Robot Dynamics: Equations and Algorithms
TL;DR A review of rigid-body robot dynamics that writes inverse dynamics (RNEA), the joint-space inertia matrix (CRBA) and forward dynamics (ABA) for tree-structured robots in one spatial-vector notation, then extends them to closed kinematic loops. It traces the move from O(N^4) Lagrangian methods to O(N) recursive algorithms; the ABA needs fewer operations than the CRBA once N > 9.
What they do
A review of robot dynamics from the recursive Newton-Euler algorithm up to 2000. It covers inverse dynamics (torques from a trajectory, for feedforward control), forward dynamics (accelerations from torques, for simulation) and the joint-space inertia matrix. Everything is written in one spatial-vector notation and applies to any tree-structured mechanism.
How it works
- Spatial notation. Velocities, accelerations and forces are 6-vectors with linear and angular parts; inertias are 6x6 matrices. Link i hangs from parent λ(i) < i through a joint with motion subspace h_i.
- RNEA (inverse dynamics). An outward pass computes link velocities and accelerations; an inward pass sums each link’s Newton-Euler force into joint forces, and τ_i = h_iᵀ f_i. Gravity enters as a fictitious base acceleration of -g.
- CRBA (inertia matrix). At zero velocity, a unit acceleration at joint i moves the subtree beyond it as one composite rigid body while the rest stays in static equilibrium. Summing composite inertias from the tips fills in column i of M.
- ABA (forward dynamics) and closed loops. Each subtree’s articulated-body inertia makes force linear in its root link’s acceleration. A tip-to-base pass computes these, a base-to-tip pass solves for joint accelerations. Closed loops use a spanning tree plus Lagrange-multiplier loop-closure forces.
Results
- Lagrangian algorithms were O(N^4). The RNEA is O(N), and later versions run up to 1.7x faster than early ones on a 6-DoF robot.
- The CRBA is O(N^3) but cheap for small N. The ABA needs fewer arithmetic operations once N > 9, and the ABA is also more accurate (Ascher et al.).
- Closed loops can be solved three ways: the full system in O((n+nc)^3), via Lagrange multipliers in O(n nc^2 + nc^3), or by eliminating constraint forces, which can be fastest when n - nc is small.
- An unbranched chain can be solved in O(log N) time on O(N) parallel processors.
Limitations
The review covers rigid bodies only; flexible links get one paragraph. Closed loops are only sketched and assume scleronomic constraints; in practice they also need constraint stabilization. Symbolic equation generation, parallel algorithms and specific applications are mentioned but not covered, and the reference list is not complete.
Abstract
This paper reviews some of the accomplishments in the field of robot dynamics research, from the development of the recursive Newton-Euler algorithm to the present day. Equations and algorithms are given for the most important dynamics computations, expressed in a common notation to facilitate their presentation and comparison.
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