Variable structure pantograph: mechanical linkage

Figure 1 shows a VSP haptic device that is composed of a variable structure parallel linkage and two SEA actuators. A brief description on the device is provided here, while more detailed information can be found in [17, 18].

The main parts of the parallel mechanism are the three joints that can be either locked or released (Figure 2(A)). By locking or releasing DOFs in these joints (I, II, III), the mechanical configuration of VSP can be changed, enabling use of the device in several operational modes shown in Figure 2. The three possible modes ("ARM", "WRIST" and "REACH") are briefly described below.

"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

The variable structure parallel linkage of VSP is actuated by a SEA based drive as shown in Figure 1. The implemented drive consists of two sets of DC motors (Maxon, RE40, 150 W) with gearheads (GP 52 C, 81:1) and incremental encoders. Torques from both gearheads' shafts impose force on the actuated bar through serially added mechanical springs and string wires, see Figure 3(A, B). The string wires are connected to the actuated bar perpendicular one to another, which enables actuation of the VSP in 2DOFs.

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

In Figure 4, the open loop model is illustrated, where M and m denotes masses, X_I and X_o positions, and F_I and F_O forces on the motor and the actuated bar, respectively. Attaching parallel mechanism on the actuated bar, significantly increases endpoint mass. The motor is coupled to the parallel mechanism via a mechanical spring K and damper b. The equivalent viscous friction in the motor and planetary gearhead is marked with B [17].

The negative sign before X_O comes from the definition of the directions of F_O and X_O. These two equations define the model of the plant to be controlled. The motion of the handle bar (X_O) is modelled as a disturbance on the output force (F_O), see shaded block in Figure 5.

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 [19]: (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 [20] proposed a control strategy for SEA with a feed-forward model and PID controller. Vallery [21] 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 (X_O). In simulation, LOW and HIGH impedance virtual force F_V was compared to calculated force on the handle bar F_O, 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

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.

Re(Z(jw)) is nonnegative for all frequencies w, if all values z0, z2 and z4 are nonnegative (see Figure 6). This gives three conservative conditions for the passivity of the system that have to be considered.

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 X_I and parallel mechanism X_O. 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 $\mathbf{(}\mathbf{P}\mathbf{\le }\frac{\mathbf{M}}{\mathbf{m}}\mathbf{\cong }\mathbf{1}\mathbf{.}\mathbf{9}\mathbf{)}$. 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 $\mathbf{V}\mathbf{\le }\mathbf{K}\frac{\mathbf{P}\mathbf{+}\mathbf{1}}{\mathbf{P}}\mathbf{1}\mathbf{2}\mathbf{0}\mathbf{0}\mathbf{0}\mathbf{N}\mathbf{/}\mathbf{m}$. Finally, from the second condition, damping of $\mathbf{\text{b}}\ge \frac{-{\mathbf{\text{B}}}^{\mathbf{2}}+\mathbf{\text{M}}\mathbf{\text{P}}\mathbf{\text{V}}+\sqrt{{({\mathbf{\text{B}}}^{\mathbf{2}}-\mathbf{\text{M}}\mathbf{\text{P}}\mathbf{\text{V}})}^{\mathbf{2}}-\mathbf{4}\mathbf{\text{B}}\mathbf{\text{K}}\mathbf{\text{P}}\mathbf{\text{m}}(\mathbf{\text{B}}+\mathbf{\text{B}}\mathbf{\text{P}})}}{\mathbf{2}(\mathbf{\text{B}}+\mathbf{\text{B}}\mathbf{\text{P}})}\cong \mathbf{8}\mathbf{0}\mathbf{0}\mathbf{\text{N}}\mathbf{\text{s}}/\mathbf{\text{m}}$ 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.

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.

Results listed in Table 1 present conservative conditions for passivity of the VSP, where all values z0, z2, and z4 are nonnegative. However, Re(Z(jw)) can be nonnegative for all frequencies w and desired V, also with suitable selection of P and b (see Figure 7). The maximal achievable virtual stiffness V in a HIGH impedance environment is 12000 N/m, which is sufficient for rehabilitation purposes. As can be seen from Figure 7, passive VSP behavior in any mode can be achieved (Re(Z(jw)) ≥ 0), if P ≤ 19 and parallel damping is at least b ≥ 780 Ns/m.

When considering technical realization, a parallel damper with damping coefficient of b ≥ 780 Ns/m would be a rather heavy duty mechanical element. For this reason, it was decided to set b ≈ 200 Ns/m, which can be technically realized, however at the expense of reduced controller's proportional gain P. By demanding VSP' passivity in a HIGH impedance virtual environment (V = 12000 N/m) where parallel damping is b = 200 Nm/s, P should not exceed a value of 0.95 (see Figure 8(A)). By reducing the virtual stiffness V, proportional gain P can be increased (Figure 8(B), while maintaining VSP passivity.

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 (X_O) 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.

Results of the VSP's haptic performance simulation with a parallel added damper (b = 200 Ns/m) are presented in Figure 9. The values of the forces presented in Figure 8 are interaction forces between the user and the handle bar and are approximately 4 times smaller than on the bottom of the actuated bar.

Variable structure pantograph: experimental evaluation

To verify results of the theoretical analysis and the simulation evaluation in previous subsections, the VSP's haptic performance was also experimentally examined on the recently developed VSP haptic robot [18]. Parallel to the spring, a damper with b = 200 Ns/m was added. The damper was technically realized by means of a pneumatic cylinder (FESTO), where damping was adjusted by setting appropriate air flow on the input and the output of the cylinder. Similar to simulation, the subject holding the handle bar imposed quasi-sinusoidal movements in the forward and backward direction for ± 8 cm at a frequency of approximately 1 Hz. Haptic performance was examined in the ARM mode by simulating LOW and HIGH impedance environments (see Figure 10). In a LOW impedance environment (V = 0 N/m), the actual interaction force between the user and the handle bar did not exceed 5 N, which is comparable to results from computer simulation. On the other hand, when simulating a HIGH impedance environment (V = 12000 N/m), the actual force corresponded to the desired values more precisely than predicted by simulation.

The passivity of the VSP was also tested experimentally. In order to destabilize haptic interaction between the human and the VSP, the handle bar was exposed to fast movements, without and with an added damper parallel to the spring. Haptic interaction in both cases is presented in Figure 11. It is obvious from Figure 11, that in the case when there is no damper added parallel to the spring, response of the VSP to fast movements (which normally do not occur in the rehabilitation movement) is much more oscillatory than in the case where a damper is added. In the oscillating interval, the subject is still holding the handle bar and thereby prevents the VSP from experiencing an unstable response. If the subjects were to release the handle bar during the oscillating interval, the VSP's response would become unstable, which could potentially lead to mechanical destruction of the device. This demonstrates that the VSP has stable behavior, if parallel to the spring, sufficient damping is presented. That was not the case when damper was omitted. Quantitative comparisons between simulations and experiments would not give meaningful results since the movement in the experimental evaluation is human-driven and therefore highly variable.