which reresents an isotroic Coulomb friction law: G(K,, ) = F K if F/K if >F/K.(3) Here, F > reresents the kinetic and maximum static friction force. Fig 3 illustrates this definition, with which Figure 2. Unilateral constraint by a frictionless flat surface. FK / = new = G( K,, ) (a) static friction FK / new = G( K,, ) (b) kinetic friction Figure 3. Isotroic Coulomb friction. new = G( K,, ) sf n tan Ä1 ñ Figure 4. Unilateral constraint by a frictional flat surface. the roxy does not move as long as the distance between the tool and the roxy is smaller than F/K, and otherwise the roxy follows the tool maintaining the distance F/K. This way of modeling Coulomb friction can be found in the literature [6, 7]. The following definition reresents a flat surface that includes the origin and has the friction coefficient of µ>: if n T > sf if n T sf sf µn T G(K,, )= sf + µn T sf sf sf sf if n T sf sf > µn T where sf = nn T, sf = nn T,andn is the outward normal vector of the surface. Fig. 4 illustrates this definition, with which the roxy osition is udated so that it remains in the friction cone. The algorithm G can be more comlex if the virtual environment is more comlex. Saving the comutational cost while maintaining the consistency of hatic rendering is the major interest of recent studies [8, 9, 1]. It nevertheless is a common rocedure to determine the current roxy osition (k) and the reaction force f(k) as described in (1). It is convenient for develoers in that a mechanistic roblem of determining reaction forces can be treated with geometric consideration. (4)

3 2.2. Limitation of Geometric Algorithms The algorithm (1) roduces an elastic force that does not include daming. It is usually referred to include some viscous daming with a feedback of the measured velocity to reduce a bouncy feeling or to revent the system from going unstable. It, however, is not straightforward to consistently combine the geometrically determined forces and viscous daming. For examle, let us consider to substitute (1) by the following rocedure: (k) =G (K, (k), (k 1)) (5a) f(k) =K((k) (k)) + B( (k) (k))/t. (5b) Here, B> denotes the viscosity of the virtual couling, T denotes the timeste size, and is the backward difference oerator, which is defined as x(k) =x(k) x(k 1). This algorithm is used in, for examle, [1]. It does not reflect the geometric relation between the roxy and the tool correctly to the reaction force f(k). For examle, when a tool in contact with a virtual surface is ulled aart from the surface, the viscous term can roduce an unnatural force sucking the tool toward the surface. This unnatural sucking (sticking) force has been recognized as one of the drawbacks of collision models based on the linear viscoelastic (Kelvin-Volgt) model [11]. Another aroach for including daming is, as described in [8], for examle, to relace (1) by (k) =G (K, (k), (k 1)) f(k) =K((k) (k)) B (k)/t. (6a) (6b) This algorithm reresents the situation in which a grounded virtual damer is connected to the roxy. Thus, the user feels viscous resistance even when the tool is in the free sace. It is not referred because it roduces unnecessary obstruction for the user s motion. Admittance-tye hatic rendering, on the other hand, reuires a force measurement as the inut. This scheme usually incororates a virtual object reresenting the tool osition in the virtual environment, and the motion of the object is udated deending on the force inut. The virtual object in this case is similar to the roxy discussed in section 2.1, but it must have a nonzero mass M >. (Hereafter, we term it as a roxy also in admittance-tye hatic rendering.) Let h(k) denote the force measured by the force sensors, and f(k) denote the force acting to the roxy from the virtual environment. Then, the roxy osition (k) can be determined by the following euation of motion. M 2 (k)/t 2 = h(k)+f(k). (7) Here, 2 satisfies 2 x(k) = x(k) x(k 1), and 2 (k)/t 2 reresents the acceleration of the roxy. The actual tool osition does not aear in this euation because it is osition-controlled to follow the roxy osition (k). The force f(k) from the virtual environment is usually determined by the roxy motion in the virtual environment. In conclusion, it is not straightforward to use the geometric algorithm G in admittance-tye hatic rendering. 3. Proosals 3.1. Mechanistic Interretation of Geometric Algorithms In order to derive new hatic rendering techniues incororating existing geometric algorithms G, we discuss mechanistic interretation of (1). Euation (1b) yields (k) =(k) f(k)/k, and substituting it into (1a) yields the following: (k) =G (K, (k) f(k)/k, (k 1)). (8) This euation reresents the algebraic constraint among K, (k), (k 1), andf(k). Here, we imose the following assumtion: Assumtion 1 If r = G (K, r f/k, ) is satisfied with vectors r R n, f R n, R n, and a ositive real number K>, thenr = G (κ, r f/κ, ) is satisfied for all ositive real numbers κ>. This means that whether (8) is satisfied or not is indeendent from K. When a ositive real number T > is given, the assumtion allows us to define a new function M : R n R n R n as follows: M(, v)=f s.t. = G (κ, f/κ, T v), κ>. (9) This allows us to remove K from (8), showing that the following exression is algebraically euivalent to (8): f(k) =M ((k), (k)/t ). (1) Thus, Assumtion 1 allows the use of the following exression as an algebraically euivalent form of (1): f(k) =M((k), (k)/t ) f(k) =K((k) (k)). (11a) (11b) Although Assumtion 1 is not guaranteed to hold, its validity can be assumed as long as the function G is mechanistically valid. The reason can be exlained as follows. Because the roxy has no mass, the total force alied to the roxy is always zero. Euation (11b) imlies that the force from the virtual couling to the roxy is f(k), and thus the sum of the forces from all other sources, i.e., the force from the virtual environment, is f(k). Therefore, the roxy

4 receives the force f(k) of (11a) from the virtual environment. Thus, (8) and (11a) can be interreted as the mechanistic constraint resulting from the virtual environment, and therefore they must be indeendent from K, which is the arameter of the virtual couling. Euation (11) algebraically constrains two unknown values: f(k) and (k). Here, the function M can be viewed as an imedance element that accets the roxy motion (osition (k) and velocity (k)/t ) as the inut and rovides the force f(k) from the virtual environment as the outut. As oosed to (9), when M and T > are given, the corresondent G can be obtained as follows: G(κ,, ) =r s.t. M (r, (r )/T )=κ(r ). (12) 3.2. Examles of Function M We describe the definitions of M that corresond to the three examles of the function G in section 2.1. In the first examle of the unilateral constraint (Fig. 2 and (2)), the force f from the virtual environment to the roxy and the roxy osition satisfy the following condition: ( f = o n T > ) ( f = n T f n T = ). (13) Here, o denotes the n-dimensional zero vector. Although (13) does not uniuely determine f when n T =,the function M can be described as the following set-valued form: { = o if n M(, ) T > {fn f>} if n T (14) =. Using (14), we can relace (13) by f = M(, ). In the case of the isotroic Coulomb friction (Fig. 3 and (3)), the relation between f and can be described as (f = F / o) ( f F = o). (15) This can be rewritten as f = M(, ) under the following definition: { = F / if o M(, ) {f R n (16) f F } if = o. Moreover, the function M that reresents the frictional flat surface (Fig. 4 and (4)) can be described as = { o ( ) if n T } > f n µ NNT M(, ) N T f> (17) if n T = N T o { f (n µt) f>, t T n =, t 1 } if n T = N T = o where N is a column full rank matrix that satisfies I = NN T + nn T and N T n = o and I is the n-dimensional identity matrix. When the function G is more comlex, it is generally difficult to derive its corresondent function M. However, the function M is only used to rovide a mechanistic interretation to the geometric algorithm G, and is not used for actual imlementation. The formulation f = M(, ) can be discussed in relation to constraint-based simulation schemes [12, 13]. In such schemes, it is usual that the constraints in the whole system is described in well-established roblem formulations such as Linear Comlementarity Problems (LCPs), which can be numerically solved. Our formulation describes only the direct interaction between the roxy and the environment. Another difference is that the most of LCPbased simulation techniues includes constraints on the acceleration while our formulation, f = M(, ), does not Combining Function G and Daming in Imedance-Tye Hatic Rendering Euation (11) describes the virtual environment and the virtual couling searately. Thus, adding a daming to the virtual couling yields f(k) =M((k), (k)/t ) (18a) f(k) =K((k) (k)) + B( (k) (k))/t. (18b) This inclusion of daming is mechanistically consistent. Euation (18), which describes algebraic constraints between f(k) and (k), has a simle closed-form solution. It is easy to find (18b) is euivalent to the following: where f(k) =(K + B/T)((k) ˆ(k)) (19) B((k 1) (k 1)) ˆ(k) =(k)+. (2) KT + B Substituting (18a) by (19) yields (K + B/T)((k) ˆ(k)) = M((k), (k)/t ). (21) Because of (12), (21) is euivalent to the following: (k) =G(K + B/T, ˆ(k), (k 1)). (22) Therefore, the analytical solution for (18) can be obtained by the following comutational rocedure: B((k 1) (k 1)) ˆ(k) =(k)+ (23a) KT + B (k) =G (K + B/T, ˆ(k), (k 1)) (23b) f(k) =(K + B/T)((k) ˆ(k)). (23c)

5 The above comutational rocedure is obtained by relacing (k) and K by ˆ(k) and K + B/T, resectively, in the basic euation (1) of the roxy-based hatic rendering. Develoers do not need to care about the velocity feedback in using G. Geometric algorithms in revious studies can be incororated in (23) as long as it can be reresented in the form of the function G. When the algorithm (23) is used with the function G of (2), no sticking force can be roduced when the tool is ulled aart from the surface. This is because the force f becomes zero when ˆ(k) comes above the surface even if the actual tool osition (k) is still below the surface. It has been ointed out that the Kelvin-Volgt model has another drawback, which is the discontinuity in the force at the time of gaining contact [11]. This characteristic still remains in our method but ractically it is not roblematic in hatic rendering Using Function G in Admittance-Tye Hatic Rendering Based on the relation between M and G discussed in section 3.1, we can derive a new techniue to incororate the geometric algorithm G in admittance-tye hatic rendering. In admittance-tye hatic rendering, the force h alied from the user to the tool is measured by a force sensor, and the roxy osition in the virtual environment is udated deending on h. In this case, the roxy has a nonzero mass M, so the discrete-time reresentation of the euation of motion of the roxy can be described as follows: M 2 (k)/t 2 = h(k)+m ((k), (k)/t ). (24) Euation (24) is euivalent to the following: where (M/T 2 )((k) ˆ(k)) = M ((k), (k)/t ) (25a) ˆ(k) =2(k 1) (k 2) + T 2 h(k)/m. (25b) By using (12), we can rewrite (25) as follows: ˆ(k) =2(k 1) (k 2) + T 2 h(k)/m (26a) (k) =G ( M/T 2, ˆ(k), (k 1) ). (26b) Thus, the geometric algorithm G is incororated in admittance-tye hatic rendering by using ˆ(k) of (26a) as the inut to the function G. 4. Imlementation The roosed method (23) of incororating daming into imedance-tye hatic rendering was demonstrated through x [m] f x [N] x [m] f x [N] x [m] f x [N] y [m] f y [N] (a) e.(1): no daming y [m] f y [N] (b) e.(5): inaroriate inclusion of daming y [m] f y [N] (c) e.(23): roosed method Figure 5. Results of Exeriment: imedancetye hatic rendering. a comarison to (5). (The other roosed techniue (26) was not demonstrated because its validity is analytically shown and because the sace of the aer is limited.) A PHANTOM R Deskto TM device of SensAble Technologies was used, which is caable of 3-DOF actuation and 6-DOF osition measurement. A virtual flat surface erendicular to y axis was constructed in the virtual environment, and the friction coefficient of the surface was chosen as µ =.3. The function G in this case is described in (2), and it was imlemented in three methods: (1), (5), and (23). Euation (1), which includes no daming, was imlemented to show the daming effect of (23) to be as significant as that of (5). The arameters were set to be T =.1 s, K = 75 N/m, and B =8Ns/m. The osition measurements were smoothed with a first-order low-ass filter with the cut-off freuency of 3 Hz. In the exeriment, the tool was first ressed onto the vir-

6 tual surface, then was moved on the surface back and forth in x direction, and finally was moved aart from the surface in +y direction. Fig. 5 shows x- andy-comonents of the tool osition and the actuator force f. When the tool collided with the virtual surface (i.e., when y reached ), the force in y direction, f y, became oscillatory with (1), while not so with (5) and (23). This shows that the viscous daming works not only with (5) but also with (23). The negative x force observed at the collision is due to the static friction and fluctuations of the exerimenter s hand intending to vertically ress the tool onto the surface. The difference between (5) and (23) aears in the gray circles in Figs. 5(b) and (c). With (5), an imulse force in y direction is roduced when the tool was moving aart from the virtual surface. Moreover, another imulse force was roduced in x direction at the transition from static friction to kinetic friction. When (23) was used, such hysically incorrect forces are not roduced. 5. Conclusions This aer has roosed two techniues for extending the scoe of alication of geometric (roxy-based) hatic rendering algorithms. One is the algorithm (23), which combines geometric algorithms and viscous daming in imedance-tye hatic rendering. The other is the algorithm (26) to incororate geometric algorithms in admittance-tye hatic rendering. The results of imlementation exeriments have been resented. The resented techniues are related to the discrete-time friction models for hatic rendering that the authors reviously roosed [14]. Substituting the function G of the definition (3) into (23) and (26) yield the imedance-tye and admittance-tye friction models described in [14]. While this aer has limited its scoe to static virtual environments with stationary virtual objects, it will also be ossible to treat a mobile virtual object by using a coordinate system attached to the object. A theoretical validation for this aroach will be a art of our future research. References [1] M. Ueberle, N. Mock, and M. Buss, VISHARD1, a novel hyer-redundant hatic interface, in Proc. of 12th Sym. on Hatic Interfaces for Virtual Environment and Teleoerator Systems, 24, [2] R. Q. Van Der Linde and P. Lammertse, HaticMaster - a generic force controlled robot for human interaction, Industrial Robot, 3(6), , 23. [3] R. Taylor, P. Jensen, L. L. Whitcomb, A. Barnes, R. Kumar, D. Stoianovici, P. Guta, Z. Wang, E. de- Juan, and L. Kavoussi, A steady-hand robotic system for microsurgical augmentation, Int. J. of Robotics Research, 18(12), , [4] C. B. Zilles and J. K. Salisbury, A constraint-based god-object method for hatic dislay, in Proc. of 1995 IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, 1995, [5] D. C. Rusini and O. Khatib, Hatic dislay for human interaction with virtual dynamic environments, J. of Robotic Systems, 18(12), , 21. [6] V. Hayward and B. Armstrong, A new comutational model of friction alied to hatic rendering, in Exerimental Robotics VI, 2, [7] S. Hasegawa, N. Fujii, Y. Koike, and M. Sato, Real-time rigid body simulation based on volumetric enalty method, in Proc. of 11th Sym. on Hatic Interfaces for Virtual Environment and Teleoerator Systems, 23, [8] L. Kim, A. Kyrikou, G. Sukhatme, and M. Desbrun, An imlicit-based hatic rendering techniue, in Proc. of 22 IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, 22, [9] K. Lundin, A. Ynnerman, and B. Gudmundsson, Proxy-based hatic feedback from volumetric density data, in Proc. of EuroHatics 22, 22, [1] M. Ikits, J. D. Brederson, C. D. Hansen, and C. R. Johnson, A constraint-based techniue for hatic volume exloration, in Proc. of 14th IEEE Visualization 23 Conference, 23, [11] K. H. Hunt and F. R. E. Crossley, Coefficient of restitution interreted as daming in vibroimact, Trans. of ASME: J. of Alied Mechanics, 7, , [12] D. Baraff, Fast contact force comutation for nonenetrating rigid bodies, Comuter Grahics Proc., Annual Conference Series, , [13] M. Anitescu and F. A. Potra, Formulating Dynamic Multi-Rigid-Body Contact Problems with Friction as Solvable Linear Comlementarity Problems, Nonlinear Dynamics, 14(3), , [14] R. Kikuuwe, N. Takesue, A. Sano, H. Mochiyama, and H. Fujimoto, Admittance and Imedance Reresentations of Friction Based on Imlicit Euler Integration, IEEE Trans. on Robotics, 22(6), , 25.

Matlab Virtual Reality Simulations for optimizations and rapid prototyping of flexible lines systems

Matlab Virtual Reality Simulations for otimizations and raid rototying of flexible lines systems VAMVU PETRE, BARBU CAMELIA, POP MARIA Deartment of Automation, Comuters, Electrical Engineering and Energetics