The stability investigation of variable viscosity control in the human‐robot interaction

Abstract Background For many co‐manipulative applications, variable damping is a valuable feature provided by robots. One approach is implementing a high viscosity at low velocities and a low viscosity at high velocities. This, however, is proven to have the possibility to alter human natural motion performance. Methods We show that the distortion is caused by the viscosity drop resulting in robot's resistance to motion. To address this, a method for stably achieving the desired behaviour is presented. It involves leveraging a first‐order linear filter to slow the viscosity variation down. Results The proposition is supported by a theoretical analysis using a robotic model. Meanwhile, the user performance in human‐robot experiments gets significantly improved, showing the practical efficiency in real applications. Conclusions This paper discusses the variable viscosity control in the context of co‐manipulation. An instability problem and its solution were theoretically shown and experimentally evidenced through human‐robot experiments.


| INTRODUCTION
Co-manipulation is a paradigm in which a robot and a person manipulate an object or a tool at the same time. [1][2][3] The comanipulation combines the advantages of human operators, such as quick learning and adaptability, 4,5 with the advantages of robots, such as the capacity to improve precision and do repetitive jobs without tiring. 6 Therefore, nowadays, the concept of co-manipulation is applied to many tasks so as to optimise ergonomics and to enhance gesture quality. [7][8][9] For example, in a human-robot co-manipulation sawing task, 10 the authors leverage a myoelectric feedback interface to online estimate human muscle fatigue level and then control the robot to adapt its physical behaviour to the human motor fatigue with the learnt task execution skill. The authors in 11 proposes a realtime model-based reinforcement learning impedance controller to assist human operators in a collaborative lifting task by online optimising the stiffness and damping impedance control parameters to minimise the human effort. The authors in 12 demonstrate that a collaborative industrial robot with active compliance controller works with a human operator to disassemble the press-fitted components of an automotive water pump.
Impedance control is a standard paradigm for controlling a comanipulator in a co-manipulative task. 13,14 While elastic fields (stiffness) are used for guidance through virtual fixtures, viscous fields provide appropriate damping or tremor filtering. 15 Furthermore, a co-manipulator may vary its impedance during manipulation activities to give adaptive assistance to changing operating circumstances.
This method was first presented in 16 for assisting point-to-point movements. It is based upon the experimental finding that when a human operator collaborates with another one to transport an object along a linear path, the operator's arm viscosity decreases at a rapid rate. A robot controller is implemented to imitate this behaviour based on the above-mentioned observation: the viscosity is set large for low velocities while it is set a small value after the velocity reaches the threshold. Consequently, human-robot co-manipulative tasks have been demonstrated to follow similar trajectories as in human-human co-manipulative tasks. According to the authors in, 17 this controller is then updated to be "optimal", that is, the value of viscosity coefficient is an exponential function with respect to time once the threshold is achieved. The resulting velocity profiles of collaborative point-to-point motions have the bell shape of minimal jerk trajectories, which, according to the authors, indicates that the movements are "natural". 18 Meanwhile, the same experiment conducted using an abrupt change of viscosity, as proposed in, 16 induces degraded control performance during point-to-point movements.
This suggests that the way the viscosity variation is dynamically programed impacts the coupled dynamics.
The variable viscosity method is also used for more advanced tasks. In, 19 an intention-driven controller is proposed for compliantly-driven robots and tested on a lower-limb exoskeleton.
The damping coefficient of the controller is online adjusted with the adaptation of the human motion intention through weighting function to suit human behaviour for better collaboration. In a manual welding task with robot assistance, 20 the damping coefficient is designed to be a piece-wise linear function of the robot velocity norm. Given the reported performance improvement, as well as the simplicity of the implementation, this controller is used as a starting point of our research.
Our investigation concerns the dynamic behaviour of the couple user + co-manipulator. In particular, we are interested in the stabilisation of a movement at a given desired velocity. In Section 2.1, we first study this question from a theoretical perspective, within a robotics framework. To this aim, we study the coupling between a conventional velocity controlled robot and the co-manipulator driven by the controller proposed in. 20 We show that, under certain conditions, the coupling may exhibit unstable behaviour. In Section 2.

| Basic control law
This paper is part of a research aimed to study co-manipulation for minimally invasive surgery. Different from tele-surgery, the surgeon and the co-manipulator together hold and maneuver the instrument during the operation. We consider a co-manipulative robot named Achilles 21 programed to assist a human subject. Achilles is a lightweight robot fabricated by Haption company, France, see its sketch in Figure 3. It has 6 revolute joints: first three are actuated and last three are passive. The last three joints intersect at a point P, thus forming a spherical passive wrist with P as the wrist centre.
When the human subject moves his/her hand, the robotic device desired behaviour consists in exhibiting: The user holds a handle connected to the end-effector through the passive spherical wrist centred at point P. The device control input is a force f ∈ R 3 exerted at point P. Its outputs consist in position x ∈ R 3 and velocity v ∈ R 3 of P, obtained from the robot sensors and used as system outputs.
With such a device, we can programme a viscosity controller as: A variable viscosity control can be implemented by computing the viscosity coefficient b ∈ R as a direct function (static map) of the norm of the velocity: with b max the maximal viscosity and 0 ≤ λ ≤ 1.
A simple implementation, proposed in, 20 consists in linearly interpolating the viscosity coefficient between two values, b min and This approach has a major issue: within a particular range of velocities, the robot-exerted resistive force exhibits a decreasing function of the velocity amplitude, as shown in Figure 2. This phenomenon is similar to the Stribeck effect, which is used in tribology to explain the decrease of friction force between the static and dynamic regimes. Taken as a local positive feedback, this behaviour can cause instability at low velocities, leading to stick-slip movements. 22 Analogously, we may anticipate the system to have unstable behaviour when the user attempts to maneuver the robot at a velocity that is in the negative slope area (illustrated as dashed red curve in Figure 2).

| Theoretical analysis of instability within a robotics framework
We first evaluate the proposed control law from a theoretical perspective. Consider such a task: a 1 dof control robot connects Achilles at its wrist centre point P and guides Achilles to move from a given starting point to a given ending point. The movement is supposed to keep a straight line path at constant velocity v d . Achilles, the co-manipulator, acts as the follower and is controlled according to Figure 3 is an conceptual illustration of the task.
The designed task is a simplified version for theoretical analysis of the practical experiment where a human subject is involved (the experiment detail is provided in Section 3). There are no consensual models available in the literature to precisely depict the complex human dynamic behaviour. Therefore, considering the task requirement of keeping at constant velocity when following a given line, we assume that the assistive control robot exploits a standard PI velocity controller: where f c is the force exerted by the assistive robot, k p (resp. k i ) is the proportional (resp. integral) velocity controller gain. We model the end-effector of the assistive robot as a pure mass m which is affected by the co-manipulator force f as well as the command force f c , and we write its closed loop behaviour as: To keep things simple, we'll use the following system state variables: Their derivatives are consequently obtained as: We now limit the study around an equilibrium point, where b can be written as linear function of v: According to Equation (7), v = v d − x 2 . Then Equation (5) becomes: Combining Equations (6)- (11), we obtain the system closed-loop state space equations: Linearising around the equilibrium point v = v d (corresponding to (x 1 , x 2 ) = (0, 0)), one gets: |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl } The eigenvalues of A(v d ) are: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Since m > 0, the real parts of e 1,2 are both negative iff We hence draw a conclusion that in a local region the system is the equilibrium becomes unstable. For the above velocity control task, the stability condition is described as: when a variable viscosity coefficient with slope α is given, to keep the velocity at v d , the stiffness k p must be high enough to ensure the system locally stable.

| Slowing down viscosity variation by adding a dynamics
In order to solve this instability problem, we apply to the viscosity coefficient a first-order low pass filter which is therefore named as viscosity coefficient filter. The purpose of adding this filter is to decelerate the viscosity coefficient variation. This further slows down the viscous force variation, giving the users more time to adjust themselves. Thus the control law writes: with b f defined as the filtered viscosity coefficient by equation: where τ is the filter time constant (unit: second), b being obtained from the static velocity-viscosity map used in the previous section.

| Theoretical investigation of stability in a robot-robot configuration
We use the same model established in Section 2.1.2 for the theoretical investigation. Now the system dynamics becomes: Keeping x 1 and x 2 the same as in Equations (6) and (7), we select x 3 as: The closed loop dynamics now writes: then the derivative of x 2 is: Equations (8), (21) and (23) together deduce the new state space equations: At the equilibrium point, we do the Jacobian linearisation: |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl } ¼:BðvÞ At the reference velocity of the PI control, the viscosity achieves the desired value. To put it another way, Thus, the system equilibrium point is (x 1 , x 2 , x 3 ) = (0, 0, 0), and around this point Equation (25) rewrites as: |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl } ¼:BðωÞ where ω = 1/τ, denotes the first-order low pass filter frequency.
We do not write down the eigenvalues of B(ω) since they are too complex to fit here. Instead, we can prove the existence of a neighbourhood around zero for ω, for any given (positive) parameters, which ensures that all the eigenvalues of B(ω) have negative real parts. We write out the characteristic polynomial of B(ω): When ω = 0, the roots are κ 1; That is to say, regardless of the controller tuning, as long as τ is large enough (i.e., variation of b f slow enough), the system is locally asymptotically stable.

| Practical analysis of instability through human-comanipulator coupled experiments
We conducted experiments to evaluate whether the robot-robot theoretical instability evidenced in Section 2.1.2 could be observed in a human-robot configuration. An experimental scene was demonstrated in Figure 4.
Achilles' control input is a pure force at point P and therefore it can be easily fitted with the variable viscosity controller given by For the experimental task a human user replaces the command robot and holds the handle of Achilles. The user is asked to move from the initial point to the final point materialised in the workspace.
The distance between the two points is 0.4 m.
The user is asked to move the robot following a straight line between the starting and ending points while at the same time concentrating on the velocity. This is assisted by an interface illustrated in Figure 5, where a gauge provides the user a visual feedback of the current velocity norm. Concretely, the user is asked to keep the velocity at a constant desired value v d during the movement, indicated by the black bold line in the graph.
As depicted in Figure 6, three velocity-to-viscosity static maps  would be obtained for both C 1 and C 3 (since α = 0) while, depending on the control parameters, instability could be observed for C 2 .
Ten naive subjects, all right-handed, participated in the experiment. Conditions are randomly loaded. Before the formal recording, subjects were suggested to take a few trials for every condition so as to get familiar with the required force level for moving Achilles at v d . Figure 7 shows the typical behaviour of a random subject. From the figure, we see that the subject is able to maintain the velocity stable under C 1 and C 3 . Further, the high viscosity coefficient under C 1 provides higher damping, leading to less error. The more important phenomenon is the instability observed under C 2 : To reach v d , the subject accelerates when initially at low velocity. The velocity acceleration corresponds to the viscosity drop, leading to a decreasing resistance to robot motion. The acceleration is easily getting higher than the subject's expectation, landing to the high velocity region (with b = b min ). The subject then adjusts the velocity to slow down, but again than expected, too much deceleration lands to the low velocity region. The final result is that the subject fails to stabilise the velocity at v d . The oscillation between two limit regions resembles the stick-slip motions observed in. 22 This observation is typical for all subjects. Figure 8 is the Rootmean-square (RMS) of velocity errors and the corresponding standard deviations under the 3 conditions. To be more informative, the error of each subject is depicted with a small red dot.
We also conducted a Student's t-test for a quantitative assess- We hence claim that the system benefits from the viscosity in terms of stability but the dropping viscosity induces the instability.
The experimental results are comparable to the theoretical results obtained with a robot-robot configuration. In the region having a large dropping viscosity, the system is prone to be unstable.

| Experimental evaluation within in a humanrobot context
Since the model for the theoretical analysis does not represent a human dynamics (whose model is not precisely known), the stability F I G U R E 6 Three designs of λ(‖v‖). The three experiments share the same desired velocity v d and the dropping slope F I G U R E 7 The velocity performance of a random subject under Average errors of all ten subjects under C 1 − C 3 condition is not appropriate for the practical controller tuning.
However, in the current experiment, we have assumed the viscosity parameters in Section 3.1 unchanged. Besides, we add a first-order low pass filter with τ = 1s or 2s.
The experiment protocol keeps the same as in Section 3.1. Specifically, we add four new conditions. C 4 − C 6 corresponds to C 1 − C 3 , but with an additional viscosity filter (τ = 1s). C 7 corresponds to C 2 , with τ = 2s. The conditions are listed in Table 1. Figure 9 displays the velocity norm w.r.t time under C 4 − C 7 for the same subject in Figure 7. Under C 4 and C 6 , the subject has similar performance as under C 1 and C 3 . Under C 5 and C 7 , the subject now has the ability to stabilise the velocity at v d , compared with the un- This observed behaviour is typical for all 10 subjects. Figure 10 is the RMS of velocity errors and the corresponding standard deviation under C 1 − C 7 (results under C 1 − C 3 reproduced from Figure 8).
Conditions are organised based on their viscosity regions for better comparison. which was expected as the viscosity is supposed to be constant (α = 0).
As to the RMS errors under C 2 with τ = 0s (μ = 13, σ = 3.5), under These observations are comparable to what was theoretically established for a robot-robot configuration: adding a filter with τ large enough leads to a locally asymptotically stable coupled system.

| CONCLUSION
This paper discusses about the variable viscosity control in the context of co-manipulation. The viscosity coefficient decreases as a function of velocity. An instability problem which has not been reported in literature, was theoretically shown using an robotics model and experimentally evidenced through human-robot experiments.
We propose to add a secondary dynamics to slow down the viscosity variation. This proposition is supported by a theoretical analysis within the robotics context, while human-robot experiments show its practical efficiency in real applications.

Conditions
High viscosity coefficient (stable area)

Low viscosity coefficient (stable area)
No filter τ ¼ 0s Filter τ ¼ 1s Filter τ ¼ 2s NA We study the instability problem of the variable viscosity control in the context of co-manipulation. Unlike tele-manipulated surgical robotic systems such as the da Vinci robot where the slave robotic arms mimic exactly the surgeon's motion, in co-manipulated robotic surgery, the involvement of human user makes the whole system dynamics different from the dynamics of the master/slave robotic system. For this reason, in the future work, more commercial co-operative robots with different mechanical structure such as Franka Emika Panda, Kuka iiwa, etc., shall replace Achilles as the co-manipulator to provide more solid experimental validation. In addition, we shall design more complicated human robot co-manipulation tasks such as following a given trajectory, virtual fixture guidance, 23 etc. to further verify the controller stability in the co-manipulated surgery.