Single Inverted Pendulum Model, Sagittal Plane

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Chapter 3 Human Models and Control

There are several difficulties in determining the basic control system for the human body. The human body has a limited number of degrees of freedom, 12 in the lower body. However, there are a much larger number of sensors and actuators in comparison to the number of degrees of freedom that the control system utilizes. Each joint is over actuated with different muscles, each with their own dynamics and responses. Additionally, the three groups of sensor inputs provide several different and sometimes redundant sets of data on the current state of the human body, each with it’s own dynamics and delays. The central nervous system (CNS) must be able to integrate each of the different inputs together and send the control signals to the muscles to actuate the body accordingly.
Kuo[48] developed an optimal control model for the human body to see if it can provide informa-tion on how the CNS integrates the information from the three sensor groups and how it controls standing. The model uses simplified sensor inputs and control signal to reduce complexity, mus-cles are modeled in groups rather than individually and the sensory information from the visual, proprioceptive and vestibular is simplified to report the body’s parameters instead of raw data. The model was calibrated using experimental data from various trials that included or eliminated combinations of the three sensor systems. Analysis of the model when compared to experimental data showed that the model accurately predicted sway when faced with degraded or eliminated sensor information. Kuo believes this demonstrates that the CNS utilizes an internal dynamic model of the body and state estimation to compensate for degraded sensor inputs and to compen-sate for disturbances and changes in the system. This interpretation is supported by several other studies.[47][7][13][63][48] However, most models do not start with the assumption of an internal model. Instead, the CNS is modeled as a controller with delayed linear feedback. As will be shown below, this assumption is adequate for quiet standing.
Normal quiet standing is the state a human is in when standing with both legs on the ground and no large disturbances are present. Quiet standing is not motionless. The body moves in distinct low frequency motions. The three sensors (proprioceptive, vestibular system, visual) each have a different effect on steadying and reducing sway while standing. Additionally, sway and responses Allan B. Cass Chapter 3. Human Models and Control 10 to disturbances in the sagittal1 and coronal2 plane exhibit distinct differences and as such need to be modeled differently. Presented here are three different sets of models for standing. For quiet standing in the sagittal plane, a single link inverted pendulum will adequately model standing. For larger disturbances in the sagittal plane, multi link inverted pendulum models are presented. For sway in the coronal plane, a two link inverted pendulum is given.

Single Inverted Pendulum Model, Sagittal Plane:

In quiet standing (standing with no disturbances) the knees and upper torso are kept stiff. Quiet standing can then be modeled as a single inverted pendulum with a single DOF (Degree of Free-dom) at the ankle joint allowing motion in the sagittal plane. The ankle is then the sole actuator used for stabilizing the body in an upright position. This model has been extensively studied to de-termine how the brain maintains balance along with the contribution from each of the three sensor groups (Vestibular, Proprioceptive and Visual). The basic control system for the inverted pendulum model involves a PID controller with time delay feedback.
In quiet standing, humans do not stand perfectly vertical; the sway in the sagittal plane is about a point 1-2 degrees forward[47]. There is also a noticeable sway in the absence of external perturba-tion. It is due to this sway that most models of human standing omit the integral portion of a PID controller and only use PD feedback as an integrative component in the feedback would generally reduce and eventually eliminate the sway.[54]
Fukuaka et al.[18] have broken down quiet standing by each of the three systems through var-ious experiments. In each experiment, two of the systems were suppressed to allow only one system as feedback. Subjects were required to maintain balance with the limited senses. It was found that each of the three systems (visual, vestibular, proprioceptive) was separately capable of stabilizing the individual. The somatosensory portion of the proprioceptive system and the vestibular system showed significant derivative gains and leading phase that has been noted in var-ious experiments.[54][52] The visual system, perhaps due to the significant lag times, displayed derivative gains but not leading phase. In the various tests, the somatosensory system had the least variation among the various test subjects, leading to the theory that it is the primary system for balance.
The leading phase from the somatosensory and vestibular system has also been cited as a reason for believing the CNS uses internal models for stability. Masani et al.[54] examined this assumption and found that large proportional and derivative gains were able to stabilize the human body during quiet standing up to 185ms with proportional and derivative Gains of 750Nm/rad and 350Nms/rad respectively. These gains also produced the leading phase observed in experiments. Kooji et al.[43] have estimated the proportional gains at 742-835Nm/rad and the derivative gains at 122-194Nms/rad with delays of 55-66ms in experiments with healthy subjects. The gains that Masani et al. found are thus in the realm of plausible gains that may be generated by healthy humans.

Multi-link models, Sagittal Plane:

When the body cannot compensate for a disturbance with just an ankle response, a hip response is used. In a hip response, the hips are brought out of line with the ankles to maintain the COM over the COP. If this fails, then a step response is used. The exact form of a hip response depends on the direction and magnitude of the disturbance. Since the hip moves independently of the ankle, the standard single link model may no longer be used. Instead two or more link inverted pendulum models are used.
Multi-link models in the sagittal plane become difficult to analyze and derive due to the high non-linearity and the increased number of states of the system. Kuo[48] uses a double link inverted pendulum, using only the ankle and hip joints. The system is modeled with lag and stabilized with a linear quadratic gaussian (LQG) controller.3 A state estimator is used in combination with a LQG in order to compensate for the delay and stabilize the system. Experiments show that the model closely simulates experimental data.
Xiang and He[83] use a more complicated model for the simulation of disturbed sagittal standing. A variable three or four link model with time delay is used. When the heels are on the ground, a three-link shank, thigh, trunk model is used. When ankle torques would force the heel off the floor, a four-link foot, shank, thigh, trunk model is used. Passive joint torques and muscle dynamics are used to increase the accuracy of the model. Xiang and He use a Smith Predictor to compensate for 100ms delay and a LQR to stabilize the linearized system. Experiments were used to calibrate the model. The LQR and Smith Predictor produce a similar response to the experimental data. However, in most cases the modeled LQR returned the COM to the equilibrium point faster than the CNS was able to in experiments.

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Double Inverted Pendulum Model, Coronal Plane

Unlike the sagittal plane, the body does not remain stiff at the hips when movement occurs. Instead, the hips move counter to the sway of the ankles, allowing the torso to remain upright in the coronal plane.[27] Thus, while the system is modeled as a double inverted pendulum, the control system may still be based around a SISO (Single Input Single Output) controller and may be derived in similar methods to the single inverted pendulum model in the sagittal plane.
Kimura and Jiang[27] use a PID model of a linear controller to approximate coronal plane control in quiet standing. Several interesting facts were discovered in experiments for deriving the coronal postural control law. Of note, there is a significant increase in sway frequency from 0.02hz to 0.12hz when the eyes were closed. Additionally, a correlation was found between age and balance where the elderly have a lower derivative gain in proportion to the product of their mass and height compared to younger adults. Proportional and integrative gains were found to be less affected by aging. This leads to the conclusion that the loss of the derivative gain as one ages leads to an increased risk of falling and impaired balance. Proportional and derivative were found to be in the range of 519Nm/rad and 72.3Nms/rad respectively.
Johansson et al.[33] have performed experiments where the eyes were closed and the use of the vestibular system was denied to healthy patients by the introduction of low current high frequency electrical noise into the inner ear. Coronal sway was significantly increased and there was a change in the sagittal sway. The noise was applied for three minutes in each trial. Of note is that the patients exhibited less sway towards the end of each trial that the beginning. The patients were adapting control strategies that relied less on the vestibular system and more on the proprioceptive system with adaptation regularly taking 40-50s.[35] However, while they were able to partially compensate for the loss of the vestibular system, sway magnitude was still larger towards the end of the trial than prior to the introduction of the noise.

Training Stability

Model Development

For full control of the lower body, a 12 link model would be necessary (see Fig#4.1)1. The 12-link model is composed of joint approximations of the ankle, knee and hip joints. The ankle joint primarily rotates in the sagittal plane2 and coronal3 plane. While the ankle does allow some movement in the transverse plane4, this movement is negligible and may be ignored. The ankle is then modeled as a two degree of freedom (DOF) joint. The first and second joints, L1 and L2, correspond to the rotation of the ankle in the coronal and sagittal plane respectively. The knee allows movement primarily in the sagittal plane. Little movement is allowed in either the coronal or transverse plane and damage may follow if the knee rotates too much in either of these planes. The knee is then modeled as a single DOF joint (L3) with rotation in the sagittal plane. The hip in the human body is a ball joint allowing for 3 DOF. The hip is capable of rotating in all three planes. The order of rotations used in the model is rotation in the transverse plane (L4), coronal plane (L5) and then the sagittal plane (L6). This will allow for the model to adequately simulate hip movement in most circumstances.
One limitation with the model is the joints are capable of moving in anatomically impossible ways. For example, the model allows the knees to bend forward. The inclusion of anatomical limits, such as the knee, would require one of two remedies that are difficult to include. The first would be a change in the model while the knee has sufficient torque to bend forward past the anatomical limits while using the standard model at all other times. The second would be to apply a compensatory torque that would prevent forward motion of the knees once the knees are sufficiently straight. The inclusion of either of these in the simulation is difficult and as such is not included.
For quiet standing, this 12-link model would involve redundant links in the system. In quiet stand-ing, both feet remain on the ground and the legs move parallel to one another. To simplify, a model based on the inverted pendulum is used instead. The joints on the right leg (L12-L7) are combined with the corresponding left leg joints (L1-L6), as the angle in each joint should always be the same in the corresponding joint. Further, joints L4 and L9 are eliminated. Joints L4 and L9 are not capa-ble of rotation while both feet remain fixed on the ground. This simplification reduces the number of links in the model from 12 to 5(see Fig#4.2). The mass and inertia of each link are recalculated for the new model. To further simplify the model, the centers of mass are placed inline with the links and are not offset. The model is fully actuated and observable. The inverted pendulum has been shown to be a viable model for modeling human balance and as such is used here.

General Model:

The dynamics of the human body are expressed as follows (See Table #4.3)
When in quiet standing, the lean possible while maintaining balance is limited from approximately 12 degrees anterior[79] to 5 degrees posterior.[18] Lean in the coronal plane is limited by the width of the stance, but is of similar limitation. This small range is nearly linear and allows us to substitute a linear model for the non-linear model for the A and G matrices in controller design.
Torques created due to the movement of the limbs are called passive torques. These torques are caused by the movement of the ligaments, fluids, muscles that are not applying torque and other portions of the body. The passive torques created act as a friction. The torques are non-linear and vary from patient to patient. Ignoring the passive torques in the system will decrease the stability. If the system can be stabilized with their absence, it can be stabilized with their presence.[54] Thus, we may safely ignore the passive torques in the model. This further simplifies the model. Additionally, the torques applied by the patient TH are ignored to derive the controller for the exoskeleton.
Experiments show that the maximum angular velocity in quiet standing is below 0.15 rad/s.[78][27] This limitation reduces the maximum effect of the coriolis and centrifugal effects in the system. The coriolis and centrifugal parameters are ignored in controller development due to the low veloc-ities and the small region of operation. The centrifugal and coriolis effects will instead be treated as disturbances in the system during operation. The system dynamics then become
Using measurements, we determine the nominal matrices for A( ) and G( ) as Ao( ) and Go( ) respectively.

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Controller

Controller Design

The mathematical model for quiet standing is a 10 variable system with non-linear elements. An LQR has been shown capable of stabilizing the system similar to the central nervous system (CNS). The LQR controller requires three parameters, the input cost matrix, the state cost weighting matrix and a linear system model. The linear system model was designed using the linearized inertia and gravity matrices, AL and GL respectively. The inertia and gravity matrices were linearized about the point = 0, corresponding to vertical standing. The linearized system is described by The LQR parameters used are listed in table 4.4. The state and input cost weighting matrices were chosen based on experiments by Tian and He[83]. The resulting feedback matrix is of size 5×10 and may be separated into proportional and derivative portions [Kp Kd] resulting in the closed loop state equations of where Kp and Kd are the proportional and derivative feedback gains of the LQR feedback matrix respectively.
This control law will stabilize the system without assistance. However, it does not allow for a different control law to be used with regards to gravity. The LQR derived feedback gains are then separated into two groups, gravity compensation and reference feedback using the linearized gravity matrix.
This design allows the gravity matrix Go( ) to be independently adapted based on what is found to be the most appropriate for the patient. For example, patients who are capable of standing autonomously, but only need training, do not need the assistance of the exoskeleton to remain upright.

Controller Adaptation

The control system for the exoskeleton has three requirements, it must be robust enough to handle uncertainties in the patient’s body, it must initially stabilize the patient and it must force the patient to learn to stabilize themself.
This leads to the requirement for an adaptable controller. Examining the equations of motion, and including the torque applied by the patient (TH ), two sets of terms are apparent where and are NxN diagonal matrices of independent adaptable gains. Substituting the torques applied by the patient (TH ) for (THG + THD) where THG is the torque the patient applies to com-pensate for gravity and THD is the torque the patient applies to change position, the equation becomes As the patient develops their own control system for TH then the parameters and will reduce to maintain the same response of the system, maintaining stability. However, if the system is completely stable, the patients will not be forced to adapt and take authority over the exoskeleton. The learning ’L’ parameter is added to the adaptive controller (see Fig#4.4). The parameter ’L’ takes on the value from 0 to 1 to adjust the optimal control signals in the adaptable control law to sub optimal levels. The desired torque will be reduced from the nominal level. By doing this, the system will become less stable or unstable as the control authority of the exoskeleton will no longer be able to stabilize the system. This parameter will have to be chosen based on experimental data with each patient as each patients learning ability will be different.
Two controllers are presented. In the first controller, the patients are trained to stabilize themselves from disturbance applied from random directions and at random amplitudes. In the second con-troller, the patients attempt to move their body in slow motions and maintain balance throughout the movements.

Reactive Balance Training

Controller Description

In this protocol, the patients are instructed to attempt to remain upright in the presence of periodic random disturbances applied through the exoskeleton. The controller is designed to assist the patient in maintaining balance by initially simulating the CNS with full authority and then steadily ceding authority to the patient. The feedback gains are determined through a LQR in order to simulate the delayed CNS response to a disturbance. As was discussed in Chapter 3, this provides an approximation of a healthy CNS. (see Fig#4.5) In Chapter 3, the model discussed included delay in the approximation of the CNS. The delay was compensated for through the use of a Smith Predictor. Here, the delay of the controller is eliminated, as it is not necessary for the training of the patient. The goal of the controller is not to perfectly mimic a healthy CNS, but to provide a stabilizing force similar to a healthy CNS that the patient may train with.
Low impulses are provided through Exoskeleton to perturb the system from equilibrium. When the impulses are applied, the exoskeleton control law is temporarily suspended in order to allow the CNS to respond to the disturbance. After a delay suitable for the CNS to react, the controller is allowed to start assisting the patient again. The controller will thus be able allow training of the CNS. This training is derived from the protocol used by Matjacic[58]. The control authority of the exoskeleton is reduced over time as the patient succeeds in remaining stable. Authority reduction may be controller manually by the patient or doctor. Alternatively, it may be reduced automatically as the patient succeeds in maintaining balance to the disturbances.
The theory behind this training protocol is that the perturbations applied to the patients will stim-ulate the brain to develop reactions that will stabilize the patient. As discussed in Chapter 2, the use of platforms to apply disturbances and support patients has proved beneficial. In this training protocol, the control system and exoskeleton do not need to be able to fully simulate a human re-sponse to the disturbance. Instead, the exoskeleton and control system are designed for the purpose of preventing injury to the patient and allow them to develop their own control system. This should allow for the development of gross in response to larger disturbances and fine motor control for smaller disturbances.
Portions of the protocol should be adaptable for fine motor control. Reduction in the disturbances combined with the addition of a stiffness in the exoskeleton and a reduction in the stiffness over time should allow the development of fine motor control. The stiffness could be applied, controlled and reduced through the use of a magneto rheological brake in substitution for actuators. As the patient is able to increase his or her own stiffness to that of a healthy individual, the stiffness would then be reduced.

Contents
1 Introduction
1.1 Exoskeleton Uses
2 Problem Overview
2.1 Cerebral Palsy
2.2 Robots in Therapy
3 Human Models and Control
3.1 Single Inverted Pendulum Model, Sagittal Plane:
3.2 Multi-link models, Sagittal Plane:
3.3 Double Inverted Pendulum Model, Coronal Plane:
4 Training Stability
4.1 Model Development
4.2 General Model:
4.3 Controller
5 System Modeling and Actuator Requirements
5.1 Model Summary
5.2 Model Testing
5.3 Robustness Modeling
5.4 Authority Reduction
5.5 Torque Requirements
5.6 Actuator Options
5.7 Actuator Saturation
6 Future Work
Bibliography
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