Graphical Abstract Figure
Graphical Abstract Figure
Close modal

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 [69], 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 [1618] 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. [3032]. 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 [3537], 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 [4043]. 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.

Fig. 1
Overview of the reconfigurable haptic device. It can be used from any direction, thanks to the HMD that displays the virtual environment. The cables linked to the end-effector delimit the polyhedron encompassing the workspace where force feedback can occur.
Fig. 1
Overview of the reconfigurable haptic device. It can be used from any direction, thanks to the HMD that displays the virtual environment. The cables linked to the end-effector delimit the polyhedron encompassing the workspace where force feedback can occur.
Close modal

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 μm 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.

Table 1

Hardware specifications

PartsSpecifications
Motors (direct drive)Maxon DCX 32 48V
EncodersMaxon ENX16 EASY 1024
Linear actuatorNippon Pulse Motor SLP-15
DC motor driversMaxon ESCON Module 50/5
Linear actuator driverPanasonic MADHT1105L01
Micro-controllerARM mbed NUCLEO-F767ZI
HMDMeta Quest 2
PartsSpecifications
Motors (direct drive)Maxon DCX 32 48V
EncodersMaxon ENX16 EASY 1024
Linear actuatorNippon Pulse Motor SLP-15
DC motor driversMaxon ESCON Module 50/5
Linear actuator driverPanasonic MADHT1105L01
Micro-controllerARM mbed NUCLEO-F767ZI
HMDMeta 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 z-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.

Fig. 2
Overview of the second prototype: (a) The system now includes 12 rotating pulley blocks and a support used as an external calibration point, (b) each cable exits the threaded shaft to reel onto a pulley block that can move freely on its axis, (c) two pulley blocks are used to ensure that the cable follows the effector while the linear actuator moves.
Fig. 2
Overview of the second prototype: (a) The system now includes 12 rotating pulley blocks and a support used as an external calibration point, (b) each cable exits the threaded shaft to reel onto a pulley block that can move freely on its axis, (c) two pulley blocks are used to ensure that the cable follows the effector while the linear actuator moves.
Close modal

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.

In Fig. 3, the model used to calculate the forward kinematics of the first prototype is depicted. The half-length, half-width, and half-height are, respectively, denoted as l, w, and h. The end-effector, with coordinates r=[xe,ye,ze]T, has its initial position at the center of the device, where r=[0,0,0]T. Note that the position of the linear actuator yl will be almost equal to ye during operation. The lengths of the cables li are calculated from the encoder’s data, and each cable’s magnitude AiE can be written as a function of ϕi.
ϕi(r,l)=(Air2)2li2
(1)
for i=1,,6, where the attachment points are
A1=[w,l,h]T,A2=[w,l,h]T,A3=[w,l,h]TA4=[w,l,h]T,A5=[w,yl,h]T,A6=[w,yl,h]T
(2)
Lastly, we define u^ as the estimation of the force applied by the user on the handle and the unit cable vectors ui:
ui=lili
(3)
These will be used in the next part for the control algorithm. Since the effector is considered as a point, the Euclidean norm of Eq. (1) yields
{l12=(xew)2+(ye+l)2+(zeh)2l22=(xe+w)2+(ye+l)2+(zeh)2l32=(xe+w)2+(yel)2+(zeh)2l42=(xew)2+(yel)2+(zeh)2l52=(xew)2+(yeyl)2+(ze+h)2l62=(xew)2+(ye+yl)2+(ze+h)2
(4)
Solving the above system of equations, given all the cable lengths, leads to obtaining r in the form:
[xeyeze]=[14w(L22L12)14l(L22L32)h±L32(xe+w)2(yel)2]
(5)
It can be noted from Eq. (5) that the configuration of the robot allows the use of only the upper cables to infer the position r. Doing so can prevent the small difference between yl and ye from adding inaccuracy in the measurements (A5 and A6 are updated in real-time). In simpler terms, the bottom part of the robot, composed of the linear actuator, is synchronized with the upper part.
Fig. 3
Simplified kinematic modeling of the hybrid SPIDAR: the anchor points of the cables are fixed except for the bottom ones and the end-effector is considered as a point
Fig. 3
Simplified kinematic modeling of the hybrid SPIDAR: the anchor points of the cables are fixed except for the bottom ones and the end-effector is considered as a point
Close modal
A more accurate approximation of r can be obtained by solving the forward kinematics using a Levenberg–Marquardt optimization algorithm. Unlike the solution from Eq. (5), this approach reduces measurement errors by incorporating data from all the encoders and not just three. The open-source solver available in Ref. [53] was used to minimize the objective function Φ derived from Eq. (1):
Φ(l)=minri=1mϕi(r,l)
(6)
This function is minimized through an iterative procedure using the equation:
(Ja(l)JaT(l)+μI)h=JaT(l)Φi(l)
(7)
where μ is an arbitrarily chosen damping parameter, h is the consecutive iteration step, and Ja(l) is the Jacobian matrix of Φ(l) defined as
Ja(l)=[ϕ1xϕ2xϕmxϕ1yϕ2yϕmyϕ1zϕ2zϕmz]T
(8)
This method has the advantage of guaranteeing a maximum computation time set by choosing a maximal number of iteration for the algorithm, thus ensuring real-time capability. Its suitability for real-time applications was demonstrated in Ref. [54], allowing the control algorithm to run at 1 kHz or higher frequencies, a necessity for haptic applications [55].

2.2.2 Extended Model.

Figure 4 shows the model chosen to represent the pulley mechanism. KA represents the entry point of the cable reeling on the pulley described by vector ai in the global coordinate system, kaz the z-axis of the winch coordinate system, DAi the detachment point of the cable, and Ci the center point of the pulley. This model was largely inspired by models proposed by Schmidt and Pott [56] and Bruckmann et al. [57]. With this model, the length li of cable i becomes
li=βi*rup+lfi
(9)
with rup as the radius of the pulley guiding the cable. Then, similarly to Eq. (1), we can define
ϕ^i(r,l)=βi*rup+lfili
(10)
To avoid having to define and compute transformation matrices, a vector from the origin KA in the direction of the center Ci is defined as
W=(kAz,i×(rai))×kAz,i
(11)
with its norm:
w=WW2
(12)
Now, the center of the ith pulley is obtained by calculating Ci:
Ci=ai+rup*w
(13)
Then, the distance between the center of the pulley and the end-effector becomes
di=rCi2
(14)
Applying Pythagoras’s theorem with the triangle EDAiCi yields:
lfi=di2rup2
(15)
Knowing the height of the end-effector:
Bz,i=kAz,i.(rai)2
(16)
we can calculate βi:
βi=Π+arccos(lfidi)arccos(Bz,idi)
(17)
Finally, all the necessary parameters of the inverse kinematics were obtained, and the LM (Levenberg–Marquardt) optimization algorithm can be applied to Eq. (18) to solve the forward kinematics:
Φ^(l)=minri=1mϕ^i(r,l,rup)
(18)
In the end, there are five methods that can be used to compute the kinematics for this system:
  • 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.

Fig. 4
Kinematic model considering the pulleys
Fig. 4
Kinematic model considering the pulleys
Close modal

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 τoffset; 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 [0,0,0]T, and for this, an additional part with dimensions known with 0.1 mm accuracy is used.

Fig. 5
Block diagram of the control scheme used to provide force feedback
Fig. 5
Block diagram of the control scheme used to provide force feedback
Close modal

3.1 Linear Actuator PID.

Using the forward kinematics described in Sec. 2, the angle θm 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.

The disturbance observer (DOB) and reaction torque observer (RTOB) were first presented in Ref. [58] by Murakami et al. Each motor is represented by an ideal model, here we chose a pure inertia I after experimental identification. The parameter identification was done by sending maximum length sequences as an input signal to the motors, then the Matlab identification toolbox estimated the parameters with a goodness-of-fit of 96% or more for each motor. The DOB suppresses the internal disturbance and parasitic noise of each motor using their speed and current input (LPF stands for low-pass filter on the diagram). Once the disturbance is suppressed, the RTOB estimates any variation of speed as an input force from the user. An estimated torque τi^ (for i=1,,6) is obtained for each motor, then the following equation gives u^:
[FxFyFz]u^=[u1xu2xu6xu1yu2yu6yu1zu2zu6z]JT[τ1^τ6^]τi^rpulley1
(19)
where JT is the transpose of the pose-specific structure matrix, τi^ is the cable force distribution, rpulley=0.0065 m is the radius of the pulley mounted on the motor shaft for the first prototype, and rpulley=0.005 m is the threaded shaft radius of the second system.

3.3 Virtual Model.

Efforts produced by the interaction between the operator and the virtual object are usually deduced using the god-object method [59]. In our work, such method is unnecessary because the object has a simple shape (sphere: see Fig. 6) and the reacting force is computed in the same thread as the control algorithm. We chose to use two virtual models: a mass-damper system model to estimate the movement of the sphere called X^object and a spring-mass-damper model to simulate the texture of the object. The movement is represented as
X^¨object=Jv1(u^Bvx˙effector)
(20)
with Jv=JvI3 representing the mass term and Bv=BvI3 representing the damper term, where I3 is the 3×3 identity matrix. By successive integration, X^object is obtained, and the actual movement of the object, considering the texture, X^, is similarly found as
X^¨=Jo1[u^Box˙effectorKo(xsurfacexeffector)]
(21)
where the mass, damper, and spring terms are, respectively, defined as Jo=JoI3, Bo=BoI3, and Ko=KoI3.
Fig. 6
Interaction between the operator and the virtual sphere. xsurface is simply calculated using the equation of a sphere in three dimensions.
Fig. 6
Interaction between the operator and the virtual sphere. xsurface is simply calculated using the equation of a sphere in three dimensions.
Close modal

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 IA, 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:
    IA=dwδxδyδzVW
    (22)
    where dw is the number of discretization points in the workspace of the robot; δx, δy, and δz are the resolution chosen for each direction (1 cm in our case) and maximal volume of the workspace.
  • The index of compactness, noted IC, is the ratio between the actual workspace and the smallest polyhedron encompassing this workspace:
    IC=dwδxδyδzVP
    (23)
    where VP is the volume of the surrounding polyhedron, in practice if the maximal workspace of our configuration is an irregular tetrahedron, VP will be the volume of a smaller tetrahedron included inside it.
  • Lastly, the index of symmetry, noted IS, represents the distortion of the polyhedron enveloping the workspace. It was redefined to fit polyhedrons with more complex shape than simple parallelepipeds:
    IS=1(i=1n|MSidesSi|nMSides)
    (24)
    where n is the number of sides of the enveloping polyhedron, MSides is the mean value of the polyhedron sides, and Si is the length of the ith side of the polyhedron.

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.

The geometric indices defined in the above section, while being useful to choose a CDPR configuration, are not sufficient to evaluate the quality of the workspace. Indeed, only focusing on increasing the workspace while reducing the number of cables could lead to poor manipulability or force exertion capabilities. A first naive approach consists of studying the resistance felt by the user when manipulating the end-effector. The manipulation resistance is defined as the Euclidean norm of the vector [Fx,Fy,Fz]T from Eq. (19) when only τoffset is applied to the handle. For the simulation, τoffset was set to 10 mN m, and then the resistance is calculated for every point inside the workspace with a 0.5 cm step using Mathematica. The same computation is done using actual data obtained by manipulating the end-effector inside the workspace. The results are presented in Fig. 7. They show a wider area of low resistance with the hybrid configuration, which could offer interesting prospects since most SPIDARs tend to use only the center of the workspace, where the tension is low, and the feedback is accurate during operation. It could be argued that cable tension distribution algorithms invalidate these results by suppressing any manipulation resistance, but these results provide insight into the isotropic properties of the system. Another approach to evaluate the quality of force closure at a specific configuration is to use the TF as a performance index (the methods and definitions proposed in Ref. [62] were followed). The TF is the ratio between the minimum and the maximum cable tension values achieved when the platform is in static equilibrium. Concretely, the tension factor depends on the structure matrix and shows the tension distribution among the cables for a specific end-effector pose inside the workspace. The TF ranges between zero and one. A TF close to zero signifies that the platform is located near the workspace limit. On the opposite, if the TF approaches one, the end-effector is far from the workspace boundary, and the CDPR behavior is nearly isotropic. To summarize, the larger the TF, the better the tension balance among the cables. To compute the TF, it is necessary to solve the optimization problem:
minJT=0tmintitmaxi=1mti
(25)
This problem can be solved using linear optimization algorithms provided by a standard Matlab function called “linprog.” Once the TF is obtained, the optimization algorithm can be applied to all the points included inside the discretized workspace. This allows us to calculate the global tension index (GTI), which is obtained by the integration of the local TFs over the whole workspace:
GTI=i=1NWsTFi(J)NWs
(26)
TFi(J) corresponds to the TF computed at the ith point of the workspace using the structure matrix J, and NWs is the number of points inside the workspace. Thus, the isotropic properties of a workspace can be evaluated, with the highest GTI being considered as the optimal configuration.
Fig. 7
Simulation of manipulation resistance on the right and actual measurements on the left. The top graphs represent the resistance for the usual configuration, while the bottom graphs are for the hybrid configuration. The scale is the same for each set of graphs to match the colors representing the effort.
Fig. 7
Simulation of manipulation resistance on the right and actual measurements on the left. The top graphs represent the resistance for the usual configuration, while the bottom graphs are for the hybrid configuration. The scale is the same for each set of graphs to match the colors representing the effort.
Close modal

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: tmin and tmax 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 IS. The high values of IA and IC show that the workspace-to-device span ratio is more optimal. Lastly, the GTI is in concordance with the expectations explained earlier.

Fig. 8
Matlab simulation of tension factor for each point of the workspace and for each configuration
Fig. 8
Matlab simulation of tension factor for each point of the workspace and for each configuration
Close modal
Table 2

Results of the quantitative and qualitative evaluation of the workspace for each configuration

ConfigurationAnchor pointsWorkspace envelopIAICISGTI
Six cables: fixed anchor pointsxyzxyz0.7520.8150.8910.243
0.2450.3850.2650.240.370.26
0.2450.3850.2650.240.370.26
0.2450.3850.2650.240.370.26
0.2450.3850.2650.240.370.26
0.24500.2650.2400.26
0.24500.2650.2400.26
Six cables: moving anchor points0.2450.3850.2650.240.380.260.8560.9000.8720.267
0.2450.3850.2650.240.380.26
0.2450.3850.2650.240.380.26
0.2450.3850.2650.240.380.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
Four cables: fixed anchor points00.3850.26500.380.26
00.3850.26500.380.260.7800.8080.9070.294
0.24500.2650.2400.26
0.24500.2650.2400.26
Four cables: moving anchor points0.2450.3850.2650.240.380.260.8410.8810.8720.344
0.2450.3850.2650.240.380.26
0.2450.3850.2650.240.380.26
0.2450.3850.2650.240.380.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
ConfigurationAnchor pointsWorkspace envelopIAICISGTI
Six cables: fixed anchor pointsxyzxyz0.7520.8150.8910.243
0.2450.3850.2650.240.370.26
0.2450.3850.2650.240.370.26
0.2450.3850.2650.240.370.26
0.2450.3850.2650.240.370.26
0.24500.2650.2400.26
0.24500.2650.2400.26
Six cables: moving anchor points0.2450.3850.2650.240.380.260.8560.9000.8720.267
0.2450.3850.2650.240.380.26
0.2450.3850.2650.240.380.26
0.2450.3850.2650.240.380.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
Four cables: fixed anchor points00.3850.26500.380.26
00.3850.26500.380.260.7800.8080.9070.294
0.24500.2650.2400.26
0.24500.2650.2400.26
Four cables: moving anchor points0.2450.3850.2650.240.380.260.8410.8810.8720.344
0.2450.3850.2650.240.380.26
0.2450.3850.2650.240.380.26
0.2450.3850.2650.240.380.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26
0.2450.250.2650.240.250.26

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.

Fig. 9
Computation time of the LM solver
Fig. 9
Computation time of the LM solver
Close modal
Fig. 10
Iterations of the LM solver
Fig. 10
Iterations of the LM solver
Close modal

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 z-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 μs.

  • 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 μs for 29 iterations for the simplified model and 52 μs 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 μs for 79 iterations for the simplified model and 151 μs for 81 iterations for the extended model.

Fig. 11
Trajectory used for the computation algorithm comparison with different calculation methods: (a) shows the trajectory of the end-effector in 3D, while (b), (c), and (d) show the displacement along the x, y, and z axis, respectively
Fig. 11
Trajectory used for the computation algorithm comparison with different calculation methods: (a) shows the trajectory of the end-effector in 3D, while (b), (c), and (d) show the displacement along the x, y, and z axis, respectively
Close modal

Although this does not serve as a demonstration of real-time (1 kHz) capability, there were no significant missteps observed. The outliers that can be seen in Figs. 9 and 10 most likely result from the purposely poor experimental conditions.

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 u^. 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

Fig. 12
Experimental evaluation of the haptic feedback. From left to right: a front view, a view from the right, and a view from 45 degrees of the object.
Fig. 12
Experimental evaluation of the haptic feedback. From left to right: a front view, a view from the right, and a view from 45 degrees of the object.
Close modal

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.

Fig. 13
Picture of the virtual environment on the left. On the right, an example of trajectory followed by the user is shown.
Fig. 13
Picture of the virtual environment on the left. On the right, an example of trajectory followed by the user is shown.
Close modal
Fig. 14
Comparison of the time (s) and distance (m) necessary to complete the picking task for two groups of users, with and without the linear actuator moving. Column (a) shows the data obtained for the untrained groups and column (b) the data for the trained group.
Fig. 14
Comparison of the time (s) and distance (m) necessary to complete the picking task for two groups of users, with and without the linear actuator moving. Column (a) shows the data obtained for the untrained groups and column (b) the data for the trained group.
Close modal

4.4.2 Results.

Paired sample t-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 (p=0.120 for the distance and p=0.404 for the completion time). The same observation can be made for trained users regarding the travel distance (with p=0.157). However, the comparison of travel time has a p-value closer to 0.05 (p=0.085). 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

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

h =

half-height of the first prototype (around 53 cm)

l =

half-length of the first prototype (around 77 cm)

n =

number of sides of the enveloping polyhedron

w =

half-width of the first prototype (around 49 cm)

r =

end-effector position vector [xe,ye,ze]T

u^ =

estimated force feedback vector

w =

unit vector

E =

end-effector position in the reference frame KO

J =

pose-specific structure matrix

F =

force vector applied by the user

W =

vector from KA to Pi

X^ =

actual movement of the object considering the texture

X^¨ =

acceleration of the actual movement of the object considering the texture

di =

distance between the center of the pulley and the end-effector

dw =

number of discretization points in the workspace

kaz =

z-axis of the winch coordinate system

li =

length of cable i

rpulley =

radius of the pulley mounted on the motor shaft

rup =

radius of the pulley guiding the cable

ti =

tension in cable i

tmin =

minimum cable tension

tmax =

maximum cable tension

yl =

linear actuator’s position

ai =

vector representing the anchor point of cable i

li =

vector from the end-effector to anchor points

ui =

unit cable vectors

xeffector =

position of the end-effector

xsurface =

position of the object’s surface

Bz,i =

height of the end-effector

Ci =

center point of pulley i

DAi =

detachment point of cable i

IA =

area index

IC =

index of compactness

IS =

index of symmetry

Ja =

analytic Jacobian matrix

KA =

entry point of the cable reeling on the pulley

MSides =

mean value of the polyhedron sides

NWs =

number of points inside the workspace

Si =

length of the ith side of the polyhedron

VP =

volume of the surrounding polyhedron

VW =

theoretical maximal value of the workspace

Ai =

anchor points of the cables

I3 =

3×3 identity matrix

X^object =

estimated movement of the sphere

X^¨object =

acceleration of the estimated movement of the sphere

GTI =

global tension index

TF =

tension factor

Jo,Bo,Ko =

mass, damper, and spring term matrix for simulating the texture of the virtual object

Jv,Bv =

mass and damper term matrix for the virtual model

βi =

angle associated with each cable in the extended kinematic model

δx, δy, δz =

resolutions in the x, y, and z directions

θi =

angle associated with each cable

θm =

angle of each motor

τi =

cable force distribution for each motor (i=1,,6)

τoffset =

minimum force of around 0.7 N used to straighten the cables

τi^ =

estimated torque for each motor (i=1,,6)

ϕi =

function representing the cable tension equations

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