Sources and References

• Winter, D. A. Biomechanics and Motor Control of Human Movement, 4th edition, Wiley.
• Zatsiorsky, V. M. Kinematics of Human Motion, Human Kinetics.
• Zatsiorsky, V. M. Kinetics of Human Motion, Human Kinetics.
• Robertson, D. G. E., Caldwell, G. E., Hamill, J., Kamen, G., and Whittlesey, S. N. Research Methods in Biomechanics, 2nd edition, Human Kinetics.
• Hall, S. J. Basic Biomechanics, McGraw-Hill.
• Nigg, B. M. and Herzog, W. Biomechanics of the Musculo-Skeletal System, 3rd edition, Wiley.
• Perry, J. and Burnfield, J. M. Gait Analysis: Normal and Pathological Function, 2nd edition, SLACK.
• Kuo, A. D. “A least-squares estimation approach to improving the precision of inverse dynamics computations.” Journal of Biomechanical Engineering, 1998.
• Zajac, F. E. “Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control.” CRC Critical Reviews in Biomedical Engineering, 1989.
• Delp, S. L. et al. “OpenSim: open-source software to create and analyze dynamic simulations of movement.” IEEE Transactions on Biomedical Engineering, 2007.
• MIT OpenCourseWare, HST.508 (Quantitative Physiology: Organ Transport Systems) and 2.183 (Biomechanics and Neural Control of Movement), open-access lecture materials.
• Stanford ME 485 (Modeling and Simulation of Human Movement), open-access lecture materials.

Chapter 1: Reference Frames, Anthropometry, and the Biomechanical Model

Human movement is organized around a hierarchy of rigid segments linked by articulations. Biomechanics proceeds by replacing the continuous, deformable body with a finite collection of idealized rigid bodies whose inertial properties are summarized by mass, center of mass location, and inertia tensor. Before any equation of motion can be written, a disciplined choice of reference frames and a careful quantification of segment parameters must be established.

1.1 Anatomical Planes and Global Frames

The standard anatomical reference assumes a subject in the erect position with palms facing forward. Three mutually orthogonal planes are defined. The sagittal plane divides left from right and contains the anterior-posterior and superior-inferior axes. The frontal (coronal) plane divides anterior from posterior. The transverse plane divides superior from inferior. Laboratory measurements are typically collected in a global frame \( \{X, Y, Z\} \) fixed to the capture volume, with \(Z\) vertical by convention. The transformation between global and anatomical frames is one of the first calibration steps in a motion capture study.

1.2 Body Segment Parameters

For each segment one requires the mass \(m_i\), the position of the center of mass in the body-fixed frame \(\mathbf{r}_{c,i}\), and the inertia tensor \(\mathbf{I}_i\) expressed about the center of mass. Three classical estimation strategies exist. Cadaver regression equations, such as those of Dempster or Chandler, give segment mass as a fraction of total body mass and locate the center of mass as a fraction of segment length. Geometric models approximate each segment as a frustum, ellipsoid, or cylinder and compute inertial properties analytically. Imaging-based methods use dual-energy X-ray absorptiometry or magnetic resonance imaging to obtain subject-specific distributions.

The following table shows typical segment mass fractions reported in Winter, normalized to total body mass \(M\).

Segment	Mass fraction	Center of mass (proximal)	Radius of gyration (proximal)
Foot	0.0145	0.50	0.475
Shank	0.0465	0.433	0.302
Thigh	0.100	0.433	0.323
Forearm	0.016	0.430	0.303
Upper arm	0.028	0.436	0.322
Head and neck	0.081	1.000	0.495
Trunk	0.497	0.500	0.406

Radii of gyration are fractions of segment length and specify the diagonal entries of the inertia tensor through \(I = m k^2 L^2\).

1.3 Markers, Clusters, and the Rigid Body Assumption

Motion capture reconstructs segment pose from surface markers. A single segment requires at least three non-colinear markers to resolve both translation and orientation. In practice a cluster of four or more markers on a rigid shell is used, and a least-squares fit between the cluster in a reference (static) trial and its configuration during movement yields the segment rotation and translation. Residual errors from soft-tissue artifact, where the skin slides over the bone, are the dominant source of kinematic error and motivate optimization-based reconstruction methods described in Chapter 3.

Chapter 2: Two- and Three-Dimensional Kinematics

Kinematics is the description of motion without reference to force. It provides the positions, velocities, and accelerations that inverse dynamics will later convert into joint moments.

2.1 Planar Kinematics

In two dimensions a segment has three degrees of freedom: two translational and one rotational. The position of the center of mass is \((x, y)\) and the orientation is a single angle \(\theta\) measured from a global axis. Joint angles are computed as differences of segment angles; for instance the knee flexion angle in the sagittal plane is

\[ \theta_{\text{knee}} = \theta_{\text{thigh}} - \theta_{\text{shank}}. \]

Signed convention must be stated: flexion positive or extension positive, right-handed or left-handed. Inconsistent conventions are the most common source of error when comparing studies.

Velocities and accelerations are obtained by numerical differentiation of marker coordinates. Because differentiation amplifies high-frequency noise, a zero-lag low-pass Butterworth filter is almost universally applied before differentiation, typically with cutoff between 4 and 10 Hz for walking and 10 and 20 Hz for running. The cutoff is chosen from a residual analysis in which the root-mean-square difference between filtered and raw signals is plotted against cutoff frequency; the knee of the curve identifies the transition from signal to noise.

2.2 Three-Dimensional Kinematics: Rotation Matrices

In three dimensions a segment has six degrees of freedom. The orientation of the body-fixed frame \(B\) relative to the global frame \(G\) is described by a proper orthogonal matrix \(\mathbf{R}_{GB} \in SO(3)\) such that

\[ \mathbf{r}_G = \mathbf{R}_{GB}\,\mathbf{r}_B + \mathbf{t}_{GB}, \]

where \(\mathbf{r}_B\) is the position of a material point in the body frame and \(\mathbf{t}_{GB}\) is the origin of \(B\) expressed in \(G\). The rotation matrix satisfies \(\mathbf{R}^\top \mathbf{R} = \mathbf{I}\) and \(\det \mathbf{R} = +1\). Composition of rotations is matrix multiplication, and \(\mathbf{R}_{GB}^{-1} = \mathbf{R}_{GB}^\top\).

2.3 Euler and Cardan Angles

A rotation matrix can be parameterized by three successive rotations about moving or fixed axes. The convention dominant in human movement studies is the Cardan \(x\)-\(y\)-\(z\) sequence proposed by Grood and Suntay for the knee, in which the first rotation is about the mediolateral axis of the proximal segment (flexion-extension), the second about a floating axis (abduction-adduction), and the third about the long axis of the distal segment (internal-external rotation). For an \(x\)-\(y\)-\(z\) sequence with angles \(\alpha\), \(\beta\), \(\gamma\) the rotation matrix is

\[ \mathbf{R} = \mathbf{R}_x(\alpha)\,\mathbf{R}_y(\beta)\,\mathbf{R}_z(\gamma). \]

2.4 Quaternions

Unit quaternions parameterize \(SO(3)\) without singularity. A quaternion is written \(q = (w, \mathbf{v})\) with \(w \in \mathbb{R}\) and \(\mathbf{v} \in \mathbb{R}^3\), subject to \(w^2 + \mathbf{v}^\top\mathbf{v} = 1\). The rotation of a vector \(\mathbf{r}\) by \(q\) is

\[ \mathbf{r}' = \mathbf{v}(\mathbf{v}^\top\mathbf{r}) + w^2 \mathbf{r} + 2w(\mathbf{v}\times\mathbf{r}) + \mathbf{v}\times(\mathbf{v}\times\mathbf{r}). \]

Quaternions are convenient for interpolation (spherical linear interpolation, slerp) and for numerical integration of angular velocity because the derivative \(\dot q = \tfrac{1}{2}\,q \otimes \omega\) avoids trigonometric functions. They must be renormalized periodically to prevent drift off the unit sphere.

2.5 Joint Coordinate Systems

A joint coordinate system (JCS) defines the axes along which clinically interpretable rotations occur. For the knee, the JCS fixes a flexion-extension axis to the femur, an internal-external rotation axis to the tibia, and lets the abduction-adduction axis float as the common perpendicular. The three angles extracted from this construction are not the same as Euler angles in general, but they correspond closely for small rotations and give anatomically meaningful values.

2.6 Inverse and Forward Kinematics

Given joint angle histories \(\mathbf{q}(t)\), the forward kinematics problem computes segment poses by successive application of rotation matrices along the kinematic chain. Given marker trajectories, the inverse kinematics problem recovers \(\mathbf{q}(t)\). Closed-form solutions exist for simple chains but most musculoskeletal models are solved numerically. A common approach minimizes

\[ J(\mathbf{q}) = \sum_{i=1}^{n} w_i \left\| \mathbf{p}_i^{\text{meas}} - \mathbf{p}_i^{\text{model}}(\mathbf{q}) \right\|^2, \]

where \(\mathbf{p}_i^{\text{meas}}\) is the measured position of marker \(i\), \(\mathbf{p}_i^{\text{model}}(\mathbf{q})\) is the position predicted by the model at joint configuration \(\mathbf{q}\), and \(w_i\) are weights chosen to emphasize reliable markers. Nonlinear least-squares or extended Kalman filtering are standard solvers.

Chapter 3: Inverse Dynamics

Inverse dynamics computes the net force and moment acting at each joint given segment kinematics, segment inertial parameters, and external forces, most often ground reaction forces measured by a force plate.

3.1 Newton-Euler Equations for a Single Segment

Consider a single rigid segment in three dimensions. Let \(\mathbf{F}_p\) and \(\mathbf{F}_d\) be the forces exerted on the segment by the proximal and distal joints respectively, \(\mathbf{M}_p\) and \(\mathbf{M}_d\) the corresponding moments, and \(\mathbf{F}_{\text{ext}}\) and \(\mathbf{M}_{\text{ext}}\) any externally applied load (including gravity through the center of mass). The linear momentum equation is

\[ m\,\mathbf{a}_c = \mathbf{F}_p + \mathbf{F}_d + \mathbf{F}_{\text{ext}}, \]

and the angular momentum equation about the center of mass is

\[ \mathbf{I}_c\,\boldsymbol{\alpha} + \boldsymbol{\omega}\times\left(\mathbf{I}_c\,\boldsymbol{\omega}\right) = \mathbf{M}_p + \mathbf{M}_d + \mathbf{r}_p\times\mathbf{F}_p + \mathbf{r}_d\times\mathbf{F}_d + \mathbf{M}_{\text{ext}}. \]

Here \(\mathbf{I}_c\) is the inertia tensor about the center of mass, expressed in the global frame (so it must be rotated from the body frame at each instant), \(\boldsymbol{\omega}\) is the angular velocity, and \(\boldsymbol{\alpha} = \dot{\boldsymbol{\omega}}\).

3.2 The Recursive Proximal-to-Distal (or Distal-to-Proximal) Algorithm

In gait analysis the ground reaction force is known at the foot and the solution proceeds distally to proximally. At each segment the unknowns are \(\mathbf{F}_p\) and \(\mathbf{M}_p\). Applying Newton’s third law, the reaction forces on the next proximal segment are \(-\mathbf{F}_p\) and \(-\mathbf{M}_p\), and the recursion continues. The procedure is:

1. Measure marker trajectories and ground reaction forces. Filter and differentiate to obtain segment kinematics.
2. Starting with the foot, substitute the known distal load (ground reaction) and solve for the ankle force and moment.
3. Use the negated ankle load as the distal input to the shank and solve for the knee force and moment.
4. Continue proximally through the thigh, pelvis, and trunk.

3.3 Indeterminacy, Dynamic Consistency, and Residuals

A common inconsistency in inverse dynamics arises because kinematics and force measurements come from different instruments with different noise characteristics. When the Newton-Euler equations are applied to the full chain with a free-floating root segment (the pelvis), the residual force and moment at the root represent the imbalance between measured and modeled quantities. Kuo’s least-squares approach distributes the residual over the chain to minimize inconsistency. Algorithms such as residual reduction in OpenSim perturb marker trajectories and body segment parameters within prescribed tolerances to drive the residuals to zero while keeping the solution close to the measurements.

3.4 From Joint Moment to Joint Power

The instantaneous mechanical power at a joint is

\[ P_j(t) = \mathbf{M}_j(t)\cdot\boldsymbol{\omega}_j(t), \]

where \(\boldsymbol{\omega}_j\) is the angular velocity of the distal segment relative to the proximal. Positive power corresponds to concentric action, negative to eccentric action. Integrating power over a phase gives the mechanical work, a useful summary measure for gait studies and prosthesis evaluation.

Chapter 4: Lagrangian Formulation for Multibody Systems

For articulated systems with many degrees of freedom the Newton-Euler approach becomes cumbersome because internal constraint forces must be eliminated. The Lagrangian formulation is preferred when one wants a minimal set of equations in independent generalized coordinates.

4.1 Generalized Coordinates and the Lagrangian

Let \(\mathbf{q} = (q_1, \ldots, q_n)\) be a set of generalized coordinates that fully specify the configuration of a kinematic chain. The Lagrangian is

\[ L(\mathbf{q}, \dot{\mathbf{q}}) = T(\mathbf{q}, \dot{\mathbf{q}}) - V(\mathbf{q}), \]

where \(T\) is the kinetic energy and \(V\) the potential energy. The Euler-Lagrange equations are

\[ \frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot q_i}\right) - \frac{\partial L}{\partial q_i} = \tau_i, \qquad i = 1, \ldots, n, \]

with \(\tau_i\) the generalized force conjugate to \(q_i\). For the human body \(\tau_i\) typically aggregates the net joint moments produced by muscles crossing that joint.

4.2 Mass Matrix, Coriolis, and Gravity

In matrix form the equations of motion read

\[ \mathbf{M}(\mathbf{q})\,\ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})\,\dot{\mathbf{q}} + \mathbf{G}(\mathbf{q}) = \boldsymbol{\tau}. \]

The mass matrix \(\mathbf{M}\) is symmetric positive-definite and encodes how kinetic energy depends on \(\dot{\mathbf{q}}\). The Coriolis-centrifugal matrix \(\mathbf{C}\) can be chosen so that \(\dot{\mathbf{M}} - 2\mathbf{C}\) is skew-symmetric, a property used in control proofs. The gravity vector \(\mathbf{G}\) is the gradient of \(V\). For inverse dynamics one specifies \(\mathbf{q}, \dot{\mathbf{q}}, \ddot{\mathbf{q}}\) and solves algebraically for \(\boldsymbol{\tau}\); for forward dynamics one specifies \(\boldsymbol{\tau}\) and integrates

\[ \ddot{\mathbf{q}} = \mathbf{M}^{-1}\!\left(\boldsymbol{\tau} - \mathbf{C}\dot{\mathbf{q}} - \mathbf{G}\right). \]

4.3 Double Pendulum as a Minimal Leg Model

4.4 Constrained Systems

Closed kinematic chains, contact between foot and ground, or two-hand manipulation introduce constraints \(\boldsymbol{\phi}(\mathbf{q}) = \mathbf{0}\). The augmented equations of motion are

\[ \mathbf{M}\ddot{\mathbf{q}} + \mathbf{C}\dot{\mathbf{q}} + \mathbf{G} + \boldsymbol{\Phi}_q^\top \boldsymbol{\lambda} = \boldsymbol{\tau}, \]

with Lagrange multipliers \(\boldsymbol{\lambda}\) and Jacobian \(\boldsymbol{\Phi}_q = \partial\boldsymbol{\phi}/\partial\mathbf{q}\). Differentiating the constraint twice yields \(\boldsymbol{\Phi}_q \ddot{\mathbf{q}} + \dot{\boldsymbol{\Phi}}_q \dot{\mathbf{q}} = \mathbf{0}\), giving a linear system for \((\ddot{\mathbf{q}}, \boldsymbol{\lambda})\). Baumgarte stabilization adds proportional and derivative terms in \(\boldsymbol{\phi}\) to prevent numerical drift.

Chapter 5: Muscle Mechanics and the Hill Model

Muscles are the actuators of the musculoskeletal system. Their force output depends on length, velocity, activation, and history in ways that cannot be captured by a simple torque source.

5.1 Structure and the Sliding-Filament Theory

A muscle fiber contains serially arranged sarcomeres, each built from interdigitating thick (myosin) and thin (actin) filaments. Force is generated by cross-bridges between myosin heads and actin binding sites. The number of attached cross-bridges depends on the overlap of thick and thin filaments, leading to the active force-length relationship. When sarcomeres shorten rapidly, the cross-bridge cycle has less time to attach, reducing force and generating the force-velocity relationship.

5.2 The Hill-Type Model

The classical Hill model lumps the muscle-tendon unit into three elements: a contractile element (CE), a parallel elastic element (PEE), and a series elastic element (SEE, primarily the tendon). The total musculotendon force is

\[ F_{mt} = F_{CE} + F_{PEE}, \]

transmitted through the tendon whose force-strain curve gives the SEE behavior. The contractile element force is

\[ F_{CE} = a(t)\,F_{\max}\,f_L(\tilde{\ell})\,f_V(\tilde{v}), \]

where \(a(t) \in [0, 1]\) is activation, \(F_{\max}\) is the maximum isometric force, \(\tilde{\ell} = \ell/\ell_0\) is normalized fiber length, and \(\tilde{v} = \dot{\ell}/v_{\max}\) is normalized fiber velocity. The active force-length curve \(f_L\) peaks near \(\tilde{\ell} = 1\) and falls to zero outside the range roughly \([0.5, 1.5]\). The force-velocity curve, following A. V. Hill’s classic equation, can be written for concentric action as

\[ f_V(\tilde{v}) = \frac{1 - \tilde{v}}{1 + \tilde{v}/k}, \qquad \tilde{v} \ge 0, \]

with empirical shape constant \(k\), and extended for eccentric action to an asymptote around \(1.5\,F_{\max}\). The parallel elastic element contributes force only beyond the slack length.

5.3 Activation Dynamics

Neural input drives calcium release and subsequent cross-bridge attachment with finite time constants. A first-order model writes

\[ \dot{a} = \frac{u - a}{\tau(u, a)}, \qquad \tau = \begin{cases} \tau_{\text{act}}, & u > a, \\ \tau_{\text{deact}}, & u \le a, \end{cases} \]

with \(u(t) \in [0,1]\) the neural excitation, \(\tau_{\text{act}} \approx 10\)-\(30\) ms, \(\tau_{\text{deact}} \approx 40\)-\(60\) ms. The asymmetry between activation and deactivation reflects calcium kinetics.

5.4 Pennation, Moment Arm, and Tendon Compliance

Muscle fibers often insert onto the tendon at a pennation angle \(\phi\). The projected fiber force along the tendon is \(F_{CE}\cos\phi\). Pennation increases packing of contractile material at the cost of reduced force transmission. The moment produced about a joint is \(M = r(q)\,F_{mt}\), where \(r(q)\) is the moment arm, the perpendicular distance from the line of action to the joint center. The moment arm is often estimated by the tendon-excursion method, using \(r(q) = d\ell_{mt}/dq\). Tendon compliance allows elastic energy storage during eccentric loading and return during concentric action, a mechanism central to efficient locomotion.

Chapter 6: Electromyography and EMG-to-Force

Surface electromyography records the summed electrical activity of motor units near an electrode. It is the most accessible non-invasive probe of muscle activation but requires careful processing before it can be related to force.

6.1 Signal Processing

Raw EMG is bandpass filtered, commonly 20-450 Hz, to remove motion artifact and high-frequency noise. Full-wave rectification yields a unipolar signal whose envelope is extracted by low-pass filtering (cutoff 3-10 Hz) to obtain a linear envelope \(e(t)\). Normalization divides \(e(t)\) by the envelope measured during a maximum voluntary contraction to yield a dimensionless activation estimate \(\tilde{e}(t) \in [0, 1]\).

6.2 EMG-Driven Muscle Models

The linear envelope is taken as an excitation input to an activation dynamics model, which in turn drives a Hill contractile element. The full pipeline is

\[ \tilde{e}(t) \;\xrightarrow{\text{activation dynamics}}\; a(t) \;\xrightarrow{\text{Hill model}}\; F_{mt}(t) \;\xrightarrow{\text{moment arm}}\; M(t). \]

Parameters such as \(F_{\max}\), \(\ell_0\), tendon slack length, and activation time constants are tuned per subject by matching predicted joint moments to inverse-dynamics moments over calibration trials.

6.3 Muscle Redundancy and Static Optimization

At most joints more muscles span the joint than are strictly necessary to produce any given moment. This muscle redundancy is resolved in simulation by solving, at each instant, an optimization problem

\[ \min_{\mathbf{a}} \;\sum_{i=1}^{n_m} \left(\frac{F_{CE,i}}{F_{\max,i}}\right)^p \quad \text{subject to} \quad \sum_{i} r_i\,F_{CE,i} = M_j, \]

with \(p\) typically 2 or 3. Low exponents spread load across muscles; higher exponents minimize peak stress. Static optimization ignores activation and tendon dynamics; dynamic optimization solves an optimal control problem across the trial.

Chapter 7: Motor Control

The central nervous system produces movement by transforming task goals into patterns of muscle activation. Several theoretical frameworks capture aspects of this transformation.

7.1 Equilibrium-Point Hypothesis

Feldman’s lambda model posits that the controller selects a threshold length \(\lambda\) for each muscle; whenever the actual length exceeds \(\lambda\), a stretch reflex produces force. The arm reaches the configuration at which all muscle forces balance the external load. The appeal is that the controller need not compute inverse dynamics; spring-like muscle properties and reflex loops handle disturbance rejection automatically. Criticisms include the observation that rapid movements produce accelerations too large to be explained by pure spring behavior without significant feedforward input.

7.2 Internal Models

An alternative framework holds that the CNS maintains an internal forward model that predicts sensory consequences of motor commands and an inverse model that computes the motor command required to achieve a desired trajectory. Evidence includes the rapid adaptation to force fields during reaching, where subjects first produce perturbed trajectories and then, after practice, produce straight ones by anticipating the perturbation.

7.3 Optimal Control and Cost Functions

Many movement regularities emerge as solutions to optimization problems. Smoothness of reaching trajectories is captured by the minimum-jerk model:

\[ J_{\text{jerk}} = \int_0^T \left\|\dddot{\mathbf{x}}(t)\right\|^2 dt, \]

whose minimizer under fixed endpoints is a fifth-order polynomial with characteristic bell-shaped velocity profile. Minimum-torque-change models predict curved paths in joint space that match observations in multi-joint tasks. Stochastic optimal feedback control frameworks, such as Todorov’s, extend these ideas to include signal-dependent noise, predicting that variability is structured along task-irrelevant directions (the minimum-intervention principle).

7.4 Reflexes and Spinal Circuits

Stretch reflexes and Golgi tendon reflexes form negative feedback loops at the spinal level. The monosynaptic Ia pathway from muscle spindle to alpha motoneuron has a loop delay of roughly 30 ms and is the fastest neural response to a perturbation. Longer-latency responses (50-100 ms) involve transcortical loops. Central pattern generators in the spinal cord produce rhythmic locomotor output even in the absence of descending command, modulated by sensory feedback from load receptors and hip proprioceptors.

Chapter 8: Gait Analysis

Walking is the most studied human movement and the benchmark application of biomechanical tools.

8.1 Phases of the Gait Cycle

One cycle, or stride, begins at heel-strike of one foot and ends at the next heel-strike of the same foot. The cycle is divided into stance (approximately 60 percent) and swing (40 percent). Stance begins with the loading response, during which the forefoot plantar-flexes to the ground while the knee flexes to absorb impact; continues through mid-stance, when the body rolls over the supporting limb; progresses to terminal stance as the heel rises; and ends at toe-off. Swing carries the limb forward through initial, mid, and terminal phases, decelerating before the next heel-strike. Double-support periods of roughly 10 percent each occur at the transitions between single-support phases.

8.2 Spatiotemporal Parameters

Parameter	Typical value (adult, comfortable walking)
Cadence	110-120 steps per minute
Stride length	1.4-1.6 m
Walking speed	1.2-1.4 m/s
Stance phase fraction	0.60
Step width	0.07-0.15 m

These quantities scale with leg length and speed, and the Froude number \(Fr = v^2 / (gL)\) predicts the transition from walking to running near \(Fr \approx 0.5\).

8.3 Ground Reaction Force and the Inverted Pendulum

Vertical ground reaction force during walking shows a characteristic double-peak pattern: a first peak near 1.1-1.2 body weights during loading response, a valley near mid-stance (the body is at its highest and momentarily unloading), and a second peak near push-off. Anterior-posterior force is braking (negative) in early stance and propulsive (positive) in late stance. The zero-crossing is near mid-stance.

The inverted-pendulum model idealizes stance as rotation of the body about the ankle or foot. Energy exchange between gravitational and kinetic pools occurs twice per step, producing high efficiency. The walking-running transition is understood as the point where the centripetal force required to keep the foot on the ground exceeds body weight.

8.4 Joint Kinematics and Kinetics in Gait

Ankle, knee, and hip flex and extend in coordinated patterns. The ankle plantarflexes slightly at heel-strike, dorsiflexes through stance, then plantarflexes forcefully at push-off, producing the single largest power burst of the stride (A2 burst in Winter’s notation). The knee flexes in the loading response, extends in mid-stance, flexes again for toe clearance, and extends in terminal swing. The hip extends during stance to translate the body forward and flexes during swing to advance the limb.

8.5 Pathological Gait

Deviations from normal are analyzed by comparison to age-matched templates. Examples include drop foot (weak dorsiflexors cause toe-dragging in swing, compensated by hip-hiking or circumduction), Trendelenburg gait (weak hip abductors cause the contralateral pelvis to drop in stance), and crouch gait in cerebral palsy (persistent knee flexion throughout stance). Quantitative measures such as the Gait Deviation Index compress multidimensional deviations into a single score for clinical tracking.

Chapter 9: Occupational Biomechanics

Occupational biomechanics applies musculoskeletal analysis to workplace tasks, especially manual lifting, to prevent injury.

9.1 Low-Back Loading During Lifting

Lumbar compression and shear forces correlate with low-back injury risk. Chaffin’s static planar model treats the lumbar spine as a fulcrum and computes the L5/S1 compression required for equilibrium:

\[ F_{\text{erector}} \cdot d_m = W_{\text{load}} \cdot d_L + W_{\text{trunk}} \cdot d_T, \]

where \(d_m\) is the short moment arm of the erector spinae (about 5 cm), and \(d_L, d_T\) are the horizontal distances from L5/S1 to the load and trunk centers of mass. Because \(d_m\) is small, the muscle force is many times larger than the load weight; the spinal compression is approximately

\[ F_{\text{comp}} \approx F_{\text{erector}} + W_{\text{load}}\cos\theta + W_{\text{trunk}}\cos\theta, \]

with \(\theta\) the trunk flexion angle. For a 20 kg load lifted with a 30 cm horizontal offset, compression can exceed 3400 N even for small subjects, near the NIOSH Action Limit.

9.2 The NIOSH Lifting Equation

The revised NIOSH equation gives a recommended weight limit as the product of a load constant \(LC = 23\) kg and six multipliers:

\[ RWL = LC \cdot HM \cdot VM \cdot DM \cdot AM \cdot FM \cdot CM, \]

accounting for horizontal distance, vertical location, vertical travel distance, asymmetry, frequency, and coupling. The Lifting Index \(LI = L/RWL\) is a normalized risk measure, with \(LI > 1\) indicating elevated risk.

9.3 Cumulative Load and Repetition

Single-lift analysis is insufficient for repetitive tasks. Cumulative-load models integrate spinal load over a shift and correlate the total with chronic injury incidence. Intradiscal pressure varies with posture (highest in seated flexion, lowest in supine) and guides ergonomic recommendations for workstation design.

Chapter 10: Assistive and Rehabilitation Devices

Biomechanical models inform the design, control, and evaluation of devices that augment or replace lost function.

10.1 Prostheses

Lower-limb prostheses must replicate stance support, controlled knee flexion during swing, and push-off. Passive energy-storing feet (carbon-fiber leaf springs) store impact energy and return it at push-off, approximating the ankle’s power profile. Microprocessor-controlled knees modulate hydraulic damping in real time, adapting to walking speed and terrain. Powered ankle-feet such as the Herr family of devices provide net positive work across a stride, approaching the energetics of intact ankles.

Upper-limb prostheses use surface EMG signals from residual muscles as control inputs. Pattern recognition classifiers map multi-channel EMG to a small set of grasp classes; targeted muscle reinnervation surgeries create additional signal sites by rerouting nerves that once innervated the missing limb.

10.2 Orthoses and Exoskeletons

Orthoses support or correct a functioning limb. Ankle-foot orthoses hold the ankle in slight dorsiflexion to compensate for drop foot. Powered exoskeletons for locomotor assistance range from full-body devices for spinal-cord-injury users to soft exosuits that apply assistive torques via cables. Key design metrics are metabolic cost reduction, wearer comfort, and stability margins. Zhang and Collins demonstrated that iterative human-in-the-loop optimization of ankle exoskeleton torque profiles can reduce the metabolic cost of walking by more than 20 percent below unassisted values.

10.3 Functional Electrical Stimulation

FES delivers short electrical pulses to peripheral nerves to elicit muscle contraction. A typical stimulation pattern uses biphasic rectangular pulses of 200-400 microseconds at 20-50 Hz. Recruitment is reverse-size-principle: large fast-fatigable fibers near the electrode recruit first, so FES-induced contractions fatigue rapidly. Applications include correction of drop foot via peroneal stimulation, cycling in paraplegia, and restoration of grasp via forearm stimulation. Closed-loop FES combines sensor feedback (accelerometers, goniometers, or implanted EMG) with controllers that modulate stimulation amplitude or pulse width to track desired joint trajectories.

10.4 Evaluation Methodology

Quantitative evaluation of assistive devices uses the same tools developed earlier: motion capture and inverse dynamics for joint-level mechanics, indirect calorimetry for metabolic cost, and EMG for user effort. Clinical outcomes such as the six-minute walk test, Timed Up and Go, and Functional Ambulation Category situate mechanical improvements in functional context. The full pipeline from segment parameters and multibody dynamics to muscle models and motor-control theory is needed because a device that improves one joint’s kinetics may shift compensatory demands elsewhere, and only a whole-body analysis reveals the net effect.