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Principle: several infrared cameras see reflective (passive) or light-emitting (active) markers attached to the skin over bony landmarks. Each camera gives a 2D image position; combining at least two calibrated views gives the 3D position of each marker by triangulation. Advantages: high accuracy (sub-millimetre), high sampling rate, no wires for passive markers. Issues: skin motion artefact (markers move relative to the bone, the largest error source), marker occlusion, marker placement errors, fixed laboratory volume, and cost. Definitions used later: a rigid body segment is described by a set of markers, from which a local (anatomical) coordinate system is built.

Electrogoniometers: two arms fixed on adjacent segments with a potentiometer or strain gauge at the joint that outputs the joint angle. Simple, but they measure one or a few angles, are sensitive to misalignment, and can hinder movement. Inertial measurement units (IMUs): combine accelerometer, gyroscope and magnetometer to estimate the orientation of the segment they are attached to. They are small, cheap and not limited to a laboratory (see the chapter on IMUs). Stereophotogrammetry is generally considered the reference standard, while wearable systems trade some accuracy for portability.

1. How is the 3D position of a marker obtained? 2. What is the main source of error with skin markers? 3. Name one advantage and one disadvantage of an electrogoniometer.

1. By triangulation from at least two calibrated camera views. 2. Skin (soft tissue) motion artefact relative to the bone. 3. Advantage: simple and direct measurement of the joint angle. Disadvantage: measures only a few angles and is sensitive to misalignment.

A rigid body keeps the distance between all its points fixed. Its pose is given by the position of a point (origin of a local frame) and the orientation of the local frame relative to the global frame, described by a 3 x 3 rotation matrix R (orthogonal, determinant 1, inverse equal to the transpose). A point with local coordinates p_local has global coordinates p = r + R p_local, where r is the origin of the local frame. Rotations do not commute, so the order of rotations matters. Composition: R_AC = R_AB R_BC. The orientation of a distal segment relative to the proximal one is R_rel = R_prox transposed times R_dist.

Anatomical frames are built from bony landmarks palpated with markers. The International Society of Biomechanics (ISB) recommends standard definitions for the ankle, hip, spine, shoulder, elbow, wrist and hand (Wu et al.). Joint angles: the relative orientation of two segments is expressed by three angles in a fixed order of rotation, for example flexion-extension, abduction-adduction and internal-external rotation (Cardan or Euler sequences; the joint coordinate system of Grood and Suntay). A different sequence gives different angle values for the same posture. Gimbal lock: when the middle rotation reaches 90 degrees, two axes align and one degree of freedom is lost, which makes Euler angles unreliable near that posture.

A quaternion q = q0 + q1 i + q2 j + q3 k has a scalar part and a vector part. Hamilton's fundamental rule: i squared = j squared = k squared = ijk = -1. The product of quaternions is not commutative. Unit quaternions (norm 1) represent rotations: a rotation by angle theta about the unit axis n is q = cos(theta/2) + n sin(theta/2). A vector is rotated by q v q*, and successive rotations are combined by quaternion multiplication. Conversions between quaternion and rotation matrix exist in both directions. Quaternions avoid gimbal lock and are efficient, so they are used for IMU orientation.

Angular velocity vector omega describes the rate of rotation of the body. The velocity of a point P of a rigid body: v_P = v_O + omega x (P - O). The acceleration: a_P = a_O + alpha x (P - O) + omega x (omega x (P - O)), where alpha is the angular acceleration (tangential term alpha x r and centripetal term omega x (omega x r)). The rotation matrix and angular velocity are related by the derivative of R equal to the skew-symmetric matrix of omega times R. These relations allow computation of segment centre-of-mass velocity and acceleration needed by inverse dynamics.

1. How many degrees of freedom has a rigid body in space? 2. Why does the order of rotations matter? 3. What is gimbal lock? 4. Write the velocity formula for two points on a rigid body. 5. How does a unit quaternion encode a rotation?

1. Six. 2. Rotations in 3D do not commute. 3. Loss of one degree of freedom when two rotation axes align (middle angle at 90 degrees) in an Euler/Cardan sequence. 4. v_P = v_O + omega x (P - O). 5. q = cos(theta/2) + n sin(theta/2) for rotation by theta about unit axis n.

Body segment inertial parameters (BSIPs): segment mass, position of the centre of mass along the segment, and moments of inertia. They cannot be measured directly in living subjects, so they are estimated from regression equations or tables from cadaver or imaging studies (for example Dumas et al.), as a fraction of total body mass and segment length, with separate values for males and females. Errors in BSIPs affect joint moments, particularly in the fast movements where inertial terms are large.

D'Alembert's principle: adding inertia force (minus mass times acceleration of the centre of mass) and inertia moment (minus the derivative of angular momentum) to the real forces turns a dynamics problem into a static equilibrium problem: the sum of forces and the sum of moments about any point are zero. Method: draw a free-body diagram of the most distal segment (for example the foot with the ground reaction force from the force plate), solve for the joint force and moment at the ankle, then apply equal and opposite loads to the next segment (shank), and continue up the chain (knee, hip). Inputs: segment kinematics (positions, linear and angular velocity and acceleration), BSIPs and external forces (force plate). Joint power: P = M x omega_joint, the product of net joint moment and joint angular velocity. Positive power means energy generation (concentric muscle action), negative power means absorption (eccentric).

1. Why are BSIPs estimated rather than measured? 2. State D'Alembert's principle. 3. In which order are segments processed in a bottom-up inverse dynamics of the leg? 4. What does negative joint power mean?

1. They cannot be measured directly in living subjects, so regression tables are used. 2. Adding inertia forces and moments to the real ones puts the body in dynamic equilibrium, so static equilibrium methods can be used. 3. Foot first, then shank, then thigh. 4. Energy absorption, with eccentric muscle action.

Biophysical signals (force, motion, EMG, pressure) are recorded as time series. A sensor output is amplified and converted by an analogue-to-digital converter (ADC). The number of bits sets the resolution: an n-bit converter has 2^n levels. Sampling theorem (Nyquist): the sampling frequency must be at least twice the highest frequency in the signal, otherwise aliasing (false low frequencies) occurs.

Noise sources: electronic noise, skin movement, vibration. Common tools: low-pass Butterworth filters (used forward and backward for zero phase shift), moving averages, and smoothing splines. The cut-off frequency is chosen by residual analysis. Too low a cut-off distorts the true signal (for example impact peaks); too high leaves noise, which is greatly amplified when the signal is differentiated to give velocity and acceleration. Signal decomposition, for instance with the Fourier transform, describes a signal as the sum of sinusoids of different frequencies.

1. How many levels does an 8-bit ADC have? 2. State the Nyquist criterion. 3. Why is noise a bigger problem in acceleration than in position?

1. 256. 2. Sample at more than twice the highest frequency in the signal. 3. Differentiation amplifies high-frequency noise.

Accelerometer: measures specific force (acceleration minus gravity). At rest it gives the direction of gravity, so it gives tilt (inclination) but not rotation about the vertical axis. Gyroscope: measures angular velocity. Integrating it gives orientation, but small biases accumulate as drift. Magnetometer: measures the local magnetic field, giving heading relative to magnetic north, but is disturbed by metal and electric devices. Sensor fusion combines them: the gyroscope is accurate in the short term, and the accelerometer and magnetometer correct long-term drift.

Stochastic errors: random noise and bias instability, characterised by the Allan variance. Deterministic errors: fixed bias (offset), scale factor error and axis misalignment. They are corrected by calibration: for accelerometers by placing the sensor in several static orientations relative to gravity, and for gyroscopes by a static recording (for bias) and known rotations (for scale factor).

The quaternion derivative is one half of the quaternion multiplied by the angular velocity quaternion, so integrating the gyroscope gives the quaternion over time. Sensor fusion filters (complementary, Kalman, Madgwick) apply corrections from the accelerometer and magnetometer. Stereophotogrammetry versus IMU: cameras give accurate absolute positions in a small volume, IMUs give orientation only (position needs double integration, which drifts) but work anywhere and are cheaper and easier to wear.

1. What does an accelerometer measure at rest? 2. Why does a gyroscope drift? 3. Which errors are deterministic? 4. What is the advantage of quaternions in IMU orientation?

1. The direction of gravity (specific force), which gives tilt. 2. Integrating angular velocity accumulates its small bias and noise. 3. Fixed bias, scale factor and misalignment. 4. No singularities (gimbal lock) and simple integration.