Maestro, "NeuroTouch: a physics-based virtual simulator for cranial microneurosurgery training," Operative. Neurosurgery, vol. 71, pp. 32-42, 2012.
Dynamic Characterization of a Parallel Haptic Device for Application as an Actuator in a Simulation Surgery Farshad Khadivar1, Soroush Sadeghnejad1*, Hamed Moradi2, Gholamreza Vossoughi2 and Farzam Farahmand2 School of Mechanical Engineering, Sharif University of Technology Tehran, Iran 1 {Khadivar_farshad, s_sadeghnejad}@mech.sharif.edu 2 {hamedmoradi, vossough, farahmand}@sharif.edu Abstract — Tactile sense is a key element is in developing virtual reality simulators or surgical training systems. In this regard, haptic interfaces as the generator of a sense of touch play a significant role in producing a realistic haptic feedback force. Since the majority of practical control theories are model based, the robot dynamic model identification is a process of high importance and application. The main concern, accordingly, is to find a precise dynamic model for the aforementioned user interfaces. Hence in this research, we proposed implementing the Lion identification method to characterize the dynamics of a parallel haptic device in actuating a surgery simulation. We chose the Novint Falcon as our haptic device which is a parallel impedance-type robot of low price, considerable load capacities, and proper workspace. In order to develop the dynamic model, we considered piecewise linear functions in different operational points in the robot workspace to cover robot nonlinearities. With respect to the Lion Identification Method, we guaranteed the stability of estimation error dynamic from the Lyapunov perspective. Thereafter, the parameter estimation dynamics and the identification cost function are derived. Then by running a single-axis haptic experimental setup the robot generated force is calibrated, and four different sets of sine wave inputs of various frequencies are imposed on the robot to calculate the required parameters by Matlab Simulink. The results reveal that the system parameters converge to the specific values while the output tracking error and its derivative behavior is reasonable, that is, the system identification is of great accuracy. Keywords— haptic interface; Lion identification method; Novint Falcon robot; dynamic model; Matlab Simulink; Lyapunov stability.
I. INTRODUCTION Haptic devices or force feedback devices are the electronic systems which are able to produce third senses to the user e.g. visual, auditory, and tactile. Nowadays, the “tactile” sense is of high necessity in developing a realistic virtual reality simulators or training systems, used for the medical as well as surgery usages. To this end, haptic interfaces have become an important part of VR simulators for minimally invasive surgery (MIS) and endoscopic surgery trainings [1-3]. The haptic interfaces are the mechanical devices which can be classified according to various criteria, one vividly
of which is based on kinematic structure and topology. Haptic devices can be divided into two general kinematic structures, parallel, and serial devices. A haptic device is considered a serial interface, or open-loop device if its kinematic structure has an open loop-chain form. A parallel interface, on the contrary, is made of a closedloop chain. The parallel haptic interfaces, in comparison with serial haptic devises, have been proved to be an excellent platform for high-speed operations due to the mechanism's lower actuated inertia, and higher stiffness, payload capability, power to weight ratio. However, there is quite complexity in the control, kinematic modeling, and dynamic modelling of such parallel mechanisms. Additionally, haptic interfaces can be divided into another general category, the impedance-type and the admittance-type haptic devices. The admittance-type haptic devices compute a displacement in response to the measurement of the interface force while the impedancetype devices generate a reaction force in response to the users’ position [4]. The widespread use of these systems is still limited by the high costs for each special purpose of haptic interfaces. Since there is no need to measure the haptic interface forces in comparison to admittance ones, the impedance type haptic interfaces are more costeffective. Different commercialized interface devices are developed recently [5, 6]. Some of the most commonly used are the PHANTOM© (SensAble Technologies, Inc., Woburn, MA) in the serial type interfaces; the delta.x, and omega.x (Force Dimension, Nyon, Switzerland); and the Novint Falcon (under present study) in parallel devices [7, 8]. Mainly designed for gaming purposes, Novint Falcon is a parallel impedance-type robot which is of low price, considerable load capacities, and proper work space. The design of the Novint Falcon is based on the 3-DOF Delta parallel robot. The Falcon’s programmable interface relieves the user from the inverse kinematics of the robot; therefore, convenient control of robot’s motion is provided on the three motion axis (x, y, z). Due to 1KH sampling frequency and smooth actuation, operators use it in precise position sensing and high fidelity of system motion control [5]. Since many developed controllers are model-based, precise system model is needed to enhance
the control performance. Thus we need to dynamically identify the haptic interfaces, by use of a calibrated force sensor. There are several methods for identifying the dynamics of a plant only by observing as well as measuring the input and output. Lion identification method has been proposed for synthesizing three different major problems. 1) linear autonomous systems, 2) a certain class of nonlinear autonomous systems, and 3) in an approximate manner, for linear non-autonomous systems [9]. This method is based on the generalized equation error, which combines the advantages of both response error and equation error systems. To so doing, the system follows a true steepest descent path in parameter space while using only the input and output of the system. Hence the system will be proved to be globally, asymptotically stable when the system input contains sufficient independent input frequencies. Enjoying the Lion identification method, this paper aims
to characterize the dynamics of a parallel haptic device for application in actuating a surgery simulation. To this end, we chose the Novint Falcon in order to develop a dynamic model, and incorporate appropriate workspace constraints for subsequent use in model-based prototype trajectory generation, estimation, and control methodology research in general tele-robotic, and virtual based haptic manipulations. Whilst the manufacturer provides some basic specifications on workspace size and forces the device is able to realize, they did not describe the sufficient details for modelling or control purposes. Therefore, the parameter estimation dynamics, and the cost function of identification are derived by use of Lion identification method firstly. Then the robot generated force is calibrated, and four different inputs with distinctive frequencies are set on robot in each linear section three times. Finally, the system parameters are calculated by means of Matlab Simulink. The rest of this paper is as follows. The characteristics of Novint Falcon as a parallel haptic device, and the experimental setup are described in Section 2. Section 3 presents the identification method, and the derivation of parameter estimation dynamics. Eventually the parameters of robot dynamic model are calculated by implementing the identification method in Section 4.
II. A PARALLEL HAPTIC DEVICE In this research, the utilized experimental set-up consists of the Novint Falcon haptic device. Three Mabuchi RS- 555PH-15280 motors are being used to actuate the Novint Falcon, the motion of which can be monitored by a coaxial 4-state encoder with 320 lines per revolution. These motors are directly coupled to a 14.25 mm diameter drum, upon which a 0.5 mm diameter cable is wrapped. The cable runs around the outer edge of the main leg links at a radius of 56.0mm, and is tensioned by a spring at the inner end of the leg [5]. This arrangement causes a low-friction, and robust-actuation mechanism which effectively provides a 7.62:1 gain for the motor output of the main leg's pivot without resorting to the use of gears [1]. A. Setting up the haptic interface Fig. 1 depicts the single-axis haptic experimental setup used in this paper. Since the Falcon robot can only present the position of the end-effector to the user, a force sensor is attached to the robot to measure applied force from the operator. The Falcon robot is equipped with a UMMA5Kg.f force sensor, produced by Dacell Company. The acquired data from sensor will be passed through DN-AM100 amplifier from Dacell Co. before passing to the acquisition system. Additionally, the system software runs on a PC platform, and the interface between the sensor and the computer is an ARDUINO chip (the baud rate value is 9600 byte/sec) which reads the sensor data via serial port, then transfers them to a computer via USB port. III.HAPTIC INTERFACE DYNAMIC MODEL IDENTIFICATION METHOD Many researches proposed some general specific dynamic model to easily simulate the behavior of robot manipulators [10, 11]. Due to the significant nonlinearities in the kinematic and dynamic properties of parallel mechanisms, there are different challenges in implementing various control strategies on them. Thus this limitation makes the robot identification process a delicate task.
Fig. 1 System setup for the dynamic characterization of a parallel haptic device
Taking into account the aforementioned issues, the use of a suitable dynamic model of parallel haptic interfaces in designing a complete tele-operated or VR based haptic system is of a paramount necessity. In spite of the simplicity in the Novint Falcon design, there are general nonlinear characteristics in both kinematics and dynamics of the system. Hence it is necessary to identify the dynamics of the product in order to develop a modelbased control for further studies. In previous researches, for simplicity, the robot dynamic has been considered as a combination of a linear set of lumped elements namely mass, spring and damper [12-14]. Nonetheless, the robot shows nonlinear behavior besides has no elastic properties caused by spring element in the dynamic model based on the robot performance. The decoupled dynamic model of the robot can be considered as:
(1)
Which 𝑀𝑞 and 𝐶𝑞 (𝑞, 𝑞̇ ) are robot inertia and damping coefficient in axis(Q) where 𝑄 ∈ {𝑋, 𝑌, 𝑍}. Parameters in (1) are generally nonlinear, and identification of them requires dealing with extreme complexities. Hence we assume that robot dynamic can be divided into some piecewise linear regions in different robot’s workspace. To this end, without loss of generality Laplace transformation is taken from (1): 𝑖 = 1, … , 𝐿
(𝑀𝑖 𝑆 2 + 𝐶𝑖 𝑆)𝑥(𝑠) = 𝐹(𝑠)
(2)
In which L is the number of assumed piecewise linear sections, and V(S) is the velocity of the robot’s endeffector in the Laplace space. 𝑎𝑖 and 𝑏𝑖 are the dynamic parameters of robot. Since we need to identify two parameters of the system, we have to calculate the tracking error signals and its derivatives. As the order of the system dynamic is two, we should define a third order filter for calculating the derivatives of the tracking error signals: (3)
Taking into account (3) with (2), we have: S2 𝑎𝑖 𝑆 𝑏𝑖 𝑋(𝑠) = − 𝑋(𝑠) + 𝐹(𝑠) Λ(𝑠) Λ(𝑠) Λ(𝑠)
(4)
Defining the input as output signals as: 𝑉𝑗 (𝑠) = −
𝑆𝑗−1 𝑋(𝑠) Λ(𝑠)
Thus (2) can be rewritten in the following form (6)
𝑉3 = −𝑎𝑖 𝑉2 + 𝑏𝑖 𝑄𝑖
If 𝛼𝑖 and 𝛽𝑖 are the estimation of 𝑎𝑖 the 𝑏𝑖 respectively, then the error and its derivation will be: 𝜀1 = 𝑥 − 𝑥̂ = 𝑉3 − 𝑉̂3
(7)
𝜀2 = 𝑆𝜀1 = 𝑉4 − 𝑉̂4
Hence: (8)
𝜀2 = 𝑉4 + 𝛼𝑖 𝑉3 − 𝛽𝑖 𝑄2
𝜃 𝑇 = [𝛼𝑖 , 𝛽𝑖 ] 𝜀 𝑇 = [𝜀1 , 𝜀2 ] 𝑊=[
𝑉2 𝑉3
𝑄1 ], 𝑄2
(9) 𝑉 𝑊0 = [ 3 ] 𝑉4
From (8), we can introduce the error dynamic and the cost function as below: 𝜀 = 𝑊𝜃 + 𝑊0
(𝑆 2 + 𝑎𝑖 𝑆)𝑥(𝑠) = 𝑏𝑖 𝐹(𝑠) → (𝑆 + 𝑎𝑖 )𝑉(𝑆) = 𝑏𝑖 𝐹(𝑠)
Λ(𝑠) = 𝑆 3 + 𝜆1 𝑆 2 + 𝜆2 𝑆 + 𝜆3
𝑗 = 1, … , 4
If we define a new notation in the vector and matrix forms, then:
𝑀𝑧 (𝑧)𝑧̈ + 𝐶𝑧 (𝑧, 𝑧̇ ) = 𝐹𝑧
𝑀𝑖 𝑥̈ + 𝐶𝑖 𝑥̇ = 𝐹 ,
𝑆𝑗−1 𝐹(𝑠), Λ(𝑠)
𝜀1 = 𝑉3 + 𝛼𝑖 𝑉2 − 𝛽𝑖 𝑄1
𝑀𝑥 (𝑥)𝑥̈ + 𝐶𝑥 (𝑥, 𝑥̇ ) = 𝐹𝑥 𝑀𝑦 (𝑦)𝑦̈ + 𝐶𝑦 (𝑦, 𝑦̇ ) = 𝐹𝑦
𝑄𝑗 (𝑠) = −
(5)
1 𝐽 = 𝜀𝑇𝜀 2
(10)
By imposing the system input, and measuring the system output, the matrix 𝑊 and the vector 𝑊0 will be calculated. Afterwards, parameters are identified by proposing an estimation dynamic for 𝜃 in a way that integral of output tracking error and it’s derivatives, or the cost function 𝐽 converge to zero. Utilizing the Lion Identification Method [9], This dynamic can be considered as: 𝜃̇ = −𝐺𝑊 𝑇 𝜀
(11)
𝜃0𝑇 = [𝛼𝑖0 , 𝛽𝑖0 ]
Where 𝐺 is a positive definite estimation gain matrix, by rising the value of which we can control the convergence rate. Also, an initial guess, 𝜃0 , is required for parameters 𝜃. As a commonplace case in all identifying methods, it is worth mentioning that the parameters will not necessarily converge to their actual values but values which minimize the cost function 𝐽. In fact, different parameter candidates are possible for the model dynamics considering a specific input and output data. the number of candidates reduces as long as rich inputs are implemented [15]. Although finding
20
IV. IDENTIFICATIO AND VERIFICATION OF A DYNAMIC MODEL Because of limitations in the Falcon operating workspace, as well as further application in surgery simulation system, the best workspace of the Falcon robot is defined in one straight direction. Hence a single-axis movement considered for the operational workspace of system in surgery simulation application. A. The robot generated force calibration A mathematical relation between the desired force and the force generated by robot is required to be derived. Therefore, the robot generated force will be calibrated before any further identification. To this end, five points of the robot in axis (X) are taking into account (Fig. 2). We just set different force values and measured the generated ones. Table 1 reveals the considered operational points for robot’s force calibration. Then, by simple data fitting methods, we reached to the following scaling factor as presented in Fig. 3. (12)
Fs = 1.75𝐹𝑑
Where F𝑑 is the desired force, and F𝑠 is the force that should be set in the robot’s program. B. The robot dynamic model identification In order to use (11) for identifying system dynamic (2), we implemented four different inputs with various frequencies in each linear section three times. Due to existence of some operational limitations on the way of experimental tests, which will not allow us to take any point as the operational points, we considered the coordinates of three different operational points, shown in (Fig. 2).
Set Force - Fs - (N)
parameter values with least output tracking error rather the exact ones is our target, we try to use the most possible rich input to find best values for the desired parameters.
15
Fs = 1.75 FitedFdline
X = +30mm X = +15mm X = 0mm X = -15mm X = -30mm
10
5
0 0
2
4
6
8
Desired Force - Fd - (N)
10
12
Fig. 3 Results of system dynamic parameters identification around operational point 𝑥2 Table 2 The robot’s operational points’ coordinate Point 𝒙𝟏 𝒙𝟐 𝒙𝟑 Distance (mm)
-20
0
+20
Considering 𝜆1 = 12, 𝜆2 = 48 and 𝜆3 = 64 for the 3rd order polynomial function defined in filter (3), the gain 𝑇 1 0 matrix as 𝐺 = 104 [ ], and the initial guess 𝜃0 = 0 1 [0 0], the system parameters are calculated by means of Matlab Simulink. Fig. 4 and Fig. 5 show the results of parameters identification, and output tracking error as well as its derivative around operational point 𝑥2 . As one can see from the results, the system parameters converge to the specific values, and the output tracking error and its derivative show reasonable behavior; that is, the system identification is working properly. The same results have been acquired for other different operational points while due to similarity, just one result has been reported in this section. Table 3 reveals the identification of robot dynamic parameters for all inputs in vicinity of all operational coordinates. The acquired values of parameters for second input are unacceptability negative since they will make the whole system unstable. The acquired parameters related to the third input, however, are quite acceptable since they have the least amount of identification error and can tolerate the higher forces. Finally, the robot dynamic model will be presented as: 𝑥̈ + 16.2𝑥̇ = 2.02𝐹 𝑥 < −10𝑚𝑚 { 𝑥̈ + 22𝑥̇ = 1.916𝐹 − 10𝑚𝑚 ≤ 𝑥 ≤ 10𝑚𝑚 𝑥̈ + 37.1𝑥̇ = 1.289𝐹 𝑥 > 10𝑚𝑚
(13)
Fig. 2 Defined points for robot’s generated vs. desired force calibration
C. The robot dynamic model verification
Table 1 The robot’s selected points’ coordinate for calibration
Points
𝑿𝟏
𝑿𝟐
𝑿𝟑
𝑿𝟒
𝑿𝟓
Distance (mm)
-30
-15
0
15
30
For verifying the presented dynamic system for the Falcon haptic interface, we utilized a high frequency input as 𝐹 = 3 cos(40𝑡 + 400) + 0.5𝑡, then recorded the robot displacement, and the displacement of (13).
Table 3 The robot dynamic parameters’ identification for all inputs in vicinity of all operational coordinates 𝒙 = −𝟐𝟎𝒎𝒎 𝒙 = 𝟎𝒎𝒎 𝒙 = +𝟐𝟎𝒎𝒎 Parameters
𝑎(
First Input
Second Input
Third Input
Fourth Input
a2 (N.s/m.kg)
𝑁. 𝑠
𝑚𝑚. 𝑘𝑔 3.09 3.74 3.74 -0.68 -0.317 -0.68 23.45 16.20 26.86 -
)
𝑏 (1/𝑘𝑔)
𝑣𝑎𝑟 (𝑚𝑚2 )
1.701 1.656 1.618 1.546 1.569 1.546 2.074 2.020 2.112 -
0.539 1.794 0.3 0.305 0.531 0.305 0.124 0.0986 0.103 -
𝑁. 𝑠
𝑎(
𝑚𝑚. 𝑘𝑔 4.43 6.15 5.22 -6.70 -8.20 -5.80 21.53 22.00 2.103 33.94 34.47 37.45
𝑣𝑎𝑟 (𝑚𝑚2 )
1.620 1.569 1.641 1.710 1.740 1.720 1.816 1.916 1.867 1.910 1.916 1.980
0.85 0.64 0.44 0.024 0.025 0.026 0.061 0.038 0.049 0.567 0.502 0.621
20 10 Paremeter a2
0 3
4
5
b2 (1/kg)
6
7
8
T (s)
1 Parameter b2
4
5
6
7
8
T (s) Fig. 4 Results of system dynamic parameters identification around operational point 𝑥2 5
x 10
𝑎(
𝑁. 𝑠
𝑚𝑚. 𝑘𝑔 14.99 14.99 14.99 -9.73 -9.74 -9.4 40.96 37.099 43.06 -
)
𝑏 (1/𝑘𝑔)
𝑣𝑎𝑟 (𝑚𝑚2 )
1.117 1.111 1.116 1.500 1.478 1.470 1.720 1.289 1.540 -
0.45 0.46 0.45 0.1143 0.1148 0.11140 0.590 0.4003 0.544 -
V.CONCLUSION AND FUTURE WORKS
0 3
-3
Tracking Error
The aim of this paper is characterizing the dynamics of a translational parallel haptic robot for application in the surgery simulation. For this purpose, the Novint Falcon haptic device has been investigated. The existing nonlinearities in the both kinematics and dynamics of the parallel robots, make the control of those aforementioned devices a challenging task. To so doing, a general model-based dynamic can be used to simulate the behavior of these types of haptic interfaces. The mass-damper (no elastic properties presented in the experimental tests) models are the common form of dynamic representation of parallel haptic devices which generally lead to proper identification of the system dynamic model. Considering piecewise linear functions in different operational pints, in the robot workspace with respect to the Lion Identification Method, plus running a single-axis haptic experimental setup, we guaranteed that the parameters estimation error dynamic of the proposed model are stable from the Lyapunov perspective.
0 -5 3.5 5
x 10
4
4.5
5
5.5
6
6.5
7
7.5
8
30
T (s)
-3
0 -5 3.5
4
4.5
5
5.5
6
6.5
7
7.5
x estimated x robot
20
Tracking Error Derivative
X (mm)
e (mm)
𝑏 (1/𝑘𝑔)
30
2
edot (mm/s)
)
8
T (s) Fig. 5 Output tracking error and its derivative for the system dynamic parameters identification
Fig. 6 represents the result of identified dynamic model versus the robot’s output which reveals satisfactory accuracy of achieved model (13) despite the high frequency of input signal.
10 0 -10 -20 -30 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
T (s) Fig. 6 Results of system dynamic parameters identification around operational point 𝑥2
Four different sets of sine wave functions as motion inputs have been provided. The results demonstrate that the system parameters converge to the specific values while the output tracking error and its derivative conduct reasonably, testifying the fact that the system identification is of desired accuracy. Ongoing works include expanding the workspace identification, and enjoying the proposed piecewise liner model in a real haptic virtual based controlling system. Furthermore, the experimental identified model can be used for other model-based controllers.
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ACKNOWLEDGMENT We are grateful to the Djavad Mowafaghian Research Center of Intelligent NeoruRehabilitation Technologies and also the Research Center of Biomedical Technology and Robotics at the Research Institute of Medical Technology of Tehran University of Medical Sciences for the support of this research and conducting the experiments. In addition, we would like to thank Ehsan Abdollahi and Mojtaba Esfandiari for helping us prepare the set-up system and their helpful comment on several other issues. REFERENCES [1]
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