Synthesis of Different Squat Strategies and Patterns With The Optimization-based Dynamic Task Controller

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Human Body Modeling

Human body is often considered as a simplified biped system and can be modeled as such. A model can be planar (2D) [Bessonnet et al., 2002, Chevallereau and Aoustin, 2001, Ren et al., 2007] or spatial (3D) [Bessonnet et al., 2010, Kim et al., 2008], skeletal [Mombaur, 2009] or musculoskeletal [Xu et al., 2015]. Generally, the system is constructed from rigid segments connected by mechanical joints. These joints are actuated by joint actuators or by line actuators representing muscle tendon units (MTU), whose model can be linear or non-linear (Fig. 1.3). Joint actuators such as torque generators are commonly used in complex movement simulation such as walking thanks to their simplicity and computational efficiency [Xiang et al., 2010]. On the other hand, some particularities of the motor system strongly influencing the way we move cannot be accounted for using such models: the unilateral nature of muscular actuation (a muscle can only pull), the actuation redundancy and the state dependant nature of muscles internal dynamics and their impact on the overall system’s dynamics [Anderson and Pandy, 2001a] as well as phenomenon such as fatigue [Xia and Law, 2008]. These aspects are represented more directly and intuitively on musculoskeletal models. These musculoskeletal models can then be used in combination with 3D experimental gait data (i.e., marker positions, and ground reaction forces) to estimate biomechanical parameters during dynamic function—for example, joint angles—but also muscle activations and forces ultimately allowing calculation of joint contact forces [Killen et al., 2020].
In the application of CP treatments, one of the major challenges perspective is to determine a reasonable level of complexity of the musculoskeletal models to represent the neuromusculoskeletal dysfunctions of the patients (i.e., the altered musculoskeletal geometry, musculoskeletal parameters, and altered neural control). These models are also subject-specific, meaning they are able to describe distinguish features of different individuals.
In parallel with the description of patient-specific features, surgical interventions are also needed to be represented in the musculoskeletal model. For tendon transferring procedures, it’s obligatory to use line actuators as they allow the demonstration of the modification of muscle attachment points and actuation path. With more complex muscle models, such Hill-type muscles [Haeufle et al., 2014], muscle tendon unit lengthening can be represented by tuning the physiological parameters of the muscle model.
Finally, the solvability of control problems to compute actuator activations depends partly on the intricacy of muscle model and the number of muscles incorporated in the human model. This is particularly problematic, as complex muscle and joint models are usually non-linear, as well as the number of degrees of freedom (DOF) and actuators can be important. This aspect is discussed more in detail in the following section.

Optimization – The Backbone of Human Movement Genera-tion

Due to the redundancy of the multibody system of human body, human movement is believed to follow a certain « principal of optimality », thanks to evolution and learning [Kulić et al., 2016]. It is, therefore, a straightforward idea to use mathematical optimization as a tool for motion analysis or motion synthesis in humans [Kulić et al., 2016]. For the sake of clarity, we use the term « motion synthesis » to indicate the generation of movement without relying on motion or force tracking.
The problem formulation of an optimization for movement synthesis consists of defining meaningful objective functions representing human performance measures and constraints. A general framework for a optimization problem can be defined as follow.
where is the optimization variables (joint angle profiles, joint torque profiles, muscle forces, etc.), f is a cost function to be minimized, hi represents the m-equality constraints, and gj represents the k-inequality constraints.
The cost function varies depending on human subjects, movements and applications. In tracking data ap-plication, the cost function is usually the squared distance between estimated values and measurements. Hypotheses about the functional control during movement can be formulated as performance measures such as dynamic effort (sum of muscle forces squared) [Schultz and Mombaur, 2009, Ozsoy, 2014], me-chanical energy [Ren et al., 2007] and metabolic energy [Wang et al., 2012, Anderson and Pandy, 2001a], and be included in the cost function with assigned priorities. Equality constraints comprise the system dynamic equations, geometry constraints, movement phase transitions. Inequality constraints contain boundary conditions of actuators and generalized coordinates, foot-ground contact behavior, foot clear-ance, self-collision and so on.
Based on the form of the cost function, the methods of optimization-based approach can be divided into two categories: static and dynamic optimization. Static optimization evaluates the performance
Figure 1.3: Muscles were modeled as massless linear actuators. (a) The model included 80 muscle-tendon units (40 per leg) actuating the lower limbs. (b) Muscles with broad attachment areas (e.g., gluteus medius) were modeled using multiple independent muscle-tendon units. (c) Muscle geometry was modeled using a set of body-fixed points (highlighted) and wrapping surfaces. (d) The force transmission mechanism between the quadriceps and patellar ligament was modeled implicitly by wrapping the quadriceps muscles over the patella and inserting the muscles directly to the tibia. Image and caption taken from [Rajagopal et al., 2016]
Figure 1.4: Comparison of (a) current and (b) future treatment design paradigms. The current paradigm relies on an implicit mental model in the mind of the clinician. Given clinical and imaging data and proposed treatment parameters as inputs, the implicit mental model produces a subjective prediction of post-treatment function, so different clinicians may propose extremely different treatment designs. The future paradigm replaces the implicit mental model with an explicit computational model that obeys laws of physics and principles of physiology. With this approach, movement data are added to the inputs, the explicit computational model produces an objective prediction of post-treatment function, and the entire process is wrapped in numerical optimization to identify the treatment design that will maximize the patient’s functional outcome. Image and caption taken from [Fregly, 2021]
Figure 1.5: Dynamic formulation workflow. Forward dynamic formulation follows the path from 0 to 6 for a neuromusculoskeletal model, taking neural controls as input to compute motion. Inverse dynamic formulation follows the path from 6 to 0, solves neural controls from motion . Image taken from [Ezati et al., 2019]
criterion limited to quantities that can be computed at any instant in time during a simulation such as total actuator forces or stability margin. On the contrary, dynamic optimization can be constructed from quantities evaluated over a period of time such as total metabolic energy over a period of time.
Optimization-based methods can also be characterized by the formulation of the dynamic: inverse or forward, as shown in Fig. 1.5. For inverse dynamics optimization, muscle and joint forces are solved from kinematic data, and eventually ground reaction forces acquired via force plates. In forward dynamics problem, muscle forces/activations are optimization variables for the optimization process to estimate model kinematics. One of the most valuable aspects of this approach is movement synthesis as it offers the potential to validate hypotheses and predict functional outcomes without requiring experimental data. However, as the dynamic of the system is integrated over time, forward dynamics optimization methods are much less computationally efficient than inverse dynamics ones.

Related Work

The combination of musculoskeletal modeling with experimental data as input of an optimization pro-cess is a prevailing method to simulate movement. There are several measurements that can be used: electromyography (EMG) signals, 3D kinematics of body segments, especially lower limbs, foot-ground reaction forces, joint torques and energy consumption [Sutherland, 1978].
Tracking captured motion data is a common optimization task that allows to produce human-like move-ment. If the tracking motion task is dominant in the cost function of the optimization problem for-mulation, meaning it is associated to a high weight relative to other tasks, the results are indepen-dent from the task conditions [Ezati et al., 2019]. These optimisation methods can be supplemented with experimentally acquired electromyography (EMG) data to constrain estimated muscle activations [Pizzolato et al., 2015, Lloyd and Besier, 2003]. Calculated muscle activation and force patterns allow researchers and clinicians to imply any changes in muscle function due to different pathologies, or loco-motion strategies [Killen et al., 2020].
Static Optimization [Rosenberg and Steele, 2017] simulated impacts of ankle foot orthoses on muscle demand and recruitment in typically-developing children and children with cerebral palsy and crouch gait by using static optimization. The optimization process estimates muscle forces by minimizing the sum of squared muscle activations required to generate experimental kinematics and ground reaction forces at each instance. Static optimization is also used in [Steele et al., 2012] to estimate muscle forces and compressive tibiofemoral forces during crouch gait. First, the generalized coordinates of the model during the movement were computed by a static optimization process to minimize the errors between computed and experimental marker positions. Then, the known motion is used as input for another static optimization to compute the joint moments and muscle forces. The cost function is a weighted sum of muscle activations to be minimized where weighting values are imposed to be between 0 and 1 while the constraint is the dynamic equation. In the same manner, [Vandekerckhove et al., 2021] also used static optimization to « explore the effect of hip muscle weakness and femoral deformities on the gait performance of CP and typical developing subjects ». The results suggested that « surgical correction of femoral deformities was more likely to be effective than strength training of hip muscles in enhancing CP gait performance ». [Thelen et al., 2003, Thelen and Anderson, 2006] designed a method called Computed Muscle Control that requires only one integration of the state equations. The method couples static optimisation with feedforward and feedback controls to estimate both muscle activations and forces allowing the tracking of experimental kinematics data.
Dynamic Optimization In the study of the interplay between the elements of the neuromusculoskele-tal system during a movement, it is desired to alter some of model parameters to determine the subse-quent consequences. This method is generally unavailable in experiments but can be easily performed with forward dynamic simulation [Thelen et al., 2003]. For example, vertical jumping was simulated in [Zajac, 1993] with forward dynamics formulation and optimal control and showed that « jump height was more sensitive to muscle strength than to muscle speed, and insensitive to musculotendon compliance » and « uniarticular muscles generate the propulsive energy and biarticular muscles fine-tune the coordination ». [Allen and Neptune, 2012] built a 3D modular human walking control based on simulated annealing algo-rithm [Goffe et al., 1994], an optimization algorithm to find global optimum. Muscle excitation patterns and initial joint velocities were computed through optimization to minimize muscle stress and the dif-ference between the simulated and experimentally measured walking data including pelvis translations, trunk, pelvis, hip, knee and ankle joint angles and GRFs, and muscle stress. Forward dynamic optimiza-tion (Fig. 1.5) is time consuming, as demonstrated in [Anderson and Pandy, 2001a] where it took a CPU time of 10,000 hours (the wall-clock time was lower as 32 processors were running in parallel) to compute muscle activations of a half-cycle human gait. In later work, [Anderson and Pandy, 2001b] found that muscle forces computed by static and dynamic optimization were « remarkably similar ».
Null space projection This technique consists of the decomposition of the total torque into two dynamically decoupled torque vectors: the torque corresponding to the commanded task behavior and the torque that only affects posture behaviors in the null space provided by the kinematic redundancy of the musculoskeletal system, as demonstrated in [Khatib et al., 2009] to synthesize human motion. However, this method presents several drawbacks, such as the difficulty in implementing time-dependant constraints, unnatural movement and unrealistic joint torque profiles.
Optimal control Optimal control drives the model from an initial state to a final state by solving for the histories of control and parameters to minimize a cost function, as defined in [Xiang et al., 2010] The prob-lem of quantifying muscular activity of the human body can be formulated as an optimal control problem [Kaplan and Heegaard, 2001]. [Pandy et al., 1995] developed an optimal control model to simulate rising from a chair and introduce the time derivation of force as a new performance criterion. This approach was found to minimize the peak forces developed by the muscles. In [Kaplan and Heegaard, 2001], neural ex-citation for steady pedaling is solved using optimal control with second-order direct collocation. For large scale systems, direct collocation can reduce considerably the amount of computation time by discretizing the differential equations. However, implementation is laborious since first and second derivatives of the state equations with respect to the control variables are formulated analytically [Thelen et al., 2003].
Predictive Control Accurately tracking trajectories of some extremities is not how humans naturally move. Unlike robots, most people are not skilled enough to follow perfectly both kinematic and force trajectories. It is suggested in [Azevedo et al., 2004] that during walking, « humans perform global pro-gression of the whole body, without trying to track a specific trajectory, while minimizing the overall energy consumption and correcting any trend to fall ». To validate these hypotheses, model predictive control were used to synthesize walking gaits for biped robots. The method is basically a constrained non-linear optimization problem with a moving horizon. A set of coherent physical constraints was established related to standing posture, locomotion rhythm, static and dynamic equilibrium. However, integrating the dynamics equation of a musculoskeletal model is computationally expensive, even with reduced num-ber of muscles [Yamaguchi and Zajac, 1990] or simplified muscle model [Neptune and Hull, 1998].
[Chung et al., 2015] predicts 3D human running (joint kinematics and torques) along curved paths with-out data-tracking. The cost function of the optimization problem consists of multiple tasks including minimizing dynamic effort and upper-body yawing moment to avoid slipping on the ground. Simula-tion results from this methodology show good correlation with experimental data obtained from human subjects.
[Falisse et al., 2019] Rapid predictive simulations with complex musculoskeletal models suggest that di-verse healthy and pathological human gaits can emerge from similar control strategies Disadvantages of optimal control approach Optimal control and dynamic optimization approaches can be used to draw insights into the principles behind human movements and muscle functions [Hamner et al., 2010]. However, defining the initial states and guesses to facilitate the convergence of optimal solutions is usually challenging, especially with models of high complexity like human bodies.

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Contributions

Our motivation behind this work is to design a tool based inspired from robotics to study how neuro-muscular impairments contribute to abnormal movement, and to predict the functional consequences of a modification of the musculoskeletal system. At present, the most common approach is using experimental captured data as input of simulations of 3D gait analysis and human movement generation. Although the results allow clinicians to identify the potential principals of a movement, the dependency on the capture data limits the prediction capacity of this approach. It is proposed in this work to address physics-based motion synthesis using a reactive optimization-based dynamic task controller. The controller architecture is inspired from whole-body controller in robotics which has shown interesting results with humanoids as well as in the other domains of robotics. Without requiring experimental data related to kinematics or contact forces, the method consists in computing joint kinematics and muscle activities by solving optimization problems describing the strategy of producing a target movement. The formulation of the optimization problems can also be modified to reflect strategy adaptation to different conditions and musculoskeletal modifications. We believe that this method has the capacity of predicting the implication of musculoskeletal modifications on movement and muscle behavior. As a result, it can be incorporated in the development of physics-based simulation tools that provide clinicians valuable information to be considered in orthopaedic treatment planning.
In chapter 2, a detailed description of the human body modeling is presented, with emphasis on actuator modeling. Linear actuators are our choice of modeling muscle actuation and their advantages are specified. Then, the review of whole-body controller is presented based on the work of [Lober, 2017], consisting of the task and constraint definitions as well as prioritization schemes.
The first contribution of this work is the development of a reactive optimization-based dynamic task controller, presented in chapter 3. We demonstrate how the robotic whole-body controller is adapted as a two-step optimization-based scheme for simulating movements with musculoskeletal models. The principal idea is to decouple to torque actuation space and the muscle actuation space by leveraging the linearity of actuator models. In this way, the controller takes advantage of both the low complexity of joint-actuated models and the characters of muscle-actuated models and demonstration through synthe-sis of two different strategies of squat under different conditions. Demonstrations of squat synthesis via the controller are performed with fixed-base and floating-base human models. The physical simulations are built and run with OpenSim, a biomechanical open-source software providing tools to model mus-culoskeletal systems and physical environments. The controller is programmed with the OpenSim C++ API. In parallel, experiments are carried out to collect data related to body kinematics, force-ground con-tact forces and electromyographic signals. The numerical models are also scaled to reflect the morphology of the subject so experimental data can be used to compare with simulated data. The proposed controller is able to generate major features of the movement strategies under different performance conditions and with different types of human models.
The second contribution is the application of our physics-based simulation method as a prediction tool of the effects of surgical interventions of lower limbs on human movements. As mentioned previously, the need of testing different surgical scenarios and their respective outcomes, on a specific patient, in order to personalize the surgical plan is of great demand. We investigate the performance of our method in two orthopaedic interventions. In the first one (Chapter 5), we assess the influence of tendon attachment points on the outcome of rectus femoris transfer surgery, a stiff-knee gait treatment. Based on the simulation results, we confirm the clinicians’ choice of attachment point in terms of related to muscle moment arms and forces. In the second one Chapter 6), we investigate the implication of the combination of patellar tendon advancement (PTA) and distal femoral extension osteotomy (DFEO), which are crouch gait treatments, on knee extensors’ activities. The outcomes give rise to a perspective of a more complete tool with the goal of optimizing this treatment method. Finally, conclusions and perspectives of our approach are discussed in chapter 7. Overall, the results are encouraging and suggest that further developments could be extended to other types of movement, notably walking, to demonstrate potential distinguished utilities of the approach.

Human Modeling and Whole-Body Control

Human movement synthesis is a predictive simulation. The approach has the potential of identifying the causal relationship between musculoskeletal elements and movement strategies without relying on experi-mental data. The methods based on this approach can offer clinicians the possibility of studying the influ-ence of isolated neuro-musculoskeletal features by adjusting model parameters [De Groote and Falisse, 2021].
To synthesize human-like movement, there are two aspects to be considered: biomechanical human mod-eling and movement synthesis method. Human modeling takes into account how skeletal and actuation structures are represented, connected and interacting with each other in accordance with the laws of physics. To model the musculoskeletal profile of a particular individual, each body segment, actua-tion element and joint are characterized by different personalized parameters related to mechanical and physiological properties. In the context of modeling patients suffering musculoskeletal deficiencies, these elements are modified to reflect not only impairments but also modifications as the outcomes of surgical interventions. For example, impaired muscles can be reflected by changing the physiological parameters of fibers and tendons; changing tendon attachment points represents tendon release and reattachment in tendon transfer procedures. With a personalized model, a synthesis method is established representing the motor strategies of the central nervous system (CNS). The common approach is to assume that the performance of a movement is optimized in terms of one or multiple criteria, e.g. the metabolic cost, while taking into account the properties of the musculoskeletal system. In this work, we develop our method based the whole-body control theory in robotics and Quadratic Program (QP). The movement is synthesized as an optimized process where multiple objectives and constraints are involved. Control objectives are corresponded to high-level physical-relevant intentions, such as lifting a body part, reaching targets or saving energy. Constraints are related to the dynamic feasibility and system intrinsic limits. This chapter presents the detailed descriptions of biomechanical human modeling and the structure of the controller. They allow us to develop our controller schemes for movement synthesis.

Musculoskeletal Modeling

Once the topology of the model is defined, the actuation system is designed in terms of complexity, along-side with the contact modeling contributing an important role of representing the interaction between the human body and the environment. In the end, the forces applied by actuators and contact wrenches produce body accelerations in accordance with the equation of motion. These aspects will be detailed in the following sections.
Figure 2.1: Kinematic representation of floating-base systems. The floating-base is connected to the inertial frame through a 6-DoF joint. Image taken from [Mistry et al., 2010]

Table of contents :

1 Introduction 
1.1 Human Body Modeling
1.2 Optimization – The Backbone of Human Movement Generation
1.3 Related Work
1.4 Contributions
2 Human Modeling and Whole-Body Control
2.1 Musculoskeletal Modeling
2.1.1 Multibody Dynamics
2.1.2 Actuator Modeling
2.1.3 Contact Modeling
2.2 Whole-Body Control
2.2.1 Constraint Formulation
2.2.2 Task Formulation
2.2.3 Prioritization strategies
3 The two-step optimization based synthesis approach of squat movement 
3.1 Overview of The Squat Movement
3.1.1 Squat Execution
3.1.2 Joint Kinetics and Kinematics During The Squat
3.2 The two-step optimization scheme
3.2.1 Step I – Reactive Optimization-based Dynamic Task Control
3.2.2 Task Switch
3.2.3 Step II – Muscle Activation Synthesis
3.3 Simulation Example: Half Squat Synthesis with A Low Number of Muscles
3.3.1 OpenSim
3.3.2 Human Model
3.3.3 Results
3.4 Simulation Example: Half Squat Synthesis with A Great Number of Muscles
3.4.1 Human Model
3.4.2 Results
3.5 Conclusions
4 Synthesis of Different Squat Strategies and Patterns With The Optimization-based Dynamic Task Controller
4.1 Synthesis of Asian Squat
4.1.1 Overview of Asian Squat
4.1.2 Method
4.1.3 Results and Discussions
4.2 Synthesis of Two Different Squat Strategies
4.2.1 Method
4.2.2 Results and Discussions
4.3 Conclusion
5 Simulation of The Effect of Different Rectus Femoris Transfer Sites on Muscle Recruitment During Knee Extension 
5.1 Method
5.1.1 Computational Modeling of RF Transfer
5.1.2 Results
5.1.3 Discussions
5.2 Conclusions
6 Prediction of Muscle Activities After Femur Osteotomy and Patellar Tendon Advancement
6.1 Computational Modeling
6.1.1 Distal Femoral Extension Osteotomy (DFEO)
6.1.2 Patellar Tendon Advancement (PTA)
6.2 Method
6.2.1 Simulation
6.2.2 Results and Discussions
6.3 Conclusion
7 Conclusions and Perspectives 
7.1 Contributions
7.2 Limitations and Perspectives
7.2.1 Muscle Modeling
7.2.2 Reactive Control Strategy
7.2.3 Experimental data
7.2.4 Towards Gait Synthesis

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