Compute 2D lower-limb joint moments with inverse dynamics

Computes sagittal-plane ankle, knee, and hip net moments from joint-center coordinates, GRF, COP, and segment inertial properties, using a recursive link-segment Newton-Euler chain (foot -> shank -> thigh).

Usage

inverseDynamics2D(
  joints,
  grf,
  sampling_rate,
  angles = NULL,
  angular_velocity = NULL,
  angular_acceleration = NULL,
  inertial = NULL,
  angle_unit = c("radian", "degree"),
  model = c("newton_euler", "quasi_static"),
  body_mass = NULL,
  body_height = NULL,
  gravity = .GRAVITY
)

Arguments

joints: Matrix/data.frame with columns ankle_x, ankle_y, knee_x, knee_y, hip_x, hip_y, and - for model = "newton_euler" - the foot distal end toe_x, toe_y.
grf: Matrix/data.frame with at least fx and fy columns, and optionally cop_x and cop_y.
sampling_rate: Sampling rate in Hz.
angles: Optional matrix/data.frame with columns ankle, knee, hip (joint angles).
angular_velocity: Optional matrix/data.frame with columns ankle, knee, hip.
angular_acceleration: Optional matrix/data.frame with columns ankle, knee, hip.
inertial: Optional data.frame from estimateSegmentInertia(). Required for model = "newton_euler" unless body_mass is given; under model = "quasi_static" it only supplies an I * alpha correction and is ignored when omitted.
angle_unit: Unit of angles: "radian" or "degree".
model: Which dynamics model to use. "newton_euler" (the default) runs the full recursive link-segment Newton-Euler chain via inverseDynamicsRNE(). "quasi_static" is the legacy massless-segment approximation, retained only for reproducing older results (see Details).
body_mass: Body mass in kg. Used to build the segment inertia table with estimateSegmentInertia() when inertial is not supplied.
body_height: Body height in m, passed to estimateSegmentInertia() when segment lengths must be estimated from stature. If NULL, segment lengths are measured from the marker data with scaleBodyModel().
gravity: Gravitational acceleration in m/s^2.

Value

A data.frame with time index and joint moments (ankle_moment, knee_moment, hip_moment). If angular velocity is available, corresponding power columns are included. model = "newton_euler" also returns the proximal joint reaction forces (ankle_fx, ankle_fy, ...).

Details

The default "newton_euler" model propagates reactions distal to proximal: the ground reaction force and free moment enter at the foot, and every segment contributes its weight \(m g\), its linear inertia \(m a_{com}\) and its angular inertia \(I \alpha\). Segment centres of mass come from the anthropometric ratios in inertial, and their linear and angular accelerations from numerical differentiation of the marker trajectories, so a segment mass source (inertial or body_mass) and the foot distal end (toe_x, toe_y) are required.

Because the recursion differentiates the marker trajectories twice, it amplifies marker noise: the joint moments are only as good as the smoothing applied beforehand. On a static limb sampled at 200 Hz, 1 mm of white marker noise produces a spurious moment about 24 times the true value; a 6 Hz zero-lag low-pass (butterworthFilter(), or filterSignals() on the positions) reduces that by more than an order of magnitude. Filter the marker data before calling this function. The quasi-static model never differentiated positions, so this exposure is new.

"quasi_static" reproduces the pre-1.0 behaviour: each joint moment is the moment of the ground reaction force alone about that joint centre, plus an optional \(I \alpha\) term. It omits segment weight and linear inertia entirely, so it returns identically zero whenever the limb is off the ground (the whole swing phase) and underestimates stance moments - at the hip by roughly 8% in quiet standing. It is kept only so that analyses published against the old implementation can be reproduced, and emits a message when selected.

References

Winter DA (2009). "Biomechanics and Motor Control of Human Movement." 4th ed. John Wiley & Sons.

Examples

n <- 200
joints <- data.frame(
  ankle_x = rep(0.00, n), ankle_y = rep(0.05, n),
  toe_x   = rep(0.15, n), toe_y   = rep(0.01, n),
  knee_x = rep(0.00, n),  knee_y = rep(0.45, n),
  hip_x = rep(0.00, n),   hip_y = rep(0.85, n)
)
grf <- data.frame(fx = rep(0, n), fy = abs(sin(seq(0, pi, length.out = n))) * 800,
                  cop_x = rep(0.02, n), cop_y = rep(0, n))
id <- inverseDynamics2D(joints, grf, sampling_rate = 100, body_mass = 70)