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A New Control Architecture for Stable and Transparent Haptic Feedback of Interactive Simulation
The International Federation of Congress Automatic Control Proceedings of 20th World Proceedings of the theJuly 20th9-14, World Congress Toulouse, France, 2017 Proceedings of the 20th World Congress The International Federation of Control Available online at www.sciencedirect.com The International Federation of Automatic Automatic Control The International of Automatic Control Toulouse, France, July Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 1346–1351 A New Control Architecture for Stable and Transparent Haptic Feedback of A New Control Architecture for StableSimulation and Transparent Haptic Feedback of Interactive A New Control Architecture for Stable and Transparent Haptic Feedback of Interactive Simulation Interactive Simulation
Myeongjin Kim*. Doo Yong Lee** Myeongjin Kim*. Kim*. Doo Doo Yong Yong Lee** Lee** Myeongjin Myeongjin Kim*. Doo Yong Lee** *Korea Advanced Institute of Science and Technology, Daejeon, Korea, (Tel: 042-350-3269; e-mail: *Korea Advanced Institute and Technology, *Korea Advanced Institute of of Science Science [email protected]). Technology, *Korea Advanced Institute ofofScience and Technology, ** Korea Advanced Institute Science and Technology, Daejeon, Korea, (Tel: 042-350-3269; e-mail: [email protected]). Daejeon, Korea, (Tel: 042-350-3269; e-mail: [email protected]). Daejeon, Korea, (Tel: 042-350-3269; e-mail: [email protected]). 042-350-3229; [email protected]) ** Korea Advanced Institute of Science and ** Korea Advanced Institute of Science and Technology, Technology, ** Korea Advanced Institute of Science and Technology, Daejeon, Daejeon, Korea, Korea, (Tel: (Tel: 042-350-3229; 042-350-3229; e-mail: e-mail: [email protected]) [email protected]) Daejeon, Korea, (Tel: 042-350-3229; e-mail: [email protected]) Abstract: This paper proposes a new control architecture for stable and transparent haptic rendering of interactive simulation involvingaaa new rigidcontrol tool andarchitecture deformable for objects. direct and rendering proxy-based Abstract: This paper proposes proposes new control architecture for stableThe andtraditional transparent haptic of Abstract: This paper stable and transparent haptic rendering of Abstract: This paper proposes a new control architecture for stable and transparent haptic rendering of haptic rendering schemes are analyzed for the absolute stability, and transmit impedance employing the interactive interactive simulation simulation involving involving aa rigid rigid tool tool and and deformable deformable objects. objects. The The traditional traditional direct direct and and proxy-based proxy-based interactive simulation involving a rigid tool and deformable objects. The traditional direct and proxy-based well-known bilateral control architecture frequently used in the field of teleoperation systems. An haptic haptic rendering rendering schemes schemes are are analyzed analyzed for for the the absolute absolute stability, stability, and and transmit transmit impedance impedance employing employing the the haptic rendering schemes are analyzed the absolute stability, transmit impedance the equivalent impedance conveyed in the for proposed controlused scheme toand the haptic insteademploying of the contact well-known bilateral architecture frequently in field of teleoperation systems. An well-known bilateral iscontrol control architecture frequently used in the the field of device teleoperation systems. An well-known bilateral control architecture frequently used in the field ofthe teleoperation systems. An force between the virtual tool and the object. The trade-off problem between absolute stability and the equivalent impedance is conveyed in the proposed control scheme to the haptic device instead of the contact equivalent impedance is conveyed in the proposed control scheme to the haptic device instead of the contact equivalent impedance isisconveyed in the proposed control scheme to the haptic device instead of the and contact transmitted impedance resolved by computing the haptic feedback using the equivalent impedance. The force between the virtual tool and the object. The trade-off problem between the absolute stability the force between the virtual tool and the object. The trade-off problem between the absolute stability and the force between the virtual toolshows and the object. Thethe trade-off problem between thehaptic absolute stabilitymethods andThe the proposed scheme transparency than the conventional rendering transmitted impedance is computing feedback using impedance. transmittedrendering impedance is resolved resolved by byhigher computing the haptic haptic feedback using the the equivalent equivalent impedance. The transmitted impedance is resolved by computing the haptic feedback using the equivalent impedance. The while maintaining absolute stability. proposed rendering scheme shows higher proposed renderingthe scheme shows higher transparency transparency than than the the conventional conventional haptic haptic rendering rendering methods methods proposed renderingthe scheme shows higher transparency than the conventional haptic rendering methods while maintaining absolute stability. while maintaining the absolute stability. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Haptic Rendering, Virtual Simulation, Bilateral Control Architecture while maintaining the absolute stability. Keywords: Haptic Haptic Rendering, Rendering, Virtual Virtual Simulation, Simulation, Bilateral Bilateral Control Control Architecture Architecture Keywords: Keywords: Haptic Rendering, Virtual Simulation, Bilateral Control Architecture 1. INTRODUCTION 2. CONVENTIONAL HAPTIC RENDERINGS FORMULATED AS BILATERAL CONTROL 1. INTRODUCTION INTRODUCTION 2. CONVENTIONAL HAPTIC RENDERINGS RENDERINGS 1. 2. CONVENTIONAL HAPTIC Virtual simulation using visual and haptic feedback has been 1. INTRODUCTION 2. CONVENTIONAL HAPTIC RENDERINGS ARCHITECTURES FORMULATED FORMULATED AS AS BILATERAL BILATERAL CONTROL CONTROL applied to medicalusing training (Kuhnapfel, MaaB, Virtual visual and feedback been FORMULATED AS BILATERAL CONTROL Virtual simulation simulation using visual and haptic hapticCakmak, feedback&has has been ARCHITECTURES ARCHITECTURES Virtual simulation visual and haptic feedback has been 2000; Mahvash &using Hayward, 2004). Maintaining virtual objects are computed using the applied to training (Kuhnapfel, Cakmak, MaaB, applied to medical medical training (Kuhnapfel, Cakmak, & &stability MaaB, Displacements of theARCHITECTURES applied to medical training (Kuhnapfel, Cakmak, &dynamic MaaB, Displacements during such simulation is difficult because of the discretized governing equation. The intersection between 2000; Mahvash & Hayward, 2004). Maintaining stability of the objects are using 2000; Mahvash & Hayward, 2004). Maintaining stability Displacements of the virtual virtual objects are computed computed using the the 2000; Mahvash & Hayward, 2004). Maintaining stability Displacements of detected the virtual objects are computed using the characteristic of the virtual environment and limited update virtual objects are using the efficient and fast collision during such simulation is difficult because of the dynamic discretized governing equation. The intersection between the during such simulation is difficult because of the dynamic discretized governing equation. The intersection between the during such simulation is computation difficult because oflimited the dynamic discretized governing equation. The intersection between the rate stemming from the load of the virtual detection algorithm (Jime, Thomas, & Torras, 2001). Contact characteristic of the virtual environment and update virtual objects are detected using the efficient and fast collision characteristic of the virtual environment and limited update virtual objects are detected using the efficient and fast collision characteristic of the virtual environment and limited update virtual objects are detected using the efficient and fast collision environment. The early research on the stability problems forces between the (Jime, virtualThomas, objects & then 2001). computed and rate algorithm Torras, Contact rate stemming stemming from from the the computation computation load load of of the the virtual virtual detection detection algorithm (Jime, Thomas, &are Torras, 2001). Contact rate stemming from theresearch computation load of the (Jime, Thomas, Torras, Contact discussed theThe haptic from static virtual detection applied toalgorithm the virtual objects. These & steps must2001). be completed environment. early the stability problems forces the objects are computed and environment. The early feedback research on on the the stability problems forces between between the virtual virtual objects are then then computed and environment. such The as early research on the the stabilityStanley, problems forces between objects are then computed and environment virtual wall & within 30the Hzvirtual update to steps provide plausible visual discussed from applied the objects. These must be discussed the the haptic hapticthefeedback feedback from(Colgate, the static static virtual virtual appliedatto toleast the virtual virtual objects.cycle These steps must be completed completed discussed the haptic feedback from the static applied to the virtual objects. These steps must be completed Brown, 1995). The stability problems often rise in the virtual feedback (Bro-Nielsen, 1998). The haptic feedback requires environment environment such such as as the the virtual virtual wall wall (Colgate, (Colgate, Stanley, Stanley, & & within within at at least least 30 30 Hz Hz update update cycle cycle to to provide provide plausible plausible visual visual1 such as the virtual wall (Colgate, Stanley, & within at least 30 Hz update cycle to provide plausible visual environment simulating dynamic behaviour of the rigid and kHz update rate (Picinbono, Lombardo, Delingette, & Ayache, Brown, Brown, 1995). 1995). The The stability stability problems problems often often rise rise in in the the virtual virtual feedback feedback (Bro-Nielsen, (Bro-Nielsen, 1998). 1998). The The haptic haptic feedback feedback requires requires 1 1 Brown, 1995). The stability problems often rise in the virtual feedback (Bro-Nielsen, 1998). The haptic feedback requires 1 deformable objects (Fierz, Spillmann, Aguinaga Hoyos, & 2002). Zero-order-holder (ZOH) is used to display the environment simulating dynamic behaviour of the rigid and kHz update rate (Picinbono, Lombardo, Delingette, & Ayache, environment simulating dynamic behaviour of the rigid and kHz update rate (Picinbono, Lombardo, Delingette, & Ayache, environment simulating dynamic behaviour ofthethesimulation, rigid and kHz update rate (Picinbono, Lombardo, Delingette, & Ayache, Harders, 2012). Many stability analyses of computed force from the virtual environment in the haptic deformable objects (Fierz, Spillmann, Aguinaga Hoyos, & 2002). Zero-order-holder (ZOH) is used to display the deformable objects (Fierz, Spillmann, Aguinaga Hoyos, & 2002). Zero-order-holder (ZOH) is used to display the deformable objects Spillmann, Aguinaga Hoyos,The & computed 2002). rate. Zero-order-holder (ZOH) is used to indisplay the however, assume that(Fierz, thestability virtual environment is passive. update Harders, Many analyses simulation, force Harders, 2012). 2012). Many stability analyses of of the the simulation, computed force from from the the virtual virtual environment environment in the the haptic haptic Harders, 2012). Many stability analyses of the simulation, computed force from the virtual environment in the haptic dynamic the virtual environment is often however, assume environment is The rate. however, characteristics assume that that the theofvirtual virtual environment is passive. passive. The update update rate.(Y. J. Kim, Otaduy, Lin, & Manocha, 2002) and the The direct however, assume theofvirtual environment analysis is passive. The rate. neglected characteristics in the that stability the update dynamic the virtual dynamic characteristics of and the performance virtual environment environment is isofoften often proxy-based rendering James, 2008) (Y. J. Otaduy, Lin, Manocha, 2002) and the The direct dynamic characteristics of the virtual environment is often simulation (Adams & Hannaford, 1999). The direct (Y.haptic J. Kim, Kim, Otaduy,(Barbič Lin, & && Manocha, 2002)are andused the neglected neglected in in the the stability stability and and performance performance analysis analysis of of the the in (Y. J. Kim, Otaduy, Lin, & Manocha, 2002) and the The direct the field of the haptic simulation. The contact force between proxy-based haptic rendering (Barbič & James, 2008) are used neglected in the stability and performance analysis of the proxy-based haptic rendering (Barbič & James, 2008) are used simulation (Adams & Hannaford, 1999). simulation (Adams Hannaford, (Hashtrudi-Zaad, 1999). proxy-based haptic rendering & James, 2008) are used The bilateral control& 2001) has in the virtual tool the simulation. virtual(Barbič object directly displayed to field the haptic The force between simulation (Adams &architecture Hannaford, 1999). in the the field of of theand haptic simulation. Theiscontact contact force between in the field of the haptic simulation. The contact force between been widely used to analyse the stability and performance of the human hands in the direct rendering. The direct rendering The displayed to to The bilateral bilateral control control architecture architecture (Hashtrudi-Zaad, (Hashtrudi-Zaad, 2001) 2001) has has the virtual virtual tool tool and and the virtual virtual object object is is directly directly displayed The bilateral control architecture (Hashtrudi-Zaad, 2001) has the virtual tool and the direct virtualrendering. object is directly displayed to the composed of theand master and slave. avoid between virtual been widely to the performance of the human hands in the direct rendering beenteleoperation widely used used system to analyse analyse the stability stability and performance of is thedifficult human to hands in the the visual direct overlapping rendering. The The directthe rendering been widelypresents used system toa analyse the stability and performance of tool the human hands in the direct rendering. The direct rendering This paper novel control architecture for transparent and the object often resulting in unrealistic visual the teleoperation composed of the master and slave. is difficult to avoid the visual overlapping between the virtual the teleoperation system composed of the master and slave. is difficult to avoid the visual overlapping between the virtual the teleoperation system composed of between the master and is difficult the to avoid the often visual resulting overlapping between virtual and of the interaction thetransparent rigidslave. tool feedback. The proxy-based employs a the proxy to This paper presents control architecture for tool object in unrealistic visual Thisstable paper simulation presents aa novel novel control architecture for transparent tool and and the object often rendering resulting in unrealistic visual This paper presents a novel control architecture for transparent tool and the object often resulting in unrealistic visual the deformable object. Limitation of the conventional provide the plausible visual feedback. The virtual tool called and stable simulation of the interaction between the rigid tool feedback. The proxy-based rendering employs a proxy to and stable simulation of the interaction between the rigid tool feedback. The proxy-based rendering employs a proxy to and stable simulation of the interaction between tool provide feedback. The proxy-based rendering atool proxy to haptic is discussed in the frame of the the rigid bilateral proxy is the connected to the haptic device The byemploys the virtual coupling and deformable object. of conventional plausible visual feedback. virtual called and the therendering deformable object. Limitation Limitation of the the conventional provide the plausible visual feedback. The virtual tool called and the deformable object. Limitation of the conventional provide the plausible visual feedback. The virtual tool called control architecture, and a method is proposed to overcome the composed of the artificial spring and damper. The virtual tool haptic haptic rendering rendering is is discussed discussed in in the the frame frame of of the the bilateral bilateral proxy proxy is is connected connected to to the the haptic haptic device device by by the the virtual virtual coupling coupling haptic rendering is and discussed in isthe frame of overcome the bilateral proxy is connected to the haptic by the virtual coupling limitation. is moved by indevice freedamper. motion. The virtual virtual tool control of the artificial spring and The tool control architecture, architecture, and aa method method is proposed proposed to to overcome the the composed composed ofthe thecoupling artificialforce spring and damper. The virtual tool control architecture, and a method is proposed to overcome the composed of the artificial spring and damper. The virtual tool remains on the surface of the virtual object by the contact force limitation. is limitation. is moved moved by by the the coupling coupling force force in in free free motion. motion. The The virtual virtual tool tool limitation. is moved bythe the coupling force in free motion. The virtual tool and the on coupling force. The virtual coupling the remains of virtual object by contact remains on the surface surface of the the virtual object by the thedecouples contact force force remains on the surface of the virtual object by thedecouples contact when force stability problem of simulation from virtual environment and the coupling force. The virtual coupling the and the coupling force. The virtual coupling decouples the and the problem coupling force. is The virtual coupling decouples less the the virtual environment passive. The transparency stability of from environment stability problem of simulation simulation from virtual virtual environmentiswhen when stability problem of simulation from virtual environment when the the virtual virtual environment environment is is passive. passive. The The transparency transparency is is less less the virtual environment is passive. The transparency is less Copyright © 2017 IFAC 1382 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 1382 Copyright 2017 responsibility IFAC 1382Control. Peer review©under of International Federation of Automatic Copyright © 2017 IFAC 1382 10.1016/j.ifacol.2017.08.224
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Myeongjin Kim et al. / IFAC PapersOnLine 50-1 (2017) 1346–1351
discussed than the stability on the simulation in the field of haptic rendering. Stabilization methods affect the transparency of the simulation. This paper uses impedances to represent characteristics of the virtual environment and analyses the effect of virtual environment to the simulation. The pose of the rigid tool is updated by the numerical integration and applied force. A tool impedance is defined to represent the relationship between the applied force and velocity of the virtual tool like the haptic device. The contact force is computed according to the contact resolution methods. The common thing is that the contact force is computed based on the penetration depth between the objects. A contact impedance is defined to represent the relationship between the penetration depth and contact force. The contact surface of the deformable object is displaced by a contact force. The relationship between displacement and contact force of the deformable object can be represented using an impedance of the virtual object.
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environment. The haptic feedback is conveyed to the haptic device using ZOH. The passivity controller is represented by the gain control. The hybrid matrix between the human arm and the contact surface of the virtual object is formulated as shown in (1).
2.1 Direct haptic rendering The time-domain passivity control (Hannaford & Ryu, 2002) has been applied to absorb the energy generated by discretization of the virtual environment. The passivity controller measures the energy flows between the haptic device and virtual environment. The artificial damping is applied to absorb the energy when the energy from the virtual environment exceeds the energy applied to the virtual environment. Several researches (J.-P. Kim, Seo, & Ryu, 2011; D. Lee & Huang, 2010; K. Lee & Lee, 2009) have tried to minimize the dissipated energy for transparent and stable simulation. Another method stabilizing the haptic feedback is to restrict stiffness and damping coefficient of the virtual coupling. Stability and performance of the simulation using the virtual coupling in the direct rendering was discussed on the previous research (Adams & Hannaford, 1999). This paper discusses the direct haptic rendering with the time-domain passivity control. Simulation using the direct rendering with the time-domain passivity controller is formulated as shown in Fig. 1. F and V indicate the force and velocity, respectively. The subscript indicates the element of the simulation. h, m, c, and e denote the human, haptic device, contact and contact surface of the virtual object, respectively. 𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 is the passivity controller. The discrete variables are indicated with a star superscript. The human generates the exogenous force input, 𝐹𝐹𝐹𝐹ℎ_𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 using to manipulate the haptic device. The input force is conveyed through the impedance of human arm. The haptic device is moved by the force from the human. The haptic device is modelled by the impedance type master. The slave is not included because the virtual tool is rigidly manipulated by the haptic device. The contact force is computed based on the penetration depth between the virtual tool and object. The interaction between the virtual tool and object is represented using the contact impedance. The contact surface of the virtual object is displaced by the contact force. The contact surface of the virtual object is modelled by impedance of the virtual
Fig. 1. Direct haptic rendering with the passivity controller. �
𝐹𝐹𝐹𝐹ℎ ℎ �=� 11 −𝑉𝑉𝑉𝑉𝑒𝑒𝑒𝑒∗ ℎ21
𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝐶𝐶𝐶𝐶pc 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 ℎ12 𝑉𝑉𝑉𝑉ℎ � � ∗� = � ℎ22 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒 −1
𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 𝑉𝑉𝑉𝑉ℎ � � � (1). 1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒∗
The absolute stability conditions are formulated as shown in (2), (3), and (4).
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍� ≥ 0 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 ) ≥ 0
2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍�𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 ) ≥ |𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍| − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍)
(2). (3). (4).
The impedance of the haptic device can be obtained by the Laplace transform (Adams & Hannaford, 1999). The impedances from the virtual environment can be obtained by the Z-transform. If the simulation satisfies the absolute stability conditions and the human arm and virtual object are passive, then the simulation is absolutely stable. The passivity controller can manage the absolute stability condition (2), (4). The condition (2) and (3) are always satisfied because the impedance of the haptic device, contact, and passivity controller are positive. The left term must be larger than the right term in the condition (4) for stable simulation. The condition (4) shows the effect of discretization on the stability. The left term decreases and the right term increases when the effect of ZOH increases. The passivity controller stabilizes the simulation by increasing 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍� on the left term.
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Proceedings of the 20th IFAC World Congress 1348 Myeongjin Kim et al. / IFAC PapersOnLine 50-1 (2017) 1346–1351 Toulouse, France, July 9-14, 2017
The performance can be formulated using the transmitted impedance, 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 to the human as shown in (5). The ideal goal is to transfer same impedance of the virtual object. The transmitted impedance is, however, highly dependent on the contact impedance and passivity controller. 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 = 𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 +
𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒
𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒
�
(5).
The transparency increases when the effect of ZOH and passivity controller decrease, the contact impedance increases. The high contact impedance decreases, however, the stability margin on the condition (3) and causes stability problems on the condition (4). Another weak point of the passivity control is that the impulsive damping force makes discontinuity on the haptic feedback because the gain of the passivity controller is determined in real time. It is difficult to achieve stable and transparent simulation using the direct haptic rendering.
2.2 Proxy-based haptic rendering
Fig. 2. Proxy-based haptic rendering.
The proxy was employed to prevent the visual overlapping on the virtual environment. The early proxy approach (McNeely, Puterbaugh, & Troy, 1999) used the constraint condition for the rigid body interaction. The pose of the proxy is determined by the geometric constraint. Dynamics of the virtual tool is used to simulate interaction between the rigid tool and the deformable object (Barbi & James, 2008). Simulation using the proxy-based rendering is formulated as a bilateral control architecture as shown in Fig. 2. The virtual tool is modelled by the tool impedance, 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 . Contact surface of the virtual object is represented using impedance of the virtual environment. The contact impedance is defined between the virtual tool and object because the contact force is computed based on the penetration between the virtual tool and object. The virtual coupling that generates a force proportional to the distance between the virtual tool and haptic device is represented using the impedance of the virtual coupling, 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 (Adams & Hannaford, 1999). The hybrid matrix between the human arm and contact surface of the virtual object is formulated as shown in (6). �
𝐹𝐹𝐹𝐹ℎ ℎ �=� 11 −𝑉𝑉𝑉𝑉𝑒𝑒𝑒𝑒∗ ℎ21
=�
ℎ12 𝑉𝑉𝑉𝑉ℎ �� � ℎ22 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒𝑝𝑝𝑝𝑝∗
𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 � −1 +
1
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �
1
𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
+
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 1 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
�
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
𝑉𝑉𝑉𝑉 � � ℎ∗ � 𝐹𝐹𝐹𝐹 � 𝑒𝑒𝑒𝑒
(6).
The absolute stability condition is formulated as shown in (7), (8), and (9). The stiffness and damping coefficients of the virtual coupling have been determined by the passivity condition (Colgate et al., 1995) or the absolute stability condition (Adams & Hannaford, 1999) without considering the dynamics of the virtual tool and contact impedance. The stability conditions show, however, that the stability is not only affected by the virtual coupling but the contact and tool impedances.
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 �𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 � 1
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 � + 𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
1
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 �𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 � �𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 �𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �
�� ≥ 0 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
� �−1 +
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 � ≥ 0
(7). (8). 1
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 � 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 � + 𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
1 𝑍𝑍𝑍𝑍 � 𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 � 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 �� 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 1 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
� �−1 +
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
1
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
+
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 ��
�� ≥ (9).
The tool impedance increases the stability margin on the condition (7) and (8) when the tool impedance is positive. The tool impedance is determined by the exploited numerical integration method for the pose update of the virtual tool. The explicit integration often used in the simulation is intuitive and easy to be implemented. The problem of the explicit integration is that the tool impedance has negative value in the certain frequency ranges. The tool impedance must be determined to satisfy the condition (7) and (8) when the explicit integration is employed. The left term must be larger than the right term of the condition (9) for the absolute stability. The right term increases when the effect of discretization increases. The left term increases when the impedance of virtual coupling and the contact decrease. The transmitted impedance to the human is determined by (10). 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 =
ℎ11 + ∆ℎ 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒 1 + ℎ22 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒
where ∆ℎ = 𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 �
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍vc𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
� + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
��
𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 +𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐 Zc
�
(10).
A perfect transparency is obtained when ℎ11 and ℎ22 are zero and ∆ℎ is one. The transparency increases when the impedance of virtual tool decreases, impedance of virtual coupling and contact increase. The requirements for
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transparency conflict with the requirements for the stability. The effect of ZOH cannot be removed in the transmitted impedance. The conventional haptic renderings have the trade-off problem between the absolute stability and transmitted impedance. The right term of the condition (4) and (9) make conflicts between the stability and transparency. If the right term is zero, than the condition (4) and (9) are always satisfied. The right term is determined by the multiplication of ℎ12 and ℎ21 . The transparency is affected by the effect of discretization because the haptic feedback is computed from the virtual environment. If the haptic feedback is computed on the continuous domain, the discretization can be removed. This paper proposes a control architecture which has zero value of ℎ12 for the stability and faithfully transfers the environment impedance. 3. PROPOSED CONTROL ARCHITECTURE USING AN EQUIVALENT IMPEDANCE The constraint-based contact resolution (Duriez, Dubois, Kheddar, & Andriot, 2006) was employed in the field of realtime simulation for realistic visual feedback. The contact force is computed from the laws of physics. The constraint-based method can guarantee the non-intersection between the virtual tool and the object. The contact force computed from the constraint is difficult to be exploited for the haptic rendering because the contact force is difficult to be computed in the haptic update rate. The constraint-based method is used with the virtual coupling in the previous research for this reason. The proposed control architecture is shown in Fig. 3, which transfers an equivalent impedance instead of the computed force from the virtual environment. The proxy is not employed in the proposed scheme. The constraint-based contact resolution is employed to guarantee the non-intersection for the visually plausible simulation. An equivalent impedance, 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 is exploited to generate the haptic feedback on the side of haptic device instead of the computed force from the virtual environment. The hybrid matrix is formulated as shown in (11). �
𝐹𝐹𝐹𝐹ℎ ℎ �=� 11 −𝑉𝑉𝑉𝑉𝑒𝑒𝑒𝑒∗ ℎ21
𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ℎ12 𝑉𝑉𝑉𝑉ℎ �� � = � ℎ22 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒∗ −1
0 𝑉𝑉𝑉𝑉 � � ℎ� 1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒∗
(11).
The absolute stability conditions are obtained as shown in (12), (13), and (14). 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 � ≥ 0
(12).
2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 �𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 ) ≥ 0
(14).
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 ) ≥ 0
Fig. 3. Proposed control architecture of the haptic rendering using an equivalent impedance. 1
𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 = �𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + 2
1
𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 � 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒 � = 𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
(15).
The contact impedance generates a force which makes the displacement on the contact surface. The displacement generated by the contact force is same with the penetration depth between the haptic device and virtual object when the constraint-based method is employed. The contact impedance is same with the impedance of the contact surface of the virtual object in (15). The transmitted impedance is equal to the summation of the haptic device and equivalent impedances. It means that the human feels the inertia of the haptic device in free motion and the inertia of the haptic device and the impedance of the virtual object in contact motion. The perfect transparency is achieved by minimizing the impedance of the haptic device and obtaining the equivalent impedance. The remaining problem is to estimate the equivalent impedance from the virtual object in real time. The contact force is computed based on the penetration depth between virtual object unlike the real environment. The human does not feels contact force by the penetration in the real environment. The equivalent impedance has to represent the contact force and the displacement on the contact surface not the penetration. The equivalent impedance will be developed in our future research.
(13).
The stability conditions are simplified than the conventional haptic renderings. The condition (12) shows the stability margin increases when the impedance of the virtual object is conveyed using the equivalent impedance. The condition (12), (13) and (14) are always satisfied because the impedances on the conditions are positive. The transparency is represented as shown in (15)
4. SIMULATION A one DOF numerical example is simulated to compare the conventional rendering methods with the proposed rendering. The absolute stability conditions in Fig. 4, 5, and 6, and the ratio of the transmitted impedance to the environment impedance in Fig. 7 are compared in the frequency domain. The semi-implicit integration is employed for the numerical integration. The normalized ZOH is used to see the effect of the discretization on the communication between the haptic
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device and the virtual environment. An ideal equivalent impedance which is equal to the environment impedance is used for the proposed rendering. Parameters used for the simulation are listed in the table 1. Table 1. Parameters of the simulation Symbol 𝑚𝑚𝑚𝑚𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 𝑚𝑚𝑚𝑚𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 𝑘𝑘𝑘𝑘𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 𝑚𝑚𝑚𝑚𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 𝑘𝑘𝑘𝑘𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 dT
Quantity Mass of the haptic device Damping of haptic device Mass of the virtual tool Damping of the virtual coupling Stiffness of the virtual coupling Mass of the virtual object Damping of the virtual coupling Stiffness of the virtual object Time step
Value 0.022 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 0.01 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 0.0001 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 0.06 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 300 𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 1 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 0.06 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 300 𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 0.03 s
Fig. 6. 2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(ℎ11 )𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(h22 ) − |ℎ12 ℎ21 | + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(ℎ12 ℎ21 ) ≥ 0 of the conventional rendering methods and the proposed rendering.
Fig. 4. 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(ℎ11) of the conventional rendering methods and the proposed rendering.
Fig. 5. 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(ℎ22) of the conventional rendering methods and the proposed rendering.
Fig. 7. Transparency of the conventional rendering methods and the proposed rendering. Figure 4 shows the left term of the conditions (2), (7), and (12). The values must be larger than zero for the absolute stability. The both direct and proxy-based rendering methods show almost zero in the figure 4. The impedance of the haptic device is low in the direct rendering method. The proxy-based rendering method has both the haptic device and tool impedance. The effect of the tool impedance, however, is lower than the proposed rendering method because the tool impedance has to be small for the transparency and the effect of the ZOH. The proposed rendering method has a larger stability margin than other rendering methods because the equivalent impedance increases the stability margin. Figure 5 shows the left term of the conditions (3), (8), and (13). The proxy-based rendering has a larger stability margin than the other rendering methods because the tool impedance increases the left term of the condition (8). Figure 6 shows the left term minus right term of the conditions (4), (9), and (14). The proposed rendering is stable when the other rendering methods are unstable with the same parameters because the proposed rendering removes the effect of the ZOH. Figure 7 shows the transparency of the conventional rendering methods and the proposed rendering. The direct rendering shows higher
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transparency than the proxy-based rendering because the proxy-based rendering has the impedance of not only the haptic device but the virtual tool. The proposed rendering shows the higher transparency than the direct rendering because the proposed rendering directly computes the haptic feedback on the haptic device.
5. CONCLUSION This paper proposes a new control architecture for the transparent and stable simulation of the interaction between a rigid tool and deformable object. The transparency is often discussed using the maximum displayable stiffness in the field of virtual simulation. The transmitted impedance is more important than the maximum displayable stiffness in the simulation of the interaction between a rigid tool and deformable objects. The conventional haptic rendering formulated as the bilateral control architecture shows the trade-off between the absolute stability and transmitted impedance. This paper removes the trade-off problem by conveying an equivalent impedance instead of the computed force from the virtual environment. The haptic feedback can be computed on the continuous time using the equivalent impedance that reflects the relationship between the contact force and displacement of the contact point not the penetration. The proposed control scheme always satisfies the absolute stability condition regardless of the transparency and shows higher transparency than the conventional haptic rendering methods.
ACKNOWLEDGEMENT This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (NRF-2015R1A2A1A10054420)
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Robots and Systems. Human Robot Interaction and Cooperative Robots, 3, 140–145. http://doi.org/10.1109/IROS.1995.525875 Duriez, C., Dubois, F., Kheddar, A., & Andriot, C. (2006). Realistic haptic rendering of interacting deformable objects in virtual environments. Visualization and Computer Graphics, IEEE Transactions on, 12(1), 36– 47. Fierz, B., Spillmann, J., Aguinaga Hoyos, I., & Harders, M. (2012). Maintaining large time steps in explicit finite element simulations using shape matching. IEEE Transactions on Visualization and Computer Graphics, 18(5), 717–28. http://doi.org/10.1109/TVCG.2011.105 Hannaford, B., & Ryu, J. (2002). Time-domain passivity control of haptic interfaces. IEEE Transactions on Robotics and Automation, 18(1), 1–10. Hashtrudi-Zaad, K. (2001). Analysis of Control Architectures for Teleoperation Systems with Impedance/Admittance Master and Slave Manipulators. The International Journal of Robotics Research, 20(6), 419–445. Jime, P., Thomas, F., & Torras, C. (2001). 3D collision detection : a survey. Computers & Graphics, 25(2), 269–285. Kim, J.-P., Seo, C., & Ryu, J. (2011). Multirate EnergyBounding Algorithm for High-Fidelity Stable Haptic Interaction Control. IEEE Transactions on Control Systems Technology, 19(5), 1195–1203. Kim, Y. J., Otaduy, M. A., Lin, M. C., & Manocha, D. (2002). Six-degree-of-freedom haptic display using localized contact computations. Proceedings - 10th Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems, HAPTICS 2002, 209–216. Kuhnapfel, U., Cakmak, H. K., & MaaB, H. (2000). Endoscopic surgery training using virtual reality and deformable tissue simulation. Computers & Graphics, 24, 671–682. Lee, D., & Huang, K. (2010). Passive-Set-PositionModulation Framework for Interactive Robotic Systems. IEEE Transactions on Robotics, 26(2), 354– 369. Lee, K., & Lee, D. (2009). Adjusting output-limiter for stable haptic rendering in virtual environments. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, 17(4), 768–779. Mahvash, M., & Hayward, V. (2004). High-fidelity haptic synthesis of contact with deformable bodies. Computer Graphics and Applications, IEEE, 24(2), 48–55. McNeely, W. a., Puterbaugh, K. D., & Troy, J. J. (1999). Six degree-of-freedom haptic rendering using voxel sampling. Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques SIGGRAPH ’99, 401–408. Picinbono, G., Lombardo, J.-C., Delingette, H., & Ayache, N. (2002). Improving realism of a surgery simulator: linear anisotropic elasticity, complex interactions and force extrapolation. The Journal of Visualization and Computer Animation, 13(3), 147–167.
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IFAC PapersOnLine 50-1 (2017) 1346–1351 A New Control Architecture for Stable and Transparent Haptic Feedback of A New Control Architecture for StableSimulation and Transparent Haptic Feedback of Interactive A New Control Architecture for Stable and Transparent Haptic Feedback of Interactive Simulation Interactive Simulation
Myeongjin Kim*. Doo Yong Lee** Myeongjin Kim*. Kim*. Doo Doo Yong Yong Lee** Lee** Myeongjin Myeongjin Kim*. Doo Yong Lee** *Korea Advanced Institute of Science and Technology, Daejeon, Korea, (Tel: 042-350-3269; e-mail: *Korea Advanced Institute and Technology, *Korea Advanced Institute of of Science Science [email protected]). Technology, *Korea Advanced Institute ofofScience and Technology, ** Korea Advanced Institute Science and Technology, Daejeon, Korea, (Tel: 042-350-3269; e-mail: [email protected]). Daejeon, Korea, (Tel: 042-350-3269; e-mail: [email protected]). Daejeon, Korea, (Tel: 042-350-3269; e-mail: [email protected]). 042-350-3229; [email protected]) ** Korea Advanced Institute of Science and ** Korea Advanced Institute of Science and Technology, Technology, ** Korea Advanced Institute of Science and Technology, Daejeon, Daejeon, Korea, Korea, (Tel: (Tel: 042-350-3229; 042-350-3229; e-mail: e-mail: [email protected]) [email protected]) Daejeon, Korea, (Tel: 042-350-3229; e-mail: [email protected]) Abstract: This paper proposes a new control architecture for stable and transparent haptic rendering of interactive simulation involvingaaa new rigidcontrol tool andarchitecture deformable for objects. direct and rendering proxy-based Abstract: This paper proposes proposes new control architecture for stableThe andtraditional transparent haptic of Abstract: This paper stable and transparent haptic rendering of Abstract: This paper proposes a new control architecture for stable and transparent haptic rendering of haptic rendering schemes are analyzed for the absolute stability, and transmit impedance employing the interactive interactive simulation simulation involving involving aa rigid rigid tool tool and and deformable deformable objects. objects. The The traditional traditional direct direct and and proxy-based proxy-based interactive simulation involving a rigid tool and deformable objects. The traditional direct and proxy-based well-known bilateral control architecture frequently used in the field of teleoperation systems. An haptic haptic rendering rendering schemes schemes are are analyzed analyzed for for the the absolute absolute stability, stability, and and transmit transmit impedance impedance employing employing the the haptic rendering schemes are analyzed the absolute stability, transmit impedance the equivalent impedance conveyed in the for proposed controlused scheme toand the haptic insteademploying of the contact well-known bilateral architecture frequently in field of teleoperation systems. An well-known bilateral iscontrol control architecture frequently used in the the field of device teleoperation systems. An well-known bilateral control architecture frequently used in the field ofthe teleoperation systems. An force between the virtual tool and the object. The trade-off problem between absolute stability and the equivalent impedance is conveyed in the proposed control scheme to the haptic device instead of the contact equivalent impedance is conveyed in the proposed control scheme to the haptic device instead of the contact equivalent impedance isisconveyed in the proposed control scheme to the haptic device instead of the and contact transmitted impedance resolved by computing the haptic feedback using the equivalent impedance. The force between the virtual tool and the object. The trade-off problem between the absolute stability the force between the virtual tool and the object. The trade-off problem between the absolute stability and the force between the virtual toolshows and the object. Thethe trade-off problem between thehaptic absolute stabilitymethods andThe the proposed scheme transparency than the conventional rendering transmitted impedance is computing feedback using impedance. transmittedrendering impedance is resolved resolved by byhigher computing the haptic haptic feedback using the the equivalent equivalent impedance. The transmitted impedance is resolved by computing the haptic feedback using the equivalent impedance. The while maintaining absolute stability. proposed rendering scheme shows higher proposed renderingthe scheme shows higher transparency transparency than than the the conventional conventional haptic haptic rendering rendering methods methods proposed renderingthe scheme shows higher transparency than the conventional haptic rendering methods while maintaining absolute stability. while maintaining the absolute stability. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Haptic Rendering, Virtual Simulation, Bilateral Control Architecture while maintaining the absolute stability. Keywords: Haptic Haptic Rendering, Rendering, Virtual Virtual Simulation, Simulation, Bilateral Bilateral Control Control Architecture Architecture Keywords: Keywords: Haptic Rendering, Virtual Simulation, Bilateral Control Architecture 1. INTRODUCTION 2. CONVENTIONAL HAPTIC RENDERINGS FORMULATED AS BILATERAL CONTROL 1. INTRODUCTION INTRODUCTION 2. CONVENTIONAL HAPTIC RENDERINGS RENDERINGS 1. 2. CONVENTIONAL HAPTIC Virtual simulation using visual and haptic feedback has been 1. INTRODUCTION 2. CONVENTIONAL HAPTIC RENDERINGS ARCHITECTURES FORMULATED FORMULATED AS AS BILATERAL BILATERAL CONTROL CONTROL applied to medicalusing training (Kuhnapfel, MaaB, Virtual visual and feedback been FORMULATED AS BILATERAL CONTROL Virtual simulation simulation using visual and haptic hapticCakmak, feedback&has has been ARCHITECTURES ARCHITECTURES Virtual simulation visual and haptic feedback has been 2000; Mahvash &using Hayward, 2004). Maintaining virtual objects are computed using the applied to training (Kuhnapfel, Cakmak, MaaB, applied to medical medical training (Kuhnapfel, Cakmak, & &stability MaaB, Displacements of theARCHITECTURES applied to medical training (Kuhnapfel, Cakmak, &dynamic MaaB, Displacements during such simulation is difficult because of the discretized governing equation. The intersection between 2000; Mahvash & Hayward, 2004). Maintaining stability of the objects are using 2000; Mahvash & Hayward, 2004). Maintaining stability Displacements of the virtual virtual objects are computed computed using the the 2000; Mahvash & Hayward, 2004). Maintaining stability Displacements of detected the virtual objects are computed using the characteristic of the virtual environment and limited update virtual objects are using the efficient and fast collision during such simulation is difficult because of the dynamic discretized governing equation. The intersection between the during such simulation is difficult because of the dynamic discretized governing equation. The intersection between the during such simulation is computation difficult because oflimited the dynamic discretized governing equation. The intersection between the rate stemming from the load of the virtual detection algorithm (Jime, Thomas, & Torras, 2001). Contact characteristic of the virtual environment and update virtual objects are detected using the efficient and fast collision characteristic of the virtual environment and limited update virtual objects are detected using the efficient and fast collision characteristic of the virtual environment and limited update virtual objects are detected using the efficient and fast collision environment. The early research on the stability problems forces between the (Jime, virtualThomas, objects & then 2001). computed and rate algorithm Torras, Contact rate stemming stemming from from the the computation computation load load of of the the virtual virtual detection detection algorithm (Jime, Thomas, &are Torras, 2001). Contact rate stemming from theresearch computation load of the (Jime, Thomas, Torras, Contact discussed theThe haptic from static virtual detection applied toalgorithm the virtual objects. These & steps must2001). be completed environment. early the stability problems forces the objects are computed and environment. The early feedback research on on the the stability problems forces between between the virtual virtual objects are then then computed and environment. such The as early research on the the stabilityStanley, problems forces between objects are then computed and environment virtual wall & within 30the Hzvirtual update to steps provide plausible visual discussed from applied the objects. These must be discussed the the haptic hapticthefeedback feedback from(Colgate, the static static virtual virtual appliedatto toleast the virtual virtual objects.cycle These steps must be completed completed discussed the haptic feedback from the static applied to the virtual objects. These steps must be completed Brown, 1995). The stability problems often rise in the virtual feedback (Bro-Nielsen, 1998). The haptic feedback requires environment environment such such as as the the virtual virtual wall wall (Colgate, (Colgate, Stanley, Stanley, & & within within at at least least 30 30 Hz Hz update update cycle cycle to to provide provide plausible plausible visual visual1 such as the virtual wall (Colgate, Stanley, & within at least 30 Hz update cycle to provide plausible visual environment simulating dynamic behaviour of the rigid and kHz update rate (Picinbono, Lombardo, Delingette, & Ayache, Brown, Brown, 1995). 1995). The The stability stability problems problems often often rise rise in in the the virtual virtual feedback feedback (Bro-Nielsen, (Bro-Nielsen, 1998). 1998). The The haptic haptic feedback feedback requires requires 1 1 Brown, 1995). The stability problems often rise in the virtual feedback (Bro-Nielsen, 1998). The haptic feedback requires 1 deformable objects (Fierz, Spillmann, Aguinaga Hoyos, & 2002). Zero-order-holder (ZOH) is used to display the environment simulating dynamic behaviour of the rigid and kHz update rate (Picinbono, Lombardo, Delingette, & Ayache, environment simulating dynamic behaviour of the rigid and kHz update rate (Picinbono, Lombardo, Delingette, & Ayache, environment simulating dynamic behaviour ofthethesimulation, rigid and kHz update rate (Picinbono, Lombardo, Delingette, & Ayache, Harders, 2012). Many stability analyses of computed force from the virtual environment in the haptic deformable objects (Fierz, Spillmann, Aguinaga Hoyos, & 2002). Zero-order-holder (ZOH) is used to display the deformable objects (Fierz, Spillmann, Aguinaga Hoyos, & 2002). Zero-order-holder (ZOH) is used to display the deformable objects Spillmann, Aguinaga Hoyos,The & computed 2002). rate. Zero-order-holder (ZOH) is used to indisplay the however, assume that(Fierz, thestability virtual environment is passive. update Harders, Many analyses simulation, force Harders, 2012). 2012). Many stability analyses of of the the simulation, computed force from from the the virtual virtual environment environment in the the haptic haptic Harders, 2012). Many stability analyses of the simulation, computed force from the virtual environment in the haptic dynamic the virtual environment is often however, assume environment is The rate. however, characteristics assume that that the theofvirtual virtual environment is passive. passive. The update update rate.(Y. J. Kim, Otaduy, Lin, & Manocha, 2002) and the The direct however, assume theofvirtual environment analysis is passive. The rate. neglected characteristics in the that stability the update dynamic the virtual dynamic characteristics of and the performance virtual environment environment is isofoften often proxy-based rendering James, 2008) (Y. J. Otaduy, Lin, Manocha, 2002) and the The direct dynamic characteristics of the virtual environment is often simulation (Adams & Hannaford, 1999). The direct (Y.haptic J. Kim, Kim, Otaduy,(Barbič Lin, & && Manocha, 2002)are andused the neglected neglected in in the the stability stability and and performance performance analysis analysis of of the the in (Y. J. Kim, Otaduy, Lin, & Manocha, 2002) and the The direct the field of the haptic simulation. The contact force between proxy-based haptic rendering (Barbič & James, 2008) are used neglected in the stability and performance analysis of the proxy-based haptic rendering (Barbič & James, 2008) are used simulation (Adams & Hannaford, 1999). simulation (Adams Hannaford, (Hashtrudi-Zaad, 1999). proxy-based haptic rendering & James, 2008) are used The bilateral control& 2001) has in the virtual tool the simulation. virtual(Barbič object directly displayed to field the haptic The force between simulation (Adams &architecture Hannaford, 1999). in the the field of of theand haptic simulation. Theiscontact contact force between in the field of the haptic simulation. The contact force between been widely used to analyse the stability and performance of the human hands in the direct rendering. The direct rendering The displayed to to The bilateral bilateral control control architecture architecture (Hashtrudi-Zaad, (Hashtrudi-Zaad, 2001) 2001) has has the virtual virtual tool tool and and the virtual virtual object object is is directly directly displayed The bilateral control architecture (Hashtrudi-Zaad, 2001) has the virtual tool and the direct virtualrendering. object is directly displayed to the composed of theand master and slave. avoid between virtual been widely to the performance of the human hands in the direct rendering beenteleoperation widely used used system to analyse analyse the stability stability and performance of is thedifficult human to hands in the the visual direct overlapping rendering. The The directthe rendering been widelypresents used system toa analyse the stability and performance of tool the human hands in the direct rendering. The direct rendering This paper novel control architecture for transparent and the object often resulting in unrealistic visual the teleoperation composed of the master and slave. is difficult to avoid the visual overlapping between the virtual the teleoperation system composed of the master and slave. is difficult to avoid the visual overlapping between the virtual the teleoperation system composed of between the master and is difficult the to avoid the often visual resulting overlapping between virtual and of the interaction thetransparent rigidslave. tool feedback. The proxy-based employs a the proxy to This paper presents control architecture for tool object in unrealistic visual Thisstable paper simulation presents aa novel novel control architecture for transparent tool and and the object often rendering resulting in unrealistic visual This paper presents a novel control architecture for transparent tool and the object often resulting in unrealistic visual the deformable object. Limitation of the conventional provide the plausible visual feedback. The virtual tool called and stable simulation of the interaction between the rigid tool feedback. The proxy-based rendering employs a proxy to and stable simulation of the interaction between the rigid tool feedback. The proxy-based rendering employs a proxy to and stable simulation of the interaction between tool provide feedback. The proxy-based rendering atool proxy to haptic is discussed in the frame of the the rigid bilateral proxy is the connected to the haptic device The byemploys the virtual coupling and deformable object. of conventional plausible visual feedback. virtual called and the therendering deformable object. Limitation Limitation of the the conventional provide the plausible visual feedback. The virtual tool called and the deformable object. Limitation of the conventional provide the plausible visual feedback. The virtual tool called control architecture, and a method is proposed to overcome the composed of the artificial spring and damper. The virtual tool haptic haptic rendering rendering is is discussed discussed in in the the frame frame of of the the bilateral bilateral proxy proxy is is connected connected to to the the haptic haptic device device by by the the virtual virtual coupling coupling haptic rendering is and discussed in isthe frame of overcome the bilateral proxy is connected to the haptic by the virtual coupling limitation. is moved by indevice freedamper. motion. The virtual virtual tool control of the artificial spring and The tool control architecture, architecture, and aa method method is proposed proposed to to overcome the the composed composed ofthe thecoupling artificialforce spring and damper. The virtual tool control architecture, and a method is proposed to overcome the composed of the artificial spring and damper. The virtual tool remains on the surface of the virtual object by the contact force limitation. is limitation. is moved moved by by the the coupling coupling force force in in free free motion. motion. The The virtual virtual tool tool limitation. is moved bythe the coupling force in free motion. The virtual tool and the on coupling force. The virtual coupling the remains of virtual object by contact remains on the surface surface of the the virtual object by the thedecouples contact force force remains on the surface of the virtual object by thedecouples contact when force stability problem of simulation from virtual environment and the coupling force. The virtual coupling the and the coupling force. The virtual coupling decouples the and the problem coupling force. is The virtual coupling decouples less the the virtual environment passive. The transparency stability of from environment stability problem of simulation simulation from virtual virtual environmentiswhen when stability problem of simulation from virtual environment when the the virtual virtual environment environment is is passive. passive. The The transparency transparency is is less less the virtual environment is passive. The transparency is less Copyright © 2017 IFAC 1382 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 1382 Copyright 2017 responsibility IFAC 1382Control. Peer review©under of International Federation of Automatic Copyright © 2017 IFAC 1382 10.1016/j.ifacol.2017.08.224
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Myeongjin Kim et al. / IFAC PapersOnLine 50-1 (2017) 1346–1351
discussed than the stability on the simulation in the field of haptic rendering. Stabilization methods affect the transparency of the simulation. This paper uses impedances to represent characteristics of the virtual environment and analyses the effect of virtual environment to the simulation. The pose of the rigid tool is updated by the numerical integration and applied force. A tool impedance is defined to represent the relationship between the applied force and velocity of the virtual tool like the haptic device. The contact force is computed according to the contact resolution methods. The common thing is that the contact force is computed based on the penetration depth between the objects. A contact impedance is defined to represent the relationship between the penetration depth and contact force. The contact surface of the deformable object is displaced by a contact force. The relationship between displacement and contact force of the deformable object can be represented using an impedance of the virtual object.
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environment. The haptic feedback is conveyed to the haptic device using ZOH. The passivity controller is represented by the gain control. The hybrid matrix between the human arm and the contact surface of the virtual object is formulated as shown in (1).
2.1 Direct haptic rendering The time-domain passivity control (Hannaford & Ryu, 2002) has been applied to absorb the energy generated by discretization of the virtual environment. The passivity controller measures the energy flows between the haptic device and virtual environment. The artificial damping is applied to absorb the energy when the energy from the virtual environment exceeds the energy applied to the virtual environment. Several researches (J.-P. Kim, Seo, & Ryu, 2011; D. Lee & Huang, 2010; K. Lee & Lee, 2009) have tried to minimize the dissipated energy for transparent and stable simulation. Another method stabilizing the haptic feedback is to restrict stiffness and damping coefficient of the virtual coupling. Stability and performance of the simulation using the virtual coupling in the direct rendering was discussed on the previous research (Adams & Hannaford, 1999). This paper discusses the direct haptic rendering with the time-domain passivity control. Simulation using the direct rendering with the time-domain passivity controller is formulated as shown in Fig. 1. F and V indicate the force and velocity, respectively. The subscript indicates the element of the simulation. h, m, c, and e denote the human, haptic device, contact and contact surface of the virtual object, respectively. 𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 is the passivity controller. The discrete variables are indicated with a star superscript. The human generates the exogenous force input, 𝐹𝐹𝐹𝐹ℎ_𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 using to manipulate the haptic device. The input force is conveyed through the impedance of human arm. The haptic device is moved by the force from the human. The haptic device is modelled by the impedance type master. The slave is not included because the virtual tool is rigidly manipulated by the haptic device. The contact force is computed based on the penetration depth between the virtual tool and object. The interaction between the virtual tool and object is represented using the contact impedance. The contact surface of the virtual object is displaced by the contact force. The contact surface of the virtual object is modelled by impedance of the virtual
Fig. 1. Direct haptic rendering with the passivity controller. �
𝐹𝐹𝐹𝐹ℎ ℎ �=� 11 −𝑉𝑉𝑉𝑉𝑒𝑒𝑒𝑒∗ ℎ21
𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝐶𝐶𝐶𝐶pc 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 ℎ12 𝑉𝑉𝑉𝑉ℎ � � ∗� = � ℎ22 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒 −1
𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 𝑉𝑉𝑉𝑉ℎ � � � (1). 1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒∗
The absolute stability conditions are formulated as shown in (2), (3), and (4).
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍� ≥ 0 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 ) ≥ 0
2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍�𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 ) ≥ |𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍| − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍)
(2). (3). (4).
The impedance of the haptic device can be obtained by the Laplace transform (Adams & Hannaford, 1999). The impedances from the virtual environment can be obtained by the Z-transform. If the simulation satisfies the absolute stability conditions and the human arm and virtual object are passive, then the simulation is absolutely stable. The passivity controller can manage the absolute stability condition (2), (4). The condition (2) and (3) are always satisfied because the impedance of the haptic device, contact, and passivity controller are positive. The left term must be larger than the right term in the condition (4) for stable simulation. The condition (4) shows the effect of discretization on the stability. The left term decreases and the right term increases when the effect of ZOH increases. The passivity controller stabilizes the simulation by increasing 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍� on the left term.
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Proceedings of the 20th IFAC World Congress 1348 Myeongjin Kim et al. / IFAC PapersOnLine 50-1 (2017) 1346–1351 Toulouse, France, July 9-14, 2017
The performance can be formulated using the transmitted impedance, 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 to the human as shown in (5). The ideal goal is to transfer same impedance of the virtual object. The transmitted impedance is, however, highly dependent on the contact impedance and passivity controller. 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 = 𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �𝐶𝐶𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 +
𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒
𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒
�
(5).
The transparency increases when the effect of ZOH and passivity controller decrease, the contact impedance increases. The high contact impedance decreases, however, the stability margin on the condition (3) and causes stability problems on the condition (4). Another weak point of the passivity control is that the impulsive damping force makes discontinuity on the haptic feedback because the gain of the passivity controller is determined in real time. It is difficult to achieve stable and transparent simulation using the direct haptic rendering.
2.2 Proxy-based haptic rendering
Fig. 2. Proxy-based haptic rendering.
The proxy was employed to prevent the visual overlapping on the virtual environment. The early proxy approach (McNeely, Puterbaugh, & Troy, 1999) used the constraint condition for the rigid body interaction. The pose of the proxy is determined by the geometric constraint. Dynamics of the virtual tool is used to simulate interaction between the rigid tool and the deformable object (Barbi & James, 2008). Simulation using the proxy-based rendering is formulated as a bilateral control architecture as shown in Fig. 2. The virtual tool is modelled by the tool impedance, 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 . Contact surface of the virtual object is represented using impedance of the virtual environment. The contact impedance is defined between the virtual tool and object because the contact force is computed based on the penetration between the virtual tool and object. The virtual coupling that generates a force proportional to the distance between the virtual tool and haptic device is represented using the impedance of the virtual coupling, 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 (Adams & Hannaford, 1999). The hybrid matrix between the human arm and contact surface of the virtual object is formulated as shown in (6). �
𝐹𝐹𝐹𝐹ℎ ℎ �=� 11 −𝑉𝑉𝑉𝑉𝑒𝑒𝑒𝑒∗ ℎ21
=�
ℎ12 𝑉𝑉𝑉𝑉ℎ �� � ℎ22 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒𝑝𝑝𝑝𝑝∗
𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 � −1 +
1
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �
1
𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
+
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 1 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
�
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
𝑉𝑉𝑉𝑉 � � ℎ∗ � 𝐹𝐹𝐹𝐹 � 𝑒𝑒𝑒𝑒
(6).
The absolute stability condition is formulated as shown in (7), (8), and (9). The stiffness and damping coefficients of the virtual coupling have been determined by the passivity condition (Colgate et al., 1995) or the absolute stability condition (Adams & Hannaford, 1999) without considering the dynamics of the virtual tool and contact impedance. The stability conditions show, however, that the stability is not only affected by the virtual coupling but the contact and tool impedances.
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 �𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 � 1
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 � + 𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
1
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 �𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 � �𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 �𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �
�� ≥ 0 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
� �−1 +
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 � ≥ 0
(7). (8). 1
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 � 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 � + 𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
1 𝑍𝑍𝑍𝑍 � 𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 � 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 �� 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 1 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
� �−1 +
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
1
�
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
+
� 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 ��
�� ≥ (9).
The tool impedance increases the stability margin on the condition (7) and (8) when the tool impedance is positive. The tool impedance is determined by the exploited numerical integration method for the pose update of the virtual tool. The explicit integration often used in the simulation is intuitive and easy to be implemented. The problem of the explicit integration is that the tool impedance has negative value in the certain frequency ranges. The tool impedance must be determined to satisfy the condition (7) and (8) when the explicit integration is employed. The left term must be larger than the right term of the condition (9) for the absolute stability. The right term increases when the effect of discretization increases. The left term increases when the impedance of virtual coupling and the contact decrease. The transmitted impedance to the human is determined by (10). 𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 =
ℎ11 + ∆ℎ 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒 1 + ℎ22 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒
where ∆ℎ = 𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 �
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐 𝑍𝑍𝑍𝑍vc𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
� + 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 �
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐
𝑍𝑍𝑍𝑍𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡
��
𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡 +𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐 Zc
�
(10).
A perfect transparency is obtained when ℎ11 and ℎ22 are zero and ∆ℎ is one. The transparency increases when the impedance of virtual tool decreases, impedance of virtual coupling and contact increase. The requirements for
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transparency conflict with the requirements for the stability. The effect of ZOH cannot be removed in the transmitted impedance. The conventional haptic renderings have the trade-off problem between the absolute stability and transmitted impedance. The right term of the condition (4) and (9) make conflicts between the stability and transparency. If the right term is zero, than the condition (4) and (9) are always satisfied. The right term is determined by the multiplication of ℎ12 and ℎ21 . The transparency is affected by the effect of discretization because the haptic feedback is computed from the virtual environment. If the haptic feedback is computed on the continuous domain, the discretization can be removed. This paper proposes a control architecture which has zero value of ℎ12 for the stability and faithfully transfers the environment impedance. 3. PROPOSED CONTROL ARCHITECTURE USING AN EQUIVALENT IMPEDANCE The constraint-based contact resolution (Duriez, Dubois, Kheddar, & Andriot, 2006) was employed in the field of realtime simulation for realistic visual feedback. The contact force is computed from the laws of physics. The constraint-based method can guarantee the non-intersection between the virtual tool and the object. The contact force computed from the constraint is difficult to be exploited for the haptic rendering because the contact force is difficult to be computed in the haptic update rate. The constraint-based method is used with the virtual coupling in the previous research for this reason. The proposed control architecture is shown in Fig. 3, which transfers an equivalent impedance instead of the computed force from the virtual environment. The proxy is not employed in the proposed scheme. The constraint-based contact resolution is employed to guarantee the non-intersection for the visually plausible simulation. An equivalent impedance, 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 is exploited to generate the haptic feedback on the side of haptic device instead of the computed force from the virtual environment. The hybrid matrix is formulated as shown in (11). �
𝐹𝐹𝐹𝐹ℎ ℎ �=� 11 −𝑉𝑉𝑉𝑉𝑒𝑒𝑒𝑒∗ ℎ21
𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ℎ12 𝑉𝑉𝑉𝑉ℎ �� � = � ℎ22 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒∗ −1
0 𝑉𝑉𝑉𝑉 � � ℎ� 1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 𝐹𝐹𝐹𝐹𝑒𝑒𝑒𝑒∗
(11).
The absolute stability conditions are obtained as shown in (12), (13), and (14). 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 � ≥ 0
(12).
2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 �𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 ) ≥ 0
(14).
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(1⁄𝑍𝑍𝑍𝑍𝑝𝑝𝑝𝑝 ) ≥ 0
Fig. 3. Proposed control architecture of the haptic rendering using an equivalent impedance. 1
𝑍𝑍𝑍𝑍𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 = �𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + 2
1
𝑍𝑍𝑍𝑍𝑐𝑐𝑐𝑐
�𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 � 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒 � = 𝑍𝑍𝑍𝑍𝑚𝑚𝑚𝑚 + 𝑍𝑍𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
(15).
The contact impedance generates a force which makes the displacement on the contact surface. The displacement generated by the contact force is same with the penetration depth between the haptic device and virtual object when the constraint-based method is employed. The contact impedance is same with the impedance of the contact surface of the virtual object in (15). The transmitted impedance is equal to the summation of the haptic device and equivalent impedances. It means that the human feels the inertia of the haptic device in free motion and the inertia of the haptic device and the impedance of the virtual object in contact motion. The perfect transparency is achieved by minimizing the impedance of the haptic device and obtaining the equivalent impedance. The remaining problem is to estimate the equivalent impedance from the virtual object in real time. The contact force is computed based on the penetration depth between virtual object unlike the real environment. The human does not feels contact force by the penetration in the real environment. The equivalent impedance has to represent the contact force and the displacement on the contact surface not the penetration. The equivalent impedance will be developed in our future research.
(13).
The stability conditions are simplified than the conventional haptic renderings. The condition (12) shows the stability margin increases when the impedance of the virtual object is conveyed using the equivalent impedance. The condition (12), (13) and (14) are always satisfied because the impedances on the conditions are positive. The transparency is represented as shown in (15)
4. SIMULATION A one DOF numerical example is simulated to compare the conventional rendering methods with the proposed rendering. The absolute stability conditions in Fig. 4, 5, and 6, and the ratio of the transmitted impedance to the environment impedance in Fig. 7 are compared in the frequency domain. The semi-implicit integration is employed for the numerical integration. The normalized ZOH is used to see the effect of the discretization on the communication between the haptic
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device and the virtual environment. An ideal equivalent impedance which is equal to the environment impedance is used for the proposed rendering. Parameters used for the simulation are listed in the table 1. Table 1. Parameters of the simulation Symbol 𝑚𝑚𝑚𝑚𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 𝑚𝑚𝑚𝑚𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 𝑘𝑘𝑘𝑘𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 𝑚𝑚𝑚𝑚𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 𝑘𝑘𝑘𝑘𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝 dT
Quantity Mass of the haptic device Damping of haptic device Mass of the virtual tool Damping of the virtual coupling Stiffness of the virtual coupling Mass of the virtual object Damping of the virtual coupling Stiffness of the virtual object Time step
Value 0.022 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 0.01 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 0.0001 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 0.06 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 300 𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 1 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 0.06 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 300 𝑁𝑁𝑁𝑁/𝑚𝑚𝑚𝑚 0.03 s
Fig. 6. 2𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(ℎ11 )𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(h22 ) − |ℎ12 ℎ21 | + 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(ℎ12 ℎ21 ) ≥ 0 of the conventional rendering methods and the proposed rendering.
Fig. 4. 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(ℎ11) of the conventional rendering methods and the proposed rendering.
Fig. 5. 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅(ℎ22) of the conventional rendering methods and the proposed rendering.
Fig. 7. Transparency of the conventional rendering methods and the proposed rendering. Figure 4 shows the left term of the conditions (2), (7), and (12). The values must be larger than zero for the absolute stability. The both direct and proxy-based rendering methods show almost zero in the figure 4. The impedance of the haptic device is low in the direct rendering method. The proxy-based rendering method has both the haptic device and tool impedance. The effect of the tool impedance, however, is lower than the proposed rendering method because the tool impedance has to be small for the transparency and the effect of the ZOH. The proposed rendering method has a larger stability margin than other rendering methods because the equivalent impedance increases the stability margin. Figure 5 shows the left term of the conditions (3), (8), and (13). The proxy-based rendering has a larger stability margin than the other rendering methods because the tool impedance increases the left term of the condition (8). Figure 6 shows the left term minus right term of the conditions (4), (9), and (14). The proposed rendering is stable when the other rendering methods are unstable with the same parameters because the proposed rendering removes the effect of the ZOH. Figure 7 shows the transparency of the conventional rendering methods and the proposed rendering. The direct rendering shows higher
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transparency than the proxy-based rendering because the proxy-based rendering has the impedance of not only the haptic device but the virtual tool. The proposed rendering shows the higher transparency than the direct rendering because the proposed rendering directly computes the haptic feedback on the haptic device.
5. CONCLUSION This paper proposes a new control architecture for the transparent and stable simulation of the interaction between a rigid tool and deformable object. The transparency is often discussed using the maximum displayable stiffness in the field of virtual simulation. The transmitted impedance is more important than the maximum displayable stiffness in the simulation of the interaction between a rigid tool and deformable objects. The conventional haptic rendering formulated as the bilateral control architecture shows the trade-off between the absolute stability and transmitted impedance. This paper removes the trade-off problem by conveying an equivalent impedance instead of the computed force from the virtual environment. The haptic feedback can be computed on the continuous time using the equivalent impedance that reflects the relationship between the contact force and displacement of the contact point not the penetration. The proposed control scheme always satisfies the absolute stability condition regardless of the transparency and shows higher transparency than the conventional haptic rendering methods.
ACKNOWLEDGEMENT This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (NRF-2015R1A2A1A10054420)
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