Model-Based Control of Biomechatronic Systems

CHAPTER FOUR

Model-Based Control of Biomechatronic Systems Naser Mehrabi*, John McPhee† *University of Washington, Seattle, WA, United States † Systems Design Engineering, University of Waterloo, Waterloo, ON, Canada

Contents 1 Biomechatronic System Models 1.1 Mechatronic System Modeling 1.2 Biomechanical Modeling 1.3 Integrated Biomechatronic Models 2 Model-Based Control Design 2.1 Model-Based Open-Loop Control 2.2 Model-Based Closed-Loop Control 3. Case Study: Design of Population-Based Electric Power Steering Systems 3.1 Introduction 3.2 Dynamic Model of Biomechatronic System 3.3 Electric Power Steering (EPS) Control Design 3.4 Simulation Results 4 Conclusions References Further Reading

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1 BIOMECHATRONIC SYSTEM MODELS Biomechatronics is an applied multidisciplinary science that integrates biology, mechanics, and electronics to develop devices that support and assist humans. Based on this broad definition, biomechatronic devices include a wide range of applications, from human prostheses and exoskeletons to driver-assist systems in vehicles. These devices usually consist of a mechanical system actuated with electrical actuators, wherein a controller coordinates the mechatronic system response based on the user’s intention and predefined logic. In this chapter, we focus on the model-based design of these controllers. Handbook of Biomechatronics https://doi.org/10.1016/B978-0-12-812539-7.00004-0

© 2019 Elsevier Inc. All rights reserved.

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1.1 Mechatronic System Modeling The first step in the design of a biomechatronic device is to understand how the device will interact with its user and the environment. A successful device considers physiology to reasonably enhance human body movements or compensate for lack of movement. A dynamic model of a biomechatronic device can provide in-depth insight into its dynamic behavior and can be used to design and evaluate model-based control systems. The system model can also be used in model-in-the-loop (MIL) simulations to improve systems design. MIL simulations accelerate the design process by saving time in developing and revising the design on the computer rather than physically creating new prototypes. MIL simulations offer many other advantages such as flexibility (i.e., allow various scenarios) and repeatability (i.e., perform the same experiments repeatedly). Various methods can be used to derive dynamic equations of motion of an multidisciplinary device such as energy-based methods, linear graph theory (McPhee, 1996), and bond graph theory (Karnopp et al., 2012).

1.2 Biomechanical Modeling To design a device for assisting human movements, it is crucial to understand how the human body works. By only contracting the skeletal muscles, our body can produce very complex and meaningful movements such as walking and reaching. All these actions are initiated by thoughts in the brain and then conveyed through the nervous system to the muscles attached to our skeleton. Some brain activities (i.e., readiness potential) can be produced up to 1 s before the actual volitional movement, and can be captured using electroencephalography (EEG) (Brinkman and Porter, 1979; Deecke and Kornhuber, 1978). EEG is a method that captures the brain’s electrical activities by placing noninvasive electrodes along the scalp. These movement initiations are transmitted through the central nervous system (CNS) to the motor neurons that innervate muscle fibers. Then, after a sequence of chemical reactions, the muscle fiber contracts and produces a change in potential in the muscle membrane. This electrical activity produced during muscle contraction can be picked up through electromyography (EMG) using an electrical sensor placed on or under the skin above the muscle of interest. EEG and EMG are windows to our brain because they record signals originating from the brain and thus can be used to capture user intention. There are several assistive devices available in the market [e.g., prostheses and brain-computer interfaces (BCIs)] that take advantage of these signals to understand user intention and control a device.

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The biomechanics of human movement can be simulated in computers through inverse and forward dynamics simulations. The natural flow of human motion starts from the motor-neuron spikes in the CNS (i.e., including the brain and spinal cord) leading to the production of muscle twitches and a force pulling the bones to reach the desired position. A forward dynamic simulation can properly capture these neuromuscular dynamics since it follows the same natural flow. Equations of motion are integrated forward in time to obtain motion trajectories from neuromuscular inputs. In contrast, an inverse dynamics approach processes information in the opposite direction: the measured joint trajectories and limb motion and external loads from a motion capture system and force sensors are the simulation inputs, and the muscle twitches are the simulation outputs. While an inverse dynamics approach is useful for clinical decision making, it cannot explain the underlying cause-and-effect relationships between motor neuron-spikes and system kinematics. The forward dynamic simulation can also be used to simulate what-if scenarios such as what happens if the stiffness of a foot-ankle orthoses increases? The biomechanical model parameters can be adjusted to represent different individuals with various physical abilities and disorders. 1.2.1 Inverse Dynamic Simulation To study the biomechanics of a task, one can measure the kinematics (motion) and perhaps a portion of kinetics (e.g., external loads) of that particular task in the laboratory. The kinematics can be measured using optical movement-monitoring systems with active or passive markers (e.g., Optotrak and Vicon motion capture systems, respectively) or with a markerless system (e.g., Microsoft Kinect), or using other movement assessment tools such as electro-goniometers and inertial measurement units (e.g., MVN suit). Force sensors can measure external loads applied to the body (e.g., foot-ground reaction forces during walking). Knowing the kinematics and external forces acting on the system, one can compute the required generalized forces (e.g., net joint torques and forces) to perform the given task by means of an inverse dynamic simulation. Before an inverse dynamic simulation can be performed, the equations of motion representing the task should be extracted using a dynamic modeling method: x_ ðt Þ ¼ f ðxðtÞ, T ðtÞ, F ðt ÞÞ gðxðtÞ, T ðt Þ, F ðtÞÞ ¼ 0

(1a) (1b)

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where x represents the model coordinates (e.g., positions and velocities) and T and F represent net joint torques and the external loads acting on the system. Eq. (1b) represents the kinematic constraints that restrict the movements of two rigid bodies relative to each other (e.g., joints). Since x and F have been measured beforehand in the laboratory, the joint torques (T) can be computed by simply substituting the measured kinematics and external loads into Eq. (1) and evaluating T at each time step. The torque and force requirement of a task are important to know when designing a system controller and its actuator capacity. For example, the maximum ankle torque during normal walking can be used to select the stiffness or the actuator power of an ankle-foot orthosis (AFO), which is a wearable assistive device that supports and corrects ankle motion. If muscle-level information is required, a static optimization can be performed to resolve the muscle indeterminacy problem and compute the share of each muscle contributing to the resultant joint torque. The muscle indeterminacy problem results from the number of muscles crossing a joint exceeding the degrees of freedom of that joint; it is difficult to identify individual muscle forces because different combinations of forces can produce the same net joint torque. To resolve this problem, the static optimization is subjected to the torque equilibrium equation: Tj ¼

n X

rim, j Fi

(2)

i¼1

where rm i,j represents the moment arm of the muscle force Fi about the joint j, and index i refers to the individual muscles crossing the joint of interest. A unique muscle activation pattern similar to that of humans can be achieved by minimizing a physiological cost function during static optimization, such as J¼

n X

p

ai

(3)

i¼1

where a is the muscle activation level at the current time step, n is the number of muscles crossing the joint, and p is an exponent (usually, p ¼ 2). The inverse dynamics can only provide insight into a task whose kinematics and kinetics have already been measured in the laboratory, and it cannot predict the dynamics of a new task based on previously measured data.

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1.2.2 Predictive Simulation A predictive simulation is a forward dynamic simulation that can predict the kinematics and kinetics of a task of interest based on the underlying physiological phenomena governing its dynamics. In these simulations, a mathematical controller representing the human CNS coordinates the movements of the biomechanical model for that task. However, to develop such a controller, we should first understand how our CNS controls our body. As first formulated by Bernstein (1967), the CNS simultaneously coordinates the kinematics and kinetics of body motions, despite uncertain (future) trajectories and the redundancy in muscle actuators. As an example, during reaching and pointing tasks, where only the final position of the hand is specified, an infinite number of hand trajectories (and muscle activation patterns) can be expected to reach the target. However, despite the possible variations, individuals usually choose a similar trajectory. The early observations of reaching and pointing tasks led to the well-known “Minimum-X” models (e.g., minimum-jerk model (Flash and Hogan, 1985; Wada et al., 2001), minimum-torque-change model (Uno et al., 1989), minimumvariance model (Harris and Wolpert, 1998), and minimum-work model (Soechting et al., 1995)) to predict the hand trajectory. These models hypothesize that the CNS coordinates the body movement such that an exertion (X) is minimized. Later, this hypothesis was extended to consider physiologically motivated exertions such as muscle activation effort (Crowninshield and Brand, 1981; Ackermann and van den Bogert, 2010; Happee and Van der Helm, 1995), metabolic energy expenditure (Anderson and Pandy, 2001; Peasgood et al., 2006), and muscle fatigue (Sharif Razavian et al., 2015). In computer simulations, the Minimum-X model has been successfully implemented using dynamic optimization (DO) to predict the normative human motion for a given task. A common DO approach parameterizes the muscle activation profiles for the period of motion and searches the feasible space to find the profiles that minimize X (Anderson and Pandy, 2001; Davy and Audu, 1987; Yamaguchi and Zajac, 1990; Neptune and Hull, 1998; Kaplan and Heegaard, 2001; Sha and Thomas, 2013). This approach provides an open-loop (feedforward) command of muscle activations to control the given task. This command can represent the descending command of a well-repeated/well-learned task (e.g., platform diving (Koschorreck and Mombaur, 2011)). In this approach, the CNS only recalls the learned information, and does not intelligently adjust the commands in real time.

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However, during conscious voluntary movements, the CNS has to continuously update the motor commands to correct for errors (Todorov, 2004). For example, previous studies on pointing and reaching (Sarlegna and Pratik, 2015) have shown that the CNS constantly updates the hand trajectory based on sensory (feedback) information. This sensory information can be received from vision, proprioception, audition, the vestibular system, and internal models that can predict the motion (Desmurget and Grafton, 2000). A few studies have used feedback controllers to coordinate the movements of a musculoskeletal model. The linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) optimal feedback control methods have been applied to a linear arm model to describe the hand trajectory (Harris and Wolpert, 1998; Todorov and Jordan, 2002; Liu and Todorov, 2007). Later, to control the nonlinear dynamics of the neuromuscular system, an iterative LQG (iLQG) controller has been developed, in which the nonlinear model is iteratively linearized (Todorov and Li, 2005). Recently, a nonlinear model predictive control (NMPC) has been used to mimic the CNS during reaching tasks (Mehrabi et al., 2017). This near-optimal controller uses a nonlinear model to predict the reaching dynamics over a finite horizon ahead of the current time, and uses the sensory information as feedback to correct the prediction errors. Depending on the application, the CNS can be modeled as either a feedforward or feedback controller, or as a combination of both. A control system with both feedforward and feedback components is preferred because it performs better and is more robust to external disturbances.

1.3 Integrated Biomechatronic Models Having a clear understanding of the dynamical system is crucial in designing a controller, since not only does it strengthen our knowledge about the system but also it reduces development time and cost. A predictive simulation of an integrated model of the biomechatronic device and its user for the task under study allows replicating the user-device interaction in silico (Ghannadi et al., 2017; Mehrabi et al., 2015a). This platform can be used to improve the device and controller design without going through the conventional and cumbersome trial and error design methods. Now that we introduced different approaches to develop and simulate biomechanical models, we will describe the benefits and deficiencies of different modelbased control techniques that can be used to operate various biomechatronic devices.

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2 MODEL-BASED CONTROL DESIGN Control systems can be categorized as either open loop or closed loop, depending on their structure. A closed-loop control system regulates control actions based on the information received through a feedback loop. There is no feedback loop in an open-loop system; thus, no further control action adjustments can be made. Both of these control systems, depending on their design methodology, can be categorized into model-based or error-based controllers. Model-based controllers exploit a physical or nonphysical model to estimate system dynamics and predict the system’s response to a control action. This category includes optimal, robust, and nonlinear control methods. Error-based controllers use only an error signal (the difference between the desired and actual trajectories) to control the system. Classic proportional-integral-derivative (PID), sliding mode, and fuzzy controllers are the most well-known controllers in this category. Since model-based control methods can consider physiological constraints and often outperform their counterparts, we will focus on model-based control methods in this section.

2.1 Model-Based Open-Loop Control An open-loop control system is a control system in which the system output does not influence the control actions [shown in Fig. 1A]. In an open-loop control system, a sequence of control actions is precomputed and stored in a feedforward controller, then executed when a trigger is activated. Once the control action is initiated, it cannot be adjusted based on the system response or external loads acting on the system. Feedforward controllers are usually used when there is no feedback available or the interaction with the

Fig. 1 Schematic representation of (A) an open-loop control system and (B) a closedloop control system.

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environment is reasonably predictable. For example, in functional electrical stimulation (FES) of foot drop, a predefined sequence of electrical impulses stimulates the appropriate muscles to raise the forefoot at the appropriate time during a gait cycle (when a trigger is activated) (Stein et al., 2006). Foot drop is a pathological gait disorder in which the forefoot drags on the ground during walking. Foot drop usually occurs because of muscle weakness or neuromuscular disorders. The control sequence can be achieved through trial and error experiments or by using a DO method. For example, for FES of gait, an optimal sequence of muscle activations can be achieved through DO of a biomechanical model of gait. The major advantage of this method over the trial and error approach is that in DO, a criterion such as applied electrical stimulation can be minimized so that the onset of muscle fatigue occurs later in therapy. A DO can be solved through direct and indirect optimal control methods. An indirect method finds an optimal solution by reformulating the original control problem such that the necessary conditions of the optimality are satisfied. In the indirect methods (optimize and then discretize), the optimal control problem is converted to a two-point boundary value problem (2PBVP) by applying Pontryagin’s minimum principle. The solution of the 2PBVP provides an optimal solution for the original problem. In a typical direct solution (discretize and then optimize), the dynamic equations are discretized using a numerical integrator; combined with the cost function, the result is a relatively large nonlinear programming (optimization) problem, or NLP. These NLPs can be solved using specially designed optimizers (e.g., IPOPT (Wachter and Biegler, 2006) and SNOPT (Gill et al., 2005)) that exploit the sparsity pattern that exists in such problems. Although this is one of the most common techniques for formulating a direct optimal control problem, there are many other methods (e.g., multipleshooting and direct collocation) that exist in the literature. Overall, indirect methods may be very sensitive to the initial values and to the changes of the unspecified boundary conditions in the 2PBVP. In contrast, direct methods usually have better convergence properties, and the user doesn’t need to worry about the costate variables that appear in indirect methods. However, in the presence of many local extrema, direct methods may converge to a local extremum (Betts, 1998). Although these approaches employ different philosophical approaches, the techniques may ultimately merge. Interested readers are referred to Rao (2009) for more information about indirect and direct optimal control techniques.

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2.2 Model-Based Closed-Loop Control As shown in Fig. 1B, in a closed-loop control system with a feedback controller, the system outputs feedback to the controller to regulate the control action. Feedback controllers based on system dynamics can be categorized into linear and nonlinear feedback controllers. 2.2.1 Linear Control Theory A linear system is a system whose dynamics obey the superposition principle and whose equations of motion are composed of linear differential equations. Optimal and robust control theories of linear systems with quadratic cost functions have been well developed over decades and have been used in many practical applications (Kirk, 2013; Doyle et al., 2013). In this section, the linear quadratic (LQ) optimal control theory is presented. Consider the linear time-varying system with a state differential equation: x_ ðt Þ ¼ AðtÞxðtÞ + Bðt Þuðt Þ zðt Þ ¼ C1 ðt ÞxðtÞ + D1 ðtÞuðt Þ

(4)

where x, z, and u are system state variables, controlled variables, and control inputs; A, B, C1, and D1 are the time-varying matrix functions of time; and x0 is the state initial condition. Linear-Quadratic Control

The LQ control law is optimal concerning a quadratic integral performance criterion, as shown below: Zt1 J ¼ x ðt1 ÞP1 xðt1 Þ + T

 T  z ðtÞR3 ðtÞzðt Þ + uT ðt ÞR2 ðt Þuðt Þ dt

(5)

t0

Here, R3(t) is a nonnegative-definite symmetric matrix that determines the weighting of each element of the controlled variable z. The quantity zT(t)R3(t)z(t) shows the error of the controlled variable z with respect to zero at time t. R2(t) is a positive-definite symmetric weighting matrix that is used to reduce the control effort. If needed, a terminal state condition can be added to the objective function with a nonnegative-definite symmetric matrix P1 [see the first term in Eq. 5] such that the state x(t) at the final time t1 is as close as possible to zero. The optimal feedback controller with respect to the performance criterion shown in Eq. (5) is in the form of a linear full-state feedback controller (Kirk, 2013) as shown in Fig. 2, and the optimal control law is

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Fig. 2 Schematic representation of a full-state feedback controller.

Fig. 3 Schematic representation of a full-state feedback controller with a state observer.

uðt Þ ¼ K ðt ÞxðtÞ

(6)

K ðtÞ ¼ R21 ðtÞBT ðtÞP ðtÞ

(7)

where

and P(t) is computed from the solution of the following matrix Riccati equation: P ðtÞ ¼ R1 ðtÞ  P ðt ÞBðtÞR21 ðtÞBT ðtÞP ðtÞ + AT ðtÞP ðtÞ + P ðt ÞAðt Þ (8) where R1 is equal to DT(t)R3(t)D(t), and the Riccati equation should be solved backward in time with the final condition of P(t1) ¼ P1. It is not easy and sometimes even infeasible to measure all the individual state variables required for a full-state feedback controller. In many cases, the measurements are restricted or are a function of a few different state variables, and they may also include measurement noise. One solution is to construct unavailable states from the available measurements (y) and controls (u) using a dynamic system called an observer (Fig. 3).

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Linear State Estimation

In this section, we introduce an optimal state observer called the Kalman filter (KF). A KF is a data processing algorithm that estimates the current value of the state variables of interest using the available information. A KF incorporates all the available measurements to estimate the current state variables by considering the system and measurement device dynamics, the statistical significance of the measurement and system noise, and the available information about the system’s initial condition. Here, consistent with continuous LQ control, a continuous KF is introduced. Consider a linear time-varying continuous-time system: x_ ðtÞ ¼ AðtÞxðtÞ + BðtÞuðtÞ + w ðtÞ yðtÞ ¼ C ðtÞxðtÞ + vðt Þ

(9)

Here, y(t) is the measurement variable, and C is a continuous time-varying matrix; w(t) and v(t) are Gaussian white noise with zero mean value and Q and R are covariance matrices that represent process noise and sensor noise, respectively. The process noise represents the uncertainty in the system model, and sensor noise is usually used to show uncertainty in the measurements. Q(t) and R(t) are symmetric and nonnegative definite matrices in which each element represents the covariance of the corresponding measurement or system noise. The initial state x(t0) is also assumed to be Gaussian random variable with a mean value of x0 and a covariance Pe0. A KF is an optimal state observer in which the state estimation x^ðtÞ is computed in a way that the expected value of the estimation error squared is minimized   (i.e., E ðxðt Þ  x^ðtÞÞðxðtÞ  x^ðtÞÞT ). The continuous-time KF observer is in the following form:  x^ðtÞ ¼ A^ xðtÞ + BuðtÞ + L ðtÞðy  C^ xðt ÞÞ (10) x^ð0Þ ¼ Efxðt0 Þg where L(t) is often called the Kalman gain from: L ðt Þ ¼ Pe ðtÞC T R1 P_ e ¼ AP e + Pe AT + Bw QBTw  Pe C T R1 CP e

(11)

which solves forward in time with the boundary condition Pe(t0) ¼ Pe0. Based on the separation principle (Kirk, 2013), the optimal control input can be determined by feeding the estimated states instead of the measurements into Eq. (6). Then, the optimal feedback control becomes: uðt Þ ¼ K ðtÞ^ xðtÞ

(12)

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Here, K(t) is the same gain array obtained from optimal control feedback in Eq. (7). With the substitution of the control law with the observer Eq. (10), the controller equations take the following form: xðt Þ + L ðt Þyðt Þ uðt Þ ¼ K ðt Þ^ xðt Þ x^_ ðt Þ ¼ ½Aðt Þ  Bðt ÞK ðt Þ  L ðt ÞC ðt Þ^

(13)

KFs are used to estimate the system state variables from indirect and noisy measurements that are common in mechatronic systems (e.g., force sensors). LQRs in conjunction with KF can be used to implement a biomechatronic system device control logic while minimizing a cost function (e.g., electrical energy consumption). As an example, this method can increase the battery life of untethered biomechatronic devices or just simply decrease the device energy consumption.

2.2.2 Nonlinear Control Theory Nonlinear control theory covers a larger class of systems and can be used for a wider range of real-life problems. Nonlinear systems do not obey the superposition principle, and the equations of motion are governed by nonlinear differential-algebraic equations (DAEs). A nonlinear system can be linearized (approximated with a linear system) by use of Taylor series expansion or perturbation methods around an operating point, and then a linear control theory can be applied to design a controller for the nonlinear system. However, the linear model is only valid if the model varies in the sufficiently small range about the operating point, while nonlinear controllers can incorporate nonlinear models to guarantee performance under nonlinear phenomena (e.g., limit cycles, multiple equilibria). In this section, we focus on the NMPC method that has attracted attention both in industry and academia in recent years. NMPC has been widely used in the chemical industry, where a lower sampling rate is required, but recently it has been applied in other industries such as automotive and assistive devices. A NMPC can be considered as the general form of the LQ control method in which the controller uses a nonlinear model and can account for constraints on inputs and states. Moreover, the NMPC is not required to have a quadratic performance criterion. The NMPC includes both feedforward and feedback control schemes. The NMPC uses a controloriented model (COM) representing the physical system to predict the optimal dynamics in a finite time interval ahead of current time called the prediction horizon, and feedback information to correct the prediction errors.

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Fig. 4 Schematic representation of the nonlinear model predictive control (NMPC).

The NMPC predicts the optimal dynamics of the system ðx, uÞ over a prediction horizon as shown in Fig. 4 by minimizing the following cost function subjected to the nonlinear dynamic equations of motion:   J ¼ Ψ t0 + tph +

tZ 0 + tph

ψ ðxðtÞ, uðt ÞÞ dt

(14)

t0

where Ψ is the cost evaluated at the end of the prediction horizon, ψ is the cost evaluated during the prediction horizon, and tph is the length of prediction horizon. As shown in Fig. 4, the state variables at the current time (t0) are obtained from the current measurements or estimated with the aid of an observer. The input ðuÞ is an optimal open-loop solution over the prediction horizon. If there are no external disturbances and no model uncertainty in the system, with infinitely long prediction horizon, the open-loop solution can be applied to the system for all time t > t0. However, for the finite horizon case and in the presence of noise and uncertainty, the open-loop solution should only be applied until the next sampling time (t0 + δ). At the new time step, the optimal solution is re-evaluated with the new initial conditions for the receding horizon and iteratively applied to the system. By incorporating the feedback information, the NMPC is converted from a completely open-loop controller to an optimal closed-loop controller. The NMPC can handle constraints on both the states and the inputs.

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Controller Reference trajectory

(A)

NMPC

Controller

State variables

Control input

Reference trajectory

Plant System output

(B)

State variables MHE NMPC Control input

System output

Plant

Fig. 5 Schematic representation of a NMPC in conjunction (A) with moving horizon estimator (MHE) and (B) without moving horizon estimator.

The optimal dynamics over the prediction horizon can be calculated using any optimal control method. Several software packages can automatically formulate and execute an NMPC controller (e.g., YANE (Grune and Pannek, 2011), MUSCOD-II (Schafer et al., 2007), ACADO (Diehl et al., 2002), MPsee (Tajeddin and Azad, 2017), SCDE (Walker et al., 2016)). In the presence of incomplete measurements and for a constrained nonlinear system, an optimization method can be used to estimate the state variables. If all the measurements from the initial to the current time are used to estimate the state at the current time, the observer is called a fullinformation estimator. However, this technique is not suitable for real-time implementation, since the computational burden grows exponentially with time. By only considering the information in a window moving behind the current time, and approximating older information by a simple function, the computation time can be significantly reduced. This so-called “moving horizon estimator” (MHE) has been shown to work for real-time vehicle dynamics applications and rehabilitation robots with current computational resources (Fig. 5). The required online solution of the optimization problem can be computationally demanding, but can provide significant benefits in estimator accuracy and rate of convergence (Soechting et al., 1995). The optimal estimations at each given horizon (window) can be computed using indirect or direct optimal control methods (Crowninshield and Brand, 1981).

3 CASE STUDY: DESIGN OF POPULATION-BASED ELECTRIC POWER STEERING SYSTEMS In this section, we examine a case study in which a systematic modelbased method to design individualized electric power steering (EPS) systems for different driver populations is introduced. An EPS system is a biomechatronic driver-assist device because it is a mechatronic system that interacts with a human driver, and supports the driver to have a better

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driving experience. The driver-assist systems receive sensory feedback from the vehicle, and commands such as acceleration, brake, and steering from the driver. Here, four neuromuscular driver models representing drivers with different physical strength, age, and gender were developed. These models were used to design a new model-based EPS controller that adjusts the steering assistance based on the driver’s physical strength. In the proposed controller, the EPS characteristic curves (determining the steering assistance) were precomputed for the predefined driver populations and stored in the controller. The characteristic curves were optimized such that the drivers within different populations performing the same steering maneuver have a similar targeted “steering feel.” The steering feel was defined by a combination of drivers’ muscular effort and road feel. Finally, the new EPS controller was evaluated in MIL simulations using a high-fidelity integrated driver-vehicle model. The results showed that the tuned EPS controller could equally assist drivers with different physical strengths and abilities.

3.1 Introduction Emerging research has resulted in new models of the interaction dynamics between the vehicle and its driver, the results of which have given rise to new driver-assistance technologies—haptic gas pedals, lane keeping, artificial steering wheel torque feedback (Abbink, 2006), and EPS systems (Mehrabi and McPhee, 2014a; Farrelly et al., 2007). Steering feel and vehicle stability are two commonly used criteria in the design of EPS controllers. Vehicle stability measures are well documented in the vehicle dynamics literature (Karnopp, 2003), while there is only a limited literature available on quantifiable steering feel measures. Previous research has found correlations between steering feel and vehicle handling characteristics; however, these investigations were limited to a specific driver population (i.e., truck drivers) (Rothh€amel et al., 2011, 2014). Vehicle manufacturers typically employ professional drivers to tune steering systems to provide “good” steering feel. However, this approach has numerous drawbacks. Such experiments can be expensive, time consuming, and are subject to human error. In addition, the preferred steering feel is different for vehicles with different handling characteristics (e.g., sport vs luxury cars) (Bertollini and Hogan, 1999), and simultaneously the optimum steering feel may vary between driver populations (i.e., drivers with different physical abilities). For example, young drivers generally have stronger muscles, and thus greater ability to overcome resistive torques at the wheel, than elderly drivers. Therefore,

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it is unlikely to find a unique steering setting that provides optimum steering feel for the general population. This work does not directly deal with setting of the preferred steering feel for a specific vehicle type. However, when a preferred steering feel is set, the proposed EPS system can provide equal steering feel across different predefined driver populations. Realistic driver models can play a major role in accelerating the development of driver-assistance technologies by reducing the cost and time associated with physical experiments. The driver models are usually developed to assess the vehicle performance and not the driver preference (e.g., pathfollowing driver model). Few studies have developed driver-centered models that consider the driver’s physiology (i.e., neuromusculoskeletal system) (Mehrabi et al., 2015a; Cole, 2012). These models can be used to give insight about how our body interacts with the steering system. Understanding and quantifying these interactions facilitates the development of the next generation of driver-assistance technologies. A forward dynamic simulation can simulate the interaction between driver and vehicle, and also provide a platform to ask “what if” questions such as “what if a stronger driver steers the same vehicle.” These predictive simulations can support the design of individualized EPS controllers for different driver populations. Accordingly, the following work presents a systematic approach to standardize EPS systems (e.g., steering feel) for various driver populations by considering the human physiology.

3.2 Dynamic Model of Biomechatronic System In this section, we present the models and methods used to develop and verify an individualized EPS system. We have two integrated models of driver and vehicle that we will refer to as (1) high-fidelity and (2) simplified models. The simplified model was used to design the EPS system, and the highfidelity model was used in MIL simulations to verify the performance of the EPS controller. Finally, the characteristic curves and the EPS controllers used in these models are presented. 3.2.1 High-Fidelity Driver-Vehicle Model The high-fidelity integrated driver-vehicle model described in Mehrabi et al. (2015a) and shown in Fig. 6A was used to simulate real-world driving conditions. This model consists of a multibody dynamic model of a vehicle and a three-dimensional (3D) neuromusculoskeletal model of a driver. The muscle activities predicted by the neuromusculoskeletal driver model were verified against the electromyographic activities of a driver’s arm muscles

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Fig. 6 (A) High-fidelity integrated driver-vehicle model and (B) variation of the shoulder and elbow angles and the rotation of the humerus about the vertical axis for a sinusoidal steering wheel angle. The presented angles are consistent with the definitions recommended by the International Society of Biomechanics (ISB) (Wu et al., 2005).

during steering experiments (Mehrabi and McPhee, 2014b). In the first experiment, the driver was instructed to hold the steering wheel stationary against external torques (indicative of on-center steering); in the second experiment, a sinusoidal steering maneuver was performed to simulate a slalom maneuver (Hayama et al., 2012). Since real-life steering usually is a combination of these two tasks, this driver model can realistically predict muscle activities during everyday steering maneuvers. The DAEs used to describe the high-fidelity integrated driver-vehicle model are very complex and computationally expensive, and thus not suitable to be used within a real-time optimal control. Therefore, a simplified version of this model that conveys the important dynamics of the system has been developed. 3.2.2 Simplified Driver-Vehicle Model The simplified integrated driver-vehicle model consists of a linear vehicle model with a column-assist EPS system and a two-dimensional (2D) neuromuscular driver model. To develop the simplified driver model, we first studied the kinematics of the high-fidelity 3D driver model performing a sinusoidal steering maneuver. The modeled driver is holding the steering wheel at the 3 and 9 o’clock positions as suggested by Hayama et al. (2012), and the steering axis is parallel to the line connecting the shoulder to the steering wheel as shown in Fig. 7A. The suggested driver’s posture can be changed without substantially affecting the method and simulation

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Fig. 7 The simplified driver-vehicle model: (A) the two-dimensional (2D) musculoskeletal driver model and (B) the simplified vehicle model with a column-assist electric power steering (EPS) system.

results. The effect of changing the grip position on the moments of muscle forces on the shoulder and elbow was briefly discussed in Mehrabi et al. (2014). Fig. 6B shows the variation of elbow and shoulder angles when the 3D driver model performs a sinusoidal steering wheel angle with an amplitude of 45 degrees. In this research, we have used the Euler XYX convention to represent the shoulder’s plane of elevation angle (PEA), elevation angle (EA), and axial rotation, respectively, where X is along the humerus and Y is normal to X and toward the humerus lateral direction. The change of shoulder’s PEA and EA is significantly larger than the elbow flexion and extension angle. The standard deviation of the elbow angle from its mean value during this simulation is about 5 degrees while it is 22 and 18 degrees for the shoulder’s PEA and EA. As expected, the standard deviation of the shoulder’s axial rotation is small, around 3 degrees. The humerus rotation about the vertical axis (i.e., parallel to torso) is less than 5 degrees when steering wheel angle varies 14 degrees, depicting a mostly planar motion of the arm for small steering angles. Therefore, the shoulder in the simplified model was reduced from a spherical joint to a revolute joint, and the elbow joint has been assumed to be fixed. Based on these assumptions, a simplified 2D driver model as shown in Fig. 7A was developed, in which the arm segments move only in the sagittal plane of the driver’s body, pivoting at the shoulder. As shown by Jonsson and Jonsson (1975), the shoulder muscles are the prime movers in steering tasks and can be classified into two groups: the muscles providing clockwise torque and muscles providing counterclockwise torque on the steering wheel (Sharif Razavian et al., 2015).

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Accordingly, in the simplified driver model, two representative muscles, one flexor and one extensor, were used to actuate each arm segment to represent the muscles producing clockwise and counterclockwise torques. A model inspired from the popular Hill muscle model (Mehrabi et al., 2014; Thelen, 2003) was used to simulate the muscle contraction dynamics. The Hill muscle model consists of a contractile element (CE) and a parallel elastic (PE) element in series with a series elastic (SE) element. In this study, the SE dynamics representing the tendon were neglected because the steering motion is relatively slow and the amount of energy transfer to tendons is small. Therefore, the muscle model was reduced to the CE element, and the muscle force (FTM) was computed as follows:    (15) FTM ¼ F0max FPE ðt, LM Þ + FCE ðt, a, LM , VM Þ cos αp where FCE represents the active force of the muscle and LM, VM, αp, and Fmax are the muscle length, contraction velocity, pennation angle, and max0 imum isometric muscle force, respectively. The muscle activation level (a) represents the number of active motor units in the muscle (between 0% and 100%), and since the SE element was removed, the pennation angle for all muscles was assumed to be zero. The force generated by FCE can be separated into force-length and force-velocity relations scaled by the muscle activation command (a): L V FCE ¼ aðtÞFCE ðt, LM ÞFCE ðt, a, LM , VM Þ

(16)

where the force-length (FLCE) and force-velocity (FV CE) relations are:  2 

LM

opt 1

=γ

L ¼ e LM FCE 8 VM max > > opt + AV M > max > V L > M M > VM < 0 > > VM > max >  + AV > opt M < V max L Af M M V FCE ¼ len VM BF max > > > + ACV max > M max opt > VM LM > > VM > 0 > > VM B > max > :  max opt + ACV M VM LM Af

(17)

(18)

where γ, A, B, and C are shape factors, Vmax M is the maximum fiber velocity, len opt LM is the optimal length of fiber at which FCE is a maximum, and F max is the

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maximum normalized muscle force during lengthening. The numerical values of the muscle parameters used in this research are reported in Mehrabi and McPhee (2014a). The PE force of muscle (FPE) is represented by an exponential function:   kpe

FPE ¼

e

LM m opt 1 =E0 LM

ekpe

1

1

(19)

where kpe (¼0.5) is a shape factor and Em 0 is passive muscle strain at maximum isometric force. For steering with two hands, the total torque Td generated at the steering wheel is as follows:  Ff ða, LM , VM Þ Td  0 Td ¼ 2 GSHS r (20) Fe ðjaj, θ, θÞ Td < 0 where Ff and Fe are flexor and extensor muscle forces that, respectively, produce a clockwise and counterclockwise torque at the steering wheel, and θ and r are the shoulder angle and the average moment arm of flexor and extensor muscles, respectively; GSHS is a fixed ratio that projects the moment of muscles produced at shoulder to the steering wheel. For simplicity, the muscle length and velocity, and moment arms, are rearranged and simplified to be only a function of shoulder angle and angular velocity _ Here, we assume that there is no muscle (i.e., LM ¼ L0  rθ and VM ¼ r θ). co-contraction between flexor and extensor muscles, and the positive and negative values of Td are produced by the flexor and extensor muscles, respectively. A simplified single-track model with a column-assist EPS steering system as shown in Fig. 7B was developed to speed up the optimization procedures. The driver torque Td transfers through a torsion bar to the steering pinion and rotates the tires. The torque sensor measures the torsion bar twist and sends it to the EPS system that regulates the assist torque (Ta). The following equation describes the steering wheel, and the torque sensor dynamics: Jsw θ€sw ¼ bsw θ_ sw + Ttb + Td   Ttb ¼ Ktb θsw  θp

(21) (22)

where Td and Ttb are the driver and the torsion bar torques, θp is the pinion angle, and θsw, Jsw, and bsw are the angle of rotation, the moment of inertia, and the viscous damping coefficient of the steering column, respectively.

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The rack and its connection to the wheel spindle, as well as the intermediate steering shaft, are combined and represented as a single inertia at the pinion. The dynamics of the steering pinion are described by Jp θ€p ¼ Kp θp  bp θ_ p + Ttb + Ta + TSAT

(23)

where θp, Jp, and bp are, respectively, angular displacement, inertia, and damping of the pinion, and Kp is the stiffness induced by the inclined kingpin axis on the rack displacement. TSAT and Ta represent the self-aligning torque (SAT) and the assist torque provided by the EPS system, respectively. In the single track, the vehicle’s center of mass velocity (V) makes an angle β with the longitudinal direction of the vehicle. Considering the sideslip angle (β) and yaw rate (ωz) of the vehicle as the state variables of the single track model, the equations of motion are expressed as follows:   (24) mvx β_ + ωz ¼ Fyf cos ðδÞ + Fyr Izz ω_ z ¼ Lf Fyf cos ðδÞ  Lr Fyr

(25)

where Fyf and Fyr are front and rear lateral force of the wheels and are approximated by a linear tire model (in contrast to a nonlinear tire model used in the high-fidelity model): Fyf ¼ Cαf αf

(26)

Fyr ¼ Cαr αr

(27)

Assuming small steer angles, the front and rear slip angles can be approximated as follows: vy + Lf ωz δ vx vy  Lr ωz αr ¼ vx

αf ¼

(28) (29)

where vx and vy, respectively, are the longitudinal [vx ¼ V cos(β)] and lateral [vy ¼ V sin(β)] components of the vehicle mass center velocity, and vx is assumed to be constant during the simulations. The steering angle of the front wheel is represented by δ ¼ θp/Gsteering, and Gsteering is the ratio of the rotation of steering wheel angle to the average value of left and right wheel steer angles. The SAT, which is created by the interaction between the tire and the road, is a linear function of slip angle (αf) for small slip angles (TSAT ¼ CTααf), where CTα is a SAT coefficient.

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3.3 Electric Power Steering (EPS) Control Design The main responsibility of an EPS system is to reduce the driver physical effort. As a result, almost all power steering systems have a component in their logic to magnify driver torque by generating an assist torque proportional to the driver torque. The relation that assigns an EPS assist torque to each driver steering torque is presented in so-called characteristic curves. Typically, the steering characteristic curves are multilinear functions of the driver steering torque at different vehicle speeds. In this research, we used a bilinear characteristic curve at each given speed as shown in Fig. 8. This characteristic curve consists of an unassisted zone to avoid the offcenter feeling, a steering assistance zone, and a maximum assist value that is restricted by maximum motor torque. The bilinear characteristic curves can be expressed as follows: 8 0 0 < Td < Td0 < Ta ¼ Ka ðTd  Td0 Þ Td0 < Td < Tdmax (30) : Tmmax Tdmax < Td where Ta, Tmax m , and Ka, respectively, represent the assist torque, the maximum torque of the motor, and the assist gain. Td, Td0, and Tmax represent d the driver’s steering torque, the driver’s steering torque when the motor begins to assist, and the driver’s steering torque when the motor assist reaches T max the maximum assistance (Tdmax ¼ Km a + Td0 ), respectively. The coefficient Ka is an adjustable shape factor that represents the rate of assist. Note that Ka reduces as vehicle speed increases. In the high-fidelity integrated

Fig. 8 Bilinear EPS characteristic curve.

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driver-vehicle model, an observer-based disturbance rejection EPS controller described in Mehrabi et al. (2015b) was used to deliver the desired assist torque to the steering system; in the simplified model, an ideal controller delivers the desired assist torque to the steering system. 3.3.1 Steering Feel Optimization Procedure In this section, a systematic approach to tune the EPS characteristic curves to provide a good steering feel is introduced. However, the word “good” is very subjective and is a function of many variables, including the driver’s physical ability. To achieve a good steering feel, the average energy transferred from road to driver (road feel) should be as strong as possible, while the physical workload of the driver should be minimized (Zaremba and Davis, 1995). The transferred torque to the steering wheel can be separated into two portions: (1) the torque due to road-tire friction and the suspension mechanism and (2) the torque due to external disturbances. Since the external disturbance is random and dependent on road conditions, this portion is neglected here. To tune the EPS characteristic curves for a particular population, the muscle parameters of the control-oriented integrated driver-vehicle model are adjusted to represent that population. Then, an optimization is performed to find the optimum EPS assist gain (Ka) for that specific population, as follows: 0 tf 1 Z  1  (31) Ka ¼ arg min @ q1 F rf + q2 GðaÞ + q3 i2 dtA tf 0

subjected to jYdesired  Yactual j2 < E

(32)

where F rf and G(a) are, respectively, the inverse of road feel and a driver’s physical measure during the steering task, and i is the EPS electric motor current. q1, q2, and q3 are the weighting factors, which have been chosen to normalize each term in the cost. The q1 and q2 weighting factors are used to adjust the steering stiffness while q3 is used to reduce the EPS electric motor size. Yactual and Ydesired are the actual and desired trajectory of the vehicle in the simulations; the desired trajectory is defined to satisfy the ISO double lane change (DLC) maneuver constraints as shown in Fig. 9. The steering assist (Ka) is tuned for an ISO double lane-change maneuver

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Fig. 9 ISO double lane change (DLC) constraint and vehicle desired trajectory.

(Forkenbrock and Elsasser, 2005) at a speed of 10 m/s, where the maximum assist torque (Tmax m ) is assumed to be 50 N m, and a value of 1 N m is selected for the no-assist zone (Td0) to avoid the off-center steering feel. The muscular effort [G(a)], defined according to a physiological cost function (Forster et al., 2004) and shown in Eq. (33), was selected to represent the driver’s physical strength. The symbol ai represents the extensor and flexor muscle activations, and the exponent p is chosen to be 2 in the simulations: GðaÞ ¼

2 X

ðai Þp

(33)

i¼1

The road feel criterion was used to quantify the intensity of feedback information (feel) from the road to the driver. To consider the nonlinearity induced by the steering system and the EPS characteristic curve for a specific maneuver, the road feel was defined in the time domain as the relationship between the resistive steering torque (SAT) to the driver torque (Td) as follows (Zaremba and Davis, 1995): 8 |Td ðtÞ| 1 < if SAT 6¼ 0 Frf ¼ ¼ |SAT ðt Þ| (34) F rf : 0 otherwise

3.4 Simulation Results In this section, the sensitivity of characteristic curves to different muscle parameters is studied. Then, the muscle parameters are set to values for

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young male, young female, old male, and old female, and the EPS characteristic curves are tuned for each driver type. Finally, the performance of the tuned controller is evaluated using the high-fidelity biomechatronic drivervehicle model. 3.4.1 Driver-Specific EPS Characteristic Curves To study the effect of variation of muscle parameters on the EPS characteristic curves, the muscle parameters are changed separately and the effect of each parameter on the curves is studied. Fig. 10A demonstrates the effect of variation of maximum isometric muscle force (Fmax 0 ) on the optimal delivered assistance. As expected, a stronger driver with a higher maximum isometric muscle force requires less assistance in steering torque. In other words, since the stronger driver has stronger muscles, the average value of muscle activations is less compared with a driver with weaker muscles. Therefore, the EPS curve stretches to reduce (slightly) the assistance. Similarly, Fig. 6B depicts that the assist gain is reduced by increasing the maximum contraction velocity (V max m ) of muscle. As shown in Fig. 10B, the amount of generated muscle force at a specific shortening velocity increases max by increasing V max m , which means that a muscle with less V m requires more muscle activation to generate the same force than a muscle with higher V max m , and more driver-assist torque. The variation of maximum muscle max force during lengthening (F len ) and passive muscle strain (Em 0 ) showed that these parameters have negligible effects on the optimal characteristic curves. Thus, the controller should target the most significant parameter F max 0 .

Fig. 10 (A) The effect of maximum isometric muscle force variation on the optimal assist curve and (B) the effect of maximum muscle contraction velocity variation on the optimal assist curve.

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Fig. 11 The optimal assist curve for the four driver types. Table 1 Optimal Characteristic Curve Parameters for Young and Old Adults # Population Bilinear Characteristic Curve (Ka)

1 2 3 4

Young male Young female Old male Old female

2.17 3.17 4.34 7.4

To find the optimum steering feel for the four predefined driver populations, the muscle parameters are adjusted in the control-oriented integrated driver-vehicle model to represent each group, and then the characteristic curves are tuned for each population. Fig. 11 presents the optimal characteristic curves for all four populations. As expected, a driver with more strength requires less assistance while perceiving more road information. Therefore, young male drivers require less assistance than young females, old male and old female drivers. Table 1 displays the optimal assist gains of the bilinear characteristic curves for each driver population. 3.4.2 Double Lane-Change Maneuver With Driver-Specific EPS Controller In this section, to study the performance of the driver-specific EPS controller, the tuned controllers are evaluated using the high-fidelity vehicle-driver model. The muscle parameters of the 3D driver model are adjusted to represent the corresponding group, that is, young male, old male, young female, and old female. Then, each group performs a DLC maneuver with

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Fig. 12 The vehicle trajectory of the four driver types performing an ISO DLC maneuver. Vehicle trajectories of all four driver types are shown.

Fig. 13 Right arm’s muscle activities during a double lane-change maneuver for the four driver types (A) anterior portion of deltoid and (B) long head of triceps. Muscle activities of all four groups are shown.

the high-fidelity vehicle model equipped with an EPS controller tuned for that specific group at the speed of 10 m/s. As shown in Fig. 12, the vehicle lateral displacements of all groups are similar to each other and to the desired trajectory, and they are all within the ISO double-lane change maneuver constraints. Therefore, the steering loads in all of the simulations are the same, since the driving conditions in all of the simulations are the same. Fig. 13 shows the predicted muscle activities of the anterior portion of deltoid and the long head of triceps of the driver’s right arm for the four

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Fig. 14 Sensitivity of the optimal characteristic curve to the variation of optimization weights, (A) variation of q1 and (B) variation of q2.

predefined driver types. Although other muscle activations are not presented here, a similar behavior can be seen in other muscles. As shown in this figure, the magnitude and trend of these patterns are very similar. Although young male drivers have higher physical strength than old female drivers, the portion of motor units that have been recruited by the CNS are the same as for other drivers. In conclusion, the drivers’ muscular efforts are equal, thereby satisfying the controller objective to provide the same targeted steering feel to all drivers. Fig. 14 shows the sensitivity of the characteristic curve to the variation of cost function weighting factors. The cost function weights are modified proportional to their nominal values. The results demonstrate that the variation of muscle fatigue weight (q2) has a greater effect on the characteristic curve’s assist gain than the variation of road feel weight (q1), because the cost function is a linear function of the road feel but a quadratic function of muscle activations. These cost function weights can be used to adjust the target steering feel. For example, for a sports car, the driver expects to have stiffer steering than in a comfortable car. Then, to have a sportier feel, the road feel weighting factor should be increased as shown in Fig. 14A, which results in less assistance and a steering system, that is, therefore more sensitive to road forces.

4 CONCLUSIONS In this chapter, we introduced various tools for the model-based design of biomechatronic systems. Included in these tools are integrated

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biomechatronic system models, model-based controllers, and inverse and predictive simulations. The biomechatronic model is an integrated model of the user’s biomechanics and a dynamic model of the assistive device, which can be used to simulate the human-machine interactions. The biomechatronic model parameters can be adjusted to represent a specific individual or groups of individuals. This biomechatronic model facilitates the design of individualized model-based controllers, and can be used to improve the device and controller design through MIL inverse or predictive simulations. In the case study, a systematic method to consider the driver’s physical characteristics in the design of a driver-specific EPS controller is proposed. To design such an EPS controller, first, the high-fidelity driver-vehicle model is simplified to reduce the computational burden associated with the multibody and biomechanical systems. The muscle parameters in the high-fidelity and simplified integrated driver-vehicle models have been adjusted to represent drivers with different physical abilities (young male, old male, young female, and old female). A steering feel optimization procedure is used to tune the EPS controller for each group. Simulation results using the high-fidelity biomechatronic driver-vehicle model showed that it is possible to develop a model-based EPS controller that considers the physical characteristics of a driver and delivers a targeted steering feel to a predefined driver population. Evaluation of the tuned EPS controller also showed that, although the EPS controller has been tuned based on the simplified model, the controller shows the same expected behavior in highfidelity simulations.

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FURTHER READING Mizuno, T., Hayama, R., Kawahara, S., Lou, L., Liu, Y., Ji, X., 2013. Research on relationship between steering maneuver and muscle activities. eb-cat.ds-navi.co.jp, 13–18. Pick, A., Cole, D., 2007. Driver steering and muscle activity during a lane-change manoeuvre. Veh. Syst. Dyn. 45 (9), 781–805.