Rigid-body dynamics for robots

Kinematics, the subject of ROB-210, describes where a robot's links are and how fast they move without asking why. Dynamics asks why: which torques and forces produce a given motion, and what motion results from a given set of torques. A controller that ignores dynamics has to discover the robot's inertia, gravity loads and coupling effects through feedback error alone, which works for slow, light robots and fails for fast or heavy ones. A controller that knows the dynamics can predict the torque each joint needs, feed it forward and leave feedback to correct only the small remaining error. This course builds that knowledge from the ground up: rigid-body mechanics in this lesson, the equations of motion of whole manipulators in lesson 2, actuators and friction in lesson 3, and then the control methods that use them. This first lesson reviews the mechanics of a single rigid body, the building block of every robot model: mass and center of mass, the inertia tensor, the parallel axis theorem, and the Newton–Euler equations that relate forces and torques to linear and angular acceleration.

Why robots need dynamic models

Consider a shoulder joint lifting an arm. At rest the motor must hold the arm's weight against gravity; to start moving it must also supply torque to accelerate the arm's inertia; and once the elbow moves as well, the two links push and pull on each other through their connection. A pick-and-place robot moving a 5 kg part at 2 m/s accelerates and decelerates several times a second, and the inertial torques can exceed the gravity torques many times over. If the controller treats every joint as an independent, constant load, it will overshoot when the arm is stretched out and inertia is high, lag when it carries a payload and wobble when two joints move together. Dynamic models allow engineers to size motors and gearboxes before any metal is cut, to simulate a design to see whether it will meet its cycle time, to write feedforward terms that cancel gravity and inertia, and to estimate contact forces without a force sensor. They are also central to safety: collaborative robots detect collisions by comparing measured motor currents with the currents the dynamic model predicts, and flag a difference as an unexpected contact.

Mass, center of mass and linear momentum

A rigid body is an idealized object whose particles keep fixed distances from one another. Its mass m is the sum of the masses of its particles, and its center of mass is the mass-weighted average position:

For a link made of several parts, such as an aluminum tube, a motor and a gearbox, the composite center of mass is the mass-weighted average of the parts' centers of mass. The linear momentum of the body is \( m\mathbf{v}_c \), where \( \mathbf{v}_c \) is the velocity of the center of mass, and Newton's second law states that the net external force equals its rate of change:

The remarkable consequence is that the center of mass moves as if all the mass were concentrated there and every external force acted there, regardless of how the body rotates. Internal forces between parts of the body cancel in pairs. For a robot link, the external forces are gravity acting at the center of mass and the constraint forces exerted by neighboring links at the joints.

Rotation and the moment of inertia

Rotation about a fixed axis is governed by the rotational analog of Newton's law. If a body spins about a fixed axis with angular acceleration \( \alpha \), the net torque about that axis is \( \tau = I\alpha \), where the moment of inertia about the axis is

with r the perpendicular distance of each mass element from the axis. Inertia grows with the square of distance, so mass far from the axis costs much more than mass near it. Standard results for simple shapes are worth knowing, because real links are usually approximated as combinations of them.

The parallel axis theorem

Robot joints rarely rotate a link about its own center of mass; they rotate it about a joint axis some distance away. The parallel axis theorem relates the two. If \( I_c \) is the moment of inertia about an axis through the center of mass and d is the distance to a parallel axis, then

The extra term \( md^2 \) is the inertia the body has simply because its center of mass travels in a circle. The rod results in the table confirm the theorem: \( mL^2/12 + m(L/2)^2 = mL^2/3 \). The theorem applies to each part of a composite link, after which the parts' inertias about the joint axis simply add.

The inertia tensor

For rotation about a fixed axis a single number suffices, but robot links rotate in three dimensions about axes that change with configuration. The general description is the inertia tensor, a symmetric 3 × 3 matrix expressed in a frame attached to the body with origin at the center of mass:

The diagonal terms are moments of inertia about the frame's axes, for example \( I_{xx} = \int (y^2 + z^2)\,dm \), and the off-diagonal terms are products of inertia such as \( I_{xy} = \int xy\,dm \). The angular momentum about the center of mass is \( \mathbf{h} = \mathbf{I}_c\boldsymbol{\omega} \). Because of the products of inertia, \( \mathbf{h} \) is generally not parallel to \( \boldsymbol{\omega} \): spinning a lopsided body about one axis can require torque about another. For every rigid body there exist principal axes in which the products of inertia vanish and the tensor is diagonal; they are the eigenvectors of \( \mathbf{I}_c \), and the principal moments are its eigenvalues. Symmetric links have their principal axes along their axes of symmetry, which is one reason CAD tools and URDF files report inertia in a frame aligned with the link.

When the body rotates relative to the frame in which you want to work, the tensor transforms with the rotation matrix R from body to world frame, using the rotation tools of ROB-210:

The full three-dimensional parallel axis theorem, for a frame displaced by vector \( \mathbf{d} \) from the center of mass, is \( \mathbf{I} = \mathbf{I}_c + m\left(\mathbf{d}^{\mathsf{T}}\mathbf{d}\,\mathbf{1} - \mathbf{d}\mathbf{d}^{\mathsf{T}}\right) \), which reduces to \( I_c + md^2 \) about any single axis.

Angular momentum and the Newton–Euler equations

The rotational counterpart of Newton's second law is Euler's equation: the net external torque about the center of mass equals the rate of change of angular momentum. Written in a frame attached to the body, where the inertia tensor is constant, the time derivative of \( \mathbf{I}_c\boldsymbol{\omega} \) picks up an extra term because the frame itself rotates:

Together with \( \sum\mathbf{F} = m\mathbf{a}_c \), these are the Newton–Euler equations of a rigid body. The first term is the familiar inertia times angular acceleration. The second, \( \boldsymbol{\omega} \times \mathbf{I}_c\boldsymbol{\omega} \), is the gyroscopic term: it vanishes when the body spins about a principal axis, but whenever angular momentum and angular velocity point in different directions, torque is needed just to keep the rotation steady. It is the reason a spinning wheel resists being tilted, and the reason a fast wrist rotation can push back on the forearm.

For a whole manipulator, the Newton–Euler equations can be applied link by link. A forward pass from the base outward computes each link's velocity and acceleration from the joint motions; a backward pass from the tool inward computes the forces and torques each joint must transmit, including the reactions from the links beyond it. This recursive Newton–Euler algorithm computes inverse dynamics in a time proportional to the number of joints, which is why it runs inside real-time controllers at kilohertz rates. Lesson 2 takes the alternative energy-based route, which gives the same answers in a form better suited to understanding and control design.

Single-link dynamics: the pendulum joint

The simplest robot joint is a single link pivoting about a horizontal axis, like a shoulder lifting a straight arm. Let q be the joint angle measured from the horizontal, I the inertia about the joint axis, m the link mass and \( l_c \) the distance from the axis to the center of mass. Summing torques about the fixed joint axis gives

The first term accelerates the link; the second holds it against gravity and is largest when the link is horizontal. This equation already shows two features of all manipulator dynamics: the gravity term depends on configuration, and the equation is nonlinear in q because of the cosine. Adding a payload of mass \( m_p \) at distance \( l_p \) adds \( m_p l_p^2 \) to the inertia and \( m_p g l_p \cos q \) to the gravity torque, which is why payload changes are a classic source of control error.

Computing link properties in software

In practice, link masses, centers of mass and inertia tensors come from CAD models or from identification experiments, and they are stored in robot description files such as URDF. It is still worth checking them with a quick independent calculation, because a mistyped inertia, often a factor of 1,000 from mixing grams and kilograms or millimeters and meters, makes simulations and model-based controllers behave strangely. The code below assembles the forearm from its parts, applies the parallel axis theorem and finds the principal axes of a body whose tensor has products of inertia.

import numpy as np
# composite link: (mass kg, distance from joint m, own inertia about parallel axis kg m^2)
parts = [(1.2, 0.20, 1.2*0.40**2/12),      # tube as slender rod
         (0.8, 0.38, 0.8*0.04**2/2)]       # wrist motor as cylinder
m = sum(p[0] for p in parts)
rc = sum(p[0]*p[1] for p in parts)/m
I_joint = sum(Ic + mi*d**2 for mi, d, Ic in parts)
print(f"m = {m:.2f} kg, r_c = {rc:.3f} m, I_joint = {I_joint:.4f} kg m^2")
print(f"torque to hold horizontal: {m*9.81*rc:.2f} N m")
# principal axes of a tensor with products of inertia
I = np.array([[2.0, -0.3, 0.0],
              [-0.3, 1.5, 0.0],
              [0.0,  0.0, 2.5]])*1e-3
vals, vecs = np.linalg.eigh(I)
print("principal moments (1e-3 kg m^2):", np.round(vals*1e3, 3))
print("first principal axis:", np.round(vecs[:, 0], 3))

m = 2.00 kg, r_c = 0.272 m, I_joint = 0.1802 kg m^2
torque to hold horizontal: 5.34 N m
principal moments (1e-3 kg m^2): [1.359 2.141 2.5  ]
first principal axis: [-0.424 -0.906 -0.   ]

The composite calculation reproduces the hand results, and the eigen-decomposition shows that the coupled x–y inertias separate into principal moments of about 1.36 and 2.14 × 10⁻³ kg·m², along axes tilted about 25° from the body x and y axes, with the z axis unchanged because it had no products of inertia.

Energy and power

Two scalar quantities connect rigid-body mechanics to the Lagrangian method of the next lesson. The kinetic energy of a rigid body is the sum of translational and rotational parts:

and its gravitational potential energy is \( V = m g h_c \), where \( h_c \) is the height of the center of mass. The mechanical power delivered by a joint torque is \( P = \tau\dot{q} \). These quantities are also practical design tools: peak power sets the motor and drive rating, and kinetic energy at full speed sets how hard a collision can be. Standards for collaborative robots limit the energy that can be transferred to a person in contact, so designers keep the moving inertia and speed low enough to stay within those limits.