Development of the Ankle Rehabilitation Robot

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Chapter 5 Computational Model of the Human Ankle

Due to significant physical interaction between the ankle rehabilitation robot and the foot of the user, it is important to obtain a better understanding of the mechanical characteristics of the human ankle and take it into consideration during design of the robot interaction controller. This chapter presents a computational ankle model developed to facilitate controller development of the ankle rehabilitation robot. To reduce its computational load, the rigid body model developed in this work is based on the biaxial ankle kinematic structure described in Chapter 4 and provides a description of the ankle mechanical characteristics through considerations of forces applied along anatomical elements around the ankle joint, which include ligaments and muscle-tendon units. The dynamics of the ankle-foot structure and its surrounding ligaments and muscle-tendon units were formulated into a state space model to facilitate simulation of the robot.

Determination of Model Complexity

Information regarding the mechanical properties of the human ankle is important for the development and evaluation of interaction control strategies for the ankle rehabilitation robot. In addition to describing the end point mechanical behaviour of the human ankle, a model with sufficient detail can also be used to estimate the forces applied to different tissues around the ankle and the reaction moments and forces exerted on the joints. This information is particularly applicable for ankle rehabilitation as injuries are typically a result of over-stretching and excessive tensioning of fibrous tissues such as ligaments and tendons. The availability of these forces can therefore facilitate the evaluation of different rehabilitation strategies. With the above in mind, the ankle model required in this research is one which can be used in controller simulations and development while also capable of providing estimates of the forces and moments acting on various ankle anatomical elements. From the review on existing computational biomechanical ankle-foot models given in Chapter 2, it can be seen that a range of computational ankle models with varying levels of complexities had been developed by researchers to advance the understanding of foot biomechanics and to study foot related pathologies. Models belonging to the lower end of the complexity spectrum mainly involve treatment of the foot and lower limb as rigid bodies while more advanced models typically utilise finite element analysis to study stresses and strains within the soft tissues [46-48], as well as three dimensional contacts to describe the ankle kinematic behaviour [45]. While finite element models can give more accurate and realistic results, they are also more computationally intensive. The same holds for the use of three dimensional contacts to describe the kinematics between foot bones. Since this research aims to incorporate the ankle-foot dynamics into an overall robot-controller model, the use of such advanced methods will result in a system which may not be suitable for dynamic simulation over longer timescales. A three dimensional rigid body based model with relatively simple kinematic constraints is therefore considered to be a more appropriate choice for this application. Of the models discussed in Chapter 2, the model proposed by Wright et al. [43, 44] and the lower limb model available in OpenSim [78] appear to be the most fitting for this application due to their lower complexity and hence lighter computational demand. However, it should be noted that since knowledge of the forces along the ankle ligaments is important for this work and that the above models does not offer such information, a computational model had been developed specifically for this research.

Modelling of Force Elements

Modelling of Ligaments

The ligaments considered in the model are fibrous tissues which connect bones to bones. Ligaments are viscoelastic and can be found at various joints in the body. The main role of ligaments is to keep the articulating bones in place during motion. A ligament can be modelled as a piece of elastic string which can only apply a resistive force while in tension. The instantaneous force-length relationship of ligaments can be represented by a piecewise function shown in (5.1)
[115], where ???? is the tension force along the ligament and ? is the strain of the ligament. Additionally, ? and ? are respectively parameters used to control the magnitude and shape the exponential relationship. The ligament force relationship shown above can produce unrealistically high tensions when the strain becomes too large. Typically, the ligaments will be damaged before they reach large forces. The use of this exponential relationship in simulation can result in an excessive increase of force for a minute increase in strain. To avoid the rapid increase of ligament forces in simulation, the exponential growth of this force is arrested when a certain force threshold is reached and replaced with a linear relationship. In this work, this threshold is set at 500N, a value which is close to the largest of the ligament failure loads reported in [115]. The actual ligament force-length relationship used in the developed model is therefore given by (5.2), where ?????? is the ligament force threshold (500N in this case) and ????? (5.3) is the strain required to produce ??????. It was also reported in literature that ligaments exhibits viscoelastic behaviour where it undergoes force relaxation after application of an initial strain. This behaviour can be modelled using a single spring in parallel to an array of serial spring-damper units. This arrangement is shown in Figure 5.1. In the case of a linear system, the sum of all the spring stiffness would yield the instantaneous stiffness while the single spring in parallel, ?? determines the force found at steady state. The dampers therefore govern the transition between the initial and steady state forces over time.The work by Funk et al. has modelled the viscoelasticity of ankle ligaments using three serial spring-damper units in parallel with another spring[115]. In their formulation, three states were required to describe the ligament model. However, since many ligaments are present around the ankle and subtalar joints, the use of four state variables for each of these ligaments will lead to a large state space model. As a result, only one serial spring-damper unit is incorporated in the ligament model in this work to reduce model complexity. Using this viscoelastic model structure, the forces along the force element can be decomposed into two components, a steady state force along the single spring and a transient force along the serial spring damper unit. The total force is then merely the sum of these two forces. The relationships between these forces are shown diagrammatically in Figure 5.2. In the diagram, ???? is the total force along the force element, ?? is the elongation along the single spring element and ?? is the elongation of the spring element in the serial spring-damper unit. The forces along these spring elements are respectively given by the nonlinear functions ?????? and ??????. The damper element is assumed to be linear with a damping coefficient of ?? while the elongation of the damper unit is represented by ???. Using the above formulation, the relationships shown in (5.4) and (5.5) can be established. The state space model which describes the viscoelastic ligament force is therefore given by (5.6). It should be noted that ?? and ?? used in the model are simply a linearly scaled version of (5.2). Specifically, these functions can be represented as (5.7) – (5.8), with ?? being the relaxed length ofthe ligament. ?? and ?? on the other hand are scaling factors which sum to unity, where a value of 0.75 has been used as the ?? parameter for all ligaments considered in this model.

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Modelling of Muscle-Tendon Units

Skeletal muscles are tissues which contract to generate force and thus motion, while tendons are fibrous tissues which connect skeletal muscles to bones to transfer the muscle forces. As with the case of ligaments, tendons can be modelled as tension only elastic strings. However, the dynamics of the muscle is more complex, as it is an active element. The Hill based muscle model has been widely used in the literature [43, 44, 116, 117] to model muscle behaviour and was also adopted in this work. The structure of the Hill based model used to describe the muscle-tendon unit is shown in Figure 5.3. The muscle-tendon unit considered in this model consists of two components (tendon and the muscle), with the tendon component modelled as a nonlinear spring ??? . The muscle component on the other hand is considered to be made up of the contractile element CE which governs the muscle’s active force characteristics and the parallel element PE which determines the passive muscle behaviour. The parallel element is in turn represented as a nonlinear spring ??? in series with a linear damper ???. The linear damper has been included to incorporate damping in the passive muscle behaviour and to ensure that a feasible solution is obtained in the state space model during simulation. It should also be noted that ? is the pennation angle of the muscle, which describes the angle between the direction of muscle fibres and the direction of force application along the muscle-tendon unit. The mathematical description of the force along the contractile element of the Hill muscle model is typically represented as (5.9). The variable ? is used to denote the extent of muscle activation and can take on values between zero and unity, while the variable ???? is the maximum active force that can be exerted by the muscle. These variables are used to scale the product of normalised tension-length relationship, ???????? and the force-velocity relationship, ????????, where is the length of the contractile element. Additionally, the normalised tendon and parallel element force-length relationships can also be respectively represented by the functions ?????? and ????????, where ?? is the tendon length. In this work, all the force-length relationships were taken to be the same as those used by default in the OpenSim software package. These functions are shown in Figure 5.4a–c and are defined through cubic spline interpolation of several known data points. In these functions, the length of the muscle/contractile element is normalised against the optimal muscle fibre length (length at which maximum active force can be produced). Although the force-length relationships used may not be exactly identical to those of the specific patient/user, the general shapes of these relationships are in line with what is typically reported in the literature [116, 118]. Consequently, they should be able to provide at least a qualitative description of the actual muscle behaviour. As the scope of this work is not to provide a patient specific ankle model, the use of these relationships can be considered acceptable. Of course, large discrepancies between the actual and model force-length relationships can still lead to significant differences in the predicted muscle forces and hence ankle motion and this is a limitation of this approach.
The force-velocity relationship of the contractile element on the other hand, is given in the form of a piecewise function shown in (5.10) [116, 117], where ?? is a parameter which is dependent on the composition of slow and fast muscle fibres in the muscle and ???? is the maximum contraction speed of the muscle being considered. Also, ? and ? are parameters used to define the forcevelocity relationship when the muscle stretch velocity is positive. These parameters were chosen so that a smooth transition is possible between the two piecewise segments. They have also been selected to provide a desired limiting value for ???????? as the muscle velocity approaches infinity.
This relationship is shown in Figure 5.4d in terms of the normalised contractile element velocity, Combination of (5.9) and (5.11) – (5.14) will then leads to (5.15), which describes how the contractile element length will evolve with time given the muscle activation and current length of the muscle-tendon unit. Although ???? is not explicitly given in (5.15), unique solutions for it can be found by first expanding (5.15) into a piecewise quadratic function (5.16). The roots of the function can then be found and ???? can be determined by selecting the solution with the appropriate sign. A first look at (5.15) suggests that the computation of ???? will not be straight forward as this quantity is also used to determine the active segment of the piecewise function (5.10). A solution to this problem was devised in this work by taking into account the fact that ???????? is greater than unity for all positive contractile element velocities and less than unity when the muscle is contracting. Additionally, since ?, ????, ??? and ??? are all positive by definition, the difference between the tendon force and the static component of the muscle force as shown in (5.17) can only be positive if ???? is positive and vice versa. This force difference was therefore used to ascertain the sign of ???? and select the appropriate segment of (5.10) to be used in (5.16).

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Chapter 1 Introduction
1.1 Rehabilitation Robots
1.2 Interaction Control
1.3 Ankle Rehabilitation
1.4 Research Objectives and Motivation
1.5 Thesis Outline
1.6 Chapter Summary
Chapter 2 Literature Review
2.1 Existing Ankle Rehabilitation Devices
2.2 Ankle Kinematics and Computational Ankle Models
2.3 Interaction Control in Rehabilitation Robots
2.4 Discussion
2.5 Chapter Summary
Chapter 3 Development of the Ankle Rehabilitation Robot
3.1 Design Requirements
3.2 Determination of a Suitable Robot Kinematic Structure
3.3 Workspace, Singularity and Force Analyses
3.4 System Description
3.5 Chapter Summary
Chapter 4 Online Identification of a Biaxial Ankle Model
4.1 Background
4.2 Mathematical Description of the Biaxial Ankle Model
4.3 Identification of the Reduced Biaxial Model
4.4 Online Identification Algorithms
4.5 Kinematic Model with Configuration Dependent Axes Orientations
4.6 Preliminary Results
4.7 RLS Algorithm with Penalty on Deviation from Nominal Parameters
4.8 Chapter Summary
Chapter 5 Computational Model of the Human Ankle
5.1 Determination of Model Complexity
5.2 Modelling of Force Elements
5.3 Definition of Force Element Parameters
5.4 Modelling of Ankle-Foot Dynamics
5.5 Chapter Summary
Chapter 6 Validation and Application of Ankle Model
6.1 Model Validation
6.2 Sensitivity Analysis
6.3 Rehabilitation Trajectory Generation
6.4 Chapter Summary
Chapter 7 Multi-Input Multi-Output Actuator Force Control
7.1 Motivation for a MIMO Actuator Force Controller
7.2 Actuator Force Control by Decoupling of Inertia Matrix
7.3 Higher Order Dynamic Model of Actuator-Sensor-Environment System
7.4 Stability Analysis of the Coupled Actuator-Sensor-Environment System
7.5 Proposed Actuator Force Controller
7.6 Simulation Results
7.7 Experimental Results
7.8 Chapter Summary
Chapter 8 Model Integration and Elementary Robot Control
8.1 Dynamic Modelling of Parallel Mechanism
8.2 Integration of Manipulator Model with Foot and Actuator Dynamics
8.3 Elementary Robot Control
8.4 Implementation Issues
8.5 Simulation Results
8.6 Experimental Results
8.7 Chapter Summary
Chapter 9 Adaptive Interaction Control via Variable Impedance Control
9.1 Biomechanical Model Based Impedance Adjustment
9.2 Simulation Results for Impedance Adjustment Module
9.3 Experimental Results for Impedance Adjustment Module
9.4 Chapter Summary
Chapter 10 Adaptive Interaction Control via Assistance Adaptation
10.1 Assistance Adaptation
10.2 Simulated Case Study for Assistance Adaptation
10.3 Experimental Results for Assistance Adaptation
10.4 Overall Control Structure and Implementation of Rehabilitation Exercises
10.5 Chapter Summary
Chapter 11 Conclusions and Future Work
11.1 Outcomes, Conclusions and Contributions
11.2 Future Work
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