Abstract
Cable robots have been used as haptic interfaces for several decades now, with the most notable examples being the SPIDAR and its numerous iterations throughout the years, as well as the more recent IPAnema 3 Mini manufactured by Fraunhofer IPA. However, these robots still have drawbacks, particularly their high number of cables required to maintain a high workspace-to-installation-space ratio. Using a hybrid structure cable robot (HSCR) could prevent some collisions that occur between the cables and the user’s body. More specifically, some applications requiring multimodal feedback could benefit from the flexibility that a reduced number of cables offers. Therefore, this paper presents a novel SPIDAR-like HSCR and its sensor-less force control method based on motor current. The purpose of this work is to clarify the advantages that a variable-structure can provide for haptic interaction. In this regard, experimental results regarding the device’s workspace and its force feedback capabilities are presented. Additionally, since real-time high-frequency updates are required for haptic display, we provide additional data regarding the control algorithm’s runtime. Lastly, another experiment was conducted to study changes in user performance when using both the variable and the usual cable configuration. The results showed that feedback accuracy is maintained, and there are no drawbacks to using hybrid configurations.
1 Introduction
The rise of technologies related to virtual reality has created a need for new ways to interact with virtual worlds. Head-mounted displays (HMDs) [1] have been the most popular type of device used to immerse the user. However, human’s complex haptic perception requires more than simple visual feedback to achieve complete immersion [2]. To allow for more complex interactions, cable-driven parallel robots (CDPRs) were used as haptic interfaces as soon as the early 1990s [3].
Indeed, thanks to their high reconfigurability, large workspace, low intrusiveness, and low inertia, CDPRs present interesting features when used as force feedback interfaces. Besides haptics, CDPRs are also used for a variety of purposes, including, for example, industrial applications [4,5], rehabilitation, other medical applications [6–9], 3D printing [10,11], or teleoperation [12,13].
Since the development of the first prototype of SPIDAR (SPace Interface Device for Artificial Reality) [3,14], CDPRs and haptic interfaces, in general, have been adapted to a wide range of applications, with even more configurations to fit each task [15]. Recent publications suggest trends toward more immersion through multimodal haptic feedback [16–18] and innovative displays (co-localization with half-tinted mirrors) [19,20] on one hand, while higher performance in terms of accuracy and haptic transparency is required for precise tasks like surgical robot systems [21,22]. Many breakthroughs have been made regarding aspects like real-time capability of control algorithms [23,24], system configurations and kinematics [25], end-effector designs [26,27], and calibration [28], so that most recent CDPRs can meet industrial standards. This development is well illustrated by the IPAnema family of cable robots manufactured by Fraunhofer IPA [5], which includes a haptic interaction system called IPAnema 3 Mini [29]. Nonetheless, cable-based haptic devices still lack diversity of feedback, and the virtual environment is usually displayed using a simple screen [7,15].
The high number of cables necessary to provide force feedback also remains a problem to overcome. It induces a poor workspace-to-installation-space ratio, interferences between the cables, and less freedom of movement for the user; finally, devices integration for multimodal haptic purposes can result in potential umbilical management issues. This applies especially for relatively small-sized interfaces or when multi-finger haptic interaction is possible, as in Refs. [30–32]. Moreover, most interfaces do not allow rotations of the effector, which keeps the control scheme and end-effector design as simple as possible but limits the possibilities of the interfaces. Progress has been made, especially using task-specific end-effectors [33,34] or configurations, but solutions like reconfigurable and variable-structure CDPRs [35–37], especially for haptic applications, were mostly left untouched until recently. Transmission systems were used to extend the workspace of CDPRs while having fewer actuators than cables [38], which optimizes the use of workspace but does not reduce possible user-cable interferences. Tan et al. proposed to reconfigure a CDPR by mounting the entire actuation unit on a mobile crane [39]. Gagliardini et al., as well as others, studied hybrid configuration strategies consisting of moving the pulley blocks of the CDPR [40–43]. A similar idea was used by Zanotto et al. with a series of cable-driven haptic interfaces called Sophia [7,44], which introduced a dynamically moved pulley block (the entire actuation unit stays fixed, note that we will use both hybrid, adaptive or reconfigurable indiscriminately to qualify such interfaces in this article). This could maximize the workspace of the device while minimizing the number of cables. However, to the author’s knowledge, only 2D haptic interfaces were built and had their studied stiffness and dexterity so far [45], while the design and properties of adaptive 3D haptic interfaces remain to be investigated.
Thus, in this paper, we present a hybrid SPIDAR-like 3D haptic interface with a hybrid adaptive structure. The aim of this interface is to act as a simple proof of concept. The number of cables is reduced to some extent (from eight to six and not four, which would be the minimum to keep an over-constrained configuration) while increasing the installation-space ratio of the robot. Besides providing more freedom of movement for the user, this could provide solutions to integrate other devices for multimodal haptic display (on the end-effector, for example). In Sec. 2, we describe the specifications of the system, the design choices, and the kinematic models. Section 3 provides a detailed explanation of the control method used to provide haptic feedback. Then, Sec. 4 presents the user-centered experiment and results obtained regarding haptic rendering, with a focus on workspace optimization; results on computing efficiency are also presented to ensure the real-time capability of the device. We conclude this paper by discussing future works and remaining challenges.
Please note that the content of this work is an extended version of a paper presented at CableCon2023 and published by Springer [46]. Several points were modified compared to the previous version: the winch system of the prototype was changed to improve accuracy, and the kinematic model was then adapted to this new configuration. An effort was made to further develop the workspace analysis since it is the main contribution of this paper, alongside the design and control of the interface. Additionally, results regarding the computation of forward kinematics were added. Any impact on previously published results or changes will be mentioned and discussed subsequently.
2 System Description
The haptic device consists of three main parts, which we will detail in this section: the actuation system controlling the cables, the linear actuator, and the virtual environment. We will also discuss the limitations and design choices, concluding with an explanation of their impact on the kinematic model of the adaptive CDPR.
2.1 Specifications and Hardware.
The layout of the first prototype is depicted in Fig. 1. It was designed to be a transportable interface, with the total dimensions of the aluminum frames measuring approximately 80 cm in length, 55 cm in width, and 160 cm in height. Six motors provide feedback through the cables: four are attached to the frame (cables on top), and two are fixed to a frame that moves dynamically via the linear actuator to follow the end-effector.
The theoretical workspace available for manipulating the end-effector, shown in Fig. 1, is delimited by the length of the actuator and the positions of the motors. It forms a pyramidal frustum or a pyramid with its top cut off, with a rectangular base. The dimensions of the workspace for the base of the pyramid (the top part of the workspace) are 77 cm in length, 49 cm in width, and 53 cm in height. The bottom part has the same dimensions except for the length, which is approximately 50 cm.
The hardware specifications are detailed in Table 1. The hardware choices were largely inspired by other existing devices [19]. The end-effector, pulleys, and motor mounts are made of 3D-printed resin. The SLP-15 linear shaft motor was chosen for its high accuracy, high speed, and lack of friction. It has a maximum speed of 3 m/s, a maximum acceleration of 3.5 G, and its encoder has a 1 m accuracy. Meanwhile, direct-drive motors like Maxon’s DCX32 provide sufficient torque (low torque is used for safety considerations) and precision for haptic applications. The low friction of both the linear actuator and the motors ensures high backdrivability, which is important for haptic interactions [47]. Furthermore, such motors have already been used [19,20], and their low cogging torque fits the sensor-less (i.e., no load cells or torque sensors) control method [35] described in Sec. 3.
Parts | Specifications |
---|---|
Motors (direct drive) | Maxon DCX 32 48V |
Encoders | Maxon ENX16 EASY 1024 |
Linear actuator | Nippon Pulse Motor SLP-15 |
DC motor drivers | Maxon ESCON Module 50/5 |
Linear actuator driver | Panasonic MADHT1105L01 |
Micro-controller | ARM mbed NUCLEO-F767ZI |
HMD | Meta Quest 2 |
Parts | Specifications |
---|---|
Motors (direct drive) | Maxon DCX 32 48V |
Encoders | Maxon ENX16 EASY 1024 |
Linear actuator | Nippon Pulse Motor SLP-15 |
DC motor drivers | Maxon ESCON Module 50/5 |
Linear actuator driver | Panasonic MADHT1105L01 |
Micro-controller | ARM mbed NUCLEO-F767ZI |
HMD | Meta Quest 2 |
The raw data from the motor encoders are sent to the microcontrollers and then to a C++ program running on a Linux OS server at a frequency of 2 kHz, using a user datagram protocol (UDP) communication protocol. The C++ program handles all the computations related to the control of the robot (kinematics) as well as the force feedback. The simulation is displayed using Meta’s Oculus Quest 2 HMD, while the simulation runs remotely on a monitoring PC. Only the position of the robot and the handle are sent from the control server through UDP to the monitoring PC, which then sends back the updated information via Wi-Fi to the HMD. The choice of this HMD was due to its simplicity of integration into the system. Also, the controllers used to superimpose the virtual environment on the actual system have an accuracy in the millimeter to sub-millimeter range [48,49]. Hand tracking was used during the experiments to help the user locate their relative position to the end-effector, although it can be relatively inaccurate (a position error of less than 1 cm can occur) and suffers from a slight delay of 38 ms [50].
2.1.1 Second Prototype and Improvements.
Following the publication of Ref. [46], several improvements have been made to the first prototype:
The position of the HMD controller used as a reference to superimpose the virtual environment onto the system was moved to face the user during operation, and one of the frames was cut to guarantee that the user’s field of view was not blocked. This change was made to prevent losing hand tracking when users move their heads, and the receptors lose track of the controller. While this was a rare occurrence experimentally, it would happen depending on the height of the operator.
Polyethylene (PE) fishing cables were replaced by Izanas cables with a diameter of 0.3 mm (also known as Dyneema cables outside of Japan).
The winch system was changed to be more accurate. The 3D-printed pulleys mounted on the motor shafts were replaced by commercially available 11 cm long threaded shafts with a pitch of 2 mm.
Pulley blocks that can rotate along the -axis were designed to guide the cables and ensure constant transmission ratio when coupled to the rotating shafts [51]. They are used to help prevent the cables from wearing out too quickly. These changes will have an impact on the kinematic model and require more complex modeling. All the implications will be detailed in the next sections.
Figure 2 shows the system after the modifications were implemented.
2.1.2 Limitations.
Please note that this device is a direct adaptation of the SPIDAR and its many variations. Hence, the kinematic model considers the end-effector as a point. The symmetric layout of the original SPIDAR was kept both for space management reasons and because the effector does not rotate which means no singularities will result from this choice (other advantages of such configurations are stated in Sec. 4.1). Even though some recent works have used more complex handles and models for dedicated applications, a simple model should be sufficient since our aim is to provide a proof of concept. A six-cable configuration might not seem sufficient to over-constrain the end-effector when considering both translational and rotational degrees-of-freedom (DoFs). However, the final aim is to reduce the number of cables to a minimum of four, with the remaining rotational DoFs being taken care of by a custom-made end-effector (in the form of an encapsulated finger cap rotated by ball bearings).
Similar considerations guided our choice of using an HMD: while achieving co-localization between the simulation and the real world is challenging, the perspective provided by the HMD will enable more complex interactions than the typical translational movements used in many studies. Nevertheless, the true impact of the display method should be investigated in future research.
2.2 Kinematic Model.
The system needs to track the position of the user’s hand, which implies that the cable lengths are given, and the pose of the effector is sought. To solve this problem, two approaches were tested. The first approach considers the cable connection points to the pulleys as fixed ideal points, leading to a rough approximation of the end-effector’s position. This approach is easy to implement, fast to compute, and sufficient to control the robot but may be inaccurate, as reported in Ref. [52]. The second method considers the pulley mechanism and provides a better estimated position but requires more computational effort.
2.2.1 Simplified Approach.
2.2.2 Extended Model.
Using the solution from Eq. (5) is the most straightforward option to compute and has been used in other works [19]. It is sufficient when accuracy is not the main goal, which can be the case for some haptic-related tasks. However, we chose to discard this method for the second prototype since superimposition discrepancies can appear between the real system and the simulation when the end-effector is close to the border of the workspace.
It is also possible to consider the small-sized pulleys as ideal fixed anchor points or cable detachment points. In this case, the forward kinematics are computed with the simplified model using the LM algorithm. Then remains another choice: using only the four upper cables with fixed actuation parts to solve the kinematics or using all six cables.
The optimal option is to use the extended kinematic model, with either four or six cables.
Obviously, these possibilities come from the fact that we consider only three DoFs and need fewer cables than when the end-effector can rotate. Nonetheless, these methods will be compared in Sec. 4 in terms of computation time using a test trajectory.
3 Control
This section focuses on the force control algorithm of the interface. An overview of the controller is shown in Fig. 5. The controller is composed of four main parts: an observer part to estimate the force applied to the effector, a Proportional–Integral–Derivative (PID) controller to control the linear actuator, a virtual model to simulate the behavior of the virtual object, and a Proportional–Derivative (PD) controller that outputs the force feedback of the virtual object through the end-effector. Note that throughout the entire manipulation, when a cable is not used for feedback, it is maintained in tension, but no real-time tension control algorithms were used otherwise. Using real-time capable algorithms like the closed-form approach proposed in Ref. [23] would be preferable, but static equilibrium was already achieved experimentally due to the choice of hardware (video evidence is available2). The minimum force of around 0.7 N used to straighten the cables is referred to as ; the total force on the end-effector is maintained under 10 N for safety considerations. Also, as explained earlier, the end-effector’s position is initialized at , and for this, an additional part with dimensions known with 0.1 mm accuracy is used.
3.1 Linear Actuator PID.
Using the forward kinematics described in Sec. 2, the angle of each motor is used to calculate the position of the handle. This position is then designated as the target position for the linear actuator, which acts as a follower system. The goal is to keep the end-effector close to the center of the workspace, where static equilibrium is achieved during manipulation. Thus, the user will feel less resistance due to cable tension when moving the effector without having to use methods to dynamically distribute the cable tension.
3.2 Disturbance Observer and Force Estimation.
3.3 Virtual Model.
3.4 Force Feedback.
After obtaining the estimated pose of the end-effector, the difference between the estimation and the actual position of the handle is sent to a PD controller. Finally, using the Moore–Penrose pseudo-inverse, the torque variation required for the force feedback is distributed to the motors (see Fig. 5).
4 Experiments and Results
Three experiments were undergone to evaluate the performance of the hybrid SPIDAR. The experimental process and the results will be presented in this section.
4.1 Workspace.
With the design of a CDPR comes the necessity to evaluate its workspace, both quantitatively and qualitatively. Intuitively, it is easy to understand that the polyhedron defined by the suspension points of the cables encompasses the actual workspace of the device (see the polyhedron in Fig. 1). However, practical limitations, such as cable tension or end-effector shape, often result in a smaller actual workspace. Furthermore, various approaches can lead to different definitions of the workspace, such as the controllable workspace or the commonly studied wrench feasible workspace. A comprehensive review of these terminologies can be found in Ref. [60].
Concurrently, a wide variety of indices were proposed to evaluate the actual quality of a cable robots’ workspace, since indices used for rigid link robots do not take into account the cable tension parameter. Among these we can cite some manipulability indices introduced by Rosati et al. [61], or the commonly used tension index or tension factor (TF) [62]. The workspace of the configurations of the device proposed in our work will be studied in the next sections. The objective is to compare configurations between each other, thus for more simplicity, all the simulations and experiment of this section were done using the simplified kinematic model.
4.1.1 Quantitative Evaluation.
Quantitative objective evaluation of workspace was undergone using three dimensionless performance indices, based on those proposed by Ferraresi et al. and used some other works [63,64]. These three indices were adapted as needed to be applied to a 3D hybrid structure cable robot.
- The area index, noted , represents the ratio between the theoretical maximal value of the workspace (spanned by the anchors points of the cables) and the actual wrench feasible workspace, which is always smaller due to tension limitations:where is the number of discretization points in the workspace of the robot; , , and are the resolution chosen for each direction (1 cm in our case) and maximal volume of the workspace.(22)
- The index of compactness, noted , is the ratio between the actual workspace and the smallest polyhedron encompassing this workspace:where is the volume of the surrounding polyhedron, in practice if the maximal workspace of our configuration is an irregular tetrahedron, will be the volume of a smaller tetrahedron included inside it.(23)
- Lastly, the index of symmetry, noted , represents the distortion of the polyhedron enveloping the workspace. It was redefined to fit polyhedrons with more complex shape than simple parallelepipeds:where is the number of sides of the enveloping polyhedron, is the mean value of the polyhedron sides, and is the length of the th side of the polyhedron.(24)
A workspace will be considered better, from a geometric point of view, the closer the indices are to 1. In other terms, a workspace, for most applications as well as for haptics, should be large, cover most of the device span, and be symmetric.
4.1.2 Qualitative Evaluation.
All the results obtained are summarized in Fig. 8 and Table 2. Note that since the final objective of our work is to reduce the number of cables, a configuration with two linear actuators moving all four anchor points of the CDPR was considered. Theoretically, it is expected that the results will be coherent with the results presented by Abdolshah et al. for planar CDPRs [45], with reconfigurability and a low number of cables yielding the better evaluation. For the computation, the parameters were set as follows: and were arbitrarily set to 1.5 N and 18 N, respectively (twice the minimum tension and an approximation of the maximum output of the motors); the tension index was calculated for points separated by 1 cm along every direction. The results show that configurations with moving anchor points are superior to usual configurations in every aspect except for a slightly lower . The high values of and show that the workspace-to-device span ratio is more optimal. Lastly, the GTI is in concordance with the expectations explained earlier.
Configuration | Anchor points | Workspace envelop | GTI | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Six cables: fixed anchor points | x | y | z | x | y | z | ||||
Six cables: moving anchor points | ||||||||||
Four cables: fixed anchor points | ||||||||||
Four cables: moving anchor points | ||||||||||
Configuration | Anchor points | Workspace envelop | GTI | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Six cables: fixed anchor points | x | y | z | x | y | z | ||||
Six cables: moving anchor points | ||||||||||
Four cables: fixed anchor points | ||||||||||
Four cables: moving anchor points | ||||||||||
From these results, it seems that reconfigurable (or adaptive) configurations are superior in almost every aspect. However, the indices cannot represent problems that could arise when designing these devices. For example, moving only pulley blocks, and not the whole actuation unit like proposed in this paper, could lead to unexpected losses in cable tension or even cables derailing from their guiding pulleys if the control method or cable management are not done properly. Such problems did not arise because the entire actuation unit is moved, but should be investigated in future works including moving pulley blocks.
4.2 Kinematics Computation.
In this section, we test the real-time capability of the LM optimization algorithm on each of the kinematics computation strategies listed in Sec. 2. Kinematics computation involves continuously re-evaluating Eq. (6) or (18) with the LM solver, and this may lead to computation times that exceed the control frequency of at least 1 kHz, which is required for haptics. The number of iterations required is highly dependent on the end-effector’s pose and the geometry of the CDPR. Therefore, the parameters evaluated in this section include computation time and the number of iterations of the algorithm. The results are reported in Figs. 9 and 10.
The simulation results reported in Ref. [54] already indicate that the algorithm is real-time capable. However, we want to verify its performance when the end-effector is moved by an operator, which produces erratic or unpredictable behaviors unaccounted for by the model. For this purpose, we conducted an experiment in which the end-effector was moved freely inside the workspace. Only the simplest kinematic computation method from Eq. (5) was run in real-time during the operation to allow the linear actuator’s PID to follow the end-effector. All the necessary parameters were recorded to run simulations with the other computation methods later. The trajectory is shown in Fig. 11. The simulations were run on the same hardware used for the control of the CDPR, a desktop computer with an Intel(R) Core(TM) i7-2600 CPU. The aim of this test was to setup a worst-case scenario to test the limits of the algorithm. Thus, the maximum number of iterations was set to 1000; no tension control was used, and an arbitrary tension was set in the cables (around 2 N); and the end-effector was slightly rotated on itself by the operator to contradict the point model used in the kinematics (effectively creating small cable collisions that can be observed in Fig. 11, where significant discrepancies appear in the -coordinate). Even with these unrealistic conditions, the algorithm performed well enough to ensure real-time capability:
As expected, the simple kinematic implementation, being fully deterministic, did not require an iteration process and had a computation time of less than 0.1 .
The method using only four cables, for both the simplified and extended kinematic models, presented a low computation time as well as a low number of iterations, with an average of 31 for 29 iterations for the simplified model and 52 for 31 iterations for the extended model.
The method using all six cables, for both the simplified and extended kinematic models, presented a higher computation time as well as a higher number of iterations, with an average of 80 for 79 iterations for the simplified model and 151 for 81 iterations for the extended model.
4.3 Haptic Feedback Evaluation.
As a proof of concept for haptic feedback capability, we conducted a simple task in which the user touches a fixed virtual sphere with a diameter of about 5 cm.3 The graphs in Fig. 12 show the sphere, the trajectory of the effector (black trajectory and colored points), and the force feedback represented using black arrows rescaled to fit inside the graph. These arrows start at the contact point between the effector and the sphere, and their direction is given by . As expected, the forces are consistent with the simulated sphere, with no noticeable errors in direction. Note that another experiment with an object being pushed along a straight line was designed, but due to the movement of the sphere, the arrows were superimposed on the object and end-effector trajectories, resulting in unreadable graphs. For similar reasons, the graphs in Fig. 12 were created using data acquired with a 20 Hz frequency. More complex tasks and a thorough survey-based evaluation of the force feedback can be found in Ref. [65].4
4.4 User-Based Evaluation.
Lastly, for the purpose of studying the influence of the hybrid configuration on free manipulation, a picking task was designed.
4.4.1 Methodology.
Users gathered 12 targets (cubes shown in Fig. 13) in a predetermined order. Each time the effector touched a target, the next target changed color to become yellow, indicating the order in which to pick the targets. This imposed a trajectory that users had to follow for each trial (see the right side of Fig. 13). To compare the hybrid and fixed configurations of the robot fairly, all the targets were situated inside the workspace of the fixed configuration, resulting in a triangular distribution. Each user operated the system at least eight times; the only instruction given was to try to maintain the same speed of manipulation for each trial. Initially, the task was performed twice to allow users to become accustomed to the interface and each configuration (once for the fixed configuration and once for the hybrid). Following this, the operator completed a first set of four trials (referred to as “untrained” in Fig. 14), alternating between each configuration. Subsequently, the user underwent training as needed to remember the order of the task picking and aimed for a completion time of around 10 s. Finally, the operator completed a second set of four trials (referred to as “trained” in Fig. 14). At this point, all the data were collected, and the experiment concluded. Regarding the participants, there were nine naive volunteers (all male) in their 20s, with both left-handed and right-handed individuals included. None of the participants reported any physical or visual impairments that could affect their performance.
4.4.2 Results.
Paired sample -tests were computed using the free software JASP [66], and the results are presented in Fig. 14. Two parameters were compared for both the “trained” and “untrained” states: the time taken by the participants to complete the task and the total travel distance of the end-effector. The two graphs on the left in Fig. 14 tend to indicate that there is no significant difference in terms of performance when using either configuration for untrained users ( for the distance and for the completion time). The same observation can be made for trained users regarding the travel distance (with ). However, the comparison of travel time has a p-value closer to 0.05 (). From these results, it can be concluded that there are no clear advantages or disadvantages to using a hybrid SPIDAR when conducting free exploration of the virtual environment with the effector.
4.4.3 Discussions.
The results presented in this section demonstrate the feasibility of using a hybrid structure cable robot to provide force feedback in three dimensions. Although this device was designed to be both simple and as a proof of concept, several points still require improvement. First, the control method should include real-time sensor-less tension control. While the current device does not require it (as shown in the video), strategies to manage tension while considering the bottom motors should be implemented. Second, the bandwidth of the system when using the end-effector as an input for impedance and admittance control should be studied to understand the impact of the linear actuator on the force feedback capabilities. Future works should also consider workspace optimization when considering the operators range of movement, as well as interferences between the operator’s limbs, the cables, and the umbilicals of the end-effector. Lastly, as shown in Table 2, fully reconfigurable symmetric configurations should be considered for future designs due to their isotropic characteristics.
5 Conclusion
In this paper, we presented a SPIDAR haptic interface with a hybrid configuration that follows the movement of the operator through the effector. We described the kinematic model of the robot and a sensor-less control method based on motor current. An analysis was conducted to show that the hybrid configuration has a larger workspace, resulting in lower manipulation resistance for the user and a better tension index. As a proof of concept, experiments were conducted to evaluate real-time capability, the accuracy of the feedback, and investigate the effect of the linear actuator during free manipulation of the effector. It was concluded that there are no apparent drawbacks to using hybrid configurations. On the contrary, an unconventional symmetric configuration with a low number of cables can present some interest for multimodal haptic applications (at the very least for non-rotating end-effectors). Future work will primarily focus on the points discussed above and on the integration of other types of feedback (e.g., vibrations and temperature) while compensating for the lack of DoFs with innovative end-effector design.
Footnotes
See Note 2.
Acknowledgment
This work was supported by JSPS KAKENHI Grant No. 22H01455.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.
Nomenclature
- =
half-height of the first prototype (around 53 cm)
- =
half-length of the first prototype (around 77 cm)
- =
number of sides of the enveloping polyhedron
- =
half-width of the first prototype (around 49 cm)
- =
end-effector position vector
- =
estimated force feedback vector
- =
unit vector
- =
end-effector position in the reference frame
- =
pose-specific structure matrix
- =
force vector applied by the user
- =
vector from to
- =
actual movement of the object considering the texture
- =
acceleration of the actual movement of the object considering the texture
- =
distance between the center of the pulley and the end-effector
- =
number of discretization points in the workspace
- =
z-axis of the winch coordinate system
- =
length of cable
- =
radius of the pulley mounted on the motor shaft
- =
radius of the pulley guiding the cable
- =
tension in cable
- =
minimum cable tension
- =
maximum cable tension
- =
linear actuator’s position
- =
vector representing the anchor point of cable
- =
vector from the end-effector to anchor points
- =
unit cable vectors
- =
position of the end-effector
- =
position of the object’s surface
- =
height of the end-effector
- =
center point of pulley
- =
detachment point of cable
- =
area index
- =
index of compactness
- =
index of symmetry
- =
analytic Jacobian matrix
- =
entry point of the cable reeling on the pulley
- =
mean value of the polyhedron sides
- =
number of points inside the workspace
- =
length of the ith side of the polyhedron
- =
volume of the surrounding polyhedron
- =
theoretical maximal value of the workspace
- =
anchor points of the cables
- =
identity matrix
- =
estimated movement of the sphere
- =
acceleration of the estimated movement of the sphere
- GTI =
global tension index
- TF =
tension factor
- =
mass, damper, and spring term matrix for simulating the texture of the virtual object
- =
mass and damper term matrix for the virtual model
- =
angle associated with each cable in the extended kinematic model
- , , =
resolutions in the , , and directions
- =
angle associated with each cable
- =
angle of each motor
- =
cable force distribution for each motor ()
- =
minimum force of around 0.7 N used to straighten the cables
- =
estimated torque for each motor ()
- =
function representing the cable tension equations