Adaptive controllers compensating communication delay

Nowadays, the teleoperation systems have been applied in many fields such as space exploration, underwater operation, tele-surgery, etc. The master and slave are often located far from each other. Hence substantial time delays occur during signal transmission and control. Consequently, the overall stability of the teleoperation systems is affected and jeopardized. The adaptive schemes used to compensate for the communication delay affecting teleoperation systems are reviewed in this section. These algorithms can be roughly classified into two groups: passivity-based adaptive controllers, and virtual internal model (VIM)-based adaptive controllers.

2.6.3.1 Passivity-based adaptive controllers

As the name suggests, passivity-based adaptive controllers for linear or nonlinear teleoperation systems in the presence of communication delays exploit the passivity of the operator defined by 𝜃�, the estimate of the control law parameters. These types of adaptive controllers improve the transparency and task performance of the teleoperation systems in the presence of communication delays via handling the system parametric uncertainty. Replacing 𝜃 with its estimate, the parameter error 𝜃� = 𝜃 − 𝜃� yields a passive map, which maintains the system passivity properties. The methods described in [104-109], [67] fit into this category of adaptive controllers.

As an example, a brief derivation of the passivity-based adaptive scheme in [104] is presented here.

In [104], an effective adaptive coordination strategy within the passivity framework is designed to achieve the following goals: (i) A feedback control law (𝜏_𝑚, 𝜏_𝑠) for the master and the slave manipulator that renders the manipulator dynamics passive with respect to an output that contains both position and velocity information; (ii) A passive coordination control law (𝜏̅_𝑚, 𝜏̅_𝑠) which uses this output from the master and the slave to kinematically lock the motion of the two mechanical systems.

Since the nonlinear dynamics in Equation (2.2) can be linearly parameterized, the master and slave torques are given as

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𝜏𝑚 = −𝜏̅𝑚− 𝑀�𝑚(𝑞𝑚)𝜆𝑞̇𝑚− 𝐶̂𝑚(𝑞𝑚, 𝑞̇𝑚)𝜆𝑞𝑚+ 𝑔�𝑚(𝑞𝑚) = −𝜏̅𝑚−

𝑌𝑚(𝑞𝑚, 𝑞̇𝑚)𝜃�𝑚, (2.89a)

𝜏𝑠 = 𝜏̅𝑠− 𝑀�𝑠(𝑞𝑠)𝜆𝑞̇𝑠− 𝐶̂𝑠(𝑞𝑠, 𝑞̇𝑠)𝜆𝑞𝑠+ 𝑔�𝑠(𝑞𝑠) = 𝜏̅𝑠 − 𝑌𝑠(𝑞𝑠, 𝑞̇𝑠)𝜃�𝑠,

(2.89b) where 𝑌_𝑚, 𝑌_𝑠, are known functions of the generalized coordinates and 𝜃�_𝑚, 𝜃�_𝑠 are the

time-varying estimates of the manipulators’ actual inertial parameters given by 𝜃𝑚, 𝜃𝑠 respectively; and 𝜏̅𝑚, 𝜏̅𝑠 are the coordinating torques.

By defining 𝑟_𝑚, 𝑟_𝑠 as 𝑟_𝑚 = 𝑞̇_𝑚+ 𝜆𝑞_𝑚, 𝑟_𝑠 = 𝑞̇_𝑠+ 𝜆𝑞_𝑠 for the master and slave transmitted signals, respectively, the master and slave dynamics (2.2) are reduced to

𝑀�𝑚(𝑞𝑚)𝑟̇𝑚+ 𝐶̂𝑚(𝑞𝑚, 𝑞̇𝑚)𝑟𝑚 = 𝑓ℎ− 𝜏̅𝑚+ 𝑌𝑚(𝑞𝑚, 𝑞̇𝑚)𝜃�𝑚, (2.90a)

𝑀�𝑠(𝑞𝑠)𝑟̇𝑠+ 𝐶̂𝑠(𝑞𝑠, 𝑞̇𝑠)𝑟𝑠 = 𝜏̅𝑠 − 𝑓𝑒 + 𝑌𝑠(𝑞𝑠, 𝑞̇𝑠)𝜃�𝑠, (2.90b)

where 𝜃�_𝑚, 𝜃�_𝑠 are the estimation errors: 𝜃�_𝑚= 𝜃_𝑚− 𝜃�_𝑚, 𝜃�_𝑠 = 𝜃_𝑠− 𝜃�_𝑠 , and 𝜏̅_𝑚, 𝜏̅_𝑠 are chosen as 𝜏̅_𝑚= 𝐾_𝑚(𝑟_𝑚−𝑟_𝑚𝑖), 𝜏̅_𝑠 = 𝐾_𝑠(𝑟_𝑠𝑖−𝑟_𝑠) where 𝑟_𝑚𝑖, 𝑟_𝑠𝑖 are the signals derived

from scattering transformation, and the gains 𝐾_𝑚, 𝐾_𝑠 are constant positive definite

diagonal matrices.

Deduced from a Lyapunov-like function, the update laws for the parameters (𝜃�_𝑚, 𝜃�_𝑠) are obtained by

𝜃�̇𝑚 = Γ𝑌𝑚𝑇𝑟𝑚, (2.91a)

𝜃�̇𝑠 = Λ𝑌𝑠𝑇𝑟𝑠, (2.91b)

where Γ, Λ are constant positive definite matrices.

2.6.3.2 Virtual Internal Model (VIM)-based adaptive controllers

VIM-based adaptive controllers are another large group of adaptive schemes addressing the communication delay issue in teleoperation systems. These controllers use a virtual internal model on the master side by estimating the geometric shape and the material properties of the objects in the remote environment, as illustrated in Figure 2.3. Therefore, the operator is haptically interacting only with a locally rendered virtual object and receives non-delayed feedback. This makes the approach robust to time delays. However, the stability of this model-mediated teleoperation system depends heavily on the accuracy of the virtual model. For a high fidelity system, the errors between virtual model and real environment should be small, i.e. the estimation has to work properly. Some references in this category are [110-117].

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Figure 2.3. The diagram of a VIM-based teleoperation system

As an example, the VIM-based adaptive scheme in [112] is described briefly. A sliding-average least-square (SALS) algorithm is adopted to identify the dynamic parameters of the remote environment. The corresponding virtual-model parameters are updated online to keep equal to the real environment. Specifically, a geometric and a dynamic model of the environment at the master site are built, and the parameters of the model are corrected online according to real information from the remote slave site. The geometric errors of the virtual model are corrected by overlaying the graphics over video images and also by fusing the position and force information from the remote site.

The dynamic of the environment is expressed by a mass-spring-damper model: 𝑓𝑒 = 𝑀𝑒𝑞̈𝑒+ 𝐷𝑒𝑞̇𝑒+ 𝐾𝑒(𝑞𝑒− 𝑞𝑖𝑛 ), (2.92)

where 𝑓_𝑒 denotes the interaction force between the slave manipulator and the

environment, 𝑞_𝑖𝑛, 𝑞_𝑒, 𝑞̇_𝑒, 𝑞̈_𝑒 represent the initial position, the actual position, velocity,

and acceleration of the contact point of the environment, and 𝑀_𝑒, 𝐷_𝑒, 𝐾_𝑒 stand for the

mass, damp, and stiffness of the environment, respectively.

Let 𝑀�_𝑒, 𝐷�_𝑒, 𝐾�_𝑒 be the estimates of 𝑀_𝑒, 𝐷_𝑒, 𝐾_𝑒 , respectively, then one can derive 𝑓̂𝑒 = 𝑀�𝑒𝑞̈𝑒+ 𝐷�𝑒𝑞̇𝑒+ 𝐾�𝑒(𝑞𝑒− 𝑞𝑖𝑛 ) , (2.93)

where 𝑓̂_𝑒 is the estimate of the interaction force 𝑓_𝑒. During the identification process, the input force and position signals are sent from the slave site to the local site to update VIM, while the dynamic of the environment is assumed to be stable. According to the SALS principle, 𝑀�_𝑒, 𝐷�_𝑒, 𝐾�_𝑒 are calculated through the following estimation algorithm: 𝐸 = ∑ [𝑓𝑁_{𝑖=1 𝑒}(𝑖) − 𝑓̂_𝑒(𝑖)]2 . (2.94a) ⎩ ⎪ ⎨ ⎪ ⎧𝜕𝑀�𝜕𝜕𝑒 = 0 . 𝜕𝜕 𝜕𝐷�𝑒 = 0 . 𝜕𝜕 𝜕𝐾�𝑒 = 0 . (2.94b)

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While dynamic parameters of the remote environment are identified by (2.94), the dynamic parameters of VIM (M_v, D_v, K_v) at the sampling time 𝑡 are updated as

�𝑀𝐷𝑣𝑣(𝑡) = 𝑀�(𝑡) = 𝐷�𝑒𝑒(𝑡).(𝑡).

𝐾𝑣(𝑡) = 𝐾�𝑒(𝑡).

(2.95)

Therefore, the virtual force 𝑓_𝑣 generated in VIM is obtained using the initial position, the actual position, velocity, and acceleration of the virtual model 𝑞_{𝑣−𝑖𝑛}, 𝑞_𝑣, 𝑞̇_𝑣, 𝑞̈_𝑣 are defined by

𝑓𝑣(𝑡) = 𝑀𝑣(𝑡)𝑞̈𝑣(𝑡) + 𝐷𝑣(𝑡)𝑞̇𝑣(𝑡) + 𝐾𝑣(𝑡)(𝑞𝑣(𝑡) − (𝑡)). (2.96)

2.6.3.3 Summary

The methods reviewed in this section are mainly aimed at ensuring stability of the overall system and synchronizing the applied commands and feedbacks from the remote environment in the presence of communication delay. The first group is only applicable to nonlinear teleoperation systems, while the second can be applied to both linear and nonlinear systems. Adaptive controllers in this group compensate for the communication delay either by estimating an accurate environment model and updating it in real-time or by suppressing uncertainties in the master and slave model. Contribution of each method and its assumptions are summarized in Table 2.3.

TABLE 2.3

A summary of adaptive controllers for communication delay compensation

Method Contribution Assumption Passivity-based adaptive controllers [104-109],[67] Robustness to master and slave model uncertainties and time delays

Additional force sensor

Virtual internal model (VIM)-based adaptive controllers [110-117]

Transparency; Robustness to time delays

Model correctness, force sensor, Persistent excitation