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Design of a static output feedback controller for bilateral teleoperation*
Design of a static output feedback controller for bilateral teleoperation ? Thomas Delwiche ∗ Samir Aberkane ∗∗ Laurent Catoire ∗ Serge Torfs ∗ Michel Kinnaert ∗ ∗
Dept. of Control Engineering and System Analysis, Universit´e Libre de Bruxelles. CP 165/55, 50 Av. F. D. Roosevelt, B-1050 Brussels, Belgium (e-mail: [email protected]). ∗∗ CRAN-UMR 7039 CNRS, Universit´e Henri Poincar´e, Nancy I, F-54506 Vandoeuvre-les-Nancy, France
Abstract: Traditional H∞ design methods available in commercial softwares lead to controllers which present the same order as the model of the plant. Considering the high order of teleoperation systems, implementing such controllers may require the use of order-reduction methods. To avoid this extra step in the design, we propose to produce fixed-order controllers directly based on a full-order model of the system. In this paper, a zero-order (static output feedback) controller is designed for a teleoperation system. The design procedure, based on a cone complementary formulation is described and applied to a 22nd -order model of the system. The resulting controller is implemented on a one degree of freedom teleoperation system. The experimental results obtained with this simple control structure validate the use of very low order control structures in the case of teleoperation systems with the extra advantage, with respect to conventional H∞ design techniques, that the involved and time-consuming order reduction step is avoided. Keywords: H∞ control, static controllers, telerobotics, matrix inequalities, MIMO. 1. INTRODUCTION Teleoperation consists in performing a remote task with an electromechanical master-slave device. The master part of the system is manipulated by the human operator while the slave part, which is located remotely, tries to reproduce the task imposed by the master. It is generally used in hazardous or narrow environments. The main applications of teleoperation include nuclear waste handling, undersea exploration and minimally invasive surgery. When force feedback is present at the master side to make the user feel the interaction forces between the slave and its environment, one refers to bilateral teleoperation. Bilateral teleoperation systems are always MIMO systems even if the master and the slave are both one degree of freedom (dof) devices. Moreover, the actuation and sensory capabilities of human beings are frequency dependent, so the design objectives are also frequency dependent. This explains why this controller design problem is often addressed in the H∞ framework (see Kim (2007), Yan (1996) and Vander Poorten (2008)). However, despite their ability to handle MIMO systems and frequency dependent ? The work of T. Delwiche is supported by a FRIA grant. The setup is financed by the FNRS (ID 1.5.031.06). This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors.The authors gratefully acknowledge the funding of the Tournesol project established in the framework of the Hubert Curien partnerships with the support of Wallonie-Bruxelles International, the FNRS and the French ministries of foreign affairs and scientific research.
performance objectives, most design methods in the H∞ framework (like the ones available in commercial softwares) produce controllers whose order is equal to that of the model of the plant which usually includes frequencydependent weights. Considering the teleoperation application, this may lead to controllers which are too complex to be implemented or at least not intuitive. Three solutions can be considered to produce a low-order controller (see Gu (2005)). The first one consists in reducing the order of the controller synthesized using a full-order model of the plant by suitable order-reduction methods. The second one relies on a reduced-order model of the plant to synthesize the controller. A third method consists in designing directly a reduced-order controller based on a full order model of the plant. The first two methods are implemented in the case of teleoperation applications in Kim (2007) and in Yan (1996) respectively. To the best of our knowledge, the implementation of the third methodology has not been reported for teleoperation applications. The first two methods require an appropriate reduction method. Once the latter is selected and mastered, the designer faces a drawback between the order of the controller and the performance degradation. Excepted under specific circumstances, the reduced controller presents reduced performance with respect to the full order controller which cannot be predicted during the design phase. In this paper, the third method is investigated: a reducedorder controller is designed directly on the basis of a fullorder model of the system, and the performance of the closed-loop system are investigated. In this framework, a static output feedback (SOF) controller, i.e. a zero-order
controller is selected. The resulting SOF controller is implemented on a one degree of freedom (dof) teleoperation system. The satisfactory experimental results motivate the use of simple, low-order control laws for teleoperation systems. The remaining of the paper is structured as follows. In section 2, the method used to compute a SOF controller is presented. Next, section 3 describes a 22nd -order model of the teleoperation setup and section 4 presents the design of the SOF controller for this application. Finally, section 5 presents the experimental results obtained with the designed controller, and section 6 gives some concluding remarks. 2. STATIC OUTPUT FEEDBACK (SOF) DESIGN In this section, a methodology to design a SOF controller in the H∞ framework is presented. It is based on a cone complementary formulation given in ElGahoui (1997). The latter problem requires a formulation in terms of matrix inequalities. The developments presented here are general and not restricted to teleoperation. First, the H∞ design problem is stated formally. Next, its matrix inequality formulation is given. Finally, the cone complementary problem is described. 2.1 Problem statement Consider the system M depicted in figure 1 and defined by the set of state-space equations (1) where x ∈ IRnx denotes the state vector, w ∈ IRnw is the vector of exogenous inputs (references, sensor noise, ...), z ∈ IRnz is a vector of controlled outputs relevant to the problem at hand, u ∈ IRnu is the vector of control inputs and finally y ∈ IRny is the vector of measured outputs. ! ! ! x˙ A B1 B2 x z = C1 D11 D12 w (1) y C2 D21 D22 u | {z } ≡M
M is connected in closed-loop with a static output feedback (SOF) controller, namely a control law u = Ky where K ∈ IRnu ×ny is a constant matrix. If D22 = 0 is assumed, the state-space equations of the closed-loop system are given by expression (2).
x˙ z
=
˜ A˜ B ˜ C˜ D
x w
(2)
A cone complementary formulation followed by a linearization algorithm (see ElGahoui (1997)) are used to solve this problem, based on a matrix inequality formulation. 2.2 Matrix inequality formulation By virtue of the Bounded Real Lemma, the suboptimal SOF H∞ problem is feasible if there exists a positive definite matrix X ∈ Rnx ×nx and a matrix K ∈ IRny ×nu such that the bilinear matrix inequality (BMI) (5) is satisfied: (A + B2 KC2 )T X + ∗ ∗ ∗ (B1 + B2 KD21 )T X −γI ∗ < 0(5) (C1 + D12 KC2 ) (D11 + D12 KD21 ) −γI where the ∗ stand for the transpose of the symmetric entry. To solve this BMI problem, the unknown K is eliminated first by using the separation lemma given in Gahinet (1994) which yields the following equivalent formulation: the suboptimal problem is feasible if there exists positive definite matrices X ∈ IRnx ×nx and Y ∈ IRnx ×nx satisfying the linear matrix inequalities (LMI) (6), (7) and (8) and the nonlinear constraint (9). T A X + XA XB1 C1T T W < 0 WPT (6) B1T X −γI D11 P C1 D11 −γI where WP is the kernel of P = [C2 D21 0] Y AT + AY B1 Y C1T T W < 0 WQT B1T −γI D11 Q C1 Y D11 −γI T where WQ is the kernel of Q = [B2T 0 D12 ] X I F (X, Y ) = ≥0 I Y
(7)
(8)
where I is the identity matrix.
˜ = B1 + B2 KD21 where A˜ = A + B2 KC2 ; B ˜ = D11 + D12 KD21 C˜ = C1 + D12 KC2 ; D
RankF (X, Y ) = nx
The H∞ problem in the case of a SOF controller consists in finding a matrix K ∈ IRnu ×ny that stabilizes the closedloop defined by expression (2) and that minimizes the following cost function: ˜ + C(sI ˜ ˜ −1 B|| ˜ ∞ ||Tzw ||∞ ≡ ||D − A) (3) This optimization problem cannot be solved. In practice, the suboptimal problem is solved, i.e. finding a stabilizing K that satisfies ||Tzw ||∞ < γ; γ ∈ IR
Fig. 1. Standard H∞ design problem.
(4)
(9)
LMIs can be solved numerically using interior-point methods. However, we need to get rid of the nonlinear constraint (9) first. Therefore, the idea is to introduce the problem min TraceXY subject to (6)-(8)
(10)
According to ElGahoui (1997), there exists a stabilizing SOF controller if and only if the global minimum of (10) is nx . This is called a cone complementary problem. To solve this problem, the trace is linearized and minimized iteratively as explained in the next section. Minimizing a linear cost under LMI constraints can be done easily.
2.3 Linearization algorithm For a given value of γ, the following steps, taken from ElGahoui (1997) lead to the design of a suboptimal SOF controller. (1) Find initial values X0 and Y0 satisfying the LMIs (6)(8). If there are none, exit. Set k = 0. (2) Set Vk = Xk and Sk = Yk and find Xk+1 and Yk+1 that minimize the linearized trace Trace(Vk Y +Sk X) subject to the LMIs (6)-(8). (3) If Trace(Xk+1 Yk+1 ) < nx + , go the step 4. Otherwise, go back to step 2 with k = k + 1. (4) Replace X by Xk+1 in (5). The latter BMI becomes an LMI in K which can be solved easily using available methods. Remark 1. The algorithm is stopped if the number of iterations exceeds a prescribed amount. In that case, it is concluded that the suboptimal problem has no solution. Remark 2. In practice, the lowest value of γ corresponding to a feasible suboptimal problem is obtained by dichotomy. However, since expression (4) is a strict inequality, γ is only an upper bound of the effective value of ||Tzw ||∞ of the closed-loop. The above algorithm will be implemented to design a SOF controller for a teleoperation setup. A 22nd -order model of this system is described in the next section. 3. MODEL OF THE TELEOPERATION SETUP An overall description of the system will be given first. Next, the model of the mechanical behavior is described and the control problem is specified. 3.1 Overall description of the system The teleoperation setup is made of two identical master and slave devices which are depicted in figure 2. The slave device is connected to a bilateral spring emulating the environment. The master and the slave consist of one degree of freedom rotating arms of a length of 95 mm. They are actuated by DC motors (Maxon RE 30) via capstan reductions with a ratio of 100 12 . Due to the presence of the capstan reduction between the motor shaft and the handle, all parameters and variables are expressed in the coordinates of the handle shaft in this paper. The arms are connected to the pulley of the capstan via torque cells (Futek TFF-325). Positions are measured by 1024 counts per turn encoders connected to the motor shafts. Because of their very structure, the strokes of the arms are limited to 100 degrees. The DC motors are driven by Maxon ADS 50/5 servoamplifiers. The teleoperation controller is implemented on a dSPACE DS1103 card at a sampling rate of 10 kHz. 3.2 Model of the mechanical behavior of the system Because the environment is a spring, it is modeled by a simple stiffness. The human operator is modeled by a second order system. Finally the master and the slave are modeled with higher order models taking into account the
Fig. 2. Upper picture: master/slave device; lower picture: slave connected to a spring environment. first vibration mode of each mechanical system. This first mode is actually due to the flexibility of the loadcell 1 . The importance of the first vibration mode on stability has been highlighted by Diaz (2008) 2 . A schematic of the mechanical behavior of the system is depicted in figure 3 and the corresponding notations are given in tables 1 and 2. The input (w, u) and output (z, y) vectors corresponding to the control problem at hand are defined in the next subsection. 1
The force measured by the loadcell is equal to its deflection multiplied by its stiffness kl . 2 Another, though more conservative, way to take this into account consists in modeling the mechanical system as a second order system with multiplicative uncertainty (see Kim (2007)). This allows to take into account higher order modes although the first one seems to be the most significant
Fig. 3. Mechanical behavior of the slave interacting with the environment and of the master grasped by the user. fs,actu /fm,actu xs1 /xm1 xs2 /xm2
actuator force (slave/master) position 1st body (slave/master) position 2nd body (slave/master)
Table 1. Variables of the model of the mechanical behavior of the system. ke kh bh mh mm1 bm1 mm2 ms1 bs1 ms2 kl
environment stiffness human stiffness human damping human mass mass master 1st body damping master 1st body mass master 2nd body mass slave 1st body damping slave 1st body mass slave 2nd body stiffness of the torque cell
2.7 N m/rad 5.37 N m/rad 0.0465 N ms/rad 0.0013 N ms2 /rad 4.25 10−4 N ms2 /rad 7 10−4 N ms/rad 9.3 10−5 N ms2 /rad 4.25 10−4 N ms2 /rad 7 10−4 N ms/rad 9.3 10−5 N ms2 /rad 125 N m/rad
Table 2. Parameters of the model of the mechanical behavior of the system and numerical values expressed in the coordinates of the handle shaft. 3.3 Statement of the control problem To define the input (w, u) and output (z, y) vectors of our control problem, the representation of figure 4 is introduced with some additional signals defined in table 3. fh fh∗ fe v1s /v1m nf e , nxs1 , nvs1 nf h , nxm1 , nvm1
force applied by the master to the user muscular force developped by the user force applied by the environment to the slave velocity of the 1st body of the slave/master sensor noises slave side sensor noises master side
Table 3. Signals introduced in figure 4. The input vector u is given by u = [fs,actu , fm,actu ]T where fs,actu and fm,actu are given by a linear combination of the components of the vector y in the case of a SOF design. The sensed outputs are denoted by the same notation as the corresponding physical variable except that a tilde is added. The vector y in our case is defined by
Fig. 4. Master-slave system y = [Wf h f˜h , Wf e f˜e , (˜ xm1 − x ˜s1 ), Wvm1 v˜m1 , Wvs1 v˜s1 ]T where Wf h , Wf e , Wvm1 and Wvs1 are second order butterworth filters with static gains equal to one processing the sensed signals. The cutoff frequency of the first two filters is equal to 120 Hz and the cutoff frequency of the last two filters is equal to 30 Hz. Position measurements are numerical signals only affected by quantization noise. Therefore they are not processed by a filter. The noise sources affecting the sensed outputs are introduced as exogenous inputs. The corresponding inputs are scaled by the constants w• where the dot stands for an index associated to the input. These shaping constants reflect the relative importance of each signal on the process. The scalings are tuned in order to make the input variables less than one in magnitude. For instance, since 7.36 10−4 rad is the peak value of the noise affecting the position sensor, wN xs will be set equal to 7.36 10−4 rad. So, when nxs1 is equal to one in magnitude, the noise affecting the position sensor at the slave side is equal to 7.36 10−4 rad. The numerical values of the shaping constants are given below: (1) wN f e = wN f h = 8.3 10−3 (2) wN xs1 = wN xm1 = 7.36 10−4 (3) wN vs1 = wN vm1 = 1.2 10−2 The muscular force developed by the user fh∗ is also modeled as an exogenous input but this time it is scaled by a frequency dependent weight Wfh∗ to reflect the physiological frequency limitation of human actuation. Wfh∗ is a second order butterworth filter with a cutoff frequency of 5 Hz and a static gain of 0.76 N m. The vector of exogenous signals is defined by w = [fh∗ , nf e , nxs1 , nvs1 , nf h , nxm1 , nvm1 ]T Finally, the design of the controller implies the definition of a vector of control outputs z. In the case of teleoperation, it is common to use the following set of four functions. (1) Position tracking weighted by a frequency dependent weighting function: W1 (s)(xm2 −xs2 ). Since we are interested in good position tracking performance in the
low frequency band, which correspond to intentional motions of the operator, W1 (s) is usually a low pass filter. In our case, it is a second order Butterworth filter with a cutoff frequency of 10 Hz and a static gain of 1/(5 10−3 ). This translates the will of the designer to have a position error lower than 5 10−3 rad up to 10 Hz. (2) Force tracking weighted by a frequency dependent weighting function: W2 (s)(fh − fe ). Again, a second order Butterworth filter with a cutoff frequency of 10 Hz is chosen but this time with a static gain of 1/0.0417. This means that a torque error lower than 0.0417 N m is wanted (this value corresponds to a torque error of 0.005 N m expressed in the coordinates of the motor shaft). (3) Actuator torque at the master side to avoid actuator saturation: w3 fm,actu with w3 = 1/0.72 since 0.72 N m at the handle side corresponds to actuator saturation. (4) Actuator torque at the slave side to avoid actuator saturation: w4 fs,actu with w3 = 1/0.72.
1/3 Hz 3 Hz
Position error 10−2 rad 10−2 rad
Force error 7 10−2 N m 0.15 N m
Table 4. Peak errors of position and force tracking obtained experimentally
The vector of controlled outputs is summarized as follows: Fig. 5. Motion tracking behavior at low frequency z = [W1 (xm2 − xs2 ), W2 (fh − fe ), w3 fm,actu , w4 fs,actu ]T With the different weights and filters, a 22th -order model of the setup is obtained. A SOF controller is synthesized for this system in the next section. 4. DESIGN OF THE CONTROLLER The design algorithm of section 2 has been implemented in Matlab. It yields the following controller: 0.0059 −0.0073 0.5322 −0.0004 0.0018 K= 0.0464 0.0161 −0.5888 −0.0009 −0.001 Its design took 153 s on a PC equipped with an Intel Core 2 QUAD CPU processor running at 2.66 GHz. The upper bound γ of ||Tzw ||∞ obtained during the synthesis is equal to 2.5 and the effective value of ||Tzw ||∞ is equal to 2.47. The effective performance of the closed-loop are investigated in the next section. Remark 3. A friction compensation method is implemented at both the master and the slave side using disturbance observers with a bandwidth of 100 rad/s. The description of this observer is beyond the scope of this paper. Interested readers can refer to Le Tien (2008) for a definition of the observer. 5. EXPERIMENTAL IMPLEMENTATION The position and force tracking behaviors obtained experimentally are depicted in figures 5 to 8. Two ranges of frequencies were considered for the movement of the operator: around 1/3 Hz (low frequency test) and around 3 Hz (high frequency test). The peak value of the position (figures 5 and 7) and force errors (figures 6 and 8) are given in table 4. The peak values of position and force errors are higher than our specifications. Indeed, a position error of 0.005 rad and a force error of 0.0417 N m are required (see filters
Fig. 6. Force tracking behavior at low frequency W1 (s) and W2 (s) respectively). This means that our specifications are too restrictive. The peak value of the position error is equal to 10−2 rad for both frequency ranges, which can still be considered as a good position tracking. The force tracking behavior is also satisfactory. At low frequency, the peak of the error is slightly above the specification but the tracking is really good. At high frequency, the tracking slightly degrades. However, despite a peak in the error of 0.15 N m during the first period, the peak value during the other periods is approximatively equal to 0.1 N m. It is interesting to note that good tracking is observed even at high frequency with respect to human operator manipulation capabilities. The design has been performed with the nominal model of the system. However, teleoperation is characterized by the fact that the environment and the user are uncertain. The design of a SOF controller with robust performance and stability properties is the object of future works. 6. CONCLUSIONS AND FUTURE WORKS The H∞ framework offers a natural way to handle MIMO control problems with frequency-dependent specifications
REFERENCES
Fig. 7. Motion tracking behavior at high frequency
Fig. 8. Force tracking behavior at high frequency like controller design for bilateral teleoperation. Unfortunately, H∞ design methods available in commercial softwares produce controllers of the same order as that of the model of the plant. In the case of bilateral teleoperation, this can lead to a controller that cannot be implemented practically. To overcome this problem, order-reduction techniques are often implemented, which implies an involved and time-consuming step for the designer. Another way to overcome the problem without requiring this extra design step consists in designing directly a reduced-order controller. The latter option is studied in this paper. A design methodology based on a matrix inequality formulation together with a cone complementary formulation is described. Its implementation yields a SOF controller that has been implemented experimentally on a one dof teleoperation system. Despite its simple structure, the implemented controller shows good position and force tracking. It is also interesting to note that good performance are maintained at high frequency with respect to human actuation capabilities. These good results motivates the use of very low-order control laws for teleoperation. It is likely that fixed-order controller of the first or second order might significantly improve the performance with respect to the SOF controller. This is the object of future works. Designing a SOF controller with robustness properties with respect to changes in the environment or human mechanical impedances is also the object of future works.
L. El Gahoui, F.Oustry and M.AitRami. A Cone Complementary Linearization Algorithm for Static OutputFeedback and Related Problems. IEEE Transactions on Automatic Control, 42:1171–1175, 1997. P. Gahinet and P. Apkarian. A Linear Matrix Inequality Approach to H∞ Control. International Journal of Robust and Nonlinear Control, 4:421–448, 1994. I. Diaz and J. Gil. Influence of internal vibration modes on the stability of haptic rendering. Proceedings of the IEEE International Conference on Robotics and Automation, 2008. E. Vander Poorten, T. Kanno and Y. Yokokohji. Robust variable-scale bilateral control for micro teleoperation. Proceedings of the IEEE International Conference on Robotics and Automation, 2008. K. Kim, M.C. C ¸ avusoglu and W.K. Chung. Quantitative comparison of bilateral teleoperation systems using µsynthesis. IEEE Transactions on Robotics, 23:776–789, 2007. J. Yan and S. Salcudean. Teleoperation controller design using H∞ -optimization with application to motionscaling. IEEE Transactions on Control Systems Technology, 4:244–258, 1996. L. Le Tien, A. Albu-Scheaffer, A. De Luca and G. Hirzinger. Friction observer and compensation for control of robots with joint torque measurements IEEE/RSJ International Conference on Intelligent Robots and Systems, 2008. D.-W. Gu, P.Hr. Petkov and M.M. Konstantinov. Robust Control Design with MATLAB. Springer-Verlag, 2005.
Dept. of Control Engineering and System Analysis, Universit´e Libre de Bruxelles. CP 165/55, 50 Av. F. D. Roosevelt, B-1050 Brussels, Belgium (e-mail: [email protected]). ∗∗ CRAN-UMR 7039 CNRS, Universit´e Henri Poincar´e, Nancy I, F-54506 Vandoeuvre-les-Nancy, France
Abstract: Traditional H∞ design methods available in commercial softwares lead to controllers which present the same order as the model of the plant. Considering the high order of teleoperation systems, implementing such controllers may require the use of order-reduction methods. To avoid this extra step in the design, we propose to produce fixed-order controllers directly based on a full-order model of the system. In this paper, a zero-order (static output feedback) controller is designed for a teleoperation system. The design procedure, based on a cone complementary formulation is described and applied to a 22nd -order model of the system. The resulting controller is implemented on a one degree of freedom teleoperation system. The experimental results obtained with this simple control structure validate the use of very low order control structures in the case of teleoperation systems with the extra advantage, with respect to conventional H∞ design techniques, that the involved and time-consuming order reduction step is avoided. Keywords: H∞ control, static controllers, telerobotics, matrix inequalities, MIMO. 1. INTRODUCTION Teleoperation consists in performing a remote task with an electromechanical master-slave device. The master part of the system is manipulated by the human operator while the slave part, which is located remotely, tries to reproduce the task imposed by the master. It is generally used in hazardous or narrow environments. The main applications of teleoperation include nuclear waste handling, undersea exploration and minimally invasive surgery. When force feedback is present at the master side to make the user feel the interaction forces between the slave and its environment, one refers to bilateral teleoperation. Bilateral teleoperation systems are always MIMO systems even if the master and the slave are both one degree of freedom (dof) devices. Moreover, the actuation and sensory capabilities of human beings are frequency dependent, so the design objectives are also frequency dependent. This explains why this controller design problem is often addressed in the H∞ framework (see Kim (2007), Yan (1996) and Vander Poorten (2008)). However, despite their ability to handle MIMO systems and frequency dependent ? The work of T. Delwiche is supported by a FRIA grant. The setup is financed by the FNRS (ID 1.5.031.06). This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors.The authors gratefully acknowledge the funding of the Tournesol project established in the framework of the Hubert Curien partnerships with the support of Wallonie-Bruxelles International, the FNRS and the French ministries of foreign affairs and scientific research.
performance objectives, most design methods in the H∞ framework (like the ones available in commercial softwares) produce controllers whose order is equal to that of the model of the plant which usually includes frequencydependent weights. Considering the teleoperation application, this may lead to controllers which are too complex to be implemented or at least not intuitive. Three solutions can be considered to produce a low-order controller (see Gu (2005)). The first one consists in reducing the order of the controller synthesized using a full-order model of the plant by suitable order-reduction methods. The second one relies on a reduced-order model of the plant to synthesize the controller. A third method consists in designing directly a reduced-order controller based on a full order model of the plant. The first two methods are implemented in the case of teleoperation applications in Kim (2007) and in Yan (1996) respectively. To the best of our knowledge, the implementation of the third methodology has not been reported for teleoperation applications. The first two methods require an appropriate reduction method. Once the latter is selected and mastered, the designer faces a drawback between the order of the controller and the performance degradation. Excepted under specific circumstances, the reduced controller presents reduced performance with respect to the full order controller which cannot be predicted during the design phase. In this paper, the third method is investigated: a reducedorder controller is designed directly on the basis of a fullorder model of the system, and the performance of the closed-loop system are investigated. In this framework, a static output feedback (SOF) controller, i.e. a zero-order
controller is selected. The resulting SOF controller is implemented on a one degree of freedom (dof) teleoperation system. The satisfactory experimental results motivate the use of simple, low-order control laws for teleoperation systems. The remaining of the paper is structured as follows. In section 2, the method used to compute a SOF controller is presented. Next, section 3 describes a 22nd -order model of the teleoperation setup and section 4 presents the design of the SOF controller for this application. Finally, section 5 presents the experimental results obtained with the designed controller, and section 6 gives some concluding remarks. 2. STATIC OUTPUT FEEDBACK (SOF) DESIGN In this section, a methodology to design a SOF controller in the H∞ framework is presented. It is based on a cone complementary formulation given in ElGahoui (1997). The latter problem requires a formulation in terms of matrix inequalities. The developments presented here are general and not restricted to teleoperation. First, the H∞ design problem is stated formally. Next, its matrix inequality formulation is given. Finally, the cone complementary problem is described. 2.1 Problem statement Consider the system M depicted in figure 1 and defined by the set of state-space equations (1) where x ∈ IRnx denotes the state vector, w ∈ IRnw is the vector of exogenous inputs (references, sensor noise, ...), z ∈ IRnz is a vector of controlled outputs relevant to the problem at hand, u ∈ IRnu is the vector of control inputs and finally y ∈ IRny is the vector of measured outputs. ! ! ! x˙ A B1 B2 x z = C1 D11 D12 w (1) y C2 D21 D22 u | {z } ≡M
M is connected in closed-loop with a static output feedback (SOF) controller, namely a control law u = Ky where K ∈ IRnu ×ny is a constant matrix. If D22 = 0 is assumed, the state-space equations of the closed-loop system are given by expression (2).
x˙ z
=
˜ A˜ B ˜ C˜ D
x w
(2)
A cone complementary formulation followed by a linearization algorithm (see ElGahoui (1997)) are used to solve this problem, based on a matrix inequality formulation. 2.2 Matrix inequality formulation By virtue of the Bounded Real Lemma, the suboptimal SOF H∞ problem is feasible if there exists a positive definite matrix X ∈ Rnx ×nx and a matrix K ∈ IRny ×nu such that the bilinear matrix inequality (BMI) (5) is satisfied: (A + B2 KC2 )T X + ∗ ∗ ∗ (B1 + B2 KD21 )T X −γI ∗ < 0(5) (C1 + D12 KC2 ) (D11 + D12 KD21 ) −γI where the ∗ stand for the transpose of the symmetric entry. To solve this BMI problem, the unknown K is eliminated first by using the separation lemma given in Gahinet (1994) which yields the following equivalent formulation: the suboptimal problem is feasible if there exists positive definite matrices X ∈ IRnx ×nx and Y ∈ IRnx ×nx satisfying the linear matrix inequalities (LMI) (6), (7) and (8) and the nonlinear constraint (9). T A X + XA XB1 C1T T W < 0 WPT (6) B1T X −γI D11 P C1 D11 −γI where WP is the kernel of P = [C2 D21 0] Y AT + AY B1 Y C1T T W < 0 WQT B1T −γI D11 Q C1 Y D11 −γI T where WQ is the kernel of Q = [B2T 0 D12 ] X I F (X, Y ) = ≥0 I Y
(7)
(8)
where I is the identity matrix.
˜ = B1 + B2 KD21 where A˜ = A + B2 KC2 ; B ˜ = D11 + D12 KD21 C˜ = C1 + D12 KC2 ; D
RankF (X, Y ) = nx
The H∞ problem in the case of a SOF controller consists in finding a matrix K ∈ IRnu ×ny that stabilizes the closedloop defined by expression (2) and that minimizes the following cost function: ˜ + C(sI ˜ ˜ −1 B|| ˜ ∞ ||Tzw ||∞ ≡ ||D − A) (3) This optimization problem cannot be solved. In practice, the suboptimal problem is solved, i.e. finding a stabilizing K that satisfies ||Tzw ||∞ < γ; γ ∈ IR
Fig. 1. Standard H∞ design problem.
(4)
(9)
LMIs can be solved numerically using interior-point methods. However, we need to get rid of the nonlinear constraint (9) first. Therefore, the idea is to introduce the problem min TraceXY subject to (6)-(8)
(10)
According to ElGahoui (1997), there exists a stabilizing SOF controller if and only if the global minimum of (10) is nx . This is called a cone complementary problem. To solve this problem, the trace is linearized and minimized iteratively as explained in the next section. Minimizing a linear cost under LMI constraints can be done easily.
2.3 Linearization algorithm For a given value of γ, the following steps, taken from ElGahoui (1997) lead to the design of a suboptimal SOF controller. (1) Find initial values X0 and Y0 satisfying the LMIs (6)(8). If there are none, exit. Set k = 0. (2) Set Vk = Xk and Sk = Yk and find Xk+1 and Yk+1 that minimize the linearized trace Trace(Vk Y +Sk X) subject to the LMIs (6)-(8). (3) If Trace(Xk+1 Yk+1 ) < nx + , go the step 4. Otherwise, go back to step 2 with k = k + 1. (4) Replace X by Xk+1 in (5). The latter BMI becomes an LMI in K which can be solved easily using available methods. Remark 1. The algorithm is stopped if the number of iterations exceeds a prescribed amount. In that case, it is concluded that the suboptimal problem has no solution. Remark 2. In practice, the lowest value of γ corresponding to a feasible suboptimal problem is obtained by dichotomy. However, since expression (4) is a strict inequality, γ is only an upper bound of the effective value of ||Tzw ||∞ of the closed-loop. The above algorithm will be implemented to design a SOF controller for a teleoperation setup. A 22nd -order model of this system is described in the next section. 3. MODEL OF THE TELEOPERATION SETUP An overall description of the system will be given first. Next, the model of the mechanical behavior is described and the control problem is specified. 3.1 Overall description of the system The teleoperation setup is made of two identical master and slave devices which are depicted in figure 2. The slave device is connected to a bilateral spring emulating the environment. The master and the slave consist of one degree of freedom rotating arms of a length of 95 mm. They are actuated by DC motors (Maxon RE 30) via capstan reductions with a ratio of 100 12 . Due to the presence of the capstan reduction between the motor shaft and the handle, all parameters and variables are expressed in the coordinates of the handle shaft in this paper. The arms are connected to the pulley of the capstan via torque cells (Futek TFF-325). Positions are measured by 1024 counts per turn encoders connected to the motor shafts. Because of their very structure, the strokes of the arms are limited to 100 degrees. The DC motors are driven by Maxon ADS 50/5 servoamplifiers. The teleoperation controller is implemented on a dSPACE DS1103 card at a sampling rate of 10 kHz. 3.2 Model of the mechanical behavior of the system Because the environment is a spring, it is modeled by a simple stiffness. The human operator is modeled by a second order system. Finally the master and the slave are modeled with higher order models taking into account the
Fig. 2. Upper picture: master/slave device; lower picture: slave connected to a spring environment. first vibration mode of each mechanical system. This first mode is actually due to the flexibility of the loadcell 1 . The importance of the first vibration mode on stability has been highlighted by Diaz (2008) 2 . A schematic of the mechanical behavior of the system is depicted in figure 3 and the corresponding notations are given in tables 1 and 2. The input (w, u) and output (z, y) vectors corresponding to the control problem at hand are defined in the next subsection. 1
The force measured by the loadcell is equal to its deflection multiplied by its stiffness kl . 2 Another, though more conservative, way to take this into account consists in modeling the mechanical system as a second order system with multiplicative uncertainty (see Kim (2007)). This allows to take into account higher order modes although the first one seems to be the most significant
Fig. 3. Mechanical behavior of the slave interacting with the environment and of the master grasped by the user. fs,actu /fm,actu xs1 /xm1 xs2 /xm2
actuator force (slave/master) position 1st body (slave/master) position 2nd body (slave/master)
Table 1. Variables of the model of the mechanical behavior of the system. ke kh bh mh mm1 bm1 mm2 ms1 bs1 ms2 kl
environment stiffness human stiffness human damping human mass mass master 1st body damping master 1st body mass master 2nd body mass slave 1st body damping slave 1st body mass slave 2nd body stiffness of the torque cell
2.7 N m/rad 5.37 N m/rad 0.0465 N ms/rad 0.0013 N ms2 /rad 4.25 10−4 N ms2 /rad 7 10−4 N ms/rad 9.3 10−5 N ms2 /rad 4.25 10−4 N ms2 /rad 7 10−4 N ms/rad 9.3 10−5 N ms2 /rad 125 N m/rad
Table 2. Parameters of the model of the mechanical behavior of the system and numerical values expressed in the coordinates of the handle shaft. 3.3 Statement of the control problem To define the input (w, u) and output (z, y) vectors of our control problem, the representation of figure 4 is introduced with some additional signals defined in table 3. fh fh∗ fe v1s /v1m nf e , nxs1 , nvs1 nf h , nxm1 , nvm1
force applied by the master to the user muscular force developped by the user force applied by the environment to the slave velocity of the 1st body of the slave/master sensor noises slave side sensor noises master side
Table 3. Signals introduced in figure 4. The input vector u is given by u = [fs,actu , fm,actu ]T where fs,actu and fm,actu are given by a linear combination of the components of the vector y in the case of a SOF design. The sensed outputs are denoted by the same notation as the corresponding physical variable except that a tilde is added. The vector y in our case is defined by
Fig. 4. Master-slave system y = [Wf h f˜h , Wf e f˜e , (˜ xm1 − x ˜s1 ), Wvm1 v˜m1 , Wvs1 v˜s1 ]T where Wf h , Wf e , Wvm1 and Wvs1 are second order butterworth filters with static gains equal to one processing the sensed signals. The cutoff frequency of the first two filters is equal to 120 Hz and the cutoff frequency of the last two filters is equal to 30 Hz. Position measurements are numerical signals only affected by quantization noise. Therefore they are not processed by a filter. The noise sources affecting the sensed outputs are introduced as exogenous inputs. The corresponding inputs are scaled by the constants w• where the dot stands for an index associated to the input. These shaping constants reflect the relative importance of each signal on the process. The scalings are tuned in order to make the input variables less than one in magnitude. For instance, since 7.36 10−4 rad is the peak value of the noise affecting the position sensor, wN xs will be set equal to 7.36 10−4 rad. So, when nxs1 is equal to one in magnitude, the noise affecting the position sensor at the slave side is equal to 7.36 10−4 rad. The numerical values of the shaping constants are given below: (1) wN f e = wN f h = 8.3 10−3 (2) wN xs1 = wN xm1 = 7.36 10−4 (3) wN vs1 = wN vm1 = 1.2 10−2 The muscular force developed by the user fh∗ is also modeled as an exogenous input but this time it is scaled by a frequency dependent weight Wfh∗ to reflect the physiological frequency limitation of human actuation. Wfh∗ is a second order butterworth filter with a cutoff frequency of 5 Hz and a static gain of 0.76 N m. The vector of exogenous signals is defined by w = [fh∗ , nf e , nxs1 , nvs1 , nf h , nxm1 , nvm1 ]T Finally, the design of the controller implies the definition of a vector of control outputs z. In the case of teleoperation, it is common to use the following set of four functions. (1) Position tracking weighted by a frequency dependent weighting function: W1 (s)(xm2 −xs2 ). Since we are interested in good position tracking performance in the
low frequency band, which correspond to intentional motions of the operator, W1 (s) is usually a low pass filter. In our case, it is a second order Butterworth filter with a cutoff frequency of 10 Hz and a static gain of 1/(5 10−3 ). This translates the will of the designer to have a position error lower than 5 10−3 rad up to 10 Hz. (2) Force tracking weighted by a frequency dependent weighting function: W2 (s)(fh − fe ). Again, a second order Butterworth filter with a cutoff frequency of 10 Hz is chosen but this time with a static gain of 1/0.0417. This means that a torque error lower than 0.0417 N m is wanted (this value corresponds to a torque error of 0.005 N m expressed in the coordinates of the motor shaft). (3) Actuator torque at the master side to avoid actuator saturation: w3 fm,actu with w3 = 1/0.72 since 0.72 N m at the handle side corresponds to actuator saturation. (4) Actuator torque at the slave side to avoid actuator saturation: w4 fs,actu with w3 = 1/0.72.
1/3 Hz 3 Hz
Position error 10−2 rad 10−2 rad
Force error 7 10−2 N m 0.15 N m
Table 4. Peak errors of position and force tracking obtained experimentally
The vector of controlled outputs is summarized as follows: Fig. 5. Motion tracking behavior at low frequency z = [W1 (xm2 − xs2 ), W2 (fh − fe ), w3 fm,actu , w4 fs,actu ]T With the different weights and filters, a 22th -order model of the setup is obtained. A SOF controller is synthesized for this system in the next section. 4. DESIGN OF THE CONTROLLER The design algorithm of section 2 has been implemented in Matlab. It yields the following controller: 0.0059 −0.0073 0.5322 −0.0004 0.0018 K= 0.0464 0.0161 −0.5888 −0.0009 −0.001 Its design took 153 s on a PC equipped with an Intel Core 2 QUAD CPU processor running at 2.66 GHz. The upper bound γ of ||Tzw ||∞ obtained during the synthesis is equal to 2.5 and the effective value of ||Tzw ||∞ is equal to 2.47. The effective performance of the closed-loop are investigated in the next section. Remark 3. A friction compensation method is implemented at both the master and the slave side using disturbance observers with a bandwidth of 100 rad/s. The description of this observer is beyond the scope of this paper. Interested readers can refer to Le Tien (2008) for a definition of the observer. 5. EXPERIMENTAL IMPLEMENTATION The position and force tracking behaviors obtained experimentally are depicted in figures 5 to 8. Two ranges of frequencies were considered for the movement of the operator: around 1/3 Hz (low frequency test) and around 3 Hz (high frequency test). The peak value of the position (figures 5 and 7) and force errors (figures 6 and 8) are given in table 4. The peak values of position and force errors are higher than our specifications. Indeed, a position error of 0.005 rad and a force error of 0.0417 N m are required (see filters
Fig. 6. Force tracking behavior at low frequency W1 (s) and W2 (s) respectively). This means that our specifications are too restrictive. The peak value of the position error is equal to 10−2 rad for both frequency ranges, which can still be considered as a good position tracking. The force tracking behavior is also satisfactory. At low frequency, the peak of the error is slightly above the specification but the tracking is really good. At high frequency, the tracking slightly degrades. However, despite a peak in the error of 0.15 N m during the first period, the peak value during the other periods is approximatively equal to 0.1 N m. It is interesting to note that good tracking is observed even at high frequency with respect to human operator manipulation capabilities. The design has been performed with the nominal model of the system. However, teleoperation is characterized by the fact that the environment and the user are uncertain. The design of a SOF controller with robust performance and stability properties is the object of future works. 6. CONCLUSIONS AND FUTURE WORKS The H∞ framework offers a natural way to handle MIMO control problems with frequency-dependent specifications
REFERENCES
Fig. 7. Motion tracking behavior at high frequency
Fig. 8. Force tracking behavior at high frequency like controller design for bilateral teleoperation. Unfortunately, H∞ design methods available in commercial softwares produce controllers of the same order as that of the model of the plant. In the case of bilateral teleoperation, this can lead to a controller that cannot be implemented practically. To overcome this problem, order-reduction techniques are often implemented, which implies an involved and time-consuming step for the designer. Another way to overcome the problem without requiring this extra design step consists in designing directly a reduced-order controller. The latter option is studied in this paper. A design methodology based on a matrix inequality formulation together with a cone complementary formulation is described. Its implementation yields a SOF controller that has been implemented experimentally on a one dof teleoperation system. Despite its simple structure, the implemented controller shows good position and force tracking. It is also interesting to note that good performance are maintained at high frequency with respect to human actuation capabilities. These good results motivates the use of very low-order control laws for teleoperation. It is likely that fixed-order controller of the first or second order might significantly improve the performance with respect to the SOF controller. This is the object of future works. Designing a SOF controller with robustness properties with respect to changes in the environment or human mechanical impedances is also the object of future works.
L. El Gahoui, F.Oustry and M.AitRami. A Cone Complementary Linearization Algorithm for Static OutputFeedback and Related Problems. IEEE Transactions on Automatic Control, 42:1171–1175, 1997. P. Gahinet and P. Apkarian. A Linear Matrix Inequality Approach to H∞ Control. International Journal of Robust and Nonlinear Control, 4:421–448, 1994. I. Diaz and J. Gil. Influence of internal vibration modes on the stability of haptic rendering. Proceedings of the IEEE International Conference on Robotics and Automation, 2008. E. Vander Poorten, T. Kanno and Y. Yokokohji. Robust variable-scale bilateral control for micro teleoperation. Proceedings of the IEEE International Conference on Robotics and Automation, 2008. K. Kim, M.C. C ¸ avusoglu and W.K. Chung. Quantitative comparison of bilateral teleoperation systems using µsynthesis. IEEE Transactions on Robotics, 23:776–789, 2007. J. Yan and S. Salcudean. Teleoperation controller design using H∞ -optimization with application to motionscaling. IEEE Transactions on Control Systems Technology, 4:244–258, 1996. L. Le Tien, A. Albu-Scheaffer, A. De Luca and G. Hirzinger. Friction observer and compensation for control of robots with joint torque measurements IEEE/RSJ International Conference on Intelligent Robots and Systems, 2008. D.-W. Gu, P.Hr. Petkov and M.M. Konstantinov. Robust Control Design with MATLAB. Springer-Verlag, 2005.