Parasitic Effects of Device Coupling on Haptic ... - Semantic Scholar
Parasitic Effects of Device Coupling on Haptic Performance Colin Gallacher1 , John Willes2 , and Jozsef K¨ovecses3
Abstract— The device dynamics of haptic systems play a crucial role in the performance of the human user. Typical operation of a haptic device during simulation requires the user to perform common tasks that constitute the bases for all user actions. These common tasks include manipulation, selection, and navigation. Navigation requires the user to transition the end-effector across regions of the device workspace moving to a new location of interest within the virtual scene. In this study we investigate the role that the inertia tensor coupling has on user performance during navigational tasks. We also adapt the operation and admissible-motion space representation for haptic systems in which forces causing deviations from a desired path can be thought of as parasitic forces that degrade a users performance. Dynamic simulations were carried out to gain insight into the effects of navigating along paths of varying coupling using a 2DOF five-bar mechanism and were experimentally validated with a Quansar 2DOF Pantograph device.
I. INTRODUCTION Performance evaluation of force feedback haptic devices presents a significant hurdle towards the general acceptance of haptic technologies by the public. A wide range of device mechanisms and uses poses a seriously challenging task of evaluating how a device will perform. Haptic systems are analysed based on the device properties in the unpowered, powered, and interactive state [9], [19], [20]. Still, we lack valuable quantitative metrics that consumers can easily understand and use to evaluate the quality of a haptic device. Progress has been made to characterize the performance of specific devices based on their ability to perform in specified tasks that are fairly universal for haptic applications based on psychophysical evaluation testbeds [4]–[6], [13], [20]. These encompassing tasks can be partitioned into navigation, manipulation, and selection tasks [4], [5], [20]. Theoretically, a device’s ability to perform well in these tasks should correspond to the overall usability of the device and thus, if quantified, could represent a means to discern a ’good’ device from a ’bad’ device. The usage of tried psychophysical evaluation techniques such as the tapping test for navigational tasks and the peg in hole test for manipulation and selection tasks, allow for the quantification of a device performance based on the users ability to perform 1 Colin Gallacher is with the Centre for Intelligent Machines, Faculty of Mechanical Engineering, McGill University, Montreal, Qc
[email protected] 2 John Willes is with the Center for Intelligent Machines, Faculty of Mechanical Engineering, McGill University, Montreal, Qc
[email protected] 3 J´ ozsef K ovecses is with the Applied Dynamics Group, Faculty of Mechanical Engineering, McGill University, Monteral, QC
[email protected]
a task of varying difficulty in a specified time [5], [7], [17], [20]. These techniques have preliminarily been shown to be an effective means of evaluating distinct devices to perform a desired task. However, these evaluations are based on an ad hoc usage of the device workspace to accomplish said tasks. In this study we perform preliminary investigations into the effects of the coupling terms of the inertia dyad on haptic device performance during navigation. We seek to evaluate the forces that a device may impart upon a user decreasing their ability to accomplish a simple task, yet nearly universally required, of moving from one area of a workspace to another along a straight line. These forces will here on out be referred to as parasitic forces due to their nature of imparting unwanted forces upon a user during haptic simulation and negatively impacting a users perceived haptic experience. The understanding of the nature of these parasitic forces will be generalized but will be further elaborated upon for the 2D case of a planar-five-bar mechanism. Finally, simulation will be compared to experimental results for the case of the five-bar-mechanism to determine if, and then to quantify by how much, a device performance for a standard navigational task can be influenced by the trajectory and orientation of the same task. The intention of this study is a desire to move towards better using device dynamics to predict user performance for haptic systems and to gain insight into understanding what dynamic effects truly impact user performance. Further motivation for this study is to formalize the notion of parasitic forces resulting from the manipulator dynamics that can decrease performance. II. THEORETICAL BACKGROUND To better familiarize the reader with the nature of the experiment, here we will elaborate on the background theory required for the study. A. Fitts’ Law Drawing from Shannon’s seminal information theory work, Mathematical Theory of Communication [21], Fitts hypothesised that the time it took for a human to accomplish a task was linearly proportional to the difficulty of the task [7]. Fitts required users to tap a stylus between two targets of varying distance and size while measuring the time taken to perform the task. He was able to devise a relationship between movement tasks and their associated difficulty (now refereed to as Fitts’ Law) analogous to Shannon’s Theorem [21]:
ID (1) IP Where, M T is the measured time, ID , is termed the Index of Difficulty and, IP , the Index of Performance. The ID was defined by Fitts as, 2A (2) ID = log2 W MT =
ID, is a way of non-dimensionalizing the difficulty of a movement task and corresponds to the ratio of the distance of an object, A, to the characteristic width of the object, W . The use of the log2 in (2) allows for the ratio of distance to object width (corresponding to the associated difficulty of a task) to be expressed in units of bits. Rearranging (1), we see the IP can be expressed in units of bits . sec The Fitts’ Law model of human response time as a function of task difficulty in 1D has held up well against scientific scrutiny [17], [20] and the results have been replicated and expanded upon for the 2D Steering Law [1], [2], [25] and even somewhat so for the 3D cases [18]. The validity of Fitts’ Law as a tool for measuring the performance of haptic devices and virtual displays has also been demonstrated [13], [20], [22], [24]. For a more detailed introduction to the intricacies of Fitts’ Law, including an attempt at a derivation of the equation from physical principals, the authors direct the reader to reference [17]. The form of Fitts’ Law we will be using is referred to as Shannon’s variation [17] as it is suggested by a direct analogy with Shannon’s information theorem and is expressed as: A + We M T = a + b log2 (3) We Where, We , is defined as the Effective Target Width [23]. This variation of Fitts’ Law has been shown to better model user performance for low ID tasks and ballistic motion [17].
x˙ = Rq˙ (5) T T and R = J B . J is where x˙ = x˙ op x˙ adm the Jacobian mapping the joint variables to the operational space task coordinates. The operation space coordinates are associated with a space of constrained motion for the device end-effector. We impose that R must be invertible through the selection of the matrix B which maps the joint velocities to the admissible-motion space. This allows us to express the joint velocities as: (6) q˙ = R−1 x˙
The variations can then be expressed as: δq T = δxT R−T
(7)
The substitution of (7) into (4) allows us to write the variational form of the system dynamics in terms of the operational and admissible-motion spaces. δxT R−T (M q¨ + c − τ ) = 0
(8)
The vector form of the operational and admissible-motion space dynamics with respect to the task space is then written as: R−T M q¨ + R−T c = R−T τ
(9)
Differentiation of (6) leads to the acceleration level relationship: ¨ + R˙−1 x˙ q¨ = R−1 x
(10)
Substitution of (10) into (9) leads to the device dynamics and kinematic parametrizations being represented in the operation and admissible-motion space minimum realization. ¨ + R−T M R˙−1 x˙ + R−T c = R−T τ (11) R−T M R−1 x
B. Dynamic Formulations and Representations In order to gain insight into the role parasitic forces may have on a haptic user, we must first develop a dynamic representation of the device. This section aims to develop a generalized minimum representation for a haptic device that partitions the forces the device imparts on the user during navigation into those that exist within the operational space and those in an orthogonal space defined as the admissiblemotion space. This representation has been described in detail [14], [15]. The decomposition of the dynamics into the operational and admissible-motion spaces as it pertains to this study is later explained in section III (A). The variational form of the system dynamics is expressed with the scalar equation: δq T (M q¨ + c − τ ) = 0
The joint space velocities are mapped to the operational and admissible-motion spaces via the transformation,
(4)
Here the inertia tensor, M , centrifugal and Coriolis forces, c, and the generalized torques, τ , are represented in the joint space.
Grouping the acceleration and velocity terms together we arrive at the equation: ¨ + R−T c¯ = R−T τ R−T M R−1 x
(12)
The first term corresponds the the inertial forces felt in the operational and admissible-motion spaces as a result of the device acceleration. ¨ facc = R−T M R−1 x
(13)
The second term corresponds to the inertial forces associated with the device velocities often called the centrifugal and Coriolis forces. fvel = R−T c¯
(14)
The term on the right hand side of (12) corresponds to the applied forces at the haptic end-effector with respect to the operational and admissible-motion spaces.
If we define W = R−T M R−1 , z = R−T c¯, and s = R τ we can rewrite the dynamic equations as: −T
¨+z =s Wx This can be represented in block matrix form as: Wop Woa x ¨op z s + op = op Wao Wa x ¨a za sa
(15)
(16)
The force that we feel in the admissible-motion space is the parasitic force that a user trying to navigate strictly in the operational space would feel for a haptic device. The dynamic equation associated with the parasitic forces is then written as: ¨ op + Wa x ¨ a + za = s a Wao x
(17)
Given the constraint forces, the current device trajectory, and the acceleration in the operational space, we can solve for the acceleration the device will undergo in the admissiblemotion space. ¨ a = Wa−1 (sa − za − W ao x ¨ op ) x
(18)
Finally, (18) is integrated to determine the device trajectory in the admissible-motion space and solve for the simulated trajectory in the base coordinate frame. III. METHODS AND ASSUMPTIONS In this study we seek to determine if a user of a haptic device travelling between two points is influenced by the coupling of the device inertia and to quantify changes in their ability to perform the task. We assume the user seeks to travel along a straight line between the starting and end point, ultimately coming to rest in the quickest time possible. This is a discrete form of a traditional tapping test and can be modelled suitably by Fitts’ law [2], [17]. The dynamics for a parallel five-bar mechanism were modelled and represented in a Cartesian coordinate frame attached to the base of the device. The dynamic parameters were produced in accordance with the specifications of the Quansar 2DOF planar pantograph (Fig. 1). Much work has already been done on geometric interpretations of the inertia tensor [3], [10], [12]. The quadratic form of the kinetic energy equation at unit magnitude is used to generate the device inertia along the principle axes with respect to the base frame. v T M v = v T P T DP v = 1
(19)
The column entries of P represent the directions in the base frame along which the device inertia is decoupled. These are significant as an acceleration in a direction that is decoupled will not result in a parasitic force or subsequent acceleration in any orthogonal space. If there is a direction of minimum coupling, so too must there be a direction along which the coupling terms are maximum. These are of particular interest in determining the effects of coupling on device performance.
Fig. 1. The kinematic representation of a planar 5-bar parallel mechanism. θ1 and θ2 are the actuated joints and the end effector position is represented by xp and yp coordinates with respect a Cartesian frame attached to the base.
By performing a similarity transformation on the device inertia, the directions along which maximum coupling occur can be determined at each point in the device workspace (Fig. 2). This is analogous to computing the magnitude and orientations of the maximum shear stress in a stress tensor analysis. Paths of maximum coupling were procedurally generated and superimposed upon a contour map of the logarithm of the coupling magnitudes(Fig. 3). This tool allowed us to select a region to study the navigation task as we wanted to minimize the effect that changes in the magnitude of the coupling had on path traversal. We felt this was important because operation within a region of too little coupling magnitude would have an insignificant impact on a users performance regardless of path orientation. Simultaneously, we wanted to reduce changes in coupling magnitude while navigating to better isolate path orientation as the independent variable. While this is not perfectly realizable, the paths we select fall within regions where the magnitude of the coupling is on the order of 10−2 -10−4 kg. This allows us to select a reasonably long path for the navigation task while in a roughly representative coupling magnitude range for the workspace; where the calculated upper and lower bounds of the magnitudes are on the order of 102 -10−8 kg , respectively. Thus, three path orientations were selected based on the directions of coupling (Fig. 3): (i) in the direction of inertia decoupling, (ii) in a direction corresponding to a high degree of inertia coupling, and (iii) a direction that represents an ’arbitrary’ path neither along a decoupled or max coupled direction. A. Simulation Simulations of a user navigating along the above selected paths were developed. The device dynamics were transformed into a minimum representation as described in section II(B). Modelling of the user’s own inertia was neglected. The minimum dynamic representation decomposes the device workspace into an operational space, along which a user wishes to travel, and an orthogonal admissible-motion space,
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0 x(m)
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Fig. 3. A color gradient is associated with the log values of the coupling terms throughout the device workspace. The paths 1,2 and 3 were selected to minimize the relative change in magnitude of coupling for a navigational task across the device workspace.
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Fig. 2. The dynamic representations of the inertia dyad at various points throughout the workspace in the decoupled directions (top) and in the directions of maximum coupling (bottom). The vector length corresponds to 1/λ, where λ is the inertia associated with the vector direction. This is equivalent the magnitude of acceleration for unit force input in the indicated direction.
into which the user’s desired trajectory may deviate as a result parasitic forces. This can be better understood by examining Fig. 4. By changing the navigational task to coincide with directions of varying coupling and comparing the IP of the user we can develop an understanding of the role of coupling on haptic performance. We aim to simulate the navigational task to theoretically predict the performance trends that may occur as well as what the largest contributing factors are to the parasitic forces experienced. Our primary assumption in simulation is that the user does not provide extensive feedback to correct for deviations from the path that may result from the device dynamics. This assumption is based on the evidence that a user performing rapid navigational tasks with a low ID may do so ballistically (i.e without visual feedback control) [8], [16]. This is potentially in violation of Fitts’ law [8], [11] though we are more concerned in the change of effective target width, We ,(and thus the ID) with respect to task orientation along directions of increased coupling. Thus we prescribe a constant acceleration in the operational space but do not constrain movement in the admissible-motion space. The accelerations were determined by performing a tap-
Fig. 4. In the above figure the ideal path of the user is along the straight path represented by the solid lined device and falls entirely within the operational space, op. This would be true for a decoupled device. The dotted lined device and corresponding path represent the true trajectory that results from the undesirable parasitic inertial effects caused by device coupling. The deviation from the desired trajectory exists within the admissible motion space, adm.
ping test with a untethered stylus. The average duration it took for participants to tap between 10 cm targets was between 0.36 and 0.44 s. We use this information and the path length to solve for a reasonable range for human acceleration. For the purposes of the simulation the operational space acceleration was kept constant and due to the restriction that the user must come to rest at the end of their journey, this acceleration is reversed after half the operational path length. This occurs when the projection of the simulated trajectory onto the operational space is greater than half the distance of the desired straight path. We perform the simulations moving along the path for each orientation angle, (90◦ , 45◦ , 0◦ ), while varying the operational space accelerations, (3.09, 2.5, 2.07) sm2 , for each test. The simulations were then carried out for the reverse directions along each path. Finally, to examine the effects of the inertial forces resulting from acceleration opposed to those resulting from velocity, the simulations were carried out again with a constant velocity for the same duration of time which corresponds to
Fig. 5. A Quansar 2DoF Pantograph was used for the experiment. When prompted with a verbal cue, users were asked to move quickly and accurately along the trajectories represented by the coloured targets. Users moved along both the forward and reverse trajectories using a target size of either 1.9cm or 0.64cm.
the average velocities from the original tests. The IP was determined for each path using eq. 3 where A and M T were specified and the We was the resulting distance error from the end point in the admissible-motion space (Fig. 4). Changes in the IP were then plotted against the path orientation for various accelerations and directions along the specified paths. B. Experiments Human experiments were preformed to validate the simulations using the Quansar 2DOF planar pantograph (Fig. 5). The testbed was set-up using the same path lengths and orientations as those simulated. Twelve undergraduate student volunteers with no prior operation of the device were selected to perform the tests. Users were asked to traverse the paths as quickly and accurately as possible. The tested trajectories included were: three path orientations ((90◦ , 45◦ , 0◦ ) located at the same positions in the workspace, as well as forward and reversed direction along the paths. The constant velocity experiment was excluded. Large and small physical target sizes, 1.9 cm, and 0.64 cm, respectively, were also included. In total these variations accounted for twelve movements per participant. The order of the motions performed were selected randomly to reduce bias from gained familiarity of the device throughout the trial. The users were provided with an auditory cue to begin the test from which a timer began and stopped when the user crossed a finish line at the leading edge of the target circle as seen in Fig. 6. This is known as a goal-crossing task and can also be modelled by Fitts’ Law [2]. The time to complete the task as well as the magnitude of the distance error from the center of the target circle was recorded for each test. The ID was then determined from the experimental data using the effective width of the target, We , for each test. Here, We , is defined as the distance from the center of the target in which all the users trajectories fell within two standard deviations (95%) of the measured error. This can
Fig. 6. The user moves from the start to stop position. Upon crossing the line tangent to the leading edge, the end-effector position was recorded and the error was calculated as the distance from the centreline. The characteristic width We is then defined as the distance from the center line that falls within two standard deviations (95%) of the measured user error (err) data.
be better illustrated graphically by examining Fig. 6 .It must be noted a linear regression was not used to calculate the IPs as is commonly done in a tapping test. The objective of this study was instead to see how the effective width, We , and thus difficulty of a task, changes along directions of varying coupling. For each of the twelve tests the measured time, MT, was evaluated as the average time of the users to complete the specified task. Thus, under the assumption that (1) holds true using the ID as defined in (3), the IPs for the device was then plotted for various path orientations and directions. IV. RESULTS A. Simulation The simulated IPs are plotted for various accelerations and orientations as seen in figure Fig. 7. The results predict that the device performance will be greatest along the decoupled directions and will suffer most along the approximate direction of maximum coupling. Along the ’arbitrary’ third direction the device performance falls, not surprisingly, between the values for the maximum and minimum coupling pathways. The simulated results more interestingly predict that the users ability to perform a translational navigation task will also be effected by their direction along the path. Moving from an area of high coupling to one of lower coupling causes a larger deviation of the end-effector from the desired trajectory during rapid navigation. Another interesting observation lies in the comparison of the work done by the inertial effects of the device on the human user in the admissible-motion space direction. By integrating the work along the simulated path and comparing the ratio of work done by the acceleration-dependent inertial force to the net work done in the direction of admissiblemotion. Σfacc ∆ua (20) ρ= Σ(facc + fvel )∆ua
1 15
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IP (bits/sec)
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0 pi/2 (Path 1) pi/4 (Path 2) Angle (rads)
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Fig. 7. Simulated Indices of Performance, IP , using a user tap test conducted before the experiment to determine a range of suitable accelerations. The decoupled path (Path 1) has the highest index of performance, the path of maximum coupling (Path 2) has the lowest index of performance and the ’arbitrary’ path (Path 3) falls in between. The indices of performance were lower in the reverse direction for each path.
we see that, even for the case of the highest magnitude of acceleration, the parasitic forces that arise from the acceleration-dependent inertia term only accounts for roughly half of the total work done in the admissiblemotion space (Fig. 8). This suggests, contrary to what the authors expected, that the velocity-dependent inertial terms contribute greatly to the parasitic inertia felt by the user. Thus seeking to greatly improve haptic device performance by way of dynamic analysis it may be necessary to consider both the velocity-dependent and acceleration-dependent inertial terms, and perhaps even just the velocity-dependent terms for simulations where high accelerations are not to be expected. B. Experiment The IPs were calculated for the Quansar 2DOF pantograph haptic device for the prescribed operational paths as well as for the forward and reverse trajectories. The ,We results can be seen in Table I and are graphically represented in Fig. 9. While there was little discernible difference in the IPs for the forward trajectories, reversal of the direction to move from regions of high coupling to low coupling closely resemble the trends predicted by simulation. TABLE I E XPERIMENTAL EFFECTIVE WIDTH , We . Direction Forward Reverse
Path 1 0.023 m 0.024 m
Path 2 0.022 m 0.059 m
Path 3 0.026 m 0.034 m
The average time that subjects completed the navigational task was substantially slower than the values used in simulation with the average time to complete the forward path task,
pi/4 (Path 2) Angle (rads) forward
0 (Path 3) reversed
Fig. 8. ρ, is defined as the ratio of work done by acceleration-dependent inertial terms to the work done by the sum of inertial acceleration-dependent and velocity-dependent terms. The work done by the parasitic forces resulting from the acceleration-dependent terms is, in the worst cast scenario (i.e the highest magnitudes of acceleration), approximately equal to the velocitydependent inertial terms. This may indicate that velocity-dependent terms play a much more significant role in performance degradation resulting from parasitic forces than their acceleration-dependent counterparts.
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Fig. 9. Experimental effective width, We , represented by circle size. This diagram corresponds to the experimental results for path traversal in the reversed direction (i.e from regions of higher coupling to lower).
tf = 0.914s (σ = 0.068s), and the average time to complete the path reversed task, tr = 0.881s (σ = 0.046s). The simulation was updated to use a new time duration of tdur = 0.881s, corresponding to the experimental data for the reversed direction, and the new IPs were subsequently plotted against the varying path directions. Without changing anything but the acceleration to more accurately represent the true performance of the human user (corresponding to an acceleration of 0.49 sm2 ) the theoretical results much better reflect the data attained (Fig. 10). V. DISCUSSION At first glance the results of this study suggest something that is seemingly intuitive, that the inertial coupling for a haptic manipulator can negatively impact the operational performance, and more specifically, navigational tasks. The motivation, however, lies in the desire to move towards better
Fig. 10. Re-simulated indices of performance using the average time of experimental users to calculate the simulated acceleration.
using device dynamics to predict user performance for haptic systems. Through proper dynamic analysis and experimental validation we can seek to design devices better suited to performing simulation specific tasks, whether they be navigation, manipulation, or selection based. This requires experimentally correlating changes in performance with varying dynamic properties. The initial results of this study suggest that rapid navigation in regions of high coupling reduce accuracy for quasi-ballistic motions. A second point of importance in this work was the adaptation of the operational and admissible-motion space dynamic formulations to be interpreted for a haptic device. Expressing the device dynamics in terms of the end-effector (where the human-device interaction occurs) with respect to a desired operational task we can meaningfully decompose the resulting inertial forces into parasitic forces that can negatively impact the performance (accuracy in the case of a navigational task) of a user. This may prove promising for reducing parasitic forces using redundant mechanisms where operation within the null space of the inertia tensor may greatly eliminate parasitic inertial effects. Interestingly in this study, the parasitic forces resulting from the velocitydependent inertial terms in simulation were found to be of greater impact on a decrease in device performance than the acceleration-dependent terms. These findings should be further investigated to determine their role on haptic performance. R EFERENCES [1] Johnny Accot and Shumin Zhai. Performance evaluation of input devices in trajectory-based tasks: An application of the steering law. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, CHI ’99, pages 466–472, New York, NY, USA, 1999. ACM. [2] Johnny Accot and Shumin Zhai. More than dotting the i’s — foundations for crossing-based interfaces. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, CHI ’02, pages 73–80, New York, NY, USA, 2002. ACM. [3] Haruhiko Asada. A geometrical representation of manipulator dynamics and its application to arm design. J. Dyn. Sys., Meas.,Control, 105(3):131–142, 1983.
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Abstract— The device dynamics of haptic systems play a crucial role in the performance of the human user. Typical operation of a haptic device during simulation requires the user to perform common tasks that constitute the bases for all user actions. These common tasks include manipulation, selection, and navigation. Navigation requires the user to transition the end-effector across regions of the device workspace moving to a new location of interest within the virtual scene. In this study we investigate the role that the inertia tensor coupling has on user performance during navigational tasks. We also adapt the operation and admissible-motion space representation for haptic systems in which forces causing deviations from a desired path can be thought of as parasitic forces that degrade a users performance. Dynamic simulations were carried out to gain insight into the effects of navigating along paths of varying coupling using a 2DOF five-bar mechanism and were experimentally validated with a Quansar 2DOF Pantograph device.
I. INTRODUCTION Performance evaluation of force feedback haptic devices presents a significant hurdle towards the general acceptance of haptic technologies by the public. A wide range of device mechanisms and uses poses a seriously challenging task of evaluating how a device will perform. Haptic systems are analysed based on the device properties in the unpowered, powered, and interactive state [9], [19], [20]. Still, we lack valuable quantitative metrics that consumers can easily understand and use to evaluate the quality of a haptic device. Progress has been made to characterize the performance of specific devices based on their ability to perform in specified tasks that are fairly universal for haptic applications based on psychophysical evaluation testbeds [4]–[6], [13], [20]. These encompassing tasks can be partitioned into navigation, manipulation, and selection tasks [4], [5], [20]. Theoretically, a device’s ability to perform well in these tasks should correspond to the overall usability of the device and thus, if quantified, could represent a means to discern a ’good’ device from a ’bad’ device. The usage of tried psychophysical evaluation techniques such as the tapping test for navigational tasks and the peg in hole test for manipulation and selection tasks, allow for the quantification of a device performance based on the users ability to perform 1 Colin Gallacher is with the Centre for Intelligent Machines, Faculty of Mechanical Engineering, McGill University, Montreal, Qc
[email protected] 2 John Willes is with the Center for Intelligent Machines, Faculty of Mechanical Engineering, McGill University, Montreal, Qc
[email protected] 3 J´ ozsef K ovecses is with the Applied Dynamics Group, Faculty of Mechanical Engineering, McGill University, Monteral, QC
[email protected]
a task of varying difficulty in a specified time [5], [7], [17], [20]. These techniques have preliminarily been shown to be an effective means of evaluating distinct devices to perform a desired task. However, these evaluations are based on an ad hoc usage of the device workspace to accomplish said tasks. In this study we perform preliminary investigations into the effects of the coupling terms of the inertia dyad on haptic device performance during navigation. We seek to evaluate the forces that a device may impart upon a user decreasing their ability to accomplish a simple task, yet nearly universally required, of moving from one area of a workspace to another along a straight line. These forces will here on out be referred to as parasitic forces due to their nature of imparting unwanted forces upon a user during haptic simulation and negatively impacting a users perceived haptic experience. The understanding of the nature of these parasitic forces will be generalized but will be further elaborated upon for the 2D case of a planar-five-bar mechanism. Finally, simulation will be compared to experimental results for the case of the five-bar-mechanism to determine if, and then to quantify by how much, a device performance for a standard navigational task can be influenced by the trajectory and orientation of the same task. The intention of this study is a desire to move towards better using device dynamics to predict user performance for haptic systems and to gain insight into understanding what dynamic effects truly impact user performance. Further motivation for this study is to formalize the notion of parasitic forces resulting from the manipulator dynamics that can decrease performance. II. THEORETICAL BACKGROUND To better familiarize the reader with the nature of the experiment, here we will elaborate on the background theory required for the study. A. Fitts’ Law Drawing from Shannon’s seminal information theory work, Mathematical Theory of Communication [21], Fitts hypothesised that the time it took for a human to accomplish a task was linearly proportional to the difficulty of the task [7]. Fitts required users to tap a stylus between two targets of varying distance and size while measuring the time taken to perform the task. He was able to devise a relationship between movement tasks and their associated difficulty (now refereed to as Fitts’ Law) analogous to Shannon’s Theorem [21]:
ID (1) IP Where, M T is the measured time, ID , is termed the Index of Difficulty and, IP , the Index of Performance. The ID was defined by Fitts as, 2A (2) ID = log2 W MT =
ID, is a way of non-dimensionalizing the difficulty of a movement task and corresponds to the ratio of the distance of an object, A, to the characteristic width of the object, W . The use of the log2 in (2) allows for the ratio of distance to object width (corresponding to the associated difficulty of a task) to be expressed in units of bits. Rearranging (1), we see the IP can be expressed in units of bits . sec The Fitts’ Law model of human response time as a function of task difficulty in 1D has held up well against scientific scrutiny [17], [20] and the results have been replicated and expanded upon for the 2D Steering Law [1], [2], [25] and even somewhat so for the 3D cases [18]. The validity of Fitts’ Law as a tool for measuring the performance of haptic devices and virtual displays has also been demonstrated [13], [20], [22], [24]. For a more detailed introduction to the intricacies of Fitts’ Law, including an attempt at a derivation of the equation from physical principals, the authors direct the reader to reference [17]. The form of Fitts’ Law we will be using is referred to as Shannon’s variation [17] as it is suggested by a direct analogy with Shannon’s information theorem and is expressed as: A + We M T = a + b log2 (3) We Where, We , is defined as the Effective Target Width [23]. This variation of Fitts’ Law has been shown to better model user performance for low ID tasks and ballistic motion [17].
x˙ = Rq˙ (5) T T and R = J B . J is where x˙ = x˙ op x˙ adm the Jacobian mapping the joint variables to the operational space task coordinates. The operation space coordinates are associated with a space of constrained motion for the device end-effector. We impose that R must be invertible through the selection of the matrix B which maps the joint velocities to the admissible-motion space. This allows us to express the joint velocities as: (6) q˙ = R−1 x˙
The variations can then be expressed as: δq T = δxT R−T
(7)
The substitution of (7) into (4) allows us to write the variational form of the system dynamics in terms of the operational and admissible-motion spaces. δxT R−T (M q¨ + c − τ ) = 0
(8)
The vector form of the operational and admissible-motion space dynamics with respect to the task space is then written as: R−T M q¨ + R−T c = R−T τ
(9)
Differentiation of (6) leads to the acceleration level relationship: ¨ + R˙−1 x˙ q¨ = R−1 x
(10)
Substitution of (10) into (9) leads to the device dynamics and kinematic parametrizations being represented in the operation and admissible-motion space minimum realization. ¨ + R−T M R˙−1 x˙ + R−T c = R−T τ (11) R−T M R−1 x
B. Dynamic Formulations and Representations In order to gain insight into the role parasitic forces may have on a haptic user, we must first develop a dynamic representation of the device. This section aims to develop a generalized minimum representation for a haptic device that partitions the forces the device imparts on the user during navigation into those that exist within the operational space and those in an orthogonal space defined as the admissiblemotion space. This representation has been described in detail [14], [15]. The decomposition of the dynamics into the operational and admissible-motion spaces as it pertains to this study is later explained in section III (A). The variational form of the system dynamics is expressed with the scalar equation: δq T (M q¨ + c − τ ) = 0
The joint space velocities are mapped to the operational and admissible-motion spaces via the transformation,
(4)
Here the inertia tensor, M , centrifugal and Coriolis forces, c, and the generalized torques, τ , are represented in the joint space.
Grouping the acceleration and velocity terms together we arrive at the equation: ¨ + R−T c¯ = R−T τ R−T M R−1 x
(12)
The first term corresponds the the inertial forces felt in the operational and admissible-motion spaces as a result of the device acceleration. ¨ facc = R−T M R−1 x
(13)
The second term corresponds to the inertial forces associated with the device velocities often called the centrifugal and Coriolis forces. fvel = R−T c¯
(14)
The term on the right hand side of (12) corresponds to the applied forces at the haptic end-effector with respect to the operational and admissible-motion spaces.
If we define W = R−T M R−1 , z = R−T c¯, and s = R τ we can rewrite the dynamic equations as: −T
¨+z =s Wx This can be represented in block matrix form as: Wop Woa x ¨op z s + op = op Wao Wa x ¨a za sa
(15)
(16)
The force that we feel in the admissible-motion space is the parasitic force that a user trying to navigate strictly in the operational space would feel for a haptic device. The dynamic equation associated with the parasitic forces is then written as: ¨ op + Wa x ¨ a + za = s a Wao x
(17)
Given the constraint forces, the current device trajectory, and the acceleration in the operational space, we can solve for the acceleration the device will undergo in the admissiblemotion space. ¨ a = Wa−1 (sa − za − W ao x ¨ op ) x
(18)
Finally, (18) is integrated to determine the device trajectory in the admissible-motion space and solve for the simulated trajectory in the base coordinate frame. III. METHODS AND ASSUMPTIONS In this study we seek to determine if a user of a haptic device travelling between two points is influenced by the coupling of the device inertia and to quantify changes in their ability to perform the task. We assume the user seeks to travel along a straight line between the starting and end point, ultimately coming to rest in the quickest time possible. This is a discrete form of a traditional tapping test and can be modelled suitably by Fitts’ law [2], [17]. The dynamics for a parallel five-bar mechanism were modelled and represented in a Cartesian coordinate frame attached to the base of the device. The dynamic parameters were produced in accordance with the specifications of the Quansar 2DOF planar pantograph (Fig. 1). Much work has already been done on geometric interpretations of the inertia tensor [3], [10], [12]. The quadratic form of the kinetic energy equation at unit magnitude is used to generate the device inertia along the principle axes with respect to the base frame. v T M v = v T P T DP v = 1
(19)
The column entries of P represent the directions in the base frame along which the device inertia is decoupled. These are significant as an acceleration in a direction that is decoupled will not result in a parasitic force or subsequent acceleration in any orthogonal space. If there is a direction of minimum coupling, so too must there be a direction along which the coupling terms are maximum. These are of particular interest in determining the effects of coupling on device performance.
Fig. 1. The kinematic representation of a planar 5-bar parallel mechanism. θ1 and θ2 are the actuated joints and the end effector position is represented by xp and yp coordinates with respect a Cartesian frame attached to the base.
By performing a similarity transformation on the device inertia, the directions along which maximum coupling occur can be determined at each point in the device workspace (Fig. 2). This is analogous to computing the magnitude and orientations of the maximum shear stress in a stress tensor analysis. Paths of maximum coupling were procedurally generated and superimposed upon a contour map of the logarithm of the coupling magnitudes(Fig. 3). This tool allowed us to select a region to study the navigation task as we wanted to minimize the effect that changes in the magnitude of the coupling had on path traversal. We felt this was important because operation within a region of too little coupling magnitude would have an insignificant impact on a users performance regardless of path orientation. Simultaneously, we wanted to reduce changes in coupling magnitude while navigating to better isolate path orientation as the independent variable. While this is not perfectly realizable, the paths we select fall within regions where the magnitude of the coupling is on the order of 10−2 -10−4 kg. This allows us to select a reasonably long path for the navigation task while in a roughly representative coupling magnitude range for the workspace; where the calculated upper and lower bounds of the magnitudes are on the order of 102 -10−8 kg , respectively. Thus, three path orientations were selected based on the directions of coupling (Fig. 3): (i) in the direction of inertia decoupling, (ii) in a direction corresponding to a high degree of inertia coupling, and (iii) a direction that represents an ’arbitrary’ path neither along a decoupled or max coupled direction. A. Simulation Simulations of a user navigating along the above selected paths were developed. The device dynamics were transformed into a minimum representation as described in section II(B). Modelling of the user’s own inertia was neglected. The minimum dynamic representation decomposes the device workspace into an operational space, along which a user wishes to travel, and an orthogonal admissible-motion space,
0.4
y(m)
0.3 0.2 0.1 0 −0.1 −0.4
−0.2
0 x(m)
0.2
0.4
Fig. 3. A color gradient is associated with the log values of the coupling terms throughout the device workspace. The paths 1,2 and 3 were selected to minimize the relative change in magnitude of coupling for a navigational task across the device workspace.
0.4
y(m)
0.3 0.2 0.1 0 −0.1 −0.4
−0.2
0 x(m)
0.2
0.4
Fig. 2. The dynamic representations of the inertia dyad at various points throughout the workspace in the decoupled directions (top) and in the directions of maximum coupling (bottom). The vector length corresponds to 1/λ, where λ is the inertia associated with the vector direction. This is equivalent the magnitude of acceleration for unit force input in the indicated direction.
into which the user’s desired trajectory may deviate as a result parasitic forces. This can be better understood by examining Fig. 4. By changing the navigational task to coincide with directions of varying coupling and comparing the IP of the user we can develop an understanding of the role of coupling on haptic performance. We aim to simulate the navigational task to theoretically predict the performance trends that may occur as well as what the largest contributing factors are to the parasitic forces experienced. Our primary assumption in simulation is that the user does not provide extensive feedback to correct for deviations from the path that may result from the device dynamics. This assumption is based on the evidence that a user performing rapid navigational tasks with a low ID may do so ballistically (i.e without visual feedback control) [8], [16]. This is potentially in violation of Fitts’ law [8], [11] though we are more concerned in the change of effective target width, We ,(and thus the ID) with respect to task orientation along directions of increased coupling. Thus we prescribe a constant acceleration in the operational space but do not constrain movement in the admissible-motion space. The accelerations were determined by performing a tap-
Fig. 4. In the above figure the ideal path of the user is along the straight path represented by the solid lined device and falls entirely within the operational space, op. This would be true for a decoupled device. The dotted lined device and corresponding path represent the true trajectory that results from the undesirable parasitic inertial effects caused by device coupling. The deviation from the desired trajectory exists within the admissible motion space, adm.
ping test with a untethered stylus. The average duration it took for participants to tap between 10 cm targets was between 0.36 and 0.44 s. We use this information and the path length to solve for a reasonable range for human acceleration. For the purposes of the simulation the operational space acceleration was kept constant and due to the restriction that the user must come to rest at the end of their journey, this acceleration is reversed after half the operational path length. This occurs when the projection of the simulated trajectory onto the operational space is greater than half the distance of the desired straight path. We perform the simulations moving along the path for each orientation angle, (90◦ , 45◦ , 0◦ ), while varying the operational space accelerations, (3.09, 2.5, 2.07) sm2 , for each test. The simulations were then carried out for the reverse directions along each path. Finally, to examine the effects of the inertial forces resulting from acceleration opposed to those resulting from velocity, the simulations were carried out again with a constant velocity for the same duration of time which corresponds to
Fig. 5. A Quansar 2DoF Pantograph was used for the experiment. When prompted with a verbal cue, users were asked to move quickly and accurately along the trajectories represented by the coloured targets. Users moved along both the forward and reverse trajectories using a target size of either 1.9cm or 0.64cm.
the average velocities from the original tests. The IP was determined for each path using eq. 3 where A and M T were specified and the We was the resulting distance error from the end point in the admissible-motion space (Fig. 4). Changes in the IP were then plotted against the path orientation for various accelerations and directions along the specified paths. B. Experiments Human experiments were preformed to validate the simulations using the Quansar 2DOF planar pantograph (Fig. 5). The testbed was set-up using the same path lengths and orientations as those simulated. Twelve undergraduate student volunteers with no prior operation of the device were selected to perform the tests. Users were asked to traverse the paths as quickly and accurately as possible. The tested trajectories included were: three path orientations ((90◦ , 45◦ , 0◦ ) located at the same positions in the workspace, as well as forward and reversed direction along the paths. The constant velocity experiment was excluded. Large and small physical target sizes, 1.9 cm, and 0.64 cm, respectively, were also included. In total these variations accounted for twelve movements per participant. The order of the motions performed were selected randomly to reduce bias from gained familiarity of the device throughout the trial. The users were provided with an auditory cue to begin the test from which a timer began and stopped when the user crossed a finish line at the leading edge of the target circle as seen in Fig. 6. This is known as a goal-crossing task and can also be modelled by Fitts’ Law [2]. The time to complete the task as well as the magnitude of the distance error from the center of the target circle was recorded for each test. The ID was then determined from the experimental data using the effective width of the target, We , for each test. Here, We , is defined as the distance from the center of the target in which all the users trajectories fell within two standard deviations (95%) of the measured error. This can
Fig. 6. The user moves from the start to stop position. Upon crossing the line tangent to the leading edge, the end-effector position was recorded and the error was calculated as the distance from the centreline. The characteristic width We is then defined as the distance from the center line that falls within two standard deviations (95%) of the measured user error (err) data.
be better illustrated graphically by examining Fig. 6 .It must be noted a linear regression was not used to calculate the IPs as is commonly done in a tapping test. The objective of this study was instead to see how the effective width, We , and thus difficulty of a task, changes along directions of varying coupling. For each of the twelve tests the measured time, MT, was evaluated as the average time of the users to complete the specified task. Thus, under the assumption that (1) holds true using the ID as defined in (3), the IPs for the device was then plotted for various path orientations and directions. IV. RESULTS A. Simulation The simulated IPs are plotted for various accelerations and orientations as seen in figure Fig. 7. The results predict that the device performance will be greatest along the decoupled directions and will suffer most along the approximate direction of maximum coupling. Along the ’arbitrary’ third direction the device performance falls, not surprisingly, between the values for the maximum and minimum coupling pathways. The simulated results more interestingly predict that the users ability to perform a translational navigation task will also be effected by their direction along the path. Moving from an area of high coupling to one of lower coupling causes a larger deviation of the end-effector from the desired trajectory during rapid navigation. Another interesting observation lies in the comparison of the work done by the inertial effects of the device on the human user in the admissible-motion space direction. By integrating the work along the simulated path and comparing the ratio of work done by the acceleration-dependent inertial force to the net work done in the direction of admissiblemotion. Σfacc ∆ua (20) ρ= Σ(facc + fvel )∆ua
1 15
0.6
10
ρ
IP (bits/sec)
0.8
0.4 0.2
5
0 pi/2 (Path 1) pi/4 (Path 2) Angle (rads)
0 (Path 3)
Fig. 7. Simulated Indices of Performance, IP , using a user tap test conducted before the experiment to determine a range of suitable accelerations. The decoupled path (Path 1) has the highest index of performance, the path of maximum coupling (Path 2) has the lowest index of performance and the ’arbitrary’ path (Path 3) falls in between. The indices of performance were lower in the reverse direction for each path.
we see that, even for the case of the highest magnitude of acceleration, the parasitic forces that arise from the acceleration-dependent inertia term only accounts for roughly half of the total work done in the admissiblemotion space (Fig. 8). This suggests, contrary to what the authors expected, that the velocity-dependent inertial terms contribute greatly to the parasitic inertia felt by the user. Thus seeking to greatly improve haptic device performance by way of dynamic analysis it may be necessary to consider both the velocity-dependent and acceleration-dependent inertial terms, and perhaps even just the velocity-dependent terms for simulations where high accelerations are not to be expected. B. Experiment The IPs were calculated for the Quansar 2DOF pantograph haptic device for the prescribed operational paths as well as for the forward and reverse trajectories. The ,We results can be seen in Table I and are graphically represented in Fig. 9. While there was little discernible difference in the IPs for the forward trajectories, reversal of the direction to move from regions of high coupling to low coupling closely resemble the trends predicted by simulation. TABLE I E XPERIMENTAL EFFECTIVE WIDTH , We . Direction Forward Reverse
Path 1 0.023 m 0.024 m
Path 2 0.022 m 0.059 m
Path 3 0.026 m 0.034 m
The average time that subjects completed the navigational task was substantially slower than the values used in simulation with the average time to complete the forward path task,
pi/4 (Path 2) Angle (rads) forward
0 (Path 3) reversed
Fig. 8. ρ, is defined as the ratio of work done by acceleration-dependent inertial terms to the work done by the sum of inertial acceleration-dependent and velocity-dependent terms. The work done by the parasitic forces resulting from the acceleration-dependent terms is, in the worst cast scenario (i.e the highest magnitudes of acceleration), approximately equal to the velocitydependent inertial terms. This may indicate that velocity-dependent terms play a much more significant role in performance degradation resulting from parasitic forces than their acceleration-dependent counterparts.
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0 pi/2 (Path 1)
0.25
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0
0.05 x(m)
0.1
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Fig. 9. Experimental effective width, We , represented by circle size. This diagram corresponds to the experimental results for path traversal in the reversed direction (i.e from regions of higher coupling to lower).
tf = 0.914s (σ = 0.068s), and the average time to complete the path reversed task, tr = 0.881s (σ = 0.046s). The simulation was updated to use a new time duration of tdur = 0.881s, corresponding to the experimental data for the reversed direction, and the new IPs were subsequently plotted against the varying path directions. Without changing anything but the acceleration to more accurately represent the true performance of the human user (corresponding to an acceleration of 0.49 sm2 ) the theoretical results much better reflect the data attained (Fig. 10). V. DISCUSSION At first glance the results of this study suggest something that is seemingly intuitive, that the inertial coupling for a haptic manipulator can negatively impact the operational performance, and more specifically, navigational tasks. The motivation, however, lies in the desire to move towards better
Fig. 10. Re-simulated indices of performance using the average time of experimental users to calculate the simulated acceleration.
using device dynamics to predict user performance for haptic systems. Through proper dynamic analysis and experimental validation we can seek to design devices better suited to performing simulation specific tasks, whether they be navigation, manipulation, or selection based. This requires experimentally correlating changes in performance with varying dynamic properties. The initial results of this study suggest that rapid navigation in regions of high coupling reduce accuracy for quasi-ballistic motions. A second point of importance in this work was the adaptation of the operational and admissible-motion space dynamic formulations to be interpreted for a haptic device. Expressing the device dynamics in terms of the end-effector (where the human-device interaction occurs) with respect to a desired operational task we can meaningfully decompose the resulting inertial forces into parasitic forces that can negatively impact the performance (accuracy in the case of a navigational task) of a user. This may prove promising for reducing parasitic forces using redundant mechanisms where operation within the null space of the inertia tensor may greatly eliminate parasitic inertial effects. Interestingly in this study, the parasitic forces resulting from the velocitydependent inertial terms in simulation were found to be of greater impact on a decrease in device performance than the acceleration-dependent terms. These findings should be further investigated to determine their role on haptic performance. R EFERENCES [1] Johnny Accot and Shumin Zhai. Performance evaluation of input devices in trajectory-based tasks: An application of the steering law. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, CHI ’99, pages 466–472, New York, NY, USA, 1999. ACM. [2] Johnny Accot and Shumin Zhai. More than dotting the i’s — foundations for crossing-based interfaces. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, CHI ’02, pages 73–80, New York, NY, USA, 2002. ACM. [3] Haruhiko Asada. A geometrical representation of manipulator dynamics and its application to arm design. J. Dyn. Sys., Meas.,Control, 105(3):131–142, 1983.
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