Dynamics Notation Glossary

July 4, 2026 · View on GitHub

Attribution: This glossary maps notation from "A Quick Tutorial on Multibody Dynamics" by C. Karen Liu and Sumit Jain to DART's C++ API. The original PDF is preserved at docs/dynamics.pdf.

Navigation: ← Recursive Inverse Dynamics | Index

Core Quantities

State Variables

PDF Symbol	Description	DART API	Return Type
$q$	Generalized positions	`Skeleton::getPositions()`	`Eigen::VectorXd`
$\dot{q}$	Generalized velocities	`Skeleton::getVelocities()`	`Eigen::VectorXd`
$\ddot{q}$	Generalized accelerations	`Skeleton::getAccelerations()`	`Eigen::VectorXd`
$\tau$ or $Q$	Generalized forces	`Skeleton::getForces()`	`Eigen::VectorXd`
$n$	Number of DOFs	`Skeleton::getNumDofs()`	`std::size_t`

Dynamics Matrices

PDF Symbol	Description	DART API	Return Type
$M(q)$	Mass matrix	`Skeleton::getMassMatrix()`	`Eigen::MatrixXd`
$M^{-1}(q)$	Inverse mass matrix	`Skeleton::getInvMassMatrix()`	`Eigen::MatrixXd`
$M_{aug}$	Augmented mass matrix	`Skeleton::getAugMassMatrix()`	`Eigen::MatrixXd`
$C(q,\dot{q})$	Coriolis/centrifugal forces	`Skeleton::getCoriolisForces()`	`Eigen::VectorXd`
$g(q)$ or $N(q)$	Gravity forces	`Skeleton::getGravityForces()`	`Eigen::VectorXd`
$C + g$	Combined bias forces	`Skeleton::getCoriolisAndGravityForces()`	`Eigen::VectorXd`
$F_k$	Constraint forces	`Skeleton::getConstraintForces()`	`Eigen::VectorXd`

Equation of Motion

The standard form in the PDF:

$M(q)\ddot{q} + C(q,\dot{q}) = Q$

In DART (with gravity and external forces):

$M(q)\ddot{q} + C(q,\dot{q}) + g(q) = \tau + \tau_{ext} + J^T \lambda$

Per-Body Quantities

Rigid Body Properties

PDF Symbol	Description	DART API	Notes
$m$	Mass	`BodyNode::getMass()`	Scalar
$I_c$	Inertia tensor (COM)	`BodyNode::getInertia()`	3×3 matrix
$I_0$	Inertia at zero rotation	Internal	$I_c = R I_0 R^T$
$x$	COM position (world)	`BodyNode::getCOM()`	`Eigen::Vector3d`
$R$	Rotation matrix	`BodyNode::getTransform().rotation()`	`Eigen::Matrix3d`
$v$	Linear velocity (world)	`BodyNode::getCOMLinearVelocity()`	`Eigen::Vector3d`
$\omega$	Angular velocity (world)	`BodyNode::getAngularVelocity()`	`Eigen::Vector3d`
$M_c$	Spatial inertia	`BodyNode::getSpatialInertia()`	6×6 matrix
$V$	Spatial velocity	`BodyNode::getSpatialVelocity()`	`Eigen::Vector6d`

Jacobians

PDF Symbol	Description	DART API	Size
$J_k$	Body Jacobian	`BodyNode::getJacobian()`	6×n
$J_{vk}$	Linear Jacobian	`BodyNode::getLinearJacobian()`	3×n
$J_{\omega k}$	Angular Jacobian	`BodyNode::getAngularJacobian()`	3×n
$J'$	Jacobian at point	`BodyNode::getWorldJacobian(offset)`	6×n
$\hat{J}_{\omega k}$	Local angular Jacobian	Internal	3×n(k)

Matrix Notation

Skew-Symmetric Matrix

The PDF uses $[a]$ to denote the skew-symmetric matrix of vector $a$ :

$[a] = \begin{bmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{bmatrix}$

In DART: dart::math::makeSkewSymmetric(a) or cross product operations.

Key Identities

Identity	Meaning
$[a]b = a \times b$	Cross product
$[a]b = -[b]a$	Anti-commutativity
$[a]^T = -[a]$	Skew-symmetric property
$-[a][a] = (a^T a)I_3 - aa^T$	Useful for inertia
$[a \times b] = [a][b] - [b][a]$	Jacobi identity
$[R\omega] = R[\omega]R^T$	Rotation transformation
$[\omega] = \dot{R}R^T$	Angular velocity from rotation

Coordinate Frames

PDF Notation	Description
World frame	Global inertial frame
$R_k^0$	Rotation from world to link k
$W_k^0$	Homogeneous transform to link k
$^\ell$ superscript	Local frame of link k
$\hat{\omega}_k$	Angular velocity in parent's frame

Tree Structure Notation

Symbol	Description	Example
$p(k)$	Parent link index	$p(4) = 2$
$p(1,k)$	Path from root to k	$p(1,4) = \{1,2,4\}$
$c(k)$	Child link indices	Used in RNEA
$n(k)$	DOFs in joint k	1 for revolute, 3 for ball
$d_k$	Joint-to-joint vector	Position offset
$c_k$	COM in local frame	Constant

Algorithm Names

PDF Name	DART Implementation	Complexity
Forward Dynamics	`computeForwardDynamics()`	$O(n)$ via ABA
Inverse Dynamics	`computeInverseDynamics()`	$O(n)$ via RNEA
Mass Matrix	`getMassMatrix()`	$O(n^2)$
RNEA	Recursive Newton-Euler	$O(n)$
ABA	Articulated Body Algorithm	$O(n)$

Python (dartpy) Equivalents

import dartpy as dart

# State
q = skeleton.getPositions()      # numpy array
qdot = skeleton.getVelocities()
qddot = skeleton.getAccelerations()
tau = skeleton.getForces()

# Dynamics matrices
M = skeleton.getMassMatrix()     # numpy 2D array
C = skeleton.getCoriolisForces()
g = skeleton.getGravityForces()

# Per-body
for body in skeleton.getBodyNodes():
    mass = body.getMass()
    com = body.getCOM()
    J = body.getJacobian()

Units Convention

DART uses SI units:

Quantity	Unit
Length	meters (m)
Mass	kilograms (kg)
Time	seconds (s)
Force	Newtons (N)
Torque	Newton-meters (N·m)
Angle	radians (rad)