Combined Adaptive and Predictive Control for a Teleoperation System with Force Disturbance and Input Delay

This work presents a new discrete-time adaptive-predictive control algorithm for a system with force disturbance and input delay. This scenario is representative of a mechatronic device for percutaneous intervention with pneumatic actuation and long supply lines which is controlled remotely in the presence of an unknown external force resulting from needle-tissue interaction or gravity. The ultimate goal of this research is the robotic-assisted percutaneous intervention of the liver under Magnetic Resonance Imaging (MRI) guidance. Since the control algorithm is intended for a digital microcontroller, it is presented in the discrete-time form. The controller design is illustrated for a 1 degree-of-freedom (DOF) system and is conducted with a modular approach combining position control, adaptive disturbance compensation, and predictive control. The controller stability is analyzed and the effect of the input delay and of the tuning parameters is discussed. The controller performance is assessed with simulations considering a disturbance representative of needle insertion forces. The results indicate that the adaptive-predictive controller is effective in the presence of a variable disturbance and of a known or variable input delay.

The proposed control scheme extends the predictive controller in (Karafyllis and Krstic, 2013) to 74 tracking tasks and removes the assumption of bounded disturbances. This is particularly 75 advantageous in the case of unstructured environments such as percutaneous intervention. To this end 76 the I&I adaptive method in its discrete-time form (Yalcin and Astolfi, 2011) is implemented for a 77 system with additive disturbance and input delay. The controller design is illustrated following a 78 modular approach and considering a 1 DOF system, which is representative of the needle insertion 79 stage in robotic devices for MRI-guided percutaneous intervention with pneumatic actuation (Yang et 80 al., 2011). The stability of the control scheme is analyzed and the effects of the input delay are 81 discussed. The performance of the controller is assessed with simulations considering a disturbance 82 representative of typical needle insertion forces. The benefits of the predictive control and of the 83 3   adaptive disturbance compensation as well as the influence of the parameters are highlighted  84 considering different scenarios. 85

Materials and Methods 86
While input delay is an ubiquitous phenomenon in systems characterized by transportation or 87 transmission, the particular case of a pneumatically actuated master-slave system for percutaneous 88 intervention operating with an impedance-control scheme is considered here as a motivating example 89 for the new adaptive-predictive control algorithm. According to this paradigm, the operator sets the 90 position of the master unit, which becomes the reference for the slave actuator that is powered and 91 controlled from a remote location. Simultaneously, the interaction force between the slave and the 92 environment is measured with a sensor and is reflected on the master actuator in order to provide 93 haptic feedback to the operator (Fig. 1) This section introduces a compliant pneumatic actuator supplied by proportional pressure regulators 105 (e.g. Tecno Basic, Hoerbiger), which is modelled as a second-order system with mass and 106 damping coefficient (Yang et al., 2011). 107 The control input corresponds to the pressure relative to atmosphere acting on the cylinder 108 chamber of surface , while the term > is the force disturbance which includes actuator specific 109 un-modelled effects, such as gravity or friction, and interaction forces with the environment (Fig. 2). 110 The term is the pressure acting in the opposite cylinder chamber of surface . In the absence of 111 disturbances, the cylinder extends if > and retracts if < . The damping coefficient 112 > is assumed known. The long transmission lines introduce an input delay due to the pressure 113 propagation at the speed of sound (Yang et al., 2011). Without loss of generality, the mass and the 114 area are assumed unitary and are omitted in the rest of the paper in order to simplify the notation. 115 Furthermore, the pressure is set constant as in (Franco et al., 2016) and the force is included 116 in the lumped disturbance . As a result of these simplifications, the system is presented in a similar 117 form to (Karafyllis and Krstic, 2013; Monaco and Normand-Cyrot, 2015). While the system (1) has a 118 linear structure, the control algorithm presented in the coming sections is designed using a general 119 approach that is valid in principle for a nonlinear system. Rewriting (1) in the state space form and 120 converting it in its discrete-time counterpart with the Euler method we obtain: 121

Combined Adaptive and Predictive Control for a System with Force Disturbance and Input Delay
4 This is a provisional file, not the final typeset article The terms , are the position and the velocity of the actuator. The input delay is assumed to be 122 a multiple of the sampling period . In the rest of the paper the time dependency is indicated as 123 follows for brevity: 124 The following sections present the position control algorithm, the adaptive controller, and the 125 predictive controller, leading to the main contribution represented by the new adaptive-predictive 126 control algorithm. 127

Position control 128
The aim of the control scheme is having the system (2) track a prescribed position . In particular, 129 the following trajectory is considered here, where and represent the prescribed velocity and 130 acceleration: 131 For better clarity, the control law is initially constructed for the ideal case of known force * and null 132 input delay = . According to (Slotine and Li, 1991), the auxiliary variable ! representing the 133 tracking error as a combination of position and velocity errors is introduced: 134 The term " > is the first design parameter in this control scheme. An analogy can be drawn 135 between the structure of (5) and that of a low-pass filter with the position error − as input 136 (Slotine and Li, 1991). In this perspective " can be interpreted as the filter bandwidth in (5) and 137 should be small in comparison to the high-frequency un-modelled system dynamics. In particular, a 138 smaller value of " results in a more damped control action (ref. Section 3). In order to design a 139 control law that satisfies the tracking condition ! = we define the following Lyapunov function 140 candidate: 141 In the continuous-time situation the control input should make the derivative of the Lyapunov 142 function candidate negative definite in order to achieve null tracking error. Similarly, in the discrete-143 time case the function # should decrease at each time step (Karafyllis and Krstic, 2013): 144 Substituting (2), (4), (5) into (7) we can rewrite it as: 145 A suitable choice of the control law that verifies (7) is: The term " > is the second design parameter. A larger value of " results in a more responsive 147 control action. Substituting (9) into (8) we obtain: 148 Since # is positive definite and decreasing at each time step, we can conclude that the tracking error 149 converges to zero. A parallel can be drawn between the parameters 1/" , " and the derivative and 150 the proportional gains in a PID controller. Consequently, the tuning methods valid for PID can be 151 used as a reference for (9). Additionally, in case of a input delay * , (Slotine and Li, 1991) suggests 152 choosing " < / * to reduce oscillations. 153

Adaptive control 154
In this section the disturbance is considered unknown and constant over the sampling interval , 155 and is estimated adaptively using the I&I method in the discrete-time form ( disturbance. This is particular advantageous if the actuator operates in an unstructured environment, 159 as in the case of percutaneous interventions. According to I&I, an unknown parameter + is estimated 160 as the sum of a state-dependent term , -and of a state-independent term + . . The estimation error /, 161 which represents the difference between the estimated parameter and its actual value, is defined for 162 system (2) as: 163 The terms 0 , 1 are the state-independent part and the state-dependent part of the disturbance 164 estimate. In this case we can already conclude that 1 does not depend on observing that the 165 disturbance only appears in the second part of (2), hence (11) is written as a function of . The 166 control objective consists in having the estimation error converge to zero. In the continuous-time 167 situation this is achieved choosing an appropriate Lyapunov function candidate # = / and ensuring 168 that the adaptation law makes its derivative negative definite. In the discrete-time case, this condition 169 corresponds to showing that # decreases at each time step. Calculating (11) for the new time step 170 we obtain: 171 Substituting (2) and (11) into (12) the convergence condition can be expressed as: 172 Exploiting the structure of (13), the state-independent term 0 is chosen as: 173 Substituting (14) into (13) we obtain: 174

Combined Adaptive and Predictive Control for a System with Force Disturbance and Input Delay
6 This is a provisional file, not the final typeset article A suitable choice of 1 that verifies (15) is 1 = −" < , where " < / is the design 175 parameter responsible for adaptation. Larger values of " result in a faster convergence of the 176 parameter estimate to the actual value, which is usually desirable. However, this is not always the 177 case in the presence of input delay, as discussed in Section 3. 178 Since # is positive definite and decreases at each time step with the chosen adaptation law, we can 179 conclude that the estimation error / converges to zero. The control law (9) with the adaptive 180 algorithm (14) becomes: 181 Although the adaptive control law (16) is in general nonlinear, it can be shown that with appropriate 182 assumptions on the disturbance it simplifies into an integral term (Franco and Ristic, 2015). 183 Consequently, an analogy can be drawn between the control scheme (16) and a PID controller, as 184 previously mentioned in Section 2.1. 185

Predictive control 186
This section considers a nominal input delay in (2) and presents a discrete-time predictive control 187 algorithm inspired by (Karafyllis and Krstic, 2013) and designed for a tracking problem. Initially, the 188 case = with known force * is considered, and the system (2) is rewritten introducing the term 189 2 as: 190 The baseline control law (9) for the un-delayed system corresponding to the first two equations in 191 (17) is: 192 The control input is computed rewriting (18) for the next time step: 193 Notably, (19) contains the future values of the states, which can however be computed form (17). The 194 convergence of the closed loop system consisting of (17), (18), (19) to the target dynamics will now 195 be verified based on Lemma 2.1 in (Karafyllis and Krstic, 2013). Considering that the un-delayed 196 system (2) is stabilized by the control law (9) according to the Lyapunov function candidate (7), the 197 following Lyapunov function candidate is proposed for the generic case = * , where 4 is an 198 arbitrary positive constant: 199 Computing (20) Substituting (18) in (20) and comparing it with (22) the convergence is proved: 202 System (17) can be expressed for the generic case = * as: 203 The control law for the system (24) is based on (18)-(19) which are expressed recursively for 204 = , . . , * + : 205 The control input is obtained at the last step with = 2 * = 2 * . Notably, the algorithm (25) 206 assumes that the target trajectory (4) is known * time steps in advance. The convergence of the 207 closed loop system (24), (25) to the target dynamics can be proved by induction, based on the case 208 = . In particular, condition (23) is verified for the Lyapunov function candidate (20) with * > . 209

Section 2.3). 253
3 Results 254 The system (2) was modelled in Matlab and the performance of the control algorithm was assessed 255 with simulations. A fix step of 10 µs was employed to simulate the system dynamics while the 256 control input was refreshed every = 1 ms. This corresponds to a sampling frequency of 1 kHz, 257 which is representative of that typically used in motion control and teleoperation (Franco et al., 2016; 258 Pan et al., 2014). The duration of all simulations was set to 3 seconds with the reference trajectory 259 starting at 0.5 seconds. 260 The nominal values of the damping coefficient , the disturbance , and the input delay * are listed 261 in Table I, together with the control parameters " , " , " . The parameters " , " , " were tuned in 262 order to achieve a rise time shorter than 0.15 s (from 10% to 90% of the step amplitude) and a 263 settling time shorter than 0.2 s (within 5% of the step amplitude) with null steady-state-error and 264 overshoot smaller than 1%. Additionally, different values of " were also employed in order to 265 highlight the effects of this parameter on the controller performance. Four different scenarios were 266 considered: the absence of disturbance and of input delay representing the baseline condition, the 267 presence of either disturbance or input delay, and the combination of both. Simulations were 268 conducted for each condition considering a step signal and a ramp as reference trajectories. In 269 particular, the step response is a standard way of comparing control algorithms, while the ramp 270 trajectory is representative of needle insertions. The slope of the ramp is 10 mm/s, corresponding to 271 typical needle insertion speeds in percutaneous interventions. 272

Combined Adaptive and Predictive Control for a System with Force Disturbance and Input Delay
10 This is a provisional file, not the final typeset article Initially, the control scheme (9) was assessed, assuming null input delay and null force * = in 273 order to establish a reference condition. The system response for both reference trajectories is 274 depicted in Figure 3. The steady state error, the settling time and the rise time for the step command 275 and the root-mean-square error (RMSE) for the ramp trajectory are reported in Table II. These values  276 meet the performance specifications previously defined, confirming that the controller parameters are 277 tuned appropriately. 278 The adaptive control scheme (16) was then assessed with a disturbance = + + sum 279 of a constant term, of a term proportional to the displacement, and of a term proportional to the 280 insertion speed. This disturbance model reflects the fact that needle insertion forces depend primarily 281 on insertion depth and in second instance on the insertion speed, while the magnitude of the 282 disturbance is representative of needle insertion forces in percutaneous interventions (van Gerwen et 283 al., 2012). The step response (Fig. 4) shows that the adaptive controller effectively compensating the 284 disturbance. The parameter " was chosen in order to achieve a similar settling time, rise time, steady-285 state error, and overshoot as the reference condition. While the step response does not change 286 noticeably with " , larger values result in a higher peak in the disturbance estimate. The results for 287 the ramp trajectory are depicted in Figure 5. In this case the peak in the disturbance estimate is lower, 288 due to the more gradual movement of the actuator. As expected, the baseline controller (9) results in 289 a noticeable steady-state error for both trajectories which increases with the magnitude of the 290 disturbance. 291 The predictive control scheme (25)  control achieves a smooth movement of the actuator (Fig. 6). Instead, the baseline controller (9) with 295 the original tuning results in oscillations in the step response. Nevertheless, the settling time, the rise 296 time, and the RMSE are larger with the predictive control (25) compared to the reference condition. 297 Consequently, the downside of the predictive control is a lower responsiveness. The performance of 298 the baseline controller (9) improves if the parameter " is tuned considering the input delay (" < 299 1/3* = 15) and becomes similar to that of the predictive control (25). Similarly, a larger delay 300 demands a smaller value of " which results in lower responsiveness. 301 The new adaptive-predictive controller (36), representing the main contribution of this work, was 302 then evaluated considering a disturbance = + + and an input delay * = . The 303 step response is depicted in Figure 7 while the ramp is in Figure 8. The settling time, the rise time, 304 and the overshoot are larger than in the reference condition but smaller compared to the predictive 305 control alone, due to the action of the adaptive algorithm. Tests with larger values of " show more 306 pronounced oscillations and suggest that a more gradual estimate of the disturbance can be preferable 307 in the presence of input delay. In this scenario the controller (25) shows a noticeable steady-state 308 error. The controller (16) becomes unstable with either " =25 or " = 15 and " = 30. Setting 309 " = 15, " = 10 in (16) results in a controlled movement but in a larger tracking error (RMSE = 310 0.77 mm instead of RMSE = 0.65 mm with (36)). 311 An additional case was considered for the adaptive-predictive controller (36) in order to test its 312 robustness. The ramp trajectory was tested with a larger disturbance including a discontinuous 313 square-wave component ( F = + + + . GHIJ GHJ K ) and with a variable input 314 delay corresponding to a random number between 1 and * (Fig. 9). Even in this case, the adaptive-315 predictive algorithm (36) achieves an acceptable tracking accuracy (RMSE = 0.74 mm) considering 316 that the disturbance is larger than in previous simulations, therefore demonstrating remarkable 317 Combined Adaptive and Predictive Control for a System with Force Disturbance and Input Delay 11 robustness. Finally, the performance of (36) remains superior to that of (16) (RMSE = 0.85 mm with 318 " = 15, " = 10). 319 4 Discussion 320 This paper presented a discrete-time control scheme that combines adaptive and predictive 321 algorithms for a system subject to force disturbance and input delay. The controller is expressed in a 322 discrete-time recursive form and, differently from (Karafyllis and Krstic, 2013), it does not rely on 323 the assumption of bounded disturbances. This is particularly beneficial in unstructured environments, 324 such as percutaneous intervention. Differently from (Pyrkin and Bobtsov, 2015), the algorithm is 325 applicable to non-linear systems and considers a generic additive disturbance. The stability and 326 convergence of the system to the target dynamics was analyzed discussing the effects of the input 327 delay and of the disturbance, and expressing criteria for the parameter selection. 328 The simulations show that the adaptive disturbance estimation is essential for compliant actuators in 329 order to achieve high tracking accuracy in the presence of a force disturbance. In the controller 330 design the disturbance was assumed independent on the system states and constant over the input 331 delay . The simulations suggest that the control scheme is also effective for variable disturbances, 332 which could be either due to external forces or that might originate as an effect of model uncertainty. 333 Furthermore, the simulations show that the predictive control scheme (25) is effective in reducing the 334 oscillations that would occur in the presence of input delay. This behavior is particularly desirable in 335 the case of teleoperated percutaneous intervention, since oscillations in the actuator position are 336 transferred to the needle insertion force and are then reflected on the operator, degrading the haptic 337 feedback. Notably, tuning the baseline controller for a more damped response can achieve a similar 338 effect to the predictive controller. In both cases, a larger input delay demands a less aggressive 339 control action that results in lower responsiveness and larger tracking error. 340 The simultaneous presence of disturbance and input delay was addressed with the adaptive-predictive 341 controller (36). In this scenario the adaptive control scheme (16) with the original tuning can result in 342 oscillations and instability. This result confirms the conclusions of (Karafyllis and Krstic, 2013), 343 indicating that parameter uncertainty can have larger effects on the system performance in the 344 presence of input delay. Conversely, the adaptive-predictive controller achieves a satisfactory step 345 response and tracking performance. In general, the simulations indicate that a larger input delay 346 could benefit from a more gradual estimate of the disturbance, while an aggressive tuning results in 347 oscillations and can be detrimental to the performance. Finally, the results show that the adaptive-348 predictive controller is also effective for a variable discontinuous disturbance and a randomly 349 variable input delay, which is a realistic operating condition for teleoperated percutaneous 350 intervention. 351 Future work will investigate the adaptive compensation of an unknown input delay. Additionally, an 352 automatic tuning procedure for the controller parameters will be explored. Furthermore, more 353 elaborated models of the interaction forces will be considered. Finally, the adaptive-predictive 354 control algorithm will be implemented on a prototype and will be validated with experiments. 355

Conflict of Interest 356
The authors declare that the research was conducted in the absence of any commercial or financial 357 relationships that could be construed as a potential conflict of interest.
This is a provisional file, not the final typeset article  Table I Adaptive-predictive control (20) Step

Ramp
Step Ramp Step Ramp Step Ramp