Feedforward and Deterministic Fuzzy Control of Balance and Posture ...
Proceedings of the 2001 IEEE International Conference on Robotics & Automation Seoul, Korea • May 21-26, 2001
Feedforward and Deterministic Fuzzy Control of Balance and Posture During Human Gait Eric Kubica Dept. of Systems Design Engineering University of Waterloo, Waterloo, ON N2L 3G1
David Wang Dept. of Electrical and Computer Engineering University of Waterloo, Waterloo, ON N2L 3G1
Abstract-The primary objective of this research is to model the biomechanical control system employed by the Central Nervous System (CNS) to maintain posture and balance of the HAT (Head-Arms-Torso) during gait. More specifically, the intent is to stabilize a model of the upper body (HAT) so that the HAT response is similar to that found experimentally in human subjects during gait. This will be accomplished by using appropriate physiological parameters as feedback to generate realistic control signals at the hip musculature. The modelling includes the HAT musculoskeletal characteristics as well as pure neural time delays. The CNS control system is modelled by a linear state feedback controller as well as a hybrid fuzzy controller that has been adapted from a linear quadratic regulator. In addition, a newly derived feedforward component is demonstrated. It is expected that a better understanding of the human gait process will prove valuable in designing assistive devices and bipedal robots.
kinematics and dynamics) of the HAT to the CNS control during gait. This work offers insight for several areas. It provides information for developing a systematic approach to model human locomotion. This could lead to improved robotic aids for the physically handicapped by using more physiologically-based prosthetic devices. As well, any progress to understanding how the CNS controls human locomotion would be of interest for those trying to create more human-like robots. 2. Experimentally Derived Performance Criteria Experimental data were analyzed from several walking trials of one subject at three different speeds (slow, normal and fast) to establish the performance characteristics of the modelled system. Linear accelerations that occur at the hip or base of the HAT during gait disturb the equilibrium of this segment. Thus, these measured accelerations are used as disturbance inputs to the HAT model in order to produce realistic simulations of walking.
1. Introduction Several different approaches can be found in the literature to quantify the ability of the CNS to maintain balance and posture. In these works, a frequently observed similarity is the implementation of some form of linear state feedback control. However, despite that it is well known that significant pure time delays are present in the CNS, these effects are typically neglected [1]. Alternate approaches have also been investigated. For example, an intelligent control approach using neural nets was presented in [2] while a robotic biped was constructed in [3] that was able to perform relatively smooth gait; however, both neglected neural delays and therefore, are not good models of the CNS control system. Often, results are obtained which appear to closely match human performance but close scrutiny often shows that the control efforts are far beyond physiological capabilities [6]. The objective of the present research is to mimic the response (both 0-7803-6475-9/01/$10.00© 2001 IEEE
David Winter Dept. of Kinesiology University of Waterloo, Waterloo, ON N2L 3G1
Data were measured by attaching 28 infrared LEDs to a human subject and recording the human gait using an OPTOtrack 3020 3-D motion measurement system. Ground reaction forces were measured using two force plates. It was found that the average angle of inclination (posture) of the HAT segment only varied about 0.5 degrees for any of the walking speeds and was roughly equal to 2 degrees. However, the amplitude of the HAT oscillations did vary with speed such that during slow, normal and fast walking the HAT angle varied a total of approximately 3.5 degrees, 2.75 degrees and, 1.5 degrees respectively (See Figure 1). These bounds were later used as guides to determine whether or not the proposed control strategy performed adequately. The reduction in the oscillation amplitude as walking speed increased was 2293
during heel contact [5] (See Figure 2 for a normal gait example). Although only one subject was used in this study, work by [7] showed that multiple walking trials from different subjects produced nearly identical perturbations and corresponding corrections throughout the gait cycle. 3. System Modelling A model of the upper body has been developed that incorporates characteristics of the actuating hip musculature, the afferent and efferent neural pathways and the inertial properties of the HAT segment itself. The HAT segment was modelled as a single rigid inverted pendulum in the sagittal plane (See Figure 4). This is a commonly accepted model [8] and has been confirmed by the experimental data collection. The dynamics of interest are the rotary dynamics involving in Figure 3. The horizontal and vertical accelerations of point H caused by the forces and were modelled as disturbances to the rotary dynamics and are taken from experimental data sets [2]. This allowed the HAT model to be exposed to the same perturbations that
HAT sagittal displacement for two slow trials
HAT sagittal displacement for two normal trials
HAT sagittal displacement for two fast trials
Figure 1: Measured HAT displacements Figure 3: Inverted pendulum HAT model occur during actual walking. The resulting inverted pendulum dynamics were modelled as:
The hip musculature was modelled as a second-order critically damped torque motor with a cut-off frequency of 4Hz. Although this type of EMG-driven model does not embody force-length or force-velocity characteristics of muscle, it has been shown that the operating range of the hip musculature during walking is small enough so that only a 10-15% error may result by neglecting these nonlinear effects [5]. Thus, the hip torque can be represented by the following second order equation:
Figure 2: Measured hip torques: left(--), right(-.), left+right(solid) unexpected although it is speculated that higher levels of co-contraction may occur at fast speeds that would increase the effective joint stiffness. Estimates of the net hip torque were also produced that fell within normally measured bounds despite large anomalies that occurred 2294
Assuming that the twitch response of the muscle can be approximated by a critically damped second order system, then the parameters of the muscle model can be determined simply by knowing the twitch time which is equal to the inverse of the low-pass filter cut-off frequency.
delays and the relatively slow muscle responses that are present in the system. However, the CNS is able to control the HAT with relatively low amplitude oscillations under the same conditions, and therefore, the CNS must be able to anticipate the onset of the disturbances that occur during gait and initiate corrective responses beforehand. This is a proactive feedforward component that must be capable of generating a motor signal that is proportional to the magnitude of the pending disturbance. In addition, it must be sufficiently in advance of the disturbance so that the corrective action can be implemented as the disturbance occurs. The vestibular system (an inner ear organ) is often credited with this function although it has been shown with actual data that the accelerations measured by the vestibular system lagged the required motor signal and therefore cannot be responsible for this type of proactive control [5].
The final component that was included in the neuromusculoskeletal model of the HAT was the pure time delay that occurs in the efferent and afferent pathways along with the corresponding computational time delays. These delays have been lumped into one latency that was estimated at 62ms assuming that the dominant HAT angle sensors were in the region of the pelvis [5]. 4. Linear Optimal State Feedback Control An initial controller design utilized full state feedback from models of the hip musculature and HAT segment dynamics and was based on linear quadratic optimal control. The basic structure of the control loop is shown in Figure 4. Note that is the gain needed so that as . This controller performed well for postural adjustments (step inputs) although this same controller was unable to regulate the HAT segment during walking as measured hip accelerations were applied to the HAT model. The only walking situation that the linear quadratic regulator could tolerate occurred when the pure time delay was cut in half to 31ms. The performance of the HAT controller with the shorter time delay was stable although the HAT still oscillated more than 7 degrees during normal walking which is more than three times that measured experimentally (this is shown as the dashed line in Figure 6). Thus, these results suggest that linear state feedback control is an unlikely candidate for controlling HAT balance.
It was then surmised that since the most significant disturbances occurred close to the push-off and heel contact phases of the gait cycle, the motion at the ankle joint may be indicative of the perturbations at the hip. A subsequent analysis revealed that, with the foot flat on the ground, the disturbance moment at the hip was approximately proportional to the angular jerk (the third derivative of angular position) at the ankle. This result was confirmed with the experimental data and it was found that a combination of the linear feedback controller and a feedforward signal proportional to the negative of the angle jerk significantly reduced the oscillations that the HAT experienced without the feedforward component. The control loop is shown in Figure 5 where B represents the feedforward ankle jerk component.
Figure 5: Feedback and feedforward control loop Note that the inverse muscle model is now required to remove the effects of the feedforward term. The resulting performance is shown in Figure 6 for normal walking where the solid line represents the combined feedforward and feedback control while the dashed line represents the response of the feedback controller alone.
Figure 4: Linear state feedback control loop
5. Feedforward Balance Control The relatively poor performance of the linear controller during gait appeared to indicate that a purely reactive control strategy could not cope adequately with the time
In addition to improving the performance of the control 2295
adequately in the neighbourhood of the origin and it would be useful to retain these characteristics. Therefore, if a fuzzy controller could be constructed with equivalent performance to the state feedback controller, an innovative advantage could be realized in the initial design of the fuzzy controller. It was found that this requirement could be fulfilled when a fuzzy controller is structured with the following general conditions: 1) The input membership functions are constructed such that the sum of the membership grades for a particular input will be unity. In other words, if the membership functions are specified as , where i
Figure 6: Sagittal HAT angle, normal speed: Feedforward control (solid), Linear feedback control (dashed) system, the feedforward component also reduced the hip torque requirements significantly. It is speculated that the feedforward signal is generated at a higher level of the CNS such that the base volitive motor commands for the entire lower limb are initiated together. Despite this improvement in the performance however, the measured CNS regulation of the HAT operates within a tighter movement envelope than that shown in Figure 6. This leads to the conclusion that the linear feedback controller should be improved upon.
designates the input, s denotes the particular fuzzy set, is the number of fuzzy sets spanning input i with n inputs, this condition can be written as:
In addition, the membership functions must vary linearly between one another. Provided that only normal fuzzy sets are used (A is a normal fuzzy set if
6. Deterministic Fuzzy Control
), these conditions can only be satisfied
Biological evidence has been identified which illustrates that some proprioceptive sensors function with an uncanny similarity to fuzzy systems. In [4], the data from several different nerve fibres from a knee joint receptor of a cat was shown as the knee was moved through 140 degrees of flexion. The output of the joint receptors bear a striking resemblance to fuzzy sets and is shown in Figure 7. Thus, it is conjectured that a fuzzy controller may be appropriate for use as a model of the control process of the CNS.
with triangular membership functions which overlap at a membership grade of 0.5. 2) The fuzzy associative memory (FAM) or rule-base is derived so that it is full or complete. This means that each combinatorially possible fuzzy association is considered. Hence, the total number of rules (M) in the FAM is equal to the product of the number of fuzzy sets spanning each of the n inputs i.e.
.
3) The output membership functions are chosen as symmetrical unimodal functions all of equal area with centres at locations where the input rules are satisfied exactly. In other words, the consequence of each part of the FAM is assigned to an output set with its centroid positioned in the output space so that it provides the same defuzzified output as the linear control law with the same inputs [5]. 4) The inference engine must utilize product inferencing and centre-of-gravity defuzzificiation must be used.
Figure 7: Joint receptor output Fuzzy control was chosen to supersede state feedback control because a fuzzy controller can be structured to incorporate heuristic knowledge in specific regions of the state-space in addition to its close association with some of the proprioceptive sensory systems. In general, however, the state feedback controller did perform
This result has been proven rigorously for a broad class of linear controllers (time invariant, time varying, lumped parameter finite-dimensional) in [5, 9] although a simple example will be given for illustrative purposes. 2296
Consider a single-input single-output system. Let the linear control law be given as such that k is a scalar gain, z is a linear function of the plant state (e.g. ), and c is an offset. A fuzzy controller
Note that (3) is general in the sense that it represents the defuzzified output for a single-input single-output fuzzy controller for the case where the input vector lies in the space mapped by the two fuzzy sets and where
which is equivalent to this linear control law can be constructed in the following manner:
such that
notice that the shape of the input membership functions has not yet been specifically incorporated into the procedure. Now, to demonstrate that the linear control law is encompassed in the fuzzy controller output, equations (1) and (2) are substituted into (3). This results in
Input Membership Functions: The contributing portion of the input space will be spanned by the two fuzzy sets and
which are constructed such that and their placement in the universe
of discourse is assumed to be
. Also
where
. The membership functions will be triangular in shape to provide a linear variation as the input moves between them and the membership grade will be computed as
which is precisely equal to the linear control law. The main benefit of a fuzzy controller that is equivalent to a linear controller is that the initial design can be performed relatively quickly and any local stability results from the linear controller can be retained. Modifications to the fuzzy controller can then be made from a known starting point.
(1) Fuzzy Associative Memory: The FAM will be constructed so that all possible contributing input combinations will be addressed. However, for this simple case there will only be two fuzzy associations: If z is If z is
The final feedback configuration is identical to that shown in Figure 5 except that the state feedback controller K is replaced with an equivalent fuzzy controller. The linear-fuzzy controller was subsequently modified so that the effective gain was slightly increased away from the origin.
then u is then u is
Output Membership Functions: One output set is utilised for each rule in the FAM so that the number of output sets is equal to 2 in this case. The output membership functions will be fuzzy singletons located at and for sets and respectively. This leads to:
Application of this method to the state feedback postural HAT controller resulted in improved performance for slow, normal and fast walking speeds although slightly different modifications of the fuzzy controller were required for each speed (See Figure 8 for normal walking). These speed sensitive modifications suggest that an additional input is required that is representative of the average walking speed. Although multiple subjects were not studied, it is expected that the general trends would be the same with different subjects while the actual numerical values would likely differ. It should be noted, however, the improved performance generally required a more aggressive torque profile than the linear state feedback controller with the feedforward component. This effect is partially the result of the feedback and feedforward controllers operating completely independently. Thus, it is expected that this effect could be alleviated in part by incorporating a fuzzy comparator that effectively regulates the combined output. However, utilizing both the feedforward balance controller and the fuzzy feedback posture controller produced results that were within the bounds of those found experimentally. Thus, it can be
(2) Defuzzification: Using the centroidal defuzzification method, the FAM is incorporated into the summation which results in:
Recall from (1) that
which leads to
(3)
2297
Subsequent phases should also incorporate balance and posture in the medial-lateral plane where the control objective is to keep the centre-of-mass of the body within the medial (inside) borders of the feet. Other future considerations will include integrating the lower limbs into the model with the corresponding muscle groups at each of the joints. When this bipedal system can be shown to act similarly to a human during gait, then the long-term objective to aid the disabled will be within sight. HAT displacement References [1] K. Barin, “Evaluation of a Generalized Model of Human Postural Dynamics and Control in the Sagittal Plane”. Biological Cybernetics, Vol. 61, 1989, pp.37-50 [2] S. Kitamura, Y. Kurematsu, M. Iwata, “Motion Generation of a Biped Locomotive Robot using an Inverted Pendulum Model and Neural Networks”, IEEE Proc. Conf. On Decision and Control, 1990, pp. 33083312
Applied hip torque
[3] J. Furusho, A. Sano, “Development of Biped Robot”, Adaptability of Human Gait, A.E. Patla (Ed.), 1991, pp. 227-303
Figure 8: HAT angle and hip torque during normal walking: Fuzzy feedback-feedforward control (solid), linear feedback-feedforward (dashed)
[4] A.C. Guyton, Textbook of Medical Physiology (6th Ed.) W.B. Saunders Company, 1981
concluded the fuzzy feedback controller with the feedforward controller are able to model the CNS HAT controller within the assumptions of this paper.
[5] E. Kubica, Feedforward and Deterministic Fuzzy Control of Balance and Posture During Human Gait, Ph.D Thesis, Dept of Elec. Eng/Dept of Kinesiology, University of Waterloo, Waterloo, ON, Canada, N2L 3G1
7. Conclusions and Future Research The research documented in this paper is the first phase in the development of a biomechanical control system that can emulate the strategies employed by the CNS to control the HAT and lower limbs during gait. The present research identified a net motor control strategy for maintaining balance and posture of the HAT during various walking speeds. Key importance should be placed on the discovery of the ankle “jerk” as an appropriate feedforward component. As well, the deterministic fuzzy control allowed the basic characteristics of the linear controller to be retained, while modifications in specific regions of the state-space improved performance.
[6] B. Bavarian, B. Wyman, H. Hemami, “Control of the Constrained Planar Simple Inverted Pendulum”, Journal of Control, Vol. 37, no. 4, 1983, pp. 741-753 [7] G.K. Ruder, Whole Body Balance During Normal and Perturbed Walking in the Sagittal Plane, Masters Thesis, University of Waterloo, ON, Canada, 1989 [8] A. Thorstensson, J. Nilsson, H. Carlson, M.R. Zomlefer, “Trunk Movements in Human Locomotion”, Acta Physiol Scand. Vol 121, 1984, pp. 9-22 [9] E. Kubica, “Method of Constructing and Designing Fuzzy Controllers”. United States Patent Number 5,796,919, Aug. 18, 1998
Future work will involve decoupling the net control strategy into constituent components to include flexor and extensor muscle groups for both the left and right limbs. Controllers for the left and right sides would initially be copies of one another although a limb coordination methodology would be required. 2298
Feedforward and Deterministic Fuzzy Control of Balance and Posture During Human Gait Eric Kubica Dept. of Systems Design Engineering University of Waterloo, Waterloo, ON N2L 3G1
David Wang Dept. of Electrical and Computer Engineering University of Waterloo, Waterloo, ON N2L 3G1
Abstract-The primary objective of this research is to model the biomechanical control system employed by the Central Nervous System (CNS) to maintain posture and balance of the HAT (Head-Arms-Torso) during gait. More specifically, the intent is to stabilize a model of the upper body (HAT) so that the HAT response is similar to that found experimentally in human subjects during gait. This will be accomplished by using appropriate physiological parameters as feedback to generate realistic control signals at the hip musculature. The modelling includes the HAT musculoskeletal characteristics as well as pure neural time delays. The CNS control system is modelled by a linear state feedback controller as well as a hybrid fuzzy controller that has been adapted from a linear quadratic regulator. In addition, a newly derived feedforward component is demonstrated. It is expected that a better understanding of the human gait process will prove valuable in designing assistive devices and bipedal robots.
kinematics and dynamics) of the HAT to the CNS control during gait. This work offers insight for several areas. It provides information for developing a systematic approach to model human locomotion. This could lead to improved robotic aids for the physically handicapped by using more physiologically-based prosthetic devices. As well, any progress to understanding how the CNS controls human locomotion would be of interest for those trying to create more human-like robots. 2. Experimentally Derived Performance Criteria Experimental data were analyzed from several walking trials of one subject at three different speeds (slow, normal and fast) to establish the performance characteristics of the modelled system. Linear accelerations that occur at the hip or base of the HAT during gait disturb the equilibrium of this segment. Thus, these measured accelerations are used as disturbance inputs to the HAT model in order to produce realistic simulations of walking.
1. Introduction Several different approaches can be found in the literature to quantify the ability of the CNS to maintain balance and posture. In these works, a frequently observed similarity is the implementation of some form of linear state feedback control. However, despite that it is well known that significant pure time delays are present in the CNS, these effects are typically neglected [1]. Alternate approaches have also been investigated. For example, an intelligent control approach using neural nets was presented in [2] while a robotic biped was constructed in [3] that was able to perform relatively smooth gait; however, both neglected neural delays and therefore, are not good models of the CNS control system. Often, results are obtained which appear to closely match human performance but close scrutiny often shows that the control efforts are far beyond physiological capabilities [6]. The objective of the present research is to mimic the response (both 0-7803-6475-9/01/$10.00© 2001 IEEE
David Winter Dept. of Kinesiology University of Waterloo, Waterloo, ON N2L 3G1
Data were measured by attaching 28 infrared LEDs to a human subject and recording the human gait using an OPTOtrack 3020 3-D motion measurement system. Ground reaction forces were measured using two force plates. It was found that the average angle of inclination (posture) of the HAT segment only varied about 0.5 degrees for any of the walking speeds and was roughly equal to 2 degrees. However, the amplitude of the HAT oscillations did vary with speed such that during slow, normal and fast walking the HAT angle varied a total of approximately 3.5 degrees, 2.75 degrees and, 1.5 degrees respectively (See Figure 1). These bounds were later used as guides to determine whether or not the proposed control strategy performed adequately. The reduction in the oscillation amplitude as walking speed increased was 2293
during heel contact [5] (See Figure 2 for a normal gait example). Although only one subject was used in this study, work by [7] showed that multiple walking trials from different subjects produced nearly identical perturbations and corresponding corrections throughout the gait cycle. 3. System Modelling A model of the upper body has been developed that incorporates characteristics of the actuating hip musculature, the afferent and efferent neural pathways and the inertial properties of the HAT segment itself. The HAT segment was modelled as a single rigid inverted pendulum in the sagittal plane (See Figure 4). This is a commonly accepted model [8] and has been confirmed by the experimental data collection. The dynamics of interest are the rotary dynamics involving in Figure 3. The horizontal and vertical accelerations of point H caused by the forces and were modelled as disturbances to the rotary dynamics and are taken from experimental data sets [2]. This allowed the HAT model to be exposed to the same perturbations that
HAT sagittal displacement for two slow trials
HAT sagittal displacement for two normal trials
HAT sagittal displacement for two fast trials
Figure 1: Measured HAT displacements Figure 3: Inverted pendulum HAT model occur during actual walking. The resulting inverted pendulum dynamics were modelled as:
The hip musculature was modelled as a second-order critically damped torque motor with a cut-off frequency of 4Hz. Although this type of EMG-driven model does not embody force-length or force-velocity characteristics of muscle, it has been shown that the operating range of the hip musculature during walking is small enough so that only a 10-15% error may result by neglecting these nonlinear effects [5]. Thus, the hip torque can be represented by the following second order equation:
Figure 2: Measured hip torques: left(--), right(-.), left+right(solid) unexpected although it is speculated that higher levels of co-contraction may occur at fast speeds that would increase the effective joint stiffness. Estimates of the net hip torque were also produced that fell within normally measured bounds despite large anomalies that occurred 2294
Assuming that the twitch response of the muscle can be approximated by a critically damped second order system, then the parameters of the muscle model can be determined simply by knowing the twitch time which is equal to the inverse of the low-pass filter cut-off frequency.
delays and the relatively slow muscle responses that are present in the system. However, the CNS is able to control the HAT with relatively low amplitude oscillations under the same conditions, and therefore, the CNS must be able to anticipate the onset of the disturbances that occur during gait and initiate corrective responses beforehand. This is a proactive feedforward component that must be capable of generating a motor signal that is proportional to the magnitude of the pending disturbance. In addition, it must be sufficiently in advance of the disturbance so that the corrective action can be implemented as the disturbance occurs. The vestibular system (an inner ear organ) is often credited with this function although it has been shown with actual data that the accelerations measured by the vestibular system lagged the required motor signal and therefore cannot be responsible for this type of proactive control [5].
The final component that was included in the neuromusculoskeletal model of the HAT was the pure time delay that occurs in the efferent and afferent pathways along with the corresponding computational time delays. These delays have been lumped into one latency that was estimated at 62ms assuming that the dominant HAT angle sensors were in the region of the pelvis [5]. 4. Linear Optimal State Feedback Control An initial controller design utilized full state feedback from models of the hip musculature and HAT segment dynamics and was based on linear quadratic optimal control. The basic structure of the control loop is shown in Figure 4. Note that is the gain needed so that as . This controller performed well for postural adjustments (step inputs) although this same controller was unable to regulate the HAT segment during walking as measured hip accelerations were applied to the HAT model. The only walking situation that the linear quadratic regulator could tolerate occurred when the pure time delay was cut in half to 31ms. The performance of the HAT controller with the shorter time delay was stable although the HAT still oscillated more than 7 degrees during normal walking which is more than three times that measured experimentally (this is shown as the dashed line in Figure 6). Thus, these results suggest that linear state feedback control is an unlikely candidate for controlling HAT balance.
It was then surmised that since the most significant disturbances occurred close to the push-off and heel contact phases of the gait cycle, the motion at the ankle joint may be indicative of the perturbations at the hip. A subsequent analysis revealed that, with the foot flat on the ground, the disturbance moment at the hip was approximately proportional to the angular jerk (the third derivative of angular position) at the ankle. This result was confirmed with the experimental data and it was found that a combination of the linear feedback controller and a feedforward signal proportional to the negative of the angle jerk significantly reduced the oscillations that the HAT experienced without the feedforward component. The control loop is shown in Figure 5 where B represents the feedforward ankle jerk component.
Figure 5: Feedback and feedforward control loop Note that the inverse muscle model is now required to remove the effects of the feedforward term. The resulting performance is shown in Figure 6 for normal walking where the solid line represents the combined feedforward and feedback control while the dashed line represents the response of the feedback controller alone.
Figure 4: Linear state feedback control loop
5. Feedforward Balance Control The relatively poor performance of the linear controller during gait appeared to indicate that a purely reactive control strategy could not cope adequately with the time
In addition to improving the performance of the control 2295
adequately in the neighbourhood of the origin and it would be useful to retain these characteristics. Therefore, if a fuzzy controller could be constructed with equivalent performance to the state feedback controller, an innovative advantage could be realized in the initial design of the fuzzy controller. It was found that this requirement could be fulfilled when a fuzzy controller is structured with the following general conditions: 1) The input membership functions are constructed such that the sum of the membership grades for a particular input will be unity. In other words, if the membership functions are specified as , where i
Figure 6: Sagittal HAT angle, normal speed: Feedforward control (solid), Linear feedback control (dashed) system, the feedforward component also reduced the hip torque requirements significantly. It is speculated that the feedforward signal is generated at a higher level of the CNS such that the base volitive motor commands for the entire lower limb are initiated together. Despite this improvement in the performance however, the measured CNS regulation of the HAT operates within a tighter movement envelope than that shown in Figure 6. This leads to the conclusion that the linear feedback controller should be improved upon.
designates the input, s denotes the particular fuzzy set, is the number of fuzzy sets spanning input i with n inputs, this condition can be written as:
In addition, the membership functions must vary linearly between one another. Provided that only normal fuzzy sets are used (A is a normal fuzzy set if
6. Deterministic Fuzzy Control
), these conditions can only be satisfied
Biological evidence has been identified which illustrates that some proprioceptive sensors function with an uncanny similarity to fuzzy systems. In [4], the data from several different nerve fibres from a knee joint receptor of a cat was shown as the knee was moved through 140 degrees of flexion. The output of the joint receptors bear a striking resemblance to fuzzy sets and is shown in Figure 7. Thus, it is conjectured that a fuzzy controller may be appropriate for use as a model of the control process of the CNS.
with triangular membership functions which overlap at a membership grade of 0.5. 2) The fuzzy associative memory (FAM) or rule-base is derived so that it is full or complete. This means that each combinatorially possible fuzzy association is considered. Hence, the total number of rules (M) in the FAM is equal to the product of the number of fuzzy sets spanning each of the n inputs i.e.
.
3) The output membership functions are chosen as symmetrical unimodal functions all of equal area with centres at locations where the input rules are satisfied exactly. In other words, the consequence of each part of the FAM is assigned to an output set with its centroid positioned in the output space so that it provides the same defuzzified output as the linear control law with the same inputs [5]. 4) The inference engine must utilize product inferencing and centre-of-gravity defuzzificiation must be used.
Figure 7: Joint receptor output Fuzzy control was chosen to supersede state feedback control because a fuzzy controller can be structured to incorporate heuristic knowledge in specific regions of the state-space in addition to its close association with some of the proprioceptive sensory systems. In general, however, the state feedback controller did perform
This result has been proven rigorously for a broad class of linear controllers (time invariant, time varying, lumped parameter finite-dimensional) in [5, 9] although a simple example will be given for illustrative purposes. 2296
Consider a single-input single-output system. Let the linear control law be given as such that k is a scalar gain, z is a linear function of the plant state (e.g. ), and c is an offset. A fuzzy controller
Note that (3) is general in the sense that it represents the defuzzified output for a single-input single-output fuzzy controller for the case where the input vector lies in the space mapped by the two fuzzy sets and where
which is equivalent to this linear control law can be constructed in the following manner:
such that
notice that the shape of the input membership functions has not yet been specifically incorporated into the procedure. Now, to demonstrate that the linear control law is encompassed in the fuzzy controller output, equations (1) and (2) are substituted into (3). This results in
Input Membership Functions: The contributing portion of the input space will be spanned by the two fuzzy sets and
which are constructed such that and their placement in the universe
of discourse is assumed to be
. Also
where
. The membership functions will be triangular in shape to provide a linear variation as the input moves between them and the membership grade will be computed as
which is precisely equal to the linear control law. The main benefit of a fuzzy controller that is equivalent to a linear controller is that the initial design can be performed relatively quickly and any local stability results from the linear controller can be retained. Modifications to the fuzzy controller can then be made from a known starting point.
(1) Fuzzy Associative Memory: The FAM will be constructed so that all possible contributing input combinations will be addressed. However, for this simple case there will only be two fuzzy associations: If z is If z is
The final feedback configuration is identical to that shown in Figure 5 except that the state feedback controller K is replaced with an equivalent fuzzy controller. The linear-fuzzy controller was subsequently modified so that the effective gain was slightly increased away from the origin.
then u is then u is
Output Membership Functions: One output set is utilised for each rule in the FAM so that the number of output sets is equal to 2 in this case. The output membership functions will be fuzzy singletons located at and for sets and respectively. This leads to:
Application of this method to the state feedback postural HAT controller resulted in improved performance for slow, normal and fast walking speeds although slightly different modifications of the fuzzy controller were required for each speed (See Figure 8 for normal walking). These speed sensitive modifications suggest that an additional input is required that is representative of the average walking speed. Although multiple subjects were not studied, it is expected that the general trends would be the same with different subjects while the actual numerical values would likely differ. It should be noted, however, the improved performance generally required a more aggressive torque profile than the linear state feedback controller with the feedforward component. This effect is partially the result of the feedback and feedforward controllers operating completely independently. Thus, it is expected that this effect could be alleviated in part by incorporating a fuzzy comparator that effectively regulates the combined output. However, utilizing both the feedforward balance controller and the fuzzy feedback posture controller produced results that were within the bounds of those found experimentally. Thus, it can be
(2) Defuzzification: Using the centroidal defuzzification method, the FAM is incorporated into the summation which results in:
Recall from (1) that
which leads to
(3)
2297
Subsequent phases should also incorporate balance and posture in the medial-lateral plane where the control objective is to keep the centre-of-mass of the body within the medial (inside) borders of the feet. Other future considerations will include integrating the lower limbs into the model with the corresponding muscle groups at each of the joints. When this bipedal system can be shown to act similarly to a human during gait, then the long-term objective to aid the disabled will be within sight. HAT displacement References [1] K. Barin, “Evaluation of a Generalized Model of Human Postural Dynamics and Control in the Sagittal Plane”. Biological Cybernetics, Vol. 61, 1989, pp.37-50 [2] S. Kitamura, Y. Kurematsu, M. Iwata, “Motion Generation of a Biped Locomotive Robot using an Inverted Pendulum Model and Neural Networks”, IEEE Proc. Conf. On Decision and Control, 1990, pp. 33083312
Applied hip torque
[3] J. Furusho, A. Sano, “Development of Biped Robot”, Adaptability of Human Gait, A.E. Patla (Ed.), 1991, pp. 227-303
Figure 8: HAT angle and hip torque during normal walking: Fuzzy feedback-feedforward control (solid), linear feedback-feedforward (dashed)
[4] A.C. Guyton, Textbook of Medical Physiology (6th Ed.) W.B. Saunders Company, 1981
concluded the fuzzy feedback controller with the feedforward controller are able to model the CNS HAT controller within the assumptions of this paper.
[5] E. Kubica, Feedforward and Deterministic Fuzzy Control of Balance and Posture During Human Gait, Ph.D Thesis, Dept of Elec. Eng/Dept of Kinesiology, University of Waterloo, Waterloo, ON, Canada, N2L 3G1
7. Conclusions and Future Research The research documented in this paper is the first phase in the development of a biomechanical control system that can emulate the strategies employed by the CNS to control the HAT and lower limbs during gait. The present research identified a net motor control strategy for maintaining balance and posture of the HAT during various walking speeds. Key importance should be placed on the discovery of the ankle “jerk” as an appropriate feedforward component. As well, the deterministic fuzzy control allowed the basic characteristics of the linear controller to be retained, while modifications in specific regions of the state-space improved performance.
[6] B. Bavarian, B. Wyman, H. Hemami, “Control of the Constrained Planar Simple Inverted Pendulum”, Journal of Control, Vol. 37, no. 4, 1983, pp. 741-753 [7] G.K. Ruder, Whole Body Balance During Normal and Perturbed Walking in the Sagittal Plane, Masters Thesis, University of Waterloo, ON, Canada, 1989 [8] A. Thorstensson, J. Nilsson, H. Carlson, M.R. Zomlefer, “Trunk Movements in Human Locomotion”, Acta Physiol Scand. Vol 121, 1984, pp. 9-22 [9] E. Kubica, “Method of Constructing and Designing Fuzzy Controllers”. United States Patent Number 5,796,919, Aug. 18, 1998
Future work will involve decoupling the net control strategy into constituent components to include flexor and extensor muscle groups for both the left and right limbs. Controllers for the left and right sides would initially be copies of one another although a limb coordination methodology would be required. 2298
