Int J of Robotics and Autonomous Systems, Elsevier publisher (accepted)

Development of Adaptive Modular Active Leg (AMAL) Using Bipedal Robotics Technology G C Nandi, A J Ijspeert*, P Chakraborty, Anirban Nandi Indian Institute of Information Technology, Allahabad-211 012 INDIA [email protected] *School of Computer and Communication Sciences, EPFL Station 14 CH 1015 Lausanne, Switzerland

Abstract The objective of the work presented here is to develop a low cost active above knee prosthetic device exploiting bipedal robotics technology which will work utilizing the available biological motor control circuit properly integrated with a Central Pattern Generator (CPG) based control scheme. The approach is completely different from the existing Active Prosthetic devices, designed primarily as standalone systems utilizing multiple sensors and embedded rigid control schemes. In this research, first we designed a fuzzy logic based methodology for offering suitable gait pattern for an amputee, followed by formulating a suitable algorithm for designing a CPG, based on Rayleigh’s oscillator. An indigenous probe, Humanoid Gait Oscillator Detector (HGOD) has been designed for capturing gait patterns from various individuals of different height, weight and age. These data are used to design a Fuzzy inference system which generates most suitable gait pattern for an amputee. The output of the Fuzzy inference system is used for designing a CPG best suitable for the amputee. We then developed a CPG based control scheme for calculating the damping profile in real time for maneuvering a prosthetic device called AMAL (Adaptive Modular Active Leg). Also a number of simulation results are presented which show the stable behavior of knee and hip angles and determine the stable limit cycles of the network. Key words: Bipedal robots, Central Pattern Generator, Fuzzy logic, Adaptive Modular Active Leg, Rayleigh’s Oscillator.

m, k, η, µ Ө0 Өi

T0 ∂ t = T0 8

T

1.

Description of symbols Mass, spring stiffness, two damping coefficients connected with linear and nonlinear velocity components respectively, in a non linear Mass-Spring-Damper model. Initial Knee angle in degrees Knee angle in degrees, at any instant of time, t measured in second. The period of a full gait cycle, sec. Delay time holding a particular damping value fixed over a gait cycle in sec. The period of the hip cycle during a full gait in sec.

ϕ

The phase difference between the hip and knee oscillation indegrees.

ts

The initial contact time of the gait from the time when the period in sec.

θi

Measured hip angle at time i ,in sec.

ti

Time when hip angle

T

is computed,

t

= t pj

θi

is measured in degree.

θi

t pj

The times ti

Vi

The voltage signal output from the potentiometer representing the hip angle

V0

Threshold voltage above the minimum voltage measured by the potentiometer. This voltage is fixed after checking the minimum voltage.

when

is maximum.

θi .

Introduction

A bipedal locomotion is one of the most complicated motions for a human body or similarly structured humanoid robots due to its complicated physics. In fact it is an inverted pendulum like structure, controlling of which, is a benchmark problem in control engineering. That is why although we have perhaps the most sophisticated brain in the animal kingdom, consisting of some 10 billions neurons and 60 trillion interconnections among them [1], a human kid needs some 8-10 months for learning how to walk while a lower level quadruped or hexapod animal whose body structure is inherently balanced, takes few hours after birth for walking and even running. It is also observed that the gait patterns vary between individuals and it is actively controlled by a control centre known as Central Pattern Generator (CPG). Over the years the neurobiology of CPGs, neurobiological models of CPGs, CPGs in robotics and design methodologies of CPGs has widely been studied. A number of

significant researches have been jotted down together in a review paper [2]. Also interesting collections of articles on biologically inspired robot locomotion can be found in [3,4,5]. In fact many routine activities in the biosphere (for our purpose, human life) like heart beat, chewing, breathing, menstrual cycle and many others show rhythmic behavior. Even in the inanimate sphere many rhythms were observed which has been discussed in [6]. A growing number of researchers are now using neural oscillators to model CPGs for controlling nonlinear systems specially robots. Oscillators are used in modeling all these cases for the very reason that they can entrain and adapt to the dynamics of the system, giving robust behavior in the face of changing system parameters or perturbations. The models are based on nonlinear dynamics using coupled oscillators. Here, an oscillator represents the activity of a complete oscillatory centre rather than a single neuron. Extensive use of such coupled oscillators includes phase oscillators [7] and Matsuoka oscillators [8]. Most of the oscillators have a fixed waveform for a given frequency. Taking these oscillators as building blocks, a number of CPG models have been developed and they are being used to control the locomotion of robots. The application domains range from biped locomotion and locomotion of multi-segmented creatures [6], to robot arm control [9]. The types of CPG models implemented in robots include connectionist models , vector maps and systems of coupled oscillators. In some rare cases spiking neural network model have been used [10]. Most of these implementations involve a sets of differential equations that are numerically integrated on a processor. In some applications CPGs have directly been realized in hardware, i.e on a chip or with analog electronics[11,12]. CPGs have also been used for controlling swimming robots and controlling walking of quadruped robots[2]. There are several reasons why the CPG based controls are becoming more popular than existing methods based on finite-state machines, sine-generators or prerecorded reference trajectories : • CPG models exhibit limit cycle behaviors to produce stable rhythmic patterns. When this is the case the system rapidly returns to its normal rhythmic behavior against perturbations. • CPGs are well suited for distributed implementation which can effectively model the morphological computations exhibited by biological creatures including human beings. • CPG models typically have fewer control parameters and therefore, reduce the dimensionality of the control problem. • CPGs are ideally suited for integrating sensory feed back signals which enables mutual entrainment between CPGs and physical structure. For more details one can see [2]. However, as pointed out in [2], there is a common drawback/challenge of CPG based systems in terms of tuning the oscillators. In spite of many theoretical work, it is extremely difficult to evolve a system which can give rise to effective determinations of oscillator parameters [13,14,15,16,17]. Somehow to mitigate these problem a purely sensor driven controller has been proposed in [18]. They implemented policy gradient reinforcement learning algorithm to tune the parameters of the

sensor driven controller in real time during walking. However, they experimented only with planar biped robot and explicit analysis of the attraction domain of its stable gaits and its relationship to the mechanical and controller parameters is missing. As discussed in [19, 20, 21], simple signals are usually sufficient to induce synchronization activity in CPGs provided the oscillator parameters are within phase-locking region. This is mainly the case because oscillators lack plasticity, they have fixed intrinsic frequencies and cannot dynamically adapt their parameters. Some solutions towards this problem has been presented in [16, 22] by using Genetic Algorithm for parameter tuning. Nevertheless, getting suitable fitness function for generalized application remains a challenge. However, while these analyses present interesting background information, our specific development need different approach for designing Prosthetic Leg as an embedded system. The reasons are manifold and here we are mentioning few of them. There is not yet a well established design method for CPG. As discussed earlier people have explored many approaches including hand coding, design exploiting the theory of non linear dynamics, and learning /optimization algorithms based on evolutionary computing. One has to confess the fact that as on date there does not exist a sound methodology for learning arbitrary limit cycles in dynamical systems. In learning and optimization algorithms, based on evolutionary computing, the main problem as mentioned earlier, is the proper codification and finding fitness function for the problem. Therefore, here we have taken a different approach- we are interested in utilizing the coupling and residual biological motor control circuits available in the higher joints (in case of above knee prosthesis it is actually hip joints) of an amputee, taping information from available biological motor controlled circuits, which exhibit synchronization and make CPG tuning parameters well bounded. This makes the CPG design process much simpler. Here we intend to use lesser and simpler sensory information in real time and extract maximum from CPG based control circuit [22,23,24]. The presentation has been arranged in the following way: First, in the section -2, we have presented the development HGOD for capturing gait data and discussed a fuzzy logic based classifier for customizing biological gait patterns, followed by a CPG design algorithm in section-3, using the gait profile obtained from section 2. In section 4 we have discussed how to use CPG generated trajectories as reference knee profile for subsequent control of AMAL-like prosthesis.

2.1 Developing HGOD, a probe for capturing Human gait For our research we needed to generate the gait data set for a large number of people. We designed our own probe, which would be simple, rugged and reliable and capable of capturing the time dependent oscillation angles of different joints of the human body. The present device is strap-on equipment, comprising rigid links interconnected by revolute joints, where each joint angle is measured by rotational sensors (single turn encoders). We name this strap-on suit as the ‘Human Gait Oscillation Detector’ (HGOD).

Potentiometer

θ

Voltage Calibrated to Angle θ

θ t

Figure2.1-1. Positioning of the sensor to measure the right knee oscillation. Stages of development of HGOD: HGOD (version 1.0), is a single joint oscillation detector. It comprised of two rigid aluminum links connected by a 100kΩ potentiometer on the pivoting point. Fig.2.1-1 illustrates HGOD-1.0, which measures only the knee angle oscillation. The two rigid aluminum links are fastened to the thigh and calf of the subject whose gait oscillation is being studied. As the knee bends, the angle between the rigid aluminum links changes, rotating the potentiometer. This changes the potentiometer resistance and is reflected in the output voltage. The video [29] shows how accurately and faithfully HGOD can capture data in real time. The sensor has been calibrated by measuring voltages for different rotation angles of the potentiometer. Fig. 2.1-2 shows the calibration plot. The calibration data points are shown with “+” sign. We see a linear relationship between angle and voltage. From the least square fit we have angle θ as a function of the voltage V. (2.1-1) θ = −65.71 V + 122.9

Figure-2.1-2 An angle vs. voltage calibration curve for the Sensor (Potentiometer). Since the relation is linear, a simple equation is required to convert the measured voltages into angles. Before the subject starts walking, a zero correction is needed and it has been done, by acquiring θ 0 for the initial position. This θ 0 is subtracted from θ . A single gait consists of two phases; The stance phase and the swing phase. The knee joint oscillates twice over one gait cycle. In the stance phase the amplitude of oscillation is less than the second oscillation, which is in the swing phase. In HGOD version 2.0, we ventured to capture many (8) joint motions simultaneously with 8 sensors. Here again the design is similar – made of rigid aluminum links connected by 100 kΩ potentiometers at appropriate locations of the suit. Figure 2.1-3 shows the HGOD2.0 suit strapped on to a subject.

a

b

c

d

Figure 2.1-3. The HGOD-2.0 Suit consisting of 8 potentiometer sensors strapped on a subject.

The circuit board provides a low pass RC filter to reduce the high frequency noises. The eight sensors are powered by 9V batteries in series, rectified to 12 V with 7812 rectifier IC. The data from the 8 sensors are transferred through a long fat cable to a remote machine. We have used dSPACE DS1104 controller board to acquire and store the HGOD data on the remote computer. dSPACE is a prototype which supports hardware DS1104 controller board and integrates with the control desk software to provide simulation platform for real time implementation. Real time applications run using user interface, dSPACE control desk which is linked to the SIMULINK package of MATLAB. Therefore the data acquisition software is built using this SIMULINK. Fig. 2.1-4 shows the Simulink model for data acquisition from the 8 potentiometer sensors of HGOD. Physically on the dSPACE connectors S1, S2, S3 and S4 are connected with a MUX ADC (DS1104MUX_ADC) while the connectors S5, S6, S7 and S8 are connected with an ADC (DS1104ADC_5 to DS1104ADC_8). Therefore in the SIMULINK model MUX ADC (DS1104MUX_ADC) is followed by a DMUX so that the eight signals S1 to S8 can be multiplexed simultaneously. Finally all the eight signals are multiplexed and the output sent to a Comma Separated Value (CSV) data file and a scope to view the data in real time.

S1 S2 S3 S4

S5 S6 S7

CSV Data File

S8

Figure 2.1-4 . A SIMULINK model for data acquisition from the 8 sensors (potentiometer) of HGOD using dSPACE. Due to hardware constraints, the signals S1 to S4 needed to be de-multiplexed so that all the eight signals S1 to S8 are simultaneously multiplexed. Fig. 2.1-5 shows the gait oscillations from the 8 joints of the body from HGOD. Figure has been arranged such that the left limb (shoulder, elbow, hip and knee) oscillations are on the left half of the figure while their right counter parts are on the right. The dotted line is a simple sine curve fit to the shoulder elbow and hip joint oscillations. (2.1-2) θ iL = θ 0Li sin(ωt + δ iL )

θ iR = θ 0Ri sin(ωt + δ iR )

(2.1-3)

Figure 2.1-5. Gait oscillations from 8 joints of the body. From the top (left and right): Left and right Shoulders, left and right elbows, left and right hips and left and right knee. As mentioned earlier, the knee joint exhibits a complex oscillation, therefore it is not fitted. Fitting is done to check consistency of the data. As expected, we notice that the periods of oscillation for the fitted joints are the same ( T = 2π ω = 1.57 sec). It is essential for the stability that all limbs should oscillate as multiples of the same frequency, while the shoulder and the hip oscillation should have the same frequency. We also notice as expected, that the phase difference between the corresponding left and right shoulders and hip joints are ∆δ i = δ iL − δ iL = π . (2.1-4) To check the correlation and coupling between the left and right of the body during a stable gait, and the consistency of the data, we plot the left with the right limb oscillations (Fig. 2.1-6). It has been observed that, the fitted sine functions in Fig. 2.1-5 describe a Lissajous curve as shown in Fig. 2.1-6 (shown with thick lines).

b) Elbows a) Shoulders

c) Hips

d) Knees

Figure 2.1-6. Correlations and coupling between the left and right limbs. In this figure, the fitted sine functions (in Fig. 2.1-5), describe a Lissajous curve, shown in thick lines. For a stable gait we expect a phase difference of π between the left and right hip joints. A good walking style, captured by HGOD has also shown, as expected, a phase difference of π between the left and right shoulder joints as well. The Lissajous curves are highly sensitive to the ratio of the amplitudes θ 0Ri θ 0Li . For a ratio of 1, the figure is an ellipse, with circles, θ 0Ri = θ 0Li , ∆δi = π 2. For a lines, ∆δ i = kπ , where k is an integer. In our case, we expected that the amplitudes of oscillations are equal between the right and the left limbs. The straight thick lines in Fig. 2.1-6 a, and Fig. 2.1-6 c are due to the phase difference ∆δ i = π . For the elbow joint (Fig. 2.1-6 b), the thick line is an ellipse. We see that the data follow considerably well the Lissajous function. It is to be noted that if gait pattern deviate considerably from the Lissajous function, strains are bound to build up between the limbs. Although the techniques we are using here to determine the correctness of our data acquisition from HGOD provides some kind of filtering, it can also be used for diagnosis and determining of illnesses due to incorrect gait pattern. However, here we are only interested in using the techniques for checking the correctness of the captured, filtered and calibrated gait data. These data are subsequently used as input data for the fuzzy classifier, the design of which is described in the next section.

special

cases

such

as

circle

and

lines.

For

2.2

Designing a Fuzzy Logic based classifier for gait synthesis

First question comes why we design Fuzzy classifier ? It is our common experience that the gait pattern varies in males and females, also it varies with body weight (heavy or light), State of mind (relaxed or nervous; happy or sad etc.). Rigorous research results in this area can be obtained in [27]. We propose a fuzzy classifier because there is enough vagueness between the gait patterns and the attributes of the creator of the gait pattern. That is the reason a simple distance measure will never work accurately and fuzzy classifier will work better. Our objective is to maintain a repository of captured gait cycles for various modes of locomotion like simple walking, brisk walking, running, stair climbing etc. from normal and healthy persons. Using Fuzzy logic based classifier a suitable gait pattern could be selected for an amputee who is incapable of producing a normal gait pattern any more. This will be used for designing a suitable CPG for the amputee. We have designed such a fuzzy classifier which at present has 8 classes (23 -3 features each having 2 states). This has been used to predict the closest gait pattern for a person from his height, weight and age. That gait pattern would be used as an input to design a CPG for controlling the damping profile of the amputee.

2.3 Development of Fuzzy logic based classifier for searching most suitable gait for an amputee We suggest that persons involved in potential risk situations should donate their various gait patterns to a central repository which could be maintained by the employer, especially in defense or risk involving professions or even they could be stored on a magnetic tape on the Identity card of an individual, using the probe similar to which has been discussed earlier. For other persons a fuzzy logic based classifier could be developed as discussed here, to select the most suitable gait for an amputee. The classifier is based on logical assumption that a morphologically similar individual can adapt a similar gait pattern easily.

2.4 Fuzzy classifier details Fuzzy logic has extraordinary expression power when applied correctly. Fuzzy logic is not a logic which is fuzzy but logic that is used to address fuzziness of the environment [25,26]. Every individual’s gait patterns are different. So the classification problem possesses enough vagueness about the class to which an individual’s gait pattern belongs. It needs expert knowledge but even then, it becomes extremely difficult for an expert to classify when the repository contains huge patterns. For example for the present problem we have eight classes based on age, height and weight for capturing gait patterns. For the present classification problem, only these three attributes have been taken since they have the most significant influence in determining the shape of a gait pattern. Each individual contributed four different kinds of gait patterns –simple walking, brisk walking, walking

with long steps and running. At present we have captured several hundred gait patterns. Rather it would be good if we could capture the commonsense (morphologically similar individual would adapt similar kind of gait patterns.) of an expert in the framework of fuzzy logic based inferencing for handling the classification tasks, for providing the most suitable gait pattern for an amputee.

2.5

Mechanism based on which the classifier works

Fuzzy logic is a set of mathematical principles for knowledge representation based on degrees of membership. Unlike two valued Boolean logic, fuzzy logic is multi valued. It deals with degrees of membership and degrees of truth with the help of continuum of logical values between completely false and completely true. Instead of just black and white, it employs the spectrum of colours, accepting that things can be partly true and partly false at the same time. The classifier developed here deals with fuzzified inputs like age, height and weight with the appropriate class in which an individual belongs, as output. The three input attributes are represented by fuzzy sets and they are manipulated by fuzzy rules formulated by the common sense reasoning of an expert. Here all the inputs U=ui, i=1,2,3 are fuzzy sets A mapped in the universe of discourse X. defined by µA(x) called membership function of set A Where, µA(x) =1 if x is totally in A; µA(x)=0 if x is not in A; 0< µA(x)<1if x is partially in A. This set allows a continuum of possible choices. AND is the fuzzy operator we have used for manipulating the fuzzy sets. The out put is the class where an individual’s CPG pattern belongs. Here we have used 8 classes (we could take more features like state of mind etc. [27] which might make gait pattern selection more appropriate but definitely would make the classifier design more complex). Moreover, A human gait pattern mainly is a function of these three attributes. As mentioned in the text section 2.3, the classification is based on logical assumption that morphologically similar individual can adapt a similar gait pattern. One must note here that the Fuzzy classifier is only providing coarse gait pattern inputs to the CPG design algorithm as described in section 3.2.In the absence of a suitable Fuzzy classifier, the tuning of the system parameters of the non linear oscillator would be extremely difficult, since in such case one will have to rely on some heuristics for selecting initial values of the non linear system parameters which may not ensure convergence of the design process as described in section 3.2.

We have considered the following linguistic sets: age- [young, old]; height-[short, tall]; weight- [light, heavy] with sigmoidal membership functions for input variables and triangular membership function for output “class” variable. The following fuzzy rules have been formulated : 1. If the amputee is young and the height is short and weight is light then his/her gait class belongs to “gait_class-1” 2. If the amputee is young and the height is long and weight is light then his/her gait class belongs to “gait_class-2” 3. If the amputee is old and the height is short and weight is light then his/her gait class belongs to “gait_class-3” 4. If the amputee is old and the height is long and weight is light then his/her gait class belongs to “gait_class-4” Like this all the 8 rules have been written exhaustively and the captured gait patterns have been classified accordingly. The fuzzified output has been shown in fig 2.3-1 whereas the defuzzzified outputs have been shown in fig 2.3-2 and fig. 2.3-3. For designing fuzzy classifier we have used the Fuzzy Logic tool box of MATLAB.

Fig.2.3-1 Shows 8 fuzzified classes.

Fig. 2.3-2 Shows defuzzified outputs on the basis of age, height and weight

Fig. 2.3-3 Shows defuzzified output through surface plot The outputs show that the gait pattern of an individual having age of 35 years, height 161 cm and weight 65 kg could actually be selected from the intersection of the two classes 2 and 3.

3.

Designing CPG

3.1 Why CPGs? As mentioned earlier, Central Pattern Generators are neural networks capable of producing coordinated patterns of rhythmic activity without any rhythmic inputs from sensory feedback or from higher control centers. As detailed in [6] they exhibit various rhythmic behaviors in the animal kingdom- both in vertebrate and invertebrate.

At the beginning of the twentieth century, there were two views among the neurobiologists. One view was that the rhythms are the results of a series of reflexes where sensory feedback plays an important role in stimulating locomotors cycles in a distributed fashion. Whereas, the other view was that, rhythms were generated centrally and they do not require any input from peripheral sensory organs. There is now very clear evidence that rhythms are generated centrally without requiring sensory information [6,20]. It is observed particularly in human locomotion, when a person is drunk which means that the central pattern generator is temporarily under sedation, and his motor control circuit is inactive. In such a situation, the rhythmic gait cycle necessary for walking can not be produced, although all his peripheral sensory organs like various tactile sensors mounted on the feet etc. are perfectly alright. They alone cannot enable the person to walk stably. So we see that, definitely our walking is not solely governed by detailed sensory feedback. That is why designing a leg as a stand alone device, which will function with the help of a detail sensory feedback based rigid controller will not be effective. Here we propose and implement our innovative ideas to integrate biological motor control circuits with our prosthetic leg controller.

3.2 Design details of Central Pattern Generator (CPG): Keeping this in mind, we first try to design a central pattern generator, which would produce the stable gait pattern and take active part in integrating motor control circuit of the higher joints to the control of the artificial joint. However, suitable design of system parameters of a CPG is a complicated problem since it involves non-linear dynamics. The parameters of these systems are notoriously difficult to tune. A common way to connect oscillators to systems (prosthetic leg controllers) is to tightly couple them using the oscillator output to drive the system and closing the loop by applying an output of the system as the oscillator input. To accomplish this task we have first taken a customized gait pattern ( obtained as an output from the fuzzy classifier which would be the desired output of the oscillator) and used it for tuning the parameters of the oscillator. Let us now consider a set of humanoid joints as shown in fig. 3.2-1 which can be modeled as a network of neurons in the form of n-coupled Rayleigh oscillators. For simplicity we have considered 3-coupled oscillators.

Fig 3.2-1 Showing placement of oscillators at various joints We are particularly interested in Rayleigh oscillators since parameter tuning for the oscillators can be derived analytically and provide us a detailed insight about the effect of the performance of the oscillators with the variations of these parameters. The British mathematical physicist Lord Rayleigh formulated this oscillator way back in the 19th century in conjunction with models of musical instruments. This oscillator can be written in the form of the following equations:

m&x& + kx = ηx& − µx& 3 ( with non linear damping speed), For modeling humanoid joints, we have used Rayleigh's equation in the following form : n

θ&&i - ε i (1 - η iθ&i 2 )θ&i + Ω i 2 (θ i − θ 0 ) − ∑ ci , j (θ& i − θ& j ) = 0,

(3.2 - 1)

j=1

i = 1 ,2 ,3...n, and ε i ,η i ≥ 0, where θ i , j are the state variables and ε i ,η i , Ω i and ci , j are the system parameters of the oscillator to be designed. Intuitively, we have placed the oscillators at the shoulders, hips and knees since these joints have considerable influence to each other in creating walking rhythm. For more accurate analysis one can incorporate the ankle and other joints in the modeling as well. However, to keep the analysis simpler, we have designed a CPG with two knees & hips and validated the design with captured data on knees. A non linear oscillator which is a network of general oscillators having frequency ω can be synchronized with another oscillator having frequency nω. In case of human walking, the

observed data shows that some degrees of freedom (for example knee) have twice the frequency of the others (hip for example).

Fig 3.2-2 Showing diagram where we have considered three joints. Rewriting the equation incorporating the coupling terms we get n

m

j=1

k =1

θ&&i - ε i (1 - η iθ&i 2 )θ&i + Ω i 2 (θ i − θ 0 ) − ∑ ci , j (θ& i − θ& j ) − ∑ ci ,k {θ&k (θ k − θ k 0 )} = 0

(3.2 - 2)

For right knee : θ&&1 − ε 1 (1 − η1θ&1 2 )θ&1 + Ω1 2 (θ1 − θ10 ) − c1, 2 (θ&1 − θ&2 ) − c1,3 {θ&3 (θ 3 − θ 30 )} = 0

(3.2 - 3)

For left knee : θ&&2 − ε 2 (1 − η 2θ&2 2 )θ&2 + Ω 2 2 (θ 2 − θ 20 ) − C 2,1 (θ&2 − θ&1 ) − c 2,3 {θ&3 (θ 3 − θ 30 )} = 0

(3.2 - 4)

For hip joint : θ&&3 − ε 3 (1 − η 3θ&3 2 )θ&3 + Ω 3 2 (θ 3 − θ 30 ) − C 3,1θ&1 (θ1 − θ10 ) − c3, 2 {θ&2 (θ 2 − θ 20 )} = 0

(3.2 - 5)

One can solve the above equations analytically with the modeling of knee gait patterns in line with [28] to get the values of the following system parameters:

4c1, 2 ( A1 − A2 ) + 4 A1ε 1 + A3 c1,3 2

η1 =

12ω 2 A1 ε 1 3

Ω1 = 2ω ; 4c 2,1 ( A2 − A1 ) + A2 ε 2 + A3 c 2,3 2

η2 =

12ω 2 A2 ε 2 3

(3.2 - 6)

Ω 2 = 2ω

η3 =

4 3ω A3 2

2

Ω3 = ω Based on these analytical solutions we have developed an algorithm for tuning the system parameters for hip-knee combinations. STEPS OF ALGORITHM Step-1

Read the data files for knee -1 and knee-2 and hip coming form the fuzzy classifier

Step-2

Calculate the parameters ω i , Ai directly form the captured gait pattern or from the Fourier series model of the gait pattern data.

Step-3

Tuning the oscillator system parameters: Step 3.1

Since there is strong coupling between the two knees compared to hip (hypothesis) select initially large values for the coupling parameters c1,2= c2,1=say, 0.5 compared to hip c1.3= c3,1= c2,3= c3,2=0.001.

Step-3.2

Start with low ε i values (say 0.005).

Step -4

Calculate system parameters using equation 3.2-6

Step-5

Solve the set of equations 3.2-3, 3.2-4, 3.2-5 and compare the results with the captured gait patterns.

Step-6

If the results are matching, display the system parameters and finalize the oscillator design.

Otherwise go to step-3.2 and try other values of ε i , if result is not reached change first ε i values and then ci,j till the results match.

Step-7 STOP

3.3

Simulation Results & Analysis

We have used several captured gait patterns to design and test the CPG. The results are shown in fig 3.3-1 to fig 3.3-18 We have used the following initial values from the captured patterns: A1=-52.57, ө10= -36, 0< ω t ≤ π A1=-12.51,

ө10= -10,

π< ω t ≤ 2π

A2=12.51,

ө20= 18,

0< ω t ≤ π

A2=52.57,

ө20 =50,

π< ω t ≤ 2π

ω = 8.4 rad/sec. In fig 3.3-1 we have kept the captured gait pattern (i.e. angle vs time) for both right and left knees after calibrating the captured data which is in the form of voltage vs time. In fig 3.3-2 we have shown the results obtained from our designed CPG which in complete form (after attaching system parameters and coupling parameters through the design process as described in the algorithm) is represented by the set of non-linear differential equations 3.2-2 and 3.2-6. In fig 3.3-3 and fig 3.3-4 the phase plots obtained from the same set of differential equations (3.2-2) for left and right knees respectively show stable limit cycle behavior, confirming the correctness of the CPG design process. In fig 3.3-5 we have shown the variation of angular velocity for both the knees during swing and stance phase. The nature of the variation would provide us very useful information for formulating the damping control strategy which has subsequently been described in the next section. In the stick diagram shown in fig 3.3-6 we have embedded the motions of knee and hip joints on the simplest possible legs and joints represented by sticks and nodes. The motion pattern of the uppermost node which represents hip shows a perfect sinusoid, which matches with the reality and confirms the correctness of the CPG design process.

In figures 3.3-7 to 3.3-12 and 3.3-13 to 3.3-18 we have shown the results of the same simulation as described above, for different individuals, having different height, weight and age. In all the three sets of figures, we have compared the experimental gait pattern with the one derived from CPG using the following parameters: C1, 2=C2, 1=0.4; ε1 , ε 2 = 0.01; Ω1 , Ω 2 = 2ω with the initial values taken experimentally from the captured gait pattern.

First candidate: age-25yrs; height-175cm; weight-58 Kg; sex-male

Fig. 3.3-1 captured gait pattern

Fig. 3.3-2 CPG generated gait pattern

Fig. 3.3-3 CPG generated Knee trajectory (left) in the phase plane. (stable limit cycle)

Fig. 3.3-4 CPG generated Knee trajectory (right)in the phase plane (stable limit cycle)

Fig. 3.3-5 CPG generated angular velocity of knees

Fig 3.3-6 Stick diagram showing gait with a step length of 0.34m

Second candidate: age-20 yrs; height-170cm; weight-55Kg; sex-male

Fig. 3.3-7 captured gait pattern

Fig. 3.3-8 CPG generated gait pattern

Fig. 3.3-9 CPG generated Knee trajectory (left) in the phase plane. (stable limit cycle)

Fig. 3.3-10 CPG generated Knee trajectory (right)in the phase plane (stable limitcycle)

Fig. 3.3-11 CPG generated angular velocity of knees

Fig 3.3-12 Stick diagram showing gait with a step length of 0.34m

Third Candidate: age-30 yrs; height-155 cm; weight-55kg; sex-female.

Fig. 3.3-13 captured gait pattern

Fig. 3.3-14 CPG generated gait pattern

Fig. 3.3-15 CPG generated Knee trajectory (left) in the phase plane. (stable limit cycle)

Fig. 3.3-16 CPG generated Knee trajectory (right)in the phase plane (stable limitcycle)

Fig. 3.3-17 CPG generated angular velocity of knees

Fig 3.3-18 Stick diagram showing gait with a step length of 0.34m

The system parameters (other than the oscillator parameters) were taken afresh every time we ran the simulations. Some small variations between captured knee profiles and CPG generated knee profiles had been observed in Figures 3.3-2, 3.3-8 and in 3.3-14 due to the existence of some errors encountered during capturing the data. However, in all the cases the pattern of oscillation (amplitude and frequency) remained same and in all the time stable limit cycles were obtained. The results confirm the convergence of CPG design process and possibilities of using mutually coupled Rayleigh oscillators in the modeling of the CPG for humanoid walking. 4. Development of Control strategy for AMAL

Hip Sensor

Fig. 4.1 a. Schematic diagram of AMAL with foot spring. b. Photograph

The Adaptive Modular Active Leg (AMAL) is fitted with an MR (Magneto-Rheological) damper that controls the knee joint rigidity (ref fig. 4.1 a and b). The beauty with the MR damper is one can change the viscosity of the magneto-rheological fluid by changing the electrical signals. However, MR damper alone cannot make the leg fully active since it cannot supply energy to the system. For that the foot has been designed in such a way that it consumes energy during stance phase and liberate the same during swing phase. The liberation of energy takes place in a controlled way as the damping is controlled in the damper as described below. A micro-controller circuit has been designed to provide a variable analog current between 0 and 2Amps to the MR damper to vary the damping profile. When a current signal of 2Amps is applied, the damping is maximized and the knee joint is held rigidly. At 0Amp signal, the knee joint is totally relaxed and can straighten by itself due to hydraulic spring action of the damper. The variation of the current is produced by the variation of voltage produced by the micro-controller and DAC circuit. We will therefore identify the damping profile as the variation of this voltage signal over for a particular period of the gait cycle. A voltage variation on 0 to 5V produces the 0-2Amps current variations. We have divided the gait cycle into 8 sections (initial contact to loading response).

Fig 4.2: Classification of eight regions over a gait cycle, superposed over the actual data of the knee angle. Fig 4.2 shows this eight divisions superposed over the actual data of knee angle variation of a single gait cycle. To begin with we have kept each of the 8 regions of equal time interval of ∂t = T0 8 , where T0 is the period of a full gait cycle. Each of the 8 regions have been associated with a specific constant damping signal value between 0 to 5V shown in the lower half of Figure 4.2. These eight voltage values has been sequentially looped with delay time of ∂t . The plan of action for activation of AMAL has been through biological entrainment using the hip joint oscillation. The hip oscillation of the amputated leg of the patient has been captured using a simple potentiometer circuit, attached near the hip of the amputated leg. The assumptions that we make are: 1) The period T of the hip cycle is the same as that of the knee. 2) The phase difference ϕ between the hip and knee oscillation is constant. Though the first assumption is one of the fundamental requirement for the stability of the gait, the phase ϕ in the second assumption, may vary between slow walking to running. However, since the human body is extremely adaptive, the hip joint oscillation will re-

adjust itself, to compensate the difference in ϕ within few steps (in all our experiments it actually took 3-4 steps). We would like to keep these operations as simple as possible with minimal sensory feedback. The micro-controller circuit along with the MR damper controller is running from a single battery of 12V (~3amp) rating. The sequences of steps that are followed are: a) Passive Mode Operation To begin with , the knee joint of AMAL is held rigidly straight by applying a constant 5V signal. We will refer to this mode of walking as Passive mode of operation. b) Identifying the Initial Contact As the patient starts walking with the passive leg, the periodic oscillations of hip movement

is being registered for determining the initial contact time t s of the gait and the period T . Since the hip joint oscillation is much simpler with a single peak per oscillation, starting point

ts

could be determined from the maximum (peak) or the minimum of the oscillation.

The starting point of the knee oscillation is then calculated by

ts + ϕ .

c) Triggering Knee Movement The knee angle movement is triggered from the next period of the hip oscillation

T + ts + ϕ .

d) Training the Hip Oscillation with Ideal Gait The hip movement of the amputee is expected to be erratic and different from the natural

oscillation of the hip with period T0 for a normal human being with both legs and having the amputee’s stature. The hip joint movement of the amputee will be forced to acquaint itself to the most suitable period T0 . The most suitable period the process, as described in section 2 and 3.

T0 has

been determined by

e) Time for Damping Sequence

The

∂t = T0 8 , is

first used for sequentially looping the damping values.

f) Correction of the Period from the Natural Period After a sufficient delay, which will allow the hip joint of the amputated leg of the patient to

train itself to most suitable period

T0 ,

a feedback from the hip is being obtained to re-

adjust the period T0 . g) Measurement of Period T of the Hip Oscillation

The hip joint oscillation of a normal human being, unlike the knee oscillation is closed to a simple sinusoidal motion. Therefore the period of oscillation T could be determined from the gap between 2 maxima of consecutive oscillations (Fig. 4.3). Since the signal measuring the hip angle

θi

t = t pj when θi

times i

with time

ti , will be noisy, the calculation to determine the

is maximum, will not be straight forward.

A healthy human being during brisk walking has roughly a period of

T = 1sec. Therefore,

we can expect the hip joint oscillation frequency to be less than ∼2Hz. The hip angle

θi

is

converted to a voltage signal of Vi by the potentiometer (POT). This signal Vi is prefiltered for higher frequencies > 5Hz electronically and through software before computing

t pj .

Fig 4.3: The hip angle θ i is measured as Vi by the potentiometer (POT). The period of hip oscillation could be determined from the time gap between 2 maxima of consecutive oscillations. We compute the following quantities

V j = ∑Vi , i

VT j = ∑Vi ⋅ t i ,

for

Vi > V0

i

where V0 is the threshold voltage, which we fix between minimum and maximum voltage output of the potentiometer. We then compute

t pj = and then obtain the period

VT j Vj

= ∑ Vi ⋅ ti i

∑V

i

i

T j = t pj − t pj −1 , the time gap between consecutive peaks

j and j-1.

h) Check for Threshold

We need to check if T0 − T j ≥ ∂t 2 , threshold is crossed, then provide new period for damping profile ( T0 = Tj ). The above tasks has been achieved using a Micro-controllers circuit (MC). The inputs to MC are: a) The analog voltage of the hip oscillations b) The period T0 as an 8-bit binary input provided through an 8 Position DIP Switch. c) A digital start triggers pulse to start knee movement of AMAL. Input(c) = 0 implies knee joint inactive and rigidly held (a +5V constant signal sent to the MR damper). Input(c) = 1 implies knee joint is allowed to be active. d) A digital start trigger pulse for adaptive mode. Input(d) = 0 implies T = T0 and output(b) ∂t = T0 8 . Input(d) =1 implies adaptive mode after checking for the threshold T0 − T j ≥ ∂t 2 . The input analog voltage (0.0-5.0V) from the potentiometer measuring the hip angle is digitized. After filtering high frequency noise, this signal is used for determining T j . The micro-controller MC stores the 8 sequential damping profile values. After computing

∂t ,

the damping profile values are sequentially sent to the output if input(a) =1; else (input(a) =0) the output is set to maximum damping (A constant +5V to the MR-damper). This output signal of digital 8-bit is converted to an analog voltage signal between 0 to 5V by a Digital to Analog Converter (DAC) and converted to a current signal (0 to 2Amps) for the MR-damper. The overall functioning of all the modules has been shown in the figure 4.5.

Fig 4.4: The actual knee angle oscillation of a healthy human being measured using a potentiometer circuit. The top panel shows the noisy data over a multiple gait oscillations. Averaging multiple gaits and convolving it with a 3value running average in 2 iterations removes the noise (shown in the lower panels).

Fig. 4.5 Shows the total integration among various modules described in this paper. 5.

Discussions and Conclusions

So far a number of active prosthetic legs have been developed. But all of them behave as stand alone devices [30, 31, 32]. They are very expensive and rely heavily on sensors and control, mounted on the artificial leg, completely ignoring the biological motor control circuits. The present investigation is an attempt to design a system integrated with biological motor control that exists within us in the form of coupling. The main challenge lies in designing a suitable model of coupling for an amputee. This challenge has been solved successfully using Rayleigh’s oscillator. An effective fuzzy logic based classifier has been designed which can provide a reasonably good initial gait pattern for an amputee. The pattern could then be used as a foundation for designing a suitable CPG for an amputee. The simulation results show that the methodology worked very successfully. Once it is designed properly, it has been used as a signal generator for generating controlled damping profile for various modes of walking like simple walking, brisk walking, running etc. An active damping profile control methodology has been developed for showing biological entrainment and adaptation. Since, for active prosthesis, energy plays an important role while a human being is walking, future research could be directed towards analyzing the energy exerted by coupling actions of the other parts of the body specially upper part of the body and how to tap it suitably and integrate it with the actuator. This would make the prosthesis energy efficient and less expensive.

ACKNOWLEDGEMENT

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[29] Field trials and Gait capturing videos: http://robita.iiita.ac.in/videos.htm [30] http://www.ottobock.in/cps/rde/xchg/ob_in_en/hs.xsl/8330.html [31] http://www.endolite.com/index.php [32] http://www.ossur.com/?pageid=2743

About Authors : 1) Dr G C Nandi G C Nandi is currently working as a Professor and Divisional head of Under Graduate Division, at the prestigious Indian Institute of Information Technology, Allahabad. He obtained his BSME degree from B E College, Shibpur, Calcutta University, MSME degree from Jadavpur University, Calcutta and Ph.D degree from Academy of Sciences, Moscow. As on date he has twenty four years of teaching and research experience and has nearly sixty five publications in the area of robotics and Artificial Intelligence. Dr Nandi acted as various program committee chairman and organized several national and international level conferences and regularly conducting summer schools for teachers and professionals in the area of robotics and artificial intelligence.

2) Dr A J Ijspeert Auke Ijspeert is a SNF (Swiss National Science Foundation) assistant professor at the EPFL (the Swiss Federal Institute of Technology at Lausanne), and head of the Biologically Inspired Robotics Group (BIRG). He has a BSc/MSc in Physics from the EPFL, and a PhD in artificial intelligence from the University of Edinburgh (with John Hallam and David Willshaw as advisors). He carried out postdocs at IDSIA and EPFL (LAMI) with Jean-Daniel Nicoud and Luca Gambardella, and at the University of Southern California (USC), with Michael Arbib and Stefan Schaal. Before returning to the EPFL, he was a research assistant professor at USC, and an external collaborator at ATR (Advanced Telecommunications Research institute) in Japan. He is still affiliated as adjunct faculty to both institutes. His research interests are at the intersection between robotics, computational neuroscience, nonlinear dynamical systems, and adaptive algorithms (optimization and learning algorithms). He is interested in using numerical simulations and robots to get a better understanding of the functioning of animals (in particular their fascinating sensorimotor coordination abilities), and in using inspiration from biology to design novel types of robots and adaptive controllers.

4) Dr P Chakraborty Currently working as senior Lecturer at IIIT, Allahabad. Earlier he obtained his B.Sc degree in Physics from Aurobindo International centre of education, pondicherry, India and M.Sc in Physics from IIT, Kanpur . He did his Ph.D in Physics from Indian Institute of Astrophysics, Bangalore. His recent area of interest is Robotics including humanoid robots.

3)Mr Anirban Nandi Currently he is pursuing B Tech degree in Information Technology from the prestigious Indian Institute of Information Technology, Allahabad. Earlier he obtained the prestigious NTSE scholarship from the government of India and was also awarded the KVPY internship by IISc Bangalore, India. He is very fond of robotics and mathematics and was awarded a medal in an International Mathematics Competition organized by City Montessori School, Lucknow, India. Apart from that he had also secured the first position in the city of Allahabad in the High school board examination conducted by Central Board of Secondary Examinations in 2003.