Model reference adaptive control of a haptic feedback device for improving force performance

In this paper, a new adaptive control algorithm of a haptic feedback device is analyzed. Forces applied to the haptic device through human hand movements are modeled as disturbances and compensated in the force control action. A model reference adaptive control (MRAC) scheme is proposed to improve force tracking performance. A separate reference model for every DOF is selected to satisfy rising time, settling time, peak time, and overshoot requirements. General adaptive control laws are developed for tuning gains in the control transfer functions based on the reference model and the force sensor and encoder readings in real time. These control gains cover force tracking performance and compensate human hand disturbances while providing robustness to sensor noise. Stability of the control system is shown analytically. Convergence and boundedness of control gains are also shown through experiments.

Model reference adaptive control of a haptic feedback device for improving force performance

ABSTRACT:
In this paper, a new adaptive control algorithm of a haptic feedback device is analyzed.Forces applied to the haptic device through human hand movements are modeled as disturbances and compensated in the force control action.A model reference adaptive control (MRAC) scheme is proposed to improve force tracking performance.A separate reference model for every DOF is selected to satisfy rising time, settling time, peak time, and overshoot requirements.General adaptive control laws are developed for tuning gains in the control transfer functions based on the reference model and the force sensor and encoder readings in real time.These control gains cover force tracking performance and compensate human hand disturbances while providing robustness to sensor noise.Stability of the control system is shown analytically.Convergence and boundedness of control gains are also shown through experiments.

INTRODUCTION
Haptic feedback devices have many useful applications such as surgical teleoperation systems.In which a surgeon can use the haptic device to operate a surgical robot working with patients.The haptic device can work as a master to provide desired trajectories and forces for a slave robot.Control of haptic feedback devices has become active research areas.The control algorithm should satisfy the objective of accurate force sensing from the desired forces.The user should feel actual forces from the desired forces not those of the structure of the haptic device.Impedance force control and admittance force control are two force control techniques used for haptic devices [1].The closed loop impedance control may improve the force performances [2-3].
Adaptive control techniques have proven their advantages with uncertain dynamic systems.Adaptive impedance control is used in haptic simulations to improve transparency and stability [4].Park and Lee [5] developed an adaptive impedance control method for a haptic device to estimate the stiffness and damping of human hand and to improve force performances.

TAÏ P CHÍ PHAÙ T TRIEÅ N KH&CN, TAÄ P 17, SOÁ K1-2014 Trang 103
Human hand and arm interact with a haptic device and may affect the force control performance.Human hand impedance can be modeled as a mass-spring-damper system [6].The human hand can be defined as an admittance model where the force input generates the motion output [7][8].This model is constructed with one mass, two springs and two dampers.Human hand and arm should be properly modeled and included in the haptic force control system.Model reference adaptive control (MRAC) is an interesting method to construct stable control systems.Design of MRAC for teleoperation system with output prediction is presented in [9].Two MRAC are designed for both master and slave devices to estimate time delay and predict output so that the transparency and stability are improved.This paper extends the preceding works of force control for a haptic device [3].Force control model of the haptic device including the human hand model is analyzed to investigate the dynamic effect caused by the human hand movements.A new adaptive impedance force control using MRAC is proposed to achieve good force tracking performances as well as compensate human hand disturbances.The reference model is selected as the third order relative one to satisfy requirements of rise time, settling time, peak time, and overshoot of the force tracking.Adaptive feedforward control is also proposed to compensate the dynamic effects caused by the human hand movements.

FORCE CONTROL MODEL
where  is a motor torque vector, h J is Jacobian and where B , K , If the estimated gravity force is perfect, the dynamic equation can be shortened as A force control model of haptic device is shown in Figure 2. The relationship between the input force h F to haptic device and its movement h x can be expressed as The relationship between contact force c F and position errors e x between user hand and haptic device is expressed as where . The motor force is The user hand keeps the steering handle of the haptic device and generates the trajectories, u x and h x .The user hand can be modeled as a simple 1-DOF mass-spring-damper model [7].
The relationship between the estimated hand trajectory u x and haptic device trajectory h x , is expressed as Where k b, are the damping and stiffness of user Equation ( 8) implies that the contact force c F is induced by two inputs of the user hand trajectory u x and the desired force d F .Equation ( 8) can be reformulated as where Equation (10) implies that the haptic device dynamic force may be reduced by the feedback control if the control gain of h K is large enough.However, the system becomes unstable if high control gains are selected [10].The force u F can also be compensated by feedforward control action if the parameters of m and c are estimated.
The control objectives in this paper are satisfying good force tracking performance as well as rejecting the undesired dynamic forces caused by the user hand movements.A model reference adaptive control is proposed to satisfy the requirements of force tracking performance criteria.An adaptive feedforward control is also designed to compensate the dynamic force caused by the user hand movements.

MODEL REFERENCE ADAPTIVE CONTROL (MRAC)
A reference model of a third order relative degree one is selected for the adaptive controller to satisfy requirements of rise time, settling time and overshoot.The reference model is described as ) ( ) ( ) Assume that ˆu ) Where ( ) ( ) ( ) ( ) ,

A ma A ma ca ba ma K ba K A ma ca ba a k a c b K ma K ba ka K A ca ba ka ka K a c b K ba ka K ba K
A ka ka K ka K ka K The closed loop transfer function of system in ( 19) is compared with the reference model m H to find ideal control gains as 1 15 where If the system parameters are known, the ideal control law is calculated as where The ideal control law also can be considered in the time domain as If the ideal control law is given, the error Since parameters of user hand and haptic device are unknown, the control gains should be updated with an adaptive law.The estimated control law is defined as where 17 ˆˆ, ...,  is the error of estimated control gains.The estimated control law is then reformulated as where The contact force c F with parameter uncertainties is then described as The error between outputs c F and reference model is then If m H is strictly positive real transfer function, the adaptive law can be selected as [11] 1 () where  is a given positive constant and 1 0 K  .The adaptive law is then obtained as Substituting the reference model The reference model m H is selected to satisfy the requirements of strictly positive real transfer function.The Kalman-Yakubovich lemma [11] indicates that there exists symmetric positive matrix P and Q so that the following equation is satisfied.
Taking its derivative to obtain 0 Therefore the dynamic system of force error is stable and K  , e X are bounded, so The transfer function Ĥ is selected to obtain reasonable errors between the user hand trajectory and haptic device trajectory as The closed loop force control algorithm using MRAC was developed and implemented in the digital controller.
The MRAC for step forces of the haptic device was first tested in order to evaluate the reduction of dynamic effects such as frictions, inertia and gravity.The desired forces Fx of 5N, Fy and Fz of 3N are applied to the haptic device while the user hand generates shaking motions working as external disturbances.The forces performances are shown in Figure 6 indicate that the contact forces Fc of haptic feedback device can track the desired force Fd.The control gains for Fx in Figure 7 can be converged into certain values so the force errors can be reduced to zeros.
The sine force experiments of MRAC are shown in Figure 8.The comparison indicates that the forces of the haptic device tracked those of desired sine forces well.However, there are small force errors along to Fz-axis because of gravity effects.Control gains of MRAC are converged and bounded as shown in Figure 9.In order to improve robustness of adaptive control, projection method in [12] can be applied to limit the control gain.Thus the adaptive law can be modified as   Điều khiển thích nghi dựa theo mô hình cho thiết bị phản hồi xúc giác để cải thiện lực bám

A 6 -
DOF haptic device shown in Figure 1 utilizes two 3-DOF parallel structures similar to the 3-DOF Delta structure.These two 3-DOF parallel structures are divided into the upper structure and the lower structure.The end effectors of the upper and lower structures are connected to a steering handle via universal joints.This haptic device has six legs controlled by six gearless DC motors fixed on the base frame.Each leg is made of hollow aluminum to meet the low weight requirement.Two weight balances are attached to the back extension of the two middle legs to minimize the effect of gravity.Each leg is composed of two links connected by two 2-DOF revolute ball bearing joints such that one revolute joint connects two links while the other revolute joint connects the link to the end effector.The haptic device can provide forces up to 30N and torque up to 2Nm.The contact forces Fc exerted by the user can be measured with two 3-DOF force sensors attached on the end effectors of the haptic device.(a) Design model (b) Manufactured model

Fig. 1 .
Fig. 1.A 6-DOF haptic feedback device coupling velocity matrix, and gravity force of the haptic device respectively.

Fig. 2 .
Fig. 2. Control model of a 6-DOF haptic feedback device   T are damping matrix, stiffness matrix, contact force vector, and position vector of the user hand respectively.The gravity force ( ) h Gx % can be estimated using a classical dynamic analysis and compensated with feed-forward control action.The dynamic equation is reorganized as The user hand has nonlinear stiffness and damping since its stiffness and damping change by the grab condition and posture of the arm.The dynamics of 6-DOF haptic device including the user hand can be decoupled under slow movements and represented as a 1-DOF dynamic model.A simple 1-DOF haptic feedback device is shown in Figure 3.The trajectory h x of haptic device can be determined by encoders while the hand trajectory u x is difficult to measure accurately.The trajectory of haptic device is used as a desired trajectory for the other slave device such as a slave robot.d F is a desired force for the control system of haptic device.The user hand may feel the force c F as the desired force d F even though the disturbance force from user hand are existed in the system.The error between the desired force d F and feedback force

cF
is controlled by a force controller h K to supply torques for motors of haptic device.The closed loop relationship between d F and c F in 1-DOF force model is described as

Fig. 4 .H 4 H 4 H
Fig. 4. A diagram of MRACThe damping ratio and natural frequency determine locations of two complex poles.The damping ratio can be increased to reduce overshoot while the natural frequency is used to adjust settling time.The rise time can be reduced when the pole

Fig. 5 .
Fig. 5.An adaptive force control model of haptic device of Jacobian matrix.Six components of desired force Fd require six model reference adaptive controllers separately.The haptic control system using MRAC is shown in Figure 5. Two different reference models are obtained for moment and force components because of different dynamic effects.A reference model for forces has rise time of 0.02 sec, settling time of 0.2 sec, and overshoot of 1.5% as

4 K
and forces of haptic device for free movements (Fd = 0) are shown in Figure 10 and 11.The control gains in Figure 12 are updated to reduce contact force Fc caused by dynamics of haptic device and user hand.The control gain decreases until it hits a limitation.This projection technique helps to improve feeling on user hand and keep the stability of system in free movements.