Model-Based Lower-Limb Powered Prosthesis Control

Introduction

Contributions

Formal stability proofs for these subsystem controllers, which guarantee full system stability under certain conditions, even in the presence of impacts, zero dynamics and measurable input estimation errors. Realization of the first and only model-based lower limb prosthetic controller, which integrates and utilizes real-time in-the-loop force sensing at the human-prosthetic interface and on the ground, resulting in improved tracking performance across subjects and terrains.

Brief Description of Chapters

For example, [70] created a neuromuscular model for the full prosthesis side limb (residual thigh and prosthesis) and included models of the prosthesis array of elastic actuators. The composite CLF of the residual system RES-CLF (blue) and equivalent subsystem RES-CLF (red) yields a RES-CLF for the entire system (purple).

A Review of Current State-of-the-Art Control Methods for Lower-

Introduction

Through the addition of net positive work, motor-driven prostheses have shown reduction in this compensatory behavior [46]. However, there are many dual-actuated prostheses used in research settings, including those illustrated in Figure 2.1.

Figure 2.1: Existing Powered Prostheses. While several dual-actuated (knee-ankle) prostheses have been developed and demonstrated in research settings, there are currently only two commercially-available powered prostheses and they are both single-actuated

Control Objectives for Prostheses

In the control architecture of the prosthesis, a low-level controller controls the desired torques of the actuators. Researchers will also compare the kinematic trajectories from the prosthesis to the healthy side to assess gait symmetry [78].

Sensing for Prosthesis Control

Broadly speaking, there are two types of EMG sensors: surface-mounted and implantable EMG sensor interfaces [98]. To our knowledge, there is only one study that used surgically implanted wireless intramuscular EMG sensors in lower limb amputees for prosthetic control [99]; these surgically implanted EMG sensors are more commonly used in upper limb prostheses.

High-Level Task Estimators

Another method of motion pattern estimation is to train machine learning classifiers using pre-collected training data. As explained in [135], there are three existing techniques for determining gait speed: 1) analytical algorithms, 2) kinematic gait modeling, and 3) regression modeling or machine learning.

High-Level Gait Phase Estimators

Any measurable amount of walker movement that is monotonic during a gait cycle can be used for the phase variable. Ideally, the phase variable will be invariant across subjects [155], so that the control strategy will generalize across all subjects.

Figure 2.3: Finite State Machine. An example of an FSM used to determine gait phase, based on [17]

Mid-Level Control via Torque Computation Without Reference Tra-

This method of control allowed the user to control the timing and amount of energy to activate the prosthesis at will. The torque, 𝜏, is calculated based on this desired trajectory and the actual trajectory, 𝑦𝑎(𝑞𝑝), of the prosthesis. in).

Figure 2.5: Common Process for Mid-Level Control via Kinematic Reference Tra- Tra-jectories

Mid-Level Control via Kinematic Reference Trajectories Tracking

These virtual constraints were all parameterized with the same phase variable, the phase of the hip on the prosthesis side. Some researchers add a feedforward term to a PD controller based on human data or the physical parameters of the prosthesis to reduce tracking error.

User-Specific Customization

The oldest and most common method of user adjustment for prosthetic control is to manually adjust various control parameters using the clinician's (or an expert operator's) perception of gait. Notably, this work is one of the first to formally incorporate subjective assessments of user preference for lower extremity prostheses using HILO.

Figure 2.6: Methods of User-Specific Customization. We present four methods of customization, each illustrated using a different color: (red) biologically-inspired tuning which leverages relationships derived from biological systems and data;

Discussion

Model-based approaches such as HZD and neuromuscular model control use analytical models of the human-prosthetic system. Model-based approaches predict the response of the human-prosthetic system to detect abnormalities.

Conclusion

To relate it to the continuous dynamics of the hybrid control system in (3.2), we write the following ODE,. The small break of the limit is due to the relaxation term in CLF-QP).

Preliminaries

Background Nonlinear Control Theory

During the stance and swing phases of bipedal walking, the dynamics of the system are continuous. This guard 𝑆 defines the states of the system when the conditions ℓ(𝑥) = 0 and ℓ¤(𝑥) < 0 are reached.

Nonlinear Control of Robotic Systems

The transition point between one domain D𝑣 and the next D𝑣+ in a directed cycle is defined by the fuse 𝑆𝑒. To prescribe outputs to this multi-domain hybrid control system, the relative outputs of level 1 and 2 for walking robots or 3.26) While the phase variable 𝜏𝑣 can be time- or state-dependent, for more robust control, the state-based phase variable 𝜏(𝑞) is used [154] and is usually defined as follows. 𝑇 is the hybrid periodic current for the complete multi-domain hybrid system (3.24), and c𝑣(C𝑣) enforces real-world robot constraints such as torque and.

Human-Prosthesis Model and Powered Prosthesis Platform

Therefore, this ESS-ID-CLF-QP prosthetic controller guarantees e-ISS of the hybrid periodic orbits of the human-prosthetic system. Output tracking of 3 simulations with variations of the ID-CLF-QP on the position of the prosthesis, plotted with the desired trajectory and the human data related to the phase variable.

Separable Subsystems

Introduction

In Section 4.2 we develop a feedback linearization control law for the isolated subsystem, a system independent of its full-order system dynamics. These new methods have the potential to construct model-dependent controllers for nonlinear subsystems where the full-order system dynamics are either unknown or computationally expensive.

Separable Subsystem Control

This enables the construction of stable model-dependent controllers for separable subsystems without knowledge of the system dynamics in full order. Although we now have a subsystem control law independent of the rest of the system dynamics, it still depends on the full-order system states𝑥 ∈ R𝑛.

Figure 4.1: Amputee-Prosthesis Separable System and Equivalent Subsystem.

Separable Robotic Control Systems

This construction therefore decouples the dynamics of one subsystem from the control input of the other so that the robot system can be in separable system form (4.1). We consider another floating base with coordinates ¯𝑞𝐵 ∈ R𝑛𝑓 at the attachment point between the subsystem denoted with coordinates 𝑞𝑠 and the rest of the system.

Figure 4.2: Robotic Models with Generalized Coordinates. (Left) Model of the full amputee-prosthesis system labeled with its generalized coordinates, where 𝑞 𝑟 is notated in blue and 𝑞 𝑠 in red

Amputee-Prosthesis Application

The prosthetic subsystem was simulated for the continuous domains with values ​​of ¯𝑞𝐵, 𝑞¤¯𝐵 and 𝐹𝑓 obtained from the full-order system simulation. In addition, under Dpns,𝜏pns,𝜏¤pns and𝜏¥pns from the system were taken in full order to the prosthesis simulation.

Figure 4.3: Control inputs of prosthesis knee (left) and prosthesis ankle (right) for subsystem control law 𝑢 𝑠 and ¯𝑢 𝑠 over stance and non-stance domains.

Conclusion

In this chapter, we exploit RES-CLFs and their formal guarantees in the context of the ID-CLF-QP framework. When no force sensing was used, all GRFs were calculated through the holonomic constraints in ID-CLF-QP.

Separable Control Lyapunov Functions

Introduction

We investigate separation of RES-CLFs for separable systems to build a RES-CLF based on the prosthesis alone with the same stability guarantees established in [199] . Here we construct separable RES-CLFs to define a class of controllers to yield provably stable hybrid periodic trajectories of nonlinear separable systems with zero dynamics.

Composite RES-CLF for Separable Systems

First, we show that given the RES-CLF for the equivalent subsystem and the rest system, there exists a RES-CLF for the entire system. In Section 5.4, we prescribe a limit cycle motion that matches the human data to the human model with the RES-CLF controller.

Figure 5.1: Human-prosthesis separable system (left) with separable prosthesis sub- sub-system (red) and remaining human sub-system (blue)

Separable CLF Construction

Through Theorem 3's proof constructions, we conclude that the summation of sub-RES-CLFs yields a RES-CLF for the whole system. By summing the sub-RES-CLFs for the separable subsystem outputs, we reach a RES-CLF𝑉𝑠.

Amputee-Prosthesis Application

Conclusion

Ankle push-off specifically contributes to the forward acceleration of the body [75] and also greatly smooths the transition from dual support to swing phase in human walking [290]. Rouse, “Mechanical impedance of the ankle during the terminal stance phase of walking,” IEEE Transactions on Neural Systems and Rehabilitation Engineering , vol.

Figure 5.4: Gait tiles of human-prosthesis system, prosthesis in red, demonstrating human-like walking in simulation.

Estimate-to-State Stability

Introduction

The human converges exponentially according to the zero dynamics Lyapunov function𝑉𝑧¯ to a set bounded by a perturbation created by the deviation of the prosthetic control input𝑢∗. This section also presents our main result establishing input-to-state stability of the hybrid periodic circuit of the zero-dynamics in the full-order dynamics in the presence of estimation errors.

Figure 6.1: Human-prosthesis system. Prosthesis exponentially converges according to CLF 𝑉 𝜀, 𝜀 ¯ to a set bounded by force estimation error Δ 𝐹

ISS and Estimate to State Stability

Taking the derivative of𝑉𝜀¯(𝑥) together (6.1) yields,. 6.7) This takes the form of (6.4) and the set size can be reduced by choosing a smaller one. We use these e-ESS constructions in the rest of the chapter to guarantee exponential stability up to a bounded region for the prosthesis when the controller includes power estimation error.

ISS and ESS Hybrid Control Systems

As discussed in 3.1, a RES-CLF, developed in [199], guarantees the stability of a zero-dynamics hybrid periodic orbit in full-order dynamics. To create the e-ISS of the full hybrid system (6.12), we construct a Lyapunov function for the stable hybrid periodic orbit 𝒪𝑍¯ of zero dynamics.

ESS-ID-CLF-QP for Human-Prosthesis System

Proof: Work [274] proved that the solution to ID-CLF-QP gives the control input 𝑢∗. By solving the prosthesis dynamics and the holonomic constraint equation in ESS-ID-CLF-QP for𝑞¥𝑠 and substituting this into the CLF equation,.

Conclusion

The numbers correspond to the phases of the walking path, shown in Figure 7.4. the interaction force between the human and the prosthesis. Sensinger, “The difference between stiffness and quasi-stiffness in the context of biomechanical modeling,” IEEE Transactions on Biomedical Engineering, vol.

Figure 6.3: Phase portraits of unactuated human torso (left purple) and prosthesis knee (right purple) from simulation using ESS-ID-CLF-QP with ¯ 𝜀 = 10

Model-Based Prosthesis Control Realization with Force Estimation117

Controller Realization for Hardware

We incorporate this estimated term into the dynamics of an ID-CLF-QP and realize this QP at sample time to implement on hardware. Find the difference between our estimated acceleration and the expected acceleration based on the dynamics from the previous time step.

Human-Prosthesis Simulation

The ID-CLF-QP+𝐹𝑓 is implemented with the exact interaction force 𝐹𝑓 calculated by (4.13), since 𝐹𝑓 ∈ 𝜆, based on a feedback-linearizing control law 𝑢 (3.6). The ID-CLF-QP+ ˆ𝐹est used the power estimator (7.3) with 𝑁 = 1, since resources are not needed in simulation.

Human-Prosthesis Experimentation

Variations of the ID-CLF-QP controller are applied to the prosthesis in position and a motion feedback linearization control law to implement the output of subsystem (4.3), where 𝑦𝑎. ID-CLF-QP regardless of the interaction force yields a very different control input and had terrible tracking, which shows the importance of force calculation.

Figure 7.3: Control Input and Force Estimation Simulation Results. (a) Prosthesis stance control input for the knee from 3 simulations of the human-prosthesis model walking with variations of the ID-CLF-QP applied to the prosthesis

Conclusion

Goldfarb, "Design and control of a powered transfemoral prosthesis," The International Journal of Robotics Research, vol. Goldfarb, "Control of stair ascent and descent with a powered transfemoral prosthesis," IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol.

Figure 7.5: Results of four phases of stance from experiment. (Top) The RES- RES-CLF derivative (blue) plotted against its bound (red)

Model-Based Prosthesis Control Realization with Real-Time

Introduction

Therefore, the lack of real-time force sensitivity necessitated the assumption of rigid ground contact (through the use of holonomic constraints). To more accurately account for the interactions between the user and the prosthesis, and the prosthesis and the environment, it is necessary to integrate real-time force sensing into the model-based controller.

Figure 8.1: Gait Tiles of Experimental Results with Force Sensor Measurements.

Controller Realization for Hardware

It is important to note that force sensing has long been used in prosthetic control, but not in the context of realizing model-based controllers via real-time force sensing. The same results generated in the previous chapter have also been used in this work.

Prosthesis Platform for Controller Realization

As an interface with the pressure sensor we use a UP Board (02/32), a small x86 single-board computer. Using the pressure of the sensor element, the surface area and the displacement from the center of rotation of the ankle, we calculate the vertical GRF 𝐹z.

Figure 8.2: Transfemoral powered prosthesis AMPRO3 with labeled hardware com- com-ponents, including newly integrated force sensors (load cell and pressure sensor).

Human-Prosthesis Experimentation

The force-sensitive ID-CLF-QP achieves better standing tracking than the ID-CLF-QP without force sensors on all terrains. These results show that the force sensing ID-CLF-QP can achieve better tracking using force sensors than without, indicating that the forces are taken into account in the dynamics.

Figure 8.4: Gait tiles of 2 subjects walking on a rubber floor with the force sensing ID-CLF-QP prosthesis controller.

Conclusion

Goldfarb, “Running with an electric knee and ankle prosthesis,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. Goldfarb, “Stair ascent and descent controller for an electrically powered ankle prosthesis,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol.

Model-Based Multi-Contact Prosthesis Walking

Introduction

Labels indicate the respective domain in which the human prosthesis is in the multi-domain hybrid system graph shown in Figure 9.2. multi-contact behavior that occurs in heel-toe roll used for ankle push-off. In this chapter, Section 9.2 describes the domains and ground model used in a multi-domain hybrid system to model multi-contact behavior.

Figure 9.1: (top) Subject 1 and (bottom) Subject 2 walking with powered prosthesis controlled by multi-domain model-based prosthesis controller with real-time force sensing following provably stable human-like walking trajectories

Generating Subject-Specific Human-Inspired Walking Trajectories . 140

Force Sensing ID-CLF-QP. To develop a hardware-implementable form of a RES-CLF, we construct a variation of the inverse dynamics quadratic CLF program (ID-CLF-QP), introduced in [37], that was developed for the Prosthetic Subsystem in [ 97]. Although the desired auxiliary control law 𝜇pd,

Figure 9.2: Six-domain directed graph of human-prosthesis hybrid system, modeling respective foot contact points for different phases of walking.

Multi-Contact Model-Based Controller Realization

Here, the ID-CLF-QP with force sensing (9.1) exhibits tight tracking to the desired trajectory, especially compared to the other controllers for the ankle. More importantly, the ID-CLF-QP with force sensing most closely matches the subject-specific human joint patterns, demonstrating that we can mimic this human-like behavior in a systematic way that does not involve tuning between subjects.

Conclusion

Sup, “Design optimization of lower extremity prostheses: a review,” IEEE Transactions on Neural Systems and Rehabilitation Engineering , vol. Herr, “Neuromuscular model-based control of an electric ankle and foot prosthesis,” IEEE Transactions on Neural Systems and Rehabilitation Engineering , vol.

Figure 9.4: Ground Reaction Forces and Output Tracking Experimental Results.

Conclusion

Future Work

Goldfarb, “Control and evaluation of a powered transfemoral prosthesis for stair ascent,” IEEE Transactions on Neural Systems and Rehabilitation Engineering , vol. Goldfarb, “Effect of a swing-assisted knee prosthesis on stair ambulation,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol.