Sliding mode control of Functional Electrical Stimulation for

knee joint angle tracking

Sašo Jezernik ¹, Philipp Inderbitzin ¹, Thierry Keller ¹, and Robert Riener ²

¹ETH Zürich, Automatic Control Laboratory, Physikstrasse 3, 8092 Zürich, Switzerland

² Technical University of Munich, Institute of Automatic Control Engineering, Arcisstr. 21, 80333 Munich, Germany

SUMMARY

An inherently robust closed-loop control strategy called sliding mode (SM) was applied to the problem of knee joint angle tracking by stimulation of the knee extensor and flexor muscles. The controller was synthesized using a state-space model of a human knee and muscles (composed of activation dynamics, muscle dynamics, and biomechanics) /1/. The model was experimentally validated in our earlier studies. The derived SM control law provided asymptotic stability of the knee joint angle and velocity, and was robust to the muscle fatigue. It was derived in a closed-form by considering only a subsystem of the muscle-knee model and using a joint angle-joint velocity sliding surface stabilized by appropriately generated muscle activation that followed from the sliding mode reachability and stabilizability conditions. In our earlier studies, instead of using a closed-form expression for the sliding mode controller, we have only used an approximate sliding mode control law, and have additionally necessitated a cascaded PI controller, which degraded the performance and sometimes caused instability /2/. Simulations with the new controller were carried out in SIMULINK with different knee angle trajectories and different controller parameters, and the results were analyzed. The controller with best parameters was able to track ramps and real knee joint angle walking trajectories with a root-mean-square (RMS) error of about 2 degrees at physiological (fast) speeds.

STATE OF THE ART

Functional Electrical Stimulation (FES) can be used to restore movement in certain paralyzed individuals by stimulation of intact peripheral nerves. Stimulation produces muscle contractions and generates joint movements. The underlying physiological/biomechanical system is highly nonlinear and time-variant, and a feedback control strategy is necessary for satisfactory control of the joint angles /3/. However, many classical closed-loop control algorithms were found unable to provide an adequate movement control. Some newer approaches gave good results though, but with no guarantees on stability /4/. Additional reasons for inadequate performance of closed-loop control strategies for FES are input constraints and bandwidth limitations due to low stimulation frequencies (<50 Hz). Our research goal is to develop new FES control strategies that insure stable angle tracking with an improved performance, and that could be applied to control several joints in order to control different movements like grasping, reaching, and walking.

In this study we have evaluated the performance of the nonlinear closed-loop control law called sliding mode /5,6/ that was used to control the knee joint angle by FES of suitable muscles. The SM control strategy was chosen as it was earlier shown to be robust to parameter variations and was already successful in controlling different complex nonlinear plants. Two variants of the sliding mode control were tested (discontinuous and continuous sliding mode control), both with the control of: (1) knee extensor muscles only, and (2), concurrent knee extensor and knee flexor muscle stimulation.

MATERIAL AND METHODS

Plant Model

The human leg model for one degree of freedom movement at the knee joint consisted of a simple leg biomechanical model and of a model describing muscular properties. The biomechanical part was basically described by the equation of motion at the knee joint (parameters: leg mass, leg inertia, leg geometry, damping, passive knee torques, active muscle torques). The muscle dynamics subject to FES excitation was described by a static recruitment curve, nervous and muscular delay, linear second order Calcium (Ca) release and reuptake model, muscle fatigue model, and a modified Hill-type model of the active muscle

contraction/force (containing force-length and force-velocity relationships). The total plant model can be written in the following state-space form:

whereby the state space variables x₁-x₄ stand for the knee angle, knee angular velocity, concentration of the Ca ions, and the derivative of the Ca ion concentration respectively. In this set of differential equations, ma stands for the muscle moment arm, fit describes the muscle fitness, g₁ and g₂ stand for the force-length and force-velocity relationships, t_g for gravity, t_p for passive knee joint torques, d for linear damping, T is the Ca release-reuptake time constant, and t_dis the delay.

Note that this model can be split into two submodels in the following way:

The state variable x₃ can now be regarded as an input to the subsystem 1 and as an output of the subsystem 2, which is driven by the input u. It is important to note that x₁ and x₂ can be measured, but x₃ and x₄ can not.

Sliding Mode Control

The basic idea behind the sliding mode control is to define a sliding surface (submanifold) in the Â^k vectorspace and to generate a control law (sliding mode control) that will force the system trajectory to reach the sliding surface in a finite time, and that will ensure that the subsequent evolution of the system trajectory will remain on the sliding surface (this latter mode is called a sliding mode). The sliding mode usually describes the error dynamics and forces the error to asymptotically decay to zero.

Let us define the sliding surface to be s(x,x_reference)=0. If we select the control law to guarantee condition ds/dt×s<0 (condition (1)) then the sliding mode surface will necessarily be reached (because (1) implies ds/dt<0 for s>0 and vice versa). The control that will furthermore ensure that s will remain 0 (sliding mode, condition (2)) can be calculated from the formula ds/dt=0. This control is called the equivalent control, u_eq. The discontinuous control law that combines (1) and (2) for the first subsystem is given by (3):

In the continuous case, the discontinuous term k×sgn(s) is replaced by a continuous one k×s.

Both laws guarantee reachability of the sliding mode AND its stability. In our two dimensional case, the sliding surface represents a line through the origin of the phase plane that has the slope of -l.

By substitution of (3) into the state-space subsystem 1 one can easily check that the conditions (1) and (2) are satisfied.

Simulations

Simulations were performed with the sliding mode control of the first subsystem only, and with the combined sliding mode control of the first subsystem and a corresponding feed-forward (FF) control of the second subsystem. The control input was always constrained and its bandwidth limited with a zero-order-hold block (f_stim=20 Hz). First we have stimulated knee extensors only (quadriceps muscles) and then we have repeated the simulations with the concurrent stimulation of the knee extensors and knee flexors (biceps femoris muscles). The control task was to track a ramp reference with a p-p amplitude of 60 degrees and a period of 2 s, and a real knee joint trajectory reference measured during walking that had a period of 2 s. Extensive simulations with the ramp reference were performed in order to obtain the optimal controller parameters l and k. The optimal values were then used in the subsequent simulations.

RESULTS

Table 1

Fig.1 A (left) and B (right).

Control of Knee Extensors

The results of tracking ramps in the knee joint angle (sliding mode control, subsystem 1, k=0.07 and l=10) are shown in Fig.1A for the continuous, and in Fig.1B for the discontinuous sliding mode control. The top panels show the control input x₃, the second panels the knee angle reference and the actual knee angle, and the bottom panels show reaching of the sliding surface and the convergence to the origin (asymptotic stability). Table 1 lists the RMS tracking errors for the continuous sliding mode for different controller parameters k and l. Best results were obtained for k=0.05 and for l=20.

k	e_RMS [deg] (l=10)	e_RMS [deg] (l=20)	e_RMS [deg] (l=30)
0.01	4.24	3.41	2.95
0.05	2.48	2.10	2.45
0.1	2.34	3.06	4.07
0.2	5.02	8.06	9.07

Fig.2A shows simulations where a real walking trajectory was tracked (cont. SM control, subsystem 1 only). Fig.2B in turn demonstrates the performance of the complete controller (FF 1^st order control of the second subsystem combined with the SM control of the first subsystem). The control input (pulse width) was limited to 100-500 µs.

Higher values of k should provide more robustness (compensate for model uncertainties), but limits on x₃ (0 and 1) lead to control signal saturation that causes performance deterioration. This can be avoided if two muscles are used so that positive and negative torques can actively be generated.

Control of Knee Extensors and Flexors

In the case of stimulation of an agonist/antagonist muscle pair, we have split the control law (3) in such a way that the extensor muscles were activated for positive s and that the flexor muscles were activated when s was negative (only the second term in (3) was split). The tracking performance improved only slightly as the flexor muscles got activated mainly at the corners of the knee joint angle ramps (Fig.3A+B). The top panels (Fig.3) show the x₃/pulse width of the extensor muscles, and the middle panels the x₃ /pulse width of the flexor muscles for the subsystem 1/combined controller respectively.

DISCUSSION

A nonlinear SM control law was derived for the knee joint angle control by stimulation of a single muscle group, and of an agonist/antagonist muscle group pair. This law guaranteed stability and achieved good results in computer simulations. Human experiments are currently being performed in order to test the developed controllers in real experiments. Next task will be to augment the SM control to two and subsequently three joints. Major limitations in the performance in simulations were due to low stimulation frequency and due to constraints on the control input signal.

REFERENCES

/1/ Riener R, Fuhr T: Patient-driven control of FES supported standing-up: A simulation study. IEEE TRE, Vol. 6, No. 2, pp. 113-124, 1998.

/2/ Jezernik S, Riener R: A computer simulation of tuned PID and continuous sliding mode FES control. In Proceedings of the International Biomechatronics Workshop, Enschede, Netherlands, pp.37-41, 1999.

/3/ Crago PE, Lan N, Veltink PH, Abbas JJ, Kantor C: New control strategies for neuroprosthetic systems. J. of Reh. Res. and Dev., Vol.33, No.2, pp.158-172, 1996.

/4/ Chang GC, Luh JJ, Liao GD, Lai JS, Cheng CK, Kuo BL, Kuo TS: Neuro-Control System for the Knee Joint Position Control with Quadriceps Stimulation. IEEE TRE, Vol.5, No.1, pp. 2-11, 1997.

/5/ Utkin VU: Sliding Modes in Control and Optimization. Springer-Verlag, Berlin, 1992.

/6/ Slotine JJE, Coetsee JA: Adaptive sliding controller synthesis for nonlinear systems. Int. J.Control, Vol.43, No.6, pp.1631-1651, 1986.

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