Design of a series visco-elastic actuator for multi-purpose rehabilitation haptic device
© Oblak and Matjačć; licensee BioMed Central Ltd. 2011
Received: 14 May 2010
Accepted: 20 January 2011
Published: 20 January 2011
Variable structure parallel mechanisms, actuated with low-cost motors with serially added elasticity (series elastic actuator - SEA), has considerable potential in rehabilitation robotics. However, reflected masses of a SEA and variable structure parallel mechanism linked with a compliant actuator result in a potentially unstable coupled mechanical oscillator, which has not been addressed in previous studies.
The aim of this paper was to investigate through simulation, experimentation and theoretical analysis the necessary conditions that guarantee stability and passivity of a haptic device (based on a variable structure parallel mechanism driven by SEA actuators) when in contact with a human. We have analyzed an equivalent mechanical system where a dissipative element, a mechanical damper was placed in parallel to a spring in SEA.
The theoretical analysis yielded necessary conditions relating the damping coefficient, spring stiffness, both reflected masses, controller's gain and desired virtual impedance that needs to be fulfilled in order to obtain stable and passive behavior of the device when in contact with a human. The validity of the derived passivity conditions were confirmed in simulations and experimentally.
These results show that by properly designing variable structure parallel mechanisms actuated with SEA, versatile and affordable rehabilitation robotic devices can be conceived, which may facilitate their wide spread use in clinical and home environments.
Rehabilitation robotics is a rapidly evolving field [1–4]. Numerous haptic robots were developed for movement training of upper extremities following neurological disorder. According to works published to date, robots for upper extremity motor rehabilitation are usually serial linkage mechanisms that can be in general divided in two groups. The first consists of serial linkage mechanisms with only 1 to 3 degrees of freedom (DOF), where the end-effector of the robot is in contact with the user's hand, making it suitable for only one activity of upper extremity movement (either arm reaching movement or wrist movements). Clinical use of such low-DOF serial mechanisms [5–10] necessitates the use of two or more devices in order to provide comprehensive upper extremity movement training. This is neither convenient from a practical nor cost-effective point of view. On the other hand, exoskeleton mechanisms may have up to 7 DOF [11–13] and can provide comprehensive upper extremity movement rehabilitation. However, such mechanisms require high quality back drivable actuators for each DOF, which necessitates complex and thus expensive design.
Few rehabilitation robots have implemented a parallel kinematic structure. Parallel mechanisms usually have mechanical linkages with many DOF that greatly exceed the resulting DOF of the whole mechanism. This property allows for a design where some of the joints may be easily locked or unlocked, thus resulting in different workspace configurations suitable for different aspects of arm or wrist movements training. Another characteristic of parallel robots is that the actuators are located at the robot's base. This feature allows the implementation of series elastic actuators (SEA) [14–16] that utilize standard off-the-shelf DC motors with suitable planetary gearheads and suitable springs, providing similar overall haptic performance as their high quality back-drivable counterparts. Universal Haptic Drive (UHD)  and Variable Structure Pantograph (VSP)  are the two devices in which mechanisms with lockable joints and SEA actuation were successfully implemented and tested in clinical practice.
However, from a control point of view, both beneficial aspects; parallel kinematic structure (such as VSP) and SEA based drive, result in a mechanical system where the reflected masses of the SEA and the parallel kinematic structure (serially connected with a spring) become comparable, resulting in a coupled mechanical oscillator. Suitable control of such a rehabilitation robot, which should provide stable haptic interaction when in contact with a human, may present a considerable challenge, not addressed in previous studies.
The aim of this paper was to investigate theoretically, through simulations and experimentally the necessary conditions that guarantee stability and passivity of a haptic device, based on a variable structure parallel mechanism driven by SEA actuators, when in contact with a human. We have analyzed an equivalent mechanical system where a dissipative element, a mechanical damper, was placed in parallel with a spring in SEA. The goal was to derive conditions that must be met in order to allow use of a SEA driven variable structure parallel mechanism as a stable haptic interface in upper extremity rehabilitation.
Variable structure pantograph: mechanical linkage
"ARM" mode: locking joint I and releasing joints II and III, results in 2 DOF quasi-planar movements in Forward/Backward/Left/Right directions, as shown in Figure 2(B). The movement prescribed by the workspace "ARM" mode is similar to required for reaching and/or moving objects on a table, desk, or countertop.
"WRIST" mode: the mechanical configuration, termed as "WRIST" mode, is achieved by releasing joint I and locking joints II and III. A subject holding on the handle bar can perform movements in wrist as shown in Figure 2(C). By setting the offset orientation of the handle bar in the horizontal or vertical position, movement of all 3-DOFs in wrist (Flexion/Extension, Radial/Ulnar deviation and Pronation/Supination as shown in Figure 2(C)) can be achieved. The resulting movement of the user's wrist is similar to what is required for performing wrist-orienting motions in the following activities: pouring from a bottle, brushing teeth, or stirring a pot.
"REACH" mode: locking joints I and III and releasing joint II results in a mechanical configuration, which allows training of Forward and Up/Lateral reach movements. These motions are therefore similar to activities such as reaching for a high drawer or cupboard, or moving objects from one side of the cupboard to the other.
Variable structure pantograph: series visco-elastic actuation
Introduction of an elastic element (mechanical spring) in series with the motor provides many benefits, including: more accurate and stable force control, attenuation of both backlash and friction nonlinear effects and the actuators' own impedance as well as providing greater shock tolerance (important for safety concerns). On the other hand, introducing SEA in haptic drive leads to reduction of the achievable force bandwidth. Since relatively slow movements can be expected during rehabilitation training, reduced force bandwidth does not present a significant problem.
Utilization of a variable structure parallel mechanism is essential in designing a versatile rehabilitation device. However, using a parallel mechanism where considerable endpoint mass is in series with both SEA's spring and motor mass, results in a coupled oscillator needing appropriate damping. Adequate dissipation of mechanical energy is needed, to achieve a stable haptic interaction when the device is in contact with a human. A convenient location for a mechanical damper may be in parallel with the SEA spring. Figure 3(A, B) presents a schematic illustration of an implemented parallel mechanism driven by a series visco-elastic actuator.
Variable structure pantograph: linearized model
The principal purpose of haptic devices is to allow human operators to touch, feel and manipulate objects in a virtual environment. For this reason, the impedance felt at the handle bar should be as close as possible to the desired virtual impedance (V, Figure 5).
Usually, three criteria are employed when designing haptic devices : (1) movement in free space (LOW impedance) should be opposed with minimal possible force, (2) solid virtual objects (HIGH impedance) must feel stiff, and (3) virtual constraints must not be easily saturated, which requires a suitable impedance- based force control. In series elastic actuators, a variety of control strategies are possible. Williamson  proposed a control strategy for SEA with a feed-forward model and PID controller. Vallery  chose the concept of cascade force control with proportional-integral controller. We decided to implement the simplest approach, which is a proportional controller, in order to have a clearer picture on the influence of various mechanical components on passivity of haptic interaction [22, 23].
Impedance based force control (Figure 5), was implemented in MATLAB (Simulink). In computer simulation, the VSP's haptic performance was investigated by simulating sinusoidal movements of the handle bar (XO). In simulation, LOW and HIGH impedance virtual force FV was compared to calculated force on the handle bar FO, for different values of damper b and proportional gain P. In order to investigate the option of using low-cost motors with potentially redundant backlash, different values of backlash were considered in the simulation model.
Conditions regarding stability and passivity of the system
Z(s) has no poles in the right half plane
Re(Z(jw)) is nonnegative for all frequencies w
If these conditions are met, the impedance is a stable function of frequency and the system exhibits passivity.
It is obvious that Hurwitz determinants of the characteristic equation for Z(s) are nonnegative for all technically realizable values of mechanical components. Henceforth, Z(s) has no poles in right half plane and the first rule is met.
First, the virtual stiffness V is limited by the stiffness of the mechanical spring K and controller's proportional gain P. However, if we look at the third condition, P is limited by the reflected motor mass M and reflected mass of the parallel mechanism m. It is straightforward, that value P can be increased by reducing m. The second condition demands that there must be sufficient damping between the relative position of motor XI and parallel mechanism XO. Usually, in technical realization of the system, damping is always present due to viscous friction in mechanical components but is not sufficient. For this reason, an additional damping element should be inserted in parallel with the spring to satisfy the damping condition for passivity of the system. The derived conservative conditions for passivity are general. In the following section, these conditions will be applied to characteristic values of the mechanical components used in technical realization of the VSP (listed in Figure 3(B)).
Variable structure pantograph: application of derived passivity conditions
In the "ARM" and "REACH" mode of the VSP, estimated reflected mass of the parallel mechanism is relatively high (m = 18 kg). From the third condition on passivity, achievable controller's proportional gain is relatively small . This is due to the high reflected mass of the parallel mechanism m. From the first condition, maximal virtual stiffness V can be determined as . Finally, from the second condition, damping of is needed, if we set virtual stiffness as V = 12000 N/m (solid virtual objects). On the other hand, if we want to emulate free space, where V = 0 N/m and all the other parameters remain the same, a much smaller damping of b ≅ 250 Ns/m satisfies the second condition for passivity.
Values of P, V and b that satisfy conditions regarding passivity.
"ARM" and "REACH" mode [m = 18 kg]
"WRIST" mode [m = 10 kg]
0 ≤ V ≤ 12000
0 ≤ V ≤ 10000
V = 0 N/m
V = 12000 N/m
V = 0 N/m
V = 10000 N/m
It is obvious from Figure 6 that it is more demanding to meet conditions of passivity in the case where the end point mass (reflected mass of parallel mechanism m) is higher. For this reason, in further analysis higher mass of parallel mechanism (m = 18 kg in the "ARM" and "REACH" modes) was considered.
Variable structure pantograph: Simulation evaluation
Based on the results obtained in the previous subsection, simulation evaluation of VSP's haptic performance was undertaken (MATLAB, Simulink). In simulation model, a damper with b = 200 Ns/m was added parallel to the spring as depicted in Figure 3(B) and Figure 4. In terms of system passivity, the proportional gain of the controller P was varied from 1.9 in a LOW impedance to P = 0.95 in a HIGH impedance simulated environment. Haptic performance was investigated by simulating sinusoidal movements of the handle bar for ± 8 cm at frequencies of 1.0 Hz, 0.5 Hz and 0.1 Hz. It is important to point out that due to the design of the VSP (see Figure 3(B)), the displacement on the bottom of the actuated bar (XO) is 4 times smaller than the movement of the handle bar. Similarly, the force on the handle bar is 4 times smaller than the force on the bottom of the actuated bar where the cable wire is attached. For this reason, impedance felt by the subject holding the handle bar is 16 times smaller than on the bottom of the actuated bar. Therefore, the impedance felt by the user on the handle bar in a HIGH impedance simulated environment should be approximately 750 N/m (12000 N/m: 16) and the maximal force while repeating sinusoidal movements with amplitude ± 8 cm should be approximately 60 N (750 N/m * 8 cm). Desired force felt by the user in a LOW impedance simulated environment (0 N/m) should be 0 N. Additionally, the influence of backlash (1 mm and 4 mm), which is typically present in DC motors with planetary gears, was investigated.
Variable structure pantograph: experimental evaluation
Actuators with series elasticity have been extensively studied in the field of robotics [14–16, 20], where they were predominantly used in actuation of walking machines. Use of these actuators in haptic devices was limited to cases where the endpoint mass of devices are negligible as compared to the reflected inertia of the actuator , (included references reflect only a portion of the relevant literature). In case of the Variable Structure Pantograph haptic device, endpoint mass is not negligible due to a variable structure parallel mechanism. The main contribution of this paper is the derivation of passivity conditions that need to be fulfilled for a rehabilitation robot with a mechanism mass comparable to the reflected mass of the geared actuator. The results show that appropriate damping must be provided parallel to the SEA spring in order to obtain stable and passive behavior of the device when it is in contact with a human.
The maximum proportional gain P of the controller must be limited by the ratio of the actuator and parallel mechanism reflected masses: . Hence, better force control can be achieved (higher P) either by use of a motor with higher reflected inertia or by use of lighter parallel mechanism.
The maximum achievable virtual stiffness V must be limited by controller's proportional gain P and the stiffness of mechanical spring K added in series to the motor: . This condition is similar to results obtained by Vallery , where it was shown that the SEA cannot display higher pure stiffness than the spring stiffness when passivity is desired.
Third, to achieve haptic device passivity, sufficient damping b should be presented in parallel to the SEA spring: . Necessity of appropriate damping was also derived by Colgate and Schenkel , where the passivity of systems comprising a continuous time plant and discrete time controller was considered. This means that a damper inserted parallel to the spring ensures required dissipation of mechanical energy.
To verify results of theoretical analysis, simulation evaluation was undertaken. Simulation results predicted stable haptic performance for both HIGH and LOW impedance simulated environments. Simulation results revealed that haptic performance is also adequate in the case when higher values of backlash are presented in the system. This means that high-cost precision motors and gearheads that are currently used in VSP haptic device may be replaced by low-cost motors with greater backlash. Experimental evaluation has confirmed the simulation results and has shown that when appropriate damping and controller's proportional gain are used, stable interaction between machine and human are achieved in LOW and HIGH impedance environments, which was not the case when the damper was omitted. Generation of a HIGH impedance environment is limited to a virtual stiffness of 750 N/m, because the impedance felt at the arm is 16 times smaller than what the actuator can provide (12000 N/m:16). However, this does not present a notable limitation to rehabilitation where more compliant and thus gentle guiding in performance of training tasks is necessary. Also, experimental evaluation revealed that the achievable impedance range is sufficient .
In most cases described in literature, discrete linear models are used when dealing with general purpose sampled haptic environments, which are characterized with high Z-bandwidth [26, 27]. It is important to point out that in this paper we utilized a continuous linear model of the studied haptic robot. In the particular case of the rehabilitation robot actuated with SEA, the typical Z-bandwidth is much lower (in our case the virtual stiffness is limited to 750 N/m). Also, the force bandwidth of the SEA as well as movement during rehabilitation are limited to app. 1 Hz , while the sampling rate is relatively high (1 kHz). Furthermore, it has been demonstrated that the effects of digitalization in conjunction with a usually high Z-bandwidth, (that a haptic interface should be able to render) can cause instabilities at frequencies of several hundred Hz, while at frequencies below 10 Hz, the effects of A/D and D/A devices placed within the control loop are negligible . This enabled the use of a continuous linear model, which is much more intuitive to comprehend. The decision to model the parallel mechanism with a simple mass is related to the fact that the range of motion of the VSP is rather limited and relatively slow, meaning that the predominant dynamics will be dominated by the mass properties of the mechanism. Finally, the use of a linear model to mimic the dynamics of a geared DC motor has been experimentally validated in our previous paper describing the UHD robot .
In conclusion the results of our study have shown that by properly designing rehabilitation device that uses a parallel mechanism and actuators with series elasticity, stability and passivity of haptic performance can be obtained. Because such a haptic system may be composed entirely of off-the-shelf mechanical components, versatile and affordable rehabilitation robotic devices can be produced, which may facilitate their wide spread use in clinical and home environments.
The authors acknowledge the financial support from FATRONIK Tecnalia and Slovenian research agency (grant no. P2-0228).
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